BACKWARD INTERGENERATIONAL GOODS AND ENDOGENOUS FERTILITY JOHN WILLIAM HATFIELD

Stanford University

Abstract This paper characterizes the consequences of introducing the public provision of intergenerational goods to the elderly in a model with endogenous fertility. With exogenous fertility, it has been shown that the government can mandate the first-best outcome by simply imposing the socially optimal transfer. By contrast, with endogenous fertility, the government can no longer enforce this outcome. This is due, in part, to the effects of mandatory provision on the birth rate. However, taxes may still have a salubrious effect on social welfare as they can eliminate particularly bad equilibria.

1. Introduction The proper level of social insurance and provision for the elderly is an important policy question, the answer to which depends on the fertility decisions made by those selfsame elderly.1 While models that have taken fertility as given have shown that the government can mandate the first-best, using only nondistortionary lump sum taxes, we find that if agents are allowed to make John William Hatfield, Graduate School of Business, Stanford University, 518 Memorial Way, Stanford, CA 94305, U.S.A. ([email protected]). I am grateful to Antonio Rangel and John Shoven for their suggestions and support during the process of writing this paper. I am also obliged to Jonathan Adams, Douglas Bernheim, Katherine Carman, Eric Hanushek, Benjamin Malin, Amalia Miller, Sita Nataraj, Azeem Shaikh, and an anonymous referee for their helpful comments and discussion. Received August 8, 2005; Accepted January 31, 2008. 1 Perhaps the best example of this type of good is risk sharing over uncertain outcomes, where asymmetric information precludes the existence of a private insurance market. The two largest uncertainties for the elderly are the cost of their health care and their length of life, and usually these two represent the largest expenditures on the elderly by the government through pension and health insurance programs. These issues have been discussed at length; see, for instance, Enders and Lapan (1982).  C 2008 Wiley Periodicals, Inc. Journal of Public Economic Theory, 10 (5), 2008, pp. 765–784.

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decisions about the number of children, this is no longer possible. In particular, provision of backward intergenerational goods by government mandate engenders an intragenerational externality, since the number of children an agent chooses to have affects all the members of his generation, since these children pay taxes for the public provision of social insurance for the elderly. This additional externality makes it impossible for a beneficent government to enforce the stationary first-best outcome without access to other instruments than a lump sum tax. In particular, mandating the first-best provision will result in a birth rate that is less than socially optimal. Many papers have considered the relationship between public and private intergenerational transfers, without considering the effect of social insurance on fertility. First and most notably, Samuelson (1958) shows that instituting a system of fixed backward intergenerational transfers may make each generation better off. Later, Hammond (1975) shows the existence of equilibria in infinitely lived economies where agents voluntarily make backward transfers, as by doing so they expect to receive transfers themselves. The most salient of these types of models for our purposes is Rangel (2002). Here, Rangel shows that in order to sustain positive levels of forward intergenerational goods in equilibrium, the equilibrium must also include positive (and welfare improving) transfers of backward intergenerational goods, or BIGs. The intuition for this result is straightforward: by linking the receivership of BIGs to providing both types of goods, positive levels of both can be sustained in equilibrium. We will use a variation of this insight and link the receivership of BIGs to one’s fertility choice. Indeed, the original insight is that by linking games where cooperation can and cannot be sustained in equilibrium, we may be able to induce more cooperation than if considering each game distinctly. 2 By endogenizing fertility choice by individuals, we obtain fundamentally different results. When fertility is a fixed parameter, the models referenced above generally have a continuum of equilibria with different welfare consequences, but if an outside entity (i.e., government) can compel transfers, it can reduce the set of stationary equilibria to only the first-best. The main contribution of this paper is to show, when fertility is endogenous, that the first-best is no longer enforceable by the government and that the policy maker must rely on private transfers in order to reach the stationary social optimum. In this case, the use of a lump sum tax to publicly provide BIGs creates a fiscal externality between members of the same generation, in that my fertility decision affects other members of my generation by affecting the number of people taxed to provide BIGs to them. Hence, the lump sum tax by itself is an insufficient instrument to achieve the first-best. If the government mandates the first-best intergenerational transfer, there are strong incentives for agents to have less than the optimal number of children.

2

See Bernheim and Whinston (1990) and Bendor and Mookherjee (1990).

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However, this does not mean there is no reason for a social insurance system, even though the first-best is attainable if no such insurance is provided. We show that the policy maker may be able to influence welfare by ensuring that particularly bad outcomes are no longer possible in equilibrium, while still preserving the first-best. Thus, in this paper, we characterize the set of stationary equilibria as a function of the tax rate. We are not the first to note that a national scheme for backward intergenerational transfers may lower the fertility rate. A number of papers have studied endogenous fertility choices in this context. In Barro and Becker (1988, 1989) and Razin and Ben-Zion (1975), agents are assumed to be maximizing a “dynastic utility function”; utility is determined by not only the agent’s consumption, but the utility of his children as well. In our work, an agent’s utility (which may include a paternalistic component that derives utility from having children) is directly affected by his fertility choices, not by the utility of his ancestors or descendants. Other more closely related work, such as Nishimura and Zhang (1992), Wigger (1992), Rosati (1996), and Boldrin and Jones (2002), has considered this question as well, but with an additional crucial assumption of altruism toward one’s parents (which drives intergenerational giving) or the enforcement of intergenerational contracts; they also do not allow decisions by children to be in any way dependent on the fertility decisions of their parents. This work does not make these assumptions. Rather, we look for subgame perfect equilibria, where children may condition their actions on those of their parents, and choose optimally given this information and their correct beliefs about how their children will act. Furthermore, we obtain different results from these earlier works, showing that a small pay-as-you-go system may not have an unfortunate effect on fertility, and furthermore may improve welfare, but a system that is too large will preclude the first-best from being an equilibrium. To study the relationship between fertility and the provision of intergenerational goods, we develop a formal model in Section 2. Section 3 presents the analysis of the benchmark case of exogenous fertility. Section 4 presents the main results. Section 5 concludes. All proofs may be found in the Appendix.

