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Rangel, Antonio. 2003. “Forward and Backward Intergenerational Goods: Why Is Social

Security Good for the Environment?” American Economic Review 93 (3): 813–34.

https://doi.org/10.1257/000282803322157106

https://www.aeaweb.org/journals/aer/about-aer/editorial-policy

https://doi.org/10.1257/000282803322157106

Forward and Backward Intergenerational Goods:Why Is Social Security Good for the Environment?

By ANTONIO RANGEL*

This paper studies the ability of nonmarket institutions to invest optimally inforward intergenerational goods (FIGs), such as education and the environment,when agents are selfish or exhibit paternalistic altruism. We show that backwardintergenerational goods (BIGs), such as social security, play a crucial role insustaining investment in FIGs: without them investment is inefficiently low, but withthem optimal investment is possible. We also show that making the provision ofBIGs mandatory crowds out the voluntary provision of FIGs, and that populationaging can increase investment in FIGs. (JEL H0, H3, H4, H5, H6, D1, D7).

“Why should I care about future generations?What have they done for me?” (Addison)

“Be nice to your children, they will pick yournursing home.” (Anonymous bumper sticker)

Every society uses a range of nonmarket in-stitutions to decide how much to invest in futuregenerations. A prominent example is the gov-ernment and the decision of how much to investon intergenerational (IG) public goods1 such as

environmental preservation and pure science.These programs entail a transfer to future gen-erations since they are financed with taxes onpresent generations and their benefits are long-lived. Another prominent example is the family.Every generation of parents decides how muchto invest in their children. Investments includethe cost of (public and private) education, andthe myriad of other sacrifices that parents makefor their children.These examples have a common structure.

First, IG exchange takes place in an infinitelylived organization that has an overlapping gen-erations (OLG) structure. Second, present gen-erations have to decide how many resources todevote to investments that disproportionatelybenefit future generations. Third, once the in-vestments are made, future generations cannotbe excluded from the benefits that are gener-ated.2 Fourth, since future generations havenot yet been born, present and future genera-tions cannot negotiate binding contracts thatreimburse present generations for the cost ofthe investment.3 Fifth, membership in the

* Department of Economics, Stanford University, 579Serra Mall, Stanford, CA 94305 (e-mail: [email protected]). I am grateful to Alberto Alesina, Doug Bernheim,Marty Feldstein, and Eric Maskin for their suggestions andsupport during the process of writing about intergenera-tional issues. I also want to thank two anonymous referees,James Alt, Gadi Barlevy, Tim Besley, Francesco Caselli,Andrew Cohen, Alejandro Cunat, David Cutler, SvenFeldman, Drew Fudenberg, Jerry Green, Oliver Hart, JamesHines, Matthew Jackson, Mireille Jacobson, Chad Jones,Jonathan Katz, David Laibson, John Ledyard, Jon Levin,Jeff Liebman, Andreu Mas-Colell, Richard McKelvey,Stephen Morris, Casey Mulligan, Dina Older-Aguilar, Tor-sten Persson, James Poterba, Luis Rayo, Todd Sandler,Andrei Schleifer, Ken Shepsle, Joaquim Silvestre, TomasSjostrom, Kent Smetters, Steve Tadelis, Jaume Ventura,Richard Zeckhauser, and seminar participants at Caltech, theUniversity of California-Davis, Harvard University, the Uni-versity of Illinois-Urbana-Champaign, Iowa State University,NBER, Northwestern University, University of Rochester, andStanford University for their helpful comments. This paper isa significantly revised version of Rangel (1997).

1 See Todd Sandler and Kerry Smith (1976) and Sandler(1978, 1982) for an early development of the concept of“intergenerational goods,” and Joaquim Silvestre (1994,1995) for a more recent reference.

2 For example, once the environmental public good hasbeen provided, all the members of future generations benefitfrom it. Similarly, once children grow into adulthood, andare in a position to compensate the parents, the investmentsreceived in childhood are sunk and cannot be removed.

3 In the case of the family, children are alive at the timeof the parent’s investment. However, they have not been“born” as economic agents since they cannot sign bindingcontracts.

813

organization is not for sale: agents are borninto the organization. As the following exam-ples show, the last three characteristics arecentral to the problem because they rule outmarket-based solutions.To see the role of excludability, consider the

case of publicly traded companies. Every periodpresent generations of stockholders decide howmuch to invest to increase future profits. Here,the benefits generated by IG investment areexcludable since the share of profits received byan agent is proportional to the amount of stockthat he owns. This organization satisfies char-acteristics (4) and (5), but not (3).4 The presenceof a stock market induces members to internal-ize the effect of investments on future profitssince they raise the price at which the stock canbe sold. As a result, this market institution typ-ically generates optimal IG investment.To see the role of exogenous membership,

consider the case of a country club, which is anorganization with an overlapping generationsstructure in which membership has to be pur-chased. This organization satisfies all of thecharacteristics listed above except exogenousmembership. In contrast to the case of the firm,members cannot be excluded from investmentssuch as golf courses. We can think of theseorganizations as IG clubs.5 In the absence ofexternalities across clubs, a market solution inwhich club membership is a tradeable assetcould generate optimal levels of investment.Finally, to see the role of incomplete IG

contracting, consider the case of the family.Even with selfish generations, an efficientamount of investment would take place if chil-dren and parents could sign binding contracts.6In the absence of “transaction costs,” an IGform of the Coase Theorem would arise: chil-

dren would commit to compensate their parentsfor the cost of the optimal investment, and par-ents would have an incentive to provide theseinvestments. The problem is that in many orga-nizations such contracts are not possible. In thecase of the family, the legal system precludesthese types of contracts since children lack theindependence and understanding required toevaluate them.This paper develops a stylized model of IG

exchange to study the conditions under whichnonmarket institutions are able to generatePareto-optimal levels of investment. Agents livefor three periods: young, middle-aged, and old.Every period the middle-aged agent decideshow much to invest in a forward intergenera-tional good (FIG) that benefits future genera-tions, but not himself. Although the role ofaltruism is studied in the paper, it is useful tostart with the case of selfish generations.We start with an immediate observation. If

the only decision made every period is howmuch to invest in FIGs, no investment takesplace. The intuition is straightforward. Agentsbenefit from investments in FIGs made by pastgenerations, but not by investments made afterthey are born. Thus, they have no incentive toinvest in FIGs, and no FIGs are produced. Wecan conclude that optimal IG investment cannotarise when the only decision made by the orga-nization is how much to invest in futuregenerations.Fortunately, in addition to choosing how

much to invest in future generations, most non-market institutions also make decisions aboutbackward IG exchange. For example, the gov-ernment transfers resources to the elderlythrough the social security system, and familiestake care of their elderly parents. The maininsight of this paper is that the presence ofbackward IG goods plays a crucial role in sus-taining investment in future generations: with-out backward exchange, investment in FIGs isinefficiently low; but with it, even optimal in-vestment by selfish generations is possible. Thecrucial insight comes from the literature on mul-timarket contact in industrial organization [seeJonathan Bendor and Dilip Mookherjee (1990)and B. Douglas Bernheim and Michael D.Whinston (1990)], which has shown that link-ages across games play an important role insustaining cooperation.

4 For the purpose of the example, we can think of mem-bership as exogenous: everyone is a “member” of the orga-nization but only agents with a positive amount of stockhave a claim on the profits.

5 See Sandler (1982).6 The “intergenerational Coasian theorem” described

here requires present and future generations to be able tobargain, face to face, and sign binding contracts. The abilityto bind future generations to transfer resources to presentgenerations is not sufficient: if present generations do nothave an incentive to invest in future generations withoutcompensation, they also have an incentive to impose trans-fers on future generations without investing in them.

814 THE AMERICAN ECONOMIC REVIEW JUNE 2003

To study the relationship between forwardand backward IG exchange, we analyze a styl-ized model in which, every period, the middle-aged agent makes two decisions: (1) how muchto invest in a FIG, and (2) how much to buy ofa backward intergenerational good (BIG) thatonly benefits the elderly. Using this framework,in Section III we show that a link between BIGsand FIGs is essential for sustaining optimallevels of investment in future generations. Wealso show that the need for a link between BIGsand FIGs has the following implications. First,the social rate of return, risk characteristics, andhorizon of the FIGs do not affect whether or notthey are financed. Second, within some limits,population aging can increase public investmentin FIGs. Finally, making the provision of BIGsmandatory can crowd out investment in futuregenerations.In Section V we explore the implications of

the analysis for two important IG organizations,the government and the family. There we an-swer the question posed in the title: why issocial security good for the environment? Hereis the bottom line. If a majority of the electoratereceives positive benefits from keeping the so-cial security system, there are voting equilibriain which even selfish generations vote to investin FIGs. In these equilibria, investment in futuregenerations is supported by a link between BIGsand FIGs: present voters correctly believe thatfuture voters’ support of social security dependson whether or not they invest in FIGs.

