Minimum wages, wage dispersion and unemployment in search models. A review

Abstract

Facing the recent German debate on the minimum wage, this paper analyses theoretical effects of minimum wages on employment and wage distribution under a frictional setting. I  review new developments in search theory and discuss the influence of the minimum wage on wages and employment under each setting. Therefore, a major theoretical focus of the paper is the integration of heterogeneity on both sides of the market in equilibrium search models. In frictional models, minimum wages are generally binding and redistribute rents from firms to workers. Employment effects are more diverse. In the homogeneous case where workers and firms are identical, minimum wages do not affect employment, while in the heterogenous case theoretical results are mixed. There is no unique connection between unemployment and minimum wages, and the effect can be negative, zero or even positive. A positive effect can arise from a reaction in labor supply. However, the most advanced models, integrating heterogeneity on both sides of the market, seem to support the hypothesis that an increase in the minimum wage generally leads to an increase in unemployment as well. In this case, a social planner faces a trade off between redistribution of rents and unemployment.

Zusammenfassung

Die Studie analysiert die theoretischen Auswirkungen von Mindestlöhnen auf Beschäftigung und Lohnspreizung in einem suchtheoretischen Bezugsrahmen. Es wird ein Überblick über neue theoretische Forschungsansätze gegeben und der Einfluss von Mindestlöhnen auf Einkommen und Beschäftigung in den unterschiedlichen Szenarien diskutiert. Dabei liegt der wichtigste theoretische Schwerpunkt auf der Integration der Heterogenität auf beiden Seiten des Marktes in Gleichgewichtsmodelle. Im Falle homogener Akteure haben Mindestlöhne keinen Einfluss auf Beschäftigung, während im Falle von heterogenen Marktteilnehmern die theoretischen Ergebnisse gemischt sind. Es gibt keine eindeutige Verbindung zwischen Beschäftigung und Mindestlöhnen, und die Auswirkungen können positiv, null oder negativ sein. Die fortgeschrittensten Modelle, die die Heterogenität auf beiden Seiten des Marktes integrieren, scheinen jedoch die Hypothese zu unterstützen, dass ein Ansteigen des Mindestlohns im Allgemeinen auch ein Ansteigen der Arbeitslosigkeit zur Folge hat.

1 Wage dispersion and unemployment: alternative views

Many economists argue that the institution of a minimum wage compresses the wage structure, thereby contributing to high unemployment, especially for low-skilled individuals. This hypothesis is common in economic literature in the context of the different experiences in the U.S. and Continental Europe regarding wage dispersion and unemployment (see, e.g., Katz and Autor 1999; Blau and Kahn 2002; Blanchard 2006). Krugman (1994, p. 62) states: “… that growing U.S. inequality and growing European unemployment are different sides of the same coin”. Many observers argue that skill-based technical progress, reorganization processes or globalization have decreased the demand for low-skilled work in industrialized countries, thereby lowering the market wage. However, in Continental Europe institutional factors have prevented wages from falling (enough), causing a reaction via the amount of laborers employed and thereby increasing unemployment of the low-skilled.¹ This view is not uncontested for several reasons. Firstly, changes in employment rates in Europe were quite similar across skill groups and changes in the employment rates of the low-skilled were quite similar in Europe and the U.S. (Acemoglu and Pischke 1999). A second reason is that institutional differences can explain differences in unemployment and inequality levels, but not in changes. Explaining changes in these variables requires changes in institutions. The decline of the minimum wage in the U.S., however, is often argued to be such an institutional change. Finally, some authors contest the view that wages in Continental Europe and the U.S. developed differently. More precisely, Dustmann et  al. (2009) argue that the development of wages in Germany was similar to that in the U.S., but that it occurred later.²

In the past, with a few exceptions, there was no state-administered minimum wage in Germany. Recently, the possibility to implement minimum wages for specific sectors has been enlarged dramatically. From a political point of view, the question of the introduction of minimum wages in Germany was a major topic of political discussion and competition between the parties of the big coalition in the summer of 2007. The result of this political process was not a general minimum wage, but basically that the possibility to declare negotiated wages to be binding by the government under certain conditions was extended from the construction sector to other sectors.

As per § 5 of the “Tarifvertragsgesetz” (Collective Agreement Law) the Federal Minister of Labour and Social Affairs can declare the negotiated wage of a collective agreement binding for not-bounded employees of the same sector if the bounded employees constitute at least 50% and the declaration is according to “public interest”. However, representatives of employees and employers of the peak organizations have to agree to it, and it is due to the resistance of the latter that the execution of this law has decreased over the last few decades.

In 1998, this order imposing extension (“Allgemeinverbindlicherklärung”) was facilitated by the adding of § 1 Sec.  3a to the “Arbeitnehmer-Entsendegesetz” (Law on the Posting of Workers). It allows the Federal Minister of Labour and Social Affairs to declare a collective agreement binding through a “Rechtsverordnung” (legal decree), without a majority of the employees being bound by this agreement and, more importantly, even against the will of the peak organizations.

However, the Arbeitnehmer-Entsendegesetz cannot be applied generally, but is restricted to specific sectors. Thus, for a minimum wage to be enforced, it has to be within a specific sector that, in turn, has to be incorporated into the law beforehand. And this was the outcome of the discussion between the parties of the big coalition in 2007: on 1  July the building cleaning sector was incorporated into the law, followed by the postal service on 28  December. Several additional sectors followed in 2008 and 2009.

Clearly, the scientific debate has been triggered by these changes in the minimum wage legislation. There has been a partly acrimonious debate about the advantages and disadvantages of this policy measure, both among politicians and among economists.³ Furthermore, the German scientific debate has gotten considerable inspiration from the contribution of König and Möller (2008), who claimed that the minimum wage in the construction sector in Western Germany was not detrimental to employment. Several papers criticizing or confirming appeared subsequently (e.g., Thum and Ragnitz 2008; Bachmann et  al. 2008 or Fitzenberger 2009).⁴ The preliminary conclusion of this debate might be that the results of König and Möller (2008) are not flawed, but it is not clear how far these results generalize (Fitzenberger 2009).

This result that, in the end, minimum wages might be not that detrimental to employment, is difficult to justify in neo-classical labor market models, except in the trivial case where they are not binding. From this point of view it is assumed that the pivotal determinant of wages is marginal productivity. If people differ in their marginal productivity, in equilibrium, they obtain different wages. Thus, the wage distribution is determined by the distribution of marginal productivity. Under these assumptions binding minimum wages generally lower the wage dispersion of those workers employed, but at the same time tend to reduce employment. This is the case, because then some individuals are too unproductive to still be employed at the higher minimum wage. Thus, a minimum wage causes structural unemployment.

The situation is different under a frictional setting. Minimum wages are not necessarily expected to increase unemployment. The reason is that frictions are a source of monopsony power for employers and that wages are below marginal productivity (Manning 2003b). Clearly, there is potential for redistribution of rents without altering employment. Do minimum wages purely redistribute rents from the firms to the workers, or do they cause structural unemployment as well? I  show that the answer to this question is ambiguous and that the discussed model variants yield different results. I  obtain mostly zero employment effects. However, there are cases where the minimum wage generates even positive employment effects, because it does not alter the incentive of the firm to employ individuals, but it does increase the incentive for individuals to work.

Search frictions in the present case arise because of incomplete information where the process of generating information is time-consuming. Under this setting, identical workers can earn different wages and the sources of wage dispersion are search duration and luck. A central result of these theories challenges the neo-classical framework: rising wage dispersion is associated with rising unemployment. Low wage dispersion is associated with low unemployment. This contradicts the basic idea of the relationship of wage dispersion and unemployment presented above.⁵

In what follows, I  present different search models of increasing complexity (see Table 1) and examine the effect of minimum wages on the realized wage distribution and employment. Since the labor market performance of individuals and the performance of firms are extremely diverse, heterogeneity is seen to be an important feature of labor markets, and thus, a focus lies on the integration of heterogeneity in search models. Sections  2.1 and  2.2 establish the theoretical basis on which most models are built. In Sect.  2.1, I  present basics from partial search theory with exogenous wage dispersion and derive the reservation wage property. In Sect.  2.2, I  establish the baseline model, a model with an endogenous wage distribution and homogeneous individuals and firms. In order to discuss more realistic settings, I  discuss model extensions that allow heterogeneity on one or the other side of the market, and which serve to check the sensitivity of the results of the baseline model.

Table 1 Heterogeneity in search models

I do not discuss the wage bargaining literature (often combined with the matching function approach) in this paper (e.g., Pissarides 2000), which has also contributed to the discussion on minimum wages (see, e.g., Flinn 2002, 2006, and Flinn and Mabli 2008 or Cahuc et  al. 2006 building upon Postel-Vinay and Robin 2002a). I  do this for two reasons: first, incorporating the wage bargaining literature here would require a lot of space. Second, I  want to concentrate on the part of the literature that permits an endogenous wage distribution for identical workers to discuss wage effects of minimum wages. I  consider this a major advantage as I  view different wages for similar workers as an important empirical fact of the labor market (see Lemieux 2006 for the opposite view). On the contrary, the wage bargaining literature mostly generates one single wage per skill group. Note, however, that the wage bargaining literature has a quite natural channel for an effect of a minimum wage on employment, namely the endogenous contact rate. This contact rate is shaped by the number of vacancies and unemployed, where vacancy creation is endogenous to the model and generally would depend upon a minimum wage. The result that a binding minimum wage increases unemployment, though, is not a unique finding in these matching models either: in Flinn (2006), “minimum wage increases may or may not lead to increases in unemployment” (p. 1013). The reason for this indeterminacy stems from the fact that both contact rates and the participation decision are affected, and that both can have counteracting effects on (un-)employment.

2 Frictional labor markets

2.1 Exogenous wage dispersion: basic results

As a reference scenario and to understand later derivations, I  briefly introduce the partial search model.

The following illustration is inspired by Cahuc and Zylberberg (2004, Chap.  3.1) and Franz (2006, Chap.  6.2).

2.1.1 Assumptions

The assumptions under which the reservation wage property and the reservation wage can be deduced are concluded in what follows.⁶

• (B0) Environment: The model is dynamic; time will be treated as continuous and the environment is stationary.

• (B1) Employees: Individuals exclusively either work or search, which precludes both on-the-job search and the existence of inactive individuals. There is no choice in the number of hours worked or searched. Individuals are risk-neutral, have rational expectations and maximize the expected present value of their life time income over an infinite time horizon. Job seekers obtain ${ z=b-a }$ per time unit, where b are unemployment benefits and a search costs. Employees obtain a wage w per time unit. The value of unemployment is called ${ W_{U} }$ (the expected income), while ${ W_{L}(\textit{w}) }$ is the value of employment at the wage w.⁷ The stationary wage offer distribution ${ H(\textit{w}) }$ is known to job seekers, while the offered wage of a specific firm is generally not known.⁸

• (B2) Search: Search is sequential, which means that if an individual has received an offer, he decides whether to accept or not and then, in the case of rejection, i.e. if the wage offer is below the reservation wage ${ \textit{w}_{\text{R}} }$, continues the search.⁹ This is an optimal stopping problem, since job offers that have been rejected once cannot be accepted later on Dixit (1990). The future is discounted at interest rate  r.

• (B3) Transition rates: At an exogenous, constant and finite rate λ, an individual samples independent wage offers from ${ H(\textit{w}) }$. Individuals leave unemployment at a rate that is the product of the job offer rate and their acceptance probability. Employees lose their job at the exogenous, constant and finite rate δ (the job destruction rate). The number of sampled job offers and the number of terminated jobs are Poisson-distributed.

2.1.2 The basic model

The value of employment ${ W_{L}(\textit{w}) }$ at wage w can be derived as follows. In a small time interval¹⁰ dt a worker obtains the wage wdt. With probability ${ {\mathrm{\delta}} {\text{d}t} }$ the worker loses her job in this time interval. If losing the job, the worker is left with value ${ W_{U} }$. With the complementary probability ${ (1-{\mathrm{\delta}} {\text{d}t} ) }$ the worker remains employed. Under stationarity the value of employment is constant over time, and, therefore, the worker is left with ${ W_{L}(\textit{w}) }$. In case of linear discounting, the Bellmann equation is:

$$ \begin{aligned} W_{L}(\textit{w}) &= \frac{1}{1+r {\text{d}t} }\left\{ \textit{w} {\text{d}t} +{\mathrm{\delta}} {\text{d}t} W_{U}+(1-{\mathrm{\delta}} {\text{d}t} )W_{L}(\textit{w})\right\} \\ rW_{L}(\textit{w}) &= \textit{w}+{\mathrm{\delta}} (W_{U}-W_{L}(\textit{w})) \,.\\[-27pt] & {} \end{aligned} $$

(1)

The value of employment at the reservation wage must equal the value of unemployment. Rewriting Eq.  (1) as ${ W_{L}(\textit{w})-W_{U}=\frac{\textit{w}-rW_{U}}{r+{\mathrm{\delta}} } }$, taking into account that ${ \frac{\partial W_{L}(\textit{w})}{\partial \textit{w}}= }$ ${ \frac{1}{r+{\mathrm{\delta}} }\,{\textgreater}\,0 }$ and that ${ W_{U} }$ is independent from the wage previously paid, then ${ \textit{w}_{\text{R}} }$ is a unique solution to ${ W_{L}(\textit{w}_{\text{R}})=W_{U} }$ and is given from ${ \textit{w}_{\text{R}}=rW_{U} }$ as the reservation wage. The offered wage must be at least as high as what the worker would have gotten if she had remained unemployed.

Using the same strategy as above, the value of unemployment can be decomposed into two components: the value in case of a job offer and the value otherwise. The first component is given by

$$ \begin{aligned} W_{{\mathrm{\lambda}}} &= H(\textit{w}_{\text{R}})W_{U}+(1-H(\textit{w}_{\text{R}}))E_{\textit{w}}[W_{L}(\textit{w})|\textit{w} \,{\textgreater}\, \textit{w}_{\text{R}}] \\ &=\int_{0}^{\textit{w}_{\text{R}}}W_{U} {\text{d}H} (\textit{w})+\int_{\textit{w}_{\text{R}}}^{\infty }W_{L}(\textit{w}) {\text{d}H} (\textit{w}) \,. \\ & {} \end{aligned} $$

(2)

The second component is ${ W_{U}=W_{\bar{{\mathrm{\lambda}}}}=\int_{0}^{\infty }W_{U} {\text{d}H} (\textit{w}) }$. Thus:

$$ \begin{aligned} W_{U} &= \frac{1}{1+r {\text{d}t} }(z {\text{d}t} +{\mathrm{\lambda}} {\text{d}t} W_{{\mathrm{\lambda}} }+(1-{\mathrm{\lambda}} {\text{d}t} )W_{U}) \\ rW_{U} &= z+{\mathrm{\lambda}} \int_{\textit{w}_{\text{R}}}^{\infty }(W_{L}(\textit{w})-W_{U}) {\text{d}H} (\textit{w}) \,. \\[7pt] & {} \end{aligned} $$

(3)

Recognizing that ${ W_{L}(\textit{w})-W_{U}=\frac{\textit{w}-rW_{U}}{ r+{\mathrm{\delta}} } }$ and ${ \textit{w}_{\text{R}}=rW_{U} }$ the reservation wage is implicitly defined as:

$$ \begin{aligned} \textit{w}_{\text{R}} &= rW_{U} = z+\frac{{\mathrm{\lambda}} }{r+{\mathrm{\delta}} }\int_{\textit{w}_{\text{R}}}^{\infty }(\textit{w}-\textit{w}_{\text{R}}) {\text{d}H} (\textit{w}) \\ &= z+\frac{{\mathrm{\lambda}} }{r+{\mathrm{\delta}} }[(1-H(\textit{w}_{\text{R}}))(E(\textit{w}|\textit{w}\,{\textgreater}\,\textit{w}_{\text{R}})-\textit{w}_{\text{R}})] \,. \\[-24pt] & {} \end{aligned} $$

(4)

Subtracting z on both sides of the equation, the left-hand side is the instantaneous cost of rejecting a wage offer ${ \textit{w}_{\text{R}} }$. At the reservation wage the cost of waiting must equal the expected gains from waiting Devine and Kiefer (1991, p. 16/23).

Optimal behavior of the individuals is completely described by Eq.  (4). A job seeker with net income z, who is confronted with a job offer rate λ, with a job destruction rate δ, with an interest rate r and a wage offer distribution ${ H(\textit{w}) }$ will accept any job offer that exceeds ${ \textit{w}_{\text{R}} }$ and reject otherwise.

