Inheritance and the Distribution of Wealth by Frank A Cowell STICERD, London School of Economics and Political Science
Discussion Paper No.DARP/34 May 1998
The Toyota Centre Suntory and Toyota International Centres for Economics and Related Disciplines London School of Economics and Political Science Houghton Street London WC2A 2AE Tel.: 020-7955 6678
Abstract The theory of functional equations is used to clarify the relationship between equilibrium distributions of wealth and population parameters such as the distribution of families by size, marriage patterns, tax mechanisms and savings behaviour within a simple model of inheritance. Keywords: Inheritance, wealth, Pareto distribution. JEL Nos.: D31, D63.
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by Frank A Cowell. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source.
Contact address: Contact address: Professor Frank Cowell, STICERD, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK. Email:
[email protected]
1
In trod uc tion
T he shape ofthe d istrib utionofw ealth isa topic that hasexerted a long f asc ination.T he em piric alregularitiesac rossd i®erent typesofec onom ieshave of ten b eenrem arked upon,and the literature c ontainsseveraltypesofd ynam ic m od el 1 that c anb e used ,inpart,to explainthe charac teristic shape ofthe d istrib ution.
O nthe strength ofthisresearc h som e have gone so f ar asto suggest \law sof d istrib ution" w hic h soc ietiesmust inexorab ly ob ey.B y c ontrast thispaper has a mod est ob jec tive: it exam inesthe e®ec t onw ealth d istrib utionofanaspec t of inheritance proc essesthat hasrec eived relatively little attention. U sing a sim ple f ram ew ork that isc onsistent w ith stand ard m od elsofsavingsand b equest b ehaviour it show sthe w ay inw hich the equilib rium d istrib utionc anb e d eterm ined over a spec i¯ed w ealth range.It also exam inesthe relationship b etw een equilib rium w ealth inequality and the d istrib utionoff am iliesb y siz e w ithina b road c lassofm od elsofthe b equest proc ess. T he principalresult isthat,und er f airly w eakc ond itions,partsofthe equilib rium d istrib utionofw ealth must b e charac terised b y a narrow c lassoff unctional f orm s.T he partsofthe d istrib utionw hich c anb e c aptured inthisw ay are d elim ited b y regionsinw hic h spec i¯c b ehaviouralc harac teristic sofw ealth-ow ners are assumed to hold .T he c ond itionsthat are required f or the m ainresultsare c onsistent w ith a numb er ofm od elsofw ealth ac c umulationand b equests.T he 1
SeeforexampleChampernowne(1 953,1 973),EichhornandG leissner(1 985),L ydall(1 968), M andelbrot(1 960 ),Sargan (1 957),V aughan (1 988),W old and W hittle(1 957).
1
f am ily off unctionalf orm sinclud esthe stand ard f ormulae that have b eend erived asequilib rium d istrib utionsina variety ofspec i¯c m od elsofthe w ealth ac c um ulationand d istrib ution, and that are of tenutilised f or ad hoc purposessuc h as c urve-¯ttingf or partic ular partsofem piric alw ealth d istrib utions. T he approac h that I use is to spec if y the c ond itions f or equilib rium ina w ealth m od eld rivenb y inheritance.T he theory off unctionalequationsisthen used to c harac terise the c lassofw ealth d istrib utionsthat are d eterm ined b y the equilib rium c ond itions. Sec tion2 outlinesthe f und am entalsofthe m od el.Sec tion3 provesthe m ain result f or a simpli¯ed versionofthe m od el,and show show the equilib rium w ealth d istrib utiond erived inthe m ainresult c anb e related to the param etersofthe system .T hensec tions4 and 5d em onstrate how the elem entary m od elc aneasily b e extend ed to a numb er ofm ore interestingc ases.
