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Harvard. Intro to Mathematical Economics. Schumpeter, Leontief 1935-42

Graduate classes in Mathematical Economics (Econ 13b in 1934-35, Econ 104b in later years) were taught every second year by Edwin Biddle Wilson (1934-35, 1936-37, 1938-39, 1940-41, 1942-43). An introduction for undergraduates and graduates was offered by Joseph Schumpeter in 1934-35 (Econ 8a), but the course was taken over and offered for nearly a decade by Wassily Leontief (new course number beginning 1936-37, Econ 4a). In this posting you will find different scraps from the Schumpeter/Leontief course over the years.

 

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[Schumpeter’s exam questions (1934-35)]

[Note these exam questions are in the Ec11 Folder. Instructor: Schumpeter according to course catalogue]
[Introduction to the Mathematical Treatment of Economic Theory, 1934/35 academic year]

 

Ec 8a
Midyear Exam Febr 4th 1935

Answer at least three of the following questions:

  1. Define elasticity of demand, and deduce that demand function, which corresponds to a constant coefficient of elasticity.
  2. Let D be quantity demanded, p price, and D = a – bp the demand function. Assume there are no costs of production. Then the price p0 which will maximize monopoly-revenue is equal to one half of that price p1, at which D would vanish. Prove.
  3. A product P is being produced by two factors of production L and C. The production-function is P = bLkC1-k , b and k being constants. Calculate the marginal degrees of productivity of L and C, and show that remuneration of factors according to the marginal productivity principle will in this case just exhaust the product.
  4. In perfect competition equilibrium price is equal to marginal costs. Prove this proposition and work it out for the special case of the total cost function
    y = a + bx, y being total cost, x quantity produced, and a and b
  5. If y be the satisfaction which a person derives from an income x, and if we assume (following Bernoulli) that the increase of satisfaction which he derives from an addition of one per cent to his income, is the same whatever the amount of the income, we have dy/dx = constant/x.
    Find y. Should an income tax be proportional to income, or progressive or regressive, if Bernoulli’s hypothesis is assumed to be correct, and if the tax is to inflict equal sacrifice on everyone?

[Following derivation added in pencil]
{{p}_{1}}=\frac{a}{b}

\frac{dp}{dD}=-\frac{1}{b}

\frac{d\,\,Dp}{dp}=D+p\frac{dD}{dp}=

=a-bp-bp=a-2bp

\therefore p=\frac{a}{2b}

Source: Harvard University Archives. Joseph Schumpeter Lecture Notes HUC(FP)–4.62, Box 9, Folder: “Ec 11 Fall 1935”.

 

Transcription of Schumpeter’s official typed version of the Economic 8a, 1934-35.

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[1935-36]

*Economics 8a 2hf. Introduction to the Mathematical Treatment of Economics
Half-course (second half-year). Mon. 4 to 6. Assistant Professor Leontief. [Course may be taken by either undergraduates or graduates for credit.]

Economics A [Principles of Economics] and Mathematics A, or their equivalents, are prerequisites for this course.

 

Source: Harvard University. Announcement of the Courses of Instruction Offered by the Faculty of Arts and Sciences for the Academic Year 1935-36 (2nd edition), p. 138.

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[Excerpt from undated lecture notes, in Folder “Introduction mathematical economics 1937”]

Intr. to Math. Ec.

Introduction

Math. Ec. Economics & Mathematics

I. The subject of m.e. and Economic Theory is the same.

MathEconVennDiagram

  1. Parts of Economics – non-quantitative in character.
  2. Parts of Economics—quantitative but can be handled without math symbols. (marg. cost [unclear word].
  3. Quantitative—of such complexity that it hardly can be handled without math. symbols (f. ex. general equilibrium distribution etc.)

Fundamental difference only in the method of handling.

Non-math economists “are mathematicians without knowing it”

 

II. Two application of math. in economics.

a) theory b) statistics

Difference in application of math to economic theory and f.ex. to physics: More general type of argument Instead of definite interrelation we have knowledge only of some characteristics.

Math economics is not imitation of physics.

 

III. Fundamental problem of math. ec.:

Translation of economic problems into mathematical terms and back. Math. economist must know economics and mathematics.

In math. econ. To formulate a problem means to solve it.

IV. The aim of this course is to

  1. teach you to apply math. to the analysis of theor. ec. problem.

Mostly we will dwell in “region 2” although some time we will advance into the “region 3”.

Main subjects.

Theory of value.

Theory of production.

  1. Procedure:

a) lecture on fundamental problem

b) Discussion of special applications

c) Solution of problems out of class.

3. Knowledge of math:

a) elementary algebra

b) elementary calculus

c) partial derivatives

Knowledge of ec.

Ec A [Principles of Economics].

4. Graphic analysis vs. calculus.

Graphic analysis is a summary which helps us to talk of.

 

V.  Literature

  1. Antoine A. Cournot (1801-1877).
    “Researches into the mathematical principles of the theory of wealth” (1838)

Léon Walras (1834-1910).
“Elements of pure political economy” (1874-1877)

Vilfredo Pareto (1848-1923)
“Cours d’Economie Politique” (1896)
“Manuale d’economiea politica” (1906)

  1. Irving Fisher
    “Mathematical Investigations in the Theory of value and prices” (1892)
    F. Y. Edgeworth (1845-1926)
    Alfred Marshall (1842-1924)  “Appendix to the Principles”

Italian School
“Econometrica”
“Review of Economic Studies”
etc.

  1. No good textbook

A. L. Bowley
“The Mathematical Groundwork of Economics”, 1924.

Evans
“Introduction into mathematical economics”.

