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Exam Questions Harvard Leontief Undergraduate

Harvard. Undergraduate mathematical economics. Schumpeter, Leontief, Goodwin. 1933-1950

 

 

Joseph Schumpeter introduced a one semester undergraduate course “Introduction to the Mathematical Treatment of Economic Theory” in the first semester of the 1933-34 academic year at Harvard. Schumpeter taught the course three times and it was taught from 1935-36 through 1947-48 by Wassily Leontief. The course was then continued by Richard Goodwin in 1949-50. This post presents a grab-bag of information that includes early and a late course description, annual enrollment data, a course outline from 1945-46 and five exams. Links to all earlier posts for the course available at Economics in the Rear-view Mirror have been included as well.

Some of the backstory to this course is included in this earlier post (memo by Crum of 4 April 1933 and a list of topics to be covered).

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Course Announcement, 1933-34

Economics 8a 1hf. Introduction to the Mathematical Treatment of Economic Theory

Half-course (first half-year). Mon., 4 to 6, and a third hour at the pleasure of the instructor. Professor Schumpeter, and other members of the Department.

Economics 8a is open to those who have passed Economics A, and Mathematics A, or its equivalent. The aim of this course is to acquaint such students as may wish it with the elements of the mathematical technique necessary to understand the simpler contributions to the mathematical theory of economics.

Source:  Announcement of the Courses of Instruction offered by the Faculty of Arts and Sciences 1933-34 (Second edition) in Official Register of Harvard University, Vol. XXX, No. 39 (September 20, 1933), p. 126.

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Course Enrollment, 1933-34

[Economics] 8a 1hf. Professor Schumpeter. — Introduction to the Mathematical Treatment of Economic Theory.

15 Graduates, 3 Seniors, 5 Others. Total 23.

Source: Harvard University. Report of the President of Harvard College and Reports of Departments for 1933-1934, p. 85.

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Exam not found for Economics 8a, 1933-34

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Course Enrollment, 1934-35

[Economics] 8a 1hf. Professor Schumpeter. — Introduction to the Mathematical Treatment of Economic Theory.

2 Seniors, 1 Junior, 1 Sophomore. Total 4.

Source: Harvard University. Report of the President of Harvard College and Reports of Departments for 1934-1935, p. 81.

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 1935 final exam questions.

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Course Enrollment, 1935-36

[Economics] 8a 2hf. Asst. Professor Leontief. — Introduction to the Mathematical Treatment of Economic Theory.

4 Juniors, 2 Sophomores. Total 6.

Source: Harvard University. Report of the President of Harvard College and Reports of Departments for 1935-1936, p. 82.

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Implicit course outline and course readings with the 1936 exam questions.

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Course Enrollment, 1936-37

[Economics] 4a 2hf. Asst. Professor Leontief. — Introduction to the Mathematical Treatment of Economic Theory.

1 Graduate, 2 Seniors, 3 Juniors, 2 Sophomores, 1 Other. Total 9.

Source: Harvard University. Report of the President of Harvard College and Reports of Departments for 1936-1937, p. 92.

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Final Examination, 1936-37
HARVARD UNIVERSITY
ECONOMICS 4a

Answer at least THREE questions: one in each group

Group I

  1. Discuss the relation between the production function of an enterprise and its cost curve.

 

Group II

  1. Given a cost of a single plant:
    C=\frac{1}{A+X}+BX
    where C indicates the total cost, X the total output, and the magnitudes of the two constants are such
    that A< 0 and B> 1/A.
    Derive the total cost curve of an enterprise which consists of two identical plants of this kind.
  2. A monopolist sells in two markets a commodity produced without costs. The total revenue, R1, obtained from the sale of q1 units of this commodity in the first market is given by:
    {{R}_{1}}=A{{q}_{1}}+Bq_{1}^{2}\text{ }\left( A>0,\text{ }B<\text{ }0 \right)
    The sale of q2 units in the second market nets:
    {{R}_{2}}=K{{q}_{2}}+Lq_{2}^{2}\text{ }\left( K>0,\text{ }L<\text{ }0 \right)
    Compute the prices which this monopolist would charge (a) with discrimination between the two markets; (b) without discrimination.

 

Group III

  1. Prove that marginal costs are increasing in the point of minimum average costs.
  2. Prove that a tax on profits cannot affect the output of an enterprise unless it induces it to suspend its operations.

