A firm has production function f(K; L) = squarerootKL. In the short run, the rm

Question

A firm has production function f(K; L) = squarerootKL. In the short run, the rm has capital K = 400;
this cannot be changed in the near future. The cost of a unit of capital is $20, while the cost of a unit of
labor is $30.
a. In the short-run, where capital is xed at K = 400, how much labor does the rm need to employ in
order to produce q = 300 output? What is the cost of producing 300 output?
b. Repeat part a for q = 400, q = 500 and q = 600.
c. What is the cost of producing q output in the short-run?
d. What is the marginal cost of producing a 301s unit? A 401st unit? If you are comfortable doing so,
you may answer this question by writing down the marginal cost function directly, rather than recalculating
total cost for q = 301 and q = 401.
e. Given your answer to part c, draw a graph with the rm's average total cost, average variable cost, and
marginal cost (hint: MC = Q(3/20) )

Explanation / Answer

a) f(K,L)=(KL)^0.5 K=400 , Q=300 => L = Q^2 / K = 300^2/400 = 225 amount labor = 225/30 = 7.5 = 8 (next integer) cost of producing 300 output = 400 + 8x30 = 640$ b)follow the above procedure c) In the short run K is fixed = 400 cost of production (C) = capital cost + labor cost = 400 + Q^2/K d) MC = dC/dQ = 2Q/K

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