Physical capital taxation and labor income taxation in an endogenous growth model with new generations

Abstract

This paper examines the growth and welfare effects of physical capital taxation and labor income taxation. It is found that the impact of a rise in the physical capital tax rate on the balanced growth rate is crucially related to the birth rate, but a rise in the labor income tax rate definitely reduces the balanced growth rate regardless of the birth rate. It is also found that an increase in the physical capital income tax rate will harm the older but will benefit the younger, while an increase in the labor income tax rate will benefit the older but will harm the younger.

Appendix

1.1 Appendix 1

This appendix provides a detailed derivation of the welfare function (i.e., the discounted sum of indirect utility function) $U*{\left( {v,t} \right)}$, the present discounted value of the household’s future lump-sum transfers G*(t), and real total assets a*(v,t) along the balanced growth path.

From Eq. 2a, we have:

$$ U{\left( {v,t} \right)} = {\int\limits_t^\infty {\ln c{\left( {v,\tau } \right)}e^{{ - {\left( {\lambda + \rho } \right)}{\left( {\tau - t} \right)}}} d\tau } } = {\int\limits_t^\infty {\frac{{ - \ln c{\left( {v,\tau } \right)}}} {{\lambda + \rho }}de^{{ - {\left( {\lambda + \rho } \right)}{\left( {\tau - t} \right)}}} } }. $$

(A1)

Using integration by parts, Eq. A1 can be expressed by:

$$ \begin{array}{*{20}c} {U{\left( {v,t} \right)} = \left. {\frac{{ - \ln c{\left( {v,\tau } \right)}}} {{\lambda + \rho }}e^{{ - {\left( {\lambda + \rho } \right)}{\left( {\tau - t} \right)}}} } \right|^{{\;\tau = \infty }}_{{\;\tau = t}} + {\int\limits_t^\infty {\frac{{e^{{ - {\left( {\lambda + \rho } \right)}{\left( {\tau - t} \right)}}} }} {{\lambda + \rho }}d} }\ln c{\left( {v,\tau } \right)}} \\ { = \frac{{\ln c{\left( {v,t} \right)}}} {{\lambda + \rho }} + {\int\limits_t^\infty {\frac{{e^{{ - {\left( {\lambda + \rho } \right)}{\left( {\tau - t} \right)}}} }} {{\lambda + \rho }}} }\frac{{{\mathop c\limits^ \cdot }{\left( {v,\tau } \right)}}} {{c{\left( {v,\tau } \right)}}}d\tau .} \\ \end{array} $$

(A2)

Substituting Eq. 3a into Eq. A2, we have:

$$ U{\left( {v,t} \right)} = \frac{{\ln c{\left( {v,t} \right)}}} {{\lambda + \rho }} + {\int\limits_t^\infty {\frac{{e^{{ - {\left( {\lambda + \rho } \right)}{\left( {\tau - t} \right)}}} {\left[ {{\left( {1 - s_{k} } \right)}r{\left( \tau \right)} - \delta _{k} - \rho } \right]}}} {{\lambda + \rho }}} }d\tau . $$

(A3)

From Eqs. 8b and 8c with x = Y ₁/K, we have the following steady-state relationship:

$$ r{\left( t \right)} = r* = \alpha x*, $$

(A4)

$$ w{\left( t \right)} = w* = \frac{{{\left( {1 - \alpha } \right)}K{\left( t \right)}}} {{{\left( {1 - u*} \right)}H{\left( t \right)}}}x*, $$

(A5)

where r* and w* denote the steady-state values of r(t) and w(t), respectively.

Let c*(v,t) denote the goods consumption at time t of a consumer born at time v along the balanced growth path, respectively. From Eq. 4a we have the following steady-state expression:

$$ c*{\left( {v,t} \right)} = {\left( {\lambda + \rho } \right)}{\left[ {a*{\left( {v,t} \right)} + G*{\left( t \right)}} \right]}. $$

(A6)

