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The political choice of social long term care transfers when family gives time and money

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Abstract

We develop a model where families consist of one parent and one child, with children differing in income and all agents having the same probability of becoming dependent when old. Young and old individuals vote over the size of a social long term care transfer (LTC hereafter) program, which children complement with help in time or money to their dependent parent. Dependent parents have an intrinsic preference for help in time by family members. We first show that low (resp., high) income children provide help in time (resp. in money), whose amount is decreasing (resp. increasing) with the child’s income. The middle income class may give no family help at all, and its elderly members would be the main beneficiaries of the introduction of social LTC transfers. We then provide several reasons for the stylized fact that there are little social LTC transfers in most countries. First, social transfers are dominated by help in time by the family when the intrinsic preference of dependent parents for the latter is large enough. Second, when the probability of becoming dependent is lower than one third, the children of autonomous parents are numerous enough to oppose democratically the introduction of social LTC transfers. Third, even when none of the first two conditions is satisfied, the majority voting equilibrium may entail no social transfers, especially if the probability of becoming dependent when old is not far above one third. This equilibrium may be local (meaning that it would be defeated by the introduction of a sufficiently large social program). This local majority equilibrium may be empirically relevant whenever new programs have to be introduced at a low scale before being eventually ramped up.

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Notes

  1. OECD (2011, chapter 7) provides a detailed taxonomy of public LTC systems within the OECD. For broader surveys on long term care, see Brown and Finkelstein (2011) and Cremer et al. (2009).

  2. It is also associated with the desire of most people to remain in their own homes for as long as possible. Recent polling by the Canadian Life and Health Insurance Association, reported in Frank (2012) found that 77 percent of Canadians would prefer to stay in their homes as they age. See also Torjman (2013).

  3. The same assumption is made by Nuscheler and Roeder (2013), which we cover in Sect. 1.1. Costa Font et al. (2014) contrast ex ante financing of LTC (such as insurance) with ex post financing (such as transfers, as in our model). They find that most OECD countries’ LTC spending is financed by close to a fifty-fifty mix of ex-ante and ex-post funding sources or that the spending relies heavily on ex-post funding sources. They also obtain that OECD countries view the two forms of funding as substitutes rather than complements.

  4. Sloan et al. (1997) find little empirical support for the hypothesis that care giving by children is motivated by the prospect of receiving bequests from their parents. Kopczuk and Lupton (2007) also find no empirical evidence of strategic bequests.

  5. We are interested in how much (more) young agents help their parent in case of dependency, so that any help to a non-dependent parent can be subsumed in the existing model. Note that we also assume away (so-called descending) altruism from parents to children to focus on help by children to parents. In the appendix, we show that introducing descending altruism would not change children preferences toward the tax rate, while parents (both autonomous and dependent) would prefer a lower value of the tax rate. Introducing descending altruism would then bias our results toward even less public LTC transfers.

  6. Pestieau and Sato (2006, 2008) make the same assumption, although it remains implicit in their setting. This assumption requires that we assume away saving, strategic interactions between parents and children (such as exchanges of help for bequests) and that the size of the public LTC program is chosen at each period. See footnote 4 for the lack of strategic motivations for family help, and footnote 14 for a defence of the last assumption above.

  7. Finkelstein et al. (2013) show that the marginal utility from consumption varies with health status.

  8. We leave for future research the case where old agents differ in income, and where children and parental income within a family are correlated.

  9. We can either assume that \(v(d,e)<0\) or subtract a constant term large enough in the definition of \(U_{Y}^{D}\) to make sure that \( U_{Y}^{N}>U_{Y}^{D}\). Note that this does not affect our analysis, since children do not choose to make their parent dependent or not.

  10. Assuming away labor income tax distortions biases the model in favor of the introduction of social LTC transfers (since distortions would decrease the most-preferred value of \(\tau \) of agents).

  11. Alternatively, we could have assumed that the benefit b decreases linearly with the child’s income w. This would have generated the same kind of redistributive effects.

  12. This formulation is made not only for analytical convenience, but also because it is more consistent with the empirical results than the alternative formulation where net income is \((1-\tau )w(1-e)\). The latter formulation means that the marginal cost of e decreases with \(\tau \), while the marginal cost of \(\tau \) decreases with e, so that e and \(\tau \) could be complements rather than substitutes in our approach. Sloan et al. (2002), Houtven and Norton (2004) and Charles and Sevak (2005) find that informal care is a substitute for home health care, nursing home care, hospital care and physician visits.