2.1.3 On-the-job search

Now, assume that employed job seekers obtain job offers at the exogenous job offer rate ${ {\mathrm{\lambda}}_{L} }$ as well, and that they do not incur search costs ${ (a_{L}=0) }$.¹¹ The assumptions (B0), (B1′), (B2) and (B3) are assumed to hold where:

• (B1′): as (B1), but: Employees search on-the-job, receive independent job offers at a constant, exogenous and finite rate ${ {\mathrm{\lambda}}_{L} }$ from the wage offer distribution ${ H(\textit{w}) }$ and do not have any search cost.

Employees accept all job offers that exceed their own wage ${ \bar{\textit{w}} }$ (Mortensen and Neumann 1988). It follows:

$$ \begin{aligned} rW_{L}(\bar{\textit{w}})=&{\,\,}\bar{\textit{w}}+{\mathrm{\delta}} (W_{U}-W_{L}(\bar{\textit{w}}))\\ &+{\mathrm{\lambda}} _{L}\int_{\bar{\textit{w}}}^{\infty }(W_{L}(\textit{w})-W_{L}(\bar{\textit{w}})) {\text{d}H} (\textit{w}) \,. \\ & {} \end{aligned} $$

(5)

The return to unemployment is still given by Eq.  (3). Evaluating Eq.  (5) at ${ \textit{w}_{\text{R}} }$, equalizing ${ rW_{L}(\textit{w}_{\text{R}}) }$ with Eq.  (3) and solving for ${ \textit{w}_{\text{R}} }$ yields:

$$ \textit{w}_{\text{R}}=z+({\mathrm{\lambda}} -{\mathrm{\lambda}} _{L})\int_{\textit{w}_{\text{R}}}^{\infty }(W_{L}(\textit{w})-W_{U}) {\text{d}H} (\textit{w}) \,. {} $$

(6)

Equation  (6) can be rewritten in terms of the parameters of the model (see Appendix 1):

$$ \textit{w}_{\text{R}}=z+({\mathrm{\lambda}} -{\mathrm{\lambda}} _{L})\int_{\textit{w}_{\text{R}}}^{\textit{w}^{o}}\frac{1-H(\textit{w})}{r+{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-H(\textit{w}))}d\textit{w} \,. {} $$

(7)

Intuitively, the possibility to search on-the-job lowers the reservation wage, since this opens the possibility to accept a low paying job and to search further on-the-job. Unemployed job seekers accept every wage offer that exceeds ${ \textit{w}_{\text{R}} }$ and continue searching otherwise, while employed job seekers accept every job offer that exceeds their current wage and continue working in their current job (and searching for a better paying job) otherwise.

2.2 Endogenous wage dispersion – the baseline model

It is not straightforward to establish dispersion of job characteristics across identical firms as an equilibrium outcome. A necessary condition for the existence of equilibrium wage dispersion is a positive connection between wage level and output (Burdett and Judd 1983). If firms have monopsony power, they make positive profits per employee. If firms can attract additional workers by setting high wages, there exists a trade-off between profits per worker and the number of workers in a firm. This means that there is indeed a positive connection between wages and production. The model of Burdett and Mortensen (1998) is based on this idea.

On-the-job searches guarantee that part of the job seekers, namely the employed, can compare wages. They compare their own wage with the wage of an alternative job offer. This insures the positive relationship between wages and output. High wage firms attract many new workers from competing firms, while losing only few. As a result they have a high employment level, making low profits per employee. In contrast to this, low wage firms have a low employment level, thereby making high profits per worker.

The assumptions of the model are given by (B0), (B1′′), (B2), (B3), (B4), (B5) and (B6), where:

• (B1′′), as (B1′), but all N individuals produce an identical amount y of the consumption good per time unit, which can be interpreted as labor productivity, where ${ y \,{\textgreater}\, \textit{w}_{\text{R}} }$ holds and z is identical across all unemployed individuals.

• (B4) firms: An infinite number of risk-neutral, c.f., identical firms on an interval [0, 1] maximizes profits. There is neither market entrance nor exit.

• (B5) wage formation: Firms determine wages ex ante from their profit maximization calculus.¹² This wage is payed forever, provided that the match does not end.

• (B6) economy: Small open economy with two goods and an exogenous interest rate r. The consumption good C is produced without capital (or with identical capital endowment in firms without depreciation), marginal productivity of labor is constant. The price of the consumption good is the numéraire, while w is the price of labor.

Starting with the reservation wage for the unemployed, it is given by Eq.  (7), assuming that ${ r=0 }$. Employed job seekers accept every wage offer that exceeds their current wage. Equilibrium unemployment follows from stationarity and equating in- and outflows to and from the pool of unemployed. In a small time interval dt ${ {\mathrm{\lambda}} {\text{d}t} (1-H(\textit{w}_{\text{R}}))U }$, unemployed individuals find a job, while ${ {\mathrm{\delta}} {\text{d}t} (N-U) }$ employees lose their job. Dividing by dt and taking the limit for ${ {\text{d}t} \rightarrow 0 }$ gives ${ \dot{U} = {\mathrm{\delta}} (N-U) - {\mathrm{\lambda}} (1-H(\textit{w}_{\text{R}}))U }$. Since firms that offer wages below the reservation wage make zero profits, since in equilibrium all firms make positive profits (see below) and since a wage distribution with mass points cannot be an equilibrium (see below) ${ H(\textit{w}_{\text{R}})=0 }$ holds. Therefore, equilibrium unemployment is

$$ U = \frac{{\mathrm{\delta}} N}{{\mathrm{\delta}} + {\mathrm{\lambda}} (1-H(\textit{w}_{\text{R}}))} = \frac{{\mathrm{\delta}} N}{{\mathrm{\delta}} +{\mathrm{\lambda}}} \,. {} $$

(8)

Let ${ g(\textit{w}) }$ denote the density of the distribution of paid wages and ${ h(\textit{w}) }$ denote the density of the distribution of wage offers. The equilibrium employment of one firm that offers wage w is given by ${ l(\textit{w})=(N-U)g(\textit{w})/h(\textit{w}) }$. Then, ${ L(\textit{w})=(N-U)G(\textit{w})=\int_{0}^{\textit{w}}l(\zeta ) {\text{d}H} (\zeta ) }$ is the amount of employees that are employed at a wage below w, and ${ G(\textit{w}) }$ is the distribution of paid wages. Concentrating on employment changes in firms in an interval dt that offer (and pay) wages above w, they have inflows from the pool of unemployed and from firms that pay wages below w, while they lose employees only through exogenous job destruction. Employees that make a job-to-job transition remain in this wage class. Unemployed job seekers obtain a wage offer with probability ${ {\mathrm{\lambda}} {\text{d}t} }$, which exceeds the wage w with probability ${ (1-H(\textit{w})) }$. Individuals that are employed at a wage below w obtain a wage offer that exceeds w with probability ${ (1-H(\textit{w})) }$ at the rate ${ {\mathrm{\lambda}}_{L} {\text{d}t} }$. From this, total inflows are ${ {\mathrm{\lambda}} {\text{d}t} U(1-H(\textit{w}))+{\mathrm{\lambda}} _{L} {\text{d}t} L(\textit{w})(1-H(\textit{w}))= {\text{d}t} ({\mathrm{\lambda}} U+{\mathrm{\lambda}} _{L}L(\textit{w}))(1-H(\textit{w})) }$. With probability ${ {\mathrm{\delta}} {\text{d}t} }$ exogenous shocks destroy existing jobs. The amount of jobs in firms that pay wages above w is given by ${ N-U-L(\textit{w}) }$. In equilibrium, employment in firms that pay wages above  w is assumed constant (stationarity).

Solving for ${ G(\textit{w})=\frac{L(\textit{w})}{N-U} }$ yields:

$$ \begin{aligned} G(\textit{w}) &= \frac{{\mathrm{\lambda}} U}{(N-U)}\frac{H(\textit{w})}{({\mathrm{\lambda}} _{L}(1-H(\textit{w}))+{\mathrm{\delta}} )} \\ &= \frac{{\mathrm{\delta}} H(\textit{w})}{({\mathrm{\lambda}} _{L}(1-H(\textit{w}))+{\mathrm{\delta}} )} \,. \\[11pt] & {} \end{aligned} $$

(9)

The second equality follows from using Eq.  (8). It shows the relationship between the distribution of wage offers ${ H(\textit{w}) }$ across firms and the distribution of paid wages in a cross section of workers ${ G(\textit{w}) }$.¹³ Since ${ L^{\prime }(\textit{w})=l(\textit{w})h(\textit{w}) }$, equilibrium employment in firms that pay wage w, is given by:

$$ \begin{aligned} l(\textit{w}) &= \frac{{\mathrm{\lambda}} U({\mathrm{\lambda}} _{L}+{\mathrm{\delta}} )}{({\mathrm{\lambda}} _{L}(1-H(\textit{w}))+{\mathrm{\delta}} )^{2}}\\ &= \frac{{\mathrm{\lambda}} {\mathrm{\delta}} N({\mathrm{\lambda}} _{L}+{\mathrm{\delta}} )}{({\mathrm{\lambda}} _{L}(1-H(\textit{w}))+{\mathrm{\delta}} )^{2}({\mathrm{\delta}} +{\mathrm{\lambda}} )} \\ &= {\mathrm{\delta}} (N-U)\frac{({\mathrm{\lambda}} _{L}+{\mathrm{\delta}} )}{({\mathrm{\lambda}} _{L}(1-H(\textit{w}))+{\mathrm{\delta}} )^{2}} \,. \\[36pt] &{} \end{aligned} $$

(10)

Because of a higher inflow rate and a smaller outflow rate, employment grows with the wage: ${ l^{\prime }(\textit{w}) \,{\textgreater}\, 0 }$. In equilibrium, every firm pays a wage from the support of the equilibrium wage offer distribution and makes expected profits of ${ \Pi (\textit{w}) = (y-\textit{w})l(\textit{w}) = \frac{{\mathrm{\lambda}} {\mathrm{\delta}} N(y-\textit{w}_{\text{R}})}{({\mathrm{\delta}} + {\mathrm{\lambda}} _{L})({\mathrm{\delta}} + {\mathrm{\lambda}} )} }$ (see Appendix  2), which are strictly positive.¹⁴ The wage offer density ${ h(\textit{w}) }$ is defined on the support ${ [\textit{w}_{\text{R}}, \textit{w}^{o}] }$, where ${ \textit{w}^{o} = y-(y-\textit{w}_{\text{R}}) }$ ${ \left( \frac{ {\mathrm{\delta}} }{{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}} \right)^2 }$ (see Appendix  2).

Taking into account that ${ \Pi (\textit{w}) = \Pi (\textit{w}^{\prime }) = (y-\textit{w})l(\textit{w}) }$ ${ \forall (\textit{w},\textit{w}^{\prime }) \in [\textit{w}_{\text{R}},\textit{w}^{o}] }$ ¹⁵ and equating the profit at the reservation wage with an arbitrary wage from the support of the wage offer distribution, one obtains ${ (y-\textit{w})\frac{{\mathrm{\lambda}} U({\mathrm{\lambda}} _{L} + {\mathrm{\delta}} )}{({\mathrm{\lambda}} _{L}(1-H(\textit{w})) + {\mathrm{\delta}} )^{2}} = (y-\textit{w}_{\text{R}}) \frac{{\mathrm{\lambda}} U({\mathrm{\lambda}} _{L} + {\mathrm{\delta}} )}{({\mathrm{\lambda}} _{L} + {\mathrm{\delta}} )^{2}} }$. Solving for ${ H(\textit{w}) }$ yields the equilibrium wage offer distribution.

$$ H(\textit{w}) = \left\{ {\!\!\!} \begin{array}{cl} 0 & {\,}\text{for}{\,\,}\textit{w} \,{\textless}\, \textit{w}_{\text{R}} \\[3pt] \frac{{\mathrm{\lambda}} _{L}+{\mathrm{\delta}} }{{\mathrm{\lambda}} _{L}}\left(1-\sqrt{\frac{y-\textit{w}}{y-\textit{w}_{\text{R}}}}\right) & {\,}\text{for}{\,\,}\textit{w}_{\text{R}}\leq \textit{w} \,{\textless}\, \textit{w}^{o} \\[6pt] 1 & {\,}\text{for}{\,\,}\textit{w} \geq \textit{w}^{o} \end{array} {\!\!\!} \right\} {} $$

(11)

The distribution H does not contain mass points or holes (Ridder and Van den Berg 1997, p. 101). The corresponding density can be calculated as ${ h(\textit{w}) = H^{\prime }(\textit{w}) = \frac{{\mathrm{\lambda}} _{L}+{\mathrm{\delta}} }{2{\mathrm{\lambda}} _{L}}\sqrt{\frac{1}{(y-\textit{w}_{\text{R}})(y-\textit{w})}} }$ and is increasing in w. Plugging ${ H(\textit{w}) }$ into Eq.  (7) yields an expression for the reservation wage for this wage distribution: ${ \textit{w}_{\text{R}} = \frac{({\mathrm{\delta}} +{\mathrm{\lambda}} _{L})^{2}z+({\mathrm{\lambda}} -{\mathrm{\lambda}} _{L}){\mathrm{\lambda}} _{L}y}{({\mathrm{\delta}} +{\mathrm{\lambda}} _{L})^{2}+({\mathrm{\lambda}} -{\mathrm{\lambda}} _{L}){\mathrm{\lambda}} _{L}} }$. Thus, the reservation wage is a weighted mean of the net unemployment benefits z and the labor productivity y.¹⁶ If individuals cannot change jobs ${ ( {\mathrm{\lambda}}_{L}=0 ) }$ the reservation wage equals z, ${ \textit{w}^{o}=\textit{w}_{\text{R}} }$ and one obtains the monopsony solution (Diamond 1971). If, on the other hand, the job offer rate on-the-job becomes big ${ ( {\mathrm{\lambda}} _{L}\rightarrow \infty ) }$, then the frictions vanish and the wage offer distribution collapses to a mass point at y. The same is true if there is no job destruction ${ {\mathrm{\delta}}=0 }$ or no search friction for the unemployed ${ {\mathrm{\lambda}} \rightarrow \infty }$ (that is no unemployment). For all intermediary cases ${ ( 0 \,{\textless}\, {\mathrm{\lambda}} _{L} \,{\textless}\, \infty , U \,{\textgreater}\, 0 ) }$ the wage offer distribution has a positive variance. The monopsony power of the firms depends on the degree of the friction. The higher the friction, i.e. the lower ${ {\mathrm{\lambda}}_{L} }$ and the higher δ, the higher is the monopsony power of the firms, and both moments of the wage distribution depend on this monopsony power (Van den Berg and Ridder 1993).

Remarkably, the model generates an equilibrium wage dispersion across identical individuals. Further, it provides arguments for the empirical facts that big firms pay higher wages than small firms, and that senior workers gain on average more than their junior counterparts. It is problematic, however, that the density of the distribution of paid wages ${ G^{\prime }(\textit{w}) = \frac{{\mathrm{\delta}} ({\mathrm{\lambda}} _{L}+{\mathrm{\delta}} )h(\textit{w})}{[{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-H(\textit{w}))]^{2}} }$ is upward sloping, which is difficult to reconcile with observed wage distributions. In addition, empirical studies point to the fact that this explanation of wage dispersion, i.e. monopsony power of firms through frictions in connection with on-the-job searches, is able to explain only a small part of the variance of an empirical wage distribution (Bontemps et  al. 2000, p. 348 f.).

An introduction or increase of a binding minimum wage alters the complete wage distribution.¹⁷ The upper bound of the wage distribution and the expectation of the wage distribution increases while the variance decreases.¹⁸

A binding minimum wage does not affect unemployment as long as the marginal productivity exceeds the minimum wage.¹⁹ Employers respond to an increase of a binding minimum wage by raising their wage offers. This reduces the profit of the firms, but not labor demand since the profit is positive for each individual employed. Even more, voluntary unemployment does not exist in this model. This result follows from the homogeneity assumption: each individual is a good allocation for each vacancy and vice versa. This result holds as long as wages are below marginal productivity, a typical result in search equilibrium models.

There is empirical evidence that, indeed, the unemployed accept every job offer, which is a central result of the model above. Many studies estimate an acceptance probability of almost one: “Il apparaît que la première offre d’emploi reçue est pratiquement toujours acceptée” (Cahuc and Zylberberg 2001, p. 77; see also Van den Berg 1999, p. F290). But this means that the mechanism that explains voluntary unemployment in the partial search model is not central for understanding unemployment. From this point of view unemployment is involuntary.

