Answer to Question #271176 in Microeconomics for Max

Question #271176

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.


1
Expert's answer
2021-11-25T10:18:37-0500

(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."


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