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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!

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On Harold W. Kuhn

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

Autobiographical sketch in WIKIMIZATION.

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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

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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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Harvard M.I.T. Math Pedagogy Princeton Teaching Wisconsin

Harvard. Draft memo on “Basic Mathematics for Economics”. Rothschild, ca. 1970

 

“These bewildering cook-books [Allen, Lancaster, Samuelson, Henderson & Quandt] are as helpful to those without mathematical training as Escoffier is to weekend barbecue chefs.”

The 1969 M.I.T. economics Ph.D. Michael Rothschild served briefly as assistant professor of economics at Harvard, a professional milestone that went somehow unmentioned in his official Princeton biography included below. He co-taught the core graduate microeconomic theory course with Zvi Griliches in the spring term of 1971 which is probably why a draft copy of his memo proposing  “a course which truly covers ‘Basic Mathematics for Economists'” is found in Griliches’ papers at the Harvard Archives.

Tip: Here is a link to an interview with Michael Rothschild posted in YouTube (Dec. 4, 2012). A wonderful conversation revealing his academic humility and wit as well as an above-average capacity for self-reflection.

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Courses Referred to in Rothschild’s Memo

Economics 199. Basic Mathematics for Economists

Half course (fall term). M., W., F., at 10. Professor G. Hanoch (Hebrew University).

Covers some of the basic mathematical and statistical tools used in economic analysis, including maximization and minimization of functions with and without constraint. Applications to economic theory such as in utility maximization, cost minimization, and shadow prices will be given. Probability and random variables will be treated especially as these topics apply to economic analysis.

Source: Harvard University, Faculty of Arts and Sciences. Courses of Instruction, Harvard and Radcliffe 1969-1970. Published in Official Register of Harvard University, Vol. LXVI, No. 12 (15 August 1969), p. 142.

*  *  *  *  *  *  *  *

Economics 201a. Advanced Economic Theory

Half course (fall term; repeated spring term). Tu., Th., (S.), at 12. Professor D. Jorgenson (fall term); Professor W. Leontief (spring term).

This course will be concerned with production theory, consumption theory, and the theories of firms and markets.
Prerequisite: Economics 199 or equivalent.

Source: Ibid., p. 143.

*  *  *  *  *  *  *  *

Economics 221a. Quantitative Methods, I

Half course (fall term; repeated spring term). Tu., Th., S., at 11. Assistant Professor A. Blackburn (fall term); Assistant Professor M. Rothschild (spring term).

Probability theory, statistical inference, and elementary matrix algebra.

Prerequisite: Economics 199 or equivalent

Source: Ibid., p. 146.

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DRAFT
[Summer or Fall 1970?]

M. Rothschild

Economics 201a, as Professor Jorgenson now teaches it1, presumes much specialized mathematical knowledge. (See attachment 1) There is no single course which covers all these topics, (chiefly the implicit function theorem, constrained maximization and Euler’s theorem), in either the economics or mathematics departments at Harvard. We are in effect demanding that our students arrive knowing these things or that they learn them on their own. The former is unlikely, the latter more so. Imagine trying to learn the mathematics necessary to follow the standard derivation of the Slutsky equation by studying the standard sources such as Allen, Mathematical Analysis for Economists, Lancaster’s Mathematical Economics or the appendices to Samuelson’s Foundations or Henderson and Quandt. These bewildering cook-books are as helpful to those without mathematical training as Escoffier is to weekend barbecue chefs. Those with some knowledge of mathematics will not find the standard sources much more helpful for they are written in a spirit alien to that of modern mathematics; they give almost no motivation or intuition for their results.

There are other bits of mathematics necessary for a thorough understanding of basic economic theory. For instance, the stability theory of difference and differential equations, the theory of positive matrices and rudiments of duality and convexity theory are required for the stability analysis of simple macro models, input output economics, and linear programming respectively. These are hardly new fangled and abstruse parts of economic theory. Indeed they are topics which should be part of every economist’s competence.

There are courses at Harvard where one can learn these things; the difficulty is that there are so many. Advanced courses in mathematical economics treat of positive matrices, duality and much more. Few students take these courses and almost no first year students do. I have no doubt that somewhere in the mathematics or applied math department, there is a course where one may learn all one would want to know and more of difference and differential equations. But all economists really need to know can be taught in three weeks or less.2

There is an obvious solution to these problems, namely for the department to offer a course which truly covers “Basic Mathematics for Economists.”3 A proposed course outline is attached. The course begins with linear algebra because most of the specialized topics needed for mathematical economics are applications of the principles of linear algebra. I know of no one semester course at Harvard which teaches linear algebra in a manner useful to economists. Another advantage to including linear algebra in this course is that it would make it possible to drop the topic from Economics 221a which is presently supposed to teach linear algebra, probability theory, and statistics in a single semester.4 I doubt this can be done. If linear algebra were excluded from the syllabus of 221a, there would be less reason for offering the course in the economics department. We could reasonably expect that our students learn statistics and probability theory from the statistics department (in Statistics 122, 123 or 190).

