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Childlessness, childfreeness and compensation

Abstract

We study the design of a fair family policy in an economy where parenthood is regarded either as desirable or as undesirable, and where there is imperfect fertility control, leading to involuntary childlessness/parenthood. Using an equivalent consumption approach in the consumption-fertility space, we show that the identification of the worst-off individuals depends on how the social evaluator fixes the reference fertility level. Adopting the ex post egalitarian criterion (giving priority to the worst off in realized terms), we study the compensation for involuntary childlessness/parenthood. Unlike real-world family policies, the fair family policy does not always involve positive family allowances, and may also include positive childlessness allowances. Our results are robust to assuming asymmetric information and to introducing Assisted Reproductive Technologies.

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Notes

  1. See Gauthier and Hatzius (1997), D’Addio and Mira d’Ercole (2005), Luci-Greulich and Thévenon (2013). One exception is Kalwij (2010), who finds that family allowances have no significant impact on fertility in Western Europe.

  2. On distributive effects, see Balestrino et al. (2002) and Pestieau and Ponthiere (2013).

  3. On population ethics and its paradoxes, classical references are Parfit (1984) and Blackorby et al. (2005). Those pieces of work question all standard social criteria in the context of varying population size, and, as such, question also a purely “productivist” evaluation of family policies in terms of their impact on the number of births.

  4. Childlessness as a whole concerns about 15 % of a women cohort, with variations across countries and epochs (Sobotka 2017). Childlessness is less widespread in France in comparison to Germany or the UK (Koppen et al. 2017; Berrington 2017; Kreyenfeld and Konietzka 2017).

  5. Toulemon (2001) shows, in the case of France, that about 5% of individuals state that childlessness is the most ideal living arrangement, whereas most men and women say that 2 or 3 is the ideal number of children. See also Kuhnt et al. (2017).

  6. The function \(v(\cdot )\) incorporates any utility gain obtained from having a (desired) child. This includes the pure joy of having a (desired) child as well as all other fertility motives.

  7. The desire for children may reinforce the marginal utility of consumption (reinforcement effect: \(u^{\prime }(c)>U^{\prime }(c)\) for a given c), or, alternatively, it may weaken the marginal utility of consumption (redundancy effect: \(u^{\prime }(c)<U^{\prime }(c)\) for a given c).

  8. Note that a model with pure random fertility would involve \(\pi =\varepsilon\) (same probability to have a child for all individuals, independently from their willingness to have a child), which is unrealistic.

  9. The assumption \(\varepsilon >0\) implies a positive number of children available for adoption.

  10. Under the Law of Large Numbers, probabilities \(\pi\) and \(\varepsilon\) will also determine, together with the parameter x, the proportions of the different types of individuals in the population.

  11. By doing so, we would obtain, in a hypothetical world where children were costless and where each individual enjoyed his ideal fertility level, that there would be no justification for transfers across individuals. While this point may support the reliance on case R3 below, there exist other arguments justifying to consider also other cases (see infra).

  12. As we already noted, one argument supporting R1 and R2 is simplicity, whereas another argument supporting R3 concerns the implications in terms of transfers in a world where all individuals would have their ideal outcomes. But other arguments supporting some reference levels exist. As we will see, one possible argument for the selection of case R1 may lie in the incentive-compatible nature of the implied optimal first-best allocations (see Sect. 5).

  13. For instance, when considering issues of health, the good health status is a salient reference level (Fleurbaey 2005). Similarly, when considering issues of life and death, the maximum lifespan is also a standard reference level (Fleurbaey et al. 2014).

  14. One may argue that progress in contraception is such that involuntary parents could be regarded as responsible for having a child (lack of prevention). Note, however, that similar preventive behaviors can give rise to distinct fertility outcomes, due to accidents or circumstances. Hence involuntary parents cannot be regarded as responsible for having a child.

  15. One could reply to this that involuntary parents knew about the imperfect reliability of contraception, a case of “option luck” instead of “brute luck” (Dworkin 2000). However, we adopt the ethical stance that involuntary parents should not be held responsible for the welfare loss due to having a child. The childfree and the involuntary parents are ex ante identical, and there is no good reason why they should have unequal welfare ex post (Fleurbaey 2010).

  16. Axiomatic foundations for maximin criteria defined on consumption equivalents can be found in Fleurbaey and Maniquet (2011). From an axiomatic perspective, each criterion based on distinct reference fertility levels (cases R1 to R3) would require a distinct characterization. The reliance on a reference fertility level is an ethical assumption on its own, which, under some conditions, follows from adopting a particular axiom defining socially desirable transfers among individuals who have distinct preferences but enjoy the reference fertility level(s).

  17. See Fleurbaey and Maniquet (2011) on the tensions between the welfarist approach and the dominance approach in social valuations.

