Profit Maximization and the Profit Function

Profit Maximization and the Profit Function Juan Manuel Puerta September 30, 2009

Profits: Difference between the revenues a firm receives and the cost it incurs Cost should include all the relevant cost (opportunity cost) Time scales: since inputs are flows, their prices are also flows (wages per hour, rental cost of machinery) We assume firms want to maximize profits. Let a be a vector of actions a firm may take and R(a) and C(a) the Revenue and Cost functions respectively. Firms seek to max a1,a 2,...,a n R(a 1, a 2,..., a n ) C(a 1, a 2,..., a n ) The optimal set of actions a are such that R(a ) a i = C(a ) a i

From the Marginal Revenue=Marginal Cost there are three fundamental interpretations The actions could be: output production, labor hiring and in each the principle is MR=MC Also, it is evident that in the long-run all firms should have similar profits given that they face the same cost and revenue functions In order to explore these possibilities, we have to break up the revenue (price and quantity of output) and costs (price and quantity of inputs).

But the prices are going to come as a result of the market interaction subject to 2 types of constraints: Technological constraints: (whatever we saw in chapter 1) Market constraints: Given by the actions of other agents (monopoly,monopsony etc) For the time being, we assume the simplest possible behavior, i.e. firms are price-takers. Price-taking firms are also referred to as competitive firms

The profit maximization problem Profit Function π(p) = max py, such that y is in Y Short-run Profit Function π(p,z) = max py, such that y is in Y(z) Single-Output Profit Function π(p, w)=max pf (x) wx Single-Output Cost Function c(w, y) = min wx such that x is in V(y). Single-Output restricted Cost Function c(w, y, z) = min wx such that (y,-x) is in Y(z).

Profit Maximizing Behavior Single output technology: p f (x ) x i = w i for i = 1,..., n Note that these are n equations. Using matrix notation pdf(x ) = w where Df(x ) = ( x x 1,..., x x n ) is the gradient of f(x), that is, the vector of first partial derivatives. Geometric interpretation of the first order conditions Intuitive/Economic interpretation of the first order condition Second-order conditions : In the two dimensional case: 2 f (x ) x 0 2 With multi-inputs: D 2 f(x ) = ( 2 f (x ) x i x j ), the Hessian matrix is negative semidefinite.

Factor Demand and Supply Functions For each vector of prices (p, w), profit-maximization would normally yield a set of optimal x Factor Demand Function: The function that reflects the optimal choice of inputs given the set of input and output prices (p, w). This function is denoted x(p, w). Supply Function: The function that gives the optimal choice of output given the input prices (p,w). This is simply defined as y(p, w) = f (x(p, w)) In the previous section (and typically) we will assume that these functions exist and are well-behaved. However, there are cases in which problems may arise.

Problems 1 and 2 The technology cannot be described by a differentiable production function (e.g. Leontieff) There might not be an interior solution for the problem. This is typically caused by non-negativity constraints (x 0) In general, the conditions to handle the boundary solutions in case of non-negativity contraints (x 0) are the following: f (x) p x i w i 0 if x i = 0 f (x) x i w i = 0 if x i > 0 p These are the so-called Kuhn-Tucker conditions (see Alpha-Chiang or appendix Big Varian)

Problems 3 and 4 There might not be a profit maximizing plan Example: If the production function is CRS and profits are positive At first sight this may seem surprising but there are some reasons why this is intuitive: 1) As a firm grows it is likely to fall into DRS, 2) price-taking behavior becomes unrealistic and 3) other firms have incentives to imitate the first one The profit maximizing plan may not be unique (conditional factor demand set).

Properties of the Demand and Supply functions The fact that these functions are the result on optimization puts some restrictions on the possible outcomes. The factor demand function is homogenous of degree 0. Fairly intuitive, if price of output and that of all inputs increase by a x%, the optimal choice of x does not change There are both theoretical and empirical reasons to consider all the restrictions derived from maximizing behavior We examine these restrictions in three ways: 1 Checking FOC 2 Checking the properties of maximizing demand and supply functions 3 Checking the properties of the associated profit and cost functions.

Comparative Statics (single input) Comparative statics refers to the study of how optimal choices respond to changes in the economic environment. In particular, we could answer questions like: how does the optimal choice of input x 1 changes with changes in prices of p or w 1? Using the FOC and the definition of factor demand function we have: pf (x(p, w)) w 0 (1) pf (x(p, w)) w 0

Comparative Statics (single input) Differentiating equation 1 with respect to w we get: pf (x(p, w)) dx(p,w) dw 1 0 And assuming f is not equal to 0, there is a regular maximum, then dx(p,w) 1 dw pf (x(p,w)) This equation tells us a couple of interesting facts about factor demands 1 Since f is negative, the factor demand slopes downward 2 If the production function is very curved then factor demands do not react much to factor prices. Geometric intuition for the 1 output and 1 input case.

