A Static Model for Voting on Social Security

Transcription

1 A Sttic Model for Voting on Socil Security Henning Bohn Deprtment of Economics University of Cliforni t Snt Brbr Snt Brbr, CA 93106, USA; nd CESifo Phone: ; Fx: E-mil: bohn@econ.ucsb.edu April 2009 JEL Code: H55. Keywords: Socil security, voting model. Abstrct This pper exmines sttic voting model for public pensions. The key premise is tht fmilies cn internlize the cost nd benefits of py-s-you-go progrms. A fmily relizes net gin if its members collectively receive more in benefits in the current period thn they py in pyroll txes. Abstrcting from differences in income, net benefits re positive if the fmily s retiree-worker rtio exceeds the ntionl verge. If sufficient frction of retirees hve suitble number of working-ge reltives not too few nd not too mny then mjority of voters belongs to fmilies with bove verge retiree-worker rtios.

2 1. Introduction Public pensions nd other py-s-you-go finnced retirement benefits enjoy strong populr support despite the esclting cost. 1 This is remrkble from the perspective of voting theory becuse retiree benefits redistribute resources from mjority of working-ge voters to minority of retirees. Following Aron 1966) nd Browning 1975), lrge literture hs exmined intertemporl models for voting on socil security the notion tht contributors my vote in fvor becuse they expect benefits in the future to explin this puzzle. 2 Others hve ppeled to income redistribution nd ltruism e.g., Tbellini 2000). This pper presents n lterntive voting model bsed on intr-fmily brgining. The key premise is tht fmilies cn internlize the cost nd benefits of retiree benefit progrms. A fmily relizes net gin from py-s-you-go progrm if its members collectively receive more in benefits tht they py in pyroll txes. Net benefits re positive if fmily s shre of retirees exceeds the ntionl verge. Preferences over tx rtes nd benefit levels re single peked in fmilies retiree shres. Retiree benefits re positive in voting equilibrium if the mjority of the popultion is in fmilies with bove verge retiree-worker rtios. This condition is plusibly stisfied if most fmilies include positive, but not too high number of intergenertionl linkges. A supporting fctor is tht if mortlity is stochstic, only the survivors vote. For ny given structure of fmily linkges, fmilies with surviving retirees hve more members thn fmilies with higher mortlity. 3 The conditions for mjority support re likely violted, however, in societies with substntil heterogeneity over fmily linkges, e.g., if most fmilies re hve either mny children or no children. 1 In the following, I use the term socil security to represent generic py-s-you-progrm. The model pplies to pensions nd to other benefit progrms, such s retiree helth cre. 2 A chllenge in dynmic models is to estblish link between current voting nd future benefits. The erly literture ignored repeted voting. More recently, Cooley nd Sorez 1999) hve modeled repeted voting with trigger strtegy rguments to tie current votes to future benefits. Applictions include my own work Bohn 1999, 2005) nd Glsso 2006). 3 With survivl uncertinty, retiree benefits cn be interpreted s mutul insurnce ginst uncertinty bout needy reltive s longevity. A difficulty with this insurnce is tht ex post, voters whose reltives hve died erly hve n incentive renege. The conditions for mjority support for socil security cn conditions my be viewed s conditions for the politicl sustinbility of survivl insurnce. 1

3 As technicl device to obtin interior solutions for tx rtes, I ssume pyroll txes distort lbor supply. This voids extreme shifts between zero txes nd confisctory txes in response to smll chnges in popultion structure.) Preferences over pyroll txes re then single-peked s function of fmilies retiree shres, or equivlently, retiree-worker rtios. In the voting equilibrium, pyroll txes re set so tht the mrginl excess burden of txes equls the net finncil benefit for the medin fmily. The model disregrds ll dynmic voting considertions nd ssumes working-ge voters tret their own future benefits s unrelted to their votes over current benefits. 4 I lso disregrd ltruism nd income heterogeneity. This is for clrity, to limit the scope of the pper, nd to void ny bis for socil security due to ltruistic nd redistribution fetures. One my suspect tht the remrkble populrity of public retirement nd old-ge helth progrms round the world is bsed on multiple fctors. This my include dynmic rguments, ltruism, nd redistribution cross heterogeneous incomes. While it would be strightforwrd to combine such rguments with the mechnism proposed here, the resulting combintion models would be unwieldy nd they would obscure the conditions under which fmily brgining per se is supportive of socil security. Tht is, I view the fmily brgining mechnism s complementry with other rguments for socil security. 5 Breyer nd v.d. Schulenburg 1987) were the first to propose fmilies s relevnt decision units for voting on socil security. In contrst to this pper, Breyer nd v.d. Schulenburg tret vote on socil security s once-nd-for-ll decision nd they follow the dynmic literture in ssuming tht voters bse their decisions on present vlue of current nd expected future txes nd benefits. They lso mke specific ssumptions bout fmily structure, nmely tht voting-ge individuls cre bout the net txes of their direct descendents children, grndchildren, etc.), ssuming sexul reproduction ech child hving single prent). 4 This is not to deny the relevnce of promised future benefits in prctice, but to meet the chllenge of constructing voter support in sttic model model tht llows repeted voting without requiring trigger strtegy rguments. Put differently, I set side the intertemporl linkges tht the dynmic literture works so hrd to estblish nd insted tke seriously the simple, non-trigger strtegy solution tht lwys exists in trigger strtegy models. 5 Understnding ll the resons why voters support socil security is importnt in prt becuse reform efforts re likely to fil unless they ttrct mjority support. The history of filed reforms in Europe nd in the U.S. suggests need for better understnding. 2

