Bequests, estate taxes, and wealth distributions

Abstract

Bossmann et al. (J Public Econ 91:1247–1271, 2007) found that estate taxes reduce the long-run wealth inequality. This result contrasts with the findings of the previous literature with idiosyncratic labor efficiency risk. We use a decomposition technique, developed by Davies (J Labor Econ 4:538–559, 1986), to reinvestigate the impact of estate taxes on the long-run wealth inequality. We find that the different results of estate taxes are due to the different redistribution effects.

A Appendix

1.1 A.1 Proof of Proposition 1

Proof

Letting $K_{t+1}=K_{t}=K$ in Eq. (11) we have

$$\begin{aligned} K=\left( \frac{1-\alpha +\varphi \alpha }{1+\tilde{\beta }^{-\frac{1}{\eta } }-\varphi (1-\delta )}A\right) ^{\frac{1}{1-\alpha }}, \end{aligned}$$

(A.1)

where $\tilde{\beta }=\beta \left[ 1+\chi ^{\frac{1}{\eta }}(1-\zeta )^{\frac{ 1-\eta }{\eta }}\right] ^{\eta }(1+r)^{1-\eta }$ and $r=\alpha AK^{\alpha -1}-\delta$.

Plugging Eq. (A.1) into $r=\alpha AK^{\alpha -1}-\delta$ we have

$$\begin{aligned} \frac{r+\delta }{\alpha }=\frac{1+\tilde{\beta }^{-\frac{1}{\eta }}-\varphi (1-\delta )}{1-\alpha +\varphi \alpha }. \end{aligned}$$

(A.2)

Plugging $\tilde{\beta }=\beta \left[ 1+\chi ^{\frac{1}{\eta }}(1-\zeta )^{ \frac{1-\eta }{\eta }}\right] ^{\eta }(1+r)^{1-\eta }=\frac{\beta }{ (1-\varphi )^{\eta }}(1+r)^{1-\eta }$ into Eq. (A.2) we have

$$\begin{aligned} \frac{1-\alpha }{\alpha }(r+\delta )+\varphi (1+r)-(1-\varphi )\beta ^{- \frac{1}{\eta }}(1+r)^{1-\frac{1}{\eta }}=1. \end{aligned}$$

We show Proposition 1 in two cases:

Case (i) $\eta >1$

Note that $0<\varphi <1$. Define

$$\begin{aligned} h(\varphi ,r)=\frac{1-\alpha }{\alpha }(r+\delta )+\varphi (1+r)-(1-\varphi )\beta ^{-\frac{1}{\eta }}(1+r)^{1-\frac{1}{\eta }}. \end{aligned}$$

The equilibrium r is determined by

$$\begin{aligned} h(\varphi ,r)=1. \end{aligned}$$

Note that $h(\varphi ,r)$ is a continuous function of r, with

$$\begin{aligned} h(\varphi ,-\delta )=\varphi (1-\delta )-(1-\varphi )\beta ^{-\frac{1}{\eta } }(1-\delta )^{1-\frac{1}{\eta }}<\varphi (1-\delta )<1 \end{aligned}$$

and

$$\begin{aligned} \lim _{r\rightarrow \infty }h(\varphi ,r)=\infty \end{aligned}$$

Also $h_{22}(\varphi ,r)=\left( 1-\frac{1}{\eta }\right) \frac{1}{\eta } (1-\varphi )\beta ^{-\frac{1}{\eta }}(1+r)^{-\frac{1}{\eta }-1}>0$ due to $\eta >1$. Thus, $h(\varphi ,r)$ is a strictly convex function of r. Therefore, there must exist a unique equilibrium $r>-\delta$.²⁴

Note that $h(\varphi ,r)$ is strictly increasing in $\varphi$. For $\varphi _{1}<\varphi _{2}<1$, suppose that

$$\begin{aligned} h(\varphi _{1},r_{1})=1\text { and }h(\varphi _{2},r_{2})=1. \end{aligned}$$

We have

$$\begin{aligned} h(\varphi _{2},r_{1})>h(\varphi _{1},r_{1})=1. \end{aligned}$$

Thus, $r_{2}<r_{1}$ since $h(\varphi _{2},-\delta )<1$ and $h(\varphi _{2},r)$ is a continuous function of r. A higher $\chi$ implies a higher $\varphi$ . Thus, a higher $\chi$ implies a lower r and a higher K.

