°
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.

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- Spring '19
- Game Theory, Supply And Demand, Utility