4 The pper is orgnized s follows. Section 2 reviews relevnt demogrphic fcts. Section 3 presents the model. Section 4 exmines voting behvior nd presents the min propositions. Section 5 provides numericl exmples. Section 6 concludes. 2. The Demogrphics of Fmilies The shre of retirees in the voting popultion is round 20% in the U.S. nd in the 25-35% rnge in continentl Europe. Thus retirees re strong minority, but fr from mjority even in Europen countries with below-replcement fertility. The effective retirement ge is bout 63 in the U.S. nd round 60 in Europe. Life expectncy t ge 60 is bout 20 yers more for femles nd somewht less for mles. This is well over hlf of the verge time spn between genertions. Using U.S. life tbles, one lso finds tht for child of 25- yer old prents, with bout 60% probbility t lest one of the prents reches ge 85. The probbility rises to 65% conditionl on both reching ge 60. This mens tht lrge mjority of working-ge voters hs living prents. For older workers, these prents re likely retired. For children nd younger workers, the sme rgument suggests tht mjority will hve retired grndprents or grndgrndprents. Thus lrge mjority of the popultion hs close reltives who benefit from py-s-yougo public pensions nd other retiree benefits e.g. helth cre). Most children re born to mothers ge 20-35, with pek round ge Hence retirees t, sy, ge 80 will likely hve children in the ge rnge nd grndchildren in the ge rnge This suggests tht the voting popultion encompsses more thn two genertions nd would not cptured dequtely by modeling nucler fmily. 6 For this reson, I tret fmily s n bstrct collection of voters, i.e., without trying to identify specific kinship reltions s necessrily more or less relevnt thn others. My bsic premise is tht fmily linkges my mtter for voting. Importntly, the rgument does not require ltruism towrds retired reltives, but merely n bility to lign members voting incentives. For exmple, college student with retired grndprents tht re supported by his prents 6 The longer time spent in working ge thn in retirement lso explins why retirees remin minority even in countries with fertility below replcement. 3

5 might vote ginst cuts in public pensions becuse such cuts might trigger shift in prentl support from the student to the grndprents, or perhps reduced gifts nd bequests. In this context, gifts, bequests, nd other intr-fmily pyments re not necessrily evidence of ltruism, but my reflect Cose-type side pyments designed to ensure coopertion in voting. 3. The Model Consider n economy with finitely lived individuls. Time is indexed by time t, nd ge is indexed by. Both re treted s discrete. Individuls born t time t form cohort t. They re eligible to vote ge =V 0, they become eligible for retirement benefits t ge =R, where R>V, nd they live up to terminl ge =A R. The lbor force consists ll non-retired voters, the ge rnge [V,R-1]. Let π be the survivl function; tht is, genertion-t individuls survive to t lest) ge with probbility π, where π >0 for A, declining in, nd π A+1 =0. Assuming children neither vote nor work, they cn be disregrded their consumption subsumed under their prents consumption); hence V=0 is without loss of generlity. Then cohort t is economiclly ctive in [t, t+a]; nd t time t, the living cohorts re those born in [t, t-a]. The economy is deterministic except for stochstic mortlity. Lbor hs n exogenous mrginl product of lbor w nd svings ern n exogenous rel interest r. Time vrition could be dded but would just clutter the nottion.) The common wge rte serves to emphsize tht the model does not rely on intr-cohort redistribution or on prticulr income distribution. At ge 0, individuls of genertion t hve preferences A U 0 t = π β 0 uc t 0 + = 0 over consumption c t 0 +,λ t 0 + not essentil for the proposed model of socil security. ) nd leisure λ t 0 +. I bstrct from ltruism to emphsize tht ltruism is Retiree benefits re structured to pose one-dimensionl voting problem: The retirement ge R is exogenous. Individul t ges <R workers) py proportionl pyroll txes t rte τ τ. Retirees ges R) receive trnsfers T t. Lbor supply l t =1 λ t is endogenous for workers <R); lt = 0 is imposed exogenously for ges R. Workers optimlity conditions for lbor nd leisure re 4

