Micro-foundation to Macroeconomics: General Equilibrium
Analysis with Production
• Price system is key to allocation of resources in the
economy.
• Markets for goods, labour, capital and financial services
are inter-linked.
• An economy is inhabited by N households and M firms, a
government which taxes and spends and the international
sector.
• A competitive equilibrium solution is outcome of the
utility maximising behaviour of households and profit
maximising activities of firms.
• General equilibrium model brings all of these elements
together.
Macroeconomic Themes: 7
1
General Equilibrium in Terms of Circular Flows
Households
C+S+T=HI
Financial
market I=S
Factor
Market
Ld =LS
Kd = KS
Government
R=G
Goods market
Y=C+I+G+M
-M
Firms
Rest of the
World
CA+KA=0
Y=F(A,K,L)
Macroeconomic Themes: 7
2
General Macroeconomic Equilibrium
LD
IS
W
i=i*
LS
M(D)
i
LM
L
Labor market
IS-LM: goods and money
Money market
(M/P)
F(Y)
Y
P
Y
Output
L
Output to output , Y
Macroeconomic Themes: 7
Price level and money supply
3
General Equilibrium Impacts of Labour
Supply Tax
Labour market
W/P
LD
LS1
LS
(a)
L1
L
D D1
S S1
Y
(c)
Y1
(b)
Employment and production
P P1 P2
Demand and Supply of Goods
Macroeconomic Themes: 7
4
Simple Micro-founded macro model on welfare impact of taxes
Max
U(C, L)
HouseholdsL-
Government
Factors -Labour
Goods consumption
Firms
Macroeconomic Themes: 7
5
Main Features of the model
Households: Representative and heterogenous cases
Fixed endowment of time
ES preferences on consumption and leisure
Utility maximisers
Firms
Profit maximisers
Cobb-Douglas technology
Government
Collects sales and income taxes
Spends on public goods
Markets:
Competitive equilibrium
Goods market clears –through price and quantity
adjustment
Labour market clears through labour-leisure choice
decision
Capital market clear: all capital is used in production
Assumes a closed economy or global economy
No money illusion
Policy evaluation criteria
Comparative static analysis with equivalent or Efficiency
effect of taxes by change in P, r and W and
reallocation C, L, LS, Y, I, R
Macroeconomic Themes: 7
6
Household’s Problem
Max U  c  l1 
Subject to:
i. l  L  1
time constraint


ii. c  w1 l   rK  
budget constraint
iii. c  0; l  0; K  0
non-negativity constraint
where c is consumption, l is leisure and
L is labour supply, K is capital stock,
p is the price of the commodity which is normalised to 1, w is the

wage rate
is the profit from owning the firm.
Macroeconomic Themes: 7
7
Firm’s Problem
Max   py  wLd  rK
subject to :
i. y  AK L1
technology
constraint
ii. y  0; L  0; K  0 non negativity constraint
where y is the output supplied by the firm and
Ld is its demand for labour,  is the share of capital or

1


 is
the elasticity of output to the capital input and




the share of labour in production. Capital stock is
constant in the short run.
Macroeconomic Themes: 7
8
Demand side of the economy
L c,l,    c  l1   w1 l   rK  c




L c,l,  


 c  1l1  
c
L c,l,   


 1  c  l   w
l
L c,l,    


 w1 l   rK  c



Macroeconomic Themes: 7


(1)
(2)
(3)
9
Demand for Consumption
Demand for consumption goods can be derived by using
the marginal rate of substitution between leisure and
consumption (dividing FOC (2) by FOC (1) and solving for
c):
L c, l,  
1  c  l  
1  
 l

w


c
l


 
w  cw
1  

 l
L c, l,  


c  1l1
c
(4)
there is more demand for the consumption goods when the
wage rate is higher or the leisure is less demanded when the
wage rate is higher.
Macroeconomic Themes: 7
10
Demand for Leisure
We obtain the demand for leisure using this result in the
budget constraint





 



w
l

w
1 l   rK  wl 1
   w  rK  =>





1   
1  



l  1w  w  rK 
(5)
 l or
Substituting (5) into (4) c  w 

1  




c  w    1w  w  rK  => c    w  rK 
1  


Macroeconomic Themes: 7
(6)
11
Supply Side of the Economy
Max   py  wL d  rK
 Max   AK L1  wL  rK
First order conditions for optimisation:
   0 AK  1L1  r

(7)
K
  0 1  AK  L  w

(8)

L
AK 1L1  r
From (7) and (8) we get 
w
1  AK  L


1  r
 Lr


L


   w K
1 K w


Macroeconomic Themes: 7
(9)
12
Market Clearing Conditions
SK
The capital market clearing condition:
N .K  M .K  K=K when N=M.
r
The labour market clearing 1 l  L
DK
Goods market clearing implies c  y
SL
K
S
w
P
D
DL
Y
L
Macroeconomic Themes: 7
13
Three Different Condition for a
General Equilibrium
1. market for goods, labour and capital clear; that means
prices (p, w, r) adjust until demand and supply are equal
in each of these market. No excess supply or excess
demand exists in any of these markets.
2.
Firms continue producing until the economic profit is
zero (competitive equilibrium)
3.
Income of household exactly balances to the expenditure
and total labour supply is exactly divided between labour
and leisure.
Macroeconomic Themes: 7
14
Labour Market Equilibrium
Using labour market clearing condition and (5) above in (9)
we have

  r K (10)
11w  w  rK   1
 w



1 




w  1   w  rK     rK


 rK  w 
 w  1  rK  1









1  











rK
This solves as w  rK 1  (11)

