5. Leontief...again Suppose that a firm’s fixed proportion production function is given by q(k, l) =...

5. Leontief...again Suppose that a firm’s fixed proportion production function is given by q(k, l) = min(5k, l)

(a) Calculate the firm’s long-run total, average, and marginal cost functions.

(b) Suppose that k is fixed at 10 in the short run. Calculate the firm’s short-run total, average, and marginal cost functions.

(c) Suppose v = 1 (cost of capital) and w = 3 (cost of labor). Calculate this firm’s long-run and short-run average and marginal cost curves.

Solutions

Expert Solution

q = Min(5K, L)

Production always take place along the ray from origin. So, q = L = 5K

L* = q

K* = q/5

Long run total cost function, LRTC = wL* + vK*

LRTC = wq + vq/5

LRTC = q*(w + v/5)

AC = LRTC/q = w + v/5

MC = dLRTC/dq = w + v/5

K = 10

q = min(50, L)

At equilibrium, L = 50 = q

SRTC = 50w + 10v

SRAC = (50w + 10v) / 50

SRAC = w + (0.2v)

SRMC = 0

In long run,

AC = w + v/5 = 3 + 0.2 = 3.2

MC = dLRTC/dq = w + v/5 = 3.2

In short run,

AC = w + (0.2v) = 3 + 0.2 = 3.2

MC = 0

Rahul Sunny answered 2 years ago

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