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.
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[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: 1st order conditions, including Lagrange multiplier case. Relation to Taylor series.
Nov. 5. 2nd order conditions. Definite & indefinite quadratic forms, with & without linear constraints.
Nov. 7 Application to obtaining 1st-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)”.
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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 ti δ qi. 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)”.
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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?
(c) …what is the revenue?
(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
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)”.
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.