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Private versus public old-age security


We directly compare two institutions, a family compact—a parent makes a transfer to her parent in anticipation of a possible future gift from her children—with a pay-as-you-go, public pension system, in a life cycle model with endogenous fertility wherein children are valued both as consumption and investment goods. Absent intragenerational heterogeneity, we show that a benevolent government has no welfare justification for introducing public pensions alongside thriving family compacts since the former is associated with inefficiently low fertility. This result hinges critically on a fiscal externality—the inability of middle age agents to internalize the impact of their fertility decisions on old-age transfers under a public pension system. With homogeneous agents, a strong-enough negative aggregate shock to middle-age incomes destroys all family compacts, and in such a setting, an optimal public pension system cannot enter. This suggests the raison d’être for social security must lie outside of its function as a pension system—specifically its redistributive function which emerges with heterogeneous agents. In a simple modification of our benchmark model—one that allows for idiosyncratic frictions to compact formation such as differences in infertility/mating status—a welfare-enhancing role for a public pension system emerges; such systems may flourish even when family compacts cannot.

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Fig. 1
Fig. 2


  1. Explicit quid pro quo, such as written provisions, were popular in colonial New England and some ethnic immigrant communities of the Midwest during the 1800s (Sundstrom and David 1988). For more generational and anecdotal evidence on the matter, see Hareven and Adams (1996).

  2. Clearly, a compact could stipulate that infertile young provide a transfer in exchange for transfers from fertile and infertile young of the next generation. But then, at what point does one draw the line between such a comprehensive FC requiring transfers to all elderly relatives over extended family lines and a PPS?

  3. As Social Security Administration administrator Altmeyer wrote (Altmeyer 1966; pp 296) about Social Security in 1946, “It is not a plan for giving everybody something for nothing but a plan for organized thrift.” Leff (1983) argues, FDR repeatedly stressed that “social security could not be allowed ‘to become a dole through the mingling of insurance and relief’ and thus ‘must be financed by contributions, not taxes.”’

  4. This externality is also discussed in other contexts in Cigno (1993) and Nishimura and Zhang (1995). See Billari and Galasso (2014) for an up-to-date discussion of the externality and its quantitative significance.

  5. The total number of people at each date t is \(N_{t}={N_{t}^{y}}+{N_{t}^{m}}+{N_{t}^{o}}\) where \({N_{t}^{i}}\) is the size of the cohort i of the same age at date t. Since \(N_{t}=\left (n_{t}n_{t-1}+n_{t-1}+ 1\right ) {N_{t}^{o}}\) and \(N_{t + 1}=\left (n_{t}n_{t + 1}+n_{t}+ 1\right ) N_{t + 1}^{o}=\left (1+n_{t}n_{t + 1}+n_{t}\right ) n_{t-1}{N_{t}^{o}}, \) the gross population growth rate is N t+ 1/N t = [(1 + n t n t+ 1 + n t )n t− 1] /[1 + n t n t− 1 + n t− 1]. In the steady state (or on a balanced growth path), N t+ 1/N t = n.

  6. We take up the issue of private savings in Appendix F.3.

  7. Since, by design, agents are not altruistic towards their children, they make no gifts or transfers to them. Cigno (2006a) additionally allows for gifts from parents to children. In his setup, an altruistic parent may choose to pass on a generous gift to her children when they are young and accept a smaller transfer from them when they become adults. The latter imperative may weaken the threat of generational autarky that keeps the FC alive and viable.

  8. A “private social security system” requires trust; there must exist rewards and punishments for such a system of trust to operate. Kandori (1992) and Salant (1991) have shown that in overlapping generations models there can be outcomes, which support such cooperative arrangements, even though the identity of players is changing over time.

  9. Why might ϕ(α t ) be convex? Differentiating \( V\left (\alpha _{t};\phi \left (\alpha _{t}\right ) \right ) =\underline {U}\) and using the envelope theorem, we obtain \(\phi ^{\prime } \left (\alpha _{t}\right ) =\frac {1}{n_{t}}\frac {u^{\prime } \left (c_{t}\right )} {\beta u^{\prime } \left (x_{t}\right )} >0, \) which, using (2) with 𝜃 = 0, and n t = n(α t ;α t+ 1), yields \(\phi ^{\prime } \left (\alpha _{t}\right ) =\frac {1}{\nu } \frac {\alpha _{t + 1}}{ n\left (\alpha _{t};\alpha _{t + 1}\right )} \). The second derivative is

    $$ \phi^{\prime \prime} \left( \alpha_{t}\right) =\frac{\alpha_{t + 1}}{\nu n^{2}}\left[ \frac{1}{\nu} \left( 1-\frac{\alpha_{t + 1}}{n}\frac{\partial n\left( \alpha_{t};\alpha_{t + 1}\right)} {\partial \alpha_{t + 1}}\right) - \frac{\partial n\left( \alpha_{t};\alpha_{t + 1}\right)} {\partial \alpha_{t}}\right] . $$

    The sign here is inconclusive; it is easy to show \(\frac {\partial n\left (\alpha _{t};\alpha _{t + 1}\right )} {\partial \alpha _{t}}<0\) but the sign of \(\frac {\partial n\left (\alpha _{t};\alpha _{t + 1}\right )} {\partial \alpha _{t + 1}}\) and the term including the elasticity, \( \left (1-\frac { \alpha _{t + 1}}{n}\frac {\partial n\left (\alpha _{t};\alpha _{t + 1}\right )} { \partial \alpha _{t + 1}}\right )\), are ambiguous. If the elasticity is negative or smaller than unity, ϕ (α t ) > 0 over the domain of α t and the function ϕ(⋅) is strictly convex.

  10. In drawing Fig. 1, we have assumed, implicitly, ϕ (0) <  1 and \(\underset {\alpha \rightarrow w}{\lim } \phi ^{\prime } \left (\alpha \right ) >1, \) which are necessary (but not sufficient) for the existence of fixed point of ϕ(⋅) if the function is strictly convex. Also note, the mapping ϕ(⋅) may not exist - its existence rests critically on the utility function, the endowment profile, the per-child cost of raising children, ν, and the discount factor β. Loosely (and not surprisingly), a mapping exists when the old-age endowment (y) is sufficiently small compared to w.

  11. A limitation of our analysis is the absence of dynamics inside the FC. This means comparison with a PPS is restricted to steady states and not along the transition. These are deeper issues worthy of further exploration.

