Tag: Hotelling

Columbia. Promotions and Memorial Minute for Abraham Wald

Post author By Irwin Collier
Post date 2020-04-06

Abraham Wald (1902-1950) got his foot into the Columbia economics department door thanks to a grant from the Carnegie Foundation arranged for him by Harold Hotelling in 1938. In this post we follow Wald’s Columbia career up through the faculty memorial minute that followed his tragic, untimely death in an airplane accident during a lecture tour in India in December 1950.

___________________

Promotion to Assistant Professor

From the November 26, 1941 letter to President Nicholas Murray Butler from Robert M. Haig, Chairman of the department of economics (pp. 3-5 and supporting annex B).

“…We feel that it is important, if at all possible, that the following action be taken.

Appoint Abraham Wald to an assistant professorship at $3,600 (Wald is now a lecturer at $3,000, of which $2,400 is financed by a special grant, the continuance of which is not assured.
(See Annex B)

A recent development in the case of Wald is an offer of a permanent post (presumably an assistant professorship) at Queens College. This post will be open to him in case it proves impossible for us to give him the status recommended. Our enthusiasm for him has increased since last year when I wrote as follows:—

“The position of this recommendation at the very head of our list is attributable primarily to a conviction that Abraham Wald is an unusually interesting gamble. By risking a moderate stake, the University can put itself in a position where it may (and in our judgment probably will) be rewarded a hundred-fold. For Wald is not only a young scholar whose attainments are of a high order of merit, but one whose potentialities are obviously large.

“When Wald came to us two years ago, as a lecturer whose stipend was supplied by a special and presumably temporary grant, we were frankly apprehensive lest we should presently find ourselves indirectly committed, without adequate consideration, to a permanent addition to our staff concerning whom we might not be enthusiastic. Consequently care was exercised to make it clear to all concerned that his appointment as a lecturer supported by a special grant carried with it no future obligation. Fortunately then, we are able to approach the consideration of his case at this juncture free from any pressure of old commitments, express or implied.

“Our recommendation of Wald should be interpreted then as a free and fresh expression of our admiration for his accomplishments and of our faith in his future. As a result of our contacts with him and with his work, we are convinced that here is a man whose contributions are reasonably certain to continue to break new ground on a section of the frontier of knowledge where notable progress seems imminent.

“We recognize that Wald’s field is one that is of interest and significance to several departments of the University and that there are unsettled questions as to whether ultimately it should be attached to our own or to some other department or whether it should constitute a separate department in its own right. Irrespective of the answers that may ultimately be given to such questions of structural organization, we, in the Department of Economics, desire to express the hope that it will prove possible for the University to provide for the further development of teaching and research in statistics on a high level and we wish to take this opportunity to make it clear that, pending a final decision as to organization, we should consider it an honor to be permitted to shelter and to stand sponsor for scientific work such as that of Wald.”

[…]

ANNEX B.
BIOGRAPHIC MEMORANDUM OF ABRAHAM WALD

I was born in Cluj, October 31, 1902. I studied at the University in Cluj and at the University of Vienna, and obtained my doctor’s degree in mathematics at the University of Vienna in 1930. The subject of my doctor’s thesis was the Hilbert system of axioms of Geometry.

For the next four years I did mathematical research at the University of Vienna and collaborated with Professor Karl Menger. I was co-editor with him of “Ergebnisse eines mathematischen Kolloquiums.” During this time my interests were chiefly in general abstract and metric geometry, theory of probability, and mathematical economics, in which fields my papers were written.

In 1934 I became in addition a research associate of the Institute for Business Cycle Research in Vienna and published several papers in mathematical economics. My interest in statistics and its application in economics dates from this time, when I became the statistical expert of the Institute.

After the annexation of Austria, I came to the United States and was for several months a fellow of the Cowles Commission for Research in Economics. Thereafter I became a lecturer at Columbia University which is the position I hold at present. Since my arrival in the United States I have been interested chiefly in statistics and mathematical economics and have published a series of papers in these fields. I have been elected a fellow of both the Institute of Mathematical Statistics and the Econometric Society.

LIST OF PUBLICATIONS

Abstract and metric geometry
1. Über den allgemeinen Raumbegriff, Ergebnisse eines math. Kolloquiums, Heft 3, Vienna. [1931]
2. Axiomatik des Zwischenbegriffes in metrischen Räumen, Mathematische Annalen, Vol. 104. [1931]
3. Der komplexe euklidische Raum [Komplexe und indefinite Räume], Erg. eines mathem. Kolloquiums, Heft 5, Vienna.
4. Indefinite euklidischen Räume, Erg. eines mathem. Kolloquiums, Heft 5, Vienna.
5. Vereinfachter Beweis des Steinitzschen Satzes, Erg. mathem. Kolloquiums, Heft 5, Vienna.
6. Bedingt konvergente Reihen von Vektoren, Erg. mathem. Kolloquiums, Heft 5, Vienna.
7. Riehen in topologischen Gruppen, Erg. mathem. Kolloquiums, Heft 5, Vienna.
8. Eine neue Definition der Flächenkrümmung, Erg. mathem. Kolloquiums, Heft 6, Vienna
9. Sur la courbure des surfaces, C. R. [Acad. Sci.] Paris, 1935.
10. Aufbau [Begründung] einer kooridinatenlosen Differentialgeometrie der Flächen, Erg. mathem. Kolloquiums, Heft 7, Vienna.
11. Eine Characterisierung des Lebesgueschen Masses, Erg. mathem. Kolloquiums, Heft 7, Vienna.

