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Harvard. A.B. Honors Degree Examination in Mathematical Economic Theory, 1939

Today’s posting is a transcription of the “correlation examination” questions for mathematical economic theory given at Harvard in May 1939.

A printed copy of questions for twelve A.B. examinations in economics at Harvard for the academic year 1938-39 can be found in the Lloyd A. Metzler papers at Duke’s Economists’ Papers Project. 

Concentrators in Economics will have to pass in the spring their Junior year a general examination on the department of Economics, and in the spring of their Senior year an examination correlating Economics with either History or Government (this correlating exam may be abolished by 1942), and a third one on the student’s special field, which is chosen from a list of eleven, including economic theory, economic history, money and banking, industry, public utilities, public finance, labor problems, international economics, policies and agriculture.
Courses in allied fields, including Philosophy, Mathematics, History, Government, and Sociology, are suggested by the department for each of the special fields. In addition, Geography 1 is recommended in connection with international policies or agriculture.
[Source: Harvard Crimson, May 31, 1938]

Economic Theory,
Economic History Since 1750
Money and Finance,
Market Organization and Control,
Labor Economics and Social Reform.

  • One of the Six Correlation Examinations given to Honors Candidates. (May 12, 1939; 3 hours)

Economic History of Western Europe since 1750,
American Economic History,
History of Political and Economic Thought,
Public Administration and Finance,
Government Regulation of Industry,
 Mathematical Economic Theory.

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DIVISION OF HISTORY, GOVERNMENT, AND ECONOMICS
CORRELATION EXAMINATION

Mathematical Economic Theory

(Three hours)

 

Answer either FOUR or FIVE questions, but not more than THREE from either group. If you answer only FOUR questions, write about one hour on ONE of the questions in Group B and mark your answer “Essay.” This question will be given double weight.

A

  1. A consumer’s indifference map for two goods X and Y is defined by \large\frac{4-\sqrt{y+1}}{x+4}=a
    Draw a graph showing the five indifference curves for the values 2, 3, 4, 5, 6 of the parameter a. Verify that they are of “normal” form.
  2. A business produces an income of $x this year and $y next year, where these values can be varied according to the relation y=1000-\frac{{{x}^{2}}}{250} . Show how \left\{ \left( -\frac{dy}{dx} \right)-1 \right\} can be interpreted as the marginal rate of return over cost. Show that the value of this marginal rate is \frac{x-125}{125} when this year’s income is $x.
  3. The market demand for a good is given by p=\beta -\alpha x . The market is supplied by two duopolists with cost functions {{\pi }_{1}}={{a}_{1}}x_{1}^{2}+{{b}_{1}}{{x}_{1}}+{{c}_{1}} and {{\pi }_{2}}={{a}_{2}}x_{2}^{2}+{{b}_{2}}{{x}_{2}}+{{c}_{2}} . Assuming that the “conjectural variations” are zero, show that the reaction curves are straight lines. Deduce the equilibrium output of each duopolist.
  4. (a) A steel plan is capable of producing x tons per day of a low grade steel and y tons per day of a high grade steel, where y=\frac{40-5s}{10-x} . If the fixed market price of low grade steel is half of that of high grade steel, show that about 5½ tons of low grade steel are produced per day for maximum total revenue.
    (b) The steel producer described in Section (a) monopolizes the sale of both quality steels. If the prices of low and high grade steel are $px and $py per ton, the demands are {{p}_{x}}=20-x and {{p}_{y}}=25-2y . Find an equation giving the output x of low grade steel for maximum total revenue. Show, by a graphical method, that just under 6 tons of this steel are produced per day.
  5. Given the marginal utility equations
    {{x}_{1}}d{{y}_{1}}+{{y}_{1}}d{{x}_{1}}\ge 0 , {{x}_{2}}d{{y}_{2}}+{{y}_{2}}d{{x}_{2}}\ge 0 ,
    for which the indifference curves are the rectangular hyperbolas {{x}_{1}}{{y}_{1}}={{c}_{1}} and {{x}_{2}}{{y}_{2}}={{c}_{2}} , given initial amounts a1, b1, and a2, b2 for the individuals (1) and (2) respectively, show that the equilibrium point for a single price transaction is
    {{x}_{1}}=\frac{{{a}_{1}}+p{{b}_{1}}}{2p} , {{y}_{1}}=\frac{{{a}_{1}}+p{{b}_{1}}}{2p} , where p=\left( \frac{{{b}_{1}}+{{b}_{2}}}{{{a}_{1}}+{{a}_{2}}} \right) .
  6. The demand for tea is {{x}_{1}}=40\frac{{{p}_{2}}}{{{p}_{1}}} and for coffee {{x}_{2}}=10\frac{{{p}_{1}}}{{{p}_{2}}} thousand lbs. per week, where p1 and p2 are the respective prices of tea and coffee in pence per lb. At what relative prices of tea and coffee are the demands equal? Draw a graph to show the shifts of the demand curve for tea when the price of coffee increases from 2s to 2s 6d and to 3s per lb.
  7. The production function is x=A{{a}^{\alpha }}{{b}^{\beta }} , where A, \alpha and \beta are constants. If the factors are increased in proportion, show that the product increases in greater or less proportion according as \left( \alpha +\beta \right)  is greater or less than unity. How is this property shown on a vertical section of the production surface through 0 and a given point on the surface? What is the special property of the case \alpha =1-\beta ?
  8. Examine the utility function u=\frac{x+a}{c-\sqrt{y+b}} , where a, b and c are positive constants, and show that the indifference map is a set of parabolic arcs and of normal form for certain ranges of values of the purchases x and y.
  9. If a{{x}^{2}}+b{{y}^{2}} = constant is the transformation function for two goods X and Y, that the marginal rate of substitution of Y production for X production is \frac{ax}{by} and that the elasticity of substitution is always unity.
  10. A monopolist produces cheap razors and blades at a constant average cost of 2s per razor and 1s per dozen blades. The demand of the market per week is {{x}_{1}}=\frac{10}{{{p}_{1}}{{p}_{2}}} thousand razors and {{x}_{2}}=\frac{20}{{{p}_{1}}{{p}_{2}}} thousand dozen blades when the prices are p1 (shillings per razor) and p2 (shillings per dozen blades). Show that the monopoly prices, fixed jointly, are 4s. per razor and 2s. per dozen blades.

