Public Employment Agency Reform, Matching Efficiency, and German Unemployment

Abstract

Our paper analyzes the role of public employment agencies in job matching, in particular the effects of the restructuring of the Federal Employment Agency in Germany (Hartz III labor market reform) for aggregate matching and unemployment. Based on two microeconomic datasets, we show that the market share of the Federal Employment Agency as job intermediary declined after the Hartz reforms. We propose a macroeconomic model of the labor market with a private and a public search channel and fit the model to various dimensions of the data. We show that direct intermediation activities of the Federal Employment Agency did not contribute to the decline in unemployment in Germany. By contrast, improved activation of unemployed workers reduced unemployed by 0.8 percentage points. Through the lens of an aggregate matching function, more activation is associated with a larger matching efficiency.

Appendices

Appendix 1: Details on Hartz Reforms

1.1 Different Reform Steps

The so-called Hartz commission (named after the head of the commission, Peter Hartz) developed recommendations how to reform the German labor market in order to reduce unemployment. The guiding principle for these reforms was “Fordern und Fördern,” the so-called principle of “rights and duties” (see Jacobi and Kluve 2007). These recommendations were implemented gradually, starting in 2003. See Hochmuth et al. (2021) or Launov and Wälde (2016) for a more detailed description:

Hartz I (implemented in 2003): The first package of the Hartz reform facilitated temporary work contracts. In addition, it introduced vouchers for training.

Hartz II (implemented in 2003): The second package introduced new types of marginal employment, with reduced social security contributions for low-income contracts. In addition, it introduced subsidies for unemployed workers to transition into self-employment.

Hartz III [implementation, started in 2004, the full roll-out ended in late 2005, see Holzner and Watanabe (2021) for details]: The core element of Hartz III was the restructuring of the Federal Employment Agency (see Launov and Wälde 2016 for details). With the introduction of Hartz III, all claims of an unemployed person were processed by the same case worker (support from a single source) and an upper limit on the number of cases handled by one single case worker was introduced. In addition, market elements for private placement services and providers of training measures were introduced.

Hartz IV (implemented in 2005 and 2006): The last step of the Hartz reforms changed the unemployment benefit system for the long-term unemployed. Before Hartz IV, long-term unemployed received benefits that were dependent on their prior net earnings. With the introduction of Hartz IV, long-term unemployed had to go through a strict means test and received a fixed transfer (independent of their prior income). See Hochmuth et al. (2021) for details.

1.2 Activation and Counseling

As part of the Hartz III reform, the Federal Employment Agency offered new services to unemployed workers, such as advising and counseling. In addition, individuals that were not placed by the PEA within six weeks received subsidies for a private placement service [see Jacobi and Kluve (2007) for institutional details, in particular, their Sect. 3. Furthermore, the Hartz reform introduced new sanctions to monitor unemployed workers’ job search activities.

We are not able to differentiate these measures in our macroeconomic matching exercise. However, all of them have in common that they stimulate private search activities of unemployed workers. In our numerical, exercise we show that activation and counseling policies play an important role to explain the macroeconomic patterns after the Hartz reforms.

Appendix 2: Model Derivations

This Appendix shows the detailed model derivations. As in the main part, we start with the household optimization, followed by the firm optimization and the derivation of wages.²¹

1.1 Household

Each unemployed worker has to make the decision whether to search privately herself or to rely only on the agency to find a job. For this decision, the probabilities of finding a job in both cases are important. If no private search is carried out, the probability of being employed in the next period is $p_t^{a}$. The worker himself cannot get in contact with two vacancies since he only searches through one channel. The worker can get in a situation, where he gets in contact with a vacancy which has a second contact from the private search market. Since firms always choose the agency contact if they have a double match this does not reduce the matching probability of a worker who only searches via the agency.

If a worker uses the private search market his probability of making a match through the private market is:

$$\begin{aligned} \frac{c_t^p-q_t^p q_t^{a}f_t}{u_t}=\frac{c_t^p-q_t^{a}\frac{c_t^p}{v_t}v_t g_t}{u_t} = p_t^p -g_t q_t^{a}p_t^p. \end{aligned}$$

If a worker gets in contact with a vacancy through the private market and this vacancy made a second contact through the public search channel, the worker will not be matched since firms prefer agency contacts. The probability that the worker himself makes two contacts (one through each search channel) is not subtracted here since workers prefer private contacts over agency contacts. The worker’s probability of being matched through the agency thus has to be reduced to $p_t^{a} -p_t^{a} p_t^{a}$.

