Optimal unemployment benefits financing scheme, search frictions and real wage rigidities Julien Albertini1 Xavier Fairise23 Juin 2008 First draft Abstract In this paper we feature the optimal unemployment benefits financing scheme when the economy is subject to labor market failures. We are particulary interested in the effects of search frictions and wages rigidity. The goal of this paper is to show how policy instruments should interact with labor market imperfections. Taking inspiration from the US unemployment insurance system, we wonder if firms should be taxed in proportion of their layoffs to finance the cost incurred by the unemployment benefit fund. The welfare gains and the stabilizing effects of such a policy are evaluated. We find that an optimal combination of unemployment benefits and layoff taxes is welfare-enhancing and can improve labor market performances. Search externalities and wage rigidities cause sizeable welfare losses and influence the optimal design of the financing scheme. Without wage rigidities the efficient layoff tax corresponds to the expected fiscal cost of an unemployed worker. Conversely, when wages are rigid, the cost firms should support is much higher.
Keys-words : DSGE models, search and matching friction, layoff tax, wage rigidities.
JEL Classification : E61; E65, J63, J65.
1
Centre d’Études des politiques Économiques de l’Université d’Evry (EPEE),
[email protected], Université d’Evry, bd François Mitterand, 91025 Evry Cedex. 2 Centre d’Études des politiques Économiques de l’Université d’Evry (EPEE),
[email protected], Université d’Evry, bd François Mitterand, 91025 Evry Cedex. 3 We are grateful to François Langot for helpful comments on a previous version of this paper.
Introduction The optimal level of unemployment benefits and the dismissal regulation is a great concern among economists and policy makers. Indeed, there is an abundant literature that aims at measuring their effects on labor market outcomes. However, only few papers study how unemployment benefits should be financed and how they interact with employment protection. In the line of Blanchard and Tirole (2008) study, we try to show how these two labor market institutions may be jointly reformed to improve labor market performances. Taking inspiration from the US unemployment insurance system, we wonder if firms should be taxed in proportion of their layoffs to finance the cost incurred by the unemployment benefit fund. We then explore the properties of an optimal unemployment benefits financing scheme. Furthermore, we introduce search frictions and wage norms. We study how an optimal tax schedule can offset these labor market failures. Our study starts from an original feature of the US unemployment insurance known as experience rating (ER thereafter). In this system individual employers’ contribution rates are varied on the basis of the firm’s history of generating unemployment. Basically, the more dismissal, the higher the firms’ contribution to unemployment insurance. According to the ETA4 “experience rating systems are designed to encourage employers to stabilize employment, equitably allocate the costs of unemployment and to encourage employers to participate in the system by providing eligibility information.”. The effects of the payroll-tax indexation on temporary layoffs (which are frequent in the US) and on unemployment has been illustrated by several works like Feldstein (1976), Topel (1983), Topel (1983) and Card and Levine (1994). They argued that a higher payroll-tax indexation lowers the incentive for firms to lay off during economic downturns and to hire during booms. On the other side, they show that unemployment insurance subsidies5 play a major role in reducing employment instead of hours in bad states. The reason is that firms pay less than the full cost of layoffs as long as experience rating is imperfect6 . To our knowledge, the first theoretical paper that deals with the optimal design of unemployment benefits and employment protection is the one written by Blanchard and Tirole (2008). In a static model, they show that employment protection is likely to be efficient in the form of a layoff tax whose level corresponds to unemployment benefits. In this line of research, Cahuc and Zylberberg (2007) 4
According to the definition provided by the Employment and Training Administration. The experience rating system is said to be perfect when an employer pays for the entire cost of unemployment benefits that are perceived by his ex-employees. When it is imperfect, an employer who fires a worker obtains an implicit subsidy which is financed through other firms. 6 In each state, regulation imposes a minimum and a maximum contribution rate. Then, if the contribution rate of an employer corresponds to the maximum, more dismissals do not increase his contribution to unemployment insurance. This is one of the reasons why an employer does not pay for the total expenditure caused by its action. See See Fougère and Margolis (2000) for more details. 5
1
conclude that the optimal layoff tax is equal to the social cost of job destruction when the government provides a public unemployment insurance and aims at redistributing incomes. Using the tractable framework of Mortensen and Pissarides (1999), Cahuc and Malherbet (2004) incorporate a simplified ER system and some features of a rigid continental European labor market to evaluate its impact on equilibrium unemployment. They show that ER reduces unemployment rate for the low-skilled workers and can improve their welfare in presence of a high minimum wage, a stringent dismissal regulation and temporary jobs. L’Haridon and Malherbet (2008) show that such a reform can reduce significatively unemployment, job creation and job destruction variability. However the current literature has largely ignored the valuable stabilizing effects of layoff taxes. They extensively focus on long-run levels leaving aside the welfare gains coming from smooth fluctuations. Moreover they do not show how policy instruments should interact with labor market failures. In this paper, we take on these tasks and we try to characterize the optimal tax schedule. Search frictions influence the average duration of unemployment and therefore the fiscal cost associated to a dismissal. The main problem is that firms do not internalize the effect their firing decisions have on others. When an employer lay a worker off, he doesn’t take into account the time spend by its ex-employee to find another job nor the decrease in the probability of finding a job for others unemployed workers. Then, the consequences on the unemployment insurance budget are not internalized. It is worth analyzing how congestion externalities caused by searching firms and workers can be reduced with an optimal tax schedule. Real wage rigidities has been pointed to by many authors (Hall 2005, Shimer 2005). They prevent wages from adjusting instantaneously to economic shocks. They support strong empirical evidences and provide a powerful propagation mechanism consistent with observed business cycle facts. On the other side, wage rigidities influence the abilities of firms and workers to use tax and benefits as a threat in the wage bargaining. Therefore, they are quite relevant to assess the quantitative effects of stabilization policies. These assumptions have been systematically ignored in the literature on optimal taxation mentioned above. Our setup allow to capture distortions caused by firm’s layoff decisions and to evaluate the welfare gains induced by an optimal labor market policy. We wonder if firms should be taxed in proportion of their layoffs and if the tax should correspond to a part or all of the fiscal cost induced by a redundancy. In other words, we ask whether firms should be responsible for the cost incurred by the unemployment benefit fund. The intuition suggest that an imperfect experience rating induces too many layoffs as firms bear a small share of the total cost of job destructions. But the existence search frictions and wage rigidities can strongly influence hiring and firing decisions. In such a context, the optimal design of employment protection and unemployment insurance is not self-evident. The question we can ask is if an appropriate financing scheme can make firms internalize the cost of separations and reduces welfare losses. 2
We use a DSGE model and borrow from Cahuc and Malherbet (2004) a simple unemployment insurance system that combines both a layoff tax and a lump-sum tax to finance unemployment benefits7 . In particular, we assume that the layoff tax is a function of the expected fiscal cost of an unemployed worker. Numerical simulations show that an optimal combination of unemployment benefits and layoff taxes is welfare-enhancing and can improve labor market performances, consistent with the previous findings. The first-best allocation (defined as the Ramsey equilibrium) allows to implement the Pareto optimality. Wage rigidities have a strong impact on both welfare losses and the optimal level of policy instruments. Indeed, without wage rigidities firms should support the entire burden of the expected fiscal cost of an unemployment worker. But when the degree of wage rigidities is high, the optimal policy display a layoff tax that is higher than the expected fiscal cost. In each case, optimality requires a replacement rate reduced by around one third. We defined a second-best allocation. It corresponds to the economy in which the levels of institutional parameters are set in such a way they maximize the welfare of agents. While the optimal financing scheme remains virtually unchanged, the labor market dynamic strongly differs from the first-best. The reason is that layoffs taxes introduce a labor hoarding phenomena and the effect of aggregate productivity shocks translate into the vacancy posting activity. However, they create a financial incentive for employers to stabilize their employment, reducing welfare losses. The rest of the paper is organized as follows. Section 2 presents the model and the unemployment insurance system. The equilibrium and the optimal policies are defined in section 3. Section 4 is devoted to simulation exercises and section 5 concludes.