Clearly, a major drawback of this model is the homogeneity assumption, especially when considering the particular minimum wage legislation in Germany, where minimum wages are different for different sectors. It is likely that homogeneity is a critical assumption for the no-employment effects result. To see why, note that when reservation wages are heterogenous it is not more clear that firms offer wages that are above the reservation wage of all individuals. On the other hand, Jolivet et  al. (2006, p. 1) conclude from an empirical implementation of the homogeneous search model in a cross country comparison that the “(…) model fits the data surprisingly well”. One possibility to enhance the results obtained for the German situation, where there are different minimum wages for different sectors, is (similarly to Van den Berg and Ridder 1998) to assume for each sector one such search model, implying counterfactually completely separated labor markets. In this case, there is more space for the minimum wage to be adapted to the specific setting in each sector. Namely, when search frictions differ across sectors, a minimum wage might make more sense in a sector with large frictions and, thus, large monopsony power. In this case there is more room for the minimum wage to be welfare-improving.²⁰

3 Heterogeneity

In reaction to the drawback of the homogeneity assumption, several extensions have been discussed in the literature that introduce heterogeneity on either side of the market. For individuals, heterogeneity can take the form of different reservation wages caused by differences in search costs (or different values of leisure) or different productivities, as implied, for example, by learning on-the-job (see, e.g., Burdett et  al. 2009). For firms the effect of different productivities has been examined.

3.1 Different search costs

This subsection presents a model extension that allows for different search costs across unemployed job seekers (Burdett and Mortensen 1998) since it is likely that this extension has a major impact on the above results for the minimum wage. If unemployed job seekers have different search costs  a, then z varies conditionally on b.²¹

First, notice that compared to the basic model different search costs have no effect on labor demand. Because of identical productivity every (or no) match is profitable. But there is an effect on offered and paid wages and, therefore, an effect on the behavior of the job seekers. Intuitively, this is the case, since for firms in this case it might be profitable to offer wages below the reservation wage of a part of the unemployed. Provided that the offered wage is still above the reservation wage of some individuals, a firm has still positive inflows. Then, for the unemployed not every contact with a firm means a profitable match. This is true, although the match is potentially profitable (in the sense that the productivity exceeds the reservation wage of the job seeker).

The resulting equilibrium wage distribution combines the effect of the informational imperfection with the effect of the heterogeneity in search costs. On the one hand, the wage distribution compensates the job seekers for different search costs. On the other hand, firms differ in size and they have to pay different wages to ensure their size.

3.1.1 The model

Assumptions (B0), (B1′′′), (B2), (B3), (B4′), (B5) and (B6) hold, where:

• (B1′′′), as (B1′′), but N individuals differ in their search costs. Their net search costs z follow a continuous distribution on ${ [ {\,} \underline{z}, \bar{z} {\,} ] }$, where ${ R(z) }$ is the share of individuals whose search costs are below z. ${ R(z) }$ is the corresponding density. In addition, ${ {\mathrm{\lambda}}_{L}={\mathrm{\lambda}} }$ holds.

• (B4′), as (B4), but firms know the distribution ${ R(z) }$, and cannot observe z individually.

Different reservation wage policies across individuals are a consequence of the assumption of different net search costs. Note that (B1′′′) implies that ${ \textit{w}_{\text{R}}=z }$. Then, equating in- and outflows to and from unemployment, yields the equilibrium unemployment rate for any z-type. It is given by ${ U_{z}=\frac{{\mathrm{\delta}} Nr(z)}{{\mathrm{\delta}} + {\mathrm{\lambda}} [ 1-H(z) ]} }$.

The optimal behavior of the firm can be deduced as follows. The amount of unemployed that accept a wage offer w is given by:

$$ \begin{aligned} S(\textit{w}) &= \int_{\underline{z}}^{\textit{w}}\left( \frac{{\mathrm{\delta}} Nr(z)}{{\mathrm{\delta}} +{\mathrm{\lambda}} [ 1-H(z) ]}\right) {\text{d}z} \\ &= \int_{\underline{z}}^{\textit{w}}\left( \frac{{\mathrm{\delta}} N}{{\mathrm{\delta}} +{\mathrm{\lambda}} [ 1-H(z) ]}\right) {\text{d}R} (z) \,. \\[12pt] & {} \end{aligned} $$

(12)

As before, let ${ G(\textit{w}) }$ be the distribution of paid wages and ${ L(\textit{w})=(N-S(\bar{z}))G(\textit{w}) }$ the amount of employees that are employed at a wage below w.²² Under stationarity, employment must remain constant in firms that pay wages below w. Inflows to this group of firms stem only from unemployment. ${ {\text{d}S} (z) }$ unemployed z-individuals obtain an acceptable job offer below w with probability ${ H(\textit{w})-H(z) }$. Thus, expected inflows are ${ {\mathrm{\lambda}} \int_{\underline{z}}^{\textit{w}}(H(\textit{w})-H(x)) {\text{d}S} (x) }$. Outflows are composed of job destruction δ and of quits towards better paying competitors that arrive at rate ${ {\mathrm{\lambda}}(1-H(\textit{w})) }$ to the amount of employees in the considered wage group ${ (N-S(\bar{z}))G(\textit{w}) }$. Taken together:

$$ \begin{aligned} &({\mathrm{\delta}} +{\mathrm{\lambda}} [ 1-H(\textit{w})])(N-S(\bar{z}))G(\textit{w})=\\ &\qquad \qquad \qquad \qquad \qquad \qquad {\mathrm{\lambda}} \int_{\underline{z} }^{\textit{w}}(H(\textit{w})-H(x)) {\text{d}S} (x) \,. {} \end{aligned} $$

Solving for ${ L(\textit{w}) }$ and differentiating with respect to w, after some simplifications²³ the equilibrium employment in a firm, offering the wage w is given by:

$$ l(\textit{w}) = \frac{(N-S(\bar{z}))G^{\prime }(\textit{w})}{h(\textit{w})} = \frac{{\mathrm{\lambda}} {\mathrm{\delta}} NR(\textit{w})}{ ({\mathrm{\delta}} +{\mathrm{\lambda}} [ 1-H(\textit{w})])^{2}} \, . {} $$

The wage offer distribution can be derived from the equality of profits ${ \Pi = (y-\textit{w})l(\textit{w}) }$ on the support of ${ H(\textit{w}) }$ in equilibrium. Let ${ \underline{\textit{w}} }$ be the lower bound of the support, then ${ (y-\underline{\textit{w}}) \frac{{\mathrm{\lambda}} {\mathrm{\delta}} NR(\underline{\textit{w}})}{({\mathrm{\delta}} +{\mathrm{\lambda}} )^{2}}=(y-\textit{w})\frac{{\mathrm{\lambda}} {\mathrm{\delta}} NR(\textit{w})}{({\mathrm{\delta}} +{\mathrm{\lambda}} [ 1-H(\textit{w})])^{2}} }$ holds and it follows:

$$ \begin{aligned} &H(\textit{w}) = \frac{{\mathrm{\delta}} +{\mathrm{\lambda}} }{{\mathrm{\lambda}} }\left[ 1-\sqrt{\frac{(y-\textit{w})R(\textit{w})}{(y- \underline{\textit{w}})R(\underline{\textit{w}})}}\right] , {\,\,\,}\text{for}{\,\,}\textit{w} \in [ \underline{\textit{w}},\textit{w}^{o}] \,. \\[-21pt] &{} \end{aligned} $$

(13)

The wage paid by a firm is given by ${ \textit{w}=\arg \max_{\textit{w}}[(y-\textit{w})R(\textit{w})] }$ and ${ \textit{w}^{o} }$ is the biggest value that satisfies ${ \frac{(y-\textit{w}^{o})R(\textit{w}^{o})}{(y-\underline{\textit{w}})R(\underline{\textit{w}})}=\frac{{\mathrm{\delta}} ^{2}}{({\mathrm{\delta}} +{\mathrm{\lambda}} )^{2}} }$ (Burdett and Mortensen 1998, p. 266).

Unemployed individuals in this extension of the basic model do not accept every wage offer they obtain, and their expected unemployment duration depends on their reservation wage. The resulting equilibrium unemployment ${ S(\bar{z}|H) }$ is higher than the unemployment that would result if all firms would pay a wage equal to marginal productivity ${ S(\bar{z}|\textit{w} }$ ${ = }$ ${ y) }$.²⁴ In addition, equilibrium unemployment depends positively on ${ \frac{{\mathrm{\delta}} }{{\mathrm{\lambda}} } }$, an indicator of the amount of frictions in the labor market. Assuming that ${ \bar{z} \,{\textless}\, y }$ and ${ H(\bar{z})\,{\textgreater}\, 0 }$, every match is potentially profitable, but not every match is formed when employers and employees meet. This is the case, since firms commit ex ante to pay some wage of the wage offer distribution and since it is optimal for a part of the firms to offer wages below ${ \bar{z} }$.

Introducing a binding minimum wage has several effects. It changes both the lower and the upper bound of the wage distribution. However, since the minimum wage shifts the wage offer distribution to the right, and since the reservation wage does not depend on the wage offer distribution for ${ {\mathrm{\lambda}}_{L}={\mathrm{\lambda}} }$, the average unemployment duration will decrease, since on average, unemployed job seekers obtain more acceptable wage offers; that is, unemployment decreases when the minimum wage increases. This is the case, since labor demand does not react, whereas job seekers accept job offers more often on average. However, one critique to this model is that it implies a counterfactual wage density, see Van den Berg (1999, p. F299). In addition, heterogeneity of workers or firms productivities is likely to have quite different effects on employment effects of minimum wages.

With respect to the specific German situation, the scenario with different search costs might be less important than when considering an economy with one single minimum wage. There, probably, different search costs are even more important. But, in the German case, it is more realistic to assume that reservation wages differ across individuals. Thus, the cited effect that minimum wages have a tendency to increase employment because job offers are more likely to be acceptable is likely to be present in the German case as well.

3.2 Heterogeneity across firms

3.2.1 A priori heterogeneity and endogenous wage dispersion

There exist several model extensions with exogenous, or even endogenous, heterogeneous productivity in the search framework. In a competitive setting with constant returns to scale, this situation could not persist, since more productive firms would pay higher wages and employees would move immediately to the better paying firm. In a market with frictions, however, this is not the case.

The following derivations are based upon Bontemps et  al. (2000). I  will assume that the assumptions (B0), (B1′′′′), (B2), (B3), (B4′′), (B5) and (B6′) hold, where:

• (B1′′′′), as (B1′′), but N identical individuals with productivity ${ \tilde{y} }$ produce unequal amounts  y of the good  C. ${ y=\tilde{y}t(k) }$ is assumed to hold, where ${ t(\cdot ) }$ is a positive function of k and displays the following properties: ${ t^{\prime }(k) \,{\textgreater}\, 0, t^{\prime \prime }(k) \,{\textless}\, 0 }$. k can be interpreted as capital intensity in a firm.

• (B4′′), as (B4), but there is an amount of M firms, whose capital intensity is distributed according to ${ {\mathrm{\Gamma}} (k)={\mathrm{\Gamma}} (y) }$. The constant, exogenous random variable K is drawn before production starts and has a finite expectation. There is a unique realization of Y that corresponds to each realization of K. Realizations of Y are continuously distributed on the support ${ {\,} [ \underline{y},\bar{y} {\,} ] }$. It is assumed that y exceeds the common reservation wage of the employees or the binding minimum wage if it exists.

• (B6′), as (B6), but the consumption good C is produced with labor and capital, and there are no depreciations.

Equilibrium unemployment is given by Eq.  (8) ${ U=\frac{{\mathrm{\delta}} N}{{\mathrm{\delta}} +{\mathrm{\lambda}} } }$. The reservation wage of the U unemployed is given by Eq.  (7) ${ \textit{w}_{\text{R}} = z + ({\mathrm{\lambda}} - {\mathrm{\lambda}} _{L}) \int_{\textit{w}_{\text{R}}}^{\textit{w}^{o}}\frac{1-H(\textit{w})}{r+{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-H(\textit{w}))}d\textit{w} }$, where ${ \textit{w}^{o} }$ is the upper bound of the wage distributions.

Describing the dynamics of employment in firms offering a wage above w ${ (\textit{w}_{\text{R}} \,{\textless}\, \textit{w} \,{\textless}\, \textit{w}^{o}) }$, imposing stationarity and using Eq.  (8) helps us deduce equilibrium employment ${ l(\textit{w}) }$ in firms offering w.

$$ \begin{aligned} \frac{L(\textit{w})}{N-U} &= G(\textit{w})=\frac{U}{(N-U)}\frac{{\mathrm{\lambda}} H(\textit{w})}{[{\mathrm{\lambda}} _{L}(1-H(\textit{w}))+{\mathrm{\delta}}]}\\[2pt] &=\frac{{\mathrm{\delta}} H(\textit{w})}{({\mathrm{\lambda}} _{L}(1-H(\textit{w}))+{\mathrm{\delta}} )} \,, \\[-10pt] &{} \end{aligned} $$

(14)

as in the homogeneous model (Eq.  9). By the same arguments as above, the reservation wage (or minimum wage, respectively) is the lower bound of the wage offer distribution (see Sect.  2.2). And, thus:

$$ l(\textit{w}) = {\mathrm{\delta}} (N-U)\frac{{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}}{[{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-H(\textit{w}))]^{2}} \,. {} $$

(15)

Note that ${ l(\textit{w}) }$ must divide by M to obtain the average amount of employment in one firm offering w: ${ \breve{l}(\textit{w})=\frac{l(\textit{w})}{M} }$.

Firms maximize expected profits:

$$ \begin{aligned} \Pi (\textit{w}|y)&=(y-\textit{w})\breve{l}(\textit{w})\\ &={\mathrm{\delta}} (y-\textit{w})\frac{(N-U)}{M}\frac{({\mathrm{\delta}} +{\mathrm{\lambda}} _{L})}{[{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-H(\textit{w}))]^{2}} \,. \\ &{} \end{aligned} $$

(16)

Again, the firm faces the trade-off between the profit per employee and the equilibrium amount of employees. In general, the wage a firm pays can depend on its productivity y. However, facing the results of Sect.  2.2, note that it is possible that firms of an identical y-type pay different wages if different wages yield identical profits. If this is the case, then a firm of type y chooses a wage randomly according to ${ H(\textit{w}|y) }$. Let

$$ K_{y} = \arg \max_{\textit{w}}\{\Pi (\textit{w}|y)|\max (\textit{w}_{\text{R}},\textit{w}_{\min }) \,{\textless}\, \textit{w} \,{\textless}\, y\} {} $$

be the entity of profit maximizing wages from which the y-firm draws one. Then, in the case of continuous productivity dispersion it can be shown that ${ K_{y}=K(y) }$ is unique (Bontemps et  al. 2000, p. 315/350). For each firm of a given y-type there is only one optimal wage, which is increasing in y. This simplifies the analysis since then the probability ${ H(\textit{w}) }$ that a firm pays a wage lower than ${ \textit{w}=K(y) }$ is determined by the probability ${ {\mathrm{\Gamma}}(y) }$ that the firm has a productivity below y. Since ${ K(y)=\textit{w} }$, and since ${ K^{\prime }(y) \,{\textgreater}\, 0 }$, the inverse ${ y=K^{-1}(\textit{w}) }$ can be calculated. The share of firms that offer wages below w equals the share of firms whose productivity is below ${ y=K^{-1}(\textit{w}) }$, or: ${ H(\textit{w}) = {\mathrm{\Gamma}} (K^{-1}(\textit{w})) }$.

The first order condition for the profit equation can be derived by differentiating Eq.  (16) with respect to w:

$$ \begin{aligned} \frac{\breve{l}^{\prime }(\textit{w})}{\breve{l}(\textit{w})} &= \frac{l^{\prime }(\textit{w})}{l(\textit{w})}\\ &= \frac{1}{y-\textit{w}} \,, {\,\,}\text{and} \\ -{\mathrm{\delta}} -{\mathrm{\lambda}} _{L}(1-H(\textit{w}))+2(y-\textit{w}){\mathrm{\lambda}} _{L}h(\textit{w}) &= 0 \,. \\[-27pt] &{} \end{aligned} $$

(17)

The first line follows from using the first equality and the second line from using the second equality in Eq.  (16). The second line determines the optimal wage for each firm ${ \textit{w}=K(y|H(\cdot )) }$ implicitly for wages above the reservation wage.

