Progressive taxation as an automatic destabilizer under endogenous growth

Abstract

It has been shown that in an otherwise standard one-sector real business cycle model with an indeterminate steady state under laissez faire, sufficiently progressive income taxation may stabilize the economy against aggregate fluctuations caused by agents’ animal spirits. We show that this previous finding can be overturned within an identical model which allows for sustained endogenous growth. Specifically, progressive taxation may operate like an automatic destabilizer that leads to equilibrium indeterminacy and sunspot-driven cyclical fluctuations in an endogenously growing macroeconomy. This instability result is obtained under two tractable progressive tax policy formulations that have been considered in the existing literature.

Appendices

Appendix A

Derivations of the dynamical system (17)–(18) Combining Eqs. (11) and (13) leads to

$$\begin{aligned} \frac{\dot{c}_{t}}{c_{t}}=\eta (1-\phi )\left( \frac{y_{t}^{*}}{y_{t}} \right) ^{\phi }\alpha h_{t}^{\left( 1-\alpha \right) \left( 1+\chi \right) }-\delta -\rho . \end{aligned}$$

(A.1)

In addition, Eqs. (15) and (16) can be rewritten as

$$\begin{aligned} \frac{g_{t}}{k_{t}}= & {} \left[ 1-\eta \left( \frac{y_{t}^{*}}{y_{t}}\right) ^{\phi }\right] h_{t}^{\left( 1-\alpha \right) \left( 1+\chi \right) }, \end{aligned}$$

(A.2)

$$\begin{aligned} \frac{\dot{k}_{t}}{k_{t}}= & {} h_{t}^{\left( 1-\alpha \right) \left( 1+\chi \right) }-z_{t}-\frac{g_{t}}{k_{t}}-\delta ,\text { where }z_{t}\equiv \frac{c_{t}}{k_{t}}. \end{aligned}$$

(A.3)

From the above (A.1)–(A.3), we obtain

$$\begin{aligned} \frac{\dot{z}_{t}}{z_{t}}=\left[ \left( 1-\phi \right) \alpha -1\right] \left[ h_{t}^{\left( 1-\alpha \right) \left( 1+\chi \right) }-\frac{g_{t}}{ k_{t}}\right] -\rho +z_{t}. \end{aligned}$$

(A.4)

Next, using the economy’s aggregate production function (3), Eq. (12) can rewritten as

$$\begin{aligned} Ah_{t}^{1+\gamma }=\frac{\eta (1-\phi )u_{t}^{\phi }\left( 1-\alpha \right) h_{t}^{\left( 1-\alpha \right) \left( 1+\chi \right) }}{z_{t}},\text { where }u_{t}\equiv \frac{y_{t}^{*}}{y_{t}}. \end{aligned}$$

(A.5)

Taking time derivatives on the both sides of (A.5) yields that

$$\begin{aligned} \frac{\dot{z}_{t}}{z_{t}}=\phi \frac{\dot{u}_{t}}{u_{t}}+[\left( 1-\alpha \right) \left( 1+\chi \right) -1-\gamma ]\frac{\dot{h}_{t}}{h_{t}}. \end{aligned}$$

(A.6)

From Eqs. (A.4) and (A.6), we find that

$$\begin{aligned} \frac{\dot{u}_{t}}{u_{t}}= & {} \frac{\left[ \left( 1-\phi \right) \alpha -1\right] \left[ h_{t}^{\left( 1-\alpha \right) \left( 1+\chi \right) }-\frac{g_{t}}{ k_{t}}\right] -\rho +z_{t}}{\phi }\nonumber \\&-\left[ \frac{\left( 1-\alpha \right) \left( 1+\chi \right) -1-\gamma }{\phi }\right] \frac{\dot{h}_{t}}{h_{t}}. \end{aligned}$$

(A.7)