Probability, Statistics and Mathematical Economics.
1. Sur la notion de collectif dans le calcul des probabilités, C.R. [Acad. Sci.] Paris, 1936.
2. Die Widerspruchsfreiheit des Kollektivbegriffes, Erg. mathem. Kolloquiums, Heft 8, Vienna.
3. Die Widerspruchsfreiheit des Kollektivbegriffes, Actualités Scientifiques, 1938, Paris.
4. Berechnung und Ausschaltung von Saisonschwankungen, Julius Springer, Vienna, 1936.
5. Zur Theorie der Preis Indexziffern, Zeitschrift für Nationalökonomie, Vienna, 1937.
6. Über die Produktionsgleichungen der ökonomischen Wertlehre, Erg. mathem. Kolloquiums, Heft 6, Vienna.
7. Über die Produktionsgleichungen der ökonomischen Wertlehre, zweite Mitteilung, Erg. mathem. Kolloquiums, Heft 7, Vienna.
8. Über einige Gleichungssysteme der mathematischen Ökonomie, Zeitschrift für Nationalökonomie, Vienna, 1936. [translated into English by Otto Eckstein, Econometrica, 1951, pp. 368-403]
9. Extrapolation des gleitenden 12-Monatsdurchschnittes, Beilage zu den Berichten des Öster. Institutes für Konjunkturforschung, Vienna, 1937.
10. Grundsaetzliches zur Berechnung des Produktionsindex, Beilage zu den Berichten des Öster. Institutes für Konjunkturforschung, Vienna, 1937.
11. Generalization of the inequality of Markoff, The Annals of Math. Statistics, December, 1938.
12. Long Cycles as a result of repeated integration, American Mathem. Monthly, March, 1939.
13. Confidence limit for continuous distribution functions (co-author J. Wolfowitz), The Annals of Math. Statistics, June, 1939.
14. Limits of a distribution function determined by absolute moments, Transact. of the Amer. Mathem. Society, September, 1939.
15. A new formula for the index of cost of living, Econometrica, October, 1939.
16. Contributions to the theory of statistical estimation, The Annals of Mathem. Statistics, December, 1939.
17. A note on the analysis of variance with unequal class frequencies, The Annals of Mathem. Statistics, March, 1940.
18. The approximate determination of indifference surfaces, Econometrica, April, 1940.
19. On a test whether two samples are from the same population (with J. Wolfowitz), The Annals of Mathem. Statistics, June, 1940.
20. Fitting of straight lines when both variables are subject to error, The Annals of Mathem. Statistics, September, 1940.
21. Asymptotically most powerful tests of statistical hypotheses, Annals of Mathem. Statistics, March, 1941.
22. Some examples of asymptotically most powerful tests will appear in the December, 1941 issue of the Annals of Mathem. Statistics.
23. Asymptotically shortest confidence intervals paper presented at the meeting of the Amer. Math. Soc., September, 1940. Accepted for publication in the Annals of Mathem. Statistics.
24. On the distribution of Wilks’ statistic for testing the independence of several groups of variates (in collaboration with R. Brookner), Annals of Mathem. Statistics, June, 1941.
25. The large sample distribution of the likelihood ratio statistics, paper presented at the meeting of the Institute of Mathem. Statistics, September, 1941, Chicago. It will be published in the Annals of Mathem. Statistics.
26. On testing statistical hypotheses concerning several unknown parameters, paper presented at the meeting of Amer. Mathem. Society, February, 1941, New York City. It will be published in the Annals of Mathem. Statistics.
27. On the analysis of variance in case of multiple classifications with unequal class frequencies, Annals of Mathem. Statistics, September, 1941.

Source: Columbia University Archives. Central Files 1890—Box 386, Folder “Haig, Robert Murray 7/1941-6/1942”

Cf: “The Publications of Abraham Wald” [1931-1952] was published in The Annals of Mathematical Statistics 23:1 (March 1952, pp. 29-33.

___________________

Aliens in the Department of Economics

December 19, 1941

Mr. Philip M. Hayden, Secretary,
213 Low Memorial Library.

Dear Mr. Hayden:

In reply to the request contained in your recent Memorandum to executive officers, I report the following aliens from the Department of Economics:

Harold Barger

29 West 8^th Street, New York City

Nationality: British
Age: 34
Alien Registration No.: 3239174

Donald Bailey Marsh

106 Morningside Drive, New York City

Nationality: Canadian
Age: 30
Alien Registration No.: 1152252

Robert Valeur

40 Barrow Street, New York City

Nationality: French
Age: 38
Alien Registration No.: 5061531

Abraham Wald

241 West 108^th Street, New York City

Nationality: born in Kolozsvar [Note: Hungarian spelling of Cluj-Napoca], Transylvania, Alien Registration officials were in doubt how to describe nationality.
Age: 39
Alien Registration No.: 4506027

Very truly yours,
Chairman, Department of Economics

Source: Columbia University Libraries, Manuscript Collections. Department of Economics Collections, Faculty. Box 2, Folder “Faculty Beginning Jan 1, 1944 [sic]”.

___________________

Promotion to Associate Professor

February 7, 1944

Professor Abraham Wald,
608 Fayerweather.

Dear Professor Wald:

I am authorized by the Provost of the University to inform you that in the provisional budget for the academic year 1944-45 you are designated associate professor of Statistics at an annual stipend of $5,000. This provisional budget goes to the Trustees with the approval of the Committee on Educational Policy. While your promotion is not final until it is adopted by the Trustees at their meeting on the first Monday in April, the Provost and I agree that there is no reason whatsoever to doubt that the recommendation for your advancement will be approved, and that you run hardly an risk in declining the offer of an associate professorship at the University of Chicago.

As an associate professor you would hold your position at the pleasure of the Trustees, i.e., you would no longer be subject to year-to-year appointment and would, in effect, have continuing tenure. The position of associate professor in this respect is the same as that of a full professor.

I should like to add my personal assurance that the Department and the Administration stand behind the recommendation for your advancement. The reputation that you have won for yourself at Columbia is a very high one indeed. You have the friendship and warm support of all of your associates in the graduate faculties. I believe that you will have here a rich and promising career of creative scholarship.

Sincerely yours,

[copy unsigned, Frederick C. Mills?]

Source Columbia University Libraries, Manuscript Collections. Columbiana. Department of Economics Collection. Box 3 Budget, 1915-1946-47, Folder “Budget Material 1944-1945”.

___________________

Promotion to Professor

April 23, 1945

President Nicholas Murray Butler,
Columbia University.

Dear Mr. President:

A recent development makes it necessary for me to supplement my letter of November 30^th, 1944, in which I submitted to you a provisional budget for the Department of Economics for the year 1945-46. Professor Abraham Wald, who occupies a place of strategic importance in our work in Mathematical Statistics, has received a very attractive offer from another institution. If we are to hold him at Columbia we must give him some advancement here. Although I am reluctant to approach you at this time, to request that you re-open the Department budget for next year, it is my strong opinion that this should be done. This opinion is shared by my colleagues who are interested in Columbia’s past and prospective accomplishments in Mathematical Statistics.

Work in Mathematical Statistics in American universities is in a pioneering stage. The fundamental bases of statistics, in mathematics and logic, have recently been materially extended. New horizons have been opened. We may expect in the next several decades further fruitful advances bearing upon the whole range of inquiry in the social and the natural sciences and in the arts of production and administration. In this field Columbia has already, through the work of Hotelling and Wald, achieved a leading position, one that is recognized throughout the world. Some indication of Columbia’s standing, and of the scientific and practical fruitfulness of our work in this field, I given by the accomplishments of the Statistical Research Group now serving the Army and the Navy as part of Columbia’s contribution to the war effort.

Columbia must hold and extend the position of preeminence we have won. We believe that in Hotelling and Wald we have men of intellectual vigor and established scientific competence who will be in the forefront of future advances in Mathematical Statistics. Their work will supplement and strengthen that of the Watson Scientific Computing Laboratory, in which Columbia will cooperate with the International Business Machines Corporation, as the work of that Laboratory will enhance the effectiveness of our efforts in Mathematical Statistics.

