See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/301286459
Advanced Macroeconomics
Book · January 2016
CITATIONS
READS
0
8,240
1 author:
Keshab Bhattarai
University of Hull
120 PUBLICATIONS 377 CITATIONS
SEE PROFILE
Some of the authors of this publication are also working on these related projects:
Dynamic cge View project
Macroeconomics View project
All content following this page was uploaded by Keshab Bhattarai on 14 April 2016.
The user has requested enhancement of the downloaded file.
ADVANCED MACROECONOMICS (56277)
Dr. Keshab Bhattarai
University of Hull Business School, Hull, England, UK.
January 12, 2016
Abstract
This monograph aims to present concisely the major elements of popular macroeconomic
models for systematic thinking about the modern economies. It contains detailed derivations
of classical, Keynesian, new Keynesian, new classical models and open economy and structural
models used for analysing short run ‡uctuations and long run growth. Tutorial problems and
assignments are provided for each sector for practice. These models could be used for advanced
policy discussions required for greater macroeconomic stability and higher rates of economic
growth.
JEL Classi…cation: E
Keywords: macroeconomic models
H U 6 7 R X , H u ll,
U K . e m a il: K .R .B h a tta ra i@ hu ll.a c .u k
1
Contents
1 L1: Classical and Keynesian Macro Models
1.1 Background: Seven Classical Macro Models . . . . . . . . . . . . . . . . . . .
1.1.1 A simple version of the classical macroeconomic model . . . . . . . . .
1.1.2 Malthusian model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Two period model of consumption and saving . . . . . . . . . . . . . .
1.1.4 Pure exchange model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.5 Ricardian Trade Model for Comparative Advantage . . . . . . . . . .
1.1.6 Intertemporal balance in budgets of households, government and …rms
1.1.7 Ramsey growth model . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Keynesian Model: Hicksian Synthesis . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Soution of the Keynesian Model . . . . . . . . . . . . . . . . . . . . .
1.2.2 Comparative Static Analysis in Keynesian Macroeconomic Model . . .
1.2.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Samuelsonian Multiplier Accelerator Model . . . . . . . . . . . . . . . . . . .
1.4 ISLM equilibrium: a new approach . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Basics of monopolistic competition in a new Keynesian model . . . . . . . . .
1.6 Micro-foundation to the Keynesian Multiplier: Mankiw (1988) . . . . . . . .
1.7 Keynesian Stochastic Macroeconomic Model and Policies . . . . . . . . . . . .
1.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.2 Stylized Facts and Macroeconomic Policies in the UK . . . . . . . . .
1.7.3 Macroeconomic Trends . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.4 Fiscal Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.5 Monetary policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.6 Trade Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.7 Modelling of the UK economy . . . . . . . . . . . . . . . . . . . . . . .
1.7.8 Keynesian Stochastic Macroeconomic Model (KSMM) . . . . . . . . .
1.7.9 Steady State in the KSMM . . . . . . . . . . . . . . . . . . . . . . . .
1.7.10 Transitional Dynamics in KSM Model . . . . . . . . . . . . . . . . . .
1.7.11 Estimation and application of the KSMM Model . . . . . . . . . . . .
1.7.12 Qualitative analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.13 A Small Model of Unemployment, In‡ation and Growth . . . . . . . .
1.7.14 Solution of the stabilisation model . . . . . . . . . . . . . . . . . . . .
1.7.15 Supply side and rational expectation . . . . . . . . . . . . . . . . . . .
1.7.16 Aggregate Demand and Aggregate Supply Model . . . . . . . . . . . .
1.7.17 Trade Policy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.18 Structural factors and the volatility of exchange rate . . . . . . . . . .
1.7.19 Monetary model of exchange rate expectation . . . . . . . . . . . . . .
1.7.20 Solving for in‡ation and exchange rate paths simultaneously . . . . .
1.7.21 Exchange rate overshooting under the ‡oating exchange rate system .
1.7.22 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.23 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.24 Identi…cation of the simultaneous equation model (SEM) . . . . . . .
1.7.25 Path of price and exchange rates in the Dornbusch model . . . . . . .
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
8
11
11
13
14
19
23
27
30
37
39
41
42
46
50
51
56
58
58
58
58
62
64
65
65
67
68
69
71
74
77
78
80
81
84
85
86
87
88
89
95
95
96
2 L2:
2.1
2.2
2.3
2.4
New Keynesian Model: Fundamentals
New Keynesian Model: a prototype example . . . . . . . . . . . . .
Two Period Model of Stabilisation: Mankiw and Weinzierl (2011) .
A DSGE Model of Macroeconomic Policy in South Asia . . . . . .
A Prototype of New Keynesian DSGE model with habit formation
2.4.1 Blanchard and Gali (2013) . . . . . . . . . . . . . . . . . .
2.4.2 Solution Procedure in the DSGE Models . . . . . . . . . . .
2.4.3 Basics of Bhattarai and Trzeciakiewicz (2012) DSGE model
2.4.4 Household problem . . . . . . . . . . . . . . . . . . . . . . .
2.4.5 Fiscal and monetary policies . . . . . . . . . . . . . . . . .
2.4.6 Log-Linearised System of Equations . . . . . . . . . . . . .
2.4.7 Households: . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.8 Firms: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.9 Government: . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.10 General equilibrium conditions: . . . . . . . . . . . . . . . .
2.4.11 Parameterisation of the model . . . . . . . . . . . . . . . .
2.4.12 Results of Hull DSGE model . . . . . . . . . . . . . . . . .
2.5 Critical assessment of the DSGE Models . . . . . . . . . . . . . . .
2.5.1 Blanchard’s New Keynesian DSGE model . . . . . . . . . .
2.5.2 Basic New Keynesian Model in logs . . . . . . . . . . . . .
2.5.3 Extended version of the New Keynesian Model . . . . . . .
2.6 Problem on Open Economy New Keynesian Model . . . . . . . . .
2.7 Stability Analysis: Illustrations . . . . . . . . . . . . . . . . . . . .
3 L3: New Classical Macro Models (Real Business Cycle)
3.0.1 Linear RBC Model . . . . . . . . . . . . . . . . . . . .
3.0.2 New Keynesian Model in relation to the RBC models
3.1 Rational Expectation . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Rational Expectation: Another example . . . . . . . .
3.2 Supply Side and Rational Expectation . . . . . . . . . . . . .
3.3 Aggregate demand and aggregate supply model . . . . . . . .
3.3.1 Estimations . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Trade Policy: Small Open Economy Macro Model . . . . . .
3.4.1 Estimations . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
97
97
99
100
104
110
112
113
115
116
117
117
118
118
118
118
121
130
131
132
132
132
134
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
143
147
148
151
154
160
161
164
164
168
4 L4: Neoclassical Growth Model
172
4.0.2 Four First Order Conditions for Dynamic Optimisation . . . . . . . . . . . . 173
4.0.3 Transitional dynamics towards steady state . . . . . . . . . . . . . . . . . . . 174
4.1 Standard macromodel of growth, …scal policy and welfare (Bruce and Turnovsky(2007))176
4.1.1 Mechanism for Poverty Alleviation (Bhattarai 2010) . . . . . . . . . . . . . . 179
4.2 Dynamic Computable General Equilibrium Model of Fiscal Policy . . . . . . . . . . 182
4.2.1 Trade arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4.2.2 Government sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4.2.3 General Equilibrium in a Growing Economy . . . . . . . . . . . . . . . . . . . 185
4.2.4 Procedure for Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.3 Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
3
5 L5: Endogenous Growth Model
5.0.1 Human capital and …nal goods sectors . . . . . . . . . . .
5.0.2 Balanced growth . . . . . . . . . . . . . . . . . . . . . . .
5.0.3 Cross country calibration of government bias in education
5.0.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .
5.0.5 China, India and SAARC Countries in the Global Growth
5.0.6 Dynamic Panel Data Model of Economic Growth . . . . .
5.0.7 GMM 2-step Estimation of Growth in South Asia . . . .
5.0.8 Dynamic Computable General Equilibrium Model . . . .
5.0.9 Macroeconomic simulation model of South Asia . . . . . .
5.1 Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
Competition
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
189
189
190
192
194
195
199
199
202
205
206
6 L6: Dynamic Programming for Macro Dynamics
6.1 Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Money in Growth Models . . . . . . . . . . . . . . . . . . . . .
6.2.1 Friedman Rule with Cash in Advance Constraint . . . .
6.2.2 Dynamic optimisation in CIA Model . . . . . . . . . . .
6.2.3 Steady State in the CIA Model . . . . . . . . . . . . . .
6.3 Money in the Utility Function and Growth . . . . . . . . . . .
6.3.1 Dynamic optimisation in the MIU model . . . . . . . .
6.3.2 Steady state in the MIU model . . . . . . . . . . . . . .
6.3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.4 Analysis of Dynamic GE Model of Financial Deepening
6.3.5 Optimal and actual …nancial deepening . . . . . . . . .
6.4 Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
7 L7:
7.1
7.2
7.3
7.4
7.5
7.6
7.7
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
208
212
214
214
216
217
223
224
224
228
230
231
233
Open Economy Model: Exchange Rate and Finance in Macro
Small Open Economy Model . . . . . . . . . . . . . . . . . . . . . . .
Global Economy model of Two Economies . . . . . . . . . . . . . . . .
Two SEctor Static Global General Equilibrium Model with Money . .
7.3.1 Outline of the Model . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.3 Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.4 Monetary Sector . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.5 Government Sector . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.6 External sector . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.7 Analytical Forms and the Solution Procedure . . . . . . . . . .
7.3.8 Parameterisation of the Model . . . . . . . . . . . . . . . . . .
7.3.9 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . .
7.3.10 Conclusion from the static two country model . . . . . . . . . .
Two Country Dynamic Global Economy Model . . . . . . . . . . . . .
7.4.1 Analytical Results of Optimisation . . . . . . . . . . . . . . . .
International macroeconomic policy coordination . . . . . . . . . . . .
Nash-VAR Policy Game . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.1 Estimates for the Nash Policy Game . . . . . . . . . . . . . . .
Multicountry macro interaction model . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
241
241
244
249
250
251
253
255
256
256
257
258
259
262
262
265
272
273
274
275
4
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
7.7.1
7.7.2
7.7.3
7.7.4
7.7.5
7.7.6
7.7.7
7.7.8
Time path in the multicountry macro interaction model . . . . . . .
Parameters of the Macroeconomic model . . . . . . . . . . . . . . . .
Results of Macro Nash Policy Game . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Growth Impacts of Foreign Direct Investment in an Open Economy
Empirical Literature on FDI and Growth . . . . . . . . . . . . . . .
Empirics of FDI in BRICS Countries . . . . . . . . . . . . . . . . . .
Empirics of FDI in OECD Countries . . . . . . . . . . . . . . . . . .
8 L8:
8.1
8.2
8.3
Business Cycles
Optimising model of business cycle . . . . . . . . . . . . . . . . . . . . .
Aggregate Demand-Aggregate Supply (AS-AD) Model of Business Cycle
AS-AD Model of Business Cycle . . . . . . . . . . . . . . . . . . . . . .
8.3.1 Role of Shocks in AD-AS Model . . . . . . . . . . . . . . . . . .
8.4 Monetary Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.1 Integration of Finance in a Macro Model . . . . . . . . . . . . . .
8.5 Policy Rule versus Optimal Discretion . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
277
277
277
278
279
281
282
283
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
291
292
294
294
297
300
304
306
9 L9: Class Test: Past Examples
308
10 L10: In‡ation and Unemployment
10.1 Natural rate of unemployment and output . . . . . . . . . . . .
10.2 Wage Price Spiral . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Equilibrium Unemployment: Matching and Bargaining Set Up
10.3.1 Markov Process of Employment and Unemployment . .
10.4 Exercise 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
313
314
315
317
321
325
11 L11: Public Debt: Impact of Taxes, Spending and De…cit on Growth
328
11.1 Classical Ricardian Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
11.2 Role of debt in the Keynesian model . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
11.3 Growth impacts of public de…cit in the Neoclassical growth model . . . . . . . . . . 333
11.4 Analysis of debt crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
12 Blake-Weale (1994) model of debt
12.1 Cole -Kehoe (2000) model of self ful…lling debt crisis .
12.2 Credibility . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Two Period Overlapping Generation Model . . . . . .
12.4 Empirical Analysis . . . . . . . . . . . . . . . . . . . .
12.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . .
12.6 International strategic policy coordination models . . .
12.7 Exercise 11 . . . . . . . . . . . . . . . . . . . . . . . .
5
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
335
337
340
342
346
348
352
360
13 Tutorial Problems
13.1 Tutorial 1: Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Tutorial 2: Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3 Tutorial 3: Open Economy DSGE Model . . . . . . . . . . . . . . . . . . . . .
13.4 Tutorial 4: Ramsey to RBC Model . . . . . . . . . . . . . . . . . . . . . . . .
13.5 Tutorial 5: Neoclassical Growth with Hamiltonian . . . . . . . . . . . . . . .
13.6 Tutorial 6: Endogenous growth model . . . . . . . . . . . . . . . . . . . . . .
13.7 Tutorial 7: Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . .
13.8 Tutorial 8: Equilibrium Unemployment Model . . . . . . . . . . . . . . . . . .
13.9 Tutorial 9: Money, In‡ation, Business Cycle and OLG Model . . . . . . . . .
13.10Tutorial 10: Small Open Economy Model . . . . . . . . . . . . . . . . . . . .
13.10.1 Monetary Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.10.2 Government Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.11Tutorial 11: New Keynesian and Newclassical Macro Models . . . . . . . . . .
13.12Tutorial 12: Real Business Cycle Model . . . . . . . . . . . . . . . . . . . . .
13.13Tutorial 13: Global Economy . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.14Tutorial 14: A Study on Housing Markets . . . . . . . . . . . . . . . . . . . .
13.15Tutorial 15: Rational Expectation . . . . . . . . . . . . . . . . . . . . . . . .
13.16Tutorial 16: Overlapping Generation Model: Impact of Taxes on Growth . . .
13.17Other Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.17.1 Problem 1: Keynesian Model . . . . . . . . . . . . . . . . . . . . . . .
13.17.2 Problem 2: Stability Analysis . . . . . . . . . . . . . . . . . . . . . . .
13.17.3 Problem 3: Neoclassical Growth with Hamiltonian . . . . . . . . . . .
13.17.4 Problem 4: Dynamic Programming . . . . . . . . . . . . . . . . . . . .
13.17.5 Problem 5: Money in utility (MIU) and cash in advance (CIA) models
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
362
362
369
370
372
373
376
378
381
382
384
385
385
390
391
393
394
396
399
401
401
402
405
406
407
14 Assignment(optional)
14.1 Best twenty articles in 100 years in the American
14.2 Other Articles . . . . . . . . . . . . . . . . . . . .
14.2.1 Useful texts . . . . . . . . . . . . . . . . .
14.2.2 Quality ranking of journals in Economics
Economic Review
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
409
415
417
422
424
15 Computation and software
15.1 GAMS . . . . . . . . . . . . . . . . . . . . .
15.2 MATLAB . . . . . . . . . . . . . . . . . . .
15.3 Dynare . . . . . . . . . . . . . . . . . . . . .
15.4 R . . . . . . . . . . . . . . . . . . . . . . . .
15.5 Econometric and Statistical Software . . . .
15.5.1 Advanced Texts in Macroeconomics
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
426
426
430
437
439
439
440
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
16 Sample Class test
442
17 Sampel Final Exam
450
6
18 Foundations
18.1 First order di¤erence equation . . . . . . . . . . . . . . . . . . . . . . . . .
18.2 First order di¤erential equation . . . . . . . . . . . . . . . . . . . . . . . .
18.3 Second order di¤erential equation: market example . . . . . . . . . . . . .
18.3.1 Higher Order Di¤erence Equations: Schurr Theorem . . . . . . . .
18.3.2 Ten Best articles in the Journal of European Economic Association
18.3.3 Best 40 articles in the Journal of Economic Perspectives . . . . . .
7
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
461
461
464
467
474
475
475
1
L1: Classical and Keynesian Macro Models
Modern macroeconomics in general is a system of thinking about the growth in the long run
and ‡uctuations in macro variables in the short run. Theories of Smith (1976), Ricardo (1817),
Walras (1874), Pigou (1917), Keynes (1936), Solow (1956), Friedman (1968), Lucas (1976), Sim
(1980), Sargent (1987), Romer (1989) and many other economists represent such thinking about
the functioning of the economy as a whole. It has four distinguishing features a) dynamic models
b) competitive equilibrium c) micro-foundation d) rational expectation. It is important to have a
clear historical perspective on these features to understand what is happening to economic growth
as well as to the output, employment, price level, interest rate, echange rate, imports and exports.
Macroeconomic models explain determinants of aggregate demand and aggregate supply in an
economy or group of economies in the global economy.
Classical view
Ideas of Adam Smith (1776) in "An Inquiry into the Nature and Causes of the Wealth of
Nations"
Economy adjusts automatically towards its long run equilibrium if the price system is perfectly
‡exible and government policy is liberal.
Division of labour creates productivity. Higher rate of saving and investment are key for capital
accumulation and rising living standards and output. The invisible hand, price system, plays
a crucial role in allocating resources. Ideas of Ricardo (1817), Say (1817), Malthus (1790) Mill
(1844), Marx (1859), Marshall (1922) Pigou (1918). General equilibrium and the real business
cycle models are based on these classical principles that inlcude:
Invisible hand sets prices to equate demand and supply.
No excess demand or no excess supply can persist.
No glut or shortages in goods market.
No unemployment or labour pressure in the labour market.
Money is neutral (quantity theory of money).
Prices proportional to money supply.
It is a long run view.
Balanced budget recommended.
Free and open market economy is competitive in the global economy.
Laisser faire: minimum government is the best government.
Downward sloping aggregate demand and vertical supply curve
8
Keynesian Revolution (Short run analysis)
Gaps between supply and demand may persist for a log time.
Markets (prices) may not work automatically itself because of de…ciency in demand: massive
unemployment labour and under utilisation of capital is possible.
Costs of waiting to return to the natural level; irresponsible to do so.
Balancing budget is stupid and dangerous policy.
Active role by government can mitigate de…ciency in private demand (consumption and investment).
Positive role of …scal policy and monetary policy.
Multiplier e¤ect of demand on output
Aggregate supply is horizontal in the short run.
Animal spirits –importance of expectations.
Consensus on the IS-LM models (Keynes-Hicks-Hansen) (1940-1970s)
Exact speci…cation of relations among economic variables discussed in Keynes; Klein’s macro
economic model, MMB, dynamic models of UK economy
IS curve thought to be too steep: active …scal policy recommended
Theory of consumption: Modigliani-(life cycle), Friedman –permanent income ; importance
of expectations
Theory of investment (Tobin’s q, user cost of capital)
Phillips curve; Okun’s law, Trade-o¤ between unemployment and in‡ation; more empirical
models adaptive expectations
Theory of unemployment (worker mis-perception, ine¢ cient markets, employer mis-perception,
E¤ectiveness of monetary policy
Mundel-Flemming open economy model
Criticism of IS-LM and new macroeconomics
Monetarist approach – Natural rate of unemployment of Milton Friedman (1968) vertical
Phillips and aggregate supply curve
Lucas critique (1976): rational expectations
No money illusion possible, people are clever and process current information to know what
the government is doing; government cannot fool people, when prices change –nominal wages
change accordingly; no trade-o¤ between in‡ation and unemployment
9
Stag‡ation –no growth and in‡ation simultaneously
Revenue may decrease if tax rate increases (La¤er curve)
Supply side policies recommended (subsidies, manpower, investment tax credit, technological
development, e-commerce, low tari¤)
IS-LM are ad-hoc models; need more micro-foundation
Real business cycle view: fully ‡exible prices
Business cycle are optimal response not due to ‡uctuations in demand
Mortensen-Pissarides search and matching theory of equilibrium unemployment
New Keynesian view: classical model may be right in the long run but the Keynesian hypothesis still valid because of contracts and staggering wages, menu costs: it takes time for prices
to adjust
Dynamic stochastic general equilibrium (DSGE) models with nominal and real frictions
Development of more powerful mathematical and computing tools being applied to analyse
and assess di¤erent versions of policy
Growth Theory
Ramsey model
Harrod-Domar model
Solow-Swan Neo-Classical Model (with …xed saving rate)
Sources of growth (labour, capital and technology)
Analysis of the salanced growth path (steady state) and transitional dynamics of the economy
Dynamically e¢ cient savings rate: golden rule
Role of human capital in enhancing technology
Optimal growth theory (Cass-Koopman)
Endogenous growth models (Lucas-Romer)
Dynamic computable general equilibrium (DCGE) models for analysis of medium and long
term growth and business cycles.
Some useful web sites:
JSTORE and Econlit database of journal articles
http://www.eea-esem.com/eea-esem/2014/prog/list_sessions.asp
http://editorialexpress.com/conference/MMF2014/program/MMF2014.html
https://www.aeaweb.org/aea/2015conference/program/preliminary.php
http://www.webmeets.com/RES/2013/prog/list_sessions.asp
10
http://www.hull.ac.uk/php/ecskrb/Confer/research.html
http://www.economicsnetwork.ac.uk/
http://www.aeaweb.org/rfe/
http://www.ssc.upenn.edu/~schorf/research.htm
http://www.webmeets.com/RES/2013/prog/
http://editorialexpress.com/conference/GAMES2012/program/GAMES2012.html
http://www.eea-esem.com/EEA-ESEM/2012/prog/list_sessions.asp
https://editorialexpress.com/cgi-bin/conference/listconfs.cgi
Key macro time series (leading lagging and coincident indicators) Above theories need
to be tested by data.
Macro data has coincident indicators (move together with the real GDP): GDP and GDP
components;
leading indicators (moves before the real GDP): inventories, capacity utilisation, stock prices,
real money balances; and ;
lagging indicators: unemployment rate, in‡ation.
Consult ESDS International, Datastream, Navidata and the central banks (http://www.bis.org/cbanks.htm)
to construct such data.
1.1
Background: Seven Classical Macro Models
Classical economists believed in the e¢ ciency of the free market mechanism, invisible hand of Adams
Smith. Believing in Say’s laws that states "supply creates its own demand" they assumed that
markets are perfectly competitive and free adjustments in relative prices guarantee full employment
in the economy (Hicks (1937)). Government should create economic and social infrastructure in
which the market institutions thrive. Money is super-neutral, it has no real impact but only prices
will increase if the government tries to raise the aggregate demand. Keynes criticised the classical
assumption of full-employment and ‡exibility of prices and o¤ered demand determined model of
an economy. Further research in macroeconomics particularly by the Real Business Cycle (RBC)
school reinstated classical theory as relevant and consistent with facts after mid 1970s. Recent
developments in the rational expectation, modelling of dynamics in the new Keynesian models have
emphasised on both real and nominal rigidities to reinstate validity of Keynesian models. This and
next two sections are going to review models underlying these four approaches to macroeconomics.
1.1.1
A simple version of the classical macroeconomic model
Output
Y = F (N )
(1)
Labour demand:
N = N(
11
W
)
P
(2)
Labour Supply
L = N(
W
)
P
(3)
Full employment
L=N
(4)
M = mP Y
(5)
S = S(i)
(6)
I = I(i)
(7)
S=I
(8)
Neutrality of money
Perfectly competitive capital market
Saving
Investment
Capital market equilibrium
Capital accumulation process
Kt = (1
) Kt
1
+ It
0<
<1
(9)
Classical Model: A Numerical Example
Output
0:05N 2
Y = 6N
(10)
Labour demand:
N = 60
10
W
P
(11)
Labour Supply
L = 20 + 5
W
P
(12)
Neutrality of money
M = 0:5P Y
(13)
First solve the above model for the real wage W
equiting demand for labour to the supply of
P
labour, then solve for equilibrium employment (N ) and then for the output and the price level;
P Y N W
. Prove neutrality of money by changing M to 150 and solving for the variables.
P
12
Table 1: Solution of the
W
N
P
Base solution
8/3 100/3
Money expansion 8/3 100/3
Classical Model
Y
P
W
144.4 1.38 3.7
144.4 2.07 5.5
M
100
150
Money is used for exchange and is completely neutral in this classical model. It does not have
any impact on real variables W
P ; N and Y . A 50 percent increase in money supply causes a 50
percent increase in price level (P ) and nominal wage rate (W ).
Output and employment change by changes in the real side of the economy, production technology or labour (and capital) market conditions. Show now the impact of an exogenous increase
in labour supply by 5 (say because of migration) to L = 25 + 5 W
P . Intuition: an increase in labour
supply, lowers the real wage rate. It raises the demand for labour and the level of output. Given
the level of money supply it lowers the price level. This is what can be seen in Table 2.
Table 2: New Scenario of
W
N
P
Base solution
7/3 36.7
Money expansion 7/3 36.7
the Classical Model
Y
P
W M
152.9 1.31 3.1 100
152.9 1.97 4.6 150
When more education (or migration) raises labour force participation, it can generate similar
e¤ects.
Can we observe classical model in real life? Take data on prices and money supply and see if
they are proportional. In most cases they are. Therefore the classical model is still very relevant in
explaining cuase of increase in prices. For instance the quantitative easing (QE) should raise prices
but the economy is operating under capacity; increase in output in compensating for price-raising
impacts of QE.
What happens to welfare? This requires evaluation of the utility function.
See: Minford P. and Z. Ou (2013) Taylor Rule or optimal timeless policy? Reconsidering the Fed’s
behavior since 1982, Economic Modelling 32 113–123
Woodford, M. (2011) Simple Analytics of the Government Expenditure Multiplier, American Economic
Journal: Macroeconomics, 3(1): 1–35
Wallis K.F. (1989) Macroeconomic Forecasting: A Survey , Economic Journal, 99, 394., 28-61
1.1.2
Malthusian model
Malthus (1778) thought that economic cycles are due to the demographic cylces. He reasoned
that given a birth rate (b), the death rate (d) depends on the level of consumption, (C). That in
e¤ect determines the number of people and workers in the economcy. Given a production function
that relate output to employment, the number of workers then determines level of output. This in
turn determines the level of consumption, which determines the death rate next period and that
determines workers next period and so on.
d (Ct ) =) N =) Y =) C =) d (C)
The sequence of Malthusian reasoning: Smaller population =) higher income =) more consumption =) healthier and larger population =) lower income =) less consumption =) smaller
population (because of higher death rates) =) higher income . Economy returns to the original
13
position where started when a cycle is complete. In general Malthusian model can be summarised
as:
Households’problem:
X
t 1
M ax
U (Ct )
(14)
Subject to income form labour (Nt ), land (Lt ) and pro…t from the ownership of the …rm (
Amount of land is constant. Number of workers depends on the level of consumption.
Ct = wt Nt + rt Lt +
t ).
(15)
t
Firms’problem:
max
Nt
t
= ALt Nt1
wt Nt
rt Lt
(16)
3
(17)
Simplify the problem (assume life of only four periods):
M ax
U (c1 ) + U (c2 ) +
2
U (c3 ) +
U (c4 )
subject to
ct = wt + rt
Lt
t
+
for t = 1; 2; 3; 4
Nt
Nt
(18)
1
Charecterisation of Malthusian equilibrium: Competitive equilibrium is path of prices fwt ; rt g0
1
given fct ; yt ; Nt ; t g0 such that
1) given wt ; rt the sequence of ct ; t solve household problem
2) given wt ; rt the sequence of yt ; nt solve …rms problem
3) markets clear ct = yt and Nt = Nt
4) population dynamics is the di¤erence of birth and death rates as Nt+1 = Nt +Nt [b d (Nt )]
wt Nt rt Lt
5) …rm’s objective t = ALt Nt 1
Steady state in this model Ct = C Yt = Y and Nt = N
In steady state Nt+1 = Nt = N SS ; this means from Nt+1 = Nt +Nt [b d (Nt )] =) b = d (N ).
Transitional dynamics when N0 6= N SS then 1) Nt+1 > Nt if b > d (N ) or 2) Nt+1 < Nt if
b < d (N ) 3) Nt+1 = Nt if b = d (N )
Explain transitional diagram in (Nt+1 ; Nt ) space.
Criticism of Malthus: too rigid in technology; does not see prospects of better technology of
production, health care and birth controls; contribution of human capital in production. However,
larger population can be burden if the economy cannot educate and provide health care to its
population.
1.1.3
Two period model of consumption and saving
How is the interest rate determined in the classical model? Two period two consumer model gives
a good example.
M ax
U (C1i ; C2i ) = ln C1i +
Subject to
14
ln C2i
i = A; B
(19)
First period budget constraint:
C1i + bi = ! i1
(20)
C2i = bi (1 + r) + ! i2
(21)
Second period budget constraint:
here C1i and C2i are consumption in period 1 and 2 by household i = A; B.
! i1 and ! i2 are endowments in period 1 and 2 of household i = A, B; r is the real interest rate
and is the discount factor.
From (21)
C2i
! i2
1+r 1+r
substituting (22) in (20) gives the intertemporal budget constraint
bi =
! i2
C2i
= ! i1 +
1+r
1+r
Lagrangian for constrained optimisation is
C1i +
Li = ln C1i +
ln C2i +
! i1 +
! i2
1+r
(22)
(23)
C1i
C2i
1+r
(24)
First order conditions for optimisation:
1
@Li
= i
i
@C1
C1
@Li
= i
@C2i
C2
=0
1+r
(25)
=0
(26)
! i2
C2i
@Li
= ! i1 +
C1i
=0
(27)
@
1+r
1+r
Now dividing (25) by (26) gives the marginal rate of substitution (MRS1;2 ) between current and
future consumption :
M U1
Ci
= 2i = (1 + r) =) C2i =
M U2
C1
(1 + r) C1i
(28)
This is the e¢ ciency condition in consumption; todays consumption is worth more than tomorrows consumption. This is re‡ected in the relative price today of one unit consumption tomorrow,
(1 + r) ; pp12 = 1+r
1 = (1 + r) .
Budget is balanced for each agent in inter-temporal sense (present value of expenditure equals
present value of income).
From (27)
C2i
! i2
(1+r)C i
C1i + 1+r
= ! i1 + 1+r
Putting (28) in budgent constraint (27) it gives C1i + 1+r 1 =
(1 + ) C1i = ! i1 +
! i2
1+r
The demand for consumption in period 1 is
15
C1i =
1
(1 + )
! i1 +
! i2
1+r
(29)
Similarly the demand for consumption in period 2 is obtained by putting (29) in (28)
C2i =
(1 + r) C1i =
(1 + r)
(1 + )
! i1 +
! i2
1+r
(30)
Consumers are taking the interest rate as given while calculating their optimal demand as
shown in above equations. These demands need to be equal to supply for an intertemporal general
equilibrium to hold. For simplicity supply of commodities given by endowments of consumers in
B
a pure exchange economy like this. Here ! A
1 and ! 1 are endowments of each consumers A and
B
B in period 1 and ! A
and
!
in
period
2.
A
consumer
which has more asset tomorrow borrows
2
2
to smooth consumption today and one who has more today than tomorrow will lend in process of
inter-temporal optimisation. This result comes from the market clearing condition in each period:
B
C1A + C1B = ! A
1 + !1
(31)
B
C2A + C2B = ! A
2 + !2
(32)
By the Walras law if one of these two markets clear, another one will automatically clear. Thus
for the market clearing interest rate, put (29) and (30) in (31) as:
C1A + C1B
=
1
(1 + )
!A
1 +
!A
2
1+r
+
1
(1 + )
!B
1 +
!B
2
1+r
B
= !A
1 + !1
(34)
1
!A
!B
2
2
B
B
!A
+
!
+
+
= !A
1
1 + !1
(1 + ) 1
1+r 1+r
1
A
B
!A + !B
2 = (1 + ) ! 1 + ! 1
1+r 2
1+r =
(33)
B
1 !A
2 + !2
;
A
!1 + !B
1
B
!A
1 + !1 =
r=
B
1 !A
2 + !2
A
!1 + !B
1
(35)
B
!A
1 + !1
(36)
1
(37)
As can be seen the gross interest rate (1 + r) is determined in terms of preference ( ) and endowment
B
A
B
parameters, ! A
1 and ! 1 and ! 2 and ! 2 .
The Lagrange multiplier ( ) is value of income in terms of utils
=
1
=
C1i
1
1
(1+ )
!A
1 +
!A
2
1+r
=
1
1
(1+ )
!B
1 +
!B
2
1+r
(38)
Proof of the Walras Law
By the Walras law when one market clears other market automatically clears. This can be
proven by equivalence by putting (29) and (30) in (32)
16
C2A + C2B
=
(1 + r)
(1 + )
!A
1 +
!A
2
1+r
(1 + r)
(1 + )
+
!B
1 +
!B
2
1+r
(39)
B
= !A
2 + !2
(40)
(1 + r) A
!B
!A
2
B
!1 + !B
= !A
+ 2
1 +
2 + !2
(1 + )
1+r 1+r
(1 + )
!A + !B
2
(1 + r) 2
1
(1 + )
(1 + r)
(1 + r) =
B
!A
1 + !1 =
1
(41)
1
!A + !B
2
(1 + r) 2
(42)
B
A
B
!A
2 + !2 = !1 + !1
B
B
1 !A
1 !A
2 + !2
2 + !2
; r=
A
B
A
!1 + !1
!1 + !B
1
(43)
1
(44)
QED
Endowments
Table 3: Summary of two period general equilibrium model
Individual A
Individual B
A
A
B
!1 ; !2
!B
1 ; !2
h
i
Equilibrium interest rate
Life time income
r=
!A
1 +
Consumption in period 1
Consumption in period 2
Saving/borrowing period 1
Saving/borrowing period 2
Life time utility
Shadow price
1
!A
2
B
!A
2 +! 2
B
!A
+!
1
1
1
!B
1 +
1+r
!A
1
A
2
(1+ ) ! 1 + 1+r
!A
(1+r)
2
!A
1 + 1+r
(1+ )
S1A = ! A
C1A
1
S2A = ! A
C2A
2
A
A
A
U (C1 ; C2 ) = ln C1 + ln C2A
1
= C1i =
A
1
A + !2
1
!
1
1+r
(1+ )
1
(1+ )
(1+r)
(1+ )
!B
2
1+r
!B
2
1+r
!B
2
!B
1 + 1+r
!B
1 +
S1 = ! 1 C1
S2 = ! 2 C2
U (C1B ; C2B ) = ln C1B +
1
=
!B
1
(1+ )
ln C2B
2
!B
1 + 1+r
Summary of results of the two period model Q2. Extend two period two individual model to
a three period economy which is inhibited by the low, middle and high income households. Again
inter temporal optimisation by each involves maximising utility subject to its life time budget
constraint.
M ax
U (C1i ; C2i ; C3i ) = ln C1i +
i
2
ln C2i +
i
3
ln C3i
i = A; B; C
subject to budget constraints while young, adult and old as following:
17
(45)
C1i + bi1 = w1i
(46)
C2i + bi2 = bi1 (1 + r) + w2i
(47)
C3i = bi2 (1 + r) + w3i
(48)
whereC1i ; C2i ; C3i are consumptions for periods 1, 2 and 3 for type i agent and i2 and i3 are
subjective discount factors for period 2 and 3 consumptions with their values between 0 and 1.
Endowment of agent i for time t is given by wti with endowments for agent A, B and C for periods
1, 2 and 3 are w1A ; w1A ; w1A ; w1B ; w1B ; w1B ; w1C ; w1C ; w1C . Again each household is allowed to borrow
and lend at the interest rate r.
Markets clear for each good for each period:
C1A + C1B + C1C = w1A + w1B + w1C
(49)
C2A + C2B + C2C = w2A + w2B + w2C
(50)
C3A + C3B + C3C = w3A + w3B + w3C
(51)
What is the interest rates and equilibrium allocations in for short, medium and long terms in
this economy? State how to extend this model to ten households.
An exercise on Ricardian equivalence
1. Consider a two period economy with the preferences of households given by
U (C1 ; C2 ) = lnC1 +
ln C2
(52)
endowments for period 1 and 2 are given by fw1 ; w2 g and the public policy is fG1 ; G2 ; T1 ; T2 ; Bg.
(a) What are the consumption in periods 1 and 2 if r = 0:01.
= 0:95; fw1 ; w2 g =
f100; 150g ; fG1 ; G2 g = f20; 30g ; fT1 ; T2 g = f20; ?g : If the budget need to be balanced
intertemporally, what should be the tax in period 2 when tax is cut in the …rst period,
fT1 ; T2 g = f10; ?g?
(b) Prove the Ricardian Equivalence. Comment what this implies to the expansionary …scal
policy observed around the world in the current recession. Is this plausible?
Answer (a)
Prove that debt-…nancing is a burden on future generation (show that T2 > T1 when
D1 = G1 T1 > 0.
L = lnC1 +
C1 =
1
1+
(w1
ln C2 +
T1 ) +
C1 +
w2 T2
1+r
C2
1+r
; C2 =
18
(w1
T1 )
(1 + r)
1+
(w1
w2 T2
1+r
T1 ) +
(53)
w2 T2
1+r
(54)
w2
G2
150
30
Here intertemporal endowment is w1 + 1+r
= 100+ 1:01
= 248:52 and G1 + 1+r
= 20+ 1:01
= 49:7.
Assume budget balance in each period
C1 =
1
1 + 0:95
(100
150 30
1 + 0:01
20) +
; C2 =
0:95 (1 + 0:01)
1 + 0:95
(100
20) +
150 30
1 + 0:01
(55)
if T1 = 0 T2 should be very high to meet the goverment expenses. Borrowing will be 20 in the
frist period and zero in the second period. T2 = 20 (1:01) + 30 = 50:2:
T2
G2
= T1 +
1+r
1+r
If G1 rises and T1 falls then either G2 should fall or T2 should rise or both.
G1 +
1.1.4
Pure exchange model
Q2. Imagine a world dominated by three large corporations each of which supply three di¤erent
products, W1 , W2 and W3 . These corporations are owned by three representative households,
indexed by i = A, B and C; their share in company supplies are given by W1i , W2i and W3i .
Demands for these products by each household are represented by X1i , X2i and X3i . Each household
i maximises its own welfare subject to its own budget constraint. Relative price of a commodity
adjusts until its demand equals its supply and thus is consistent with the household optimisation.
Households prefer each three goods equally. Preferences and constraints for household type i are
given by
M ax
U (X1i ; X2i ; X3i ) = X1i X2i X3i
i = A; B
(56)
Subject to the budget constraint:
I i = P1 W1i + P2 W2i + P3 W3i = P1 X1i + P2 X2i + P3 X3i
(57)
Markets clear (demand equals supply):
X
X
X
X
X
X
X1i =
W1i ;
X2i =
W2i ;
X3i =
W3i
(58)
Welfare maximising demand functions for given preferences and budget constraints (derive them
using Lagrangian constrained optimisation):
X1i =
Ii
Ii
Ii
; X2i =
; X3i =
3P1
3P2
3P3
(59)
Demand-supply equilibrium condition for each market is:
3
3
3
3
3
X
X
X
X
X3
X
X3
Ii
X1i =
=
W1i ;
X2i =
W2i ;
X3i =
W3i
3P
1
i=1
i=1
i=1
i=1
i=1
i=1
i=1
(60)
In the …xed supply and distribution situation as above, endowments W1i ,W2i ,W3i for each agent
i is taken as given by ownership agreements. Relative prices can be obtained by solving above three
equations. It also determines the income of each consumer-producer household, I i .
19
1
I A + I B + I C = W1A + W1B + W1C
3P1
A
B
C
I +I +I
1
=
P1 =
3 W1A + W1B + W1C
3
3
P
(61)
Ii
i=1
3
P
i=1
(62)
W1i
When the price of good 1 is considered as a numeraire: i.e. P1 = 1 then the relative prices of
good 2 and 3 can be determined using above conditions
3
3
3
3
X
X
X
X
i
i
i
I = 3 W1 = 3P2 W2 = 3P3 W3i
i=1
i=1
3
P
P1
= i=1
3
P
P2
i=1
i=1
3
P
W2i
(63)
i=1
3
P
W3i
W3i
P1
P2
i=1
i=1
;
;
= 3
= 3
P i P3
P i
P3
W1i
W1
W2
i=1
P1 = 1; P2 =
3
P
i=1
3
P
W1i
i=1
3
P
(64)
W1i
i=1
3
P
; P3 =
W2i
i=1
(65)
W3i
i=1
Now it is easy to evaluate income and demand accurately for each household i:
i
I =
P1 W1i
+
P2 W2i
+
P3 W3i
=
W1i
+
3
P
i=1
3
P
i=1
W1i
W2i
+
W2i
3
P
i=1
3
P
i=1
W1i
W3i
(66)
W3i
Demand for each household for each of three goods is evaluated as:
X1i =
X2i =
0
1B i
Ii
= B
W +
3P1
3@ 1
1
Ii
=
3P2
3
1
3
P
W1i
i=1
3
P
W2i
i=1
0
3
P
W1i
i=1
W2i
3
P
W2i
i=1
B i
BW1 +
@
20
3
P
+
W1i
i=1
W2i
3
P
W2i
i=1
1
3
P
W1i
C
i=1
W3i C
A
3
P i
W3
i=1
+
3
P
i=1
3
P
i=1
W1i
W3i
(67)
1
C
W3i C
A
(68)
X3i =
Ii
1
=
3P3
3
1
3
P
W1i
i=1
3
P
W3i
i=1
0
3
P
3
P
W1i
W1i
1
C
B i i=1
BW1 +
W2i + i=1
W3i C
A
@
3
3
P i
P i
W2
W3
i=1
(69)
i=1
It is possible now to evaluate the welfare of each household:
U (X1i ; X2i ; X3i ) = X1i X2i X3i
U (X1i ; X2i ; X3i )
= X1i X2i X3i =
1
3
1
3
P
W1i
i=1
3
P
W2i
i=1
1
3
1
3
P
W1i
i=1
3
P
W3i
i=1
Parameterisation and solution
following table:
0
0
i = A; B; C
3
P
(70)
3
P
W1i
W1i
1
C
1B
BW1i + i=1
W2i + i=1
W3i C
A
@
3
3
P i
P i
3
W2
W3
i=1
3
P
W1i
i=1
3
P
W1i
1
(72)
1
(73)
B i i=1
C
BW1 +
W2i + i=1
W3i C
A
@
3
3
P i
P i
W2
W3
0
i=1
i=1
3
P
3
P
W1i
W1i
B i i=1
C
BW1 +
W2i + i=1
W3i C
@
A
3
3
P i
P i
W2
W3
i=1
(71)
i=1
The ownership composition of households were given as in the
Table 4: Endowment Structure of Households
W1 W2 W3
A
20
70
60
B
30
50 100
C
50
80 140
Total 100 200 300
Find the equilibrium prices and allocations and utility for each household.
With these endowments and the share parameters the relative prices that equate demand and
supply in each market from the solution of the model are as given in the following table.
Decimals matter.
Following observations can be made from above solutions of the model:
1) Given that households prefer each good equally goods with larger amount of supplies have
lower price. Relative price of good 3 is 0.33 compared to 0.5 for good 2 and 1 for good one. This
21
Table 5: Market Price and Optimal Allocations
Market
Goods 1
2
3
Price
1
0.5
0.3333
Supply 100
200
300
Household Demand
1
2
3
A
25.0
50.0
75.0
B
29.444 58.889 88.333
C
45.556 91.111 136.667
Total
100
200
300
Table 6: Income and utilities
A
B
C
Income
75
88.333
136.667
Utility 93750 153165.6 567251
price structure would change with di¤erences in preferences of households and the endowments of
goods in the economy.
2) Income and expenditure of households is balanced in the equilibrium.
3)Welfare of households under market based allocations depends not only on their endowments
and preferences but also that of others in the economy. Economic agents are interdependent, choices
of one household a¤ects possibilities of choices by others.
4) How would these prices change if there is a 20 percent tax on the richest household and
60 percent and 40 percent of the collected revenue are distributed to bottom and middle income
households.
Decimals matter.
Who gains and who loses from the tax reform?
22
Table 7: Allocations after tax
Household Demand
1
2
A
30.4667 60.9333
B
33.0889 66.1778
C
36.4444 72.8889
Total 100
200
Income
Utility
1.1.5
and transfer
3
91.400
99.2667
109.3333
300
Table 8: Income and utilities
A
B
C
91.4
99.267
109.333
169678.2097 217369.0975 290432.5266
Ricardian Trade Model for Comparative Advantage
There are two countries indexed by j, producing two goods, manufacturing and services. Each of
them have an option to be self reliant or to trade on the basis of comparative advantage. The
exchange rate is determined by the relative prices of two commodities in the global market.
Preferences in country j are expressed by its utility function in consumption of good 1 and 2 ,
C1j and C2j respectively:
j
max
U j = C1j
C2j
1
j
(74)
Income of country j is obtained from the wage income in sector 1 and sector 2 plus the transfers
to country j
I j = w1j Lj1 + w2j Lj2 + T Rj
(75)
where Lj1 and Lj2 are labour employed in sector 1 and sector 2 w1j and w2j are corresponding
wages respectively and T Rj is the transfer income.
Technology constraints in sector 1 in country j
X1j = aj1 :Lj1
(76)
where aj1 is the productivity of labour in sector 1 in country j.
Technology constraints in sector 2 in country j
X2j = aj2 :Lj2
(77)
aj2
is the productivity of labour in sector 2 in country j.
where
Resource constraint in country j de…ned by the labour endowment as:
Lj = Lj1 + Lj2
Production possibility frontier of country j now can be de…ned as
23
(78)
Table 9: Percentage gains and loses from Tax-Transfer System Compared to No Tax and Transfer
Base Case
A
B
C
Income 17.9% 11.01% -25.0%
Utility 44.7% 29.5% -95.3%
Lj =
1
aj1
:X1j +
1
:X2j
aj2
(79)
Given above preferences the demand for good 1 in country j is
j
:I j
P1
C1j =
(80)
the demand for good 2 therefore is:
j
1
C2j =
:I j
(81)
P2
Theoretically two trade arrangements are possible in this model. First one is an autarky equilibrium in which each country is separate and isolated from another. It produces just for its own
consumption and no trade take place between these two countries. Such autarky solution is close
to the production arrangement when countries were adopting ISI trade strategy.
Proposition 1 Autarky solution is Pareto dominated by trade equilibrium for reasonable parameters of preferences and technology.
This is proven below by analytical and numerical solutions.
A Lagrangian function is used to express how each country j maximises welfare subject to its
production possibility frontier constraint under the autarky equilibrium as:
"
#
1
1
(1
j
j)
$j = X1;j X2;j
+
Lj
X1;j
X2;j
(82)
aj1
aj2
First order conditions with respect to X1j and X2j and
@$j
=
@X1;j
j
j X1;j
@$j
= (1
@X2;j
@$j
= Lj
@
1
(1
X2;j
j)
aj1
(
j
j ) X1;j X2;j
1
aj1
X1;j
(1
24
=0
j)
1
aj2
aj2
(83)
=0
(84)
X2;j = 0
(1
)
X2;j j
( j)
j
j )X1;j X2;j
j
j X1;j
From the …rst two …rst order conditions
as:
(85)
1
=
j
(1
j)
X2;j
X1;j
=
aj2
aj1
X2;j =
(1
j)
aj2
X1;j
(86)
aj1
is found now putting this condition in the production possibility frontier
j
optimal value of X1;j
constraint.
1
1
1 (1
X1;j + j X2;j = j X1;j + j
j
a1
a2
a1
a2
aj2
j)
1
aj1
j
X1;j =
1
aj1
X1;j 1 +
(1
j)
j
j a1 Lj
X1;j =
= Lj
(87)
j
(88)
Similarly the optimal value of X2;j is found by
(1
j)
aj2
(1
j)
aj2
j
j
(89)
j a1 Lj = (1
j ) a2 Lj
a1
aj1
j
For each of j country amount produced depends on productivity and preferences parameters
and the endowment of its labour input. The autarky welfare level is:
X2;j =
j
U j = (X1;j )
j
X1;j =
j
(X2;j )
1
j
=
j
j
j a1 Lj
(1
j
j ) a2 Lj
(1
j)
(90)
Thus the level of welfare in country j is determined in terms of its preferences for consumption
of good 1 and 2 as re‡ected by j and its own production technology as re‡ected in aj1 and aj2 .
Numerical version of this model is applied to China and the US taking the population as rough
indicator of its resource in production. US has 365 million population and China has 1200 million
population. US is more productive in producing services goods X2 whereas China has more advantage in producing manufacturing goods X1 . Preferences are similar but technologies are di¤erent.
These parameters are set out in Table 1.
Table 10: Parameters of the Autarky Model
a1 a2
L
US
China
0.6
0.6
2
5
5
2
365
1200
Under the autarky equilibrium these two economies are completely isolated and produce only
for domestic consumption. The optimal production and consumption and employment of labour
for both sectors, prices of commodities and labour, and utility for the representative household are
as given in Table 2. In per capita terms citizens of the US and China have welfare of 1.46 and
1.76 respectively.
Table 11: Parameters of the Autarky Model
X1
X2 L1
L2
U
p2
US
China
438
3600
730
960
219
720
146
480
537.3
2121.7
1.67
0.27
Each country produces both goods in no trade equilibrium which as explained here is very
ine¢ cient. Welfare can be improved by making these countries trade.
25
Analytical Solutions for Trade Equilibrium A representative household in each country
maximises its welfare subject to its budget constraint. Demand for goods are derived by standard
constrained optimisation on supply side for each country j . Under trade equilibrium it is optimal
for each country to specialise in goods in which it has comparative advantage. The optimisation
problem and the …rst order conditions for constrained optimisation are given as follows:
(1
j)
$j = X1;jj X2;j
+ [Ij
P1 X1;j
P2 X2;j ]
(91)
First order conditions:
@$j
=
@X1;j
j
j X1;j
@$j
= (1
@X2;j
j
(1
1
(1
X2;j
P1 X1;j
(1
(
j ) X1;j X2;j
X2;j =
=
j)
(1
j)
j
j
(1
j)
X1;j =
P1
P2 X1;j
j Ij
P2 = 0
P2 X2;j = 0
j
(1
P1 = 0
j)
j)
X2;j
j
P1 X1;j + P2 X2;j = P1 X1;j + P2
j)
(
j
j ) X1;j X2;j
@$j
= Ij
@
j X1;j
1
X2;j
P1
=
P2
j ) X1;j
P1
X1;j
P2
(92)
(93)
(94)
(95)
(96)
= Ij
; X2;j =
(1
j ) Ij
P1
P2
Global market clearing conditions for goods 1 and 2 are
(97)
N
X
X1;j = X1
(98)
N
X
X2;j = X2
(99)
j
j
Prices adjust until this equilibrium condition holds.
Under complete specialisation, country 1 US specialises in services X2 and produces 1825 units
of it. China specialises in manufacturing X1 goods and produced 6000 units of it. It is easy to
determine China’s income if we choose good 1 as numeraire setting P1 = 1.
I c = P1 X1 = 1
6000 = 6000
(100)
Relative price of good 2, P2 need to be determined to …nd the level of income in the US. This
can be done using the global market clearing condition
c c
:I
:I u
+
= 0:6 (1825
P1
P1
u
26
P2 ) + 0:6 (6000) = 6000
(101)
6000 3600
= 2:192
1095
Now it is easy to determine the income of the US as:
P2 =
I u = P2 X2 = 365
5
P2 = 1825
(102)
P2 = 1825
2:192 = 4000:4
(103)
Since income level for both China and the US are determined, it is now easy to determine the
level of demand in both countries:
X1;u =
X2;u =
(1
u Iu
= 0:6 (4000:4) = 2400:2; X1;ch =
P1
u ) Iu
P2
=
(1
0:4 (4000:4)
= 730; X2;ch =
2:192
ch Ich
= 0:6 (6000) = 3600
P1
ch ) Ich
P2
=
0:4 (6000)
= 1094:9
2:192
(104)
(105)
Solutions of both autarky and trade equilibria are given in Table 3 and 4. Given the preferences and technology speci…cations, with complete specialisation both countries gain from trade.
Comparative static analysis of trade can be done changing the preference or technology parameters.
Table 12: Comparing Specialisation and Autarky Regimes
Production
Autarky
Trade
US
China
Consumption
Autarky
Trade
X1
X2
X1
X2
C1
C2
C1
C2
438
3600
730
960
0
6000
1825
0
438
3600
730
960
2400.2
3600
730
1094.9
Table 13: Comparing Employment and Welfere under Specialisation and Autarky
Employment
Autarky
Trade
US
China
Uitlity
Autarky Trade
L1
L2
L1
L2
U
U
219
720
146
480
0
1200
365
0
537.7
2121.7
1490.9
2236.3
Gains from trade may be distributed di¤erently across countries (Bhattarai and Whalley (2006)).
Further there are opportunities for bargaining on the share of those gains particularly from dynamic
strategic considerations and the basic elements required for such dynamic model is provided in the
next section.
1.1.6
Intertemporal balance in budgets of households, government and …rms
Classical model depends on economic disciplines by not only households but by governments, …rms
and economy as a whole; it is subject to No-Ponzi conditions for each agent (that means not default
and no bankrupcy in the entire model horizon).
27
Household accumulates debt whenever more debt f(Bt
servicing (rBt 1 ) is above current saving (Yt Ct ).
Bt
Bt
1
=
Bt
Bt
1
= Yt
Bt
Ct + rBt
1)
> 0g if the amount for debt
1
(106)
Yt + Ct
Ct Yt
Bt
=
+
1+r
1+r
1+r
(107)
Ct+1 Yt+1
Bt+1
Yt+1 + Ct+1
=
+
1+r
1+r
1+r
(108)
By successive iteration forward
Bt =
Bt+1
Bt+2
Bt+1 =
(109)
Bt+2
Ct+2 Yt+2
Ct+1 Yt+1
+
+
2
2
1+r
(1 + r)
(1 + r)
(110)
Ct+1 Yt+1
Bt+n
Ct+2 Yt+2
+ :::: +
+
n
2
1+r
(1
+ r)
(1 + r)
(111)
Bt =
Bt =
Yt+2 + Ct+2
Ct+2 Yt+2
Bt+2
=
+
1+r
1+r
1+r
For the budget balance over the life time
1
X
Ct+2
t=1 (1
+ r)
i
= B0 (1 + r) +
1
X
Yt+2
t=1 (1
i
+ r)
; )
Bt+n
Lim
=0
t ! 1 (1 + r)n
(112)
The life time budget should balance, though the household is free to borrow and lend in the
…nancial markets. The present value of expenditure should equal the present value of income
(endowments). When productive …rms are included this life time budget set is slightly modi…ed
1
P
t+1
and includes value of the …rm V =
which is derived from the Tobin’s q.
(1+r)i
t=1
1
X
Ct+2
t=1 (1
i
+ r)
= B0 (1 + r) +
1
X
Yt+2
t=1 (1
+ r)
i
+V;
Bt+n
Lim
=0
t ! 1 (1 + r)n
(113)
Similar logic applies for the governments debt dynamics:
Dt
Dt
Dt =
1
=
Dt
Dt
1
= Tt
1
Tt + Gt
Gt Tt
Dt
=
+
1+r
1+r
1+r
Gt+1 Tt+1
Dt+1
+
;
1+r
1+r
Dt+1 =
Gt+2 Tt+2
Dt+2
+
1+r
1+r
(114)
(115)
(116)
Gt+1 Tt+1
Gt+2 Tt+2
Dt+2
+
+
2
2
1+r
(1 + r)
(1 + r)
(117)
Gt+1 Tt+1
Gt+2 Tt+2
Dt+n
+
+ :: +
n
2
1+r
(1 + r)
(1 + r)
(118)
Dt =
Dt =
Gt + rDt
28
1
X
Gt+2
t=1 (1
i
+ r)
= D0 (1 + r) +
1
X
Tt+2
t=1 (1
i
+ r)
;
Dt+n
Lim
=0
t ! 1 (1 + r)n
(119)
Tobin’s q and investment For …rms investment decisions are guided by inter temporal optimisation. It depends on the ratio of market value to the cost of capital assets (Tobin’s q):
1
I1 = K 1
No investment occurs if q
(q
1) ;
I1 = 0 if q
1
(120)
1. Here qt represents excess return on investment:
Lt (It ; Kt ; qt ) =
1
X
t
t=0 (1
t
+ r)
1
X
qt
t [Kt+1
t=0 (1 + r)
(1
) Kt
It ]
(121)
In multisectoral and multi-household model the inter-temporal balance conditions usually are
explained as following:
Households: prevent value of expenditure = present value of income
N
T X
X
h
Pi;t 1 + thci Ci;t
=
t=0 i=1
T
X
rt (1
tk ) Kth + Rth + wh Lh
(122)
t=0
Firms: Present value of revenue = present value of cost
"
#
T
T
H
X
X
X
h h
Pi;t Yi;t =
rt (1 tk ) Ki;t +
wt Li;t
t=0
t=0
Government: present value of public spending = present value of revenue
!
H
T
T
X
X
X
Rth
RVt +
Gt =
G=
t=1
(123)
h=i
t=1
(124)
h=1
where RVt is the total tax revenue from direct and indirect taxes:
RVt =
T X
N
h
X
X
h
Pi;t thci Ci;t
+
rt tk Kth + wh Lh
t=0 i=1
(125)
h=1
Economy: present value of exports = present value of imports
T X
N
X
P Ei;t Ei;t
t=0 i=1
T X
N
X
=
P Mi;t Mi;t
(126)
t=0 i=1
For any period: current account de…cit (surplus) = capital in‡ow (out‡ow)
N
X
P Ei;t Ei;t
i=1
N
X
P Mi;t Mi;t =
F
(127)
i=1
Exchange rate appreciation (depreciations) should follow trade surplus (de…cit) but
not all countries follow this rule.
29
1.1.7
Ramsey growth model
In Ramsey (1928), a benevolent social planner or a representative household optimises the lifetime
utility from consumption in each period and his saving equals investment in equilibrium. Investment
generates additional capital stock and enhances the productive capacity of the economy. The basic
Ramsey model can be expressed in …ve functions expressing the utility of a representative household,
production of a …rm, the process of capital accumulation, conditions for market clearing and the
initial state of the economy:
The solutions for optimal consumption from this in…nite horizon problem can be obtained by
substituting the consumption term from the market clearing condition in the utility function and
using the standard …rst order conditions for utility maximisation and by analysing optimal conditions for any two periods in terms of control and state variables that apply to all other periods in
the model as illustrated below.
max
Uo =
Ct
1
X
t
ln (Ct )
(128)
t=0
Subject to production technology:
Yt = At Kt ; 0 <
<1
(129)
+ It
(130)
It = St
(131)
Capital accumulation:
Kt = (1
) Kt
1
Market clearing:
Yt = Ct + St
Initial (boundary) condition:
Ko = Ko
(132)
Solution of the Ramsey Model
max
Uo =
Kt
1
X
t
ln [AKt
fKt+1
t=0
(1
) Kt g]
(133)
In…nite horizon problem can be solved only for two periods to give Euler equations
Uo
= ::::: +
+
t+1
@Uo
=
@Kt+1
t
ln [AKt
ln AKt+1
t
Ct
Ct+1
=
Ct
fKt+1
fKt+2
(1
(1
) Kt g]
) Kt+1 g + ::
(134)
t+1
+
AKt+11 + (1
Ct+1
) = 0:
(135)
t+1
t
AKt+11 + (1
In steady state Ct+1 = Ct = C; Kt+1 = Kt = K
30
)
(136)
h
1=
1
AK
+ (1
)
Steady State Capital Stock in Ramsey Model
K
1
1 1
A
=
(1
1
K=
1
) =
(1
A
i
(137)
(1
A
)
(138)
1
)
1
(139)
Steady State Output in Ramsey Model
Y =A
1
(1
A
)
1
(140)
Steady state investment
I=K
C=Y
(1
I=A
1
)K = K =
1
(1
A
)
1
(1
A
)
1
(1
A
Utility of the representative household in the steady state:
"
1
1
X
1
1
(1
)
t
Uo =
ln A
A
C
t=0
(1
A
1
1
(141)
1
)
)
1
(142)
1
1
#
b
=U
(143)
Transitional dynamics explains how an economy moves towards the steady state if it o¤ this
path Ct+1 7 Ct and C and Kt+1 7 Kt :
Exercise: First solve this model in excel . Then in GAMS.
Table 14: Solution of the Classical Model
L
Base parameters 0.4 1.5 0.99 0.02 0.7 100
gl
0.03
0.3
See the steady state soution of basic Ramsey model
Table 15: Solution of the Ramsey Model
K
Q
R
C
I
Base solution 0.31 4.70 0.99 4.6 0.01
Where there is a positive shock to the technology, it raises output, cpaital stock, investment,
consumption as shown in the imulse response (generated by RAMSEY_DEMO.MOD, see software
section).
MATLAB/dynare code is popular for DSGE model but it use square system and linearisation
to solve this. GAMS seems better for to solve a non-linear Rasey problem.
31
See: Ramsey, F.P. (1928) “A Mathematical Theory of Saving,” Economic Journal 38, 543-559.
Aghion P. and P. Howitt (1998) Endogenous Growth Theory, MIT Press, Cambridge MA.
Barro R. J. and Sala-i-Martin (1995) Economic Growth, McGraw Hill.
Romer D. (2008) Advanced Macroeconomic Theory, McGraw Hill.
Impacts of Cost of Financial Intermediation The above two scenarios assume an ideal
benevolent social planner and a smooth and e¢ cient …nancial market. The …nancial markets are
incomplete in the real world and the investments are not equal to saving because of intermediation
cost. When represents a charge imposed by …nancial institutions in the intermediation process
with a higher value of representing more ine¢ ciency in the …nancial system then = (1 + )
of saving is required for one unit of investment. Higher intermediation cost, (higher values of )
reduces capital, output, consumption and investment and modi…es the macroeconomic balance to:
Ct = Yt
It = AKt
fKt+1
Kt (1
)g
(144)
Now the …nancial system deviates from the standard Arrow-Debreau competitive equilibrium;
1
only (1+
) fraction of saving is channelled onto investment; thus the investment equals savings net
of intermediation cost St = It = (1 + ) It as in Pagano (1993); It amount of savings is wasted
in process of …nancial intermediation. As such a higher value represents more ine¢ ciency in the
…nancial system. Solutions of models with are further modi…ed as:
max
Uo =
Kt
1
X
t
ln [AKt
t=0
fKt+1
(1
) Kt g]
(145)
In…nite horizon problem can be solved only for two periods to give Euler equations. As before
part of this in…nite sum actually can be written as:
Uo
=
::::: +
+
t+1
t
ln [AKt
fKt+1
ln AKt+1
fKt+2
32
(1
(1
) Kt g]
) Kt+1 g + ::
(146)
First order conditions or the Euler equation in terms of consumption as a control and capital
stock as a state variable changes to
@Uo
=
@Kt+1
t
Ct
Ct+1
=
Ct
t+1
+
AKt+11 + (1
Ct+1
) = 0:
(147)
t+1
t
AKt+11 + (1
)
(148)
As above this equation implies that the ratio of consumption between two periods should equal
the discounted value of the marginal product of capital in the next period, in a competitive equilibrium this should equal the gross interest rate. Again for an optimal allocation between consumption
and savings, loss in utility by not consuming now should equal gain from production (consumption)
next period.
As above in a steady state both capital and consumption remain constant; Kt+1 = Kt = K and
alsoCt+1 = Ct = C; . Like above the term in the parenthesis of becomes a constant number and
can be taken out of the summation sign
max
Kt
h
Uo = ln AK
K + (1
)K
1
iX
t
(149)
t=0
This again gives a constant utility in the steady state. The steady state capital stock K can be
found using steady state
of consumption
in equation.
h values
i
t+1
1
1
Ct+1
C
=
=
AK
+
(1
)
=)
AK
+ (1
)=
t
Ct
C
K=
1
1
A
1
(1
)
(150)
Thus in case of …nancial intermediation the output, investment and consumption in the steady
state are:
Y = AK = A
I=K
C=Y
I=A
(1
1
A
1
1
A
)K = K =
1
(1
1
A
1
1
(1
)
(151)
)
1
1
(1
1
A
)
(152)
1
1
(1
)
(153)
In addition to , , A and the steady state consumption, investment and capital stock also
depend on the cost of intermediation = (1 + ).
See: in Bhattarai K (2005) Consumption, Investment and Financial Intermediation in Ramsey
Models, Applied Financial Economics Letters 1:6:1-5; See model in excel and GAMS.
1. Consider a standard version of Ramsey’s optimal growth model
33
max
U=
1
X
t
ln(ct )
0<
<1
(154)
t=0
subject to:
Yt = AKt
0<
Kt+1 = Kt (1
<1
) + It
(155)
(156)
Yt = Ct + It
(157)
K0 = K0
(158)
(a) Solve this model for the capital stock, output, consumption and investment in the steady
state.
(b) In what sense is this model di¤erent from the Solow growth model?
(c) How would you solve this model if the technology A is given by a stochastic process?
At+1 = At + "t where "t ~ N (0; 2 ):
(d) Financial intermediaries take away a certain fraction of saving. Let
represent the
fraction of savings taken away (wasted) by them while (1
) fraction of saving is
channelled to investment. As such a higher value represents more ine¢ ciency in the
…nancial system. How does a¤ect the saving and investment and capital accumulation
in this economy?
(e) Study the impacts of capital income taxation in economic growth using Ramsey’s model
of optimal growth. Use GAMS program Captax.gms to compute the optimal growth.
Exercise on Multihousehold Multisectoral General Equilibrium Model with Money
The representative household for a country receives utility from consuming both goods X1 and X2
given by a Cobb-Douglas utility function as:
U = X1 1 X21
1
(159)
Technology of production of goods Y1 and Y2 are respectively
Y1 = L1 K11
(160)
Y2 = L2 K21
(161)
Resources of …rm 1 and 2 are
C1 = w1 L1 + r1 K1
(162)
C2 = w2 L2 + r2 K2
(163)
Households receive income from labour and capital, from transfers (T R) and net borrowing (B) as:
34
I = w1 L1 + r1 K1 + w2 L2 + r2 K2 + T R + B
(164)
Market clearing conditions in goods market are:
X1 = Y1
G1
(EX1
IM P1 )
(165)
X2 = Y2
G2
(EX2
IM P2 )
(166)
Labour market clearing implies:
L1 + L2 = L
(167)
K1 + K2 = K
(168)
P1 Y1 + P2 Y2 = P:Y
(169)
P:Y = M S:V
(170)
Capital market clearing implies:
Aggregate volume of output:
Monetary Sector
Quantity theory of money implies
where P is price level, Y national income, M S money supply and V the velocity of circulation.
Initial reserve (R) of the banking system constitutes of currency (C) and initial demand deposit
(D0)
R = C + D0
Currency in circulation is
(171)
fraction of total reserves
C = :R
Initial deposit is the remaining (1
(172)
) of initial reserve
D0 = (1
) :R
(173)
Total deposit (T D) is inversely related to the required reserve (rr) ratio
D0
(174)
rr
Aggregate money supply in the economy constitutes of currency in circulation plus the total deposit
TD =
MS = C + TD
(175)
Government Sector
Government collects revenue (RV ) from direct taxes on capital (tr1: , tr2: ), labour (tw1: ; tw2: )
and indirect tax on commodities (t1: ; t2: ) as:
35
RV = t1: P1: X1 + t2: P2: X2 + tr1: r1: K1 + tw1: w1 :L1 + tr2: r2 K2 + tw2: w2 :L2 + T R + B
(176)
Aggregate government expenditure (G) is spent in public consumption from both sectors (G1 ; G2 )
G = G1 + G2
(177)
Government expenditure on sector 1 and 2 goods are g1 and g2 fractions of its revenue:
G1 = g1 :RV
(178)
G2 = g2 :RV
(179)
Budget de…cit is the di¤erence between government spending and the revenue:
B=G
RV
(180)
External sector
Exports from sector 1 and 2 ,EX1 and EX2 and imports e1 and e2 .fractions of sectoral output
as:
EX1 = e1 :Y1
(181)
EX2 = e2 :Y2
(182)
Imports by sector 1 and 2, IM P1 and IM P2 are m1 and m2 .fractions of sectoral output as:
IM P1 = m1 :Y1
(183)
IM P2 = m2 :Y2
(184)
The real exchange rate is given by the ratio of total value of exports to the total value of imports:
ER =
P1 :EX1 + P2: EX2
P M1 :IM P1 + P M2: IM P2
(185)
Solve this model numerically for plausible values of parameters. Do sensitivity tests with respect
of changes in preferences, technology, …scal and monetary policy variables. Analyse the macro and
microeconomic impacts of devalutaion or depreciation of home currency on the basis of a well
speci…ed small open economy model. Provide real world examples (Bhattarai and Okyere (2013)
have multi-household multisectoral dynamic general equilibrium model for Ghana).
References
[1] Bhattarai K (2011) General Equilibrium Impacts of Monetary and Fiscal Policies on Welfare
of Households in South Asia, Review of Development Economics, 15:4:745-757, October.
[2] Hicks, J. R. (1937) Mr. Keynes and the “Classics”; A Suggested Interpretations, Econometrica
5:147-159.
36
[3] Keynes J.M. (1936) The General Theory of Employment, Interest and Money, MacMillan and
Cambridge University Press.
[4] Pigou A.C. (1947) A Study in Public Finance, Macmillan, London.
[5] Ramsey, F.P. (1928) A Mathematical Theory of Saving, Economic Journal 38, 543-559.
[6] Samuelson P.A. (1939) Interaction between the Multiplier Analysis and the Principle of Acceleration, Review of Economic Statistics, May, pp. 75-78.
[7] Sorensen PB and H. J Whitta-Jacobsen (2010) Introducing Advanced Macroeconomics, McGraw Hill.
[8] Minford P. and D. Peel (2002) Advanced Macroeconomics: A Primer, Edward Elgar Publishing.
[9] Simon and Blume (1994) Mathematics for Economists, Norton.
[10] Shone Ronald (2002) .Economic Dynamics, Cambridge.
1.2
Keynesian Model: Hicksian Synthesis
Aggregate demand:
Y = C +I +G
(186)
C = C (Y
(187)
Consumption function
T)
Investment
I= I(r)
(188)
M s = M (Y; R)
(189)
Money Market
Above four equations are further reduced into two equations for the goods and money markets
as:
Y
C (Y
T)
I(r) = G
(190)
M s = M (Y; R)
(191)
Y and r are implicit functions of G, T and M s :Take total di¤erentiation of these two equations
dY
C 0 (Y
T ) dY
I(r)dr = dG + C 0 (Y
@M
@M
dY +
dr = dM s
@Y
@r
Solve this model for dY and dr
37
T ) dT
(192)
(193)
1
C 0 (Y
T)
I(r)
@M
@Y
dY
dr
=
dY
dr
@M
@r
C 0 (Y
1
T)
I(r)
dr =
1
D
1
@M
@r
@M
@Y
dY =
dG + C 0 (Y T ) dT
dM s
=
1
D
dG + C 0 (Y T ) dT
dM s
1
C 0 (Y
dG + C 0 (Y T ) dT
dM s
I(r)
@M
@r
T ) dG + C 0 (Y T ) dT
dM s
@M
@Y
(194)
(195)
(196)
(197)
Impacts of monetary and …scal policies on output and interest rates can be evaluated using
above two equations. Money supply is a¤ected by the central bank by in‡uencing the interest rates
or other credit channels and …scal policy is a¤ected by changing various components of taxes or
public spending. It is a comparative static analysis thus far. Hicks (1937) had integrated Keynesian
ideas nicely like this. Samuelson (1939) multiplier-accelerator model provides good dynamics in the
system.
Another example
Y = F (K; N )
Fk > 0; FN > 0; Fkk < 0; FN N < 0:
(198)
Consumption
C = c Y d ; Y d = (1
)Y
(199)
Investment
Labour demand
I = I(r)
(200)
W
= FN (N; K)
P
(201)
W = W0 + W (N )
(202)
Labour supply
W (N ) =
Z
0 for N 5 N
(203)
+for N > N
Money market equilibrium conditions:
M
= M (Y; r)
P
My > 0; Mr < 0
(204)
Net exports
NX = X
IM
(205)
Equilibrium condition
Y = C + I (r) + G + N X
38
(206)
1.2.1
Soution of the Keynesian Model
First …nd the reduced form of the system
F (N; K ) = c (1
) F (N; K) + I (r) + G + N X
W
= FN (N; K)
P
(208)
M
= M (F (N; K ) ; r)
P
(209)
Take the total di¤erentiation
FN dN + FK dK = c (1
) dF (N; K) + d (c (1
FN dN + FK dK = c (1
) FN dN + c (1
dW
P
(207)
)) F (N; K) + Ir dr + dG + d (N X)
) FK dK
cd F (N; K) + Ir dr + dG + d (N X) (210)
W
dP = FN N dN + FN K dK
P2
(211)
dM
M
dP = My FN dN + My FK dK + Mr dr+
(212)
P
P2
By further expansion and rearrangement for endogenous variable labour (dN), price (dP) and
interest rate (dr) this model is succinctly written as:
FN dN
c (1
) FN dN
Ir dr = c (1
My FN dN +
) FK dK
cd F (N; K)
M
dM
dP + Mr dr =
2
P
P
dW
W
dP =
P2
P
Or this can be written in a matrix notation
FN N dN +
2
4
2
= 4
(1
c (1
c (1
)) FN
My FN
FN N
) FK dK
0
M
P2
W
P2
FK dK + dG + d (N X) (213)
My FK dK
FN K dK
32
3
Ir
dN
Mr 5 4 dP 5
dr
0
3
cd F (N; K) FK dK + dG + d (N X)
dM
5
My FK dK
P
dW
FN K dK
P
(214)
(215)
(216)
This matrix can be solved for the changes in the employment, price level and the interest rate
if the determinant of the coe¢ cients of endogenous variable in the left side (Jacobian matrix) is
non-singular; the determinant of this matrix should be non-zero:
39
(1
=
=
=
c (1
)) FN 0
Ir
M
M y FN
M
r
P2
W
FN N
0
P2
W
M
W
My FN 2 Ir + FN N 2 Ir Mr (1 c (1
)) FN 2
P
P
P
W
W
M
Mr 2 [1 c (1
)] FN
My FN 2
FN N 2 I r
P
P
P
(217)
The …rst term of the determinant is positive since slope of money demand function Mr is negative
FN is positive. The second term also is positive since the slope of the investment function Ir is
negative, the production function is subject to the diminishing returns, FN N < 0. This means that
determinant is non-vanishing and it is possible to …nd a solution for this model. The Cramer’s rule
can be applied to …nd out the solution for each endogenous variable.
dN =
1
c (1
) FK dK
cd F (N; K) FK dK + dG + d (N X)
dM
My FK dK
P
dW
FN K dK
P
0
M
P2
W
P2
Ir
Mr
0
(218)
My FK dK PW2 Ir +
FN K dK
fc (1
) FK dK cd F (N; K) FK dK + dG + d (N X)g
As can be seen the change in the employment depends upon the monetary and …scal policy
variables as well as the structural parameters of the model. Impact on output can be found using
the total derivative of the production function. dy = FN dN + FK dK: But the capital stock is
constant in the short run, dK = 0 . The above value of dN can be used to solve for dy.
dM
My FK dK PW2 Ir +
P
dy = dN
dW
M
W
FN K dK P 2 Ir Mr P 2 fc (1
) FK dK cd F (N; K) FK dK + dG + d (N X)g
P
This equation can be used to …nd the output multiplier of change in tax, or money supply or the
government expenditure, or the because of the changes in the structural features of the economy.
For instance a multiplier e¤ect of the change in the marginal income tax is given by
dN =
1
dW
P
M
P 2 Ir
dy
=
@
Mr PW2
dM
P
cd F (N; K)
Mr
W
P2
(219)
Thus increase in the tax rate will reduce the level of income. The size of such reduction depends
upon the value of c, Mr and PW2 .
(1 c (1
)) FN c (1
) FK dK cd F (N; K) FK dK + dG + d (N X)
Ir
dM
1
M
F
M
F
dK
M
dp =
y N
y K
r
P
dW
FN N
FN K dK
0
P
(1 c (1
)) FN 0 c (1
) FK dK cd F (N; K) FK dK + dG + d (N X)
M
dM
My FN
My FK dK
dp = 1
2
P
P
dW
W
FN K dK
FN N
2
P
P
It is even simpler to …nd the solution of the system for the short run. For empirical analysis a
standard modelling approach is to estimate the structural parameters using time series data, and
40
make these parameters as reliable as possible and compute the values of multiplier and accelerator
coe¢ cients of interest and …nd out the impacts of changes in government spending or tax rates or
money supply in output, employment and prices. The major issue, however, remains about the
stability of these parameters. A policy change is not only likely to change the levels of variables but
also the behavior of people which further might change the value of those parameters itself. The
policy analyses based on a given set of parameters, therefore, are less likely to be accurate though
their value in providing a benchmark scenario is unquestionable.
1.2.2
Comparative Static Analysis in Keynesian Macroeconomic Model
The regular ISLM model based on ideas of Keynes (1936) and Hicks (1937) is the …rst step in
understanding the economic challenges. They have been extended in several direction in terms of
multiplier accelerator model of Samuelson (1939), Mundell-Flemming open economy model (1961),
Donrnbush (1981), Rankin (1992), McCallum and Nelson(1999). Ash and Smyth (1973) Desai and
Weber (1988) and Wallis (1989) were early studies for the UK. Contributions by Cook, Holly and
Turner (2000), Greensdale, Hall, Henry and Nixon (2000), Mellis and Whittaker (2000), Leith and
Wren-Lewis (2000) Blake, Weale and Young (2000) in Holly and Weale (2000) show how optimal
monetary policy could be designed taking account of structural changes and unemployment in‡ation trade o¤ existing in the UK economy. Church, Mitchel, Sault and Wallis (1997), Bean (1998),
Hendry and Clement (2000) Garratt, Lee,Pesaran and Shin (2003) had compared forecasting capability with structural and VAR models. This section presents analytical forms and empirical
evidences to a) simultaneous IS-LM model, b) in‡aton- unemployment stabilisation model, c) aggregate demand and aggregate supply model and d) exchange rate overshooting models based on
quarterly time series from 1967:1 to 2011:1.
The basics of the ISLM model can be explained in terms of the following ten equations:
Consumption function:
Ct =
0
+
It =
0
1
(Yt
Tt )
(220)
Investment:
1 Rt
(221)
Tt = T0 + t1 Yt
(222)
M t = m0 + m1 Yt + m2 Et
(223)
Taxes:
Imports:
Macro balance:
Yt = Ct + It + Gt + Xt
Mt = Ct + Tt + St
(224)
Keynes-Hicks Macroeconomic Model: Money Market:
MM
P
= b0 + b1 Yt
t
41
b2 Rt
(225)
Money Market Equilibrium:
Rt =
b0
b2
1
b2
MM
P
+
t
b1
Yt
b2
(226)
Equilibrium in Goods Market (IS curve):
Yt =
1 T0
0
+
1
Economy wide equilibrium:
0 +Gt +Xt
Yt = 0 11 T0 ++0 tm1 +m
1
1
1
Equilibrium output:
Yt
=
m0 + Gt + Xt + m2 Et
+
1
1 t1 + m1
0
1
1
1 + 1 t1 +m1
h
b0
b2
1
b2
MM
P
1
1
1
t
+
+
1 t1
b1
b2 Yt
+ m1
Rt
i
1
b1
+
t
+
m
1 b2
1
1 1
1
1
1 T0
0
+
1
0
(227)
(228)
m0 + Gt + Xt + m2 Et
1 + 1 t1 + m1
b0
b2
1
b2
MM
P
t
Equilibrium interest rate:
Rt
=
b0
b2
1
b2
0
MM
P
1 T0
+
1
+
t
0
b1
b2
b1
+
t
+
m
b
1 2
1
1 1
1
1
1
m0 + Gt + Xt + m2 Et
1 + 1 t1 + m1
b0
b2
(229)
1
b2
MM
P
t
Exogenous policy variables Gt ; Xt ; MPM ; Et along with the behavioral parameters: 0 ; 1
T0 ; 0 ; m0 ; t1 ; m1 ; m2 ; b0 ; b1 ; and b2 determine endogenous variables Yt ; Rt ; Ct ; It ; Tt ; Mt . Quarterly time series data on these variables for the UK economy from 1967:1 to 2011:1 was downloaded
from the ONS.
Implement Keyns.mod to get the stochastic version of this model.
1.2.3
Estimation
Once parameters are estimated using the full or limited information likelihood or other methods and exogenous policy variables are projected, it is possible to forecast pathh of endogenous
i
variables Yt ; Rt ; Ct ; It ; Tt ; Mt and multiplier e¤ect of …scal and monetary polices Gt ; Xt ; MPM ; Et
in the economy. Empirically estimated structural parameters of the model are then used to assess
the impacts of changes in government spending or money supply, exchange rates, exports as in
Table 41 .
Estimates of the simultaneous equation presented in Table 4 leads to sensible results as follows:
1 PcGive
is used for estimations and forecasting.
42
Table 16: Macro simultaneous equation model of UK (1967:1-2011:1)
Consumption
Investment
Imports
Tax
Treasury bills rate
t-prob
t-prob
t-prob
t-prob
t-prob
G
1.87
0.00
0.684
0.00
0.310
0.00
1.210
0.00
0.0002
0.01
E
1221.6
0.39
665.0
0.27 2827.0
0.00 -952.5
0.53
-1.91
0.00
M4
-0.017
0.00
-0.011
0.00 -0.001
0.29 -0.013
0.00
-3.2 6
0.00
X
1.171
0.00
0.280
0.00
0.900
0.00
0.637
0.00 -0.0001
0.00
Const -4262.4
0.17 -1057.4
0.426
-7713
0.00 2745.7
0.39
13.76
0.00
F(20,554) = 247.032 [0.0000] **; N =176; R^2(LR) 0.999502; R^2(LM) 0.364715
1. On …scal policy side government spending (G) has positive and signi…cant e¤ect in consumption, investment, imports, tax revenue and the interest rate. Thus data shows evidence for
the Keynesian multiplier. The crowding out e¤ect due to increase in the interest rate is very
minimal.
2. On monetary policy side increase in money supply (M4) had negative and signi…cant impacts
in consumption, investment, tax revenues and treasury bills rate. These must have been due
to the in‡ationary impacts of increase in money supply. It did not have signi…cant e¤ect on
imports.
3. On the trade front only imports and the interest rates are signi…cantly in‡uenced by the
exchange rate (E) depreciation, its e¤ect in comprehension, investment and tax revenue were
not statistically signi…cant though with expected signs. Impacts of expansion in exports were
similar to that of government spending but smaller in magnitude for consumption, investment,
imports and tax revenue. It had small but negative impacts on the interest rates as more
export earning takes o¤ some pressure from the …nancial system.
Histoical Simulation of the UK Economy
CONS_HH
Fitted
GFCF
200000
50000
100000
25000
1970
Imports
1980
1990
2000
Fitted
2010
1970
Revenue
150000
100000
Fitted
1980
1990
2000
2010
1990
2000
2010
Fitted
100000
50000
50000
1970
15
Treasury
1980
1990
2000
2010
1990
2000
2010
1970
1980
Fitted
10
5
1970
1980
Figure 1:
Model does quite well in historical simulation for consumption, investment , imports, revenue
and treasury bills rate (Figure 6). It is does well in picking up the trends as well as the turning
points of variables. This gives con…dence in using model for forecasting.
43
Whether the future of the economy is pessimistic or optimistic depends upon the trajectory
of …scal, monetary or trade policies. It is illustrated below by considering three di¤erent policies.
In the …rst scenario government spending, exports and exchange rate decrease by 1 percent each
quarter but the money supply increases by 2 percent. This gives a very pessimistic forecast, all
model variables have downward trend then as in Figure 7.
Figure 2:
Exports are growing positively in recent quarters and can be expected to do so in coming years.
If it increases by 1 percent per quarter it slightly modi…es the declining trend as in Figure 8.
The forecast of the economy becomes very optimistic when there is inertia in the treasury
bills rate. If the monetary policy tie the current interest rate to the previous quarter by one
autoregression then consumption, investment, imports and revenue all have positive growth rates
(Figure 9). Interest rate rise is slow and steady.
Thus this sort of results demonstrates that good coordination between the …scal and monetary
policies is essential for smooth functioning of the economy. These long run impacts are subject to
the error system summarised in the follow correlation sof errors:
Table 17: Residual Correlations
Consumption Investment
Imports
Tax
consumption
5598.1
0.59175
0.12632
0.38084
Investment
0.59175
2386.1
0.33727
0.44213
Imports
0.12632
0.33727
1760.1
0.11488
Tax
0.38084
0.44213
0.11488
5758.5
Interest rate
-0.28689
0.27594 0.011569 0.070281
Error standard deviations are in the diagonal.
44
Interest rate
-0.28689
0.27594
0.070281
0.070281
2.5324
Figure 3:
Optimistic Forecasts of UK Economy
275000
Forecasts
CONS_HH
80000
Forecasts
GFCF
70000
250000
60000
50000
225000
2010
140000
2011
Forecasts
2012
2013
2014
2015
Imports
2010
2011
Forecasts
2012
2013
2014
2015
2013
2014
2015
Revenue
160000
130000
120000
140000
110000
2010
2011
Forecasts
2012
2013
2014
2015
2013
2014
2015
Treasury
5
0
2010
2011
2012
Figure 4:
45
2010
2011
2012
Table 18: Static long run relation in the macro simultaneous equation model of UK
Fiscal (G) Monetary (r) Exchange rate (e) Constant
Consumption
2.8302
-538.01
-6633.4
21380
GFCF
0.76632
396.34
-1590.5
1875.7
Imports
1.4104
-245.96
-245.20
1558.6
Revenue
1.6826
238.74
-4864.7
11189
M4
1.0701
6484.5
56510
151550
For stochastic Keynesian model see
http://www.hull.ac.uk/php/ecskrb/Confer/Policy_Challenges_BOE_2012.pdf
1.3
Samuelsonian Multiplier Accelerator Model
Macro balance
Yt = Ct + It + G0
(230)
Consumption function
Ct =
Yt
1;
0<
<1
(231)
Investment
It =
(Ct
Ct
1) ;
>1
(232)
Equilibrium (putting Ct and It in Yt ): second order di¤erence equation
Yt =
(1 + ) Yt
Yt
1
2
+ G0
(233)
Income is constant in the steady state
Yt = Yt
= Yt
1
2
=Y
(234)
Samuelsonian Multiplier Accelerator Model
Macro balance
Yt = Ct + It + G0
(235)
Consumption function
Ct =
Yt
1;
0<
<1
(236)
Investment
It =
(Ct
Ct
1) ;
>1
(237)
Equilibrium (putting Ct and It in Yt ): second order di¤erence equation
Yt =
(1 + ) Yt
1
46
Yt
2
+ G0
(238)
Business Cycle in Samuelsonian Multiplier Accelerator Model
Income is constant in the steady state
Yt = Yt
Y
1
= Yt
2
(1 + ) Y +
=Y
(239)
Y = G0
(240)
Steady state output
Y =
G0
(1 + ) +
1
=
G0
(241)
1
Transitional dynamics depends on the homogenous part of this second order di¤erence equation
Yt
(1 + ) Yt
1
+
Yt
2
=0
(242)
Solution of the Samuelsonian Multiplier Accelerator Model
Transitional dynamics (replace Yt = Abt in homogenous equation).
Yt
Abt
(1 + ) Yt
(1 + ) Abt
b2
1
1
+
Yt
Abt
+
(1 + ) b +
2
2
=
0
=
0
=0
Cycle depends on roots of the quadratic equation
q
2 (1 + )2
(1 + )
b1 ; b2 =
2
Three Cases in Samuelsonian Multiplier Accelerator Model
Distinct real root case (no cycle)
(243)
(244)
4
(245)
2
(246)
2
(247)
(1 + ) < 4
2
(248)
Yt = A1 bt1 + A2 bt2 + Y
(249)
2
(1 + ) > 4
2
(1 + ) = 4
2
Repeated real root case (no cycle)
Complex root case (cycle)
Complete solution
Yt = A1 Rt (cos
t + i sin
t) + A2 Rt (cos
t
47
i sin
t) + Y
Exercise: use a simple excel …le to compute business cycle using this model.
Two roots of a characteristic equation are related as:
b1 + b2 =
(1 + )
(250)
b1 b2 =
(251)
Using these consider how the values of characteristic root compare to the values of
(1
(1
b1 ) (1
b1 ) (1
b2 ) = 1
b2 ) = 1
(b1 + b2 ) + b1 b2
(1 + ) +
and .
(252)
=1
(253)
Condition 0 < (1 b1 ) (1 b2 ) < 1 is necessary to remain consistent with the potential value of
the marginal propensity to consume, 0 < < 1.
Two roots of a characteristic equation are related as:
b1 + b2 =
(1 + )
(254)
b1 b2 =
(255)
Using these consider how the values of characteristic root compare to the values of
(1
(1
b1 ) (1
b1 ) (1
b2 ) = 1
b2 ) = 1
(b1 + b2 ) + b1 b2
(1 + ) +
and .
(256)
=1
(257)
Condition 0 < (1 b1 ) (1 b2 ) < 1 is necessary to remain consistent with the potential value of
the marginal propensity to consume, 0 < < 1.
There are …ve possible con…gurations of and their implications on multiplier and acceleration
terms and are as following:
1. 0 < b2 < b1 < 1 =) 0 < < 1 and
<1 Convergent
2.0 < b2 < b1 = 1 =)
=1
MPC should be less than 1
3. 0 < b2 < 1 < b1 =)
>1
MPC should be less than 1
4. 0 < b2 = 1 < b1 =)
>1
MPC should be less than 1
5.1 = b2 < b1 =) 0 < < 1 and
>1 Divergent
Multiplier Accelerator Model: Real Distinct Roots:
The convergence of the system depends on term
. System is convergent
< 1, has steady if
= 1or divergent
> 1.
4
relative to . The system is explosive with no
The degree of ‡uctuation depends on (1+
)2
oscillation if
>
4
;
(1+ )2
4
4
it is recurrent if = (1+
and has damped oscillations if < (1+
)2
This last case requires solving the model using a complex root.
Multiplier Accelerator Model: Repeated Roots
2
2
(1 + ) = 4
When there is only one solution for both b1 and b2 .
48
)2
.
There can be three cases in this situation.
q
(1 + )
b1 ; b2 =
2
2
(1 + )
4
2
(1 + )
2
=
(258)
1 = b2 < b1 =) 0 < < 1 and
>1
1. 0 < b1 < 1 =) < < 1 and
< 1 convergence no oscillation.0 < (1 b1 ) (1 b2 ) < 1
2. 0 < b1 = 1 =) = 1 violates condition 0 < (1 b1 ) (1 b2 ) < 1
3. 0 < b1 = 1 =)< < 1
> 1 divergent no oscillation
Multiplier Accelerator Model: Complex Root
2
2
(1 + ) < 4
Need to consider the algebra for the imaginary number and some trigonometric functions in
this case. Using Pythagorean in an imaginary axis is used to derive the roots of the characteristic
equation.
s
2 (1 + )2
4
(1 + )
i
(259)
b1 ; b2 = (h v i) =
2
2
t
t
Yt = A1 bt1 + A2 bt2 = A1 (h + v i) + A2 (h
v i)
(260)
t) for Rt > 0:
(261)
Using DeMoivre’s theorem
(h
v i) = Rt (cos
t
i sin
Multiplier Accelerator Model: Complex Root Imaginary axis (Pithagorus Theorem)
p
R = h2 + v 2 =
(262)
Coe¢ cient of Yt-2
Yt = A1 Rt (cos
Yt = A1 Rt cos
2
t + i sin
t + i sin
2
t) + A2 Rt (cos
t + A2 Rt cos
t
2
i sin
t
i sin
Three possibilities:
i) Rt > 1;
> 1 ii) Rt = 1
= 1 and ii) Rt < 1
< 1 Only the
other two cases are divergent.
Show stability analysis in ( ; ) space, See: Samuel.mod
49
t)
2
(263)
t
(264)
< 1 case is convergent
R1
B
r=R0
A
Y1
1.4
Y0
ISLM equilibrium: a new approach
Jones (2014) simpli…es macroeconomic analysis for steady state analysis. As before the starting
point is macroeconomic balance:
Yt = Ct + It + Gt + Xt
Mt
(265)
Devide both sides of this equation by a steady state output as:
Yt
Ct
It
Gt
Xt
Mt
=
+ +
+
= ac + ai b (R r) + ag + ax am
(266)
Y
Y
Y
Y
Y
Y
De…ne a = ac + ai + ag + ax am ; in (R r) ; R is the real interest rate and r is the marginal
productivity of capital in equilibrium, b is the slope of investment with respect to excess real return.
Yt
Y
1 = Ybt = a
b (Rt
r) ; a = a
1
(267)
In steady state Yb = 0 and a = 0 =) b (R r) = 0 =) R = r. Thus in equilibrium the real
interest rate equals the marginal productivity of capital where demand for and supply of capital
are equal. In a diagram:
When the real interest (R1 ) is above the equilibrium real interest rate (R0 ) then it is contractionary and output (Y1 ) is below the steady state equilibrium (R0 ).
On the supply side the in‡ation is positively linked to output gap Ybt and is subject to supply
shock ("t ) as:
t
Ybt + "t
=
This assumes a backward looking in‡ation expectation
.
(268)
t
=
e
t+
Ybt +"t or
t
=
t 1+
Ybt +"t
Monetary policy rule is to alter the real interest rate to control in‡ation towards its target ( ),
50
(Rt
r) = m (
)
t
Putting this rule in the IS curve gives the aggregate demand equation as:
Ybt = a
b (Rt
r) = a
bm (Rt
r)
(269)
Nominal interest rate is sum of real interest rate and in‡ation.
i = Rt +
t
=r+
t
+ m(
t
)
(270)
When in‡ation is above the target in‡ation interest rate should rise. How much will output
contract depends on parameters b and m and the di¤erence between the current real interest rate
(Rt ) long run average marginal product of capital (r). Stabilisation policy can also be activated
changing components of aggregate demand ac , ai , ag , ax and am .
1.5
Basics of monopolistic competition in a new Keynesian model
There are i:::n …rms each with technology
Yi = AL1i
M P Li =
;
0<
@Yi
= (1
@Li
<1
(271)
) ALi
(272)
Relation to aggregate output (new Keynsian supply function):
Yi = A
Pi
P
Y
n
(273)
Each …rm has some market power that is related to the price elasticity of demand for its product
T Ri = Pi Yi ;
M Ri =
=
@Yi Pi
@Pi Yi
(274)
@ (T Ri = Pi Yi )
@Pi
@Pi Yi
= Pi +
Yi = Pi 1 +
@Yi
@Yi
Pi @Yi
= Pi 1
1
(275)
Demand for labour and output under monopolistic competition; marginal cost has mark up over
wage rate:
M Ci =
Wi
=
M P Li
(1
Wi
) ALi
(276)
Firm’s equilibrium condition
M Ri = M C i
=) Pi 1
1
Price charged by …rm with mark up on price (mp )
51
=
(1
Wi
) ALi
(277)
Pi = mp
Yi =
Pi
P
Wi
) ALi
(1
Y
n
; mp =
mp
=) Yi =
(278)
1
Wi
) ALi P
(1
Y
n
(279)
Demand for labour and its elasticity under the monopolistic competition:
Yi = AL1i
=
mp
mp
Wi 1
) P A
Wi
) ALi P
(1
Y
n
(280)
Solve for labour input (Li )
1+ (
Li
1)
=
Y
nA
Li =
(1
"
Y
nA
; "=
(1
)A
"
mp
1+
Wi
P
(
1)
>0
(281)
"
(282)
Elasticity of labour demand to the real wage is thus:
Wi =P
@Li
=
@ (Wi =P ) Li
Y
nA
"
(1
Y
nA
)A
mp
"
"
(
(1
)
)A
mp
" 1
Wi
P
"
"
Wi
P
Wi =P
=
"
(283)
Thus higher marp up by …rms results in lower demand for labour. Hence lower employment
and output. The expansionary monetary policy can still have any positive impacts when prices or
nominal wages are sticky.
See:
Blanchard O. and J. Galí (2013) Labor Markets and Monetary Policy: A New Keynesian Model with
Unemployment American Economic Journal: Macroeconomics 2 (April 2010): 1-30
Smet F. and R. Wouters (2003) An estimated dynamic stochastic general equilibrium model of the Euro
Area, Journal of European Economic Association, Sept, 1(5):1123-1175.
See: website of Dynare programs a number of applications of DSGE models http://www.douglaslaxton.org/dynare.html.
Gri¤oli TM (2007) Dynare v4 - User Guide: An introduction to the solution & estimation of DSGE
models.
Collard F and M Julliard (2009) Stochastic simulations with DYNARE: A practical guide.
Wickens M. (2012) How Useful Are Dsge Macroeconomic Models For Forecasting? CEPR Discussion
Paper No. 9049.
Programming Exercises 1) Practice Example0.m, EX0_Hand.m from the Herold Uhlig’s toolkit
for macroeconomic analysis.
http://www2.wiwi.hu-berlin.de/institute/wpol/html/toolkit.htm
2) Practice some dynare programs.
http://www.douglaslaxton.org/dynare.html
3) Write some GAMS and MPSGE programs to compute dynamic models.
http://www.gams.com/ and http://www.mpsge.org/mainpage/mpsge.htm
52
See how applied general equilibrium models are constructed using the input-output table. Particularly study static and dynamic multisectoral models for UK, India, China, Germany, France
and Hull.
Exercise 2 1. Consider the Keynesian model with the production function as following
Y = F (K; N )
Fk > 0; FN > 0; Fkk < 0; FN N < 0:
(284)
Consumption
C = c Y d ; Y d = (1
Labour demand
)Y
(285)
W
= FN (N; K)
P
(286)
W = W0 + W (N )
(287)
Labour supply
W (N ) =
Z
0 for N 5 N
(288)
+for N > N
money market equilibrium conditions:
M
= M (Y; r)
P
My > 0; Mr < 0
(289)
Equilibrium condition
Y =C +I +G+X
IM
(290)
derive the income tax multiplier for this model and determine its sign.
derive the income tax multiplier for this model when the money demand depends upon the
disposable income and determine its sign.
Linearise the model for comparative static analysis and determine the employment and output
impacts of changes in the government spending, tax rates the …xed nominal wage rate.
2. Multiplier accelerator model of Samuelson (1939) applies the second order di¤erence equation
for analysis of the business cycle.
2
Solve the complex root case of this model ( 2 (1 + ) < 4 ):
Yt = Ct + It + Gt
Ct = Yt
It =
(Ct
(292)
1
Ct
(291)
1)
(293)
[Hint: use De Moivre and pythagorian theorems.]
Comment on applicability of this model to analyse macroeconomic event in the current context.
3. What is the mark up if demand of a …rm is given by:
53
p = 130
5q
(294)
and its cost function is:
c = 10q
(295)
How much does this …rm charge and how much is the mark up?
monopolistic competition
perfect competition
p
70
10
q
12
24
c
120
240
m
60
0
720
0
What causes entry barrier and price rigidity here? How should a regulator react?
Q1. Consider and contrast classical and Keynesian macro models expressed in terms of equations as given below.
Classical model Output (Y )
Y = F (N )
(296)
Labour demand (N ) :
N = N(
W
)
P
(297)
Labour Supply (L) :
W
)
P
Labour market equilibrium condition as a function of real wage rate ( W
P ):
L = L(
W
W
) = N( )
P
P
Neutrality of money (M ) to price level (P ) with given velocity of circulation (m) :
L(
(298)
(299)
M = mP Y
(300)
S = S(i)
(301)
I = I(i)
(302)
S=I
(303)
Savings (S)
Investment (I)
Capital market equilibrium
Capital (K) accumulation process
Kt = (1
) Kt
1
+ It
54
0<
<1
(304)
Keynesian model Output:
Y = F (K; N )
Fk > 0; FN > 0; Fkk < 0; FN N < 0; FkN > 0:
(305)
Labour demand (real wage function of marginal productivity of labour):
W
= FN (N; K)
P
(306)
Consumption:
C = c Y d ; Y d = (1
)Y
(307)
Investment:
I = I(r)
(308)
Nominal wage(W ) and labour supply (N ):
W = W0 + W (N )
W (N ) =
Z
(309)
0 for N 5 N
(310)
>0 for N > N
where N is the labour supply at the full employment.
Money market equilibrium conditions with supply of real balances M
equal to money demand
P
M (Y; r):
M
= M (Y; r) My > 0; Mr < 0
(311)
P
Net exports as a di¤erence between exports (X) and imports (IM ):
NX = X
IM
(312)
Goods market equilibrium condition:
Y =C +I +G+X
IM
(313)
1. Determine the level of employment, output and price level in the classical model.
2. Determine the tax and government spending multipliers in the Keynesian model.
3. Assess the impacts of changes in government spending and taxes on the output, consumption
and price level in the Keynesian model using comparative static analysis.
4. Assess strengths and weakness of the classical and Keynesian models based on above analysis.
[Hints: Linearise the model for comparative static analysis and determine the corresponding
multipliers.]
55
1.6
Micro-foundation to the Keynesian Multiplier: Mankiw (1988)
Mankiw (1988) tried to rescue the Keynesian multiplier analysis against critics who disbelieved in
them due to lack of micro-foundation in the Keynesian model. He provides utility maximising set
up for households and labour markets on the supply side providing a simple Walrasian model for
comparative static multiplier analysis. The structure of this model is as follows:
max u =
log C + (1
) log L
subject to (with L as a numeraire and L as time endowment):
PC = L
PC + L = L +
T.
L +
T
With the Cobb-Douglas preference; the demand for consumption is:
PC =
L+
T
Government gets revenue from the lump sum taxes (T) and spends it in purchasing public goods
and payings public employees
T =G+W
Aggregate expenditure (Y) is sum of private and public spending
L+
Y = PC + G =
T +G
Output in the economy (Q) is given by aggregate demand divided by the price level.
Y
P
There are N number of …rms each producing q output and the total cost is sum of …xed and
marginal costs as:
Q=
T C(q) = F + cq
Pro…t margin of a …rm shows how much the market prices above the …rm’s cost of production:
p
=
c
p
This implies p = 1 c ; therefor Q = 1 c Y:
Pro…t of …rms in the economy equal
= PQ
=
=
c
1
c
1
Q
NF
NF
1
c
Y
cQ
cQ =
c
1
NF = Y
56
Q
NF
NF
Labour market clearing:
LS = L
(1
) L+
T = L
(1
)[
T]
(1
)[
Now the labour demand should equal to this labour supply:
LD
=
N F + cQ + W = (Y
=
L+
T +G
) + (T
+ (T
G)
G) = L
T]
In aggregate the labour demand equals labour supply.
Now the multiplier analysis follows:
L+
T + G as above
Given = Y N F and Y = P C + G =
Y =
@Y
@G
=
1
1
L+
T +G=
@Y
@T
and
=
1
L+ Y
NF
T + G; Y =
L
NF
1
T +G
This implies:
@Y
1
@Y
+
=
@G
@T
1
1
=
1
1
The degree of market power (or imperfect competition) as measured by is very crucial in
this multiplier analysis. If there is no market power ( = 0) this multiplier stops at the initial
iteration. If ( = 1) the multiplier is 1. When the competition is in between (0 < < 1) then the
net multiplier is 11
:
Welfare analysis:
Take the budget constraint P C + L = L +
T . substitue the demand for consumption
assuming P =1 in P C =
L+
T ; This become
L+
T +L = L+
T . This means
L+
T = LS +
T.
L+
T = (Y
) + (T G) +
T = (Y G) = 1 c Q G
This should mean
[
T ] = (1 c ) Q G L = cQ (1 (1 )G) (1 )L
@[
T]
@G
=
(1
)
(1
)
This means increase is government spending reduces welfare of households by
(1
1
)
(1
)
(1
) :
[note this
derivation is
as printed in Mankiw (1988)]. Main point of the analysis is that marginal
propensity to consume and share in consumption in the utility function linked to the Keynesian
multiplier in a similar way. This provides micro-foundation to the Keynesian multiplier analysis.
Mankiw G N (1988) Imperfect Competition And The Keynesian Cross, Economics Letters 26
(1988) 7-13
Two period model of Mankiw and Weinzierl (2011) more comprehensive in analysing impacts
of …scal and monetary polcies in stabilisation.
57
1.7
Keynesian Stochastic Macroeconomic Model and Policies
"The General Theory of Employment is a useful book; but it is neither the beginning nor the end
of Dynamic Economics." J. R. HICKS (1937)
"No rule is likely to remain optimal for long" Mervyn King (2004); "Monetary stability means
stable prices and con…dence in the currency." MPC (2014).
1.7.1
Introduction
UK economy growing at around 2.2 percent annually since 1967 suddenly entered into the deepest
recession in the last quarter of 2008, resulting in a …ve percent decline in GDP in 2009. Labour
government brought a heavy …scal stimulus package to …ght recession in the last quarter of 2008
…nanced by borrowing £ 175 billion (13% GDP) in the budget of 2009. The Bank of England brought
£ 200 billion package (£ 375 billion now) for quantitative easing (QE). The coalition government
formed after the general election of May 2010 continued this stimulus slightly toning down the
de…cit to 11 percent of GDP. These measures were very consistent to the concept of cyclical …ne
tuning of the economy and macroeconomic stabilisation role of …scal policy as proposed in Keynes
(1936) and Hicks (1937).
Process of recovery has been very slow. With a high liquidation rates of companies, the unemployment rate had increased up to 8.1 percent; rates of saving and investment have fallen. Flow of
credits to private sector dwindled. It took long time to restore con…dence in the British …nancial
and business services sector, which is almost one third of the GDP (32 percent). Pound Sterling,
that depreciated by almost 25 percent since January 2008 took time to gain strength. Despite great
moderations in macroeconomic volatilities after the independence of the Bank of England the general level of prices rose well above the stipulated 2 percent target mainly due hike in prices of fuel,
food and commodities and imported products. Real wages fell and were below the pre-crisis level in
2014 causing deteriorating living standards of the majority of people. When will the UK economy
be out of slow growth and high unemployment situation is a question bothering all concerned. The
ongoing process of globalisation has further added volatility and constraints on the trade of goods
and services. How should these challenges of …scal, monetary and trade policies be analysed and
solved using a stochastic Keynesian model and wisdom in tradition of Hicks-Stone–Meade-MirrleesPissarides in this situation? This paper aims to provide some empirical evidence based analysis of
this issue.
1.7.2
Stylized Facts and Macroeconomic Policies in the UK
This section aims to provide a basic idea on stylized facts and trends in the UK economy. These
show how the …scal, monetary and trade policies interact to the underlying structural features of
the economy and what sort of policy options are available to the policy makers to …nd solutions to
macroeconomic problems. This sets a nice background of four distinct empirical macro modelling
exercises in the coming sections.
1.7.3
Macroeconomic Trends
Quarterly GDP was £ 382 billion in the 3rd quarter of 2013. Among its components, the consumption, public expenditure, exports have been growing faster than investment as presented in Fig.1.
This is one of the cause of lower growth rate of the UK economy (Fig. 2).
58
F ig u re 1 : G D P a n d its c o m p o n e nts in U K
F ig u re 2 : G row th ra te o f G D P in U K
Fluctuations in aggregate demand impacts on unemployment and in‡ation. When economy is
depressed unemployment rises (Fig. 3). In‡ation has strong seasonal as well as cyclical components
(Fig.4). Relationship between unemployment rate and in‡ation, Phillips curve, has changed over
time (see Figures 27 and 28). PPI in‡ation is slightly more volatile than the CPI in‡ation.
F ig u re 3 : U n e m p loy m e nt ra te in U K
F i g u r e 4 : I n ‡a t i o n o f C P I a n d P P I
Fiscal policy aims to stabilise the economy and smooth the economic growth by manipulating
the size and components of revenue, spending and de…cit. In general the sizes of revenue and
spending have grown with the economy (Fig. 5). Government’s e¤orts to …ne tune the economy is
clear from the ‡uctuations in the amount borrowed over time (Fig. 6).
59
F ig u re 5 : G ove rn m e nt re ve nu e a n d sp e n d in g
F ig u re 6 : C e ntra l G ove rn m e nt N e t B o rrow in g : £ m C P N S A
Ratio of outstanding public debt to the GDP rises as government borrows more from the central
bank or the private sector to …nance its de…cit as shown in Fig. 7. It is a matter of concern as it
may reduce the capability of government to …ne-tune,redistribute or reallocate resources ultimately
making the economy less competitive and ine¢ cient in coming years.
F ig u re 7 : R a tio o f d e b t to G D P in U K
The Bank of England uses monetary policy to regulate the economy more e¤ectively by actively
changing the baseline interest rate according to how the aggregate demand and supply situations
are away from the steady state capacity level of the economy. Generally accommodative monetary
policy means lower interest rates during the …scal austerity and higher rates during the …scal
expansions. This is clear from the path of the interest rates in Fig. 8. By changing the discount
factor between the current and the future periods, the interest rate impacts on the accumulation
and composition of the …nancial assets and stock of money in the economy (Fig. 9).
60
F ig u re 8 : T h re e m o nth s tre a su ry b ills ra te in U K
F ig u re 9 : M 4 a n d G D P in U K (M illio n £ )
Interest policy plays discernible impacts not only in the ‡ow of domestic credits but also in the
‡ow of international assets mainly through its impacts on the exchange rates. It is clear from the
trends of exchange rate of Sterling pounds with respect to the Euro and the US dollars as given
in Figures 10 and 11. These two rates do not seem to move in tandem as the o¤setting changes
in exchange rates (Fig. 12) also re‡ects the movements in the domestic prices of these economies
(Fig. 13).
Figure 10: Exchange rate Pound Sterling to Euro
61
Figure 11: Exchange rate Pound Sterling to US dollars
F ig u re 1 2 : E x ch a n g e ra te s o f S te rlin g to E u ro a n d U S D o lla rs
Figure 13: Domestic and foreign price indices
Changes in domestic demands due to such changes in the interest and exchange rates impact on
the volume of exports and imports and resulting imbalances in the current account, which normally
are between 2 to 3 percents in the UK (Figure 12).
