Skip to main content

Governmental transfers and altruistic private transfers

Abstract

If an altruist is expected to aid a person with low utility, that person may be induced to save little. Such behavior generates a good Samaritan dilemma, in which welfare is lower than when no one is altruistic. Governmental transfers, which restrict reallocation from a person who saves much to one who saves little, reduce the effect and can lead to an outcome which is Pareto-superior to the outcome under a Nash equilibrium with no government taxation and transfers.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. Conditions under which the rotten kid theorem holds are given by Bergstrom (1989): with more than two goods, consumption in two periods, transferable utility, and utility of a recipient of a normal good for the donor, the donor’s most preferred outcome is realized as a subgame-perfect equilibrium. Utility is transferable if whenever a distribution of utilities is possible, any distribution of utilities in which the sum of the utilities is unchanged is also possible. The utility function must be quasi-linear.

  2. Work in development economics also considers the behavior of recipients anticipating aid from a wealthy country. In particular, the donor may induce the recipient countries to invest by delegating the allocation of aid to an agent with altruistic preferences different from his own (Svensson 2000; Hagen 2006).

  3. Bernheim and Bagwell (1988) consider two families linked through their children, showing that public policies such as interfamily transfers are neutralized and thus have no real effect. In our model, in contrast, each of two linked families will want to save little, so as to have the other family make transfers to the children. Public policies can reduce this incentive.

  4. An overlapping generations model where not altruism but the positive externalities that the young enjoy from the elderly’s consumption induce the young to make transfers to the old is considered by Veall (1986). The transfers in turn induce the old to save less, relying on their children for retirement assistance. Hence, a funded public pension plan is subject to crowding out of private transfers. A pay-as-you-go public pension is not subject to crowding out, and is Pareto improving. Homburg (2000) shows that in a model with three goods (current consumption, future consumption, and labor), compulsory savings alleviate the problem of insufficient saving, but reduces labor supply.

  5. We can think of N identical families, each consisting of a donor and of two recipients. The government’s total tax revenue is NT, which it distributes among 2N recipients in the population.

  6. In Bruce and Waldman (1990, 1991) an altruistic parent makes transfers in both periods 1 and 2. The parent might make transfers in period 1 that induce the child to increase his savings; transfers with this effect may be payment for a college education or a down payment on a house. Under-saving is then ameliorated (Bruce and Waldman 1991). The analysis, however, becomes formidably complicated even in the case of one recipient. The model in this paper has two recipients. Instead of the parent’s choice in period 1, Coate (1995) introduces government’s choice of a transfer. In this paper, we use the same timeline as that of Coate (1995).

  7. In contrast, if recipient 2 saves much, the private transfer to recipient 1 is less sensitive to a change in recipient 2’s savings. If recipient 2 saves much, the donor makes no transfer to that recipient. Reduced savings by recipient 1 induces the donor to increase the private transfer to that recipient, but without reallocating the private transfer from the high-saver to the low-saver. Therefore, a change in recipient 2’s saving does not affect recipient 1’s decision.

  8. Under the constraint, their utilities are equal. Without this constraint, the maximized sum of their utilities is larger. For example, recipient 1 saves a sufficiently large amount, whereas recipient 2 saves much less and gets a large private transfer. However, recipient 1 will not cooperate with recipient 2 unless recipient 1 can expect a side payment from recipient 2. Yet, if the donor expects the side payment, the donor will not make a transfer to only one recipient. Therefore, the maximized sum of the utilities without the constraint S 1 = S 2 cannot be attained.

  9. The transfer to recipients can be in the form of a local public good. For example, transfers to middle-aged children can take the form of day care, mortgage interest deduction, or good roads.

  10. The result appears when the other recipient saves only \(\left (\frac {\beta -1 }{2+\beta } \right ) w\). In contrast, if the other recipient saves \(\frac { \beta w -\frac {1}{2}T}{1+\beta }\), a recipient who saves little, in the amount \(\left (\frac {\beta -1}{2+\beta } \right ) w\), gets a private transfer. Increased savings thus reduce the private transfer, though the transfer is small. Nevertheless, in this range, increased savings, though they reduce the private transfer, increase utility by improving the intertemporal allocation of consumption. The recipient increases savings up to \(\frac { \beta w -\frac {1}{2}T}{1+\beta }\). See Appendix 4.

  11. Appendix 2 gives details about the savings of two recipients and the derivations when the government makes a transfer to only one of them.