  12. The optimal stationary compact, α, must also satisfy \(u\left (w-\nu n-\alpha -\tau \right ) +\beta u\left (y+\alpha n+\tau n\right ) +\theta g\left (n\right ) \geq u\left (w-\nu \underline {n}-\tau \right ) \) \(+\beta u\left (y+\tau \underline {n}\right ) +\theta g\left (\underline {n}\right ) \) where \(\underline {n}\) satisfies \(\nu u^{\prime } \left (w-\nu \underline {n} -\tau \right ) =\theta g^{\prime } \left (\underline {n}\right ) \). Once again, note that given u (0) = , if y = 0 and τ = 0, a viable stationary compact will always survive since there is no other means to acquire the consumption good. Finally, note if 𝜃 = 0, no one has children outside of the FC and the economy ends; as such, if 𝜃 = 0, and FCs are not viable, a PPS cannot emerge. In other words, if we are to study the possible emergence of a PPS in a world where FCs have ceased to be viable, we have to impose 𝜃 >  0.

  13. If private agents did internalize the budgetary implications of the public scheme—as is true in the “large economy” case studied in Kolmar (1997)—their fertility decision and the accompanying pension would exactly match what obtains under the compact. Laffont (1975) calls such an equivalence, Kantian, one in which socially desirable outcomes can follow from private decisions when the government makes people aware of a macroeconomic constraint. If the PPS could be redesigned to link some part of the individual benefit entitlements to the number of children (and as Cigno and Werding 2007 argue), and/or to the income of each child, then the fiscal externality can be curtailed automatically.

  14. In Von Auer and Büttner (2004), if the young at t − 1 choose a lower fertility rate than that of the preceding generation, then the current young at t have to pay higher implicit taxes, the source of an intergenerational fiscal externality. In van Groezen et al. (2003), children exert two opposing externalities—an additional child implies a higher future output, but a lower capital–labor ratio—such that laissez faire is Pareto-efficient as both externalities exactly cancel out.

  15. For simplicity, we assume no change in old-age endowment, y. Conditions for non-existence can be suitably modified to incorporate changes in the entire income profile (w,y)

  16. We also have u (w ν n α ) > (𝜃/ν)g (n ) + (α /ν)β u (y + α n ), which means if the transfer α were to remain, agents would choose to have fewer children.

  17. In Section 4, we consider a modification of the existing model where agents are heterogenous in terms of their infertility/match status. In that setting, it may be possible for an optimal PPS to obtain even in the presence of a shock that destroys all FCs.

  18. As it turns out, the assumption regarding the costless replacement of an existing compact places an important constraint on the originator of the FC, as will be apparent in our discussion in Section 4.

  19. Moreover, if the government provided positive net transfers to those hit by the shock to their income when of working age—think of it as stylized unemployment insurance—these individuals might be willing to support intergenerational transfers. Indeed, when the USA introduced social security, they coupled the legislation with a provision for federal unemployment compensation.

  20. This type of agent heterogeneity is distinct from the form suggested at the close of the previous section. There, agents that find their income w falling below \(\hat {w}(y)\) are not open to transfers of income from middle to old age, so a PPS would not find support among this group. However, if some experienced an income shock where \(\hat {w}(y)<\) w <w, while others do not, all agents are willing to participate in an transfer program – they differ on the amount of the transfer. A richer design of the FC may be able to accommodate this type of agent heterogeneity and risk by making contributions contingent on the agent’s income. By contrast, heterogeneity along the lines of infertility/match status draws out a complete separation from the FC system that cannot be readily addressed by considering a richer compact design, since some middle age agents are left without the means to provide a critical ingredient to the FC system, namely children.

  21. See Cigno et al. (2017) for a more detailed treatment of matching and decision-making within a household in the presence of a FC. In comparison with our present model, Cigno et al. (2017) provides a richer matching environment but with an exogenous reproduction process. However, rather than using an intensive margin to increase the number of children they have in order to increase the amount they receive in transfers in old age—as we had done here—parents can increase their expected transfers from their children by investing in their children’s education.

  22. It is simple enough to allow all couples (fertile or not) to have an endowment stream different from unmatched agents, keeping in spirit with Ecclesiastes 4:9-10: “Two are better than one, because they have a good return for their labor: If either of them falls down, one can help the other up. But pity anyone who falls and has no one to help them up.”

    Doing so divides the population into three well-defined groups (fertile and infertile couples, and unmatched agents). As it stands, our assumption that all agents have the same endowment profile and that couples act similar to individual decision-makers means infertile agents in marriages act no different from unmatched agents.

  23. We presume the government can readily identify fertile agents at each date. In practice, in most countries, such subsidies are written into the tax code and are given to fertile/matched couples ex post, after a child is born.

  24. 24We thank an anonymous referee for pointing this out to us.

  25. This consideration is motivated by Hill (2015) which suggests the economic collapse in the USA in the early 1930s brought about large and temporary decline in marriage rates.

  26. When we say “nothing to prevent...,” we acknowledge of course, the ever-present constraint the planner faces that the compact must dominate autarky.

  27. It is not difficult to establish directly that Λ > 0 for small enough 𝜃.

  28. Of course, alternatively, we can compute the optimal compact by differentiating lifetime utility V (α) (37)with respect to α.

  29. Technically, an agent can pass along a transfer \(\widehat {\alpha }_{t}\) to her parent that is greater than her familial obligation α t , but in this model with no altruistic tie running from child to parent, there is no incentive for her to give any more than α t .

  30. This is reminiscent of the problem associated with second-order conditions in the context of the Serendipity Theorem; see Jaeger and Kuhle (2009) for a recent discussion.

  31. An important rationale at the time of its inception (the Great Depression) was the need to provide sustenance to the existing elderly who were confronted with the disappearance of family compacts they had erstwhile paid into. To be fair, this may have been an overiding concern of FDR.


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Puhakka thanks the Yrjö Jahnsson Foundation and the OKO Bank Foundation for support. The authors thank two anonymous referees for their generosity and patience, and the editor-in-charge, Alessandro Cigno, along with Costas Azariadis, Susan Barnett, and Huberto Ennis for valuable feedback.

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Correspondence to Joydeep Bhattacharya.

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Responsible editor: Alessandro Cigno



A Proof of Lemma 1

To see that marginal conditions (10) and (11) cannot be supported with equality for some other configuration of (α, n), when the endowment profile is \(\left (\hat {w} (y),y\right )\), note that for any α > 0,

$$\beta \hat{n}(y)u^{\prime} \left( y+\alpha \hat{n}(y)\right) <\left( \theta/\nu \right) g^{\prime} \left( \hat{n}(y)\right) +\left( \beta \alpha /\nu \right) u^{\prime} \left( y+\alpha \hat{n}(y)\right) . $$

Since (𝜃/ν)g (n) + (α/ν)β u (y + α n) is decreasing in n, we need a value n >\(\hat {n}(y)\) for (12) to hold.

