Optimal income taxation and migration

Abstract

We tackle the issue of optimal dynamic taxation of capital income in an economy with disconnection as in Weil (J Public Econ 38:183–198, 1989), generated by migration and intra-family altruism. We show that, when the government aims at correcting such a disconnection using time-varying weights in the social welfare function, then there is room for nonzero capital income taxation, both in the short and in the long run.

Appendix

1.1 Derivation of the implementability constraint

In order to obtain the implementability constraint, write Eq. (2) in its intertemporal form:

$$\begin{aligned} \sum _{t=s}^\infty {\frac{({c_{s,t} -\tilde{w}_{s,t} l_{s,t}})}{\prod \limits _{i=s+1}^t {({1+\tilde{r}_{s,i}})}} =0}. \end{aligned}$$

(21)

Since:

$$\begin{aligned} \frac{U_{c_{s,t}}}{U_{c_{s,t+1}}} \frac{({1+\beta })}{({1+n})} = \frac{p_{s,t}}{p_{s,t+1}} =\frac{({1+\tilde{r}_{t+1}})}{({1+n})}, \end{aligned}$$

we have

$$\begin{aligned} \frac{1}{({1+n})^{t-s}}\prod \limits _{i=s+1}^t {({1+\tilde{r}_{s,i}})} =\frac{p_{s,s}}{p_{s,s+1}} \frac{p_{s,s+1}}{p_{s,s+2}} ...\frac{p_{s,t-1}}{p_{s,t}}. \end{aligned}$$

By substituting into Eq. (21), we obtain

$$\begin{aligned} \sum _{t=s}^\infty {\frac{({c_{s,t} -\tilde{w}_{s,t} l_{s,t}}) p_{s,t}}{p_{s,s}} =0} \end{aligned}$$

and exploiting the FOCs from the individual maximization problem, we get

$$\begin{aligned} \sum _{t=s}^\infty {\left( {\frac{1+n}{1+\beta }} \right) ^{t-s} ({U_{c_{s,t}} c_{s,t} +U_{l_{s,t}} l_{s,t}})=0}. \end{aligned}$$

which is Eq. (13) in the text.

1.2 Derivation of the feasibility constraint

To derive the feasibility constraint, first aggregate Eq. (2) over population at time t (notice that $a_{s,s-1} = 0$):

$$\begin{aligned} \sum _{s=0}^t {P_{s,t} a_{s,t} =\sum _{s=0}^t {P_{s,t} \left[ {\frac{({1+r_t})}{({1+n})}a_{s,t-1} +w_t l_{s,t} -c_{s,t} -\frac{\tau _{s,t}^k r_t a_{s,t-1}}{1+n}-\tau _{s,t}^l w_t l_{s,t}} \right] }}. \end{aligned}$$

(22)

and by recalling that $A_t \equiv \sum \limits _{s=0}^t {P_{s,t} a_{s,t}}$, $P_{s,t} =P_{s,t-1} ({1+n})$ and $P_{t,t-1} =0$, so that $\sum \limits _{s=0}^t {P_{s,t-1} a_{s,t-1}} =A_{t-1}$ we can rewrite Eq. (22) as follows

$$\begin{aligned} A_t = ({1+r_t})A_{t-1} +w_t L_t -C_t -T_t \end{aligned}$$

where $C_{t}$ is aggregate consumption. Finally, by subtracting Eq. (10) and exploiting the market clearing condition we obtain

$$\begin{aligned} K_t = ({1+r_t})K_{t-1} +w_t L_t -C_t -G_t \end{aligned}$$

which, in per capita terms, becomes

$$\begin{aligned} k_t =\frac{({1+r_t})}{({1+\gamma })}k_{t-1} +w_t l_t -c_t -g_t \end{aligned}$$

where $\frac{r_t k_{t-1}}{({1+\gamma })} +w_t l_t \equiv y_t$ due to CRS.