1
The economic environment and the model
Following Den Haan, Ramey, and Watson (2000), our DSGE model includes Non-Walrasian labor market with endogenous job destruction in the spirit of Mortensen and Pissarides (1994). According to Shimer (2005), we focus on workers flows between employment and unemployment. Workers “out of the labor force” are thus not taken into account. Time is discrete and our economy is populated by ex ante homogeneous workers and firms. Endogenous separations occur because of firms specific productivity shocks. There are search and matching frictions in the labor market, wages are determined through a Nash bargaining process. There is no other market failures. 7
Layoff taxes can be viewed as an ersatz of experience rating because firms are in charged of the benefits payments they create through their dismissal decisions. We discuss later the differences between the system we use and the regulations in force.
3
1.1
The labor market
Search process and recruiting activity are costly and time-consuming for both, firms and workers. To produce, a firm needs to hire one worker, thus, each firm offers one job. A job may either be filled and productive or unfilled and unproductive. To fill its vacant job, the firm posts a vacancy and incurs a cost κ. Workers are ex ante identical, they may either be employed or unemployed. Unemployed workers are engaged in a search process. The number of matches Mt is given by the following Cobb-Douglas matching function : Mt = χ(1 − Nt )ψ Vt1−ψ with ψ ∈]0, 1[, χ > 0
(1)
with Vt the vacancies and 1 − Nt the unemployed workers. The labor force is normalized to 1, the number of unemployed workers Ut thus satisfied Ut = 1−Nt . The matching function (1), satisfying the usual assumptions, is increasing, concave and homogenous of degree one. A vacancy is filled with probability qt = Mt /Vt . Let θt = Vt /(1 − Nt ) be the labor market tightness, the probability an unemployed worker finds a job is θt qt = Mt /(1 − Nt ). It is useful to rewrite these probabilities as follows : qt = χ θt qt = χ
1 − Nt Vt Vt 1 − Nt
ψ (2) 1−ψ (3)
At the beginning of each period, separations occur for two reasons. Firstly, some separations occur at an exogenous rate ρx . Secondly, firms productivity is subject to idiosyncratic shocks i.i.d. drawn from a time-invariant distribution G(.) defined on [0, ε]. If the firm specific productivity component εt falls below an endogenous threshold εt , the job is destroyed and the employment relationship ceases. Endogenous separations occur at rate : ρnt = P (εt < εt ) = G(εt )
1.2
(4)
The sequence of events
At each date, a firm is characterized by its specific productivity level εt drawn from the distribution G(.). The firm productivity is also subject to an aggregate productivity shock zt . The production level is given by : yt = zt εt 4
(5)
We now describe the sequence of events and the labor market timing, we mainly follow Zanetti (2007). Employment in period t has two components : new and old workers. New employment relationship are formed through the matching process. Matches formed at period t − 1 contribute to period t employment. New jobs begin with the highest productivity level ε, thus, all the new employment relationship are productive (at the first period). Let NtN = Mt−1 denote the new employment relationships. At the beginning of period t, Nt−1 jobs are inherited period t − 1 and ρx Nt−1 jobs are exogenously destroyed. Then after, idiosyncratic shocks are drawn and firms observe their specific component εt . If the specific component is below the threshold εt , the employment relationship is severed. Otherwise, the employment relationship goes on. A fraction ρnt of the remaining jobs (1−ρx )Nt−1 is destroyed. Let nC t (ε), with ε ∈ [εt , ε], denote the number of continuing productivity level ε employment relationships. It satisfies : x nC t (ε) = (1 − ρ )Nt−1 G (ε)
(6)
The total number of continuing employment relationships is thus given by NtC = ε C n (ε)dε = (1 − ρx )(1 − ρnt )Nt−1 and the total separation rate is defined as εt t follows : ρt = ρx + (1 − ρx )ρnt
(7)
Finally, the employment law of motion is described by the following equations : N Nt+1 = Mt ε N nC Nt = t (ε)dε + Nt
(8) (9)
εt
Nt = (1 − ρx )(1 − ρnt )Nt−1 + NtN
1.3
(10)
The large family program
To avoid heterogeneity, we suppose that infinitely lived households are members of a large family. There is a perfect risk sharing, family members pool their incomes (labor incomes and unemployment benefits) that are equally redistributed. Following Algan (2004), the large family assumption allows to assess the own impact of layoff taxes on the cost of aggregate fluctuations. Contrarily to Andolfatto (1996) the large family model allows to distinguish unemployed workers and tenured workers trajectories. The expected intertemporal utility of the large family writes : VtM
= Et
∞ s=t
β s−t
(Cs + (1 − Nt )h)1−σ 1−σ
5
(11)
β ∈]0, 1[ is the discount factor and σ ∈]0, 1[∪]1, ∞[ is the intertemporal elasticity of substitution. h denotes unemployed workers home production. Family consumption is thus the sum of the total home production (1 − Nt )h and of the market consumption goods Ct . The dynamic optimization problem consists of choosing a sequence of consumption {Cs }∞ t maximizing the expected intertemporal utility subject to the budget constraint and a set of equations describing the employment motion. The large family’s choice problem takes the following recursive form : M
(Ct + (1 − Nt )h)1−σ M + βEt V (Θt+1 ) 1−σ
(Θt ) = max Ct ⎧ ε N N ⎪ −Ct + ε nC t (ε)wt (ε)dε + Nt wt (ε) + (1 − Nt )bt + Πt = 0 ⎪ ⎪ εt C ⎨ −Nt + ε nt (ε)dε + Ntn = 0 s.t. t N ⎪ ⎪ ⎪ −Nt+1 + θt qt (1 − Nt ) = 0 ⎩ (1 − ρx )Nt−1 G (ε) − nC t (ε) = 0
V
(12) (λt ) (μ1t ) (μ2t ) (μt (ε)), ∀ε ∈ [εt , ε]
with the state vector Θt = (Nt−1 , NtN ; zt ). bt is the unemployment benefit perceived by an unemployed worker and wt (ε) denotes the wage associated to the productivity level ε. New jobs begin with the highest productivity level, the associated wage writes wtN (ε). Finally, the large family receives instantaneous profits for an amount Πt . The fist constraint is the budget constraint and the three other constraints describe the employment motion. The third constraint expresses the fact that the large family takes as given the job finding probability. The consumption optimality condition writes : (Ct + (1 − Nt )h)−σ = λt
(13)
The envelop conditions can be expressed as follows : V2M (Θt ) = λt wtN (ε) − λt bt − λt h − θt qt βEt V2M (Θt+1 ) ε x + (1 − ρ )βEt μt+1 (ε)dG(ε)
(14)
εt+1
μt (ε) = λt wt (ε) − λt bt − λt h − θt qt βEt V2M (Θt+1 ) ε x + (1 − ρ )βEt μt+1 (ε)dG(ε)
(15)
εt+1
Equations (14) and (15) respectively provide the family’s marginal values of a new job and of a continuing job of productivity ε.
