## 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.

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## Notes

According to McDaniel (2007) and Rogerson (2008), in the early 1970s the average tax rate on labor income in Belgium, France, Germany, Italy, and the Netherlands was about 39 % which was higher than the 18 % average rate in the US. In the early 2000s, the average tax rate on labor in these European countries was about 51 % which was still much higher than the 22 % average rate in the US.

Alesina et al. (2006) suggested that Prescott’s results held true only under implausibly high labor supply elasticities. These authors and other recent contributors like Eugster et al. (2011), Giavazzi et al. (2013) and Azariadis et al. (2013) suggested an alternative explanation based on different leisure preferences between Europeans and Americans. Moreover, some economists argued Europe’s low labor supply and high unemployment based on stronger labor unions (e.g., Alesina et al. 2006) and higher labor market regulations like generous unemployment compensation (e.g., Ljungqvist and Sargent 2007b, 2008a). There are also other explanations based on differences in entry costs (Fonseca et al. 2001), changes in technology and government (Rogerson 2006), and home production (Ngai and Pissarides 2008; Olovsson 2009). Since our paper focuses on the explanation based on high labor taxes in Europe, in order to simply the model we abstract from these alternative explanations.

Available data in OECD countries indicate that in some countries, relative to the US, declining hours per person from the early 1970s to the early 2000s come essentially from decreasing hours per worker which accounts for more than 90 %. Conversely, in other countries, declining hours per person in the same period basically are from falling employment rates which also accounts for more than 90 %. See Appendix Table 6 for details.

There is a recent literature that compares labor market adjustments on the intensive vs. the extensive margin in different institutional settings during the Great Recession (e.g. US vs. Germany) put forth by Merkl and Wesselbaum (2011) and Burda and Hunt (2011). Arpaia and Mourre (2012) reviewed related literature that analyzes the effect of institutional differences on labor market outcomes in OECD countries. Siebert (1997) offered a nice survey concerning comparative institutional factors at the root of unemployment in Europe. See also Acemoglu (2001) that found that the institutional factors of minimum wages and unemployment benefits shift the composition of employment toward high-wage jobs, which, if it operates in the European labor market, may suggest that institutional factors may dominate market forces (employers exercising their bargaining power under unemployment).

See also Walque et al. (2009) which formalized different bargaining powers for hours bargaining and wage bargaining.

While the first hour-shaping mechanism has been taken up by Fang and Rogerson (2009), the second and the third were used by Prescott (2004) and Marimon and Zilibotti (2000), respectively. Fang and Rogerson (2009) set up a matching model of labor supply and examined the effects of tax and transfer policies on the employment and the hour margins of labor supply. Marimon and Zilibotti (2000) envisaged employment and distributional effects of regulating (reducing) working time in a general equilibrium model with search-matching frictions. While Fang and Rogerson (2009) and Marimon and Zilibotti (2000) studied labor search models, Prescott (2004) analyzed a neoclassical growth model. Prescott (2004) and Marimon and Zilibotti (2000) considered capital adjustment, whereas there is no capital in Fang and Rogerson (2009).

They also found that a rise in labor tax progressivity decreased unemployment rates and in-work effort but increased participation rates provided that the unemployment rate was inefficiently high.

Saez (2002) discovered that the optimal transfer program was characterized by a classical negative income tax program when labor supply are along the intensive margin and by a earned income tax credit when behavioral responses are concentrated along the extensive margin. Laroque (2005) uncovered that, given an income guarantee, a feasible allocation was second-best optimal if and only if the associated taxes are lower than the Laffer bound, determined by the joint distribution of the agents’ productivities and work opportunity costs.

An example is the separable utility \(u(c_t )-g(h_t )\) used in Fang and Rogerson (2009) wherein taking an average over all members in the large household gives \(e_t [u(c_t )-g(h_t )]+(1-e_t )[u(c_t )-g(0)]=u(c_t )-e_t g(h_t ).\)

Working hours may be bargained by the two sides of a successful match, completely determined by the household, or regulated by the authority. Details will be offered in Sect. 3 below.