But by construction, \( u^{\prime } \left (\hat {w}(y)-\nu \hat {n}\right ) =\hat {n} \beta u^{\prime } \left (y\right ) \) (Definition 2). Our assumption u ″′(c) > 0, along with \(-\nu u^{\prime \prime } \left (\hat {w}(y)-\nu \hat {n}(y)\right ) =\beta u^{\prime } \left (y\right ) \) ensures \(-\nu u^{\prime \prime } \left (\hat {w}(y)-\nu n-\alpha \right ) >\beta u^{\prime } \left (y\right ) \) for all \(n\geq \hat {n}(y)\); in other words, the cost curve \(u^{\prime } \left (\hat {w}(y)-\nu n-\alpha \right ) \) grows faster than n β u (y) for \(n\geq \hat {n}(y)\). Since

$$u^{\prime} \left( \hat{w}(y)-\nu \hat{n}(y)-\alpha \right) >u^{\prime} \left( \hat{w}(y)-\nu \hat{n}(y)\right) =\hat{n}(y)\beta u^{\prime} \left( y\right) $$

it follows that for any α > 0, \( u^{\prime } \left (\hat {w}(y)-\nu n-\alpha \right ) >n\beta u^{\prime } \left (y\right ) >n\beta u^{\prime } \left (y+\alpha n\right ) \) for all \(n\geq \hat {n}(y)\). Additionally, the marginal cost curve \(u^{\prime } \left (\hat {w}(y)-\nu n-\alpha \right ) \) must also lie above the benefit curve n β u (y + α n) for \(0<n<\hat {n}(y), \) since the slope of the curve \(u^{\prime } \left (\hat {w}(y)-\nu n\right ) \) is less than β u (y) for \(n<\hat {n}(y)\) due to the fact we have assumed these are equal at n =\(\hat {n}(y)\). In sum, the marginal cost curve \(u^{\prime } \left (\hat {w}(y)-\nu n-\alpha \right ) \) for any α > 0 lies above the marginal benefit n β u (y + α n), for all n > 0, ensuring at any point n where (11) holds, (10) must hold with strict inequality.

B Properties of optimal n and α

First-order conditions

$$ n:V_{n}=-\nu u^{\prime} \left( w-\nu n-\alpha -\tau \right) +\theta g^{\prime} (n)+\beta \alpha u^{\prime} (y+\alpha n+b) $$
$$ \alpha :V_{\alpha} =-u^{\prime} \left( w-\nu n-\alpha -\tau \right) +\beta nu^{\prime} \left( y+\alpha n+b\right) $$

Second-order conditions:

  1. 1.

    V n n = ν 2 u (wν nατ) + 𝜃 g (n) + β α 2 u (y + α n + b) < 0

  2. 2.

    V α α = β n 2 u (y + α n + b) + u (wν nατ) < 0

  3. 3.

    \(V_{nn}V_{\alpha \alpha } -V_{\alpha n}^{2}>0, \) where V α n = ν u (wν nατ) + β u (y + α n + b) + β α n u (y + α n + b)

Using budget constraint b = n τ along with the first-order conditions (28) and (29), we can use the implicit function theorem to determine how b changes with a change in τ. We have

$$\left[ \begin{array}{ccc} 1 & -\tau & 0 \\ \beta \alpha u^{\prime \prime} (y+\alpha n+b) & V_{nn} & V_{\alpha n}\\ \beta nu^{\prime \prime} (y+\alpha n+b) & V_{\alpha n} & V_{\alpha\alpha} \end{array} \right] \left[ \begin{array}{c} db \\ dn \\ d\alpha \end{array} \right] =\left[ \begin{array}{c} n \\ -\nu u^{\prime \prime} \left( w-\nu n-\alpha -\tau \right)\\ -u^{\prime \prime} \left( w-\nu n-\alpha -\tau \right) \end{array} \right] d\tau $$

Let Λ denote the determinant of the coefficient matrix,

$${\Lambda} \equiv V_{nn}V_{\alpha \alpha} -V_{\alpha n}^{2}+\tau \left( V_{\alpha \alpha} \beta \alpha u^{\prime \prime} (y+\alpha n+b)-V_{\alpha n}\beta nu^{\prime \prime} (y+\alpha n+\tau n)\right) $$

We assume Λ does not vanish over the interval. The sign of Λ is ambiguous. However, note that with the same system of equations, we can differentiate with respect to 𝜃 and obtain

$$\frac{dn}{d\theta} =\frac{-\nu g^{\prime} V_{\alpha \alpha} }{\Lambda} $$

The numerator of this expression is positive. Hence, adopting Assumption 2 is equivalent to assuming Λ > 0.Footnote 27

We are interested in the derivative d b/d τ. We have

$$\begin{array}{@{}rcl@{}} \frac{db}{d\tau} &=&\frac{\left\vert \begin{array}{ccc} n & -\tau & 0 \\ -\nu u^{\prime \prime} \left( w-\nu n-\alpha -\tau \right) & V_{nn} & V_{\alpha n}\\ -u^{\prime \prime} \left( w-\nu n-\alpha -\tau \right) & V_{\alpha n}&V_{\alpha \alpha} \end{array} \right\vert} {\Lambda} \\ &=&\frac{n\left( V_{nn}V_{\alpha \alpha} -V_{\alpha n}^{2}\right) +\tau u^{\prime \prime} \left( w-\nu n-\alpha -\tau \right) \left( -\nu V_{\alpha \alpha} +V_{\alpha n}\right)} {\Lambda} \end{array} $$

This is used in the proof in the following section below.

C Proof of Lemma 2

Using (30)

$$-n+\frac{db}{d\tau} =\tau \beta u^{\prime} \left( y+\alpha n+\tau n\right) \left[ \frac{\beta n^{2}u^{\prime \prime} (y\,+\,\alpha n\,+\,b)+u^{\prime \prime} \left( w\,-\,\nu n-\alpha -\tau \right)} {\Lambda} \right] \leq 0 $$

using Assumption 2 to sign Λ.

D Addendum to Proposition 4

As noted in Section 3.4, if the condition \(-\nu u^{\prime \prime } \left (\hat {w}(y)-\nu \hat {n}(y)\right ) =\beta u^{\prime } \left (y\right ) \) is not met, we can find an endowment \(\tilde {w}\left (y\right ) <\hat {w}(y)\) and child decision \(\tilde {n}\left (y\right ) \) such that the cost curve is repositioned in such a way as to make

$$u^{\prime} \left( \tilde{w}\left( y\right) -\nu \tilde{n}\left( y\right) \right) =\beta \tilde{n}\left( y\right) u^{\prime} \left( y\right) $$


$$-\nu u^{\prime \prime} \left( \tilde{w}\left( y\right) -\nu \tilde{n}\left( y\right) \right) =\beta u^{\prime} \left( y\right) $$

hold. This new cost curve \(u^{\prime } \left (\tilde {w}\left (y\right ) -\nu n\left (y\right ) -\alpha \right ) \) lies everywhere above the marginal benefit curve n β u (y + α n) for any α > 0. It then follows for any permanent shock to first period endowment w with \(w^{\prime } <\tilde {w}\left (y\right ) \) will nullify the existing compact as well as prevent any new family compact from emerging. The proof is as follows.

