2.1 Historical Background

At the beginning of the 2000s, the dismal performance of the German labor market and a tightening of the social security budget convinced the German government that a drastic policy reversal had to take place. As a consequence, the German government enacted in 2003-2005 a number of far-reaching labor market reforms, the so-called Hartz reforms. These reforms consisted of four laws that were implemented in three steps in January 2003 (Hartz I+II), January 2004 (Hartz III), and January 2005 (Hartz IV). The main objective of the Hartz reforms was simple yet ambitious: improve the process of moving workers from unemployment to employment.¹⁰ To achieve their objective, the reforms used a multi-layered strategy that had three core elements: i) increase the matching efficiency by providing job search assistance, ii) increase labor supply by activating the unemployed, and iii) increase labor demand by deregulating the market for temporary work and providing employment subsidies. At the risk of over-simplification, we can say that the Hartz III was mainly about the first point, the Hartz IV reform was concerned with the second point, and the Hartz I and Hartz II reforms were mainly dealing with the last point. In this paper, we confine attention to the Hartz III and Hartz IV reforms, to which we turn next.

2.2 Hartz III Reform (Improvement in Job Search Assistance)

On January 1, 2004, Hartz III was enacted with the goal to improve the efficiency of the job placement services for the unemployed. To this end, the Public Employment Agency was restructured and transformed from a strongly centralized and bureaucratic institution with little quality control into a decentralized, customer-oriented organization with a high degree of responsibility and accountability of local employment offices (called job centers after the reform). Further, in the wake of the reform many services were streamlined and heavy emphasis was placed on job search assistance to improve the process of matching unemployed workers with vacant jobs. In addition, the reform broke the de facto monopoly of the Public Employment Agency, which introduced competition in the market for job placement services. In particular, a voucher system was introduced providing individual job seekers with the opportunity to choose private placement agencies.¹¹

In the Online Appendix we review the evidence regarding the effect of Hartz III on the efficiency of job search. The available evidence suggests that the restructuring of the Public Employment Agency (Hartz III) increased the quality of JSA leading to an increase in the efficiency parameter of the aggregate matching function by at least 5 percent and perhaps up to 10 percent (Fahr and Sunde, 2009, and Klinger and Rothe, 2012). Note that by estimating a matching function this literature controls for variations in vacancy rates and therefore controls for variations in this labor demand channel. The evidence also suggests that the introduction of vouchers for placement services improved the job finding rate of affected workers by about 10 − 30 percent (Winterhager, Heinze, and Spermann, 2006). On average, 20 percent of unemployed workers receive a voucher (Pfeiffer and Winterhager, 2006), which translates into an increase in the unconditional job finding rate by 2 − 6 percent for all unemployed workers. Thus, the available evidence suggests that Hartz III in conjunction with the introduction of the voucher system improved the quality of JSA substantially and led to an increase in search efficiency by at least 7 − 16 percent. Based on this evidence, in our quantitative analysis we simulate the effect of the Hartz III reform as an increase in search efficiency by 8 percent – see Section 6 for details.

2.3 Hartz IV (Reduction in Unemployment Benefits)

The Hartz IV legislation was enacted in January 1, 2005, and constituted a radical overhaul of the German unemployment insurance system. The main objective of Hartz IV was to activate unemployed job seekers. Before the reform, the unemployment system was characterized by a very long period of Unemployment Benefit entitlement followed by a relatively generous and essentially unlimited, means-tested Social Welfare Program (consisting of a combination of Unemployment Assistance and/or Social Assistance). The Hartz IV reform resulted in a simple two-tier unemployment insurance system in which most unemployed workers with unemployment spells less than one year (short-term unemployed) receive unemployment benefits that are proportional to earnings at the last job (called Unemployment Benefit I) and most unemployed workers with an unemployment spell of more than one year (long-term unemployed) receive means-tested payments that heavily depend on household composition (called Unemployment Benefit II).¹²

The Hartz IV reform reduced unemployment payments for many households, but the extent of the reduction varies substantially across household groups and length of unemployment spell. One way to aggregate this heterogeneity is to use the OECD data on net replacement rates for different household groups and to compute an average net replacement rate as described in more detail in Krebs and Scheffel (2013). In Figure 2 we plot the resulting net replacement rate over the period 2000-2012. Clearly, Hartz IV had almost no effect on the net replacement rate of the short-term unemployed (unemployment spell less than one year), but a very large effect on the net replacement rate of the long-term unemployed (unemployment spell more than one year). Specifically, Hartz IV reduced the net replacement rate from 0.57 in the period 2000-2004 to 0.46 after the reform in 2005 – a reduction by 11 percentage points. Based on this evidence, in our quantitative analysis conducted in Section 6 we simulate the effects of Hartz IV assuming that it reduced the net replacement rate for the long-term unemployed by 11 percentage points and that it left the net replacement rate for the short-term unemployed unchanged.

