Prospect Theory and consumer behavior: Goals and Tradeoffs

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Prospect Theory and consumer behavior: Goals and

Tradeoffs

Florent Buisson

To cite this version:

Florent Buisson. Prospect Theory and consumer behavior: Goals and Tradeoffs. 2013.

�halshs-00820722�

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Documents de Travail du

Centre d’Economie de la Sorbonne

Prospect Theory and consumer behavior: Goals and Tradeoffs

Florent B

UISSON

(3)

FlorentBuisson

∗

PSE and Université Paris1

23mars 2013

∗

UniversitéParis1;mail:MaisondesS ien esÉ onomiques,106-112bddel'Hpital,75647 Paris edex13,

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Jemontrequ'un onsommateuraverseàlapertequidoitrépartirsonbudgetentredeux

bienspréfèredesallo ationspourlesquellesla onsommationestégaleaupointderéféren e

pouraumoinsundesbiens, oudessolutionsen oin.L'intensitéduphénomène dépend de

la ourburedelafon tiond'utilité.Ces résultatssont ohérentsave plusieursfaitsstylisés

quinepeuventpasêtreexpliquésparlathéoriestandarddu onsommateur.

Mots- lés : Aversion pour laperte, théoriedesperspe tives.

Abstra t

Ishowthatalossaverse onsumerwhomustshareherbudgetbetweentwogoodsprefer

allo ations for whi h onsumption equals referen e point for at least one good. The

phe-nomenonintensitydependsonthe urvatureoftheutility urve. Theseresultsare onsistent

withseveralstylizedfa ts whi h annotbeexplainedbythestandard onsumertheory

JEL lassi ation numbers: D03, D11,D12.

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The standard onsumer theoryis basedon the notion of utility maximization, through the

sear h for the highest possible onsumption, but in the reality, we set ourselves intermediary

goals,whi h make de ision-making andtradeoseasier. Thesegoalsserve asreferen epointsto

assess our progress. It is therefore important to assess how these goals inuen e our behavior

; in parti ular, it turns out that agents behavior is often quitedierent depending on whether

their onsumption isaboveor below agiven target.

Thesedieren es anbeexplainedbytheProspe tTheory. Prospe tTheorywasinitially

de-veloppedfor risky hoi es (Kahnemanand Tversky(1979)), thenoneofits omponents,namely

loss aversion, was extended to ertain hoi es (Tversky and Kahneman (1991)). But this

ex-tension wasbuilt mainly for labor supply ( f. infra) or for dis rete hoi es (and more pre isely

binary hoi es, whi harethe losest to the traditionalframework ofrisky hoi es), whereasthe

theoryof the onsumerbehaviorrelieson ontinuous hoi es undera budgetary onstraint.

Thus, Köszegi and Rabin (2006) deal with the issueof the referen e point formation when

thereexistsaninitialun ertainty,whi hdisappearsbeforethenal onsumption de ision. From

this standpoint, their model is an intermediary step between risky and ertain hoi e, not an

analysis of ertain hoi e stri to sensu ;inparti ular, their model ontinues to fo uson binary

hoi es, between buying and not buying fora singlegood,or between two lump goods.

In the next se tion, I study the optimal hoi e of a onsumer who must share her budget

between two goods andwho islossaverse. Isu essively onsider threefun tionalforms for the

utilityfun tion :

linear,

onvexinlosses and on ave ingains,

and nally on ave everywhere.

In the third se tion, I dis uss several behaviors whi h annot be explained in the standard

onsumer theory,but whi hare onsistent withlossaversion

2 Models

Prospe t Theory endows the onsumer with a utility fun tion whi h is slightly on ave in

gains (when the onsumption is higher than the referen e point), and slightly onvex in losses,

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the referen e point.

Onthe ontrary,standard onsumer theory relies on utility fun tionswhi h are everywhere

on ave,andwhi h have no referen epoint (i.e. utilityisequal to zerofor anull onsumption).

Iwillthereforepro eedinthree steps,and onsidertheee tof lossaversioninthree ases:

First,inthe simplest ase, namelya linearutility fun tion ;

Se ond, whenthe utilityfun tion isS-shapedasintheProspe tTheory;

Finally, whentheutilityisstandard, thatis on ave,but hasareferen e point.

2.1 Linear utility

Ibegin withthe aseof alinear utility fun tionwithtwo goods,

c

1

and

c

2

:

U (c

1 , c

2 ) = c

1 + β ∗ c

2 s.t. p

1 c

1 + p

2 c

2 = W

with

β

therelative weight of thegood 2intermsof thegood 1intheutilityfun tion.

Figure 1 shows the graphof the utility fun tion and the orresponding indieren e urves.

Ona side note,for all guresinthis arti le,I took

β = p

1 = p

2 = 1

withoutlossof generality. Figure1: Linear Utility

(a)

0

10

20

30

40

50

0

10

20

30

40

50

0

10

20

30

40

50

60

70

80

90

100 c1

Fonction utilité

c2

(b)

Courbes indifférence

c2

c1

10

20

30

40

50

5

10

15

20

25

30

35

40

45

50

Thesolutionofthe onsumer'sutilitymaximizationprogram isa ornersolution iftheslope

of the budget onstraint line is not equal to the slope of theindieren e urves. Ifboth slopes

areequal, the onsumeris indierent between all possible allo ationson herbudget onstraint.

Figure2shows anexample of apossiblebudgetline. Theoptimalsolution isto allo atethe

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Courbes indifférence

c1

c2

5

10

15

20

25

30

35

40

45

50

5

10

15

20

25

30

35

40

45

I now introdu e loss aversion for ea h of the two goods. That is to say, for ea h good, the

onsumer hasareferen e point, atwhi h utilityisequal to zero.