2. Model In order to compare our work more easily with the existing literature, we will extend the model of Rangel (2002) to the case where agents make fertility decisions along with their decisions about intergenerational transfers. Consider an overlapping generations economy where each agent lives for three periods. Each generation consists of a continuum of identical agents. An agent of generation t is young at time t − 1, middle-aged at time t, and old at time t + 1. At time 1 there is a middle-aged generation that lives for only two periods, as well as an old generation that lives for only that period. Each agent is endowed with w units of a storable consumption good upon reaching middle age. In middle age, the agent makes two decisions: first, how

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many backward intergenerational goods (BIGs) to privately provide for his parents, g t , and second, how many children to have, nt . The endowment w is used for consumption in time periods t and t + 1, as well as the provision of intergenerational goods and the care of children. (c τt is the amount consumed by the agent born at time t − 1 in time period τ .) These children make up generation t + 1. In addition, there is a pay-as-you-go social insurance system, financed by a lump-sum tax T t . Since agents are simply endowed with their wealth, there is no loss of generality in using lump-sum taxes; we use nondistortionary taxes in order to abstract from the usual inefficiency arguments due to taxation. Thus the budget constraint for the agent at time t is given by c tt +

c tt+1 + g t + ν(nt ) = w − Tt , 1+r

where r is the exogenous interest rate, and ν(·) designates the cost of having children. ν(·) is strictly increasing, twice continuously differentiable, and convex. Given these public and private transfers, the total number of BIGs provided for an agent of generation t when old is nt g t+1 + β t T t+1 , where β t is the birth rate of generation t. 3 However, we shall restrict attention to equilibria where agents of the same generation behave identically, so that βt = nt in equilibrium. This is only an equilibrium condition; when considering deviations, we must consider changes in nt while holding β t fixed. The preferences of the agent at time t are given by   U c tt , c tt+1 + P (nt ) + B(nt g t+1 + βt Tt+1 ). U (·) is strictly increasing, twice continuously differentiable, homothetic and strictly concave. P (·) denotes the direct utility that parents receive from having a number of children nt . B(·) denotes the utility the agent receives from BIGs provided for him by the next generation when he is old. Note that while the agent chooses his number of children nt , he takes the birth rate of his generation, β t , as a given parameter; the agent receives BIGs both directly from his own children (nt g t+1 ) and also from the next generation as a whole (β t T t+1 ). P (·) and B(·) are both strictly increasing, twice continuously differentiable and concave. To simplify the examination of the model, let V (x) ≡ max{U (x − z, (1 + r )z)} z

3

We assume that public and private BIGs are perfectly substitutable. This is done for simplicity; it is our goal to show that even if public and private BIGs are perfectly substitutable, our main result, that the first-best can not be enforced as a unique equilibrium in a model with endogenous fertility, still holds.

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so that V (x) denotes the indirect utility from consuming an amount of wealth x. Note that V (·) is twice continuously differentiable, strictly increasing and strictly concave. Finally, we assume that each family can observe the entire history of backward intergenerational goods and children, h t = ((g 1 , n1 ), . . . , (g t−1 , nt−1 )). An agent of generation t chooses a strategy s t (h t ) = (g t (h t ), nt (h t )) that specifies the number of children, as well as the private intergenerational transfer for any possible history. Note again that while the agent chooses his number of children, he takes the birth rate β t as a given. Thus the payoff for an agent of generation t given a history h t is given by V (w − Tt − g t (h t ) − ν(nt (h t ))) + P (nt (h t )) + B(g t+1 (h t , s t (h t ))nt (h t ) + βt (h t )Tt+1 ). The equilibrium concept is subgame perfection. The path γ = {(g t , nt )} is an equilibrium outcome if there exists a strategy for each generation such that those strategies form a subgame perfect Nash equilibrium and produce as an outcome the transfers and birth rates described by that path.

