Answer to Question #121230 in Microeconomics for Ali

Question

QUIZ PRODUCTION

Ikami Bhd, a manufacturer of home furniture, has the following production function:
Q = 35K0.6 L0.8
Q = units of home furniture produced
L = units of labor
K = unit of capital
The current input mix employed by Ikami Bhd is 60 units of capital and 80 units of labor at the price is RM95 and RM110 per unit of capital and labor respectively.

a) Using the current input mixed, calculate the maximum output Ikami can produce and the total cost it has to pay for this input mixed. (3 marks)
b) Derived the expansion path for Ikami Bhd. (5 marks)
c) Is Ikami Bhd operating efficiently with the current input mixed? (3 marks)
d) Determine the least cost input combination given a budget of RM15 000. (4 marks)
e) Based on existing production function and input mixed, what would be the effect on output level if labor input increase by 5% and capital decrease by 10%? (3 marks)
f) Determine the return to scale for Ikami Bhd. Justify your answer. (2 marks)

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

c) Ikami Bhd is not operating efficiently with the current input mixed since the input combination of (60units of capital and 80units of labor) is not optimized at the total cost of 14,500RM. 

d)Budget (TC)= 15,000RM

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

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

                               TC = 95K + 110L = 15,000

L(λ, k, l) = 35K^0.6L^0.8  - λ(95K + 110L = 15,000)

∂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 = 15,000) = 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 = 15,000

95K + 110*1.151515K = 15,000

95K + 126.6666K = 15,000

221.6666K = 15,000

K = 15,000 / 221.6666

K = 67.66919

L = $1.151515 \times 67.66919$

L = 77.92209

e)Production function: Q = 35K^0.6L^0.8

Increase by 5%: L = 105% = 1.05L

Decrease by 10%: K = 90% = 0.9K

Change in production function: Q = $35\times (0.9K)^{0.6} \times (1.05L)^{0.8}$

Q = $35\times 0.93874\times 1.039804K^{0.6}L^{0.8}$

Q = 34.1637K^0.6L^0.8

Q = 34.1637*(60^0.6)*(80)^0.8

Q = 34.1637* 11.66516*33.30213

Q =    $ 13,271.73

Change in output level:

Initial output level: $ 13,596.61

New output level: $13,271.73

Change = 13,271.73 - 13,596.61

Change = -324.88

The output level decreases by 324.88 units. 

f)Return to scale = -324.88/ 13,596.61

Return to scale = -2.39%