For each firm, equilibrium profits ${ \Pi (\cdot ) }$ and employment ${ \breve{l}(\cdot ) }$ can be obtained as a function of y. Using this, an explicit expression for ${ K(y) }$ can be obtained. y-firms make profits of ${ \Pi (y)=(y-K(y))\breve{l} (K(y)) }$. Differentiating with respect to y yields ${ \Pi ^{\prime }(y)=(1-K^{\prime }(y))\breve{l}(K(y))+(y-K(y))\breve{l}^{\prime }(K(y))K^{\prime }(y) }$. Using optimality and the envelope theorem, the following result is obtained:

$$ \Pi ^{\prime }(y) = \breve{l}(K(y)) \,. {} $$

(18)

Thus, profits of the firms are increasing with productivity. Integrating Eq.  (18) yields an explicit expression for ${ \Pi (y) }$ (see Appendix  3). Using this and a suitable form of the equilibrium employment equation, I  obtain the following expression for ${ K(y) }$:

$$ K(y) = y-[ {\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-{\mathrm{\Gamma}} (y))]^{2}\int_{\underline{\textit{w}}}^{y}\frac{1 }{[{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-{\mathrm{\Gamma}} ({\mathrm{\varrho}} ))]^{2}}d{\mathrm{\varrho}} \,, {} $$

(19)

where ${ \underline{\textit{w}} = \max (\textit{w}_{\text{R}},\textit{w}_{\min }) }$.

The wage offer distribution follows from ${ H(\textit{w})={\mathrm{\Gamma}} }$ ${ (K^{-1}(\textit{w})) }$, but in general there is no closed form expression ${ K^{-1}(\textit{w}) }$.

The equilibrium wage offer distribution ${ H(\textit{w})={\mathrm{\Gamma}} }$ ${ (K^{-1}(\textit{w})) }$ uniquely determines the distribution of paid wages ${ G(\textit{w}) }$ in Eq.  (14). It also determines the equilibrium profit of firms ${ \Pi (\textit{w}|y) }$ in Eq.  (16) depending on y. Profit is maximized if firms choose the wage according to Eq.  (19) and is given by Eq.  (39) in Appendix  3. Equilibrium unemployment is given by Eq.  (8).

Wage dispersion in this model arises as a result of the interaction of both search frictions and productivity dispersion. Productivity dispersion itself is not sufficient for wage dispersion. It has been shown above, however, that informational frictions alone with on-the-job searches are a sufficient condition for wage dispersion. However, first, the integration of different productivities across firms is an important ingredient empirically. Second, it has been mentioned before that the homogeneous model implies counterfactual wage distributions and it only provides an explanation for part of the variance of wages between individuals. In this context, the resulting wage distributions depend on the productivity distribution. If, for example, a Pareto-distribution for the productivity is assumed, a realistic shape for the wage distribution can be obtained (Bontemps et  al. 2000). Indeed, the model generates wage distributions that are in accordance with the data. This is astonishing, especially since the assumption of homogeneous workers has been retained.

Since ${ K^{\prime }(y) \,{\textgreater}\, 0 }$, an increasing variance of the distribution of the productivities, increases the variance of the wage distribution, whereas equilibrium unemployment remains unchanged. A minimum wage affects both the lower and the upper bounds of the wage distribution. As in the homogeneous model, equilibrium unemployment is, in general, not affected while the monopsony power of the firms is affected. Rents can be redistributed from firms to workers. Although, in general, an increasing minimum wage drives unproductive firms out of the market, labor demand does not react, because the missing demand of the less productive firms is fully compensated for by their more productive counterparts. However, the assumption that there is always a continuum of firms that demands labor is critical in that context. The fact that firms are driven out of the market stipulates another point, namely that a minimum wage has dynamic effects on the composition of the firms. This aspect is, however, beyond the scope of this paper.²⁵

With respect to the specific German situation, one would like to consider not one single minimum wage, but several minimum wages according to sectors. So, one could imagine for each sector a model as presented above, but then clearly the continuum of firms assumption is even more unrealistic. It is more imaginable that with a given number of firms, the minimum wage is detrimental to employment; for example, the job offer rate might decrease in this case. But also in the case of a limited number of firms, for firms that are profitable enough, it is worthwhile to try to increase their employment since they make profits on each worker employed. Secondly, a sectoral minimum wage might have effects on the product market, as has been suggested in the case of the minimum wage for the yellow post sector. Again, modeling monopoly power on the product market is beyond the scope of this paper.

3.3 Heterogeneity on both sides of the market

If trying to explain the variance of paid wages between observationally equivalent workers, basically two components are required: first, there are firm effects on the wage, and, second, there is an effect of the degree of frictions on wages. In addition, wages vary considerably between workers with different observed characteristics, controlling for firm characteristics and search frictions. Summing up, three factors are needed to explain empirical wage distributions: heterogeneous firms, heterogeneous workers and search frictions (see Bontemps et  al. 2000; Abowd et  al. 1999). So far, the presented models explain wage variation by search frictions (Burdett and Mortensen 1998), by search frictions and heterogeneity of the employees (Burdett and Mortensen 1998), and by search frictions and exogenous technology differences (Bontemps et  al. 2000). The model presented in this section integrates the three important factors for the explanation of an empirical wage distribution. The model is due to Postel-Vinay and Robin (2002a).

3.3.1 Assumptions

Heterogenous productivity of the individuals is integrated into the model in the following way. Consider the labor market for a specific homogeneous professional group, where all homogenous job seekers are substitutable to a certain degree, however, individuals differ in their productivity as measured by an index ${ {\mathrm{\varepsilon}} }$. Individuals determine their productivity by drawing a value from the continuous productivity distribution ${ {\mathrm{\Omega}} ({\mathrm{\varepsilon}} ) }$ on an interval ${ [{\mathrm{\varepsilon}} _{\min },{\mathrm{\varepsilon}} _{\max }] }$ with density  ${ {\mathrm{\omega}} ({\mathrm{\varepsilon}} ) }$. It is assumed that unemployed job seekers of type  ${ {\mathrm{\varepsilon}} }$ obtain a net unemployment income of ${ z({\mathrm{\varepsilon}} )={\mathrm{\varepsilon}} b }$. w brings the individual the utility ${ {\mathrm{\Xi}} (\textit{w}) }$ and individuals maximize the present value of their expected utility over an infinite time horizon. Leisure does not enter the utility function of the individuals.

Each firm produces with technology y which is distributed according to a CDF ${ {\mathrm{\Gamma}} (y) }$ with density ${ {\mathrm{\gamma}} (y) }$ on a bounded support ${ [\underline{y},\bar{y}] }$ and which is determined by an ex ante random draw. A firm maximizes the present value of its expected profits over an infinite time horizon. It is assumed that the “home productivity” b exceeds ${ \underline{y} }$. Marginal productivity of an efficiency unit of labor is constant given the y-type of the firm. That is, an individual of type ${ {\mathrm{\varepsilon}} }$, and a firm of type y produce together an output ${ y{\mathrm{\varepsilon}} }$, a production function with homogeneity degree of  2.

The sequential process of contacts between employers and employees is as follows. Unemployed job seekers contact firms at rate λ, while employed job seekers contact firms at ${ {\mathrm{\lambda}}_{L} }$. It is assumed that each firm of type y makes wage offers to individuals with a specific probability that is identical over all ${ {\mathrm{\varepsilon}} }$-types. The contact probability for a type-y firm follows a distribution function ${ {\mathrm{\Psi}} (y) }$ with density ${ {\mathrm{\psi}} (y) }$. Postel-Vinay and Robin (2002a) argue that the relative frequency of contacts for a y-firm ${ {\mathrm{\psi}} (y)/{\mathrm{\gamma}} (y) }$ is determined by the search intensity of this firm. There is, however, no microfoundation for the ratio of ${ {\mathrm{\psi}} (y) }$ and ${ {\mathrm{\gamma}} (y) }$.

Upon meeting, both sides have complete information about all relevant characteristics of the other side. Therefore, the wage offer of the firm can condition on the type ${ {\mathrm{\varepsilon}} }$ of the job seeker. In addition, if an employee gets an outside job offer from a competing firm, the employing firm can make a binding counteroffer. Implications of this assumption are detailed in Postel-Vinay and Robin (2002b). In a model with endogenous technology dispersion, they show that job-to-job transitions depend basically on the productivities of the competing firms. Before discussing their derivations, assumptions (B0), (B1′′′′′), (B2), (B3′), (B4′′′), (B5′) and (B6′′) summarize the foundation of the model:

• (B1′′′′′), as (B1′), but N individuals maximize their utility function ${ {\mathrm{\Xi}} (\textit{w}) }$ and enter and leave the labor market at finite rate n. Newcomers enter the labor market as unemployed job seekers. Individuals differ in their productivity ${ {\mathrm{\varepsilon}} }$, according to a distribution function ${ {\mathrm{\Omega}} ({\mathrm{\varepsilon}} ) }$ on ${ [{\mathrm{\varepsilon}} _{\min },{\mathrm{\varepsilon}} _{\max }] }$. ${ {\mathrm{\varepsilon}} }$-type unemployed obtain ${ z={\mathrm{\varepsilon}} b }$. ${ W_{U}(\cdot ) (W_{L}(\cdot )) }$ are the values of unemployment and employment, respectively. When a job seeker and a firm meet, the probability that the productivity of the firm is below y is ${ {\mathrm{\Psi}} (y) }$.

• (B3′), as (B3), but matches dissolve at rate ${ {\mathrm{\delta}}+n }$, where δ is the finite job destruction rate and n is the rate at which individuals leave (and enter) the market.

• (B4′′′), as (B4), but firms differ in their productivity y, distributed according to ${ {\mathrm{\Gamma}} (y) }$ on ${ [\underline{y},\bar{y}] }$, with ${ \underline{y} \,{\textgreater}\, b }$. Upon meeting, the firm observes both the type ${ {\mathrm{\varepsilon}} }$ and the productivity y of the firm that presently employs the individual.

• (B5′), as (B5) but firms condition their wage offer ${ \textit{w}({\mathrm{\varepsilon}} ,y,\cdot ) }$ that is nonnegotiable on the type ${ {\mathrm{\varepsilon}} }$ of the individual and on the productivity y of the firm that presently employs the individual.

• (B6′′): Specific labor market, where r is the discount rate in this market and where a homogeneous product is produced from heterogeneous agents. A type y firm and a type ${ {\mathrm{\varepsilon}} }$ individual produce together the output ${ y{\mathrm{\varepsilon}} }$ of the homogeneous good. The price of the produced good is the numéraire.

3.3.2 The model

Let ${ W_{U}({\mathrm{\varepsilon}} ,b,{\mathrm{\varepsilon}} b)=W_{U}({\mathrm{\varepsilon}} ) }$ be the value of unemployment of an individual of type ${ {\mathrm{\varepsilon}} }$ and let ${ W_{L}({\mathrm{\varepsilon}} ,y,\textit{w}) }$ be the value of employment of the same ${ {\mathrm{\varepsilon}} }$-type, depending on the productivity of the employing firm and the paid wage. The value equation depends not only on the wage that the firm pays but also on the productivity of this firm. This is the case, since the productivity of the firm determines the career opportunities (in the sense of potential wage gains) in this firm. If an employed individual of type ${ {\mathrm{\varepsilon}} }$ obtains a wage offer from a competing firm, the upper bound of the wage increase for the individual is determined by the productivity of the employing firm. To see this, notice that the maximal counteroffer that the employing firm can make is bound by its productivity. However, because of perfect information, the competing firm will choose its wage offer such that the individual is indifferent between changing firms and not. Job seekers always obtain exactly their reservation wage upon engagement. For unemployed job seekers, this can be calculated from ${ W_{L}({\mathrm{\varepsilon}} ,y,\textit{w}_{\text{R}})=W_{U}({\mathrm{\varepsilon}} ) }$, where ${ \textit{w}_{\text{R}}=\textit{w}_{\text{R}}({\mathrm{\varepsilon}} ,y,z)=\textit{w}_{\text{R}}({\mathrm{\varepsilon}} ,y) }$. The reservation wage of the unemployed job seekers, like the reservation wage of employed job seekers depends on the career opportunities in the firm that makes the offer.

If a ${ y^{\prime } }$-firm with ${ y^{\prime } \,{\textgreater}\, y }$ makes an offer, it chooses the wage such that the individual is indifferent between the value of the highest wage ${ \textit{w}={\mathrm{\varepsilon}} y }$ he can get in his firm without career opportunities,²⁶ and the value of the wage with the positive career opportunities if changing to the ${ y^{\prime} }$-firm. Let ${ \textit{w}_{\textit{w}}({\mathrm{\varepsilon}} ,y,y^{\prime }) }$ be the wage that makes the ${ {\mathrm{\varepsilon}} }$-individual indifferent between the firms ${ y,y^{\prime } }$. Then: ${ W_{L}({\mathrm{\varepsilon}} ,y,{\mathrm{\varepsilon}} y)=W_{L}({\mathrm{\varepsilon}} ,y^{\prime },\textit{w}_{\textit{w}}) }$. If the competing firm has a lower productivity than the employing firm ${ y^{\prime } \,{\textless}\, y }$, then the firm is ready to pay at most ${ {\mathrm{\varepsilon}} y^{\prime } }$. The counteroffer that inhibits the individual from changing the firm is because of the better career opportunities smaller than ${ {\mathrm{\varepsilon}} y^{\prime } }$ and given by ${ \textit{w}_{\textit{w}}({\mathrm{\varepsilon}} ,y^{\prime },y) }$, where ${ W_{L}({\mathrm{\varepsilon}} ,y,\textit{w}_{\textit{w}})=W_{L}({\mathrm{\varepsilon}} ,y^{\prime },{\mathrm{\varepsilon}} y^{\prime }) }$ holds.

The value of unemployment can be derived as usual from the no-arbitrage condition, where the instantaneous income has to be replaced by the instantaneous utility ${ {\mathrm{\Xi}} ({\mathrm{\varepsilon}} b) }$. The value of a job offer is given by the value of unemployment since firms offer exactly the reservation wage to the individual. Future utility flows are discounted by the discount rate r plus the instantaneous mortality rate n. It follows ${ W_{U}({\mathrm{\varepsilon}} ) = \frac{1}{1+(r+n) {\text{d}t} } \{{\mathrm{\Xi}} ({\mathrm{\varepsilon}} b) {\text{d}t} +{\mathrm{\lambda}} {\text{d}t} W_{U}({\mathrm{\varepsilon}} )+(1-{\mathrm{\lambda}} {\text{d}t} )W_{U}({\mathrm{\varepsilon}} ) \} }$, or:

$$ W_{U}({\mathrm{\varepsilon}} )=\frac{{\mathrm{\Xi}} ({\mathrm{\varepsilon}} b)}{r+n} \,. {} $$

The value of employment contains several components. If ${ {\mathrm{\varepsilon}} }$-individuals that are employed at wage w in a y-firm obtain an offer from a competing firm, three possibilities arise. First, if the productivity ${ y^{\prime } }$ from the competing firm is so small that the employing firm could poach ${ {\mathrm{\varepsilon}} }$-employees from the ${ y^{\prime } }$-firm for a wage ${ \textit{w}_{\textit{w}}({\mathrm{\varepsilon}} , y^{\prime },y) \,{\textless}\, \textit{w} }$, nothing changes. Let the critical productivity of a competing firm for which ${ \textit{w}_{\textit{w}}({\mathrm{\varepsilon}} ,\check{y},y)=\textit{w} }$ holds, be ${ \check{y} ({\mathrm{\varepsilon}} ,y,\textit{w}) }$. Then the probability that the offer does not change anything is given by ${ {\mathrm{\Psi}} (\check{y}) }$. The second possibility is that ${ \check{y} \,{\textless}\, y^{\prime } \,{\textless}\, y }$. That is, the competing firm cannot win the Bertrand-competition, but is able to offer employees a higher value than in the current firm with current wage. That is, these employees get the wage increase ${ \textit{w}_{\textit{w}}({\mathrm{\varepsilon}} ,y^{\prime },y)-\textit{w} }$ and their new value of work is given by ${ W_{L}({\mathrm{\varepsilon}} ,y^{\prime },{\mathrm{\varepsilon}} y^{\prime }) }$. This happens with probability ${ {\mathrm{\Psi}} (y)-{\mathrm{\Psi}} (\check{y}) }$. The value equation must account for the expected value of labor over the productivity of the competing firms in this case. Finally, if the productivity of the competing firm ${ y^{\prime } }$ is higher than the productivity of the employing firm y, it wins the Bertrand-competition, employees change the firm and the wage changes by the amount ${ \textit{w}_{\textit{w}}({\mathrm{\varepsilon}} ,y,y^{\prime })-\textit{w} }$.²⁷ With the corresponding probability ${ (1-{\mathrm{\Psi}} (y)) }$ the new value of work is then ${ W_{L}({\mathrm{\varepsilon}} ,y^{\prime },\textit{w}_{\textit{w}}(\cdot ))=W_{L}({\mathrm{\varepsilon}} ,y,{\mathrm{\varepsilon}} y) }$. The value equation summarizes these possibilities:

$$ \begin{aligned} &[ r + {\mathrm{\delta}} + n + {\mathrm{\lambda}} _{L}(1-{\mathrm{\Psi}} (\check{y}(\cdot ))) ] W_{L}({\mathrm{\varepsilon}} ,y,\textit{w}) = {\mathrm{\Xi}} (\textit{w}) \,\,+ \\ &{\mathrm{\delta}} W_{U}({\mathrm{\varepsilon}} )+ {\mathrm{\lambda}} _{L}[{\mathrm{\Psi}} (y)-{\mathrm{\Psi}} (\check{y}(\cdot ))]E_{{\mathrm{\Psi}} }[W_{L}({\mathrm{\varepsilon}} ,x,{\mathrm{\varepsilon}} x)|\check{y} \,{\textless}\, x \,{\textless}\, y] \\ &{\!\!}+{\mathrm{\lambda}} _{L}(1-{\mathrm{\Psi}} (y))W_{L}({\mathrm{\varepsilon}} ,y,{\mathrm{\varepsilon}} y) \,. \\[-27pt] &{} \end{aligned} $$

