Question

Harpic Corp, manufactures plastic broom and mop holders, and the production function for these holders is expressed as

Q=30L^0.85K^0.20
where, Q=no of plastic holders produced per day
K= units of capital input per day
L= labor input in working hours per day
Assuming K= 1000 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 a 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 firm operating 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

If K=1000, L=100, then

$$Q=50 \times 1000^{0.5}\times 100^{0.5}=15811.4$$

The average productivity of a unit of labor is the ratio manufactured product in the amount of labor expended:

$$AQ_L=\frac {Q}{L}=\frac {15811.4}{100}=1581.14$$

Average return on assets is the ratio of output product to fixed assets:

$$AQ_K=\frac {Q}{K}=\frac {15811.4}{1000}=15.8$$

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

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

$$MQ_L=\frac {25 \times 1000^{0.5}}{100^{0.5}}=79$$

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

$$MQ_K=\frac{ 25 \times 100^{0.5}}{1000^{0.5}}=7.9$$

If K=20 and $p_K=75$, L=25 and $p_L=50$

a)

$$Q=50 \times 20^{0.5}\times 25^{0.5}=1118$$

b)

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

c)

$$\frac {{\delta}^2Q}{\delta L^2}=-0.44$$

$$\frac {{\delta}^2 Q}{\delta L \delta K}=0.56$$

$$\frac {{\delta}^2Q}{\delta K^2}=-0.7$$

$$-0.44\times (-0.7)-0.56^2=-0.0056$$

The company does not work effectively.

d)

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

e) The company is at the stage of growth.(0.56>0)

f) if $K=1.15 \times 20=23$ , $L=1.15 \times 25=28.75$

$$Q=50 \times23^{0.5}\times28.75^{0.5}=1285.7$$