Рrofit function

Дата: 21.05.2016
```		PROFIT FUNCTION
Given any production set Y, we have seen how to calculate theprofit function. 7г(р), which gives us the maximum profitattainable at prices p. The profit function possesses several importantproperties that follow directly from its definition. These properties are veryuseful for analyzing profit-maximizing behavior.
Recall that the profit function is, by definition, the maximum profitsthe 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 isimportant is that the objective function in this problem is a linear functionof prices.
Properties of the profit function
We begin by outlining the properties of the profit function. It isimportant to recognize that these properties follow solely from the assumptionof profit maximization. No assumptions about convexity, monotonicity, or othersorts of regularity are necessary.
Properties of the profit function
1) Nondecreasingin output prices, nonincreasing in input prices. If’ p\ > pi for all outputsand p«- < pj for all inputs, then 7r(p») > 7r(p).
2) Homogeneousof degree 1 in p. 7r(£p) — £тг(р) for all t > 0.
3) Convexin p. Letp« = 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 fromthe definition of the profit function alone and do not rely on any propertiesof the technology.
1)Let у be a profit-maximizing net output vector atp, 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 > 0and p\ < pi for all i for which y^ < 0, we also havep«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 outputvector 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’ maximizeprofits 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 twoinequalities 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 theMaximum de scribed in Chapter 27. page 506. I
The facts that the profit function is homogeneous of degree 1 andincreasing in output prices are not terribly surprising. The convexityproperty, on the other hand, does not appear to be especially intuitive.Despite this appearance there is a sound economic rationale for the convexityresult, 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 pricevector (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 sameproduction plan (y*,x.*). Call the profits yielded by this passivebehavior the «passive profit function» and denote it by П(р) = py* — w*x*. This is easily seen to be a straight line. Theprofits from pursuing an optimal policy must be at least as large as theprofits from pursuing the passive policy, so the graph of ir(p) must lieabove the graph of H(p). The same argument can be repeated for any pricep, so the profit function must lie above its tangent lines at everypoint. 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 functionincreases at an increasing rate.
The properties of the profit function have several uses. At this point wewill satisfy ourselves with the observation that these properties offer severalobservable implications of profit-maximizing behavior. For example,suppose that we have access to accounting data for some firm and observe thatwhen all prices are scaled up by some factor t > 0 profits do notscale up proportionally. If there were no other apparent changes in theenvironment, 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 pricefor its output. For simplicity we imagine that the price of output will be piwith probability q and pi with probability (1 — q). Ithas been suggested that it may be desirable to stabilize the price of output atthe average price P — 4P\ + (1 ~ 4)P2- How would this affectprofits of a typical firm in the industry?
We have to compare average profits when p fluctuates to theprofits 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 aswith a stabilized price.
At first this result seems counterintuitive, but when we remember theeconomic reason for the convexity of the profit function it becomes clear. Eachfirm will produce more output when the price is high and less when the price islow. The profit from doing this will exceed the profits from producing a fixedamount 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 theprofit 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 tofind the net supply functions. How can that be done? It turns out that there isa very simple way to solve this problem: just differentiate the profitfunction. 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 supplyfunction 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 beat least as profitable as the production plan y*. However, the plan y* will bea profit-maximizing plan at prices p*, so the function g reaches aminimum value of 0 at p*. The assumptions on prices imply this is an interiorminimum.
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 relationshipsdepicted in Figure 3.1. Since the graph of the «passive» profit linelies below the graph of the profit function and coincides at one point, the twolines must be tangent at that point. But this implies that the derivative ofthe profit function at p* must equal the profit-maximizing factor supplyat 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 isgoing on.
Let us consider the case of a single output and a single input. In thiscase 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-ordercondition. 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, wehave
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 aninput—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 twoeffects. First, there is a direct effect: because of the price increase thefirm will make more profits, even if it continues to produce the same level ofoutput.
But secondly, there will be an indirect effect: the increase in theoutput price will induce the firm to change its level of output by a smallamount. However, the change in profits resulting from any infinitesimal change inoutput must be zero since we are already at the profit-maximizing productionplan. Hence, the impact of the indirect effect is zero, and we are left onlywith the direct effect.