  18. For instance, in the consumption-lifetime space, Fleurbaey et al. (2014) provide a characterization of the maximin on equivalent consumptions, based on the Pareto Principle, Hanson Independence, and two transfer axioms: the Pigou–Dalton axiom for same preferences, and the Pigou–Dalton axiom for different preferences and reference lifetime.

  19. The reason why we allow for the reallocation of children in our benchmark case lies in the fact that adoption policies have been widely used across countries and epochs, as instruments allowing to reduce the prevalence of involuntary childlessness. For the sake of robustness to ethical foundations, Sect. 4.5 characterizes a constrained social optimum where the reallocation of children (adoption) is prohibited.

  20. Surrogacy is not allowed here, so that the social planner takes the total number of children as given. The consequences of introducing surrogacy are studied in Sect. 4.4.

  21. The case of excess supply of children is excluded under our assumption \(0<\varepsilon \le \frac{\left( 1-\pi \right) x}{\left( 1-x\right) }\).

  22. This assumption is elicited in Sect. 4.3 for particular forms for \(u\left( \cdot \right)\), \(U(\cdot )\), \(v(\cdot )\) and \(V\left( \cdot \right) .\)

  23. Alternatively, if there were perfect complementarity between consumption and parenthood, transfers would not allow to achieve full compensation.

  24. Those assumptions are discussed below.

  25. Under Assumption A1, the involuntary childless are better off adopting a child, and involuntary parents are better off leaving their child (since \(U(w(1-q)-g)-V(1)<U(w)\)).

  26. Note that case R3, which accounts for some form of “neutrality” of the social planner towards preferences (Fleurbaey and Maniquet 2019), still involves transfers affecting voluntary parents and childfree individuals.

  27. Indeed, in case of an individual cost, this would require to give additional monetary transfers to involuntary childless who would engage in the process of adopting a child. If it is a cost to the society as a whole, this would enter the government budget constraint and modify the amounts of lump-sum transfers to be made.

  28. Similar conditions could be derived for the case R3.

  29. Given that the interests of children are not easily observable, prohibiting all reallocations of children is a way to avoid reallocations that could worsen the situation of some of them.

  30. Quite realistically, we assume that having a child or not is observable, so that mimicking can never happen on this dimension.

  31. In other words, a pure adoption system (stages 1 to 4) would be incentive-compatible.

  32. Under R3, since in the first-best, \(c_{1}=c_{4}\), \(c_{1}<c_{4}<c_{2}\) is the unique second-best solution.

  33. The same rankings of consumption equivalents are obtained under R3.

  34. On ART, see Trappe (2017). This section abstracts from surrogacy (see Sect. 4.4).

  35. This amounts to assume that \(\varepsilon =0\) (i.e. perfect contraception).

  36. We use, here again, the Law of Large Numbers.

  37. That result, which is close to the result of zero prevention (against mortality) in Fleurbaey and Ponthiere (2013), is due to the fact that there is a conflict between the goal of ex post compensation and the goal of investing in costly prevention that may be unsuccessful.

  38. If we fixed ART to zero, we would be left with three types: voluntary parents, childless individuals and childfree persons, which would be a reduced form of the model studied above.

  39. For simplicity, we assume here that childfree individuals have the same disutility of the ART treatment as other individuals.

  40. One could oppose that the relevant incentive-compatibility constraint is:

    $$\begin{aligned} U(c_{4})\ge {\tilde{p}}[U(c_{2})-V(1)-\varphi (\ell )]+(1-{\tilde{p}} )[U(c_{3})-\varphi (\ell )] \end{aligned}$$

    where the RHS is the expected utility of investing in ART, and \({\tilde{p}}\) is the probability of successful ART for individuals who do not want the treatment to be successful. Assuming that a type 4 can make the treatment inoperative (leading to \({\tilde{p}}\rightarrow 0\)), the two formulations are equivalent.

  41. One could oppose that the incentive-compatibility constraint has to include the expected utility of investing in ART. If we assume that \(\tilde{ p}\rightarrow 0\) for type 4 (i.e. sabotage of ART), the two formulations are equivalent.

  42. When the sign of the transfer is ambiguous, its sign depends on the specific forms of u(.), U(.), v(.), V(.) and on parameter values.

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Correspondence to Marie-Louise Leroux.

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The authors would like to thank A. Baurin, D. de la Croix, J. Drouard, M. Fall, M. Fleurbaey, P. Gobbi, B. Le Maux, E. Malin, F. Maniquet, F. Moizeau, Y. Nishimura, T. Penard, L. Pensionero, M. Shrime, two anonymous reviewers and participants of seminars at Princeton, CREM (Rennes), IRES (Louvain) and Osaka (SOPE), for their suggestions and comments on this paper.