Substituting Eqs. A4 and A6 into Eq. A3, we have the welfare function of cohort v evaluated at time t, namely, U*(v,t):

$$ \begin{array}{*{20}c} {U*{\left( {v,t} \right)} = \frac{{\ln c*{\left( {v,t} \right)}}} {{\lambda + \rho }} + {\int\limits_t^\infty {\frac{{e^{{ - {\left( {\lambda + \rho } \right)}{\left( {\tau - t} \right)}}} {\left[ {{\left( {1 - s_{k} } \right)}\alpha x* - \delta _{k} - \rho } \right]}}} {{\lambda + \rho }}} }d\tau } \\ { = \frac{{\ln c*{\left( {v,t} \right)}}} {{\lambda + \rho }} - \left. {\frac{{e^{{ - {\left( {\lambda + \rho } \right)}{\left( {\tau - t} \right)}}} {\left[ {{\left( {1 - s_{\operatorname{k} } } \right)}\alpha x* - \delta _{k} - \rho } \right]}}} {{{\left( {\lambda + \rho } \right)}^{2} }}} \right|^{{\;\tau = \infty }}_{{\;\tau = t}} } \\ { = \frac{{\ln {\left[ {G*{\left( t \right)} + a*{\left( {v,t} \right)}} \right]}}} {{{\left( {\lambda + \rho } \right)}}} + \frac{{{\left( {1 - s_{\operatorname{k} } } \right)}\alpha x* - \delta _{k} - \rho }} {{{\left( {\lambda + \rho } \right)}^{2} }} + \frac{{\ln {\left( {\lambda + \rho } \right)}}} {{\lambda + \rho }}.} \\ \end{array} $$

(A7)

Substituting Eq. 10 into Eq. 4b, we have:

$$ G{\left( t \right)} = {\int\limits_t^\infty {{\left[ {s_{k} r{\left( \tau \right)}\frac{{K{\left( \tau \right)}}} {{L{\left( \tau \right)}}} + s_{\operatorname{h} } w{\left( \tau \right)}\frac{{H{\left( \tau \right)}}} {{L{\left( \tau \right)}}}} \right]}\;} }e^{{ - {\int_t^\tau {{\left[ {{\left( {1 - s_{k} } \right)}r{\left( z \right)} - \delta _{k} + \lambda } \right]}dz} }}} d\tau . $$

(A8)

Substituting Eqs. A4 and A5 into Eq. A8 yields the following steady-state value of G(t):

$$ G*{\left( t \right)} = {\int\limits_t^\infty {{\left[ {s_{\operatorname{k} } \alpha x*\frac{{K{\left( \tau \right)}}} {{L{\left( \tau \right)}}} + s_{h} x*\frac{{{\left( {1 - \alpha } \right)}K{\left( \tau \right)}}} {{{\left( {1 - u*} \right)}L{\left( \tau \right)}}}} \right]}\;} }e^{{ - {\left[ {{\left( {1 - s_{k} } \right)}\alpha x* - \delta _{k} + \lambda } \right]}{\left( {\tau - t} \right)}}} d\tau . $$

(A9)

In the balanced growth equilibrium, we have ${{\mathop K\limits^ \cdot }} \mathord{\left/ {\vphantom {{{\mathop K\limits^ \cdot }} K}} \right. \kern-\nulldelimiterspace} K = {{\mathop H\limits^ \cdot }} \mathord{\left/ {\vphantom {{{\mathop H\limits^ \cdot }} H}} \right. \kern-\nulldelimiterspace} H = Bu* + \beta \theta - \delta _{\operatorname{h} }$ and ${{\mathop L\limits^ \cdot }} \mathord{\left/ {\vphantom {{{\mathop L\limits^ \cdot }} L}} \right. \kern-\nulldelimiterspace} L = \beta - \lambda$. As a result, we can infer:

$$ K{\left( \tau \right)} = K{\left( t \right)}e^{{{\left( {Bu* + \beta \theta - \delta _{h} } \right)}{\left( {\tau - t} \right)}}} \,, $$

(A10)

$$ L{\left( \tau \right)} = L{\left( t \right)}e^{{{\left( {\beta - \lambda } \right)}{\left( {\tau - t} \right)}}} . $$

(A11)