Theutilityfun tion isthen

U (c

1 , c

2 ) = u

1 (c

1 ) + β ∗ u

2 (c

2 )

s.t. p

1 c

1 + p

2 c

2 = W

with

β

therelative weight of good 2 withrespe tto the good 1,

λ

the lossaversion oe ient,

W

the onsumer'sbudget, and

u

i

(c

i

) = c

i

−

r

i

si

c

i

≥

r

i

u

i

(c

i

) = −λ(r

i

−

c

i

)

si

c

i

< r

i

where

r

i

is the referen epoint for good

i

,and

c

i

isthe onsumption of good

i

.

Figure3shows the utilityfun tion andthe orrespondingindieren e urves, for ave torof

referen e points

(20, 20)

.

Astheutilityfun tionforea hgoodisstepwiselinear,theindieren e urvesarealsostepwise

linear, andtheMarginalRateofSubstitution isastepfun tion(i.e. stepwise onstant). We an

noti e that MRS are equal in quadrants South-West and North-East, and they arethe inverse

of ea hother inquadrants South-Eastand North-West.

Therefore, there always exists orner solutions, when the relative pri e is very low or very

high. Formally, we have a orner solution when

p

2 p

1 <

β

λ

or

p

2 p

1 > βλ

. But when the relative

pri e is between thesetwo boundaries, thereexistsa new type ofsolutions, whi h we ould all

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(a)

0

10

20

30

40

50

0

10

20

30

40

50 −100

−80

−60

−40

−20

0

20

40

60 c1

Fonction utilité

c2

(b)

Courbes indifférence

c1

c2

5

10

15

20

25

30

35

40

45

50

5

10

15

20

25

30

35

40

45

50

good,and thusitisoptimal for the onsumerto setthe onsumption ofone ofthetwogoodsat

the orrespondingreferen epoint, andtoadjustthe onsumptionoftheothergooda ordingly.

Figure 4 shows an example of a lassi al orner solution, and an example of the new solution

type.

Figure 4: LinearUtilitywithLoss AversionOptimal Solutions

(a)

Courbes indifférence

c1

c2

5

10

15

20

25

30

35

40

45

50

5

10

15

20

25

30

35

40

45

50

(b)

Courbes indifférence

c1

c2

5

10

15

20

25

30

35

40

45

50

5

10

15

20

25

30

35

40

45

50

2.2 Prospe t Theory

Iwill now onsiderthe ase whenthe onsumerhasa Prospe t Theoryutilityfun tion :

U (c

1 , c

2 ) = u

1 (c

1 ) + β ∗ u

2 (c

2 )

(9)

with

β

therelative weight of the good 2 interms of the good 1,

λ

the lossaversion oe ient,

W

u

i

(c

i

) = (c

i

−

r

i

)

α

si

c

i

≥

r

i

u

i

(c

i

) = −λ(r

i

−

c

i

)

α

si

c

i

< r

i

Figure5shows the utilityfun tion andthe orrespondingindieren e urves, for ave torof

referen e points

(20, 20)

. I took

α = 0.5

to make more salient the onvex and on ave parts of theutilityfun tion,even thoughthestandard valuein theliteratureis

α = 0.88

.

Figure 5: UtilityFun tion of theProspe t Theory

(a)

0

10

20

30

40

50

0

10

20

30

40

50 −25

−20

−15

−10

−5

0

5

10

15 c1

Fonction utilité

c2

(b)

Courbes indifférence

c1

c2

5

10

15

20

25

30

35

40

45

50

5

10

15

20

25

30

35

40

45

50

The solution of the onsumer's program is not obvious : the shape of the utility fun tion

hangesa ordingtothepositionofthe onsumptionve tor

(c

1 , c

2 )

As the shape of the utility fun tion hanges from one quadrant to the next, instead of being

everywhere on ave,we annotdeterminetheoptimalsolutionforthe onsumerprogramdire tly

through variational methods (rst and se ond order onditions). For ea h quadrant, we need

to al ulate the lo ally optimal solutions,whether theyareinterior, orner, or "interior orner"

solutions ; then we will be able to determine the global optimum by omparing all the lo al

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2.2.1 North East Quadrant :

c

1 ≥

r

1

and

c

2 ≥

r

2

Interior Solution IntheNE quadrant ofthe onsumption spa e, we have

U (c

1 , c

2 ) = (c

1 −

r

1 )

α

+ β(c

2 −

r

2 )

α

s.t. p

1 c

1 + p

2 c

2 = W

Or, ifwe rempla e

c

2

byits expressioninthe budget onstraint

U (c

1 ) = (c

1 −

r

1 )

α

+ β

W − p

1 c

1 p

2 −

r

2 α

Wetake the rstderivative :

∂U

∂c

1 (c

1 ) = α(c

1 −

r

1 )

α−1

+ αβ

−p

1 p

2 W − p

1 c

1 p

2 −

r

2 α−1

Thisderivative isequal to zero if

α(c

∗

1 −

r

1 )

