Question

12.5 Unanswered 2 attempts left A firm's production function is x = k0.541.5. We have p=500,w=10 and r=10. In the short-run if capital is fixed at k=27, how much labor does the firm hire to produce 27 units of output? Enter a number only, round to two decimal places. If the answer is op enter 999. If the answer is a range (any range) e.g. [0,2] enter 111. Type your response

Answer 1

We are supposed to do only these many questions. For solution to others please post as separate question.

12.5)

Production function: x = k^0.5l^1.5

Given k = 27 and output x = 27

Use the production function to solve for l we get

27 = (27)^0.5(l)^1.5

(27)^0.5 = l^1.5, solve for l we get

l = (27)^1/3 = 3

12.6)

To obtain the labor that the firm will hire in the long run the firm will maximize the profit.

Π = 500(k^0.5l^1.5) – 10l -10k

Differentiate wrt to l and k respectively to obtain the first order condition.

dΠ/dl = 500*1.5*k^0.5l^0.5 -10 = 0---------------------------------1)

dΠ/dk = 500*0.5*k^-0.5l^1.5 -10 = 0----------------------------------------2)

Divide equation 1) by 2) we get

3k = l , put l = 3k in the production function we get,

x = (3)^1.5k²

[x/(3)^1.5]^0.5 = k

l = (3)^0.25x^0.5