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Average Net Replacement Rate 2001 – 2010

Citation: IMF Working Papers 2017, 087; 10.5089/9781475592245.001.A001

Notes: Own calculation based on OECD data. Net replacement rates conditional on family types are taken from the OECD Tax-Benefit Models and averages are computed using population weights taken from the OECD Family Database.

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We note that a net replacement rate of 57 percent, the value for the long-term unemployed in Germany before the reform, is exceptionally high. In comparison, using comparable OECD data and methodology, we find a value of 27 percent in Spain and 38 percent in France for the long-term unemployed. Indeed, even after the reform, the net replacement rate for the long-term unemployed in Germany, which lies at 46 percent, is still higher than in Spain and France. In contrast, net replacement rates for short-term unemployed are very similar for all three countries: 68 percent for France, 66 percent for Spain, and 63 percent for Germany. Finally, using comparable OECD data and methodology, for the US we find a net replacement rate of 55 percent for the short-term unemployed and 14 percent of the long-term unemployed. Thus, compared to the US, the unemployment insurance and welfare system in post-reform Germany is still relatively generous.

In Section 5 we survey the empirical literature on the effect of unemployment benefits on job finding rates of unemployed workers. This literature has shown that the generosity of unemployment benefits is an important determinant of unemployment duration, and the results of this literature will be used to calibrate the model economy – see Section 5 for details. There is very little micro-econometric work evaluating the effect of the benefit reductions associated with the Hartz IV reform on job-finding rates of the unemployed. One reason for this lack of evidence is that the Hartz IV reform entailed a significant change in the official measurement of unemployment, which substantially changed the composition of the pool of unemployed. However, a recent paper by Engbom, Detragiache, and Raei (2015) provides a difference-in-difference analysis of the labor market effect of the Hartz reforms that takes into account these composition effects. They find that those unemployed workers who experienced the largest reform-induced reduction in unemployment benefits were also the workers whose job finding rates increased the most. Clearly, this empirical finding is in line with the view that the Hartz IV reform increased search incentives.¹³

2.4 Time Series Evidence

This paper emphasizes two channels through which the Hartz III reform and the Hartz IV reform improved the fluidity of the German labor market: improvements in matching efficiency and increases in search effort. The previous two sections argued that the empirical literature provides strong evidence in support of the view that these two reforms increased job finding rates by increasing matching efficiency and search effort. In this section, we argue that the aggregate time series evidence is also in line with this view.

Figure 3 depicts the job finding rates for both short-term unemployed and long-term unemployed workers in the period 2000-2012. The figure shows that these job finding rates have been relatively stable before the implementation of the two reform packages and then began to rise steadily until the years 2007, at which stage they remained relatively stable at a significantly higher level.¹⁴ For the long-term unemployed, the average job finding rate in the period 2000-2003 is 6.3 percent and the average job finding rate in the period 2007-2012 is 9.3 percent – an increase by almost 48 percent. For the short-term unemployed, the corresponding numbers are 24 and 37 percent – an increase by 54 percent. Thus, the time series evidence supports the main economic channel emphasized in this paper, namely that the Hartz III reform and the Hartz IV reform increased job finding rates thereby making the German labor market more fluid.

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Quarterly Job Finding Rates by Duration of Unemployment Spell 2000 – 2011

Citation: IMF Working Papers 2017, 087; 10.5089/9781475592245.001.A001

Notes: Monthly job finding rates by duration of unemployment spell provided by the German Employment Agency. Data are seasonally adjusted and aggregated to quarterly frequency.

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In principle, the increase in job finding rates depicted in Figure 3 can be caused by an increase in matching efficiency, an increase in search effort (labor supply) or an increase in vacancy posting (labor demand). To disentangle the first two channels from the vacancy channel, in Figure 4 we show the vacancy-unemployment relationship (Beveridge curve) for Germany in the period 2000 − 2015. Figure 4 shows that the German Beveridge curve experienced a substantial inward shift shortly after the implementation of the Hartz III and Hartz IV reforms, which is in line with our view that the two reforms led to an increase job finding rates by increasing matching efficiency and search effort. According to the quantitative analysis conducted in Section 6, the two reforms taken together account for most of the non-cyclical rise in job finding rates depicted in Figure 3 and two-thirds of the shift in the Beveridge curve depicted in Figure 4 – see Section 6 for details.

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Beveridge Curve 2000 – 2015

Citation: IMF Working Papers 2017, 087; 10.5089/9781475592245.001.A001

Notes: Vacancies and unemployment rates provided by the German Employment Agency. Data are seasonally adjusted.

