Fernando de Holanda Barbosa
Luiz Antônio de Lima Junior
Workbook
for Macroeconomic
Theory
Fluctuations, Inflation and Growth in
Closed and Open Economies
Workbook for Macroeconomic Theory
Fernando de Holanda Barbosa
Luiz Antônio de Lima Junior
Workbook for
Macroeconomic Theory
Fluctuations, Inflation and Growth
in Closed and Open Economies
Fernando de Holanda Barbosa
Brazilian School of Economics
and Finance
Fundação Getulio Vargas
Rio de Janeiro
Rio de Janeiro, Brazil
Luiz Antônio de Lima Junior
Faculty of Economic Sciences
Federal University of Juiz de Fora,
campus Governador Valdares (Brazil)
Governador Valadares
Minas Gerais, Brazil
ISBN 978-3-030-61547-5
ISBN 978-3-030-61548-2 (eBook)
https://doi.org/10.1007/978-3-030-61548-2
© The Editor(s) (if applicable) and The Author(s), under exclusive license to
Springer Nature Switzerland AG 2020
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Preface
This book presents the answers to the exercises in Macroeconomic Theory, Fluctuations, Inflation and Growth in Closed and Open Economies by Fernando de Holanda
Barbosa (Cham, Switzerland: Springer, 2018), hereafter referred as Macro Theory.
Altogether, there are 166 exercises in eleven chapters and three appendices.
Macro Theory points out that many of these exercises are based on, or inspired in,
the literature listed in the bibliography, although the sources are not documented.
We would like to thank the authors of these papers for shaping our understanding of
the issues addressed in Macro Theory.
It is a science tenet that models should be falsifiable representation of a
phenomenon. The goal of a good number of exercises is to help the student to
develop the skills necessary to obtain the models’ empirically testable predictions.
You should try to solve each exercise by yourself, but do not be upset if you cannot.
Some exercises are very hard and take time to work out. However, in order to learn,
you should persevere and try again and again. We hope that this workbook will help
you in the learning process of macroeconomic theory.
Macro Theory presents almost all models with continuous variables because it
is very easy to derive the qualitative results with phase diagrams. There are a few
exceptions, like Appendix C, where we present the new Keynesian model using
discrete variables.
Macro Theory departs from the praxis of using the representative agent model
to analyze the small open economy. Simpler models should be preferred to more
complex models, according to Occam’s razor. Thus, why should one not use
the very simple representative agent model to analyze the small open economy?
The answer to this question is based on an empirical stylized fact, namely that
some small economies are creditor countries, while others are debtor countries.
The representative agent model is unable to provide a long run net foreign asset
equilibrium either as a creditor or as a debtor country.
Appendix C of Macro Theory takes stock of the new Keynesian model. Its
IS curve turns the consumption Euler equation into a level curve. It is common
knowledge (so, we do not have to provide references) that the Euler equation is
a statement about the slope (smoothing) of the optimal consumption path. It does
v
vi
Preface
not say anything about the level of consumption but just states that given the level
of current consumption the Euler equation can be used to forecast the expected
level of future consumption. The new Keynesian IS curve turns this feature on its
head: the level of consumption today depends on the expected level of consumption
tomorrow. This IS curve is solved forward and implies that the effect of the rate
of interest on output gap is the same today, tomorrow and at any time in the future.
Empirical observation rejects this hypothesis. A very clever, but flawed, idea to solve
this “puzzle” is the discounted Euler equation, which transforms this equation into
a statement about the level of consumption. Exercise 14 of Chap. 7 deals with this
issue.
The organization of this workbook is the same as that of Macro Theory.
The first part deals with flexible price models and has five chapters and 41
exercises. Chapter 1 presents the representative agent model, Chap. 2 analyzes
the open economy representative agent model, Chap. 3 addresses the overlapping
generations model, Chap. 4 presents the Solow growth model and Chap. 5 introduces
endogenous savings and endogenous growth in models of economic growth.
The second part covers sticky price models, both Keynesian and new Keynesian,
and has four chapters. Chapter 6 presents the IS, LM and Phillips curves and the Taylor monetary policy rule. Chapter 7 analyzes models of economic fluctuations and
stabilization in closed economies as well as deals with the issue of chronic inflation.
Chapter 8 introduces the basic concepts of open economy macroeconomics such
as arbitrage in markets for goods and services and in asset markets. This chapter
also presents the specifications of the IS curve, Phillips curve and monetary policy
rules in an open economy. Chapter 9 deals with the models of fluctuations and
stabilization in open economies. This part has 55 exercises.
The third part has 2 chapters and 33 exercises. Chapter 10 introduces the
government budget constraint and analyzes the following topics: (1) public debt
sustainability, (2) hyperinflation, (3) Ricardian equivalence and (4) the fiscal theory
of the price level. Chapter 11 addresses several monetary theory issues, such as
price level determination, the optimum quantity of money, dynamic inconsistency,
smoothing of the interest rate by central banks, inflation targeting, operational
procedures of monetary policy and the term structure of interest rates.
Macro Theory as well as this workbook has three appendices with 37 exercises.
These appendices make the Macro Theory book self-contained. Appendix A deals
with differential equations, Appendix B presents the essential of optimal control
theory and Appendix C gives the basic tools of difference equation needed to
understand the new Keynesian model.
There are two types of exercises in Macro Theory. The first type aims to
provide the student with material to practice for a full understanding of the subjects
presented in the book. The second type of exercises addresses issues that were left
out of the book because we chose to limit the size of Macro Theory to be less than
500 pages. Those exercises can be solved using the tools presented in the chapter, or
appendix, where they belong. The exercises marked with an asterisk were prepared
for this workbook. They cover the topics that are not dealt within Macro Theory, but
we have decided to include them for the sake of completeness.
Preface
vii
The topics covered in the second type of exercises in each chapter are:
Chapter 1: Discount rate and time inconsistency; cash-in-advance (CIA) constraint
and money superneutrality; unpleasant monetarist arithmetic; incorrect specification
of the Ramsey/Cass/Koopmans model.
Chapter 2: Variable rate of interest; habit formation and small open economy model;
intertemporal approach to the balance of payments.
Chapter 3: Diamond growth model; social security system: fully funded versus payas-you-go.
Chapter 4: CES production function, Inada conditions and endogenous growth;
Solow growth model with money.
Chapter 5: Human capital growth model with leisure in the utility function; human
capital model with externalities.
Chapter 6: Calvo Phillips curve in continuous time with discount.
Chapter 7: Discounted Euler equation in the new Keynesian model.
Chapter 8: Monetary approach to the balance of payments with fixed and flexible
exchange rates; Harberger–Laursen–Metzler effect; portfolio balance approach to
exchange rates.
Chapter 9: Incorrect specification of a small open economy new Keynesian model;
tradable and nontradable goods.
Chapter 10: Tax smoothing.
Chapter 11: Consumption asset pricing model.
Appendix A: Present value models: fundamentals and bubbles; housing model;
Tobin’s q model.
Appendix B: Tobin’s q model with installation costs; dynamic inconsistency in
monetary models.
Appendix C: Taking stock of the new Keynesian model.
We would like to thank the excellent work of LATEX expert Cristina Maria Igreja
for processing and editing our original files.
Rio de Janeiro, Brazil
Fernando de Holanda Barbosa
Governador Valadares, Brazil
Luiz Antônio de Lima Junior
Contents
Part I Flexible Price Models
1
The Representative Agent Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2
The Open Economy Representative Agent Model . . . . . . . . . . . . . . . . . . . . . .
29
3
Overlapping Generations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4
The Solow Growth Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
5
Economic Growth: Endogenous Saving and Growth . . . . . . . . . . . . . . . . . . .
81
Part II Sticky Price Models
6
Keynesian Models: The IS and LM Curves, the Taylor Rule
and the Phillips Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
7
Economic Fluctuation and Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8
Open Economy Macroeconomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9
Economic Fluctuation and Stabilization in an Open Economy. . . . . . . . 161
Part III Monetary and Fiscal Policy Models
10
Government Budget Constraint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
11
Monetary Theory and Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Appendices
A
Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
B
Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
C
Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
ix
x
Contents
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Macro Theory: Errata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Author Biographies
Dr. Fernando de Holanda Barbosa is Professor of Economics at the FGV EPGE
Brazilian School of Economics and Finance. He has a Ph.D. in economics from the
University of Chicago (US). He is the author of many academic articles in monetary
economics and some are collected in the book Exploring the Mechanics of Chronic
Inflation and Hyperinflation (Springer, 2017).
Dr. Luiz Antônio de Lima Junior is Adjunct Professor of Economics, Federal
University of Juiz de Fora, campus Governador Valdares (Brazil).
xi
Part I
Flexible Price Models
Chapter 1
The Representative Agent Model
(1) The representative agent maximizes the objective function:
∞
β(t)u(c)dt
0
subject to the constraints
ȧ = ra + y − c
a(0) = a0 given
The Hamiltonian is:
H = β(t)u(c) + λ (ra + y − c)
The first-order conditions are:
∂H
= β(t)u (c) − λ = 0
∂c
(1.1)
∂H
= λr = −λ̇
∂a
(1.2)
∂H
= ra + y − c = ȧ
∂λ
(1.3)
The derivative of (1.1) with respect to time is:
β̇u (c) + βu (c)ċ = λ̇
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_1
3
4
1 The Representative Agent Model
By taking into account (1.2) we get:
−
β̇
u (c)
ċ = r +
u (c)
β
Now let’s think about a second agent that maximizes, for s > 0:
∞
β (t − s) u(c)dt
s
subject to the constraints
ȧ = ra + y − c
a(s) = as given
The Hamiltonian is:
H = β (t − s) u(c) + λ (ra + y − c)
The first-order conditions are:
∂H
= β (t − s) u (c) − λ = 0
∂c
(1.4)
∂H
= λr = −λ̇
∂a
(1.5)
∂H
= ra + y − c = ȧ
∂λ
(1.6)
The derivative of (1.4) with respect to time is:
β̇ (t − s) u (c) + β (t − s) u (c)ċ = λ̇
By taking into account (1.2) and rearranging the terms of the equation we get:
−
β̇ (t − s)
u (c)
ċ = r +
u (c)
β (t − s)
The solution for the two agents is the same when:
β̇ (t − s)
β̇(t)
=
β(t)
β (t − s)
Thus:
1 The Representative Agent Model
5
β(t) = e−ρt
β (t − s) = e−ρ(t−s)
ρ = constant
(2) The representative agent maximizes the objective function:
∞
e−ρt [u(c) + υ(m)] dt
0
subject to the constraints
y+τ =c+
Ṁ
P
M(0) given
Since m = M
P , the derivative with respect of time of real per capita money stock
is:
ṁ =
Ṁ
− πm
P
π = PṖ (inflation rate). The constraint can be rewritten:
ṁ = y − c + τ − mπ
(a) The current value Hamiltonian is:
H = u(c) + υ(m) + λ (y − c + τ − π m)
The first-order conditions are:
∂H
= u (c) − λ = 0
∂c
ρλ −
∂H
= ρλ − υ (m) + ρπ = λ̇
∂m
∂H
= y − c + τ − π m = ṁ
∂λ
∂c
(b) From u (c) = λ we may write c as a function of λ : c = c(λ), ∂λ
< 0,
since u (c) < 0. The dynamical system is:
6
1 The Representative Agent Model
Fig. 1.1 Phase diagram for
the λ and m system
λ̇ = λ (ρ + π ) − υ (m)
ṁ = y − c(λ) + τ − π m
The Jacobian of the system is:
J =
∂ λ̇
∂λ
∂ ṁ
∂λ
∂ λ̇
∂m
∂ ṁ
∂m
ρ + π −υ (m)
=
∂c
−π
− ∂λ
The determinant of this Jacobian is:
|J | = − (ρ + π ) π −
∂c υ (m) < 0
∂λ
∂c
< 0 and υ (m) < 0. Thus, the steady-state equilibrium is a
because ∂λ
saddle point. The ṁ = 0 curve slopes upward and λ̇ = 0 slopes downward,
as shown in Fig. 1.1. The arrows show the dynamics of the system and SS
is a saddle path.
(c) The marginal utility of consumption is equal to the costate u (c) = λ. By
taking derivatives with respect to time we get:
u (c)ċ = λ̇
Therefore, the equation for λ̇ can be written as:
u (c)ċ = (ρ + π ) u (c) − υ (m)
The dynamical system is given by:
υ (m)
ċ = uu(c)
(c) (ρ + π ) − u (c)
ṁ = y − c + τ − π m
1 The Representative Agent Model
7
The Jacobian of the system is:
J =
∂ ċ
∂ ċ
∂c ∂m
∂ ṁ ∂ ṁ
∂c ∂m
− υu(m)
(c)
=
−1 −π
∂ ċ
∂ ċ
The determinant of this Jacobian is:
|J | = −π
∂ ċ υ (m)
− ∂c
u (c)
The derivative of ċ with respect to c is:
∂ ċ
= (ρ + π )
∂c
u (c)
2
− u (c) u (c)
(u (c))2
+
υ (m)u (c)
(u (c))2
which can be written as:
(ρ + π ) u (c)
∂ ċ
=
∂c
2
− u (c) (ρ + π ) u (c) − υ (m)
(u (c))2
In steady-state:
(ρ + π ) u (c) = υ (m)
Thus
∂ ċ
= (ρ + π ) ≥ 0 if π ≥ −ρ
∂c
It follows that the determinant of the Jacobian, evaluated at the steady-state
point is negative. Thus, the system has a saddle point.
The ċ = 0 curve slopes upward and ṁ = 0 slopes downward, as depicted
in Fig. 1.2. The arrows show the dynamics of the system and SS is the
saddle path.
Fig. 1.2 Phase diagram for
the λ and m system
8
1 The Representative Agent Model
(3) Since c = y, u (c) = λ = constant. Thus λ̇ = 0 and υu(m)
(c) = (ρ + π ).
(a) The derivative with respect to time of the real quantity of money m = M
P
is:
ṁ
Ṁ
=
=μ−π
m
P
It follows that:
ṁ = (μ − π ) m = [(μ + ρ) − (ρ − π )] m
By taking into account the previous equilibrium conditions we get:
ṁ = (μ + ρ) m −
mυ (m)
u (c)
If lim mυ (m) = 0 we get two equilibria, m = 0 and m = m̄, as
shown in Fig. 1.3. The path EA is a hyperdeflation bubble. The path E0
is a hyperinflation path.
If limm→0 mυ (m) > 0 there is one equilibrium, as shown in Fig. 1.4. The
path EA is a hyperdeflation bubble and the path EB is a hyperinflation
bubble.
(b) When Ṁ
P = constant = f , the previous real quantity of money equation
can be written:
ṁ = f − mπ − ρm + ρm
which is equivalent to:
ṁ = ρm + f − m (ρ + π )
Fig. 1.3
limm→0 mυ (m) = 0
1 The Representative Agent Model
Fig. 1.4
limm→0 mυ (m) > 0
Fig. 1.5
limm→0 mυ (m) = 0
Fig. 1.6
limm→0 mυ (m) > 0
9
10
1 The Representative Agent Model
By taking into account the equilibrium condition we get:
ṁ = ρm + f −
mυ (m)
u (c)
If limm→0 mυ (m) = 0 we may get two equilibrium points as shown in
Fig. 1.5. There is no hyperinflation (bubble) but there is a hyperdeflation
bubble path, as shown by the arrows. There is a stable low equilibrium
(point m1 ) and an unstable high equilibrium (point m2 ).
If limm→0 mυ (m) > 0 there is just one equilibrium, which is unstable
(Fig. 1.6). There are a hyperinflation and a hyperdeflation, both are bubbles.
(4) The Leibnitz rule (Macro Theory, p. 337) for the particular case where
V (r) =
β(r)
f (x)dx
α
is given by:
dV (r)
dβ(r)
= f (β(r), r)
dr
dr
Applying this rule to:
F (θ ) =
t+θ
c(s)ds
t
we obtain:
dF (θ )
d (t + θ )
= c (t + θ )
= c (t + θ )
dθ
dθ
It follows that
d 2 F (θ )
dc (t + θ )
=
= ċ (t + θ )
dθ
dθ
The Taylor Expansion of F (θ ) around the point θ = 0 is:
F (θ ) = F (0) + F (0)θ +
F (0) 2
θ + ···
2
Thus
2
(a) F (θ ) = c(t)θ + ċ(t)
2 θ + ...
(b) M(t) ≥ F (θ )
From (a) we can write the second-order approximation:
1 The Representative Agent Model
11
M(t) ≥ c(t)θ +
c(t)θ 2
θ + ···
2
It follows that:
M(t) ≥ c(t)θ
(5) Total assets are given by:
a =k+m
Thus the dynamic equation can be written as:
ȧ = f (a − m) + τ − c − δ (a − m) − π m
The cash-in-advance constraint (CIA) is m = c. The ȧ equation becomes:
ȧ = f (a − c) + τ − c − δ (a − c) − π c
The Hamiltonian of this problem is:
H = u(c) + λ [f (a − c) + τ − (1 − δ + π ) c − δa]
(a) The first-order conditions are:
∂H
= u (c) + λ f (a − c) (−1) − (1 − δ + π ) = 0
∂c
ρλ −
∂H
= ρλ − λ f (a − c) − δ = λ̇
∂a
∂H
= f (a − c) + τ − (1 − δ + π ) c − δa = ȧ
∂λ
(b) In the steady-state equilibrium:
λ̇ = ρ − f (k̄) + δ λ = 0
Thus:
ρ = f (k̄) − δ
Therefore, money is superneutral in this model because k̄ does not depend
on the rate of growth of money.
12
1 The Representative Agent Model
(6) By taking into account the CIA and the fact that:
Ṁ
= ṁ + mπ
P
we can write:
f (k) + τ = c + k̇ + δk +
Ṁ
= m + ṁ + mπ
P
and
ṁ = f (k) + τ − (1 + π ) m
The CIA can be written as:
k̇ = m − c − δk
The representative agent maximizes the objective function:
∞
e−ρt u(c)dt
0
subject to the constraints
ṁ = f (k) + τ − (1 + π ) m
k̇ = m − c − δk
k(0) and M(0) given
(a) The Hamiltonian of this problem is:
H = u(c) + λ (m − c − δk) + μ [f (k) + τ − (1 + π ) m]
The first-order conditions are:
∂H
= u (c) − λ = 0
∂c
λ̇ = ρλ −
∂H
= ρλ − −λδ + μf (k)
∂k
μ̇ = ρμ −
∂H
= ρμ − [λ − μ (1 + π )]
∂m
In steady-state equilibrium
1 The Representative Agent Model
13
λ̇ = 0
μ̇ = 0
Thus:
λ̇ = (ρ + δ) λ − μf (k) = 0
μ̇ = μ (ρ + 1 + π ) − λ = 0
From the first expression we get:
ρ+δ =
μ f (k)
λ
and from the second expression, we obtain:
(1 + ρ + π ) =
λ
μ
The marginal product of capital is:
f (k) = (ρ + δ) (1 + i)
where i = ρ + π is the nominal interest rate. The real rate of interest is:
f (k) − δ = ρ + i (ρ + δ)
(b) Money is neutral since a change of its level does not change the nominal
interest rate.
(c) Money is not superneutral because the rate of growth of the stock of money
changes the nominal interest rate, and a change of the nominal interest rate
affects the quantity of capital in steady-state equilibrium.
(7) First, let us change notation. We denote the nominal interest rate by i instead of
r. Thus, the representative agent budget is:
(1 − τ ) (ib + y) = c +
Ṁ
Ḃ
+
P
P
and the government budget constraint is:
g + ib − τ (ib + y) =
Total assets are defined by:
Ḃ
Ṁ
+
P
P
14
1 The Representative Agent Model
a =m+b
By using the a equation we can rewrite the agent budget constraint as:
(1 − τ ) (ib + y) = c + ḃ + π b + ṁ + π m = c + ȧ + π a
Taking into account that b = a − m, we may write:
ȧ = (1 − τ ) [i (a − m) + y] − π a − c
or:
ȧ = [(1 − τ ) i − π ] a − (1 − τ ) im + (1 − τ ) y − c
The representative agent maximizes the objective function:
∞
e−ρt [u(c) + υ(m)] dt
0
subject to the previous transition equation. The Hamiltonian of this problem is:
H = u(c) + υ(m) + λ [((1 − τ ) i − π ) a − (1 − τ ) im + (1 − τ ) y − c]
The first-order conditions are:
∂H
= u (c) − λ = 0
∂c
∂H
= υ (m) − λ (1 − τ ) i = 0
∂m
ρλ −
∂H
= ρλ − λ [(1 − τ ) i − π ] = λ̇
∂a
The goods and services market is in equilibrium. Thus, c is constant and the
costate variable is constant, so λ̇ = 0. Therefore:
(ρ + π ) = (1 − τ ) i
From the second first-order condition equation we get:
υ (m) = λ (1 − τ ) i
Since u (c) = λ, we obtain:
1 The Representative Agent Model
15
υ (m)
= (1 − τ ) i
u (c)
From the differential equation of the real stock of money, we get:
ṁ = μ̄m − mπ − ρm + ρm
or:
ṁ = μ̄m − (π + ρ) m + ρm = (μ̄ + ρ) m −
υ (m)m
u (c)
The government budget constraint can be written as:
g + ib − τ (ib + y) = ṁ + mπ + ḃ + bπ
From the monetary policy rule:
ṁ = m (μ − π )
Thus, we get the following equation for the government budget constraint:
ḃ = g − τy + [(1 − τ ) i − π ] b − μ̄m
or
ḃ = ρb − μ̄m + g − τy
The dynamical system has two equations, one for m and another for b, that is:
ṁ = (μ̄ + ρ) m − m υu(m)
(c)
ḃ = ρb − μ̄m + g − τy
The Jacobian of the system is:
J =
∂ ṁ
∂m
∂ ḃ
∂m
∂ ṁ
∂b
∂ ḃ
∂b
=
∂ ṁ
0
−μ̄ ρ
∂m
where:
υ (m) + mυ (m)
υ (m) mυ (m)
∂ ṁ
= (μ̄ + ρ) −
=
μ̄
+
ρ
−
−
∂m
u (c)
u (c)
u (c)
Since:
16
1 The Representative Agent Model
υ (m)
= (1 − τ ) i = ρ + π
u (c)
we get
∂ ṁ
mυ (m)
= μ̄ + ρ − (ρ + π ) −
∂m
u (c)
In equilibrium μ̄ = π , therefore
∂ ṁ
mυ (m)
=− ∂m
u (c)
(a) The determinant of this Jacobian is positive:
|J | =
∂ ṁ
ρ>0
∂m
The trace of J is positive:
trJ =
∂ ṁ
+ρ >0
∂m
We may conclude that the steady-state equilibrium is unstable. Figure 1.7
shows the phase diagram for the dynamical system of two differential
equations, one for the real quantity of public debt (b) and the other for the
real quantity of money (m). The arrows in this figure show the dynamics of
the system.
(b) Figure 1.8 shows the monetary experiment whereby the central bank
reduces the monetary expansion rate from μ0 to μ1 at instant zero. The
adjustment of the economy is shown in Fig. 1.9. The path E0 ET shows
the system adjustment to the tight monetary policy with a ceiling of the
Fig. 1.7 The phase diagram
for the b and m system
1 The Representative Agent Model
17
Fig. 1.8 Experiment of a
tight monetary policy
Fig. 1.9 Dynamic
adjustment of the system
real stock of debt bs . The real quantity of money decreases and inflation
increases since the announcement of the tight monetary policy.
(8) When ḃ = f + ρb we may write:
d −ρs
be
= f e−ρs
ds
or:
dbe−ρs = f e−ρs ds
(a) By integrating this equation, we obtain:
T
t
dbe−ρs =
T
f e−ρs ds
t
or:
b(T )e−ρT − b(t)e−ρt =
T
t
f e−ρs ds
18
1 The Representative Agent Model
It follows that:
b(T ) = b(t)eρT e−ρT +
T
f e−ρs ds eρT
t
(b) From this expression we get:
lim λb(T )e
−ρT
T →∞
Since
= lim λb(t)e
−ρt
T →∞
∞
+ lim λ
T
T →∞
e−ρs ds =
t
fe
−ρs
ds
t
e−ρt
ρ
the former becomes:
lim λb(T )e−ρT = lim λb(t)e−ρt + lim λ
T →∞
T →∞
T →∞
e−ρt
ρ
which is equal to:
lim λb(T )e−ρT = λe−ρt b(t) +
T →∞
e−ρt
ρ
= 0
(c) The real deficit is constant:
ḃ = f
Thus,
T
t
db =
T
f ds
t
and
b(T ) − b(t) = f (T − t)
or
b(T ) = b(t) + f (T − t)
The transversality condition is:
lim λb(T )e−ρT = lim λb(t)e−ρT + lim λf (T − t) e−ρT
T →∞
T →∞
T →∞
1 The Representative Agent Model
19
It is easy to verify that:
lim λb(t)e−ρT = 0
T →∞
and
lim λ
T →∞
f (T − t)
=0
eρT
We conclude that
lim λb(T )e−ρT = 0
T →∞
(d) The conclusions drawn from items (b) and (c) are that a constant primary
deficit does not obey the transversality condition, but a constant real deficit
satisfies the transversality condition.
(9) The real business cycle model with a government is given by the dynamical
system:
k̇ = Ak α − ρ+δ
α k
k α − g − δk
K̇ = AKk α−1 − A(1−α)
β
(a) The first experiment is an unanticipated permanent increase in government
spending as described in Fig. 1.10. The k̇ = 0 curve does not shift but the
K̇ = 0 curve shifts downward as Fig. 1.11 shows.
The variable K is predetermined but the variable k can jump. At the instant
of the increase in the government expenditure k jumps to the point I , with
k(0+ ), in saddle path SS, and then converges on the new equilibrium (point
Ef ).
(b) The second experiment is an anticipated permanent increase in government
spending as described in Fig. 1.12. The k̇ = 0 will shift at instant t.
Fig. 1.10 An unanticipated
permanent increase in
government spending
20
1 The Representative Agent Model
Fig. 1.11 Dynamic
adjustment of the economy to
unanticipated change in
government spending
Fig. 1.12 Anticipated
permanent increase in
government spending
Fig. 1.13 Dynamic adjustment of the economy to anticipated change in government spending
1 The Representative Agent Model
21
However the economy jumps at instant zero in such way that at instant T
will be at the saddle path SS, as shown in Fig. 1.13. The economy converges
on the point Ef , the new steady-state equilibrium.
(c) The third experiment is an unanticipated transitory increase in government
spending as described in Fig. 1.14.
This transitory increase last until time T , when government goes back to the
previous level. Figure 1.15 shows the dynamic adjustment of the economy.
The variable k jumps to point I in such a way that by time T it will reach
the point IT , and then converges back on the previous equilibrium E0 .
(d) The fourth experiment is an anticipated transitory increase in government
spending as described in Fig. 1.16.
The k̇ = 0 will change at time T1 and it will be back to the original position
at T2 . Figure 1.17 shows the dynamic adjustment of the economy. The
variable k jumps to point I when the fiscal policy is announced. The capital
Fig. 1.14 An unanticipated transitory increase in government spending
Fig. 1.15 Dynamic adjustment of the economy to an unanticipated transitory change in government spending
22
1 The Representative Agent Model
Fig. 1.16 An anticipated transitory increase in government spending
Fig. 1.17 Dynamic
adjustment of the economy to
an anticipated transitory
change in permanent
spending
stock decreases until time T1 and starts increasing in such a way to arrive
at the saddle path at instant T2 . Then, it converges on the initial equilibrium
point E0 .
(10) The representative agent maximizes:
∞
e−ρt u(c)dt
subject to the constraints
k̇ = f (k) − (η + δ) k − c
k(0) given
The Hamiltonian of this problem is:
H = u(c) + λ [f (k) − (η + δ) k − c]
1 The Representative Agent Model
23
The first-order conditions are:
∂H
= u (c) − λ = 0
∂c
ρλ −
∂H
= ρλ − λ f (k) − (η + δ) = λ̇
∂k
∂H
= f (k) − (η + δ) k − c = k̇
∂λ
In equilibrium λ̇ = 0 we get:
ρλ = λ f (k) − (η + δ)
This is equivalent to:
f (k) − δ = ρ + n
(a) The net marginal product is not equal to the rate of time preference.
(b) The previous items result was obtained because the objective function
does not take into account the fact that population is growing at a rate
n. The objective function should be:
∞
ent e−ρt u(c)dt
0
(11) *(Real Business Cycle (RBC)). The dynamical RBC system is given from
equations (1.38) and (1.39). [Macro Theory, p. 24] by:
⎧
⎨ k̇ = Ak α − ρ+δk
(1.7)
α
⎩ K̇ = AKk α−1 −
A(1−α)k α
β
− δK
(1.8)
where k = K
L and the steady-state values for k, K and L are:
k̄ =
αA
ρ+δ
1
1−α
;
α (1 − α) A
K̄ =
β [ρ + (1 − α) δ]
L̄ =
αA
ρ+δ
α
1−α
;
(1 − α) (ρ + δ)
β [ρ + (1 − α) δ]
The steady-state value for labor (L̄) does not depend on the technology index
(A).
24
1 The Representative Agent Model
(a) Solve the RBC model using a dynamical system for k and L such as:
k̇ = G (k, L, · · · )
(1.9)
L̇ = H (k, L, · · · )
(1.10)
(b) Linearizes the dynamical system of item (a) taking a first-order Taylor
expansion around the steady-state values.
(c) Analyze an unanticipated permanent increase in the technological index
A.
(d) Analyze an unanticipated permanent decrease in the technological index
A.
(a) The first equation of the dynamical system is Eq. (1.7). To obtain the
second equation we use the definition:
L̇
K̇
k̇
=
−
L
K
k
By substituting (1.7) and (1.8) and rearranging terms results:
L̇ =
ρ + δ (1 − α) L A (1 − α) k α−1
−
α
β
(1.11)
Therefore, the RBC dynamical system is given by:
k̇ = Ak α − ρ+δ
α k
α−1
L̇ = ρ+δ(1−α)L
− A(1−α)k
α
β
(b) We linearize this system around the steady-state point k̄, L̄ as follows.
When k̇ = 0:
0 = Ak̄ α −
ρ+δ
k̄
α
Subtracting from the differential equation for k this expression yields:
k̇ = A k α − k̄ α −
ρ+δ
k − k̄
α
A first-order Taylor approximation of the first term in the right-hand side
gives:
k α = k̄ α + α k̄ α−1 k − k̄
By combining the two last expressions we obtain:
1 The Representative Agent Model
25
k̇ = −
(1 − α) (ρ + δ)
k − k̄
α
(1.12)
By the same token, when L̇ = 0
0=
ρ + δ (1 − α) L̄ A (1 − α) k̄ α−1
−
α
β
Subtracting this from the L̇ equation yields:
A (1 − α) α−1
ρ + δ (1 − α)
L − L̄ −
k
− k̄ α−1
α
β
L̇ =
By using the first-order Taylor approximation:
k α−1 = k̄ α−1 + (α − 1) k̄ α−2 k − k̄
we obtain the linear differential equation for labor:
L̇ =
A (1 − α)2 k̄ α−2 k − k̄
ρ + δ (1 − α)
L − L̄ +
α
β
Therefore, the linearized dynamical RBC system of differential equations
is given by:
k̇ = − (1−α)(ρ+δ)
k − k̄
α
2 α−2
k̄
L − L̄ + A(1−α)
L̇ = ρ+δ(1−α)
α
β
k − k̄
It should be noticed that both variables, k and L, are not predetermined.
They are jump variables related by:
kL = K
and K is a predetermined variable. The Jacobian of this system is:
J =
∂ k̇
∂k
∂ L̇
∂k
∂ k̇
∂L
∂ L̇
∂L
=
(1−α)(ρ+δ)
−
α
0
A(1−α)2 k̄ α−2 ρ+δ(1−α)
β
α
The determinant of this matrix is negative:
|J | = −
(1 − α) (ρ + δ)
α
ρ + δ (1 − α)
α
<0
Thus, the steady-state is a saddle point as depicted in Fig. 1.18. The phase
diagram of this figure shows the saddle path SS, which is downward
sloping. Labor and the capital-labor ratio should be negatively correlated.
26
1 The Representative Agent Model
Fig. 1.18 The phase diagram for the L and k system
Fig. 1.19 An unanticipated permanent increase in the technological index
(c) Figure 1.19 depicts the experiment of an unanticipated permanent increase
in the technological index A. At moment zero it increases from A0 to A1
and stays at this level forever. The steady-state equilibrium of labor will
not change as shown in Fig. 1.20.
The initial point, immediately after the change is obtained by the
intersection of the curve LK = k0 with the saddle path SS. The quantity
of labor increases from L̄ to L(0+ ), and the capital-labor ratio decreases
from k0 to k(0+ ). From that point onward labor decreases and the capitallabor ratio increases until the new steady-state is reached. The stock of
capital increases and the quantity of labor remains the same.
(d) Figure 1.21 depicts the experiment of unanticipated permanent decreases
in the technological index A.
The steady-state of labor market does not change with this experiment, as
shown in the phase diagram of Fig. 1.22. The initial point after the change
of A is given by the intersection of the curve kL = K0 with the saddle
path SS, because both variables, L and k, are jump variables, but they are
related by the curve kL = K.
1 The Representative Agent Model
27
Fig. 1.20 Dynamic adjustment of the economy to an unanticipated permanent increase in the
technological index
Fig. 1.21 An unanticipated permanent decrease in the technological index
Fig. 1.22 Dynamic adjustment of the economy to an unanticipated decrease in the technological
index
28
1 The Representative Agent Model
Thus, when the parameter A changes the quantity of labor decreases
and the capital-labor ratio increases. From that point onward the quantity
of labor increases and the capital-labor ratio decreases. The end result of
this experiment is that the capital stock decreases, but the quantity of labor
does not have a permanent change with the experiment.
Chapter 2
The Open Economy Representative
Agent Model
(1) Price index Pt is defined by
1
1−n
1−n 1−n
Pt = (1 − γ ) PH,t
+ γ PF,t
which can be written as
1−n
1−n
+ γ PF,t
Pt1−n = (1 − γ ) PH,t
or
1−n
Pt1−n = PH,t
1−γ +γ
PF,t
PH,T
1−n Taking natural logarithm in both sides we get
(1 − n) log(Pt ) = (1 − n) log PH,t + log 1 − γ + γ
PF,t
PH,t
1−n By using the notation x = log(X), we get:
(1 − n) pt = (1 − n) pH,t + log 1 − γ + γ
PF,t
PH,t
1−n (a) From this expression it follows that:
1
log (1 − γ ) + γ
pt = pH,t−1 +
1−n
PF,t
PH,t
1−n © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_2
29
30
2 The Open Economy Representative Agent Model
(b) Let us use the fact that
x = elog(x)
Thus
⎡
log ⎣(1 − γ ) + γ e
log
P
F,t
PH,t
1−n ⎤
⎦
which can be written as:
log (1 − γ ) + γ e(1−n)(pF,t −pH,t )
We can use the approximation ex ∼
= 1 + x, to write:
log (1 − γ ) + γ (1 − n) pF,t − pH,t + 1
=
log 1 + γ (1 − n) pF,t − pH,t
Since log(1 + x) ∼
= x, this expression can be approximated by:
log 1 − γ + γ
PF,t
PH,t
1−n ∼
= γ (1 − n) pF,t − pH,t
(c) From the equation of item (a) we get:
pt = pH,t +
1
γ (1 − n) pF,t − pH,t
1−n
which can be written as:
pt = (1 − γ ) pH,t + γpF,t
(2) Price index of imported good (F ) is given by:
PF,t = St Pt∗
or
pF,t = st + pt∗
2 The Open Economy Representative Agent Model
31
From the last expression of the first problem we obtain:
pt = (1 − γ ) pH,t + γ st + pt∗
The real exchange rate is defined by:
qt = st + pt∗ − pt
By combining the two equations we get:
pt = (1 − γ ) pH,t + γ (qt + pt )
which is equivalent to:
pt = pH,t +
γ
qt
1−γ
Alternative Solution
The price index Pt can be written as:
1−n
1−n
+ γ PF,t
Pt1−n = (1 − γ ) PH,t
Let’s use the following approximation:
X = elog(X) = ex = x + 1
Therefore:
e(1−n)pt = (1 − γ ) e(1−n)pH,t + γ e(1−n)pF,t
or:
(1 − n) pt + 1 = (1 − γ ) (1 − n) pH,t + 1 + γ (1 − n) pF,t + 1
This expression can be written as:
(1 − n) pt + 1 = (1 − γ ) (1 − n) pH,t + 1 − γ + γ (1 − n) pF,t + γ
or:
pt = (1 − γ ) pH,t + γpF,t
32
2 The Open Economy Representative Agent Model
(3)
(I) Show how the consumption index:
η
η−1
η−1 η−1
1
1
η
η
η
η
Ct = (1 − γ ) CH,t
+ γ CF,t
,η > 0
can be used to obtain the price index:
1
1−η
1−η 1−η
Pt = (1 − γ ) PH,t
+ γ PF,t
First, to obtain this index the following problem has to be solved.
minimize E = PH,t CH,t + PF,t CF,t
subject to the constraint:
n
n−1
n−1 n−1
1
1
η
n
n
n
+ γ CF,t
Ct = (1 − γ ) CH,t
The Lagrangian of this problem is given by:
L = PH CH,t +PF,t CF,t +λ Ct − (1 − γ )
1
n
CH,t
n−1
n
The first-order conditions are:
n
1
∂L
n
A n−1 −1 (1 − γ ) n
= PH,t −λ
∂CH,t
n−1
n
1
∂L
n
A n−1 −1 γ n
= PF,t − λ
∂CF,t
n−1
+γ
η−1
η
1
n
CF,t
CH,t
n−1
n
n n−1
n−1
n −1
=0
η−1
−1
n−1
CF,tη
=0
n
where:
η−1
η−1
1
1
η
η
η
η
A = (1 − γ ) CH,t
+ γ CF,t
By dividing the first expression by the second we get:
PH,t
=
PF,t
1−γ
γ
1
η
CH,t
CF,t
− 1
η
In order to obtain each good’s demand equation, we have to solve a
two-equation system given by the previous expression and the aggregator
2 The Open Economy Representative Agent Model
33
function of Ct , which can be written as:
η−1
η−1
η−1
1
1
Ct η = (1 − γ ) η CH,t η + γ η CF,t η
From the first-order condition, we have:
−η
CH,t = (1 − γ ) PH,t
CF,t
−η
γ PF,t
Combining the two previous equations yields:
η−1
η
Ct
1−η
1−η
= (1 − γ ) PH,t + γ PF,t
η−1
CF,tη
η−1
1−η
γ η PF,t
or:
η−1
η
η−1
1−η
Ct γ η PF,t
η
1−η
1−η η−1
= (1 − γ ) PH,t + γ PF,t
CF,t
Thus:
−η
γ PF,t Ct
CF,t = η
1−η
1−η η−1
(1 − γ ) PH,t + γ PF,t
which can be written as:
⎤−η
⎡
⎥
⎢
PF,t
⎥
CF,t = γ ⎢
1 ⎦
⎣
1−η
1−η 1−η
(1 − γ ) PH,t + γ PF,t
Ct
It is straightforward to obtain:
⎡
⎤−η
⎥
⎢
PH,t
⎥
CH,t = (1 − γ ) ⎢
1 ⎦
⎣
1−η
1−η 1−η
(1 − γ ) PH,t + γ PF,t
To obtain the price level we use the budget constraint:
Et = PF,t CF,t + PH,t CH,t
Ct
34
2 The Open Economy Representative Agent Model
and the two previous expression for CF,t and CH,t . This yields:
Et = 1−η
1−η
1−η
1−η
γ PF,t + (1 − γ ) PH,t
η Ct
1−η
1−η − 1−η
(1 − γ ) PH,t + γ PF,t
When Ct = 1, Et = Pt . Thus:
γ PF,t + (1 − γ ) PH,t
Pt = −η
1−η
1−η 1−η
(1 − γ ) PH,t + γ PF,t
Since 1 −
−η
1−η
1
= 1−η
, it follows that:
1
1−η
1−η 1−η
Pt = (1 − γ ) PH,t
+ γ PF,t
and each good’s demand equation can be written as:
PH,t −η
Ct
Pt
PF,t −η
Ct
Pt
CH,t = (1 − γ )
CF,t = γ
which solves item (b) of the problem.
(II) Show how consumption index:
CH,t =
1
CH,t (j )
θ−1
θ
θ
θ−1
dj
θ >1
,
0
can be used to obtain:
PH,t =
1
PH,t (j )1−θ dj
1
1−θ
0
First, we solve the problem:
1
minimize:
PH,t (j )CH,t (j )dj
0
subject to the constraint:
2 The Open Economy Representative Agent Model
CH,t =
35
1
CH,t (j )
θ−1
θ
θ
θ−1
dj
0
The Lagrangean of this problem is:
L=
1
0
⎧
⎨
PH,t (j )CH,t (j )dj + λ Ct −
⎩
1
CH,t (j )
θ−1
θ
⎫
θ
θ−1
⎬
dj
0
⎭
The first-order conditions is
∂L
θ
= PH,t (j ) − λ
∂CH,t (j )
θ −1
1
CH,t (j )
θ−1
θ
θ
θ−1
−1
dj
0
θ−1
θ −1
CH,t (j ) θ −1 = 0
θ
Thus:
PH,t (j ) = λ
1
CH,t (j )
θ−1
θ
1
θ−1
dj
−1
CH,t (j ) θ
0
which can be written as:
PH,t (j )−θ λθ
θ
θ−1 − θ−1
θ
C
(j
)
H,t
0
CH,t (j ) = "1
which is equivalent to:
CH,t (j ) = PH,t (j )−θ λθ CH,t
By using the aggregate CH,t , we get:
CH,t =
1
PH,t (j )
−θ θ−1
θ
λ
θθ−1
θ
0
θ−1
θ
CH,t dj
which can be simplified as:
CH,t =
1
PH,t (j )1−θ λθ−1 dj CH,t
0
Thus:
λ=
1
PH,t (j )
0
1−θ
1
1−θ
dj
θ
θ−1
36
2 The Open Economy Representative Agent Model
The demand equation for the goods are:
CH,t (j ) = PH,t (j )
1
−θ
PH,t (j )
1−θ
1
1−θ
dj
CH,t
0
CH,t =
1
PH,t (j )
1
1−θ
1−θ
θ−1
1
dj
PH,t (j )
0
1−θ
1
1−θ
dj
0
which can be written as:
⎞−θ
⎛
⎜
CH,t (j ) = ⎜
⎝
PH,t (j )
"1
1−θ dj
0 PH,t (j )
⎟
⎟
1 ⎠
CH,t
1−θ
The expenditure is given by:
Et =
1
PH,t (j )CH,t (j )dj
0
Using the previous expression yields:
Et =
1
PH,t (j ) "1
0
PH,t (j )−θ
1−θ dj
0 PH,t (j )
−θ dj Ct
1−θ
When Ct = 1, Et = Pt . Thus:
Pt =
1
PH,t (j )1−θ dj
θ
1+ 1−θ
0
or
Pt =
1
PH,t (j )1−θ dj
1
1−θ
0
Therefore, the demand equations can be written as:
CH,t (j ) =
PH,t (j )
Pt
which answers point (b) of the question.
−θ
Ct
CH,t
2 The Open Economy Representative Agent Model
37
Item (III) has the same solution of item (II).
(4) The representative agent maximizes the objective function:
∞
"t
e− 0 ρ(s)ds u(c)dt
0
subject to constraints
ȧ = ra + y − c
a(0) given
(a) Define:
δ(t) =
t
ρ(s)ds
0
Applying Leibnitz rule, we get:
δ̇ =
dδ
=ρ
dt
(b) Maximize:
∞
e−ρt e−δ+ρt u(c)dt
0
subject to the constraints:
ȧ = ra + y − c
δ̇ = ρ
a(0) = a0 given
The Hamiltonian of this problem is:
H = e−δ+ρt u(c) + λ (ra + y − c) + μρ
where λ and μ are the costate variables. The first-order conditions are:
∂H
= e−δ+ρt u (c) − λ = 0
∂c
ρλ −
∂H
= ρλ − λr = λ̇
∂a
38
2 The Open Economy Representative Agent Model
ρμ −
∂H
= ρμ − −e−δ+ρt u(c) = μ̇
∂δ
∂H
= ra + y − c = ȧ
∂λ
∂H
= ρ = δ̇
∂μ
(c) The dynamical system for the consumption and wealth variables are
obtained as follows. The costate variable is given by:
e−δ+ρt u (c) = λ
By taking derivatives with respect to time for both sides yields:
λ̇ = e−δ+ρt −δ̇ + ρ u (c) + e−δ+ρt u (c)ċ
Since ρ = δ̇ and e−δ+ρt = uλ(c) we obtain:
u (c)
λ̇
= ċ
λ
u (c)
Because λ̇λ = ρ − r we can write:
u (c)
ċ = ρ − r
u (c)
or
u (c)
ċ = − (r − ρ)
u (c)
The dynamical system is given by:
ċ = − uu(c)
(c) (r − ρ)
ȧ = ra + y − c
(d) The system is autonomous in spite of the fact that the discount rate (δ)
depends on time.
(e) The system has a stationary equilibrium.