References

  • Altonji JG, Hayashi F, Kotlikoff LJ (1997) Parental altruism and inter vivos transfers: theory and evidence. J Polit Econ 105(6):1121–1166

    Article  Google Scholar 

  • Becker GS (1974) A theory of social interactions. J Polit Econ 82(6):1063–1094

    Article  Google Scholar 

  • Bergstrom TC (1989) A fresh look at the rotten kid theorem—and other household mysteries. J Polit Econ 97(5):1138–1159

    Article  Google Scholar 

  • Bernheim BD, Bagwell K (1988) Is everything neutral? J Polit Econ 96(2):308–338

    Article  Google Scholar 

  • Bernheim BD, Shleifer A, Summers LH (1985) The strategic bequest motive. J Polit Econ 93(6):1045–1076

    Article  Google Scholar 

  • Bruce N, Waldman M (1990) The rotten-kid theorem meets the Samaritan’s dilemma. Q J Econ 105(1):155–165

    Article  Google Scholar 

  • Bruce N, Waldman M (1991) Transfers in kind: why they can be efficient and nonpaternalistic? Am Econ Rev 81(5):1345–1351

    Google Scholar 

  • Buchanan J (1977) The Samaritan’s dilemma. In: Phelps ES (ed) Altruism, morality and economic theory. Russell Sage Foundation, New York

    Google Scholar 

  • Cigno A, Rosati FC (1992) The effect of financial markets and social security on savings and fertility behaviour in Italy. J Popul Econ 5(4):319–341

    Article  Google Scholar 

  • Cigno A (1993) Intergenerational transfers without altruism: family, market and state. Eur J Polit Econ 9(4):505–518

    Article  Google Scholar 

  • Coate S (1995) Altruism, the Samaritan’s dilemma, and government transfer policy. Am Econ Rev 85(1):46–57

    Google Scholar 

  • Cox D, Jimenez E (1990) Achieving social objectives through private transfers: a review. World Bank Res Obser 5(2):205–218

    Article  Google Scholar 

  • Cox D, Jakubson G (1995) The connection between public transfers and private interfamily transfers. J Public Econ 57(1):129–167

    Article  Google Scholar 

  • Cox D, Eser Z, Jimenez E (1998) Motives for private transfers over the life cycle: an analytical framework and evidence for Peru. J Dev Econ 55(1):57–80

    Article  Google Scholar 

  • Cox D, Hansen BE, Jimenez E (2004) How responsive are private transfers to income? Evidence from a laissez-faire economy. J Public Econ 88(9–10):2193–2219

    Article  Google Scholar 

  • Diamond PA (1977) A framework for social security analysis. J Public Econ 8(3):275–298

    Article  Google Scholar 

  • Hagen RJ (2006) Samaritan agents? On the strategic delegation of aid policy. J Dev Econ 79(1):249–263

    Article  Google Scholar 

  • Hochguertel S, Ohlsson H (2009) Compensatory inter vivos gifts. J Appl Econom 24(6):993–1023

    Article  Google Scholar 

  • Hochman HM, Rodgers JD (1969) Pareto optimal redistribution. Am Econ Rev 59(4):542–557

    Google Scholar 

  • Homburg S (2000) Compulsory savings in the welfare state. J Public Econ 77(2):233–239

    Article  Google Scholar 

  • Jensen RT (2004) Do private transfers ’displace’ the benefits of public transfers? Evidence from South Africa. J Public Econ 88(1–2):89–112

    Article  Google Scholar 

  • Laitner J, Ohlsson H (2001) Bequest motives: a comparison of Sweden and the United States. J Public Econ 79(1):205–236

    Article  Google Scholar 

  • Lindbeck A, Weibull JW (1988) Altruism and time consistency: the economics of Fait Accompli. J Polit Econ 96(6):1165–1182

    Article  Google Scholar 

  • McGarry K (2012) Dynamic aspects of family transfers. Working paper no. 18446. National Bureau of Economic Research, Cambridge

    Book  Google Scholar 

  • McGarry K, Schoeni RF (1995) Transfer behavior in the health and retirement study: measurement and the redistribution of resources within the family. J Hum Resour 30:S184–226

    Article  Google Scholar 

  • Nishimura K, Zhang J (1992) Pay-as-you-go public pensions with endogenous fertility. J Public Econ 48(2):239–258

    Article  Google Scholar 

  • Roberts RD (1984) A positive model of private charity and public transfers. J Polit Econ 92(1):136–148

