Question

A firm produces toasters using capital (K) and labor (L). The price of capital is r > 0 and the price of labor is w > 0. The quantity Q of toasters produced is given by the function: Q = f(L, K) = L^(1/2) K^(1/3)

(a) What type of returns to scale does the firm have?

(b) Assume that the firm minimizes costs and that all factors are variable

. i. Explain the conditions that hold when the plant produces a fixed number of toasters at least cost. Illustrate your answer with a graph. What shape do you expect the long run cost curve to have? Sketch it and explain.

(c) Assume now that K is fixed at K ̄ = 8.

i. Calculate the plant’s short-run cost function.

ii. Plot the average total cost curve, the average variable cost curve and the marginal cost for w = 4 and r = 5.

iii. Plot the firm’s short run supply of toasters, assuming that the market for toasters is perfectly competitive.

Answer 1

a.) f(L,K)= 1/2 kl 3 f (AL, AK) = (2)! (OK) 3 = 1/2+1/3 f(L,K) = 15/6 f(L,KS thus decreasing Retums to scale Ang & b.) Long Run Cost Min at Eqm. MAIS = w/8. MATS = MPLIMPK = 1/2 ullakl 11 112 k213 = 3K QL. at Eqm 96 ) 3k = who SL 3 k8 - 20L 80 L= 368/2w from puodo fune", q= (38k ) 12 k. 9= 3o) 2w) 12 Sto. Bet = [ (au) Wallets Kch = 9615 aw 345 Am 10 5 ) ( 9615) (20 96 - 945 (95 38

Now Long Run cost cune c = wicht 8 kct 15 = when tok* (aw 315 of C®= 9615 als 6,7l5 [Ele95 +213) 876) ac = 6/5 (A) q115 A. Now a = 6 (A) (1) 15 y so it is convex in shape. opp 1, Cast ↑ at Increasing speed. An Bo2 as C) then K=8, . q = 1/2(8) Q113 q = 12.2. Lc" = (a ) 2 = 9/4 then SR Cost furch (= Whch tok - SR? C= wq2 + 8011 4 - I - 055, 64, C= 92+40. ATC= c/ q = q +40/9) y AUC= 9/9 = q MC= alaq = 29 Jhong (ili X= pq- c. = pq- q2 40. 09 -> p q-29_= 0.. SR ss quove 12 = Ple] Ang

C(ii) graph

ATC, MC, AVC MO -40 ATC -20- AVC 10 Output q 15
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short run supply Curve Price -40 -20- 10 Q output 20

Part b) TC function

Cost Output