Case (ii) $\eta =1$

In this case $\tilde{\beta }=\beta (1+\chi )$ and $\varphi =\frac{1}{1+\frac{1 }{\chi }}$, Eq. (A.1) implies

$$\begin{aligned} K= & {} \left( \frac{1-\alpha +\chi }{1+\frac{1}{\beta }+\delta \chi }A\right) ^{\frac{1}{1-\alpha }} \\= & {} \left( \left[ \frac{1}{\delta }-\frac{\frac{1}{\delta }\left( 1+\frac{1}{ \beta }\right) -(1-\alpha )}{1+\frac{1}{\beta }+\delta \chi }\right] A\right) ^{\frac{1}{1-\alpha }}. \end{aligned}$$

Thus, a higher $\chi$ implies a higher K. $\square$

1.2 A.2 Proof of Proposition 2

Proof

Obviously $c_{4}\ge 0$. From Eq. (A.2) we have

$$\begin{aligned} 1+r=\frac{\left( 1+\tilde{\beta }^{-\frac{1}{\eta }}\right) \alpha +(1-\delta )(1-\alpha )}{(1-\alpha )+\varphi \alpha } \end{aligned}$$

Thus,

$$\begin{aligned} c_{4}=\frac{\left( 1-\zeta \right) \varphi (1+r)}{1+\tilde{\beta }^{-\frac{1}{ \eta }}}=\left( 1-\zeta \right) \frac{\alpha +\frac{1-\delta }{1+\tilde{\beta }^{-\frac{1}{\eta }}}(1-\alpha )}{\alpha +\frac{1}{\varphi }(1-\alpha )}<1 \end{aligned}$$

since $\tilde{\beta }=\beta \left[ 1+\chi ^{\frac{1}{\eta }}(1-\zeta )^{\frac{ 1-\eta }{\eta }}\right] ^{\eta }(1+r)^{1-\eta }>0$ and $0<\varphi <1$. $\square$

1.3 A.3 Proof of Proposition 3

Proof

From Eq. (13) we have

$$\begin{aligned} a_{t+1}=c_{3}l_{t}+c_{4}a_{t}+c_{5}, \end{aligned}$$

where $c_{3}=\frac{1}{1+\tilde{\beta }^{-\frac{1}{\eta }}}w$, $c_{4}=\frac{ \left( 1-\zeta \right) \varphi (1+r)}{1+\tilde{\beta }^{-\frac{1}{\eta }}}$, and $c_{5}=\frac{\zeta \varphi (1+r)}{1+\tilde{\beta }^{-\frac{1}{\eta }}}K$. Let $B_{t}=c_{5}+c_{3}l_{t}$. We have

$$\begin{aligned} a_{t+1}=c_{4}a_{t}+B_{t}. \end{aligned}$$

(A.3)

Note that $\{B_{t}\}$ is stationary and ergodic since $\{l_{t}\}$ is stationary and ergodic by Assumption 1. We have $-\infty \le \log c_{4}<0$. Also $E(B_{t})=c_{5}+c_{3}<\infty$, since $E(l_{t})=1$ by Assumption 2. Thus, $E(\log B_{t})^{+}\le E(B_{t})<\infty$. By Theorem 1 of Brandt (1986) we know that $a_{t}$ converges to $\sum _{j=0}^{\infty }c_{4}^{j}B_{t-j-1}$ almost surely as t approaches infinity. Thus, we have

$$\begin{aligned} a_{t}\rightarrow _{st}\sum _{j=0}^{\infty }c_{4}^{j}B_{t-j-1}\text { as } t\rightarrow \infty . \end{aligned}$$