(5) The dynamical system for the representative agent model with variable rate of
time preference is:
2 The Open Economy Representative Agent Model
39
˙ = (ρ(c) − r)
ċ = α (c, ) [ρ (c, ) − r]
ȧ = ra + y − c
where
α (c, ) =
c
ucc + −U
ρc ρcc
uc − ρc u(c)/ρ(c)
ρ (c, ) = ρ(c)
(a) Show that:
α (c, ) < 0,
∂ρ (c, )
<0
∂c
From the fisrt order conditions (p. 40):
μ̇ = rμ + e−S u(c)
Solving this differential equation, we obtain:
μ=−
∞
e−rt e−S u(c)dt
0
By hypothesis u(c) > 0, then μ < 0. From the Hamiltonian,
∂ 2H
= e−s ucc + μρcc
∂c2
2
Since μ < 0, if ρcc > 0, then ∂∂cH2 < 0.
and M are related by (p. 41):
− uc
= M = μeS < 0
ρc
c
Since −u
ρc < 0 and ρcc > 0 it follows that:
α (c, ) =
Let us show that:
c
ucc + −u
ρc ρcc
<0
40
2 The Open Economy Representative Agent Model
∂ρ (c, )
<0
∂c
The expression for ρ(c, ) (p. 41) is:
ρ (c, ) = ρ(c)
uc − ρc u(c)/ρ(c)
The derivative of ρ(c, ) with respect to c is:
u(c)
uc − ρc ρ(c)
∂ρ (c, )
∂
= ρc (c)
+ ρ(c)
∂c
∂c
uc − ρc u(c)/ρ(c)
The derivative of the second expression on the right-hand side of this
equation is:
u(c)
∂
uc − ρc
∂c
ρ(c)
= ucc − ρcc
u(c)
(ρ(c) − uc (c) − u(c)ρc (c))
− ρc
ρ(c)
ρ(c)2
After some algebra we obtain:
∂ρ (c, )
ρ(c)ucc − u(c)ρcc
=
<0
∂c
(b) Calculate the determinant of the Jacobian matrix
J =
∂ ˙
∂
∂ ċ
∂
∂ ˙
∂c
∂ ċ
∂c
The dynamical system is:
˙ = (ρ(c) − r)
ċ = α (c, ) [ρ (c, ) − r]
It follows that:
∂ ˙
= ρ(c) − r
∂
∂ ˙
∂ρ(c)
=
= ρc
∂c
∂c
∂ ċ
∂α (c, )
∂ρ (c, )
=
[ρ (c, ) − r] + α (c, )
∂
∂
∂
2 The Open Economy Representative Agent Model
41
∂α (c, )
∂ ċ
∂ρ (c, )
=
[ρ (c, ) − r] + α (c, )
∂c
∂
∂
In steady-state: ˙ = ċ = 0. Thus:
ρ(c) = r
and
ρ (c, r) = r
Therefore, the components of the Jacobian evaluated at steady-state are:
∂ ˙ ))
) =0
∂ ss
∂ ċ ))
∂ρ (c, )
) = α (c, )
∂ ss
∂
∂ ċ ))
∂ρ (c, )
) = α (c, )
∂c ss
∂c
The Jacobian evaluated at steady-state is:
0
ρc
J =
∂ρ(c,)
α (c, ) ∂ρ(c,)
α
)
(c,
∂
∂c
Thus, the determinant of this matrix is:
|J | = −ρc α (c, )
∂ρ (c, )
<0
∂
since α (c, ) < 0 and ∂ρ(c,)
< 0.
∂
(c) What happens in this economy when the foreign interest rate rises? In
steady-state ρ(c) = r. Since ρc > 0, when r rises consumption increases.
(6) The dynamical system for the representative agent model with interest rate risk
premium:
uc
∂r
ċ =
ρ−r −b
ucc
∂b
ḃ =
pc c − y
+ rb
Q
42
2 The Open Economy Representative Agent Model
∗
r = r + pr
Qb
, b > 0, pr > 0, pr < 0
y
(a) Analyze this model’s dynamics on a phase diagram with consumption (c)
on the vertical axis and the debt stock (b) on the horizontal axis. The
Jacobian of the dynamical system is:
J =
∂ ċ
∂c
∂ ḃ
∂c
∂ ċ
∂b
∂ ḃ
∂b
The derivatives of this Jacobian are:
∂ ċ
∂r
∂
uc
uc ∂
∂r
ρ−r −b
=
+
ρ−r −b
∂c
∂c ucc
∂b
ucc ∂c
∂b
∂
∂ ċ
=
∂c
∂c
uc
ucc
∂ ċ
uc ∂
∂r
=−
b
∂b
ucc ∂b
∂b
ρ−r −b
uc
=−
ucc
∂ ḃ
pc
=
∂c
Q
∂ ḃ
=r
∂b
In steady-state: ċ = ḃ = 0. Thus:
ρ−r −b
∂r
=0
∂b
and
pc c − y
+ rb = 0
Q
It follows that in steady-state:
∂ ċ
=0
∂c
∂r
∂b
∂r
∂ 2r
+b 2
∂b
∂b
2 The Open Economy Representative Agent Model
43
Fig. 2.1 The phase diagram
for the c and b system
The Jacobian evaluated at this point is:
JSS =
0 − uuccc pr + bpr
pc
r
Q
The determinant of this matrix is:
|Jss | =
pc
uc
<0
pr + bpr
ucc
Q
since ucc < 0, pr > 0 and pr > 0. Thus, the steady-state is a saddle point.
∂r
When ċ = 0, ρ − r − b ∂b
= 0. The solution of this equation gives b̄, the
steady-state of this variable. Thus, when b > b̄, ċ > 0, and b < b̄, ċ < 0,
as shown in phase diagram of Fig. 2.1. When ḃ = 0, pc c = y − rbQ, and
the ḃ = 0 curve is downward sloping. Below this curve ḃ < 0 and above
ḃ > 0. The SS curve is the saddle path of the dynamical system.
(b) Anticipated permanent change in the real exchange rate depicted in Fig. 2.2.
When Q increases, both curves, ċ = 0 and ḃ = 0, shift, as shown in
the phase diagram of Fig. 2.3.Consumption is a jump variable and b is a
predetermined variable. At the time the permanent change is announced,
consumption decreases but b stays at the same value. The curve ċ = 0 and
ḃ = 0 do not shift at moment t = 0. They will shift at moment T . The
dynamics of the model is given by the curves ċ(Q0 ) = 0 and ḃ(Q0 ) = 0.
At time T the economy will have to reach a new saddle path S1 S1 , and then
converge on the new equilibrium Ef .
(c) Anticipated transitory change in the real exchange rate depicted in Fig. 2.4.
44
2 The Open Economy Representative Agent Model
Fig. 2.2 An an anticipated
permanent increase in the real
exchange rate
Fig. 2.3 Dynamic
adjustment of the economy to
an anticipated permanent
increase in the real exchange
rate
At time zero a transitory change is announced. The real exchange rate will
increase from Q0 to Q1 at time T1 and returns to its previous value at time
T2 . Figure 2.5 shows the phase diagram of this experiment. At time zero
consumption jumps and the stock b stays the same. The economy travels
southwest according to the arrows of the ċ(Q0 ) = 0 and ḃ(Q0 ) = 0 system.
At time T1 the ċ = 0 and ḃ = 0 curves shift to ċ(Q1 ) = 0 and ḃ(Q1 ) = 0,
respectively, and the dynamics change. The economy goes northwest until
time T2 when it reaches the saddle path S0 S0 and converges on the original
equilibrium (Fig. 2.5).
2 The Open Economy Representative Agent Model
45
Fig. 2.4 An anticipated
transitory increase in the real
exchange rate
Fig. 2.5 Dynamic
adjustment of the economy to
an anticipated transitory
increase in the real exchange
rate
(7) The functional U is defined by:
∞
"υ
u(c)e− t ρ(c)ds dυ
U=
t
(a) Show that
U̇ = ρ(c)U − u(c)
Leibnitiz rule (Macro Theory, p. 137) is given by:
d
dr
β(r)
α(r)
f (x, r) dx = f (β(r), r)
β(r)
∂f (x, r) dx
dβ(r)
dα(r)
− f (α(r), r)
+
dr
dr
∂r
α(r)
Thus
"t
dU
= −u(c)e− t ρ(c)ds +
dt
∞
t
"υ
u(c) e− t ρ(c)ds ρ(c) dυ
46
2 The Open Economy Representative Agent Model
which can be written as:
∞
"υ
dU
= −u(c) + ρ(c)
u(c)e− t ρ(c)ds dυ
dt
t
Therefore:
U̇ = −u(c) + ρ(c)U
(b) What is the economic interpretation of this differential equation?
This differential equation can be interpreted as an arbitrage:
U̇ + u(c)
U
ρ(c) =
where U̇ is the capital gain (or loss), u(c) is the cash flow and U is the
“price”.
(8) The representative agent maximizes:
∞
e
"
t
− 0 ρ(c)ds−nt
u(c)dt
0
subject to:
k̇ = f (k) − (n + δ) k − c
k(0) = k0
(a) Define S =
given
"t
0 ρ(c)ds − nt. Show that:
Ṡ = ρ(c) − n
d
dS
=
dt
dt
t
ρ(c)ds − nt
0
Applying the Leibnitiz rule is straightforward to obtain:
dS
= ρ(c) − n
dt
(b) Solve the representative agent’s problem using the new state variable S.
∞
max
0
e−S u(c)dt
2 The Open Economy Representative Agent Model
subject to:
k̇ = f (k) − (n + δ) k − c
Ṡ = ρ(c) − n
k(0) and S(0) given.
The Hamiltonian is given by:
H = e−S u(c) + λ [f (k) − (n + δ) k − c] + u [ρ(c) − n]
The first-order conditions are:
∂H
= e−S u (c) − λ + μρ (c) = 0
∂c
λ̇ = −
μ̇ = −
∂H
= −λ f (k) − (n + δ)
∂k
∂H
= − (−1) e−S u(c) = e−S u(c)
∂S
∂H
= f (k) − (n + δ) k − c = k̇
∂λ
∂H
= ρ(c) − n = Ṡ
∂μ
The first-order conditions can be written as:
⎧
⎪
λ = e−S u (c) + μρ (c)
⎪
⎪
⎪
⎪
⎨ λ̇ = −λ f (k) − (n + δ)
μ̇ = e−S u(c)
⎪
⎪
⎪ k̇ = f (k) − (n + δ) k − c
⎪
⎪
⎩
Ṡ = ρ(c) − n
Let us define:
= λeS
M = μeS
We take the time derivative of to obtain:
˙ = λ̇eS + λeS Ṡ
47
48
2 The Open Economy Representative Agent Model
We substitute out for ˙ and Ṡ from the first-order conditions to get:
˙ = ρ(c) − f (k) − δ
We take the time derivative of M to obtain:
Ṁ = μ̇eS + μeS Ṡ
We substitute out for μ̇ and Ṡ from the first-order conditions to get:
Ṁ = u(c) + M [ρ(c) − n]
The first equation of the first-order conditions can be written as:
λeS = u (c) + μeS ρ (c)
or:
= u (c) + Mρ (c)
We take the time derivative of this expression to obtain:
˙ = u (c)ċ + Ṁρ (c) + Mρ (c)ċ
which can be written as:
˙ = u (c) + Mρ (c) ċ + Ṁρ (c)
We substitute out for Ṁ from the Ṁ equation to get:
˙ = u (c) + Mρ (c) ċ + [u(c) + M (ρ(c) − n)] ρ (c)
From:
= u (c) + Mρ (c)
we have:
M=
− u (c)
ρ (c)
we substitute out M to get:
− u (c) ˙
= u (c) +
ρ (c) ċ + u(c)ρ (c) + − u (c) (ρ(c) − n)
ρ (c)
2 The Open Economy Representative Agent Model
49
Since:
˙ = ρ(c) − f (k) − δ
we substitute out ˙ in the former equation to obtain the differential equation
for c:
ċ =
ρ(c) − f (k) − δ − u(c)ρ (c) + − u (c) (ρ(c) − n)
(c)
(c)
u (c) + −u
ρ
ρ (c)
Therefore, we obtain the dynamical system for c, k and :
⎧
[ρ(c,)−(f (k)−δ )]
⎪
⎪
−u (c) ⎨ ċ = u (c)+
ρ (c)
ρ (c)
k̇ = f (k) − (n + δ) k − c
⎪
⎪
⎩˙
= ρ(c) − f (k) − δ
where:
ρ (c, ) = ρ(c) −
u(c)ρ (c) + − u (c) (ρ(c) − n)
(c) This model cannot be reduced to a two of differential equations. Thus, we
cannot use a phase diagram with the two variables, c and k.
(d) In steady-state ρ(c) = n. When n decreases the steady-state capital stock
increases because ρ(c) = f (k) − δ.
(e) In steady-state ρ(c) = f (k) − δ. If the depreciation rate δ increases the
capital stock, at steady-state, decreases.
(9) The representative agent maximizes the functional:
∞
U=
e−ρt u (c, z) dt
0
where utility function u(c, z) depends on consumption (c) and on a past
consumption index z, according to:
t
z(t) =
βe−β(t−τ ) c(τ )dτ
−∞
(a) Show that:
ż = β (c − z)
Applying Leibnitiz rule we get:
t
t
d
−β(t−τ )
−β(t−t)
βe
c(τ )dτ = βe
c(t) +
β (−β) eβ(t−t) c(t)dt
dt −∞
−∞
50
2 The Open Economy Representative Agent Model
which can be written as:
ż = βc(t) − βz(t) = β (c − z)
(b) Establish the first-order condition of the following problem:
∞
max
e−ρt u (c, z) dt
0
subject to the constraints:
ȧ = ra + y − c
ż = β (c − z)
a(0) = a0 and z(0) = z0 given
The current value Hamiltonian of this problem is:
H = u (c, z) + λ (ra + y − c) + μβ (c − z)
The first-order conditions are:
∂H
= uc (c, z) − λ + μβ = 0
∂c
λ̇ = ρλ −
μ̇ = ρμ −
∂H
= ρλ − λr
∂a
∂H
= ρμ − [uz (c, z) − μβ]
∂z
∂H
= ra + y − c = ȧ
∂λ
∂H
= β (c − z) = ż
∂μ
The first-order conditions can be written as:
⎧
⎪
uc (c, z) = λ − μβ
⎪
⎪
⎪
⎪
⎨ λ̇ = λ (ρ − r)
μ̇ = (ρ + β) μ − uz (c, z)
⎪
⎪
⎪
ȧ = ra + y − c
⎪
⎪
⎩ ż = β − z)
(c
(c) From the first equation of the first-order condition, we have:
2 The Open Economy Representative Agent Model
51
λ = ucc (c, z) + μβ
Taking the time derivative of this expression yields:
λ̇ = ucc ċ + ucz ż + μ̇β
For stationary consumption we should have:
λ̇ = ż = μ̇ = 0
Thus, from
λ̇ = 0
we conclude that:
ρ = r,
the rate of the preference should be equal to the real interest rate.
(10) The representative agent maximizes:
∞
e−ρt u(c)dt
0
subject to the constraints:
ȧ = ra + y − c
a(0) = a0 given
Rate r is the foreign interest rate.
(a) Assume that r = ρ. Show that:
ȧ = y − y p
where:
yp = r
∞
e−rt ydt
0
How would you interpret this result?
(b) Assume that r = ρ and u(c) = c
where:
1− σ1
1− σ1
show that ȧ = y − y p + σ (r − ρ) W
52
2 The Open Economy Representative Agent Model
W =
∞
ye−rt dt + a0
0
(c) The Hamiltonian of this problem is:
H = u(c) + λ (ra + y − c)
The first-order conditions are:
∂H
= u (c) − λ = 0
∂c
λ̇ = ρλ −
∂H
= ρλ − λr
∂a
If r = ρ, then λ̇ = λ (ρ − r) = 0, and λ is constant. Thus, consumption
is also constant. From the budget constraint we obtain:
a0 =
∞
e−rt (c − y) dt =
0
c
−
r
∞
e−rt ydt
0
Thus:
c = ra0 + r
∞
e−rt ydt = ra0 + y p
0
Taking into account the definition of permanent income:
yp = r
∞
e−rt (y)dt
0
and by using the budget constraint it follows that:
ȧ = ra + y − c = ra + y − ra + y p
Thus:
ȧ = y − y p
The intertemporal approach to the balance of payments states that the
current account of the balance of payments is equal to the difference
between current income and permanent income. If
ȧ = y − y p > 0
2 The Open Economy Representative Agent Model
53
there is a surplus, and a deficit when
ȧ = y − y p < 0
(d) From the first-order condition and the utility function we obtain:
ċ
= σ (r − ρ)
c
and we can write:
c(t) = c0 e(r−ρ)t
Inserting this expression into the budget constraint we obtain:
a0 +
∞
ye−rt dt =
0
∞
e−rt c0 eσ (r−ρ)t dt
0
or:
a0 +
∞
ye−rt dt = c0
∞
0
e−(r−σ (r−ρ))t dt
0
We assume (r − σ (r − ρ)) > 0. Thus:
∞
e−(r−σ (r−ρ))t dt =
0
1
r − σ (r − ρ)
and the consumption c0 is:
∞
e−rt ydt
c0 = (r − σ (r − ρ)) a0 +
0
Substituting the consumption c0 into the budget constraint results:
∞
e−rt ydt
ȧ = ra0 + y − (r − σ (r − ρ)) a0 +
0
which can be written as:
∞
∞
e−rt ydtr + σ (r − ρ) a0 +
e−rt ydt
ȧ = y − r
0
0
or:
ȧ = y − y p + σ (r − ρ) W
54
2 The Open Economy Representative Agent Model
where:
y =r
p
∞
e−rt ydt
0
and:
W = a0 +
∞
e−rt ydt
0
The current account (ȧ) depends on two factors. The first is the discrepancy between income and permanent income because the agent wants
to smooth consumption. The second factor depends on the degree of
impatience of the agent. If the agent is patient (r > ρ) there will be a
current account surplus. On the other hand if the agent is impatient (ρ > r)
this factor will contribute to a deficit in the current account.
Chapter 3
Overlapping Generations
(1) The budget constraint in this two-periods-of-life model is:
c1,t +
c2,t+1
= wt
1 + rt+1
(a) The individual maximizes:
u c1,t +
1
u c2,t+1
1+ρ
subject to the previous budget constraint. The Lagrangean of this problem
is:
1
c2,t+1
L = u c1,t +
u c2,t+1 + λ wt − c1,t −
1+ρ
1 + rt+1
The first-order conditions are:
∂u c1,t
∂L
=
−λ=0
∂c1,t
∂c1,t
∂L
1 ∂u c2,t+1
=
−λ=0
∂c2,t+1
1 + ρ ∂c2,t+1
(b) Consumption for each period can be obtained by solving the system of two
equations:
∂u(c1,t )
∂c1,t
∂u(c2,t+1 )
∂c2,t+1
=
1 + rt+1
1+ρ
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_3
55
56
3 Overlapping Generations
Fig. 3.1 Intertemporal
allocation of consumption
c1,t +
c2,t+1
= wt
1 + rt+1
Figure 3.1 shows the solution of the problem, with c2,t+1 in the vertical
axis, and c1,t+1 in the horizontal axis. When c2,t+1 = 0, c1,t = wt , and
when c1,t = 0, c2,t+1 = (1 + rt+1 )wt . Both c1,t and c2,t+1 are functions of
wt , rt+1 and ρ. Thus, savings is a function of wt , rt+1 and ρ. That is:
st = s (wt , rt+1 )
(c) Assuming that consumption today and consumption tomorrow are normal
goods:
∂st
>0
∂wt
(d) When the rate of interest shifts it is common knowledge that there are an
income effect and a substitution effect. Thus:
∂st
0
∂rt+1
(e) When the utility function is isoelastic:
1
if σ = 1, u (c) = c− σ
ifσ = 1, u (c) = c−1
3 Overlapping Generations
57
The first-order condition is:
u c1,t
u c2,t+1
−1
=
c1,tσ
=
− σ1
c2,t+1
1 + rt+1
1+ρ
Thus:
c2,t+1 =
1 + rt+1
1+ρ
σ
c1,t
If we substitute out c2,t+1 in the budget constraint we obtain:
c1,t =
wt
1 + (1 + ρ)
−σ
(1 + rt+1 )σ −1
Therefore, the savings function is given by:
st = 1 −
1
1 + (1 + ρ)−σ (1 + rt+1 )σ −1
wt
Taking the derivative of st with respect to rt+1 we obtain:
∂st
(σ − 1) (1 + ρ)−σ (1 + rt+1 )σ −2
=
w
2 t
∂rt+1
1 + (1 + ρ)−σ (1 + rt+1 )σ −1
Therefore,
∂st
> 0, if σ > 1
∂rt+1
∂st
= 0, if σ = 1
∂rt+1
∂st
< 0, if σ < 1
∂rt+1
(2) In the previous question’s OLG economy, the supply side is specified as follows:
Production function: Y = F (K, L)
Population growth: Lt = (1 + n) Lt−1
Wages: wt = f (kt ) − kt f (kt )
Interest rate: rt = f (kt ) − δ
where:
58
3 Overlapping Generations
kt =
Kt
Lt
Savings equals investment:
Kt+1 − Kt + δKt = Lt st
(a) Show that:
kt+1 =
(1 − δ) kt + st (wt , rt+1 )
1+n
By dividing both sides by the working population Lt yields:
Kt
δKt
Lt s t
Kt+1
−
+
=
Lt
Lt
Lt
Lt
Kt+1 Lt+1
− kt + δkt = st
Lt+1 Lt
kt+1 (1 + n) = (1 − δ) kt + st
Thus:
kt+1 =
(1 − δ) kt + st (wt , rt+1 )
1+n
(b) Analyze the model’s equilibrium and dynamics. Since:
wt = f (kt ) − kt f (kt )
and
rt+1 = f (kt+1 ) − δ
we can write:
kt+1 =
(1 − δ) kt + st f (kt ) − kt f (kt ), f (kt+1 ) − δ
1+n
This is a nonlinear difference equation in the stock of capital goods per
worker. For every value of kt we can obtain kt+1 , but, in general, it cannot
be solved in closed form. We take the differential for both sides of this
difference equation:
(1 + n) − sr f (kt+1 ) dkt+1 = (1 − δ) − sw kt f (kt ) dkt
3 Overlapping Generations
59
Thus:
1 − δ − sw kt f (k)
dkt+1 ))
0
) =
dkt k
1 + n − sr f (k)
The model is stable if:
−1 <
dkt+1 ))
) <1
dkt k
(c) The economy resource constraint is given by:
Lt c1,t + Lt−1 c2,t + Kt+1 = F (Kt , Lt ) + (1 − δ) Kt
Dividing both sides by the number of workers at period t we obtain:
ct + (1 + n)kt+1 = f (kt ) + (1 − δ) kt
In steady-state:
kt = kt+1
Thus:
c = f (k) + (1 − δ)k − (1 + n)k
or:
c = f (k) − (n + δ) k
The golden-rule occurs when c is maximum:
dc
= f (k) − (n + δ) = 0
dk
Thus, for k = k ∗ , consumption is maximum:
f k∗ − δ = n
In the nonlinear difference equation of the OLG model the stock of capital
per worker can be larger than k ∗ . Therefore, it is possible to have overaccumulation of capital, in which case the economy will be inefficient.
(3) The utility and output functions are respectively specified by:
60
3 Overlapping Generations
1
u(c) =
c1− σ
1 − σ1
f (k) = Ak α
Savings is given by:
st = 1 −
1
1 + (1 + ρ)−σ (1 + rt+1 )σ −1
wt
Let us specialize to the case where the elasticity of substitution is equal to one:
σ = 1. Let us define:
β=
1
1+ρ
Saving is then given by:
st =
β
wt
1+β
The wage is expressed by:
wt = f (kt ) − kt f (kt ) = Aktα − kt Aαktα−1 = (1 − α) Aktα
Therefore:
st =
β
(1 − α) Aktα
1+β
The difference equation of the model is:
(1 + n) kt+1 = (1 − δ) kt +
β
(1 − α) Aktα
1+β
Let us specialize for δ = 1. Therefore:
kt+1 =
β (1 − α) A α
k
(1 + β) (1 + n) t
1
In steady-state, kt+1 = k:
β (1 − α) A
k=
(1 + β) (1 + n)
1−α
3 Overlapping Generations
61
The equilibrium is stable because:
dkt+1 ))
) =α<1
dkt SS
The golden-rule capital per worker is:
f k∗ − δ = n
In this example we assume δ = 1. Therefore:
Aαk α−1 = 1 + n
and:
k∗ =
Aα
1+n
1
1−α
The steady-state capital stock k relative to the golden-rule capital is:
k=
β (1 − α)
(1 + β) α
1
1−α
k∗
Thus, k k ∗ depending on α and β.
(4) Consider the infinite-life OLG model:
ċ = σ (r − ρ) c − nθ a
ȧ = (r − n) a + w − c
Hypothesis:
r<ρ
The Jacobian matrix is given by:
J =
∂ ċ ∂ ċ ∂c ∂a
∂ ȧ ∂ ȧ
∂c ∂a
σ (r − ρ) −nθ
=
−1
r −n
The determinant of this Jacobian is
|J | = σ (r − ρ) (r − n) − nθ
In steady-state ċ = ȧ = 0 Thus:
62
3 Overlapping Generations
Fig. 3.2 The phase diagram
for c and a system
Fig. 3.3 An unanticipated
permanent decrease in the
international interest rate
c=
nθ a
σ (r − ρ)
c = (r − n) a + w
Evaluated at steady-state the determinant of the Jacobian is negative:
|J |ss =
nθ (c − w) − nθ c
−nθ w
nθ a (r − n)
− nθ =
=
<0
c
c
c
Thus, the steady-state is a saddle point. Figure 3.2 depicts the phase diagram
of the dynamical system. The diagram was drawn assuming r < ρ. Thus, the
country is a debtor country.
(a) Unanticipated permanent decrease in the international interest rate as
depicted in Fig. 3.3. Figure 3.4 shows the phase diagram for this experi-
3 Overlapping Generations
63
Fig. 3.4 Dynamic
adjustment of the economy to
an unanticipated permanent
decrease in the international
interest rate
ment. Consumption is a jump variable and foreign debt is a predetermined
real variable. At the time of the announcement consumption jumps to the
new saddle path, and then the economy converges on the new equilibrium.
(b) Anticipated permanent decrease in the international interest rate depicted
in Fig. 3.5. At the moment of the announcement consumption jumps and
then decreases towards the new saddle path. At moment T the economy
starts to converge on the new equilibrium along the saddle path, as shown
in Fig. 3.6.
Fig. 3.5 An anticipated
permanent decrease in the
intertemporal interest rate
64
3 Overlapping Generations
Fig. 3.6 Dynamic
adjustment of the economy to
an anticipated permanent
decrease in the international
interest rate
(c) Unanticipated transitory decrease in the international interest rate as
depicted in Fig. 3.7. Figure 3.8 shows the phase diagram of the adjustment
of the economy. International debt is a predetermined variable and
consumption is a jump variable. Therefore, when the international interest
rate decreases consumption jumps to c(0+ ) and starts to decrease in such a
way as reach the saddle path SS at time T . Then, it converges back on the
old equilibrium.
Fig. 3.7 An unanticipated
transitory decrease in the
international interest rate
(d) Anticipated decrease in the international interest rate as depicted in Fig. 3.9.
Figure 3.10 shows the dynamic adjustment of the economy to the transitory
change in the rate of interest. At the moment that the change is announced
consumption will jump in such way that by time T2 the economy will be on
the saddle path SS, otherwise it will not be in equilibrium.
3 Overlapping Generations
Fig. 3.8 Dynamic
adjustment of the economy to
an unanticipated transitory
decrease in the international
interest rate
65
c
.
c(r0) = 0
S
.
c(r1) = 0
c(0+)
.
a(r1) = 0
c0
E0
S
.
a(r0) = 0
cT
a
a0
Fig. 3.9 An anticipated
transitory decrease in the
international interest rate
Fig. 3.10 Dynamic
adjustment of the economy to
an anticipated transitory
decrease in the international
interest rate
c
.
c(r0) = 0
S
.
a(r0) = 0
.
c(r1) = 0
c(0+)
c(T1)
E
.
a(r1) = 0
c0
c(T2)
S
a0
a
66
3 Overlapping Generations
(5)
(a) Consider the following model of a finite-life OLG economy with production:
ċ = σ f (k) − δ − ρ c − φθ k
k̇ = f (k) − δk − c
The Jacobian matrix is given by:
J =
∂ ċ
∂c
∂ k̇
∂c
∂ ċ
∂k
∂ k̇
∂k
=
σ f (k) − δ − ρ σf (k)c − φθ
−1
f (k) − δ
In steady-state ċ = k̇ = 0. Thus:
φθ k
σc
f (k) − δk = c
f (k) − δ − ρ =
The determinant of this Jacobian is:
|J | = σ f (k) − δ − ρ
f (k) − δ + σf (k)c − φθ
In steady-state:
φθ k f (k) − δ + σf (k) − φθ = φθ
|J |ss =
c
k f (k) − δ
− 1 + σf (k)
c
which can be written as:
|J |ss = φθ
k f (k) − δ − c
+ σf (k)
c
Since f (k) < 0 and:
w + kf (k) = c + δk
in steady-state, therefore:
|J |ss < 0
and the equilibrium is a saddle point. When:
ċ = 0
3 Overlapping Generations
67
Fig. 3.11 The phase diagram
for the consumption equation
Fig. 3.12 The phase diagram
for the c and k system
c=
φθ k
σ [f (k) − δ − ρ]
Figure 3.11 shows the phase diagram of this curve. Figure 3.12 shows the
phase diagram for the dynamical system.
(b) Is there dynamic inefficiency in the model?
When k̇ = 0, c = f (k) − δk. The golden-rule is the solution of the
equation:
dc
= f (kGR ) − δ = 0
dk
In equilibrium, ċ = 0, the rate of interest is given by:
68
3 Overlapping Generations
Fig. 3.13 An unanticipated
decrease in mortality rate
Fig. 3.14 Dynamic
adjustment of the economy to
an unanticipated decrease in
the mortality rate
f k̄ − δ = ρ +
φθ k
σ c̄
Thus,
k̄ < kGR
and there is no dynamic inefficiency in this model.
(c) Analyze the effects of an unanticipated change in mortality rate on consumption and capital. Figure 3.13 depicts an unanticipated decrease in
mortality rate φ.
3 Overlapping Generations
69
The change of mortality rate shifts ċ = 0 as shown in Fig. 3.14. Consumption is a jump variable and k is a predetermined variable.
Consumption jumps to c(0+ ), the point of the new saddle path corresponds
to k0 . Then, the economy converges on the new equilibrium Ef , where
both consumption and the stock of capital are larger than the previous
equilibrium.
(6) *(Social Security Systems) There are two canonical systems of social security:
(1) fully funded, and (2) pay-as-you-go (PAYG). In the fully funded system each
young person pays an amount It of taxes and receives when old the proceeds
of the investment. In the PAYG system each young person pays an amount It
of taxes and receives when old an amount It+1 = (1 + n) It , paid by the new
generation of young people. Will the allocation of resources be the same under
the two systems?
To answer this question, we use the model of exercise 2, which is specified by
the following equations:
Lt = (1 + n) Lt−1
(3.1)
st = s (wt , rt+1 )
(3.2)
rt = f (kt )
(3.3)
wt = f (kt ) − kt f (kt )
(3.4)
Kt+1 = Lt st
(3.5)
We assume, for simplicity, that the production function is net of depreciation.
The dynamic equation of this model is:
(1 + n) kt+1 = st f (kt ) − kt f (kt ) , f (kt+1 )
Equations (3.1), (3.3), and (3.4) do not depend on the social security system.
Equations (3.2) and (3.5) can be affected by the social security system.
Fully Funded System
The budget constraint under this hypothesis is given by:
c1t + τt +
c2t
τt (1 + rt+1 )
= wt +
1 + rt+1
1 + rt+1
or:
c1t +
c2t
= wt
1 + rt+1
70
3 Overlapping Generations
Therefore, the consumption decision will not change when social security is
fully funded. However, saving is now given by:
f
st = wt − ct − τt
Thus, for each dollar of social security contribution private savings decreases
by the same amount. The accumulation equation becomes:
f
Kt+1 = Lt st + τt = Lt (wt − ct ) Lt st
We conclude that a fully funded system will not change the allocation of
resources since total savings will remain the same.
PAYG System
When the PAYG system is in place the budget constraint becomes:
c1t + τt +
c2t
τt (1 + η)
= wt +
1 + rt+1
1 + rt+1
which can be written as:
c1t +
c2t
τt (n − rt+1 )
= wt +
1 + rt+1
1 + rt+1
There are three possibilities depending on n rt+1 . If n = rt+1 the second
term on the right-hand side of the budget constraints is equal to zero. In this
case the allocation of resources will not change.
When n = rt+1 saving will be a function of the social contribution and the
rate of growth of population:
stP AY G = s (wt , rt+1 , τt , n)
and the accumulation equation will be given by:
Kt+1 = Lt stP AY G
When n > rt+1 , consumption c1t will increase and stP AY G will decrease. In
both cases the allocation of resources will change.
Chapter 4
The Solow Growth Model
(1) Solve the Solow Model for production functions given by the following
specifications:
I. Cobb-Douglas
Y = K α (AL)1−α
This function can be written as:
Y
=
AL
K
AL
α
Thus:
f (k) = k α
The Solow model is:
k̇ = sk α − (g + n + δ) k
Multiplying both sides by
(1 − α) k −α
we obtain:
(1 − α) k −α k̇ = (1 − α) s − (1 − α) (g + n + δ) k 1−α
We use the following transformation:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_4
71
72
4 The Solow Growth Model
z = k 1−α
Taking the derivative with respect to time of this expression yields:
ż = (1 − α) k −α k̇
The Solow equation model can be written as:
ż = (1 − α) s − (1 − α) (g + n + δ) z
The solution of this linear differential equation is:
z=
s
+ Ce−(1−α)(n+δ+g)t
n+δ+g
where C is a constant that depends on initial conditions. Thus:
s
s
+ z(0) −
z=
n+δ+g
n+δ+g
e−(1−α)(n+δ+g)t
Since z = k 1−α , we obtain:
k
1−α
s
s
+ k01−α −
=
n+δ+g
n+δ+g
e−(1−α)(n+δ+g)t
II. CES:
Y = δK −θ + (1 − δ) (AL)−θ
− θ1
, θ > −1
1
The elasticity of substitution σ is given by σ = 1+θ
. When θ → −1, σ →
∞ and production is linear, when θ = 0, σ = 1 and the production function
is Cobb-Douglas, and if θ → ∞, the production function is Leontief with
fixed proportions as technology.
The CES function can be written as:
Y
= δ
AL
K
AL
−θ
− 1
θ
+ (1 − δ)
Therefore,
f (k) = δk −θ + (1 − δ)
The Solow model equation is:
− θ1
4 The Solow Growth Model
73
k̇ = s δk −θ + (1 − δ)
− θ1
− (n + δ + g) k
which can be written as:
s δk −θ + (1 − δ)
k̇
=
k
k
− θ1
− (n + δ + g)
The first term in the right-hand side that multiplies the saving parameter is
the average product of capital.
δk −θ + (1 − δ)
f (k)
=
k
k
− θ1
which can be written as:
f (k)
−1
= δ + (1 − δ) k θ θ
k
when σ > 1, −1 < θ , and limk→∞ k θ = 0. Therefore: limk→∞ f (k)
=
k
1
δ − θ > 0.
It is straightforward to show that the marginal product of capital
f (k) = δ + (1 − δ) k θ
− 1+θ
θ
Thus, when σ > 1
1
lim f (k) = δ − θ = lim
k→∞
k→∞
f (k)
k
For the elasticity of substitution greater than one (σ > 1) the Inada condition
is not satisfied.
(a) The short run rate of growth of labor productivity is given by (4.14,
Macro Theory, p. 96). The long run rate of labor productivity is equal
to g.
(b) The short run rate of growth of labor productivity is given by expression
(4.14, Macro Theory, p. 96). The long run rate of growth of labor
productivity depends on the value of the elasticity of substitution.
(c) When σ > 1, the CES production function yields the same result as an
endogenous growth model. There is no difference between the short and
long run rates of labor productivity.
(2) The Cobb-Douglas production function can be written as
74
4 The Solow Growth Model
Y
= kα
AL
or:
Y
= k α A0 egt
L
Taking the logs of both sides:
log
Y
= log (A0 ) + gt + α log(k)
L
In steady-state
sk α = (g + n + δ) k
or:
s
= k 1−α
g+n+δ
Taking the logs of both sides
log(s) − log (g + n + δ) = (1 − α) log(k)
or:
log(k) =
1
log(s) − log (g + n + δ)
1−α
Substitution for log(k) in the equation for the log of the labor productivity
yields:
Y
log
L
= log A0 + gt +
α
α
log(s) −
log (g + n + δ)
1−α
1−α
(3) Assume a Cobb-Douglas (intensive form) production function: y = Ak α hβ ,
where α is the share of capital in output and β is the share of human capital in
output. The economy, in the model including human capital, is at steady-state:
sk f (k, h) = (n + g + δk ) k
sh f (k, h) = (n + g + δk ) h
Show that the log of labor productivity is given by:
4 The Solow Growth Model
log
75
Y
α
β
= log A0 + gt +
log sk +
log sh
L
1−α−β
1−α−β
−
α
β
log (n + g + δk ) −
log (n + g + δh )
1−α−β
1−α−β
The production function can be written as:
Y
= Ak α hβ
L
By hypothesis
A = A0 egt
Thus:
Y
log
L
= log A0 + gt + α log(k) + β log(h)
At steady-state:
sk k α hβ = (n + g + δk ) k
sh k α hβ = (n + g + δh ) h
which can be written as:
sk
+ β log(h) = (1 − α) log(k)
log n+g+δ
k
sh
log n+g+δ
+ α log(k) = (1 − β) log(h)
h
This system of two equations has the solutions:
log(k) =
sh
sk
1−β
β
log
log
+
1−α−β
n + g + δh
1−α−β
n + g + δk
α
sk
sh
1−α
log(h) =
log
log
+
1−α−β
n + g + δk
1−α−β
n + g + δh
If we substitute out for log(k) and log(h) in the equation of productivity of labor
then we get
log
Y
α
= log A0 + gt +
log sk
L
1−α−β
+
β
β
log sh −
log (n + g + δh )
1−α−β
1−α−β
76
4 The Solow Growth Model
−
α
log (n + g + δk )
1−α−β
(4) The differential equations of the exogenous growth model with human capital
are given by:
k̇ = sk f (k, h) − (g + n + δk ) k
ḣ = sh f (k, h) − (g + n + δh ) h
(a) Deduce this system’s Jacobian matrix:
The Jacobian matrix is given by:
J =
∂ k̇
∂k
∂ ḣ
∂k
∂ k̇
∂h
∂ ḣ
∂h
sk fh
sk fk − (n + g + δk )
=
sh fk
sh fh − (n + g + δh )
In steady-state:
sk f (k, h) = (n + g + δk ) k
sh f (k, h) = (n + g + δh ) h
Therefore:
sk f (k, h)
k
sh f (k, h)
(n + g + δh ) =
h
(n + g + δk ) =
Thus, we can write:
sk fk − (n + g + δk ) = sk fk −
f (k, h) − kf (k)
sk f (k, h)
= −sk
<0
k
k
and:
sh fh − (n + g + δh ) = sh fh −
f (k, h) − hsh
sh f (k, h)
= −sh
<0
h
h
If we substitute out for these expressions in the Jacobian matrix, we get:
Jss =
−sk
f (k,h)−kfk
k
sh fk
sk fh
n
−sh f (k,h)−hf
h
(b) The determinant of the Jacobian Matrix is:
4 The Solow Growth Model
77
|Jss | = sk sh
f − kfk
k
f − hfh
h
− sk fh − sh fk
which can be written as:
|Jss | =
sk sh
[f − hfh − kfk ] > 0
kh
where f = f (k, h), and f − hfh − kfk > 0.
The trace of the Jacobian matrix is:
trJss = −
sk (f − kfk ) sh (f − hfh )
−
<0
k
h
(5) (a) Consider the following model:
Production Function: Y = AK + γ K α L1−α
Investment=Saving: K̇ − δK = sY
Population: L̇ = nL
The production function can be written as:
Y
K
Kα
=A +γ α
L
L
L
or:
f (k) = Ak + γ k α
The Solow model differential equation is:
k̇ = s Ak + γ k α − (n + δ) k
The economy’s growth rate of the capital stock is:
k̇
= s A + γ k α−1 − (n + δ)
k
(b) The economy’s long run growth rate of capital stock is:
k̇
= sA − (n + δ) > 0
k
and we assume that sA > n + δ.
(6) The exogenous growth model with government is specified by the following
equations:
Production Function: Y = F (K, AL)
Saving: S = s (Y − T )
Investment: S = I − K + δK
78
4 The Solow Growth Model
Government: G = T
Technological Progress: Ȧ = gA
Population: L̇ = nL
(a) Deduce the model’s differential equation for capital accumulation measured
in labor efficiency units. The production function can be written as:
Y
=F
AL
K
, 1 = f (k)
AL
From the saving-investment equation we get:
s (Y − T )
K̇
K
=
+δ
AL
AL
AL
or:
s (f (k) − τ ) = k̇ + (g + n) k + δk
Thus, the model’s differential equation for capital is:
k̇ = s [f (k) − τ ] − (g + n + δ) k
(b) The growth rate of capital is:
s [f (k) − τ ]
k̇
=
− (g + n + δ)
k
k
Thus, the government affects the short run rate of growth of output.
(c) In the long run the government affects the level of income.
(7) The Solow model with money is specified by the following equations:
Production function: y = f (k)
Assets: a = m + k
Saving: S = s (y + τ − mπ )
Investment-Saving: S = k̇ + δk + ṁ
Money demand: m = L(r)k, L < 0
Real interest rate: ρ = f (k) − δ
Monetary policy: ṁ = m (μ − π − n)
Ṁ
= constant
μ= M
Y
M
where y = L , k = K
L , m = PL
We assume that population grows at a constant rate: L̇ = nL. The
government makes lump sum (τ ) transfers of money to the private sector.
τ=
Ṁ M
Ṁ
=
= μm
PL
M PL
4 The Solow Growth Model
79
where μ denotes the growth rate of the stock of money.
The asset decision of this model is given by:
s (y + τ − mπ ) = k̇ + (n + δ) k + ṁ + nm
Since:
ṁ = m (μ − π − n)
the capital accumulation equation can be written as:
k̇ = sf (k) − (n + δ) k − (1 − s) (μ − π ) m
The inflation rate is the difference between the nominal and real rates of interest
according to the Fisher equation:
π =r −ρ
The inverse of the money demand equation is:
r=l
m
k
, l < 0
and the real rate is equal the marginal product of capital less the depreciation
rate. Thus:
m
− f (k) − δ
π =l
k
The dynamical system of this model is given by:
k̇ = sf (k) − (n + δ) k − (1 − s) m (μ − π )
ṁ = m (μ − π − n)
Where π is a function of m, k and δ as specified above. In steady-state k̇ = ṁ =
0, and μ − π = n. Thus:
sf k̄ = (n + δ) k̄ − (1 − s) m̄n
Thus, monetary policy affects the stock of capital and real income. Monetary
policy is not superneutral in this model.
(8) The Cobb-Douglas production function is given by:
Y = AK α L1−α
80
4 The Solow Growth Model
(a) Show that:
log
1
α
K
Y
=
log A +
log
L
1−α
1−α
Y
We can write the production function as:
Y α Y 1−α = AK α L1−α
Thus:
Y
L
1−α
=A
K
Y
α
Taking logs of both sides and rearranging terms we obtain:
log
1
α
K
Y
=
log A +
log
L
1−α
1−α
Y
(b) In the short run the productivity of labor increases when the capital-output
ratio increases. In the long run the productivity of labor depends on the
parameter A.
Chapter 5
Economic Growth: Endogenous Saving
and Growth
(1) Show that in the RCK model:
(a) The saving (investment) rate at stationary equilibrium is given by:
∗
s =
I
Y
∗
=
α (g + n + δ)
ρ + δ + σ1 g
The dynamical system of the RCK model is given by equation ((5.8), Macro
Theory, p. 122):
k̇ = f (k) − c − (g + n + δ) k
ċ
g
= σ f (k) − δ − ρ −
c
σ
and the transversality condition (Macro Theory, p. 123) is:
ρ−n>g−
g
σ
The saving rate in steady-state is:
s∗ =
(g + n + δ) k
(f (k) − c)
=
f (k)
f (k)
or:
s∗ =
g+n+δ
f (k)
k
From the Cobb-Douglas production function:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_5
81
82
5 Economic Growth: Endogenous Saving and Growth
f (k) = k α
the marginal product of capital is:
f (k) = αk α−1 = α
f (k)
k
thus:
f (k)
f (k)
=
k
α
In steady-state, k̇ = ċ = 0, and:
f (k) = ρ + δ +
g
σ
therefore:
s∗ = 1
g+n+δ
g
α ρ+δ+ σ
=
α (g + n + δ)
ρ + δ + σg
(b) The stationary equilibrium saving rate is lower than the share of capitaloutput:
s∗ < α
From the transversality condition, we get:
ρ+
g
>g+n
σ
We add δ to both sides to get:
ρ+
g
+δ >g+n+δ
σ
thus:
g+n+δ
<1
ρ + σg + δ
therefore:
s∗
<1
α
5 Economic Growth: Endogenous Saving and Growth
83
(2) Show that, in the Solow, RCK, and OLG models, the saving (investment) rate
s, the capital-output ratio (υ), the growth rate of output (g + n), and the
depreciation rate (δ) are connected by the following equation in the long run:
s
=g+n+δ
υ
From the previous exercise, we get:
s
=g+n+δ
k
f (k)
Denoting:
k
=υ
f (k)
we obtain:
s
=g+n+δ
υ
(3) The representative agent maximizes:
∞
e−(ρ−n)t u (c, l) dt
0
subject to the following constraints:
k̇ = k α (uh)1−α − c − (n + δ) k
ḣ = λth h
l + u + th = 1
The utility function is given by:
1
u (c, l) =
1
c1− σ + φl 1− σ
1 − σ1
(a) Write down the first-order conditions for this problem.