    Article  Google Scholar 

  • Sinn HW (1996) Social insurance, incentives and risk taking. Int Tax Public Finan 3(3):259–280

    Article  Google Scholar 

  • Svensson J (2000) When is foreign aid policy credible? Aid dependence and conditionality. J Dev Econ 61(1):61–84

    Article  Google Scholar 

  • Veall MR (1986) Public pensions as optimal social contracts. J Public Econ 31(2):237–251

    Article  Google Scholar 

  • Warr PG (1982) Pareto optimal redistribution and private charity. J Public Econ 19(1):131–138

    Article  Google Scholar 

  • Wolff FC (2012) Les parents sont-ils vraiment si peu altruistes? Working paper, Laboratoire d’Economie et de Management Nantes-Atlantique Universite de Nantes

Download references

Acknowledgments

We are indebted to comments made by the participants at the Seventh Japan/Irvine Conference on Public Policy, the anonymous referees of this journal, and to the editor, Alessandro Cigno, whose suggestions much improved the paper. Horoki Kondo is grateful for the support of the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (C) number 22530241.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hiroki Kondo.

Additional information

Responsible editor: Alessandro Cigno

Appendices

Appendix 1: Donor’s choice of private transfer

The values of d 1 and d 2 that maximize (2) satisfy the following:

$$ \frac{-1}{w-d_1-d_2-T} + \frac{\alpha \beta}{S_{1} + d_{1} + t_{1}} \leq 0, $$
(9)
$$ d_{1} \geq 0, $$
(10)
$$ \left(\frac{-1}{w-d_1-d_2-T} + \frac{\alpha \beta}{S_{1} + d_{1} + t_{1}} \right) d_{1} = 0, $$
(11)
$$ \frac{-1}{w-d_1-d_2-T} + \frac{\alpha \beta}{S_{2} + d_{2} + t_{2}} \leq 0, $$
(12)
$$ d_{2} \geq 0, $$
(13)

and

$$ \left(\frac{-1}{w-d_1-d_2-T} + \frac{\alpha \beta}{S_{2} + d_{2} + t_{2}} \right) d_{2} = 0, $$
(14)

where t i is the governmental transfer to recipient i, and t 1 + t 2 = T.

If both d 1 and d 2 are positive, from (11) and (14), we can see that both (9) and (12) hold as equalities. Solving them for d 1 and d 2 yields

$$ d_{i} = \left(\frac{\alpha \beta }{1 + 2 \alpha \beta}\right)(w-T+S_j+t_j) - \left(\frac{1 + \alpha \beta}{1 + 2 \alpha \beta}\right)(S_{i} + t_i) \geq 0, \ \ \ i, j = 1, 2, \ \ i \neq j. $$
(15)

From (15), d 1 and d 2 are positive if

$$ S_{i} \leq \left(\frac{\alpha \beta}{1+ \alpha \beta } \right) (w - T+ S_{j} + t_j) - t_{i}, \ \ \ i, j = 1, 2, \ \ i \neq j. $$
(16)

If d 2 is positive yet d 1 = 0, inspection of (14) shows that (12) holds as an equality. Solving (12) as an equality and d 1 = 0 yields

$$ d_{2} = \left(\frac{\alpha \beta}{1 + \alpha \beta} \right) (w - T) - \left(\frac{1}{1 + \alpha \beta} \right) (S_{2} + t_2) \geq 0. $$
(17)

Expression (17) is positive if

$$ S_{2} \leq \alpha \beta (w - T) - t_2. $$
(18)

Substituting d 2 and d 1 in (9) using (17) and d 1 = 0 and rearranging yields the condition under which d 1 = 0. This condition is the inverse of the condition (16) (with i = 1 and j = 2).

Lastly, if d 1 = 0 and d 2 = 0, from (11) and (14), we can see that (9) and (12) hold. Rearranging them yields

$$ S_{i} \geq \alpha \beta (w - T) - t_{i}, \ \ \ \ i = 1, 2. $$
(19)

When \(t_1=t_2=\frac {1}{2}T\), expression (15) becomes (4), and (16) becomes (3). That is, if \(t_1=t_2=\frac {1}{2}T\) and if both S 1 and S 2 are sufficiently small to satisfy (3), then d 1 and d 2 are (4).