6
1.4
The large firm program
The expected discount sum of instantaneous profits of the la firm writes :
ε ε ∞ F s−t λs C N β zs εns (ε)dε + Ns zs ε − ws (ε)nC Vt = Et s (ε)dε λ t ε ε s s s=t − NsN wsN (ε) − (1 − ρx )ρns Ns−1 (F + τsE ) (16) New jobs begin with the highest specific productivity level and are always productive, obviously no separation occurs. Conversely, old jobs do not continue (recall the decision to continue is taken after observing the specific productivity shock) if their specific productivity level is below a threshold εt . If the large firm terminates an employment relationship, it has to support a cost F induced by the employment protection legislation and to pay a firing tax τtE , this is expressed by the term (1 − ρx )ρnt Nt−1 (F + τtE ). The dynamic optimization problem consists of choosing sequences of vacancies, thresholds and the number of continuing employment relationships, that is Ct = (Vt , εt , {nC t }ε∈[εt ,ε] ), maximizing the expected discount sum of instantaneous profit subject to the constraints describing the employment motion. The large firm problem takes the following recursive form :
F
V (Δt ) = max Ct
−
x
(1 − ρ )Nt−1 (F +
ε
εt
N zt εnC t (ε)dε + Nt zt ε −
τtE )
+ (F +
τtE )
ε
εt
ε
εt
nC t (ε)dε
N N wt (ε)nC t (ε)dε − Nt wt (ε)
λt+1 F + βEt V (Δt+1 ) λt
(17)
⎧ ε n ⎪ (Λ1t ) ⎨ −Nt + εt nC t (ε)dε + Nt = 0 N s.t. (Λ2t ) −Nt+1 + qt Nt = 0 ⎪ ⎩ (1 − ρx )N G (ε) − nC (ε) = 0 (ζ (ε)), ∀ε ∈ [ε , ε] t−1 t t t
with the state vector Δ = (Nt−1 , NtN ; zt ). The second constraint means that the large firm take the probability to fill a job. The optimality conditions may be written as follows : λt+1 F κ + βEt V (Δt+1 ) = 0 qt λt 2 −V2F (Δt ) + zt ε − zt εt − (wt (ε) − wt (εt )) − (F + τtE ) = 0 ε λt+1 (zt+1 ε − zt+1 εt+1 wt (εt ) − zt εt − F − τtE − (1 − ρx )βEt λt εt+1 E −(wt+1 (ε) − wt+1 (εt+1 ))dG(ε) − (F + τt+1 ) = 0 −
(18) (19)
(20)
Equations (18) and (19) provide the employment creation condition whereas equation (20) is the destruction condition. The envelop condition write :
7
V2F (Δt )
= zt ε −
wtN (ε)
− (1 − ρx )βEt
λt+1 + (1 − ρ )βEt λt x
ε
ζt+1 (ε)dG(ε)
εt+1
λt+1 E (F + τt+1 ) λt
ζt (ε) = zt ε − wt (ε) + F + − (1 − ρx )βEt
τtE
(21)
λt+1 + (1 − ρ )βEt λt x
ε
ζt+1 (ε)dG(ε)
εt+1
λt+1 E (F + τt+1 ) λt
(22)
Equations (21) and (22) respectively give the large firm’s marginal values of a new job and of a continuing job.
1.5
Wage setting mechanism
At equilibrium, filled jobs generate a return (the value of the job plus the corresponding employed worker value) greater than the sum of values of a vacant job and of an unemployed worker. The net gain issued from a filled job is the total surplus of the match. In our model, we have to distinguish two surplus, the surplus of a new job and the surplus of a continuing job, that is : V2M (Θt ) + V2F (Δt ) λt μt (ε) St (εt ) = + ζt (ε) λt
StN (¯ ε) =
(23) (24)
The two wages are determined through an individual Nash bargaining process between a worker and a firm who share the total surplus. Each participant threat point corresponds to the value of the alternative option, that is the value of being unemployed or the value of a vacant job. The outcome of the bargaining process is given by the solution of the following maximization problems :
wtN (¯ ε)
= arg max
wtN (¯ ε)
wt (εt ) = arg max
wt (εt )
1−ξ F ξ V2M (Θt ) V2 (Δt ) λt 1−ξ μt (ε) (ζt (ε))ξ λt
(25) (26)
where ξ ∈]0, 1[ and 1 − ξ denote the bargaining power of firms and workers respectively. Using the free entry condition, the optimality conditions of the
8
above problems may be written as follows : V2M (Θt ) = (1 − ξ)V2F (Δt ) (27) λt μt (ε) ξ = (1 − ξ)ζt(ε) (28) λt Using equations (14), (15), (18), (21) and (22) to substitute values in (27) and (28) by their expression, wages are given by : λt+1 N x E F + τt+1 ε) = (1 − ξ) zt ε + κθt − β(1 − ρ )Et wt (¯ λt + ξ(bt + h) (29) ξ
λt+1 E x E wt (εt ) = (1 − ξ) zt εt + κθt + F + τt − β(1 − ρ )Et F + τt+1 λt + ξ(bt + h) (30) The structure of the wage equations is the same as in the standard matching theory. It contains the weighted contribution of both parties. Both equations take into account the expected firing costs (F and τtE ). During the bargaining, firms internalized that hiring a worker may be costly if the job is destroyed. The burden of the expected firing costs is thus subtracted from the worker’s contribution to firm’s output. Equations (29) and (30) differ because of the firing costs. Concerning an old job, firing costs should be paid in case of separation. Each party may use the cost of layoffs as a threat. Real wage rigidity Following Shimer (2005) and Hall (2005), real wage rigidities are introduced. There exists a wage norm w t constraining wage adjustment. The real wage paid for a given productivity level job is a weighted average of the Nash bargaining process wage and the wage norm w t . One has : λt+1 x E F + τt+1 = γ (1 − ξ) zt ε + κθt − β(1 − ρ )Et λt + ξ(bt + h) + (1 − γ)w t
ε) wtN (¯
(31)
λt+1 E x E wt (εt ) = γ (1 − ξ) zt εt + κθt + F + τt − β(1 − ρ )Et F + τt+1 λt t (32) + ξ(bt + h) + (1 − γ)w with γ ∈ [0, 1]. The higher 1−γ, the higher the real wages are rigid. Following Hall (2005), we set the wage norm equal to the steady state average wage (the average wage is defined latter by equation (37)), that is w t = w. 9
1.6
Job creation and job destruction
Job creation and job destruction are determined by equations (18) — (19). Using the wage setting equations and , the job creation and job destruction conditions respectively write : λt+1 κ E zt+1 ε − zt+1 εt+1 − (F + τt+1 + (1 − γ(1 − ξ))βEt ) =0 (33) qt λt (1 − γ(1 − ξ))(zt εt + F + τtE ) − γ(1 − ξ)κθt − (1 − γ)w t − γξ(bt + h) ε λt+1 x E +(1 − γ(1 − ξ))(1 − ρ )βEt (zt+1 ε − zt+1 εt+1 )dG(ε) − F − τt+1 = 0(34) λt εt+1
−
Equation (33) says that the expected gain from hiring a new worker is equal to the expected cost of search (which is κ times the average duration of a vacancy 1/qt ). It defines the relationship between the labor market tightness and the threshold value of idiosyncratic productivity. Equation (34) is the job destruction condition, it teaches us that the critical value of a job productivity depends on the reservation wages and on firing costs. It states that higher firing costs lower the reservation productivity because separations are more costly.