F ig u re 1 2 : Va lu e o f im p o rts a n d e x p o rts in U K
1.7.4
Fiscal Challenges
Challenges of the …scal policy in the UK are clear from the ratios of revenue, spending, de…cit
and debt in Table 1. Public debts and de…cits are becoming more acute than what Prest (1968)
or Pain, Weale and Young (1997) observed earlier. These problems add dilemmas that are much
more serious than any political business cycle models justify for it (Price (1997)). Cutting public
spending under the austerity reduces aggregate demand severely but the collection of revenue cannot
increase unless GDP growth rates are up. Full con…dence of consumers and producers to generate
62
adequate demand is key for it (Hicks 1990). Excessive de…cit …nancing has raised the debt/GDP
ratio. Alarmed by this the government has put forward a de…cit reduction plan in place so that the
UK does not become as insolvent as Greece, Ireland, Portugal and Spain in recent years (OBR).
A long run view like this is essential not only because the direct and indirect sources of revenue
have already been stretched to the limit of public tolerance (Tables 2) but also for the fact that
the ‡exibility on spending side is limited as UK is committed to maintain generous social security
system, in provision of universal health care, good standards in education and other public services
(Table 3). UK remains one of the high tax-spend economy in the global economy already making
…rms here less competitive to their global counterparts, a great deviation from the Ramsey-Mirrlees
optimal tax rules (Ramsey (1927), Mirrlees (1971), Mirrlees et al. (2010)).
Table 19: Ratios of Revenue, Speding and De…cit to GDP
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
R evenue/G D P
3 7 .2
3 6 .0
3 7 .0
3 7 .2
3 7 .4
3 7 .0
3 7 .0
3 7 .4
3 7 .8
3 8 .0
3 8 .1
S p e n d in g / G D P
4 4 .1
4 7 .0
4 6 .2
4 4 .8
4 4 .7
4 3 .5
4 2 .5
4 1 .6
4 0 .2
3 8 .8
3 7 .8
D e …c t/ G D P
-6 .9
-1 1 .0
-9 .2
-7 .6
-7 .3
-6 .5
-5 .5
-4 .2
-2 .4
-0 .8
0 .3
D ebt/G D P
5 2 .8
6 0 .5
6 7 .5
7 0 .1
7 0 .1
7 3 .6
7 7 .4
7 9 .7
7 9 .6
7 8 .1
7 5 .7
S o u rce: O B R , M a rch 2 0 1 4
Excessive de…cit is dangerous but the raising tax rates during the period of weak economic
growth is an unpleasant and infeasible task. Reducing budget de…cit from currently 11 percent of
GDP to a surplus of 0.3 percent by 2018-19 (Table 1), as proposed by the O¢ ce of the Budget
Responsibility (OBR) is not an easy task. The optimal taxes and growth concept of Ramsey (1927,
1928) or Mirrlees (1971) need further attention2 .
Table 20: Source of Revenue in UK (St. Pounds, Billion)
Sources of R evenue
2009
2010
2011
2012
2013
2014
In co m e ta x
1 4 6 (0 .2 7 )
1 5 0 (0 .2 7 )
1 5 8 (0 .2 7 )
1 5 0 (0 .2 7 )
1 5 5 (0 .2 7 )
1 6 7 (0 .2 6 )
N a tio n a l in su ra n c e
9 7 (0 .1 8 )
9 9 (0 .1 8 )
1 0 1 (0 .1 8 )
9 9 (0 .1 7 )
1 0 7 (0 .1 7 )
1 1 0 (0 .1 7 )
C o rp o ra tio n ta x
4 2 (0 .0 8 )
4 3 (0 .0 8 )
4 8 (0 .0 8 )
4 3 (0 .0 8 )
3 9 (0 .0 8 )
4 1 (0 .0 6 )
E x c ise ta x
4 6 (0 .0 9 )
4 6 (0 .0 8 )
4 6 (0 .0 8 )
4 6 (0 .0 8 )
4 7 (0 .0 8 )
4 7 (0 .0 7 )
VA T
7 8 (0 .1 4 )
8 1 (0 .1 5 )
1 0 0 (0 .1 5 )
8 1 (0 .1 7 )
1 0 3 (0 .1 7 )
1 1 1 (0 .1 7 )
B u sin e ss ta x
2 5 (0 .0 5 )
2 5 (0 .0 5 )
2 5 (0 .0 5 )
2 5 (0 .0 4 )
2 7 (0 .0 4 )
2 7 (0 .0 4 )
C o u n c il ta x
2 6 (0 .0 5 )
2 5 (0 .0 5 )
2 6 (0 .0 5 )
2 5 (0 .0 4 )
2 7 (0 .0 4 )
2 7 (0 .0 4 )
O ther
8 1 (0 .1 5 )
7 9 (0 .1 4 )
8 5 (0 .1 4 )
7 9 (0 .1 4 )
1 0 7 (0 .1 7 )
1 1 8 (0 .1 8 )
541
548
589
548
548
648
To ta l
S o u r c e : B u d g e t R e p o r t ( M a r c h 2 0 1 4 ) H M T r e a s u r y, h t t p : / / w w w . h m - t r e a s u r y. g o v ; ; % i n ( ) .
2 Taxes can be optimal instrument for transferring resources from one generation to the next (Samuelson (1954),
Modigliani (1961), Diamond (1965), Atkinson and Stern (1974), Feldstein (1985, 1982), Auerbach and Kotliko¤
(1987) ) or for correcting ine¢ ciencies due to negative or positive externalities (Buchanan (1958)). Public debt
rightly used could help the government to maintain the intertemporal balance in resources available to it (Meade
(1956), Barro(1974), Besley (2001), Ni Shawn and Wang (1995), Basu (1996), Burnside, Eichenbaum and Fisher
(2004), Brauninger (2005), Fisher and Ryan (2010)).
63
Table 21: Elements of Public Expenditure in UK (St.Pounds, Billion))
E x p e n d itu re Ite m s
2009
2010
2011
2012
2013
2014
S o c ia l p ro te c tio n
1 9 0 (0 .2 8 )
1 9 4 (0 .2 8 )
2 0 0 (0 .2 8 )
1 9 4 (0 .2 8 )
2 2 0 (0 .3 1 )
2 2 2 (0 .3 0 )
2 9 (0 .0 4 )
3 2 (0 .0 4 )
3 2 (0 .0 5 )
3 2 (0 .0 4 )
3 1 (0 .0 4 )
3 1 (0 .0 4 )
1 1 9 (0 .1 8 )
1 2 2 (0 .1 8 )
1 2 6 (0 .1 8 )
1 2 2 (0 .1 8 )
1 3 7 (0 .1 9 )
1 4 0 (0 .1 9 )
E d u c a tio n
8 8 (0 .1 3 )
8 9 (0 .1 3 )
8 9 (0 .1 3 )
8 9 (0 .1 3 )
9 7 (0 .1 3 )
9 8 (0 .1 3 )
Tra n sp o rt
2 3 (0 .0 3 )
2 2 (0 .0 3 )
2 3 (0 .0 3 )
2 2 (0 .0 3 )
2 1 (0 .0 3 )
2 3 (0 .0 3 )
D efen ce
3 8 (0 .0 5 )
4 0 (0 .0 6 )
4 0 (0 .0 6 )
4 0 (0 .0 6 )
4 0 (0 .0 6 )
3 8 (0 .0 5 )
I n d u s t r y, A g r , E m p l o y m e n t
2 1 (0 .0 3 )
2 0 (0 .0 3 )
2 0 (0 .0 3 )
2 0 (0 .0 3 )
1 9 (0 .0 2 )
1 7 (0 .0 2 )
H o u sin g a n d E nv iro n m e nt
3 0 (0 .0 4 )
2 7 (0 .0 4 )
2 4 (0 .0 3 )
2 7 (0 .0 4 )
2 3 (0 .0 3 )
2 5 (0 .0 3 )
P u b lic o rd e r a n d s a fe ty
3 6 (0 .0 5 )
3 5 (0 .0 4 )
3 3 (0 .0 5 )
3 3 (0 .0 5 )
3 3 (0 .0 4 )
3 2 (0 .0 4 )
D e b t a n d inte re st
4 3 (0 .0 6 )
4 4 (0 .1 1 )
5 0 (0 .0 7 )
5 0 (0 .0 7 )
5 0 (0 .0 7 )
5 3 (0 .0 7 )
O thers
7 4 (0 .1 1 )
7 4 (0 .1 0 )
7 3 (0 .1 0 )
4 3 (0 .1 0 )
5 3 (0 .0 7 )
5 3 (0 .0 7 )
704
696
711
696
720
732
P e rso n a l so c ia l se rv ic e s
H e a lth
To ta l
S o u r c e : O B R M a r c h 2 0 1 4 ; H M T r e a s u r y, h t t p : / / w w w . h m - t r e a s u r y. g o v . ; % i n ( ) .
The fact that both taxes and spending policies have wide ranging reallocative as well as redistributive implications as the taxes create distortions in consumption, production and trade is well
recognised in the literature (Meade et al. 1971 and Mirrlees et al. (2010), IFS (2014)). There is
a perception that over the years these public policies have led to the disparity in income not only
among rich and poor households but also across and within the regions of England, Scotland, Wales
and Northern Ireland (Sawyer 2003). These regions are facing a situation of two speed economy as
the households in the lower income deciles are lagging far behind those in the upper income groups
and the regional and sectoral disparities exist in allocation of public resources. Ricardian equivalence of public debt as proved in Barro (1974, 1989) does not seem to apply in credit constrained
and regionally imbalanced economies like this (Spencer 1998, Besley 2001).
1.7.5
Monetary policy
Price stability is the most important role of the monetary policy (Friedman (1968), Goodhart (1989),
Bernanke and Mishkin (1997)). Interest rates are raised during the booms and reduced during the
recessions as guided by an interest rate rule. The BOE’s has …xed the bank rate at 0.5 percent,
lowest bank rate in the Bank’s history (Fig. 3) and remained at that rate for more than four years
now. However, it is less clear whether 0.5 percent rate was optimal in the recession as the impacts
of such move are unclear for three reasons. First, the …rst, second or tertiary rounds of transmission
mechanisms of monetary policy as stated in Monetary Policy Committee (1999) becomes less clear
when interest rates are very low in this way. When in‡ation rose above 4 percent, higher than
the 2 percent target (Fig. 4), the In‡ation Report of the Bank of England (BOE) attributed it to
rising energy, food and commodity prices and increase in VAT. Secondly these are severely a¤ecting
the rate of saving and the UK economy is in a situation of liquidity trap. Thirdly when there are
uncertainties in the demand side of the economy, the lower cost of capital is not able to restore
con…dence among investors at the desirable level despite more than three years of historically low
interest rate and Fund for Lending Schemes being in operation (BOE 2012). Uncertainties of future
not only deter businesses to invest but policy makers are not sure about the future state of the
64
economy. In the Ely lecture the Governor Mervyn King (2004) had stated that "we cannot fully
describe an optimal monetary arrangements because we do not know all possible states of the world
and hence the policy rule to which we would like to commit". Households do not …nd remunerative
enough to save and lenders are reluctant to advance at a lower interest rate. This has squeezed the
credit market. The unexpectedly sluggish growth realised in last three years suggests that under
the "constrained discretion" the actual shocks hitting the supply and demand sides of the economy
are much bigger than policy measures taken to raise the ‡ows of credit aiming to mitigate the
consequences of such crises3 . Given "sustained period of tight credit conditions" the MPC predicts
that the level of output is less likely to "surpass pre-crisis level till 2014" (BOE 2012).
1.7.6
Trade Issues
Trade is essential for growth and stability. No country is self su¢ cient in producing goods and
services that it needs. Ideas of free and liberal trade originated from the UK as the Ricardian
theory of comparative advantage is still one of the most important concepts guiding trade practices around the world. External demand can mitigate consequences of reduction in the internal
demand. Opening the economy also makes it easier to transmit problems originated elsewhere in
the world. The process of globalisation has added further challenges to the UK economy as in
the majority of other advanced economies (Haskel and Slaughter (2001), Monacelli and Perotti
(2010)). Trade balance deteriorates when exportable goods and services become more expensive
than imports. The trade creation and trade diversion e¤ects increased due to regional blocks such as
European Union and NAFTA, ASEAN and Mercosur impact on comparative advantage of British
…rms. The emergence of global multinational corporations after the revolutions in transport and
communications and development of emerging markets including China and India, Russia, Brazil
and South Africa in recent years have put further pressures in the trade policies of the UK (Miller
and Spencer (1977)). Firms in these new countries can supply goods and services at lower costs
than by home …rms. Modelling frameworks and arguments found in Mundell (1962), Dornbusch
(1976), Krugman (1979), Taylor (1995) and Gali and Monacelli (2005) are far from complete to
handle complications arising from a need to comply to global environmental agreements such as the
Kyoto protocol, WTO regulations on fair trade or migration of skilled and unskilled workers, tari¤
and non-tari¤ barriers in movement of goods and services.
1.7.7
Modelling of the UK economy
The tradition of economic modelling for policy analysis started with the pioneering ideas of classical
economists including Smith, Ricardo, Malthus, Pigou and Marshall in UK believing that the market system is dynamically stable. They argued for policies that would promote the free enterprise
economy under the competitive markets where the ‡exible relative prices of commodities would
guarantee full employment equilibrium. Keynesian and New Keynesian economists prefer to stick
to the macro-modelling framework that Keynes (1936) had proposed in which rigidities in prices of
3 While the crisis in the monetary general equilibrium contexts analysed in Diamond, Douglas and Dybvig (1983),
Rankin (1992), Altig, Carlstrom and Lansing (1995), Ghosal and Miller (2003) Angelopoulou and Gibson (2009)
provide useful frameworks to think about the real side causes of …nancial disturbances, the new Keynesian solutions
to the problems are provided by Barro and Gordon (1983), Pain, Weale and Young (1997, 2000),Clarida, Gali and
Gertler (1999, 2001), Dri¢ ll and Snell (2003), Martin and Milas (2004), Christiano, Eichenbaum and Evans (2005),
Rudebusch (2006), Chadha and Nolan (2007), Blinder et al. (2008), Johnson (2009), Tillmann (2009)) or new classical
models like King and Plosser (1984), Wickens (1995), Holland and Scott (1998) which assume super-neutrality of
money and ignore the modelling of the monetary sector.
65
commodities cause de…ciencies in aggregate demand. Wallis (1989) mentions that macroeconometric
models of the UK improved substantially under the auspices of ESRC’s Macroeconomic Modelling
Bureau that coordinated research activities across research and policy institutions in the UK including the Bank of England, the HM-Treasury, the LBS, the Liverpool, Edinburgh, Cambridge, Oxford
or Exeter. Modelling activities for policy analysis of UK economy became more comprehensive and
realistic after modelling innovations following from ability to compute (Church et al. (1997)). For
instance Hendry and Doornik (1994) formulated "a linear dynamic system, noting closed, open,
complete and incomplete systems for both stationary and integrated l(1) data.. adopting general to
simple modelling of the joint data density function . . . ”. In the meanwhile the latest new classical
models in works of Kydland and Prescott (1977) or Prescott (1986) or Plosser (1989) resurrected
almost all elements of the classical general equilibrium models to explain growth and ‡uctuations
simultaneously attributing ‡uctuations in macroeconomic activities to shocks to the preferences
and technology of production. For them the collapse of consumer and investor con…dence are root
causes of instability and active …scal and monetary policies are required to restore con…dence as in
Hicks (1937, 1990). Dynamic Stochastic General Equilibrium (DSGE) and Dynamig Computatble
General Equilibrium (DCGE) models have been developed in UK in recent years (Den Haan and
Marcet (1990), Holland and Scott (1998) Bhattarai (2007) and Gai, Kapadia, Millard and Perez
(2008), Liu and Mumtaz (2011), Bhattarai and Dixon (2014)). Despite that the big slump started
in 2008 proved to be a big challenge to both theoreticians and practitioners.
Impossibility and undesirability to predict the future state of the economy might have been the
main cause of failure of the popular macro models in analysing economic choices available in the
context deep contraction (King (2004)). Controversies among di¤erent paradigms (Dixon(1988))
made it further complex. Progress in macro economic theory from the classical to Keynesian to New
Keynesian and rational expectation to the real business cycle theories has been quite extra-ordinary
and so has been the modelling technology including on the issue of coordination of budgetary
and monetary policies (Wilson(1949), Meade (1956), Diamond (1965), Blake and Weale (1998),
Turnovsky and Miller (1984), Woodford (2011), Nordhaus (1995)). These analysis underpin the
decisions that made the independence of the Bank of England in 1997 producing greater stability
by reducing the volatility of in‡ation, unemployment and growth rate as illustrated by trends of
volatilities in Figures 13 and 14 as observed by Bean (1998, 2009).
F i g u r e 1 3 : V o l a t i l i t y o f i n ‡a t i o n a n d u n e m p l o y m e n t
66
F ig u re 1 4 : Vo la tility o f g row th ra te
Given above experiences and controversies the main objective here is to apply simple macroeconomic models that are easy enough to understand but generate scenarios that are helpful to
analyse ways out of the challenges that the economy is facing today. We formulate, empirically
estimate and apply the Keynesian Stochastic Macroeconomic Model (KSMM), Stochastic Phillips
curve model, AS-AD with rational expectation model and small open economy macro model in line
of theoretical "deconstruction" and "reconstruction" approaches taken by Wren Lewis et al. (1996)
in analysing above issues in the UK economy in the next sections4 .
1.7.8
Keynesian Stochastic Macroeconomic Model (KSMM)
Let us consider a version of dynamic Keynesian stochastic macroeconomic model (KSMM) to assess
impacts of shocks in the consumer and producer con…dences. It also includes shocks in trade as
well as to …scal and monetary policy instruments as anticipated by Meade (1951). The structural
features of the stochastic dynamic Keynesian economy are given by following six equations for the
consumption (F.589), investment (315), tax revenue (316), imports (317), macro balance (318) and
the money market equilibrium (319).
Ct =
+
1
It =
0
0
(Yt
Tt
1)
+
2 Xt
+ "C ;
+ Yt
1
+ "I ;
"I
1
1 Rt
Tt = T0 + t1 Yt + t2 Mt + "T ;
"T
"C
N 0;
N 0;
M t = m0 + m1 Yt + m2 Rt + m3 Tt + m4 Et + "M
Yt = Ct + It + Gt + Xt
Rt =
b0
b2
1
b2
MM
P
+
t
N 0;
2
C
(314)
2
I
(315)
2
T
(316)
"M
N 0;
2
M
Mt = Ct + Tt + St
b1
Yt + "M P
b2
"M P
N 0;
(317)
(318)
2
MP
(319)
This interest rate is solution to the money market equilibrium condition given by:
MM
P
= b0 + b1 Yt
b2 Rt
(320)
t
4 Pain, Weale and Yong (1998) had attributed de…cit problems to increasing commitments of UK government for
social securities and transfer programmes rather than to slow growth of revenue. Contributions by Cook, Holly and
Turner (2000) on monetary policy, Greensdale, Hall, Henry and Nixon (2000) on natural rate of unemployment,
Mellis and Whittaker (2000) on forecasting, Leith and Wren-Lewis (2000) on in‡ation, Blake, Weale and Young
(2000) on optimal monetary policy in Holly and Weale (2000) show how monetary policy could be designed taking
account of structural changes and unemployment in‡ation trade o¤ existing in the UK economy. Analyses of Church,
Mitchel, Sault and Wallis (1997) on the role of technology in the economy, Hendry and Clement (2000) on failures of
forecast, Garratt, Lee,Pesaran and Shin (2003) and Bernanke et al. (2005) on structural and VAR models focused
on forecasting capability of models. General equilibrium models have been developed to complement above analysis
with explicit introduction of the hetorogeniety of …rms and households in the economy for analysis of growth and
redistribution the UK (Bhattarai and Whalley (2000) in Holly and Weale (2000) and Bhattarai (2012)).
67
1.7.9
Steady State in the KSMM
Steady state equilibrium in goods and money markets from this IS-LM analysis, is given by (322)
and (323) respectively, where mean of idiosyncratic shocks are zero, E ("i ) = 0:
Yt =
1 T0
0
+
1
0
m0 + Gt + Xt + m2 Et
1 + 1 t1 + m1
1
1
1
+
1 t1
+ m1
Rt
(321)
For economy wide equilibrium use the
h money market steadyistate condition in the IS curve Yt =
b0
1
MM
1 T0 + 0 m0 +Gt +Xt
1
+ bb21 Yt to …nd output and the interest rate
1
1
b2
b2
P
1 + 1 t1 +m1
1 + 1 t1 +m1
t
at equilibrium as:
0
Yt
=
1
1 T0
0
Rt
=
1
b1
+
t
+
m
1 b2
1
1 1
1
b0
b2
1
b2
0
+
1
0
MM
P
1 T0
+
1
m0 + Gt + Xt + m2 Et
1 + 1 t1 + m1
+
t
0
(322)
b1
b2
b0
b2
b1
+
t
+
m
b
1 2
1
1 1
1
b2
MM
P
t
1
1
1
m0 + Gt + Xt + m2 Et
1 + 1 t1 + m1
b0
b2
(323)
1
b2
MM
P
t
Exogenous policy variables Gt ; Xt ; MPM and Et along with the behavioral parameters 0 ; 1 T0 ;
0 ; m0 ; t1 ; m1 ; m2 ; b0 ; b1 and b2 determine endogenous variables Yt ; Rt ; Ct ; It ; Tt and Mt (see
Peacock and Shaw (1979), McCallum and Nelson (1999) for this type of models). For parameters
in Table 4, this model generates the steady state as given in Table 5.
Table 22: Parameters
Parameters
0
1
Values
600
0.65
Parameters b0
b1
Values
10000 0.15
of the dynamic stochatic Keynesian model
T0
m0
m1
0
1
8000
4000
0.1
10000 0.2
b2
M4
X
G
P
30000000 2157.3 123.50 879.87 1.2
m2
-50
E
1.5
Parameters in a Keynesian macroeconomic simulation model like this should be chosen so that
they generate the steady states of output, consumption, investment, revenue and imports. Values
of these endogenous variables should be close enough to current levels in the economy.
Variables
Values
Table 23: Macro Variables in the steady state
C
I
T
M
Y
R
64072 26639 53278 40111 150928 0.00102805
These steady state values implied in the current model are close to the actual quarterly values
of the UK economy.
68
1.7.10
Transitional Dynamics in KSM Model
The next step is to explain how this economy ‡uctuates around that steady state due to exogenous
shocks in consumer and investor con…dences or to imports or revenues or to the interest rate.
Stochastic assumptions are similar to Den Haan and Marcet (1990), Holland and Scott (1998) and
Gai, Kapadia, Millard and Perez (2008), Liu and Mumtaz (2011) but the Keynessian structure here
di¤ers from the new Keynesian or the RBC frameworks in those models. Underlying structural
parameters are assumed to be stable though it is possible to update them each period using a
Bayesian MCMC algorithm as in the fashion of Benati (2008). Schocks to Keynesian models
generate as intuitive results as in any other classes of models.
First consider impacts of stochastic positive shock to the consumer con…dence. This makes consuers more optimistic exerting expansionary impacts that raises the overall demand in the economy
shown in Figure 15. Increase in consumption raises the levels of investment and output and revenue.
It also raises the rate of interest via increase in demand for money.
C
I
C
0.1
0.01
0.02
0.05
0.005
0.01
0
5
10
15
0
20
5
10
T
15
0
20
I
0.05
5
10
M
15
0
20
0.01
0.02
0.01
0.01
0.005
0.01
0.005
0
0
0
10
15
20
5
10
-10
Y
0.05
4
15
20
5
10
R
x 10
15
0
20
5
10
15
0.05
5
-3
C
0
10
15
20
5
20
15
20
15
20
R
10
15
0
20
5
10
F ig u re 1 6 : Im p u lse s o f a sh o ck to inve sto r c o n …d e n c e
I
x 10
C
-2
-0.02
10
x 10
4
0
F ig u re 1 5 : Im p u lse s o f a sh o ck to c o n su m e r c o n …d e n c e
0
15
2
0
20
5
-10
Y
2
0
10
M
0.02
5
5
T
I
0
0
-0.02
-0.01
-4
-0.04
5
10
15
20
5
T
10
15
-0.04
20
5
M
10
15
20
-0.02
0
0
0.1
0
-0.005
-0.02
0
5
10
15
20
-0.01
5
-10
Y
0
0
-0.5
-0.02
x 10
10
15
-0.04
20
5
R
10
15
20
-0.1
5
-10
Y
-1
10
15
20
15
20
15
20
M
0.1
-0.1
5
T
0
0
-0.05
-2
x 10
10
R
-1.5
-0.04
5
10
15
20
5
10
15
-0.1
20
F ig u re 1 7 : Im p u lse s o f a sh o ck to ta x re ve nu e
5
10
15
20
-4
5
10
F ig u re 1 8 : Im p u lse s o f a sh o ck to im p o rts
Macro impacts of shocks to investment, shown in 16, are of similar magnitute on economic
69
activities as those from shocks in consumption despite the size of investment demand being less
than one third of the demand for consumption. The investment demand is a lot more volatile than
any other components of aggregate demand as more con…dent investors drive the rate of capital
accumulation and growth process in the economy causing more expansionary impacts than a similar
increase in con…dence of consumers. Bad news lowers the level of investment more than similar
negative shock in consumption (Blanchard and Kiyotaki, 1987). The long run equilibrium point
where the marginal e¢ ciency of capital equals the user cost of capital is disturbed by these positive
or negative shocks in investment.
Taxes are designed to achieve a number of social and economic objectives. As mentioned above
government is bound to change taxes to meet expenditure plans that need to be optimal for voters
(Meade et al. (1978), Mirrlees et al. (2010)). While the ‡exibility of government to raise or lower
the level of taxes either to …nance more public consumption or in order to redistribute income are
political economic questions (Price 1997) these decisions can themselves be cause of business cycles
as they in‡uence the optimal conditions of consumers and producers and search and matching
processes in labour markets (Pissarides,1985). While the net e¤ects of …scal policy in consumption
and investment are obtained by deducting the contractionary impact of taxes from the positive
multiplier e¤ects of public spending, shocks to policies can alter magnitudes of these multipliers.
Thus changes in policy regimes create shocks that disturb and distort the system (Woodford (2011)).
Negative …scal shocks depress output, consumption, investment and imports and the interest rate
as shown in Figure 17 and these results are consistent to stories of Fisher and Whitley (2000),
Fisher and Ryan (2010) or Feldstein (1982).
External demand can create expansionary impacts in the UK but an increase in the level of
imports causes leakage of resources from the economy and hence reduces the multiplier impacts
from net exports as is evident from the response of consumption, investment, output tax and
interest rate to a unit shock in imports as shown in Figure 18. Increase in the interest rate raises
the cost of investment and consumption and thus contributes to a reduction in demand as shown
by responses of macro variables to interest rate shocks in Figure 19.
-3
0
C
x 10
I
0
-2
-0.005
-4
5
-3
0
10
15
20
-0.01
x 10
0
-2
-4
5
-3
T
10
15
20
15
20
15
20
M
x 10
-1
5
10
15
20
-2
5
Y
10
R
0
0.2
-0.005
0
-0.01
-0.2
5
10
15
20
5
10
F ig u re 1 9 : Im p u lse s o f a sh o ck s to inte re st ra te (M P )
Let us move to estimation of the KSMM model taking dataset on the exogenous policy variables
Gt ; Xt ; MPM ; Et and endogenous variables Yt ; Rt ; Ct ; It ; Tt ; and Mt . The behavioral parameters of the model 0 ; 1 T0 ; 0 ; m0 ; t1 ; m1 ; m2 ; b0 ; b1 ; and b2 can be retrieved from the reduced form
estimates when this system is identi…ed by order (K k > m 1) and rank ( (A) > (M 1) (M 1))
70
conditions. As each equation of above the system is identi…ed (see appendix for details of identi…cation) we take the quarterly time series data from 1967:1 to 2011:1 available from the ONS to
estimate the model empirically. Details on estimation and application of this model for analysis of
…scal, monetary and exchange rate policies is provided in the next section.
1.7.11
Estimation and application of the KSMM Model
The reduced form parameters of the KSMM model estimated using the full or limited information likelihood method in Table 45 . As these empirically estimated structural parameters of the
model are signi…cant and have theoretically expected sings, these are used to assess the impacts of
changes in government spending, money supply, exchange rates, exports. The multiplier e¤ects of
…scal and monetary policies in the economy are computed with these parameters which along with
hypothetical predicted paths of exogenous policy variables Gt ; Xt ; MPM ; Et forecast the path of
endogenous variables Yt ; Rt ; Ct ; It ; Tt ; Mt in the model.
Table 24: Macro simultaneous equation model of UK (1967:1-2011:1)
Consumption
Investment
Imports
Tax
t-prob
t-prob
t-prob
t-prob
G
1.87
0.00
0.684
0.00
0.310
0.00
1.210
0.00
E
1221.6
0.39
665.0
0.27 2827.0
0.00
-952.5
0.53
M4
-0.017
0.00
-0.011
0.00
-0.001
0.29
-0.013
0.00
X
1.171
0.00
0.280
0.00
0.900
0.00
0.637
0.00
Const -4262.4
0.17 -1057.4
0.426
-7713
0.00 2745.7
0.39
F(20,554) = 247.032 [0.000] **; N =176; R^2(LR) 0.99; R^2(LM) 0.36
Treasury bills rate
t-prob
0.0002
0.01
-1.91
0.00
-3.2 6
0.00
-0.0001
0.00
13.76
0.00
Main points emerging from the estimates of the simultaneous equation presented in Table 6 are
as follows:
1. The government spending (G) has positive and signi…cant e¤ect in consumption, investment,
imports, tax revenue and the interest rate. The sign and the magnitude of multipliers are as
one would expect from a Keynesian model. There is very small crowding out e¤ect due to
increase in the interest rate following an increase in public spending.
2. Increase in money supply (M4) had negative and signi…cant impacts in consumption, investment, tax revenues and treasury bills rate but did not have signi…cant e¤ect on imports. These
must have been due to the in‡ationary impacts of increase in money supply and consequences
in QE policies.
3. On the trade front only imports and the interest rates are signi…cantly in‡uenced by the
exchange rate (E ) depreciation; its e¤ect in consumption, investment and tax revenue were
not statistically signi…cant though with expected signs. Impacts of expansion in exports were
similar to that of government spending but smaller in magnitude for consumption, investment,
imports and tax revenue. It had small but negative impacts on the interest rates as more
export earnings takes o¤ some pressure from the …nancial system.
5 PcGive is used for estimations and forecasting (see Wallis (1989), Hendry (1997), Church et al. (1997)
and Holly and Weale (2000) for more extensive analysis of macroeconometric forecasting).
71
Under the current structure of the model, whether the future of the economy is pessimistic or
optimistic depends upon the trajectory of …scal, monetary or trade policies. Let us consider three
di¤erent policy options for macroeconomic …ne tuning. In the …rst scenario government spending,
exports and exchange rates decrease by 1 percent each quarter but the money supply increases by
2 percent. This gives a very pessimistic forecast, all model variables have downward trend as in
Figure 21. Second option is to promote export to compensate for decrease in domestic demand.
If exports could increase by 1 percent per quarter it slightly modi…es the declining trend as in
Figure 22. Thirdly, the forecast of the economy becomes very optimistic when there is inertia in
the treasury bills rate. If the monetary policy ties the current interest rate to the previous quarter
by one autoregression then consumption, investment, imports and revenue all have positive growth
rates (Figure 23). Interest rate rises slowly but steadily.
Performance of the model is judged by studying how well the historical simulations for consumption, investment, imports, revenue and treasury bills rate match to past trends. Current model does
it well in picking up the trends as well as the turning points of variables (Figure 20). This gives
con…dence in using model for forecasting.
Histoical Simulation of the UK Economy
CONS_HH
Fitted
GFCF
200000
50000
100000
25000
1970
Imports
1980
1990
2000
Fitted
2010
150000
100000
1970
Revenue
Fitted
1980
1990
2000
2010
1990
2000
2010
Fitted
100000
50000
50000
1970
15
Treasury
1980
1990
2000
2010
1990
2000
2010
1970
1980
Fitted
10
5
1970
1980
Figure 20: Hostorical simulation
Figure 21: Pessimistic Forecasts
72
Optimistic Forecasts of UK Economy
275000
Forecasts
CONS_HH
Forecasts
80000
GFCF
70000
250000
60000
50000
225000
2010
140000
2011
Forecasts
2012
2013
2014
2015
2010
Imports
2011
Forecasts
2012
2013
2014
2015
2013
2014
2015
Revenue
160000
130000
120000
140000
110000
2010
2011
Forecasts
2012
2013
2014
2015
2013
2014
2015
2010
2011
2012
Treasury
5
0
2010
2011
Figure 22: Intermediate forecast
2012
Figure 23: Optimistic forecast
The simulation results demonstrate that good coordination between the …scal and monetary
policies is essential for smooth functioning of the economy (Blake, Weale (1998)) to avoid noncooperative Nash results rather than cooperative outcome in the policy games between …scal and
monetary authorities engaged in the least-square learning process. Model estimated so far implicitly
assumes that the policy makers are free to choose …scal, monetary and trade policy measures such
as the level of public spending, exports, money supply or the exchange rate in order to achieve
desired or optimal values of target variables like consumption, investment, revenue, interest rate or
imports. In fact the policy makers are not free to choose but are constrained by the level of GDP,
money supply, tax revenue or imports that are acceptable to the general public in the country. As
King (2004) states, "no rule is likely to remain optimal for long" applies in this context. It is
sensible to re-specify the above simultaneous equation model by endogenising …scal and monetary
policy instruments as the function of macro target variables including the levels (or growth rates)
of GDP, imports, money supply and revenue. Tinbergenian matching of instrument and policy
targets elaborated by Meade (1951, 1956) in the form of targeting nominal GDP or imports to
achieve internal or external balance as in Bean (1998, 2009). The simultaneous equation model
discussed here is used to estimate policy response parameters, presented in Table 7, necessary to
achieve target values of macro variables.
Table 25: Macro simultaneous equation model of UK (1967:1-2011:1)
GDP
Imports
M4
Revenue
Const
Government Expenditure
t-prob
0.460
0.00
0.108
0.07
-5.072
0.00
0.517
0.05
-16606.4
0.00
= 1804
In‡ation
t-prob
-2.253
0.00
5.164
0.006
0.001
0.00
7.284
0.351
1.353 -0.061
= 0:5479
Exports
t-prob
-0.121
0.00
1.008
0.00
10.054
0.00
-0.018
0.38
18356.0
0.00
= 1455:7
Investment
t-prob
1.122
0.00
-0.011
0.53
-10.093
0.00
0.0184
0.00
-18408.6
0.39
= 1464:86
F(20,554) = 247.032 [0.0000] **; N =176; R^2(LR) 0.999502; R^2(LM) 0.364715
73
Interest rate
t-prob
9.6 6
0.05
3.9 5
0.31
-0.006
0.00
4.760
0.77
19926
0.00
=1.161
Model …ts to the past series (Fig. 24) and predicts these policy instruments quite well (Fig. 25).
Figure 24: Histoical simulations of policy instruments.
Figure 25: Forecasting of policy instruments.
The de…cit forecasts implied by above parameters is given in Fig. 26.
Figure 26: Forecsting public borrowing
1.7.12
Qualitative analysis
Hicks (1937) had suggested comparative static analyses to measure the impacts of exogenous variables in the employment (N ), price level (P ) and interest rate (r) using three equations showing
goods (324), labour (325) and money market(G.653) equilibrium conditions while synthesising the
Keynesian model to supply side in the classical system.
F (N; K ) = c (1
) F (N; K) + I (r) + G + N X
74
(324)
W
= FN (N; K)
P
(325)
M
= M (F (N; K ) ; r)
P
Implicit solution of the model requires linearising by the total di¤erentiation as:
FN dN + FK dK = c (1
) FN dN + c (1
dW
P
) FK dK
(326)
cd F (N; K) + Ir dr + dG + d (N X) (327)
W
dP = FN N dN + FN K dK
P2
(328)
M
dM
dP = My FN dN + My FK dK + Mr dr
(329)
P
P2
By further expansion and rearrangement for endogenous variable labour (dN ), price (dP ) and
interest rate (dr), this model is succinctly written as:
FN dN
c (1
) FN dN
Ir dr = c (1
My FN dN +
) FK dK
cd F (N; K)
M
dM
dP + Mr dr =
2
P
P
W
dW
dP =
P2
P
Or this can be written in a matrix notation:
FN N dN +
2
4
2
(1
c (1
)) FN
My FN
FN N
c (1
= 4
) FK dK
0
M
P2
W
P2
FK dK + dG + d (N X) (330)
My FK dK
(331)
FN K dK
(332)
3
32
Ir
dN
Mr 5 4 dP 5
dr
0
(333)
3
cd F (N; K) FK dK + dG + d (N X)
dM
5
My FK dK
P
dW
FN K dK
P
This matrix can be solved for changes in the employment (dN ), price level (dP ) and the interest
rate (dr) if the determinant of the coe¢ cients of endogenous variables in the left side (Jacobian
matrix) is non-singular; the determinant of this matrix should be non-zero:
(1
=
=
=
c (1
)) FN 0
Ir
M
My FN
M
2
r
P
W
FN N
0
P2
W
M
My FN 2 Ir + FN N 2 Ir Mr (1 c (1
P
P
W
M
Mr 2 [1 c (1
)] FN + FN N 2 Ir
P
P
75
(334)
)) FN
W
P2
The …rst term of the determinant (334) is positive since slope of money demand function Mr is
negative FN is positive. The second term also is positive since the slope of the investment function
Ir is negative, the production function is subject to the diminishing returns, FN N < 0. This means
that determinant is non-vanishing and it is possible to …nd a solution for this model. The Cramer’s
rule can be applied to …nd out the solution for each endogenous variable.
dN =
1
c (1
) FK dK
2
1 4
dN =
cd F (N; K) FK dK + dG + d (N X)
dM
My FK dK
P
dW
FN K dK
P
dM
P
My FK dK
W
P 2 Ir
+
dW
FN K dK
P
) FK dK
0
M
P2
W
P2
M
P 2 Ir
c (1
cd F (N; K) FK dK + dG + d (N X)
Mr PW2
Ir
Mr
0
3
5
(335)
(336)
As can be seen the change in the employment (335) depends upon the monetary (My ) and …scal
policy variables ( ; G) as well as the structural parameters of the model. Impact on output can be
found using the total derivative of the production function, dy = FN dN + FK dK: But the capital
stock is constant in the short run, dK = 0. The above value of dN can be used to solve for the
change in output, dy.
dy =
dM
P
dN
Mr PW2
fc (1
My FK dK PW2 Ir + dW
FN K dK PM2 Ir
P
) FK dK cd F (N; K) FK dK + dG + d (N X)g
(337)
This equation (337) can be used to …nd the output multiplier of change in tax, or money supply
or the government expenditure, or the because of the changes in the structural features of the
economy. For instance a multiplier e¤ect of the change in the marginal income tax is given by
dy
=
@
cd F (N; K)
Mr
W
P2
(338)
Thus increase in the tax rate will reduce the level of income. The size of such reduction depends
upon the value of c, Mr and PW2 . Price changes can be computed similarly.
Ir
Mr
0
(339)
Estimates of the parameters for empirical comparative static analysis are made applying the generalised unrestricted model (GUM) estimation routine of Castle, Doornik and Hendry (2011) in the
PcGive on macro quarterly time series for 1997:1 to 2012:1 of the UK. This model determines the
changes in employment, price level and the interest rate (dN , dP and dr (333)) endogenously in
terms of changes in net exports (DN X) and net investment (DInv) as shown in Table 8. Changes
in government consumption or change in money supply were insigni…cant and were automatically
deleted by the GUM system.
dp =
1
(1
c (1
)) FN
My FN
FN N
c (1
) FK dK
cd F (N; K) FK dK + dG + d (N X)
dM
My FK dK
P
dW
FN K dK
P
76
Table 26: Comparative Static: GUM analysis 1997(2) - 2011(2)
Change in employment
t-prob
DN X 0.0203
2.05
DInv 0.0221
4.49
Const
-3.882
-0.195
= 98
F(6,104) = 13.5931 [0.0000]**; T
Change in price level Treasury bills rate
t-prob
t-prob
0.0002
5.19
-3.129 05 -0.613
8.5 05
4.68
7.670 05
3.02
0.257
3.49
-0.348
-3.39
= 0.363
=0.508
=52; R2 (LR)= 0.686; R2 (LM )=0.263
There are two main criticisms against the analysis above. First by assuming rigidity of nominal wages it overestimates the impact of changes in public spending in output and employment.
Secondly it does not provide a good transitional dynamics. Under perfect ‡exibility of wages and
prices as one …nds in the classical and new classical system, aggregate demand management policies
do not have real impacts in the economy. Growth and employment are only driven by the accumulation of capital, TFP and growth in the labour force. New Keynesian economists equipped with
the Phillips curve and rational expectation argue that expansionary monetary policies by raising
aggregate demand can have a signi…cant role in reducing unemployment (Dixon (1988), Dixon and
Rankin (1994)) because of rigidities. Prices adjust at slower rate than the wage rates, breaching
the homogeneity of degree zero in output and input prices required for the classical system, after
the launch of an expansionary programme (Wilson (1949), Phillips (1958), Phelps (1968), Taylor (1977), Ball (1999), Ball and Romer (1990), Mankiw (1989) Blanchard and Summers (1986)
and Rankin (1992)). Their arguments are tested using a simple stochastic stabilisation model of
unemployment rate and in‡ation and growth rates of output and money in UK in the next section.
1.7.13
A Small Model of Unemployment, In‡ation and Growth
The basic mechanism of stabilisation program can be explained by a simple model that involves
use of stochastic versions of the Phillips curve, Okun’s law and the growth rate of money supply
(gm;t ) with given natural growth rates of output (gy;n ) and natural rate of unemployment (un ).
ut
ut
t
1
=
t 1
a (gy;t
=
gy;n ) + bgm;t + "u
b (ut
gm;t = gy;t +
ut
t
1)
+ "m
+ "p
N 0;
N 0;
N 0;
2
m
2
p
2
u
(340)
(341)
(342)
Okun’s law (340) establishes link between unemployment (ut ) and output gap (gy;t gy;n ) : Then
the expectation augmented Phillips curve (341) shows a trade-o¤s between in‡ation ( t ) and unemployment rate (ut ) linking the nominal to the real side of the economy. Third equation (342) is
an identity, equivalent to the classical quantity theory of money relating growth of money supply
(gm;t ) to the growth rates of output (gy;t ) and in‡ation. In‡ation should be lower than the growth
rate of money supply when the economy is growing. Despite a signi…cant reduction in the volatility
of unemployment rate, in‡ation and the growth rate of output in UK as shown in Figures 13 and 14
after the independence of the Bank of England, events after the …nancial crisis of 2008 is creating
77
doubts in such claims. For instance a closer look at the cross plot between the unemployment rate
and in‡ation shows some trade-o¤ in line of the Phillips curve over 1967:1 to 2011:1(Fig.27) but
this negative relation has turned to be positive after 1997, as shown in Fig.28. This stabilisation
model does not seem to work well when the e¢ ciency of matching of vacancies to jobs and productivity growth are weak as in the current recession. E¢ ciency of job search and matching process
that determines the ins and outs of unemployment not only depends on economic activities (Smith
(2011)) but also in the way taxes, subsidies and transfers are used on wages, employment or job
creating enterprises (Pissarides (1984)).
Figure 27: Phillips’curve for the full sample
1.7.14
Figure 28: Phillips curve after 1997
Solution of the stabilisation model
This small stabilisation policy model is handy in linking four macro variables relating to the stability
in the real and nominal sides of the economy. When in‡ation is up to 5 percent, above the 2 percent
target the central bank should raise basic interest rate to release pressure o¤ the demand but that
is likely to cut in demand and raise the cost of production causing increase in the unemployment
rate as households postpone purchasing expensive items and …rms will layo¤ workers due to a fall
in the demand for products. In stable scenario given the in‡ation target the central bank should
equate the growth rate of money supply to the growth rate of output plus in‡ation. Once the
equilibrium is disturbed the transition paths of these variable, shown in Fig.29 and Fig.30, are
found by simulation using the realistic values of parameters of the model as given in Table 9.
Table 27: Estimated parameters of the stabilisation model
a
b
un
gy;n
u1
1
values -0.3 -0.09 0.05 0.022 0.051 0.02 0.08
The stabilisation model is simulated to trace the path of in‡ation during the stabilisation period
starting from 5.1 percent in‡ation that was observed in the fourth quarters of 2011. In‡ation is
reduced by 0.2 percent each quarter until it reaches it target 2 percent. This lowers demand and
raises unemployment rate above its natural rate of …ve percent till the in‡ation target is met as
shown in Fig. 29 with implied growth rates of money and output as given in Fig.30.
78
F i g u r e 2 9 : S i m u l a t e d p a t h o f i n ‡a t i o n a n d u n e m p l o y m e n t r a t e
F ig u re 3 0 : S im u la te d p a th o f g row th ra te s o f o u tp u t a n d m o n e y
The transitional path suggested above can be subject to shocks to the Okun, Phillips or money
supply equations. These would create impulses to above four variables as shown in Figures 31 to
33.
-4
5
-3
u
x 10
3
0
2
-5
1
-10
0
-15
-1
pi
x 10
-3
5
u
x 10
0
-5
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
pi
5
-3
15
10
15
20
5
-3
gm
x 10
0.01
15
10
15
20
0
gy
x 10
-0.01
10
10
5
5
0
0
0
-5
-0.01
10
15
20
4
6
8
10
gy
-5
5
2
0.01
5
10
15
20
Figure 31: Impulses of Okun shocks
2
4
6
8
10
Figure 32: Impulses of Phillips’curve schocks
79
-3
1
u
x 10
0
-1
2
4
6
8
-3
2
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
pi
x 10
0
-2
2
4
6
8
10
gy
0.01
0
-0.01
2
4
6
8
10
Figure 33: Impulses of money supply shocks
1.7.15
Supply side and rational expectation
Keynesian economists argue that increase in demand has real e¤ect on output and employment
because of rigidity in prices and wages in the short run. Increase in the aggregate demand either
by increase in the government spending or by a reduction in the interest rate (increase in money
supply) would have permanent impacts on output and employment (Bean (2009), Rudebusch(2006),
Greensdale at al. (2000)). The price level would not increase when an economy is below full
employment6 . Under the rational expectation, workers are fully informed, nominal wage rate rises
according to the expected in‡ation. Workers demand higher wage rate to compensate fully for
higher anticipated changes in prices. Thus there is no real impact of an increase in demand even
in the short run as it is anticipated by workers. Only unanticipated policy measures can have real
impacts as explained above in the short run7 . Higher aggregate demand puts upward pressure in
prices and …rms can reduce their markups without altering market prices. Additional workers could
be hired to supply additional output without changing prices when there is a pool of unemployed
workers (Boinet and Martin (2008), Johnson (2009), Monacelli and Perotti (2010), Nelson (2009),
Fisher and Ryan (2010)). Thus an expansionary monetary policy can raise the level of output and
employment in the economy in the short run though the economy tends to return to its natural
rate in the long run.
Despite criticism on the stability of parameters set as above (Lucas critique), it is di¢ cult to
6 In contrast to this, the classical or the new classical proposition remains that the prices and wages are perfectly
‡exible and economy is always in full equilibrium (Kydland and Prescott (1982)). Consequently it is impossible to
arti…cially increase real output by increasing demand. Real drivers of the economy are capital accumulation and
increase in human capital and increase in work hours and technological progress. Monetary policy is super neutral.
Price system that guarantees general equilibrium in goods and factor markets matter for the e¢ cient allocation of
resources (Bhattarai and Whalley (1999)) and should be dynamically e¢ cient in terms of growth and redistribution
(Bhattarai (2012)).
7 New Keynesian synthesis …nds a more realistic middle path between the Keynesian and real business cycle schools
(Arestis et al. (2010), Gali and Monacelli (2005), Kirsanova, Leith and Wren-Lewis (2009)). These features rest
on the monopolistic competition and staggering wage contracts (Taylor (1972), Rankin (1992)). Firms with market
power under the monopolistically competitive markets are able to absorb demand shocks (Dixon and Rankin (1994),
Nickel (1990), Dixon (1988), Blanchard and Kiyotaki (1986)). These issues are further assessed in Angelopoulou and
Gibson (2009), Arnold et al. (2011), Gemmell et al. (2011), Beetsma and Giuliodori (2011) in recent years.
80
get an alternative framework of analysis that is as transparent as this one and would provide a
benchmark scenario for policy analysis. Rational expectation models developed in Sargent and
Wallace (1976) are theoretically very convincing but di¢ cult to implement as correct expectation
formation under uncertain economy is a very challenging task. Unanticipated policy changes are
likely to have macroeconomic impacts. These issues are better analysed by aggregate demand and
aggregate supply models with rational expectation under the new Keynesian framework developed
by Lucas (1973) and Mankiw (1989). In fact the another way to study the unemployment in‡ation
problem in UK is to consider a popular version of aggregate supply aggregate demand model in
line of Lucas (1973), Taylor (1973), Bean (1998), Woodford and Taylor (1999) and Sorensen and
Whitta-Jacobsen (2010). It relates output gap to in‡ation instead of the unemployment rate.
1.7.16
Aggregate Demand and Aggregate Supply Model
The AS-AD is our third model to study the impacts of …scal, monetary and trade policies in output
and price level in the economy. From the Fisher equation the real interest rate (rt ) is the nominal
interest rate (ipt ) adjusted for the risk ( t ) and the expected in‡ation et+1 as:
rt = ipt +
t
e
t+1
(343)
Aggregate demand (yt ) is subject to the …scal policy shock (gt ) and monetary policy (rt
the demand shock (vt ).
yt
y=
1
(gt
g)
2
(rt
r) + vt ;
vt v N 0;
2
v
;r = r +
r) and
(344)
Nominal interest rate is set by the monetary authority to close the in‡ation and output gap in a
policy rule of the form:
ipt = r +
e
t+1
+ h(
t
) + b (yt
y)
(345)
The aggregate supply function with the supply shock (st ) is given by:
t
=
e
t+1
+ (yt
y) + st ; st v N 0;
2
s
(346)
With a backward looking in‡ation expectation as et = t 1 the aggregate demand equation could
be derived using the Fisher equation and the interest rate rule in the demand function with rt
e
e
) + b (yt y) or rt r = t
+ h( t
) + b (yt y) : The
t + t+1 = r + t+1 + h ( t
aggregate demand yt y = 1 (gt g)
[
+
h
(
)
+
b
(y
y)]
+
v
2 t
t
t
t could be written
as yt y = 1+2 h2 b (
)
+
z
.
t
t
yt
y=
(
t)
+ zt ;
=
2h
1+
2b
(347)
Where zt term includes …scal policy shock (gt ), risks ( t ) and random shocks (vt ); zt = 1+ 12 b (gt g)
2
) + 1+vt 2 b : Aggregate demand is downward slopping; higher rate of in‡ation requires
1+ 2 b ( t
central bank to increase the interest rate, that raises the cost of capital, thus causes lower investment and hence lower output. After putting the in‡ation expectation into the supply function, it
becomes:
t
=
t 1
+ (yt
81
y) + st
(348)
It is upward slopping; larger output requires employers to hire more workers, that lowers the
productivity of labour. The cost of production rises resulting in in‡ation. Term st includes trade,
exchange rate, technology or other shocks. De…ne deviation from the steady state as bt = t
and ybt = yt y when there are no further shocks zt = 0 and st = 0. Then the aggregate demand
is bt+1 = 1 ybt+1 :The aggregate supply (by iterating forward and di¤erencing) can be written as
bt+1 = bt + ybt+1 =) bt = ybt . There is empirical evidence for such relation in the UK time series
(1967: 3 to 2011:1):
bt = 0:001096 + 0:6241 ybt
(SE) : (0:00193) (0:877)
The estimated b is slightly higher than 0.48 contained in Bean (1998). Equilibrium in terms of
the …rst di¤erences of output and in‡ation can be found by equating AD and AS curves:
1
ybt+1 =
1
bt+1 = bt + (
ybt + ybt+1 =) ybt+1 =
bt+1 ) =) bt+1 =
1
1+
ybt =) ybt+1 = ybt
1
1+
bt =) bt+1 = bt
(349)
(350)
Starting from initial states yb0 and b0 both ybt and bt converge to their stationary state as 1 <
1
< 1. This condition is essential for stabilisation of output and in‡ation.
1+
t
ybt = yb0
t
and bt = b0
for t = 0; 1; 2; :::::
=
(351)
The parameters ; and could be calibrated from the time series to study the impulse responses
from demand and supply shocks, zt and st respectively when they are not zero, zt 6= 0 and st 6= 0.
Setting demand equals to supply equilibrium condition the stochastic time paths of ybt in (357) and
bt in (361) result in autoregressive processes as following:
yt
y=
t)
AD :
bt =
ybt =
1
=)
ybt +
1
1+
ybt =
bt = bt
1
ybt
(zt
ybt = zt
1
ybt
1
bt = bt
AS :
AD = AS
+ zt =) ybt =
(
+
1
1
ybt ) =
zt
1
1+
+
(zt
+ ybt + st = bt
82
zt
1
(zt
+ ybt
(zt
(zt
ybt )
+ ybt + st
1
1
bt + zt
zt
1)
+ (
1
1)
(353)
(354)
ybt
1)
st
1
(352)
1+
+ ybt + st
st
st
bt + zt ) + st
(355)
(356)
(357)
(358)
(359)
bt =
1
1+
bt =
bt
1
bt
+
1
zt +
1+
+
1
1+
st
zt + st
(360)
(361)
Thus solutions of the dynamic aggregate demand and aggregate supply model results in a …rst
order autoregressive time path of output ybt and in‡ation bt which are subject to demand and
supply shocks, zt and st . This theoretical justi…cation for using AR(1) model is similar to that
one would get under the rational expectation models of Sargent and Wallace (1975) , Calvo (1983),
Taylor (1987), Dri¢ ll and Schultz (1992). Either the DSGE models of Uhling (1995),Blanchard
and Perotti (2002), Smet and Wouters (2003), Nelson (2009), Iacoviello and Neri (2010) or the
RBC models of Wickens (1995), Minford and Peel (2002) generate such impulse responses from
demand, supply or TFP shocks in the economy. Empirically autoregressive terms are not only
signi…cant but also explain 82 percent of the growth rate and 97 percent of in‡ation in the UK
(see estimates in Table 10). While the AR(1) coe¢ cients measure persistency, the intercept terms
indicate to other regular structural features including …scal, monetary and trade factors.
Table 28: AR(1) model of growth rate and in‡ation in UK
Intercept
AR(1) term
R2
F
DW
2
N
Growth equation
Coe¢ cient T-value
0.410
2.93
0.817
18.7
0.67
381( 0.00)
1.97
87.1(0.00)
176:q11967-q12011
In‡ation equation
Coe¢ cient T-value
0.172
1.19
0.972
54.7
0.95
2292(0.00)
1.02
102.6(0.00)
176:q11967-q12011
The impulse responses to demand and supply shocks output and in‡ation in a VAR are as shown
in Fig. 34.
83
Figure 34: Impulse responses of output and price shocks
1.7.17
Trade Policy Model
Net exports represent the external demand for domestic products, particularly helpful when the
other domestic components of aggregate demand as in the previous recession. When growth rate of
the UK economy was negative 5 percent at the end of 2009 policy makers thought about creating
more external demand to compensate de…ciencies in internal demand. On the face of it this appears
to be a plausible and sensible strategy considering the trends of exports and imports, exchange rate,
growth rates of money and output, in‡ation and the interest rate in the UK. Three fundamental
questions arise in this context: 1) Is there any cause-e¤ect relation between the net exports and the
exchange rate? 2) What are the determinants of the exchange rate? 3) Are the economic e¤ects of
the exchange rate predictable? The …rst issue has been extensively discussed in the literature since
the seminal works of Meade (1951, 1955). Second issue were analysed in Mundell (1962), Flemming
(1962), Krugman (1979), Taylor (1995, 2010), Holly and Weale (2000), Clarida, Gali and Gertler
(2001), and Gali and Monacelli (2005) with a small open economy or a global economy model
of interdependent economies. Thirdly Dornbusch (1976), Taylor (1995) and Taylor (2010) have
discussed reasons for the overshooting of exchange rate, lack of PPP relation and unpredictability
of spill over e¤ects of it in modern economies. Our empirical …ndings suggests two things. First
there is a very thin relation between the trade and exchange rate in recent years. Secondly the
export promoting e¤ects of changes in the exchange rates are very unreliable due not only to
overshooting of the exchange rate but also because of violations of PPP and UIP fundamentals
in the short run. Thus income and employment generating e¤ects of additional external demand
are unpredictable. In theory the price of foreign currency relative to the domestic currency relates
essentially to the relative prices of goods at home and abroad determined by real factors including
preferences of consumers, technology of producers and endowments of factors of production in these
economies (Obstfeld and Rogo¤ 1996, Eaton and Kortum 2002). In practice adjustment towards
84
such equilibrium takes longer as shown by empirical …ndings in PPP (Taylor 2010).
1.7.18
Structural factors and the volatility of exchange rate
In theory total output of an economy produced from employing labour and capital can either
be consumed (C) domestically or exported (E) as in (363). The level of exports in (364) not only
depends on the real exchange rate e P
P
with the nominal exchange rate (e), indices of domestic (P )
and foreign P prices but also in the elasticity of exports ( ). Imports in (365) similarly depend
on the elasticity of imports ( ), exchange rate (e) that mingles with price indices of domestic
(P ) and imported commodities, (Pm ). Any discrepancy between the domestic income P Y and
total expenses (P C + ePm M ) is met by external lending or borrowing eB as in (366). A
set of structural features underpin the trade and the exchange rate relationship (Johnson 195354). In the classical Ricardian theory, the terms and patterns of trade are based on comparative
advantage, expressed in terms of ratio of domestic to foreign prices. In the Mundell-Fleming set
up of the Keynesian model such exchange rate (e) can be …xed or ‡exible policy instrument used
to determine the volume of net exports (N X = a0 a1 e) and the net ‡ows of capital. Net exports
are larger when the home currency depreciates and lower when it appreciates.
N X = a0
a1 e; a0 > 0; a1 > 0
(362)
Y = f K; L = C + E
E = E0 e
(363)
P
P
(364)
C
P
= K0 e m
M
P
P Y + eB = P C + ePm M =) eB = ePm M
(365)
PE
(366)
In an Armington set-up the elasticities of exports ( ) and imports ( ) are crucial parameters
that measure the ‡exibility of trading system and indicate the impact of nominal exchange rate
(e) on next exports. Putting all these together implicitly the exchange rate is function of trade
elasticity and other parameters of the trading system of the economy as:
e = f X0 ; C; K0 ; ; ; K; L; Y ; P ; P; B
(367)
Thus the endogenous variables of this trade sub-model C; E; M; P; e depend on the parameters of
the model E0 ; K0 ; ; ; K; L; Y ; ; B and Pm but the exchange rate and the prices are the main
variables determining the distribution of gains from trade from this model. The rapid space of globalisation has caused swift changes in these parameters, particularly export and import elasticities,
and ; which have become larger making UK very vulnerable to the international economy.
Depreciation lowers the foreign price of domestic goods P , it raises supply of exports (E),
makes foreign goods more expensive and reduces the amount of imports (M ) and raises the production of import substitute goods at home. Depreciation thus can raise both domestic and foreign
demand for home products. The Marshall-Lerner condition implies that depreciation is expansionary when the elasticity of exports to the exchange rate is higher than the elasticity of imports.
Positive growth rate of exports during the current recession in UK provides support for this theory.
85
If the exchange rate overshoots in‡ation as in Dornbusch (1976) it may cause greater volatility in
net exports. It is often di¢ cult to disentangle the impacts of global shocks using the exchange rate
instrument as the relation of next exports and exchange rate are unreliable when fundamentals of
PPP or UIP do not hold (Taylor 2010).
Free monetary policy, ‡oating exchange rate and free capital in‡ow and out‡ow are three pillars
that charaterise the exchange rate, trade and balance of payment system in the UK. Change in
exchange rate not only re‡ects the underlying changes in trade ‡ows and capital movements but
also is an adjustment mechanism of the economy towards rapidly changing system of global trade
and payments. Since London is one of the most important …nancial centre of the global economy
a steady and stable exchange rate of Sterling Pound to major currencies is in the interest of the
UK economy. Available evidence suggests that UK has been quite successful in maintaining stable
but with depreciating Pound in recent years (Figures 10 to 12 and 35) tolerating greater volatilities
in net exports (Figures 13 and 36). In welfare terms quick adjustments of exchange rates bu¤er
consumers and producers from large swings in real variables in order to adjust to the external
shocks8 . Mundell (1962) and Gali and Monacelli (2005) open up the basic Keynesian model for
trade where net exports (N X) are larger and when the nominal exchange rates (e) of home currency
depreciates as in (362). We adopt Dornbusch (1976) and empirical evidence from UK to illustrate
this point in this section.
F i g u r e 3 5 : R e d u c e d v o l a t i l i t y o f i n ‡a t i o n a n d e x c h a n g e r a t e
1.7.19
F i g u r e 3 6 : I n c r e a s i n g ‡u c t u a t o n s i n t r a d e ( n e t e x p o r t s )
Monetary model of exchange rate expectation
It is important to understand the dynamic path of in‡ation and exchange rate while assessing the
role of external sector in the economy. Under Dornbusch (1976) and Taylor (1987) monetary model
of exchange rate with rational expectation swiftly adjusting exchange rate overshoots price level
8 While
the Keynesian trade multipliers can provide a preliminary estimate on the impact of external shocks to the
real and BOP conditions in the UK, it is important to consider standard classical comparative advantage arguments
of Ricardo and subsequent theories of trade developed in Meade (1951), Miyazawa (1960), Mundell (1962), Dornbusch
(1976), Krugman (1979), Wilson (1979) and Taylor (1995) in studying bilateral and multi-lateral relations of trade
to get more precise understanding of the impact of external sector in the economy.
86
because of inertia in prices. Under the uncovered interest parity conditions (370) in style of Hoy et
al. (2001) the change in the exchange rate equation is obtained with the money demand function
(368) the money market equilibrium condition (G.657), interest rate parity (370) and the exchange
rate expectation ( e = E e ) as:
mD =
m
ar + by
p=
(368)
ar + by
(369)
r =r +E e
e =
p by m
+
a
a
(370)
r
(371)
yD yS ;
> 0; the aggregate
While in‡ation is positively related to the excess demand p =
demand is determined by the real exchange rate (e p) and other demand factors (u)
y D = u + v (e
p)
(372)
S
Assuming the steady state supply y = y to be equal to the demand, the in‡ation can be
expressed as:
p =
vp + ave + a (u y)
(373)
1.7.20
Solving for in‡ation and exchange rate paths simultaneously
The dynamic system of in‡ation and exchange rates relate to the monetary policy instrument,
foreign interest rate and the supply capacity as:
!
a (u y)
v
v
p
p
=
+
(374)
by m
1=a
0
e
r
e
a
The explicit time path of price and exchange rate thus is given by:
p(t) = C1 exp
e(t) =
With
p
e
!
=
0
0
1
+ v
C1 exp
v
1t
+C2 exp
1t
+
2
2t
(375)
+p
+ v
C2 exp
v
2t
+e
(376)
condition, the steady state price level is obtained when e = 0 as p =
1
m by + ar and steady state exchange rate when p = 0 as e = p
y) : The constant
v (u
terms C1 and C2 can be evaluated with initial conditions p0 and e0 and the initial speed p1 and
e1 . Qualitatively above results could be presented using a phase diagram in in (e, p) space for p
=
vp + ave + a (u y) and e = ap + by a m r equations as presented in Figure A1 in the
appendix.
87
1.7.21
Exchange rate overshooting under the ‡oating exchange rate system
Analysis of quarterly time series of UK reveals three empirical facts to justify above model. Firstly
there is a good evidence of exchange rate overshooting in UK. Sterling dollar exchange rate responds
immediately to any shocks in the market (but is less volatile compared to that with Euro in recent
years) but the in‡ation is more rigid (less volatile) as shown by conditional volatility of GARCH
(1,1) models of the exchange rate and in‡ation in Fig. 37. Secondly the interest rate seems to
Granger cause changes in both in‡ation and exchange rates but there is no signi…cant causality
from in‡ation to the exchange rate. In fact exchange rate seems to be more persistent as shown
by the coe¢ cient of its lagged term in Table 11. Thirdly there is a good empirical support for the
long run relationship between in‡ation and the exchange rate; they are cointegrated on the basis of
trace test at 3 percent level of signi…cance as shown in Table 13 (at 7.6 percent by the max test).
Perhaps this long run should mean to be the time span for the entire business cycles.
Figure 37: Conditional volatility of exchange rate and in‡ation
Table 29: Simultaneous equation model of in‡ation and exchange rate
In‡ation
Exchange rate
Exogenous variables Coe¢ cient tvalue prob Coe¢ cient tvalue prob
In‡ation (-1)
0.952
49.4
0.00 -0.0002
-0.176 0.861
Exchange rate (-1)
0.733
2.53
0.01 0.951
45.1
0.00
Constant
-1.06
-2.10
0.04 0.086
2.33
0.02
In fact exchange rate seems to be explained by growth of money supply and its lagged term.
These empirical facts imply that ‡oating exchange rate system is optimal for the UK as it lowers
88
Table 30: Correlation among residuals of in‡ation and exchange rate equations (standard deviations
on diagonal)
Correlation among errors
In‡ation Exchange rate
In‡ation
1.266
0.095
Exchange rate
0.095
0.0921
Table 31: Cointegration between in‡ation and exchange rate
rank-order Trace test [prob] max-test [prob]
0
16.69 [0.031]*
13.03 [0.076]
1
3.65 [0.056]
3.65 [0.056]
the real side adjustments due to external shocks to the economy. These …ndings are consistent
to theoretical analysis of Miller and Weller (1991) that the exchange rates and the impacts of
exchange rates in the economy are very unpredictable. Therefore greater reliance should be in
creating internal demand to combat the sluggish growth problem.
1.7.22
Conclusion
Deepest recession since the World War II in 2009 has created serious challenges to …scal, monetary
and trade policies in UK. While the government is struggling to implement debt reduction plan
that aims to reduce de…cit that rose to 13 percent of GDP in 2009 to 2 percent by 2018-19, options
available on revenue and spending sides are very limited. With the lowest 0.5 percent bank rate in
the history, the Bank of England is facing a liquidity trap and credit ‡ows to private sectors are
very slow but the in‡ation went up to 5 percent, well above the target rate of 2 percent. Challenges
in trade appear as the Ricardian comparative advantage gradually is in more favour of emerging
economies including Brazil, China, India and South Korea though around 25 percent devaluation
of Sterling Pound has contributed a bit in growth of exports. With econometric estimation of parameters based on time series data of the closed and open economy models with the basic Keynesian
models this paper tries to provide answers to questions relating to appropriate models of business
cycles and impulse response analyses in the short run of real and nominal shocks for analysing the
dynamics of output, in‡ation, exchange rate and other macro variables in the UK economy.
Stochastic Keynesian IS-LM, stabilisation, AS-AD and open economy models are found signi…cant and e¤ective in evaluating the impacts of …scal, monetary and trade policy shocks in
macro variables in the UK. They are simpler and more transparent than equivalent DSGE or
RBC models and should be complemented by dynamic general equilibrium models in Hicks-Stone–
Meade-Mirrlees-Pissarides tradition that takes account of heterogeneity of …rms and households for
analysing challenges of stability and slow recovery facing the UK economy after the slump that
started with the …nancial crisis 2008.
References
[1 ]
A i y a g a r i S . R a o , A l b e r t M a r c e t , T h o m a s J . S a r g e n t , J u h a S e p p ä l ä ( 2 0 0 2 ) O p t i m a l T a x a t i o n w i t h o u t S t a t e - C o n t i n g e n t D e b t , Journal of P olitical
E conom y, 1 1 0 : 6 : D e c . , 1 2 2 0 - 1 2 5 4
[2 ]
A l t i g D E , C . T . C a r l s t r o m a n d K . L . L a n s i n g ( 1 9 9 5 ) C o m p u t a b l e G e n e r a l E q u i l i b r i u m M o d e l s a n d M o n e t a r y P o l i c y A d v i c e , Journal of M oney
C redit an d B an kin g, 2 7 : 4 : N o v . : 1 4 7 2 - 1 4 9 3 .
89
[3 ]
A n g e lo p o u lo u E ., H . D . G ib so n (2 0 0 9 ) T h e B a la n c e S h e e t C h a n n e l o f M o n e ta ry P o lic y Tra n sm issio n : E v id e n c e fro m th e U n ite d K in g d o m ,
E conom ica, 7 6 : 3 0 4 : O c t . : 6 7 5 - 7 0 3 .
[4 ]
A r e s t i s P a n d M . S a w y e r ( 1 9 9 8 ) K e y n e s i a n E c o n o m i c P o l i c i e s f o r t h e N e w M i l l e n n i u m , E conom ic Journ al, 1 0 8 , 4 4 6 , 1 8 1 - 1 9 5
[5 ]
A r e s t i s P , G . C h o r t a r e a s a n d J D T s o u k a l a s ( 2 0 1 0 ) M o n e y a n d i n f o r m a t i o n i n a N e w N e o c l a s s i c a l S y n t h e s i s F r a m e w o r k , E conom ic Journal,
1 2 0 : 5 4 2 : F e b r u a r y.
[6 ]
A r n o l d J . M . , B . B r y s , C . H e a d y, C . S c h w e l l n u s a n d L V a r t i a ( 2 0 1 1 ) T a x p o l i c y f o r E c o n o m i c R e c o v e r y a n d G r o w t h , E conom ic Journal,
1 2 1 :5 5 0 :F 5 9 -F 8 0
[7 ]
A s h J . C . K a n d D . J . S m y t h ( 1 9 7 3 ) F o r e c a s t i n g t h e U n i t e d K i n g d o m E c o n o m y, S a x o n H o u s e , L o n d o n .
[8 ]
A t k i n s o n A . B . a n d N . H . S t e r n ( 1 9 7 4 ) P i g o u , T a x a t i o n a n d P u b l i c G o o d s , Review of E con om ic Studies, 4 1 : 1 : 1 1 9 - 1 2 8 .
[9 ]
A u e r b a c h A . J . a n d L . K o t l i k o ¤ ( 1 9 8 7 ) D y n a m i c F i s c a l P o l i c y, C a m b r i d g e U n i v e r s i t y P r e s s .
[1 0 ]
B a n k o f E n g l a n d ( 2 0 1 2 ) I n ‡a t i o n R e p o r t , A u g u s t , 2 0 1 2 .
[1 1 ]
B a r r o , R . J . ( 1 9 7 4 ) , A r e G o v e r n m e n t B o n d s N e t W e a l t h ? , Journal of P olitical E con om y, 8 2 , 1 0 9 5 - 1 1 1 7 .
[1 2 ]
B a r r o R . J . ( 1 9 8 9 ) T h e R i c a r d i a n A p p r o a c h t o B u d g e t D e … c i t s , Journal of E con om ic P erspectives, 3 , 2 , 3 7 - 5 4
[1 3 ]
B a r r o R . J . a n d G o r d o n ( 1 9 8 3 ) A P o s i t i v e T h e o r y o f M o n e t a r y P o l i c y i n a N a t u r a l R a t e M o d e l , Journal of P olitical E con om y, 9 1 : 4 : 5 8 9 - 6 1 0 .
[ 1 4 ] B a s u P . ( 1 9 9 6 ) P r o p o r t i o n a l I n c o m e T a x a n d t h e R i c a r d i a n E q u i v a l e n c e i n a N o n - E x p e c t e d U t i l i t y M a x i m i z i n g M o d e l S outhern E conom ic
Journal, 6 3 , 1 , 2 3 3 - 2 4 3 .
[1 5 ]
B e a n C . ( 2 0 0 9 ) T h e m e a n i n g o f i n t e r n a l b a l a n c e ’ T h i r t y y e a r s o n , E conom ic Journ al, 1 1 9 ( N o v e m b e r ) , F 4 4 2 –F 4 6 0 .
[1 6 ]
B e a n C . ( 1 9 9 8 ) T h e N e w U K M o n e t a r y A r r a n g e m e n t s : A V i e w f r o m t h e L i t e r a t u r e , E conom ic Journ al, 1 0 8 , 4 5 1 , 1 7 9 5 - 1 8 0 9
[1 7 ]
B e e t s m a R . a n d M . G i u l i o d o r i ( 2 0 1 1 ) T h e E ¤ e c t s o f g o v e r n m e n t p u c h a s e s s h o c k s a n d e s t i m a t e s f o r E U , E conom ic Journ al, 1 2 1 : 5 5 0 : F 4 - F - 3 2
[1 8 ]
B e n a t i , L u c a . ( 2 0 0 8 ) “ T h e ‘G r e a t M o d e r a t i o n ’ i n t h e U n i t e d K i n g d o m . ” Journal of M on ey,C redit an d B an king, 4 0 , 1 2 1 –4 7 .
[1 9 ] B e r n a n k e B e n S ., J . B o iv in a n d P. E lia s z ( 2 0 0 5 ) M e a s u r in g t h e E ¤ e c t s o f M o n e t a r y P o lic y : A Fa c t o r -A u g m e n t e d Ve c t o r A u t o r e g r e s s iv e
( FAVA R ) A p p r o a c h , Q uarterly Journ al of E conom ics, 1 2 0 , 1 , 3 8 7 - 4 2 2 .
[ 2 0 ] B e r n a n k e B . S a n d F . S . M i s h k i n ( 1 9 9 7 ) I n ‡a t i o n T a r g e t i n g : A N e w F r a m e w o r k f o r M o n e t a r y P o l i c y ? Journal of E con om ic P erspectives, 1 1 2 ,
9 7-11 6
[2 1 ]
B e r n h e i m B . D . ( 1 9 8 9 ) A N e o c l a s s i c a l P e r s p e c t i v e o n B u d g e t D e … c i t s , Journal of E con om ic P erspectives, 3 : 2 S p r i n g : 5 5 - 7 2
[2 2 ]
B e n a s s y J . P . ( 2 0 0 2 ) T he m acroeconom ics of Im perfect C om petition an d N on clearin g M arkets, M I T P r e s s .
[2 3 ]
B e s l e y T . ( 2 0 0 1 ) F r o m m i c r o t o m a c r o : p u b l i c p o l i c i e s a n d a g g r e g a t e e c o n o m i c p e r f o r m a n c e , F iscal S tudies, 2 2 , 3 , 3 5 7 –3 7 4 .
[2 4 ] B h a tta ra i K a n d H . D ix o n (2 0 1 4 ) E q u ilib riu m U n e m p loy m e nt in a G e n e ra l E q u ilib riu m M o d e l w ith Ta x e s, T h e M a n ch e ste r S ch o o l, fo rth c o m in g .
[ 2 5 ] B h a t t a r a i K ( 2 0 1 2 ) F i s c a l p o l i c y g r o w t h a n d i n c o m e d i s t r i b u t i o n i n U K , paper for the A E A an d E E A /E S E M conferences, U n i v e r s i t y o f H u l l ,
B u sin e ss S ch o o l.
[ 2 6 ] B h a t t a r a i K ( 2 0 1 1 ) E c o n o m e t r i c a n d S t o c h a s t i c G e n e r a l E q u i l i b r i u m M o d e l s f o r E v a l u a t i o n o f E c o n o m i c P o l i c i e s , In tern ation al Journ al of
Trade and G lobal M arkets, 4 : 2 : 2 1 2 - 2 4 1 .
[2 7 ]
B h a t t a r a i , K . ( 2 0 0 8 ) A n E m p i r i c a l S t u d y o f I n t e r e s t D e t e r m i n a t i o n R u l e s , A pplied F in an cial E con om ics, 1 8 : 4 : 3 2 7 - 3 4 3 .
[ 2 8 ] B h a t t a r a i K . a n d J . W h a l l e y ( 2 0 0 0 ) G e n e r a l E q u i l i b r i u m M o d e l l i n g o f U K T a x P o l i c y, i n H o l l y S a n d M W e a l e E d s . " E conom etric M odel ling:
Techniques an d A pplications " , p p . 1 3 2 - 1 5 7 , C a m b r i d g e U n i v e r s i t y P r e s s .
[ 2 9 ] B h a t t a r a i K . a n d J . W h a l l e y ( 1 9 9 9 ) R o l e o f l a b o u r d e m a n d e l a s t i c i t i e s i n t a x i n c i d e n c e a n a l y s i s w i t h h e t e r o g e n e i t y o f l a b o u r , E m pirical
E conom ics, 2 4 ( 4 ) . 5 9 9 - 6 2 0 .
[3 0 ]
B l a k e A . P . , M . W e a l e ( 1 9 9 8 ) C o s t s o f S e p a r a t i n g B u d g e t a r y P o l i c y f r o m C o n t r o l o f I n ‡a t i o n : A N e g l e c t e d A s p e c t o f C e n t r a l B a n k I
[3 1 ]
n d e p e n d e n c e , O xford E conom ic P apers, 5 0 , 3 , 4 4 9 - 4 6 7
[3 2 ]
B la ck b u r n K e it h a n d M o r t e n O . R a v n ( 1 9 9 2 ) B u s in e s s C y c le s in t h e U n it e d K in g d o m : Fa c t s a n d F ic t io n s , E c o n o m ic a , 5 9 , 2 3 6 , 3 8 3 -4 0 1
[ 3 3 ] B l a k e A . P . , M . W e a l e a n d G . Y o u n g ( 2 0 0 0 ) O p t i m a l m o n e t a r y p o l i c y i n H o l l y S a n d M W e a l e E d s . " E conom etric M odel lin g: Techn iques and
A pplications " , p p . 2 0 9 - 2 3 6 , C a m b r i d g e U n i v e r s i t y P r e s s .
[3 4 ]
B l a n c h a r d , O a n d C . M . K a h n ( 1 9 8 0 ) , T h e S o l u t i o n o f L i n e a r D i ¤ e r e n c e M o d e l s u n d e r R a t i o n a l E x p e c t a t i o n s , E conom etrica 4 8 , 1 3 0 5 - 1 3 1 3 .
[ 3 5 ] B l a n c h a r d O . J . a n d N . K i y o t a k i ( 1 9 8 7 ) M o n o p o l i s t i c c o m p e t i t i o n a n d t h e e ¤ e c t s o f a g g r e g a t e d e m a n d , Am erican E conom ic Review , 7 7 : S e p t . :
6 47-6 6.
[3 6 ] B la n ch a rd , O . a n d P e ro tti, R . (2 0 0 2 ). A n e m p iric a l ch a ra c te riz a tio n o f th e d y n a m ic e ¤ e c ts o f ch a n g e s in g ove rn m e nt sp e n d in g a n d ta x e s
o n o u t p u t , Q uarterly Journ al of E conom ics, 1 1 7 ( 4 ) : N o v . : 1 3 2 9 –6 8 .
90
[ 3 7 ] B l a n c h ‡o w e r , D . G a n d A . B r y s o n ( 2 0 1 0 ) T h e W a g e I m p a c t o f T r a d e U n i o n s i n t h e U K P u b l i c a n d P r i v a t e S e c t o r s , E conom ica, 7 7 , 3 0 5 ,
92 -10 9
[3 8 ] B lin d e r A .S ., M . E h r m a n n , M . Fr a t z s ch e r , d e H a a n J ., D . J . J a n s e n ( 2 0 0 8 ) C e n t r a l B a n k C o m m u n ic a t io n a n d M o n e t a r y P o lic y : A S u r v e y
o f T h e o r y a n d E v i d e n c e , Journal of E con om ic L iterature, 4 6 : 4 : 9 1 0 - 9 4 5 .
[ 3 9 ] B o i n e t V . , C . M a r t i n ( 2 0 0 8 ) , T a r g e t s , Z o n e s , a n d A s y m m e t r i e s : A F l e x i b l e N o n l i n e a r M o d e l o f R e c e n t U K M o n e t a r y P o l i c y, O xford E con om ic
P apers, 6 0 , 3 , J u l . , 4 2 3 - 4 3 9 .
[4 0 ]
B u c h a n a n , J . , ( 1 9 5 8 ) P ublic prin ciples of public debt ( I r w i n : H o m e w o o d , I l l i n o i s ) .
[4 1 ]
B u rn s, A a n d W . M ich e ll, (1 9 4 6 ), “M e a su rin g B u sin e ss C y c le s” N B E R , N e w Yo rk .
[4 2 ]
B u r n s i d e , C . , E i c h e n b a u m , M . a n d F i s h e r , J . D . ( 2 0 0 4 ) . F i s c a l s h o c k s a n d t h e i r c o n s e q u e n c e s , Journal of E con om ic T heory, 1 1 5 : 8 9 –1 1 7 .
[4 3 ]
B r a u n i n g e r M . ( 2 0 0 5 ) T h e b u d g e t d e … c i t , p u b l i c d e b t , a n d e n d o g e n o u s g r o w t h , Journal of P ublic E con om ic T heory, 7 ( 5 ) , 2 0 0 5 , p p . 8 2 7 –8 4 0 .
[4 4 ]
C a l v o , G . ( 1 9 8 3 ) S t a g g e r e d c o n t r a c t s i n a u t i l i t y - m a x i m i z i n g f r a m e w o r k , Journal of M on etary E con om ics, 1 2 , 3 8 3 - 9 8 .
[4 5 ] C a u c u tt E .M , S . Im ro h o ro g lu a n d K . B . K u m a r (2 0 0 6 ) D o e s th e P ro g re s s iv ity o f In c o m e Ta x e s M a tte r fo r H u m a n C a p ita l a n d G row th ?
Journal of P ublic E con om ic T heory, 8 : 1 : 9 5 - 1 1 8
[ 4 6 ] C a s t l e , J . L . , D o o r n i k , J . A . , a n d H e n d r y, D . F . ( 2 0 1 1 ) . E v a l u a t i n g a u t o m a t i c m o d e l s e l e c t i o n , Journal of T im e S eries E con om etrics, 3 : 1 : 1 9 4 1 1928.
[ 4 7 ] C l a r i d a R . , G a l i J . , a n d G e r t l e r M . ( 1 9 9 9 ) T h e S c i e n c e o f M o n e t a r y P o l i c y : A N e w K e y n e s i a n p e r s p e c t i v e , Journal of E con om ic L iterature ; 3 7
(4), 1 66 1-70 7.
[ 4 8 ] C l a r i d a R . , G a l i J . , a n d G e r t l e r M . ( 2 0 0 1 ) O p t i m a l M o n e t a r y P o l i c y i n C l o s e d V e r s u s O p e n E c o n o m i e s : A n i n t e g r a t e d a p p r o a c h . Am erican
E conom ic Review , 9 1 ( 2 ) ; M a y ; 2 4 8 - 2 5 2 .
[ 4 9 ] C h a d h a , J . S . a n d C . N o l a n ( 2 0 0 7 ) O p t i m a l S i m p l e R u l e s f o r t h e C o n d u c t o f M o n e t a r y a n d F i s c a l P o l i c y, Journal of M acroecon om ics, D e c ,
2 9 :4 : 6 6 5 -8 9 .
[ 5 0 ] C h r i s t i a n o , L , M . E i c h e n b a u m a n d C . E v a n s ( 2 0 0 5 ) N o m i n a l r i g i d i t i e s a n d t h e d y n a m i c e ¤ e c t s o f s h o c k s t o m o n e t a r y p o l i c y, Journal of
P olitical E conom y, 1 1 3 : 1 : 1 - 4 5 .
[ 5 1 ] C h u r c h K . B . , P . M i t c h e l , J . S a u l t a n d K . W a l l i s ( 1 9 9 7 ) C o m p a r a t i v e p r o p e r t i e s o f m o d e l s o f t h e U K e c o n o m y, N ation al In stitute E conom ic
Review , 1 6 1 : 9 1 - 1 0 0 .
[ 5 2 ] C o b h a m D . ( 1 9 9 7 ) T h e P o s t - E r m F r a m e w o r k f o r M o n e t a r y P o l i c y i n t h e U n i t e d K i n g d o m : B o u n d e d C r e d i b i l i t y, E conom ic Journal, 1 0 7 , 4 4 3 :
J u l.:1 1 2 8 -1 1 4 1 .
[5 3 ] C o o k S . S . H o lly a n d P. Tu rn e r ( 2 0 0 0 ) T h e c o n d u c t o f m o n e t a ry p o lic y w h e n t h e b u s in e s s c y c le is n o n -lin e a r in H o lly S a n d M W e a le
E d s . ( 2 0 0 0 ) E conom etric M odel ling:Techniques an d A pplications, C a m b r i d g e U n i v e r s i t y P r e s s .
[5 4 ]
D a v i d s o n P . ( 1 9 8 8 ) P o s t K e y n e s i a n E m p l o y m e n t A n a l y s i s a n d t h e M a c r o e c o n o m i c s o f O E C D U n e m p l o y m e n t , E conom ic Journ al, M a y 8 1 7 - 8 3 1 .
[ 5 5 ] d e n H a a n W . J . a n d A M a r c e t ( 1 9 9 0 ) S o l v i n g t h e s t o c h a s t i c g r o w t h m o d e l b y p a r a m e t e r i s i n g e x p e c t a t i o n s , Joun ral of B usin ess and E conom ic
S tatistics, 8 : 1 : 3 1 - 3 4 .
[5 6 ]
D e s a i M . , a n d G . W e b e r ( 1 9 8 8 ) A K e y n e s i a n M a c r o - E c o n o m e t r i c M o d e l o f t h e U K : 1 9 5 5 - 1 9 8 4 , Journal of A pplied E con om etrics, 3 , 1 1 - 3 3 .
[5 7 ]
D i a m o n d , P . A . , ( 1 9 6 5 ) , N a t i o n a l d e b t i n a n e o c l a s s i c a l g r o w t h m o d e l , Am erican E con om ic Review , 5 5 , 5 , 1 2 2 6 - 1 1 5 0 .
[5 8 ]
D i a m o n d , D o u g l a s W . a n d D y b v i g , P h i l i p H . ( 1 9 8 3 ) B a n k R u n s , D e p o s i t I n s u r a n c e , a n d L i q u i d i t y, Journal of P olitical E con om y, 4 0 1 - 4 1 9 .
[5 9 ]
D i x o n H . ( 1 9 8 8 ) C o n t r o v e r s y : t h e m a c r o e c o n o m i c s o f u n e m p l o y m e n t i n t h e O E C D , E conom ic Journ al, 1 0 8 : 7 7 9 - 7 8 1 , M a y
[6 0 ]
D i x o n H a n d N R a n k i n ( 1 9 9 4 ) I m p e r f e c t C o m p e t i t i o n a n d M a c r o e c o n o m i c s : A S u r v e y, O xford E conom ic P apers, 4 6 , 2 , 1 7 1 - 1 9 9 9 .
[6 1 ]
D o r n b u s c h R . ( 1 9 7 6 ) E x p e c t a t i o n s a n d E x c h a n g e R a t e D y n a m i c s , Journal of P olitical E con om y, 8 4 : 6 : 1 1 6 1 - 1 1 7 6
[6 2 ]
D r i ¢ l l J . , A . S n e l l ( 2 0 0 3 ) W h a t M o v e s O E C D R e a l I n t e r e s t R a t e s ? Journal of M on ey, C redit and B an king, 3 5 : 3 : J u n . : 3 7 5 - 4 0 2
[6 3 ]
D r i ¢ l l J . , C . S c h u l t z ( 1 9 9 2 ) W a g e S e t t i n g a n d S t a b i l i z a t i o n P o l i c y i n a G a m e w i t h R e n e g o t i a t i o n , O xford E conom ic P apers, 4 4 , 3 , 4 4 0 - 4 5 9 .
[6 4 ]
E a t o n J a n d S K o r t u m ( 2 0 0 2 ) T e c h n o l o g y, G e o g r a p h y a n d T r a d e , E conom etrica, 7 0 : 5 : 1 7 4 1 - 1 7 7 9 .
[6 5 ]
E l l i s o n M . a n d J . P e a r l m a n ( 2 0 1 1 ) S a d d l e p a t h l e a r n i n g , Journal of E con om ic theory 1 4 6 , 1 5 0 0 - 1 5 1 9 .
[ 6 6 ] G a i P . , S . K a p a d i a , S . M i l l a r d , A . P e r e z ( 2 0 0 8 ) F i n a n c i a l I n n o v a t i o n , M a c r o e c o n o m i c S t a b i l i t y a n d S y s t e m i c C r i s e s , E conom ic Journal, 1 1 8 ,
5 27, 40 1-42 6
[ 6 7 ] G a l i J . a n d T . M o n a c e l l i ( 2 0 0 5 ) . M o n e t a r y P o l i c y a n d E x c h a n g e R a t e V o l a t i l i t y i n a S m a l l O p e n E c o n o m y, Review of E con om ic Studies ; 7 2 ( 3 ) :
7 07-7 34
[ 6 8 ] G a r r a t t A . , K . L e e , M . H . P e s a r a n a n d Y . S h i n ( 2 0 0 3 ) A S t r u c t u r a l C o i n t e g r a t i o n VA R A p p r o a c h t o M a c r o e c o n o m e t r i c M o d e l l i n g , E conom ic
Journal, 1 1 3 : 4 8 7 : 4 1 2 - 4 4 5 .
[ 6 9 ] G h o s a l S . a n d M . M i l l e r ( 2 0 0 3 ) C o - O r d i n a t i o n F a i l u r e , M o r a l H a z a r d a n d S o v e r e i g n B a n k r u p t c y P r o c e d u r e s , E conom ic Journal, 1 1 3 , 4 8 7 :
A p r.: 2 7 6 -3 0 4
91
[7 0 ]
G ilh o o ly R ., M . W e a le a n d T . W ie la d e k (2 0 1 2 ) D isa g g re g a tin g th e inte rn a tio n a l b u sin e ss c y c le , D isc u ssio n P a p e r N o . 3 7 , B a n k o f E n g la n d .
[7 1 ]
G o o d h a r t C . ( 1 9 8 9 ) T h e C o n d u c t o f M o n e t a r y P o l i c y, E conom ic Journ al, 9 9 , 3 9 6 , 2 9 3 - 3 4 6 .
[ 7 2 ] G o o d h a r t C . ( 1 9 9 4 ) W h a t S h o u l d C e n t r a l B a n k s D o ? W h a t S h o u l d B e T h e i r M a c r o e c o n o m i c O b j e c t i v e s a n d O p e r a t i o n s ? E conom ic Journal,
1 0 4 , 4 2 7 ,1 4 2 4 -1 4 3 6
[7 3 ]
F e l d s t e i n M ( 1 9 8 5 ) D e b t a n d t a x e s i n t h e t h e o r y o f p u b l i c … n a n c e , Journal of P ublic E con om ics, 2 8 : 2 3 3 - 4 6
[7 4 ]
F e l d s t e i n M ( 1 9 8 2 ) G o v e r n m e n t d e … c i t s a n d a g g r e g a t e d e m a n d , Journal of M on etary E conom ics 9 , 1 , 1 - 2 0 .
[ 7 5 ] F i s h e r P . a n d J . W h i t l e y ( 2 0 0 0 ) M a c r o e c o n o m i c m o d e l s a t t h e B a n k o f E n g l a n d i n H o l l y S a n d M W e a l e E d s . E conom etric M odel lin g: Techn iques
an d A pplication s, C a m b r i d g e U n i v e r s i t y P r e s s .
[7 6 ]
F i s h e r J . D . M . a n d R y a n P . ( 2 0 1 0 ) U s i n g S t o c k R e t u r n s t o I d e n t i f y G o v e r n m e n t S p e n d i n g S h o c k s , E conom ic Journ al, 1 2 0 : 5 4 4 , M a y : 4 1 6 - 4 3 6 .
[7 7 ]
F l e m i n g , J . M . ( 1 9 6 2 ) D o m e s t i c … n a n c i a l p o l i c i e s u n d e r … x e d a n d u n d e r ‡o a t i n g e x c h a n g e r a t e s , IM F S ta¤ P apers, N o v . , 9 , 3 6 9 - 7 9 .
[7 8 ]
F r i e d m a n , M . ( 1 9 6 8 ) , T h e R o l e o f M o n e t a r y P o l i c y, Am erican E con om ic Review , 5 8 : M a r c h , 1 - 1 7 .
[7 9 ] G e m m e ll N ., R . K n e lle r a n d I. S a n z (2 0 1 1 ) T h e tim in g p e rsiste n c e o f …sc a l p o lic y im p a c ts o n g row th : E v id e n c e fro m O E C D , c o u ntrie s,
E conom ic Journ al, 1 2 1 : 5 5 0 : F 3 3 - F - 5 8
[8 0 ] G re e n sd a le J . V ., S . H a ll, S .G .B . H e n ry a n d J N ix o n (2 0 0 0 ) U n e m p loy m e nt, th e n a tu ra l ra te a n d stru c tu ra l ch a n g e in H o lly S a n d M W e a le
E d s . " E conom etric M odel ling: Techn iques and A pplication s " , p p . 1 3 2 - 1 5 7 , C a m b r i d g e U n i v e r s i t y P r e s s .
[8 1 ]
H a s k e l J . a n d M . J . S l a u g h t e r ( 2 0 0 1 ) T r a d e , T e c h n o l o g y a n d U . K . W a g e I n e q u a l i t y, E conom ic Journ al, 1 1 1 , 4 6 8 , 1 6 3 - 1 8 7
[8 2 ]
H e n d r y D . F . ( 1 9 9 7 ) T h e E c o n o m e t r i c s o f M a c r o e c o n o m i c F o r e c a s t i n g , E conom ic Journ al, 1 0 7 , 4 4 4 . , 1 3 3 0 - 1 3 5 7 .
[ 8 3 ] H e n d r y D . F . a n d M . P . C l e m e n t ( 2 0 0 0 ) E c o n o m i c f o r e c a s t i n g i n t h e f a c e o f s t r u c t u r a l b r e a k s , i n H o l l y S a n d M W e a l e E d s . E conom etric
M odel lin g: Techniques an d A pplications, C a m b r i d g e U n i v e r s i t y P r e s s .
[8 4 ]
H e n d r y D . F a n d J . A . D o o r n i k ( 1 9 9 4 ) M o d e l l i n g L i n e a r D y n a m i c E c o n o m e t r i c S y s t e m s , S cottish Journ al of P olitical E conom y, 4 1 , 1 - 3 3
[8 5 ]
H i c k s J o h n ( 1 9 9 0 ) T h e U n i … c a t i o n o f M a c r o - E c o n o m i c s , E con om ic Journ al, 1 0 0 , 4 0 1 , 5 2 8 - 5 3 8
[8 6 ]
H i c k s , J . R . ( 1 9 3 7 ) M r . K e y n e s a n d t h e “ C l a s s i c s ”; A S u g g e s t e d I n t e r p r e t a t i o n s , E conom etrica 5 : 1 4 7 - 1 5 9 .
[8 7 ]
H M Trea su ry (2 0 1 1 ) B u d g et R ep o rt, M a rch 2 0 1 1 , L o n d o n .
[8 8 ]
H o l l a n d A . a n d A . S c o t t ( 1 9 9 8 ) D e t e r m i n a n t s o f U K B u s i n e s s C y c l e , E conom ic Journal, 1 0 8 : 4 4 9 : 1 0 6 7 - 1 0 9 2 .
[8 9 ]
H o y M . , J L i v e r n o i s , C M c K e n n a , R R e e s a n d T . S t e n g o s ( 2 0 0 1 ) M athem atics for E conom ics, 2 n d e d . , M I T P r e s s
[ 9 0 ] I a c o v i e l l o , M . , a n d S . N e r i ( 2 0 1 0 ) H o u s i n g M a r k e t S p i l l o v e r s : E v i d e n c e f r o m a n E s t i m a t e d D S G E M o d e l , Am erican E con om ic Journ al M acro,
(A p ril), 1 -4 4 . http :/ / w w w 2 .b c .e d u / ~ ia c ov ie l/
[9 1 ]
In s t it u t e o f F is c a l S t u d ie s ( 2 0 1 4 ) T h e IF S G r e e n B u d g e t , Fe b r u a r y 2 0 1 4 , IF S L o n d o n .
[9 2 ]
J o h n s o n H . G . ( 1 9 5 3 - 5 4 ) O p t i m a l t a r i ¤ s a n d R e t a l i a t i o n , Review of E con om ic Studies, 2 1 ( 2 ) , p p . 1 4 2 - 4 3 .
[9 3 ]
Johnson T . C .
[9 4 ]
K e y n e s J . M . ( 1 9 3 7 ) T h e G e n e r a l T h e o r y o f E m p l o y m e n t , Q uarterly Journ al of E conom ics, 5 1 , 2 , F e b . , 2 0 9 - 2 2 3 .
( 2 0 0 9 ) L i q u i d C a p i t a l a n d M a r k e t L i q u i d i t y, E conom ic Journal, 1 1 9 , . 5 4 0 : O c t . : 1 3 7 4 - 1 4 0 4 .
[9 5 ] K irs a n ova T ., C . L e ith a n d S . W re n -L e w is ( 2 0 0 9 ) M o n e ta ry a n d …s c a l p o lic y in te ra c tio n : T h e c u rre n t c o n s e n s u s a s s ig n m e n t in th e lig h t
O f r e c e n t d e v e l o p m e n t s , E conom ic Journ al, 1 1 9 , N o v , F 4 8 2 –F 4 9 6 .
[ 9 6 ] K l e i n , L . ( 1 9 7 1 ) ‘F o r e c a s t i n g a n d p o l i c y e v a l u a t i o n u s i n g l a r g e s c a l e e c o n o m e t r i c m o d e l s ’, i n I n t r i l i g a t o r , M . D . ( E d . ) : Fron tiers of Q uan titative
E conom ics, N o r t h H o l l a n d .
[9 7 ] K o ch e rla ko ta N . R . a n d K . M Y i (1 9 9 6 ) A sim p le tim e se rie s te st o f e n d o g e n o u s a n d e x o g e n o u s g row th m o d e ls: a n a p p lic a tio n to th e U n ite d
S t a t e s , Review of E con om ic Studies, 7 8 : 1 : 1 2 6 - 1 3 4
[9 8 ]
K r u g m a n P . ( 1 9 7 9 ) A M o d e l o f B a l a n c e o f P a y m e n t C r i s i s , Journal of M on ey C redit an d B an kin g, 1 1 , 3 , 3 1 1 - 3 2 5 .
[ 9 9 ] K y d l a n d F . E a n d E . C . P r e s c o t t ( 1 9 7 7 ) R u l e s r a t h e r t h a n d i s c r e t i o n s : t h e I n c o n s i s t e n c y o f O p t i m a l P l a n s , Journal of P olitical E conom y, 8 5 : 3 :
4 73-4 91 .
[1 0 0 ]
K i n g M e r v y n ( 2 0 0 4 ) T h e I n s t i t u t i o n s o f M o n e t a r y P o l i c y, Am erican E con om ic Review - P r o c e e d i n g s , 9 4 , 2 , 1 - 1 3
[1 0 1 ]
K i n g R . G . a n d P l o s s e r C . I . ( 1 9 8 4 ) M o n e y C r e d i t a n d P r i c e s i n a R e a l B u s i n e s s C y c l e , Am erican E con om ic Review , 6 4 ( J u n e ) 2 6 3 - 3 8 0 .
[ 1 0 2 ] L e i t h C . B a n d S . W r e n - L e w i s ( 2 0 0 0 ) H o w t o u g h m o n e t a r y p o l i c y b e i f i n ‡a t i o n i s f o r w a r d l o o k i n g , i n H o l l y S a n d M W e a l e E d s . E conom etric
M odel lin g: Techniques an d A pplications, C a m b r i d g e U n i v e r s i t y P r e s s .
[1 0 3 ]
L i m G . C . a n d M c N e l i s ( 2 0 0 8 ) , C o m p u t a t i o n a l M a c r o e c o n o m i c s f o r t h e O p e n E c o n o m y, M I T P r e s s .
[1 0 4 ] L iu , P h ilip H . M u m ta z (2 0 1 1 ) E vo lv in g M a c ro e c o n o m ic D y n a m ic s in a S m a ll O p e n E c o n o m y :
M o d e l f o r t h e U K , J ournal of M on ey, C redit, an d B ankin g, 4 3 , 7 , 1 4 4 3 - 7 4 .
92
A n E stim a te d M a rkov S w itch in g D S G E
[1 0 5 ]
L u c a s R . E . ( 1 9 7 3 ) S o m e I n t e r n a t i o n a l E v i d e n c e o n O u t p u t I n ‡a t i o n T r a d e - O ¤ s , Am erican E con om ic Review , 6 3 : 3 : 3 2 6 - 3 3 4 .
[1 0 6 ]
M a n k i w N . G . ( 1 9 8 9 ) R e a l B u s i n e s s c y c l e : A N e w K e y n e s i a n P e r s p e c t i v e , Journal of E con om ic P erspectives, 3 , 3 : 7 9 - 9 0 .
[1 0 7 ]
M a r t i n , C . a n d M i l a s , C . ( 2 0 0 4 ) M o d e l l i n g m o n e t a r y p o l i c y : i n ‡a t i o n t a r g e t s i n p r a c t i c e , E conom ica, 7 1 , 2 0 9 - 2 1 .
[ 1 0 8 ] M c C a l l u m , B . a n d N e l s o n , E . ( 1 9 9 9 ) A n o p t i m i z i n g I S - L M S p e c i … c a t i o n f o r m o n e t a r y p o l i c y a n d b u s i n e s s c y c l e a n a l y s i s , J o u r n a l o f M o n e y,
C re d it, a n d B a n k in g , 3 1 , 2 9 6 -3 1 6 .
[ 1 0 9 ] M e a d e J . E . , a n d I r o n s i d e , J o n e s , B e l l , F l e m m i n g , K a y, K i n g , M a c d o n a l d , S a n d f o r d a n d W h i t t i n g t o n , W i l l i s . ( 1 9 7 8 ) . T he Structure and
Reform of D irect Taxation. I F S , L o n d o n : G e o r g e A l l e n a n d U n w i n .
[1 1 0 ]
M e a d e , J .E . (1 9 5 1 ), T h e T h e o ry o f In te ra n tio n a l E c o n o m ic P o lic y : B a la n c e o f P ay m e n ts , L o n d o n : O x fo rd U n ive rs ity P re s s .
[1 1 1 ]
M e a d e , J . E . ( 1 9 5 8 ) , I s t h e n a t i o n a l d e b t a b u r d e n ? O xford E conom ic P apers, 1 0 , 2 , 1 6 3 - 1 8 3 .
[ 1 1 2 ] M e l l i s C a n d R W h i t t a k e r ( 2 0 0 0 ) T h e T r e a s u r y ’s f o r e c a s t s o f G D P a n d t h e R P I : h o w h a v e t h e y c h a n g e d a n d w h a t a r e u n c e r t a i n t i e s i n
H o l l y S a n d M W e a l e E d s . ( 2 0 0 0 ) E conom etric M odel ling: Techn iques and A pplication s, C a m b r i d g e U n i v e r s i t y P r e s s .
[ 1 1 3 ] M i l l e r , M . H . ; J . E . S p e n c e r ( 1 9 7 7 ) T h e S t a t i c E c o n o m i c E ¤ e c t s o f t h e U K J o i n i n g t h e E E C : A G e n e r a l E q u i l i b r i u m A p p r o a c h , Review of
E conom ic S tudies, 4 4 , 1 , 7 1 - 9 3 .
[1 1 4 ]
M i l l e r M . a n d P . W e l l e r ( 1 9 9 1 ) E x c h a n g e R a t e B a n d s w i t h P r i c e I n e r t i a , E conom ic Journ al, 1 0 1 : 4 0 9 : 1 3 8 0 - 1 3 9 9 .
[ 1 1 5 ] M i r r l e e s J . , a n d S . A d a m , T . B e s l e y, R . B l u n d e l l , S . B o n d , R . C h o t e , M . G a m m i e , P . J o h n s o n , G . M y l e s , J . P o t e r b a . 2 0 1 0 . D im en sion s of
tax design : the M irrlees review , O x f o r d : O x f o r d U n i v e r s i t y P r e s s .
[1 1 6 ]
M i r r l e e s , J . A . ( 1 9 7 1 ) A n e x p l o r a t i o n i n t h e t h e o r y o f o p t i m u m i n c o m e t a x a t i o n , Review of E con om ic Studies, 3 8 : 1 7 5 - 2 0 8 .
[1 1 7 ]
M i n f o r d P . a n d D . P e e l ( 2 0 0 2 ) Advan ced M acroeconom ics: A P rim er, E d w a r d E l g a r P u b l i s h i n g .
[ 1 1 8 ] M i y a z a w a K . ( 1 9 6 0 ) F o r e i g n T r a d e M u l t i p l i e r , I n p u t - O u t p u t A n a l y s i s a n d t h e C o n s u m p t i o n F u n c t i o n , Q uarterly Journal of E conom ics, 7 4 , 1
, 5 3-64
[ 1 1 9 ] M o d i g l i a n i , F . , ( 1 9 6 1 ) , L o n g r u n i m p l i c a t i o n s o f a l t e r n a t i v e … s c a l p o l i c i e s a n d t h e b u r d e n o f t h e n a t i o n a l d e b t , E conom ic Journ al 7 1 ,
7 28-7 55 .
[1 2 0 ]
M o n a c e l l i T a n d R . P e r o t t i ( 2 0 1 0 ) F i s c a l P o l i c y, t h e R e a l E x c h a n g e R a t e a n d T r a d e d G o o d s , E conom ic Journal, 1 2 0 ( M a y ) , 4 3 7 - 4 6 1 .
[ 1 2 1 ] M u n d e l l R . A ( 1 9 6 2 ) C a p i t a l m o b i l i t y a n d s t a b i l i s a t i o n p o l i c y u n d e r … x e d a n d ‡e x i b l e e x c h a n g e r a t e s , C anadian Journal of E con om ic and
P olitical S cien ce, 2 9 , 4 7 5 - 8 5 .
[1 2 2 ]
M o n e t a r y P o l i c y C o m m i t t e e B a n k o f E n g l a n d ( 1 9 9 9 ) T h e T r a n s m i s s i o n M e c h a n i s m o f M o n e t a r y P o l i c y, w w w . b a n k o f e n g l a n d . c o . u k .
[1 2 3 ]
N e l s o n E . ( 2 0 0 9 ) A n O v e r h a u l o f D o c t r i n e : T h e U n d e r p i n n i n g o f U K I n ‡a t i o n T a r g e t i n g , E conom ic Journal, 1 1 9 : 5 3 8 , F 3 3 3 - F 3 6 8 .
[1 2 4 ]
N e l s o n , E d w a r d , a n d K a l i n N i k o l o v . ( 2 0 0 4 ) , M o n e t a r y P o l i c y a n d S t a g ‡a t i o n i n t h e U K , Journal of M on ey, C redit an d B an king, 3 6 , 2 9 3 –3 1 8 .
[1 2 5 ]
N i c k e l S ( 1 9 9 0 ) U n e m p l o y m e n t S u r v e y, E conom ic Journ al, 1 0 0 : 4 0 1 : J u n e : 3 9 1 - 4 3 9 .
[ 1 2 6 ] N i S h a w n a n d X W a n g ( 1 9 9 5 ) B a l a n c e d g o v e r n m e n t b u d g e t s v e r s u s d e … c i t … n a n c e i n a g r o w t h e c o n o m y, C anadian Journal of E conom ics, 2 8 ,
4 b ,1 1 2 0 -1 1 3 4 .
[1 2 7 ] N o rd h a u s W .D . (1 9 9 5 ) P o lic y G a m e s:
Activities 2 : 1 9 9 4 : 1 3 9 - 2 1 6 .
C o - o r d i n a t i o n a n d I n d e p e n d e n c e i n M o n e t a r y a n d F i s c a l P o l i c e i s , B rookin gs P apers on E con om ic
[1 2 8 ]
O b s t f e l d M . a n d K . R o g o ¤ ( 1 9 9 6 ) E x c h a n g e r a t e d y n a m i c s r e d u x , Journal of P olitical E con om y, 1 0 3 : 3 : J u n e : 6 2 4 - 6 6 0 .
[1 2 9 ]
O ¢ c e o f N a tio n a l S ta tistic s (2 0 1 0 ) In p u t-O u tp u t Ta b le o f U K , http :/ / w w w .sta tistic s.g ov .u k / sta tb a se / .
[1 3 0 ]
P a i n N i g e l ; M a r t i n W e a l e ; G a r r y Y o u n g ( 1 9 9 7 ) B r i t a i n ’s F i s c a l P r o b l e m s , E conom ic Journ al, 1 0 7 , 4 4 3 , 1 1 4 2 - 1 1 5 6 .
[1 3 1 ]
P e a c o c k A . a n d G . K . S h a w ( 1 9 7 9 ) T he E con om ic T heory of F icals P olicy, G e o r g e A l l e n a n d U n w i n .
[1 3 2 ]
P h e l p s , E d m u n d S . ( 1 9 6 8 ) , M o n e y - W a g e D y n a m i c s a n d L a b o r - m a r k e t e q u i l i b r i u m , Journal of P olitical E con om y, 7 6 : 6 7 8 - 7 1 0 .
[1 3 3 ] P h illip s , A . W . (1 9 5 8 ) T h e R e la tio n B e tw e e n U n e m p loy m e n t a n d th e R a te o f C h a n g e o f M o n e y W a g e R a te s in th e U n ite d K in g d o m ,
1 8 6 1 - 1 9 5 7 , E conom ica, 2 5 : N o v . : 2 8 3 - 2 9 9 . .
[1 3 4 ]
P i s s a r i d e s C . A . ( 1 9 8 5 ) : T a x e s , S u b s i d i e s a n d E q u i l i b r i u m U n e m p l o y m e n t , Review of E con om ic Studies, 5 2 , 1 , J a n . , 1 2 1 - 1 3 3
[1 3 5 ]
P l o s s e r C . I . ( 1 9 8 9 ) U n d e r s t a n d i n g R e a l B u s i n e s s C y c l e , Journal of E con om ic P erspectives, 3 , 3 , 5 1 - 7 7 .
[1 3 6 ]
P r e s c o t t , E . C . ( 1 9 8 6 ) , T h e o r y A h e a d o f B u s i n e s s C y c l e M e a s u r e m e n t , F e d e r a l R e s e r v e B a n k o f M i n n e a p o l i s , Q uarterly Review ; F a l l , 9 - 2 2 .