To ease analysis, we present a model without capital adjustment. In the Appendix, we present a model with capital accumulation and the production function \(y_t =f(k_t ,h_t ), f_x >0>f_{xx} ,\) where \(x=k\), \(h\).

The sufficient condition for any equilibrium with positive employment is that the vacancy creation cost be not too large. The surplus from a match is always positive under our assumptions on the functions \(u\) and \(f\).

Under the Hosios (1990) rule and thus \(\beta =\gamma \), the bargaining is efficient. The results in our paper hold no matter whether the bargaining is efficient or not.

The household takes profits and future values as given when bargaining over current values. An individual worker also takes all other members’ bargains in the current period as given. See Fang and Rogerson (2009).

We note that a low-skilled worker is different from the case of \(\beta \) going to 0. Unless

*AP*is zero, it is difficult to justify the case of \(\beta \) going to 0 in that a worker would not accept a job offer if he/she is paid only a wage equal to the value of leisure hours.This is a property in search and match models; see, for example, Cheron and Langot (2004).

To obtain the expression, we follow Fang and Rogerson (2009, p. 1158) and consider the case with finite family members. Let \(E_{t}\) denote the number of members that are employed in period \(t\). In the bargaining over hours, we take the derivatives of \(U(E_{t})-U(E_{t}-1\)) with respect to the current hours of the \(E\)th worker, taking as given the hours of all other (\(E_{t}-1\)) workers in the family. Thus, working hours of the \(E\)th worker only enter into the current period utility in \(U(E_{t})\) and do not enter into \(U(E_{t}-1\)). Therefore, if the \(E\)th worker works one more hour, consumption is increased by the unit of (\(1-\tau \))\(w_{t}\) while leisure is decreased by \(g'(h_{t})\) which would change the value of \(U(E_{t})\) by \(u\)’(\(c_{t})(1-\tau )w_{t}- g'(h_{t})\).

See the “Appendix” for the derivation.

The sign holds when \(\beta \) is not too small or the curvature of the production \(f(h)\) is not too flat.

If \(e=1\), there is no friction in the labor market and the wage rate is determined solely by the marginal product of labor as it is in Prescott (2004).

Our results in this subsection indicate that in a search model, when hours worked per worker is completely determined by the household, a rise in labor taxes does not have an adverse effect on employment. All the detrimental effects of labor taxes are on hours per worker. Intuitively, as the after-tax wage per hour is decreased, the employed worker chooses more leisure hours and less work hours and thus, less consumption. As in a frictionless neoclassical growth model, the adverse effect on work hours emerges here because we assume that the substitution effect dominates the income effect. In the case when the substitution effect is small, it is possible that the income effect dominates the substitution effect and there is thus a backward bending portion of the labor supply curve. Then a rise in labor tax would increase rather than decrease hours worked per worker.

It is worth noting that when regulated hours are reduced, say from \(\bar{h}\) to \(h_{2}\) in Fig. 4, with other things being equal, the steady state changes from E\(_{0}\) to E\(_{2}\). Thus, a working time reducing policy can increase employment that achieves the goal “work less, work all.”

Under the assumption \({u}'''=0.\)

Under the assumption \({u}'''=0.\)

We will do robustness analysis when a smaller value of

*LSE*is used.While the case with capital adjustment in Marimon and Zilibotti (2000) was carried out in a small open economy when the interest rate is taken as given, we will maintain the closed-economy setup and thus the interest rate is endogenously determined.