2
T he M od el
Consid er a populationthat ismad e up ofa sequence ofgenerations. Let tim e b e d isc rete and ind exed b y t= :::;0 ;1;2 ,...and assum e that each generationis uniquely assoc iated w ith one c ontiguouspair ofperiod s: those w ho are child ren at tim e tb ec om e ad ultsat tim e t+ 1.Assum e that at any tim e the population c onsistsofa numb er off am ilieseac h ofw hic h hasa d eterm inate,¯nite numb er of
2
c hild ren,and that there are no c hild lessf am ilies:2 apart f rom thisthe d istrib ution off am iliesinthe populationisarb itrary.Let the proportionoff am iliesw ith k c hild renb e p k ¸ 0 , k = 1;2 ;:::;K , and w rite the vec tor (p 1;p 2 ;:::p K ) asp.B y d e¯nition: K X
p k = 1;
(1)
k= 1
and f or populationstationarity p must satisf y:
K X
kp k = 2 :
(2 )
k= 1
Inanin¯nite populationthisassum ptionc anb e relaxed . Imagine the ec onomy at any m om ent t: w e c anc onceive ofthe population asb eing c om posed off am iliesc harac terised b y their joint w ealth leveland the numb er ofc hild renw ho w illeventually inherit that w ealth.T he w ealth d istrib utionofsuc h f am iliesd epend suponthe spec i¯c assum ptionsm ad e ab out the w ay that w ealth grow sineac h period ,people'ssavingsb ehaviour,the f orm ofw ealth taxation, the w ay inw hic h new f am iliesare f orm ed ineac h generation, and the w ay inw hic h parentsd istrib ute their w ealth.Ishallassum e the f ollow ing: Axiom 1 B equeathable w ealth grow severyw here byanexogenousfac tor ¯ d uring one generation. 2
Contrast this with, for example, the model of W old and W hittle (1 957)and Eichhorn and G leissner(1 985),in which thewealth ofeach vanishinghousehold is distributed amongn bene¯ciaries,wheren is simplytheaveragenumberofinheritors in theeconomy¶ .
3
For exam ple, ifthere is anexogenous rate ofgrow th oftotalw ealth g, a unif orm average propensity to c onsum e out ofw ealth3 c , and a taxonb equests ¿,then ¯ = [1 + g][1 ¡c ][1 ¡¿]:
(3)
Axiom 2 Allparentsw hose ind ivid ualw ealth satis¯ esW 2 I , w here I isa proper intervalthat d oesnot containzero,4 follow a polic yofequald ivisionam ongst their k kid s.5 It isc onvenient to summ arise the param etersc harac terising any spec i¯c im plem entationofthese assum ptionsthus:
¼ := (¯ ;p)
(4 )
Let the w ealth d istrib utioninany generationtb e d enoted b y a d istrib ution f unctionFt:< 7 ! [0 ;1]. B ec ause the d istrib utionw illb e c ond itionaluponthe partic ular value ofthe set ofparam etersw e shallw rite it asFt(W ;¼ ). For a given¼ equilib rium isd e¯ned asa situationw here,f or any tand f or allW 2 I :
Ft+ 1 (W ;¼ ) = Ft(W ;¼) = F (W ;¼): 3
(5)
T his behaviouris consistentwith utility maximisation where preferences are homothetic; see,forexample,B eckerandT omes (1 979). 4 T hatis,Imay be aclosed interval, [W 0 ;W 1 ],oran open interval,ormay be unbounded above. 5 T his behaviourwouldbetheconsequenceofutilitymaximisationunderhomotheticpreferences-B eckerandT omes(1 979).H oweverwedonotneedtoappealtothisspeci¯cassumption: insomesomesocieties equaldivisionmaybeimposedbylaw-seeKesslerand M asson(1 988).
4
W e shallalso assum e: Axiom 3 F iscontinuousover I . T he f und am entalprob lem isto ¯nd the f am ilyoff unctionsF (W ;¼) giventhe set ofparam eters¼ . Inprinciple a d istrib utionf unctionofw ealth ought to b e d e¯ned onthe w hole realline - people c anhave negative asw ellaspositive net w orth - b ut it w ould b e a d em and ingand perhapsunillum inatingtask to try to spec if y every d etailofthe d istrib utionf unctionover itsentire range.How ever, f or theoretic aland em piric alreasons,it isof tenec onom ic ally interestingto f oc us onc ertainpartsofthe d istrib ution,f or exam ple the upper tail.So,w hat w e w ill d o isc harac terise the shape ofthe f unctionF over the restric ted d om ainI .