I. Relation with the Literature

Using the BIGs and FIGs framework, theliterature on nonmarket IG organizations can bedivided into three strands.The first strand studies organizations in

which there is only intragenerational exchange.Jacques Cremer (1986), David Salant (1991),Michihiro Kandori (1992), Lones Smith (1992),and Kenneth Shepsle (1999) study OLG orga-nizations in which every period all of the agentssimultaneously take an action that affects everyone alive at the time, but has no effect on futuregenerations. For example, in Cremer (1986),agents simultaneously choose how much effortto exert in production, and total output dependson the sum of the efforts. The main insight fromthis literature is that the standard “Folk Theo-

rem” results extend to the OLG context: coop-eration can be sustained as long as agents arepatient and/or live long enough. The last condi-tion is needed because agents in the last periodof life cannot be given an incentive to cooper-ate. By contrast, the results developed in thispaper are not limit results.7The second strand of the literature studies

organizations in which the exchange problemlooks like a BIG. Here the organization chooseshow much to produce of a good that is basicallya transfer from the younger to the older gener-ations. Consider, for example, the “PensionGame” in Hammond (1975),8 in which he stud-ies a standard OLG economy with two-periodlifetimes. Agents have an endowment whenyoung, but not when old, and have no access toa savings technology. Hammond shows thatthere are equilibria, similar to the one developedin Section III, subsection B, that sustain Pareto-improving transfers from young to old in everyperiod. To study the political economy of pay-as-you-go social security, Hansson and Stuart(1989), Henning Bohn (1998), and Cooley andSoares (1999) extend this model to a setting inwhich agents live for more than two periods anddecisions are made by majority rule. The in-sights from all of these papers are similar to theresults for BIGs in Sections III and V.Two important papers in this literature are

Kotlikoff et al. (1988) and David Kreps (1990),who show that the presence of a sustainableBIG can be used to solve inefficiencies in theeconomy. Kotlikoff et al. (1988) study a stan-dard OLG economy with two-period lifetimesin which every generation elects its own sepa-rate government. Each generational governmentfaces a standard commitment problem: it wouldlike to choose low capital tax rates but cannotcredibly commit to do so. They show that thecommitment problem can be overcome throughthe introduction of a self-sustainable “IG com-pact” in which every generation agrees to trans-fer a large sum to the previous generation aslong as it has followed the compact and haschosen low capital tax rates for itself. Kreps

7 This is also true in Peter Hammond (1975), LaurenceKotlikoff et al. (1988), Ingemar Hansson and Charles Stuart(1989), and Thomas Cooley and Jorge Soares (1999), whichare described below.

8 See also Narayana Kocherlakota (1998).

815VOL. 93 NO. 3 RANGEL: FORWARD AND BACKWARD INTERGENERATIONAL GOODS

(1990) shows that the transfers can be used toovercome moral hazard problems. As we dohere, these papers show that linking the provisionof a BIG with something else can be beneficial; inour case to provide investment in future genera-tions, in Kotlikoff et al. and Kreps to solve anintragenerational incentive problem.Next, several papers have studied organiza-

tions in which there are only FIGs and shownthat underinvestment must take place. For ex-ample, Jacobus Doeleman and Sandler (1998)study investment in IG public goods in a finiteOLG model and conclude that, with selfish gen-erations, underinvestment takes place [see alsoKotlikoff and Robert Rosenthal (1993) andDavid Collard (2000)]. By contrast, in this pa-per we study organizations in which both BIGsand FIGs are provided.Finally, two other papers have argued that

there is a link between forward and backwardIG exchange. Gary Becker and Kevin Murphy(1988) suggest that it is possible to think of oldage social insurance and education as a tradeamong generations: children receive an educa-tion from their parents and in exchange pay fortheir retirement benefits. However, their’s ismostly an accounting argument. They do notstudy the sustainability of these arrangements,which is the focus of this paper. This is prob-lematic because when children grow up theycan default on their obligations. Michele Bold-rin and Ana Montes (1998) have independentlydeveloped an analysis that is closely related.9They study the majority rule politics of pay-as-you-go social security and public educationusing an overlapping generations economy. Al-though there are some differences in the detailsof the model, both papers arrive at similar in-sights. In particular, their main result is analo-gous to Proposition 3 in this paper.

II. Model

Consider a simple model of an infinitely livedorganization with an overlapping generationsdemographic structure. Each period t a newmember, called generation t, enters the organi-zation and stays there for three periods: t, t �1, and t � 2. We say that the agent is young in

the first period, middle-aged in the second, andold in the third. Time is indexed by t � 1,2, ... . At time 1 there is also an old generation�1 that stays in the organization only for thatperiod, and a middle-aged generation 0 thatbelongs to the organization in periods 1 and 2.Every period t, the middle-aged generation

t � 1 has to make two decisions: (1) how muchto invest on a FIG that only benefits generationt � k, where k � 0; and (2) how much to spendon a BIG that only benefits the current old. k �0 denotes the lag between the time the FIG isproduced and the time the benefits are received.Let ft denote the investment in FIGs in period t,and bt the amount spent on BIGs. These costsare paid by generation t � 1.Every generation receives endowments wy,

wm, and wo of a private consumption good ineach of the three stages of their lives. Agentscan borrow and lend at the economy’s interestrate r. The preferences of generation t are given by

(1) U�cy, cm, co� � F� ft� k � � B�bt� 2 �,

where cy, cm, co denote their consumptionwhen young, middle-aged, and old. In this spec-ification of the model, the young of generationt � k are the only agents who benefit from theFIG produced at time t, and there is no IGaltruism. We start with this stark case for expo-sitional purposes. More complex FIGs and IGaltruism are introduced later on. All the functionsare twice continuously differentiable and increas-ing,U satisfies the usual strict concavity and Inadaconditions, and F and B are strictly concave.The following notation greatly simplifies the

analysis. Let

(2) V�x� � arg maxcy,cm,co

U�cy, cm, co�

s.t. cy�1 � r� � cm �co

�1 � r� � x,

which denotes the indirect utility from the con-sumption of private goods for an agent whospends an amount of wealth x, where wealth ismeasured at middle-age.The actions that are taken in the organization

define an infinite game with overlapping gener-ations of players. Every period t, generation t �1 chooses9 This paper is a revised version of Rangel (1997).


(3) �bt , ft � � ��b, f ��0 � b � f � w� ,

b � 0, and f � 0�,

where w� wy(1 � r) � wm �wo

�1 � r� denotes

the lifetime wealth of every generation (mea-sured at middle-age). Let ht � ((b1, f1), ... ,(bt�1, ft�1)) denote the history of actionstaken up to period t. A strategy for generationt � 1, which takes an action in period t, is afunction st(ht) � (stB(ht), stF(ht)) that specifiesthe amount of BIGs and FIGs purchased at anypossible history. Let s � (s1, s2, ...) denote aprofile of strategies for every generation. Thepayoff for generation t � 1, conditional onhistory ht, is given by:

(4) t �s�ht � � V�w� � stB�ht � � stF�ht ��

� F� ft� �k� 1� �

� B�st� 1B �ht , st �ht ��.

Note that the payoff of generation t � 1 isaffected only by a small number of the decisionstaken in the organization: ft� (k�1), bt, ft, andbt�1. We use subgame-perfect equilibrium asthe solution concept.

III. Results

A. A Useful Tool

We start the analysis by deriving a usefulresult. Given any path � � {(bt, ft)}t�1�

of BIGs and FIGs, define the following profileof simple trigger strategies (STSs):

(5) st �ht � � � �b t , f t � if ��ht � � C�0, f t � if ��ht � � P.

� is defined recursively as follows: �(h1) � Cand

(6) ��ht � � �C if ��ht� 1 � � C and

�bt� 1 , ft� 1 � � �bt� 1 , f t� 1 �C if ��ht� 1 � � P and

�bt� 1 , ft� 1 � � �0, f t� 1 �P otherwise.

