Question

The production of golf balls takes the following form:
Q = 3(KL) 1/3
With the price of capital (K) being $120 per day and the price of labor $30 per day,
a) What is the minimum cost of producing 10,000 packs of golf balls?
b) With the budget of the production manager set at $60,000, what combination of inputs
must be purchased to produce the maximum level of output?
c) If the price of a golf ball is $50 per pack, what is the optimum combination of K and L
that maximizes profit or minimizes loss?

Answer 1

Production function is Q = 3(KL)^(1/3)

Rental price r = $120, wage rate w = $30

MRTS = MPL/MPK

= 3*(1/3)K^(1/3)L^(-2/3) / 3*(1/3)*K^(-2/3)L^(1/3)

= K/L

Wage rental ratio = w/r = 30/120 = 0.25

At the optimal choice of inputs MRTS = w/r

K/L = 0.25

K = 0.25L

Production function becomes Q = 3(0.25L^2)^(1/3)

L = 2(Q/3)^1.5

K = 0.5(Q/3)^1.5

These are the demand functions for labor and capital

Cost function is C = 30*(2(Q/3)^1.5) + 120*(0.5(Q/3)^1.5)

C = 120(Q/3)^1.5

a) C = 120(10000/3)^1.5 = 23,094,010.80

b) C = 60000 so we have 60000 = 120(Q/3)^1.5

Q = 188.988

Then K = 10000 and L = 250.

c) Profit = revenue – cost

= 50Q - 120(Q/3)^1.5

Profit is maximum when 50 – 120*1.5*(Q/3)^0.5 = 0

This gives Q = 0.231481

Then K = 0.010716 and L = 0.0428669