( Theory of the Firm) Profits University produces student credit hours (y) with
ID: 1232137 • Letter: #
Question
( Theory of the Firm) Profits University produces student credit hours (y) with two inputs: Professors" hours of work (X1) and TAs' hours of work (x2;) according to the production function f(x1,x2)=10x1 .5 x2.25. Both inputs arc variable Suppose professors are paid $80 per hour and TAs are paid $2.50 per hour. Profits receive $40 per credit hour. Compute the two first order conditions for profit maximization What do they mean? Compute the inputs and outputs that will maximize Profit's profits. What is the maximum profit that Profits can earn?Explanation / Answer
y=10x1^.5x2^.25 Suppose a firm produces output y from two inputs x1 and x2. The price of the output is p and the prices of the two inputs are w1 and w2. Let f(x1, x2) b e the production function. Let f1 = ?f(x1,x2) ?x1 and f2 = ?f(x1,x2) ?x2 denote the marginal products. Assume that all prices are positive and so are both the marginal products. The firms’ profit maximization problem can then be written as follows: Choose y, x1, x2 to maximize py - w1x1 - w2x2 subject to y = f(x1, x2). (1) Since prices are positive, and marginal products are positive, the constraint will always be binding, and the problem can be rewritten as: Choose x1, x2 to maximize pf(x1, x2) - w1x1 - w2x2. (2) If x* 1, x* 2 solve problem (2), the first order conditions are: pf1(x* 1, x* 2) = w1, pf2(x* 1, x*2) = w2. Notice that these two equations should provide a solution to (x* 1, x* 2) as functions of p,w1, w2 – the input demand functions. And once we have found (x* 1, x* 2) we can plug these into f to find the optimal supply of y, and also compute the maximized level of profit. Cost Minimization: An intermediate step in profit maximization Now consider a different problem, that of minimizing the cost of producing the level of output y, i.e., Given y, choose (x1, x2) to minimize w1x1 + w2x2 subject to f(x1, x2) = y. (3) This determines the input levels which minimize the cost of producing y units of the output. Let c(y;w1, w2) denote the result of this minimization exercise. Thus c(y;w1, w2) refers to the minimum cost of producing y. Of course, this is not the 1 primary interest of the firm but cost minimization can be seen as one auxiliary part of the more important problem of profit maximization. Having solved problem (3), problem (1) can now be rewritten in the following form: choose y to maximize py - c(y;w1, w2). (4) Clearly, the first order condition for problem (4) is: p = dc(y;w1, w2) dy , in other words, if y is the profit maximizing level of output then the marginal cost (at y) must equal the price of the output. Problem (4) is an alternative way of writing problem (1). It seems like a longwinded way of stating (1) since it goes through the intermediate step of first solving the optimization problem (3). The advantage of this approach is that even if there are several inputs, (4) is a problem that involves only one variable – y. And if our primary interest is in studying the optimal level of y, this is the most convenient formulation of the firm’s problem. It allows us to see the problem of choosing the profit maximization level of y in a two-dimensional graph. And from this analysis, it is very easy to derive the firm’s supply curve of y. Clearly, if there is only input, then the cost minimization step should be trivial, and the two approaches should then be very similar. If there is only one input, then there will be, according to the production function, a unique level of x1 which results in output y. And this is simply the inverse of the production function, f-1(y). The cost function is simply c(y;w1) = w1f-1(y), and problem (4) becomes: choose y to maximize py - w1f-1(y). (5) The corresponding first order condition is p = w1 dx1 df which is the same as the first order condition corresponding to (1) in the one input case. 2 Conditions for Profit Maximization Suppose the input prices are fixed, and given these prices the (long-run) cost curve is c(y). Suppose c(y) is well-behaved in the sense that it exhibits increasing returns to scale at low levels of output and decreasing returns at high levels of output. The rules for profit maximization are then the following: (1) Find y* such that p = c(y*); the first order condition. (2) Make sure that c(y*) > 0; second order condition. (3) Check that py* - c(y*) = 0. Otherwise set y = 0; global condition. In other words, the rules are as follows. Find a level of output at which price equals marginal cost and marginal cost is rising. This is the profit maximizing level of output if the resulting profits are non-negative. Otherwise, the profit maximizing level of output is 0. Notice that y* satisfying rules 1 and 2 has the property that it provides the maximum profit among all output plans that involve positive production. Comparing the profit of y* to the profit at 0 is then the only other condition left to be checked. This comparison can be rewritten as the condition: minAC = p. These three conditions immediately allow us to derive the supply curve of the competitive firm. The Short-Run This analysis can now be extended to the short-run problem of maximizing profits. Notice that the long-run cost function c(y) subsumes within it the optimization problem of cost minimization. In particular, there are optimal demands for x1 and x2 which are functions of y (we are assuming w1 and w2 fixed). Thus c(y) = w1x1(y)+w2x2(y). For a certain level of output, ¯y, let ¯x2 = x2(¯y). In other words, ¯x2 is the long-run (optimal) demand for input 2 if the output level is ¯y. Suppose input 2 is fixed in the short run while input 1 is variable. Deriving the short-run cost (when there are 2 inputs) is an easy exercise. Suppose input 2 is fixed at level ¯x2. Then corresponding to any y there is a unique level x1 such that f(x1, ¯x2) = y. The short run cost, cs(y, ¯x2) can now be written simply as cs(y, ¯x2) = w1x1 + w2¯x2 where f(x1, ¯x2) = y. Note that there is a short-run cost function corresponding to each level at which input is fixed. There must be a relationship between short-run and long-run costs. 3
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