6 u λ c t,1 l t ) u c c t,1 l t ) ) = w1 τ t ) < R. To simplify, ssume u hs the form u c t,λ t ) = uc t + φλ t )), where u nd φ re incresing nd concve. Then lbor supply for ll <R is chrcterized by φ'λ t ) = w1 τ t ) nd cn be written s declining function of the current tx rte, l t = Lτ t ). Assume φ hs shpe such tht tx revenues τlτ) re strictly incresing nd strictly concve in τ up to n interior mximum t some vlue τ mx <1. Tht is, I ssume Hrberger tringle nd Lffer curve intuition pplies. This simple tretment of leisure suffices becuse the tx distortions only serve s device to void extreme solutions to the voting problem.) Then τ mx stisfies Lτ mx ) τl'τ mx ) = 0; nd Lτ) τl'τ) is positive nd strictly decresing τ in for τ < τ mx. Let N t denote the number of individuls of ge living in period t. Then N W R 1 t = N t, N R A R 1 N t = N t, nd L t = t W l N t = Lτ t ) =0 =R =0 t re the working-ge popultion, the retired popultion, nd the per-worker lbor supply, respectively. re the working-ge popultion nd the retired popultion, respectively. Let denote per worker lbor supply t time t. 7 Assuming txes nd trnsfers re dministered py-s-you-go, the government s budget constrint is T t N t R = τ t N t W wlτ t ). Every period, the tx rte τ t is determined by voting decisions, ssuming voters understnd tht trnsfers T t = τ t Lτ t )w N t W /N t R depend on the tx rte τ t nd on the ggregte worker-retiree rtio. Retiree preferences over τ t re strictly incresing in τ t wheres worker preferences re strictly decresing in τ t, i.e., both re single peked. If there re more workers thn retirees, N t W > N t R, the unique equilibrium with individulistic voting would be τ t =0. 8 This insight genertes the bsic puzzle of socil security in democrcy: With mjority of voters in working-ge, how cn one explin the existence of socil security? 7 If ϕ denotes the frction of ge- individuls hving child fertility), the popultion dynmics cn be described by N +1 t+1 = N 0 t π +1 /π < A nd N t+1 A = ϕ N t =0 8 In the empiriclly unrelistic cse N t W < N t R, one would obtin τ t=τ mx.. The dynmics re inessentil, however. 5

7 The economic literture on socil security hs tried to resolve the puzzle in vrious directions, notbly by considering dynmic/intertemporl rguments reputtion, genertionl contrcts, trigger strtegies), ltruism, linkges between intr- nd inter-genertionl redistribution, public choice rguments the old hving more time to lobby), or combintions thereof. For clrity, none of these ssumptions is invoked here. 4. Fmily Preferences The defining feture of fmily for the purposes of this pper is the bility to internlize the cost nd benefits of government progrms tht ccrue to its members. Tht is, fmily is ny collection of individuls with the bility to coordinte its ctions in wy tht ttins the Preto frontier of its members utilities. Fmilies my include direct child-prent-grndprent linkges, higher-order kinship reltions e.g. nephews, uncles), or linkges by mrrige. I do not pply specific lbels to reltionships to void d hoc ssumptions bout which linkges mtter nd to keep the nottion simple. Let the economy s popultion be prtitioned into fmilies nd consider the Preto problem of generic fmily with n t members of ge nd n t = totl members. Preto-optiml fmily lloctions cn be obtin by mximizing weighted verge of member utilities, W = A ω n t U t, =0 for some collection of weights ω >0. With or without intertemporl commitments, the period-t decision problem reduces to mximizing A W t ω n t uc t + φλ t )) subject to =0 A n t =0 R 1 A =0 c t + S t = n t w1 τ t )1 λ t ) + n t T t, =0 for ny given level of fmily svings S t. Becuse problem is sttic, time subscripts re dropped in the following. While the resulting intr-fmily lloction of consumption depends on the welfre weights, the conclusions bout voting do not, nor do they depend on fmily welth or on decisions bout svings. n t A =R 6