If \(-\nu u^{\prime \prime } \left (\hat {w}(y)-\nu \hat {n}(y)\right ) >\beta u^{\prime } \left (y\right ) \) or \(-\nu u^{\prime \prime } \left (\hat {w} (y)-\nu \hat {n}(y)\right ) <\beta u^{\prime } \left (y\right ) \) holds, we know, by the convexity of the cost curve \(u^{\prime } \left (\tilde {w}\left (y\right ) -\nu n\left (y\right ) \right )\), there exists two values for n such that \(u^{\prime } \left (\tilde {w}\left (y\right ) -\nu n\left (y\right ) \right ) =\) n β u (y); one of them, by definition, is \(\hat {n}\left (y\right ) \). Call the smaller one, \( \tilde {n}_{0}^{A}(y), \) and \(\tilde {n}_{0}^{B}(y)\) the larger, and for convenience, let \(\tilde {w} _{0}\left (y\right ) \equiv \hat {w}(y)\).By the convexity of

$$ u^{\prime} \left( w-\nu n\right) ,u^{\prime} \left( \tilde{w}_{0}\left( y\right) -\nu n\right) <n\beta u^{\prime} \left( y\right) \forall n,n\in \left( \tilde{n}_{0}^{A}(y),\tilde{n}_{0}^{B}(y)\right) . $$

At these endpoint values, we must have

$$-\nu u^{\prime \prime} \left( \tilde{w}_{0}\left( y\right) -\nu \tilde{n} _{0}^{A}(y)\right) <\beta u^{\prime} \left( y\right) $$


$$-\nu u^{\prime \prime} \left( \tilde{w}_{0}\left( y\right) -\nu \tilde{n} _{0}^{B}(y)\right) >\beta u^{\prime} \left( y\right) . $$

Since \(-\nu u^{\prime \prime } \left (\tilde {w}_{0}\left (y\right ) -\nu n\right ) \) is a continuous function of n, we can find an \(\tilde {n} _{1}^{B}(y)\in \left (\tilde {n}_{0}^{A}(y),\tilde {n}_{0}^{B}(y)\right ) \) such that

$$-\nu u^{\prime \prime} \left( \tilde{w}_{0}\left( y\right) -\nu \tilde{n} _{1}^{B}(y)\right) =\beta u^{\prime} \left( y\right) . $$

As noted in (31), \( u^{\prime } \left (\tilde {w}_{0}\left (y\right ) -\nu \tilde {n}_{1}^{B}(y)\right ) <\tilde {n}_{1}^{B}(y)\beta u^{\prime } \left (y\right ) \).

Now define \(\tilde {w}_{1}\left (y\right ) \) implicitly by the condition,

$$ u^{\prime} \left( \tilde{w}_{1}\left( y\right) -\nu \tilde{n} _{1}^{B}(y)\right) =\tilde{n}_{1}^{B}(y)\beta u^{\prime} \left( y\right) . $$

Since u (⋅) < 0, we have \(\tilde {w} _{1}\left (y\right ) <\) \(\tilde {w}_{0}\left (y\right )\), and since (−ν u (wν n))/ w = −ν u ″′(⋅) < 0, we have

$$ -\nu u^{\prime \prime} \left( \tilde{w}_{1}\left( y\right) -\nu \tilde{n} _{1}^{B}(y)\right) >\beta u^{\prime} \left( y\right) . $$

The convexity of the cost curve, along with (32) and (33) and the fact \(\lim \limits _{n\rightarrow 0}u^{\prime } \) \(\left (\tilde {w}_{1}\left (y\right ) -\nu n\right ) =u^{\prime } \left (\tilde {w}_{1}\left (y\right ) \right ) >0\) ensures there is another value for n, \( \tilde {n}_{1}^{A}(y)< \tilde {n}_{1}^{B}(y)\) with

$$ u^{\prime} \left( \tilde{w}_{1}\left( y\right) -\nu \tilde{n} _{1}^{A}(y)\right) =\tilde{n}_{1}^{A}(y)\beta u^{\prime} \left( y\right) $$


$$-\nu u^{\prime \prime} \left( \tilde{w}_{1}\left( y\right) -\nu \tilde{n} _{1}^{A}(y)\right) <\beta u^{\prime} \left( y\right) . $$

Continuing the process, we generate sequences \(\left \{ \tilde {w}_{k}\left (y\right ) \right \} \), \( \left \{ \tilde {n}_{k}^{A}\left (y\right ) \right \} ,\left \{ \tilde {n}_{k}^{B}\left (y\right ) \right \}\), with limits \(\tilde {w} \left (y\right ) \) , \( \tilde {n}^{A}\left (y\right ) \) , \( \tilde {n}^{B}\left (y\right )\), respectively, and

$$\tilde{n}^{A}\left( y\right) =\tilde{n}^{B}\left( y\right) =\tilde{n}\left( y\right) $$
$$u^{\prime} \left( \tilde{w}\left( y\right) -\nu \tilde{n}\left( y\right) \right) =\tilde{n}\left( y\right) \beta u^{\prime} \left( y\right) $$
$$-\nu u^{\prime \prime} \left( \tilde{w}\left( y\right) -\nu \tilde{n}\left( y\right) \right) =\beta u^{\prime} \left( y\right) . $$

Figure 3 illustrates the first few iterations of the process described above.

Fig. 3
figure 3


E Proof of Proposition 7

At a point where \(w\leq \hat {w}(y), \) the FC system breaks down and α = 0. Assuming the fertile still choose to have children, the fertility choice n satisfies

$$ u^{\prime} \left( w-\tau -\nu n\right) =\left( \theta /\nu \right) g^{\prime} \left( n\right) $$

Differentiating totally this expression, we obtain

$$dn/d\tau =-\frac{u^{\prime \prime} \left( w-\tau -\nu n\right)} {\nu u^{\prime \prime} \left( w-\tau -\nu n\right) +\left( \theta /\nu \right) g^{\prime \prime} \left( n\right)} <0 $$

Let n(w) be defined implicitly as the value of n such that u (wν n) = (𝜃/ν)g (n). Since, by definition, \( u^{\prime } \left (\hat {w}(y)-\nu n\left (y\right ) \right ) =\left (\theta /\nu \right ) g^{\prime } \left (n\left (y)\right ) \right ) \) and \(w\leq \hat {w}(y), \) we have (𝜃/ν)g (n(w)) ≥ (𝜃/ν)g (n(y))) → n(w) ≤ n(y) by the concavity of g(⋅).

