Tough Love and Intergenerational Altruism

Abstract

This chapter develops and studies a tough love model of intergenerational altruism. We model tough love by modifying the Barro-Becker standard altruism model in two ways. First, the child’s discount factor is endogenously determined, so lower childhood consumption leads to a higher discount factor later in life. Second, the parent evaluates the child’s lifetime utility with a constant high discount factor. Our model predicts that parental transfers will fall when the child’s discount factor falls. This is in contrast with the standard altruism model, which predicts that parental transfers are independent of exogenous changes in the child’s discount factor.

The original article first appeared in the International Economic Review 53, 791–814, 2012. A newly written addendum has been added to this book chapter.

Appendices

Appendices

1.1 A Proof for Proposition 1

In this section of the appendix we provide an analytical proof of our main result specified in Eq. (2.13):

$$ \displaystyle\begin{array}{rcl} \frac{\partial T^{{\ast}}} {\partial \beta _{0}} \!\!>\!\! 0\ \ \mathit{iff }\ \ {\biggl [1 + R(\beta _{p} -\beta _{3,k})\frac{u^{{\prime\prime}}(R(y_{2} - C_{2}^{{\ast}}))} {u^{{\prime}}(R(y_{2} - C_{2}^{{\ast}}))} \frac{\partial C_{2}^{{\ast}}} {\partial \beta _{0}} - (\beta _{p} -\beta _{3,k})\frac{\partial ^{2}C_{2}^{{\ast}}} {\partial \beta _{0}\partial T} \biggr ]}> 0.& & {}\\ \end{array} $$

We start the derivation of the above result by noting that the parent accounts for C ₂ ^∗ when maximizing utility by choosing transfers, T. From the first-order condition for the parent’s problem described in Eqs. (2.10) and (2.11), we obtain:

$$\displaystyle\begin{array}{rcl} V ^{{\prime}}(R(y_{ p} - T^{{\ast}}),\tilde{\beta })& =& {}\\ & & {\biggl (\frac{1-\eta } {\eta } \biggr )} \frac{\tilde{\beta }} {R}{\biggl (u^{{\prime}}(y_{ 1} + T^{{\ast}})\! +\!\beta _{ p}u^{{\prime}}(C_{ 2}^{{\ast}})\frac{\partial C_{2}^{{\ast}}} {\partial T} -\beta _{p}^{2}Ru^{{\prime}}(R(y_{ 2} - C_{2}^{{\ast}}))\frac{\partial C_{2}^{{\ast}}} {\partial T} \biggr )},{}\\ \end{array}$$

where T ^∗ denotes optimal parental transfers, $V ^{{\prime}}(.) = \frac{\partial V (.)} {\partial T}$, and $u^{{\prime}}(.) = \frac{\partial u(.)} {\partial T}.$

Now consider an exogenous change in the child’s discount factor (β _k ) captured by a change in β ₀. From the parent’s first-order condition described above we obtain:

$$\displaystyle\begin{array}{rcl} \frac{\partial T^{{\ast}}} {\partial \beta _{0}} = A {\ast}{\biggl ( \frac{\partial } {\partial \beta _{0}}{\biggl [\beta _{p}u^{{\prime}}(C_{ 2}^{{\ast}})\frac{\partial C_{2}^{{\ast}}} {\partial T} -\beta _{p}^{2}Ru^{{\prime}}(R(y_{ 2} - C_{2}^{{\ast}}))\frac{\partial C_{2}^{{\ast}}} {\partial T} \biggr ]}\biggr )},& & {}\\ \end{array}$$

where:

$$\displaystyle{A = - \frac{{\biggl (\frac{1-\eta } {\eta } \biggr )}\tilde{\beta }} {V ^{{\prime\prime}}(R(y_{p} - T^{{\ast}}),\tilde{\beta }) +{\biggl ( \frac{1-\eta } {\eta } \biggr )}\tilde{\beta }u^{{\prime\prime}}(y_{1} + T^{{\ast}})}.}$$

