Question

2 Long-run production (6 points) Another firm in the same industry, Cake, considers setting up a plant in Canada and is thus evaluating its long-run production possibilities there. Besides using machinery K, it can hire labor L. The production function is q = f(K, L) = K0.5 L0.5 (a) What is the equation of the isoquant, i.e. K as a function of labor (and output)? (2) (b) What is the marginal rate of technical substitution? (2) (c) The manufacturer produces 12 units of its product. How many additional units of capital do you need to substitute for one marginal unit of labor when L 6? How many do you need when you want to substitute for one marginal unit of labor when L = 12? In which case do you need more additional capital?1 (2)

Answer 1

q = K0.5L0.5

(a)= From production function we get

K0.5 = L0.5 / q

Squaring both sides,

K = L / (q2) [Equation of isoquant]

(b)= MRTS = MPL/MPK

MPL = q/L = 0.5 x (K/L)0.5

MPK = q/K = 0.5 x (L/K)0.5

MRTS = MPL/MPK = [0.5 x (K/L)0.5] / [0.5 x (L/K)0.5] = K/L

(c)= When q = 12,

(i) When L = 6,

K = 6 / (12)2 = 6/144 = 1/24

Additional units of capital for one marginal unit of labor = MRTS = (1/24) / 6 = 1/144 (= 0.0069)

(ii) When L = 12,

K = 12 / (12)2 = 1/12

Additional units of capital for one marginal unit of labor = MRTS = (1/12) / 12 = 1/144 (= 0.0069)

(iii) I need equal amount of additional capital in both cases.