Suppose the production function for high-quality brandy is given by

q = √K ∙ L

Where q is the output of brandy per week and L is labor hours per week. In the short run, K is fixed at 100, so the short-run production function is

q = 10 √L

a. If capital rents for $10 and wages are $5 per hour, show that short-run total costs are

STC = 1,000 + 0.05q2

b. Given the short-run total cost curve in part a, short-run marginal costs are given by SMC = 0.1q

With this short-run marginal cost curve, how much will the firm produce at a price of $20 per bottle of brandy? How many labor hours will be hired per week?

c. Suppose that, during recessions, the price of brandy falls to $15 per bottle. With this price, how much would the firm choose to produce, and how many labor hours would be hired?

d. Suppose that the firm believes that the fall in the price of brandy will last for only one week, after which it will wish to return to the level of production in part a. Assume also that, for each hour that the firm reduces its workforce below that described in part a, it incurs a cost of $1. If it precedes as in part c, will it earn a profit or incur a loss? Explain.