Appendix

Appendix

Proof of Proposition 2

Excess demand of children

Under R1, and once type 3 has disappeared thanks to the reallocation of children, the problem of the planner is:

$$\begin{aligned}&\max _{c^{1},c^{2},c^{4}} {\hat{c}}^{2} \\&\quad \text {s.t. }(\pi x+\varepsilon (1-x))c_{1}+(x(1-\pi )-\varepsilon (1-x))c_{2}+(1-x)c_{4}+(\pi x+\varepsilon (1-x))g \\&\quad =(\pi x+\varepsilon (1-x))w(1-q)+(x(1-\pi )+(1-\varepsilon )(1-x))w \\&\quad \text {s.t. }{\hat{c}}^{1}\ge {\hat{c}}^{2}\text {, }{\hat{c}}^{4}\ge {\hat{c}}^{2} \text {, }{\hat{c}}^{1}\ge {\hat{c}}^{4} \end{aligned}$$

where consumption equivalents \({\hat{c}}_{i}\) are defined by

$$\begin{aligned} u({\hat{c}}^{1})+v(0)&= {} u(c^{1})+v(1)\iff {\hat{c}}^{1}=u^{-1}\left( u(c^{1})+v(1)\right) \\ u({\hat{c}}^{2})+v(0)&= {} u(c^{2})+v(0)\iff {\hat{c}}^{2}=c^{2} \\ U({\hat{c}}^{4})-V(0)&= {} U(c^{4})-V(0)\iff {\hat{c}}^{4}=c^{4} \end{aligned}$$

At the optimum, the egalitarian constraints are binding so that:

$$\begin{aligned} {\hat{c}}^{1}={\hat{c}}^{2}\implies c^{2}>c^{1}\text { and }{\hat{c}}^{2}={\hat{c}} ^{4}\implies c^{2}=c^{4}. \end{aligned}$$

Under R2, the problem of the planner is the same as under R1, except that equivalent consumptions are defined as \({\tilde{c}}^{i}\) and are given by:

$$\begin{aligned} u\left( {\tilde{c}}^{1}\right) +v(1)&= {} u(c^{1})+v(1)\iff {\tilde{c}}^{1}=c^{1} \\ u\left( {\tilde{c}}^{2}\right) +v(1)&= {} u(c^{2})+v(0)\iff {\tilde{c}}^{2}=u^{-1} \left[ u(c^{2})-v(1)\right] \\ U({\tilde{c}}^{4})-V(1)&= {} U(c^{4})-V(0)\iff {\tilde{c}}^{4}=U^{-1}\left[ U(c^{4})+V(1)\right] \end{aligned}$$

At the optimum, the egalitarian constraints are binding so that:

$$\begin{aligned} {\tilde{c}}^{1}={\tilde{c}}^{2}\implies c^{2}>c^{1}\quad \text {and}\quad {\tilde{c}}^{4}= {\tilde{c}}^{1}\implies c^{1}>c^{4}. \end{aligned}$$

Under R3, the problem of the planner is the same as under R1, except that equivalent consumptions are defined as \({\bar{c}}^{i}\) and satisfy:

$$\begin{aligned} u\left( {\bar{c}}^{1}\right) +v(1)&= {} u(c^{1})+v(1)\iff {\bar{c}}^{1}=c^{1} \\ u\left( {\bar{c}}^{2}\right) +v(1)&= {} u(c^{2})+v(0)\iff {\bar{c}}^{2}=u^{-1} \left[ u(c^{2})-v(1)\right] \\ U({\bar{c}}^{4})-V(0)&= {} U(c^{4})-V(0)\iff {\bar{c}}^{4}=c^{4} \end{aligned}$$

The egalitarian constraints are binding so that:

$$\begin{aligned} {\bar{c}}^{1}={\bar{c}}^{2}\implies c^{2}>c^{1}\quad \text {and}\quad {\bar{c}}^{1}={\bar{c}} ^{4}\implies c^{1}=c^{4}. \end{aligned}$$

Equal supply and demand for children

The economy is, after reallocation of children, composed of voluntary parents and of childfree individuals. The resource constraint is: \(xc_{1}+(1-x)c_{4}+xg=xw(1-q)+(1-x)w\). Under R1, we can show that the optimal allocation is \(c^{4}>c^{1}\). Under R2, the optimal allocation is \(c^{1}>c^{4}\). Under R3, the optimal allocation is \(c^{1}=c^{4}\).

Proof of Proposition 3

Let us consider four lump-sum transfers \(b_{i}\) given to our types \(i=\{1,2,3,4\}\).

