A zero fixed-cost firm has a Cobb-Douglas production function F (L, K)=LK with >0 and >0.

 A zero fixed-cost firm has a Cobb-Douglas production function F (L, K)=LαKβ with α>0 and β >0.

(a) Show that this firm’s conditional factor demand functions can be written as L(q, w, r)=γL (w, r)q1/(α+β)and K(q, w, r)=γK (w, r)q1/(α+β)for some positive values γL (w, r)and γK (w, r).
(b) Show that the firm’s cost function can be written as C (q, w, r)=γC (w, r)q1/(α+β)for some positive value γC (w, r).
(c) Show that this cost function is strictly convex with respect to q if and only if α+β <1. (remember, a function is strictly convex if it’s the second derivative is positive).
(d) Assume α =β < 1/2 and w =r =1. Solve the profit max problem and show that their supply function can be written as qS (p)=(αp)γ for some positive value γ.
(e) Assume α=β =1/2 and w =r =1. Argue that if p >2, then the firm has an “infinite” (or maximum capacity) supply, whereas if p <2, the supply is 0.
(f) What is the relationship between the firm’s supply and the returns to scale of the production function?