[1 3 7 ]
P r e s t A . R . ( 1 9 6 8 ) S e n s e a n d N o n s e n s e i n B u d g e t a r y P o l i c y, E conom ic Journ al, 7 8 , 3 0 9 , 1 - 1 8
[1 3 8 ]
P r i c e S i m o n ( 1 9 9 7 ) P o l i t i c a l B u s i n e s s C y c l e s a n d M a c r o e c o n o m i c C r e d i b i l i t y : A S u r v e y, P ublic C hoice, 9 2 , 3 / 4 , 4 0 7 - 4 2 7
[1 3 9 ]
R a m s e y, F . P . ( 1 9 2 8 ) A M a t h e m a t i c a l T h e o r y o f S a v i n g , E conom ic Journ al 3 8 , 1 5 2 , D e c , 5 4 3 - 5 5 9 .
[1 4 0 ]
R a m s e y, F r a n k P . ( 1 9 2 7 ) A C o n t r i b u t i o n t o t h e T h e o r y o f T a x a t i o n , E conom ic Journ al 3 7 ( 1 4 5 ) : 4 7 - 6 1 .
93
[1 4 1 ]
R a n k i n N . ( 1 9 9 2 ) I m p e r f e c t c o m p e t i t i o n , e x p e c t a t i o n s a n d t h e m u l t i p l e e ¤ e c t s o f m o n e t a r y g r o w t h , E conom ic Journ al 1 0 2 : 7 4 3 - 7 5 3 .
[1 4 2 ]
R o m e r P . ( 1 9 9 0 ) E n d o g e n o u s T e c h n o l o g i c a l C h a n g e , Journal of P olitical E con om y, 9 8 : 5 : 2 : . s 7 1 - s 1 0 2 .
[1 4 3 ]
R u d e b u s c h , G . ( 2 0 0 6 ) M o n e t a r y p o l i c y i n e r t i a : f a c t o r … c t i o n ? In tern ation al Journ al of C en tral B ankin g, 2 , 8 5 - 1 3 5 .
[ 1 4 4 ] S a m u e l s o n P . A . ( 1 9 3 9 ) I n t e r a c t i o n B e t w e e n t h e M u l t i p l i e r A n a l y s i s a n d t h e P r i n c i p l e o f A c c e l e r a t i o n , Review of E conom ics and S tatistics,
2 1 : 2 : 7 5 - 7 8 , M a y.
[1 4 5 ]
S a m u e l s o n P . A . ( 1 9 5 4 ) P u r e T h e o r y o f P u b l i c E x p e n d i t u r e , Review of E con om ic Statistics, 3 6 : 3 8 7 - 9 /
[1 4 6 ] S a rg e nt, T .J . a n d N . W a lla c e (1 9 7 5 ) " R a tio n a l" E x p e c ta tio n s, th e O p tim a l M o n e ta ry In stru m e nt, a n d th e O p tim a l M o n e y S u p p ly R u le ,
Journal of P olitical E con om y, 8 3 : 2 : 2 4 1 - 2 5 4 .
[1 4 7 ]
S a w y e r M ( 2 0 0 3 ) E m p l o y e r o f L a s t R e s o r t : C o u l d I t D e l i v e r F u l l E m p l o y m e n t a n d P r i c e S t a b i l i t y ? Journal of E con om ic Issues, 3 7 , 4 , 8 8 1 - 9 0 7
[ 1 4 8 ] S m e t F . a n d R . W o u t e r s ( 2 0 0 3 ) A n e s t i m a t e d d y n a m i c s t o c h a s t i c g e n e r a l e q u i l i b r i u m m o d e l o f t h e E u r o A r e a , Journal of E uropean E con om ic
A ssociation , S e p t , 1 ( 5 ) : 1 1 2 3 - 1 1 7 5 .
[1 4 9 ]
S m i t h J . C . ( 2 0 1 1 ) T h e i n s a n d o u t s o f U K u n e m p l o y m e n t , E conom ic Journal, 1 2 1 , 4 0 2 –4 4 4 .
[1 5 0 ]
S o r e n s e n P B a n d H . J . W h i t t a - J a c o b s e n ( 2 0 1 0 ) In troducin g Advan ced M acroecon om ics, M c G r a w H i l l .
[ 1 5 1 ] S p e n c e r P e t e r D . ( 1 9 8 4 ) T h e E ¤ e c t o f O i l D i s c o v e r i e s o n t h e B r i t i s h E c o n o m y –T h e o r e t i c a l A m b i g u i t i e s a n d t h e C o n s i s t e n t E x p e c t a t i o n s ,
E conom ic Journ al, 9 4 , 3 7 5 , 6 3 3 - 6 4 4
[1 5 2 ]
S t o n e R i c h a r d . 1 9 4 2 - 4 3 . N a t i o n a l I n c o m e i n t h e U n i t e d K i n g d o m a n d t h e U n i t e d S t a t e s o f A m e r i c a , Am erican E con om ic Review , 1 0 ( 1 ) : 1 - 2 7 .
[1 5 3 ]
T a y l o r M . P . e d . ( 2 0 1 0 ) , P urchasin g P ow er P arity an d Real E xchange Rates, L o n d o n : Routledge.
[1 5 4 ]
T a y l o r M . P . ( 1 9 9 5 ) T h e E c o n o m i c s o f E x c h a n g e R a t e s , Journal of E con om ic L iterature, M a r c h , 3 3 : 1 : 1 3 - 4 7 .
[1 5 5 ]
T a y l o r M P ( 1 9 8 7 ) O n t h e l o n g r u n s o l u t i o n t o d y n a m i c e c o n o m e t r i c e q u a t i o n s u n d e r r a t i o n a l e x p e c t a t i o n , E conom ic Journ al, 9 7 : 3 8 5 : 2 1 5 - 2 1 8 .
[1 5 6 ]
T i l l m a n n P . ( 2 0 0 9 ) R o b u s t M o n e t a r y P o l i c y w i t h t h e C o s t C h a n n e l , E conom ica, N e w S e r i e s , 7 6 : 3 0 3 : 4 8 6 - 5 0 4 .
[ 1 5 7 ] T u r n o v s k y S . J . , M . H . M i l l e r ( 1 9 8 4 ) T h e E ¤ e c t s o f G o v e r n m e n t E x p e n d i t u r e o n t h e T e r m S t r u c t u r e o f I n t e r e s t R a t e s , Journal of M oney,
C redit an d B an kin g, 1 6 : 1 : F e b . : 1 6 - 3 3
[1 5 8 ] U h lin g H . (1 9 9 5 ) A to o lk it fo r a n a ly z in g n o n lin e a r e c o n o m ic d y n a m ic m o d e ls e a sily : M AT L A B
Reserve B ank of M inn eapolis, M i n n e s o t a , U S A .
p r o g r a m s , d i s c u s s i o n p a p e r 1 0 1 , Federal
[1 5 9 ] W icke n s M . (1 9 9 5 ) R e a l B u sin e ss C y c le A n a ly sis: A N e e d e d R e vo lu tio n in M a c ro e c o n o m e tric s (in C o ntrove rsy : B u sin e ss C y c le E m p iric s)
E conom ic Journ al, 1 0 5 , 4 3 3 . , 1 6 3 7 - 1 6 4 8 .
[1 6 0 ]
W a l l i s K . F . ( 1 9 8 9 ) M a c r o e c o n o m i c F o r e c a s t i n g : A S u r v e y , E conom ic Journ al, 9 9 , 3 9 4 . , 2 8 - 6 1
[1 6 1 ]
W i l s o n J . S . G . ( 1 9 4 9 ) I n v e s t m e n t i n a M o n e t a r y E c o n o m y, E conom ica, 1 6 , 6 4 , 3 2 1 - 3 3 5
[1 6 2 ]
W i l s o n C . A . ( 1 9 7 9 ) A n t i c i p a t e d S h o c k s a n d E x c h a n g e R a t e D y n a m i c s , Journal of P olitical E con om y, 8 7 , 3 , 6 3 9 - 6 4 7 .
[1 6 3 ]
W o o d f o r d , M . ( 2 0 1 1 ) S i m p l e A n a l y t i c s o f t h e G o v e r n m e n t E x p e n d i t u r e M u l t i p l i e r , Am erican E con om ic Journ al: M acroecon om ics, 3 ( 1 ) : 1 –3 5 .
[1 6 4 ]
W o o d f o r d M . a n d J . T a y l o r ( 1 9 9 9 ) H andbook of M acroeconom ics, E l s e v i e r , N o r t h - H o l l a n d .
[ 1 6 5 ] W r e n - L e w i s S . , J . D a r b y, J . I r e l a n d , O . R i c c h i ( 1 9 9 6 ) T h e M a c r o e c o n o m i c E ¤ e c t s o f F i s c a l P o l i c y : L i n k i n g a n E c o n o m e t r i c M o d e l w i t h
T h e o r y, E conom ic Journ al, 1 0 6 , 4 3 6 , 5 4 3 - 5 5 9
94
1.7.23
Appendix
1.7.24
Identi…cation of the simultaneous equation model (SEM)
Each equation in the SEM is identi…ed by both order and rank conditions. For instance, with nine
exogenous variables in the model including the intercept term the consumption function has only
two exogenous variables. It is over identi…ed by order condition, K k > m 1 =) 9 k > 5 1: All
other equations similarly satisfy order conditions, which is a necessary but not su¢ cient condition
for identi…cation. Each equation is identi…ed by the rank condition when a rank of the coe¢ cients
of the matrix of dimension of (M 1) (M 1) order exists for that equation in a model with M
endogenous variables. This matrix is formed from the coe¢ cients in the model for both endogenous
and exogenous variables excluded from that particular equation but included in other equations of
the model. Here the rank condition, (A) > (M 1) (M 1) = 4 is used to …nd out whether a
particular equation is identi…ed or not and involves following steps (Bhattarai (2011)): .
1.
Write down the system in the tabular form.
2.
Strike out all coe¢ cients in the row corresponding to the equation to be identi…ed.
3.
Strike out the columns corresponding to non-zero coe¢ cients in that particular equation.
4.
Form matrix from the remaining coe¢ cients. It will contain only the coe¢ cients of the
variables included in the system but not in the equation under consideration.
Table 32: Identi…cation in macro simultaneous equation model of UK
const Y
C
M
I
R
T
G
X
(M M=P )
C
0
T
M
I
R
Y
1
1
0
0
0
t0
t1
0
t2
0
0
1
0
m0
m1
0
1
0
m2
m3
0
0
0
1
1
0
0
0
0
b0
b2
0
-
b1
b2
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
b2
0
0
0
0
0
1
0
Yt
1
2
1
1
From these coe¢ cients form all possible A matrices of order (M 1) (M 1) and ascertain that
determinant of order (M 1) (M 1) exist for each equation in this system. If at least one of these
determinants is non-zero then that equation is identi…ed. Model is identi…ed when all equations are
identi…ed.
For consumption
function:
2
3
t1
t2 0
0
6 1
0
0
0 7
7 ; jAc j = 1 t2 6= 0; (Ac ) = 4:
Ac = 6
b2
4 0
5
0
0
1
0
0
0
b2
Tax function:
2
3
1 0
0 0
6 1
7
m2 0 0
7 ; jAT j = 1 m2 6= 0; (AT ) = 4:
AT = 6
b2
4 0
5
1 0
1
1
0 0
0
b2
Import function:
95
2
3
1 0
0
0
2
6 0
t3 0
0
0 7
7 ; jAM j = 1 t2 2 6= 0; (AM ) = 4:
AM = 6
b2
4 0 1
0
0
- 5
1
0 0
0
0
b2
Investment
function:
2
3
0
0
1
1
6 t1
t2 1
0 7
7 ; jAI j = 1 t2 1 + 1 t2 m1 6= 0; (AI ) = 4:
AI = 6
b2
b2
4 m1 1
m3 0 5
b1
1
0
0
b2
b2
Interest
2 rate function:
3
1 0
0 0
6 0
t
0
0 7
2
7 ; jAR j = t2 6= 0; (AR ) = 4:
AR = 6
4 0 1
1 0 5
0 0
0 Thus each of above equations are identi…ed by order and rank conditions and model is identi…ed.
Estimates of the model could be used for policy analysis.
1.7.25
Path of price and exchange rates in the Dornbusch model
First …nd roots for stability
analysis using two di¤erential equations:
p
tr(A)
1
2
;
=
tr(A)
4 jAj; tr(A) = (a11 + a22 ) ; jAj = (a11 a22 a12 a21 )
1
2
2
2
The roots of the equation depend on the behavioral parameters , v and a These values
determine the path of price level and the exchange rate consistent to the demand and supply sides
of the economy.
v
v
Here A =
; tr(A) =
v and jAj = av .
1=a
0
r
tr(A) 1 p
v 1
v
2
tr(A)
4 jAj =
( v)2 + 4
(A.1)
1; 2 =
2
2
2
2
a
p
p
v
v
1
v
1
v
2
2
p(t) = C1 exp( 2 + 2 ( v) +4 a )t +C2 exp( 2 2 ( v) +4 a )t +p
(A.2)
v
e(t)
2
=
+
1
2
v
+
2
p
p
v)2 + 4 av + v
v
1
C1 exp( 2 + 2 (
p v
1
p
( v)2 + 4 av + v
v
1
2
C2 exp( 2 2 (
v
(
As can be seen below when p = 0; p = e +
it; when e = 0; p =p =m
by
(u y)
v ,
)t
v)2 +4
v
a
)t +e
(A.3)
p rises above p = 0 isocline and falls below
ar here e rises and falls.
96
v
a
v)2 +4
2
L2: New Keynesian Model: Fundamentals
New Keynesian models have all features of modern macroeconomics. These are
a) dynamic models b) have competitive equilibrium c) based on micro-foundation d) have rational expectation.
Most new Keynesian models include consumption saving decision in Ramsey type intertemporal
dynamics, include leisure/labour supply decisions, have money/bonds as …nancial assets and include
nominal rigidities - Calvo type price setting mechanism. By putting nominal and real rigidities in
the RBC models the new Keynesian dynamic stochastic general equilibriums (DSGE) are able to
generate Keynesian features in otherwise standard RBC models of modern economies.
2.1
New Keynesian Model: a prototype example
New Keynesian business cycle model in which output equal employment
Qi = Li
(B.4)
(This example is based on Romer D. (2008) Advanced Macroeconomic Theory, McGraw Hill).
Utility is positive from consumption and negative from work:
1
Ui = Ci
Li ;
>1
(B.5)
Consumption equals real income:
Ci =
Pi Qi
P
(B.6)
In terms of labour input:
max Ui =
Pi Qi
P
Pi
@Ui
=
@Li
P
1
Li
1
Li =
P i Li
P
1
Li
Pi
P
= 0 =) Li =
(B.7)
1
1
(B.8)
In logs:
li =
1
1
(pi
p)
(B.9)
labour supply and production depends on relative price.
Demand with shock
qi = y + zi
n (pi
mean: q i = y; z i ; pi = p: output y = m
Equilibrium supply equals demand:
1
1
(pi
p) ;
n>0
(B.10)
p
p) = y + zi
97
n (pi
p)
(B.11)
Solve for pi
1
1
1
pi + npi = y + zi +
1+n
n
1
1+n
pi = (y + zi ) +
1
1
pi =
1+n
p + np
(B.12)
n
p
1
(y + zi ) + p
n
(B.13)
(B.14)
Given Y = 1; devition from the steady state y = 0 =) m = p:
Consumer has pro…t and labour income:
1
Ui = Ci
Li =
* qi = y n (pi p) =) Qi = Y
Choice variables are Pi and Li :
Pi
P
n
n (pi
Pi
P
=n
=Y
Pi
P
1
n 1 1
P
Pi
P
w) Y
p
w
@Ui
=
@Li
P
w
P
+ wLi
=
Li
(B.15)
=0
(B.16)
n
nw
=)
p
Price set by the …rm depends on the markup
Labour supply Li = Qi :
n
Pi
P
w) Y
p
Pi
P
Pi
P
Y
@Ui
=
@Pi
(pi
1
Li
Pi
P
n
n 1
n
=
n
w
1p
(B.17)
.
1
w
P
= 0 =) Li =
1
(B.18)
1
n
n
Pi
P
w
=)
1p
=
n
n
1
Y
1
(B.19)
Taking logs
pi
p = ln
n
n
1
+(
if pi = p =) Y =
1) ln Y = c + y
n
1
n
(B.20)
1
1
(B.21)
Y should be 1 but equilibrium output is less than optimal when producers have mark up power
n 1
< 1.
n
If the aggregate demand equals the real money balances, then Y = M
P then price level os proportional to stock of money and inversly related to the market power of the …rm:
as
P =
M
=
Y
98
M
n 1
n
1
1
(B.22)
2.2
Two Period Model of Stabilisation: Mankiw and Weinzierl (2011)
This is a basic new Keynesian monetary model in which consumers and producers have horizon of
two periods. It explores analysis of …scal and monetary policy under …xed and ‡exible price set up.
It is simple analytically tractable and easier to compute policy scenarios.
Representative households maximise utility subject to intertemporal budget constraint as:
M ax
U = fu (C1 ) + v (G1 )g +
fu (C2 ) + v (Gv )g
(B.23)
Subject to the intertemporal budget constraint:
P1 [
1
T1
P2 [
C1 ] +
T2
1 + i1
2
C2 ]
=0
(B.24)
Representative …rms maximise pro…t subject to technology constraint as:
max P1
1
+
P2 2
1 + i1
(B.25)
subject to
1
= Y1
I1 ; K2 = I1 given K1
Yt = At Kt ;
(B.26)
At > 0
Money:
M t = P t Ct ; M t =
A high implies high velocity of money and
Fiscal policy:
Mt
= Pt Ct
! 0 implies a cashless economy.
gt =
Gt
At Kt
and the government budget constraint:
P1 [T1
G1 ] +
P2 [T2 C2 ]
=0
1 + i1
(B.27)
Macrobalance:
Yt = Ct + It + Gt
Demand is less or equal to the capacity:
Yt
At Kt
max P1 (A1 K1
K2 ) +
Firms problem can be restated as:
99
P2
A2 K2
1 + i1
(B.28)
Firm chooses capital stock for period two K2 to maximise pro…t:
P1 +
P2
A2 = 0 =)
1 + i1
(1 + i) =
P2
P1
(B.29)
Lagrangian for the household problem:
L = fu (C1 ) + v (G1 )g +
fu (C2 ) + v (Gv )g +
P1 [
T1
1
C1 ] +
P2 [
T2
1 + i1
2
C2 ]
(B.30)
Consumer chooses C1 and C2 to maximise consumption (FOC):
u0 (C1 )
u0 (C2 )
P1 = 0
P2
=0
1 + i1
u0 (C1 )
P1
= (1 + i)
u0 (C2 )
P2
(B.31)
Market clearing:
Y1 = C1 + I1 + G1 = A1 K1
Y2 = C2 + G2 = A2 K2
C1 = A1 K1
Solve C1 , C2 , K2 and
P2
P1
I1
G1 ;
C2 = A2 K2
G2
(B.32)
by using (B.29), (B.31) and (B.32).
C
1
1
1
2
Now specialise the utility function to U (C1 ) = 11 1 and 1) solve for C1 , C2 , K2 and P
P1 2)
study the impacts of …scal and monetaries policis on allocation under the ‡exible price system 3)
show how the …scal and monetary policies a¤ect the economy under the …xed price system. Study
the following paper carefully to answer these questions.
Mankiw N.G., M. Weinzierl, O. Blanchard and G. Eggertsson (2011) An Exploration of Optimal
stabilization Policy [with Comments and Discussion], Brookings Papers on Economic Activity,
Spring, 209-272
2.3
A DSGE Model of Macroeconomic Policy in South Asia
Micro-foundations, dynamics and rational expectations, stochastic shocks to preferences, technologies and policies along with the nominal and real rigidities underpin the business cycle analyses
in DSGE models. Analysis of short or long run multipliers, variance decompositions and impulse
responses to changes in policies and shocks on the deviations of model variables from the steady
state is often the focus of such analysis. Computations have become easier for such models after
development of Sim’s BVAR algorithm in the MATLAB and dynare.
100
Let us consider a DSGE model of growth in India (I) and neighbouring (n) countries. We
consider stochastic shocks eg_I;t and eg_n;t for growth in two countries as a function of relative
prices, exchange rates, trade balance and relative interes trates:
gI;t = b0 + b1
gn;t =
0
+
1
pI;t
pn;t
+ b2
eI;t
en;t
+ b3
iI;t
in;t
+ b4
pn;t
pI;t
+
en;t
eI;t
+
in;t
iI;t
+
2
3
T BI;t
T Bn;t
4
T Bn;t
T BI;t
+ eg_I;t
+ eg_n;t
(B.33)
(B.34)
Adjustment in the relative price lags by one period
pI;t
pn;t
eI;t
en;t
=
=
p
(gn;t
r
iI;t
in;t
gI;t )
+
pI;t
pn;t
Global growth rate is
gt = gI;t + gn;t
Relative interest rate responds to the global
iI;t
in;t
=
pI;t
pn;t
+ gt
We apply the Metropolis-hasting algorithm of the Bayesian VAR methodology (in dynare 4) to
compute this BVAR DSGE with the quarterly dataset for advanced and emerging economies (based
on Bhattarai and Mallick (2014)). Initial values of parameters are assinged to simulate the model
as in Table
Param eters
b0
Va lu e s
0 .0 1
Table 33: Parameters of the DSGE Model
b1 b2 b3 b4
1
0
1
2
0 .3
0 .4
0 .5
0 .0 3
0 .4
0 .1
0 .3
0 .6
3
0 .2
4
0 .0 3
DSGE model of growth in India and neighbouring SAARC countries: priors
101
2
0 .8
b va r_ d c a d v _ d s g e _ In d ia _ P rio rs 1 .e p s
S E _eg_a
P rio rs a n d p o ste rio rs
S E _eg_d
150
SE_eg_I
b1
80
4
60
3
40
2
20
1
SE_eg_n
b1
4
100
50
0
0
0.02
0.04
0.06
0
0
0.1
g2
4
3
3
2
2
1
0
0.5
0
-0.5
0
0.5 1 1.5 2 2.5
0.5
0
0.05
g2
6
4
4
2
2
0
-1
1
0
0
-0.8-0.6-0.4-0.2 0 0.2
0.1
b3
rho1
rho1
1
0
-0.5
b3
4
50
2
0
-1
0.2
100
0
0
0.5
5
10
0
0
0.5
0
-0.6
0.2 0.4 0.6 0.8 1
-0.4
-0.2
0
0
0 0.2 0.4 0.6 0.8
rho2
1
rho2
4
6
2
4
0
2
0
0.5
1
0.4 0.6 0.8 1 1.2 1.4
1.5
DSGE model of growth in India and SAARC countries: shocks
b va r_ d c a d v _ d s g e _ In d ia _ S m o o th e d S h o ck s 1 .e p s
b va r_ d c a d v _ d s g e _ In d ia _ H is to ric a lA n d S m o o th e d Va ria b le s 1 .e p s
eg_a
g_a
15
15
10
10
5
5
0
0
-5
-5
-10
20
40
60
-4
1.5
80
100
20
120
40
60
eg_d
x 10
80
100
120
80
100
120
g_d
10
1
8
0.5
6
0
4
2
-0.5
-1
0
-1.5
-2
-2
-4
20
40
60
80
100
20
120
40
60
DSGE model of growth in India and SAARC countries: Impulses
b va r_ d c a d v _ d s g e _ In d ia _ IR F _ e g _ I.e p s
b va r_ d c a d v _ d s g e _ In d ia _ IR F _ e g _ n
-3
0
g_ a
g_I
x 10
1.4
1.2
-2
1
-4
0.8
0.6
-6
0.4
-8
0.2
0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
10
g_n
0.01
g_ d
0
-0.5
0.005
-1
-1.5
-2
0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
10
DSGE model of growth in India and SAARC countries: estimation of time varying parameters
102
b va r_ d c a d v _ d s g e _ In d ia _ u d ia g 1 .e p s
b va r_ d c a d v _ d s g e _ In d ia _ u d ia g 2 .e p s
-3
g2 (Interv al)
SE_ e g _ a (In te rv a l )
SE_ e g _ a (m 2 )
0 .7
0 .6
0 .5
0.26
SE_ e g _ a (m 3 )
0 .0 6
0 .0 5
0 .0 2
0 .0 4
0 .0 1 5
0 .0 3
0 .0 1
0.24
0.22
SE_ e g _ d (In te rv a l )
6
-4
x 1 0 SE_ e g _ d (m 2 )
1 .5
0 .0 4
4
1
0 .0 2
2
0 .5
0
1000 2000 3000 4000 5000
0
1000 2000 3000 4000 5000
-3
b 1 (In te rv a l )
0 .3
12
0 .2 8
-3
g2 (m 2)
x 10
1.6
x 10
8
1.2
x 10
g2 (m 3)
1
6
1000 2000 3000 4000 5000
-4
0.05 5
4.5
0.8
1000 2000 3000 4000 5000
-5
b3 (m 2)
x 10
1.5
3.5
0.04 5
x 10
b3 (m 3)
1
3
0.04
1000 2000 3000 4000 5000
1000 2000 3000 4000 5000
-3
rho1 (In terv al )
b 1 (m 3 )
x 10
4
0.05
-3
2
1.4
b3 (Interv al)
-5
x 1 0 SE_ e g _ d (m 3 )
0
1000 2000 3000 4000 5000
b 1 (m 2 )
9
7
0.2
1000 2000 3000 4000 5000
0 .4
0 .0 2
0 .0 0 5
1000 2000 3000 4000 5000
1000 2000 3000 4000 5000
1000 2000 3000 4000 5000
0 .0 6
10
0 .0 2 5
x 10
0.5
1000 2000 3000 4000 5000
-3
rho1 (m 2)
0.35
12
0.3
10
2
0.25
8
1
0.2
6
0.5
x 10
rho1 (m 3)
1.5
10
0 .2 6
1 .5
8
0 .2 4
0 .2 2
1000 2000 3000 4000 5000
6
1000 2000 3000 4000 5000
1
1000 2000 3000 4000 5000
1000 2000 3000 4000 5000
4
1000 2000 3000 4000 5000
0
1000 2000 3000 4000 5000
DSGE model of growth in India and SAARC countries: historical and smoothed variables
b va r_ d c a d v _ d s g e _ In d ia _ H is to ric a lA n d S m o o th e d Va ria b le s 1 .e p s
b va r_ d c a d v _ d s g e _ In d ia _ S m o o th e d S h o ck s 1 .e p s
eg_I
g_a
10
15
5
10
0
5
-5
0
-10
-5
20
40
60
80
100
20
40
60
80
100
120
80
100
120
120
-4
2
g_d
eg_n
x 10
10
5
0
0
-5
-2
-10
-15
-4
20
-20
-25
20
40
60
80
100
40
60
120
DSGE model of growth in India and SAARC countries : variance decomposition
Va ria n c e d e c o m p o sitio n r
Va ria n c e d e c o m p o sitio n g t
15
10
10
Initial values
5
5
Initial values
0
0
-5
-5
eg_n
-10
-15
-15
-20
-25
eg_d
-10
-20
eg_I
0
20
40
60
80
100
120
eg_a
-25
140
0
20
40
60
80
100
120
140
see data and code for the South Asia model.
Study MATLAB programme DSGEM.m with chapter2_netfun.m.
This is based on Lim G. C. and McNelis (2008), Computational Macroeconomics for the Open
Economy, MIT Press.
103
2.4
A Prototype of New Keynesian DSGE model with habit formation
The new Keynesian models introduce nominal and real frictions in the standard RBC models. Real
frictions can be habit formation in consumption or cost of investment or …scal or monetary policy
rules. Nominal frictions occur through price and wage rigidities.
Habit formation h (for a represenative houshold and …rm case) is in the form of:
Uth
here
=U
t
Cth ; lth
=
h
(Ct
(1
1)
hCt
)
(1
1
Lt )
c
i1
c
1
(B.35)
is the preference shock
h
Uc;t
=
t
(1
h
UL;t
=
) (Ct
t
(Ct
hCt 1 )
(Ct
hCt
(1
)(1
Ct
(1
1)
Euler equation (with nominal interest rate,Rt
"
h
Uc;t
1
(1
Lt )
(
(
1) 1
c
1)
1)
)(1
(1
c)
c)
(1
Lt )
c
Lt )
1)
:
Rt
=
t
1
h
Uc;t
#
Marginal rate of substitution:
h
UL;t
h
Uc;t
=
Wt
Pt
Stochastic discount factor:
t
h
Uc;t
h
UL;t
1
=
!
wholesale retail sector relation:
Yt =
c) YtW
(1
(B.36)
t
Price dispersion:
t
=
t 1
+ (1
)
Whole sale production function:
YtW =
(At Lt ) Kt1
t
FOC for the labour market:
104
1
J
H
PtW YtW
Wt
=
Lt
Pt
Yt = Ct + It + Gt
Kt =
where
t
t It
(1
(B.37)
2
(Xt
1) + (1
) Kt
(B.38)
1
is the investment shock.
z1t = 2:0
Qt
t
(1
x Xt
2
x ) (Xt 1 )
z2t =
(1
2
(Xt ) Qt
2:0
x Xt
t
(Xt
1)
) PtW YtW
+ (1
Kt 1
Rt
=
t+1
(B.39)
t
+ z1t+1 = 1
) Qt
z2t+1
Qt
Budget of the government:
Gt =
w Wt Lt
Real interest rate
rt =
Rt
t
Then in‡ation dynamics:
h
e t+1 = Yt UL;t
H
Ht
Jt
Jet+1 = (1= (1
et
Taylor rule:
1
h
)) Yt UC;t
mc; mc = PtW = 1
e t+1 = e(t
H
1)
Ht
Jet+1 = et Jt
+ (1
et =
Jt
Ht
)
t
t 1
105
(1
)
=1
1
ln
Rt
Rt 1
=
r
ln
Rt 1
R
+ (1
r)
t
ln
+ (1
r)
y
ln
Yt
Y
+
m
Processes of shocks to technology, public spending, preference and investment:
log At
log At =
A (log At 1
log At ) +
A
t
log Gt
log Gt =
G (log Gt 1
log Gt ) +
G
t
log
t
log
t
=
(log
t 1
log
t
t
log
t
=
(log
t 1
log
log
Variables in the deviation form:
t
; it =
yt = YYt ; kt = K
K
It
I
; ct =
(1
)
Ct
C
; wpt =
t)
t)
+
+
t
W Pt
WP
; lt =
Lt
L
; rrt =
rt
r
;
Rt = RRt :
Steady state ratios
iy =
1
; iy = 1
iy
gy
1+
A typical parameterisation
Parameters gy
c
h
c
x
values
0.2 0.7 0.7 7.0 1/
0.9871 0.025 2
2
.75 0:7 0.5
Parameters on Taylor rule and persistency of shocks
Parameters
Ass
r
y
r
y
I
Values
0.7 1.5 0.3 1
0.7 0.7 0.7 0.7
Model is computed in the dynare. The impulse responses to technology, investment, public
spending and preference shocks presented in the following diagrams.
Literature
Wickens M. (2012) Macroeconomic Theory: A Dynamic General Equilibrium Approach, 2nd edition,Princeton University Press.
Levine P, J. Pearlman, G. Perendia and B Yang (2013) Endogenous persistence in an estimated DSGE
model under imperfect information, Economic Journal, 122 (December), 1287–1312.
Practical work:
More extensieve literature on this topic can be found on the web http://economics.sas.upenn.edu/~schorf/research.htm.
NK_hab.mod RBCInvcost.mod and RBCSummer.mod.which includes habit formation A simple model
from the CIMS Univeristy of Surrey.
106
Figure 5: A1
107
108
109
2.4.1
Blanchard and Gali (2013)
Blanchard and Fisher (1990) and Blanchard and Gali (2013) consider provide a more extensive
version of the new Keynesian model as:
Problem of Households i
"( 1
)
#
X
Mit+k+1
k
Q (Nit+k ) = t
max E
U (Cit+k ) + V
(B.40)
P t+k
t=0
Subject to:
Cit =
Z
1
0
and the budget constraint
Z
1
1
Cijt dj
;
Pt =
Z
1
0
1
Pjt
1
dj
1
(B.41)
L
Pjt Cijt + Mit+1 + Bit+1 = Wt Nit + (1 + it ) Bit + Mit +
it
+ Xit
(B.42)
0
Cit+k is consumption, Mit+k money, Nit+k labour supply P t aggregate price, Pjt price of
sector j , Bit+1 bonds, it pro…t from …rms Xit public transfer.
First order conditions of household optimisation
Pjt
Pt
Cijt =
Cit
(B.43)
U 0 (Cit ) = E [f (1 + rt+1 ) U 0 (Cit+1 )g =
V0
Mit+1 0
=U (Cit )
Pt
=
t]
(B.44)
it+1
1 + it+1
(B.45)
Wt 0
U (Cit ) = Q0 (Nit )
Pt
(B.46)
Firms’problem assuming a linear production technology
Yjt = Zt Njt
Pjt
Pt
Cijt =
(B.47)
Cit
(B.48)
Firms take wage rates as given and set prices a la Calvo with probability of changing it every
period. Then Yjt is solution to the …rms pro…t mamimization problem
max E
kU
0
(Ct+1 )
(1
U 0 (Ct )
k
)
Pjt
Yjt+k
Pt
Subject to:
110
Wt+K Yjt+k
P t+1 Zt+1
=
t
(B.49)
Pjt
Pt
Yjt+k =
Yt+1
(B.50)
FOC of …rm implies
E
Pjt = Pt =
1
hX
t+k
=
A (k) W
Zt+1
hkX
i
E
A (k) = t
t
i
(B.51)
k
kU
0
(Ct+1 )
(1
U 0 (Ct )
A (k) =
k
1
) P t+k Yt+k
(B.52)
Aggregate price level is CES of prices of t and t-1 with probability
h
P t = (1
1
)Pt
1
1
+ Pt
i11
(B.53)
General equilibrium implies clearing of goods, labour and money markets Cit = Ct ; Nt =
Mit = Mt ;
Above equations now give the IS and LM curves as:
IS: demand for goods depends on expected future income and the real interest rate
U 0 (Yt ) = E [f (1 + rt+1 ) U 0 (Yt+1 )g =
t]
Yt
Zt ;
(B.54)
LM: the …nancial market determines the equilibrium interest rate
V0
Mit+1
Pt
=U 0 (Cit ) =
it+1
1 + it+1
(B.55)
LS: Real wage is determined by the marginal productivity of labour
Wt 0
U (Cit ) = Q0
Pt
Yt
Zt
= Q0 (Nt )
(B.56)
Price settings
E
Pjt = Pt =
1
hX
t+k
=
A (k) W
Zt+1
k
hX
i
E
A (k) = t
t
i
(B.57)
k
A (k) =
kU
0
(Ct+1 )
(1
U 0 (Ct )
k
1
) P t+k Yt+k
(B.58)
Aggregate price level is CES of prices of t and t-1 with probability
h
P t = (1
1
)Pt
Derivation of market clearing wage rate
Production function:
Nt =
1
1
+ Pt
Yit
Zt
111
i11
(B.59)
(B.60)
When there are no nominal rigidities
Wt
1 Zt
Pt =
(B.61)
Yt
Q0 Z
Wt
t
=
= 0
U (Yt )
Pt
1
Zt
(B.62)
This can implicitly determine the output as a result of technological shocks.
Y
Q0 ( t )
In a logarithmic utility function U 0 (YZtt ) =
1 Zt implies
Q0
Yt
Zt
Yt
= Q0
Zt
Yt
Zt
Nt =
1
(B.63)
Employment remains constant function of the elasticity of markets.
Output varies one to one with Zt :
Q0
Yt
Zt
Yt =
1
Zt
(B.64)
Current output (deviation from the steady state) is just a function of the shock. The RBC
model also generates the same result, only di¤erence is that RBC model has technological shock
(See Blanchard and Fisher (1990) Lectures on Macroeconomics, MIT Press).
2.4.2
Solution Procedure in the DSGE Models
Most DSGE models are solved using the log-linearization procedure discussed in Campbell (1994)
and Uhlig (1995). Deviation of actual capital (Kt ) from steady state capital (K) equals:
^ t = ln Kt
K
ln K
(B.65)
where K without subscript t denote steady state value; Kt with subscript t is the actual value
^ t with subscript t and hat above denote log deviations of particular variable
of capital at time t; K
from steady state.
^t
ln Kt = ln K + K
(B.66)
Taking out log both sides one gets:
^
^
eln Kt = eln K+Kt = eln K eKt
(B.67)
Thus:
^
^
Kt = KeKt ) eKt =
Kt
K
(B.68)
^
Next step is to take the …rst order Taylor approximation of eKt around the steady state thus
^ t = 0, though we get:
K
^
^t
eKt = e0 + e0 (K
thus:
112
^t
0) = 1 + K
(B.69)
^t =
1+K
Kt
^ t)
) Kt = K(1 + K
K
(B.70)
^ t = Kt
K
(B.71)
or:
K
K
^ t multiplied by 100 informs by what percentage capital at time t diverges from
The variable K
^ t is equal 0:2 we interpret that capital is 20% above the steady
the steady state. So for example if K
state. Apply this method to all equations to log-linearize a model.
Then log-linearized system is solved by means of Method of Undetermined Coe¢ cients (Uhlig
1995). In order to do so, log-linearized system has to be transformed to the following form:
0 = Et [F F xt+1 + GGxt + HHxt
z(t+1) = N N z +
t+1 ; and
1
+ LLzt+1 + M M zt ]
Et [
(t+1) ]
= 0;
(B.72)
(B.73)
Using method of undetermined coe¢ cients the following recursive law of motion is estimated:
xt = P xt
1
+ Qzt
(B.74)
so that the equilibrium solution is stable. Following Uhlig (1995) matrix P satis…es the following
equation:
0 = F F P 2 + GGP + HH
(B.75)
Subsequently denoting V as:
V = NN0
F F + Ik
(F F P + G)
(B.76)
Q can be obtained from
VQ=
vec (LLN N + M M )
(B.77)
where ’vec’denotes vectorization. Quadratic equation is solved as proposed by Uhlig (1995).
Read Uhlig, H. (1995). A Toolkit for Analyzing Nonlinear Dynamic Stochastic Models Easily.
2.4.3
Basics of Bhattarai and Trzeciakiewicz (2012) DSGE model
DSGE models are used by the New Keynesian (NK) macroeconomists in analysing and explaining causes and consequences business cycles and to assess the role of policy instruments used for
stabilising ‡uctuations in macro variables. While the micro-foundations to the macro model and
inter-temporal optimisation of households and …rms, and shocks to technology, preferences and …scal policy instruments are similar to those in the real business cycle models in tradition of Kydland
and Prescott (1992), the NK models show important roles that economic policies play in containing
such ‡uctuations. They integrate nominal rigidities in prices and real rigidities in institutional
setup such of Calvo (1993), Rankin (1992), Mankiw (2000) or Campbell Mankiw(2003), to generate
Keynesian features of …scal and monetary policies on output, employment and prices. Starting from
113
DSGE exercises of Smets and Wouters (2003), Christiano, Eichenbaum and Evans (2005) and Iacoveillo (2005), these models have become increasingly popular in analysing causes of volatilities in
prices and output particularly among the Federal Reserve System in the US, the Bank of England,
the European Central Bank, the IMF and other central banks around the world. Many studies
based on DSGE models have appeared in the literature in recent years.
The new Keynesian DSGE models became more popular after the 2008 crisis as these models
could justify large …scal stimulus and credit or quantitative easing (QE) by accommodating nominal
and real rigidities in the economic system (Benati (2008), Zanetti (2010), Blanchard and Galí
(2012), Levine, Pearlman,Perendia and Yang (2012)). These models are applied also to analyse
consequences of search and matching frictions in the labour market, frictions in the …nancial and
commodity markets (Blanchard and Gali (2013), Faccini, Millard and Zanetti (2013)). These have
also been extended to study open economic issues
From policy perspective most of the DSGE models have focused on how monetary policy could
be designed to mitigate the consequences of rigidity on wages and prices (Ascari and Rankin (2013)).
Very few studies were designed to evaluate the role of …scal policy in the business cycle ). This thesis
aims to contribute this lacuna in the literature by introducing a structure of …scal policy rules to
enrich existing DSGE models. It also contains a simple form of the monetary policy rule appropriate
to study interaction between the …scal and monetary policies ((Blake and Weale (1998), Bouakez
and Rebei (2007), Mirrlees et al. (2010), Woodford (2011), Coenen, Straub and Trabandt (2013),
Coenen, Erceg, Freedman, Furceri, Kumhof, Lalonde, Laxton, Linde, Mourougane, Muir, Mursula,
de Resende, Roberts, Roeger, Snudden,Trabandt and in’t Veld (2012)). While the dynamic general
equilibrium models aim to …nd out the dynamically e¢ cient and equitable paths for the allocation
of resources in an economy, the DSGE models focus mainly on measuring the short and long run
multipliers and impulse responses of various shocks in the economy. They show the pattern of
deviations macroeconomic variables from the steady state originating from stochastic shocks in
preferences, technology or the policy instruments. Results from these models have short run rather
than long-run focus. These models are computationally intensive as they have rich Bayesian statespace for model parameters generated from the application of Metropolis-Hastings algorithm (Gibbs
sampling) in traditions of Sims (1980) or Doan, Litterman, and Sims (1984; see also Collard and
Juillard (2001)).
This is a DSGE model with two types of households with and without the borrowing constraints. Households and …rms solve inter-temporal optimisation problems subject to those constraints. Model constains all …rst order conditions requited for optimisation of economic agents,
the revenue and spending structure of the government, a set of …scal policy rules and a monetary
policy rule. It discusses calibration and computation procedure. The model is applied in order
to assess the present value multipliers of six tax instruments (taxes on consumption, labour and
capital income, transfer, public consumption and investment). Then it shows impulse responses
to other …ve structural shocks. It provides some discussions on the Metropolis-Hestings algorithm
(as given below). Application of DSGE model in computing multipliers, variance decompositions
and impulse responses of …scal policy instruments in the context of the UK economy is speci…c
to this model as Batini, Harrison and Millard (2003) Harrison and Oomen (2010) to Faccini,
Millard and Zanetti (2013), Krause and Lubik (2007), Zanetti (2010), Woodford (2011), Ravn,
Schmitt-Grohé and Uribe (2012), Levine, Pearlman, Perendia and Yang (2012) have not considered
the analysis of …scal policy explicitly in this manner. Early version of this paper is avialable at
http://www.hull.ac.uk/php/ecskrb/Confer/Dawid_May_2012.pdf.
114
2.4.4
Household problem
Households maximize utility subject to the ‡ow budget constraint, capital accumulation function
and the demand for labour they faced from labour unions. Lagrangian takes the following form:
L
where:
t
=
X
"B
t
(Ct
t
hCt
1
t=0
+
t
+
k
t
Rt
1 bt
t
(1
1
1)
1
"L
t
1+
c
c
+ 1
a(! t )Kt
))Kt
1
Lc;t
L
l
t
k
t
wt Lt + 1
bt
1 + divt + tt
+ 1
"It It
It 1
S
k
t
denotes marginal utility of income;
It
1+
L
!
rk;t ! t Kt
It Ct
1
!
(B.78)
Kt
stands for the shadow price of capital; and
Wt
Wt
Lt =
Nt . The wages are set to balance the productivity marginal utility from work and
disutility of working as:
(
)
1
X
~ t Xtl
W
Ul;t+l
l
( $W ) Lt+l t+l
=0
(B.79)
l
Pt+l
(1
) Uc;t+l 1
t+l
l=0
When wages are ‡exile i.e. when all households are able to negotiate their wages each period,
~t
Ul;t
$W = 0, wage then becomes: W
l :
Pt =
(
1) Uc;t (1
t)
Index of wages is given by:
1
Wt = (1
1
~t
$W ) W
+ $W (
Pc;t
Pc;t
1
w
1
Wt
1
1)
(B.80)
2
Final goods are produced using di¤erentiated intermediate goods: Yt =
in new Keynesian supply function
Pt {{ 1
Yt
)
Yj;t = (
Pj;t
hR
1
0
1
{
Y;j;t
dj
i{
. It results
(B.81)
These intermedate goods are used using private and public capital and labour as:
Yj;t = At (! t Kj;t
1)
1
G
Nj;t
Kj;t
(B.82)
1
where K G is the public capital, and At is a total factor productivity shock and follows a …rstA
order autoregressive process: ln At = ln At 1 + A
N 0; 2A : The …rst order
t , where t
conditions yield to:
Kj;t
=
Nj;t
(1
Wt
) ! t Rk;t
(B.83)
Nominal marginal costs become:
Pt mct = (
1
1
)1
1
( ) At 1 Kj;t
115
1 (Wt )
G
1
(Rk;t )
(B.84)
Marginal cost increases as wages and return on capital increase. Higher amount of government’s
capital along with positive TFP shock along with an increase in public capital will lead to a decrease
in marginal costs.
1 {
R 1 1 1{
is derived
dj
Calvo(1983) pricing rules apply to this economy the priceindex Pt = 0 Pj;t
as:
"
1
1
b
$) P
t
Pt = (1
1
Pt
Pt
{
+$
p
1
1
Pt
1
2
{
#1
{
(B.85)
This economy is subject to the market clearing conditions:
Yt
a(! t )Kt
1
= Ct + Gt + It + ItG
R1
Ct = 0 ct d = (1
) Ctr + Ctnr
R1
) Lrt + Lnr
Lt = 0 Lt d = (1
t
R1
R1
Kt = 0 Kt d = Ktr It = 0 It d = Itr
R1
Bt = 0 Bt d = Btr
R1
T Rt = 0 T Rt d
2.4.5
Fiscal and monetary policies
Model evaluates the impact of …scal and monetary policy shocks that a¤ect the government budget
constraint:
c
t Ct
+
l
t wt Lt
+
k
t rk;t ! t Kt 1
+ bt
trt =
(1 + Rt 1 )
bt
1+ t
1
+ gt + ItG
(B.86)
where b denotes public borrowing, g stands for government spending, and I g stands for government investment. Public capital follows according to the following law of motion:
G
G
k ))Kt 1
KtG = (1
+ ItG
(B.87)
In setting the …scal rules we follow Leeper (2010). Fiscal shocks a¤ect the revenue and spending
sides of the government. All shocks follow a …rst-order autoregressive process: ex;t = x ex;t 1 + x;t
where x;t N 0; 2x :are i:i:d:- normally distributed errors.
Government spending and government investment respond to the output and public borrowing
(hats over variables denote deviations from the steady state):
^
g;y Yt 1
^
ig;y Yt 1
g^t =
b;g bt 1
ct =
Ig
b;ig bt 1
^
+ eg;t
^
+ eig;t
(B.88)
(B.89)
Tax rates respond positively to the debt to output ratio:
^ct =
b;
c
^bt
1
+
y;
116
c
Y^t
1
+e
c ;t
(B.90)
^lt =
b;
^kt =
b;
l
^bt
1
+
y;
l
k
^bt
1
+
y;
k
Y^t
1
+e
l ;t
(B.91)
Y^t
1
+e
k ;t
(B.92)
Finally transfers in this model are de…ned as lump sum taxes minus transfers. Transfers respond
to the output public debt and hours worked:
^
bt =
tr
^
b;tr bt 1
^ + etr;t
y;tr Yt 1
l;tr Lt
(B.93)
Nominal interest rate follows a Taylor rule version given by:
^t = R
^t
R
where
2.4.6
m
t
2
m
N 0;
1
+ (1
)
^
^t +
y Yt
m
t
+
is an i:i:d:- normally distributed error.
Log-Linearised System of Equations
This model is solved by a log-linearised system of equations. The capital letters without subscript
t denote the steady state values, whereas a hat over a variable denotes its log-deviations from the
steady state.
2.4.7
Households:
^
^
^t = Et Ct+1 + hC t 1
C
1+h
1+h
^t=
Q
^ t +Et ^ t+1 +
R
1
1
+ (1
k) r
u
^t =
w
^t
1+
c
k
1
"
k
r^k;t
wL
(1 + c ) Cnr
=
1
k
^
)K
t
1+
l
1
w
1+
^t
1
l
^lt
^kt
k
^"It+1
k
r^k;t+1
117
1
k
^kt+1
c
^ct
^"It
#
I^t
+
c
TR
d
T
Rt
c
(1 + ) Cnr
1+
1+ w
1
w
^t 1 +
Et ^ t+1
^t
1+
1+
1+
$w ) (1 $w )
1 (1
(Xtw +"w
t )
(1+& w ) l
1+
w
1+ & w
$
Et w
^t+1 +
1+
+
^t
w
^ t +L
(^ct Et ^ct+1 ) +^"B
"B
t+1 ^
t
^ +rk 1
)Q
t+1
Et (1
^ t = (1
K
l
^ t+1 +
^t
I^t 1
E t I^t+1
1
Q
+
+
+
Et
(1 + ) 1 +
1+
1+
I^t =
^nr;t = 1
C
c
1 1 h
Et Rt
c 1+h
1
^
Xtw = w
^t
l Lt
1
l
^t bC
^t
C
b
1
c
^ c;t = ^ t +
2.4.8
Firms:
h
^
Y^t = 'y ^"A
t + Kt
1+
c
1+
(^ct ^ct
^t=
2.4.9
1+
Et ^ t+1 +
p
p
^t
1+
(1
p
c CT
Y
^ T;t +
^ct +C
^ rev;t = R b
G
Y
^t
R
1
l wL
Y
^ g;t = (1
K
1
c
t
^
g Kg;t
1
i
^
g Kg;t 1
$) (1
$ 1+
$)
p
k rk K
(mc
c t +^"pt )
^t
^kt +^
rk;t +^
ut +K
Y
1
b ^ G ^ IG c T R d
bt + Gt +
IGt +
T Rt
Y
Y
Y
Y
^
)K
g;t
General equilibrium conditions:
1+
ct
IG
CT ^
I
G ^ IG c
Y^t =
CT;t + I^t + G
IGt +(1
t+
Y
Y
Y
Y
CT C^T;t = (1
c
1+
w
^t
^t +
^lt +w
^ t +L
^ t +^bt
c
1)
Government:
^ rev;t =
G
2.4.10
1
)w
^t + r^t ^"A
t
1+
^lt +^"L
t
^t+
)L
^t + (1
1+ u
^t= u
^t
L
^t +^
r t +K
mc
c t = (1
l
k
)
rk K
u
^t
Y
) C C^ t + C nr C^nr;t
The above equations plus the equations specifying …scal and monetary policy in the text above
(which are already in the log-linear form) comprise the system of equations which is subsequently
solved and estimated using Dynare routine for a DSGE model.
2.4.11
Parameterisation of the model
Parameters of this model are calibrated from the time series data as follows:
118
Table 34: Steady state ratios and calibrated parameters
Discount factor
0.990
Private capital depreciation rate
0.025
Puplic capital depreciation rate
0.020
Share of capital in production function
0.31
Steady-state wage markup parameter
0.15
Capital tax rate
40.71%
Labour tax rate
28.44%
Consumption tax rate
20.08%
Transfer to GDP ratio
0.14
Private investment to GDP ratio
0.15
Private consumption to GDP ratio
0.63
Gov. consumption to GDP ratio
0.20
Gov. investment to GDP ratio
0.02
Gross nominal interest rate
1.0101
Return on capital
0.0518
Gov. debt to GDP ratio
0.6
Note on Bayesian Estimation Applied in this Model
The Bayesian VAR became popular after seminal works of Sim (1980) and Doan, Litterman and
Sims (1984). In the context of DSGE models the Bayesian estimation can be perceived as a
combination of maximum likelihood estimation and calibration. Calibration because of presence
of priors, which comprise weights on likelihood function so more importance is given to particular
areas in parameters subspace.
According to An and Schorfeide (2007), Smets and Wouters (2003) and Mancini (2010) …rst the
prior probability distribution is denoted by p ( ), where denotes parameters of the model and p ( )
stands for probability distribution function, likelihood function as L jY T ; where Y T denotes the
complete sample of data, and …nally p jY T , as a posterior distribution. Secondly the likelihood
T
Q
can be formulated as: L jY T = p Y T j = p (Y0 j )
p (Yt jYt 1 ; ). Finally the Bayes’theorem
t=1
is used in order to get posterior, p jY T , which can be derived from the de…nition of conditional
probability:
p( ;Y T )
p( ;Y T )
p Y T j = p( ) ; p jY T = p(Y T ) ; ) p ; Y T = p jY T p Y T Thus: p jY T =
p(Y T j )p( )
.
p(Y T )
Since p Y T is constant, Bayes’ theorem can be written as: p jY T / p Y T j p ( )
| jY T where p Y T j stands for maximum function and p ( ) stands for prior probability distributions, and | jY T stands for posterior kernel. Likelihood function is estimated with help of
119
the Kalman …lter.
Estimation of Likelihood Function of the Model
The state space representation of the solution to the model can be rewritten in the following way:
x
^t+1 = A^
xt + Bvt+1
y^t = C x
^t + wt
where …rst equation is the equation comprising the solution of the model (measurement equation)
and the second equation is the observation equation (transition equation) i.e. y^ is an observable
variable, and wt is an measurement error. Hats over variables denote that the solution is in the
deviation from the steady state form in case of model solution, and in case of observable variable
0
vt
vt
Q V
it means that data are detrended by a linear trend. Moreover, E
=
, vt
wt
wt
V R
and wt are uncorrelated and orthogonal to yt . Kalman …lter recursion is the following:
b
y~t+1 = y^t+1
Et x
^t+1 = AEt
CEt x
^t+1
^t
1x
+ Kt b
y~t
Kt = (APt C 0 + BV ) (CPt C 0 + R)
Pt+1 = APt A0 + BQB 0
1
0
Kt (APt C 0 + BV )
Subsequently from the Kalman …lter recursion log-likelihood is derived. With the assumption of
0
b
y~t b
y~t
1
exp
normal distribution which has the probability distribution function: p (Yt j ) = p
(CPt C 0 +R) ;
0
2 (CPt C +R)
the log-likelihood is given by:
L
jY T =
T
ln (2 )
2
T h
X
(CPt C 0 + R)
t=1
0
b
y~b
y~ (CPt C 0 + R)
1
i
(B.94)
The log posterior kernel then becomes: ln | jY T = ln L jY T + ln p ( ). Subsequently,
maximizing the above log posterior kernel with respect to the mode of the posterior distribution
is found.
Derivation of the Posterior Distribution
At this stage only the mode of posterior distribution is known. In order to simulate posterior
distribution a particular version of Markov Chain Monte Carlo (MCMC) algorithm i.e. Metropolis
algorithm is employed and involves follwoing steps:
1. Choose a starting point - posterior mode.
P
P
2. Draw
from the distribution f
j i = N i ; c m , where
m is the inverse of the
Hessian matrix computed at the mode of the posterior distribution.
is a candidate for
i+1
with the probability of q i+1 j i , and i is a candidate for i+1 with probability of
p( ;Y T )
p( ;Y T )
1 q i+1 j i , where q i+1 j i = min 1; p i ;Y T , where p i ;Y T is an acceptance ratio.
(
)
(
)
120
3. Accept, or discard the proposed
:
4. Update mean of the drawing distribution, retain value of the parameter.
5. Repeat steps 2,3, and 4 for a chosen number of times.
6. Plot histogram of the retained values.
According to An and Schorfeide (2007), Smets and Wouters (2003) and Mancini (2010) the
Bayesian estimation process involves search through the space of using appropriate size of steps.
This is why the variance of and in particular the scaling parameter are of special interest. Increase
in the scaling parameter will cause acceptance rate to decrease and decrease in it will that to
increase. In case of too high acceptance ratio the Metropolis algorithm would never visit the tails
of the distribution and in case of too low acceptance ratio it would take long time to converge since
it can easily get stuck in the local subspaces. Literature proposes acceptance ratio in a range of
0.2-0.4 (Roberts, Gelman and Gilks (1997) get an optimal acceptance rate of 0.234).
2.4.12
Results of Hull DSGE model
Results of this Bayesian DSGE model for UK is calibrated to quarterly timeseries data of the UK
economy. It generates interesting results on multipliers. Impact investment shocks output multiplier
of 0.96 that of government spending is 0.8225. Increase in consumption tax has long run multiplier
-0.8277. Model also is applied to compute the imulse responses to 11 di¤erent shocks in the model.
Table 35: Prior and posterior distributions of estimated parameters
P rio r d istrib u t
E st m a x p o ste rio r
P o ste rio r d istrib u tio n M H
P a r a m e t e r ’s n a m e
ty p e
m ean
st. error
m ode
st. error
m ean
5%
95%
p ro d u c tiv ity s h o ck
inv _ g
0 .0 1
0 .0 1
0 .0 0 7 8
0 .0 0 0 6
0 .0 0 8 0
0 .0 0 6 9
0 .0 0 9 0
p referen ces sh o ck
inv _ g
0 .0 1
0 .0 1
0 .0 0 4 8
0 .0 0 1 4
0 .0 0 6 5
0 .0 0 3 2
0 .0 0 9 8
w a g e m a rk u p sh o ck
inv _ g
0 .0 1
0 .0 1
0 .0 0 3 2
0 .0 0 0 3
0 .0 0 3 2
0 .0 0 2 7
0 .0 0 3 6
p ric e m a rk u p sh o ck
inv _ g
0 .0 1
0 .0 1
0 .0 0 8 0
0 .0 0 0 8
0 .0 0 8 0
0 .0 0 6 8
0 .0 0 9 2
c o n su m p tio n ta x sh o ck
inv _ g
0 .0 1
0 .0 1
0 .0 3 2 9
0 .0 0 2 4
0 .0 3 3 5
0 .0 2 9 6
0 .0 3 7 4
c a p ita l ta x sh o ck
inv _ g
0 .0 1
0 .0 1
0 .0 5 1 3
0 .0 0 3 7
0 .0 5 2 4
0 .0 4 6 0
0 .0 5 8 5
la b o u r ta x sh o ck
inv _ g
0 .0 1
0 .0 1
0 .0 1 4 9
0 .0 0 1 1
0 .0 1 5 4
0 .0 1 3 5
0 .0 1 7 3
inve stm e nt sh o ck
inv _ g
0 .0 1
0 .0 1
0 .0 4 7 1
0 .0 2 0 1
0 .0 9 3 4
0 .0 2 8 6
0 .1 6 1 6
tra n sfers sh o ck
inv _ g
0 .0 1
0 .0 1
0 .0 4 8 8
0 .0 0 3 6
0 .0 5 0 2
0 .0 4 3 8
0 .0 5 6 0
g ov . inve stm e nt sh o ck
inv _ g
0 .0 1
0 .0 1
0 .3 0 6 4
0 .0 2 3 3
0 .3 1 2 8
0 .2 7 2 5
0 .3 5 1 6
m o n e ta ry p o lic y sh o ck
inv _ g
0 .0 1
0 .0 1
0 .0 0 4 3
0 .0 0 1 0
0 .0 0 4 9
0 .0 0 3 0
0 .0 0 6 7
g ov . c o n su m p tio n sh o ck
inv _ g
0 .0 1
0 .0 1
0 .0 0 9 5
0 .0 0 0 7
0 .0 0 9 8
0 .0 0 8 6
0 .0 1 1 0
A R (1 ) p ro d u c tiv ity s h o ck
b eta
0 .8
0 .1
0 .6 6 2 9
0 .0 8 5 2
0 .6 7 3 6
0 .5 3 7 9
0 .8 1 7 0
A R (1 ) p referen ces sh o ck
b eta
0 .8
0 .1
0 .7 6 3 0
0 .1 0 9 6
0 .7 2 8 9
0 .5 5 5 2
0 .9 0 2 2
121
Table 36: Priors and posteriors of estimated parameters
P rio r d istrib u t
E st m a x p o ste rio r
P o ste rio r d istrib u tio n M H
m ean
5%
95%
P a r a m e t e r ’s n a m e
ty p e
m ean
st. error
m ode
st. error
A R (1 ) inve stm e nt sh o ck
b eta
0 .8
0 .1
0 .2 7 9 0
0 .0 8 0 8
0 .2 9 1 5
0 .1 6 6 4
0 .4 0 9 4
A R (1 ) n o m . inte re st ra te
b eta
0 .8
0 .1
0 .6 1 8 4
0 .0 7 5 5
0 .6 0 0 4
0 .4 6 4 1
0 .7 3 3 0
i n ‡a t i o n r e s p o n s e
norm al
1 .5
0 .1
1 .5 2 7 2
0 .0 9 7 6
1 .5 3 4 4
1 .3 7 1 0
1 .6 8 8 4
output resp onse
norm al
0 .1 2 5
0 .1
0 .0 7 2 3
0 .0 3 8 4
0 .0 7 7 9
0 .0 2 0 0
0 .1 3 4 5
inve st. a d j. c o st
norm al
4 .0
1 .5
0 .9 0 5 8
0 .4 3 7 3
1 .9 6 5 6
0 .4 8 9 4
3 .5 5 4 7
c a p ita l u til. a d j. c o st
norm al
0 .2
0 .1
0 .6 8 0 5
0 .0 7 9 8
0 .6 7 6 8
0 .5 6 0 3
0 .8 0 4 5
inv . e la st. o f la b o u r
norm al
2 .0
0 .2
2 .0 2 2 1
0 .1 9 6 5
2 .0 1 0 9
1 .6 8 9 9
2 .3 4 0 1
C R R A co e¤.
norm al
0 .6 6
0 .1
0 .6 7 5 5
0 .0 8 7 3
0 .6 9 0 0
0 .5 4 4 9
0 .8 3 3 0
p ric e in d e x .
b eta
0 .5
0 .1 5
0 .1 4 5 3
0 .0 6 9 8
0 .1 7 0 6
0 .0 5 4 9
0 .2 7 9 3
w a g e in d e x .
b eta
0 .5
0 .1 5
0 .1 6 9 6
0 .0 6 9 0
0 .1 9 0 1
0 .0 7 9 1
0 .2 9 9 7
c a lvo p ric e s
b eta
0 .7 5
0 .1
0 .7 4 1 2
0 .0 4 0 6
0 .7 5 1 0
0 .6 8 2 8
0 .8 1 5 8
c a lvo w a g e s
b eta
0 .7 5
0 .1
0 .4 5 4 4
0 .0 6 4 8
0 .5 0 2 2
0 .3 8 8 7
0 .6 1 4 4
h a b it
b eta
0 .7
0 .1
0 .4 7 5 1
0 .0 6 7 7
0 .5 0 8 4
0 .3 9 6 2
0 .6 2 4 3
…x e d c o st
norm al
1 .4
0 .1
1 .5 6 6 0
0 .0 8 7 6
1 .5 7 1 5
1 .4 3 3 0
1 .7 1 4 6
Impulse responses to the government spending shock
-3
0
x 10
-3
c
0
x 10
-4
cr
5
x 10
-4
cnr
4
x 10
-4
R
2
-0.5
-0.5
0
2
1
-1
-1
-5
0
0
5
-3
2
x 10
10
15
5
-3
y
0
x 10
10
15
5
-4
I
0
x 10
10
15
5
-4
w
5
x 10
10
15
x 10
-4
pi
2
5
mc
10
15
0
-1
0
0.005
0
-2
-5
0
-0.01
10
15
L
5
5
10
15
-4
x 10 omega
5
-3
1
x 10
10
15
5
-3
ro
0
x 10
10
15
5
-3
Ig
0
x 10
10
15
1
0
0
0.5
-2
-2
0
-5
0
-4
-4
-1
10
15
5
10
15
5
10
15
5
10
15
Impulse responses to the government investment shock
122
5
10
5
-3
trans
-2
5
15
10
15
0.01
-4
5
10
g
0.01
-2
-3
5
b
0
x 10
pip
1
-2
2
x 10
15
x 10
5
g_rev
10
15
Table 37: Priors and posteriors of estimated parameters parameters
P rio r d istrib u t
-3
1
x 10
E st m a x p o ste rio r
P o ste rio r d istrib u tio n M H
m ean
5%
P a r a m e t e r ’s n a m e
ty p e
m ean
st. error
m ode
sh a re o f n o n -R ic a rd ia n
b eta
0 .5
0 .1
0 .1 4 9 5
0 .0 3 3 9
0 .1 6 4 0
0 .1 0 3 8
0 .2 1 8 9
A R (1) gov. sp en d .
b eta
0 .8
0 .1
0 .7 4 3 9
0 .0 7 9 0
0 .7 8 0 5
0 .6 6 3 9
0 .9 0 4 0
A R (1 ) g ov . inv .
b eta
0 .8
0 .1
0 .3 7 1 2
0 .0 8 2 2
0 .3 7 8 2
0 .2 4 2 8
0 .5 1 3 2
A R (1) gov. tr.
b eta
0 .8
0 .1
0 .4 1 6 0
0 .0 7 9 0
0 .4 4 0 2
0 .3 1 0 3
0 .5 7 6 4
A R (1) cap. tax.
b eta
0 .8
0 .1
0 .6 0 4 7
0 .0 7 0 5
0 .6 0 8 0
0 .4 8 9 1
0 .7 1 8 3
A R (1 ) la b . ta x .
b eta
0 .8
0 .1
0 .4 8 1 2
0 .0 8 0 9
0 .5 1 1 6
0 .3 7 0 3
0 .6 5 1 2
A R (1) con. tax.
b eta
0 .8
0 .1
0 .6 0 5 6
0 .0 7 5 4
0 .6 2 6 8
0 .5 0 0 2
0 .7 5 0 0
gov. sp en d to ou tp u t.
norm
0 .1 5
0 .1
0 .1 7 6 6
0 .0 9 6 5
0 .1 6 9 9
0 .0 1 1 6
0 .3 3 5 4
g ov . inv to o u tp u t.
norm
1 .5
0 .1
1 .5 0 8 1
0 .1 0 0 0
1 .5 1 0 6
1 .3 4 3 1
1 .6 7 6 1
gov. tran s. to ou tp u t.
norm
1 .5
0 .1
1 .5 1 4 4
0 .0 9 9 0
1 .5 1 8 4
1 .3 6 2 0
1 .6 8 3 8
cons. tax to output.
norm
0 .5
0 .1
0 .5 3 6 3
0 .0 9 9 3
0 .5 3 0 6
0 .3 7 1 3
0 .6 9 4 0
cap. sp end to output.
norm
1 .5
0 .1
1 .5 4 3 0
0 .0 9 9 5
1 .5 3 9 6
1 .3 6 7 9
1 .7 0 0 7
la b . sp e n d to o u tp u t.
norm
0 .5
0 .1
0 .4 7 2 7
0 .0 3 3 9
0 .4 8 0 5
0 .3 2 3 9
0 .6 3 2 0 0
-3
c
1
x 10
-3
cr
2
x 10
cnr
-4
5
x 10
st. error
-4
R
5
x 10
-4
pi
2
0
0
0
0
0
0
-1
-1
-2
-5
-5
-2
5
10
15
5
-3
y
0.01
2
x 10
10
15
2
0
0
0
-0.01
-2
-2
5
-3
5
x 10
10
15
5
-3
L
2
x 10
5
-4
I
10
15
x 10
4
15
5
-3
2
x 10
10
15
5
x 10
10
15
5
ro
10
15
0
0
-0.5
-0.01
15
5
10
15
5
10
15
x 10
-4
10
15
10
15
g
10
5
10
5
-3
2
0
0
15
5
4
2
10
5
-4
trans
0
5
15
0.01
-2
15
10
b
0
Ig
0.5
0
10
x 10
pip
-2
-2
-5
5
5
-3
mc
0
5
-3
omega
10
w
x 10
95%
15
x 10
0
g_rev
5
10
15
10
15
10
15
Impulse responses to the transfer shock
-3
2
x 10
-4
c
5
x 10
cr
-4
cnr
0.02
2
x 10
-4
R
4
0
0
0
1
2
-2
-5
-0.02
0
0
5
-3
1
x 10
10
15
5
-3
y
1
x 10
10
15
5
-4
I
0
x 10
10
15
5
-4
w
5
x 10
10
15
0
0.01
-4
-5
0
10
15
L
0
5
10
15
-4
x 10 omega
-2
0
5
-3
1
x 10
10
15
5
ro
10
15
5
Ig
0
15
0
pip
5
-3
-0.5
-2
5
10
0
0
-3
5
0.02
-1
x 10
0.5
b
0
x 10
-4
pi
1
mc
-1
1
x 10
10
15
x 10
-1
5
-3
trans
0
0.05
1
-0.005
0
0.5
x 10
g
g_rev
-4
-1
5
10
15
5
10
15
-1
5
10
15
-0.01
123
5
10
15
-0.05
5
10
15
0
5
10
15
Table 38: Priors and posteriors of estimated parameters
P rio r d istrib u t
E st m a x p o ste rio r
P o ste rio r d istrib u tio n M H
m ean
5%
95%
P a r a m e t e r ’s n a m e
ty p e
m ean
st. error
m ode
st. error
gov. sp en d to d eb t
norm
0 .2
0 .1
0 .0 4 9 5
0 .0 2 6 1
0 .0 7 0 9
0 .0 1 2 6
0 .1 3 2 0
g ov . inv to d e b t
norm
0 .4
0 .1
0 .4 2 5 2
0 .0 9 4 8
0 .4 2 9 7
0 .2 7 1 4
0 .5 8 1 9
gov. tran s. to d eb t
norm
0 .2
0 .1
0 .0 4 7 3
0 .0 5 0 4
0 .0 6 9 0
-0 .0 1 4 2
0 .1 5 6 2
cons. tax to debt
norm
0 .2
0 .1
0 .0 9 7 9
0 .0 4 7 7
0 .1 0 7 7
0 .0 2 3 4
0 .1 9 5 2
cap. tax to debt
norm
0 .2
0 .1
0 .0 2 6 1
0 .0 6 2 0
0 .0 2 4 7
-0 .0 7 6 5
0 .1 2 3 2
la b . ta x to d e b t
norm
0 .2
0 .1
0 .0 0 7 0
0 .0 1 7 1
0 .0 1 1 4
-0 .0 2 1 5
0 .0 4 5 3
g ov . sp en d to d ef.
norm
0 .1 5
0 .1
0 .1 5 0
0 .0 8 3 1
0 .1 0 1 8
-0 .0 3 4 4
0 .2 3 8 0
g ov . inv to d e f.
norm
0 .1 5
0 .1
0 .1 5 8 7
0 .0 9 9 7
0 .1 6 2 4
-0 .0 0 9 4
0 .3 1 6 9
g ov . tra n s. to d ef.
norm
0 .1 5
0 .1
0 .1 3 6 9
0 .0 9 2 5
0 .1 4 6 2
-0 .0 0 5 8
0 .3 0 2 6
co n s. ta x to d ef.
norm
0 .2
0 .1
0 .2 0 2 4
0 .0 9 2 0
0 .2 1 7 4
0 .0 6 1 1
0 .3 6 7 5
ca p . sp en d to d ef.
norm
0 .2
0 .1
0 .2 0 1 6
0 .0 9 6 5
0 .2 0 1 7
0 .0 4 5 3
0 .3 6 0 4
la b . sp e n d to d e f.
norm
0 .2
0 .1
0 .2 2 9 7
0 .0 6 2 8
0 .2 3 2 4
0 .1 2 6 6
0 .3 4 5 7
tr. resp to hours
norm
1 .5
0 .1
1 .5 2 0 1
0 .0 9 9 2
1 .5 1 8 5
1 .3 6 2 9
1 .6 9 0 0
Table 39: Present Value Government Consumption Multiplier
Variable
Impact
4 quarters
8 quarters
12 quarters
PV (
PV (
PV (
PV (
PV (
PV (
0.8225
0.5252
0.3288
0.2130
-0.1211
-0.2678
-0.3330
-0.3652
-0.0719
-0.1797
-0.2890
-0.3579
Y)
G)
C)
G)
I)
G)
Impulse responses to the consumption tax shock
-3
5
x 10
-3
c
5
0
-3
cr
5
0
-5
5
-3
5
x 10
x 10
10
15
5
-3
5
-3
cnr
5
0
-5
y
x 10
x 10
10
15
x 10
R
0
-5
5
I
10
15
-5
5
w
10
5
0
0
-0.01
15
5
-3
mc
0.01
-4
pi
0.01
0.01
5
x 10
10
15
-5
5
0
0
0
0
0
0
-5
-0.01
15
-0.01
-5
-5
5
-3
x 10
10
15
5
-3
L
5
x 10
10
5
-3
omega
5
x 10
10
15
5
-3
ro
5
x 10
10
15
0
0
0
-5
-5
-0.01
15
5
10
15
5
10
15
Impulse responses to the labour tax shock
124
5
10
5
10
x 10
5
-3
-0.5
0
-5
15
15
0
0
10
10
trans
0.01
-2
5
5
Ig
15
-1
pip
5
-4
b
-5
2
x 10
x 10
5
10
15
10
15
g
g_rev
10
15
Table 40: Present Value Government Investment Multiplier
Variable
Impact
4 quarters
8 quarters
12 quarters
PV (
PV (
PV (
PV (
PV (
PV (
0.9591
0.7282
0.6584
0.6646
-0.0463
-0.1600
-0.1601
-0.1332
-0.0242
-0.0794
-0.1173
-0.1195
Y)
Ig)
C)
Ig)
I)
Ig)
Table 41: Present Value Transfers Multiplier
-3
1
Impact
4 quarters
8 quarters
12 quarters
PV (
PV (
PV (
PV (
PV (
PV (
0.1275
0.0045
-0.1179
-0.2065
0.1516
0.1182
0.1165
0.1298
-0.0096
-0.0341
-0.0564
-0.0620
-4
c
x 10
Variable
2
Y)
tr)
C)
tr)
I)
tr)
-3
cr
x 10
5
x 10
-5
cnr
5
x 10
-5
R
5
x 10
-5
pi
5
0.5
1
0
0
0
0
0
0
-5
-5
-5
-5
5
10
-4
5
15
2
0
10
15
5
10
-4
0
5
-4
0
x 10
10
15
x 10
10
15
0
5
-4
5
x 10
10
15
x 10
10
-2
15
5
-3
ro
0
x 10
5
mc
10
15
10
0.01
0
0.005
-2
0
15
5
-3
Ig
0
x 10
10
0
-2
0
-2
-1
0
-4
-5
-4
-2
-5
10
15
5
10
15
5
10
15
5
10
15
5
10
15
x 10
10
15
10
15
g
5
-3
5
-5
5
x 10
-4
15
trans
pip
5
-4
b
-1
-4
omega
5
-4
w
-2
-2
15
L
x 10
5
-4
I
x 10
0
-5
5
5
-4
y
x 10
x 10
g_rev
5
10
15
Impulse responses to the capital tax shock
-3
1
x 10
-3
c
2
0
1
-1
0
-3
cr
x 10
0
x 10
-4
cnr
5
-2
x 10
-3
R
1
x 10
-3
pi
1
0
0
0
-5
-1
-1
x 10
pip
-4
5
-3
5
x 10
10
15
5
-3
y
4
10
15
5
-4
I
x 10
5
x 10
10
15
5
w
10
15
0
-0.005
0.01
0
-5
-0.01
0
0
10
15
L
5
10
15
5
omega
-2
10
15
5
ro
10
15
5
-3
Ig
2
x 10
5
-3
-1
2
5
15
0
0
-3
10
b
0.02
-5
x 10
5
mc
0
10
15
-2
x 10
5
trans
10
15
10
15
g
g_rev
0.02
0
0
0.01
0
-0.02
-0.005
0
0
-0.02
-0.04
-0.01
-2
-0.01
-4
5
10
15
5
10
15
5
10
15
5
10
15
5
10
15
5
10
15
Smoothed shocks 2000:Q1 until 2011:Q1
Model is then applied to compute the variance decomposition. The private and public investment
accounts for around 50 per cent of the ‡uctuations, in‡ation and monetary shocks account for around
30 per cent and taxes account for around 8 per cent.
125
Table 42: Present Value Consumption Tax Multiplier
Variable
Impact
4 quarters
8 quarters
12 quarters
PV (
PV (
PV (
PV (
PV (
PV (
-0.8277
-0.7001
-0.4347
-0.3417
-0.6369
-0.5624
-0.3852
-0.3520
-0.0936
-0.0809
0.0026
0.0552
Y)
tc)
C)
tc)
I)
tc)
Table 43: Present Value Capital Tax Multipliers
Variable
Impact
4 quarters
8 quarters
12 quarters
PV (
PV (
PV (
PV (
PV (
PV (
-0.1382
-0.0708
-0.0127
0.0380
-0.1607
-0.1616
-0.1891
-0.2122
0.0020
-0.0004
-0.0150
0.0306
Y)
tc)
C)
tc)
I)
tc)
0.06
Initial values
tr
0.04
ig
it
0.02
wt
pit
0
tl
tk
-0.02
tc
vt
nt
-0.04
lt
-0.06
at
0
20
40
60
80
100
References
[1] An,S., Schorfheide, F.,(2007) Bayesian analysis of DSGE models, Econometric Reviews, 30,
889-920.
[2] Ascari, G., Rankin, N. (2013) The e¤ectiveness of government debt for demand
management:Sensitivity to monetary policy rules, Journal of Economic Dynamics and
126
Table 44: Present Value Capital Tax Multipliers
Variable
Impact
4 quarters
8 quarters
12 quarters
PV (
PV (
PV (
PV (
PV (
PV (
-0.2242
-0.2643
-0.2606
-0.2337
0.0086
-0.0313
-0.0515
-0.0614
-0.0139
-0.0408
-0.0649
-0.0746
Y)
tc)
C)
tc)
I)
tc)
Table 45: Variance Decomposition in Percents
tfp .
wage
p ref.
g_ sp.
c_ tx.
k_ tx.
l_ tx .
in f.
R.
p r_ inv .
g _ inv .
tr.
y
3 .0 5
1 .6 3
1 .5 8
0 .9 1
4 .4 3
3 .8
0 .0 7
1 5 .5
1 6 .5
4 4 .2 4
7 .6 2
0 .6 7
c
5 .9 9
1 .6 7
1 0 .3 5
0 .6 5
8 .3 2
0 .5 4
0 .2 1
2 1 .7 7
2 0 .2 7
2 8 .8 9
0 .5 4
0 .8 1
I
1 .8 4
0 .8 1
1 .6 6
0 .2 7
0 .2 2
0 .2 1
0 .0 0
1 .0 4
1 .9 9
9 1 .8 5
0 .0 8
0 .0 3
L
3 2 .8 5
2 .4 8
1 .3 4
0 .7 3
2 .0 7
9 .4
0 .0 8
6 .6 6
1 1 .9
2 5 .1 7
6 .9 6
0 .3 7
w
1 .7 4
1 2 .7 8
0 .4 4
0 .3 6
1 4 .1 1
0 .2 7
0 .2 2
4 2 .7 7
5 .0 4
2 1 .6 3
0 .0 9
0 .5 4
R
5 .0 4
1 .5 9
0 .9 7
0 .2 6
1 6 .8 2
0 .5 0
1 .0 1
3 3 .7 1
3 3 .9 1
6 .4 8
0 .5 5
0 .1 6
pi
3 .6
1 .5
0 .3
0 .0 9
3 6 .4 2
0 .6 8
0 .0 0
5 4 .0 9
0 .9 6
2 .4 9
0 .2 1
0 .1 1
b
0 .7 8
1 .7 9
0 .7 6
1 .4
0 .3 5
4 .7 8
0 .6 7
1 5 .2 6
2 1 .7 4
4 8 .0 0
0 .3 2
4 .1 5
Control,37(8),1544-1566.
[3] Batini, N., Harrison, R., Millard, S. P. (2003) Monetary policy rules for an open economy,
Journal of Economics Dynamics and Control, 27(11-12), 2059–2094.
[4] Basu, P., Kollmann, R. (2013) Productive Government Purchases and the Real Exchange Rate,
Manchester School, 81(4), 461-469.
[5] Basu, P., (1987) An adjustment cost model of asset pricing. International Economic Review 28
(3), 609–621.
[6] Benati, Luca. (2008) “The ‘Great Moderation’in the United Kingdom.”Journal of Money,Credit
and Banking, 40, 121–47.
[7] Bhattarai K. (2011) Econometric and Stochastic General Equilibrium Models for Evaluation of
Economic Policies, International Journal of Trade and Global Markets, 4:2: 212-241
[8] Bhattarai K. (2007) " Welfare Impacts of Equal-Yield Tax Experiment in the UK Economy,
Applied Economics, 39,1545-1563.
[9] Blanchard O. and Jordi Galí (2012) Labor Markets and Monetary Policy: A New Keynesian
Model with Unemployment, American Economic Journal: Macroeconomics 2:1–30
[10] Blake A. P., M. Weale (1998) Costs of Separating Budgetary Policy from Control of In‡ation:
A Neglected Aspect of Central Bank Independence, Oxford Economic Papers, 50, 3, 449-467
127
tfp
transfers
gov. inv
0.02
0.5
5
0
0
0
-0.02
20
priv. inv
40
0.2
-0.5
20
40
monet. policy
0.02
-5
0
0
0
-0.02
-0.02
40
20
cap. tax
40
0.05
0.2
0.2
0
0
0
-0.05
20
40
gov. spend
-0.2
20
pref.
40
-0.2
0.05
0.01
0.01
0
0
0
-0.05
20
40
-0.01
40
20
con. tax
40
20
wages
40
0.02
-0.2
20
lab. tax
20
inflation
20
40
-0.01
0
50
[11] Bouakez, H., Rebei , N. (2007) Why Does Private Consumption Rise After A Government
Spending Shock?, Canadian Journal of Economics, 40(3), 954–979.
[12] Campbell, J. Y., Mankiw, N. G. (1989) Consumption, income, and interest rates: reinterpreting
the time series evidence, NBER Macroeconomics Annual 1989, 4, 185–216.
[13] Calvo, G. A. (1983) Staggered prices in a utility-maximizing framework, Journal of Monetary
Economics, 12(3), 383-398
[14] Chadha, J. S. and C. Nolan (2007) Optimal Simple Rules for the Conduct of Monetary and
Fiscal Policy, Journal of Macroeconomics, Dec, 29:4: 665-89.
[15] Clarida R., Gali J., and Gertler M. (1999) The Science of Monetary Policy: A New Keynesian
perspective, Journal of Economic Literature; 37 (4), 1661-707.
[16] Coenen, G., Straub, R., Trabandt, M. (2013) Gauging the e¤ects of …scal stimulus packages
in the euro area, Journal of Economic Dynamics and Control, 37(2), 367-386.
[17] Coenen, G., Erceg, C. J., Freedman, C., Furceri, D., Kumhof, M., Lalonde, R., Laxton, D.,
Linde, J., Mourougane, A., Muir, D., Mursula, S., de Resende, C., Roberts, J., Roeger, W.,
Snudden, S., Trabandt, M., in’t Veld, J.,(2012) E¤ects of …scal stimulus in structural models,
American Economic Journal: Macroeconomics, 4(1), 22–68.
[18] Christiano, L., Eichenbaum, M., Rebelo, S. (2011) When is the government spending multiplier
large?, Journal of Political Economy, 119(1), 78- 121.
[19] Christo¤el, K.,Linzert,T.,2010.Theroleofrealwagefrictionsandlabormarketfrictionsforin‡ationpersistence.Journal
of Money,Credit,and Banking 42,1435–1446.
128
[20] Collard, F. and M. Juillard (2001) Accuracy of stochastic perturbation methods: The case of
asset pricing models, Journal of Economic Dynamics and Control, 25, 979–999.
[21] Doan, T, R. Litterman, and C. Sims (1984) Forecasting and Conditional Projections Using
Realistic Prior Distributions", Econometric Reviews, 3, 1-100
[22] Ellison M. and J. Pearlman (2011) Saddlepath learning, Journal of Economic theory 146,
1500-1519.
[23] Faccini. R, S. Millard and F. Zanetti (2013) Wage rigidities in an estimated dynamic stochastic
general equilibrium model of the UK labour market, Manchester School, 81, 66-99.
[24] Harrison, R., Oomen, O. (2010) Evaluating and estimating a DSGE model for the United
Kingdom, Bank of England Working Paper, no. 433, Bank of England.
[25] Iacoviello, M. (2005) House Prices, Borrowing Constraints, and Monetary Policy in the Business Cycle. American Economic Review, 25(3), 739-764.
[26] Iacoviello, M., Neri, S. (2010) Housing Market Spillovers: Evidence from an Estimated DSGE
Model, American Economic Journal: Macroeconomics, 2(2), 125-164.
[27] Koop G and D. Korobilis (2010) Bayesian Multivariate Time Series Methods for Empirical
Macroeconomics, now Publishers Inc., Hanover MA, USA
[28] Gary Koop, Dale J. Poirier, Justin L. Tobias (2010) Bayesian Econometric Methods, Cambridge
University Press.
[29] Krause, M. and Lubik,T. (2007).The(ir)relevance of real wage rigidity in the new Keynesian
model with search frictions.Journal of Monetary Economics 54, 706–727.
[30] Levine P, J. Pearlman, G. Perendia and B. Yang (2012) Endogenous Persistence in an estimated
DSGE model under imperfect information, Economic Journal,122,1287–1312
[31] Mirrlees J., and S. Adam, T. Besley, R. Blundell, S. Bond, R. Chote, M. Gammie, P. Johnson, G. Myles, J. Poterba.2010. Dimensions of tax design: the Mirrlees review, Oxford: Oxford
University Press.
[32] Monacelli, T., Perotti, R. (2010) Fiscal Policy, The Real Exchange Rate and Traded Goods,The
Economic Journal, 120, 437–461.
[33] Pissarides, C. A. (2013) Unemployment in the Great Recession, Economica, 80: 385–403
[34] Pissarides, C. A. (2011) Equilibrium in the Labor Market with Search Frictions, American
Economic Review, 101(4): 1092-1105.
[35] Ramey, V. A. (2011) Can government purchases stimulate the economy?, Journal of Economic
Literature, 49(3), 673-685.
[36] Rankin N. (1992) Imperfect competition, expectations and the multiple e¤ects of monetary
growth, Economic Journal 102 (1992): 743-753.
[37] Ravn, M. O., Schmitt-Grohé, S. and Uribe, M. (2012) Consumption, government spending,
and the real exchange rate, Journal of Monetary Economics, 59(3) 215-234.
129
[38] Rotemberg, J.J.,(1982) Sticky prices in the United States. Journal of Political Economy
90,1187–1211.
[39] Sim C.A (1980) Macroeconomics and Reality, Econometrica, 48:1 Jan.1 -45.
[40] Sims, C.A. and T. Zha (1999) Error Bands for Impulse Responses, Econometrica, 67 (5):
1113-1155
[41] Smets, F., Wouters, R. (2007) Shocks and frictions in US business cycles: a Bayesian DSGE
approach, American Economic Review, 97(3), 586-606.
[42] Smet F. and R. Wouters (2003) An estimated dynamic stochastic general equilibrium model
of the Euro Area, Journal of European Economic Association, Sept, 1(5):1123-1175.
[43] Wickens M. (1995) Real Business Cycle Analysis: A Needed Revolution in Macroeconometrics
(in Controversy: Business Cycle Empirics, Economic Journal, 105, 433., 1637-1648.
[44] Woodford, M. (2011) Simple Analytics of the Government Expenditure Multiplier, American
Economic Journal: Macroeconomics, 3(1): 1–35.
[45] Zanetti, F. (2010), ‘Labor market institutions and aggregate ‡uctuations in a search and
matching model’, European Economic Review, 55,644-658.
2.5
Critical assessment of the DSGE Models
Bhattarai and Dixon (2013) opine that for almost …ve decades economists have tried to incorporate
unemployment as a feature of the equilibrium process in the modelling of an economy. Job matching
and search models developed by Phelps (1968), Mortensen (1970), Lucas and Prescott (1974),
Pissarides (1979, 1985, 1986), Mortensen and Pissarides (1994, 1999) have signi…cantly contributed
to the analysis of unemployment dynamics based on bargaining and matching for wages and work by
workers and …rms. In the last decade, the new Keynesian DSGE model has been extended to assess
the impacts on in‡ation dynamics of both …rm-union bargaining (Zanetti (2007), Gertler, Sala and
Trigari (2008), Gertler and Trigari (2009)) and search-matching a la Mortensen and Pissarides (see
for example Kraus and Lubik (2007), Christo¤el and Kuester (2008), Trigari (2009), Christo¤el and
Kuester (2008), Krause, Lopez-Salido and Lubik (2008), Christo¤el, Kuester and Linzert (2009),
Zanetti (2011)). However, whilst these recent non-Walrasian developments are very much to be
welcomed, in standard DSGE models the focus is on short run "business cycle" ‡uctuations around
a steady-state in a representative household setting (for an evaluation of these models in this context,
see Christo¤el et al. (2009)).
Bhattarai and Dixon (2014) observe that the DGSE framework is lacking in certain key dimensions. First, these models shed little or no light on the long run general equilibrium impacts
of tax-transfer policies on equilibrium unemployment, growth, capital accumulation among various
sectors of the economy and the utility, wages and labour supply of households into the future.
Secondly, the micro-foundations are very simple and abstract from diversity across households and
sectors, with no role for relative prices and wages by sectors of production and skill categories of
labour They extend the computable general equilibrium (CGE) framework to allow a full analysis of
equilibrium unemployment in a dynamic general equilibrium model with heterogenous households
and …rms, providing the structural details required for more realistic analysis of the allocation
130
mechanism in the economy. They provide a medium and long-term framework for evaluating and
understanding economic policy as opposed to the short-run focus of standard new Keynesian DGSE
models. They also assess the full impacts of the equilibrium rate of unemployment on labour supply,
consumption and saving behavior of households, alongside the investment and capital accumulation
behavior of …rms and the resulting relative prices of commodities and factors of production in the
broader economy.
2.5.1
Blanchard’s New Keynesian DSGE model
Problem of household i that maximises expected utility from consumption (Cit+k ), accumulation
of money (Mit+k+1 ) and labour supply (Nit+k ) taking account of all information ( t ) available up
to period t is given as:
"1
#
X
Mit+k+1
k
Q (Nit+k ) j t
max E
U (Cit+k ) + V
(B.95)
P t+k
t=0
subject to:
a) CES aggregation of consumption (Cit ) and price level P t over j commodities:
Cit =
Z
0
1
1
1
Cijt dj
;
Pt =
Z
1
0
1
Pjt
1
dj
1
(B.96)
b) the budget constraint
Z
1
Pjt Cijt + Mit+1 + Bit+1 = Wt Nit + (1 + it ) Bit + Mit +
it
+ Xit
(B.97)
0
where Bit , it and Xit denote bonds held, pro…ts earned and transfer received by the household
i ; Wt is wage earned for supplying labour (Nit ) :
c) demand for a product Cijt relates to composite demand as:
Cijt =
Pjt
Pt
Cit
(B.98)
Firms take wage rates as given and set prices a la Calvo with probability of changing it every
period. Then Yj;t is the solution to the …rms’pro…t maximization problem:
#
"
X U 0 (Ct+1 )
Pjt
Wt+k Yjt+k
k
k
max E
(1
)
Yjt+k
j t
(B.99)
U 0 (Ct )
P t+k
P t+k Zt+k
k
subject to:
a) a linear production technology
Yjt = Zt Njt
(B.100)
b) supply
Yjt+k =
Pjt
P t+k
131
Yt+k
(B.101)
1. Write …rst order conditions for optimisation by households and …rms in this model.
2. Solve for the price level, employment and output at the steady state.
3. Prove that volatility of output is generated from the technological shock. Comment on how
does it compare to a standard RBC model.
2.5.2
Basic New Keynesian Model in logs
IS: goods market equilibrium (demand)
yt = Eyt+1
art+1 ;
rt+1 = it+1
E
(B.102)
t+1
LM: money market equilibrium (…nancial markets)
pt = b yt
mt+1
c:it+1
(B.103)
Phillips curve (Supply)
t
2.5.3
= E
t+1
+ d:xt ;
ybt ; xt = yt
xt = yt
zt ;
(B.104)
Extended version of the New Keynesian Model
IS: goods market equilibrium (demand)
yt = (1
) yt
1
+ Et yt+1
rt+1 = it+1
art+1 ;
E
rt+1 = it+1
it+1 =
t+1
E
t+1
(B.105)
(B.106)
t
LM: money market equilibrium (…nancial markets)
mt+1
pt = b yt
c:it+1
(B.107)
Phillips curve (Supply)
t
= E
t+1
+ d:xt ;
xt = yt
Zt =
2.6
Zt
1
+ "zt
ybt ; xt = yt
zt ;
(B.108)
(B.109)
Problem on Open Economy New Keynesian Model
1. Consider a standard open economy optimal growth model with
Household problem:
max
U = E0
1
X
t
Ut (Ct ; Lt )
t=0
132
0<
<1
(B.110)
Ut (Ct ; Lt ) =
Ct1
1
L1+!
t
1+!
(B.111)
subject to budget constraint as given by
Wt Lt +
t
+ Pt Kt = Pt Ct + Ptf It + Bt + Tt + 1 + Rt
1
+
t 1
e t Ft
1
(B.112)
Aggregation of di¤erentiated goods from the monopolistically competitive …rms as
Ct =
Z
1
1
1
(Cj;t )
d:j
(B.113)
0
1.
Z
Pt =
1
1
(Pj;t )
1
(B.114)
d:j
0
Firm’s problem
M ax
t
= Pt Yt
Wt Lt
PtK Kt
(B.115)
Subject to the CES production technology and stochastic TFP growth constraints as:
1
Yj;t = Zt (1
1 ) Lj;t
Zt = ln Zt
Yt =
Z
1+
1
(Yj;t )
+
(1
1
1 Kj;t
(B.116)
)Z
(B.117)
1
d:j
(B.118)
0
Lt =
Z
1
(Lj;t )
1
1
d:j
(B.119)
0
Kt =
Z
1
(Kj;t )
1
1
d:j
(B.120)
0
(a) Write the Lagrangian function for constrained dynamic optimisation by households and
derive the Euler equations for optimisation.
(b) Write the Lagrangian function for constrained dynamic optimisation by …rms and derived
the demand functions for labour and captial.
(c) Solve the model using the projection method and numerical optimisation using MATLAB
routines. Write approximation functions and Euler errors functions
(d) Derive impulse responses for shock to the …scal policy.
(e) Include Taylor Rule in the above problem for analysis of monetary policy.
133
(f) Conduct dynamic simulations to analyse the impact of demand shocks and supply shocks.
Reference: Lim G. C. and McNelis (2008), Computational Macroeconomics for the Open
Economy, MIT Press.
Study MATLAB programme DSGEM.m with chapter2_netfun.m.
See working papers of
Bank of England http://www.bankofengland.co.uk/publications/Pages/workingpapers/default.aspx
ECB: http://www.ecb.europa.eu/pub/html/index.en.html;
IMF: http://www.imf.org/external/pubs/cat/wp1_sp.aspx
Conference papers from EEA/ESEM, ASSA, Econometric Society and others such as http://www.hull.ac.uk/php/ecskrb/Confer
Uhlig’s toolkit: http://www2.wiwi.hu-berlin.de/institute/wpol/html/toolkit.htm
Schmitt-Grohe, S.,M. Uribe, 2004. Solving dynamic general equilibrium models using a second-order
approximation to the policy function. Journal of Economic Dynamics and Control 28, 755–775.
Thes study example1.mod, exmaple2.mod, bvar_and_dsge.mod; RBC_Summer.mod; RBC_Invcost.mod;
NK_hab.mod.
Blanchard O. and J. Galí (2013) Labor Markets and Monetary Policy: A New Keynesian Model with
Unemployment American Economic Journal: Macroeconomics 2 (April 2010): 1-30
Chadha, J, S, & Nolan, C, (2002) In‡ation and price level targeting in a new Keynesian model,
Manchester School, 70(4), pp. 570-595.
Holly S and M Weale Eds. "Econometric Modelling: Techniques and Applications", Cambridge University Press.
Juillard, M. (1996), Dynare: A Program for the Resolution and Simulation of Dynamic Models with
Forward Variables through the Use of a Relaxation Algorithm, CEPREMAP, Couverture Orange, 9602.
See: website of Dynare programs a number of applications of DSGE models http://www.douglaslaxton.org/dynare.html.
see BB_Er_…nal.mod programme from Basu and Bhattarai (2012); Jules1.mod by Iacoviello and Neri
(2005).
http://www.hull.ac.uk/php/ecskrb/Confer/Fina_crisis_criese012_rev.pdf
2.7
Stability Analysis: Illustrations
Di¤erence and di¤erential equations are necessay to model the macroeconomic dynamics. The
convergence or divergence of these systems depend on the roots (eigen values) of system. Exercise
here illustrates on these.
1. Solutions of system of linear di¤erential equations:
a) Distinct and real roots case
y1 (t) = C1 er1 t + C2 er2 t + y 1
y2 (t) =
r1
a11
a12
C 1 e r1 t +
r2
a11
a12
C2 er2 t + y 2
(B.121)
(B.122)
b) real and equal roots case
y1 (t) = C1 er1 t + C2 er2 t + y 1
134
(B.123)
y2 (t) =
r1
a11
a12
(C1 + C2 t) er1 t +
C2 rt
e + y2
a12
(B.124)
c) Complex roots
y1 (t) = eht [A1 cos(vt) + A2 sin(vt)] + y 1
(h
y2 (t) = eht
a11 ) A1 + vA2
(h
cos (vt) +
a12
a11 ) A2
a12
solution diverges if h > 0 and converges if h < 0 where h =
q
2
2
v = 12 4 (a11 a22 a12 a21 )
(a11 + a22 )
vA1
1
2
(B.125)
sin(vt) + y 2
(B.126)
(a11 + a22 ) ;
1. Solutions of system of linear di¤erence equations:
yt+1 = a11 yt + a12 xt
(B.127)
xt+1 = a21 yt + a22 xt
(B.128)
yt = C1 r1t + C2 r2t + y
(B.129)
a) Distinct and real roots case
xt =
r1
a11
r2 a11
C1 r1t +
C2 r2t + x
a12
a12
(B.130)
yt = (C1 + C2 t) rt + y
(B.131)
b) real and equal roots case
xt =
where r1; r2 =
tr(A)
2
1
2
c) Complex roots
r1
a11
a12
p
tr(A)2
(C1 + C2 t) er1 t +
C2 t
r +x
a12
4 jAj; tr(A) = (a11 + a22 ) ; jAj = (a11 a22
yt = Rt [C1 cos( t) + C2 sin( t)] + y
xt
where R =
2.
p
=
C1 R cos( t) + C2 R sin( t) a11 C1
cos ( t)
a12
C2 R cos( ) + C1 R sin( ) a11 C2
sin ( t) +x
+Rt
a12
Rt
jAj; cos( ) =
4
4
1
4
a12 a21 )
(B.133)
tr(A)
2R ; sin(
q
) = 4 jAj
; b) A =
1
1
4
3
3
135
(B.134)
tr(A)2
2R
y = Ay
where a) A =
(B.132)
;c) A =
(B.135)
2
2
5
4
[Hint: for a second order di¤erence equation r2
p
1
tr(A)2 4 jAj
2
tr(A)r + jAj = 0 ; or r1; r2 =
tr(A)
2
Answer a)
4
4
y =
1
4
y
(B.136)
let y = kert then y = rkert Also y = Ay implies rkert = A kert jA
4
4
1
4
Evaluating the determinants (4
r
1
0
r) (4
4
6
1
4 4 6
k1 = 1 then k2 =
k1
1
=
k2
2
4
2
4 4
k1 = 1 then
k1
=
k2
k1
k2
0
0
=
r
4
1
r
0
1
=
r)
4 = 0; r1 = 6 r2 = 2.
4
Find the eigen vectors for both roots as
4
rIj = 0
or
r
4
4
2
4
1
2
4
1
r
=0
k1
k2
(B.137)
0
0
=
k1
k2
=
0
0
; 2k1 + k2 = 0; if
k1
k2
=
0
0
; 2k1
2
when r1 = 6 Similarly when r2 = 2.
1
k1
=
2
k2
k2 = 2
1
when r2 = 2
2
0
0
2
4
or
1
2
k2 = 0; if
y 1 (t) =
k1
k2
e r1 t =
1
2
e6t
(B.138)
y 2 (t) =
k1
k2
e r2 t =
1
2
e2t
(B.139)
e6t + C2
1
2
e2t + y
(B.140)
y(t) = C1
1
2
De…nitising C1 and C2 constant require two initial conditions such as
initial position
y 1 (0)
y 2 (0)
=
2
3
; and initial speed
y 1 (1)
y 2 (1)
=
3
4
:y =
0
0
3. Solve the following system of equations and represent solutions in a phase diagram
a)
y1 =
2y1 + 2
(B.141)
y2 =
3y2 + 6
(B.142)
Answer: Steady state y1 = 1; and y2 = 2:
136
2
0
Here A =
2 r
0
0
3
0
3 r
;apply the above rule.
= (2 + r) (3 + r)
0 = 0; r1 =
2 r2 =
3.
y 1 (t) = C1 er1 t + y 1 = C1 e
2t
+1
(B.143)
y 1 (t) = C2 er1 t + y 1 = C2 e
3t
+2
(B.144)
use initial conditions such as y 1 (0) = 3 and y 2 (0) = 1=2 to de…nitise the values of C1 and C2 :
Study the motion of y1 and y2 for phase diagrams. This generates stable trajectory.
b)
2
0
Here A =
2
r
0
0
3
r
0
3
y1 = 2y1
2
(B.145)
y2 = 3y2
6
(B.146)
;apply the above rule.
= (2
r) (3
r)
0 = 0; r1 = 2 r2 = 3.
y 1 (t) = C1 er1 t + y 1 = C1 e2t + 1
(B.147)
y 1 (t) = C2 er1 t + y 1 = C2 e3t + 2
(B.148)
use initial conditions such as y 1 (0) = 3 and y 2 (0) = 1=2 to de…nitise the values of C1 and C2 :
Study the motion of y1 and y2 for phase diagrams. This give unstable trajectory.
c)
y1 = y2
2
(B.149)
y1
4
1
2
(B.150)
y2 =
Here A =
0
0
1
4
1
0
;apply the above rule.
r
1
= (0 r) (0 r) 41 = 0; r1 =
0 r
equilibrium as two roots are of opposite signs.
1
4
1
2
r2 =
1
2 ..
This gives saddle point
y 1 (t) = C1 er1 t + y 1 = C1 e2t + 1
(B.151)
y 1 (t) = C2 er1 t + y 1 = C2 e3t + 2
(B.152)
use initial conditions such as y 1 (0) = 3 and y 2 (0) = 1=2 to de…nitise the values of C1 and C2 :
Study the motion of y1 and y2 for phase diagrams. This give unstable trajectory.
d)
y1 =
137
y2 + 2
(B.153)
y2 = y1
0
1
Here A =
1
1
1
2
tr(A)r + jAj = 0 and r1; r2 =
p
3
2 i
y2 (t) = e
"
tr(A)
2
1
2
p
tr(A)2
p #
3
3
y1 (t) = e
A1 cos(
t) + A2 sin(
t) + 1
2
2
!
!
p
p !
p
p #
A1
3
3
3
3
A2
+
A2 cos
t +
A1 sin(
t) + 2
2
2
2
2
2
2
1
2t
1
2t
(B.154)
;apply the above rule.
This is a complex root case; by r2
r1; r2 =
y2 + 1
"
p
4 jAj
(B.155)
(B.156)
4. Apply above techniques to
a) Dornbusch model of exchange rate overshooting
mD =
m
(B.157)
ar + by
p=
(B.158)
ar + by
r =r +E e
(B.159)
e =E e
(B.160)
The exchange rate equation is obtained from these four equations as
e =
p =
y
D
p by m
+
a
a
r
(B.161)
yD
>0
(B.162)
yS
= u + v (e
p)
(B.163)
yS = y
p =
p =
p
e
!
=
v
1=a
(u + v (e
(B.164)
p)
y)
vp + ave + a (u
v
0
p
e
+
(B.165)
y)
(B.166)
a (u
by m
a
y)
r
(B.167)
p(t) = C1 er1 t + C2 er2 t + p
(B.168)
r1 + v
r2 + v
e(t) =
C1 er1 t +
C 2 e r2 t + e
v
v
(B.169)
138
p
e
Steady state is obtained when
!
=
0
0
; From the exchange rate equation given above
when e = 0 steady state price level is
p=m
by
ar
(B.170)
Similarly when p = 0 from the price equaution
e=p
1
(u
v
y)
(B.171)
For dynamics solve
pthe transitional dynamics
tr(A)
1
tr(A)2 4 jAj; tr(A) = (a11 + a22 ) ; jAj = (a11 a22 a12 a21 )
r1; r2 = 2
2
v
v
Here A =
; tr(A) =
v and jAj = av .
1=a
0
r
tr(A) 1 p
v 1
v
2
r1; r2 =
tr(A)
4 jAj =
( v)2 + 4
2
2
2
2
a
p
p
v
v
v
v
1
1
2
2
p(t) = C1 e( 2 + 2 ( v) +4 a )t + C2 e( 2 2 ( v) +4 a )t + p
v
e(t)
2
=
+
v
+
2
1
2
p
v)2 + 4 av + v
C1 e(
v
p
1
( v)2 + 4 av v
2
C 2 e(
v
(
v
2
v
2
+ 21
1
2
p
p
(
(
v)2 +4
v)2 +4
v
a
v
a
(B.172)
(B.173)
)t
)t + e
(B.174)
Given the initial conditions p(t = 0) and e(t = 0) the constant terms C1 and C2 can be evaluated.
For phase diagram from p =
vp + ave + a (u y) when p = 0; p = e + (u v y) , p rises above
p = 0 isocline and falls below it.
From e = ap + by a m r when e = 0; p =p =m
space.
b) Markov model of employment and Layo¤
et+1 = (1
by
=
(1
(B.175)
) ut
(B.176)
)
(1
et
ut
)
1. Apply r2
)r + (1
tr(A)r + jAj = 0 r2 (2
)r + (1
)=0
p
1
tr(A)2 4 jAj; r1; r2 = (2 2
r1; r2 = tr(A)
2
2 p
(2
)
1
r1; r2 =
( + )2
2
2
r1; r2 = 1
( + )
2
1
2(
now explain the diagram in (e, p)
) et + ut
ut+1 = et + (1
et+1
ut+1
ar
+ ); r1 = 1; r2 = (1
(B.177)
) (1
)
1
2
p
)
(2
= 0 ; r2
)2
4 (1
(2
);
)
By rule
yt = C1 r1t + C2 r2t + y
139
(B.178)
xt =
r1
a11
r2 a11
C1 r1t +
C2 r2t + x
a12
a12
et = C1 + C2 (1
xt =
C1
)
t
C2 (1
)
(B.179)
(B.180)
t
(B.181)
De…nitising the solution using initial conditions the time path of …nding jobs and unemployment are:
e0
u0
t
et =
+
(1
)
(B.182)
( + )
( + )
xt =
e0
u0
(1
( + )
( + )
)
t
(B.183)
c) Model of price war
yt+1
xt+1
r1 = 1; r2 = (1
yt+1 = yt
(yt
xt )
(B.184)
xt+1 = xt
(xt
yt )
(B.185)
(1
)
(1
=
)
yt
xt
)
yt = C1 + C2 (1
C1
xt =
)
C2 (1
(B.186)
t
)
t
(B.187)
De…nitising the solution using initial conditions the time path of …nding jobs and unemployment are:
y0 + x0
y0 x0
t
yt =
+
(1
)
(B.188)
2
2
y0 + x0
y0 x0
t
xt =
(1
)
(B.189)
2
2
d) Entry adjustment model
qD
p =
p =
(a + bp
N =
qS
mN )
(p
(B.190)
>0
c)
(B.191)
>0
(B.192)
Applying
y1 (t) = C1 er1 t + C2 er2 t + y 1
r1 a11
r2 a11
y2 (t) =
C 1 e r1 t +
C2 er2 t + y 2
a12
a12
(B.193)
(B.194)
Solution to this problem is:
p
N
!
=
b
m
0
140
p
N
+
a
c
(B.195)
P (t) = C1 er1 t + C2 er2 t + p
N (t) =
r1
b
m
r2
C1 er1 t +
b
m
(B.196)
C 2 e r2 t + N
(B.197)
Now draw the phase diagram for stability analysis.
Reference: Hoy et al. (2001) Mathematics for Economics, MIT Press.
Exercise 4: Stability Analysis
1. Solve the following system of di¤erential equations.
y = Ay
where a) A =
4
4
1
4
; b) A =
1
3
3
1
4
(B.198)
;c) A =
[Hint: for a second order di¤erence equation r2
p
1
tr(A)2 4 jAj]
2
2
2
5
4
tr(A)r + jAj = 0 ; or r1; r2 =
tr(A)
2
2. Solve the following system of equations and represent solutions in a phase diagram
a)
y1 =
2y1 + 2
y2 =
3y1 + 6
(B.199)
y1 =
2y1
(B.200)
b)
2
y2 = 3y1
6
(B.201)
y1 = y2
2
(B.202)
y1
4
1
2
(B.203)
y2 + 2
(B.204)
c)
y2 =
d)
y1 =
y2 = y1
y2 + 1
(B.205)
3. Apply above techniques to
a) Dornbusch model of exchange rate overshooting
e =E e
(B.206)
r =r +E e
(B.207)
141
mD =
m
ar + by
p=
(B.208)
ar + by
(B.209)
) et + ut
(B.210)
b) Markov model of employment and Layo¤
et+1 = (1
ut+1 = et + (1
) ut
(B.211)
yt+1 = yt
(yt
xt )
(B.212)
xt+1 = xt
(xt
yt )
(B.213)
c) Model of price war
d) Entry adjustment model
p =
p =
(a + bp
N =
(p
qD
qS
(B.214)
mN )
c)
>0
>0
(B.215)
(B.216)
References
[1] Hoy et al. (2001) Mathematics for Economics, MIT Press.
[2] Mankiw N.G. (1989) Real Business cycle: A New Keynesian Perspective, Journal of Economic
Perspectives, vol. 3, no. 3 pp. 79-90.
[3] Phelps E. S. (1968) Money wage dynamics and labour market equilibrium, Journal of Political
Economy, 76 , 678-711.
[4] Plosser Charles I (1989) Understanding Real Business Cycle, Journal of Economic Perspectives,
vol. 3, no. 3 pp. 51-77.
[5] Cooly Thomas F (1995) Frontiers of Business Cycle Research, Princeton.
[6] Romer D. (2006) Macroeconomics, McGraw Hill.
[7] Minford P. and D. Peel (2002) Advanced Macroeconomics: A Primer, Edward Elgar Publishing.
[8] Sorensen PB and H. Jl Whitta-Jacobsen (2010) Introducing Advanced Macroeconomics, McGraw Hill.
