Tag: Quandt

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

Give formulas for the

1. 4. marginal revenue
   5. 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.

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

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?

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

a ≦ x ≦ b. Let a < c < b.

1. 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.

4. Lagrange multipliers are used to solve what class of calculus problems? Give at least one example from economic theory.
5. 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.)
6. 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

7. Consider the following economic model: A set of n goods are produced (jointly by m activities. The i^th activity requires a_ij units of good j and produces b_ik units of good k.
   Let x = (x₁,…,x_m) represent the levels of the activities
   and y^t = (y₁,…,y_n) 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.