Queen's University Archives - Economics in the Rear-View Mirror

Your curator of the economic artifacts so lovingly transcribed here for future digital generations of historians of economics, human or artificially intelligent, has a weakness for orphaned examinations that he discovers upon riffling through the archival papers of defunct professors of economics. While I have no illusions about being able to give each worthy artifact I stumble upon a warm post in this collection, perhaps others will be inspired by my efforts to join in. Transcribe and upload!

About a dozen years ago I found a folder of University of Chicago economics exams in George Stigler’s papers in which rested a forlorn ten-page Ph.D. comprehensive examination in economic theory from Queen’s University at Kingston, Ontario, Canada (May 1974). I found no clue to the backstory of how this exam happened to have found its way to George Stigler’s papers. Was it from Stigler demand (“Could you send me a copy of your most recent Ph.D. exam in economic theory”) or was it a Queen’s colleague’s pride of workmanship that resulted in the supply? My maximum-likelihood-gut-instinct estimate is that the source was Richard Lipsey, then Sir Edward Peacock Professor of Economics, but it would come as no surprise if there was some Chicago Ph.D. alum on the faculty then at Queen’s who shared the examination with Stigler out of fealty. But the backstory is unimportant for our purposes here, below you will find an excellent comprehensive exam that expands our international cross-section of economic theory questions. Economics in the Rear-view Mirror hereby raises the following artifact from literal archival obscurity to virtual visibility.

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QUEEN’S UNIVERSITY AT KINGSTON
Department of Economics

Ph.D. Comprehensive Examination
ECONOMIC THEORY
May, 1974

Answer FOUR (4) questions — one from each of the four parts of the examination. All questions are of equal value.

PART I

Consider a world in which there are two commodities, beer and whisky and four individuals described by the following utility functions:

Tom: U^T = 2B + W
Sally: U^S = B + W
Dick: U^D = min (B, W)
Jane: U^J = min (B/2 , W)

where utility (U) is measured in utils, beer (B) in bottles and whisky (W) in shots. Bottles of beer and shots of whisky are perfectly divisible.

1. Draw the one-util indifference curve for each of these individuals.
2. Construct Tom’s and Dick’s uncompensated demand curves for beer under the assumption that each has a money income of $10 and the price of whisky is 50¢ a shot.
3. How much beer would Tom and Dick each consume if their money incomes were each $10 and a bottle of beer and a shot of whisky cost 25¢ and 50¢ respectively? Construct each of their compensated demand curves for beer through this point.
4. Suppose that Tom and Sally are stranded on an island with 15 shots of whisky and 10 bottles of beer. Use an Edgeworth box diagram to derive the set of efficient (Pareto optimal) allocations of beer and whisky between them.
5. Suppose that Tom had been stranded with Jane rather than Sally in Part (d). Derive the set of efficient allocations of the 15 shots of whisky and 10 bottles of beer between Tom and Jane.
6. Suppose finally that Dick and Jane had been caught in the predicament described in Part (d). Derive the set of efficient allocations of the beer and whisky in this case.
7. Suppose that Dick and Jane had decided to use a price system to sustain one of the efficient allocations in Part (f). What would be the effect on the price of whisky of a decrease in the endowment of whisky from 15 to 12 shots?

Assume a simple Keynesian type model in equilibrium with unemployed resources. Consider the effects on the velocity of circulation of money of (a) an upward shift in the consumption function, and (b) an open market purchase by the central bank. Answer the question under each of the following alternative assumptions.
1. 1. The money supply is fully exogenous.
  2. The money supply is partly exogenous and partly endogenous, the endogenous part being a function of national income and “the” rate of interest.
  3. The money supply is wholly endogenous because the monetary authorities are stabilizing domestic interest rates.

In answering this question assume a general demand for money function of the form

M^D = L(Y,r)

in which the only restrictions are L_Y > 0, L_r ≤ 0.

You should also pay attention to the following two special cases:

a) L_r = 0, L_Y > 0 and b) L_r > 0, (Y/M)L_Y = 1.

