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

by Antonio Rangel
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Title:
Forward and Backward Intergenerational Goods: Why Is Social Security Good for the Environment?
Author:
Antonio Rangel
Year: 
2003
Publication: 
The American Economic Review
Volume: 
93
Issue: 
3
Start Page: 
813
End Page: 
834
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English
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Abstract:

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

This paper studies the ability of nonmarket institutions to invest optimally in forward intergenerational goods (FIGs), such as education and the environment, when agents are seljish or exhibit paternalistic altruism. We show that backward intergenerational goods (BIGs), such as social security, play a crucial role in sustaining investment in FIGs: without them investment is inefJiciently low, but with them optimal investment is possible. We also show that making the provision of BIGs mandatory crowds out the voluntary provision of FIGs, and that population aging can increase investment in FIGs. (JEL HO, H3, H4, H5, H6, Dl, D7).

"Why should I care about future generations? environmental preservation and pure science. What have they done for me?'(Addison) These programs entail a transfer to future gen- erations since they are financed with taxes on "Be nice to your children, they will pick your present generations and their benefits are long- nursing home." (Anonymous bumper sticker) lived. Another prominent example is the family. Every generation of parents decides how much

Every society uses a range of nonmarket in- to invest in their children. Investments include stitutions to decide how much to invest in future the cost of (public and private) education, and generations. A prominent example is the gov- the myriad of other sacrifices that parents make ernment and the decision of how much to invest for their children. on intergenerational (IG) public goods' such as These examples have a common structure.

First, IG exchange takes place in an infinitely lived organization that has an overlapping gen-

* Department of Economics, Stanford University, 579 erations (OLG) structure. Second, present generations have to decide how many resources to

edu). I am grateful to Alberto Alesina, Doug Bernheim,

to investments that dis~ro~ortionatel~

Maw Feldstein, and Eric Maskin for their suggestions and suppofi during the process of writing about intergenera- benefit future generations. Third, once the in- tional issues. 1 also want to thank two anonymous referees, vestments are made, future generations cannot James Alt, Gadi Barlevy, Tim Besley, Francesco Caselli,

be excluded from the benefits that are gener-

Andrew Cohen, Alejandro Cunat, David Cutler, Sven

ated.2 ~ ~ since future generations have ~ ~ h ,

Feldman, Drew Fudenberg, Jerry Green, Oliver Hart, James

yet been present and future genera-

Hines, Matthew Jackson, Mireille Jacobson, Chad Jones, Jonathan Katz, David Laibson, John Ledyard, Jon Levin, tions Cannot negotiate binding contracts that Jeff Liebman, Andreu Mas-Colell, Richard McKelvey, reimburse present generations for the cost of Stephen Monis, Casey Mulligan, Dina Older-Aguilar, Tor-

the investment.3 ~if~h, membership in the

sten Persson, James Poterba, Luis Rayo, Todd Sandler, Andrei Schleifer, Ken Shepsle, Joaquim Silvestre, Tomas Sjostrom, Kent Smetters, Steve Tadelis, Jaume Ventura, Richard Zeckhauser, and seminar participants at Caltech, the University of California-Davis, Harvard University, the Uni- For example, once the environmental public good has versity of Illinois-UrbanaChampaign, Iowa State University, been provided, all the members of future generations benefit NBER, Northwestern University, University of Rochester, and from it. Similarly, once children grow into adulthood, and Stanford University for their helpful comments. This paper is are in a position to compensate the parents, the investments a sigdicantly revised version of Rangel (1997). received in childhood are sunk and cannot be removed.

'See Todd Sander and Kerry Smith (1976) and Sandler In the case of the family, children are alive at the time (1978, 1982) for an early development of the concept of of the parent's investment. However, they have not been "intergenerational goods," and Joaquim Silvestre (1994, "born" as economic agents since they cannot sign binding 1995) for a more recent reference. contracts.

813

organization is not for sale: agents are born into the organization. As the following exam- ples show, the last three characteristics are central to the problem because they rule out market-based solutions.

To see the role of excludability, consider the case of publicly traded companies. Every period present generations of stockholders decide how much to invest to increase future profits. Here, the benefits generated by IG investment are excludable since the share of profits received by an agent is proportional to the amount of stock that he owns. This organization satisfies char- acteristics (4) and (3,but not (3).4The presence of a stock market induces members to internal- ize the effect of investments on future profits since they raise the price at which the stock can be 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 an organization with an overlapping generations structure in which membership has to be pur- chased. This organization satisfies all of the characteristics listed above except exogenous membership. In contrast to the case of the firm, members cannot be excluded from investments such as golf courses. We can think of these organizations as IG clubs.5 In the absence of externalities across clubs, a market solution in which club membership is a tradeable asset could 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 efficient amount of investment would take place if chil- dren and parents could sign binding contract^.^ In the absence of "transaction costs," an IG form of the Coase Theorem would arise: chil-

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 stock have a claim on the profits.

See Sandler (1982).

The "intergenerational Coasian theorem" described here requires present and future generations to be able to bargain, face to face, and sign binding contracts. The ability to bind future generations to transfer resources to present generations is not sufficient: if present generations do not have an incentive to invest in future generations without compensation, they also have an incentive to impose trans- fers on future generations without investing in them.

dren would commit to compensate their parents for the cost of the optimal investment, and par- ents would have an incentive to provide these investments. The problem is that in many orga- nizations such contracts are not possible. In the case of the family, the legal system precludes these types of contracts since children lack the independence and understanding required to evaluate them.

This paper develops a stylized model of IG exchange to study the conditions under which nonmarket institutions are able to generate Pareto-optimal levels of investment. Agents live for three periods: young, middle-aged, and old. Every period the middle-aged agent decides how much to invest in a forward intergenera- tional good (FIG) that benefits future genera- tions, but not himself. Although the role of altruism is studied in the paper, it is useful to start with the case of selfish generations.