When \(t_1=t_2=\frac {1}{2}T\), (17) is written as follows:

$$ d_{2} = \frac{\alpha \beta w - S_{2} - \left(\frac{1}{2} + \alpha \beta \right) T}{1+ \alpha \beta} \geq 0, $$
(20)

and (18) is written as follows:

$$ S_{2} \leq \alpha \beta w - \frac{1}{2} (1 + 2 \alpha \beta) T. $$
(21)

If S 1 is too large to satisfy (3) (with i = 1), yet S 2 is sufficiently small to satisfy (21), then d 1 = 0 and d 2 is (20).

Lastly, when \(t_1=t_2=\frac {1}{2}T\), (19) is written as follows:

$$ S_{i} \geq \alpha \beta w - \frac{1}{2} (1 + 2 \alpha \beta) T, \ \ \ \ i = 1, 2. $$
(22)

That is, if both S 1 and S 2 are sufficiently large to satisfy (22), then d 1 = d 2 = 0.

Appendix 2: Recipient 1’s savings as a function of recipient 2’s savings

From (16) and (19), the interval of (S 1, S 2) in which recipient 1 gets no transfer is as follows:

$$ S_{1} \geq \text{Min} \left[ \left(\frac{\alpha \beta}{1+ \alpha \beta } \right) (w - T + S_{2} + t_{2} ) -t_{1}, \alpha \beta (w - T) - t_{1} \right]. $$
(23)

When \(t_{1} =t_{2} =\frac {1}{2}T\), (23) is written as follows:

$$ S_{1} \geq \text{Min} \left[ \left(\frac{\alpha \beta}{1+ \alpha \beta } \right) w + \left(\frac{\alpha \beta}{1+ \alpha \beta } \right) S_{2} - \frac{ 1}{2} \left(\frac{1 + 2 \alpha \beta}{1+ \alpha \beta } \right) T, \alpha \beta w - \frac{1}{2} (1 + 2 \alpha \beta) T \right]. $$
(24)

Appendix 2.1: Recipient 1’s optimal savings limited to the interval (23)

Differentiating (1) with respect to S 1 with d 1 = 0 yields

$$ \frac{-1}{w-S_{1}} + \beta \left(\frac{1}{S_{1} + t_{1}} \right). $$
(25)

If S 1 in the interval (23) makes (25) zero, this S 1 is recipient 1’s optimal savings in this interval. From (25), this S 1 is

$$ S_{1} = \frac{\beta w - t_{1}}{1 + \beta}. $$
(26)

When \(t_1=t_2=\frac {1}{2}T\), this expression becomes

$$ \frac{\beta w -\frac{1}{2}T}{1+\beta}. $$
(27)

When the government makes a transfer only to recipient 2 (t 1 = 0 and t 2 = T), (26) becomes

$$ \frac{\beta w}{1+\beta}. $$
(28)

Inversely, when the government makes a transfer only to recipient 1 (t 1 = T and t 2 = 0), (26) becomes

$$ \frac{\beta w - T}{1+\beta}. $$
(29)

This value minus α β(wT) − t 1 yields

$$ \frac{\beta(1 - \alpha (1 + \beta )) w + \beta (t_{1} + \alpha (1 + \beta)T) }{1+\beta }. $$
(30)

As 0 < β < 1 and 0 < α < 1 / 2, this expression is positive. Savings \(\frac {\beta w - t_{1}}{1 + \beta }\) are higher than the border of (23), and thus it is the optimal choice.

Appendix 2.2: Recipient 1’s optimal savings limited to the interval of S 1 is too small to satisfy (23)

The interval in which S 1 is smaller than the right-hand side of (23) is divided into two intervals. In one of the intervals, S 1 is smaller than the border of (16) (with i = 1 and j = 2), but larger than the border of (16) (with i = 2 and j = 1), so that the donor makes positive transfers to both recipients 1 and 2, as in (15). That is,

$$ S_{1} \in \left[ -(w - T + t_1) + \left(\frac{1+ \alpha \beta}{\alpha \beta } \right) (S_{2} + t_2) , \left(\frac{\alpha \beta}{1+ \alpha \beta } \right) (w - T + S_{2} + t_{2} ) - t_{1} \right]. $$
(31)

In the other interval, S 1 is smaller than the border of (16) (with i = 2 and j = 1) and the border of (19) (with i = 1). When (S 1, S 2) lies in this interval, the donor makes a transfer only to recipient 1. This interval is

$$ S_{1} \leq \text{Min} \left[ -(w - T + t_1) + \left(\frac{1+ \alpha \beta}{ \alpha \beta } \right) (S_{2} + t_2) , \alpha \beta (w - T) - t_{1} \right]. $$
(32)