Under excess demand for children, there remain, after the reallocation of children, three types: voluntary parents who receive \(b_{1}\), involuntary childless with \(b_{2}\) and voluntary childless with \(b_{4}\). The budget constraint is:

$$\begin{aligned} b_{1}(x\pi +\varepsilon (1-x))+(x(1-\pi )-\varepsilon (1-x))b_{2}+(1-x)b_{4}=0. \end{aligned}$$
(6)

Under R1, the decentralization requires \(c^{4,D}=c^{2,D}>c^{1,D}\), where D stands for Decentralization, so as to achieve \({\hat{c}}^{1}={\hat{c}}^{2}=\hat{c }^{4}\). Transfers \(b_{i}\) satisfy:

$$\begin{aligned} u(w(1-q)-g+b_{1})+v(1)= {} u(w+b_{2}) \end{aligned}$$
(7)
$$\begin{aligned} w+b_{2}= {} w+b_{4} \end{aligned}$$
(8)

The first equation ensures that \(c^{1,D}=w(1-q)-g+b_{1}\) and \(c^{2,D}=w+b_{2}\) are set such that \({\hat{c}}^{1}={\hat{c}}^{2}\). The second equality ensures that \({\hat{c}}^{2}={\hat{c}}^{4}\). Equation (8) yields \(b_{2}=b_{4}=b\). Using equation (7) together with Assumption A1, we obtain that \(b_{1}<b\). For the budget constraint to be satisfied, only one solution is possible: \(b_{1}<0<b\).

Under R2, the ranking of consumptions at the decentralized optimum is: \(c^{2,D}>c^{1,D}>c^{4,D}\), so as to achieve \({\tilde{c}}^{1}={\tilde{c}}^{2}= {\tilde{c}}^{4}\). Transfers satisfy:

$$\begin{aligned} u(w(1-q)-g+b_{1})+v(1)= {} u(w+b_{2}) \end{aligned}$$
(9)
$$\begin{aligned} w(1-q)-g+b_{1}= {} U^{-1}[U(w+b_{4})+V(1)] \end{aligned}$$
(10)

so that there is equalization of consumption equivalents. Equation (9) yields: \(b_{1}<b_{2}\) and Eq. (10) that \(b_{4}<b_{1}\). Together with the budget constraint, two sets of solutions \(b_{4}<0<b_{1}<b_{2}\) and \(b_{4}<b_{1}<0<b_{2}\) are possible.

Under R3, the ranking of consumptions is \(c^{1,D}=c^{4,D}<c^{2,D}\), so that we achieve \({\bar{c}}^{1}={\bar{c}}^{2}={\bar{c}}^{4}\). Transfers \(b_{1}\), \(b_{2}\) and \(b_{4}\) satisfy:

$$\begin{aligned} w(1-q)-g+b_{1}&= {} w+b_{4}\implies b_{1}>b_{4} \\ u(w(1-q)-g+b_{1})+v(1)&= {} u(w+b_{2})\implies b_{1}<b_{2} \end{aligned}$$

Together with the budget constraint, only two solutions are possible: \(b_{4}<0<b_{1}<b_{2}\) or \(b_{4}<b_{1}<0<b_{2}\).

When there is equality between supply and demand of children, there remain only two types of individuals: voluntary parents who receive \(b_{1}\) and the childfree who receive \(b_{4}\). These transfers satisfy: \(xb_{1}+(1-x)b_{4}=0\) . The solutions are: under R1, \(b_{1}<0<b_{4}\); under R2, \(b_{4}<0<b_{1}\); under R3, \(b_{4}<0<b_{1}\).

Proof of Proposition 4

Assume quasi-linear preferences: \(u(x)=x\) and \(U(x)=x-\alpha\). Take Proof 9.2 of Proposition 3.

Consider case R1. Using the budget constraint and \(b_{2}=b_{4}=b\), we have:

$$\begin{aligned} b=\frac{-\left[ x\pi +\varepsilon (1-x)\right] b_{1}}{\left[ x(1-\pi )+(1-x)(1-\varepsilon )\right] }. \end{aligned}$$

This defines locus A in the \(\left( b_{1},b\right)\) space. Using condition ( 7), we obtain locus B, defined by

$$\begin{aligned} b=-qw-g+b_{1}+v(1) \end{aligned}$$

This is a 45\({{}^\circ }\) line with strictly positive value at \(b_{1}=0\) since under Assumption 1, \(-qw-g+v(1)>0\). We also have \(b=0\) at \(b_{1}=qw+g-v(1)<0\).

The decentralization by a mixed adoption-transfer scheme is possible only if the two loci A and B intersect. One can see that, when an intersection takes place between locus A and B, it must be for \(b_{1}<0,b>0.\) In addition, consumptions cannot be negative, so that \(-\left[ w(1-q)-g\right] <b_{1}\) and \(b>-w\).