3. Benchmark Case: Exogenous Fertility In order to understand the role that fertility plays in the analysis, we begin by reexamining the case of exogenous fertility. The results in this section are a straightforward extension of results from previous papers (see, for instance, Rangel 2002). For the rest of the section assume that nt (h t ) = n¯ t for all histories and time. Thus, the strategy space of an agent is simply how many BIGs to provide privately. 3.1. Characterization of the Equilibrium Set

Our first task is to characterize the equilibrium set for a given tax sequence {T t }∞ t=1 . Fortunately, a simple algebraic condition allows us to check whether a path of transfers {g t }∞ t=1 is an equilibrium. The following proposition gives this condition. PROPOSITION 1: For a given sequence of tax rates {T t }∞ t=1 and birth rates ∞ {¯nt }∞ t=1 , γ = {g t }t=1 is an equilibrium outcome if and only if the condition V (w − Tt − ν(¯nt )) V (w − Tt − g t − ν(¯nt )) ≤ + P (¯nt ) + B(¯nt Tt+1 ) + P (¯nt ) + B(¯nt g t+1 + n¯ t Tt+1 )

(1)

holds for all t. A class of strategies which support these equilibria are simple trigger strategies. Using a simple trigger strategy, an agent plays the strategy which gives the outcome described by the path unless their parents deviate. If their

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parents deviate, they provide no BIGs to them. So an agent privately providing positive amounts of BIGs at any other level than the amount specified by the path γ will result in the agent losing consumption good while gaining nothing in return. Hence, it must be optimal for the agent either to play the equilibrium strategy (i.e., provide g t ), which results in a utility equal to the right-hand side of (1), or privately provide no BIGs at all. Providing no BIGs gives a utility equal to the left-hand side of (1). This strategy will be referred to as the deviation strategy, and the resulting utility as the deviation utility. Thus, the equilibrium condition above is simply that the maximal utility that can be attained from deviating is no greater than the utility along the equilibrium path. 3.2. The Equilibrium Set and Taxation

We now characterize the set of stationary equilibrium allocations as a function of the lump sum tax. Let n¯ t = n¯ for all t. The stationary socially optimal level of BIGs is b ∗ (¯n) ≡ arg max{V (w − b − ν(¯n)) + P (¯n) + B(¯nb)}. b

∗

Let u (¯n) denote the utility of an agent who gives b ∗ (¯n) to his parents and receives n¯ b ∗ (¯n) in BIGs. Note that b ∗ (¯n) is the total transfer of BIGs, including both public and private transfers. Thus, as T increases, the socially optimal private transfer g ∗ (¯n) = b ∗ (¯n) − T falls. Now we wish to characterize the set of steady state equilibria, and in particular look at how the equilibrium set changes with the lump sum tax. Let b max (T ; n¯ ) and b min (T ; n¯ ) be the maximum and minimum total transfer in equilibrium for a given tax rate. PROPOSITION 2: (1) For T ∈ [0, b ∗ (¯n)), b ∈ [T, b max (T ; n¯ )] is sustainable in equilibrium, where b max (T ; n¯ ) is a decreasing function of T . (2) For T ≥ b ∗ (¯n), the only sustainable level of private transfer is 0, so b max (T ; n¯ ) = b min (T ; n¯ ) = T . This result characterizes the entire set of static equilibrium outcomes for a given fertility rate. Zero private transfers trivially satisfies the equilibrium condition (1), so that as the mandatory provision rises, b min (T ; n¯ ) = T must rise. Also, b max (T ; n¯ ) can be higher than the socially optimal transfer; we can calculate b max (T ; n¯ ) using (1); it is the largest transfer that satisfies V (w − T − ν(¯n)) + B(¯nT ) ≤ V (w − b max (T ; n¯ ) − ν(¯n)) + B(¯nb max (T ; n¯ )). Hence, as long as the agents are weakly better off following the equilibrium strategy than deviating (and having transfers of 0 between generations), the transfer prescribed by that path is an equilibrium. Note that as long as

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T < b ∗ (¯n), b min (T ; n¯ ) = T < b max (T ; n¯ ). Furthermore, as the lump sum tax rises, the deviation utility rises, as a larger mandatory provision increases the utility of the deviation payoff, as long as T < b ∗ (¯n). Thus, for a given b, the utility of the equilibrium remains the same, while the deviation utility rises, making it harder to sustain equilibrium. Since utility is a concave function of the total transfer, b max (T ; n¯ ) must fall as the tax increases. However, the social optimum always remains an equilibrium if T < b ∗ (¯n), since it must have a higher utility than that of any other stationary path, including that of the deviation point. For T ≥ b ∗ (¯n), the socially optimal equilibrium mandates a private transfer of 0, and hence, no other steady state equilibrium is possible, since an agent can always attain at least the payoff provided by the socially optimal equilibrium by providing no BIGs.

3.3. Utility Outcomes and Taxation

Proposition 2 also shows that for T < b ∗ (¯n), there are multiple equilibria. This introduces a role for mandatory social provision, as by raising taxes to b ∗ (¯n) we collapse the equilibrium set to a single point. Furthermore, that point is socially optimal. Let umax (T ; n¯ ) and umin (T ; n¯ ) be the maximum and minimum utility attainable in equilibrium for a given tax rate. PROPOSITION 3: For T < b ∗ (¯n), umax (T ; n¯ ) = u∗ and umin (T ; n¯ ) is an increasing function of T . At T = b ∗ (¯n), umax (T ; n¯ ) = umin (T ; n¯ ) = u∗ (¯n), and for T > b ∗ (¯n), umax (T ; n¯ ) = umin (T ; n¯ ) < u∗ (¯n) and is a decreasing function of T . With no lump sum tax, the first best is sustainable as an equilibrium, but many other equilibria are sustainable as well. However, by increasing the tax rate we cut down on the size of the equilibrium set; furthermore, the equilibria lost by increasing the tax rate are those with the lowest utility. By mandating the optimal transfer b ∗ (¯n), the government enforces the social optimum, and no other equilibria are sustainable, since the deviation point has the highest possible utility, as it is the social optimum. Thus, in the case of exogenous utility, the first-best can be enforced as the unique equilibrium outcome. Enforcing transfers is valuable because it reduces the equilibrium set so that only the best outcome is possible.