With these probabilities we can define the value of only searching through the agency as

$$\begin{aligned} S_{t}^{a}=p_{t}^{a}W_{t}^{a}+\left( 1-p_{t}^{a}\right) U_{t}^{a} \end{aligned}$$

(45)

and the value of using both channels as

$$\begin{aligned} \begin{aligned} S_{it}^{p}&=-e_{it}+\left( p_{t}^{a}-p_{t}^{a}p_{t}^{p}\right) W_{t}^{a}+\left( p_{t}^{p}-g_{t}q_{t}^{a}p_{t}^{p}\right) W_{t}^{p} \\&\quad +\left( 1-\left( p_{t}^{a}-p_{t}^{a}p_{t}^{p}\right) -\left( p_{t}^{p}-g_{t}q_{t}^{a}p_{t}^{p}\right) \right) U_{t}^{p}. \end{aligned} \end{aligned}$$

(46)

The worker will only use both channels if the value from doing so is higher than the value of searching through the agency only.

$$\begin{aligned} S_{it}^{p}\ge S_{t}^{a}\Rightarrow S_{it}^{p}- S_{t}^{a}\ge 0. \end{aligned}$$

(47)

The worker with the highest individual search cost who is still searching through the private market is indifferent between searching privately and not searching privately. Thus, for this worker equation (47) holds with equality. Using equations (45) and (46), we can derive the cutoff search costs:

$$\begin{aligned} \tilde{e}_{it}&=\left( p_{t}^{a}-p_{t}^{a}p_{t}^{p}\right) W_{t}^{a}+\left( p_{t}^{p}-g_{t}q_{t}^{a}p_{t}^{p}\right) W_{t}^{p} \nonumber \\&\quad +\left( 1-\left( p_{t}^{a}-p_{t}^{a}p_{t}^{p}\right) -\left( p_{t}^{p}-g_{t}q_{t}^{a}p_{t}^{p}\right) \right) U_{t}^{p}-p_{t}^{a}W_{t}^{a}-\left( 1-p_{t}^{a}\right) U_{t}^{a} \end{aligned}$$

(48)

$$\begin{aligned} \tilde{e}_{it}&= -p_{t}^{a}p_{t}^{p}W_{t}^{a}+\left( p_{t}^{p}-g_{t}q_{t}^{a}p_{t}^{p}\right) W_{t}^{p} \nonumber \\&\quad +\left( 1-\left( p_{t}^{a}-p_{t}^{a}p_{t}^{p}\right) -\left( p_{t}^{p}-g_{t}q_{t}^{a}p_{t}^{p}\right) \right) U_{t}^{p}-\left( 1-p_{t}^{a}\right) U_{t}^{a} \end{aligned}$$

(49)

Using

$$\begin{aligned} U_{t}^{p}-U_{t}^{a}=b-b^{r}, \end{aligned}$$

(50)

we obtain equation (11) stated in the main text:

$$\begin{aligned} \tilde{e}_{it}=\left( 1-p_{t}^{a}\right) \left( b-b^{r}\right) +\left( p_{t}^{p}-g_{t}q_{t}^{a}p_{t}^{p}\right) \left( W_{t}^{p}-U_{t}^{p}\right) -p_{t}^{a}p_{t}^{p}\left( W_{t}^{a}-U_{t}^{p}\right) . \end{aligned}$$

(51)

Every job seeker who draws a value of $e_{it}\le \tilde{e}_t$ uses the private market. Thus the share of privately searching job seekers is

$$\begin{aligned} \xi _{t}=\int _{-\infty }^{\tilde{e}_{t}}h\left( e_{t}\right) {\text {d}} e_{t}\text {.} \end{aligned}$$

(52)

The conditioned expected value of search costs is

$$\begin{aligned} \widehat{e}_{t}=\frac{\int _{-\infty }^{\tilde{e}_{t}}e_{t}h\left( e_{t}\right) d e_{t}\text {{}}}{\xi _{t}}. \end{aligned}$$

(53)

1.2 Firm

The probability of matching with a worker through the private search market is

$$\begin{aligned} \frac{c_t^{p}-q_t^{a}q_t^{p}f_t}{v_t} = q_t^{p}-q_t^{a}q_t^{p}g_t. \end{aligned}$$

The probability of matching with a worker through the public search market is

$$\begin{aligned} \frac{c_t^{a}-p_t^{a}p_t^{p}u_t}{f_t} = \frac{c_t^{a}-c_t^{a}p_t^{p} \xi _t}{f_t} =q_t^{a} -q_t^{a}p_t^{p}\xi _t. \end{aligned}$$

The representative firm solves the following maximization problem:

$$\begin{aligned} \max _{m_{t}^{a},n_{t}^{p},v_t,g_t} E_0 \sum _{t=0}^{\infty }\beta ^{t} \{(a_{t}-w^{p}_{t})n^{p}_{t}+(a_{t}-w^{a}_{t})m^{a}_{t}-v_{t}^{p}( \kappa ^{p}+\kappa ^{a}g_{t})\}, \end{aligned}$$

