Bequest motives and fertility decisions B

Transcription

1 Economics Letters 92 (2006) Bequest motives nd fertility decisions B Ritsuko Futgmi, Kimiyoshi Kmd b, *, Tkshi Sto c Deprtment of Mngement Informtion Systems, Chubu Gkuin College, Jpn b Deprtment of Economics, Chukyo University Show, Ngoy , Jpn c Deprtment of Economics, Shimonoseki City University, Jpn Received 22 October 2005; received in revised form 4 Mrch 2006; ccepted 10 Mrch 2006 Avilble online 14 August 2006 Abstrct We compre ltruistic nd strtegic bequest motives nd exmine which one induces the higher demnd for children. Although prents with strtegic bequests hve the stronger demnd for filil ttention, they choose to hve fewer children thn those with ltruistic bequests. D 2006 Elsevier B.. All rights reserved. Keywords: Fertility; Altruism; Bequests; Attention JEL clssifiction: D11; J13 1. Introduction The purpose of this pper is to clrify the interction between bequest motives nd fertility decisions. As discussed in the literture on bequest motives, wht cuses prents to leve bequests depends on ssumptions bout prents preferences nd the economic environment. Since such ssumptions lso ffect prents decisions on fertility, the type of bequest motives nd the demnd for children must be closely relted. Given the lck of consensus on bequest motives fter numbers B We re grteful to the Institute of Economics, Chukyo University nd the Ministry of Eduction, Science, Sports nd Culture (Grnt-in-Aid for Scientific Reserch No ) of Jpn for finncil support. * Corresponding uthor. Tel.: ; fx: E-mil ddress: kkmd@mecl.chukyo-u.c.jp (K. Kmd) /$ - see front mtter D 2006 Elsevier B.. All rights reserved. doi: /j.econlet

2 R. Futgmi et l. / Economics Letters 92 (2006) of empiricl works, it is worth exmining how different bequest motives ffect fertility. One of few studies ddressing this issue is Zhng nd Zhng (2001), who compre fertility nd other vribles under four different bequest motives. While severl importnt results re obtined, the reltive mgnitude of the fertility rte between different bequest motives is indeterminte in most cses in their study. In this pper, to drw explicit reltionships between fertility nd bequest motives, we focus on two well-known hypotheses, i.e., ltruistic bequest motives (Brro, 1974) nd strtegic bequest motives (Bernheim et l., 1985). Considering fmily consisting of n ltruistic prent nd selfish children, we exmine two different ssumptions bout the prentl nd filil preferences for ttention the children give to the prent, i.e., the optiml ttention level for the prent is either equl to or greter thn tht for the children. Under the first ssumption, since there is no conflict between prents nd children, prentl bequests would be motivted solely by ltruism, s in Becker nd Brro (1988). Under the second ssumption, prentl bequests would be used strtegiclly in order to induce the children to enhnce their ttention, s in Cremer nd Pestieu (1991). 1 We will show tht prents with ltruistic bequest motives choose the greter number of children thn those with strtegic bequest motives, lthough the ltter hve the stronger demnd for filil ttention. The reson is s follows. Prents with strtegic bequest motives choose higher bequests thn those with ltruistic ones in order to elicit more ttention from their children. This implies tht strtegic bequest motives involve the higher mrginl cost of child. Furthermore, with strtegic bequest motives, the optiml ttention level for prents cnnot be chieved while the children re plced on the reservtion utility level. This implies tht the mrginl benefit of child is lower with strtegic bequest motives. While Cremer nd Pestieu (1991) hve shown tht there is bis towrds high fertility in the strtegic bequest equilibrium, our results further imply tht such bis is greter in the ltruistic bequest equilibrium thn in the strtegic one. 2. The model Consider fmily consisting of prent nd children, ll of whom re eqully potentil heirs. The timing is s follows: (i) the prent chooses the number of children n (z0); (ii) the prent chooses the bequests to ech child b (z0) (or commits to bequest rule); (iii) the children observe b (or the bequest rule) nd then choose the ttention level (z0). The utility function for the prent is given by U p c p ; i ; n; Ui k X n ¼ up y p nbþ ð bþ þ i¼1 w i ð i Þþ Xn i¼1 d i ðþu n i k ; ð1þ where c p (=y p n(b +b)) is the prent s consumption, U i k is the utility of ech child, y p is the prent s income, b is the cost of rering child, nd d i (n) is the weight the prent ttches to ech child s utility. 1 In strtegic bequest model without ltruism, Cremer nd Pestieu (1991) show tht, without the cost of rering children, the prent my choose to hve s mny children s possible. This pper introduces the rering cost, which is ssumed to be high enough to ensure n interior solution for the number of children.

