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Treat this problem as an unconstrained maximization problem. A perfectly-competitive firm produces output using three factors of production: capital (K), labor (L), and land (N). The firm’s production function is Q = F(K, L, N) = 9 ln(K + 1) + 5 ln(L + 1) + 4 ln(N + 1).
The firm rents capital for $10 per unit; it hires labor for $25 per unit; and it rents land for $40 per unit. It sells its output for $300 per unit. a. Let π(K, L, N) represent the firm’s profit function, and use the first-order conditions to find the values of K, L, and N that produce a stationary point for the profit function. b. Use the second-order conditions for a maximum to show that your stationary point from part (a) does, in fact, maximize the firm’s profit.
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