The idea behind STSs is straightforward. � is aflag that keeps track of whether the organizationis in a cooperative or in a punishment phase. Inthe cooperative phase agents produced the levelof BIGs and FIGs prescribed by �. In a punish-ment phase they only produce the prescribedlevel of FIGs. Note that if every generationplays the STS then � is the outcome of thegame.

PROPOSITION 1: A path � of BIGs and FIGscan be sustained as a subgame-perfect equilib-rium if and only if it can be sustained as asubgame-perfect equilibrium using STSs.10

This result is useful because it shows that tocheck if a particular path of BIGs and FIGs canbe sustained, it is enough to test if it can besustained using STSs. Note that STSs are notthe only strategies that can be used. For exam-ple, if � can be sustained using STSs, then it canalso be sustained using “grim strategies” inwhich failure to produce the prescribed level ofBIGs or FIGs ends cooperation forever. Wefocus on STSs because they have two appealingproperties. First, they are simple. Second, thepunishment phase lasts for only one period,and only the generation that failed to pro-duce the prescribed level of BIGs and FIGs ispunished.11Using this tool it is easy to characterize the

set of paths that can be sustained as a subgame-perfect equilibrium.

PROPOSITION 2: A path � of BIGs and FIGscan be sustained as a subgame-perfect equilib-rium if and only if

(7) V�w� � b t � f t � � B�b t� 1 �

� V�w� � � B�0� for all t.

10 All of the proofs are in the Appendix.11 Venkataraman Bhaskar (1998) shows that, in this

type of overlapping generations game, the existence ofcooperative equilibria depends crucially on the observ-ability of the entire history of play. In particular, nocooperation is the unique equilibrium in pure strategieswhen generations can observe at most the actions of thelast n predecessors.


B. Characterization of Equilibria

We start with an immediate observation: noFIGs are produced in an organization in which noother decisions are made.12 Thus, the presence ofBIGs, or other forms of exchange that will bediscussed below, are essential to generate positiveinvestments in FIGs. This section explores in de-tail the relationship between BIGs and FIGs.It is useful to start with an organization in

which there are only BIGs. Consider for a mo-ment a version of the model in which the onlyaction chosen every period t is how much tospend on BIGs ( ft � 0 for all t). Everythingelse remains unchanged. This generates a gamein which {b�0 � b � w� } is the action set foreach generation.Consider any path � {bt}t�1� of BIGs, and

define a continuation surplus function given by

(8) StB�� V�w� � b t � � B�b t� 1 �

� �V�w� � � B�0� .

This measures the surplus generated by the IGtrade implicit in : the first term measures thepayoff for generation t � 1 of producing bt andreceiving bt�1, the second term measures itspayoff at generational autarky where no BIGsare produced.Note that the continuation surplus function is

positive for all t if and only if the path satisfies condition (7). Thus, we conclude that apath of BIGs can be sustained as a subgame-perfect equilibrium if and only if StB() � 0 forall t. This characterization is very intuitive.Every generation needs to decide how manyBIGs to give to the old, and in exchange itreceives some BIGs from the next generation.In a STS, a generation that does not provide theprescribed amount of BIGs for the previousgeneration is punished by not receiving anyBIGs when it becomes old. Thus, the cost of notcooperating is equivalent to returning to gener-ational autarky. This characterization says that apath can be sustained only if the amount ofBIGs received in old age outweighs, for every

generation, the cost of financing the BIGs forthe previous generation.Now consider the level of BIGs that can be

sustained as a stationary equilibrium. By Prop-osition 2, a stationary level of BIGs b can besustained if and only if

(9) SstB �b� � �V�w� � b� � B�b�

� �V�w� � � B�0� � 0.

If B�(0) � V�(w� ), SstB (�) takes positive valuesin the interval [0, bBmax], where bBmax denotes themaximum level of BIGs that can be sustained.Furthermore, bBmax � b*B, where b*B denotes theoptimal stationary level of BIGs in the model inwhich there are no FIGs.13 This has two impli-cations that will be useful below. First, for anylevel of BIGs b � (0, bBmax), the stationary pathgenerates a positive surplus: every generation isbetter off producing and receiving this level ofBIGs than in generational autarky. Second, in-efficient overproduction and underproduction ofBIGs is possible.Using these insights we now characterize the

level of FIGs that can be sustained in the fullmodel. For any path � define the followingcontinuation surplus function:

(10) St �� V�w� � b t � f t � � B�b t� 1 �

� �V�w� � � B�0�

� StB�� V�w� � bt �

� V�w� � bt � ft �].

This function measures the value of the ex-change implicit in � over generational autarky.It is equal to the continuation surplus of theexchange in BIGs implicit in �, minus the utilitycost of financing an amount of FIGs ft. Thus,St(�) � StB(�) whenever ft � 0. Note that thecontinuation surplus does not measure “socialsurplus” since it excludes the benefits generatedby FIGs.

12 This can be seen by setting B(�) � 0 in condition (7),which implies that the only path that satisfies the inequalityis � � {(0, 0)}t�1� . 13 b*B arg maxb V(w� � b) � B(b).


PROPOSITION 3:

(1) A path � of BIGs and FIGs can be sus-tained as an equilibrium if and only ifSt(�) � 0 for all t.

(2) A positive level of FIGs can be sustained inevery period if and only if there exists apath of BIGs such that StB() � 0 for all t.

(3) If B�(0) � V�(w� ), then there are equilibriain which positive levels of FIGs are pro-duced in every period.

Proposition 3 provides a simple but importantinsight. Positive investments in FIGs can takeplace even when present generations are selfish.However, three conditions are necessary. First,the members of the organization must also faceanother exchange problem that requires cooper-ation. BIGs are one such possibility, but as wewill see in Section III, subsection E, not the onlyone. Second, the non-FIG dimension must becapable of generating cooperative trades thatgenerate a positive continuation surplus. Third,the generations must play strategies that linkcooperation in the non-FIG dimension with suf-ficient investment in future generations. Withthis link, an agent receives a BIG in old age onlyif he purchases the right level of BIGs for theprevious generation and invests enough on FIGs.Now consider the levels (b, f ) of BIGs and

FIGs that can be sustained as stationary equi-libria. By Proposition 3, (b, f ) can be sustainedif and only if

(11) Sst �b, f� � �V�w� � b � f� � B�b�

� �V�w� � � B�0� � 0.

Let fmax(b) denote the maximum amount ofFIGs that can be sustained with a level of BIGsequal to b. This function is defined implicitly bythe equation Sst(b, f ) � 0. By the ImplicitFunction Theorem, the function is defined forany b � 0, is continuously differentiable, andits derivative is given by

(12) f�max�b� �B��b� � V��w� � b � fmax�b��

V��w� � b � fmax�b��.

Figure 1 characterizes the set of stationaryequilibrium outcomes. The bell-shaped curvedepicts the function fmax(b) for an organization

in which B�(0) � V�(w� ). [If this is not the case,fmax(b) � 0 for all b]. The figure also plots thelocus B�(b) � V�(w� � b � f ), which isdownward sloping and intersects the horizontalaxis at b*B. The function fmax increases until itintersects the locus, and then decreases untilbBmax, which denotes the maximum level ofBIGs that can be sustained. Since f � fmax(b)implies that S(b, f ) � 0, the area under thecurve provides a complete characterization ofthe set of stationary equilibrium outcomes.Note that the benefits generated by FIGs play

no role in the characterization of this set.Changes in F(�) affect the optimal level ofFIGs, but not the amount of FIGs that can besustained. This is somewhat counterintuitive.One would expect IG investments with a moreadvantageous benefit-cost ratio to be morelikely to be financed; but this is not the case. Asa result, investments in future generations thatare relatively inexpensive are more likely to befunded than programs that generate larger netsocial benefits but are also more costly.Can the stationary Pareto-optimal level of

investment in FIGs be sustained? To answer thisquestion it is useful to introduce a parameter that affects the benefits generated by FIGs. Let

(13) �b* , f* � � arg maxb,f

V�w� � b � f�

� B�b� � F� f�

denote the optimal stationary level of BIGs andFIGs. As long as B and F satisfy the Inada

FIGURE 1. SET OF STATIONARY EQUILIBRIUM OUTCOMES


condition, the solution is interior and satisfiesthe FOCs

(14) B��b* � � V��w� � b* � f * � � F�� f * �.

Furthermore, we have that

(15)�f *�

� 0,�b*�

0,

and �b*0 , f *0 � � �b*B , 0�.