The envelope theorem
The derivative property of the profit function is a special case of amore general result known as the envelope theorem, described in Chapter27, page 491. Consider an arbitrary maximization problem where the objectivefunction depends on some parameter a:
M(a) = max f(x,a).
X
The function M(a) gives the maximized value of the objectivefunction as a function of the parameter a. In the case of the profitfunction 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 theprice.
Let x(a) be the value of x that solves the maximizationproblem. Then we can also write M(a) = f(x(a), a). This simply says thatthe optimized value of the function is equal to the function evaluated at theoptimizing 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 ais given by the partial derivative of / with respect to a, holding xfixed at the optimal choice. This is the meaning of the vertical bar to theright of the derivative. The proof of the envelope theorem is a relativelystraightforward calculation given in Chapter 27, page 491. (You should try toprove the result yourself before you look at the answer.)
Let’s see how the envelope theorem works in the case of a simpleone-input, one-output profit maximization problem. The profit maximizationproblem 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 of7r(p, w) with respect to p is simply the partial derivativeof 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 mustsatisfy certain properties. We have just seen that the net supply functions arethe derivatives of the profit function. It is of interest to see what theproperties of the profit function imply about the properties of the net supplyfunctions. 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 negativeif good i is an input and positive if good i is an output. Thisis simply the sign convention for net supplies that we have adopted.
Second, the profit function is homogeneous of degree 1 in the prices. Wehave seen that this implies that the partial derivatives of the profit functionmust be homogeneous of degree 0. Scaling all prices by a positive factor t won’tchange the optimal choice of the firm, and therefore profits will scale by thesame factor t.
Third, the profit function is a convex function of p. Hence, the matrixof second derivatives of 7r with respect to p—the Hessian matrix—must be apositive semidefinite matrix. But the matrix of second derivatives of theprofit function is just the matrix of first derivatives of the netsupply 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 netsupply of good г changes as the price of good j changes.It follows from the properties of the profit function that this must be asymmetric, positive semidefinite matrix.
The fact that the net supply functions are the derivatives of the profitfunction gives us a handy way to move between properties of the profit functionand properties of the net supply functions. Many propositions aboutprofit-maximizing behavior become much easier to derive by using thisrelationship.
EXAMPLE: The LeChatelier principle
Let us consider the short-run response of a firm’s supply behavior ascompared to the long-run response. It seems plausible that the firm willrespond more to a price change in the long run since, by definition, it hasmore factors to adjust in the long run than in the short run. This intuitiveproposition can be proved rigorously.
For simplicity, we suppose that there is only one output and that theinput prices are all fixed. Hence the profit function only depends on the(scalar) price of output. Denote the short-run profit function by ns(p,z) wherez is some factor that is fixed in the short run. Let the long-runprofit-maximizing demand for this factor be given by z(p) so that thelong-run profit function is given by itl(p)= ^s{Pi z{p))- Finally, let p* be somegiven output price, and let z* = z(p*) be the optimal long-run demandfor the 2-factor at p*.
The long-run profits are always at least as large as the short-runprofits since the set of factors that can be adjusted in the long run includesthe subset of factors that can be adjusted in the short run. It follows that
h(p) = nL(p) — ns(p, 2*) = ns(p, z(p))  —7TS(p, z*) > 0
for all prices p. At the price p* the difference betweenthe short-run and long-run profits is zero, so that h(p) reaches aminimum at p = p*. Hence, the first derivative must vanish at p*. ByHotelling’s lemma, we see that the short-run and the long-run net supplies foreach good must be equal at p*.
But we can say more. Since p* is in fact a minimum of h(p),the second derivative of h(p) is nonnegative. This means that
d2nL(p*) d2ns(p*,z*) >Q
dp2 dp2~
Using Hotelling’s lemma once more, it follows that
dVLJP*) dys(p*,z*) = d2irL{p*)d2ns(p*,z*) >Q
dp dp dp2dp2 ~
This expression implies that the long-run supply response to a change inprice is at least as large as the short-run supply response at z* = z(p*).
Notes
The properties of the profit function were developed by Hotelling (1932),Hicks (1946), and Samuelson (1947).
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