We start with an immediate observation. If the only decision made every period is how much to invest in FIGs, no investment takes place. The intuition is straightforward. Agents benefit from investments in FIGs made by past generations, but not by investments made after they are born. Thus, they have no incentive to invest in FIGs, and no FIGs are produced. We can conclude that optimal IG investment cannot arise when the only decision made by the orga- nization is how much to invest in future generations.

Fortunately, in addition to choosing how much to invest in future generations, most non- market institutions also make decisions about backward IG exchange. For example, the gov- ernment transfers resources to the elderly through the social security system, and families take care of their elderly parents. The main insight of this paper is that the presence of backward IG goods plays a crucial role in sus- taining investment in future generations: with- out backward exchange, investment in FIGs is inefficiently low; but with it, even optimal in- vestment by selfish generations is possible. The crucial insight comes from the literature on mul- timarket contact in industrial organization [see Jonathan 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 in sustaining cooperation.

To study the relationship between forward and backward IG exchange, we analyze a styl- ized model in which, every period, the middle- aged agent makes two decisions: (1) how much to invest in a FIG, and (2) how much to buy of a backward intergenerational good (BIG) that only benefits the elderly. Using this framework, in Section I11 we show that a link between BIGS and FIGS is essential for sustaining optimal levels of investment in future generations. We also show that the need for a link between BIGS and FIGS has the following implications. First, the social rate of return, risk characteristics, and horizon of the FIGS do not affect whether or not they are financed. Second, within some limits, population aging can increase public investment in FIGS. Finally, making the provision of BIGS mandatory can crowd out investment in future generations.

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 is social security good for the environment? Here is the bottom line. If a majority of the electorate receives positive benefits from keeping the so- cial security system, there are voting equilibria in which even selfish generations vote to invest in FIGs. In these equilibria, investment in future generations is supported by a link between BIGS and FIGs: present voters correctly believe that future voters' support of social security depends on whether or not they invest in FIGs.

I. Relation with the Literature

Using the BIGS and FIGS framework, the literature on nonmarket IG organizations can be divided 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 agents simultaneously take an action that affects every one alive at the time, but has no effect on future generations. For example, in Cremer (1986), agents simultaneously choose how much effort to exert in production, and total output depends on the sum of the efforts. The main insight from this literature is that the standard "Folk Theo- rem" results extend to the OLG context: coop- eration can be sustained as long as agents are patient and/or live long enough. The last condi- tion is needed because agents in the last period of life cannot be given an incentive to cooper- ate. By contrast, the results developed in this paper are not limit results.'

The second strand of the literature studies organizations in which the exchange problem looks like a BIG. Here the organization chooses how much to produce of a good that is basically a transfer from the younger to the older gener- ations. Consider, for exam le, the "Pension

A'

Game" in Harnmond (1975), in which he stud- ies a standard OLG economy with two-period lifetimes. Agents have an endowment when young, but not when old, and have no access to a savings technology. Hammond shows that there are equilibria, similar to the one developed in Section 111,subsection B, that sustain Pareto- improving transfers from young to old in every period. To study the political economy of pay- as-you-go social security, Hansson and Stuart (1989), Henning Bohn (1998), and Cooley and Soares (1999) extend this model to a setting in which agents live for more than two periods and decisions are made by majority rule. The in- sights from all of these papers are similar to the results for BIGS in Sections I11 and V.

Two important papers in this literature are Kotlikoff et al. (1988) and David Kreps (1990), who show that the presence of a sustainable BIG can be used to solve inefficiencies in the economy. Kotlikoff et al. (1988) study a stan- dard OLG economy with two-period lifetimes in which every generation elects its own sepa- rate government. Each generational government faces a standard commitment problem: it would like to choose low capital tax rates but cannot credibly commit to do so. They show that the commitment problem can be overcome through the introduction of a self-sustainable "IG com- pact" in which every generation agrees to trans- fer a large sum to the previous generation as long as it has followed the compact and has chosen low capital tax rates for itself. Kreps

'This is also true in Peter Hammond (1975), Laurence Kotlikoff et al. (1988), Ingemar Hansson and Charles Stuart (1989). and Thomas Cooley and Jorge Soares (1999), which are described below.

See also Narayana Kocherlakota (1998).

(1990) shows that the transfers can be used to overcome moral hazard problems. As we do here, these papers show that linking the provision of a BIG with something else can be beneficial; in our case to provide investment in future genera- tions, in Kothkoff et al. and Kreps to solve an intragenerational incentive problem.

Next, several papers have studied organiza- tions in which there are only FIGs and shown that underinvestment must take place. For ex- ample, Jacobus Doeleman and Sandler (1998) study investment in IG public goods in a finite OLG model and conclude that, with selfish gen- erations, underinvestment takes place [see also Kotlikoff and Robert Rosenthal (1993) and David Collard (2000)l. By contrast, in this pa- per we study organizations in which both BIGS and FIGS are provided.

Finally, two other papers have argued that there is a link between forward and backward IG exchange. Gary Becker and Kevin Murphy (1988) suggest that it is possible to think of old age social insurance and education as a trade among generations: children receive an educa- tion from their parents and in exchange pay for their retirement benefits. However, their's is mostly an accounting argument. They do not study the sustainability of these arrangements, which is the focus of this paper. This is prob- lematic because when children grow up they can default on their obligations. Michele Bold- rin and Ana Montes (1998) have independently developed an analysis that is closely related.9 They study the majority rule politics of pay-as- you-go social security and public education using an overlapping generations economy. Al- though there are some differences in the details of the model, both papers arrive at similar in- sights. In particular, their main result is analo- gous to Proposition 3 in this paper.

11. Model

Consider a simple model of an infinitely lived organization with an overlapping generations demographic structure. Each period t a new member, 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

This paper is a revised version of Rangel (1997)

the first period, middle-aged in the second, and old 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 that period, and a middle-aged generation 0 that belongs to the organization in periods 1 and 2.

