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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.

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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.

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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).

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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).

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Chicago Exam Questions

Chicago. First price theory course exams. Hanoch, 1964

Giora Hanoch graduated with a doctorate in economics from the University of Chicago in December 1965 with his dissertation “Personal earnings and investment in schooling.” He held the rank of assistant professor of economics for the academic year 1964-65, after which, according to his entry in the AEA 1969 Biographical Listing of Members,  he returned to Hebrew University, Jerusalem, in 1965. With many visiting appointments throughout his career, his academic home was Hebrew University.

Clearly the faculty thought highly enough of him in his fourth year at Chicago to entrust him with the first quarter of the 300-level price theory sequence (Autumn Quarter, 1964).

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Giora Hanoch, Professor Emeritus of Economics
Hebrew University of Jerusalem

1932. Born in Haifa, Israel
1960. A.B. Hebrew University.
1961. A.M. Hebrew University.
1965. Ph.D. University of Chicago. Thesis “Personal Earnings and Investment in Schooling”
1970. Visiting Lecturer, Harvard University.
1974. Visiting Lecturer, Harvard University.
1975. Visiting Lecturer, University of California, Los Angeles
1975— Fellow of the Econometric Society.

Source: “Giora Hanoch, economist”, Prabook website.

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Economics 300
G. Hanoch

Mid-Term Examination
November 18, 1964

I. (60 points)

Answer the following True, False, or Uncertain. Explain your answer briefly.

  1. If two individuals engage in barter, or direct exchange of goods, then always either: a) One individual benefits by the transaction while the other one is hurt; or b) Both are neither benefited nor hurt.
  2. In a perfect market economy, each consumer participates equally in determining what is produced.
  3. If an increase in the demand for X results in an increase i n the price of X, the demand for X is upward sloping.
  4. If the demand for X has unitary elasticity (η = -1), changes in the price of X will not affect the total expenditures on all other goods.
  5. If one good is inferior, at least one other good purchased by the consumer has to be income-elastic (ηxI> 1).
  6. If the marginal revenue is decreasing with an increase in the quantity X, the demand for X is inelastic.
  7. The substitution effect of a decrease in price, as defined by Slutsky, is positive for a normal good and negative for an inferior good.
  8. If the market for beef is in a stable equilibrium, changes in the supply of beef will have little or no effect on its price.
  9. It is possible for a consumer to buy a fixed positive) quantity of X every month, whatever the price of X may be. (i.e., his demand for X has zero elasticity for all prices).
  10. The demand for agricultural products is inelastic; hence plentiful harvests result in lower incomes for farmers, in a free market economy.
  11. In view of (10), each individual farmer can improve his own position by destroying a part of his production in good years.
  12. A linear and downward-sloping demand curve is always elastic at high prices and inelastic at low prices.
  13. If the Laspeyres quantity index between two periods is 1.10 and the Paasche index is 0.90, the consumers’ taste must have changed,
  14. The cross-elasticity of demand for left shoes with respect to the price of right shoes is zero.
  15. A consumer with a utility function is in equilibrium if the marginal utility of each good is proportional to its price.
  16. If all prices increase by 10%, but money income remains the same, the quantity of each good purchased will decrease.
  17. The demand of a consumer for X cannot be infinitely elastic at every quantity of X, because of the budget constraint.
  18. In an economy where the king distributes all the goods and services as free gifts to the consumers, all the prices are zero. Hence there is no place for price theory in that country.
  19. The demand for X is of unitary elasticity, and 200 similar firms sell X. A reduction of 1% in the price PX charged by one firm will result in doubling that firm’s sales, if other firms sell the same quantity at any price.
  20. Because of transportation costs, prices will differ in different geographical locations, whether or not there exists free competition in the market.

II. (40 points)

Two consumers, A and B, have equal and stable tastes and incomes. In December, each spent his entire monthly income on x units of X and y units of Y, when the prices in the market were $2.00 for X and $5 for Y. Consumer A accepted an offer of his employer to be paid in kind, by receiving the same quantities y and y every month directly. (He could still exchange any quantity of X and Y at the market, for the current market prices). B’s money income remained the same.

The following prices prevailed in the market during the next few months:

Month

 $ per unit of X $ per unit of Y
1 2.00 5.00
2 2.20 5.50
3 2.00 5.50
4 2.00 4.50
5 1.80 4.50
6 2.20 4.50

1) Compare consumer A’s position in each of these months with his position in December (was he better-off, worse-off, or indifferent?)