4. The Case of Endogenous Fertility In this section, we show the conclusion above, that the first-best is implementable as a unique equilibrium, is not robust to the inclusion of endogenous fertility. If agents are able to make both private transfers of BIGs and fertility decisions, the government will not be able to reduce the equilibrium set to a point containing only the social optimum.

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Consider first the autarkic path for a given tax rate. (We will use autarky to refer to a path where no BIGs are provided privately, but government may still provide them by using a lump sum tax.) For the autarkic path, g t = 0 for all t. The autarkic birth rate is defined as the solution to na (Tt , Tt+1 ) ≡ arg max{V (w − Tt − ν(n)) + P (n) + B(βTt+1 )} n

= arg max{V (w − Tt − ν(n)) + P (n)}. n

a

In other words, n (T t , T t+1 ) is the personal fertility rate the agent would choose if he expects to receive no private support from his children (i.e., g t = 0) but does expect to receive some BIGs through taxes levied on the children of all agents. 4 Note that the autarkic birth rate does not depend on the provision of backward intergenerational goods. Thus, we will call the autarkic birth rate simply na (T t ). Since all BIGs are provided by the entire next generation in autarky, the agent’s level of these goods is not changed by his personal fertility decision. We will assume that na (T t ) > 0. 4.1. Characterization of the Equilibrium Set

As before, our first task is to characterize the equilibrium set for a given tax sequence {T t }∞ t=1 . Fortunately, the necessary and sufficient condition for a path to be an equilibrium is yet again a simple algebraic expression. ∞ PROPOSITION 4: For a given sequence of tax rates {T t }∞ t=1 , γ = {(g t , n t )}t=1 is an equilibrium outcome if and only if the condition    V w − Tt − ν na (Tt ) V (w − Tt − g t − ν(nt ))  a  ≤ (2) + P (nt ) + B(nt g t+1 + nt Tt+1 ) + P n (Tt ) + B(nt Tt+1 )

holds for all t. This proposition verifies whether a path is an equilibrium. Simple trigger strategies can again be used to support any equilibrium and are useful in gaining an intuition for the results. Agents now play the strategy prescribed by the path unless their parents deviate by changing their backward intergenerational transfer or their fertility level. If their parents deviate, they provide no BIGs to them. Thus, if an agent deviates from the equilibrium strategy, he will deviate to maximize his utility assuming that he will be provided with no BIGs from his children. The best deviation for an agent, then, is to provide no BIGs and to have na (T t ) children. Providing no BIGs and having na (T t ) children will be referred to as the deviation strategy, and the resulting utility 4

Note that, when the agent is making his decision when expecting no direct support from his children, he treats society’s birth rate as given, and hence the amount of BIGs he receives as independent of his actions. Since B(βT t+1 ) is constant with respect to n, it drops out of the maximization problem.

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as the deviation utility. It is important here that the agent is affected not only by the decisions of his children, but also by the decisions of other members of his generation. If other members of his generation have many children, this raises his utility, as he receives BIGs from these children through the social insurance system. Note that the set of equilibrium outcomes (for a given birth rate) cannot be larger than the set of equilibrium outcomes if fertility was exogenously fixed at that birth rate. Formally, this follows as the left-hand side of (1) is smaller than the left-hand side of (2), as the agent can now choose any fertility rate for himself, not just the nt prescribed by the path. In other words, the agent’s choice set has been expanded, as he can now change his fertility decision. Thus, the set of private transfers that can be sustained at that birth rate is smaller, as the agent now can deviate by changing his fertility choice, in addition to changing the number of BIGs provided to his parents. 4.2. The Equilibrium Set and Taxation

We now characterize the set of stationary equilibrium allocations. The stationary social optimum is now defined by (b ∗ , n∗ ) ≡ arg max{V (w − b − ν(n)) + P (n) + B(nb)}. (b,n)

The utility attained with these stationary allocations is u ∗ . 5 Figure 1 depicts the equilibrium set for three levels of taxes. If T = 0, by Proposition 4 the equilibrium set is all (n, g ) satisfying V (w − ν(na )) + P (n ) + B(0) a

≤

V (w − g − ν(n)) + P (n) + B(ng).

The equilibrium set is the interior of the utility contour of the autarkic equilibrium. Thus the social optimum must be contained in this set. The no tax case is special in that the utility of an agent does not depend on the childbearing decisions of other agents. Since taxes are 0, the agent receives BIGs only from his own children, not from the children of others. Note also that both very high and very low birth rates are not sustainable, even with no taxes. Consider the case of a high birth rate. The direct consumption of the agent is very small, since he has a large number of children. The agent could deviate to having only na (T t ) children, and providing no BIGs to his parents. By doing so, he reduces the number of BIGs he receives to 0 and reduces the direct utility he obtains from having children, but is able to greatly increase his consumption. For high enough birth rates, this effect

We will assume here that b ∗ > 0 in order to make the problem interesting, i.e., so that the socially optimal path includes intergenerational transfers. This will be the case if, for instance, B(·) satisfies the Inada conditions.