In so far as you have time, give explanations of your results.

Make the following assumptions about the market for cloth and yarn in a small country:
1. One spool of yarn is required for the manufacture of one bolt of cloth.
2. The world price of yarn is $1 per spool and the world price of cloth is $3 per bolt.
3. The domestic demand curve for cloth, the domestic supply curve of yarn, and the domestic supply curve of factors of production (other than yarn) in the manufacture of cloth have constant elasticities, and these elasticities are -1, 1 and 2 respectively.
4. In free trade, domestic consumption of cloth is 1,000,000 bolts, domestic production of yarn is 100,000 spools, and domestic production of cloth is 200,000 bolts.
  1. What tariff on cloth would just compensate factors of production in the manufacture of cloth for the imposition of a 15 percent tariff on yarn?
  2. What is the total welfare loss of imposing both tariffs?
  3. What would be your answer to (ii) above if the 15 percent tariff on yarn were imposed without the compensating tariff on cloth?
  4. In general, if there is to be a tariff on yarn, how does one determine the welfare implications of imposing a tariff on cloth, too?
  5. How would your answers to Parts (i) and (ii) above be affected if the elasticity of substitution between yarn and other factors in the manufacture of cloth were greater than zero?

PART II

Consider an economy producing two goods, X₁ and X₂, according to constant returns to scale production functions of the following form:

(1) X₁ = F¹(L₁)
(2) X₂ = F²(L₂, X₁₂)

where X_i is the gross output of the i-th good, X₁₂ is the amount of X₁ used as an input in the production of X₂, L_i is the amount of labour used in the production of X_i and the total supply of labour is fixed

${L_{1}+L_{2}=\bar{L}}$

Tastes in this economy can be represented by the utility function

U = AX_i^.25 X₂^.75

A) Suppose that equations (1) and (2) are of the particular form:

(1′) X₁ = 2L₁(2′) X₂ = L₂

and the total labour supply is 200 units.

1. Construct the economy’s production possibilities set.
2. What will be the effect on the equilibrium relative price of X₁ of:
  1. an increase of the labour force to 225 units,
  2. a doubling of the efficiency of labour in the X₁ sector?

B) Suppose now that equations (1) and (2) are of the particular form:

(1”) X₁ = 2L₁(2”) X₂ = min (L₂, 4X₁₂)

Answer Questions A(a) and A(b) under these circumstances.

C) Answer the same questions in the case where a second production process is made available to the X₂sector so that the production functions are of the form:

(1”’) X₁ = 2L₁(2”’) X₂ = (min (L₂, 4X₁₂) or min (.75L₂, 8X₁₂)

D) What would be the general shape of the production possibility curve if X₂ could be produced according to a differentiable concave constant returns to scale production function of the general form of (2) (e.g., Cobb-Douglas) while the production function for X₁ remained X₁ = 2L₁?

Standard demand theory predicts that if a man buys more apples when his income goes up he buys less apples when their price goes up. Say whether this prediction continues to follow, and why or why not, under each of the following circumstances.
1. 1. The quality of apples is judged by their price.
  2. The man is upset by his neighbour’s discarded apple cores.
  3. The man owns an apple orchard.
  4. The man loves the scent of apple blossoms.
  5. Apples are storable cheaply enough that one can speculate in them.
  6. The man drinks cider, the market price of which is positively related to the price of apples.
  7. Apples are a heavily advertised good.
  8. Apples and flour are only consumed in fixed proportions, in the form of apple pies.
  9. The price of apples divided by the price of pears is not equal to the man’s marginal rate of substitution between apples and pears.
  10. All goods are rationed, in such a way that they must be paid for not only with money but also with ration coupons, each consumer having a fixed total coupon allowance. (Coupons may not be bought and sold.)