The scholarly record of Professor Wald is set forth in an attached statement. I need only add here that Wald’s researches in statistical theory have been fundamental in character and seminal in their influence. A recent outstanding example of the fertility of his thought is provided by his contribution of a new mathematical basis for techniques of quality control in manufacturing production, techniques that have been widely adopted in the control of war production. The sequential methods utilized in Dr. Wald’s procedures are capable of application in scientific experiments, and in a wide variety of other fields.

In the conviction that Columbia should reinforce success, in planning for the future, and should build where firm foundations have already been laid, I urge that the position of the University in the field of Mathematical Statistics be maintained, and strengthened. Dr. Abraham Wald’s continuance here is crucial in such a program. I recommend, therefore, that Dr. Wald, who is now Associate Professor of Statistics at an annual salary of $5,000, be appointed Professor of Mathematical Statistics, at a salary of $7,500 a year, the appointment to be effective July 1, 1945.

Respectfully submitted,

FREDERICK C. MILLS

Source Columbia University Libraries, Manuscript Collections. Columbiana. Department of Economics Collection. Box 3 Budget, 1915-1946-47, Folder “Budget Material 1944-1945”.

___________________

April 27, 1951

Memorial Minute for Professor A. Wald

Professor Wolfowitz then presented a minute memorializing the late Professor Abraham Wald. It was adopted by a rising vote, and a copy was ordered sent to Professor Wald’s family.

ABRAHAM WALD

Abraham Wald, Professor of Mathematical Statistics and a distinguished member of this Faculty, was killed in an airplane accident in India on December 13, 1950. He had been on a lecture tour of Indian universities and research centers. Mrs. Wald was killed in the same accident.

Dr. Wald arrived in the United States in the summer of 1938, a refugee from Nazi persecution. In the fall of 1938 he came to Columbia as a fellow of the Carnegie Corporation. He became a member of this Faculty in 1942 and professor of Mathematical Statistics in 1945. Much of his statistical work was done here. It shed luster on Columbia and largely helped to make Columbia an important center of mathematical statistics. His work changed the whole course and emphasis of modern mathematical statistics. In addition to many other contributions the theory of statistical decision functions and the theory of sequential analysis were founded by him. He also made important contributions to mathematical economics, the theory of probability, and metric geometry.

He was a good friend to many, a genial colleague, and an inspiring teacher. By his death the University and science have sustained a grievous loss.

Source: Columbia University Archives. Minutes of the Faculty of Political Science, 1950-1962.

Image Source: Naval Ordnance Test Station, Inyokern, California from the obituary by J. Wolfowitz published in The Annals of Mathematical Statistics 23:1 (March, 1952), pp. 1-13.

Tags Haig, Hotelling, MillsFC, Wald, Wolfowitz

Bibliography Columbia Courses Economists Suggested Reading

Columbia. Friedman’s lecture notes to first Hotelling lecture in Mathematical Economics, 1933

Post author By Irwin Collier
Post date 2017-10-06

On October 3, 2017, Antoine Missemer tweeted an image of an undated examination question by Harold Hotelling “Describe two mathematical contributions to economics published before 1910”. One should note that asking students to talk about work published at least a quarter century before the current academic year is not necessarily a deep dive into the history of economics, though of course Cournot, Bertrand and Edgeworth had achieved “historical” fame by 1933.

From Harold Hotelling’s course in Mathematical Economics taught in the first semester of 1933/34 at Columbia, Milton Friedman kept about forty-five 3 by 5 inch index cards worth of notes (both sides). From his first lecture, we can put together a convenient “short list” of Hotelling’s chosen greatest hits in mathematical economics. I have taken the liberty of expanding Friedman’s abbreviations, figuring the main purpose of transcribing archival material is to ease digital search down the road.

Earlier postings include a list of Hotelling’s courses and his class rolls at Columbia as well as an outline and exam for his course in mathematical economics offered at North Carolina (1946, 1950).

___________________________

Milton Friedman’s student notes to Harold Hotelling’s first lecture in Mathematical Economics (1933)

9/2/33 (1)

Hotelling, Harold on Mathematical Economics

Has been stated that methodological difference between economics + natural sciences is that in former cannot + in latter do experiments

Not entirely true: in econonomics may experiment, + in some physical sciences (e.g. astronomy, meteorology etc.) do not experiment.

Better dividing line to be found in number of relevant factors

Use of Mathematics in Economics:

A. Cournot 1838

J. Bertrand 1883 Journal des Savants (reviewed Cournot)

F. Y. Edgeworth 1881 Math. Psychics. Papers relating to Pol. Economy.

Pareto

Alfred Marshall Principles of Economics

(Edgeworth laid foundation of many theories more modern than Marshall

Using higher Mathematics in Economics

G. C. Evans

C. F. Roos

Zeuthen

Pareto in Encyclopedie des Science Math, Vol I, Tome IV part 4 (Tome I, Vol. IV)

[Yes, that is all that Friedman wrote down for that lecture]

Source: Hoover Institution Archives. Milton Friedman Papers, Box 120, Class note cards.

___________________________

Links to Works Referred to by Hotelling

Cournot, Augustin. Recherches sur les Principes Mathématiques de la Théorie des Richesses. Paris: Hachett, 1838.

Nathaniel T. Bacon translation: Researches into the Mathematical Principles of the Theory of Wealth with a bibliography of Mathematical economics by Irving Fisher. New York: Macmillan, 1897.

Bertrand, J. (Review of) Théorie Mathématique de la Richesse Sociale par Léon Walras: Recherches sur les Principes Mathématiques de la Théorie des Richesses par Augustin Cournot. Journal des Savants 67 (1883), 499-508.

Edgeworth, F. Y. Mathematical Psychics. An Essay on the Application of Mathematics to the Moral Sciences. C. Kegan Paul & Co., 1881.

Edgeworth, F. Y. Papers Relating to Political Economy. Volume I; Volume II; Volume III. London: Macmillan, 1925.

Pareto, Vilfredo. Économie mathématique, —in Encyclopédie des sciences mathématique, Tome I, vol. 4 (Fascicule 4, pp. 590-640), 1906 [?].

Marshall, Alfred. Principles of Economics (8^th edition). London: Macmillan, 1920.

Griffith C. Evans. Mathematical Introduction to Economics. New York: McGraw-Hill Book Co., 1930.

Reviewed by Hotelling in Journal of Political Economy, 39, no. 1 (Feb 1931) pp. 107-09.

F. Zeuthen Problems of Monopoly and Economic Warfare. London: Routledge, 1930.

Reviewed by Corwin D. Edwards (New York University) in AER, 21, no. 4 (December, 1931), pp. 701-704.

Charles Frederick Roos. Dynamic Economics—Theoretical and Statistical Studies of Demand, Production and Prices. Monographs of the Cowles Commission for Research in Economics, No. 1. Bloomington, Indiana: Principia Press, 1934.

Image source: From a photo of the Institute of Statistics leadership around 1946: Gertrude Cox, Director, William Cochran, Associate Director-Raleigh and Harold Hotelling, Associate Director-Chapel Hill. North Carolina State University.