α−1

= αβ

p

1 p

2 _{W −p}

1 c

∗

1 p

2 −

r

2 α−1

c

∗

1 −

r

1 = β

1 α−1

p

1 p

2 _α−1

1 _{W −p}

1 c

∗

₁

p

2 −

r

2 c

∗

1 −

r

1 = β

1 α−1

_p

1 p

2 _α−1

1 W

p

2 −

r

2 −

β

α−1

1 _p

1 p

2 _α−1

1 _p

1 p

2 c

∗

1 c

∗

1 + β

1 α−1

p

1 p

₂

_α−1

1 p

₂

c

∗

1 = r

1 + β

1 α−1

p

1 p

₂

_α−1

1 W

p

₂

−

r

2 1 + β

α−1

1 p

₁

p

₂

α

α−1

c

∗

1 = r

1 + β

1 α−1

p

₁

p

₂

1 α−1

W

p

₂

−

r

2 c

∗

1 =

1 + β

α−1

1 p

2 _α−1

α

−1

r

1 +

β

α−1

1 p1

p2

1 α−1

1+β

α−1

1 p1

p2

α

α−1

W

p

2 (r

1 −

r

2 ) +

W

2 ,

andthus

c

∗

1 −

r

1 = c

2 −

r

2 =

W

2 −

1

2 (r

1 + r

2 )

That is to say, the onsumer has an equal "net onsumption" ( onsumption minus referen e

point)for ea h good.

Let'stake the se ond orderderivative :

∂

2 U

∂(c

1 )

2 (c

1 ) = α(α − 1)(c

1 −

r

1 )

α−2

+ α(α − 1)β

−p

1 p

2

2 W − p

1 c

1 p

2 −

r

2 α−2

(11)

The orrespondingse ond order ondition (aftersimplifying by

α(α − 1) < 0

) is

(c

∗

1 −

r

1 )

α−2

+ β

p

1 p

2

2 W − p

1 c

∗

1 p

2 −

r

2 α−2

≥

0

This ondition isfullled for

c

1 ≥

r

1

,

c

2 ≥

r

2

and

α ≤ 1

.

Thismeanswe an ex lude"frontier" solutions

c

1 = r

1 , c

2 > r

2

et

c

1 > r

1 , c

2 = r

2

. Indeed, ifwe onsider the rstpossibility(

c

1 = r

1 , c

2 > r

2

),we have

U (r

1 ) = U (c

∗

1 ) + U

′

(c

∗

1 ) (r

1 −

c

∗

1 ) + U

′′

(c

∗

1 ) (r

1 −

c

∗

1 )

2 = U (c

∗

1 ) + U

′′

(c

∗

1 ) (r

1 −

c

∗

1 )

2 < U (c

∗

1 )

by on avity of the utility fun tion in this quadrant. Therefore, we annot have a frontier

solution between this quadrant and another one, as the interior solution gives a higher utility.

We an ex ludethe se ondtype offrontier solutions witha similarline ofreasoning.

2.2.2 South-East Quadrant:

c

1 ≥

r

1

and

c

2 < r

2

Interior Solution IntheSE quadrant, we have

U (c

1 , c

2 ) = (c

1 −

r

1 )

α

−

βλ(r

2 −

c

2 )

α

s.t. p

1 c

1 + p

2 c

2 = W

or, ifwe repla e

c

2

byits expressioninthe budget onstraint :

U (c

1 ) = (c

1 −

r

1 )

α

−

βλ

r

2 −

W − p

1 c

1 p

2 α

Wetake the rstderivative :

∂U

∂c

1 (c

1 ) = α(c

1 −

r

1 )

α−1

−

αβλ

p

1 p

2 r

2 −

W − p

1 c

1 p

2 α−1

The orrespondingrst order ondition (aftersimplifying by

α > 0

) is

(c

1 −

r

1 )

α−1

= βλ

p

1 p

2 r

2 −

W − p

1 c

1 p

2 α−1

thatis

r

2 −

W −p

p

2

1 c

1 1−α

= βλ

p

1 p

2 (c

1 −

r

1 )

1−α

r

2 −

W −p

_p

₂

1 c

1 =

h

βλ

p

1 p

2 i

1 1−α

(c

1 −

r

1 )

r

2 +

h

βλ

p

1 p

2 i

_1−α

1 r

1 −

W

_p

₂

=

h

βλ

p

1 p

2 i

_1−α

1 c

1 +

p

1 p

2 c

1

(12)

c

∗

1 =

"

p

1 p

2 +

βλ

p

1 p

2

1 1−α

#

−1

"

r

2 +

βλ

p

1 p

2

1 1−α

r

1 −

W

p

2 #

Let'stakethese ondorderderivativeand al ulatethe se ondorder ondition forthis equation

to dene a lo almaximum:

∂

2 _U

∂(c

1 )

2 (c

1 ) = α(α − 1)(c

1 −

r

1 )

α−2

−

α(α − 1)βλ

p

1 p

2

2 r

2 −

W − p

1 c

1 p

2 α−2

The SOC in

c

∗

1

is

∂

2 _U

∂(c

1 )

2 (c

∗

1 ) ≤ 0

,or ifwe simplify by

α(α − 1) < 0

,

(c

1 −

r

1 )

α−2

−

βλ

p

1 p

2

2 r

2 −

W − p

1 c

1 p

2 α−2

≥

0

We an repla e

(c

∗

1 −

r

1 )

α−1

byits expressionintheFOC :

(c

∗

1 −

r

1 )

−1

"

βλ

p

1 p

2 r

2 −

W − p

1 c

∗

1 p

2 α−1

#

−

βλ

p

1 p

2

2 r

2 −

W − p

1 c

∗

1 p

2 α−2

≥

0

whi h means we angreatly simplify the inequality:

(c

∗

1 −

r

1 )

−1

−

p

1 p

2 r

2 −

W − p

1 c

∗

1 p

2 −1

≥

0

or

r

2 −

W − p

1 c

∗

1 p

2 ≥

p

1 p

2 (c

∗

1 −

r

1 )

And eventually (aftermultiplying by

p

2

) :

p

2 r

2 + p

1 r

1 ≥

W

This inequality is the SOC for the result of the FOC to be a lo al maximum. If it is not

fullled, thelo ally optimal solution isa orner solution. This inequalityhas a straightforward

interpretation: fortheoptimalsolutiontoimply

c

2 < r

2

,ane essary onditionisthatthebasket

(r

1 , r

2 )

isnot partof the budgetsetfor pri es

p

1

and

p

2

,and wealth

W

.