3 350 R. Futgmi et l. / Economics Letters 92 (2006) Assuming tht the children re identicl, we hve w i =w, i =, d i (n)=d(n), nd U k i =U k for ll i. Hence, Eq. (1) cn be rewritten s U p c p ; ; n; U k ¼ up y p nbþ ð bþ þ nwðþþd ðþnu n k : ð2þ We ssume tht u p N0, u p Wb0, 0bd(n)b1, d(n)b0, d(n)+d(n)n N0, 2d(n)+dU(n)n b0, nd w() first increses nd then decreses in. The utility function for ech child is given by U k ðc k ; Þ ¼ u k ðy k þ bþþmðþ; ð3þ where c k (=y k +b) is the child s consumption, nd y k is the child s income. We ssume tht u k N0, u k Ub0, nd m() first increses nd then decreses in. 3. Altruistic nd strtegic bequest motives Even though the prent is ltruistic in tht her utility directly depends on her children s utility, conflict occurs between the prent nd her children regrding the children s ttention when the optiml ttention level for the prent * p fils to coincide with tht for ech child *. k Since * p nd * k re respectively derived from BU p /B =w()+d(n)v()=0 nd BU k /B =v()=0, given n, we hve *p p * k if rgmx w ðþprgmx mðþ. This implies tht the prent hs n incentive to use bequests strtegiclly in order to mnipulte the children s behvior. On the other hnd, if rgmx w ðþ¼rgmx mðþ, then *= p *. k In this cse, the prent need not influence the children s decisions, nd thus behves s pure ltruist in choosing bequests Altruistic bequest motives In the cse with ltruistic bequests, we replce w() with w A () in Eq. (2), nd ssume tht rgmx w A ðþ¼rgmx mðþ. Given n, the equilibrium of the subgme beginning t the second stge (b A (n), A ) is chrcterized by mðþ¼0; ð4þ u p y p nbþ ð bþ þ dðþu n k ðy k þ bþ ¼ 0: ð5þ In the first stge, the prent chooses n so s to stisfy ðþ=dnu n b A n A þ b u p y p n A b A n A þ b þ w A A þ d n A A n A uk y k þ b A n A þ m A ¼ 0: ð6þ 3.2. Strtegic bequest motives In the cse with strtegic bequests, we replce w() with w S () in Eq. (2), nd ssume tht w S ()=w A ( ), where N0. This implies tht rgmx w S ðþn rgmx mðþ.

