Answer to Question #272493 in Microeconomics for TIBAWO GOAT

Question

The output of a brick maker can be determined by the following production function:

B=K0.75L0.25

Where:

B =number of bricks produced in a given period

L= amount of labour employed

K= amount of capital employed.

If the price of capital is PK, the price of labour is PL and the firm’s total expenditure is E,

i.State the firm’s optimization problem.

ii.Set up the Lagrangian function based on the information provided.

iii.Obtain the first order conditions for optimization.

iv.Derive the optimum combination of inputs.

v.If the price of capital is P20, the price of labour is P30 and the brick maker’s total expenditure on inputs in P120, calculate the firm’s output maximizing input combination.

vi.How may bricks will the brick layer produce in a given period?

Answer 1

Maximum output

Q = 35K^0.6L^0.8

Q = $35 \times 60^{0.6}*80^{0.8}$

Q = 35* 11.66516*33.30213

Q =   $ 13,596.61

Total Cost (TC) =$P_{K}\times K +P_{L} \times L$

TC = $(95 \times 60) + (110 \times 80)$

TC = 5700 + 8800

TC = 14,500

Objective: Maximise Q = 35K^0.6L^0.8

Subject to: Total Cost (TC) = $P_{K}\times K +P_{L}\times L$

                               TC = 95K + 110L = 14,500

L(λ, k, l) = 35K^0.6L^0.8 - λ(95K + 110L = 14,500)

∂L / ∂K = 21K^-0.4L^0.8 - λ(95) = 0

∂L / ∂L = 28K^0.6L^-0.2 - λ(110) = 0

∂L / ∂ λ = -1(95K + 110L = 14,500) = 0

Ratio of FOC

(∂L / ∂K) / (∂L / ∂L) = λ(95)/ λ(110) = 21K^-0.4L^0.8 / 28K^0.6L^-0.2 

95 / 110 = 21/28K^-0.4-0.6L^{0.8- - 0.2}

0.863636 / 0.75 = K^-1L¹

L = 1.151515K

 

Solving for the equations:

95K + 110L = 14,500

95K + 110*1.151515K = 14,500

95K + 126.6666K = 14,500

221.6666K = 14,500

K = 14,500 / 221.6666

K = 65.41355

L = 1.151515 *65.41355

L = 75.32468