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Figure 5 shows the unemployment rate in Germany in the period 1970-2015 and lends additional support to the view that the German labor market reforms of 2003-2005 caused a break in the long-run trend. Specifically, the graph suggests that the unemployment rate has a strong cyclical component, but also a trend component that has been rising since the 1970s until the mid 2000s. This trend was then reversed in the mid 2000s, and the unemployment rate fell from its peak of almost 11 percent in 2005 to 7.5 percent in 2008, barely increased during the Great Recession, and then continued its downward trend reaching 4.4 percent at the end of 2015.¹⁵ Figure 5 suggests that since 2005 the non-cyclical component of the German unemployment rate has declined by at least 3 percentage points and perhaps even 5 percentage points. Our quantitative analysis implies that the Hartz III reform and the Hartz IV reform made a substantial contribution to this decline in structural unemployment – see Section 6 for details.

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Unemployment Rate 1970 – 2015

Citation: IMF Working Papers 2017, 087; 10.5089/9781475592245.001.A001

Notes: The blue line shows the unemployment rate reported by the OECD and the green line shows the unemployment rate reported by the German Employment Agency (IAB).

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3.1 Workers

Time is discrete and open ended. There is a unit mass of infinitely-lived workers. The employment status of a worker in period t is denoted by s_t and can take on three values, s_t ∊ {e, su, lu}, where e stands for employed, su for short-term unemployed, and lu for long-term unemployed. Employed individuals work a fixed and common number of hours in each period. In other words, there is no labor-leisure choice. Employed workers receive a before-tax wage wh, where h is the level of human capital of the individual worker and w is the common wage rate per unit of human capital. The assumption that all employed workers receive the same wage per unit of human capital rules out on-the-job search. Unemployed workers search for jobs and choose a search intensity l. In each period, an unemployed worker either receives a job offer (success), which promises per period wage payment wh for a worker with human capital h, or the worker does not receive an offer (no success) and remains unemployed. We confine attention to the case in which unemployed workers always accept job offers – below we provide conditions on fundamentals that ensure that this is indeed an equilibrium outcome.

The exogenous source of business cycle fluctuations is defined by a Markov process with state S and stationary transition probabilities, which we denote by π(S’|S). For simplicity, we assume that there are only two aggregate states S = R, N, where S = R denotes recession and S = N normal times. Business cycle conditions S affect the labor market in two ways. First, employed workers can become unemployed with exogenous probability π_su|e(S). We call the transition probability π_su|e(S) the job destruction/loss rate. Note that the job destruction rate is independent of effort, that is, there is no moral hazard problem during employment.

The second way S affects the labor market is through its effect on the efficiency of job search. Specifically, the probability that the job search of an unemployed worker is successful (the job finding rate) depends on individual search effort l, a cycle-independent but type-dependent efficiency parameter x_s, with s = su, lu, as well as business cycle condition S. The efficiency variables x_su and x_lu are determined by the level and quality of public job search assistance (JSA) for the two types of unemployed workers. To achieve search efficiency x_s, the government has to spend resources λ_s(x_s) with s = su, lu, where λ_s(.) is an increasing and convex function. We denote the job finding rate of an unemployed worker of type s = su, lu who exerts search effort l and receives JSA x_s by π_e|s(l, x_s, S) with s = su (short-term unemployed) and s = lu (long-term unemployed). We assume that for given S, the job finding rate π_e|s(l, x_s, S) is an increasing and concave function in (l, x_s). An unemployed worker begins the unemployment spell as short-term unemployed and becomes long-term unemployed with exogenous probability π_lu|su.

Workers have different skills and abilities that determine their productivity. The human capital of an individual worker, h_t, is an aggregate of all those worker characteristics that affect a worker’s labor productivity. Employed workers receive after-tax labor income (1 − τ)wh_t, where τ is a linear tax on labor income (social security tax). Human capital of a worker evolves exogenously, that is, it is not a choice variable. The law of motion for individual human capital is

In (1), the variable ∊_t+1 is a random variable that captures wage risk of employed workers. We assume that {∊_t} is i.i.d. over time and across workers and allow for the possibility that E [∊] > 0, which captures the productivity effect of learning-by-doing or on-the-job training. Note that the wage per unit of human capital, w, in conjunction with the process of individual human capital (1) define the individual wage/earnings process, and that the mean of ∊_t+1 defines the average growth rate of wages/earnings. The parameters δ_su and δ_lu capture skill depreciation during unemployment for workers who are short-term unemployed, respectively long-term unemployed. Equation (1) implies that unemployment has permanent effects on wages if E[∊] > δ_su,δ_lu. Note that such a scarring effect of unemployment can also occur if δ_su = δ_lu = 0 as long as there is wage growth for employed workers, E [∊ > 0]. Finally, note that our formulation implies that the scarring effect of unemployment increases during a typical recession due to the increase in unemployment duration, but that we rule out the possibility that δ_su, δ_lu, or E [∊] depend on aggregate labor market conditions S.