Since $\{B_{t}\}$ is stationary, we know that the sequence of $(B_{t-1},B_{t-2}, \ldots ,B_{t-j-1}, \ldots )$ has the same distribution as the sequence of $(B_{0},B_{1}, \ldots ,B_{s}, \ldots )$. Thus, we have

$$\begin{aligned} \sum _{j=0}^{\infty }c_{4}^{j}B_{t-j-1}=_{st}\sum _{s=0}^{\infty }c_{4}^{s}B_{s},\text { }\forall t\in \mathbb {Z} . \end{aligned}$$

Let

$$\begin{aligned} a_{\infty }=_{st}\sum _{s=0}^{\infty }c_{4}^{s}B_{s}=_{st}c_{3}\sum _{s=0}^{\infty }c_{4}^{s}l_{s}+\frac{c_{5}}{ 1-c_{4}}. \end{aligned}$$

Thus, we know that

$$\begin{aligned} a_{t}\rightarrow _{st}a_{\infty }\text { as }t\rightarrow \infty . \end{aligned}$$

$\square$

1.4 A.4 Proof of Theorem 1

Proof

Theorem 3.A.36 of Shaked and Shanthikumar (2010) shows that

Lemma 2

Let $X_{1}$, $X_{2}$, $\ldots$, $X_{n}$ and Y be $n+1$ random variables. If $X_{i}\preceq _{cx}Y$, $i=1$, 2, $\ldots$, n, then

$$\begin{aligned} \sum _{i=1}^{n}a_{i}X_{i}\preceq _{cx}Y, \end{aligned}$$

whenever $a_{i}\ge 0$, $i=1$, 2, $\ldots$, n, and $\sum _{i=1}^{n}a_{i}=1$.²⁵

Theorem 3.A.10 of Shaked and Shanthikumar (2010) states that

Lemma 3

Let X and Y be two nonnegative random variables with equal means. Then, $X\preceq _{cx}Y$ if and only if $L_{X}(p)\ge L_{Y}(p)$ for all $p\in [0,1]$.

Note that $a_{\infty }^{A}$ has the same Lorenz curve as $l_{1}$. We only need to show that $a_{\infty }^{B}\succeq _{L}l_{1}$.

In economy B, pick $a_{1}=\frac{c_{3}}{1-c_{4}}$.²⁶ Thus,

$$\begin{aligned} a_{1}\preceq _{cx}\frac{c_{3}}{1-c_{4}}l_{1} \end{aligned}$$

since $a_{1}=E\left( \frac{c_{3}}{1-c_{4}}l_{1}\right)$.²⁷

Suppose that

$$\begin{aligned} a_{t}\preceq _{cx}\frac{c_{3}}{1-c_{4}}l_{1}\text {.} \end{aligned}$$

Thus, $\frac{1-c_{4}}{c_{3}}a_{t}\preceq _{cx}l_{1}$.²⁸

And

$$\begin{aligned} a_{t+1}= & {} c_{3}l_{t}+c_{4}a_{t} \\= & {} \frac{c_{3}}{1-c_{4}}\left( (1-c_{4})l_{t}+c_{4}\frac{1-c_{4}}{c_{3}} a_{t}\right) . \end{aligned}$$

Note that $(1-c_{4})l_{t}+c_{4}\frac{1-c_{4}}{c_{3}}a_{t}$ is a weighted average of $l_{t}$ and $\frac{1-c_{4}}{c_{3}}a_{t}$. For $\forall t\ge 1$, $l_{t}$ and $l_{1}$ have the same distribution. We have $l_{t}\preceq _{cx}l_{1}$, $\forall t\ge 1$. By Lemma 2 we have

$$\begin{aligned} (1-c_{4})l_{t}+c_{4}\frac{1-c_{4}}{c_{3}}a_{t}\preceq _{cx}l_{1}. \end{aligned}$$

Thus,

$$\begin{aligned} a_{t+1}\preceq _{cx}\frac{c_{3}}{1-c_{4}}l_{1}. \end{aligned}$$

By mathematical induction we have

$$\begin{aligned} a_{t}\preceq _{cx}\frac{c_{3}}{1-c_{4}}l_{1}, \ \ \forall t\ge 1. \end{aligned}$$