2) Compare the positions of A and B in each month.

NOTE: Use budget lines (and, if necessary, indifference curves) for your analysis. Do not attempt to answer more questions than you were asked. Be brief and clear.

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ECONOMICS 300
G. Hanoch

FINAL EXAMINATION
December 14, 1964

(two hours)

I. (40 points)

Mark the following True, False, or Uncertain. Explain your answers very briefly.

  1. A monopolist can afford to pay wages below the market wage rates.
  2. A rise in the price of gasoline will lead to a rise in the price of tires.
  3. In a long run competitive equilibrium, the marginal firms produce where marginal costs equal average total costs.
  4. If a firm is in long run equilibrium, it is also in short-run equilibrium, whether it is a competitive or a monopolistic firm.
  5. If a production function is characterized by constant returns to scale, an increase in the use of one factor by 10% will increase output by less than 10%.
  6. A rise in the price of any factor used by the firm (other things unchanged) will always lead to a decrease in production by the firm.
  7. A firm producing the same product in many plants will determine the quantity produced in each plant so that average costs will be equal in all the plants.
  8. If a firm has zero variable costs, then its best profit output is where the elasticity of demand for the product is unitary.
  9. A firm will carry production to the point where the marginal productivities of all variable factors are equal.
  10. In a competitive industry with external economies, the total short run supply curve of the industry shifts to the left when there is a permanent decrease in demand for the product.

Il. (30 points)

A monopolist is faced with the following stable demand schedule for his patented machines:

Price per machine
(thousand dollars)

 Quantity
per month		 TR MR TC MC
40 1
35 2
30 3
25 4
20 5
15 6
10 8
5 10

The Costs of production are $5000 per machine, and the fixed costs are $16000 per month.

1.) Compute total and marginal revenue and total and marginal costs in the table above.

2.) Find the equilibrium price, quantity and profits of this firm.

3.) A tax of $60,000 per month is imposed on the firm. Find the new price, quantity and profits.

4.) Instead, a tax of 60% of the market price is imposed on the machines. What will be the monopolist price, output, profits? The tax revenues?

5.) Alternatively, a tax of $24000 per machine is levied. What are the equilibrium price, quantity, profits and tax revenues? What will be the long-run equilibrium quantity?

6.) If no tax is imposed, but a maximum price of $10,000 is enforced, what will be the quantity sold? The Profits?

7.) State your preference among the 5 alternatives ((2) – (6)) above, and justify your choice briefly.

III (30 points)

The current charge for telephone service in city C is $6.40 per month, allowing the consumer 80 free local calls every month, Each additional call costs five cents. Installation is free, and no long-distance calls are available.

NOTE: In the following, assume that each consumer behaves rationally, has constant money income and tastes, with convex indifference curves and no saturation in the relevant range.

Use separate diagrams for each sub-problem. Be precise.

  1. Use a diagram with money-income Y and phone calls X on the axes, to show a consumer’s budget constraint. Be careful to show all the combinations of X and Y available to him, including the case where no service is installed.
    (This portion is crucial for the rest of the problem).
  2. Use indifference curves between Y and X to analyze the consumer’s decision whether to have a telephone installed or not.
  3. Consumer A chooses to have a telephone, and he uses 120 calls every month. Show his equilibrium position geometrically. What is the average price (in cents) of a phone call for him? What is his marginal rate of substitution between money and phone calls?
  4. If the current rates are replaced by a flat rate of 7 cents a call for any number of calls,

(a) Show consumer A’s new budget line, compared with the current position.
(b) Would he now use more or less than 120 calls per month?
(c) Would he be better-off, indifferent, or worse-off relative to the current position?

  1. Consumer A claims that he would prefer to pay a flat rate of 84 per call rather than the current rates. Could he be rational? (demonstrate your answer geometrically).
  2. Consumer B uses only the 80 “free” calls every month, given the current rates. Compare (as in (4)) his consumption and welfare positions with the alternative of being charged a flat rate of 8¢ per call for any number of calls. Could he be indifferent with respect to the two alternative rates?

 

Source: Harvard University Archives. Papers of Zvi Griliches. Box 130, Folder “Syllabi and exams, 1961-1969”.

Image Source: Giora Hanoch in “These Israelis Were Present at the Declaration of Independence.”  Haaretz. Apr. 17, 2018