Month: November 2024

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Exam Questions Math Princeton

Princeton. Mathematics for Economics Grad Students Exam. 1960

Before one gets too smug about the modest level of mathematical sophistication revealed in the following examination that was taken in 1960 by ten Princeton economics graduate students and only passed by half of them, it is important to keep in mind that the purpose of the examination appears only to have been to permit economics students to substitute mathematics for a foreign language as a formal requirement to be awarded a Ph.D. degree. As far as I am aware, by 1960 the exams to test a reading knowledge of a foreign language (at least those administered by an economics department itself) were rather low hurdles hardly capable of tripping any diligent student and generally a waste of time for all but the area specialists and economic historians. Still five of the ten economics grad students at Princeton failed the mathematics exam transcribed below!

__________________________

On Harold W. Kuhn

Princeton University obituary for Harold W. Kuhn (1925-2014).

Autobiographical sketch in WIKIMIZATION.

__________________________

MEMORANDUM

To: Members of the Economics Department
From: H. W. Kuhn
Re: Mathematics Examination for graduate students.

Attached is a copy of the first Mathematics Examination for graduate students in Economics which, as you know, can substitute for one language examination. This memorandum is to describe what the examination was intended to test, report on the performance of the students who took it, and invite comments from you concerning the design of future examinations. (Will Baumol is writing the next one now.)

By agreement of those charged with the conduct of the examination (Baumol, Coale, Kuhn, Okun, and Quandt), it deals only with two subjects, calculus and matrix algebra. The level of the calculus that is assumed is thoroughly elementary and could be acquired in a one-year course. However, it should be augmented by those calculus tools peculiar to economics such as Lagrange multipliers, partial derivatives, and optimization conditions. Study of R. G. D. Allen’s “Mathematical Analysis for Economists” is recommended. The level of matrix algebra is harder to specify. Almost any standard course is too much. Two indications of the level of proficiency demanded are the matrix algebra sections of “Finite Mathematics” by Kemeny, Snell, and Thompson or the Appendix to Dorfman, Samuelson, and Solow. Another book appropriate for study would be “Mathematical Economics” by R. G. D. Allen

The following is an explanation of the first test, question by question, with remarks on the performance of the ten students who took it.

  1. Straightforward translation of economic terms from words to formulas and back. Four parts out of five was par for the course.
  2. The definition of matrix multiplication and of a production matrix. All answers were correct.
  3. A test of understanding of the first and second order conditions for a maximum. Very poor performance; much confusion between necessary and sufficient conditions.
  4. A test of their acquaintance with an indispensable mathematical tool, the Lagrange multiplier. The first pages of “Value and Capital” will give an example. Good performance.
  5. This was intended to draw out the linear case in which solvability is stated in matrix terms. Good performance.
  6. The proper method was by means of partial differentiation. From the variety of answers (mostly weak), this should have been clued.
  7. This model is reproduced almost verbatim from “Finite Mathematics.” The question is intended to test the ability to translate matrix relations into meaningful economic conditions. The average was about half right.

The test was graded on a strict percentage basis, with 70% a passing grade. Five passed and five failed. This may be somewhat hard on those who failed but reflects my own belief that requirements are better too hard than so easy as to be meaningless.

COMMENTS INVITED

__________________________

PRINCETON UNIVERSITY

Department of Economics
Mathematics Examination

October 26, 1960

Please spend no more than two hours on this examination. No books or papers may be consulted. Please attempt all of the questions.

  1. Let y = f(z) be a production function, where y denotes the quantity of output for a quantity of input z. Let c = g(y) be the associated cost function. Let P = F(y) define the demand schedule.

Give the common names for

    1. dy/dz
    2. dc/dy
    3. Py

Give formulas for the

    1. marginal revenue
    2. price elasticity of demand.
  2. The number of tubes and the number of speakers used in assembling three different models (a), (b), (c) of TV sets are specified by a parts-per-set matrix.

\begin{gathered}\\ \begin{matrix}(a)&(b)&(c)&\  \  \  \  \  \  \  \ \end{matrix}\\ \left[ \begin{matrix}13&18&20\\ 2&3&4\end{matrix} \right] \begin{matrix}\text{tubes}\\ \text{speakers}\end{matrix}\end{gathered}

The number of orders received for the three different models in January and February are specified in a sets-per-month matrix

\begin{gathered}\begin{matrix}\  \  \ &\text{Jan.}&\text{Feb.} \  \ \end{matrix}\\ B=\  \left[ \begin{matrix}12&6\\ 24&12\\ 12&9\end{matrix} \right] \begin{matrix}(a)\\ (b)\\ (c)\end{matrix}\end{gathered}

Express the number of parts used per month as a matrix C in terms of A and B. How many tubes were used in February?

  1. Let y = f (x) be a differentiable function defined for

a ≦ x ≦ b. Let a < c < b.

    1. The conditions f'(c)=0 and f”(c)< are necessary and sufficient for f(c) to be a local maximum value for f. True or false? (Give explanation.)
    2. Describe a method for finding the absolute maximum value of f.
  1. Lagrange multipliers are used to solve what class of calculus problems? Give at least one example from economic theory.
  2. Discuss the assertion: Every system of n equations in n unknowns has a unique solution. (It is clearly false; show this by example and modify the statement to be useful.)
  3. The following formula gives the profit P in dollars as a function of the quantities x1, and x2 of two commodities.