5

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g +T

Equilibria for no taxes Equilibria for low taxes Equilibria for high taxes Social Optimum Autarkic paths

Birth Rate Figure 1: The lines denote the boundaries of the endogenous fertility equilibria for three different tax rates. Note that for the autarkic paths, the total transfer equals T, the transfer mandated by the government.

will dominate. Hence, very high birth rates will not be sustainable. Now consider the case of a low birth rate. Here an agent may deviate to having na (T t ) children and providing no BIGs, as by doing so he increases the amount of utility he receives from children. Also, by providing no BIGs, he increases the amount he has to spend on direct consumption and children. The agent receives no BIGs if he deviates, but if the birth rate is small, this effect is small, since he did not receive many BIGs even in equilibrium. In the limiting case where the path prescribes a birth rate of 0, the agent would receive no BIGs in any case, and so could deviate to providing no BIGs and the autarkic number of children with no unfortunate consequences. Figure 1 also depicts the equilibrium set with positive taxes. Here, the set of sustainable birth rates is much smaller, so much so that for high taxes the socially optimal path is no longer an equilibrium path. (This high tax rate, however, is still less than b ∗ .) By increasing the enforced transfer, the deviation payoff will rise, making an equilibrium harder to sustain. Consider the case where the equilibrium birth rate n is greater than the autarkic birth rate. Since the equilibrium (n, b = g + T ) exists, with g > 0, it is clear that an agent with wealth w − T − ν(n) and nT BIGs prefers w − g − T − ν(n) wealth and n(g + T ) BIGs, by revealed preference. Hence, since V and B are concave, an agent with wealth w − T − ν(n) and nT BIGs prefers w − ε − T − ν(n) wealth and n(ε + T ) for any small ε > 0. However, the agent who

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deviates to (na (T ), 0) has more wealth than w − T − ν(n), as he has less children. Thus the agent who deviates (na (T ), 0) also improves his welfare by converting direct consumption into BIGs. Now consider raising the tax rate by ε. As raising taxes by ε converts consumption to BIGs at the rate n, it must improve the utility of the deviating agent. This will make the (n, b = g + T ) equilibrium harder to sustain, since the utility of this equilibrium stays the same. (For b to remain constant, the size of the private transfer must fall the same amount the tax increases.) The point in the equilibrium set with the largest population growth is on the boundary of the equilibrium set, and so, if this is the equilibrium, an agent has the same utility whether he deviates to (n a (T ), 0) or plays the equilibrium strategy. Hence, using the logic in the preceding paragraph, this point ceases to be an equilibrium. Summarizing the above arguments in a theorem, we have, (letting n max (T ) be the maximum birth rate sustainable in equilibrium): PROPOSITION 5: (1) na (T ) is a decreasing function of the tax rate. (2) nmax (T ) is a decreasing function of the tax rate.

4.3. Utility Outcomes and Taxation

With these tools, we can now consider the central questions of the paper, how the tax rate effects the set of utilities which can be attained in equilibrium, and how these results differ from the case with exogenous fertility. Figure 2 depicts the stationary utility possibility set in the cases of exogenous and endogenous fertility as a function of the lump sum tax. There are two important differences compared to the exogenous fertility case. First, although the set of utility possibilities still collapses to a single point for a large enough tax, it is no longer the socially optimal point. The socially optimal equilibrium is no longer enforceable, as tax rates corresponding to such a high level of intergenerational transfer lead individuals to deviate by having too few children, so the birth rate is less than n ∗ . This can be seen in Figure 2, as if the mandatory social provision is b ∗ , then the only equilibrium left is autarky, and this equilibrium does not attain u ∗ . Second, it is no longer the case that autarky is necessarily the worst possible equilibrium for T > 0. For positive tax rates, consider the equilibrium (n min (T ), g ), where nmin (T ) < na (T ). This equilibrium is on the boundary of the equilibrium set, so we have from the equilibrium condition 2 that V (w − T − g − ν(nmin (T ))) + P (nmin (T )) + B(nmin (T )g + nmin (T )T )

=

V (w − T − ν(na (T ))) . + P (na (T )) + B(nmin (T ))

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Utility

Extremal Utility of Exogenous Fertility Equilibria

Autarkic Utility Extremal Utility of Endogenous Fertility Equilibria

T*

Ta

b*

T

Figure 2: Graph of minimal and maximal equilibrium provision of BIGs as a function of the tax rate. The exogenous birth rate chosen was n∗ . T ∗ is the highest tax rate at which the social optimum is sustainable. T a is the tax rate that maximizes the utility of the autarkic equilibrium. b∗ is the optimal transfer.

However, since nmin (T ) < na (T ), B(nmin (T )T ) < B(na (T )T ), as B is an increasing function. Hence V (w − T − g − ν(nmin (T ))) + P (nmin (T )) + B(nmin (T )g + nmin (T )T )

<

V (w − T − ν(na (T ))) + P (na (T )) + B(na (T )T ).

