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
1) 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).
2) Homogeneous of degree 1 in p. 7r(£p) — £тг(р) for all t > 0.
3) Convex in p. Let p" = fp + (1 - t)p' for 0 < t < 1. Then тг(р") < *7г(р) + (1-*)тг(р').
4)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.
1) 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.
2) 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).
3) 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)
J U(p)
= py-
- w'x'
MP')
^j^\
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. ............