Every period t, the middle-aged generation t -1 has to make two decisions: (1) how much to invest on a FIG that only benefits generation t + k, where k r0; and (2) how much to spend on a BIG that only benefits the current old. k r 0 denotes the lag between the time the FIG is produced and the time the benefits are received. Letf,denote the investment in FIGs in period t, and b, the amount spent on BIGS. These costs are paid by generation t -1.

Every generation receives endowments w", wm, and wOof a private consumption good in each of the three stages of their lives. Agents can borrow and lend at the economy's interest rate r. The preferences of generation tare given by

where cY, cm, c0 denote their consumption when young, middle-aged, and old. In this spec- ification of the model, the young of generation t + k are the only agents who benefit from the FIG produced at time t, and there is no IG altruism. We start with this stark case for expo- sitional purposes. More complex FIGS and IG altruism are introduced later on. All the functions are twice continuously differentiable and increas- ing, Usatisfies the usual strict concavity and Inada conditions, and F and B are strictly concave.

The following notation greatly simplifies the analysis. Let

(2) V(x) = arg max U(cj,cm, cO)

C',Cm,C"

which denotes the indirect utility from the con- sumption of private goods for an agent who spends an amount of wealth x, where wealth is measured 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 chooses

b r0, andf 20),

wO where ii, = WY(1 + r) + M))I + ---(I + r) denotes

the lifetime wealth of every generation (mea- sured at middle-age). Let h, = ((b,, f,), ... , (b,-,, f, -,)) denote the history of actions taken up to period t. A strategy for generation t -1, which takes an action in period t, is a function s,(h,) = (s:(h,), sf(h,)) that specifies the amount of BIGs and FIGS purchased at any possible history. Let s = (s,, s,, ...) denote a profile of strategies for every generation. The payoff for generation t -1, conditional on history h,, is given by:

Note that the payoff of generation t -1 is affected only by a small number of the decisions taken in the organization: f, -(,+ ,,,b,, f,, and b,+ ,. We use subgame-perfect equilibrium as the solution concept.

m. Results

A. A Usefil Tool

We start the analysis by deriving aAuseful result. Given any path y = {(b,, f,)}y= "=, of BIGS and FIGs, define the following profile of simple trigger strategies (STSs):

p is defined recursively as follows: p(h,) = C and

C if~(h,-~)

= Cand (bt-195-1) = (6,-,,j1-1)

(6) p(h,) = C if p(h, -,) = P and @I-1,h-1)= (0,j-1)

P otherwise.

The idea behind STSs is straightforward. p is a flag that keeps track of whether the organization is in a cooperative or in a punishment phase. In the cooperative phase agents produced the level of BIGs and FIGs prescribed by y. In a punish- ment phase they only produce the prescribed level of FIGs. Note that if every generation plays the STS then y is the outcome of the game.

PROPOSITION 1: A path y of BIGs and FIGs can be sustained as a subgame-perfect equilib- rium if and only if it can be sustained as a subgame-perfect equilibrium using STSs.1°

This result is useful because it shows that to check if a particular path of BIGS and FIGS can be sustained, it is enough to test if it can be sustained using STSs. Note that STSs are not the only strategies that can be used. For exam- ple, if y can be sustained using STSs, then it can also be sustained using "grim strategies" in which failure to produce the prescribed level of BIGS or FIGS ends cooperation forever. We focus on STSs because they have two appealing properties. First, they are simple. Second, the punishment phase lasts for only one period, and only the generation that failed to produce the prescribed level of BIGs and FIGS is punished.

Using this tool it is easy to characterize the set of paths that can be sustained as a subgame- perfect equilibrium.

PROPOSITION 2: A path y of BIGs and FIGs can be sustained as a subgame-perfect equilib- rium if and only if

V(G)+ B(0) for all t.

lo All of the proofs are in the Appendix.

l1 Venkataraman Bhaskar (1998) shows that, in this type of overlapping generations game, the existence of cooperative equilibria depends crucially on the observ- ability of the entire history of play. In particular, no cooperation is the unique equilibrium in pure strategies when generations can observe at most the actions of the last n predecessors.

B. Characterization of Equilibria

We start with an immediate observation: no FIGSare produced in an organization in which no other decisions are made." Thus, the presence of BIGs, or other forms of exchange that will be discussed below, are essential to generate positive investments in FIGs. This section explores in de-tad the relationshp 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 only action chosen every period t is how much to spend on BIGs (f, = 0 for all t). Everything else remains unchanged. This generates a game in which {blO 5 b 5 G] is the action set for each generation.

Consider any path P = {it}:=, of BIGs, and define a continuation surplus function given by

This measures the surplus generated by the IG trade implicit in P: the first term measures the payoff for eneration t -1 of producing 6, and receiving d,+ ,,the second term measures its payoff at generational autarky where no BIGs are produced.

Note that the continuation surplus function is positive for all t if and only if the path /3 satisfies condition (7).Thus, we conclude that a path of BIGs P can be sustained as a subgarne- perfect equilibrium if and only if s:(P) 20 for all t. This characterization is very intuitive. Every generation needs to decide how many BIGs to give to the old, and in exchange it receives some BIGs from the next generation. In a STS, a generation that does not provide the prescribed amount of BIGs for the previous generation is punished by not receiving any BIGs when it becomes old. Thus, the cost of not cooperating is equivalent to returning to gener- ational autarky. This characterization says that a path P can be sustained only if the amount of BIGs received in old age outweighs, for every

l2 This can be seen by setting B(.) = 0 in condition (7), which implies that the only path that satisfies the inequality is y = ((0,O))~='=,.

generation, the cost of financing the BIGs for the 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 be sustained if and only if

If B' (0) > V' (G), s:,(-) takes positive values in the interval [0, b:"], where b,"" denotes the maximum level of BIGs that can be sustained. Furthermore, b,"" > b*,, where b*, denotes the optimal stationary level of BIGs in the model in which there are no FIGs.'~ This has two impli- cations that will be useful below. First, for any level of BIGs b E (0, b;"), the stationary path generates a positive surplus: every generation is better off producing and receiving this level of BIGs than in generational autarky. Second, in- efficient overproduction and underproduction of BIGs is possible.

