Answer to Question #123814 in Microeconomics for NOOR HAZLIN BINTI MOHD AMIN

Question

Harpic Corp manufactures plastic broom and mop holders, and the production function for these holders is expressed as :
Q = 30L^(0⋅85) K^0.20

Where, Q = number of plastic holders produced per day
K = units of capital input per day
L = labour input in working hours per day

The firm currently employs 20 units of capital at cost of RM75 per unit and 25 units of labour at a cost of RM50 per unit

a) Based on the current inputs used, compute the level of output

b) Compute the current total costs

c) Given the current input usage, is the first operation efficiently?

d) Derive the equation path equation

e) Does the production function exhibits increasing, constant or decreasing returns to scale? Explain.

f) Determine the percentage increase in output if both labour and capital are each increased by 15%

Answer 1

a) Based on the current inputs used, compute the level of output

The production function is:

$Q = 30L^{0⋅85}K^{0.20}\\[0.3cm] Q = 30(25)^{0.85}(20)^{0.20}\\[0.3cm] Q = 842.5123$

b) Compute the current total costs

The total cost equation is:

$C = 75K + 50L\\[0.3cm] C = 75(20) + 50(25) C = 2,750$

c) Given the current input usage, is the first operation efficiently?

At the optimal production:

$\dfrac{MP_L}{MP_K} = \dfrac{w}{r}\\[0.3cm] MP_L = \dfrac{\delta Q}{\delta L} = 25.5L^{-0.15}K^{0.20}\\[0.3cm] \rightarrow MP_L = 25.5(25)^{-0.15}(20)^{0.20} = 26.6454\\[0.3cm] MP_K = \dfrac{\delta Q}{\delta K} = 6L^{0.85}K^{-0.80}\\[0.3cm] \rightarrow MP_K = 6(25)^{0.85}(20)^{-0.80} = 8.4251$

Therefore:

$\dfrac{MP_L}{MP_K} = \dfrac{26.6454}{8.4251} = 3.1626\\[0.3cm] \dfrac{w}{r} = \dfrac{50}{75} = 0.6667$

Since $\dfrac{MP_L}{MP_K}>\dfrac{w}{r}$ , the firm is not producing the efficient level of output.

d) Derive the equation path equation

The equation path or the expansion path is derived from the cost minimization problem.

$\dfrac{MP_L}{MP_K} = \dfrac{w}{r}\\[0.3cm] \dfrac{25.5L^{-0.15}K^{0.20}}{6L^{0.85}K^{-0.80}} = \dfrac{50}{75}\\[0.3cm] \color{red}{\dfrac{K}{L} = \dfrac{8}{51}}$

e) Does the production function exhibits increasing, constant or decreasing returns to scale? Explain.

For a Cobb-Douglas production function $Q = aL^aK^b$ , the firm experiences increasing returns to scale if $a + b = 1$ , decreasing returns to scale if $a + b<0$ and constant returns to scale if $a + b = 1$ .

Our production function is:

$Q = 30L^{0⋅85}K^{0.20}\\[0.3cm] \rightarrow a = 0.85,\; b = 0.20$

Therefore:

$\rightarrow a + b = 0.85 + 0.20 = 1.05>1$

The production function exhibits increasing returns to scale.

f) Determine the percentage increase in output if both labour and capital are each increased by 15%

$Q = 30L^{0⋅85}K^{0.20}\\[0.3cm] Q_1 = 30(1.15L)^{0.85}(1.15K)^{0.20}\\[0.3cm] Q_1 = 1.1581(30L^{0⋅85}K^{0.20})\\[0.3cm] Q_1 = 1.1581Q$

This means that the quantity has increased by a percentage of:

$\Delta Q = 1.1581 - 1\\[0.3cm] \Delta Q = 0..1581 = 15.81\%$