Corner Solutions The only possible orner solution in this quadrant is

c

1 ≥

r

1

and

c

2 = 0

. Let us al ulate the rst orderderivative oftheutilityfun tion in

c

1

for

c

1 = W/p

1

,

c

2 = 0

:

∂U

∂c

1 W

p

1 = α

W

p

1 −

r

1 α−1

−

αβλ

p

1 p

2 (r

2 )

α−1

A ne essary onditionfor a ornersolution isthat thisderivative bepositive:

∂U

∂c

1 W

p

1 ≥

0

(13)

α

W

_p

1 −

r

1 α−1

−

αβλ

p

1 p

2 r

₂

α−1

≥

0 W

p

1 −

r

1 α−1

≥

βλ

p

1 p

2 r

₂

α−1

r

₂

1−α

≥

βλ

p

1 p

2 W

p

1 −

r

1 1−α

r

2 ≥

h

βλ

p

1 p

2 i

1 1−α

W

p

1 −

r

1

we an seethat for

β = p

1 = p

2 = 1

,this resultyields

(c

1 −

r

1 )

α−1

≥

λr

2 α−1

This on lusionhasan interpretationintermsoftheutilityfun tion urvature: inthestandard

ase,utilityis on avewithrespe ttoea hgood,thereforethemorewediminishthe onsumption

of one good, the more the marginal utility orresponding to this good in reases, and themore

painfulanyadditionalde reaseinthe onsumptionofthisgood. Buthere,utilityis onvexwith

respe ttothegood2. Thismeans thatthemore wediminish the onsumptionofthis good,the

more themarginal utilityofthis goodde reases. Ifwe rossthethreshold whenmarginalutility

for both goods is equal (e.g. when the referen e point for good 2 is very high), the onsumer

wouldbe better osuppressing all onsumption of good 2 and allo ating all her budget to the

onsumption of good1. Figure 6illustratesthis phenomenon.

Figure6: Cornersolution inthe SEquadrant

c1

c2

10

20

30

40

50

5

10

15

20

25

30

35

40

45

50

(14)

quadrants. Indeed, even if this frontier solution grants the onsumer a higher utility than the

interior solution of the SE quadrant, the interior solution of the NE quadrant dominates this

frontier solution, whi h therefore annotbea global maximum.

The only remaining possible frontier solution is

c

1 = r

1 , c

2 < r

2

. We an apply the same line of reasoningasfor theNE quadrant :

U (r

1 ) = U (c

∗

1 ) + U

′

(c

∗

1 ) (r

1 −

c

∗

1 ) + U

′′

(c

∗

1 ) (r

1 −

c

∗

1 )

2 = U (c

∗

1 ) + U

′′

(c

∗

1 ) (r

1 −

c

∗

1 )

2 < U (c

∗

1 )

if the previous SOC is fullled. In this ase, there annot be anyfrontier solution be ause the

interior solutiongrants the onsumera higherutility.

2.2.3 North-West quadrant:

c

1 < r

1

et

c

2 ≥

r

2

The treatment of the NW quadrant is similar to the treatment of the SE quadrant. The

utilityfun tion is

U (c

1 , c

2 ) = −λ(r

1 −

c

1 )

α

+ β(c

2 −

r

2 )α s.t. p

1 c

1 + p

2 c

2 = W

Whi h means that

U (c

1 ) = −λ(r

1 −

c

1 )

α

+ β

W − p

1 c

1 p

2 −

r

2 α

The rstderivative ofthe utilityfun tion is

∂U

∂c

1 (c

1 ) = λα(r

1 −

c

1 )

α−1

−

α

p

1 p

2 β

W − p

1 c

1 p

2 −

r

2 α−1

The orrespondingFOC is

λ(r

1 −

c

1 )

α−1

=

p

1 p

2 β

W −p

1 c

1 p

2 −

r

2 α−1

W −p

1 c

1 p

2 −

r

2 1−α

=

p

1 p

2 β

λ

(r

1 −

c

1 )

1−α

W −

p

1 p

2 c

1 −

r

2 =

p

1 p

2 ∗

β

λ

_1−α

1 (r

1 −

c

1 )

W − r

2 −

p

1 p

2 ∗

β

λ

_1−α

1 r

1 =

p

_p

1 ₂

c

1 −

p

1 p

2 ∗

β

λ

_1−α

1 c

∗

1 =

p

₁

p

₂

−

_p

1 p

₂

∗

β

λ

_1−α

1 −1

W − r

2 −

_p

1 p

₂

∗

β

λ

_1−α

1 r

1

(15)

∂

2 _U

∂(c

1 )

2 (c

1 ) = −λα(α − 1)(r

1 −

c

1 )

α−2

+ α(α − 1)

p

1 p

2

2 β

W − p

1 c

1 p

2 −

r

2 α−2

The orrespondingSOC is

λ(r

1 −

c

1 )

α−2

−

p

1 p

2

2 β

W − p

1 c

1 p

2 −

r

2 α−2

≤

0

Here again, we anrepla e

λ(r

1 −

c

1 )

α−1

by itsexpresssion intheFOC when

c

1 = c

∗

1

:

p

1 p

2 β

W − p

1 c

∗

1 p

2 −

r

2 α−1

(r

1 −

c

∗

1 )

−1

−

p

1 p

2

2 β

W − p

1 c

∗

1 p

2 −

r

2 α−2

≤

0

Then we simplify:

(r

1 −

c

∗

1 )

−1

−

_p

1 p

2 _{W −p}

1 c

∗

1 p

2 −

r

2 −1

≤

0 W −p

1 c

∗

1 p

2 −

r

2 ≤

p

1 p

2 (r

1 −

c

∗

1 )

W − p

1 c

∗

1 −

p

2 r

2 ≤

p

1 (r

1 −

c

∗

1 )

W

≤

p

1 r

1 + p

2 r

2

Wend thesame SOCasfor theSE quadrant,whi h islogi al asthe 2goods are symmetri al.