Using these insights we now characterize the level of FIGs that can be sustained in the full model. For any path y define the following continuation surplus function:

This function measures the value of the exchange implicit in y over generational autarky. It is equal to the continuation surplus of the exchange in BIGs implicit in y, minus the utility cost of financing an amount of FIGs ft. Thus, S,(y) < whenever?, > 0.Note that the continuation surplus does not measure "social surplus" since it excludes the benefits generated by FIGs.

l3 hi -arg max, V(G -b) + B(b).

condition, the solution is interior and satisfies the FOCs

(14) B1(b*,)= V1(ii,-b*, -f*,) = 8F1(f$).

Furthermore. we have that

and (bz, f;) = (b*,,0).

The locus (b*,, f *,) is depicted in Figure

1.14 For small 8, (b*,, f *,) lies in the interior of the sustainable set and thus optimal production, and even inefficient overproduction, are possi- ble. As 6 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: InefJicient overproduction of FIGs can be sustained as a stationary equi- librium if and only if S(b*, f*) > 0. In this case, any level of FIGs f E [O, max, f,,(b)], including the optimal stationary level f *, can be sustained as a stationary equilibrium.

As Figure 1 starkly illustrates, the model gen- erates a large number of equilibria: in some of them IG cooperation takes place, in others it does not. This is common in models of long- lived institutions that use noncooperative game theory. The equilibrium set could be reduced by imposing equilibrium refinements like Markovian equilibrium or renegotiation proofness. However, given that so far game theory has not provided a fully satisfactory justification for such refinements, we proceed by characterizing the entire equilibrium set. Future theoretical de- velopments might be able to identify variables that affect the coordination of expectations across generations, and thus rule out some of the equilibria.

We conclude this section with a comparison of forward and backward IG exchange. From a technological point of view, BIGS and FIGS are not that different. Both types of exchange re-

l4 It is given by the line B'(.) = v'(.).

quire agents to provide a good that is valuable for another generation, and in exchange benefit from a good that is provided for them. However, from an incentive point of view, they are sig- nificantly different. First, even when BIGS and FIGs generate identical benefits [i.e., when B(.) = F(.)],BIGS can be sustained in organi- zations that only make this type of decisions, but FIGS cannot be sustained in isolation. A positive level of investment in FIGs can arise only by linking them with BIGs. Second, the cost and benefits of BIGS are crucial in deter- mining the level of BIGs and FIGs that can be sustained. By contrast, the benefits generated by FIGs play no role. Third, the optimal stationary level of BIGs is always sustainable. This is not the case for FIGs.

C. What Type of FIGs Can Be Sustained?

The basic model ignores some important properties of FIGs. For example, environmental programs often generate benefits for multiple generations, including those making the invest- ment, and have uncertain returns. In this section we 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 FIG for which the preferences of generation t are given by

where o,is a random shock realized at the beginning of period t. In this case, the FIG produced at time t can benefit every future generation, including generation t.

It is straightforward to see that the path of sustainable FIGS has not changed. Given that FIGS are nonexcludable, past investments in FIGS do not affect the incentive constraint of the decision maker. As a result, the characterization provided in Propositions 3 and 4 still holds, and the set of stationary equilibrium outcomes is still the one depicted in Figure 1. The lesson is clear: risks, lags, and multiple beneficiaries affect the optimal level of investment in FIGs, but not the level that can be sustained.

Thus, when the surplus generated by BIGs is large relative to the value of FIGs, there are pathological equilibria in which no generation invests in FIGs even though they benefit di- rectly from those investments.

D. The Role of Altruism

Another concern with the basic model is the assumption of selfish generations. The effect of altruism on the analysis depends on the specific form that it takes. In particular, one needs to distinguish between paternalistic and nonpaternalistic altruism. With paternal- istic altruism, the level of FIGs that future generations consume enters as an argument in the utility function of present generations. With nonpaternalistic altruism, it is the utility level of future generations that is internalized by present generations.

The analysis of paternalistic altruism is equivalent to the case discussed at the end of the previous section. The only difference is that the function G is now interpreted as altruism, in- stead of a direct benefit from consuming the FIG. Therefore, in this case a link between BIGs

E. Investment in Future Generations
Without BIGs

This section shows that BIGs are not the only type of exchange that can be used to sustain FIGs. To see this, consider the following gen- eralization of the model. Every period t, generation t -1 makes two choices: (1) how much to invest in FIGs, just like before, and (2) how many resources to spend in another good. De- note the second decision by e,. The preferences of generation t -1 are now given by

Note that a generation could be affected by all the choices made in the non-FIG dimension after it is born. This includes, as special cases, the previous model of BIGs, and the case in which the non-FIG dimension only affects those alive at the time.

Let ea" denote the action taken in genera- tional autarky, and e* the optimal stationary action. Suppose that ea" # e* so that cooper- ation is required to sustain e*. For any path & =

and FIGs is still needed to generate optimal levels 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 exchange problem. The organization behaves like an infinitely lived agent that perfectly internal- izes the future spillovers of investing in FIGs. Thus, in applications for which the dynastic model is a good description of behavioral motives, the issues studied in this paper do not arise.

Casual evidence suggests that IG altruism is at work in most of the applications of this model: voters care about the future of human- ity and parents care about their children. However, the key question is which is the form that their altruism takes. Although a lot of work remains to be do done in this area, and a con- sensus does not exist yet, some existing evi- dence suggests paternalistic altruism might be a better approximation. l5

IS See Joseph Altonji et al. (1992, 1997).

define the continuation surplus function :=8,)(

that measures the value of the allocation E over generational autarky. It is straightforward to extend the previous arguments to show that a positive amount of investment in FIGs can be sustained if and only if there exists a path E such that SF(&) > 0 for all t.