The Hamiltonian is:
H = u (c, l) + θ1 k α (uh)1−α − c − (n + δ) k + θ2 λ (1 − l − u) h
The first-order conditions are:
84
5 Economic Growth: Endogenous Saving and Growth
∂H
= uc − θ1 = 0
∂c
∂H
= ul + θ2 λh (−1) = 0
∂l
∂H
= θ1 k α (1 − α) (uh)−α h + θ2 λh (−1) = 0
∂u
θ̇1 = (ρ − n) θ1 −
∂H
= (ρ − n) θ1 − θ1 αk α−1 (uh)1−α − (n + δ)
∂k
θ̇2 = (ρ − n) θ2 −
∂H
= (ρ − n) θ2
∂h
− θ1 k α (1 − α) (uh)−α h + θ2 λ (1 − l − u)
∂H
= k α (uh)1−α − c − (n + δ) k = k̇
∂θ1
∂H
= λ (1 − l − u) h = ḣ
∂θ2
(b) What is the long-term growth rate of this economy’s output? By taking into
account the utility function the first-order conditions can be written as:
⎧ 1
⎪
c− σ = θ1
⎪
⎪
⎪ −1
⎪
⎪
φl σ = θ2 λh
⎪
⎪
⎪
−α
α
⎪
⎪
⎨ θ1 k (1 − α) (uh) = θ2 λ
θ̇1
α−1 (uh)1−α
θ1 = ρ + δ − αk
⎪
⎪
θ̇
⎪ 2 = (ρ − n − λ (1 − l − u)) − θ1 k α (1 − α) (uh)−α h
⎪
⎪
θ2
θ2
⎪
⎪
α (uh)1−α − c − (n + δ) k
⎪
k̇
=
k
⎪
⎪
⎪
⎩ ḣ = λ − l − u)
(1
h
Ż
Let us denote a variable with a hat its rate of growth: Ẑ = Z
.
From the first equation of the first-order conditions we have:
1
− ĉ = θ̂1
σ
From the second equation
1
− lˆ = θ̂2 + ĥ
σ
5 Economic Growth: Endogenous Saving and Growth
85
Since lˆ = 0, we obtain:
θ̂2 = −ĥ = −λ (1 − l − u)
From the fourth equation we have that:
θ̂1 = constant
Since in long run growth, k̂ = ĥ. From the third equation of the first-order
conditions we get:
θ̂1 + α k̂ − α ĥ = θ̂2
Thus:
θ̂1 = θ̂2
because k̂ = ĥ.
From the fifth equation of the first-order condition we obtain:
θ̂2 = (ρ − n) − λ (1 − l − u) −
θ1 α
k (1 − α) (uh)−α h
θ2
Since θ̂2 = −λ (1 − l − u), it follows that:
ρ−n=
θ1 α
k (1 − α) (uh)−α h
θ2
Thus:
θ̂1 − θ̂2 + α k̂ − α ĥ + ĥ = 0
we conclude that:
ĥ = 0
because θ̂1 = θ̂2 , and k̂ = ĥ. Therefore, in this model there is no long-term
output growth rate.
(c) If the parameter φ = 0 the solution is given by the Lucas model (Macro
Theory, p. 144):
g = σ (λ − ρ)
(4) The representative agent maximizes:
86
5 Economic Growth: Endogenous Saving and Growth
∞
e−(ρ−n)t u(c)dt
0
subject to the constraint:
k̇ = k α + αk − c − (n + δ) k
The utility function is given by:
1
u(c) =
c1− σ
1 − σ1
(a) Write down the first-order conditions for this problem. The Hamiltonian of
this problem is:
H = u(c) + λ k α + αk − c − (n + δ) k
The first-order conditions are:
1
∂u
∂H
=
− λ = c− σ − λ
∂c
∂c
λ̇ = (ρ − n) λ −
∂H
= (ρ − n) λ − λ αk α−1 + α − (n + δ)
∂k
∂H
= k α + αk − c − (n + δ) k = k̇
∂λ
(b) What is the long-term growth rate of the economy’s output? From the first
equation of the first-order condition we obtain:
λ̇
ċ
= −σ
c
λ
From the second equation of the first-order condition we get:
λ̇
=ρ−a+δ
λ
because:
lim k α−1 = 0
k→∞
Therefore:
5 Economic Growth: Endogenous Saving and Growth
87
ẏ
ċ
= σ (a − δ − ρ) =
c
y
which is the growth rate of output per capita.
(c) What would the answer to the previous item be if the parameter a is equal
to zero?
When the parameter a is equal a zero the dynamical system is given by:
k̇ = k α − c − (n + δ) k
ċ
α−1 − δ − ρ
c = σ αk
This system has a steady-state k̄ and c̄. Thus, the long run rate of growth of
output is equal to the rate of growth of the population.
(d) Calculate the share of labor in output for k → ∞. The share of labor in
output is given by:
w
f (k) − kf (k)
=
f (k)
f (k)
which can be written as:
w
kf (k)
=1−
f (k)
f (k)
The production function is:
f (k) = k α + ak
and the marginal product of capital is:
f (k) = αk α−1 + a
Therefore, the share of labor in output is given by:
k αk α−1 + a
w
=1−
f (k)
k α + ak
or:
w
αk α + ak
=1− α
f (k)
k + ak
It is straightforward to show that:
88
5 Economic Growth: Endogenous Saving and Growth
αk α
ak
αk α + ak
= lim
+ α
lim
k→∞ k α + ak
k→∞ k α + ak
k + ak
α
1
= lim
+ α−1
=1
k
k→∞ 1 + ak 1−α
+1
α
Therefore,
w
lim
k→∞ f (k)
=1−1=0
(5) The representative agent maximizes:
∞
e−ρt u(c)dt
0
subject to the constraint:
k̇ = k α (uh)1−α h̄γ − c
ḣ = λ (1 − u) h
(a) Write down the problem’s first-order conditions and the equations for the
costate variables. The Hamiltonian of this problem is given by:
H = u(c) + θ1 k α (uh)1−α h̄γ − c + θ2 λ (1 − u) h
The utility function is given by:
1
u(c) =
c1− σ
1 − σ1
The first-order conditions are:
1
∂u
∂H
=
− θ1 = c− σ − θ1 = 0
∂c
∂c
(5.1)
∂H
= θ1 k α (1 − α) (uh)−α hh̄γ − θ2 λh = 0
∂u
(5.2)
θ̇1 = ρθ1 −
∂H
= ρθ1 − θ1 αk α−1 (uh)1−α h̄γ
∂k
(5.3)
5 Economic Growth: Endogenous Saving and Growth
θ̇2 = ρθ2 −
89
∂H
= ρθ2 − θ1 k α (1 − α) (uh)−α (uh)−γ − θ2 λ (1 − u)
∂h
(5.4)
∂H
= k α (uh)1−α h̄γ − c = k̇
∂θ1
(5.5)
∂H
= λ (1 − u) h = ḣ
∂θ2
(5.6)
(b) What is the economy’s growth rate of labor productivity?
From (5.1):
ĉ = g = −σ θ̂1
(5.7)
A hat over a variable denotes its rate of growth:
x̂ =
ẋ
x
From (5.3):
θ̂1 = ρ − αk α−1 (uh)1−α h̄γ
which can be written as:
θ̂1 = ρ − αk α−1 (u)1−α h1+γ −α
(5.8)
g = σ αk α−1 u1−α h1+γ −α − ρ
(5.9)
From (5.7) and (5.8)
or:
k α−1 u1−α h1+γ −α =
ρ + σg
α
(5.10)
c
k
(5.11)
From (5.5)
k̂ = k α−1 u1−α h1+γ −α −
The first term of the right-hand side of (5.11) is given by (5.10). Thus:
k̂ =
ρ + σ1 g
c
−
α
k
90
5 Economic Growth: Endogenous Saving and Growth
If k̂ is constant it follows that kc is constant. Therefore:
ĉ = k̂ = g
From (5.10) we obtain:
(α − 1) k̂ + (1 + γ − α) ĥ = 0
Thus,
ĥ =
(1 − α) g
(1 − α) k
=
1+γ −α
1+γ −α
(5.12)
Next, we determine g as a function of parameters of the model. From (5.2):
θ1
−1
= λ k α (1 − α) u−α hγ −α
θ2
(5.13)
θ1 α
k (1 − α) u−α uhγ −α − λ (1 − u)
θ2
(5.14)
From (5.4),
θ̂2 = ρ −
The ratio θθ12 is given by (5.13), thus:
θ̂2 = ρ − λ k α (1 − α) u−α hγ −α
−1 α
k (1 − α) u−α uhγ −α − λ (1 − u)
Thus, we get:
θ̂2 = ρ − λu − λ (1 − u) = ρ − λ
(5.15)
θ̂1 − θ̂2 = − α k̂ + (γ − α) ĥ
(5.16)
From (5.13):
We substitute for θ̂1 and θ̂2 from (5.7) and (5.15) to get:
1
− g − (ρ − λ) = −αg − (γ − α) ĥ
σ
and we obtain:
ĥ =
(1 − ασ ) g + σ (ρ − λ)
(γ − α) σ
(5.17)
5 Economic Growth: Endogenous Saving and Growth
91
From (5.12) and (5.17) we obtain the following equation to determine g:
(1 − ασ ) g + σ (ρ − λ)
[(1 − α) g]
=
1+γ −α
(γ − α) σ
Thus,
g=
σ (λ − ρ) (1 + γ − α)
(1 + γ − α) − σ γ
(5.18)
We substitute for g in (5.12) to obtain:
ĥ =
(1 − α) σ (λ − ρ)
(1 + γ − α) − σ γ
(5.19)
If γ = 0, the rate g growth of human capital is:
ĥ = σ (λ − ρ)
which is the equation (5.63) of Macro Theory, p. 144. Equation (5.19) can
be written as:
ĥ =
(1 − α) σ (λ − ρ)
1 − α + (1 − σ ) γ
(5.20)
Thus,
ĥ σ (λ − ρ) , if σ 1
(c) What is the difference between the rates of growth of private and social
labor productivity in this economy?
When the human capital externality is present (γ > 0) the representative
agent problem is different from the social planner problem because the
planner internalizes the externality. The current-value Hamiltonian is the
same of item a:
H = u(c) + θ1 k α u1−α h1+γ −α − c + θ2 λ (1 − u) h
The first-order conditions of the social planner problem are the same of the
agent, except for the costate equation of the price of human capital, which
is given by:
∂H
= ρθ2 − θ1 k α u1−α (1 + γ − α) hγ −α − θ2 λ (1 − u)
∂h
(5.21)
We use Eq. (5.13):
θ̇2 = ρθ2 −
92
5 Economic Growth: Endogenous Saving and Growth
θ1
−1
= λ k α (1 − α) u−α hγ −α
θ2
to write Eq. (5.21) as:
θ̂2 = ρ − λ −
γ λu
1−α
To get rid of the variable u we use the human capital production function:
ĥ = λ (1 − u)
Thus,
λu = λ − ĥ
Therefore:
θ̂2 = ρ − λ −
γ λ − ĥ
1−α
We know, from (5.13), that:
θ̂1 − θ̂2 = −αg − (γ − α) ĥ
We write this equation as:
γλ g
λ − ĥ = −αg − (γ − α) ĥ
− − ρ−λ−
σ
1−α
taking into account (5.22) and (5.7). From this equation we obtain:
ĥ =
(1 − α) g α − σ1 + (1 − α) (λ − ρ) + γ λ
α (1 + γ − α)
From (5.12),
ĥ =
(1 − α) g
1+γ −α
Therefore:
(1 − α) g
=
1+γ −α
(1 − α) g α − σ1 + (1 − α) (λ − ρ) + γ λ
α (1 + γ − α)
(5.22)
5 Economic Growth: Endogenous Saving and Growth
93
Solving this equation we get the expression for g:
g = σ (λ − ρ) +
σγλ
1−α
Thus, the rate of growth of human capital is:
ĥ =
σγλ
(1 − α) g
(1 − α)
=
σ (λ − ρ) +
1+γ −α
1−α
(1 + γ − α)
which can be written as:
ĥ = σ
(1 − α) (λ − ρ) + γ λ
1+γ −α
or:
ĥ = σ λ −
(1 − α) ρ
1+γ −α
When γ = 0, we obtain:
ĥ = σ (λ − ρ)
(d) How the government might solve the previous item’s distortion: The market
solution is not optimal because the agents do not take into account the
externality of investing in human capital. To correct this distortion a subsidy
to human capital can induce the agents to invest more and to provide the
optimal amount of capital.
(6) *(Increasing Returns with Productive Externalities). The production function in
the Ramsey/Cass/Koopmans model takes the form F (K, L, K) for each of the
N firms, where F has constant returns to scale in own capital and labor, taking
aggregate capital as given. Assume the following specification for F :
F (K, L, K) = K α L1−α K η
and the utility function is:
u(c) = log(c)
(a) What is the rate of growth of the economy when α + η < 1?
(b) What is the rate of growth of the economy when α + η = 1?
The agent maximizes:
94
5 Economic Growth: Endogenous Saving and Growth
∞
e−ρt u(c)dt
0
subject to:
k̇ = f (k, K) − c
k(0) given
The current-value Hamiltonian is:
H = u(c) + λ [f (k, K) − c]
The first-order conditions are:
∂H
= u (c) − λ = 0
∂c
λ̇ = ρλ −
∂H
= ρλ − λf (k, K)
∂k
∂H
= f (k, K) − c = k̇
∂λ
and the transversality condition is:
lim λke−ρt = 0
t→∞
The production function, net of depreciation, in intensive form is:
f (k, K) = k α K η
The marginal product of capital from the point of view of the firm is:
f (k, K) = αk α−1 K η
Since there are N firms, K = Nk. Thus:
f (k, K) = αk α+η−1 N η
Thus, the costate equation is:
λ̇
= ρ − αk α+η−1 N η
λ
From the logarithmic utility function it is straightforward to obtain:
5 Economic Growth: Endogenous Saving and Growth
95
Fig. 5.1 The phase diagram
for the c and k system when
α+η <1
λ̇
ċ
=−
c
λ
Combining those two expressions gives:
ċ
= αk α+η−1 N η − ρ
c
Therefore, the differential equations for k and c are:
k̇ = N η k α+η − c
ċ
α+η−1 N η − ρ
c = αk
(a) Hypothesis: α + η < 1. The Jacobian of the system under this hypothesis
is:
∂ k̇ ∂ k̇
(α + η) N η k α+η−1 −1
∂k
∂c
J = ∂ ċ ∂ ċ =
cα (α + η − 1) k α+η−2 0
∂k ∂c
The determinant of this matrix is negative if α + η − 1 < 0,
|J | = cα (α + η − 1) k α+η−2 < 0
and the equilibrium is a saddle point. Figure 5.1 depicts the phase diagram
of the model.
(b) Hypothesis: α + η = 1. Under this hypothesis the rate of growth of
consumption per capita g = ċc is given by
g = αN η − ρ
96
5 Economic Growth: Endogenous Saving and Growth
Therefore, if αN η − ρ > 0 there is endogenous growth and there is no
stationary point. From the transition equation results:
c
k̇
= Nη − = g
k
k
and:
k(t) = k0 egt
c(t) = c0 egt
Therefore, the ratio kc is equal to:
c0 egt
c(0)
c
=
=
gt
k
k0 e
k(0)
It follows that:
k̇
c(0)
= Nη −
=g
k
k(0)
From this equation we obtain the initial value of consumption:
c(0) = k(0) N η − g
Part II
Sticky Price Models
Chapter 6
Keynesian Models: The IS and LM
Curves, the Taylor Rule and the Phillips
Curve
(1) Suppose that investment depends on the real income level according to:
i = i r − π e, y
Does the IS curve always have a negative slope?
In this question r is the nominal rate of interest. Let us denote by ρ the real rate
and assume perfect foresight:
ρ = r − πe
The IS curve assumes that output is given by demand:
y = c(y) + i (ρ, y)
Differentiating this equation, we get:
dy = cy dy + iρ dρ + iy dy
Therefore:
iρ
dy
=
dρ
1 − cy − iy
Thus, if: 1 − cy − iy < 0,
dy
>0
dρ
We conclude that the IS curve may have an upward slope.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_6
99
100
6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve
(2) Assume that consumption depends on disposable income (y d ), c = c(y d ), and
that disposable income is defined by:
yd = y − g
where y is the real income and g is government spending.
(a) Why could you define disposable income in this way?
This definition assumes Ricardian equivalence, e.g., government expenditures have to be paid now or in the future.
(b) Would a tax cut, for a given level of g, affect expenditure in this economy?
No, because consumption depends on disposable income, and disposable
income, as defined in this exercise, is not affected by tax.
(3) Assume that consumption depends on disposable income (y d = y − τ ) and the
real quantity of money, m = M
P : c = c (y − τ, m).
(a) Is the IS curve independent from monetary policy?
No. Monetary policy affects the real quantity of money. An easy monetary
policy increases m and thus consumption. This is known in the literature as
the real cash balance effect due to Pigou.
(b) Is the full-employment real interest rate independent from monetary policy?
The full-employment real interest rate (r̄), the natural rate of interest, is
obtained from the IS curve when output is at its full-employment level,
y = ȳ. Thus:
ȳ = c (ȳ − τ, m) + i (r̄)
Therefore, r̄ depends on the real quantity of money m.
(4) Consider the model:
I S : y = c (y − τ ) + i(i) + g
LM :
M
= L (y, i)
P
MP R : i = ī
When the Central Bank sets this economy’s interest rate according to this
monetary policy rule, is the economy’s price level determined?
This model assumes that the expected rate of inflation is equal to zero. Thus, the
nominal rate of interest is equal to the real rate of interest. Output is determined
by the IS curve:
I S : y = c (y − τ ) + i(ī) + g
6
Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .
101
Therefore, given this output and the fixed interest rate, the LM curve determines: m = M
P :
M
= L y, ī
P
and not the price level. Neither the stock of money nor the price level are
determined. The economy has no anchor.
(5) Assume that the utility function is:
1
u (ct ) = − e−αc , a > 0
α
(a) How do you interpret parameter α?
The absolute risk premium is defined by:
Eu (x + ) = u (x − pr )
where , a stochastic variable has mean zero, E = 0, and variance E 2 =
σ 2 . A Taylor expansion of u (x + ) is given by:
u (x + ) = u(x) + u (x) +
u (x) 2
2
and a Taylor expansion of u (x − pr ) is:
u (x − pr ) = u(x) + u (x) (−pr )
By using the definition of the risk premium we obtain:
u(x) +
u (x) 2
σ = u(x) + u (x) (−pr )
2
and the risk premium is:
pr = −
u (x) 2
σ
2u (x)
This premium is known as the absolute risk premium:
ar = −
For the utility function:
u (x)
u (x)
102
6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve
1
u(c) = − e−αc
α
the two derivatives are given by:
u (c) = e−αc
u (c) = −αe−αc
and the absolute risk premium is:
−
−αe−αc
u (c)
=
−
=α
u (c)
e−αc
(b) Use the Euler equation to deduce the equation of the IS curve associated
with this utility function.
The Euler equation is given by:
u (ct ) = β (1 + rt ) u (ct+1 )
and
u (ct ) = e−αct , u (ct+1 ) = e−αct+1
Substituting this expression into the Euler equation results:
e−αct = β (1 + rt ) e−αct+1
By taking the log of both sides we get:
−αct = log β (1 + rt ) − αct+1
The economy only has consumer goods. Thus,
yt = ct , yt+1 = ct+1
Therefore:
yt = yt+1 −
1
log β (1 + rt )
α
which can be written as:
yt − ȳt = yt+1 − ȳt+1 + ȳt+1 −
1
(rt − ρ)
α
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Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .
103
1
where β = 1+ρ
, and ȳ denotes potential output. We can write this IS curve
as:
xt = xt+1 −
1
(rt − r̄)
α
where the natural rate of interest is:
r̄ = ρ + αȳt+1
where ȳt+1 is the rate of growth of potential output per-capita.
(6) The marginal utility of consumption in period t + 1 may be written as a
function of the marginal utility in period t, of the derivative of the marginal
utility of period t, and of the difference between consumption tomorrow and
consumption today, according to the following Taylor expansion:
u (ct+1 ) ∼
= u (ct ) + u (ct ) (ct+1 − ct )
This expansion disregards second-order terms. Show that the Euler equation
with continuous variable is:
ċ = −
u (c)
(r − ρ)
cu (c)
The Euler equation is given by:
u (ct ) = β (1 + rt ) u (ct+1 )
Substituting the Taylor expansion into this equation results:
u (ct ) =
1 + rt u (ct ) + u (ct ) (ct+1 − ct )
1+ρ
which can be written as:
1+ρ
u (ct )
= 1+ (ct+1 − ct )
1 + rt
u (ct )
Taking the logs of both sides and using the approximation log (1 + x) ∼
= x, we
obtain:
ρ − rt = ct
u (ct ) (ct+1 − ct )
u ((ct )
ct
Taking into account that the rate of growth of consumption:
104
6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve
(ct+1 − ct ) ∼ d
log ct = ċ
=
ct
dt
the Euler equation becomes:
ċ = −
u (c)
(r − ρ)
cu (c)
(7) Suppose that the output gap depends on a lagged output gap and on the interest
rate gap according to:
xt = λxt−1 − α (rt − r̄t )
xt = λxt+1 − α (rt − r̄t )
The parameter λ lies between zero and one and the parameter α is positive.
(a) Write down the output gap, respectively, as:
xt = −α
−∞
+
λi (rt−i − r̄t−i )
i=0
xt = −α
∞
+
λi (rt+i − r̄t+i )
i=0
We use the log operator L, Lzt = zt−1 , to write the first equation as:
(1 − λL) xt = −α (rt − r̄t )
which can be written as:
xt = −
α
(rt − r̄t ) = −α 1 + λL + λ2 L2 · · · · · · (rt − r̄t )
(1 − λL)
By using the operator L we get:
xt = −α (rt − r̄t ) − αλ rti − r̄t−1 − α 2 λ (rt−2 − r̄t−2 )
or:
xt = −α
−∞
+
i=0
λi (rt−i − r̄t−i )
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Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .
105
We use the forward operator F , F zt = zt+1 , to write the second equation
as:
(1 − λF ) xt = −α (rt − r̄t )
which can be written as:
xt = −
α
xt = −α 1 + λF + λ2 F 2 · · · · · · (rt − r̄t )
(1 − λF )
By using the operator F we obtain:
xt = −α (rt − r̄t ) − αλ (rt+1 − r̄t+1 ) − αλ2 (rt+2 − r̄t+2 )
or:
xt = −α
∞
+
λi (rt+i − r̄t+i )
i=0
(b) What happens when the parameter λ is equal to one? Can the output gap be
written as a function of the past (future) interest rate gaps?
When λ = 1, the first equation is:
xt = xt−1 − α (rt − r̄t )
This equation can have
either a backward or a forward solution:
,−∞
backward: xt = −α
i=0 (rt−i − r̄t−i )
,
forward: xt = α ∞
i=0 (rt+1+i − r̄t+1+i )
To check that both are solutions just subtract xt−1 from xt to find that:
xt − xt−1 = −α (rt − r̄t )
When λ = 1, the second equation is:
xt = xt+1 − α (rt − r̄t )
This equation can ,
have either a backward or a forward solution:
backward: xt = α ,−∞
i=0 (rt−1−i − r̄t−1−i )
forward: xt = −α ∞
i=0 (rt+i − r̄t+i )
To check that both are solutions just subtract xt+1 from xt to find that:
xt − xt+1 = −α (rt − r̄t )
(8) The consumer’s utility function is:
106
6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve
1
u(c) =
c1− σ − 1
1 − σ1
Deduce the money demand equation when the utility function of money is
specified by:
(a) υ(m) = m 1−λ−1 , λ = 1, and υ(m) = log(m), λ = 1
The first-order condition to obtain the demand for money is:
1−λ
υ (m)
=i
u (c)
From the utility function we get:
υ (m) = m−λ
1
u (c) = c− σ
Therefore:
i=
υ (m)
m−λ
=
1
u (c)
c− σ
Taking the logs of both sides we obtain the following demand for money
equation:
log(m) =
1
1
log(c) − log(i)
σλ
λ
(b) υ(m) = m (α − β log(m)), β > 0. The marginal utility of money is:
υ (m) = α − β log(m) + m −
β
m
or:
υ (m) = α − β log(m) − β
Therefore:
i=
which can be written as:
α − β − β log(m)
υ (m)
=
1
u (c)
c− σ
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Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .
107
1
c− σ
α−β
−
i
log(m) =
β
β
When consumption is constant this is the Cagan money demand equation.
(9) When uncertainty is introduced into the transaction cost model, the bank solves
the following problem:
min {E [iR + c (t, T )]}
t
(a) Show that this problem’s first-order condition implies the following
turnover ratio:
γ
K
∗
t =
αET δ
1
where K = E{iT } and γ = 1+β
From Macro Theory (p. 174),
t=
α β
T
, C (t, T ) =
t −1 Tδ
R
β
The problem is:
α β
T
δ
t −1 T
min E i +
t
t
β
The first-order condition is:
α δ β−1
1
=0
E iT − 2 + T βt
β
t
Which can be written as:
iT
t2
E
= αt β−1 ET δ
It is straightforward to obtain the optimum turnover ratio:
∗
t =
E (iT )
αET δ
1
β+1
(b) Assume, for simplicity, that T is log-normally distributed. For a normal X,
it is known that:
108
6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve
1
E exp (τ X) = exp μτ + σ 2 τ 2
2
Show that this expression can be used to calculate the mathematical
expectation of T δ : ET δ = E exp (δ log(T )) and obtain:
log(t) = γ log
K
α
1
− γ δE log(T ) − γ δ 2 Var log(T )
2
where Var represents variance. T is log-normally distributed. We may write
the expectation of T δ as:
δ
ET δ = Eelog T = Eeδ log T = E exp (δ log T )
T is log-normally distributed. Thus, log T is normally distributed, hence:
1
ET δ = eδE log T + 2 γ δ Var log(T )
2
From the first-order condition:
1 2
E (iT ) = αt β+1 eδE log T + 2 δ Var log T
Taking the logs of both sides:
log
E (it)
1
= (β + 1) log t + δE log T + δ 2 Var log T
α
2
By rearranging terms of this expression we obtain the t ratio equation:
K
1
log
log t =
1+β
α
δ
1
−
E log T −
1+β
2
δ2
1+β
Var log(T )
(c) In this model, does the volatility of parameter (T ) affect the turnover? From
the last equation we obtain:
∂ log t
1 δ2
=−
<0
∂Var log T
21+β
Hence, the volatility of payments (t) affects the turnover rate.
(10) Consider the model:
Aggregate demand: m + υ = p + y
Aggregate supply: p = pe + δ (y − ȳ)
Assuming rational expectations, what would the expected value of the price
index be?
6
Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .
109
The two equations of the model can be written in matrix form as
1 1
p
m+υ
= e
1 −δ y
p − δ ȳ
The solution is:
1
δ 1
p
m+υ
=
y
1 + δ 1 −1 pe − δ ȳ
Hence:
p=
1
δ (m + υ) + pe − δ ȳ
1+δ
Rational expectations imply that:
pe = p
Therefore,
pe = m + υ − ȳ
(11) The income velocity of money is the ratio of nominal output to the nominal
stock of money. That is:
Py
Y
=
M
M
(a) When the income elasticity of money is equal to one, does its velocity
depend on real output?
The velocity of money is equal to the real output/real cash balance ratio:
V =
y
y
Py
= M =
M
m
P
The demand for money is a function of real income and nominal interest
rate. Hence:
V =
y
y
=
m
L (y, i)
or:
log V = log y − log L (y, i)
110
6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve
The elasticity of velocity with respect to real income is:
∂ log L
∂ log V
=1−
∂ log y
∂ log y
Therefore, if:
∂ log L
=1
∂ log y
it follows that:
∂ log V
=0
∂ log y
(b) Does the interest rate affect the velocity?
The elasticity of the velocity of money with respect to the nominal interest
rate is:
∂ log V
∂ log L (y, i)
=−
>0
∂ log i
∂ log i
This elasticity is positive because the elasticity of the real quantity of money
with respect to the nominal interest rate is negative.
(c) Define k = V1 . What is the unit of k?
k=
M
1
=
V
Y
M is measured, say, in dollars and Y in dollars per unit of time. Hence, the
unit of k is time.
(12) The former German Central Bank (the Bundesbank) carried out monetary
programming based on the following identity:
MV = P y
Assume that potential output of the German economy grows at an annual rate
of 2.5%. The Bundesbank annual inflation target was 2.5%.
(a) What information did the Bundesbank need to calculate the corresponding
growth rate of M?
Let a hat denote the rate of growth: xẋ = x̂. From the identity MV = P y
we get:
M̂ = P̂ + ŷ − V̂
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Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .
111
The assumptions are P̂ = 2.5% and ŷ = 2.5%. Hence the Bundesbank
would need information to predict the rate of growth of the velocity of
money.
(b) How would you obtain this information?
From exercise (11) the velocity of money is a function of real income and
the nominal interest rate:
V = V (y, i)
To predict V we would need to estimate this function.
(c) Assume unstable income velocity. Would you adopt the same method?
If V = V (y, i) is unstable we would not have a good forecast for V . Hence,
we would not adopt the Bundesbank’s method.
(13) Assume that the elasticity of money demand with respect to the interest rate is
equal to minus infinity (liquidity trap).
(a) Use the identity MV = P y to show why the monetary policy does not
affect the economy’s output.
Let us assume that the money demand function is specified by:
log m = δ +
β
, β > 0, γ > 0
i−γ
The elasticity of money demand with respect to the interest rate is given by:
ηm,i = i
β
∂ log m
= −
2
∂i
i−γ
i
i
lim ηm,i = −∞
i→γ
Hence, i = γ is a liquidity trap as shown in Fig. 6.1. Any change in
the money stock will be held by the public, without changing the interest
rate. Hence, M does not change, if M increases V decreases in the same
proportion and vice-versa.
(b) Some claim that the liquidity trap is a reasonable assumption when the
nominal interest rate approaches zero. Others claim that, under these
circumstances, the elasticity must equal zero. How might the issue be
resolved?
The issue may be resolved by specifying a demand for money function that
encompasses the two hypotheses, such as:
log m = δ − αi +
β
i−γ
112
6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve
Fig. 6.1 Money demand: a
liquidity trap
The elasticity of money with respect to the nominal interest rate is given
by:
ηm,i = i
β
∂ log m
= −αi +
∂i
i−γ
If β = 0, then ηm,i = 0 when i = 0. On the other hand, if α = 0, and
γ = 0, ηm,i = −∞ when i = 0.
(14) Right, wrong, or maybe. Justify your answer:
(a) The balanced budget multiplier (increased government spending=increased
taxes) equals zero.
The Keynesian IS curve is specified by [Macro Theory, p. 159]:
x = −α (r − r̄) + β f − f¯ + g − ḡ
The balanced budget multiplier assumes f = f¯. Thus, the multiplier is the
coefficient of g, that is, equal to one.
(b) In the short term, the inflation rate depends on monetary policy alone.
In the short term demand shocks as well as supply shocks affect the inflation
rate. In the long run inflation depends on monetary policy alone.
(c) When the monetary policy is expansionary the economy’s real liquidity
decreases.
When the monetary policy is expansionary the economy’s real liquidity
increases in the short run. In the long run, as the nominal rate of inflation
increases, real liquidity decreases.
(d) Inflation inertia increases the social cost of fighting inflation.
If there is inflation, inertia is a backward-looking variable. Hence, the only
way to bring inflation down is through recession. The social cost of fighting
inflation, under this environment, is unavoidable.
(e) The real interest rate is independent from the public deficit in the case of
Ricardian equivalence.
6
Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .
113
Under Ricardian equivalence the public deficit does not affect any real
variable. Hence, the real interest rate does not depend on the public deficit.
(f) Increased government spending increases the economy’s real output in both
the short and the long run.
When there is slack in the economy increased government spending
increases real output in the short term. In the long-term there will be
crowding out of either consumption or investment.
(15) Suppose that an economic model can be represented by the following finite
difference equation:
yt = αE (yt+1 |It ) + βxt ,
|α| < 1
Show how to obtain this model’s fundamental and bubble solutions, and apply
the method to the following cases.
(a) Arbitrage between fixed income and variable income (riskless).
The difference equation can be written with the forward operator F ,
F xt = xt+1 , as
Et [(1 − αF ) yt − βxt ] = 0
or:
Et yt −
β
xt = 0
1 − αF
The fundamental solution is:
f
yt = Et
∞
+
αβ i xt+i
i=0
It is straightforward to verify that:
ytb = Cα −t
is a solution. This is the bubble solution. The general solution is:
f
yt = yt + ytb = Et
∞
+
αβ i xt+i + Cα −t
i=0
The arbitrage equation:
E (pt+1 |It ) − pt + dt
= rt
pt
114
6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve
can be written as:
pt =
1
dt
E (pt+1 |It ) +
1+r
1+r
The general solution is:
pt = Et
∞
+
i=0
1
1+r
dt+i
(1 + r)i
+ C (1 + r)t
The first component is the fundamental solution and the second component
is the bubble solution.
(b) Determination of the price level according to the Cagan model of money
demand:
mt − pt = −γ [E (pt+1 |It ) − pt ]
Cagan’s demand for money equation can be written as an equation that
determines the price level:
pt =
γ
mt
E (pt+1 |It ) +
1+γ
1+γ
The solution of this finite difference equation is:
pt = Et
∞
+
mt+i
i=0
(1 + γ )i
+C
γ
1+γ
−t
The first part is the fundamental solution and the second part is the bubble
solution.
(16) Consider the model:
IS: y = −αi + u
LM: m = −βi + γ y + υ
Where u and v υ are random, non-correlated, variables with averages equal to
zero and the respective variances σu2 and συ2 .The Central Bank’s loss function
is given by:
L = y2
The Central Bank may choose between the interest rate (i) or the quantity of
money (m) as a policy instrument.
(a) What value of m minimizes the expected value of the loss function?
Solving the IS and LM equations we obtain the equation for output:
6
Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .
α
m+
y=
β + αγ
β
β + αγ
α
u− υ
β
115
If y is output gap m = 0 (deviation from the trend) minimizes the expected
value of the loss function. The expected value of the loss function is:
β
β + αγ
Lm = Ey = Et
2
α
u− υ
β
2
which can be written as:
Lm =
β 2 σu2 + α 2 συ2
(β + αγ )2
(b) What value of i minimizes the expected value of the loss function?
The IS equation is:
y = −αi + u
If y is output gap i = 0 (deviation from trend) minimizes the expected
value of the loss function, which is given by:
L = Ey 2 = σu2
(c) What instrument should the Central Bank choose?
The Central Bank should choose the interest rate as its instrument if:
σu2 <
β 2 σu2 + α 2 συ2
(β + αγ )2
Otherwise, the Central Bank should choose the money supply (m).
(17) The price level pt is a weighted average of the price vt and of the price in period
t − 1, according to:
pt = λυt + (1 − λ) pt−1
(a) Use recursive (back) substitution to show that:
pt =
∞
+
λ (1 − λ)i υt−i
i=0
The price level for period t − 1 is:
pt−1 = λυt−1 + (1 − λ) pt
116
6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve
Substituting this price level into the price level for period t results:
pt = λυt + (1 − λ)2 pt−2
The price level for period t − 2 is:
pt−2 = λυt−2 + (1 − λ) pt−3
Substituting this into the previous expression we get:
pt = λυt + (1 − λ) λυt−1 + (1 − λ)2 λυt−2 + (1 − λ)3 λυt−3 + · · ·
Therefore:
pt =
∞
+
λ (1 − λ)i υt−i
i=0
(b) Solve the same exercise using the lag operator L, Lzt = zt−1 , and the
property:
1
= 1 + αL + λ2 L2
1 − αL
The price level for period t can be written as:
[1 − (1 − λ) L] pt = λυt
or:
pt =
λ
υt
[1 − (1 − λ) L]
Taking into account that:
1
= 1 + (1 − λ) L + (1 − λ)2 L2 + · · ·
1 − (1 − λ) L
we obtain:
p = λυt + (1 − λ) λυt−1 + (1 − λ)2 λυt−2 · · ·
or:
pt =
∞
+
i=0
λ (1 − λ)i υt−i
6
Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .
117
(18) The optimum price υt depends on price pt∗ and on the mathematical expectation
Et υt+1 , according to:
υt = [1 − β (1 − λ)] pt∗ + β (1 − λ) Et υt+1
(a) Use recursive (forward) substitution to show that
υt = [1 − β (1 − λ)]
∞
+
∗
[β (1 − λ)]j Et pt+j
j =0
The optimum price for period t + 1 is:
∗
+ β (1 − λ) Et+1 υt+2
υt+1 = [1 − β (1 − λ)] pt+1
The expected value of υt+1 given the information in t is:
∗
+ β (1 − λ) Et υt+2
Et υt+1 = [1 − β (1 − λ)] Et pt+1
Substituting this into the optimum price in period t results
∗
+ β (1 − λ) Et υt+2
υt = 1 − β (1 − λ) pt∗ + β (1 − λ) Et pt+1
The optimum price in t + 2 is:
∗
+ β (1 − λ) Et+2 υt+3
υt+2 = [1 − β (1 − λ)] pt+2
Taking expectation and using the property Et (Et+1 Zt+1 ) = Et Zt+1 , we
get
∗
Et υt+2 = [1 − β (1 − λ)] Et pt+2
+ β (1 − λ) Et υt+3
Substituting this into the optimum price υt results:
υt = [1 − β (1 − λ)] pt∗
∗ + β (1 − λ)2 [1 − β (1 − λ)] E p ∗
+ β (1 − λ) [1 − β (1 − λ)] Et pt+1
t t+2
+ [β (1 − λ)]2 Et υt+3
This iteration yields the expression for υt .
(b) Solve the same exercise using lag operator F , F zt = zt+1 , and the property:
1
= 1 + αF + α 2 F 2 + · · ·
1 − αF
118
6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve
The optimum price can be written as:
Et (1 − β (1 − λ) F ) υt − [1 − β (1 − λ)] pt∗ = 0
or:
Et υt −
(1 − β (1 − λ))
1 − β (1 − λ) F
pt∗
=0
Using the lag operator property results:
Et υt − [1 − β (1 − λ)] (1 + β (1 − λ)] F + (β (1 − λ))2 F 2 · · · pt∗ = 0
Hence,
υt = [1 − β (1 − λ)]
∞
+
∗
[β (1 − λ)]j Et pt+j
j =0
(19) Consider the Phillips curve:
πt − πt∗ = βEt πt+1 − π ∗ + δ (yt − ȳt )
where π ∗ is the trend inflation. Is this curve vertical in the long run?
In the long run:
πt = πt+1 = πt∗
Thus,
yt = ȳt
Therefore, in the long run there is no trade-off between inflation and output gap.
(20) Each firm sets the price of its product only when it receives a random,
exponentially distributed sign. That is, the probability that the sign will be
received in h periods from today is given by:
δe−δh , δ > 0
The (log of) the price set by the firm in t, when it receives the sign is:
υt =
∞
t
(ps + αxs ) δe−δ(s−t) ds, α > 0
6
Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .
119
where ps is the price level and xs is excess demand, both in period s. The (log
of) price level (p) is defined by the formula:
pt =
T
−∞
υs δe−δ(t−s) ds
(a) Show that the average of the random variable H is given by:
EH =
1
δ
The expected value of H is defined by:
EH =
∞
hf (h)dh =
∞
0
hδe−δh dh
0
We use integration by parts:
udυ = uυ −
υdu
to compute the integral:
∞
hδe
−δh
dh = −he
)∞ ∞ −e−δh dh
) −
−δh )
0
0
0
Since:
)∞
)
−he−δh ) = 0
0
we may write:
∞
hδe−δh dh =
∞
0
0
e−δh =
1
δ
Therefore,
EH =
1
δ
(b) Make explicit the arguments that justify the expression of υt , and pt .
The price υt is given by:
υt =
∞
t
(ps + αxs ) δe−δ(s−t) ds, α > 0
120
6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve
The probability that the sign will be received in s − t periods from t is
δe−δ(s−t) and ps + αxs is the price set by the firm that receives the sign.
The price pt is given by:
pt =
t
−∞
υs δe−δ(t−s) ds,
where υs is the price set at s and e−δ(t−s) is the percentage of firms that set
that price.
(c) Take the derivative of υt and pt with respect to time, applying Leibnitiz
rule, and show that:
υ̇ = δ (υ − p − αx)
π = ṗ = δ (υ − p)
The Leibnitz rule (Macro Theory, p. 337) is given by:
V (r) =
β(r)
f (x, r) dx
α(r)
and:
dV (r)
dβ(r)
dα(r)
= f (β(r), r)
− f (α(r), r)
+
dr
dr
dr
β(r)
α(r)
∂f (x, r) d
∂r
Applying this rule to υt we obtain:
dυt
= − (p(t) + αx(t)) δ +
dt
∞
δ (ps + αxs ) δe−δ(s−t) ds
t
or:
dυt
= − (p(t) + αx(t)) δ + δυt
dt
By rearranging terms, we get:
dυt
= δ (υ(t) − p(t) − αx(t))
dt
Applying Leibinitz rule to pt we obtain:
dpt
= υ(t)δ +
dt
t
−∞
υs δe−δ(t−s) (−δ) ds
6
Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .
which can be written as:
dpt
= υ(t)δ − δp(t)
dt
Hence,
π=
dpt
= δ (υ(t) − p(t))
dt
(d) Take the derivative of π with respect to time and show that:
π̇ = −αδ 2 x
The rate of inflation obtained in the previous item is:
π = δ (υ − p)
We take the time derivative of this expression to get:
π̇ = δ (υ̇ − ṗ)
The derivative of υ (item c) is:
υ̇ = δ (υ − p − αx)
and
ṗ = δ (υ − p)
We substitute these two expressions in the equation of π̇ to obtain:
π̇ = δ (υ̇ − ṗ) = δ (δ (υ − p − αx) − δ (υ − p))
By simplifying yields:
π̇ = −αδ 2 x
This is the well known Calvo Phillips curve as derived by him.
(e) Suppose that υt is given by:
υt =
∞
e−ρ(s−t) (ps + αxs ) δe−δ(s−t) ds, α > 0
t
where ρ is the discount rate. Show that:
121
122
6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve
π̇ = ρδp + ρπ − αδ 2 x
We take the time derivative of υt applying Leibinitz rule to obtain:
dυt
= −δ (pt + αxt ) +
dt
∞
e−ρ(s−t) (ps + αxs ) δe−δ(s−t) (ρ + δ) ds
t
or:
υ̇ = −δ (p + αx) + (ρ + δ) υ
The inflation rate change and the inflation rate are given by:
π̇ = δ (υ̇ − ṗ)
and
π = ṗ = δ (υ − p)
Hence:
υ̇ − ṗ = ρυ − αδx
Therefore:
π̇ = δρυ − αδ 2 x
We use the inflation rate equation to get rid of υ, to obtain:
π̇ = ρδp + ρπ − αδ 2 x
(f) Is the previous item’s Phillips curve vertical in the long run?
When π̇ = 0:
x=
ρ
ρ
p + 2π
αδ
αδ
Therefore, the Phillips curve is not vertical in the long run.
(21) Consider the Phillips curve model:
π = π e + δ (y − ȳ)
Assume that the expected inflation rate follows the adaptive expectations
mechanism:
6
Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips. . .
123
π̇ e = β π − π e
Show that the acceleration of the inflation rate is:
π̇ = βδ (y − ȳ) + δ ẏ
when the growth rate of potential output is equal to zero ȳ˙ = 0. We take the
derivative of both sides of the Phillips curve to obtain:
π̇ = π̇ e + δ ẏ
By combining the Phillips curve and the adaptive expectation mechanism
results:
π̇ e = β π − π e = βδ (y − ȳ)
We substitute this into the previous expression to obtain:
π̇ = βδ (y − ȳ) + δ ẏ
We may conclude that the acceleration of inflation depends on the output gap
and also on the rate of change of the output gap.
(22) The new Keynesian Phillips curve with perfect foresight is:
πt = βπt+1 + δxt
Show that the New Keynesian curve with continuous variable is:
π̇ = ρπ − kx
δ
where ρ = 1−β
β and k = − β .
The New Keynesian Phillips curve can be written as:
πt = β (πt+1 − πt ) + βπt + δxt
which can be rewritten as:
(1 − β) πt − δxt = βπt+1
Hence:
πt+1 =
1−β
δ
πt − xt
β
β
124
6 Keynesian Models: The IS and LM Curves, the Taylor Rule and the Phillips Curve
1
δ
The discount factor β = 1+ρ
. Thus, 1−β
β = ρ. Let us denote β = k. Hence, the
New Keynesian Phillips curve with continuous variable is:
π̇ = ρπ − kx
Chapter 7
Economic Fluctuation and Stabilization
(1) Consider the model:
IS: ẋ = σ (i − π − r̄)
PC: π̇ = δx
MPR: i = r̄ + π + φ (π − π̄ ) + θ x
IC: Given p(0) and π(0)
Analyze the consequences of an unanticipated permanent change in the
inflation target.
The dynamical system of this model is given by the two linear differential
equations:
ẋ = σ φ (π − π̄ ) + σ θ x
π̇ = δx
The first equation was obtained by combining the IS curve and the MPR. The
Jacobian of this system is:
J =
∂ ẋ ∂ ẋ ∂x ∂π
∂ π̇ ∂ π̇
∂x ∂π
σφ σθ
=
δ 0
The determinant of this matrix is given by:
|J | = −σ θ δ < 0
and it is negative. Thus, the steady-state is a saddle point. Figure 7.1 depicts
the phase diagram of the model with the inflation rate on the vertical axis and
the output gap on the horizontal axis. The saddle path SS is downward sloping.
