Fiscal policy reforms and dynamic Laffer effects

Abstract

This paper looks at the conditions under which a dynamic Laffer effect occurs. Using a basic model, we explain and reconcile selected findings in the literature. We numerically show that a lower tax rate on capital income is the best candidate for obtaining a dynamic Laffer effect—here defined as an improvement in the long-run budget balance of the government. Moreover, ignoring the stock of initial debt and changes in labor supply lead to an overestimation and underestimation of the effect, respectively. Finally, we show that when lower taxes on factor income are financed by higher taxes on consumption, there exists a wide array of combinations for which there is an improvement in both the long-run government budget balance and lifetime welfare. These combinations, however, differ in their implications for labor supply and immediate welfare effects.

Appendix

1.1 Existence of the equilibrium

Both (16a) and (16b) are defined for $L\in (0,1]$. Over this interval, it holds that

$$\begin{aligned} \gamma _{P}(L)&=\frac{(1-\tau _{A})(aL^{\beta }-\delta )-\rho }{1-(1-\sigma )(\phi +\theta )},\\ \frac{\partial \gamma _{P}(L)}{\partial L}&=\frac{(1-\tau _{A})a\beta L^{\beta -1}}{1-(1-\sigma )(\phi +\theta )}>0,\\ \frac{\partial ^{2}\gamma _{P}(L)}{\partial L^2}&=-\frac{(1-\tau _{A})a\beta (1-\beta )L^{\beta -2}}{1-(1-\sigma )(\phi +\theta )}<0,\\ \gamma _{Q}(L)&=L^{\beta }\left[ 1-\omega _{G}-\frac{1-\tau _{L}}{1+\tau _{C}}\frac{b}{\beta }\frac{b\phi }{\eta }\frac{1-L}{L}\right] -\delta ,\\ \frac{\partial \gamma _{Q}(L)}{\partial L}&=L^{\beta }\left[ \frac{(1-\omega _{G})\beta }{L} +\frac{1-\tau _{L}}{1+\tau _{C}}\frac{b\phi }{\eta }\frac{1-L}{L}\left( \frac{1-\beta }{L}+\frac{1}{1-L}\right) \right] >0,\\ \frac{\partial ^{2}\gamma _{Q}(L)}{\partial L^2}&=L^{\beta }\left[ \frac{(1-\omega _{G})\beta ^{2}}{L^{2}}+\frac{1-\tau _{L}}{1+\tau _{C}}\frac{b\phi }{\eta }\frac{1-L}{L}\left( \frac{\beta (1-\beta )}{L^{2}}+\frac{2\beta }{L(1-L)}\right) \right] \\&\quad -L^{\beta }\left[ \frac{(1-\omega _{G})\beta }{L^{2}}+\frac{1-\tau _{L}}{1+\tau _{C}}\frac{b\phi }{\eta }\frac{1-L}{L}\left( \frac{2(1-\beta )}{L^{2}}+\frac{2}{L(1-L)}\right) \right] <0, \end{aligned}$$

where we used that $0<\beta <1$. Moreover

$$\begin{aligned}&\lim _{L\rightarrow 0}\gamma _{P}(L)=\frac{-\delta (1-\tau _{A})-\rho }{1-(1-\sigma )(\phi +\theta )},&\lim _{L\rightarrow 0}\gamma _{Q}(L)=-\infty ,\\&\gamma _{P}(1)=\frac{(a-\delta )(1-\tau _{A})-\rho }{1-(1-\sigma )(\phi +\theta )},&\gamma _{Q}(1)=1-\omega _{G}-\delta . \end{aligned}$$

Now, a unique equilibrium exists if $\gamma _{P}(1)<\gamma _{Q}(1)$. This is the case if

$$\begin{aligned} \sigma >1-\frac{1-\omega _{G}-(a-\delta )(1-\tau _{A})+\rho }{(1-\omega _{G})(\phi +\theta )}. \end{aligned}$$