The intuition is simple. The central feature of FIGs is that present generations do not care about how many FIGs are produced by future decision makers. This is what makes FIGs dif- ficult to sustain, and the reason a link is needed. If there exists another dimension of exchange in the organization that requires cooperation, and if the cooperative allocation generates a positive continuation surplus, then that surplus can be used to sustain investment in FIGS using strat- egies that link FIGs and non-FIGS. It does not matter what the other dimension of exchange is as long as: (1) it requires IG cooperation, and

(2) it generates a positive continuation surplus. For this to be the case, present decision makers must care about future decisions.

This argument can be pushed even further. In organizations in which more than two decisions are made, say if there are several BIGs, it is possible to link several of these decisions to FIGs. If each individual BIG generates a sur- plus, each additional BIG provides additional incentives to provide FIGs. In the context of the political economy of IG public goods discussed in Section V, pay-as-you-go social insurance, the choice of capital tax rates, and the decision to honor the national debt can be used simulta- neously to sustain investment in IG public goods.

IV. The Effect of Mandatory Provision

The previous analysis has shown that there are always equilibria in which FIGs and/or BIGs are not provided. A natural question to ask is whether the introduction of minimal provi- sion constraints can improve the outcomes gen- erated by these organizations. These types of constraints are common. For example, tax-financed public education places a constraint on the minimum amount of educational expendi- tures that a parent can give to his child, and mandatory old age social insurance im oses a similar constraint for the case of BIGs. P6

In this section we study the effect that mini- mal provision constraints have on voluntary provision. Denote the minimum constraint by (b,f). These constraints have no effect on the payoffs, but shrink the action sets to {(b, f)lb + f 5 w, f rf, and b r b}.

To characterize thi equilibrium set for this case we need to define a new continuation sur- plus function. Let y be any path of BIGs and FIGS satisfying the constraints and define

l6 Another natural institution is a constitutional con- straint requiring the provision of exactly the optimal level of BIGs and FIGS in every period. This institution, how- ever, does not work well when there is considerable uncertainty about future parameters. For example, what will be the optimal amount of R&D in nanotechnology in

2050?

It is straightforward to extend the arguments in Propositions 1 and 2 to show that a path y can be sustained as an equilibrium if and only if slbf)(y) r 0 for all t. This provides a full characterization of the equilibrium set. The only difference is that now the payoff of generational autarky has changed to V(% -f -b) + B(b), which has implications for thelevels of BIGs and FIGs that can be sustained.

Letf m"(b, f),fmin(b,f),brn"(b, f 1, and bm"(b, f) denote, respectively, the maximum and rnhhum level of FIGS and BIGs that can be sustained as a stationary equilibrium in aninstitution with constraints (b,f ). The following propo- sition describes the effkt of (1) introducing a minimum constraint only on BIGs (f = O), and (2) introducing a minimum constraint-only on FIGS (b = 0).The effect of introducing a constraint in both goods is discussed below.

PROPOSITION 6: Suppose that B1(0) > V1(0), then: l7

(I)

The introduction of a minimum provision constraint b E [O, G) on BIGs has the following effects: f m"(b) decreases as b increases between 0 and b*, and equals 0 aferwards, f min(b)= 0 for all b, bm"(b) decreases between 0 and b*, and equals b aferwards, and b"'"(b) = b.

(2)

The introduction of a minimum provision constraint f E [O, w) on FIGs has the following Zffects: f f) is increasing, f -( f) = f, bm"( f) ii decreasing, and b"'"(7)-= -0.

Figure 3 summarizes the effects described in this result. Figures 4 (left-hand side) depicts the effect of b on the equilibrium set. For b E [0, b*,] the set shrinks towards (b*,, 0) as b increases, bmin(.)increases, and bm"(-) decreases. The intuition for the latter effect follows from

l7 This condition can be relaxed to B'(0) > V1(G). But in that case there may exist 3such that, for all f >3,f"(f) = fh(f) and bm"(f) = bd"(f) = 0:

the fact that the payoff at generational autarky, which is the worse feasible punishment for a generation that fails to cooperate, increases in this range. This reduces the incentives to coop- erate. At b = b*,, the equilibrium set shrinks to a single point: (b*,, 0). Further increases in b generate the equilibrium set { (b, 0) ). Since the BIG 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 reaches b*,, the BIGs generate no surplus and thus no FIGs 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 provision constraint in BIGs eliminates the possibility of the bad equilibrium in which no BIGS are pro- duced, but it also reduces the maximum level of investment in future generations that can be sustained."

The constraint for FIGs f has a different ef- fect. Increases in f have a positive effect on the maximum and minimum level of FIGs that can be sustained, and a negative effect on the max- imum amount of BIGs. The result is also driven by the effect off on generational autarky pay- off, given by V(G -f) + B(O), which is decreasing in f. As a result, as shown in Figure 4 (right-hand side), bundles (b, f) that were not sustainable before, now become sustainable.

Now consider the effect of introducing a min- imum provision constraint in both FIGs and BIGs. Since the analysis is very similar to the previous two cases, the details are omitted. Once more, the impact on FIG provision de- pends on the payoff at generational autarky. If V(G -b -f)+ B(b) > V(G) + B(0) the equilibrium sit with constraints is a subset of the equilibrium set for the case of no constraints, and is similar to the one depicted in Figure 4 (left-hand side) except for one minor change: the points (b, f) for which f <f-have

l8 Bernheim and Whinston (1998) obtain results with a similar flavor for nonintergenerational contracting prob- lems. In many economic relationships complete contracts are impossible. In this case, voluntary cooperation is needed in the dimensions where the contract is incomplete. In this context, they show that it might be advantageous to pur- posefully leave some dimensions out of the contract to increase the incentive to cooperate in the dimensions that cannot be included.

to be removed. As a result, the constraint de- creases the maximum level of FIGs that can be sustained. By contrast, if V(G -b -f) + B(b) < V(G) + B(O), the equilibrium set resembles the one depicted in Figure 4 (righthand side) with one minor change: the points (b, f) for which b < b have to be removed from the set. In this case, the maximum level of FIGs 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 out voluntary cooperation in BIGs and FIGs, in- creases in the constraint for FIGs increase the total amount of FIGs that can be sustained. This asymmetry is interesting because, to the extent that the minimal provision constraints are de- termined endogenously, present generations have 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 a theory of IG exchange within a family that is either selfish or exhibits paternalistic altruism. Families exchange two types of IG goods: (1) FIGs, that are provided by parents to their young children in the form of education and parental care, and (2) BIGs, that adult children provide to their aging parents in the form of care, insurance, and status. In this interpretation k = 0.

