I need my answers checked for this question, thanks.

Question

A firm's production function:
Q = 100 K^0.5 L^0.5
During the last production period, the firm operated efficiently and used input rates of 100 and 25 for labor and capital respectively.
(a) What is the marginal product of capital and the marginal product of labor based on the input rates specified?
-marginal product of capital=100
-marginal product of labor=25

(b) If the price of capital was $20 per unit, what was the wage rate?
-wage rate=5

(c) For the next production period, the price per unit of capital is expected to increase to $25 while the wage rate and the labor input will remain unchanged. If the firm maintains efficient production, what input rate of capital will be used?
-i need help on this question.
Thanks.

A good problem. Unfortunately, my calculas skills are not what they once were. Fortunately, this particular problem has a particular feature which makes the solution almost obvious (once you see it).

Assume, this firm operates in a perfectly competitive market. So, the price of its output is given. Step 1, determine the price is sells its output. A firm will expand production until the marginal cost of producing an additional unit equals the marginal revenue (price) of that unit. You know that marginal product of labor is 25 unit and the cost of one unit of labor is $5. So, the marginal cost of an additional unit is 5/25 = $0.20. (Similarly, the mp of K is 100 and one K costs 20. So the MC of and additional unit is 20/100= $0.20 again.)

So, the price of the output is $0.20 per unit.

Now calculate total revenue and net profit under the initial conditions. Q=100K^.5L^.5 = 100*(25)^.5 * (100).5 = 5000. Total revenue is 0.2*5000=1000. Total cost is 100*$5 + 25*$20 = 1000. Profit = 0.

Under price of capital increase, the problem becomes:
max(profit) = .2*(100K^.5 L^.5) - 25K - 5L. (here you have a two-variable maximization calculas problem. I am on shaky ground here).

However, one can see the solution ahead of this step. By operating efficiently, given the initial input prices, the firms best solution was to earn zero profits. It barely covered its variable costs. With the increase in the cost of capital, the firm cannot even hope to break even. So, the optimal solution is to shut down.

Of course, this assumes the firm is operates in a competitive environment. If the firm is a monopolist or has some monopoly power, then, I believe, you need more information about the marginal revenue obtained by the firm.

Lotsa luck