Tags Friedman, Hotelling

Columbia Economists

Columbia. Arrow on the Subordination of Price Theory, 1940-42

Post author By Irwin Collier
Post date 2017-08-30

Reading this account by Kenneth Arrow, I wondered why the lecturer in his history of economic thought course was not identified by name and who the lecturer was. In the Arrow papers at Duke’s Economists’ Papers Archive one finds his notes to John Maurice Clark’s course “On Current Types of Economic Theory” so for now I’ll presume that the son of the great John Bates Clark was the unknown lecturer of Arrow’s anecdote.

_________________________

Kenneth Arrow Recalls the Subordination of Price Theory at Columbia

The intellectual environment at Columbia University when I was a graduate student in 1940-1942 was far different from that in which the modern graduate student in economics finds himself. Neoclassical price theory now holds pride of place, as all students will acknowledge, some joyfully, some ruefully. But at Columbia at that period there was no required course in price theory. Indeed there was no course at all offered which gave a systematic exposition of microeconomics, except for Harold Hotelling’s one term offering of mathematical economics, the content of which would today be more or less standard for a general course but which was then regarded as highly esoteric indeed. The one required course which was most nearly equivalent to price theory was a course on the history of economic thought, where the lecturer gave potted summaries of everyone from the mercantilists on. Walras was barely mentioned and certainly was much less prominent than H. J. Davenport. Keynes was not mentioned (for that matter the General Theory was not mentioned even in the course on business cycles, though there were some glancing references to the Treatise on Money).

But the work of Thorstein Veblen was indeed prominently displayed in the course on economic thought, and it was no accident. The corrosive skepticism of Veblen towards “received” theory had, belatedly and even posthumously, under mined the never-very-secure hold of neoclassical thought on teaching of American economics. Of course he was not alone in effecting the change; the more benign, but equally negative, judgments of John R. Commons, in whose name we are gathered, shaped a generation of economists trained under him at the University of Wisconsin. At Columbia, the channel of influence was Wesley C. Mitchell, creator of the National Bureau of Economic Research. His version of the attack upon neoclassical economics was an insistence on the large-scale accumulation of data. It was in large part his direct influence plus the general background created by Veblen and Commons that led to the subordination of price theory at Columbia.

Source: From Kenneth J. Arrow, “John R. Commons Award Paper: Thorstein Veblen as an Economic Theorist.” The American Economist 19, no. 1 (1975): 5-9.

Image Source: Kenneth J. Arrow as Guggenheim Fellow (1972) John Simon Guggenheim Memorial Foundation.

Tags Arrow, Commons, Davenport, Hotelling, Mitchell

Courses North Carolina

North Carolina. Course outline and exam for mathematical economics, Hotelling. 1946 and 1950

Post author By Irwin Collier
Post date 2017-05-24

Harold Hotelling continued teaching courses in mathematical economics after leaving Columbia [here an earlier posting that lists all his courses taught at Columbia] for the University of North Carolina in 1946. From two folders in Hotelling’s papers in the Columbia University archives we can piece together the week-by-week list of topics he covered for the Fall quarter of 1946 and 1950. Note that the record for 1946 begins as a typed document that then is corrected and extended by hand-written additions.

______________________________

[Begin typed]

MATHEMATICAL ECONOMICS COURSE

Fall quarter, 1946. Two hours each Tuesday and Thursday.

Oct. 1. General scope of economic theory as treated mathematically. Static and dynamic economics. Greater superficial interest in dynamic economics but greater body of definite knowledge in the static field. Moreover an understanding of static theory is important for an understanding and elucidation of dynamic phenomena.

List of important mathematical writers on economic theory, especially static economic theory: Cournot, Dupuit, Walras, Edgeworth, Pareto, I. Fisher, Schultz. Also Amoroso, Pantaleoni, Slutzky. Bibliography of Hotelling writings on economic theory.

Oct. 3. Demand curves and functions. Classification of exchanges by double dichotomy according as buyer and seller are monopolistic or competitive. Indeterminacy of the case of monopoly vs. monopsony. Other cases classically regarded as determinate. But beware of assuming determinacy if there are as many as 2 monopolies in the system. Incidence of taxation on a monopolist of just one commodity (classical treatment). Proof that a tax of t per unit on monopolist leads him to increase the price.

Assignment for Oct. 8: Find 3 demand functions such that for monopoly and zero cost the increase in price will be (1) greater than t, (2) = t, (3) less than t. Be sure the functions are realistic in the sense of being monotonically decreasing, and that both price and quantity are positive.

Oct. 8: Discussion of problems assigned. (2 out of the 8 enrolled came forward with solutions.) Demand functions for 2 related commodities. The nature of cost functions for a single commodity: total cost as a function of quantity produced tends to rise stepwise in industrial production. Average cost varies in a quasi-periodic fashion, usually tending to decrease. Marginal cost usually low but with short periods of very high values as production increases.

[Handwritten insert:]

Oct. 10. Indifference loci. Utility, ophelimity, pleasure, non-measurable but ordinable.

[Last typed item:]

Oct. 15. Duopoly (“Stability in Competition,” Econ. J. 1929). Contributions of Cournot, Bertrand, Edgeworth.

[Handwritten items follow:]

Oct. 17. Bilateral monopoly, etc. Problem: Prove competitive price in Cournot’s duopoly < monopoly price with same demand function (Costs zero)

Oct. 22. Further discussion of problem. More on indifference curves & demand functions. Emphasis derived from “stability in competition” upon need for expressing quantities of cont[inuou]s functions of more than one price. Does Cournot duopoly imply a lower price than monopoly?

Oct. 24. Demand function with limited budgets.

Oct. 24. Equations of general equilibrium. Indeterminacy of price ratios.

Oct. 29. Bartlett gives proof (which the class could not find previously) that Cournot duopoly implied lower price than monopoly.
Equation of exchange: MV +M´V´= Σpq

Oct. 31. Theory of maxima & minima: 1^st order conditions, including Lagrange multiplier case. Relation to Taylor series.

Nov. 5. 2^nd order conditions. Definite & indefinite quadratic forms, with & without linear constraints.

Nov. 7 Application to obtaining 1^st-order conditions on demand functions unlimited budget case): symmetry conditions and inequalities. (Case of soap manufacturer)

Nov. 12. Conditions on supply functions in unlimited budget case. Demand functions with limited budgets. 6-term integrability conditions.

Nov. 14. Inequalities on demand functions with limited budgets; on supply functions with limited budgets

Nov. 19. Construction of suitable utility functions for general-equilibrium illustrations. Schultz attempted verifications. Problems of demand-function fitting. (Loaned Schultz book to [illegible word]) Taxation in general-equilibrium theory.

Nov. 21 Omitted

Paper will be required in lieu of exam.

Nov. 26. Edgeworth’s taxation paradox.