As shown in Section 111, subsections B and D, in a selfish family investment in children can be positive only if there is a link between BIGs and FIGs, and in families with paternalistic altruism the link is required to generate investments in excess of what parents are willing to invest on their own. For example, without BIGs, a parent might 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 by strategic considerations: parents believe that (excess) investments in FIGs are the price that they pay for getting the BIGs that they desire in old age.

The model predicts the existence of three types of families. First, families that link BIGs and FIGS sustain a high level of investment in both. Second, gerontocratic families with high levels 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 variation within and across cultures. Endogenous prefer- ences are likely to be an important part of the explanation, specially if culture and institutions influence the amount of altruism within fami- lies. But family norms might also play an im- portant role: some societies and families converge to cooperative codes of behavior, oth- ers do not.

As shown in Section 111,in order for FIGS to be provided, there must be a BIG that generates a positive surplus. This is guaranteed as long as the elderly place a marginal value on the first unit of the BIG that exceeds the marginal cost for the middle-aged. Although in our stylized model this is imposed as an assumption [condi- tion B1(0)> Vr(%)in Proposition 31, several of the BIGs exchanged within families satisfy these characteristics. Consider, for example, the case of insurance. Retirees face risks that are not insurable through financial markets such as a collapse of the stock market, or a crime that significantly reduces their wealth. As long as the serial correlation of the adverse shocks is low, there are gains from exchange between the middle-age and the elderly. Kotlikoff and Avia Spivak (1981) show that a similar argument holds for the provision of "annuity insurance" within the family when annuity markets are imperfect.

The results on mandatory provision also have interesting implications for the family. Suppose that the government introduces a law that forces the middle-aged to provide a BIG that they used to provide voluntarily. This would be the case, for example, if the good is financed with taxes on the middle-aged. This has three types of effects (see Figure 3). First, it increases the minimal amount of BIGs that any elderly person receives. If a fraction of the families in the economy are in a bad equilibrium in which BIGs are not provided, the policy improves the welfare of these elderly. Second, it decreases the maximum amount of voluntary provision of BIGS that can be sustained. Thus, if the public program is not large enough to fully crowd out family care, the welfare of the elderly who belong to families that are in a good equilibrium can go down. l9 Finally, it reduces the maximum amount of FIGS that can be sustained, and thus can crowd out investment in children within the family.20 These mechanisms could contribute to our understanding of some trends that have taken place in the last few decades of the twen- tieth century: (1) an increase in the generosity of government transfers to the elderly, (2) an in- crease in measures of family disintegration, (3) a decrease in the amount of time that parents spend with their children, (4) a decrease in educational performance, and (5) a decrease in the birth rate.

B. Why Is Social Security Good for the Environment?

Now consider the political economy of IG exchange. In this case the FIG is an IG public good, such as the environment or R&D, and the BIG 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 that now decisions are made by majority rule, and thus many agents participate in the decision- making process. Also, to further explore the nature of BIGs, we model the social security program explicitly, and consider a more realis- tic demographic structure.'' As we will see, the basic insights remain unchanged.

l9 Whether or not there is a reduction on the total level of BIGs consumed by this type of families depends on the equilibrium that was played originally. As can be seen from Figure 4, a minimum provision constraints eliminates some but not all equilibria. A similar comment applies to the next statement.

20 Of course, this assumes that the government inter- venes only in BIGs. One could argue that the problem disappears if the government intervenes in both BIG and FIGS. But, at best, this can only be a partial solution. First of all, government programs require revenue that must be raised with costly distortionary taxes. Second, these pro- grams are not likely to be tailored optimally to the specific needs of each family. Finally, and perhaps most impor- tantly, the nature of some BIGS and FIGS is such that they can only be provided in the context of the family. For example, there does not seem to be a substitute for the impact that parents' care and love have on the emotional and character development of children.

21 Another reason for complicating the demographic structure is that with voting the three-period model is a knife-edge case. It generates results that do not hold as long as agents can live for five or more periods.

negative. Let ii(T) denote the smallest age at which the continuation value becomes positive. Workers with a < ii(T")always vote against social security. Similarly, retirees always vote for the system since they do not have to pay more payroll taxes. This implies that social se- curity's fate depends on the vote of the middle- aged group, with ages between ii(T") and 7. Since seven generations (a = 3, ... , 9) cast a vote every period, the system passes with at least four votes as long as the middle-aged vote positively and ii(Ts)5 6. Note that the middle- aged vote for social security not because they care about current retirees, but because they correctly believe that otherwise they will not be able to receive benefits.

The second part of the result provides neces- sary and sufficient conditions for a positive level of investment in FIGs to be sustained by majority rule. The forces at work are also sim- ilar to the ones for the basic model. First, present voters care about social security but not about FIGS such as the environment. Thus, they vote against the environment unless future vot- ers play a voting strategy that links BIGS and FIGs: a generation is punished in retirement if, during its voting years, society failed to provide social security for its parents or to invest suffi- ciently in FIGs. Note, however, that the punish- ment is conditioned on the outcome of the election, and not on the voting behavior of particular generations, since individual votes are not observable. Second, there is a limit to how much investment in BIGs can be sustained. A generation is willing to vote for FIGs only if the taxes that it has to pay, (~,/7),

are less than the continuation value of the system, CV:(~).