The locus (b*, f *) is depicted in Figure1.14 For small , (b*, f *) lies in the interior ofthe sustainable set and thus optimal production,and even inefficient overproduction, are possi-ble. As increases, the optimal level of FIGs f *eventually increases beyond what can be sus-tained with the BIGs available in the organiza-tion. This establishes the following result.

PROPOSITION 4: Inefficient overproductionof FIGs can be sustained as a stationary equi-librium if and only if S(b*, f*) � 0. In thiscase, any level of FIGs f � [0, maxb fmax(b)],including the optimal stationary level f*, canbe sustained as a stationary equilibrium.

As Figure 1 starkly illustrates, the model gen-erates a large number of equilibria: in some ofthem IG cooperation takes place, in others itdoes not. This is common in models of long-lived institutions that use noncooperative gametheory. The equilibrium set could be reduced byimposing equilibrium refinements like Mar-kovian equilibrium or renegotiation proofness.However, given that so far game theory has notprovided a fully satisfactory justification forsuch refinements, we proceed by characterizingthe entire equilibrium set. Future theoretical de-velopments might be able to identify variablesthat affect the coordination of expectationsacross generations, and thus rule out some ofthe equilibria.We conclude this section with a comparison

of forward and backward IG exchange. From atechnological point of view, BIGs and FIGs arenot that different. Both types of exchange re-

quire agents to provide a good that is valuablefor another generation, and in exchange benefitfrom a good that is provided for them. However,from an incentive point of view, they are sig-nificantly different. First, even when BIGs andFIGs generate identical benefits [i.e., whenB(�) � F(�)], BIGs can be sustained in organi-zations that only make this type of decisions,but FIGs cannot be sustained in isolation. Apositive level of investment in FIGs can ariseonly by linking them with BIGs. Second, thecost and benefits of BIGs are crucial in deter-mining the level of BIGs and FIGs that can besustained. By contrast, the benefits generated byFIGs play no role. Third, the optimal stationarylevel of BIGs is always sustainable. This is notthe case for FIGs.

C. What Type of FIGs Can Be Sustained?

The basic model ignores some importantproperties of FIGs. For example, environmentalprograms often generate benefits for multiplegenerations, including those making the invest-ment, and have uncertain returns. In this sectionwe show that these issues do not alter the in-sights obtained above.Consider first the case of multiple beneficia-

ries and risk. For concreteness, consider a FIGfor which the preferences of generation t aregiven by

(16) U�cy, cm, co� � F�� t , �ft� k�k� 0t� 1 �

� B�bt� 2 �,

where �t is a random shock realized at thebeginning of period t. In this case, the FIGproduced at time t can benefit every futuregeneration, including generation t.It is straightforward to see that the path of

sustainable FIGs has not changed. Given thatFIGs are nonexcludable, past investments inFIGs do not affect the incentive constraint of thedecision maker. As a result, the characterizationprovided in Propositions 3 and 4 still holds, andthe set of stationary equilibrium outcomes isstill the one depicted in Figure 1. The lesson isclear: risks, lags, and multiple beneficiaries af-fect the optimal level of investment in FIGs, butnot the level that can be sustained.14 It is given by the line B�(�) � V�(�).


Now consider the case in which the genera-tion investing in FIGs also benefits from them.For concreteness, suppose that the preferencesof generation t � 1, the one making decisionsin period t, are given by

(17) U�cy, cm, co� � F� ft� �k� 1� �

� G� ft � � B�bt� 1 �,

where G is concave, continuously differentia-ble, and increasing. The only difference with thebasic model is that generation t � 1 now gets adirect benefit G( ft) from investing in FIGs.To analyze this case define a new continua-

tion surplus function given by

(18) StG�� V�w� � b t � f t �

� B�bt� 1 � � G� f t �]

� �maxfV�w� � f � � B�0� � G� f � .

The key difference with the previous case is thatnow generations want to produce some of theFIG even if cooperation breaks down. This isreflected in the second term, which is the payoffat generational autarky.

PROPOSITION 5: Consider the case in whichFIGs generate benefits for the generation mak-ing the investment:

(1) A path � of BIGs and FIGs can be sus-tained as an equilibrium if and only ifStG(�) � 0 for all t.

(2) If G�(0) � V�(w� � bBmax), any stationaryequilibrium that does not link BIGs andFIGs involves inefficient underproductionof FIGs.

Proposition 5 shows that underinvestment inFIGs still takes place unless two conditions aremet. First, the agents must play strategies thatlink BIGs and FIGs. Second, the BIGs mustgenerate a large enough continuation surplusthat can be used to give incentives to investbeyond the short-sighted level.A characterization of the set of stationary

equilibrium outcomes illustrates this point. Fig-ure 2 depicts the set for an environment in

which G�(0) � V�(w� ) and B�(0) � V�(w� �f au). The first condition guarantees that f au, theamount of FIGs produced in generational au-tarky, is positive. The second condition guaran-tees that the maximum amount of FIGs that canbe sustained is larger than f au. Since the char-acterization is similar to the one for Figure 1 thedetails are omitted.Note a few properties of the equilibrium set.

First, (0, 0) is no longer an equilibrium. Thisfollows from the fact that agents invest in FIGsin the absence of cooperation. Second, considera parameterization of the preferences given by

(19) U�cy, cm, co� � F� ft� �k� 1� �

� G� ft � � B�bt� 1 �,

where measures the relative fraction of thebenefits that are internalized by the decisionmaker. Let (b*, f*) denote the optimal station-ary level of production. As before, the optimallevel of FIGs is sustainable when is small butnot when it is large. Finally, the equilibrium setcuts the horizontal axis (i.e., there is an equilib-rium in which no FIGs are produced) if and onlyif

(20) maxbV�w� � b� � B�b� � G�0�

� maxfV�w� � f � � B�0� � G� f �.

FIGURE 2. SET OF STATIONARY EQUILIBRIUM OUTCOMESWHEN FIGS BENEFIT PRESENT GENERATIONS


Thus, when the surplus generated by BIGs islarge relative to the value of FIGs, there arepathological equilibria in which no generationinvests in FIGs even though they benefit di-rectly from those investments.

D. The Role of Altruism

Another concern with the basic model isthe assumption of selfish generations. Theeffect of altruism on the analysis depends onthe specific form that it takes. In particular,one needs to distinguish between paternalisticand nonpaternalistic altruism. With paternal-istic altruism, the level of FIGs that futuregenerations consume enters as an argument inthe utility function of present generations.With nonpaternalistic altruism, it is the utilitylevel of future generations that is internalizedby present generations.The analysis of paternalistic altruism is

equivalent to the case discussed at the end of theprevious section. The only difference is that thefunction G is now interpreted as altruism, in-stead of a direct benefit from consuming theFIG. Therefore, in this case a link between BIGsand FIGs is still needed to generate optimallevels of production.The case of nonpaternalistic altruism is

qualitatively different. Consider, in particu-lar, the dynastic model in Robert Barro(1974). In this model there is no IG exchangeproblem. The organization behaves like aninfinitely lived agent that perfectly internal-izes the future spillovers of investing in FIGs.Thus, in applications for which the dynastic modelis a good description of behavioral motives, theissues studied in this paper do not arise.Casual evidence suggests that IG altruism

is at work in most of the applications of thismodel: voters care about the future of human-ity and parents care about their children.However, the key question is which is the formthat their altruism takes. Although a lot of workremains to be do done in this area, and a con-sensus does not exist yet, some existing evi-dence suggests paternalistic altruism might be abetter approximation.15

E. Investment in Future GenerationsWithout BIGs

This section shows that BIGs are not the onlytype of exchange that can be used to sustainFIGs. To see this, consider the following gen-eralization of the model. Every period t, gener-ation t � 1 makes two choices: (1) how muchto invest in FIGs, just like before, and (2) howmany resources to spend in another good. De-note the second decision by et. The preferencesof generation t � 1 are now given by

(21) U�cy, cm, co� � F� ft� �k� 1� �

� B��ek �k� t�1,t,t� 1, ...�.