Dec. 3. Rent. Site rental & capital values. Benefit. Consumers’ & producers’ surpluses.

Dec. 5. Proof (à la “General Welfare” paper) that sales should be at marginal cost; also net loss given by a quadratic form.

Dec. 10. Connection of above with pp. 606ff. of “Edgeworth’s Taxation Paradox.” Taxation of site rentals; of scarce things such as space in crowded trains; on inheritances; of incomes; of nuisances. Minimization of net loss consistently with raising a specified revenue.

Dec. 12. Discussion. Index numbers of prices.

Source: Columbia University. Rare Book & Manuscript Library. Papers of Harold Hotelling, Courses Taught M-S (partial) Box 48, Folder “Mathematical Economics (2)”.

______________________________

Statistics/Economics 182. Mathematical Economics.
Autumn, 1950.
Harold Hotelling

September 21. General introduction. List of references. “Comparative statics,” as distinguished from study of transitional conditions. Edgeworth’s taxation paradox as a historic demonstration for need of calculus and algebra, not merely wordy or geometric arguments. Effect of tax on monopoly price with 1 commodity—the classical graphic argument. Assignment: (1) Prove that dp/dt, the rate of increase of monopoly price with tax t, is positive; (2) determine whether dp/dt has any positive limits which are the same for all monotonically decreasing demand functions having derivatives.

Sept. 26. Further discussion of taxation of monopoly. Assignment to calculate the effect on prices of 2 commodities controlled by 1 monopolist of a tax on 1 of these commodities, with a specified pair of demand functions.

Sept. 28. Preview of conditions under which Edgeworth’s taxation paradox may hold, and of nature of demand and supply functions.

Oct. 3. Duopoly. Cournot’s treatment. Duality with double monopoly (by different producers of parts of 1 final product). The 1929 “Stability in Competition” treatment. Mutual gravitation of competitors.

Oct. 5. Double dichotomy of markets, with extension to duopoly, oligopoly, duopsony, oligopsony; also to a multiplicity of commodities. Location of industry. Von Thünen, Goodrich. Problems of shape of a city; of layout of a railroad on a homogeneous plain for the purpose of bringing grain to one city.

Oct. 10. Holiday.

Oct. 12. Review of previous work. Classic supply and demand curves, with generalization to 2 commodities. illustration with linear demand and supply functions of effects of taxation of 2 commodities.

Oct. 17. Cost –total, marginal, average. Indeterminacy of average cost. Joint costs. Allocation logically impossible without consideration of demand. “Cost-finding systems” and cost accounting. Relative precision of marginal cost.

Oct. 19. Consumers’ surplus, producers’ surplus, benefits, effects of excise taxes—all for one isolated commodity; graphic and algebraic treatments. Distribution of excise taxes among independent commodities; but the necessity of replacing this result by something based on relations between commodities. The need of algebra and calculus rather than geometry for this.

Oct. 24. Demand functions for multiple commodities with unlimited budgets. Theory of maxima and minima.

Oct. 26, 31; Nov. 2, 7. Theory of maxima and minima; demand and supply functions with unlimited budgets. Symmetry-integrability conditions; inequalities

Nov. 9. Demand functions with limited budgets. Indifference curves. Utility.

Nov. 14. Further developments à la Slutsky and Hicks.

Nov. 16. Giffen phenomenon, exhibited by means of the utility function Ø = x – e^-y.

Nov. 21. Equations of general equilibrium, approximately according to Irving Fisher. Need of monetary equation to fix general level of prices.

Nov. 23. Thanksgiving holiday.

Nov. 28. Assignment: Work out and bring in next time (if not too hard) solution of equations of general equilibrium for 2 groups, farmers and fishermen, of equal numbers, large and competitive, with respective utility functions Ø = x – e^-y; and the smaller root of (x- Ø)(y- Ø) = 1. Calculus of variations in the small and in the large. Formulae for variations of prices and quantities in terms of excise tax rates for a group of commodities for which demand and supply maximize profits without budgetary limitations.

Nov. 30, Dec. 5, 7. Incidence and effects of taxation with unlimited and with limited budgets. Net loss from excise taxes is positive and approximately equal to ½ Σ t_iδ q_i. Criterion for social value of investment. Economy of making all sales at marginal cost. Index numbers of prices.

Source: Columbia University. Rare Book & Manuscript Library. Papers of Harold Hotelling, Courses Taught M-S (partial) Box 48, Folder “Mathematical Economics (1)”.

______________________________

Final Examination
Mathematical Economics. Math./Stat. 182
December 13, 1950

I.

A monopolist sells quantities x, y of two commodities at prices p₁, p₂ and pays taxes t₁, t₂ per unit sold respectively. His costs amount to*

C = 20 – x – y

and the demand functions are

x = 5 – 2p₁ + p₂,

y = 10 + p₁ – 3p₂.

Determine as functions of the tax rates (a) the prices and (b) the quantities yielding maximum revenue.

*(This cost function is unrealistic but the students were told to use it)

II.

If a toll of $p is levied for each crossing of a certain bridge, the number of crossings per year is q = 10,000 (9 -3p-p²) when this expression is positive, and is otherwise zero.

(a) What toll yields the maximum revenue?

When this toll is charged, …

(b) …how many crossings will be made per year?

(d) …what is the consumers’ surplus?

(e) …what is the total benefit from the bridge?

(f) What is the maximum possible total benefit?

III.

For a class of people all having the preference function

$\Phi =q_{1}^{{{\alpha }_{1}}}q_{2}^{{{\alpha }_{2}}}\cdots q_{n}^{{{\alpha }_{n}}}$

prove that a suitable index number of the cost of living is a certain weighted geometric mean.

Source: Columbia University. Rare Book & Manuscript Library. Papers of Harold Hotelling, Courses Taught M-S (partial) Box 48, Folder “Mathematical Economics (1)”.

Tags Hotelling

Columbia

Columbia. Preparation for Graduate Economics, 1941

Post author By Irwin Collier
Post date 2016-07-09

Up through the 1941-1942 Course Announcements of the Columbia University Faculty of Political Science did not provide prospective graduate students of economics any guidance with respect to their undergraduate preparation. Late in the Fall of 1941 the Executive Officer of the Department of Economics, i.e. chairman, Robert Murray Haig received suggestions and comments that were discussed at the December 2, 1941 faculty meeting that resulted in the insertion of two paragraphs into the Course Announcements that address undergraduate preparation in general and mathematical preparation in particular. Horace Taylor’s suggestion for the general preparation was taken over with only minor revisions. However we can see that the suggestion for mathematical preparation by Harold Hotelling and Frederick Mills was significantly toned down.

_________________________________

12 copies send around
Nov.

Dear Colleague:

Will you kindly examine the attached exhibits relating to material to be inserted in the announcements of the Faculty of Political Science and of Columbia College and be prepared to pass judgment at the meeting on December 2^nd?