8 To see this, differentite W with respect to τ noting tht effects on leisure nd on svings cncel out by the envelope theorem) to obtin A W τ =Φ [ n T R 1 t τ n t w1 λ t )]. = R =0 where Φ>0 is the Lgrnge multiplier on the period-t resource constrint. Note tht 1 λ t = Lτ) nd T τ = [Lτ) + τl'τ)]w N W t R+ = [Lτ) + τl'τ)]w 1 η, where N t η η = N t R+ /N t R+ + N t W ) 0,1) denotes the ggregte popultion shre of retirees. Similrly, let A =R η= n t /n t [0,1] denote the fmily s own shre of retirees. Then fmily s optimlity condition for txes cn be written s W τ =Φw1 η)n [Lτ ) + τ L 'τ )]η η t Lτ )1 η 1 η = 0, 1) where the second equlity must hold t interior solutions. For η<1, η/1 η) cn be interpreted s the retiree-worker rtio, more commonly used vrible. I use the retiree-shre for the nlysis becuse the retiree shre is finite nd bounded even for fmilies without workers, for which there is no finite retiree-worker rtio.) Noting tht W / τ is incresing inη, one finds: Proposition 1: Fmily preferences over pyroll txes re single-peked function of the retiree shre. The preferred vlues stisfy τ *η) = 0 for fmilies with η η ; τ *η) 0,τ mx ) for η < η < 1 with τ *η) strictly incresing in η; nd τ *η) = τ mx for η = 1. Proof: From 1), n interior solution for the preferred tx rte τ* must stisfy [Lτ) + τl'τ)] η 1 η = Lτ). For η =1, τ* this reduces to Lτ) + τl'τ) = 0, which mens η 1 η τ* = τ mx. For η = η <1, the condition reduces to L'τ) = 0, which implies τ* = 0. Becuse L'τ) < 0 for ll τ > 0, τ* = 0 pplies s corner solution for ll η < η. For η < η < 1, [1 + τ L'τ ) Lτ ) ] = η /1 η ) 0,1), where the r.h.s. is decreses strictly in η. Becuse the l.h.s. is η /1 η) strictly decresing in τ, the preferred vlue is unique nd incresing in η. For ll η, preferences re single peked becuse W / τ chnges sign only once, from positive to negtive t τ*. QED. 7

9 Proposition 1 implies tht fmily preferences re lso single peked in the fmily s retiree-worker rtio. The economic intuition why only the retiree shre mtters is bsed on Cosin resoning. In countries where retirees re not entitled to fmily support, retirees my hve to mke side-pyments to induce working-ge reltives to endorse positive pyroll txes. In countries where children re obliged to support needy prents by lw or perhps by custom), no side pyments my be needed, or pyments my go from working-ge to retired members. 9 In the limiting cse of lmost) inelstic lbor supply, L'τ) 0 would imply τ mx 1 nd τ *η) 1 for ll η >η. Tht is, preferred tx rtes would diverge to the extremes of the unit intervl. The ssumption of qusi-liner preferences over leisure is ment s smoothing device, to rule out such extreme solutions nd to ensure continuous preferences over the size of the socil security system. To define voting equilibrium, let individuls be sorted by the retiree shres of the fmilies they belong to. Let Fη) denote the resulting cumultive distribution function of voters. Tht is, in multi-person fmilies, the fmily s η vlue is counted s mny times s the fmily hs members. Single, unttched workers nd retirees re interpreted s fmilies of size one with retiree shres η=0 nd η=1, respectively. Let η med = F -1 1/2) denote the retiree shre of the medin voter s fmily. Then: Proposition 2: ) A unique voting equilibrium exists nd is given by τ * = τ *η med ). b) The equilibrium pyroll tx rte τ * is strictly positive if nd only if η med > η. Proof: Follows from Prop.1 nd the definition of η med. QED. 9 Note tht W / τ does not depend on welfre weights. The irrelevnce of welfre weights lso implies tht the sign of W / τ would remin unchnged if individuls hd preferences tht put positive weight on their reltives U 0 t functions. It would be strightforwrd to include this type of ltruism formlly, but requiring tedious nottion nd regulrity conditions. Altruism would influence the mount nd direction of side-pyments. 8