  13. Karagyozova and Siegelman (2012) review the empirical literature on relative risk aversion. They report several studies which find a coefficient of relative risk aversion lower than one. Hansen and Singleton (1983) evaluate it to be between [0.35, 1]; Holt and Laury (2002) elicit the distribution of coefficients of relative risk aversion and find that \(64\,\%\) of respondents had a coefficient comprised between 0.15 and 0.97. Chetty (2006) finds that “using 33 sets of estimates of wage and income elasticities, the mean implied value of (the coefficient of relative risk aversion) is 0.71, with a range of 0.15 to 1.78 in the additive utility case.” Assumption 2 is then reasonable.

  14. The alternative assumption that the result of a vote would hold for decades does not seem reasonable to us. The literature on the political economy of pensions often assumes that voting takes place once and for all (see for instance, Casamatta et al. (2000) and Cremer et al. (2007)), but this assumption is a pis-aller to explain the emergence of pay-as-you-go social transfer schemes in the absence of altruism. The presence of altruism in our model makes the unpalatable assumption of voting once and for all not necessary.

  15. The boundaries of this “middle class” are obtained analytically by (i) setting \(V_{Ye}=0\) for \(f=e=0\) and solving for \( w<\beta \) (to obtain its lowerbound), and (ii) by setting \(V_{Yf}=0\) for \( f=e=0\) and solving for \(w>\beta \) (to obtain its upperbound). All figures are based on the following assumptions: \(u(x)=2\sqrt{x},\) \(v(d,e)=ln[d+\beta e]\), \(\pi =0.5\), \({\bar{w}}=5\), \(T=2\), \({\bar{p}}=4\), \(\beta =5\). In Fig. 1, we assume moreover that \(\tau =0\) and \(\alpha =1\).

  16. Proceeding simultaneously proves much easier to obtain the individually optimal amounts of \(\tau \), e and f than proceeding sequentially and using the envelope theorem, because of the frequent corner solutions for the variables.

  17. In order to focus on empirically relevant situations and not to multiply cases, we assume from now on that no young agent with a dependent parent most prefers \(\tau _{Y}^{*}\ge 1\), which could in theory occur in our setting with untaxed labor income \(w(T-1)\). This condition is satisfied if

    $$\begin{aligned} wu^{\prime }\left( w\left( T-1\right) \right) >\alpha \frac{{\bar{w}}}{\pi }\phi ^{\prime }\left( {\bar{p}}+ \frac{{\bar{w}}}{\pi }\right) . \end{aligned}$$
  18. Figs. 2 and 3 differ only in the value of \(\alpha \). In both figures, all agents prefer either \(\tau ^{*}>0\) or \(f^{*}>0\). The curve \(\beta e^{*}(0,w)\) (as in Fig. 1) is only depicted here to allow the comparison with \(\tau ^{*}{\bar{w}}/\pi \) when \(\tau ^{*}\) is set at its most-preferred level for individual w.

  19. The case where \(\beta >{\bar{w}}/\pi \) is available upon request but not reported here since the majority chosen value of \(\tau \) is zero in that case.

  20. Assuming that they throw a dice to determine which value of \(\tau \) to vote for would not affect our results. Alternatively, we could assume that autonomous parents vote in favor of the best alternative for their child, that is for \(\tau ^{*}=0\). In that case, no group forms a majority by itself with \(0<\pi <1\), and the necessary condition for \(\tau ^{*}>0\) becomes \(\pi >1/2\), a more stringent condition than the one we obtain after Lemma 1.

  21. We assume throughout the paper that dependency does not prevent old agents from voting (because of cognitive issues, for instance). If all dependent agents were prevented from voting, the Condorcet winning value of the tax rate would be zero if \(F({\bar{w}}/\pi )<1/2\pi \) (which always holds if \(\pi <1/2\)), thanks to an alliance of young agents with autonomous parents, and of young agents with dependent parents and \(w\ge {\bar{w}}/\pi \). If \(F(\bar{w }/\pi )>1/2\pi \), then the Condorcet winning value of \(\tau \) is positive, and the decisive voter is a young agent with a dependent parent and a productivity \(w<{\bar{w}}/\pi \) such that \(F^{-1}(w)=1/2\pi \).