(20)

Evaluating this formula at ${ \textit{w}={\mathrm{\varepsilon}} y }$, I  find that:

$$ W_{L}({\mathrm{\varepsilon}} ,y,{\mathrm{\varepsilon}} y)=\frac{{\mathrm{\Xi}} ({\mathrm{\varepsilon}} y)+{\mathrm{\delta}} W_{U}({\mathrm{\varepsilon}} )}{r+{\mathrm{\delta}} +n} \,. {} $$

(21)

This is true, since ${ \check{y}({\mathrm{\varepsilon}} ,y,{\mathrm{\varepsilon}} y)=y }$. Using Eq.  (21) for the conditional expectation in Eq.  (20) together with the conditional distribution ${ {\mathrm{\Psi}} (x|\check{y} \,{\textless}\, x \,{\textless}\, y) = \frac{{\mathrm{\Psi}} (x)}{{\mathrm{\Psi}} (y)-{\mathrm{\Psi}} (\check{y})} }$, using in addition integration by parts, the fact that ${ W_{L}({\mathrm{\varepsilon}} ,y,\textit{w})=W_{L}({\mathrm{\varepsilon}} ,\check{y},{\mathrm{\varepsilon}} \check{y}) }$ and the relationship ${ {\mathrm{\lambda}} _{L}\left( \frac{{\mathrm{\Xi}} ({\mathrm{\varepsilon}} y)-{\mathrm{\Xi}} ({\mathrm{\varepsilon}} \check{y})}{r+{\mathrm{\delta}} +n}\right) =\frac{{\mathrm{\lambda}} _{L}{\mathrm{\varepsilon}} }{r+{\mathrm{\delta}} +n}\int_{\check{y}}^{y}{\mathrm{\Xi}} ^{\prime }({\mathrm{\varepsilon}} x)dx }$, a new expression for the value of work is obtained.

$$ \begin{aligned} (r+{\mathrm{\delta}} +n)W_{L}({\mathrm{\varepsilon}} ,y,\textit{w}) =&{\,\,}{\mathrm{\Xi}} (\textit{w})+{\mathrm{\delta}} W_{U}({\mathrm{\varepsilon}} ) +\frac{{\mathrm{\lambda}} _{L}{\mathrm{\varepsilon}} }{r+{\mathrm{\delta}} +n}\\ &\times \int_{\check{y}}^{y}(1-{\mathrm{\Psi}} (x)){\mathrm{\Xi}} ^{\prime }({\mathrm{\varepsilon}} x)dx \end{aligned} $$

(22)

The return to working at wage w can be decomposed in the instantaneous utility of the wage minus the loss in the case where the job is lost ${ -({\mathrm{\delta}} (W_{L}(\cdot )-W_{U}(\cdot ))+nW_{L}(\cdot )) }$ plus the instantaneous probability to get an offer of a competing firm times the expected discounted utility gain in this case.²⁸ Using that by definition ${ W_{L}({\mathrm{\varepsilon}} ,y,\textit{w})=W_{L}({\mathrm{\varepsilon}} ,\check{y},{\mathrm{\varepsilon}} \check{y}) }$ and Eq.  (21) on the left-hand side of Eq.  (22), ${ {\mathrm{\Xi}} (\textit{w})={\mathrm{\Xi}} ({\mathrm{\varepsilon}} \check{y})- \frac{{\mathrm{\lambda}} _{L}{\mathrm{\varepsilon}} }{r+{\mathrm{\delta}} +n}\int_{\check{y}}^{y}(1-{\mathrm{\Psi}} (x)){\mathrm{\Xi}} ^{\prime }({\mathrm{\varepsilon}} x)dx }$ is obtained. If a firm with ${ y^{\prime } \,{\textgreater}\, y }$ makes an offer to an employee, then the employee changes the firm and obtains the wage ${ \textit{w}_{\textit{w}}({\mathrm{\varepsilon}} ,y,y^{\prime }) }$. Plugging this into the last formula and using that ${ \check{y}({\mathrm{\varepsilon}} ,y^{\prime },\textit{w}_{\textit{w}}(\cdot ))=y }$, one obtains an implicit characterization of the wage an individual obtains when changing jobs.

$$ \begin{aligned} {\mathrm{\Xi}} (\textit{w}_{\textit{w}}({\mathrm{\varepsilon}} ,y,y^{\prime })) =&{\,\,}{\mathrm{\Xi}} ({\mathrm{\varepsilon}} y) - \frac{{\mathrm{\lambda}} _{L}{\mathrm{\varepsilon}} }{r+{\mathrm{\delta}} +n}\\ &\times \int_{y}^{y^{\prime }}(1-{\mathrm{\Psi}} (x)){\mathrm{\Xi}} ^{\prime }({\mathrm{\varepsilon}} x)dx \end{aligned} $$

(23)

Analogously, the reservation wage of an unemployed job seeker when obtaining an offer from a ${ y^{\prime } }$-firm ${ ( y^{\prime } \,{\textgreater}\, b ) }$ is given implicitly by using ${ \textit{w}_{\text{R}}({\mathrm{\varepsilon}} ,b,y^{\prime })=\textit{w}_{\textit{w}}({\mathrm{\varepsilon}} ,b,y^{\prime }) }$ and ${ \check{y}({\mathrm{\varepsilon}} ,y^{\prime },\textit{w}_{\text{R}}(\cdot ))=b }$.

$$ \begin{aligned} {\mathrm{\Xi}} (\textit{w}_{\text{R}}({\mathrm{\varepsilon}} ,b,y^{\prime })) =&{\,\,}{\mathrm{\Xi}} ({\mathrm{\varepsilon}} b) - \frac{{\mathrm{\lambda}} _{L}{\mathrm{\varepsilon}} }{r+{\mathrm{\delta}} +n}\\ &\times \int_{b}^{y^{\prime }}(1-{\mathrm{\Psi}} (x)){\mathrm{\Xi}} ^{\prime }({\mathrm{\varepsilon}} x)dx {} \end{aligned} $$

(24)

Both wages are reservation wages in the sense that they correspond to the minimum wage offer that a type ${ y^{\prime } }$-firm must make ${ {\mathrm{\varepsilon}} }$-individuals to induce them to work in this firm. In both cases this reservation wage depends on the current productivity, either of the employing firm or the home productivity. Since a firm with ${ y^{\prime } \,{\textgreater}\, y }$ offers career opportunities, the wage ${ \textit{w}_{\textit{w}}(\cdot ) }$ it pays is lower than the maximal wage a y-firm can afford. The discounted value of the career opportunities is given by the second addend in Eq.  (23). Thus, the model generates voluntary job-to-job transitions under wage cuts. The analog holds for the reservation wage of the unemployed; it is lower than the value of their home production.

Paid wages are either the first wage ${ \textit{w}_{\text{R}}({\mathrm{\varepsilon}} ,b,y^{\prime }) }$ or a wage that results from a Bertrand-competition between two firms ${ y,y^{\prime } }$, that is, ${ \textit{w}_{\textit{w}}({\mathrm{\varepsilon}} ,y^{\prime },y) }$ with ${ \check{y} \,{\textless}\, y^{\prime } \,{\textless}\, y }$ (or ${ \textit{w}_{\textit{w}}({\mathrm{\varepsilon}} ,y,y^{\prime }) }$, if ${ y^{\prime } \,{\textgreater}\, y }$). So there are always three components contained in the wage: individual productivity, firm productivity and luck. For a CRRA-utility function (such as ${ {\mathrm{\Xi}} (\textit{w})=\ln \textit{w} }$) the reservation wage from Eq.  (23) can be decomposed additively into its three components, ${ \ln \textit{w}_{\textit{w}}({\mathrm{\varepsilon}} ,y,y^{\prime })=\ln {\mathrm{\varepsilon}} +\ln \textit{w}_{\textit{w}}(1,y,y^{\prime })=\ln {\mathrm{\varepsilon}} +\ln y+\frac{{\mathrm{\lambda}} _{L}}{r+{\mathrm{\delta}} +n}\int_{y}^{y^{\prime }}\frac{1}{x}(1-{\mathrm{\Psi}} (x))dx }$ (see Postel-Vinay and Robin 2002a, p. 2305). From this, the decomposition of the variance of paid wages can be derived the model from which it was motivated (see Appendix  4).

$$ \begin{aligned} var_{\textit{w}} (\ln \textit{w}) =&{\,\,}var_{{\mathrm{\varepsilon}} } (\ln {\mathrm{\varepsilon}} ) + var_{y} [E_{(y^{\prime }|y)} (\ln \textit{w}_{\textit{w}}(1,y,y^{\prime })|y)] \notag \\ &+ E_{y} [var_{(y^{\prime }|y)} (\ln \textit{w}_{\textit{w}} (1,y,y^{\prime })|y)] \end{aligned} $$

The variance of wages can be decomposed into a component that is attributable to individual productivity differences  (${ {\mathrm{\varepsilon}} }$), a component that comes from different firm productivities  (y) and a component that comes from labor market frictions (see also Appendix  4).

Before characterizing the equilibrium, some additional definitions must be made. Let ${ L({\mathrm{\varepsilon}} ,y) }$ be the share of individuals whose productivity is below ${ {\mathrm{\varepsilon}} }$ and that are employed in firms with productivity below y. Then, ${ L_{y}(y)=\int_{{\mathrm{\varepsilon}} _{\min }}^{{\mathrm{\varepsilon}} _{\max }}L({\mathrm{\varepsilon}} ,y)d{\mathrm{\varepsilon}} }$ is the share of individuals that are employed in firms with productivity below y. Let ${ l({\mathrm{\varepsilon}} ,y) }$ and ${ l_{y}(y) }$ be the corresponding densities. Further, let ${ G(\textit{w}|{\mathrm{\varepsilon}} ,y) }$ be the conditional distribution of paid wages. Equilibrium unemployment follows from ${ {\mathrm{\lambda}} u=(1-u)({\mathrm{\delta}} +n) }$, or

$$ \begin{aligned} &u = \frac{{\mathrm{\delta}} +n}{{\mathrm{\lambda}} +{\mathrm{\delta}} +n} \,.\\[-12pt] & {} \end{aligned} $$

(25)

This is the usual condition for unemployment, accounting, however, for the fact that there is turnover in the population.

The conditional wage distribution in equilibrium is characterized by the following. There are ${ G(\textit{w}|{\mathrm{\varepsilon}} ,y)l({\mathrm{\varepsilon}} ,y) }$ ${ N(1-u) }$ employees of type ${ {\mathrm{\varepsilon}} }$, who are employed in a type y-firm at wage below w. Employees of this category leave the class either because of job destruction δ, death n, or because they obtain an offer from a firm whose productivity is above ${ \check{y}(\cdot ) }$. So, outflows amount to ${ ({\mathrm{\delta}} +n+{\mathrm{\lambda}} _{L}(1-{\mathrm{\Psi}} (\check{y} )))G(\textit{w}|{\mathrm{\varepsilon}} ,y)l({\mathrm{\varepsilon}} ,y)N(1-u) }$. Inflows come from the pool of unemployed ${ {\mathrm{\lambda}} {\mathrm{\psi}} (y){\mathrm{\omega}} ({\mathrm{\varepsilon}} ) }$, since firms offer the reservation wage, which is always acceptable to the individuals.²⁹ On the other hand, y-firms can poach ${ {\mathrm{\varepsilon}} }$-individuals to a wage below w from firms, whose productivity is below ${ \check{y}({\mathrm{\varepsilon}} ,y,\textit{w}) }$. The expected flow is given by ${ {\mathrm{\lambda}} _{L}N(1-u){\mathrm{\psi}} (y)\int_{\underline{y}}^{\check{y}({\mathrm{\varepsilon}} ,y,\textit{w})}l({\mathrm{\varepsilon}} ,x)dx }$. In equilibrium, using Eq.  (25) and canceling out ${ N(1-u) }$, it follows:

$$ \begin{aligned} &({\mathrm{\delta}} +n+{\mathrm{\lambda}} _{L}(1-{\mathrm{\Psi}} (\check{y})))G(\textit{w}|{\mathrm{\varepsilon}} ,y)l({\mathrm{\varepsilon}} ,y) =\\ &\qquad \qquad \left[ ({\mathrm{\delta}} +n){\mathrm{\omega}} ({\mathrm{\varepsilon}} )+{\mathrm{\lambda}} _{L}\int_{\underline{y}}^{ \check{y}({\mathrm{\varepsilon}} ,y,\textit{w})}l({\mathrm{\varepsilon}} ,x)dx\right] {\mathrm{\psi}} (y) \,. \\[-21pt] & {} \end{aligned} $$

(26)

Evaluating this equality at ${ \textit{w} }$ ${ = }$ ${ {\mathrm{\varepsilon}} y }$, and using that ${ G({\mathrm{\varepsilon}} y|{\mathrm{\varepsilon}} ,y) }$ ${ = }$ ${ 1 }$ and ${ \check{y}({\mathrm{\varepsilon}} ,y,\textit{w}) }$ ${ = }$ ${ y }$, ${ ({\mathrm{\delta}} +n+{\mathrm{\lambda}} _{L}(1-{\mathrm{\Psi}} (y)))l({\mathrm{\varepsilon}} ,y) = \left[ ({\mathrm{\delta}} +n){\mathrm{\omega}} ({\mathrm{\varepsilon}} )+{\mathrm{\lambda}} _{L}\int_{\underline{y}}^{y}l({\mathrm{\varepsilon}} ,x)dx\right] {\mathrm{\psi}} (y) }$ is obtained. The solution to this differential equation is (ibid., p. 2341):

$$ \begin{aligned} l({\mathrm{\varepsilon}} ,y)&=\frac{(1+\kappa _{L}){\mathrm{\psi}} (y)}{[1+\kappa _{L}(1-{\mathrm{\Psi}} (y))]^{2}}{\mathrm{\omega}} ({\mathrm{\varepsilon}} )\\ &=l_{y}(y){\mathrm{\omega}} ({\mathrm{\varepsilon}} )\,, {\,\,}\text{with}{\,\,} \kappa _{L}=\frac{{\mathrm{\lambda}} _{L}}{{\mathrm{\delta}} +n} \,. {} \end{aligned} $$

(27)

Using the primitive ${ L({\mathrm{\varepsilon}} ,y)=\frac{{\mathrm{\Psi}} (y)}{1+\kappa _{L}(1-{\mathrm{\Psi}} (y))}{\mathrm{\Omega}} ({\mathrm{\varepsilon}} ) = L_{y}(y) }$ ${ {\mathrm{\Omega}} ({\mathrm{\varepsilon}} ) }$ and ${ l({\mathrm{\varepsilon}} ,y) }$ this is easily checked. Equation  (27) basically says that the employment of type ${ {\mathrm{\varepsilon}} }$ individuals is independent of the type y of the firm, i.e., there is no sorting.

