∗
Abstract
Labour supply in the market for small jobs in Germany is strongly influenced by nonlinearities in the tax schedule - even for individuals to whom this tax schedule does not apply. Tax regimes not only distort labour supply incentives, but also labour demand incentives. We present a simple equilibrium job search model which implies incentives for firms to select contracts as a function of their expected attractiveness to workers. Every worker’s labour supply thereby becomes a function of preferences among the population of job-seekers. We apply our model to the market for small jobs in Germany to identify the unintended effects of the nonlinearities in the tax schedule on unaffected worker. We also plan to use our model to investigate a reform which extended the tax breaks to workers taking on low-paying second jobs.
Keywords : Job Search, Peer Effects, Labour Supply Elasticities
∗
We are grateful for funding via the DFG ORA grant “Social Security Contributions”. Subject to the usual disclaimer we would like to thank seminar participants in Mannheim (EEA), Muenster
(VfS) and at DIW Berlin.

It is largely recognized that workers face important constraints in choosing labour supply. Employed workers cannot easily change hours, and unemployed workers do not know about all jobs on offer. It has been recognized that this importantly influences estimates of labour supply elasticities and policy implications (e.g.
Chetty et al. (2011)).
Job search models provide a way of explaining some of these constraints without resorting to ad-hoc restrictions on the set of labour supply choices available to individuals: Following Burdett and Mortensen (1998) wage offers are endogenously generated by firms. As a result, labour demand is a function of average workers’ preferences. Firms offer contracts of (fixed) hours and wages that maximize their profits taking into account the likelihood of acceptance by job-seeking workers. This results in an additional “population” labour supply effect on individuals’ labour supply: Because firms package their hours-wage bundles according to average preferences of job-seeking workers, the restricted set of any particular individual worker will depend on other workers’ preferences.
This has some important implications. First, less dominant groups in the labour market might face working hours constraints when their hours preferences differ from average preferences.
Population labour supply effects might, for example, play a role in reinforcing low levels of labour supply of women. Second, population labour supply effects induce differences between individual- and market-level labour supply elasticities. Chetty et al. (2011), therefore, adjust their bunching estimator of individual elasticity by subtracting these aggregate effects. It also contributes to explain differences in micro and macro estimates. Third, policy needs to take into account that tax incentives for one group of workers might have unintended effects to other groups of workers.
We find that a disposition in the German tax law that provided for tax reductions for specific workers affected the hours choices of workers who were not directly affected by the measure, in line with similar evidence presented by Chetty et al. (2011). We interpret this as a population labour supply effect, which we
2

identify in a simple equilibrium job search model. A reform in the tax incentives for the specific subgroup of workers (with second jobs) will allow us to validate this labour supply reaction. Besides evaluating this reform, we can estimate labour supply reactions to counterfactual changes in taxation of marginal employment taking into account this labour demand channel.
In section (2) we review other relevant recent contributions. Section (3) briefly discusses the market for marginal employment in Germany. Section (4) presents our model, its identification and simulation results. Section (5) discusses estimation which is work in progress. In section (6) we present our data and some descriptive statistics supporting our basic modelling choices. Section (7) concludes.
We first review the literature modelling constrained labour supply choices in section
(2.1), then turn to job search models which can be viewed as expressing a specific type of constraint. We believe that there are interesting dynamics between labour supply and demand in our model, thus section (2.2) focuses mainly on equilibrium job search models. Whilst population labour supply effects have not been studied in job search models explicitly, the phenomenon is not new and part (2.3) reviews previous literature in this area.
It is well-known that workers are constrained in their choice of work hours, and that desired and actual hours deviate substantially (e.g.
Stewart and Swaffield
(1997)). There are two interrelated causes: First, firms face organizational costs when adapting workers’ weekly hours. Hence, as is standard in the wage posting literature, we here assume that jobs come with fixed weekly hours attached, and changing them elicits adjustment costs. These adjustment costs are so high that many researchers specify labour supply as binary or ternary choices (no employment, part-time at 20h or full-time at 40h).
3

Second, there are informational frictions in the labour market. Workers are confronted with a limited number of job offers in any period, and firms receive a finite number of applications when posting a job offer. Finding a job hence requires search costs. A large strand of literature explicitly incorporates such constraints in labour supply models, we focus on the equiilbrium search literature in section (2.2) below.
Similar to the job search literature (see below), Van Soest et al. (1990) augment a Hausman (1980) type neoclassical labor supply model with hours constraints by letting individuals choose between a finite set of wage-hours packages. Similar approaches are, for example, followed by Tummers and Woittiez (1991), Dickens and
Lundberg (1993) and Bloemen (2000). However, separately identifying preferences as well as the job offer distribution requires strong assumptions (Beffy et al., 2014).
Therefore, Beffy et al. (2014) exploit non-convexities in the budget sets and Bloemen
(2008) makes use of stated desired hours of work. In a different approach Peichl and
Siegloch (2012) combines structural labour demand and supply models, others estimate computable general equilibrium (CGE) models (Peichl (2009); Davies (2009)).
In Chetty et al. (2011) firms commit in advance to a production technology which requires employees to work a fixed amount of hours. Cogan (1981) was the first to include fixed job entry costs in a labour supply model and found that these are particularly crucial for determining the labour supply behaviour of married women.
The job search frameworks naturally generate constraints as a result of informational deficiencies - individuals are only aware of specific job offers. The restriction on the set of available jobs is modelled as a stochastic process.
In equilibrium job search models these offers are endogenously generated by profit maximizing firms that typically choose wages taking into account the benefits of higher wages in attracting more workers: First, higher wages may induce certain workers with higher reservation wages to pick up work, thereby increasing firms’ workforce and turnover as in Eckstein and Wolpin (1990) who extend a model by
Albrecht and Axell (1984). Wage dispersion is here explained by different reservation wages - however, the fit of the model is poor, suggesting that different reservation
4

wages may not be the most important mechanism explaining wage dispersion. Second, higher wages will enduce workers to move to a firm from other firms in models which allow for on-the-job search. Both sources of attraction may be combined as in Bontemps et al. (1999) and Burdett and Mortensen (1998).
The population labour supply effect has been described and documented by Chetty et al. (2011) who analyze the top tax cutoff in Denmark where the net-of-tax rate falls by 30% and which applies to the sum of individual wage and non-wage earnings.
As 60% of the population do not have any non-wage income, they share the same threshold. As predicted by their model, workers over-proportionally locate at the threshold. Importantly, however, the earnings distribution of the remaining population also exhibits bunching at the threshold - a population labour supply effect.
Best (2014) documents the same phenomenon in Pakistan and shows that this effect mitigates the impact of adjustment costs and information frictions on workers’ responsiveness of taxes.
Effects of others’ labour supply might also result not from frictions via labour demand, but from the need of coordination between workers (Weiss (1996), Rogerson
(2011)) or social interactions (Woittiez and Kapteyn (1998), Weinberg et al. (2004)).
Ignoring such interdependent behaviour can result in biased estimates of labour supply responses to tax changes (Blomquist, 1993).
This study focuses on low-paid marginal employment characterized by regular low earnings 1 . Since certain jobs in this sector are subject to special tax and social security treatment, it is important to clarify the institutional setting. In 2003, a major reform to the regulations of marginal employment was introduced.
This version of the paper focuses on the period before 2003.
1
Marginal employment can be separated into two types: low-paid marginal employment and short-term employment. Whereas low-paid marginal employment covers all those employments that yield regular earnings below a certain threshold, short-term employment comprises all jobs that last for less than a certain number of days per year or calender year.
5

Marginal employment contracts qualify for special tax treatment if yearly earnings do not exceed a threshold which increased from 3900 e before to 4800 e after 2003
2
. We designate these employment relations as minijobs. The most important feature is that minijob earnings are not subject to SSC for employees.
As employees’ SSC amount to approximately 20 % of earnings, this constitutes significant savings at the expense of no social security. This generated a notch in the budget set because earnings above the threshold are entirely subject to usual
SSC. Employers’ SSC were 10% (health) and 12% (pension) of minijob earnings before 2003.Minijobs are also subject to a specific treatment of income taxation.
Before 2003, minijob earnings are either taxed by a 20 % flat rate or by including the earnings in the yearly tax return. If there is no other income, the latter option results in no taxes because of the general income tax allowance.
Important for this study, before 2003, a side job in addition to a main job subject to SSC could not qualify as a minijob and was, therefore, not exempted from SSC.
Holding two minijobs simultaneously was possible, though.
We estimate the importance of the effect of aggregate preferences for labour supply.
To do this, we exploit a peculiarity in the German tax system relating to earnings from second jobs as well as a reform to it.
In this, we identify the effect of aggregate preferences on labour supply by using information on hours choices of German workers who were not subject to discontinuous treatment by the tax authorities, but nevertheless located over-proportionally at points in the earnings distribution where a large fraction of the population was subject to discontinuous treatment. More precisely, earnings from first jobs are exempted from SSC given they do not exceed a monthly earnings threshold. Prior to 2003, second jobs could not benefit from this exemption from SSC. As opposed
2
The thresholds are actually defined in terms of monthly earnings. These monthly thresholds can, however, be exceeded for two months within a year as long as yearly earnings are below the yearly threshold.
6

