Suppose that the firm is paid a positive price p for each

Question

Question: (a) A firm uses capital K, labour L, and land T to produce Q units of a commodity, where

Q = K^2/3 + L^1/2 + T^1/3

Suppose that the firm is paid a positive price p for each unit it produces, and that the positive prices it pays per unit of capital, labour, and land are r, w, and q, respectively. Express the firm's profits as a function π of (K, L, T ), then find the values of K, L, and T (as functions of the four prices) that maximize the firm's profits (assuming a maximum exists).

(b) Let Q∗ denote the optimal number of units produced and K∗ the optimal capital stock. Show that ∂Q∗/∂r = -∂K∗/∂p.