We first consider recipient 1’s utility-maximizing savings limited to the interval (31). Differentiating (1) with respect to S 1, with d 1 that is equal to (15), yields

$$ \frac{-1}{w-S_{1}} + \beta \left(\frac{1}{S_{1} + d (S_{1}, S_{2}, T) + t_{1}} \right) \left[ 1 - \left(\frac{1 + \alpha \beta}{1 + 2 \alpha \beta} \right) \right]. $$
(33)

If S 1 in the interval (31) makes (33) zero, this S 1 is recipient 1’s optimal savings in this interval. From (33) and (15) (with i = 1), this S 1 is calculated as (5). If (5) is larger than the upper limit in the interval (31), recipient 1’s optimal S 1 limited to this interval is the upper limit of this interval. The value of (5) exceeds the upper limit if

$$ S_{2} \leq \left(\frac{\beta - 1 - 2 \alpha \beta}{1 + 2 \alpha \beta + \alpha \beta^{2}} \right) w + \left[ \frac{\alpha \beta (1 + \beta)}{1 + 2 \alpha \beta + \alpha \beta^{2}} \right] (T - t_{2} ) + \left[ \frac{(1 + \alpha \beta)(1 + \beta)}{1 + 2 \alpha \beta + \alpha \beta^{2}} \right] t_1. $$
(34)

In contrast, if (5) is smaller than the lower limit in the interval (31), recipient 1’s optimal S 1 in this interval is the lower limit of this interval. The value of (5) is lower than the lower limit if

$$ S_{2} \geq \left(\frac{2 \alpha \beta^{2}}{1 + \beta + 2 \alpha \beta + \alpha \beta^{2}} \right) w - \left[ \frac{\alpha \beta (1 + \beta)}{1 + \beta + 2 \alpha \beta + \alpha \beta^{2}} \right] (T - t_1) - \left[ \frac{(1 + \beta) (1 + \alpha \beta)}{1 + \beta + 2 \alpha \beta + \alpha \beta^{2}} \right] t_2. $$
(35)

Consider next recipient 1’s optimal savings limited to the interval (32). Differentiating (1) with respect to S 1 when d 1 equals the value of (17) for recipient 1 yields:

$$ \frac{-1}{w-S_{1}} + \beta \left(\frac{1}{S_{1} + d (S_{1}, S_{2}, T) + t_{1}} \right) \left[ 1 - \left(\frac{1}{1 + \alpha \beta} \right) \right]. $$
(36)

If S 1 in the interval (32) makes the value of (36) zero, this S 1 is recipient 1’s optimal savings in this interval. From (36) and (17) for recipient 1, this S 1 is

$$ S_{1} = \frac{(\beta - 1) w + T - t_{1}}{1+\beta} . $$
(37)

If (37) exceeds αβ(wT) − t 1, that is, if

$$ \frac{(\beta - 1) w + T - t_{1} }{1+\beta} \geq \alpha \beta (w - T) - t_{1}, $$
(38)

then recipient 1’s optimal savings limited to this interval is the upper limit of this interval. If (37) is smaller than αβ(wT) − t 1 but greater than \(-(w - T + t_1) + \left (\frac {1+ \alpha \beta }{\alpha \beta } \right ) (S_{2} + t_2)\), the last value is recipient 1’s optimal savings limited to this interval. The value of (37) exceeds \(-(w - T + t_1) + \left (\frac {1+ \alpha \beta }{\alpha \beta } \right ) (S_{2} + t_2)\) if:

$$ S_{2} \leq \left[ \frac{2 \alpha \beta^{2}}{(1 + \alpha \beta)(1 + \beta)} \right] w - \left[ \frac{\alpha \beta^{2}}{(1 + \alpha \beta)(1 + \beta)} \right](T - t_{1} ) - t_2. $$
(39)

The right-hand side of (35) is necessarily larger than the right-hand side of (34), and the right-hand side of (39) is necessarily larger than the right-hand side of (35). Therefore, to summarize, recipient 1’s optimal savings limited to the interval of S 1 that is too small to satisfy (23) are as follows:if \(S_{2} \leq \left (\frac {\beta - 1 - 2 \alpha \beta }{1 + 2 \alpha \beta + \alpha \beta ^{2}} \right ) w + \left [ \frac {\alpha \beta (1 + \beta )}{1 + 2 \alpha \beta + \alpha \beta ^{2}} \right ] (T - t_{2} ) + \left [ \frac {(1 + \alpha \beta )(1 + \beta )}{1 + 2 \alpha \beta + \alpha \beta ^{2}} \right ] t_{1}\),