(54)

subject to the constraints:

$$\begin{aligned}{} & {} n_{t}^{p} =(1-\phi )n_{t-1}^{p}+(1-\phi )m_{t-1}^{a}+v_{t}(q_{t}^{p}-q_{t}^{a}q_{t}^{p}g_t), \end{aligned}$$

(55)

$$\begin{aligned}{} & {} m_{t}^{a} = v_t g_t(q_{t}^{a}-q_{t}^{a}p_{t}^{p}\xi _t). \end{aligned}$$

(56)

The corresponding Lagrangian is:

$$\begin{aligned} L&=E_0 \sum _{t=0}^{\infty }\beta ^{t} \{(a_{t}-w^{p}_{t})n^{p}_{t}+(a_{t}-w^{a}_{t})m^{a}_{t}-v_{t}^{p}(\kappa ^{p}+\kappa ^{a}g_{t})\} \\&\quad -\lambda _t^{p}(n_{t}^{p} -(1-\phi )n_{t-1}^{p}-(1-\phi )m_{t-1}^{a}-v_{t}(q_{t}^{p}-q_{t}^{a}q_{t}^{p}g_t)) \\&\quad -\lambda _t^{a} (m_{t}^{a} - v_t g_t(q_{t}^{a}-q_{t}^{a}p_{t}^{p}\xi _t) \} \end{aligned}$$

The first-order conditions are:

$$\begin{aligned}{} & {} \frac{\partial L}{\partial m_{t}^{a}} = a_t-w^{a}_{t} -\lambda _t^{a}+E_t\beta (1-\phi )\lambda _{t+1}^{p} = 0 \nonumber \\{} & {} \quad \Rightarrow \lambda _t^{a} = a_t-w^{a}_{t} +E_t\beta (1-\phi )\lambda _{t+1}^{p} \end{aligned}$$

(57)

$$\begin{aligned}{} & {} \frac{\partial L}{\partial n_{t}^{p}} = a_t-w^{p}_{t} -\lambda _t^{p}+E_t\beta (1-\phi )\lambda _{t+1}^{p} = 0 \nonumber \\{} & {} \quad \Rightarrow \lambda _t^{p} = a_t-w^{p}_{t} +E_t\beta (1-\phi )\lambda _{t+1}^{p} \end{aligned}$$

(58)

$$\begin{aligned}{} & {} \frac{\partial L}{\partial v_{t} } = -(\kappa ^p +g_t \kappa ^{a}) + \lambda _t^{a} g_t(q_{t}^{a}-q_{t}^{a}p_{t}^{p}\xi _t) +\lambda _t^{p} (q_{t}^{p}-q_{t}^{a}q_{t}^{p}g_t) =0 \nonumber \\{} & {} \quad \Rightarrow (\kappa ^p +g_t \kappa ^{a}) = \lambda _t^{a} g_t(q_{t}^{a}-q_{t}^{a}p_{t}^{p}\xi _t) +\lambda _t^{p} (q_{t}^{p}-q_{t}^{a}q_{t}^{p}g_t) \end{aligned}$$

(59)

$$\begin{aligned}{} & {} \frac{\partial L}{\partial g_{t} } = - v_t \kappa ^{a} + \lambda _t^{a} v_t (q_{t}^{a}-q_{t}^{a}p_{t}^{p}\xi _t) - \lambda _t^{p} q_{t}^{a}q_{t}^{p} v_t =0\nonumber \\{} & {} \quad \Rightarrow \kappa ^{a} = \lambda _t^{a} (q_{t}^{a}-q_{t}^{a}p_{t}^{p}\xi _t) - \lambda _t^{p} q_{t}^{a}q_{t}^{p} \nonumber \\{} & {} \quad \Rightarrow \frac{\kappa ^{a}+\lambda _t^{p} q_{t}^{a}q_{t}^{p}}{(q_{t}^{a}-q_{t}^{a}p_{t}^{p}\xi _t)} = \lambda _t^{a} \end{aligned}$$

(60)

Substituting into first-order condition for $v_t$:

$$\begin{aligned} (\kappa ^p +g_t \kappa ^{a})= & {} \frac{\kappa ^{a}+\lambda _t^{p} q_{t}^{a}q_{t}^{p}}{(q_{t}^{a}-q_{t}^{a}p_{t}^{p}\xi _t)} g_t(q_{t}^{a}-q_{t}^{a}p_{t}^{p}\xi _t) +\lambda _t^{p} (q_{t}^{p}-q_{t}^{a}q_{t}^{p}g_t) \end{aligned}$$

