Question

1.A price-taking firm Bikes Inc makes and sells bicycles. Bikes Inc uses the production function q=(k+2l)^(1/3). Input prices are w=2 and v=3  for labor and capital respectively. Given that the firm uses initial amount of capital k=12 .

1. compute the firm's short-run cost function. To verify that your have correct short-run cost function, calculate the short-run cost of producing 8 bicycles.
2. Consider that the firm uses initial amount of capital k = 12 , and the firm faces the price P = $ 108. What is the optimal output to produce for Bikes Inc.?

Answer 1

a)

q = (k + 2l)^1/3

w = 2

v = 3

q = (k + 2l)^1/3

k = 12

q = (12 + 2l)^1/3  

so short run production function is q = (12 + 2l)^1/3  

q = (12 + 2l)^1/3   

k = 12

q³ = 12 + 2l

q³ - 12 = 2l

q³/2 - 12/2 = l

l = q³/2 - 6  

short run cost function

c = wl + vk

c = 2( q³/2 - 6) + 3(12)

c = q³ - 12 + 36

c = q³ + 24

Thus, short run cost function is c(q) = q³ + 24

c(q) = q³ + 24

q = 8  

c = (8)³ + 24  

c = 512 + 24

c = 536

b)

P = 108

since firm is price taking therefore optimal output is given by

P = MC

c(q) = q³ + 24

c'(q) = MC = 3q²  

MC = 3q²  

P = MC  

108 = 3q²  

36 = q²  

q² = 36

q² - 36 = 0

(q + 6)(q -6) = 0

q = 6 or - 6

sufficient condition or SOC is

At optimal level of output MC must be increasing  

MC = 3q²  

dMC/dq = 6q

at q = 6

dMC/dq = 6(6) = 36 > 0

at q = - 6

dMC/dq = 6(-6) = - 36 < 0

Thus, optimal level of output is q = 6