Differentiating U(τ) (with the proviso that α = 0) we have

$$\begin{array}{ll} U^{\prime} \left( \tau \right) =-\pi u^{\prime} \left( w-\tau -\nu n\right) +\pi^{2}n\beta u^{\prime} \left( y+\pi \tau n\right) + \\ -\left( 1-\pi \right) u^{\prime} \left( w-\tau \right) +\left( 1-\pi \right) \pi \beta nu^{\prime} \left( y+\pi \tau n\right) + \\ \left[ \pi \beta u^{\prime} \left( y+\pi \tau n\right) +\left( 1-\pi \right) \beta u^{\prime} \left( y+\pi \tau n\right) \right] \pi \tau dn/d\tau \end{array} $$

The limit of U (τ) as τ → 0 is

$$\begin{array}{cc}-\pi u^{\prime} \left( w-\nu n\left( w\right) \right) +\pi^{2}n\left( w\right) \beta u^{\prime} \left( y\right) -\left( 1-\pi \right) u^{\prime} \left( w\right) +\left( 1-\pi \right) \pi \beta n\left( w\right) u^{\prime} \left( y\right) \\ =-\pi u^{\prime} \left( w-\nu n\left( w\right) \right) -\left( 1-\pi \right) u^{\prime} \left( w\right) +\pi \beta n\left( w\right) u^{\prime} \left( y\right) \\ =\pi \lbrack -u^{\prime} \left( w-\nu n\left( w\right) \right) +\beta n\left( w\right) u^{\prime} \left( y\right) ]-\left( 1-\pi \right) u^{\prime} \left( w\right) \end{array} $$

Since it is not optimal to initiate a family compact when the first period endowment is \(w\leq \hat {w}(y)\), − u (wν n(w)) + π n(w)β u (y) < 0. However, if the shock to w is not too large, it can be that π[−u (wν n(w)) + β n(w)u (y)] > (1 − π)u (w) > 0.

F Discussion and extensions

F.1 An example with logarithmic preferences

Let u(c) = ln c and \(\beta w>2\sqrt {\left (1 + 2\beta \right ) \nu y}\). For this example, an interior solution for the number of children, n, ((2) above) satisfies

$$\frac{\nu} {w-\alpha_{t}-\nu n_{t}}=\frac{\beta \alpha_{t + 1}}{y+\alpha_{t + 1}n_{t}} $$

which implies a solution

$$ n\left( \alpha_{t};\alpha_{t + 1}\right) =\frac{\beta \alpha_{t + 1}w-\beta \alpha_{t}\alpha_{t + 1}-\nu y}{\alpha_{t + 1}\nu \left( 1+\beta \right)} . $$

Here, α min = ν y/β w, and, for a given α t , the set \(\left \{ \alpha _{t + 1}:\alpha _{t + 1}\geq \frac {\alpha _{\min } w}{ w-\alpha _{t}}\right \} \) defines the set of permissible future transfers that ensure n(α t ; α t+ 1) ≥ 0. (For any α t+ 1 not in this set, the solution to the agent’s problem at date t is n t = 0).

With α t = α t+ 1 = α for all t, n is increasing in α, for \(\alpha <\sqrt {y\nu /\beta } \) and decreasing thereafter. The set of stationary compacts which ensure \(n\left (\alpha ;\alpha \right ) 0, A=\left \{ \alpha :\frac {1}{2\beta } \left (w\beta -\sqrt {w^{2}\beta ^{2}-4y\beta \nu } \right ) <\alpha <\frac {1}{2\beta } \left (w\beta +\sqrt { w^{2}\beta ^{2}-4y\beta \nu } \right ) \right \}\).

An agent’s utility under a compact is:

$$\begin{array}{@{}rcl@{}} V\left( \alpha_{t};\alpha_{t + 1}\right) &=&\ln \left( w-\nu \frac{\beta \alpha_{t + 1}w-\beta \alpha_{t}\alpha_{t + 1}-\nu y}{\alpha_{t + 1}\nu \left( 1+\beta \right)} -\alpha_{t}\right) \\ &&+\beta \ln \left( y+\alpha_{t + 1}\frac{ \beta \alpha_{t + 1}w-\beta \alpha_{t}\alpha_{t + 1}-\nu y}{\alpha_{t + 1}\nu \left( 1+\beta \right)} \right) \\ &=&\left( 1+\beta \right) \ln \left( \alpha_{t + 1}\left( w-\alpha_{t}\right) +y\nu \right) -\ln \alpha_{t + 1}+C, \end{array} $$

where Cβ ln β −(1 + β) ln (1 + β) − β ln ν, is a constant. The expression for lifetime utility, V(α t ; α t+ 1), is valid for all pairs (α t ; α t+ 1), α t ,α t+ 1w, with n(α t ; α t+ 1) ≥ 0. In particular, when α t = α t+ 1 = α, V(α) is defined for all αA. (Note that in this case, the expression, wν n(α; α) − α, which represents first period consumption, equals \(\frac {\left (w\alpha -\alpha ^{2}+y\nu \right )} {\alpha \left (1+\beta \right )} \); this term is positive for all compacts 0 < α < w). We have

$$ V\left( \alpha \right) =\left( 1+\beta \right) \ln \left( w\alpha +y\nu -\alpha^{2}\right) -\ln \alpha +C. $$

The limit of lifetime utility V(α) as α approaches the lower bound of A,α̲ A is \(\ln \left (w-\underline {\alpha }_{A}\right ) +\beta \ln y, \) and at its upper bound, \( \overline {\alpha }_{A}, \) is \(\ln \left (w-\overline {\alpha } _{A}\right ) +\beta \ln y\); both are less than the benchmark (no-compact) alternative ln w + β ln y. As we indicated above, the set A of transfers that equal or beat the benchmark alternative is contained in A.

For a given α t , the mapping ϕ(α t ) is defined implicitly in the following manner:

$$\left( 1+\beta \right) \ln \left( \phi \left( \alpha_{t}\right) \left( w-\alpha_{t}\right) +y\nu \right) -\ln \phi \left( \alpha_{t}\right) +C=\ln w+\beta \ln y, $$

and the points \(\underline {\alpha } \) and \(\overline {\alpha } \) satisfy

$$\left( 1+\beta \right) \ln \left( \underline{\alpha} \left( w-\underline{ \alpha} \right) +y\nu \right) -\ln \underline{\alpha} +C=\ln w+\beta \ln y $$
$$\left( 1+\beta \right) \ln \left( \overline{\alpha} \left( w-\overline{ \alpha} \right) +y\nu \right) -\ln \overline{\alpha} +C=\ln w+\beta \ln y. $$

The set of stationary compacts \(A^{\prime } \equiv \left \{ \alpha : \underline {\alpha } \leq \alpha \leq \overline {\alpha } \right \} \).

Above, we showed that the optimal stationary compact satisfies the condition, \( n=\frac {\alpha } {\nu } \). Using the expression n(α; α), we can solve directly for α.Footnote 28

$$ \alpha =\frac{w\beta \pm \sqrt{w^{2}\beta^{2}-4y\nu -8\beta y\nu} }{4\beta + 2}. $$

Call these two roots α and α +. The assumption \(\beta w>2\sqrt {\left (1 + 2\beta \right ) \nu y}\) ensures the solutions (38) are real valued.