10 The key condition for the existence of socil security is tht the mjority of the popultion lives in fmilies with retiree shre greter thn the ggregte retiree shre; or equivlently, tht the mjority lives in fmilies with retiree-worker rtios greter thn the ggregte retiree-worker rtio. 5. Exmples A nturl next question is which ssumptions bout demogrphics re likely to stisfy the condition η med > η. A study of the empiricl distribution of fmily sizes is beyond the scope of this pper, nd it my not be persusive nywy becuse one would hve to mke d hoc ssumptions bout the strengths of vrious linkges. Hence I focus on exmples tht illustrte the bsic forces driving the voting outcomes. Exmple #1 [Two point distribution] Suppose frction f of working-ge individuls forms lsting bonds with m>0 fmily members who will be voting ge when the person is in retirement clled intergenertionl bonds in the following. Nturl exmples would be bonds with children some or ll but the bonds could be with other younger reltives like grndchildren, nieces or nephews, or in-lws. Assume frction π R 0,1] of the working-ge popultion lives to the mximum ge A, nd frction 1-π R dies t the retirement ge R. Let ˆ η = η /π R be the hypotheticl retired popultion if ll retirees hd survived, expressed s shre of ctul popultion. Assume retirees re not linked to ech other nd tht ech working-ge voter is linked to t most one retiree. Then f nd m re bounded by fm 1 η )/ˆ η, or fm π R 1 η )/η. Also, ssume η <1/2 so retirees re minority. The exmple economy hs three types of fmilies. Retirees without intergenertionl bonds form single-member fmilies; they ccount for shre π R ˆ η 1 f ) = η 1 f ) of the popultion. A frction f of retirees lives in fmilies consisting of the retiree plus m workers. Such fmilies hve retiree shre η=1/1+m)>0 nd their popultion shre is π R 1 + m) ˆ η f = 1+ m)η f. Finlly, fmilies hve workers nd no retirees, either becuse the workers never bonded with retiree popultion shre 1 η fm ˆ η ) or becuse the retired member hs died popultion shre 1 π R ) fm ˆ η ). The first type 9

11 prefers τ *1)= τ mx but is minority. The second type prefers τ *η) > 0 if nd only if η =1/1+ m) > η. The third type prefers τ *0)= 0 nd constitutes mjority if nd only if F0) =1 η fm ˆ η + 1 π R ) fm ˆ η =1 η fmη >1/2. 10 With this popultion structure, socil security txes re positive if nd only if A) the medin voter s fmily is fmily of retirees with workers: 1 η fmη <1/2; nd B) this type of fmily prefers positive txes: 1/1+ m) > η. Condition A) imposes lower bound on the number of workers linked to retirees, fm > 1/2 η )/η. Becuse fm π R 1 η )/η, A) lso requires sufficiently high survivl rte, nmely π R > 1/2 η )/1 η ). Condition B) imposes n upper bound on the number of workers in ech worker-retiree fmily, m < 1 η )/η. For η <1/2, the intervl [1/2 η )/η,1 η )/η ] hs length exceeding one nd therefore includes one or more integers m. For integers m in this intervl, A) holds if π R is sufficiently close to one, nd B) holds if f is sufficiently close to one. Thus, voting solutions with positive txes do exist. Quntittively, retiree shres in developed countries re in the 0.2 to 0.35 rnge. Consider η = 0.3 s specific exmple. One obtins 1 η )/η = 7/3 nd 1/2 η )/η = 2/3. Hence conditions A) nd B) re stisfied for m=1 with f>2/3 nd for m=2 with f>1/3. Tht is, fmily linkges support socil security if there re sufficient number of smll fmilies either t lest 2/3 of retirees in fmilies with one working-ge member or t lest 1/3 of retirees in fmilies with exctly two workingge members. Socil security does not hve mjority support if either too few retirees hve intergenertionl bonds, or if retirees with intergenertionl bonds re linked to too mny workers. Tble 1 displys, for rnge of retiree shres, combintions of fmily sizes m), miniml survivl rtes π R ), nd miniml frequency of fmily bonds f) tht support socil security. For retireeshres in the over-1/4 to ner-1/2 rnge, significnt frction of one-worker-one-retiree linkges suffices, or smller number of multiple linkges. When retiree shres re lower, socil security requires lrger number of fmily linkges higher m) to bind sufficient number of workers into 10 Throughout, I disregrd the cse of tied, 50:50 voters, becuse they would distrct from substntive insights nd becuse such cses re non-generic in the prmeter spces considered. 10