  22. It is easy to show, using the implicit function theorem on (1), that \(\breve{w}\) decreases with \(\pi \).

  23. The assumption mentioned in footnote 17 implies that this threshold is lower than 1.

  24. The assumption mentioned in footnote 17 implies that \(f^{*}(\tau ,w)=0\) for \(\tau >{\tilde{\tau }}\) with \({\tilde{\tau }}<1\).

  25. It is immediate that \(f^{*}(0,{\bar{w}}/\pi )<{\bar{w}}/\pi .\)

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Correspondence to Marie-Louise Leroux.

Additional information

The authors thanks participants to the 55th SCSE annual congress in Montreal, CORE 50th anniversary conference as well as E. Bonsang, P. Pestieau and two anonymous reviewers for their comments and suggestions. Financial support from the Chaire “Marché des risques et création de valeur” of the FdR/SCOR is gratefully acknowledged. Part of the research has been done while the first author visited Université du Québec à Montréal, whose hospitality is gratefully acknowledged.

Appendix

Appendix

1.1 Proof of Proposition 4

  1. (a)

    \(e>0\) cannot be part of the individually optimal allocation since \( V_{Ye}=0\) implies \(V_{Yf}>0\) with \(w>\beta \). Also, \(\tau >0\) cannot be part of the individually optimal allocation since \(V_{Y\tau }=0\) implies \( V_{Yf}>0 \) with \(w>{\bar{w}}/\pi \). Alternatively, \(V_{Yf}\le 0\) is consistent with the individually optimal allocation with \(f\ge 0\), \(e=0\) and \(\tau =0\), since it implies that \(V_{Ye}<0\) (since \(w>\beta \)) and that \( V_{Y\tau }<0\) (since \(w>{\bar{w}}/\pi \)).

  2. (b)

    \(e>0\) cannot be part of the individually optimal allocation since \( V_{Ye}=0\) implies \(V_{Y\tau }>0\) with \(\beta <{\bar{w}}/\pi \). Also, \(f>0\) cannot be part of the individually optimal allocation since \(V_{Yf}=0\) implies \(V_{Y\tau }>0\) with \(w<{\bar{w}}/\pi \). Alternatively, \(V_{Y\tau }\le 0\) is consistent with the individually optimal allocation with \(\tau \ge 0\) , \(e=0\) and \(f=0\), since it implies that \(V_{Ye}<0\) (since \(\beta >{\bar{w}} /\pi \)) and that \(V_{Yf}<0\) (since \(w<{\bar{w}}/\pi \)).

  3. (c)

    \(\tau >0\) cannot be part of the individually optimal allocation since \( V_{Y\tau }=0\) implies \(V_{Ye}>0\) with \(\beta >{\bar{w}}/\pi \). Also, \(f>0\) cannot be part of the individually optimal allocation since \(V_{Yf}=0\) implies \(V_{Ye}>0\) with \(w<\beta \). Alternatively, \(V_{Ye}\le 0\) is consistent with the individually optimal allocation with \(e\ge 0\), \(\tau =0\) and \(f=0\), since it implies that \(V_{Y\tau }<0\) (since \(\beta >{\bar{w}}/\pi \) ) and that \(V_{Yf}<0\) (since \(w<\beta \)).

  4. (d)

    If \(\beta >{\bar{w}}/\pi \), then \(V_{Ye}>V_{Y\tau }\) \(\forall (\tau ,f,e)\) and no child ever most-prefer \(\tau _{Y}^{*}>0\). Propositions 1, 2 and 3 have jointly established that the amount of (financial or informal) family help is minimum for the child with productivity \(w=\beta \). The condition for this child to give a strictly positive amount of help is

    $$\begin{aligned} u^{\prime }(T\beta )<\alpha \phi ^{\prime }({\bar{p}}). \end{aligned}$$

    If \(\beta <{\bar{w}}/\pi \), then \(V_{Y\tau }>V_{Ye}\) \(\forall (\tau ,f,e)\) and no child ever most-prefers \(e^{**}>0\). In that case, children with \( w<{\bar{w}}/\pi \) most-prefer \(\tau _{Y}^{*}\ge 0\) and \(f^{**}=0\) (see (b) above) while children with \(w>{\bar{w}}/\pi \) most-prefer \(f^{**}>0\) and \(\tau _{Y}^{*}=0\) (see (a) above). Proposition 5(a) will show that \(\tau ^{*}\) is decreasing in w while Proposition 3 has established that \(f^{**}\) is increasing in w. Hence, the child who gives the lowest amount of aid has a productivity level of \(w={\bar{w}}/\pi \), and the condition for this child to give a strictly positive amount is

    $$\begin{aligned} u^{\prime }(T{\bar{w}}/\pi )<\alpha \phi ^{\prime }({\bar{p}}). \end{aligned}$$

