Answer to Question #123363 in Microeconomics for NOOR HAZLIN BINTI MOHD AMIN

Question

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

Assuming K = 1,000 and L = 100, calculate both the average and marginal products of
Q = 50K^(0⋅6) L^0.5

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

Since by the condition of the problem K=1,000 and L=100, then we have

$$Q=50 \times 1000^{0.6}\times 100^{0.5}=31,548$$

We can take the average productivity of a unit of labor as the ratio of the produced product to the amount of labor expended.

$$AQ_L=\frac {Q}{L}=\frac {31,548}{100}=315.48$$

We can take average return on assets is the ratio of output product to fixed assets.

$$AQ_K=\frac {Q}{K}=\frac {31,548}{1000}=31.548$$

Marginal products characterize the effect as volume products obtained from increased resource costs.

$$MQ_L=\frac{\delta Q}{\delta L}=\frac{25K^{0.6}}{L^{0.5}}$$

$$MQ_L=\frac {25 \times 1000^{0.6}}{100^{0.5}}=158$$

$$MQ_K=\frac {\delta Q}{\delta K}=\frac {30 L^{0.5}}{K^{0.4}}$$

$$MQ_K=\frac{ 30 \times 100^{0.5}}{1000^{0.4}}=19$$

A)

$$Q=50 \times 20^{0.6}\times 25^{0.5}=1509$$

B)

$$TC=p_K\times K+p_L \times L=75\times20+50\times25=1500+1250=2750$$

C)

$$\frac {{\delta}^2Q}{\delta L^2}=-0.6$$

$$\frac {{\delta}^2 Q}{\delta L \delta K}=0.9$$

$$\frac {{\delta}^2Q}{\delta K^2}=-18.1$$

$$-0.6 \times (-18.1)-0.9^2=10.05$$

The company is located at the point of maximum.

D)

$$dQ=\frac {\delta Q}{ \delta L}dL+\frac {\delta Q}{\delta K}dK$$

E)

$$\frac {{\delta}^2 Q}{\delta L \delta K}=0.9$$

The company is at the stage of growth.(0.9>0)

F)

if  

$$K=1.15 \times 20=23K=1.15×20=23$$

and

$$L=1.15 \times 25=28.75L=1.15×25=28.75$$

then

$$Q=50 \times23^{0.6}\times28.75^{0.5}=1759$$