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Appendix to Chapter Production and Cost Theory— A Mathematical Treatment This appendix presents a mathematical treatment of the basics of production and cost theory As in the appendix to Chapter 4, we use the method of Lagrange multipliers to solve the firm’s cost-minimizing problem Cost Minimization 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 factor in the production process has positive but decreasing marginal products Therefore, writing the marginal product of capital and labor as MPK(K, L) and MPL(K, L), respectively, it follows that MPK(K,L) = 0F(K, L) 0, 0K 02F(K, L) MPL(K,L) = 0F(K, L) 0, 0L 02F(K, L) 0K 0L2 6 A competitive firm takes the prices of both labor w and capital r as given Then the cost-minimization problem can be written as Minimize C = wL + rK (A7.1) subject to the constraint that a fixed output q0 be produced: F(K, L) = q0 (A7.2) C represents the cost of producing the fixed level of output q0 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 constrained optimization problem in three steps using the method discussed in the appendix to Chapter 4: • Step 1: Set up the Lagrangian, which is the sum of two components: the cost of production (to be minimized) and the Lagrange multiplier l times the output constraint faced by the firm: ⌽ = wL + rK - l[F(K, L) - q0] (A7.3) • Step 2: Differentiate the Lagrangian with respect to K, L, and l Then equate the resulting derivatives to zero to obtain the necessary conditions for a minimum.1 These conditions are necessary for a solution involving positive amounts of both inputs 273
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