CHAPTER • The Cost of Production 275 Because the left side of (A7.8) 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) We can look at this result another way by rewriting (A7.9): MPL/w = MPK/r (A7.10) Equation (A7.10) is the same as (A7.5) and 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 Duality in 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 Suppose we wish to spend C0 on production The dual problem asks what combination of K and L will let us produce the most output at a cost of C0 We can see the equivalence of the two approaches by solving the following problem: Maximize F(K, L) subject to wL + rL = C (A7.11) We can solve this problem using the Lagrangian method: • Step 1: We set up the Lagrangian ⌽ = F(K, L) - o(wL + rK - C0) (A7.12) where μ is the Lagrange multiplier • Step 2: We differentiate the Lagrangian with respect to K, L, and o and set the resulting equation equal to zero to find the necessary conditions for a maximum: 0⌽ = MPK(K, L) - or = 0K 0⌽ = MPL(K, L) - ow = 0L 0⌽ = wL - rK + C = 0l (A7.13) • Step 3: Normally, we can use the equations of (A7.13) to solve for K and L In particular, we combine the first two equations to see that o = MPK(K, L) r MPL(K, L) w MPK(K, L) MPL(K, L) = w r o = (A7.14)