ECON 1110
Rajiv Vohra
Profit Maximization under Perfect Competition
Perfect Competition
• A firm in a perfectly competitive market knows it has no
influence over the market price for its product because it is
small compared to the market. The firm is a price-taker.
• Firms sell homogeneous products, e.g. wheat.
• Perfect Information among buyers and sellers.
• Free entry and exit very long-run.
Suppose a firm produces output y from two inputs x1 and x2 .
The price of the output is p and the prices of the two inputs are
w1 and w2 .
Let f (x1 , x2 ) be the production function.
Let f1 =
@f (x1 ,x2 )
@x1
and f2 =
@f (x1 ,x2 )
@x2
denote the marginal products.
Assume that all prices are positive and so are both the marginal
products.
The firms’ profit maximization problem can then be written as
follows:
Choose x1 , x2 to maximize py
w 1 x1
w2 x2 , subject to
y  f ( x1 , x2 ) .
(1)
Since prices are positive, and marginal products are positive, the
constraint will always be binding (it holds with an equality), and
the problem can be rewritten as:
Choose x1 , x2 to maximize pf (x1 , x2 )
w 1 x1
w 2 x2
If x⇤1 , x⇤2 solve problem (2), the first order conditions are:
pf1 (x⇤1 , x⇤2 ) = w1 ,
pf2 (x⇤1 , x⇤2 ) = w2 .
(2)
Notice that these two equations should provide a solution to
(x⇤1 , x⇤2 ) as functions of p, w1 , w2 – the input demand functions. And
once we have found (x⇤1 , x⇤2 ) we can plug these into f to find the
optimal supply of y, and also compute the maximized level of profit.
Cost Minimization: An intermediate step in profit maximization
Now consider a di↵erent problem, that of minimizing the cost
of producing the level of output y, i.e.,
Given y choose (x1 , x2 ) to minimize w1 x1 + w2 x2
subject to f (x1 , x2 ) = y.
(3)
This determines the input levels which minimize the cost of producing y units of the output.
Let c(y; w1 , w2 ) denote the result of this minimization exercise.
It refers to the minimum cost of producing y.
Cost minimization is not the main objective of a firm but it can
be seen as one auxiliary part of the more important problem of
profit maximization.
Having solved the cost minimization problem, (3), the profit
maximization problem, (1), can now be rewritten as:
choose y to maximize py
c(y; w1 , w2 ).
(4)
Clearly, the first order condition for problem (4) is:
p=
dc(y; w1 , w2 )
,
dy
in other words, if y is the profit maximizing level of output then
the marginal cost (at y) must equal the price of the output.
Problem (4) is an alternative way of writing problem (1).
It seems like a long-winded way of stating (1) since it goes
through the intermediate step of first solving the optimization problem (3).
The advantage of this approach is that even if there are several
inputs, (4) is a problem that involves only one variable: y.
And if our primary interest is in studying the optimal level of y,
this is the most convenient formulation of the firm’s problem.
It allows us to see the problem of choosing the profit maximization level of y in a two-dimensional graph. And from this analysis,
it is very easy to derive the firm’s supply curve.
If there is only one input, then the cost minimization step is
trivial. Then there will be, according to the production function,
a unique level of x1 which results in output y. And this is simply
the inverse of the production function, f 1 (y ). The cost function
is simply c(y; w1 ) = w1 f 1 (y ), and problem (4) becomes:
choose y to maximize py
w1 f
1 (y ).
(5)
1
The first order condition is p = w1 dx
which is the same as the first
df
order condition corresponding to (1) in the one input case.
Conditions for Profit Maximization
Suppose the input prices are fixed, and given these prices the
(long-run) cost curve is c(y ). Suppose c(y ) is well-behaved in the
sense that it exhibits increasing returns to scale at low levels of
output and decreasing returns at high levels of output. The rules
for profit maximization are then the following:
(1) Find y ⇤ such that p = c0 (y ⇤ ); the first order condition.
(2) Make sure that c00 (y ⇤ ) > 0; second order condition.
(3) Check that py ⇤
dition.
c(y ⇤ )
0. Otherwise set y = 0; global con-
Find a level of output at which price equals marginal cost and
marginal cost is rising. This is the profit maximizing level of output if the resulting profits are non-negative. Otherwise, the profit
maximizing level of output is 0.
Notice that y ⇤ satisfying rules 1 and 2 has the property that it
provides the maximum profit among all output plans that involve
positive production. Comparing the profit of y ⇤ to the profit at 0 is
then the only other condition left to be checked. This comparison
can be rewritten as the condition: min AC  p.
These three conditions immediately allow us to derive the longrun supply curve of the competitive firm.
The Firm s Long-Run Supply
Decision
$/output unit
MC(y)
AC(y)
y
The Firm s Long-Run Supply
Decision
$/output unit
MC(y)
p > AC(y)
AC(y)
y
The Firm s Long-Run Supply
Decision
$/output unit
MC(y)
p > AC(y)
AC(y)
y
The Firm s Long-Run Supply
Decision
$/output unit
The firm s long-run
supply curve
MC(y)
AC(y)
y