Relative effects of labor taxes on employment and working hours: role of mechanisms shaping working hours

Abstract

High labor income taxes are one of the most important explanations advanced for the large decline of labor supply in Europe over the past 30 years. While in some countries the decline comes evenly from employment and hours per worker, in others it comes mostly from hours per worker, or predominantly from employment. This paper studies why labor taxes have different relative effects on employment and hours per worker. We show that different hour-shaping mechanisms are one of the underlying reasons. In the mechanism when hours per worker are bargained by matched job-worker pairs, a higher labor income tax would reduce both employment and hours per worker. As the worker’s hour-bargaining share is larger, hours per worker are decreased by more and employment is decreased by less. In the mechanism when hours per worker are determined exclusively by the household, this goes to the case when the worker has a one-hundred percent hour bargaining power. In this situation, when the leisure utility is linear in hours, the effect on employment is zero and all negative effects are on hours per worker. At the other extreme, in the mechanism when hours per worker are effectively regulated, a higher labor tax only reduces employment with a zero effect on hours. We calibrate the model and show that the quantitative effects of Europe’s increases in average effective tax rates in the past 30 years are consistent with the theoretical predictions.

Appendices

Appendix A: This appendix derives matching rates, vacant jobs, and the values of the change in total household utility, a filled job and a vacant job in the steady state

We use the matching relationships to solve the two matching rates and the vacant jobs as functions of $e$.

$$\begin{aligned} p(e)= & {} \frac{\lambda e}{1-e},\quad \hbox {where}\,{p}'(e)>0,\end{aligned}$$

(19a)

$$\begin{aligned} q(e)= & {} m^{\frac{1}{1-\gamma }}\left( \frac{\lambda e}{1-e}\right) ^{\frac{-\gamma }{1-\gamma }}=m^{\frac{1}{1-\gamma }}[p(e)]^{\frac{-\gamma }{1-\gamma }},\quad \hbox {where}\,{q}'(e)<0,\end{aligned}$$

(19b)

$$\begin{aligned} v(e)= & {} \left[ {\frac{\lambda e}{m(1-e)^\gamma }} \right] ^{\frac{1}{1-\gamma }}=m^{\frac{-1}{1-\gamma }}[p(e)]^{\frac{\gamma }{1-\gamma }}\lambda e,\quad \hbox {where}\,{v}'(e)>0. \end{aligned}$$

(19c)

Then, from (4), the change in total household utility in a steady state is

$$\begin{aligned} U_e (e)=\frac{1+\rho }{\rho +\lambda +p}\left[ {{u}'(c)( {1-\tau })wh-g(h)} \right] . \end{aligned}$$

(20a)

From (5a), the value of a filled job in a steady state is

$$\begin{aligned} \pi _e =\frac{1+\rho }{\rho +\lambda }( {f(h)-wh}). \end{aligned}$$

(20b)

Obviously, $f(h)>$ wh if employment is positive.

Moreover, in a steady state, the free-entry condition implies that a firm will create vacant jobs until $\pi _{v}=\phi$. Using (5b) and (20b), the free-entry condition implies

$$\begin{aligned} \pi _v =\frac{q}{\rho +q}\frac{1+\rho }{\rho +\lambda }( {f(h)-wh})=\phi . \end{aligned}$$

(20c)

Appendix B: This appendix derives the effects of labor taxes

The stead-state conditions are (12) and (15a). Differentiating these two conditions gives

$$\begin{aligned} \Gamma _e =MRS_e \cdot {g}'-\frac{\beta _h -\beta }{\beta _h ( {1-\beta })}( {1-\tau })w_e \mathop =\limits ^{\beta _h =\beta } MRS_e \cdot {g}'>0, \end{aligned}$$

$$\begin{aligned} \begin{array}{l} \Gamma _h =MRS_h \cdot {g}'+MRS\cdot {g}''-\frac{\beta }{\beta _h }\frac{1-\beta _h }{1-\beta }( {1-\tau }){f}''-\frac{\beta _h -\beta }{\beta _h ( {1-\beta })}( {1-\tau })w_h \\ \mathop =\limits ^{\beta _h =\beta } \underbrace{MRS_h \cdot {g}'+}_{\Gamma _h^1 >0}\underbrace{MRS\cdot {g}''-( {1-\tau }){f}''}_{\Gamma _h^2 >0}>0,\\ \end{array} \end{aligned}$$