to workers with minijobs as first jobs, their budget set, therefore, did not exhibit a notch. However, we observe that a large number of individuals locate at the earnings threshold relevant for individuals with only one job even if the earnings in question are derived from a second job. We use this to identify labour demand constraints on hours choices generated by the effect of other workers’ preferences.
Population labour supply effects are labour demand effects on labour supply. To model these effects, an equilibrium search model with wage posting is the most appropriate. By contrast to simple search models, equilibrium models endogenize the job offer distribution such that it reflects the preference distribution of potential workers. If earnings were totally flexible (in a simple bargaining framework), we should expect no bunching at thresholds that do not apply to individuals personally.
Our assumption, rather, is that firms make take-it-or-leave-it offers along the lines of Burdett and Mortensen (1998) and Bontemps et al. (1999). As we are mainly interested in the responses to the special German tax treatment of minijobs, we only model the market of jobs with less than full-time hours and assume that these jobs form a separate labour market. Section 6.1 shows that the industry sector distributions of marginal and full-time employments are very different and that the former are mainly concentrated in household and economic services.
The labour market is composed of a continuum of workers and firms. Some workers have another - typically full-time - job but are nonetheless active in the market and are thus seeking a second job. We refer to these as “types ” workers, of which there are n s .
These workers are not qualified for the tax exemption, which is available for workers seeking a first job, i.e. “typef ” workers. Thus the budget set of type f workers exhibits a notch at z = z
∗ while it is smooth for types workers.
The labour supply incentives of typef workers imply that certain workers have a particular incentive of remaining below the threshold of z
∗
. Note that a precise quantification of the incentive is not feasible using German administrative data, since spousal earnings are not available. On an individual level we cannot
7

differentiate between an individual with a rich spouse who faces a high marginal tax rate from an individual for who disincentives will be much less important.
Instead of attempting to precisely determine the set of different attitudes towards jobs above and below the threshold, we simply model two populations of typef job seekers: one group accepts all jobs and prefers higher earnings.
A second fraction of typef employees, θn f , do not accept any jobs with earnings above the threshold ( z > z
∗
). The parameter θ describes the fraction of workers who only enter the market due to the tax exemption below the threshold. In the following, this group of workers is indexed by f 2. The remaining n f worker of type f , who do not account for the notch, is indexed by f 1. In sum there are thus n f (1 + θ ) typef workers in the market. This group generates the incentive for firms to offer earnings below or at the threshold. A precise identification of θ is thus essential to pin down the labour demand constraints on hours choices generated by the effect of other workers’ preferences.
Workers search when unemployed and when employed with uniform search intensity by drawing from a known job offer distribution F ( .
). The exogenous job offer arrival rate ( λ ) is allowed to differ between type f and type s workers but is independent from employment status. Workers lose their job at an exogenous rate δ . Workers seek to maximize the expected steady-state future utility. Utility increases in earnings. This implies that either workers do not care about hours of work or that hours of work are fixed and homogeneous - in the following section we relax this assumption. The strategy of worker type s and f 1 is to accept every job with earnings exceeding homogeneous 3 reservation earnings z r . Workers of type f 2 accept all jobs with earnings in the interval [ z r
, z
∗
].
Firms seek to maximize profits π = [ p − z ] l ( z ) with p being turnover per worker and l ( z ) the size of a firm’s labour force which offers jobs with earnings z (we assume that within firms job offers are identical). This already implies that in equilibrium no firm will offer earnings smaller than z r . In the following, we use our information about worker mobility to establish the firm size distribution l ( z ),
3 For the moment, we do not allow for heterogeneity in reservation earnings and productivity of firms.
8

critical in determining firms’ optimal choice of z .
4.3.1
Worker mobility
In equilibrium, the flows of workers of each type j who move from unemployment to employment and vice versa must balance.
δ ( n j − u j
) = λ j u j u j n j
=
1 + κ j for j ∈ ( s, f 1)
(1) u j denotes the number of unemployed type j workers and κ j
=
λ j
δ j
. As type f 2 workers do not accept jobs with earnings higher than z
∗
, the flow between employment and unemployment becomes:
δ ( θn f
− u f 2
) = λ f u f 2
F ( z
∗
) u f 2
=
θn f
1 + κ f F ( z ∗ )
The number of employed individuals, thus, are: n j − u n f 2
− u f 2 j
= n j κ j for j ∈ ( s, f 1)
1 + κ j n f 2
κ f
F ( z
∗
)
=
1 + κ f F ( z ∗ )
(2)
Similarly, in steady-state the flow of workers of each type entering jobs with earnings no greater than z must equal the flow of workers of the same type leaving that group. The left-hand side of equation (3) consists of employees of type j ∈ ( s, f 1) moving from a job with value no greater than z to unemployment ( δG j
( z )( n j − u j
)) or to a better job ( λ (1 − F ( z )) G j ( z )( n j − u j )) with G j ( .
) denoting the cumulative density function of realized job values for workers of type j . The right-hand side represents the flow into jobs with value no greater than z consisting of unemployed individuals receiving a job offer with value no greater than z .
[ δ + λ j
(1 − F ( z ))] G j
( z )( n j
− u j
) = λ j
F ( z ) u j for j ∈ ( s, f 1) (3)
9

κ j F ( z ) u j
G j
( z ) =
(1 + κ j (1 − F ( z )))( n j − u j )
F ( z )
=
1 + κ j (1 − F ( z )) for j ∈ ( s, f 1)
(4)
Equations (5) and (6) are the corresponding relationships of type f 2 workers for z ≤ z
∗
.
[ δ + λ f
( F ( z
∗
) − F ( z ))] G f 2
( z )( n f 2 − u f 2
) = λ f
F ( z ) u f 2 for z ≤ z
∗
(5)
κ j F ( z ) u f 2
G f 2
( z ) =
(1 + κ f ( F ( z ∗ ) − F ( z )))( n f 2 − u f 2 )
F ( z )
=
(1 + κ f ( F ( z ∗ ) − F ( z ))) F ( z ∗ ) for z ≤ z
∗
4.3.2
Firm size
(6)
In the steady-state the amount of workers of type j which is employed at a firm offering jobs with earnings z can be expressed by equation (7) (Burdett and Mortensen,
1998).
l j
( z ) = lim
→ 0
( G j
( z ) − G j
( z − ))( n j
F ( z ) − F ( z − )
− u j
) for j ∈ ( s, f 1 , f 2) (7)
For workers of type j ∈ { s, f 1 } we can then give firm size as a function of offered wage. Using (4), (7) and (1) – resp. (6), (7), (2) and for type f 2 workers
– appendix (E) shows that firm size is increasing both below and above z
∗
, but with a discontinuity at z
∗
, since typef 2 workers do not accept these jobs above this threshold value: l ( z ) = l s
( z ) + l f 1
( z ) + l f 2
( z )
=







(1+ κ s n s
κ s
(1 − F ( z )))(1+ κ s
(1 − F ( z − )))






(1+ κ f
( F ( z
θn f
κ f
∗
) − F ( z )))(1+ κ f
( F ( z
∗
) − F ( z − )))
(1+ κ s n s
κ s
(1 − F ( z )))(1+ κ s
(1 − F ( z − )))
+
+
(1+ κ f n f
κ f
(1 − F ( z )))(1+ κ f
(1 − F ( z − )))
(1+ κ f n f
κ f
(1 − F ( z )))(1+ κ f
(1 − F ( z − )))
+
∀ z ≤ z
∀ z > z
∗
∗ (8)
10

4.3.3
Equilibrium wage offer distribution
As firms are identical, it must hold that in equilibrium profits are equal for different z . In the following this is used to analyze the equilibrium wage offer distribution.
The mechanism follows Burdett and Mortensen (1998): Firms offering jobs with low earnings achieve higher profits per worker but attract fewer workers than firms offering jobs with higher earnings. However, when earnings exceed z
∗
, θn j individuals drop out. The endogenous offer distribution might, therefore, include a mass point because profits are not necessarily increased by offering slightly more than z
∗
.
Proposition (I) If we observe offers above z
∗
, there must be a mass point of job offers at z
∗
. The wage offer distribution above z
∗ is continuous up to some ¯ .
Proof: See appendix (A).
Given proposition (I), the data implies that there is a mass point at z = z
∗
(i.e. that f ( z
∗
) > 0). A mass point in our setting implies that any job offers with wages just below the mass point (at z
∗ − ε ) will earn less profits, since margins per worker are only slightly higher, but firm size will be discontinuously lower since there is a mass of firms (offering z
∗
) that can poach a worker employed at wage z
∗ − ε .
Proposition (II) If there is a mass point at z
∗
, there will be a gap in the offer distribution just below the threshold.
Proof: See appendix (B)
Given the lack of offers just below the mass point, i.e.
a gap in the offer distribution, the question arises whether any wages lower than z
∗ are offered in equilibrium. Is there a wage offer z
00
< z
∗ − consistent with the equal profit condition?
Proposition (III) There may or may not exist wage offers below the threshold z
∗ in equilibrium.
The wage offer distribution will then be continuous between z ∈ [ z, z
00
] for z
00
< z
∗
.
11