At the beginning of life, workers have no financial wealth but they can save at the risk free rate r_t and borrow at the rate r_t + α, where α is a parameter that describes an exogenous cost of financial intermediation. Clearly, α is also the equilibrium spread between lending rate, r_t, and borrowing rate, r_t + α. In addition, we assume that for long-term unemployed workers a fraction ψ of financial assets are confiscated, which represents the fact that in many countries, and in particular in Germany, unemployment benefits are only paid to long-term unemployed workers who have depleted their assets (means-tested program).¹⁷ Finally, we assume that unemployed workers receive unemployment benefits b(s_t)h_t with s_t = su,lu.¹⁸ Thus, the worker’s budget constraint reads

where c_t denotes consumption, a_t denotes financial asset holdings at the beginning of period t, and $\overline{D}>0$ is an explicit debt constraint. Note that our equilibrium characterization result (proposition 1) remains valid even if we replace the debt constraint in (2) by the debt constraint $a_{t+1}\ge -\overline{D}$ or by a no-Ponzi scheme condition. In (2) each c_t is a function mapping histories of individual shocks, s^t, and aggregate shocks, S^t, into consumption choices c_t(s^t, S^t). A similar comment applies to a_t+1 and h_t+1.

In this paper, we consider two types of labor market institutions. The first labor market institution is the Public Employment Agency that provides job search assistance (JSA) and job placement services, which improves the job finding rates through its effect on the efficiency parameters x = (x_su, x_lu). The resource cost of providing JSA, λ_s(x_s), is paid by the government and enters into the government budget constraint, which we define below. The second labor market institution is the unemployment insurance system (UI) defined by the benefit levels b = (b_su, b_lu), which enter into the worker’s budget constraint and the government’s budget constraint. The benefit levels b affect search incentives of unemployed workers and therefore their job finding rates. Unemployment benefits, b, and job search assistance, λ(x_s), are financed through the labor income (social security) tax, τ, which also affects search incentives of unemployed workers by affecting the take-home pay of employed workers. Note that labor market institutions (b, x, τ), only affect job finding rates, but have no effect on job destruction rates in our formulation. Note further that b, x, and τ do not depend on business cycle conditions S. We make this assumption in order to focus on labor market institutions, i.e. labor market policies that do not respond to business cycle conditions.

Workers are risk-averse and have identical preferences that allow for a time-additive expected utility representation. The one-period utility function depends on consumption, search effort, and the employment status. We confine attention to utility functions that are additive over consumption and search effort, and logarithmic over consumption, u_s(c, l) = ln c − d_s(l), where d_s is an increasing and strictly convex function in l. This formulation entails the case in which a type-independent disutility function is defined over time spent searching l hours and working n hours: $\tilde{d}=\tilde{d}(l,n)$. To see this, note that for unemployed workers, s = su, lu, we have l > 0 and n = 0. Thus, the term $d_s\left(l\right)=\tilde{d}(l,0)$ represents the disutility of searching l hours and not spending any time working, n = 0. For the employed workers, we have l = 0 (there is no on-the-job search) and n > 0, and $d_e\left(0\right)=\tilde{d}(0,\tilde{n})$ represents the disutility from working a fixed number of hours, $n=\tilde{n}$, and not searching.

Expected life-time utility (welfare) associated with a consumption-effort plan {c_t, l_t} for a worker of initial type s₀ is given by

where β is the pure discount factor of workers. Note that the expectations E[.] in (3) is taken with respect to a joint distribution over idiosyncratic and aggregate shocks that depends on the effort plan {l_t} through its effect on transition probabilities, π.

The initial type of a worker is defined by a pair (s₀, h₀) – recall that a₀ = 0 for all workers. The initial aggregate state of the economy is defined by S₀ = N, R and M₀, where M₀ is a joint distribution over initial employment states, s₀, and initial human capital levels, h₀. Note that we do not include the initial distribution over asset holdings into M₀ since we have a₀ = 0 for all workers. Note also that the marginal distribution of M₀ over employment states is simply the vector of initial unemployment rates, U₀ = (U_0,su, U_0,lu). For given initial state (s₀, h₀, S₀, M₀), workers choose a plan {c_t, l_t, a_t} so as to maximize (3) subject to (2), where a plan is a sequences of (arbitrary) functions of the history of individual employment states and aggregate productivity shocks: c_t = c_t(s^t, S^t), l_t = l_t(s^t, S^t), and a_t+1 = a_t(s^t, S^t). Aggregate variables are denoted by upper case letters and are obtained from individual plans by taking the expectations over individual histories and initial types conditional on the history of aggregate shocks: C_t ≐ E_t[c_t] = ∫_h0,s0 Σ_s^t c_t(h₀, s₀, M₀, S₀; s^t, S^t)π_t(s^t, S^t|s₀, S₀)dM₀(h₀, s₀). This notation linking individual plans to aggregate variables is used throughout the paper.¹⁹ Note that we have C_t = C_t(M₀, S₀, S^t), that is, aggregate variables depend on the initial state, (S₀, M₀), and the (partial) history of aggregate shocks, S^t.