1.2 Stability of the equilibrium

We define $\varOmega _{P}(L)\equiv \partial \gamma _{P}(L)/\partial L$ and $\varOmega _{Q}(L)\equiv \partial \gamma _{Q}(L)/\partial L$. The sign of the derivative of the rate of growth of labor supply (14) with respect to labor, valued at $\tilde{L}$, is given by

$$\begin{aligned} {\text {sgn}}\left\{ \frac{\text {d}\dot{L}/L}{\text {d}L}\biggl |_{L=\tilde{L}}\right\} ={\text {sgn}}\left\{ \frac{1-(1-\sigma )(\phi +\theta )}{\varTheta ^{D}(\tilde{L})}\ \biggl [\varOmega _{P}(\tilde{L})-\varOmega _{Q}(\tilde{L})\biggr ]\right\} . \end{aligned}$$

(28)

If (28) is positive, then small deviations in labor supply lead to a permanent deviation from $\tilde{L}$ so that the equilibrium is locally unstable and said to be locally determinate. If (28) is negative, then small deviations of the supply of labor will force labor supply back to $\tilde{L}$ so that the equilibrium is locally stable and said to be locally indeterminate. Since $\varOmega _{Q}(\tilde{L})>\varOmega _{P}(\tilde{L})$ when a unique equilibrium exists, and $1-(1-\sigma )(\phi +\theta )>0$ by the concavity of the felicity function, the equilibrium is locally unstable if $\varTheta ^{D}(\tilde{L})<0$.

1.3 Comparative static effects

The comparative static effects of the various tax rates are as follows:

$$\begin{aligned} \frac{\text {d}\tilde{L}}{\text {d}\tau _{A}}&=\left[ \varOmega _{P}(\tilde{L})-\varOmega _{Q}(\tilde{L}) \right] ^{-1}\frac{a\tilde{L}^{\beta }-\delta }{1-(1-\sigma )(\phi +\theta )}<0,\\ \frac{\text {d}\tilde{L}}{\text {d}\tau _{L}}&=\left[ \varOmega _{P}(\tilde{L})-\varOmega _{Q}(\tilde{L}) \right] ^{-1}\frac{b\phi }{\eta }\tilde{L}^\beta \frac{1-\tilde{L}}{\tilde{L}}\frac{1}{1+\tau _{C}}<0,\\ \frac{\text {d}\tilde{L}}{\text {d}\tau _{C}}&=\left[ \varOmega _{P}(\tilde{L})-\varOmega _{Q}(\tilde{L}) \right] ^{-1}\frac{b\phi }{\eta }\tilde{L}^\beta \frac{1-\tilde{L}}{\tilde{L}}\frac{1}{1+\tau _{C}}\frac{1-\tau _{L}}{1+\tau _{C}}<0,\\ \frac{\text {d}\tilde{L}}{\text {d}\omega _{G}}&=\left[ \varOmega _{P}(\tilde{L})-\varOmega _{Q}(\tilde{L}) \right] ^{-1}(-\tilde{L}^{\beta })>0, \end{aligned}$$

where $\varOmega _{P}(\tilde{L})>\varOmega _{Q}(\tilde{L})$ in the equilibrium that is determinate. In an indeterminate equilibrium, it holds that $\varOmega _{P}(\tilde{L})<\varOmega _{Q}(\tilde{L})$ so that the effects are reversed.

$$\begin{aligned} \frac{\text {d}\gamma (\tilde{L})}{\text {d}\tau _{A}}&=\varOmega _{P}(\tilde{L})\frac{\text {d}\tilde{L}}{\text {d}\tau _{A}}-\frac{a\tilde{L}^{\beta }-\delta }{1-(1-\sigma )(\phi +\theta )}<0,\\ \frac{\text {d}\gamma (\tilde{L})}{\text {d}\tau _{L}}&=\varOmega _{P}(\tilde{L})\frac{\text {d}\tilde{L}}{\text {d}\tau _{L}}<0,\\ \frac{\text {d}\gamma (\tilde{L})}{\text {d}\tau _{C}}&=\varOmega _{P}(\tilde{L})\frac{\text {d}\tilde{L}}{\text {d}\tau _{C}}<0,\\ \frac{\text {d}\gamma (\tilde{L})}{\text {d}\omega _{G}}&=\varOmega _{P}(\tilde{L})\frac{\text {d}\tilde{L}}{\text {d}\omega _{G}}>0. \end{aligned}$$