$$\begin{aligned} \Gamma _\tau =\frac{\beta }{\beta _h }\frac{1-\beta _h }{1-\beta }{f}'+\frac{\beta _h -\beta }{\beta _h ( {1-\beta })}\frac{\beta ( {\rho +\lambda +p})}{\rho +\lambda +\beta p}AP\mathop =\limits ^{\beta _h =\beta } {f}'>0, \end{aligned}$$

$$\begin{aligned} \Omega _e= & {} \underbrace{-\frac{q}{\rho +q}\frac{1+\rho }{\rho +\lambda +\beta p}\frac{( {1-\beta })}{1-\tau }MRS_e g}_{\Omega _e^1 <0}\\&+\underbrace{\frac{1+\rho }{\rho +q}\left[ {\frac{\rho {q}'}{\rho +q}\frac{f-wh}{\rho +\lambda }-\frac{q( {1-\beta })\beta {p}'}{( {\rho +\lambda +\beta p})^2}( {AP-\frac{MRS}{1-\tau }\frac{g}{h}})h} \right] }_{\Omega _e^2 <0}<0, \end{aligned}$$

$$\begin{aligned} \Omega _h =\frac{q}{\rho +q}\frac{1+\rho }{\rho +\lambda +\beta p}\frac{( {1-\beta })}{1-\tau }\left[ {\underbrace{( {1-\tau }){f}'-MRS\cdot {g}'}_{=0\,\mathrm{if}\,\beta _h =\beta }-MRS_h \cdot g} \right] <0, \end{aligned}$$

$$\begin{aligned} \Omega _\tau =-\frac{q}{\rho +q}\frac{1+\rho }{\rho +\lambda +\beta p}\frac{( {1-\beta })}{1-\tau }\frac{MRS\cdot g}{1-\tau }<0. \end{aligned}$$

Note that $\Omega _e^2 <0,$ because the surplus of a filled job net of the labor cost $( {AP-\frac{MRS}{1-\tau }\frac{g}{h}})$ must be positive in order for a vacant job to fill a worker. Under a linear leisure utility, $( {AP-\frac{MRS}{1-\tau }\frac{g}{h}})= =AP-MP.$ Since $-\frac{\Gamma _e }{\Gamma _h }<0$ and $-\frac{\Omega _e }{\Omega _h }<0$, the BH and the FE curves are both downward sloping in the $h$ – $e$ space. Moreover, by noting that $\Gamma _e \Omega _h =\Gamma _h^1 \Omega _e^1$, we have $\Gamma _e \Omega _h -\Gamma _h \Omega _e =-\Gamma _h^1 \Omega _e^2 -\Gamma _h^2 \Omega _e >0,$ which implies $-\frac{\Gamma _e }{\Gamma _h }>-\frac{\Omega _e }{\Omega _h}.$ Since the BH curve is always flatter than the FE curve at any point of intersection, there is at most one intersection.

For a given $e$, when $\tau$ increases, the BH and the FE curves shift downward as follows.

$$\begin{aligned} \left. {\frac{dh}{d\tau }} \right| _{LD} =-\frac{\Omega _\tau }{\Omega _h }=-\frac{MRS\cdot g/( {1-\tau })}{MRS_h \cdot g+MRS\cdot {g}'-( {1-\tau }){f}'}\mathop =\limits ^{g( h)=\bar{g}h} -\frac{{f}'}{MRS_h \cdot \bar{g}}<0, \end{aligned}$$

$$\begin{aligned} \begin{array}{l} \left. {\frac{dh}{d\tau }} \right| _{LS} =-\frac{\Gamma _\tau }{\Gamma _h }=-\frac{{\beta \over {\beta _h }}{{1-\beta _h } \over {1-\beta }}{f}'+{{\beta _h -\beta } \over {\beta _h ( {1-\beta })}}{{\beta ( {\rho +\lambda +p})} \over {\rho +\lambda +\beta p}}AP}{MRS_h \cdot {g}'+MRS\cdot {g}''-{\beta \over {\beta _h }}{{1-\beta _h } \over {1-\beta }}( {1-\tau }){f}''-{{\beta _h -\beta } \over {\beta _h ( {1-\beta })}}( {1-\tau })w_h } \\ \mathop =\limits ^{\beta _h =\beta } -\frac{{f}'}{MRS_h \cdot {g}'+MRS\cdot {g}''-( {1-\tau }){f}''}\mathop =\limits ^{g( h)=\bar{g}h} -\frac{{f}'}{MRS_h \cdot \bar{g}-( {1-\tau }){f}''}<0. \\ \end{array} \end{aligned}$$