(61)

$$\begin{aligned} (\kappa ^p +g_t \kappa ^{a})= & {} \kappa ^{a}g_t+\lambda _t^{p} q_{t}^{a}q_{t}^{p} g_t+\lambda _t^{p} (q_{t}^{p}-q_{t}^{a}q_{t}^{p}g_t) \nonumber \\ \kappa ^{p}= & {} q_{t}^{p} \lambda _t^{p} \nonumber \\ \frac{\kappa ^{p}}{q_{t}^{p}}= & {} \lambda _t^{p} \end{aligned}$$

(62)

Substituting back into first-order conditions for $g_t$:

$$\begin{aligned}{} & {} \frac{\kappa ^{a}+\frac{\kappa ^{p}}{q_{t}^{p}} q_{t}^{a}q_{t}^{p}}{(q_{t}^{a}-q_{t}^{a}p_{t}^{p}\xi _t)} = \lambda _t^{a}\nonumber \\{} & {} \quad \Rightarrow \frac{\kappa ^{a}+\kappa ^{p} q_{t}^{a}}{(q_{t}^{a}-q_{t}^{a}p_{t}^{p}\xi _t)} = \lambda _t^{a} \end{aligned}$$

(63)

Plug into the FOCs for $m_t^{a}$ and $n_t^{p}$:

$$\begin{aligned}{} & {} \frac{\kappa ^{a}+\kappa ^{p} q_{t}^{a}}{(q_{t}^{a}-q_{t}^{a}p_{t}^{p}\xi _t)}= a_t-w^{a}_{t} +E_t\beta (1-\phi )\frac{\kappa ^{p}}{q_{t+1}^{p}} \end{aligned}$$

(64)

$$\begin{aligned}{} & {} \frac{\kappa ^{p}}{q_{t}^{p}} = a_t-w^{p}_{t} +E_t\beta (1-\phi )\frac{\kappa ^{p}}{q_{t+1}^{p}} \end{aligned}$$

(65)

1.3 Wage Bargaining

A worker’s expected value of a match via the private market is:

$$\begin{aligned} \begin{aligned} W_{t}^{p}&=w_{t}^{p} +\beta (1-\phi )E_{t}W_{t+1}^{p}\\ {}&\quad +\beta \phi E_{t}\xi _{t+1}\left[ \begin{array}{c} -\hat{e}_{t+1}+\left( p_{t+1}^{p}-g_{t+1}q_{t+1}^{a}p_{t+1}^{p}\right) W_{t+1}^{p} \\ +\left( p_{t+1}^{a}-p_{t+1}^{a}p_{t+1}^{p}\right) W_{t+1}^{a}+ \\ \left( \begin{array}{c} 1-\left( p_{t+1}^{p}-g_{t+1}q_{t+1}^{a}p_{t+1}^{p}\right) \\ -\left( p_{t+1}^{a}-p_{t+1}^{a}p_{t+1}^{p}\right) \end{array} \right) U_{t+1}^{p} \end{array} \right] \\&\quad +\beta \phi E_{t}\left( 1-\xi _{t+1}\right) \left[ p_{t+1}^{a}W_{t+1}^{a}+ \left( 1-p_{t+1}^{a}\right) U_{t+1}^{a}\right] . \end{aligned} \end{aligned}$$

(66)

$U_{t}^{p}$ is the average value of being unemployed after having used the private market:

$$\begin{aligned} \begin{aligned} U_{t}^{p}&= b +\beta E_{t}\xi _{t+1}\left[ \begin{array}{c} -\hat{e}_{t+1}+\left( p_{t+1}^{p}-g_{t+1}q_{t+1}^{a}p_{t+1}^{p}\right) W_{t+1}^{p} \\ +\left( p_{t+1}^{a}-p_{t+1}^{a}p_{t+1}^{p}\right) W_{t+1}^{a}+ \\ \left( \begin{array}{c} 1-\left( p_{t+1}^{p}-g_{t+1}q_{t+1}^{a}p_{t+1}^{p}\right) \\ -\left( p_{t+1}^{a}-p_{t+1}^{a}p_{t+1}^{p}\right) \end{array} \right) U_{t+1}^{p} \end{array} \right] \\&\quad +\beta E_{t}\left( 1-\xi _{t+1}\right) \left[ p_{t+1}^{a}W_{t+1}^{a}+ \left( 1-p_{t+1}^{a}\right) U_{t+1}^{a}\right] . \end{aligned} \end{aligned}$$

(67)

