Categories
Exam Questions M.I.T.

M.I.T. Core Micro Theory. Exam Questions for Resource Allocation/Price System. Weitzman, 1973

 

In earlier posts I provided transcriptions of the course outline and readings and the final examination questions for Martin Weitzman’s 1974 course on resource allocation and the price system that was the second of four half-term courses that made up the required core graduate microeconomic theory sequence at M.I.T. back then. In the approximately one hundred page typescript that he distributed for his course Weitzman also included three earlier exams from his course. These exams are transcribed below. The September exam was offered for people to place out of taking the course, the November exam would have been the actual course final examination, and I presume the following February exam was offered as a make-up examination.

An expression that Martin Weitzman was fond of saying to get us to think about the economic intuition behind some result or to motivate an argument was for us to first consider “a quick-and-dirty banker’s calculation”. The diagram above was just borrowed from a random blog post because of its “quick-and-dirty” theme and was never used by Weitzman.

_________________

Typically three questions are answered in an hour and a half.

Exam
(September, 1973)

  1. There are n “investment opportunities” indexed by i = 1,2,…, n. Up to ci dollars can be invested in the ith investment opportunity, and each dollar so invested yields ai present value dollars back (ai > 1). There is a budget constraint limiting investment spending to M dollars.
    1. Specify the programming problem of picking investments to maximize returns subject to a budget constraint.
    2. Describe precisely the solution to the problem.
    3. What is the shadow price of an extra dollar of budget?
    4. If a new project is discovered which pays an+1 per dollar invested, should it be undertaken? Show how the shadow price of an extra dollar of budget can be used to easily give an answer. (Hint: if you are having trouble, try a numerical example.)
  2. There are n ways of combining capital with labor to produce output. The ith technique (i = 1,…,n) has coefficients ai for labor per unit of output and bi for capital per unit of output. If K units of capital and L units of labor are available, we can define a production function F(K,L) as follows:

\begin{array}{c}F\left( K,L \right)=\max \sum\limits_{i=1}^{n}{{{y}_{i}}}\\\text{subject to}\\\sum\limits_{i=1}^{n}{{{a}_{i}}{{y}_{i}}}\le L\\\sum\limits_{i=1}^{n}{{{b}_{i}}{{y}_{i}}}\le K\\{{y}_{i}}\ge 0.\end{array}

a. Prove that F(K,L) is concave and homogeneous of degree one in K and L.
b. Interpret the shadow prices associated with the inequalities in labor and capital, for a given K and L.

  1. State clearly…
    1. …the conditions under which it is possible to decentralize allocation decisions and yet achieve efficiency in a production system by using prices.
    2. Prove the theorem you have just stated, or at least sketch an outline of it.
    3. Comment on the importance of the convexity assumption. Will and non-convexity spoil the situation irreparably? What about externalities?
  2. A village owns given amounts of grade A land, and grade B. The total labor supply of \overline{L}=5 is used solely for growing corn. The production function for grade A land is {{Y}_{A}}=12{{L}_{A}}-2L_{A}^{2} and for grade B land is {{Y}_{B}}=7{{L}_{B}}-\left( 1/2 \right)L_{B}^{2}. Naturally {{L}_{A}}+{{L}_{B}}=\overline{L}.
    1. The village uses the land as communal property to give each man an equal income, i.e., everyone gets an amount of A land or an amount of B land which yields each one the same output when he works it. Spell out in detail what this solution implies. Is this an efficient social production scheme? Why or why not?
    2. Suppose the village chief hires a consultant from 14.122. His problem is to maximize total output (YA+ YB) subject to production constraints. Then returns are equally distributed. What is the solution? What is the marginal product of labor?
    3. Let the marginal product of labor from (b) be w. Now show that if plots A and B are run as profit-maximizing firms with w being the wage rate (maximize separately {{Y}_{A}}-w{{L}_{A}} and {{Y}_{B}}-w{{L}_{B}}), that society ends up with the same solution as in (b). What does this mean? What is the difference between (b) and (c) especially as regards workers’ welfare? Are the workers better off under (a) or (c)? Would your answer have to be the same if the production functions changed? Explain.
  3. The following table specifies four processes for the production of steel. Assume
    1. factors of production are perfectly divisible and transferable among processes,
    2. all processes exhibit constant returns to scale,
    3. the wage rate is $10 per man/day,
    4. production is always efficient.
Factor requirements/ton of steel/day
Process No. Man-days Machine days
1 2 5
2 4 3
3 2 2
4 6 1