## References

Acemoglu D (2001) Good jobs versus bad jobs. J Labor Econ 19:1–21

Andolfatto D (1996) Business cycles and labor-market search. Am Econ Rev 86:112–132

Alesina A, Glaeser E, Sacerdote B (2006) Work and leisure in the U.S. and Europe: why so different? In: Gertler M, Rogoff K (eds) NBER Macroeconomics Annual 2005. MIT Press, Cambridge

Arpaia A, Mourre G (2012) Institutions and performance in European labour markets: taking a fresh look at evidence. J Econ Surveys 26:1–41

Azariadis C, Chen B-L, Lu C, Wang Y (2013) A two-sector model of endogenous growth with leisure externalities. J Econ Theory 148:843–857

Bils M, Chang Y, Kim S-B (2011) Worker heterogeneity and endogenous separations in a matching model of unemployment fluctuations. Am Econ J Macroecon 3:128–154

Burda MC, Hunt J (2011) What Explains the German labor market miracle in the great recession. Brookings Papers Econ Activity 42:273–335

Calmfors L (1985) Work sharing, employment and wages. Eur Econ Rev 27:293–309

Cheron A, Langot F (2004) Labor market search and real business cycles: reconciling Nash bargaining with the real wage dynamics. Rev Econ Dyn 7:476–493

Chetty R, Guren A, Manoli D, Weber A (2011) Are micro and macro labor supply elasticities consistent? A review of evidence on the Intensive and extensive margins. Am Econ Rev Papers Proc 101:471–475

Diamond PA (1982) Wage determination and efficiency in search equilibrium. Rev Econ Stud 49:217–227

Eugster B, Lalive R, Steinhauer A, Zweimuller J (2011) The demand for social insurance: does culture matter? Econ J 121:F413–F448

Fang L, Rogerson R (2009) Policy analysis in a matching model with intensive and extensive margins. Int Econ Rev 50:1153–1168

Fonseca R, Lopez-Garcia P, Pissarides CA (2001) Entrepreneurship, start-up costs and unemployment. Eur Econ Rev 45:692–705

Giavazzi F, Schiantarelli F, Serafinelli M (2013) Attitudes, policies, and work. J Eur Econ Assoc 11:1256–1289

Hoel M, Vale B (1986) Effects on unemployment of reduced working time in an economy where firms set wages. Eur Econ Rev 30:1097–1104

Hosios AJ (1990) On the efficiency of matching and related models of search and unemployment. Rev Econ Stud 57:279–298

Imai S, Keane MP (2004) Intertemporal labor supply and human capital accumulation. Int Econ Rev 45:601–641

Jacobs B (2009) Is Prescott right? welfare state policies and the incentives to work, learn, and retire. Int Tax Public Financ 16:253–280

Jones S, Riddell C (1999) The measurement of unemployment: an empirical approach. Econometrica 67:147–161

King RG, Rebelo ST (1999) Restructuring real business cycles. In: Taylor JB, Woodford M (eds) Handbook of Macroeconomics 1. Elsevier Science B.V. G. KING, pp 927–1007

Krusell P, Mukoyama T, Rogerson R, Sahin A (2010) Aggregate labor market outcomes: the role of choice and chance. Quant Econ 1:97–127

Krusell P, Mukoyama T, Rogerson R, Sahin A (2011) A three state model of worker flows in general equilibrium. J Econ Theory 146:1107–1133

Kydland FE, Prescott EC (1991) Hours and employment variation in business cycle theory. Econ Theor 1:63–81

Laroque G (2005) Income maintenance and labor force participation. Econometrica 73:341–37

Lehmann E, Lucifora C, Moriconi S, Van der Linden B (2014) Beyond the labour income tax wedge: the unemployment-reducing effect of tax progressivity, IZA Discussion Paper No. 8276

Ljungqvist L, Sargent TJ (2007a) Do taxes explain European employment? indivisible labor, human capital, lotteries, and savings. In: Acemoglu D, Rogoff K, Woodford M (eds) NBER Macroeconomics Annual 2006. MIT Press, Cambridge, pp 181–224

Ljungqvist L, Sargent TJ (2007b) Understanding European unemployment with matching and search-island models. J Monetary Econ 54:2139–2179