Figure 7.2 shows an unanticipated permanent change in the inflation target,
from π̄0 to π̄1 < π̄0 . Figure 7.3 shows the dynamic adjustment of the economy
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_7
125
126
7 Economic Fluctuation and Stabilization
Fig. 7.1 The phase diagram
for the π and x system
Fig. 7.2 An unanticipated
permanent decrease in the
inflation target
Fig. 7.3 Dynamic
adjustment of the economy to
an unanticipated permanent
decrease in the inflation target
to the inflation target change. To simplify the figure, we just draw the saddle
path SS after the inflation target change. The initial conditions of the model
state that inflation is a predetermined variable. Thus, the economy jumps from
the initial equilibrium (E0 ) to the point Ei on the saddle path. From then
onwards the economy converges on the new equilibrium (Ef ).
(2) Consider the model:
IS: ẋ = −λx − α (r − r̄)
PC: π̇ = δx
7 Economic Fluctuation and Stabilization
127
MPR: i = r̄ + π + φ (π − π̄ ) + θ x
IC: Given p(0) and π(0)
(a) Analyze the model’s equilibrium and dynamics on a phase diagram, with
the inflation rate (π ) on the vertical axis and the output gap (x) on the
horizontal axis.
The dynamical system of this model is given by the two linear
differential equations:
ẋ = − (λ + αθ ) x − αφ (π − π̄ )
π̇ = δx
The first equation was obtained by combining the IS curve and the MPR.
The Jacobian of this system is:
J =
∂ ẋ ∂ ẋ ∂x ∂π
∂ π̇ ∂ π̇
∂x ∂π
=
− (λ + αθ ) −αφ
δ
0
The determinant and the trace of this matrix are given by:
|J | = αφδ > 0
trJ = − (λ + αθ ) < 0
The determinant is positive and the trace is negative. Thus the model is
stable. Figure 7.4 depicts the phase diagram of the model with inflation on
the vertical axis and the output gap on the horizontal axis.
(b) Use the phase diagram from item (a) to show what happens in this model
when the Central Bank raises the inflation rate target.
Fig. 7.4 The phase diagram
for the π and x system
128
7 Economic Fluctuation and Stabilization
Figure 7.5 depicts the phase diagram when the inflation target increases
to π̄1 from π̄0 < π̄1 . In this model the initial conditions stipulate
that inflation is a predetermined variable. Thus, at the initial moment
when the change occurs, inflation does not change and the output gap
becomes positive. The phase diagram of Fig. 7.5 assumes inflation rate
overshooting. In the long run it converges on the new inflation target.
(c) What happens in this model if the parameter φ is negative? How do
you interpret the monetary policy rule in this case? If the parameter φ
is negative the determinant of the Jacobian matrix is negative:
|J | = αφδ < 0
The equilibrium of the model is a saddle point as depicted in Fig. 7.6.
The MPR can be written as:
i = r̄ + π̄ + (1 + φ) (π − π̄ ) + θ x
Fig. 7.5 The dynamic
adjustment of the economy to
an unanticipated increase in
the inflation target
Fig. 7.6 The phase diagram
for the π and x system when
φ<0
7 Economic Fluctuation and Stabilization
129
In spite of the fact that 1 + φ is less than one, the economy converges on
the inflation target.
(3) Consider the model:
IS: x = −α (r − r̄) + β f − f¯ + g − ḡ
PC: π̇ = δx
MPR: i = r̄ + π + φ (π − π̄ ) + θ x
IC: Given p(0) and π(0)
Fiscal policy rule:
f − f¯ = −ϕ1 x, g − ḡ = −ϕ2 x
(a) How would you interpret the monetary and fiscal policy rules?
The monetary policy rule is a conventional Taylor rule. The fiscal policy
rule is a counter-cyclical rule: (1) the fiscal deficit should increase
(decrease) when the output gap is negative (positive); and (2) the government expenditure should increase (decrease) when the output gap is
negative (positive).
(b) Analyze the model’s equilibria and dynamics on a diagram with π on the
vertical axis and x on the horizontal axis.
The dynamical system of the model is given by a system of two equations:
αφ(π −π̄ )
x = − 1+βϕ
1 +ϕ2 +αθ
π̇ = δx
The first equation was obtained by combining the IS curve, the monetary
policy and the fiscal policy rule. Figure 7.7 depicts the phase diagram of
the model, with the inflation rate on the vertical axis and the output gap on
the horizontal axis.
Fig. 7.7 The phase diagram
for the π and x system
130
7 Economic Fluctuation and Stabilization
The curve DD is the equation that relates the output gap with the inflation
gap: any point on this curve converges on the equilibrium E, where π = π̄
and x = 0.
(c) What could happen in this model if the parameter φ were negative?
As shown in Fig. 7.8 the curve DD would be upward sloping and no point
on this curve would converge on the equilibrium. The economy would be
unstable.
(d) What would happen in this economy if the government raised the inflation
target to π̄1 > π̄0 from π̄0 ?
Figure 7.9 depicts the phase diagram of the adjustment of the economy
when the inflation target increases. The economy jumps from point E0 to
point Ei because the inflation rate is a predetermined variable. From then
onwards the economy converges on the new equilibrium (π = π̄1 and
x = 0).
Fig. 7.8 The phase diagram
for the π and x system when
φ<0
Fig. 7.9 The dynamic
adjustment of the economy to
an unanticipated increase in
the inflation target
7 Economic Fluctuation and Stabilization
131
(4) Consider the model:
IS: x = −α (r − r̄) , α > 0
PC: π̇ = −δx, δ > 0
MPR: i = r̄ BC + π + φ (π − π̄ ) + θ x
IC: Given p(0)
(a) The dynamical system of this model is given by a system of two equations:
αφ
α
x = − 1+αθ
r̄ BC − r̄ − 1+αθ
(π − π̄ )
π̇ = δx
The first equation of this model was obtained by combining the IS curve
and the MPR. In the steady-state equilibrium the output gap is zero (x =
0), but the inflation rate π ∗ is given by:
π ∗ = π̄ −
1 BC
r̄ − r̄
φ
Thus, if r̄ BC = r̄, π ∗ = π̄ , as shown in the phase diagram of Fig. 7.10.
(b) How would you interpret r̄ = r̄ BC ?
There are two ways to interpret this hypothesis. One is that the Central
Bank estimate of the natural rate of interest is biased. The other is that the
operational procedure uses a real interest rate.
(5) Consider the model:
IS: x = −α (r − r̄) , α > 0
PC: π̇ = −δx, δ > 0
MPR: i = r̄ + π + φ (π − π̄ ) + θ x, φ > 0, θ > 0
IC: Given p(0) and π(0)
This exercise is a particular case of exercise (2) when λ = 0.
Fig. 7.10 The phase diagram
for the π and x system
132
7 Economic Fluctuation and Stabilization
(6) Consider the model:
IS: u − ū = α (r − r̄) , α > 0
PC: π̇ = −δ (u − ū) , δ > 0
MPR: i̇ = ϕ (π − π̄) − θ u̇
IC: Given p(0) and π(0)
The symbols have the following meanings: u = unemployment rate; ū =
du
natural unemployment rate, i = di
dt , u̇ = dt .
(a) How would you obtain this model’s IS curve?
Okun’s Law (Macro Theory, p. 180, equation 6.66) is given by:
y − ȳ = x = −b (u − ū)
The IS curve is specified as:
x = −a (r − r̄)
By combining Okun’s Law with the traditional IS curve we obtain:
−b (u − ū) = −a (r − r̄)
which can be written as:
u − ū = a (r − r̄) , α =
a
>0
b
(b) The MPR, di
dt = ϕ (π − π̄ ) − θ u̇, does not need any information on nonobservable variables, such as the natural rate of interest and the natural
unemployment rate.
(c) Analyze the model’s equilibrium and dynamics on a phase diagram with
inflation on the vertical axis and the unemployment rate on the horizontal
axis. Is this model stable?
We start taking derivatives with respect to time of the IS curve:
di
u̇ = α
− π̇
dt
Next, we use the PC and the MPR to obtain:
u̇ = αϕ (π − π̄ ) − αθ u̇ + αδ (u − ū)
Therefore, the dynamical system of this model is given by:
u̇ =
αϕ
αδ
(u − ū) +
(π − π̄ )
1 + αδ
1 + αθ
π̇ = −δ(u − ū)
The Jacobiano of this system is:
∂ u̇ ∂ u̇ αδ
αϕ
1+αθ
1+αθ
∂u
∂π
J = ∂ π̇ ∂ π̇ =
−δ
0
∂u ∂π
7 Economic Fluctuation and Stabilization
133
Fig. 7.11 The phase diagram
for the π and x system
The determinant and trace of this matrix are:
|J | =
δαϕ
>0
1 + αθ
trJ =
αδ
>0
1 + αθ
Both the determinant and the trace are positive. Therefore, the system is
unstable. Thus, albeit the MPR is interesting for not requiring information
on non-observable variables, if the Central Bank does use this MPR the
economy will not reach the inflation target. Figure 7.11 depicts the phase
diagram of the model.
(d) Assume that the model’s initial inflation is an endogenous variable and the
parameter δ is negative. Analyze the model’s equilibrium and dynamics
under these circumstances. If δ is negative, the determinant of the Jacobian
matrix is negative:
δαϕ
<0
1 + αθ
The steady-state is a saddle path. Therefore, the model’s equilibrium is
not determined because the initial inflation can be any value. The Phillips
curve in this item, is assumed to be forward looking; However, the IS
curve is a traditional Keynesian Curve. If we assume a New Keynesian IS
Curve [Macro Theory, p. 167, equation (6.30), ẋ = α (r − r̄). The New
Keynesian model would have the following dynamical system.
u̇ = −γ (i − π − r̄)
π̇ = δ (u − ū)
134
7 Economic Fluctuation and Stabilization
i̇ =
di
= ϕ (π − π̄ ) + γ θ (i − π − r̄)
dt
The Jacobian matrix of this system is:
⎡ ∂ u̇ ∂ u̇ ∂ u̇ ⎤
∂u
J = ⎣ ∂∂uπ̇
∂i
∂u
∂π
∂ π̇
∂π
∂i
∂π
⎡
⎤
0
γ
−γ
∂i
∂ π̇ ⎦ = ⎣
δ
0
0 ⎦
∂i
∂i
0 (ϕ − γ θ ) γ θ
∂i
The determinant and trace of this matrix are:
|J | = −γ δϕ < 0
trJ = γ θ > 0
This model has two positive characteristic roots and one negative root. The
model is unstable and has just one equilibrium.
(e) Would you recommend using this monetary policy rule?
If the world were a Keynesian world I would not recommend the monetary
rule because the Central Bank would not be able to deliver its inflation
target. On the other hand, if the world were New Keynesian the monetary
policy rule would achieve the goal of the Central Bank. We would not
recommend this monetary policy rule because we do not know what the
true model is.
(7) Consider the model:
IS: x = −α (r − r̄)
PC: π̇ = δx
MPR: i = r̄ + π + φ (π − π̄ ) + θ x
(a) In this model, is the long run inflation a monetary phenomenon?
In strict sense the long run inflation (π̄ ) is not a monetary phenomenon
because we do not need the stock of money to determine inflation, either
in the short run or in the long run.
(b) Assume that LM curve is specified by:
LM : m − m̄ = λx − β i − ī
where m is the real quantity of money, m = M
P . Does the inflation rate
target equal the growth rate of monetary base in the long run? In the long
run x = 0 and i̇ = ī. Thus m = m̄ =constant. Thus, the rate of growth of
money is equal to the rate of growth of the price index.
π̄ =
Ṁ
Ṗ
=
P
M
7 Economic Fluctuation and Stabilization
135
(c) Does the monetary policy rule that sets the interest rate imply that inflation
is not a monetary phenomenon in the long run?
The monetary policy rule that sets the interest rate implies that money
is an endogenous variable in the long run. In the long run inflation is a
monetary phenomenon in the sense that inflation is equal to the rate of
growth of money.
(d) Based on this model, might one argue that if the society’s objective is to
reduce the interest rate, the interest rate must rise initially?
Society does not control the long run real interest rate, the natural rate
of interest. Society does control the long run nominal rate of interest.
According to the MPR if society desires to have less inflation in the long
run it has to increase the interest rate initially.
(8) Consider the model:
IS: x = −α (r − r̄)
PC: π̇ = δx
MPR: i s = r̄ s + π + θ (π − π̄ ) + φx
Credit: i = i s + sp
¯ + β (sp − sp)
IC: Given p(0) and π(0)
Definitions: i = r − π, r̄ = r̄ s + sp
The symbol sp represents the spread of the interest rate on the credit market,
and sp is the long run equilibrium spread.
Substituting the MPR equation into the credit equation we obtain:
i = r̄ + π + θ (π − π̄ ) + φx + β (sp − sp)
The interest rate gap provided by this equation can be substituted into the IS
curve:
x=−
αθ
αβ
(π − π̄) −
(sp − sp)
1 + αθ
1 + αφ
(a) Show that this model’s aggregate demand equation is given by:
π = π̄ −
1 + αφ
β
x − (sp − sp)
αθ
θ
From the previous equation it is straightforward to obtain the aggregate
demand equation.
(b) Use a phase diagram (π on the vertical axis and x on the horizontal axis)
to show what happens to the output gap and the inflation rate when a credit
market shock causes sp − sp > 0. Figure 7.12 depicts the phase diagram
of the model.
The credit shock shifts the curve D0 D0 to D1 D1 . The rate of inflation is
a predetermined variable. Thus, at the moment of the shock the economy
enters into a recession. The initial output gap becomes negative (x0+ ).
136
7 Economic Fluctuation and Stabilization
Fig. 7.12 Dynamic
adjustment of the economy to
an unanticipated credit shock
Then, from that point onward the economy converges on full employment
(Ef ).
(9) Consider the model:
IS: ẋ = −α (r − r̄) , α > 0
PC: π̇ = −γ (π − π̄ ) + δx, δ > 0
MPR: i = r̄ + π + φ (π − π̄ ) + θ x, φ > 0, θ > 0
IC: Given p(0) and π(0)
(a) Analyze the model’s equilibrium and dynamics on a phase diagram with
π on the vertical axis and x on the horizontal axis.
The dynamical system of this model is given by a system of two linear
differential equations:
ẋ = −αφ (π − π̄ ) − αθ x
π̇ = −γ (π − π̄ ) + δx
The first equation of this system was obtained by combining the IS curve
and the MPR. The Jacobian of this system is:
J =
∂ ẋ ∂ ẋ ∂x ∂π
∂ π̇ ∂ π̇
∂x ∂π
=
−αθ −αφ
δ −γ
The determinant and the trace of this matrix are:
7 Economic Fluctuation and Stabilization
137
Fig. 7.13 The phase diagram
for the π and x system
Fig. 7.14 Dynamic
adjustment of the economy to
an unanticipated decrease in
the inflation target
p
p0 E0
Ef
.
x=0
x
|J | = αθ γ + αφδ > 0
trJ = − (αθ + γ ) < 0
The system is stable because the determinant is positive and the trace is
negative. Figure 7.13 depicts the phase diagram of the model.
(b) Use the previous item’s phase diagram to show this model’s dynamics
when the Central Bank reduces the inflation target.
Figure 7.14 shows the phase diagram describing the economy adjustment
to a reduction of the inflation target.
The economy enters into a recession and the inflation rate decreases. It
is possible that there will be an undershooting of the inflation rate as the
economy converges on full employment and the new inflation target.
(c) What would happen in this model if the parameter φ were negative?
The determinant of the Jacobian matrix is given by:
138
7 Economic Fluctuation and Stabilization
|J | = α (θ γ + φδ)
If φ < 0 in such way that:
θ γ + φδ > 0
The model is stable as shown in Fig. 7.13. If φ < 0 and |J | =
α (θ γ + φδ) < 0, the steady-state is a saddle point.
(10) Consider the model:
IS: ẋ = −α (r − r̄) , α > 0
PC: π̇ = δx, δ > 0
MPR: i = ī
IC: Given p(0) and π(0)
(a) Analyze the model’s equilibrium and dynamics on a phase diagram with
π on the vertical axis and x on the horizontal axis.
The dynamical system of this model is given by a system of two linear
differential equations:
ẋ = −α ī − π − r̄
π̇ = δx
The Jacobian of this system is:
J =
∂ ẋ ∂ ẋ ∂x ∂π
∂ π̇ ∂ π̇
∂x ∂π
0α
=
δ 0
The determinant of this matrix is:
|J | = −αδ < 0
The steady-state of this model is a saddle point because the determinant is
negative. Figure 7.15 depicts the phase diagram of the model.
The implicit inflation target is given by the difference between the fixed
nominal rate of interest and the natural rate of interest π̄ = ī − r̄.
(b) Use the previous item’s phase diagram to show the model’s dynamics
when the Central Bank reduces the inflation rate.
The Central Bank reduces the nominal rate of interest, implying a cut in
the inflation target. The inflation rate is a predetermined variable.
The economy jumps from E0 to Ei (Fig. 7.16) yields a recession, the output gap becomes x(0+ ). From that point onwards the economy converges
on the new equilibrium (Ef ), with full employment and inflation given
by: π̄1 = ī1 − r̄ (Fig. 7.16).
7 Economic Fluctuation and Stabilization
139
Fig. 7.15 The phase diagram
for the π and x system
Fig. 7.16 Dynamic
adjustment of the economy to
an unanticipated decrease in
the nominal interest rate
(11) Consider the model:
IS: ẋ = γ x + σ (r − r̄)
PC: π̇ = δx
MPR: i = r̄ + π + φ (π − π̄ ) + θ x
IC: Given p(0)
The dynamical system of this model is given by a system of two differential
equations:
ẋ = (γ + σ θ ) x + σ φ (π − π̄ )
π̇ = δx
The Jacobian of this system is:
140
7 Economic Fluctuation and Stabilization
J =
∂ ẋ ∂ ẋ ∂x ∂π
∂ π̇ ∂ π̇
∂x ∂π
γ + σθ σφ
=
δ
0
The determinant and the trace of this Jacobian are:
|J | = −σ φδ
trJ = γ + σ θ
(a) In equilibrium: π̇ = 0 = ẋ. This implies π = π̄ and x = 0. If all
parameters were positive, the equilibrium would be a saddle point, and
π(0) would be a predetermined variable.
(b) What are the model’s properties when γ = 0 and δ < 0 (new Keynesian
model)?
Under these assumptions the determinant and the trace of the Jacobian
matrix are:
|J | = −σ φδ > 0
trJ = σ θ
The model would be unstable and the steady-state would be a unique
equilibrium.
(c) What are the model’s properties when δ < 0, δ > 0, and σ < 0 (Keynesian
model)?
Under these assumptions the determinant and the trace of the Jacobian
matrix are:
|J | = −σ φδ > 0
trJ = σ θ < 0
The model is stable because |J | > 0 and trJ < 0. The inflation rate is a
predetermined variable.
(12) Consider the model:
IS: ẋ = −α (r − r̄) , α > 0
PC: π̇ = δx
MPR: di
dt = φ (π − π̄ ) + θ x
(a) What condition must the parameter θ meet for the model to be stable?
The dynamical system of this model is given by a system of three linear
differential equations:
7 Economic Fluctuation and Stabilization
141
⎧
⎨ ẋ = −α (i − π − r̄)
π̇ = δx
⎩ di
dt = φ (π − π̄ ) + θ x
The Jacobian of this system is given by:
⎡ ∂ ẋ ∂ ẋ ∂ ẋ ⎤
∂x
J = ⎣ ∂∂xπ̇
∂i
∂x
∂π
∂ π̇
∂π
∂i
∂π
⎡
⎤
0 α −α
∂i
∂ π̇ ⎦ = ⎣
δ 0 0 ⎦
∂i
∂i
θ
φ 0
∂i
The determinant and trace of this matrix are:
|J | = −γ δφ < 0
trJ = 0
As you may recall the determinant is equal to the product of the characteristic roots. The determinant is negative. Thus, the system has either three
negative roots or two positive roots and one negative root. The trace, which
is equal to the sum of the characteristic roots, is equal to zero. Thus, we
can rule out three negative roots. The system has a saddle path because it
has two positive roots and one negative root. On the other hand, the model
has two predetermined variables, the inflation rate and the nominal interest
rate. Therefore, the dynamical system is unstable regardless of the sign of
the parameter θ .
(b) Assume that α < 0 and δ < 0. Does the model have multiple equilibria?
Under these assumptions the determinant of the Jacobian is negative and
the trace is equal to zero. Therefore, two roots are positive and one root
is negative. Because δ < 0 the inflation rate is a jump variable. The only
predetermined variable is the nominal interest rate. Therefore, the model
has a stable path and no multiple equilibria.
(c) Would you recommend that the Central Bank should use this monetary
policy?
No, we would not recommend this monetary policy rule because it is not
robust to the hypothesis of the model.
(13) Consider the New Keynesian model:
IS: ẋ = σ (r − r̄)
PC: π̇ = ρπ − kx
MPR: r − r̄ = φ (π − π̄ ) + θ (x − x̄)
(a) The inflation target π̄ and the output gap x̄ are such that x̄ = ρkπ̄ .
The dynamical system of this model has two linear differential equations:
142
7 Economic Fluctuation and Stabilization
ẋ = σ φ (π − π̄ ) + σ θ (x − x̄)
π̇ = ρπ − kx
The first equation was obtained by combining the IS curve and the MPR.
The Jacobian of this system is given by:
J =
∂ ẋ ∂ ẋ ∂x ∂π
∂ π̇ ∂ π̇
∂x ∂π
=
σθ σφ
−k ρ
The determinant and trace of this matrix are:
|J | = σ (θρ + kφ) > 0
trJ = σ θ + ρ > 0
The system is unstable and it has two jump variables, the inflation rate and
the output gap. Therefore, the model has a unique equilibrium.
(b) Is the monetary policy neutral in the long run?
In the long run when π̇ = 0, the output gap depends on the inflation rate
according to:
x̄ =
ρ π̄
k
Therefore, for each inflation target there is an output gap. The trade-off
depends on the ratio of the rate of time preference (ρ) and the output gap
coefficient of the Phillips curve.
(14) *(Discounted Euler Equation): The canonical new Keynesian model implies
that forward guidance interest rate policy has implausible effects on current
output gap. The “discounted Euler equation” introduces a coefficient less than
one in the expected future output gap that discounts the effects of future
interest rates.
(a) Show how this specification solves the “forward guidance puzzle”.
(b) Specify this new Keynesian IS curve using continuous variables and show
the difference between forward looking and backward looking behavior.
(c) The “discounted Euler equation” approach assumes that there are heterogeneous agents and the Euler equation is given by:
−γ
−γ
CH,t = β (1 + rt ) Et wCH,t+1 + (1 − w) M −γ
where the index H denotes one type of agent, w is probability of the agent to
be employed, M is the amount of consumption goods received by the agent
7 Economic Fluctuation and Stabilization
143
when unemployed, and γ is the coefficient of relative risk aversion. Aggregate
consumption Ct is given by:
Ct = wCH,t + (1 − w) M
What is the level of aggregate consumption in stationary equilibrium
CH,t = CH,t+1 = CH ?
(a) The discounted specification of the IS curve is given by:
xt = αEt xt+1 − σ (rt − r̄) , 0 < α < 1
(7.1)
The solution of this difference equation is:
xt = −σ Et
∞
+
α i (rt+i − r̄)
i=0
The weight α i goes to zero as i → ∞. When α = 1 the solution of this IS
curve is not given by:
xt = −σ Et
∞
+
(rt+i − r̄)
i=0
as shown in exercise C of the Appendix C.
(b) With continuous variables the IS curve (7.1) becomes:
ẋ = λx + φ (r − r̄) , λ > 0, φ > 0
(7.2)
The solution of this differential equation is:
x=−
∞
φe−λ(τ −t) (r − r̄) dτ
t
When λ < 0 and φ < 0 the IS is the old Keynesian curve, which is
backward looking:
x=
t
−∞
φeλ(t−τ ) (r − r̄) dτ
We conclude that IS reduced form (7.2) can be used to test whether the IS
curve is forward or backward looking.
In the canonical new Keynesian model λ = 0 and the interest rate
affects not the level but the change of output gap. In the new specification
the interest rate affects both the level and the change in the output gap,
like the old Keynesian IS curve.
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7 Economic Fluctuation and Stabilization
(c) The discounted Euler equation can be written as:
M −γ
1+ρ
CH,t+1 −γ
= Et w
+ (1 − w)
1 + rt
CH,t
CH,t
(7.3)
Next, we use the following approximations:
CH,t+1
CH,t
−γ
M
CH,t
=e
−γ
−γ log
=e
CH,t+1
CH,t
−γ log CM
H,t
= e−γ (cH,t+1 −cH,t ) ∼
= 1−γ cH,t+1 − cH,t
= e−γ (m−cH,t ) ∼
= 1 − γ m − cH,t
Substituting these approximations into (7.3) results:
1+ρ
= Et w 1 − γ cH,t+1 − cH,t + (1 − w) 1 − γ m − cH,t
1 + rt
By simplifying and rearranging terms we obtain:
cH,t = wEt cH,t+1 + (1 − w) m −
1
(rt − ρ)
γ
In steady-state,
rt = ρ, cH,t = cH,t+1 = cH
Therefore:
cH = m ⇒ CH = M
Aggregate consumption is:
Ct = wCH,t + (1 − w) M
In steady-state:
C=M
This conclusion is at odds with consumption theory. The traditional Euler
equation is a statement about change in levels, but not about levels. The
discounted Euler equation is a statement about level. The former needs the
budget constraint to determine the level of consumption. The discounted
Euler equation rules out the budget constraint. At best it could be used to
determine M.
Chapter 8
Open Economy Macroeconomics
(1) Consider the following model of a small open economy (monetary approach of
the balance of payments with a fixed exchange rate):
Ms = C + R
M d = P L(y, i)
i = i∗
y = ȳ
P = EP ∗ , E = Ē = constant
(a) What is the effect of a foreign exchange devaluation on the balance of
payments? This model assumes that the money market is in equilibrium:
Md = Ms
Therefore,
C + R = P L (y, i)
By using the hypothesis that P = EP ∗ , i = i ∗ and y = ȳ, we obtain:
R = EP ∗ L ȳ, i ∗ − C
Therefore,
∂R
= P ∗ L ȳ, i ∗
∂E
The effect of a foreign exchange devaluation on the balance of payments is
to increase the stock of international reserves.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_8
145
146
8 Open Economy Macroeconomics
(b) What is the effect of an increase in net domestic credit on the balance of
payments?
The partial derivative of R with respect to C is:
∂R
= −1 < 0
∂C
Therefore, the effect of an increase in net domestic credit on the balance of
payments is to decrease the stock of international reserves.
(2) Consider the following model of a small open economy (monetary approach of
the balance of payments with a flexible exchange rate):
Equilibrium in Country A’s money market:
M
= L (y, i)
P
Equilibrium in Country B’s money market:
M∗
= L y∗, i∗
P∗
Exchange rate:
E=
P
P∗
Comment on the following:
(a) The exchange rate depreciates when the country grows more quickly than
others. We use the exchange rate definition and the two money market
equilibrium equations to write:
E=
M
M L (y ∗ , i ∗ )
L(y,i)
=
∗
M
M ∗ L (y, i)
L(y ∗ ,i ∗ )
(8.1)
∗
y
When y
y > y ∗ the answer to this item depends on the income elasticities
in the two countries. If the income elasticities are equal, E decreases
(appreciate).
(b) The exchange rate appreciates when the money stock grows more quickly
M ∗
than other countries. When M
M > M ∗ it is straightforward to see from
Eq. (8.1) that E increases, e.g., it depreciates.
(3) Consider the following model of a small open economy (monetary approach of
the balance of payments with a flexible exchange rate):
Equilibrium in Country A’s money market:
8 Open Economy Macroeconomics
147
m − p = αy − βi
Equilibrium in Country B’s money market:
m∗ − p∗ = αy ∗ − βi ∗
Exchange rate: e = p − p∗
Uncovered interest parity: i = i ∗ + ė
(a) Deduce the differential equation for exchange rate determination.
We use the exchange rate definition and the two money market equilibrium
equations to write:
e = m − αy + βi − m∗ + αy ∗ − βi ∗
which can be written as:
e = β i − i ∗ + m − m∗ − α y − y ∗
We substitute the uncovered interest parity condition into this equation to
obtain:
e = β ė + m − m∗ − α y − y ∗
which can be written as:
ė =
1
e−f
β
where:
f =
m − m∗ − α (y − y ∗ )
β
(b) Does the solution to this equation include a bubble component?
The solution of the exchange rate differential equation is:
e=
∞
e
− β1 (τ −t)
f dτ
t
and the second component is the bubble solution:
1
eb = Ce β
(4) Consider the regression:
t
148
8 Open Economy Macroeconomics
et+1 − et = a0 + a1 (ft − et ) + t
or:
et+1 − et = a0 + a1 it − it∗ + t
(a) Is the forward market’s exchange rate, or the differential of the interest rate,
a good predictor of the future exchange rate on forward market?
The UIP is given by:
it = it∗ + et+1 − et
and the CIP is expressed by:
it = it∗ + ft − et
By comparing the two equations it follows that:
et+1 = ft
Therefore, the coefficient a1 of the expression should be equal to one:
a1 = 1
Therefore, the forward rate or the interest rate gap should be a good
predictor of the future exchange rate.
(b) Several empirical studies have obtained negative values for the a1 parameter. How would you interpret this result?
One possibility to explain why a1 < 0 is to include a risk premium in the
UIP, such as:
it = it∗ + et+1 − et + ρt
where ρt is a risk premium that changes over time. Thus, if this is the
case, the regressions of this exercise have a specification error that bias
the estimates.
(5) Harberger-Laursen-Meltzler (HLM) Effect. In an open economy, the national
product (Y ) equals the sum of absorption (A) and the balance of the current
account on the balance of payments (X − Z):
Y =A+X−Z
The price index of absorption is a geometric average of the price of the
domestically product (P ) and the price of the foreign product, converted into
8 Open Economy Macroeconomics
149
the domestic currency at the exchange rate (SP ∗ ):
Pa = P 1−α EP ∗
α
=P
EP ∗
P
α
= P Sα
∗
where α is the share of the imported good in absorption and S = EP
P is the
terms of trade. The product may be written in real terms as:
y = d + x − Sz
Z
where d = PPa a , X = xP and z = SP
∗ . Real absorption (a) depends on the real
income as defined by: ya = P y/Pa
(a) Show that the elasticity of expenditure with respect to the terms of trade
ηd,s is:
ηd,s = α 1 − ηa,ya
where na,ya is the elasticity of absorption with respect to the real income.
National output is equal to absorption added to net-exports:
Y =A+X−M
which can be written as:
P y = Pa a + P x − EP ∗ cm
or:
y=
EP ∗
Pa a
+x−
cm
P
P
d=
EP ∗
Pa a
and S =
P
P
We define:
Therefore:
y = d + x − Scm
Domestic expenditure is the sum of domestically produced consumer goods
and imported goods:
d = cd + Scm
150
8 Open Economy Macroeconomics
The absorption price index is a geometric average of domestic and imported
goods:
Pa
=
P
EP ∗
P
α
= Sα
Using the price index real domestic expenditure can be written as:
d=
Pa
a = Sα a
P
ya =
Py
= S −α y
Pa
Real income is defined by:
and we take derivative of d with respect to S to obtain:
∂d
∂a
= αS α−1 a + S α
∂S
∂S
By taking into account that:
∂a ∂ya
∂a
=
∂S
∂ya ∂S
we write the previous expression as:
d
∂d
∂a ∂ya
= α + Sα
∂s
S
∂ya ∂S
or:
∂d
d
aS α ∂a ya ∂ya S
=α +
∂s
S
S ∂ya a ∂S ya
By rearranging the terms of this expression we obtain:
d
d ∂a ya ∂ya S
∂d
=α +
∂s
S
S ∂ya a ∂S ya
From the definition of ya we get:
∂ya S
= −α
∂S ya
and we define:
8 Open Economy Macroeconomics
151
ηa,ya =
∂a ya
∂ya a
to write the previous expression as:
S ∂d
= α + ηa,ya (−α) = α 1 − ηa,ya
d ∂S
Therefore:
ηd,s = α 1 − ηa,ya
where:
ηd,s =
S ∂d
d ∂S
(b) Show that saving (s = y − d − τ , where τ represents taxes) varies with the
terms of trade:
∂s
αd
=
ηa,ya − 1
∂S
S
The partial derivative of savings with respect to the terms of trade is:
∂y
∂d
∂s
=
−
∂S
∂S
∂S
S
d
However,
∂y
=0
∂S
and
∂s
d
= − ηd,S
∂S
S
From the previous item we get:
d
∂s
= − α 1 − ηa,ya
∂S
S
Therefore:
αd
∂s
=
ηa,ya − 1
∂S
S
d
S
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8 Open Economy Macroeconomics
(c) According to the HLM effect, worsening (improving) terms of trade
decrease (increase) the economy’s real income, reducing (increasing)
saving. For a given level of investments, the reduction (increase) in saving
deteriorates (improves) the current account on the balance of payments.
What happens to the current account on the balance of payments if ηa,ya <
1 and the country’s terms of trade improve?
From item (b), if ηa,ya < 1,
αd
∂s
=
ηa,ya − 1 < 0
∂S
S
Therefore, in this case, if the country’s terms of trade improve the current
account on the balance of payments deteriorates.
(6) Consider the model:
⎧
∗
⎪
⎪ M = m (i, ∗i + ė) W
⎪
⎪
⎪
⎨ B = b (i, i + ė) W
EF = f (i, i ∗ + ė) W
⎪
⎪
W = M + B+ EF
⎪
⎪
⎪
⎩ Ḟ = ϕ EP ∗ + i ∗ F
P
(a) Discuss the specification of each of the model’s equations and analyze
its equilibrium. This exercise deals with the portfolio balance model. It
assumes that foreign assets and domestic bonds are not perfect substitutes.
Net financial wealth has three components money (M), domestic bonds (B)
and foreign assets (F in foreign currency and EF in domestic currency, i is
the domestic interest rate, i ∗ is the foreign interest rate and ė is the expected
d
rate of change of the exchange rate ė = dt
log E .
The asset demand equations are represented by m( ), b( ) and f . Its partial
derivatives have the following signs:
mi < 0, m∗i < 0
bi > 0, bi∗ < 0
fi < 0, fi∗ > 0
where the index denotes a partial derivative, e.g., mi = ∂m
∂i . To simplify the
solution, we specify demand equations as:
M
∗
= e−αi e−β (i +ė)
W
8 Open Economy Macroeconomics
153
EF
∗
= e−γ i eδ (i +ė)
W
B
M
EF
=1−
−
W
W
W
Let us denote letters m and f the following ratios (in logs):
log
M
W
= m; log
EF
W
=f
In this model one equation is redundant due to the constraint. Therefore,
we use the two equations, one for money and the other for foreign assets,
to determine i and i ∗ + ė:
−αi − β i ∗ + ė = m
−γ i + δ i ∗ + ė = f
The solution of this linear system is:
1
i
−δ −β m
=
i ∗ + ė
αδ + βγ −γ −α f
Thus:
i=−
δm + αf
αδ + βγ
i ∗ + ė =
−γ m + αf
αδ + βγ
The symbol e is the log of the exchange rate, e = log E. Thus ė = Ė
E . From
the last equation we obtain the differential equation for the exchange rate:
Ė =
−γ m + αf
αδ + βγ
E
where we use the simplifying hypothesis i ∗ = 0. Therefore, the dynamical
system of this model has two equations:
Ė =
−γ m + αf
αδ + βγ
E
154
8 Open Economy Macroeconomics
EP ∗
P
Ḟ = ϕ
+ i∗F
In equilibrium, Ė = Ḟ = 0, and
−γ m + αf = 0
EP ∗
P
∗
i F = −ϕ
The Jacobian of this system is:
J =
∂ Ḟ
∂E
∂ Ė
∂E
∂ Ḟ
∂F
∂ Ė
∂F
=
i∗ ϕ
∂ Ė
∂F
P∗
P
∂ Ė
∂E
For the equilibrium of this model to be a saddle point the following
inequality has to be satisfied:
i
∗ ∂ Ė
E
<ϕ
P∗
P
∂ Ė
∂F
and
E
∂ Ė
=
∂F
αδ + βγ
−γ
∂m
∂f
+α
∂F
∂F
∂ Ė
E
=
∂E
αδ + βγ
∂m
∂f
−γ
+α
∂E
∂E
E
αδ + βγ
γ EF + α (W − EF )
WF
E
=
αδ + βγ
γ EF + α (W − EF )
WE
=
It is straightforward to obtain:
∂ Ė
∂F
∂ Ė
∂E
=
E
F
Thus, the inequality becomes:
ϕ P ∗
E>1
i∗F P
In equilibrium i ∗ F = −ϕ. If F > 0, −ϕ > 0. Therefore, the inequality can
be written as an elasticity:
ηT B,S =
∂ϕ S
>1
∂S (−ϕ)
8 Open Economy Macroeconomics
155
This inequality is the Marshall-Lerner condition for the trade balance (TB)
to increase when the terms of trade increase.
Figure 8.1 depicts the phase diagram of the model with the exchange rate
on the vertical axis and foreign assets on the horizontal axis. We assume a
creditor country.
(b) Show what happens to E and F in each of the following circumstances: (1)
an increase in M and (2) an increase in B.
Figure 8.2 shows what happens when the stock of money increases and
Fig. 8.3 what happens to E and F when the stock of bonds increases. In
both cases the curve Ḟ = 0 does not shift.
Figure 8.2 depicts an increase in the money stock that shifts upward the
Ė = 0 curve. There is an overshooting of the exchange rate, which then
converges on its long run equilibrium.
Figure 8.3 depicts an increase in the stock of bonds that shifts downward
the Ė = 0. There is an undershooting of the exchange rate, which then
converges on its long run equilibrium along the saddle path.
Fig. 8.1 The phase diagram
for the E and F system
Fig. 8.2 Dynamic
adjustment of the economy to
an unanticipated increase in
the stock of money
156
8 Open Economy Macroeconomics
Fig. 8.3 Dynamic
adjustment of the economy to
an unanticipated increase in
the stock of bonds
(7) Consider the following model of a flexible exchange rate portfolio:
E = g F, M, B, i ∗ ,
E
Ḟ = X
P
∂E
<0
∂F
+ i∗F
Based on this model comment on the following proposition: “A country with a
deficit on the current account on the balance of payments tends to depreciate the
exchange rate, while one with a surplus tends to appreciate the exchange rate.”
Figure 8.4 shows the phase diagram of the differential equation for F . The
equation Ḟ = 0 slopes downward and the arrows denotes the dyamics of the
variables:
The first equation of the model states that E and F are negatively correlated
∂E
because ∂F
< 0 (Fig. 8.5).
Fig. 8.4 The phase diagram
of the differential equation for
F
8 Open Economy Macroeconomics
157
Fig. 8.5 E and F are
negatively correlated
according to the first equation
of the model
Fig. 8.6 (a) Model is unstable (b) Model is stable
Figure 8.6 shows two possibilities in case (a) the model is unstable and in case
(b) the model is stable.
In the case (b) of Fig. 8.6 a country with a deficit depreciates and with a surplus
appreciates the exchange rates.
(8) In a small open economy, the Central Bank sets the nominal interest rate. That
is:
i = ī
(a) Is this economy’s nominal exchange rate determined (Hint: use the uncovered interest rate parity).
The UIP is given by:
i = i ∗ + ė
Therefore,
ė = ī − i ∗
158
8 Open Economy Macroeconomics
Fig. 8.7 Demand for money
We may conclude that the exchange rate e is not determined.
(b) Compare your response with what would happen in a closed economy if
Central Bank used the same monetary policy rule. Figure 8.7 shows the
demand for money equation. The nominal interest rate is on the vertical
axis and real cash balance (M/P ) on the horizontal axis:
The Central Bank fixes the nominal interest rate. Thus, m̄ is given but not
the price level, because:
P =
M
m̄
and M is not determined. This monetary policy rule does not provide a
nominal anchor for the economy.
(9) The Taylor rule adopted by a small open economy is:
i = r̄ + π + φ (π − π̄ ) + θ x + λq
The coefficients φ, θ and λ are positive and assume, for simplicity, that the
long-run foreign exchange rate is equal to zero: q̄ = 0.
The foreign country’s Taylor rule is:
i ∗ = r̄ ∗ + π ∗ + φ π ∗ − π̄ + θ x ∗
The asterisk (*) represents the foreign variables and the two rule’s parameters
are equal to simplify the algebra.
The two countries’real interest rates and related by means of the real interest
rate parity:
r = r ∗ + q̇
8 Open Economy Macroeconomics
159
(a) Show that the real exchange rate follows the differential equation:
q̇ = −f + λq
Subtracting the foreign Taylor rule from the domestic one we obtain:
r − r ∗ = r̄ − r̄ ∗ + φ (π − π̄ ) + θ x + λq − φ π ∗ − π̄ ∗ − θ x ∗
which can be written as:
r − r ∗ = r̄ − r̄ ∗ + φ (π − π̄ ) − π ∗ − π̄ ∗
+ θ x − x ∗ + λq
Substituting this interest rate gap into the UIP yields:
q̇ = λq − f
where f is given by:
f = r̄ ∗ − r̄ − φ (π − π̄ ) − π ∗ − π̄ ∗
− θ x − x∗
(b) Show the solution to this differential equation with the two components,
fundamentals and the bubble.
The solution of the differential equation:
q̇ = λq − f
has two components, the fundamentals component qf (t) and the bubble
component qb (t): q = qf (t) + qb (t)
qf (t) =
∞
e−λ(τ −t) f dτ
t
and
qb (t) = ceλt
Chapter 9
Economic Fluctuation and Stabilization
in an Open Economy
(1) Consider the following model of a small open economy (Mundell-FlemingDornbusch model):
ṗ = δ (d − y)
d = k + α e + p∗ − p − βi
m − p = −γ i + φy
i = i ∗ + ėe
ėe = ė
By combining the two first equations of this model we obtain:
ṗ = δ k + α e + p∗ − p − βi − y
In equilibrium ṗ = 0, ė = 0. Thus, p = p̄ and e = ē. We may write
0 = δ k + α ē + p∗ − p̄ − βi ∗ − y
Subtracting this equation from the previous one yields:
ṗ = δα (e − ē) − δα (p − p̄) − δβ i − i ∗
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_9
161
162
9 Economic Fluctuation and Stabilization in an Open Economy
In equilibrium the money demand equation becomes:
m − p̄ = −γ i ∗ + φy
By subtracting this equation from the money demand we obtain:
− (p − p̄) = −γ i − i ∗
Since i − i ∗ = ė, we get
i − i ∗ = ė =
1
(p − p̄)
γ
Thus, the equation for ṗ can be written as:
β
ṗ = δα (e − ē) − δ α +
γ
(p − p̄)
The model of this exercise can be collapsed into a system of two linear
differential equations:
ṗ = δα (e − ē) − δ α +
ė =
β
γ
(p − p̄)
1
(p − p̄)
γ
The Jacobian of the system is:
∂ ṗ ∂ ṗ J =
∂p ∂e
∂ ė ∂ ė
∂p ∂e
=
−δ α + γβ αδ
1
γ
0
The determinant of this Jacobian is:
|J | = −
αδ
<0
γ
and the steady-state is a saddle point. The phase diagram of this model is
depicted in Fig. 9.1, with p on the vertical axis and e on the horizontal axis.
SS is a saddle path.
(a) What are the short and long run effects of a public deficit increase on the
exchange rate? The public deficit can be represented by the parameter k. It
is straightforward to see that the steady-state price level does not change and
9 Economic Fluctuation and Stabilization in an Open Economy
163
Fig. 9.1 The phase diagram
for the p and e system
Fig. 9.2 Dynamic
adjustment of the economy to
an unanticipated increase of
the public deficit
the equilibrium exchange rate decreases. Figure 9.2 depicts this permanent,
non-anticipated, change in k. The exchange rate jumps from ē(0) to ē(1).
(b) What are the short and long run effects of an increase in the price index of
goods and services produced abroad on the exchange rate?
The price level p does not change when p ∗ increases, but the steadystate exchange rate will decrease. Figure 9.3 depicts the dynamics of the
economy when this non-anticipated permanent increase in p∗ occurs. The
exchange rate jumps immediately from ē(0) to ē(1).
164
9 Economic Fluctuation and Stabilization in an Open Economy
Fig. 9.3 Dynamic
adjustment to an
unanticipated increase of the
foreign price index
(2) Consider the model:
ṗ = φ (y − ȳ)
y = α0 + α1 e + p∗ − p − α2 i
m − p = β0 + β1 y − β2 i
i = i ∗ + ė
Substituting the second equation of this model into the first equation we obtain:
ṗ = φ α0 + α1 e + p∗ − p − α2 i − ȳ
In equilibrium ṗ = ė = 0. Thus, p = p̄, e = ē and i = i ∗ . When ṗ = 0 we
may write:
0 = φ α0 + α1 ē + p∗ − p̄ − α2 i ∗ − y
Subtracting from the ṗ equation this equation yields:
ṗ = φα (e − ē) − φα1 (p − p̄) − φα2 i − i ∗
(9.1)
In steady-state the demand for money equation is:
m − p̄ = β0 + β1 ȳ − β2 i ∗
Subtracting this equation from the money demand equation of the model we
obtain:
p − p̄ = −β1 (y − ȳ) − β2 i − i ∗
(9.2)
9 Economic Fluctuation and Stabilization in an Open Economy
165
In steady-state the second equation of the model is given by:
ȳ = α0 + α1 ē + p∗ − p̄ − α2 i ∗
Therefore, subtracting this equation from this second equation of the model we
obtain:
y − ȳ = α1 [(e − ē) − (p − p̄)] − α2 i − i ∗
(9.3)
By substituting Eq. (9.3) into Eq. (9.2) yields:
p − p̄ = −β1 α1 [(e − ē) − (p − p̄)] + (β1 α2 + β2 ) ė
taking into account that i − i ∗ = ė. Thus, the differential equation for e is:
ė =
α1 β1
1 + α1 β1
(e − ē) +
(p − p̄)
β1 α2 + β2
β1 α2 + β2
(9.4)
By substituting (9.4) into (9.1) we get the differential equation for p:
ṗ =
α1 β2 φ
φ (α1 β2 + α2 )
(e − ē) −
(p − p̄)
β1 α2 + β2
β1 α2 + β2
(9.5)
The differential equation system given by (9.4) and (9.5) has the following
Jacobian matrix:
∂ ṗ ∂ ṗ −φ(α β +α ) α β φ 1 2
J =
∂p ∂e
∂ ė ∂ ė
∂p ∂e
=
2
β1 α2 +β2
1 α1 −1)
− (β
β1 α2 +β2
1 2
β1 α2 +β2
α1 β1
β1 α2 +β2
The determinant of this Jacobian is given by:
|J | = −
α1 φ
<0
β1 α2 + β2
Therefore, the equilibrium is a saddle point. Figures 9.4 and 9.5 depict the phase
diagram for this model. Figure 9.4 assumes that α1 β1 < 1. In this case the
ė = 0 curve is downward sloping and the saddle path is a downward sloping.
