Рrofit function
PROFIT FUNCTION
Given any production set Y,
we have seen how to calculate the
profit function. 7г(р),
which gives us the maximum profit attainable at prices p. The profit
function possesses several important properties that follow directly
from its definition. These properties are very useful for analyzing
profit-maximizing behavior.
Recall that the profit function is, by definition, the
maximum profits the firm can make as a function of the vector of
prices of the net outputs:
7г(р)
= max py у
such that у
is in Y.
From the viewpoint of the
mathematical results that follow, what is important is that the
objective function in this problem is a linear
function of prices.
Properties of the profit function
We begin by outlining the properties of the profit
function. It is important to recognize that these properties follow
solely from the assumption of profit maximization. No assumptions
about convexity, monotonicity, or other sorts of regularity are
necessary.
Properties of the profit function
Nondecreasing in output prices, nonincreasing in
input prices. If' p > pi for all outputs and p'- < pj for all
inputs, then 7r(p') > 7r(p).
Homogeneous of degree 1 in p.
7r(Јp) — Јтг(р)
for all t > 0.
Convex in p.
Let p"
= fp + (1 - t)p' for 0
< t < 1.
Then тг(р")
< *7г(р) + (1-*)тг(р').
Continuous in p. The function ir(p) is
continuous, at least when 7r(p) is well-defined and pt
> 0 for i = 1,..., n.
Proof. We emphasize once more that the proofs of
these properties follow from the definition of the profit function
alone and do not rely on any properties of the technology.
Let у
be a profit-maximizing net output vector at p, so that 7r(p) = py
and let y' be a profit-maximizing net output vector at p' so that
7r(p') = p'y'. Then by definition of profit maximization we have
p'y' > p'y. Since p'i >
Pi for
all i for
which j/i > 0 and p < pi for
all i for
which y^ < 0,
we also have p'y > py. Putting these two inequalities together,
we have " (p') =
р'у'
> РУ
= it(p), as
required.
Let у
be a profit-maximizing net output vector at p, so that py > py'
for all y' in Y. It
follows that for t >
0, tpy > tpy' for
all y' in Y. Hence
у also maximizes profits at prices
tp. Thus ir(tp) = tpy
= tn(p).
Let у
maximize profits at p, y' maximize profits at p', and y"
maximize profits at p". Then we have
тг(р") = p"y"
= (tp + (1 - t)p')y"
= tpy" + (1 -
t)p'y".
(3.1)
By the definition of profit maximization, we know that
фу" <
Фу =
Мр)
(l-t)p'y"<(l-t)p'y'
= (l-t)n(p').
Adding these two inequalities and using
(3.1). we have
TT(p")<tTr(p) +
(l-t)7T(p').
as required.
4) The continuity of 7r(p) follows from the Theorem of
the Maximum de scribed in Chapter 27. page 506. I
The facts that the profit function is homogeneous of
degree 1 and increasing in output prices are not terribly surprising.
The convexity property, on the other hand, does not appear to be
especially intuitive. Despite this appearance there is a sound
economic rationale for the convexity result, which turns out to have
very important consequences.
Consider the graph of profits versus
the price of a single output good, with the factor prices held
constant, as depicted in Figure 3.1. At the price vector (p*,w*) the
profit-maximizing production plan (y*,x*) yields profits p*y*
— w*x*. Suppose that p
increases, but the firm continues to
use the same production
plan (y*,x.*). Call
the profits yielded by this passive behavior the "passive profit
function" and denote it by П(р)
= py* — w*x*.
This is easily seen to be a straight line. The profits from pursuing
an optimal policy
must be at least as large as the profits from pursuing the passive
policy, so the graph of ir(p) must
lie above the graph of H(p). The
same argument can be repeated for any price p,
so the profit function must lie above
its tangent lines at every point. It follows that n(p)
must be a convex function.
| PROFITS |
|
MP) |
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|
J U(p)
|
= py- |
- w'x'
|
| MP') |
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|
^j^ |
|
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S^ P*
|
|
OUTPUT PRICE |
Figure The profit function. As the output price
increases, the profit 3.1 function increases at an increasing rate.
