Question

Let K be the level of capital in the short run. The prices of capital and labor are r = 10 and w-5, respectively. For each of the following production functions, (1) derive the short-run cost function (5 pts), (2) derive the long-run cost function (5 pts), and (3) illustrate how you derive the long-run cost function with isoquants and isocosts (5 pts). Hint: for the long-run cost function, first set up the cost minimization problem. (a) (15 pts) f(K, L) = K1/2L1/4

Answer 1

Q = f(K, L) = K^1/2L^1/4

(1) When K = 1,

Q = L^1/4

L = Q⁴

Short run total cost (STC) = wL + rK = 5Q⁴ + (10 x 1) = 5Q⁴ + 10

(2) In long run, cost is minimized when MPL/MPK = w/r = 5/10 = 1/2

MPL = $\partial$ Q/$\partial$L = (1/4) x K^1/2 / L^3/4

MPK = $\partial$ Q/$\partial$L = (1/2) x L^1/4 / K^1/2

MPL/MPK = [(1/4) x K^1/2 / L^3/4] / [(1/2) x L^1/4 / K^1/2] = (1/2) x (K/L) = 1/2

K/L = 1

K = L

Q = L^1/2L^1/4

L^3/4 = Q

L = Q^4/3

K = L = Q^4/3

Substituting in long run cost function (LTC),

LTC = wL + rK = 5 x Q^4/3 + 10 x Q^4/3 = 15 x Q^4/3

(3) Graphically, long-run optimal input mix is obtained at the tangency point between isoquant and isocost lines. In following graph, AB is the isocost line and Q0 is the isoquant which is tangent to AB at point E with optimal amount of capital and labor being K0 and L0 respectively.

Ko Lo