(A) The Proposed Statement on Mathematical Preparation to be Inserted in the “Announcement of Courses” of the Faculty of Political Science.
Hotelling, Mills. October 13, 1941

Mathematical Preparation. The use of mathematics, including higher mathematics, has become important in several branches of economics and advanced statistics. Calculus, probability, the algebra of matrices and quadratic forms and the calculus of variations, for example, have important applications in economic study. Since the acquisition of an adequate mathematical training requires several years, students planning work that entails the use of advanced mathematics should include in their undergraduate studies courses providing the mathematical foundation essential to these advanced studies.

(B) Suggestion from Horace Taylor for Paragraph to be Inserted in the Columbia College Announcement and Comment on Hotelling and Mills’ Statement

October 20, 1941

Professor Robert M. Haig,
Fayerweather Hall.

Dear Professor Haig:

I enclose* two copies of a tentative paragraph intended to give effect in the Announcement to the recommendation made at our last departmental dinner. I would be glad to amend or amplify this in any way that seems desirable.

I have one or two misgivings as to the statement on mathematical preparation that has been prepared for the Announcement. In the first place it almost never happens that an undergraduate student decides to study economics in the graduate school earlier than the end of his junior year. Very often it happens at the end of his senior year. This lateness makes it impossible for such students to get the amount of mathematical training that is presented as desirable in this statement. In the second place, even those students who do decide to go in for graduate study at some point fairly early in their college careers are not likely to refer to our Announcement until a very short time before their actual application for admission as graduate students. Consequently the message presented in this statement would not reach them until too late. In the third place, I believe that the indefiniteness of the statement as it now stands might serve to frighten well qualified people away from graduate study of economics – at least in our department. Perhaps this difficulty would be relieved by making it more explicit as to just the fields of work in which such intensive mathematical preparation is a desirable prerequisite.

I doubt if we can accomplish very much in this regard by our own individual effort. I wonder if a broader attack in which it would be attempted to get the understanding and support of collegiate departments of economics would not be more successful. If, for example, the economics departments at Columbia, Chicago, Harvard, and perhaps two or three other principal graduate schools would agree on a general statement of what is desirable in the way of mathematical training and would publicize this through one or another of the Journals or by some other means, I think that better results would ensue. As I understand it, this question may come up for consideration at one of our later dinner meetings.

Sincerely,

HORACE TAYLOR

* “Undergraduate preparation. Since graduate study in economics necessarily entails a high degree of concentration in this field, students planning to enter graduate work are advised not to specialize narrowly in economics during their undergraduate study. Basic training in economics and a knowledge of its general literature and methods is desirable, but for the purposes of the more advanced work in graduate school, there is greater advantage in the study of history, philosophy, modern languages and mathematics than in narrowly specialized courses in economics taken as undergraduates.”

(C) Memorandum to Professor Haig from Professor Wolman: November 11, 1941

Professor Wolman agrees to the last paragraph typed on the page containing the memorandum from Professor Taylor. Doesn’t care how much Mathematics they are getting, no time to scare students away.

(D) Comment of Dean Calkins

Columbia University
in the City of New York

School of Business
Local

November 11, 1941

Professor R. M. Haig
Fayerweather

Dear Professor Haig

Your request for my comments on the proposed recommendation of undergraduate preparation for graduate study in economics and on Professor Taylor’s observations with respect to it prompts the following response:

I am impressed by the three points raised by Professor Taylor. They represent my own views after experience at California and Stanford. No satisfactory system now exists for detecting undergraduates who will later pursue graduate work in economics, and hence advice can rarely be given in time to be effective. It is my impression that most students who undertake graduate work in economics are as undergraduates either unacquainted with the opportunities in the field, unaware of their own interest in it, uncertain of their academic abilities to pursue graduate work, without prospects of financing graduate study, or forced by financial circumstances to utilize their four years of undergraduate study for instruction which might lead to employment upon graduation. Moreover many of these conditions also apply to first year graduate students and candidates for the master’s degree.

That there is no easy way to overcome the foregoing conditions is evident. Ordinarily a student needs to proceed some distance in the subject as an undergraduate to convince himself that he wishes to go on for graduate study, that he has the ability to go on, and that his opportunities in the field are sufficiently promising to justify the effort. I am impressed, too, with the number of cases in which graduate students receive their first impulse to go on for advanced study from an interest in a specialized course.

No statement in the Columbia College catalog alone can produce more than a small effect on the preparation of your graduate students, who are recruited so largely from other institutions. It is too vague to mean very much to the average undergraduate and will be interpreted by advisers according to their own predilections.
While I agree that more graduate students ought to have more of the sort of preparation recommended, we cannot be certain that this prescription is the only, best, or preferred preparation for either the students who may wish to undertake the graduate study of economics or who should be encouraged to do so.

In guiding the preparation of students who will be able to excel in economics we seek to produce graduates who can maintain high standards of competence, not standardized products.

I have no serious objection to the statement as a guide for one type of preparation, but this is clearly not the only desirable type of preparation. Its value probably lies in the prospect that a few will heed it, and that may be desirable, and the great majority will ignore it and that may also be desirable.

I shall be glad to discuss this with you if you desire an amplification of these opinions.

Sincerely yours,
[signed] Robert D. Calkins
Dean

Source: Columbia University Libraries, Manuscript Collections. Columbiana, Department of Economics Collection, Faculty. Box 2, Folder “Department of Economics—Faculty Beginning Jan 1, 1944”.

_________________________________

Recommended preparations printed in the 1942-43 Course Announcements

General Undergraduate Preparation. Since graduate study in economics necessarily entails a high degree of concentration in this field, students planning to enter graduate work are advised not to specialize narrowly in economics during their undergraduate study. Basic training in economics and a knowledge of its general literature and methods is desirable, but for the purposes of the more advanced work on the graduate level, there is greater advantage in the study of history, philosophy, modern languages and mathematics than in narrowly specialized courses in economics taken as undergraduates.

Mathematical Preparation. The use of mathematics, including higher mathematics, has become important in several branches of economics and statistics. Much of the recent important literature of general economics is written in a language not easily understood without some knowledge of the differential and integral calculus. Students planning to work for the degree of Doctor of Philosophy in economics will therefore find it advantageous to acquire familiarity with the calculus and with higher algebra before beginning their graduate studies in economics.

Source: History, Economics, Public Law, and Sociology. Courses Offered by the Faculty of Political Science for the Winter and Spring Sessions 1942-1943. Columbia University, Bulletin of Information, Forty-second Series, No. 24, May 23, 1942, p. 18.

_________________________________

The 1948 Directory of the American Economic Association. American Economic Review, Vol. 39, No. 1 (January 1949).