1.7
The unemployment insurance financing
An unemployed worker receives a benefit bt . Unemployment benefits are financed through a layoff tax and a lump sum tax paid by the large family. The layoff tax is paid by employers when an endogenous separation occurs. We impose the unemployment benefits may not be financed by debt. The unemployment insurance fund budget constraint is thus balanced every period : = (1 − Nt ) bt Unemployment benefits
Tt + (1 − ρx ) ρnt Nt−1 τtE Lump sum Experience tax rating part
(35)
The sequences followed by Tt , τtE and bt may be chosen following different ways, provided they satisfy the above budget constraint. Our aim is to evaluate some rules close to the US labor market institutions and to study their optimality in the presence of real wage rigidities or labor market inefficiencies, that is if the Hosios condition is not satisfied. Experience rating system Here, we describe an institutional rule setting taxes and unemployment benefits levels, it is close to the US system. We follow
10
Cahuc and Malherbet (2004) to represent an experience rating system. An unemployed worker receives a benefit bt equal to a proportion of the average wage wt , that is : bt = ρR wt
(36)
ρR < 1 is the average replacement rate. The average wage of the economy w¯t is given by : ε NtN N Nt−1 wt = w (¯ ε) + (1 − ρx ) wt (˜ ε) dG(˜ ε) (37) Nt t Nt εt Using equations equations (29) and (30), the above equation may be rewrite as follows : λt+1 x E (F + τt+1 Nt wt = Nt γξ(bt + h) + Nt γ(1 − ξ) κθt − β(1 − ρ )Et λt +γ(1 − ξ)
NtN zt ε
+ Nt−1 (1 − ρ )zt x
ε
εdG(ε) εt
+Nt−1 (1 − ρx )(1 − ρnt )γ(1 − ξ)(F + τtE ) + Nt (1 − γ)w t
(38)
The experience rating system works as follows : the lay off tax is proportional to the expected fiscal cost of an unemployed worker Qt+1 . Let e > 0 be the experience rating index (ERI), the firing tax τtE satisfies : τtE = eQt
(39)
Qt = bt + βEt θt qt × 0 + (1 − θt q(θt ))Qt+1
(40)
where
The above equation recursively determines the expected cost of an unemployed worker. The lay off tax corresponds to a share of the expected fiscal cost of an unemployed worker paid by the firm. The higher the ERI, the higher the firm contribution to the unemployment insurance. Its a very simple way to represent the US experience rating system. Its consistency may be questionable considering the complexity of current regulations. However, our representation may be viewed as an approximation of the US unemployment insurance system. As emphasized by Cahuc and Malherbet (2004) and L’Haridon and Malherbet (2008), it is a convenient mean to make firms contribute to the fiscal cost they induce. The rule previously described embodies some important features such that : • The higher the experience rating index, the higher the firms’ contribution to the unemployment insurance fund. If e = 1, firms fully take care the expected fiscal cost of an unemployed worker. 11
• The experience rating tax is increasing in the replacement rate and decreasing in the labor market tightness. The first one raises the expected fiscal cost of an unemployed worker while the second moves it in the opposite direction, indeed, it reduces the average unemployment duration.
1.8 1.8.1
The equilibrium The aggregate resource constraint
The aggregate output Yt is obtained through the sum of individual productions : ε¯ Yt = (1 − ρx )Nt−1 zt εdG( ε) + NtN zt ε (41) εt
The aggregation of the individual profits provides the amount of profits Πt received by the large family, that is : Πt = Yt − wt Nt − κVt − (F + τtE )(1 − ρx )ρnt Nt−1 The above equation together with the large family budget constraint (program (12) )and the government budget constraint (equation 35) gives the aggregate ressource constraint : Yt = Ct + κVt + F (1 − ρx )ρnt Nt−1 1.8.2
(42)
Definition of the equilibrium
Definition 1 For given lump sum tax rate Tt and firing tax τtE processes, and for a given exogenous stochastic process zt , the competitive equilibrium is a sequence of prices and quantities Nt , NtN , Ct , Vt , εt , θt , λt , qt , w t , wtN , Yt , ρnt , Mt and bt satisfying equations (1)-(4), (8),(10), (13), (1.6), (33)-(35),(38), (41) and (42). If taxes and benefits are set as described in subsection 1.7, the equilibrium definition writes as follows : Definition 2 (Experience rating system) For given parameters ρR and e and for a given exogenous stochastic process zt , the competitive equilibrium is a sequence of prices and quantities Nt , NtN , Ct , Vt , εt , θt , λt , qt , w t , wtN , Yt , ρnt , Mt , bt , τtE , Tt and Qt satisfying equations (1)-(4), (8),(10), (13), (1.6), (33)-(35),(38), (41), (42), (36), (39) and (40). t = w t−1 . The two above definitions apply for w t = w and w
12
1.8.3
The Ramsey allocation
As shown by equation (35), unemployment benefit may be financed through two ways : an experience rating tax (τtE ) and a lump-sum tax (Tt ). The lump-sum tax adjusts to equilibrate, at each date, the unemployment benefit fund.If the Hosios condition is not satisfied or if the real wage is rigid, the decentralized equilibrium of the economy without unemployment benefit and taxes is not optimal. Our aim is to determine an optimal unemployment benefit financing scheme and to compare the equilibrium allocation obtained with the Pareto allocation. The Ramsey policy is the taxation policy under commitment maximizing the intertemporal welfare of the representative household. Definition 3 (The Ramsey allocation) The Ramsey equilibrium is a sequence of prices, quantities and taxes Nt , NtN , Ct , Vt , εt , θt , λt , qt , w t , wtN , Yt , ρnt , Mt , bt , Tt , τtE maximizing the representative agent life-time utility : Et
∞ j=0
βj
(Ct+j + (1 − h)Nt+j )1−σ 1−σ
subject to the equilibrium conditions (1)-(4), (8),(10), (13), (1.6), (33)-(35),(38), (41) and (42) and given the exogenous stochastic processes zt . 1.8.4
The Pareto allocation and the equivalence with the Ramsey allocation
Consider equations (33) and (34), suppose the Hosios condition (ξ = 1 − ψ) be satisfied and there is no wage rigidities (γ = 1) and set the unemployment benefit and taxes equal to 0. One gets the following equations : −
κ Vt λt + βEt λt+1 zt+1 ε − zt+1 εt+1 − F = 0 1 − ψ Mt
ψ Vt λt λt (zt εt + F − h) − κ 1 − ψ 1 − Nt ε +β(1 − ρx )Et λt+1 (zt+1 ε − zt+1 εt+1 )dG( ε) − F =0
(43)
(44)
εt+1
Definition 4 (The Pareto allocation) For a given exogenous stochastic process, the Pareto allocation is a sequence of quantities Nt , NtN , Ct , Vt , εt , λt , Yt , ρnt , Mt satisfying equations satisfying equations (1),(4), (8), (10), (13) and (41)-(44). Result 1 (The Pareto allocation implementation) The optimal unemployment benefit financing scheme [definition 3] allows to implement the Pareto allocation. [This result holds w t = w and w t = wt−1 .] 13
Proof See appendix. The above result provides a simple way to determine the taxes and unemployment benefit processes implementing the Pareto allocation. The equilibrium values of Nt , NtN , Ct , Vt , εt , θt , λt , qt , Yt , ρnt and Mt are determined using equations (1),(4), (8), (10), (13) and (41)-(44). The exogenous stochastic process being given. The processes followed by the average wage, the taxes and unemployment benefits Tt , τtE and bt are then easily deduced from equations (38) and (33)-(35). Finally, wtN is provided by equations (29). In our economy, there is no reason the equilibrium be a Pareto optima. This occurs if there is real wage rigidity or, without wage rigidity, if the Hosios condition is not satisfied, that is if 1 − ψ = ξ. How do the firing tax and the unemployment benefit work to restore Pareto optimality? To begin, suppose there is no wage rigidity, that is γ = 1. The firing tax and the unemployment benefit restoring Pareto optimality can be determined using equations (33), (34), (43) and (44). To simplify, we consider these equations at the steady state, one gets : κ V ξ − (1 − ψ) β M ξ(1 − ψ) ψ − (1 − ξ) κV V ψ − (1 − ξ) b = κ + (1 − β(1 − ρx )) ξ(1 − ψ) 1 − N ξ(1 − ψ) βM
τE =