Since $a_{t}\rightarrow _{st}a_{\infty }^{B}$ as t approaches infinity, we have

$$\begin{aligned} a_{\infty }^{B}\preceq _{cx}\frac{c_{3}}{1-c_{4}}l_{1}, \end{aligned}$$

by part (c) of Theorem 3.A.12 of Shaked and Shanthikumar (2010). By Lemma 3 we have $a_{\infty }^{B}\succeq _{L}\frac{c_{3}}{1-c_{4}} l_{1}$ since $E\left( a_{\infty }^{B}\right) =E\left( \frac{c_{3}}{1-c_{4}} l_{1}\right) =\frac{c_{3}}{1-c_{4}}$. Thus, $a_{\infty }^{B}\succeq _{L}l_{1}$. $\square$

1.5 A.5 Proof of Lemma 1

Proof

Let

$$\begin{aligned} g(x)=(1-\zeta )x+\zeta E\left( X\right) , \ \ x\in [0,+\infty ) \end{aligned}$$

and

$$\begin{aligned} h(x)=(1-\hat{\zeta })x+\hat{\zeta }E\left( X\right) , \ \ x\in [0,+\infty ) \end{aligned}$$

Note that $g(\cdot )$ and $h(\cdot )$ are nonnegative increasing functions defined on $[0,+\infty )$, since $0\le \hat{\zeta }\le \zeta <1$. Also $g(x)>0$ and $h(x)>0$ for $x>0$. Note that $\frac{h(x)}{g(x)}$ is increasing in $x\in (0,+\infty )$, since

$$\begin{aligned} \frac{h(x)}{g(x)}= & {} \frac{(1-\hat{\zeta })x+\hat{\zeta }E\left( X\right) }{ (1-\zeta )x+\zeta E\left( X\right) } \\= & {} \frac{1-\hat{\zeta }}{1-\zeta }\left[ 1-\frac{\zeta -\hat{\zeta }}{\left( 1- \hat{\zeta }\right) \left( 1-\zeta \right) }\frac{E(X)}{x+\frac{\zeta }{ 1-\zeta }E(X)}\right] . \end{aligned}$$

By Theorem 3.A.26 of Shaked and Shanthikumar (2010) we have $g(X)\succeq _{L}h(X)$, i.e., $(1-\zeta )X+\zeta E\left( X\right) \succeq _{L}(1-\hat{\zeta })X+\hat{\zeta }E\left( X\right)$. By Lemma 3 we have $(1-\zeta )X+\zeta E\left( X\right) \preceq _{cx}(1-\hat{\zeta })X+\hat{\zeta }E\left( X\right)$ since $E\left[ (1-\zeta )X+\zeta E\left( X\right) \right] =E(X)=E \left[ (1-\hat{\zeta })X+\hat{\zeta }E\left( X\right) \right]$. $\square$

1.6 A.6 Proof of Theorem 2

Proof

Note that $a_{\infty }^{\zeta }$ is the stationary distribution of the stochastic process $\{a_{t}^{\zeta }\}$ which is generated by

$$\begin{aligned} a_{t+1}^{\zeta }=c_{6}l_{t}+c_{7}\left[ (1-\zeta )a_{t}^{\zeta }+\zeta \bar{K }\right] \end{aligned}$$

and a given $a_{1}^{\zeta }$. And $a_{\infty }^{\hat{\zeta }}$ is the stationary distribution of the stochastic process $\{a_{t}^{\hat{\zeta }}\}$ which is generated by

$$\begin{aligned} a_{t+1}^{\hat{\zeta }}=c_{6}l_{t}+c_{7}\left[ (1-\hat{\zeta })a_{t}^{\hat{\zeta }}+\hat{\zeta }\bar{K}\right] \end{aligned}$$

and a given $a_{1}^{\hat{\zeta }}$.

Let $a_{1}^{\zeta }=_{st}a_{1}^{\hat{\zeta }}$. Thus, $a_{1}^{\zeta }\preceq _{cx}a_{1}^{\hat{\zeta }}$ by the definition of the convex order.