1.2 Proof of Proposition 5

  1. (i)

    Applying the implicit function theorem on (6) together with \(f=e=0\), we obtain

    $$\begin{aligned} sign\left( \frac{d\tau _{Y}^{*}}{dw}\right) =sign\left( -u^{\prime }\left( c\right) -cu^{\prime \prime }\left( c\right) \right) \le 0 \end{aligned}$$

    under Assumption 2(a). Part (b) is proved similarly. Part (c) is obtained applying the implicit function theorem together with Assumption 2(b) so that

    $$\begin{aligned} sign\left( \frac{d\tau _{Y}^{*}}{d\pi }\right) =sign\left( -\frac{\alpha {\bar{w}}}{\pi ^{2}}\phi ^{\prime }(x)\left[ 1-\frac{{\bar{w}}\tau _{Y}^{*}/\pi }{x}\left( -\frac{\phi ^{\prime \prime }(x)x}{\phi ^{\prime }(x)} \right) \right] \right) <0 \end{aligned}$$

    where \(x={\bar{p}}+\tau _{Y}^{*}{\bar{w}}/\pi \).

  2. (ii)

    Observe that the FOCs for \(\tau \) when \(e=f=0\) and for e when \(\tau =f=0\) can both be written as

    $$\begin{aligned} -wu^{\prime }\left( w(T-x)\right) +\alpha y\phi ^{\prime }\left( {\bar{p}} +yx\right) =0, \end{aligned}$$

    where \((x,y)\in \{(\tau ,{\bar{w}}/\pi ),(e,\beta )\}.\) Applying the implicit function theorem, we obtain that

    $$\begin{aligned} sign\left( \frac{dx}{dy}\right) =sign\left( \alpha \phi ^{\prime }\left( {\bar{p}}+yx\right) +\alpha yx\phi ^{\prime \prime }\left( {\bar{p}}+yx\right) \right) >0 \end{aligned}$$

    by Assumption 2(b). Hence, \(\beta<{\bar{w}}/\pi \Rightarrow e^{*}(0,w)<\tau _{Y}^{*}\Rightarrow \beta e^{*}(0,w)<\tau _{Y}^{*}{\bar{w}}/\pi .\)

1.3 Proof of Proposition 6

We prove this proposition by looking successively at parents whose children productivity w is (a) lower than \(\beta \), (b) in between \(\beta \) and \( {\bar{w}}/\pi \) and (c) larger than \({\bar{w}}/\pi \).

  1. (a)

    If \(w<\beta \), then \(f^{*}=0\) and \(e^{*}\ge 0\). Assuming \( e^{*}>0\), the FOC for \(\tau \) is given by

    $$\begin{aligned} \frac{\partial U_{O}^{D}}{\partial \tau }= & {} v_{d}\left[ \frac{{\bar{w}}}{\pi } +\beta \frac{de^{*}}{d\tau }\right] \nonumber \\= & {} \frac{\phi ^{\prime }\left( {\bar{p}}+\tau {\bar{w}}/\pi +\beta e^{*}\right) }{w^{2}u^{\prime \prime }\left( w\left( T-\tau -e^{*}\right) \right) +\alpha \beta \phi ^{\prime \prime }\left( {\bar{p}}+\tau {\bar{w}}/\pi +\beta e^{*}\right) }\nonumber \\&\times w^{2}u^{\prime \prime }\left( w\left( T-\tau -e^{*}\right) \right) \left[ \frac{{\bar{w}}}{\pi }-\beta \right] \end{aligned}$$
    (11)

    where we replaced for the expression for \(de^{*}/d\tau \) in (3). We know from Proposition 2(c) that \(de^{*}/d\tau <0\) so that pushing \(\tau \) above some threshold results in \(e^{*}=0\).Footnote 23 In that case, we have that

    $$\begin{aligned} \frac{\partial U_{O}^{D}}{\partial \tau }=\frac{{\bar{w}}}{\pi }\phi ^{\prime }\left( {\bar{p}}+\tau {\bar{w}}/\pi \right) >0. \end{aligned}$$
    (12)

    When \(\beta <{\bar{w}}/\pi \), then (11)\(>0\), so that \( U_{O}^{D}\) is increasing in \(\tau \) (whether \(e^{*}\) is positive or nil) and \(\tau ^{*}=1.\)