If locus B is above locus A at \(b_{1}=-\left[ w(1-q)-g\right]\), the decentralization of the optimum through the mixed system does not hold. That case arises when:

$$\begin{aligned}&-qw-g-\left[ w(1-q)-g\right] +v(1)>\frac{-\left[ x\pi +\varepsilon (1-x) \right] \left[ -\left[ w(1-q)-g\right] \right] }{\left[ x(1-\pi )+(1-x)(1-\varepsilon )\right] } \nonumber \\&\quad \iff w<\frac{g\left[ x\pi +\varepsilon (1-x)\right] +v(1)\left[ x(1-\pi )+(1-x)(1-\varepsilon )\right] }{1-q\varepsilon (1-x)-qx\pi }\equiv {\tilde{w}} _{1} \end{aligned}$$
(11)

Thus only when \(w\ge {\tilde{w}}_{1}\), the mixed system can decentralize the social optimum.

Consider now case R2. Using Eq. (6), we obtain: \(b_{4}=-b_{1}\frac{\left[ x\pi +(1-x)\varepsilon \right] }{(1-x)}-\frac{\left[ x(1-\pi )-(1-x)\varepsilon \right] }{(1-x)}b_{2}\). Moreover, Eq. (9) can be rewritten as

$$\begin{aligned} b_{2}=-wq-g+b_{1}+v(1) \end{aligned}$$

which defines the locus I in the \(\left( b_{1},b_{2}\right)\) space. It is a 45 degree line which crosses the x axis at \(b_{1}=wq+g-v(1)<0\).

Also, replacing for the expression of \(b_4\) into Eq. (10 ), we obtain that:

$$\begin{aligned} w(1-q)-g+b_{1}&= {} w-b_{1}\frac{\left[ x\pi +(1-x)\varepsilon \right] }{(1-x)} -\frac{\left[ x(1-\pi )-(1-x)\varepsilon \right] }{(1-x)}b_{2}+V(1) \\ \rightarrow b_{2}&= {} -b_{1}\frac{1-x(1-\pi )+(1-x)\varepsilon }{\left[ x(1-\pi )-(1-x)\varepsilon \right] }+\frac{(1-x)\left( V(1)+wq+g\right) }{ \left[ x(1-\pi )-(1-x)\varepsilon \right] } \end{aligned}$$

That equation defines locus II.

Consumption cannot be negative so that \(-(w(1-q)-g)<b_{1}\). Moreover, \(b_{2}>-w\). Hence, the transfer system does not decentralize the optimum when the locus I remains above the locus II at \(b_{1}=-(w(1-q)-g)\), that is, when:

$$\begin{aligned} w<\frac{g\left( x\pi +(1-x)\varepsilon \right) -(1-x)V(1)+v(1)\left[ x(1-\pi )-(1-x)\varepsilon \right] }{1-(1-x)\varepsilon q-qx\pi }\equiv {\tilde{w}}_{2} \end{aligned}$$
(12)

The mixed system decentralizes the optimum only if \(w>{\tilde{w}}_{2}\).

Let us finally show that \({{\tilde{w}}}_1>{{\tilde{w}}}_2\). We do so by comparing the RHS of (11) and (12) and acknowledging that \(v(1)>0>-V(1)\).

Proof of Proposition 6

Under R1, the social planner’s problem is:

$$\begin{aligned}&\max _{c^{1},c^{2},c^{3},c^{4}} {\hat{c}}^{3} \\&\quad \text {s.t. }x\pi c_{1}+x(1-\pi )c_{2}+(1-x)\varepsilon c_{3}+(1-x)(1-\varepsilon )c_{4}+\pi xg+\varepsilon (1-x)g \\&\quad =\pi xw(1-q)+(1-\pi )xw+\varepsilon (1-x)w(1-q)+(1-\varepsilon )(1-x)w \\&\quad \text {s.t. }{\hat{c}}^{1}\ge {\hat{c}}^{3}\text {, }{\hat{c}}^{2}\ge {\hat{c}}^{3} \text {, }{\hat{c}}^{4}\ge {\hat{c}}^{3}\text {, }{\hat{c}}^{2}\ge {\hat{c}}^{1} \end{aligned}$$

Assuming that the egalitarian constraints are binding, we have:

$$\begin{aligned} {\hat{c}}^{1}={\hat{c}}^{2}\implies c^{2}>c^{1}\text { and }{\hat{c}}^{4}={\hat{c}} ^{3}\implies c^{3}>c^{4}\text { and }{\hat{c}}^{2}={\hat{c}}^{4}\implies c^{2}=c^{4} \end{aligned}$$

At the optimum under R1, one should implement: \(c^{3}>c^{4}=c^{2}>c^{1}\), so that \({\hat{c}}^{1}={\hat{c}}^{2}={\hat{c}}^{3}={\hat{c}}^{4}\). Similar proofs can be carried out for cases R2 and R3.