Note that a generation could be affected by allthe choices made in the non-FIG dimensionafter it is born. This includes, as special cases,the previous model of BIGs, and the case inwhich the non-FIG dimension only affects thosealive at the time.Let eau denote the action taken in genera-

tional autarky, and e* the optimal stationaryaction. Suppose that eau � e* so that cooper-ation is required to sustain e*. For any path � �{et}t�1� define the continuation surplus function

(22) StE�� V�w� � e t �

� B��ek �k� t�1,t,t� 1, ...�]

� �V�w� � � B�eau, eau, ...� ,

that measures the value of the allocation � overgenerational autarky. It is straightforward toextend the previous arguments to show that apositive amount of investment in FIGs can besustained if and only if there exists a path � suchthat StE(�) � 0 for all t.The intuition is simple. The central feature of

FIGs is that present generations do not careabout how many FIGs are produced by futuredecision makers. This is what makes FIGs dif-ficult to sustain, and the reason a link is needed.If there exists another dimension of exchange inthe organization that requires cooperation, andif the cooperative allocation generates a positivecontinuation surplus, then that surplus can beused to sustain investment in FIGs using strat-15 See Joseph Altonji et al. (1992, 1997).


egies that link FIGs and non-FIGs. It does notmatter what the other dimension of exchange isas long as: (1) it requires IG cooperation, and(2) it generates a positive continuation surplus.For this to be the case, present decision makersmust care about future decisions.This argument can be pushed even further. In

organizations in which more than two decisionsare made, say if there are several BIGs, it ispossible to link several of these decisions toFIGs. If each individual BIG generates a sur-plus, each additional BIG provides additionalincentives to provide FIGs. In the context of thepolitical economy of IG public goods discussedin Section V, pay-as-you-go social insurance,the choice of capital tax rates, and the decisionto honor the national debt can be used simulta-neously to sustain investment in IG publicgoods.

IV. The Effect of Mandatory Provision

The previous analysis has shown that thereare always equilibria in which FIGs and/orBIGs are not provided. A natural question to askis whether the introduction of minimal provi-sion constraints can improve the outcomes gen-erated by these organizations. These types ofconstraints are common. For example, tax-financed public education places a constraint onthe minimum amount of educational expendi-tures that a parent can give to his child, andmandatory old age social insurance imposes asimilar constraint for the case of BIGs.16In this section we study the effect that mini-

mal provision constraints have on voluntaryprovision. Denote the minimum constraint by(b� , �f ). These constraints have no effect on thepayoffs, but shrink the action sets to {(b,f )�b � f � w� , f � �f, and b � b�}.To characterize the equilibrium set for this

case we need to define a new continuation sur-plus function. Let � be any path of BIGs andFIGs satisfying the constraints and define

(23) St�b� , �f �� V�w� � b t � f t � � B�b t� 1 �

� �V�w� � �f � b� � � B�b� � .

It is straightforward to extend the arguments inPropositions 1 and 2 to show that a path � canbe sustained as an equilibrium if and only ifSt(b� ,f� )(�) � 0 for all t. This provides a fullcharacterization of the equilibrium set. The onlydifference is that now the payoff of generationalautarky has changed to V(w� � �f � b� ) � B(b� ),which has implications for the levels of BIGsand FIGs that can be sustained.Let f max(b� , �f ), f min(b� , �f ), bmax(b� , �f ), and

bmin(b�, �f ) denote, respectively, the maximumand minimum level of FIGs and BIGs that can besustained as a stationary equilibrium in an institu-tion with constraints (b�, �f ). The following propo-sition describes the effect of (1) introducing aminimum constraint only on BIGs (�f� 0), and (2)introducing a minimum constraint only on FIGs(b� � 0). The effect of introducing a constraint inboth goods is discussed below.

PROPOSITION 6: Suppose that B�(0) �V�(0), then:17

(1) The introduction of a minimum provisionconstraint b� � [0, w� ) on BIGs has thefollowing effects: f max(b� ) decreases as b�increases between 0 and b*B and equals 0afterwards, f min(b� ) � 0 for all b� , bmax(b� )decreases between 0 and b*B and equals b�afterwards, and bmin(b� ) � b� .

(2) The introduction of a minimum provisionconstraint �f � [0, w� ) on FIGs has thefollowing effects: f max( �f ) is increasing,f min( �f ) � �f, bmax( �f ) is decreasing, andbmin( �f ) � 0.

Figure 3 summarizes the effects described inthis result. Figures 4 (left-hand side) depicts theeffect of b� on the equilibrium set. For b� � [0,b*B] the set shrinks towards (b*B, 0) as b� in-creases, bmin(�) increases, and bmax(�) decreases.The intuition for the latter effect follows from16 Another natural institution is a constitutional con-

straint requiring the provision of exactly the optimal levelof BIGs and FIGs in every period. This institution, how-ever, does not work well when there is considerableuncertainty about future parameters. For example, whatwill be the optimal amount of R&D in nanotechnology in2050?

17 This condition can be relaxed to B�(0) � V�(w� ).But in that case there may exist f such that, for all �f �f, f max( �f ) � f min( �f ) and bmax( �f ) � bmin( �f ) � 0.


FIGURE 4. EFFECT OF MINIMAL PROVISION CONSTRAINTS ON THE SET OF STATIONARY EQUILIBRIUM OUTCOMES

FIGURE 3. EFFECT OF MINIMAL PROVISION CONSTRAINTS ON MAXIMAL AND MINIMAL PROVISION


the fact that the payoff at generational autarky,which is the worse feasible punishment for ageneration that fails to cooperate, increases inthis range. This reduces the incentives to coop-erate. At b� � b*B, the equilibrium set shrinks toa single point: (b*B, 0). Further increases in b�generate the equilibrium set {(b� , 0)}. Since theBIG surplus is needed to sustain FIGs, the max-imum level of FIGs that can be sustained de-creases with b� . In fact, by the time b� reachesb*B, the BIGs generate no surplus and thus noFIGs can be sustained.This result shows that there is a perverse

institutional trade-off in this class of organiza-tions. The imposition of a minimal provisionconstraint in BIGs eliminates the possibility ofthe bad equilibrium in which no BIGs are pro-duced, but it also reduces the maximum level ofinvestment in future generations that can besustained.18The constraint for FIGs �f has a different ef-

fect. Increases in �f have a positive effect on themaximum and minimum level of FIGs that canbe sustained, and a negative effect on the max-imum amount of BIGs. The result is also drivenby the effect of �f on generational autarky pay-off, given by V(w� � �f ) � B(0), which isdecreasing in �f. As a result, as shown in Figure4 (right-hand side), bundles (b, f ) that were notsustainable before, now become sustainable.Now consider the effect of introducing a min-

imum provision constraint in both FIGs andBIGs. Since the analysis is very similar to theprevious two cases, the details are omitted.Once more, the impact on FIG provision de-pends on the payoff at generational autarky. IfV(w� � b� � �f) � B(b� ) � V(w� ) � B(0) theequilibrium set with constraints is a subset ofthe equilibrium set for the case of no con-straints, and is similar to the one depicted inFigure 4 (left-hand side) except for one minorchange: the points (b, f ) for which f � �f have

to be removed. As a result, the constraint de-creases the maximum level of FIGs that can besustained. By contrast, if V(w� � b� � �f ) �B(b� ) � V(w� ) � B(0), the equilibrium setresembles the one depicted in Figure 4 (right-hand side) with one minor change: the points(b, f ) for which b � b� have to be removedfrom the set. In this case, the maximum level ofFIGs that can be sustained increases.Minimal provision constraints for BIGs and

FIGs have very different effects. Whereas in-creases in the constraint for BIGs crowd outvoluntary cooperation in BIGs and FIGs, in-creases in the constraint for FIGs increase thetotal amount of FIGs that can be sustained. Thisasymmetry is interesting because, to the extentthat the minimal provision constraints are de-termined endogenously, present generationshave an incentive to introduce minimal provi-sion constraints in BIGs but not on FIGs.