Now suppose that $a_{t}^{\zeta }\preceq _{cx}a_{t}^{\hat{\zeta }}$. By Lemma 1 we have

$$\begin{aligned} (1-\zeta )a_{t}^{\zeta }+\zeta \bar{K}\preceq _{cx}(1-\hat{\zeta } )a_{t}^{\zeta }+\hat{\zeta }\bar{K} \end{aligned}$$

since $E\left( a_{t}^{\zeta }\right) =\bar{K}$.

By Corollary 3.A.22 of Shaked and Shanthikumar (2010) we have $(1-\hat{\zeta } )a_{t}^{\zeta }\preceq _{cx}(1-\hat{\zeta })a_{t}^{\hat{\zeta }}$ since $\left( 1-\hat{\zeta }\right)$ is independent of $a_{t}^{\zeta }$ and $a_{t}^{ \hat{\zeta }}$. By Part (d) of Theorem 3.A.12 of Shaked and Shanthikumar (2010) we have

$$\begin{aligned} (1-\hat{\zeta })a_{t}^{\zeta }+\hat{\zeta }\bar{K}\preceq _{cx}(1-\hat{\zeta } )a_{t}^{\hat{\zeta }}+\hat{\zeta }\bar{K}, \end{aligned}$$

since $\hat{\zeta }\bar{K}$ is independent of $(1-\hat{\zeta })a_{t}^{\zeta }$ and $(1-\hat{\zeta })a_{t}^{\hat{\zeta }}$. By the transitivity of the convex order we have

$$\begin{aligned} (1-\zeta )a_{t}^{\zeta }+\zeta \bar{K}\preceq _{cx}(1-\hat{\zeta })a_{t}^{ \hat{\zeta }}+\hat{\zeta }\bar{K}. \end{aligned}$$

Thus, we have $c_{7}\left[ (1-\zeta )a_{t}^{\zeta }+\zeta \bar{K}\right] \preceq _{cx}c_{7}\left[ (1-\hat{\zeta })a_{t}^{\hat{\zeta }}+\hat{\zeta }\bar{K }\right]$ by the property of the convex order in Footnote 28. Note that $c_{6}l_{t}$ and $c_{7}\left[ (1-\zeta )a_{t}^{\zeta }+\zeta \bar{K}\right]$ are independent. And $c_{6}l_{t}$ and $c_{7}\left[ (1-\hat{\zeta })a_{t}^{ \hat{\zeta }}+\hat{\zeta }\bar{K}\right]$ are independent. Thus, by part (d) of Theorem 3.A.12 of Shaked and Shanthikumar (2010), we have

$$\begin{aligned} c_{6}l_{t}+c_{7}\left[ (1-\zeta )a_{t}^{\zeta }+\zeta \bar{K}\right] \preceq _{cx}c_{6}l_{t}+c_{7}\left[ (1-\hat{\zeta })a_{t}^{\hat{\zeta }}+\hat{\zeta } \bar{K}\right] . \end{aligned}$$

Thus, we have

$$\begin{aligned} a_{t+1}^{\zeta }\preceq _{cx}a_{t+1}^{\hat{\zeta }}. \end{aligned}$$

By mathematical induction we have

$$\begin{aligned} a_{t}^{\zeta }\preceq _{cx}a_{t}^{\hat{\zeta }},\text { }\forall t\ge 1. \end{aligned}$$

Since $a_{t}^{\zeta }\rightarrow _{st}a_{\infty }^{\zeta }$ and $a_{t}^{\hat{ \zeta }}\rightarrow _{st}a_{\infty }^{\hat{\zeta }}$ as t approaches infinity, we have

$$\begin{aligned} a_{\infty }^{\zeta }\preceq _{cx}a_{\infty }^{\hat{\zeta }}, \end{aligned}$$

by part (c) of Theorem 3.A.12 of Shaked and Shanthikumar (2010). By Lemma 3 we have

$$\begin{aligned} a_{\infty }^{\zeta }\succeq _{L}a_{\infty }^{\hat{\zeta }}, \end{aligned}$$

since $E\left( a_{\infty }^{\zeta }\right) =E\left( a_{\infty }^{\hat{\zeta } }\right) =\bar{K}$. $\square$

1.7 A.7 An alternative setup of the model

Here we investigate an alternative setup of our benchmark model. The main difference is that $b_{t}$ in our benchmark model is the before-tax bequest. In this alternative setup, $b_{t}$ is the after-tax bequest.

