Progressive taxation and macroeconomic stability in two-sector models with social constant returns

Abstract

It has been shown that, in the two-sector Benhabib–Farmer–Guo model with technologies of social increasing returns that exhibits indeterminacy, progressive income taxes de-stabilize the economy. This paper revisits the robustness of the tax implication in the two-sector Benhabib–Nishimura model with technologies of social constant returns that exhibits indeterminacy. We show that a progressive income tax stabilizes the economy against sunspot fluctuations, and thus the tax implication based on the two-sector Benhabib–Farmer–Guo model is not robust.

Appendix

1.1 A. The trace and determinant of the Jacobian matrix

Using a caret to denote a variable in logarithmic deviations from the steady-state value, the log-linear approximations give⁹$^{,}$¹⁰:

$$\begin{aligned}&\displaystyle \left[ {{\begin{array}{l} {\hat{{K}}_{t+1} } \\ {\hat{{P}}_{t+1} } \\ \end{array} }} \right] =\left[ {{\begin{array}{ll} {J_{11} }&{}\quad {J_{12} } \\ {J_{21} }&{}\quad {J_{22} } \\ \end{array} }} \right] \left[ {{\begin{array}{l} {\hat{{K}}_t } \\ {\hat{{P}}_t } \\ \end{array} }} \right] , \end{aligned}$$

(A1)

$$\begin{aligned} J_{11}= & {} \tilde{J}_{11} -\frac{dY_I }{dK}\left[ {\frac{a_c }{b_c }\Psi L_K -[1-\eta (1-\varphi )]} \right] ;\nonumber \\ J_{12}= & {} \frac{1}{\Delta }\frac{r}{p}\left[ {a_c (W_P +L_P )\Psi -b_c R_P } \right] -\delta<0\quad \hbox {if }\theta _c<\theta _I \hbox { and }\Delta >0;\nonumber \\ J_{21}= & {} \frac{\Lambda }{\Gamma }J_{11} ;\nonumber \\ J_{22}= & {} \frac{-1}{\Gamma }+\frac{\Lambda }{\Gamma }J_{12} ,\nonumber \\ \Lambda\equiv & {} -\varphi [1-\rho (1-\delta )]\left[ {\frac{\Xi }{\Psi }+\Xi L_K } \right] \le \hbox {0, and }\Lambda =0\quad \hbox {if }\varphi =0;\nonumber \\ \Gamma\equiv & {} -1+\frac{[1-\rho (1-\delta )](1-\theta _I )}{\theta _c -\theta _I }+\varphi \Omega ;\nonumber \\ \Omega\equiv & {} [1-\rho (1-\delta )]\left[ {\frac{\Xi }{\Psi }R_P +\Xi (W_P +L_P )} \right]<0\quad \hbox {if }\theta _c <\theta _I .\nonumber \end{aligned}$$

1.2 B. Determinacy in the model with income taxes

In the presence of progressive income taxes, $\eta <1$ and $\varphi >0$. The two eigenvalues are determined by:

$$\begin{aligned} \lambda _1 +\lambda _2= & {} Trace(J)=J_{11} +J_{22} =J_{11} +\frac{-1}{\Gamma }+\frac{\Lambda }{\Gamma }J_{12} , \end{aligned}$$

(B1a)

$$\begin{aligned} \lambda _1 \lambda _2= & {} Det(J)=J_{11} J_{22} -J_{21} J_{12} =\left( {J_{11} } \right) \left( {\frac{-1}{\Gamma }} \right) . \end{aligned}$$

(B1b)

Solving conditions (B1a) and (B1b) gives the following two eigenvalues:

$$\begin{aligned} \lambda _1= & {} \frac{1}{2}\left[ {\left( {J_{11} -\frac{1}{\Gamma }} \right) +\frac{\Lambda J_{12} }{\Gamma }+\sqrt{\left[ {\left( {J_{11} -\frac{1}{\Gamma }} \right) +\frac{\Lambda J_{12} }{\Gamma }} \right] ^{2}+4J_{11} \frac{1}{\Gamma }}} \right] ,\qquad \end{aligned}$$

