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.

Appendix

1.1 Proof of Proposition 2

1.1.1 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 $c^{1}=c^{4}$.

1.2 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, $b_{4}<0<b_{1}$.

1.3 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 $v(1)>0>-V(1)$.

1.4 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.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. $\square$

1.5 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).