(i) Initially only processes 1 and 2 are known and both are in use. Draw one diagram showing the unit isoquant for steel production and another showing the long-run average total cost curve for steel production. Indicate the magnitude of long-run average total cost per ton of steel on your cost diagram.
(ii) Show on the relevant diagrams the unit isoquant and long-run average total cost curve which prevail after the discovery of processes 3 and 4.
(iii) Suppose that prior to the change in technology some firm was producing 10 tons of steel per day by operating each process at a level of 5. On your cost curve diagram show this firm’s short-run average total cost curve before and after the change in technology. Indicate the magnitude of any shifts in the cost curve on your diagram. (Note: in the short-run labor can be hired in any desired amount but the total amount of machinery a firm possesses is fixed.).

 

Exam
(November, 1973)

  1. There are n types of machine, indexed by i= 1,2,…, The ith type of machine is designed to work with {{\alpha }_{i}} workers, in which case it produces {{\beta }_{i}} units of output. There are {{\gamma }_{i}} machines of type i available.
    1. Formulate the problem of finding the minimum amount of labor to produce a given amount of output, say .
    2. Describe precisely the solution to the problem for arbitrary .
    3. What is the shadow cost of an extra unit of output and why?
    4. What is the shadow rental on a machine of type iand why?
    5. If a new machine is discovered of type n+1 with specification {{\alpha }_{n+1}}, {{\beta }_{n+1}}, {{\gamma }_{n+1}}, should it be used? Show how the shadow cost of an extra unit of output can be used to easily give an answer.
  2. There are n plots of land. Land of type i (i= 1,…,n) can be used to grow either ai bushels of wheat or bi bushels of rye (or the appropriate combination).
    1. Characterize efficient specialization patterns for these n plots of land. Which land should grow which crop and why?
    2. Suppose (as in a competitive market economy) a price of wheat is announced and also a price of rye. Then every plot grows the profit maximizing crop. Is this efficient? Why or why not? (show directly).
    3. Show how a supply of wheat curve would be derived in a market economy. Is it upward sloping? Why or why not?
  3. A firm consists of two plants. For each of statements (a) and (b), either prove the statement or give a counterexample. Be precise.
    1. The firm’s overall production plan will be efficient whenever each plant is producing efficiently.
    2. If both plants are maximizing profits at the same prices, the resulting overall plan will be efficient for the firm.
  4. There are two products in an economy (guns and cars) and two basic factors of production (A and B). There are n techniques for producing guns: the ith technique uses ai units of A and bi units of B per gun. Similarly there are m techniques for producing cars: the ith technique uses up ai units of A and bi units of B per automobile. The economy has a total availability of units of A and units of B.
    1. Define carefully the production possibilities set for this economy. Sketch roughly what it looks like in the space of cars and guns.
    2. Prove that it must be convex (what does this mean geometrically?).
    3. State carefully why (b) is important for the possibility of effective indirect or decentralized control (using prices) of the way guns and cars will be produced.
  5. We derived the result that a free market competitive allocation of resources is efficient. Does this mean that a market system is the best way to organize an economy in the sense that it gives the greatest benefits? Why or why not? What are the major real world situations which would cause an exception to the above cited “result” in a real life market economy? Try to be as specific as possible in showing why and where the efficiency result breaks down for each cited example.
  6. Suppose a firm or economy consists of a number of divisions or subsectors, each of which is characterized by a convex technology. Then any efficient plan will have associated with it a set of prices (one for each good, the same prices for each subsector) such that the efficient plan is sustained by having each subsector maximize its own profits. Outline a detailed sketch of how the proof of such an assertion goes. Why is this sometimes called a “decentralization theorem”. What does it suggest as a way of efficiently organizing production?
  7. An organization has 2 inputs and 2 outputs, and a production set consisting of the following 7 activities, all scalar multiples of activities in the set, and all sums of activities in the set.
Activity a b c d e f g
Good 1 2 1 2 1 1 3 1
Good 2 0 2 1 1 1 1 3
Good 3 -2 -4 -3 -1 -3 -5 -6
Good 4 -2 -3 -4 -3 -1 -5 -4

a. Consider a plan to run activities (a, d,e) at a positive level. If this were an efficient plan, what shadow prices would be associated with it?
b. Is this plan efficient, in fact? Prove your answer by appealing to particular theorems or results.
c. Consider a plan to run activities (a, b,d) at a positive level. Is this an efficient? Why, or why not?