Figure 9.5 shows the phase diagram of the model when α1 β1 > 1. In this
case the saddlepath is upward sloping. It should be noticed that the angular
coefficient of the ė = 0 curve is greater than the coefficient of the ṗ = 0 curve
because:
β2
α1 β2 +α2
α1 β1
α1 β1 −1
α1 β1 − 1
β2
β2
=
=
α1 β2 + α2 α1 β1
α1 β2 + α2
1
1−
α1 β1
<1
166
9 Economic Fluctuation and Stabilization in an Open Economy
Fig. 9.4 The phase diagram
for the p and e system when
α1 β1 < 1
Fig. 9.5 The phase diagram
for the p and e system when
α1 β1 > 1
and
α1 β1
β2
<
α1 β2 + α2
α1 β1 − 1
(a) What is the effect of an increase in the quantity of money on exchange rate?
When the quantity of money increases both the steady-state price level
and the steady-state exchange rate increase say from p̄(0), ē(0) to p̄ (1),
ē(1). Both curves ė = 0 and ṗ = 0, shift. Figures 9.6 and 9.7 show the
new saddle path. When the saddle path is downward sloping there is an
overshooting of the exchange rate. But, when the saddle path is upward
sloping there is no overshooting of the exchange rate.
9 Economic Fluctuation and Stabilization in an Open Economy
167
Fig. 9.6 Dynamic
adjustment to an
unanticipated increase of the
money stock when the saddle
path is downward sloping
Fig. 9.7 Dynamic
adjustment to an
unanticipated increase of the
money stock when the saddle
path is upward sloping
(b) Compare your response to the previous item with the one that you would
obtain replacing the ṗ equation with ṗ = φ(d −y) with the second equation
becoming the expenditure equation: d = α0 + α1 (e + p∗ − p) − k2 i. Is
there a chance that a monetary expansion may cause undershooting instead
of overshooting?
Figure 9.1 depicts the phase diagram of the model of Exercise 1. When m
has a permanent unanticipated increase, both curves, ė = 0 and ṗ = 0,
shift as shown in Fig. 9.8. The new saddle path SS slopes downward and
the exchange rate will overshoot to e(0+ ) at the moment when the money
supply increases, because the price level is a predetermined variable. The
economy then converges through the saddle path on the new equilibrium.
168
9 Economic Fluctuation and Stabilization in an Open Economy
Fig. 9.8 Dynamic
adjustment to an
unanticipated increase of the
stock of money when output
is given
(3) Consider the following model of a small open economy:
IS: x = −α (r − r̄) + β (s − s̄)
PC: π̇ = γ ṡ + δx
UIP: r = r̄ + ṡ, r̄ = r ∗
MPR: i = r̄ + π + φ (π − π̄ ) + θ (s − s̄)
IC: Given p(0) and π(0)
(a) Analyze the model’s equilibrium and dynamics on a phase diagram with the
real interest rate (r) on the vertical axis and the terms of trade (s) on the horizontal axis. To obtain the differential equation for r we write the MPR as:
r − r̄ = φ (π − π̄) + θ (s − s̄)
and we take the time derivatives of both sides:
ṙ = φ π̇ + θ ṡ
We substitute the PC into this equation to obtain:
ṙ = φγ ṡ + φδx + θ ṡ
which can be written as:
ṙ = (φγ + θ ) ṡ + φδx
Next, we substitute the UIP and the IS equations in this expression to get:
ṙ = (φγ + θ ) (r − r̄) + φδ [−α (r − r̄) + β (s − s̄)]
By rearranging terms we obtain the differential equation for r:
ṙ = (φγ + θ − φδα) (r − r̄) + φδβ (s − s̄)
9 Economic Fluctuation and Stabilization in an Open Economy
169
The differential equation for s is the UIP equation:
ṡ = r − ṙ
The Jacobian of the linear system of differential equations is:
∂ ṙ ∂ ṙ φγ + θ − φδα φδβ
∂r
∂s
J = ∂ ṡ ∂ ṡ =
1
0
∂r ∂s
The determinant of this Jacobian is:
|J | = −φδβ < 0
The steady-state equilibrium is a saddle point. Figure 9.9 depicts the phase
diagram of the model with the interest rate on the vertical axis and the
exchange rate on the horizontal axis.
(b) Are the real interest rate and the terms of trade negatively correlated
regardless of the values of the model’s parameters?
The phase diagram of Fig. 9.9 assumes that:
φγ + θ > φαδ
When
φγ + θ < φαδ
the phase diagram of the model is depicted in Fig. 9.10. The ṙ = 0 curve
is upward sloping. However, the saddle path is downward sloping as in
Fig. 9.9. Thus, the terms of trade (the real exchange rate) and the real
interest are negatively correlated regardless of the values of the parameters.
(c) Use the phase diagram from item (a) to show what happens in this model
when the public deficit increases.
The public deficit is an argument of the IS curve:
Fig. 9.9 The phase diagram
for the r and s system when
φγ + θ > φαδ
170
9 Economic Fluctuation and Stabilization in an Open Economy
x = −α (r − r̄) + β (s − s̄) + λf
where λ > 0 and f stands for the public deficit. Thus, the differential
equation for ṙ is now specified as:
ṙ = (φγ + θ − φδα) (r − r̄) + φδβ (s − s̄) + f
In the phase diagram of Fig. 9.10 the ṙ = 0 curve shifts up as depicted in
Fig. 9.11. The terms of trade (real exchange) jumps from s̄(0) to s̄(1), due
to the jump of the nominal exchange rate.
Fig. 9.10 The phase diagram
for the r and s system when
φγ + θ < φαδ
Fig. 9.11 Dynamic
adjustment to an
unanticipated increase of the
public deficit
9 Economic Fluctuation and Stabilization in an Open Economy
171
Fig. 9.12 Dynamic
adjustment to an
unanticipated increase of the
foreign interest rate
(d) Use the phase diagram from item (a) to show what happens in this model
when the real foreign interest rate increases.
When the real foreign interest rate increases the central bank increases
the domestic interest rate according to the MPR. The ṙ = 0 curves shifts
up. Assuming that the term of trade (real exchange rate) does not change
Fig. 9.12 depicts the phase diagram of the model. The economy goes from
point E0 to E1 .
(4) Consider the model:
IS: x = −α (r − r̄) + β (s − s̄)
PC: π̇ = γ ṡ + δx
UIP: r = r̄ + ṡ, r̄ = r ∗
MPR: e = ē
IC: Given p(0) and π(0)
We use the same letters of exercise 3 for the parameters of the Phillips Curve
(PC). To obtain the dynamical system of this model we start with the terms of
trade definition:
s = e + p∗ − p
We take the time derivatives of both sides of this expression:
ṡ = ė + ṗ∗ − ṗ = π ∗ − π
We use the MPR so that the nominal exchange rate is constant. Thus, ė = 0. It
follows from UIP that:
ṡ = r − r̄ = π ∗ − π
Taking the time derivative of this expression we obtain:
ṙ = π̇ ∗ − π̇ = −π̇
because we assume π̇ ∗ = 0. Next, substitution for π̇ from the PC curve implies:
ṙ = −γ ṡ − δx
172
9 Economic Fluctuation and Stabilization in an Open Economy
Substituting for ṡ and x for UIP and the IS curve, respectively, yields the
following differential equation after rearranging terms.
ṙ = (−γ + αδ) (r − r̄) − βδ (s − s̄)
Together with the UIP equation,
ṡ = r − r̄
we have a system of two linear differential equations, with the following
Jacobian matrix:
∂ ṙ ∂ ṙ αδ − γ −βδ
∂r
∂s
J = ∂ ṡ ∂ ṡ =
1
0
∂r ∂s
The determinant of this Jacobian is positive:
|J | = βδ > 0
The trace of this Jacobian is:
trJ = αδ − γ
and it can be either positive or negative. We assume the trace to be negative:
αδ − γ < 0,
for the system to be stable.
Use a phase diagram with r in the vertical axis and s on the horizontal axis to
show this model’s equilibrium and dynamics.
Figure 9.13 depicts the phase diagram of the model. The ṙ = 0 curve is
downward sloping. The arrows show the dynamics of the model.
Fig. 9.13 The phase diagram
for the r and s system
9 Economic Fluctuation and Stabilization in an Open Economy
173
(5) Consider the model:
IS: x = −α (r − r̄) + β (s − s̄)
PC: π̇ = γ ṡ + δx
UIP: r = r̄ + ṡ
MPR: i = r̄ + π + θ (π − π̄ ) + φ ṡ
IC: Given p(0) and π(0)
(a) Analyze the model’s equilibrium and dynamics on a phase diagram with
inflation on the vertical axis and the terms of trade on the horizontal axis.
From the MPR we obtain the interest rate gap:
r − r̄ = θ (π − π̄ ) + φ ṡ
Substituting for the interest rate gap into the UIP yields:
ṡ = θ (π − π̄ ) + φ ṡ
which can be written as:
ṡ =
θ
(π − π̄ )
1−φ
Substituting for the output gap from the IS curve into the Phillips Curve
results:
π̇ = γ ṡ + δ [−α (r − r̄) + β (s − s̄)]
which can be written as:
π̇ = γ ṡ − αδ (r − r̄) + βδ (s − s̄)
Substituting for ṡ from the previous equation into the π̇ equation after
taking into account the UIP equation we get:
π̇ =
(γ − αδ)
θ (π − π̄ ) + βδ (s − s̄)
1−φ
The dynamical system of the model is:
π̇ =
(γ − αδ)
θ (π − π̄ ) + βδ (s − s̄)
1−φ
ṡ =
θ
(π − π̄ )
1−φ
174
9 Economic Fluctuation and Stabilization in an Open Economy
Fig. 9.14 The phase diagram
for the r and s system
Fig. 9.15 Dynamic
adjustment to an
unanticipated decrease in the
inflation target
The Jacobian of this system is:
J =
∂ π̇ ∂ π̇ ∂π ∂s
∂ ṡ ∂ ṡ
∂π ∂s
=
(γ −αδ)θ
βδ
1−φ
θ
0
1−φ
The determinant of this Jacobian is:
|J | = −
βδθ
1−φ
This determinant is negative if 1 − φ > 0. We assume that the monetary
policy rule is such that φ < 1. Thus, the equilibrium is a saddle path.
Figure 9.14 depicts the phase diagram of the model. The inflation rate, a
9 Economic Fluctuation and Stabilization in an Open Economy
175
predetermined variable is on the vertical axis. The terms of trade, a jump
variable, is on horizontal axis. SS, the saddle path, is downward sloping.
(b) Use the previous item’s phase diagram to show what happens when the
Central Bank lowers the inflation target to π̄1 (< π̄0 ) from π̄0 . The dynamic
adjustment of the economy for this experiment is depicted in Fig. 9.15. The
point E0 is the initial equilibrium of the model before the Central Bank
changes its inflation target. When the Central Bank changes its inflation
target the curve π̇ = 0 shifts down. The new long run equilibrium of the
economy is given by point Ef . Because π is a predetermined variable there
is a jump from E0 to E1 when the new policy is announced. Then the
economy converges on Ef through the saddle path.
(6) Consider the model:
IS: x = −α (r − r̄) + β (q − q̄)
PC: π̇ = γ q̇ + δx
UIP: r = r̄ + q̇
MPR: i = r̄ + πc + θ (πc − π̄c )
w
CPI: πc = π + χ q̇, χ = 1−w
FE: i = r + πc
IC: Given p(0) and π(0)
(a) Analyze the model’s equilibrium and dynamics on a phase diagram with
the real interest rate (on the vertical axis) and the real exchange rate (on the
horizontal axis).
In this Keynesian model the real exchange rate is defined by
q = e + p ∗ − pc
and the consumer price is a weight average of the domestic price and the
imported price:
pc = (1 − w) p + w e + p∗
From the MPR we can write:
r = r̄ + θ (πc − π̄c )
Taking the derivatives with respect to time of both sides of this expression
results in:
ṙ = θ π̇c
From the consumer price index (CPI) definition we get
π̇c = π̇ + χ q̈ = π̇ + χ ṙ
176
9 Economic Fluctuation and Stabilization in an Open Economy
since the UIP implies
ṙ = q̈
If we substitute out for π̇c in the ṙ equation we get
ṙ = θ π̇ + θ χ ṙ
which can be written as
ṙ =
θ
π̇
1 − θχ
If we substitute out π̇ from the Phillips equation
ṙ =
θ
[γ q̇ + δx]
1 − θχ
If we substitute out q̇ and x from the UIP and the IS curve, respectively, we
get the differential equation for the real interest rate:
ṙ =
θ (γ − αδ)
βθ δ
(r − r̄) +
(q − q̄)
1 − θχ
1 − θχ
The second differential equation of the model is the UIP equation:
q̇ = r − r̄
The Jacobian of this system of differential equations is given by:
J =
∂ ṙ
∂r
∂ q̇
∂r
∂ ṙ
∂q
∂ q̇
∂q
=
θ(γ −αδ) βθδ
1−θχ 1−θχ
1
0
The determinant of this Jacobian is negative,
|J | = −
βθ δ
1 − θχ
if 1 − θ χ > 0. We assume that the MPR is such that this assumption holds.
Figure 9.16 depicts the phase diagram of this model. SS is the saddlepath.
It should be noticed that both variables, r and q, are jump variables.
9 Economic Fluctuation and Stabilization in an Open Economy
177
Fig. 9.16 The phase diagram
for the r and s system
(b) Use the previous item’s phase diagram to show what happens in this
economy when the real interest rate increases to r1∗ > r0∗ from r0∗ .
Figure 9.17 depicts the adjustment of the economy to an increase in
the international rate of interest. The real exchange rate depends on the
international real rate of interest.
When the international rate of interest increases the real exchange rate
decreases. The economy jumps from E0 to E1 because the nominal
exchange rate adjusts instantaneously.
Fig. 9.17 Dynamic
adjustment to an
unanticipated increase in the
foreign interest rate
(c) Analyse the model’s equilibrium and dynamics with the inflation rate on
the vertical axis and real output on the horizontal axis.
178
9 Economic Fluctuation and Stabilization in an Open Economy
Substitution of the UIP equation into the PC curve results
π̇ = γ (r − r̄) + δx
We substitute the interest rate gap given by MPR into this equation to
obtain:
π̇ = γ θ (πc − π̄c ) + δx
From the definition of the consumer price index it is straightforward to show
that
π̇c =
1
π̇
1 − χθ
Combining the two last equations yields the differential equation for the
consumer inflation rate:
π̇c =
γθ
δ
x
(πc − π̄c ) +
1 − χθ
1 − χθ
Taking the derivatives with respect to time of the IS curve we get
ẋ = −α ṙ + β q̇ = −α ṙ + β (r − r̄)
The expression after the second sign of this equality was obtained using the
UIP equation.
We substitute the interest rate gap from the MPR into the ẋ equation to
obtain
ẋ = −α ṙ + βθ (πc − π̄c )
Taking the time derivatives of the MPR yields
ṙ = θ π̇c
Thus, we substitute this expression into the ẋ equation to get:
ẋ = −αθ π̇c + βθ (πc − π̄c )
By substituting the π̇c equation into this expression and rearranging terms
we obtain the equation for ẋ. Therefore, the dynamical system is given by:
αθ δ
αθ γ
ẋ = −
x+θ β −
1 − χθ
1 − χθ
(πc − π̄c )
9 Economic Fluctuation and Stabilization in an Open Economy
π̇c =
179
γθ
δ
x
(πc − π̄c ) +
1 − χθ
1 − χθ
The Jacobian matrix of this system is given by:
J =
∂ π̇c ∂ π̇c
∂πc ∂x
∂ ẋ ∂ ẋ
∂πc ∂x
γθ
δ
1−χ θ 1−χ θ
=
αθγ
αθδ
θ β − 1−χ
θ − 1−χ θ
The determinant of this Jacobian is:
|J | =
Fig. 9.18 The phase diagram
for the πc and x system
Fig. 9.19 Dynamic
adjustment to an
unanticipated decrease in the
inflation target
−θ δβ
<0
1 − χθ
if
1 − χθ > 0
180
9 Economic Fluctuation and Stabilization in an Open Economy
We assume that 1 − χ θ > 0. Figure 9.18 depicts the phase diagram of the
model with the inflation rate on the vertical axis and the output gap on the
horizontal axis. SS is the saddle path, which is downward sloping.
(d) Use the previous item’s phase diagram to show what happens when the
Central Bank lowers the inflation target to π̄1 (< π̄0 ) from π̄0 .
Phase diagram 9.19 shows what happens to the inflation rate and to the
output gap. The inflation rate is a predetermined variable while the output
gap is a jump variable.
Figure 9.19 does not show the complete phase diagram, just the saddle path
SS after the inflation target change. The inflation rate is a predetermined
variable. Therefore, at the initial moment, the economy jumps to point E0
yielding a recession, the output gap becomes negative, and the inflation rate
converges on the new equilibrium at π̄1 .
(7) Consider the model:
IS: x = −α (r − r̄) + β (q − q̄)
PC: π̇ = γ q̇ + δx
UIP: r = r̄ + q̇
MPR: e = ē
CPI: πc = π + χ q̇
FE: i = r + πc
IC: Given p(0) and π(0)
(a) Analyze the model’s equilibrium and dynamics on a phase diagram with
the real interest rate (on the vertical axis) and the real exchange rate (on the
horizontal axis).
Substituting the IS curve and the UIP equation into the Phillips Curve we
obtain:
π̇ = (γ − αδ) (r − r̄) + βδ (q − q̄)
From the definition of the real exchange rate, q = e + p∗ − pc , we get:
q̇ = π ∗ − πc
Substituting the CPI definition in this equation results:
q̇ = π ∗ − π − χ q̇
or
(1 + χ ) q̇ = π ∗ − π
Taking the time derivatives of both sides of this expression yields:
(1 + χ ) q̈ = −π̇
9 Economic Fluctuation and Stabilization in an Open Economy
181
assuming that π̇ ∗ = 0. From the UIP it follows that:
ṙ = q̈
By combining the two last equations we obtain:
ṙ = −
1
π̇
1+χ
Plugging the π̇ equation, derived at the beginning of this item, into this
expression we obtain the differential equation for the real interest rate:
ṙ =
αδ − γ
βδ
(r − r̄) −
(q − q̄)
1+χ
1+χ
together with the UIP differential equation for q
q̇ = r − r̄,
we obtain the dynamical system of the model. The Jacobian matrix is:
J =
∂ ṙ
∂r
∂ q̇
∂r
∂ ṙ
∂q
∂ q̇
∂q
=
αδ−γ
1+χ
1
βδ
− 1+χ
0
The determinant and the trace of this matrix are given by:
|J | =
βδ
>0
1+χ
trJ = αδ − γ
If αδ − γ < 0 the system is stable. We assume that this inequality holds.
Figure 9.20 is the phase diagram of the model:
(b) Use the previous item’s phase diagram to show what happens in this
economy when the real interest rate increases to r1∗ > r0∗ from r0∗ .
Figure 9.21 depicts the phase diagram of the model when the real interest
rate increases. The arrowed path shows the dynamic adjustment of the
economy towards the new equilibrium (point Ef ). The real exchange rate
starts to fall until it reaches the new q̇ = 0 curve. From that point it rises
and then, in a cyclical way, converges on the new equilibrium where the
real interest rate is equal to the new world rate of interest.
(c) Analyze the model’s equilibrium and dynamics with the inflation rate on
the vertical axis and real output gap on the horizontal axis.
From the definition of the real exchange rate, q = e + p∗ − pc , we obtain
182
9 Economic Fluctuation and Stabilization in an Open Economy
Fig. 9.20 The phase diagram
for the r an q system
Fig. 9.21 Dynamic
adjustment to an
unanticipated increase in the
foreign interest rate
Ef
1*
q̇̇ 1* 0
0*
E0
q̇̇ 0* 0
˙̇ 0
q
q̇ = ė + π ∗ − πc = π ∗ − πc
The last equality is due to the fact the exchange rate is fixed: ė = 0.
Substituting the consumer price into this expression results:
q̇ = π ∗ − π − χ q̇
Thus,
q̇ =
π∗ − π
1+χ
Substituting this into the Phillips curve we obtain the differential equation
for the rate of inflation:
π̇ = −
γ
π − π ∗ + δx
1+χ
9 Economic Fluctuation and Stabilization in an Open Economy
183
The differential equation for the output gap can be obtained as follows.
First, we take the derivatives with respect to time of the IS curve:
ẋ = −α ṙ + β q̇
From the UIP equation we get:
ṙ = q̈ = −
π̇
1+χ
because
π ∗ − π̇
1+χ
q̈ =
and we assume π̇ ∗ = 0. Thus the differential equation for the output gap is:
αδ
1
ẋ =
x−
1+χ
1+χ
αγ
β+
1+χ
π − π∗
The system of linear differential equation has the following Jacobian:
J =
∂ π̇ ∂ π̇ γ
− 1+χ
δ
=
αγ
1
αδ
− 1+χ β + 1+χ 1+χ
∂π ∂x
∂ ẋ ∂ ẋ
∂π ∂x
The determinant and the trace of this matrix are:
|J | =
βδ
>0
1+χ
trJ =
αδ − γ
1+χ
We assume that αδ − γ < 0 for for the system to be stable. The phase
diagram of this dynamical system is depicted in Fig. 9.22. The slope of the
ẋ = 0 line is less than the slope of π̇ = 0 because
αδ(1+χ )
β(1+χ )+αγ
δ(1+χ )
δ
=
αγ
<1
αγ + β (1 + χ )
(d) Use the previous item’s phase diagram to show what happens when the
international inflation rate decreases to π1∗ (< π0∗ ) from π0∗ .
Figure 9.23 depicts the dynamic adjustment of the economy. The inflation
rate is a predetermined variable. Thus, at the initial moment the inflation
rate does not change. The economy enters into recession; the output gap is
184
9 Economic Fluctuation and Stabilization in an Open Economy
Fig. 9.22 The phase diagram
for the π and x system
Fig. 9.23 Dynamic
adjustment to an
unanticipated increase in the
foreign inflation rate
π˙̇ 0
π
E0
x˙̇ 0
Ef
x
negative. The inflation rate decreases and in the case drawn in Fig. 9.23
there is an undershooting, but the inflation rate converges on the new
equilibrium.
(8) Consider the model:
IS: x = −α (r − r̄) + β (q − q̄)
PC: π̇ = γ q̇ + δx
UIP: r = r̄ + q̇
MPR: ė = π − π ∗
IC: Given p(0) and π(0)
(a) Analyze the model’s equilibrium and dynamics.
9 Economic Fluctuation and Stabilization in an Open Economy
185
The real exchange and the consumer price index are defined by:
q = e + p ∗ − pc
pc = (1 − ω) p + ω e + p∗
It is straightforward to show that:
q̇ = ė + π ∗ − πc = π − π ∗ + π ∗ − πc = π − πc
where we have used the monetary policy rule. From the consumer price
index, we obtain:
πc = π +
ω
q̇
1−ω
Therefore, we conclude by comparing these expressions that:
q̇ = 0
The UIP implies that:
r = r̄
Thus, the IS curve becomes:
x = β (q − q̄)
and the Phillips curve states that:
Fig. 9.24 Inflation
acceleration and the natural
real exchange rate gap
186
9 Economic Fluctuation and Stabilization in an Open Economy
π̇ = βδ (q − q̄)
Figure 9.24 shows that if q > q̄ inflation increases and if q < q̄ inflation
decreases. Thus, the fact that q̇ = 0 does not imply that q = q̄, it says that
q is constant.
(b) Show what happens in this economy when the natural interest rate
increases.
If the natural rate of interest (which is equal to the foreign rate of interest)
increases, the natural real exchange rate will decrease. If q were equal
to q̄ before the change in foreign interest rate, the MPR does not yield
adjustment in the real exchange rate.
(9) Assume that imports and labor are used as inputs in the production of a domestic
product.
The economy’s real output is then equal to the sum of consumption and exports:
yt = ω1 ct + ω2 ext
The variables are in log form and the weight ωt is the respective variable
proportion in the steady-state. The demand for export equation is the demand
equation for an input that depends on the world output and the relative price:
ext = yt∗ + ηst + k
The parameter η is elasticity of substitution between the input and labor, k is a
constant and s is the terms of trade as defined by: st = et + pt∗ − pt . The Euler
equation is given by:
ct = −σ (rt − ρ) + ct+1
(a) Show that the IS curve for the open economy model is the following:
xt = xt+1 − ω1 σ (rt − r̄t ) − ω2 η (st+1 − s̄t+1 )
By substituting the Euler equation into the output equation we obtain:
yt = −ω1 σ (r + ρ) + ω1 ct+1 + ω2 ext
The output equation for next period is:
yt+1 = ω1 ct+1 + ω2 ext+1
9 Economic Fluctuation and Stabilization in an Open Economy
187
Subtracting this equation from the previous one results:
yt = yt+1 − ω1 (rt − ρ) − ω2 ext+1 + ω2 ext
The export equation for period t and t + 1 are:
ext = yt∗ + ηst + κ
∗
+ ηst+1 + κ
ext+1 = yt+1
Thus,
∗
− yt∗ + η (st+1 − st )
ext+1 − ext = yt+1
Taking into account this expression the output equation is given by:
∗
− ω2 ηst+1
yt = yt+1 − ω (rt − ρ) − ω2 yt+1
where
∗
∗
= yt+1
− yt∗
yt+1
and
st+1 = st+1 − st
This output equation can be written as:
∗
yt − ȳt = yt+1 − ȳt+1 + (ȳt+1 − ȳt ) − ω1 σ (rt − ρ) − ω2 yt+1
− ω2 η (st+1 − s̄t+1 ) − ω2 ηs̄t+1
where ȳ is potential output, and we added and subtracted the term
ω2 ηs̄t+1 , where the bar over the variable is its full employment value. By
using the notation x = y − ȳ , we write the IS curve as:
∗
xt = xt+1 − ω1 σ (rt − ρ) + ȳt+1 − ω2 yt+1
− ω2 ηs̄t+1
−ω2 η (st+1 − s̄t+1 )
This equation can be rearranged as:
ω2 yt+1
ω2 η
ȳt+1
+
+
s̄t+1
xt = xt+1 − ω1 σ rt − ρ −
ω1 σ
ω1 σ
ω1 σ
− ω2 η (st+1 − s̄t+1 )
188
9 Economic Fluctuation and Stabilization in an Open Economy
(b) Show that the natural rate of interest is given by the expression:
r̄t = ρ +
∗
ω2 yt+1
ω2 η
ȳt+1
−
−
s̄t+1
ω1 σ
ω1 σ
ω1 σ
This expression is inside the terms in brackets in the equation above. Thus,
the IS curve of this model is:
xt = xt+1 − ω1 σ (rt − r̄t ) − ω2 η (st+1 − s̄t+1 )
(c) Show that this expression of the natural rate can be simplified, using a little
algebra, to obtain the same formula as in a closed economy:
r̄t = ρ +
1
ȳt+1
σ
If the natural rate of interest is given by this expression and by the previous
one, it follows that:
∗
ω2 yt+1
ω2 ηs̄t+1
1
ȳt+1
ȳt+1 =
−
−
σ
ω1 σ
ω1 σ
ω, σ
It follows that:
∗
ω1 ȳt+1 = ȳt+1 − ω2 yt+1
− ω2 ηs̄t+1
which can be written as:
∗
+ ω2 ηs̄t+1
(1 − ω1 ) ȳt+1 = ω2 yt+1
We can write the output equation for full employment in periods t and t +1:
ȳt = ω1 c̄t + ω2 ex̄t
ȳt+1 = ω1 c̄t+1 + ω2 ex̄t+1
Subtracting the former from the latter we obtain:
ȳt+1 = ω, c̄t+1 + ω2 ex̄t+1
The export equations for periods t and t + 1 are:
ex
¯ t = yt∗ + ηs̄t + κ
∗
ex
¯ t+1 = yt+1
+ ηs̄t+1 + κ
9 Economic Fluctuation and Stabilization in an Open Economy
189
thus,
∗
+ ηs̄t+1
ex̄t+1 = yt+1
In full employment:
ȳt+1 = c̄t+1
By taking into account these two expressions we can write the rate of
growth of potential output as:
∗
+ ηs̄t+1
ȳt+1 = ω1 ȳt+1 + ω2 yt+1
which can be written as:
∗
+ ω2 ηs̄t+1
(1 − ω1 ) ȳt+1 = ω2 yt+1
Therefore, the natural rate of interest in this open economy model is the
same rate of the closed economy.
(d) What is your verdict regarding this IS curve model of a small open
economy?
The verdict is that this open economy IS curve has no theoretical coherence
because according the UIP the natural rate of interest is equal to the
international rate of interest plus the change in the real exchange rate:
r̄ = rt∗ + q̄t+1
(10) *(Tradable and nontradable goods). In a small open economy there are two
goods, a tradable and a nontradable good. The tradable good price is given
and fixed in international markets. The endowments of the two goods are given:
yN = ȳN , yT = ȳT , where yN and yT denote output of nontradable and tradable
goods, respectively, and a bar over a variable stands for the given endowment.
The demand for nontradable goods relative to tradable goods is specified:
cN
=
cT
PN
PT
−η
= S −η
where PN and PT are, respectively, the prices of nontradable and tradable
goods, the letter c denotes consumption with indexing stating for the type of
good. We assume both goods are normal. S is the relative price of nontradable
goods in terms of tradable goods, and η is the elasticity of substitution between
the two goods.
190
9 Economic Fluctuation and Stabilization in an Open Economy
The price index is approximated by:
P = EPT∗
1−γ
γ
PN
where PT = EPT∗ , E is the nominal exchange rate and PT∗ is the international
price of tradable goods. For simplicity we normalize PT∗ = 1. Thus, PT =
E. We assume perfect international capital markets. This small open economy
takes as given the international interest rate.
(a) Assume that prices are flexible. What is the natural exchange rate?
(b) Assume that prices of nontradable goods are sticky and change according
to a Phillips Curve. Suppose that the Central Bank uses a Taylor rule to set
the interest rate. Analyze the equilibrium and stability of this model under
two hypotheses: (i) the inflation rate is predetermined, and (ii) the inflation
rate is free to change.
(a) The accumulation of a foreign asset, the current account of the balance of
payments, is:
Ḃ = iB + PT yT + PN yN − PT cT − PN cN
where B is the stock of foreign assets, the current account of the balance of
payments, is:
iB
Ḃ
=
+ yT + SyN − cT − ScN
PT
PT
We define the stock of foreign assets in terms of tradable goods by:
b=
B
PT
We take the time derivatives of both sides of this equation to obtain:
ḃ =
Ḃ
− bπT
PT
where πT is the inflation rate of tradable goods:
πT =
ṖT
PT
By substituting the equation for ḃ into the current account yields:
ḃ = πT b + yT + SyN − cT − ScN
9 Economic Fluctuation and Stabilization in an Open Economy
191
where the real interest rate in terms of tradable goods is:
rT = i − πT
The real interest rate in terms of prices of both goods is:
r =i−π
The price index can be written as:
1−γ
P = PT
γ
PN = PT
PN
PT
γ
= PT S γ
Therefore
π = πT + γ ṡ
where s = log(S). It is straightforward to show that:
r = rT − γ ṡ
The equilibrium of the nontradable goods market is given by:
cN = ȳN
The domestic output of tradable goods is equal to the endowment:
yT = ȳT
The last two equations imply that the current account is given by:
ḃ = rT b + ȳT − cT
Integrating forward this international flow constraint yields the international
stock budget constraint:
b(0) =
∞
e−rT t (cT − ȳT ) dt
0
In equilibrium the consumption of tradables is constant. Thus, from this
stock constraint results:
c̄T = rT b(0) + ȳT
192
9 Economic Fluctuation and Stabilization in an Open Economy
The relative price of nontradable goods in equilibrium is obtained from the
demand equation for nontradable:
ȳN
= S̄ η
rT b(0) + ȳT
which can be written in logs as:
s̄ =
1
log (rT b(0) + ȳT ) − log (ȳN )
η
The real exchange rate is defined by:
Q=
EP ∗
P
We assume, for simplicity, that the foreign price level has the same weights
of the domestic price level, thus:
∗1−γ
∗γ
EPT
PN
EPT∗
EP ∗
=
Q=
=
1−γ
P
PT
P
T
S∗
S
γ
=
S∗
S
γ
By taking logs results:
q = γ s∗ − s
Again, for simplicity, and no loss of generality, s ∗ = log S ∗ = 0. Thus:
q = −γ s
By substituting s̄ into this expression we obtain the natural exchange rate:
q̄ =
γ
log (ȳN ) − log (rT b(0) + ȳT )
η
The small open economy takes the international interest rate as given. Thus,
uncovered real interest rate parity yields:
r = r ∗ + q̇
In equilibrium q̇ = ṡ = 0, and
r̄T = r̄ = r ∗
9 Economic Fluctuation and Stabilization in an Open Economy
193
Therefore, we conclude that the natural exchange rate is given by:
q̄ =
γ
log (ȳN ) − log r ∗ b(0) + ȳT
η
The natural exchange rate depends on ȳN , r ∗ , b(0) and ȳT with the
following partial derivatives:
∂ q̄
∂ q̄
∂ q̄
∂ q̄
< 0,
> 0,
< 0 and ∗ 0 if b(0) 0
∂ ȳN
∂b(0)
∂ ȳT
∂r
(b) Nontradable goods prices are sticky and change according to the Phillips
curve:
π̇ = κ (cN − ȳN )
The parameter κ can be either positive or negative. It is positive in the
Keynesian model, which is backward looking. It is negative in the New
Keynesian model, which is forward looking.
From the demand curve for nontradables we obtain:
cN − cT = −ηs
In full employment this equation becomes:
ȳN − cT = −ηs̄
Subtracting one expression from the other results:
cN − ȳN = −η (s − s̄)
and taking into account the relationship between the relative price s and the
real exchange rate we obtain:
cN − ȳN =
η
(q − q̄)
γ
Therefore, the Phillips curve depends on the exchange rate gap:
π̇N =
κη
(q − q̄)
γ
The Central Bank sets interest rates following a simplified Taylor rule:
i = r ∗ + π + φ (πN − π̄N )
194
9 Economic Fluctuation and Stabilization in an Open Economy
By combining this rule with the interest rate arbitrage condition results:
q̇ = r − r ∗ = φ (πN − π̄N )
The model of this item can be abridged into a system of two differential
equations:
π̇N = κη
γ (q − q̄)
q̇ = φ (πN − π̄N )
The Jacobian matrix of this system is given by:
∂ π̇
J =
N
∂πN
∂ q̇
∂πN
∂ π̇N
∂q
∂ q̇
∂q
0 κη
γ
=
φ 0
The determinant and the trace of J are:
|J | = −
φκη
γ
trJ = 0
If κ > 0, |J | < 0. Thus, the equilibrium of the Keynesian model is a saddle
point, as depicted in the phase diagram of Fig. 9.25.
If κ < 0, |J | > 0, trJ = 0. Thus the equilibrium is unstable, as depicted
in phase of diagram of Fig. 9.26.
It should be mentioned that the trace is equal to zero and the determinant of
the Jacobian matrix is positive. Thus the two roots are complex conjugates
and the real part is equal to zero, such as ±bi, where i 2 = −1.
Fig. 9.25 The phase diagram
for the πN and q system
when k > 0
9 Economic Fluctuation and Stabilization in an Open Economy
195
Fig. 9.26 The phase diagram
for the πN and q system
when k < 0
(c) (Natural exchange rate: Keynesian and Australian models). Compare the
natural exchange rates obtained from the Keynesian model and the Australian model and interpret the differences between the two rates.
The current account (ca) is defined as the sum of the trade balance (tb)
and the income balance (ib):
ca = tb + ib
The trade balance is the net exports of goods and services. The income
balance is the net factor payment received from (or paid to) abroad. Net
assets can be either positive or negative, depending on the country being a
creditor or a debtor country. Thus, we can write:
tb = ca − r ∗ b
where r ∗ is the foreign interest rate and b is the total stock of foreign assets
(b > 0) or foreign debt (b < 0).
The Keynesian model has two goods, a domestic one and an imported
good. When the Marshall-Lerner condition [Macro Theory, p. 249] holds
the trade balance depends on the real exchange rate according to:
tb = tb (q) ,
∂tb
>0
∂q
The economy is in equilibrium when savings (s) is equal to the sum of
investment (i), public deficit (f ) and the current account [Macro Theory,
p. 254]:
s (ȳ − τ̄ ) = i r ∗ + f¯ + tb (q̄) + r ∗ b
196
9 Economic Fluctuation and Stabilization in an Open Economy
Fig. 9.27 The natural real exchange rate
Savings depends on the real disposable income and investment depends on
the real interest rate. A bar over a variable denotes full employment. The
domestic real interest rate is, by arbitrage, equal to the real foreign interest
rate. It follows from this expression that the trade balance in the long run
equilibrium, can be written as:
tb (q̄) = s (ȳ − τ̄ ) − i r ∗ − f¯ − r ∗ b = tb
Figure 9.27 depicts this equation, with the real exchange rate on the
vertical axis and the trade balance on the horizontal axis. Given the full
employment trade balance we obtain the natural exchange rate. The full
employment trade balance depends on the foreign interest rate (r ∗ ), on the
full employment fiscal deficit (f¯), on the net foreign assets (debt) (b), and
the full employment real disposable income.
The Australian model has two goods, a tradable good and a nontradable
good. The trade balance is the difference between the domestic production
of tradable goods (yT ) and the domestic consumption of tradable goods
(cT ). Thus, the current account is given by:
ca = tb + r ∗ b = yT − cT + r ∗ b
The domestic production of tradable goods depends on the relative price
of nontradable goods in terms of tradable goods since its marginal cost
of production is upward sloping. The consumption of tradable goods is a
function of the relative price of nontradable goods. Therefore, the trade
balance depends on the relative price of nontradable goods according to:
tb = yT (S) − cT (S) = tb(S)
9 Economic Fluctuation and Stabilization in an Open Economy
197
and
∂tb
<0
∂S
because
∂yT (S)
∂cT (S)
< 0,
>0
∂S
∂S
In the first part of this exercise we have shown that the real exchange rate
and the relative price of nontradable goods in terms of tradable goods are
related by:
Q=
S∗
S
γ
It follows that Q and S are negatively correlated and the trade balance and
the real exchange rate are positively correlated:
tb = tb (q) ,
∂tb
>0
∂q
where q = logQ.
The natural real exchange rate is the rate consistent with the long run
equilibrium of the current account:
ca = tb (q̄) + r ∗ b
Thus,
tb (q̄) = ca − r ∗ b = tb
Figure 9.27 shows this equation curve. Both models, the Keynesian and the
Australian, yield a positive correlation between the trade balance and the
real exchange rate. The framework is very much alike as Fig. 9.27 depicts
albeit the different specification of each model.
Part III
Monetary and Fiscal Policy Models
Chapter 10
Government Budget Constraint
(1) The fiscal policy rule is given by:
f = g − τ + ib = a + αb > 0
From the budget constraint it follows that
ḃ + (n + π ) b + μm = a + αb
which can be written as:
ḃ = a − μm + (α − n − π ) b
If α − n − π > 0 and μm − a > 0 we can write:
∞
b=
e−(α−n−π )t (μm − a) dt
0
(2) Consider the model:
Bonds-and-money-financed public deficit:
g − τ + rb = ḃ + μm
Tax dependent on public debt:
τ = τ (b), τ (b) > 0
Money demand:
m = f (i), f < 0
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_10
201
202
10 Government Budget Constraint
From this demand we can express the services of money as a function of the
real quantity of money:
s(m) = im = f −1 (i)m = s(m)
and:
s (m) =
ds(m)
0
dm
Monetary policy rule:
ṁ = m (μ − π ) , μ = constant
Constant real interest rate:
r = constant
The monetary policy rule can be written as:
ṁ = mμ − mπ − rm + rm
which is equivalent to:
ṁ = (r + μ) m − (r + π ) m = (r + μ) m − im = (r + μ) m − s(m)
The model can be reduced to two differential equations:
ḃ = g − τ (b) + rb − μm
ṁ = (r + μ) m − s(m)
The Jacobian of this dynamical system is:
J =
∂ ḃ
∂b
∂ ṁ
∂b
∂ ḃ
∂m
∂ ṁ
∂m
=
−μ
−τ (b) + r
0
(r + μ) − s (m)
The determinant and the trace of this matrix are:
|J | = r − τ (b) r + μ − s (m)
trJ = r − τ (b) + (r + μ) − s (m)
There are four possibilities for stability of this model as shown in Table 10.1:
10 Government Budget Constraint
203
Table 10.1 Stability of the
dynamical system
r + μ − s (m)
r − τ (b)
+
+
|J | > 0
trJ > 0
|J | < 0
−
−
|J | < 0
|J | > 0
trJ < 0
The equilibrium can be unstable (fourth cell) or a saddle point (second and
third cells), according to the sign of r + μ − s (m) and r − τ (b) .
(3) Consider the model:
Money demand: m = α −βπ e , α > 0, β > 0, m ≤ m̄. Money-financed public
deficit:
f =
Ṁ
, f constant, ṁ = f − mπ
P
π̇ e = θ π − π e
(a) We take the time derivative of the money demand equation and we take
into account the expected inflation rate to obtain:
ṁ = −β π̇ e = −βθ π − π e
and we use the money demand function to get:
ṁ = −βθ π +
βθ (α − m)
β
which can be written as:
ṁ = −βθ π + θ (α − m) = f − mπ
where we use the hypothesis of money-financed public deficit. From this
expression we obtain the inflation rate as:
π=
f − θ (α − m)
,
m − βθ
m = βθ
Substituting the inflation rate into the public deficit financed by money
yields:
f − θ (α − m)
ṁ = f − mπ = f − m
m − βθ
204
10 Government Budget Constraint
which can be written as:
ṁ = −
θ m2 − αm + βf
m − βθ
This is the dynamic equation of the model. In steady-state ṁ = 0. Thus:
m2 − αm + βf = 0
The roots of this equation are:
α
α
4βf
m= ±
1− 2
2
2
α
We assume:
1−
4βf
≥ 0 → α 2 − 4βf ≥ 0
α2
Figure 10.1 shows the phase diagram of the model. The high inflation (H )
equilibrium is stable and the low-inflation equilibrium is unstable.
(b) When θ → ∞, π e = π . The money demand equation is:
m = α − βπ
and:
π=
Fig. 10.1 The phase diagram
of the model: high inflation
equilibria is stable and the
low inflation equilibria is
unstable
α−m
β
10 Government Budget Constraint
205
Fig. 10.2 The phase diagram
of the model: high inflation
equilibrium is stable and the
low inflation equilibria is
unstable
Substituting this inflation rate into the public deficit financed by money
equation yields:
ṁ = f − mπ = f −
m (α − m)
β
which can be written as:
ṁ =
m2 − αm + βf
β
This is the dynamic equation of this model under the hypothesis π e =
π . When ṁ = 0, the two roots are the same as in the previous item.
Figure 10.2 depicts the phase diagram of the model. The high inflation
(H ) equilibrium is stable and the low inflation equilibrium is unstable.
(c) The inflation tax revenue of the item (a) model is:
τ = πm =
f − θ (α − m)
m
m − βθ
which can be written as:
τ=
θ m2 + (f − αθ ) m
m − βθ
The real quantity of money that maximizes the inflation tax revenue is
obtained by solving the equation
θ m2 − 2βθ m + β (αθ − f )
dτ
=
=0
dm
(m − βθ )2
206
10 Government Budget Constraint
The inflation tax revenue of the item (b) model is given by:
τ = π m = π (α − βπ )
which can be written as:
τ = απ − βπ 2
The inflation rate that maximizes the inflation tax revenue is obtained by
solving the equation:
dτ
= α − 2βπ = 0
dπ
(4) Assume an economy described by:
π̇ = F (π, m, α)
and two economic policy regimes:
MP Ra : ṁ = m (μ − π )
MP Rb : ṁ = f − mπ
(a) The specification of the equation for π̇ is given in (Macro Theory,
page 228), by combining the PC, IS and LM curves. It should be noticed
that:
∂F
= Fπ > 0
∂π
and:
∂F
= Fm > 0
∂m
(b) In the first regime we have the following model:
π̇ = F (π, m, α)
ṁ = m (μ − π )
The Jacobian of this dynamical system is:
J =
∂ π̇ ∂ π̇ ∂π ∂m
∂ ṁ ∂ ṁ
∂π ∂m
Fπ Fm
=
−m μ − π
10 Government Budget Constraint
207
In the steady-state μ − π = 0, and the determinant and trace of the matrix
are:
|J | = mFm > 0
trJ = Fπ > 0
Thus, this model is unstable. The second monetary regime is given by the
two differential equations system:
π̇ = F (π, m, α)
ṁ = f − mπ
The Jacobian of this dynamical system is:
J =
∂ π̇ ∂ π̇ ∂π ∂m
∂ ṁ ∂ ṁ
∂π ∂m
=
Fπ Fm
−m −π
In the steady-state μ − π = 0, and the determinant and trace of the matrix
are:
|J | = −Fπ π + mFm 0
trJ = Fπ − π 0
Thus, this system can be either stable or unstable depending on parameters
and inflation rate.