Using Eq.  (27), Eq.  (26) can be solved for ${ G(\textit{w}|{\mathrm{\varepsilon}} ,y) }$. The conditional distribution of wages for ${ {\mathrm{\varepsilon}} }$-individuals in y-firms is given by:

$$ G(\textit{w}|{\mathrm{\varepsilon}} ,y)=\left( \frac{1+\kappa _{L}(1-{\mathrm{\Psi}} (y))}{1+\kappa _{L}(1-{\mathrm{\Psi}} (\check{y}({\mathrm{\varepsilon}} ,y,\textit{w})))}\right) ^{2} \,. {} $$

(28)

The equilibrium size of a y-firm can be derived by the equilibrium conditions as ${ N\frac{l_{y}(y)}{{\mathrm{\gamma}} (y)}=\frac{N(1+\kappa _{L})}{ [1+\kappa _{L}(1-{\mathrm{\Psi}} (y))]^{2}}\frac{{\mathrm{\psi}} (y)}{{\mathrm{\gamma}} (y)} }$. The first term implies that the size of a firm increases with the productivity of the firm since, upon meeting, they are more often capable of attracting individuals from competing firms than low productivity firms. The second term reflects by assumption the search intensity of a firm, which can increase or decrease with firm productivity. So, firm size does not uniquely depend on productivity. Since the random variable ${ {\mathrm{\varepsilon}} }$ does not depend on the random vector ${ (y,y^{\prime }) }$ and since this implies independence between ${ {\mathrm{\varepsilon}} }$ and y, the conditional distribution of ${ (y^{\prime }|y) }$ can be derived. Let ${ y^{\prime } \,{\textless}\, y }$, then ${ G(\textit{w}_{\textit{w}}({\mathrm{\varepsilon}} ,y,y^{\prime })|{\mathrm{\varepsilon}} ,y)=G({\mathrm{\varepsilon}} y^{\prime }|{\mathrm{\varepsilon}} ,y)=\frac{{\mathrm{\Omega}} ({\mathrm{\varepsilon}} ) \breve{G}(y,y^{\prime })}{{\mathrm{\Omega}} ({\mathrm{\varepsilon}} )L_{y}(y)}=\tilde{G}(y^{\prime }|y)=\left( \frac{1+\kappa _{L}(1-{\mathrm{\Psi}} (y))}{1+\kappa _{L}(1-{\mathrm{\Psi}} (y^{\prime }))}\right)^{2} }$. This is true since ${ \check{y}({\mathrm{\varepsilon}} ,y^{\prime },\textit{w}_{\textit{w}})=y^{\prime } }$.

Summarizing, it is to be noticed that the model integrates the three factors that are empirically important for explaining wage dispersion into a theoretical framework. An attractive feature of the model is that it provides a rationale for voluntary job-to-job transitions under wage cuts, since this seems to be a phenomenon that is empirically important (for Germany, see Fitzenberger and Garloff 2007; Pfeiffer 2003). On the other hand, (real) wage cuts for job stayers seem to be empirically important too (Postel-Vinay and Robin 2002a; Pfeiffer 2003, p. 2313f., p. 40ff.). Of course, this cannot be explained by the model. A further interesting result is that ${ {\mathrm{\varepsilon}} }$ and y are distributed independently, which implies that there is no sorting within professional groups. This is implied by Eq.  (27).³⁰ Although it is true that more productive firms prefer employing more productive individuals, and although they can attract them if competing with low productive firms, firms can earn positive profits for each employee. Therefore, in the model, they employ everybody and in equilibrium there is no sorting. The limitation to professional groups is important, since there is evidence for positive assortative matching in labor markets (Van den Berg and Van Vuuren 2006).

Now, introducing a binding minimum wage has multiple effects. Assume that ${ \textit{w}_{\min } \,{\textgreater}\, \underline{y} {\mathrm{\varepsilon}} _{\min } }$. There is a set of matches that are no longer profitable and they are not performed. For workers with productivity ${ {\mathrm{\varepsilon}} _{\text{crit}}<\frac{\textit{w}_{\min }}{\underline{y}} }$, i.e. workers that are affected by the minimum wage, the specific unemployment rate is higher. However, those workers that are still employed and earn higher wages. Let ${ y_{\text{crit}}=\frac{\textit{w}_{\min }}{{\mathrm{\varepsilon}} _{\min }} }$ denote the productivity boundary, above which all matches are profitable. Then firms with ${ y \,{\textless}\, y_{\text{crit}} }$ have less employees and make lower profits than in the absence of the minimum wage. For all other firms there is a number of matches that are in fact profitable, but which require the firms to pay the obligatory minimum wage instead of the reservation wage. This reduces profits of the firms but leads to higher average wages for the employees. Summarizing, in this model, the effects of a binding minimum wage on employment is negative, while rents are distributed from firms to workers.

With respect to the German situation, this model is clearly the most realistic of those seen so far. Imagine many of these models, one for each sector. This corresponds to the German situation of multiple minimum wages according to sectors. Then, the models clearly show that we will have negative employment effects for the lowest productive individuals and that there is some redistribution from the firms towards the individuals that are still employed. How large these effects are in the end is then an empirical question, depending on the relative size of the minimum wage as compared to productivities and, thus, also on search frictions.

3.4 Production functions and marginal productivity

So far, the models discussed allow for differences both on the side of the firms and on the side of the individuals. They maintain, however, the assumption that the productivity of an individual does not depend on the number of individuals employed. Yet another possibility to introduce heterogeneity is the assumption of non-constant marginal productivities. Again, this difference is likely to be crucial for the effects of minimum wages on wages and employment. Spill-over effects between different skill-groups might become relevant. So far, there are only little attempts in the literature to incorporate this production function view in the search framework. One such attempt is Ridder and Van den Berg (1997), which draws on Manning (1993) and Mortensen and Vishwanath (1993, 1994). This model assumes a non-linear production function with one production factor, only. Introducing several skill groups and linking them over a production function is analytically very demanding. There are not many models performing such a task. One such model with several production factors linked over a production function will be discussed below and is due to Holzner and Launov (2010).

I  start by discussing the model where one production factor is allowed to have non-constant marginal productivity in a production function, depending on its use. For the presentation, I  refer to Ridder and Van den Berg (1997). Consider the framework as outlined in Sect.  2.2. As before, let ${ l(\textit{w}) }$ be the steady-state number of individuals employed in a firm that pays a wage w, and let ${ y(l(\textit{w})) }$ denote the output depending on the employment ${ l(\textit{w}) }$.³¹ Assume that y is concave ${ [ y^{\prime}(l) \,{\textgreater}\, 0,y^{\prime\prime}(l) \,{\textless}\, 0 ] }$ and that ${ y(0)=0 }$. Assume, in addition, that the job offer rate off-the-job and on-the-job are identical ${ {\mathrm{\lambda}}={\mathrm{\lambda}}_{L} }$.

The reservation wage is given by Eq.  (7) and reduces to the monetary value of obtaining unemployment benefits minus search costs, because the option value of unemployment vanishes through the equality of job offer rates, i.e. ${ \textit{w}_{\text{R}}=z }$. Equilibrium unemployment is unchanged and follows from the steady-state condition for inflows into and outflows from unemployment ${ u }$ ${ = }$ ${ \frac{{\mathrm{\delta}}}{{\mathrm{\delta}}+{\mathrm{\lambda}}} }$ since firms do not offer wages below the reservation wage. The objective function of the firms³² is given by

$$ \pi(\textit{w}) = y(l(\textit{w})) - \textit{w}l(\textit{w}) \,. {} $$

(29)

Concavity of y implies that there is a size ${ l(\textit{w}) }$ where ${ y^{\prime}(l(\textit{w})) \leq \textit{w} }$. This implies that, at the upper bound of the wage distribution, a mass point can be obtained. In this case it is present both in the wage offer distribution ${ H(\textit{w}) }$ and in the distribution of paid wages ${ G(\textit{w}) }$. It arises since at the employment level where the wage equals marginal productivity, every additional worker (even when obtained at that wage) would contribute a negative amount to the objective function and, thus, there is no incentive to pay a higher wage.³³ Still, firms paying a marginal productivity wage make positive profits because of the concavity.

Depending on the parameters, this model has several possible solutions. One solution is an equilibrium where all firms pay a common wage, which can be either equal to the reservation wage or equal to marginal productivity and guarantees employment ${ \frac{N{\mathrm{\lambda}}}{{\mathrm{\lambda}}+{\mathrm{\delta}}} }$.³⁴ The reservation wage solution is obtained if marginal productivity in the symmetric equilibrium is below or equal to z, i.e. ${ y^{\prime} \big( \frac{N{\mathrm{\lambda}}}{{\mathrm{\lambda}}+{\mathrm{\delta}}} \big) \leq z }$. In this case, it does not pay to deviate: paying a lower wage guarantees zero employees, whereas paying a higher wage increases the number of employees thereby decreasing marginal productivity. This makes the contribution of a an additional worker negative. So deviating does not pay and this is an equilibrium if profits are positive. Otherwise, there is no production. A dispersed equilibrium cannot exist in this case. To see this, recognize that for employment to be positive all wages must be above or equal to z. However, assume that z was the upper bound of the wage distribution (if it is higher, profits are even lower). Employment in the continuous part of the wage distribution is given by Eq.  (10). Employment at the upper bound of the wage distribution is higher than in the equal wages equilibrium ${ l(\textit{w}^{o}) = \frac{N{\mathrm{\lambda}}}{{\mathrm{\delta}}} \,{\textgreater}\, \frac{N{\mathrm{\lambda}}}{{\mathrm{\lambda}}+{\mathrm{\delta}}} }$. Thus, marginal productivity is strictly lower than z. Firms paying wage z want to shrink. Thus, there is no such equilibrium.

Similarly, unique equilibria with dispersed wages or partly dispersed wages are obtained under the following parameter constellations. If the profit at z for a dispersed equilibrium is higher than the profit at a common wage equilibrium (which equals marginal productivity) ${ \bar{\textit{w}}=y^{\prime} \big( \frac{N{\mathrm{\lambda}}}{{\mathrm{\lambda}}+{\mathrm{\delta}}} \big) \,{\textgreater}\, z }$, then an equilibrium with dispersed wages is obtained, since deviating from the common wage to the reservation wage pays directly. Otherwise, if

$$ y \left(\frac{N{\mathrm{\lambda}}}{{\mathrm{\delta}}+{\mathrm{\lambda}}}\right)-\bar{\textit{w}}\frac{N{\mathrm{\lambda}}}{{\mathrm{\delta}}+{\mathrm{\lambda}}} \,{\textgreater}\, y\left(\frac{N {\mathrm{\delta}} {\mathrm{\lambda}}}{({\mathrm{\delta}}+{\mathrm{\lambda}})^2}\right)-z\frac{N {\mathrm{\delta}} {\mathrm{\lambda}}}{({\mathrm{\delta}}+{\mathrm{\lambda}})^2} {} $$

(30)

the common wage equilibrium is obtained, since deviating does not pay (neither below, nor above).

Now, consider the dispersed equilibrium. Still, it is possible that there is a both a dispersed part and a mass point having a probability mass ${ {\mathrm{\gamma}}=Prob(\textit{w}=\bar{\textit{w}}) \,{\textless}\, 1 }$. Steady-state employment in these firms can be calculated by equating in- and outflows.³⁵

$$ {\mathrm{\lambda}}(N-{\mathrm{\gamma}} l(\bar{\textit{w}})) = {\mathrm{\delta}} l(\bar{\textit{w}}) {} $$

(31)

This yields ${ l(\bar{\textit{w}})=\frac{N{\mathrm{\lambda}}}{{\mathrm{\delta}}+{\mathrm{\lambda}} {\mathrm{\gamma}}} }$ as employment in one firm that offers the wage at the mass point. Clearly, the wage at the mass point must correspond to the marginal productivity given the employment, i.e. ${ y^{\prime} \big( \frac{N{\mathrm{\lambda}}}{{\mathrm{\delta}}+{\mathrm{\lambda}} {\mathrm{\gamma}}} \big)=\bar{\textit{w}} }$.³⁶ Paying the mass point or being in the continuous part must yield identical profits and thus:

$$ \begin{aligned} y \left(\frac{N{\mathrm{\delta}} {\mathrm{\lambda}}}{({\mathrm{\delta}}+{\mathrm{\lambda}})^2}\right)-z\frac{N{\mathrm{\delta}} {\mathrm{\lambda}}}{({\mathrm{\delta}}+{\mathrm{\lambda}})^2} =&{\,\,}y\left(\frac{N{\mathrm{\lambda}}}{{\mathrm{\delta}}+{\mathrm{\lambda}} {\mathrm{\gamma}}}\right) \\ &-y^{\prime}\left(\frac{N{\mathrm{\lambda}}}{{\mathrm{\delta}}+{\mathrm{\lambda}} {\mathrm{\gamma}}}\right)\frac{N{\mathrm{\lambda}}}{{\mathrm{\delta}}+{\mathrm{\lambda}} {\mathrm{\gamma}}} \,. \\[-24mm] & {} \end{aligned} $$

(32)

If there is a value between  0 and  1 for γ that solves this equation, then a mass point is obtained. Otherwise, there is no mass point in the wage distribution.

For the general case (unspecified production), it is not possible to obtain a closed form solution for H and G. Note, however, that the counterfactual shape of the wage density does not get lost. On the contrary, to compensate for the decreasing profit per worker because of a decreasing marginal productivity, employment must grow even faster with the wage than in the linear model.³⁷ Then, Eq.  (10) implies that the density is even steeper.

Increasing minimum wages might change the equilibrium obtained and compress the wage distribution. The probability of obtaining a mass point at the upper bound of the wage distribution increases. In general, however, this does not result in lower employment, since all equilibria but one yield the same employment. If minimum wages have employment effects, however, that means shifting from an equilibrium with production to an equilibrium where there is no production at all. Clearly, this is not a very realistic possibility.

With respect to the German situation, the model is about as appealing as the previous model. Here as well, one could imagine such a model for each sector, neglecting the possibility that individuals wander between sectors. From this, we would expect wage compression and no employment effects.

An extension of the baseline model to a production function with several skill groups i, each of size ${ q_{i} }$ with ${ \sum{q_{i}} }$ ${ = }$ ${ N }$, and heterogeneous production technologies as indexed by j is due to Holzner and Launov (2010). Note, that this is the most general model discussed in this paper: it allows for differences between firms employing different technologies and for differences in workers that vary in productivity a priori, depending on their use. Let me introduce some notation first. ${ \textit{w}_{ij} }$ is the wage offer a firm of type j offers to an individual of type i. Let ${ H_{ij} }$ be the wage offer distribution for firms that produce with a technology j for skill group i. In other words, ${ H_{ij} ( \textit{w}_{ij} ) }$ is the amount of type j firms that offer a wage below ${ \textit{w}_{ij} }$ for type i individuals. ${ H_{i} }$ is the wage offer distribution for skill group i aggregated over all firm types, i.e. the distribution individuals care about, when looking for a job. Finally, ${ H_{j} }$ is the I-dimensional wage offer distribution of type j firms to all skill groups. Similarly, ${ l_{i}(\textit{w}_{ij}) }$ gives the employment of skill group i in a type j firm that offers ${ \textit{w}_{ij} }$, while ${ l(\textit{w}_{j}) }$ is the I-dimensional employment of all skill groups in this firm.

The reservation wage ${ \textit{w}^{R}_{i} }$ for individuals of skill group  i is given by Eq.  (7) and is indexed by i. Skill-specific unemployment is given by equality of in- and outflows and determined by the skill-specific friction parameter ${ {\mathrm{\lambda}}_{i} }$.³⁸ In addition, the dynamics for each skill group is similar as in the standard model, meaning that Eq.  (10) holds for each skill group, indexed by i, except for the following modification. In the denominator ${ ({\mathrm{\delta}}+{\mathrm{\lambda}}_{L}(1-H(w)))^2 }$ must be replaced by ${ ({\mathrm{\delta}}+{\mathrm{\lambda}}_{L}(1-H_{i}(\textit{w}_{i})))({\mathrm{\delta}}+{\mathrm{\lambda}}_{L}(1-H_{i}(\textit{w}_{i}^{-}))) }$. The modification follows from the fact that ${ H_{i} }$ is allowed to contain mass points and thus ${ H_{i}(\textit{w}_{i})=H_{i}(\textit{w}_{i}^{-})+{\mathrm{\gamma}}_{i}(\textit{w}_{i}) }$. If there are no mass points, the original employment equation is obtained. Recognize that ${ {\mathrm{\lambda}}_{L},{\mathrm{\delta}} }$ are assumed identical across skill groups.

