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Harvard. Final examination for graduate mathematical economics. Goodwin, 1950-51

 

 

 

The reading list for the one-term course Economics 204b. Mathematical Economics, taught by Richard Goodwin during the first term of 1948-49 was posted earlier.

This same course number, Economics 204b, was later assigned to a Goodwin’s seminar in mathematical economics in 1950-51 (that covered “General interdependence systems: in particular, Leontief linear systems”), so we can assume that much of the reading list for his course Economics 204a in 1950-51  would have been very similar to the earlier list linked above. In any event, here are the exam questions for the 1950-51 course.

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Course Description

Economics 204a. Mathematical Economics

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

Micro-and macro-dynamical economic systems.

Prerequisites: one course in economics and one in college mathematics. Properly qualified undergraduates will be admitted to the course.

Source: Final Announcement of the Courses of Instruction Offered by the Faculty of Arts and Sciences During 1950-51. Published in Official Register of Harvard University, Vol. XLVII, No. 23 (September, 1950), p. 83.

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HARVARD UNIVERSITY
ECONOMICS 204a
Final. January, 1951

Please use right hand side of page only, leaving the left hand side for scratchwork.

I
Required hour question.

  1. Write an essay on the role of stability in economic science. Include some discussion of various definitions of it, how to measure it, and its use by Walras and by Marshall. Would you be inclined to accept as valid the following type of proposition: we may assume that all or most markets have stable equilibria, for “the plausibility of such a stability hypothesis is suggested by the consideration that positions of unstable equilibrium, even if they exist, are transient, nonpersistent states, and hence on the crudest probability calculation would be observed less frequently than stable states” (Samuelson)?

II
Take any TWO questions, allowing about 40 minutes for each.

  1. State concisely the economic assumptions and derivation of any one macrodynamic model of an inventory cycle. In terms of the model discuss the factors governing the period of its cycle.
  2. Develop briefly a simply macrodynamic model based on ‘fixed’ capital theory. Indicate the important non-linearities and give a graphical discussion of the nature of the solutions and their limit cycle. What are the principal characteristics of the system’s behavior?
  3. Two different economic situations are described by the somewhat similar equations:
    1. \ddot{x}+.12\dot{x}+2.3x=10;
    2. x\left( t+2 \right)+.12x\left( t+1 \right)+2.3x\left( t \right)=10.
      Compare and contrast the behavior implied in the two cases.

 

III
Take any TWO questions, allowing about 20 minutes for each.

  1. If the multiplier mechanism is represented by
    {1}/{v}\;\dot{y}+\left( 1-\alpha \right)y=i\left( t \right)+H ,
    and if
    i\left( t \right)=\kappa \dot{y}+I\cos \omega t,
    Obtain the complete solution for y.
  2. Given a Marshallian type market, specified by
    \begin{array}{l}{{p}_{d}}={{p}_{d}}\left( q \right),\\{{p}_{s}}={{p}_{s}}\left( q \right),\\\text{and}\\\dot{q}=f\left( {{p}_{d}}-{{p}_{x}} \right),\end{array}

    1. State the meaning and use of its phase line, \psi \left( q \right);
    2. Derive the relationship between the slope of \psi \left( q \right) and the slopes of the supply, the demand, and the reaction curves;
    3. Show what conditions on the slopes of the other curves are implied by a stable equilibrium point.
    4. If all curves are linear for small variations, show how the time constant, \tau , depends on the slopes of the supply, the demand, and the reaction curves.
    5. With the help of graphs, discuss the various possible behavior types of a ‘cob-web’ market with general, non-linear supply and demand curves. Explain briefly in words the meaning of your results.

Source:  Harvard University Archives. Final examinations, 1853-2001. Box 17, Bound volume: Papers Printed for Final Examinations. History, History of Religions, …, Economics, …, Military Science, Naval Science. January, 1951.

Image: Harvard Class Album 1951.