12 fmilies with retirees. In ddition, the number of links must not be too high becuse fmily with too mny workers would hve retiree shre below the ntionl verge nd prefer zero txes. Note tht survivl probbility of less thn ½ suffices in ll cses. Becuse the survivor him/herself cn vote, fmilies with retirees cn hve mjority of votes even if retirees hve died in mjority of fmilies. If socil security is interpreted s insurnce ginst the cost of supporting fmily members who survive into retirement, Tble 1 cn be interpreted s providing conditions under which fmilies with erly deths cnnot renege on their insurnce pyments. Overll, Tble 1 suggests tht fmily support for socil security requires blnce between retiree shres nd fmily sizes. Societies with high fertility rtes tend to hve lrge fmilies nd low retiree shres. In such societies, sttic support for socil security requires tht most retiree re linked to substntil number of working-ge fmily members. In societies with low fertility, where fmilies tend to be smll nd retiree shres high, support for socil security requires ppropritely smller number nd/or frequency of fmily bonds. Prior to the demogrphic trnsition, survivl rtes into retirement were probbly too low to support socil security even with lrge fmilies. Thus the model is consistent with the introduction of socil security during the demogrphic trnsition, when retiree shres nd survivl rtes incresed while fmily size decresed. Note tht the equilibrium tx rte τ * 1 ) is incresing s m decreses. Hence socil 1+m security system bcked by size m=1 fmilies hs higher tx rte thn socil security system supported by size m>>1 fmilies. This suggests tht the historicl growth of socil security my be relted to the historicl trend towrds smller fmilies nd not only) to the ging process per se. Exmple #2 [Generl distribution] The ssumption in Exmple 1 of common fmily size for ll worker-nd-retiree fmilies is clerly restrictive. Consider now the sme setting, but with distribution over the number of fmily bonds. Let π m be the frction of retirees with m links, with m rnging from zero to some mximum vlue M. Let m be the verge number of fmily bonds per retiree. Note tht, nlogous to Exmple 1, 11

13 m 1 η )/ˆ η = π R 1 η )/η, nd tht socil security requires less thn 50% popultion shre of workers-only fmilies, 1 η m η <1/2. Among fmilies with retirees, those with η =1/1+ m) > η which lwys includes m=1) will prefer τ *η) > 0. Fmilies with 1/1+ m) < η prefer τ *η) = 0. Support for socil security is thus concentrted mong reltively smll fmilies with retiree, wheres lrger fmilies would be better off supporting their retiree privtely. Let m = 1 η )/η denote the highest m-vlue for which fmilies support socil security. Then voting equilibrium hs positive socil security txes if nd only if 11 Σ 0 = m π m m=0 1+ m) η >1/2. 2) Condition 2) ensures tht fmilies with retiree nd up to m workers constitute mjority. Unless the jump from 1 η )/η to the next lower integer exceeds 1 η )/η m, the m exceeds m. If 2) is stisfied, let m* {1,..., m m* }be the lowest integer for which 1+ m) η >1/2. Then the π m m=0 equilibrium tx rte is τ * 1 ) nd it is decresing in m *. 1+m* For quntittive illustrtion, consider gin η = 0.3, which implies m = 2. Condition 2) then specilizes to Σ 0 = 0.3 π π π 2 > 0.5. Sufficient conditions re π 2 > 5/9, or π 1 > 5/6, or liner combintion where ny gp between π 2 nd 5/9 is covered by sufficient π 1 nd π 0 vlues. The medin voter s fmily size is m* =1 if 0.3 π π 1 > 0.5, which mens tht π 1 plus hlf of π 0 must exceed 5/6; otherwise m* = 2. Note tht fmilies with m > m vote ginst socil security. The model is thus consistent with the bsence of socil security systems in societies with mny lrge fmilies. The lest supportive environment for socil security would be combintion of disconnected retirees m=0) nd fmilies of size m = M = m +1. This scenrio would mximize the number of retirees in fmilies tht vote for no txes. Only the frction π 0 of retirees would support socil security. Retiree-only fmilies do help to stisfy condition 2), but becuse the weight on π 0 is less thn ½, their support is insufficient. Fmilies ties between retirees nd smll but positive number of workers re therefore centrl to generting voting support for socil security. 11 To cover cses with m > M, formlly define π m =0 for m>m. 12