 

Source: Harvard University Archives. Examination Papers. Finals 1937. (HUC 7000.28) Vol. 79. Faculty of Arts and Sciences. Papers Printed for Final Examinations. History, History of Religions, …, Economics, …, Military Science, Naval Science. January-June, 1937.

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Course Enrollment, 1937-38

[Economics] 4a 2hf. Asst. Professor Leontief. — Introduction to the Mathematical Treatment of Economic Theory.

2 Graduates, 2 Seniors, 6 Juniors, 1 Sophomore. Total 11.

Source: Harvard University. Report of the President of Harvard College and Reports of Departments for 1937-1938, p. 85.

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Final Examination, 1937-38
HARVARD UNIVERSITY
ECONOMICS 4a2

Answer THREE questions including question 1. Devote to discussion of question 1 about one hour and a half.

  1. Discuss fully the relation between the production function and the cost curve of an enterprise.
  2. Given:
    1. The cost curve of a monopolist:
      C= A+ BQ+ CQ2
      C indicates the total cost, Q the total output, A, B, C,are given constants.
    2. The demand function for his product in Market I.
      q1= a1– b1p1
      q1 is the quantity consumed for his product in Market I at the price p1.
      a1and b1 are given constants
    3. The demand function for his product in Market II.
      q2= a2– b2p2
      q2is the quantity taken in at the price p2;
      a2 and b2 are given constants.
      The monopolist is able to discriminate between the two markets provided the difference between the two prices is not larger than K
      Find (and express in terms of the given constants) that the value of Kwhich would maximize the sales q1 in the first market.
  3. Given:
    1. A, monopolist’s cost curve:
      C = A+ BQ+ CQ
    2. The demand curve for his product:
      p= a – bQ
      C stands for total costs, Q for total output, p for the market price, A, B, C, d, and b are constants.
      A subsidy at S dollars is paid to the monopolist per unit of output.
      Find how large the subsidy S must be in order to induce him to produce and sell twice as much as he would without the subsidy.
  4. Is it possible that the average costs of an enterprise are increasing with the output while the marginal costs are decreasing at the same time?
    Give and answer and demonstrate that it is correct.

 

Source: Harvard University Archives. Harvard University Final Examinations, 1853-2001. (HUC 7000.28) Box 4. Faculty of Arts and Sciences. Papers Printed for Final Examinations. History, History of Religions, …, Economics, …, Military Science, Naval Science. January-June, 1938.

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Course Enrollment, 1938-39

[Economics] 4a 2hf. Asst. Professor Leontief. — Introduction to the Mathematical Treatment of Economic Theory.

2 Graduates, 2 Seniors, 2 Juniors, 1 Sophomore. Total 7.

Source: Harvard University. Report of the President of Harvard College and Reports of Departments for 1938-1939, pp. 97-98.

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Exam not found for Economics 4a, 1938-39

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Course Enrollment, 1939-40

[Economics] 4a 2hf. Associate Professor Leontief. — Introduction to the Mathematical Treatment of Economic Theory.

1 Graduate, 1 Sophomore. Total 5.

Source: Harvard University. Report of the President of Harvard College and Reports of Departments for 1939-1940, p. 98.

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Final Examination, 1939-40
HARVARD UNIVERSITY
ECONOMICS 4a2

Answer four questions including question 1.

  1. Discuss the relation between the marginal costs of an enterprise and the marginal productivities of the factors used in production.
  2. An enterprise manufactures two commodities X and Y, using two factors of production, V and W. The production function is x(yb– 1) = vnwm.
    Given the prices px, py, pvand pwwrite down the equations which determine the most profitable outputs of X and Y and the corresponding inputs of V and W.
  3. Given:
    1. The total cost curve of a monopolist
      C = A + Kx2 and
    2. the market demand curve for his product
      p = B – Lx,
      p is the price and x the quantity of the commodity produced and sold. A, K, B and L are positive constants.
      An excise tax of z dollars per unit of output is being levied.
      What magnitude of z (expressed in terms of the given constants) would maximize the total tax receipts?
  4. Prove that the price of labor will exceed its marginal value productivity if
    1. labor is the only factor of production used in manufacture of a given commodity,
    2. the producer of this commodity sells his output on a purely competitive market, but is the only (“monopsonistic”) buyer of the particular kind of labor used in his plant,
    3. The supply curve of labor is negatively inclined.
  5. Discuss the problem of price discrimination by a monopolist.

