Harvard. Graduate Mathematical Economics. Syllabus and Final Exam. Chipman, 1952

 

 

John S. Chipman was born in Montreal, Canada, in 1926. He received his Ph.D. from the Johns Hopkins University in 1951, and taught at the University of Minnesota from 1955 to his retirement as Regents’ Professor in 2007.

Before going to Minnesota Chipman was assistant professor of economics at Harvard from 1951-55. This post provides a transcription of the course syllabus and final examination for Chipman’s “General Interdependence Systems”, a name he chose for the course he inherited bearing the nominal title of “mathematical economics”.

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Course Announcement

Economics 214b (formerly Economics 204b). Mathematical Economics

Half-course (spring term). Tu., Th., 2:30-4. Assistant Professor Chipman.

General interdependence systems; in particular, Leontief linear systems. Properly qualified undergraduates will be admitted to the course.

Source: Harvard University Archives. Courses of Instruction, Box 6. Final Announcement of the Courses of Instruction Offered by the Faculty of Arts and Sciences during 1951-52 published in Official Register of Harvard University Vol. XLVIII, No. 21 (September 10, 1951), pp. 80-81 .

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Economics 214b
General Interdependence Systems

Second Semester, 1951-52
Syllabus

*Texts

I STATIC LEONTIEF MODEL

*Wassily W. Leontief: The Structure of American Economy, 1919-1939, New York, Oxford University Press, 1951.

II DYNAMIC MODELS

*John S. Chipman: The Theory of Inter-Sectoral Money Flows and Income Formation, Baltimore, The John Hopkins Press, 1951

Richard M. Goodwin: “The Multiplier as Matrix,” Economic Journal, December 1949

________________: “Does the Matrix Multiplier Oscillate?” December 1950

________________: “Static and Dynamic Linear General Equilibrium Models,” (mimeographed, Littauer Library)

David Hawkins and Herbert A. Simon, “Note: Some Conditions of Macroeconomic Stability,” Econometrica, July-October 1949

Oscar Lange, Price Flexibility and Employment, Bloomington, Indiana, Principia Press, 1944, Appendix.

Lloyd A. Metzler: “Stability of Multiple Markets: The Hicks Conditions,” Econometrica, October 1945

________________: “A Multiple Region Theory of Income and Trade,” Econometrica, October 1950

________________: “A Multiple-Country Theory of Income Transfers,” Journal of Political Economy, February 1951.

Paul A. Samuelson: Foundations of Economic Analysis, Ch. IX and Appendix B

________________: “A Fundamental Multiplier Identity,” Econometrica, July-October 1943

Arthur Smithies: “The Stability of Competitive Equilibrium,” Econometrica, July-October 1942

Robert Solow: “On the Structure of Linear Models,” Econometrica, January 1952

Frederick V. Waugh: “Inversion of the Leontief Matrix by Power Series,” Econometrica, April 1950

III ALLOCATION OF RESOURCES (“LINEAR PROGRAMMING”)

*Tjalling C. Koopmans (ed.): Activity Analysis of Production and Allocation, New York, John Wiley & Sons, Inc., 1951.

________________: “Efficient Allocation of Resources,” Econometrica, October 1951.

Nicholas Georgescu-Roegen: “Leontief’s System in the Light of Recent Result,” Review of Economics and Statistics, August 1950

John von Neumann: “A Model of General Economic Equilibrium,” Review of Economic Studies, October 1945.

 

MATHEMATICAL REFERENCES

(1) Matrices

R. A. Frazer, W. J. Duncan and A. R. Collar, Elementary Matrices, New York, Macmillan, 1947

C. C. MacDuffee, Vectors and Matrices, Menasha, Wisconsin, 1943

A. C. Aitken, Determinants and Matrices, Edinburgh, Oliver and Boyd, 1948

(2) Difference and Differential Equations

P. A. Samuelson, “Dynamic Process Analysis,” in A Survey of Contemporary Economics, Philadelphia, Blakiston, 1948

R. G. D. Allen, “Mathematical Foundations of Economic Theory,” Q.J.E. February 1949

F. R. Moulton, Differential Equations, New York, Macmillan, 1930, Ch. XV

W. J. Baumol, Economic Dynamics, New York, Macmillan, 1951.

(3) Set Theory and Abstract Algebra

F. P. Northrup and Associates, Fundamental Mathematics, Vol. I, Chicago, Univ. of Chicago Press, 1948

Richard Courant and Herbert Robbins, What is Mathematics?, New York, Oxford University Press, 1941

Garrett Birkhoff and Saunders MacLane, A Survey of Modern Algebra, New York, Macmillan, 1950

Paul Halmos, Finite Dimensional Vector Spaces, Princeton, Princeton University Press, 1948

 

Source: Harvard University Archives. Syllabi, course outlines and reading lists in Economics, 1895-2003. Box 5. Folder: “Economics, 1951-1952 (2 of 2)”.

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1951-52
HARVARD UNIVERSITY
ECONOMICS 214b
[Final examination]

Answer both questions

1. Consider an economy composed of n sectors (say industries and households) and described by the model
   
   where y_i(t) is the output of sector i at time t, of which b_i is the exogenous component, and where the input-output coefficients a_ij are all taken to be non-negative. Assuming the economy to be initially in a state of equilibrium, show under what conditions an autonomous rise in all the b_i
   1. implies an increase in the equilibrium values of all the sector outputs;
   2. causes all sector outputs to approach, with time, new equilibrium values.
      Compare and interpret these conditions.
2. “A possible activity is efficient if and only if there exists a set of positive prices for all commodities, which give rise to zero profits on this activity and non-positive profits on all other possible activities.”
   1. Prove this theorem, keeping in mind the following two properties of convex polyhedral cones:
      1. The negative polar of the intersection of two cones is equal to the sum of their negative polars;
      2. The sum of two cones is equal to the sum of their dimensionality spaces if and only if the relative interior of the one cone intersects the negative of the relative interior of the other.
   2. Indicate a modification of the theorem, with the appropriate modification of the definition of efficiency, when availability limitations are specified on certain primary commodities, which are no longer regarded as “undesirable.”
   3. Outline the properties of the efficient set in (b) when there is only one primary commodity and there is no joint production.
   4. Discuss the usefulness, significance, and validity of the notion of an efficient set of activities as employed in (a), (b), and (c).

Source: Harvard University Archives. Harvard University, Final examinations, 1853-2001. Box 27, Papers Printed for Final Examinations [in] History, History of Religions,…Economics,…, Air Sciences, Naval Science, June 1952.

Image Source: September 1961 entry card for John Somerset Chipman (b. 28 June 1926 in Montreal).  Rio de Janeiro, Brazil, immigration Cards, 1900-1965 at ancestry.com.