14 Exmples 1-2 do not llow for fmily bonds between retirees. Given the institution of mrrige, this is restrictive. Turning to fmily structures with multiple retirees, key questions re how the number of working-ge members grows with the number of retired members, nd to wht extent the deth of retiree breks linkges between surviving retirees nd working-ge members. In some cses, one my suspect multiple bonds tht survive the deth of one retiree e.g., mrried retirees with common children), wheres in other cses ech retiree my contribute seprte set of intergenertionl bonds. If mrrige correltes with children or otherwise helps to coordinte voting mong lrger number of members, two-retiree fmilies my even hve more thn twice the number of fmily bonds thn one-retiree fmilies. Exmple #3 [Multiple retirees] Consider the sme setting s in Exmple 2, but llow for fmilies with two retirees. A lrger number could be modeled long the sme lines, but the cse of one- nd two-retiree fmilies suffices to illustrte the new issues.) Suppose frction μ of prospective retirees estblishes, during working-ge, fmily bond with sme-ge prtner lbeled mrrige for brevity plus up to M bonds to younger members who will be working-ge when the senior members re retired, i.e., intergenertionl bonds. A frction 1-μ is never mrried nd forms m intergenertionl bonds with probbility π m, s in Exmple 2. Assume survivl into retirement is independent of prtner s survivl. Then frction μ π R 2 ˆ η = μ π R η of the popultion re mrried retirees, frction μ 1 π R ) η re widowed retirees, nd 1 μ) η re never-mrried retirees. Socil security is gin supported by fmilies with retirees nd with not too mny working-ge m 2 = 21 η )/η denote the highest m-vlues for which onend two-retiree fmilies support socil security, respectively. If m 2 is even, m 2 = 2 m, otherwise members. Let m = 1 η )/η nd m 2 = 2 m +1. A voting equilibrium hs positive socil security txes if single-retiree fmilies with m m plus two-retiree fmilies with m m 2 constitute mjority. 13

15 As benchmrk, suppose intergenertionl bonds in two-retiree fmilies re dditive in the sense tht ech prtner contributes bonds ccording to the distribution π m nd tht these links re broken when the contributing prtner dies. Then intergenertionl bonds in two-retiree fmilies re mxm,m ) distributed π 2,m = i=minm,m ) π i π m i for m 2M, the distribution of the sum. The condition for positive socil security txes is m m 2 Σ 1 = π m 1 + m) [1 μ + μ1 π R )]η + π 2,m 1 + m /2) μπ R η >1/2. 3) m=0 m=0 Compred to Exmple 2, one finds m 2 m Σ 1 =Σ 0 + π 2,m 1 + m /2) π m 1 + m) m=0 m=0 μπ Rη. 4) Note tht children per retiree in two-retiree fmilies hve distribution with the sme men m s one-retiree fmilies, but with only hlf the vrince. Recll tht the cutoff m tends to be bove the men. This suggests tht the frction of two-retiree fmilies with m m 2 will exceed the frction of one-retiree fmilies with m m, nd thus mrrige increses the support for socil security. This intuition does not lwys pply, however, becuse voting support is lost whenever retiree with m m is pired with retiree who contributes more thn m 2 m dditionl intergenertionl bonds. For quntittive illustrtion, suppose η = 0.3, which implies m = 2 nd m 2 = 4. In ddition, ssume μ = π R = 0.7 so 49% of retirees re surviving mrried nd 21% widowed. 12 For the distribution π 0 =.4 nd π 1 = π 2 =.30, one finds Σ 1 =Σ 0 =0.57. Mrrige does not ffect voting outcomes. One cn show tht this irrelevnce pplies in generl whenever m M.) For π 0 =.35, π 1 = π 2 =.30, nd π 3 =.05, Σ 1 = exceeds Σ 0 =0.5555, so voter support is incresed. While the support condition m 2 fils for 5% of one-retiree fmilies, condition m 4 fils for only 3.25% of two-retiree fmilies. For π 0 =.35, π 1 = π 2 =.30, nd π 5 =.05, in contrst, Σ 1 = is less thn Σ 0 =0.5555, so voter support is decresed. This is becuse fmilies with m=5 will voter ginst socil security even with two retirees. Different ssumptions bout two-retiree fmilies yield similrly mixed results. For exmple, suppose mrrige does not dd intergenertionl bonds but cretes linkges tht survive the other 12 Note tht becuse μ nd π R only determine the weight on the brcket expression in 4) nd not the sign of Σ 1 -Σ 0, they do not mtter for the key question if multiple-retiree fmilies retiree prtnerships increses or reduces voter support; lso note tht μ nd π R enter only through their product. The key issue is thus the distribution of fmily bonds the shpes of π m nd π 2m nd their interction. 14

