In: Economics
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.
q = Min(5K, L)
a.
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
b.
K = 10
q = min(50, L)
At equilibrium, L = 50 = q
SRTC = 50w + 10v
SRAC = (50w + 10v) / 50
SRAC = w + (0.2v)
SRMC = 0
c.
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