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Exam Questions M.I.T.

M.I.T. General Examination in Advanced Economic Theory. Sept 1962 and May 1963

 

 

Edwin Burmeister received an M.A. from Cornell in September 1962 before going on to M.I.T. to complete his Ph.D. in economics in 1965. His papers at the Duke Economists’ Papers Archive include a folder of advanced economic theory general examinations at M.I.T. (May and September 1962; May 1963). The copy of the May 1962 exam has been transcribed and posted earlier. This post adds the remaining two exams to the collection of artifacts. Pro-tip:  Burmeister’s papers includes his solutions to the September 21, 1962 exam, most likely prepared during his preparation for the May 1963 exam.

I should mention that on none of the three exams is “M.I.T.” actually written. However, since Samuelson and Solow’s names are typed on the copy of the Sept 1962 exam and since Burmeister was a M.I.T. graduate student  for certainly the May 1963 examination (and, like many before and after him, cast an eye on previous exam questions), it is pretty obvious where the exam questions must have come from.

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General Examination
Advanced Economic Theory

Professors P. A. Samuelson and R. M. Solow
Friday, September 21, 1962

Do as many problems as you have time.

  1. Derive the demand function, Xi = Di(I, p1, …, pn) ≥ 0 for a consumer with income I and having positive prices and having respectively preferences satisfying the following utility functions:
      1. U = k1 log X1 + … + kn log Xn
      2. U = mX0 +logX1 [Be careful!]
      3. U = a1X1 + a2X2 + … + anXn [where] ai≥0

Extra credit

      1. U = Min (X1/b1, X2/b2, …, Xn/bn)
  1. A firm owning some fixed and non-transferable “capital” has a production function

Q = f(labor, land) = 20L.5T.25

It sells in a competitive market at $Pq. It rents labor in a competitive factor market at $W and rents land at $R.
What are its demand relations for factors, and its supply relation for output? What are its “profits” or “quasi-rents to owned capital.”
It will suffice for you to write down all the relations that define these desired functions and describe how they could be solved. (In other words, you don’t have to do the explicit solving.)

  1. In a Hicksian general equilibrium model all income effects turn out to be negligible. Comment decisively on its

(a) Property of dynamic stability (or possible instability)
(b) Property of imperfect stability (or possible instability)
(c) Property of perfect stability (or possible instability)

  1. Let H(X,y) be a function of non-negative vectors X(of dimension m) and y (of dimension n). Define X*, y* as a saddle point of H if

H(X*,y) ≥ H(X*,y*) ≥H(X,y*)

For all non-negative (X,y).
Prove that X* and y* are optimal vectors for a pair of dual linear programs if and only if they provide a saddle point for the function

H(X,y) = C’X+b’y – y’AX.

Show that a simple Leontief model is capable of producing any positive vector final demands (given enough labor) if and only if (I-A)-1 is non-negative.

  1. Consider the von-Neumann model with 3 activities and 4 commodities and with input matrix

\text{A}=\left[ \begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{matrix} \right] and output matrix \text{B}=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right]

Find the optimal activity and price vectors in the von-Neumann sense, and the associated expansion rate.

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General Examination in Advanced Economic Theory: May 1963

Answer any 4 questions.

  1. Suppose all of the N people in a market have identical indifference maps, that are homothetic (i.e., with unitary income elasticities everywhere). Let each jth man have his endowment

\left( \bar{Q}_{1}^{j},\bar{Q}_{2}^{j},\ldots ,\bar{Q}_{r}^{j} \right)

      1. Show that the final equilibrium of exchange is quite independent of the distribution among men of the fixed totals

    \begin{array}{l}\bar{Q}_{1}^{1}+\bar{Q}_{1}^{2}+\ldots +\bar{Q}_{1}^{N}={{A}_{1}}\\...................................\\\bar{Q}_{r}^{1}+\bar{Q}_{r}^{2}+\ldots +\bar{Q}_{r}^{N}={{A}_{r}}\end{array}

    1. Show that the equilibrium prices can be found by treating any man as the single Robinson-Crusoe living under autarky.
    2. What can you, therefore, state about the i) Imperfect, ii) Perfect, and iii) Dynamic stability of the equilibrium?
  2. A Kaldor-Goodwin model defines[sic]
    \text{a}\frac{\text{dK}}{\text{dt}}=\beta \text{Y}-\text{K, }\left( \text{a,b,}\beta \right)>0
    \text{b}\frac{\text{dY}}{\text{dt}}=\frac{\text{dK}}{\text{dt}}-\text{S}\left( \text{Y} \right)
    (i) Explain the meaning of each equation. (ii) Give an equation for its stationary equilibrium solution. (iii) What does its local stability and oscillation depend on? (iv) What shape for the only arbitrary function will give rise to unique-amplitude oscillation?
  3. In Mitopia
    \text{C}+\frac{\text{dK}}{\text{dt}}=\sqrt{\text{KL}}\text{ and L = }{{\text{L}}_{0}}{{\text{e}}^{\text{gt}}}.
    How must K(t) grow if C/L, per capita consumption, is to remain at a maximum constant level? What will then be the interest rate, and the relative share of labor?
  4. A machine with a length of life T costs $f(T). The machine is known with certainty to yield a net income stream of $a per year steadily throughout its lifetime. Find the equation determining the optimal length of life of a machine under each of the following assumptions.
    1. The instantaneous rate of interest in a perfect capital market is r; the length of life is chosen to maximize the present value of net cash flow (including initial cost).
    2. The interest rate r is used to discount net income, and durability is chosen to maximize the capital value of a new machine per dollar of initial cost.
    3. The internal rate of return (i.e. the discount rate that equates capital value and initial cost) is maximized.

Suppose that in cases (a) and (b) the interest rate is such that the capital value of the machine equals its initial cost. Show that all three solutions then coincide. Which is the “right” way to look at the problem?

  1. In a Leontief system with n commodities and one primary factor, labor, let Pi be the money price of commodity i, P0 the money wage, aoi the direct labor input per unit output of commodity i, Xi the output of commodity i, and Ci the final demand for commodity i. Show that the increase in Pj/P0 resulting from a unit increase in a0i equals the increase in Xi needed for a unit increase in Cj.
  2. Consider an individual whose life is divided into two periods, Present and Future. He is endowed with some physical good in each period.
    1. Show how to construct a supply curve relating the amount of saving he will do in the Present as a function of the rate of interest.
    2. Show that in a society of identical individuals with no time preference, the equilibrium rate of interest is zero if corresponding to each individual with endowment X in the Present and Y in the Future, there is another individual with endowment Y in the Present and X in the Future.

 

Source: Duke University. David M. Rubenstein Rare Book & Manuscript Library. Economists’ Papers Archive. Edwin Burmeister papers. Box 23, (unlabeled) Folder.