Gifts bequests and family incentives

Munich Personal RePEc Archive

Gifts bequests and family incentives

Jellal, Mohamed and wolff, François charles

Al Makrîzî Institut d’Economie

2007

Online at

https://mpra.ub.uni-muenchen.de/38420/

Gifts, bequests and family inentives

∗

Mohamed Jellal

†

François-Charles Wol

‡

Revised, september 2004

Abstrat

In this note, we use the theory of inentives ontrating to haraterize the pattern ofnanial transfers within the family. Usingan altruisti modelwithone parent andtwohildren,wendthatthe parentmayprovidealowergifttotheless well-o hild,while bequests areompensatory.

JEL lassiation: D62, J2

Key words: Familyinentives, gifts,bequests

1 Introdution

Two main theoretial models have been suggested to explain intergenerational transfers within the family, i.e. altruism and exhange (see Laferrère and Wol, 2004). In order to dierentiate between these two motives, most empirial studies fous on the relation-ship between the level of transfers and the reipient's inome, whih should be negative under altruism. Reallingthatfamily transfersmay bemadeeither asinter vivosgiftsor bequests, some preditions of the exhange motive t with the data, but some ndings onerning the provision of transfers amongsiblings are more onsistent withaltruism.

On the one hand, bequests are often equally divided aross the hildren, espeially for large estates. Wilhelm (1996)observes equal sharing inabout two-thirdsof the ases

∗

Weareindebtedto ananonymousrefereeforveryhelpfulommentsandsuggestions.

†

UniversitÚMohammedV,Rabat,Morroo;ESCToulouseandConseils-Eo,Frane. E-mail: jellal-mohamedyahoo.fr

‡

BehrmanandRosenzweig(2004)ndsigniantdierenesinreportsofbequestsbetween siblings,butthisismainlyduetomeasurementerrors. Ontheotherhand,intrahousehold evideneonintervivostransferssuggeststhatthelesswell-ohildrenreeivemoremoney (MGarry, 1999,MGarry and Shoeni, 1995),but Cox(1987) andCox and Rank(1992) reah the opposite onlusion. Another nding is that inter vivos gifts are very often direted towards high-eduated hildren, even after ontrolling for permanent inome eets (MGarry, 1999).

As these ndingshallengethe validity of existing theories regarding the motives for family transfers, some papers have proposed alternative explanations to rationalize the patternof giftsandbequests. Parents maysuer fromapsyhiostif they deviatefrom equal division of bequests (Wilhelm, 1996). Gifts are likely to be private information (Lundholm and Ohlsson, 2000), and parents may know the permanent inomes of the hildren only with unertainty (MGarry, 1999). Cremer and Pestieau (1996) develop a modeloftransfersinasettingofmoral hazardand adverse seletionand showthat altru-istiparentsuseamixofintervivosgiftsandbequests. Thehildmaybemoredeprivated when reeiving less than siblings (Stark, 1998), and the division of bequests provides a signal about parental preferenes (Bernheim and Severinov, 2003). Interestingly, these theoretial explanations predit the use of a mix of gifts and bequests, but the expeted eets are ambiguousonerning their redistributiveeets

1 .

Inthispaper,wedrawonthetheoryofinentivesontratingandpresentanaltruisti modelinvolvingthe use ofa gift-bequest mix. Weprovethat bequests are ompensatory owingtoparentalaltruism,whiletherelationshipbetweeninter vivosgiftsand the reipi-ent'sinome maybe positiveornegativedependingonthe hild'sdegreeof riskaversion. The remainder of this note is organized as follows. In setion 2, we desribe the model. The optimalpatternof transfers is derived in setion3. Setion4 onludes.

2 The model

Let us onsider a family with one parent and two hildren, respetively denoted by up-sripts

p

and

i

= 1

,

2

. Childrenare heterogeneousand thetypeofhildisgivenby ability

w

i

, where

w

i

is the hild's wage rate on the labor market. We assume that

w

1

_{> w}

2

, meaning that hild1 ismore able than hild2. There are twoperiodsinthe model.

In the rst one, hildren are young and they hoose a level of eort

e

i

, whih aets inome expetations. The hildwill reeive more money inthe seond period with more

1

opportunies. However, eort is ostly for the hildren. The parent who behaves in an altruistially way seeks to ompensate the hildren for disutility involved by eort. Let

m

i

be the prie per unit of eort, so that the gift made by the parent to the hild

i

is simply

e

i

_m

i

. Importantly, we assume that eort is perfetly observable, so that there is noproblemofadverse seletion. Inthe seondperiod,hildrenreeiveanativityinome whih is proportionalto previous eort. They also reeive a bequest

b

i

from the parent. The budgetonstraints are (

∀

i

= 1

,

2

):

C

₁

i

=

m

i

e

i

(1)

C

₂

i

=

w

i

e

i

+

b

i

(2)

where

C

i

1

and

C

i

2

are the levels of onsumption per period for hild

i

. In period 1, the problem for the hild

i

is to maximize the utility funtion

v

i

, whose arguments are

C

i

1

and

e

i

2

. In the sequel, we assume that both hildrenhave idential preferenes, so that

v

1

₌

_v

2

₌

_v

. The maximizationprogramis forhild

i

given by:

max

e

i

v

(

C

i

1

, e

i

)