Firms with production technology j maximize their expected profit by choosing the wage vector ${ \textit{w}_{j} }$ ${ = }$ ${ (\textit{w}_{1j}, }$ ${ \textit{w}_{2j},{\ldots},\textit{w}_{Ij}) }$, ${ j=1,{\ldots},J }$ i.e.:

$$ \pi _{j}=\underset{\textit{w}_{j}}{\max }E \big[ y_{j}(l(\textit{w}_{j}))-\textit{w}_{j}^{\prime }l(\textit{w}_{j}) \big] \,. {} $$

(33)

Using a second order Taylor-approximation, ${ E[y_{j}(l(\textit{w}_{j}))-\textit{w}_{j}^{\prime }l(\textit{w}_{j})] }$ can be rewritten as ${ y_{j}(E(l(\textit{w}_{j})))-\textit{w}_{j}^{\prime }E(l(\textit{w}_{j})) }$. Note that, due to tractability reasons, it is assumed that firms do not react on short-run variations in employment, implying that firms specify their wage policy at the outset and do not change it. Holzner and Launov (2010) assume complementarity between the production factors (supermodularity) in the production function ${ y_{j} }$. This guarantees, provided a continuous distribution, that (type j) firms cover exactly the same position in the wage offer distribution of each skill group.³⁹ As above, if there exists a mass point, it exists at the upper bound ${ \bar{\textit{w}}_{ij} }$ of the skill-specific wage distribution  ${ H_{ij} }$ for firms with technology j. At this point, marginal productivity given the employment at that wage equals the wage ${ y^{\prime}_{j}(l_{ij}(\textit{w}_{ij}))=\textit{w}_{ij} }$.

It is assumed that firms' profits differ according to the employed technology j. In this case, firms sort according to their profitabilities in the skill-specific wage offer distributions, meaning that more profitable firms pay higher wages. Thus, the share of firms that offer wages below the upper bound ${ \bar{\textit{w}_{ij}} }$ of a skill-specific type j firm wage offer distribution equals the share of firms ${ s_{j} }$ with technology j and less profitable ${ H_{ij}(\bar{\textit{w}}_{ij})=s_{j} }$. The resulting skill-specific wage offer distributions ${ H_{i} }$ have no holes (connected support) and the reservation wage (of skill group i) is the lower bound of the wage offer distribution (of skill group i) as in the standard model. Excluding mass points, Holzner and Launov (2010) are able to derive an analytical form for the wage offer distribution. They show that depending on the degree of homogeneity of the production function, the model generates increasing or decreasing skill-specific wage offer densities ${ h_{ij} }$.⁴⁰ That means that they do not require differences in technologies to generate a well-shaped wage density, as opposed to the models discussed so far. In addition, they show that for higher wages decreasing densities are more likely.

Introducing a binding minimum wage has the following effects. Assume that a binding minimum wage is introduced for one skill group only. This compresses the wage distribution for this skill group from the left. In addition, the complete skill-specific wage distribution ${ H_{i} }$ is shifted to the right. This follows from the fact that the upper bound of the skill-specific wage offer distributions ${ \bar{\textit{w}}_{ij} }$ for each technology j depends positively on the lower bound. As long as the skill-specific wage offer distribution does not contain a mass point, the other wage offer distributions remain unchanged and firms still cover the same position in the wage distributions for each skill group. It is possible, however, that the increase of the binding minimum wage leads to a mass point. Increasing the minimum wage above marginal productivity of the most unprofitable technology would make it optimal for the firms to employ less individuals. This is, however, not allowed by the model. It is possible that the minimum wage increases to a level where firms with the most unprofitable technology make negative profits and, thus, are driven out of the market.

Independently of the precise effects on the wage distribution, employment effects of increasing a minimum wage are zero, since labor demand as represented by ${ {\mathrm{\lambda}}_{i} }$ does not react, even when firms go bankrupt. The reason is that ${ {\mathrm{\lambda}}_{i} }$ does not depend on the measure of active firms nor does it depend on the profitability of the use of a specific skill group (as long as the productivity of this skill group is high enough to guarantee some employment). Thus, the minimum wage redistributes rents from firms to workers and might eliminate unprofitable technologies but does not increase unemployment.

Now, with respect to the German institutional setting, introducing a binding minimum wage for several skill groups simultaneously is similar in its effects, with the modification that the point where a technology becomes unprofitable is attained faster.

Note that the existence of mass points in the wage distribution makes it reasonable to think about the rationing of jobs (Ridder and Van den Berg 1997), because there are cases where profits at the mass point are higher with lower employment. The model of Holzner and Launov (2010) does not account for this possibility, which must be seen as a drawback.

4 Conclusion

The aim of this paper is twofold. On the one hand, it presents models of the labor market that give ample consideration to the frictional character of the labor market. A number of models that explicitly take account of the incompleteness of information and the process of acquiring information are introduced. In the basic model, an equilibrium wage distribution, is derived for ex ante homogeneous employees and employers. In extensions to the model, the impact of differences between the employer and employee sides of the resulting equilibrium wage distribution are examined. Because heterogeneity of the actors in the labor market is seen as an important factor, it is sensible to study models that explicitly model heterogeneity on both market sides and that are still analytically tractable. All these equilibrium search models are seen to generate residual wage dispersion, which is an important component of wage dispersion as a whole (see, e.g. Juhn et  al. 1993; Katz and Autor 1999). Thus, these models contribute to our understanding of wage inequality and its changes.

Recently, the possibility to implement minimum wages in Germany has been extended dramatically in Germany. In addition, the scientific debate about employment effects of minimum wages in Germany has obtained new impulses by the paper of König and Möller (2008).⁴¹ So, the second aim of this paper was to discuss the impact of minimum wages on employment and realized wages under this frictional setting. It turns out that the compression hypothesis, i.e. the assumption that institutional wage compression leads to higher unemployment, is not supported by all models analyzed. I  obtain all sorts of employment effects: positive, zero, and negative ones. For example, in the case of continuous search costs, a minimum wage can even lead to a reduction of unemployment. The mechanism is that individuals in this case find more wage offers acceptable. Where I  obtain zero employment effects, this stems from the fact that search frictions guarantee firms a monopsonistic position on the labor market. A minimum wage then restricts the monopsony power of firms and redistributes rents from firms to workers. Labor demand effects do not occur or only in the unlikely case where the minimum wage increases so strong that all matches (for a certain skill group) become unprofitable. Only one of the proposed models support the compression hypothesis: namely, the model with heterogeneity on both sides of the market. In this model, this is the case, since the minimum wage makes a part of the matches unprofitable and, thus, not every meeting does result in a match. A further possibility to incorporate labor demand effects in a reasonable way in a model with search frictions is to endogenize λ. This is done in Fitzenberger and Garloff (2009) and is shown to imply negative employment effects of minimum wages. Note, however, that even with endogenous job offer rates, as implied by most models if wage bargaining, employment effects can be indeterminate as is shown, e.g., in Flinn (2006). The reason is that participation decisions might counteract such labor demand effects.

Summarizing, in the search context, no definite answer is available to the question of the influence of a binding minimum wage on unemployment. In the spirit of Koning et  al. (1995), one could argue that as long as the minimum wage is not too high, there are no employment effects of minimum wages and these only redistribute rents. However, considering the labor market as an ensemble of segmented specific labor markets suggests that a too high minimum wage could make a whole segment unprofitable (see Van den Berg and Ridder 1998). In this case there are pronounced employment effects. Note that when search frictions or more general monopsonistic structures are important, from the point of view of maximizing social welfare it can be desirable to introduce or increase a minimum wage in order to redistribute rents from firms to workers without incurring the cost of increasing unemployment (see, e.g. Manning 2003a).

Again, with respect to the particular German setting I  would argue that sector-specific minimum wages offer a possibility to introduce minimum wages that are not particularly harmful to employment. From the point of view of search theory, search frictions might differ across sectors and, thus, monopsony power and the desire to redistribute rents will differ. Sector-specific minimum wages could be set to this end. Whether they are indeed set to this end is another question.

In summary, search approaches offer a good alternative and complement to neo-classical model frameworks. The frictional framework provides a basis for a better understanding of labor market mechanisms in a world of imperfect information. It adds to our understanding as far as labor market dynamics is concerned and as far as the determinants of residual wage dispersion are concerned. Thus, the model is a valuable alternative framework, for evaluating labor market policies, as, for example, minimum wages. It turns out from our analysis above that the impact of minimum wages is complex. Further theoretical and empirical work is necessary to decide on the issue of employment and wage effects of a minimum wage legislation.

Executive summary

In the recent German debate about the introduction of legislated minimum wages, which often cites the argument from marginal productivity theory, was that minimum wages would always increase unemployment as soon as they are binding. However, contributions for the German constructions sector have shown that this is not necessarily the case (König and Möller 2008).

This paper analyses theoretical effects of minimum wages on employment and the wage distribution in an environment where wages are not set according to marginal productivity. This arises as a natural result in a model with search frictions (it takes time to find a job), as search frictions lend market power to firms. The paper relies on the assumption that this market imperfection, search frictions, is important and has to be taken into account.

I  review new developments in the field of equilibrium search theory, which has undergone a dynamic development in recent years. Equilibrium search theory, as compared to partial search theory, explicitly takes into account wage setting as a decision variable to the firms, thus allowing a firm reaction to any change of institutions. The special theoretical focus of the review is on the integration of heterogeneities on both sides of the market. Typically such individual firm heterogeneities are seen to be important as can be seen, for example, by the large amount of literature dealing with human capital endowments and the like. I  consider them also to be an important empirical fact of the labour market, and, thus, they have to be included in the model, if the models can be kept tractable. There are considerable advances in this field and heterogeneities have been incorporated successfully on both sides of the market (Burdett and Mortensen 1998; Bontemps et  al. 2000; Postel-Vinay and Robin 2002a). I  discuss each of these models in detail, considering both the assumptions and results.

In addition, I  discuss minimum wage effects and try to link them to the specific German institutional situation with sector specific minimum wages, rather than economy wide ones to contribute theoretical arguments to the above cited debate. In the homogeneous baseline case, i.e. the case where agents on both sides of the market are homogenous, I  show that a binding minimum wage, which is below the highest paid equilibrium wage, does not affect employment. This is due to the fact that the monopsony power of firms, caused by search frictions induces wages below marginal productivity. Increasing the lower bound of the wage distribution shifts the whole distribution but does not affect labour demand, since for every additional worker the profit is always positive.

In the heterogenous case theoretical results are mixed. There are cases where a binding minimum can induce a positive employment effect, since wage offers then tend to be more often acceptable. So is the case when reservation wages or search costs differ. There are also cases where employment effects can be negative, when firms with too low productivity are driven out of the market, or if a part of the worker-firm meetings do not yield profitable matches given the minimum wage. So, unfortunately theory does provide a clear guideline: There is no unique connection between unemployment and minimum wages, and the effect can be positive, zero or negative.

Summing up the paper warns to predict immediately negative employment effects of minimum wages and, thus, provides a theoretical underpinning for the results of König and Möller (2008). More careful empirical research is needed to decide the theoretical ambiguity.

Kurzfassung

Weil Mindestlöhne die Lohnverteilung stauchen, tragen sie zu einer hohen Arbeitslosigkeit insbesondere im Bereich der niedrig qualifizierten Arbeitnehmer bei, so eine gängige Hypothese in der wissenschaftlichen und wirtschaftspolitischen Diskussion. Diese Hypothese wurde wiederholt auch in der jüngsten Diskussion um die Ausweitung des deutschen Arbeitnehmerentsendegesetzes (ein branchenspezifischer Mindestlohn) aufgestellt.

Theoretisch beruht diese Folgerung auf dem Paradigma der Grenzproduktivitätsentlohnung. Häufig wird argumentiert, dass sich die Nachfrage nach gering qualifizierter Arbeit aufgrund von qualifikationsverzerrtem technischen Fortschritt und globalisierten Märkten verändert habe. Daher müssten Löhne in diesem Bereich flexibler sein als dies in vielen kontinentaleuropäischen Ländern der Fall ist. Gestützt wird diese Analyse durch die Beobachtung, dass in den USA in den vergangenen 30 Jahren eine deutliche Zunahme der Lohnungleichheit zu verzeichnen war, während bei den US-amerikanischen Arbeitslosenquoten kein eindeutiger Trend zu erkennen ist. In demselben Zeitraum ist in vielen europäischen Ländern eine deutliche Zunahme der Arbeitslosenquoten, insbesondere im Bereich der gering Qualifizierten, zu konstatieren, wohingegen sich bei den relativen Löhnen nur geringe Verschiebungen ergeben haben. Die in Europa vergleichsweise rigiden Mindestlöhne und/oder starke Gewerkschaften, so diese Argumentation, sind mitverantwortlich, dass das notwendige Fallen der Löhne im Niedriglohnsektor verhindert wurde und damit einer deutlichen Zunahme der Arbeitslosigkeit im Bereich gering qualifizierter Arbeit Vorschub geleistet wurde.

Diese Schlussfolgerung ist jedoch im theoretischen Rahmen der Suchtheorie, die der Existenz von imperfekten Märkten (in der Form zeitaufwändiger Suchprozesse) Rechnung trägt, keineswegs zwingend. Dort kann, im Gegenteil, ein Mindestlohn sogar dazu führen, dass die gleichgewichtige Beschäftigung steigt, da Arbeitslose Jobangebote dann häufiger akzeptieren und die Arbeitsnachfrage nicht reagiert.

Diesem Arbeitspapier liegt die Hypothese zugrunde, dass Suchfriktionen im Arbeitsmarktkontext relevant und somit zu modellieren sind. Das Ziel der Studie ist dabei zweigeteilt: Einerseits liefert das Arbeitspapier einen Überblick über neuere Entwicklungen im sich dynamisch entwickelnden Bereich der Suchtheorie. Hierbei wird ein besonderes Augenmerk darauf gerichtet, inwieweit es gelingt, bedeutenden Heterogenitäten (Stichwort: Humankapital) im Arbeitsmarktkontext Tribut zu zollen. Andererseits wird vor dem Hintergrund friktioneller Arbeitsmärkte untersucht, welchen Einfluss Mindestlöhne auf die gleichgewichtige Beschäftigung ausüben. Aufgrund der Nachfragemacht von Firmen in solchen Märkten liegt die Vermutung nahe, dass staatlich administrierte Mindestlöhne mindestens die beiden folgenden Effekte hat: Erstens sind Mindestlöhne in der Lage, durch Nachfragemacht entstehende Renten umzuverteilen und somit einen Effekt zu erzielen, der unabhängig von der Beschäftigungshöhe ist. Zweitens wird auch in einem Umfeld, das durch Suchfriktionen gekennzeichnet ist, die Arbeitsnachfage auf die Lohnhöhe reagieren, sodass auch hier Beschäftigungskonsequenzen zu erwarten sind.

Die dargestellten theoretischen Ansätze zeigen, dass es in der Literatur gelingt, Heterogenitäten auf beiden Marktseiten in den Kontext der gleichgewichtigen Suchtheorie zu integrieren, ein wichtiger Fortschritt in diesem Bereich.

Mit der gleichzeitig erfolgten empirischen Umsetzung dieser Modelle ist damit ein wichtiger Schritt hin zu einer empirisch validen, gemeinsamen Modellierung von Lohnbildung und Verweildauern in Beschäftigung und Arbeitslosigkeit erfolgt. Hinsichtlich des Einflusses von Mindestlöhnen auf die gleichgewichtige Beschäftigung sind die Ergebnisse der theoretischen Analyse uneindeutig.

Einerseits zeigen gleichgewichtige Suchmodelle, die auf der Annahme homogener Firmen und Arbeitnehmer beruhen, dass Mindestlöhne ausschließlich Renten umverteilen, ohne dass Beschäftigungseffekte zu verzeichnen sind. Damit wird zu Lasten von Unternehmensgewinnen ein Teil der Arbeitnehmerschaft besser gestellt. Jedoch tragen einfache Suchmodelle, die auf der Homogenitätsannahme basieren, dem tatsächlichen Arbeitsmarktgeschehen nur unzureichend Rechnung. In den Modellen andererseits, die der Heterogenität von Arbeitnehmern und Arbeitgebern Rechnung tragen, sind die Konsequenzen von Mindestlöhnen komplizierter. Hier kann nicht ausgeschlossen werden, dass der Einfluss gewerkschaftlicher Verhandlungsmacht auf die Löhne auch Beschäftigungskonsequenzen hat, die, sofern vorhanden, zumeist negativ ausfallen.

Die Hypothese, dass Mindestlöhne für die hohe Arbeitslosigkeit der Geringqualifizierten verantwortlich sein könnten, wird so von einem Teil der suchtheoretischen Literatur gestützt, während ein anderer Teil ihr widerspricht. Es ist festzuhalten, dass der theoretische Rahmen der Suchtheorie zumindest ein Warnschild aufstellt, bei Ursachen für die hohe Arbeitslosigkeit niedrig qualifizierter Arbeitnehmer, ausschließlich an Mindestlöhne zu denken.