    (b and c) If \(w>\beta \), then \(f^{*}\ge 0\) and \(e^{*}=0\). Assuming \( f^{*}>0\), the FOC for \(\tau \) is given by

    $$\begin{aligned} \frac{\partial U_{O}^{D}}{\partial \tau }= & {} v_{d}\left[ \frac{{\bar{w}}}{\pi } +\frac{df^{*}}{d\tau }\right] \nonumber \\= & {} \frac{\phi ^{\prime }\left( {\bar{p}}+\tau {\bar{w}}/\pi +f^{*}\right) }{ u^{\prime \prime }\left( w(T-\tau )-f^{*}\right) +\alpha \phi ^{\prime \prime }\left( {\bar{p}}+\tau {\bar{w}}/\pi +f^{*}\right) }u^{\prime \prime }\nonumber \\&\times \left( w(T-\tau )-f^{*}\right) \left[ \frac{{\bar{w}}}{\pi }-w\right] \end{aligned}$$
    (13)

    where we replaced for the expression for \(df^{*}/d\tau \) in (4). We know from Proposition 3(c) that \(df^{*}/d\tau <0\) so that pushing \(\tau \) above some threshold may result in \(f^{*}=0\).Footnote 24 In that case, we have that \(\partial U_{O}^{D}/\partial \tau \) is given by (12).

  2. (b)

    If \(w<{\bar{w}}/\pi \), then (13)\(>0\), so that \( U_{O}^{D} \) is increasing in \(\tau \) (whether \(f^{*}\) is positive or nil) and \(\tau ^{*}=1\).

  3. (c)

    If \(w>{\bar{w}}/\pi \), then (13)\(<0\), so that zero is the agent’s most-preferred value of \(\tau \) among the values of \(\tau \) low enough that \(f^{*}(\tau ,w)>0\). Alternatively, one is the agent’s most-preferred value of \(\tau \) among the values of \(\tau \) large enough that \(f^{*}(\tau ,w)=0\). Comparing the elderly’s utility level with \( \tau =0\) (given by \(v({\bar{p}}+f^{*}(0,w))\)) and with \(\tau =1\) (given by \(v({\bar{p}}+{\bar{w}}/\pi )\)), we obtain that the elderly prefers \(\tau =0\) if

    $$\begin{aligned} f^{*}(0,w)>\frac{{\bar{w}}}{\pi }. \end{aligned}$$

    Since we know from Proposition 3(b) that \(f^{*}(0,w)\) is increasing in w, we define \(\breve{w}\) in the following way: (i) If \( f^{*}(0,w_{+})>{\bar{w}}/\pi \) then \(\breve{w}\) is such that \(f^{*}(0, \breve{w})={\bar{w}}/\pi \),Footnote 25 (ii) If \( f^{*}(0,w_{+})<{\bar{w}}/\pi \) then \(\breve{w}=w_{+}\).

1.4 Proof of Proposition 8

  1. (a)

    We have already established that preferences of dependent parents with \( w>{\bar{w}}/\pi \) are not single-peaked (see part (c) of the proof of Proposition 6), so that we cannot apply the usual median voter theorem. Donder (2013) shows that, if voters can be grouped into exogenous categories (i.e., here, categories not depending on \(\tau \)), and if the single-crossing property à la Gans and Smart (1996) is satisfied inside each group of voters, then a global Condorcet winning value of \(\tau \) exists. We then form four exogenous groups of voters, according to their age (child or parent) and to whether the child in the family prefers informal to direct financial help (\(w<\beta \)) or the opposite (\( w>\beta \)). We then check that the preferences of all groups satisfy the single-crossing property (i.e., monotonicity of the marginal rate of substitution in w) in the (\(\tau ,b\)) space.

    1. (i)

      We first show that preferences of young agents with dependent parents are indeed single-crossing. Their indirect utility function of \(\tau \) and b is

      $$\begin{aligned} I_{Y}^{D}=u\left( w\left( T-\tau -e^{*}-f^{*}\right) \right) +\alpha \phi \left( {\bar{p}}+b+\beta e^{*}+f^{*}\right) \end{aligned}$$

      where \(e^{*}=e^{*}(\tau ,w)\) and \(f^{*}=f^{*}(\tau ,w)\). Using the envelope theorem, the marginal rate of substitution between b and \(\tau \) is

      $$\begin{aligned} MRS=-\frac{\partial I_{Y}^{D}/\partial \tau }{\partial I_{Y}^{D}/\partial b}= \frac{wu^{\prime }(c)}{\alpha \phi ^{\prime }(x)} \end{aligned}$$

      where \(c=w(T-\tau -e^{*})\) and \(x=b+{\bar{p}}+\beta e^{*}\) for \( w<\beta \), and \(c=w(T-\tau )-f^{*}\) and \(x=b+{\bar{p}}+f^{*}\) when \( w>\beta \).