Fig. 1: Hours distribution - sidejob less than 800 euros
Fig. 2: Hours distribution - main job less than threshold
Proof: See appendix (C)
The equilibrium offer distribution in the interval z ∈ [ z, z
00
] is determined by
π ( z ) = π ( z ) with z being the lowest earnings offered in the market, see appendix (C).
The model outlined in the previous section assumes that workers always prefer jobs with higher total (net) earnings, neglecting the fact that working hours may differ.
Note that the minijob-threshold is based on monthly earnings, not hourly wages.
The data shows that there is some variation in hours for the jobs we consider (figure
1 and 2), this section thus considers how an equilibrium model can include workers and firm behaviour in the face of such variation. We thus assume that hours vary across firms and workers care about hours and earnings. We assume, however, that hours of work is not a choice variable of firms but predetermined by, for example, technology.
12

Including hours in the model is not only attractive in theory: The case of homogeneous hours demonstrates how we can rationalize a mass point. In fact, following Proposition (1) if there are any observed earnings above the threshold earnings level we require a mass point.
Since we observe earnings above the threshold value, we followed this route in the previous section. However, any mass point also requires a gap in the offer distribution below this mass point. This is less consistent with the data 4 . We can now allow both for the existence of earnings offers above the threshold and a lack of a gap below the threshold if we include variation in working hours in our model.
The attractiveness of a wage offer to workers depends crucially on how many wage offers are viewed as superior and could poach a worker away from a firm.
In order to determine this measure, firms need to determine the implied utility distribution of offers. The same strategic arguments as in the case of homogeneous hours will prevent firms from locating just below a mass point in this utility distribution.
Including variation in hours is then a non-trivial complication since there are now several thresholds, corresponding to the mini-job threshold with different working hours. We demonstrate this in a version of the model with two hours sectors in appendix (D) and note that the model becomes intractable for many different hours offers. We here focus on the case in which hours vary continuously, and every job has a different hours requirement. We plan to estimate this model.
4.4.1
Continuous variation in working hours
Consider a scenario where every firm has a different requirement of weekly hours.
Equal wage offers of different firms will correspond to different utility levels 5 . This will also mean that every firm will face a different wage threshold w
∗ corresponding
4
We could rationalize this lack of gap below the threshold via measurement error also, but the distribution appears to be increasing continuously.
5 This is true unless workers’ marginal utility of leisure is zero, such that the utility function is independent of hours. This case naturally generates the same firm strategies as the case of homogeneous hours and is thus excluded here.
13

to the wage level that generates monthly earnings of z
∗
.
As the amount of weekly hours is not a choice variable, the offered utility at z
∗ will similarly differ between firms.
The reasoning for a mass point in the offer distribution given homogeneous hours can, therefore, not be applied to a case with a continuous variation of hours (although there might still be bunching at the earnings threshold).
Proposition (IV) : There will be no mass point in the utility offer distribution.
Sketch of Proof: If there was a mass point in the utility distribution, a firm could increase profits by slightly increasing the offered wage (and therefore utility).
The loss in margin would be small while the firm size would increase discontinuously.
The best response of any firm will be to take the utility distribution of employed workers (that are poachable) as given, responding only to the individual threshold as in the homogeneous hours case.
In order to study worker flows, we need to make assumptions about preferences about jobs with different wages and hours. We assume utility v be generated by v = z
α
(24 − h )
(1 − α )
, with 24 − h denoting leisure and α determining the elasticity of v with respect to earnings and leisure.
In the case of homogeneous hours we had established the size of the firm as a function of the earnings offer, as a result of the flows of workers into these jobs.
Workers’ objective function is now to maximize utility, thus the flows now depend on the utility of an offered job. Equation (7) is thus valid in terms of utility: l j
( v ) = lim
→ 0
( G j ( v ) − G j ( v − ))( n j
F ( v ) − F ( v − )
− u j ) for j ∈ ( s, f 1 , f 2) (9)
For an individual firm with hours h , the strategic choice is then whether to offer utility v ( z, h ) or utility v ( z
0
, h ) where strictly speaking the choice variable is the wage rate, but for fixed hours different wage rates obviously directly translate to different earnings. We have a wage threshold function w
∗
( h ), where w
∗
( h ) h ≡ z
∗ ∀ h .
The attractiveness of firms’ offers will thus depend on whether or not a
14

given wage offer corresponds to an earnings level that lies above or below the earnings threshold z
∗
- since a discontinuous amount of workers ( j = f 2) do not accept offers above. Given proposition (IV) this translates into two more propositions:
Proposition (V) : There will be a gap in the distribution of earnings above z
∗
.
Sketch of Proof : Firms’ profit drops discontinuously between offering v ( z
∗
, h ) and v ( z
∗
+ ε, h ), so that no profit-maximizing firm will offer z
∗
( h ) + ε .
Proposition (VI) : There will be no gap in the distribution of earnings below z
∗ although there may or may not be a mass point at z
∗
.
The structure of the proofs (not provided here formally) follows the strategy established above of comparing profits at different earnings levels. The intuition for the gap in earnings is the same as before. However, there are two key differences, both directly resulting from proposition (IV) establishing that there is no mass point in the utility distribution: First, there is no longer a rationale for a gap to the left of the threshold z
∗
. It is no longer the case that firms offering z
∗ − ε will attract discontinuously less workers than those offering z
∗
, since l ( v ( z
∗ − ε, h ) , h ) is only marginally smaller than l ( v ( z
∗
, h ) , h ) but margins are higher. Similarly, offers above z
∗ no longer necessarily imply the existence of a mass point in earnings: A firm offering a job with earnings z > z
∗ still run the risk to lose workers to firms offering z ≤ z
∗ as utility depends on the combination of hours and wage rates instead of earnings only. By contrast to the case of homogeneous hours, a firm might, therefore, ascend in the utility ranking by increasing earnings above the threshold. A firm has the incentive to offer earnings z > z
∗ if the higher amount of firms offering less utility compensates for forgoing all type f 2 workers as well as the loss in margin.
This is most likely the case for firms where the marginal utility at z = z
∗ is large.
For which firms this applies depends on the parameters of the model, including in particular the relative importance of earnings and hours.
15

Since 2003, earnings from at most one second job are subject to the advantageous tax treatment of minijobs. Individuals who prior to the reform had only one job now have an incentive to take up a second job given the increase in marginal return to working an extra hour in form of a second job. Although the reform resulted in a sustained increase in number of second jobs, most individuals still just work in one job. We believe that set-up costs of a second job (bureaucracy etc.) prevent many individuals from doing this. Additional utility derived from additional earnings must thus at least cover the set-up costs. Subject to assumptions about the distribution of these costs we can use the earnings distribution of actual new second jobs to identify set-up costs which deter marginal second jobs.
The change in tax rules also created an incentive to reduce earnings in the existing job and take on a second job with earnings up to the maximum tax-free threshold
(or increase earnings of an existing second job). However, adjusting hours is often not easy. The likelihood of observing such adjustments will depend on the benefits minus the costs - thus conditional on observing the benefits of this choice (taking into account leisure, consumption and job-seeking parameters), the distribution of the earnings adjustments in the first job then identifies the size of the adjustment costs.
This version of the paper does not exploit this reform and focuses only on the period before 2003. In future versions, though, we plan to use the reform in order to either validate our model, relax restrictive assumptions concerning the comparability of workers looking for first and second jobs or to estimate additional structural parameters like adjustment or setting-up costs.
This section simulates the model described in the previous section maintaining the assumption of homogeneous hours. We assume the following parameter values: p = 800 e
/month; z
∗
= 325 e
/month; z r = 10 e
/month; λ s
= 0 .
3; λ f
= 0 .
5;
δ = 0 .
2; θ = 0 .
1; n s
= 0 .
5; n f
= 0 .
7
The resulting equilibrium cumulative offer distribution equals zero for z < z r =
16

Fig. 3: Cumulative offer distribution
10, increases smoothly in the interval z ∈ [10 , 170] and is constant in the interval z ∈ [170 , 325) (figure 3). That is, the offer distribution exhibits a gap in the latter interval. The additional margin of reducing offered earnings compensates for the discontinuously lower firm size below the mass point not until a firm reduces its earnings to 170 e monthly. The jump at z = z
∗
= 325 implies that there is a mass point at z
∗
. For z > z
∗ the cumulative offer distribution again increases smoothly until z = 706, the highest level of earnings offered.
This shape of the offer distribution is reflected in the corresponding earnings distributions of the three types of workers (figure 4 to 6). Although type s workers do not have an incentive to bunch at z
∗ and type f 1 workers do not care about the tax notch, the earnings distributions of all three types exhibit a mass point at and a gap below this threshold. For type f 2 workers who actually have an incentive to bunch at the threshold the mass point is most distinct and there is no mass above.
Figure 7 plots the joint earnings distribution of all types.
17

Fig. 4: Distribution of earnings, type s
Fig. 5: Distribution of earnings, type f1
18

Fig. 6: Distribution of earnings, type f2
Fig. 7: Distribution of earnings, all types
19