A worker’s expected value of a match via the employment agency:

$$\begin{aligned} \begin{aligned} W_{t}^{a}&=w_{t}^{a} +\beta (1-\phi )E_{t}W_{t+1}^{p}\\ {}&\quad +\beta \phi E_{t}\xi _{t+1}\left[ \begin{array}{c} -\hat{e}_{t+1}+\left( p_{t+1}^{p}-g_{t+1}q_{t+1}^{a}p_{t+1}^{p}\right) W_{t+1}^{p} \\ +\left( p_{t+1}^{a}-p_{t+1}^{a}p_{t+1}^{p}\right) W_{t+1}^{a}+ \\ \left( \begin{array}{c} 1-\left( p_{t+1}^{p}-g_{t+1}q_{t+1}^{a}p_{t+1}^{p}\right) \\ -\left( p_{t+1}^{a}-p_{t+1}^{a}p_{t+1}^{p}\right) \end{array} \right) U_{t+1}^{p} \end{array} \right] \\&\quad +\beta \phi E_{t}\left( 1-\xi _{t+1}\right) \left[ p_{t+1}^{a}W_{t+1}^{a}+ \left( 1-p_{t+1}^{a}\right) U_{t+1}^{a}\right] . \end{aligned} \end{aligned}$$

(68)

$U_{t}^{a}$ is the average value of being unemployed after having used the agency only (and not the private market):

$$\begin{aligned} \begin{aligned} U_{t}^{a}&=b^{r} +\beta E_{t}\xi _{t+1}\left[ \begin{array}{c} -\hat{e}_{t+1}+\left( p_{t+1}^{p}-g_{t+1}q_{t+1}^{a}p_{t+1}^{p}\right) W_{t+1}^{p} \\ +\left( p_{t+1}^{a}-p_{t+1}^{a}p_{t+1}^{p}\right) W_{t+1}^{a}+ \\ \left( \begin{array}{c} 1-\left( p_{t+1}^{p}-g_{t+1}q_{t+1}^{a}p_{t+1}^{p}\right) \\ -\left( p_{t+1}^{a}-p_{t+1}^{a}p_{t+1}^{p}\right) \end{array} \right) U_{t+1}^{p} \end{array} \right] \\&\quad +\beta E_{t}\left( 1-\xi _{t+1}\right) \left[ p_{t+1}^{a}W_{t+1}^{a}+ \left( 1-p_{t+1}^{a}\right) U_{t+1}^{a}\right] . \end{aligned} \end{aligned}$$

(69)

A firm’s value of a matched job depends on whether the match was established via the private market $J_{t}^{p}$ or the agency $J_{t}^{a}$:

$$\begin{aligned}{} & {} J_{t}^{p}=a_{t}-w_{t}^{p}+\beta E_{t}\left( 1-\phi \right) J_{t+1}^{p}, \end{aligned}$$

(70)

$$\begin{aligned}{} & {} J_{t}^{a}=a_{t}-w_{t}^{a}+\beta E_{t}\left( 1-\phi \right) J_{t+1}^{p}, \end{aligned}$$

(71)

The Nash bargaining problem can be written as for workers that matched via the agency:

$$\begin{aligned} w^{a}_t \in argmax\left( W^{a}_{t}-U^{a}_{t}\right) ^{\gamma }\left( J^{a}_{t}\right) ^{1-\gamma }, \end{aligned}$$

(72)

which results in the following sharing rule:

$$\begin{aligned} \gamma J^{a}_{t}=(1-\gamma )(W^{a}_{t}-U^{a}_{t}). \end{aligned}$$

(73)

Equivalently, the Nash bargaining problem for workers that matched via the private market is:

$$\begin{aligned} w^{p}_t \in {\text {argmax}}\left( W^{p}_{t}-U^{p}_{t}\right) ^{\gamma }\left( J^{p}_{t}\right) ^{1-\gamma }, \end{aligned}$$

(74)

which results in the following sharing rule:

$$\begin{aligned} \gamma J^{p}_{t}=(1-\gamma )(W^{p}_{t}-U^{p}_{t}). \end{aligned}$$

(75)

The term on the right-hand side includes

$$\begin{aligned} \begin{aligned} W_{t}^{p}-U_{t}^{p}&=w_{t}^{p}-b+\beta (1-\phi )E_{t}W_{t+1}^{p} \\&\quad -\beta \left( 1-\phi \right) E_{t}\xi _{t+1}\left[ \begin{array}{c} -\hat{e}_{t+1}+\left( p_{t+1}^{p}-g_{t+1}q_{t+1}^{a}p_{t+1}^{p}\right) W_{t+1}^{p} \\ +\left( p_{t+1}^{a}-p_{t+1}^{a}p_{t+1}^{p}\right) W_{t+1}^{a}+ \\ \left( \begin{array}{c} 1-\left( p_{t+1}^{p}-g_{t+1}q_{t+1}^{a}p_{t+1}^{p}\right) \\ -\left( p_{t+1}^{a}-p_{t+1}^{a}p_{t+1}^{p}\right) \end{array} \right) U_{t+1}^{p} \end{array} \right] \\&\quad -\beta \left( 1-\phi \right) E_{t}\left( 1-\xi _{t+1}\right) \left[ p_{t+1}^{a}W_{t+1}^{a}+\left( 1-p_{t+1}^{a}\right) U_{t+1}^{a}\right] \end{aligned} \end{aligned}$$