3.2 Firms

There is a continuum of firms producing a single consumption good using labor as the only input factor. A firm is defined by its productivity parameter that takes on two values, Z > 0 or 0. One firm is matched with one worker and a worker-firm match produces Z x h or 0. We assume that a firm with productivity 0 will have productivity 0 forever, and therefore shuts down. Thus, the job destruction rate π_u|e(S’) introduced in the previous section is equal to the probability that firm-level productivity changes from Z in period t to 0 in period t + 1. Note that these transitions rates, π_u|e(S’), are exogenous. We assume that the matching of workers with firms is random so that aggregate output produced in period t is equal to ${\tilde{Y}}_t=Z{H}_{t,e}$, where H_te = E[h_t|s_t = e] is the aggregate human capital stock of employed workers. The labor market is competitive in the sense that the real wage per unit of human capital, w, is equal to the marginal product of labor: w = Z. This means that workers receive the entire match surplus and firms make zero profit.

The assumption that all producing firms have a common productivity, Z, rules out the possibility that UI has a positive effect on productivity and wages by encouraging unemployed workers to search for high productivity jobs (Acemoglu and Shimer, 1999, 2000). The empirical literature has found mixed results regarding the effect of UI on post-displacement wages (Card, Chetty, and Weber, 2007, and Schmieder, Wachter, and Bender, 2016). We return to this issue in our robustness analysis in Section 7, where we consider an extension of the baseline model in which UI affects the quality/productivity of newly formed worker-firm matches.

We assume that lnZ — d_e(0) > lnb_s — d_s(0) with s = su, lu, which ensures that in equilibrium it is always optimal for unemployed workers to accept job offers. Note that even though employed workers are paid a common wage per unit of human capital, w = Z, the model implies a non-degenerate, endogenous cross-sectional distribution of wages/earnings since the cross-sectional distribution of human capital is non-degenerate and endogenous. This cross-sectional distribution fans out over time given that workers are infinitely lived and wages/earnings follow a logarithmic random walk with drift – see equation (1). Thus, there is no stationary cross-sectional distribution of wages/earnings. However, it is straightforward to extend the model to the case in which workers face a constant probability of death and a stationary cross-sectional distribution of wages/earnings exists (Constantinides and Duffie, 1996).

3.3 Equilibrium

We assume that the risk-free asset that can be traded by workers is in zero net supply. Thus, market clearing in the (domestic) asset market reads

Denote by $A_t^{-}$ the total debt of workers and $A_t^{+}$ the total asset holdings of workers. Note that $A_t^{-}=A_t^{+}$ because of (4). We define output net of the cost of financial intermediation as $Y_t={\tilde{Y}}_t-A_t^{-}$.

We next define feasible labor market institutions (b, x, τ). To this end, note first that any government choice of a labor market institution (b, x, τ) induces a stochastic process of government spending, {G_t}, and government revenues, {T_t}, where G_t and T_t are defined as

The first term in the definition of G is government spending on short-term unemployed workers (unemployment benefits plus jobs search assistance) and the second term is spending on long-term unemployed workers (recall that the functions λ_s measure the resource cost of providing job search assistance). Government revenue, T_t, is equal to the revenue from the flat-rate labor income tax (social security tax).

A labor market institution, (b, x, τ), is feasible if the induced fiscal policy, {G_t, T_t}, satisfies a present value budget constraint. More precisely, for any fiscal policy {G_t, T_t} define the associated expected present value of government deficits/surpluses as

where ν_t is the t-period discount factor used by the government and the expectation indicates that we average over histories of aggregate states S^t conditional on the initial aggregate state: E [G_t|M₀, S₀] = Σ_S^t G_t(M₀, S₀; S^t)π_t(S^t|S₀) and E[T_t|M₀, S₀] = Σ_S^t T_t(M₀, S₀; S^t)π_t(S^t|S₀). We use the deficit-to-output ratio, instead of the deficit, in order to avoid technical issues that arise when taking the limit for sequences of random variables that grow in expectations, which occurs naturally in the equilibria considered in this paper. A labor market institution (b, x, τ) is feasible if it induces a fiscal process {G_t, T_t} that satisfies the present value budget constraint