V. Applications

A. Investment in Children Within the Family

A natural interpretation of the model is as atheory of IG exchange within a family that iseither selfish or exhibits paternalistic altruism.Families exchange two types of IG goods: (1)FIGs, that are provided by parents to theiryoung children in the form of education andparental care, and (2) BIGs, that adult childrenprovide to their aging parents in the form ofcare, insurance, and status. In this interpretationk � 0.As shown in Section III, subsections B and D,

in a selfish family investment in children can bepositive only if there is a link between BIGs andFIGs, and in families with paternalistic altruismthe link is required to generate investments inexcess of what parents are willing to invest ontheir own. For example, without BIGs, a parentmight be willing to finance a high school edu-cation, but not college. In both types of families,at least part of the investments are driven bystrategic considerations: parents believe that(excess) investments in FIGs are the price thatthey pay for getting the BIGs that they desire inold age.The model predicts the existence of three

types of families. First, families that link BIGsand FIGs sustain a high level of investment in

18 Bernheim and Whinston (1998) obtain results witha similar flavor for nonintergenerational contracting prob-lems. In many economic relationships complete contractsare impossible. In this case, voluntary cooperation is neededin the dimensions where the contract is incomplete. In thiscontext, they show that it might be advantageous to pur-posefully leave some dimensions out of the contract toincrease the incentive to cooperate in the dimensions thatcannot be included.


both. Second, gerontocratic families with highlevels of provision for the elderly, but low in-vestment in children. Third, “dysfunctional”families that underprovide both. Casual obser-vation suggests that there is significant variationwithin and across cultures. Endogenous prefer-ences are likely to be an important part of theexplanation, specially if culture and institutionsinfluence the amount of altruism within fami-lies. But family norms might also play an im-portant role: some societies and familiesconverge to cooperative codes of behavior, oth-ers do not.As shown in Section III, in order for FIGs to

be provided, there must be a BIG that generatesa positive surplus. This is guaranteed as long asthe elderly place a marginal value on the firstunit of the BIG that exceeds the marginal costfor the middle-aged. Although in our stylizedmodel this is imposed as an assumption [condi-tion B�(0) � V�(w� ) in Proposition 3], severalof the BIGs exchanged within families satisfythese characteristics. Consider, for example, thecase of insurance. Retirees face risks that arenot insurable through financial markets such asa collapse of the stock market, or a crime thatsignificantly reduces their wealth. As long asthe serial correlation of the adverse shocks islow, there are gains from exchange between themiddle-age and the elderly. Kotlikoff and AviaSpivak (1981) show that a similar argumentholds for the provision of “annuity insurance”within the family when annuity markets areimperfect.The results on mandatory provision also have

interesting implications for the family. Supposethat the government introduces a law that forcesthe middle-aged to provide a BIG that they usedto provide voluntarily. This would be the case,for example, if the good is financed with taxeson the middle-aged. This has three types ofeffects (see Figure 3). First, it increases theminimal amount of BIGs that any elderly personreceives. If a fraction of the families in theeconomy are in a bad equilibrium in whichBIGs are not provided, the policy improves thewelfare of these elderly. Second, it decreasesthe maximum amount of voluntary provision ofBIGs that can be sustained. Thus, if the publicprogram is not large enough to fully crowd outfamily care, the welfare of the elderly whobelong to families that are in a good equilibrium

can go down.19 Finally, it reduces the maximumamount of FIGs that can be sustained, and thuscan crowd out investment in children within thefamily.20 These mechanisms could contribute toour understanding of some trends that havetaken place in the last few decades of the twen-tieth century: (1) an increase in the generosity ofgovernment transfers to the elderly, (2) an in-crease in measures of family disintegration, (3)a decrease in the amount of time that parentsspend with their children, (4) a decrease ineducational performance, and (5) a decrease inthe birth rate.

B. Why Is Social Security Goodfor the Environment?

Now consider the political economy of IGexchange. In this case the FIG is an IG publicgood, such as the environment or R&D, and theBIG is a pay-as-you-go social insurance pro-gram such as social security or Medicare.This application requires a slight specializa-

tion of the model. The key difference is thatnow decisions are made by majority rule, andthus many agents participate in the decision-making process. Also, to further explore thenature of BIGs, we model the social securityprogram explicitly, and consider a more realis-tic demographic structure.21 As we will see, thebasic insights remain unchanged.

19 Whether or not there is a reduction on the total level ofBIGs consumed by this type of families depends on theequilibrium that was played originally. As can be seen fromFigure 4, a minimum provision constraints eliminates somebut not all equilibria. A similar comment applies to the nextstatement.

20 Of course, this assumes that the government inter-venes only in BIGs. One could argue that the problemdisappears if the government intervenes in both BIG andFIGs. But, at best, this can only be a partial solution. Firstof all, government programs require revenue that must beraised with costly distortionary taxes. Second, these pro-grams are not likely to be tailored optimally to the specificneeds of each family. Finally, and perhaps most impor-tantly, the nature of some BIGs and FIGs is such that theycan only be provided in the context of the family. Forexample, there does not seem to be a substitute for theimpact that parents’ care and love have on the emotionaland character development of children.

21 Another reason for complicating the demographicstructure is that with voting the three-period model is aknife-edge case. It generates results that do not hold as longas agents can live for five or more periods.


Agents live for nine periods, each period rep-resenting a decade of life. They are dependentchildren in the first two periods, workers in thenext five, and retirees in the last two. There is nopopulation growth. Workers receive a wage w;everyone else has no income. Agents can bor-row and save at the constant interest rate r � 0,which implies that the economy is dynamicallyefficient.Every period society needs to choose the size

of a balanced pay-as-you-go social security sys-tem. Let Tts denote the lump-sum payroll taxpaid by workers in period t, and Bts � 5⁄2 Ttsdenote the benefits for retirees.22 Society alsochooses the level Et of expenditures in a FIGthat benefits children and future generations.The revenue for the FIG is raised using equallump-sum taxes on the workers and elderly.Preferences are now given by U(c1, ... , c9) �F({Et� k}k� �1). Note that the social securityBIG is a transfer, instead of a commodity thatenters directly in the utility function.Consider a social security system Ts �

{Tts}t�1� . The continuation value of the systemfor generation t, at age a, is given by

(24) CVta�Ts� � �k� 8

9 Bt� k�1s

�1 � r�k� a

� �k� a

7 Tt� k�1s

�1 � r�k� a ,

for a worker of age a, and

(25) CVta�Ts� � �k� a

9 Bt� k�1s

�1 � r�k� a

for a retiree of age a � 8, 9. At its nameindicates, CVta(�) measures the value for anagent of age a of keeping the social securitysystem going. The continuation value will playa role analogous to the surplus generated byBIGs in the standard model: investment in FIGs

can be sustained only as long as the continua-tion value is positive for a majority of voters inevery period.To simplify we discretize the policy space.

Let � � {(Tts, Et)}t�1� denote a path for fiscalpolicy. We assume that every period society canonly choose between two levels of the BIG, Tts� {0, Tts}, and two levels of the FIG, Et � {0,Et}.Every adult agent (a � 3) casts a vote

�as (ht) � {0, Tts} for social security, and avote, �f(ht) � {0, Et} for the FIG. ht denotesthe history of policy outcomes. Individual votesare not observable. Decisions are made by issue-by-issue direct democracy: in each dimension,the level chosen by the majority is implemented.23However, to deal with the well-known prob-lem of indifference among nonpivotal voters,we assume as-if-pivotal-voting: agents al-ways cast their vote for the option that yieldstheir preferred continuation outcome. This re-finement rules out unreasonable equilibriain which agents cast votes against theirpreferences.

PROPOSITION 7:

(1) A path � with a positive level of socialsecurity benefits can be sustained as anequilibrium as long as CVta(Ts) � 0 for alla � 6, ... , 9 and t.

(2) A path � with a positive level of investmentin FIGs can be sustained as long as thefollowing conditions are true:

(i) CVta(Ts) � (Et/7) � 0 for all a � 5, ... ,8 and t.

(ii) agents play voting strategies that link socialsecurity and FIGs.

The first part of the result provides necessaryand sufficient conditions for social security tobe sustainable by majority rule. As before, it isuseful to focus first in the case in which thereare only BIGs. For the purpose of buildingintuition consider stationary policies in whichTts � Ts for all t. Since the economy is dynam-ically efficient, the continuation value of so-cial security for sufficiently young voters is

