Partial equilibrium analysis:
Competitive markets
Lectures in Microeconomic Theory
Fall 2008, Part 16
30.09.2008
G.B. Asheim, ECON4230-35, #16
1
Partial vs. general equilibrium analysis
„
In a partial equilibrium model, all prices other than
the price of the good studied are assumed to remain
Price
fixed.
Aggregate supply
given factor prices.
Equilibrium
price.
Aggregate demand
given prices of other
goods and given income.
Equilibrium quantity.
„
Quantity
In a general equilibrium model, all prices are
variable. A general equilibrium requires that all
markets clear.
30.09.2008
G.B. Asheim, ECON4230-35, #16
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Overview of lectures on partial equilibrium
„
23 Oct: Competitive markets
Quasi-linear utility functions. Welfare analysis.
„
30 Oct: Monopoly
Price discrimination.
„
6 Nov: Oligopoly and game theory
Cournot competition. Nash equilibrium
„
(13 & 20 Nov: Repetition of the theory of the firm,
consumer theory, and general equilibrium analysis)
30.09.2008
G.B. Asheim, ECON4230-35, #16
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1
A competitive market is characterized by
„
Sellers and buyers take the market price as given and
determine their supply and demand accordingly.
„
The market price is determined so that
market supply = market demand.
„
A good is transferred if and only if the price is paid.
„
Sellers and buyers have the same information about
the transferred good.
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G.B. Asheim, ECON4230-35, #16
Demand facing each competitive firm
if p > p
⎧0
⎪
D ( p ) = ⎨any amount if p = p
⎪∞
if p < p
⎩
p
p
D( p )
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G.B. Asheim, ECON4230-35, #16
Supply of a competitive profit maximizing firm, if convex costs
π ( p ) = max py − c ( y )
c′( y )
y ( p)
p
y
y ( p ) = arg max py − c ( y )p
y
If interior solution:
FOC : p = c ′( y ( p ))
SOC : c ′′( y ( p )) ≥ 0
p
y( p)
How does a competitive
firm respond to a change in
the price of output?
30.09.2008
y
By differentiating the FOC:
1 = c ′′( y ( p )) y ′( p )
c ′′( y ) > 0 ⇒ y ′( p ) > 0
G.B. Asheim, ECON4230-35, #16
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2
Supply of a competitive
profit maximizing firm,
if non-convex costs
(1, p′′)
(1, p′)
OutputOutput
Profit-max.
output
Profit in
terms of
output
Cost
Cost at profit-maximizing output
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Interior solution
if price exceeds
minimal average
variable cost.
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G.B. Asheim, ECON4230-35, #16
Supply of a competitive profit maximizing firm, if
non-convex costs (cont.)
Price
y( p)
AC
AVC
p′′
Min.
AVC
p′
MC
Output
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G.B. Asheim, ECON4230-35, #16
Further analysis of the profit function
π ( p ) = py ( p ) − c ( y ( p ))
π ′( p ) = y ( p ) +
Hotelling’s lemma
y ( p)
p′′
( p − c ′( y ( p )) ) y ′( p )
= y ( p ) if
p
p> p
∗
π ( p′′) − π (0)
p∗
π ′( p ) = 0 = y ( p )
if
y
p < p∗
π ( p)
π ( p ′′) − π ( 0 )
=
∫
30.09.2008
p ′′
0
π ′( p ) dp =
∫
p ′′
0
y ( p ) dp
G.B. Asheim, ECON4230-35, #16
p
p∗
9
3
Firm 2
Price
Firm 1
The industry supply function with a given
number of firms
Market
supply
m
S ( p) = ∑ y j ( p)
j =1
Output
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G.B. Asheim, ECON4230-35, #16
Market equilibrium
Price
Market
supply
Equil.
price
n
m
∑ x ( p) = ∑ y ( p)
i =1
i
j =1
j
Market
demand
Output
Equil. output
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G.B. Asheim, ECON4230-35, #16
Entry with non-convex costs
Price
Long-run
market supply
Long-run
equil. price
Market demand
Long-run
equil. output
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G.B. Asheim, ECON4230-35, #16
Output
12
4
Welfare economics
„
Assume that lump-sum transfers are available, and
that the distribution is optimal. This allows us to
adopt the representative consumer approach.
„
Assume that there are no income effects; i.e., the
Mashallian and Hicksian demand functions coincide.
This means that preferences are represented by a
quasi-linear utility function:
Utility = u ( x) + z
Utility derived from good in question
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Money spent
on other goods
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G.B. Asheim, ECON4230-35, #16
Quasi-linear utility
u ′( x)dx
+ dz = 0
Utility = u ( x) + z
x
Marg. willingness to pay
u′(x)
u ′(x′)
dz
−
= u′(x)
dx
x
x′
v
MRS is
independent x′
of money
spent on
other goods
Demand fn is
determined by:
p = u′( D( p))
z
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Welfare analysis
p
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G.B. Asheim, ECON4230-35, #16
p
Efficiency gain
c′(x)
c′(x)
S ( p)
S ( p)
D( p)
D( p )
u′(x)
x′
u′(x)
x
x′′
x′
x′′
x
x′′
x′′
u ( x′′) − u ( x′) = ∫ u ′( x)dx
c( x′′) − c( x′) = ∫ c′( x) dx
Willingness to pay for
increased quantity
Cost to produce
increased quantity
x′
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x′
G.B. Asheim, ECON4230-35, #16
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5
Welfare analysis (cont.)
p
In equilibrium:
u ′( D( p )) = p = c′( S ( p))
c′(x)
D( p) = S ( p)
1st welfare thm
S ( p)
2nd welfare thm
p̂
D( p )
u′(x)
Welfare maximization:
max u ( x) + z s.t. z = ω − c( x)
x
x
x̂
FOC : u′( x ) = c′( x)
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G.B. Asheim, ECON4230-35, #16
Welfare analysis (cont.)
Total surplus maximization … p
CS ( xˆ ) = u ( xˆ ) − pˆ xˆ
c′(x)
arg max{CS ( x) + PS ( x)}
x
S ( p)
= arg max{(u ( x) − px )
x
p̂
D( p )
+ ( px − c ( x) )}
u′(x)
= arg max{u ( x ) − c( x)}
x
x
x̂
… entails maximization of u(x)−c(x), PS ( xˆ ) = pˆ xˆ − c( xˆ )
which is achieved in equilibrium.
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G.B. Asheim, ECON4230-35, #16
Taxes and subsidies
Market equilibrium with a tax. p
Tax revenue
D ( pˆ d ) = S ( pˆ s )
pˆ d = pˆ s +t
Alternatively:
D ( pˆ s +t ) = S ( pˆ s )
p̂d
c′(x)
S ( p)
t
D( p )
p̂s
u′(x)
Or: D ( pˆ d ) = S ( pˆ d −t )
x
x̂
Tax revenue is smaller than
the reduction in total surplus.
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G.B. Asheim, ECON4230-35, #16
Deadweight
loss
18
6