[9] Simon and Blume (1994) Mathematics for Economists, Norton.
[10] Shone Ronald (2002) .Economic Dynamics, Cambridge.
142
3
L3: New Classical Macro Models (Real Business Cycle)
Neither Keynesian nor the rational or adaptive expectation models include explicit optimisation by
households and …rms. Prices in those models are either sticky or have a very limited role in economic
allocation. New classical models have tried to overcome this short coming by introducing alternative
models in which demand and supply functions for goods and services are derived explicitly from the
optimising behaviour of economic agents as in a Walrasian general equilibrium system. These are
dynamic and stochastic general equilibrium models with perfect ‡exibility in prices. Recently more
development has taken place on decentralised applied general equilibrium economy with multiple
consumers, producers and traders.
Simple speci…cation of a real business cycle (RBC) model is similar to the perfect foresight models but it includes stochastic technology to explain macro ‡uctuations of output and employment
that is observed in real economy. These technological shocks a¤ect productivity and income and
result in intertemporal and intratemporally substitutions (Prescott and Kydland (1982), Prescott
(1986)) by households and …rms. Prices are ‡exible clear markets in Walrassian way. Demand
and supply in goods, factor and …nancial markets re‡ect optimising behaviour of households and
…rms. Stochastic process of technology or public policy such as the government spending causes
‡uctuations of output employment and prices around the trend. More recent versions of RBC
models includes non-Walrassian features, such as imperfect competition, externalities, assymetric
information, departure from rationality and failure of market to clear- while explaining economic
‡uctuations (Black (1995), Cooley (1995)).
Technical innovation (Shumpetarian) leads to a productivity shock, investments become profitable. Demand for investment goods rises along with output and interest rates and savings. Economy slows down with slow down in productivity. A simple RBC model can be illustrated as follows.
Output
1
Yt = Kt (At Lt )
0<
<1
(C.217)
Capital Accumulation
Kt = (1
) Kt
1
+ It
0<
<1
(C.218)
Market Clearing
Yt = Ct + It
(C.219)
Wage rate
wt = (1
) Kt (At Lt )
At
(C.220)
Interest rate
rt =
At Lt
Kt
1
use lower case letter for per capita variables and upper case for aggregate variable
Representative Consumer’s Problem
143
(C.221)
max
1
X
t
U (ct ; 1
lt )
(C.222)
t=0
subject to
ct + kt+1 = wt lt + (1 + rt ) kt
(C.223)
Lagrangian
L=
1
X
t
U (ct ; 1
lt ) +
t=0
1
X
t
[wt lt + (1 + rt ) kt
ct
kt+1 ]
(C.224)
t=0
First order conditions wrt ct ; lt ;and
t
@L
=
@ct
@L
=
@lt
t
t
Uc (ct ; 1
Ul (ct ; 1
@L
=
@kt+1
t
+
lt )
t
lt ) +
t wt
t+1
=0
(C.225)
=0
(C.226)
(1 + rt ) = 0
(C.227)
Plugging the C.225 into C.227
Uc (ct ; 1
lt ) = Uc (ct+1 ; 1
lt+1 ) (1 + rt+1 )
(C.228)
From C.225 and C.226
1
lt )
=
lt )
wt
Uc (ct ; 1
Ul (ct ; 1
wt Uc (ct ; 1
(C.229)
lt ) = Ul (ct ; 1
lt )
(C.230)
Technology of t+1 period is unknown at period t; that is At+1 is uncertain. At+1 is a stochastic
process. With this C.228 should be written with expectation (Et ) in the right hand side
Uc (ct ; 1
lt ) = Et Uc (ct+1 ; 1
lt+1 ) (1 + rt+1 )
(C.231)
RBC model with a Cobb-Douglas preferences:
U (ct ; 1
lt ) = ln ct + b ln (1
L = ln ct + b ln (1
lt )
lt ) + [wlt
1
@L
=
@ct
ct
@L
b
=
+
@lt
1 lt
144
t
ct ] ; p = 1
=0
tw
b>0
=0
(C.232)
(C.233)
(C.234)
(C.235)
@L
= wlt
@ t
ct
1
=
lt
ct = 0
(C.236)
wt
b
(C.237)
1
1 + rt+1
= Et
ct
ct+1
(C.238)
from the interest rate then
1 + rt+1 = 1
+
Yt+1
Kt+1
(C.239)
=1
(C.240)
Putting this back in C.238
Yt+1
1
Kt+1
= Et
ct
ct+1
Optimal consumption and investment rules:
with representative agents and 100% depreciation per capita and aggregate variables are the
same
Ct + Kt+1 = Yt =)
Yt
=
Ct
Kt+1
Ct
=)
Et 1 +
Kt+2
Ct+1
1+
Yt+1
=
Ct+1
1+
Kt+2
Ct+1
(C.241)
Plugging this in C.240
Kt+1
=
Ct
Let Zt =
(C.242)
Kt+1
Ct
Zt =
+
Et Zt+1
(C.243)
Iterate forward
Zt
=
=
+
Et Zt+1 =
2
+(
) +(
Zt =
+(
Zt =
Ct = (1
+
2
) (
+
2
) +(
=
1
(
+
Et Zt+3 ) :::
3
) + :::
(C.244)
(C.245)
(C.246)
Kt+1
Ct
) Kt+1 = (1
(C.247)
) (Yt
(Ct + Kt+1 ) = Kt+1
145
Et Zt+2 )
Ct )
(C.248)
(C.249)
Kt+1 =
Ct = (1
Yt
(C.250)
) Kt+1 = (1
Ct = (1
Equilibrium employment use wt = (1
) (Yt
(C.252)
Lt =
At ; ; Ct = (1
ct
1 lt
=
wt
b
(C.253)
b
(1
)
) + (1
(1
) Yt and
Yt
)L
t
(1
) Yt
=
Lt
1
(C.251)
) Yt
) Kt (At Lt )
(1
Ct )
(C.254)
)b
Reduced form of output and consumption
ln Yt = (1
) ln At +
ln Kt + (1
ln Kt+1 = ln
) ln Lt
(C.255)
+ ln Yt
ln Kt = ln
+ ln Yt
(C.256)
(C.257)
1
BY substitution
ln Yt = (1
L=
(1
(1
)
)+(1
)b
and
) ln At +
=
ln
+ (1
ln Yt =
+
[ln
+ ln Yt
1]
+ (1
) ln L
(C.258)
) ln L
ln Yt
+ (1
1
) ln At
(C.259)
fAt g is an autoregressive process of order 1, AR(1); Using lag operator B
(1
B) yt =
yt =
De…ne "t = (1
) ln At and vt =
+ (1
(1
) ln At ; yt = ln Yt
+
)
(1
(1
) ln At
B)
(C.260)
(C.261)
"t
(1
B)
yt =
(1
Output series is thus divided into permanent
h
)
(1
vt = avt
1
+ vt
)
i
+ "t
(C.262)
and transitory components [vt ].
(C.263)
Productivity shock
the sequence of output fyt g though eventually it reverts to its
h in‡uences
i
:
(1
)
unconditional mean
146
3.0.1
Linear RBC Model
Yt = rKt + Ut
(C.264)
Yt = Ct + It
(C.265)
Process fUt g is shock to output.
It = Kt+1
(1
) Kt
(C.266)
Ct + It = rKt + Ut
Ct + Kt+1
(C.267)
Kt = rKt + Ut ;
=0
Ct + Kt+1 = (1 + r) Kt + Ut
(C.268)
(C.269)
Linear RBC Model
Uc (ct ) =
with
(1 + rt+1 ) Et Uc (ct+1 )
(C.270)
(1 + rt+1 ) = 1
Uc (ct ) = Et Uc (ct+1 )
When utility function is a linear in C as U = C
Uc (ct ) = 1 2bCt and Uc (Ct+1 ) = 1 2bCt+1
bC
(C.271)
2
optimal consumption is C =
Ct = Et Ct+1
1
2b
and
(C.272)
Future consumption is the best predictor of the current consumption. Let there be a shock to
the future consumption as
Ct+1 = Ct +
(C.273)
t+1
E t+1 = 0; and and constant variance 2 :
Present value of consumption equals present value of income
Ct +
Ct+1
Ct+1
Ct+2
Ut+1
Ut+1
Ut+2
+
+
+ ::: = (1 + r) Kt + Ut +
+
+
+ ::: (C.274)
1 + r (1 + r)2 (1 + r)3
1 + r (1 + r)2 (1 + r)3
Present value of consumption equals present value of income
Ct +
Ct+1
Et Ct+1
Ct +
+
Ct+1
Ct+2
Ut+1
Ut+1
Ut+2
+
+ ::: = (1 + r) Kt + Ut +
+
+
+ :::
(1 + r)2
(1 + r)3
1+r
(1 + r)2
(1 + r)3
(C .2 7 5 )
+
Et Ct+1
Et Ct+2
Et Ut+1
Et Ut+1
Et Ut+2
+
+ ::: = (1 + r) Kt + Ut +
+
+
+ :::
(1 + r)2
(1 + r)3
1+r
(1 + r)2
(1 + r)3
(C .2 7 6 )
1+r
1+r
Ct = Et Ct+1 = Et Ct+1 = ::::
Ct
"
1
1+
1
+
1+r
(1 + r)2
1
+
(1 + r)3
+ :::
#
= (1 + r) Kt + Ut +
147
(C .2 7 7 )
Ut+1
1+r
+
Ut+1
Ut+2
+
+ :::
(1 + r)2
(1 + r)3
(C .2 7 8 )
r
Ct =
1+r
"
(1 + r) Kt + Ut +
Ut+1
+
1+r
Ut+1
Ut+2
+
+ :::
(1 + r)2
(1 + r)3
#
r
[(1 + r) Kt + Ut ]
1+r
Derivation of optimal investment rule
Ct =
It = rKt + Ut
It = rKt + Ut
It = Ut
From It = Kt+1
(1
) Kt when
(C .2 7 9 )
(C.280)
Ct
(C.281)
r
[(1 + r) Kt + Ut ]
1+r
(C.282)
r
Ut
Ut =
1+r
1+r
(C.283)
=0
Kt+1 = Kt +
Ut
1+r
(C.284)
Process for changes in output
Yt+1
Yt+1
Yt = r (Kt+1
Yt = r Kt +
Kt ) + Ut+1
Ut
1+r
Ut
rKt + Ut+1
(C.285)
Ut
Ut
1+r
Change in output depends purely on current and previous shocks.
Yt+1
Yt = Ut+1
(C.286)
(C.287)
Derive the optimal consumption and investment rules and output process when Ut = Ut+1 +
et
et
3.0.2
1:
New Keynesian Model in relation to the RBC models
The RBC models are the better starting point for the New Keynesian models.
Standard RBC
models have labour-leisure choice and preference shocks in the demand side. Households get utility
from consumption and leisure:
Uth = U Cth ; lth
The optimal conditions are expressed by an Euler equation as:
148
(C.288)
h
Uc;t
= Rt Et [UC;t+1 ]
(C.289)
This can be further written as Rewrite Euler equations as: 1 = Rt Et [Ut+1 ] ; Ut+1 =
h
Uc;t+1
:
h
Uc;t
MRS between labour and leisure (labour supply):
h
lc;t
Lhc;t
= h =
h
Uc;t
Uc;t
Wt
Pt
It is a full employment model in which the total time is allocated between the labour supply
and leisure.
h
Lht = Lt
lth
(C.290)
On the supply side output is produced using technology, labour and capital inputs and could
be devided between the whole sale and retail output as:
W
Yi;t
= Fi [Ai;t ; Li;t ; Ki;t
1]
(C.291)
Retail output:
Yi;t = (1
W
c) Yi;t
; 0<c<1
(C.292)
Marginal productivity of baour equals to the real wage rate:
Wt
PtW
Fi;Lt =
Pt
Pt
Similarly the marginal productivity of capital equals to the user cost of capital as:
W
Pt+1
Fi;K;t+1 = Rt
Pt+1
1+
Production is subject to a price markup rules:
1
PtW
1
Market clearing is essential to obtain general equilibrium in the economy:
Pt =
Yi;t =
H
X
h
Ci;t
+ Ii;t + Gi;t
(C.293)
h=1
Capital accumulation:
Ii;t = Ki;t
(1
149
) Ki;t
1
(C.294)
In this simple version public sector balances with amount of government spending to be equal
to the lumpsum taxes:
Gt = Tt
(C.295)
It is important to decide on the functional forms in order to be able to compute equilibrium in
such economy.
F (At ; Lt ; Kt
1)
= (At Lt ) Kt1
YtW
Lt
FL (:) =
) YtW
(1
FK (:) =
1
Kt
1
The ‡uctions around the trend occur becuause of shockes to the technology and public spending
as:
log At
log At =
A (log At 1
log At ) +
A
t
log Gt
log Gt =
G (log Gt 1
log Gt ) +
G
t
A CES utility function is derived as:
(1
)
Cth
Uth =
1
c
lth
1
c
Marginal utility of consumption:
(
(1
h
Uc;t
= (1
)
) Cth
c
1
)
1
( c
lth
1)
Marginal utility of leisure (lalbour supply)
h
Ul;t
=
(1
Cth
)
(
c
1
)
1
Lht
(
c
1)
t
The ratio of marginal utility of leisure and consumption should equal the real wage rate:
h
UL;t
h
Uc;t
=
Wt
Pt
The Basic RBC model equilibrium means …nding the solution of endodengous variables Uth ; Cth ; lth ; Lht ; Kt ; It ; Pt ; PtW ; R
in terms of the model parameters ( ;
A;
c;
; ; ) and the technological shocks
150
Here positive technological shock is raising output, capital accumulation and then consumption.
Impact of this on interest and output is short-lived as households manage to intertemporally balance
their income and expenditures.
A positive investment cost modi…es the capital accumulation equation as:
Ki;t = (1
) Ki;t
1
+ (1
S(Xi;t ) Ii;t ; Xi;t =
Ii;t
Ii;t 1
(C.296)
See programmes Ramsey_demo.mod; RBC-Summer.mod and RBC_Invcost.mod, graphs_rbc.m
from the CIMS Univeristy of Surrey.
A typical parameterisation
Parameters gy
c
values
3.1
0.2
0.7
1/
c
0.7
0.99
0.025
Rational Expectation
Rational Expectation
Aggregate demand:
151
2
x
2
e
A
G
.68
0.7
0.7
152
yt
y = vt
(rt
r) ; vt v N 0;
2
v
(C.297)
Real interest rate:
e
t+1
rt = it
(C.298)
Aggregate supply (price formation):
t
e
t+1
=
y) + st ; vt v N 0;
+ (yt
2
s
(C.299)
Monitory policy rule:
rt = r + h (
) + b (yt
t
y)
(C.300)
Expectation
e
t;t 1
= E [ t =It
1]
(C.301)
Underlying assumptions on demand and supply shocks
E [vt ] = 0; E vt2 =
2
v;
E [vt vs ] = 0; E [st ] = 0; E s2t =
2
s;
E [st ss ] = 0; E [vt ss ] = 0;
Thee steps of solving a rational expectation model
1. Solve for endogenous variables as a function of exogenous variables and parameters;
e
2. Find the solutions for yt;t
3. Insert values of
e
yt;t
1
and
1
and
e
t;t 1
e
t;t 1
by taking the conditional expectation
into the expression found in step 1.
Insert (C.299) into (C.297) to get
yt = y + vt
h
e
t;t 1
e
+ b yt;t
1
y
e
t;t 1
e
+ b yt;t
1
y
(C.302)
substitute (C.297) into (C.299) to get
t
=
e
t+1
+ vt
h
Expressions (C.302) and (C.303) have yt and
t
+ st
(C.303)
e
in terms of expected values yt;t
1
and
e
t+1
Take expectations of these two equations.
e
yt;t
1
=y
h
e
t;t 1
e
+ b yt;t
1
y
(C.304)
substitute (C.302) into (C.303) to get
t
=
e
t+1
h
e
t;t 1
e
+ b yt;t
1
153
y
=h
e
t;t 1
e
+ b yt;t
1
y =0
e
yt;t
=y
(C.305)
=
(C.306)
1
e
t;t 1
Substituting (C.305) and (C.306) into (C.302) and (C.303)
yt = y + vt
t
Policy parameter
=
(C.307)
+ vt + st
(C.308)
does not enter in output yt equation; policy does not a¤ect the real output.
Rearrange (Q.1514)
1
yt = y +
e
t+1
t
+
1
st
(C.309)
Here the monetary policy can have e¤ects in the real output only by in‡uencing the unanticipated
in‡ation
t
e
t+1
Systematic monetary policy rules cannot generate surprises and hence cannot
cause in‡ation and output away from the natural rate of output.
3.1.1
Rational Expectation: Another example
Expected in‡ation next period (t + 1) based on information at period t depends on di¤erences on
expected and actual prices
Et
t+1
=Et pt
pt
(C.310)
a0 > 0 a1 > 0
(C.311)
Demand
ytd = a0 + a1 (mt
pt ) + t ;
yts = yn + b1 pt
Et
1
pt + vt
a1 > 0
(C.312)
Demands equals supply in equilibrium
ytd = yts = yt
t
N 0;
2
t
154
N 0;
(C.313)
2
(C.314)
First solve two endogenous variables yt pt
taking expectations
Et p t
exogenously. Find the
reduced form..
yt + a1 pt = a0 + a1 mt + t ;
(C.315)
yt
(C.316)
b1 pt = yn
b1Et 1 pt + vt
In matrix notation
1
a1
1
yt
pt
b1
!
!
yt
1
a1
=
!
pt
1
b1
a0 + a1 mt +
=
!
yn
1
t
b1Et 1 pt + vt
a0 + a1 mt +
yn
!
t
b1Et 1 pt + vt
!
(C.317)
(C.318)
Easy to solve this by Cramer’s rule
a0 + a1 mt +
yt =
yn
b1Et 1 pt + vt
1
b1
(C.319)
a1
1
pt =
a1
t
b1
1
a0 + a1 mt +
1
yn
t
b1Et 1 pt + vt
1
1
(C.320)
a1
b1
Evaluate the determinants
yt =
b1 (a0 + a1 mt + t ) a1 yn
( b1 a1 )
pt =
yn
b1Et 1 pt + vt
( b1
b1Et 1 pt + vt
(a0 + a1 mt + t )
a1 )
(C.321)
(C.322)
Upon further simpli…cation
yt =
a b mt
b1 a0 + a1 yn
+ 1 1
a1 + b1
a1 + b1
pt =
a1 b1Et 1 pt
b t + a1 vt
+ 1
a1 + b1
a1 + b1
b1Et 1 pt
a 0 yn
a mt
vt
t
+
+ 1
+
a1 + b1
a1 + b1
a1 + b1
a1 + b1
155
(C.323)
(C.324)
How to form the expectation
Et
Simpli…cation
Et
Et
1
1
pt =
t
1
Et
b1Et 1 pt
a E mt
a0 yn
+
+ 1 t 1
+
a1 + b1
a1 + b1
a1 + b1
= 0 and
pt
pt ? .. For this take conditional expectation of pt at (t
1
Et
1
yt =
Et
1
b1Et 1 pt
=Et
a1 + b1
1
b1
a0 yn
+
a 1 + b1
a1 + b1
pt =
pt =
a0
yn
a1
a1 b1
a1 + b1
a0
yn
=
+Et
a0
pt
1
a1 (mt mt
a1 + b1
=
yn
1)
yn
a1
+
b1
t
=
1
+
t
+
Et
Use the money supply rules mt
yn
Et
1
vt
(C.325)
a1 )
1
a E mt
a0 yn
+ 1 t 1
a1 + b1
a1 + b1
(C.326)
mt
(C.327)
1
mt
mt
mt
+
b1
+ a1 vt
a1 + b1
t
(C.328)
vt
a1 mt
t
+
a1 + b1
( b1 a1 )
a1 mt + b1Et 1 mt
vt
t
+
a1 + b1
a1 + b1
(C.331)
1
pt
+
b1
(C.329)
(C.330)
= pt
1
+ b1Et 2 mt
a1 + b1
1
m t Et
a1 + b1
1
+
1
t + a1 vt
a1 + b1
a1 mt
mt
+Et
a1
a1 b1 mt Et
a1 + b1
a1
a0
a0
+Et
a1
Iterate backward to get in‡ation as
=
t
pt could be put in the reduced form equations.
yt = yn +
t
b1
a1 + b1
pt 1
1
b1 a0 + a1 yn
a b mt
+ 1 1
a1 + b1
a1 + b1
pt =
1
( b1
vt = 0;
Et
This value of
Et
1)
2
=
mt
1
or
Et
a1
b1
(
+
+
a1 + b1
a1 + b1
t
+
1
(
1
t 1)
(vt
a 1 + b1
t
mt
vt
a1 + b1
t 1
+
Et
2
mt
t 1)
(vt
a1 + b1
vt
1
1
(C.332)
vt
1)
(C.333)
=
1)
(C.334)
Rational Expectation: An example
t
If there is no shock then
Et
1
=
t
+
(
t
= 0 and
t 1)
(vt
a1 + b1
Et
1
in‡ation simply equals money growth
156
vt
v t = 0 ; Et
1)
1
t 1
(C.335)
= 0 and
Et
1
vt
1
= 0 Then
=
t
(C.336)
Conclusion of this model: only positive shocks in demand or supply in‡uence the level of prices
or output.
Backward looking expectation Start from a model with the backward looking expectation
Aggregate demand:
y = vt
yt
r) ; vt v N 0;
(rt
2
v
(C.337)
Real interest rate:
e
t+1
rt = it
(C.338)
Aggregate supply (price formation):
t
e
t+1
=
+ (yt
y) + st ; vt v N 0;
2
s
(C.339)
Monitory policy rule:
e
t+1
it = r +
+ h(
) + b (yt
t
y)
(C.340)
Expectation
e
t+1
=
(C.341)
t 1
Solution of the model
Substitute (C.338), (C.340) and (C.341) in (C.337) (assuming vt = st = 0 )
(rt
r) (1 + b) = h (
);
t
yt
where
=
1+
(rt
h( t
)
(1+ b)
r) =
this yields
h( t
)
=
(1 + b)
y=
h(
)
t
(C.342)
:Substitute (C.341) into (C.341)
t
=
t 1
+ (yt
y)
(C.343)
Insert (C.343) into (C.342) to get
t
=
t 1
+
(
t
t
) =)
=
1
1+
=
t
(
t 1
t 1
) ; =)
157
+
t
(
) and this yields
t
=
(
t 1
)
(C.344)
where
=
1
1+
Here (C.344) is the …rst order di¤erence equation in
target of the central bank
the time path of
t
This is a converging sequence since
t
.Given initial in‡ation
e
t+1
=
t
t 1
Compute time paths of
=
t
(
t
=
+
0<
< 1 for t = 0; 1; 2; ::::::
t ; rt ; yt ; it
t 1
)
0
and in‡ation
is given by
t
(
)
0
(C.345)
t
t!1
In‡ation forecast error for backward looking in‡ationary expectation is:
t
0
(
)=
0
t 1
(
0
!
)(
.
1)
(C.346)
and plot them into a …gure.
Q4. This exercise relates to adaptive and partial adjustment and combinations of these two.
First consider a adaptive expectation model:
Let yt be the growth rate and xt be the optimal long run equilibrium interest rate
yt = b0 + b1 xt + t ;
(C.347)
Adaptive expectation
xt
xt
1
=
xt
xt
1
(C.348)
Prove that this results into an autoregressive process of order 1 for yt .
Then consider the partial adjustment model as:
Let desired long run growth rate of economy be yt and that depend on a number of explanatory
variables as
yt =
0
+
1
xt +
(C.349)
t
Partial adjustment hypothesis emplies
yt
yt
1
=
(yt
yt
1)
(C.350)
Derive the partial adjustment model from using these two equations.
Combine adaptive and partial adjustment elements of above two models to derive a model when
both yt and xt are not observable as in
yt =
0
+
1
158
xt +
t
(C.351)
with
xt
xt
yt
yt
1
=
xt
=
1
xt
(yt
yt
1
(C.352)
1)
(C.353)
Answer
From growth equation
xt =
xt
1
=
1
(yt
b1
1
(yt
b1
b0
t)
b0
1
(C.354)
t 1)
(C.355)
substitute these values in adaptive expectation equation
xt = xt + (1
1
(yt
b1
b0
t)
) xt
= xt + (1
1
(yt
b1
)
(C.356)
1
b0
1
t 1)
(C.357)
t
(C.358)
now by reorganisation:
yt = b1 xt + (1
yt = b1 xt + (1
) (yt
) (yt
b0
1
1)
t 1)
+ b0 +
(1
) (b0 +
t 1)
+ b0 +
xt +
t
into partial adjustment equation
yt
1)
(C.360)
) yt
1
(C.361)
t
(C.359)
This is clearly an autoregressive equation.
For partial adjustment substitute yt =
yt
yt
0
+
1
=
1
(yt
yt = yt + (1
yt =
yt =
0
0
+
+ (1
1
xt +
) yt
159
+ (1
t
1
+
) yt
1
xt +
1
(C.362)
t
(C.363)
Combination of both adaptive and partial adjustment requires using both of the above solutions
yt =
0
+
1
xt +
(C.364)
t
with
xt
xt
yt
yt
1
1
=
=
xt
xt
(yt
yt
1
(C.365)
1)
(C.366)
From partial adjustment we had
yt =
0
+ (1
) yt
1
+
1
xt +
(C.367)
t
from adaptive expectation we had
yt = b1 xt + (1
) (yt
1)
(1
) (b0 +
t 1)
+ b0 +
(C.368)
t
lag it by one period then plug into the former equation
yt
yt =
3.2
0
+(1
1
= b1 xt
) [b1 xt
1
1
+ (1
+ (1
) (yt
) (yt
2)
2)
(1
(1
) (b0 +
) (b0 +
t 2)
t 2)
+ b0 +
+ b0 +
(C.369)
t 1
t 1 ]+
1
xt +
t
(C.370)
Supply Side and Rational Expectation
Keynesian economists argue that increase in demand has real e¤ect on output and employment
because of rigidity in prices and wages in the short run. Increase in the aggregate demand either by
increase in the government spending or by a reduction in the interest rate (increase in money supply)
would have permanent impacts on output and employment. The price level would not increase
when an economy is below full employment. In contrast to this, the classical or the new classical
proposition remains that the prices and wages are perfectly ‡exible and economy is always in full
equilibrium (Kydland and Prescott (1982)). Consequently it is impossible to arti…cially increase real
output by increasing demand. Real drivers of the economy are capital accumulation and increase
in human capital and increase in work hours and technological progress. Monetary policy is super
neutral. Price system that guarantees general equilibrium in goods and factor markets matter for
the e¢ cient allocation of resources (Bhattarai and Whalley (2000)). Under the rational expectation,
workers are fully informed, nominal wage rate rises according to the expected in‡ation. Workers
160
demand higher wage rate to compensate fully for higher anticipated changes in prices. Thus there
is no real impact of an increase in demand even in the short run as it is anticipated by workers.
Only unanticipated policy measures can have real impacts as explained above in the short run. New
Keynesian synthesis …nds a more realistic middle path between the Keynesian and real business
cycle schools. These features rest on the monopolistic competition and staggering wage contracts
(Taylor (1972)). Firms with market power under the monopolistically competitive markets are able
to absorb demand shocks (Dixon (1988), Blanchard and Kiyotaki (1986), Rankin (1992)). These
issues are further assessed in Angelopoulou and Gibson (2009), Arnold, Brys, Heady, Schwellnus
and Vartia (2011), Gemmell, Kneller and Sanz (2011), Arestis, Chortareas and Tsoukalas (2010),
Beetsma and Giuliodori (2011), Bean (2009), Boinet and Martin (2008), Gali and Monacelli (2005),
Johnson (2009), Kirsanova , Leith and Wren-Lewis ( 2009), Monacelli and Perotti (2010), Nelson
(2009), Fisher and Ryan (2010) in recent years. Higher aggregate demand puts upward pressure in
prices and …rms can reduce their markup without altering market prices. Additional workers could
be hired to supply additional output without changing prices when there is a pool of unemployed
workers. Thus an expansionary monetary policy can raise the level of output and employment in
the economy in the short run though the economy tends to return to its natural rate in the long
run.
3.3
Aggregate demand and aggregate supply model
Here is a popular version of Lucas (1973) aggregate supply aggregate demand function model
(Taylor (1973), Woodford and Taylor (1999), Sorensen and Whitta-Jacobsen (2010)). Real interest
rate (rt ) from the Fisher equation is the nominal interest rate (ipt ) adjusted for the risk ( t ) and
the expected in‡ation
e
t+1
as:
rt = ipt +
t
e
t+1
(C.371)
Aggregate demand is determined by the …scal policy shock (gt ) and monetary policy (rt
r)
and the demand shock (vt )
yt
y=
1
(gt
g)
2
(rt
r) + vt ;
vt v N 0;
2
v
;r = r +
(C.372)
Nominal interest rate is set by the monetary authority following a policy rule of the form:
ipt = r +
e
t+1
+ h(
t
) + b (yt
y)
The aggregate supply function with the supply shock (st ) is given by:
161
(C.373)
t
=
e
t+1
y) + st ; st v N 0;
+ (yt
2
s
(C.374)
In‡ation expectation:
e
t
=
(C.375)
t 1
Derivation of aggregate demand just requires using the Fisher equation and the interest rate rule
in the demand function. With some rearrangement this generates the aggregate demand function
as:
yt
yt
y=
y=
2h
1+
(
2b
t)
(
t)
+ zt ;
=
+ zt
(C.376)
2h
1+
(C.377)
2b
Where zt term includes …scal policy shock (gt ), risks ( t ) and random shocks (vt )
vt
2
1
)+
(gt g)
(
(C.378)
1 + 2b
1 + 2b t
1 + 2b
Aggregate demand is downward slopping; higher rate of in‡ation requires central bank to inzt =
crease the interest rate, that raises the cost of capital, thus causes lower investment and hence lower
output. The term zt includes …scal policy shock (gt ), risks ( t ) and random shocks (vt ) : Aggregate
supply function is derived by putting the in‡ation expectation into the supply function as:
t
=
t 1
+ (yt
y) + st
(C.379)
It is upward slopping; larger output requires employers more people, that lowers the productivity
of labour and the cost rises and in‡ation has to rise. Term st includes trade, exchange rate,
technology or other shocks.
De…ne deviation from the steady state as bt =
t
and
further shocks zt = 0 and st = 0. Then the aggregate demand is:
The aggregate supply:
bt+1 =
1
ybt = yt
y when there are no
ybt+1
bt+1 = bt + ybt+1
Output and prices in the equilibrium can be obtained by solving these equations:
162
(C.380)
(C.381)
1
1
ybt+1 =
bt+1 = bt + (
ybt + ybt+1 =) ybt+1 =
bt+1 ) =) bt+1 =
1
1+
ybt =) ybt+1 = ybt
1
1+
bt =) bt+1 = bt
Solutions of the …rst order di¤erence equations:
Since 1 <
=
The parameters
1
1+
;
t
ybt = yb0
t
and bt = b0
for t = 0; 1; 2; :::::
(C.383)
(C.384)
< 1 both ybt and bt converge to the steady state values y and
and
(C.382)
.
could be calibrated from the time series to study the impulse
responses from de…cit when shocks zt 6= 0 and st 6= 0
yt
y=
t)
AD :
bt =
1
AS :
ybt =
(zt
ybt +
AS :
1
1+
ybt =
1
bt = bt
AS :
AS :
+ zt =) ybt =
(
ybt
1
ybt ) =
ybt
1
(zt
1
(zt
ybt = zt
1
+
+
(C.385)
ybt )
(C.386)
+ ybt + st
ybt
1
zt
1
1+
(zt
bt + zt
1
1)
+ ybt
(zt
zt
(C.387)
+ ybt + st
1)
st
1
zt
(C.388)
1)
1+
st
(C.389)
st
(C.390)
(C.391)
Similar process follows for the derivation of the price level:
bt = bt
bt =
1
+ ybt + st = bt
1
1+
bt
1
+
1
1+
163
+ (
zt +
bt + zt ) + st
1
1+
st
(C.392)
(C.393)
bt =
bt
1
+
zt + st
(C.394)
Thus solutions of the dynamic aggregate demand and aggregate supply model provides us with
a …rst order autoregressive time path of output ybt and in‡ation bt which are also in‡uenced by
demand and supply shocks, zt and st . Taking note of the wage in‡ation dynamics, price formation
and rational expectations in the form of Phillips (1958), Phelps (1968), Lucas (1973), Mankiw
(1989) Sargent and Wallace (1975) , Calvo (1983), Taylor (1987) , Dri¢ ll and Schultz (1992),
Wickens (1995), Minford and Peel (2002) the DSGE models of Prescott (1986), Uhling (1995),
Smet and Wouters (2003), Nelson (2009), Iacoviello and Neri (2010) generate impulse responses to
assess impact of demand, supply or TFP shocks in the economy.
3.3.1
Estimations
Estimates of above equations using the quarterly data is as given in the following table. Autoregressive terms are signi…cant for both growth and in‡ation equations; intercept terms represent …scal,
monetary and trade factors and the shocks to demand and supply.
Table 46: AR(1) model of growth rate and in‡ation in UK
Growth equation
In‡ation equation
Coe¢ cient
T-value
Coe¢ cient
T-value
Intercept
0.410
2.93
0.172
1.19
AR(1) term
0.817
18.7
0.972
54.7
R
2
0.67
0.95
381( 0.00)
2292(0.00)
DW
1.97
1.02
2
87.1(0.00)
102.6(0.00)
F
N
176:q11967-q12011
176:q11967-q12011
These results provide good empirical support for the theoretical derivations of the demand and
supply models as above. Thus 82 percent of the growth rate and 97 percent of in‡ation are persistent
in the UK economy.
3.4
Trade Policy: Small Open Economy Macro Model
Policy options available to policy makers are often studied using a small open economy trade model.
In addition to the standard classical Ricardian and neoclassical theories of trade, contributions of
164
Mundell (1962), Dornbusch (1976), Krugman (1979), Taylor (1995) and Gali and Monacelli (2005)
focus on basics of a small open economy macro model. Mundell-Fleming model opens up the basic
Keynesian model for trade including equations for net exports (N X) that depend on nominal
exchange rates (e).
N X = a0
a1 e
(C.395)
Next exports are larger when the currency depreciates.
In simple formulation total output of an economy produced from employing labour and capital
can either be consumed (C) domestically or exported (E) as:
Y = f K; L = C + E
(C.396)
Exports depends on exchange rate (e), domestic price (P ) and foreign price for domestic goods P
and the elasticity of exports ( )
E = E0 e
P
P
(C.397)
Imports depends on exchange rate (e), domestic price (P ) and price of imported goods (Pm ) and
the elasticity of exports ( )
C
P
= K0 e m
M
P
(C.398)
Resource balance with foreign lending (borrowing) B
P Y + eB = P C + ePm M
(C.399)
Depreciation lowers the foreign price of domestic goods P , it raises supply of exports (E), it
reduces the amount of imports (M ) and raises the production of import substitute goods. Thus
both domestic and foreign demand for home products rise. Thus depreciation is expansionary when
the elasticity of exports to the exchange rate is higher than the elasticity of imports (MarshallLerner condition). This component of growth was positive in UK during this recession. Changes in
the capital account balance is given by the excess of the value imports over the value of exports as:
eB = ePm M
PE
(C.400)
Variables of this sub-model C; E; M; P; e depend on the parameters of the model E0 ; K0 ; ; ;
K; L; Y ; ; B and Pm .
165
Redistribution impacts of depreciation depends on the composition exports and imports. Dynamics of exchange rates and prices under rational expectation in the Mundell-Fleming framework
as …rst formulated in the exchange rate overshooting model of Dornbusch (1979) with uncovered
interest parity conditions with following four equations.
money demand function:
mD =
(C.401)
ar + by
money market equilibrium:
m
p=
ar + by
(C.402)
interest rate parity:
r =r +E e
(C.403)
e =E e
(C.404)
change in the exchange rate:
The exchange rate equation is obtained from these four equations as (Hoy et al. (2001)) :
e =
p by m
+
a
a
r
(C.405)
>0
(C.406)
In‡ation is positively related to the excess demand as:
p =
yD
yS ;
Demand is determined by the real exchange rate and the demand factors:
y D = u + v (e
p)
(C.407)
Assuming supply to be at the steady state:
yS = y
p =
(C.408)
vp + ave + a (u
y)
(C.409)
Thus the system of in‡ation and exchange rates relate to …scal and monetary choices:
p
e
!
=
"
v
v
1=a
0
#
p
e
!
+
Time path of price and exchange rate thus is given by:
166
a (u
y)
by m
a
r
!
(C.410)
p(t) = C1 exp
e(t) =
Steady state is obtained when
1
1t
+C2 exp
2t
+p
(C.411)
+ v
2 v
C1 exp 1 t +
C2 exp 2 t +e
(C.412)
v
v
!
!
p
0
=
; From the exchange rate equation given above
e
0
when e = 0 steady state price level is
p=m
by + ar
(C.413)
Similarly when p = 0 from the price equation
e=p
1
(u
v
y)
(C.414)
Figure 6:
For dynamics solve the transitional dynamics
p
tr(A)
1
tr(A)2 4 jAj; tr(A) = (a11 + a22 ) ; jAj = (a11 a22
1; 2 =
2
2
a12 a21 )
The roots of the equation depend on the behavioral parameters and these in‡uence the path of
price level and the exchange rate in the economy which a¤ect both demand and supply sides of the
economy.
Here A =
"
v
v
1=a
0
1;
2
#
; tr(A) =
tr(A)
=
2
v and jAj =
1p
tr(A)2
2
4 jAj =
167
v
a .
v
2
1
2
r
(
v)2 + 4
v
a
(C.415)
v
p(t) = C1 exp(
2
v
e(t)
+
2
=
+ 12
v
+
2
p
1
2
v)2 +4
(
p
v
a
)t +C exp(
2
v
2
1
2
p
v)2 + 4 av + v
C1 exp(
v
p
1
( v)2 + 4 av v
2
C2 exp(
v
(
v)2 +4
(
v
2
+ 21
v
2
1
2
p
p
(
(
v
a
)t +p
v)2 +4
v)2 +4
(C.416)
v
a
v
a
)t
)t +e
(C.417)
Given the initial conditions p(t = 0) and e(t = 0) the constant terms C1 and C2 can be
evaluated. Qualitatively above results could be presented using a phase diagram in in (e, p) space;
from p =
vp + ave + a (u
= 0; p = e +
(u y)
v ,
y) and e =
p
a
+ by a m
r equations. As can be seen below when p
p rises above p = 0 isocline and falls below it; when e = 0; p =p =m
by
ar
here e rises and falls.
3.4.1
Estimations
There is good support for the exchange rate overshooting model in the data. Exchange rates
change immediately when fundamentals change in the market but there is some rigidity in in‡ation.
In‡ation volatility is much greater than the that in the exchange rate as is evident from Figure 16.
Table 47: Simultaneous equation model of in‡ation and exchange rate
In‡ation
Exchange rate
Exogenous variables
Coe¢ cient
tvalue
prob
Coe¢ cient
tvalue
prob
In‡ation (-1)
0.952
49.4
0.00
-0.0002
-0.176
0.861
Exchange rate (-1)
0.733
2.53
0.01
0.951
45.1
0.00
Constant
-1.06
-2.10
0.04
0.086
2.33
0.02
Table 48: Correlation among residuals of in‡ation and exchange rate equations (standard deviations
on diagonal)
Correlation among errors
In‡ation
Exchange rate
In‡ation
1.266
0.095
Exchange rate
0.095
0.0921
168
Figure 7:
169
There is good empirical support for the long run relationship between in‡ation and the exchange
rate, they are cointegrated on be basis of trace at 3 percent level of signi…cance (at 7.6 percent by
the max test).
Table 49: Cointegration between in‡ation and exchange rate
Cointegration tests
Order
Trace test [prob]
max-test [prob]
0
16.69 [0.031]*
13.03 [0.076]
1
3.65 [0.056]
3.65 [0.056]
Further empirical analysis suggests us that the interest rate seem to cause changes in both
in‡ation and exchange rates in UK but there is no signi…cant causality from in‡ation to the exchange
rate. In fact exchange rate seems to be explained by growth of money supply and its lagged term.
References
[1] Lucas R.E. (1976) Econometric Policy Evaluation: A Critique, Carnegie Rochester Conference
Series on Public Policy 1: 19-46.
[2] Muth J. (1961) Rational expectations and the theory of price movements, Econometrica, 29,
315-335.
[3] Sargent, T. J. (1986) Rational Expectation and In‡ation, Harper and Row Publishers, New
York.
[4] Sargent, T.J. and N. Wallace (1975) "Rational" Expectations, the Optimal Monetary Instrument, and the Optimal Money Supply Rule, Journal of Political Economy, pp. 241-254.
[5] Taylor M P (1987) On the long run solution to dynamic econometric equations under rational
expectation, Economic Journal, 97:385:215-218.
[6] Wallis Kenneth (1980) Econometric Implications of the Rational Expectations Hypothesis,
Econometrica,48:1, pp, 48-71.
[7] Sorensen PB and H. Jl Whitta-Jacobsen (2010) Introducing Advanced Macroeconomics, McGraw Hill.
170
[8] Minford P. and D. Peel (2002) Advanced Macroeconomics: A Primer, Edward Elgar Publishing.
[9] Simon and Blume (1994) Mathematics for Economists, Norton.
[10] Shone Ronald (2002) .Economic Dynamics, Cambridge.
171
4
L4: Neoclassical Growth Model
Many growth models use dynamic optimisation tool to analyse the capital accumulation process and
to identify a set of parameters that are critical to the balanced growth path (Ramsay (1928), Cass
(1965), Koopman (1965), Dorfman (1969), Lucas (1988), Romer (1989), (Parente (1994)). Such
model involves maximising the utility of the in…nitely lived household subject to the technology
constraint and capital accumulation process. When simpli…ed the optimisation problem is often
formulated in the form of a current value Hamiltonian as follows:
max
Uo =
Ct
Z
1
1
t Ct
e
Subject to technology constraint ( 0 <
dt
1
t=0
(D.418)
< 1)
Yt = At Kt Nt1
(D.419)
Capital Accumulation
K t = Yt
Nt Ct
Kt
(D.420)
It = St
(D.421)
Market Clearing
Yt = Ct + St
Initial (boundary) condition
Ko = Ko ; Assume (At = 1; Nt = 1)
(D.422)
Current Value Hamiltonian
Ct1
+
1
where C is consumption, a control variable;
H (C; K; ) =
t
[Kt
Ct
Kt ]
K is the capital stock, a state variable,
t is
the shadow price of the capital stock in terms of the utility, a co-state variable.
Market clearing, implicit in the budget constraint, implies that output is either
consumed or invested.
172
(D.423)
4.0.2
Four First Order Conditions for Dynamic Optimisation
The optimal path of capital accumulation is found using four …rst order conditions:
H (C; K; )
= 0 =) Ct
@Ct
t
=
t
H (C; K; )
=)
@Kt
K t = Yt
t
=
t
Nt C t
Lim e
t
t !1
=
(D.424)
t
t
Kt
1
Kt
t Kt
(D.425)
(D.426)
(D.427)
Optimality conditions in the Neoclassical Model
The …rst equation denotes the shadow price of capital in terms of the marginal utility of
consumption.
The second shows how the shadow price is sensitive to subjective discount factor and accumulation constraint.
Third condition is simply the accumulation equation
The …nal terminal condition implies no need for capital accumulation at the end of the planning horizon.
Capital stock, consumption and the shadow price of capital remain constant in the balanced
growth path
K
C
= gc ;
= gk ;
C
K
Balanced Growth Path (Steady State)
Kt
1
=
=g ;
t
+
(D.428)
(D.429)
t
This is the most important equation for deriving the equilibrium in this model.
It simply states that the marginal productivity of capital should equal the cost of capital, where
the shadow price measures the opportunity cost of capital.
173
By assumption the RHS in it is constant. This implies that the LHS also should be a constant,
therefore,
K
K
= 0.
Then from the production function
Y
Y
= 0. From the budget constraint when output and capital
stocks are not growing the consumption is also not growing; thus
C
C
=0
The shadow price also is not changing in the steady state as is obvious by the log di¤erentiation
of …rst FOC
Ct
=0
Ct
t
Capital stock, output and consumption in the steady state:
t
1
Kt
t
=
t
=
+ =) Kt
==) K =
+
Y =
C =Y
1
+
1
1
(D.431)
1
(D.432)
1
+
K =
(D.430)
1
+
(D.433)
Saving Rate in the Steady State
1
K
s=
=
Y
+
1
=
1
+
+
(D.434)
Thus the saving rate is determined in terms of parameters of preferences and technology
rather than being assumed as in the Solow model.
The higher discount rate for future consumption implies lower saving rate and more productive
capital implies higher saving rate.
Higher discount rate of capital reduces the steady state capital but raises the level of saving
in the steady state.
4.0.3
Transitional dynamics towards steady state
The transitional dynamics show a process where by the economy converges towards the steady state
once it is disturbed from that path.
174
(
)
In θ t , K t space the transition dynamics of the shadowprice θ t relative to
the steady state capital stock is that
θ&t = 0
θ&t > 0
θ&t < 0
(θ t )
K*
K > K * > K'.
K& = 0
K& > 0
θ
K& < 0
K*
K'
K
K > K * > K'.
K& = 0
K& > 0
θ
K& < 0
K*
K'
Transitional dynamics towards steady state
175
K
Growth Panel Regression
Mirrlees J. and editors (2010) Dimensions of tax design: the Mirrlees review, Oxford: Oxford
University Press.
4.1
Standard macromodel of growth, …scal policy and welfare (Bruce
and Turnovsky(2007))
Traditional Macromodel for Fiscal Policy and Growth (Bruce and Turnovsky(2007)) framework
with
U=
Z1
1
t
(CGc ) e
dt
(D.435)
0
Production function with public (Gc ) and private capital (K)
Y = GP K 1
H=
1
(CGc ) +
K +B
(1
;
0
1
) rB + GP K 1
(D.436)
(1
!) C
T
(D.437)
Standard macromodel economic growth, …scal policy and welfare: Optimisation by Households
@H
= (CGc )
@C
1
Gc
176
(1
!) = 0
(D.438)
Table 50: Determinants of growth rate of per capita income
Growth Regression
External Factor Model
Determinants
Coe¢ cient
t-prob
Coe¢ cient
t-value
Investment ratio
0.1820
.00060
-
-
Export Ratio
0.0257
.3830
-
-
Exchange rate -1
-
-
0.9710
0.00
Real Interest rate
-
-
-0.0290
0.00
Population growth rate
-0.8849
0.1540
0.7917
0.00
Constant
3.0116
0.1780
0.3400
0.00
Nepal
-3.0341
0.0000
0.0662
0.00
India
-2.0244
0.0000
0.0496
0.00
Bangladesh
-2.6448
0.0000
0.0568
0.00
Pakistan
-1.6057
0.0840
0.0735
0.00
South Africa
-5.1070
0.0000
0.0709
0.00
Brazil
-4.5529
0.0000
-0.0324
0.00
UK
-4.5630
0.0020
0.0031
0.00
Japan
-5.9846
0.0000
-0.0422
0.00
USA
-3.7902
0.0000
0.0295
0.00
Australia
-4.8616
0.0000
0.0351
0.00
Germany
-5.6408
0.0000
-0.0074
2
N =324
@H
=
@K
(1
R = 0:46
) GP (1
N =312
)K
=0
0.00
2
R = 0:9857
(D.439)
@H
=
(1
)r = 0
(D.440)
@B
Standard macromodel economic growth, …scal policy and welfare: Optimisation by Firms.
Firms optimal conditions:
Y =
(1
)
GP
K
K=
r
1
K
(D.441)
Solving this equilibrium results in:
(1
) r = (1
) (1
177
)
GP
K
=
(D.442)
Transversality conditions:
t
Lim Be
t!1
= Lim Ke
t
t!1
=0
(D.443)
Steady State equilibrium
Y = C + Gc + GP + K
(
(
(D.444)
1) ln C + n ln Gc = ln + ln (1
1)
C
Gc
+ n
=
C
Gc
=
!)
r (1
(D.445)
)
(D.446)
Balanced growth:
K
B
Gc
C
GP
=
=
=
=
=
C
K
B
Gc
GP
(D.447)
Steady state growth
(
1) + n =
=
r (1
1
r (1
)
(D.448)
)
(D.449)
n
Consumption to capital ratio:
Y
K
GP
C
Gc
GP
K
+
+
+ ; Gc = gc Y and
K
K
K
K
= gP Y ; 0 < gc < 1; 0 < gP < 1;
=
C
Y
=
K
K
Impact of tax on consumption ratios
C
Y
=
K
K
Gc
K
GP
K
Gc
K
K
=
K
=r
1
GP
K
=
gC
1
178
r
1
gP
(D.450)
K
=
K
gC
(D.451)
r
1
r
1
(D.452)
(D.453)
Increase in ince tax ( ) reduces growth rate but raises the private consumption ratio
C
K
with no e¤ect in the interest rate.
Consumption tax (!) does not a¤ect growth rate, .
Increase in government consumption (gc ) has no e¤ect on growth rate or interest rate but
crowds out private consumption.
Spending on infrascture (GP ) raises growth rate.
References
[1 ]
A c e m o g lu D . (2 0 0 9 ) Intro d u c tio n to M o d e rn E c o n o m ic G row th , P rin c e to n .
[2 ]
A g h i o n P . a n d P . H o w i t t ( 1 9 9 8 ) E n d o g e n o u s G r o w t h T h e o r y, M I T P r e s s , C a m b r i d g e M A .
[3 ]
B a rro R . J . a n d S a la -I-M a rtin (1 9 9 5 ) E c o n o m ic G row th , M c G raw H ill.
[4 ]
B a s u P. a n d K . B h a t t a r a i ( 2 0 1 2 ) G ov e r n m e n t B ia s in E d u c a t io n , S ch o o lin g A t t a in m e n t a n d L o n g -r u n G r ow t h , S o u t h e r n E c o n o m ic J o u r n a l,
7 9 (1 ), 1 2 7 -1 4 3 .
[5 ]
B a s u P. a n d K . B h a t t a r a i ( 2 0 1 2 ) C o g n it iv e S k ills , O p e n n e s s a n d G r ow t h , t h e E c o n o m ic R e c o r d , 8 8 : 2 8 0 : 1 8 -3 8 , M a r ch .
[6 ]
B h a t t a r a i K ( 2 0 1 2 ) F i s c a l p o l i c y, g r o w t h a n d r e d i s t r i b u t i o n i n U K , p a p e r t o t h e E S E M / E E A ( A u g . ) a n d A E A ( J a n . ) C o n f e r e n c e s . ( c o n t a i n s
d e ta ile d re fe re n c e s to th e re le va n t lite ra tu re ).
[7 ]
B h a t t a r a i K ( 2 0 1 0 ) S t r a t e g i c a n d g e n e r a l e q u i l i b r i u m m o d e l s o f p o v e r t y, Rom an ian Journ al of E con om ic Forecastin g, 1 3 : 1 : 1 3 7 - 1 5 0 .
[8 ]
B ru c e N e il
a n d S t e p h e n J . T u r n o v s k y ( 1 9 9 9 ) B u d g e t B a l a n c e , W e l f a r e , a n d t h e G r o w t h R a t e : " D y n a m i c S c o r i n g " o f t h e L o n g - R u n , Journal
of M oney, C redit and B an king, 3 1 , 2 , 1 6 2 - 1 8 6 .
[9 ]
C a s s , D . ( 1 9 6 5 ) O p t i m u m G r o w t h i n A g g r e g a t i v e M o d e l o f C a p i t a l A c c u m u l a t i o n , Review of E con om ic Studies, 3 2 : 2 3 3 - 2 4 0 .
[1 0 ]
M a d d iso n A . (1 9 9 1 ) D y n a m ic o f C a p ita l A c c u m u la tio n a n d E c o n o m ic G row th , O x fo rd .
[1 1 ]
M i r r l e e s J . a n d e d i t o r s ( 2 0 1 0 ) D im en sion s of tax design : the M irrlees review , O x f o r d : O x f o r d U n i v e r s i t y P r e s s .
[1 2 ]
P a re nte S .L .a n d E .C . P re sc o tt (2 0 0 2 ) B a rrie rs to R ich e s, M IT P re ss.
[1 3 ]
R a m s e y, F r a n k P . ( 1 9 2 8 ) A M a t h e m a t i c a l T h e o r y o f S a v i n g , E conom ic Journal 3 8 , 5 4 3 - 5 5 9 .
[1 4 ]
S o low , R . M .(1 9 5 6 ) A C o ntrib u tio n to th e T h e o ry o f E c o n o m ic G row th , Q u a rte rly J o u rn a l o f E c o n o m ic s, 7 0 :1 :6 5 -9 5 .
4.1.1
Mechanism for Poverty Alleviation (Bhattarai 2010)
There are three players in the poverty game -poor, rich and government; each has three
strategies available to it to play, s, l, and k , cooperation, indi¤erence and non cooperation.
The outcome of the game is the strategy contingent income for poor and rich, ytp (s; l; k) and
ytR (s; l; k) with the probability of being in particular state like this is given by
R
t (s; l; k)
p
t (s; l; k)
respectively and tax and transfer pro…les associated to them.
The state-space of the game rises exponentially with the length of time period t. T
179
and
he objective of these rich and poor households is to maximize the expected utility that is
assumed to be concave in income.
The government can in‡uence this outcome by choices of taxes and transfers that can be
liberal, normal or conservative.
Literature on poverty
S m ith (1 7 7 6 ), R ow ntre e , (1 9 0 2 ) H a n se n (1 9 2 6 )
K e e z e r (1 9 4 3 ) D av is (1 9 4 5 ) A tk in so n (1 9 7 0 ) , S e n (1 9 7 6 ) B e cke rm a n (1 9 7 9 ) S chu ltz (1 9 7 9 )
Tow n se n d (1 9 7 9 ) K a k w a n i (1 9 8 0 ) D a n z ig e r a n d G o ttsch a lk ,(1 9 8 3 ) C u tle r (1 9 8 4 ), B a su ,(1 9 8 5 ) P ia ch a u d (1 9 8 7 ) P ya tt (1 9 8 7 ) S w into n ,(1 9 8 7 )
A tk in so n (1 9 8 7 ) K n ie sn e r, M c E lroy a n d W ilc ox (1 9 8 8 ) L e w is a n d U lp h (1 9 8 8 ) H a g e n a a rs a n d Vo s, K la a s d e (1 9 8 8 ) D av id so n (1 9 8 8 ) W e b b (1 8 8 9 )
B row n (1 9 9 0 ) J e n k in s(1 9 9 1 ) S h a rif (1 9 9 1 ) G a u d e a n d W a tz law ick (1 9 9 2 ) K e e n (1 9 9 2 ) B la ck b u rn (1 9 9 4 ) Z h e n g (1 9 9 4 ) P re sto n (1 9 9 5 ) B a rd h a n
(1996)
R ava llio n (1 9 9 6 ) W h ite h o u s e (1 9 9 6 ) B a rrin g to n (1 9 9 7 )
B e tso n a n d W a rlick (1 9 9 8 ) Trie st (1 9 9 8 )
B e sle y , B u rg e ss (2 0 0 3 ) B la u g (1 9 6 3 )
C a sp e r (1 9 9 4 ) S h o rro ck s (1 9 9 5 ) S le sn ick (1 9 9 6 ) D e a to n (1 9 9 8 )
H a v e m a n B e r s h a d k e r ( 1 9 9 8 ) Fo s t e r ( 1 9 9 8 )
(2000)
Fo s t e r a n d S h o r r o ck s ( 1 9 8 8 ) G a r …n k e l ( 1 9 9 4 )
M ick le w rig ht a n d S te w a rt (1 9 9 9 ) G u o a n d H a rris
S u th e rla n d a n d P ia ch a u d (2 0 0 1 ) H illm a n (2 0 0 2 )
S t i f e l a n d T h o r b e c k e ( 2 0 0 3 ) , B a n e r j e e a n d D u ‡o ( 2 0 0 7 , 2 0 0 8 ) , B h a t t a r a i ( 2 0 1 0 )
W e lfa re re fo rm s: S n ow d e n (1 9 0 7 ), K in g (1 9 8 3 ),M o rte n se n a n d P issa rid e s (1 9 9 4 ),M e ye r a n d R o se nb a u m (2 0 0 1 ), B lu n d e ll (2 0 0 1 ),M o ¢ tt
( 2 0 0 3 ) , L o c k w o o d a n d M a n n i n g ( 1 9 9 3 ) , B h a t t a r a i a n d W h a l l e y ( 2 0 0 9 ) , B e a u d r y, B l a c k o r b y a n d S Z a l a y ( 2 0 0 9 ) .
Mechanism for Poverty Alleviation: Proposition 1
Proposition 1: The state contingent expected money metric utility of poor is less than that of
rich, which can be expressed as:
s X
l X
k X
T
X
p
p
p
t (s; l; k) t u (yt (s; l; k))
s=1 l=1 k=1 t
<
T
k X
l X
s X
X
R
R
t (s; l; k) t u
ytR (s; l; k)
(D.454)
s=1 l=1 k=1 t
where
p
t (s; l; k)
gives the probability of choosing one of strategies by poor given that the rich
and the government has chosen l and k strategies. Utility is derived from income as given by
u (ytp (s; l; k)) and
p
t
=
1
(1+rtp )
is the discount factors for poor and
R
t
=
1
(1+rtR )
the discount factor
for rich.
Mechanism for Poverty Alleviation: Proposition 2
Proposition 2: Transfer raises money metric expected utility of poor and reduces the utility of
rich.
180
s X
l X
k X
T
X
s=1 l=1 k=1 t
<
s X
l X
k
X
s=1 l=1 k=1
"
"
p
p
p
t (s; l; k) t u (yt (s; l; k))
+
T
X
Ttp (s; l; k)
t
T
X
R
R
t (s; l; k) t u
ytR (s; l; k)
t
#
T
X
+
TtR (s; l; k)
t
#
(D.455)
Mechanism for Poverty Alleviation: Proposition 3
Proposition 3: Incentive compatibility requires that
s X
l X
k X
T
X
s=1 l=1 k=1 t
>
"
s X
l X
k X
T
X
p
p
p
t (s; l; k) t u (yt (s; l; k))
+
T
X
t
Ttp (s; l; k)
#
p
p
p
t (s; l; k) t u (yt (s; l; k))
(D.456)
s=1 l=1 k=1 t
and
T
k X
l X
s X
X
R
R
t (s; l; k) t u
ytR (s; l; k)
s=1 l=1 k=1 t
>
" T
l X
k
s X
X
X
s=1 l=1 k=1
R
R
t (s; l; k) t u
ytR (s; l; k)
T
X
TtR (s; l; k)
+
t
t
#
(D.457)
Mechanism for Poverty Alleviation:Proposition 4
Proposition 4: Growth requires that income of both poor and rich are rising over time:
p
p
p
(s; l; k)
(s; l; k) < Tt+1
(s; l; k) < ::::: < Tt+T
Ttp (s; l; k) < Tt+1
(D.458)
p
p
p
Ytp (s; l; k) < Yt+1
(s; l; k) < Yt+1
(s; l; k) < ::::: < Yt+T
(s; l; k)
(D.459)
R
R
R
YtR (s; l; k) < Yt+1
(s; l; k) < Yt+1
(s; l; k) < ::::: < Yt+T
(s; l; k)
(D.460)
Mechanism for Poverty Alleviation:Proposition 5
Proposition 5: Termination of poverty requires that every poor individual has at least the level
of income equal to the poverty line determined by the society. When the poverty line is de…ned one
half of the average income this can be stated as:
Ytp (s; l; k)
1
>
2
N
1X h
Yt (s; l; k)
N
h=1
181
!
(D.461)
Above …ve propositions comprehensively incorporate all possible scenarios in the poverty game
mentioned above. Propositions 2-5 present optimistic scenarios for a chosen horizon T .
Mechanism for Poverty Alleviation: Tests
Testing above propositions in a real world situation is very challenging exercise.
It requires modelling of the entire state space of the economy.
Moreover in real situation consumers and producers are heterogeneous regarding their preferences, endowments and technology and economy is more complicated than depicted in the
model above.
In essence it requires a general equilibrium set up of an economy where poor and rich households participate freely in economic activities taking their share of income received from
supplying labour and capital inputs that are a¤ected by tax and transfer system as illustrated
in the next section.
4.2
Dynamic Computable General Equilibrium Model of Fiscal Policy
Most ot the models reviewed so far abtract away from more complex relations of productions and
consumption in the economy and thus are of limited use in formulating economic policies at sectoral
and household levels. Dynamic Computable General Equilibrium (DCGE) Models developmed in
the last two decades have been phenomenal in creating an analytical and modelling structure that
contains consumption, production and trade as in the real economies. These models are applied
to assess the impact of tax, transfer, spending and trade policies not only on e¢ cient economic
growth but also for evaluating the distribution of income over time. How can a set of policies be
more e¢ cient in terms of welfare to one household rather than to another is evaluated with a social
welfare function. Model is good for analysing available alternatives for long run growth prospects
from the accumulation of physical and human capital or for evaluating the e¢ ciency gains from
inter-temporally balanced budget or from the tax-transfer system or welfare reforms or from the
low-carbon growth strategy. Short run ‡uctuations often studied in the Keynesian or the new
Keynesian type economy could be introduced incorporating stochastic shocks to the production or
the consumption sides of the economy (see Stern 1992 for desirable properties of this type of model).
The comparative static frameworks in the pioneering work of Whalley (1975) has been improved
signi…cantly in recent years. The general features of these models from Bhattarai (2007 and 2013)
stated in this section as a brief introduction to this topic.
Preferences Model adopts a standard Ramsey (1928) type time separable constant elasticity
of substitution (CES) utility function to measure the welfare of households in each period. They
182
engage in the intra-period and inter-temporal substitution between consumption and leisure on
relative prices, interest rate, wage rate, tax rates and spending allocations in the economy. It
contains AIDS demand similar to that in Deaton and Muellbauer (1980) and has multiple nests.
h
The …rst stage of it is the aggregation at the level of goods and services Ci;t
, next stage of the
nest is the choice between that composite goods and leisure Cth ; lth and …nally choice is over
consumption-saving decisions across various periods based on Euler conditions. Thus the problem
of household h is:
max U0h =
1
X
t;h
Uth Cth ; lth
(D.462)
t=0
Subject to an intertemporal budget constraint of the form:
"1
X
Pi;t 1 +
tchi
h
Ci;t
+ wj;t 1
twih
h
li;t
t=0
#
"
1
X
h
wi;t
1
twih
h
Li;t
+ rj;t (1
h
tki ) Ki;t
t=0
#
(D.463)
here tax rates on consumption and income tchi ; twih ; tki are set by the policy makers who aim for
optimality and revenue neutrality in process of tax reform.
Production Technology Each …rm in the model has a unit pro…t function (
i;t )
which is the
di¤erence between aggregate composite market price - the composite of prices of domestic sales
(P Di;t ) and exports (P Ei;t ), and prices of primary inputs (P Yi;t ) and intermediate inputs (Pi;t ).
Thus the problem of a …rm i is:
1
y
max
i;t
=
(1
i ) P Di;t
y
+
i P Ei;t
1
y 1
1
y
y
i P Yi;t
d
i
1
X
ai;t Pi;t
(D.464)
t=0
Subject to production technology:
1
p
Yj;t = (1
i ) Ki;t
p
1
p
+
i Li;t
p
1
p 1
p
(D.465)
Sector speci…c capital (Ki;t ) accumulation:
Ii;t = Ki;t
Here
i
and
i
are share parameters,
y
(1
and
) Ki;t
p
1
(D.466)
are elasticities of substitution in trade and
production, ai;t are the input-output coe¢ cients giving the economy wide forward and backward
linkages.
183
The real returns (rj;t ) from investments across sectors are determined by the marginal productivity of capital that adjust until the net of business tax returns are equal across sectors. The
nominal interest rates set by the central bank should converge to these real rates in the long run.
Wage rate of household h; wth , equals its marginal productivity (Becker et al. 1990, Meyer and
Rosenbaum 2001).
4.2.1
Trade arrangements
Economy is open for the trade. Domestic …rms supply products di¤erentiated from corresponding
foreign goods. Traders decide on how much to buy (Di;t ) in the domestic markets and how much to
import (Mi;t ) while supplying goods (Ai;t ) to the economy. Choice of consumers between imports
and domestic consumption depend on the elasticity of substitution (
m)
between domestic supplies
and imported commodities in line of Krugman (1980) and Armington (1969). UK exports products
that she produces at lower cost and imports products in which she has no comparative advantage.
m
Ai;t =
1
X
t=0
d
m
i Di;t
1
+
P Ei;t Ei;t =
m 1
m
m
i Mi;t
1
X
P Mi;t Mi;t
m
y 1
(D.467)
(D.468)
t=0
UK economy, being one of the most liberal economies in the world, has almost no tax on exports
and has very minimal tari¤s and non-tari¤ barriers on imports.
4.2.2
Government sector
Government receives revenues from direct and indirect taxes and tari¤s. These taxes are distortionary and a¤ect the marginal conditions of allocation in consumption, production and trade
causing widespread shifts in the demand and supply functions of commodities.Which ones of these
tax instruments are optimal sources of revenue and which ones are the most ine¢ cient for it and in
generating growth process of the economy is a very important question but could be set following
the logic of micro level incentive compatible mechanism of Mirrlees (1971, 2011) or in DiamondMirrlees (1971). It can adopt a balanced budget or a de…cit budget or a cyclically balanced budget
or inter temporally balanced budget or it may simply peg de…cit to a …xed debt/GDP ratio. Which
one of these strategies is adopted may depend on circumstances of the economy, policy debates and
rules based on conventions and international commitments made in the treaties or agreements (i.e.
EU or G20).
184
Rt =
H X
N
X
h
tchi Pi;t Ci;t
h=1 i=1
H X
N
N
X
X
h h
h
+
twi wj;t LSi;t +
tki ri Ki;t
h=1 i=1
Gt
(D.469)
i=1
Ideally people’s preference for public good should decide the degree of freedom the government
is given in determining the size public sector relative to the aggregate economic activities (Devereux
and Love 1995, Barro (1990), Jensen and Rutherford (2002)).
4.2.3
General Equilibrium in a Growing Economy
General equilibrium is a point of rest, where the opposing forces of demand and supply balance
across all markets in each period and over the entire model horizon. It is given by the system of prices
of commodities and services, wage rate and interest rate in which demand and supply balance for
each period (Hicks 1939). When a model is properly calibrated to the benchmark micro-consistent
data set, such prices re‡ect the scarcity for those goods in the economy. Cost bene…t analysis or
economic decisions can be based on real level of welfare for a set of alternatives available to the
households, …rms and the government. Theoretically there has been much work, since the time of
Walras, in …nding whether such equilibrium exists, or is unique or is stable (Scarf 1973, Feenberg
and Poterba 2000, Feldstein 1985, Friedman 1962, Lee and Gordon 2005, Hines and Summers 2009,
Naito 2006, Lockwood and Manning 1993, Bovenberg and Sørensen 2009). Uniqueness is guaranteed
by the properties of preferences, technology and trade, such as continuity, concavity or convexity
or twice di¤erentiability of functions. Explicit analytical solutions are possible only for very small
scale models that are instructive but hardly representative of the economy (Heckman, Lochner and
Taber 1998,García-Peñalosa and Turnovsky 2007). It is common to apply numerical methods to
…nd the solutions of these models for a realistic policy analysis.
Yi;t =
H
X
h
Ci;t
+ Ii;t + Ei;t + gi;t
(D.470)
h=1
h
h
Lt = L0 en
h
Gt =
;t
= LSth + lth
N
X
gi;t
(D.471)
(D.472)
i=1
Markets for goods clear but the economy may not always be in equilibrium. Imperfections either
in goods or input markets are common giving rise to monopolistic or oligopolist situations. Such
imperfections in the markets are often represented by appropriately designed mark-up schemes
(Dixit and Stiglitz 1977). These mark ups may be sensitive to strategic interactions between
consumers and producers, …rms and government or between the national economy and the Rest of
185
the World. With widening gap between number of vacancies and unemployed workers it is possible
to incorporate the equilibrium unemployment features of Mortensen and Pissarides (1994) in the
model.
4.2.4
Procedure for Calibration
Computation and calibration of dynamic models like this are discussed in greater details in the
literature (Blanchard and Kahn 1980, Sims 1980, Rutherford 1995, Smet and Wouters 2003, den
Haan and Marcet 1990, Sims 1980, Kehoe 1985, Taylor and Uhlig 1990, Harrison and Vinod 1992).
This model is calibrated to the reference path of the economy using the arbitrage condition in the
capital market:
k
t
Pi;t
= Ri;t
t
Ri;t
= (r +
k
Pi;t+1
k
Pi;t
(1
i ) Pi;t
=
k
i ) Pi;t+1
= (r +
1
1 + ri
(1
(D.473)
k
i ) Pi;t+1
i)
(D.474)
(D.475)
This helps to calibrate the capital stock and the level of investment in equilibrium path:
V i;t = (r +
k
i ) Pi;t+1 Ki;t ;
Ki;t =
V i;t
;
ri + i
k
Pi;t = Pi;t+1
(D.476)
gi + i
V i;t
(D.477)
ri + i
Even a small reform in the public policy of a sector can have a large impact on the welfare and
Ii;t =
growth over time if such policy has larger knock on e¤ects in the wider economy and removes
the root source of the distortions that can have a detrimental impact on output, employment and
investment levels in the economy.
Most important aspect of DCGE model is that these provide an evolution of the economy along
with essential structures that we observe in the real economies. Paths of the relative prices are such
that all households and …rms are making optimal choices regarding their economic decisions. Model
simulations based on the solutions with these parameters are compared for alternative policies under
considerations. These provide basis for selecting the best policy that are dynamically prudent on
for growth across sector and more equal distribution of income across households.
See GAMS/MPSGE programmes and solutions in excel spreadsheets for a general understanding
of the evolution of economies over time.
186
4.3
Exercise 6
1. An economy has to decide how much to consume today and how much save and invest to add
into the capital stock that can help produce goods for future consumption. The optimal capital
stock maximises the present value of utility from consumption. Problem of this economy is:
M ax U0 =
Z
T
e
rt
C (t) dt
(D.478)
0
subject to the production technology:
Q = Q(K)
(D.479)
Capital accumulation constraint:
Kt =
@K
=Q
@t
C
K
(D.480)
Write the current value Hamiltonian for dynamic optimisation in this model.
Discuss …rst order conditions and the terminal conditions required for dynamic optimisation
Use a phase diagram to determine the convergence process towards the optimal capital stock.
Apply this model for determining the optimal pricing strategy for exhaustible resources (nonrenewable resources) such as oil and gas in a competitive economy.
2. Consider a dynamic economy with
Preference:
M ax U0 =
Z
T
e
0
(1
Technology: Yt = At Kt Nt
)
1
t Ct
1
dt
(D.481)
assume At = 1 and Nt = 1
Capital accumulation:
K t = Yt
Nt C t
Kt
All of the above notations have usual meaning.
Write the current value Hamiltonian for this problem.
Give four …rst order conditions for the dynamic optimisation in this economy.
Characterise the balanced growth path using those conditions for this economy.
Discuss the transitional dynamics in space when and when .
187
(D.482)
References
[1] Acemoglu D. (2009) Introduction to Modern Economic Growth, Princeton.
[2] Aghion P. and P. Howitt (1998) Endogenous Growth Theory, MIT Press, Cambridge MA.
[3] Barro R. J. and Sala-I-Martin (1995) Economic Growth, McGraw Hill.
[4] Bhattarai K. (2007) Welfare Impacts of Equal-Yield Tax Experiment in the UK Economy,
Applied Economics, 39, 10-12, 1545-1563, June-July.
[5] Cass, D. (1965): Optimum Growth in Aggregative Model of Capital Accumulation, Review of
Economic Studies, 32:233-240.
[6] Maddison A. (1991) Dynamic of Capital Accumulation and Economic Growth, Oxford.
[7] Solow, R. M.(1956) A Contribution to the Theory of Economic Growth, Quarterly Journal of
Economics, 70:1:65-95.
188
5
L5: Endogenous Growth Model
(This model is based on Basu and Bhattarai (2012) that has adapted Lucas-Uzawa (Lucas, 1988)
model for analyis of government bias in on economic growth).
Issues
The e¤ect of public expenditure on educational attainment and growth is an unresolved issue.
in majority of the cases, the active involvement of the government in the education sector is
deemed to be a failure.
If the government involvement in education has such questionable e¤ects on pupil’s educational
attainment, the spillover e¤ect of this on economic growth thus also becomes debatable.
Two E¤ects
First is a positive complementarity e¤ect that arises because of the government provision of
intermediate inputs in the form of teachers and other school aids.
Second is a distortionary e¤ect that comes into play when the government taxes resources
away from the non-education sector to …nance education spending. Such a spending based
public education policy could fail if the latter negative e¤ect is stronger.
Thus contrary to conventional wisdom, a blanket increase in government spending on education
may not necessarily promote growth and welfare in all countries.
5.0.1
Human capital and …nal goods sectors
Human capital sector
ht+1 = (1
h )ht
+ AH gt (lHt ht )1
(E.483)
Final goods sector
yt = AG kt (lG ht )1
(E.484)
Capital accumulation
kt+1 = (1
Financing education
189
k )kt
+ ikt
(E.485)
gt =
t yt
(E.486)
Social Planners Problem
M ax
1
X
t
ln(ct )
t=0
subject to the resource constraint:
ct + it = (1
t )yt
(E.487)
and (E.483) through (E.485).
Proposition 2 Along the balanced growth path, the optimal share of public spending in GDP is
given by:
=
1
1
1+
:
1
1
lH
lG
:
(E.488)
lH
lG
In economies where private schooling e¤orts (lHt ) are higher, it is optimal to tax the goods
sector more.
Balanced Growth Properties
the steady state government spending share in GDP is given by:
gt
=
yt
5.0.2
(E.489)
Balanced growth
De…ne the gross balanced growth rate as : There are three key balanced growth equations. Based
on the …rst order condition for the physical capital stock we get:
=
[(1
)( yt =kt ) + 1
k]
(E.490)
Based on the …rst order condition for the human capital stock, one gets:
= [1
h
+ AH (1
)
lH (yt =ht ) ]
(E.491)
Finally, using the human capital technology (E.483), we get a third balanced growth equation:
=1
h
+ AH
(1
1
lH
AG lG
190
)
(kt =ht )
(E.492)
Return to Schooling
It is easy to verify that this value of human capital is the same as the ratio of the shadow price
of consumption to that of investment in schooling. In other words,
t
qth =
where
t
and
t
(E.493)
t
are the Lagrange multipliers associated with the schooling technology (E.483) and
the ‡ow resource constraint (see E.487). Using the Euler equation for human capital (see (E.506),
one gets the following valuation equation for the human capital:
h
qth = mt+1 [fqt+1
f1
)(1 lGt+1 )1
h +AH gt+1 (1
ht+1
g+fAG (1
t+1 )(1
1
)kt+1 ht+1 lGt+1
g]
where mt+1 is the intertemporal marginal rate of substitution in consumption given by
(E.494)
t+1 = t :
Next verify from (E.504) in the appendix that
qth =
(1
G
t )M P Ht
E
M P Ht
(E.495)
Return to Schooling
Rewrite (E.494) as
h
qth = mt+1 qt+1
(1
h
E
+ lHt+1 M P Ht+1
) + lGt+1 (1
G
t+1 )M P Ht+1
(E.496)
h
The return to schooling (Rt+1
) is thus given by:
h
Rt+1
=
h
qt+1
(1
h
E
+ lHt+1 M P Ht+1
) + lGt+1 (1
qth
Rh = 1
h
G
t+1 )M P Ht+1
+ M P HE
(E.497)
(E.498)
Using (E.498) one can rewrite the balanced growth equation (M.1196) as follows:
1 + g = Rh
(E.499)
Comparison of (M.1195) with (E.499) immediately reveals a familiar arbitrage condition that
the return on human capital must balance the after tax return on physical capital. In other words,
Rh = (1
)( y=k) + 1
191
k
(E.500)
Table 51: Cross country steady state distribution of the education technology
lH
AH
k=y
Mean
0.47
0.15
0.07
1.91
Std Deviation
0.07
0.02
0.03
0.21
Table 52: Regional Features of the Government Bias in Education
Asia
0.057
5.0.3
Europe
0.078
Latin America and
Middle East
Carribean
and North Africa
0.068
OECD
0.063
0.08
North
South
America
Asia
0.096
0.036
Africa
0.077
Cross country calibration of government bias in education
Cross country calibration of government bias in education
Let
First order conditions
t;
t ;be
the Lagrangian multipliers associated with the ‡ow budget
constraint (N.1408), human capital technology.
The Lagrange is:
1
1
P
P
t
L=
U (ct ) +
t=0
1
P
+
t=0
t [AG (1
t )kt
t=0
t [(1
(lGt ht )1
+ AH gt (lHt ht )1
h )ht
+ (1
k )kt
ct
kt+1 ]
ht+1 ]
First order conditions are:
t
ct :
kt+1 :
ht+1
:
t
t
=
+
lGt :
t (1
+
t+1 [(1
t+1 [1
:
t yt
=
yt+1
+1
kt+1
t (1
1
t
192
k]
)ht+1
=0
(E.502)
1
lHt+1
]
(E.503)
1
)kt+1 ht+1 lGt+1
]
kt h1t
t AH
(E.501)
t
+ AH gt+1 (1
t+1 )(1
t )AG lGt
t
t+1 )
h
t+1 [AG (1
)(1
U 0 (ct ) =
(ht lHt )1
)gt AH h1t
yt
lHt = 0
(E.504)
(E.505)
Table 53: Cross country correlations of the key macroeconomic varaibles
lH
AH
k=y
Rh
lH
1
AH
0.92
1
-0.64
-0.39
1
-0.94
-0.96
0.35
1
-0.14
0.12
0.81
-0.19
1
0.93
0.99
-0.46
-0.95
0.01
1
0.93
0.99
-0.46
-0.95
0.01
1
k=y
R
h
1
The expression for the optimal tax rate in proposition 1 immediately
Proof of Proposition 1
follows after substituting out
t= t
from (E.504) and (E.505). One gets the optimal tax rate:
t
=
1
1
1+
:
1
1
lHt
lGt
Ht
: llGt
Next, we exploit the fact that along the balanced growth path, the time allocations to goods
and schooling (lGt and lHt ) are constants. Unless the time allocations are constant, a constant
balanced growth rate does not exist because the marginal product of capital will be time varying
(see (E.502)). Since lGt is a constant, this means that the optimal tax rate
t
is also stationary.
Derivation of the Balanced Growth Equations
Hereafter we drop time subscripts for variables which are stationary along the balanced growth
path. To prove (M.1195), use (E.501) and (E.502).
To get (M.1196), rewrite (E.503) as:
t
t+1
=
t
t+1
:
t+1
+
t+1
t
t
t+1
t
h
fAG (1
t+1
Using (E.501), check that
balanced growth condition
[1
t
=
+ AH gt+1 (1
)(1
ct
ct+1 :
)(1
lGt+1 )1
ht+1
]
(E.506)
1
t+1 )kt+1 ht+1 lGt+1 g
Use (E.505) to substitute out
t
t
and also use the
= =(1 + g) which upon substitution in (E.506) yields:
= [1
h
+ AH (1
)
To get (M.1197) use (E.483), (N.1405) and (M.1192).
Proposition
193
lH (yt =ht ) ]
(E.507)
Proposition 3 The tax rate that maximizes growth also maximizes the long run welfare.
Proof. The steady state welfare can be written as:
Wt
=
1
X
j
ln ct+j
j=0
=
=
ln ct
ln
+
1
(1
)2
ln kt
ln(ct =kt )
+
+
1
1
(1
(E.508)
)2
ln
Use the resource constraint (N.1408) and the balanced growth condition to very that
Proposition
ct
(1
)yt
=
+ (1
kt
kt
Proof. Next plug (M.1195) into (E.509) to …nd
1
ct
=
kt
which upon substitution in (??) yields
(1
k)
)(1
(E.509)
)
ln kt
1
+ ln(
(1
)) +
ln + ln
2
1
(1
)
This shows that the steady state welfare is positively related to growth rate.
Wt =
(E.510)
(E.511)
Thus the growth maximizer tax rate is also a welfare maximizer.
5.0.4
Conclusion
The e¤ect of public education spending on growth is an empirically unsettled issue.
A
plethora of studies document that public education spending does not help promote growth.
Our cross country stylized facts also support this …nding. Growth and schooling returns are
in fact lower in countries with a higher ratio of public spending to GDP except for very high
education spenders.
In this paper, we reopen this issue and investigate this within an endogenous growth framework. Public spending on education appears directly in the human capital technology.
The relative intensity of public and private spending on education in the human capital production, which we call government bias in education, appears to be a fundamental determinant
of cross country dispersion in long run growth and schooling returns.
194
Conclusion
A higher government bias has con‡icting e¤ects on growth. On the one hand, it lowers growth
by crowding out private schooling e¤orts. On the other hand, it promotes growth through
the complementarity channel.
The latter e¤ect is stronger in countries which have historically a greater government bias in
education.
Based on our growth model, we estimate this government bias parameter for a wide range of
countries and …nd that the government bias in education is generally higher in rich countries.
The policy implications of our analysis is that an increase in public spending on education
without adequate infrastructural support may not necessarily be bene…cial for the society.
For the complementarity e¤ect of public spending to dominate, a nation may need a greater
educational infrastructure. This infrastructural role of the government in education is an area
worth exploring in future research.
see: Basu P. and K. Bhattarai (2012) 1) Cognitive Skills, Openness and Growth, the Economic Record,
88: 280: 18-38, March; 2) Government Bias in Education, Schooling Attainment and Long-run Growth,
Southern Economic Journal, 79(1), 127-143.
See dynare programme:BB_Er_…nal.mod and GAUSS programme growth.g and data…le EDU_GDP_EXP_IMP_gr_panel.cs
5.0.5
China, India and SAARC Countries in the Global Growth Competition
The process of convergence and divergence has been going on in the global economy in the last
three hundred years after the scienti…c discoveries and technical innovations that have fundamentally changed the nature of production, exchange and consumption. Industrialisation came to the
current stage going through stages of development from 18th to the last quarter of 20th century.
This process has further intensi…ed in the last six decades. Every country in the world wants to
achieve a higher rate of growth of GDP per capita. While the countries in the West were successful
in achieving higher growth till 1980s the growth pole has now gradually shifted towards the countries in developing Asia including India in the South Asia. Stylized facts of growth and economic
development presented here are based on the data sets from the World Economic Outlook of the
IMF and World Bank Development Indicators (WBDI).
Economists generally agree on the factors that lead to economic growth as above based on
experienced of Western Europe, North America, Japan and other advanced economies. Policies that
raise the rate of accumulation of physical and human capital and advancement in the production
195
technology lead to higher economic growth (Madison (1995)). Classical, neoclassical and endogenous
growth models have been constructed to show the precise relationships among these factors and
economic growth. Early versions of South Asian growth models used by the Planning Commission
of these nations were based on basic Harrod-Domar set up where given the capital output ratio
increasing growth required just increasing the rate of national saving. Then there were various
sectoral decomposition exercises aimed to …t the aggregate target. Big gaps remained between
targets and accomplishments. Levels of per capita income were similar across all SAARC countries
till 1980 but these started to di¤er substantially following the economic reforms and liberalisations
that started in India in late 1980s (after the success of similar trend in China). Kotwal, Ramaswami
and Wadhwa (2011) explain how the recent growth in India was spurred by exports of high tech
services rather than manufacturing products as in China.
The average growth rate in developing Asia has been 7 to 8 percent in the last 30 years, twice the
global average and three times or more of that in the EU economies. After decades of sluggishness,
growth rates in South Asian countries have been higher than those in other regions of the world;
particularly very impressive in India (5.5 to 7.0 percents) and china (8.5 to 10.3 percents). Bosworth
and Collins (2008) provide growth accounting at aggregate and sectoral levels of the extraordinarily
growth occurring in China and India, residence of over one third of the global population; less than
20 percent population reside now in advanced countries. Thus a higher growth rate in China and
India in next two three decades is likely to tranform the structure of the global economy.
Table 1: GDP growth rates around the globe
ASEAN-5
ADV Econ
5.30
5.03
4.87
5.61
3.12
2.78
1.78
1.88
1980-89
1990-99
2000-09
2010-14
CIS
CE Europe
-4.26
5.98
3.72
2.11
1.70
3.90
3.30
DevAsia EmDevEcon.
6.79
7.36
8.31
7.37
EuroA
3.51
3.67
6.15
5.66
1.97
1.35
0.68
EU Majadv (G7)
2.15
2.16
1.75
0.93
3.03
2.55
1.45
1.87
MENA
MENAP
OthAdv
SSA
WestHm
World
1.47
4.35
5.42
3.99
1.99
4.37
5.34
3.94
4.73
4.33
3.37
3.28
2.60
2.23
5.53
5.39
2.12
2.97
3.18
3.86
3.24
3.09
3.62
3.75
Table 2: Average annual growth rate of GDP in SAARC countries (%)
Afghanistan
1980-89
1990-99
2000-09
2010-14
9.23
6.72
Bangladesh
3.28
4.80
5.82
6.15
Bhutan
9.37
5.33
8.10
8.66
China
9.76
10.00
10.29
8.46
India
5.54
5.63
7.00
5.81
Maldives
10.52
6.61
7.10
4.33
Nepal Pakistan Sri Lanka
4.10
6.59
4.21
4.87
4.50
5.61
4.14
4.69
4.64
4.25
3.34
7.13
Table 2: GDP per capita, current prices ($)
Afghanistan
1980
1990
2000
2010
641
2014
641
Bangladesh
236
284
355
703
1006
Bhutan
321
544
802
2063
3042
China
307
341
946
4423
7138
India
277
386
461
1432
1389
Maldives
413
1092
2967
6668
7501
Nepal Pakistan Sri Lanka
138
374
301
215
483
509
247
581
917
596
1034
2429
703
1234
3360
By maintaining average 8 percent growth, it is possible that India will catch up the advanced
countries in the West and the East in per capita income within a generation. Other SAARC (South
196
Asian Assotiation for Regional Cooperation) member countries, may be able to converge to India in
per-capita income taking appropriate actions to create stable institutions and socioeconomic conditions required for growth. By the size of the economy and manpower-strength, India is the centre of
the economic gravity with seven smaller economies surrounding it. Considering the growth success
story of China since 1980s, which is in the eastern neighbor of this region, it is very essential and
bene…cial to India to have an integrated approach for the development of these countries in South
Asia. Modi’s recent proposal for HIT-ways9 (highways, information technology and transmission
ways) for the region is a timely and visionary proposal for growth. In an address on the Independence Day 2014 he has proposed new strategies including i) "no defect" and "zero e¤ect" approach
to manufacturing, ii) a model village in each constituency iii) new initiative for expanding bank
accounts to million of poor households, iv) massive investment on skills and sanitation iv) …ght
against poverty in all SAARC countries and v) an open approach to the foreign direct investment
or "make in India". Several strategic points for growth emerging from the analysis of facts in this
paper are worth considering in this context. These are as follows:
1. Given the 20 percent population residing in South Asia this region should push for growth
and increase its share of global GDP up to 20 percent from roughly 6.5 percent in 2014.
2. Such growth requires increasing the ratio of saving and investment about 10 percent above
the current averages around 35 percent.
3. Process of structural transformation should continue so that output and employment increases
substantially in industrial and services sectors and till both output and employment in the
agriculture sector are less than 5 percent from around 17 and 50 percent in recent years.
4. Such transformation will occur as this region moves towards urbanisation so than about 90
percent of the population starts living in urban area with facilities. Building mega cities like
this will create not only employment but also income. It also will gradually free up rural
lands for more scienti…c cultivations and other meaningful economic uses.
5. On manpower issues it is important to reduce the student teacher ratio from 40 to close to
16 to raise the quality of education and cognitive skill among children. This is essential for
human capital required for science and technology and for improving the PISA scores.
6. Revenue and spending of government should balance at least in the medium term and debt
to GDP ratio should not increase over 50 percent of GDP; the size of the public sector is not
over 30 percent of GDP.
9 It
is very appropriate for India’s new government to take extra initiative on forming growth links with China
(including Xi Jinpin’s announcement for building industrial parks Gujarat and improving railnetworks in South
India), Japan (making Varanasi a smart city) and other advanced countries including Germany and United states.
197
7. Trade ratio should increase to around 100 percent from the 50 percent at this time. Free trade
regimes can enhance both the supply and demand side of the economy.
8. Liquidity of the …nancial system need at least to treble to have a smooth ‡ow of credits
required for new and existing enterprises.
9. Free convertibility of currency is essential to protect this region from international shocks.
10. A high 8 percent growth strategy is consistent with all above and requires …rm commitment,
e¢ cient and strong public administration. Gini coe¢ cient should not be above 35 percent for
social integrity and cohesion.
Size of the SAARC region has increased to around 7 percent of global GDP in PPP which
more has more than doubled since 1980. However this growth in global share pales when compared
to China which raised its global share to 16.5 in 2014 percent compared to 6 percent of India.
Srinivasan (2005) reports on TFP growth rates underlying these trends.
Economic integration of the South Asian region must base on the strength of its members. India
is stable, dynamic and economic power of the region. Bhutan and Maldives two tiny countries of
the region are doing better economically by pursuing strategies appropriate to the vastly growing
production sectors and middle classes in India. Bhutan is bene…ting by proximity of India by
developing a number of hydro power stations generating electricity to sell to India. Maldives is
developing fast by tourism aiming at individuals in the growing middle income class in India.
Bangladesh is achieving higher growth rates than before by exporting textiles but still caught in
natural disasters and political problems. War torn Afghanistan and Pakistan could not emerge
above the ethnic con‡icts to focus on economic growth. Despite uprooting the age old monarchy
and being able to restore the peace with Maoists it is an irony that Nepal is yet struggling to form a
political consensus to draft a new constitution for the republic of Nepal. Given above potentials and
absurdities a systematic study, particularly focusing on the role that India can play in development
of the South Asia region has become an interesting topic of research, apparently very little is found
on this in the existing literature.
There is no single economic model that is perfect and …t for analysis of all important issues
relating to growth and development. Each type of model has its strength and limitations. Since
the overall objective is having a comprehensive understanding of underlying factors that in‡uence
on growth and development it is essential to consider each of these models and appreciate how
it can contribute to our understanding of the economy. We illustrate this by applying a panel
data model of growth, dynamic CGE model with …nancial deepening, macroeconometric model for
macroeconomic forecastging and a policy coordination model to analyse gains from cooperation to
enhance growth and development in India and SAARC countries in this section.
198
5.0.6
Dynamic Panel Data Model of Economic Growth
Growth models show how the output per capita increases over time with accumulation of physical
and human capital and improvement in technology (Solow (1956), Lucas (1988), Romer (1990)).
However the growth rates di¤er signi…cantly by countries and the degree of convergence in per capita
income varies substantially across nations. Frustrated from the dismal growth performance from
1950-1980s Malenbaum (1982) even stated pessimistically that "decades of slow growth lie ahead
before either nation emerges as a modern industrial state of developed-nation status". Fortunately
there occurred a structural break in the growth process around mid 1980s in India motivating
Rodrik and Subramanian (2005) to assess policy and structural factors that caused a surge from
"Hindu growth" to productivity surge. These surges occurred because of the reforms of the labour
market giving freedom in hiring and …ring of workers to …rms, end of reservation in small scale
industries, reforms of the banking sector, simpli…cation of FDI rules, improvement in infrastructure
and reduction of debt. These policy factors accelerated growth in India starting in early 1990s (Kaur
(2007)). Agrawal (2010) empirically establishes causality between savings and economic growth in
India. Bosworth and Collins (2008) provided growth accounting at aggregate and sectoral levels
of the extraordinarily growth occurring in China and India. From the panel data analysis and
endogenous growth models Basu and Bhattarai (2012a) found that cognitive skill and openness to
be factors of higher economic growth. Shocks to the technology sectors caused more macroeconomic
‡uctuations than the total productivity shocks in the short run in their models. Education is the
key for growth but it is the joint responsibility of public and private sectors to educate children.
Public bias to education does not produce desired results (Basu and Bhattarai (2012b)). South
Asia forms the part of global economy in both of these endogenous growth models. We estimate
coe¢ cient the dynamic panel data model of growth for the South Asian economies report results
in Table 14. This shows in general trade ratio and investment ratios contribute signi…cantly and
positively on the growth rates of per capita income but the higher population growth rates reduced
output growth rates signi…cantly. However there are country and time speci…c factors at play as
growth rate vary signi…cantly across countries and time years.
5.0.7
GMM 2-step Estimation of Growth in South Asia
Consider a dynamic panel data model of the form where growth rate of output of country i at time
t, yi;t is explained by its lagged values and a set of exogenous explanatory variables xi;t . Here
individual speci…c e¤ects and
t
i
is
represents the time speci…c e¤ects.
yi;t = yi;t
1
+
i
+
i xi;t
+
t
+ +ei;t
<1
(E.512)
A generalised method of moments (GMM) as proposed by Hansen (1982) for a panel data model
199
generates the unbiased estimate of
and
solving endogeneity and bias in estimation due to the
i
presence of correlation between the lagged values of dependent variables yi;t
Right instrument for lagged yi;t
estimator (ignoring xi;t and
1
say by yi;t
2
and errors terms ei;t .
solves this inconsistency and generates unbiased
2
t ):
bIV =
where yi;t
1
T P
N
P
yi;t
t i
T P
N
P
t
1
y i;t
2
2 (yi;t
1
yi;t
2)
yi;t
2 ).
yi;t
2
(E.513)
yi;t
i
is used as instrument of (yi;t
1
GMM method includes the most e¢ cient instrument, Zi :
GM M
N
X
=
yi;t Zi
i=1
N
X
yi;t Zi
i=1
!
!
WN
N
X
0
Zi yi;t
i=1
WN
N
X
0
Zi yi;t
i=1
!!
1
!!
(E.514)
Arrelano and Bond (1995), Wijndmeir (2000), Blundell and Smith (1989) and Verbeek (2004),
Wooldridge (2002) among others have more extensive analysis of the GMM estimation. The essence
of the GMM estimation remains in …nding a weighting matrix that can guarantee the most e¢ cient
estimator. This should be inversely proportional to transformed covariance matrix.
WNopt
=
1
N
N
X
0
Zi ei;t e;i;t Zi
i=1
!
1
(E.515)
The GMM estimator with instrument (levels, …rst di¤erences, orthogonal deviations, deviations
from individual means, combination of …rst di¤erences and levels) used in PcGive is:
b=
where AN =
N
X
Wi Zi
i=1
N
P
!
AN
i=1
1
0
Z i Hi Z i
N
X
0
Z i Wi
!!
1
N
X
Wi Z i
i=1
!
AN
N
X
0
Zi yi
i=1
is the individual speci…c weighting matrix.
i=1
200
!!
(E.516)
Ta b le 1 4 : P a n e l e stim a te s o n th e g row th ra te o f p e r c a p ita in c o m e in In d ia a n d S o u th A sia
1 -S te p E stim a tio n
D e te rm in a nts
2 -ste p e stim a tio n
C o e ¢ c ie nt
t-p rob
C o e ¢ c ie nt
t-p rob
Tra d e ra tio
0 .0 0 2 5
0 .0 5 0 0
0 .0 0 3 8
0 .0 2 0 0
Inve stm e nt ra tio
0 .0 0 8 6
0 .0 0 8 0
0 .0 0 8 9
0 .0 0 4 0
G D P grow th rate
0 .9 7 4 9
0 .0 0 0 0
0 .9 7 7 3
0 .0 0 0 0
P o p u la tio n g row th ra te
-2 .9 0 5 5
0 .0 0 0 0
–0 . 1 2 7 5
0 .0 0 0 0
C onstant
-0 .3 5 7 5
0 .0 0 0 0
0 .3 4 2 4
0 .0 0 0 0
T 2005
-0 .1 7 9 7
0 .0 5 9 0
-0 .1 3 9 0
0 .0 7 9 0
T 2006
-0 .0 4 3 5
0 .4 0 0 0
0 .1 0 7 5 0
0 .4 0 9 0
T 2007
-0 .0 5 1 6
0 .0 3 2 0
-0 .0 2 9 9
0 .3 7 9 0
T 2008
0 .0 2 0 8
0 .0 1 2 0
0 .0 4 3 9
0 .0 0 0 0
T 2009
0 .1 7 6 4
0 .0 0 6 0
0 .1 0 0 1
0 .2 2 1 0
T 2010
0 .1 3 5 3
0 .0 0 2 0
-0 .1 4 9 3
0 .0 2 0 0
T 2011
0 .1 3 0 1
0 .0 0 0 0
-0 .1 6 6 4
0 .0 0 0 0
T 2012
0 .2 7 9 1
0 .0 0 0 0
0 .2 7 9 1
0 .0 0 0 0
A fg h a n ista n
0 .2 1 9 1
0 .0 0 0 0
0 .0 4 5 4
0 .0 0 0 0
B hutan
0 .0 6 9 4
0 .0 0 0 0
0 .0 2 3 7
0 .0 0 2 0
B a n g la d e sh
0 .2 2 0 0
0 .0 0 0 0
0 .0 5 5 4
0 .0 0 0 0
In d ia
0 .3 3 7 1
0 .0 0 0 0
0 .0 3 5 0
0 .0 0 0 0
M a ld ive s
0 .1 6 9 9
0 .0 0 0 0
0 .0 6 0 5
0 .0 0 0 0
N epal
0 .2 8 2 3
0 .0 0 0 0
0 .0 1 3 5
0 .0 0 0 0
P a k ista n
0 .3 4 6 6
0 .0 0 0 0
0 .0 7 4 2
0 .0 0 0 0
S ri L a n ka
base
base
base
N = 8; T = 9
D a ta so u rce: W B D I,
R
2
= 0 .9 9
N = 8; T = 9
base
R
2
= 0 .9 9
IF S o f IM F a c c e sse d fro m D a ta A rch ive U K
Doornik and Hendry (2001, chap. 7-10) provide a procedure on how to estimate coe¢ cients using
…xed e¤ect, random e¤ect and the GMM methods including a lagged terms of dependent variable
p
P
among explanatory variables for a dynamic panel data model: yi;t =
ak yi;t s + t (L) xi;t + t +
i=1
i + ei;t
or in short yi;t = Wi + i ai + ei . It will be relevant to study process of convergence among
states in India and SAARC countries using this type of growth model in coming years (see Brandt,
Ma, Rawski (2014) for China).
201
5.0.8
Dynamic Computable General Equilibrium Model
One sector growth models presented above are analytically tractable but practically they are not
designed to answer questions relating to sectoral structure of production, issue of structural transformation and distribution of income as an outcome of the general equilibrium process in the
economy. This requires a full dynamic computable general equilibrium (DCGE) model for a decentralised economy. DCGE models contain the relative price system and intertemporal choices of
…rms and households as key factors determining the growth of various sectors of the economy and
distribution of income among households while studying the long run cycles of model economies
(Bhattarai (2010)). The main equations for a typical DCGE model are as follows:
h
1) Demand side: welfare of households U0h given by consumption Ci;t
and leisure Lht :
U0h
M ax
=
1
X
t h
h Ut ;
0<
t
h
<1
(E.517)
t=1
h
Uth = U Ci;t
; Lht ;
(E.518)
c
Subject to budget constraints:
I0h
=
=
"1
X
e
t=0
1
X
e
t
N
X
i=1
t h
It
t=0
h
Pi;t (1 + ti ) Ci;t
+ wth (1
"
1
X
=
wth (1
h
tl ) Lt
tl ) Lht
#
tk ) Kth
+ rt (1
t=0
(E.519)
#
2) Supply: production, …nance and accumulation:
Yi;t = Fi Ki;t ri;t ; wth ; pi;t ; p; Li wth ; pi;t ; Ai ;
T
X
t=0
Savings (Yt
Pi;t Yi;t =
T
X
t=0
"
rt (1 + tk ) Ki;t +
H
X
wth
(1 +
c
tl ) Lhi;t
h=i
#
(E.520)
(E.521)
Ct ) adds to the accumulation of assets (At ) in the economy:
At (1 + rbt ) + Yt
At rt + Yt
Ct
fAt+1
Ct = At+1
(1
) At g = 0
(E.522)
(E.523)
In equilibrium there is equivalence between …nancial assets (At ) and physical capital (Kt ) ;
replace At by Kt :
202
Yt
Ct
(Kt+1
(1
) Kt ) = 0;
=)=) Yt = Ct + It
(E.524)
This the optimal …nancial deepening at the sectoral and aggregate levels:
Ft =
Kt
;
Yt
Ki;t
;
Yi;t
Fi;t =
Ft =
N
X
Fi;t ; Kt =
i=1
N
X
Ki;t ; Yt =
i=1
N
X
Yi;t
(E.525)
i=1
3) Intetemporal balance:
T X
N
T
X
X
h
Pi;t 1 + thci Ci;t
=
rt (1
t=0 i=1
tk ) Kth + Rth + wth (1
tl ) LSth
(E.526)
t=0
"
T
T
X
X
Pi;t Yi;t =
rt (1
t=0
H
X
tk ) Ki;t +
wth Lhi;t
t=0
h=i
T
T
H
X
X
X
Gt 7
RVt +
Rth
t=1
t=1
h=1
#
!
(E.527)
(E.528)
4) Trade and …nance:
N
T X
N
T X
X
X
P Mi;t Mi;t
P Ei;t Ei;t =
(E.529)
t=0 i=1
t=0 i=1
N
X
P Ei;t Ei;t
i=1
N
X
P Mi;t Mi;t =
F Lt
(E.530)
i=1
5) Public sector and …nancial deepening:
1
X
e
t=0
RVt =
t
RVt 7
1
X
e
t
Gt + Rth
(E.531)
t=0
H X
N
N X
H
X
X
h
Pi;t thci Ci;t
+
wth tl Lhi;t + rt (1 + tk ) Ki;t
h=1 i=1
i=1 h=i
The general equilibrium is achieved when the excess demand are zero in each market for each
period representing balance between demand and supply in each market. Households and producers
optimise given their budget constraints. Relative price adjustment mechanisms guarantee the most
e¢ cient outcome in these markets. The existence of the general equilibrium is guaranteed by …xed
point theorems and solved using the dynamic routines in the GAMS/MPSGE software. Given the
properties of demand and supply functions equilibrium is stable and unique and gives the evolution
of the model economies from 2006 to 2101(see Bhattarai (2007) and Bhattarai (2011)).
203
This model has been applied to China, India to study optimal and actual capital deepening
ratios (OFDR and AFDR) and the results are summarised in Table 15. These show that the
optimal capital intensity in China at 0.81 is much lower than in India’s 1.54. This implies India
economy being more capital intensive than the Chinese economy in production technology. However
the ratio of actual stocks of the …nancial assets to GDP is much higher in China at 1.88 compared
to 0.78 in India. Thus China is over-…nanced with over …nancing ratio (OFR) at 2.3 and India is
under-…nanced with the OFR at 0.49. This result implies speedy growth in India requires a rapid
growth of its …nancial sector (see Dougas and Rajan (2008) and Kawai (2011)).
Ta b le 1 5 : O p tim a l a n d a c tu a l …n a n c ia l d e e p e n in g ra tio s a n d G row th R a te s fo r 2 0 0 8 -1 2
Countries
OFDR
AFDR
OFR
GR 2008-12
China
0.81
1.88
2.3
9.30
India
1.54
0.78
0.49
6.50
N o te : O F D R a n d A F D R a re o p tim a l a n d a c tu a l …n a n c ia l d e e p e in g ra tio s; O F R ove r …n a n c in g ra tio
B a s e d o n B h a tta ra i (2 0 1 4 ); M o re d e ta ils ava ila b le u p o n re q u e s t.
Main focus of this DCGE model is to study the long run growth in output and employment
across sectors given endogenous or exogenous changes in the rate of taxes and tari¤s. Comparative
static features of Parikh, Narayana, Panda and Kumar (1995) could be put in such dynamic frameworks to study the evolution of Indian economy in coming decades. GTAP and GTAPinGAMS
models also could be applied for empirical investigation on equilibrium relations among all South
Asian economies to test theories of Bhagwati and Srinivasan (2002), Panagaria (2006), Neary (1998)
for assessing how these countries bene…t from inter and intra regional trade. Various arrangements
for creating free trade area (FTA) under the SAPTA or other bilateral agreements can be studied
constructing small open economy or multicountry trade models. Opening economies for trade with
specialisation based on comparative advantages are essential features of the growth competition.
A free trade association (FTA) under the South Asia Free Trade Association (SAFTA) can open
such opportunities of cross boarder production and trade. India can sell skill, technology and manufacturing goods to its neighbors; it can buy cheaper hydro electricity from Nepal and Bhutan and
agricultural products from Pakistan. Gains from cooperative rather than discriminatory approach
with respect to the rest of the world could be used for the development of the region. Given the
development of the GTAP/Unido/STAN databases it is possible now to analyse the signi…cance
of bilateral and multilateral trade relations among these countries. As opening intra-regional FDI
could increase productivities, it is essential to remove limited product coverage, existence of negative lists and restrictive rules of origin that are becoming obstacle in such settings (Taneja and
Sawhney (2007)).
204
5.0.9
Macroeconomic simulation model of South Asia
With time series on major components of aggregate demand, price levels, interest rate and exchange
rates presented above it is possible to construct a macroeconometric model to forecast macro variables of India and South Asian economies. Essentially these models are helpful in studying trends
and forecasts in the short run specially useful for annual projection of macro quantities such as consumption, investment, imports or exports or public spending and prices in the private and public
sectors given projections of the public …nance or the BOP conditions of the economy. Each South
Asian economy have some sorts of open economy IS-LM model underlying their policy decisions and
assessing the macroeconomic ‡uctuations. These basically Keynesian demand driven models are
popular as they are easier to compute and implement because of recent innovations in econometric techniques (Hendry and Doornik (1994), Bhattarai (2008) and Bhattarai and Mallick (2013)).
We estimate simultaneous equations models of India, China and SAARC countries to study how
in‡ation, current account balance and growth rates relate to …scal and monetart policy variables
represented by the size of the government (g_y) ad liquidity ratio (M2_y) and structural facture
(a_g). Again results presented in tables 16 to 19 below show signi…cance (t_prob) and sign of
coe¢ cients ( ) on them vary tremendously across these countries. This means markets and policies
are very di¤erent among these countries.
