A Diagrammatic Proof That Indirect Utility Functions Are Quasi-Convex
Author(s): Wing Suen
Source: The Journal of Economic Education, Vol. 23, No. 1 (Winter, 1992), pp. 53-55
Published by: Heldref Publications
Stable URL: http://www.jstor.org/stable/1183478
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A
DiagrammaticProof
Utility
Functions
Are
That
Indirect
Quasi-Convex
WingSuen
Duality theory has become part of the standardtrainingof economics
graduatestudentsand many upper-levelundergraduates.My teachingexperiencesuggeststhat studentshave little problemunderstandingwhy the
cost function is concave or why the indirectprofit function is convex,
becausea simpleenvelope-typediagrammaticargumentis readilyavailable
(e.g., Kreps1990;Silberberg1990;Varian1984).Manystudents,however,
have difficultygraspingthe quasi-convexityof the indirectutilityfunction.
Althoughan algebraicproof can be found in textbookssuchas Kreps(1990)
and Varian(1984),those who are uninitiatedin mathematicallanguagemay
find the proof unilluminatingif not unintelligble.I have constructeda diagrammaticproof for use in class, and studentsseem to be quite receptive.
As a bonus, the proof leads to Roy's identity.The studentshould be reminded, however,that the diagrammaticproof is meant to supplementnot to replace-the algebraictreatment.
The indirectutility function, v(p,,p2,y), gives the maximumachievable
utilitywhenpricesarep,,p2 (in the two-goodcase)and whenmoneyincome
is y. A convenientway to representan indirectutilityfunctionis the priceindifferencediagram.A price-indifference
curveshows combinationsof p,
and p2 such that the consumeris indifferent.It is downwardsloping because, whenp, is increased,the consumercan maintainthe sameutilitylevel
only if p2 is reduced. Unlike ordinary indifference curves, however,
the directionof preferencepointsto the southwest:lowerpricesarepreferred
to higherprices.This is a reflectionof the fact that indirectutilityis a decreasingfunctionin prices.
A function v(p,,p2,y)is quasi-convexin (p,,p2) if and only if its lower
contourset-the set of prices(p,,p2)suchhtat v(p,,p2,y)5 k-is convexfor
any k. Because combinations of prices to the northeast of a priceindifferencecurveyieldlowerutilitythanthose on the curve,they constitute
a lower contour set for the indirectutility function. To prove that the indirectutilityfunctionis quasi-convex,it is thus necessaryand sufficientto
show that the price-indifference
curvesare convex to the origin. (When a
utilityfunctionis quasi-concave,the associatedindifferencecurvesare also
convexto the origin.The apparentinconsistencyin terminologyis resolved
if one notices that ordinaryindifferencediagramsand price-indifference
diagramshave oppositepreferencedirections.)
Wing Suen is a lecturer in the Department of Economics, University of Hong Kong.
Winter1992
53
In Figure1, the line VV'showsa price-indifference
curve.Anypointin
the shadedregionis preferred
to anypointon line VV'.As drawnin Figure
curveis convexto the origin.To seewhythismust
1, theprice-indifference
be the case,consideran arbitrary
Let(4,x') be the oppointE = (pO,pO).
timalconsumption
bundleassociatedwiththat set of pricesand income.
Drawa lineAA ' passingthroughE withslope-'/x. Anycombination
of
thatthebundle(x?,x2)
is affordable
with
priceson lineAA ' hastheproperty
theinitialbudget.(Moreprecisely,
the equationforlineAA ' ispl + p2X
= plx + p22. The slopeof the line is therefore
dp2/dp,= - x/x, as
stated.)Thus,if indirectutilityat E is vo,thelevelof utilityat anypointon
AA ' mustbe at leastvo.WhenpriceschangefrompointE to someother
canmaintain
the sameutilitylevelby keeping
pointon AA ', the consumer
hisconsumption
bundleunchanged.
Usuallyhe cando betterby substituting awayfromthe good that is now moreexpensive.It followsthat the
curvefor utilitylevelvomustlie withinareaAECto the
price-indifference
leftof E andwithinA 'EC'to therightof E. VV'is therefore
boundedbelow by AA ' andis tangentto AA ' at E. (Thetangencyconditionobtains
whenthe consumer
maximizes
utilityat a regularinteriorsolution.Thestudentmaybe invitedto guesswhatthe price-indifference
curvewouldlook
likewhenx, andx2 areperfectsubstitutes
or perfectcomplements.)
Thisin
turnimpliesthatthe priceindifference
curveis convexin the neighborhood
of E. Becausethe pointE is arbitrary,
the sameargument
can be usedto
showthatprice-indifference
curvesareeverywhere
convexto the origin.In
otherwords,the indirectutilityfunctionis quasi-convex.
FIGURE
Convex
A
I
Price-Indifference
Curve
P2
A
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.
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.
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.
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.
*
.
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P2??
........
........
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....,....,.... ........
..................?
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......~?
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0
0 ?? ?
?????
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A
54
A
JOURNAL OF ECONOMIC EDUCATION
Our argumentalso establishesthat, for the regularcase, the priceindifference curve has the same slope as AA' at point E. Thus,
- (6v/6p,)/(6v/6p2) = - x'/xl. As an exercise, the student can be asked to
show that this tangencycondition is just a restatementof Roy's identity.
(Hint:Apply Euler'stheoremto the indirectutilityfunction. Then plug in
the tangencyconditionand use the budgetconstraint.)
REFERENCES
Kreps, D. 1990. A course in microeconomic theory. Princeton: Princeton University Press.
Silberberg, E. 1990. The structure of economics: A mathematical analysis. 2nd ed. New York:
McGraw-Hill.
Varian, H. 1984. Microeconomic analysis. 2d ed. New York: Norton.
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