Question

You just purchased an existing business that produces solar panels according to the following production function: Q=K^.5L^.5

The business currently has 25 units of capital, which cost $1, and no workers. Your first order of business is to hire workers and begin production. Workers can be hired for $1 each.  

The market price for the panels that you produce just fell to $100. How much profit, if any, will you make if you produce solar panels given the existing capacity (short run)?

Answer 1

Q = K^0.5L^0.5

When K = 25,

Q = (25)^0.5L^0.5 = 5L^0.5

Profit is maximized when VMPL = w, where VMPL = MPL x Output price = (dQ/dL) x $100.

MPL = dQ/dL = 5 x 0.5 / L^0.5 = 2.5 / L^0.5

Therefore,

(2.5 / L^0.5) x $100 = $1

250 / L^0.5 = 1

L^0.5 = 250

Squaring,

L = 62,500

Q = 5 x 250 = 1,250

Revenue (R) = Output price x Q = $100 x 1,250 = $125,000

Total cost (C) = wL + rK = $1 x 62,500 + $1 x 25 = $62,500 + $25 = $62,525

Profit = R - C = $(125,000 - 62,525) = $62,475