HAIG, Robert Murray, Columbia Univ., Fayerweather Hall, New York 27, N.Y. (1911) Columbia Univ., McVikar Prof. of Polit. Econ., teach., res.; b. 1887; A.B., 1908, LL.D., 1925, Ohio Wesleyan; M.A., 1909, Illinois; Ph.D., 1914, Columbia; LL.D., 1944, Rollins. Field 9 [Public Finance]. Doc.dis. History of general property tax in Illinois ([Flanigan-Pearson Company, Printers] Univ. of Illinois, 1914). Pub. “Taxation of excess profits in Great Britain,” A.E.R., 1920; Economic factors in metropolitan growth and arrangement (Russell Sage Found., 1927); Sales tax in American states (with Shoup) (Columbia Univ. Press, 1929). Res. Concept of taxable income; federal state financial relations. Dir. W.W. in Amer., Dir. of Schol., Lead. in Educa. Int. W.W. [p. 77]

MILLS, Frederick Cecil, Columbia Univ., New York 27, N.Y. (1920) Columbia Univ., prof of econ. and statis.; Nat. Bur. of Econ. Res., memb. res. staff; teach., res., govt. serv.; b. 1892; B.L., 1914, M.A., 1916, LL.D., 1947, California; Ph.D., 1917, Columbia; 1919, London School of Econ. Fields 3 [Statistics and Econometrics], 6 [Business Fluctuations], 5 [National Income and Social Accounting]. Doc dis. Contemporary theories of unemployment (Columbia Univ. Press, 1917). Pub. Behavior of prices (1927), Economic tendencies in U.S. (1932) (Nat. Bur. of Econ. Res.); Statistical Methods (Holt, 1924, 1938). Res. Prices in business cycles; industrial productivity. Dir. W.W. in Amer., Dir. of Schol. [p. 129]

WOLMAN, Leo, 993 Park Ave., New York 28, NY. (1915) Columbia Univ., prof. of econ.; Nat. Bur. of Econ. Res., res. staff; b. 1890; A.B., 1911, Ph.D., 1914, LL.D., 1948, Johns Hopkins. Fields 16 [Labor], 6 [Business Fluctuations], 3c [Economic Measurements]. Doc. dis. Boycott in American trade unions (Johns Hopkins Press, 1916). Pub. Growth of American trade unions, 1880-1923 (1924), Planning and control of public works (1930), Ebb and flow in trade unionism (1936) (Nat. Bur. of Econ. Res.). Res. Wages in U. S. since 1860; changes in union membership. Dir. W.W. in Amer., Dir. of Schol. Lead in Educa. [p. 204]

CALKINS, Robert D., 445 Riverside Dr., New York 27, N.Y. (1930) Gen. Educa. Bd., dir.; B.S., 1925, LL.B., 1942, William and Mary; M.A., 1929, Ph.D., 1933, Stanford. Fields 11a [Business Organization, Administration, Methods, and Management], 12a [Industrial Organization and Market Controls; Policies Concerning Competition and Monopoly], 14a [Industry Studies: Manufacturing]. Doc. dis. Price leadership among major wheat futures markets (Wheat Studies, Nov., 1933). Dir. W.W. in Amer. [p. 30]

_________________________________

The 1942 Directory of the American Economic Association. American Economic Review, Vol. 33, No. 1, Part 2, Supplement (March 1943).

TAYLOR, Horace, Columbia Univ., New York City. (1924) A [Institution, rank, nature of activity]Columbia Univ., prof. of econ., TRA [teaching, research, administration]. B [Degrees] A.B., 1922, Oklahoma; A.M., 1924, Ph.D., 1929, Columbia. C [doctoral dissertation]Making goods and making money (Macmillan, 1929). D [Fields] 1 [Economic theory; general works], 10 [Public control of business; public administration; national defense and war], 3 [Economic systems; national economics]. E [Research projects underway] Systematic economic theory. F [Most significant publications]Main currents in modern economic life (Harcourt, Brace, 1941). G [Directories cross referenced] SE [Biographical Directory of American Scholars, Leaders in Education]. [p. 11]

_________________________________

Harold Hotelling. Professor of Economics

A.B. Washington, 1919; M.S., 1921; Ph.D., Princeton, 1924.

_________________________________

Image Source: Columbia Spectator Archive. Left: Horace Taylor (14 April 1959). Right: Frederick C. Mills (11 February 1964)

Tags Calkins, Haig, Hotelling, Mills, TaylorH, Wolman

Columbia Courses Economists

Columbia. Course Descriptions. Hotelling. 1931-1945

Post author By Irwin Collier
Post date 2015-07-15

Class rolls from Hotelling’s courses on mathematical economics he taught at Columbia have been posted, as have course outlines and a final exam from the course as taught at the University of North Carolina in 1946 and 1950.

_________________________

[1931-32]

Economics 311-312—Statistical inference. 3 points each session. Professor H. Hotelling.
M. and W. at 10. 412 Fayerweather.

Summarizing and interpretation of data; probability, mathematical and philosophical; the normal law of error; probable errors; student’s distribution and the work of R. A. Fisher; least squares; fitting trend lines and other empirical curves; their reliability; accuracy of forecasts; tests of stability, homogeneity, and goodness of fit; analysis of variance; simple, partial and multiple correlation coefficients and their interpretation in terms of probability; periodogram analysis.

Prerequisite: Calculus. A knowledge of determinants is desirable.

Economics 313-314—Mathematical economics. 3 points each session. Professor H. Hotelling.
M. and W. at 4:10. 302 Fayerweather.

Supply and demand functions; monopoly; competition; duopoly; utility; taxation; tariffs; index numbers; exhaustible resources; dynamical economics.

Prerequisite: Calculus.

[1932-33]

Economics 311—Statistical inference. 3 points Winter Session. Professor H. Hotelling.
M. and W. at 10. 412 Fayerweather.

Summarizing and interpretation of data; probable errors; significance of means, of differences of variances; and of least-square determinations; accuracy of forecasts; Student’s distribution and R. A. Fisher’s extensions; comparison of observed with theoretical frequencies; tests of independence, homogeneity and goodness of fit. Proofs of the formulae are considered. Examples are drawn from a variety of fields, both within and outside of the social sciences.

Prerequisite: Calculus is a prerequisite. Mathematics 101 (Probability. Professor B. O. Koopman. Tu. and Th., 2:10-3) must be taken simultaneously. Mathematics 57 (Higher algebra. Professor L. P. Siceloff. M., W., and F. at 11) should also be taken at the same time if the student is to go on with Economics 312 or undertake research in statistics, unless he is already familiar with determinants and quadratic forms.

Graduate credit in economics will be allowed for Mathematics 57 and 101, which are required for this work in statistics.

Economics 312—Modern statistical theory. 3 points Spring Session. Professor H. Hotelling.
M. and W. at 10. 412 Fayerweather.

Correlation, simple, partial and multiple, with exact and approximate tests of significance; comparison and analysis of variances; the theory of estimation and efficiency; frequency curve fitting; analysis of time series; periodicity. Recent discoveries will be discussed.