(3)

so that the marginal ost of eortis equalizedwith its marginalbenet:

−

m

i

v

1

(

m

i

e

i

∗

, e

i

∗

) =

v

2

(

m

i

e

i

∗

, e

i

∗

)

(4)

The optimal level of eort

e

i

∗

may be expressed as

e

i

∗

₌

_e

i

∗

₍

_m

i

₎

. When dierentiating

∂v/∂e

i

_{= 0}

, we nd that

e

i

∗

depends on the hild's degree of relative risk aversion

σ

. Indeed, we have sgn

de

i

∗

_/dm

i

₌

sgn

(

v

1

+

m

i

e

i

v

11

+

e

i

v

21

)

. For the sake of simpliity, we assume separability for

v

suh that

v

21

= 0

. It follows that :

sgn

de

i

∗

_/dm

i

₌

sgn

(1

−

σ

)

wherethemeasureofrelativeriskaversionis

σ

=

−

m

i

_e

i

_v

11

(

.

)

/v

1

(

.

)

. Clearly,

de

i

∗

_/dm

i

_>

₀

when

σ <

1

. Eortisaninreasingfuntionoftheparentalintervivostransferonlywhen the hildisharaterizedby a lowrisk aversion. Finally, let

V

(

m

i

_{) = sup}

e

i

v

(

m

i

e

i

, e

i

)

be the hild's indiretutility,with

V

′

₌

_e

i

∗

_v

1

(

.

)

>

0

.

The parent proposes an inentives ontrating menu that aounts for the hildren's well-being. We assume that the parent is motivated by paternalisti altruism. Children behave in a myopi way and are unable to perfetly foresee the onsequenes of their urrent behavior. Hene, the parent's objetive funtion is given by a weighted sum of his own (linear) utility

Y

p

₋

2

i

=1

(

m

i

_e

i

∗

₊

_b

i

₎

and of the hildren's well-being given by

2

i

=1

(

V

(

m

i

_{) +}

_u

₍

_w

i

_e

i

∗

₊

_b

i

₎₎

,

Y

p

beingtheparentalinome. Thefuntion

u

indiateshow 2

Theutilityfuntion

v

i

the parentevaluatesthe hildren'sationsthrough hisownpreferenes . Theproblemfor the parent is:

max

m

i

_,b

i

W

=

Y

p

₋

2

i

=1

m

i

_e

i

∗

₊

_b

i

+

β

2

i

=1

V

(

m

i

_{) +}

_u

₍

_w

i

_e

i

∗

₊

_b

i

₎

(5)

where

β

is the degree of parental altruism (with

0

< β <

1

). From the two rst-order onditions

∂W/∂m

i

_{= 0}

and

∂W/∂b

i

_{= 0}

, weobtain (

∀

i

= 1

,

2

):

−

(

e

i

₊

_m

i

_e

i

′

) +

β

(

V

i

′

+

w

i

_e

i

′

u

′

_{) = 0}

(6)

−

1 +

βu

′

_{= 0}

(7)

Sine

u

′

_{= 1}

_/β

and using(6), weget

−

(

e

i

₊

_m

i

_e

i

′

) +

βV

′

₊

_w

i

_e

i

′

= 0

. Then,reallingthat

V

′

₌

_e

i

∗

_v

1

(

.

)

, we nallydedue:

βe

i

∗

_v

1

(

m

i

e

i

∗

, e

i

∗

) +

e

i

′

(

w

i

−

m

i

∗

₎

₋

_e

i

∗

_{= 0}

(8)

3 The optimal pattern of transfers

We now haraterize the distribution of parental transfers, gifts and bequests. Given the separability assumption for

v

, we rely on the following form

v

(

m

i

_e

i

₎

₋

_ψe

i

. Thus,

ψ

is simply the ost per unit of eort. Condition (4) is now

m

i

_v

′

₍

_m

i

_e

i

∗

_{) =}

_ψ

(with

v

′

₌

_v

1

)

, so that we obtain

1 +

m

i

_e

′

₍

_m

i

₎

_/e

₍

_m

i

_{) = 1}

_/σ

after some manipulations. Using

βe

i

_ψ/m

i

₊

_e

′

₍

_w

i

₋

_m

i

₎

₋

_e

i

_{= 0}

and

m

i

_e

′

_/e

_{= (1}

₋

_σ

₎

_/σ

, itfollows that :

−

m

i

∗

_{+ (}

_w

i

−

m

i

∗

₎

1

−

σ

σ

+

βψ

= 0

∀

i

= 1

,

2

(9)

from whih we dedue the optimal inter vivos gifts

m

i

. The bequest values

b

i

are then obtained using

u

′

₍

_w

i

_e

i

∗

₊

_b

i

∗

_{) = 1}

_/β

.