Corner Solutions The onlypossible ornersolution is

c

1 = 0

. Letus al ulate thederivative of theutility fun tion w.r.t.

c

1

when

c

1 = 0, c

2 = W/p

2

:

∂U

∂c

1 (0) = λαr

₁

α−1

−

αβ

p

1 p

2 W

p

2 −

r

2 α−1

The ne essary onditionfor a ornersolution isthat thisderivative be negative :

∂U

∂c

1

(16)

λαr

α−1

₁

≤

αβ

p

1 p

2 W

p

2 −

r

2 α−1

r

α−1

₁

≤

β

_λ

×

p

1 p

2 W

p

2 −

r

2 α−1

W

p

2 −

r

2 1−α

≤

β

_λ

×

p

1 p

2 r

1−α

₁

W

p

2 −

r

2 ≤

β

λ

×

p

1 p

2

1 1−α

r

1

This ondition has the same interpretation in terms of the utility fun tion urvature as the

ondition inthe SEquadrant.

FrontiersSolutions Theonlyfrontiersolutionweneedto onsideris

c

1 < r

1 , c

2 = r

2

,be ause wehave seenthatthe frontier solutions withtheNE quadrant annotbe global maxima.

We anapply thesame lineof reasoningasintheSEquadrant (with

c

∗

1

beingnow thelo al

maximuminthe NO quadrant) :

U

W −p

2 r

2 p

1 = U (c

∗

1 ) + U

′

(c

∗

1 )

W −p

2 r

2 p

1 −

c

∗

1 + U

′′

_(c

∗

1 )

W −p

2 r

2 p

1 −

c

∗

1

2 = U (c

∗

1 ) + U

′′

(c

∗

1 )

W −p

2 r

2 p

1 −

c

∗

1

2 ≤

U (c

∗

1 )

iftheformer SOC isfullled. Inthis ase, there annotbeanyfrontier solution.

2.2.4 SO quadrant :

c

1 < r

1

et

c

2 < r

2

Interior Solution IntheSOquadrant,wehave

U (c

1 , c

2 ) = −λ(r

1 −

c

1 )

α

−

βλ(r

2 −

c

2 )

α

s.t. p

1 c

1 + p

2 c

2 = W

Then Irepla e

c

2

byits expression inthe budgetary onstraint :

U (c

1 ) = −λ(r

1 −

c

1 )

α

−

βλ

r

2 −

W − p

1 c

1 p

2 α

I take thederivative:

∂U

∂c

1 (c

1 ) = αλ(r

1 −

c

1 )

α−1

−

αβλ

p

1 p

2 r

2 −

W − p

1 c

1 p

2 α−1

(17)

∂

2 _U

∂(c

1 )

2 (c

1 ) = −α(α − 1)λ(r

1 −

c

1 )

α−2

−

α(α − 1)βλ

p

1 p

2

2 r

2 −

W − p

1 c

1 p

2 α−2

The asso iated SOC (after simplifyingby

−α(α − 1)λ > 0

) is

(r

1 −

c

1 )

α−2

+ β

p

1 p

2

2 r

2 −

W − p

1 c

1 p

2 α−2

≤

0

This ondition is not veried for

c

1 ≤

r

1

utility than the interior solution. To determine the optimal solution, we only have to ompare

thetwo orner solutions : theoptimal solution dependson therelative pri e and

β

. Indeed, we have :

U (0) = −βλ

r

2 −

W

p

2 α

et

U

W

p

1 = −λ

r

1 −

W

p

1 α

2.2.5 Global Optima

Graphi al Analysis We an analyze graphi ally the onsumer's optimal behavior, by

on-sidering that she possesses two "utility pumps", one for ea h good, and that she allo ate her

budgetbyintrodu ing1-euro oins onebyoneinone ofthetwopumps. Atthebeginning, both

pumps are empty, and the onsumer must hoose to whi h pumpallo ate herrst euro. If she

allo ates itto pump1,thegenerated utility willbe approximately equal to

u

′

1 (0) = λα (r

1 )

α−1

, and similarlyfor pump2.

Letus assumethat

u

′

1 (0) ≥ u

′

2 (0)

and thatthe onsumer allo ate her rst euroto pump 1. As the fun tion

u

1

is onvex in losses, marginal utility is in reasing, so it is in the onsumer's bestinteresttoalsoallo ateherse ondeurotopump1,andsoon,asthe onsumer" limbs"the

urveoftheutilityfun tion. On e the onsumerrea hesthegainzone,marginalutilitybe omes

de reasing, but it is still high enough sothat the onsumer ontinues to allo ate her budget to

pump1. When the onsumerrea hes thepoint

c

¯

1 (r

2 )

su h that

u

′

1 ( ¯

c

1 (r

2 )) = u

′

2 (0)

then the marginal utility of pump 2 be omes higher than themarginal utility of pump 1, and

(18)

u

r

1 ¯

c

1 (r

2 )

su ession of euros to pump 2. She will start allo ate euros to pump 1 again only when the

marginal utility for both pumps will be equal. From this point on, she will allo ate her euros

alternatively to ea hpumpuntil shehasexhaustedherbudget.