      If \(w<\beta \), then \(f^{*}=0\) and \(e^{*}\ge 0\). If \(e^{*}>0\), then using the FOC for e (Eq. (2)), we obtain that \( MRS=\beta \). If \(e^{*}=0\), we make use of the fact that the same FOC for \(e^{*}\) is negative for \(e=0\) to obtain that the MRS is larger than \( \beta \) and is weakly increasing in w given Assumption 2. Observe that the government budget constraint \(b(\tau )\) is linear in \(\tau \) with a slope \({\bar{w}}/\pi \). Tangency between indifference curve and budget constraint then occurs on the strictly concave part of the indifference curve, where \(e^{*}=0\) for the individual (see Proposition 4(b)). Preferences are single-crossing over these parts (see Epple and Romano (1996, Figure 4)).

      If \(w>\beta \), then \(e^{*}=0\) and \(f^{*}\ge 0\). If \(f^{*}>0\), then using the FOC for f (Eq. (1)), we obtain that \( MRS=w \). If \(f^{*}=0\), we make use of the fact that the same FOC for \( f^{*} \) is negative for \(f=0\) to obtain that the MRS is larger than w and is weakly increasing in w given Assumption 2. The strictly concave parts of the indifference curves are then single-crossing (agents with \(w>{\bar{w}}/\pi \) most prefer \(\tau ^{*}=0\)).

      We now study the preferences of dependent individuals. The indirect utility function of a dependent elderly whose child has \(w<\beta \) is \( I_{O}^{D}=\phi ({\bar{p}}+b+\beta e^{*})\), which is monotonically increasing in \(\tau \) for all values of \(w<\beta \) (see the proof Proposition 6(a)). The single-crossing condition (see Gans and Smart (1996), Sect. 2.1) is then trivially satisfied, since all agents with \(w<\beta \) prefer any larger to any smaller value of \(\tau \) (graphically, indifference curves all cut the budget constraint from above in the \((\tau ,b)\) space, since \(\beta \mid de^{*}/d\tau \mid <{\bar{w}} /\pi \) as shown in the proof of Proposition 6(a)).

      The indirect utility function of a dependent elderly whose child has \( w>\beta \) is \(I_{O}^{D}=\phi ({\bar{p}}+b+f^{*})\) so that the MRS between \( \tau \) and b is equal to \(MRS=-df^{*}/d\tau \) and \(dMRS/dw=-d^{2}f^{ *}/d\tau dw.\)

      So, if \(d^{2}f^{*}/d\tau dw<0\), the preferences of the four exogenous sets of voters are single-crossing and, by Donder (2013), the global Condorcet winner exists and corresponds to the median most-preferred value of \(\tau \).

    2. (ii)

      We show that logarithmic preferences satisfy the single crossing property for dependent parents with \(w>\beta \). Observe that

      $$\begin{aligned} \frac{d^{2}f^{*}}{d\tau dw}= & {} \frac{-1}{[u^{\prime \prime }\left( c\right) +\alpha \phi ^{\prime \prime }\left( x\right) ]^{2}}\left[ \phantom {\left. +\alpha \left( w-\frac{{\bar{w}}}{\pi }\right) \left( \frac{dc}{dw}\phi ^{\prime \prime }\left( x\right) u^{\prime \prime \prime }\left( c\right) -\frac{dx}{ dw}\phi ^{\prime \prime \prime }\left( x\right) u^{\prime \prime }\left( c\right) \right) \right] } u^{\prime \prime }\left( c\right) ^{2}+\alpha \phi ^{\prime \prime }\left( x\right) u^{\prime \prime }\left( c\right) \right. \\&\left. +\,\,\alpha \left( w-\frac{{\bar{w}}}{\pi }\right) \left( \frac{dc}{dw}\phi ^{\prime \prime }\left( x\right) u^{\prime \prime \prime }\left( c\right) -\frac{dx}{ dw}\phi ^{\prime \prime \prime }\left( x\right) u^{\prime \prime }\left( c\right) \right) \right] \end{aligned}$$

      which is obtained from (4) after tedious computations. In the case of a logarithmic utility, the expression inside brackets becomes

      $$\begin{aligned} \frac{1}{c^{4}}+\frac{\alpha }{c^{2}x^{2}}+\left( w-\frac{{\bar{w}}}{\pi }\right) \alpha \left[ \frac{-2}{x^{2}c^{3}}\frac{dc}{dw}+\frac{2}{x^{3}c^{2}}\frac{dx}{dw} \right] . \end{aligned}$$
      (14)