Proposition 1 The optimal pattern of family transfers is given by (

∀

i

= 1

,

2

):

i)

m

i

∗

_{= (1}

₋

_σ

₎

_w

i

₊

_σβψ

ii)

b

i

∗

₌

_u

′−

1

₍₁

_/β

₎

₋

_w

i

_e

i

∗

₍

_m

i

∗

₎

In our framework, the inentive rate

m

i

is a onvex ombination of

w

i

and

βψ

, with weightsgivenby

σ

and

1

−

σ

. Resultsfromomparativestatisimplythat

∂m

i

_∗

∂β

=

σψ >

0

,

∂m

i

_∗

∂ψ

=

σβ >

0

,

∂m

i

_∗

∂σ

=

−

(

w

−

βψ

)

<

0

(sine

β <

1

and

ψ < w

i

), and

∂m

i

_∗

∂w

i

= (1

−

σ

)

. Let us briey interpret these results. The inter vivos transfer is larger when the parent

3

funtion of the hild's risk aversion. Finally, the gift value may be either inreasing or dereasing with the hild's wage rate, depending on the hild's measure of relative risk aversion. With

σ <

1

, the parent provides more inentives to the hild and a riher hild is expeted to reeive more money. Conversely, when the hild is risk averse, less money is given by the parent, whih is usually expeted in an altruisti model. Hene, our inentivesontratingmodelshows that itis importantto aountfor risk attitudes.

Proposition 2 The distribution of gifts and bequests between hildren is suh that:

i)

b

2

∗

_{> b}

1

∗

_∀

₀

_{< σ}

ii)

m

1

∗

>

<

=

m

2

∗

⇔

σ <

>

= 1

So, we nd that hild 2is expeted to reeivea higher amount of bequests that hild 1

4

. Given heterogeneity in wages, our model leads to a ompensatory motivation for bequests. However, results are dierent for inter vivos transfers sine the intrafamily distribution of giftsdepend onthe hildren's riskaversion. Despite of parentalaltruism, inter vivos transfers are not neessarily direted to the less well-o hild. Gifts may be either ompensatory or unompensatory, depending on the hildren's degree of relative risk aversion. With a low risk aversion, the model predits that the most able hild(i.e. hild1)willreeivealargergift,whilehild2reeivesalargerbequest. Conversely, when

σ >

1

,the less able hildreeives larger amountsof gift and bequest fromthe parent.

4 Conlusion

In this paper, we have onsidered an inentives ontrating model of family transfers. An altruisti parent provides inter vivos transfers to ompensate the hildren for disu-tility involved by observable eort and post mortem bequests to equalize the hildren's marginal utilitiesof onsumption. When haraterizingthe optimalpattern of transfers, we nd that bequests are ompensatory, while the relationship between inter vivos gifts and hildren's inomes an be either positive or negative. Thus, aounting for family inentives may explain why inter vivos transfers are most ofen direted towards more eduated hildren. Also, our theoretial framework suggests that it would be useful to introdue the hildren'srisk attitudein empirialanalyses of familytransfers.

4

Using Proposition 1, wededuethat

b

2

∗

_{> b}

1

∗

if

w

2

_e

2

₍

_m

2

₎

_{< w}

1

_e

1

₍

_m

1

₎

, whih always holds. The proofis similarwhenomparing

m

1

∗

and

m

2

∗

Behrman, J.R. andRosenzweigM.R.(2004),Parental alloationsto hildren: Newevidene on bequestdierenesamong siblings, Review ofEonomis and Statistis86:637-640.

Bernheim, B.D. and S. Severinov (2003), Bequests as signals : An explanation for the equal division puzzle,Journal of Politial Eonomy 111:733-764.

Bisin, A. and T. Verdier (2001), The eonomis of ultural transmission and the dynamis of preferenes, Journal of EonomiTheory97:298-19.

Cox,D. (1987),Motivesfor privateinome transfers, Journal ofPolitialEonomy95:508-46. Cox, D. and M.R. Rank(1992), Inter-vivos transfers and intergenerational exhange, Review of Eonomis andStatistis 74:305-314.

Cremer, H.andP.Pestieau (1996),Bequestsasa heirdisipline devie,Journal ofPopulation Eonomis 9:405-414.

Dunn,T.A.andJ.W.Phillips(1997),Thetiminganddivisionofparentaltransferstohildren, Eonomis Letters 54:135-138.

Laferrère, A. and F.C. Wol (2004), Miroeonomi models of family transfers, in J. Merier Ythier, S.C. Kolm (eds), Handbook on the Eonomis of Giving, Reiproity and Altruism, North-Holland, forthoming.

Lundholm, M. and H. Ohlsson (2000), Post mortem reputation, ompensatory gifts and equal bequests, EonomisLetters 68:165-171.

MGarry,K. (1999),Intervivostransfers andintendedbequests,Journal of PubliEonomis 73:321-351.

MGarry, K. andR.F. Shoeni (1995),Transfer behaviorinthe Health and Retirement Study. Measurementandtheredistributionofressoureswithinthefamily,JournalofHumanResoures 30:S185-S226.

Stark, O. (1998), Equal bequests and parental altruism : Compatibility or orthogonality ?, Eonomis Letters 60:167-171.