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^aL^b, where A, a, and b are positive parameters satisfying a + b ≤ 1. Suppose that the unit prices per unit of

Question

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 [latex]Q = F(K, L) = AK^{a}L^{b},[/latex] 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
[latex]min rK + wL s.t. AK^{a}L^{b} = Q[/latex]
Explain why the Lagrangian is convex, so that a critical point of the Lagrangian must minimize costs. (Hint: See Exercise 13.2.8.)

Accepted Answer

The Lagrangian is [latex]\mathcal{L}=r K+w L-\lambda(A K^{a}L^{b}-Q),[/latex] and the first-order conditions are [latex]r = λAaK^{a−1}L^{b}[/latex] and [latex]w = λAbK^{a}L^{b−1},[/latex] implying that λ > 0. From Exercise 13.2.8, we see that − [latex]\mathcal{L}[/latex] is concave, so [latex]\mathcal{L}[/latex] is convex.

Question 14.5.1: Consider a firm that uses positive inputs K and L of capital......

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