Why do People Buy Lottery Tickets? Choices Involving Risk and the Indivisibility of
Expenditure
Author(s): Ng Yew Kwang
Source: Journal of Political Economy, Vol. 73, No. 5 (Oct., 1965), pp. 530-535
Published by: The University of Chicago Press
Stable URL: http://www.jstor.org/stable/1829141 .
Accessed: 30/01/2015 02:35
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp
.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
.
The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to Journal
of Political Economy.
http://www.jstor.org
This content downloaded from 155.69.24.171 on Fri, 30 Jan 2015 02:35:30 AM
All use subject to JSTOR Terms and Conditions
WHY DO PEOPLE BUY LOTTERY TICKETS? CHOICES INVOLVING
RISK AND THE INDIVISIBILITY OF EXPENDITURE'
NG YEW KWANG
Nanyang University,Singapore
visible. In fact,consumptionexpenditures
UNDER
the assumptionsthat, first,the are not infinitelydivisible; for example,
whilea personmay choosebetweenhaving
pleasuresof gamblingmay be neglected and, second,the marginalutilityof one motor car or two, or even between
incomeis diminishing,
it can easilybe shown having a Volkswagonand having a Merthatfairgamblingis an economicblunder.2 cedes 220,he cannothave halfa car or one
Thus, thefactthatpeopleengagein unfair and one-thirdcars.4 The indivisibilityof
is a recognized
fact,demanding
as wellas fairgamblesis clearlyinconsistent expenditures
of
qualification
the
equimarginal
principle;
eitherwithutilitymaximization
or withthe
and
its
presence
implies
a
peculiar
behavior
assumptionofdiminishing
marginalutility.
Marshallresolvedthiscontradiction
by re- of theutilityfunction.
Considera studentjust graduatingfrom
jectingutilitymaximization
as an explanaschooland contemplating
high
a university
tionofchoicesinvolvingrisk.MiltonFriedmanand L. J.Savagepointedout thatMar- education,whichconstitutesan indivisible
Let OX in Figure 1 represent
shall need not have done so, and suggested expenditure.
his
income
and
and OY repreexpenditure,
the hypothesisthat marginalutilitysucsent
the
income
to him.
marginal
utility
of
cessivelydecreases,increases,and decreases;
the total utility-curve
concave
from
being
3 See "The UtilityAnalysisof Choices Involving
ofactual income Risk," Journalof Political Economy,LVI (August,
above in theneighborhood
and convexfromabove forhigherand lower 1948), 279-304. Their hypothesisis based on the
thevalidity analysis of J. von Neumann and 0. Morgenstern,
incomes.'Whilenotchallenging
Theoryof Gamesand EconomicBehavior(Princeton,
oftheiranalysis,thisnotesuggestsan alter- N.J.: PrincetonUniversityPress, 1947), especially
native resolutionof the problem,which pp. 15-31, 617-32. The idea that choices involving
ofexpenditure
and risk can be explained by the maximizationof exrestson theindivisibility
its influenceon the utilityfunctionof the pectedutilitydates back at least to Daniel Bernoulli's
celebrated analysis of the St. Petersburgparadox
consumerunit.
(Versucheeiner neuen Theorieder Wertbestimmung
I. THE PROBLEM
II.
INDIVISIBILITY
OF EXPENDITURE
AND THE UTILITY
FUNCTION
The assumptionthatthemarginalutility
is deof incomeis continuallydiminishing
rivedfromtheassumption
thattheexpendiditure of the consumerunit is infinitely
von Gliicksfallen["Sammlung alterer und neuer
SchriftenNo. 9], LeipStaatswissenschaftlicher,"
zig: Grundlage der modernen Wertlehre, 1896;
translated by AlfredPringsheimfrom "Specimen
theoriae novae de mensura sortis," Commentarii
academiae scientiarumimperialisPetropolitanae,V,
forthe years 1730-31,publishedin 1738).
