ECON 607 Lecture 11
Professor Gronberg
Department of Economics
Texas A& M University
November, 13, 2013
Profit-Maximizing Firm Model
Profit Maximization: Cost Function Approach
Profit Maximization: One Output Case
First Order Condition
Second Order Condition
Marginal Revenue/Marginal Cost Analysis
Short-Run Supply Curve for a Price-Taking Firm
Avoidable versus Unavoidable Fixed Costs
Shutdown Decision (Nicholson Figure 11.3)
Producer Surplus
Definition
Short Run Example
Competitive Industry/Market Supply with a Fixed Number
of Firms
Key Assumptions:
1. Large number of profit-maximizing firms
producing homogeneous product.
2. Each firm is a price-taker.
3. Information is perfect and transactions are
costless.
Competitive Industry/Market Supply with a Fixed Number
of Firms
The industry supply function is simply the sum
of the individual j firm supply functions.
If qj (p) is the supply function of the jth firm in
an industry with J firms, the industry supply
function is given by
Q(p, J) =
qj (p).
Competitive Industry/Market Supply with a Fixed Number
of Firms
The inverse supply function for the industry is
just the inverse of this function: it gives the
minimum price at which the industry is willing
to supply a given amount of output.
Example: Identical cost functions
Suppose that there are J firms that have the
common (short run) cost function c(q) = q 2 + 1.
Competitive Industry/Market Supply with a Fixed Number
of Firms
The marginal cost function is simply
MC(q) = 2q, and the average variable cost
function is AV C(q) = q.
Since in this example the marginal costs are
always greater than the average variable costs,
the inverse supply function of the firm is given
by p = MC(q) = 2q.
Competitive Industry/Market Supply with a Fixed Number
of Firms
It follows that the supply function of the firm is
q(p) = p/2 and the industry supply function is
Q(p, J) = Jp/2.
The inverse industry supply function is therefore
p = 2Q/J.
Competitive Industry/Market Supply with a Fixed Number
of Firms
Example: Different cost functions
Consider a competitive industry with two firms,
one with cost unction c1 (q) = q 2 , and other
with cost function c2 (q) = 2q 2 .
The supply functions are given by
q1 = p/2
q2 = p/4.
Competitive Industry/Market Supply with a Fixed Number
of Firms
The industry supply curve is therefore
Q(p) = 3p/4.
For any level of industry output Q, the marginal
cost of production in each firm is 4Q/3.
Industry producer surplus
Competitive Industry/Market Supply with a Fixed Number
of Firms
Following figures from Microeconomics by
Goolsbee, Levitt, and Syverson, Worth
Publishing, 2013
Competitive Market Equilibrium with Fixed Number of
Firms
The industry supply function measures the total
output supplied at any price.
The industry demand function measures the
total output demanded at any price.
An equilibrium price is a price where the amount
demanded equals the amount supplied.
Competitive Market Equilibrium with Fixed Number of
Firms
If we let x(p) be the demand function of
individual i for i = 1, ..., n and q(p) be the
supply function of firm j for j = 1, ..., J, then
an equilibrium price is simply a solution to the
equation
qj (p).
xi(p) =
Competitive Market Equilibrium with Fixed Number of
Firms
Example: Identical firms
Suppose that the industry demand curve is
linear, X(p) = a − bp, and the industry supply
curve is that derived in the last example,
Q(p, J) = Jp/2.
Competitive Market Equilibrium with Fixed Number of
Firms
The equilibrium price is the solution to
a − bp = Jp/2,
which implies
p∗ = a/(b + J/2).
Note that in this example the equilibrium price
decreases as the number of firms increases.
Competitive Market Equilibrium with Fixed Number of
Firms
For an arbitrary industry demand curve,
equilibrium is determined by
X(p) = Jq(p).
How does the equilibrium price change as J
changes?
Competitive Market Equilibrium with Fixed Number of
Firms
We regard p as an implicit function of J and
differentiate to find
X (p)p (J) = Jq (p)p (J) + q(p)
which implies
p (J) = q(p)/(X (p) − Jq (p)).
Assuming that industry demand has a negative
slope, the equilibrium price must decline as the
number of firms increases.
Free Entry and Long Run Competitive Equilibrium
4 Key Assumptions
1. Infinite number of potential firms
2. Access identical “best” technology
3. No Technological Externalities
4. Input Prices Fixed to Industry (horizontal
industry input supply curves)
Free Entry and Long Run Competitive Equilibrium
Given X(p), c(q) with c(0) = 0, (p∗, q ∗, J ∗) is a
Long Run Competitive Equilibrium if
i. q ∗ solves Max p∗ q − c(q)
ii. X(p∗) = J ∗ q ∗
iii. p∗q ∗ − c(q ∗ ) = 0
At p∗ , X(p) = Q(p), where Q(p) is the long run
industry/market supply curve.
Free Entry and Long Run Competitive Equilibrium
Standard Case:
Let c(q) = K + f (q) if q > 0, c(q) = 0 if q = 0,
where f (0) = 0, f > 0, f > 0.
Then there exists a unique efficient scale qm
where long run average cost is minimized at a
value cm .
Free Entry and Long Run Competitive Equilibrium
Then
Q(p) = ∞ if p > cm
= Q ≥ 0 : Q = Jqm for some integer J ≥ 0 if p = cm
= 0 if p < cm
So Long Run Competitive Industry Supply Curve
is “approximately” horizontal at the entry price
p = cm .
More on Long Run Competitive Industry Supply
Conclude that, in general, with homogeneous
firms, free entry long run competitive equilibrium
implies that supply curves are perfectly elastic.
Three common avenues to upward-sloping
long-run aggregate supply:
1. Assume that the industry faces an
upward-sloping supply curve for some input.
2. Assume heterogeneity in costs across firms.
3. Assume that entry is limited (and assume
increasing marginal costs)
More on Long Run Competitive Industry Supply
Case 1: Endogenous factor price
This is the Increasing Cost Industry case on
pp.428-29 of N& S, 11th edition.
More on Long Run Competitive Industry Supply
Case 2: Heterogeneity
Suppose firms differ in cost efficiency.
Figure 7.7 from Layard and Walters
More on Long Run Competitive Industry Supply
Long Run Producer Surplus and Economic Rents
This material goes with the Ricardian Rent case
on pp. 436-438 of N& S, 11th edition.
More on Long Run Competitive Industry Supply
Case 3: Limited Entry
If we maintain the price-taking behavioral
assumption, and if a limited number of firms
have convex costs, then industry has upward
sloping supply with
dQ/dp = J/c (q)
where J is the number of firms and C (q) is the
second derivative of the firm cost function.
Are profits zero?
Application: U.S. Sugar Market Program Analysis
This application is based on Lecture Note 8 for
the course Applied Microeconomics and Public
Policy (MIT 14.03) authored by Professor David
Autor.
The material is available for use under the MIT
Open Course Ware initiative and distributed to
you under the agreements described in the
following link: http://ocw.mit.edu/terms.
Autor: Lecture Note 8:
http://tinyurl.com/AutorLec8