22 The benefit formula follows from the fact that thereare five workers for every two retirees.

23 The results below hold for any political institution inwhich Condorcet winners are selected whenever they exist.


negative. Let a� (Ts) denote the smallest age atwhich the continuation value becomes positive.Workers with a � a� (Ts) always vote againstsocial security. Similarly, retirees always votefor the system since they do not have to paymore payroll taxes. This implies that social se-curity’s fate depends on the vote of the middle-aged group, with ages between a� (Ts) and 7.Since seven generations (a � 3, ... , 9) cast avote every period, the system passes with atleast four votes as long as the middle-aged votepositively and a� (Ts) � 6. Note that the middle-aged vote for social security not because theycare about current retirees, but because theycorrectly believe that otherwise they will not beable to receive benefits.The second part of the result provides neces-

sary and sufficient conditions for a positivelevel of investment in FIGs to be sustained bymajority rule. The forces at work are also sim-ilar to the ones for the basic model. First,present voters care about social security but notabout FIGs such as the environment. Thus, theyvote against the environment unless future vot-ers play a voting strategy that links BIGs andFIGs: a generation is punished in retirement if,during its voting years, society failed to providesocial security for its parents or to invest suffi-ciently in FIGs. Note, however, that the punish-ment is conditioned on the outcome of theelection, and not on the voting behavior ofparticular generations, since individual votesare not observable. Second, there is a limit tohow much investment in BIGs can be sustained.A generation is willing to vote for FIGs only ifthe taxes that it has to pay, (Et/7), are less thanthe continuation value of the system, CVta(Ts).As before, the benefits that the FIGs generate

on future generations play no role on their sus-tainability, only the direct benefits for the gen-erations making the investment do. This impliesthat programs such as the Clean Air Act, whichgenerate benefits in the short term, are morelikely to be financed than programs like globalwarming prevention, where most of the benefitsappear only in the very long run. In particular,there could be programs that generate muchlarger benefits in the long run, and have a bettersocial rate of return, but that are not producedbecause those benefits only accrue to unborngenerations.Bohn (1998) has studied the political econ-

omy of pay-as-you-go social insurance in theUnited States. He calculates the continuationvalue of social security for voters of differentages and shows that it is negative for youngvoters, but strictly positive for voters at orabove the median age. As a result, social secu-rity is sustainable and it generates surplus thatcan be used to sustain investments in the envi-ronment. This is the reason why social securitycan be good for the environment.Other public BIGs that could be used to sustain

FIGs include the choice of capital tax rates and thedecision to honor the national debt. Consider thefirst example. Every generation needs to save forretirement and can do so only if future generationsrefrain from expropriating its savings. However,every generation would like to expropriate thecurrent elderly through a 100-percent capital tax,but not to be expropriated in old age. In this caseproducing the BIG takes the form of selecting alow capital tax for the current period.Propositions 3 and 7 provide a different per-

spective on the political economy of logrolling.The prevailing view in the literature is thatlogrolling is often a cause of inefficiencies be-cause it allows inefficient “pork barrel” projectsto be enacted.24 One can think of the link be-tween BIGs and FIGs as an IG and dynamicform of logrolling in which agents who favorthe environment are willing to vote for socialsecurity, but only if current retirees invested infuture generations when they were young. Thisform of logrolling is beneficial since it is essen-tial to sustain investment in future generations.The results on mandatory provision have in-

teresting political economy implications. Con-sider a constitutional reform requiring that asufficiently large minimum level of social secu-rity benefits be paid every period unless a su-permajority votes against it. If the supermajorityrequirement is strong enough, the reform givesveto power to the elderly, who always vote forsocial security. In this case, middle-aged work-ers know that social security cannot be voteddown and thus have no incentive to invest inFIGs. Similarly, some analysts have proposedeliminating the current system and moving to asystem of personal savings accounts. If expro-priation of the balances in these accounts

24 For example, see Gordon Tullock (1998).


through taxation is very unlikely (perhaps becauseof constitutional or other legal restrictions), theretirement benefits of current workers do not de-pend on the actions of future generations. Thiseliminates the need for IG cooperation in socialsecurity, and thus a source of surplus that could beused to sustain investment in future generations.A surprising implication of the model is that

the aging of the electorate can be beneficial forfuture generations. To see this, consider a smallcomplication of the political economy model inwhich, for exogenous reasons, only a fraction ofthe population in each age group votes.25 Prop-osition 7 then changes as follows. First, a pay-as-you go social security system is sustainable aslong as it generates a positive continuation valuefor a majority of the population that actuallyvotes. Second, a positive level of FIGs can besustained as long as if CVta(T ts) � (Et /7) for amajority of the population that actually votes.Suppose, for the purposes of this example,

that the continuation value of social securitybecomes positive at age 6. If the share of agentsthat vote is constant across age groups, socialsecurity can be sustained but FIGs cannot (theyonly get three votes, those of ages 6 to 8, sincethe eldest always vote against FIGs). Comparethis with a situation in which voters in ages 6 to 9become twice more likely to vote than youngervoters in ages 3 to 5. In this case, the voters in ages6 to 8 constitute a majority and investment in FIGscan be sustained. Intuitively, any demographicchange that increases the continuation surplus ofthe “median voter” increases the amount of FIGsthat can be sustained.

VI. Concluding Remarks

This paper has studied the ability of nonmar-ket institutions, such as the government and thefamily, to invest optimally in future genera-tions. We have shown that BIGs, such as socialsecurity, play a crucial role in sustaining invest-ment in FIGs: without them, investment in FIGsis inefficiently low; with them, even optimalinvestment by selfish generations is possible.We have shown that IG organizations can con-

verge to three types of equilibria: underprovisionof BIGs and FIGs, provision of BIGs but notFIGs, and provision of both. This multiplicity ofequilibria has normative and positive implications.From a normative point of view, the multi-

plicity represents an opportunity. The link be-tween BIGs and FIGs is a mechanism that couldbe harnessed by institutional designers to sus-tain investment in future generations. Consider,for example, the introduction of a constitutionalconstraint that requires a minimal amount ofexpenditure in FIGs for every dollar spent onthe elderly. This constraint forces a link be-tween BIGs and FIGs analogous to the one thatarises in the equilibrium with positive invest-ment in future generations. As long as the re-quired amount of FIG expenditures do notexceed the surplus that the “median voter” getsfrom pay-as-you-go social insurance, the reformkills the bad equilibrium in which BIGs areprovided but FIGs are not.26 Another poten-tially useful reform would make the link be-tween BIGs and FIGs more transparent byrequiring legislation in BIGs and FIGs to bedebated and voted on together, as a package.From a positive point of view, the multiplic-

ity of equilibria calls for empirical study. Casualobservation suggests that the political processhas not coordinated to the “good equilibrium,”since social security and the environment do notseem to be linked in the public debate. How-ever, the link might play an important role inother organizations such as the family. After all,the quid pro quo nature of intergenerationalexchange is more transparent within a familythan at the social level. Parents seem to under-stand that their behavior towards their childreninfluences their emotional development andhow they are treated in old age. By contrast, itmight be more difficult for a voter to understandthat present policies could affect the votingattitudes of future generations. Nevertheless,given the lack of a solid theoretical foundationfor choosing one equilibrium over another, thebest the theory can do for now is characterizethe entire set of possible organizational out-comes, and provide guidelines for how to bringthe theory to the data.

25 Say, if agents have heterogeneous preferences for po-litical participation and/or heterogeneous costs of voting.

26 See Rangel (2002) for an institution that generates thelink using market forces.


APPENDIX

PROOF OF PROPOSITION 1:Sufficiency is obvious, now look at necessity. Consider any path � � {(bt, ft)}t�1� that can be

sustained as a subgame-perfect equilibrium. It must be the case that, for all t,

(A1) V�w� � b t � f t � � F� f t� �k� 1� � � B�b t� 1 � � V�w� � � F� f t� �k� 1� � � B�0�.

This follows because (bt, ft) must be a best response along the equilibrium path. If this inequalityis violated, generation t � 1 is better off choosing (0, 0) in history ht � ((b1, f1), ... , (bt�1,ft�1))—a contradiction.Let st(ht) � (stB(ht), stF(ht)) be the STSs associated with �. Since the STSs generate � as the

outcome path, it suffices to show that they are a subgame-perfect equilibrium. We need to considertwo types of histories.

(i) �(ht) � C. In this case, generation t � 1 plays st(ht) if and only if

V�w� � b t � f t � � F�ft� �k� 1� � � B�b t� 1 �

� arg max�b,f��bt , ft �

V�w� � b � f� � F� ft� �k� 1� � � B�st� 1B �ht , �b, f��

� V�w� � � F� ft� �k� 1� � � B�0�.

The equality follows because (0, 0) is the best possible deviation. The claim then follows from(A1).

(ii) �(ht) � P. Here, generation t � 1 plays st(ht) if and only if

V�w� � f t � � F�ft� �k� 1� � � B�b t� 1 �

� arg max�b,f��bt , ft�

V�w� � b � f� � F� ft� �k� 1� � � B�bt� 1 �stB, �b, f��

� V�w� � � F� ft� �k� 1� � � B�0�.

Once more, (0, 0) is the best possible deviation. Equation (A1) implies that the inequality issatisfied.