Figure7 illustratesthisreasoning.

Global Optima Based on the former graphi al analysis, Iwill determine global maxima, by

examiningthedierentpossiblesituations,dependingonthequadrantsa rosswhi hthebudget

onstraint passes. Indeed, the budget onstraint an pass :

1. only a rossthe SOquadrant,

2. a rossthe SOand NOquadrants,

3. a rossthe SOand SE quadrants,

4. a rosstheSO, NOand SEquadrants,

5. a rosstheNE, NOandSE quadrants,

Figure 8illustratesthese dierent ases. Let us onsider themsu essively.

SOquadrant Ifthebudget onstraintpassesonlya rosstheSOquadrant,theoptimalsolution

(19)

c1

c2

10

20

30

40

50

10

20

30

40

50 c1

c2

10

20

30

40

50

10

20

30

40

50 c1

c2

10

20

30

40

50

10

20

30

40

50 c1

c2

10

20

30

40

50

10

20

30

40

50 c1

c2

10

20

30

40

50

10

20

30

40

50

the onsumer is indierent between both orner solutions. If

β = 1

,

p

1 = p

2

but

r

1 6= r

2

, the onsumer devotes all herbudget to the good with the lowest referen e point : by onvexity of

theutilityfun tion,it isthe goodwiththehighestmarginal utility.

SO and NOquadrants Ifthe budget onstraint passesa rosstheSOandNOquadrants,we

have three andidate solutions :

the orner solutionof the SOquadrant,

the orner solutionof the NOquadrant,

theinteriorsolution of theNOquadrant .

We annoti eimmediatelythatifthebudget onstraintpassesa rosstheSOandNOquadrants,

thepoint

(r

1 , r

2 )

isnot partofthe budgetset, andthereforetheSOC for theinteriorsolution is fullled.

The onsumer hoosestheoneofthetwo goodswiththehighestmarginal utilityper euro. If

it isthegood 1,asthe onsumer'sbudgetis insu ient torea h the on ave zoneofherutility

fun tion, the onsumer will devote all her budget to good 1, whi h orresponds to the orner

solution inthe SOquadrant.

Ifitisthegood2,itis heapenoughforthe onsumertorea hthe on avezoneofherutility

fun tion. If the onsumer's budget is insu ient to rea h the point

c

¯

1 (r

2 )

, she will allo ate all her budget to good 1,whi h orrespondsto the orner solution inthe NO quadrant. If the

(20)

to theinterior solutioninthe NOquadrant.

SO andSE quadrants When thebudget onstraintpassesa rosstheSOandSE quadrants,

the reasoning isexa tlythesame asinthe previous ase, withgood1 and 2inversed.

SO,NOandSEquadrants Inthis ase,we animmediatelyeliminatetheinteriorextremum

ofthe SOquadrant,be ausewehaveseenthatitisaminimumandnotamaximum. The hoi e

between theNOand SEquadrants isbasedon therelative pri ebetween thetwo goods andthe

onsumer'spreferen e. On ethe onsumerhassele tedher"favorite"quadrant (i.e. thegoodto

whi h the onsumer will rstallo ate hermoney), the hoi e between the interiorsolution and

the orner solution isbasedon the same riterion asintheprevious ases.

NE, NO and SE quadrants Inthis ase, the onsumer'sbehavior will depends on whether

her budgetis su ient to rea h the point

c

¯

1 (r

2 )

(if sheallo ates her budget rst to thegood 1 ; the reasoning is reversed if she hooses the good 2 rst). If it is insu ient, the onsumer

allo ates all her budget to good 1, and the optimal solution is the orner solution of the NO

quadrant. If it is su ient, the onsumer will start allo ating some of her budget to good 2,

whi h orrespondsto theinteriorsolution ofthe NO quadrant. Beyond a ertainthreshold, the

onsumer will rea h her on ave zone for the se ond good, whi h orresponds to the interior

solution ofthe NE quadrant.

Con lusion We an sum up the general behavior of the onsumer by plotting the in ome

expansion path, i.e. the set of onsumption ve tors hosen for all the possible levels of wealth,

for given relativepri es. Thegure 9represents thein omeexpansionpath when the onsumer

onsumes inprioritythe good1.

In other words, the lassi al result of the onsumer theory stating that onsumers have a

taste for variety and prefer splitting their onsumption rather homogeneously between goods

appears as a parti ular ase, whi h applies only when onsumers are ri h enough to keep their

onsumption above their referen epoint for allgoods.

2.3 Con ave Utility

Finally, I onsider the ase of a utility fun tion whi h is everywhere on ave, but with loss

(21)

c

2 c

1 r

2 r

1 ¯

c

1 (r

2 )

¯

c

2 (r

1 )

L`Haridon (2008) in risky hoi es, even though most empiri al results for hoi es under risk

are rather heading in the dire tion of a utility fun tion whi h is slightly onvex in losses (e.g.

Abdellaoui, Blei hrodt, and Paras hiv (2007)). As far as I know, this possibility has not been

onsidered for hoi es under ertainty in the behavioral e onomi s literature, and loss aversion

hasalwaysbeen asso iatedwithalinearorS-shaped(rst onvexthen on ave)utilityfun tion.

