On the distribution of lifetime wealth accumulation

Abstract

I derive a stationary distribution of lifetime wealth accumulation factor in a model featuring inheritance of productivity, wealth, and health condition, where lifetime wealth is the sum of financial wealth and human wealth. Assuming ex-ante heterogeneity in the death rate, I show that the distribution of the lifetime wealth accumulation factor is constituted by a weighted sum of shape-differing Pareto distributions. It is shown that raising the wealth tax reduces inequality of lifetime wealth not only within a death-rate type but also across all the death-rate types.

Supplementary Appendix

1.1 A Proof of Lemma 1

For convenience, we now work with the effective time preference rate $\rho ={\tilde{\rho }}+\delta$. The household’s problem reads

$$\begin{aligned} \max _{\{c(t)\}_{t=0}^{\infty }}U(0)&=\int _{0}^{\infty }e^{-\rho t}u\left( c(t)\right) dt\end{aligned}$$

(A.1)

$$\begin{aligned}&\text {subject to}\nonumber \\ dw(t)&=gw(t)dt\end{aligned}$$

(A.2)

$$\begin{aligned} \dot{a}\left( t\right)&=(r-\tau ^a)a(t)+w\left( t\right) -c\left( t\right) . \end{aligned}$$

(A.3)

The household chooses a consumption path to maximize her intertemporal utility (A.1) subject to the law of motion of wage (A.2) and the budget constraint (A.3). Let us denote the value function by V(a, w) that maps the two dimensional state space (a, w) into real numbers through a mapping V. The Bellman equation for the household’s problem therefore reads

$$\begin{aligned} \rho V(a(t),w(t))=\max _{c\left( t\right) }\left\{ u(c(t))+\frac{dV(a(t),w(t))}{dt}\right\} . \end{aligned}$$

Computing the differential dV(a(t), w(t)), taking the constraints (A.2) and (A.3) into account yields, suppressing the time argument t for simplicity,

$$\begin{aligned} \rho V(a,w)=\max _{c}\left\{ u(c)+V_{1}(a,w)((r-\tau ^a)a+w-c)+gwV_{2}(a,w)\right\} \end{aligned}$$

(A.4)

The first-order condition for this problem reads

$$\begin{aligned} u^{\prime }(c(a,w))=V_{1}(a,w). \end{aligned}$$

(A.5)

We guess $V(a,w)=\frac{\Gamma _{1}(a+\Gamma _{2}w)^{1-\sigma }-\Gamma _{3} }{1-\sigma }$, $\Gamma _{1}\ne 0$. Using the first order condition (A.5) yields

$$\begin{aligned} c(a,w)=\Psi (a+\Gamma _{2}w), \end{aligned}$$

(A.6)

where $\Psi \equiv \Gamma _{1}^{-\frac{1}{\sigma }}$. Plugging (A.6) into the Bellman equation (A.4) yields

$$\begin{aligned}&(a+\Gamma _{2}w)^{1-\sigma }\left[ \sigma \Gamma _{1}^{\frac{\sigma -1}{\sigma }}+\Gamma _{1}\left[ (1-\sigma )(r-\tau ^a)-\rho \right] \right] \nonumber \\&+w(a+\Gamma _{2}w)^{-\sigma }\Gamma _{1}(1-\sigma )(g\Gamma _{2}+1-(r-\tau ^a)\Gamma _{2})+\rho \Gamma _{3}-1=0. \end{aligned}$$

(A.7)

As Equation (A.7) must hold for any a and w, it implies

$$\begin{aligned} \Psi&=\frac{\rho -(1-\sigma )(r-\tau ^a)}{\sigma },\end{aligned}$$

(A.8)

$$\begin{aligned} \Gamma _{2}&=\frac{1}{r-\tau ^a-g}, \end{aligned}$$

(A.9)

$$\begin{aligned} \Gamma _{3}=\frac{1}{\rho }. \end{aligned}$$

(A.10)