 

Exam
(February, 1974)

  1. A firm consists of two plants. There are no externalities. For each of statements (a) and (b), either prove the statement or give a counterexample. Be precise.
    1. The firm’s overall production plan will be efficient whenever each plant is producing efficiently.
    2. If both plants are maximizing profits at the same positive prices, the resulting overall plan will be efficient for the firm.
        An organization has a production set consisting of the following 7 activities, all scalar multiples of activities in the set, and all sums of activities in the set.
  2. An organization has a production set consisting of the following 7 activities, all scalar multiples of activities in the set, and all sums of activities in the set.
Activity a b c d e f g
Good 1 2 1 2 4 0 3 0
Good 2 2 5 1 0 -1 1 2
Good 3 -1 -3 -1 -1 1 -2 0
Good 4 -2 -3 -4 -4 -3 -3 -1

a. Consider a plan to run activities (a,d,e) at a positive level. If this were an efficient plan, what shadow prices would be associated with it?
b. Is this plan efficient, in fact? Prove your answer by using the prices from (a) and appealing to particular theorems or results.
c. Consider a plan to run activities (a,d,g) at a positive level. Is this an efficient plan? Why, or why not?

  1. State…
    1. …clearly the conditions under which it is possible to decentralize allocation decisions and yet achieve efficiency in a production system by using prices.
    2. Prove the theorem you have just stated, or at least sketch an outline of it.
    3. Comment on the importance of the convexity assumption. Will any non-convexity spoil the situation irreparably? What about externalities?
  2. The following table specifies four processes for the production of steel. Assume
    1. factors of production are perfectly divisible and transferable among processes,
    2. all processes exhibit constant returns to scale,
    3. the wage rate is $10 per man/day,
    4. production is always efficient.
Factor requirements/ton of steel/day
Process No. Man-days Machine days
1 1 2.5
2 2 1.5
3 1 1
4 3 .5

(i) Initially only processes 1 and 2 are known and both are in use. Draw one diagram showing the unit isoquant for steel production and another showing the long-run average total cost curve for steel production. Indicate the magnitude of long-run average total cost per ton of steel on your cost diagram.
(ii) Show on the relevant diagrams the unit isoquant and long-run average total cost curve which prevail after the discovery of processes 3 and 4.
(iii) Suppose that prior to the change in technology some firm was producing 10 tons of steel per day by operating each process at a level of 5. On your cost curve diagram show this firm’s short-run average total cost curve before and after the change in technology. Indicate the magnitude of any shifts in the cost curve on your diagram. (Note: in the short-run labor can be hired in any desired amount but the total amount of machinery a firm possesses is fixed.)

  1. Consider the following production function:Y={{T}^{\alpha }}{{L}^{1-\alpha }}
    where Y is output, T and L are inputs and 0 < {{\alpha }} < 1.

a. Prove that this production function has constant returns to scale.
b. Describe the production set that corresponds to this production function. Which points are efficient?
c. Given an arbitrary efficient point (Y*, T*, L*), find the shadow or efficiency prices associated with that point.
d. Prove that the production set is convex.

  1. Let i be an index of land plots running from 1 to 10. The higher the index, the more moisture in the soil. Either rye or wheat can be grown. Land of type i (i=1,…,10) can costlessly produce either 5 units of rye or i units of wheat (or an appropriate combination).
    1. If a total of 30 units of rye is to be grown, what is the most wheat that can also be obtained?
    2. For the same situation as in (a), what are the shadow prices of wheat and rye? Explain fully what these shadow prices mean. What is the shadow rent on land of various types?
    3. Describe in general the efficient patterns of specialization. Which land should grow which crop and why?
    4. Derive the supply of wheat curve and show what it looks like.
    5. Verify that any profit maximizing output combination is efficient.

Source:  Martin Weitzman’s Notes to 14.122, personal copy of Irwin Collier, pp. 88-97.

Image Source: Blogpost: “The truth about quick and dirty” POSTED BY NIK ⋅ 10 JULY 2007 at niksilver.com