The only arti le whi h dis uss this possibility is Lapidus and Sigot (2000) in the History of

E onomi Thought, inananalysisof thebenthamian theoryof pleasureand pain.

Therefore,Isuggestthefollowingfun tionalform,whi hyieldsthedesiredshapewhilestaying

lose to the spirit ofthe initial Prospe tTheory.

U (c

1 , c

2 ) = u

1 (c

1 ) + β ∗ u

2 (c

2 )

s.t. p

1 c

1 + p

2 c

2 = W

(22)

W

u

i

(c

i

) = λ ∗ c

α

i

si

c

i

< r

i

u

i

(c

i

) = (λ − 1) ∗ r

α

+ c

α

i

si

c

i

≥

r

i

The gure 10 shows the utility fun tion and the orresponding indieren e urves, with a

ve tor of referen epoints

(20, 20)

.

Figure 10: Con ave Utilityand LossAversion

(a)

0

10

20

30

40

50

0

10

20

30

40

50

5

10

15

20

25

30

35

40 c1

Fonction utilité

c2

(b)

Courbes indifférence

c1

c2

5

10

15

20

25

30

35

40

45

50

5

10

15

20

25

30

35

40

45

50

Letus solve the onsumer's program. The utility fun tion is not everywhere dierentiable:

for

c

i

= r

i

,the utilityfun itonis left-and right-dierentiable, but thederivativesarenot equal. Asa result, the optimalsolution is hara terized by

u

′

1 (c

∗

1 )

βu

′

2 (c

∗

2 )

=

p

1 p

2

iftheutilityfun tion isdierentiable in

c

∗

1

and

c

∗

2

,but

u

′

1 d

(c

∗

1 )

βu

′

2 d

(c

∗

2 )

≤

p

1 p

2 ≤

u

′

1 g

(c

∗

1 )

βu

′

2 g

(c

∗

2 )

otherwise.

Astheutilityfun tioniseverywhere on ave,the onsumer'sbehaviorbe omesagainroughly

" lassi al": ex ept if the preferen es or relative pri es are extreme, the optimal solutions are

interior solutions. However, thereis still some "rigidity": asthe slope of theindieren e urves

is dierent on theleft andon the right of areferen e point, iftheoptimal solutionimplies that

the onsumption isequal tothereferen epoint forone ofthetwogoods,there existsaninterval

for therelativepri e su hthatthe onsumption ofthisgoodremains onstant atthelevelofthe

referen epoint,andonlythe onsumptionoftheothergoodadjusts. Thissituationisillustrated

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(a)

Courbes indifférence

c1

c2

5

10

15

20

25

30

35

40

45

50

5

10

15

20

25

30

35

40

45

50

(b)

Courbes indifférence

c1

c2

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

2.4 Dis ussion

I onsideredtheee tsoflossaversionfordierentfun tionalforms. Thisee tsare

qualita-tivelyhomogeneous: lossaversionindu essolutions where onsumptionisequaltothereferen e

point for (atleast) one of thegoods. But these ee ts hange quantitatively depending on the

fun tional form : for a on ave utilityfun tion, there exists only a rigidity of the onsumption

intheneighbourhoodofareferen epoint,buttheoptimalsolutions arealwaysinteriorsolutions

and they aregenerallyresponsiveto a relative pri e variation. For a linearutilityfun tion,and

a fortiori for a onvexutilityfun tion (atleastinlosses), theonly possible solutions are orner

solutions or solutions where onsumption equals thereferen e point.

We an re all that the shape of the utility fun tion has two possible interpretations. From

the standpoint ofthe onsumertheory, on avitymeans ade reasingmarginal utility;fromthe

standpoint of the theoryof hoi e underun ertainty, on avitymeans risk aversion. Therefore,

our urrent results suggest that loss aversion should not be onsidered inand by itself, but in

onjon tion withan analysisof the ontext and of the onsumer'spsy hologi al hara teristi s,

inordertodeterminetheappropriatefun tionalformoftheutilityfun tion(atleasttemporarily

and lo ally).

The standard theory of the onsumer's behavior relies on on ave utility fun tions, whi h

indu es a preferen e for mixing rather than for extreme allo ations, but some empiri al results

suggest that onsumers might a tually demonstrate mixing aversion and a taste for extreme

allo ations(e.g. Dharand Simonson (1999)). Inthenext se tion, Ipresent several examplesin

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3.1 Cabdrivers labor supply

Inan oftenquoted arti le, Camerer, Bab o k, Loewenstein, and Thaler (1997) observe that

New-York abdriversshowa"wage"-elasti ityoftheir laborsupplywhi hisnegative: theydrive

lessongooddays(i.e. dayswhenthedemandfortaxisishigh, hen eadriverspendsmoreofhis

time a tually driving lientsinsteadof lookingfor one,whi h impliesahigher in omeper hour)

than on bad days

1

. This behavior annot be explained if drivers maximize a standard utility

fun tion: itwouldbemoree ienttoworkmorehoursongooddaysandlessonbaddays. Even

if thein ome ee t is stronger than the substitution ee t, assoonas thetime horizont of the

driversex eedsoneday,itwouldbebettertoworkmoreongooddaysandtakemu hmoreleisure

on bad days. This result has been onrmed by Fehr and Goette (2007) during a randomized

eldexperiment, whi h oersa more ontrolled environment than a natural experiment.

