Assignment
Q1. Sarah's Lawn Mowing Service is a small firm that takes prices as given. The prevailing market price for lawn mowing is $20 per acre. Although Sarah can use the family mower for free, she has other costs given by TC = 0.1q2 + 10q + 50, where q is the number of acres Sarah chooses to mow in one week.
a) Do these TC refer to the short run or the long run? Why? What is Sarah's marginal cost?
b) How many acres should Sarah choose to mow in order to maximize her profit?
c) Calculate Sarah's maximum weekly profit.
d) Graph average total cost and marginal cost for q=0, 10, 20, 30, 40, 50, and 60. Label Sarah's supply curve and show her maximum profits.
Q2. Consider again the profit-maximizing decision of Sarah's Lawn Mowing Service from the problem above. Suppose Sarah's father decides to charge her for the use of the family lawn mower.
a) If Sarah's father charges her $100 per week, how will this affect Sarah's total costs? Her marginal costs? The acres of lawn Sarah decides to mow? What will her profits be now?
b) Suppose instead that Sarah's father requires her to pay 50% of weekly profits for the use of the family mower. How will this affect Sarah's profit-maximizing decision?
c) Suppose now that Sarah's father imposes a charge of $2 per acre. How will this affect Sarah's marginal cost function? Her profit-maximizing decision? What will her profits be now? How much will her father get?
d) Finally, suppose Sarah's father decides to collect his $2 per acre by collecting 10% of the revenue from each acre Sarah mows. How will this affect Sarah's profit-maximizing decision? Explain why you get the same result here as in part c.
Q3. Sam's Pizza, the local pizza shop, has hired you as a consultant to help it compete with national chains in the area. Because most business is handled by these national chains, Sam's Pizza operates as a price taker. Using historical data on costs, you find that short run total costs each day are given by STC = 10 + q + 0.1q2, where q is daily pizza production.
a) What is Sam's short-run supply curve? Write its algebraic expression.
b) Does Sam have a shutdown price? That is, what is the lowest price at which Sam will produce any pizza?
c) Sam tells you that his short-run (total) average cost (SAC) reaches a minimum at q=10. How would you verify his claim without using calculus?
d) Sam also claims that any price for pizza of less than $3 will cause his shop to lose money. Is he correct? Explain.
e) Currently, the price of pizza is low ($2) because one major pizza chain is having a sale. Does this price cover Sam's average costs? Would you recommend Sam cease operations until the sale is over? Why or why not?
Q4. Suppose the production function for high-quality wine is given by q = √KL, where q is the output of wine per week (in bottles) and L is labor hours per week. In the short run, K is fixed at 100.
a) What is the short-run production function?
b) If capital rents for $10 and wages are $5 per hour, what is the algebraic expression for the total cost function? For the short-run total cost function? Write short-run total costs as a function of q. Hint: use your answer in (a).
c) Given the short-run total cost curve in (b), short-run marginal costs are given by SMC = 0.1q. How much will the firm produce at a price of $20 per bottle of wine? How many labor hours will be hired per week?
d) Suppose that, during recessions, the price of wine falls to $15 per bottle. Whith this price, how much would the firm choose to produce? How many labor hours would be hired?
e) Suppose that the firm believes that the fall in the price of wine will last for only one week, after which it will wish to return to the level of production in (b). Assume also that, for each hour that the firm reduces its workforce below that described in (b), it incurs a cost of $1 (think of it as severance costs). If it proceeds as in part (d), will it earn a profit or incur a loss? Explain.