3 Assortative M ating Asa ¯rst step c onsid e the c ase ofpositive assortative m ating.T hisisa situation ofstric t \c lassmarriage": a personat tw ith w ealth W seeksm arriesanother personw ith w ealth W and f ormsa new f am ily int+ 1.G iventhe b ehavioural assum ptionsoutlined ab ove, the issue ofthe d istrib utionofw ealth thenf oc uses onthe d istrib utionoff amiliesb y size. Inthisc ase m em b ersofthe population c om pletely arec harac terised b y the pair (W ;k).A (W ;k)-f am ily c onsistsof2 + k persons: tw o parents, eac h ofw hom possessesonm arriage w ealth W , and w ho d ivid e their w ealth equally amongtheir kc hild ren.
5
3. 1
M ainresul t
E ac h c hild ofa (W ;k)-f am ily inheritsw ealth
2 ¯W k
.So the equilib rium c ond ition
(5) requires K X 1
F (W ) =
k= 1
2
kp kF
Ã
kW 2¯
!
(6)
E quilib rium c ond ition(6) impliesthat f or any tw o d istinct valuesW ,W 0 2 I the unknow nf unctionF must satisf y:
F (W ) =
K X
a k F (W k);
(7)
K X
a k F (W k0);
(8)
1 kp k; 2
(9)
kW 2¯
(10 )
k= 1
0
F (W ) =
k= 1
w here a k := W k :=
Inview of(2 ), 0 · a k < 1 f or non-trivialf am ily struc tures.B y d e¯nitionof a d istrib utionf unctionW
0
¸ W () F (W 0;¼) ¸ F (W ;¼). T here are tw o
c ases: (1) F is c onstant over I ; (2 ) there exist som e W ;W 0 2 I suc h that F (W 0;¼) > F (W ;¼ ).Case (1) istrivialsince it m eansthat there isno-one w ith w ealth inI . Inc ase (2 ), b ec ause ofthe assum ed c ontinuity ofF over I , there must b e anintervalI 0 µI f or w hic h F isincreasing.For c onvenience introd uc e
6
the f ollow ingc hangesofvariab le:6
1 [inf(I ) + sup (I )]; 2
°:=
X :=
(
W x :x = ; 8W 2 I °
G :X 7 ! [0 ;1]; G
Ã
)
(11)
;
!
W = F (W ;¼ ); 8W 2 I : °
(12 )
(13)
It isthenevid ent that (6) im plies
x= G
¡1
Ã
K X
!
ak G (xk) ;
k= 1
(14 )
Inother w ord sthe equilib rium c ond itionf or the w ealth intervalisequivalent to requiring that the younger generation'sw ealth b e a quasilinear w eighted m ean7 ofK valuesofw ealth inthe old er generation,w here the quasilinear m eanisc onstruc ted using the (transf orm ed ) d istrib utionf unction.How ever w e c anf urther restric t the f unctionG ,and hence the d istrib utionf unctionF . Lem m a 1 T he functionG must satisfy either
G (x) = Alog(x) + B ; x 2 X 6
N oticethat,byde¯nition,1 2 X . SeeA cz¶ el(1 966)page240 ,D hombres (1 984).
7
7
(15)
or G (x) = Axµ + B ; x 2 X
(16)
w here A isnonzero. P roof .See Append ix. Lem m a 1 lead sim med iately to a result that haspartic ularly interestingec onom ic im plic ations. T heorem 2 Inthe case ofstric t c lassm arriage, the equilibrium w ealth d istributionm ust belongto the extend ed P areto T ype Ifam ily, throughout the region w here the equal-d ivisioninheritance rule applies.Inother w ord sF m ust satisfy
W ¡® (¼ ) ¡1 F (W ;¼ ) = a + b ® (¼ )
(17)
w here a and b are constantsand ® (¼ ) 2