PROOF OF PROPOSITION 2:By Proposition 1, it suffices to show that the STSs associated with � are satisfied if and only if

(7) is satisfied. Let st(ht) � (stB(ht), stF(ht)) be the STS associated with �. For this strategy profileto be a subgame-perfect equilibrium it must be the case that, for all histories ht with �(ht) � C,

V�w� � b t � f t � � F�ft� �k� 1� � � B�b t� 1 �


V�w� � b � f� � F� ft� �k� 1� � � B�st� 1B �ht , �b, f��

� V�w� � � F� ft� �k� 1� � � B�0�.

Similarly, for the STS to be an equilibrium, it must be the case that for all histories ht with �(ht) � P,


V�w� � f t � � F�ft� �k� 1� � � B�b t� 1 �


V�w� � b � f � � F� ft� �k� 1� � � B�st� 1B �ht , �b, f ��

� V�w� � � F� ft� �k� 1� � � B�0�.

If (7) is satisfied, then these two conditions hold and the STS is a subgame-perfect equilibrium. If(7) is violated, the first condition cannot hold and the STS is not an equilibrium.

PROOF OF PROPOSITION 3:(1) Follows directly from Proposition 2.(2) Follows from part 1 plus the fact that St(�) � StB(�) whenever ft � 0. (3) B�(0) � V�(w� )implies that SstB (b) � 0 for some b � 0. The result then follows from (2).

PROOF OF PROPOSITION 5:(1) This part of the proof is very similar to the proof of Propositions 1 and 2 and thus is omitted.(2) Consider any stationary equilibrium in which BIGs and FIGs are not linked. That means that,

for all t, generation t � 1 plays a strategy of the form st(ht) � (stB(b1, ... , bt�1), stF( f1, ... ,ft�1)). Let b denote the level of BIGs generated by such a strategy. Clearly, b � bBmax, themaximum level of BIGs that can be sustained when there are only BIGs. Since generation t �1 is not affected by decisions about FIGs taken after period t, it does not care about how futureagents respond to its choice of ft. Thus, along the equilibrium path, every generation solves

maxfV�w� � b � f � � B�b� � G� f �.

Then, G�(0) � V�(w� � bBmax) implies that the equilibrium level of FIGs must be f satisfyingV�(w� � b � f) � G�( f).To conclude the proof, consider a marginal increase in the production of FIGs in every period.

The impact on the utility of generations born after period k � 1 is given by

�V��w� � b � f � � G�� f � � F�� f � � 0.

The impact on generations 0 to k � 1 is given by

�V��w� � b � f � � G�� f � � 0.

PROOF OF PROPOSITION 6:(1) Consider first the introduction of a minimum provision constraint b� � [0, w� ) on BIGs. The set

of bundles (b, f ) that can be sustained as a stationary equilibrium is given by:

��b, f ��V�w� � b � f � � B�b� � V�w� � b� � � B�b� �, f � 0, and b � b�.

(The proof of this step is almost identical to the proofs of Propositions 1 and 2 and is thereforeomitted.)This set satisfies the following properties: (1) for b� � [0, b*B], the set shrinks with b� as

depicted in Figure 4 (left-hand side); and (2) for b� � [b*B, w� ), the set equals {(b� , 0)}. Let �b� (b)denote the level of FIGs that defines the upper boundary of the equilibrium set. This boundaryis implicitly defined by V(w� � b � �) � B(b) � V(w� ) � B(b� ). By the IFT, �b� (b) is a


continuously differentiable function, and ��b��b� �B� � V�

V�. Thus, the boundary is increasing

to the left of the locus {(b, f )�B�(b, f ) � V�(b, f )}, and decreasing to the right. Given thatB�(0) � V�(w� ), this locus intersects the horizontal axis at b*B � 0 and has the shape depictedin Figure 4. The properties of f max, f min , bmax, and bmin then follow directly.

(2) Now consider the introduction of a minimum provision constraint �f � [0, w� ) on FIGs. The setof bundles (b, f ) that can be sustained as a stationary equilibrium is given by:

��b, f ��V�w� � b � f � � B�b� � V�w� � �f � � B�0�, b � 0, and f � �f �.

(The proof of this step is also almost identical to the proofs of Propositions 1 and 2 and isomitted.)Consider the equilibrium set defined by �f which is depicted in Figure 4 (right-hand side). It

is easy to see that f min( �f ) � �f, and bmin( �f ) � 0. Let ��f (b) denote the level of FIGs that definethe upper boundary of the set, which is implicitly defined by V(w� � b � f ) � B(b) �V(w� � �f ) � B(0). By the IFT, ��f(b) is a continuously differentiable function and

��f �b� �

B� � V�

V�. B�(0) � V�(0) implies that ��f (0) � 0 for all �f � [0, w� ). Also, since

��f (0) � �f, we get that, f max( �f ) � f min( �f ). Finally, as shown in the figure, since the autarkypayoff V(w� � �f ) � B(0) is strictly decreasing in f, f max( �f ) must be increasing in �f.Finally look at bmax( �f ), which is defined implicitly by the equation

V�w� � b � �f � � B�b� � V�w� � �f � � B�0�.

By the IFT, bmax( �f ) is continuously differentiable with

�bmax� �f ��f

�V��w� � bmax� �f � � �f � � V��w� � �f �

�V��w� � bmax� �f � � �f � � B��bmax� �f ��.

The sign of this derivative is equal to the sign of the denominator. But since bmax( �f ) occursto the right of the locus {(b, f )�B�(b, f ) � V�(b, f )}, the sign is negative.

PROOF OF PROPOSITION 7:(1) Let � � {(Tts, Et)}t�1� be a policy path in which there are no FIGs (Et � 0 for all t) and in

which payroll taxes are positive in every period. For every period t, let a� t(Ts) denote thesmallest age at which the continuation value of social security becomes positive. Consider thefollowing voting strategies for any period t:

(A2) �as �ht � � Tts for a � 8, 9;

(A3) �as �ht � � 0 for a � 3, ... , a� t �Ts� � 1; and

(A4) �as �ht � � � Tts if Tks � Tks for all k t or t � 1

0 otherwise for a � a� t �Ts�, ... , 7.

We claim that if a� t(Ts) � 6 for all t, these strategies are an equilibrium in which social securityis implemented every period. We need to consider two types of histories: those in which socialsecurity has always won (Tk � Tks for all k � t), and those in which it has been defeated at leastonce in the past.Consider the second type of histories first. Given (A2) to (A4), everyone knows that social


security will not exist in the future. As a result, the best response of the workers is to vote for0 payroll taxes, and the best response of the retirees is to vote for Tts.Now consider the first type of histories. Clearly, retirees must always vote as in (A2) since

they benefit from the positive payroll taxes. Voters in ages a � a� t(Ts), ... , 7 know that if socialsecurity wins, it will be there for the rest of their lives, and that if it is defeated the systemcollapses forever. Since they have a positive continuation value, they vote for Tts. For ananalogous reason, voters younger than a� t(Ts) always vote for 0. As long as a� t(Ts) � 6, Tts winswith four or more votes.

(2) Let � � {(Tts, Et)}t�1� be a policy path in which Et � 0 for all t. Let ht � {(Tks , Ek)}k�1t�1 denotethe history of public policy. Suppose that agents play voting strategies of the form �a(ht) �(�as (T1s , ... , Tt�1s ), �a

f (ht)). In this case agents know that their vote on FIGs have noeffect on how future generations care about BIGs. Since they do not benefit from future FIGs,their optimal response must be to vote for zero FIG provision. Thus, no FIGs are provided.To show that � can be sustained as long as CVta(Tts) � (Et/7) � 0 for all t and a � 5, ... ,

8 define the following strategies:

(A5) �a �ht � � �Tts, 0� for a � 9;

(A6) �a �ht � � � �Tts, Et � if Tks � Tks and Ek � Ek for all k t or t � 1�Tts, 0� otherwise for a � 8;

(A7) �a �ht � � �0, 0� for age a � 3, ... , a� t �Ts� � 1; and

(A8)

�a �ht � � � �Tts, Et � if Tks � Tks and Tkf � Tkf for all k t or t � 1�0, 0� otherwise for a � a� t �Ts�, ... , 7.

A repetition of the arguments in step 1 shows that these strategies are an equilibrium and thatthey generate positive provision of social security and FIGs along the equilibrium path.

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Copyright Notice · 2020. 7. 15. · Serra Mall, Stanford, CA 94305 (e-mail: rangel@stanford. edu). I am grateful to Alberto Alesina, Doug Bernheim, Marty Feldstein, and Eric Maskin