4It may be suggested that expenditureon an
automobile may be made divisible by resortingto
1 I am gratefulto Larry Sjaastad for much de- renting.The possibilityof rentingthe services of
the presentationof the indivisible goods may reduce the influenceof intailed assistancein clarifying
elimidivisibilitybut in practicedoes not effectively
argumentof thisnote.
nate it. Moreover,rentingand owningare notidenti2 See A. Marshall, Principles of Economics(8th
in the
cal formsof consumption;and imperfections
ed.; New York: Macmillan Co., 1920), Mathemati- capital marketfrequentlydebar the consumerfrom
the possibilityof renting.
cal Notes IX, p. 843.
530
This content downloaded from 155.69.24.171 on Fri, 30 Jan 2015 02:35:30 AM
All use subject to JSTOR Terms and Conditions
WHY DO PEOPLE BUY LOTTERY TICKETS?
531
Excludingthepossibilityofuniversity
edu- added to theincomeO12,themarginalutilcation,hismarginalutilityfunction
is repre- ity of incomewill be greaterthan 12C, for
sentedby the continuously
decliningcurve thisincrement
will enabletheindividualto
MNPQ. Let the fixedcost of a university purchaseitems of consumptionpreviously
educationbe OH and its totalutilityto the forgoneto permitpaymentofhiseducationstudentbe JOHK. Then if his incomeis al expenses.If thesatisfaction
derivedfrom
highenough(higherthanOI3, whereI113 =
theseotheritemsis assumedforsimplicity
OH and 1,N = HK) he willcertainly
choose to be independent
ofwhetheror nothe goes
the university
education,and his marginal to university,
D willbe at thesameheightas
utility-curve
will seem' to assumethe path A, and DEF will be a rightwardshiftof
MNEF, whereNE = OH and is parallel ANP. The individual's marginal utility
to the X-axis. On the otherhand, if his functionwill be MNPDEF if he purchases
incomeis less thanOIj, he will not choose theindivisiblegood,university
training;the
y
M
GIS
I
1.
I%*,,I
education,sincehe willderive
theuniversity
greaterutilityfromspendinghis moneyon
otheritemsofconsumption.
Somewherebetween0I1 and 013 there
existsa level ofincomeat whichhe willbe
aboutchoosing
on themarginofindifference
education;let thisincomebe
theuniversity
educa012. If he does notchooseuniversity
tion, his marginal utility-curvewill be
MNP; if he does, it will be MABC. By
he gainsutilitymeasured
goingtouniversity
by thearea NPC and losesutilitymeasured
of
by the area ABN, and by the definition
12 theseareas are just equal.
For a marginal incrementof income
5 It
is shownbelowthattheactual path is MPDF.
upwardstep fromP to D is attributableto
the utilityderivedfromthe consumption
of thisgood.'
education"were
If the good "university
on it would
divisible,expenditure
perfectly
have a marginalutilitydiminishingconin
tinuouslyin the usual way, represented
the diagramby the curveG'LF' (the area
G'JL being equal to the area LF'K). The
individualwouldbeginto consumethegood
6 Consideringthat BC is spent on university
education, the marginal utility curve could be
thoughtof as MABCDEF; but since the marginal
utilityofincomeis pJ2 in the absence and D12 in the
presenceof the universityeducation,it is moreappropriateto take MNPDEF as the marginalutilityof-incomecurve.
This content downloaded from 155.69.24.171 on Fri, 30 Jan 2015 02:35:30 AM
All use subject to JSTOR Terms and Conditions
532
NG YEW KWANG
at income OIo (where G'O = GIO) and marginalutilityofincomewouldhave to be
would consumeone unit of it at income increased.Now in termsof Figure1, as inOI4 (whereFI4 = F'H). His marginalutil- come is increasedfromthe initiallevel of
ity-curvein this case would follow the 1o, the consumerwill initiallyspend all of
dotted curve MGUF, being higherthan that increaseon good Y (when good X is
MNPDEF in the segmentUF (the area indivisible),whichcauses themarginalutility of Y to fallmorerapidlythanif he alGPU beingequal to the area UDF).
