Econ 400 Chapter 12-15.pdf - Chapter 12 The Partial...

Chapter 12 The Partial Equilibrium Competitive Model
What is a partial equilibrium?
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 for good X .
Market demand function: X ( p
X
) =
∑
n
i = 1
x
i
( p
X
, p
Y
, m
i
) ,
where x
i
( p
X
, p
Y
, m
i
) is individual i °s demand for good X .
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
) .
Q. Wen (UW) Econ 400 University of Washington 1 / 54

Chapter 12 The Partial Equilibrium Competitive Model
Elasticities are de±ned 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 ±rm°s supply functions.
Short-run market supply: Q
S
( p ) =
∑
m
i = 1
q
i
( p , v , w ) .
In short-run, the number of ±rms, m , is ±xed, and use SR supply
functions.
Example 12.2 A Short-run Supply Function
there are 100 identical ±rms, 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 .
Q. Wen (UW) Econ 400 University of Washington 2 / 54

Chapter 12 The Partial Equilibrium Competitive Model
Equilibrium price determination: Q
D
(
p
±
) =
Q
S
(
p
±
)
.
In long run, each ±rm mostly follows its long-run supply function.
If the number of ±rms is ±xed exogenously,
then the analysis is the same as in short run (non-negative pro±t).
If ±rms are free to enter/exist the market.
then the number of ±rms is determined endogenously !
When market price is higher than a ±rm°s min average cost,
then a typical ±rm will have positive pro±t,
and more ±rm will enter the market.
the market price decreases,
entry stops when it is no longer pro±table.
Additional to price and quantity, the number of ±rms varies!
Q. Wen (UW) Econ 400 University of Washington 3 / 54

Chapter 12 The Partial Equilibrium Competitive Model
Q
p
Market demand (Blue), MC (Red), 6 ±rms
Q. Wen (UW) Econ 400 University of Washington 4 / 54

Chapter 12 The Partial Equilibrium Competitive Model
Example 12.4 In±nitely Elastic Long-run Supply (simpli±ed)
Market demand: Q
D
(
p
) =
25 , 000
²
3 p .
A representative ±rm°s cost function: C ( q ) = 40 q
2
+ 8 , 000.
Average cost function: AC
(
q
) =
40 q
+
8 , 000
q
.
Marginal cost function: MC ( q ) = 80 q .
To ±nd min AC, set MC ( q ) = AC ( q ) :
80 q
=
40 q
+
8 , 000
q
)
q
=
20.
So the min of AC is AC
(
20
) =
1 , 600.
A ±rm°s supply function is
q
(
p
) =
°
p
80
when p
³
1 , 600 ,
0 when p
´
1 , 600 .
Q. Wen (UW) Econ 400 University of Washington 5 / 54

Chapter 12 The Partial Equilibrium Competitive Model
Example 12.4 In±nitely Elastic Long-run Supply
When there are n ±rms, the market supply function is
Q
S
(
p
) =
n
°
q
(
p
)
.