Although it is difficult to compare the relative downward shift of these two curves, a linear leisure utility helps pin down the relative magnitude. Under $g( h)=\bar{g}h,$ since ($1-\tau )f''(h)<0$, the BH curve is unambiguously shifted downward less than the FE curve. The following comparative state confirms this conjecture. As $\Gamma _\tau \Omega _h =\Gamma _h^1 \Omega _\tau$ and $\Gamma _e \Omega _\tau =\Gamma _\tau \Omega _e^1,$ we have $\Gamma _\tau \Omega _h -\Gamma _h \Omega _\tau = -\Gamma _h^2 \Omega _\tau >0$ and $\Gamma _e \Omega _\tau -\Gamma _\tau \Omega _e =-\Gamma _\tau \Omega _e^2 >0.$ Thus, $\frac{de}{d\tau }=-\frac{\Gamma _\tau \Omega _h -\Gamma _h \Omega _\tau }{\Gamma _e \Omega _h -\Gamma _h \Omega _e }<$ and $\frac{dh}{d\tau }=-\frac{\Gamma _e \Omega _\tau -\Gamma _\tau \Omega _e }{\Gamma _e \Omega _h -\Gamma _h \Omega _e }<0.$

Appendix C: The model with capital adjustments

In this appendix, we show that the results in Sect. 3 are robust if capital is adjustable in the same way as was in Marimon and Zilibotti (2000).³¹ We assume that the production function is now $y_t =k_t^\alpha ( {h_t })^{1-\alpha }=f( {h_t, k_t })$ and capital $k_{t}$ is accumulated by firms. As in Marimon and Zilibotti (2000), we think of final-goods producing firms that take the output from worker firms and combine it with capital. Hence, capital is separate from the wage bargaining process. The result will be the same if the firm rents capital from the household since the capital market is perfect. By assuming that capital $k$ does not depreciate in order to simplify our analysis, then the interest rate equals the marginal product of capital: $r_t =f_k ( {h_{t+1}, k_{t+1} })$.

The representative household’s problem and the optimization conditions all remain the same as the model above. The government’s behavior remains the same as (9). While the lifetime value of an unfilled job is also the same as (5b), the lifetime value of a filled job in (5a) is modified as

$$\begin{aligned} \pi _{et} =[f( {h_t, k_t })-w_t h_t +k_t] +\frac{1}{1+r_t }( {1-\lambda })\pi _{e( {t+1})}. \end{aligned}$$

(21)

Note that different from (5a), here the flow value in $t$ includes the value of capital. In a steady state, the interest rate satisfies $r=\rho$ and hence$f_k ( {h,k})=\rho$. This implies that the capital-hour ratio in a steady state is constant and thus $k$ is in proportion to $h$,

$$\begin{aligned} k=\kappa h\equiv k( h),\quad \hbox {where } \kappa =\left( {\frac{\alpha }{\rho }}\right) ^{{1 \over {1-\alpha }}}. \end{aligned}$$

The goods market clearing condition is now

$$\begin{aligned} c=e( {f(h,k(h))-\phi \lambda })\equiv \mathop {c}\limits ^{\frown } (e,h), \end{aligned}$$

(22)

where ${\mathop {c}\limits ^{\frown }}_e =f( {h,k( h)})-\phi \lambda >0$ and ${\mathop {c}\limits ^{\frown }}_h =e( {f_h +f_k k_h })>0.$

From (21), the value of a filled job in a steady state is

$$\begin{aligned} \pi _e =\frac{1+\rho }{\rho +\lambda }( {f({h,k( h)})-wh+k(h)}), \end{aligned}$$

and then the free-entry condition in a steady state is

$$\begin{aligned} \pi _v =\frac{q}{\rho +q}\frac{1+\rho }{\rho +\lambda }( {f(h,k(h))-wh+k(h)})=\phi . \end{aligned}$$

