MIT. Final Examination in 2nd Core Microeconomics. Martin Weitzman, 1974

The theory core at MIT in the mid-1970s consisted of four half-semester courses in microeconomics and four half-semester courses in macroeconomics. For reasons unknown to me, Microeconomic Theory I (A) taught by Martin Weitzman was scheduled to follow Microeconomic Theory II (A) taught by Robert Bishop for the First Term of 1974-75. I guess I should really say, there was no good reason not to simply reverse the numbering of the courses since Weitzman’s course was in most respects the more advanced of the two. The course featured the economic intuition behind some “quick and dirty bankers’ calculations”, an introduction to linear models, and the first essay of Koopmans’ Three Essays on the State of Economic Science.

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Microeconomic Theory I (A)

14.121 December 1974                 Final Exam                M. Weitzman

Instructions:

1. Be sure you have picked a number and identify yourself by that number only on each blue book you use.
2. Try to answer any three out of the following six questions.
3. Complete answers on all three questions are not required for passing. Two well answered questions would easily be enough to pass, for example.
4. Total time: one and a half hours.
5. Answer each question in a separate blue book.
6. Try to be concise and to the point. Wordiness is not going to help anyone.

 

1. Explain carefully why the following three features of the American economy lead to productive inefficiency. Say what might be done to rectify the inefficiency.
   1. water pollution
   2. existence of “free” fishing grounds.
   3. the fact that the price of certain raw materials (Like natural gas) is artificially suppressed.

2. Suppose there are a total of I tasks to be accomplished. A limited number of labor saving machines are available to help out. Task i can be performed by using a_i units of labor alone, or b_i (< a_i) units of labor along with e_i machines, or the appropriate combination. There are a total of M machines available.
   1. Formulate the problem of using the available machines to minimize the amount of labor required to perform the tasks.
   2. Describe the optimal solution.
   3. What is the value or shadow price of an extra machine? Show directly that minimizing shadow costs at shadow prices yields the right answer.

3. A particular “two-armed” model of a drill-press can be worked by either one or two operators. With one operator it produces U units of output per unit time; with two operators it produces V units per unit time.
   1. With L laborers and M machines available, describe precisely how to calculate how many machines should be operated by one worker and how many by two in order to maximize output. What is the marginal rate of substitution between machines and laborers? (Hint: Try to get an answer using “common sense.” If that doesn’t work, draw isoquants.)
   2. Suppose total output is fixed in the long run. As many machines can be rented and workers hired as desired at the going rates. How do you decide whether it is better to operate machines with one or with two laborers?
   3. In the short run the number of machines is fixed but as many workers as desired can be hired at the going wage rate. The output is variable. How is the short run supply of output schedule determined?

4. There are two farm plots, A and B. Both have identical production functions. If x units of labor is applied to A (or to B) it results in f(x) units of output. A total of L units of labor is available for application to both farm plots.
   1. Formulate the planning problem of allocating labor to A and to B so as to maximize total output from both farms when a total of L units of labor are available.
   2. Assuming f’(x) > 0, f”(x) < 0, characterize exactly the solution to problem (a) above and show why it is optimal.
   3. Show directly that there is an efficiency price of labor relative to output which supports the optimal solution of (b).
   4. Assuming f’(x) > 0, f” > 0, characterize exactly the solution to problem (a) above. Does (c) hold now? Why or why not?

5. A firm or economy consists of a number of divisions or subsectors. There are no externalities. From first principles, prove rigorously the following result: If each subsector is maximizing profits at the same positive prices, the firm’s overall mixture of inputs and outputs is being efficiently produced.

6. Suppose modern low-cost shell housing is made according to the following production formula:

H = (A + T)^αL^1-α

A,T,L ≥ 0

Where H is housing, A is aluminum, T is tin, and L is labor. Tin is produced by a perfect competitor, so there is free entry into the tin industry. One unit of labor produces a unit of tin. Aluminum, on the other hand, is a monopolistic industry which can charge any price it wants to, and can restrict entry. One unit of labor produces b > 1 units of aluminum. The aluminum, tin, and housebuilding industries have competitive labor supplies. For simplicity, suppose that the total budget of all the housebuilders is fixed and aluminum has no other uses.

a. What will be the competitive price of tin? The monopoly price of aluminum? Why?

b. What input mix will the home builders select and why?

c. Referring to question (b), are houses being produced efficiently? Why or why not? Give as precise an answer as you can. If you find that houses are produced inefficiently, give the efficient way to produce them.

 

Source: Personal copy of Irwin Collier.

Image Source: Detail from 1976 MIT economics department group picture.