      Fully differentiating (1) with respect to w, we obtain

      $$\begin{aligned} \frac{dc}{dw}=\frac{\alpha c^{2}}{x^{2}}\frac{dx}{dw}. \end{aligned}$$

      Replacing this expression into (14), we obtain

      $$\begin{aligned} \frac{1}{c^{4}}+\frac{\alpha }{c^{2}x^{2}}+\left( w-\frac{{\bar{w}}}{\pi }\right) \alpha \frac{dx}{dw}\frac{2}{x^{3}c}\left[ \frac{1}{c}-\frac{\alpha }{x}\right] \end{aligned}$$

      with \([\frac{1}{c}-\frac{\alpha }{x}]=0\) from the FOC with respect to f, ( 1).

      Hence

      $$\begin{aligned} \frac{d^{2}f^{*}}{d\tau dw}=\frac{-1}{[u^{\prime \prime }(c)+\alpha \phi ^{\prime \prime }(x)]^{2}}\left[ \frac{1}{c^{4}}+\frac{\alpha }{c^{2}x^{2}} \right] <0. \end{aligned}$$
    3. (iii)

      Since condition (8) is not satisfied, we have that

      $$\begin{aligned} \pi \left( F({\bar{w}}/\pi )+F(\breve{w})\right) >\frac{1+\pi }{2}, \end{aligned}$$

      where the LHS represents the measure of children with a dependent parent and of old dependents who most prefer a strictly positive value of \(\tau \). At the same time, we have that

      $$\begin{aligned} \pi F(\breve{w})<\frac{1+\pi }{2}, \end{aligned}$$

      since \(F(\breve{w})<1\) and \(\pi \le (1+\pi )/2\,\,\forall \pi \le 1\). Thus, old dependents who prefer the maximum tax rate cannot form a majority by themselves. The decisive voter is therefore a young agent with a dependent parent and with a productivity level, denoted by \(w^{GCW},\) such that \(w^{GCW}<{\bar{w}}/\pi \). Voters who want a larger value of \(\tau \) are young agents with dependent parents and \(w<w^{GCW}\) together with old dependent agents with \(w<\breve{w}\) (the other groups of agents want a lower value of \(\tau \)) and must represent one half of the voters. This is how we implicitly define \(w^{GCW}\):

      $$\begin{aligned} \pi \left( F\left( \breve{w}\right) +F\left( w^{GCW}\right) \right) =\frac{1+\pi }{2}. \end{aligned}$$
    4. (b)

      Condition \(f^{*}(\tau ^{LCW},w_{+})=0\) implies that \(f^{*}(\tau ^{LCW},w)=0\) for every child of a dependent parent with \({\bar{w}}/\pi<w<w_{+} \), since \(f^{*}(\tau ,w)\) is increasing in w, \(\forall \tau \). Then, every dependent parent with \(w>{\bar{w}}/\pi \) has locally increasing utilities around \(\tau ^{LCW}\) as \(db/d\tau >0\) (see Eq. (13)). We already know that the utility of all dependent agents with \( w<{\bar{w}}/\pi \) monotonically increases with \(\tau \) (see the proof of Proposition 6(a) and (b)). The dependent parents form a mass of measure \(\pi \). As for children with dependent parents, we know from Proposition 5(a) that their most-preferred value of \(\tau \) is decreasing with w, so that all of them with \(w<w^{LCW}\) prefer a larger-than-\(\tau ^{LCW}\) value of \(\tau \). The mass \(\pi \left( 1+F(w^{LCW})\right) \) then represents the mass of voters who prefer a slightly larger value of \(\tau \) than \(\tau ^{LCW}\), so that \(\tau ^{LCW}\) is a local Condorcet winner if this mass represents exactly one half of voters.

