Question

Suppose a firm’s production function is q = f(K,L) = (K)1/3 (L)1/3

(a) Set up the firm’s problem and solve for K∗ and L∗ here. Show your work to derive the value of K∗ and L∗ otherwise no marks will be awarded. Note: your solution

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should be:

∗ K = P^3/27r2w L = P^3/27w2r

How much does the firm produce (i.e. what is q∗)? What is the profit earned by this firm (i.e. what is π∗)?

(b) The firm could produce using a different technology: q = g(K, L) = min{.5K, 3L}. Suppose the firm wants to continue to produce the same q∗ from part (a), what is the new K∗ and L∗? How do the profits compare with part (a)? Assuming there is no additional cost, when would the firm want to switch to this technology (Note: Your answer will be a relationship between P, w, and r)?

(c) SupposeK ̄ =1.5,r=4,w=2,andP =6,showthefirm’sshortrunproblem and solve for L∗. No marks will be awarded for just giving the answer. Check your answer using part (a). Derive the short run cost function C(q) = wL(q) + rK ̄ , your solution should be:

4q3 C(q)= 3 +6

What is the short run average cost? What is the short run marginal cost? For what q∗ does the firm make zero profits (round to 2 digits)?

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(d) Rearranging the function from part (a) gives a cost function of C(q) = 2(wr)2 q2 .

Set up the firm’s problem and solve for q∗ and π∗. How does this compare with part (a)? (Bonus marks for showing the derivation of C(q). Helpful hint: Use MRTS and the fact that q = f(K,L) to solve C(q)).

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Answer 1