$$ S_{1} = \left(\frac{\alpha \beta}{1+ \alpha \beta } \right) (w - T + S_{2} + t_{2} ) - t_{1} , $$
(40)

if \(\left (\frac {\beta - 1 - 2 \alpha \beta }{1 + 2 \alpha \beta + \alpha \beta ^{2}} \right ) w + \left [ \frac {\alpha \beta (1 + \beta )}{1 + 2 \alpha \beta + \alpha \beta ^{2}} \right ] (T - t_{2} ) + \left [ \frac {(1 + \alpha \beta )(1 + \beta )}{1 + 2 \alpha \beta + \alpha \beta ^{2}} \right ] t_1 \leq S_{2} \leq \left (\frac {2 \alpha \beta ^{2}}{1 + \beta + 2 \alpha \beta + \alpha \beta ^{2}} \right ) w - \left [ \frac {\alpha \beta (1 + \beta )}{1 + \beta + 2 \alpha \beta + \alpha \beta ^{2}} \right ] (T - t_1) - \left [ \frac {(1 + \beta ) (1 + \alpha \beta )}{1 + \beta + 2 \alpha \beta + \alpha \beta ^{2}} \right ] t_{2}\), S 1 is (5),if (38) holds and if \(\left (\frac {2 \alpha \beta ^{2}}{1 + \beta + 2 \alpha \beta + \alpha \beta ^{2}} \right ) w - \left [ \frac {\alpha \beta (1 + \beta )}{1 + \beta + 2 \alpha \beta + \alpha \beta ^{2}} \right ] (T - t_1) - \left [ \frac {(1 + \beta ) (1 + \alpha \beta )}{1 + \beta + 2 \alpha \beta + \alpha \beta ^{2}} \right ] t_{2} \leq S_{2}\),

$$ S_{1} = \text{Min} \left[ -(w - T + t_1) + \left(\frac{1+ \alpha \beta}{\alpha \beta } \right) (S_{2} + t_2) , \alpha \beta (w - T) - t_{1} \right]. $$
(41)

if (38) does not hold and if \(\left (\frac {2 \alpha \beta ^{2}}{1 + \beta + 2 \alpha \beta + \alpha \beta ^{2}} \right ) w - \left [ \frac {\alpha \beta (1 + \beta )}{1 + \beta + 2 \alpha \beta + \alpha \beta ^{2}} \right ] (T - t_1) - \left [ \frac {(1 + \beta ) (1 + \alpha \beta )}{1 + \beta + 2 \alpha \beta + \alpha \beta ^{2}} \right ] t_{2} \leq S_{2} \leq \left [ \frac {2 \alpha \beta ^{2}}{(1 + \alpha \beta )(1 + \beta )} \right ] w - \left [ \frac { \alpha \beta ^{2}}{(1 + \alpha \beta )(1 + \beta )} \right ](T - t_{1} ) - t_{2} \),

$$ S_{1} = -(w - T + t_1) + \left(\frac{1+ \alpha \beta}{\alpha \beta } \right) (S_{2} + t_2), $$
(42)

if (38) does not hold and if \(\left [ \frac {2 \alpha \beta ^{2}}{(1 + \alpha \beta )(1 + \beta )} \right ] w - \left [ \frac {\alpha \beta ^{2}}{(1 + \alpha \beta )(1 + \beta )} \right ](T - t_{1} ) - t_{2} \leq S_{2}\), then S 1 is (37).

When t 1 = t 2 = T / 2, (37) becomes

$$ \frac{(\beta - 1) w + \frac{1}{2}T}{1+\beta}. $$
(43)

When the government makes a transfer only to recipient 2 (t 1 = 0 and t 2 = T), (37) becomes

$$ \frac{(\beta - 1)w+T}{1+\beta}. $$
(44)

Inversely, only recipient 1 gets a governmental transfer (t 1 = T and t 2 = 0), expression (37) becomes

$$ \left(\frac{\beta -1 }{1+\beta} \right) w. $$
(45)

Appendix 3: Proof of Proposition 1

Suppose that recipient 2 saves (6). Recipient 1’s best response is to save little in the amount (6) and get private transfer in the amount (7), or to save more in the amount (27) and have no private transfer. From (1), the recipient’s utility when he saves (6) is as follows:

$$ \ln \left(\frac{3}{2+\beta} \right) w + \beta \ln \left[ \frac{3 \alpha \beta^{2}}{(1+2 \alpha \beta )(2+\beta)} \right] w. $$
(46)

and that when he saves (27) is as follows:

$$ \ln \left(\frac{1}{1+\beta} \right) \left(w+\frac{1}{2} T \right) + \beta \ln \left(\frac{\beta}{1+\beta} \right) \left(w+\frac{1}{2} T \right). $$
(47)