The estimation procedure we present here is based on the model with homogeneous hours. Estimation of the model with heterogeneous hours is work in progress.
t ub
= elapsed unemployment duration t uf
= residual unemployment duration d ub
= 1 if unemployment duration left-censored, otherwise 0 d uf
= 1 if unemployment duration right-censored, otherwise 0 t eb
= elapsed unemployment duration t ef
= residual unemployment duration d eb
= 1 if employment duration left-censored, otherwise 0 d ef
= 1 if employment duration right-censored, otherwise 0 z u
= earnings accepted by unemployed individuals z e
= earnings of employees in first period j 2 j = 1 if job-to-job transition takes place
An individual of type j who is initially unemployed and observed to start a job paying z u has the following likelihood contribution (this follows Bontemps et al.
(1999)):
L j u
( t ub
, t uf
, z u
) =( D j u
)
2 − d ub
− d uf exp [ − D j u
( t ub
+ t uf
)] ∗ f ( z u
)
1 − d uf u j n j
(10)
The first part of equation (10) pertains to the duration of the unemployment spell.
It is the joint density of elapsed and residual time in the spell, f ( t ub
, t uf
) = f ( t uf
| t ub
) f ( t ub
) which both are exponentially distributed with the exit rate D j u
.
Therefore, f ( t uf
| t ub
) = D j u
1 − d uf exp [ − D j u t uf
] and f ( t ub
) = D j u
1 − d ub exp [ − D j u t ub
].
The second part is the probability that the accepted job has earnings z u
.
This part vanishes if the unemployment spell is right-censored (i.e.
d uf
= 1).
The third part ( u j
) is the probability that an
20

unemployed individual is drawn in the initial state.
For an individual who is initially employed the likelihood contribution is
L j e
( t eb
, t ef e j
, z e
) = n j g j
( z e
)( D j e
( z ))
2 − d eb
− d ef exp [ − D j e
( z )( t eb
+ t ef
)] ∗
( δ
1 − j 2 j
λ (1 − F ( z e
)) j 2 j
)
1 − d ef (11)
The first part is the probability to be employed in a certain sector in the first period with earnings w e or q ( w e
). The second part pertains to the duration of the employment spell which is derived parallely to before. The last part is the probability of a transition to unemployment ( δ ) or to a better job ( λ (1 − F ( w ))). It vanishes if the employment spell is right-censored (i.e.
d ef
= 1).
The likelihood contributions differ between the worker types because of D j u
,
D j e
( .
), u j and e j k
. While we observe if an individual is of type s or f , we cannot distinguish between type f 1 and f 2 workers if observed wages are below the threshold or not observed. If a type f worker is observed to have higher wages than the threshold, she must be of type f 1. For s = ( u, e ) and z s
≤ z
∗ 6 the likelihood for type f workers, therefore, is
L f s
( t sb
, t sf
, z s
| z s
≤ z
∗
) =
θn f
(1 + θ ) n f
L f 2 s
( t sb
, t sf
, z s
| z s n f
≤ z
∗
)+
(1 + θ ) n f
L f 1 s
( t sb
, t sf
, z s
| z s
(12)
≤ z
∗
)
Note further that the presented likelihood contributions are only based on the workers’ flow equations and only consider the initial state and the first transition.
For the moment the following description of the estimation process is restricted to the case with homogeneous hours. Similar to Bontemps et al. (1999) we do not have an analytical expression for F ( .
) which only depends on structural parameters.
Bontemps et al. (1999) apply a two-step procedure in which they first estimate the cdf and pdf of the realized wage distribution ( G and g ) non-parametrically
6 Conditioning on z ≤ z
∗ mainly impacts the probability that an (un-)employed person is drawn in the initial state.
21

by kernel density estimators.
Using these estimates and equations (4) and (6), the likelihood contributions can be expressed in terms of ˆ g and the model parameters. Once the transition parameters are estimated from workers’ mobility patterns, these can be used to transform the observed distribution G ( .
) to the offer distribution F ( .
) which is the object about which theory provides us with predictions concerning the likelihood.
As this strategy only exploits equations with respect to workers’ behaviour, the offer distribution is basically treated as exogenous. Firms’ behaviour (described by the equal profit conditions above) which lead to the endogenous offer distribution is not exploited.
However, in our setting, besides the transition parameters, we need to identify
θ , the unobserved fraction of type f workers who accept jobs with z > z
∗
(type f 1) or those who don’t (type f 2). Intuitively it makes sense that the size of the mass point at the earnings threshold will be informative of the amount of individuals who are subject to the incentives generated by the threshold, i.e.
θ . Thus it makes sense to use the theoretical restrictions on the wage distribution in estimation.
Note that - in our simple model at least - we can identify θ without the earnings distribution akin to Bontemps et al. (1999) based solely on workers’ behaviour.
Without worker heterogeneity 7 , i.e. every job offer is accepted by type s and f 1 individuals, the only driver for a correlation between unemployment duration and accepted wage are the type f 2 workers. The reason is that the latter do not accept jobs with w > w
∗ which is why the probability of a match in one period is λ f
F ( w
∗
).
For other workers this probability is simply λ f or λ s
, respectively. The extent of correlation between the unemployment duration and the wage, thus, is informative about θ .
7
Worker heterogeneity would also introduces such a correlation (albeit in the opposite direction). If the distribution of reservation wages is assumed to be identical for the different groups, identification is enhanced by comparing the durations above and below the threshold and between type s and type f workers. Further, while the reservation wage distribution is arguably smooth around the threshold, durations might change discontinuously.
22

Although not necessary in our simple model, we think that making use of the restrictions put on the wage distribution (by the equal profit conditions) will create a more robust estimate of θ . We exploit that the relative size of the mass point in the offer distribution at the threshold contains information about the fraction of type f 2 workers (and, thus, θ ). By contrast to the two-step process proposed by
Bontemps et al. (1999) we, therefore, replace g ( .
) in the likelihood expressions by a function of F or f and solve in every iteration the system of equal profit conditions which defines the offer distribution F ( .
) subject to the structural parameters.
The Sample of Integrated Labour Market Biographies (SIAB) is a representative two percent sample of all individuals for whom an employer’s report to the social security system exists 8 . For the present analysis, its specific characteristics makes the SIAB the most suited data set. First, exact total gross earnings of a period of an employment report are observed 9 . This information is very accurate due to the administrative character of the data. Second, with approximately 1.6 million sampled employees, the sample size is comparatively large and, therefore, includes a substantial amount of individuals holding second jobs. It does not include civil servants and self-employed.
Third, the SIAB consists of complete employment biographies of the sampled individuals. We can, therefore, differentiate between first and second employments and observe, for example, the exact day a second employment starts as well as earnings changes over time. Fourth, we observe several characteristics important for labour market outcomes.
On individual level this particularly includes age, sex, occupation and education. Further, experience and tenure can be derived. On firm level, industry sector, size and wage structure are especially noteworthy. Furthermore, a regional unemployment rate can be matched based on the district of workplace.
The SIAB has three main limitations. First, minijobs are only included as of the middle of 1999. Second, working time is only differentiated between full- and
8 See Dorner et al. (2011) for a detailed description of the data.
9 A period lasts at most until the end of the calender year. Gross earnings are capped at the earnings cap of pension insurance which is not problematic here as we focus on low-paid jobs.
23

part-time employment. The amount of hours worked is not exactly observed. Hours information only exists in three broad categories. Third, we do not observe the household context like marriage status, the income of a potential spouse or other income sources which precludes the calculation of the exact income tax rates.
We restrict the sample to full-time employees between 18 and 62 years and exclude old-age pensioners. Spells with daily gross earnings below one Euro as well as one-time payments are excluded. If an individual has two parallel full-time employments the employment with less earnings is excluded.
As the education variable is not necessary for the administrative process, the quality is not as good as for most other variables (Dorner et al., 2011). Therefore, it is improved by the first imputation procedure proposed by Fitzenberger et al. (2006) which is shown to perform best (Wichert and Wilke, 2012).
Second jobs are identified by the following procedure. Parallel spells 10 are ranked by earnings with highest earnings as first rank. Yearly second job earnings, then, are the sum of earnings of all employments of second rank. That is, yearly second job earnings are not necessarily generated by a single employment relation because its rank can vary. Therefore, the sample is (so far) restricted to employees holding their first jobs the entire year. We, further, exclude employees if first and second jobs switch.
The response of first job earnings to the notch in the budget set is dramatic
(Figure 8).
There is a huge peak at the threshold for minijobs implying that individuals adjust their labour supply to their individual incentives. Figure 9 plots the empirical density of daily second job earnings for the year 2002 given yearly first job earnings exceed 5000 e
. The density features a clear peak at the minijob threshold and hardly any mass beyond. As in 2002 second jobs were not eligible for the favourable tax treatment of minijobs (they had to be taxed jointly with first job earnings), this is strong evidence that other workers’ preferences matter as well.
Firms seem to cater offered wage hours packages to employees looking for minijobs
10 Note that in the SIAB data set only completely parallel spells exist. If two spells partly overlaps, original spells are split.
24

