Question

A firm uses two types of inputs, labor (L) and capital (K), to produce an output, which is sold in a perfectly competitive market. The production function is given by y = f(L, K) = L 1 6 K 1 6 for all L, K ≥ 0. The price of labor is w > 0 and the price of capital is 1. Each unit of the output is sold at price p > 0. First, we consider the short-run decision problem for the firm, with capital fixed at K = 1.

(a) Find the short-run cost function cs(y, K).

(b) Find the short-run average cost, short-run average variable cost, and short-run supply function. Sketch their graphs in a diagram.

(c) From now on, we consider the long-run decision problem. Find the longrun cost function c(y).

(d) Find the long-run average cost and long-run supply function. Sketch their graphs in a diagram.

Answer 1

15-1, Р q= LY6K%6, W. r=1, q=Ll6 (1) % R al Now [*= 16 9 80 SR cost funco C = COLA tok c=w (9)6 + IF Ang b) Now SRAC= cla = wq5 + 119 SRAUC= wq5 MC: a139 = 6695. 80 SS cwewe, q= Plow) 15 . c) Long Run 2 at eam MRTS = MPL IMPK = w/ KIL = w/y. k=WLlo brom puodo funch, q = 1/6 (WL) 1/6 q= L (olo) 16 L = [q (rw)o] 3 L93 (0)42 K= 98 (wlo) 12 C(y)= WLtok = q3 2(wo)l2 12 LRACE Cly/y = y22 Jwr= 29² Jur 80 MC= ALR gaq = 692 Jug 80 supply funen 6Jwr ) Any Cow Touth wal 143

Short run Short run supply cost AC AVC Q

Long run supply curve cost LRAC Q