Ljungqvist L, Sargent TJ (2008a) Two questions about European unemployment. Econometrica 76:1–29

Ljungqvist L, Sargent TJ (2008b) Taxes, benefits, and careers: complete versus incomplete markets. J Monetary Econ 55:98–125

Manning A (1987) An integration of trade unions models in a sequential bargaining framework. Econ J 97:121–139

Marimon R, Zilibotti F (2000) Employment and distributional effects of restricting working time. Eur Econ Rev 44:1291–1326

McDaniel C (2007) Average tax rates on consumption, investment, labor and capital in the OECD 1950–2003. Arizona State University working paper

Merkl C, Wesselbaum D (2011) Extensive versus intensive margin in Germany and the United States: any differences? Appl Econ Lett 18:805–808

Ngai LR, Pissarides CA (2008) Trends in hours and economic growth. Rev Econ Dyn 11:239–256

OECD (2010a) Labour productivity growth, OECD Productivity Statistics (database)

OECD (2010b) Labour market statistics: labour force statistics by sex and age, Employment and Labour Market Statistics (database)

Ohanian L, Raffo A, Rogerson R (2008) Long-term changes in labor supply and tax: evidence from OECD countries, 1956–2004. J Monetary Econ 55:1353–1362

Olovsson C (2009) Why do Europeans work so little? Int Econ Rev 50:39–61

Prescott EC (2002) Prosperity and depression. Am Econ Rev 92:1–15

Prescott EC (2004) Why do Americans work so much more than Europeans? Fed Reserve Bank Minneap Q Rev 28:2–13

Prescott EC (2006) Nobel lecture: the transformation of macroeconomic policy and research. J Polit Econ 114:203–235

Rocheteau G (2002) Working time regulation in a search economy with worker moral hazard. J Public Econ 84:387–425

Rogerson R (2006) Understanding differences in hours worked. Rev Econ Dyn 9:365–409

Rogerson R (2008) Structural transformation and the deterioration of European labor market outcomes. J Polit Econ 116:235–259

Rogerson R, Wallenius J (2009) Micro and macro elasticities in a life cycle model with taxes. J Econ Theory 144:2277–2292

Saez E (2002) Optimal income transfer programs: intensive versus extensive labor supply responses. Quart J Econ 117:1039–1073

Shimer R (2005) The cyclical behavior of equilibrium unemployment and vacancies. Am Econ Rev 95:25–49

Shimer R (2008) Comment on: new Keynesian perspectives on labor market dynamics by Tommy Sveen and Lutz Weinke. J Monetary Econ 55:931–935

Siebert H (1997) Labor market rigidities: at the root of unemployment in Europe’. J Econ Perspect 11:37–54

Walque G, Pierrard O, Sneeseens H, Wouters R (2009) Sequential bargaining in a neo-classical model with frictional unemployment and staggered negotiations. Annu Econ Stat 95(06):223–250

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We thank three anonymous referees for valuable comments and suggestion. Earlier versions have benefited from discussions with Gary Hansen and Roger Farmer and comments made by Shun-Fa Lee and Charles Leung.

## 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 \(e\).

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

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

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

### Appendix B: This appendix derives the effects of labor taxes

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

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.

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).^{Footnote 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

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 \(h\),

The goods market clearing condition is now

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

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

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

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

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

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

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.

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

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.

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

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 \(\Gamma _e =MRS_e \cdot \bar{g}>0\), \(\Gamma _h =MRS_h \cdot \bar{g}-( {1-\tau })AP_h >0\), \(\Gamma _\tau =AP>0\),

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

Differentiating (25) and (29) gives

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.

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Chen, BL., Lai, CF. Relative effects of labor taxes on employment and working hours: role of mechanisms shaping working hours.
*J Econ* **117**, 49–84 (2016). https://doi.org/10.1007/s00712-015-0440-x

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DOI: https://doi.org/10.1007/s00712-015-0440-x