The properties of the profit function
have several uses. At this point we will satisfy ourselves with the
observation that these properties offer several observable
implications of profit-maximizing
behavior. For example, suppose that we have access to accounting data
for some firm and observe that when all prices are scaled up by some
factor t >
0 profits do not scale up proportionally. If there were no other
apparent changes in the environment, we might suspect that the firm
in question is not maximizing profits.
EXAMPLE: The effects of price stabilization
Suppose that a competitive industry
faces a randomly fluctuating price for its output. For simplicity we
imagine that the price of output will be pi
with probability q
and pi
with probability (1 — q).
It has been suggested that it may be
desirable to stabilize the price of output at the average price P
— 4P + (1 ~ 4)P2-
How would this affect profits of a
typical firm in the industry?
We have to compare average profits
when p fluctuates
to the profits at the average price. Since the profit function is
convex,
Qir(pi) + (1
- 9)t(P2) >
f(<7Pi + (1 - q)P2) =
7
Thus average profits with a fluctuating price are at
least as large as with a stabilized price.
At first this result seems counterintuitive, but when we
remember the economic reason for the convexity of the profit function
it becomes clear. Each firm will produce more output when the price
is high and less when the price is low. The profit from doing this
will exceed the profits from producing a fixed amount of output at
the average price.
Supply and demand functions from the profit function
If we are given the net supply function y(p), it is easy
to calculate the profit function. We just substitute into the
definition of profits to find
t(p) =
РУ(Р)-
Suppose that instead we are given the profit function
and are asked to find the net supply functions. How can that be done?
It turns out that there is a very simple way to solve this problem:
just differentiate the profit function. The proof that this works is
the content of the next proposition.
Hotelling's lemma. (The
derivative property) Let j/j(p) be
the firm's net supply function for good i. Then
ViV =
—r
for г = l,...,n,
OPi
assuming that the derivative
exists and that pi > 0.
Proof. Suppose
(y*) is a profit-maximizing net output vector at prices (p*). Then
define the function
g(p) =
тг(р)
-ру*.
Clearly, the profit-maximizing
production plan at prices p will always be at least as profitable as
the production plan y*. However, the plan y* will be a
profit-maximizing plan at prices p*, so the function g
reaches a minimum value of 0 at p*. The
assumptions on prices imply this is an interior minimum.
The first-order conditions for a minimum then imply that
дд(р*) Этг(р*)
—Ј
= —и Vi = 0 for
г = 1,..., п.
dpi дрг
Since this is true for all choices of p*, the proof is
done. I
The above proof is just an algebraic
version of the relationships depicted in Figure 3.1. Since the graph
of the "passive" profit line lies below the graph of the
profit function and coincides at one point, the two lines must be
tangent at that point. But this implies that the derivative of the
profit function at p* must
equal the profit-maximizing factor supply at that price: y(p*)
= дтт(р*)/др.
The argument given for the derivative property is
convincing (I hope!) but it may not be enlightening. The following
argument may help to see what is going on.
Let us consider the case of a single output and a single
input. In this case the first-order condition for a maximum profit
takes the simple form
/Ш
_ w
=
o. (3.2)
ax
The factor demand function x(p, w) must satisfy
this first-order condition. The profit function is given by
7r(p, w) = pf(x(p, w)) — Wx(p, w).
Differentiating the profit function with respect to w,
say, we have
дтт
df(x(p,w)) дх дх
-к- = Р я я w-
x(p,
w)
aw ox
ow aw
df{x(p,w))
P я
w
ox
Ox
Substituting from (3.2), we see that
дтт
- = -x(p,u,)-
The minus sign comes from the fact that we are
increasing the price of an input—so profits must decrease.
This argument exhibits the economic rationale behind
Hotelling's lemma. When the price of an output increases by a small
amount there will be two effects. First, there is a direct effect:
because of the price increase the firm will make more profits, even
if it continues to produce the same level of output.