The right-hand side of the above equation, however, is simply the utility of the agent if everyone has the autarkic number of children and provides no BIGs privately. In other words, the utility of the equilibrium (nmin (T ), g) is the same as that of the deviation point, which has a lower utility than autarky, since in autarky the other members of the agent’s generation are having more children, and so the agent is better off as he receives more BIGs. This can be seen in Figure 2, as with low (but positive) taxes, the autarkic utility is higher than the minimal utility attained in equilibrium. PROPOSITION 6: With endogenous fertility, umax (T ) is a decreasing function with umax (0) = u∗ . Further, if umax (T ) = u∗ , umin (T ) = u∗ . The key insight here is that with positive taxes, an agent can expropriate from other people of his own generation. A deviating agent does this as he benefits through taxes from the entire next generation, while bearing none

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of the costs of having a greater than autarkic number of children. Note that this option was not available to him in the case with exogenous birth rates; there, he simply decided between no private transfer or the social norm of society. This new externality is why, with large positive taxes, it can now be impossible to sustain the social optimum. A deviating agent can do better than the social optimum, by obtaining a large number of BIGs through taxation, while increasing his wealth by having less children and not engaging in an intergenerational transfer. Proposition 6 says, in considering optimal levels of BIGs, a policy maker can not simply ask what is the optimal level of intergenerational transfer, and then enforce that via taxation. Rather, the policy maker must ask what level of public provision of BIGs is most likely to lead to good private equilibria. Further, the tax rates that satisfy this property will be less than the optimal transfer level. We see that there exists a trade-off for the policy maker; by increasing taxes, he may raise the utility of the worst possible outcome, but may also decrease the utility of the best possible outcome. This was not the case with exogenous fertility, as Proposition 3 showed that the government could mandate the first-best. 6 Note that this result does not depend on linking fertility and intergenerational transfers. If we only allow agents to condition their choices on the previous generation’s decisions about BIGs, and not on their fertility choices, the first-best can only be sustained as an equilibrium if T = 0. This follows from the logic above. With positive taxes, agents do not fully internalize the cost to other people in their generation of having too few children. (If an agent has fewer children, the other agents of his generation receive less BIGs.) Due to this externality, if we have a positive tax rate, they will choose to have less children than is socially optimal, so (n ∗ , b ∗ ) will be unsustainable.

5. Conclusions We have argued that policy makers must consider models with both fertility and private transfers as choice variables in order to characterize the set of possible outcomes after instituting a social insurance system. A model with exogenous fertility implies that the tax rate should be equal to the optimal intergenerational transfer, as this ensures the unique equilibrium is the socially optimal outcome. This paper has shown this insight no longer holds in a world where agents also make fertility decisions. In the case of endogenous fertility, as the level 6

Note that this result is not driven solely by the autarkic equilibrium. Although in Figure 2 the autarkic equilibrium is improving with the tax rate for all T such that the social optimum is sustainable, this is not always the case. For some preferences, one can improve the utility possibility set by raising taxes even if it causes the autarkic utility to fall. Thus, a policy maker may wish to raise taxes even if it lowers the utility of generational autarky, if it raises the minimal equilibrium utility by making particularly bad equilibria unsustainable.

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of social insurance rises, the maximum sustainable birth rate falls, and eventually falls below the optimal birth rate. Thus the government can no longer mandate the first-best. Note that if the link between the BIGs provision and fertility decisions is broken, then the result is even more stark: the first-best can only be sustained as an equilibrium if T = 0. One could also consider a model where the government could make the amount of taxes an agent pays dependent on his number of children, and through this enforce the first-best. Yet this depends crucially on the assumption of nondistortionary taxation. In a more realistic model, the deadweight losses from these taxes would be nontrivial, and the government would need to balance the effects of improving the fertility rate with the costs due to tax distortions. Furthermore, optimal fertility decisions will be different for different families depending on their individual characteristics, which may well be observable to their children but not to the government; in this world, a government wishing to enforce a unique equilibrium will once again only achieve second-best. Considerations of tax distortions and redistributional effects make for a difficult problem for policy makers. However, absent these issues, with exogenous fertility, the debate centered around the optimal level of intergenerational transfer, and setting the tax rate equal to it. In this paper, we find that the optimal level of transfer in this context is not even the correct question for policy makers to be asking. Rather, they must ask a much more nuanced question: what tax rates will allow the best equilibrium, while making it less likely that low utility equilibria will be coordinated upon?

Appendix: Proofs A.1. Exogenous Fertility

The proofs of Propositions 1 and 2 can essentially be found in Rangel (2002). We shall proceed in a like manner. Before beginning the proof of Proposition 1, define the following simple trigger strategies (STSs):  s t (h t ) =

gˆ t 0

if µ(h t ) = C if µ(h t ) = P,

where µ is defined as  C    C µ(h t ) = C    P

if t = 1 if µ(h t−1 ) = C and g t−1 = gˆ t−1 if µ(h t−1 ) = P and g t−1 = 0 otherwise.