The idea underlying the present value budget constraint (6) is that the government can borrow and lend in international capital markets, and that it uses this ability to finance the budget deficits and surpluses induced by cycle-independent labor market institutions (b, x, τ). Specifically, consider the sequential government budget constraint

where D_t is government debt and r_w is the interest rate at which the government can borrow or lend (the world interest rate). It is straightforward to show that for D₀ = 0 the sequential budget constraint (7) in conjunction with the no-Ponzi scheme condition lim_t→∞ E[(1 + r_w)^-tD_t|M₀,S₀] = 0 implies the present value budget constraint (6) if we choose ${\nu }_t=(1+r_w{)}^{-t}{\mathrm{\Pi }}_{n=1}^t(1+g_n)$ in (6), where g_n is the growth rate of output. Note that our equilibrium characterization result (see proposition 1) also holds if r_w = r, that is, if the world interest rate is equal to the domestic lending/saving rate. Note further that we impose the no-Ponzi-scheme condition only at the initial state (M₀, S₀). If the no-Ponzi scheme condition is imposed for all histories S^t, then the equivalence between sequential budget constraint and present-value budget constraint only holds in the case in which the government has access to a sequentially complete set of financial contracts.

For the fiscal policies considered in this paper, the debt process may have an unbounded debt-to-output ratio and such government policies are sometimes considered problematic – see, for example, Bohn (1995) for a general discussion. To address this issue, we show in the Online Appendix that for any economy with labor market institution (b, x, τ) and corresponding fiscal policy {G_t, T_t} with unbounded debt-to-output ratio there is an economy with the same labor market institution (b, x, τ) and a fiscal policy $\{G_t^{\prime },T_t^{\prime }\}$ with bounded government debt-to-output process so that equilibrium effort choices are the same and the welfare cost of recessions are arbitrarily close in the two economies. The $\{G_t^{\prime },T_t^{\prime }\}$ in the new economy is defined by the labor market institution (b, x, τ) plus a consumption tax/subsidy that is only non-zero when government debt is very high – see the Online Appendix for details. In this sense the labor market institutions considered in this paper also satisfy a bounded debt-to-output condition.

Given the concepts introduced so far, our definition of sequential equilibrium is standard:

Definition. For given labor market institution (b, x, τ,) and initial state (M₀, S₀), a sequential equilibrium is a wage rate w, a sequence of interest rates, {r_t}, and a family of worker plans, {c_t, a_t, l_t}, so that

i) For all worker types (h₀, s₀), the individual plan {c_t, a_t, l_t} maximizes expected lifetime utility (3) subject to (2).

ii) Market clearing condition (4) holds in each period t.

The labor market institution (b, x, τ) is feasible if the government budget constraint (6) is satisfied.

The household budget constraint (2) in conjunction with the asset market clearing condition (4) and the government budget constraint (6) imply the aggregate resource constraint (goods market clearing condition)

where C_t is aggregate consumption and Λ_t is aggregate spending on job search assistance for unemployed. Note that the aggregate resource constraint (8) is an expected present value constraint, where the expectation is taken over all histories S^t. This has to be the case since the government budget constraint (6) is an expected present value constraint.

3.4 Equilibrium Characterization

The previous section defined sequential equilibria. Recursive equilibria are defined accordingly. We now turn to the characterization of recursive equilibria confining attention to recursive equilibria with minimal state space. We show below that it suffices to use the exogenous state S as the relevant aggregate state variable. In other words, neither the cross-sectional distribution of assets nor the cross-sectional distribution of human capital are relevant state variables. Anticipating this result, we can write the worker’s maximization problem as

where the human capital “shocks” η are defined as η_s’,∊’ = ∊’ if s^’ = e and ηs’,∊’ = –δ_s’ for s’ = su, lu. The correspondence Γ is defined by the individual budget set (2). Note that effort choice l is only relevant for the transition probability π_s’|s(l, S’) if s = su, lu.

We show below that in equilibrium workers do not trade the risk-free asset (do not self-insure) and consume their current income: a_t = 0 and c_t = ϕ(s_t)h_t for all histories, where we define ϕ(e) = (1 — τ)w and ϕ(s) = b(s) for s = su, lu. In this case, the value function of workers is given by

where v together with the optimal effort choice are the solution to the intensive-form Bellman equation

The solution to equation (11) determines the equilibrium effort choice of workers, l = l(S). We have a_t = 0 and c_t = ϕ(s_t)h_t with h_t+1 = (1+η(s_t+1, ∊_t+1))h_t in equilibrium, which implies that the solution to (11) also pins down the equilibrium plan {c_t, a_t, l_t} of workers.