Suppose now that 1 − ψ < ξ, that is the bargaining process is in the favor of firms. The firing tax τ E and the unemployment benefit b are positive. The labor market is characterized by trade externalities. A greater number of vacancies increases the probability an unemployed worker finds a job and reduces the probability a firm fills a vacancy. Similarly, a greater number of unemployed increases the probability a firm fills a vacancy and reduces the probability a worker finds a job. If the bargaining power of workers 1−ξ is weak, that is less than ψ, the wage is low and firms post a lot of vacancies. In this case, without taxes and benefits, there are congestion externalities caused by searching firms posting a great number of vacancies, unemployment is below its optimal level. There exits a firing taxes and unemployment benefits scheme allowing to ensure optimality. Firing taxes reduce job creation, there are less searching firms. Unemployment benefits allow to strengthen the threat point of workers. Wage is thus set at a higher level, which reduces job creations. The optimal unemployment benefit financing scheme works like the Hosios condition. The negative intra-group externalities and the positive inter-group externalities just offset. The distorsion comes from a too strong firms bargaining power and firing taxes allow to ensure optimality. Conversely, if the bargaining process is at the advantage of workers, that is if 1 − ψ > ξ, firing taxes must be negative. Consider now there is wage rigidities (γ < 1). Using equations (33) and (43) 14
taken at the steady state, the optimal value of the firing tax is obtained, that is : τE =
ψ − γ(1 − ξ) κV β M (1 − γ(1 − ξ))(1 − ψ)
Concerning the optimal value of the unemployment benefit, it is not possible to obtain an expression being easily interpreted. To provide some interpretation, we consider our benchmark calibration with 1 − ψ < ξ and we compute the steady state optimal values of τ E and b for values of γ lying between 0 and 1. Results are presented in figure 1. 0.65
0.244
0.6
0.2435 0.243
unemployment benefit
layoff tax
0.55
0.5
0.45
0.4
0.2425 0.242 0.2415 0.241
0.35
0.2405
0
0.5 gamma
1
0.24
0
0.5 gamma
1
Figure 1: Optimal layoff tax and unemployment benefit. An increase in the wage rigidity (that is a decrease in γ) induces a layoff tax increase and an unemployment benefit decrease. The existence of a wage norm tends to reduce wages dispersion. Other things being equal, the large firm marginal value of a new job is increased (equation (19)), it follows that more vacancies are posted. Furthermore, wages associated to low productivity jobs are higher than their level under flexible wages. The threshold ε and the destruction rate are enhanced. We conclude that wage rigidities magnify the initial labor market trade externalities. To restore optimality, the layoff tax must increase and the unemployment benefit must decrease as the wage rigidity increases. 1.8.5
The Second best allocation
The equilibrium allocation (definition 1) is defined conditionally to the unemployment benefits financing scheme (equations (36), (39) and (40)). This unemployment benefits financing scheme is a proxy of the American system. The key parameters, that is the replacement rate ρR and the ERI e, are set by the authorities. Thereafter, quantitative evaluations are made using a benchmark 15
calibration based on US data, but there is no reason these two parameters be optimal. Here, we define a second best allocation where ρR and e are chosen to maximize the conditional welfare. Given initial conditions N−1 and N0N and given parameters ρR and e, let t the consumption and employment equilibrium t and N denote respectively by C allocation. The conditional welfare under the equilibrium allocation writes : R , e; N−1 , N0N ) = E0 W(ρ
∞
βt
t=0
t + (1 − N t )h)1−σ (C 1−σ
Optimal values for ρ∗R and e∗ are obtained solving the following problem : R , e; N−1 , N N ) {ρ∗R , e∗ } = arg max W(ρ 0 ρR ,e
The second best allocation is given by definition 2, knowing that ρR = ρ∗R and e = e∗ . 1.8.6
The welfare costs
In order to compare the different alternative allocations with the Ramsey allocation, we compute their welfare costs. We evaluate the fraction of the consumption stream from an alternative policy needed to be added to achieve the Ramsey allocation welfare. Let W0∗ be the conditional welfare under the Ramsey allocation and let Cta and Nta denote an alternative allocation. The welfare cost Ψ is obtained by solving the following equation : W0∗
= E0
∞
β
t [(1
t=0
+ Ψ) (Cta + (1 − Nta )h)]1−σ 1−σ
(45)
Ψ can be written as follows : Ψ =
W0∗ W0a
1 1−σ
−1
with :
W0a = E0
∞ t=0
βt
(Cta + (1 − Nta )h)1−σ 1−σ
Ψ is numerically computed using a second order approximation (see SchmittGrohé and Uribe (2004)). 16
2 2.1
Quantitative evaluation of the model Calibrating and solving the model
The benchmark economy is calibrated according to quarterly frequencies over the period 1951Q1-2004Q4. We follow Shimer (2005) to set the US labor market parameters. His approach concern only transitions between employment and unemployment and start from a simple measure of the job finding and separation probabilities. There is an unemployment insurance. Baseline parameter are reported in table 1. We set the discount factor to 0.99 to have an annual steady state interest rate close to 4%. The aggregate productivity shock follows a first order autoregressive process : log zt+1 = ρz log zt + εzt+1 . ρz corresponds to the autocorrelation coefficient; it is equal to 0.95 as in Den Haan, Ramey, and Watson (2000). zt+1 is a random variable whose realization are i.i.d. from a time-invariant Gaussian distribution H(.) with mean zero and whose standard deviation (σz ) is 0.007. The distribution G(.) of idiosyncratic productivity shock is i.i.d. and log-normal with mean zero and whose upper bound is equal to 95 percentile as in Zanetti (2007). The risk aversion coefficient is set to 2. The probability of being unemployed is 3.51 percent on average in the US. We suppose as in Den Haan, Ramey, and Watson (2000), Zanetti (2007) and Algan (2004) that exogenous separations are two times higher than endogenous ones. Consequently, ρx = 0.0236 and ε is fixed in such a way that ρn ≡ 1/2ρx = 0.0118 at the steady state. We keep the traditional value of 0.5 for the workers bargaining power. Following Shimer’s estimations, the elasticity of the matching function with respect to unemployment is 0.7. The equilibrium unemployment rates U is calibrated to 5.5%. At the steady state, the number of match must be equal to the number of separations: M = ρN. Following Andolfatto (1996), the rate at which a firm fills a vacancy is 0.9. Therefore it takes 1 quarter and one week to fill a vacancy. We can deduce the number of vacancy V = M/qt and the job finding probability of about 0.61. Then, it takes a little bit more than one and a half quarter on average for an unemployed worker to find a job. χ is calculated in such a way that M = χ(1 − N)ψ V 1−ψ . Statistics from the Census Bureau of labor exhibit an average ERI across states and over the period 1988-2007 of about 0.65. According to the OECD, the US net replacement rate is 0.32. The remaining parameters κ and h are only given by solving the system of three equations (33), (34) and (37) in three unknown (κ, h and w). ¯ In this way, the expected cost of a vacant job κ/qt represent 6% of the average annual wage which is broadly consistent with empirical finding. Finally we set σε and the rigid wage parameter to catch up with the observed cyclical properties of labor market outcomes. We target the ratio of the labor market tightness standard deviation on the output standard deviation (σθ /σY ). The obtained γ is 0.35 and σε = 0.144. Some US business cycle properties are 17
report on table 3. Variables Symbol Discount factor β Autocorrelation coefficient ρz Std. dev. of aggregate shock σz Std. dev. of idiosyncratic shock σε Risk aversion coefficient σ 95 percentile upper bound ε¯ Matching elasticity ψ Total separation rate ρ Exogenous separation rate ρx Endogenous separation rate ρn Worker bargaining power ξ Replacement rate ρR Experience rating index e Vacancy cost κ Wage rigidity parameter γ
Value 0.99 0.95 0.007 0.144 2 1.2672 0.7 0.0351 0.0236 0.0118 0.5 0.32 0.65 0.11 0.35
Table 1: Baseline parameters. We solve the model with a second order perturbation method. State variables are Nt , NtN and zt . Changing parameters lead up to a new steady state. It is calculated with a Newton algorithm. To evaluate integrals we use GaussChebyshev quadratures with 100 nodes to compute the grid.