*  *  *  *  *  *

1…and, I hasten to say, as it should be taught

2A word must be said here about Mathematics 21. This excellent full year course in linear algebra and the calculus of several variables contains all the insights, and almost none of the material, which economists should know. With a slight rearrangement of topics, principally the addition of the implicit function theorem, constrained maximization, and the spectral theory of matrices this would be a great course for economists. As it is now it is a good, but rather time consuming, way to develop mathematical maturity which should make it easy to learn the mathematical facts economists need to know.

3The present title of Economics 199 which is a remedial calculus course taken only by those students with almost no mathematical training.

4I became aware of the need for such a course while teaching 221a. After spending three very rushed weeks developing some of the basic notions of linear algebra I had to drop the subject just when it would have been easy to go on and explain the mathematics behind basic economic theory. The desire of the students that I do so is indicated by the fact that most of them were enticed to sit through a second (optional) hour of lecture on a Saturday by the promise that I would unravel the mysteries of the determinental second order conditions for maximization of a function of several variables.

*  *  *  *  *  *

Proposed course outline:
  1. Linear Algebra, vector spaces, linear independence, bases, linear mappings, matrices, linear equations, determinants.
  2. Cursory review of the calculus of several variable from the vector space point of view, the implicit function theorem, Taylor’s theorem.
  3. Quadratic forms and maximization with and without constraints; diagonalization, orthogonality and metric concepts, projections.
  4. The Theory of Positive matrices; matrix power series.
  5. Linear Difference Equations, stability.
  6. Linear Differential Equations, stability.
  7. Convex sets and Duality. (If time permits.)

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Michael Rothschild

Mike Rothschild first came to Princeton in 1972 as a lecturer in economics and quickly rose to the rank of professor three years later. Mike is an economist with broad interests in social science. His 1963 B.A. from Reed College was in anthropology, his 1965 M.A. from Yale University was in international relations, and his 1969 Ph.D. from the Massachusetts Institute of Technology was in economics.

In the early 1970s, Mike published a string of ground- breaking papers studying decision making under uncertainty and showing the effects of imperfect and asymmetric information on economic outcomes. With Joseph Stiglitz, Mike proposed now- standard definitions of what it means for one random variable to be “riskier” than another random variable. He studied consumer behavior when the same good is offered at different prices and when the consumer does not know the distribution of prices. He studied the pricing behavior of fi when they are uncertain about demand and showed that a fi may end up setting the wrong price even when it optimally experiments to learn about the demand for its product. Arguably, Mike’s most important early work was a 1976 paper with Stiglitz on insurance markets in which insurance companies did not know the heterogeneous risk situations of their customers. Mike and Stiglitz showed that under certain circumstances a market equilibrium exists in which companies offer a menu of policies with different premiums and deductibles that separate customers into appropriate risk groups. This research is one of the landmarks in the field of information economics.

Mike left Princeton in 1976 for the University of Wisconsin and moved to the University of California–San Diego (UCSD) seven years later. His research over this period included papers on taxation, investment, jury-decision processes, and several important papers in finance. Mike’s research contributions led to recognition and awards: he became a fellow of the Econometric Society in 1974, received a Guggenheim Fellowship in 1978, became a fellow of the American Academy of Arts and Sciences in 1994, and in 2005 was chosen as a distinguished fellow of the American Economic Association.

In 1985, Mike decided to branch out from teaching and research, and he spent the next 17 years in university administration. Shortly after arriving at UCSD he became that university’s first dean of social sciences. Under his watch, the division grew dramatically in the number of students, faculty, departments, and programs. He presided over the launching of cognitive science, ethnic studies, and human development. During his deanship, the UCSD social sciences soared in the national rankings, reaching 10th nationally in the last National Research Council tally for 1996.

Mike was lured back to Princeton in 1995 to become the dean of the Woodrow Wilson School of Public and International Affairs. During his seven-year tenure as dean, Mike started the one-year Master in Public Policy program for mid-career professionals; the Program in Science, Technology, and Environmental Policy; the Center for the Study of Democratic Politics; and the Center for Health and Wellbeing. Under his leadership, the Wilson School added graduate policy workshops to the curriculum, expanded course offerings, added multi-year appointments of practitioners to the faculty, and enhanced professional development. Mike shared his dean duties with his trusted and loyal dog, Rosie, who became an important part of the school’s community and accompanied Mike throughout campus.

Finally, Mike likes to wear a hardhat. At UCSD he oversaw the planning and construction of the Social Sciences Building, and at Princeton he built Wallace Hall and renovated Robertson Hall. The Princeton community may remember Mike most for turning Scudder Plaza from the home of a formal reflecting pool where guards kept people out of the fountain into a community wading pool that welcomes and attracts students, families, and children (many under the age of three) each summer evening.

Source: Princeton University Honors Faculty Members Receiving Emeritus Status (May 2009), pp. 18-20.

Image Source: Screenshot from the interview (Posted Dec. 4, 2012 in YouTube).