4 R. Futgmi et l. / Economics Letters 92 (2006) The equilibrium of the subgme beginning t the second stge (b S (n), S (n)) cn be obtined s b nd tht mximize the prent s utility subject to the children s prticiption constrint U 0 k =u k ( y k +b)+v(), where U 0 k uu k ( y k, *). k The first-order conditions for the problem re nu p y p nbþ ð bþ þ ½dðÞn n þ kšu k ðy k þ bþ ¼ 0; ð7þ nw S ðþþd ½ ðþn n þ kšmðþ¼0; ð8þ u k ðy k þ bþþmðþu k 0 ¼ 0: where k is the Lgrnge multiplier ssocited with the children s prticiption constrint. Note tht S (n) is lower thn the prent s optimum when the children s prticiption constrint is binding. In the first stge, the prent chooses n so s to stisfy d S ðþ=dnu n b S n S þ b up y p n S b S n S þ b þ w S S n S þ d n S S n S uk y k þ b S n S þ m S n S ¼ 0: ð10þ ð9þ 3.3. Comprison of the two bequest motives We now contrst the strtegic bequest equilibrium (SBE) with the ltruistic bequest equilibrium (ABE). The number of children in SBE (n S ) nd tht in ABE (n A ), respectively, derived from Eqs. (10) nd (6). The first term in Eqs. (10) nd (6) is the mrginl disutility of n from the decrese in the prent s consumption, nd cn be defined s the mrginl cost of child. The second nd third terms re the increse in utility (derived from ttention nd the children s welfre, respectively) from dding n dditionl child for given mounts of consumption. The sum of these two terms cn be defined s the mrginl benefit of child. Proposition 1. n S bn A. Proof. Replcing n A with n S in Eq. (6), we hve ðþ=dn n ¼ b n¼n A n S þ b up y S p n S b A n S þ b þ w A A þ d n S S n S uk y k þ b A n S þ m A : ð11þ Compring the first terms in Eqs. (10) nd (11), since, b A (n)bb S (n), 2 we hve b S n S þ b up y p n S b S n S þ b b b A n S þ b up y p n S b A n S þ b : ð12þ As to the second terms in Eqs. (10) nd (11), we hve A b S (n) since the prent should set S (n) t level higher thn the children s optimum k *, which is equl to A. However, S (n) is still smller thn the 2 Define F(b, h)uu p ( y p n(b +b))+d(n)u k ( y k +b)+h[(k /n)u k ( y k +b)]=0. Note tht this eqution reduces to Eq. (7) when h =1, wheres it reduces to Eq. (5) when h =0. We hve BF /Bh =(k /n)u k ( y k +b)n0 nd the second-order condition implies tht BF /Bb b0. Using the implicit function theorem yields db /dh =(BF /Bh)/(BF /Bb)N0, hence, b A (n)bb S (n), given n.

5 352 R. Futgmi et l. / Economics Letters 92 (2006) prent s optimum in SBE while A coincides with the prent s optimum in ABE. Noting tht w S ()=w A ( ), this implies tht w S S n S bw A A : ð13þ Compring the third terms in Eqs. (10) nd (11), the children s utility is plced on the reservtion level in SBE. With ny positive bequests, therefore, the children obtin higher utility s long s they cn choose their optiml ttention. This implies tht y k þ b A n S þ m A zuk y k þ b S n S þ m S n S ¼ U 0 k : ð14þ u k From Eqs. (10) (11) (12) (13) nd (14), we hve ðþ=dn n N d S ðþ=dn n ¼ 0: n¼n S n¼n S Therefore, the second-order condition d 2 A (n)/(dn) 2 b0 implies tht n S bn A Conclusions Investigting the effect of bequest motives on fertility decisions, we show tht the prent with ltruistic bequest motives chooses greter number of children thn the prent with strtegic bequest motives. In SBE, the prent, who hs n incentive to elicit higher level of ttention from her children, leves higher bequests, nd thus fces higher mrginl cost of child thn in ABE. Furthermore, in SBE, the optiml ttention level for the prent cnnot be chieved, lthough the children s utility remins t its reservtion level. This implies tht, the mrginl benefit of child is lower in SBE thn in ABE. Therefore, the prent with ltruistic bequest motives fces higher mrginl net benefit from child reltive to the prent with strtegic bequest motives. References Brro, R.J., Are government bonds net welth? Journl of Politicl Economy 82, Becker, G.S., Brro, R.J., A reformultion of the economic theory of fertility. Qurterly Journl of Economics 103, Bernheim, B.D., Schleifer, A., Summers, L.H., The strtegic bequest motive. Journl of Politicl Economy 93, Cremer, H., Pestieu, P., Bequests, filil ttention nd fertility. Economic 58, Zhng, J., Zhng, J., Bequest motives, socil security, nd economic growth. Economic Inquiry 39,