2.2
The optimal labor market policy
The optimal labor market policy (first-best allocation) solve the definition 3 problem8 . The second best allocation is obtained by setting the two institutional parameters (e and ρR ) at a value maximizing the large family welfare (sub-section 1.8.5). We quantitatively evaluate the welfare gains induced by reforms of the US under labor market failures. Labor market frictions involves the non-optimality of the competitive equilibrium. In our model, non-optimality occurs because the Hosios condition is not satisfied. Furthermore, non-optimality is strengthened by real wage rigidities. We evaluate the consequences of wage rigidities on the optimal unemployment benefit financing scheme. 8 We solve the model with a second-order approximation method around the steady state to make welfare comparisons (Schmitt-Grohé and Uribe (2004)).
18
Experience rating index Replacement rate Output Consumption Employment Vacancies Welfare Job finding rate Separation rate Welfare loss
Benchmark economy 0.6500 0.3200 100.00 100.00 100.00 100.00 100.00 60.23 % 3.51 % 0.0547
1st best allocation Pareto Ramsey 0 0.9682 0 0.2445 100.56 100.39 100.57 100.32 100.06 70.93 % 2.36 % 0.0000
Second best allocation 1.0299 0.2415 100.57 100.38 100.59 108.64 100.05 73.35 % 2.36 % 0.0003
Table 2: Optimal labor market policy (No wage rigidity). Output, consumption, employment and welfare have been standardized. e and ρR have been recalculated when we compute the Ramsey. Percentage welfare losses are relative to the Ramsey allocation.
Experience rating index Replacement rate Output Consumption Employment Welfare Vacancies Job finding rate Separation rate Welfare loss
Benchmark economy 0.6500 0.3200 100.00 100.00 100.00 100.00 100.00 60.23 % 3.51 % 0.4440
1st best allocation Pareto Ramsey 0 1.5790 0 0.2429 102.06 101.96 102.64 100.46 100.32 70.93 % 2.36 % 0.0000
Second best allocation 0.9662 0.2187 102.10 101.96 102.67 100.44 108.24 73.24 % 2.36 % 0.0184
Table 3: Optimal labor market policy (Wage rigidity w t = w and γ = 0.35). Output, consumption, employment and welfare have been standardized. e and ρR have been recalculated when we compute the Ramsey. Percentage welfare losses are relative to the Ramsey allocation. Numerical investigations concerning the US economy are reported in tables 2 and 3. In the two cases, compared to the benchmark economy, the first best exhibits two features : (i) the optimal (apparent) experience rating index9 is increased and (ii) an average replacement rate that is roughly thirty percent lower. The second best allocation displays similar features, the experience rating 9
To allow comparisons, values of e and ρR implied by the first best policy are computed using the steady state values of unemployment benefit and taxes.
19
index and the replacement rate have the same order of magnitude than in the first best case. The main difference between the two case concerns the optimal experience rating index. Without wage rigidities, the optimal (first and second best) experience rating index is close to 1. It follows that the layoff tax is close to the fiscal cost of an unemployed worker. In case of wage rigidities, the optimal experience rating index is sharply increased and takes a value around 1.5. Wage rigidities reduce wages dispersion, the destruction threshold ε and the destruction rate are increased. To restore optimality, it is necessary to increase the layoff-tax which now represent 1.5 times the expected fiscal cost of an unemployed worker. As a whole, optimal financing schemes (first and second-best) sharply depart from the benchmark one. Labor market failures are strongly reduced when the second-best allocation is implemented. The case with wage rigidities needs some comments because implementing an optimal policy has more sensitive effects. The optimal layoff tax is sharply greater than the expected fiscal cost of an unemployed worker. In the Ramsey allocation, equilibrium worker flows are reduced by around 28%. As mentioned Algan (2004) and L’Haridon and Malherbet (2008), turnover costs introduce a labor hoarding phenomenon. As long as firing is costly, firms prefer continue the relation with a low productivity level than pay for the layoff tax. They cut back the reservation productivity to reduce endogenous separations. The reservation productivity falls up to a point where endogenous separations are close to zero. Then, ins and outs of employment are almost only governed by exogenous separations. In that case, an higher index doesn’t reduce labor market flows anymore10 . Output and employment increase by 2.06% and 2.64% respectively. The welfare is enhanced by 0.46% compare to the benchmark. The welfare loss (Ψ) of the benchmark economy is of about 0.44% relative to the optimal policy. The alternative policy (second-best) displays a very weak loss (0.0184%). Policy implementation Welfare may be enhanced through a labor market reform. Our numerical experiments suggest the experience rating index and the replacement rate of the US economy are away from their optimal levels. In terms of welfare cost, the second-best appears to be a good approximation of the first-best. The second-best thus provides a good way to implement a labor market reform allowing to enhance welfare. A welfare improvement may be achieve through a change in parameters e and ρR . 10
The reason come from the reservation productivity. When τ E increase, ε strongly decreases to balance the job destruction rule (34). According to the shape of the distribution, a small negative change in ε lead to an important decrease of the endogenous separation rate G(ε). These results remain virtually unchanged with a capital accumulation or/and a uniform distribution.
20
2.3
Business cycle analysis
Let us now investigate how labor market policies affect the propagation of shocks. We simulate a one percent negative aggregate productivity shock and compute impulse response functions (see Fig. 4-6) and the cyclical properties of the model. We carry out this exercise for the benchmark economy and the two optimal allocations. Simulations show that optimal policies strongly influence the propagation of shocks and especially separations. Implementing the second-best allocation strongly affect the labor market dynamic.
Output Consumption Employment Unemployment Vacancy Tightness Job finding rate Separation rate Output Unemployment Job finding rate Separation rate Ut , Vt
US Economy Benchmark 1st best Standard Deviations 1.58 1.51 0.93 0.78 0.96 0.99 0.63 0.52 0.02 7.83 9.00 0.62 8.83 10.09 1.94 16.31 16.82 2.31 6.79 5.05 0.69 3.58 6.05 0.01 Autocorrelation (1) 0.845 0.830 0.720 0.866 0.895 0.820 0.801 0.722 0.718 0.480 0.703 0.709 Correlation -0.916 -0.580 -0.513
2nd best 1.01 0.93 0.13 3.99 12.20 14.60 4.38 0.01 0.767 0.815 0.720 0.709 -0.520
Table 4: Cyclical properties In the US economy, statistics are computed using a quarterly HP-filtered data from 1951Q1:2004Q4. Data are constructed by the BLS from the CPS. The help-wanted advertising index is provided by the Conference Broad. Job finding and separation probability are build by Shimer (2005). All standard deviations are relative to output (except output). The model is simulated 500 times over 120 quarters horizon. Results are report in logs as deviations from an HP trend with smoothing parameter 1600 and ignoring the first 1000 observations.