1.3 Proof of Proposition 1

Proof

Since a competitive equilibrium (or implementable allocation) satisfies both the feasibility and the implementability constraints by construction, in this “Appendix” we demonstrate the reverse of Proposition 1: any feasible allocation satisfying implementability is a competitive equilibrium.

Suppose that an allocation satisfies the implementability and the feasibility constraints. Then, define a sequence of after tax prices as follows: $\tilde{w}_{s,t} =-\frac{U_{l_{s,t}}}{U_{c_{s,t}}}$, $\tilde{r}_{s,t} =\frac{p_{s,t}}{p_{s,t+1}} ({1+n})-1$, with $p_{s,t} =U_{c_{s,t}} ({\frac{1+n}{1+\beta }})^{t-s}$, $\forall s$ and $\forall t$, and a sequence of before tax prices: $f_{k_{t-1}} =\frac{r_t}{1+\gamma }$, $f_{l_t} =w_t$. Therefore, by construction such allocation satisfies both the consumers’ and firms’ optimality conditions.

The second step is to show that the allocation satisfies the consumer budget constraint. Take the implementability constraint and substitute $U_{c_{s,t}}$ and $U_{l_{s,t}}$ by using the expressions above:

$$\begin{aligned} \sum _{t=s}^\infty {({p_{s,t} c_{s,t} -\tilde{w}_{s,t} p_{s,t} l_{s,t}})=0} \quad \forall s. \end{aligned}$$

Then, by recursively using the expression $p_{s,t} =\frac{p_{s,s} ({1+n})^{t-s}}{\prod \limits _{i=s+1}^t {({1+\tilde{r}_{s,i}})}}$ we get

$$\begin{aligned} \sum _{t=s}^\infty {\frac{({1+n})^{t-s}}{\prod \limits _{i=s+1}^t {({1+\tilde{r}_{s,i}})}} p_{s,s} ({c_{s,t} - \tilde{w}_{s,t} l_{s,t}})=0}. \end{aligned}$$

Finally, by eliminating $p_{s,s}$ and defining $c_{s,t} -\tilde{w}_{s,t} l_{s,t} = -\,({1+n}) q_{s,t} + ({1+\tilde{r}_{s,t}}) q_{s,t-1}$, we get

$$\begin{aligned} \sum _{t=s}^\infty {\frac{({1+n})^{t-s}}{\prod \limits _{i=s+1}^t {({1+\tilde{r}_{s,i}})}} [{- ({1+n})q_{s,t} + ({1+\tilde{r}_{s,t}}) q_{s,t-1}}]=0} \end{aligned}$$

that turns out to be:

$$\begin{aligned} -\lim \frac{({1+n})^{t-s}}{\prod \limits _{i=s+1}^t {({1+\tilde{r}_{s,i}})}} q_{s,t} +q_{s,s} =0 \end{aligned}$$

which holds if $q_{s,t} = a_{s,t}$ and $\lim \frac{({1+n})^{t-s}}{\prod \limits _{i=s+1}^t {({1+\tilde{r}_{s,i}})}} a_{s,t} =0$.

As for the public sector budget constraint, by aggregating the individuals’ budget constraints over population at time t and expressing them in per capita terms, we get

$$\begin{aligned} a_t =\frac{({1+r_t})}{({1+\gamma })}a_{t-1} +w_t l_t -c_t -\tau _t. \end{aligned}$$

(23)

Finally, by subtracting the feasibility constraint

$$\begin{aligned} k_t =\frac{({1+r_t})}{({1+\gamma })}k_{t-1} +w_t l_t -c_t -g_t \end{aligned}$$

and defining $b_t =-\,k_{t-1} +a_t$, we obtain

$$\begin{aligned} b_t =\frac{({1+r_t})}{({1+\gamma })}b_{t-1} +g_t -\tau _t \end{aligned}$$

which is Eq. (11) in the text. $\square$