T a b l e 1 6 : M a c r o e c o n o m i c m o d e l o f i n ‡a t i o n , c u r r e n t a c c o u n t a n d g r o w t h i n I n d i a a n d N e p a l
In d ia
I n ‡a t i o n
N epal
C A b a la n c e
t_ prob
G row th
t_ prob
I n ‡a t i o n
t_ prob
C A b a la n c e
t_ prob
G row th
t_ prob
t_ prob
g_ y
0 .0 8 4
0 .2 2 3
-0 .0 0 7
0 .9 8 5
0 .0 0 7
0 .9 0 4
-0 .4 9 4
0 .0 1 7
0 .0 1 7
0 .1 9 2
-0 .1 1 1
0 .4 6 3
a_ g
1 .4 1 7
0 .0 0 1
-6 .2 7 5
0 .0 0 8
-0 .1 3 8
0 .6 6 7
0 .2 5 8
0 .0 4 9
-0 .0 0 2
0 .8 1 2
-0 .0 3 8
0 .6 9 3
M 2_ y
0 .4 5 5
0 .0 0 2
-3 .3 3 7
0 .0 0 0
-0 .0 0 2
0 .9 8 5
0 .2 5 1
0 .0 3 0
0 .0 0 3
0 .6 7 2
0 .0 3 2
0 .7 1 4
const
-0 .5 3
0 .0 0 5
3 2 1 .9
0 .0 0 3
9 .6 2 6
0 .5 1 8
-1 0 .4 4
0 .2 2 9
-0 .1 6 4
0 .7 8 0
5 .3 7 7
0 .4 4 1
R
2
= 0 .6 6 ; N = 3 6 ; F (9 ,6 5 ) = 5 .9 9 [0 .0 0 0 0 ] * *
R
2
= 0 .5 9 ; N = 3 6 ; F (9 ,6 5 ) = 3 .2 9 5 1 7 [0 .0 0 2 3 ] * *
T a b l e 1 7 : M a c r o e c o n o m i c m o d e l o f i n ‡a t i o n , c u r r e n t a c c o u n t a n d g r o w t h i n B a n g l a d e s h a n d C h i n a
B a n g la d e sh
I n ‡a t i o n
C h in a
C A b a la n c e
t_ prob
G row th
t_ prob
I n ‡a t i o n
t_ prob
C A b a la n c e
t_ prob
G row th
t_ prob
t_ prob
g_ y
1 .7 1 3
0 .0 0 3
-0 .0 0 6
0 .9 6 2
-0 .4 3 6
0 .0 0 3
-0 .0 6 6
0 .6 8 9
3 .0 5 8
0 .2 4 7
0 .1 7 8
0 .0 3 1
a_ g
0 .9 2 5
0 .0 0 3
-0 .0 5 0
0 .5 6 8
-0 .2 1 3
0 .0 2 1
-1 .3 5 7
0 .0 5 7
-1 3 .2 4
0 .2 3 2
-0 .7 7 5
0 .0 2 6
M 2_ y
0 .1 3 3
0 .1 8 9
0 .0 1 3
0 .5 9 5
0 .0 1 8
0 .4 8 9
-0 .2 1 0
0 .0 7 5
-0 .6 0 3
0 .7 3 9
-0 .1 2 7
0 .0 2 7
const
-4 4 .4 6
0 .0 0 4
0 .6 5 5
0 .8 5 8
1 5 .6 6
0 .0 0 0
5 4 .4 9
0 .0 3 6
3 3 9 .3 1
0 .3 4 9
3 5 .9 3
0 .0 0 5
R
2
= 0 .8 7 ; N = 3 6 ; F (9 ,6 5 ) = 9 .9 3 0 9 1 [0 .0 0 0 0 ] * *
R
205
2
= 0 .6 6 ; N = 3 6 ;F (9 ,6 5 ) = 4 .0 7 8 9 8 [0 .0 0 0 3 ] * *
T a b l e 1 8 : M a c r o e c o n o m i c m o d e l o f i n ‡a t i o n , c u r r e n t a c c o u n t a n d g r o w t h i n P a k i s t a n a n d S r i L a n k a
P a k ista n
I n ‡a t i o n
S ri L a n ka
C A b a la n c e
t_ prob
G row th
t_ prob
I n ‡a t i o n
t_ prob
C A b a la n c e
t_ prob
G row th
t_ prob
t_ prob
g_ y
0 .1 8 7
0 .0 5 5
-0 .1 2 9
0 .1 4 5
-0 .1 8 0
0 .0 0 0
0 .0 2 3
0 .8 1 2
0 .0 0 7
0 .6 9 6
0 .0 3 5
0 .4 0 9
a_ g
0 .3 2 1
0 .3 6 8
0 .0 6 4
0 .8 4 4
-0 .1 3 1
0 .4 1 6
-0 .0 7 4
0 .7 7 2
0 .0 5 8
0 .0 0 3
-0 .1 3 1
0 .2 8 8
M 2_ y
-0 .2 8 4
0 .1 7 8
0 .0 7 7
0 .6 1 2
0 .2 3 8
0 .0 1 6
-0 .5 8 2
0 .1 8 4
0 .0 3 1
0 .0 9 4
-0 .1 5 4
0 .4 2 2
const
1 0 .1 8 4
0 .4 7 8
-6 .4 4 6
0 .6 2 4
0 .0 5 1
0 .9 6 9
3 2 .9 2 9
0 .1 0 3
-8 .9 4 0
0 .0 1 5
1 2 .5 6 0
0 .1 5 2
R
2
= 0 .5 6 ; N = 3 6 ; F (9 ,6 5 ) = 2 .9 2 6 1 [0 .0 0 5 7 ] * *
R
2
= 0 .4 2 ; N = 3 6 ; F (9 ,6 5 ) = 1 .8 6 8 8 3 [0 .0 7 2 6 ]
T a b b l e 1 9 : M a c r o e c o n o m i c m o d e l o f i n ‡a t i o n , c u r r e n t a c c o u n t a n d g r o w t h i n B h u t a n a n d M a l d i v e s
B hutan
I n ‡a t i o n
M a ld ive s
C A b a la n c e
t_ prob
G row th
t_ prob
I n ‡a t i o n
t_ prob
C A b a la n c e
t_ prob
G row th
t_ prob
t_ prob
g_ y
-0 .0 8 8
0 .1 5 7
-0 .0 0 2
0 .3 2 4
0 .1 8 5
0 .0 5 3
0 .1 1 6
0 .2 0 3
-0 .0 0 4
0 .0 0 8
-0 .1 6 4
0 .0 3 9
a_ g
-0 .1 6 7
0 .2 0 6
0 .0 0 7
0 .0 7 3
-0 .2 7 0
0 .1 7 7
-0 .9 9 6
0 .0 0 9
0 .0 0 8
0 .1 4 6
0 .2 6 0
0 .3 9 9
M 2_ y
-0 .1 5 7
0 .0 2 0
0 .0 0 1
0 .4 5 2
-0 .1 1 8
0 .2 3 3
0 .0 4 7
0 .7 1 3
-0 .0 0 9
0 .0 0 0
0 .0 0 9
0 .9 3 2
const
2 2 .3 4
0 .0 0 1
-0 .2 6 5
0 .1 6 1
1 2 .8 1
0 .1 7 1
6 .0 7 3
0 .1 6 9
0 .3 1 8
0 .0 0 0
1 0 .3 8 5
0 .0 0 9
R
2
= 0 .5 5 ; N = 3 3 ; F (9 ,6 5 ) = 2 .8 9 2 8 1 [0 .0 0 6 1 ] * *
R
2
= 0 .7 9 ; N = 3 3 ; F (9 ,6 5 ) = 6 .7 8 8 4 3 [0 .0 0 0 0 ] * *
The business cycle analyses in DSGE models contrain micro-foundations, dynamics and rational
expectations, stochastic shocks to preferences, technologies and policies along with the nominal and
real rigidities than present in above models. Analysis of short or long run multipliers, variance
decompositions and impulse responses to changes in policies and shocks on the deviations of model
variables from the steady state are often the focus of such analysis. Computations have become
easier for such models after development of Sim’s BVAR algorithm in the MATLAB and dynare.
However we skip this model here as the growth and redistribution analysis in the DCGE model
presented above is better suited for analysis of structural features of the South Asian economies
than these DSGE models. See details in Bhattarai (2014) paper for the Institute for Economic
Growth, New Delhi, August.
5.1
Exercise 7
1. The optimization problem facing the social planner of an economy is:
M ax
1
X
t=0
206
t
U (ct )
s.t.
ct + xt = yt = AGt kt (lG ht )1
ht+1 = (1
h )ht
+ Aht (1
: Resource constraint
lG )ht : Law of motion of human capital
xt = pkt ikt : Current account constraint
ikt = kt+1
(1
k )kt
(E.532)
(E.533)
(E.534)
: Investment
Formulate the constrained optimisation form of this problem. Derive the balanced growth using
the standard optimal conditions of this model.
207
6
L6: Dynamic Programming for Macro Dynamics
1. Consider a version of Brock-Mirman type dynamic programming problem
max
U=
1
X
t
ln(Ct )
0<
<1
(F.535)
t=0
subject to market clearing condition
Kt+1 + Ct = AKt
0<
<1
(F.536)
Here output (AKt ) is either consumed (Ct ) or invested (Kt+1 ) :
1. What are the control and state variables in this model and why?
2. Explain the meaning of the value function (Bellman equation) and the policy functions of this
problem, V1 (K 0 ) = ln C + V0 (K 0 ); where K 0 amount of optimal capital stock.
3. Assume K 0 = 0 for the last period. Demonstrate a recursive solution method of this problem
using three iterations of the policy and value functions and characterise the rest of the solution.
4. Use limit theorem to …nd the explicit solution of the value function.
5. Introduce a stochastic technology At+1 =
At + "t and explain conjectures to solve this
problem.
Iteration Procedure Starts from the Terminal Period
Bellman (1957) and Sargent (1987) technique can be used to solve for state and co-state variables
iteratively from the following value function.
V1 (K) = max fln C + V0 (K 0 )g
c;k
When the capital stock at the terminal period is made zero. all is consumed at the last period
thus KT +1 + CT = AKT
CT = AK 0
(F.537)
Household utility is given by
V1 (K) = ln C = ln A +
Sinceln (AK 0 ) = ln A +
ln K
(F.538)
ln K: Now use this for the second last period
V2 (K) = ln C + V1 (K 0 ) = ln (AK
208
K 0 ) + V1 (ln A +
ln K)
(F.539)
Iteration Procedure in Dynamic Optimisation
K 0 ) + V1 (ln A +
V2 (K) = ln (AK
ln K)
(F.540)
Now the optimal value of , the capital to be saved for the next period in the second last period
can be obtained using the …rst order conditions
@V2 (K)
=
@K
1
K0 =
K0 =
K0
=0
(F.541)
K 0)
(AK
1+
+
K0
AK
(F.542)
AK
(F.543)
Iteration Procedure in Dynamic Optimisation
Again consumption is total output minus the savings for the next period
K 0 = AK
C=Y
1+
AK =
1
1+
AK
V2 (K) = ln C + V1 (K 0 ) now can be written with V1 (K) = ln A +
V2 (K) = ln
1
1+
AK
+
ln A +
(F.544)
ln K as
ln K
(F.545)
ln K)
(F.546)
This now contains only one state variable, the capital stock.
V2 (K 0 ) = ln
1
1+
AK
+
1
1+
AK
+
ln A +
V2 (K 0 ) = ln
(ln A +
ln
1+
AK
(F.547)
Iteration Procedure in Dynamic Optimisation
V2 (K 0 ) = ln
1
1+
A +
ln A +
ln
1+
A +
(1 +
) ln K 0
(F.548)
In a similar fashion consider the problem in the third last period given in (17)
V3 (K) = ln C + V2 (K 0 )
Using the market clearing condition and above solution becomes:
209
(F.549)
2
6
6
4
K 0) +
max V3 (K) = ln (AK
K
1
1+
ln
ln A +
3
A +
ln
7
7
5
A
1+
) ln K 0
+ (1 +
Iteration Procedure in Dynamic Optimisation: Period 3
(F.550)
The optimal …rst order conditions on control (consumption) and state (capital stock) variables.
@V3 (K)
=
@K
1
K0 =
(1 +
K0 =
=0
(F.551)
(F.552)
2 2
+
1+
)
K 0)
) (AK
+
2 2 AK
(F.553)
2 2
+
K 0 = AK
C=Y
(1 +
K0
+
K0
AK
1+
+
Iteration Procedure in Dynamic Optimisation: Period 3
2 2 AK
(F.554)
1
AK
1+
+ 2 2
Combining solutions for periods
h two and three return function for period 3 is
0
0
V3 (K ) = ln (AK
K ) + ln 1+1 A + ln A +
ln 1+ A + (1 +
C=
V3 (K 0 ) = ln
1
+
1+
2 2 AK
+
2
ln
1
1+
(F.555)
) ln K 0
3
A
6
6 + ln A +
ln 1+ A
4
+ (1 + ) ln K 0
7
7
5
i
(F.556)
Upon further simpli…cation by collecting terms involving capital stock it becomes
Iteration Procedure in Dynamic Optimisation: Period 3
Upon further simpli…cation by collecting terms involving capital stock it becomes
V3 (K 0 )
=
ln
+ ln
+
1
1+
1+
1+
A +
2
ln A +
1
A +
+ 2 2
+ 2 2 ln K 0
210
2
ln
(1 +
1+
) ln
A
+
1+
2 2
+
A
2 2
(F.557)
Iteration Procedure in Dynamic Optimisation: Period 4
K 0) +
max V4 (K) = ln (AK
1+
2 2
+
ln K 0
(F.558)
K
Again using the …rst order conditions
@V4 (K)
=
@K
1
K0
AK
K0 =
K0 =
C=Y
1+
+
1+
2 2
+
3 3 AK
(F.561)
2 2
+
K 0 = AK
(F.560)
3 3
+
2 2
1+
(F.559)
K 0)
) (AK
+
=0
K0
(1 +
+
2 2
+
2 2
+
3 3
+
+
3 3 AK
(F.562)
Iteration Procedure in Dynamic Optimisation: Period 4
C=
1
1+
+
2 2
+
3 3 AK
(F.563)
Using the optimal solution for the next period the value function of the 4th last period is:
V4 (K 0 )
=
ln
1
1+
2
6
6 +
6
+ 6
6
4
+
2 2
+
A + 2 ln A
h
i
ln 1+ A + ln 1+ 1+ 2 2 A +
h
i
2 2
(1 + ) ln 1+ + + 2 A2 +
ln
2
3 3 AK
1
1+
1+
+
2 2
Iteration Procedure in Dynamic Optimisation: Period 4
Optimal capital accumulation after 4th iteration becomes:
211
ln K 0
3
7
7
7
7
7
5
(F.564)
V4 (K 0 )
=
1
ln
1+
+
2
+
3
ln
+
+
1+
(1 +
2
3 3A
ln
+
ln
1+
1
+
2 2A
A
1+
ln A +
+
+
+
2 2
+
+
) ln
1+
+
ln
2 2
+
"
+
1+
2 2
+
+
3 3
2 2
+
A
3 3
#
A
2 2
A
1+
1+
2 2
2 2
+
3 3
ln K 0
(F.565)
Conclusion
Thus in a dynamic economy accumulation should depend on preferences of households, their
discount factors, technology of …rms and initial and terminal conditions.
Economy may experience various trajectories of growth paths depending on con…gurations of
these parameters.
One numerical example of a dynamic applied general equilibrium model for the Humberside
economy is in Bhattarai (2007); in stochastic settings these solutions can be approximated by
more robust estimation as in Bhattarai(2010).
Solve the dynamic programming problem of the habit persistent model
1
P
t
ln(ct + ct
Kt+1 + Ct = AKt
0<
1. max
U=
1)
0<
<1
t=0
subject to
6.1
< 1; A > 1; K0 > 0; c
1
given
Exercise 8
1. Consider a version of Brock-Mirman type dynamic programming problem
1
P
t
ln(ct )
0< <1
max U =
t=0
subject to
Kt+1 + Ct = AKt
0<
<1
212
(a) what are the control and state variables in this model and why?
(b) Explain the meaning of the value function (Bellman equation) and the policy functions
of this problem
V1 (K) = ln C + V0 (K):
(c) Assume V0 (K) = 0: Demonstrate a recursive solution method of this problem using four
iterations of the policy and value functions.
(d) Use limit theorem to …nd the explicit solution of the value function.
(e) Introduce a stochastic disturbance term for the state variable and show how it can be
solved.
Exercise
2. Solve the dynamic programming problem of the habit persistent model
1
P
t
max U =
ln(ct + ct 1 )
0< <1
t=0
1. subject to
Kt+1 + Ct = AKt
0<
< 1; A > 1; K0 > 0; c
1
given
3. What is dynamic programminsg squared?
References
References
[1] Bellman R. (1957) Dynamic Programming, Princeton University Press, New Jersey.
[2] Sargent T. J. (1987) Dynamic Macroeconomic Theory, Chapter 1, Harvard University
Press,Cambridge, Mass.
[3] Ljungqvists L. and Sargent T.J (2012) Recursive Macroeconomic Theory, MIT Press, 3rd edition.
[4] http://ideas.repec.org/s/dge/qmrbcd.html
Bewley T (1994) A Di¢ culty with the Optimum Quantity of Money, Econometrica, 51, 5,
1485-1504
Aiyagari S. R. (1994) Uninsured Idiosyncratic Risk and Aggregate Saving, Quarterly Journal
of Economics, 109, 3, 659-684
Following is based on Bhattarai K (2014) Money and economic growth, Journal of Economic
Asymmetries, 11 (2014), 8-18.
213
6.2
Money in Growth Models
It has been agreed for long that money serves economic purposes as a medium of exchange and
unit of standard as well as a standard of di¤ered payment and means of a store. Yet there remains
a substantial debate about the neutrality and non-neutrality of money in the long run growth.
While Tobin (1965) viewed that money in the form of public debt could be instrumental in channeling savings to investment and hence lead to the higher growth rate, Friedman (1968) opined
the growth rate of money should not be greater than that of output for a smooth functioning
economy. In theory Sidrauski (1967) showed the role of money in growth, putting money in the
utility (MIU) function of an intertemporally optimising representative household and came to the
conclusion that money is super neutral, will not have any real impacts and higher rate of growth of
money only causes in‡ation. Similarly Brock (1973) had provided more extensive perfect foresight
model to show contribution of money in economic growth. These early views on relations between
money and growth are endorsed in subsequent works by Hayakawa (1986), Rankin (1992), Gomme
(1993), Balasko (2003), Berentsen et al. (2012) and Aruoba et al (2011). However there seem to
be no explicit numerical analysis on showing ‡uctuations in macro economic variables under these
theoretical exercises. Purpose of this paper is to assess how these theoretical propositions may
be brought into numerical analysis and whether conclusions reached in those studies are robust
enough to the way money is introduced in these models. Growth of money is exogenous in models
with cash in advance (CIA) constraints or is endogenous in the models with money in the utility
(MIU) functions. Is the super- neutrality proposition of money independent of the way money is
introduced in a model? If so super-neutrality should hold in all models whether they have the
CIA or the MIU forms. This issue is illustrated with simulations of popular CIA and MIU models
discussed in Williamson (2008) and Walsh (1998) to the reasonable set of parameters characterising
the three economies. These simulations provide some insights on the role of money in the business
cycle and the growth of the model economies.
6.2.1
Friedman Rule with Cash in Advance Constraint
Friedman is known for his statement that the stock of money should equal the growth rate of output.
Money is introduced exogenously in a cash in advance monetary economy where a representative
household maximises lifetime utility U ( ) from consumption (Ct ) but experiences disutility from
labour e¤orts put in work, V (Lt ). The problem of the economy is to maximize this utility (F.566)
with the technology (F.567), cash in advance (F.568) and lifetime budget constraints (F.569) as:
max
1
X
t
[U (Ct )
t=0
214
V (Lt )]
(F.566)
subject to a technology constraint:
Yt = zLt
(F.567)
Pt Ct + qt Bt+1 + Pt st Xt+1 + Pt Tt = Mt + Bt + Pt Xt
(F.568)
and a cash in advance constraint:
where Pt Ct is consumption expenditure, Pt price of goods, Ct consumption, Bt+1 is the amount of
nominal bonds, qt is the price of nominal bonds, Xt+1 real bonds, st prices of real bonds, Tt lump
sum tax payment and Mt the stock of money. Budget constraint of the consumer include income
from production and allocation of money for the next period.
Pt Ct + qt Bt+1 + Pt st Xt+1 + Pt Tt + Mt+1 = Mt + Bt + Pt Xt + Pt zLt
(F.569)
Government controls the money supply and engages itself in an in‡ationary tax. Its budget constraint for a particular time t is:
M t+1
Mt =
The stock of money grows at a constant rate
Mt =
Pt Tt
(F.570)
, thus M t+1 = (1 + ) M t . With this provision,
Pt Tt : Normalising the cash in advance and budget constraint by
1
Mt
and denoting the
real values in small case letters, the cash in advance constraint and budget constraints become:
pt Ct + qt bt+1 (1 + ) + pt st Xt+1 + pt Tt = mt + bt + pt Xt
(F.571)
and
pt Ct + qt bt+1 (1 + ) + pt st Xt+1 + pt Tt + mt+1 (1 + ) = mt + bt + pt Xt + pt zLt
(F.572)
The representative agent in the economy chooses Ct , Lt , bt+1 , Xt+1 , mt+1 from t = 0; 1; 2; : to 1:
The Bellman value function for this problem is:
v (mt ; bt ; Xt ; pt ; qt ; st )
max
Ct ;Lt ;bt+1 ;Xt+1 ;mt+1
[U (ct )
V (Lt )] + v [mt+1 ; bt+1 ; Xt+1 ; pt+1 ; qt+1 ; st+1 ]
(F.573)
215
6.2.2
Dynamic optimisation in CIA Model
It is easier to solve the above Bellman problem if it is written in a Lagrangian constrained optimisation problem as:
L (Ct ; Lt ; bt+1 ; Xt+1 ; mt+1 ;
t;
t)
1
X
=
t=0
+
+
t
t
t
[U (Ct )
"
V (Lt )]
(F.574)
mt + bt + pt Xt
qt bt+1 (1 + )
"
pt Ct
pt st Xt+1
pt Tt
mt + bt + pt Xt + pt zLt
qt bt+1 (1 + )
pt st Xt+1
pt Tt
#
pt Ct
mt+1 (1 + )
#
This CIA model is solved analytically with the …rst order conditions for optimisations as:
Ct : U 0 (Ct )
Lt :
bt+1 :
(
V 0 (Lt ) +
qt (1 + ) (
Xt+1 :
mt+1 :
+
t
pt st (
t
t
(1 + )
t pt z
+
+
t)
t)
t
t ) pt
+
=0
=0
+
@v
=0
@bt+1
@v
=0
@Xt+1
@v
=0
@mt+1
+
(F.575)
(F.576)
(F.577)
(F.578)
(F.579)
By the envelop theorem on di¤erentiating the Bellman equation:
@v
=(
@bt
t
@v
= pt (
@Xt
+
t
+
t)
t)
(F.580)
(F.581)
@v
= ( t + t)
(F.582)
@mt
Combining above last three and the …rst two …rst order conditions, the middle three …rst order
conditions can be expressed as:
qt (1 + ) U 0 (Ct )
U 0 (Ct+1 )
+
=0
pt
pt+1
216
(F.583)
sU 0 (Ct ) + U 0 (Ct+1 ) = 0
(F.584)
(1 + ) V 0 (Lt )
U 0 (Ct+1 )
+
=0
pt z
pt+1
(F.585)
Higher productivity lowers the level of employment:
V 00
dL
=
00
d
(1 + ) V
z 2 U 00
<0
(F.586)
Here
can be set to achieve the optimal in‡ation in in‡ation targeting regimes to maximize the
1
P
t
level of welfare in the economy,
maxt 1
[U (Ct ) V (Lt )]. The optimal employment (L ) is
fCt ;Lt gt+0 t=0
obtained implicity
0
zU (zL )
0
V (L ) = 0
(F.587)
The optimal growth rate of
1 where the
h money supply given by ithe Friedman rule is =
1
nominal interest rate is zero R = q 1 = 0 =) q = 1 , the real interest rate is r = 1 1, cash
in advance constraint does not bind
=
=
6.2.3
U 0 (C)
p
C:U 0 (C)
1+
=
1 because
= 0.
C:U 0 (C) V 0 (L)
C:U 0 (C)
U 0 (C)
=
1+
pz
1+
1+
p
C:U 0 (C)
C:U 0 (C)
C:U 0 (C)
=
1
=
(1
1+
1+
1+
1+
1+
=
q) (F.588)
Steady State in the CIA Model
With the …rst order conditions for dynamic optimisation, as given above; the steady state levels of
prices and quantities are obtained in terms of parameters
;
and z.
First simplify the steady
state with mt = 1; bt = 0; Xt = 0: Then the above equilibrium conditions ( M t =
Pt Tt ) with
and the budget constraint become:
pt Ct = 1 +
(F.589)
This shows that in CIA model like this money is held only for consumption which equals total
output, Ct = zLt . Setting steady state variables to constant values, Ct = C, Lt = L, pt = p, qt = q,
st = s, analytical solutions for prices and quantities are then expressed in terms of subjective
discount factor ( ) and the growth rate of money supply ( ).
217
Price of nominal bond from (F.583) is given in terms of
q=
and :
(F.590)
1+
Price of real bond from (F.584) is:
s=
(F.591)
The level of employment is given implicitly by (F.585)
(1 + ) V 0 (Lt )
zU 0 (zL) = 0
(F.592)
Given the steady state (C) the price of commodity is directly proportional to the growth rate of
money supply and inversely to the level of output and the productivity of the labour:
1+
1+
=
(F.593)
C
zL
Nominal interest rate (R) depends on the price of nominal bonds (q), directly on the growth rate
p=
of money ( ) and inversely on the subjective discount factor ( ).
R=
1
q
1=
1+
1
(F.594)
Real interest rate (r) inversely relates to the price of real bond (s) and the subjective rate of time
preference ( ):
r=
1
s
1=
1
1
(F.595)
In‡ation (i) equals the growth rate of money supply in the steady state:
i=
Pt+1
Pt
1=
pt+1M t+1
pt M t
1=1+
1=
(F.596)
Fisher equation implies gross real interest (1 + r) rate to be equal to the ratio of gross nominal
interest rate (1 + R) to the in‡ation factor (1 + i), which gives analytical expression that the gross
real interest rate is the inverse of the subjective discount factor as:
1+r =
Thus the prices q; s; p; R; r; i and
1+R
1+
=
1+i
1+
=
1
(F.597)
are all solved in terms of growth rate of money ( ) and
the discount rate ( ). From the equilibrium condition it is clear that Y = C = zL =
L=
1+
zP
. Thus the level of output, consumption and employment increase with
1+
P
and
and decline with
in‡ation. While the greater liquidity helps to mobilise resources, the higher in‡ation distorts the
218
Table 54: Parameters of CIA Model
Parameters
L0
z
m
b
X
CIA
0
0
0:03
0:99
100
(1; 0:05)
1
intertemporal decisions. Higher growth rate of money supply lowers the level of employment by
causing distortions through in‡ation (see Ellison and Pearlman (2011) for saddle point solutions o¤
the steady state).
Now let us perturb this model around this steady state and show how the shocks in growth rate
of money supply or the level of technology can impact on the transitional dynamics of the economy.
These are shown in a series charts that represent solutions of this model to the shocks in
or z for
given values of parameters in Table 1, as shown in Figures 1 to 5 with time period indexed from t1
to t99 in their horizontal axes.
yy
yy
110
108
106
104
102
100
98
96
94
Figure 1: Fluctuations in output
219
t97
t94
t91
t88
t85
t82
t79
t76
t73
t70
t67
t64
t61
t58
t55
t52
t49
t46
t43
t40
t37
t34
t31
t28
t25
t22
t19
t16
t13
t10
t7
t4
88
t1
90
t100
92
zz
zz
1.1
1.08
1.06
1.04
1.02
1
0.98
0.96
0.94
0.92
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0.88
0
0.9
Figure 2: Idiosyncratic technological shocks with unpredictable and long lasting consequences.
cc
cc
cc
cc
110
108
106
104
102
100
98
96
94
92
Figure 3: Consumption ‡uctuations follow from ‡uctuations in output and income
220
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
88
0
90
Table 55: Parameters of CA Model
a
gk
L0
v
Parameters
ln (z)
M0
Country 1
0.05
0.5
0.95
0.01
100
1
0.01
(1; 0:05)
100
Country 2
0.05
0.4
0.99
0.02
100
2
0.02
(1; 0:05)
100
Country 3
0.05
0.45
0.98
0.015
100
3
0.015
(1; 0:05)
100
ush
ush
4.7
4.68
4.66
4.64
4.62
4.6
4.58
4.56
4.54
t97
t94
t91
t88
t85
t82
t79
t76
t73
t70
t67
t64
t61
t58
t55
t52
t49
t46
t43
t40
t37
t34
t31
t28
t25
t22
t19
t16
t13
t10
t7
t4
t1
4.5
4.48
t100
4.52
Figure 4: Utility of households ‡uctuates along with variations in consumption and liesure
This means under the Friedman rule the cash in advance constraint does not bind. There are
no distortions between the real and nominal assets; the rate of return in all assets are equal in
equilibrium.
With parameter sets in Table 2, a simple three country version of this model is solved subject
to idiosyncratic technological shocks for t1 to t15 years to generate time pro…les of capital, output,
prices, money, consumption, investment, labour supply and lifetime utilities of households as shown
by multiple bars for three interdependent economies, i = 1; 2; 3. Countries vary in labour productivity (a), discount rates ( ), terminal growth rates of capital stock (gk ) and velocity of money
(v) and the growth rate of money ( ) : Even a small di¤erence in these parameters cause a huge
di¤erence in their respective positions. Fluctuations in these economies originate in the …nancial
sector and can have signi…cant consequences in the levels of welfare in these economies.
221
Figure 5: Country two (i2) which discounts less its future accumulates more capital and is better o¤ in the long run.
welfare
168.052 i1
130.316 i2
137.152 i3
i1
i2
i3
Figure 6: Less prudent country gains more than more prudent ones when risks are pooled together in a coalition.
Main lessons that can be drawn from the CIA model is that the …nancial crises occur because of
shifts in the investor and consumer con…dences, changes in perceptions and beliefs and technological
shocks that hit the system. Impacts of such changes can be very sudden which a¤ects the velocity of
circulation of money, technological progress, discount factors or the beliefs in the underlying growth
rates of the economy (Rankin (1992) and Spencer (2008)). These factors impact on prices, trend
of output, employment, consumption and other macro variable in the model economy as shown by
the path of model variables and welfare solutions as presented in above …gures. It is clear that a
222
balanced path of …nancial depth enhances welfare of households but this depends on the attitudes
of consumers towards the future of the economy. These features are not typical of an economy with
exogenous money but can persist even with the endogenous growth rate of money. This is shown
using a solution of the money in utility function model in the next section.
6.3
Money in the Utility Function and Growth
Role of money was for pure exchange in the cash in advance model and the growth rate of money
was exogenous. There are circumstances when household prefer to store more or less cash depending
on expected utilities from it. Thus the stock of money they like to keep is endogenous and is a part
of utility maximising choice of households. This feature is captured by the money in the utility
function model of Sidrauski (1967). When this desire is excessive it causes a crisis in the system as
observed during the recession that started in 2008. The problem of household as in the CIA model
is to maximise the lifetime welfare (W ) from consumption (ct ) and possessing the stock of money
(mt ).
max W =
1
X
t
U (ct ; mt )
(F.598)
t=0
subject to the production (Yt ) technology constraint with capital (Kt ) and labour
(Lt ) inputs and the technological shock (zt ):
Yt = zt F (Kt ; Lt )
Under constant returns to scale yt = f (kt ) where yt =
(F.599)
Yt
Lt
and kt =
Kt
Lt .
Economy wide budget
constraint is given by:
Mt 1
Mt
= Ct + Kt +
Pt
Pt
where Yt is output, Pt price of goods, Ct consumption, Kt+1 is capital stock,
Yt +
t Lt
+ (1
) Kt
1
+
each individual, Mt money, Lt employment and
(F.600)
t
is net transfer for
is the rate of depreciation of capital. In per
capita terms:
! t = f (kt ) +
t
+
1
1+n
kt
1
+
(1 +
mt 1
= ct + kt + mt
t ) (1 + n)
(F.601)
The recursive dynamic program of this household is:
V (! t ) = u (ct ; mt ) + V (! t+1 )
223
(F.602)
V (! t ) = max u (ct ; mt ) + V f (! t
6.3.1
ct
mt ) +
t+1
1
1+n
+
(! t
ct
mt ) +
mt
(1 + t+1 ) (1 + n)
(F.603)
Dynamic optimisation in the MIU model
Again using the Lagrange multiplier ( t ) to simplify this constrained optimisation problem:
L (ct ; mt ;
1
X
t
t)
=
1
X
t
u(ct ; mt ) +
t=0
f (! t
ct
mt ) +
t+1
1
1+n
+
t=0
(! t
ct
mt ) +
mt
(F.604)
(1 + t+1 ) (1 + n)
As before solving MIU model explicitly means expressing the prices and quantities like yt ; kt ; ct ; mt
in terms of the preference and technology parameters as ; ;
and n. In other words the optimal
values of variables are determined by subjective discount factor ( ), depreciation ( ), productivity
of capital ( ) and growth rate of population (n). This is done using the …rst order conditions:
ct : uc (ct ; mt )
fk (kt ) +
Here marginal utility of holding capital
utility of consumption uc (ct ; mt ):
m : um (ct ; mt )
h
fk (kt ) +
1
1+n
fk (kt ) +
1
1+n
1
1+n
V! (! t+1 ) = 0
i
V! (! t+1 ) +
(F.605)
V! (! t+1 ) should equal the marginal
(1 +
V! (! t+1 )
=0
t+1 ) (1 + n)
(F.606)
Transversality conditions
t
lim
t!1
t kt
= 0; lim
t!1
t
t mt
=0
(F.607)
By envelop theorem:
t
6.3.2
= V! (! t ) = uc (ct ; mt )
(F.608)
Steady state in the MIU model
Dynamic optimisation with the …rst order conditions:
um (ct ; mt ) +
uc (ct+1 ; mt+1 )
= uc (ct ; mt )
(1 + t+1 ) (1 + n)
224
(F.609)
Left hand side gives the total marginal bene…t of holding money; the …rst term in it is the direct
utility of money and the second term denotes the real balance e¤ect of holding money mt at
time t for t + 1. Thus the marginal utility of holding money should equal the marginal utility
of consumption. By constant returns to scale assumption the income of households is function of
capital stock rk k + w = fk k + (f (k)
fk k) = f (k). Financial crises promote hoarding of money,
this means less capital, more in‡ation and less growth. Consider a steady state with n = 0 and
V! (! t ) = V! (! t+1 ) = V! (! ss ). From the …rst …rst order conditions 1
zfk (k ss ) + (1
[fk (k ss ) + (1
)] = 0
1
)=
Assuming a Cobb-Douglas production function, f (k) = zk ; this condition converts to
+ (1
)=
zk
1
1
k ss =
1+
1
z
(
1
(F.610)
1)
Consumption in the steady state:
ss
c
ss
= zf (k )
Steady state in‡ation (
ss
k
ss
=
1+
z
(
where
1)
1+
) equals growth rate of money supply (
ss
mss
=
mss
(1 +
z
(
1
ss
1
1
1)
(F.611)
):
ss
ss )
mss = 0 implies growth rate of money supply,
=0
ss
=
M ss
M ss ,
and equal in‡ation,
ss
=
ss
.
Stock of money in excess of the amount required for transactions reduces the amount of capital
stock, hence output in the economy. Excessive supply of money manifests in in‡ation and is not
good for the economy. As in the CIA model the transitional dynamics of the MIU model is found
numerically for the set of parameters in Table 3. The response of yt ; rt ; zt ; ct , ut to shocks are
represented in Figures 7 to 10 with time periods t1 to t99 in the horizontal axis.
225
yy
2.056
yy
2.055
2.054
2.053
2.052
2.051
2.05
2.049
2.048
2.047
t97
t94
t91
t88
t85
t82
t79
t76
t73
t70
t67
t64
t61
t58
t55
t52
t49
t46
t43
t40
t37
t34
t31
t28
t25
t22
t19
t16
t13
t7
t4
2.043
t1
2.044
t10
2.045
t100
2.046
Figure 7: Technological shock directly e¤ects output but indirectly through its e¤ect in capital stock
rr
rr
0.057
0.057
0.056
0.056
0.056
0.056
t100
t97
t94
t91
t88
t85
t82
t79
t76
t73
t70
t67
t64
t61
t58
t55
t52
t49
t46
t43
t40
t37
t34
t31
t28
t25
t22
t19
t16
t13
t7
t4
t1
0.056
t10
0.056
Figure 8: Real interest rate ‡uctuates due to ‡uctuations in the marginal productivity of capital
226
cc
cc
1.503
1.503
1.503
1.503
1.503
1.503
1.503
1.503
1.503
Figure 9:
Consumption ‡uctuates with output and income.
Figure 10: A country with lower discount rate accumulates more capital and is better o¤ in the long run
Again scenarios are derived for three economies with di¤erent values of labour productivity (a),
discount rates ( ), terminal growth rates of capital stock (gk ) and velocity of money (v) as given in
Table 3, the time path of variables yt ; kt ; ct ; mt , ut are easily computed based on model solutions.
Country two with lower discount factor than countries 1 and 3 grows more becomes larger within
a reasonable time (Figures 10). However such austerity or thriftiness cause lower level of welfare
gain for country 2 than other two countries (Figure 11). Cost of …nancial adjustments in the global
economy is born by a country which is more prudent, though it is clear that such a perversity in
distribution of gains are less likely to destabilise and cause a …nancial crises as was observed in
2008.
227
t100
t97
t94
t91
t88
t85
t82
t79
t76
t73
t70
t67
t64
t61
t58
t55
t52
t49
t46
t43
t40
t37
t34
t31
t28
t25
t22
t19
t16
t13
t7
t4
1.502
t1
1.502
1.502
1.502
t10
1.503
1.502
Table 56: Parameters of MIU Model
Parameters
z
MIU
0:3
0:99
0:05
(1; 0:05)
Table 57: Parameters of MIU Model
a
gk
L0
v
Parameters
ln (z)
M0
Economy 1
0:05
0:5
0:95
0:01
100
1
0:01
(1; 0:05)
100
Economy 2
0.05
0.45
0.99
0.02
100
2
0.02
(1; 0:05)
100
Economy 3
0.05
0.45
0.98
0:015
100
1.5
0:015
(1; 0:05)
100
welfare
i1 275.516
i2 203.96
i3 219.6
Figure 11: Small country gains more than larger countries when risks are pooled together through a coalition.
The CIA and MIU models provide intuition about the nature of ‡uctuations that a¤ect interdependent economies and allocation of welfare. Policy analyses could be based in more detailed
assessment of the structural features of the economy as found in the micro-consistent dataset for
consumption, production and trade but these two models show how the growth of money should
be set according the real growth rate of the economy.
6.3.3
Conclusion
We study results of the cash in advance and money in utility models to examine how the nature
of ‡uctuations in economic activities and welfare in three interdependent economies relate to stock
and growth rate of money. When the money is introduced exogenously in the form of cash in
228
advance constraint, it serves as a medium of exchange and the rate of return in real and nominal
assets become equal. Idiosyncratic technological shocks generate ‡uctuations in the growth rates
of capital, output, prices, money, consumption, investment, labour supply and lifetime utilities of
households. When households have money endogenously in their utility functions, the stock of
money in excess of the amount required for transactions causes in‡ation and reduces the amount
of capital stock and output in these economies. Both CIA and MIU models support for a steady
growth rate of money according to the smooth growth rate of output. While targeting in‡ation by
manipulating the interest rate for macroeconomic stability is theoretically a prudent policy move, it
seems impossible for a central bank to eliminate business cycles that arise from shocks to production
technology or structural features of the economy.
References
[1] Aruoba S.B., C. J. Waller and R. Wright (2011) Money and capital, Journal of Monetary
Economics, 58, 2,98-116.
[2] Bank of England (1999) The Transmission Mechanism of Monetary Policy, London, UK.
www.bankofengland.co.uk
[3] Bank of England (2001) Financial Stability Review, London, UK. www. Bankofengland.co.uk.
[4] Balasko Y. (2003) Temporary Financial Equilibrium, Economic Theory, 21, 1, 1-18.
[5] Benati, Luca (2008) The ‘Great Moderation’in the United Kingdom, Journal of Money,Credit
and Banking, 40, 121–47.
[6] Berentsen A., M. R. Breu, S Shi (2012) Liquidity, innovation and growth, Journal of Monetary
Economics, 59, 8, 2012,721-737
[7] Bean C. (2009) The meaning of internal balance’ Thirty years on, Economic Journal, 119,
F442–F460.
[8] Brock W. A. (1974) Money and Growth: The Case of Long Run Perfect Foresight, International
Economic Review, 15, 3, 750-777
[9] Diamond P, D. W. Douglas and Dybvig, P. H. (1983) Bank Runs, Deposit Insurance,and Liquidity, Journal of Polotical Economy, 91:3: 401-419.
[10] Ellison M. and J. Pearlman (2011) Saddlepath learning, Journal of Economic theory 146,
1500-1519.
229
[11] Friedman, M. (1968), The Role of Monetary Policy, American Economic Review, 58: March,
1-17.
[12] GAMS Development Corporation (1999) GAMS: A User’s Guide, Washington DC 20007, USA.
[13] Gomme P (1993) Money and growth revisited: Measuring the costs of in‡ation in an endogenous growth model, Journal of Monetary Economics, 32, 1, 51-77
[14] Hayakawa H (1986) Intertemporal optimization and neutrality of money in growth models,
Journal of Monetary Economics, 18, 3, 323-328
[15] Rankin N. (1992) Imperfect competition, expectations and the multiple e¤ects of monetary
growth, Economic Journal 102: 743-753.
[16] Sinn H.W (2009) Risk Taking, Limited Liability and the Banking Crisis, Ifo Institute, University of Munich.
[17] Sidrauski M. (1967) Rational Choice and Patterns of Growth in a Monetary Economy, American Economic Review, 57,2,534-544
[18] Spencer P. D. (2008) Stochastic Volatility in a Macro-Finance Model of the U.S. Term Structure
of Interest Rates 1961-2004, Journal of Money, Credit and Banking, 40, 6,1177-1215
[19] Sproul A. (1947) Monetary Management and Credit Control, American Economic Review, 37,
3:Jun., 339-350
[20] Tobin J. (1965) Money and Economic Growth, Econometrica, 33, 4, 671-684
[21] Williamson S.D. (2008) Macroeconomics, 3rd Edition, Pearson International.
[22] Walsh C.E (1998) Monetary Theory and Policy, MIT Press.
6.3.4
Analysis of Dynamic GE Model of Financial Deepening
The micro-consistent data for this model is taken from the input output table published by the
OECD in 2006 for Germany, France and UK (Appendix Tables C1 - C3 available upon request).
This data set provides information on the actual values for demand supply balances of …rms, revenue
and expenditure of the government, saving and investment balance for the private sector and the
export-import balance for the economy.
A number of assumptions are made regarding the nature of the steady states among these
economies. First, the bench mark rate of return on capital stock is chosen to be the natural
rate of interest (r) for each country. Information about the rate of deprecation of capital ( i ) in
230
Table 58: Optimal and actual …nancial deepening in Frnace, Germany and the UKl
Parameters Optimal Financial Deepening Actual Financial Deepening Over Financing
France
3.16
10.98
3.5
Germany
3.31
8.02
2.4
UK
3.24
19.12
5.1
each sector is obtained from the historical data and tested with sensitivity analyses. The steady
state growth rates (gi ) are made consistent with the historical growth rates for each sector. The
parametric values of r;
in consumption (
c)
i
and gi de…ne the reference path of the economy. Elasticities of substitution
and production (
p)
are based on the literature. Fundamentals to all these rest
on the optimising behavior of households regarding the division of labour between leisure Lht and
work and division of income between consumption Cth and saving Sth . Tax rates tc ; tw ; tk ; Rth
are retained for all sectors except for the …nancial and real estate sectors in the counter factual
analyses. Model is applied for policy analysis only after the calibration of the benchmark economies.
6.3.5
Optimal and actual …nancial deepening
The general equilibrium theory provides a very solid framework for analysis of results obtained by
solving more than 14 thousands equations simultaneously for France, Germany and UK. Results
on optimal and actual …nancial deepening, the ratios of …nancial assets to GDP, relevant for this
paper are summarised in Table 710 .
The overall optimal real …nancial deepening ratios from the general equilibrium models are
consistent across countries; these are found to be around 3.16 in France, 3.31 in Germany and
3.24 for the UK. These are sensible results and consistent to the converging patterns of economic
growth across these countries. The actual ratios of …nancial deepening reported in the OECD nonconsolidated balance sheets of 10.98, 8.02 and 19.12 exceed by factor of 3.5, 2.4 and 5.1 than the
optimal ratios computed from the solutions of the general equilibrium models of France, Germany
and the UK respectively as shown in Table 7. The discrepancy between the real and the nominal
magnitudes of …nancial deepening gives credibility to the hypothesis that UK economy is more vulnerable to …nancial crises as it has more assets originating from the …nancial derivatives and more
subject to the problems caused by asymmetric information. Sectoral impacts of …nancial sector
reforms are di¤erent for each of three countries. Despite this, economic growth rates in these models are driven by fundamentals of the …nancial markets based on the net present value calculation,
1 0 Detailed
solutions of these models to be available upon request.
231
portfolio selection satisfying the arbitrage across markets, risk-return analysis to minimise risks and
maximise returns and insurances to cover unforeseen contingencies. Supply of funds arises from
inter-temporal utility maximising consumers and demand for funds for investment originates from
pro…t maximising producers. Subjective discount factors of consumers and depreciation rates of
capital is balanced by the real interest rates so that funds are allocated according to the marginal
utilities of households or productivities across various sectors leaving regulatory roles to the government for maintaining law and order to create opportunities for the participants from the private
sector.
On-going …nancial sector reforms can be expected to make these economies more e¢ cient so
that the costs of funds decline in the counter factual experiments, where the taxes on the …nancial
sectors are set to minimise distortions relative to the benchmark. Such measures will then result in
the higher rate of growth of output, employment and capital stock in almost all sectors even with
lower capita output ratios. The …nancial liberalisation is paying for itself, welfare of consumers
improves with reforms rather than without it.
The proper reforms of …nancial markets improve e¢ ciency of …nancial intermediation and brings
speedier rate of economic growth by linking the lending and borrowing rates to the fundamentals of
demand and supply of funds; removing controls on credits; by creating right structure of incentives
for investors and depositors and freeing up the foreign exchange market from arbitrary decisions
and by making it subject to fundamentals of domestic and foreign markets. These mechanism
remove repressionary regimes with non in‡ationary public …nance for smooth process of capital accumulation, increased liquidity, technical advancement and economic growth, elimination of parallel
markets and reducing the proportion of toxic non-performing assets. Liberation and reform mechanisms thus are instrumental in reversing repressionary …nancial regimes towards more classical free
enterprise economy that would promote accumulation and growth in these model economies.
The general equilibrium model results presented above rely on classical economic principles in
which the self-adjusting mechanism of the real interest rates would balance demand for and supply
of …nancial assets in a market driven economy and do not contain liquidity trap and credit crunch
situations as imagined by Keynes (1936). These results are consistent to literature that has emerged
since late 1960s on harmful impacts of …nancial repressions in works of McKinnon (1968), Shaw
(1968), Roubini and Sala-i-Martin (1992) and Stiglitz and Weiss (1981) and more recently in Boyd
and Jalal (2012).
Competitive …nancial markets are perfect in allocating assets as all agents that have complete
information and are e¢ cient in processing such information. This assumption, however, is far from
perfect. Financial markets are full of asymmetric information, activities of one set of players depend
on actions taken by another set of players and the amount of information they have impacts on
the likely choices of others. This requires state contingent incentive compatible mechanisms in this
232
general equilibrium system and is an issue for further investigation.
See full paper (Bhattarai (2013)).
6.4
Exercise 9
I. Consider a money in utility function model
max W =
1
X
t
U (ct ; mt )
(F.612)
t=0
Subject to:
Yt = F (Kt ; Nt )
(F.613)
Under constant returns to scale yt = f (kt ) where yt =
Yt
Nt
and kt =
Kt
Nt
.
Cash in advance constraint
Yt +
t Nt
+ (1
) Kt
1
+
Mt
Pt
1
= Ct + Kt +
1
Mt
Pt
where Yt is output, Pt price of goods, Ct consumption, Kt+1 is capital stock,
for each individual, Mt money, Yt , output, Nt employment and
(F.614)
t
is net transfer
is the rate of depreciation of
capital.
1) set up the constrained optimisation functions and derive the …rst order conditions of maximisation.
2) Solve the model for its steady state. Express consumption, output, capital stock and money
in terms of model parameters.
3) Characterise the transitional dynamics if the economy is not in the steady state in the
beginning.
II. Consider the problem of a cash in advance economy given below.
max
1
X
t
U (Ct )
V (Nt )
(F.615)
t=0
Subject to:
Yt = zNt
Cash in advance constraint
233
(F.616)
Pt Ct + qt Bt+1 + Pt st Xt+1 + Pt Tt = Mt + Bt + Pt Xt
(F.617)
where Pt Ct is consumption expenditure Pt price of goods, Ct consumption, Bt+1 is the amount
of nominal bonds qt is the price of nominal bonds, Xt+1 real bonds st prices of real bonds, Tt lump
sum tax payment, Mt money.
Budget constraint of the consumer:
Pt Ct + qt Bt+1 + Pt st Xt+1 + Pt Tt + Mt+1 = Mt + Bt + Pt Xt + Pt zNt
(F.618)
Government’s budget constraint:
M t+1
Mt =
Assuming a constant rate of money growth
Pt Tt
(F.619)
and M t+1 = M t
Mt =
Pt Tt
(F.620)
1) Write the appropriate Lagrangian for constrained optimisation and derive the optimal …rst
order conditions
2) Solve for the steady state of the model
3) what are the values of price of goods, bonds, stocks in the steady state? What are the
corresponding values of consumption, labour supply, money and the interest rate.
References
[1] Altig D E, C.T. Carlstrom and K.L. Lansing (1995) Computable General Equilibrium Models
and Monetary Policy Advice, Journal of Money Credit and Banking, 27, 4, Nov., 1472-1493.
[2] Arestis P., P. O. Demtriades and K. B. Luintel (2001) Financial Development and Economic
Growth: the Role of Stock Markets, Journal of Money, Credit and Banking, 33:1:16-41.
[3] Arestis P., A. Charles and G. Fontana (2013) Financialisation, the ‘Great Recession’ and the
Strati…cation of the US Labor Market, Economics Seminar, HUBS, June.
[4] Bacchetta P. (1992) Liberalization of Capital Movements and of the Domestic Financial System
Economica, New Series, 59, 236 Nov., 465-474
[5] Bank
of
England
(1999)
The
Transmission
www.bankofengland.co.uk
234
Mechanism
of
Monetary
Policy,
[6] Bank of England (2001) Financial Stability Review, www. Bankofengland.co.uk.
[7] Balasko Y. (2003) Temporary Financial Equilibrium, Economic Theory, 21, 1, 1-18.
[8] Balasko Y. (1984) The Size of Dynamic Econometric Models, Econometrica, 52, 1, 123-142.
[9] Benati, Luca (2008) The ‘Great Moderation’in the United Kingdom, Journal of Money,Credit
and Banking, 40, 121–47.
[10] Banerji S and P Basu (2009) Universal Banking and Equity Risk Premium, April; Paper
presented in CEP Seminar in Hull Business School.
[11] Bean C. (2009) The meaning of internal balance’ Thirty years on, Economic Journal, 119
(November), F442–F460.
[12] Beaudry P and M. Poitevin (1995) Competitive Screening in Financial Markets when Borrowers
can Recontract, Review of Economic Studies, 62, 3 , 401-423
[13] Bhattarai K. (2013) Financial Deepening and Economic Growth, Munich Personal RePEc
Archive, Paper No. 48696.
[14] Bhattarai K. (2005) Consumption, Investment and Financial Intermediation in Ramsey Models, Applied Financial Economics Letters, 1(6):1-5.
[15] Bhattarai K. R. (1997) Financial Deepening and Economic Development in Nepal: A Forward
Looking CGE Model with Financial Intermediation, Ph.D. dissertation Northeastern University,
Boston, Massachussetts.
[16] Bolnick B. R. (1987) Financial Liberalization with Imperfect Markets: Indonesia during the
1970s Economic Development and Cultural Change, 35, 3 :Apr., 581-599
[17] Boycko M, A. Shleifer and R. W. Vishny (1996) A Theory of Privatisation The Economic
Journal, 106, 435 :Mar., , 309-319
[18] Boyd J. H and A. M. Jalal (2012) A new measure of …nancial development: Theory leads
measurement, Journal of Development Economics 99, 341–357.
[19] Boyd, J.H. and Prescott, E. C. (1986) Financial Intermediary-Coalitions, Journal of Economic
Theory, 38: 211-232..
[20] Brunnermeier, M. K. (2009.) Deciphering the Liquidity and Credit Crunch 2007-2008, Journal
of Economic Perspectives, 23(1): 77–100.
235
[21] Cecchetti, S. G. (2009). Crisis and Responses: The Federal Reserve in the Early Stages of the
Financial Crisis. Journal of Economic Perspectives, 23(1): 51–75.
[22] Chadha J. S. & C. Nolan (2002) In‡ation and price level targeting in a new Keynesian model,
Manchester School, 70(4), pp. 570-595.
[23] Champ B. , B. D. Smith and S. D. Williamson (1996) Currency Elasticity and Banking Panics:
Theory and Evidence The Canadian Journal of Economics / Revue canadienne d’Economique,
29, 4 :Nov., 826-824.
[24] Chari V.V., P. J. Kehoe and E. R McGrattan (2000) Sticky price models of the business cycle:
Can the contract multiplier solve the persistence problem? Econometrica, 66:5:1151-1179.
[25] Chiang A. C. (1959) Instalment Credit Control: A Theoretical Analysis The Journal of Political
Economy, 67, 4 :Aug., 363-376
[26] Covas F. and W. J. Den Haan (2012) The role of debt and equity …nance over the business
cycle, Economic Journal, 122, 565 , 1262-1286
[27] Cripps M. W. (1997) Bargaining and the Timing of Investment International Economic Review,
38, 3 :Aug., 527-546
[28] Dasgupta P. and E. Maskin (2000) E¢ cient Auctions The Quarterly Journal of Economics,
115, 2 :May, 341-388
[29] Diamond P, D. W. Douglas and Dybvig, P. H. (1983) Bank Runs, Deposit Insurance,and
Liquidity, Journal of Polotical Economy, 91:3: 401-419.
[30] Davies R. and P. Richardson, V. Katinaite and M. Manning (2010) Evolution of the UK
banking system, Bank of England Quarterly Bulletin 2010 Q4, 321-332.
[31] De Fraja Giovanni‘(1991) E¢ ciency and Privatisation in Imperfectly Competitive Industries
The Journal of Industrial Economics, 39, 3 :Mar., 311-321
[32] Epstein L. G. and S E. Zin (1989) Substitution, risk aversion and the temporal behaviour of
consumption and asset returns: A theoretical Framework, Econometrica, 57:4:937-969.
[33] Fama, E. F. (1980) Banking in Theory of Finance, Journal of Monetary Economics, 6- 39-57.
[34] Farmer REA (2013) Animal Spirits, Financial Crises and Persistent Unemployment,Economic
Journal, 123,586,317-340
236
[35] Friedman M. (2005) A Natural Experiment in Monetary Policy Covering Three Episodes of
Growth and Decline in the Economy and the Stock Market, The Journal of Economic Perspectives, 19, 4 :Autumn, 145-150
[36] Fry M. J. (1978) Money and Capital or Financial Deepening in Economic Development? Journal of Money, Credit and Banking, 10, 4, 464-475.
[37] Gai P., S. Kapadia, S. Millard, A. Perez (2008) Financial Innovation, Macroeconomic Stability
and Systemic Crises, Economic Journal, 118, 527, 401-426.
[38] GAMS Development Corporation (1999) GAMS: A User’s Guide, Washington DC 20007, USA.
[39] Ginsburgh V and M. Keyzer (1997) The Structure of Applied General Equilibrium Models,
MIT Press.
[40] Greenwood J, and B. Javanovic (1990) Financial Development, Growth, and the Distribution
of Income, Journal of Political Economy, 98:5: pt.1:1076-1107.
[41] Giovannini A. and M. de Melo (1993) Government Revenue from Financial Repression The
American Economic Review, 83, 4 Sept., 953-963
[42] Hansen L. P., T. J. Sargent and T. D. Tallarini (1999) Robust permanent income and pricing,
Review of Economic Studies, 66:4:873-907, October.
[43] Hills S. and R. Thomas and N. Dimsdale (2010) The UK recession in context-what do three
centuries of data tell us? Bank of England Quarterly Bulletin 2010 Q4, 277-291
[44] King M., E. Sentana and S. Wadhwani (1994) Volatility and Links between National Stock
Markets Econometrica, 62, 4 ,901-933
[45] King R. G. and R. Levine (1993) Finance and Growth: Shumpeter Might Be Right, Quarterly
Journal of Economics, Aug., 717-737
[46] Kiyotaki N. and J. Moore (2006) Financial Deepening, Journal of the European Economic
Association, 3, 2/3, Apr.-May, 701-713.
[47] Klein M. A.(1971) A Theory of the Banking Firm, Journal of Money, Credit and Banking, 3,
2, Part 1 May, 205-218
[48] Krugman P. (1979) A Model of Balance of Payment Crisis, Journal of Money Credit and
Banking, 11, Aug,3, 311-325.
237
[49] Levine P, J. Pearlman, G. Perendia and B Yang (2013) Endogenous persistence in an estimated
DSGE model under imperfect information, Economic Journal, 122 (December), 1287–1312.
[50] Levine R (1997) Financial Development and Economic Growth: Views and Agenda, Journal
of Economic Literature, 35:2:688-726.
[51] Lim G.C. and P D. McNeilis (2008) Computaitonal Macroeconomics for the Open Economy,
MIT Press.
[52] Mayer, C., K. Pence, and S. M. Sherlund. (2009) The Rise in Mortgage Defaults, Journal of
Economic Perspectives, 23(1): 27–50.
[53] Maskin E. and J. Tirole (1990) The principal-agent relationship with an informed principle,
Econometrica, 58:379-410.
[54] McKinnon R. I. (1973) Money and Capital in Economic Development, the Brookings Institution, Washington D. C.
[55] Mercenier J. and T.N. Srinivasan ed. (1994) Applied General Equilibrium and Economic Development: Present Achievement and Future Trends, University of Michigan Press, Ann Arbor.
[56] Milde H. and J. G. Riley (1988) Signaling in Credit Markets The Quarterly Journal of Economics, 103, 1 :Feb, 101-129
[57] Miller M and J. Stiglitz (2010) Leverage and Asset Bubbles: Averting Armageddon with
Chapter 11?, Economic Journal 120, 544, 500–518.
[58] Nash J. (1951) The Bargaining Problem, Econometrica, 18:155-162.
[59] Nordhaus W.D. (1995) Policy Games: Co-ordination and Independence in Monetary and Fiscal
Policies, Brookings Papers on Economic Activities 2:1994: 139-216.
[60] Pagano Marco (1993) Financial Markets and Growth: An Overview, European Economic
Review, 37(2-3, April) 613-622.
[61] Pilbeam K, J.Olmo and W.Pouliot (2011) Detecting the Presence of Insider Trading via Structural Break Tests, Journal of Banking and Finance, 35,11, 2820-2828.
[62] Price Simon (1997) Political Business Cycles and Macroeconomic Credibility: A Survey, Public
Choice, 92, 3/4, 407-427
[63] Prescott E. C. and R. M. Townsend (1984) Pareto Optima and Competitive Equilibria with
Adverse Selection and Moral Hazard Econometrica, 52, 1 :Jan, 21-45
238
[64] Rasmusen E. (1987) Moral Hazard in Risk-Averse Teams The RAND Journal of Economics,
18, 3 :Autumn, 428-435
[65] Radelet S. , J. D. Sachs, R. N. Cooper and B. P. Bosworth (1998) The East Asian Financial
Crisis: Diagnosis, Remedies, Prospects Brookings Papers on Economic Activity, 1998, 1, 1-90
[66] Raghuram G. R. and L. Zingales (1998) Financial Dependence and Growth, American Economic Review, 88:3:559-586.
[67] Riley J. G (2001) Silver Signals: Twenty-Five Years of Screeding and Signalling, Journal of
Economic Literature, 39:2:432-478
[68] Rogerson W. P. (1985) Repeated Moral Hazard Econometrica, 53, 1 :Jan., 69-76
[69] Roth A. E. (2008) What have we learned from market design?, Economic Journal, 118, 285–
310.
[70] Roubini N. and Sala-i-Martin (1992) Financial Repression and Economic Growth, Journal of
Development Economics 39: 5-30: July.
[71] Rutherford T.F. (1998) GAMS/MPSGE Guide, GAMS Corporation , Washington DC.
[72] Sargent T J (1987) Dynamic Macroeconomics, Oxford University Press.
[73] Scarf H. (1967) Core of n Person Game, Econometrica, 35:50-69.
[74] Scarf H.E and Shoven J. B. (1986) Applied General Equilibrium Analysis, Cambridge University Press.
[75] Schmitt-Grohé, S. and Uribe, M. (2014), Liquidity Traps: an Interest-rate-based Exit Strategy.
The Manchester School, 82: 1–14.
[76] Shapley L (1953) A Value for n Person Games, Contributions to the Theory of Games II,
307-317, Princeton.
[77] Shapley L and M. Shubik (1969) On Market Games, Journal of Economic Theory, 1:9-25.
[78] Shaw, E.S. (1973) Financial Deepening in Economic Development, Oxford University Press,
New York.
[79] Sinn H.W (2009) Risk Taking, Limited Liability and the Banking Crisis, Ifo Institute, University of Munich
[80] Sinn H.W (2010) Casino Capitalism, Oxford.
239
[81] Smith Warren L.(1958) The Discount Rate as a Credit-Control Weapon The Journal of Political
Economy, 66, 2 :Apr., 171-177
[82] Sidrauski M. (1967) Rational Choice and Patterns of Growth in a Monetary Economy, American Economic Review, 57,2,534-544
[83] Spencer P. D. (2008) Stochastic Volatility in a Macro-Finance Model of the U.S. Term Structure
of Interest Rates 1961-2004, Journal of Money, Credit and Banking, 40, 6,1177-1215
[84] Spencer P. D. (1984) Precautionary and Speculative Aspects of the Behaviour of Banks in the
United Kingdom Under Competition and Credit Control 1972-1980, Economic Journal, 94, 375
:Sep.,554-568
[85] Sproul A. (1947) Monetary Management and Credit Control The American Economic Review,
37, 3:Jun., 339-350
[86] Stiglitz, J.E. and A. Weiss (1981) Credit Rationing in Markets with Imperfect Competition,
American Economic Review, June, 393-410.
[87] Townsend, R. M. (1983) Financial Structure and Economic Activity, American Economic Review
895-911.
[88] Townsend R.M. and K. Ueda (2006) The Financial Deepening, Inequality, and Growth: A
Model-Based Quantitative Evaluation, Review of Economic Studies, 73, 1, 251-280
[89] Ventura, J. (1997) Growth and Interdependence, Quarterly Journal of Economics, 112 (1):
57-84.
[90] Williamson S.D. (2008) Macroeconomics, 3rd Edition, Pearson International.
[91] Walsh C.E (1998) Monetary Theory and Policy, MIT Press.
[92] Wickens M. (1995) Real Business Cycle Analysis: A Needed Revolution in Macroeconometrics
(in Controversy: Business Cycle Empirics) Economic Journal, 105, 433, 1637-1648.
Web pages
Bank of England: http://www.bankofengland.co.uk
European Central Bank: http://www.ecb.int/home/html/index.en.html
Hubs of central banks: http://www.bis.org/
International Monetary Fund: http://www.imf.org
240
7
L7: Open Economy Model: Exchange Rate and Finance
in Macro
Global economy is interdependent; growth rates of output in one economy a¤ect that in other
economies. Policy makers like to introduce economic policies to achieve a higher rate of growth at
home country given their understanding of the structure of the global economy and the interaction
of home economy with other economies. This section builds on four di¤erent modelling frameworks
widely used in the international macroeconomics literature to explain such interdependence. First
one is a small open economy model solved numerically for assessing the impacts of expansionary
…scal, monetary and …scal-monetary policy mix scenarios. The next one is the model for an interdependent global economy in which policies of one economy can in‡uence another economy. Third
model is based on actual time series data set of …ve major industrial economies on growth rates
estimated using single equation and simultaneous equation econometric techniques. Final one is
a micro founded dynamic general equilibrium model formulated and solved to explain how prices,
exchange rates, real exchange rates a¤ect major macroeconomic variables in a global economy. In
each case assessment is made on how policy actions can alter the course of an economy by discussing
model generated results in response to speci…c policy chosen by policy makers.
7.1
Small Open Economy Model
Economic activities of a small open economy have very limited impacts on other economies. According to the standard Keynesian and Mundell-Flemming (Keynes (1936), Mundell (1962), Fleming
(1962)) models external in‡uences occur through the in‡ow and out‡ow of capital as dictated by the
movements in the foreign trade and exchange rates in response to the underlying trends of in‡ation
and the interest rates. Such a small open economy macroeconomic model includes consumption,
investment, tax or spending, net exports, real exchange rate, money demand and money supply
and aggregate supply. As a work-horse model for analysing open economy issues, this model can
be illustrated using following eleven equations.
Consumption is determined by the disposable income as:
C = a + b (Y
Tax revenues are proportional to income:
T = tY
Investment is inversely related to the interest rate:
241
T)
I = I0
I1 r
Imports respond positively to the domestic income and negatively to the exchange rate:
IM =
0
+
1Y
+
2 ER
Exports relate positively to the exchange rate and foreign income:
X=
1 ER
+
2Y
Money demand depends positively on income and negatively to the interest rate:
M
= f Y kr
P
Using the market clearing condition Y = C + I + G + (X
Y =
a + I0 +
I1 M
k P
+G+
1
1 ER
+
b + bt +
1
2Y
I1
k f
m0
M ) this model is solved as:
2 ER
and r =
1
fy
k
M
P
Trade balance de…ned as the di¤erence between exports and imports:
TB = X
M
Budget de…cit is the excess of government spending over the tax revenue:
BD = T
G
Savings represents residual income after meeting the consumption and tax spending:
S=Y
C
T
Private plus public saving equals net exports equals net capital ‡ow:
(S
I) + (T
G) = (X
M ) = KA
Prices move according to the classical money market:
MV = PY
Above eleven equations are regular text book versions of an open economy macroeconomic
model (Blanchard (2003), Mankiw and Taylor (2008), Miles and Scott (2002)). Model is demand
determined and closed by the saving investment identity as given in equation (10). In order to solve
242
all eleven equations simultaneously, model parameter, a, b, t, i0, i1, m0, f, k , v, G, M/P and Y*
need to be speci…ed as I have presented in Table below.
Table 59: Parametric Speci…cation of the Small Open Economy ISLM model
Demand Trade
Money
a
200
0
20
M
P
100
b
0.75
1
0.2
P
1
t1
0.3
2
-3
f
0.2
i0
50
1
5
k
-60
i1
-5
2
0.2
v
5
G
100
Y
200
Key Model variables : Consumption (C), Income (y ), Interest rate (r), Imports (m ), Tax
revenue (T), Investment (I ), Trade-gap (X-M), Budget-gap (T-G), Saving-investment gap (S-I)
and Exchange rate (ER).
A small open economy model is solved for the base case using above values of parameters. Then
model is solved for three di¤erent policy scenarios (a) …scal expansion (b) monetary expansion and
(c) both …scal and monetary mix. Model assumes a ‡exible exchange rate regime in all above
scenarios. The classical scenario in which prices are perfectly ‡exible is also computed at the end.
The results of the model are given in the Table below.
Table 60: Impacts of Fiscal and Monetary Policy in the small open economy ISLM model
Variables Base (G=100; M=100) Fiscal(G=200) Monetary (M=300) Fiscal and Monetary(G=200; M=300)
C
742.8
1222.8
956.0
1100.0
Y
1033.9
1604.4
1260.0
1500.0
r
1.8
3.7
0.8
0.0
M
100.5
171.8
128.8
158.8
T
310.2
240.7
252.0
300.0
I
310.2
240.7
252.0
300.0
X
M
150.0
150.0
150.0
150.0
T
G
210.2
40.7
152.0
100.0
S
R
-60.2
109.3
-2.0
50.0
42.1
56.4
47.8
53.8
ER
(Simulation calculations are done in Excel and GAMS)
All of these results make intuitive sense. Increase in public spending raises income and con243
sumption but crowds out the private investment. Levels of tax collection and saving rise in response
to increase in income but the budget surplus signi…cantly deteriorates as expansion in spending is
much higher than extra revenue raised after increase in income. When capital out‡ow is …xed at
150, increase in public spending puts pressure in the exchange rate.
The monetary expansion lowers both the interest rate and the exchange rates but has less impact
on output and consumption than of the expansionary …scal policy. Though the lower interest rate
has impact on investment, tax collection is more sensitive to income than the investment is to
the interest rate. Consequently tax revenue rises and dampens the expansionary impact of the
monetary policy.
The accommodative …scal and monetary policies generate better results both in terms of increase
in consumption and income as well as in terms of saving, investment gaps and the trade gaps.
7.2
Global Economy model of Two Economies
Small open economy model presented above assumes foreign income as given, which may be realistic
for a very small economy whose trade links are trivial in comparison to its trading partners. In most
cases one economy has discernible in‡uence in another economy. Such inter dependency requires a
model for inter-dependent economies. Similarly time factors need to be taken into account as it takes
time for dynamic adjustment. The static model presented above is not enough in analysing these
dynamic issues. More realistic modelling requires capturing the inter-dependency among economies
over time. These dimensions can be extended to the small open economy model presented above
by adding time and country speci…c subscripts to the variables and parameters of the above model.
This section illustrates how these features can be added to make the model more appropriate for
analysis of the interdependent global economy. For simplicity the global economy is assumed to
consist of two inter-dependent economies and the economic horizon is of twenty seven years from
2003 to 2027. Both economies have standard macroeconomic set up with product, labour and
money markets. Phillip’s curve explains the trade-o¤ between unemployment and in‡ation. Fiscal
and monetary policies can be adopted to in‡uence the growth path of the economy. This model is
in spirit of Mundell-Fleming speci…cation of an open economy with ‡exible exchange rate regime.
Two economies are linked by international trade in goods and capital. Subscript i in a variable
denote country i, and subscript t denotes time period t. Symbol “*” is used to represent a foreign
variable to economy i. Model is kept simple to make results more appealing.
The …rst equation gives the national income identity in terms of consumption, investment,
government spending and net exports:
Yi;t = Ci;t + Ii;t + Gi;t + (Xi;t
244
Mi;t )
where Y is output, C consumption, I investment, G government spending, X is exports and M
is imports and T is tax revenue.
Current consumption depends on current disposable income:
Ci;t = a + b (Yi;t
Ti;t )
Tax revenue at period t depends on income at period t:
Ti;t = tYi;t
Investment of country I is inversely related to interest in country i:
Ii;t = I0
I1 ri;t
I assume that the public spending grows by rate gr in each period:
Gi;t = Gi;1 (1 + gi;t )t
Exports depend on current exchange rate and current foreign income:
Xi;t =
1 ERi;t
+
2 Yi;t
Similarly imports depend on current domestic output and current exchange rate:
IMi;t =
0
+
1 Yi;t
+
2 ERi;t
Net exports is the di¤erence between current exports and imports:
T Bi;t = Xi;t
Mi;t
This model includes a version of the Lucas aggregate supply function with actual and expected
price levels
Yi;t = Y i;t +
e
Pi;t
Pi;t
e
here P is domestic price Pi;t
expected in‡ation.
Output is linked to the employment in a Keynesian fashion:
Yi;t = Li;t
The model assumes a standard money demand function:
M
P
= f Yi;t
i;t
245
kri;t
Budget de…cit is the excess of government spending over tax revenue:
BDi;t = Ti;t
Gi;t
Level of savings in period t is residual of income after consumption spending and tax collection:
Si;t = Yi;t
Ci;t
Ti;t
Saving investment identity
(Si;t
Ii;t ) + (Ti;t
Gi;t ) = (Xi;t
Mi;t ) = KAi;t
The trade o¤ between unemployment in in‡ation is given by the Phillips curve:
Pi;t =
(Ui;t
Ui;t
1)
The external debt accumulates if the trade balance is positive:
Di;t = Di;0 + T Bi;t + 0:1(1 + ri;t )Di;t
1
Real exchange rate:
i;t
where ER nominal exchange rate,
=
i;t
ERi;t Pi;t
Pi;t
real exchange rate, Pi;t foreign price level
Financial integration condition:
ii;t = ii;t
where i interest rate, i* is the foreign interest rate .
Prices move according to the classical money market:
Mi;t Vi;t = Pi;t Yi;t
The in‡uence of the external sector at the home economy occurs through foreign income, real
exchange rate, the interest rate that is determined in the global market. The level of consumption,
investment and net exports and the national income are determined once values of the E, Pi;t and
Yi;t , and the domestic and foreign interest rates are known. The model is closed by making net
national saving equal to net exports. This model can be applied to assess impacts of increase in
the domestic price level, a rise in government spending, increase or decrease in real money balances
246
Table 61: Parametric Speci…cation of the two coutry global economy ISLM model
Economy 1
Economy 2
Demand
Trade
Money
Demand
Trade
Money
100
a
20
0
20
M
P
20
a
200
0
10
M
P
b
0.75
1
0.3
v
3
b
0.8
1
0.3
v
3
t1
0.3
2
-1.3
f
0.2
t1
0.05
2
-1.3
f
0.3
i0
50
1
1.55
k
-30
i0
55
1
1.5
k
-50
i1
-5
2
0.2
lf
1000
i1
-5
2
0.2
lf
500
G
100
-150
gr
0.05
G
20
KA
-70
gr
0 03
KA
and the interest rate, the real exchange rate ,increase in the capital and labour inputs a rise in the
level of foreign income, change in both taxes and spending.
Two country interdependent global economy model is twice as complex as the small open economy model to solve. More often it is important to consider more than two countries either to
analyse the international policy coordination issue or to assess impacts of development in the international economies at home. Models become analytically intractable as more and more countries
are added into it. Simultaneous solution of all of the model equations requires numerical techniques
as illustrated below. Mixed complementarity solver in GAMS is used to solve this model.
Summary
Global economy is interdependent.
A small dynamic general equilibrium model of the global economy is solved analytically and
numerically.
Long run relationship obtained in the dynamic general equilibrium are tested by the GMM
estimation of dynamic panel model.
Empirical evidences are based on recent developments in the exchange rate markets and trade
balances.
Literature
Abdel-Rahman H.M. and G Norman and P Wang (2002),Allen, Gasiorek, Smith, Flam,
Sørensen (1998) Baldwin, Francois, Portes, Rodrik, (1997)
Barro R. J.(1991) ,Basu S. and D N Weil (1998),Blundell R W and R J. Smith (1989), Broer,
Westerhout, Bovenberg (1994),Beaulieu E, M. Benarroch and J. Gaisford (2004), Bhattarai
Francois (1996)
247
Demidova S. (2008),Dixit A K and J E. Stiglitz (1977), Doornik J A and Hendry D.F.
(2001),Dornbusch R. (1976),Epifani, P. and G. Gancia (2008),Fleming J. Marcus (1962),Frankel
J. A and D. Romer (1999),Francois J. F. (1996),Greenaway D. W. Morgan and P. Wright
(2002) ,Hansen L.P. (1982),Helpman E (1976),Hine R.C. and P.W. Wright (1998),Holly S and
M Weale Eds.(2000) ,
Haaland (1992) Harrison., Rutherford, Tarr (1997)
Haaland, Norman, Wergeland, Rutherford (1987), Kohler, Casella, Sapir (1996),Krugman P.
(1980),Melitz M. T. (2003),Mausz S J (1994),Miller, Marcus H.; Spencer, John E. (1977),Mundell
R. A (1962),Naylor R (2000) , Nordhaus and Yang (1996)
Piazolo (2001), Roe T and H. Mohladi (2001), Saito (2004) , Saarenheimo T (1993), WrenLewis, Darby, Ireland, Ricchi (1996) Wright (1988)
Krusell, Ohanian Ríos-Rull, and Violante (2000),Rogo¤ K and M Obstfeld (1996), Taylor
Mark (1995),Temple J. (1999),Uzawa, H. (1962),Verbeek M. (2004),Wooldridge J. M. (2002)
Abrego and Whalley(2000), Touhami, (1998) ,Armington (1969), Bhattarai (2007, 2001,1999)
Bhattarai and Whalley (2006, 2003), Daniel and Blanchard (1988) ,Maureen and Oates (1992)
,Edwards and Whalley (2007). Haskel, and Slaughter (2001),Winchester, Greenaway and Reed
(2006), Wright (1988), Robinson (2008) have modelled this issue from many di¤erent angles.
Ecomod conference series have made good contributions.
Piazolo (2001), Roe T and H. Mohladi (2001), Saito (2004) , Saarenheimo T (1993), WrenLewis, Darby, Ireland, Ricchi (1996) Wright (1988)
Krusell, Ohanian Ríos-Rull, and Violante (2000),Rogo¤ K and M Obstfeld (1996), Taylor
Mark (1995),Temple J. (1999),Uzawa, H. (1962),Verbeek M. (2004),Wooldridge J. M. (2002)
Abrego and Whalley(2000), Touhami, (1998) ,Armington (1969), Bhattarai (2007, 2001,1999)
Bhattarai and Whalley (2006, 2003), Daniel and Blanchard (1988) ,Maureen and Oates (1992)
,Edwards and Whalley (2007). Haskel, and Slaughter (2001),Winchester, Greenaway and Reed
(2006), Wright (1988), Robinson (2008) have modelled this issue from many di¤erent angles.
Ecomod conference series have made good contributions.
Literature on policy coordination
Cooper (1969) and Hamada (1976)
Kydland (1975) shows the inferiority of the non-cooperative Nash equilibrium compared to a
cooperative solution.
248
Lucas (1976), and Kydland and Prescott (1977) use the concept of rational expectations
and argue for the advantage of rule-based policies to create rational expectations equilibrium
solution.
Petit (1989): di¤erential games
Obstfeld (2001) and Rogo¤ (2002) provide an excellent review of some of the models used for
policy coordination with Mundell-Fleming-Dornbush type models.
Aarle et.al. (2002) examine the coalition formation in EMU; Conzoneri et. al.(2005)
Obstfeld (1994), Sutherland (1996), Senay (1998), Martin and Rey (2000), Evans and Hnatkovska
(2007), douglaslaxton-dynare
Aarle et.al. (2002) examine the coalition formation in EMU; Conzoneri et. al.(2005); Bhattarai and Mallick (2013), Taylor (2013) and Bhattarai (2014), Hirose (2014)
7.3
Two SEctor Static Global General Equilibrium Model with Money
This section is taken from Bhattarai K. (2011) General Equilibrium Impacts of Monetary and Fiscal
Policies on Welfare of Households in South Asia, Review of Development Economics, 15:4:745-757,
October
"Relatively few studies have integrated money and …nancial markets in general equilibrium
framework. Tobin (1969) put …nancial sector in the macro general equilibrium framework and had
found that equilibrium in the markets of stocks of assets conditional upon the assumed values of
outputs, incomes, and other ‡ows. Equilibrium of the …nancial and real sectors were mutually
consistent . McKinnon (1973) and Shaw (1973) provided qualitative and descriptive overview of
the contribution of the …nancial sector on economic development. Then Altig et al. (1995) argued
for a computable general equilibrium model to forecast macroeconomic variables taking account of
structural features of the economy. Such models could capture the …rst, second and third rounds of
impacts of monetary policies with proper accounting of the economy-wide income and substitution
e¤ects as outlined in the Bank of England (1999). While money is neutral in the ideal classical and
new classical worlds, it has real e¤ects under the New Neoclassical Synthesis (NNS) when prices are
assumed to be sticky (Arestis, Chortareas and Tsoukalas (2010), Rankin (1992), Patinkin (1989)).
Asymmetric information among lenders and borrowers, not only raises risks in investment from
credit market imperfections and ine¢ ciency as stated in Stiglitz and Weiss (1983) but also results
in panics and crises in the …nancial system as seen in advanced countries in 2008 and 2009. Such
incidents increase when macroeconomic policies fail to consider general equilibrium interactions in
the economy (Fama (1980), Diamond et.al. (1983) and Burnmeir (2009), Sinn (2009)). Underlying
249
root causes of all these volatilities and ‡uctuations are issue of redistribution of income and wealth,
the proceeds that a healthy …nancial system can generate. In my view earlier studies have not
properly integrated multiple households into the monetary general equilibrium model to analyse
redistribution impacts of …scal and monetary policies on income and welfare among households. A
simple AK type model, as found in Pagano (1993) shows how the growth of the …nancial sector
contributes to growth of output but is abstracts away from the reality as it neither has …scal
and monetary instruments nor the multiplicity of households and …rms to make it interesting for
policy analysis. In the …nancial CGE literature more concerns have been in the real side of the
economy leaving monetary and redistribution side aside( Robinson (1991) articles in Mercinier
and Srinivasan (1994)). This paper takes a step towards …lling this gap in the literature. The
household sector is integrated into two sector open economy model in the next section including
the speci…cations of preferences and technologies of the economy, monetary, government and the
external sectors. Reduced form expressions are derived for key variables in section four. Then
comes the parameterisation of the model and a brief discussion on comparative static results from
the model. Conclusion and list of references are at the end.
7.3.1
Outline of the Model
Utility measures the welfare of the households from consumption of goods and services purchased
from product markets 1 and 2. Producers demand capital and labour to produce and supply those
commodities. They pay remunerations to factors and aim to maximise pro…ts from those sales.
Households spend their labour and non-labour income on goods and services and pay direct and
indirect taxes to the government. Exports and imports link the South Asian economy to the rest
of the global economy11 . Fiscal and monetary policies aim to contain these ‡uctuations that arise
either from the demand or the supply sides of the economy.
1 1 Codes
for general equilibrium model with money written in GAMS, are available upon request. Model
is general enough to be applied to any other regions.
250
Open Economy Two Sector Multi-household General Equilibrium
Tax Model with Money
Money
Utility
U(X1,X2)
P.Y
H6, H7, H8, H9 H10
Market
MV
Product market 1
Product Demand
X1(I,P1)
X2(I,P2)
t1
Indirect taxes
t2
Production
Y1(K1,L1)
Y2(K2,L2)
Factor Demand
w1, r1
Wage and interest rate
w2, r2
tw1, tr1
Tax instruments
tw2, tr2
Trade
Exp1, Imp1
Labour
Resources
Product market 2
H1, H2, H3, H4 H5
Exp2, Imp2
Capital
TFP
Figure 1
For simplicity this model of South Asian economy has ten households ordered by income and
skills into deciles, two goods X1 and X2 representing industries (agriculture, manufacturing, machineries) and services (transport, communication, education, health,housing), two types of labour
L1 and L2 representing skilled and unskilled labour. It has two types of capital stock,K1 and K2
durable and non durable respectively. Economy is linked to the rest of the world by the ‡ows of
exports EX1 and EX2 and imports IM P1 and IM P2 . Government receives revenue from direct
taxes of labour and capital income and indirect taxes on consumption of X1 and X2 . Monetary
side of the economy are …rst characterised by the classical quantity theory of money where the total
money supply consists of both currency and demand deposits. Prices are proportional to money
supply and also in‡uenced by the volume of transaction and the velocity of money. The exchange
rate is determined by the value of the ratio of exports to imports.
7.3.2
Households
This model categorizes households in ten deciles who form a representative household sector of the
entire South Asian economy. E¢ cient allocations of resources means maximisation of welfare of
each of these households and the representative household sector of South Asia.
Aggregate utility of the economy is composite of household utilities
251
U=
H
X
uh
(G.621)
h=1
Utility of individual household h is
h
share of aggregate utility U given by uh .
uh =
h
:U
(G.622)
Aggregate demand for good X1 and X2 are total of demand for the commodities by households.
X1 =
H
X
xh1
(G.623)
h=1
X2 =
H
X
xh2
(G.624)
h=1
Share in demand for X1 and X2 of household h
xh1 =
h
:X1
(G.625)
xh2 =
h
:X2
(G.626)
Aggregate of labour demand of type L1 and L2 in the economy is total of labour supply of all
households:
L1 =
H
X
Lh1
(G.627)
h=1
L2 =
H
X
Lh2
(G.628)
h=2
Share of household h in total labour supply of type L1 and L2 then are:
Lh1 =
h
:L1
(G.629)
Lh2 =
h
:L2
(G.630)
Demand for leisure by households are determined through preference parameter for leisure,
LE h =
252
h
L
h
(G.631)
Aggregate of capital demand of type K1 and K2 of total of capital supplied by households
H
X
K1 =
K1h
(G.632)
K2h
(G.633)
h=1
H
X
K2 =
h=2
Share of type K1 and K2 capital of household h is:
K1h =
h
K2h =
h
:K1
(G.634)
:L2
(G.635)
Aggregate income is the total of the household income
I=
H
X
Ih
(G.636)
h=1
Income of household h is then given by
Ih =
h
:I
(G.637)
Households are heterogenous here as they di¤er by levels of income, preferences and endowments,
and hence by the amount of utility they are entitled to. Distribution of income and consumption
goods and services, labour supply, tax payments and transfer receipts and welfare across these
households are sensitive to aggregate levels of economic activities reported in the next section.
7.3.3
Economy
The representative household for the South Asia receives utility from consuming both X1 and X2
goods representing bundles of agricultural and manufacturing products and services, given by a
Cobb-Douglas utility function as:
U = X1 1 X21
1
(G.638)
This utility function is modi…ed to consider impacts of public goods on household utility in
additive form as U = X1 1 X21
where
G
H
1
+
G
H
represents utility to the representative household from public goods where G is govern-
ment spending and H is the number of households in the economy. Amount of G is determined by
253
the ability of government to …nance it through tax revenues. Theoretically government can provide
more public goods to households only if they are ready to pay more taxes.
Demand for X1 and X2 are derived from the standard optimality conditions as:
X1 =
1 :I
P1 (1 + t1 )
(G.639)
X2 =
(1
1 ) :I
P2 (1 + t2 )
(G.640)
Technology of production of goods Y1 and Y2 are respectively
Y1 = L1 K11
(G.641)
Y2 = L2 K21
(G.642)
Resources that …rm 1 and 2 employ in production process are related to the quantity and price of
inputs as:
C1 = w1 L1 + r1 K1
(G.643)
C2 = w2 L2 + r2 K2
(G.644)
Optimal conditions for pro…t maximising …rm 1 and 2 , given above technologies of production
and resource constraints are given by
(1
K1
w1 (1
=
) L1
r1 (1
(1
tw1 )
; K1 =
tr1 )
) w1 (1
r1 (1
tw1 )
L1
tr1 )
(G.645)
(1
K2
w2 (1
=
) L2
r2 (1
(1
tw2 )
; K2 =
tr2 )
) w2 (1
r2 (1
tw2 )
L2
tr2 )
(G.646)
Households receive remunerations for labour and capital from …rms and transfers (T R) from
the government and are allowed to engage in borrowing and lending resulting in net borrowing (B).
Thus their income is:
I = w1 L1 + r1 K1 + w2 L2 + r2 K2 + T R + B
(G.647)
General equilibrium requires clearing of goods, factors and …nancial markets. Market clearing
conditions in commodity markets are:
254
X1 = Y1
G1
(EX1
IM P1 )
(G.648)
X2 = Y2
G2
(EX2
IM P2 )
(G.649)
Labour market clearing implies total demand for labour equal to endowments net of leisure of
households:
L1 + L2 = L
H
X
h
L
(G.650)
h=1
Capital market clearing implies (investment process drives the capital accumulation process in the
dynamic version of the model):
K1 + K2 = K
(G.651)
Aggregate volume of output is total of sectoral volumes of output.
P1 Y1 + P2 Y2 = P:Y
7.3.4
(G.652)
Monetary Sector
Quantity theory of money implies balance between nominal demand and aggregate money supply
P:Y = M S:V
(G.653)
where P is price level, Y national income, M S money supply and V the velocity of circulation.
Initial reserve (R) of the banking system constitutes of currency (C) and initial demand deposit
(D0)
R = C + D0
Currency in circulation is
(G.654)
fraction of total reserves
C = :R
Initial deposit is the remaining amount (1
(G.655)
) of initial reserve
D0 = (1
) :R
Total deposit (T D) is inversely related to the required reserve (rr) ratio
255
(G.656)
D0
(G.657)
rr
Aggregate money supply in the economy constitutes of currency in circulation plus the total deposit
TD =
MS = C + TD
(G.658)
Monetary policy can have impact on real macro variables when price level P are sticky.
7.3.5
Government Sector
Government collects revenue (RV ) from direct taxes on capital (tr1 , tr2 ), labour (tw1 ; tw2 ) and
indirect tax on commodities (t1 ; t2 ) as:
RV = t1: P1: X1 + t2: P2: X2 + tr1: r1: K1 + tw1: w1 :L1 + tr2: r2 K2 + tw2: w2 :L2 + T R + B
(G.659)
Aggregate government expenditure (G) is spent in public consumption from both sectors (G1 ; G2 )
G = G1 + G2
(G.660)
Government expenditure on sector 1 and 2 goods are g1 and g2 fractions of its revenue:
G1 = g1 :RV
(G.661)
G2 = g2 :RV
(G.662)
Budget de…cit is the di¤erence between government spending and the revenue:
B=G
7.3.6
RV
(G.663)
External sector
Exports from sector 1 and 2, EX1 and EX2 are e1 and e2 fractions of sectoral output as:
EX1 = e1 :Y1
(G.664)
EX2 = e2 :Y2
(G.665)
Amount of exports from these sectors are made sensitive to prices making export demand
inversely sensitive to prices as EX1 = e1 : PY11 and EX2 = e1 : PY12 .
256
Imports by sectors 1 and 2, IM P1 and IM P2 are m1 and m2 fractions of sectoral output as:
IM P1 = m1 :Y1
(G.666)
IM P2 = m2 :Y2
(G.667)
The real exchange rate is given by the ratio of total value of exports to the total value of imports:
P1 :EX1 + P2: EX2
(G.668)
P M1 :IM P1 + P M2: IM P2
This is a small open economy model where trade does not automatically balance under the …xed
ER =
exchange rate regime. Therefore de…cit or surplus in the current account must be met by in‡ows
and out‡ows of capital.
7.3.7
Analytical Forms and the Solution Procedure
Interconnections between the various components of this model can be appreciated by …nding
reduced forms, derived below for few key variables of the model. By substitution of components
(G.648) and (G.649)
1 :I
= L1 K11
P1 (1 + t1 )
(1
e1
m1 )
g1 :RV
(G.669)
(1
1 ) :I
= L2 K21
P2 (1 + t2 )
(1
e2
m2 )
g2 :RV
(G.670)
Relative prices clearing both market then can be found by taking ratios of these two equations
(1 + t1 )
P2
=
P1
(1 + t2 ) (1
1
1)
L1 K11
(1
e1
m1 )
g1 :RV
L2 K21
(1
e2
m2 )
g2 :RV
!
(G.671)
Similarly (G.645) and (G.646) can be substituted into (G.643) and (G.644) to …nd analytical
expressions of labour and capital
L1 =
K1 =
(1
w1 + r1
) w1 (1
r1 (1
(1
tw1 )
tr1 )
) w1 (1
r1 (1
w1 + r1
(1
tw1 )
tr1 )
1
) w1 (1
r1 (1
C1
tw1 )
tr1 )
(G.672)
1
C1
(G.673)
Similar functions can be obtained for L2 and K2 . If we continue in this manner the relative
prices have to be determined by the parameters of preferences, technology and other institutional
factors in the economy. Impact of monetary factors in the economy follows from money market
equations as:
257
P: (Y1 + Y2 ) = P [F (L1 ; K1 ) + F (L2 ; K2 )] =
:R +
D0
rr
:V
(G.674)
Importance of …scal and monetary policies in the production occurs through taxes on labour and
capital income. Parameters determining the volume labour and capital also determine the amount
of output in the economy. When aggregate price is sticky and velocity is constant there is one to
one correspondence between the money supply and the level of output. Demand for good i thus
becomes a function of parameters of both the real and monetary sides the economy as:
Xih = Xi
P1 ; P2 ;
1;
!
; ; t1 ; t2 ; w1 ; w2 ; tw1 ; tw2 ; r1 ;
h
r2 ; e1 ; e2 ; m1 ; m2 ; g1 ; g2 ; C1 ; C2 ; :; R; D0; V;
h
;
h
;
(G.675)
Equilibrium demand found this way is optimal and is in‡uenced by the real and nominal sectors
of the economy. When prices are sticky increase is money supply is bound to have real impact in
the economy as the outputs Y1 or Y2 have to increase or decrease with amount of money supply
to balance the equation. With multiple households in the model it also could provide scope for
a signi…cant degree of redistribution of resources in the economy. Economy wide income and
substitution e¤ects continue till economy reaches a long run balanced equilibrium. Ultimately the
welfare of individual household depends on how much is produced in the economy and what share
they have out of total output and how much they contribute in production by supplying labour and
capital. Thus the household welfare which directly depends on household consumption is ultimately
determined by these deep structural parameters representing preferences, technology, policy and
institutional features in the model. While the ‡exibility of prices guarantee smooth functioning of
the economy generating e¢ ciency, rigidity in prices create distortions and hence reduce output and
welfare in the economy.
"
U h = U X2
P1 ; P2 ;
1;
; ; t1 ; t2 ; w1 ; w2 ; tw1 ; tw2 ; r1 ;
r2 ; e1 ; e2 ; m1 ; m2 ; g1 ; g2 ; C1 ; C2 ; :; R; D0; V;
h
;
h
;
h
!#
(G.676)
Integrating money into the general equilibrium in this way is a good way to trace out the impacts
of both …scal and monetary policy factors on output, employment and welfare in the economy.
7.3.8
Parameterisation of the Model
The open economy model presented in equations (1) to (56) above has 139 variables. In theory it
is possible to solve all 139 endogenous variables in terms of the parameters as illustrated above. In
practice it is di¢ cult to produce analytical solutions for so many variables simultaneously. Therefore
this model is solved using numerical non-linear programming technique in GAMS. Model variables
are uniquely determined in terms of parameters of preferences
258
h
, share of labour
h
, and
h
share of capital income
of households as given in Table 3. Income distribution structure are
taken from the last column of Table 2. Thus this model is computed for a representative South
Asian economy with multiple households with these structural features.
Table 62: Preference and share parameters of households
h1
h2
h3
h4
h5
h6
h7
h8
h9
h10
h
0.44
0.48
0.49
0.41
0.47
0.57
0.47
0.55
0.48
0.47
h
0.03
0.05
0.06
0.06
0.07
0.08
0.10
0.11
0.14
0.30
h
0.02
0.03
0.04
0.05
0.06
0.07
0.09
0.10
0.11
0.43
This model includes six tax policy instruments: tax on goods, wages and interest rates (t1 ,t2 ,tw1 ,
tw2 , tr1 , tr2 ) ,two export shares e1 and e2 two propensities to import m1 and m2 , preference and
technology parameter for the aggregate economy
, ,and
and endowments L and K aggregate
resources available for …rms C1 and C2 (which is in fact equity plus debt) , velocity of money v,
reserve requirement ratio rr; transfer (T R), share of public spending in sectors 1 and 2, g1 and g2
reserves and share of currency in initial reserves
t1
t2
0.25
0.15
0.4
0.6
as shown in Table 4.
Table 63: Policy and Technology Parameters
tw1
tw2
tr1
tr2
e1
e2
m1
m2
0.40
0.5
0.10
0.05
0.3
0.25
0.30
L
K
C1
C2
v
580
2068
2000
1600
1
0.15
0.03
R
g1
0.50
100
0.25
rr
TR
g2
0.05
0
0.1
Plausible values of these parameters given in Tables 1 and 2 are used for simulation in the next
section. Results are compared between the baseline model with ‡exible prices but without leisure
and public goods in utility functions of households to the results from the counterfactual model
where changes in monetary and …scal policies occur under the sticky prices with public and leisure
being taken into account into household’s utility function.
7.3.9
Numerical Example
The optimal allocation of resources in the economy are given by the equilibrium prices of goods
P1 and P2 , wage rates w1 and w2 , rental rates r1 and r2 and the exchange rate ER consistent
to optimisation by households and …rms, budgetary process of the public sector as given above,
market clearing conditions for goods and factors and trade balance conditions as explained above.
This is a macroeconomic model with detailed speci…cation of households and …rms. The results of
the model show a wide gap between the poorest and the richest categories of households in the level
259
of utility they receive as shown in Tables 5 and 6. Sensitivity of model solutions are tested for the
structural and policy parameters. Scenario (a, b and gm) relate to minor changes to preferences
( ,) , technologies of sector 1 and 2 given by ,and
respectively. These parameters are increased
by 0.02 for successive scenarios. Scenario R represents change in reserve requirement (R increased
by 0.02). Scenario F is for changes in …scal policy (g1 and g2 increased by 0.02). Scenario M for
changes in monetary policy (rr reduced by 0.01) and the scenario F M is for the mix of monetary
and …scal policy. Only levels of welfare of the lowest (poorest) and highest (richest) deciles for
scenario 1 and 6 are reported skipping intermediate households and scenarios. Detailed output on
all 139 model variables and all scenarios cannot be discussed here because of space limitations.
Table 64: Welfare Scenarios for the Poorest Decile in the Flexible Price Model
Scenario
a
b
gm
R
F
M
FM
1
6.34
6.21
4.69
4.68
5.39
5.39
5.39
.
.
.
.
.
.
.
.
6
6.28
5.57
5.04
5.39
9.01
5.39
5.25
Table 65: Welfare Scenarios for the Richest Household in the Flexible Price Model
Scenario
a
b
gm
R
F
M
FM
1
136.37
133.53
4.69
100.90
115.84
115.84
115.84
.
.
.
.
.
.
.
.
6
135.10
119.77
5.04
108.33
193.71
115.84
112.92
Fiscal policy can operate either by changing the tax instruments or by setting the level of public
spending or by its allocation in sector 1 and 2 according to spending objectives or by changes in
the borrowing requirement resulting from the combination of …scal operations. When price level is
perfectly ‡exible, particularly considering the smaller amount of tax revenue in this region relative
to the demand for public services, one may argue that households adjust to the tax, spending and
borrowing plan of the public sector in spirit of the Ricardian equivalence. Higher budget de…cit
means less income left for the households who internalise the public de…cit by saving more in
anticipation of future rise in taxes for redeeming accumulated debt as a result of de…cit …nancing.
However the model results show that a prudent …scal policy would be to leave prices ‡exible so that
markets could allocate resources e¢ ciently. Public spending should gear towards more productive
activities such as creation of human capital through better education system. Should government
choose a path of higher taxes it should make sure that prices are ‡exible because sticky prices
with higher taxes would distort enormously and will cause massive deterioration in allocation of
260
resources. Public spending used in an unbalanced fashion will bring ine¢ ciencies in allocation of
resources and reduce the welfare of households. The golden rule of public budget is to make it
sustainable and leave plenty of scope for the private sector to operate adhering to the ability and
bene…t principles of taxation and minimum interventions in the functioning of markets.
Monetary policy operates through a number of channels. Most important is the credit market
channel that is controlled by the rate of deposit creation and credit expansion in the economy. In the
recent economic crises the rate of deposit creation has rather been slow sending economy towards
credit crunch. This is equivalent to loss in con…dence and increase in the reserve requirement in
the banking system. Thus parameter rr and
are crucial for the operation of monetary policy.
The consequences of the monetary policy in the economy are given by the classical quantity theory
of money contained in the model where the aggregate prices are proportional to the total supply of
money. This model is rich enough to assess the impacts of …scal and monetary policies in all other
variables such as output, prices, employment , investment and exchange rates.
The base model was modi…ed in a number of directions taking account of suggestions by the
referee of the paper. Both public goods and leisure enter into the utility function of households.
Exports and imports are sensitive to price signals. Aggregate price level is made sticky. Policy
parameters are kept as same as in the base version of the model. Summary of welfare results of
the poorest and richest households from this version are reported in Tables 7 and 8.
Table 66: Welfare Scenarios for the Poorest Household in the Sticky Price Model
Scenario
a
b
gm
R
F
M
FM
1
5.69
5.76
5.33
4.97
5.33
5.33
5.33
.
.
.
.
.
.
.
.
6
5.71
5.37
5.04
5.33
2.79
5.33
5.34
Table 67: Welfare Scenarios for the Richest Household in the Sticky Price Model
Scenario
a
b
gm
R
F
M
FM
1
92.28
95..06
87.89
82.04
87.89
87.89
87.89
.
.
.
.
.
.
.
.
6
94.20
88.66
83.10
87.89
44.39
87.89
88.17
Complementarity between …scal and monetary policies generate better redistribution results in
comparison to earlier scenarios. However expansion in …scal policy transfers resources from private
to the public sector which seems to be less e¢ cient than the private sector in producing output.
Sharp redistribution of income is noticed from the richer to the poorer households. As the market
261
is distorted by higher taxes and …xed prices, richer households withdraw their labour. Such policy
distorts incentives of the private sector and both rich and poor households are worse o¤. Rich
household value leisure more and supply less labour and that reduces aggregate output. Economywide income and substitution e¤ects have worked in reverse directions by imposition of sticky prices.
Model results show that incentive e¤ects must be considered while considering policy reforms.
7.3.10
Conclusion from the static two country model
Impacts of …scal and monetary policies are assessed in the an open economy two sector multihousehold general equilibrium tax model with money and applied to the South Asia. The impacts
of …scal expansions are generally positive for all categories of households under the ‡exible price
system but the gains are much higher for households in the upper income group than for those in the
bottom. The ‡exible relative prices guarantee the optimal allocation of resources in such economy.
Simulation results show that demand, output and employment are sensitive to the preferences of
consumers, con…dence of producers and sector speci…c production technologies. Monetary policy
is super-nuetral under the ‡exible price regime but can complement the …scal policy well when
aggregate prices are made sticky. Combination of monetary and …scal policies in this manner can
have extensive impacts in the e¢ ciency and redistribution as higher taxes distort incentives to
work and investment from richer households. This slows down the economy. Stickiness of prices
generates similar results. Higher taxes or sticky price level cause distortions and reduce welfare level
of both rich and poor households despite some desirable redistributive e¤ects. Stickiness distorts
economy as much as higher rates of taxes. Flexibility in prices leads to super neutrality of money
but enhances the market mechanism and makes the …scal policy more e¤ective and e¢ cient."
Dynamic two country model is sased on Bhattarai K. and S. Mallick (2013) Impact of China’s
currency valuation and labour cost on the US in a trade and exchange rate model. North American
Journal of Economicsand Finance, 25:40-59
7.4
Two Country Dynamic Global Economy Model
Taking intuitive lessons from above theoretical developments regarding impacts of factor prices on
real exchange rates and growth rate of trading nations we propose a dynamic two country open
economy model of trade to ascertain factors that determine mutual gains from free trade and cause
‡ows of capital when trade does not balance. Then we test the model with empirical evidences on
relative growth rates, wages, the interest rate and the real exchange rates and trade balances of the
US and China to explain recent developments and to speculate what might happen to them in the
future. Each country consumes goods produced at home and produced in the partner country and
uses labour along with capital in production. Domestic and foreign wage, interest rates and relative
262
prices are determined by the optimality conditions when economic activities of two countreis are
connected through the real exchange rate. Those optimal conditions which are subject to shocks
of technological progress, preferences and policy from time to time will cause a deviation from the
optimal equilibrium as capital is mobile across countries but not the labour.
Our model consists of a home country i and a foreign country j. The utility function of a representative household in country i contains goods produced at home (Ci;t ), imported from abroad
(Mi;t ) and the leisure (li;t ). Government uses taxes on consumption (tci;t ), imported goods (tmi;t )
and labour income (twi;t ) to provide for public consumption (Gi;t ). With the Cobb-Douglas utility
function and the subjective discount factor (0 <
i
< 1), the intertemporal problem of the repre-
sentative household in home country i can be stated as:
max U0i =
1
X
t
i
Ci;ti Mi;ti li;ti
(G.677)
t=0
subject to its intertemporal bugdet constraint:
"
1
X
Pi;t (1 + tci ) Ci;t + Pj;t (1 + tmi ) Mi;t + wj;t (1
t=0
"1
X
wi;t (1
twi ) Li;t + rj;t (1
tki ) Ki;t
t=0
where share parameters, each bewteen zero and one (0 <
i;
#
i;
twi ) li;t
#
(G.678)
i
< 1); sum to one ( i + i +
i
= 1).
Shocks to the preferences in this modle occur either with changes in the subjective discount factor
i
or in share parameters
i;
i
and
i.
The representative households in the foreign contry j solves
similar intertemporal problem.
A representtive …rm in home country i maximises pro…t (
i;t )
in a similar way supplying output
(Yi;t ) with labour (LSi;t ) and capital inputs (Ki;t ) as in Eaton (1985) and Grossman and Helpman
(1990):
max
i;t
= Pi;t Yi;t
ri;t Ki;t
wi;t LSi;t
(G.679)
subject to the technology and accumulation constraints:
(1
Yi;t = Ai;t Ki;tt Li;t
Ii;t = Ki;t
(1
t)
) Ki;t
(G.680)
Random productivity shocks Ai;t with costant mean Ai and variance
Investment (Ii;t ) net of of depreciation ( Ki;t
1)
(G.681)
1
2
i
in‡uence output of …rms.
contributes to the accumulation of capital stock.
263
Government receives revenue (Ri;t ) from taxes on consumption and imports as well as in labour
and capital income and spends on public services (Gi;t ) as:
Ri;t = tci Pi;t Ci;t + tmi Pj;t Mi;t + twi wj;t LSi;t + tki ri Ki;t
Gi;t
(G.682)
Markets for goods clear but can be segmented across borders (Gopinath et al. (2011)):
Yi;t = Ci;t + Ii;t + Xi;t
Mi;t + Gi;t
(G.683)
Labour market clears at national level as in Markusen and Svensson (1985):
Li;t = LSi;t + li;t
(G.684)
The foreign country j has similar speci…cation of technology and labour markets.There can be
two di¤erent ways of trade balance. First one where trade is balanced period by period in the sense
that value of export and imports are the same for country i as:
Pi;t Xi;t = Pj;t Mi;t
(G.685)
Trade is …nanced by the ‡ow of credits which is subject to trade-…nance shocks as in Ahn et al.
(2011):
(Si;t
Ii;t ) + (Xi;t
Mi;t ) = 0
(G.686)
Another way to make trade balanced intertemporally (in the present value terms) as:
1
X
t
(Pi;t Xi;t
Pj;t Mi;t ) =
1
X
t
(T Bi;t ) =
1
X
t
( Fi;t ) = 0
(G.687)
Imbalances result in the accumulation of foreign assets temporarily but should disappear in the
long run though this may last far long in the future. A country with trade surplus (T Bi;t > 0)
accumulates foreign assets (Ft ) and one with de…cit (T Bi;t < 0) decummulates it. The dynamics
of foreign asset accumulation is by the real interest rate as:
Fi;t+1 = Fi;t (1 + ri;t ) +
Fi;t
(G.688)
Stocks of these assets increase options available to an economy in investment (Ii;t ) or adoption of a
new technology (Ai;t ) which determine growth of output and welfare of households in it. Depletion
of these assets persistently can in‡uence on con…dence of consumers and producers, lower growth
rate and can cause …nancial and economic crisis.
264
Current price of commodity in country i is linked to the future price and the interest rate
through an inter-temporal arbitrage condition as:
Pi;t =
Pi;t+1
1 + ri;t
(G.689)
Bilateral real exchange rate for country i is expressed in ratios of domestic and foreign prices for
countries i and j respectively as:
Ei;t =
Pi;t
Pj;t
(G.690)
A competitive equilibrium in this two country trade model is given by the sequence of prices
fPi;t ; Pj;t g interest rates, fri;t ; rj;t g wage rates fwi;t ; wj;t g , the real exchange rates,fEi;t ; Ej;t g
such that given public policies that include taxes in consumption ftci;t ; tcj;t g labour income
ftwi;t ; twj;t g and capital income ftri;t ; trj;t g and imports tari¤s ftmi;t ; tmj;t g the allocations of
consumption, imports, leisure, fCi;t ; Mj;t ; li;t ; Cj;t ; Mi;t ; lj;t g that maximise the lifetime utility
of households U0i and U0j in home and the foreign countries. The choices labour and capital inputs
fLSi;t ; Kj;t ; Ki;t ; LSj;t g maximise pro…t of …rms and the government expenditures fGi;t ; Gj;t g
are compatible with the government revenue, fRi;t ; Rj;t g and exports fXi;t ; Xj;t g are compatible
with importsfMi;t ; Mj;t g in both countries. Market mechanism in‡uences allocations of resources
in both coutries through the real exchang rate that depend on relative prices.
The in…nite horizon problem is reduced to …nite horizon by …xing the terminal period T to the
far distance in the future. Similarly the labour endowments Li;t ; Lj;t grow exogenously. Policy
shocks to these economies occur through tax and tari¤ instruments such as ftci;t ; tcj;t ; tmi;t ; tmj;t ; twi;t ; twj;t ; tri;t ; trj
that are determined by the policy makers taking national and international circumstances. The
model parameters
7.4.1
i;
i;
i
and
i
are estimated from the data.
Analytical Results of Optimisation
Since the in…nite horizon problem is analytically intractable model is solved using the …rst order
intertemporal optimisation conditions for any two time intervals as these optimality should hold
for any other periods. First order conditions for households with respect to consumption, imports,
leisure and shadow prices for t and t + 1 periods are:
Ci;t :
Ci;t+1 :
i
i
t+1
t
Ct i
1
M t i lt i =
i 1
i
i
Ct+1
Mt+1
lt+1
=
265
t Pi;t
(1 + tci )
t+1 Pi;t+1
(1 + tci )
(G.691)
(G.692)
t
Mi;t :
i
t+1
Mi;t+1 :
i
i;t
:
:
=
1
C t i M t i lt i
i
i
i
Ct+1
Mt+1
lt+1
t Pj;t
(1 + tmi )
t+1 Pj;t+1
=
1
t wi;t
=
twi ) Li;t + rj;t (1
(G.693)
(1 + tmi )
(G.694)
twi )
(G.695)
(1
t+1 wi;t+1
(1
Pi;t (1 + tci ) Ci;t + Pj;t (1 + tmi ) Mi;t + wj;t (1
= wi;t (1
i;t+1
lt i =
1
t+1
i i
li;t+1 :
1
i
i
i
Ct+1
Mt+1
lt+1
=
t
i i
li;t :
Ct i Mt i
twi )
twi ) li;t
tki ) Ki;t
(G.697)
Pi;t+1 (1 + tci ) Ci;t+1 + Pj;t+1 (1 + tmi ) Mi;t+1 + wj;t+1 (1
wi;t+1 (1
(G.696)
twi ) Li;t+1 + rj;t+1 (1
twi ) li;t+1
tki ) Ki;t+1
(G.698)
Above …rst order conditions result in the Euler equations as follows:
Ci;t
:
Ci;t+1
1
Mi;t
:
Mi;t+1
1
li;t
1
li;t+1
:
i
i
Ci;t+1
:
Mi;t+1
li;t+1
:
Mi;t+1
Mi;t+1
:
li;t+1
Ci;t
Ci;t+1
(
i
Ci;t
Ci;t+1
i
Ci;t
Ci;t+1
i
1)
Mi;t
Mi;t+1
Mi;t
Mi;t+1
Mi;t
Mi;t+1
i
i
i
i
i
i
(
li;t
i
li;t+1
i
1)
li;t
li;t
li;t+1
Pi;t
P
i;t+1 i;t+1
(G.699)
=
Pi;t
P
i;t+1 i;t+1
(G.700)
i
li;t+1
i
=
i
i
1
=
i;t
i;t
wi;t
w
i;t+1 i;t+1
i;t
(G.701)
Mi;t+1
Ci;t+1
=
Pi;t+1 (1 + tci )
Pj;t+1 (1 + tmi )
(G.702)
li;t+1
Ci;t+1
=
Pi;t+1 (1 + tci )
wi;t+1 (1 twi )
(G.703)
li;t+1
Mi;t+1
=
Pj;t+1 (1 + tmi )
wi;t+1 (1 twi )
(G.704)
266
Similarly the …rst order conditions for …rms are:
i;t
= Pi;t Yi;t
ri;t Ki;t
Ki;t :
i;t Pi;t Ki;t
1
Kj;t :
j;t Pj;t Kj;t
1
t
t
Li;t :
1
Lj;t :
1
j;t
i;t
(1
Li;t
(1
Lj;t
t)
t)
wi;t LSi;t
Pj;t Kj;tt Lj;t t = wj;t or
i;t Pi;t Yi;t
= ri;t or
Ki;t
j;t Pj;t Yj;t
= rj;t or
Pi;t Ki;tt Li;t t = wi;t or
(1
1
(G.705)
Kj;t
Pi;t Yi;t
i;t
Li;t
)Pj;t Yj;t
Lj;t
i;t
= ri;t
(G.706)
= rj;t
(G.707)
= wi;t
(G.708)
= wj;t
Initial capital stocks and the terminal investment conditions for country i and j are:
Ki;0 and Kj;0
Ii;t = (gi +
i ) Ki;T
1;
Ij;t = (gj +
(G.709)
j ) Kj;T
1
(G.710)
Whether the wage rates and the interest rates are same or di¤er from one country to another depend
partly upon the marginal productivity and mobility of factors and partly on the tari¤ rates across
countries as mentioned in the literature above. If labour and capital are perfectly mobile then ratios
of marginal productivities across two countries in equilibrium are same as the ratios of rental rates
and wage rates as:
j;t Pj;t Yj;t
Ki;t
rj;t
=
P
Y
K
ri;t
i;t
i;t
j;t
i;t
1
j;t
1
i;t
(G.711)
Pj;t Yj;t Li;t
wj;t
=
wi;t
Pi;t Yi;t Lj;t
(G.712)
These conditions give us the equilibrium real exchange rate in terms of relative prices of commodities
between two countries, which further relate to marginal productivies of labour and capital, rental
rates and ratio of imports to domestic consumption as follows:
Ei;t =
Pi;t
=
Pj;t
i;t Yi;t
1
Kj;t rj;t
=
1
j;t Yj;t Ki;t ri;t
i;t
j;t
Yi;t Lj;t wj;t
=
Yj;t Li;t wi;t
i
i
Mi;t (1 + tmi )
Ci;t (1 + tci )
(G.713)
These theoretical derivations, similar to those in Bhattarai (2011), show interdepedence of the
exchange rates, relative output, relative wage rate, relative interest rate, consumption taxes and
267
tari¤ rates between two trading nations. The model solutions can di¤er remarkably when two
countries di¤er in productivites of capital
i;t ;
or interest rates (ri;t ; rj;t ) or the wage
j;t
rates (wi;t ; wj;t ) or in the stock of capital (Ki;t ; Kj;t ) or endowments of labour (Li;t ; Lj;t ) or in
tari¤s and tax rates (tmi;t ; tcj;t ) or in the preferences and technologies ( i ;
i;
i) :
Cooperation
in policies of home and foreing countries can result in mutually bene…cial in‡ows and out‡ows or
the retaliation could result in the collapse of trade as seen in 2008-09 recession when international
demand or supply shocks had reduced the global trade by up to 14 percent. How such structural
features of the real exchange rates underpin the patterns of the nominal exchange rates is well
explained in the studies of Mundell (1957), Meade(1978), Miller and Spencer (1977), Eaton (1987),
Neary (1988), Taylor (1995), Eaton and Kortum (1999). In short, the long run equilibrium real
exchange rate is a consequence of the balancing forces of the demand and supply for home and
foreing products.