We first look on the dynamic of the benchmark economy. On impact, firms post fewer vacancies while the size of unemployment increases with a one-lag period, reproducing the Beveridge curve. The labor market tightness and the number of matches both jump below their steady state level. The probability of finding a job falls while the jump in the reservation productivity raises the job separation rate instantaneously. As in Den Haan, Ramey, and Watson (2000), the increase in unemployment after the shock drives new matches (hiring rate) 21
above their initial level (known as the "echo effect"). Output and consumption decline following the shock and return gently to their equilibrium value. In the first-best allocations, the jump of the job finding rate and the separation rate is strongly reduced. The last one does not seem to fluctuate following the shock and its standard deviation is close to zero. The reason is that the policy strongly influences the steady state reservation productivity which is now located at the bottom of the distribution11 . Therefore, a shift of εt does not generate a strong increase of endogenous separations. The initial impact on hirings, measured by the variable NtN /Nt , is lower in the Ramsey economy. The optimal allocation seems to reduce the incentive to hire during expansions and to fire during recessions. However, these results no longer hold when implementing the second-best allocation. Indeed, the initial falls of hirings is 2 times higher compared to the benchmark but the "echo effect" is reduced. Firms cut back vacancies during the recession rather than firing an inefficient worker. The standard deviation is 21% higher than in the benchmark. The reason is that layoffs taxes introduce a labor hoarding phenomena. Then, the effect of the aggregate shock translate into the job posting activity when the cost of separation is high. This magnifies the initial effect on vacancies while it dampens unemployment fluctuations. The overall effect on the labor market tightness is amplified but its volatility remains weaker than the benchmark economy. In the two optimal policies, output and consumption decline with a less pronounced hump-shaped response. Their persistence are reduced by around 13.3% in Ramsey economy and 7.59 % in the second-best economy compare to the benchmark12 . We now scrutinized the dynamic effects of the financing scheme. We first deal with the benchmark economy and discuss later how the optimal policies affect benefits and taxes fluctuations. In the benchamrk economy, the unemployment compensation is reduced according to its wage indexation. But the probability of finding a job falls, leaving the overall effect on the expected fiscal cost of an unemployed worker undetermined. Simulations show that the increase in the average duration of unemployment has a higher impact on the fiscal cost Qt (measured by the IRF of the layoff tax which is proportional to Qt ) than the decrease of benefits per unemployed worker. Consequently, the layoff tax jumps above its steady state level to cut back on the cost incurred by the unemployment benefit fund. The lump-sum tax decreases following the shock and overtakes its initial value as soon as unemployment increases. Since unemployment is persistent, the fiscal cost of an unemployed worker remains high for a long time. Taxes slowly converge to their equilibrium value. 11
On the bottom, the slope of the log-normal cumulative distribution function is nearly horizontal. 12 Consumption and output autocorrelation coefficient is identical.
22
In the Ramsey allocation, it is shown that taxes jump in the opposite direction. One can explain it by the dampened fluctuations of unemployment and the strong sensitivity of wages (and therefore of the unemployment compensation). As a consequence, unemployment insurance expenditures go down following the shock. Taxes have to decrease in order to balance the budget. The main difference between the Ramsey and the second-best allocation is the path followed by taxes. Once again, the reason is that total benefits paid increase with the rise of unemployment and decrease with the fall of benefits per unemployed worker. The overall effect depends on the sensitivity of the two key variables. In the secondbest allocation unemployment benefits respond little to shifts in productivity compare to the Ramsey economy while the rise of unemployment is stronger.
2.4
Conclusion
In this paper, we use a DSGE model to study the properties of an optimal unemployment benefits financing scheme. We wonder if firms should be taxed in proportion of their separations and if such a tax should correspond to a part or all of the cost incurred by the unemployment insurance. In particular, we investigate how the optimal policy can offset labor market failures generated by search frictions and wages rigidities. In our framework, we find that the optimal unemployment benefits financing scheme require that employers should be fully responsible for their dismissal decisions. When an employer lay a worker off he should pay the entire expected fiscal cost of an unemployed worker. This result is magnified in the presence of wage rigidities. In each case, optimality imposes a replacement rate reduced by around one third. The optimal combination of unemployment benefits and layoff taxes is welfare-enhancing and can improve labor market performances. Furthermore, it is fund that layoff taxes induce a labor hoarding phenomenon by increasing the cost of separations. They create a financial incentive for employers to stabilize their employment, reducing welfare losses. However, the model remains limited and can be extended in several directions. First, throughout this paper we use a simple unemployment insurance system, borrowed from Cahuc and Malherbet (2004), as a proxy of current regulations. However, the experience system in force exhibits wide differences, leaving the comparison between an improvement in the US system difficult. In the US, employer contribution rates depend on the firm layoff history and unemployment benefits that are perceived by its ex-employees. Here we consider a combination of a layoff tax and a lump-sum tax which are forward looking variables. To catch up with current regulations, it will be worth introducing a better approximation that takes into account the past record of insured unemployed as in Topel (1983). Second, it will be worth introducing the idea of temporary layoffs and a pool of unemployed attached to the firm as highlight Feldstein (1976) to evaluate whether these institutions affect the optimum. Third, one can ask the following 23
questions: what are the consequences of an imperfect experience rating system when labor turnover is heterogenous among firms. Does the implicit subsidy financed through other firms induce too many layoffs ? To answer these questions, ex-ante heterogeneity among firms have to be considered. These issues remain interesting topics for future research but are beyond the scope of this paper.