Lemma 1

Assuming that the economy is sufficiently productive, the ex post egalitarian optimum involves equal equivalent consumption levels for the four types. Under R1 (\({\bar{n}}=0\)), we have: \(c^{1}<c^{2}=c^{4}<c^{3}\). Under R2 (\({\bar{n}}=1\)), we have \(c^{4}<c^{1}=c^{3}<c^{2}\). Under R3 (\({\bar{n}}_{1,2}=1,\) \({\bar{n}}_{3,4}=0\)), we have \(c^{1}=c^{4}<c^{2},c^{3}\).

Proof

See above. \(\square\)

Let us now consider the decentralization by means of 4 monetary transfers \(b_{i}\) given to types \(i=\{1,2,3,4\}\). The government’s budget constraint is:

$$\begin{aligned} b_{1}x\pi +x(1-\pi )b_{2}+b_{3}(1-x)\varepsilon +(1-x)(1-\varepsilon )b_{4}=0 \end{aligned}$$
(13)

Under R1, the decentralization of the optimum requires: \(c^{3,D}>c^{4,D}=c^{2,D}>c^{1,D}\) where D stands for Decentralization, so as to achieve \({\hat{c}}^{1}={\hat{c}}^{2}={\hat{c}}^{3}={\hat{c}}^{4}\). Transfers \(b_{i}\) satisfy:

$$\begin{aligned} u(w(1-q)-g+b_{1})+v(1)= u(w+b_{2}) \end{aligned}$$
(14)
$$\begin{aligned} U(w(1-q)-g+b_{3})-V(1)= U(w+b_{4}) \end{aligned}$$
(15)
$$\begin{aligned} w+b_{2}= w+b_{4} \end{aligned}$$
(16)

Equation (14) ensures that \(c^{1,D}=w(1-q)-g+b_{1}\) and \(c^{2,D}=w+b_{2}\) are set such that \({\hat{c}}^{1}={\hat{c}}^{2}\). Equation (15) ensures that \(c^{3,D}=w(1-q)-g+b_{3}\) and \(c^{4,D}=w+b_{4}\) are set such that \({\hat{c}}^{3}={\hat{c}}^{4}\), while the third equality ensures that \({\hat{c}}^{2}={\hat{c}}^{4}\). Together with (14) and (15), this implies that \({\hat{c}}^{1}={\hat{c}}^{3}\) so that all consumption equivalents are equalized. Equation (16) yields: \(b_{2}=b_{4}=b\). Using Eq. (14) with Assumption A1, we obtain that \(b_{1}<b\) and using Eq. (15), we have that \(b<b_{3}\). Let us now find the signs of \(\{b,b_{1},b_{3}\}\). Situations where \(0<b_{i}\forall i\) or \(b_{i}<0\forall i\) would not be possible as they do not satisfy (13). Yet, using Eqs. (14), (15) and (16), both \(b>0\) or \(b<0\) are possible solutions.

Similar proofs exist for cases R2 and R3. Lemma 2 summarizes our results.Footnote 42

Lemma 2

Assume that the economy is sufficiently productive. The decentralization of the ex post egalitarian optimum can be achieved by means of the following instruments:

Reference fertility Monetary transfers
R1 (\({\bar{n}}=0\)) \(b_{1}<0<b_{2}=b_{4}<b_{3}\) or \(b_{1}<b_{2}=b_{4}<0<b_{3}\)
R2 (\({\bar{n}}=1\)) \(b_{4}<0<b_{3}=b_{1}<b_{2}\) or \(b_{4}<b_{3}=b_{1}<0<b_{2}\)
R3 (\({\bar{n}}_{1,2}=1,\) \({\bar{n}}_{3,4}=0\)) \(b_{4}<b_{1}<0<b_{2},b_{3}\) or \(b_{4}<0<b_{1}<b_{2},b_{3}\)

Proof

See above. \(\square\)

Finally, let us consider the quasi-linear case. In case R1, transfers \(b_{1},b_{2},b_{3},b_{4}\) satisfy Eqs. (13)–(16), where we have replaced for the quasi-linear utilities. Equation (16) leads to \(b_{2}=b_{4}=b\) so that the budget constraint leads to:

$$\begin{aligned} b_{3}=\frac{-x\pi b_{1}-\left[ (1-x)(1-\varepsilon )+x(1-\pi )\right] b}{ (1-x)\varepsilon } \end{aligned}$$

Equation (14) leads to: \(b=-qw-g+b_{1}+v(1).\) This defines the locus I, i.e. the set of pairs \(\left( b_{1},b\right)\). This can be represented by an increasing line, with slope 1 and with a positive intercept (when \(b_{1}=0\)) at \(v(1)-qw-g>0\) (Assumption A1).