Fig. 8: Distribution of first job earnings in 2002 as first jobs.
Figures 8 and 9 contain the same graphs for 2005, two years after the reform which, among others, increased the threshold to 400 e per month and extended the coverage to second jobs. The red lines mark the old threshold. For both, first and second jobs, the peak moved to the new threshold. While for first jobs this seems to be the only noteworthy change, the amount of second jobs increased tremendously.
As our data differentiates between minijobs and employments subject to SSC we can actually test whether misclassifying first as second jobs can be the reason for the discontinuity in the density of the latter. Table 1 shows that this is not the case.
Almost all first (second) jobs with daily earnings below the threshold are classified as minijobs (employment subject to SSC).
Table 2 shows the different sectors in which minijobs are found - according to whether these are first jobs (first column), second jobs (second column) or first jobs of employees holding second jobs. The distributions are consistent with the idea
25

Fig. 9: Distribution of second job earnings in 2002
Fig. 10: Distribution of first job earnings in 2005
26

Fig. 11: Distribution of second job earnings in 2005
Tab. 1: Job type if earnings are below the threshold in 2002
First job
Second job*
Marginal employment Employment s.t. SSC
68,595
529
Source: SIAB (strongly anonymized)
* If first job earnings exceed 5000 e
.
4,016
9,421
27

that offers for first and second jobs are drawn from the same wage offer distribution:
The main sectors are economic and household services as well as cleaning. The only striking difference between minijobs as first and as second jobs is the particularly high incidence of retail industry in the former group. The sector distribution of all first jobs (of second job holders) is much more different. Only 20 % of employees with second jobs work in the same sector in both jobs. With respect to occupations this is true for 23 %.
11
32 % work in both jobs in either the same sector or the same occupation. That is, it seems that the lion’s share of second jobs are not advisory activities of professors or the like.
Tab. 2: Distribution of sectors, 2000-2002
Sector
1 Agriculture, energy, mining
2 Primary production, prod. of goods
Marginal Employment
1st job
1,591 2
1,396 1
3 Facture of structural metal products 2,081 2
4 Steel deformation, vehicle constr.
2,465 2
2nd job
344
1st jobs of
2nd job hold.
454 1 1,536
834
720
1
2
2
519
2,445
2,340
2
5
7
7
5 Consumer goods industry
6 Food and luxury food industry
7 Main construction industry
8 Finishing trade
3,283
3,783
1,244
1,933
3
4
1
2
819 2 1,675
629 2
449 1
510 1
892
721
615
2
2
5
3
9 Wholesale trade
10 Retail industry
11 Transport and communication
12 Economic services
13 Household services
14 Education, social and health-care
15 (Street)Cleaning, organisations
16 Public admin., social security
5,732
17,923
3,246
21,549
12,209
7,022
6
18
3
22
12
7
2,066
2,540
2,136
10,907
4,990
2,993
6
7
6
30
14
8
1,685
2,524
1,806
4,564
1,483
4,952
12,395 12 4,443 12 2,664
1,941 2 1,215 3 2,717
4
15
8
8
5
14
5
8
Source: SIAB (strongly anonymized)
Table 3 compares the difference between desired and actual (or contracted) hours of work for employees with and without a second job 12 . The fraction of underemployed workers is larger for holders of second jobs, especially if desired are compared to contractual instead of actual hours of work. This implies that second jobs are used to compensate for potential hours restrictions in the first job. However, most holders of second jobs do not state that they are underemployed. Although the robustness
11 These numbers depend on the aggregation level.
Here, sectors are aggregated to 17 and occupations to 120 groups.
12 This information is taken from the German Socio Economic Panel which cannot be matched
(on individual level) to our main data set.
28

of such survey data is arguably not very high, this suggests that other motivations for second jobs - like for example the heterogeneity of jobs - are important as well
(which is for example also found by Smith Conway and Kimmel (1998)).
Tab. 3: Differences between actual and desired hours of work, 2002
Actual hours Contracted hours
With second job Without second job With second job Without second job
N % N % N % N %
Underemployed 45
No difference 243
Overemployed 132
11 520
58 5484
31 2266
6 84
66 292
27 44
20 1035
70 6420
10 815
12
78
10
Notes: No difference is defined with a tolerance of +-5
Source: SOEP wave 2002
Tab. 4: Socio-demographic characteristics in 2000-2002 female age
German
East German
Education
Intermediate
Voc. training
Grammar
University
With second job Without second job
.56
40.74
.86
.07
.22
.62
.04
.05
Source: SIAB (strongly anonymized)
.47
41.27
.92
.18
.12
.65
.05
.09
At first glance, the mean hourly wage 13 is much larger in the second (22.5
e
) than the first job (15.5
e
). For the median wage rate, however, the opposite is true
(12.5 vs. 14 e
). Further, restricting second jobs to those with more than 5 work days a month also decreases the hourly wage (mean=14.8
e
).
We present a job search model in which firms tailor their offers to the preferences of the most likely job seeking candidates. As a result, workers’ labour supply depends on other workers’ preferences. Given that labour supply in the market for marginal
13 This information is taken from the German Socio Economic Panel which cannot be matched
(on individual level) to our main data set.
29

employment in Germany is strongly influenced by non-linearities in the tax schedule induced by special treatment of marginal employment below an earnings threshold, so-called minijobs, we can identify the importance of this channel by considering the labour supply of individuals who are not affected by specific non-linearities. The effect of aggregate preferences of individual labour supply will mean that standard analyses of labour market policy are not reliable, since workers’ choice over hours is not independent of other workers. This provides a rationale for differences between micro- and macro labour supply elasticities. Whilst this possibility has already been voiced in the literature, we are the first to present an estimable structural model.
Estimation is work in progress. The model will allow us to perform counterfactual policy analyses, including new estimates of tax incidence.
Proposition (I) If we observe offers above z
∗
, there must be a mass point of job offers at z
∗
. The wage offer distribution above z
∗ is continuous up to some ¯ .
Assume there exists no mass point (i.e.
f ( z
∗
) = 0), the offer distribution for z < z
∗ is then continuous and profits at the mass point are n s
κ s
π ( z
∗
) = ( p − z
∗
)
(1 + κ s
(1 − F ( z
∗
)))
2
+ n f
κ f
(1 + κ f
(1 − F ( z
∗
)))
2
+ θn f
κ f
.
(13)
The profit of a firm offering jobs with earnings slightly above the threshold (for
→ 0) is:
π ( z
∗
+ ) = ( p − ( z
∗ n s
κ s
+ ))
(1 + κ s
(1 − F ( z
∗
+ )))(1 + κ s
(1 − F ( z
∗ − + ))) n f
κ f
+
(1 + κ f
(1 − F ( z
∗
+ )))(1 + κ f
(1 − F ( z
∗ − + ))) n s
κ s
= ( p − z
∗
)
(1 + κ s
(1 − F ( z
∗
)))
2
+ n f
κ f
(1 + κ f
(1 − F ( z
∗
)))
2
(14)
Comparing equations (13) and (14) and assuming f ( z
∗
) = 0, it is obvious that
π ( z
∗
+ ) < π ( z
∗
) implying that there is a gap to the right of the threshold. Is there an earnings level z
0
> z
∗
+ where the equal profit condition holds again?
Equation (15) makes use of F ( z
0
) = F ( z
∗
) which holds if there is a gap in the interval z ∈ ( z
∗
, z
0
).
30

n s
κ s
π ( z
0
) = ( p − z
0
)
(1 + κ s (1 − F ( z
0
)))(1 + κ s (1 − F ( z
0
− )))
+ n f
κ f
(1 + κ f (1 − F ( z
0
)))(1 + κ f (1 − F ( z
0
− ))) n s
κ s
= ( p − ( z
0
))
(1 + κ s
(1 − F ( z
∗
)))
2 n f
κ f
+
(1 + κ f
(1 − F ( z
∗
)))
2
(15)
As ( p − z
0
) < ( p − z
∗
) it holds that π ( z
0
) < π ( z
∗
) implying that no job with earnings z > z
∗ will be offered if there is no mass point at z
∗
.
Allowing for the existence of a mass point at z
∗
,
∂π ( z
∗
)
∂f ( z ∗ )
< 0 and
∂π ( z
∗
+ )
∂f ( z ∗ )
= 0 imply that there might be a value for f ( z
∗
) for which the equal profit condition between z
∗ and z
∗
+ holds ( π ( z
∗
+ ) = π ( z
∗
)).
The equilibrium offer distribution for the interval z
0 ∈ [ z
∗
+ , z ] is, then, continuous and determined by π ( z
0
) = π ( z
∗
+ ).
Proposition (II) If there is a mass point at z
∗
, there will be a gap in the offer distribution just below the threshold.
We need to show that a mass point in the wage offer distribution is only consistent with equal profits if there is a gap in the wage offering distribution. We compare profits π ( z
∗
) with profits π ( z
∗ − ).
The profit of a firm offering jobs with earnings at the threshold is
π ( z
∗
) = ( p − z
∗
) l ( z
∗
) n s
κ s
= ( p − z
∗
)(
(1 + κ s
(1 − F ( z
∗
)))(1 + κ s
(1 − F ( z
∗ − )))
+ n f
κ f
(1 + κ f
(1 − F ( z
∗
)))(1 + κ f
(1 − F ( z
∗ − )))
+
θn f
κ f
(1 + κ f
( F ( z
∗
) − F ( z
∗
)))(1 + κ f
( F ( z
∗
) − F ( z
∗
− )))
) n s
κ s
= ( p − z
∗
)(
(1 + κ s (1 − F ( z
∗
)))(1 + κ s (1 − F ( z
∗
) + f ( z
∗
)))
+ n f
κ f
(1 + κ f (1 − F ( z
∗
)))(1 + κ f (1 − F ( z
∗
) + f ( z
∗
)))
+
θn f
κ f
(1 + κ f f ( z
∗
)))
(16)
Given proposition (I), the data implies that there is a mass point at z = z
∗
(i.e.
that f ( z
∗
) > 0). Assuming, thus, that f ( z
∗
) > 0 and z < z
∗
, the profit slightly below the threshold can be given by (for → 0):
31