It is possible to make theoretical predictions using the derivations of the dynamic two country
trade model as presented in Table 1.
Exchange rate (Ei;t ) rises when the domestic goods are sold at higher price than comparable
foreign goods, (@Ei;t " if @Pi;t > @Pj;t ). Term of trade is in favour of the home country. Higher domestic wage rate makes home country less competitive causing a fall in the exchange rate but it rises
when wage rate increases in the foreign country, (@Ei;t # if @wi;t > @wj;t ). Similar arguments apply
to the interest rate. Higher domestic interest rate pushes exchange rate up by raising the the cost
of capital at home but higher interest rate abroad makes the foreign country less competitive and
raises the value of home currency (@Ei;t # if @ri;t > @rj;t ). Increase in labour and capital at home
lowers the price of commodity and hence puts downward pressure on the exchange rate but these are
sensitive to productivity of labour and capital inputs, (@Ei;t " if @Li;t > @Lj;t or @Ki;t > @Kj;t ).
Table 1: Theoretical Predictions from Dynamic Two Country Trade Model
Pi;t
Pj;t
wi;t
wj;t
ri;t
rj;t
Yi;t
Yj;t
Li;t
Lj;t
Ki;t
Kj;t
Ei;t
Ci;t
Mi;t
Ei;t
+
-
-
+
-
+
+
-
-
+
-
+
+
-
1
-
+
Yi;t
+
-
-
+
+
-
1
+
+
-
+
-
-
+
+
-
+
ri;t
-
+
+
-
1
+
+
-
+
-
-
+
+
-
-
+
-
wi;t
-
+
1
+
+
-
+
-
-
+
+
-
-
+
-
+
-
Ci;t
-
+
+
-
+
-
-
+
+
-
+
-
-
+
-
1
+
+
1
i;t
j;t
Mi;t +
+
+
+
+
+
+
+
Similar arguments can be made to other model variables including the GDP (Yi;t ) ;wage rate
(wi;t ) ;interest rate (ri;t ) ; consumption (Ci;t ) ; and imports (Mi;t ) as illustrated in Table 1.
The model solutions can di¤er remarkably when two countries di¤er in productivites of capital
i;t ;
j;t
or interest rates (ri;t ; rj;t ) or the wage rates (wi;t ; wj;t ) or in the stock of capital
(Ki;t ; Kj;t ) or endowments of labour (Li;t ; Lj;t ) or in tari¤s and tax rates (tmi;t ; tcj;t ) or in the
268
preferences and technologies ( i ;
i;
i) :
Cooperation in policies of home and foreign countries can
result in mutually bene…cial in‡ows and out‡ows or the retaliation could result in the collapse
of trade as seen in 2008-09 recession when international demand or supply shocks had reduced
global trade by up to 14 percent. How such structural features of the real exchange rates underpin
the patterns of the nominal exchange rates is well explained in the studies of Mundell (1957),
Meade(1978), Miller and Spencer (1977), Eaton (1987), Neary (1988), Taylor (1995), and Eaton
and Kortum (1999). In short, the long run equilibrium real exchange rate is a consequence of the
balancing forces of the demand and supply for home and foreign products.
References
[1] Ahn J.B, M. Amity and D.E. Weinstein (2011) Trade …nance and the greate trade collapse,
American Economic Review, 101:3:298-302.
[2] Baak, S.J. (2008) The bilateral real exchange rates and trade between China and the U.S.,
China Economic Review, 19 (2): 117-127.
[3] Bahmani-Oskooee, M., and A. Ratha (2010), S-Curve dynamics of trade between U.S. and
China, China Economic Review, 21 (2): 212-223.
[4] Bahmani-Oskooee, M. and Wang, Y. (2007), United States-China Trade at the Commodity
Level and ther Yuan-Dollar Exchange Rate. Contemporary Economic Policy, 25: 341–361.
[5] Balassa Bela (1967) Trade Creation and Trade Diversion in the European Common Market,
Economic Journal, 77:305:Mar.:1-21
[6] Baldwin R. E. (1992) Measurable Dynamic Gains from Trade, Journal of Political Economy,
100:1 :Feb: 162-174
[7] Bergin, P.R., and R.C. Feenstra (2009) Pass-Through of Exchange Rates and Competition
between Floaters and Fixers, Journal of Money, Credit and Banking, 41 (s1): 35-70.
[8] Bhagwati J. (1964) The Pure Theory of International Trade: A Survey, Economic Journal,
74:293:1-84.
[9] Bhattarai K. (2011) Econometric and stochastic general equilibrium models for evaluation of
economic policies, Int. J. Trade and Global Market, 4:2:212-241.
[10] Bhattarai, K.R. and J. Whalley (2006) Division and Size of Gains from Liberalization of Trade
in Services, Review of International Economics, 14 (3): 348-361, August.
269
[11] Bhattarai K. and S. Mallick (2013) Impact of China’s currency valuation and labour cost on
the US in a trade and exchange rate model. North American Journal of Economicsand Finance,
25:40-59.
[12] Corden, W.M. (2009), China’s Exchange Rate Policy, Its Current Account Surplus and the
Global Imbalances. The Economic Journal, 119: F430–F441.
[13] Dekle R, J. Eaton, and S. Kortum (2007) New approaches to international trade: Unbalanced
Trade, American Economic Review, 70:2:351-355.
[14] Eaton J. (1987) Dynamic speci…c factor model of international trade, Review of Economic
Studies, 54:2: 325-338, April.
[15] Eaton J and S. Kortum (2002) Technology, Geography, and Trade, Econometrica, 70:
5:Sep:1741-1779
[16] Frenkel J. A. and M. L. Mussa (1981) Monetary and …scal policies in an open economy, American Economic Review, 71:2:PP:May 253-258.
[17] Fry R. and A. Pagan (2011) Sign restrictions in structural vector autoregressions: a critical
survey, Journal of Economic Literature, 49:4:938-960.
[18] Gopinath G., P.O Gourinchas, C.T Hsieh and N. Li (2011) International prices, costs and
markup di¤erences, American Economic Reivew, 101 (6): 2450-2486.
[19] Granville, B., S. Mallick, and N. Zeng (2011), Chinese exchange rate and price e¤ects on G3
import prices, Journal of Asian Economics, 22 (6): 427-440.
[20] Grosssman G.M. and E. Helpman (1990) Comparative Advantage and Long-Run Growth,
American Economic Reivew, 80:4:796-815.
[21] Imbs, J.M., Mumtaz, H., Ravn, M.O., Rey, H. (2010), One TV, One Price? Scandinavian
Journal of Economics, 112 (4): 753-781.
[22] Johnson H. G. (1951-1952). Optimum welfare and maximum revenue tari¤s, Review of Economic Studies, 19:1:28-35.
[23] Krugman P. (1980) Scale Economies, Product Di¤erentiation and the Pattern of Trade, American Economic Review, 70:5:950-959.
[24] Krugman P. and A. J. Venables (1995) Globalization and the Inequality of Nations, Quarterly
Journal of Economics, 110:4: Nov: 857-880.
270
[25] Markusen J. R. and L. E. O. Svensson (1985) Trade in Goods and Factors with International
Di¤erences in Technology, International Economic Review, 26:1:Feb:175-192.
[26] Melitz M. T. (2003) The Impact of Trade on Intra-Industry Reallocations and Aggregate
Industry Productivity, Econometrica, 71:6:1695-1725.
[27] Marquez, J. and Schindler, J. (2007), Exchange-rate E¤ects on China’s Trade. Review of International Economics, 15: 837–853.
[28] Meade, James (1978) The Meaning of Internal Balance, Economic Journal, 88 (351): Sep
423-435.
[29] Mehrotra, A.N. (2007) Exchange and interest rate channels during a de‡ationary era— Evidence
from Japan, Hong Kong and China, Journal of Comparative Economics, 35 (1): 188-210.
[30] Miller, M. H.; Spencer, J. E. (1977)The Static Economic E¤ects of the UK Joining the EEC:
A General Equilibrium Approach Review of Economic Studies, 44:1: 71-93, Feb
[31] Mundell R. A. (1957) International Trade and Factor Mobility, American Economic Review,
47: 3:Jun: 321-335.
[32] Naylor R (2000) Trade and Wages When the Trade Regime is Determined Endogenously,
Review of International Economics, 8:3:556-565.
[33] Neary P.J. (1988) Determinants of the Equilibrium Real Exchange Rate, American Economic
Review, 78:1:Mar.: 210-215.
[34] Norman V. D. and A. J. Venables (1995) International trade, factor mobility and trade costs,
Economic Journal, 105:433:1488-1504.
[35] Ra…q, M.S. and S.K. Mallick (2008) The e¤ect of monetary policy on output in EMU3: A sign
restriction approach, Journal of Macroeconomics, 30 (4): 1756-1791.
[36] Sim C.A (1980) Macroeconomics and Reality, Econometrica, 48:1 Jan.1 -45.
[37] Sims, C.A. and T. Zha (1999) Error Bands for Impulse Responses, Econometrica, 67 (5):
1113-1155
[38] Stock, J.H., and M.W. Watson (2001). Vector Autoregressions, The Journal of Economic
Perspectives, 15 (4): 101-115.
[39] Turnovsky S. (1999) International Macroeconomic Dynamics, MIT Press.
271
[40] Taylor, M. (1995) The Economics of Exchange Rates, Journal of Economic Literature, March,
33,1: 13-47.
[41] Ventura, J. (1997) Growth and Interdependence, Quarterly Journal of Economics, 112 (1):
57-84.
[42] Xu Chenggang (2011) The fundamental Institutions of China’s reforms and Development,
Journal of Economic Literature, 49:4:1076-1151.
[43] Xu, Zhenhui (2008), China’s Exchange Rate Policy and Its Trade Balance with the US, Review
of Development Economics, 12 (4): 714-727.
[44] Witte, M.D., (2009) Pricing to Market: Chinese Export Pricing to the USA after the Peg,
China & World Economy, 17 (2): 65-78.
7.5
International macroeconomic policy coordination
Basics of the Nash policy game
Let us consider three countries aiming for a policy coordination with the Nash utility frontier
Nt = U1;t U2;t U3;t
(G.714)
Each receive utility from consuming products produced in each country:
Ui;t = F (y1;t; y2;t ; y3;t )
(G.715)
Goods supply process is determined simultaneously as
y1;t =
1;0
+
1;2
y2;t +
1;3
y3;t +
1;1
y1;t
1
+
1;2
y2;t
1
+
1;3
y3;t
1
+ e1;t
(G.716)
y2;t =
2;0
+
2;1
y1;t +
2;3
y3;t +
2;1
y1;t
1
+
2;2
y2;t
1
+
2;3
y3;t
1
+ e2;t
(G.717)
+
3;1
y1;t +
3;2
y2;t +
3;1
y1;t
1
+
3;2
y2;t
1
+
3;3
y3;t
1
+ e3;t
(G.718)
Nash to VAR
y3;t =
3;0
272
0
1
B
B
@
3;1
B
= B
@
1;0
2;0
3;0
1;3
1
2;1
0
7.6
1;2
2;3
1
3;2
0
1
C B
C+B
A @
10
1;1
1;2
2;1
2;2
3;1
3;2
y1;t
1
C
CB
CB y C
A @ 2;t A
y3;t
10
1;3
2;3
3;3
y1;t
CB
CB y
A @ 2;t
y3;t
0
e1;t
10
y1;t
1
1
C
C B
C+B e C
A @ 2;t A
e3;t
1
1
1
(G.719)
Nash-VAR Policy Game
0
=
1
y1;t
C
B
B y C
@ 2;t A
y3;t
0
1
B
B
2;1
@
3;1
0
B
B
@
1
B
+B
@
2;3
1
3;2
C
C
A
2;3
1
3;2
1
2;1
1
1;3
1
3;1
C
C
A
1;3
1
1;2
2;1
0
1;2
1
1;2
1;3
1
3;1
3;2
2;3
1
Paramters of Nash Policy Game
1
1
1
0
1
C
C+
A
B
B
@
1;0
B
B
@
1;1
1;2
1;3
2;1
2;2
2;3
3;2
3;3
2;0
3;0
0
1
C
C
A
0
3;1
1
CB
CB y
A @ 2;t
y3;t
e
B 1;t C
B e C
@ 2;t A
e3;t
1
1
1
1
C
C
A
(G.720)
In common meetings or summits they decide policies given by
1;0
;
2;0 ;
3;0
but each of them
face idiocyncratice shocks e1;t ; e2;t ; e3;t
Then each country determine its action yi;t taking account of actions taken by others yj;t and
such response patterns are given by parameters
1;2
;
1;3 ;
2;1
;
2;3
;
3;1
;
3;2
,
1;2
;
1;3 ;
2;1
;
and shocks e1;t ; e2;t ; e3;t :
Each would like to get more utility and this opens the bargain.
The optimal solution of this game should ful…ll four properties of Nash bargaining game.
This must be symmetric, e¢ cient, linear invariance and IIA.
273
2;3
;
3;1
;
3;2
7.6.1
Estimates for the Nash Policy Game
Estimates for the Nash Policy Game between Advanced and BRIC Countries
Table 68: Interdependence in Economic Growth between US, EU and BRIC Countries
USGR
EUGR
JPGR
CHGR
INGR
BRGR
RUGR
USGR_1
0.903 (10.1)
0.222 (1.06)
0.323 (1.81)
-0.153 (-1.39)
-0.203(-1.26)
0.004(0.02)
0.184(0.58)
EUGR_1
-0.049 (-0.45)
0.388(1.54)
-0.241 (-1.22)
0.118 (0.89)
0.167(0.86)
-0.046(-0.20)
-0.358(-0.94)
JPGR_1
0.187 (1.91)
0.538(2.34)
0.682 (3.48)
0.153(1.27)
-0.023(-0.13)
0.084(0.40)
0.880(2.54)
CHGR_1
0.071 (0.79)
0.543(2.59)
0.194 (1.09)
0.645(5.84)
0.138(0.85)
0.027(0.18)
0.798(2.52)
INGR_1
0.072(1.07)
-0.052(-0.34)
0.167(1.24)
0.251(3.01)
0.562(4.60)
0.479(3.11)
-0.193(-0.81)
BRGR_1
-0.135 (-1.91)
-0.356(-2.14)
-0.031 (-0.22)
-0.095(-1.08)
-0.052(-0.40)
0.479(3.11)
-0.543(-2.16)
RUGR_1
-0.499(-0.76)
0.086(-0.56)
0.117(0.88)
-0.095(-1.17)
-0.077(-0.64)
0.060(0.42)
0.719(3.08)
Constant
-0.270(-0.36)
-2.323(-1.28)
-2.322(-1.28)
2.119(2.22)
2.137(1.53)
-2.675(-1.60)
-2.706(-0.99)
RSq (Adj)
0.84
0.75
0.61
0.71
0.45
0.55
0.69
F-stat
45.9
24.9
13.5
20.7
7.6
10.7
18.7
T-values are in the parentheses.
Impulse Responses in Growth between US, EU and BRIC Countries
Estimates for the Nash Policy Game between Advanced Country Club
Table 69: Rich Country Growth Club
USGR
EUGR
JPGR
CHGR
USGR_1
0.961(12.8)
0.287 (1.06)
0.242 (1.67)
-0.103(-1.06)
EUGR_1
-0.103(-2.39)
0.617(1.54)
-0.046(-0.55)
-0.038(-0.68)
JPGR_1
0.045(0.57)
0.130(2.34)
0.624(4.13)
0.067(0.66)
CHGR_1
0.085(1.19)
0.336(2.59)
0.235(1.70)
0.813(8.73)
Constant
-0.401(-0.54)
-2.425(-1.32)
-2.473(-1.72)
2.147(2.21)
RSq (Adj)
0.84
0.71
0.61
0.65
F-stat
77.1
37.4
23.9
28.3
T-values are in the parentheses.
274
US-EU and Japan Growth Club
Accumulated Response to Cholesky One S.D. Innov ations ± 2 S.E.
Accumulated Response of USGR to USGR
Accumulated Response of USGR to EUGR
Accumulated Response of USGR to CHGR
12
12
12
8
8
8
8
4
4
4
4
0
0
0
0
-4
-4
-4
-4
-8
-8
1
2
3
4
5
6
7
8
9
-8
10
1
Accumulated Response of EUGR to USGR
2
3
4
5
6
7
8
9
10
-8
1
Accumulated Response of EUGR to EUGR
2
3
4
5
6
7
8
9
10
1
Accumulated Response of EUGR to JPGR
15
15
15
10
10
10
10
5
5
5
5
0
0
0
-5
1
2
3
4
5
6
7
8
9
1
Accumulated Response of JPGR to USGR
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
1
Accumulated Response of JPGR to JPGR
12
12
12
8
8
8
8
4
4
4
4
0
0
0
-4
1
2
3
4
5
6
7
8
9
1
Accumulated Response of CHGR to USGR
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
1
Accumulated Response of CHGR to JPGR
8
8
8
6
6
6
6
4
4
4
4
2
2
2
0
0
0
0
-2
-2
-2
-2
-4
-4
-4
-4
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
8
9
10
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
Accumulated Response of CHGR to CHGR
8
1
7
-4
1
Accumulated Response of CHGR to EUGR
6
0
-4
10
5
Accumulated Response of JPGR to CHGR
12
-4
4
-5
1
Accumulated Response of JPGR to EUGR
3
0
-5
10
2
Accumulated Response of EUGR to CHGR
15
-5
[Figure]
Accumulated Response of USGR to JPGR
12
2
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Estimates for the Nash Policy Game in the BRIC Club
Table 70: BRIC Country Growth Club
CHGR
INGR
BRGR
RUGR
CHGR_1
0.679(6.35)
0.170(1.08)
0.409(2.23)
0.790(2.34)
INGR_1
0.272(3.36)
0.597(5.02)
0.025(0.18)
-0.247(-0.97)
BRGR_1
-0.014(-0.21)
-0.034(-0.33)
0.504(4.19)
-0.255(-1.15)
RUGR_1
-0.009(-0.31)
-0.013(-0.30)
0.053(1.03)
0.750(7.88)
Constant
1.225(1.62)
1.440(1.30)
-2.820(-2.17)
-3.822(-1.60)
RSq (Adj)
0.71
0.45
0.57
0.62
F-stat
35.9
12.6
19.7
24.3
T-values are in the parentheses.
7.7
Multicountry macro interaction model
Yi;t = Ci;t + Ii;t + N Xi;t + Gi;t
(G.721)
Consumption function for this country is
Ci;t =
i Yi;t 1 ;
275
0<
i
<1
(G.722)
Investment
Ii;t =
i
(Ci;t
Ci;t
1) ;
i
>1
(G.723)
<1
(G.724)
Exports:
Xi;t =
i Yi;t
+
x;t
0<
x;t
Mi;t = mi Yi;t +
m;t
0<
m;t
Imports
<1
(G.725)
Net exports:
N Xi;t =Xi;t -Mi;t
(G.726)
Objective of policy coordination is to maximise the global output:
N
X
YT =
Yi;t
(G.727)
i=1
Reduced form of this model for country i takes the form:
(1
i
+ mi ) Yi;t
i
(1 +
i ) Yi;t 1
i i Yi;t 2
= Gi +
i;x
i;m
(G.728)
From the reduced form of this model the steady state output for country i is obtained as:
Yi =
Gi +
+
m)
i
[(1
x
m
(1 + ) +
(G.729)
]
The transitional dynamics of the model is obtained solving the homegenous part of equations(replace Yt = Abt in homogenous equation).
(1
(1
i
i
+ mi ) Yi;t
+
mi ) Abti
(1
i
i
i
(1 +
i ) Yi;t 1
(1 +
t 1
i ) Abi
+ mi ) b2i
i
(1 +
+
i i Yi;t 2
=
0
+
t 2
i i Abi
=
0
i ) bi
+
i i
=0
(G.730)
(G.731)
Cycle depends on quadratic roots of this equation
bi;1 ; bi;2 =
i (1 +
i)
q
2
i
(1 +
2 (1
276
2
i)
i
4
+ mi )
i i
(1
i
+ mi )
(G.732)
The transitional dynamics of the model is obtained solving the homegenous part of equations(replace Yt = Abt in homogenous equation).
(1
(1
i
+ mi ) Yi;t
+
mi ) Abti
(1
i
i
i
i
(1 +
i ) Yi;t 1
(1 +
t 1
i ) Abi
+ mi ) b2i
i
(1 +
+
i i Yi;t 2
=
0
+
t 2
i i Abi
=
0
i ) bi
+
i i
=0
(G.733)
(G.734)
Cycle depends on quadratic roots of this equation
i (1 +
bi;1 ; bi;2 =
7.7.1
q
i)
2
i
2
i)
(1 +
2 (1
i
4
i i
(1
i
+ mi )
+ mi )
(G.735)
Time path in the multicountry macro interaction model
Di¤erential Policy Game: multicountry macro
Complete solution
Yi;t = Ai;1 bti;1 + Ai;2 bti;2 + Y i
(G.736)
and the de…nite time path:
Yi;t
=
0
@
2
6
4
Yi;1
Yi;0
bi;1
i 1+
i +
0
B
+ @ Yi;0
2
6
4
Yi;1
i 1+
q
q
Gi +
+
1
i +m
Yi
bi;2
2 1+
i
i
2 1
2
i 1
Yi;0
Yi;1
bi;1
2 1+
i
2 1
x
i
2
4 i
i + mi
m
(1 +
Results of Macro Nash Policy Game
Results of Macro-Nash Policy Game
277
bi;2
i 1
3t
i + mi 7
5
b 2
i;
Yi
1
C
A
3t
i + mi 7
5
(G .7 3 7 )
)+
Parameters of the Macroeconomic model
Parameters of the Macro Policy Game Model
7.7.3
A
4 i
B h a t t a r a i ( 2 0 1 1 ) A d v a n c e d M a c r o e c o n o m i c T h e o r y, U n i v e r s i t y o f H u l l .
7.7.2
1
i + mi
Yi;1
Yi;1
i
bi;2
Table 71: Parameters of the Macro Policy Game Model: Homegeneous Case
US
EU
Japan China India
Brazil Russia
0.845
0.8
0.724
0.562
0.743
0.815
0.68
11.7
15.6
6.2
10.3
3.03
8.2
3.4
0.106
0.333
0.118
0.255
0.184
0.11
0.327
m
0.136
0.321
0.105
0.221
0.248
0.105
0.237
Yi;0
7.0346
6.9549
4.3520
0.4563
0.3059
0.5339
0.2819
Yi;1
7.0293
7.1615
4.4067
0.4999
0.3280
0.5644
0.2367
G
1.4571
1.6872
0.7664
0.2083
0.573
0.1320
0.411
i;x
N 0;
= M P C;
2
i;x
and
i;m
N 0;
= accelerator;
2
i;m
= exp ratio
m=imp ratio; Yi;0 = initial Y; G=pub spend
Source: Constructed from the World Development Indicators, 2012
Table 72: Outcome of Nash Cooperative and Non-Cooperative Solution
Cooperative Solution Non-Cooperative Solution
Scenario 1
Nash product
Scenario 2
Nash product
7.7.4
= 0:65;
= 1:5
44141
= 0:7;
= 0:65;
= 0:9
3176
= 1:2
56468
= 0:7;
= 0:8
9829
Conclusions
Strategic models show gains from macroeconomic policy coordination
Estimates show that there is a considerable growth competition among these countries.
An algorithm of the di¤erential policy game is outlined
International dependence occurs through trade and trade risks are important sources of macroeconomic ‡uctuations.
Macor di¤erential policy game could be evaluated using a multiplier-accelerator model with
the second order quadratic di¤erence equations
278
7.7.5
Growth Impacts of Foreign Direct Investment in an Open Economy
The in…nite horizon utility maximisation problem subject technology, domestic and foreign capital
accumulation and market clearing conditions can be written as:
max U0 =
Z
1
e
t
U (Ct ) dt ; U (Ct ) =
0
Subject to
Ct1
1
(G.738)
Yt = At Kt Ft1
(G.739)
Net domestic investment that causes a change in physical capital:
K t = Ik
Kt
(G.740)
1
Dynamic Optimisation
Net foreign investment similarly causes accumulation of foreign capital:
F t = IF
f Ft 1
(G.741)
Market clearing requires in each period requires that total output should equal total demand
Yt = Ct + Ik;t + IF;:t
(G.742)
This in…nite horizon constrained dynamic optimisation problem is solved using the current value
Hamiltonian function as:
J
=
Ct1
1
t
e
+ [Ik
Kt 1 ] + [IF
f Ft
h
i
+! At Kt Ft1
Ct
Ik;t IF;:t
1]
(G.743)
Hamiltonian First Order Conditions
First order conditions with respect to consumption, domestic and foreign capital and shadow
prices are:
@J
=C
@C
e
@J
=
@IK
279
t
!=0
!=0
(G.744)
(G.745)
@J
=
@IF
!=0
(G.746)
=
@J
=0
@K
(G.747)
=
@J
=0
@F
(G.748)
Hamiltonian First Order Conditions
These …ve equations can be used to solve the values of K; F; Y; C;
and
to show analytically
how such an economy can grow at a constant growth rate over time.
=
=
@J
=
@K
! AK
@J
=
@F
! (1
1
F1
(G.749)
) AK F
(G.750)
From equation (9) and (10)
C
e
t
=
(G.751)
Taking log both sides
ln C
t = ln
(G.752)
Hamiltonian First Order Conditions
By di¤erentiating both sides with respect to time:
C
C
=
(G.753)
Substituting
gc
=
=
Similarly from
C
=
C
1
1
AK
+
1
!
1
=
1
! AK
F1
F1
+
(G.754)
=!
=
! AK
1
F1
280
=
AK
1
F1
(G.755)
Hamiltonian First Order Conditions
=
1
! AK
=
F1
=
(1
1
) AK
F1
(G.756)
implies
1
AK
K
K
1
F1
F
=
1F 1
=
(1
(1
)
) AK
K
=
F
(1
or
1
F
(G.757)
(G.758)
)
Thus the ratio of domestic and foreign capital is constant. Putting this value in the production
function:
Y = AK F 1
= AK
K
F1
K
= AK
F1
K1
= AK
1
1
(G.759)
Hamiltonian First Order Conditions
So, despite the diminishing rate of return on domestic and foreign capital individually, the
complementarity between them makes the marginal productivity of domestic capital [K] equal to
A
1
1
. It does not diminish and may increase with technology. Adding domestic or foreign
capital generates economic growth at a constant rate in the manner close to the AK endogenous
growth model as:
g=
7.7.6
Y
C
K
F
=
=
=
=
Y
C
K
F
=
= gA + gK + (1
) ln
1
(G.760)
Empirical Literature on FDI and Growth
Among empirical studies on FDI, Wallis (1968) had looked at increase in in‡ows of FDI from
the US to the EU and assessed the importance of FDI in enhancing economic growth. Then
Feldstein and Horioka (1980) had estimated impacts of FDI on saving and investment. Desai, Foley
and Hines (2005) had found an almost one to one positive relationship between FDI in‡ows and
saving GDP ratios and investment, and negative relation between FDI out‡ows and reduction in
investment among OECD countries in the 1990s. Borensztein et al. (1998) found the need of
domestic absorptive capacity to make FDI important factor on economic growth in a study of FDI
‡ows from industrialised countries to 69 developing countries. In a recent study de Mello (1999)
used the panel data model to conclude that growth and FDI nexus are sensitive to country speci…c
factors and generally supports a positive relationship between FDI and growth in the long run.
Balasubramanyn et al. (1999) use panel data study of 46 developing countries to …nd support for
281
the Bhagwati hypothesis that the impact of FDI is larger in countries that have adopted export
led growth strategies. Similar …ndings are reported in country speci…c studies such as Ram and
Zhang (2000) and Binh and Haughton (2002). Wang and Zhao (2008) look at the technology
spillover e¤ect across vertically and horizontally integrated …rms and industries in China and …nd
ownership of FDI an important variable in assessing externalities of FDI. Helpman (2006) looks at
the role foreign aid to be similar to FDI. Lencik and Morrissey (2006) have shown how the volatility
of investment has detrimental impacts on economic growth. More recently, using a sample of 84
countries from 1987 to 2001, it is shown that the e¤ects of green…eld-investment and merger and
acquisition (M&A) have di¤erent impacts on actual economic growth. It is observed that, in most
cases, green…eld-investment raises economic growth whereas M &A can be bene…cial only when the
host country has adequate human capital [Wang and Wong (2009)].
Since none of above studies have explicitly tested the growth proposition developed above using
empirical data aim of this section is to illustrate the positive contribution of FDI on growth of
BRICS countries with the panel data from 2001 to 2011 constructed from the WDI (see Appendix
A for generic speci…cations of panel modes used for this study).
7.7.7
Empirics of FDI in BRICS Countries
FDI is a signi…cant factor in explaining the size of GDP in BRICS countries as is evident from the
following estimate based on panel data.
Table 73: GDP on FDI in BRICS countries
Level of GDP Constant $ in 2000
In‡ows of FDI
Constant
2
R = 0:87,
2
D a ta so u rce: W B D I;
Coe¢ cient
t-value
17.408
17.4
3.99621e+011
4.19
= 303:7 [0:000]
T = 1 1 (2 00 1-20 11 ),
N = 5 (B R IC S )
.
Similarly the size of the market, as measured in the GDP and the per capita income, is empirically robust factor both for the level of FDI and its ratio to GDP. It is evidence from the estimates
of the cross country panel of BRICS shown below.
The FDI ‡ows were not much sensitive to the ratios taxes, public spending and trade to GDP.
282
Table 74: Size of the economy and FDI in BRICS countries
In‡ows of FDI
Out‡ows of FDI
Coe¢ cient
t-value
Coe¢ cient
t-value
0.0519
22.6
0.0629
17.0
GDP Per Capita
4.77053e+006
3.28
6.22732e+006
2.05
Constant
-3.01516e+010
-9.76
-2.82548e+010
-3.03
GDP
2
R = 0:89,
2
2
= 575:4 [0:000]
R = 0:87,
2
= 303:0 [0:000]
Datasource: WBDI; T=11 (2001-2011), N=5 (BRICS).
Table 75: Size of the economy and Ratios of FDI In‡ows and Out‡ows in BRICS Countries
Ratio of FDI in‡ows
Ratio of FDI Out‡ows
Coe¢ cient
t-value
Coe¢ cient
t-value
Man va GDP
0.07606
8.70
-0.0205
-2.09
De…cit ratio
0.0683
2.07
0.1402
9.66
1.088
3.43
Constant
2
R = 0:22,
2
1.396
2
= 75:8 [0:000]
R = 0:15,
3.24
2
= 99:8 [0:000]
Datasource: WBDI; T=11 (2001-2011), N=5 (BRICS).
7.7.8
Empirics of FDI in OECD Countries
Taking account of these …ndings in the literature this section aims to test the predictions of the
above theories, particularly the impact of FDI in‡ows and out‡ows in investment and growth in
31 OECD countries for the period of 1990 to 2004.The data on GDP and GFCF is taken using
the currency for each country and then converting to US dollars using the exchange rate of the
national currency to the US dollar. Growth rates of GDP, investment and FDI are computed by
the authors. Variables used in this analysis were stationary (Table 2). We do not …nd any evidence
of reverse causality from growth to investment ratio as indicated by Blomstrom et al. (1996).
We have obtained the data for our analysis from the OECD database on FDI available from the
economic and social database for the UK ( http://www.esds.ac.uk/International/ international) for
years 1990 to 2004.
In‡ows and out‡ows relative to total domestic investment were extra-ordinarily high in Luxembourg (forty times higher) and noticeably higher in the South Korea (two to three times higher)
than in other countries. All variables used here were stationary on the basis of Levin, Lin and Chu
test statistics in Eviews as:
Results presented in Tables 4 to 9, estimated using the PcGive, reveal several interesting facts
283
Table 76: Common Unit Root Test of Panel Data with Levin, Lin, Chu (LCC) Test
Test-statistics Probability
Growth
-10.60
0.00
Investment ratio
-5.01
0.00
In‡ow ratio
-2.11
0.02
Out‡ow ratio
-2.05
0.02
Cross sections: 30; No of observations: 390
regarding the impact of FDI on growth and investment.
Firstly, the ratio of investment to GDP is a signi…cant determinant of growth rates across
OECD countries as shown in Table 5. This is exactly what is expected from the theory of economic
growth. Net investment adds to capital accumulation and more capital associated with given labour
generates more output. The negative sign in the lagged term shows cyclical pattern of investment
ratio. FDI contributes positively to growth. Higher tax rates cause lower growth rates which is
very intuitive. Overall …t of the model is good as R2 is 42 percent.
Table 77: Growth rate of output on investment Ratio in OECD Countries
Coe¢ cient Standard Error t-value t-prob
Growth (-1)
0.30686
0.130
-2.360
0.019
FDI ratio
0.00049
Tax rate
-0.00042
0.000
4.680
0.000
0.000
-2.010
0.045
Invratio
0.86255
0.202
4.270
0.000
Invratio (-1)
-0.85115
0.182
-4.670
0.000
Constant
0.03319
0.014
2.400
0.017
R2 = 0:42,
2
= 399:2 [0:000] ; T = 14; N = 31:
Table 78: Determinants of FDI in‡ows in OECD Countries
Coe¢ cient Standard Error t-value t-prob
openk
0.047
0.018
2.550
0.011
intract
0.001
0.000
2.440
0.015
-8.732
3.276
-2.670
0.008
Constant
2
R = 0:59,
2
= 7:68 [0:021] ; T = 14; N = 31:
In‡ows of FDI in OECD countries relates positively to the openness of the country (openk) and
284
Table 79: Determinants of FDI out‡ows in OECD Countries
Coe¢ cient Standard Error t-value t-prob
openk
0.051
0.021
2.460
0.014
intract
0.001
0.000
2.320
0.021
-9.366
3.690
-2.540
0.011
Constant
R2 = 0:57,
2
= 7:09 [0:029] ; T = 14; N = 31:
the size of the country (interaction of investment ratio and per capita GDP in PPP) as shown in
Table 6. Openness (openk) and size (intract) are also signi…cant determinant of out‡ows as shown
in Table 7.
In Table 6 we show that domestic investment ratio falls with a rise in the tax rate (taxrate)
but responds positively to share of capital in output (ki) and the ratios out‡ows to investment
(o‡winvratio) and are a bit lower for countries with higher per capita income (GDP_PPP). All
these …ndings correspond to the neoclassical theory of capital accumulation and are consistent to the
…ndings of Desai, Foley and Hines (2005). Thus, the panel regression analysis clearly reveals very
little in‡uence of FDI out‡ows on aggregate investment ratios and but good in‡uence on growth
rates from analysis of the results as in table 5.
Table 80: Contribution of FDI In‡ows and Out‡ows to Domestic Investment in OECD Countries
Coe¢ cient Standard Error t-value t-prob
Invratio (-1)
0.881728
0.01695
52.00
0.019
ki
0.000476
0.00011
4.170
0.000
Tax rate
-0.000185
0.00009
-2.110
0.045
o‡winvratio
0.000212
0.00007
3.190
0.000
GDP_PPP
-0.00000
0.00000
-2.160
0.000
0.020043
0.00567
3.540
0.017
Constant
2
R = 0:89,
2
= 1068 [0:000] ; T = 14; N = 31:
Foreign investment substitutes domestic investment a bit and there is some justi…cation on
popular sentiments against foreign capital in this empirical analysis. The contribution of FDI to
economic growth is direct and indirect. In‡ows or out‡ows make economy more sensitive to the
foreign capital, hence domestic …rms have to be more competent. This enhances economic growth.
Similarly the amount of investment and the growth rate are in‡uenced through indirect channels.
This is clear from the result in table 8 where in‡ows seem to in‡uence growth rates in the similar way
as the domestic investment having both positive and cyclical e¤ects. Country size and investment
285
interaction e¤ect (intract) is positive but tax GDP interaction term is negative as expected.
Table 81: Contribution of FDI in‡ows and out‡ows to growth rate of output in OECD Countries
Coe¢ cient Standard Error t-value t-prob
growth(-1)
0.214
0.110
1.940
0.053
in…nnvratio
0.006
0.002
3.760
0.000
in…nnvratio (-1)
-0.004
0.001
-4.520
0.000
in…nnvratio (-3)
-0.002
0.001
-2.070
0.039
intract
0.000
0.000
-3.170
0.002
intract (-1)
0.000
0.000
3.350
0.001
tax*GDP
0.001
0.000
2.790
0.006
tax*GDP (-1)
-0.001
0.000
-2.890
0.004
invratio
1.558
0.380
4.100
0.000
invratio(-1)
-1.576
0.357
-4.410
0.000
0.045
0.016
2.750
0.006
Constant
2
R = 0:49,
2
= 233:2 [0:000] ; T = 14; N = 31:
We also tried to disentangle the country and time speci…c e¤ects of FDI on investment and
growth rates across OECD countries. When controlled for time speci…c and country speci…c factors,
out‡ows had negative impacts on domestic investment ratio but the corresponding impacts of in‡ows
were not very signi…cant (Table 9). Country speci…c and time speci…c factors were more dominant
in determining the investment ratio or growth rates than in‡ows or out‡ows of FDI. Countries with
more liberal FDI policies such as Ireland, South Korea, Slovakia and Spain had positive impacts
of FDI on growth rate than in other OECD countries. FDI contributed positively on growth rates
from 1994 to 2001 but had either positive or negative e¤ects on growth in other years.
These time and country speci…c e¤ects are found to be consistent with the stylized facts relating
to the growth rates of output, investment ratios and in‡ows and out‡ows of FDI.
References
[1] Barro R.J. and D. B. Gordon (1983) Rules, Discretion and Reputation in a Model of Monetary
Policy, Journal of Monetary Economics, 12: 101-121, North-Holland.
[2] Breitner M. H. (2005) Genesis of Di¤erential Games in Light of Isaac’s Contributions, Jounral
of Optimisation Theory and Applications, 124:3:529-559
286
[3] Burda M. and C. Wyplosz (2010) Mundell-Flemming model, Macroeconomics, Oxford University Press.
[4] Canzoneri M. B. and J A Gray (1985) “Monetary Policy Games and the Consequences of NonCooperative Behaviour”, International Economic Review, 26:3:1985, pp. 547-567.
[5] Canzoneri M. B., R. E. Combey and B. T. Diba (2005) The need for International monetary
policy coordination,: what is old, what is new and what is yet to come, Journal of International
Economics, 66:363-384
[6] Chakravarty S. (1965) Optimal Programme of Capital Accumulation in a Multi-Sector Economy,
Econometrica, 33, 3 ,Jul. 557-570
[7] Cooley T. and Hansen (1989) The in‡ation tax in real business cycle models, American Economic
Review, 79,3:733-748
[8] Cooper R. N. (1969) Macroeconomic Policy Adjustment in Interdependent Economies, Quarterly
Journal of Economics, 83: 1:Feb.:1-24
[9] Corden W.M. (1985) Macroeconomic Policy Interaction under Flexible Exchange Rates: A
Two-Country Model, Economica, New Series, 52:205:Feb. 9-23
[10] Cruz J.B. (1975) Survey of Nash and Stackeberg equilibrium strategies in dynamic games,
Annals of Economic and Social Measurement, 4:2:339-344.
[11] Cruz J.B. and Chen (1971) Series Nash Solution of Two Person Non-Zero-Sum Linear
Quadratic Di¤erential Games, Jounral of Optimisation Theory and Applications, 7:4:240-257
[12] Currie D, P. Levine (1985) Simple Macropolicy Rules for the Open Economy The Economic
Journal, 95, 60-70.
[13] Ellsberg D. (1968) Theory and practicel of blackmail, Rand Corporation.
[14] Dri¢ l J. (1988) Macroeconomic Policy Games with Incomplete Information: A Survey, European Economic Review, 32 (2-3) 513-41.
[15] Frankel J. (1992) Obstacles to the macroeconomic policy coordination with an analysis of
international policy targets, UC, Berkley.
[16] Frankel J. A., K. E. Rockett (1988) International Macroeconomic Policy Coordination When
Policymakers Do Not Agree on the True Model, American Economic Review, 78, 3,Jun.,: 318-340.
287
[17] Freiling Jank and Abou-Kendill (1996) On global existence solutions to coupled matrix Riccati
equations in closed loop Nash games, memio?
[18] Hamada K (1976) Strategic Analysis of Monetary Interdependence, Journal of Political Economy, 84 August.
[19] Ivanov I. G. (2008) A method to solve discrete time coupled algebraic Ricaati equations,
Applied Mathematics and Computations, 206, 34-41
[20] Jank G and Kremer ( ) Open loop Nash games and positive systems: solvability conditions for
non-symmetric Riccati equations, memio?
[21] Jorgenson D. W. (1961) Stability of a Dynamic Input-Output System Stability of a Dynamic
Input-Output System, Review of Economic Studies, 28,2, Feb., 105-116
[22] Kehoe P.J. (1989) Policy Cooperation Among Benevolent Governments May Be Undesirable
,The Review of Economic Studies, 56, 2, Apr., 289-296
[23] Krugman Paul (1979) A Model of Balance of Payment Crisis, Journal of Money Credit and
Banking, 11,Aug.
[24] Kumar T. K. (1969) The Existence of an Optimal Economic Policy Econometrica, 37:4 :Oct.:
600-610.
[25] Kydland F. (1975) Equilibrium Solutions in Dynamic Dominant Player Models, Journal of
Economic Theory, 15:307-324
[26] Kydland F. (1975) Non-cooperative and dominant player solutions in dynamic discrete games,
International Economic Review, 16:2:321-335
[27] Kydland F.E and E.C. Prescott (1977) Rules rather than discretion: the Inconsistency of
Optimal Plans, Journal of Political Economy, 85:3: 473-491.
[28] Li M., J.B. Cruz, M. A. Simaan (2002) An approach to discrete-time incentive feedback stackelberg games, IEEE Transactions on System, Man and Cybernatics- Part A: Systems and Humans,
32:4:472-481.
[29] Miller M, Mark Salmon (1985) Dynamic Games and the Time Inconsistency of Optimal Policy
in Open Economies, Economic Journal, 95, Conference Papers:124-137.
[30] Miller, Marcus; Salmon, Mark When Does Coordination Pay? Journal of Economic Dynamics
and Control, July-Oct. 1990, v. 14, iss. 3-4, pp. 553-69
288
[31] Mundell R. A (1962) Capital mobility and stabilisation policy under …xed and ‡exible exchange
rates,Canadian Journal of Economic and Political Science, 29, 475-85.
[32] Mukaidani H. (2004) Proceedings of the American Control Conference, Boston
[33] Nordhaus W.D. (1995) Policy Games: Co-ordination and Independence in Monetary and Fiscal
Policeis, Brookings Papers on Economic Activity 2:1994: 139-216.
[34] Obstfeld M. (2001) International macroeconomics: beyond the Mundell-Flemming model,
Cebter for International and Development Economics Research, UC Berkeley.
[35] Oudiz G. and J Sachs (1984) Macroeconomic policy coordination among industrial economies,
Brookings papers in economic activities, 1:1-64
[36] Papavassiloupoulos G. P. and Olsder (1984) On the linear-quadratic, closed loop, no-memory,
Nash game, Journal of Optimisation Theory and Applications, 42:4:551-560
[37] Parikh A. (1979) Forecasts of Input-Output Matrices Using the R.A.S. Method, Review of
Economics and Statistics, 61, 3 Aug., 477-481.
[38] Petit M. S. (1989) Fiscal and Monetary Policy Co-Ordination: A Di¤erential Game Approach,Journal of Applied Econometrics, 4:2:Apr. - Jun.:161-179
[39] Rogo¤ K. (2002) Dornbusch’s Overshooting Model After 25 Years, IMF Sta¤ Papers, 49:1-35.
[40] Starr A. W. and Y. C. Ho (1969) Non-zero sum di¤erential games, Jounral of Optimisation
Theory and Applications, 3:3:184-206
[41] Taylor J (1993) Discretion versus policy rules in practice, Carnegie Rochester Conference,Series
on Public Policy 29 Amsterdam.
[42] Turnovsky S.J., T. Basar and V. d’Orey (1988) Dynamic Strategic Monetary Policies and
Coordination in Interdependent Economies, American Economic Review, 78, 3,Jun., 341-361
[43] Van Long N. and G. Sorger (2009) A dynamic principle agent as a stacklebege di¤erential
game, working paper 0905, Department of Economics, University of Vienna.
[44] Van Aarle B, J. Engwerda and J. Plasmann (2002 ) Monetary and Fiscal policy interaction in
the EU: A Dynamic Game Approach, Annals of Operation Research, 109, 229-264
[45] Van Aarle B, J. Engwerda and J. Plasmann (2001) Monetary and Fiscal policy interaction in
the EU: A Dynamic Game Approach, Open Economies Review, 12:1, 29-60
289
[46] van de Klundert T. and F. van der Ploeg (1989)Fiscal Policy and Finite Lives in Interdependent
Economies with Real and Nominal Wage Rigidity, Oxford Economic Papers, New Series, Vol. 41,
No. 3 (Jul., 1989), pp. 459-489
290
8
L8: Business Cycles
Measurement of Business Cycle
Decomposing trends and cycles
Growth rate with a straight line trend imposes a constant slope (for any log trend ln yt )
ln yt
ln yt
1
=g'
yt
yt
yt
1
(H.761)
1
Hodrick-Prescott …lter generates trend instead by minimises following function
T
X
ln yt
ln yt
2
1
t=1
+
T
X
ln yt
ln yt
2
1
(H.762)
t=1
= 1600 standard for quarterly series. Get trends and HP tred for a GDP variable and see the
di¤erence.
Statistical tools for business cycle measurement:
mean, variance and covariance
Business cycle models aim to capture mean, variance and covariance of the actual GDP series
by model generated series.
1. Pro-cyclical variables move along with the GDP. GDP and GDP components move together.
Increase in consumption, investment, exports and government expenditure raises GDP.
2. Anti-cyclical variables move in the opposite direction of the GDP. In‡ation and Unemployment
rate, and unemployment rate and the GDP are anti-cyclical variables.
3. Acyclical variables do not have any relation with GDP.
Y is GDP and X any onother variable.
x;y
P
X X Y Y
p
= p
V ar(X) V ar(Y )
Procyclical, Anticyclical and Acyclical Variables
1. Variable is pro-cyclical if
2. counter-cyclical if
3. acyclical if
x;y
x;y
x;y
>0
< 0 and
=0
4. agging procyclical if
x
1 ;y
>0
291
(H.763)
Table 82: Percentage standard deviation of macro variables
GDP
Consumption
Investment
Hours worked
% standard deviation
Relative % standard deviation
Table 83: Lag, contemporaneous and lead correlations among macro variables
GDPt ; xt
1
GDPt ; xt
GDPt ; xt+1
GDP
Consumption
Investment
Hours worked
5. leading procyclical if
x;y
1
>0
Correlations to …nd out cyclical, procyclical and anticyclical macro variables
Examine these features in macro series in PcGive using data …les: macro_OECD.csv; GDP_Components_UKUS.
Standard deviation of growth rate of output, consumption, investment and hours worked.
S e e e x c e l …le s fo r U K 1 9 7 1 :q 1 to 2 0 1 0 :q 2
8.1
Optimising model of business cycle
Aggregate demand is derived from the optimisation behavior of the household as (see next
slide for derivation)
Ct+1
=
Ct
(1 + rbt+1 )
(H.764)
where rbt+1 is the interest rate net of tax and depreciation is rbt+1 = (1
k ) (rt+1
)
shocks to technology in‡uences supply (zt )
Table 84: Percentage standard deviation of macro variables
GDP
Consumption
Investment
Hours worked
% standard deviation
0.0097
0.0109
0.0279
0.0060
Relative % standard deviation
1
1.1251
2.8786
0.6142
292
Table 85: Lag, contemporaneous and lead correlations among macro variables
GDPt ; xt
1
GDPt ; xt
GDPt ; xt+1
GDP
0.9995
1
0.9995
Consumption
0.9988
0.9988
0.9986
Investment
0.9955
0.9860
0.9856
Hours worked
0.6628
0.6728
0.6832
Yt = zt Kt L1t
(H.765)
interest rate is given by
1
rt+1 = zt Kt
L1t
(H.766)
A positive technological shock (zt ) raises output and interest income rt+1 that raises consumption; business cycle is caused by erratic nature of these shocks.
Optimising model of business cycle: an example
L a g ra n g ia n
ln (Ct ) +
2
+ t+1 4
F irst o rd e r c o n d itio n s fo r th is:
ln Ct+1
+
h
bt ) + w
bt
t At (1 + r
Ct
At+1 1 + r
bt+1 + dt+1 + pt+1 Ht+1
+w
b t+1
Ct+1
At+2
pt+1 Ht+2
At+1
3
pt Ht+1
i
5
(H .7 6 7 )
1
=
Ct :
Ct+1;i :
At+1 :
Ht+1 :
=
Ct+1
t =
t pt =
t+1
t+1
(H .7 6 8 )
t
Ct
t+1
(H .7 6 9 )
1+r
bt+1
(H .7 7 0 )
dt+1 + pt+1
Substitution …rst two …rst order conditions in the third
1
Ct pt
=
Ct+1
(dt+1 + pt+1 )
1=
Ct
(1 + rbt+1 )
Ct+1
293
(H .7 7 1 )
and fourth
1
Ct
=
Ct+1
(1 + rbt+1 );
(H.772)
8.2
Aggregate Demand-Aggregate Supply (AS-AD) Model of Business
Cycle
Aggregate demand is in‡uenced by
– lagged demand
– growth rate of real money balances and
– shocks to the demand
Aggregate supply is in‡uenced by
– core in‡ation
– deviation of output from its trend
– and supply shock
Model solutions very similar to the Multiplier-Accelerator model
8.3
AS-AD Model of Business Cycle
Aggregate demand as a function of lagged output, growth rate of money supply and in‡ation:
Yt = a1 Yt
1
+ a2 (mt
t)
+ dt ;
a1 > 0; a2 > 0
(H.773)
Aggregate supply as a function of core in‡ation and output gap and supply shock
t
=
t
+ b1 Yt
Y t + st
(H.774)
backward and forward looking aspects of core in‡ation (with 0 <
t
=
t
+ (1
)
< 1):
(H.775)
t 1
These three equations are enough to generate business cycles in output and in‡ation.
Substitute H.775 into H.774 to get
t
t
=
t
+ (1
t 1
=
)
t 1
b1
(1
)
Yt
Lag H.773
294
+ b1 Yt
Yt +
Y t + st
1
(1
)
st
(H.776)
(H.777)
Yt
1
= a1 Yt
2
+ a2 (mt
t 1)
1
+ dt
(H.778)
1
and deduct it from H.773
Yt
Yt
1
=
a1 Yt
a1 Yt
1
a2 (
2
t 1)
t
+ a2 (mt
+ dt
mt
dt
1)
(H.779)
1
substitute H.777
Yt
Yt
=
1
a1 Yt
a1 Yt 2 + a2 (mt mt 1 )
1
b1
Yt Y t +
st
(1
)
(1
)
dt 1
1
a2
+dt
Yt 1 +
a2 b1
(1
)
=
(1 + a1 ) Yt
+
Yt
=
a1 Yt
1
2
a2
a 2 b1
Yt
(1
)
(1
)
+ a2 (mt
st + dt
(H.780)
mt
dt
1
(1 + a1 ) (1
)
a1 (1
)
Yt 1
Yt 2
1
+ a2 b1
1
+ a2 b1
a2 (1
)
a2 b1
Yt
+
(mt mt 1 ) +
1
+ a 2 b1
1
+ a2 b1
a2
(1
)
st +
(dt dt 1 )
1
+ a 2 b1
(1
) + a2 b1
1)
(H.781)
(H.782)
Rede…ning the reduced form coe¢ cients:
Yt =
1 Yt 1
2 Yt 2
+
3 4mt
+
4Y t
5 st
+
6 4dt
(H.783)
This is a second order di¤erence equation on output and can be solved by quadratic roots.
Then solve this system for in‡ation.
Deduct Y t from H.773 to generate Yt
Yt
Y t = a1 Yt
Y t term and then substitute it into supply equation.
1
Y t + a2 (mt
295
t)
+ dt
(H.784)
t
t 1
b1
=
(1
t
a1 Yt
)
b1 a 1
Yt
(1
)
b1 a2
(1
)
=
t 1
Y t + a2 (mt
1
t)
+ dt +
1
(1
)
st
(H.785)
b1
b1 a2
Yt+
mt
(1
)
(1
)
b1
1
dt +
st
t+
(1
)
(1
)
1
(H.786)
AS-AD Model of Business Cycle
Now need to eliminate Yt
1;
t 1
t 2
Yt
t
t 1
t
t 1
for this lag H.777 and extract Yt
1
=
=
=
b1
(1
(1
)
b1
b1 a1
(1
)
b1
(1
1
+
(1
= a1 (
t 1
b1
(1
+
t
=
(
Yt
)
)
Yt+
Yt
t 2)
b1
)
1
t 1
(1
(
+
1
+Yt
t 1
b1 a2
mt
(1
)
1
1
(1
)
1
st
b1
st
(H.787)
1
(H.788)
1
1
st 1
b1
b1 a2
b1
dt
t+
(1
)
(1
)
t 2)
+Yt
1
st
(H.789)
b1 a1
b1 a1
Yt 1
st 1
(1
)
(1
)
b1 a2
b1
b1 a2
mt
dt
Yt+
t+
)
(1
)
(1
)
(1
)
t 2)
1
(1
)
1
)
+
st
(H.790)
(1 + a1 ) (1
)
a1 (1
)
a1 (1
)
Yt
t 1
t 2+
1
+ a2 b1
1
+ a2 b1
1
+ a2 b1
b1 a1
b1
b1 a2
st 1
Yt+
mt
1
+ a2 b1
1
+ a2 b1
1
+ a2 b1
b1
1
+
dt +
st
1
+ a2 b1
1
+ a2 b1
1
(H.791)
This again is a second order di¤erence equation, which gives cycles. Backward looking element
vanishes when
! 1 and the forward looking element is weaker when
296
! 0:
Measurement of business cycles
Business cycles are measured as deviations of actual series from their trends. There are various
methods of taking trends as:
Linear trend
Yt = a + bt
(H.792)
ln (Yt ) = a + bt
(H.793)
log linear trend
Polynomial trends (e.g. with 4 order pol).
Yt = a + bt + bt2 + bt3 + bt4
(H.794)
Hodrick-Prescott Filter ( = 1600 is normal)
F
=
min
T
X
2
ln (y t )] +
[ln (yt )
t=1
T
X1
t=1
8.3.1
fln (yt )
ln (y t )g
ln (yt
1)
ln y t
2
1
(H.795)
Role of Shocks in AD-AS Model
Consider an AS-AD model with following equations:
Real interest rate (Fisher equation)
rt = ipt +
t
e
t+1
(H.796)
Aggregate demand:
yt
y=
1
(gt
g)
2
(rt
r) + vt ;
vt v N 0;
2
v
;r = r +
(Ad_r)
Interest rate rule
ipt = r +
e
t+1
+ h(
t
) + b (yt
y)
(MP_r)
Aggregate supply (price formation):
t
=
e
t+1
+ (yt
y) + st ; st v N 0;
297
2
s
(H.797)
In‡ation expectation
e
t
=
(H.798)
t 1
Derivation of aggregate demand
Put Fisher equation and interest rate rule in the aggregate demand function
rt
h(
t
+
e
t+1
= r +
+ h(
t
) + b (yt
r=
t
+ h(
y) or
rt
t
+
e
t+1
= r
+
e
t+1
+
y)
) + b (yt
t
e
t+1
rt
t
) + b (yt
+ h(
t
y)
(MP_r2)
put this in AD equation
yt
yt
y=
yt
y=
y=
1
1+
2b
1
1+
2b
1
(gt
(gt
g)
(gt
g)
g)
2
[
2
1+
2b
2
1+
yt
2b
y=
t
(
t
)
(
t
)
2h
1+
2b
(
) + b (yt
2h
1+
2b
2h
1+
2b
t)
y)] + vt
(H.799)
(
t
)+
vt
1 + 2b
(H.800)
(
t
)+
vt
1 + 2b
(H.801)
+ zt
(H.802)
Where zt term includes …scal policy shock(gt ), risks ( t ) and random shocks (vt )
zt =
1
1+
2b
(gt
g)
2
1+
2b
(
)+
t
vt
1+ 2 b
Putting the in‡ation expectation into the supply function
t
=
t 1
+ (yt
y) + st
(H.803)
Property of AS: Upward slopping; larger output requires employers more people, that lowers
the productivity of labour and the cost rises and in‡ation has to rise. Term st includes trade,
exchange rate, technology or other shocks.
2h
2h
(
=
; yt y = (
(H.804)
t ) + zt ;
t ) + zt
1 + 2b
1 + 2b
Property of AD: Downward slopping; higher rate of in‡ation requires central bank to increase
yt
y=
the interest rate, that raised the cost of capital thus causes lower investment and hence lower output.
zt term includes …scal policy shock(gt ), risks ( t ) and random shocks (vt ) :
298
De…ne deviation from the steady state as bt =
and ybt = yt
t
AS-AD Model of Business Cycle
when there are no further shocks zt = 0 and st = 0
bt+1 =
AD :
AS :
Solve these equations
1
Output :
1
ybt+1 =
ybt+1
bt+1 ) =) bt+1 =
1
1+
1
1+
Solutions of the …rst order di¤erence equations
Since 1 <
t
ybt = yb0
t
(H.805)
bt+1 = bt + ybt+1
ybt + ybt+1 =) ybt+1 =
bt+1 = bt + (
Inf lation :
1
y
and bt = b0
t
(H.806)
ybt =) ybt+1 = ybt
(H.807)
bt =) bt+1 = bt
(H.808)
for t = 0; 1; 2; :::::
(H.809)
< 1 both ybt and bt converge to the steady state values y and
.
Time taken to converge to the steady state
First calibrate parameters (borrowed from SW text):
=
=
1
1+
=)
2h
1+
= 0:742;
2b
=
= 0:3;
( 2 = 5:76) (h = 0:5)
= 0:742
1 + ( 2 = 5:76) (b = 0:5)
=4 = 0:3=4 = 0:075 =)
=
1
1 + 0:742
(H.810)
0:075
= 0:947
(H.811)
If the objective is to close half of the initial gap
ybt = yb0
t
=
1
yb0 =)
2
t
=
1
=) t ln
2
= ln
1
2
=) t =
0:693
0:693
=t =
' 13 (H.812)
ln
ln (0:947)
It takes approximately 13 quarters to close half of the gap.
Impulse Response in AS-AD Model
when shocks zt 6= 0 and st 6= 0
yt
y=
(
t)
+ zt =) ybt =
299
bt + zt
(H.813)
bt = bt
AS :
1
AS :
1
1+
ybt =
Impulse Response in AS-AD Model
1
bt =
ybt
ybt
1
1
1
(zt
+
+ ybt + st = bt
1
1+
bt =
bt
1
+
bt
+
1
1
1)
+ ybt
(zt
zt
1+
1
ybt
1
zt
(zt
(H.814)
+ ybt + st
1
1+
+
ybt )
(zt
ybt + ybt = zt
ybt =
bt = bt
1
ybt ) =
(zt
AS :
AS :
1
bt =
AD :
(H.815)
+ ybt + st
st
1
zt
1)
(H.816)
1)
(H.817)
st
1+
(H.818)
st
(H.819)
bt + zt ) + st
+ (
zt +
1
1+
(H.820)
st
(H.821)
zt + st
(H.822)
Table 86: Parameters of AS-AD model
Values
1
2
0.4
5.76
b
0.5
h
0.3
0.5
0.2
0.742
0.947
zt
st
1
1
This can be computed in Excel.
8.4
Monetary Policy
Interest Rate Determination Rule: Taylor Rule
Output gap and interest rate
(yt
yt ) =
d (it
1
300
i );
d>0
(H.823)
In‡ation and output (Supply or Phillips curve)
(
t)
t
= c yt
yt
1
1
;
c>0
(H.824)
a > 0; b > 0
(H.825)
Interest rate determination rule
it = i + a (yt
yt ) + b (
t
t);
yt ) + b (
t
Solution of the Interest Rate Rule Model
it
= i + a (yt
t)
= i
ad (it
1
i ) + cb yt
= i
ad (it
1
i )
1
cbd (it
yt
1
i )
(H.826)
= i + ad:i + cbd:i
(H.827)
it+2 + ad:it+1 + cbd:it = i + adi + cbdi
(H.828)
2
Collecting terms
it + ad:it
1
+ cbd:it
2
Iterating forward by two periods
Long run natural rate of interest: steady state
it = it
1
= it
2
= bi
(H.829)
(1 + ad + cbd)bi = i (1 + ad + cbd)
(H.830)
bi = i
(H.831)
it+2 + ad:it+1 + cbd:it = 0
(H.832)
Fluctuations around this long run interest rate depends on homogenous part of the second order
di¤erence equation
Transitional dynamics (replace it = A
A
t+2
t
in homogenous equation).
+ ad:A
t+1
301
+ cbd:A
t
=0
(H.833)
2
+ ad: + cbd = 0
(H.834)
Three Cases in Samuelsonian Multiplier Accelerator Model
Cycle depends on roots of the quadratic equation
1;
2
q
ad
=
2
(ad)
4cbd
(H.835)
2
Distinct real root case (no cycle)
2
(H.836)
2
(H.837)
2
(H.838)
(ad) > 4cbd
Repeated real root case (no cycle)
(ad) = 4cbd
Complex root case (cycle)
(ad) < 4cbd
Complete solution
t
1
it = A1
it = A1 Rt (cos
t + i sin
t) + A2 Rt (cos
+ A2
t
t
2
+ bi
i sin
Parameters and solutions of the model
(H.839)
t) + bi
Table 87: Parameters of the Interest Rate Rule Model
a
b
c
d
i0
i
y0
0
values
1.5
0.25
0.4
-0.25
0.010
0.0575
0.02
0.02
yt
-0.05
Solution of the Interest Rate Rule Model
Example of Complex Root Case: Example
Preliminaries: Exponential forms and polar coordinates
sin =
p
h2 + v 2 = bcd
(H.840)
v
=) v = R:sin
R
(H.841)
R=
302
cos =
h
=) h = R:co
R
ei = cos + i Si n
h
@ sin
@
= cos ;
@ cos
@
vi = R:co
=
e
i
(H.842)
= cos
R:i sin = R: (co
i Si n
(H.843)
i sin ) = Re
i
(H.844)
sin ;
2
Example of Complex Root Case: Example (ad) < 4cbd
Need to consider the algebra for the imaginary number and some trigonometric functions in
this case. Using Pythagorean in an imaginary axis is used to derive the roots of the characteristic
equation.
1;
2
Yt = A1
= (h
v i) =
t
1
t
2
+ A2
ad
2
s
i
2
4cbd (ad)
2
t
(H.845)
t
(H.846)
t) for Rht > 0:
(H.847)
= A1 (h + v i) + A2 (h
v i)
Using DeMoivre’s theorem
(h
v i) = Rht (cos
t
i sin
Imaginary axis (Pithagorus Theorem)
R=
it = A1 Rht (cos
it = A1 Rht cos
2
p
t + i sin
t + i sin
2
h2 + v 2 = bcd
(H.848)
t) + A2 Rht (cos
t + A2 Rht cos
t
2
i sin
t
i sin
t)
2
(H.849)
t
(H.850)
Three possibilities:
i) Rht > 1; bcd > 1 ii) Rht = 1
bcd = 1 and ii) Rht < 1 bcd < 1 Only the bcd < 1 case is
convergent other two cases are divergent.
Principles of Finance
Maximisation of return and minimisation of risk given the arbitrage opportunities in the
economy.
303
Essence: discounting and net present value, capital asset pricing (CAPM) model, e¢ cient
market hypothesis -arbitrage, life cycle decisions, options.
Intertemporal balance, mobilisation of saving and investment; borrowers and lenders.
Risk pooling and sharing by the economy as a whole.
Bad …nancial system very harmful for the economy: bubbles and crises.
E¢ ciency of the …nancial system is important for real economic growth.
Mechanism required to correct moral hazard and adverse selection: e¢ cient regulation.
Miles, D. (2014), Monetary Policy and Forward Guidance in the UK. The Manchester School,
82: 44–59
James H. Stock and Mark W. Watson (2005) Understanding changes in international business
cycle dynamics,Journal of European Economic Association, 3:5:968-1006.
8.4.1
Integration of Finance in a Macro Model
Technology
Yt = At Kt
(H.851)
Capital Accumulation
It = Kt+1
Yt = Ct + St
with
I
Y
=
S
Y
(1
) Kt
It = St
(H.852)
0 < <1
(H.853)
and balance growth condition Kt+1 = (1 + g)Kt
I
=A s
(H.854)
Y
E¢ ciency of the …nancial system ( ) along with technical knowledge determines the growth rate
g=A
of output
Integration of …nance and real economy
M ax U (Ct ; Ct+1 ) = ln Ct +
Ct ;Ct+1
Subject to
304
ln Ct+1
(H.855)
At (1 + rbt ) + Wt
Ct = At+1
(H.856)
Ct is consumption At assets Wt endowment, and rbt+1 return to asset net of tax and depreciation
rate.
is the subjective discount factor.
Integration of …nance and real economy
Iterate (H.856) one period forward
1
(At+2
1 + rbt
Put this in the original budget constraint
At+1 =
At (1 + rbt ) + Wt
Wt+1 + Ct+1 )
1
(At+2
1 + rbt
Ct
(H.857)
Wt+1 + Ct+1 ) = 0
(H.858)
The Lagrangian for the constrained optimisation
L
=
ln Ct +
+
ln Ct+1
At (1 + rbt ) + Wt
1
(At+2 + Wt+1
1 + rbt
Ct
Integration of …nance and real economy
From the assets to the real economy
rbt = (1
assuming
k ) (r
k
) with r real interest rate,
rate of depreciation and
Ct+1 )
k
(H.859)
capital income tax
=0
At+1 =
At rt + Wt
Replace At by Kt
1
(At+2
1 + rbt
Ct
Wt+1 + Ct+1 )
fAt+1
(1
) At g = 0
(H.860)
(H.861)
and Yt by At rt + Wt = Ct + It
Yt
Ct
(Kt+1
(1
) Kt ) = 0
Yt = Ct + It
(H.862)
(H.863)
This shows integration of asset markets with the real economy for an economy with the representative agent. Asset market model is consistent with the GDP and national accounting.
305
8.5
Policy Rule versus Optimal Discretion
Objective of a policy maker is to minimise loss function Subject to the aggregate supply constraint.
M in S ( ) = b (y
y )
a
2
;
a>0 b>0
(H.864)
Subject to
y = y + c(
E ( )) ;
c>0
(H.865)
where y is actual output y is the natural level of output and (y
y ) is the output gap and
is the actual in‡ation rate.
Aggregate Supply: Output responds to higher level of in‡ation
Using the value of from the constraint in the objective function
M in S ( ) = bc (
E ( ))
a
2
(H.866)
E ( ))
a
2
(H.867)
Optimal In‡ation Under the Policy Rule
M in S ( ) = bc (
Let in‡ationary expectation of people,E ( ) to be a constant.
Policy maker have two choices: stick to a policy rule or use optimal discretion
If they stick to policy rule; people know this, actual in‡ation equals expected in‡ation.
= E ( );;y = y
S( )=
a
2
@S
=
@
2a
(H.868)
Optimal In‡ation in the policy rule:
=0
(H.869)
Optimal In‡ation Under the Discretion
M in S ( ) = bc (
E ( ))
a
2
(H.870)
Policy makers choose the in‡ation rate to minimise the loss function .
First order condition of wrt
@S
= bc
@
2a = 0;
Conclusion:
306
=
bc
>0
2a
(H.871)
In‡ation rate under discretion is higher than the in‡ation under the policy rule; it depends on
a, b and c, the parameters of the loss function (a, b) and the slope of the supply (c). This is the
main reason for the argument for central bank independence and the policy rule. These conclusions
are for normal times.
However,many economists agree that recession like 2008-2009 requires …scal stimulus and quantitative easing.
Minford P and Zhirong Ou (2013) Taylor Rule or optimal timeless policy? Reconsidering the
Fed’s behavior since 1982,Economic Modelling 32 (2013) 113–123
307
9
L9: Class Test: Past Examples
Questions are given in sections A and B. Answer two questions, at least one from each section. Each
question is worth 100 marks. Each subquestion within a question is of equal value. Use diagrams
to illustrate your answers.
Section A
Q1 Consider the basic IS-LM model as given in the following equations
Y = C +I +G
(I.872)
C = C (Y
(I.873)
Consumption function
T)
Investment
I = I(r)
(I.874)
M s = M (Y; r)
(I.875)
Money Market
where Y is output, C consumption, I investment and G public spending, T tax revenue, M
money and r the interest rate.
1. Derive separate equations to demonstrate the equilibrium in goods and money market.
2. Take the total di¤erentiation of the system of those two equations and …nd out how changes
in the output and interest rate could be determined in terms of the structural features of the
economy.
3. Find the expression of total change in output
4. Solve the equation to …nd the total change in the interest rate.
5. What are the multipliers with respect to the government spending and taxes?
6. What is the multiplier with respect to changes in the money supply?
7. How can this model be applied to analyse impacts of …scal and monetary policies in an
economy?
308
Q2. Consider a Markov model of employment and layo¤
et+1 = (1
) et + ut
ut+1 = et + (1
(I.876)
) ut
(I.877)
Where et and ut are the levels employment and the unemployment.
1) What is the level of employment and unemployment in the steady state.
2) Find the transitional path towards the steady state.
Q3. Consider a version of the Brock-Mirman type dynamic programming problem
max
U=
1
X
t
ln(Ct )
0<
<1
(I.878)
t=0
subject to market clearing condition
Kt+1 + Ct = AKt
0<
<1
(I.879)
Here output (AKt ) is either consumed (Ct ) or invested (Kt+1 ) :
1. What are the control and state variables in this model and why?
2. Explain the meaning of the value function (Bellman equation) and the policy
functions of this problem, V1 (K 0 ) = ln C + V0 (K 0 ); where K 0 is the amount of
optimal capital stock.
3. Assume K 0 = 0 for the last period. Demonstrate a recursive solution method of
this problem using three iterations of the policy and value functions. Characterise
the rest of the solution.
4. Use the limit theorem to …nd the explicit solution of the value function.
5. Introduce a stochastic technology At+1 = At +"t and examine conjectures to solve
this problem.
Section B
Q4. Consider a monopolistic competition model with i:::n …rms each with technology
Yi = AL1i
309
;
0<
<1
(I.880)
1. Find the marginal revenue product for a …rm i and show how its level of output (Yi ) depends
on its own price, aggregate output and general price level (P ).
2. De…ne the total revenue of the …rm and show how the price (Pi ) this …rm relates to the
elasticity of output to the price level.
3. Show explicitly how the price charged by the …rm and its output are related to the mark up,
wage rate and marginal product of labour.
4. Determine the demand for labour by this …rm.
5. Derive the expression for the elasticity of labour to the real wage rate.
Q5. Consider the New Keynesian model in which the problem of household i is to maximise
expected utility from consumption (Cit+k ), accumulation of money (Mit+k+1 ) and labour
supply (Nit+k ) taking account of all information (
t)
available up to period t and is given
as:
max
E
"
subject to:
1
X
k
U (Cit+k ) + V
t=0
Mit+k+1
P t+k
Q (Nit+k ) j
t
#
(I.881)
a) CES aggregation of consumption (Cit ) and price level P t over j commodities:
Cit =
Z
1
0
b) the budget constraint
Z
1
Cijt dj
1
;
Pt =
Z
1
0
1
Pjt
1
dj
1
(I.882)
1
Pjt Cijt + Mit+1 + Bit+1 = Wt Nit + (1 + it ) Bit + Mit +
it
+ Xit
(I.883)
0
where Bit ,
it
and Xit denote bonds held, pro…ts earned and transfer received by the household
i ; Wt is wage earned for supplying labour (Nit ) :
c) demand for a product Cijt relates to composite demand as:
Pjt
Cit
Pt
Firms take wage rates as given and set prices a la Calvo with
(I.884)
Cijt =
probability of changing it every
period. Then Yj;t is the solution to the …rms’pro…t maximization problem:
max E
"
X
k
kU
0
(Ct+1 )
(1
U 0 (Ct )
k
)
310
Pjt
Yjt+k
P t+k
Wt+k Yjt+k
P t+k Zt+k
j
t
#
(I.885)
subject to:
a) a linear production technology
Yjt = Zt Njt
(I.886)
b) supply
Yjt+k =
Pjt
P t+k
Yt+k
(I.887)
1. Write …rst order conditions for optimisation by households and …rms in this
model.
2. Solve for the price level, employment and output at the steady state.
3. Prove that volatility of output is generated from the technological shock. Comment how it compares to a standard RBC model.
Q6. Expected in‡ation next period (Et
t+1 )
based on information at period t depends on di¤er-
ences on expected and actual prices as:
Et
t+1
=Et pt
pt
Demand ytd is function of real money balances (mt
ytd = a0 + a1 (mt
pt ) + t ;
(I.888)
pt ) as:
a0 > 0 a1 > 0;
t
2
N 0;
(I.889)
Actual output (yts ) deviates from the natural rate of output when actual prices are not equal to
expected prices pt 6=Et
1
pt as:
yts = yn + b1 pt
Et
1
p t + vt ;
a1 > 0 ;
t
N 0;
2
(I.890)
Demand equals supply in equilibrium as:
ytd = yts = yt
(I.891)
Consider a money supply rule given by:
mt
mt
1
=
(I.892)
1. Use rational expectation method to solve for equilibrium output and prices in
this model.
311
2. Show that under the rational expectation average in‡ation equals the growth rate
of money supply but only the unanticipated shocks to demand or supply in‡uence
the level of output.
3. Critically assess the policy irrelevant propositions (PIP) under the rational expectation hypothesis.
Q7. Consider a three period economy which is inhibited by the low, middle and high income
households. Inter-temporal optimisation by each involves maximising utility subject to its life
time budget constraint.
M ax
U (C1i ; C2i ; C3i ) = ln C1i +
i
2
ln C2i +
i
3
ln C3i
i = A; B; C
(I.893)
subject to budget constraints while young, adult and old as follows:
C1i + bi1 = w1i
(I.894)
C2i + bi2 = bi1 (1 + r) + w2i
(I.895)
C3i = bi2 (1 + r) + w3i
(I.896)
where C1i ; C2i ; C3i are consumptions for periods 1, 2 and 3 for type i agent and
i
2
and
i
3 are
subjective discount factors for period 2 and 3 consumptions with their values between 0 and 1.
Endowment of agent i for time t is given by wti with endowments for agent A, B and C for periods
1, 2 and 3 are w1A ; w1A ; w1A ; w1B ; w1B ; w1B ; w1C ; w1C ; w1C . Again each household is allowed to borrow
and lend at the interest rate r. Markets clear for each good for each period:
C1A + C1B + C1C = w1A + w1B + w1C
(I.897)
C2A + C2B + C2C = w2A + w2B + w2C
(I.898)
C3A + C3B + C3C = w3A + w3B + w3C
(I.899)
What is the interest rate and equilibrium allocations in this economy? State how to extend this
model to ten households.
312
10
L10: In‡ation and Unemployment
In‡ation is increase in the general level of prices. Classical quantity theory of money takes it as a
monetary phenomenon
MV = PT
(J.900)
M
V
P
T
+
=
+
M
V
P
T
(J.901)
P
M
V
T
=
+
(J.902)
P
M
V
T
As the velocity is constant, in‡ation is caused by growth of money supply in excess of the growth
rate of the economy.
= gm
gy
(J.903)
Monetarist’s policy rule for stable prices gm = gy Growth in money is linked to budget de…cit
(also trade)
B
Y
=
G
T
Y
+ (i
g)
B
Y
M
PY
(J.904)
In‡ation and seigniorage
Money demand is determined by the private sector and money supply more by the public sector.
Government can transfer resources from the private sector through in‡ation tax.
M
= L (i; Y )
P
M
= L (r +
P
S=
e
(J.905)
;Y )
M
MM
=
= gm L (r +
P
M P
@S
= L (r +
@gm
e
(J.906)
e
; Y ) + gm L1 (r +
First term is positive and second term is negative;
gm .
313
@S
@gm
;Y )
(J.907)
e
(J.908)
;Y )
> 0 for small gm but
@S
@gm
< 0 for a large
(see Romer (2006))
Cagan hypothesis on in‡ation
Speci…c money demand function
M
=a
P
ln
M
= ea e
P
S
=
C
=
M
= gm ea e
P
ea e br Y
b(r+ )
gm
@S
= Ce
@gm
bi + ln Y
(J.909)
b(r+ )
(J.910)
Y
Y = gm ea e
b(r+gm )
Y = Cgm e
bgm
;
(J.911)
bgm
bCgm e
bgm
= (1
bgm ) Ce
This is at maximum when gm = 1b . S is positive until gm <
1
b
bgm
(J.912)
and negative there after.
Exercise: Calculate optimal in‡ation that maximises seniorage, assuming
= gm and using
the following function.
S =a+b
10.1
c
2
(J.913)
Natural rate of unemployment and output
There are n number of …rms each producing Yi and employing Li workers with Y = nYi and
L = nLi
Total supply of the economy with N number of economically active population and unemployment rate u is
Y = nA
where L = (1
L
n
u) N and ln(1
1
= n AL1
u) =
= n A ((1
u) N )
1
(J.914)
u
ln Y = ln n + ln A + (1
y = ln n + ln A
) ln (1
(1
314
u) + (1
) u + (1
) ln N
) ln N
(J.915)
(J.916)
u = ln N +
ln n + ln A
(1
)
y
(J.917)
Natural rate of unemployment and output
Natural output is given as
1
Y = n AL
L = (1
u) N ;
ln(1
u) =
u; y = ln Y
u = ln N +
u
u=
1
(1
)
(y
y) +
1
(1
)
(J.918)
ln
A
A
=) u
ln n + ln A
(1
)
u=
1
(1
)
y
(y
(J.919)
y) +
1
(1
)
ln
A
A
Use this result into the Phillips curve
st =
1
(1
)
ln
t
=
e
t
b (ut
u) + st
(J.920)
t
=
e
t
+ a (yt
y) + st
(J.921)
A
A
More output requires hiring extra workers, this raises wage rate and this causes in‡ation. Natural
rate of output is independent of in‡ation.
10.2
Wage Price Spiral
Price setting
Pt = (1 + ) Wt
(J.922)
Wt = (1 + ) Pte
(J.923)
Firm’s mark up ( > 1)
Wage setting
Unions’mark up ( > 1)
Wage price spiral
315
Pt = (1 + ) (1 + ) Pte
(J.924)
Derivation of expectation augmented Phillips curve
Pt
Pe
= (1 + ) (1 + ) t
Pt 1
Pt 1
De…ne in‡ation as
t
=
Pt Pt
Pt 1
1
(1 +
Both
and
(J.925)
t)
= (1 + ) (1 + ) (1 +
e
t)
(J.926)
y)
(J.927)
rise during boom period
( + )=
b (ut
u) = a (yt
where u natural rate of unemployment and y = natural rate of output.
Derivation of expectation augmented Phillips curve
By using the approximation of small numbers rule
(1 +
t)
t
=1+
+
=
+
+
=
b (ut
+
e
t
(J.928)
e
t
(J.929)
Phillips curve
t
e
t
=
+
u) = a (yt
y)
(J.930)
Phillips curve is considered a missing link between demand side and supply side as it brings
supply side (labour market) and the demand side together. Most important point in
macroeconomics.
Non-accelerating In‡ation Rate of In‡ation (NAIRU): Natural Rate of Unemployment
Where actual in‡ation equals to what is expected
e
t
t
Unemployment is at its natural rate (NAIRU)
316
=0
(J.931)
ut
u=0
(J.932)
Output gap is zero (y natural rate of output ; u natural rate of unemployment)
yt
[yt > y
t
>
e
t
y=0
(J.933)
ut < u] or [yt < y
t
<
e
t
ut > u]
(J.934)
Labour market e¢ ciency is important for higher level of output, employment and lower level of
prices.
10.3
Equilibrium Unemployment: Matching and Bargaining Set Up
Matching function aggregates vacancies and unemployment with job creation as:
M = M (V; U ) = V U (1
)
(J.935)
M denote the number of matching of vacancies and job seekers, V is number of vacancies and
U the number of unemployed,
is the parameter between zero and one
Nash-product of the bargaining game over the di¤erence between the earnings from work (W )
rather than in being unemployed (U ) and earnings to …rms from …lled and vacant jobs.
(Wi
U ) (Ji
1
V)
(J.936)
Symmetric solution of this satis…es joint pro…t maximisation condition for worker as:
(Wi
Parameter
U) =
(Ji + Wi
V
U)
which is the ratio of vacancy to job seeking workers
(J.937)
=
V
U.
The probability …lling a vacancy is given then by f ( ) and not …lling it by 1
f( ).
probability of …nding a job by an unemployed worker is q ( ) @t and the not …nding is 1
q ( ) @t; job creation occurs when matching takes place between …rms with vacancies and
workers seeking the job.
With labour force L and the unemployment rate u, the number of workers who enter unemployment is
(1
u) L@t.
317
There is a balance between job creation, mL@t = q ( ) L@t; and job destruction, (1
u) L@t.
in the steady state.
The term q ( ) measures the transition probability from unemployed to employed.
Normalising L to 1 the dynamics of unemployment is explained by transition dynamics
between the job destruction and job creation u =
u=
(1
u)
q ( ) u and in equilibrium
+ q( ) .
1) Dynamics of unemployment depends on the rate of job destruction, (1
u) , and the rate
of job creation, q ( ) u .
u=
(1
u)
q( )u
(J.938)
and in equilibrium
u=
where
(J.939)
+ q( )
is the rate of idiosyncratic shock of job destruction and
is the ratio of vacancy to the
unemployment and q ( ) is the probability of …lling a job with a suitable candidate through the
matching process.
Matching Model of Unemployment
2) Optimal job creation or (demand for labour curve) shows how …rms balance the marginal
revenue product of labour to wage and hiring and …ring costs in ( ; w) space
p
w
(r + )
pc
=0
q( )
(J.940)
where p is the price of product, w the wage rate, and (r + ) q(pc) is the cost of hiring and …ring
3) With 0 <
< 1 the wage curve shows positive links between the reservation wage (z) the
price of product p and costing of hiring ( c)
wi = z (1
) + p (1 + c)
(J.941)
Matching Model of Unemployment
Return from vacancy
rV =
pc + q ( ) (J
V)
(J.942)
where V denotes the value of vacancy and J the expected value for occupied jobs, pc the cost
of vacancy.
318
In equilibrium V = 0 and thus J =
pc
q( )
.
Returns from an occupied job is given by
rJ = p
w
J;
(J.943)
where a job generates revenue p against the cost of wage rate w and loss due to the stochastic
job termination J,
being the ratio of idiosyncratic shocks.
Thus the optimal condition for employment is given by equality between price of the product,
wage rate and the hiring cost of the job,
p
w
(r + )
pc
= 0:
q( )
(J.944)
Price of a product should cover wage payment and the expected hiring costs. Firms take price
and interest rate as given in the market, parameters
and
are set exogenously. Value of
unemployment and wage rate
rU = z + q ( ) (Wi
U)
(J.945)
or rU + q ( ) U = z + q ( ) Wi .
Return for employed worker is
rW = w + (U
or (r + ) W = w + U or W =
w
(r+ )
+
(r+ ) U
W):
(J.946)
.
Putting this in unemployment equation
rU + q ( ) U = z + q ( )
rU
rU (r + )
w
w
+
U = z + q( )
+ q( )
U
(r + ) (r + )
(r + )
(r + )
q( )
(r + )
U + q( )U = z + q( )
w
(r + )
(J.947)
(J.948)
q ( ) U + q ( ) (r + ) U = z (r + ) + q ( ) w
rU [(r + ) + q ( )] = z (r + ) + q ( ) w
rU =
z (r + ) + q ( ) w
[(r + ) + q ( )]
Similarly (r + ) W = w + U or
319
(J.949)
(r + ) W = w +
rW =
z (r + ) + q ( ) w
[(r + ) + q ( )]
r
=
wr [(r + ) + q ( )] + fz (r + ) + q ( ) wg
r [(r + ) + q ( )]
(J.950)
rw
rw
z (r + ) + q ( ) w
+
rU =
+
(r + ) (r + )
(r + ) (r + ) [(r + ) + q ( )]
(J.951)
Wage bargaining between …rms and workers
rJ = p
w
J or J =
rW = w + (U
p w
(r+ )
W ) or Wi =
w
(r+ )
+
(r+ ) U
Nash-product of the bargaining game
(Wi
U ) (Ji
1
V)
(J.952)
Symmetric solution of this satis…es value maximisation jointly by …rms and workers
(Wi
U) =
(Ji + Wi
V
wi
r+
(1
) wi + U (1
(1
) wi =
(p
wi = p + (1
U ) with V = 0 Wi (1
+
)=
U (1
r+
(p
wi ) + (1
U) =
(Ji + Wi
(Wi
Therefore
wi = p + (1
)U
)U
(J.953)
) (r + ) U
) rU
) rU
wi = p + (1
From (Wi
p wi
+ (1
r+
)=
wi ) + (1
) = Ji + (1
V
U) =
n
) z + q( ) 1
) fz + q ( ) (Wi
U )g
(J.954)
U) ;
Ji () (Wi
1
pc
q( )
U) =
1
pc
q( )
(J.955)
o
wi = p + (1
wi = z (1
)z +
pc
(J.956)
) + p (1 + c)
(J.957)
Thus wage rate includes reservation wage (z) and average hiring costs . Putting the wage curve
in job creation curve
p
w
(r + ) q(pc) = 0 or
320
p
z (1
(1
)
)( p
p (1 + c)
z)
(r + )
pc
=0
q( )
q( )+r+
q( )
pc
(J.958)
=0
(J.959)
This analysis is based on constant labour supply assumption though could be extended to a
growing economy.
Adding sectoral and structural features of the economy makes equilibrium unemployment theory
even closer to the real economy as presented in this paper.
10.3.1
Markov Process of Employment and Unemployment
In a simple dynamic model of transition from employment (et ) to unemployment (ut ) and then to
the bene…t could be explained by a Markov process of the system as by Ljungqvist and Sargent
(2008) or by Hoy et al. (2001) as:
et+1 = (1
) et + ut
ut+1 = et + (1
et+1
ut+1
Here (1
) and (1
!
=
(1
)
(1
)
!
(J.960)
) ut
et
ut
(J.961)
!
;
0<
<1
0<
<1
!
(J.962)
) are measures persistency of employment and unemployment rates. Using
the undetermined coe¢ cient method and using the initial conditions the complete time path of et
and ut are given by (see the derivations for this in the appendix):
et =
ut =
( + )
+
( + )
By in‡uencing the behavioral parameters
e0
u0
(1
( + )
) +
e0
u0
(1
( + )
) +
and
t
e
(J.963)
u
(J.964)
t
of the transition equations the bene…t system
in‡uences the course of unemployment and in‡ation. In theory it is possible to go back to 1942 and
study all transition paths by calibrating the historical time series of et and ut (Card, Chetty, and
Weber 2007).
Redundancies (negative shocks to employment) and vacancies (shock to unemployment) cause
‡uctuations in the transitional paths of employment and unemployment as shown in the charts
below. With the set of parameters in Table 2 this model provides the evolution of employment and
unemployment rates over time then is shown in Figure 13 and 14 respectively.
321
Table 88: Markov process for employment and unemployment
Parameters
e0
u0
e
u
Values
0:05
0:2
56:9%
9:6%
N (0; 0:5)
N (0; 0:5)
Figure 13: Equilibrium employment rate
u1
13
12.5
12
11.5
11
10.5
10
Figure 14: Equilibrium unemployment rate
Bhattarai and Dixon (2014) further extend this model considering the state space model of
the labour market as given by the transition proabilities between employment, unemployment and
inactive states in the labour market in a general equilibrium model with multiple sectors and
households in the economy as:
h
t;se
=
h
t 1;se
h
t 1;se
sht + ssht + fth
322
h
t 1;su
+ ieht
h
t 1;si
(J.965)
t100
t97
t94
t91
t88
t85
t82
t79
t76
t73
t70
t67
t64
t61
t58
t55
t52
t49
t46
t43
t40
t37
t34
t31
t28
t25
t22
t19
t16
t13
t7
t4
t1
9
t10
9.5
Probability of being in unemployment state is:
h
t;su
h
t 1;su
=
h
t 1;su
fth + f fth + sht
h
t 1;se
+ iuht
h
t 1;si
(J.966)
h
t 1;su :
(J.967)
and probability of being in the inactive state is:
h
t;si
=
h
t 1;si
h
t 1;si
ieht + iuht + ssht
h
t 1;se
+ f fth
with hetoregeniety in skills these translate to active states as in …gure 15 and 16.
0.75
0.74
0.73
0.72
0.71
0.7
0.69
t2099
t2096
t2093
t2090
t2087
t2084
t2081
t2078
t2075
t2072
t2069
t2066
t2063
t2060
t2057
t2054
t2051
t2048
t2045
t2042
t2039
t2036
t2033
t2030
t2027
t2024
t2021
t2018
t2015
0.66
t2012
0.67
t2010
0.68
0.65
Figure 16: Equilibrium unemployment rates by skills of individuals
323
t2101
t2099
t2097
t2095
t2093
t2091
t2089
t2087
t2085
t2083
t2081
t2079
t2077
t2075
t2073
t2071
t2069
t2067
t2065
t2063
t2061
t2059
t2057
t2055
t2053
t2051
t2049
t2047
t2045
t2043
t2041
t2039
t2037
t2035
t2033
t2031
t2029
t2027
t2025
t2023
t2021
t2019
t2017
t2015
t2013
t2011
0.15
0.145
0.14
0.135
0.13
0.125
0.12
0.115
0.11
0.105
0.1
0.095
0.09
0.085
0.08
0.075
0.07
t2009
Figure 15: Equilibrium employment rates by skills of individuals
143.5
143
142.5
142
141.5
141
140.5
Figure 16: Fluctuations in productivity of households in the model
References
[1] Bhattarai K and H. Dixon (2014) Equilibrium Unemployment in a General Equilibrium Model
with Taxes, The Manchester School, 82, S1, 90-128
[2] Dixon H and N Rankin (1994) Imperfect Competition and Macroeconomics: A Survey, Oxford
Economic Papers, 46:2:171-1999.
[3] Dixon Huw (1988) A Simple Model of Imperfect Competition with Walrassian Features in
R.Cross ed. Unemployment, Hysteresis and the Natural Rate Hypothesis, Basil Blackwell.
[4] Faccini. R, S. Millard and F. Zanetti (2013) Wage rigidities in an estimated dynamic stochastic
general equilibrium model of the UK labour market, Manchester School, 81, 66-99.
[5] Gertler, M. and Trigari, A. (2009), ‘Unemployment ‡uctuations with staggered Nash bargaining’, Journal of Political Economy, Vol. 117(1), pages 38-86
[6] Layard R and S. Nickell (1986) Unemployment in Britain, Economica, 53: S121-69.
[7] Ljungqvist L and T J. Sargent (2007) Two Questions about European Unemployment, Econometrica.
[8] Phelps, Edmund S. (1968), Money-Wage Dynamics and Labor-market equilibrium, Journal of
Political Economy, vol. 76, pp. 678-710.
[9] Phillips, A. W., (1958) The Relation Between Unemployment and the Rate of Change of Money
Wage Rates in the United Kingdom, 1861-1957, Economica, pp.283-299.
[10] Pissarides, C. A. (2013) Unemployment in the Great Recession, Economica, 80: 385–403.
324
h99
h97
h95
h93
h91
h89
h87
h85
h83
h81
h79
h77
h75
h73
h71
h69
h67
h65
h63
h61
h59
h57
h55
h53
h51
h49
h47
h45
h43
h41
h39
h37
h35
h33
h31
h29
h27
h25
h23
h21
h19
h17
h15
h13
h9
h7
h5
h3
h1
139
h11
140
139.5
[11] Pissarides, C. A. (2011) Equilibrium in the Labor Market with Search Frictions, American
Economic Review, 101(4): 1092-1105.
[12] Pissarides C A (2000) Equilibrium Unemployment Theory, MIT Press.
[13] Rankin Neil (1992) Imperfect competition, expectations and the multiple e¤ects of monetary
growth, the Economic Journal 102: 743-753.
[14] Taylor J B (1972) Staggered Wage Setting in a Macro Model, American Economic Review, 62,
pages 1-18.
[15] Yellen J. L (1984) E¢ ciency wage models of unemployment, American Economic Review,
Papers and Proceedings vol.74 No.2, May, pp. 199-205.
10.4
Exercise 10
Consider a standard open economy optimal growth model with
Household problem:
max
U = E0
1
X
t
Ut (Ct ; Lt )
0<
<1
(J.968)
t=0
Ut (Ct ; Lt ) =
Ct1
1
L1+!
t
1+!
(J.969)
1. subject to budget constraint as given by
Wt Lt +
t
+ Pt Kt = Pt Ct + Ptf It + Bt + Tt + 1 + Rt
1
+
t 1
et Ft
1
(J.970)
Aggregation of di¤erentiated goods from the monopolistically competitive …rms as
Z
Ct =
1
(Cj;t )
1
1
d:j
(J.971)
0
Z
Pt =
1
(Pj;t )
1
1
d:j
(J.972)
0
Firm’s problem
M ax
t
= Pt Yt
Wt Lt
PtK Kt
(J.973)
Subject to the CES production technology and stochastic TFP growth constraints as:
325
1
Yj;t = Zt (1
1 ) Lj;t
Zt = ln Zt
Yt =
Z
1+
+
(1
1
(Yj;t )
1
1 Kj;t
(J.974)
)Z
(J.975)
1
d:j
(J.976)
0
Lt =
Z
1
(Lj;t )
1
1
d:j
(J.977)
0
Kt =
Z
1
(Kj;t )
1
1
d:j
(J.978)
0
(a) Write the Lagrangian function for constrained dynamic optimisation by households and
derive the Euler equations for optimisation..
(b) Write the Lagrangian function for constrained dynamic optimisation by …rms and derived
the demand functions for labour and capital.
(c) Solve the model using the projection method and numerical optimisation using MATLAB
routines. Write approximation functions and Euler errors functions
(d) Derive impulse responses for shock to the …scal policy.
(e) Include Taylor Rule in the above problem for analysis of monetary policy.
(f) Conduct dynamic simulations to analyse the impact of demand shocks and supply shocks.
Reference: Lim G. C. and McNelis (2008), Computational Macroeconomics for the Open
Economy, MIT Press.
References
[1] Bank of England (1999) Monetary Policy Committee Transmission Mechanism of Monetary
Policy.
[2] Bernanke B. S. and F.S. Mishkin (1997) In‡ation Targeting: A New Framework for Monetary
Policy, Journal of Economic Perspectives, vol. II, no.2, Spring, pp. 97-116.
[3] Bhattarai, K., An Empirical Study of Interest Determination Rules, Applied Financial Economics, 18:4:(2008) 327-343, March.
326
[4] De Grauwe P (1997) The Economics of Monetary Integration, Oxford.
[5] De Grauwe P (1996) International Money, Oxford.
[6] Dri¢ l J. (1988) Macroeconomic Policy Games with Incomplete Information: A Survey, European Economic Review, 32 (2-3) 513-41.
[7] Friedman, M. (1968)The Role of Monetary Policy,American Economic Review, No.1 vol. LVIII
March.
[8] HM Treasury (2004) Microeconomic Reforms in Britain, Palgrave.
[9] HM Treasury (2002) Reforming Britain’s Economic and Financial Policies, Palgrave.
[10] Ghatak S. (2008) Monetary Economics, Palgrave.
[11] Goodhart Charles (1989) The Conduct of Monetary Policy, Economic Journal, 99, June, pp.
293-346.
[12] Hicks, J. R. (1937): Mr. Keynes and the “Classics”; A Suggested Interpretations, Econometrica
5:
[13] Kydland, F. E. and Prescott, E. C. (1977): Rules Rather than Discretion: The Inconsistency
of Optimal Plans, Jornal of Political Economy 85: 3: 473-491.
[14] Nelson E (2009) An Overhaul of Doctrine: The Underpinning of UK In‡ation Targeting 119:
538, June
[15] Laidler D and M Parkin (1975) In‡ation: A Survey, The Economic Journal, 85:340:741-809.
327
11
L11: Public Debt: Impact of Taxes, Spending and De…cit
on Growth
The major objectives of …scal policy in any country include 1) macroeconomic stabilisation for higher
growth rate of output, full employment, stable prices, interest and exchange rates and low in‡ation
2) attaining horizontal and vertical equity through tax/transfers and achieving e¢ ciency in resource
allocation and provision of public goods; 3) maximizing positive externality by investing in public
services such as health and education and minimising the negative externality through appropriate
taxes and subsidies. Direct taxes on income, pro…t and wealth and indirect taxes including VAT,
tari¤, excise, business and subsidies on goods/services and for use of inputs and spending on pure
public goods (defence, law / order, national parks and semi-public goods) including education,
health and Rn&D are major instruments to achieve these objectives. When the revenue from taxes,
the compulsory payments from citizens to the government, in return of public services are not
enough to meet public spending government borrows from the private sector. It crowds out private
investment raising interest rate, in‡ation a well as the current account de…cit while it borrows from
the central banks.
There is a controversy in the literature about the economic contribution of public de…cit. Keynesian economists generally argue that by spending more on goods and services and infrastructure,
budget de…cit is helpful in creating more jobs, reducing the unemployment rate and raising the economic growth rate of the economy. Neoclassical economists are more concerned about the adverse
consequences of public de…cit on capital accumulation and economic growth rate. Classical economists under Ricardian equivalence proposition argue that private saving and public dis-saving o¤set
each other. Despite this all recognise the adverse consequences of excessive budget de…cit on in‡ation, current account balances and redistribution of income. How much budget de…cit in‡uences real
choices of people through its impact on economic growth is essentially an empirical issue. Enough
debates have taken place regarding the optimal size of the government (Pigou (1947), Samuelson
(1954), Buchanan (1965), Atksinson and Stern (1974), Feldstein (1974), Whalley (1975), Boadway
(1979), Summer (1980), Blomquest (1985), Bovenberg (1989), Benabou (2002) and Taveres (2004).
Whether de…cit is good, bad or insigni…cant partly depends on which of these paradigms tends
to believe. Barro (1974, 1989) argues for the Ricardian equivalence theory - households with
perfect foresight maintain balance between the present value of their income and expenditure and
internalise the public de…cit through intertemporal optimisation raising savings to make up for
anticipated higher tax rates in the future though this result may not apply when households face
lending and borrowing constraints. Aiyagari et. al. (2002) using stochastic Ramsey model proves
that intertemporal balance is essential for maximising welfare but budget need not to balanced
328
on continual basis. They favour tax and expenditure smoothing policies when both of these are
subject to random shocks. Burnheim (1989) denounces Ricardian view in favour of New-Keynesian
propositions. He draws parallels between these two and suggests decomposing de…cit into permanent
and temporary parts. In the neoclassical model where farsighted individuals plan consumption over
lifetime, budget de…cit raises lifetime consumption by shifting taxes to the next generation; this
raises consumption and lowers savings and raises interest rate. Public sector de…cit then crowds
out private investment. As Diamond (1965) and Auerback and Kotliko¤ (1986) demonstrated
high debt/GDP ratio depresses capital labour ratio. Ni and Wang (1995) have proven how high
saving …scal policy regime with lower public sector de…cit enhances long run growth rate of the
economy. In contrast Keynesian models show positive multiplier e¤ect of budget de…cit on income
and consumption- which is just inverse of the marginal propensity to save. Beetsma and Giuliodori
(2011) using VAR impulse response analysis have found positive impacts of government purchases
among EU countries. Based on major theoretical paradigms this paper aims to provide empirical
evidence to support in favour or in against these theories and reexamine the claim that there is a
weak link between de…cit and income.
11.1
Classical Ricardian Equivalence
Ricardian equivalence means that individual households save more in response to a rise in the
budget de…cit now so that they will be able to pay higher taxes rates of taxes when the government
imposes on them when repaying those debts in the future. The household budget constraint shows
how the accumulation of public debt (Bt+1 ) and private asset (At+1 ) in t + 1 period relate to the
current income from wages (Wt Nt ), pro…ts (
t ),
interest income on bonds (1 + Rt ) Bt and income
on assets (1 + RAt ) At and expenses on consumption (Ct ) and taxes (Tt ).
Bt+1 + At+1 = Wt Nt +
t
Tt
Ct + (1 + Rt ) Bt + (1 + RAt ) At
(K.979)
Change in government borrowing occurs due to di¤erence in government spending and taxes and
the interest rate payment on outstanding debt. Thus the government’s budget constraint becomes:
Bt+1
Bt = Gt
Tt + Rt Bt =) (1 + Rt ) Bt = Bt+1
Gt + Tt
(K.980)
Gt + Tt + (1 + RAt ) At
(K.981)
Putting government budget into the household budget constraint
Bt+1 + At+1 = Wt Nt +
t
Tt
Ct + Bt+1
Which yields to Ricardian Equivalence (Only Gt a¤ects household budget not Tt ):
329
At+1 = Wt Nt +
Ct
t
Gt + (1 + RAt ) At
(K.982)
Thus in the classical spirit the larger public sector (Gt ) implies smaller private sector (At+1 ).
Then the dynamic equilibrium with this constraint implies market clearing in each period:
Yt = Ct + Gt
(K.983)
Neo-Keynesian business cycle model with leisure and consumption in the utility functions and
a stochastic technology of production is expressed in the following form:
max
E
"1
X
t=0
subject to:
t
U (Ct+i ; Lt+i ) j
t
#
Nt+i + Lt+i = 1
(K.985)
Ct+i + St+i = Zt+i F (Kt+i ; Nt+i )
Kt+i
1
= (1
(K.984)
Gt+i
) Kt+i + St+i
(K.986)
(K.987)
First order conditions imply that the disutility from labour should equal the marginal utility
from work as:
Wt
Ct
Satisfaction of the …rst order condition for optimisation.
V 0 (Lt+i ) =
E
(1 + rt+1 )
Cit
Cit+1
j
(K.988)
t
=1
(K.989)
Ct
j t =1
(K.990)
Ct+1
The New Keynesian model thus suggests that the higher government spending leads to lower
E
Rt+1
private consumption but the decrease is less than one to one; it raises output and employment. Taxes
go up if increase in Gt is permanent and investment is lower. Higher the transitory component
of Gt lower will be its in‡uence in output. Substitution and income e¤ects work; taxes are highly
discretionary and distortionary. Optimal size of public sector is very much a political issue. Higher
the transitory component of output smaller the decrease in consumption and greater the impact on
output. Ricardian equivalence fails.
330
11.2
Role of debt in the Keynesian model
Marginal propensity to consume with lump-sum or proportional taxes are key components in a
Keynesian model of government spending.
Y =C +I +G
C = a + b(Y
(K.991)
T ); a > 0 , 0 < b < 1
Assume that tax (T ) is collected lump sum and de…cit (G
(K.992)
T ) is …nanced by borrowing (B)
when tax is not enough to meet expenses (G).
G=T +B
(K.993)
C =I +T +B
(K.994)
Rearrange for a matrix:
Y
bY + C = a
Y
C
!
=
"
1
b
1
1
#
bT
1
(K.995)
I +T +B
a
bT
!
(K.996)
Using Cramer’s rule
Y =
C=
Y =
(I + T + B) + (a
1 b
(a
bT )
bT ) + (I + T + B)
1 b
a + I + (1 b) T
B
+
1 b
1 b
331
(K.997)
(K.998)
(K.999)
Thus the budget de…cit will have direct impact on output and consumption by the Keynesian
multiplier,
@Y
@B
=
1
1 b
> 0 or
@C
@B
=
1
1 b
> 0:; in this set up
@Y
@T
= 1 and
@C
@T
= 1 a balanced budget
multiplier e¤ect is achieved when budget is exactly balanced, B = 0. By log di¤erentiation it can
be shown that growth rate of GDP depends on the percentage change in the public borrowing:
gY =
+
1
2 gB
(K.1000)
This model can be extended to an open economy model by adding exports and imports in the
aggregate demand function. It can include in‡ation making the interest rate subject to the real
interest rate and using the Fisher equation. With these modi…cations the model becomes:
Y = C + I (r) + G + X
r=
IM
(K.1001)
i
(K.1002)
IM = mY
(K.1003)
Y =C +I(
i) + T + B + CA
(K.1004)
Central bank determines the nominal interest rate and then the in‡ation is determined from
the money market where the demand for money for money equals the supply of money, which is
in…nitely elastic given the central bank’s commitment to a certain interest rate.
M
= kL (
i) + f Y
(K.1005)
P
Taking log di¤erentiation of this function in‡ation is the di¤erence between the growth rate of
money supply and the sum of growth rate of output and liquidity as:
= gm
gy
gL
(K.1006)
From this equation one could link in‡ation, current account de…cit and de…cit to the growth
rate of the economy.
gY =
X
IM = Y
1
C
+
2 gB
I (r)
+
2
+
G=Y
2 gCA
C
+e
I (r)
(K.1007)
T
B
If the private sectors investment and savings are balanced this simply becomes:
332
(K.1008)
X
IM =
(T + B)
(K.1009)
From this equation one could argue that higher government de…cit will lead to larger current
account de…cit.
11.3
Growth impacts of public de…cit in the Neoclassical growth model
Growth impacts of public de…cit in a neoclassical growth model could be based on studies of
Feldstein (1974), Whalley (1975), Boadway (1979), Summer (1980), Blomquest (1985), Bovenberg
(1989), Rankin (1992) Ni and Wang (1995), Benabou (2002). Larger public sector de…cit is found
to be harmful for long term growth in neoclassical growth models where households choose the
1
optimal path of consumption and accumulation of capital fct ; kt gt=1 in response to public policy that
1
includes plan of taxes and public expenditure f ; ggt=1 . Particularly the household’s optimisation
problem is:
max
1
X
t
U (ct )
(K.1010)
t=0
subject to
ct + kt+1 = (1
Uc ((1
t) f
(kt )
kt+1 ) = Et (1
t) f
t+1 ) Uc
(kt )
((1
0
t
t+1 ) f
(kt+1 )
(K.1011)
kt+2 ) f 0 (kt+1 ) (K.1012)
When government is forced to operate a balanced budget every period the link between tax
revenue and public spending is given by:
tf
(kt ) = g
(K.1013)
When government is allowed to operate a structural balance it is permitted to intertemporally
balance the budget
bt+1
=b+g
1 + rt
Balancing the budget in the entire model horizon would imply
f (kt ) +
f (k0 )
g+
8
1 >
<
X
t=1
>
:
tf
t 1
t=0
333
(kt )
9
>
g=
>
(1 + rt ) ;
=0
(K.1014)
(K.1015)
Uc ((1
t) f
(kt )
kt+1
g) = Et (1
t+1 ) Uc
((1
t+1 ) f
(kt+1 )
kt+2
g) f 0 (kt+1 )
(K.1016)
In steady sate
) f 0 (k) = 1
(1
g
f 0 (k)
1
f 0 (k) = 1
(K.1017)
(K.1018)
g
1
f 0 (k) =
(K.1019)
(k)
Positive e¤ect of public sector …nances is possible only when ratio of tax rates to the marginal
G(k) =
1
productivity of capital is less than one, 1
11.4
f0
g
f 0 (k)
> 0:
Analysis of debt crisis
Let R be the risk free payo¤ for investors and R be the return on government bonds. Let
be the
probability of default. Then an arbitrage condition implies
(1
)R = R
(K.1020)
Some arrangement yields:
=
R
R
R
334
(K.1021)
As the probability of default rises the government need to pay higher interest rate, as shown by
line D in the graph.
Then the government retire debt if T = RD . This implies
T
D
= R. When the interest rate is
low, as at point R, the collected tax revenue is likely to be enough to serve the debt and therefore
probability of default ( ) on public debt is zero. Then
0<
< 1 between A and B points and
probability of default line is shown by line T. After point T the probability of default is 1 therefore
the government cannot borrow even paying very high interest rate and R =) 1 .