Substituting Eqs. A10 and A11 into Eq. A9 gives rise to:

$$ \begin{array}{*{20}c} {G*{\left( t \right)} = {\int\limits_t^\infty {{\left[ {s_{k} \alpha x*\frac{{K{\left( t \right)}}} {{L{\left( t \right)}}} + s_{h} x*\frac{{{\left( {1 - \alpha } \right)}K{\left( t \right)}}} {{{\left( {1 - u*} \right)}L{\left( t \right)}}}} \right]}\;} }e^{{{\left[ {Bu* - \beta {\left( {1 - \theta } \right)} - {\left( {1 - s_{k} } \right)}\alpha x* - {\left( {\delta _{h} - \delta _{k} } \right)}} \right]}{\left( {\tau - t} \right)}}} d\tau } \\ {\left. { = \frac{{{\left[ {{s_{k} \alpha + s_{\operatorname{h} } {\left( {1 - \alpha } \right)}} \mathord{\left/ {\vphantom {{s_{k} \alpha + s_{\operatorname{h} } {\left( {1 - \alpha } \right)}} {{\left( {1 - u*} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 - u*} \right)}}} \right]}x*K{\left( t \right)}e^{{{\left[ {Bu* - \beta {\left( {1 - \theta } \right)} - {\left( {1 - s_{k} } \right)}\alpha x* - {\left( {\delta _{\operatorname{h} } - \delta _{k} } \right)}} \right]}{\left( {\tau - t} \right)}}} }} {{{\left[ {Bu* - \beta {\left( {1 - \theta } \right)} - {\left( {1 - s_{\operatorname{k} } } \right)}\alpha x* - {\left( {\delta _{h} - \delta _{k} } \right)}} \right]}L{\left( t \right)}}}} \right|^{{\tau = \infty }}_{{\tau = t}} } \\ { = \frac{{{\left[ {{s_{\operatorname{k} } \alpha + s_{h} {\left( {1 - \alpha } \right)}} \mathord{\left/ {\vphantom {{s_{\operatorname{k} } \alpha + s_{h} {\left( {1 - \alpha } \right)}} {{\left( {1 - u*} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 - u*} \right)}}} \right]}x*K{\left( t \right)}}} {{{\left[ {\beta {\left( {1 - \theta } \right)} + {\left( {1 - s_{\operatorname{k} } } \right)}\alpha x* - Bu* + {\left( {\delta _{h} - \delta _{k} } \right)}} \right]}L{\left( t \right)}}}.} \\ \end{array} $$

(A12)

From Eqs. 8c and 9b we can further infer the following:

$$ P{\left( t \right)} = \frac{{w{\left( t \right)}}} {B} = \frac{{{\left( {1 - \alpha } \right)}D}} {B}{\left[ {\frac{{K{\left( t \right)}}} {{{\left( {1 - u} \right)}H{\left( t \right)}}}} \right]}^{\alpha } . $$

(A13)

Equation 15a leads to the following steady-state relationship:

$$ \frac{{K{\left( t \right)}}} {{{\left( {1 - u*} \right)}H{\left( t \right)}}} = {\left[ {\frac{{{\left( {1 - s_{\operatorname{k} } } \right)}\alpha D}} {{{\left( {1 - s_{h} } \right)}B - {\left( {\delta _{h} - \delta _{\operatorname{k} } } \right)}}}} \right]}^{{\frac{1} {{1 - \alpha }}}} . $$

(A14)

Substituting Eq. A14 into Eq. A13 yields:

$$ P{\left( t \right)} = P* = \frac{{{\left( {1 - \alpha } \right)}D}} {B}{\left[ {\frac{{{\left( {1 - s_{\operatorname{k} } } \right)}\alpha D}} {{{\left( {1 - s_{h} } \right)}B - {\left( {\delta _{h} - \delta _{k} } \right)}}}} \right]}^{{\frac{\alpha } {{1 - \alpha }}}} , $$

(A15)

where P* denotes the steady-state value of P(t).

Moreover, by substituting Eq. A15 into Eq. 2c, we obtain the following relationship along the balanced growth path:

$$ a*{\left( {v,t} \right)} = k{\left( {v,t} \right)} + \frac{{{\left( {1 - \alpha } \right)}D}} {B}{\left[ {\frac{{{\left( {1 - s_{\operatorname{k} } } \right)}\alpha D}} {{{\left( {1 - s_{h} } \right)}B - {\left( {\delta _{h} - \delta _{\operatorname{k} } } \right)}}}} \right]}^{{\frac{\alpha } {{1 - \alpha }}}} h{\left( {v,t} \right)}. $$

(A16)

Equations A7, A12, A16, and A4 are identical to Eqs. 21a, 21b, 21c, and 21d in the main text, respectively.