(23)

From the first order condition of the wage bargaining problem, the bargained wage rate is

$$\begin{aligned} w= & {} \mathop {w}\limits ^{\frown } \left( {e,h;\tau } \right) \equiv \frac{{\beta \left( {\rho + \lambda + p} \right) }}{{\rho + \lambda + \beta p}}\left[ {AP\left( h \right) } \right] + \frac{{\left( {1 - \beta } \right) \left( {\rho + \lambda } \right) }}{{\rho + \lambda + \beta p}}\nonumber \\&\times \left[ \frac{{MRS( {\mathop {c}\limits ^{\frown },1 - eg\left( h \right) })}}{{1 - \tau }}\frac{{g\left( h \right) }}{h}\right] , \end{aligned}$$

(24)

where $AP(h)\equiv \frac{1}{h}( {f( {h,k(h)})+k(h)-\frac{\rho +\lambda }{1+\rho }\phi })$. Notice that the steady-state matching relationships given in the text still hold. Thus, $p$ and $q$ both are functions of $e$ as stated in (19a) and (19b).

Substituting the bargained wage rate in (24) into (23) yields the free-entry condition

$$\begin{aligned} \Omega \left( \mathop e\limits _{\left( - \right) } ,\mathop h\limits _{\left( - \right) } ;\mathop \tau \limits _{\left( - \right) } \right) \equiv \frac{{q(e)}}{{\rho + q\left( e \right) }}\frac{{1 + \rho }}{{\rho + \lambda }}\left( {f\left( {h,k\left( h \right) } \right) - \mathop {w}\limits ^{\frown } \left( {e,h} \right) h + k\left( h \right) } \right) - \phi = 0,\nonumber \\ \end{aligned}$$

(25)

which relates employment negatively to hours. As in Sect. 3, (25) is referred to as the LD curve.

1.1 C.1 Hours bargained by job-worker pairs

First, consider the mechanism when a worker’s hours is determined by a matched pair in a bargaining game. When the laborer’s hour bargaining power is $\beta _{h}$, the hour is determined by

$$\begin{aligned} -\beta _h \frac{{u}'( c)( {1\!-\!\tau })w\!-\!{g}'( h)}{\frac{1+\rho }{\rho \!+\!\lambda \!+\!p}\left[ {{u}'( c)( {1\!-\!\tau })wh-g( h)} \right] }=( {1-\beta _b })\frac{f_h ( {h,k( h)})\!-\!w}{\frac{1+\rho }{\rho +\lambda }( {f( {h,k( h)})-wh+k( h)})-\phi }. \end{aligned}$$

Substituting (24) and (22) into the above expression yields

$$\begin{aligned}&\Gamma (e,h;\tau )\equiv MRS(\mathop {c}\limits ^{\frown },1-eg(h)){g}'(h)-\frac{\beta }{\beta _h }\frac{1-\beta _h }{1-\beta }( {1-\tau })MP(h)\nonumber \\&\quad -\frac{\beta _h -\beta }{\beta _h ( {1-\beta })}( {1-\tau })\mathop {w}\limits ^{\frown } (e,h)=0, \end{aligned}$$

(26)

which is the same as (15a) in Sect. 3.1 except $MP( h)\equiv f_h ( {h,k( h)}).$

When $\beta _{h}=\beta$, (26) yields the LS curve like (15b) in Sect. 3.1 as follows.

$$\begin{aligned} \Gamma ( {e,h;\tau })\equiv MRS(\mathop {c}\limits ^{\frown }({e,h}),1-eg(h)){g}'(h)-({1-\tau })MP(h)=0. \end{aligned}$$

(27)

