Optimal asymmetric sector-specific labour taxation in an overlapping generations model

Abstract

This paper presents a simple rule for optimal asymmetric labour taxation and subsidization in a two-sector model with logarithmic utilities and Cobb–Douglas production functions, linked to demographic factors: fertility rate and longevity. The paper shows that depending on whether the economy is dynamically efficient or inefficient, it may be optimal to tax or subsidize labour in the sectors. Under dynamic inefficiency, it is optimal to tax the investment-goods sector and a Pareto-improving tax reform is possible. Larger output elasticities of capital in the sectors reduce the possibilities of a Pareto-improving reform, while population ageing in terms of higher longevity enhances the possibilities of welfare improvement for all generations. Fertility rates do not affect optimal taxation. In appendix, we also address the cases of capital taxation/subsidisation and value-added taxes.

Appendices

Appendix 1: derivation of equation (24)

1.1 Utility maximization

We plug Eqs. (22) and (23) into the utility function (7), and differentiate it with respect to $\tau _I$. As Eqs. (22) and (23) contain three multipliers depending on $\tau _I$, it is convenient to express the first order condition in the following form:

$$\begin{aligned} \frac{\partial U}{\partial \tau _I}= \bigg (1+\frac{\psi }{1+\rho } \bigg )\big (D_1+D_2 \big )+D_3 =0. \end{aligned}$$

(28)

where

$$\begin{aligned} D_1= & {} -\frac{\alpha }{1-\tau _I} \nonumber \\ D_2= & {} \frac{\psi (1-\alpha )(1-\beta )}{1+\rho +\beta \psi }\bigg (1+\frac{\tau _I\psi (1-\beta )}{1+\rho +\beta \psi } \bigg )^{-1} \nonumber \\ D_3= & {} \bigg (-\frac{\alpha }{1-\beta } +\frac{\psi }{1+\rho } \frac{1-\alpha -\beta }{1-\beta } \bigg )\frac{\partial D_4}{\partial \tau _I}. \nonumber \\ D_4= & {} log\bigg (1+\frac{\alpha }{\beta (1-\alpha )(1+\frac{\tau _I\psi (1-\beta )}{1+\rho +\beta \psi })} \bigg [\frac{1+\rho +\psi }{\psi }-(1-\beta )(1-\tau _I)\bigg ] \bigg )\nonumber \\ \end{aligned}$$

(29)

and

$$\begin{aligned} \frac{\partial D4}{\partial \tau _I}= & {} \bigg (\frac{\alpha (1-\beta )}{\beta (1-\alpha )\big (1+\frac{\tau _I \psi (1-\beta )}{1+\rho +\beta \psi }\big )}\nonumber \\&- \frac{\alpha \psi (1-\beta ) \big ((1+\rho +\psi )/\psi -(1-\beta )(1-\tau _I)\big ) }{\beta (1-\alpha )\big (1+\frac{\tau _I\psi (1-\beta )}{1+\rho +\beta \psi }\big )^2(1+\rho +\beta \psi )}\bigg )D_4^{-1}. \end{aligned}$$

(30)

After a number of simple algebraic manipulations we get that $\partial D4 / \partial \tau _I=0$. Hence, $D_3=0$. Substituting $D_1$ and $D_2$ into equation (28), the derivation of equation (24) is straightforward.

1.2 $Y_C$ maximization

Instead of $Y_C$ maximization in respect to $\tau _I$, it is more convenient to maximise $\log {Y_C}$, which gives an identical result, because logarithmic function is a strictly increasing function on the interval $(0,\infty )$. The first order condition is:

$$\begin{aligned} \frac{\partial Y_C}{\partial \tau _I}=\frac{\partial \log {L_C}}{\partial \tau _I} +\alpha \frac{\partial \log {k_C}}{\partial \tau _I}=0. \end{aligned}$$

(31)