Appendices

Appendix 1

1.1 Derivation of the reservation wage as a function of the parameters

Consider the derivative ${ \frac{\partial W_{L}(\bar{\textit{w}})}{\partial \bar{\textit{w}}} }$ from Eq.  (5). Rewriting Eq.  (5) as ${ W_{L}(\bar{\textit{w}})=\frac{\bar{\textit{w}}+{\mathrm{\delta}} W_{U}+{\mathrm{\lambda}} _{L}\int_{\bar{\textit{w}}}^{\infty }W_{L}(\textit{w}) {\text{d}H} (\textit{w})}{[r+{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}\int_{\bar{ \textit{w}}}^{\infty } {\text{d}H} (\textit{w})]} }$, the resulting derivative is given by ${ W_{L}^{\prime }( \bar{\textit{w}})=\frac{\partial W_{L}(\bar{\textit{w}})}{\partial \bar{\textit{w}}}=\frac{1}{r+{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-H(\bar{\textit{w}}))} }$.⁴² Denote with ${ \textit{w}^{o} }$ the upper limit of ${ H(\textit{w}) }$, then integration by parts yields ${ \int_{\textit{w}_{\text{R}}}^{\textit{w}^{o}}(W_{L}(\textit{w})-W_{U}) {\text{d}H} (\textit{w})=[(W_{L}(\textit{w})- }$ ${ W_{U})H(\textit{w})]_{\textit{w}_{\text{R}}}^{\textit{w}^{o} }-\int_{\textit{w}_{\text{R}}}^{\textit{w}^{o}}H(\textit{w})W_{L}^{\prime }(\textit{w})d\textit{w} }$. This leads to:

$$ \begin{aligned} \textit{w}_{\text{R}} &= z+({\mathrm{\lambda}} -{\mathrm{\lambda}} _{L}) \left[ {\!} W_{L}(\textit{w}^{o})-W_{U}- {\!\!} \int_{\textit{w}_{\text{R}}}^{\textit{w}^{o}} {\!\!\!}{\kern-0.5pt} H(\textit{w})W_{L}^{\prime }(\textit{w})d\textit{w} {\!} \right] \\ &= z+({\mathrm{\lambda}} -{\mathrm{\lambda}} _{L})\left[\int_{\textit{w}_{\text{R}}}^{\textit{w}^{o}}(1-H(\textit{w}))W_{L}^{ \prime}(\textit{w})d\textit{w}\right] \\ &= z+({\mathrm{\lambda}} -{\mathrm{\lambda}} _{L})\int_{\textit{w}_{\text{R}}}^{\textit{w}^{o}}\frac{1-H(\textit{w})}{r+{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-H(\textit{w}))}d\textit{w} \,. {} \end{aligned} $$

The second row follows by using ${ W_{L} }$ ${ (\textit{w}^{o}) }$ ${ - }$ ${ W_{U} }$ ${ = }$ ${ \int_{\textit{w}_{\text{R}}}^{\textit{w}^{o}} }$ ${ W_{L}^{\prime }(\textit{w})d\textit{w} }$ and the above expression for ${ W_{L}^{\prime }(\textit{w}) }$.

Appendix 2

1.1 Derivation of the equilibrium employment at wage w

Starting point for the derivation of Eq.  (10), is the following equation which describes inflows and outflows to firms paying wages above w, ${ ({\mathrm{\lambda}} U+{\mathrm{\lambda}} _{L}L(\textit{w}))(1-H(\textit{w}))={\mathrm{\delta}} (N-U-L(\textit{w})) }$.

Differentiating both sides of the equation with respect to the wage, substituting ${ h(\textit{w}) }$ for ${ H^{\prime }(\textit{w}) }$, using that ${ L^{\prime }(\textit{w})=l(\textit{w})h(\textit{w}) }$, and dividing by ${ h(\textit{w}) }$ yields:

$$ [ {\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-H(\textit{w}))]l(\textit{w})={\mathrm{\lambda}} U+{\mathrm{\lambda}} _{L}L(\textit{w}) \,. {} $$

(34)

Firms maximize expected profits ${ \Pi (\textit{w})=(y-\textit{w})l(\textit{w}) }$. The first-order condition for a profit maximum yields the following differential equation:

$$ \frac{l^{\prime }(\textit{w})}{l(\textit{w})}=\frac{1}{y-\textit{w}} \,. {} $$

(35)

This equation holds for all firms that pay wages above ${ \textit{w}_{\text{R}} }$. With the help of Eq.  (35) ${ l(\textit{w}) }$ can be determined explicitly. Integrating both sides ${ \int \frac{l^{\prime }(\textit{w})}{ l(\textit{w})}d\textit{w}=\int \frac{1}{y-\textit{w}}d\textit{w} }$ or ${ \log l(\textit{w})+ }$ ${ d_{1} }$ ${ = }$ ${ - }$ ${ \log }$ ${ (y-\textit{w})+d_{2} }$ is obtained, where ${ d_{1} }$ and ${ d_{2} }$ are integration constants. Letting ${ d=d_{2}-d_{1} }$ and exponentiating both sides yields:

$$ l(\textit{w})=\frac{\exp (d)}{y-\textit{w}}=\frac{D}{y-\textit{w}} \,. {} $$

(36)

The integration constant D can be derived through the constraint that Eq.  (34) imposes on the above equation. Evaluating Eqs.  (34) and  (36) at ${ \textit{w}_{\text{R}} }$ and imposing equality, I  obtain ${ l(\textit{w}_{\text{R}})=\frac{{\mathrm{\lambda}} U}{{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}}=\frac{D}{y-\textit{w}_{\text{R}}} }$ or ${ D=\frac{{\mathrm{\lambda}} U}{{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}}(y-\textit{w}_{\text{R}}) }$. Using D and U in Eq.  (36) provides the solution to the differential equation: ${ l(\textit{w})=\frac{{\mathrm{\lambda}} {\mathrm{\delta}} N}{({\mathrm{\delta}} +{\mathrm{\lambda}} _{L})({\mathrm{\delta}} +{\mathrm{\lambda}} )}\cdot \frac{y-\textit{w}_{\text{R}}}{y-\textit{w}} }$.

The equilibrium profits of a firm that pays a wage from the support of the wage distribution is given by ${ \Pi (\textit{w})=(y-\textit{w})l(\textit{w})=\frac{{\mathrm{\lambda}} {\mathrm{\delta}} N(y-\textit{w}_{\text{R}})}{({\mathrm{\delta}} +{\mathrm{\lambda}} _{L})({\mathrm{\delta}} +{\mathrm{\lambda}} )} }$.

The upper limit of the support of wage distribution ${ \textit{w}^{o} }$ can be calculated by inserting ${ \textit{w}^{o} }$ in Eq.  (34) and in ${ l(\textit{w})=\frac{{\mathrm{\lambda}} {\mathrm{\delta}} N}{({\mathrm{\delta}} +{\mathrm{\lambda}} _{L})({\mathrm{\delta}} +{\mathrm{\lambda}} )}\cdot \frac{y-\textit{w}_{\text{R}}}{y-\textit{w}} }$, and solving for ${ l(\textit{w}^{o}) }$, respectively. Noting that ${ L(\textit{w}^{o})=N-U }$, I  obtain

$$ \textit{w}^{o}=y-(y-\textit{w}_{\text{R}})\left( \frac{{\mathrm{\delta}} }{{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}}\right)^{2} \,. {} $$

(37)

as upper limit of the support of the wage distribution and the distribution of paid wages. Note that the highest paid wage is below the marginal productivity of the employees.

When the lower bound of the wage offer distribution is given by a binding minimum wage ${ \textit{w}_{\text{min}} }$ rather than by the reservation wage, we obtain the partial derivative with respect to the minimum wage:

$$ \frac{\partial \textit{w}^{o}}{\partial \textit{w}_{\text{min}}}=\left(\frac{{\mathrm{\delta}} }{ {\mathrm{\delta}} +{\mathrm{\lambda}}_{L} }\right) ^{2} \,. {} $$

(38)

which is uniquely positive as long as ${ \textit{w}_{\text{min}} \,{\textless}\, y }$.

The effect of an increasing minimum wage on the moments of the wage distribution can be calculated as follows. The moments are derived in Van den Berg and Ridder (1993), the expectation is given by:

$$ E_{G}(\textit{w}-\textit{w}_{\text{min}})=\left( y-\textit{w}_{\text{min}}\right) \left(1-\frac{ \protect{\mathrm{\delta}} }{\protect{\mathrm{\delta}} +\protect{\mathrm{\lambda}}_{L}}\right) \,. {} $$

We obtain

$$ \begin{aligned} &\frac{\partial }{\partial \textit{w}_{\text{min}}}E_{G}(\textit{w}-\textit{w}_{\text{min}})=-1\left(1-\frac{ {\mathrm{\delta}} }{{\mathrm{\delta}} +{\mathrm{\lambda}}_{L} }\right) \,{\textless}\, 0 \,, \\[-10pt] &{} \end{aligned} $$

which is negative, but its absolute value smaller than  1. That is, the expectation of w increases, but it increases by less than ${ \textit{w}_{\text{min}} }$.

For the variance, we have:

$$ var_{G}(\textit{w}) = \frac{1}{3}\left( y-\textit{w}_{\text{min}}\right) ^{2}\frac{{\mathrm{\lambda}}_{L} ^{2}{\mathrm{\delta}} }{\left( {\mathrm{\delta}} +{\mathrm{\lambda}}_{L} \right) ^{3}} \,. {} $$

The partial derivative is calculated as:

$$ \frac{\partial var_{G}(\textit{w})}{\partial \textit{w}_{\text{min}}}=-\frac{2}{3}\left( y-\textit{w}_{\text{min}}\right) \frac{{\mathrm{\lambda}}_{L} ^{2}{\mathrm{\delta}} }{\left( {\mathrm{\delta}} +{\mathrm{\lambda}}_{L} \right)^{3}} \,, {} $$

which is clearly negative as long as ${ y \,{\textgreater}\, \textit{w}_{\text{min}} }$.

Appendix 3

1.1 Profits with continuous productivity dispersion

The solution of Eq.  (18) ${ \Pi ^{\prime }(y)=\breve{l}(K(y)) }$ is obtained when integrating ${ \Pi (y)=\int_{\underline{y} }^{y}\Pi ^{\prime }({\mathrm{\varrho}} )d{\mathrm{\varrho}} =A+\int_{\underline{y}}^{y}\breve{l} (K({\mathrm{\varrho}} ))d{\mathrm{\varrho}} }$. A  is the integration constant and follows from Eq.  (16), when evaluated at ${ (\underline{y},\underline{\textit{w}}) }$, where ${ \underline{\textit{w}}=\max \{\textit{w}_{\text{R}},\textit{w}_{\min }\} }$. Therewith, ${ A= \frac{{\mathrm{\delta}} }{{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}}\frac{N-U}{M}(\underline{y}-\underline{\textit{w}}) }$. Furthermore, the share of firms that pay wages below ${ K(y) }$ is equal to the share of firms whose productivity is below y: ${ H(K(y))={\mathrm{\Gamma}} (y) }$. Using Eq.  (15), it is ${ \breve{l}(K(y))={\mathrm{\delta}} \frac{(N-U)}{M}\frac{{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}}{[{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-{\mathrm{\Gamma}} (y))]^{2}} }$ and thus the profit function becomes:

$$ \begin{aligned} \Pi (y) =&{\,\,}\frac{{\mathrm{\delta}} }{{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}}\frac{N-U}{M}(\underline{y}- \underline{\textit{w}})\\ &+\int_{\underline{y}}^{y}{\mathrm{\delta}} \frac{(N-U)}{M}\frac{{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}}{[{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-{\mathrm{\Gamma}} ({\mathrm{\varrho}} )]^{2}}d{\mathrm{\varrho}} \\ \Pi (y) =&{\,\,}\int_{\underline{\textit{w}} }^{y}{\mathrm{\delta}} \frac{(N-U)({\mathrm{\delta}} +{\mathrm{\lambda}} _{L})}{M}\frac{1}{[{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-{\mathrm{\Gamma}} ({\mathrm{\varrho}} ))]^{2}}d{\mathrm{\varrho}} \,. \\[-21pt] & {} \end{aligned} $$

(39)

The second row follows from the fact that ${ {\mathrm{\Gamma}} (y)=0 }$ for ${ y\in [ \underline{\textit{w}},\underline{y}] }$ and thus the integral on the interval ${ [\underline{\textit{w}},\underline{y}] }$ in the second row, equals the first summand in the first row. This equation yields the profit of a type y firm depending on the model parameters and on the distribution of firm productivities. Solving ${ \Pi (y)=(y-K(y))\breve{l}(K(y)) }$ with respect to ${ K(y)=\textit{w} }$ yields an expression for the wage as a function of the productivity y: ${ \textit{w}=K(y)=y-\frac{\Pi (y)}{\breve{l}(K(y))} }$. Using the corresponding expressions yields: ${ K(y)=y-[{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-{\mathrm{\Gamma}} (y))]^{2}\int_{\underline{\textit{w}}}^{y}\frac{1}{[{\mathrm{\delta}} +{\mathrm{\lambda}} _{L}(1-{\mathrm{\Gamma}} ({\mathrm{\varrho}} ))]^{2}}d{\mathrm{\varrho}} }$.

Appendix 4

1.1 Variance analysis

Start by assuming a utility function with constant relative risk aversion (CRRA) and consider a worker who is employed in a firm of type y, then: ${ \ln \textit{w}_{\textit{w}}({\mathrm{\varepsilon}} ,y,y^{\prime })=\ln {\mathrm{\varepsilon}} +\ln \textit{w}_{\textit{w}}(1,y,y^{\prime }) }$. The conditional (on y) expectation of the log-wage is given by Postel-Vinay and Robin (2002a, p. 2310): ${ E_{({\mathrm{\varepsilon}} ,y^{\prime }|y)}(\ln \textit{w}|y)=E_{{\mathrm{\varepsilon}} }(\ln {\mathrm{\varepsilon}} )+E_{(y^{\prime }|y)}(\ln \textit{w}_{\textit{w}}(1,y,y^{\prime }) }$ ${ |y) }$.⁴³ Using the independence of ${ {\mathrm{\varepsilon}} }$ and ${ (y,y^{\prime }) }$, the conditional (on y) variance is given by ${ var_{({\mathrm{\varepsilon}} ,y^{\prime }|y)}(\ln \textit{w}|y)=var_{{\mathrm{\varepsilon}} }(\ln {\mathrm{\varepsilon}} )+var_{(y^{\prime }|y)}(\ln \textit{w}_{\textit{w}}(1,y,y^{\prime })|y) }$. Applying this variance decomposition, the variance of wages can be decomposed in the variance of the conditional expectation of wages and in the expectation of the conditional variance:

$$ \begin{aligned} var_{\textit{w}}(\ln \textit{w}) =&{\,\,}var_{y}[E_{({\mathrm{\varepsilon}} ,y^{\prime }|y)}(\ln \textit{w}|y)]\\ &+E_{y}[var_{({\mathrm{\varepsilon}} ,y^{\prime }|y)}(\ln \textit{w}|y)] \\ =&{\,\,}var_{y}[E_{{\mathrm{\varepsilon}} }(\ln {\mathrm{\varepsilon}} )+E_{(y^{\prime }|y)}(\ln \textit{w}_{\textit{w}}(1,y,y^{\prime })|y)]\\ &+E_{y}[var_{{\mathrm{\varepsilon}} }(\ln {\mathrm{\varepsilon}} ) \\ &+var_{(y^{\prime }|y)}(\ln \textit{w}_{\textit{w}}(1,y,y^{\prime })|y)] \,. {} \end{aligned} $$

Thus, the following decomposition of the wage variance is obtained:

$$ \begin{aligned} var_{\textit{w}}(\ln \textit{w}) =&{\,\,}var_{{\mathrm{\varepsilon}} }(\ln {\mathrm{\varepsilon}} )+var_{y}[E_{(y^{\prime }|y)}(\ln \textit{w}_{\textit{w}}(1,y,y^{\prime })|y)]\\ &+E_{y}[var_{(y^{\prime }|y)}(\ln \textit{w}_{\textit{w}}(1,y,y^{\prime })|y)] \,. {} \end{aligned} $$

The first summand of this formula results from productivity differences among employees. The second summand reflects the effect of different firm productivities on the variance of paid wages. The expected wage changes along with y, the productivity of the firm. The variance of the conditional expectation reflects the variance of wages between firms of different productivities. Note, that the conditional expectation of the log-wage and, thus, the variance of the second summand depends on the joint distribution of ${ y,y^{\prime } }$. The third summand reflects wage fluctuations for firms and workers whose productivity is identical. Thus, the wage fluctuations among identical individuals in identical firms are contained in this part.⁴⁴ From the point of view of an individual it is explained by the luck of receiving a valuable job offer that implies pay raises. The extent of this variance is explained by frictions because frequent job offers lead to a faster adjustment of wages to the marginal productivity and thus lowers the variance.