6. [Capital theory and functional income distribution]

1. You are dealing with an economy in which there is only one product which is used either as a consumption good or as a capital good. Each period’s output is divided in some proportion between consumption use and additions to the capital stock. Capital goods do not depreciate. There are two factors of production, labour and capital goods, available in fixed supply in the short period. There is no unemployment of any factor that has a positive price.
  1. In what circumstances will the functional distribution of income be determined by conditions in the factor market only?
  2. Given that investment is exogenously determined and consumption is functionally related to incomes, in what circumstances might the functional distribution of income be determined from conditions in the product market?
  3. In what circumstances would the distribution of income be indeterminate?
  4. In what circumstances might conditions in the factor and product markets be inconsistent with a determinate distribution of income?
2. Now consider an economy with two consumer goods, one of which can also be used as the single capital good, both products being produced by labour and capital goods. Again there can be positive investment in each period and there is no depreciation.
  What would your answers to questions (i), (ii), (iii), and (iv) of part (a) be now?

PART III

An economy producing only one commodity, which may be used either as a consumer good or as a capital good, has a production function of the form

${Y_{t}=e^{\lambda t}K^{\alpha }_{t}L^{1-\alpha }_{t}}$

and we are given that

${L_{t}=e^{gt}L_{0}}$

where Y_t is the quantity of output; K_t is the capital stock; L_t the number of units of labour; g is the rate of growth of the labour force; ${\lambda}$ is the rate of Hicks neutral technological change; and ${\alpha}$ takes the values 0 < ${\alpha}$ < 1.

a) [“Golden-rule”]

1. 1. Derive the general formula that determines the golden rule capital-labour ratio when there is no technical change.
  2. Let g = 0.02, ${\lambda}$ = 0, ${\alpha}$ = 0.25. Calculate the golden rule level of the capital-labour ratio. (Carry your calculation to the last stage in which the solution is set up in the form of logs or as a power of a number.)
  3. Now let the growth rate of the labour force rise to .03. Find the ratio of the new golden rule capital-labour ratio, k₁, to the old golden rule capital-labour ratio, k₀.
  4. Maintain the growth rate of the labour force, g₁ at .03 and make ${\lambda}$ = .0225. Obtain the golden rule values of the path of the capital-labour ratio, labour being measured in natural units.

b) [Optimal steady state growth]

1. 1. Derive the rule that determines the capital-labour ratio of the optimal steady state growth path when there is positive subjective discounting of future per capita utilities at the rate, ${\rho}$ .
  2. Assume that g = .02, ${\lambda}$ = 0, ${\alpha}$ = 0.25 and ${\rho}$ = .04. Calculate the capital-labour ratio of the optimal steady state growth path.

8. [Keynesian macroeconomic theory]

1. Investigate the consequences for fiscal and monetary policy in a simple Keynesian model of assuming that consumption expenditure is unrelated to current income (because it is related to permanent income which is assumed constant over the time period in which we are interested). Would it matter whether or not there were some other element of aggregate expenditure that was related to current national income?
2. Explain how the permanent income hypothesis reconciles the observations drawn from long and short period time series and cross section studies that appeared inconsistent with the Keynesian consumption function. Is the weight of evidence such that we can have substantial confidence in the truth of the general concept that consumption relates to permanent rather than current income?
3. Briefly discuss the significance for the efficacy of short-term fiscal policy of establishing that consumption is a function of permanent income rather than current income.

Examine the validity of the following statements: a) when wages are only flexible upwards and b) when wages are fully flexible.
1. 1. Government spending in excess of tax revenue is inflationary in the long run.
  2. Union wage pressure can reduce unemployment due to the stimulating effects of higher wages on aggregate demand.
  3. Sooner or later automation will cause unemployment.

“Thinking about saving and investment from this technocratic point of view has convinced me that the central concept in capital theory should be the rate of return on investment. In short we really want a theory of interest rates, not a theory of capital.” (Solow: Capital Theory and the Rate of Return, p. 16. Italics in the original.)
1. What is the relationship between the two sentences in Solow’s quotation that makes the second sentence follow from the first? In your answer elaborate on the concept of the rate of return.
2. Can stocks of capital goods (or the aggregate of asset values) be more or less excluded from capital theory, which among other things must include a theory of intertemporal choice?