How can we be assured that lifetime utility V(α) under one of these solutions exceeds the benchmark u(w) + β u(y)? First, note that

$$V^{\prime} \left( \alpha \right) =-\frac{y\nu +\alpha^{2}+ 2\alpha^{2}\beta -w\alpha \beta} {\alpha \left( \alpha \left( w-\alpha \right) +\nu y\right)} $$

which is negative for low α, and, in particular, for α < α . Second, we note that at the boundaries of the set of permissible stationary compacts with n ≥ 0, we have \(V\left (\overline {\alpha }_{A}\right ) <V\left (\underline {\alpha }_{A}\right ) <u\left (w\right ) +\beta u\left (y\right ) \). Finally, we show \(\underline {\alpha }_{A}<\alpha ^{-}<\alpha ^{+}<\overline {\alpha }_{A}:\)

$$\begin{array}{cc}\overline{\alpha}_{A}-\alpha^{+}=\frac{1}{2\beta} \left( w\beta +\sqrt{ w^{2}\beta^{2}-4y\beta \nu} \right) -\frac{w\beta +\sqrt{w^{2}\beta^{2}-4y\nu -8\beta y\nu} }{4\beta + 2} \\ =\frac{1}{2\beta \left( 2\beta + 1\right)} \left( w\beta \,-\,\beta \sqrt{ w^{2}\beta^{2}-8y\nu \beta -4y\nu} + 2\beta \sqrt{w^{2}\beta^{2}\,-\,4y\beta \nu} +w\beta^{2}+\sqrt{w^{2}\beta^{2}-4y\beta \nu} \right) >0, \end{array} $$

since \(w\beta -\beta \sqrt {w^{2}\beta ^{2}-8y\nu \beta -4y\nu } >w\beta -\beta \sqrt {w^{2}\beta ^{2}}=w\beta \left (1-\beta \right ) >0\).


$$\begin{array}{cc}\alpha^{-}-\underline{\alpha}_{A}=\frac{w\beta -\sqrt{w^{2}\beta^{2}-4y\nu -8\beta y\nu} }{4\beta + 2}-\frac{1}{2\beta} \left( w\beta -\sqrt{ w^{2}\beta^{2}-4y\beta \nu} \right) \\ =\frac{1}{2\beta \left( 2\beta + 1\right)} \left( \left( 2\beta + 1\right) \sqrt{w^{2}\beta^{2}-4y\beta \nu} -\beta \sqrt{w^{2}\beta^{2}-8y\nu \beta -4y\nu} -w\beta \left( 1+\beta \right) \right) \end{array} $$

This latter expression is decreasing in y. Using the assumption \(\beta w>2\sqrt {\left (1 + 2\beta \right ) \nu y}\Rightarrow \frac { \left (\beta w\right )^{2}}{4\left (1 + 2\beta \right ) \nu } >y, \) we evaluate at \(y=\frac {\left (\beta w\right )^{2}}{4\left (1 + 2\beta \right ) \nu } :\)

$$\begin{array}{cc}\alpha^{-}-\underline{\alpha}_{A}>\frac{w\beta} {2\beta \left( 2\beta + 1\right)} \left( \left( 2\beta + 1\right) \sqrt{\frac{1+\beta} {2\beta + 1}} -\left( 1+\beta \right) \right) \\ =\frac{w\beta} {2\beta \left( 2\beta + 1\right)} \left( \sqrt{2\beta + 1}\sqrt{ 1+\beta} -\left( 1+\beta \right) \right) \\ >\frac{w\beta} {2\beta \left( 2\beta + 1\right)} \left( \sqrt{1+\beta} \sqrt{ 1+\beta} -\left( 1+\beta \right) \right) = 0 \end{array} $$

The stationary compact and the optimal compact, α + are depicted below in the figure below:

F.2 The compact’s fine print

An efficient way to characterize a compact is to make the distinction between obligations, α t ,actions, \( \widehat { \alpha }_{t}, \) and contingent obligations \(\widetilde {\alpha }_{t}\), of which the latter can be thought of as a description of how a child, in accordance with the compact, should respond to the actions of a parent. Define a history of the family obligations as a list of the amount of obligatory donations up to the date t, t ≥ 2 as H t ≡ (α 1,α 2,...,α t− 1), and likewise, for actions \(\widehat {H}^{t}\) and contingent obligations \(\widetilde {H}^{t}\).

We envision the matriarch of this family compact, the middle age decision-maker at date 1, proposing a sequence of family transfer obligations for all future generations \(\left \{ \alpha _{t}\right \}_{t = 2}^{\infty } \) , as well as a transfer to her own parents, α 1 > 0 at date 1. The compact also lays out how individuals should act (contingent obligations \(\left \{ \widetilde {\alpha }_{t}\right \}_{t = 2}^{\infty } \). Trivially, we set \(\alpha _{1}=\widehat {\alpha }_{1}=\widetilde {\alpha }_{1}\).

At date 2, a middle age decision-maker faces obligation α 2 and contingent obligation \(\widetilde {\alpha }_{2}, \) which, in this case, since \(\alpha _{1}=\widehat {\alpha }_{1}=\widetilde {\alpha }_{1}\), \( \widetilde {\alpha }_{2}\) is equal to α 2. In other words, since her parent at date 1 selected action \(\widehat {\alpha }_{1}=\widetilde {\alpha }_{1}, \) the minimum transfer she needs to make at date 2 to be in compliance with the compact is the obligation α 2.Footnote 29 At this time, she chooses an action \(\widehat {\alpha }_{2}\) (this choice is described in detail further below), which may or may not be in compliance with the compact.

The compact at this stage becomes more descriptive. A middle age decision-maker at date 2 is obliged to pass along a transfer \(\widetilde { \alpha }_{2}=\alpha _{2}\). At date 3, the parent’s child knows whether the parent’s action \(\widehat {\alpha }_{2}\) lived up to the compact or not. Therefore, the contingent obligation facing this child, at date 3, is

$$ \widetilde{\alpha}_{3}=\left\{ \begin{array}{cc} \alpha_{3} & \text{if}\ \widehat{\alpha}_{2}=\widetilde{\alpha}_{2} \\ 0 & \text{otherwise} \end{array} \right. . $$

More generally, given \(H^{t},\widetilde {H}^{t}\) and \(\widehat {H}^{t}, \) for t ≥ 2, a contingent obligation facing a middle age agent at date t consists of a transfer plan \(\widetilde {\alpha }_{t}\), with

$$ \widetilde{\alpha}_{t}=\left\{ \begin{array}{cc} \alpha_{t} & \text{if} \ \widehat{\alpha}_{t-1}=\widetilde{\alpha}_{t-1} \\ 0 & \text{otherwise} \end{array} \right. . $$