Hence, the steady-state condition includes the LD curve (25) and the LS curve (27) and determines $e$ and $h$. Differentiating (25) and (27) gives

$$\begin{aligned} \Gamma _e =MRS_e \cdot {g}'-\frac{\beta _h -\beta }{\beta _h ( {1-\beta })}( {1-\tau }){\mathop {w}\limits ^{\frown }}_e \mathop =\limits ^{\beta _h =\beta } MRS_e \cdot {g}'>0, \end{aligned}$$

$$\begin{aligned} \begin{array}{l} \Gamma _h =MRS_h \cdot {g}'+MRS\cdot {g}''-\frac{\beta }{\beta _h }\frac{1-\beta _h }{1-\beta }( {1-\tau })\underbrace{( {f_{hh} +f_{hk} k_h })}_{=0}-\frac{\beta _h -\beta }{\beta _h ( {1-\beta })}( {1-\tau }){\mathop {w}\limits ^{\frown }}_h \\ \mathop =\limits ^{\beta _h =\beta } \underbrace{MRS_h \cdot {g}'+}_{\Gamma _h^1 >0}\underbrace{MRS\cdot {g}''}_{\Gamma _h^2 >0}>0, \\ \end{array} \end{aligned}$$

$$\begin{aligned} \Gamma _\tau =\frac{\beta }{\beta _h }\frac{1-\beta _h }{1-\beta }f_h +\frac{\beta _h -\beta }{\beta _h ( {1-\beta })}\frac{\beta ( {\rho +\lambda +p})}{\rho +\lambda +\beta p}AP\mathop =\limits ^{\beta _h =\beta } f_h >0, \end{aligned}$$

$$\begin{aligned} \Omega _e= & {} \underbrace{-\frac{q}{\rho +q}\frac{1+\rho }{\rho +\lambda +\beta p}\frac{( {1-\beta })}{1-\tau }MRS_e \cdot g}_{\Omega _e^1 <0}\\&+\underbrace{\frac{1+\rho }{\rho +q}\left[ {\frac{\rho {q}'}{\rho +q}\frac{f-wh+k}{\rho +\lambda }-\frac{q( {1-\beta })\beta {p}'}{( {\rho +\lambda +\beta p})^2}( {AP-\frac{MRS}{1-\tau }\frac{g}{h}})h} \right] }_{\Omega _e^2 <0}<0, \end{aligned}$$

$$\begin{aligned} \Omega _h= & {} \underbrace{\frac{q}{\rho +q}\frac{( {1+\rho })^2( {1-\beta })}{\rho +\lambda +\beta p}k_h }_{\Omega _h^1 >0}\\&+\underbrace{\frac{q}{\rho +q}\frac{1+\rho }{\rho +\lambda +\beta p}\frac{( {1-\beta })}{1-\tau }\left[ {\underbrace{( {1-\tau })f_h -MRS\cdot {g}'}_{=0}-MRS_h \cdot g} \right] }_{\Omega _h^2 <0},\\ \Omega _\tau= & {} -\frac{q}{\rho +q}\frac{1+\rho }{\rho +\lambda +\beta p}\frac{( {1-\beta })}{1-\tau }\frac{MRS\cdot g}{1-\tau }<0. \end{aligned}$$

By noting that $\Gamma _e \Omega _h^2 =\Gamma _h^1 \Omega _e^1,$ we have $\Gamma _e \Omega _h -\Gamma _h \Omega _e =\Gamma _e \Omega _h^1 -\Gamma _h^1 \Omega _e^2 -\Gamma _h^2 \Omega _e >0,$ which implies $-\frac{\Gamma _e }{\Gamma _h }>-\frac{\Omega _e }{\Omega _h }$. Since the LS curve is always flatter than the LD curve at any point of intersection, there is a unique steady state.

Given $e$, when $\tau$ is increased, the LS and the LD curves shift downward, respectively, as follows.

Thus, even under a linear leisure utility, the LS curve is shifted downward less than the LD curve and thus hours and employment both are reduced.