π ( z
∗
− ) = ( p − ( z
∗ n s
κ s
− ))(
(1 + κ s (1 − F ( z
∗
− )))(1 + κ s (1 − F ( z
∗
− 2 )))
+ n f
κ f
(1 + κ f (1 − F ( z
∗
− )))(1 + κ f (1 − F ( z
∗
− 2 )))
+
θn f
κ f
(1 + κ f ( F ( z
∗
) − F ( z
∗
− )))(1 + κ f ( F ( z
∗
) − F ( z
∗
− 2 )))
) n s
κ s
= ( p − z
∗
)(
(1 + κ s
(1 − F ( z
∗ − )))
2 n f
κ f
+
(1 + κ f
(1 − F ( z
∗ − )))
2
+
θn f
κ f
(1 + κ f
( f ( z
∗
)))
2
) n s
κ s
= ( p − z
∗
)(
(1 + κ s
(1 − F ( z
∗
) + f ( z
∗
)))
2 n f
κ f
+
(1 + κ f
(1 − F ( z
∗
) + f ( z
∗
)))
2
+
θn f
κ f
(1 + κ f
( f ( z
∗
)))
2
)
Given the assumption that f ( z
∗
) > 0, it holds that (1 + κ j (1 − F ( z
∗
) + f ( z
∗
))) >
(1 + κ j (1 − F ( z
∗
))) and (1 + κ f ( f ( z
∗
))) > 1. Therefore, π ( z
∗
) > π ( z
∗ − ) implying that there will be a gap to the left of the threshold.
Proposition (III) There may or may not exist wage offers offers below the threshold z
∗ in equilibrium. The wage offer distribution will then be continuous between z ∈ [ z, z
00
] for z
00
< z
∗
.
Define the highest wage offer below the threshold as z
00
, such that F ( z
00
) =
F z
00
( z
∗ − ). Note that since there is a gap left of the threshold, if in equilibrium a
-offer exists, it may be significantly below z
∗
. In equilibrium any make the same amount of profits as the threshold wage offer z
∗
.
z
00
-offer must n s
κ s
π ( z
00
) = ( p − z
00
)
(1 + κ s (1 − F ( z
00
)))(1 + κ s (1 − F ( z
00
− )))
+ n f
κ f
(1 + κ f (1 − F ( z
00
)))(1 + κ f (1 − F ( z
00
− )))
+
θn f
κ f
(1 + κ f ( F ( z
∗
) − F ( z
00
)))(1 + κ f ( F ( z
∗
) − F ( z
00
− ))) n s
κ s
= ( p − z
00
)
(1 + κ s
(1 − F ( z
∗
) + f ( z
∗
)))
2 n f
κ f
+
(1 + κ f
(1 − F ( z
∗
) + f ( z
∗
)))
2
θn f
κ f
+
(1 + κ f
( f ( z
∗
)))
2
(17)
Comparing equations (16) and (17) illustrates that π ( z
00
) = π ( z
∗
) can hold as
π ( z
00
) increases with decreasing z
00
. That is, there might be a certain size of the gap where π ( z
00
) = π ( z
∗
) holds.
Using that F ( z ) = 0, we now determine the lowest wage offer z that will be made (if there are wage offers below z
∗
).
n s
κ s
π ( z ) = ( p − z )
(1 + κ s (1 − F ( z )))(1 + κ s (1 − F ( z − )))
+
(1 + κ f n f
κ f
(1 − F ( z )))(1 + κ f (1 − F ( z − )))
+
θn f
κ f
(1 + κ f ( F ( z
∗
) − F ( z )))(1 + κ f ( F ( z
∗
) − F ( z − ))) n s
κ s
= ( p − z )
(1 + κ s
)
2
+ n f
κ f
(1 + κ f
)
2
+
θn f
κ f
(1 + κ f
( F ( z
∗
)))
2
(18)
If z
00
< z
∗ exists and z < z
00 there will then be a continuity of wage offers between these two values, generating equal profits with the standard trade-off between margins and firm-size.
32

For simplification, we allow for two different amount of hours h k where k ∈ [0 , 1].
In the market for small jobs, thes may correspond to 10 and 20 hours of working.
We later consider how this model generalizes to three or more hours sectors, and consider continuous variation in hours in the following section.
Firms set wage rates, w , workers care for wage rates as well as hours.
In order to simplify notation, we follow Shephard (2012) and define q
1
( w ) = w and
U ( q
0
( w ) , h
0
) = U ( q
1
( w ) , h
1
) = U ( w, h
1
), so q
0
( w ) is a function that denotes the wage rate that makes individuals indifferent between working with few ( k = 0) hours at q
0
( w ) or working more ( k = 1) hours at w . Depending on preferences, individuals may require a low-hours premium or accept a low-hours wage penalty.
The flow from and into unemployment is equivalent to before (see equations (1) and
(2)). The flow of workers of type j ∈ ( s, f 1 , f 2) from and into jobs with hours h k and wage rate w is
D j
( w ) g k j
( q k
( w )) e j k
= λ j f k
( q k
( w ))( u j
+ G j
0
( q
0
( w ) − ) e j
0
+ G j
1
( w − ) e j
1
) (19) with D j ( w ) = [ δ + λ j ((1 − F
1
( w )) + (1 − F
0
( q
0
( w ))))] for j ∈ ( s, f 1). Equation (20) states the corresponding definition for workers of type f 2 who do not accept jobs with wage rates larger than w
∗ k
.
D f 2
( w ) =


[ δ + λ j
(( F
1
( w
∗
1
) − F
1
( w )) + ( F
0
( w

[ δ + λ j
( F
1
( w
∗
1
) − F
1
( w ))]
∗
0
) − F
0
( q
0
( w ))))] ∀ w ≤ w
∗
1
∀ w ≤ w
∗
1 and q
0
( w ) ≤ w
∗
0 and q
0
( w ) > w
∗
0
(20)
The LHS of equation (19) pertains to workers who leave a job in a sector k with wage q k
( w ). For k = 1 this group consists of workers who move from sector 1 to sector 0 ( λ j
(1 − F
0
( q
0
( w ))) g j
1
( w ) e j
1 for j ∈ ( s, f 1)), who move to a better paying job within sector 1 ( λ j (1 − F
1
( w )) g
1 j
( w ) e j
1 for j ∈ ( s, f 1)) and who become unemployed
( δ j g j
1
( w ) e j
1
). The RHS pertains to workers who start a job in sector k with wage rate q k
( w ). For k = 1 this consists of workers who move from sector 0 to sector 1
( λ j f
1
( w ) G j
0
( q
0
( w ) − ) e j
0
), who changes jobs within sector 1 ( λ j f
1
( w ) G j
1
( w − ) e j
1
)
33

and who come from unemployment ( λ j f
1
( w )). The overall flow (i.e. both sectors) between jobs with wage rate of no greater than w and unemployment is:
( G j
0
( q
0
( w )) e j
0
+ G j
1
( w ) e j
1
) D j
( w ) = λ j u j
F
0
( q
0
( w )) + λ j u j
F
1
( w )
= λ j u j
+ λ j u j − λ j u j
(1 − F
0
( q
0
( w ))) − λ j u j
(1 − F
1
( w )) (21)
Using equation (1) gives
G j
0
( q
0
( w )) e j
0
+ G j
1
( w ) e j
1
=
δn j − u j D j ( w )
D j ( w )
By combining equations (19) and (22) we obtain g j k
( q k
( w )) e j k
=
λ j f k
( q k
( w ))( u j
+
δn j
− u j
( D j
( w − ))
D j ( w − )
D j ( w )
(22)
(23)
D.1.1
Firm size
The amount of workers of type j in steady-state employed at a firm in sector k which offers wage rate q k
( w ) is l j k
( q k
( w )) = g k j
( q k
( w )) e j k f k
( q k
( w ))
λ j δn j
=
D j ( w ) D j ( w − )
(24)
The steady state firm size is then l k
( q k
( w )) = l s k
( q k
( w )) + l f 1 k
( q k
( w )) + l f 2 k
( q k
( w ))
=



D s (
λ w ) s
δn s
D s ( w − )