Equation (15) together with the budget constraint

$$\begin{aligned} b=-\frac{x\pi }{1-\pi x}b_{1}-\frac{V(1)+qw+g}{\frac{1-\pi x}{\varepsilon (1-x)}} \end{aligned}$$

It defines the locus II, i.e. the set of pairs \(\left( b_{1},b\right)\) such that \({\hat{c}}_{3}={\hat{c}}_{4}\) and the budget constraint is satisfied. Since \(\pi x<1\), it has a negative slope, less than the 45\({{}^\circ }\)line. When \(b_{1}=0\), we have \(b=-\frac{V(1)+wq+g}{\frac{1-\pi x}{ \varepsilon (1-x)}}<0\). In addition, consumptions cannot be negative, so that \(-\left[ w(1-q)-g\right] <b_{1}\) and \(b>-w\).

Non-existence arises when locus I is above locus II at \(b_{1}=-\left[ w(1-q)-g\right]\), that is, when:

$$\begin{aligned} w<\frac{g\left( x\pi +\varepsilon (1-x)\right) +v(1)\left( 1-\pi x\right) +V(1)\varepsilon (1-x)}{1-x\pi q-\varepsilon q(1-x)}\equiv {\bar{w}}_{1} \end{aligned}$$
(17)

The decentralization through a pure transfer system exists if and only if \(w\ge {\bar{w}}_{1}\).

Consider now case R2. The proof is similar to the one of case R1. Since under R2, \(b_{1}=b_{3}=b\), the (im)possibility to decentralize the optimum with equal consumption equivalent can be studied by examining the (non)intersection of two loci in the \(\left( b,b_{2}\right)\) space:

$$\begin{aligned} b_{2}&= b-wq-g+v(1)\text { (locus I)} \\ b_{2}&= -\frac{1-x+x\pi }{x(1-\pi )}b+\frac{(1-x)(1-\varepsilon )}{x(1-\pi ) }\left( V(1)+wq+g\right) \text { (locus II)} \end{aligned}$$

Consumptions cannot be negative, so that \(c_{1}>0\) implies \(b\ge -w(1-q)+g.\) Moreover, \(c_{2}>0\) implies \(b_{2}\ge -w\). Decentralization cannot take place when the locus I is above the locus II at \(b=\left( -w(1-q)+g\right)\) , that is, when:

$$\begin{aligned} w<\frac{g\left( x\pi +\varepsilon (1-x)\right) +v(1)x(1-\pi )-(1-x)(1-\varepsilon )V(1)}{1-qx\pi -q\varepsilon (1-x)}\equiv {\bar{w}}_{2} \end{aligned}$$
(18)

Thus the decentralization of the constrained optimum is possible only if \(w\ge {\bar{w}}_{2}\).

Comparing the RHS of (17) with (18), we can show that \({\bar{w}}_{1}>{\bar{w}}_{2}\), since \(v(1)>0>-V(1)\). We can also show that \(\bar{w }_{1}>{\tilde{w}}_{1}\) by comparing the RHS of (11) and (17), and that \({\bar{w}}_{2}>{\tilde{w}}_{2}\) by comparing the RHS of (12) and (18). Finally, comparing the RHS of (11) and (18), we obtain that \({\tilde{w}}_{1}>{\bar{w}}_{2}\) so that: \({\bar{w}}_{1}>{\tilde{w}}_{1}>{\bar{w}}_{2}>{\tilde{w}}_{2}\). Lemma 3 sums up our findings.

Lemma 3

Assume quasi-linear utility and excess demand for children. Define \({\bar{w}}_{1}\equiv \frac{g\left( x\pi +\varepsilon (1-x)\right) +v(1)\left( 1-\pi x\right) +V(1)\varepsilon (1-x)}{ 1-\varepsilon q(1-x)-qx\pi }\); \({\bar{w}}_{2}\equiv \frac{g\left( x\pi +\varepsilon (1-x)\right) +v(1)x(1-\pi )-(1-x)(1-\varepsilon )V(1)}{ 1-\varepsilon q(1-x)-qx\pi }\). Threshold wage levels satisfy: \({\tilde{w}}_{2}< {\bar{w}}_{2}<{\tilde{w}}_{1}<{\bar{w}}_{1}\).

Under R1, (i) if \(w>{\bar{w}}_{1}>{\tilde{w}}_{1}\), equalizing \({\hat{c}}^{i}\) can be done by a pure transfer system or a mixed system; (ii) if \({\bar{w}}_{1}>w> {\tilde{w}}_{1}\), equalizing \({\hat{c}}^{i}\) can only be done by a mixed system; (iii) if \({\bar{w}}_{1}>{\tilde{w}}_{1}>w\), equalizing \({\hat{c}}^{i}\) cannot be done.