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LEMMA 1: A path of BIGs can be sustained as a subgame perfect equilibrium if and only if it can be sustained as a subgame perfect equilibria using STSs. Proof: It is clear that if a path can be sustained using STSs; it is a subgame perfect equilibrium. Now consider any path that is sustained by some subgame perfect equilibrium. It must be the case that V (w − Tt − ν(nt )) + P (nt ) V (w − Tt − g t − ν(nt )) + P (nt ) ≤ + B(nt Tt ) + B(nt g t+1 + nt Tt+1 ) holds for all t, otherwise the generation for which this is violated could do better by deviating to g t = 0. Let s t (h t ) be the STSs associated with this path. Clearly, they do generate the path, so we need to show this is a subgame perfect equilibrium. For each generation, if µ(h t ) = C, then, since not providing any BIGs is the best possible deviation, as long as the above inequality holds, the agent does not wish to deviate. If µ(h t ) = P , then we need V (w − Tt − ν(nt )) + P (nt ) V (w − Tt − ν(nt )) + P (nt ) ≤ + B(nt Tt ) + B(nt g t+1 + nt Tt ), which is obvious.
We now prove Proposition 1: Proof: By the Lemma, it is enough to show that the STSs associated with the path are satisfied if and only if the inequality condition holds. Let s t (h t ) be the STSs associated with this path. For this strategy to be subgame perfect, we need that for all histories such that µ(h t ) = C, he does not wish to deviate. But the best deviation is deviating to 0, and so if the condition is not satisfied, we have found a profitable deviation, thus proving necessity. If it is satisfied, then we also need that for all histories such that µ(h t ) = C, he does not wish to deviate. But V (w − Tt − ν(nt )) + Y (nt ) V (w − Tt − g t − ν(nt )) + Y (nt ) ≤ + B(nt Tt ) + B(nt g t+1 + nt Tt ) ≤

V (w − Tt − ν(nt )) + Y (nt ) + B(nt g t+1 + nt Tt ),

which is obvious.
To see that the social optimum is unique consider the second-order condition for the social optimum: V  (w − b − ν(n)) + n2 B  (nb), which is negative as both V and B are concave.

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Journal of Public Economic Theory We now prove Proposition 2:

Proof: Consider a tax rate T ≤ b ∗ (n) and a tax rate T¯ < T . Suppose a total level of transfer b = g + T , with g = 0 is sustainable with tax rate T . We know from Proposition 1 that V (w − T − ν(n)) + P (n) V (w − T − g − ν(n)) + P (n) ≤ + B(nT ) + B(ng + nT ) ≤

V (w − b − ν(n)) + P (n) + B(nb).

We need to show that a private transfer g¯ = b − T¯ is sustainable at a tax rate T¯ , so by Proposition 1 we need to show V (w − T¯ − ν(n)) + P (n) V (w − T¯ − g¯ − ν(n)) + P (n) ≤ + B(nT¯ ) + B(n¯g + nT¯ ) ≤

V (w − b − ν(n)) + P (n) + B(nb ).

So it is enough to show that V (w − T¯ − ν(n)) + P (n) V (w − T¯ − ν(n)) + P (n) ≤ + B(nT¯ ) + B(nT¯ ) but here we can consider the tax as the total transfer. Since we know the utility level as a function of total transfer is increasing if the transfer is less than b ∗ (n), the above inequality follows. Thus, b max (T ) is decreasing with the tax rate, if there are possible levels of private transfer at that rate. Since g = 0 is always an equilibrium, the minimal provision is simply the tax rate. If T ≥ b ∗ (n), then for g > 0 V (w − T − ν(n)) V (w − T − g − ν(n)) > + B(nT ) + B(ng + nT ) due to the concavity of the representative agent’s utility function with respect to T . Thus no equilibrium with g > 0 can be sustained.
We now prove Proposition 3: Proof: It is clear from the above that the highest private level of transfer sustainable is falling with the tax rate and that the government can impose the social optimum by setting T = b ∗ (n). Since that level of total transfer must then be sustainable for any lesser tax rate, we have that umax (T ;n) = u∗ . The minimal level is rising with the tax rate, and utility is rising with the

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transfer from the concavity of the social planner’s problem so we can see that umin (T ;n) is rising with the tax rate. Finally, since from Proposition 2 no private transfer is sustainable with T > b ∗ (n), the transfer in this case is just the tax. But then, the utility must fall due to the concavity of the socially planners problem, so umax (T ;n) = umin (T ;n) is a declining function of the tax rate.
A.2. Endogenous Fertility

Before beginning the proof of Proposition 4, define the following STSs:  (ˆg t , nˆ t ) if µ(h t ) = C s t (h t ) = (0, nˆ t ) if µ(h t ) = P, where µ is defined as  C    C µ(h t ) =  C    P

if t = 1 if µ(h t−1 ) = C and (g t−1 , nt−1 ) = (ˆg t−1 , nˆ t−1 ) if µ(h t−1 ) = P and (g t−1 , nt−1 ) = (0, nˆ t−1 ) otherwise.

LEMMA 2: A path of BIGs and children can be sustained as a subgame perfect equilibrium if and only if it can be sustained as a subgame perfect equilibria using STSs. Proof: It is clear that if a path can be sustained using STSs, it is a subgame perfect equilibrium. Now consider any path that is sustained by some subgame perfect equilibrium. It must be the case that      V (w − Tt − g t − ν(nt )) + P (nt ) V w − Tt − ν nat + P nat ≤ + B(nt g t+1 + nt Tt ) + B(nt Tt ) holds for all t, otherwise a member of the generation for which this is violated could do better by deviating to (0, nat ). Let s t (h t ) be the STSs associated with this path. Clearly, they do generate the path, so we need to show this is a subgame perfect equilibrium. For each generation, if µ(h t ) = C, then, since not providing any BIGs, and having the autarkic number of children is the best possible deviation, as long as the above inequality holds, the agent does not wish to deviate. If µ(h t ) = P , then we need      V (w − Tt − ν(nt )) + P (nt ) V w − Tt − ν nat + P nat ≤ + B(nt g t+1 + nt Tt ), + B(nt Tt ) which is satisfied if the original inequality is satisfied.
We now prove Proposition 4:

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Proof: By the Lemma, it is enough to show that the STSs associated with the path are satisfied if and only if the inequality condition holds. Let s t (h t ) be the STSs associated with this path. For this strategy to be subgame perfect, we need that for all histories such that µ(h t ) = C, he does not wish to deviate. But the best deviation is deviating to (0, nat ), and so if it is not satisfied, we have found a profitable deviation, thus proving necessity. If it is satisfied, then we also need that for all histories such that µ(h t ) = C, he does not wish to deviate. But      V (w − Tt − g t − ν(nt )) + P (nt ) V w − Tt − ν nat + P nat ≤ + B(nt g t+1 + nt Tt ) + B(nt Tt ) ≤

V (w − Tt − ν(nt )) + P (nt ) + B(nt g t+1 + nt Tt )

so as long as the original inequality holds, he will not wish to deviate.
The uniqueness of the social optimum can be seen by looking at the second-order Hessian

V  (·) + n2 B  (·) ν  (n)V  (·) + nbB  (·) . ν  (n)V  (·) + nbB  (·) −ν  (n)V  (·) + (ν  (n))2 V  (·) + P  (·) + b 2 B  (·) It is clear from the concavity of V and B that the top diagonal element is negative. The determinant is −ν  (n)V  (·)(V  (·) + n2 B  (·)) + V  (·)B  (·)(ν  (n)n − b)2 >0 + Y  (·)(V  (·) + n2 B  (·)) as V , Y , and B are concave, V is increasing, and ν is convex. We now prove Proposition 5: Proof: (1) Consider the problem of deciding how many children to have in autarky: max{V (w − T − ν(n)) + P (n)}. The maximand has the single crossing property as ∂2 (V (w − T − ν(n)) + P (n)) = −ν  (n)V  (·) > 0 ∂n∂(w − T ) so by Topkis’s Theorem the autarkic birth rate, if >0, is decreasing. Before proving the next part, note that for any given n, the equilibrium set must be convex in g . This follows since by changing g without changing the birth rate, we are not changing the deviation utility. Thus, due to the concavity of the utility function, we have that the set of private transfers sustainable must be convex.

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(2) Assume that at the tax rate T , the birth rate nmax is the highest sustainable in equilibrium. Then, we know that (g ∗ (nmax ), nmax ) is a sustainable equilibrium, where g ∗ (n) is the best possible private transfer given that tax rate and birth rate. We also know that if n = nmax (T ), that the utility of this equilibrium must be exactly that from deviating, as if not, we can find an equilibrium (g ∗ (n), n + ε) for ε > 0 sufficiently small, so n is not the maximum birth rate. So we have that V (w − τ − ν(na (T ))) + P (na (τ )) + B(nT )

=

V (w − T − g ∗ (n) − ν(n)) + P (n)B(ng + nT ).

Now consider raising the tax rate by ε > 0. Now, the best private transfer is g ∗ (n) − ε, and so the utility attainable at that birth rate is the same as before. (If ε is sufficiently small, the new private transfer is feasible.) Now consider deviating to (0, na (T )), which is less than the true deviation utility: V (w − T − ν(na )) + P (na ) + B(nT ). A rise in the tax rate must make the utility of this choice higher, as even for an agent with a lower wealth and more intergenerational transfers, i.e., the equilibrium agent above, the optimal transfer is greater than 0. Thus, due to the concavity of V and B, it must also make the deviating agent better off. But then V (w − T − ν(na (T + ε))) + P (na (T + ε)) + P (na (T + ε)) + B(nT + ε)

>

V (w − (T + ε) − g ∗ (n) − ν(n)) + P (n) + B(ng + n(T + ε))

and so no equilibrium at that birth rate can be sustained.
We now prove Proposition 6: Proof: (1) To see that the social optimum cannot be the only nonautarkic equilibrium, suppose, by way of contradiction, that it was. Now consider (n∗ − ε, g ∗ ). The change in utility is of order ε 2 . However, the deviation payoff has decreased by n∗ B  (n∗ τ )ε, which is first order in ε. Thus, for small enough ε, (n∗ − ε, g ∗ ) must be an equilibrium. For zero taxes, it is clear from the discussion of Figure 1 that the utility contour of the autarkic point encompasses the equilibrium set. Since the social optimum must have this utility or greater, it must be an equilibrium at T = 0. To see that the maximal utility is decreasing with the tax rate, consider two cases: nmax (T ) > n∗ and nmax (T ) ≤ n∗ . If nmax (T ) > n∗ , then we know the social optimum is an equilibrium, as it has a higher utility than (nmax (T ), g(nmax (T ))). Thus umax (T ) is constant. If nmax (T ) ≤ n∗ ,

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Journal of Public Economic Theory then we know that the best equilibrium is (g(nmax (T )), nmax (T )), but nmax (T ) must be decreasing with T by Proposition 5. Therefore, umax (T ) is decreasing in this region.


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