Question

Hannah and Sam run Moretown Makeovers, a home remodeling business. The number of square feet they can remodel in a week is described by the Cobb-Douglas production function Q=F(L,K) Q=10L0.5K0.5, where L is their number of workers and K is units of capital. The wage rate is $200 per week and a unit of capital costs $200 per week. Suppose that when initially producing 100 square feet a week, they use 10 units of capital. a. What is their short-run cost of remodeling 200 square feet per week? b. What is their long-run cost of remodeling 200 square feet per week? (ELI5, please)

Answer 1

a. The Cobb-Douglas production function is

Q= 10L^0.5 K^0.5 ........(i)

Wage rate (w) = $200 and rate of capital (r) = $200

The Cobb douglas equilibrium condition is

    MPL/MPK= w/r

or, 10*0.5K^0.5 /L^0.5 /10*0.5L^0.5 /K^0.5 =200/200=1

or, K/L=1 or, L=K

Putting the value of L in (i) we get,

Q= 10L^0.5 L^0.5 = 10L

or, Q= 10L

Again 10 units of capital (K) is used for remodeling 100 sq. ft per week. Then K=L=10

Therefore, for remodeling of 200 sq.ft per week, the total units of capital or workers required is

10L=200 or, L=20 and K=20

Therefore, short run cost of remodeling 200 sq. ft per week = wL+rK = 20*200+20*200= $8000

b) In long run there is no fixed factor and therefore, fixed costs.

Therefore, long run cost of remodeling 200 sq. ft per week = wL= 20*200= $4000