Given concavity of V(.) and u(.), we know that A > 0. Hence:

$$\displaystyle\begin{array}{rcl} \mathit{sign}{\biggl (\frac{\partial T^{{\ast}}} {\partial \beta _{0}} \biggr )} = \mathit{sign}{\biggl ( \frac{\partial } {\partial \beta _{0}}{\biggl [\beta _{p}u^{{\prime}}(C_{ 2}^{{\ast}})\frac{\partial C_{2}^{{\ast}}} {\partial T} -\beta _{p}^{2}Ru^{{\prime}}(R(y_{ 2} - C_{2}^{{\ast}}))\frac{\partial C_{2}^{{\ast}}} {\partial T} \biggr ]}\biggr )}.& & {}\\ \end{array}$$

Then $\frac{\partial T^{{\ast}}} {\partial \beta _{0}}> 0$ if and only if:

$$\displaystyle{ \frac{\partial } {\partial \beta _{0}}{\biggl [\beta _{p}u^{{\prime}}(C_{ 2}^{{\ast}})\frac{\partial C_{2}^{{\ast}}} {\partial T} -\beta _{p}^{2}Ru^{{\prime}}(R(y_{ 2} - C_{2}^{{\ast}}))\frac{\partial C_{2}^{{\ast}}} {\partial T} \biggr ]}> 0.}$$

Now using the first-order condition for the child’s problem, we can rewrite the above condition as follows.

$$\displaystyle\begin{array}{rcl} \frac{\partial } {\partial \beta _{0}}{\biggl [ - (\beta _{p} -\beta _{3,k})Ru^{{\prime}}(R(y_{ 2} - C_{2}^{{\ast}}))\frac{\partial C_{2}^{{\ast}}} {\partial T} \biggr ]}> 0& & {}\\ \end{array}$$

It is straightforward to show that the LHS of the above expression is given by:

$$\displaystyle\begin{array}{rcl} & & Ru^{{\prime}}(R(y_{ 2} - C_{2}^{{\ast}}))\frac{\partial C_{2}^{{\ast}}} {\partial T} \biggl (1 + R(\beta _{p} -\beta _{3,k})\frac{u^{{\prime\prime}}(R(y_{2} - C_{2}^{{\ast}}))} {u^{{\prime}}(R(y_{2} - C_{2}^{{\ast}}))} \frac{\partial C_{2}^{{\ast}}} {\partial \beta _{0}} {}\\ & & \qquad \qquad \qquad \qquad \qquad - (\beta _{p} -\beta _{3,k})\frac{\partial ^{2}C_{2}^{{\ast}}} {\partial \beta _{0}\partial T} \biggr ). {}\\ \end{array}$$

Since $\frac{\partial C_{2}^{{\ast}}} {\partial T}> 0$ and u ^′(. ) > 0, the above expression is strictly positive if and only if:

$$\displaystyle\begin{array}{rcl} {\biggl [1 + R(\beta _{p} -\beta _{3,k})\frac{u^{{\prime\prime}}(R(y_{2} - C_{2}^{{\ast}}))} {u^{{\prime}}(R(y_{2} - C_{2}^{{\ast}}))} \frac{\partial C_{2}^{{\ast}}} {\partial \beta _{0}} - (\beta _{p} -\beta _{3,k})\frac{\partial ^{2}C_{2}^{{\ast}}} {\partial \beta _{0}\partial T} \biggr ]}> 0& & {}\\ \end{array}$$

This establishes our claim in Eq. (2.14). Now, given positive marginal utility, concavity of u(. ), and $\frac{\partial C_{2}^{{\ast}}} {\partial \beta _{0}} <0$, the sufficient conditions for the above expression to be strictly positive are:

• (i) β _p  ≥ β _3, k and

• (ii) $\frac{\partial ^{2}C_{2}^{{\ast}}} {\partial \beta _{0}\partial T} \leq 0$.

1.2 A Proof for Proposition 2

In this section we show that $\frac{\partial ^{2}C_{2}^{{\ast}}} {\partial \beta _{0}\partial T} \leq 0$ will depend on the convexity of the marginal utility (as captured by the positive third derivative of the utility function) and the impatience level of the child.