The agent’s problem is

$$\begin{aligned}&\max _{c_{t}^{y},s_{t},c_{t+1}^{o},b_{t+1}}\log c_{t}^{y}+\beta \left( \log c_{t+1}^{o}+\chi \log b_{t+1}\right) \\&\quad \hbox {s.t.} \ \ c_{t}^{y}+s_{t} =w_{t}l_{t}+b_{t}+g_{t}, \\&\quad c_{t+1}^{o}+\left( 1+\zeta \right) b_{t+1} =(1+r_{t+1})s_{t}. \end{aligned}$$

The agent’s optimal policy functions are

$$\begin{aligned} c_{t+1}^{o}= & {} \frac{1}{1+\chi }(1+r_{t+1})s_{t},\\ b_{t+1}= & {} \frac{\chi }{\left( 1+\chi \right) \left( 1+\zeta \right) } (1+r_{t+1})s_{t},\\ c_{t}^{y}= & {} \frac{1}{1+\beta \left( 1+\chi \right) }\left( w_{t}l_{t}+b_{t}+g_{t}\right) , \end{aligned}$$

and

$$\begin{aligned} \quad \, s_{t}=\frac{\beta \left( 1+\chi \right) }{1+\beta \left( 1+\chi \right) } \left( w_{t}l_{t}+b_{t}+g_{t}\right) . \end{aligned}$$

From the government’s budget constraint we have

$$\begin{aligned} g_{t}=\zeta \int b_{t}di, \end{aligned}$$

where $\int di$ denotes the aggregation of old agents.

Thus, the aggregate capital follows

$$\begin{aligned} K_{t+1}= & {} \int s_{t}di \\= & {} \frac{\beta \left( 1+\chi \right) }{1+\beta \left( 1+\chi \right) }\left[ w_{t}+\frac{\chi }{1+\chi }(1+r_{t})K_{t}\right] , \end{aligned}$$

where $w_{t}=(1-\alpha )AK_{t}^{\alpha }$ and $r_{t}=\alpha AK_{t}^{\alpha -1}-\delta$.

In the steady-state aggregate economy we have $K_{t+1}=K_{t}=\bar{K}$. Thus, we have

$$\begin{aligned} \bar{K}=\left( \frac{1-\alpha +\chi }{1+\frac{1}{\beta }+\delta \chi } A\right) ^{\frac{1}{1-\alpha }}. \end{aligned}$$

The estate tax does not affect the aggregate capital. Then, $\bar{w} =(1-\alpha )A\left( \bar{K}\right) ^{\alpha }$ and $\bar{r}=\alpha A\left( \bar{K}\right) ^{\alpha -1}-\delta$.

Let $a_{t+1}=s_{t}$. From the agent’s policy functions we have the individual wealth accumulation equation,

$$\begin{aligned} a_{t+1}=c_{6}l_{t}+c_{7}\left[ \frac{1}{1+\zeta }a_{t}+\frac{\zeta }{1+\zeta }\bar{K}\right] , \end{aligned}$$

(21)

with $c_{6}=\frac{1}{1+\frac{1}{\beta \left( 1+\chi \right) }}\bar{w}$ and $c_{7}=\frac{1}{\left( 1+\frac{1}{\beta \left( 1+\chi \right) }\right) \left( 1+\frac{1}{\chi }\right) }\left( 1+\bar{r}\right)$. Both $c_{6}$ and $c_{7}$ do not depend on the estate tax $\zeta$.