Source: Harvard University Archives. Wassily Leontief Papers. HUG 4517.30, Box 5, Folder “Introduction to Mathematical Economics (notes)”.

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[Reading Period assignment: 1936, Leontief ]

Economics 8a: Evans, G. C., Mathematical Introduction to Economics, Chs. I and II.

 

Source: Harvard University Archives. Syllabi, course outlines and reading lists in Economics, 1895-2003. HUC 8522.2.1, Box 2, Folder “1935-1936”.

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[Course Final Exam 1936, Leontief]

[carbon copy]

1935-36

HARVARD UNIVERSITY
ECONOMICS 8a2

 

Answer THE FIRST and at least THREE of the subsequent questions:

  1. Discuss the relation between the cost function and the production function of a single enterprise.
  2. Prove that in the point where the average unit costs are the smallest, they are equal to the marginal costs.
  3. Given a total revenue curve, R = Aq –Bq2, and a total cost curve, C= K + Lq, find the monopoly output, the monopoly price and the net revenue of monopolist. (A, B, K, and L are constants.)
  4. Discuss Cournot’s analysis of competition between two monopolies (duopoly).
  5. Given the production function Z = x½ y½ find out whether the two factors x and y are complementary or competing.
  6. Derive the relation between factor prices and marginal productivities under conditions of free competition (fixed prices).

Final   1936

 

Source: Harvard University Archives. Wassily Leontief Papers. Box 5, Folder “(notes Introduction to Mathematical Economics”

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[Reading Period assignment: 1937, Leontief ]

 Economics 4a:

A. Cournot, Researches into Mathematical Economics.

Ch. IV, pp. 44-55;
Ch. V, pp. 56-61.
Ch. IX, pp. 99-107.

Source: Harvard University Archives. Syllabi, course outlines and reading lists in Economics, 1895-2003. HUC 8522.2.1, Box 2, Folder “1936-1937”.

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[Reading Period assignment: 1938, 1939, 1940, Leontief]

Economics 4a: Read the following

  1. Cournot, Researches into Mathematical Economics. Chs. IV, V, VII, VIII, IX.
  2. Evans, Mathematical Introduction to Economics, Ch. II.

Source: Harvard University Archives. Syllabi, course outlines and reading lists in Economics, 1895-2003. HUC 8522.2.1, Box 2, Folders “1937-1938”, “1938-39”, “1939-40”.

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[Course Outline, Leontief]

Economics 4a
1939-40
Introduction to the Mathematical Treatment of Economic Theory

 

Introductory remarks. The profit function and the profit tax. The cost function; total, fixed, variable, marginal and average costs. Minimum average total and minimum average variable costs. General properties and the cost function. Aggregate cost function of a multiple plant enterprise.

The revenue function, the demand function and the price. Marginal revenue and elasticity concept. Principle of dimensional transformation. Conditions for the existence of an individual supply function.

Introduction into the theory of the markets. Necessary and sufficient conditions for the existence of market supply and market supply functions. Competition and monopoly. Theory of discrimination.

Introduction into the study of the production function. Marginal productivity, increasing and diminishing returns. Complementary and competing factors. Principle of minimum costs. Cost function and production function.

Introduction into the theory of consumers behavior: Concept of the indifference varieties.

Introduction into the analysis of dynamic economies. The cobweb problem and basic equilibrium concepts.

Introduction into the theory of general interdependence. Data and variables, basic equations and unknowns.

 

Source: Harvard University Archives. Syllabi, course outlines and reading lists in Economics, 1895-2003. HUC 8522.2.1, Box 2, Folder “1939-1940”.

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[Course Outline, Leontief]

Economics 4a
1941-42 [also for 1942-43]

 

  1. Introductory remarks.
    The profit function.
    Maximizing profits.
  2. The cost functions: Total costs, fixed costs, variable costs, average costs, marginal costs, increasing and decreasing marginal costs.
    Minimizing average total and average variable costs.
  3. The revenue function.
    Price and marginal revenue.
    Demand function
    Elasticity and flexibility.
  4. Maximizing the net revenue (profits).
    Monopolistic maximum.
    Competitive maximum.
    Supply function.
  5. Joint costs and accounting methods of cost imputation.
    Multiple plants.
    Price discrimination.
  6. Production function.
    Marginal productivity.
    Increasing and decreasing productivity.
    Homogeneous and non-homogeneous production functions.
  7. Maximizing net revenue, second method.
    Minimizing costs for a fixed output.
    Marginal costs and marginal productivity.
  8. Introduction into the theory of consumers’ behavior.
    Indifference curves and the utility function.
  9. Introduction to the theory of the market.
    Concept of market equilibrium.
    Duopoly, bilateral monopoly.
    Pure competition.
  10. Cobweb problem.
  11. Introduction into the theory of general equilibrium.

 

Reading: R. G. D. Allen, Mathematical Analysis for Economists.

Evans, Introduction into Mathematical Economics.

Antoine Cournot, Researches into the Mathematical Principles of the Theory of Wealth.

Weekly problems.

Source: Harvard University Archives. Syllabi, course outlines and reading lists in Economics, 1895-2003. HUC 8522.2.1, Box 3, Folders “1941-1942”. “1942-1943 (1 of 2)”.

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[Reading Period assignment: 1942, Leontief]

 

Evans, Introduction into Mathematical Economics. Ch’s I, II, III

Antoine Cournot, Researches into the Mathematical Principles of the Theory of Wealth. pp. 44-55, 56-66, 99-107.

Econ. 4.[a]

 

Source: Harvard University Archives. Syllabi, course outlines and reading lists in Economics, 1895-2003. HUC 8522.2.1, Box 3, Folder “1941-1942”.