Answer to Question #270278 in Microeconomics for Max

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.

3. Please show the cost function is concave in v.

Suppose now that capital used for producing hockey sticks is fixed at “k1” in the short run.

4. Calculate the firm’s short-run total costs as a function of q, w, v, and k1.

Answer 1

(1)

Given the production function:

$q=2(kl)^\frac{1}{2}$

rental rate, $v=$ $1

wage rate,$w=$ $4

In the short run, $k=100$

hence short run production function, $q=2(100\times l)^\frac{1}{2}$ $=20(l)^\frac{1}{2}$

A firm's short run total cost:

$STC=SFC+SVC$

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

Now, $SFC=v\times k=1\times 100=$ $100 (In the short run, capital is fixed)

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

$STC=SFC+SVC=100+4l$ =$(100+4$l$)

Hence, firm's short run total cost curve is $100+4l.$

Again, firm's short run average cost, $SAC=\frac{STC}{q}$

$SAC=\frac {(100+4l)}{20(l)^\frac{1}{2}}$

$=\frac {5}{l^\frac{1}{2}}+\frac{l^\frac{1}{2}}{5}$

(2)

Firm's short run marginal cost, $SMC=\frac{d(STC)}{dq}$

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

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

$=\frac{4}{(\frac{10}{l^{1}{2}})}$

$\implies SMC=4\times \frac{l^\frac{1}{2}}{10}=\frac{2l^\frac{1}{2}}{5}$

(3)

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

So, $\frac{5}{l^\frac{1}{2}+\frac{l^\frac{1}{2}}{5}}=\frac{2l^\frac{1}{2}}{5}$

$\implies\frac{(25+l)}{5l^\frac{1}{2}}=\frac{2l^\frac{1}{2}}{5}$

$\implies125+5l=10l$

$\implies l=25$

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