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
Vi\V = —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*)
—£ = —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.
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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.