the located part of his additionalexpenditure
An alternativeway of constructing
in the presenceof on X as well. By our assumptions,the
marginalutility-curve
has been suggestedto me in marginalutilityof incomeis equal to the
indivisibility
by LarrySjaastad, to whom marginalutilityof Y. Hence ifGF describes
correspondence
I am indebted for the followingthree the marginalutilityof incomewhen both
goods can be varied continuously,then
paragraphs:
to Figure1, let MGF be a seg- marginalutilityis describedby something
Referring
with likeGP whenX can be variedonlydiscretementof the marginalutilityfunction,
incomeon the horizontalaxis, when there ly.The areaGPU is thelossofutilitycaused
on the mannerin which by theindivisibility.
are no restrictions
At somepointbetweenIo and I4, income
the consumerallocateshis income,that is,
Now let goodX be willbe just largeenoughthat theutilityof
thereis no indivisibility.
indivisible,and let F representall other consuminganotherunit of good X is just
goods which are assumed to be perfectly equal to theutilityassociatedwiththeunits
divisible.Assumethatthereare at leasttwo of Y whichmust be givenup to buy X.
incomelevels,1o and I4, such thatindivisi- Justbeyondthatpoint,the individualwill
bilitiesdo not matter-that at 1o the con- change his consumptionpattern-he will
sumerwould demand exactlyN units (N buy anotherunitofX and willconsumeless
may be zero) of good X ifit wereperfectly Y. As a result,theratioof Y to X is sharply
divisible,and that at 14 he would demand reduced, and consequentlythe marginal
exactlyN + 1 unitsofX ifit wereperfectly utilityof Y is sharplyincreased.Since the
divisible.Thus,hismarginalutilityfunction marginalutilityof Y is also the marginal
must pass throughpoints G and F even utilityof income,we have theverticalsegare introduced.It will mentPD in Figure 1. Moreover,point D
whenindivisibilities
not followthe smoothcurve fromG to F mustlie above thecurveGF, becauseat the
as incomeis increasedfromIo to 14, but incomeleveljust beyondI2, theconsumeris
ratherthebrokencurveGPDF.
takingmoreX, henceless Y, thanifX were
of X at
For simplicity,let us defineour units divisible.That is, his consumption
suchthatthepricesofall divisiblegoodsare incomeI2 is the same as it wouldbe at the
unity. In equilibriumthen, the marginal largerincomeI4 whenX is divisible.Thus
utilityofincomewillbe exactlyequal to the his consumptionof Y must be less at the
marginalutilityofthegoodswhichthecon- incomelevel just beyondI2 thanis implied
sumercan vary. If the marginalutilityof by the smoothcurve GF, hence Y has a
incomeis declining,the marginalutilityof largermarginalutilityat thisincomelevel
the marall goodsmustdeclineas theyare increased when X is indivisible;therefore,
in quantityrelativeto othergoods. If the ginalutilityof incomejust beyondincome
contrarywere true-if even one good had I2 is actuallygreaterin theindivisiblecase.
increasingmarginalutility-an increasein Moreover,this remainstrue until income
incomewouldlead to increasedconsumption reachesI4. The gain in utilityis the area
in theconsump- UDF, whichis exactlyequal to the area
ofthatgoodand a reduction
tionofall others,and themarginalutilityof GPU because total utilityat incomeI4 is
ofX, as has
each good wouldhave increased,hencethe unaffected
by theindivisibility
This content downloaded from 155.69.24.171 on Fri, 30 Jan 2015 02:35:30 AM
All use subject to JSTOR Terms and Conditions
WHY DO PEOPLE BUY LOTTERY TICKETS?
533
been assertedabove. The slope of the seg- ies and otherformsofgamblingthereis such
ment DF must be greaterthan the cor- a contradiction
if the pleasureof gambling
respondingsegmentof GF for the same is ignored,as is reasonablein the case of a
reasonthat theslope ofGP is greaterthan lottery.The questionis: Whydo peoplebuy
the corresponding
partofGF-all increases lotterytickets?The analysisoftheprevious
in incomeare beingspenton Y.
sectionprovidesa possibleexplanation.