From Eq. (21) we have the long-run wealth distribution,

$$\begin{aligned} a_{\infty }=_{st}\sum _{s=0}^{\infty }\left( \tilde{c}_{8}\right) ^{s}\left( c_{6}l_{s}+c_{7}\frac{\zeta }{1+\zeta }\bar{K}\right) , \end{aligned}$$

where $\tilde{c}_{8}=c_{7}\frac{1}{1+\zeta }$. Comparing Eqs. (21 ) and (15) we find that all the theoretical results of the long-run wealth inequality in the benchmark model still hold in this alternative setup.

1.8 A.8 The Becker–Tomes model

Here we briefly review some main results of Becker–Tomes models by Becker and Tomes (1979) and Davies (1986). As in Becker and Tomes (1979) and Davies (1986), we assume that each agent only lives for one period. At the end of the period, the agent dies and gives birth to one child. The prices of r and w are exogenous and constant. Davies (1986) explained that the aim of using exogenous prices of r and w in his paper is exactly close to the general equilibrium effect of the estate tax.²⁹

As in Becker and Tomes (1979) and Davies (1986) we assume that the agent can correctly anticipate the labor efficiency of his child. The agent’s problem is

$$\begin{aligned}&\max _{c_{t},b_{t+1},I_{t+1}}\frac{c_{t}^{1-\eta }-1}{1-\eta }+\chi \frac{ I_{t+1}^{1-\eta }-1}{1-\eta }\\&\quad \hbox {s.t.} \ \ c_{t}+b_{t+1}=I_{t},\\&\quad I_{t+1}=wl_{t+1}+(1+r)\left( 1-\zeta \right) b_{t+1}+g, \end{aligned}$$

where $I_{t+1}$ is the total wealth of the child. The agent’s optimal policy functions are

$$\begin{aligned} c_{t}= & {} \frac{1}{1+\left[ (1+r)\left( 1-\zeta \right) \right] ^{\frac{1-\eta }{ \eta }}\chi ^{\frac{1}{\eta }}}\left( I_{t}+\frac{wl_{t+1}+g}{(1+r)\left( 1-\zeta \right) }\right) ,\\ b_{t+1}= & {} \frac{1}{1+\left[ (1+r)\left( 1-\zeta \right) \right] ^{\frac{\eta -1 }{\eta }}\chi ^{-\frac{1}{\eta }}}I_{t}-\frac{1}{1+\left[ (1+r)\left( 1-\zeta \right) \right] ^{\frac{1-\eta }{\eta }}\chi ^{\frac{1}{\eta }}} \frac{wl_{t+1}+g}{(1+r)\left( 1-\zeta \right) }, \end{aligned}$$

and

$$\begin{aligned} I_{t+1}=\frac{(1+r)\left( 1-\zeta \right) }{1+\left[ (1+r)\left( 1-\zeta \right) \right] ^{\frac{\eta -1}{\eta }}\chi ^{-\frac{1}{\eta }}}\left( I_{t}+\frac{wl_{t+1}+g}{(1+r)\left( 1-\zeta \right) }\right) . \end{aligned}$$

(22)

From the government’s budget constraint we have

$$\begin{aligned} g=\zeta (1+r)\int b_{t}di, \end{aligned}$$

where $\int di$ denotes the aggregation of young agents.

In the steady-state aggregate economy we have $\int I_{t+1}di=\int I_{t}di= \bar{I}$. Thus, we have

$$\begin{aligned} \bar{I}=\int I_{t}di=w+(1+r)\left( 1-\zeta \right) \int b_{t}di+g=w+\frac{g}{ \zeta }. \end{aligned}$$

(23)

From Eq. (22) we have

$$\begin{aligned} \bar{I}=\frac{(1+r)\left( 1-\zeta \right) }{1+\left[ (1+r)\left( 1-\zeta \right) \right] ^{\frac{\eta -1}{\eta }}\chi ^{-\frac{1}{\eta }}}\left( \bar{ I}+\frac{w+g}{(1+r)\left( 1-\zeta \right) }\right) . \end{aligned}$$

(24)

Combining Eqs. (23) and (24) we have

$$\begin{aligned} \bar{I}=\frac{1}{1-(1+r)\left( 1-\left[ (1+r)\left( 1-\zeta \right) \right] ^{-\frac{1}{\eta }}\chi ^{-\frac{1}{\eta }}\right) }w, \end{aligned}$$

and

$$\begin{aligned} g=mw, \end{aligned}$$

where $m=\frac{\zeta }{\frac{1}{(1+r)\left( 1-\left[ (1+r)\left( 1-\zeta \right) \right] ^{-\frac{1}{\eta }}\chi ^{-\frac{1}{\eta }}\right) }-1}$.

