1. A firm’s production function for good X is given by
q = 100(KL)0.5
(where q is the firm’s output, K is the quantity of capital, and
L is the quantity of labor)
a. Assume that q = 1000. Derive an equation for the isoquant related to this level of output and draw this isoquant on the a graph.
b. Discuss how the marginal product of each factor changes (in relative terms) as you move along this isoquant, and provide an explanation for these relative changes.
c. If K = 10, how much labor is needed to produce 1000 units of output?
d. Given your answer in part c, what is the average product of labor?
e. Assume that technological change causes the production function
to become q = 200(KL)0.5. How is the isoquant from part a affected?
2. John sells CDs with pirated music from his Dad’s computer. It’s a small business and we’ll assume that many other college students do the same thing, so John must operate as a price taker. The current (market) price for a full CD is $20.
In a given time period:
a. How many CDs should John sell at the current market price to
maximize his profits?
b. What are John’s profits from part a?
Up until now, John’s father didn’t know about John’s little underground business. John’s father wants to make some money himself, but the only problem is that he isn’t sure about how to charge John for the use of the computer.
Figure out how your results in parts a and b change if his father decides to use one of the following "payment plans":
c. Just to access the computer, John must pay a flat fee of $100 at the start of each time period.
d. John must pay 50% of his profits to his father.
e. John must pay a $2 fee to his father, whenever he sells a CD.
3. Gasoline is sold through local gas stations under perfectly competitive conditions. All gas station owners face the same long run average cost curve given by
And the same long run marginal cost curve given by
where q is the quantity of gas sold each day.
a. Assuming the market is in long run equilibrium, how much
gas will each individual owner sell per day? What are the long run
average and marginal costs associated with this output level?
b. The market demand for gasoline is given by
Where QD is the quantity of gas demanded each day and P is the price of gas. Given your answer in part a, what is the long run equilibrium price of gas each day?
c. Given your answer in part b, how much gas is demanded in this market?
d. Given your answers in part a and part b, how many gas stations will there be?