Optimal taxes and pensions with myopic agents

Abstract

So far the economic literature has concentrated on analyzing the income tax and pension scheme in isolation. The present paper asks how both transfer schemes should be optimally designed in a society where individuals differ in productivity and rationality. Rational agents (if not liquidity constrained) smooth consumption over lifetime while myopic agents suffer from self-control problems and overconsume in young age, even though, ex post they regret their earlier decisions. The government’s aim is twofold. On the one hand, it wants to redistribute income from high- to low-productivity agents. On the other hand, it aims to secure old-age consumption possibilities for myopic agents. Analytical and numerical results show that, redistribution of the income tax scheme is increasing in society’s prevalence of self-control problems while it is decreasing in the pension scheme. The two transfer schemes are perfect substitutes when individuals do not suffer from self-control problems.

Appendix

1.1 Comparative statics

$$\begin{aligned} \frac{\partial s_i}{\partial t}\!&= \!\frac{-u^{\prime \prime }(x_i)w_i\ell _i}{u^{\prime \prime }(x_i)+\beta _i u^{\prime \prime }(d_i)},\quad \frac{\partial s_i}{\partial \tau }\!=\!\frac{u^{\prime \prime }(x_i)}{u^{\prime \prime }(x_i)+\beta _i u^{\prime \prime }(d_i)},\quad \frac{\partial s_i}{\partial B}=\frac{-\beta _i u^{\prime \prime }(d_i)(1-\alpha )}{u^{\prime \prime }(x_i)+\beta _i u^{\prime \prime }(d_i)},\nonumber \\ \frac{\partial s_i}{\partial b}&= \frac{-w_i\ell _i(u^{\prime \prime }(x_i)+\beta _i u^{\prime \prime }(d_i)\alpha )}{u^{\prime \prime }(x_i)+\beta _i u^{\prime \prime }(d_i)}, \quad \frac{\partial s_i}{\partial \alpha }=\frac{-\beta _i u^{\prime \prime }(d_i)(bw_i\ell _i-B)}{u^{\prime \prime }(x_i)+\beta _i u^{\prime \prime }(d_i)}\nonumber \\ \frac{\partial \tilde{s_i}}{\partial t}&= \frac{\partial s_i}{\partial t}+\frac{\partial s_i}{\partial \tau }\frac{\partial \tau }{\partial t}+\frac{\partial s_i}{\partial B}\frac{\partial B}{\partial t}\nonumber \\ \frac{\partial \tilde{s_i}}{\partial b}&= \frac{\partial s_i}{\partial b}+\frac{\partial s_i}{\partial B}\frac{\partial B}{\partial b}+\frac{\partial s_i}{\partial \tau }\frac{\partial \tau }{\partial b}\nonumber \\ \frac{\partial \tilde{s_i}}{\partial \alpha }&= \frac{\partial s_i}{\partial \alpha }+\frac{\partial s_i}{\partial B}\frac{\partial B}{\partial \alpha }+\frac{\partial s_i}{\partial \tau }\frac{\partial \tau }{\partial \alpha } \end{aligned}$$

(28)

With the above equations and noting that $\frac{\partial \ell _i}{\partial t}=\frac{\partial \ell _i}{\partial b}$ for $\alpha =0$:

$$\begin{aligned} \frac{\partial \tilde{s_i}}{\partial b}-\frac{\partial \tilde{s_i}}{\partial t}&= \frac{-w_i\ell _i u^{\prime \prime }(x_i)-\beta _i u^{\prime \prime }(d_i)[b\text{ E }w_i\frac{\partial \ell _i}{\partial b}+\text{ E }w_i\ell _i]+u^{\prime \prime }(x_i)t\text{ E }w_i\frac{\partial \ell _i}{\partial b}}{u^{\prime \prime }(x_i)+\beta _i u^{\prime \prime }(d_i)}\nonumber \\&\quad -\frac{-w_i\ell _i u^{\prime \prime }(x_i)+u^{\prime \prime }(x_i)[t\text{ E }w_i\frac{\partial \ell _i}{\partial t}+\text{ E }w_i\ell _i]-\beta _i u^{\prime \prime }(d_i)b\text{ E }w_i\frac{\partial \ell _i}{\partial t}}{u^{\prime \prime }(x_i)+\beta _i u^{\prime \prime }(d_i)}\nonumber \\&= -\text{ E }w_i\ell _i. \end{aligned}$$

(29)

1.2 Proof of Proposition 1

Setting $\beta _i=1$ and noting that then $u^{\prime }(x_i)=u^{\prime }(d_i)$ (see Eq. (6)), the FOCs of the government reduce to

$$\begin{aligned} t+(1-\alpha )b&= \frac{\text{ Cov }(u^{\prime }(x_i),w_i\ell _i)}{\text{ E }u^{\prime }(x_i)\text{ E }w_i\frac{\partial \ell _i}{\partial t}}\nonumber \\ t+(1-\alpha )b&= (1-\alpha )\frac{\text{ Cov }(u^{\prime }(d_i),w_i\ell _i)}{\text{ E }u^{\prime }(d_i)\text{ E }w_i\frac{\partial \ell _i}{\partial b}}\nonumber \\ t+(1-\alpha )b&= -b\frac{\text{ Cov }(u^{\prime }(d_i),w_i\ell _i)}{\text{ E }u^{\prime }(d_i)\text{ E }w_i\frac{\partial \ell _i}{\partial \alpha }} \end{aligned}$$

(30)

Making use of (9) and noting that $\text{ Cov }(u^{\prime }(x_i),w_i\ell _i)=\text{ Cov }(u^{\prime }(d_i),w_i\ell _i)$ for rational unrestricted individuals, each equation in (30) can be written as

$$\begin{aligned} t+(1-\alpha )b=\frac{\text{ Cov }(u^{\prime }(x_i),w_i\ell _i)}{\text{ E }u^{\prime }(x_i)\text{ E }w_i\frac{\partial \ell _i}{\partial t}}. \end{aligned}$$

(31)

The right hand-side of (31) depends on the expression $t+(1-\alpha )b$; equalizing first- and second-period consumption defined in (3) and (4) and solving for savings amounts to

$$\begin{aligned} u^{\prime }(x_i)=u^{\prime }(0.5\left(\left(1-\left(t+(1-\alpha )b\right)\right)w_i\ell _i-v(\ell _i)+ \left(t+(1-\alpha )b\right)\text{ E }w_i\ell _i\right)) \end{aligned}$$

With the above equation and by defining $\xi \equiv t+(1-\alpha )b$, (31) can be rewritten as

$$\begin{aligned} \xi =-\frac{\text{ Cov }(u^{\prime }(0.5\left(\left(1-\xi \right)w_i\ell _i-v(\ell _i)+\xi \text{ E }w_i\ell _i\right)),w_i\ell _i)}{\text{ E }u^{\prime }\left(0.5\left(\left(1-\xi \right)w_i\ell _i-v(\ell _i)+\xi \text{ E }w_i\ell _i\right)\right),w_i\ell _i)\text{ E }\frac{w_i^2}{v^{\prime \prime }(\ell _i)}}. \end{aligned}$$

(32)

Since $\ell _i=v^{\prime -1}((1-\xi )w_i)$ the solution to (30) is solely determined by $\xi$. Thus, the tax and pension system are perfect substitutes and either $t$ or $b$ can be set equal to zero. $\square$