Thus the introduction
of indivisibility Considerthe exampleof thehigh-school
ofexpenditure
givesrisetopeculiarbehavior graduatepreviouslydiscussed,and assume
of the marginalutilityof incomefunction, that his ordinaryincomeis insufficient
to
the smoothcurveGF beingconvertedinto pay fora university
education,but thathe
thebrokencurveMNPDEF.
would go to university
if he won a lottery
M
15-
0
10
20 24 25
X2X,
30
37.6 40
50
12
60
67.6
X3
FIG. 2
prize. Let his situationbe presentedby
Figure2, wherehis ordinary
incomeis OX1,
less
than
the
critical
level
I2;
and let X1X2
FollowingFriedmanand Savage,choices
the
of
be
a
price
and X2X3
lottery
ticket,
by the
involvingriskmay be represented
thatis,"thecertain be the (single)prizein thelottery.
buyingoffireinsurance,
If he is notgoingto enteruniversity,
then
lossofa smallsum (theinsurancepremium)
in
terms
of
total
utility
he
is
gambling
the
of a small
in preference
to thecombination
A
area
a
small
against
chance
of
winning
the
chanceof a muchlargerloss (the value of
if all goods
the house) and a largerchanceof no loss," area B1 + B2. Alternatively,
coursearedivisible,
and thepurchaseof a lotteryticket,thatis, includingtheuniversity
"a large chance of losinga small amount he is gamblingthe area A + A' againsta
(thepriceof thelotteryticket)plus a small small chance of winningB1 + B2 + B +
chanceofwinninga largeamount(a prize) B'. In eithercase, sincemarginalutilityis
in preference
to avoidingbothrisks."In the diminishing,
he wouldnot be behavingaccase of insurancethereis no contradiction cordingto the rulesof utilitymaximization
and diminish- if he boughtthe lotteryticket(assuming
betweenutilitymaximization
ingmarginalutility,butin thecase oflotter- that the lotteryis no betterthan fair).If,
III.
GAMBLING AND THE INDIVISIBILITY
OF EXPENDITURE
This content downloaded from 155.69.24.171 on Fri, 30 Jan 2015 02:35:30 AM
All use subject to JSTOR Terms and Conditions
NG YEW KWANG
534
edu- Hence if the chance of winningis betterthan
on university
however,theexpenditure
cation is indivisible,he is gamblingA 1/48.6, it will pay the student to buy the
lotteryticket.
againsta smallchanceofgainingB1 + B2 +
Thus it has been shown that the introducB + C; and it may pay himto do so even
tion of indivisibilityof expenditurepermits
if thelotteryis less thanfair.
This can be shownby a mathematical the rationalization of gambling according to
example.Suppose that the student'smar- the principlesof utilitymaximization,while
ginal utilityfunction(the curveGPQ) ex- maintaining the assumption of diminishing
is y = marginal utility of consumption.8
cludingthe possibilityof university
Acceptance of the indivisibility of exf(x) = 100/(x+ 1), his existingincome25
units, the minimumcost of a university penditure illuminates the analysis of beeducation30 units,and the totalutilityof havior in the face of risk. In the firstplace,
the education150 units;the priceof a lot- the fact that many people both buy insurteryticket1 unit,and the singleprize of ance and gamble, that is, choose certainty
thelottery43.6 units.Undertheseassump- and at the same time subject themselves
tions,the areas A, B1,B2, B2, and C in the to risk, can be explained without either
destroyingthe assumption of utility maxidiagramcan be calculatedto be'
-25
A=
=
00
Kx-i)
10)dx
X + 1)
(log 38.6 -log 26) = 39.5,
-00
B2+ B2+ C =?
=
25) = 39.2,
100(log 26-log
jJ2
B1 =
dx
dx
100 (log 38.6 -log 8.6)
A
B1+B2[+B2-+[C
=
150.2
3.92
150.2 + 39.5
3.92
1 89./
1
48.6'
mization or introducingthe supplementary
assumption of increasingmarginal utility of
income over a certainrange of income variation.