D s (
λ w ) s
δn s
D s ( w − )
+
+
D f 1 (
λ w f
)
δn f 1
D f 1 ( w − )
D f 1 (
λ w f
)
δn f 1
D f 1 ( w − )
+
D f 2 (
λ w f
)
δn f 2
D f 2 ( w − )
∀ w ≤ w
∗
∀ w > w
∗
(25)
Following the standard arguments of profit equalization, we find the following
(the reasoning is parallel to the case without hours variation):
Proposition (A1) There can be (at most one) mass point in the wage offer distribution at the threshold in each sector, i.e. at wages w
∗ k
≡ z
∗ h k
.
Sketch of Proof : The following argument closely mirrors the argument in the case of homogeneous hours. We compare profits at the threshold value with profits above. We find that if there are offers above, there must be a mass point at the
34

threshold.
The profit of a sector k firm offering wage rate q k
( w ) can be expressed as
π k
( q k
( w )) = ( ph k
− q k
( w ) h k
) l k
( q k
( w )). We first state the profits of a type-1-firm, assuming that q
0
( w
∗
1
) ≤ w
∗
0
.
π k
( w
∗
1
) =
=
+
D s ( w
∗ k
λ s
δn
) D s ( s w
∗ k
− )
[ δ + λ s ((1 − F
1
( w
∗
1
+
λ f
δn f 1
D f 1 ( w
∗ k
) D f 1 ( w
∗ k
− )
)) + (1 − F
0
( q
0
( w
∗
1
+
λ s
δn s
))))][ δ + λ
D f 2 ( w
∗ k
) D f 2 ( w
∗ k s ((1 −
λ f
δn f 2
F
1
( w
∗
1
− )
− )) + (1 − F
0
( q
0
( w
∗
1
) − )))]
+
λ f
δn f 1
[ δ + λ f
((1 − F
1
( w
∗
1
)) + (1 − F
0
( q
0
( w
∗
1
))))][ δ + λ f
((1 − F
1
( w
∗
1
− )) + (1 − F
0
( q
0
( w
∗
1
) − )))]
+
λ f
δn f 2
+
[ δ + λ j
( F
0
( w
∗
0
) − F
0
( q
0
( w
∗
1
)))][ δ + λ j
( f
1
( w
∗
1
) + ( F
0
( w
∗
0
) − F
0
( q
0
( w
∗
1
) − )))]
(26)
Evaluated slightly above the threshold, profits are
π k
( w
∗
1
+ ) =
=
D s ( w
∗ k
[ δ + λ s
λ s
δn s
+ ) D s ( w
∗ k
)
((1 − F
1
( w
∗
1
λ f
δn f 1
+
D f 1 ( w
∗ k
+ ) D f 1 ( w
∗ k
)
λ s
δn s
+ )) + (1 − F
0
( q
0
( w
∗
1
+ ))))][ δ + λ s ((1 − F
1
( w
∗
1
)) + (1 − F
0
( q
0
( w
∗
1
))))]
+
λ f
δn f 1
+
[ δ + λ f
((1 − F
1
( w
∗
1
+ )) + (1 − F
0
( q
0
( w
∗
1
+ ))))][ δ + λ f
((1 − F
1
( w
∗
1
)) + (1 − F
0
( q
0
( w
∗
1
))))]
(27)
Equations (26) and (27) shows that the equal profit condition can only hold if there is a mass point in the offer distribution of sector 1 at w
∗
1
. By symmetry, note that the same argument can be made with respect to a type-0 firm. However, if the utility of a threshold offer lies in the “gap area” due to a threshold in another sector, it may be the case that there is no mass point in that sector. This explains the restriction “at most one” in Proposition (IV) and completes this sketch of a proof.
We now consider the influence of thresholds in other hours sectors on the wage distribution. Consider a firm of type-1, i.e. seeking a worker to work for h
1 hours.
The impact of a potential mass point in the offer distribution of sector 0 at w
∗
0 depends on the relation between w
∗
1
, q
0
( w
∗
1
) and w
∗
0
.
Proposition (A2) There will be no wage offers at wage levels (and in a certain interval below this level) that offer the same utility as is available at threshold wages w
∗ j = k in other sectors.
The intuition for Proposition (A2) is the following: It is a dominated strategy to offering a wage rate that is equal in utility to an offer made by several other firms.
A slightly higher offer will attract all workers from these firms at only marginal
35

cost. By Proposition (IV), wage offers at earnings thresholds generate mass points in the wage offer distributions. Thus for example a type-1 firm will offer a wage w
1
(where U ( ˜
1
, h
1
) = U ( w
∗
0
, h
0
).) in order to additionally attracts workers from this positive mass of sector 0 firms. This implies that there w
1
. How much below this utility value an offer can be sustained in equilibrium will depend on the parameters of the model in an analogous way to the potential existence of offers below the threshold offer in the homogeneous case.
Sketch of proof of Proposition (A2) w
1 denote the wage rate which satisfies profits of a sector 1 firm offering wage rate ˜
U w
1
( ˜
1
, h
1
) = U ( w
∗
0
, h
0 and slightly above are: w
1
> w
∗
1 the
π k w
1
) =
=
D s w k
λ s
) D
δn s s
( ˜ k
− )
+
D f 1 w
λ f
δn f 1 k
) D f 1 w k
− )
λ s
δn s
+
D f 2 w
λ k f
)
δn
D f 2 f 2 w k
− )
[ δ + λ s
((1 − F
1 w
1
)) + (1 − F
0
( w
∗
0
)))][ δ + λ s
((1 − F
1 w
1
− )) + (1 − F
0
( w
∗
0
− )))]
+
+
λ f
δn f 1
[ δ + λ f ((1 − F
1 w
1
)) + (1 − F
0
( w
∗
0
)))][ δ + λ f ((1 − F
1 w
1
λ f
δn f 2
− )) + (1 − F
0
( w
∗
0
− )))]
+
+
=
[ δ + λ j
(( F
1
( w
∗
1
) − F
1 w
1
)) + ( F
0
( w
∗
0
) − F
1
( w
∗
0
))))][ δ + λ j
(( F
1
( w
∗
1
) − F
1 w
1
− )) + ( F
0
( w
∗
0
) − F
1
( w
∗
0
− )))]
λ s
δn s
[ δ + λ s ((1 − F
1 w
1
)) + (1 − F
0
( w
∗
0
)))][ δ + λ s ((1 − F
1 w
1
)) + (1 − F
0
( w
∗
0
) + f
0
( w
∗
0
))))]
+
+
λ f
δn f 1
[ δ + λ f
((1 − F
1 w
1
)) + (1 − F
0
( w
∗
0
)))][ δ + λ f
((1 − F
1
λ f
δn f 2
+ w
1
)) + (1 − F
0
( w
∗
0
) + f
0
( w
∗
0
)))]
+
[ δ + λ j (( F
1
( w
∗
1
) − F
1 w
1
)))][ δ + λ j (( F
1
( w
∗
1
) − F
1 w
1
)) + f
0
( w
∗
0
))]
(28)
π k w
1
− ) =
=
D s w k
−
λ s
)
δn
D s s
( ˜ k
− 2 )
λ f
δn f 1
+
D f 1 w k
− ) D f 1 w k
λ s
δn s
− 2 )
+
D f 2
( ˜ k
λ f
δn f 2
− ) D f 2 w k
− 2 )
[ δ + λ s
((1 − F
1 w
1
− )) + (1 − F
0
( w
∗
0
− )))][ δ + λ s
((1 − F
1 w
1
− 2 )) + (1 − F
0
( w
∗
0
− 2 )))]
+
+
[ δ + λ f ((1 − F
1
λ f
δn f 1 w
1
− )) + (1 − F
0
( w
∗
0
− )))][ δ + λ f ((1 − F
1 w
1
− 2 )) + (1 − F
0
( w
∗
0
− 2 )))]
+
+
=
λ f
δn f 2
[ δ + λ j
(( F
1
( w
∗
1
) − F
1 w
1
− )) + ( F
0
( w
∗
0
) − F
1
( w
∗
0
λ s
δn s
+
[ δ + λ s ((1 − F
1 w
1
)) + (1 − F
0
( w
∗
0
) + f
0
( w
∗
0
)))] 2
− )))][ δ + λ j
(( F
1
( w
∗
1
) − F
1 w
1
− 2 )) + ( F
0
( w
∗
0
) − F
1
( w
∗
0
− 2 )))]
+
[ δ + λ f
((1 − F
1
λ f
δn f 1 w
1
)) + (1 − F
0
( w
∗
0
) + f
0
( w
∗
0
)))]
2
+
λ f
δn f 2
+
[ δ + λ j (( F
1
( w
∗
1
) − F
1 w
1
)) + f
0
( w
∗
0
))] 2
(29)
36