 

Source: Harvard University Archives. Harvard University Final Examinations, 1853-2001. (HUC 7000.28) Box 5. Faculty of Arts and Sciences. Papers Printed for Final Examinations. History, History of Religions, …, Economics, …, Military Science, Naval Science. June, 1940.

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Economics 4a not offered in 1940-41

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Course Enrollment, 1941-42

[Economics] 4a 2hf. Associate Professor Leontief. — Introduction to the Mathematical Treatment of Economic Theory.

1 Graduate, 5 Seniors, 8 Juniors, 3 Sophomores, 1 Freshman. Total 18.

Source: Harvard University. Report of the President of Harvard College and Reports of Departments for 1941-1942, p. 62.

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Course Outline Economics 4a 1941-42 (and 1942-43)

https://www.irwincollier.com/harvard-intro-to-mathematical-economics-schumpeter-leontief-1935-42/

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Final Examination, 1941-42
HARVARD UNIVERSITY
ECONOMICS 4a

Answer one question in each of the following three groups:

(a) 1 or 2
(b) 3 or 4
(c) 5 or 6

  1. Describe in detail the relation between a production function and the corresponding cost function.
  2. Show that the slope of a supply curve of a single enterprise is positive.
  3. Show that a total cost curve can be of such a shape that the marginal costs are increasing but the average costs decreasing throughout its whole length. Give example.
  4. The cost curve of an enterprise is
    C = A + x + Bx2+ Kx3
    (C are the total costs, x – the output, A, B, and K – constants).
    What is the lowest competitive price at which the owner will find it profitable to operate the plant rather than to cease production entirely?
  5. An enterprise consists of two identical plants. Each has a following cost curve:
    C = A + Bx2+ x3
    (C are the total costs, x – the output, A and B are constants).
    Compute the combined cost curve of the whole enterprise.
  6. Given a production function y = f(x,z)
    (y is the amount of product, py – its price, x and z inputs of two factors, px and pz – their respective prices.)
    The producer maximizes his profits under conditions of pure competition. Show that an increase of the price px of factor x will reduce the amount (x) of this factor used in the process of production.

 

Source: Harvard University Archives. Harvard University Final Examinations, 1853-2001. (HUC 7000.28) Box 6. Faculty of Arts and Sciences. Papers Printed for Final Examinations. History, History of Religions, …, Economics, …, Military Science, Naval Science. June, 1942.

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Course Description, 1942-43

Economics 4a 1hf. Introduction to the Mathematical Treatment of Economic Theory. Half-course (first half-year). Mon., 4 to 6. Associate Professor Leontief.

Economics A and Mathematics A, or their equivalents, are prerequisites for this course.
The course is intended to instruct beginners in economic theory (having had elementary mathematical training) in the application of elementary mathematical methods in economics and at the same time to enable them to understand some of the major contributions to economic theory made by such writers as Marshall, Cournot, Walras, and Edgeworth.

Source:  Official Register of Harvard University, Vol. XXXIX, No. 45 (June 30, 1942). Division of History, Government, and Economics Containing an Announcement for 1942-43. 

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Course Enrollment, 1942-43

[Economics] 4a 1hf. Associate Professor Leontief. — Introduction to the Mathematical Treatment of Economic Theory.

1 Graduate, 2 Seniors, 4 Juniors, 2 Sophomores, 1 Public Administration. Total 10.

Source: Harvard University. Report of the President of Harvard College and Reports of Departments for 1942-1943, p. 46.

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Exam not found for Economics 4a, 1942-43

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Course Enrollment, 1943-44

[Economics] 4a. (winter term) Associate Professor Leontief. — Introduction to the Mathematical Treatment of Economic Theory.

2 Juniors in ROTC, 1 Radcliffe, 3 Seniors, 4 Navy (V-12). Total 10.

Source: Harvard University. Report of the President of Harvard College and Reports of Departments for 1943-1944, p. 56.

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Exam not found for Economics 4a, 1943-44

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Economics 4a not offered in 1944-45

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Course Enrollment, 1945-46

[Economics] 4a. (fall term) Associate Professor Leontief. — Introduction to the Mathematical Treatment of Economic Theory.

1 Senior, 2 Juniors, 3 Sophomores, 2 Radcliffe. Total 8.

Source: Harvard University. Report of the President of Harvard College and Reports of Departments for 1945-1946, p. 58.

