Problem sets for Microeconomics II [110051-0471]

3 Production and cost analysis
Exercise 3.1. Suppose that production function of the steelworks is given by Q =
5LK, where L is the number of workers, and K is the amount of capital used to
give Q units of steel daily. Steelwork can hire labor at $10 per unit, and the cost
of capital is $20 per unit.
What is the optimal mix of inputs to produce 40 units of steel
Exercise 3.2. The owner of the “Magic car wash” describes the relation between
number of cars washed and labor input as follows: Q = −.8 + 4.5L − .3L 2 , where
Q is the number of cars washed per hour, and L is the number of employees. For
each car washed the owner gets $5, and he pays $4.5 per hour to his employees.
a. How many persons should the owner employ to maximize profit
b. What is the profit per hour
c. Is the above labor to cars washed relation true for all L Explain.
Exercise 3.3. Firm Z is developing a new product. An early introduction (beating
rivals to market) would greatly enhance the company’s revenues. However, the
intensive development effort needed to expedite the introduction can be very
expensive. Suppose total revenues and costs associated with the new product’s
introduction are given by R = 720 − 8t and C = 600 − 20t + .25t 2 , where t is the
introduction date (in months from now).
a. Some executives have argued for an expedited introduction date 12 months
from now (t = 12). Do you agree
b. How about instant introduction (t = 0) What introduction date is most profitable
Explain.
Exercise 3.4. In a particular region, there are two lakes rich in fish. The quantity
of fish caught in each lake depends on the number of persons who fish in each,
according to Q 1 = 10N 1 − .1N1 2 and Q 2 = 16N 2 − .4N2 2, where N 1 and N 2 denote
the number of fishers at each lake. In all, there are 40 fishers.
a. Suppose N 1 = 16 and N 2 = 24. At which lake is the average catch per fisher
greater In light of this fact, how would you expect the fishers to redeploy
themselves
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