Thus, the optimal consumption reads

$$\begin{aligned} c\left( t\right) =\frac{\rho -\left( 1-\sigma \right) (r-\tau ^a)}{\sigma }\left( a\left( t\right) +\frac{w\left( t\right) }{r-\tau ^a-g}\right) . \end{aligned}$$

(A.11)

Inserting the closed form solution (A.11) into the budget constraint (A.3) yields

$$\begin{aligned} \dot{a}\left( t\right) -g^{c}a\left( t\right) =\eta w_{0}e^{gt}, \end{aligned}$$

(A.12)

where

$$\begin{aligned} g^{c}&\equiv \frac{r-\tau ^a-\rho }{\sigma },\\ \eta&\equiv \frac{r-\tau ^a-(\rho +\sigma g)}{\sigma (r-\tau ^a-g)}. \end{aligned}$$

Multiplying (A.12) by $e^{-g^{c}t}$ and integrating gives

$$\begin{aligned} a(t)e^{-g^{c}t}=\frac{\eta w_{0}}{g-g^{c}}e^{(g-g^{c})t}+k, \end{aligned}$$

where k is the constant of integration. Further simplification gives

$$\begin{aligned} a(t)=-\frac{w_{0}}{r-g}e^{gt}+ke^{g^{c}t}. \end{aligned}$$

(A.13)

Evaluating (A.13) at $t=0$ gives $k=a_{0}+\frac{w_{0}}{r-g}$. Thus, the evolution of (financial) wealth is described by

$$\begin{aligned} a(t)=\left[ a_{0}+\frac{w_{0}}{r-\tau ^a-g}\left( 1-e^{\left[ g-g^{c}\right] t}\right) \right] e^{g^{c}t}. \end{aligned}$$

(A.14)

This can be written as

$$\begin{aligned} a(t)=\,&a_{0}e^{g^{c}t}+\frac{w_{0}}{r-\tau ^a-g}e^{g^{c}t}-\frac{w_{0}}{r-\tau ^a-g}e^{gt}\\ a(t)+\frac{w_{0}}{r-\tau ^a-g}e^{gt}=\,&a_{0}e^{g^{c}t}+\frac{w_{0}}{r-\tau ^a-g}e^{g^{c}t}\\ a(t) + \frac{w(t)}{r-\tau ^a-g}=\,&\left[ a(0) + \frac{w(0)}{r-\tau ^a-g}\right] e^{g^{c}t}. \end{aligned}$$

Using the definitions of W(t), W(0), and $g^c$ gives

$$\begin{aligned} W(t)=W_{0}e^{\frac{\left( r-\tau ^{a}\right) - \rho }{\sigma }t}. \end{aligned}$$

(A.15)

1.2 B Continuous distribution of the death rate

Now let us assume the death rate $\delta$ follows a continuous distribution with a density function $f(\delta )$ and support $[\underline{\delta },\overline{\delta }]$.

Consider the joint probability

$$\begin{aligned} \Pr \left( q<Q,\delta =\hat{\delta }\right)&=\Pr \left( q<Q|\delta =\hat{\delta }\right) \Pr \left( \delta =\hat{\delta }\right) \\&=\Pr \left( q<Q|\delta =\hat{\delta }\right) f(\hat{\delta })\\&=q^{-\frac{\sigma \hat{\delta }}{r-\tau ^{a}-{\tilde{\rho }}-\hat{\delta }}}f(\hat{\delta }). \end{aligned}$$

The probability distribution function of q is therefore given by

$$\begin{aligned} F\left( q\right) \equiv \Pr \left( q<Q\right) =\int ^{\bar{\delta }}_{\underline{\delta }}\Pr \left( q<Q|\delta \right) f({\delta })d \delta =\int ^{\bar{\delta }}_{\underline{\delta }}q^{-\frac{\sigma {\delta }}{r-\tau ^{a}-{\tilde{\rho }}-{\delta }}}f({\delta })d \delta . \end{aligned}$$

F(q) is the ‘weighted sum’ of Pareto distributions, which can be interpreted as an ‘average’ Pareto distribution.