When more than one period is involved it beliefs of other people about the possibility of default
in the next period a¤ects the decision whether to purchase a bond at the current period. Beliefs
about beliefs about beliefs and thus leads to a self ful…lling crisis.
One could apply above model in the context of current debt crises faced by Greece, Spain or
Portugal in recent years.
This is one of the reason why the UK government would like to limit
debt GDP ratio at the reasonable rate of around 76 percent (See Romer (2006), Calvo (1988), Cole
and Keheo (2000)).
12
Blake-Weale (1994) model of debt
Fiscal policy makers choose the tax rate that is consistent to the target level of debt and take the
actions of central bank as given; the monetary policy makers choose the interest rate in order to
stabilise the price level taking the choice of the …scal authority as given. This is a simple but very
powerful model to explain the time path of debt in the economy.
Dt = Rt Dt
1
+ Et
Tt
(L.1022)
Expenditure (Et ) is proportional to income
Et = Yt
(L.1023)
Expenditure (Et ) is proportional to income
Tt = St Yt
(L.1024)
Output:
Yt = Y t Rt
Rt
Phillip’s curve
335
St
St
(L.1025)
t
=
e
t
t
=
Yt
Yt
+
(L.1026)
In‡ation expectation
(L.1027)
t 1
Steady state output
Y t = Y 0 egt
(L.1028)
By substitutions
t
=
st
t 1
rt
(L.1029)
log of expenditure, tax revenue and output functions:
et =
+ yt
(L.1030)
tt = st + yt
(L.1031)
yt = g
rt
st
By de…ning ratios of debt and tax revenue de…ne B =
dt =
+
1+r
dt
1+g
1
+
1
(et
B
(L.1032)
D
E;K
Ktt ) +
=
T
E
and log of debt as:
1
rt
1+g
(L.1033)
r
1+g
(L.1034)
where
=
gB + (1
K) (b
B
g) + kK
Proof for this statement:
Dt = Rt Dt
1
+ Et
Dt
Dt 1
(1 + rt ) =
Dt
Dt 1
(1 + rt ) =
Tt = (1 + rt ) Dt
Et + Et
Et 1
Et + Et
Et 1
336
1
1
Et
Dt
Et
Dt
1
1
+ Et
1
(L.1035)
1
Tt
Et
(L.1036)
1
Tt
Et
(L.1037)
1
1
Tt
(1 + g)
(1 + r) = (1 + g)
r
(1
1
(1
B
K)
(L.1038)
g
(1 + g)
=
K)
B
(L.1039)
Dynamic e¢ ciency requires that r > g.
Taylor approximation:
g
b+
1+g
( dt
g r
g) +
dt =
1
g
+
r
(rt
1+r
dt
1+g
r) +
1
+
K
1
1
(et
B
K
(tt
et
Ktt ) +
k) ' et
dt
1
1
rt
1+g
(L.1040)
(L.1041)
Stochastic optimal control method and learning
12.1
Cole -Kehoe (2000) model of self ful…lling debt crisis
Cole and Kehoe (2000) use a dynamic stochastic general equilibrium model in which self-ful…lling
crisis may arise. They say that "Because of the government’s need to roll over its debt, a liquidity
crunch induced by the inability to sell new debt can lead to a self-ful…lling default" and "if
fundamentals like the level of the government’s debt, its maturity structure, and the private capital
stock, lie within a particular range (the crisis zone), then the probability of default is determined
by the beliefs of market participants."
It is "also related to the literature on how the government’s inability to commit to future policy
choices can generate multiple equilibria."
Household:
E
1
X
t
(Ct + V (gt )))
(L.1042)
t=0
ct + kt+1 < (1
)at f (kt )
(L.1043)
Banker:
E
1
X
t
xt
(L.1044)
t=0
xt 5 x + zt bt
337
qt bt+1
(L.1045)
Government budget constraint:
gt + zt Bt 5 at t f (kt ) + qt Bt+1
(L.1046)
Timing. The timing of actions within each period is the following.
1. The sunspot variable
t
is realized, and the aggregate state is st = (Bt ; Kt ; at
1; t)
2. The government, taking the price schedule qt = q(st ; Bt+1 ) as given, chooses Bt+1 .
3. The international bankers, taking qt as given, choose bt .
4. The government chooses whether or not to default, zt , and how much to consume, gt
5. The consumers, taking at as given, choose ct and kt+1 .
Consumer’s dynamic problem:
Vc (k; s; B0; g; z) = max c + v(g) + EVc (k0; s0; B0(s0); g0; z0)
c;k0
(L.1047)
subject to
c + k0 5 (1
)a(s; z)f (k)
(L.1048)
c; k0 > 0
(L.1049)
s = (B0; K0(s; B0; g; z); a(s; z); c0);
(L.1050)
g0 = g(s0; B0(s0); q(s0; B0(s0)));
(L.1051)
z = z(s0; B0(s0); q(s0; B0(s0)))
(L.1052)
The representative banker’s value function is de…ned by the functional equation
Vb (b; s; B0) = max x + z(s; B0; q(s; B0))b
b0
q(s; B0)b0 + EVb (b0; s0; B0(s0));
(L.1053)
subject to
q(s; B0)b0 5 x
b0 >
A;
s = (B0; K0(s; B0; g; z); a(s; z); c0)
338
(L.1054)
(L.1055)
(L.1056)
The government’s value function is de…ned by the functional equation
Vg (s) = max c(K; s; B0; g; z) + v(g) + EVg (s0);
(L.1057)
g = g(s; B0; q(s; B0));
(L.1058)
z = z(s; B0; q(s; B0))
(L.1059)
s = (B0; K0(s; B0; g; z); a(s; z); c0)
(L.1060)
B0
subject to
Later in the period, the government makes its default choice z, which in turn determines the
level of productivity a and, through its budget constraint, the level of government spending g.
Given the government’s initial value function, Vg (s), they de…ne the policy functions g(s; B0; q) and
z(s; B0; q) as the solutions to the problem
max c(K; s; B0; g; z) + v(g) + EVg (s0)
(L.1061)
g + zB 5 a(s; z)f (K) + qB0;
(L.1062)
z = 0 or z = 1
(L.1063)
g>0
(L.1064)
s0 = (B0; K0(s; B0; g; z); a(s; z); 0)
(L.1065)
g;z
subject to
De…nition of an equilibrium. An equilibrium is a list of value functions Vc for the representative
consumer, Vb for the representative banker, and Vg for the government;policy functions c and k0 for
the consumer, b0 for the banker, and B0, g, and z for the government; a price function q; and an
equation of motion for the aggregate capital stock K0 such that:
1. Given B0, g, and z, Vc is the value function for the solution to the representative consumer’s
problem, and c and k0 are the maximizing choices;
2. Given B0, q, and z, Vb is the value function for the solution to the representative banker’s
problem, and the value of B0 chosen by the government solves the problem whenb = B;
339
3. Given q; c; K0; g; and z; Vg is the value function for the solution to the government’s …rst
problem (), and B0 is the maximizing choice. Furthermore, given
C; K0; Vg ; and B0; g and z solve the government’s second problem ();
4. B0(s) 2 b0(B; s; B0);
5. K0(s; B0; g; z) = k0(K; s; B0; g; z).
Cole H. L. , T. J. Kehoe (2000) Self-Ful…lling Debt Crises, Review of Economic Studies, 67, 1,
91-116.3. Given q; c; K0; g; and z; Vg is the value function for the solution to the government’s …rst
problem (), and B0 is the maximizing choice. Furthermore, given
C; K0; Vg ; and B0; g and z solve the government’s second problem ();
4. B0(s) 2 b0(B; s; B0);
5. K0(s; B0; g; z) = k0(K; s; B0; g; z).
Cole H. L. , T. J. Kehoe (2000) Self-Ful…lling Debt Crises, Review of Economic Studies, 67, 1, 91-116.
Tamai, T.,(2013) The macroeconomic e¤ects of …scal policy in a stochastically growing economy, EconomicModelling (2013), Economic Modelling 35 xxx–xxx
12.2
Credibility
Two types of time protocol:
1. Chooses sequence of
t+j
once and walks away
2. Chooses sequence of
t+j
in each period
this requires ideas of game in the modelling. Can reputation be subject to ability to commit.
Need to form a strategy space that is history dependent. Reputation could be based on the rational
expectation. Credibility is based on beliefs and it leads to the theory of government. They will do
as this is in their interest and feasible. Motives of the government is included in the model.
Model speci…cation
Household h chooses consumption
2 X and the private sector average x 2 X. The public
sector chooses y, e.g. in‡ation. Utility is ( ; x; y) ;
when x = Q
y=
t+j
Choice problem:
max ( ; x; y)
2X
where choice of household depends on average choice
= f (x; y)
Rational expectation equilibrium is equivalent to competitive equilibrium: REE s CE;
f (x; y)
Set of competitive equilibrium
C =) f(x; y) ; X = h (g)g
340
x=
Ramsey problem:
Government chooses y knowing x = ln (y)
max u (h (y) ; h (y) ; y) = max
y2Y
(x;y)2C
u (x; x; y) =) V R ; y R
Nash equilibrium X N ; y N satis…es that:
1. X N ; y N 2 C
2. G, X N , u X N ; X v ; y G = max u (x0 ; x0 ; ) =) V N ; y N and V N < V R
2Y
Example
u (l; c; g) = l + lg ( + c) + lg ( + g) ;
l+g = 1+l ;
l ( ) = f11
History
t
if
Vg x ; y
1
2
( ; g) s y
if 2(0;1
>1
=
1
1
X
t
r (xt ; yt ) ;
t=0
x; y
0;
)
2 X 8t ; xt 2 X 8t ; yt 2 X 8t for t 1 1
! !
! !
2
2 (0; 1)
1
= f(xt ; yt )gt=0
Reputation means choice at t is a function of t-1
yt = X t
1
;Y t
1
Dynamic programming square
Let V be the value to government in the …rst period of following the policy that the private
sector had expected.
Let V1 be the continuation value of known policy.
Let V2 be the continuation value if the private sector believes that the government choice is not
what they expect.
V = (1
) u (x; x; y) + V1
>
(1
(x;y)2C
) u (x; x; ) + V2 ; 8
2Y
A strategy pro…le implies a trajectory of outcome (x; y) and a value function
Vg ( ) = Vg [x ( ) ; y ( )]
341
and continuation pro…le j(x;y) ; j(x
;y ) :
A strategy pro…le is a subgame perfect equilibrium (SPE) of in…nitely repeated economy if 8
t > 1 and 8 (xt ; yt ) 2 X t
a) xt =
b) 8
h
t
Xt
1
;Y t
1
1
;Y t
1
is consistent with the competitive equilibrium where
g
t
Xt
1
;Y t
1
2Y
(1
) (xt ; yt ) + Vg
j(x;y)
>
(1
) (xt ; ) + Vg
(x;y)2C
j(x;
)
Ljungqvist L. and T. J. Sargent (2012) Recursive Macroeconomic Theory, 3rd ed. MIT Press.
12.3
Two Period Overlapping Generation Model
Impacts of de…cit spending on individuals vary by the age group they belong to. Overlapping
generation models as in Samuelson (1958) and Auerbach and Kotliko¤ (1987) provide framework to
evaluate such age speci…c impacts. For instance consider an economy, inhibited by two generations,
young and old. Young ones work, earn , consume and save and old ones stay at home in retirement
and consume out of their past savings. Economy is continuum of generations such as gi;t where
i = 1; 2; ::::N refers to the generations and t = 1; 2; ::::T refers to the time period. Each agent is
assumed to live for two periods - as a young worker …rst and then as an old retiree. For instance,
person in generation 1, g1;1 is born and young in t = 1 and becomes old in t = 2 and is succeeded
by g2;1 who is young in t = 2 , becomes old one in period t = 3 and dies at the end of that period.
In this manner new generations continuously replace the old generations but the economy continues
without any interruption with these two types of people forever. Behavior of each type is similar
to their types in earlier periods; young ones work, earn, save part of their income and make families
and get children and old ones retire and consume their savings and leave some bequest to their
children.
Production is function of capital, labour and technology and is subject to constant return to
scale with here
+
= 1:
Yt = AKt Lt
(L.1066)
In terms of income per e¤ective worker:
yt = Ak
Market clears in each period, whatever is produced is either consumed or invested.
342
(L.1067)
Yt = Ct + It
(L.1068)
Equilibrium conditions in overlapping generation model requires aggregate consumption be
total of the consumption of young and old
Ct = N cyt + N cot
(L.1069)
Net of tax wage income is given by the labour share in production
(1
l )Wt
= AKt Lt
(L.1070)
Net of tax interest rate equals the marginal product of capital
(1
Agents consume
k )rt
estate tax
v)
(L.1071)
Lt
fraction of their income in period 1 and pay a VAT rate at
cyt = (1
Young save (1
1
= AKt
wt
v
(L.1072)
) share of wt and invest it in assets for consumption at the old age subject
A:
at = (1
A ) (1
cot = at (1 + rt ) = (1 +
) wt
A ) (1
) wt (1 + rt )
(L.1073)
(L.1074)
Law of accumulation of capital stock, with no depreciation is:
Kt+1 = Kt + It
(L.1075)
From L.1068 and L.1066
Ct = AKt Lt
It
(L.1076)
Kt+1 + Kt
(L.1077)
Then substituting L.1075 and L.1069 in L.1076
N cyt + N cot = AKt Lt
343
Capital Accumulation in Overlapping Generation Model
Further substituting ?? and ?? for consumption of young and old
N (1
v )(1
l)
wt +N (1
v )(1
l )(1
k ) (1
) wt (1 + rt )+g = AKt Lt
Kt+1 +Kt (L.1078)
substituting L.1070
AKt Lt
Kt+1 +Kt = (1
v )(1
l)
AKt Lt +(1
v )(1
l )(1
k ) (1
) (1
) AKt Lt (1 + rt )
(L.1079)
By further re-arrangement
Kt+1 Kt = AKt Lt
(1
v )(1
l)
AKt Lt
(1
v )(1
l )(1
k ) (1
) (1
) AKt Lt (1 + rt )
(L.1080)
Parameters and results in Overlapping Generation Model
Table 89: Parameters of the Two Period OLG Model
K 0 k0 N
l
k
Parameter
Value
0.5
0.5
0.5
300
3
100
0.35
0.28
Table 90: Results of the Two Period OLG Model
k
K
Y
w
r
cy
c0
Variables
v
0.2
S=I
Solution without tax
Initial condition
1.5
150
1129.3
7.90
2.26
3.95
4.9
245.3
Steady State
1.78
178.5
1189.8
8.3
2.0
4.16
5.8
191.8
Solution with tax
Initial condition
1.5
150
1129.3
6.3
1.8
3.2
4.2
166.2
Steady State
1.78
178.5
1189.8
6.7
1.6
4.2
5.4
0
Larger de…cits raise consumption of old generation but lower the consumption of younger generations if such de…cit is used mainly for transfer but can improve living standards of young if spent
on creation of physical and human capital.
344
Three types of people exist every year in the economy: young ones, adults and old ones.
Young ones go to the school, adults work, and old ones stay at home in retirement.
In g11, …rst subscript refers to the generation and second subscript to the period.
Person in g11 is born in period 1, becomes adult in period 2 and becomes old in
period3 and dies at the end of period 3.
Economy continues with these three types of people forever. It never stops. new generations
continuously replace the old generations
Behavior of each type is very di¤erent.
1. Young ones borrow to fund their education;
2. adult ones work, earn and save part of their income and make families and get children;
3. old ones retire and consume their saving and leave some bequest to their children.
Three Period Overlapping Generation Model
Summary of the OLG model
First order di¤erential equation in Kt and can be solved iteratively using a numerical method
starting from initial condition where K0 is given. System converges to the steady state when
Kt+1 = Kt .
A numerical method is adopted to solve the model using Excel for tax and no tax scenarios. Labour income and capital income taxes distort the …rst order conditions (1
AKt
1
Lt and (1
l )Wt
= AKt Lt .
345
k )rt
=
This raises the cost of capital and labour to the producer and reduces the capital stock and
output as the level of welfare of the households. Net investment and savings are zero in the
steady state. Solutions of the model for parameter values given in Table 9 is given Table 10.
As expected capital and labour income taxes have signi…cantly reduced the capital stock,
output, wage rate, saving and investment and consumption of young and old in the model.
12.4
Empirical Analysis
Economic and political believes and circumstances keep changing in response to new opportunities
and di¢ culties which augment theoretical controversy regarding the relationship between growth
and public debt. As the public decisions a¤ect millions of households and …rms and their reactions
to announced or anticipated policies vary the empirical analysis of the link becomes of great public
interest. Here data on growth,de…cit, current account and several macroeconomic variables are obtained for advanced countries from the World Economic Outlook database of the IMF from 2000 to
2010 including the IMF forecasts for up to 2015. ( http://www.imf.org/external/pubs/ft/weo/2010/01/weodata/index.asp
This data set is used here to examine whether the public de…cit helpful or harmful for economic
growth and whether de…cit stabilises or destabilises an economy in terms of its impact on in‡ation
and current account de…cit. Regression coe¢ cients of de…cit or a set of variables including de…cit
multiple explanatory variables are estimated using the OLS or GLS models and examining their
validity on the basis of t; F ,
2
and R2 tests. These estimates are tested for heteroskedasticity,
autocorrelations and any restrictions as appropriate.
Regresses growth rate of output (Yi ) on net borrowing (Xi ) as:
Yi =
1
+
2 Xi
+ ei
i = 1 :::T
Following the OLS technique to …nd estimators of b 1 and b 2 .
b = (X 0 X)
1
X 0Y
(L.1081)
These estimates are subject to standard OLS assumptions on error terms normality ei
N 0;
2
, homoskedaticity, non- autororrelation (E ("i "j ) = 0)and independence of errors from
the dependent variables, (E ("i Xi ) = 0).
"
b
b
1
2
#
==
"
N
P
Xi
P
P
Xi
Xi2
#
1
"
b
b
1
2
#
=
"
12
51:92
346
51:92
413:52
#
1
"
21:3
26:23
#
=
"
3:283
#
0:349
(L.1082)
Table 91: Testing overall signi…cance by F-test
Source of Variance
Sum
Degrees of freedom Mean
Total sum square (TSS)
56.597
12
5.145
Regression Sum Square (RSS)
22.967
1
22.967
Sum of square error
33.629
10
3.737
F-value
6.147
Where k = number of parameters in the regression; N = number of observations
Table of results summarising all above calculations are presented as:
Table 92: Growth on net borrowing
Coe¢ cient Standard Error
t-value
Intercept
3.283
0.783
4.191
Net borrowing
0.349
0.133
2.613
2
R = 0.406 , F = 6:147 ;
N = 12:
Coe¢ cients as well as t-statistics are signi…cant. Autocorrelation is positive because d = 1:74 < 2
but that autocorrelation is not statistically signi…cant. The calculated DW value, d = 1:74 is clearly
out of the inconclusive region as it does not fall in the range of [0:971; 1:331] of the Durbin-Watson
table. White test or ARCH and AR test suggest there is slight problem of heteroskedasticity in
the errors in this model. However, heteroskedasticity is more serious for cross section than for time
series. Therefore conclusion of above model are still valid. One way is to regress predicted square
2
errors eb2 in predicated square of y, Yb 2 . The test statistics for normality of errors is nR2
with
i
i
df
df =1.
eb2i =
eb2i =
0
+
1 X1;i
+
0
+
b2
1 Yi
2 X2;i
+
+ vi ; n:R2 = 6:089
2
3 X1;i
+
2
4 X2;i
+
(L.1083)
5 X1;i X2;i
Null hypothesis of homoskedasticity is rejected as nR2 = 6:089 >
2
df
+ vi
= 2:7055.
Table 93: Price index on net borrowing
Coe¢ cient Standard Error t-value
Intercept
102.5
1.603
63.9
Net borrowing
-1.85
0.273
-6.76
2
R = 0.82 , F = 45:7 [0:00] ;
347
N = 12 ; DW = 1:09
(L.1084)
Table 94: Current account balance on net borrowing
Coe¢ cient Standard Error t-value
Intercept
-2.44
0.225
-10.8
Net borrowing
-0.008
0.038
-2.20
2
R = 0.33 , F = 4:9 [0:05] ;
N = 12 ; DW = 1:03
Prices were relatively stable despite …scal expansion during the study period as the monetary
policy mainly concerned in achieving the target in‡ation, had been complementary to the …scal
policy in UK in the period of study as shown in above Tables. However higher borrowing had
caused slight deterioration in the current account, as both consumers and producers tend to import
more in response to higher income they received from …scal expansion. There is weak evidence
on simultaneity between growth and de…cit in UK in last ten years. Past records like this may
or may not apply for projecting the impacts of current debt reduction plans in the future years;
these require analysis of the impacts of such de…cit in the path of economy under dynamic general
equilibrium system or under the DSGE or VAR frameworks. These tasks have been analysed in my
other papers.
12.5
Conclusion
There is a controversy in the literature about the economic contribution of public de…cit. Keynesian
economists generally argue that by spending more on goods and services and infrastructure possible,
the public de…cit is helpful to create more jobs, reduce unemployment rate and raise the economic
growth rate of the economy. Neoclassical economists are worried about the adverse consequences of
public de…cit on capital accumulation and the long run growth rate. Classical Ricardian equivalence
proposition does not match well with the empirical evidences on adverse consequences of budget
de…cit on in‡ation, current account balances and redistribution of income. In practice this is
essentially an empirical issue, evidence suggests that the role of de…cit largely depends on economic
circumstances. Empirical estimates in this paper show that de…cit has contributed for growth in
UK; 1 percent increase in net borrowing would raise growth rate by 0.34 percent between 2000 and
2010. In other words statistical and econometric evidence clearly suggests that reducing de…cit will
lower the growth rate; proposed de…cit reduction plan will clearly slow down the growth rates.
348
References
[1] Aiyagari S. Rao, Albert Marcet, Thomas J. Sargent, Juha Seppälä (2002) Optimal Taxation
without State-Contingent Debt, Journal of Political Economy, Vol. 110, No. 6 Dec., pp. 1220-1254
[2] Atkinson A. B.and N. H. Stern (1974) Pigou, Taxation and Public Goods, Review of Economic
Studies, 41:1:119-128.
[3] Atkinson A. B. (1971) Capital Taxes, the Redistribution of Wealth and Individual Savings ,The
Review of Economic Studies, 38: 2: 209-227.
[4] Auerbach A.J.and L. Kotliko¤ (1987) Dynamic Fiscal Policy, Cambridge University Press.
[5] Bank of England (2010) In‡ation Report, August, 2010.
[6] Barro, R. J. (1974), Are Government Bonds Net Wealth?, Journal of Political Economy, 82:
Nov., 1095-1117.
[7] Barro Robert J. (1989) The Ricardian Approach to Budget De…cits, Journal of Economic Perspectives, 3: 2: Spring, 37-54
[8] Bean Charles (2009) the meaning of internal balance’Thirty years on, Economic Journal, 119
(November), F442–F460.
[9] Bean Charles (1998) The New UK Monetary Arrangements: A View from the Literature Economic Journal, 108:451: Nov. 1795-1809
[10] Beetsma R and M. Giuliodori (2011) The e¤ects of government purchanges shocks: review and
estimates for the EU, Economic Journal, 121-550: F4-F32.
[11] Bernheim B. Douglas (1989) A Neoclassical Perspective on Budget De…cits, The Journal of
Economic Perspectives, 3:2:Spring:55-72
[12] Besley Tim (2001) From micro to macro: public policies and aggregate economic performance,
Fiscal Studies, 22: 3: September.
[13] Bhattarai K. (2012) Advanced Macroeconomics, University of Hull, Business School.
[14] Bhattarai K (2010) Taxes, public spending and growth in OECD countries, Problems and
Perspective in Management, 8:1: 14-30.
[15] Bhattarai K. (2007) Welfare Impacts of Equal-yield Tax Reforms in the UK Economy, Applied
Economics, 39:10-12: 1545-1563.
349
[16] Blanchard, O. and Perotti, R. (2002). _An empirical characterization of the dynamic e¤ects of
changes in government spending and taxes on output_, Quarterly Journal of Economics, 117(4)
: Nov.:1329–68.
[17] Boadway Robin (1979) Long-Run Tax Incidence: A Comparative Dynamic Approach The
Review of Economic Studies, 46:3:505-511.
[18] Bovenberg L.A. (1989) The E¤ects of Capital Income Taxation on International Competitiveness and Trade Flows, The American Economic Review, 79,: 5 :1045-1064.
[19] Buchanan, J., 1958, Public principles of public debt (Irwin: Homewood, Illinois).
[20] Burnside, C., Eichenbaum, M. and Fisher, J. D. (2004). Fiscal shocks and their consequences,
Journal of Economic Theory,115: 89–117.
[21] Brauninger Michael (2005) The budget de…cit, public debt, and endogenous growth ,Journal
of Public Economic Theory, 7 (5): 827–840.
[22] Caucutt E.M, S. Imrohoroglu and K. B. Kumar (2006) Does the Progressivity of Income Taxes
Matter for Human Capital and Growth? Journal of Public Economic Theory, 8:1:95-118
[23] Diamond, P.A., 1965, National debt in a neoclassical growth model, American Economic Review 55, 112551150.
[24] Diamond P, Douglas W. and Dybvig, Philip H. (1983) Bank Runs, Deposit Insurance,and
Liquidity, Journal of Political Economy, 401-419.
[25] Fisher Jonas D.M. and Ryan Peters (2010) Using Stock Returns to Identify Government Spending Shocks, Economic Journal, 120: :544: May,
[26] Feldstein M (1985) Debt and taxes in the theory of public …nance, Journal of Public Economics,
28:233-46
[27] Feldstein M (1982) Government de…cits and aggregate demand, Journal of Monetary Economics 9 (1982) 1-20
[28] Fullerton, D., J. Shoven and J. Whalley (1983) Dynamic General Equilibrium Impacts of
Replacing the US Income tax with a Progressive Consumption Tax, Journal of Public Economics
38, 265-96.
[29] Holly S and M Weale (Eds.) Econometric Modelling: Techniques and Applications, pp.69-93,
the Cambridge University Press, 2000.
350
[30] HM Treasury (2010) Pre-Budget Report, October 2010.
[31] Kirsanova Tatiana, Campbell Leith and Simon Wren-Lewis ( 2009) Monetary and …scal policy
interaction: The current consensus assignment in the light Of recent developments, Economic
Journal,119:Nov,F482–F496.
[32] Kydland F.E and E.C. Prescott (1977) Rules rather than discretions: the Inconsistency of
Optimal Plans, Journal of Political Economy, 85:3: 473-491.
[33] Meade, J.E. (1958), Is the national debt a burden? Oxford Economic Papers IO, 1633183.
[34] Mendoza E. G and V. Z. Yue (2012) A general equilibrium model of sovereign default and
business cycles, Quarterly Journal of Economics 127, 889–946.
[35] Mirlees, J.A. (1971) An exploration in the theory of optimum income taxation, Review of
Economic Studies, 38:175-208.
[36] Modigliani, F., 1961, Long run implications of alternative …scal policies and the burden of the
national debt, Economic Journal 71, 728755.
[37] Monacelli T and R. Perotti (2010) Fiscal Policy, the Real Exchange Rate and Traded Goods,
Economic Journal, 120(May), 437-461.
[38] Ni Shawn and X Wang ( 1995) Balanced government budgets versus de…cit …nance in a growth
economy, Canadian Journal of economics, XXVIII: 4b:1120-1134.
[39] Obstfeld, Maurice, Jay C. Shambaugh and Alan M. Taylor. (2010) Financial Stability, the
Trilemma, and International Reserves, American Economic Journal: Macroeconomics: 2:2, April
,
[40] Perroni, C. (1995), Assessing the Dynamic E¢ ciency Gains of Tax Reform When Human
Capital is Endogenous, International Economic Review 36:907-925.
[41] Phelps, Edmund S. (1968), Money-Wage Dynamics and Labor-market equilibrium, Journal of
Political Economy, 76: 678-710.
[42] Rankin Neil (1992) Imperfect competition, expectations and the multiple e¤ects of monetary
growth, Economic Journal 102: 743-753.
[43] Rebelo S. (1990) Long run policy analysis and long run growth, Journal of Political Economy
vo. 99 no. 3 pp. 500-521.
351
[44] Samuelson P. A. (1954) Pure Theory of Public Expenditure, Review of Economic Statistics,
36:387-9/
[45] Sargent, T. J. (2011), Where to Draw Lines: Stability Versus E¢ ciency. Economica, 78: 197–
214.
[46] Schmitt-Grohe´ S.,M. Uribe, Balanced-budget rules,distortio nary taxes and aggregate instability,
[47] J. Polit. Econ. 105 (1997) 976–1000.
[48] Sorensen PB and H. Jl Whitta-Jacobsen (2010) Introducing Advanced Macroeconomics, McGraw Hill.
[49] Tekin-koru Ayc a and Erdal O¨ zmen (2003) Budget de…cits, money growth and in‡ation: the
Turkish evidence, Applied Economics, 35, 591–596
[50] Whalley J. (1975) “A General Equilibrium Assessment of the 1973 United Kingdom tax reform”, Economica, 42:166:139-161.
12.6
International strategic policy coordination models
Economic crisis very often is contegious. It transmits from one economy to another. Policy coordinations can mitigate adverse consequences of these crisis. This requires studying how one economy
is linked to the another in the regional or global economy settings.
Interdependence among economies and interactions could be studied using bargaining, signalling
and mechanism designing concepts. Cooperative and non-cooperative games with complete and
incomplete information among nations, households and …rms could be used to conceptualize the
issues and solutions to the problems of growth and development in these economies. There are three
generations of literature in the policy coordination. First generation models include studies such
as Kydland and Prescott (1977), Dri¢ l (1988), Currie and Levine (1986) and Obstfeld and Rogo¤
(2000). These had found gains from coordination to be small. Cooper (1969) and Hamada (1976)
and Kydland (1975) showed inferiority of the non-cooperative Nash equilibrium compared to a
cooperative solution. Lucas (1976), and Kydland and Prescott (1977) used rational expectations and
argued for the advantage of rule-based policies to create rational expectations equilibrium solution.
Petit (1989) used di¤erential games as did the studies of Obstfeld (1994), Sutherland (1996), Senay
(1998), Martin and Rey (2000). Obstfeld (2001) and Rogo¤ (2002) provide an excellent review of
some of the models used for policy coordination with Mundell-Fleming-Dornbush type models with
little gains from coordination. Second generation models of policy coordination in Pappa (2004),
352
Canzoneri, Cumby and Diba (2005), Clerc, Dellas and Loisel (2011), Juillard and Villemot (2011)
and Goyal (2007) …nd pay o¤ from monetary and …scal policy coordination to be bigger. Supply
and strategic modelling has much improved in recent literature on the policy coordination showing
more gains from coordination as stated by Conzoneri et. al.(2005), Evans and Hnatkovska (2007),
Douglas and Laxton in dynare. Aarle et.al. (2002) examine the coalition formation in EMU. Recent
models such as Kempf and von Thadden (2013), Dedola et al. (2013) add asymmetric information
and commitment where the welfare gains can be bigger as the number of countries increase in such
deals. Given this literature let us consider three countries aiming for a policy coordination with the
Nash utility frontier:
Nt = U1;t U2;t U3;t
(L.1085)
Each receive utility from consuming products produced in each country:
Ui;t = F (y1;t; y2;t ; y3;t )
(L.1086)
Goods supply process is determined simultaneously as:
y1;t =
1;0
+
1;2
y2;t +
1;3
y3;t +
1;1
y1;t
1
+
1;2
y2;t
1
+
1;3
y3;t
1
+ e1;t
(L.1087)
y2;t =
2;0
+
2;1
y1;t +
2;3
y3;t +
2;1
y1;t
1
+
2;2
y2;t
1
+
2;3
y3;t
1
+ e2;t
(L.1088)
y3;t =
3;0
+
3;1
y1;t +
3;2
y2;t +
3;1
y1;t
1
+
3;2
y2;t
1
+
3;3
y3;t
1
+ e3;t
(L.1089)
Coe¢ cient of a VAR model estimated from the time series data provides information on interactions
among model economies as:
0
B
B
@
0
B
= B
@
1
1;2
2;1
3;1
1;0
2;0
3;0
1
1
2;3
1
3;2
0
C B
C+B
A @
10
1;3
1;1
1;2
2;1
2;2
3;1
3;2
y1;t
1
C
CB
CB y C
A @ 2;t A
y3;t
10
353
1;3
2;3
3;3
y1;t
CB
CB y
A @ 2;t
y3;t
1
1
1
1
0
e1;t
1
C B
C
C+B e C
A @ 2;t A
e3;t
(L.1090)
0
=
1
y1;t
C
B
B y C
@ 2;t A
y3;t
0
1
B
B
2;1
@
3;1
0
B
B
@
1
1
1
1;2
2;1
3;1
C
C
A
2;3
3;2
1
1
1;3
1
3;1
B
+B
@
2;3
3;2
1
3;2
1
C
C
A
1;3
1
1;2
2;1
0
1;2
1
1;3
2;3
1
1
1
0
1
B
B
@
1;0
B
B
@
1;1
1;2
1;3
2;1
2;2
2;3
3;2
3;3
2;0
3;0
0
1
C
C
A
0
3;1
C
C+
A
1
e
B 1;t C
B e C
@ 2;t A
e3;t
10
y1;t
CB
CB y
A @ 2;t
y3;t
1
1
1
1
C
C
A
(L.1091)
Paramters of VAR could be interpreted in the context of Nash Policy Game as:1) In common
meetings or summits they decide policies given by
1;0
;
2;0 ;
3;0
but each of them face idiocyn-
cratice shocks e1;t ; e2;t ; e3;t ; 2) Then each country determine its action yi;t taking account of actions
taken by others yj;t and such response patterns are given by parameters
,
;
1;2
1;3 ;
;
2;1
;
2;3
;
3;1
3;2
1;2
;
1;3 ;
2;1
;
2;3
;
3;1
;
3;2
and shocks e1;t ; e2;t ; e3;t ; 3)Each would like to get more utility and this
opens the bargain; 4) The optimal solution of this game should ful…ll four properties of Nash bargaining game; 5) This must be symmetric, e¢ cient, linear invariance and IIA. Extention of this
model for the many countries case is very obvious.
Beetsma R. M.W.J. and H. Jensen (2005) Monetary and …scal policy interactions in a microfounded model of a monetary union Journal of International Economics, 67, 2, 320-352
Bullard J and ASingh (2008) Worldwide macroeconomic stability and monetary policy rules
Journal of Monetary Economics, 55, S34-S47
Chang, Roberto. (1997) Financial integration with and without international policy coordination ,International Economic Review, 38, 3, 547. 18p.
Canzoneri M. B., R. E. Cumby and B.T. Diba (2005) The need for international policy coordination: what’s old, what’s new, what’s yet to come? Journal of International Economics,
66, 2, 363-384
Clerc L, H. Dellas, O. Loisel (2011) To be or not to be in monetary union: A synthesis Journal
of International Economics, 83, 2, 154-167
354
Clarida R, Jordi Galí and M. Gertler (2002) A simple framework for international monetary
policy analysis, Journal of Monetary Economics, 49, 5, 87–904
Conconi P and Carlo Perroni (2009) Do credible domestic institutions promote credible international agreements? Journal of International Economics, 79, 1, 160-170
Cooper, R, D. DeJong , R Forsythe and T. W. Ross(1992) Communication in coordination
games, Quarterly Journal of Economics. 107 2, p739. 33p.
Currie D and P Levine (1986) Time inconsistency and optimal policies in deterministic and
stochastic worlds Journal of Economic Dynamics and Control, 10, 1–2,191-199
D. Luca , P Karadi and G. Lombardo (2013) Global implications of national unconventional
policies Journal of Monetary Economics, 60, 1, 66-85
Fratzscher M (2009) How successful is the G7 in managing exchange rates? Journal of International Economics, 79, 1, 78-88
Goodfriend, M.; R. G. King (1997) The New Neoclassical Synthesis and the Role of Monetary
Policy NBER/Macroeconomics Annual (MIT Press). 12 1, 231-283.
Juillard M, S. Villemot (2011)Multi-country real business cycle models: Accuracy tests and
test bench Journal of Economic Dynamics and Control, 35, 2„178-185
Levine P, A. Brociner (1994) Fiscal policy coordination and EMU: A dynamic game approach
Journal of Economic Dynamics and Control, 18, s 3–4, 699-729
Hansen L. P. and T. J. Sargent (2003) Robust control of forward-looking models Journal of
Monetary Economics, 50, 3, 581-604
Kempf H. and L. von Thadden (2013) When do cooperation and commitment matter in a
monetary union? Journal of International Economics, 91, 2, 252-262
Liu Z and E. Pappa (2008) Gains from international monetary policy coordination: Does it
pay to be di¤erent? Journal of Economic Dynamics and Control, 32, 7, 2085-2117
Marquez J (1988) International policy coordination and the reduction of the US trade de…cit,
Journal of Economic Dynamics and Control, 12, 1, 19-25
Gar…nkel M.R. (1989) Global macroeconomics: Policy con‡ict and cooperation: A review
essay Journal of Monetary Economics, 23, 2, 345-352
355
Pappa E. (2004) Do the ECB and the fed really need to cooperate? Optimal monetary policy
in a two-country world Journal of Monetary Economics, 51, 4, 753-779
Kose Ayhan K.M, C. Otrok and C. H. Whiteman (2008) Understanding the evolution of world
business cycles Journal of International Economics, 75, 1, 110-130
Je¤rey S. (1992) International monetary and …scal policy cooperation in the presence of wage
in‡exibilities: Are both counterproductive? Journal of Economic Dynamics and Control, 16,
2, 359-387
References
[1] Barro R.J. and D. B. Gordon (1983) Rules, Discretion and Reputation in a Model of Monetary
Policy, Journal of Monetary Economics, 12: 101-121, North-Holland.
[2] Bhattarai K. (2014) Money and economic growth, Journal of Economic Asymmetries 11, 8–18
[3] Bhattarai K. and S. Mallick (2013) Impact of China’s currency valuation and labour cost on
the US in a trade and exchange rate model. North American Journal of Economics and Finance,
25, 40-59.
[4] Bhattarai K and J Whalley (2006), Division and Size of Gains from Liberalization of Trade in
Services, Review of International Economics, 14:3:348-361, August.
[5] Breitner M. H. (2005) Genesis of Di¤erential Games in Light of Isaac’s Contributions, Jounral
of Optimisation Theory and Applications, 124:3:529-559
[6] Burda M. and C. Wyplosz (2010) Mundell-Flemming model, Macroeconomics, Oxford University Press.
[7] Canzoneri M. B. and J A Gray (1985) “Monetary Policy Games and the Consequences of NonCooperative Behaviour”, International Economic Review, 26:3:1985, pp. 547-567.
[8] Canzoneri M. B., R. E. Combey and B. T. Diba (2005) The need for International monetary
policy coordination,: what is old, what is new and what is yet to come, Journal of International
Economics, 66:363-384
[9] Canzoneri Matthew B. , Robert E. Cumby, Behzad T. Diba (2005) The need for international
policy coordination: what’s old, what’s new, what’s yet to come? Journal of International
Economics, 66, 2, 363-384
356
[10] Conconi P, Carlo Perroni (2009) Do credible domestic institutions promote credible international agreements? Journal of International Economics, 79, 1,160-170.
[11] Currie D, P. Levine (1985) Simple Macropolicy Rules for the Open Economy The Economic
Journal, 95, 60-70.
[12] Currie D adn P Levine (1986) Time inconsistency and optimal policies in deterministic and
stochastic worlds Journal of Economic Dynamics and Control, 10, 1–2,191-199
[13] Chakravarty S. (1965) Optimal Programme of Capital Accumulation in a Multi-Sector Economy, Econometrica, 33, 3 ,Jul. 557-570
[14] Cooley T. and Hansen (1989) The in‡ation tax in real business cycle models, American Economic Review, 79,3:733-748
[15] Cooper R. N. (1969) Macroeconomic Policy Adjustment in Interdependent Economies, Quarterly Journal of Economics, 83: 1:Feb.:1-24
[16] Corden W.M. (1985) Macroeconomic Policy Interaction under Flexible Exchange Rates: A
Two-Country Model, Economica, New Series, 52:205:Feb. 9-23
[17] Cruz J.B. (1975) Survey of Nash and Stackeberg equilibrium strategies in dynamic games,
Annals of Economic and Social Measurement, 4:2:339-344.
[18] Cruz J.B. and Chen (1971) Series Nash Solution of Two Person Non-Zero-Sum Linear
Quadratic Di¤erential Games, Jounral of Optimisation Theory and Applications, 7:4:240-257
[19] Ellsberg D. (1968) Theory and practicel of blackmail, Rand Corporation.
[20] Datt G, Martin Ravallion (2011) Has India’s Economic Growth Become More Pro-Poor in the
Wake of Economic Reforms? World Bank Economic Review, 25, 2 ,157-189
[21] Dedola Luca , Peter Karadi, Giovanni Lombardo (2013) Global implications of national unconventional policies, Journal of Monetary Economics, 60, 1, 66-85
[22] Dri¢ l J. (1988) Macroeconomic Policy Games with Incomplete Information: A Survey, European Economic Review, 32 (2-3) 513-41.
[23] Evans, M. D., and V. V. Hnatkovska (2007): “Financial Integration, Macroeconomic Volatility,
and Welfare,”Journal of the European Economic Association, 5(2-3), 500–508.
[24] Frankel J. (1992) Obstacles to the macroeconomic policy coordination with an analysis of
international policy targets, UC, Berkley.
357
[25] Frankel J. A., K. E. Rockett (1988) International Macroeconomic Policy Coordination When
Policymakers Do Not Agree on the True Model, American Economic Review, 78, 3,Jun.,: 318-340.
[26] Freiling Jank and Abou-Kendill (1996) On global existence solutions to coupled matrix Riccati
equations in closed loop Nash games, memio?
[27] Goyal A (2007) Tradeo¤s, Delegation, and Fiscal-Monetary Coordination in a Developing
Economy, Indian Economic Review,42, 2,141-164
[28] Hamada K (1976) Strategic Analysis of Monetary Interdependence, Journal of Political Economy, 84 August.
[29] Ivanov I. G. (2008) A method to solve discrete time coupled algebraic Ricaati equations,
Applied Mathematics and Computations, 206, 34-41
[30] Jank G and Kremer ( ) Open loop Nash games and positive systems: solvability conditions for
non-symmetric Riccati equations, memio?
[31] Jorgenson D. W. (1961) Stability of a Dynamic Input-Output System Stability of a Dynamic
Input-Output System, Review of Economic Studies, 28,2, Feb., 105-116
[32] Kehoe P.J. (1989) Policy Cooperation Among Benevolent Governments May Be Undesirable
,The Review of Economic Studies, 56, 2, Apr., 289-296
[33] Kawai M (2011) G-20 Financial Reforms and Emerging Asia’s Challenges, in Asia and Policymaking for the Global Economy, Asian Development Bank.
[34] Kempf Hubert , Leopold von Thadden (2013) When do cooperation and commitment matter
in a monetary union? Journal of International Economics, 91, 2, 252-262
[35] Kotwal A, Bharat Ramaswami, and Wiliam Wadhwa (2011) Journal of Economic Literature
2011, 49:4, 1152–1199.
[36] Krugman Paul (1979) A Model of Balance of Payment Crisis, Journal of Money Credit and
Banking, 11,Aug.
[37] Kumar T. K. (1969) The Existence of an Optimal Economic Policy Econometrica, 37:4 :Oct.:
600-610.
[38] Kydland F. (1975) Equilibrium Solutions in Dynamic Dominant Player Models, Journal of
Economic Theory, 15:307-324
358
[39] Kydland F. (1975) Non-cooperative and dominant player solutions in dynamic discrete games,
International Economic Review, 16:2:321-335
[40] Kydland F.E and E.C. Prescott (1977) Rules rather than discretion: the Inconsistency of
Optimal Plans, Journal of Political Economy, 85:3: 473-491.
[41] Li M., J.B. Cruz, M. A. Simaan (2002) An approach to discrete-time incentive feedback stackelberg games, IEEE Transactions on System, Man and Cybernatics- Part A: Systems and Humans,
32:4:472-481.
[42] Martin, P., and H. Rey (2000): “Financial integration and asset returns,”European Economic
Review, 44(7), 1327–1350.
[43] Miller M, Mark Salmon (1985) Dynamic Games and the Time Inconsistency of Optimal Policy
in Open Economies, Economic Journal, 95, Conference Papers:124-137.
[44] Miller, Marcus; Salmon, Mark When Does Coordination Pay? Journal of Economic Dynamics
and Control, July-Oct. 1990, v. 14, iss. 3-4, pp. 553-69
[45] Mundell R. A (1962) Capital mobility and stabilisation policy under …xed and ‡exible exchange
rates,Canadian Journal of Economic and Political Science, 29, 475-85.
[46] Mukaidani H. (2004) Proceedings of the American Control Conference, Boston
[47] Nordhaus W.D. (1995) Policy Games: Co-ordination and Independence in Monetary and Fiscal
Policeis, Brookings Papers on Economic Activity 2:1994: 139-216.
[48] Obstfeld M. (2001) International macroeconomics: beyond the Mundell-Flemming model,
Cebter for International and Development Economics Research, UC Berkeley.
[49] Obstfeld, M. (1994): “Risk-Taking, Global Diversi…cation, and Growth,” American Economic
Review, 84(5), 1310–29.
[50] Oudiz G. and J Sachs (1984) Macroeconomic policy coordination among industrial economies,
Brookings papers in economic activities, 1:1-64
[51] Papavassiloupoulos G. P. and Olsder (1984) On the linear-quadratic, closed loop, no-memory,
Nash game, Journal of Optimisation Theory and Applications, 42:4:551-560
[52] Parikh A. (1979) Forecasts of Input-Output Matrices Using the R.A.S. Method The Review of
Economics and Statistics, 61, 3 Aug., 477-481.
359
[53] Petit M. S. (1989) Fiscal and Monetary Policy Co-Ordination: A Di¤erential Game Approach,Journal of Applied Econometrics, 4:2:Apr. - Jun.:161-179
[54] Rogo¤ K. (2002) Dornbusch’s Overshooting Model After 25 Years, IMF Sta¤ Papers, 49:1-35.
[55] Senay, O. (1998): “The Ex
oects of Goods and Financial Market Integration on Macroeconomic
Volatility,” Manchester School of Economic & Social Studies, 66(0), 39–61.
[56] Sutherland, A. (1996):
“Financial Market Integration and Macroeconomic Volatil-
ity,”Scandinavian Journal of Economics, 98(4), 521–39.
[57] Starr A. W. and Y. C. Ho (1969) Non-zero sum di¤erential games, Jounral of Optimisation
Theory and Applications, 3:3:184-206
[58] Taylor J (1993) Discretion versus policy rules in practice, Carnegie Rochester Conference,Series
on Public Policy 29 Amsterdam.
[59] Turnovsky S.J., T. Basar and V. d’Orey (1988) Dynamic Strategic Monetary Policies and
Coordination in Interdependent Economies,The American Economic Review, 78, 3,Jun., 341-361
[60] Van Long N. and G. Sorger (2009) A dynamic principle agent as a stacklebege di¤erential
game, working paper 0905, Department of Economics, University of Vienna.
[61] Van Aarle B, J. Engwerda and J. Plasmann (2002 ) Monetary and Fiscal policy interaction in
the EU: A Dynamic Game Approach, Annals of Operation Research, 109, 229-264
[62] Van Aarle B, J. Engwerda and J. Plasmann (2001) Monetary and Fiscal policy interaction in
the EU: A Dynamic Game Approach, Open Economies Review, 12:1, 29-60
[63] van de Klundert T. and F. van der Ploeg (1989)Fiscal Policy and Finite Lives in Interdependent
Economies with Real and Nominal Wage Rigidity, Oxford Economic Papers, New Series, Vol. 41,
No. 3 (Jul., 1989), pp. 459-489
[64] Xu C (2011) The Fundamental Institutions of China’s Reforms and Development Journal of
Economic Literature 2011, 49:4, 1076–1151
12.7
Exercise 11
1. Consider the macroeconomic system in two interdependent economies, i.e. Europe and the
ROW
Economy 1
Y1 = C1 + I1 + G1 + N X1
360
(L.1092)
C1 = a1 + b1 (Y1
T1 )
(L.1093)
I1 = k1 + d1 r1
(L.1094)
N X1 = Y2
(L.1095)
Y2 = C2 + I2 + G2 + N X2
(L.1096)
Economy 2
C2 = a2 + b2 (Y2
T2 )
(L.1097)
I2 = k2 + d2 r2
(L.1098)
N X2 = Y1
(L.1099)
(a) Solve for the national income of both economies simultaneously.
(b) Determine how public spending of economy 1 would impact economy 2.
(c) How would the monetary policy one economy a¤ect the monetary policy of another
economy?
361
13
Tutorial Problems
13.1
Tutorial 1: Comparative Statics
Q1. Keynesian Model: Hicksian Synthesis
Y = C +I +G
(M.1100)
C = C (Y
(M.1101)
Consumption function
T)
Investment
I = I(r)
(M.1102)
M s = M (Y; R)
(M.1103)
Money Market
Reduced form for goods and money markets
Y
C (Y
T)
I(r) = G
(M.1104)
M s = M (Y; R)
(M.1105)
Y and r are implicit functions of G, T and Ms
Derive comparative static equations for for dY and dr. Find the expression to analyse the
impacts of …scal and monetary policy instruments in output and the interest rate.
Hint: Take total de¤erentition of these two equations
dY
C 0 (Y
T ) dY
I(r)dr = dG + C 0 (Y
T ) dT
(M.1106)
@M
@M
dY +
dr = dM s
(M.1107)
@Y
@r
Using the time series contained in the Workhours.csv …le estimate this Keynesian
model and use it for policy analysis. Be able to execute the programmes written in
MATLAB, dynare, GAMS, Oxmetrics 7 and Eviews 8.
Find the absolute and relative standard deviation of growth rate of output, consumption,
investment and hours worked observed in the data.
Test the validity of the quantity theory of money MV = PY with appropriate data.
362
Table 95: Percentage standard deviation of macro variables
GDP
Consumption
Investment
Hours worked
% standard deviation
Relative % standard deviation
Table 96: Lag, contemporaneous and lead correlations among macro variables
GDPt ; xt
GDPt ; xt
1
GDPt ; xt+1
GDP
Consumption
Investment
Hours worked
Q2. Hicks (1937) had integrated Keynesian ideas nicely like this.
Output
Y = F (K; N )
Fk > 0; FN > 0; Fkk < 0; FN N < 0:
(M.1108)
Consumption
C = c Y d ; Y d = (1
)Y
(M.1109)
Investment
I = I(r)
(M.1110)
W
= FN (N; K)
P
(M.1111)
W = W0 + W (N )
(M.1112)
Labour demand
Labour supply
W (N ) =
Z
0 for N 5 N
(M.1113)
+for N > N
money market equilibrium conditions:
M
= M (Y; r)
P
My > 0; Mr < 0
363
(M.1114)
Net exports
NX = X
IM
(M.1115)
Equilibrium condition
Y = C + I (r) + G + N X
(M.1116)
Q3. Samuelsonian Multiplier Accelerator Model (1939) provides good dynamics in the system.
Macro balance
Yt = Ct + It + G0
(M.1117)
Consumption function
Ct = Yt
1;
0<
<1
(M.1118)
Investment
It =
(Ct
Ct
1) ;
>1
(M.1119)
Equilibrium (putting Ct and It in Yt ): second order di¤erence equation
Yt =
(1 + ) Yt
1
Yt
2
+ G0
(M.1120)
a) What is the level of income in the steady state?
b) Analyse the dynamic properties of the system
Distinct real root case (no cycle)
2
(M.1121)
2
(M.1122)
2
(M.1123)
2
(1 + ) > 4
2
(1 + ) = 4
2
(1 + ) < 4
Repeated real root case (no cycle)
Complex root case (cycle)
Complete solution
364
Yt = A1 bt1 + A2 bt2 + Y
(M.1124)
Practice with stochastic Keynesian and Samuelsonian models.
Q4.Imagine an economy inhabited by rich, middle income and poor households, indexed by i
= A, B and C. There are three types of goods in the economy. Endowments of these three goods
to three categories of households are W1 , W2 and W3 respectively. Each household prefers to
consume all three goods, j = 1; 2;and 3. The demand of household i for good j , is denoted by
Xji ; i.e. X1i ; X2i and X3i . Each household i maximises its own welfare subject to its own budget
j
P
constraint, I i =
Pj Wji , where I i is the total income of the household, Pj is the relative price
j=1
of a commodity and Wji is the endowment of commodity j of household i. Price of good j adjusts
until demand for it equals its supply. For simplicity assume that each household is endowed only
with one type of good but prefers to consume each of three goods equally. Thus preferences and
constraints for household type i are given by following equations:
U (X1i ; X2i ; X3i ) = X1i X2i X3i
M ax
i = A; B; C
(M.1125)
subject to
Ii =
j
X
Pj Wji = P1 X1i + P2 X2i + P3 X3i
(M.1126)
j=1
Markets clear (only A is endowed by W1 ; only B is endowed by W2 and only C is endowed by
W3 )
X
X1i = W1A ;
X
X2i = W2B ;
X
X3i = W3C
(M.1127)
The endowments of households were as given in Table 1.
Table 97: Endowment Structure of Households
W1 W2 W3
A
100
0
0
B
0
200
0
C
0
0
300
100
200
300
Total supply
a. Derive demand functions, X1i ; X2i and X3i consistent with utility maximisation by each
household. Find equilibrium prices, optimal allocations and utility for each household.
365
Table 98: Optimal Consumption of Households
X1 X2 X3 U
A
B
C
Total
100
200
300
Price
b. Record the quilibrium solutions of the model in respective cells of Table 2.
c. How would these prices change if there is a 20 percent tax on income of each household and
all revenue collected are distributed equally among them.
Q5. Three period model of consumption
Extend two period two individual model to a three period economy which is inhibited by the low,
middle and high income households. Again inter temporal optimisation by each involves maximising
utility subject to its life time budget constraint.
M ax
U (C1i ; C2i ; C3i ) = ln C1i +
i
2
ln C2i +
i
3
ln C3i
i = A, B,C
(M.1128)
subject to budget constraints while young, adult and old as following:
C1i + bi1 = w1i
(M.1129)
C2i + bi2 = bi1 (1 + r) + w2i
(M.1130)
C3i = bi2 (1 + r) + w3i
(M.1131)
whereC1i ; C2i ; C3i are consumptions for periods 1, 2 and 3 for type i agent and
i
2 and
i
3 are
subjective discount factors for period 2 and 3 consumptions with their values between 0 and 1.
Endowment of agent i for time t is given by wti with endowments for agent A, B and C for periods
1, 2 and 3 are w1A ; w1A ; w1A ; w1B ; w1B ; w1B ; w1C ; w1C ; w1C . Again each household is allowed to borrow
and lend at the interest rate r.
Markets clear for each good for each period:
C1A + C1B + C1C = w1A + w1B + w1C
366
(M.1132)
C2A + C2B + C2C = w2A + w2B + w2C
(M.1133)
C3A + C3B + C3C = w3A + w3B + w3C
(M.1134)
What is the interest rate and equilibrium allocations in this economy? State how to extend this
model to ten households.
Q6. Consider a New Keynesian business cycle model in which
Qi = Li
Ui = Ci
(M.1135)
Li ;
>1
(M.1136)
Consumption equals real income
Ci =
Pi Qi
P
(M.1137)
with demand shocks as given by
qi = y + zi
n>0
(M.1138)
Prove that equilibrium output is less than optimal when producers have mark up power.
Q7. Consider a new Keynesian structure with are i:::n …rms each with technology
Yi = AL1i
M P Li =
;
0<
@Yi
= (1
@Li
<1
) ALi
(M.1139)
(M.1140)
Relation to aggregate output (new Keynsian supply function):
Yi = A
Pi
P
Y
n
(M.1141)
Each …rm has some market power that is related to the price elasticity of demand for its product
T Ri = Pi Yi ;
=
@Yi Pi
@Pi Yi
Prove that oputput of the (Yi ); employment (Li ) in this model are given by
367
(M.1142)
Yi =
mp
Wi
) ALi P
(1
Y
n
(M.1143)
and
Li =
with the mark-up mp =
(
1) :
Y
nA
"
(1
)A
mp
"
Wi
P
"
(M.1144)
Show that higher marp up by …rms results in lower demand for
labour. What is the expression for the elasticity of labour demand to the real wage in this model.
368
13.2
Tutorial 2: Stability Analysis
Solve the following system of di¤erential equations
1.
where a) A =
"
4
4
y = Ay
"
#
"
#
1
1
3
2
; b) A = 1
;c) A =
3
4
2
4
[Hint: for a second order di¤erence equation r2
p
1
tr(A)2 4 jAj]
2
(M.1145)
5
4
#
tr(A)r + jAj = 0 ; or r1; r2 =
tr(A)
2
2. Solve the following system of equations and represent solutions in a phase diagram
a)
y1 =
2y2 + 2
(M.1146)
y2 =
3y1 + 6
(M.1147)
y1 =
2y2
(M.1148)
b)
2
y2 = 3y1
6
(M.1149)
y1 = y2
2
(M.1150)
y1
=
4
1
2
(M.1151)
y2 + 2
(M.1152)
c)
y2
d)
y1 =
y2 = y1
y2 + 1
(M.1153)
3. Apply above techniques to
a) Dornbusch model of exchange rate overshooting
e =E e
(M.1154)
r =r +E e
(M.1155)
mD =
(M.1156)
m
p=
369
ar + by
ar + by
(M.1157)
yD
p =
yS
>0
y D = u + v (e
(M.1158)
p)
(M.1159)
yS = y
(M.1160)
b) Markov model of employment and Layo¤
et+1 = (1
) et + ut
ut+1 = et + (1
(M.1161)
) ut
(M.1162)
Reference: Hoy et al. (2001) Mathematics for Economics, MIT Press.
13.3
Tutorial 3: Open Economy DSGE Model
Q1 Consider a standard open economy optimal growth model with
Household problem:
max
U = E0
1
X
t
Ut (Ct ; Lt )
0<
<1
(M.1163)
t=0
Ut (Ct ; Lt ) =
Ct1
1
L1+!
t
1+!
(M.1164)
subject to budget constraint as given by
Wt Lt +
t
+ Pt Kt = Pt Ct + Ptf It + Bt + Tt + 1 + Rt
1
+
t 1
et Ft
1
(M.1165)
Aggregation of di¤erentiated goods from the monopolistically competitive …rms as
Z
Ct =
1
(Cj;t )
1
1
d:j
(M.1166)
0
Z
Pt =
1
(Pj;t )
1
1
d:j
(M.1167)
0
Firm’s problem
M ax
t
= Pt Yt
Wt Lt
PtK Kt
(M.1168)
Subject to the CES production technology and stochastic TFP growth constraints as:
370
1
Yj;t = Zt (1
1 ) Lj;t
Zt = ln Zt
Yt =
Z
1+
1
(Yj;t )
+
(1
1
1 Kj;t
(M.1169)
)Z
(M.1170)
1
d:j
(M.1171)
0
Lt =
Z
1
(Lj;t )
1
1
d:j
(M.1172)
0
Kt =
Z
1
(Kj;t )
1
1
d:j
(M.1173)
0
1. Write the Lagrangian function for constrained dynamic optimisation by households and derive
the Euler equations for optimisation..
2. Write the Lagrangian function for constrained dynamic optimisation by …rms and derived the
demand functions for labour and capital.
3. Solve the model using the projection method and numerical optimisation using MATLAB
routines. Write approximation functions and Euler errors functions
4. Derive impulse responses for shock to the …scal policy.
5. Include Taylor Rule in the above problem for analysis of monetary policy.
6. Conduct dynamic simulations to analyse the impact of demand shocks and supply shocks.
Refer Lim G. C. and McNelis (2008), Computational Macroeconomics for the Open Economy,
MIT Press for this problem.
Q2. Introduce habit formation in consumer preferences and mark-behaviour of …rms as in the
CISM model. Evaluate responses on output, consumption, investment, hours worked, real
wage and real interest rate due to shocks on technology, public spending, investment and
preferences. Study NK_hab.mod for this.
Q3. Consider learning model from Martin Elison of Oxford available at http://users.ox.ac.uk/~exet2581/.
371
13.4
Tutorial 4: Ramsey to RBC Model
Q1 Consider a standard version of Ramsey’s optimal growth model
max
U=
1
X
t
ln(ct )
0<
<1
(M.1174)
t=0
subject to:
Yt = AKt
0<
Kt+1 = Kt (1
<1
(M.1175)
) + It
(M.1176)
Yt = Ct + It
(M.1177)
K0 = K0
(M.1178)
(a) Solve this model for the capital stock, output, consumption and investment in the steady
state.
(b) In what sense is this model di¤erent from the Solow growth model?
(c) How would you solve this model if the technology A is given by a stochastic process?
At+1 = At + "t where "t ~ N (0;
2
):
(d) Financial intermediaries take away a certain fraction of saving. Let
fraction of savings taken away (wasted) by them while (1
represent the
) fraction of saving is
channelled to investment. As such a higher value represents more ine¢ ciency in the
…nancial system. How does
a¤ect the saving and investment and capital accumulation
in this economy?
(e) Study the impacts of capital income taxation in economic growth using Ramsey’s model
of optimal growth. Use GAMS program Captax.gms to compute the optimal growth.
(f) Introduce the labour leisure choice, investment cost, shocks to preferences, investment
and techology in the model as discussed in the lectures. Write the dynare …le and
compare solutions with and without investment costs.
Study RBC_Summer.mod and RBCInvcost.mod to answer this question.
372
13.5
Tutorial 5: Neoclassical Growth with Hamiltonian
Q1 An economy has to decide how much to consume today and how much save and invest to add
into the capital stock that can help produce goods for future consumption. The optimal capital
stock maximises the present value of utility from consumption. Problem of this economy is:
M ax U0 =
Z
T
e
rt
C (t) dt
(M.1179)
0
subject to the production technology:
Q = Q(K)
(M.1180)
Capital accumulation constraint:
Kt =
@K
=Q
@t
C
K
(M.1181)
1. Write the current value Hamiltonian for dynamic optimisation in this model.
2. Discuss …rst order conditions and the terminal conditions required for dynamic optimisation
3. Use a phase diagram to determine the convergence process towards the optimal capital stock.
4. Apply this model for determining the optimal pricing strategy for exhaustible resources (nonrenewable resources) such as oil and gas in a competitive economy.
Q2 Consider a dynamic economy with
Preference:
M ax U0 =
Z
T
e
0
(1
Technology: Yt = At Kt Nt
)
1
t Ct
1
dt
(M.1182)
assume At = 1 and Nt = 1
Capital accumulation:
K t = Yt
Nt C t
Kt
All of the above notations have usual meaning.
1. Write the current value Hamiltonian for this problem.
2. Give four …rst order conditions for the dynamic optimisation in this economy.
3. Characterise the balanced growth path using those conditions for this economy.
4. Discuss the transitional dynamics in space when and when .
373
(M.1183)
Q3. Consider a life time discounted utility (U0 ) maximisation from consumption (C) problem of
a representative household in an economy subject to production technology, domestic and foreign
capital accumulation and market clearing conditions as stated below.
max U0 =
Z
1
t
e
U (Ct ) dt; U (Ct ) =
0
subject to:
Ct1
1
;
>0
(M.1184)
a) technological constraint with domestic capital (K) and foreign capital (F ) and the stock of
technical knowledge (A):
Yt = At Kt Ft1
;
>0
(M.1185)
b) net domestic investment (Ik ) adds to the stock of domestic physical capital (K):
K t = Ik
Kt
1;
0<
<1
(M.1186)
c) net foreign investment (IF ) raises the accumulation of foreign capital (Ft ):
F t = IF
f Ft 1 ;
0<
f
<1
(M.1187)
d) market clears when total output (Yt ) equals total demand for consumption (Ct ) and investment (Ik;t + IF;:t ):
Yt = Ct + Ik;t + IF;:t
(M.1188)
Role of subjective discount factor ( ) ; costant ealsticity of risk aversion ( ), productivity of
capital ( ), depreciation rates
and
f
for the domestic and foreign capital is obvious from the
context they appear in above equations.
1. Formulate the current value Hamiltonian function suitable for in…nite horizon optimisation by
the representative household in this economy. Use shadow prices on domestic investment ( ) ;
foreign investment ( ) and ‡ow (material) balance condition (!) as required to incorporate
all above constraints for dynamic optimisation.
2. Derive the …rst order conditions with respect to domestic capital (K) ; foreign capital (F ) ;
domestic and foreign investments (Ik ; IF ) ; consumption (C) ; shadow prices on domestic investment ( ) ; foreign investment ( ) and ‡ow (material) balance condition (!).
3. Find the growth rate of consumption gc =
C
C
as well as of the relevant shadow prices
that are consistent to above …rst order conditions.
374
and
4. Show that foreign direct investment (IF ) and foreign capital (F ) in aggregate lead to increasing
returns to scale with respect to domestic physical capital (K) in this economy. Can it support
more globalisation than what we have today?
5. Prove that the balanced growth rate of output can be expressed explicitly in terms of growth
rate of technology, capital and its productivity( ) from this analysis.
375
13.6
Tutorial 6: Endogenous growth model
Q1. Consider an endogenous growht model developed in Basu and Bhattarai (2010)
Human capital sector
ht+1 = (1
h )ht
+ AH gt (lHt ht )1
(M.1189)
Final goods sector
yt = AG kt (lG ht )1
(M.1190)
Capital accumulation
kt+1 = (1
k )kt
+ ikt
(M.1191)
Financing education
gt =
t yt
(M.1192)
Social Planners Problem
M ax
1
X
t
ln(ct )
t=0
subject to the resource constraint:
ct + it = (1
t )yt
(M.1193)
and (M.1189) through (M.1191).
1) Prove that along the balanced growth path, the optimal share of public spending in GDP is
given by:
=
1
1
1+
:
1
1
lH
lG
:
(M.1194)
lH
lG
In economies where private schooling e¤orts (lHt ) are higher, it is optimal to tax the goods
sector more.
2) De…ne the gross balanced growth rate as : Solving the …rst order conditions prove that
there are three key balanced growth equations. Based on the …rst order condition for the physical
capital stock we get:
=
[(1
)( yt =kt ) + 1
376
k]
(M.1195)
Based on the …rst order condition for the human capital stock, one gets:
= [1
h
+ AH (1
)
lH (yt =ht ) ]
(M.1196)
Finally, using the human capital technology (E.483), we get a third balanced growth equation:
=1
h
(1
1
lH
AG lG
+ AH
)
(kt =ht )
(M.1197)
3) Prove that the tax rate that maximizes growth also maximizes the long run welfare.
Study results from the GAMS programme LDC.gms and dynare programme BB_Er_…nal.mod.
GAUSS programme growth.g Explain them reading relevant papers.
Q2. The optimization problem facing the social planner of an economy is:
M ax
1
X
t
U (ct )
t=0
s.t.
ct + xt = yt = AGt kt (lG ht )1
ht+1 = (1
h )ht
+ Aht (1
: Resource constraint
lG )ht : Law of motion of human capital
xt = pkt ikt : Current account constraint
ikt = kt+1
(1
k )kt
(M.1198)
(M.1199)
(M.1200)
: Investment
international borrowing constraint.
xt + bt+1 = (1 + r )bt + pk ikt
(M.1201)
The home country faces a borrowing constraint. The amount that it can borrow in the international market is constrained by the current capital stock which means:
bt
kt
Formulate the constrained optimisation form of this problem.
Derive the balanced growth using the standard optimal conditions of this model.
377
(M.1202)
13.7
Tutorial 7: Dynamic Programming
Q1. Consider a version of Brock-Mirman type dynamic programming problem
1. max
U=
1
P
t
ln(ct )
0<
<1
t=0
subject to
Kt+1 + Ct = AKt
0<
<1
(a) what are the control and state variables in this model and why?
(b) Explain the meaning of the value function (Bellman equation) and the policy functions
of this problem
V1 (K) = ln C + V0 (K):
(c) Assume V0 (K) = 0: Demonstrate a recursive solution method of this problem using four
iterations of the policy and value functions.
(d) Use limit theorem to …nd the explicit solution of the value function.
(e) Introduce a stochastic disturbance term for the state variable and show how it can be
solved.
Q2. Consider a money in utility function model
max W =
1
X
t
U (ct ; mt )
(M.1203)
t=0
Subject to:
Yt = F (Kt ; Nt )
(M.1204)
Under constant returns to scale yt = f (kt ) where yt =
Yt
Nt
and kt =
Kt
Nt
.
Cash in advance constraint
Yt +
t Nt
+ (1
) Kt
1
+
Mt
Pt
1
= Ct + Kt +
1
Mt
Pt
where Yt is output, Pt price of goods, Ct consumption, Kt+1 is capital stock,
for each individual, Mt money, Yt , output, Nt employment and
(M.1205)
t
is net transfer
is the rate of depreciation of
capital.
1) set up the constrained optimisation functions and derive the …rst order conditions of maximisation.
378
2) Solve the model for its steady state. Express consumption, output, capital stock and money
in terms of model parameters.
3) Characterise the transitional dynamics if the economy is not in the steady state in the
beginning.
Q3. Consider the problem of a cash in advance economy given below.
max
1
X
t
[U (Ct )
V (Nt )]
(M.1206)
t=0
Subject to:
Yt = zNt
(M.1207)
Pt Ct + qt Bt+1 + Pt st Xt+1 + Pt Tt = Mt + Bt + Pt Xt
(M.1208)
Cash in advance constraint
where Pt Ct is consumption expenditure Pt price of goods, Ct consumption, Bt+1 is the amount
of nominal bonds qt is the price of nominal bonds, Xt+1 real bonds st prices of real bonds, Tt lump
sum tax payment, Mt money.
Budget constraint of the consumer:
Pt Ct + qt Bt+1 + Pt st Xt+1 + Pt Tt + Mt+1 = Mt + Bt + Pt Xt + Pt zNt
(M.1209)
Government’s budget constraint:
M t+1
Mt =
Assuming a constant rate of money growth
Pt Tt
(M.1210)
and M t+1 = (1 + ) M t
Mt =
Pt Tt
(M.1211)
1) Write the appropriate Lagrangian for constrained optimisation and derive the optimal …rst
order conditions
2) Solve for the steady state of the model
3) what are the values of price of goods, bonds, stocks in the steady state? What are the
corresponding values of consumption, labour supply, money and the interest rate.
Q4. Show how the FDI promotes economic growth solving the in…nite horizon utility maximisation problem subject technology, domestic and foreign capital accumulation and market clearing
conditions can be written as:
379
max U0 =
Z
1
e
t
U (Ct ) dt ; U (Ct ) =
0
Subject to
Ct1
1
Yt = At Kt Ft1
(M.1212)
(M.1213)
Net domestic investment that causes a change in physical capital:
K t = Ik
Kt
1
(M.1214)
Dynamic Optimisation
Net foreign investment similarly causes accumulation of foreign capital:
F t = IF
f Ft 1
(M.1215)
Market clearing requires in each period requires that total output should equal total demand
Yt = Ct + Ik;t + IF;:t
380
(M.1216)
13.8
Tutorial 8: Equilibrium Unemployment Model
Consider 2010 Nobel laureate Pissarides (2000) model of equilibrium unemployment.
Matching function aggregates vacancies and unemployment with job creation as:
M = M (V; U ) = V U (1
)
(M.1217)
M denote the number of matching of vacancies and job seekers, V is number of vacancies and
U the number of unemployed,
is the parameter between zero and one
Nash-product of the bargaining game over the di¤erence between the earnings from work (W )
rather than in being unemployed (U ) and earnings to …rms from …lled and vacant jobs.
(Wi
U ) (Ji
1
V)
(M.1218)
Symmetric solution of this satis…es joint pro…t maximisation condition for worker as:
(Wi
Let parameter
U) =
(Ji + Wi
V
U)
be the ratio of vacancy to job seeking workers
vacancy be given then by f ( ) and not …lling it by 1
unemployed worker is q ( ) @t and the not …nding is 1
(M.1219)
=
V
U;
the probability …lling a
f ( ) ; probability of …nding a job by an
q ( ) @t; job creation occurs when matching
takes place between …rms with vacancies and workers seeking the job.
a) Derive the equilibrium unemployment in the system.
b) Derive optimal job creation or (demand for labour curve)
c) Establish links between the reservation wage (z) the price of product p and costing of hiring
( c).
381
13.9
Tutorial 9: Money, In‡ation, Business Cycle and OLG Model
Q1. Using classical quantity theory of money prove that in‡ation is always a monetary phenomenon
as stated by Friedman.
Q2. Prove that natural rate of output is independent of in‡ation.
Q3. Derive business cycle in the AD-AS model given below.
Aggregate demand as a function of lagged output, growth rate of money supply and in‡ation:
Yt = a1 Yt
1
+ a2 (mt
t)
+ dt ;
a1 > 0; a2 > 0
(M.1220)
Aggregate supply as a function of core in‡ation and output gap and supply shock
t
=
t
+ b1 Yt
Y t + st
backward and forward looking aspects of core in‡ation (with 0 <
t
=
t
+ (1
)
(M.1221)
< 1):
t 1
(M.1222)
These three equations are enough to generate business cycles in output and in‡ation.
Q4. Assume an economy, inhibited by two generations, young and old. Young ones work, earn
, consume and save and old ones stay at home in retirement and consume out of their past savings.
Economy is continuum of generations such as gi;t where i = 1; 2; ::::N refers to the generations and
t = 1; 2; ::::T refers to the time period. Each agent is assumed to live for two periods - as a young
worker …rst and then as an old retiree. For instance, person in generation 1, g1;1 is born and young
in t = 1 and becomes old in t = 2 and is succeeded by g2;1 who is young in t = 2 , becomes old one in
period t = 3 and dies at the end of that period. In this manner new generations continuously replace
the old generations but the economy continues without any interruption with these two types of
people forever.Behavior of each type is similar to their types in earlier periods; young ones work,
earn, save part of their income and make families and get children and old ones retire and consume
their savings and leave some bequest to their children. The simplest version of this model can be
explained in …fteen equations as following (see Samuelson, 1958; Auerbach and Kotliko¤,1987 for
details).
Production is function of capital, labour and technology and is subject to constant return to
scale with here
+
= 1:
Yt = AKt Lt
In terms of income per e¤ective worker:
382
(M.1223)
yt = Ak
(M.1224)
Market clears in each period, whatever is produced is either consumed or invested.
Yt = Ct + It
(M.1225)
Aggregate consumption is total of the consumption of young and old
Ct = N cyt + N cot
Agents consume
(M.1226)
fraction of their income in period 1
cyt = wt
Young save (1
(M.1227)
) share of wt and invest it in assets for consumption at the old age:
at = (1
) wt
(M.1228)
a. Labour income and capital income taxes distort the …rst order conditions (1
AKt
1
Lt and (1
l )Wt
k )rt
=
= AKt Lt . Find the output, consumption, saving, wages and interest
rate in the steady state for this economy with following parameters
Table 99: Parameters of the Two Period OLG Model
K0 k0 N
l
k
Parameter
Value
0.5
0.5
0.5
300
3
100
0.1
0.1
Fill up the following table with solution of the model.
Table 100: Results of the Two Period OLG Model
k K Y w r cy c0 S = I
Variables
Solution without tax
Initial condition
Steady State
Solution with tax
Initial condition
Steady State
383
I
13.10
Tutorial 10: Small Open Economy Model
The representative household for a country receives utility from consuming both goods X1 and X2
given by a Cobb-Douglas utility function as:
U = X1 1 X21
(M.1229)
1
Technology of production of goods Y1 and Y2 are respectively
Y1 = L1 K11
(M.1230)
Y2 = L2 K21
(M.1231)
Resources of …rm 1 and 2 are
C1 = w1 L1 + r1 K1
(M.1232)
C2 = w2 L2 + r2 K2
(M.1233)
Households receive income from labour and capital, from transfers (T R) and net borrowing (B) as:
I = w1 L1 + r1 K1 + w2 L2 + r2 K2 + T R + B
(M.1234)
Market clearing conditions in goods market are:
X1 = Y1
G1
(EX1
IM P1 )
(M.1235)
X2 = Y2
G2
(EX2
IM P2 )
(M.1236)
Labour market clearing implies:
L1 + L2 = L
(M.1237)
K1 + K2 = K
(M.1238)
Capital market clearing implies:
384
Aggregate volume of output:
13.10.1
P1 Y1 + P2 Y2 = P:Y
(M.1239)
P:Y = M S:V
(M.1240)
Monetary Sector
Quantity theory of money implies
where P is price level, Y national income, M S money supply and V the velocity of circulation.
Initial reserve (R) of the banking system constitutes of currency (C) and initial demand deposit
(D0)
R = C + D0
Currency in circulation is
(M.1241)
fraction of total reserves
C = :R
Initial deposit is the remaining (1
(M.1242)
) of initial reserve
D0 = (1
) :R
(M.1243)
Total deposit (T D) is inversely related to the required reserve (rr) ratio
D0
(M.1244)
rr
Aggregate money supply in the economy constitutes of currency in circulation plus the total deposit
TD =
MS = C + TD
13.10.2
(M.1245)
Government Sector
Government collects revenue (RV ) from direct taxes on capital (tr1: , tr2: ), labour (tw1: ; tw2: ) and
indirect tax on commodities (t1: ; t2: ) as:
RV = t1: P1: X1 + t2: P2: X2 + tr1: r1: K1 + tw1: w1 :L1 + tr2: r2 K2 + tw2: w2 :L2 + T R + B
(M.1246)
Aggregate government expenditure (G) is spent in public consumption from both sectors (G1 ; G2 )
385
G = G1 + G2
(M.1247)
Government expenditure on sector 1 and 2 goods are g1 and g2 fractions of its revenue:
G1 = g1 :RV
(M.1248)
G2 = g2 :RV
(M.1249)
Budget de…cit is the di¤erence between government spending and the revenue:
B=G
RV
(M.1250)
External sector
Exports from sector 1 and 2 ,EX1 and EX2 and imports e1 and e2 .fractions of sectoral output
as:
EX1 = e1 :Y1
(M.1251)
EX2 = e2 :Y2
(M.1252)
Imports by sector 1 and 2, IM P1 and IM P2 are m1 and m2 .fractions of sectoral output as:
IM P1 = m1 :Y1
(M.1253)
IM P2 = m2 :Y2
(M.1254)
The real exchange rate is given by the ratio of total value of exports to the total value of imports:
P1 :EX1 + P2: EX2
P M1 :IM P1 + P M2: IM P2
Solve this model numerically for plausible values of parameters.
ER =
(M.1255)
Do sensitivity tests with respect of changes in preferences, technology, …scal and monetary policy
variables.
See Bhattarai (2011); study results to twosector_hh_gov_saa.gms Do interesting plots to explain results.
Q2. Consider a global economy model that consists of N number of countries. Let i be index for
home country and j for foreign countries. Household utility function in country i contains
home produced consumption goods (Ci;t ), imported goods (Mi;t ) and leisure (li;t ) :
With the Cobb-Douglas production function, the household problem can be stated as:
386
max U0i =
1
X
t
(M.1256)
Ci;t Mi;t li;t
t=0
subject to its intertemporal budget constraint :
1
X
t=0
1
X
[Pi;t (1 + tci ) Ci;t + Pj;t (1 + tmi ) Mi;t + wi;t (1
wi;t (1
twi ) Li;t + ri;t (1
twi ) li;t ]
tki ) Ki;t
(M.1257)
t=0
Notations: prices Pi;t and Pj;t , interest rates, ri;t and rj;t wage rates wi;t and wj;t , taxes in
consumption tci;t and tcj;t in imports tmi;t and tmj;t , labour income taxes twi;t and twj;t ;tax on
capital income tri;t and ttj;t ; parameters satisfy 0 < ; ;
< 1; 0 <
< 1;
+
+
= 1).
The pro…t maximisation problem of the representative …rm for each period is given by the
di¤erence in its revenue and cost as:
max
i;t
= Pi;t Yi;t
ri;t Ki;t
wi;t LSi;t
(M.1258)
and is subject to technology and capital accumulation constraint as:
(1
t)
Yi;t = Ki;tt Li;t
Ii;t = Ki;t
(1
) Ki;t
(M.1259)
1
(M.1260)
While the labour and capital inputs Li;t , Kj;t , Ki;t and Lj;t determine the levels of output, the
accumulation of capital stock depends on investment, Ii;t and Ij;t .
Revenue from the taxes and tari¤s could be just enough, or more or less than the government
expenditure:
Ri;t = tci Pi;t Ci;t + tmi Pj;t Mi;t + twi wi;t LSi;t + tki ri Ki;t ? Gi;t
(M.1261)
Any discrepancy between the government revenue, Ri;t and Rj;t and the government expenditures Gi;t and Gj;t represents the level of de…cit implied by the existing measures of …scal policy
with consequences on markets for public debt and gilts.
Market clear for goods and labour each period as:
387
Yi;t = Ci;t + Xi;t + Gi;t
(M.1262)
Li;t = LSi;t + li;t
(M.1263)
Labour inputs LSi;t and LSj;t and demand for leisure li;t and lj;t have to balance to time
endowments Li;t and Lj;t .
Domestic and international balance requires that net saving (Si;t
(Xi;t
Ii;t ) o¤sets net exports
Mi;t ) as:
(Si;t
Ii;t ) + (Xi;t
Mi;t ) = 0
(M.1264)
The levels of savings are Si;t and Sj;t and exports Xi;t and Xj;t relate to optimal choice of
consumption Ci;t and Cj;t , imports Mi;t and Mj;t and the demand for leisure li;t and lj;t .
The external sector may balance either period by period or intertemporally over the model
horizon as:
Xi;t = Mi;t
1
X
t
(Pi;t Xi;t
Pj;t Mi;t ) =
(M.1265)
1
X
t
(T Bi;t )
(M.1266)
While the net saving a¤ects the …nancial markets the net external current account balance has
implications of in‡ows or out‡ows of capital from or to the international capital markets.
Prices from the inter-temporal arbitrage condition:
Pi;t =
Pi;t+1
1 + ri;t
(M.1267)
Bilateral exchange rate between economies i and j is given by:
Ei;t =
Pi;t
Pj;t
(M.1268)
The exchange rates Ei;t and Ej;t in this economy is given by the ratio of domestic to foreign prices
and depend on behavioral and policy instruments including tci;t ,tcj;t ,tmi;t ,tmj;t ,twi;t ,twj;t ; tri;t and
trj;t .
388
1. De…ne the competitive equilibrium for this economy.
2. Derive or characterise the …rst order conditions for households optimisation with
respect to Ci;t ,Mj;t ,li;t ,Cj;t ,Mi;t and lj;t and shadow prices
i;t
and
i;t+1 .
3. Obtain the …rst order conditions for …rms to derive the input demands LSi;t , Kj;t
, Ki;t , and LSj;t :
4. Find the equilibrium exchange rate that regulates ‡ows of imports and exports
between these two economies. Relate analysis of model to the international …nancial crisis that shook the global economy in 2008.
see.Bhattarai and Mallick (2013). Do theoretical predictions from the dynamic model.
389
13.11
Tutorial 11: New Keynesian and Newclassical Macro Models
Q1. Consider the New Keynesian model given below
Problem of Houheholds i
max
E
Subject to:
"
1
X
k
Mit+k+1
P t+k
U (Cit+k ) + V
t=0
Cit =
Z
0
1
1
1
Cijt dj
;
Pt =
Z
1
0
and the budgent constraint
Z
Q (Nit+k ) =
t
#
(M.1269)
1
1
Pjt
dj
1
(M.1270)
L
Pjt Cijt + Mit+1 + Bit+1 = Wt Nit + (1 + it ) Bit + Mit +
it
+ Xit
(M.1271)
0
Firms’problem assuming a linear production technology:
Yjt = Zt Njt
Cijt =
Pjt
Pt
(M.1272)
Cit
Firms take wage rates as given and set prices a la Calvo with
(M.1273)
probability of chaning it every
period. Then Yjt is solution to the …rms pro…t mamimization problem
max E
kU
0
(Ct+1 )
(1
U 0 (Ct )
k
)
Pjt
Yjt+k
P t+k
Wt+K Yjt+k
P t+k Zt+1
=
t
(M.1274)
Subject to:
Pjt
P t+k
Yjt+k =
Yt+k
(M.1275)
1. Write …rst order conditions for optimisation of this model.
2. Solve for the steady state price level, employment and output.
3. Prove that volatility of output is generated from the technological shock.
Q2. Solve the real business cycle model
Output
1
Yt = Kt (At Lt )
0<
390
<1
(M.1276)
Capital Accumulation
Kt = (1
) Kt
1
+ It
0<
<1
(M.1277)
Market Clearing
Yt = Ct + It
(M.1278)
Show how the optimal output and cosumption could be derived as an autoregressive process.
13.12
Tutorial 12: Real Business Cycle Model
Kydland and Prescott won Nobel prize of 2004 for developing a real business cycle model to explain
that explains long run growth and short run ‡uctuations in output, employment and other macro
variables as equilibrium phenomenon. For this consider the consumer and producer maximisation
problems as following
max
ln (Ct ) + ln (Ct+1 )
(M.1279)
Ct +
Wt+1
Ct+1
= Wt +
(1 + rt )
(1 + rt )
(M.1280)
subject to
Producer’s problem
max
t
= Pt Yt
wt Lt
rt Kt
(M.1281)
subject to
Yt = zt Kt L1t
(M.1282)
zt = exp(e)
(M.1283)
where e ~ N (0; 1) :
1. show that positive technological shock raises output, consumption and interest rates in period
t + 1.
Ct+1
Ct
=
(1 + rbt+1 ) and rbt+1 = zt Kt
1
L1t
2. Illustrate how the negative technological shocks lead to fall in output, consumption and
interest rates in period t + 1.
391
3. Explain how series of technological shocks are responsible for short run ‡uctuations with long
run growth.
4. Find the percentage standard deviation of growth rate of output, consumption, investment
and hours worked
Table 101: Percentage standard deviation of macro variables
GDP
Consumption
Investment
Hours worked
% standard deviation
Relative % standard deviation
Table 102: Lag, contemporaneous and lead correlations among macro variables
GDPt ; xt
1
GDPt ; xt
GDPt ; xt+1
GDP
Consumption
Investment
Hours worked
5. Test the validity of the quantity theory of money MV = PY with appropriate data.
392
13.13
Tutorial 13: Global Economy
1. Consider the macroeconomic system in two interdependent economies, i.e. Europe and the
ROW
Economy 1
Y1 = C1 + I1 + G1 + N X1
C1 = a1 + b1 (Y1
T1 )
(M.1284)
(M.1285)
I1 = k1 + d1 r1
(M.1286)
N X1 = Y2
(M.1287)
Y2 = C2 + I2 + G2 + N X2
(M.1288)
Economy 2
C2 = a2 + b2 (Y2
T2 )
(M.1289)
I2 = k2 + d2 r2
(M.1290)
N X2 = Y1
(M.1291)
(a) Solve for the national income of both economies simultaneously.
(b) Determine how public spending of economy 1 would impact economy 2.
(c) How would the monetary policy in one economy a¤ect the monetary policy of another
economy?
393
13.14
Tutorial 14: A Study on Housing Markets
The major aim of this exercise is to understand the microeconomic aspects of housing markets in
European countries required for constructing a macro model with housing assets. Model is simple
enough to organise stylized facts of the housing market.
A household consumes housing (H) and other goods (C), his problems is
U = H C1
max
(M.1292)
subject to
Y = C + (r + ) PH H
(M.1293)
Substituting the constraint in the objective function
U = H (Y
(r + ) PH H)
1
(M.1294)
Get the …rst order conditions with respect to H to get housing demand as
In equity premium literature
H=
Y
(r + ) PH
(M.1295)
Demand for housing increases with an increase in income (Y ) and decreases with an increase in
the real interest rate (r) and prices of housing (PH ). It also is lower for higher maintenance costs
( ).
Accumulation of housing stock is given by
Ht+1 = (1
) Ht + ItH
(M.1296)
Additional supply of houses owe to the investment in the housing market.
ItH = AX ;
0<
<1
(M.1297)
Housing sector production input (X) includes labour (L) and material inputs (Q) and input
aggregations are given by
L = aX
394
(M.1298)
Q = bX
(M.1299)
P = aW + bP Q
(M.1300)
Therefore problem of the housing supply company is to do determine optimal investment in
housing taking account of these input an output prices as given in the market:
= PH I
Di¤erentiating
PH
P
P
IH
A
1
(M.1301)
with respect to I H gives the demand function for investment as:
ItH =
Here
H
A1
1
1
PH
P
1
(M.1302)
can be considered as the Tobin’s q of housing.
Substitute earlier derivation PH =
ItH =
Y
(r+ )H
1
A1
1
in it
Y
1
(r + ) H P
1
(M.1303)
Investment in housing is driven positively by income but negatively by the interest rate, maintenance cost and general price level. Technology of housing construction (A; ) and preference of
household ( ) matter.
Things to be done:
1. get the data on Y; P; PH ; IH ; r; ; H and a A for a number countries/regions of
interest from 1980s onward. Annual or quarterly data is …ne. Construct a panel data.
Make few tables showing the stylized facts of the housing market.
2.
Estimate the parameters
(You may need to compute
3.
;
and A using …xed or random e¤ect panel model
parametricallyin case data is not available on maintenance.)
Interpret the results whether they makes empirical sense; put results in nice
tables.
4. Do counter-factual simulations on demand supply imbalances bootstrapping on ;
5. Do a short literature review on housing market and relate the conclusions of this study to
those in the literature.
6. Explain way forward for putting housing model in a New Keynesian dynamic stochastic
general equilibrium model
395
13.15
Tutorial 15: Rational Expectation
Q1. Consider an AS-AD model with following equations:
Real interest rate (Fisher equation)
rt = ipt +
e
t+1
t
(M.1304)
Aggregate demand:
yt
y=
1
(gt
g)
2
(rt
vt v N 0;
r) + vt ;
2
v
;r = r +
(M.1305)
y)
(M.1306)
Interest rate rule
ipt = r +
e
t+1
+ h(
t
) + b (yt
Aggregate supply (price formation):
t
=
e
t+1
y) + st ; st v N 0;
+ (yt
2
s
(M.1307)
In‡ation expectation
e
t
=
(M.1308)
t 1
1. Derive aggregate demand (AD) and aggregate supply (AS) from above equations.
2. Find expressions for the deviation of in‡ation and output from the steady state bt =
and ybt = yt
t
y when there are no further shocks to the AD or AS zt = 0 and st = 0.
3. Find the time path for output and in‡ation given their initial values yb0 and b0 .
4. Calculate time taken for yt and
bt = 0) when
= 0:742;
t
to converge to the steady state y and
= 0:3 ;
2
= 5:76; b = 0:5
( ybt = 0 and
Q2. Expected in‡ation next period (t+1) based on information at period t depends on di¤erences
on expected and actual prices
Et
t+1
=Et pt
pt
(M.1309)
a0 > 0 a1 > 0
(M.1310)
Demand
ytd = a0 + a1 (mt
pt ) + t ;
396
yts = yn + b1 pt
Et
pt + vt
1
a1 > 0
(M.1311)
Demands equals supply in equilibrium
ytd = yts = yt
t
N 0;
2
t
(M.1312)
N 0;
2
(M.1313)
1. Solve this model for output and prices.
2. Show that only positive shocks in demand or supply in‡uence the level of prices or output.
Q3. Consider a model with backward looking expectation given as following
Aggregate demand:
yt
y = vt
r) ; vt v N 0;
(rt
2
v
(M.1314)
Real interest rate:
e
t+1
rt = it
(M.1315)
Aggregate supply (price formation):
t
=
e
t+1
+ (yt
y) + st ; vt v N 0;
2
s
(M.1316)
Monitory policy rule:
it = r +
e
t+1
+ h(
) + b (yt
t
y)
(M.1317)
Expectation
e
t+1
Show that it is important to have 0 <
=
t 1
(M.1318)
< 1 for the convergence of prices to the steady state
in this model.
Q4. This exercise relates to adaptive and partial adjustment and combinations of these two.
First consider a adaptive expectation model:
Let yt be the growth rate and xt be the optimal long run equilibrium interest rate
397
yt = b0 + b1 xt + t ;
(M.1319)
Adaptive expectation
xt
xt
1
=
xt
xt
1
(M.1320)
Prove that this results into an autoregressive process of order 1 for yt .
Then consider the partial adjustment model as:
Let desired long run growth rate of economy be yt and that depend on a number of explanatory
variables as
yt =
0
+
1
xt +
(M.1321)
t
Partial adjustment hypothesis emplies
yt
yt
1
=
(yt
yt
1)
(M.1322)
Derive the partial adjustment model from using these two equations.
Combine adaptive and partial adjustment elements of above two models to derive a model when
both yt and xt are not observable as in
yt =
0
+
1
398
xt +
t
(M.1323)
13.16
Tutorial 16: Overlapping Generation Model: Impact of Taxes on
Growth
here
+
Yt = AKt Lt
(M.1324)
yt = Ak
(M.1325)
Yt = Ct + It
(M.1326)
= 1:
Percapita income
Market clearing condition
Aggregate consumption is total of the young and old
Ct = N cyt + N cot
(M.1327)
Wage income is given by the labour share in production
Wt = (1
) AKt Lt
(M.1328)
Interest rate equals the marginal product of capital
rt = AKt
Agents consume
1
Lt
(M.1329)
fraction of their income currently
cyt = wt
Saving equation (1
(M.1330)
) share of wt
at = (1
cot = at (1 + rt ) = (1
) wt
) wt (1 + rt )
(M.1331)
(M.1332)
Law of accumulation of capital stock is
Kt+1 = Kt + It
From ?? and L.1066
399
(M.1333)
Ct = AKt Lt
It
(M.1334)
Kt+1 + Kt
(M.1335)
Then substituting ?? and L.1069
N cyt + N cot = AKt Lt
Further substituting ?? and ?? for consumption of young and old
N wt + N (1
) wt (1 + rt ) = AKt Lt
Kt+1 + Kt
(M.1336)
) AKt Lt + (1
) (1
) AKt Lt (1 + rt )
(M.1337)
) AKt Lt
) (1
) AKt Lt (1 + rt )
(M.1338)
substituting L.1070
AKt Lt
Kt+1 + Kt = (1
By further re-arrangement
Kt+1
Kt = AKt Lt
(1
(1
This is a …rst order di¤erential equation in Kt and can be solved iteratively using a numerical
method starting from initial condition where K0 is given. System converges to the steady state
when Kt+1 = Kt
Solve this model using numerical method in Excel with the following set of parameters.
Table 103: Parameters of the OLG Model
Parameters
K0 k0 N
Values
0.5
0.5
0.5
300
3
100
Impact of taxes on economic growth can be simulated by using the tax induced …rst order
conditions in the above model.
400
13.17
Other Problems
13.17.1
Problem 1: Keynesian Model
1. Consider the Keynesian model with the production function as following
Y = F (K; N )
Fk > 0; FN > 0; Fkk < 0; FN N < 0:
(M.1339)
C = C (Y
(M.1340)
Consumption function
Labour demand
T)
W
= FN (N; K)
P
(M.1341)
1. Labour supply
W = W0 + W (N )
W (N ) =
Z
(M.1342)
0 for N 5 N
(M.1343)
+for N > N
money market equilibrium conditions:
M
= M (Y; r)
P
My > 0; Mr < 0
(M.1344)
Equilibrium condition
Y =C +I +G+X
IM
(M.1345)
(a) derive the income tax multiplier for this model and determine its sign.
(b) derive the income tax multiplier for this model when the money demand depends upon
the disposable income and determine its sign.
(c) Linearise the model for comparative static analysis and determine the employment and
output impacts of changes in the government spending, tax rates the …xed nominal wage
rate.
2. (a) Multiplier accelerator model of Samuelson (1939) applies the second order di¤erence
equation for analysis of the business cycle.
Solve the complex root case of this model (
2
401
2
(1 + ) < 4
):
Yt = Ct + It + Gt
Ct = Yt
It =
(Ct
(M.1346)
(M.1347)
1
Ct
1)
(M.1348)
[Hint: use De Moivre and pythagorian theorems.]
Comment on applicability of this model to analyse macroeconomic event in the current context.
3. Again inter temporal optimisation by each involves maximising utility subject to its life time
budget constraint.
max
U (C1i ; C2i ; C3i ) = ln C1i +
i i
2 C2
+
i i
3 C3
i = A; B; C
(M.1349)
(a) subject to budget constraints while young, adult and old as following:
C1i + bi1 = w1i
(M.1350)
C2i + bi2 = bi1 (1 + r) + w2i
(M.1351)
C3i = bi2 (1 + r) + w3i
(M.1352)
where C1i ,C2i and C3i are consumptions for periods 1, 2 and 3 for type i agent and
i
3 are
i
2 and
subjective discount factors for period 2 and 3 consumptions with their values
between 0 and 1. Endowment of agent i for time t is given by wti with endowments .
Each household is allowed to borrow and lend at the interest rate r.
Determine the equilibrium rate of interest in equilibrium that solves the problems of each
of these households and is consistent to the resource constraint of the economy.
13.17.2
Problem 2: Stability Analysis
1. Solve the following system of di¤erential equations
where a) A =
"
4
4
y = Ay
#
"
#
"
1
1
3
2
; b) A = 1
;c) A =
4
3
2
4
[Hint: for a second order di¤erence equation r2
p
1
tr(A)2 4 jAj]
2
402
(M.1353)
5
4
#
tr(A)r + jAj = 0 ; or r1; r2 =
tr(A)
2
2. Solve the following system of equations and represent solutions in a phase diagram
a)
y1 =
2y2 + 2
(M.1354)
y2 =
3y1 + 6
(M.1355)
y1 =
2y2
(M.1356)
b)
2
y2 = 3y1
6
(M.1357)
y1 = y2
2
(M.1358)
y1
4
1
2
(M.1359)
y2 + 2
(M.1360)
c)
y2 =
d)
y1 =
y2 = y1
y2 + 1
(M.1361)
3. Apply above techniques to
a) Dornbusch model of exchange rate overshooting
e =E e
(M.1362)
r =r +E e
(M.1363)
mD =
(M.1364)
m
p =
yD
ar + by
p=
ar + by
(M.1365)
yS
>0
(M.1366)
y D = u + v (e
p)
yS = y
(M.1367)
(M.1368)
b) Markov model of employment and Layo¤
et+1 = (1
) et + ut
ut+1 = et + (1
403
) ut
(M.1369)
(M.1370)
1. c) Model of price war
yt+1 = yt
(yt
xt )
(M.1371)
xt+1 = xt
(xt
yt )
(M.1372)
d) Entry adjustment model
qD
p =
p =
(a + bp
N =
(p
qS
(M.1373)
mN )
>0
c)
>0
(M.1374)
(M.1375)
Reference: Hoy et al. (2001) Mathematics for Economics, MIT Press.
Q3. Consider a standard version of Ramsey’s optimal growth model
max
U=
1
X
t
ln(Ct )
0<
<1
(M.1376)
t=0
subject to
a) production technology:
Yt = AKt
0<
<1
(M.1377)
b) capital accumulation:
Kt+1 = Kt (1
) + It
(M.1378)
c) market clearing:
Yt = Ct + It
(M.1379)
K (0) = K0
(M.1380)
d) initial condition:
1. Solve this model for the capital stock, output, consumption and investment in the steady
state.
2. Characterise the transitional dynamics of the model and explain in what sense this model is
di¤erent from the Solow growth model.
3. How would you solve this model if the technology A is given by a stochastic process At+1 =
At + "t where "t
N (0;
2
)?
404
4. Financial intermediaries take away a certain fraction of saving. Let
of savings taken away (wasted) by them while (1
investment. As such a higher value of
How does
represent the fraction
) fraction of saving is channelled into
represents more ine¢ ciency in the …nancial system.
a¤ect the saving and investment and capital accumulation in this economy?
5. Suggest modi…cation in the Ramsey model to study the impacts of capital income taxation
in economic growth.
6. Study the impacts of capital income taxation in economic growth using Ramsey’s model of
optimal growth. Use GAMS program Captax.gms to compute the optimal growth.
13.17.3
Problem 3: Neoclassical Growth with Hamiltonian
1. An economy has to decide how much to consume today and how much save and invest to add
into the capital stock that can help produce goods for future consumption. The optimal capital
stock maximises the present value of utility from consumption. Problem of this economy is:
Z
M ax U0 =
T
e
rt
C (t) dt
(M.1381)
0
subject to the production technology:
Q = Q(K)
(M.1382)
Capital accumulation constraint:
Kt =
@K
=Q
@t
C
K
(M.1383)
1. (a) Write the current value Hamiltonian for dynamic optimisation in this model.
(b) Discuss …rst order conditions and the terminal conditions required for dynamic optimisation
(c) Use a phase diagram to determine the convergence process towards the optimal capital
stock.
(d) Apply this model for determining the optimal pricing strategy for exhaustible resources
(non-renewable resources) such as oil and gas in a competitive economy.
2. Solve for the steady state and characterise the transitional dynamics
in the
following neo-classical growth model
max
Ct
Uo =
Z
1
t=0
405
e
1
t Ct
1
dt
(M.1384)
Subject to technology constraint
Yt = At Kt Nt1
;
0<
<1
(M.1385)
Capital accumulation process
K t = Yt
Nt Ct
Kt
(M.1386)
It = St
(M.1387)
Market clearing:
Yt = Ct + St
Initial (boundary) condition:
K (0) = Ko
(M.1388)
Here Uo life time utility of the consumer, Ct is consumption, Yt output, Kt capital stock, Nt
labour input, K t change in capital stock each period;
substitution, and
is discount parameter
eslatisicity of
rate of depreciation. Assume At = 1 and Nt = 1 for simplicity.
1. Set up the current value Hamiltonian function for this problem.
2. State four …rst order conditions for optimisation and write meanings of each.
3. Compute the steady state of the model.
4. Show the transitional dynamics of the shadow price and the capital stock.
5. Represent the saddle path solution in a set of nicely labelled diagrams in (K; ) space where
is the shadow price of capital (K)
13.17.4
Problem 4: Dynamic Programming
1. Consider a version of Brock-Mirman type dynamic programming problem
1
P
t
max U =
ln(ct )
0< <1
t=0
subject to
Kt+1 + Ct = AKt
0<
<1
(a) what are the control and state variables in this model and why?
406
(b) Explain the meaning of the value function (Bellman equation) and the policy functions
of this problem
V1 (K) = ln C + V0 (K):
(c) Assume V0 (K) = 0: Demonstrate a recursive solution method of this problem using four
iterations of the policy and value functions.
(d) Use limit theorem to …nd the explicit solution of the value function.
(e) Introduce a stochastic disturbance term for the state variable and show how it can be
solved.
2. The optimization problem facing the social planner of an economy is:
M ax
1
X
t
U (ct )
t=0
s.t.
ct + xt = yt = AGt kt (lG ht )1
ht+1 = (1
h )ht
+ Aht (1
: Resource constraint
lG )ht : Law of motion of human capital
xt = pkt ikt : Current account constraint
ikt = kt+1
(1
k )kt
(M.1389)
(M.1390)
(M.1391)
: Investment
Formulate the constrained optimisation form of this problem.
Derive the balanced growth using the standard optimal conditions of this model.
13.17.5
Problem 5: Money in utility (MIU) and cash in advance (CIA) models
I. Consider a money in utility function model
max W =
1
X
t
U (ct ; mt )
(M.1392)
t=0
Subject to:
Yt = F (Kt ; Nt )
407
(M.1393)
Under constant returns to scale yt = f (kt ) where yt =
Yt
Nt
and kt =
Kt
Nt
.
Cash in advance constraint
Yt +
t Nt
+ (1
) Kt
1
+
Mt
Pt
1
= Ct + Kt +
1
Mt
Pt
where Yt is output, Pt price of goods, Ct consumption, Kt+1 is capital stock,
for each individual, Mt money, Yt , output, Nt employment and
(M.1394)
t
is net transfer
is the rate of depreciation of
capital.
1) set up the constrained optimisation functions and derive the …rst order conditions of maximisation.
2) Solve the model for its steady state. Express consumption, output, capital stock and money
in terms of model parameters.
3) Characterise the transitional dynamics if the economy is not in the steady state in the
beginning.
II. Consider the cash in advance model and characterise the transitional dynamics of this economy. Problem of the household now becomes:
max
1
X
t
[U (Ct )
V (Nt )]
(M.1395)
t=0
a) Subject to the technology constraint:
Yt = zNt
(M.1396)
Pt Ct + qt Bt+1 + Pt st Xt+1 + Pt Tt = Mt + Bt + Pt Xt
(M.1397)
b) Cash in advance constraint:
where Pt Ct is consumption expenditure Pt price of goods, Ct consumption, Bt+1 is the amount
of nominal bonds qt is the price of nominal bonds, Xt+1 real bonds, st prices of real bonds, Tt lump
sum tax payment, Mt money. Budget constraint of the consumer:
Pt Ct + qt Bt+1 + Pt st Xt+1 + Pt Tt + Mt+1 = Mt + Bt + Pt Xt + Pt zNt
(M.1398)
c) Government’s budget constraint:
M t+1
Assuming a constant rate of money growth
Mt =
Pt Tt
and M t+1 = (1 + ) M t
408
(M.1399)
Mt =
Pt Tt
(M.1400)
The representative agent chooses Ct , Nt ,bt+1 ,Xt+1 ,mt+1 from t = 0; 1; 2; :::: to 1: Normalising
the cash in advance and budget constraints by
1
Mt
and denoting the real values in small case letters,
the cash in advance and budget constraints become
pt Ct + qt bt+1 (1 + ) + pt st Xt+1 + pt Tt = mt + bt + pt Xt
pt Ct + qt bt+1 (1 + ) + pt st Xt+1 + pt Tt + mt+1 (1 + ) = mt + bt + pt Xt + pt zNt
(M.1401)
(M.1402)
1. Set up the Lagrange multiplier function for this problem.
2. Derive the …rst order conditions with respect to Ct , Nt ,bt+1 ,Xt+1 and mt+1 .
3. Show the solution procedure using the envelop theorem and market clearing conditions.
4. Determine the prices of goods (P ) and nominal and real bonds (q; s) and interest rate (R)
and the Fisher equation in terms of the model parameters.
5. Find the steady state values of Ct , Nt ,bt+1 ,Xt+1 and mt+1 .
6. Characterise the transitional dynamics of the system.
14
Assignment(optional)
Q1. Developing a macro model for policy analysis.
a. Specify a macroeconomic model for an economy of your choice..
Keynesian open economy simulation model to
DSGE model
VAR Model (classical or Bayesian)
Multi country policy coordination model
b. Estimate parameters of this using available time series data (quarterly preferred)
c. Forecast scenarios showing impacts of contractionary …scal policy and accomodating expansionary monetary policy.
409
d. Modify this model and show how the e¢ ciency of …nancial sector a¤ects the economy.
e. Write a short essay based on above model and analysis.
Q2. Consider an endoengous growth model developed in Basu and Bhattarai (2011) in which
the home country produces the output in the goods sector (yt ) with physical capital (kt ) and home
grown intangible or human capital (ht ). The human capital evolves following the linear technology:
ht+1 = (1
where
h
h )ht
+ Qt ht
(N.1403)
2 (0; 1) is the rate of depreciation and Qt is a crucial human capital fundamental called
cognitive skills of the home country’s population. Given the current level of human capital (ht ), the
human capital achieved in the following period will be greater if the cognitive skills, Qt are higher.
Cognitive skill implies the learning ability of pupils. This learning ability could depend partly on
parent’s and pupil’s schooling e¤orts is produced by the following technology.
Qt = AHt :lHt
where
(N.1404)
> 0 and lHt is the fraction of raw labour time (inelastically supplied at unity) allocated
to schooling. We do not impose any restriction such as diminishing returns to schooling e¤orts in
augmenting cognitive skill as the nature of returns to scale in human capital is a debatable question.
In fact, increasing returns to cognitive skill are quite plausible ( exceeding unity) if there is family
based externality. For example, in addition to parent’s own e¤ort, the child can additionally bene…t
if other family members such as grandparents could spend time on the child’s education. This is
akin to what Friedman (1962) calls "neighbourhood e¤ect" of education in a free society. In our
calibration exercise, we allow a range of variation of
around the baseline value of unity. The
variable AHt is an exogenous educational total factor productivity (TFP) variable that depends on
a host of institutional and public policy factors including positive externality and social returns of
public spending on education.
Final goods (yt ) are produced with the help of human and physical capital via the Cobb-Douglas
production technology:
yt = AGt kt (lGt ht )1
with 0 <
(N.1405)
< 1: The variable AGt is the date t exogenous total factor productivity (TFP) in
the goods sector, and lGt (= 1
lHt ) is the fraction of raw labour directed to the goods sector
production.
Assume the following stationary stochastic processes for these two TFP shocks around the steady
state:
410
AGt
AG =
G (AGt 1
AG ) +
G
t
(N.1406)
AHt
AH =
H (AHt 1
AH ) +
H
t
(N.1407)
where AG and AH are the steady state TFP of the goods and education sectors. Autocorrelation
coe¢ cients
G
and
H
are positive fractions and
G
t
and
H
t
are white noises.
Final goods are used for consumption (ct ), domestic investment (idt ) and export (xt ).
The
resource constraint facing the home country is:
ct + idt + xt = yt
(N.1408)
The home country imports raw materials (rmt ) at a …xed price pk : Examples of these imported
raw materials are machine tools, technology blueprints, patents etc.
Investment goods (ikt ) are produced combining domestic nontraded investment goods (idt ) and
imported raw materials (rmt ) in …xed proportions using the following Leontief production function:
ikt = min idt ; rmt
(N.1409)
which means that ikt = idt = rmt along an e¢ cient production frontier.12
The domestic physical capital stock evolves following the standard linear depreciation rule:
kt+1 = (1
k )kt
+ ikt
(N.1410)
The home country …nances these imported raw materials by a combination of export and foreign
borrowing (bt ) at a …xed world interest rate, r . The current account equation is given by:
xt + bt+1 = (1 + r )bt + pk rmt
(N.1411)
The home country faces a borrowing constraint. The amount that it can borrow in the international market is constrained by the current capital stock of home country, which means:
bt
1 2 An
kt
(N.1412)
example could help to motivate such a technological environment. Suppose the home country produces
an extra computer (ikt ). It requires a home produced mother board (idt ) and an imported co-processor (rmt
). Thus an increase in investment in physical capital necessitates an equi-proportionate increase in imported raw
materials/intermediate input.