$\partial \log {L_C}/\partial \tau _I$ can be easily found from Eqs. (14) and (17). $k_C$ is given by Eq. (12), it depends on $\tau _I$, $\tau _C$ and $k_I$, and the last two variables depend on $\tau _I$ themselves. As $\partial D4 / \partial \tau _I=0$, it is easy to see that $\partial k_I / \partial \tau _I=0$. Hence, Eq. (31) simplifies to

$$\begin{aligned} \frac{\psi (1-\beta )}{(1+\rho +\psi )\big (1-\frac{\psi (1-\beta )(1-\tau _I)}{1+\rho +\psi } \big )}-\frac{\alpha }{1-\tau _I}-\frac{\alpha \psi (1-\beta )}{(1+\rho +\beta \psi )\big (1+\frac{\tau _I\psi (1-\beta )}{1+\rho +\beta \psi }\big )}=0. \end{aligned}$$

Denominators of the first and the third fractions can be simplified. Furthermore, these fractions can be combined together. Then, the derivation of Eq. (24) is simple.

Appendix 2: Proof of proposition 1

Proof

Suppose that $r^*=n$. Then, Eq. (25) can be rewritten as

$$\begin{aligned} (1+n)(\alpha \rho +\beta \psi +\alpha -\psi -\alpha \psi )=0. \end{aligned}$$

(32)

As it is assumed that $n>-1$, $\alpha \rho +\beta \psi +\alpha =\psi (1-\alpha )$ and condition (27) holds.

If we assume that (27) holds, Eq. (25) can be rewritten as

$$\begin{aligned} r^*=r^*+\frac{\alpha \psi +\alpha \rho +\beta \psi +\alpha -\psi }{\psi (1-\alpha )} \end{aligned}$$

(33)

Hence, the nominator of the second term in the right hand side of Eq. (33) is equal to zero, i.e., $\alpha \rho +\beta \psi +\alpha =\psi (1-\alpha )$ and condition (27) simplifies to $r^*=n$. $\square$

Appendix 3

1.1 Capital taxation

When capital is taxed instead of labour, the real interest rate, which affects consumption when old (Eq. 9) changes to: $1+R_x(t)=(1+r_x(t))(1-\tau _x)/\psi$, $x\in \{I, C\}$; labour market equilibrium is given by an equality of wages in the sectors: $(1-\alpha )k_C^\alpha =p(t)(1-\beta )k_I^\beta$; capital market equilibrium is given by an equality of net capital incomes: $\alpha k_C^{\alpha -1}(t)(1-\tilde{\tau }_C)=p(t)\beta k_C^{\beta -1}(t)(1-\tilde{\tau }_I)$, where $\tilde{\tau }_x$, $x\in \{I,C\}$ denote sector-specific taxes. Dividing labour market equilibrium by capital market equilibrium we get

$$\begin{aligned} k_C(t)=\frac{\alpha (1-\beta ) (1-\tilde{\tau }_C)}{\beta (1-\alpha )(1-\tilde{\tau }_I)}k_I(t). \end{aligned}$$

(34)

Equation for savings (10) changes to

$$\begin{aligned} s_x(t)=\frac{\psi w_x(t)}{1+\rho +\psi }, \quad x\in \{C,I\}, \end{aligned}$$

(35)

leading to a different expression for share of agents working in the investment-good sector:

$$\begin{aligned} \frac{L_I(t)}{\Lambda (t)}=\frac{\psi (1-\beta )}{1+\rho +\psi }. \end{aligned}$$

(36)

Equalizing total savings in the economy to the amount of capital, we get an expression for capital–labour ratio in sector I:

$$\begin{aligned} k_I^{*\beta -1}=(1+n)\bigg [1+\frac{\alpha (1-\tilde{\tau }_I)}{\beta (1-\alpha )(1-\tilde{\tau }_I)} \bigg (\frac{1+\rho +\psi }{\psi }-1+\beta \bigg ) \bigg ] . \end{aligned}$$

(37)