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 implies that programs such as the Clean Air Act, which generate benefits in the short term, are more likely to be financed than programs like global warming prevention, where most of the benefits appear only in the very long run. In particular, there could be programs that generate much larger benefits in the long run, and have a better social rate of return, but that are not produced because those benefits only accrue to unborn generations.

Bohn (1998)has studied the political econ- omy of pay-as-you-go social insurance in the United States. He calculates the continuation value of social securitv for voters of different ages and shows that it is negative for young voters, but strictly positive for voters at or above the median age. As a result, social secu- rity is sustainable and it generates surplus that can be used to sustain investments in the envi- ronment. This is the reason why social security can be good for the environment.

Other public BIGs that could be used to sustain FIGS include the choice of capital tax rates and the decision to honor the national debt. Consider the first example. Every generation needs to save for retirement and can do so only if future generations refram from expropriating its savings. However, every generation would like to expropriate the current elderly through a 100-percent capital tax, but not to be expropriated in old age. In this case producing the BIG takes the form of selecting a low 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 that logrolling is often a cause of inefficiencies be- cause it allows inefficient "pork barrel" projects to be enacted.24 One can think of the link be- tween BIGS and FIGS as an IG and dynamic form of logrolling in which agents who favor the environment are willing to vote for social security, but only if current retirees invested in future generations when they were young. This form 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 a sufficiently large minimum level of social secu- rity benefits be paid every period unless a su- permajority votes against it. If the supermajority requirement is strong enough, the reform gives veto power to the elderly, who always vote for social security. In this case, middle-aged work- ers know that social security cannot be voted down and thus have no incentive to invest in FIGs. Similarly, some analysts have proposed eliminating the current system and moving to a system 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 because of constitutional or other legal restrictions), the retirement benefits of current workers do not de- pend on the actions of future generations. This eliminates the need for IG cooperation in social security, and thus a source of surplus that could be used to sustain investment in future generations.

A surprising implication of the model is that the aging of the electorate can be beneficial for future generations. To see this, consider a small complication of the political economy model in which, for exogenous reasons, only a fraction of the 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 as long as it generates a positive continuation value for a majority of the population that actually votes. Second, a positive level of FIGs can be sustained as long as if CV;(~)r (ktn)for a majority of the population that actually votes.

Suppose, for the purposes of this example, that the continuation value of social security becomes positive at age 6. If the share of agents that vote is constant across age groups, social security can be sustained but FIGS cannot (they only get three votes, those of ages 6 to 8, since the eldest always vote against FIGs). Compare this with a situation in which voters in ages 6 to 9 become twice more likely to vote than younger voters in ages 3 to 5. In this case, the voters in ages 6 to 8 constitute a majority and investment in FIGS can be sustained. Intuitively, any demographic change that increases the continuation surplus of the "median voter" increases the amount of FIGs that can be sustained.

VI. Concluding Remarks

This paper has studied the ability of nonrnar- ket institutions, such as the government and the family, to invest optimally in future genera- tions. We have shown that BIGs, such as social security, play a crucial role in sustaining invest- ment in FIGs: without them, investment in FIGs is inefficiently low; with them, even optimal investment by selfish generations is possible.

We have shown that IG organizations can con-

25 Say, if agents have heterogeneous preferences for political participation andlor heterogeneous costs of voting.

verge to three types of equilibria: underprovision of BIGS and FIGs, provision of BIGs but not FIGS, and provision of both. This multiplicity of equilibria 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 could be harnessed by institutional designers to sus- tain investment in future generations. Consider, for example, the introduction of a constitutional constraint that requires a minimal amount of expenditure in FIGs for every dollar spent on the elderly. This constraint forces a link be- tween BIGS and FIGs analogous to the one that arises in the equilibrium with positive invest- ment in future generations. As long as the re- quired amount of FIG expenditures do not exceed the surplus that the "median voter" gets from pay-as-you-go social insurance, the reform kills the bad equilibrium in which BIGS are provided but FIGs are not.26 Another poten- tially useful reform would make the link be- tween BIGS and FIGS more transparent by requiring legislation in BIGS and FIGS to be debated and voted on together, as a package.

From a positive point of view, the multiplic- ity of equilibria calls for empirical study. Casual observation suggests that the political process has not coordinated to the "good equilibrium," since social security and the environment do not seem to be linked in the public debate. How- ever, the link might play an important role in other organizations such as the family. After all, the quid pro quo nature of intergenerational exchange is more transparent within a family than at the social level. Parents seem to under- stand that their behavior towards their children influences their emotional development and how they are treated in old age. By contrast, it might be more difficult for a voter to understand that present policies could affect the voting attitudes of future generations. Nevertheless, given the lack of a solid theoretical foundation for choosing one equilibrium over another, the best the theory can do for now is characterize the entire set of possible organizational out- comes, and provide guidelines for how to bring the theory to the data.

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

PROOF OF PROPOSITION 1:

Sufficiency is obvious, now look at necessity. Consider any path y

=

( (if,

f,) }

:=

that can be

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

This follows because (it,ff) must be a best response along the equilibrium pa$. Ifnthis ineqqality is violated, generation t -1 is better off choosing (0, 0) in history h, = ((b,,f,),... , (b,-, , f,-,))-a contradiction.

Let s,(h,) = (s:(h,), s:(h,)) be the STSs associated with y. Since the STSs generate y as the outcome path, it suffices to show that they are a subgame-perfect equilibrium. We need to consider two types of histories.

(i) p(h,) = C. In this case, generation t -1 plays s,(h,) if and only if

2 argmax V(G-b -fl + Fur-(k+l))+ B(sLl(h,, (b,fl>)

(b,fJ+(6r,ft)

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

(A1).