16 retiree s deth. Then the distribution π m pplies to ll fmilies with retirees, nd the condition for positive socil security txes is m m 2 Σ 2 = π m 1+ m) [1 μ + μ1 π R )]η + π m 1 + m /2) μπ R η >1/2. 5) m=0 m=0 Compred to the previous cses, one finds m 2 m Σ 2 =Σ 0 + π m 1 + m /2) π m 1 + m) m=0 m=0 μπ Rη m 2 =Σ 1 + π m π 2,m )1+ m /2) m=0 μπ Rη In this cse, mrrige turns fmilies with m [ m +1, m 2 ] workers into socil security supporters. But in fmilies with m [1, m ] workers, where one retiree would hve been enough to gin the workers support, second retiree my represent lost opportunity to form seprte fmily tht includes dditionl workers. The net effect cn be positive or negtive, depending on how much support is gined for m [ m +1, m 2 ] versus support lost for m [1, m ]. 13 Overll, Exmple 3 suggests tht multi-retiree fmilies complicte the nlysis without necessrily ffecting the results in either direction. Figure 1 plots the cutoff vlues m nd m 2 ginst the retiree shre η. Two specific scenrios my be instructive, one for the U.S., the other for continentl Europe. First, suppose η is equl or slightly below 0.2, the relevnt rnge for the U.S. Then fmilies with one retiree nd up to 4 workingge members plus fmilies with two retirees nd up to 8 working-ge members will support socil security. It seems plusible tht these conditions cover the vst mjority of Americn fmilies ll but very lrge extended fmilies. Second, suppose η is round 0.3, the relevnt rnge in continentl Europen countries. Then fmilies with one retiree nd up to two working-ge members, plus fmilies with 2 retirees nd up to 4 working-ge members, will support socil security. Becuse fmilies with more thn two children hve become rre in Europe, nd becuse even fmilies with two retirees, two children, nd two grndchildren would lso stisfy these conditions, the conditions re plusibly stisfied in Europen countries, too. 6) 13 For exmple, consider gin π 0 =.4, π 1 = π 2 =.30, nd π 3 =.05, with the sme other ssumptions s bove. Then Σ 2 = is less thn Σ 0 nd Σ 1, but still bove 50%. 15

17 6. Conclusions The pper provides conditions for py-s-you-go benefits to retirees to be supported s voting equilibrium in sttic model. The key requirement is tht sufficient frction of retirees re in fmilies with positive, but not excessively lrge number of working-ge members. Then the medin voter lives in fmily with bove-verge retiree shre. Numericl exmples suggest tht this condition is stisfied in Europe nd in the United Sttes. The pper s focus on sttic fmily brgining is not to dispute the relevnce of other voting rguments. It would strightforwrd, for exmple, to mke lbor productivity heterogeneous nd to show tht skewed wge distribution would increse voter support for public pensions with progressive benefits. The model is lso potentilly complementry to dynmic models; mny such models rely on trigger strtegies tht rbitrrily) select one of mny equilibrium pths tht could hve been supported by lterntive beliefs bout voting in the future. The sttic rgument for socil security provides nturl lower bound for expecttions bout future benefits. Overll, the mechnism proposed here helps to explin the remrkble populrity of public retirement nd retiree helth cre progrms. This is importnt becuse understnding the populrity of such progrms is rgubly pre-condition for designing politiclly fesible reforms. 16

18 References Aron, H.J., The Socil Insurnce Prdox, Cndin Journl of Economics nd Politicl Science 32, Bohn, Henning, Will Socil Security nd Medicre Remin Vible s the U.S. Popultion is Aging? Crnegie-Rochester Conference Series on Public Policy 50, 1-53., Will Socil Security nd Medicre Remin Vible s the U.S. Popultion is Aging? An Updte. In: Robin Brooks nd Assf Rzin eds.), The Politics nd Finnce of Socil Security Reform, Cmbridge University Press, Breyer, Friedrich nd J-Mtthis Grf v. d. Schulenburg, Voting on Socil Security: The Fmily s Decision-Mking Unit. Kyklos 404), Browning, E.K. 1975). Why the Socil Insurnce Budget is too lrge in Democrtic Society. Economic Inquiry 13, Cooley, Thoms, nd Jorge Sorez, A Positive Theory of Socil Security Bsed on Reputtion, Journl of Politicl Economy 107, Glsso, Vincenzo, The Politicl Future of Socil Security in Aging Societies. MIT Press. Tbellini, Guido, 2000, A Positive Theory of Socil Security. The Scndinvin Journl of Economics 1023),

19 Tble 1: Conditions for Socil Security in Exmple 1 Retiree Shre η ) Given Survivl Prob. π R ) Lower bound Conditions on the number of fmily links m) Lower bound Frction linked f) Upper bound Integer m Lower bound

20 Figure 1: Mximum number of working-ge members for fmilies with one nd two retirees to support socil security Retiree shre 1-retiree fmilies 2-retiree fmilies