Equation (11) determines the household choices for given labor market institutions, (b, x, τ). To find feasible labor market institutions, we need to ensure that the government budget constraint (6) is satisfied. For the recursive equilibria considered here, the present value budget constraint (6) can be rewritten in a more convenient way. Specifically, define Ω as the vector of human capital shares of workers of type s defined as Ω_s = H_s/H. We show in this paper that the equilibrium law of motion for Ω, which we denote by Φ, is

where ${\overline{\eta }}_{s\prime }={\mathrm{\Sigma }}_{\epsilon \prime }{\eta }_{s\prime ,\epsilon \prime }{\pi }_{\epsilon \prime }$. Further, the present value of government deficits/surpluses becomes a function PV_g, = PV_g (Ω₀, S_o), and can be found as the solution to the recursive equation

where Φ is the law of motion for Ω given by (12) and g is the growth rate of output specified in equation (18) below. Note that in (13) we assume that the discount factor in (6) is given by ${\nu }_t={(1+r_w)}^{-t}{\mathrm{\Pi }}_{n=1}^t(1+g_n)$, that is, we assume that the government can borrow and lend in international financial markets at the world interest rate r_w.

In summary, we have the following equilibrium characterization result:

Proposition 1. Suppose that the cost of financial intermediation α (the interest rate spread) is greater or equal to α_min given in (A2). Then the solution to (11) defines a recursive equilibrium with aggregate state S for given labor market institution (b, x, τ). The labor market institution is feasible if the government budget constraint (6) holds, where the present value function PV_g is the solution to the recursive equation (13) using the law of motion (12).

Proof: Online Appendix.

The proof of proposition 1 is based on the idea that workers will choose a_t = 0 if the spread between the borrowing rate and lending rate is sufficiently large. Note that the proof is non-standard since first-order conditions may not be sufficient due to the non-concavity of the objective function, a problem that is well-known in the literature on multi-period moral hazard problems with (unobserved) asset trade. See, for example, Abraham, Pavoni, and Koehne (2011) for a survey of the literature and a fairly general treatment of the two-period case. We show that in our model the first-order conditions are indeed sufficient, and we use this property of our model to prove that no-trade is an equilibrium with sufficiently high cost of financial intermediation. Our proof heavily relies on three key assumptions: i) homothetic preferences, ii) non-asset income that is proportional to human capital of workers, and iii) human capital that evolves subject to shocks that are proportional to human capital (proportional skill depreciation during unemployment).

To gain a better understanding of proposition 1, consider the special case in which there is only one type of unemployment, s = u, e, there are no wages shocks, $\epsilon =\overline{\epsilon }$, and {S_t} is i.i.d. In this case, equation (11) becomes

where ${\overline{\pi }}_{u|e}={\pi }_{u|e}\left(N\right)\pi \left(N\right)+{\pi }_{u|e}\left(R\right)\pi \left(R\right)$ denotes the mean job destruction rate and ${\overline{\pi }}_{e|u}\left(l\right)={\pi }_{e|u}(l,N)\pi \left(N\right)+{\pi }_{e|u}(l,R)\pi \left(R\right)$ is the mean job finding rate. Thus, the solution to (14) only depends on the mean of the job destruction rate and the job finding rate, but not their volatilities. In other words, aggregate fluctuations in worker flow rates have no effect on equilibrium choices and welfare, a result that is driven by the property that expected utility is linear in probabilities – see Krebs (2007) for a general discussion of such neutrality results. Note further that the solution is independent of the aggregate state S, that is, the equilibrium effort choice, l, is constant over the cycle. Equation (14) is the Bellman equation associated with the maximization problem of a long-lived worker who moves between unemployment and employment, chooses search effort when unemployed, and consumes b when unemployed and (1 — τ)w when employed.

3.5 Computation and Tractability

The equilibrium analysis conducted in this paper is based on proposition 1. Specifically, we use proposition 1 to compute and analyze recursive equilibria in two steps. First, for given labor market institution (b, x, τ) we solve the worker problem (11). Second, we vary the tax rate τ until the government budget constraint (6) is satisfied using the recursive equation (13) and the law of motions (12). Note that this approach simplifies our analysis for two reasons. First, it reduces the problem of solving the Bellman equation (9) to the much simpler problem of solving the intensive-form Bellman equation (11). Note that in (9) the individual worker makes a consumption-saving choice and an effort choice, whereas in (11) the worker only makes an effort choice. In other words, the fact that a_t = 0 in equilibrium means that we have found the optimal consumption-saving choice. Second, neither the cross-sectional distribution over asset holdings nor the cross-sectional distribution of human capital are relevant state variables. Clearly, these distributions are in general infinite-dimensional objectives, which means that in general approximation methods along the lines of Krusell and Smith (1998) are required for the equilibrium analysis. In contrast, in the current paper we do not rely on approximation methods since we compute the solution to the exact equilibrium conditions.