Expression for government budget balance (21) changes as well. Now government budget is balanced when $K_C (1+r_C) \tilde{\tau }_C=- K_I (1+r_I) \tilde{\tau }_I$, implying that

$$\begin{aligned} \tilde{\tau }_C=-\frac{\tilde{\tau }_I \psi (1-\alpha )\beta }{(1+\rho +\beta \psi )\alpha }. \end{aligned}$$

(38)

Having derived capital labour ratios and government budget constraint, we solve for the other variables and maximize the utility functions. We received the same optimal tax for the investment goods sector as in (24); however, taxes for the consumption goods sector are slightly different (compare Eqs. (21) and (38)).

1.2 Value added taxes

If VAT taxes are analysed instead of labour taxes, Eqs. (1–11) do not change; however, in equation (9) $R_x(t), x\in \{I, C\}$ changes to $1+R_x(t)=(1+r_x(t))(1-\tau _x)/\psi$, which does not affect savings.

Capital market equilibrium changes to

$$\begin{aligned} \alpha k_C^{\alpha -1}(t)(1-\tilde{\tau }_C)=p(t)\beta k_C^{\beta -1}(t)(1-\tilde{\tau }_I). \end{aligned}$$

(39)

As a result, Eq. (12) simplifies to

$$\begin{aligned} k_C(t)=\frac{\alpha (1-\beta )}{\beta (1-\alpha )}k_I(t). \end{aligned}$$

(40)

Equations (17)–(18) do not change, but Eq. (20) changes to

$$\begin{aligned} k_I^{*\beta -1}=(1+n)\bigg [1+\frac{\alpha }{\beta (1-\alpha )} \bigg (\frac{1+\rho +\psi }{\psi (1-\tau _I)}-(1-\beta ) \bigg ) \bigg ] \end{aligned}$$

(41)

Government budget constraint changes to $Y_C(t)\tau _C=-p Y_I \tau _I$ resulting in

$$\begin{aligned} \tau _C=-\frac{\psi \tau _I(1-\alpha )}{1+\rho +\beta \psi +\psi \tau _I (\alpha -\beta )}. \end{aligned}$$

(42)

Having derived capital–labour ratio and government budget constraint, we express all the other variables of interest. We have two opportunities: to maximize agents’ utilities with respect to $\tau _I$ directly, or to maximize $Y_C$. In contrast to the cases of separate labour and capital taxation, these two methods do not give the same results, because agent’s income is taxed/subsidized twice: first it is taxed/subsidized when young agents receive their wages; next, the income of the old generation is taxed/subsidized. Therefore, a part of the tax for the young, working in one sector, returns as a subsidy for the old, invested to another sector, or vice versa. This produces an intergenerational reallocation of income, comparable to a pay-as-you-go pension scheme and such a reallocation mechanism affects agents’ budget constraints in a way different from separate labour or capital taxation/subsidization.

Indeed, direct maximization of agents’ utilities has not given us an analytical solution. But it can be made numerically. For example, for parameter values used in Sect. 5 we received $\tau _I=10.76\%$, $\tau _C=-\,4.41\%$ for the case $\alpha =0.2$ and $\tau _I=-\,26.64\%$, $\tau _C=8.63\%$ when $\alpha =0.3$.

The government may choose to maximize $Y_C$. Then, the optimization problem gives $\tau _I$ exactly as in the case of labour taxation (Eq. 24), and $\tau _C$ is given by Eq. (42). For the parameter values used in Sect. 5 this results in $\tau _I=14.20\%$, $\tau _C=-\,5.84\%$ for the case $\alpha =0.2$ and $\tau _I=-\,11.06\%$, $\tau _C=5.97\%$ when $\alpha =0.3$. Intuitively, it is clear, that if $Y_C$ is maximized, the government may reallocate consumption goods with lump-sum taxes and subsidies in such a way, that consumption of all the agents increases after an introduction of such a tax, resulting in a higher utility level.