We start our derivation with the partial derivative of the optimal second period consumption with respect to parental transfers:

$$\displaystyle\begin{array}{rcl} \frac{\partial C_{2}^{{\ast}}} {\partial T} = \frac{\beta ^{{\prime}}(y_{1} + T)Ru^{{\prime}}(R(y_{2} - C_{2}^{{\ast}}))} {[u^{{\prime\prime}}(C_{2}^{{\ast}}) +\beta _{3,k}R^{2}u^{{\prime\prime}}(R(y_{2} - C_{2}^{{\ast}}))]},& & {}\\ \end{array}$$

where

$$\displaystyle{\beta _{3,k} =\beta _{0} +\psi (y_{1} + T).}$$

Differentiating with respect to β ₀

$$\displaystyle\begin{array}{rcl} \frac{\partial ^{2}C_{2}^{{\ast}}} {\partial \beta _{0}\partial T} = \frac{N} {D},& & {}\\ \end{array}$$

where

$$\displaystyle{D = [u^{{\prime\prime}}(C_{ 2}^{{\ast}}) +\beta _{ 3,k}R^{2}u^{{\prime\prime}}(R(y_{ 2} - C_{2}^{{\ast}}))]^{2}> 0.}$$

and

$$\displaystyle\begin{array}{rcl} N = [u^{{\prime\prime}}(C_{ 2}^{{\ast}}) +\beta _{ 3,k}R^{2}u^{{\prime\prime}}(R(y_{ 2} - C_{2}^{{\ast}}))][-\psi ^{{\prime}}(y_{ 1} + T)R^{2}u^{{\prime\prime}}(R(y_{ 2} - C_{2}^{{\ast}}))\frac{\partial C_{2}^{{\ast}}} {\partial \beta _{0}} ]& & {}\\ -\psi ^{{\prime}}(y_{ 1} + T)Ru^{{\prime}}(R(y_{ 2} - C_{2}^{{\ast}}))[u^{{\prime\prime\prime}}(C_{ 2}^{{\ast}})\frac{\partial C_{2}^{{\ast}}} {\partial \beta _{0}} + R^{2}u^{{\prime\prime}}() -\beta _{ 3,k}R^{3}u^{{\prime\prime\prime}}(R(y_{ 2} - C_{2}^{{\ast}}))\frac{\partial C_{2}^{{\ast}}} {\partial \beta _{0}} ].& & {}\\ \end{array}$$

Since, D is always positive, the sign of $\frac{\partial ^{2}C_{2}^{{\ast}}} {\partial \beta _{0}\partial T}$ depends on the sign of N. Now, the sign of N will be the same as the sign of:

$$\displaystyle{\mathit{sign}([u^{{\prime\prime\prime}}(C_{ 2}^{{\ast}})\frac{\partial C_{2}^{{\ast}}} {\partial \beta _{0}} + R^{2}u^{{\prime\prime}}(R(y_{ 2} - C_{2}^{{\ast}})) -\beta _{ 3,k}R^{3}u^{{\prime\prime\prime}}(R(y_{ 2} - C_{2}^{{\ast}}))\frac{\partial C_{2}^{{\ast}}} {\partial \beta _{0}} ]).}$$

If the above expression is negative then N < 0.

Hence, the condition for N < 0 is:

$$\displaystyle{[u^{{\prime\prime\prime}}(C_{ 2}^{{\ast}})\frac{\partial C_{2}^{{\ast}}} {\partial \beta _{0}} + R^{2}u^{{\prime\prime}}(C_{ 3}^{{\ast}})\{1 - (\beta _{ 3,k}R\frac{u^{{\prime\prime\prime}}(C_{3}^{{\ast}})} {u^{{\prime\prime}}(C_{3}^{{\ast}})} \frac{\partial C_{2}^{{\ast}}} {\partial \beta _{0}} \}] <0.}$$

The above condition holds if,

• (i) $u^{{\prime\prime\prime}}(.) \geq 0$ and

• (ii) $\beta _{3,k}\ R\ G \leq 1$,

where

$$\displaystyle{G ={\biggl ( \frac{u^{{\prime\prime\prime}}(C_{3}^{{\ast}})} {u^{{\prime\prime}}(C_{3}^{{\ast}})} \biggr )}\frac{\partial C_{2}^{{\ast}}} {\partial \beta _{0}}.}$$