(76)

for the private wage and

$$\begin{aligned} \begin{aligned} W_{t}^{a}-U_{t}^{a}&=w_{t}^{a}-b^{r}+\beta (1-\phi )E_{t}W_{t+1}^{p} \\&\quad -\beta \left( 1-\phi \right) E_{t}\xi _{t+1}\left[ \begin{array}{c} -\hat{e}_{t+1}+\left( p_{t+1}^{p}-g_{t+1}q_{t+1}^{a}p_{t+1}^{p}\right) W_{t+1}^{p} \\ +\left( p_{t+1}^{a}-p_{t+1}^{a}p_{t+1}^{p}\right) W_{t+1}^{a}+ \\ \left( \begin{array}{c} 1-\left( p_{t+1}^{p}-g_{t+1}q_{t+1}^{a}p_{t+1}^{p}\right) \\ -\left( p_{t+1}^{a}-p_{t+1}^{a}p_{t+1}^{p}\right) \end{array} \right) U_{t+1}^{p} \end{array} \right] \\&\quad -\beta \left( 1-\phi \right) E_{t}\left( 1-\xi _{t+1}\right) \left[ p_{t+1}^{a}W_{t+1}^{a}+\left( 1-p_{t+1}^{a}\right) U_{t+1}^{a}\right] \end{aligned} \end{aligned}$$

(77)

for the agency wage. The two terms can be written in a compact way using the definition of the variable $V_{t+1}$

$$\begin{aligned} \begin{aligned} V_{t+1}&= W_{t+1}^{p}-\xi _{t+1}\left[ \begin{array}{c} -\hat{e}_{t+1}+\left( p_{t+1}^{p}-g_{t+1}q_{t+1}^{a}p_{t+1}^{p}\right) W_{t+1}^{p} \\ +\left( p_{t+1}^{a}-p_{t+1}^{a}p_{t+1}^{p}\right) W_{t+1}^{a} +\\ \left( \begin{array}{c} 1-\left( p_{t+1}^{p}-g_{t+1}q_{t+1}^{a}p_{t+1}^{p}\right) \\ -\left( p_{t+1}^{a}-p_{t+1}^{a}p_{t+1}^{p}\right) \end{array} \right) U_{t+1}^{p} \end{array}\right] \\&\quad -\left( 1-\xi _{t+1}\right) \left[ p_{t+1}^{a}W_{t+1}^{a}+\left( 1-p_{t+1}^{a}\right) U_{t+1}^{a}\right] \\ \end{aligned} \end{aligned}$$

(78)

such that

$$\begin{aligned} W_{t}^{a}-U_{t}^{a}&=w_{t}^{a}-b^{r}+\beta (1-\phi )E_{t} V_{t+1} \end{aligned}$$

(79)

and

$$\begin{aligned} W_{t}^{p}-U_{t}^{p}&=w_{t}^{p}-b+\beta (1-\phi )E_{t} V_{t+1}. \end{aligned}$$

(80)

Combining these expressions with sharing rules (73) and (75) as well as firms’ values (71) and (70) we get the wage equations stated in the main text

$$\begin{aligned}{} & {} w_{t}^{p}=\gamma a_{t}+\left( 1-\gamma \right) b+\beta E_{t}\left( 1-\phi \right) \left( \gamma J_{t+1}^{p}-\left( 1-\gamma \right) V_{t+1}\right) , \end{aligned}$$

(81)

$$\begin{aligned}{} & {} w_{t}^{a}=\gamma a_{t}+\left( 1-\gamma \right) b^r+\beta E_{t}\left( 1-\phi \right) \left( \gamma J_{t+1}^{p}-\left( 1-\gamma \right) V_{t+1}\right) . \end{aligned}$$

(82)

Appendix 3: Details on Bargaining

Figure 8 shows the model assumptions and implications on the out-of-equilibrium outcomes for bargaining. Workers receive a lower starting wage if they are matched via the public employment agency than via the private channel.

Fig. 8

Different Bargaining Outcomes. Note: This figure shows fallback options and resulting wages for different scenarios

Appendix 4: Robustness of Numerical Results

1.1 Reform Complementarities or Substitutabilities

In addition to the effect of one reform step as the only change (Table 5), we calculate the effect of each reform step if it was the last change to happen (Table 8). To do this, we simulate the model with all three reform steps, remove one reform step and then calculate the respective difference for the three outcome variables. It can be seen that this reduces the unemployment effects for sanctions and the surplus shock. This can at least partly be explained by nonlinearities in the model resulting from the cost shock distribution. Although the cutoff point for search costs moves by a similar amount for the exercises in Tables 5 and 8, the additional number of workers who use the private channel increases by less if the other exercise is in place. With a larger share of private searchers, the cutoff point moves into a thinner part of the idiosyncratic distribution.