Under R2, (i) if \(w>{\bar{w}}_{2}>{\tilde{w}}_{2}\), equalizing \({\tilde{c}}^{i}\) can be done by a pure transfer system or a mixed system; (ii) if \({\bar{w}} _{2}>w>{\tilde{w}}_{2}\), equalizing \({\tilde{c}}^{i}\) can only be done by a mixed system; (iii) if \({\bar{w}}_{2}>{\tilde{w}}_{2}>w\), equalizing \({\tilde{c}} ^{i}\) cannot be done.

Proof

See above. \(\square\)

Proof of Proposition 8

Assuming \({\bar{n}}_{1,2,3}=1\) and \({\bar{n}}_{4}=0\), as well as \({\bar{e}} _{1,4}=0\) and \({\bar{e}}_{2,3}=\ell\), we have:

$$\begin{aligned} u\left( C^{1}\right) +v(1)&= u\left( c^{1}\right) +v(1)\implies C^{1}=c^{1} \\ u\left( C^{2}\right) +v(1)-\varphi \left( \ell \right)&= u(c^{2})+v(1)-\varphi \left( \ell \right) \implies C^{2}=u^{-1}\left[ u\left( c^{2}\right) \right] =c^{2} \\ u\left( C^{3}\right) +v(1)-\varphi \left( \ell \right)&= u(c^{3})+v(0)-\varphi \left( \ell \right) \implies C^{3}=u^{-1}\left[ u(c^{3})-v(1)\right] \\ U(C^{4})-V(0)&= U(c^{4})-V(0)\implies C^{4}=c^{4} \end{aligned}$$

If egalitarian constraints are binding at the optimum (i.e., a sufficiently productive economy), we have \(C^{1}=C^{2}=C^{3}=C^{4}\), so that \(c^{1}=c^{2}=c^{4}<c^{3}\).

As to the decentralized solution, consumption equivalents can be written as:

$$\begin{aligned} u\left( C^{1}\right)&= u\left( w(1-q)-g-\ell +b_{1}\right) \text { and } u\left( C^{2}\right) =u\left( w(1-q)-g-\ell +b_{2}\right) \\ u\left( C^{3}\right)&= u(w-\ell +b_{3})-v(1)\text { and }U(C^{4})=U(w+b_{4}) \end{aligned}$$

From \(C^{1}=C^{2}=C^{4}\), we obtain: \(b_{2}>b_{1}>b_{4}\). We also have that \(b_{3}>b_{2}\), by contradiction as we show now. When condition (5) is satisfied and starting from \(b_{2}=b_{3}=0\), we have: \(u\left( w(1-q)-g-\ell +b_{2}-\ell \right) >u(w-\ell +b_{3})-v(1)\). If \(b_{2}\) increases, \(b_{2}>b_{3}\) and the LHS becomes larger than the RHS, so that equivalent consumption levels cannot be equalized. Hence, the only possible solution to ensure that \(C_{2}=C_{3}\), and thus that \(u\left( w(1-q)-g-\ell +b_{2}-\ell \right) =u(w-\ell +b_{3})-v(1)\), consists in setting \(b_{3}>b_{2}\).

Furthermore, the budget constraint of the government is balanced when \(x\pi b_{1}+x(1-\pi )pb_{2}+x(1-\pi )(1-p)b_{3}+(1-x)b_{4}=0\). This implies, with the relations above, that \(b_{3}>0\) and \(b_{4}<0\). \(b_{2}\) and \(b_{3}\) can be positive or negative depending on the value of v(1) and on \(\{\pi ,x,p\}\).

Transfers must ensure that individuals who cannot have children but want one decide to invest in ART (\(e=\ell\)) after the State’s intervention, that is:

$$\begin{aligned}&p[u(w(1-q)-g-\ell +b_{2})+v(1)-\varphi (\ell )]\nonumber \\&\quad +(1-p)[u(w-\ell +b_{3})-\varphi (\ell )]\ge u(w+b_{4}) \end{aligned}$$
(19)

where the RHS is the utility the individual if he does not invest in ART and remains childless with probability 1. Using the definitions of \(C_2, C_3\) and \(C_4\), we show that this condition is satisfied whenever \(v(1)\ge \varphi (\ell )\), which is always the case when the involuntary childless invests in ART (see condition 5).

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Leroux, ML., Pestieau, P. & Ponthiere, G. Childlessness, childfreeness and compensation. Soc Choice Welf 59, 1–35 (2022). https://doi.org/10.1007/s00355-021-01379-y

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