1.3 Power Utility Function

In this section of the appendix we use the power utility function to interpret the following condition:

$$\displaystyle{\beta _{3,k}\ R\ G \leq 1,}$$

where

$$\displaystyle{G ={\biggl ( \frac{u^{{\prime\prime\prime}}(C_{3}^{{\ast}})} {u^{{\prime\prime}}(C_{3}^{{\ast}})} \biggr )}\frac{\partial C_{2}^{{\ast}}} {\partial \beta _{0}}.}$$

We assume that the period utility function is given by:

$$\displaystyle{u(c) = \frac{c^{1-\sigma }} {1-\sigma }.}$$

Using the above specification of the utility function, from the child’s optimization problem, we get:

$$\displaystyle\begin{array}{rcl} C_{2}^{{\ast}} = \frac{\mathit{Ry}_{2}} {R + (\beta _{3,k}R)^{\frac{1} {\sigma } }} & & {}\\ \end{array}$$

$$\displaystyle\begin{array}{rcl} C_{3}^{{\ast}} = R(y_{ 2} - C_{2}^{{\ast}}) = (\beta _{ 3,k}R)^{\frac{1} {\sigma } }C_{2}^{{\ast}}& & {}\\ \end{array}$$

$$\displaystyle\begin{array}{rcl} \frac{\partial C_{2}^{{\ast}}} {\partial \beta _{0}} = -\frac{R^{2}y_{2}(\beta _{3,k}R)^{\frac{1-\sigma } {\sigma } }} {\sigma [R + (\beta _{3,k}R)^{\frac{1} {\sigma } }]^{2}}& & {}\\ \end{array}$$

$$\displaystyle\begin{array}{rcl} \frac{u^{{\prime\prime\prime}}(C_{3}^{{\ast}})} {u^{{\prime\prime}}(C_{3}^{{\ast}})} = -(\sigma +1)(\beta _{3,k}R)^{-\frac{1} {\sigma } }{\biggl ( \frac{R + (\beta _{3,k}R)^{\frac{1} {\sigma } }} {\mathit{Ry}_{2}} \biggr )}& & {}\\ \end{array}$$

Hence, for the power utility case we get:

$$\displaystyle\begin{array}{rcl} G ={\biggl ( \frac{\sigma +1} {\sigma } \biggr )} \frac{1} {\beta _{3,k}[R + (\beta _{3,k}R)^{\frac{1} {\sigma } }]}& & {}\\ \end{array}$$

Using the above expression for G we can rewrite the inequality of interest as:

$$\displaystyle{\beta _{3,k}R{\biggl (\frac{\sigma +1} {\sigma } \biggr )} \frac{1} {\beta _{3,k}[R + (\beta _{3,k}R)^{\frac{1} {\sigma } }]} \leq 1}$$

Rearranging, we get the following condition:

$$\displaystyle{\beta _{3,k}R \leq {\biggl ( \frac{\sigma } {R}\biggr )}^{\frac{1} {\sigma } }}$$

Addendum: Recent Developments

This addendum has been newly written for this book chapter.

Two directions of research that are closely related to the tough love model have been developing since the working paper version (Bhatt and Ogaki 2008) of this chapter was published. One direction is for normative economics and the other is for positive economics. The first direction is to add an element of virtue ethics to welfarism that has been the basis of normative economics. The second direction is to empirically evaluate the tough love model.