P = x150 x235 + x185

When x1 = x2 = 100, P = 2 • 10170
Approximate P when x1 = 101 and x2 = 100

  1. Consider the following economic model: A set of n goods are produced (jointly by m activities. The ith activity requires aij units of good j and produces bik units of good k.
    Let x = (x1,…,xm) represent the levels of the activities
    and yt = (y1,…,yn) represent the prices of the goods, while A and B denote the input and output matrices. Suppose α and β are non-negative numbers. Give common English interpretations of the following equilibrium conditions:

    1. x (B – α A) ≧ 0
    2. (B – β A) y ≦ 0
    3. x (B – α A) y = 0
    4. x (B – β A) y = 0
    5. x B y > 0

What condition on A would insure that every process uses some good as input?
What condition on B would insure that every good can be produced in the economy?

Source:  Duke University. David M. Rubenstein Rare Book & Manuscript Library. Economists’ Papers Archive.  William J. Baumol Papers, Box 10, Folder “Princeton University 1952-69”.

Image Source: Harold W. Kuhn, ca. 1961. Wikimization website.

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Exam Questions Money and Banking UCLA

UCLA. Monetary Economics, PhD qualifying exam. 1971

Having just spent nearly a month travelling along the East Coast of the U.S., it is great to get back to posting new content. On this trip I was able to get in three fine days of work in the Economists’ Papers Archive at Duke. While there I found much useful material for Economics in the Rear-view Mirror in the Robert W. Clower papers. A copy of his UCLA obituary can still be found at the Wayback Machine internet archive.

In 1971 Clower joined the UCLA economics department so it is unclear whether he actually contributed to the Ph.D. preliminary examination in monetary economics transcribed below 

_______________________

Ph.D. Qualifying Examination
Four Hours

May, 1971

Monetary Economics

Answer five of the following seven questions.

  1. [On the concept of money]
    1. “Contemporary monetary theory analytically treats money as merely another commodity.” State if (and why) you agree or disagree with this proposition.
    2. Money is sometimes distinguished from commodities by the following assertions. Briefly discuss the meaning of each assertion and whether you agree or disagree with it.
      1. Money has no “intrinsic value”; it cannot be enjoyed directly, but must first be converted into something else.
      2. Money is used but is not used up.
      3. Money buys goods and goods do not buy money.
      4. Money has superior liquidity than other goods.
      5. The value of money is fixed in terms of the unit of account.
      6. Money is traded directly for every commodity and vice versa, while commodities are not traded for one another.
    3. Discuss the limitations placed on research in monetary theory if money is considered merely as a commodity.
  2. Many writers have asserted in the press that the recent international currency “crisis” points up the unique role of the dollar in present international monetary arrangements. Discuss the international role of the dollar with reference to each of the following statements taken discussions of the crisis.
    1. Over the past couple of years the U.S. has been exporting an unwanted inflation to the countries of Europe, especially Germany.
    2. The immediate cause of the crisis was the presence of interest rates in the U.S. which were too low relative to those in Europe and therefore initiated massive capital flows from the U.S. to Europe.
    3. The massive accumulation by foreigners of dollars underlined the fact that the dollar has become de facto inconvertible into gold and was now little more than an unbacked IOU.
    4. The U.S. should be unconcerned with its balance of payments deficit. Under present arrangements any adjustments to international disequilibrium must be made by foreigners; and all the options available to foreign surplus countries, assuming moderately rational behavior on their part, should be acceptable to the U.S.
    5. The recent crisis points up the inherent instability of current international monetary arrangements. The increase in foreign short-term claims upon U.S. gold reserves and the revaluation of currencies in terms of the dollar will undermine the employment of the dollar as the banking currency of the world and speed the development of a unified European currency.
    6. The recent crisis has strengthened the world monetary system by bringing closer the day when the dollar-gold fixed exchange rate standard is replaced by a system of floating exchange rates.
  3. Discuss the following three propositions. (State whether they are true or false and explain why) .
    1. Legal reserve requirements are unnecessary to place a finite limit on the quantity of commercial bank deposits if the deposits are convertible into the government supplied dominant money.
    2. Elimination of the convertibility requirement would lead to an unlimited expansion of deposits.
    3. There is no limit on the extent to which the government can expand the supply of dominant money.
  4. An economist recently wrote a letter to the Wall Street Journal complaining that much discussion of how to control inflation has been based on a neo-quantity theory which emphasizes “the quantity of money” while ignoring “the quality of credit”. The Federal Reserve was established, he noted, to regulate commercial bank assets while current discussion (and policy) concentrates on the liability side of the commercial bank balance sheet and entirely ignores the asset side. He maintained that if, for example, commercial banks were forced to limit their lending activity to short-term, self-liquidating business loans, inflation would quickly be controlled. Evaluate this argument.
  5. [Monetary vs. fiscal policy.]
    1. It is sometimes argued that fiscal policy should be used to maintain domestic full employment while monetary policy should be used to maintain balance of payments equilibrium. Present this argument and clearly state the assumptions upon which it is based.
    2. Summarize and evaluate the existing empirical evidence on the effectiveness of monetary versus fiscal policy as a stabilization device
  6. [Inflation]
    1. Inflation is often considered to be a tax. In what sense is this correct? What is the magnitude of the tax? Who pays and who collects the tax?
    2. What are the effects of inflation on real resource allocation.
      [In (a) and (b) make sure you distinguish between anticipated and unanticipated inflation.]
  7. [The Gibson Paradox]
    1. What is the Gibson Paradox?
    2. Why is it considered to be a paradox?
    3. What theoretical explanations have been advanced to explain the phenomenon?
    4. What is the existing state of the evidence concerning these explanations?

Source: Duke University. David M. Rubenstein Rare Book & Manuscript Library, Economists’ Papers Archive. Robert W.Clower Papers, Box 4, Folder: “Monetary Economics PhD exams, Reading List, Exams. UCLA, 1971-1988”.

Image Source: Screen shot from Abba—Money, Money, Money karaoke video.