Table 8 Policy responses: last reform

1.2 Homogeneous Labor Market

As a second exercise, we look at a version of the model with only one search channel which we interpret as the agency. We then check if the results based on this version of the model are comparable to the results of Launov and Wälde (2016) if we follow their methodology in simplified manner. If we restrict workers to search via the public agency only, there is no second private search market. In different words, the Federal Employment Agency is the only intermediary/matching function in the economy. The model equations are:

$$\begin{aligned} m_t&= \psi s_t^{1-\alpha } v_t^{\alpha }\\ p_t&= \psi \theta _t^\alpha \\ q_t&= \psi \theta _t^{\alpha -1} \\ \theta&=\frac{v_t}{s_t}\\ \frac{\kappa }{q_t}&= a_{t}-w_{t} +E_t \beta (1-\phi ) \frac{\kappa }{q_{t+1}} \\ w_t&= \gamma (a_t + \kappa E_t \beta \theta _{t+1} )+(1- \gamma )b \\ n_t&= (1-\phi ) n_{t-1} +m_t \\ s_{t}&=1-n_{t-1}+\phi n_{t-1} \\ s^{u}_{t}&=1-n_{t-1} \end{aligned}$$

In this homogeneous model, we have to redesign our calibration exercise, as targeting the vacancy share and matching is not possible (by definition, they are equal to 100%). Note that we replicate the parametrization from the main part.

We follow Launov and Wälde (2016)’s numerical strategy in simplified fashion (as our homogeneous model is less complex than theirs, e.g., in terms of unemployment durations). We take the increase of the matching efficiency from our reduced-form matching function in Table 11, namely 7%. We impose this increase in the matching efficiency on our model. This exercise yields a decline in unemployment of 0.65 pp (see Table 9). This number is comparable to Launov and Wälde (2016) who find a 0.88 pp decline (see their Table 2). In a second step, we use the 7% benefit reduction for long-term unemployed from Launov and Wälde (2016). To simulate this in our model, we multiply this change by 0.36 which corresponds to the average share of long-term unemployed in West Germany in the years from 1998 to 2003 (Hochmuth et al. 2021; vom Berge et al. 2013). The result is a reduction of the unemployment rate by 0.11 pp which is comparable to the results of Launov and Wälde (2016) who find a 0.08 pp decline (see their Tables 2, 9).

Table 9 Policy responses in a simple model

Appendix 5: Data and Further Empirical Facts

1.1 Data

1.1.1 German Socio-Economic Panel

As stated before, we construct the matching share of the agency from the German Socioeconomic Panel (SOEP). We also use it to get our target for the share of privately searching unemployed. The SOEP is a longitudinal survey covering approximately 30,000 individuals. For further descriptions of the SOEP, see Goebel et al. (2019). Since we use wave 35, we have observations from the starting year of the SOEP 1984 up to the year 2018. However, due to variations in the questionnaires, the time period of the data used is restricted depending on the variable constructed from the SOEP. For our calibration, we use observations from individuals living in West Germany.

The basis for the share of privately searching unemployed is the question whether a non-employed individual has been actively searching for employment in the last four weeks. To stay close to the model, we only use individuals registered as unemployed at the agency. Since the question whether an active search is being carried out includes the search via the employment agency as active search, a further adjustment is necessary. For the years 2003-2007, additional information is available on the channels through which employment is searched for. For these years, the share of actively searching, registered unemployed who are not only searching through the agency is calculated using the cross-sectional individual weights. The corresponding value for West Germany for the year 2003 is the stated target.

For the matching share, we use the question how an individual found out about her new position. This question is only answered by individuals who started their current employment in the year of the questionnaire or in the year before. The construction of the time series shown in Sect. 3 and used as a target in Sect. 5 takes into account the possibility that the employment started in the year before the questionnaire. In addition, we exclude individuals who claim to have become self-employed, who have changed jobs in the same firm, and who have stated multiple channels. We also add job centers to the agency and exclude personnel service agencies. Finally, we also count individuals who found their job with the help of a voucher from the agency to the matches of the agency. The survey also contains the question what type of occupational change occurred. Based on this question, we again exclude individuals who change their job in a firm and individuals who switch to self-employment as well as individuals for whom this information is missing. We also exclude apprenticeship positions, individuals who are employed in a sheltered workshop, 1 Euro jobs, and public job creation schemes (ABM) positions as well as returnees from parental leave for all years with the respective information. Finally, employees older than 65 are excluded. Based on these adjustments we calculate the matching share of the agency using the cross-sectional individual weights. Not all necessary questions were asked before the time period considered in the main text. That is why the corresponding adjustments were not possible in the longer time series in Fig. 4. The time series in Fig. 11 is based on the same adjustments. Additionally, information from the SOEP spell data is used to obtain the information in which month unemployment spells end. For this, the do-files by Hamjediers et al. (2018) are used. The shown times series is the matching share if an unemployment spell ended in the month in which the new position started or one month before.