But secondly, there will be an indirect effect: the
increase in the output price will induce the firm to change its level
of output by a small amount. However, the change in profits resulting
from any infinitesimal change in output must be zero since we are
already at the profit-maximizing production plan. Hence, the impact
of the indirect effect is zero, and we are left only with the direct
effect.
The envelope theorem
The derivative property of the profit
function is a special case of a more general result known as the
envelope theorem, described
in Chapter 27, page 491. Consider an arbitrary maximization problem
where the objective function depends on some parameter a:
M(a) = max
f(x,a).
X
The function M(a)
gives the maximized value of the
objective function as a function of the parameter a.
In the case of the profit function a
would be some price, x
would be some factor demand, and M(a)
would be the maximized value of profits
as a function of the price.
Let x(a)
be the value of x
that solves the maximization problem.
Then we can also write M(a) = f(x(a),
a). This simply says that the optimized
value of the function is equal to the function evaluated at the
optimizing choice.
It is often of interest to know how
M(a) changes
as a changes.
The envelope theorem tells us the answer:
dM(a) <9/(x,
a)
da
da=x(a)
This expression says that the
derivative of M with
respect to a is
given by the partial derivative of / with respect to a,
holding x fixed at the optimal choice. This
is the meaning of the vertical bar to the right of the derivative.
The proof of the envelope theorem is a relatively straightforward
calculation given in Chapter 27, page 491. (You should try to prove
the result yourself before you look at the answer.)
Let's see how the envelope theorem works in the case of
a simple one-input, one-output profit maximization problem. The
profit maximization problem is
7r(p, w)
= max pf(x)
— wx.
46 PROFIT FUNCTION (Ch. 3)
The a in the envelope theorem is p or w,
and M(a) is n(p, w). According to the envelope
theorem, the derivative of 7r(p, w) with respect to p is
simply the partial derivative of the objective function,
evaluated at the optimal choice:
9^1 =f(X)
= /(*(*«,)).
Op x=x(p,w)
This is simply the profit-maximizing supply of the firm
at prices (p, w).
Similarly,
dn{p, w)
—я =
~x
=-x(p,w),
aw x=x(p,w)
which is the profit-maximizing net supply of the factor.
Comparative statics using the profit function
At the beginning of this chapter we proved that the
profit function must satisfy certain properties. We have just seen
that the net supply functions are the derivatives of the profit
function. It is of interest to see what the properties of the profit
function imply about the properties of the net supply functions. Let
us examine the properties one by one.
First, the profit function is a
monotonic function of the prices. Hence, the partial derivative of
7r(p) with respect to price i will
be negative if good i is
an input and positive if good i is
an output. This is simply the sign convention for net supplies that
we have adopted.
Second, the profit function is
homogeneous of degree 1 in the prices. We have seen that this implies
that the partial derivatives of the profit function must be
homogeneous of degree 0. Scaling all prices by a positive factor t
won't change the optimal choice of the
firm, and therefore profits will scale by the same factor t.
Third, the profit function is a
convex function of p. Hence, the matrix of second derivatives of 7r
with respect to p—the Hessian matrix—must be a positive
semidefinite matrix. But the matrix of second derivatives of the
profit function is just the matrix of first
derivatives of the net supply
functions. In the two-good case, for example, we have
(dy,дш
=
dpidp2
I dy2dy2
I "
dp{op2
'
The matrix on the right is just the
substitution matrix—how the net supply of good г
changes as the price of good j
changes. It follows from the properties
of the profit function that this must be a symmetric, positive
semidefinite matrix.
The fact that the net supply functions are the
derivatives of the profit function gives us a handy way to move
between properties of the profit function and properties of the net
supply functions. Many propositions about profit-maximizing behavior
become much easier to derive by using this relationship.
EXAMPLE:
The LeChatelier principle
Let us consider the short-run response of a firm's
supply behavior as compared to the long-run response. It seems
plausible that the firm will respond more to a price change in the
long run since, by definition,