The interpretation that the authors oer is that, rst, the drivers's time horizont is a day,

and se ond, they have a daily in ome target that they try to rea h, and they feel lossaversion

when they an't rea hit. Thisinterpretationis onsistent withmyresults,but isisin omplete,

asitdoesn'ttakeinto a ountthe urvatureoftheutilityfun tion. Andyet,Ihaveshowed that

the urvature of the utility fun tion strongly ae t the onsumers's behavior. Moreover, both

Camerer, Bab o k, Loewenstein, and Thaler (1997) and Fehr and Goette (2007) onsider only

lossaversionwithrespe ttothein ome,and noteort (orreversely,leisure),whi h restrainthe

generalityof their analysis.

Letusgetba ktothemodelsIintrodu edsupra,and onsiderthatonegoodrepresentsleisure

and the other one represents aggregate onsumption. We also assume, in line with Camerer,

Bab o k, Loewenstein, and Thaler (1997), that the referen e points orrespond to the agent's

average leisure and average onsumption in the long run. We observe that indeed, whatever

the shape of the utility fun tion, a ounter lo kwise rotation of the budget line around the

point

L

max

(i.e. the agent's initial endowment of time) leads the agent to keep onstant her onsumption and to de reaseonly herleisure. Butifthehourlylabor in omede rases strongly,

the agent's optimal behavior will depend on the urvature of the utility fun tion in losses: if

the utility fun tion islinear or onvex, the agent will try to keep her onsumption equal to the

referen epoint aslongaspossible,butifthehourlylaborin omegetsundera ertainthreshold,

(25)

bystronglyde rasingher onsumption. Onthe ontrary,fora on aveutilityfun tion,whenthe

hourly labor in ome gets lower, the agent will start to de rease progressively her onsumption

while alsode reasing herleisure.

Similarly, with a lo kwise rotation of the budget line (i.e. the hourly labor in ome gets

aboveaverage), theagentsoptimalbehaviorwill depend onthe urvatureoftheutilityfun tion.

If utility is linear, the agent will in rease her onsumption while keeping her leisure equal to

the referen e point. If utility is on ave (whether it isglobally on ave, or only on ave in the

gains à la Prospe t Theory), the agent will split her additional in ome between onsumption

and leisure,and both will in rease.

Asa on lusion, thisparti ular ase illustrateshowmu h theimpa toflossaversionon the

onsumer's behaviordependson the urvatureof the utility fun tion.

3.2 The "What the Hell" Ee t

People on a diet often set themselves numeri al targets (e.g. a ertain number of alories

perday). Their situation an be understood asa tradeo between two goals: on the one hand

alori restri tion(inorderto leanout),andon theotherhand thepleasure oming fromeating

tasteful food andtaming their hunger. If these people had utility fun tionsfor these goals that

were on ave, a modi ation of the "relative pri e" between the two goals or of the "budget

onstraint",su h asbeingfor edunwillinglyinto thelosszonefor oneofthegoals,wouldimply

only small hanges in behavior. Yet, as suggestCo hran and Tesser(1996), a minor hange in

the situation an leadto a major hange inbehavior:

Gunilla foundthe beautiful formalshe wanted for the prom, but she neededto lose

some weight. Her friend, fresh from a so ial psy hology ourse, re ommended that

shesetaspe i ,daily alori goal. Onthe thirdday,aftereatingaserving of"lite"

spaghetti, Gunilla readthe pa kage and realizedthat shewasslightly over herdaily

goal. Her response was interesting: She said to herself "What the hell. Sin e I'm

already overmygoal it doesn't matter what I eat". And shepro eeded to onsume

halfof hermom's applepie.

Co hran andTesser(1996) allthis phenomenon the"What theHell Ee t".

This ee t is not restrained to food diets, but also applies to all situations when a pre ise

(26)

more prone to superuous expenses ompared to onsumers who are still in the limits of their

budget. AndWil ox, Blo k,and Eisenstein(2011) ndthatfor onsumers witha higherdegree

of self- ontrol (and hen e onsumers who an refrain most of the time from using their redit

ard), having anoutstanding rediton their ardleadsthem to usetheir ardmore.

These dierent empiri al results are onsistent with an utility fun tion whi h is linear, or

onvex in losses. Moreover, when a onsumer ends up just below a numeri al goal with no

possibility to getba k into the gain zone for this dimension, she has a tenden y to go further

awayfromthegoalbydeepeningherlosses,whi hsuggeststhatinriskless hoi e,the onvexity

of the utilityfun tion hasa stronger ee t than lossaversion. Thisbehavioris the ounterpart

for riskless hoi eof the lower risk-aversioninlossesfor risky hoi e.

3.3 Convexity and Self-Control

The examplesI presentedin the previous se tion ould also be interpreted interms of

self- ontrol. An argument against this interpretation and infavour of my interpretation is that in

othersituationswhereself- ontroldoesnotintervene,weobservesimilarbehaviors. Forexample,

Dörner (1989)ndsthatwhenfa inga omplexsituationwithmultipleand ontradi torygoals,

where itisnot possibleto rea hallofthem,de ision-makersoften hooseto on entrateononly

one goal, even ifitmeans higher lossesfor the other goals.

4 Con lusion

In this arti le, I showed that loss aversion has observable onsequen es whi h depends on

the urvatureof the utilityfun tion. Theseobservable onsequen esare onsistent with several

empiri alphenomenawhi h annotbeexplained bythestandardtheoryof onsumer'sbehavior.

Referen es

Abdellaoui, M., H. Blei hrodt, and O. L`Haridon (2008): Atra tablemethodto

mea-sureutilityand lossaversion underprospe t theory, Journal of Risk and Un ertainty,36(3),

24566.

Abdellaoui,M., H.Blei hrodt,andC.Paras hiv(2007): LossAversionUnderProspe t

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New York City Cab Drivers: One Day at a Time, The Quarterly Journal of E onomi s,

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