(B2a)

$$\begin{aligned} \lambda _2= & {} \frac{1}{2}\left[ {\left( {J_{11} -\frac{1}{\Gamma }} \right) +\frac{\Lambda J_{12} }{\Gamma }-\sqrt{\left[ {\left( {J_{11} -\frac{1}{\Gamma }} \right) +\frac{\Lambda J_{12} }{\Gamma }} \right] ^{2}+4J_{11} \frac{1}{\Gamma }}} \right] .\qquad \end{aligned}$$

(B2b)

If $0<\lambda _{1}<1$ and $\lambda _{2}<-1$, the steady state is a saddle. The part $0<\lambda _{1}<1$ is proved in the text. Here we prove $\lambda _{2}<-1$.

To show $\lambda _{2}<-1$, we will find a threshold value so that if $\varphi$ is larger than the threshold value, then $\lambda _{2}<-1$. To derive the threshold value, it suffices to impose $\lambda _{2}<-1$ in (B2b). This gives the following cubic equation in terms of $\varphi$:

$$\begin{aligned} m_3 \varphi ^{3}+m_2 \varphi ^{2}+m_1 \varphi +m_0 >0, \end{aligned}$$

(B3)

where

$$\begin{aligned} m_3= & {} -(2-\delta )\delta (\Delta \rho \eta )^{2}[(1+\chi )\theta _c H+M]<0,\nonumber \\ m_2= & {} [M+(1+\chi )\theta _c H]\Delta \rho \eta [(b_c H+2\delta \Delta \rho \eta )(2-\delta )-(a_c +b_c )\delta H]\nonumber \\&-\,M(2-\delta )\delta \Delta (\rho \eta )^{2}\chi (a_c -a_I ) \nonumber \\&-\,[2(1-\theta _c )-\delta (1-\theta _I )]H^{2}\Delta \rho \eta \chi a_c , \nonumber \\ m_1= & {} [M+(1+\chi )\theta _c H][(a_c +b_c )H-(2-\delta )\Delta \rho \eta ][b_c H+\delta \Delta \rho \eta ]\nonumber \\&+\,M(2-\delta )\Delta \rho \eta \chi [H+(a_c -a_I )\delta \rho \eta ] \nonumber \\&+\,[2(1-\theta _c )-\delta (1-\theta _I )]H^{2}\Delta \rho \eta \chi a_c \nonumber \\&-\,\chi M(a_c -a_I )\delta \rho \eta [b_c H+(2-\delta )\Delta \rho \eta ], \nonumber \\ m_0= & {} \chi M[H+(a_c -a_I )\delta \rho \eta ][b_c H+(2-\delta )\Delta \rho \eta ]>0,\nonumber \\ M= & {} 2(\theta _c -\theta _I )+(1-\theta _I )H,\nonumber \\ H\equiv & {} [1-\rho (1-\delta )]. \end{aligned}$$

Note that the condition $\theta _{I}-\theta _{c}>0$ is used in signing $m_{0}$ and $m_{3}$.

If we set (B3) equal to 0, we obtain $(\varphi -\varphi _1 )(\varphi -\varphi _2 )(\varphi -\varphi _3 )=0,$ and there are three critical values: $\varphi _{1},\,\varphi _{2}$ and $\varphi _{3}$. Let ${ max}\{\varphi _{1},\,\varphi _{2},\,\varphi _{3}\}$ be $\varphi _{1}$, the largest value of the three critical values.

To solve the three critical values, we first notice that the product of these three critical values is $\varphi _{1}\,\varphi _{2}\,\varphi _{3}=-m_{0}/m_{3}>0$. This indicates that there are either case (i) with three positive critical values or case (ii) with one positive and two negative critical values.

Next, we order these three critical values in a way such that $\varphi _{1}> \varphi _{2}> \varphi _{3}$. Thus, the largest value is $\varphi _{1}$. Hence, no matter whether it is case (i) or case (ii), the largest value is $\varphi _{1}$. As a result, for the steady state to be a saddle point, the required condition is $\varphi >\varphi _{1}$.