Question 14.5.1: Consider a firm that uses positive inputs K and L of capital......
Consider a firm that uses positive inputs K and L of capital and labour, respectively, to produce a single output Q according to the Cobb–Douglas production function Q = F(K, L) = AK^{a}L^{b}, where A, a, and b are positive parameters satisfying a + b ≤ 1. Suppose that the unit prices per unit of capital and labour are r > 0 and w > 0, respectively. The cost-minimizing inputs of K and L must solve the problem
min rK + wL s.t. AK^{a}L^{b} = Q
Explain why the Lagrangian is convex, so that a critical point of the Lagrangian must minimize costs. (Hint: See Exercise 13.2.8.)
The Lagrangian is \mathcal{L}=r K+w L-\lambda(A K^{a}L^{b}-Q), and the first-order conditions are r = λAaK^{a−1}L^{b} and w = λAbK^{a}L^{b−1}, implying that λ > 0. From Exercise 13.2.8, we see that − \mathcal{L} is concave, so \mathcal{L} is convex.