Cost Minimization

Chapter 2 Cost Minimization 2.1 Introduction In this chapter we begin the study of the firm's market behavior. We will begin by studying how a firm behaves in competitive factor markets. We will derive the set of conditional or derived factor demand func-tions. Conditional demand functions are functions of factor prices and a target level of output. Notice that regardless of the behavior of the firm in the goods market, it will always strive to minimize its cost in the factor market (question: are monopolies more or less efficient than competitive firms). Therefore, the set of conditional demands does not depend on the price of the output. The cost function is the maximum value function we derive from the from the cost minimization problem. It is also a function of factor prices and target output. In this chapter we will assume that the firm produces its output with two variable factors: labor and capital. That is, the firm can select the amounts of both capital and labor that it uses in production. We will see later on that we can interpret all the results of this chapter as the long run behavior of the firm. The cost minimization problem is, mathematically speaking, a problem in constrained optimization. The firm wishes to minimize the cost of pro-ducing a certain level of output, but it is constrained by its technological possibilities, as summarized by the production function. Consequently, we will obtain a Lagrange multiplier as part of the answer. The Lagrange mul-tiplier, u", is the shadow value of the constraint. In this case, u tells us the increase in the firm's cost that would arise if we expand output just a little bit. This quantity is known as marginal cost. Finally we will study the properties of the cost function and of condi-tional factor demands. 2.2 Optimal Choice 2.2.1 The Firm's Cost Minimization Problem As we stated in the introduction, firms seek to minimize cost subject to the constraint that they produce y units of output. That is, the firm's cost minimization problem, CMP), is given by: min wlrk s.t. f (k,l) 120 = y k≥0 In general we will ignore the non-negatvity constraints on factor inputs by arguing that the firm is not endowed with any factors and so must be a net buyer in factor markets. We will rule out corner solutions by plac-ing sufficient structure on the production function (basically that marginal products of both factors are always strictly positive. These conditions are due to Inada.). Notice that in the [CMP) parameters are given by factor prices, w and r, and the target level of output, y. If we were given values for these parameters we could obtain the actual quantities demanded. Since we are using arbitrary quantities, our solution will be quantities of I and kas functions of factor prices and target output levels: l (rw,y) and k" (r, w, y). This set of demand functions is known as conditional or derived de-mand functions. Notice that conditional demand functions are functions of target output y. This means that as we vary the relative factor cost, the firm's optimal quantities will always be on the same isoquant. 2.2.2 Solving the Firm's Problem As we saw in Chapter xxxx, we can solve a constrained optimization prob-lem either via the substitution method or the method of Lagrange multi- b. Find the conditional demand functions. Label them l" (w,r.y) and k" (w,r,y). c. Find the cost function: C (w.r.y). What is particular about the expression you obtained. That is discuss the economic intuition of your result. d. Find ". What is its interpretation? e. Find and show that it is equal to p f. Explain why the expression that you obtain makes intuitive sense. Exercise 2.8.3 Suppose that the firm's utility function is given by: f (k.l) = kl. Also suppose that factor prices are given by wand and that the target output parameter is given by y. Denote the Lagrunge multiplier by p. a. Write down the agent's cost minimization problem. Be very precise. b. Find the conditional demand curves, I" (w,r,y) and r" (w, r, y)and show that they are homogeneous of degree 0 in w and r. c. Find μ". d. Find the cost function, C (w.r.y). e. Show that the cost function is homogeneous of degree 1 in w and r. What is the intuition? f. Show that≥0 and 20. What is the intuition? dC dr d g. Find. dy h. Show that => 0. What is the intuition? dy i. Show that the expenditure function is concave in w. Do this graphi-cally and mathematically. What is the intuition?