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Course Outline, 1945-46

INTRODUCTION TO THE MATHEMATICAL TREATMENT OF ECONOMIC THEORY
Economics 4a
1945-46, Fall Term

  1. Introductory remarks.
    Profit function.
    Maximizing profits.
  2. 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. 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.
    Monopoly.
  10. Time lag and time sequences.
  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.

Jacob L. Mosak, General Equilibrium Theory in International Trade.

Weekly problems.

Source: Harvard University Archives. Syllabi, course outlines and reading lists in Economics, 1895-2003. HUC 8522.2.1, Box 3, Folders “1945-1946 (1 of 2)”.

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Final Examination, 1945-46
1945-46
HARVARD UNIVERSITY
ECONOMICS 4a
Introduction to Mathematical Economics

Answer any three questions.

  1. Show the relationship between the total cost curve and the supply curve of an enterprise.
  2. Show that, at the point of optimum output, the marginal costs of an enterprise are equal to the price of any cost factor divided by its marginal productivity.
  3. A consumer has an income of q1 dollars in the first and of y2 dollars in the second year. Although the combined expenditures in the two years equal y1+ y2, he can spend more than y1 in the first year, and correspondingly less in the second year or vice versa. In both years, he purchases one kind of consumers’ goods, its price being p1 dollars in the first and p2 dollars per unit in the second year. The utility function which the consumer maximizes is u= f(x1, x2) where u is the utility level, x1 and x2 the quantities consumed in the first and second year respectively.
    1. Derive the equations which determine the optimum magnitudes of x1 and x2.
    2. Show that an increase of the price p1, with p2, y1,y2 remaining constant, might increase x1.
  4. The demand, q, for the product of a monopolist depends upon the price, p, of his produce and the amount of money, y, which he spends on advertising. The total production cost, c, depends upon the quantity of output, q. Given the demand function: q=\frac{A}{p}+{{y}^{{1}/{4}\;}}-p
    and the total (production) cost function c = q
    where A is a positive constant;
    Determine the output, the price, and the advertising outlay which would maximize the profits (total revenue minus total outlay) of this enterprise.
  5. The well-being, u, of a worker depends upon the amount, x, of consumers’ goods which he can buy with his daily wage, and the number of hours of leisure, y, which remain to him after he finishes his daily work:
    u= f(x, y)

    1. Derive the equations determining the number of hours (call it l) of daily work which he will be willing to do at the wage of w dollars per hour, if the price of the consumers’ good is p dollars per unit.
    2. Show that an increase of the hourly wage rate might reduce the number of hours which the worker will choose to work.

 

Source: Harvard University Archives. Harvard University Final Examinations, 1853-2001. (HUC 7000.28) Box 11. Faculty of Arts and Sciences. Papers Printed for Final Examinations. History, History of Religions, …, Economics, …, Military Science, Naval Science. January, 1946.

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Economics 4a not offered in 1946-47

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Course Enrollment, 1947-48

[Economics] 4a. Professor Leontief. — Introduction to the Mathematical Treatment of Economic Theory (Sp).

2 Graduates, 6 Seniors, 8 Juniors, 1 Sophomore, 2 Public Administration, 1 Radcliffe. Total 20.

Source: Harvard University. Report of the President of Harvard College and Reports of Departments for 1947-1948, p. 89.

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Reading list and midterm and final examination question, 1947-48

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Economics 4a not offered in 1948-49

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Course Enrollment, 1949-50

[Economics] 104 (formerly Economics 4a). Assistant Professor Goodwin. — Introduction to the Mathematical Treatment of Economic Theory (Sp).

3 Graduates, 6 Seniors, 1 Junior, 2 Sophomores, 1 Public Administration, 1 Radcliffe. Total 14.

Source: Harvard University. Report of the President of Harvard College and Reports of Departments for 1949-1950, p.72.

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Course Texts on Library Reserve, 1945-46

R.G.D. Allen. Mathematical analysis for economists

W.L. Crum. Rudimentary mathematics for economists and statisticians

P.A. Samuelson. Foundations of economic analysis.

Source: Harvard University Archives. Syllabi, course outlines and reading lists in Economics, 1895-2003. HUC 8522.2.1, Box 4, Folders “1949-1950 (1 of 3)”.

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Image Sources: Schumpeter and Leontief from Harvard Class Album 1950, Goodwin from Harvard Class Album 1951.