The optimal inheritance tax in the presence of investment in education

Abstract

This paper provides an example aimed at calculating the optimal inheritance tax in a model in which inheritances are used to finance investment in education. Two results are obtained: (1) The optimal inheritance tax schedule includes a threshold, estimated between 2.5 and 5.5 times per-capita GDP. This result holds for a Rawlsian social planner that maximizes the welfare of the poorest individual, who does not leave bequests. (2) Contrary to the result of a 100 % tax on pure accidental bequests, the optimal simulated tax rates are between 28 %, for the case of educational bequests, and 57 %, for the case where educational and accidental bequests interact. This range is in line with existing schedules in developed economies.

Appendices

Appendix 1: Investment in education

1.1 The decision about investing in education

In Fig. 3, I use subindexes 1, 2, and 3 to identify the cost parameters ($\lambda _{1} >\lambda _{2} > \lambda _{3})$:

Fig. 3

Investment in education

Let us start by characterizing the poor dynasty. Since individuals that belong to this dynasty do not receive a bequest, they do not invest in education and their net wage is $(1-\tau )$ $n_\mathrm{u}$. The corresponding point in Fig. 3 is A, which implies not investing in education at all. Theoretically parents from this dynasty could ask for a loan. However, as shown by Galor and Zeira (1993), the cost of a loan increases with the amount borrowed, since the tracking costs go up with the size of the loan. In my model, this feature implies that $\lambda _{1}$ is higher than $\lambda _{2}$. Thus, asking for a loan would allow individuals from the poor dynasty to arrive to point C, which implies an investment in education that is lower than $X^{*}$. Consequently, this investment is not enough for receiving the hourly wage of skilled labor; even if they borrow and invest on education, they would continue to receive the unskilled hourly wage. Therefore, this dynasty does not invest in education.

For the middle dynasty, the education costs include borrowing costs that are not present for the rich dynasty; this implies that $\lambda _{2} > \lambda _{3}$. Thus, the middle dynasty is at point B and invests $X^{*}$, which is on the edge of indifference to investing in education. Clearly, the second dynasty would be affected by the imposition of the inheritance tax at a rate t, causing it to move to point F; this move would imply an optimal investment in education that is lower than $X^{*}$, and consequently it would not be enough for getting the skilled hourly wage. Thus, in this case the second dynasty would choose avoiding investment in education.

Note that for the second dynasty there is a difference between the extensive and intensive margin. At the extensive margin, the existence of an inheritance tax implies receiving an amount that is lower than $X^{*}$, and consequently the investment would result in the unskilled hourly wage. This return is obtained also if the amount of money invested equals zero. Thus, at the extensive margin the second dynasty prefers not to invest in education. Concerning the intensive margin, the decision of the invested amount varies depending on the amount of available resources; in particular, if available resources are higher than $X^{*}$, the second dynasty would invest in education even in the presence of an inheritance tax at the first dollar. Moreover, its investment would be higher than $X^{*}$.

The rich dynasty, receiving a high bequest, would be at point D; this implies investing in education. Moreover, note that this dynasty would invest in education even if there were a tax on bequests²² (f.e., in point E), represented by t, as long as the tax did not reduce the return to education to a desired level of investment lower than $X^{*}$.

1.2 Interpretation of the steady state

In the model presented above I have assumed that the first generation of the second and third dynasties had enough resources so as to invest in education. This assumption seems trivial concerning the rich (third) dynasty. Concerning the middle class (second dynasty), I could have assumed alternatively that it starts the world as unskilled; i.e., they receive a bequest that is lower than $X^{*}$. Thus, a detailed discussion of my assumptions is at place.

Galor and Zeira (1993) show that when the bequest received by the middle class is lower than a threshold, it eventually converges to the poor dynasty in the long-run, deriving in a two-dynasties economy: the poor and the rich. Under the alternative assumption that they receive a bequest that is higher than the threshold, it would eventually arrive at the rich dynasty, converging again to a two-dynasties economy.

