Question

A firm producing hockey sticks has a production function given by

q=2√kl

The price of labor is “w”, the price of capital is “v”. For any given level of output “q”:

1. Calculate the firm’s long-run total, average and marginal cost function.

2. Please show the cost function is homogeneous of degree 1 in input prices.

Answer 1

(1)

Production function:

$q=2\sqrt {kl}$

rental rate, $v$ =$1

wage rate,$w=$ $4

In the short run,$k=100$, hence short run production function:

$q=2\sqrt{100\times l}=20\sqrt{l}$

A firm's short run total cost is given by:

$STC=SFC+SVC$

where $SFC$ is the short run fixed cost and $SVC$ is the short run variable cost.

$SFC=v\times k=1\times 100=$ $100.

$SVC=w\times l=4\times l=$ $$4l$

$STC=SFC+SVC$

$=100+4l=$ $$(100+4l)$

Hence, the firm's short run total cost curve is $400+l.$

Again, the firm's short run average cost:

$SAC=\frac{STC}{q}$

$SAC=\frac{(100+4l)}{20\sqrt l}$

$SAC =\frac{5}{\sqrt l}+\frac{\sqrt l}{5}$

The firm's short run marginal cost:

$SMC=\frac{d(STC)}{dq}$

$SMC=\frac{d(100+4l)}{dq}$

$dq=\frac{[\frac{d(100+4l)}{dl}]}{[\frac{dq}{dl}]}$

$=\frac{4}{\frac{10}{\sqrt l}}$

$\implies SMC=4\times \frac{\sqrt l}{10}$

$=\frac{2\sqrt l}{5}$ .

(2)

SMC C=curve intersects SAC curve at the point where $SAC=SMC$ .

So,

$\frac {5}{\sqrt l +\frac{\sqrt l}{5}}=\frac {2\sqrt l}{5}$

$\implies \frac{(25+l)}{5\sqrt l}=\frac {2\sqrt l}{5}$

$\implies 125+5l=10l$

$\implies l=25$

Thus, $SMC$ curve intersects $SAC$ curve at $l=25.$