(ii) p(h,) = P. Here, generation t -1 plays s,(h,) if and only if

2 arg max V(G-b -fl + FV,-(k+ + B(b,+1 (s:, (b, fl))

(b,fJ#(&,fr)

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

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

(7) is satisfied. Let s,(h,) = (sf(h,), s:(h,)) be the STS associated with y. For this strategy profile to be a subgame-perfect equilibrium it must be the case that, for all histories h, with p(h,) = C,

L arg max V(iD-b -fl + Fur-(k+1)) + B(sL,(Af, (b,A)) (bJI'(6,,?,)

Similarly, for the STS to be an equilibrium, it must be the case that for all histories h, with p(h,) = P, 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.

(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, gensration t -1 plays a strategy of the form s,(h,) = (s:(bl, ... , b,,), $6, ... , ft-,)). Let b denote the level of BIGS generated by such a strategy. Clearly, b 5 bTax, the maximum 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 future agents respond to its choice off,. Thus, along the equilibrium path, every generation solves

(2)

Follows from Bart 1 plus the fact that S,(y) < s:(~) whenever?, > 0. (3) Br(0) > Vr(G) implies that S,,(b) > 0 for some b > 0. The result then follows from (2).

PROOF OF PROPOSITION 5:

max V(i4 -6-f) + ~(6)+ GCf).

Then, Gr(0) > Vr(i4 -b;") implies that the equilibrium level of FIGS must be f satisfying V'(G -6 -3 =

GI@.

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

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

PROOF OF PROPOSITION 6:

(1) Consider first the introduction of a minimum provision constraint b E [0, 6) on BIGs. The set of bundles (b, f) that can be sustained as a stationary equilibrium is given by:

{(b, f)I~(w -b -f) + B(b) r V(i4 -b) + B(b), fro, andbrb).

(The proof of this step is almost identical to the proofs of Propositions 1 and 2 and is therefore omitted.)

This set satisfies the following properties: (1) for b E [0, b*,], the set shrinks with b as depicted in Figure 4 (left-hand side); and (2) forb E [b*,, w), the set equals {(b, 0)). Let +,(b) denote the level of FIGS that defines the upper boundary of the equilibrium set. This boundary is implicitly defined by V(iG -b -4) + B(b) = V(G) + B(b). By the IFT, +,(b) is a

B' -V' continuously differentiable function, and +bib) = ----. Thus, the boundary is increasing

v' to the left of the locus {(b,f )l~'(b,f) = V1(b,f )}, and decreasing to the right. Given that B1(0)> V1(@),this locus intersects the horizontal axis at b*, > 0 and has the shape depicted in Figure 4. The properties off "", f bm", and bminthen follow directly.

(2) Now consider the introduction of a minimum provision constraint f E [0,6)on FIGS. The set of bundles (b, f) that can be sustained as a stationary equilibrium is given by:

(The proof of this step is also almost identical to the proofs of Propositions 1 and 2 and is omitted.)

Consider the equilibrium set defined by f which is depicted in Figure 4 (right-hand side). It is easy to see that f m'n( f) =f, and bmin( f j = 0. Let +,(b) denote the level of FIGS that define the upper boundary of the set, which is implicitly defined by V(G -b -f) + B(b) = V(G -f) + B(0). By the IFT, +&b) is a continuously differentiable function and

YR' -17'

4) = 7' . B1(0)> V1(0)implies that cbj(0) > 0 for all f E [0, G). Also, since @(O)=J we get that, f f) > f m'n( f ). Finally, as shown in the figure, since the autarky payoff V(G -f) + B(0) is strictly decreasing in f, f """( f ) must be increasing in f.

-

Finally look-at bmax(f ), which is defined implicitly by ?he equation

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

The sign of this derivative is equal to the sign of the denominator. But since bmax(f) occurs

-

to the right of the locus {(b, f )l~'(b, f) = V1(b,f )), the sign is negative.

PROOF OF PROPOSITION 7:

(1) Let y = {(c,Pt)};"=,be a policy path in which there are no FIGS (E, = 0 for all t) and in which payroll taxes are positive in every period. For every period t, let Z,(T") denote the smallest age at which the continuation value of social security becomes positive. Consider the following voting strategies for any period t:

(A2) ai(h,) = for a = 8, 9;

(A3) a;l(h,)= 0 for a = 3, ... , Z,(T)-1; and

if T;. = c.for all k < t or t = 1

for a = Z,(T), ... , 7

(A41

We claim that if Z,(T")5 6 for all t, these strategies are an equilibrium in which social security is implemented every period. W: need to consider two types of histories: those in which social security has always won (T, = TL for all k < t),and those in which it has been defeated at least once 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 for 0 payroll taxes, and the best response of the retirees is to vote for c.

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 = Zi,(T),... ,7 know that if social security wins, it will be there for the rest of their lives, and that if it is defeated the system collapses forever. Since they have a positive continuation value, they vote for c. For an analogous reason, voters younger than ii,('l'F) always vote for 0. As long as ii,(P) 5 6, wins with four or more votes.

(2) Let y

Let h, t.0 for all >I?,a policy path in which & }:=(c, fit) {=

= { (T",, E,) }&',

denote

the history of public policy. Suppose that agents play voting strategies of the form u,(h,) = (<(G, ... , c-,),af,(h,)). In this case agents know that their vote on FIGS have no effect 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 y can be sustained as long as CT(~) r (fi47) r 0 for all t and a = 5, ... , 8 define the following strategies:

(-45) ua(h,) = (c, 0) for a = 9;

(c, I?,) if z = fi and E, = I?, for all k < t or t = 1

for a = 8;(A6) ua(hf)= {(c, 0) otherwise

(-47) ua(h,) = (0, 0) for age a = 3, ... , ii,(T") -1; and

, ) if T;= cand fi= qforall k< t or t = 1 for a = iif(Ts)),... , 7(0, 0) otherwise

A repetition of the arguments in step 1 shows that these strategies are an equilibrium and that

they generate positive provision of social security and FIGS along the equilibrium path.

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