    5. (c)

      Result obtained after long and tedious but straightforward manipulations of the FOCs for financial and informal family help (Eqs. (1 ) and (2)), available from the authors.

1.5 Descending altruism

Let us assume in this section that parents exhibit altruism toward their children, independently of whether they are dependent or autonomous. In that situation, the utility of young agents does not change and is represented by \(U_{Y}^{N}\) and \(U_{Y}^{D}\) as before, implying that the results of Sect. 3.1 are maintained. Young agents with an autonomous parent prefer a zero tax rate while the preference of children with a dependent parent are defined by Propositions (1)–(4).

The preferences of parents are now given by

$$\begin{aligned} v(d,e)+\gamma u(c), \end{aligned}$$

where \(\gamma >0\) represents their degree of altruism toward their child. Autonomous parents are now in favor of \(\tau =0\) (because of their concern for their child’s consumption) while in the absence of descending altruism they were indifferent as to the value of \(\tau \).

We assume as previously that \(\beta <{\bar{w}}/\pi \). As for dependent parents, we first obtain that introducing descending altruism reinforces the impact of \(\tau \) on their utility as long as social insurance does not totally crowd out family help. If \(w<\beta \) (so that \(e^{*}\ge 0\) while \(f^{*}=0\)), the FOC for \(\tau \) is now given by

$$\begin{aligned} \frac{\partial U_{O}^{D}}{\partial \tau }=v_{d}\left[ \frac{{\bar{w}}}{\pi } +\beta \frac{de^{*}}{d\tau }\right] -\gamma wu^{\prime }(c)\left[ \frac{ de^{*}}{d\tau }+1\right] , \end{aligned}$$

where the first term is positive (see Proposition 6(a)) and where the second term, which represents the marginal utility cost of taxation for a young with a dependent parent, is positive when \(\beta <{\bar{w}}/\pi \) (see Proposition 2). The intuition is that \(e^{*}\) decreases so fast with \( \tau \) that a parent who cares for his child favors social insurance in order to increase the child’s consumption level. Descending altruism then reinforces our result that dependent parents wish to completely crowd out informal help with social insurance.

A similar phenomenon occurs when \(w>\beta \) (so that \(e^{*}=0\) while \( f^{*}\ge 0\)). The FOC for \(\tau \) is now given by

$$\begin{aligned} \frac{\partial U_{O}^{D}}{\partial \tau }=v_{d}\left[ \frac{{\bar{w}}}{\pi }+ \frac{df^{*}}{d\tau }\right] -\gamma u^{\prime }(c)\left[ w+\frac{ df^{*}}{d\tau }\right] , \end{aligned}$$
(15)

where the last square bracket can be rewritten as

$$\begin{aligned} \frac{\phi ^{\prime \prime }({\bar{p}}+\tau {\bar{w}}/\pi +f^{*})\alpha }{ u^{\prime \prime }(c)+\alpha \phi ^{\prime \prime }({\bar{p}}+\tau {\bar{w}}/\pi +f^{*})}\left[ w-\frac{{\bar{w}}}{\pi }\right] . \end{aligned}$$

The proof of Proposition 6(b) shows that the first term in (15) is positive when \(w<{\bar{w}}/\pi \) , which is then also the case for the second term. Alternatively, if \(w>{\bar{w}}/\pi \), then both terms in (15) are negative. In both cases, descending altruism reinforces the impact of \(\tau \) on dependent parents’ utility.

Whatever the value of w, pushing \(\tau \) above a threshold totally crowds out family help. From that point on, the derivative of \(U_{O}^{D}\) with respect to \(\tau \) becomes

$$\begin{aligned} v_{d}({\bar{p}}+\tau {\bar{w}}/\pi )\frac{{\bar{w}}}{\pi }-\gamma wu^{\prime }(w(T-\tau )), \end{aligned}$$

which is decreasing in \(\tau \), and equal to zero when \(\tau ^{*}<1\) (if \(T=1\) and \(\lim u^{\prime }(c)=\infty \) when c tends toward zero). We also have that the value of \(\tau \) maximizing \(U_{O}^{D}\) when \(e^{*}=f^{*}=0\) is decreasing in \(\gamma \).

We then obtain that parents (dependent or autonomous) have a lower most-preferred value of \(\tau \) in the presence of descending altruism.

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De Donder, P., Leroux, ML. The political choice of social long term care transfers when family gives time and money. Soc Choice Welf 49, 755–786 (2017). https://doi.org/10.1007/s00355-016-0999-3

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