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)
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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