(c) When the fiscal deficit f decreases, the inflation rate decreases and this
can affect the stability of the model.
(d) When the monetary expansion rate (μ) decreases the steady-state inflation
rate decreases.
(5) Consider the model:
Aggregate demand: y = k + α (m − p) + βπ
PC: π = π e + δ (y − ȳ)
Expectations: π e = μ
Ṁ
= μ = constant or: ṁ = m (μ − π )
MPR: M
(a) In this economy the Phillips curve is:
π = μ + δ (y − ȳ)
Thus, when π = μ, y = ȳ, and the output differs from full employment
output.
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10 Government Budget Constraint
(b) By taking the time derivatives of the aggregate demand and the Phillips
curve, we obtain:
ẏ = α (μ − π ) + β π̇ = α (μ − π ) + βδ ẏ
π̇ = δ ẏ
The first equation can be written as:
ẏ =
α
(μ − π )
1 − βδ
Substituting this value into the equation of π̇ we obtain:
π̇ =
αδ
(μ − π )
1 − βδ
From the Phillips curve μ − π = −δ (y − ȳ). Thus the differential
equation of ẏ is:
ẏ = −
αδ
(y − ȳ)
1 − βδ
The Jacobian of the dynamical system is:
∂ ẏ ∂ ẏ J =
∂y ∂π
∂ π̇ ∂ π̇
∂y ∂π
αδ
− 1−αδ
0
=
αδ
0 − 1−βδ
The determinant and trace of this matrix are:
αδ
|J | = −
1 − βδ
trJ =
2
,
2αδ
1 − βδ
Thus the stability of the system depends on the values of the parameters.
(c) The monetary policy rule can be written as:
ṁ = f − mπ
From the aggregate demand curve we have:
y − ȳ = α log
M
P
− log
M̄
P
+ β (π − π̄ ) = c + α log m + βπ
10 Government Budget Constraint
209
Fig. 10.3 The phase diagram of the model: a) 1 − δβ < 0 b) 1 − δβ > 0
From the Phillips curve and taking into account that
get:
π=
Ṁ
M
f
=μ= m
, we
f
+ α (y − ȳ)
m
By combining the three previous equations, we obtain:
ṁ = −
δ
f + αm log m
1 − δβ
Figure 10.3 shows the phase diagram of model.
(6) Consider the following budget constraint:
f + rb = ḃ +
Ṁ
P
and the following rule:
Ṁ
= αf
P
(a) By substituting this rule into the budget constraint we obtain:
f + rb = ḃ + αf
or:
ḃ = rb + (1 − α) f
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10 Government Budget Constraint
Solving this differential equation, we get:
b=
∞
e−rt (α − 1) f dt
0
Thus, if α > 1 the public debt can be sustainable.
(b) From the fiscal rule we obtain:
ṁ = αf − mπ + rm − rm
or:
ṁ = rm + αf − (π + r) m
where r is the real rate of interest, i = r + π is the nominal rate of
interest and s(m) = im is the value of the services of money. We write the
differential equation as:
ṁ = rm + αf − s(m)
Solving this differential equation, we obtain:
m=
∞
e−rt (s(m) − αf ) dt
0
If a financial innovation occurs in such way that
s(m) − αf
becomes negative a hyperinflation will occur.
(7) The government budget constraint is:
Gt + it−1 Bt−1 = Tt + Bt − Bt−1
Define:
dt =
dt−1
, t = 1, 2, · · ·
1 + rt−1
d0 = 1
(a) Show that:
+
dt Tt =
+
dt Gt + B0 − lim dT BT
T →∞
10 Government Budget Constraint
211
From the budget constraint we can write:
B0 =
T1
B1
G1
+
−
1 + i0
1 + i0
1 + i0
Taking into account that:
d0
1
=
1 + i0
1 + i0
d1 =
the previous expression becomes:
B0 = d1 T1 + d1 B1 − d1 G1
The stock B1 is given by:
B1 =
T2
B2
G2
+
−
1 + i1
1 + i1
1 + i1
Substituting this expression in the former equation yields:
B0 = d1 T1 − d1 G1 +
d1 T2
d1 G2
d1 B2
−
+
1 + i1
1 + i1
1 + i1
since:
d2 =
d1
1 + i1
we can write:
B0 = d1 T1 + d2 T2 − d1 G1 − d2 G2 + d2 B2
It follows from recursive substitution that:
B0 =
T+
−1
dt Tt −
t=1
T+
−1
dt Gt + dt BT
t=1
Thus,
B0 =
∞
+
t=1
dt Tt −
∞
+
t=1
dt Gt + lim dT BT > 0
T →∞
If limT →∞ dT BT > 0 the public debt is not sustainable and I would not
buy government bonds.
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10 Government Budget Constraint
(8) The government finances public deficit by printing money according to:
Gt − Tt = Mt − Mt−1
We divide both sides by the price index Pt ,
Gt − Tt
Mt
Mt−1
Mt−1
Mt−1
=
−
−
+
Pt
Pt
Pt
Pt−1
Pt−1
t−1
and we added and subtracted M
Pt−1 . We use the notation:
Gt − Tt
Mt
Mt−1
= ft ;
= mt ;
= mt−1
Pt
Pt
Pt−1
and we write:
ft = mt − mt−1 +
Mt−1
Mt−1
−
Pt−1
Pt
The rate of inflation is defined by:
Pt
= 1 + πt
Pt−1
therefore:
ft = mt + mt−1 −
Mt−1 Pt−1
Pt−1 Pt
which can be written as:
ft = mt + mt−1 −
mt−1
1 + πt
or:
ft = mt +
πt mt−1
1 + πt
The inflation tax with discrete variables is measured by:
inflation tax =
πt mt−1
1 + πt
(9) The balance of the current account of the balance of payments (Ḃ) is given by:
Ḃ
TB
=−
+i
Y
Y
B
Y
10 Government Budget Constraint
213
We define b = BY and we take the time derivatives of both sides:
Ḃ
ḃ = − B
Y
1
Y2
dY
dt
The gross domestic product is equal to the price level P ∗ times the real
product y. Thus:
ḃ =
Ḃ
− b π∗ + g
Y
where:
π∗ =
Ṗ ∗
ẏ
and g =
∗
P
y
When we combine this expression with the current account balance we obtain:
ḃ = −tb + i − π ∗ − g b
where:
tb =
TB
Y
The real rate of interest r ∗ is defined by:
r∗ = i − π∗
Thus the differential equation can be written as:
ḃ = r ∗ − g b − tb
The solution of this differential equation is:
∗
b = lim e−(r −g )T b(T ) +
∞
T →∞
∗
e−(r −g )t tbdt
0
The foreign debt is sustainable when:
∗
lim e−(r −g )T b(T ) = 0
T →∞
This is equivalent to saying that the foreign debt should grow at a rate less than
the difference between the international rate of interest and the rate of growth
of the economy.
ḃ
< r∗ − g
b
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10 Government Budget Constraint
(10) According to the Ricardian equivalence theorem the “public debt is not
regarded as part of the private sector’s wealth because the value of public debt
corresponds to a liability equal to the present value of future taxes needed to
repay the debt” (Macro Theory, p. 330). Thus, if the government defaults on
its public debt, there will not be future taxes to be paid. Therefore, consumer
spending will not be affected and the default has no consequence for the
economy.
(11) The government budget constraint is given by:
G T
B
Ḃ
Ṁ
− +i = +
Y
Y
Y
Y
Y
The government program has the following objectives:
(a) To keep the tax burden stable:
T
= stable
Y
(b) To increase the ratio of government spending to output:
G
= increases
Y
(c) To reduce the debt service:
i
B
= reduces
Y
(d) To keep the inflation rate stable.
Ṁ
Ṁ
=
Y
M
M
Y
= stable
Thus the deficit will be financed by issuing treasury debt:
Ḃ
= increases
Y
At the same time the government wants to reduce the debt service. Since
B
Y will increase, the nominal rate of interest has to decrease. On the
other hand, the government wants to keep inflation stable. Therefore, this
government program is not consistent.
(12) The government budget constraint is given by:
G−T
B
Ḃ
Ṁ
+i = +
Y
Y
Y
Y
10 Government Budget Constraint
215
(a) The government has a nominal deficit of 3% as an objective. The value of
the primary surplus should be:
G−T
B
+ i = 0.03
Y
Y
thus:
B
T −G
= i − 0.03
Y
Y
(b) The government sets its primary surplus according to the following rule:
T −G
= αrb
Y
Public debt is sustainable when:
ḃ
<r −n
b
From equation (10.13), [Macro Theory, p. 312], we have:
ḃ − (r − n) b = −
T −G
= −αrb
Y
thus:
ḃ = (r − n − αr) b
and:
ḃ
= r − n − αr
b
Therefore, if α > 0 public debt is sustainable.
(c) Suppose that the primary surplus is given by:
T −G
= αb
Y
From equation (10.13), [Macro Theory, p. 312], we have:
ḃ − (r − n) b = −
T −G
= −αb
Y
thus:
ḃ
=r −n−α
b
If α > 0, public debt is sustainable.
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10 Government Budget Constraint
(13) Consider the model:
Bonds-and-money-financed public deficit:
ḃ = f + rb − μm
Monetary policy:
ṁ = m (μ − π )
Money demand:
m < 0
m = m(i),
Fisher equation:
i =r +π
Assumption: public deficit is constant.
From the money demand equation, we obtain the services of money as a
function of the real quantity of money:
im = s(m)
From the Fisher Equation we write the monetary policy as:
ṁ = μm − (i − r) m = (μ + r) m − im
which can be written as:
ṁ = (r + μ) m − s(m)
We have the following dynamical system:
ḃ = f + rb − μm
ṁ = (r + μ) m − s(m)
(a) The Jacobian of this dynamical system is given by:
J =
∂ ḃ
∂b
∂ ṁ
∂b
∂ ḃ
∂m
∂ ṁ
∂m
r
−μ
=
0 r + μ − s (m)
The determinant and trace of this Jacobian are:
|J | = r μ + r − s (m)
10 Government Budget Constraint
217
Fig. 10.4 The phase diagram
of the b and m system
trJ = r + μ + r − s (m)
If s (m) < 0 both the determinant and the trace of the Jacobian are
positive. Under this hypothesis the model is unstable.
When ḃ = 0, we can write:
b=−
μ
f
+ m
r
r
This curve is depicted in Fig. 10.4. When ṁ = 0,
(r + μ) m̄ = s(m̄)
and the real quantity of money is constant and equal to m̄, as shown in
Fig. 10.4, which shows the phase diagram of the dynamical system, with
b on the vertical axis and m on the horizontal axis.
(b) Figures 10.5 and 10.6 depict this question’s experiment. Figure 10.5 shows
that the Central Bank reduces the monetary expansion, at time zero, from
μ0 to μ1 until time T . Figure 10.6 shows that there is a public debt ceiling
equal to b(T ).
In equilibrium, ḃ = ṁ = 0. Thus,
f + r b̄ = μm̄
(μ + r) m̄ = s (m̄)
By combining these two equations, we obtain:
f + r b̄ = s (m̄) − r m̄
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10 Government Budget Constraint
Fig. 10.5 Unanticipated
decrease in the rate of growth
of the stock of money
μT ?
μ
μ0
μ1
T
Fig. 10.6 Public debt ceiling
at time T : b(T )
b
bT
T
or:
b̄ =
1
[−f + s (m̄) − r m̄]
r
We take the derivative of b̄ with respect to m̄:
1 d b̄
=
s (m̄) − r
d m̄
r
The sign of this derivative can be positive, zero or negative according to:
s (m̄) 0
We assume that
d b̄
<0
d m̄
and government indexed bonds. Thus, b(0) is given, a predetermined
variable and m is a jump variable. Figure 10.7 shows the phase diagram
to answer item (b) of this exercise. The curves ḃ = 0 and ṁ = 0 shift as
depicted in Fig. 10.7.
10 Government Budget Constraint
219
Fig. 10.7 Dynamic of the
perverse monetarist
arithmetic
The curve EE shows the locus of the steady-state points on the assumption
d b̄
d m̄ < 0. At the time of the monetary policy change the real quantity of
money jumps to m0 (+) and b0 stays the same. The economy starts moving
from point I to point ET . The real quantity of money decreases. Thus, the
price index grows faster than the stock money. Therefore, we have the
perverse monetarist arithmetic because the tight monetary policy caused
more inflation.
(14) Consider the limit:
lim m(T )e−r(T −t) = Cert
T →∞
where:
C = lim m(T )e−rT
T →∞
The solution of the differential equation:
ṁ = rm + f − s(m)
is given by:
m(t) = m(T )e
−r(T −t)
+
T
e−r(T −t) [s(m) − f ] dt
t
when C = 0 the solution of the differential equation has a bubble component.
220
10 Government Budget Constraint
(15) Take the derivative of b(t) with respect to time:
b(t) =
∞
fs e−(r−n)(υ−t) dυ
t
to obtain:
ḃ = (r − n) b − fs
Applying the Leibinitz rule to:
V (r) =
β
f (x, r) dx
α(r)
yields:
dV (r)
dα(r)
= −f (α(r), r)
+
dr
dr
β
∂f (x, r)
dx
∂r
α(r)
Thus:
db
= −fs e−(r−n)(t−t) +
dt
∞
(r − n) fs e−(r−n)(υ−t) dυ
t
or:
ḃ = −fs + (r − n)
∞
fs e−(r−n)(υ−t) dυ
t
which can be written as:
ḃ = (r − n) b − fs
(16) The derivative of m(t) with respect to time:
m(t) =
∞
[s(m) − f ] e−r(υ−t) dυ
t
is:
ṁ = rm + f − s(m)
Applying the Leibnitiz rule:
dm(t)
= − [s(m) − f ] e−r(t−t) +
dt
∞
t
[s(m) − f ] e−r(υ−t) dυ
10 Government Budget Constraint
221
which can be written as:
ṁ = f − s(m) + r(m)
(17) The derivative of a(t) with respect to time:
a(t) =
∞
[τ + s(m) − g] e−r(υ−t) dυ
t
is:
ȧ = ra + g − τ − s(m)
Applying Leibnitiz rule:
da(t)
= − [τ + s(m) − g] e−r(t−t) +
dt
∞
r [τ + s(m) − g] e−r(υ−t) dυ
t
which can be written as:
ȧ = − [τ + s(m) − g] +
∞
r [τ + s(m) − g] e−r(υ−t) dυ
t
or:
ȧ = ra − [τ + s(m) − g]
(18) Given the budget constraint:
−fs + (r − n) b − (π + n) m = ḃ + ṁ
(a) Show that it may be written as:
a(t) =
∞
[fs + s(m)] e−(r−n)(υ−t) dυ
t
where:
a =b+m
We can add and subtract rm to the flow budget constraint:
−fs + (r − n) b − (π + n) m + rm − rm = ḃ + ṁ
222
10 Government Budget Constraint
which can be written as:
−fs + (r − n) b − (r − n) m − (π + r) m = ḃ + ṁ
or:
−fs + (r − n) (b + m) − im = ḃ + ṁ
since
b+m=a
and
(r + π ) m = im = s(m)
we obtain:
−fs − s(m) + (r − n) a = ȧ
Solving this differential equation, we obtain the stock constraint:
a(t) =
∞
[fs + s(m)] e−(r−n)(υ−t) dυ
t
(b) Show that:
∞
∞
s(m)e−(r−n)(υ−t) dυ − m(t) =
t
(r − n) [m(υ) − m(t)] e−(r−n)(υ−t) dυ
t
+
∞
(π + n) m(υ)e−(r−n)(υ−t) dυ
t
Taking into account that:
s(m) = im = (r + π ) m = [(r − n) + (π + n)] m
we write:
∞
s(m)e
t
+
∞
t
−(r−n)(υ−t)
dυ =
∞
(π + n) m(υ)e−(r−n)(υ−t) dυ
t
(r − n) m(υ)e−(r−n)(υ−t) dυ
10 Government Budget Constraint
223
which can be written as:
∞
s(m)e−(r−n)(υ−t) dυ − m(t) =
t
∞
(r − n) me−(r−n)(υ−t) dυ − m(t)
t
+
∞
(π + n) me−(r−n)(υ−t) dυ
t
(c) How do you interpret the two components on the right side of the above
equation?
The equation above states that the present value of the services of money
that exceeds the real quantity of money, at a given point of time, should
be equal to the sum of the real return of money (net of population growth)
plus the inflation tax collected from people holding money added to the
rate of population growth.
(19) The public deficit is financed with treasury sales of bonds denominated in local
and in foreign currency, according to:
G − T + iB + i ∗ SB ∗ ≡ Ḃ + S Ḃ ∗
We divide this budget constraint by nominal product:
Y = Py
where P is the price index and y is real output, and we define:
f =
B
SB ∗
G−T
, b = , b∗ =
Y
Y
Y
to write the budget constraint as:
f + ib + i ∗ b∗ =
S Ḃ ∗
Ḃ
+
Y
Y
Taking the time derivatives of b and b∗ yields:
ḃ =
ḃ∗ =
Ḃ
− (π + n) b
Y
S Ḃ ∗
+ ṡb∗ − (π + n) b∗
Y
224
10 Government Budget Constraint
(a) Thus, the flow budget constraint can be written as:
f + ib + i ∗ b∗ = ḃ + (π + n) b + ḃ∗ + b∗ (π + n) − b∗ ṡ
where ṡ = ṠS . This equation can be written as:
f + (i − π − n) b + i ∗ − π − n + ṡ b∗ = ḃ + ḃ∗
(b) The coefficient of b∗ can be expressed as:
i ∗ − π − n + ṡ = i ∗ − π ∗ + π ∗ − π − n + ṡ
The real exchange rate is defined by:
Q=
SP ∗
P
thus:
q̇ = ṡ + π ∗ − π
where q = log Q and s = log S. Therefore:
i ∗ − π − n − ṡ = i ∗ − π ∗ + q̇ − n
According to uncovered interest rate parity:
r ∗ + q̇ = i ∗ − π ∗ + q̇ = r
Therefore, the flow budget constraint can be written as:
f + (r − n) b + b∗ = ḃ + ḃ∗
(20) (a) The flow budget constraint is:
G−T
B
Ḃ
+i =
Y
Y
Y
we define b = BY and take the time derivative:
ḃ =
B dY
Ḃ
Ḃ
−
= − b (π + n)
Y
Y dt
Y
10 Government Budget Constraint
225
If ḃ = 0, it follows that:
Ḃ
= b (π + n) = f
Y
where f is the public deficit as a proportion of nominal output.
(b) For b = 0.60, n = 0.03, π = 0.02 the fiscal deficit should be:
f = b (π + n) = 0.60 (0.02 + 0.03) = 0.03
(21) Consider the model:
Production function: y = f (k)
Government budget constraint: g + rb = τ (y + rb)
Gross interest rate: r = f (k)
Net interest rate: ρ = (1 − τ ) r
The notation is standard. Taxes are a proportional income tax with rate τ .
The model describes a steady-state. The question we want to answer is: does
government debt (b) crowd out capital (k)?
(a) Show that:
dk
τ
ff −1
=
+ 2
db
1−τ
(f )
We differentiate the system of equations to obtain:
dy = f (k)dk
dg + bdr + rdb = (y + rb) dτ + τ (dy + bdr + rdb)
(10.3)
0 = −rdτ + (1 − τ ) dr
(10.4)
dτ = (1 − τ )
dr
(1 − τ ) dr
=
r
f (k)
When we substitute (10.1), (10.3) and (10.4) into (10.2) we get:
db =
f τ
bf f +f b
+
− dk
1−τ
f
(f )2
which can be written as:
(10.2)
dr = f (k)dk
It is straightforward to get from Eq. (10.4)
(10.1)
226
10 Government Budget Constraint
db =
ff τ
+
1−τ
(f )2
2
where
f = f (k), f = f (k), f = f (k)
Therefore:
τ
ff −1
dk
=
+ 2
db
1−τ
(f )
dk
(b) Assume that τ = 13 , α = 13 , y = k α . Show that db
= − 23 .
α
From f = k , we obtain:
f = αk α−1
f = α (α − 1) k α−2
Therefore:
k α α (α − 1) k α−2
α (α − 1)
α−1
ff =
=
=
2
2
2
α
(f )
α
αk α−1
Thus:
dk
=
db
−1 1
3 −1
3 −1
+
=
−
1
2
1 − 13
3
1
3
We conclude that:
2
dk
=−
db
3
(22) *(Tax Smoothing): Distortion cost of taxes is given by the convex function:
c = c(τ ), c(0) = 0, c > 0, c > 0
where τ stands for taxes and c for cost. The government chooses the path of
taxes to minimize the present value of the costs:
∞
0
e−rt c(τ )dτ
10 Government Budget Constraint
227
subject to the flow budget constraint:
ḃ = rb + g − τ, b(0) given
where b is the stock of public debt, r the real rate of interest and g government
expenditures. The real rate of interest is constant and the path of government
expenditures is exogenous.
(a) What is the optimal path of taxes?
(b) Define permanent government expenditures and show how transitory
government expenditures are financed.
(a) The government minimizes:
∞
e−rt c(τ )dτ
0
subject to:
ḃ = rb + g − τ
b(0) given
The current-value Hamiltonian is:
H = c(τ ) + λ (rb + g − τ )
The first-order conditions are:
∂H
= c (τ ) − λ = 0
∂c
λ̇ = rλ −
∂H
= rλ − λr
∂b
∂H
= rb + g − τ = ḃ
∂λ
From the second equation λ is constant:
λ̇ = rλ − λr = 0
Thus, the marginal cost of distortions should be constant:
c (τ ) = λ = constant
228
10 Government Budget Constraint
We conclude that the best tax policy for the government is to smooth
taxes, namely the tax rate should be constant in this perfect foresight
environment.
(b) The stock government budget constraint is obtained by integrating the flow
budget constraint, which yields:
b(0) +
∞
e−rt τ dt =
∞
0
ert gdt
0
Permanent government spending (ḡ) is defined as the annuity value that
has the same present value of the path of government expenditures,
∞
e
−rt
ḡdt =
0
∞
e−rt gdt
0
It follows that:
ḡ = r
∞
e−rt gdt
0
By combining this equation with the stock government budget constraint,
taking into account that tax rate is constant and rearranging terms, results:
τ = ḡ + rb(0)
Therefore, the optimum tax rate should be set equal to the sum of the
permanent expenditure with the interest payment on public debt.
By substituting the optimal tax value in the flow government budget
constraint we obtain:
ḃ = r (b − b(0)) + g − ḡ
This equation states that transitory government expenditures should be
financed by issuing government securities.
Chapter 11
Monetary Theory and Policy
(1) Consider the model:
IS-Curve: x = −α (r − r̄) , α > 0
Phillips Curve: π̇ = δx δ > 0
MPR: i̇ = λ (i ∗ − i) , λ > 0, i ∗ = r̄ + π + φ (π − π̄ )
From the Fisher equation i = r + π , taking the derivatives with respect to time
yields:
ṙ = i̇ − π̇
We substitute in this expression the MPR and PC to obtain:
ṙ = λ [r̄ + π + φ (π − π̄ ) − (r + π )] − δx
Using the IS and collecting terms yields
ṙ = (αδ − λ) (r − r̄) + λφ (π − π̄ )
The second equation of the dynamical system combines the Philips Curve and
the IS equation:
π̇ = −αδ (r − r̄)
The Jacobian of this dynamical system is:
J =
∂ ṙ
∂ ṙ
∂r ∂π
∂ π̇ ∂ π̇
∂r ∂π
=
(αδ − λ) λφ
−αδ
0
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2_11
229
230
11 Monetary Theory and Policy
Fig. 11.1 The phase diagram
for the π and r system
Fig. 11.2 Dynamic
adjustment to an decrease in
the inflation target
π
π0
π–1
π˙̇ 0
E0
˙̇ 0
Ef
The determinant and trace of this matrix are:
|J | = αδλφ > 0
trJ = (αδ − λ) < 0, if λ > αδ
Since the determinant is positive and the trace is negative the dynamical system
is stable.
(a) Figure 11.1 depicts the phase diagram of the dynamical system, with π on
the vertical axis and r on the horizontal axis.
(b) Figure 11.2 shows what happens when the inflation target is lowered from
π̄0 to π̄1 . The economy converges on the new equilibrium along the path
E0 EF . First, inflation starts declining and the real rate of interest increases.
The path depicted Fig. 11.2 assumes an oscillating path.
11 Monetary Theory and Policy
231
(2) Consider the model:
IS-Curve: x = −α (i − π − r̄) , α > 0
Phillips Curve: π̇ = δx
MPR: i = ī
(a) Substituting the MPR into IS equation, and this into the Phillips Curve we
obtain:
π̇ = −αδ ī − r̄ + αδπ
Figure 11.3 shows the phase diagram of this equation. This system is
unstable.
(b) When the parameter δ is negative, the equation of π̇ is downward sloping
as shown in Fig. 11.4. The system is stable, but in this forward looking
Phillips Curve inflation at the initial moment, π0 is not given. Thus, there
are infinite solutions.
(c) This monetary policy rule should not be recommended because inflation
would not converge on the implicit inflation target.
Fig. 11.3 The phase diagram
of the model: δ > 0
Fig. 11.4 The phase diagram
of the model: δ < 0
232
11 Monetary Theory and Policy
(3) The Central Bank’s loss function is:
L=
α 2
π − y + ȳ,
2
α>0
This economy’s Phillip curve is given by:
π = π e + β (y − ȳ) ,
β>0
(a) The Central Bank minimizes:
L=
α 2
π − (y − ȳ)
2
subject to:
π = π e + β (y − ȳ)
When we substitute out y − ȳ in the loss function we obtain
L=
α 2 1
π −
π − πe
2
β
The first-order condition of this problem is:
1
∂L
= απ − = 0
∂π
β
Thus:
π∗ =
1
αβ
(b) Would a monetary policy rule that could be enforced be better for this
economy?
Yes, if a monetary policy rule has a target inflation π = 0.
(c) Would a conservative central banker produce better results than an ad-hoc
one at the Central Bank?
A conservative Central Bank would have a very larger parameter
α (α → ∞). Thus:
lim π =
α→∞
1
=0
αβ
(d) Would a conservative central banker’s loss function be the one specified in
this exercise?
11 Monetary Theory and Policy
233
No, the loss function specified in this exercise is appropriate for a populist
government that wants to use monetary policy to increase output beyond
potential output. A conservative central bank would have a loss function
such as:
L=
α 2
π + (y − ȳ)2
2
(4) Consider the model:
IS: yt = ȳt − α (rt − r̄) + t
PC: πt = πt−1 + β (yt − ȳ) + υt
The random variables t and υt are uncorrelated zero average, constant
variance.
The Central Bank’s loss function is:
L = γ (πt − π̄ )2 + (yt − ȳ)2
When we substitute out for yt − ȳt in the PC equation from the IS equation we
get:
πt = πt−1 − αβ (rt − r̄t ) + βt + υt
Using this equation and the IS equation the loss function can be written as:
L = γ [πt−1 − αβ (rt − r̄t ) + βt + υt − π̄ ]2 + [−α (rt − r̄t ) + t ]2
or:
L = γ (πt−1 − π̄ − αβ (rt − r̄t ))2 + (βt + υt )2
+ 2 (πt−1 − π̄ − αβ (rt − r̄)) (βt + υt )
+ α 2 (rt − r̄)2 + t2 − 2α (rt − r̄) t
The expected value of this loss-function is:
.
/
Et L = Et γ (πt−1 − π̄ − αβ (rt − r̄t ))2 + α 2 (rt − r̄)2 + constants
This result is obtained taking into account that:
Et (βt + υt )2 = β 2 σ2 + στ2
Et (πt−1 − π̄ − αβ (rt − r̄t )) (βt + υt ) = 0
Et t2 = σ2
Et (rt − r̄t ) t = 0
234
11 Monetary Theory and Policy
The first-order condition to minimize the expected value of the loss function is:
∂Et Lt
= 2γ [(πt−1 − π̄ ) − αβ (rt − r̄t )] (−αβ) − 2α 2 (rt − r̄t ) = 0
∂rt
Collecting terms and simplifying, we obtain:
rt − r̄t =
βγ
(πt−1 − π̄ )
α β 2γ + 1
(5) Consider the following model of the bank reserves market:
R d = R0 − αi
R S = BR + NBR
BR = β i − i d
Rd = RS
Figure 11.5 shows the model with the interest rate on the vertical axis and total
reserves on the horizontal axis.
(i) Consider the following operational procedure: setting i = ī. Figure 11.6
shows how this operational procedure works, when the demand curve shifts
from D0 D0 to D1 D1 .
¯ This
(ii) Consider the following operational procedure: setting BR = BR.
procedure is equivalent to set i = ī, of the first item.
¯
(iii) Consider the following operational procedure: setting NBR = NBR.
Figure 11.7 shows how this operational procedure works when the demand
curve shifts from D0 D0 to D1 D1 .
Fig. 11.5 The market bank
reserves
11 Monetary Theory and Policy
235
Fig. 11.6 The market for
bank reserves operational
procedure: fixing the rate of
interest
Fig. 11.7 The market for
bank reserves operational
procedure: sitting
non-borrowed reserves
(NBR)
(6) Consider the following model of the bank reserves market:
e
Rtd = α − βrt + δit+1
RtS = R
Rtd = RtS
Graphically show what happens today when the market expects the Central
Bank to raise the interest rate tomorrow.
When the market expects the Central Bank to raise the interest rate tomorrow
the demand curve shifts from D0 D0 to D1 D1 , as depicted in Fig. 11.8. The
interest rate rises from r0 to r1 .
236
11 Monetary Theory and Policy
Fig. 11.8 The market for
bank reserves and expected
interest rates
(7) Assume that the long-term security is a perpetuity paying $ 1 per period. This
security’s price P is the inverse of the long-term interest rate:
P =
1
iL
(a) Show why the short-term interest rate must satisfy the equation:
iS =
1 + Ṗ
P
This equation is an arbitrage equation. The short-term security return is is .
The long-term security return is:
1 + Ṗ
P
Thus, the two returns should be equal.
(b) Show that:
i̇L
= i L − iS
iL
The yield on the long-term security is:
iL =
1
P
Thus:
i̇L
Ṗ = − 2
iL
11 Monetary Theory and Policy
237
Which can be written as:
i̇L
Ṗ
=−
P
iL
From the arbitrage condition (a)
1
Ṗ
Ṗ
+
= iL +
= iS
P
P
P
or:
Ṗ
= iS − iL
P
Thus:
i̇L
= i L − iS
iL
(8) Consider the following model:
IS: x = −α (r − r̄) , α > 0
PC: π̇ = δx, δ > 0
TSIR: i̇s = β (r − rs ) , β > 0, rs = is − π
MPR: is = r̄ + π + φ (π − π̄ ) + θ x
IC: given p(0) and π(0)
(a) How do you interpret the TSIR (term structure of interest rates)?
The TSIR equation states that when the short-term interest rate is expected
to rise the long-term is greater than the short-rate. Thus,
i̇s > 0 ⇒ r > rs
and:
i̇s < 0 ⇒ r < rs
(b) Analyze the model’s equilibrium and dynamics on a phase diagram with π
on the vertical axis and x on the horizontal axis.
We start by taking the time derivatives of both sides of the MPR:
i̇s = π̇ + φ π̇ + θ ẋ = (1 + φ) π̇ + ẋθ
when we substitute out i̇s from the TSIR we get
β (r − rs ) = (1 + φ) π̇ + θ ẋ
238
11 Monetary Theory and Policy
which can be written as:
β (r − (is − π )) = (1 + φ) π̇ + θ ẋ
We substitute out is from the monetary policy rule (MPR) to obtain:
β [r − r̄ − φ (π − π̄ ) − θ x] = (1 + φ) π̇ + θ ẋ
Next, we substitute out r − r̄ from the IS curve and π̇ from the Phillips
Curve (PC) to get:
ẋ = −
β
θ
1
(1 + φ) δ
+θ +
α
β
x−
βφ
(π − π̄)
θ
The dynamical system of this model has two equations, this equation for ẋ
and the Phillips Curve:
π̇ = δx
The Jacobian of this system is given by:
J =
∂ π̇ ∂ π̇ ∂π ∂x
∂ ẋ ∂ ẋ
∂π ∂x
=
0
β
− βφ
θ −θ
δ
(1+φ)δ
1
α +θ +
β
The determinant and trace of this matrix are:
|J | =
trJ = −
β
θ
βφδ
>0
θ
1
(1 + φ) δ
+θ +
α
β
<0
Therefore, the system is stable. Figure 11.9 shows the phase diagram of the
model.
(c) Show what happens when the inflation target is lowered to π̄1 from π̄0 in
the following situations:
(i) Unanticipated reduction, as depicted in Fig. 11.10.
(ii) Anticipated reduction, as depicted in Fig. 11.12.
Figure 11.11 shows the dynamics of the adjustment when an unanticipated reduction occurs in the inflation target. At the time of the
announcement the economy jumps from E0 to the point A. Thus the
arrowed path describes the adjustment of the economy towards the
new equilibrium (point Ef ).
11 Monetary Theory and Policy
239
Fig. 11.9 The phase diagram
for the π and x system
Fig. 11.10 An unanticipated
decrease in the inflation target
Fig. 11.11 Dynamic
adjustment to an
unanticipated decrease in the
inflation target
π π˙̇ 0
E0
Ef
˙̇ π–0 0
˙̇ π–1 0
When an anticipated reduction occurs in the inflation target the longterm interest rate is affected and the economy jumps from E0 to
point A, and starts moving southwest. At point A0 when the inflation
240
11 Monetary Theory and Policy
Fig. 11.12 An unanticipated
decrease in the inflation target
Fig. 11.13 Dynamic
adjustment to an anticipated
decrease in the inflation target
π π˙̇ 0
E0
π–0
˙̇ π–0 0
Ef
π T ˙̇ π–1 0
target changes, the economy starts moving southwest as depicted in
Fig. 11.13.
(9) Consider the model:
IS: x = −α (R − π − r̄)
PC: π̇ = −δx
ISIR: R = i + λi̇
MPR: i = r̄ + π + φ (π − π̄ )
IC: given p(0)
(a) Show what conditions the models parameters must satisfy for a unique
equilibrium to exist.
We start by taking the time derivatives of both sides of the MPR to obtain:
11 Monetary Theory and Policy
241
i̇ = (1 + φ) π̇
Substitution for i and i̇ in the TSIR equation from the monetary policy rule
(MPR) equation and the previous equation yields:
R = π + r̄ + +φ (π − π̄ ) + λ (1 + φ) π̇
Substitution for π̇ from the PC equation we get:
R − π − r̄ = φ (π − π̄ ) − λ (1 + φ) δx
Substituting for R − π − r̄ in the IS equation from this expression, we
obtain:
x=
−αφ
(π − π̄ )
1 − αλ (1 + φ) δ
Substituting for x in the PC equation we get:
π̇ =
αφδ
(π − π̄ )
1 − αλ (1 + φ) δ
Figure 11.14 shows the phase diagram of this equation when:
1 − αλ (1 + φ) δ > 0
Otherwise there will be no unique equilibrium because π(0) is not given.
(b) When δ < 0 the PC curve is backward looking and π(0) is given. Thus, if:
1 − αλ (1 + φ) δ < 0
Fig. 11.14 The phase
diagram of the model when
1 − αλ(1 + φ)δ > 0
π˙̇
π–
π
242
11 Monetary Theory and Policy
Fig. 11.15 The phase
diagram of the model when
1 − αλ(1 + φ)δ < 0
π˙̇
A
π–0
π–
π
the phase diagram is depicted in Fig. 11.15. When the initial inflation is
π(0), the economy will not converge on π̄ . In this case, the model has no
equilibrium, unless π0 = π̄ .
(10) *(Consumption Asset Pricing Model): The Euler equation under uncertainty
states that the agent is indifferent between consuming one unit of the good today
or saving to consume the expected proceeds of the investment tomorrow:
u (ct ) = βEt (1 + rt ) u (ct+1 )
1
and Et is the expected value of the variable conditional
where β = 1+ρ
on information available at time t. Assume that the agent’s utility function is
isoelastic:
u(c) =
c1−γ
1−γ
where γ is the relative risk aversion coefficient.
(a) The risk-free rate of interest is given by the Euler equation:
f
u (ct ) = β 1 + rt Et u (ct+1 )
What is the expression for the risk-free rate of interest?
(b) The rate of returns on stocks is uncertain and given by the Euler equation:
u (ct ) = βEt 1 + rts u (ct+1 )
What is the expression for the rate of return on stocks?
(c) What is the equity premium puzzle and the risk-free rate puzzle?
11 Monetary Theory and Policy
243
(d) In a stochastic environment the risk-free rate is the natural rate of interest.
How can the risk-free rate explain a negative natural rate of interest?
Note: To answer this question assume that the distribution of the variables
can be described by a multivariate normal distribution and its moment
generating function is given by:
1 ,
M(t) = Eet X = eμ t+ 2 t
t
where t =
=
,[t1 , 0t2 , . 1. . , . . . , tn ] is a vector, μ = EX and VarX
a matrix:
= σij , i, j = 1, . . . , n, σij = σj i , σii = σi2 .
, ,
,
is
(a) The Euler equation for the risk-free rate can be written as:
f
1 + ρ = 1 + rt Et
ct+1
ct
−γ
To compute the expected value in this expression we write:
Et
ct+1
ct
−γ
= Et e
log
c
t+1
ct
−γ
= Et e
c
−γ log t+1
c
t
has a normal distribution. It follows from
We assume that gt = log ct+1
ct
the moment generating function that:
1
Et e−γ gt = e−γ Egt + 2 γ σg
2 2
where σg2 is the variance of gt . Substituting this result into the Euler
equation we obtain:
1 2 2
f
1 + ρ = 1 + rt e−γ Egt + 2 γ σg
We take lags of both sides of this equation, use the approximation
log (1 + χ ) ∼
= χ , and rearrange terms to get the expression for the risk-free
rate of interest:
1
f
rt = ρ + γ Et gt − γ 2 σg2
2
(b) When the rate of interest is uncertain at time t, the Euler equation is:
1 + ρ = Et 1 + rts
ct+1
ct
−γ
The expected value of this equation can be written as:
244
11 Monetary Theory and Policy
Et 1 + rts
ct+1
ct
−γ
= Et e
c
log(1+rts )−γ log t+1
c
t
, has a bivariate normal
We assume that log (1 + rt ) , gt , gt = log ct+1
ct
distribution. Thus, we use the multivariate normal moment generating
function to compute this expected value. The bivariate normal distribution
moment generating function is:
M (t1 , t2 ) = Eet1 X1 +t2 X2 = et1 μ1 +t2 μ2 +
t1 σ12 +2t1 t2 σ12 +t22 σ22
2
Therefore
Et elog(1+rt )−γ gt = eE log(1+rt )−γ Egt +
s
s
σr2 −2γ σr,g +γ 2 σg2
2
Substituting this expression into the Euler equation, taking logs and
rearranging yields:
E log 1 + rts = ρ + γ Et (gt ) −
γ 2 σg2
2
+ γ cov (rt , gt ) −
σr2
2
where σg2 is the variance of gt , σr2 is the variance of rts , cov rts , gt = σr,g
is the covariance between rts and gt . The first three terms on the right-hand
side of this expressions are the risk-free rate of interest. Therefore:
f
E log 1 + rts = rt + γ cov rts , gt −
σr2
2
By Jensen’s inequality:
E log 1 + rts +
σr2 ∼
= Erts
2
Thus, the rate of return on stocks is:
f
Erts = rt + γ cov rts , gt
Note: Jensen’s inequality. If f is concave (convex) Ef (X) ≤
f (EX) [Ef (X) ≥ f (EX)]. The function f (X) = log (1 + X) is
1
concave, f (X) = 1+X
, f (X) = − 1 2 < 0. A first-order expansion,
(1+X)
around the point X = 0, gives:
1
f (X) = f (0) + f (0)X + f (0)X2
2
11 Monetary Theory and Policy
245
which is equal to:
2
X
log (1 + X) ∼
=X−
2
Thus,
E log (1 + X) = EX −
Var(X) EX2
EX2
= EX −
+
2
2
2
and we can use the following approximation:
E log (1 + X) = EX −
VarX
2
(c) The risk premium between the risky asset (stock) and the risk-free asset
(bonds) is given by:
f
Erts − rt = γ cov rts , gt
which depends on the risk aversion coefficient (γ ) and the covariance
between the rate of return on the risky asset and the rate of growth of
consumption.
There are two stylized facts about the financial market prices: (i) the rate of
return on risky assets (stocks) is in the range of 6–8% per year, and (ii) the
risk-free rate of return on government treasury bills is 1% per year.
The premium risk is around 6% per year. This premium risk should be
explained by two parameters, the risk aversion coefficient (γ ) and the
covariance between rts and gt . If this covariance were 0.03 the risk aversion
coefficient would have to be equal to 20. This is considered too high, to
describe economic behavior. The equity premium puzzle is a misnomer for
the rejection of this model to explain stylized facts of the financial markets.
The risk-free rate puzzle is the difficulty of the consumption asset pricing
model, with isoelastic utility function, to match the data, reproducing a low
rate, such as 1% per year. This match would occur if one is willing to accept
unreasonable parameters of the model.
The risk-free rate is the natural rate in a stochastic environment. For this
rate to be negative the following condition has to be satisfied:
1 2 2
γ σg > ρ + γ Eg
2
Likewise, the equity premium and the risk rate puzzles, this condition
would be satisfied for unreasonable values of the parameters.
246
11 Monetary Theory and Policy
(11) *(FED Operational Procedure After the 2007/2008 Financial Crisis). Before
the 2007/2008 financial crisis the Federal Open Market Committee (FOMC) of
the American Central Bank (FED) set a target for the Fed funds, implemented
by the operational desk (The Desk) at the Federal Reserve Bank of New York.
After the crisis, it has changed the operational procedure due a superabundant
level of reserves balance in the banking system as a result of the huge amount
of securities bought by the FED.
Since October 2008 the FED pays interest rate on reserves balances as follows:
(i) interest rate on required reserves (IORR), and (ii) interest rates on excess
reserves (IOER). These interest rates have been kept at the same level. The Fed
has a primary credit rate (PCR), which is the upper bound of the FED funds
rate. Thus, the interest rate corridor of the reserves market is given by:
i s = P CR and i l = I OER
(a) Show the operational procedure in the market for federal funds prior to the
2007/2008 financial crisis.
(b) Show the operational procedure in the market for federal funds after the
2007/2008 financial crisis.
(a) Figure 11.16 shows the demand for FED funds with three segments: (i) the
upper bound, (ii) the lower bound, which was zero until 2008, and (iii) the
segment in between. The supply of funds is vertical (SS) because the
Central Bank is monopolistic in this market. The DESK would implement
the interest rate target (Target) by buying and selling securities.
Fig. 11.16 The demand for FED reserves when reserves were not paid
(b) Figure 11.17 shows the demand for FED funds with three segments. Now,
the difference is that the lower bound interest rate can be different from
11 Monetary Theory and Policy
247
Fig. 11.17 The demand for FED reserves when reserves are paid
zero (i l = I OER). With abundant excess reserves the main tool of the
American Central bank is the lower bound interest rate. It can shift the
lower bound to raise (i1e ) or to lower (i2e ) the FED funds as depicted in
Fig. 11.17.
Appendix A
Differential Equations
(1) With help from operator Dx = ẋ, the differential equation system can be
written as:
Dx = a11 x + a12 y + b1
Dy = a21 x + a22 y + b2
Collecting terms on the two variables, the system becomes:
(D − a11 ) x − a12 y = b1
−a21 x + (D − a22 ) y = b2
or using matrix notation:
D − a11 −a12
x
b
= 1
−a21 D − a22 y
b2
Thus:
−1 b1
x
D − a11 −a12
=
−a21 D − a22
b2
y
The inverse of the matrix in the left-hand side of this equation is:
1
D − a22 a12
a21 D − a11
(D − a11 ) (D − a22 ) − a21 a12
From this inverse it is straightforward to obtain the answer to item (a). The
solution of item (a) can be written as two equations:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2
249
250
A Differential Equations
[(D − a11 ) (D − a22 ) − a21 a12 ] x = (D − a22 ) b1 + a12 b2
[(D − a11 ) (D − a22 ) − a21 a12 ] y = a21 b1 + (D − a11 ) b2
Since
(D − a11 ) (D − a22 ) − a21 a12 = D 2 − (a11 + a22 ) D + a11 a22 − a21 a21
and:
tr(A) = a11 + a22
|A| = a11 a22 − a21 a12
where:
A=
a11 a12
a21 a22
It follows that:
D 2 − (a11 + a22 ) D + a11 a22 − a21 a12 = D 2 − tr(A)D + |A|
Using this expression, we obtain the system:
ẍ − tr(A)ẋ + |A|x = a12 b2 − a22 b1
ÿ − tr(A)ẏ + |A|y = a21 b1 − a11 b2
taking into account that:
Db1 = 0
Db2 = 0
since b1 and b2 are constants.
(2) A good’s market model is exemplified by the equations:
Demand: q d = α − βp
Supply: q s = γ + δp
Adjustment: ṗ =
dp
= φ qd − qs , φ > 0
dt
A Differential Equations
251
The differential equation of the model is:
ṗ = φ (α − βp − γ − δp)
or
ṗ = φ (α − γ ) − φ (β + δ) p
(a) When ṗ = 0, we obtain the equilibrium price
φ (α − γ ) − φ (β + δ) p̄ = 0
Thus
p̄ =
α−γ
(β + δ)
α>γ
Fig. A.1 The phase diagram
of the model
(b) The solution of the differential equation is given by:
p = p̄ + Ce−φ(β+δ)t
Given the initial price p(0) = p0 , we obtain the solution
p(t) = p̄ + (p0 − p̄) e−φ(β+δ)t
(c) The phase diagram of the model is shown in Fig. A.1.