π k w
1
+ ) =
=
D s k
[ δ + λ s
λ s
δn s
+ ) D s w k
)
((1 − F
1 w
1
+
+
D f 1 w
)) + (1
λ k
− f
δn f 1
+ ) D f 1 w k
)
λ s
δn s
F
0
( w
∗
0
+
D f 2 w
λ f
δn f 2 k
+ ) D f 2 w k
)
+ )))][ δ + λ s
((1 − F
1 w
1
)) + (1 − F
0
( w
∗
0
)))]
+
+
[ δ + λ f
((1 − F
1 w
1
+ )) + (1 − F
0
( w
∗
0
λ f
δn f 1
+ )))][ δ + λ f
((1 − F
1 w
1
)) + (1 − F
0
( w
∗
0
)))]
+
λ f
δn f 2
+
=
[ δ + λ j (( F
1
( w
∗
1
) − F
1 w
1
λ s
δn s
+ )))][ δ + λ j (( F
1
( w
∗
1
) − F
1
[ δ + λ s
((1 − F
1 w
1
)) + (1 − F
0
( w
∗
0
)))]
2
+ w
1
)))]
+
+
λ f
δn f 1
[ δ + λ f ((1 − F
1 w
1
)) + (1 − F
0
( w
∗
0
)))] 2
+
λ f
δn f 2
[ δ + λ j
(( F
1
( w
∗
1
) − F
1 w
1
)))]
2
(30)
(31)
As f
0
( w
∗
0
) > 0 and → 0, it holds that π k w
1
− ) < π k w
1
) < π k w
1
+ ). This w
1
.
As ( ph − wh ) increases with decreasing w, there might be a wage rate w
0 where it holds that π k
( w
0
) = π k w
1
+ ).
This implies f
1
( .
) exhibit a gap in the interval (w’, ˜
1 w
1
< w
∗
1
, the terms of equations (28) and (30) referring to workers of type f 2 drop out. Although this might reduce the extent of the gap, π k w
1
) < π k
( ˜
1
+ ) still holds. If ˜
1
= w
∗
1 the necessary size of the mass point at w
∗
1 to balance the loss of type f 2 workers w
1
= w
∗
1
). How large the gap is, i.e. whether any offers w
1 will depend on the economic environment captured by the parameters of the model.
The following derivation uses equations (4), (7) and (1).
37

get l j
( z ) =
=
=
=
F ( z )
1+ κ j (1 − F ( z ))
−
F ( z − )
1+ κ j (1 − F ( z − ))
( n j
F ( z ) − F ( z − )
− u j
)
F ( z )(1+ κ j
(1 − F ( z − ))) − F ( z − )(1+ κ j
(1 − F ( z )))
(1+ κ j (1 − F ( z )))(1+ κ j (1 − F ( z − )))
( n j
F ( z ) − F ( z − )
− u j
)
F ( z )+ F ( z ) κ j − F ( z ) κ j
F ( z − ) − ( F ( z − )+ F ( z − ) κ j − F ( z − ) κ j
F ( z ))
(1+ κ j
(1 − F ( z )))(1+ κ j
(1 − F ( z − )))
( n j
F ( z ) − F ( z − )
− u j
)
F ( z )+ F ( z ) κ j
− ( F ( z − )+ F ( z − ) κ j
)
(1+ κ j (1 − F ( z )))(1+ κ j (1 − F ( z − )))
( n j
F ( z ) − F ( z − )
− u j
)
=
=
( F ( z ) − F ( z − ))(1+ κ j
)
(1+ κ j
(1 − F ( z )))(1+ κ j
(1 − F ( z − )))
( n j
F ( z ) − F ( z − )
− u j
)
(1 + κ j
)
(1 + κ j (1 − F ( z )))(1 + κ j (1 − F ( z − )))
( n j − u j
)
(1 + κ j ) n j κ
=
(1 + κ j (1 − F ( z )))(1 + κ j (1 − F ( z − ))) 1 + κ j n j κ
=
(1 + κ j (1 − F ( z )))(1 + κ j (1 − F ( z − )))
For type f 2 workers, we use equations (6), (7) and (2). For earnings z < z
∗ we l f 2
( z ) =
=
=
(1+ κ f
F ( z )
( F ( z
∗
) − F ( z ))) F ( z
∗
)
−
(1+ κ f
F ( z − )
( F ( z
∗
) − F ( z − ))) F ( z
∗
)
( θn f
F ( z ) − F ( z − )
− u f 2
)
F ( z )((1+ κ f
( F
((1+ κ f
( z
∗
) − F ( z − ))) F ( z
∗
( F ( z
∗
) − F ( z ))) F ( z
∗
)) − F ( z − )((1+ κ f
))((1+ κ f
( F ( z
∗
( F ( z
∗
) − F (
) − F ( z − ))) F ( z z
∗
)))
))
F ( z
∗
))
( θn f
F ( z ) − F ( z − )
− u f 2
)
( F ( z )+ F ( z ) κ f
F ( z
∗
) − F ( z ) κ f
F ( z − ) − ( F ( z − )+ F ( z − ) κ f
F ( z
∗
)
2
(1+ κ f
( F ( z
∗
) − F ( z )))(1+ κ f
F ( z
∗
) − F ( z − ) κ
( F ( z
∗
) − F ( z − ))) f
F ( z ))) F ( z
∗
)
( θn f
F ( z ) − F ( z − )
=
=
( F ( z )+ F ( z ) κ f
F ( z
∗
)(1+ κ f
( F (
F z
( z
∗
) − ( F ( z − )+ F ( z − ) κ
∗
) − F ( z )))(1+ κ f
( F ( z
∗ f
F ( z
∗
)))
) − F ( z − )))
( θn f
F ( z ) − F ( z − )
F ( z
∗
)(1+ κ f
( F ( z ) − F ( z − ))(1+ κ f
( F ( z
∗
) − F ( z )))(1+ κ f
F ( z
∗
))
( F ( z
∗
) − F ( z − )))
( θn f
F ( z ) − F ( z − )
−
− u u f 2 f 2
)
)
=
F ( z ∗ )(1 + κ f
(1 + κ f F ( z
∗
))
( F ( z ∗ ) − F ( z )))(1 + κ f ( F ( z ∗ ) − F ( z − )))
( θn f − u f 2
)
(1 + κ f
F ( z
∗
)) θn f
κ f
F ( z
∗
)
=
F ( z ∗ )(1 + κ f ( F ( z ∗ ) − F ( z )))(1 + κ f ( F ( z ∗ ) − F ( z − ))) 1 + κ f F ( z ∗ )
θn f κ f
=
(1 + κ f ( F ( z ∗ ) − F ( z )))(1 + κ f ( F ( z ∗ ) − F ( z − )))
− u f 2
)
The total firm size, is then the sum of the amount of workers of each type, which gives expression (8).
38

The profit of a firm offering jobs with earnings at the threshold is
π ( z
∗
) = ( p − z
∗
) l ( z
∗
) n s
κ s
= ( p − z
∗
)(
(1 + κ s
(1 − F ( z
∗
)))(1 + κ s
(1 − F ( z
∗ − )))
+ n f
κ f
(1 + κ f
(1 − F ( z
∗
)))(1 + κ f
(1 − F ( z
∗ − )))
+
θn f
κ f
(1 + κ f ( F ( z
∗
) − F ( z
∗
)))(1 + κ f ( F ( z
∗
) − F ( z
∗
− )))
) n s
κ s
= ( p − z
∗
)(
(1 + κ s (1 − F ( z
∗
)))(1 + κ s (1 − F ( z
∗
) + f ( z
∗
)))
+ n f
κ f
(1 + κ f (1 − F ( z
∗
)))(1 + κ f (1 − F ( z
∗
) + f ( z
∗
)))
+
θn f
κ f
(1 + κ f f ( z
∗
)))
(32)
Appendix (A) shows that a continuous distribution without a mass point ( f ( z
∗
) = 0) can be rejected since it implies no job offers above z
∗
.
Assuming, thus, that f ( z
∗
) > 0 and z < z
∗
, the profit slightly below the threshold can be given by (for → 0):
π ( z
∗
− ) = ( p − ( z
∗ n s
κ s
− ))(
(1 + κ s (1 − F ( z
∗
− )))(1 + κ s (1 − F ( z
∗
− 2 )))
+ n f
κ f
(1 + κ f (1 − F ( z
∗
− )))(1 + κ f (1 − F ( z
∗
− 2 )))
+
θn f
κ f
(1 + κ f ( F ( z ∗ ) − F ( z ∗ − )))(1 + κ f ( F ( z ∗ ) − F ( z ∗ − 2 )))
) n s
κ s
= ( p − z
∗
)(
(1 + κ s
(1 − F ( z
∗ − )))
2
+ n f
κ f
(1 + κ f
(1 − F ( z
∗ − )))
2
+
θn f
κ f
(1 + κ f
( f ( z
∗
)))
2
) n s
κ s
= ( p − z
∗
)(
(1 + κ s
(1 − F ( z
∗
) + f ( z
∗
)))
2 n f
κ f
+
(1 + κ f
(1 − F ( z
∗
) + f ( z
∗
)))
2
+
θn f
κ f
(1 + κ f
( f ( z
∗
)))
2
)
Given the assumption that f ( z
∗
) > 0, it holds that (1 + κ j (1 − F ( z
∗
) + f ( z
∗
))) >
(1 + κ j
(1 − F ( z
∗
))) and (1 + κ f
( f ( z
∗
))) > 1. Therefore, π ( z
∗
) > π ( z
∗ − ) implying that there will be a gap to the left of the threshold.
39

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