Finally, as $\Gamma _\tau \Omega _h^2 =\Gamma _h^1 \Omega _\tau$ and $\Gamma _e \Omega _\tau =\Gamma _\tau \Omega _e^2,$ then $\Gamma _\tau \Omega _h -\Gamma _h \Omega _\tau =\Gamma _\tau \Omega _h^1 -\Gamma _h^2 \Omega _\tau >0$ and $\Gamma _e \Omega _\tau -\Gamma _\tau \Omega _e =-\Gamma _\tau \Omega _e^2 >0.$ Thus, a higher labor tax reduces both hours per worker and employment.

$$\begin{aligned} \frac{de}{d\tau }=-\frac{\Gamma _\tau \Omega _h -\Gamma _h \Omega _\tau }{\Gamma _e \Omega _h -\Gamma _h \Omega _e }<0\quad \hbox {and}\quad \frac{dh}{d\tau }=-\frac{\Gamma _e \Omega _\tau -\Gamma _\tau \Omega _e }{\Gamma _e \Omega _h -\Gamma _h \Omega _e }<0. \end{aligned}$$

When $\beta _{h}> \beta$ , we find that the LD curve is not affected. Moreover, ${{d\Gamma _h } \over {d\beta _h }}=-{\beta \over {1-\beta }}{{1-\tau } \over {\beta _h ^2}}{\mathop {w}\limits ^{\frown }}_h <0$ and ${{d\Gamma _\tau } \over {d\beta _h }}=-{\beta \over {1-\beta }}{1 \over {\beta _h ^2}}( {{{\beta ( {\rho +\lambda +p})} \over {\rho +\lambda +\beta p}}AP-f_h })>0$ if $\beta$ is not too small or productivity diminishes more in $h$. Thus, the BH curve is shifted downward more. Hence, even though capital is adjustable, when the worker’s supply of hours is determined by a bargaining game, the relative effect of a higher labor tax on the intensive and extensive margins of labor supply in Proposition 1 continues to hold.

1.2 C.2 Hours determined by households

Next, consider the mechanism wherein, given employment, the supply of hours is exclusively decided the household. The leisure-consumption tradeoff condition is (18a). By using the bargained wage in (24), consumption in (22) and the utility of leisure linear in hours $\tilde{g}( {h_t })=gh_t,$ (18a) is rewritten to yield the following FH curve

$$\begin{aligned} \Gamma \left( \mathop e\limits _{( +)}, \mathop h\limits _{( +)}; \mathop \tau \limits _{( +)}\right) \equiv MRS(\mathop {c}\limits ^{\frown }({e,h}),1-eg(h))\bar{g}-( {1-\tau })AP( h)=0. \end{aligned}$$

(28)

The steady-state conditions of model are (25) and (28) wherein $p$ and $q$ are functions of $e$, defined by (19a) and (19b), and $c$ is a function of $e$ and $h$, given by (22). Differentiating (25) and (28) gives $\Gamma _e =MRS_e \cdot \bar{g}>0$, $\Gamma _h =MRS_h \cdot \bar{g}-( {1-\tau })AP_h >0$, $\Gamma _\tau =AP>0$,

$$\begin{aligned} \Omega _e =&\underbrace{-\frac{q}{\rho +q}\frac{1+\rho }{\rho +\lambda +\beta p}\frac{( {1-\beta })}{1-\tau }MRS_e \cdot \bar{g}h}_{\Omega _e^1<0}\\&+\underbrace{\frac{1+\rho }{\rho +q}\left[ {\frac{\rho {q}'}{\rho +q}\frac{f-wh+k}{\rho +\lambda }-\frac{q( {1-\beta })\beta {p}'}{( {\rho +\lambda +\beta p})^2}\overbrace{( {AP-\frac{MRS\cdot \bar{g}}{1-\tau }})}^{=0}h} \right] }_{\Omega _e^2 <0}<0, \end{aligned}$$

$$\begin{aligned} \Omega _h= & {} \frac{q}{\rho +q}\frac{1+\rho }{\rho +\lambda +\beta p}\frac{( {1-\beta })}{1-\tau }\\&\times \left[ {( {1-\tau })( {f_h +f_k k_h +k_h }) -MRS\cdot \bar{g}-MRS_h \cdot \bar{g}h} \right] <0,\\ \Omega _\tau= & {} -\frac{q}{\rho +q}\frac{1+\rho }{\rho +\lambda +\beta p}\frac{( {1-\beta })}{1-\tau }\frac{MRS\cdot \bar{g}}{1-\tau }h<0, \end{aligned}$$

where $AP_h =\frac{1}{h}( {f_h +f_k k_h +k_h -AP})$. Since $\Gamma _e \Omega _h =\Gamma _h \Omega _e^1,$ we have $\Gamma _e \Omega _h -\Gamma _h \Omega _e =-\Gamma _h \Omega _e^2 >0,$ which implies $-\frac{\Gamma _b }{\Gamma _e }>-\frac{\Omega _h }{\Omega _e }$. Since the FH curve is always flatter than the FE curve at any point of intersection, there is at most one intersection.