The value of (46) minus the value of (47) is

$$ (1+\beta) \ln \left[ \frac{3(1 + \beta)}{2+\beta} \right] + \beta \ln \left(\frac{\alpha \beta}{1 + 2 \alpha \beta} \right). $$
(48)

If (48) is positive, in the Nash equilibrium, each recipient saves little, as in (6). When β = 0 and α = 1 / 2, the value of (48) is positive. Differentiating (48) with respect to α yields

$$ \frac{\beta}{\alpha (1+2 \alpha \beta)}. $$
(49)

This expression is positive, and thus large α generates the Nash equilibrium in which each recipient saves little. Differentiating (48) with respect to β yields

$$ \frac{3 + \beta + 2 \alpha \beta }{(2 + \beta) (1+2 \alpha \beta)} + \ln \left[ \frac{3 (1 + \beta)}{2 + \beta} \right] \left(\frac{\alpha \beta}{1 + 2 \alpha \beta} \right). $$
(50)

This expression is negative when β is small. Hence, the smaller the β, the wider is the range of parameters for which the Nash equilibrium has each recipient saving little.

Appendix 4: Proof of Proposition 3

We show that if \(t_1=t_2= \bar {d}\) (\(T=2 \bar {d}\)) and if recipient 2 saves \(\frac {\beta w -t_{1}}{1+\beta }\), as in (26), recipient 1 will save the same amount.

From Appendix 2.2, if recipient 2 saves \(\frac {\beta w -\frac {1}{2}T}{1+\beta }\) and if (38) holds, the optimal S 1 limited to the interval of S 1α β(wT) − t 1 is S 1 = α β(wT) − t 1. In other words, in this interval, recipient 1’s utility increases with S 1. Hence, he gains by choosing S 1 in the interval of (23). From Appendix 2.1, recipient 1 will choose (26).

It therefore suffices to show that (38) holds when \(t_1=t_2= \bar {d}\). From (7), \(\bar {d}\) is

$$ \bar{d} = \left[ \frac{1 - \beta + \alpha \beta^{2} + 2 \alpha \beta}{(1 + 2\alpha \beta )(2 + \beta )} \right]w. $$
(51)

Using \(t_1=t_2= \bar {d}\), \(T=2 \bar {d}\), and the equation above, \(\frac { (\beta - 1) w + T - t_{1} }{1+\beta } - \left [ \alpha \beta (w - T) - t_1 \right ]\) becomes as follows:

$$ \begin{array}{ll} & \left[ \frac{ \beta -1 - \alpha \beta - \alpha \beta^{2} }{1 + \beta} \right] w + \left[ \frac{ \beta +2 + 2 \alpha \beta + 2 \alpha \beta^{2} }{1 + \beta} \right] \bar{d} \\ =& \left[ \frac{ \beta -1 - \alpha \beta - \alpha \beta^{2} }{1 + \beta} \right] w + \left[ \frac{ \beta +2 + 2 \alpha \beta + 2 \alpha \beta^{2} }{1 + \beta} \right] \left[ \frac{1 - \beta + \alpha \beta^{2} + 2 \alpha \beta}{ (1 + 2\alpha \beta )(2 + \beta ) } \right]w \\ = & \left[ \frac{3 \alpha \beta^{2}}{(1 + \beta)(2 + \beta)(1 + 2 \alpha \beta) } \right]w > 0 \end{array} $$
(52)

Hence, (38) holds.

Appendix 5: Proof of Proposition 4

First, we consider recipient R NG’s best response when recipient R G saves \(\frac {\beta w -\bar {d}}{1+\beta }\).

If recipient R NG saves the same amount that he saves in the equilibrium with no governmental transfer, his utility is (46) and the same as it is in that equilibrium. In the equilibrium with no governmental transfer, a donor makes private transfer \(\bar {d}\) to both of the two recipients saving in the amount \(\left (\frac {\beta -1}{2+\beta } \right )w\). In contrast, now the donor makes no direct private transfer to recipient R G. Yet the donor is taxed in the amount \(\bar {d}\), and this is transferred to recipient R G by government. Then, the donor will make a private transfer in the same amount \(\bar {d}\) to recipient R NG, if that recipient saves \(\left (\frac {\beta -1}{2+\beta }\right )w\). We can confirm \(d_i= \bar {d}\) by calculating (15) in which \(T=t_j= \bar {d}\) is (51), t i = 0, and \(S_i=S_j=\left (\frac {\beta -1}{2+\beta }\right )w\).

