From the two focs we can solve the demand functions

° From the two FOCs, we can solve the demand functions for capital and labor.
° Second order condition..
Chapter 12 Competitive Markets
° The Partial Equilibrium Competitive Model
° Focus on the market for one good only.
° Not a complete economy!
° The market/aggregate demand for good X is the sum of all individual demands.
° Market demand function: X ( p
X
) =
P
n
i =1
x
i
( p
X
; p
Y
; m
i
) ; where x
i
( p
X
; p
Y
; m
i
) is
individual i ³s demand for good X .
34

° X ( p
X
) obviously depends p
X
( endogenous variable). X ( p
X
) also depends on
other prices and incomes ( exogenous) .
° It is the horizontal sum of all individual demand curves.
° Market demand curve shifts when any exogenous variable changes.
° For simplicity, denote the market demand as Q
D
( p ) = X ( p
X
) .
° Elasticities are de²ned similarly
° Price elasticity: e
Q;p
=
@Q
D
( p )
@p
p
Q
D
( p )
:
° In general, market demand cannot be written as a function of aggregate income.
° In short run, the market supply is the sum of all individual ²rm³s supply functions.
° Short-run market supply: Q
S
( p ) =
P
m
i =1
q
i
( p; v; w ) , where the number of ²rms,
m , is ²xed, and use SR supply functions.
° Example 12.2 A Short-run Supply Function
° there are 100 identical ²rms, each has supply function q
i
( p ) =
10
3
p .
° The market supply function is Q
S
( p ) = 100 ¶
10
3
p =
1 ; 000
3
p .
° Equilibrium price determination: Q
D
( p
°
) = Q
S
( p
°
) .
° In long run, each ²rm mostly follows its long-run supply function.
° If the number of ²rms is ²xed exogenously, then the analysis is the same as in
short run (non-negative pro²t).
° If ²rms are free to enter/exist the market, then the number of ²rms is deter-
mined endogenously !
° When market price is higher than a ²rm³s min average cost, then a typical ²rm
will have positive pro²t, and more ²rm will enter the market, the market price
decreases, entry stops when it is no longer pro²table.
° Additional to price and quantity, the number of ²rms varies!
° Example 12.4 In²nitely Elastic Long-run Supply (simpli²ed)
° Market demand: Q
D
( p ) = 25 ; 000 ´ 3 p:
° A representative ²rm³s cost function: C ( q ) = 40 q 2 + 16 ; 000 .
° Average cost function: AC ( q ) = 40 q +
16 ; 000
q
:
° Marginal cost function: MC ( q ) = 80 q:
35

° To ²nd min AC, set MC ( q ) = AC ( q ) :
80 q = 40 q +
16 ; 000
q
) q = 20 .
So the min of AC is AC (20) = 1 ; 600 .
° A ²rm³s supply function is
q ( p ) =
³
p
80
when p ¸ 1 ; 600 ;
0 when p · 1 ; 600 :
° When there are n ²rms, the market supply function is Q
S
( p ) = n ¶ q ( p ) .
° To ²nd the competitive equilibrium, set Q
S
( p ) = Q
D
( p )
n ¶
p
80
= 25 ; 000 ´ 3 p:
° Find the market price: p ( n ) =
2 ; 000 ; 000
n + 240
° To ²nd the number of ²rms, p ( n ) ¸ 1600 > p ( n + 1)
2 ; 000 ; 000
n +240
¸ 1 ; 600 ) n = 1 ; 010
If n = 1 ; 011 , then p (1 ; 011) = 1 ; 599 < 1 ; 600 :
° A good approximation of the market supply function is the in²nitely elastic supply
function p = min AC ( q ) :
Chapter 13 General Equilibrium and Welfare
° A general equilibrium model describes a complete economy.
° Every consumer maximizes his/her utility. Every ²rm maximizes its pro²t.
° At a general equilibrium , demand = supply in every market.