PART IV

Various officials and ministers of the Canadian government have expressed concern and implemented or at least suggested policies in response to recent increases in the world prices of certain commodities. One instance has been the imposition of a tax on exports of Canadian oil. This move has been hailed by members of the government on various grounds: its “anti-inflationary consequences”, its “benefit to the poor and to fixed income classes to whom cheap energy is so vital” and “the stimulus it will provide to the Canadian manufacturing sector”. Some observers have been obviously impressed with the success of this policy and have suggested a similar two-price system for certain Canadian agricultural exports because “there is no reason the Canadian consumer should suffer due to a world scarcity of these commodities when there is an abundant supply in Canada.”
1. Attempt to give a precise formulation to the exact policy goals which might be implied by each of the above quotations.
2. How would an export tax help to achieve each of these goals?
3. What would be the nature of the economic costs resulting from the use of an export tax to achieve these goals and what would determine the magnitude of these costs?

A certain country, previously without an unemployment insurance system, sets one up. Its rules are as follows:
1. 1. A minimum of 10 weeks work must be done before any benefits can be drawn.
  2. For each week worked above 10, a person is entitled to two-thirds of a week on unemployment benefits, except that no more than 12 weeks of benefit can be received in a year.
  3. While on benefit the weekly payment is three-quarters of the weekly wage previously earned.

To simplify the problem, the following assumptions may be made: how the system is financed is irrelevant; each year is self-contained, in the sense that benefits earned must be used up in the year or lost; one can draw benefit even if one quits voluntarily; no time is ever lost in finding a job if one wants one; a year is 50 weeks long.

1. Establish that a person’s income-leisure opportunity locus consists of four line segments, and draw them on the accompanying graph paper.
2. Will anyone ever work exactly 10 weeks under the new unemployment insurance system? Explain.
3. If leisure is a superior good, prove that those who used to work between 28 and 38 weeks a year (before the new unemployment insurance system) will now work less.
4. Define a “full-time worker” as one who works more than 38 weeks a year. Would you expect such a worker to work less after the scheme is introduced?
5. Demonstrate the possibility that people who did not work before will now work. Could they conceivably become “full-time” workers? Why or why not?

Consider a country with the following attributes:
1. 1. It has a floating exchange rate.
  2. It has a substantial rate of inflation which it is trying to lower.
  3. It has recently taken monetary action that has raised interest rates relative to those of other countries.
  4. It is buying quite large amounts of foreign exchange, apparently to have a dampening effect on the rise in the exchange value of its currency.
  5. This foreign exchange is acquired by use of government cash which requires extra government borrowing to the extent that the exchange accumulation was not foreseen (the country has been lucky in its foreign exchange receipts and disbursements for oil roughly matching).
  6. The country has outstanding large amounts of non-negotiable savings bonds which are redeemable on demand at face value, which have coupon rates considerably lower than rates of comparable securities, and which are being cashed in substantial quantities.

1. Are the objectives implied by the above actions mutually consistent? Always? In some circumstances? Never? In your answer elaborate on the relevance of some of the parameters of the system to it.
2. The federal government of this country will be bringing down its annual budget soon. What policies would you recommend (i) to alleviate inconsistencies in the implied objectives noted above, and (ii) to reinforce the impact of the actions described above?

Source: University of Chicago Archives. George Stigler Papers, Addenda. Box 33. Folder “Exams & Prelim Questions.”

Image Source: “Coat of Arms” in the on-line Queen’s Encyclopedia.

QUEEN’S UNIVERSITY AT KINGSTON Department of Economics

Ph.D. Comprehensive Examination ECONOMIC THEORY May, 1974

QUEEN’S UNIVERSITY AT KINGSTON
Department of Economics

Ph.D. Comprehensive Examination
ECONOMIC THEORY
May, 1974