Second, indivisibilityof expenditure explains why lotteries generally have several
or many prizes. The Social and Welfare
Services Lottery in Malaya, for example,
offersthe followingprizes:9
Firstprize............$..
Second prize. .
Thirdprize.
Fourthprize.
5 fifthprizes,each .......
15 sixthprizes,each......
30 seventhprizes,each. . .
40 eighthprizes,each....
75 ninthprizes,each ..
Gift,each ..............
Lucky,each.............
Consolation,each ...... ..
375,000
125,000
60,000
30,000
10,000
5,000
3,000
2,000
1,000
500
250
100
7 The criticalincomelevel 12is obtainedby solving the followingequation:
j-30(X
+ ) dx = 5X30
30
-log(I-29)]
[log(I +
I + 1 _e.
I
-
e
29
I
=
37.6
=
1.5
8 The departure
frombehaviorin accordance
with this assumptionis the consequenceof the
ofexpenditure
and is notinassumedindivisibility
itself.
herentin thenatureofutility
9A Malayandollaris worthapproximately
onedollar.
thirdofan American
This content downloaded from 155.69.24.171 on Fri, 30 Jan 2015 02:35:30 AM
All use subject to JSTOR Terms and Conditions
WHY DO PEOPLE BUY LOTTERY TICKETS?
535
The explanationof the offerof multiple the curve ONQ, the interpretationof which
prizes suggestedby indivisibilityof ex- is "to regard the two convex segments as
penditureis, first,thatformostpeople the correspondingto qualitatively differentsoinvolvedis substan- cioeconomic levels, and the concave segamountof expenditure
tiallysmallerthanthetotalsum to be paid ment to the transition between the two
potentialpur- levels. On this interpretation,increases in
out; second, that different
indivisible income that raise the relative position of the
chasersof ticketshave different
expendituretotals in mind,and even the consumer unit in its own class but do not
sumsin shiftthe unit out of its class yield diminishsameindividualmayhave different
uses.10
mindfordifferent
ing marginal utility, while increases that
help to shiftit into a new class, that give it a new
indivisibilities
Third,expenditure
explain the fact that lotterytickets are social and economic status, yield increasing
eagerlyboughtby thepoor,and the "num- marginal utility."11
bers game" and similargambles flourish
classes;
especiallyamongthe lower-income
ofexpenditure Y
ofindivisibility
fortheeffect
in thecase of
itselfmoststrongly
manifests
low-incomeconsumerunits. The cost of
edua car,a house,a university
purchasing
cation,or a businessappearsfarbeyondthe
means of a poor family,unless the money
is obtained by gambling;but the same
is only a small fractionof the
expenditure
incomeof therichand raiseslittleproblem
ofindivisibility.
IV. COMPARISON WITH THE FRIEDMANSAVAGE HYPOTHESIS
The analysis presentedhere resembles
that of Friedmanand Savage in some respects,but differsfromit in others.The
betweenthe twomay be clarirelationship
to Figure3, in whichinfiedby reference
comeis measuredalongtheX-axisand total
utility along the Y-axis. The FriedmanSavage utilityfunctionis representedby
x
o
FIG. 3
The total utility functionof the present
analysis is representedby OKQ; the kink at
K correspondsto the broken marginal utility-curveof Figures 1 and 2, and is caused
by the existence of indivisibilities rather
than inherent in the nature of utility. It
differsin shape from the Friedman-Savage
10In addition,of course,the offerof manyprizes
functionin that the latter is a smooth curve
may lead the potential purchaser irrationallyto
believe that his chances of winningare higherthan with no kink. The curve OPQ representsthe
they reallyare; or the offerof prizes whose magni- orthodox total utility function,which canindividuals not explain gambling behavior.
tudes match the amountsthat different
would considera fortunemay exercisea psychological appeal.
11 Op.
cit.,Sec. 5b.
This content downloaded from 155.69.24.171 on Fri, 30 Jan 2015 02:35:30 AM
All use subject to JSTOR Terms and Conditions