3 Relative Effects of Labor Taxes on Working Hours and Employment: Role of
3.3 Effects of Labor Taxes on Employment and Working Hours
In (16), the marginal cost of a vacant job is constant at ϕ while the marginal benefit is decreasing in employment e. Thus, with given hours, there is a unique employment level in (16). Moreover, a longer work hour h increases the output and thus the marginal benefit, but a longer work hour also increases the labor cost and reduces the marginal benefit. As the effect on the labor cost dominates, the marginal benefit is decreasing in h (See Appendix A). Thus, (16) is downward-sloping in the h - e space, referred to as the free-entry (FE) curve in Figure 2, indicating a trade-off between e and h from the firm perspective. The net marginal benefit of vacancies is decreasing in τ as a higher labor tax increases the bargained wage and pushes up the labor cost.
In addition to the free-entry condition, we need the worker’s supply of hours in order to solve employment and hours in a steady state. Three kinds of mechanisms are studied in the next section.
3.3 Effects of Labor Taxes on Employment and Working Hours
In this section, we explore the relative effects of a higher labor tax on employment and hours. We start with the general mechanism when the worker’s supply of hours is negotiated by a matched job-worker pair. We then envisage the mechanism when the
32 This is a property in search and match models; see, for example, Cheron and Langot (2004).
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household chooses the supply of hours per worker which is the special case when the worker has a one-hundred percent share in the labor hour bargaining game. Finally, we study the mechanism in the other extreme when the supply of hours per worker is regulated effectively.
As the curvature of the leisure utility affects relative effects, to ease analysis we use the following form in both employment and working hours.
3.3.1 Hours Bargained by Job-Worker Pairs
When a worker’s supply of hours is governed by a matched job-worker pair, it is negotiated in a cooperative bargaining game. Fang and Rogerson (2009) and others used bargaining to determine hours.33 In this mechanism, a matched job-worker pair negotiates not only the wage but also hours. Although they bargain wage and hours simultaneously, we assume they could have different bargaining power across wage and hours.34 While the wage bargaining gives the wage rate in (15), the hour bargaining leads to
33 See also Rocheteau (2002) and Shimer (2008), among others.
34 Since the government has different regulations on price and quantity, workers may have different bargaining power across wage and hours.
where e( )t c( , ) 1t t
( )
t l( , ) ( ),t t tdenote the laborer’s power in the hour bargaining in (18a) as βh, which may or may not be the laborer’s power β in the wage bargaining in (15).
In (18a), the left-hand side is the household’s utility from supplying a marginal unit of hours per worker, and the right-hand side is the firm’s gain from recruiting the marginal unit of hours. Condition (18a) will be replaced by other conditions later when a worker’s supply of hours is determined by other mechanisms. In steady state, by using (13a)-(13c), equation (18a) becomes
Substituting the bargained wage rate in (15) into (18b) and rearranging terms yields
( )
( )( )
the wage bargaining. Then, (19a) is( )
( ) ( ) ( )
( ,e h; ) MRS c e h( ( , ),1 eg h( )) g h( ) 1 MP h( ) 0.
+ + + ′
Γ τ = − ⋅ − −τ = (19b)
Then, the supply of hours is determined by the marginal cost of hours MRS g′⋅ equal the after-tax marginal gain of hours. Eq. (19b) is referred to as the bargained hour (BH) curve. In (19b), given employment, the net marginal cost of hours is increasing in hours (∂Γ/∂h>0) as longer hours increase the marginal cost but decrease the marginal
35 To obtain the expression, we follow Fang and Rogerson (2009, p. 1158) and consider the case with finite family members. Let Et denote the number of members that are employed in period t. In the bargaining over hours, we take the derivatives of U(Et)-U(Et-1) with respect to the current hours of the Eth worker, taking as given the hours of all other (Et-1) workers in the family. Thus, working hours of the Eth worker only enter into the current period utility in U(Et) and do not enter into U(Et-1). Therefore, if the Eth worker works one more hour, consumption is increased by the unit of (1-τ)wt while leisure is decreased by g h′( ),t which would change the value of U(Et) by uc(1−τ)wt−u g hl′( ).t
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gain. Moreover, the net marginal cost of hours is increasing in employment (∂Γ/∂e>0) because higher employment augments the marginal cost, as resulted from higher pooled consumption and lower pooled leisure in the large household which increases the marginal utility of leisure relative to the marginal utility of consumption (See Appendix A). Therefore, the BH curve is downward-sloping in the h - e space, indicating an underlying trade-off between e and h from the household perspective.