1.1.2 IAB Job Vacancy Survey

The data we use for the vacancy share, for the additional time series of the matching share, and for vacancies come from the IAB Job Vacancy Survey (Bossler et al. 2020a, b). The Job Vacancy Survey is a repeated cross section. It was carried out for the first time in the year 1989 and covers up to around 14,000 establishments.

The vacancy share is based on the question how many vacancies an establishment has. In parallel, it is asked how many of these have been reported to the agency. The ratio of the two, each weighted by the weighting factors, gives the vacancy share. In addition, more detailed questions are asked on the last successful hire. Two of these questions are, which search channels were used, and which of those led to the hiring. The latter is the question used for the Job Vacancy Survey time series on the matching share. From 2004 onward, the agency’s internet services are listed as a separate response option in the questionnaire. We add the matches resulting from this option to the matches of the agency. The share of hires for which the agency was stated as the recruitment channel is the matching share. The corresponding weighting factors have been used. When we construct qualification groups, the group university includes degrees from universities and universities of applied sciences.

We use the number of vacancies from the Job Vacancy Survey for our target of the public labor market tightness. The number of unemployed as well as the job finding rates is from the Integrated Labour Market Biographies (vom Berge et al. 2013). For more details see Appendix B in Hochmuth et al. (2021).

1.1.3 Baseline Sample

Data from the IAB Job Vacancy Survey for West and East Germany is only available from 1992 onward. In addition, one key question for the construction of the matching share (on the job transition type, see Appendix “Data”) in the SOEP was rephrased in 1994. Because of the backward-looking nature of the underlying questions and the relatively large sample size in these years, we start calculating the matching share from 1993. Given these two restrictions, we chose the time period from 1993 to 2018 as our baseline. In addition, we do robustness checks based on a long sample from 1985 to 2018. Technically, it is possible to calculate the matching share also for the year 1984 since a share of respondents refers to positions started in the year 1984 when they are surveyed in the year 1985. Because of the small number of observations resulting from this, we chose to start the long time series in 1985, the first year in which the questions necessary were asked.

1.2 Vacancy and Matching Share: Robustness

Figure 9 shows the matching share for different education requirements for Germany and West Germany. This figure suffers from a small number of micro-observations per aggregate data point and is therefore somewhat more noisy that the aggregated figures. However, it is visible that the matching share (within certain groups) peaks in times of high unemployment.

Fig. 9

Matching Share for different Education Requirements. Note: The figure shows the matching share (based on SOEP) for positions with different education requirements

Figure 10 shows the matching share based on the Job Vacancy Survey. Although the level of this matching share is somewhat larger, the dynamics is very similar to the SOEP-based matching share.

Fig. 10

IAB Job Vacancy Matching Share. Note: The figure shows the matching share of the agency based on IAB JVS

Figure 11 and Table 10 show further robustness checks with alternative definitions. The key patterns in the data are very robust.

Fig. 11

Matching Share: Ending Unemployment Spell. Note: The figure shows the matching share based on all observations in blue and the matching share that is restricted to observation where an unemployment spell ended in the month of the match or the month before in red. Both time series are normalized to a mean of one. On average the restricted matching share was roughly 5 percentage points lower for Germany and 2.5 percentage points lower for West Germany in the post-reform period. (Colour figure online)

Table 10 Matching shares for loosely connected unemployed

1.3 Matching Function Estimations

Table 11 shows matching function estimations with shift dummies for the Hartz III reform. The left column shows a standard matching function estimation, based on the aggregate job-finding rate and market tightness. The right column shows the estimation for a public matching, based on agency matches and agency market tightness. While there is a positive and statistically significant coefficient on the aggregate shift dummy, the estimated coefficient is negative and not statistically significant for the agency.

Table 11 Estimated matching functions

1.4 Probability of Being Matched via the PEA

Table 12 shows how the individual-level probability of being matched via the agency shifted after the reforms (Hartz III dummy). It controls for aggregate and individual-level observables. The estimations are based on individual-level data from SOEP.

In line with our descriptive evidence from the main part, the probability of being matched via the agency drops in the aftermath of the Hartz reforms. Thus, this fact is robust to controlling for individual-level characteristics.

Table 12 Probability of being matched via the agency