 

B

  1. (a) A radio manufacturer produces x sets per week at a total cost of \$\left( \tfrac{1}{25}{{x}^{2}}+3x+100 \right) . He is a monopolist and the demand of his market is x=75-3p , when the price is $p per set. Show that the maximum net revenue is obtained when about 30 sets are produced per week. What is the monopoly price? Illustrate by drawing an accurate graph.
    (b) In the case of Section (a), a tax of $k per set is imposed by the government. The manufacturer adds the tax to his cost and determines the monopoly output and price under the new conditions. Show that the price increases by rather less than half the tax. Find the decrease in output and monopoly revenue in terms of k.
    Express the receipts from the tax in terms of k and determine the tax for maximum return. Show that the monopoly price increases by about 33 per cent. when this particular tax is imposed.
  2. (a) If u={{x}^{\alpha }}{{y}^{\beta }} is an individual’s utility function for two goods, show that his demands for the goods are x=\frac{\alpha }{\alpha +\beta }\frac{\mu }{{{p}_{x}}} and y=\frac{\beta }{\alpha +\beta }\frac{\mu }{{{p}_{y}}} where px and py are the fixed prices and \mu the individual’s fixed income. Deduct that the elasticity of demand for either good with respect to income or to its price is equal to unity.
    (b) The incomes of an individual in two years are {{x}_{0}} and and {{y}_{0}} his utility function for incomes is u={{x}^{\alpha }}{{y}^{\beta }} . Show that the demand \left( x-{{x}_{0}} \right) for loans this year decreases as the given market rate of interest 100r per cent. increases. Deduce that the individual will not borrow this year at any (positive) rate of interest if {{y}_{0}}<\frac{\beta }{\alpha }{{x}_{0}} .
  3. Discuss this quotation from Chamberlin: “We must conclude that the problems of proportion [among the factors] and of size cannot ordinarily be separated. The goal of the entrepreneur is not to discover the most efficient proportions and then to reproduce these continuously until the most efficient size is secured.”
  4. Discuss this quotation from Douglas: “Since the demand curves for labor and capital tend to approximate and to conform to the respective marginal productivity curves, it follows that an increase of 1 per cent in the quantity of labor would, other things being equal, normally tend to be followed by a decrease of ¼ per cent in the rate of wages.”
    [He uses a production function P=1.01\,\,{{L}^{{\scriptstyle{}^{3}\!\!\diagup\!\!{}_{4}\;}}}{{C}^{{\scriptstyle{}^{1}\!\!\diagup\!\!{}_{4}\;}}} ]
  5. Discuss this quotation from Broster: “Secondly, the chief concern of the railways is the maximization not of gross but of net revenue the maximum values of which are not the simultaneous product of the same level of fares. As is well known, the fare that attracts the former is that which corresponds to unit elasticity. It is perhaps not so well known—especially amongst the managers of sales departments—that except where the total cost remains constant for different rates of output of services, the fare that attracts the maximum net revenue is necessarily higher.”
  6. Discuss this quotation from Hicks: “The elasticity of substitution of labor for capital is the same as the elasticity of substitution of capital for labor.”

 

Source: Duke University. David M. Rubenstein Rare Book & Manuscript Library. Lloyd Appleton Metzler Papers. Box 7. [Harvard University], Division of History, Government and Economics. Division Examinations for the Degree of A.B., 1938-39.