The BH curve (19b) and the FE curve (16) together characterize the allocation of e and h in steady state. Although both the FE and the BH curves are downward-slopping in the h - e space, we have shown that the BH curve is always flatter than the FE curve at any point of intersection, implying that there is at most one intersection. See E0 in Figure 2. Once we determine the unique pair of hours (h0) and employment (e0) in a steady state, we can use other conditions to solve for other variables. In particular, the product of employment and hours per worker gives working hours per person (e0h0).
We now analyze the effect of a higher labor tax on hours in both curves, given employment.
A higher labor tax shifts the FE curve downward, because it increases the bargained wage and decreases a firm’s marginal benefit of vacant jobs. In optimum, given employment, hours per worker need to lower in order to increase the marginal
benefit of vacant jobs thereby shifting the FE curve downward. Moreover, a higher labor tax also shifts the BH curve downward, since it decreases a household’s after-tax marginal gain of hours. In optimum, given employment, the household needs to decrease hours per worker in order to decrease the net marginal cost of hours per worker.
First, note that a linear utility in leisure helps to pin down the relative shift. With ε=0 and thus g h( )=gh, the FE curve is shifted downward more than the BH curve, as MRSh·g-(1-τ)f″>MRSh·g>0. The relative shifts are similar when g h( ) is concave. See E2 in Figure 2. It follows that a higher labor income tax reduces both hours and employment.
Intuitively, given employment, a higher labor tax increases the hourly bargained wage and lowers the firm’s demand for hours per worker. Similarly, given employment, a higher labor tax reduces the net hourly bargained wage and lowers the supply of hours per worker. Now, the household cannot flexibly change but needs to negotiate hours per worker with the matched firm. Then, the household is not able to reduce the supply of hours sufficiently even if the leisure utility is linear in hours. As a result, a higher labor tax reduces both hours per worker and employment, like those in Fang and Rogerson (2009).
When βh≠β, different laborer’s shares in the hour bargaining βh affect the relative effects a higher labor tax has on employment and hours. To see this, we carry out the exercise of increasing βh while holding the laborer’s share in the wage bargaining fixed at β. While such a change does not influence the FE curve, the effect on the BH curve is affected as follows.
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Obviously, as the value of βh is increased from β, the value in the denominator in (20b) is reduced and the value in the numerator in (20b) is increased. Thus, with a larger laborer’s share in the hour bargaining, a higher labor tax unambiguously shifts the BH curve downward more. See E4 in Figure 2. Thus, with a larger laborer’s share in the hour bargaining, the negative effect of a higher labor tax on employment is diminished while the negative effect on hours is enhanced. Intuitively, with a larger laborer’s share in the hour bargaining, the household is able to reduce more of the supply of hours per worker. To summarize our finding,
Proposition 1. Let the worker’s supply of hours be determined by a cooperative bargaining game. Then,
(i) a higher labor tax reduces both hours per worker and employment;
(ii) as the laborer’s share in the hour bargaining increases, the negative effect on employment is smaller and the negative effect on hours per worker is larger.
3.3.2 Hours Determined by Households
When the supply of hours is determined exclusively by the household, the household takes wage as given and trades off between consumption and leisure. Prescott (2004) and many other studies in the neoclassical growth model use the same mechanism to determine the supply of hours.37 By substituting (1) and (2) into (3) and taking derivatives with respect to ht, we obtain
( )
In the left-hand side of (21a) is the marginal rate of substitution between leisure
37 See also Rogerson (2008) and Azariadis et al. (2012), among others.
and consumption (hereafter, MRS) which is the marginal cost of hours. The right-hand side is the after-tax wage rate which is the marginal gain of hours. It is clear that (21a) is a special case of (18b) in subsection 3.1 which emerges when βh=1. As βh is the largest, the negative effect of labor taxes on employment is minimum and the negative effect on hours per worker is maximum. By using the bargained wage (15) and consumption (12), (21a) is rewritten as a form of a zero net marginal cost of hours as in (19b).38
( )
( ) ( ) ( )
(e h, ; ) MRS c e h( ( , ),1 geh g) 1 AP h( ) 0.
+ + +
Γ τ ≡ − − −τ = (21b)
which is referred to as the flexible hours (FH) curve.