From Eq. (22) we have the individual wealth accumulation equation,

$$\begin{aligned} I_{t+1}=c_{12}I_{t}+\frac{1}{1+\left[ (1+r)\left( 1-\zeta \right) \right] ^{ \frac{\eta -1}{\eta }}\chi ^{-\frac{1}{\eta }}}\left( wl_{t+1}+g\right) , \end{aligned}$$

(25)

where $c_{12}=\frac{(1+r)\left( 1-\zeta \right) }{1+\left[ (1+r)\left( 1-\zeta \right) \right] ^{\frac{\eta -1}{\eta }}\chi ^{-\frac{1}{\eta }}}$.

For simplicity we assume that $\{l_{t}\}$ is i.i.d. Thus, we have

$$\begin{aligned} \hbox {Var}(I_{t})=\frac{c_{12}^{2}\hbox {Var}(l_{t})}{\left( 1-c_{12}^{2}\right) \left[ (1+r)\left( 1-\zeta \right) \right] ^{2}}w^{2}. \end{aligned}$$

The impact of $\zeta$ on $c_{12}$ in the individual wealth accumulation Eq. (25) represents the inheritance effect of the estate tax on the stationary wealth distribution. The higher the estate tax $\zeta$, the lower the $c_{12}$.³⁰ Thus, the inheritance effect of the estate tax increases the long-run wealth inequality. The impact of $\zeta$ on the lump-sum transfer g in the individual wealth accumulation Eq. (25) represents the redistribution effect of the estate tax.

We can calculate the coefficient of variation,

$$\begin{aligned}&\hbox {CV}(I_{t}) \\= & {} \frac{\sqrt{\hbox {Var}(I_{t})}}{\bar{I}} \\= & {} \frac{1-(1+r)\left( 1-\left[ (1+r)\left( 1-\zeta \right) \right] ^{-\frac{ 1}{\eta }}\chi ^{-\frac{1}{\eta }}\right) }{(1+r)\left( 1-\zeta \right) } \frac{c_{12}}{\sqrt{1-c_{12}^{2}}}\sqrt{\hbox {Var}(l_{t})}. \end{aligned}$$

To illustrate the impact of the estate tax $\zeta$ on the lump-sum transfer g and that of the estate tax $\zeta$ on the long-run wealth inequality, we implement a simple calibration exercise. We pick $\eta =2$, $\chi =0.8$, $r=2$, and $w=1$. We assume that one generation lasts for 30 years. Thus, $r=1$ corresponds to the annual interest rate of $3.7\%$. We increase the estate tax $\zeta$ from 0.1 to 0.5. Figure 2 shows that the higher the estate tax $\zeta$, the lower the g. Thus, the redistribution effect of the estate tax increases the long-run wealth inequality.

Fig. 2

The impact of the estate tax on the transfer

We also investigate the net effect of the estate tax on the long-run wealth inequality. We assume that $l_{t}\sim U[0,2]$. Thus, $E(l_{t})=1$ and $\hbox {Var}(l_{t})=\frac{2}{3}$. Figure 3 shows that the higher the estate tax $\zeta$, the higher the CV of the long-run wealth inequality.

Fig. 3

The impact of the estate tax on the CV

Figure 4 shows that the higher the estate tax $\zeta$, the higher the Gini coefficient of the long-run wealth inequality. Figures 3 and 4 show that the estate tax increases the long-run wealth inequality. This result is reasonable since both the inheritance effect and the redistribution effect of the estate tax increase the long-run wealth inequality.

Fig. 4

The impact of the estate tax on the Gini coefficient