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A Differential Equations
(3) The Harrod-Domar economic growth model has three equations:
⎧
⎨ S = sY
I = υ Ẏ
⎩
I =S
Thus:
sY = υ Ẏ
The rate of growth of the economy is given by:
s
Ẏ
=
Y
υ
(a) The path of this economy’s output is:
s
Yt = Y0 e υ t
(b) The phase diagram of Harrod-Domar is given by Fig. A.2.
(4) The asset price model of this question is given by:
r=
υ + ṗ
p
which yields the differential equation:
ṗ = rp − υ
(a) When ṗ = 0, the equilibrium price is:
p̄ =
Fig. A.2 The phase diagram
of the Harrod-Domar model
υ
r
A Differential Equations
253
(b) The solution of the differential equation is given by:
p = p̄ + Cert
This solution has two components: (i) the fundamental solution p̄ and
(ii) the bubble component Cert . When C = 0 the market price and the
equilibrium price differ.
(c) This model with discrete variables is given by:
r=
e
− pt
υt + pt+1
pt
e
pt+1
= pt+1
which yields:
pt =
vt
pt+1
+
1+r
1+r
(5) An economy’s model is specified by the equations:
⎧
⎪
ẏ = α (d − y)
⎪
⎪
⎪
⎪
⎨d = c + i + g
c = βy
⎪
⎪
⎪
i = ī
⎪
⎪
⎩ ġ = −γ − ȳ)
(y
(a) In equilibrium ẏ = ġ = 0. Thus, y = ȳ and:
d = β ȳ + ī + ḡ = ȳ
It follows that
ȳ =
ī + ḡ
,
1−β
β<1
(b) The dynamical system is given by:
ẏ = −α (1 − β) y + α ī + αg
ġ = −γ (y − ȳ)
The Jacobian of this system is:
∂ ẏ ∂ ẏ J =
∂y ∂g
∂ ġ ∂ ġ
∂y ∂g
=
−α (1 − β) α
−γ
0
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A Differential Equations
The determinant and the trace of this matrix are:
|J | = αγ > 0
trJ = −α (1 − β) < 0
Thus, the system is stable and the fiscal policy rule will lead its real output
to full employment.
(6) An economy’s model is specified by the equations:
⎧
⎪
ẏ = α (d − y)
⎪
⎪
⎪
⎪
⎨d = c + i
c = βy
⎪
⎪
⎪
ṙ = γ md − m
⎪
⎪
⎩ d
m = δy − λr
(a) Assume that investment is constant: i = ī. The dynamical system is
given by:
ẏ = −α (1 − β) y + α ī
ṙ = γ δy − γ λr − γ m
In equilibrium ẏ = ṙ = 0. Thus:
ȳ =
ī
1−β
and
r̄ =
m
δ
ȳ −
λ
λ
The Jacobian of the dynamical system is:
∂ ẏ ∂ ẏ J =
∂y ∂r
∂ ṙ ∂ ṙ
∂y ∂r
=
−α (1 − β) 0
γδ
−γ λ
The determinant and the trace of this matrix are:
|J | = α (1 − β) γ λ > 0
trJ = − [α (1 − β) + γ λ] < 0
We assume β < 1. Since the determinant is positive and the trace is negative
the system is stable.
A Differential Equations
255
(7) Consider the model:
⎧
⎪
Ṗ = α (d − y) , α > 0
⎪
⎪
⎨ ṙ = β L (y, r) − M , β > 0
P
> 0, ∂d
d = d (y, r) , ∂d
⎪
⎪
∂y
∂r < 0
⎪
⎩
y = ȳ
(a) The dynamical system of this model is given by:
Ṗ = α [d (y, r) − ȳ]
ṙ = β L (ȳ, r) − M
P
In equilibrium Ṗ = ṙ = 0. Thus, the interest rate and the price level are
given by:
d (ȳ, r̄) = ȳ
M
L (ȳ, r̄)
P̄ =
The Jacobian of the dynamical system is:
J =
∂ Ṗ ∂ Ṗ
∂P ∂r
∂ ṙ ∂ ṙ
∂P ∂r
α ∂d
∂r
=
β PM2 β ∂L
∂r
0
The determinant and the trace of this matrix are:
|J | = −α
∂d M
β
>0
∂r P 2
trJ = β
∂L
<0
∂r
∂L
We assume ∂d
∂r < 0 and ∂r < 0. Since the determinant is positive and the
trace is negative the dynamical system is stable.
(8) Consider the model:
⎧
M
⎪
⎪ Ṗ = β P − L (y, r) , β > 0
⎪
⎪
⎪
⎨ ṙ = α [i + g − t − s] , α > 0
∂i
<0
i = i(r), ∂r
⎪
∂s
⎪
⎪ s = s(y), ∂r
>0
⎪
⎪
⎩
y = ȳ
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A Differential Equations
The dynamical system of the model is:
Ṗ = β M
P − L (ȳ, r)
ṙ = α [i(r) + g − t − s(ȳ)]
In equilibrium Ṗ = ṙ = 0, and the steady-state for P and r, P̄ and r̄ are given
by:
M
L (ȳ, r̄)
P̄ =
i(r̄) + g − t = s(ȳ)
The Jacobian of the dynamical system is:
J =
∂ Ṗ ∂ Ṗ
∂P ∂r
∂ ṙ ∂ ṙ
∂P ∂r
−β PM2 −β ∂L
∂r
=
∂i
0
α ∂r
The determinant and the trace of this matrix are:
|J | = −β
trJ = −β
M ∂i
α
>0
P 2 ∂r
∂i
M
+α
<0
2
∂r
P
The determinant is positive and the trace is negative. Therefore, the dynamical
system is stable.
(9) Consider the model:
⎧
⎪
⎪ ẏ = α (d − y) , α > 0,
⎪
⎪
M
⎪
⎨ P = L (y, r)
∂d
d = d (y, r) , ∂d
∂y > 0, ∂r < 0
⎪
⎪
⎪ M = M̄
⎪
⎪
⎩
P = P̄
This model can be expressed as two equations:
ẏ = α [d (y, r) − y]
M
P = L (y, r)
In equilibrium ẏ = 0:
d (y, r) = y
A Differential Equations
257
The LM curve of this model jointly with IS curve can be solved for y and r. The
LM curve has an implicit function for the rate of interest:
r=f
y,
M
P
∂r
>0
∂y
,
From the differential equation for y we obtain:
∂ ẏ
∂d
= −α 1 −
∂y
∂y
+α
∂d ∂r
<0
∂r ∂y
We assume 1 > ∂d
∂y . Thus the model is stable.
(10) Consider the model:
⎧
⎪
ẏ = φ (d − y)
⎪
⎪
⎨ π = π e + δ (y − ȳ)
π̇ e = θ (π − π e )
⎪
⎪
⎪
⎩ d = d (y, π e ) , ∂d > 0, ∂d > 0
∂y
∂π e
Combining the Phillips curve and the adaptive expectation mechanism we
obtain:
π̇ e = θ δ (y − ȳ)
The differential equation of output is given by:
ẏ = φ d y, π e − y
In equilibrium π̇ e = ẏ = 0. Thus, y = ȳ and:
d (ȳ, π̄ ) = ȳ
which gives the inflation rate of equilibrium (π̄). The Jacobian of the
dynamical system is:
J =
∂ π̇ e ∂ π̇ e
∂π e ∂y
∂ ẏ ∂ ẏ
∂π e ∂y
0
θδ =
d
φ π e φ ∂d
∂y − 1
The determinant of this matrix is:
|J | = −θ δφ
∂d
<0
∂π e
The determinant is negative. Thus, the system has a saddle point.
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A Differential Equations
(11) Consider the model:
⎧
e
⎪
⎪ π = π + φ (d − y)
⎪
⎨ ẏ = Ψ (ȳ − y)
⎪ π̇ e = θ (π − π e )
⎪
⎪
⎩ d = d (y, π e ) , ∂d > 0, ∂d > 0
∂y
∂π e
Combining the first, third and fourth equation we obtain the differential
equation for the expected rate of inflation.
π̇ e = θ φ d y, π e − y
The differential equation for real output is given by:
ẏ = (ȳ − y)
In equilibrium π̇ e = ẏ = 0. It follows that y = ȳ and π = π e , d(ȳ, π̄ ) = ȳ.
From this equation we obtain the steady-state rate of inflation (π̄ ).
The Jacobian of the dynamical system is:
J =
∂ π̇ e ∂ π̇ e
∂π e ∂y
∂ ẏ ∂ ẏ
∂π e ∂y
=
∂d
θ φ ∂π
e −θ φ
0
−
The determinant of this matrix is:
|J | = −θ φ
∂d
<0
∂π e
Thus, the system has a saddle point, π e is a predetermined variable and output
is a jump variable.
(12) Consider the model:
d = α0 + α1 y − α2 R − π e + α3 f
ẏ = φ (d − y)
m − p = β0 + β1 y − β2 r
Ṙ = R − r
(a) What is the effect of an increase in government spending on real output?
First we will set up the dynamical system of this model. By combining
the first two equations we obtain the differential equation:
ẏ = −φ (1 − α1 ) y − φα2 R + φα0 + φα2 π e + φα3 f
A Differential Equations
259
Fig. A.3 The phase diagram
of the R and y system
The last two equations of the model can be combined to yield the
differential equation:
Ṙ = R −
β1
β0 − (m − p)
y−
β2
β2
The Jacobian of the dynamical system is given by:
J =
∂ Ṙ
∂R
∂ ẏ
∂R
∂ Ṙ
∂y
∂ ẏ
∂y
1
− ββ12
=
−φα2 −φ (1 − α1 )
The determinant of this Jacobian is negative:
|J | = −φ (1 − α1 ) −
β1
φα2 < 0
β2
We assume α1 < 1. Therefore, this dynamical system has a saddle path as
depicted in Fig. A.3, with R on the vertical axis and y on the horizontal
axis.
The government spending increases from f0 to f1 > f0 . This increase
was not anticipated. This increase shifts upward the ẏ = 0 curve as shown
in Fig. A.4.
The long run interest rate is a jump variable, and at the announcement
of the new fiscal policy it jumps to the new saddle path. From that point
onward the economy converges on the new equilibrium (point Ef ).
(b) What is the effect of an increase in the quantity of money on real output?
The quantity of money increases from m0 to m1 > m0 . This increase shifts
downward the curve Ṙ = 0 as depicted in Fig. A.5. The variable R jumps
at the time of the announcement of the new monetary policy to R(0+ )
in such way that the economy will be in the new saddle path. Then, the
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A Differential Equations
Fig. A.4 Dynamic
adjustment to an
unanticipated increase in
government spending
Fig. A.5 Dynamic
adjustment to an unaticipated
increase in the money stock
economy converges on the new equilibrium where output is larger than in
the previous equilibrium.
(13) Consider the model:
⎧
dM
⎨ I − S = −α (r − r ∗ ) = dt , α > 0
ṗ = β (I − S) , β > 0
⎩
ṙ = γ (p − p̄) + δ ṗ, δ > 0, γ > 0
This model can be reduced to two differential equations:
ṗ = −βα (r − r ∗ )
ṙ = γ (p − p̄) − αβδ (r − r ∗ )
The Jacobian of the dynamical system is:
∂ ṗ ∂ ṗ J =
∂p ∂r
∂ ṙ ∂ ṙ
∂p ∂r
0 −βα
=
γ −αβδ
A Differential Equations
261
The determinant and the trace of this matrix are:
|J | = βαγ > 0
trJ = −βαδ < 0
The determinant is positive and the trace is negative. Thus, the model is stable.
In equilibrium ṗ = ṙ = 0. Therefore, r = r ∗ , p = p̄ and dM
dt = 0.
(14) The model has three equations: an aggregate demand, a Phillips curve and
perfect foresight according to:
⎧
M
⎨ y = k + α log P + βπ e + γf
e
π = π + δ (y − ȳ)
⎩ e
π =π
From the Phillips curve and the hypothesis of perfect foresight, it follows that
output is equal to potential output:
y = ȳ
Thus, the aggregate demand curve can be written as:
ȳ = k + α (m − p) + β ṗ + γf
where m = log(M) and p = log(P ). This is a first-order differential equation
in the price level:
ṗ =
α
p−υ
β
where:
υ=
αm + γf + k − ȳ
β
The solution of this first-order differential equation is given by:
p(t) =
∞
e
− βα (τ −t)
υdτ
t
assuming that there is no bubble.
(a) Analyze the effects of a change in the government fiscal policy in the
following situations:
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A Differential Equations
(i) Permanent, unanticipated;
(ii) Permanent, anticipated;
(iii) Transitory, unanticipated;
(iv) Transitory, anticipated.
(i) Permanent, unanticipated
The fiscal policy change is given by:
υ0 , τ ≤ t
υ1 , τ ≥ t
υ=
The price level is obtained by:
p(t) =
∞
e
− βα (τ −t)
υ1 dτ = υ1 e
α
βt
∞
e
t
− βα τ
dτ
t
Thus:
p(t) = υ1 e
α
βt
e
− βα t
=
α
β
β
υ1
α
where:
υ1 =
αm + γf1 + k − ȳ
β
(ii) Permanent, anticipated
The fiscal policy change is given by:
υ=
υ0 , t ≤ τ ≤ T
υ1 , τ ≥ T
The price level is obtained by:
p(t) =
T
e
− βα (τ −t)
υ0 dτ +
∞
t
e
− βα (τ −t)
υ1 dτ
T
The first integral on the right-hand side is:
T
e
t
− βα (τ −t)
α
βt
υ0 dτ = e υ0
T
e
− βα τ
α
βt
dt = e υ0
t
α
β − βα T
t
−αt
= e β υ0
−e
+e β
α
−e
− βα τ )T
α
β
)
)
t
A Differential Equations
263
Therefore:
T
e
− βα (τ −t)
υ0 dτ =
t
β − α −t)
υ0 1 − e β (T
α
The second integral on the right-hand side is:
T
e
− βα (τ −t)
α
βt
υ1 dτ = e υ1
∞
e
t
− βα τ
α
βt
dτ = e υ1
−e
α
β
T
α
t
=
β
− α −t)
υ1 e β (T
α
υ1 dτ =
β
− α −t)
υ1 e β (T
α
= e β υ1 e
− βα T
− βα τ )∞
)
)
T
Thus,
∞
e
− βα (τ −t)
T
Collecting the two integrals just obtained we get the price level:
p(t) =
β
β − α −t)
− α −t)
υ0 1 − e β (T
+ υ1 e β (T
α
α
(iii) Transitory, unanticipated
υ=
υ1 , t ≤ τ ≤ T
υ0 , τ ≥ T
The price level is obtained by the following integral:
p(t) =
T
e
− βα (τ −t)
υ1 dτ +
t
∞
e
− βα (τ −t)
υ0 dτ
T
From the previous item it is straightforward to obtain:
T
e
− βα (τ −t)
υ1 dτ =
t
β − α −t)
υ1 1 − e β (T
α
and:
∞
e
T
− βα (τ −t)
υ0 dτ =
β − α −t)
υ0 1 − e β (T
α
The price level is obtained adding these two expressions:
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A Differential Equations
p(t) =
β
α
β − α (T −t)
−t)
+ υ0 e β (T
υ1 1 − e p
α
α
(iv) Transitory, anticipated
⎧
⎨ υ0 , t ≤ τ ≤ T1
υ = υ1 , T1 ≤ τ ≤ T2
⎩
υ0 , τ ≥ T2
The price level is obtained by:
p(t) =
T1
e
− βα (τ −t)
T2
∞
− α (τ −t)
− α (τ −t)
υ0 dτ +
e β
υ1 dτ +
e β
υ0 dτ
t
T1
T2
From the previous item, it is easy to get:
T1
e
− βα (τ −t)
υ0 dτ =
t
β −α
−t)
υ0 1 − e β (T1
α
and:
∞
e
− βα (τ −t)
υ0 dτ =
T2
β
−α
−T
υ0 e β (T2 )
α
The second integral on the right-hand side of the price level equation
is given by:
T2
e
− βα (τ −t)
υ1 dτ = υ1 e
− βα t
T1
T2
e
− βα τ
dτ = υ1 e
α
βt
−e
T1
=
− βα τ )T
α
β
) 2
)
T1
β − βα (T1 −t)
−α
−t)
υ1 e
− e β (T2
α
By collecting the three integrals we obtain the price level:
β
β
β
− α (T −t)
− α (T −t)
− α (T −t)
− α (T −t)
+ υ1 e β 1 −e β 2
+ υ0 e β 2
p(t)= υ0 1−e β 1
α
α
α
(b) Analyze
the effect of a change in the growth rate of money stock
d log M
in the following situations:
μ = dt
(i) Permanent, unanticipated;
(ii) Permanent, anticipated;
(iii) Transitory, unanticipated;
(iv) Transitory, anticipated.
A Differential Equations
265
The stock (log) of money grows at a constant rate μ according to:
m = μτ
When τ is equal to zero, m = 0(M = 1 and log(M) = 0).
(i) Permanent, unanticipated
The change in the growth rate of money stock is specified as follows:
μ0 , τ ≤ t
μ=
μ1 , τ ≥ t
The fundamentals of the price is given by:
αm + γf + k − ȳ
β
υ=
we choose units in such a way that:
γf + k − ȳ = 0
Thus,
υ=
α
α
m = μτ
β
β
The price level is given by the following integral:
p(t) =
∞
e
− βα (τ −t) α
β
t
μ1 τ dτ
which can be written as:
α
α
t
p(t) = μ1 e β
β
∞
e
− βα τ
τ dτ
t
"
"
We use the method of integration by parts udυ = uυ − υdu to
solve the integral in the price level formula:
∞
e
t
− βα τ
τ dτ = −
e
− βα τ )∞
α
β
)
)
t
−
∞
t
−ατ
e β
β2 α
β α
dτ = e− β t t + 2 e− β t
α
α
− βα
By substituting this integral into the equation for p(t) we obtain:
p(t) = μ1 t + μ1
β
α
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A Differential Equations
(ii) Permanent, anticipated
The monetary policy change is given by:
μ=
μ0 , τ ≤ T
μ1 , τ ≥ T
The price level is obtained by:
p(t) =
T
e
− βα (τ −t) α
β
t
μ0 τ dτ +
∞
e
− βα (τ −t) α
β
T
μ1 τ dτ
which can be written as:
α
t
p(t) = e β μ0
T
α
β
e
− βα τ
α
t
τ dτ + e β μ1
t
α
β
∞
e
− βα τ
τ dτ
T
To compute the second integral on the right-hand side we use the result
from the previous item to write:
∞
e
− βα τ
T
β
T +
α
β −αT
τ dτ = e β
α
The first integral on the right-hand side is:
T
e
t
− βα τ
β − α τ ))T
β 2 − α τ ))T
τ dτ = − e β ) − 2 e β )
t
t
α
α
It is straightforward to verify that:
T
e
t
− βα τ
β
β −αt
τ dτ = e β t +
α
α
β
β −αt
− e β T +
α
α
Substituting the two integrals into the price level formula, we obtain,
after some algebra, the following expression:
p(t) = μ0 t +
β
α
+ (μ1 − μ0 ) e
− βα (T −t)
(iii) Transitory, unanticipated
⎧
⎨ μ0 , τ ≤ t
μ = μ1 > μ0 , t ≤ τ ≤ T
⎩
μ0 , τ ≥ T
T +
β
α
A Differential Equations
267
The price level is given by:
p(t) =
T
e
− βα (τ −t) α
β
t
μ1 τ dτ +
∞
e
− βα (τ −t) α
β
T
μ0 τ dτ
From the previous item the first integral on the right-hand side of this
expression is:
T
e
− βα (τ −t) α
β
t
μ1 t +
β
α
− μ1 dτ = e
− βα t
T +
β
α
By the same token, the second integral of the right-hand side of the
price level expression was derived in the previous item. Thus:
∞
e
− βα (τ −t) α
β
T
μ0 τ dτ = μ0 T +
β
α
Adding the two last integrals we obtain the price level:
p(t) = μ1
β
t+
α
β
− βα (T −t)
+ μ0 − μ1 e
T +
α
(iv) Transitory, anticipated
The monetary policy change is specified as:
⎧
⎨ μ0 , t ≤ τ ≤ T1
μ = μ1 , T1 ≤ τ ≤ T2
⎩
μ0 , τ ≥ T2
The price level is given by:
p(t) =
T1
e
t
+
∞
e
T2
− βα (τ −t) α
β
− βα (τ −t) α
β
μ0 τ dτ +
T2
e
T1
− βα (τ −t) α
β
μ1 τ dτ
μ0 τ dτ
From the previous item, the first and the third integral on the right-hand
side are:
T1
β
β
− βα (τ −t) α
− βα (T1 −t)
T1 +
μ0 τ dτ = μ0 t +
− μ0 e
e
β
α
α
t
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A Differential Equations
and:
∞
e
− βα (τ −t) α
β
T2
μ0 τ dτ = μ0 e
β
T2 +
α
− βα (T2 −τ )
It takes just a little bit of algebra to obtain the second integral on righthand side of the price level expression. This integral is:
T2
e
T1
− βα (T −t) α
β
μ1 τ dτ = μ1 e
− βα (T1 −t)
T1 +
β
α
− μ1 e
− βα (T 2−t)
T2 +
β
α
By collecting the three integrals we write the price level as:
β
−α
−t)
+ μ1 − μ0 e β (T1
T1 +
α
β
−α
−T
− μ0 − μ1 e β (T2 1 ) T2 +
α
p(t) = μ0 t +
β
α
(15) The housing market model has three equations:
⎧
⎨ A = (δ + r − π + τ ) P − Ṗ
dA
<0
A = A(H ), dH
⎩
Ḣ = S(P ) − δH
This model can be written as a system of two differential equations:
Ṗ = (δ + r − π + τ ) P − A(H )
Ḣ = S (P ) − δH
The Jacobian of this system is:
∂ Ṗ
J =
∂ Ṗ
∂P ∂H
∂ Ḣ ∂ Ḣ
∂P ∂ Ḣ
=
∂A
δ + r − π + τ − ∂H
∂S
−δ
∂P
The determinant of this matrix is:
|J | = − (δ + r − π + τ ) δ +
∂S ∂A
<0
∂P ∂H
The model has a saddle point because the determinant is negative.
(a) Figure A.6 shows the equilibrium and dynamics of the model, and SS is
the saddle path, which slopes downward.
A Differential Equations
269
Fig. A.6 Dynamical system
for the housing market model
Fig. A.7 Permanent
unanticipated increase in the
property tax rate
(b) When the property tax rate (τ ) increases the Ṗ = 0 curve shifts
downward. The housing stock is a predetermined variable and the housing
price is a jump variable.
(i) Figure A.7 shows a permanent anticipated increase in the property tax
rate. Figure A.8 shows the dynamic adjustment of the housing market.
First, price jumps to P (0+ ) and converges along the new saddle path
on the new equilibrium.
(ii) Figure A.9 shows a permanent anticipated increase in the property
tax rate. Figure A.10 shows the dynamic adjustment of the housing
market to a permanent anticipated increase in the tax rate. First prices
jumps to P (0+ ) and then it starts decreasing until it reachs the new
saddle path at time T . The economy converges on the new equilibrium
point Ef along the new saddle path.
(iii) Figure A.11 shows a transitory unanticipated increase in the property
tax rate. Figure A.12 shows the dynamic adjustment of the economy.
The price jumps to P (0+ ) and then starts increasing until it reachs the
saddle path SS at time T . Then, it converges on the old steady-state.
270
A Differential Equations
P
Fig. A.8 Dynamic
adjustment of the housing
market to a permanent
unanticipated increase in the
property tax rate
H˙̇ 0
S
Ef
Pf
E0
Ṗ̇ τ0 0
P 0+ S
Ṗ̇ τ1 0
Fig. A.9 A permanent
anticipated increase in the
property tax rate
τ
τ1
τ0
T
Fig. A.10 Dynamic
adjustment of the housing
market to a permanent
anticipated increase in the
property tax rate
P
Ḣ̇ 0
E0
S
P 0+
Pf
Ef
Ṗ̇ τ0 0
P T S
Ṗ̇ τ1 0
(iv) Figure A.13 shows a transitory anticipated increase in the property
tax rate. It will increase at time T1 and return to its former value at
time T2 . Figure A.14 shows the dynamic adjustment of the economy
A Differential Equations
271
Fig. A.11 A transitory
unanticipated increase in the
property tax rate
τ
τ1
τ0
T
Fig. A.12 Dynamic
adjustment of the housing
market to a transitory
unanticipated increase in the
property tax rate
Ḣ̇ 0
P
S
E0
P T P 0+
S
Ṗ̇ τ0 0
Ṗ̇ τ1 0
Fig. A.13 A transitory
anticipated increase in the
property tax rate
to the transitory anticipated increase in the property tax rate. First, the
price of housing jumps to P (0+ ) and then it decreases until time T1 .
From this time it increases until time T2 when it reachs the saddle path
SS, and converges on the former steady-state (point E0 ).
272
A Differential Equations
Fig. A.14 Dynamic
adjustment of the housing
market to a transitory
unanticipated increase in the
property tax rate
P
.
H=0
S
E0
P (T2)
P (0+ )
S
.
P(t 0) = 0
P (T1)
Ef
.
P(t1) = 0
H
(16) Consider the model:
q̇ = (r + δ) q − F (K) , F (K) < 0
K̇ = I (q) − δK, I (q) > 0
(a) From the first equation the market price of capital (q) can be written as:
q(t) =
∞
e−(r+δ)(τ −t) F (K)dτ
t
The second equation governs the accumulation of capital. The Jacobian of
the differential equation system is:
∂ q̇
J =
∂q
∂q ∂K
∂ K̇ ∂ K̇
∂q ∂K
=
r + δ −F (K)
−δ
I (q)
The determinant of this matrix is negative:
|J | = − (r + δ) δ + I (q) F (K) < 0
Thus, the steady-state is a saddle point. Figure A.15 shows the dynamics
of the system.
(b) On the saddle path SS the price of capital (q) and the stock of capital are
negatively correlated.
(c) Figure A.16 shows an unanticipated permanent increase in the interest
rate. Figure A.17 describes the dynamic adjustment of the economy to
the permanent increase in interest rate, which was unanticipated. First, the
interest rates jump to the new saddle path and then converge on the new
equilibrium. In the long run both the price q ∗ capital and the stock of
capital decrease.
A Differential Equations
273
Fig. A.15 The phase
diagram of the q and K
system
Fig. A.16 An unanticipated
permanent increase in the
interest rate
Fig. A.17 Dynamic
adjustment to an
unanticipated permanent
increase in the interest rate
q
.
K=0
E0
S
q0
qf
.
q(t0) = 0
Ef
q(0+)
S
.
q(t1) = 0
Kf
K0
K
Figure A.18 shows a permanent, unanticipated increase in the interest rate.
Figure A.19 describes the dynamic adjustment of the economy. As a result
of the permanent, unanticipated increase in the interest rate, the price of
capital jumps to P (0+ ) and then it starts decreasing until time T when it
reachs the new saddle path (SS).
274
A Differential Equations
Fig. A.18 A permanent
anticipated increase in the
interest rate
Fig. A.19 Dynamic
adjustment to a permanent
anticipated increase in the
interest rate
.
K=0
q
E0
S
q(0+)
qf
.
q(t0) = 0
Ef
q(T )
S
.
q(t1) = 0
Kf
K0
K
The economy converges on the new steady-state through the saddle
path SS.
Figure A.20 shows a transitory, unanticipated, increase in the interest
rate. Figure A.21 shows the dynamic adjustment of the economy to the
transitory, unanticipated increase in the interest rate.
The price of capital jumps to P (0+ ) since the capital stock is a predetermined variable. The price of capital increases until it reachs the saddle
path (SS) at time T . From that point on the price of capital decreases along
the saddle path towards the original steady-state (point E0 ).
Figure A.22 shows an anticipated, transitory, increase in the interest
rate. Figure A.23 shows the dynamic adjustment of the economy to the
anticipated, transitory, increase in the interest rate. First, the price of
capital jumps to P (0+ ). Then, the price decreases until time T1 . From this
time on the price increases until it reachs the saddle path SS, to converge
on the previous steady-state.
A Differential Equations
275
Fig. A.20 A transitory
unanticipated increase in the
interest rate
Fig. A.21 Dynamic
adjustment to a transitory
unanticipated increase in the
interest rate
.
K=0
q
S
q(T)
q0
E0
S
.
q(t0) = 0
q(0+)
.
q(t1) = 0
0
Fig. A.22 A transitory
anticipated increase in the
interest rate
K0
K
276
Fig. A.23 Dynamic
adjustment to a transitory
anticipated increase in the
interest rate
A Differential Equations
q
.
K=0
S
q(T2)
q(0+)
E0
S
q(T1)
.
q(t0) = 0
.
q(t1) = 0
0
K0
K
Appendix B
Optimal Control Theory
(1) (Calculus of Variations):
(a)
t1
max
F (t, x, u) dt
t0
ẋ = u
x (t0 ) = x0 ,
x (t1 )
given
free
(b) The Hamiltonian of this problem is:
H = F (t, x, u) + λu
The first-order conditions are:
∂F
∂H
=
+λ=0
∂u
∂u
∂F
∂H
=
= −λ̇
∂x
∂x
∂H
= ẋ = u
∂λ
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2
277
278
B Optimal Control Theory
Taking the time derivatives of the first equation yields:
λ̇ = −
d
dt
∂F
∂u
=−
d
dt
∂F
∂ ẋ
From the second equation of the first-order condition it follows that:
∂F
∂ ẋ
d
dt
=
∂F
∂x
This is called the Euler equation. Taking into account that:
d
dt
∂F
∂ ẋ
=
∂ 2F
∂ 2F
∂ 2F
ẍ +
ẋ +
∂ ẋ∂ ẋ
∂x∂ ẋ
∂t∂ ẋ
the Euler equation can be written as
∂ 2F
∂ 2F
∂ 2F
∂F
ẍ +
ẋ +
−
=0
∂ ẋ∂ ẋ
∂x∂ ẋ
∂t∂ ẋ
∂x
(c) The transversality condition of the contral problem is:
λ (t1 ) = 0
Since λ = − ∂F
∂u it follows that:
∂F ))
=0
)
∂ ẋ t=t1
(2) The representative agent maximizes:
∞
e−ρt u(c)dt
0
subject to:
k̇ = f (k) − c − δk
k(0) = k0 ,
given
(a) The Hamiltonian is:
H = u(c) + λ (f (k) − c − δk)
B Optimal Control Theory
279
The first-order conditions are:
∂H
= u (c) − λ = 0
∂c
λ̇ = ρλ −
∂H
= ρλ − λ f (k) − δ
∂k
∂H
= f (k) − c − δk = k̇
∂λ
From the equation u (c) = λ, we can write c as a function of λ,
c = c(λ),
∂c
<0
∂λ
Thus, we have the dynamical system:
λ̇ = λ ρ − f (k) − δ
k̇ = f (k) − c(λ) − δk
The phase diagram of this model is depicted in Fig. B.1, with λ on the
vertical axis and k on the horizontal axis. The equilibrium is a saddle point
and SS is the saddle path.
(b) Substituting the accumulation of capital equation into the utility function
we get:
∞
max
0
Fig. B.1 The phase diagram
of the λ and k system
e−ρt u f (k) − δk − k̇ dt
280
B Optimal Control Theory
The calculus of variations equation is given by:
F t, k, k̇ = e−ρt u f (k) − δk − k̇
The Euler equation can be written as
∂ 2F
∂ 2F
∂ 2F
∂F
=0
k̈ +
k̇ +
−
∂k
∂k∂ k̇
∂k∂ k̇
∂t∂ k̇
It is straightforward to obtain the following partial derivatives:
∂F
= e−ρt u (c) (−1) = −e−ρt u (c)
∂ k̇
∂ 2F
= −e−ρt u (c) (−1) = e−ρt u (c)
∂ k̇∂ k̇
∂ 2F
= −e−ρt u (c) f (k) − δ
∂k∂ k̇
∂ 2F
= − (−ρ) e−ρt u (c) = ρe−ρt u (c)
∂t∂ k̇
∂F
= e−ρt u (c) f (k) − δ
∂k
Substituting these partial derivatives into the Euler equation yields:
k̈ − f (k) − δ k̇ +
u (c)
ρ − f (k) + δ = 0
u (c)
Using the optimal control theory we obtain two differential equations:
ċ =
u (c)
ρ + δ − f (k)
u (c)
k̇ = f (k) − c − δk
We take the derivatives with respect to time of this accumulation equation:
k̈ = f (k) k̇ − ċ − δ k̇
Substituting the equation for ċ into this equation results in the second-order
differential equation for k.
B Optimal Control Theory
281
(3) The individual solves the following problem:
∞
ẋ 2
−ρt α
2
min
e
dx
(x − x̄) +
2
2
0
subject to:
x(0) = x0 , given
The calculus of variations equation is given by:
ẋ 2
−ρt α
2
F (t, x, ẋ) = e
(x − x̄) +
2
2
The Euler equation is:
∂ 2F
∂ 2F
∂F
∂ 2F
ẍ +
ẋ +
−
=0
∂ ẋ∂ ẋ
∂x∂ ẋ
∂t∂ ẋ
∂x
It’s straightforward to obtain the following partial derivatives:
∂F
= e−ρt ẋ
∂ ẋ
∂ 2F
= e−ρt
∂ ẋ∂ ẋ
∂ 2F
=0
∂x∂ ẋ
∂ 2F
= −ρe−ρt ẋ
∂t∂ ẋ
∂F
= e−ρt α (x − x̄)
∂x
Thus, the Euler equation is given by:
e−ρt ẍ + 0ẋ − ρe−ρt ẋ − e−ρt α (x − x̄) = 0
or:
ẍ − ρ ẋ − α (x − x̄) = 0
The coefficient of x is negative. Thus, this equation has one negative and one
positive root. We take the negative root (−k, k > 0) to write:
x = x̄ + Ce−kt
282
B Optimal Control Theory
where C is a constant. Taking the derivative with respect to time yields:
ẋ = C (−k) e−kt
Since x − x̄ = Ce−kt , we write:
ẋ = −k (x − x̄) = k (x̄ − x)
(4) The Central Bank determines the nominal interest rate by solving the following
problem:
∞
min
e−δt
0
1
ϕ
(π − π̄ )2 + (y − ȳ)2 dt
2
2
subject to
π̇ = β (y − ȳ)
p(0) and π(0) given
The Hamiltonian of this problem is
H =
1
ϕ
(π − π̄)2 + (y − ȳ)2 + λβ (y − ȳ)
2
2
The first-order conditions are
∂H
= ϕ (y − ȳ) + λβ = 0
∂y
λ̇ = ρλ −
∂H
= ρλ − (π − π̄ )
∂π
∂H
= β (y − ȳ) = π̇
∂λ
From the first equation
λ=−
ϕ
(y − ȳ)
β
Taking the derivative with respect to time yields
ϕ
λ̇ = − ẏ
β
B Optimal Control Theory
283
Then, using the second equation of the first-order conditions we get:
ϕ
ϕ
− ẏ = ρ − (y − ȳ) − (π − π̄ )
β
β
or:
β
(π − π̄ )
ϕ
ẏ = ρ (y − ȳ) +
The dynamical system has two differential equations:
ẏ = ρ (y − ȳ) + βϕ (π − π̄)
π̇ = β (y − ȳ)
The Jacobian of this system is:
∂ ẏ ∂ ẏ J =
∂y ∂π
∂ π̇ ∂ π̇
∂y ∂π
ρ βϕ
=
β 0
The determinant of this matrix is negative:
|J | = −
β2
<0
ϕ
Thus, the equilibrium is a saddle point. Figure B.2 shows the phase diagram of
the dynamical system. The saddle path equation can be written as:
π − π̄ = −φ (y − ȳ) , φ > 0
Fig. B.2 The phase diagram
of the π and y system
284
B Optimal Control Theory
where the coefficient φ depends on the parameters of the model. Assume that
the IS equation is given by:
y − ȳ = −α (i − π − r̄)
where i is the nominal interest rate and r̄ is the natural rate of interest. We use
this equation to write:
i = r̄ + π −
1
(y − ȳ)
α
Substituting the output given by the saddle path equation into this equation
yields the rule for nominal interest rate:
i = r̄ + π +
1
(π − π̄ )
αφ
(5) A consumer maximizes:
∞
e−ρt u(c)dt
0
subject to:
ȧ = ra + y − c
a(0) = a0 ,
given
The Hamiltonian of this problem is:
H = u(c) + λ (ra + y − c)
The first-order conditions are:
⎧ ∂H
∂u
⎨ ∂c = ∂c − λ = 0
λ̇ = ρλ − ∂H
∂a = ρλ − λr
⎩ ∂H
∂λ = ra + y − c = ȧ
As shown in (Macro Theory, p. 37) the determinant of the Jacobian is equal to
zero. Thus, one root is equal to zero. The dynamical system is given by:
ċ = uu(c)
(c) (ρ − r)
ȧ = ra + y − c
B Optimal Control Theory
285
Fig. B.3 The phase diagram
of the c and a system
Fig. B.4 A permanent
unanticipated increase in real
income
(a) When there is a stationary equilibrium r = ρ. Figure B.3 shows the phase
diagram of the system:
(i) Permanent, anticipated increase in real income, as shown in Fig. B.4.
Consumption jumps to the new equilibrium as shown in Fig. B.5.
(ii) Permanent, anticipated increase in real income as shown in Fig. B.6: at
time T income increases.
At time zero consumption jumps to the new equilibrium and total assets
decrease to the final equilibrium at time T as shown in Fig. B.7.
(iii) A transitory, unanticipated, increase in real income as shown in
Fig. B.8. Figure B.9 describes the adjustment of the economy. Consumption jumps to the new equilibrium and total assets increase until
time T when the transitory increase dies out.
(iv) Figure B.10 shows a transitory, anticipated increase in real income.
Figure B.11 describes the dynamic adjustment of the economy. First,
consumption jumps and it is financed by decreasing total assets. At
time T 1 when income increases total assets starts increasing until it
286
Fig. B.5 Dynamic
adjustment to a permanent
unanticipated increase in real
income
Fig. B.6 A permanent
anticipated increase in real
income
Fig. B.7 Dynamic
adjustment to a permanent
anticipated increase in real
income
B Optimal Control Theory
B Optimal Control Theory
287
Fig. B.8 Transitory
unanticipated increase in real
income
Fig. B.9 Dynamic
adjustment to a transitory
unanticipated increase in real
income
Fig. B.10 A transitory
anticipated increase in real
income
reachs time T 2. Consumption stays constant at the new level all the
time, as shown in Fig. B.11.
(b) In both cases of transitory changes, either unanticipated or anticipated, the
transitory changes have permanent effects as described in Figs. B.9 and
B.11.
288
B Optimal Control Theory
Fig. B.11 Dynamic
adjustment to a transitory
anticipated increase in real
income
(c) Assume no stationary equilibrium. Deduce the consumption equation when
1
u(c) =
c1− σ
1 − σ1
, σ = 1
u(c) = log c, σ = 1
When the utility function is isoelastic the Euler equation is given by
ċ
= σ (r − ρ)
c
Thus,
c(t) = c(0)eσ (r−ρ)t
From the flow budget constraint,
ȧ = ra + y − c
we obtain the stock constraint:
∞
∞
a(0) +
e−rt ydt =
e−rt cdt
0
0
If y is constant,
∞
0
and
e−rt ydt =
y
r
B Optimal Control Theory
289
∞
e
−rt
cdt =
0
∞
e−rt c(0)eσ (r−ρ)t dt
0
which can be written as
∞
e−(r−σ (r−ρ))t c(0)dt =
0
c(0)
r − σ (r − ρ)
we assume r − σ (r − ρ) > 0, otherwise the integral would not exist. It
follows that
y
c(0) = [r (1 − σ ) + σρ] a(0) +
r
when σ = 1,
y
c(0) = ρ a(0) +
r
(6) (Tobin’s q) The firm solves the following problem:
∞
max
0
αI 2
dt
e−ρt pQ (K, L) − wL − I −
K
subject to:
K̇ = I − δK
K(0) = K0 ,
given
The Hamiltonian of this problem is:
H = Q (K, L) − wL − I −
αI 2
+ q(I − δK)
2K
We assume, to simplify, that p ≡ 1 and we use α2 instead of α.
The first-order conditions are:
⎧ ∂H
⎪
∂L = QL − w = 0
⎪
⎪ ∂H
αI
⎨
0
∂I = −1 − K + q = α
q̇ = ρq − ∂H
⎪
∂K = ρq − QK + 2
⎪
⎪
⎩ ∂H
∂q = I − δK = K̇
I 2
− qδ
K
∂Q
where QK = ∂K
and QL = ∂Q
∂L . These first-order conditions can be written as:
QL ≡ w
290
B Optimal Control Theory
q =1+
αI
K
α
q̇ = (ρ + δ) q − QK −
2
I
K
2
K̇ = I − δK
By substituting q into the first and fourth equation we obtain the dynamical
system with two differential equations:
⎧
⎨ q̇ = (ρ + δ) q − Q − α q−1 2
K
2
α
⎩ K̇ = q−1 − δ K
α
In equilibrium q̇ = K̇ = 0, and:
q̄ = 1 + αδ
Q̄K = (ρ + δ) q −
α
2
q −1
α
2
The Jacobian of this dynamical system is:
∂ q̇
J =
∂ q̇
∂q ∂K
∂ K̇ ∂ K̇
∂q ∂K
K
ρ + δ − dQ
dK
=
q−1
K
α
α −δ
In steady-state the determinant of the Jacobian is:
|J | =
K dQK
α dK
K
This determinant is negative if dQ
dK < 0. We show below that for a constant
return to scale production function this determinant is equal to zero. We proceed
assuming that such is not the case.
(a) The phase diagram of the dynamical system with q on the vertical axis and
k on the horizontal axis is depicted in Fig. B.12. The equilibrium is a saddle
point and the saddle path is downward sloping.
K
To obtain the derivative dQ
dK we take into account that QL = w. The
differential of this expression is:
QLL dL + QLK dK = dw
and the differential of QK is:
B Optimal Control Theory
291
Fig. B.12 The phase
diagram of the Tobin’s q
model with q on the vertical
axis and the stock of capital
K on the horizontal axis
dQK = QKK dK + QKL dL
Combining these two expressions, we obtain:
dQK = QKK dK +
QKL (dw − QLK dK)
QLL
which can be written as:
dQKK =
QKK QLL − Q2KL dK
QKL
+
dw
QLL
QLL
The production function Q(K, L) is concave and obeys the following
property:
QKK < 0, QLL < 0, QKL > 0, QKK QLL − Q2KL > 0
When the production function Q(K, L) has constant returns to scale its
Hessian H is singular:
H =
QKK QKL
QLK QLL
|H | = 0. Therefore, the model in this case cannot be analyzed on a phase
diagram with q on the vertical axis and K on the horizontal axis:
(b) The Marginal Value of Capital.
292
B Optimal Control Theory
Let:
t1
max
f (t, x, u) dt = V (t0 , x0 )
t0
subject to:
ẋ = g (t, x, u)
The maximum value can be written as:
t1
V (t0 , x0 ) =
[f (t, x, u) dt + λ (g (t, x, u) − ẋ)] dt
t0
We integrate by parts:
t1
t0
λẋdt = λx|tt10 −
t1
x λ̇dt = λ (t1 ) x (t1 ) − λ (t0 ) x (t0 ) −
t0
t1
x λ̇dt
t0
Substituting this expression into the V (t0 , x0 ) equation, we obtain:
V (t0 , x0 ) =
t1
H + x λ̇ dt + λ (t0 ) x (t0 )
t0
where H is the Hamiltonian and we use the transversality condition λ(t1 ) =
0. Taking the derivative with respect to x0 yields:
∂V (t0 , x0 )
=
∂x0
t1
Hx
t0
λ̇d ẋ
dx
du
+ Hu
+
dx0
dx0
dx0
dt + λ (t0 )
∂H
where Hx = ∂H
∂x , Hu = ∂u . This expression can be written as:
∂V (t0 , x0 )
=
∂x0
t1
Hx + λ̇
t0
du
dx
dt + λ (t0 )
+ Hu
dx0
dx0
From the first-order condition:
Hx + λ̇ = 0
Hu = 0
B Optimal Control Theory
293
we obtain:
∂V (t0 , x0 )
= λ (t0 )
∂x0
Therefore, Tobin’s q can be interpreted as the marginal value of capital:
∂V
= q0
∂K0
The Average Value of Capital
The differential equation for q is given by:
I
K
α
q̇ = (ρ + δ) q − QK −
2
2
We multiply both sides of this equation by the stock of capital K,
I2
K
α
q̇K = (ρ + δ) qK − QK K −
2
and we assume a constant return to scale production function. Thus:
Q = QK K + QL L
Therefore:
QK K = Q − QL L = Q − wL
taking into account the fact that the marginal product of labor is equal to
the real wage. Substituting this expression into the q̇K equation yields:
α
q̇K − ρqK + q K̇ = δqK − Q + wL −
2
I2
K
+ q K̇
where we added q K̇ to both sides. Since K̇ = I − δK we can write this
equation as:
q̇K − ρqK + q K̇ = δqK − Q + wL −
In equilibrium:
αI
q = 1+
K
α
2
I2
K
+ qI − qδK
294
B Optimal Control Theory
and:
qI = I +
αI 2
K
By substituting this expression into the former equation we obtain:
α
q̇K − ρqK + q K̇ = −Q + wL +
2
I2
K
+I
Next, we use the fact that:
d −ρt
e qK = −ρe−ρt qK + e−ρt q̇K + e−ρt K̇q
dt
to write:
−
α
d −ρt
e qK = e−ρt Q − wL − I −
dt
2
I2
K
Thus:
−
∞
de
−ρt
qK =
0
∞
e
−ρt
0
−
∞
α
Q − wL − I −
2
I2
K
dt
de−ρt qK = q(0)K(0) − lim e−ρt qK
t→∞
0
From the transversality condition:
lim e−ρt qK = 0
t→∞
Thus, average value of capital is equal to Tobin’s q:
q(0) =
V (0)
K(0)
(c) When the parameter α equals zero, q is equal to one, and there is no
investment because capital adjusts instantaneously.
