B.1 German wages: Imputed hourly wages from SIAB and GSOEP

Previous studies analyzed hourly wages for Germany by using the earnings surveys, the micro census (both provided by the Federal Statistical Office), or, more commonly, the German Socio-Economic Panel (GSOEP). Unfortunately, at the individual-level, neither the quarterly earnings survey, nor the micro census are freely available for research. Social security records, namely the Sample of Integrated Labour Market Biographies (SIAB), contains an imputed daily wage at the individual level, but lacks information about working hours. The GSOEP is used frequently because it is freely accessible for research and it contains information about monthly wages and hours worked per week.

In both micro data sets, all socially insured employees in full-time or part-time work are grouped by the following categories: age groups, gender, place of residence, and qualification. In addition to the previously described differences in the two surveys, variables such as employment status, qualification or wage have different definitions. To aggregate both micro data sets in groups is meaningful only, if these aggregates are conditional on corresponding characteristics in both data sources.

The conceptual discrepancy, i. e. definitions of variables and respondents of the two data sources, lead to differences in the wage measures in levels and its growth rates.^[21] However the two wage measures are highly correlated in levels as well as growth rates. An hourly wage measure is imputed in a synthetic panel based in group means of working hours from the GSOEP and median gross daily wages from the SIAB. The synthetic panel fills a lack in limited availability of hourly wage information for socially insured employees of all firm sizes.

B.2 Construction of relative wages and relative employment

The relative wage is defined as:

which is weighted by the relative employment γit=Lit/∑i=1NLit. The relative, weighted employment is defined as:

with average relative weighted wage of group i: ω¯i=∑t=1Tωit/T. For the description of relative participation, see Section 4.3 footnote 14.

C.1 Marshall: Market clearing

The model can be extended in a straightforward way to include endogenous labor supply with an analogue of Equation (5). Let labor supply be LS=S(W,Z) and assume that it is “everywhere upward-sloping” in the sense that Z contains the marginal utility of wealth (the Lagrange multiplier from the canonical labor supply problem), and that SW contains only substitution effects. Market clearing dLS=dLD=dL implies

and for the case of stable demand (dX=0)

which is a quadratic form in the k×1 vector DW−SW−1SZdZ. The fact that DW is negative definite plus dX=0 imply that dW′dL<0.

C.2 Pigou: Introducing wage rigidity

The wage is a linear combination of market clearing and exogenous rigid wage dW‾:

where ϕ is a constant 0<ϕ<1 that controls the extent of wage rigidity in the economy. The change in the market-clearing wage W∗ is the change in the Marshallian outcome, so after substitution

Because labor force participation equals labor supply, we have

The inner product dW′dP is given by

This will form the basis of predictions regarding the correlation of wages and participation.

D.1 Basic structure and continuation values for labor force participants

The exposition follows Mortensen and Pissarides (1999) and Pissarides (2000) and omits well-established proofs. The mass of total working population is fixed at 1, and can be in one of three labor market states: unemployment (u), nonparticipation (ℓ) and employment (e=1−u−ℓ). When out of the labor force, the worker receives monetary equivalent flow bε at each point in continuous time. Each worker in the labor force ℓ∈[0,1] has a valuation of nonparticipation described by a continuous cdf G(). b is the unemployment benefit paid to those searching for work (Arbeitslosengeld I), and ε measures the flow value of being outside of the labor force – leisure, value of education, social welfare (Arbeitslosengeld II), plus cost of active job search, all measured as a fraction of the unemployment benefit.

First, we study the sub-system conditional on labor force participation. In the steady state, the continuation valuations of the two possibilities for workers participating in the labor market (ILO definition) U and W are defined by functional equation

for unemployed workers. The prime (′) refers to the state in the next instant (after dt has transpired) conditional on a new draw of x (a shock actually having occurred). Similarly, for firms producing with a worker of productivity x

The reservation level of productivity R is the level of the idiosyncratic productivity shock x below which mutually agreed dissolution of the match occurs; there are no involuntary separations in this model. Shocks occur in the interval (t,t+dt) with Poisson incidence rate λ and x is distributed according to a time-invariant c. d. f. F(z).

The participation constraint implies

The wage, which serves to divide the match surplus, is determined by Nash bargaining:

with first order condition

The solution for the (state contingent) wage w(x) is

which, given the reservation level of utility can be rewritten as

A free-entry condition for vacancy posting by firms:

Constant unemployment implies du=0:

which completes the model. The unemployment rate is u=s(1−ℓ)s+θq(θ), where the separation rate s and nonparticipation ℓ are yet to be determined.

D.2 Modeling the participation margin

Following Pissarides (2000, Chapter 7), stock equilibrium between states of nonparticipation and unemployment is determined by indifference between participation and unemployment for the marginal worker with identity ℓ∈0,1, there are no frictions between the two states.^[22] While all workers value the state of unemployment at rU and receive identical income in nonparticipation (εb), each worker is heterogenous with respect to nonparticipation, described by a cumulative distribution function G(.), which also gives the worker’s unique identity on 0,1, with ℓ=0 having the highest, and ℓ=1 having the lowest (periodic) monetary valuation of nonparticipation. Aggregate equilibrium reflects indifference at the margin between participation and unemployment for the marginal worker, satisfying

The mass of nonparticipation ℓ is thus given by

Workers with low values of ℓ have the highest valuation of non-work time and are least likely to participate. Note that the right hand side is either parametric (b, β, c) or is endogenous and determined by free entry and zero profit condition on vacancies (θ).

We consider comparative-static changes in parameters b, β, and A; b describes the monetary periodic flow value to job searchers relative to ε. Because 0<ε<1, non-participation in equilibrium will be interior: 1>ℓ>0.

D.3 Equilibrium

In steady-state, W=W′, U=U′,J=J′ and V=V′. The firm’s valuation equations for the two states plus the free entry/exit condition for vacancies V=0 imply J=cq(θ)=p−wr+s; labor market tightness θ is fully determined by model parameters and the matching function according to

and RU into LP leads to the LP curve.

D.4 Comparative statics

We seek expressions for the following derivatives of equilibrium reservation productivity R, labor market tightness θ, and labor force nonparticipation ℓ with respect to the “Hartz-parameters” matching efficiency A and income of searching unemployed b, as well as worker bargaining power β. Total differentiation of JD, JC and LP, eliminating rU using RU and substituting q(θ)=Ax(θ−1,1), with dA,db,dβ≠0 and all other model parameters held constant, yields the following linearized system in dθ, dR, and dℓ:

where the functions g and x are evaluated at steady state values. Defining

we can use Cramer’s Rule to derive the following comparative statics results:

Using the JC curve cq(θ)=(1−β)1−Rr+λp, the last expression can be rewritten as