Since these contingent plans are defined recursively, it is useful to express them in terms of actions and obligations, as well as updated one period:

$$ \widetilde{\alpha}_{t + 1}=\left\{ \begin{array}{ccc} \alpha_{t + 1} & \text{if} \ \widehat{\alpha}_{t}=\alpha_{t} & \text{(\textbf{A)}} \\ \alpha_{t + 1} & \text{if} \ \widehat{\alpha}_{t}= 0\text{ and} \ \widehat{\alpha} _{t-1}\neq \widetilde{\alpha}_{t-1} & \text{(\textbf{B)}} \\ 0 & \text{otherwise} & \text{(\textbf{C)}} \end{array} \right. . $$

Note that in this form, an offspring of a middle age decision-maker at date t is obliged to follow the compact provided her parent passes along a sufficient transfer to her parent at date t (A). The compact also stipulates that the child, at t + 1, is obliged to pass on α t+ 1 in the event that her parent, in fulfilling her compact compact commitments, does not leave a transfer to her parent (the child’s grandparent) at t in the event that the grandparent failed to carry out the contingent obligation to her parent (the child’s great grandparent) at t − 1 (B) Any transfer action \(\widehat {\alpha }_{t}\) not meeting the conditions laid out in (A) or (B) of (41) constitutes a failure to live up to the compact at date t and confirms onto the child the obligation to pass along a 0 transfer at t + 1.

The force of these contingency plans (and especially, B) ensures that if the child follows the compact and punishes the parent for noncompliance of the compact at date t, the child will not face a withholding of a transfer from her child at t + 2. Another way to put this is that she is assured the transfer α t+ 2 > 0 even if she withholds a transfer from the wayward parent at t + 1 (and, by (C), she is assured punishment at t + 2 if she fails to punish her wayward parent at t + 1). Together, these contingency plans ensure the threat of punishment is credible - the parent’s child has the incentive to mete out a punishment at date t + 1 if the parent fails to live up to her contingent obligation at t - provided the compact obligations α t+ 1,α t+ 2 offer the child something better than what she can do on her own (which we assume they do). Any deviation by one generation simply results in a response by the next that reverts back to the original compact plan (Fig. 4).

Fig. 4
figure 4

Lifetime utility V(α) and the no-compact benchmark

F.3 Private savings and compacts

Throughout, we have assumed, intensionally, the absence of any private vehicle for old-age support (such as saving) other than the family compact. Naturally, such an assumption creates a favorable environment for the existence of a family compact. Below, we explore briefly how the introduction of private savings comes to bear on our analysis of family compacts. We leave out PPS; the reason will be apparent.

Assume at each date, agents have access to a safe savings technology which allows them to convert a unit of time t good into r units of time t + 1 good. The fact that children provide utility to their parents, in addition to serving as an investment good, means the return on saving must dominate that of the family compact if the two are to coexist. Additional conditions on the size of r are discussed below.

Let s t denote the savings of a middle age agent at date t. The agent’s problem is maximize (1) subject to

$$\begin{array}{c}c_{t}+s_{t}+\alpha_{t}+\nu n_{t}\leq w \\ x_{t}\leq rs_{t}+\alpha_{t + 1}n_{t} \\ c_{t}\geq 0;\;x_{t}\geq 0;\;s_{t}\geq 0;\text{} n_{t}\geq 0. \end{array} $$

The first-order conditions for this problem are

  1. d1

    n t : ν u (c t ) − 𝜃 g (n t ) ≥ α t+ 1 β u (x t ) (= if n t > 0),

  2. d2

    s t : u (c t ) ≥ r β u (x t ) (= if s t > 0).

If both solutions are interior, we have the standard intertemporal condition, r = u (c t )/β u (x t ), and α t+ 1/ν = r𝜃 g (n t )/β u (x t ). (As noted, the return on the family compact, α t+ 1/ν, is less than r, since 𝜃 g (n t )/β u (x t ) > 0). Keeping with our focus of stationary compacts, we dispense with time subscripts; the two first-order conditions can be written as

$$\nu u^{\prime} \left( w-s-\nu n\right) -\theta g^{\prime} \left( n\right) -\alpha \beta u^{\prime} \left( y+rs+\alpha n\right) = 0 $$
$$u^{\prime} \left( w-s-\nu n\right) -r\beta u^{\prime} \left( y+rs+\alpha n\right) = 0 $$

and together,

$$ r=u^{\prime} \left( c\right) /\beta u^{\prime} \left( x\right) =u^{\prime} \left( w-s-\nu n\right) /\beta u^{\prime} \left( y+rs+\alpha n\right) $$
$$ \alpha /\nu =r-\theta g^{\prime} \left( n\right) /\beta u^{\prime} \left( x\right) =r-\theta g^{\prime} \left( n\right) /\beta u^{\prime} \left( y+rs+\alpha n+b\right) $$

Taking the derivative of the indirect, steady-state utility with respect to α yields the first-order condition

$$ -u^{\prime} \left( c\right) +\beta nu^{\prime} \left( x\right) \leq 0 $$

If the family compact and private saving are to be jointly operative, then (44) equals 0; together with (42), we would have n = r, i.e., the compact would require agents to have r children. In that case, the optimal family compact would line up the intertemporal transfer of resources in exactly the same way as what the individual would choose to do via private saving. The problem with this solution is that the individual may wish to have more or less children than r, a problem that manifests itself in the fact that the second-order condition required for such an optimum is violated.Footnote 30 Formally, we have

Proposition 8

There is no stationary equilibrium with positive saving s > 0and an optimal family transfer α > 0.

We illustrate this point with the case where g(a) = u(a) = ln a for a = c,x,n.

If s > 0 and n > 0, the agent faces the lifetime budget constraint (for an arbitrary compact transfer α):

$$ c+x/r+\nu n=w+y/r-\alpha +\alpha n/r. $$

Together with the marginal conditions, c = n(ν rα)/ (𝜃 r), x = β r c, one can solve for n as a function of α :

$$ n\left( \alpha \right) =\frac{w+y/r-\alpha} {\left( \left( \nu r-\alpha \right) /\left( \theta r\right) \right) \left( 1+\beta +\theta \right)} $$

The agent’s indirect utility is written as

$$ W\left( \alpha \right) =\left( 1\,+\,\beta \,+\,\theta \right) \ln \left[ \frac{ w+y/r-\alpha} {\left( \left( \nu r\,-\,\alpha \right) /\left( \theta r\right) \right) \left( 1\,+\,\beta \,+\,\theta \right)} \right] +\left( 1+\beta \right) \ln \left[ \left( \nu r\,-\,\alpha \right) /\left( \theta r\right) \right] . $$

We then have:

$$W^{\prime} \left( \alpha \right) =\frac{r\alpha \left( 1+\beta \right) +\theta \left( rw+y\right) -r^{2}\left( 1+\theta +\beta \right) \nu} {\left( r\nu -\alpha \right) \left( rw+y-r\alpha \right)} $$

Setting W (α) = 0 and solving for α, we have

$$\alpha^{\ast} =\frac{\nu r^{2}\left( 1+\beta +\theta \right) -\theta \left( wr+y\right)} {\left( 1+\beta \right) r}. $$

This solution assumes \(r\geq \frac {w\theta +\sqrt {w^{2}\theta ^{2}+ 4\nu y\theta \left (1+\beta +\theta \right )} }{2\nu \left (1+\beta +\theta \right )}\), and it also presumes that savings s(α )≥ 0; for the moment, assume both conditions hold. Evaluating the second derivative of the agent’s indirect utility W(α) at α yields

$$W^{\prime \prime} \left( \alpha^{\ast} \right) =\frac{\left( 1+\beta \right)^{3}r^{2}}{\theta \left( 1+\beta +\theta \right) \left( rw+y-\nu r^{r}\right)} >0; $$

hence, W(α) cannot attain a maximum at α .The following numerical example illustrates this general point:

Example 1

Letg(a) = u(a) = ln a fora = c,x,n,andw = 10; y = 2; β = 0.85; 𝜃 = 0.25; ν = 2; r = 1.5.