Further, by noting that $\Gamma _e \Omega _\tau =\Gamma _\tau \Omega _e^1,$ we obtain $\Gamma _\tau \Omega _h -\Gamma _h \Omega _\tau =0$ and $\Gamma _e \Omega _\tau -\Gamma _\tau \Omega _e = -\Gamma _\tau \Omega _e^2 >0.$ Thus, ${{de} \over {d\tau }}=-{{\Gamma _\tau \Omega _h -\Gamma _h \Omega _\tau } \over {\Gamma _e \Omega _h -\Gamma _h \Omega _e }}=0$ and ${{dh} \over {d\tau }}=-{{\Gamma _e \Omega _\tau -\Gamma _\tau \Omega _e } \over {\Gamma \Gamma _e \Omega _h -\Gamma _h \Omega _e }}<0.$ Hence, even though capital is adjusted, when the supply of hours is determined exclusively by the household, under a linear leisure utility, a higher labor income tax only reduces hours per worker without affecting employment, a result the same as Proposition 2.

1.3 C.3 Hours regulated by authorities

Finally, when the worker’s supply of hours is regulated effectively, the hour curve is replaced by $h=\bar{h}.$ Then, the steady state is characterized by $h=\bar{h}$ and (25). The conditions are (25) and

$$\begin{aligned} \Gamma (e,h;\bar{h},\tau )\equiv h-\bar{h}=0. \end{aligned}$$

(29)

Differentiating (25) and (29) gives

$$\begin{aligned}&\Gamma _e =0, \Gamma _h =1, \Gamma _{\bar{h}} =-1, \Gamma _\tau =0, \Omega _{\bar{h}} =0, \\&\Omega _\tau =-\frac{q}{\rho +q}\frac{1+\rho }{\rho +\lambda +\beta p}\frac{( {1-\beta })}{1-\tau }\frac{MRS\cdot g}{1-\tau }<0, \end{aligned}$$

$$\begin{aligned} \Omega _e =&\underbrace{-\frac{q}{\rho +q}\frac{1+\rho }{\rho +\lambda +\beta p}\frac{( {1-\beta })}{1-\tau }MRS_e \cdot g}_{\Omega _e^1 <0}\\&+\underbrace{\frac{1+\rho }{\rho +q}\left[ {\frac{\rho {q}'}{\rho +q}\frac{f-wh+k}{\rho +\lambda }-\frac{q( {1-\beta })\beta {p}'}{( {\rho +\lambda +\beta p})^2}( {AP-\frac{MRS}{1-\tau }\frac{g}{h}})h} \right] }_{\Omega _e^2 <0}<0, \end{aligned}$$

$$\begin{aligned} \Omega _h =&\underbrace{\frac{q}{\rho +q}\frac{( {1+\rho })^2( {1-\beta })}{\rho +\lambda +\beta p}k_h }_{\Omega _h^1 >0}\\&\underbrace{+\frac{q}{\rho +q}\frac{1+\rho }{\rho +\lambda +\beta p}\frac{( {1-\beta })}{1-\tau }\left[ {\underbrace{( {1-\tau })f_h -MRS\cdot {g}'}_{=0}-MRS_h \cdot g} \right] }_{\Omega _h^2 <0}. \end{aligned}$$

The comparative-static results are as follow: ${{de} \over {d\tau }}=-{{\Gamma _\tau \Omega _h -\Gamma _h \Omega _\tau } \over {\Gamma _e \Omega _h -\Gamma _h \Omega _e }}=-{{-\Omega _\tau } \over {-\Omega _e }}<0, {{dh} \over {d\tau }}=-{{\Gamma _e \Omega _\tau -\Gamma _\tau \Omega _e } \over {\Gamma _e \Omega _h -\Gamma _h \Omega _e }}=0.$ Thus, a higher labor tax rate only reduces employment. Hence, the results in Sect. 3.3 continue to hold.