Daniel l. Rubinfeld

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246 Part 2 Producers, Consumers, and Competitive Markets Chapter 7 The Cost of Production 247 Step 3. In general, these equations can be solved to obtain the optimizing valli ues of L, K, and A. It is particularly instructive to combine the first hvo conditions in (A7.4), to obtain (A7.5) This appendix presents a mathematical treatrnent of the basics of production and cost theory. As in the appendix to Chapter 4, vve use the method of Lagrange multipliers to solve the firm's cost-minimizing problem. The theory of the firm relies on the assumption that firms choose inputs to the production process that minimize the cost of producing output If there are two inputs, capital K and labor L, the production function F(K, L) describes the maximum output that can be produced for every possible combination of inputs. We assume that each of the factors in the production process has positive but decreasing marginal products. Writing the marginal product of capital as MPK(K, L) = aF(K, L)/aK, we assume that MPK(K, L) > 0 and aMPK(K, L)/aK < O. Similarly, if the marginal product of labor is given by MPL(K, L) = aF(K, L)!aL, we assume that MPL(K, L) > 0 and aMPL(K, L)/aL < O. A competitive firm takes the prices of both labor wand capital r as given. 111en the cost-minimization problem can be written as Minimize C = wL + rK subject to the consh'aint that a fixed output Qo be produced: F(K, L) = Qo (A7.1) (A7.2) C represents the cost of producing the fixed level of output Qo. To determine the firm's demand for capital and labor inputs, we choose the values of K and L that minimize (A7.1) subject to (A7.2). We can solve this cansh'ained optimization problem in three steps using the method discussed in the Appendix to Chapter 4: II Step 1. Set up the Lagrangian, which is the sum of hvo components: the cost of production (to be minimized) and the Lagrange multiplier A times the output constraint faced by the firm: cI:> = wL + rK A[F(K, L) - QoJ (A7.3) II Step 2. Differentiate the Lagrangian with respect to K, L, and A. Then equate the resulting derivatives to zero to obtain the necessary conditions for a minimum:! (lcI:>/aK = r - AMPK(K, L) = 0 acI:>/aL = W - AMPL(K, L) = 0 acI:>/aA = F(K, L) Qo = 0 1 These conditions are necessary for a solution im'olving positive amounts of both inputs (A7.4) Equation (A7.5) tells us that if the firm is 1:1inimizing costs it \:'ill chos.e its factor inputs to equate the ratio of the margmal product of each factor dIvIded bv its price. To see that this makes sense, suppose MPK/r were greater than Ml\/U'. Then the firm could reduce its cost while still producing the same output by using more capital and less labor. Finally, we can combine the first hvo conditions of (A7.4) in a different way to evaluate the Lagrange multiplier: (A7.6) Suppose output increases by one Lmit Because the marginal product of capital measures the extra output associated with an additional input of capital, l/MP.;;(K, L) measures the extra capital needed to produce one unit of output. Therefore, r/MPK(K, L) measures the additional input cost of producing an additional unit of output by increasing capital. Likewise, w/MPL(K, L) measures the additional cost of producing a unit of output using additional labor as an input. In both cases, the Lagrange multiplier is equal to the marginal cost of production, because it tells us how much the cost increases if the amount is increased by one unit. Marginal Rate of Technical Substitution Recall that an iSOqllllllt is a cun'e that represents the set of all input combinations that gi\'e the firm the same level of output-say, Q*. Thus the condition that F{K, L) = Q* represents a production isoquant. As input combinations are changed along an isoquant, the change in output, given by the total derivative of F{K, L) equals zero (i.e., dQ = 0). Thus It follows by rearrangement that (A7.7) (A7.8) where MRTS LK is the firm's marginal rate of technical substitution between labor and capital. Now, rewrite the condition given by (A7.5) to get (A7.9) Because the left side of (A7.S) represents the negative of the slope of the isoquant, it follows that at the point of tangency of the isoquant and the isocost line, the firm's marginal rate of technical substitution (which trades off inputs while keeping output constant) is equal to the ratio of the input prices (which represents the slope of the firm's isocost line).
248 Part 2 Producers, Consumers, and Competitive Markets We can look at this result another way be rewriting (A7.9) again: (A7.10) Equation (A7.10) tells us that the marginal products of all production inputs must be equal 'when these marginal products are adjusted by the unit cost of each input. If the cost-adjusted marginal products ,,,'ere not equal, the firm could change its inputs to produce the saIne output at a lower cost. Duality Production and Cost Theory As in consumer theory, the firm's input decision has a dual nature. The optimum choice of K and L can be analyzed not only as the problem of choosing the lowest isocost line tangent to the production isoquant, but also as the problem of choosing the highest production isoquant tangent to a given isocost line. To verify this, consider the following dual producer problem: subject to the cost constraint that Maximize F(K, L) wL + rK = Co The corresponding Lagrangian is given by (A7.11) ct> = F(K, L) - f.L(wL + rK - Co) (A7.12) where f.L is the Lagrange multiplier. The necessary conditions for output maximization are MP}.:(K, L) jJ.I = 0 MPL(K, L) - jJ.W = 0 wL + rK Co = 0 By solving the first two equations, we see that (A7.13) (A7.14) We assume that 0' < 1 and j3 < 1, so that the firm has decreasing marginal products of labor and capitaL 2 If 0' + j3 = 1, the firm has constallt returns to scale, because doubling K and L doubles F. If 0' + j3 > 1, the firm has illcreasing retums to scale, and if 0' + j3 < 1, it has decreasing retums to scale. As an application, consider the carpet industry described in Example 6.4. The producti~n of ~)oth small and lar:ge firms can~e described by Cobb-Douglas production functions. For small fums, 0: = .7/ and j3 = .23. Because 0' + j3 = 1, there is constant returns to scale. For larger firms, however, 0: = .83 and j3 = .22. Thus 0: + j3 = 1.05, and there is increasing returns to scale. To find the am.ounts of capital and labor that the firm should utilize to minimize the cost of producing an output Qo, \ve first write the Lagrangian Chapter 7 The Cost of Production 249 (A7.15) Differentiating with respect to L, K, and A, and setting those derivatives equal to 0, we obtain From equation (A7.16) we have act>/aL = w - A(j3AKoLf3- 1 ) 0 act>/aK = r - A(O'AKC< lL(3) = 0 act>/aA = AKC
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Daniel l. Rubinfeld

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