Figure 5 below shows a plot of W(α) as a function of the compact transfer α; for illustrative purposes, the graph includes values for α for which s(α) < 0. In this instance, the compact that sets n = r (that is, at α ≈ 1.87) minimizes the function W(α). Here savings becomes negative for values of α > 2.06. Incorporating, then, the non-negativity constraint on s(α), and since W(0) > W(2.06), the best compact – in terms of maximizing steady-state per capita utility – sets α = 0, effectively destroying the family compact altogether. At α = 0, savings s > 0, the number of children, n < r, and children are solely consumer goods.

Fig. 5
figure 5

Indirect utility W(α)

What happens in instances where α < 0? Consider the following modification of the previous example:

Example 2

Let g(a) = u(a) = ln a for a = c,x,n,and w = 10; y = 2; β = 0.85; 𝜃 = 1; ν = 2; r = 1.5.

Note the only distinction between these examples is the size of the parameter, 𝜃, governing the individual’s preference towards having children. In this instance, α ≈− 1.50. Figure 6 below shows a plot of W(α).

Fig. 6
figure 6

Indirect utility W(α)

Here, savings becomes negative for α > 0.98. At α = 0.98, W(0.98) ≈ 3.58. However, the best compact, with s = 0, sets α ≈ 1.86, and steady-state utility equals 3.67. In other words, the best family compact is determined along the lines we have presented in the version of our model without savings, since the best compact will offer a transfer α that is sufficiently high that agents choose not to save.

In summary, savings and family compacts can coexist. Yet, if the compact is optimally chosen, one of two possibilities emerge: either children no longer serve as investment goods and the family transfer is 0, or, old-age security is provided exclusively by the family compact and s = 0. The underlying trade-off is simple. If children do not provide much utility as consumption goods, then agents, from Generation 1 onward, are better off if old-age consumption is financed out of savings. By way of contrast, if children are highly valued as consumption goods, then it is efficient to use them exclusively to facilitate old-age support. And, it is in the latter case that our original omission of savings in Section 3 is of little consequence.

F.4 Enhancing the PPS: Numerical Examples

Having shown a basic public pension plan can be welfare-enhancing even in the presence of a thriving family compact, we turn to the question of whether adornments to that plan (such as more favorable payments to infertile old agents or child rearing subsidies for fertile/matched middle agents) have merit. We are drawn to these possibilities for two reasons. First, those charged with crafting the US social security felt it necessary to go beyond a simple plan of elderly income support, embellishing it with other features, such as redistributive returns, in order to make it more acceptable to the general electorate.Footnote 31 Second, as noted in our introduction to this section, Cigno (2010) outlines how a hybrid public-private pension system might work. Our examples below lend support to a kind of subsidy-redistributive pension plan envisioned by Cigno.

The government’s problem, at this point, has policy variables b,b , λ and τ which conform to the constraint (26). We adjust the redistributive nature of the social security payment by setting b = ρ b, where 1 ≤ ρ; when ρ > 1, the system transfers relatively more resources to each childless elderly at each date (what we referred to in the Introduction as redistributive returns). The budget constraint then reduces to

$$ \tau =\left( \frac{\pi +\left( 1-\pi \right) \rho} {\pi} \right) \frac{b}{n} +\left( 1-\lambda \right) \pi \nu n. $$

The utility of the agent in the steady state is

$$\hat{U}\left( \tau ;\lambda ;\rho ;b\right) \equiv \pi U^{f}\left( \tau ;\lambda ;b\right) +\left( 1-\pi \right) U^{i,n}\left( \tau ;\rho b\right) $$

where U f(τ; λ; b), (U i,n(τ; ρ b)) is the indirect lifetime utility of a fertile/matched (infertile/unmatched) agent.

Our main objective in this section is to highlight the point that once we step beyond a pension-only role for social security, a publicly funded system can be welfare enhancing. Towards this end, policies such as subsidies to child-rearing (λ) and redistributive returns to the public system (ρ) are of secondary importance. With that in mind, we focus on optimal (τ,b) pairs for given policies (λ,ρ). This in turn will allow us to see how the public system fares, alone, without the adornments (λ,ρ), as well as emphasize how attaching these other features to the public system can improve upon the overall pension system. The following example illustrates these points.

Example 3

Let u(c) = ln c; 𝜃 = 0; w = 5; y = 1/4; β = .95; π = .85; ν = 1; r = .80.

The table below provides summaries of the key variables of this model, for different configurations of the pair (λ,ρ). The last three columns indicate the percentage over baseline (no public system).

Table 1

The table above illustrates a number of key features of these initiatives. First, as confirmed in Proposition 6, a traditional pension system without the additional embellishments (ρ = 1,λ = 1) will raise steady-state ex ante lifetime utility—intuitively, this is not too surprising, since there is a reasonably chance (15%) an agent is born infertile/unmatched, and the public system provides some cushion against low old age consumption due to low income when old and low return to saving (y is 95% lower than w and r = .80, respectively). Of course, ex post, fertile agents prefer no PPS, since the family compact provides higher returns. A similar situation emerges when the returns to social security become more redistributive, but no tax revenue is diverted to subsidizing child rearing (ρ = 1.25,λ = 1).

Likewise, if sufficient tax revenue is returned to fertile couples in the form of subsidies for children, the PPS fails to improve the lot of the childless, ex post(ρ = 1,λ = .75). More progressive policies (ρ = 1.25,λ = .75) and (ρ = 1.50,λ = .50) provide gains ex ante and ex post for both types of agents. These gains, however, come at the price of lower transfers to the initial old, as can be seen by comparing the b and b columns for row 1 with that of rows 4 and 5.

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Barnett, R.C., Bhattacharya, J. & Puhakka, M. Private versus public old-age security. J Popul Econ 31, 703–746 (2018).

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  • Fertility
  • Family compacts
  • Social security
  • Intergenerational cooperation
  • Pensions
  • Self-enforcing constitutions

JEL Classification

  • E 21
  • E 32