ECON 603
LECTURE 12
GENERAL EQUILIBRIUM (REVISITED)
Two consumers: A & B
Two goods: x & y (Total X,Y)
For consumer A
For consumer B
Max U A x A , y A 
Max U B x B , y B 
s.t. x A  x B  X
s.t. x A  x B  X
y A  yB  Y
y A  yB  Y
xA , yA  0
xB , y B  0
Consumer A
LA  U A x A , y A   x  X  x A  xB    y Y  y A  y B 
Assuming both constraints are binding
L A U A

  x  0 1
x A x A
L A U A

  y  0 2
y A y A
U A
x A
U A
y A

x
y
MRS

for A 3
Similarly for B
MRS for B
If
x  x

y  y
U B
x B
U B
y B

x
y
4


 say  x  4,  x  2 


y
y


Consumer A finds relative value of x greater than Consumer B does.
They can gain from trade
A: 4 y  1x
B: 2 y  1x
For example, 3 y  1x will benefit both.
Same is true for
For
x  x

y  y
x  x

 trading off is not advantageous.
y  y
U A
x A
U A
y A

U B
x B
U B
y B
 MRS of both consumers are equal.
xB
OB
C
D
Y=YA+YB
E
yB
yA
IB
IA
OA
xA
X=XA+XB
At “C”, MRS of both are not equal. Process goes on until MRS are equal  Indifference curves are tangent to
each other.
OB
Locus of such
equilibrium points.
Contract curve (or
conflict curve)
OA
Assume goods are owned by these individuals. They can stay at null – trade, or trade off and be better off.
xBF
yAF
OB
F
yBF
H
G
OA
xAF
Initial resource allocation is at F. consumers can trade of only if they are better off both (in the GH region).
Now, trade off takes place through the prices Px,Py. the value of final bundle consumed can not exceed the
value of initial allocation (expenditure cannot be grater).
Then problem becomes
A  max U A x A , y A 
s.t. Px x AF  Py y AF  Px x A  Py y A , x A , y A  0
B  max U B x B , y B 
s.t. Px xBF  Py y BF  Px xB  Py y B , xB , y B  0
where x AF  x BF  X  x A  x B
y AF  y BF  Y  y A  y B


LA  U A x A , y A    Px x AF  x A   Py  y AF  y A 
From first order conditions
U A
x A
P
MRS A 
 x
U A
Py
y A
U B
x B
P
MRS A 
 x
U B
Py
y B
MRS are equal to price ratios.
xB
OB
C
H
J
G
IB
IA
Px/PY
OA
for these two consumers MRS are equal and they are equal to price ratio.
Consumption & Production
Assume A & B has initial allocation of resources L & Km which are used to produce X & Y, that are sold to
A & B.
LX
OB
KX
KY
L
M
OA
LY
Firms will agree to move from L to M if both will produce a higher product.
OY
Y
N
qx
N
qy
Contract –
conflict curve
PPF
X
OX
then for q x  q x Lx , K x 

q y  q y Ly , K y

this implies that, on the contract curve
q x
Lx
q x
K x
q y
L y

q y
 MRTS must be equal for both.
K y
Similar to the argument on consumption side, if we introduce the price of resources
MRTS 
PL
PK
X producer wants to maximize profits
Max  X  PX q X  PL L X  PK K X
 x
q
 Px x  PL  0
Lx
Lx

q x PL

L x Px

q x PK

K x Px
1
2
total differential
dq x 
q x
q
dLx  x dK x
Lx
K x
substitute (1) and (2)
dq x 
PL
P
dLx  K dK x 3
Px
Px
similarly dq y 
PL
P
dL y  K dK y 4
Py
Py
on PPF
dLx  dLy
dK x  dK y
(4) dqY  
PL
P
dLx  K dK x  0 5
Py
Py
5  dq y
3
dq x

Px
 Slope of the PPF is equal to price ratio.
Py
Y
N
YN
E1
E2
Slope  
XN
Px
Py
X
Production price ratio at N = consumption price ratio at E1.
Amount of x & y for consumption are given by the appropriate point on PPF.
So for each of the points on the PPF, we can draw a new Edgeworth – Box inside the PPF.
If consumer A has extra income to spend, he can move up on the contract curve and B must retreat. When
reach E2, slope of indifference curves are no longer equal to price ratio.
Then at the new price ratio, both consumers regard y as over price.
Then there will be a pressure to increase Px/Py. then producers react to this change by moving along the PPF
where the slope will reveal the new price ratio.
Y
N1
E2
new
Px
Py
X
EQUILIBRIUM CONVERGENCE
Resource endowments
L  L A  LB 

K  K A  KB 
Exogenous resource endowments.
PL , PK  0
Initial resource prices
q y
q x
Lx
q x
K x

L y
q y

PL
PK
Point N on producers contract curve.
K y
Px dq y

Py dqx
Slope of the PPF
q x  x A  xB  X N 

q y  y A  y B  YN 
Edgeworth – Box for consumers
U A
U B
x A
x B
P

 x
U A
U B
Py
y A
y B
Point on consumer contract curve
PL L A  PK K A  Px x A  Py y A
PL LB  PK K B  Px x B  Py y B
NO
Adjust
Budget Constraint
YES
PL
PK
Equilibrium
General Revenue Max. Problem
Assumption. Production functions are linear homogenous (constant returns to scale)
f i tLi , tK i   tf i Li , K i   tyi t
for t 
1
yi
L L 
f i  i , i   f i a Li , a Ki   1 technological coefficients, not constants but functions (variables).
 yi yi 
Max z  P1 y1  P2 y2
s.t. a L1 y1  a L 2 y2  L
a K1 y1  a K 2 y2  K
f i a Li , a Ki   1 i  1,2
**




L  P1 y1  P2 y2  wL  a L1 y1  a L 2 y2   r K  a K1 y1  a K 2 y2   1 f 1 a L1 , a K1   1  2 f 2 a L 2 , a K 2   1
first order conditions  10 equations (Page 552).
It is possible thatthis revenue max can be conceived in 2 stages.
1. first, minimum cost of any output level
2. with this minimum cost, determine the output level that maximizes revenue (profit).
For this rearrange (**) (Page 554, equation 18 – 42)
Then we have 2 minimization problem.
Min wL1  rK1  C1 s.t. f1 L1 , K1   y1 and
Min wL2  rK 2  C2 s.t. f 2 L2 , K 2   y2
First order conditions of each are 3, so there will be 6 equations.
From them we get
 w
 technological coefficients are functions of factor prices.
r
1. aij  aij 
2. MC functions are also functions of factor prices.
j
yj

j
yj
w, r 
3. Unit I – O factor levels are downward sloping in their own prices.
a Lj
w
 0,
a Kj
r
0
there remain only 4 equations.
a *L1w  a *K 1r  P1 

 A *
 zero profit conditions.
a L 2 w  a *K 2 r  P2 

a *L1 y1  a *L 2 y 2  L 

B  *
 resource constraints.
*
a K1 y1  a K 2 y 2  K 

Also,
w  w* P1 , P2 
 functions of output prices only
*
r  r P1 , P2  
Also,
y1* y 2*
,
 0 upward sloping supply curve.
P1 P2