(7) The agent maximizes:
∞
max
0
e−ρt μmdt
B Optimal Control Theory
295
subject to constraint:
ṁ = μm − τ (m)
The current value Hamiltonian is given by:
H = μm + λ (μm − τ (m))
which is linear in the rate of growth of money, the control variable:
H = (1 + λ) μm − λτ (m)
A singular control requires that:
1+λ=0
The costate equation is given by:
λ̇ = ρλ −
∂H
= ρλ − (1 + λ) μm − λτ (m)
∂m
or:
λ̇ = ρλ + λτ (m)
Taking into account that 1 + λ = 0, λ = −1. Thus, λ̇ = 0, and ρ + τ (m) = 0.
(a) From this equation we obtain m̄, as:
τ (m̄) = −ρ
Since m = m̄, μ = π and:
μ̄ =
τ (m̄)
m̄
The initial price level is given by:
P (0) =
M(0)
m̄
Figure B.13 depicts the inflation, tax curve and the equilibrium value of
m, m̄. This policy is inconsistent because the Central Bank can increase
seigniorage by moving to point A of Fig. B.13.
296
B Optimal Control Theory
Fig. B.13 The inflation tax
curve and the inconsistent
monetary policy
(8) Using the constraint:
ṁ = μm − τ (m)
we can write:
∞
e−ρt μmdt =
∞
0
e−ρt (ṁ + τ (m)) dt
0
By integration by parts we obtain:
∞
0
e−ρt ṁdt = e−ρt m|∞
0 −
∞
m (−ρ) e−ρt dt
0
Thus:
∞
e−ρt ṁdt = lim e−ρt m − m(0) +
t→∞
0
∞
ρme−ρt dt
0
The limit in this expression is equal to zero. Therefore:
∞
e−ρt ṁdt =
0
∞
ρ (m − m(0)) e−ρt dt
0
It follows that:
∞
∞
∞
e−ρt μmdt =
e−ρt τ (m)dt +
ρ (m − m(0)) e−ρt dt
0
0
0
B Optimal Control Theory
297
(9) The Central Bank maximizes:
∞
0
e−ρt μmdt + m(0) − m(0− )
where m(0) − m(0− ) is the change of the real stock of money at the initial
moment. From the previous exercise we have:
∞
e
−ρt
μmdt =
0
∞
e
−ρt
τ (m)dt +
0
∞
ρ (m − m(0)) e−ρt dt
0
By substituting this expression in the Central Bank’s objective function, we
obtain:
∞
∞
−ρt
e τ (m)dt +
e−ρt ρ (m − m(0)) dt + m(0) − m(0− )
0
0
Due to the fact that:
m(0) − m(0− ) =
∞
0
e−ρt ρ (m(0) − m(0− )) dt
we get:
∞
e
0
−ρt
μmdt+m(0) − m(0− )=
∞
e
−ρt
∞
τ (m)dt+
0
e−ρt ρ (m−m(0)) dt
0
Thus, the Central Bank does not take into account the instantaneous change in
the real stock of money at the initial moment in its new monetary policy. Thus,
there is no dynamic inconsistency.
(10) (a) Taking the derivatives with respect to time of the money demand equation
we obtain:
ṁ
= −α π̇ e = −αβ π − π e
m
where we use the adaptive expectation mechanism. From the definition of
the real quantity of money: m = M
P , we get:
Ṁ
Ṗ
ṁ
=
−
=μ−π
m
M
P
Thus,
π =μ−
ṁ
m
298
B Optimal Control Theory
and
πe = −
1
log m
α
Taking this expression into the first equation yields:
ṁ
ṁ
1
= −αβ μ −
+ αβ − log m
m
m
α
It is straightforward to obtain:
ṁ = −
β
αβ
μm −
m log m
1 − αβ
1 − αβ
(b) The optimal control problem is to maximize:
∞
e−ρt μmdt
max
0
subject to the constraint:
ṁ = −
αβ
β
μm −
m log m
1 − αβ
1 − αβ
The current value Hamiltonian is:
β
αβ
μm +
m log m
H = μm − λ
1 − αβ
1 − αβ
which can be written as:
H = 1−
λαβ
1 − αβ
μm −
λβ
m log m
1 − αβ
A singular control exists when:
1−λ
αβ
=0
1 − αβ
or:
1 − αβ
=λ
αβ
The first-order condition for the costate variables is:
λ̇ = ρλ −
∂H
λβ
= ρλ +
(1 + log m)
∂m
1 − αβ
B Optimal Control Theory
299
Since λ is constant, λ̇ = 0. Thus:
β
λ ρ+
(1 + log m) = 0
1 − αβ
and:
(1 + log m) = −
ρ (1 − αβ)
β
When:
ṁ = 0,
−
βm log m
αβ
μm −
=0
1 − αβ
1 − αβ
It follows that:
μ=−
1
log m
α
or:
ρ (1 − αβ)
1
−1 −
μ=−
α
β
Therefore:
μ̄ = α1 + ρ
m̄ = e−α μ̄
1
αβ − 1
If m0 = m̄ there is no optimal control.
(c) If the Central Bank can inject or remove money at the initial moment
such that m(0) = m(0− ) there is an optimal monetary policy, which is
inconsistent.
(11) (a) From the government budget constraint
ṁ =
Ṁ
− mπ = −x − mπ
P
and from the money demand equation
π =−
1
log m
α
It follows that
ṁ =
m log m
−x
α
300
B Optimal Control Theory
The representative agent maximizes
∞
e−ρt [u(c) + υ(m)] dt
0
subject to
c = f (x)
ṁ =
m log m
−x
α
Thus, the control problem is:
∞
max
e−ρt [u (f (x)) + υ(m)] dt
0
subject to:
ṁ =
m log m
−x
α
The initial condition for m is not given since the price level at moment
zero, P (0), is not a predetermined variable. We assume that the initial
stock of money is given:
M(0) given
The current-value Hamiltonian is:
m log m
−x
H = u (f (x)) + υ(m) + λ
α
The first-order conditions are:
∂u ∂c
∂H
=
−λ=0
∂x
∂c ∂x
∂H
λ
λ̇ = ρλ −
= ρλ − υ (m) + (log m + 1)
∂m
α
m log m
∂H
=
− x = ṁ
∂λ
λ
and the transversality condition is:
lim λme−ρt = 0
t→∞
B Optimal Control Theory
301
From the first-order condition:
∂u ∂c
=λ
∂c ∂x
We obtain x as a function of λ:
x = x(λ)
The dynamical system for the two variables, x and λ, is given by:
m log m
− x (λ)
α
1 + log m
λ̇ = −υ (m) − λ
−ρ
α
ṁ =
The Jacobian of this system is:
J =
∂ ṁ
∂m
∂ λ̇
∂m
∂ ṁ
∂λ
∂ λ̇
∂λ
=
1+log m
−x (λ)
α
λ 1+log m
−ρ
−υ (m) − αm
α
In steady-state, λ̇ = 0, ṁ = 0, and:
m log m
= x (λ)
α
υ (m)
λ = − 1+log m
−ρ
α
The determinant of the Jacobian evaluated at the steady-state point is
negative:
|J | =
1 + log m
α
−υ (m)
λ
λ
− x (λ) υ (m) +
mα
The dynamical system equilibrium is a saddle point.
(b) The phase diagram of the dynamical system is depicted in Fig. B.14 with
costate variable λ on the vertical axis and the real quantity of money on
the horizontal axis. SS is the saddle path.
(c) The initial real quantity of money is not given. Thus, the costate variable
must be zero at the initial moment, λ(0) = 0, and the economy converges
on the equilibrium point E. Time inconsistency arises in this model
because if government wants to solve the maximization in the future, the
302
B Optimal Control Theory
Fig. B.14 Time inconsistent
monetary policy: the phase
diagram of the model
initial point of the real quantity of money will be m(0) and not the level
prevailing at that time.
(12) (a) The Central Bank minimizes:
∞
0
e−ρt
ϕ 2 1 2
x + π dt
2
2
subject to:
π̇ = α (π − μ) + βx
ẋ = γ (π − μ) + δx
π(0) = μ0
x(0) = 0
For a given rate of growth of the monetary base we assume that the
dynamical system for π and x is stable. The matrix of this system
A=
αβ
γ δ
is such that its determinant is positive and its trace is negative:
|A| = αδ − γβ > 0
trA = α + β < 0
B Optimal Control Theory
303
The current-value Hamiltonian of the control problem is given by:
H =
ϕ 2 1 2
x + π + λ [α (π − μ) + βx] + θ [γ (π − μ) + δx]
2
2
where λ and θ are costate variables. This Hamiltonian can be written as:
H =
ϕ 2 1 2
x + π + λ [απ + βx] + θ [γ π + δx] − (αλ + γ θ ) μ
2
2
The Hamiltonian is linear on μ. Thus, to have a singular optimal control
the coefficient of μ should be zero:
αλ + γ θ = 0
The first-order conditions are:
∂H
= ρλ − π
∂π
∂H
θ̇ = ρθ −
= ρθ − ϕx − βλ − δθ
∂x
∂H
= π̇ = α (π − μ) + βx
∂λ
∂H
= ẋ = γ (π − μ) + δx
∂θ
λ̇ = ρλ −
We have to solve the following system:
⎧
⎪
⎪ αλ + γ θ = 0
⎪
⎪
⎪
⎨ λ̇ = ρλ − π
θ̇ = ρθ − ϕx − βλ − δθ
⎪
⎪
⎪ π̇ = α (π − μ) + βx
⎪
⎪
⎩ ẋ = γ
(π − μ) + δx
Eliminating λ and θ
We start by taking the time derivative of the first equation:
α λ̇ + γ θ̇ = 0
Then, substitute λ̇ and θ̇ , the second and the third equations to obtain:
(αδ − γβ) λ = απ + γ ϕx
Next, we take the derivative with respect to time of this equation:
(αδ − γβ) λ̇ = α π̇ + γ ϕ ẋ
304
B Optimal Control Theory
When we substitute the expression of λ̇ in this equation we obtain:
[α (ρ − δ) + γβ] π + ργ ϕx = α π̇ + γ ϕ ẋ
Thus, we have a three-equation system:
⎧
⎨ [α (ρ − δ) + γβ] π + ργ ϕx = α π̇ + γ ϕ ẋ
π̇ = α (π − μ) + βx
⎩
ẋ = γ (π − μ) + δx
Eliminating μ
We multiply the π̇ equation by γ and the ẋ equation by α, and subtract
one equation from the other to obtain:
γ π̇ − α ẋ = (γβ − αδ) x
Therefore, we have a two-equation system:
α π̇ + γ ϕ ẋ = [α (ρ − δ) + γβ] π + ργ ϕx
γ π̇ − α ẋ = (γβ − αδ) x
This system is solved to obtain:
π̇
π
=J
ẋ
x
where the matrix J is:
2
1
α (ρ − δ) + αγβ αργ ϕ + γ ϕ (γβ − αδ)
J = 2
α + γ 2 ϕ γ α (ρ − δ) + γ 2 β γργ ϕ − α (γβ − αδ)
The determinant of this matrix is negative:
|J | =
− (αδ − γβ − αρ) (αδ − γβ)
α2 + γ 2ϕ
if αδ − γβ − αρ > 0.
(b) The two equations of the dynamical system are:
π̇ =
−α (αδ − γβ − αρ) π
γ ϕ (αδ − γβ − γρ) x
−
2
2
α +γ ϕ
α2 + γ 2ϕ
ẋ =
γ 2 ρϕ + α (αδ − γβ) x
−γ (αδ − γβ − αρ) π
+
2
2
α +γ ϕ
α2 + γ 2ϕ
B Optimal Control Theory
305
Fig. B.15 The phase
diagram of the π and x
system
The phase diagram of this system is depicted on Fig. B.15, with π on the
vertical axis and x on the horizontal axis. SS is the saddle path.
(13) (a) The consumer maximizes:
∞
e−ρt u(c)dt
T
subject to the constraints:
ẋ = −c
x(0) = S
x(T ) = 0
The current value-Hamiltonian is given by:
H = u(c) − λc
The first-order conditions are:
λ̇ = ρλ −
∂H
= ρλ − 0 = ρλ
∂x
∂H
= u (c) − λ = 0
∂c
∂H
= ẋ = −c
∂λ
Taking derivative with respect to time of the second equation yields:
u (c)ċ = λ̇
306
B Optimal Control Theory
which can be written as:
u (c)ċ = ρλ
or:
ċ =
ρu (c)
u (c)
The rate of growth of consumption is:
ċ
u (c)
= ρ
c
cu (c)
if the utility function is given by:
1
u(c) =
c1− σ
1 − σ1
the rate of growth of consumption is:
ċ
= −σρ
c
and:
c(t) = c(0)e−σρt
We know that:
T
cdt = S
0
Thus:
T
c(0)e−σρt dt = S
0
It is straightforward to obtain:
c(0) =
σρS
1 − e−σρT
B Optimal Control Theory
307
(b) What happens to the path when ρ = 0?
From the equation:
ċ =
ρu (c)
u (c)
when ρ = 0, ċ = 0, e.g., consumption is constant:
ct = c̄
Thus:
T
c̄dt = S
0
and:
c̄ =
S
T
(14) The firm maximizes:
T
e−ρt (pq − c (q, x)) dt
0
subject to the constraints:
ẋ = −q
x(0) = S
0≤q≤S
(a) When c(q, x) = 0, the current value-Hamiltonian is:
H = (p(x) − λ) q
which is linear in q. The switching function σ (x, λ) is:
σ (x, λ) = p − λ
Thus, the optimal control is given by:
308
B Optimal Control Theory
⎧
⎨ qmin , if p − λ < 0
q = q̄ ∈ [qmin , qmax ] , if p − λ = 0
⎩
qmax , if p − λ > 0
The other necessary conditions are:
λ̇ = ρλ −
∂H
= ρλ − 0 = ρλ
∂x
and the transversality condition:
λ(T ) = 0
If there is a singular control p − λ = 0. In this case, λ is constant and
λ̇ = 0. It follows from the first-order condition, that λ = 0. Thus, there
is no singular control. If p > λ, the solution is q = qmax = S, and the
interest rate is not relevant for the firms decision.
(b) When c (q, x) = c(x)q the current-value Hamiltonian is:
H = pq − c (x) q − λq
which is linear in the control variable:
H = (p − c (x) − λ) q
The switching function σ (x, λ) gives the following solution
σ (x, λ) = p − c(x) − λ
Thus, the optimal control is given by:
⎧
⎨ qmin , if p − c(x) − λ < 0
q = q̄ ∈ [qmin , qmax ] , if p − c(x) − λ = 0
⎩
qmax , if p − c(x) − λ > 0
The other necessary conditions are:
λ̇ = ρλ −
∂H
= ρλ − −c (x)q = ρλ + c (x)q
∂x
and the transversality condition:
λ(T ) = 0
B Optimal Control Theory
309
Fig. B.16 The singular
central solution
If there is a singular control p − c(x) − λ = 0. We take the derivatives
with respect to time of this expression to obtain:
−c (x)ẋ − λ̇ = 0
which can be written as:
λ̇ = c (x)q
by taking into account the transition equation:
ẋ = −q.
Equating the two expression for λ̇, yields:
λ̇ = ρλ + c (x)q = ρλ + λ̇
Therefore,
λ=0
and there is a stock of non-renewable resource x ∗ such that:
p = c x∗
Since c (x) < 0 and c (x) > 0, Fig. B.16 shows that p > c(x) if x > x ∗ ,
and p < c(x) if x < x ∗ .
(15) A firm solves the following problem:
∞
max
0
e−ρt [(p − c(x))] qdt
310
B Optimal Control Theory
subject to the following constraints:
ẋ = f (x) − q
x(0) = 0
0≤q≤S
(a) The current value Hamiltonian is:
H = (p − c(x)) q + λ (f (x) − q)
which is linear in the control variable:
H = (p − c(x) − λ)q + λf (x)
The switching function σ (x, λ) gives the following solutions:
⎧
⎨ qmax , if σ (x, λ) > 0
q = q̄ ∈ [qmin , qmax ] , if σ (x, λ) = 0
⎩
qmin , if σ (x, λ) < 0
There is a singular control when the switching control is equal to zero for
some time interval.
p − c (x) − λ = 0
The other necessary conditions are:
λ̇ = ρλ −
∂H
= ρ − −c (x)q + λf (x)
∂x
∂H
= f (x) − q = ẋ
∂λ
and the transversality condition:
lim xe−ρt = 0
t→∞
Taking derivatives with respect to time of the switching function we
obtain:
λ̇ = −c (x)ẋ = −c (x) f (x) − q
B Optimal Control Theory
311
By equating two expressions for λ̇ yields:
λ̇ = ρ − f (x) (p − c(x)) + c (x)q = −c (x) f (x) + c (x)q
which can be simplified as:
ρ = f (x) −
c (x) f (x)
p − c (x)
The solution of this equation is x ∗ , which is constant. Thus ẋ = 0 and the
optimal control is:
q ∗ = f (x ∗ )
The initial condition x(0) may be different from x ∗ . If x(0) > x ∗ ,
σ (x, λ) > 0 and q = qmax = q̄. If x(0) < x ∗ , σ (x, λ) < 0 and
q = qmin = 0. Figure B.17 shows the solutions for the three cases:
x(0) = x ∗ , x(0) > x ∗ , and x(0) < x ∗ .
(b) When
x
f (x) = αx 1 −
S
the optimal solution is obtained by solving the equation:
ρ=α
Fig. B.17 The three
solutions of the central
problem
2x
1−
S
c (x) x
−
1−
p − c (x)
S
Appendix C
Difference Equations
(1) Consider the model:
I S : xt = Et xt+1 − σ (it − Et πt+1 − r̄t )
P C : πt = βEt πt+1 + kxt
MP R : it = r̄t
The model is specified by the two equations:
xt = Et xt+1 + σ Et πt+1
πt = βEt πt+1 + kxt
which can be written in matrix notation:
1 σ Et xt+1
1 0 xt
=
0 β Et πt+1
−k 1 πt
or
Ex
xt
= A t t+1
πt
Et πt+1
where:
−1 1σ
1 0
1 σ
A=
=
0β
−k 1
k σk + β
The trace and the determinant of the matrix A are given by:
trA = 1 + σ k + β
|A| = σ k + β − σ k = β
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2
313
314
C Difference Equations
Since 1 − trA + |A| = −σ k < 0, 1 + trA + |A| = 1 + σ k + β + β > 0 and
|A| < 1, the roots will not be less than one in absolute value.
(a) The model is unstable.
(b) The solution is not unique.
(2) Consider the model:
P C : πt = βEt πt+1 + kxt
I S : xt = Et xt+1 − σ (it − Et πt+1 − r̄t )
MP R : it = r̄t + πt + φπt + θ xt
The model is specified by the two equations:
xt = Et xt+1 − σ (πt + φπt + θ xt − Et πt+1 )
πt = βEt πt+1 + kxt
or:
(1 + σ θ ) xt + σ (1 + φ) πt = Et xt+1 + σ Et πt+1
−kxt + πt = βEt πt+1
This system can be written in matrix notation:
1 + σ θ σ (1 + φ) xt
1 σ Et xt+1
=
−k
1
πt
0 β Et πt+1
The solution of this system is given by:
xt
Et xt+1
=A
πt
Et πt+1
where the matrix A
A=
−1 1 1 σ − βσ (1 + φ)
1σ
1 + σ θ σ (1 + φ)
=
0β
−k
1
k kσ + (1 + σ θ ) β
and = 1 + σ θ + σ k (1 + φ). The trace and the determinant of the matrix A
are:
trA =
1 + σ k + β (1 + σ θ )
Δ
|A| =
β
C Difference Equations
315
(a) Since 1 + trA + |A| > 0, and |A| < 1, 1 − trA + |A| = σ [θ(1−β)+kφ]
> 0,
the model has a unique equilibrium.
(b) The condition θ (1 − β) + kφ > 0 does not require that the parameter φ
must always be positive. It can be negative, but must satisfy the condition:
kφ > −θ (1 − β).
(3) Consider the new Keynesian model:
P C : πt = βEt πt+1 + kxt
I S : xt = Et xt+1 − σ (it − Et πt+1 − r̄t )
MP R : it = r̄t + πt + φEt πt+1 + θ xt
After substituting the MP R into the I S cure the model can be reduced to a
system of two equations:
(1 + σ θ ) xt + σ πt = Et xt+1 + σ (1 − φ)Et πt+1
−kxt + πt = βEt πt+1
which can be written in matrix form as follows:
1 + σ θ σ xt
1 σ (1 − φ) Et xt+1
=
0
β
Et πt+1
−k 1 πt
The solution of this system is given by:
xt
Et xt+1
=A
πt
Et πt+1
where A
1 + σθ σ
A=
−k 1
−1 1 1
1 σ (1 − φ)
σ (1 − φ) − σβ
=
0
β
Δ k kσ (1 − φ) + (1 + σ θ )β
and = 1 + σ θ + σ k The trace and the determinant of the matrix are:
trA =
|A| =
1 + kσ (1 − φ) + (1 + σ θ ) β
Δ
kσ (1 − φ) + (1 + σ θ ) β − k [σ (1 − φ) − β]
β
=
Δ
Δ2
> 0,
(a) Since 1 + trA + |A| > 0, and |A| < 1, 1 − trA + |A| = σ [θ(1−β)+kφ]
the model has a unique equilibrium.
(b) Therefore, θ (1 − β) + kφ > 0 even if φ is negative such that kφ >
−θ (1 − β).
(c) Assume that monetary policy rule is given by:
it = r̄t + πt + φEt πt+1 + θ Et xt+1
316
C Difference Equations
We use this MP R to eliminate the nominal interest rate in the I S equation
to obtain the first equation of the system:
xt + σ πt = (1 − σ θ ) Et xt+1 + σ (1 − φ) Et πt+1
−kxt + πt = βEt πt+1
which can be written in matrix form as follows:
1 − σ θ σ (1 − φ) Et xt+1
1 σ xt
=
0
β
Et πt+1
−k 1 πt
The solution of this system is given by:
xt
Ex
= A t t+1
πt
Et πt+1
where the matrix A is given by:
A=
1 σ
−k 1
−1 1
1 − σ θ σ (1 − φ)
1 −σ 1 − σ θ σ (1 − φ)
=
0
β
0
β
1 + σk k 1
Thus
1
A=
1 + σk
1 − σ θ σ (1 − φ) − σβ
k (1 − σ θ ) kσ (1 − φ) + β
The trace and the determinant of A are:
trA =
1 − σ θ + kσ (1 − φ) + β
1 + σk
|A| =
(1 − σ θ )β
1 + σk
>
We assume 1+trA+|A| > 0, and |A| < 1, 1−trA+|A| = σ [θ(1−β)+kφ]
0. Thus, φ can be negative and still we can get: 1 − trA + |A| > 0.
(4) (a) The equation xt = Et xt+1 − σ (it − Et πt+1 − r̄t ) has two solutions, one
forward looking:
xt = −σ Et
∞
+
i=0
it+j − Et πt+1+j − r̄t+j
C Difference Equations
317
and another backward looking:
xt = −σ
−∞
+
it−j − Et πt+1−j − r̄t−j
i=0
(b) The forward solution states that when the elasticity of substitution is equal
to one, consumption in t will be affected by present and future interest
rates. This is an incorrect statement because consumption in period t does
not depend on the interest rate.
(5) Consider the new Keynesian Phillips Curve:
πt = Et πt+1 + kxt
It is important to notice that in this specification β = 1 and the future is not
discounted.
(a) There are two solutions to this difference equation. One is forward looking:
πt = Et
∞
+
(kxt+i )
i=0
and one backward looking:
πt =
−∞
+
kxt−i
i=0
(b) There are no criteria to choose the solution.
(6) Consider the new Keynesian model:
I S : xt = Et xt+1 − σ (it − Et πt+1 − r̄)
P C : πt − π̄ = βEt (πt+1 − π̄ ) + kxt
MP R : it = r̄t + πt + φEt (πt+1 − π̄ ) + θ xt
(a) This Phillips curve is vertical in the long run because, when:
πt = πt+1 = π̄
the output gap is equal to zero.
(b) Eliminating the nominal interest rate of the I S curve with the MP R we
obtain the first equation of the following system of difference equations:
(1 + σ θ ) xt + σ πt = Et xt+1 + σ (1 − φ)Et πt+1 + σ φ π̄
−kxt + πt = βEt πt+1 − β π̄
318
C Difference Equations
This system can be expressed in matrix form as:
1 + σ θ σ xt
1 σ (1 − φ) Et xt+1
σφ
=
+
π̄
0
β
Et πt+1
−k 1 πt
−β
which can be solved to get:
xt
Et xt+1
=A
+b
πt
Et πt+1
where
−1 1 + σθ σ
1 σ (1 − φ)
A=
0
β
−k 1
1
1
σ (1 − φ) − σβ
=
1 + σ θ + kσ k kσ (1 − φ) + (1 + σ θ ) β
b=
1 + σθ σ
−k 1
−1 1
σφ
σ (φ + β)
π̄
π̄ =
−β
1 + σ θ + kσ θ + kσ kσ φ − β (1 + σ θ )
The trace and the determinant of A are given by:
trA =
1 + kσ (1 − φ) + (1 + σ θ ) β
1 + σ θ + kσ
|A| =
β
1 + σ θ + kσ
The model has a unique solution when |A| < 1, 1 + trA + |A| > 0, and
1 − trA + |A| = σ [θ(1−β)+kφ]
> 0. Thus, the following restriction has to
1+σ θ+kσ
satisfied: θ (1 − β) + kφ > 0. This condition does not imply that kφ > 0.
Indeed kφ can be negative as long as kφ > −θ (1 − β).
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Macro Theory: Errata
Chapter 1
(1) Page 31, 4th line from bottom should read:
“b(t) = b(T )e−ρ(T −t) +
T
f e−ρ(τ −t) dτ ”
t
(2) Page 32, 2nd line of text should read:
“where f is now the real deficit. Show that: b(T ) − b(t) = f T and that”
(3) Page 32, 4th line of text should read:
“(d) what conclusion can you draw from items (b) and (c).”
Chapter 2
(1) Page 42, 2nd line from bottom should read:
“Thus, a variable rate of time preference cannot solve the representative agent
model in a small open economy.”
(2) Page 58, 2nd line of the text should read:
“qt = st + pt∗ − pt . Use this definition to show that:”
(3) Page 61, 4th line" from the bottom should read:
∞
“where y p = r 0 e−rt ydt. How would you interpret this result?”
Page 61, 3rd line from the bottom should read:
“(b) Assume that r = ρ and u(c) = c
1− σ1
1− σ1
. Show that:”
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2
321
322
Macro Theory: Errata
Chapter 3
(1) Page 83, 9th line from the bottom should read:
r = f (k) − δ
(2) Page 85, 2nd line from the bottom should read:
1
“u c1,t + 1+ρ
u c2,t+1 , u > 0, u < 0”
(3) Page 86, 8th line from the bottom should read:
“wt = f (kt ) − kt f (kt )”
(1−δ)kt +St (wt ,rt+1 )
Page 86, 2nd line from the bottom should read: “kt+1 =
”
1+n
Chapter 4
(1) Page 100, 7th line from the bottom should have
a space between line 7 and
(n+g+δ)f (k ∗ )k ∗
−
+
g
+
δ)
−
line 8: “k̇ =
(n
(k k ∗ ) placing (n + g + δ) in
f (k ∗ )
evidence yields”
(2) Page 115, 2nd line from bottom should read:
“It measures the workforce’s human capital. Thus, the production function is
given by:”
(3) Page 115, Table 4.1 the 5th and the 6th line of this table should read:
“
K/Y
L̂
2.5
1.5%
(4) Page 117, 7th line of the text should read:
“log YL = log A0 + gt + 1−ααkk−αh log Sk + 1−ααkh−αh log Sh − 1−ααkk−αh
log (n + g + δk ) − 1−ααkh−αh log (n + g + δh )”
(5) Page 117, 11th and 12th line of the text should read:
k̇ = sk f (k, h) − (n + g + δk ) k
ḣ = sh f (k, h) − (n + g + δh ) h
(6) Page 118, 8th line of the text should read:
Ṁ
= constant”
“Monetary policy: ṁ = m (μ − π ) , μ = M
Macro Theory: Errata
323
Chapter 5
(1) Page 132, 16th line of the text should read:
“a public good.”
(2) Page 133, 13th line of the text should read:
“Y = K α (φ (K/L) L)1−α = φ 1−α K
(3) Page 138, 6th line of the text should read:
“For the consumption-capital ratio not to be negative, the following inequality
must be satisfied:”
(4) Page 140, equation (5.50) should read:
⎡
BI = φ −1 γ 1−τ ⎣
M
p(z)1− dz
1
1−
⎤τ
⎦
(5.50)
0
(5) Page 150, 7th line of the text should read:
“(c) What would be the answer to the previous item be if the parameter φ were
equal a zero?”
Chapter 6
(1) Page 169, table Central Bank Balance Sheet, item (b) liabilities, should read:
“Reserves (R)”
(2) Page 175, Fig. 6.6. The reserve market:
Fig. 6.6 The reserve market
(3) Page 177, 1st line from the bottom should read:
dy P
dy P
“ε = dP
y ; |ε| = − dP y ”
(4) Page 179, equations (6.63) and (6.64) should read: “
324
Macro Theory: Errata
π=
Ẇ
Ṗ
=
P
W
(6.63)
Ẇ
= π e − a (u − ū)
W
(6.64)
(5) Page 190, 13th line of the text should read:
u (c)
ċ
= − (r − ρ)
c
cu (c)
(6) Page 193, 1st line of the text should read:
E (pt+1 /It ) − pt + dt
= rt
pt
(7) Page 195, 2nd line of exercise (22) should read:
πt = βπt+1 + δxt
(8) Page 195, the last line of exercise (22) should read:
−δ
“Where ρ = 1−β
β and k = β ”
Chapter 7
(1) Page 197, equation (7.2) should read:
“PC:π̇ = δx”
(2) Page 198, 1st line of the text should read:
“IC: given p(0) and π(0)”
(3) Page 201, equation (7.7) should read:
“IC:i = r̄ BC + π + φ (π − π̄ ) + θ x”
(4) Page 214, Fig. 7.19 a) and b) should read the figure below.
Fig. 7.19 (a) Keynesian
Fig. 7.19 (b) New Keynesian
Macro Theory: Errata
325
(5) Page 223, equation (7.31) should read:
M
“MPR:μ = d log
= constant”
dt
(6) Page 229, 12th line of the text should read:
“π̇ = −k + β log m + γ π ”
(7) Page 230, Fig. 733
(8) Page 231, 10th line from the bottom:
“real if > 0 and complex if < 0. If the roots are complex and the trace is
equal to”
(9) Page 235, 5th line of the text should read:
“IC: Given p(0)”
(10) The symbols have the following meanings: u = unemployment rate;
Chapter 8
(1) Page 241, 4th line from the bottom should read:
P = PT1−w PN1−w = PT
PN
PT
ω
(8.12)
(2) Page 251, 9th line of the text should read:
“Thus, the deviation of the log of terms of”
(3) Page 254, 13th line of the text should read:
“At the long run equilibrium of an open economy with perfect capital”
(4) Page 255, text of Fig. 8.4 should read:
“The effects of an increase in the public deficit on the current account: the twin
deficits”
(5) Page 263, 3rd line from the bottom should read:
“(b) Show that saving (s = y − d − τ where, τ represents taxes) varies with the
terms of”
(6) Page 264, 20th line of the text should read:
“(b) show what happens to E and F in each of the following circumstances: (i)
an”
Chapter 9
(1) Page 270, 5th line from the bottom of the text should read:
“Figure 9.4 describes an experiment in which the foreign inflation rate rises to
π1∗ from π0∗ .”
(2) Page 275, 15th line of the text should read:
“the components of the characteristic vector determined by the solutions of the
linear system:”
326
Macro Theory: Errata
(3) Page 278, 2nd line from the bottom should read:
“given by [equation (3.22), p.77]:”
(4) Page 280, 2nd line of the text should read:
“(r − n) ā + Ēȳx = 0”
(5) Page 280, 11th line of the text should read:
“αa = ρ̄ − n, αr = ᾱ, αs = ε”
(6) Page 301, exercise 4, this equation should be read:
“UIP: r = r̄ + ṡ, r̄ = r ∗ ”
(7) Page 302, exercise 6, third line of the text should read:
dq
“=inflation rate,π̇ = dπ
dt ; π̄c =inflation rate target; q̇ = dt ; inominal”.
Chapter 10
(1) Page 308, equation (10.2) should read:
“Ṁ = Ḃ BC ”
(2) Page 328, 2nd line from the bottom should read:
“For a Ponzi game not to occur, the following limit must be satisfied:”
(3) Page 333, exercise 2, the monetary policy rule should read:
“Monetary policy rule: ṁ = m (μ − π ) , μ constant”
(4) Page 334, exercise 7, 2nd of the bottom should read:
dt−1
“dt = 1+r
, t = 1, 2, · · · ”
t−1
(5) Page 335, 1st line of exercise 8 should read:
“8. The government finances public deficit by printing money according to:”
(6) Page 336, 2nd line of exercise 14 should read:
“lim m(T )e−r(T −t) = ert ”
(7) Page 337," exercise 18, equation of the item (a) should read:
∞
“a(t) = t [fs + s(m)] e−(r−n)(υ−t) dυ”
(8) Page 338, 7th line of the text should read:
“economy’s nominal output Y = P y, P is the price level and y is real output.”
(9) Page 338, 11th line of the text should read:
“where π = PṖ , ṡ = ṡs , n = yẏ ”
Chapter 11
(1) Page 341, 9th line from the bottom should read:
“both on the legal framework that establishes the legal condition for the”
(2) Page 364, 4th line of the test should read:
“interest rates may be easily deduced from the above expression, when iL >
iS , i̇L > 0.”
Macro Theory: Errata
327
(3) Page 367, exercise 1, 3rd equation should read:
“MPR i = λ (i ∗ − i) , λ > 0, i ∗ = r̄ + π + φ (π − π̄ )”
(4) Page 369, exercise 8, 3rd equation should read:
“TSIR: i̇S = β (r − rS ) , β > 0, rS = iS − π ”
Appendix A
(1) Page 374, 9th line of the text should be read:
“root is positive and the other is negative r1 > 0 > r2 ”.
(2) Page 374, 2nd line from the bottom should read:
“Let r1 = α + βi and r2 = α − βi be the two complex roots where i 2 = −1”.
(3) Page 389, exercise 8, first equation should read:
“Ṗ = β
M
− L (y, r) , β > 0
P
(4) Page 391, exercise 391, 4th line should read:
“perfect foresight: π e = π ”
(5) Page 392, 4th line should read:
“Ḣ = S(P ) − δH,
Appendix B
(1) Page 405, Fig. B.8 caption:
“An unanticipated transitory increase in parameter α”
(2) Page 409, 2nd line should read:
“L =
1
ϕ
(π − π̄)2 + (y − ȳ)2
2
2
(3) Page 411, 1st line should read:
∞
“ max
e−ρt μmdt 0
(4) Page 411, exercise 8, 2nd equation should read:
∞
“
0
e−ρt μmdt =
∞
0
e−ρt τ (m)dt +
∞
0
e−ρt (m − m(0)) dt 328
Macro Theory: Errata
(5) Page 412, 11th and 12th lines should read:
“(b) Show . . . .
(c) Suppose . . . .”
(6) Page 413, 6th line of the text should read:
∞
“
e−ρt [u (f (x)) + υ(m)] dt 0
(7) Page 415, 5th line from the bottom should read:
“0 ≤ q ≤ S”
(8) Page 415, 1st line from the bottom should read:
x “f (x) = αx 1 −
S
Appendix C
(1) Page 418, 9th line of the text should read:
“solution of Eq. (C.1) is backward looking. When |α| = 1, Eq. (C.1) has two
solutions,”
(2) Page 419, 8th line from the bottom should read:
“|λ2 | > 1”
(3) Page 422, equation (C.22) of the text should read:
“IS:xt = Et xt+1 − σ (rt − r̄t )”
(4) Page 424, equation (C.32) of the text should read:
1
(1 + αθ ) β + σ k k
“A =
1 + σ θ + σ (1 + φ) k −σ (1 + φ) β + σ 1
(5) Page 427, equation (C.41) of the text should read:
“yt = af Et yt+1 + ab Et yt−1 + bxt , yt−1 given, y0 free (C.41)”
(6) Page 435, exercise 2, 3rd equation of the text should read:
“MPR:it = r̄t + πt + φπt + θ xt ”
(7) Page 435, exercise 3, the 1st and 3rd equation of the text should read:
“PC:πt = βEt πt+1 + kxt
MPR:it = r̄t + πt + φEt πt+1 + θ xt ”
(8) Page 435, exercise 6, the 3rd equation of the text should read:
“MPR:it = r̄t + πt + φEt (πt+1 − π̄ ) + θ xt ”
Index
A
Absolute risk premium utility function, 101
Adaptive expectations, 122, 123, 257, 297
Aggregate demand curve, 208, 261
Aggregate supply curve, 108
AK Model Share of Labor in Output, 87, 89
Asset demands, 152
Asset pricing model/present value model, vii,
242, 245
Average value of capital, 293, 294
Consumption habit formation, vii
Covered interest parity (CIP), 148
Credit market, 135
Crowding Out, 225
Current value Hamiltonian, 5, 50, 91, 94, 227,
295, 298, 300, 303, 305, 307, 308, 310
B
Backward and forward IS curve, 143
Backward and forward Philips curve (PC), 317
Backward IS curve, 142, 143
Balanced budget multiplier, 112
Bank reserves market, 234, 235
Bubbles, vii, 8, 10, 113, 114, 147, 159, 219,
253, 261
E
Endogenous growth, vi, vii, 73, 81–96
Endogenous rate of time preference, 23, 38,
142, 321
Equity premium puzzle, 242, 245
Euler equation, v, vi, vii, 102–104, 142, 144,
186, 242–244, 278, 280, 281, 288
Exchange rate determination, 147
Exchange rate overshooting, 155, 166, 167
Exogenous growth model, 76, 77
C
Cagan money demand equation, 107
Calculus of variations, 277, 280, 281
Calvo Philips curve, vii, 121
Canonical new Keynesian model, 142, 143
Cash in advance constraint (CIA), vii, 11, 12
Central bank operational procedure, 131
Central bank’s loss function, 114, 232, 233
Cobb-Douglas production function, 71–74, 79,
81
Conservative central bank, 232, 233
Consumption asset pricing model, vii, 242, 245
D
Dynamic inefficiency, 67, 68
F
FED operational procedure, 246
Finite-life OLG model, 66
Fiscal policy rule, 21, 129, 201, 254, 259, 261,
262
Fiscal rule, 129, 201, 210, 254, 259, 261, 262
Fisher equation, 79, 216, 229
Fixed exchange rate, 145
Fixed exchange rate MFD model, 145
Flexible exchange rate MFD model, 146
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
F. de Holanda Barbosa, L. Antônio de Lima Junior, Workbook for
Macroeconomic Theory, https://doi.org/10.1007/978-3-030-61548-2
329
330
Foreign interest rate premium, 41, 51, 152,
171, 186
Forward exchange rate market, 148
Forward guidance puzzle, 142
Forward IS curve, 143
Fully funded social security system, vii, 69
Fundamental and bubble solutions of finite
difference equations, 113
G
Golden-rule, 59, 61, 67
H
Hamiltonian, 3–5, 11, 12, 14, 22, 37, 39, 47,
50, 52, 83, 86, 88, 91, 94, 227, 277,
278, 282, 284, 289, 292, 295, 298, 300,
303, 305, 307, 308, 310
Harberger-Laursen-Meltzer (HLM) effect, vii,
148, 152
Harrod-Domar model, 252
Housing market model, 268, 269
Human capital with externality, vii, 91, 93
Hyperdeflation bubbles, 10
Hyperinflation, vi, 8, 10, 210
Hyperinflation bubble, 8, 10
I
Increasing Returns with Endogenous Growth,
93–96
Inflation target, 110, 125, 126, 128–130, 133,
137, 138, 141, 142, 174, 177, 178, 180,
230, 231, 238–240
Inflation target change, 125, 126, 175, 180
Inflation target monetary policy rule, 133, 231
Inflation tax, 205, 206, 212, 223, 295, 296
Intertemporal approach to the balance of
payments, vii, 52
IS curve, v, vi, 99, 100, 103, 112, 125, 127,
129, 131, 132, 136, 142, 143, 169, 172,
173, 176, 178, 180, 183, 185–189, 229,
231, 238, 257, 317
IS curve in an open economy, 186, 189
IS curve with positive slope, 99
J
Jensen’s inequality, 244
Index
K
Keynesian model, v, vi, vii, 99–124, 133,
141–143, 175, 193, 194, 315, 317
Keynesian Phillips curve, 123, 124, 317
L
Leibinitz rule, 120, 122, 220
Linear optimal control, 277–311
Liquidity trap, 111
LM curves, 99–124, 134, 206, 257
Long run money neutrality, 142
Lucas model with leisure, 85
M
Marginal value of capital, 291, 293
Marshall-Lerner condition, 155
Monetary approach to the balance of payments,
vii, 145, 146
Monetary policy rule (MPR), vi, 15, 100, 125,
127–129, 131–136, 138–142, 158, 168,
171, 173–176, 178, 180, 184–186, 202,
207, 208, 229, 231, 232, 237, 238, 240,
241, 313–317, 322, 325, 327, 328
Monetary regime, 207
Money and the natural rate of interest, 100,
103, 131, 132, 135, 138, 186, 188, 189,
242, 284
Money financed public deficit, 201, 203, 216
Money neutrality, 13, 142
Money superneutrality, vii
Multiple equilibria, 141
Mundell-Fleming-Dornbusch (MFD) model,
161
N
Natural exchange rate, 190, 192, 193
Natural rate of interest in a small open
economy, vii, 157, 190, 192
Negative natural rate of interest puzzle, 242
Net domestic credit, 146
New Keynesian model, v–vii, 141–143, 315,
317
New Keynesian Phillips curve, 123, 124, 317
O
Okun’s Law, 132
Old Keynesian IS curve, 143
OLG models, 59, 61, 83
Index
Open economy representative agent model, v,
vi, 29–54
Optimal control, vi, 277–311
Optimal monetary policy, 299
Overlapping generations diamond model
(OLG), 55–70
P
Pay-as-you-go (PAYG) social security system,
69
Perfect foresight, 99, 123, 228, 261, 327
Permanent anticipated change, 262, 264, 266,
269
Permanent government expenditures, 227, 228
Permanent income, 52, 54
Permanent unanticipated change, 262, 264, 265
Perpetuity, 236
Perverse monetarist arithmetic, 219
Phillips curve (PC), vi, vii, 99–126, 129, 131,
132, 134–136, 138–142, 168, 171, 173,
175, 178, 180, 182, 184, 185, 190, 193,
207–209, 229, 231, 233, 237, 238, 240,
241, 257, 261, 313–315, 317, 324, 328
Phillips curve in an open economy, vi, 207
Portfolio balance model, 152
Present value Hamiltonian, 5, 50, 91, 94, 227,
295, 298, 300, 303, 305, 307, 308, 310
Price index, 29–32, 134, 148, 150, 163, 175,
178, 185, 190, 191, 212, 219, 223
Price Level Indeterminacy, 158, 162, 163
Primary deficit, 19
Private and social labor productivity, 91
R
Ramsey/Cass/Koopmans model, vii, 93
Rational expectations, 108, 109
RCK Model, 81
Real business cycle (RBC), 19, 23–25
Real cash balance effect, 100, 109, 158
Real deficit, 18, 19, 321
Real interest rate and the real exchange rate,
175, 180
Representative agent model, v, vi, 3–27, 29–54
Ricardian equivalence, vi, 100, 112, 113, 214
Risk-free rate puzzle, 242, 245
331
S
Short Run Money Neutrality, 13
Singular optimal control, 303
Social security systems, vii, 69, 70
Solow model with CES production function,
vii, 71
Solow model with Cobb-Douglas production
function, 71–74, 79, 81
Solow model with endogenous growth, vii,
73
Solow model with money, 78
Sustainability of foreign debt, 213
T
Tax smoothing, vii, 226
Taylor rule, 99–124, 129, 158, 159, 190, 193
Taylor rule in an open economy, 158
Terms of trade, 149, 151, 152, 155, 168–171,
173, 175, 186
Term structure of interest rates (TSIR), vi, 237,
241, 327
The “Discounted Euler Equation”, vi, vii, 142,
144
Time inconsistency, vii, 301, 302
Time Inconsistent Monetary Policy, 223, 301
Tobin’s q, vii, 289, 291, 293, 294
Tobin’s q with installation costs, vii
Tradable and nontradable goods, vii, 189–193
Transaction cost model, 107
Transitory anticipate change, 262, 264, 267,
270, 271
Transitory unanticipated change, 262–266,
269, 271, 272, 274, 275, 285
U
UIP risk premium, 148
Uncovered interest parity (UIP), 147, 148, 157,
159, 168, 169, 171–173, 175, 176, 178,
180, 181, 183–185, 189, 326
V
Variable discount rate, 38, 121
Velocity of money, 109–111