Comparative Statics (multiple inputs) In the case of two-inputs, the factor demand functions must satisfy the following first order conditions f (x 1 (w 1, w 2 ), x 2 (w 1, w 2 )) x 1 w 1 (2) f (x 1 (w 1, w 2 ), x 2 (w 1, w 2 )) x 2 w 2 (3) Differentiating these two equations with respect to w 1 we obtain f 11 x 1 w 1 + f 12 x 2 w 1 = 1 (4) f 21 x 1 w 1 + f 22 x 2 w 1 = 0 (5)

and similarly, differentiating these two equations with respect to w 2 x 1 x 2 f 11 + f 12 = 0 (6) w 2 w 2 writing equations (4)-(7) in matrix form, ( ) ( ) x f11 f 1 x 1 12 w 1 w 2 = f 21 f 22 f 21 x 1 w 2 + f 22 x 2 w 2 = 1 (7) x 2 w 1 x 2 w 2 ( ) 1 0 0 1 If we assume a technology that has a regular maximum, then the Hessian is negative definite and, thus, the matrix is non-singular.

Comparative statics (cont.) Inverting the Hessian, ( x1 w 1 x 1 w 2 x 2 w 1 x 2 w 2 ) ( f11 f = 12 f 21 f 22 ) 1 Factor demand functions (substitution matrix) under (strict) conditions of optimization implies: 1 The substitution matrix is the inverse of the Hessian and, thus, negative definite. 2 Negative Slope: xi w i 0 3 Symmetric Effects: xi w j = xj w i These derivations were done for the 2-input case, it turns out that it is straightforward to generalize it to the n-input case using matrix algebra.

Substitution matrix for the n-input case 1 The first order conditions are Df(x(w)) w 0 2 Differentiation with respect to w yields D 2 f(x(w))dx(w) I 0 3 Which means that Dx(w) = (D 2 f(x(w))) 1 What is the empirical meaning of a negative semi-definite matrix? 1 Changes in input demand if prices vary a bit: dx = Dx(w)dw 2 Use the Hessian formula above: dx = (D 2 f(x(w))) 1 dw 3 Premultiply by dw, dwdx = dw(d 2 f(x(w))) 1 dw 4 Negative semi-definiteness means: dwdx 0

Properties of the Profit Function The profit function π(p) results from the profit maximization problem. i. e. π(p) = max py, such that y is in Y. This function exhibits a number of properties: 1 Non-increasing in input prices and non-decreasing in output prices. That is, if p i p i for all the outputs and p i p i for all the inputs, π(p ) π(p) 2 Homogeneous of degree 1 in p: π(tp) = tπ(p) for t 0 3 Convex in p. If Let p = tp + (1 t)p. Then π(p ) tπ(p) + (1 t)π(p ) 4 Continuous in p. When π(p) is well-defined and p i 0 for i = 1, 2,..., n

Proof All these properties depend only on the profit maximization assumption and are not resulting from the monotonicity, regularity or convexity assumptions we discussed earlier. The first 2 properties are intuitive, but convexity a little bit so. Economic intuition in the convexity result This predictions are useful because they allow us to check whether a firm is maximizing profits.

Supply and demand functions The net supply function is the net output vector that satisfy profit maximization, i.e. π(p) = py(p) It is easy to go from the supply function to the profit function. What about the other way around? Hotelling s Lemma. Let y i (p) be the firm s net supply function for good i. Then, y i (p) = π(p) p i for i=1,2,...,n 1 input/1 output case. First order conditions imply, pf (x) w = 0. By definition, the profit function is π(p, w) pf (x(p, w)) wx(p, w) Differentiation with respect to w yields, = p π(p,w) w f (x(p,w)) x(p,w) x(p,w) w [x(p, w) + w x(p,w) w ]

grouping by x(p,w) w f (x(p,w)) x π(p,w) w = [p w] x(p,w) w x(p, w) = x(p, w) The last equality follows from the first order conditions, which f (x(p,w)) imply p x w = 0 There are 2 effects when prices go up: 1 Direct effect: x is more expensive due to the increase in prices. 2 Indirect effect: Change in x due re-optimization. This effect is equal to 0 because we are at a profit maximizing point. Hotelling s Lemma is an application of a mathematical result: the envelope theorem

The envelope theorem Assume an arbitrary maximization problem M(a) = max x f (x, a) Let x(a) be the value that solves the maximization problem, M(a) = f (x(a), a). It is often interesting to check dm(a)/da The envelope theorem says that dm(a) da = f (x,a) a x=x(a) This expression means that the derivative of M with respect to a is just the derivative of the function with respect to a where x is fixed at the optimal level x(a).

In the 1 input/1 output case, π(p, w) = max x pf (x) wx. The a in the envelope theorem is simply (p,w), which means that π(p, w) p = f (x) x=x(p,w) = f (x(p, w)) (8) π(p, w) w = x x=x(p,w) = x(p, w) (9)

Comparative statics with the profit function What do the properties of the profit function mean for the net supply functions 1 π increasing in prices means net supplies positive if it is an output and negative if it is an input (this is our sign convention!). 2 π homogeneous of degree 1 (HD1) in p, means that y(p) is HD0 in prices 3 Convexity of π with respect to p means that the hessian of π ( 2 π p 2 1 2 π p 2 p 1 2 π p 1 p 2 2 π p 2 2 ) = ( y1 y 1 p 1 p 2 y 2 y 2 p 1 p 2 is positive semi-definite and symmetric. But the Hessian of π is simply the substitution matrix. )

Quadratic forms and Concavity/Convexity A function is convex (concave) if and only if its Hessian is positive (negative) semi-definite A function is strictly convex (concave) if (but not only if) its Hessian is positive (negative) definite A negative (positive) definite Hessian is a sufficient second order condition for a maximum (minimum). However, a less stringent necessary SOC are often used: negative (positive) semi-definite.