Prerequisites: Economics 311 and Mathematics 57 and 101.

Economics 313-314—Mathematical economics. 3 points each session. Professor H. Hotelling.
M. and W. at 4:10. 302 Fayerweather.

Supply and demand functions; monopoly; competition; duopoly; utility; taxation; tariffs; index numbers; exhaustible resources; dynamical economics.

Prerequisite: Calculus. Mathematics 57 will also be found helpful, though it is not a required prerequisite.

[1933-34]

[Starting this year the two courses in Statistics were moved from “Research Courses” to “General Courses” as reflected in the course numbering. Note the label “Statistics” instead of “Economics” before the course numbers]

Statistics 111—Statistical inference. 3 points Winter Session. Professor H. Hotelling.
M. and W. at 10. 412 Fayerweather.

Calculus is a prerequisite. Mathematics 101 (Probability. Professor B. O. Koopman. Tu. and Th., 2:10-3) should be taken simultaneously. Mathematics 57 (Higher algebra. Professor L. P. Siceloff. M., W., and F. at 11) should also be taken at the same time if the student is to go on with Statistics 312 or undertake research in statistics, unless he is already familiar with determinants and quadratic forms.

Graduate credit in economics will be allowed for Mathematics 57 and 101, which are required for this work in statistics.

Statistics 112—Statistical inference. 3 points Spring Session. Professor H. Hotelling.
M. and W. at 10. 412 Fayerweather.

Prerequisites: Economics 111 and Mathematics 57 and 101 or equivalent knowledge.

Economics 117-118—Mathematical economics. 3 points each session. Professor H. Hotelling. [Starting this year Mathematical Economics was moved from “Research Courses” to “General Courses” as reflected in the course numbering.]
M. and W. at 4:10. 418 Business.

Supply and demand functions; monopoly; competition; duopoly; utility; taxation; tariffs; index numbers; exhaustible resources; dynamical economics.

Prerequisite: Calculus. Mathematics 57 will also be found helpful, though it is not a required prerequisite. Graduate credit in economics is allowed for Mathematics 57 and Mathematics 101.

[1934-35]

Statistics 111—Statistical inference. 3 points Winter Session. Professor Hotelling.
M. and W. at 10. 412 Fayerweather.

Summarizing and interpretation of data; probable errors; significance of means, of differences of variances; and of least-square determinations; accuracy of forecasts; student’s [sic] distribution and R. A. Fisher’s extensions; comparison of observed with theoretical frequencies; tests of independence, homogeneity and goodness of fit. Proofs of the formulae are considered. Examples are drawn from a variety of fields, both within and outside of the social sciences.

Graduate credit in economics will be allowed for Mathematics 57 and 101, which are required for this work in statistics.

Statistics 112—Statistical inference. 3 points Spring Session. Professor Hotelling.
M. and W. at 10. 412 Fayerweather.

Prerequisites: Economics [sic, Statistics] 111 and Mathematics 57 and 101 or equivalent knowledge.

Statistics 301—Seminar in advanced mathematical statistics. 3 points Winter Session. Professor Hotelling.
[According to Bulletin 1935-36 this course was given in 1934-1935 but not given in 1935-1936]

Economics 117-118—Mathematical economics. 3 points each session. Professor Hotelling.
M. and W. at 4:10. 418 Business.

The fundamentals of economics as a set of problems in maxima and minima. The maximizing of utility or of profits by individuals and the consequent equations of general equilibrium; the nature and interrelations of utility, curves and surfaces of indifference, demand and supply functions, consumers’ surplus, and welfare. Monopoly and various forms of competition. The extent to which selfish activities of individuals promote the general welfare; contrasts and resemblances of a planned society with the outcome of competition. Interrelation of prices. Taxation. Dynamical economics: cycles, lagging effects, and exhaustible resources.

The theory of maxima and minima of functions of n variables, with and without restraining conditions; the elements of the calculus of variations; tensors. The effects of variation of parameters on maximizing conditions are applied to discover consequences of taxation and other interferences.

A thorough knowledge of calculus, with something of differential equations, is an essential prerequisite. Mathematics 57 (Higher algebra; M., W., and F. at 11) is highly desirable. Graduate credit in economics is allowed for Mathematics 57 and Mathematics 101.

[1935-36]

Statistics 111—Statistical inference. 3 points Winter Session. Professor Hotelling.
M. and W. at 4:10 611 Business.

Summarizing and interpretation of data. Frequency distributions. Significance of the normal distribution. Accuracy of means and of differences of means. Relations of statistics and probability.

Prerequisites: Calculus and Mathematics 101 (probability); but the latter may be taken simultaneously. Mathematics 57 (higher algebra) is recommended.

Graduate credit in economics will be allowed for Mathematics 57, and Mathematics 101.

Statistics 112—Least squares and the treatment of time series. 3 points Spring Session. Professor Hotelling.
M. and W. at 10. 412 Fayerweather.

The classical method of least squares and modern modifications and developments, with stress on the interpretation of results in terms of probability. Diverse applications, both to social and to natural sciences.

The problems of observations ordered in time. Correlation and regression of time series. Seasonal variation and secular trend. Methods of correcting lack of independence and avoiding fallacies. Periodogram analysis. Recent discoveries and improvements.

Prerequisites: Statistics 111, Mathematics 57, and Mathematics 101 or equivalent knowledge.

Statistics 114 [new course]—Correlation, analysis of variance, and the χ² test. 3 points Spring Session. Professor Hotelling.
M. and W. at 4:10. 611 Business.

The multivariate normal distribution. Simple, partial, multiple, and vector correlation. Rank correlation and the problem of non-normal populations. Tests of independence, homogeneity, and goodness of fit for tables of frequencies. The analysis of variance and covariance to segregate factors producing significant variation. Recent discoveries in statistical theory.

Prerequisites: Statistics 111, Mathematics 57 and Mathematics 101.

[Statistics 301—Seminar in advanced mathematical statistics. 3 points Winter Session. Professor Hotelling.
Given in 1934-1935; not to be given in 1935-1936]

Economics 117 [Course reduced to a single semester]—Mathematical economics. 3 points Winter Session. Professor Hotelling.
M. and W. at 10. 507 Business.

The consequences of the simultaneous attempts by different persons to maximize their respective profits or degrees of satisfaction. Utility, indifference curves, demand, supply and cost functions, monopoly and various forms of competition, interrelations of prices, taxation. Theory of interest, depreciation, exhaustible resources. Contrasts and resemblances of a planned society with the outcome of competition. Dynamical economics: cycles and lagging effects.

The theory of maxima and minima of functions of n variables, with and without constraining conditions, is developed beyond the treatments in calculus books to include the second-order conditions. Elements of the calculus of variations.

Prerequisite: A thorough knowledge of calculus. Mathematics 57 (higher algebra) is highly desirable. Graduate credit in economics is allowed for Mathematics 57 and Mathematics 101.

[1936-37]