The solution to (11), (12), (13) determines the equilibrium values of search intensity and job finding rates, the tax level τ ensuring budget feasibility, and the equilibrium law of motion for Ω. The equilibrium evolution of unemployment and output (growth) can be computed from these variables as follows. The law of motion for the unemployment rate is

Further, aggregate output is given by Y_t = ZH_t,e, where aggregate human capital, H_t = (H_t,e, H_t,su, H_t,lu), evolves according to

Note that equation (16) determines the equilibrium growth rate of aggregate human capital of employed workers, g,_t+1 = H_e,t+1/H_et — 1, which is equal to the equilibrium growth rate of output: H_e,t+1/H_et = Y_t+1/Y_t. It follows from the definition of Ω and the equilibrium law of motion for Ω that this equilibrium growth rate is given by

To see that expression (17) is the growth rate of output, consider the special case of s = u, e (only one type of unemployment), δ_u = 0 (no skill depreciation) and ∊ = 0 (no wage growth). In this case, we have Ω_u = U and Ω_e = (1 — U), and (17) reduces to $1+g={\pi }_{e|e}+{\pi }_{e|u}\frac{U}{1+U}$, which is simply the change in the employment rate, that is, output growth equals employment growth. Further, if we set U equal to its steady state value, then we find g = 0. Thus, in an economy without wage growth for employed workers and without skill depreciation during unemployment, there is no steady state growth and changes in the steady-state values of the job destruction or job finding rate have no effect on steady state growth. This result is no longer valid in the general economy with wage growth for employed workers and/or skill depreciation, and this non-neutrality property is an important source of the output cost of recessions discussed in this paper – see also our discussion in Section 4.

4.1 Welfare Cost of Recessions

In analogy to the welfare cost of business cycles (Lucas 1987, 2003), we define the welfare cost of recessions as the gain in social welfare that accrues when recessions are eliminated. More precisely, we consider a thought experiment in which we move from an economy with recessions and S’-dependent job destruction rates π_su,e and job finding rates π_e|_s to an economy without recessions and job flow rates, ${\hat{\pi }}_{su|e}$ and ${\hat{\pi }}_{e|s}$, that are equal to the transition rates that prevail in normal times. Thus, we assume that we remove recessions without affecting the labor market in normal (or boom) times. We also assume that human capital shocks and the job finding function, which do not depend on macroeconomic shocks S, remain the same. In sum, the elimination of recessions amounts to:

Equation (18) shows how the elimination of recessions affects job destruction and job finding rates, and proposition 1 provides an algorithm to compute equilibria before and after the elimination of recessions. Let Δ stand for the welfare cost of recessions, that is, the difference, expressed in lifetime consumption units, between social welfare in the economy without recessions and social welfare in the economy with recessions. More precisely, let {c_t, l_t} stand for the equilibrium allocation in the economy with recessions and $\{{\hat{c}}_t,{\hat{l}}_t\}$ for the equilibrium allocation in the economy without recessions. We then define the welfare cost of recessions as the number Δ solving

where μ(h₀, s₀) is the welfare weight the social planner assigns to workers of initial type (h₀, s₀) and W({c_t, l_t}|h₀, s₀, Ω₀ S₀) is expected lifetime utility of a worker of type (h₀, s₀) defined in (3). Note that (19) takes into account the transition path, that is, we compare two economies with different future aggregate shock processes (different worker flow rates) and different tax rates τ necessary to satisfy the government budget constraint (6), but the same initial aggregate state (Ω₀, S₀) – recall that the expected present discounted value of government surpluses/deficits is a function of the initial state: PV_g, = PV_g((Ω₀, S₀). Note further that S₀ is a relevant state because it affects both future worker flow rates and the tax rate, whereas Ω₀ is a relevant state variable only because it affects the tax rate necessary to satisfy (6). Finally, note that for the general analysis we do not need to impose the normalization ∫dμ(h₀, s₀) = 1, that is, we only need to assume that μ is a measure, but not necessarily a probability measure. If the social planner assigns equal weight to each individual worker, we have μ_su = U_0,su, = U_0,lu, and μ_e = 1 — U_0,su — U_0,lu.

For given initial state (h₀, s₀, Ω₀, S₀), expected lifetime utility is W({c_t, l_t}|h₀, s₀, Ω₀, S₀) = V_s0 (h₀, Ω₀, S₀), where V is the value function determined by the solution to the Bellman equation (9). Note that V depends on Ω₀ because Ω₀ affects the present value of government deficits/surpluses, PV_g, and therefore also affects the tax rate τ that is required to satisfy the government budget constraint (6). From proposition 1 we know that the equilibrium value function has the functional form V_s0(h₀, Ω₀, S₀) = ν_so(Ω₀, S₀) + 1/(1 — β) ln h₀. Using this representation, we find that the welfare cost of recessions (19) is given by the following expression: