A firms production function is q5L05 K05 The firms capital i

A firm\'s production function is q=5L^0.5 K^0.5. The firm\'s capital is fixed at ¯K=400 units in the short run. The rental rate of a unit of capital is $9 and the wage rate is $100 in the short run. Derive the firm\'s fixed cost and average fixed cost in the short run. Does the average fixed cost increase when the firm\'s output increases? Derive the firm\'s variable cost, average variable cost, and marginal cost (as a function of output). Derive the firm\'s total cost and average cost (as functions of output, q).

Solution

q = 5L^0.5K^0.5

In short run, K = 400, w = 100, r = 9

q = 5L0.5 x 20 = 100L^0.5

(a)

Total cost, TC = wL + rK = 100L + (9 x 400) = 100L + 3,600

Fixed cost = 3,600

Average fixed cost = FC / Q = 3,600 / 100L^0.5

(b)

From part (a), we find that as Q increases, AFC decreases.

(c)

Since Q = 100L^0.5, We get

L^0.5 = Q / 100

Squaring both sides,

L = Q² / 10,000

From total cost function,

TC = 100L + 3,600 = 100 x ( Q² / 10,000) + 3,600 = ( Q² / 100) + 3,600

VC = 100L = 100 x (Q² / 10,000) = Q² / 100

AVC = VC / Q = ( Q² / 100) / Q = Q / 100

MC = dTC / dQ = 2Q / 100 = Q / 50

Since TC = ( Q² / 100) + 3,600, Average cost (AC) = TC / Q = (Q / 100) + (3,600 / Q)