Like the BH curve in (19b), here the FH curve is also downward-sloping in the h - e space.39 The FE curve in (16) and the FH curve in (21b) together determine a unique pair of e and h in a steady state. See E0 in Figure 3. As in subsection 3.1, a higher labor tax shifts both the FE and the FH curves downward (See Appendix B). In particular, when ε=0, given employment, both curves decrease hours at the same level and as a result, a higher labor tax reduces only hours without affecting employment.
(
1 1)
1(
1) (
1( ))
( ) 0,The result can be understood. The Frisch hour elasticity is 1/ε>0. In the case when ε=0, the Frisch hour elasticity is the infinite. Now, the household can flexibly choose hours per worker. When the Frisch hour elasticity is infinite, given employment, the
38 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).
39 Note the difference between the condition MRS g⋅ = −′ (1 τ)MP h( ) in (19b) and the condition (1 ) ( )
MRS g⋅ = −′ τ AP h in (21b). Under a utility of leisure linear in hours, g h′( )=g.
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household would reduce the supply of hours per worker exactly to the level the firm demands. As a result, employment is not changed and all the effects are on hours.
Conversely, when ε>0, the Frisch hour elasticity is less than infinite. Then, given employment, the household will not reduce the supply of hours per worker to exactly the level the firm demands. Thus, a higher labor tax also lowers hours in a steady state.
Nevertheless, when ε is smaller, the Frisch hour elasticity is larger. The household reduces more of the supply of hours and thus the negative effect on hours is larger and the negative effect on employment is smaller. To summarize results, we obtain
Proposition 2. Let the supply of hours be determined by the household via the leisure-consumption tradeoff. Then,
(i) under a linear utility of leisure in hours, a higher labor tax reduces only hours with no impact on employment;
(ii) with a strictly convex utility of leisure in hours, the flatter the utility in hours, the smaller the negative effect of a higher labor tax on employment.
3.3.3 Hours Regulated by Authorities
When working hours are regulated by the union and the government, a worker’s supply of hours is fixed. Marimon and Zilibotti (2000) and others used regulation to determine hours.40 Following Marimon and Zilibotti (2000), we assume that there is a regulation of maximum work time which is reduced from the level of bargained hours in subsection 3.1 and the regulation is effectively enforced. Suppose that hours are initially at the level of the market equilibrium. In this case, ht =h is referred to as the regulated hour (RH) curve. In steady state, while the horizontal RH curve h=h determines
40 See Calmfors (1985), Hoel and Vale (1986) and Marimon and Zilibotti (2000), among others.
hours, the downward-slopping FE curve (16) determines employment e. In Appendix C, we have shown that there exists a unique steady state. With h=h in Figure 4, then E0
is the steady state and e=e0.
To analyze the effect of a higher labor income tax, it is obvious that the RH curve is not affected while the FE curve is shifted downward in the h - e space. As a result, hours are unchanged but the employment is reduced. See E1 in Figure 4. Intuitively, a higher labor tax drives up the bargained wage and thus depresses the value of an unfilled job. As hours are fixed, firms will respond to a higher labor income tax by creating less vacant jobs so employment is reduced in steady state. Indeed, this model matches exactly the standard Mortensen-Pissarides search-and-match model. Thus, when hours are regulated, the labor tax only reduces employment without affecting hours.41
In Ljungqvist and Sargent (2007a, 2008b), the labor supply is represented by jobs with some fixed hours and thus the labor supply is adjusted only by employment. Thus, when the labor income tax is increased, firms respond only by adjusting job creation and households only by changing job search. As a result, the effect of a higher labor tax is entirely on employment.
To recapitulate our results in this section, we find that the relative effects of a higher labor income tax on employment vis-à-vis depends on the mechanism determining the supply of hours per worker. The effect on employment changes from a small negative effect when the worker’s supply of hours is determined exclusively by the household, to a partial negative effect when the worker’s supply of hours is governed by a bargaining game, and finally to a full negative effect when the worker’s
41 It is worth noting that when regulated hours are reduced, say from h to h5 in Figure 4, with other things being equal, the steady state changes from E0 to E5. Thus, a working time reducing policy can increase employment that achieves the goal “work less, work all.”
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supply of hours is effectively regulated. Our analysis above abstracts from the capital adjustment. We have shown that these results are robust if capital is adjustable (See Appendix E).