In the model presented by Galor and Zeira (1993), the parents or the children can invest an additional amount, $Z-X^{*}$, in education. The reason for this additional investment is that the return to education is the skilled wage, $n_\mathrm{s}$, and consequently it is worth to invest an amount that is higher than $X^{*}$. Note that in my model the education decision differs from Galor and Zeira (1993) in two aspects: (i) the return to education is dependent on the invested amount X, and it is equal to $Xn_\mathrm{s}$; (ii) investing in education implies a cost that depends on X, as described in the function $g(x)$, that is defined in Eq. 3. Consequently, in my model the optimal investment in education is obtained by equalizing the marginal benefit to the marginal cost. Using this rule, the parents decide about the optimal bequest. The third dynasty has a lower cost parameter $\lambda _{3}$, and consequently its investment in education is higher. As a consequence of that, the consumption of its kids is higher than the one of kids from the second dynasty (see the last line of Table 3).

Concerning the second dynasty, I have assumed particular parametric assumptions, which merit a detailed discussion. For this purpose, let us characterize the steady state. By elevating both sides of Eq. 6 to $\lambda _{2}$, I obtain

$$\begin{aligned} \ln [(1-\tau )n_\mathrm{s} ]=\lambda _2 \ln X^*\end{aligned}$$

(39)

Which implies

$$\begin{aligned} \lambda _2 =\frac{\ln [(1-\tau )n_\mathrm{s} ]}{\ln X^*}=B. \end{aligned}$$

(40)

Since $n_\mathrm{s}$ is determined at the labor market equilibrium, there are three values of $\lambda _2$ that fulfill my basic assumption: (i) $\lambda _2 =B$, as assumed above; (ii) if $\lambda _2 >B$, the second dynasty would not invest in education, causing a decrease in the supply of skilled workers. Since demand is constant, the equilibrium at the labor market implies that the hourly wage $n_\mathrm{s}$ increases. By plugging the new value of $n_\mathrm{s}$ in 40, I obtain a higher value, $\lambda ^\prime _2$, that satisfies my basic assumption; (iii) if $\lambda _2 <B$, the second dynasty would invest a higher amount in education, which causes an increase in the supply of educated workers (or supplied hours); for a given demand, $n_\mathrm{s}$ declines. Using 40 I obtain a lower value, $\lambda ^{\prime \prime }_2$, that satisfies my basic assumption.

Clearly, the assumption that $\lambda _2$ equals exactly one of these three values does not necessarily hold, and thus a discussion of the characteristics of the steady state is needed.

For the case in which $\lambda _2 >B$, if the equality is not restored and $\lambda _{2} >\lambda ^\prime _{2}$, the second dynasty would converge to the poor dynasty—ending in a two-dynasties economy (poor and rich). This scenario is contrary to existing evidence on the persistence of the middle class, as found by Milanovic and Yitzhaki (2002) and by Loayza et al. (2012).

If $\lambda _2 <\lambda ^{\prime \prime }_2$, the economy is still composed by the three dynasties, but the second dynasty behavior becomes similar to the third dynasty. Note that in this case, a small positive inheritance tax would be optimal from the first dollar.

Finally, let us discuss the implications for the steady state when the demand is not constant. A decrease in demand would cause a decline in $n_\mathrm{s},$ causing the second dynasty to become unskilled which would imply in the long run a two-classes economy. An increase in demand implies a rise in $n_\mathrm{s}$, which would not affect the steady state and would allow the second dynasty to invest in education, even when a small inheritance tax exists.

Appendix 2: The derivation of accidental bequests

1.1 Actuarially fair annuities

The problem is to maximize the function given in 15, subject to the budget constraint stated in Eq. 16. Substituting the budget constraint into the utility function I get

$$\begin{aligned}&\mathrm{MAX}\left\{ \ln (c_{1i} )+\delta (1-p)\ln \left[ R(n_\mathrm{s} (1-\tau )X_i -X_i-c_{1i}-a_i )+Aa_i )\right] \right. \nonumber \\&\left. \quad +\left[ \ln (1-l_i )\right] ^{\theta }+\ln (\bar{c}_K )\right\} , \end{aligned}$$

(41)

where the parent decides about the first-period consumption, $c_{1i}$. Note that, since the return to annuities is higher than the one of savings, in this case the parent chooses to annuitize all his second period savings, and consequently:

$$\begin{aligned} n_\mathrm{s} (1-\tau )X_i -X_i -c_{1i} =a_i. \end{aligned}$$

(42)