If, however, recipient R G saves as much as \(\frac {\beta w -\bar {d}}{ 1+\beta }\), (29) where \(T=\bar {d}\), and gets no private transfer, recipient R NG can enjoy utility higher than (46) by increasing his savings to \(\frac {(\beta -1) w + \bar {d}}{1+\beta }\), (44) where \(T= \bar {d}\).

The utility of recipient R NG is highest when he saves \(\frac {(\beta -1) w +\bar {d}}{1+\beta }\). This utility exceeds that described by (46), and (46) is higher than the maximized utility with no private transfer (47) attained by saving the amount \(\frac {\beta w}{1+\beta }\). The results hold because we are now considering outcomes when α and β lie in the range for which (46) is higher than (47), so that (46) is the maximized utility in the case with no governmental transfer, as Proposition 1 indicates. Recipient R NG therefore saves \(\frac {(\beta -1) w +\bar {d}}{1+\beta }\).

Next, we confirm that recipient R G’s best response is to save \(\frac { \beta w -\bar {d}}{1+\beta }\) when recipient R NG saves \(\frac {(\beta -1) w +\bar {d}}{1+\beta }\).

Recipient R G chooses between the values of expressions \(\frac {\beta w - \bar {d}}{1+\beta }\) and (5) in which the amount that recipient R NG saves is \(\frac {(\beta -1) w +\bar {d}}{ 1+\beta }\) to give him the highest utility. From (29), \(T=\bar {d}\), (51), and d G = 0, recipient R G’s utility when he saves \(\frac {\beta w - \bar {d}}{1+\beta }\) is calculated as follows:

$$ \begin{array}{ll} & \ln \left(\frac{1}{1+\beta} \right)+\beta \ln \left(\frac{\beta}{1+\beta} \right) + (1+\beta) \ln (w+\bar{d}) \\ =& \ln \left(\frac{1}{1+\beta} \right)+\beta \ln \left(\frac{\beta}{1+\beta } \right) + (1+\beta) \ln \left[\frac{3(1+\alpha \beta^{2} + 2 \alpha \beta)}{ (1+2 \alpha \beta)(2+\beta)}\right]w. \end{array} $$
(53)

From (5) in which the amount that recipient R NG saves is \(\frac {(\beta -1) w +\bar {d}}{1+\beta }\), \(T=\bar { d}\), (51), and (15), recipient R G’s utility when he saves (5) is

$$ \begin{array}{ll} & \ln \left(\frac{1}{1+\beta} \right)+\beta \ln \left(\frac{\beta}{1+\beta} \right) +\beta \ln \left(\frac{\alpha \beta}{1+2 \alpha \beta}\right) \\ &+ (1+\beta) \ln \left[2w + \frac{(\beta - 1)w + \bar{d}}{1+\beta} \right] \\ =& \ln \left(\frac{1}{1+\beta} \right)+\beta \ln \left(\frac{\beta}{1+\beta } \right) +\beta \ln \left(\frac{\alpha \beta}{1+2 \alpha \beta}\right) \\ &+(1+\beta) \ln \left[\frac{3(1+2 \beta + 2 \alpha \beta + 5 \alpha \beta^2 + \beta^{2} +2 \alpha \beta^3)}{(1+\beta)(1+2 \alpha \beta)(2+\beta)}\right]w \end{array} $$
(54)

The value of (53) minus (54) is calculated as follows, and this is positive.

$$ \beta \ln \left(\frac{1+2 \alpha \beta}{\alpha \beta}\right)-(1+\beta)\ln \left(\frac{1+2 \beta + 2 \alpha \beta + 5 \alpha \beta^{2} + \beta^{2} + 2 \alpha \beta^{3}}{1+\beta + 2 \alpha \beta + 3\alpha \beta^{2} + \alpha \beta^{3}} \right) $$
(55)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Glazer, A., Kondo, H. Governmental transfers and altruistic private transfers. J Popul Econ 28, 509–533 (2015). https://doi.org/10.1007/s00148-014-0503-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00148-014-0503-2

Keywords

  • Social security
  • Moral hazard
  • Savings
  • Altruism