In Bhatt et al. (2014), we proposed an approach of normative economics for models with endogenous preferences that adds an element of virtue ethics to the traditional formulation based on the Bergson-Samuelson social welfare function (SWF). Our approach is based on the moral evaluation function (MEF) that evaluates different endogenous preferences in terms of moral virtue and on the social objective function (SOF), which is a function of SWF and MEF in order to express a balanced value judgment. This is in a sense a response to Sandel’s (2013) call to introduce more value judgment into economics based on his ideas of political philosophy explained for wide audience in Sandel (2009, 2012). In the same issue of the Journal of Economic Perspectives as in Sandel (2013) and Bruni and Sugden (2013) argued that neoclassical economics has already incorporated virtue ethics if we think about market virtues. Our approach can incorporate market virtues, but can also incorporate other virtues such as altruism toward a disabled stranger as shown in Bhatt et al. (2014). In Bhatt and Ogaki (2014, Rational addiction and optimal taxation: a reexamination based on the social objective function, unpublished), we applied our approach to the rational addiction model. If a society finds that preferences with drug addiction caused by past drug consumption is less virtuous, preferences affected by drug addiction are less virtuous than preferences that are not.

For the purpose of illustrating our approach by an example, Bhatt et al. (2013) applies it to a version of Bhatt and Ogaki’s tough love model. In this version, the original model is extended by bequest and bequest tax. In the model, the parent thinks that he should not spoil the child so that the child will grow to be patient, but is tempted to spoil the child because he enjoys watching his child having higher childhood utility. With a bequest motive, the parent can use the money saved by lowering childhood transfers in order to increase his bequest given to the child after she grows up. In this version of the model, the government has a policy tool of the bequest tax rate that can be used to influence the optimizing behaviors of the parent and the child. When the bequest tax rate is higher, the parent is more tempted to spoil the child because the money he saves by giving less transfers to the child will be taxed away when he gives the bequest.

If our informal discussions with many economists give a good guidance, many economists seem to think that moral value ethics is not desirable for public policy evaluations because they do not want the government to influence people’s preferences. However, in our model, the government does influence the child’s preferences as long as the bequest tax rate is not zero. The optimum tax rate is positive when the SWF is maximized. On the other hand, the optimum tax rate is zero when the SOF is maximized with α = 0. 3. Thus, introducing moral virtue ethics may result in a policy that does not affect people’s preferences. This illustrates that introduction of moral virtue ethics does not necessarily mean that the government starts to influence people’s preferences. Because any government policy may be influencing people’s preferences even when the government does not intentionally do so, it seems important to examine how each policy is influencing people’s preferences and think about how desirable such influences are for the society.

This line of thoughts leads us to the second direction of research. In order to examine whether or not any policy is influencing people’s preferences, we need empirical work on models with endogenous preferences in which such policy can affect preferences. For the tough love model with the bequest tax rate, there already exit some empirical work. A starting point of any model with endogenous time discounting is that genetic factors do not completely determine time discounting. Using a unique data set of twins in Japan, Hirata et al. (2010) found empirical evidence in favor of this.

Kubota et al. (2013) examined how parents’ tendencies for tough love behavior depend on various measures of time discounting for parents’ own lending and borrowing over different time horizons. Using the Osaka University Global COE survey data for Japan and the United States, they found evidence that is consistent with the tough love model. They also found one empirical puzzle that proportionately more U.S. parents show tough love to young children before the school age than Japanese parents even after controlling for time discounting and other economic and demographic factors. The is especially puzzling because more patient parents tend to show tough love, and Japanese parents are estimated by Kubota et al. (2013) to be more patient than U.S. parents.

Kubota et al. (2013, Time discounting and intergenerational altruism, unpublished) examined a possible solution to this puzzle is cultural differences between the two countries. They used a framework in anthropology that a worldview is behind each culture. Using the Osaka University Global COE survey data for Japan and the United States, they found that differences in the distributions of the confidence in worldview beliefs can explain a substantial portion of the international differences in the parental attitudes.

Akabayashi et al. (2014) reported evidence that is more consistent with the tough love model than with the standard Barro-Becker model from experiments with parent-child pairs. In their time preference experiments, each parent-child pair makes individual and joint decisions about how much and when the child receives a payment (e.g., 1,000 yen now versus 1,001 yen in 2 months). The Barro-Becker model in which the parent does not know the child’s time discount rate predicts that the only reason for the parent to make a different decision from the child when s/he makes an individual decision is the lack of knowledge about the child’s time discount factor. Hence the joint decision is predicted to be the same as the child’s individual decision. However, they find that about half of the pairs do not follow this prediction.