This equality means that the first term of the right parenthesis that appears in 41 equals zero. Thus, we are left with the term $Aa_{i}$: It is optimal to annuitize all savings, so as to spend them on second period consumption. Since $A=R/(1-p)$, the first-order condition is

$$\begin{aligned} \frac{1}{c_{1i} }=\frac{R\delta }{[n_\mathrm{s} (1-\tau )-1]X_i -X_i -c_{1i}}. \end{aligned}$$

(43)

From this equation, after assuming as before that R=1, I immediately obtain the consumption functions shown in Eq.  17:

$$\begin{aligned} c_{1i} =\frac{[n_\mathrm{s} (1-\tau )-1]X_i -X_i }{1+\delta }; \quad c_{2i} =\frac{\delta [(n_\mathrm{s} (1-\tau )-1)X_i -X_i ]}{1+\delta }. \end{aligned}$$

1.2 Actuarially unfair annuities (no annuities case)

I look at the maximization problem given in Eq. 15, subject to the budget constraint given in Eq. 16, but this time taking into account that $A<R/(1-p)$; i.e., annuities are actually unfair.

Thus, the parent does not use annuities and consequently the budget constraint given in 16 becomes

$$\begin{aligned} c_{2i} =R[n_\mathrm{s} (1-\tau )X_i -X_i -c_{1i} ]=R[(n_\mathrm{s} (1-\tau )-1)X_i -c_{1i}]. \end{aligned}$$

(44)

I substitute 44 in Eq. 15, and I get

$$\begin{aligned} \mathrm{MAX}\left\{ \ln (c_{1i} )\!+\!\delta (1\!-\!p)\ln [R[(n_\mathrm{s} (1\!-\!\tau )-1)X_i \!-\!c_{1i}]+\left[ \ln (1-l_i )\right] ^{\theta } +\ln (\bar{c}_K )\right\} . \end{aligned}$$

(45)

The maximization that appears in 45 is performed over the first period consumption, $c_{1}$.

Thus, the first-order condition is

$$\begin{aligned} \frac{1}{c_{1i}}=\frac{R\delta (1-p)}{[n_\mathrm{s} (1-\tau )-1]X_i -c_{1i}}. \end{aligned}$$

(46)

From 46, after assuming that $R=1$, I obtain the first-period consumption:

$$\begin{aligned} c_{1i} =\frac{1}{1+\delta (1-p)}\left[ n_\mathrm{s} (1-\tau )X_i -[n_\mathrm{s} (1-\tau )]^{\frac{1}{\lambda _i }}\right] . \end{aligned}$$

(47)

By substituting this result in the budget constraint, I obtain an expression for the accidental bequest:

$$\begin{aligned} b_i =\frac{\delta (1-p)}{1+\delta (1-p)}\left[ n_\mathrm{s} (1-\tau )X_i -[n_\mathrm{s} (1-\tau )]^{\frac{1}{\lambda _i }}\right] . \end{aligned}$$

(48)

Since the marginal propensity to leave bequests is constant in all periods, it immediately shows that for dynasties 2 and 3, in the case in which the parent died young in m periods, the accidental bequests equal, respectively, the expressions shown in Eqs. 19 and 20.

Appendix 3: Accidental bequests of the poor dynasty

In this appendix I use plausible empirical values for the parameters, so as to check whether accidental bequests of the poor dynasty are lower than the minimum level of education.

For this purpose I assume that the ratio of skilled and unskilled wage equals 3 and N equals 3 (which is conservative for poor dynasties).

Concerning the additional parameters, I use the same values as used for simulations with skilled individuals. By applying Eq. 22, I get the following results:

$$\begin{aligned}&c_1^\mathrm{POOR} =0.67 ,\\&\frac{b_1}{N}=0.11. \end{aligned}$$

The size of the bequest must be compared with the minimum level of education, $X^{*}$, which appears in the following table:

	$\lambda _{2}$= 4	$\lambda _{2}$= 2	$\lambda _{2}$= 1	$\lambda _{2}$= 0.8
$X^{*}$	3.95	3	1.73	1.32

Assume now that the parent of the poor dynasty allocates all the accidental bequest to one of his children so as to maximize the chance of using the accidental bequest for investing in education. In this case, the results are

$$\begin{aligned}&c_1^\mathrm{POOR} =0.67, \\&b_1 =0.33, \end{aligned}$$

i.e., the bequest is still lower than the threshold level in all four simulations. In summary, these results confirm that under the parameters used in the simulation, accidental bequests of the poor dynasty are too low, and thus they do not allow investing in education.