On the other hand, we use Eqs. (3) and (A.3) to obtain

$$\begin{aligned} \frac{\dot{u}_{t}}{u_{t}}=\theta -\frac{\dot{y}_{t}}{y_{t}}=\theta -h_{t}^{\left( 1-\alpha \right) \left( 1+\chi \right) }+z_{t}+\frac{g_{t}}{ k_{t}}+\delta -\left( 1-\alpha \right) \left( 1+\chi \right) \frac{\dot{h} _{t}}{h_{t}},\text { where }\theta =\frac{\dot{y}_{t}^{*}}{ y_{t}^{*}}. \end{aligned}$$

(A.8)

Equating (A.7) and (A.8) leads to

$$\begin{aligned} \frac{\dot{h}_{t}}{h_{t}}=\frac{\left( 1-\phi \right) \left\{ z_{t}+\left( 1-\alpha \right) \left[ \frac{g_{t}}{k_{t}}-h_{t}^{\left( 1-\alpha \right) \left( 1+\chi \right) }\right] \right\} -\rho -\phi \left( \theta +\delta \right) }{\left( 1-\phi \right) \left( 1-\alpha \right) \left( 1+\chi \right) -1-\gamma }. \end{aligned}$$

(A.9)

By utilizing (A.5), we can express Eq. (A.2) as

$$\begin{aligned} \frac{g_{t}}{k_{t}}=h_{t}^{\left( 1-\alpha \right) \left( 1+\chi \right) }- \frac{Az_{t}h_{t}^{1+\gamma }}{\left( 1-\phi \right) \left( 1-\alpha \right) }. \end{aligned}$$

(A.10)

Finally, substituting (A.10) into (A.9) leads to Eq. (17); and substituting (A.10) into (A.4) leads to Eq. (18) in the main text.

Appendix B

Derivations of the Jacobian’s determinant (24) and trace (25) The dynamical system (17)–(18) can be approximated by the following linearized equations around a balanced-growth equilibrium characterized by $\left( h^{*},z^{*}\right)$:

$$\begin{aligned} \left[ \begin{array}{c} \dot{h}_{t} \\ \dot{z}_{t} \end{array} \right] =\underbrace{\left[ \begin{array}{c@{\quad }c} \mathbf {J}_{11} &{} \mathbf {J}_{12} \\ \mathbf {J}_{21} &{} \mathbf {J}_{22} \end{array} \right] }_{\mathbf {J}}\left[ \begin{array}{c} h_{t}\quad -h^{*} \\ z_{t}\quad -z^{*} \end{array} \right] ,\, \, \ \end{aligned}$$

(B.1)

where

$$\begin{aligned} \mathbf {J}_{11}&=\frac{\left( 1-\phi \right) \left( 1-\alpha \right) \left( 1+\gamma \right) \left( \rho -z^{*}\right) }{\left[ \left( 1-\phi \right) \left( 1-\alpha \right) \left( 1+\chi \right) -1-\gamma \right] \left[ 1-\left( 1-\phi \right) \alpha \right] }, \end{aligned}$$

(B.2)

$$\begin{aligned} \mathbf {J}_{12}&=\frac{\left[ \rho +\phi \left( \theta +\delta \right) \right] h^{*}}{\left[ \left( 1-\phi \right) \left( 1-\alpha \right) \left( 1+\chi \right) -1-\gamma \right] z^{*}}, \end{aligned}$$

(B.3)

$$\begin{aligned} \mathbf {J}_{21}&=\frac{\left( 1+\gamma \right) \left( \rho -z^{*}\right) z^{*}}{h^{*}}, \end{aligned}$$

(B.4)

$$\begin{aligned} \mathbf {J}_{22}&=\rho . \end{aligned}$$

(B.5)

It is then straightforward to derive the determinant $\left( =\mathbf {J}_{11} \mathbf {J}_{22}-\mathbf {J}_{12}\mathbf {J}_{21}\right)$ and trace $\left( = \mathbf {J}_{11}+\mathbf {J}_{22}\right)$ of the model’s Jacobian matrix $\mathbf {J}$ given by Eqs. (24)–(25) in the main text.