411
The time-line is as follows. At date t, the state of the economy is characterized by kt , ht and
bt . The home country after realizing the TFP shocks,
G
t
and
H
t ,
makes decisions about goods
production (yt ), schooling time (lHt ), exports (xt ); external borrowing (bt+1 ) and consumption (ct )
which maximizes the following expected utility functional:
E0
1
X
t
U (ct )
t=0
subject to (N.1403) through (P.1456).
Assuming that the borrowing constraint binds, plugging (N.1409), (N.1410) , (N.1411) and
(P.1456) into (N.1408) one gets the combined resource constraint:
ct + pk kt+1
f(1 + pk )(1
k)
1
r gkt = yt :
(N.1413)
a) Prove that
Growth Rate:
kt+1
ct+1
ht+1
=
=
= [1
ht
kt
ct
Export Share in GDP:
1+g =
xt
=
GDPt
(1
h
+ AH )(pk
1) + (1 + r )
MPK
+ AH lH 1 (lH + lG )]
h
k
k )p
(1
:
lG
lG + (1
(N.1414)
)lH
(N.1415)
Import Share in GDP:
Denote the import bill of raw materials as mt . By de…nition, mt = pk :rmt : Thus, import share
in GDP is given by:
mt
=
GDPt
pk f (1 + AH
h)
MPK
(1
k )g
:
lG
lG + (1
)lH
(N.1416)
where M P K denotes the marginal product of physical capital.
Education Share in GDP:
Educ =
(1
)lH
lG + (1
)lH
(N.1417)
where GDP at date t is de…ned as:
GDPt =
t yt
412
+
t Qt ht
(N.1418)
b) Illustrate the if
= 1, along the balanced growth path, the following results hold:
(1
lH =
)(1
h)
(N.1419)
AH
Educ =
2
k 4
=
h
(1
1
)lH
lH
(N.1420)
AG
pk (1
h
+ AH ) + (1 + r )
M P K = pk (1
h
(1
k )(1
+ AH ) + (1 + r )
+
pk )
31 1
5
(1
(1
k )(1
lH )
(N.1421)
+ pk )
(N.1422)
c) Replicate the comparative static results reported in this paper with the following sets of
parameters
Table 104: Baseline Parameters
p
0.65
k
6.00
AH
AG
r
0.172
1.2
0.04
h
0.9
k
0.020
0.011
G
1.00
0.962
H
0.962
G
0.032
H
0.032
d) Derive the short run equations to show the transitional dynamics of the system
e) Use the dynare routines to generate the impulse response functions and explain the resutls.
Write one essay in any one of the following topics:
Q2. Describe a dynamic open economy model with households, …rms and a government that
operates …scal and monetary policies to stabilise the economy. Particularly focus on the following:
1. (a) State the problems of households using time separable utility functions and appropriate
budget constraints.
(b) Explain the problem of producers for both when …rms operate under the perfect competition and under the monopolistic competition with a stochastic technology.
(c) De…ne the steady state of the model and the characterise the general equilibrium in the
economy.
(d) Derive reduced form of the system and solve the model where appropriate log-linearising
to the steady state.
(e) Prove that equilibrium exists, is unique and stable for this economy.
(f) Solve the model with calibrated values of parameters and elasticities based on the literature.
413
(g) Identify features of the model that make it classical, Keynesian, new Keynesian or new
classical.
(h) Construct policy rules that the government has to adopt according to a well speci…ed
social welfare function.
(i) Consider expansionary monetary policy and prove that money is super-neutral in the
classical and new classical system but could have short run impacts in output under the
Keynesian and new Keynesian framework.
(j) Evaluate the impacts of contractionary …scal policy with reduction in public spending
by 20 percent holding tax rates unchanged.
Table 105: Percentage standard deviation of macro variables
GDP
Consumption
Investment
Hours worked
% standard deviation
Relative % standard deviation
Table 106: Lag, contemporaneous and lead correlations among macro variables
GDPt ; xt
1
GDPt ; xt
GDPt ; xt+1
GDP
Consumption
Investment
Hours worked
Q2. Analyse growth around the world using appropriate exogenous or endogenous growth
models. Assess e¤ectiveness of policies aimed to raise investment and accumulation along with
e¢ cient distribution of income and welfare of households.
Q3. Develop a stochastic dynamic open economy general equilibrium model and apply it for
analysing the shocks to trade, technology and …scal policies.
Q4. Assess impacts of human capital in economic growth and who how foreign capital can
complement to domestic capital in the growth process.
Note: Each part of this question is very much interconnected and forms a coherent macro dynamic general equilibrium model of an economy. You may need to write codes in MATLAB, Dynare,
GAMS/MPSGE for numerical implementation part of the model. Lecture notes and tutorials from
414
macroeconomics can be used for analysis. Student should submit both hard copy and electronic
copy of 2000 word essay by the due date. Submission procedure through Turnitin is outlined in the
module handbook.
14.1
Best twenty articles in 100 years in the American Economic Review
Arrow, Kenneth J., B. Douglas Bernheim, Martin S. Feldstein, Daniel L. McFadden, James M. Poterba, and
Robert M. Solow. 2011. "100 Years of the American Economic Review: The Top 20 Articles." American
Economic Review, 101(1): 1–8.
1. Alchian, Armen A., and Harold Demsetz. 1972. “Production, Information Costs, and Economic
Organization.”American Economic Review, 62(5): 777–95.
2. Arrow, Kenneth J. 1963. “Uncertainty and the Welfare Economics of Medical Care.” American
Economic Review, 53(5): 941–73.
3. Cobb, Charles W., and Paul H. Douglas. 1928. “A Theory of Production.” American Economic
Review, 18(1): 139–65.
4. Deaton, Angus S., and John Muellbauer. 1980. “An Almost Ideal Demand System.” American
Economic Review, 70(3): 312–26.
5. Diamond, Peter A. 1965. “National Debt in a Neoclassical Growth Model.” American Economic
Review, 55(5): 1126–50.
6. Diamond, Peter A., and James A. Mirrlees. 1971. “Optimal Taxation and Public Production I:
Production E¢ ciency.” American Economic Review, 61(1): 8–27.
7. Diamond, Peter A., and James A. Mirrlees. 1971. “Optimal Taxation and Public Production II: Tax
Rules.” American Economic Review, 61(3): 261–78.
8. Dixit, Avinash K., and Joseph E. Stiglitz. 1977. “Monopolistic Competition and Optimum Product
Diversity.” American Economic Review, 67(3): 297–308.
9. Friedman, Milton. 1968. “The Role of Monetary Policy.” American Economic Review, 58(1): 1–17.
10. Grossman, Sanford J., and Joseph E. Stiglitz. 1980. “On the Impossibility of Informationally E¢ cient
Markets.” American Economic Review, 70(3): 393–408.
11. Harris, John R., and Michael P. Todaro. 1970. “Migration, Unemployment and Development: A
Two- Sector Analysis.” American Economic Review, 60(1): 126–42.
415
12. Hayek, F. A. 1945. “The Use of Knowledge in Society.” American Economic Review, 35(4): 519–30.
13. Jorgenson, Dale W. 1963. “Capital Theory and Investment Behavior.” American Economic Review,53(2): 247–59.
14. Krueger, Anne O. 1974. “The Political Economy of the Rent-Seeking Society.” American Economic
Review, 64(3): 291–303.
15. Krugman, Paul. 1980. “Scale Economies, Product Di¤erentiation, and the Pattern of Trade.” American Economic Review, 70(5): 950–59.
16. Kuznets, Simon. 1955. “Economic Growth and Income Inequality.”American Economic Review,45(1):
1–28.
17. Lucas, Robert E., Jr. 1973. “Some International Evidence on Output-In‡ation Tradeo¤s.”American
Economic Review, 63(3): 326–34.
18. Modigliani, Franco, and Merton H. Miller. 1958. “The Cost of Capital, Corporation Finance and the
Theory of Investment.” American Economic Review, 48(3): 261–97.
19. Mundell, Robert A. 1961. “A Theory of Optimum Currency Areas.” American Economic Review,51(4): 657–65.
20. Ross, Stephen A. 1973. “The Economic Theory of Agency: The Principal’s Problem.” American
Economic Review, 63(2): 134–39.
21. Shiller, Robert J. 1981. “Do Stock Prices Move Too Much to Be Justi…ed by Subsequent Changes in
Dividends?” American Economic Review, 71(3): 421–36.
IMF Lists 25 Brightest Young Economists, August 27, 2014 (source IMF.org)
1. Nicholas Bloom, Stanford, Uncertainty
2. Amy Finkelstein, MIT , healthcare
3. Raj Chetty, Harvard, tax policy
4. Melissa Dell, Harvard Poverty
5. Kristin Forbes, BOE and MIT International macro
6. Roland Fryer, Harvard, Randomised experiment
7. Xavier Gabaix, New York, finance and macro
8. Gita Gopinath, Harvard, exchange rate
9. Esther Duflo, MIT microeconomics issues in developing countries
10. Matthew Gentzkow, Chicago, empirical micro and media
11. Emmanuel Farhi, Harvard, Macro
12. Oleg Itskhoki, Princeton, globalisation and inequality
416
1. Hélène Rey, LBS, international macro
2. Emmanuel Saez, California, income inequality
3. Jonathan Levin, Stanford, market design
4. Atif Mian, Princeton, Debt
5. Emi Nakamura, Columbia, business cycle
6. Nathan Nunn, Harvard, economic development
7. Parag Pathak, MIT, market design
8. Thomas Philippon, NYU, risk and financial intermediation
9. Amit Seru, Chicago, regulation and financial intermediation
10. Amir Sufi, Chicago, house price
11. Iván Werning, MIT, macro prudential policy
12. Justin Wolfers, Peterson Institute, political economy
13.
Thomas Piketty, Paris, income inequality
14.2
Other Articles
References
[1] Abbott A, J.Easaw, T. Xing (2008) Trade Integration and Business Cycle Convergence: Is the
Relation Robust across Time and Space, Scand. J. of Economics 110(2), 403–417.
[2] Altig D E, C.T. Carlstrom and K.L. Lansing (1995) Computable General Equilibrium Models
and Monetary Policy Advice, Journal of Money Credit and Banking, vol. 27, no. 4, Nov., pp.
1472-1493.
[3] Arestis P , G. Chortareas and J D Tsoukalas (2010) Money and information in a New Neoclassical Synthesis Framework, Economic Journal, 120:542: February.
[4] Ascari G. and N Rankin (2013) ’The E¤ectiveness of Government Debt for Demand Management: Sensitivity to Monetary Policy Rules’Journal of Economic Dynamics and Control, 37, 8,
1544–1566.
[5] Basu, P.,1987.Anadjustment cost model of asset pricing. International Economic Review 28(3),
609–621.
[6] Basu P. and K. Bhattarai (2012) Government Bias in Education, Schooling Attainment and
Long-run Growth, Southern Economic Journal, 79(1), 127-143.
[7] Basu P. and K. Bhattarai (2012) Cognitive Skills, Openness and Growth, the Economic Record,
88: 280: 18-38, March.
[8] Basu, P., Gillman, M. and Pearlman, J. (2012). In‡ation, Human Capital and Tobin’s q. Journal
of Economic Dynamics & Control, Vol. 36, No.7, pp. 1057-1074.
[9] Bhattarai K (2011) General Equilibrium Impacts of Monetary and Fiscal Policies on Welfare of
Households in South Asia, the Review of Development Economics, 15:4:745-757.
[10] Bhattarai K (2011) Econometric and stochastic general equilibrium models for evaluation of
economic policies, International Journal of Trade and Global Markets, 4:2: 212-241.
[11] Bhattarai K. and S. Mallick (2013) Impact of China’s currency valuation and labour cost on
the US in a trade and exchange rate model. North American Journal of Economicsand Finance,
25:40-59.
[12] Blanchard O. and J. Galí (2013) Labor Markets and Monetary Policy: A New Keynesian Model
with Unemployment American Economic Journal: Macroeconomics 2: 1-30
417
[13] Blanchard, O and C.M. Kahn (1980), The Solution of Linear Di¤erence Models under Rational
Expectations, Econometrica 48, pp. 1305-1313.
[14] Blanchard O.J.and Kiyotaki (1987) Monopolistic competition and the e¤ects of aggregate demand, American Economic Review, 77: September, pp 647-66/
[15] Barro and Gordon (1983) A Positive Theory of Monetary Policy in a Natural Rate Model,
Journal of Political Economy, 91:4: 589-610.
[16] Benassy Jean Pascal (2002) The macroeconomics of Imperfect Competition and Nonclearing
Markets, MIT Press.
[17] Campbell,J. (1994) Inspecting the mechanism: an analytical approach to the stochastic growth
model. Journal of Monetary Economics vol 33(4), 463-506.
[18] Cass, David, (1965) Optimum Growth in Aggregative Model of Capital Accumulation,
Re-
view of Economic Studies, 32:233-240.
[19] Chakraborty S. , C. Papageorgiou, F. P. Sebastian (2010) Diseases,infection dynamics,anddevelopment, Journal ofMonetaryEconomics 57 (2010) 859–872
[20] Clarida R., Gali J., and Gertler M. (1999) The Science of Monetary Policy: A New Keynesian
perspective. Journal of Economic Literature ; 37 (4), 1661-707.
[21] Clarida R., Gali J., and Gertler M. (2001). Optimal Monetary Policy in Closed Versus Open
Economies: An integrated approach. American Economic Review; 91 (2); May; 248-252
[22] Chadha, J, S, & Nolan, C, (2002) In‡ation and price level targeting in a new Keynesian model,
Manchester School, 70(4), pp. 570-595.
[23] Christiano, L, M. Eichenbaum and C. Evans (2005) Nominal rigidities and the dynamic e¤ects
of shocks to monetary policy, Journal of Political Economy, 113:1:1-45.
[24] Davidson P. (1988) Post Keynesian Employment Analysis and the Macroeconomics of OECD
Unemployment, Economic Journal, May 817-831.
[25] Diamond, Peter A. (1965). “National Debt in a Neoclassical Growth Model.” American Economic Review, 55(5): 1126–50.
[26] Dixon Huw (1988) Controversy: the macroeconomics of unemployment in the OECD, the
Economic Journal, 108:779-781, May
418
[27] Dixon H and H. Le Bihan (2012) Generalised Taylor And Generalised Calvo Price And Wage
Setting: Micro-Evidence With Macro Implications Economic Journal, 122 (May), 532–554
[28] Ellison M. and A. Scott (2000) Sticky prices and volatile output, Journal of Monetary Economics 46 (2000) 621-632.
[29] Gali J. and T. Monacelli. (2005) Monetary Policy and Exchange Rate Volatility in a Small
Open Economy. Review of Economic Studies; 72(3): 707-734
[30] Farmer REA (2013) Animal Spirits, Financial Crises and Persistent Unemployment,Economic
Journal, 123,586,317-340
[31] Friedman, Milton. (1968). “The Role of Monetary Policy.”American Economic Review, 58(1):
1–17.
[32] Gomme,P.,Rupert,P.,2007.Theory,measurementandcalibration
of
macroeconomic
models.
Journal of Monetary Economics 32(2), 51–77.
[33] Harris, John R., and Michael P. Todaro. 1970. “Migration, Unemployment and Development:
A Two-Sector Analysis.” American Economic Review, 60(1): 126–42.
[34] Hayek, F. A. 1945. “The Use of Knowledge in Society.” American Economic Review, 35(4):
519–30.
[35] Hicks, J. R. (1937) Mr. Keynes and the “Classics”; A Suggested Interpretations, Econometrica
5:147-159.
[36] Iacoviello, M., and S. Neri (2010) Housing Market Spillovers: Evidence from an Estimated
DSGE Model, American Economic Journal Macro, (April), 1-44. http://www2.bc.edu/~iacoviel/
[37] Kocherlakota N. R. and K. M Yi (1996) A simple time series test of endogenous and exogenous
growth models: an application to the United States, Review of Economic Studies, 78:1:126-134
[38] Kydland F.E and E.C. Prescott (1977) Rules rather than discretions: the Inconsistency of
Optimal Plans, Journal of Political Economy, 85:3: 473-491.
[39] King R.G.and Plosser C.I. (1984) Money Credit and Prices in a Real Business Cycle, American
Economic Review, 64 (June) 263-380.
[40] Kuznets, Simon. 1955. “Economic Growth and Income Inequality.” American Economic Review,45(1): 1–28.
419
[41] Levine P, J. Pearlman, G. Perendia and B Yang (2013) Endogenous persistence in an estimated
DSGE model under imperfect information, Economic Journal, 122 (December), 1287–1312.
[42] Ljungqvist, Lars, and Thomas J. Sargent. 2011. "A Labor Supply Elasticity Accord?" American
Economic Review, 101(3): 487-91.
[43] Lucas, Robert E., Jr. 1973. “Some International Evidence on Output-In‡ation Tradeo¤s.”
American Economic Review, 63(3): 326–34.
[44] Miller M and J. Stiglitz (2010) Leverage and Asset Bubbles: Averting Armageddon with
Chapter 11?, Economic Journal 120, 544, 500–518.
[45] Modigliani, Franco, and Merton H. Miller. (1958) “The Cost of Capital, Corporation Finance
and the Theory of Investment.” American Economic Review, 48(3): 261–97.
[46] Mundell, Robert A. (1961). “A Theory of Optimum Currency Areas.” American Economic
Review,51(4): 657–65.
[47] Monetary Policy Committee Bank of England (1999) The Transmission Mechanism of Monetary Policy, www.bankofengland.co.uk.
[48] Pissarides, C. A. (2013) Unemployment in the Great Recession, Economica, 80: 385–403.
[49] Pissarides, C. A. (2011) Equilibrium in the Labor Market with Search Frictions, American
Economic Review, 101(4): 1092-1105.
[50] Phelps, Edmund S. (1968), Money-Wage Dynamics and Labor-market equilibrium, Journal of
Political Economy, 76: 678-710.
[51] Plosser Charles I (1989) Understanding Real Business Cycle, Journal of Economic Perspectives,
vol. 3, no. 3 pp. 51-77.
[52] Prescott, E.C. (1986), Theory Ahead of Business Cycle Measurement, Federal Reserve Bank
of Minneapolis, Quarterly Review, Fall: 9-22.
[53] Rankin N. (2012) ’Maximum Sustainable Government Debt in the Perpetual Youth Model’,
September, Bulletin of Economic Research
[54] Rankin Neil (1992) Imperfect competition, expectations and the multiple e¤ects of monetary
growth, the Economic Journal 102: 743-753.
[55] Romer P. (1990) Endogenous Technological Change, Journal of Political Economy, 98:5:2,
pp.s71-s102.
420
[56] Samuelson P. A. (1939) Interaction Between the Multiplier Analysis and the Principle of Acceleration, Review of Economics and Statistics, 75-78.
[57] Ramsey, F.P. (1928) A Mathematical Theory of Saving, Economic Journal 38, 543-559.
[58] Sargent, T.J. and N. Wallace (1975) "Rational" Expectations, the Optimal Monetary Instrument, and the Optimal Money Supply Rule, Journal of Political Economy, pp. 241-254.
[59] Smet F. and R. Wouters (2003) An estimated dynamic stochastic general equilibrium model
of the Euro Area, Journal of European Economic Association, Sept, 1(5):1123-1175.
[60] Schmitt-Grohe S, Martin Uribe (2007) Optimal simple and implementable monetary and …scal
rules,Journal of Monetary Economics 54 1702–1725
[61] Schmitt-Grohe S, Martin Uribe (2004) Solving dynamic general equilibrium models using a
second-order approximation to the policy function, Journal of Economic Dynamics & Control 28
755 –775
[62] Taylor M P (1987) On the long run solution to dynamic econometric equations under rational
expectation, Economic Journal, 97:385:215-218.
[63] Uhlig, H. (1995). A Toolkit for Analyzing Nonlinear Dynamic Stochastic Models Easily.
[64] Wickens M. (2012) How Useful Are DSGE Macroeconomic Models For Forecasting? CEPR,
DP 9049.
[65] Wickens M. (1995) Real Business Cycle Analysis: A Needed Revolution in Macroeconometrics
(in Controversy: Business Cycle Empirics) Economic Journal, 105, 433., 1637-1648.
[66] Plosser Charles I (1989) Understanding Real Business Cycle, Journal of Economic Perspectives,
v3, 3 51-77.
[67] Wallis K.F. (1989) Macroeconomic Forecasting: A Survey , The Economic Journal, 99, 394.,
28-61
[68] Woodford M. and J. Taylor (1999) Handbook of Macroeconomics, Elsevier, North-Holland.
[69] Wolpin K (1996) Public-Policy Uses of Discrete-Choice Dynamic Programming Models, AEA
Papers and Proceedings, May, 427-432.
421
14.2.1
Useful texts
References
[1] Acemoglu Darren (2009) Introduction to Modern Economic Growth, Princeton University Press.
[2] Aghion P. and P. Howitt (1998) Endogenous Growth Theory, MIT Press, Cambridge MA.
[3] Barro R. J. and Sala-I-Martin (1995) Economic Growth, McGraw Hill.
[4] Bhattarai K. (2008) Static and Dynamic Applied General Equilibrium Models Tax and Trade
Policy
Models of the UK Economy, Serials Publications, New Delhi, ISBN: 978-81-8387-138-
9.
[5] Bhattarai K. (2009) Allocations and Welfare in General Equilibrium Tax Model with
Money,EEFS 2009, Warsaw, Poland.
[6] Blanchard O. J. and S. Fisher (1990) Lectures on Macroeconomics, MIT Press.
[7] Blundell R, W. K. Newey and T Persson (2007) Advances in Economics and Econometrics,
Cambridge.
[8] Chada J. and S. Holly (2012) Interest Rates, Prices and Liquidity, Cambridge.
[9] Co¤man, Leanard and Neal (2012) Questioning Credible Committment, Cambridge.
[10] Cooley T F (1995) Frontiers of Business Cycle Research, Princeton.
[11] Dixon D H and N Rankin (1995) The new macroeconomics: imperfect markets and policy
e¤ectiveness, Cambridge University Press.
[12] Dreze J (2003) Advances in Macroeconomic Theory, New York:Palgrave.
[13] Fontana G. (2009) Money, Uncertainty and Time, Routledge. London
[14] Fry, M. J. (1995) Money, Interest, and Banking in Economic Development, 2nd editiion, The
JHUP, Baltimore.
[15] Gillman,M.,2011.AdvancedModernMacroeconomics:AnalysisandApplication.PearsonEducation,FinancialTimesPress
[16] Heijdra B J and F. Van der Ploeg (2002) Foundations of Modern Macroeconomics, Oxford
[17] Holly S and M Weale Eds.(2000) Econometric Modelling: Techniques and Applications, Cambridge University Press.
422
[18] Hoy M., J Livernois, C McKenna, R Rees and T. Stengos (2001) Mathematics for Economics,
2nd ed., MIT Press.
[19] Jones C. (2010) Macroeconomics, Norton
[20] Jones C. I. (2002) Economic Growth, W.W. Norton, 2nd edition.
[21] Lim G. C. and McNelis (2008), Computational Macroeconomics for the Open Economy, MIT
Press.
[22] Ljungqvists L and T.J. Sargent (2012) Recursive Macroeconomic Theory, MIT Press.
[23] Maddison A. (1991) Dynamic of Capital Accumulation and Economic Growth, Oxford.
[24] Mankiw N.G. and D. Romer (1993) New Keynesian Economics, vol. 1, 2, the MIT Press.
[25] Minford P. and D. Peel (2002) Advanced Macroeconomics: A Primer, Edward Elgar Publishing.
[26] Obstfeld M. and K. Rogo¤ (1996) Foundation of International Macroeconomics, MIT Press.
[27] Parente, S.L. and E.C. Prescott (2002). Barriers to Riches. MIT Press, Cambridge.
[28] Peacock A. and G. K. Shaw (1979) The Economic Theory of Ficals Policy, George Allen and
Unwin.
[29] Pissarides C A (2000) Equilibrium Unemployment Theory, MIT Press.
[30] Meade J.E. (1961) A Neoclassical Theory of Economic Growth, London: George Allen &
Unwin.
[31] Romer D. (2008) Advanced Macroeconomic Theory, McGraw Hill.
[32] Sargent T. J. (1987) Macroeconomic Theory, Academic Press.
[33] Sargent T. J. (1987) Dynamic Macroeconomic Theory, Harvard University Press.
[34] Shone Ronald (2002) .Economic Dynamics, Cambridge.
[35] Simon C. P. and L. Blume (1994) Mathematics for Economists, Norton.
[36] Snell Karl ed. (1967) Essays on the Theory of Optimal Economic Growth, London: MIT Press.
[37] Sorensen PB and H. J. Whitta-Jacobsen (2010) Introducing Advanced Macroeconomics, McGraw Hill.
[38] Surrey M.J.C. (1977) Macroeconomic Themes, Oxford: Oxford University Press.
423
[39] Taylor JB and M Woodford ( 1999) Handbook of Macroeconomics, Volumes 1A-1C.
[40] Wan H. (1971) Economic Growth, Harcourt Brace Jovanovich, Inc, New York.
[41] Weale M., A. Blake, N. Christodoulakis, J. Meade and D. Vines (1989) Macroeconomic Policy:
In‡ation, Wealth and Exchange Rate, Unwin Hayman, London.
[42] Wickens M. (2012) Macroeconomic Theory: A Dynamic General Equilibrium Approach, 2nd
edition, Princeton University, Press.
[43] Woodford, M. (2003) Interest and Prices: Foundations of a Theory of Monetary Policy. NJ:
Princeton University Press.
14.2.2
Quality ranking of journals in Economics
Findings of theoretical and applied research are published in journals. Better the quality of a paper,
more likelihood that it will be published in highly ranked journals, though this relationship is not
always perfect one. It is instructive to look into the Association of Business School (ABS) ranking
on quality of journals given below in process of reviewing the literature as well as in writing a paper.
ABS 4* Journals American Economic Review; Economic Journal; Econometrica; Journal of
Labour Economics; Rand Journal of Economics; Journal of Political Economy; Journal of Monetary
Economics; International Economic Review; Quarterly Journal of Economics; Review of Economic
Studies; Journal of Econometrics; Journal of Economic Literature; Journal of Economic Perspective;
Journal of Economic Theory; Journal of Economic Geography; Journal of Environmental Economics
and Management; Journal of Financial Economics.
ABS 3* Journals
Brookings Economics Papers; Journal of Economic Growth; Economic Let-
ters; Econometric Theory; European Journal of Political Economy; European Economic Review;
Journal of Development Economics; Canadian Journal of Economics; European Review of Agricultural Economics; Cambridge Journal of Economics; Journal of Applied Econometrics ; Journal
of Comparative Economics; Journal of Development Studies;Journal of Economic Dynamics and
Control; Journal of Health Economics; Journal of Economic Behaviour and Organisation; Journal
of Economics and Management Strategy; Journal of Economics of Law and Organisation; Journal
of Evolutionary Economics; Journal of Industrial Economics; Economica; Journal of Public Economics; Journal of European Economic Association; Journal of Urban Economics; Kyklos; Labour
Economics; Ecological Economics; Land Economics; Oxford Bulletin of Economics and Statistics;
Oxford Economics Papers;Oxford Review of Economic Policy; Review of Economics and Statistics;
424
Review of International Economics;Social Choice and Welfare; Southern Economic Journal; World
Bank Economic Review; Journal of International Economics..
ABS 2* Journals Advances in Econometrics; Agricultural Economics; Applied Economics;
Applied Economics Letters; Annals of Public and Cooperative Economics; Applied Financial Economics; Australian Economic Review; Australian Journal of Agricultural and Resource Economics;
Bulletin of Economic Research; Bulletin of Indonesian Economic Studies; Canadian Journal of
Agricultural Economics; Contemporary Economic Policy; Contributions to the Political Economy;
Defence and Peace Economics; Econometric Reviews; Economics of Education Review; Economics
of Innovation and New Technology; Economics of Planning;Economics of Transition; EconomistNetherlands;Environmental Resource Economics; Fiscal Studies; Global Business and Economic
Review; History of Political Economy; IMF Sta¤ Papers; Insurance Mathematics and Economics;
International Journal of Game Theory;International Journal of Economics of Business; International Review of Applied Economics; International Review of Economics and Finance; Journal of
Agricultural and Resource. Economics; World Economy.
ABA 1* Journals Business Economics; Eastern European Economics; Economy and Society;
Empirical Economics; Employee Relations Europe Asian Studies; Hitsubashi Journal of Economics;
Information Economics and Policy;International Journal of Social Economics; Journal of Economic
Methodology; Journal of Economic Psychology; Journal of Economics;Journal of Industry, competition and Trade; Journal of interdisciplinary Economics; Macroeconomic Dynamics.
For the latest version visit: http://www.associationofbusinessschools.org/node/1000257.
Note also that there are many journals which have not been ranked by the ABS.
425
15
Computation and software
Macroeconomic theories after detailed optimisation procedure express variables in terms of behavioral parameters. Application of these model requires calibration or estimation of these parameters
with the real world data and computation of alternative scenarios according to …nd out the impacts of economic policies or changes in behavior. Solving a simultaneous equations becomes more
complicated as number of equations increase in the model.
Excel is good for small scale examples. Special software such as General Algebraic Modelling
System (GAMS) or MATLAB are used for solving bigger models. GAMS/MPSGE is very e¤ective
in solving large scale models. Econometrics often involves with estimation parameters using cross
section or time series data; PcGive/Stamp/GiveWin, Eviews , STATA, Shazam, Limdep are good
software for this. SPSS good for processing large scale survey and statistical analysis.
15.1
GAMS
GAMS is good particularly in solving linear and non-linear system of equations. It has widely been
used to solve general equilibrium models with many linear or non-linear equations on continuous
or discrete variables. It comes with a number of solvers that are useful for numerical analysis such
as CONOPT, DICOPT, MILES, MINOS, DNLP, PATH. It can solve very large scale models using
detailed structure of consumption, production and trade arrangements on unilateral, bilateral or
multilateral basis in the global economy where the optimal choices of consumers and producers are
constrained by resources and production technology or arrangements for trade. It is a user friendly
software.
Any GAMS programme involves declaration of set, parameters, variables, equations, initialisation of variables and setting their lower or upper bounds and solving the model using Newton or other methods for linear or non-linear optimisation and reporting the results in tables or
graphs see examples below. GAMS/MPSGE software is good for large scale standard general equilibrium models. GAMS programme can be downloaded from demo version of GAMS free from
www.gams.com/download.
Learn GAMS by practicing following examples. First write them using a text editor and save
…le *.gms. Then execute the program and study the result and then revise the model as necessary.
$Title overlapping generation model
Set t time /t1*t30/;
Parameters al, b, K0, d, A(t), N;
al = 0.5;
b = 0.3;
426
N(t) = 100;
A(t) = 10;
K0 = 300;
K0 = 3;
d = 0.05;
Variables
Y(t) total output at time t
K(t) total capital stock
S(t) total saving
I(t) total investment
yk(t) percapita output
kl(t) percapita capital
r(t) rental rate
w(t) wage rate
Co(t) consumption of old
Cy(t) consumption of young
YT total output
;
Equations EY(t), EK(t),EKK(t), EKLast(t),Eyk(t), Ekl(t), ew(t), er(t), eco(t), ecy(t), ES(t), EI(t),
obj;
* separate equations for the …rst, intermediate and last periods
EKK(t)$(ord(t) eq 1)..
K(t)$(ord(t) eq 1) =e= K0;
EK(t+1)$(ord(t) ne 1)..
K(t) =e=(K(t-1)+I(t-1)) ;
EKLast(t)$(ord(t) eq card(t))..
K(t) =e=(K(t-1)+I(t-1));
EY(t)..
Y(t) =e=A(t)*K(t)**b*N(t)**(1-b);
eyk(t)..
yk(t) =e= (Y(t)/N(t));
ekl(t)..
kl(t) =e= (K(t)/N(t));
ew(t)..
w(t)=e=(1-b)*A(t)*kl(t)**(b);
er(t)..
427
r(t) =e= b*A(t)*kl(t)**(b-1);
eco(t)..
cy(t) =e= al*w(t);
ecy(t)..
co(t)=e=kl(t)*(1+r(t));
ES(t)..
S(t) =e= Y(t)-N(t)*cy(t) - N(t)*co(t);
EI(t)..
I(t) =e= S(t);
obj..
YT =e= sum(t, yk(t));
Model OLG /all/;
Y.L(t)=0.01; K.L(t)=0.01; S.L(t)=0.01; I.L(t)=0.01;yk.l(t)=0.01;
kl.L(t)=0.01; r.L(t)=0.01; w.L(t)=0.01; Co.L(t)=0.01; Cy.L(t)=0.01;
*solve olg using mcp;
solve olg maximising YT using nlp;
Parameter report;
report(t, "Y") = Y.L(t);
report(t, "K") = K.L(t);
report(t, "I") = I.L(t);
report(t, "S") = S.L(t);
report(t, "yk") = yk.L(t);
report(t, "kl") = kl.L(t);
report(t, "r") = r.L(t);
report(t, "w") = w.L(t);
report(t, "co") = co.L(t);
report(t, "cy") = cy.L(t);
Display Y.L, K.L, S.L, I.L,yk.L, kl.L,r.L, w.L,co.L, cy.L, report;
Ramsey model
SET T TIME PERIODS /1*50/;
SCALAR
A LABOR VALUE SHARE /0.75/
PHI SCALE PARAMETER /1 /
LGR LABOR GROWTH RATE /0.02/
D DEPRECIATION /0.05/
B RATE OF TIME PREFERENCE /0.02/
428
ITR INCOME TAX RATE /0.3/;
PARAMETER
L(T) LABOR FORCE;
L(T) = (1+LGR)**(ORD(T)-1);
VARIABLES
K(T) CAPITAL STOCK
I(T) INVESTMENT
C(T) CONSUMPTION
Q(T) OUTPUT
R(T) RETURN TO INVESTMENT
Z(T) EXCESS DEMAND
OBJ SUM OF SQUARED EXCESS DEMANDS;
EQUATIONS
KACC(T) CAPITAL ACCUMULATION
KTERM(T) TERMINAL CONDITION FOR INVESTMENT
QREF(T) PRODUCTION
DR(T) RETURN TO INVESTMENT DEFINATION
FOCS(T) FOCS FOR UTILITY MAXIMIZATION
RC(T) RESOURCE CONSTRAINT
OBJDEF OBJECTIVE DEFINATION;
KACC(T)$(ORD(T) LT CARD(T))..
K(T+1) =E= K(T)*(1-D) + I(T);
KTERM(T)$(ORD(T) EQ CARD(T))..
I(T) =E= K(T)*(D + LGR);
QREF(T)..
Q(T) =E= PHI*L(T)**A*K(T)**(1-A);
DR(T)..
R(T) =E= (PHI*(1-A)*L(T)**A*K(T)**(-A) -D)*(1-ITR);
FOCS(T)$(ORD(T) LT CARD(T))..
C(T)*(1+R(T+1)) =E= C(T+1)*(1+B);
RC(T)..
Z(T) =E= I(T) +C(T) -Q(T);
OBJDEF..
OBJ =E= SUM(T,Z(T)*Z(T));
MODEL CAPTAX /ALL/;
K.LO(T) = 0.0000001;
429
K.FX("1") = 1;
C.LO(T) = 0.0000001;
I.LO(T) = 0;
I.L(T) =0;
K.L(T) =1;
Q.L(T) = PHI*L(T)**A;
R.L(T) =(1-A)*PHI*L(T)**A -D;
C.L(T) = Q.L(T) -I.L(T);
SOLVE CAPTAX MINIMIZING OBJ USING NLP;
PARAMETER
GR(T) GROWTH RATE
KL(T) CAPITAL LABOUR RATIO;
GR(T)$(ORD(T) LT CARD(T)) = Q.L(T+1)/Q.L(T) -1;
KL(T) = K.L(T)/L(T);
DISPLAY L, K.L, I.L, C.L, Q.L, Z.L, GR, KL;
parameter report, reportgr;
report(t, "y") = q.l(t);
report(t, "c") = c.l(t);
report(t, "I") = I.l(t);
report(t, "k") = k.l(t);
reportgr(t, "y")$(ORD(T) LT CARD(T)) = Q.L(T+1)/Q.L(T) -1;
GAMS/MPSGE systax see http://www.mpsge.org.
The check whether the results are consistent with the economic theory underlying the model such
as ISLM-ASAD analysis for evaluating the impacts of expansionary …scal and monetary policies.
Use knowledge of growth theory to explain results of the Solow growth model from Solow.gms.
Consult GAMS and GAMS/MPSGE User Manuals, GAMS Development Corporation, 1217
Potomac Street, Washington D.C or www.gams.com.
For other relevant software visit: http://www.feweb.vu.nl/econometriclinks/ or
https://www.aeaweb.org/rfe/
15.2
MATLAB
MATLAB is widely used for solving models. It has script and function …les used in computations.
Both have *.m extensions. Its syntax are case sensivite.
Solving a system of linear equations and handling matrices
Example 1
430
Write a programme …le matrix.m like the following and try run.
% now solve a linear equation
% 5x1 + 2x2 =20
% 3x2 + 4x2 =15
k =[5 2;3 4];
n = [20 15];
kk = inv(k)
x = kk*n’
One more example of system of equation and factorisation of matrices
A=[1 2 3; 3 3 4; 2 3 3]
b=[1; 2; 3]
%solve AX=b
X = inv(A)*b
%eigen value and eigenvectors of A
[V,D]=eig(A)
%LU decomposition of A
[L,U]=lu(A)
%orthogonal matrix of A
[Q,R]=qr(A)
%Cholesky decomposition (matrix must be positive de…nite)
%R = chol(A)
%Singular value decomposition
[U,D,V]=svd(A)
%simple Markov Chain
a = [0.2 0.8];
%transition matrix
b = [0.9 0.1; 0.7 0.3];
%initial state
c0 = a*b
%subsequent states
c1 = c0*b
c2 = c1*b
c3 = c2*b
c4 = c3*b
c5 = c4*b
c6 = c5*b
431
%stationary state
c7 = c6*b
%eigen values
d = eig(b)
[V,D]=eig(b)
Example 3
Solving …rst order ordinary di¤erential equation
write the function simpode.m which has just two lines
function xdot = simpode(t,x);
xdot = x+t;
simpode(t,x) is MATLAB function.
Write a scrit …rst_ode.m
tspan =[0,2], x0=0;
[t,x]=ode23(’simpode’,tspan,x0);
plot(t,x)
xlabel(’t’), ylabel(’x’)
then execute program in the commandline to get the graph
>>…rst_ode
Example 3
%solving system of ordinary di¤erential equations
type the following and save in regid.m functiion …le
function dy = rigid(t,y)
dy = zeros(3,1); % a column vector
dy(1) = y(2) * y(3);
dy(2) = -y(1) * y(3);
dy(3) = -0.51 * y(1) * y(2);
end
Then script …le second_ode.m
tspan =[0,20], z0=[1;0];
[t,z]=ode23(’pend’,tspan,z0);
x=z(:,1); y=z(:,2);
plot(t,x,t,y)
xlabel(’t’), ylabel(’Consumption and Income’)
…gure(2)
plot(x,y)
xlabel(’consumption’), ylabel(’income’)
432
title(’consumption and income’)
then execute the program typing
>>second_ode
This will give two …gures from the solution.
Example 3
Evaluating an integral
de…ne the function
function y =erfcousin(x);
y = exp(-x^2)
end
write script integral.m
y = quad(’erfcousin’,1/2,3/2)
>>integral
Example 3
Evaluating a double integral
write a script …le
%evaluating double integral
F = inline(’1-6*x.^2*y’);
I = dblquad(F,0,2,-1,1)
>>integral_d
For more see MATLAB help/ examples and documentation/Mathematics.
Try sample programs provided there and have a tiny model to practice.
Stochastic General Equilibrium model: an example
clear all
global eta omega beta alpha1 kappa rho phi0 phi1 chis
global Rstar PFstar PXstar
global Blam_ss C_ss F_ss G_ss K_ss L_ss P_ss Pk_ss R_ss S_ss W_ss X_ss Y_ss Z_ss
global nstart T1 T2 zshock nstatevar neuronx neuler
fun =’chapter2_netfun’;
Rstar = 1.01;
PFstar = 1.0;
PXstar = 1.0;
eta = 1.5;
omega =0.25;
beta = 1/1.01;
alpha1 = 0.15;
433
kappa=0.1;
chis=0.1;
rho=0.9;
phi0=0.9;
phi1=1.5;
Blam_ss=3.22696890007;
C_ss=0.457934;
F_ss=0;
G_ss=0;
K_ss=0.02729049;
L_ss = 0.7473200;
P_ss = 0.999999998539410;
Pk_ss = 1.0;
R_ss = 0.0100;
S_ss=0.999999998539410;
W_ss = 0.28812562001981;
X_ss=0.027290490;
Y_ss=0.485225112772;
Z_ss=1;
nstatevar =3; neuler = 2; neuronx =1;
neuronx1= neuronx +1;
nparm = nstatevar *neuler *neuronx;
T1 = 20;
T2 = 5;
randn(’state’,888);
se_shock = 0.01;
zshock = randn(T1,T2)*se_shock;
nstart=4;
options = optimset(’Display’,’iter’,’MaxFunEvals’,100,’MaxIter’,100,’TolFun’,0.0001);
gammaf=[2.8342,-1.6406,-0.0868, 0.5264,- 0.0000, 3.0591];
gammaf = fminsearch(fun,gammaf,options);
[ERROR,C,F,K,L,P,R,S,W,Pk,Y,Z,trade,ERR_C,ERR_S]=feval(fun,gammaf);
…gure(1);
subplot(5,2,1);plot(Z);title(’Z’)
subplot(5,2,2);plot(C);title(’C’)
subplot(5,2,3);plot(S);title(’S’)
434
subplot(5,2,4);plot(Y);title(’Y’)
subplot(5,2,5);plot(K);title(’K’)
subplot(5,2,6);plot(L);title(’L’)
subplot(5,2,7);plot(W./P);title(’W./P’)
subplot(5,2,8);plot(P);title(’P’)
subplot(5,2,9);plot(R);title(’R’)
subplot(5,2,10);plot(F);title(’F’)
function [ERROR,C,F,K,L,P,R,S,W,Pk,Y,Z,trade,ERR_C,ERR_S] = chapter2_netfun(gamax);
global eta omega beta alpha1 kappa rho phi0 phi1 chis
global Rstar PFstar PXstar
global Blam_ss C_ss F_ss G_ss K_ss L_ss P_ss Pk_ss R_ss S_ss W_ss X_ss Y_ss Z_ss
global nstart T1 T2 zshock nstatevar neuronx neuler
Blam=Blam_ss*ones(T1,T2); %200x50 3.22696890007
C=C_ss*ones(T1,T2); %200x50 0.457934
F=F_ss*ones(T1,T2); %200x50 0
K=K_ss*ones(T1,T2); %200x50 0.02729049
L=L_ss*ones(T1,T2); %200x50 0.7473200
P=P_ss*ones(T1,T2); %200x50 0.999999998539410
R=R_ss*ones(T1,T2); %200x50 0.01
S=S_ss*ones(T1,T2); %200x50 0.999999998539410
W=W_ss*ones(T1,T2); %200x50 0.28812562001981
Pk=Pk_ss*ones(T1,T2); %200x50 1
Y=Y_ss*ones(T1,T2); %200x50 0.485225112772
Z=Z_ss*ones(T1,T2); %200x50 1
Zrisk=zeros(T1,T2); %200x50 0
ERR_C=zeros(T1,T2); %200x50 0
ERR_S=zeros(T1,T2); %200x50 0
jk = nstatevar*neuler*neuronx; %6
jj = 1:nstatevar:jk; %1 4
kk = nstatevar:nstatevar:jk; %3 6
for j=1:T2; %1:50
for i = nstart +1:T1; %4:200
Zz= rho*log(Z(i-1,j))+ (1-rho)*log(Z_ss)+zshock(i,j); %0,9*0+0,1*0+0,01=0,01 ?= -0.0021
Z(i,j)= exp(Zz); %200X50 1.0101 4 pierwsze wiersze to 1
ZZ(i,j)= Z(i,j) - Z_ss; %200x50 0.0101 4 pierwsze wiersze to 0
435
FF(i,j)= F(i-1,j) - F_ss; %200x50 0
RR(i,j)= R(i-1,j) - R_ss; %200x50 0
xstate = [ZZ(i,j) FF(i,j) RR(i,j)]; % -0.0021 0 0
for nn = 1: neuler*neuronx; %1:2
neuron(1,nn) = 1./(1+exp(-gamax(jj(nn):kk(nn))*xstate’))-0.5; %-0.0015 -0.0003
end
pea_C=neuron(1,1:neuronx);
pea_S=neuron(1,neuronx+1:2*neuronx);
C(i,j) = exp(pea_C)*C_ss; %200x50 *
S(i,j) = exp(pea_S)*S_ss; %200x50 *
Y(i,j)= C(i,j)+G_ss+X_ss; %200x50 *
LL=0.5*(1-alpha1)*(Z(i,j)^kappa)*(Y(i,j)^(1-kappa))*(C(i,j)^-eta);
L(i,j)=LL^(1/(1-kappa+omega));
L(i,j)=real(L(i,j));
KK=((Y(i,j)/Z(i,j))^kappa)-(1-alpha1)*L(i,j)^kappa;
K(i,j)=(KK/alpha1)^(1/kappa);
K(i,j)=real(K(i,j));
mpl=(1-alpha1)*(Z(i,j)^kappa)*(Y(i,j)/L(i,j))^(1-kappa);
mpk=(alpha1)*(Z(i,j)^kappa)*(Y(i,j)/K(i,j))^(1-kappa);
mpl=real(mpl);
mpk=real(mpk);
Pk(i,j)=S(i,j)*PFstar;
P(i,j)= 2*Pk(i,j)/mpk;
W(i,j)= (L(i,j)^omega)*(C(i,j)^eta)*P(i,j);
W(i,j)= real(W(i,j));
Zinf(i,j)= 0.25*((P(i,j)/P(i-4,j))-1);
R(i,j)= phi0*R(i-1,j)+(1-phi0)*(Rstar+phi1*Zinf(i,j));
trade(i,j)=P(i,j)*X_ss-S(i,j)*PFstar*K(i,j);
trade1=trade(i,j)/S(i,j);
F(i,j)=F(i-1)*(1+Rstar+Zrisk(i-1,j))-trade1;
Blam(i,j)=(C(i,j)^-eta)/P(i,j);
Blam(i,j)= real(Blam(i,j));
Zrisk(i,j)=sign(F(i-1,j))*chis*(exp(abs(F(i-1,j)))-1);
Zder(i,j)=chis*(exp(abs(F(i-1,j))));
MUC= Blam(i,j)*(beta*(1+R(i-1,j)));
MUCLAG=Blam(i-1,j);
436
ERR_C(i,j)=(MUC/MUCLAG)-1;
MUS=S(i,j)*(1+Rstar+Zrisk(i-1,j)+Zder(i-1,j)*F(i-1,j));
MUSLAG=(1+R(i-1,j))*S(i-1,j);
ERR_S(i,j)=(MUS/MUSLAG)-1;
end;
end;
err1=reshape(ERR_C,T1*T2,1);
err2=reshape(ERR_S,T1*T2,1);
ERROR=mean(err1.^2)+mean(err2.^2)+2*mean(err1.*err2);
References
[1] Uhling’s toolkie Examples in http://www2.bc.edu/~iacoviel/.
[2] Cleve Moler’s Numerical Computing with MATLAB or Experiments with MATLAB available
at http://www.mathworks.com/moler/index.html.
[3] Michael Ferris has developed Interface between GAMS and MATLAB. The details of the new
package can be found at:
http://www.cs.wisc.edu/math-prog/matlab.html. For this a) install a new version of GAMS
(23.4) b) put the system directory of GAMS into your MATLAB path .
Contents.m for list of …les in MATLAB demo. MATLAB demo available in http://www.youtube.com/.
15.3
Dynare
Dynare has been developed to solve the stochastic general equilibrium model. It generates MATLAB
codes and easy to implement.
It can be downloaded free from http://www.dynare.org/. See a simple example Ramsey_demo.mod.
var A K R I C Q;
varexo e_A;
Parameters Al PHI LGR D B ITR L ro;
Al = 0.4;
PHI = 1.5;
LGR = 0.03;
D = 0.02;
B = 0.99;
437
ITR = 0.3;
L =100;
ro = 0.7;
model;
K = K(-1)*(1 - D) + I;
Q = PHI*(L^Al)*K(-1)^(1-Al);
R = (PHI*(1-Al)*(L^A)*K(-1)^(-Al) -D)*(1-ITR);
C*(1+R(+1)) = C(+1)*(1+B);
I + C = Q;
A = ro*A(-1)+ e_A;
end;
initval;
A =1.5;
I =0.25;
K =1;
R = 0.06;
Q =1;
C =0.75;
end;
shocks;
var e_A;
stderr 0.09;
end;
steady;
stoch_simul(irf=20,order=1) Q K I C R A;
See website of Dynare programs a number of applications of DSGE models http://www.douglaslaxton.org/dynare.html
Frank Schorfheide’s web page: http://economics.sas.upenn.edu/~schorf/research.htm
References
[1] Juillard, M. (1996), Dynare: A Program for the Resolution and Simulation of Dynamic Models
with Forward Variables through the Use of a Relaxation Algorithm, CEPREMAP, Couverture
Orange, 9602.
438
15.4
R
For non-parametric and nonlinear estimation.see at http://www.r-project.org/
R programming: https://www.coursera.org/course/rprog ; and this for R learning resources:
http://www.ats.ucla.edu/stat/r/
15.5
Econometric and Statistical Software
Excel
OXMetrics7-GiveWin/PcGive/GARCH/STAMP
Eviews8 (contain Bayesian VAR)
Shazam
micro…t
RATS
GAUSS
STATA/SPSS
http://www.feweb.vu.nl/econometriclinks/; https://www.aeaweb.org/rfe/
OX-GiveWin/PcGive/STAMP (www.oxmetrics.net) is a very good econometric software for
analysing time series and cross section data. This software is available in all labs in the network
of the university by sequence of clicks Start/applications/economics/givewin. Following steps are
required to access this software.
a. save the data in a standard excel …le. Better to save in *.csv format .
b. start give win at start/applications/economics/givewin and pcgive (click them separately)
c. open the data …le using …le/open data…le command.
d. choose PcGive module for econometric analysis.
e. select the package such as descriptive statistics, econometric modelling or panel data models.
d. choose dependent and independent variables as asked by the menu. Choose options for
output.
e. do the estimation and analyse the results, generate graphs of actual and predicted series.
A Batch …le can be written in OX for more complicated calculations using a text editor such as
pfe32.exe. Such …le contains instructions for computer to compute several tasks in a given sequence.
See: Doornik J A and D.F. Hendry ((2003) PC-Give Volume I-III, GiveWin Timberlake Consultants Limited, London
439
15.5.1
Advanced Texts in Macroeconomics
References
[1] Acemoglu D. (2009) Introduction to Modern Economic Growth, Princeton.
[2] Aghion P. and P. Howitt (1998) Endogenous Growth Theory, MIT Press, Cambridge MA.
[3] Allen RGD ( 1956) Mathmatical Economics, ELBS and McMillan, London.
[4] Barro R. J and Sala-i-Martin (1995) Economic Growth, McGraw Hill.
[5] Benassy Jean Pascal (2002) The macroeconomics of Imperfect Competition and Nonclearing
Markets, MIT Press.
[6] Bhattarai K. (2008) Economic Theory and Models: Derivations, Computations and Applications
for Policy Analyses, Serials Publications, New Delhi.
[7] Bhattarai K. (2008) Static and Dynamic Applied General Equilibrium Models Tax and Trade
Policy Models of the UK Economy, Serials Publications, New Delhi
[8] Blanchard O. J. and S. Fisher (1990) Lectures on Macroeconomics, MIT Press.
[9] Blundell R, W. K. Newey and T Persson (2007) Advances in Economics and Econometrics,
Cambridge.
[10] Cooley T F (1995) Frontiers of Business Cycle Research, Princeton.
[11] Dreze Jacque (2003 ) Advanced Macroeocnomic, Palgrave.
[12] Fair R. C. (1994) Testing Macroeconometric Models, Harvard.
[13] Fair R. C. (1984) Speci…cation, Estimation and Analysis of Macroeconometric Models, Harvard.
[14] Gali Jordi (2008) Monetary Policy, In‡ation and the Busines Cycle: An Introduction to the
New Keynesian Framework, Princeton University Press.
[15] Hicks J R (1939) Value and Capital, ELBS, MacMillan.
[16] Heijdra B J and F. Van der Ploeg (2002) Foundations of Modern Macroeconomics, Oxford.
[17] Holly S and M Weale Eds.(2000) Econometric Modelling: Techniques and Applications, pp.6993, the Cambridge University Press.
440
[18] Jones C. (2010) Macroeconomics, Norton.
[19] Keynes J.M. (1936) The General Theory of Employment, Interest and Money, MacMillan and
Cambridge University Press.
[20] Kocherlakota N. R. (2010) The New Dynamic Public Finance, Princeton University Press.
[21] Lavoie M. (2009) Introduction to post-keynesian economics, Basingstoke: Palgrave Macmillan.
[22] Lim G. C. and McNelis (2008), Computational Macroeconomics for the Open Economy, MIT
Press.
[23] Ljungqvists L. and Sargent T.J (2012) Recursive Macroeconomic Theory, MIT Press, 3rd
edition.
[24] Malthus T. R (1798) An Essay on the Principle of Population, J. Johnson, London, 1798.
[25] Mankiw N.G and D Romer ed. (1993) New Keynesian Economics, MIT Press.
[26] Miller P.J. (1994) The Rational Expectation Revolution, MIT Press.
[27] Minford P. and D. Peel (2002) Advanced Macroeconomics: A Primer, Edward Elgar Publishing.
[28] Morris Davis (2009) Macroeconomics, Cambridge University Press.
[29] Obstfeld M. and K. Rogo¤ (1996) Foundation of International Macroeconomics, MIT Press.
[30] Parente S.L.and E.C. Prescott (2002) Barriers to Riches, MIT Press.
[31] Pigou A.C. (1947) A Study in Public Finance, Macmillan, London.
[32] Romer D. (2008) Advanced Macroeconomic Theory, McGraw Hill, 3rd ed..
[33] Samuelson P. (1947) Foundation of Economic Analysis, Harvard University Press.
[34] Sargent T. J. (1987) Macroeconomic Theory, Academic Press.
[35] Sargent T. J. (1987) Dynamic Macroeconomic Theory, Harvard University Press.
[36] Shone Ronald (2002) Economic Dynamics, Cambridge.
[37] Smith Adam (1776) In Inquiry into the Nature and Cause of Wealth of Nations, vol I and II,
Liberty Fund, Indianapolis, Indiana.
[38] Sorensen PB and H. J. Whitta-Jacobsen (2010) Introducing Advanced Macroeconomics, McGraw Hill.
441
[39] Stokey N L and R E Lucas (1989) Recursive Methods in Economic Dynamics, Harvard University Press.
[40] Surrrey M.J.C. (1976) Macroeconomic Themes, Oxford University Press, London.
[41] Taylor JB and M Woodford ( 1999) Handbook of Macroeconomics, Volumes 1A-1C.
[42] Turnovsky S J (1999) International Macroeconomics, MIT Press.
[43] Walsh, C.E (1998), Monetary Theory and Policy, MIT Press .
[44] Weale W. et al. (1989) Macroeconomic policy : in‡ation, wealth and the exchange rate, London
: Unwin Hyman.
[45] Wickens M. (2012) Macroeconomic Theory: A Dynamic General Equilibrium Approach, 2nd
edition,Princeton University Press.
16
Sample Class test
(Time allowed: 50 minutes)
Answer any two questions, one from each section. Each question is worth 100
marks. Each subquestion has equal value in any question.
Section A
Q1 Consider the Dornbusch model of exchange rate overshooting in following equations:
1) change in the exchange rate:
e =E e
(P.1423)
r =r +E e
(P.1424)
mD =
(P.1425)
2) interest rate parity
3) money demand function
ar + by
4) money market equilibrium
m
p=
442
ar + by
(P.1426)
5) process of price adjustment
p =
yD
yS
>0
(P.1427)
6) aggregate demand
y D = u + v (e
p)
(P.1428)
7) demand supply balance
yS = y
(P.1429)
1. What are the steady state values of exchange rate and price level in this economy?
2. Find the time paths of the exchange rate and price level solving di¤erential equations simultaneously. Explain the convergence or divergence properties of the
system.
3. Illustrate the transitional dynamics in a phase diagram in (e; p) space.
4. Discuss why the exchange rate overshoots in the short run using the above derivations and analysis.
Q2. Imagine an economy inhabited by rich, middle income and poor households, indexed by i =
A, B and C. There are three types of goods in the economy. Endowments of these three
goods to three categories of households are W1 , W2 and W3 respectively. Each household
prefers to consume all three goods, j = 1; 2;and 3. The demand of household i for good j , is
denoted by Xji ; i.e. X1i ; X2i and X3i . Each household i maximises its own welfare subject
j
P
to its own budget constraint, I i =
Pj Wji , where I i is the total income of the household, Pj
j=1
is the relative price of a commodity and Wji is the endowment of commodity j of household
i. Price of good j adjusts until demand for it equals its supply. For simplicity assume that
each household is endowed only with one type of good but prefers to consume each of three
goods equally. Thus preferences and constraints for household type i are given by following
equations:
M ax
U (X1i ; X2i ; X3i ) = X1i X2i X3i
subject to
443
i = A; B; C
(P.1430)
i
I =
j
X
Pj Wji = P1 X1i + P2 X2i + P3 X3i
(P.1431)
j=1
Markets clear (only A is endowed by W1 ; only B is endowed by W2 and only C is endowed by
W3 )
X
X1i = W1A ;
X
X2i = W2B ;
X
X3i = W3C
(P.1432)
The endowments of households were as given in Table 1.
Table 107: Endowment Structure of Households
W1 W2 W3
A
100
0
0
B
0
200
0
C
0
0
300
100
200
300
Total supply
a. Derive demand functions, X1i ; X2i and X3i consistent with utility maximisation
by each household. Find equilibrium prices, optimal allocations and utility for each
household.
b. Record the quilibrium solutions of the model in respective cells of Table 2.
Table 108: Optimal Consumption of Households
X1 X2 X3 U
A
B
C
Total
100
200
300
Price
c. How would these prices change if there is a 20 percent tax on income of each
household and all revenue collected are distributed equally among them.
Q3. A representative household in a economy has to decide on how much to consume today and
how much to save and invest to add to the capital stock to produce more goods for future
consumption. The optimal capital stock maximises the present value of utility (U0 ) from
consumption (C (t)). Problem of this representative household is:
444
M ax U0 =
Z
T
e
rt
C (t) dt
(P.1433)
0
subject to:
1) the production technology relates how output (Q) relates to capital stock (K) as:
Q = Q(K)
(P.1434)
The …rst and the second order derivatives of output w.r.t capital (K) are:
@Q
@K
@2Q
@K 2
> 0 and
<0
.
2) capital accumulation constraint with depreciation rate of capital ( )
@K
=Q C
K
@t
3) Initial and terminal (transversality) conditions Ko and KT :
Kt =
(P.1435)
1. Write the current value Hamiltonian for dynamic optimisation in this model.
[Note: current valued costate
and the present valued costate
= e
t
; H =He
t
]
2. Discuss the …rst order conditions and the terminal conditions required for dynamic optimisation.
3. Use a phase diagram to show the convergence process towards the optimal capital
stock.
4. Apply this model to determine the optimal pricing strategy for exhaustible resources (non-renewable resources) such as oil and gas in a competitive economy
considering following set up:
a ) Inverse demand for such resources
P = P (c)
U (c) =
Z
1
(P.1436)
P (c) dc
(P.1437)
0
b) Longrun objective
M ax U0 =
Z
T
0
c) Resource deaccumulation constraint:
445
e
rt
U (c) dt
(P.1438)
Kt =
C
4) Transverality conditions Ko and KT
[ Hint: Hamiltonian H(C; K; ) = e
rt
(P.1439)
:
:
U (Ct ) + K + K]
Q4. Consider a problem of a representative household whose problem is to maximise utility from
consumption (ct ) and real money balances (mt ) as:
max W =
1
X
t
U (ct ; mt )
(P.1440)
t=0
subject to:
1) Technology constraint:
Yt = F (Kt ; Nt )
(P.1441)
2) Cash in advance constraint
Mt 1
Mt
= Ct + Kt +
Pt
Pt
where Yt is output, Pt price of goods, Ct consumption, Kt+1 is capital stock,
Yt +
t Nt
+ (1
) Kt
1
+
for each individual, Mt money, Yt , output, Nt employment and
(P.1442)
t
is net transfer
is the rate of depreciation of
capital.
Under constant returns to scale yt = f (kt ) where yt =
Yt
Nt
and kt =
Kt
Nt .
You may de…ne the
constraint in per capita terms as:
! t = f (kt
1)
+
t
+
1
1+n
kt
1
+
(1 +
mt 1
= ct + kt + mt
t ) (1 + n)
Use the Bellman equation V (! t ) = u (ct ; mt )+ V (! t+1 ) with 0 <
with multiplier
t
(P.1443)
< 1or Lagrangian function
for dynamic optimisation.
1. Set up the relevant functions for constrained dynamic optimisation and derive the
…rst order conditions of maximisation with respect to choice variables (ct ; mt ;
t ).
2. Solve the model for optimal consumption, output, capital stock and money stock
in the steady state.
3. Explain the results and discuss how the economy would move towards the steady
state if it is above or below from it.
446
Section B
Q5. Consider an AS-AD model of an economy given be …ve equations. First one is the Fisher
equation that relates the real interest rate (rt ) to the nominal interest rate (ipt ) , the exchange
rate risk ( t ) and expected in‡ation
e
t+1
as:
rt = ipt +
e
t+1
t
(P.1444)
y) to
Second equation is for the aggregate demand function and that relates output gap (yt
the …scal policy gap (gt
rate (rt
g), to the deviation of real interest rate from the long run average interest
r) and the demand shock vt .as:
yt
y=
1
(gt
g)
(rt
2
r) + vt ;
vt v N 0;
2
v
(Ad_r)
The aerage interest rate (r) is aggregate of natural rate and average risk, r = r + .
Third equation is the interest rate rule adopted by the central bank which relates the nominal
interest rate (ipt ) to the in‡ation gap (
ipt = r +
) and the output gap (yt
t
e
t+1
+ h(
t
) + b (yt
y)
y)
(MP_r)
Forth equation is for aggregate supply (price formation) function relates expected in‡ation to
the output gap and the supply shock st .as:
t
=
e
t
y) + st ; st v N 0;
+ (yt
2
s
(P.1445)
For simplicity model assumes backward looking in‡ation expectation:
e
t
=
(P.1446)
t 1
1. Derive reduced form functions for the aggregate demand (AD) and the aggregate
supply (AS) using all above equations.
2. Find expressions for the deviation of in‡ation and output from the steady state
bt =
t
and
ybt = yt
y when there are no further shocks to the AD or AS;
i.e. when zt = 0 and st = 0.
3. Find the time path for output and in‡ation given their initial values yb0 and b0 .
4. Calculate time taken for yt and
ybt = 0 and bt = 0) when
t
= 0:742;
to converge to the steady state y and
= 0:3;
447
2
= 5:76; b = 0:5:
(
5. Derive impulse response functions for yt and
to compute impacts of unit shocks
t
to aggregate demand and supply.
Q6. Expected in‡ation next period (Et
t+1 )
based on information at period t depends on di¤er-
ences on expected and actual prices as:
Et
t+1
=Et pt
pt
Demand ytd is function of real money balances (mt
ytd = a0 + a1 (mt
pt ) + t ;
(P.1447)
pt ) as:
a0 > 0 a1 > 0;
t
2
N 0;
(P.1448)
Actual output (yts ) deviates from the natural rate of output when actual prices are not equal to
expected prices pt 6=Et
1
pt as:
yts = yn + b1 pt
Et
1
p t + vt ;
a1 > 0 ;
t
N 0;
2
(P.1449)
Demands equals supply in equilibrium as:
ytd = yts = yt
(P.1450)
Consider a money supply rule given by:
mt
mt
1
=
(P.1451)
1. Use rational expectation method to solve for equilibrium output and prices in
this model.
2. Show that under the rational expectation average in‡ation equals growth rate of
money supply but only the unanticipated shocks to demand or supply in‡uence
the level of output.
Q7. Consider a two sector endogenous growth model in which output (yt ) is produced using
physical capital (kt ) and human capital (ht ). This output is either consumed (ct ) or exported
(xt ). Part of the human capital (lG share, 0 < lG < 1) is used in producing …nal goods and
remaining (1
lG ) of it is used to produce more human capital. The technical progress in the
…nal goods sector is AGt and that in the human capital sector is Aht . The physical capital
depreciates at
k
rate and the human capital at
purchase investment goods
ikt ,
xt =
pkt ikt
where
pkt
h:
Proceedings from exports are used to
is the price of capital good. International
borrowing (bt ) is permited at the interest rate r but being a small open economy it faces
448
borrowing constraints, it can borrow only up to its physical capital, bt
kt :More speci…cally
the optimization problem faced by the benevolent social planner of this economy is:
M ax
1
X
t
U (ct );
0<
<1
t=0
subject to
1) resource constraint
ct + xt = yt = AGt kt (lG ht )1
;0<
<1
(P.1452)
2) human capital accumulation constraint
ht+1 = (1
h )ht
+ Aht (1
lG )ht ; 0 <
h
<1
(P.1453)
3) physical capital accumulation constraint
ikt = kt+1
(1
k )kt ;
0<
k
<1
4) current account constraint
xt = pkt ikt
(P.1454)
xt + bt+1 = (1 + r )bt + pk ikt
(P.1455)
5) capital account constraint
6) The amount that home country can borrow in the international market is constrained by the
current capital stock:
bt
Parameters ; ;
h; k
kt
(P.1456)
and AGt and Aht are set exogenously.
a. Formulate the Lagrangian for constrained in…nite horizon dynamic optimisation
for this social planner.
b. Derive the balanced growth rate using the standard …rst order conditions for
optimisation.
Q8. Consider Mortensen and Pissarides (1994) model of equilibrium unemployment in which the
matching function aggregates vacancies and unemployment with job creation as:
449
M = M (V; U ) = V U (1
)
(P.1457)
Here M denotes the number of matching between vacancies and job seekers, V is the number of
vacancies and U the number of unemployed,
the parameter between zero and one . Nash-product
of the bargaining game over the di¤erence between the earnings from work (W ) rather than in
being unemployed (U ) and earnings to …rms from …lled rather than vacant jobs is given by:
(Wi
U ) (Ji
1
V)
(P.1458)
Symmetric solution of this satis…es joint pro…t maximisation condition in which share of workers
becomes:
(Wi
Let parameter
U) =
(Ji + Wi
V
U)
be the ratio of vacancy to job seeking workers
a vacancy be given then by f ( ) and not …lling it by 1
unemployed worker is q ( ) @t and the not …nding is 1
(P.1459)
=
V
U;
the probability …lling
f ( ); probability of …nding a job by an
q ( ) @t; job creation occurs when matching
takes place between …rms with vacancies and workers seeking the job. With labour force L and the
unemployment rate u, the number of workers who enter unemployment is
(1
u) L@t where
is
the rate of idiosyncratic shock of job destruction.
1. Determine the equilibrium unemployment in the system.
2. Derive optimal job creation or (demand for labour curve) by …nding the optimal
returns from vacancy [rV =
pc + q ( ) (J
V )] ; returns from jobs [rJ = p
returns from unemployment [rU = z + q ( ) (Wi
returns from work [rW = w + (U
w
J] ;
U )] and
W )] :
3. Establish links between the reservation wage (z) the price of product p and costing
of hiring ( c)
=== End ===
17
Sampel Final Exam
Time Allowed: Two Hours
Q1. Consider and contrast classical and Keynesian macro models expressed in terms of equations as given below.
450
Classical model Output (Y )
Y = F (N )
(Q.1460)
Labour demand (N ) :
N = N(
W
)
P
(Q.1461)
Labour Supply (L) :
W
)
P
Labour market equilibrium condition as a function of real wage rate ( W
P ):
L = L(
W
W
) = N( )
P
P
Neutrality of money (M ) to price level (P ) with given velocity of circulation (m) :
L(
(Q.1462)
(Q.1463)
M = mP Y
(Q.1464)
S = S(i)
(Q.1465)
I = I(i)
(Q.1466)
S=I
(Q.1467)
Savings (S)
Investment (I)
Capital market equilibrium
Capital (K) accumulation process
Kt = (1
) Kt
1
+ It
451
0<
<1
(Q.1468)
Keynesian model Output:
Y = F (K; N )
Fk > 0; FN > 0; Fkk < 0; FN N < 0; FkN > 0:
(Q.1469)
Labour demand (real wage function of marginal productivity of labour):
W
= FN (N; K)
P
(Q.1470)
Consumption:
C = c Y d ; Y d = (1
)Y
(Q.1471)
Investment:
I = I(r)
(Q.1472)
Nominal wage(W ) and labour supply (N ):
W = W0 + W (N )
W (N ) =
Z
(Q.1473)
0 for N 5 N
(Q.1474)
>0 for N > N
where N is labour supply at the full employment.
Money market equilibrium conditions with supply of real balances
M
P
equal to money demand
M (Y; r):
M
= M (Y; r) My > 0; Mr < 0
P
Net exports as a di¤erence between exports (X) and imports (IM ):
NX = X
IM
(Q.1475)
(Q.1476)
Goods market equilibrium condition:
Y =C +I +G+X
IM
(Q.1477)
1. Determine the level of employment, output and price level in the classical model.
2. Determine the tax and government spending multipliers in the Keynesian model.
3. Assess the impacts of changes in government spending and taxes on the output,
consumption and price level in the Keynesian model using comparative static
analysis.
452
4. Assess strengths and weakness of the classical and Keynesian models based on
above analysis.
[Hints: Linearise the model for comparative static analysis and determine the corresponding
multipliers.] [Continued...[56277]].
453
Q2. Consider a standard version of Ramsey’s optimal growth model
max
U=
1
X
t
ln(Ct )
0<
<1
(Q.1478)
t=0
subject to
a) production technology:
Yt = AKt
0<
<1
(Q.1479)
b) capital accumulation:
Kt+1 = Kt (1
) + It
(Q.1480)
c) market clearing:
Yt = Ct + It
(Q.1481)
K (0) = K0
(Q.1482)
d) initial condition:
1. Solve this model for the capital stock, output, consumption and investment in
the steady state.
2. Characterise the transitional dynamics of the model and explain in what sense
this model is di¤erent from the Solow growth model.
3. How would you solve this model if the technology A is given by a stochastic
process At+1 = At + "t where "t
N (0;
2
)?
4. Financial intermediaries take away a certain fraction of saving. Let
the fraction of savings taken away (wasted) by them while (1
saving is channelled into investment. As such a higher value of
ine¢ ciency in the …nancial system. How does
represent
) fraction of
represents more
a¤ect the saving and investment
and capital accumulation in this economy?
5. Suggest modi…cation in the Ramsey model to study the impacts of capital income
taxation in economic growth.
[Continued...[56277]]
454
Q3. Consider a version of the Brock-Mirman type dynamic programming problem
max
U=
1
X
t
ln(Ct )
0<
<1
(Q.1483)
t=0
subject to market clearing condition
Kt+1 + Ct = AKt
0<
<1
(Q.1484)
Here output (AKt ) is either consumed (Ct ) or invested (Kt+1 ) :
1. What are the control and state variables in this model and why?
2. Explain the meaning of the value function (Bellman equation) and the policy
functions of this problem, V1 (K 0 ) = ln C + V0 (K 0 ); where K 0 is the amount of
optimal capital stock.
3. Assume K 0 = 0 for the last period. Demonstrate a recursive solution method of
this problem using three iterations of the policy and value functions. Characterise
the rest of the solution.
4. Use the limit theorem to …nd the explicit solution of the value function.
5. Introduce a stochastic technology At+1 = At +"t and examine conjectures to solve
this problem.
[Continued...[56277]]
455
Q4. Consider the New Keynesian model in which the problem of household i is to maximise
expected utility from consumption (Cit+k ), accumulation of money (Mit+k+1 ) and labour supply
(Nit+k ) taking account of all information (
max
E
"1
X
k
t)
U (Cit+k ) + V
t=0
subject to:
available up to period t is given as:
Mit+k+1
P t+k
Q (Nit+k ) j
t
#
(Q.1485)
a) CES aggregation of consumption (Cit ) and price level P t over j commodities:
Cit =
Z
0
b) the budget constraint
Z
1
1
1
Cijt dj
;
Pt =
Z
1
1
Pjt
0
1
dj
1
(Q.1486)
1
Pjt Cijt + Mit+1 + Bit+1 = Wt Nit + (1 + it ) Bit + Mit +
it
+ Xit
(Q.1487)
0
where Bit ,
it
and Xit denote bonds held, pro…ts earned and transfer recieved by the household
i ; Wt is wage earned for supplying labour (Nit ) :
c) demand for a product Cijt relates to composite demand as:
Cijt =
Pjt
Pt
Cit
(Q.1488)
Firms take wage rates as given and set prices a la Calvo with
probability of changing it every
period. Then Yj;t is the solution to the …rms’pro…t maximization problem:
max E
"
X
k
subject to:
kU
0
(Ct+1 )
(1
U 0 (Ct )
)
k
Pjt
Yjt+k
P t+k
Wt+k Yjt+k
P t+k Zt+k
j
t
#
(Q.1489)
a) a linear production technology
Yjt = Zt Njt
(Q.1490)
b) supply
Yjt+k =
Pjt
P t+k
Yt+k
(Q.1491)
1. Write …rst order conditions for optimisation by households and …rms in this
model.
2. Solve for the price level, employment and output at the steady state.
456
3. Prove that volatility of output is generated from the technological shock. Comment how it compares to a standard RBC model.
Q5.
Solve for the steady state, and characterise the transitional dynamics in the
following neo-classical growth model.
max
Uo =
Ct
Z
1
e
1
t Ct
dt
(Q.1492)
<1
(Q.1493)
1
t=0
Subject to technology constraint
Yt = At Kt Nt1
;
0<
Capital accumulation process
K t = Yt
Nt C t
Kt
(Q.1494)
It = St
(Q.1495)
Market clearing:
Yt = Ct + St
Initial (boundary) condition:
K (0) = Ko
(Q.1496)
Here Uo life time utility of the consumer, Ct is consumption, Yt output, Kt capital stock, Nt
labour input, K t change in capital stock each period;
substitution, and
is discount parameter
eslatisicity of
rate of depreciation. Assume At = 1 and Nt = 1 for simplicity.
1. Set up the current value Hamiltonian function for this problem.
2. State four …rst order conditions for optimisation and write meanings of each.
3. Compute the steady state of the model.
4. Show the transitional dynamics of the shadow price and the capital stock.
5. Represent the saddle path solution in a set of nicely labelled diagrams in (K; )
space where
Q6.
is the shadow price of capital (K) :
Consider the cash in advance model and characterise the transitional dynamics of this
economy. Problem of the household now becomes:
457
max
1
X
t
[U (Ct )
V (Nt )]
(Q.1497)
t=0
a) Subject to the technology constraint:
Yt = zNt
(Q.1498)
Pt Ct + qt Bt+1 + Pt st Xt+1 + Pt Tt = Mt + Bt + Pt Xt
(Q.1499)
b) Cash in advance constraint:
where Pt Ct is consumption expenditure Pt price of goods, Ct consumption, Bt+1 is the amount
of nominal bonds qt is the price of nominal bonds, Xt+1 real bonds, st prices of real bonds, Tt lump
sum tax payment, Mt money. Budget constraint of the consumer:
Pt Ct + qt Bt+1 + Pt st Xt+1 + Pt Tt + Mt+1 = Mt + Bt + Pt Xt + Pt zNt
(Q.1500)
c) Government’s budget constraint:
M t+1
Mt =
Pt Tt
(Q.1501)
and M t+1 = (1 + ) M t
Assuming a constant rate of money growth
Mt =
Pt Tt
(Q.1502)
The representative agent chooses Ct , Nt ,bt+1 ,Xt+1 ,mt+1 from t = 0; 1; 2; :::: to 1: Normalising
the cash in advance and budget constraints by
1
Mt
and denoting the real values in small case letters,
the cash in advance and budget constraints become
pt Ct + qt bt+1 (1 + ) + pt st Xt+1 + pt Tt = mt + bt + pt Xt
pt Ct + qt bt+1 (1 + ) + pt st Xt+1 + pt Tt + mt+1 (1 + ) = mt + bt + pt Xt + pt zNt
(Q.1503)
(Q.1504)
1. Set up the Lagrange multiplier function for this problem.
2. Derive the …rst order conditions with respect to Ct , Nt ,bt+1 ,Xt+1 and mt+1 .
3. Show the solution procedure using the envelop theorem and market clearing conditions.
458
4. Determine the prices of goods (P ) and nominal and real bonds (q; s) and interest
rate (R) and the Fisher equation in terms of the model parameters.
5. Find the steady state values of Ct , Nt ,bt+1 ,Xt+1 and mt+1 .
6. Characterise the transitional dynamics of the system.
Q7. Consider the real business cycle model in which producers maximise pro…t subject to
technology and accumulation constraints and the households maximise lifetime utility subject to
their budget constraints. Goods and labour markets clear.
Firms’problem:
max
= Yt
t
wt Lt
rt Kt
(Q.1505)
subject to
a) technology
1
Yt = Kt (At Lt )
0<
<1
(Q.1506)
b) capital accumulation
Kt = (1
) Kt
1
+ It
0<
<1
(Q.1507)
Representative consumer’s problem:
a) maximises lifetime utility
max
1
X
t
U (ct ; 1
lt )
(Q.1508)
t=0
subject to budget constraint
ct + kt+1 = wt lt + (1 + rt ) kt
(Q.1509)
Goods market clears for every period t : output is either consumed or invested.
Yt = Ct + It
(Q.1510)
Consider Cobb-Douglas preferences:
U (ct ; 1
lt ) = ln ct + b ln (1
lt )
b>0
(Q.1511)
1. Find wage rate and the interest rate consistent with the producer’s optimisation.
459
2. Derive the Euler equation for optimisation by households using the …rst order
conditions that equate the current marginal utility to the expected marginal
utility by consumers.
3. Show how consumption, capital stock and labour supply are related to the output
in equilibrium.
4. Demonstrate how the optimal output and consumption could be derived as an
autoregressive process using above demand and supply side solutions.
5. Decompose output process into transitory and permanent components.
Q8. Consider the following macroeconomic model of an economy where term yt
the deviation of actual output (yt ) from its trend (y) ,
t
is the actual in‡ation and
e
t+1
y denotes
expected
in‡ation; rt ; r and it are real, natural and nominal interest rates respectively; error terms vt and
st denote demand and supply shocks respectively; E [ t jIt
t based on information set (It
1)
available up to t
1]
denotes the expected in‡ation at time
1 period.
Aggregate demand:
yt
y = vt
r) ; vt v N 0;
(rt
2
v
(Q.1512)
Fisher equation for the real interest rate:
e
t+1
rt = it
(Q.1513)
Aggregate supply (price formation):
t
=
e
t 1
y) + st ; vt v N 0;
+ (yt
2
s
(Q.1514)
y)
(Q.1515)
Monetary policy rule:
rt = r +
e
t+1
+h
e
t+1
+ b (yt
Expectation
e
t;t 1
=E[
t 1 jIt 1 ]
(Q.1516)
Critically assess the policy irrelevant propositions (PIP) under the rational expectation hypothesis.
460
18
18.1
Foundations
First order di¤erence equation
Supply
Qd;t =
Pt ;
( ;
> 0)
(R.1517)
( ; > 0)
(R.1518)
Demand
QS;t =
+ Pt
1;
Qd;t = QS;t =) Pt + Pt
1
=
+
(R.1519)
Steady state or intertemporal solution
P =
+
+
(R.1520)
Complete solution Pt = PC + PP
Complementary solution
Pt + Pt
1
= 0 =) Pt+1 +
Pt = 0
(R.1521)
Let Pt = Abt and Pt+1 = Abt+1
Abt+1 +
Abt = 0
(R.1522)
Steady state or intertemporal solution
b=
(R.1523)
t
Pt = PC + PP = Abt + PP = A
+
+
+
(R.1524)
Determiner A from the initial condition P0
P0 = Ab0 + PP = A
0
+
+
+
=)
A = P0
Complete and de…nite solution
461
+
+
(R.1525)
t
Pt = A
+
+
+
=
t
+
+
P0
+
+
+
(R.1526)
Application of First Order Di¤erence Equation: Inventory and Price Adjustment Model
Consider a demand supply model with inventory and price adjustments.
Demand depends on current price:
Xtd =
Pt
(R.1527)
Supply depends on current price:
XtS =
+ Pt
(R.1528)
Price adjustment process:
Xtd
Pt+1 = Pt +
XtS
(R.1529)
Equilibrium conditions without inventory would be Xtd = XtS but here prices do not clear market
instantly. Therefore prices adjust according to:
Pt+1 = Pt + (
Pt +
Pt )
(R.1530)
This is a …rst order di¤erence equation in prices.
Application of First Order Di¤erence Equation: Inventory and Price Adjustment Model
Pt+1
(1
( + )) Pt =
( + )
(R.1531)
Intertemporal solution
P =
( + )
( + )
(R.1532)
Whether prices converge to this stationary solution depends on solutions to the complementary
part
Pt+1
(1
( + )) Pt = 0
Abt+1
b = (1
(1
( + )) Abt = 0
(R.1533)
( + ));
Application of First Order Di¤erence Equation: Inventory and Price Adjustment Model
The general solution for price is:
462
t
Pt = PC + PP = A ((1
( + ))) +
( + )
( + )
(R.1534)
Value of A can be obtained by assuming initial price at time t = 0 , P0
implies A = P0
+
+
Pt =
+
+
P0
t
(1
( + )) +
( + )
( + )
(R.1535)
Inventory and Price Adjustment Model: Dynamic Properties
Pt =
b = (1
+
+
P0
t
(1
( + )) +
( + )
( + )
( + ))
1. 0 < b < 1 convergent and non-oscillatory,
<
1
( + ).
1
( + )
2.b = 0 solution is convergent to the steady state,
=
3. 1 < b < 0 gives oscillating but convergent path,
1
( + )
4. b =
1 case of regular oscillation,
5. b <
1 divergent oscillations,
>
=
<
2
( + )
2
( + )
Inventory and Price Adjustment Model: Dynamic Properties
463
<
2
( + )
(R.1536)
18.2
First order di¤erential equation
Di¤erence equations are used to denote the time path of a variable when variables change continuously not discretely.
First order di¤erential equation only involved di¤erential term of order one.
y + ay = b or
@y
+ ay = b
@t
(R.1537)
Solution of a di¤erential equation includes complementary and particular (steady state) parts
yt = yc + yp
(R.1538)
For the steady state equilibrium y = 0: This implies
yp =
b
a
Solve the homogeneous system for the complementary solution:
y + ay = 0
y
=
y
(R.1539a)
a
(R.1540)
Solution of Di¤erential Equation
Integrate both sides with respect to t
Z
y
@t =
y
Z
ln (yt ) + c1 =
a@t
(R.1541)
at + c2
(R.1542)
Taking anti-log both sides
yc = e
at c2
e
yc = Ce
where C = ec2
c1
(R.1543)
at
(R.1544)
c1
Complete solution
yt = yc + yp = Ce
at
+
b
a
(R.1545)
The time path of yt converges if a > 0 .
First Economic Example of the …rst order di¤erence equation (IS-LM Model):
464
Consumption function :
C = a + BY
nR
(R.1546)
Let investment and government spending be as given at I = Iand G = G
Goods markets does not balance automatically, it take time for adjustment as given by the
following equation ( < 1):
@y
= (a + by nR + I + G
@t
Money market is assumed to balance instantaneously
L = ky
y)
(R.1547)
hR
(R.1548)
L=M
(R.1549)
Money market equilibrium implies
R=
k
y
h
1
M
h
(R.1550)
Putting the money market equilibrium in the goods market gives the economywide equilibrium
process as:
@y
=
@t
nk
y
h
a + by
nM
h
+I +G
y
(R.1551)
nM
+I +G
h
(R.1552)
By rearrangement
@y
+
@t
1
b+
nk
h
y=
a+
@y
+ Ay = B
@t
where A =
by yp =
B
A
=
1
b+
a+ nM
h
(1
nk
h
+I+G
b+ nk
h )
and B =
a+
nM
h
(R.1553)
+ I + G The steady state equilibrium is given
and the complementary path is given by
yc = Ce
At
= Ce
(1
b+ nk
h )t
(R.1554)
Complete income path from solving the di¤erence equation is given by combinations of these
two:
yt = Ce
At
465
+
B
A
(R.1555)
De…nite solution requires getting value of C using the initial conditions yt=0 = y0 as C = y0
yt
=
B
e
A
y0
2
At
nM
M
a+
= 4y0
+
1
B
A
3
+I +G
b+
(1
5e
nk
h
a+
b+ nk
h )t
+
1
Convergence to the steady state requires that A > 0. This implies 1
.The slope of the LM curve
k
h
nM
h
B
A
+I +G
b+
(R.1556)
nk
h
k
h
b + nk
h > 0 or
should be greater than the slope of the IS curve
1 b
n
>
1 b
n
:
Consider a market price adjustment model where it takes time for demand and supply to adjust
towards equilibrium. Starting from an initial point, does market prices converge to the long run
equilibrium or not depends on the roots of the equations. These provide stability conditions for the
system:
demand
Supply
QD =
P with
QS =
;
+ P with
>0
(R.1557)
; >0
(R.1558)
price adjustment process
@P
= k(
@t
by rearranging
@P
@t
+ k( + )P = k(
P +
P)
(R.1559)
+ )
+
+
The steady state equilibrium is P =
Homogeneous equation for complementary solution is given by:
Z
@P
+ k( + )P = 0
@t
(R.1560)
Z
k ( + ) dt
(R.1561)
k ( + ) t + c2
(R.1562)
@P
@t
P
dt =
ln (Pt ) + c1 =
Taking anti-log both sides
Pt = e
k( + )t c2
e
Pt = Ce
k( + )t
466
c1
(R.1563)
(R.1564)
Complete solution
Pt = Pc + Pp = Ce
18.3
k( + )t
+
+
+
(R.1565)
Second order di¤erential equation: market example
In addition to the structure of market above let the speculations in the demand side market
determined by the …rst and second order conditions as following
demand
Supply
P + mP 0 + nP 00
QD =
(R.1566)
+ P + uP 0 + wP 00
QS =
(R.1567)
for a while assume that u = 0 and w = 0
Let market …nd its equilibrium in each period QD = QS : This implies
P + mP 0 + nP 00 =
nP 00 + mP 0
( + )P =
The steady state equilibrium like before is : Pp =
+ P
(R.1568)
( + )
(R.1569)
+
+
For complementary solution derive the homogenous equation
P 00 +
m 0
P
n
+
n
P =0
(R.1570)
Let P = Aert sot that P 0 = rAert and P 00 = r2 Aert : and r2 Aert +
+
n
m
rt
n rAe
Aert = 0:
The corresponding characteristic equations is:
r2 +
m
r
n
+
n
=0
(R.1571)
Roots of this equations are given by:
m
n
r1 ; r2 =
r
m 2
n
2
+4
+
n
1
=
2
"
m
n
General solutions in the distinct real roots case when
s
h
m
n
m 2
n
Pt = Pc + P p = A 1 e r 1 t + A 2 e r 2 t +
467
2
+4
>4
+
+
+
n
+
n
i
#
(R.1572)
:
(R.1573)
It requires two initial conditions for de…nite solution
P t = A1 e
1
2
m
n
q
In case of repeated root
(m
n )
2
4(
m 2
n
+
n
=
)
t
+ A2 e
+
n
4
q
1
2
m
n +
2
(m
n )
+4(
parts as:
h
m 2
n
<
4
+
n
i
t
+
+
+
m
2n
(R.1574)
m
2n
+
+
(R.1575)
the roots are divided between the real and imaginary
r1 ; r2 = h
where the real part in this case is h =
)
there is only one root r1 ; r2 =
Pt = Pc + Pp = A3 er1 t + A4 ter2 t +
for complex root case
+
n
vi
and the v =
h
4
+
n
m 2
n
i
(R.1576)
p
1.
and i =
Substituting real and imaginary parts and using the Euler equation and DeMoivre theorems:
Pt = P c + Pp = e
m
2n t
[A5 cos (vt) + A6 Si n (vt)] +
+
+
(R.1577)
Second order di¤erential equation only involved di¤erential term of order two. The procedure
is similar to the second order di¤erence equation. As before yt = yc + yp and yp =
complementary solution y = Ae
rt
rt
sot that y = rAe
2
b
a2
. For
rt
and y = r Ae :
y + a1 y + a2 y = b
(R.1578)
r2 Aert + a1 rAert + a2 Aert = 0
(R.1579)
r2 + a1 r + a2 = 0
(R.1580)
p
a21 4a2
(R.1581)
r1 ; r2 =
2
There can be three cases in the solution of this equation depending on the value of the term
( a1 )
under the square root
I. Distinct real root if a21 > 4a2
II. Repeated real root if a21 = 4a2
III. Complex real root a21 < 4a2 This requires use of the imaginary number, De Moivre theorem
and trigonometry.
These cases is illustrated below by two examples:
468
Consider a market price adjustment model where it takes time for demand and supply to adjust
towards equilibrium. Starting from an initial point, does market prices converge to the long run
equilibrium or not depends on the roots of the equations. These provide stability conditions for the
system:
Preliminaries
Example of Complex Root Case: Example
Exponential forms and polar coordinates
p
h2 + v 2
(R.1582)
sin n =
v
=) v = Rsin
R
(R.1583)
cos =
h
=) h = Rco
R
(R.1584)
R=
ei = cos + i Si n
h
@ sin
@
= cos ;
@ cos
@
vi = Rco
=
e
i
= cos
Ri sin = R (co
i Si n
i sin ) = Re
(R.1585)
i
(R.1586)
sin ;
Thus the Cartesian coordinates of the complex numbers have been transformed to polar coordinates R and
and also expressed as exponential form Re
i
:
Give the Cartesian form of the complex number 5e
3i 2
R (co
i ( 1)) =
i sin ) = 5 cos 3 2
i sin 3 2 = 5 (cos 0
: Here R = 5,
5i = h
= 32
vi
By De Moivre’s theorem
n
(h + vi) = Rn ein
and
n
(h
vi) = Rn e
in
(h
vi) = Rn (cos n
n
i sin n )
(R.1587)
Solving a di¤erential equation with complex roots
Table 109: Values of Trigonometric Ratios
0
300 450 600 900 1200 1800 2700
0
0
sin
cos
0
1
6
1
2
p
3
2
4
p1
2
p1
2
3
p
3
2
1
2
2
1
0
469
3
4
p1
2
- p12
3
2
0
-1
2
1
0
3600
0
1
Example of Complex Root Case: Example
An Example
y + 2y + 17y = 34
(R.1588a)
34
b
=
=2
a2
17
(R.1589)
Steady state
yp =
This is a complex root case because (a1 = 2; a2 = 17; b = 34)
a21
4a2 = 22
4
17 = 4
68 =
64 < 0
Use the formula explained above
h
h=
vi = Rco
1
2 a1
=
Ri sin = R (co
i sin ) = Re i
p
p
p
1 v = 12 4a2 a21 = 21 4 (17) 22 = 12 64 =
1
2
(8) = 4
In case of the complex root
yc
=
eht A1 evit + A2 e
vit
=
eht [A1 (cos vt + i sin vt) + A2 (cos vt
(R.1590)
i sin vt)]
(R.1591)
For this problem complementary solution
yc
=
eht A1 e4it + A2 e
=
e
yt = yc + yp = e
t
yt = e
t
(R.1592)
[A1 (cos 4t + i sin 4t) + A2 (cos 4t
[A1 (cos 4t + i sin 4t) + A2 (cos 4t
t
[(A1 + A2 ) cos 4t + (A1
yt = e
where A5 = (A1 + A2 )
4it
t
A6 = (A1
i sin 4t)]
i sin 4t)] + 2
A2 ) i sin 4t] + 2
[A5 cos 4t + A6 sin 4t] + 2
(R.1593)
(R.1594)
(R.1595)
(R.1596)
A2 ) i
Use two initial conditions to de…nitize the values of A5 and A6 .
y0 = 3 and y = 11:
When t = 0
y0 = 3 = e
t
[A5 cos 4t + A6 sin 4t] + 2 = [A5 cos 0 + A6 sin 0] + 2 = A5 + 2
470
(R.1597)
Thus A5 = 1
take the …rst derivative of with respect to time
y
y
=
=
@y
e t [A5 cos 4t + A6 sin 4t] + 2
@t
e t [A5 cos 4t + A6 sin 4t] + e t [ 4A5 sin 4t + 4A6 cos 4t]
(R.1598)
(R.1599)
Evaluated when t = 0
y=
e
t
[A5 cos 0 + A6 sin 0] + e
t
[ 4A5 sin 0 + 4A6 cos 0]
11 =
(A5 + 0) + [0 + 4A6 ]
(R.1600)
A6 = 3
Thus the complete solution of this equation is:
t
yt = e
[cos 4t + 3 sin 4t] + 2
(R.1601)
The …rst trigonometric function gives the cycle and second part is the steady state.
Numerical example 1 for SODE
demand
QD = 42
Supply
QS =
4P + 4P 0 + P 00
6 + 8P
(R.1602)
(R.1603)
Initial conditions P0 = 6 and P 0 (t = 0) = 4:
Let market …nd its equilibrium in each period QD = QS : This implies
42
4P + 4P 0 + P 00 =
The steady state equilibrium like before is : Pp =
For homogenous solution rearrange P
00
P 00
4P
0
4P 0
471
46
12
6 + 8P
(R.1604)
=4
4P + 42 =
8P = 0
6 + 8P to
(R.1605)
Numerical example 1 for SODE Corresponding quadratic equation is given by
( 4)
r1 ; r2 =
q
2
( 4)
4:1:( 12)
2
=
4
p
16 + 46
= 6; 2
2
Pt = Pc + Pp = A1 er1 t + A2 er2 t + 4 = A1 e6t + A2 e
2t
+4
(R.1606)
(R.1607)
Use two initial conditions for the complete solution
P0 = 6 = A1 e6:0 + A2 e
P 0 = 4 = 6A1 e6:0
2:0
+ 4 = A1 + A2 + 4
2:0
2A2 e
= 6A1
(R.1608)
2A2
(R.1609)
+4
(R.1610)
Solving these equations A1 = 1 and A2 = 1:
Pt = A1 er1 t + A2 er2 t + 4 = e6t + e
2t
This path is dynamically unstable because of r1 = 6: This gives divergent Oscillations.
Numerical example 2 for SODE
demand
QD = 40
QS =
Supply
2P 0
2P
P 00
(R.1611)
5 + 3P
(R.1612)
Initial conditions P0 = 12 and P 0 (t = 0) = 1:
Let market …nd its equilibrium in each period QD = QS : This implies
40
2P 0
2P
P 00 =
45
5
The steady state equilibrium like before is : Pp =
For homogenous solution rearrange 40
2P
p
22 4:1:5
2
=
r1 ; r2 =
2
This is complex root case with h + vi =
2
p
2
1
2P
4
20
0
5 + 3P
=9
P 00 =
1
( 2
2
2i where h =
=
(R.1613)
5 + 3P toP 00 + 2P 0 + 5P = 45
4i) =
1
2i
(R.1614)
1 and v = 2
The general solution of this model is
Pt = Pc + Pp = e
t
[A5 cos (2t) + A6 Si n (2t)] + 9
472
(R.1615)
Using the initial conditions
0
P0 = 12 = e
Pt0 =
t
e
[A5 cos (0) + A6 Si n (0)] + 9 = A5 (1) + A6 :0 + 9 = A5 + 9
[A5 cos (2t) + A6 Si n (2t)] + e
0
Pt=0
=
t
[ 2A5 sin (2t) + 2A6 Cos (2t)]
(R.1616)
(R.1617)
1
e
0
[A5 cos (2:0) + A6 Si n (2:0)]
+e
0
[ 2A5 sin (2:0) + 2A6 Cos (2:0)]
=
= A5 + 0 + 0 + 2A6
(R.1618)
Solving A5 + 9 = 12 and A5 + 2A6 = 1 we get A5 = 3 and A6 = 2. Thus the de…nite solution
path of the system is
t
Pt = e
Pt ‡uctuates in each period of
2
v
=
[3 cos (2t) + 2 Si n (2t)] + 9
(R.1619)
= 3:1452. when t increases 3.1452 the Pt completes one
cycle.
This cycle is damped because of the multiplicative term e t .
That means this path Pt starts at 12 and gradually converges to 9 in a cyclical fashion.
Generic Di¤erential Equations In a higher order di¤erential equation Routh theorem is applied to …nd whether time path converges to long run equilibrium:
Take a polynomial of the form
a0 rn + a1 rn
1
+ a2 rn
2
+ ::: + an 1 r + an = 0
the real parts of all the roots of nth degree polynomial are negative when …rst n sequence of
determinants are positive. Therefore above equation is convergent.
Routh Theorem
Routh Matrix is formed by letting odd coe¢ cients head a row and succes-
sively reducing the subscripts and writing zero for negative coe¢ cients (Samuelson (1947) Foundations of Economic Analysis).
ja1 j ;
a1
a3
a0
a2
a1
a3
a5
; a0
a2
a4
0
a1
a3
a1
a3
a5
a7
a0
a2
a4
a6
0
a1
a3
a5
0
a0
a2
a4
473
000
00
0
Numerical example y 4 (t) + 6y (t) + 14y (t) + 16y (t) + 8y = 24
a0 = 1; a1 = 6; a2 = 14; a3 = 16; a4 = 8; a5 = 0; a6 = 0;
0
= ja1 j = j6j > 0;
6
16
0
0
1
14
8
0
0
6
16
0
1
=
a1
a3
a0
a2
=
6
16
1
14
6
16
0
= 84 16 = 68 > 0; 1
14
8
0
6
16
= 800 > 0;
= 6400 > 0
0 1 14 8
The …rst n sequence of determinants are positive, the real parts of all the roots of nth degree
polynomial are negative . Therefore the time path of y(t) in above equation is convergent.
Higher Order Di¤erence Equations: Schurr Theorem
Checking convergence of a di¤erence equation (Schur determinants approach)
1
Yt+2 + Yt
6
This is a second order di¤erence equation
a0 = 1; a1 = 16 ; a2 =
;
1
2
18.3.1
=
=
a0
a2
a2
a0
1
Yt = 2
6
1
(R.1620)
1
6
> 0;
a0
0
a2
a1
a1
a0
0
a2
a2
0
a0
a1
a1
a2
0
a0
1
1
6
1
=
1
6
1
=
1
6
1
1
6
1
1
6
1
1
6
1
=
35
36
>0
1
6
1
1
6
1
1
6
1
1
6
= 0907407 > 0
1
Higher Order Di¤erence Equations: Schurr Theorem
Divide the matrix in four parts:
A
B
C
D
Start with a0 in diagonal at the upper left matrix (A), put zeros above the diagonal and successively higher subscripts down the column (A)
Matrix at the southeast corner (D) is the transpose of the northwest corner (A’);
Put an in the diagonal of the south west corner (C) and zeros above the diagonal and successively
smaller subscripts down the column of (C)
The matrix at northeast corner (B) is transpose of matrix at the southwest corner (C)
474
Roots of the polynomial are less than unity when Schur determinants are positive. Therefore
above di¤erence equation gives a convergent path.
Routh theorem used for di¤erential equations.
18.3.2
Ten Best articles in the Journal of European Economic Association
1. Frank Smets and Raf Wouters (2003) An Estimated Dynamic Stochastic General Equilibrium Model
of the Euro Area", Journal of European Economic Association, 1:5:1123-1175.
2. Jean-Charles Rochet and Jean Tirole (2003) Platform Competition in Two-Sided Markets" Journal
of European Economic Association, 1:4:990-1029.
3. Daron Acemoglu, Philippe Aghion and Fabrizio Zilibotti (2006) Distance to Frontier and Economic
Growth",Journal of European Economic Association, 4:1:37-74.
4. Alberto Alesina, Filipe R. Campante and Guido Tabellini (2008) Why is …scal policy often procyclical?Journal of European Economic Association, 6:5:1006-1036.
5. Richard Blundell, Monica Costa Dias and Costas Meghir, (2004) Evaluating the employment impact
of a mandatory job search program,Journal of European Economic Association, 2:4:569-606.
6. Ernst Fehr and John List,(2004) The hidden costs and returns of incentives— trust and trustworthiness among CEOs, Journal of European Economic Association, 2:5:743-771.
7. Jordi Galí, J. David López-Salido and Javier Vallés (2007) Understanding the e¤ects of government
spending on consumption, Journal of European Economic Association, 5:1:277-270.
8. Thomas Laubach New Evidence on the Interest Rate E¤ects of Budget De…cits and Debt, Journal of
European Economic Association, 7:4:858-885.
9. James H. Stock and Mark W. Watson (2005) Understanding changes in international business cycle
dynamics,Journal of European Economic Association, 3:5:968-1006.
10. Guido Tabellini (2010) Culture and institutions: economic development in the regions of Europe,Journal
of European Economic Association, 8:4:677-716.
18.3.3
Best 40 articles in the Journal of Economic Perspectives
David Autor (2012) The Journal of Economic Perspectives at 100, Journal of Economic Perspectives, 26,
2,Spring, 3–18
1. Porter, Michael E.;van der Linde,Claas 1995 Toward a New Conception of the Environment-Competitiveness
Relationship 9(4) 657
475
2. Kahneman, Daniel; Knetsch, Jack L.; Thaler, Richard H. 1991 Anomalies: The Endowment E¤ect,
Loss Aversion, and Status Quo Bias 5(1) 572
3. Diamond, Peter A.; Hausman, Jerry A. 1994 Contingent Valuation: Is Some Number Better than No
Number? 8(4) 524
4. Fehr, Ernst; Gächter,Simon Fairness and Retaliation: The Economics of Reciprocity 2000 14(3) 490
5. Katz, Michael L.; Shapiro, Carl 1994 Systems Competition and Network E¤ects 8(2) 448
6. North, Douglass C. 1991 Institutions 5(1) 395
7. Koenker, Roger; Hallock, Kevin F. 2001 Quantile Regression 15(4) 375
8. Markusen, James R. 1995 The Boundaries of Multinational Enterprises and the Theory of International Trade 9(2) 375
9. Bernanke, Ben S.; Gertler, Mark 1995 Inside the Black Box: The Credit Channel of Monetary Policy
Transmission 9(4) 365
10. Romer, Paul M. 1994 The Origins of Endogenous Growth 8(1) 365
11. Brynjolfsson, Erik; Hitt, Lorin M. 2000 Beyond Computation: Information Technology, Organizational Transformation and Business Performance14(4) 350
12. Nickell, Stephen 1997 Unemployment and Labor Market Rigidities: Europe versus North America
11(3) 344
13. Machina, Mark J. 1987 Choice under Uncertainty: Problems Solved and Unsolved 1(1) 338
14. Hanemann, W. Michael 1994 Valuing the Environment through Contingent Valuation 8(4) 332
15. Camerer, Colin; Thaler, Richard H. 1995 Anomalies: Ultimatums, Dictators, and Manners 9(2) 316
16. Ostrom, Elinor 2000 Collective Action and the Evolution of Social Norms 14(3) 313
17. Smith, James P. 1999 Healthy Bodies and Thick Wallets: The Dual Relation between Health and
Economic Status 13(2) 311
18. Jarrell, Gregg A.; Brickley, James A.; Netter, Je¤ry M. 1988 The Market for Corporate Control:
The Empirical Evidence since 1980 2(1) 295
19. Andrade, Gregor; Mitchell, Mark; Sta¤ord, Erik 2001 New Evidence and Perspectives on Mergers
15(2) 290
476
20. Scotchmer, Suzanne 1991Standing on the Shoulders of Giants: Cumulative Research and the Patent
Law 5(1) 280
21. Simon, Herbert A. 1991 Organizations and Markets 5(2) 278
22. Bikhchandani, Sushil; Hirshleifer,David; and Welch, Ivo 1998 Learning from the Behavior of Others:
Conformity, Fads, and Informational Cascades 12(3) 273
23. Elster, Jon 1989 Social Norms and Economic Theory 3(4) 272
24. Feenstra, Robert C. 1998 Integration of Trade and Disintegration of Production in the Global Economy 12(4) 272
25. Frank, Robert H.; Gilovich, Thomas; Regan, Dennis T. 1993 Does Studying Economics Inhibit Cooperation? 7(2) 272
26. Kirman, Alan P. 1992 Whom or What Does the Representative Individual Represent? 6(2) 272
27. Jensen, Michael C. 1988 Takeovers: Their Causes and Consequences 2(1) 268
28. Przeworski, Adam; Limongi, Fernando 1993 Political Regimes and Economic Growth 7(3) 268
29. Newhouse, Joseph P. 1992 Medical Care Costs: How Much Welfare Loss? 6(3) 265
30. Dixit, Avinash 1992 Investment and Hysteresis 6(1) 259
31. Oliner, Stephen D.; Sichel, Daniel E.2000 The Resurgence of Growth in the Late 1990s: Is Information
Technology the Story? 14(4) 257
32. Cutler, David M; Glaeser, Edward L.; Shapiro, Jesse M. 2003 Why Have Americans Become More
Obsese? 17(3) 250
33. Milgrom, Paul 1989 Auctions and Bidding: A Primer 3(3) 242
34. Portney, Paul R. 1994 The Contingent Valuation Debate: Why Economists Should Care 8(4) 239
35. Babcock, Linda; Loewenstein,George 1997 Explaining Bargaining Impasse: The Role of Self-Serving
Biases 11(1) 231
36. Grossman, Gene M.; Helpman, Elhanan 1994 Endogenous Innovation in the Theory of Growth 8(1)
225
37. Palmer, Karen; Oates, Wallace E.; Portney, Paul R. 1995 Tightening Environmental Standards: The
Bene… t-Cost or the No-Cost Paradigm 9(4) 222
477
38. Angrist, Joshua D.; Krueger, Alan B. 2001 Instrumental Variables and the Search for Identi… cation:
From Supply and Demand to Natural Experiments 15(4) 221
39. Pritchett, Lant 1997 Divergence, Big Time 11(3) 209
40. Dawes, Robyn M.; Thaler, Richard H. 1988 Anomalies: Cooperation 2(3) 206
41. Lundberg, Shelly; Pollak, Robert A. 1996 Bargaining and Distribution in Marriage10(4) 206
Reading for policy coordintion
Beetsma R. M.W.J. and H. Jensen (2005) Monetary and …scal policy interactions in a microfounded model of a monetary union Journal of International Economics, 67, 2, 320-352
Bullard J and ASingh (2008) Worldwide macroeconomic stability and monetary policy rules
Journal of Monetary Economics, 55, S34-S47
Chang, Roberto. (1997) Financial integration with and without international policy coordination ,International Economic Review, 38, 3, 547. 18p.
Canzoneri M. B., R. E. Cumby and B.T. Diba (2005) The need for international policy coordination: what’s old, what’s new, what’s yet to come? Journal of International Economics,
66, 2, 363-384
Clerc L, H. Dellas, O. Loisel (2011) To be or not to be in monetary union: A synthesis Journal
of International Economics, 83, 2, 154-167
Clarida R, Jordi Galí and M. Gertler (2002) A simple framework for international monetary
policy analysis, Journal of Monetary Economics, 49, 5, 87–904
Conconi P and Carlo Perroni (2009) Do credible domestic institutions promote credible international agreements? Journal of International Economics, 79, 1, 160-170
Cooper, R, D. DeJong , R Forsythe and T. W. Ross(1992) Communication in coordination
games, Quarterly Journal of Economics. 107 2, p739. 33p.
Currie D and P Levine (1986) Time inconsistency and optimal policies in deterministic and
stochastic worlds Journal of Economic Dynamics and Control, 10, 1–2,191-199
D. Luca , P Karadi and G. Lombardo (2013) Global implications of national unconventional
policies Journal of Monetary Economics, 60, 1, 66-85
478
Fratzscher M (2009) How successful is the G7 in managing exchange rates? Journal of International Economics, 79, 1, 78-88
Goodfriend, M.; R. G. King (1997) The New Neoclassical Synthesis and the Role of Monetary
Policy NBER/Macroeconomics Annual (MIT Press). 12 1, 231-283.
Juillard M, S. Villemot (2011)Multi-country real business cycle models: Accuracy tests and
test bench Journal of Economic Dynamics and Control, 35, 2„178-185
Levine P, A. Brociner (1994) Fiscal policy coordination and EMU: A dynamic game approach
Journal of Economic Dynamics and Control, 18, s 3–4, 699-729
Hansen L. P. and T. J. Sargent (2003) Robust control of forward-looking models Journal of
Monetary Economics, 50, 3, 581-604
Kempf H. and L. von Thadden (2013) When do cooperation and commitment matter in a
monetary union? Journal of International Economics, 91, 2, 252-262
Liu Z and E. Pappa (2008) Gains from international monetary policy coordination: Does it
pay to be di¤erent? Journal of Economic Dynamics and Control, 32, 7, 2085-2117
Marquez J (1988) International policy coordination and the reduction of the US trade de…cit,
Journal of Economic Dynamics and Control, 12, 1, 19-25
Gar…nkel M.R. (1989) Global macroeconomics: Policy con‡ict and cooperation: A review
essay Journal of Monetary Economics, 23, 2, 345-352
Pappa E. (2004) Do the ECB and the fed really need to cooperate? Optimal monetary policy
in a two-country world Journal of Monetary Economics, 51, 4, 753-779
Kose Ayhan K.M, C. Otrok and C. H. Whiteman (2008) Understanding the evolution of world
business cycles Journal of International Economics, 75, 1, 110-130
Je¤rey S. (1992) International monetary and …scal policy cooperation in the presence of wage
in‡exibilities: Are both counterproductive? Journal of Economic Dynamics and Control, 16,
2, 359-387
479
View publication stats