Substitute (11) in (7) to get the equilibrium interest rate
1  AK  L  w






1  AK  1  r K 
 


  w 





Macroeconomic Themes: 7
w
15
Determination of equilibrium interest and other quantities in
the equilibrium
1  AK 1










rK 





 w1
1
AK    r   w1
 1 
1



AK     r  w 1
r  1 

1
1  1
w
r  A 1












Macroeconomic Themes: 7
(12)
16
Equilibrium Relative Wage Rate
Substituting (12) into (11) yields
w  rK 1  

1
1  1 K 

w  A 1
w
1  

















1
11  
1  K 


w
  A1 
1 


 





  
1 
K 


*
w    1   A1 

(13)

 





Macroeconomic Themes: 7
17
Determination of Equilibrium
Quantities
Use (13) to find the optimal values of labour, capital stock
and output and consumption in this economy. The optimal
capital stock is obtained by using r and w in equation (10).
The optimal labour supply is obtained by using that value
of K in (9).
Once K, and L are determined output is determined from
the production function.
By market clearing condition the consumption equals
output.
Macroeconomic Themes: 7
18
Simulation of the Model to a Change in
Capital Stock and Technology
Parameterisation and base case soultion
K-stock
alpha
0.5
beta
0.3
Technolog
y
0.7
4
Sensitivity of the General Equilibrium Results to Capital Stock and the Technlology
wage rate
interest rate
Labour
Output
Profit
Leisure
Consumption
Utility
L-income
K-income
Total income
Impact of increasing capital stock(K)
0.500
1.000
2.000
0.656
0.808
0.995
0.349
0.215
0.132
0.620
0.620
0.620
0.581
0.716
0.881
0.000
0.000
0.000
0.380
0.380
0.380
0.581
0.716
0.881
0.512
0.592
0.685
0.407
0.501
0.617
0.174
0.215
0.264
0.581
0.716
0.881
4.000
1.224
0.081
0.620
1.085
0.000
0.380
1.085
0.792
0.759
0.325
1.085
Impact of technological improvement (A)
base case
2.000
3.000
4.000
0.656
1.616
2.424
3.231
0.349
0.085
0.050
0.034
0.620
0.123
0.048
0.024
0.581
0.461
0.357
0.298
0.000
0.177
0.192
0.185
0.380
0.877
0.952
0.976
0.581
1.191
1.731
2.286
0.512
1.086
1.447
1.770
0.407
0.199
0.116
0.079
0.174
0.085
0.050
0.034
0.581
0.284
0.165
0.113
Macroeconomic Themes: 7
19
Tax distortions
1 t c  1 t w1 l   1 t rK 
l
k





(14)
t is the value added tax rate, tl is the labour income tax
rate and tk is the capital income tax rate


L c, l,    c  l1    1 t w1 l   1 t rK  1 t c (15)


l
k

y c g
(16)
g  t wL  t rK  tc
l
k
(17)
Now apply the first order conditions (1)-(3) in order to get the
demand side of the tax distorted economy. Also consider that the
market clearing condition now should include both private and
public demand goods.
Macroeconomic Themes: 7
20
Reference:
1. Arrow, K.J. and G. Debreu (1954) “Existence of an Equilibrium for a Competitive
Economy” Econometrica 22, 265-90.
2. Bhattarai K. (2002) Welfare Impacts of Equal-yield Tax Reforms in the UK Economy, mimio,
University of Hull.
3. Bhattarai K. (2002) Macroeconomic impacts of taxes, mimio, University of Hull.
4. Bhattarai K. (2001) A Prototype Multi-Sectoral Multi-household General equilibrium Tax Model, Hull
Advances in Policy Economics Working paper no. 9.
5. Bhattarai K. and J. Whalley (1999) “Role of labour demand elasticities in tax incidence analysis with
hetorogeneous labour” Empirical Economics, 24:4, pp.599-620.
6. Bhattarai and J Whalley (2000) “General Equilibrium Modelling of UK Tax Policy” in S. Holly and M
Weale (Eds.) Econometric Modelling: Techniques and Applications, pp.69-93, the Cambridge
University Press, 2000.
7. Lau M.L. A. Pahlke and T.F. Rutherford(2002), Approximating infinite-horizon models in a
complementarity format: A primer in dynamic general equilibrium analysis, Journal of Economic
Dynamics and Control 26 577-609.
7. Parente Stephen L. 1994, Technology Adoption, Learning-by-Doing, and Economic Growth,
Journal of Economic Theory, 63, pp. 346-369.
8. Ramsey, F.P. (1928) “A Mathematical Theory of Saving,” Economic Journal 38, 543-559.
9. Rutherford, T. F. (1995) “Extension of GAMS for Complementary Problems
Arising in applied Economic Analysis” Journal of Economic Dynamics and Control 19 12991324.
10. Shoven, J.B. and J.Whalley (1992) Applying General Equilibrium, Cambridge University
Press, 1992.
Macroeconomic Themes: 7
21