24
A
Proof of result 1
Let’s write the lagrangian of the Ramsey allocation problem : L =
E0
+
Ω1t
∞ t=0
+
β
t
(Ct + (1 − Nt )h)1−σ 1−σ
Vt e − β(1 − γ(1 − ξ))λt+1 (zt+1 ε − zt+1 εt+1 − F − τt+1 κ Mt Vt Ω2t γ(1 − ξ)κλt + γξ (bt + h) λt − (1 − γ(1 − ξ)) (zt εt + F + τte ) λt 1 − Nt
+
(1 − γ)w t λt − (1 − γ(1 − ξ))β(1 − ρx )ξλt+1
+
Λ1t (Yt − Ct − κVt − F (1 − ρx )ρnt Nt−1 )
+ +
Λ6t −Yt + (1 − ρx )Nt−1 zt
+
Λ7t
+ +
zt+1
εt+1
(ε − εt+1 )dG(ε) − F −
e τt+1
N Λ2t (−Nt+1 + Mt ) + Λ3t (−Nt + (1 − ρx )(1 − ρnt )Nt−1 + NtN ) εt 4 ϕ 1−ϕ 5 n Λt (−Mt + χ(1 − Nt ) Vt ) + Λt −ρt + dG(ε) 0
+
+
ε
ε
εt
εdG(ε) + NtN zt ε
(Ct + (1 − Nt )h)−σ − λt + Λ8t (−(1 − Nt )bt + Tt + (1 − ρx )ρnt Nt−1 τtE ) Vt E Λ9t −Nt w t λt + Nt γξ(bt + h)λt + Nt γ(1 − ξ) κ λt − β(1 − ρx )λt+1 (F + τt+1 ) 1 − Nt
γ(1 − ξ) NtN zt ε + Nt+1 (1 − ρx )zt x
(1 − γ)((1 − ρ )(1 −
ρnt )Nt−1
+
ε
εt
εdG(ε) λt + Nt−1 (1 − ρx )(1 − ρnt )γ(1 − ξ)(F + τtE )λt
NtN )w t λt
The optimality conditions with respect to Tt , bt and τtE write : ∂L = Λ8t = 0 ∂Tt ∂L = Ω2t γξλt − Λ8t (1 − Nt ) + Λ9t Nt γξλt = 0 ∂bt
(46) (47)
Knowing that w t = w, w being the steady state real wage, the optimality condition with respect to w t writes : ∂L = −Λ9t Nt λt = 0 ∂w t It follows that Λ8t = Λ9t = Ω2t = 0. Consider now the optimality condition with respect to τtE , one gets : 25
∂L ∂τtE
= Ω1t−1 (1 − γ(1 − ξ))λt = 0
The others optimality conditions may then be written as follows : ∂L ∂Ct ∂L ∂λt ∂L ∂Yt ∂L ∂Vt ∂L ∂ρnt ∂L ∂Nt
= (Ct + (1 − Nt )h)−σ − Λ1t − Λ7t σ(Ct + (1 − Nt )h)−σ−1 = 0
(48)
= −Λ7t = 0
(49)
= Λ1t − Λ6t = 0
(50)
= −Λ1t κ + Λ4t χ(1 − Nt )ϕ (1 − ϕ)Vt−ϕ = 0
(51)
= −Λ1t F (1 − ρx )Nt−1 − Λ3t (1 − ρx )Nt−1 − Λ5t = 0
(52)
= −βEt Λ1t+1 F (1 − ρx )ρnt+1 − Λ3t + βEt Λ3t+1 (1 − ρx )(1 − ρnt+1 ) ε 4 ϕ−1 1−ϕ 6 − Λt χϕ(1 − Nt ) Vt + βEt Λt+1 (1 − ρx )zt+1 εdG(ε) εt+1
+
Λ7t σh(Ct
+ (1 − Nt )h)
−σ−1
− h(Ct + (1 − Nt )h)−σ = 0
∂L = −Λ2t + βEt Λ3t+1 + βEt Λ6t+1 zt+1 ε = 0 N ∂Nt+1 ∂L = Λ2t − Λ4t = 0 ∂Mt ∂L = Λ5t − Λ6t (1 − ρx )Nt−1 zt εt = 0 ∂εt
(53) (54) (55) (56)
The system formed by equations 48 — 56 seems untractable. However, it can easily be showed that it reduces to the equations system defining the Pareto allocation. It immediately follows from equation 49, 48 and 50 that Λ7t = 0, Λ1t = (Ct + (1 − Nt )h)−σ = λt and Λ6t = λt . >From equations 51, 52, 55 and 56, is is easily deduced that : κ Vt λt 1 − ϕ Mt = (1 − ρx )Nt−1 λt εt = −λt F − λt zt εt = Λ4t
Λ4t = Λ5t Λ3t Λ2t
26
Substituting in equations 53 and 54 provides : κ Vt λt − βEt {λt+1 (zt+1 (ε − εt ) − F )} = 0 1 − ϕ Mt Vt ϕ λt λt (zt εt + F − h) − κ 1 − ϕ 1 − Nt ε +β(1 − ρx )Et λt+1 zt+1 (ε − εt+1 )dG(ε) − F =0 −
εt+1
The above equations are exactly equations 44 and 43. We thus have verified that the Ramsey allocation corresponds to the Pareto one.
27
References Algan, Y. (2004): “La protection de l’emploi : des vertus stabilisatrices ?,” working paper. Andolfatto, D. (1996): “Business cycles and labor market search,” The american economic review, 86(1), 112–132. Blanchard, O., and J. Tirole (2008): “The joint design of unemploymment insurance and employment protection. A First Pass,” Journal of European Economic Association, 6(1), 45–77. Cahuc, P., and F. Malherbet (2004): “Unemployment compensation finance and labor market rigidity,” Journal of Public Economics, 88, 481–501. Cahuc, P., and A. Zylberberg (2007): “Optimum income taxation and layoff taxes,” Journal of Public Economics, Forthcoming. Card, D., and P. Levine (1994): “Unemployment insurance taxes and the cyclical and seasonal properties of unemployment,” Journal of Public Economics. Den Haan, W., G. Ramey, and J. Watson (2000): “Job Destruction and Propagation of Shocks,” American economic review, 90(3), 482–498. Feldstein, M. (1976): “Temporary Layoffs in the Theory of Unemployment,” Journal of political economy, 84(5), 937–957. Hall, R. (2005): “Employment fluctuations with equilibrium wage stickiness,” American Economic Review, 95(1), 50–65. L’Haridon, O., and F. Malherbet (2008): “Employment protection reform in search economies,” European economic revew, Forthcoming. Mortensen, D., and C. Pissarides (1994): “Job creation and job destruction in the theory of unemployment,” The review of economic studies, 61(3), 397– 415. (1999): “Job reallocation, employment fluctuations and unemployment,” in Handbook of Macroeconomics, vol. 1, chap. 18, pp. 1171–1228. Elsevier Science, New York. Schmitt-Grohé, S., and M. Uribe (2004): “Solving dynamic general equilibrium models using a second-order approximation to the policy function,” Journal of economic dynamics and control, 28, 755–775. Shimer, R. (2005): “The Cyclical Behavior of Equilibrium Unemployment and Vacancies,” American Economic Review, 95(1), 25–49. 28
Topel, R. (1983): “On Layoffs and Unemployment Insurance,” American economic review, 73(4), 541–559. Zanetti, F. (2007): “Labor market institutions and aggregate fluctuations in a search and matching model,” working paper 333, Bank of England.
29
8
10
6 4 % Deviation
% Deviation
5 0 −5
−10
Q5
Q10
Q15
Q20
Q25
−2 −4
Vacancies 0
0 Job finding rate
Unemployment
−15
2
Separation rate Hiring rate
−6 Q30
0
Q5
Q10
Q15
0.2
Q25
Q30
Layoff tax
0
6
Lump−sum tax Unemp. Benefits
% Deviation
−0.2 % Deviation
Q20
−0.4 −0.6 −0.8
2 0
Output
−1
4
Consumption
−1.2
−2
Employment 0
Q5
Q10
Q15
Q20
Q25
Q30
0
Q5
Q10
Q15
Q20
Q25
Q30
Figure 2: Impulse response functions - Benchmark economy. We simulate a one percent negative aggregate productivity shock.
30
0.1
0.5
0 −0.1
0
% Deviation
% Deviation
1
−0.2
−0.5
−0.3
−1
Job finding rate
−0.4 Unemployment
−1.5 −2
0 0.1
Q5
Q10
Q15
Q20
Q25
Separation rate
−0.5
Vacancies
Hiring rate 0
Q30
Q5
Q10
Q15
Q20
Q25
Q30
0
0 −2
% Deviation
−0.1 % Deviation
−0.2 −0.3 −0.4
−4 −6 −8
−0.5 Output
−0.6
Consumption
−0.7 −0.8
−10
Q5
Q10
Q15
Q20
Q25
Lump−sum tax
−12
Employment 0
Layoff tax
Q30
Unemp. Benefits 0
Q5
Q10
Q15
Q20
Q25
Q30
captionImpulse response functions - Ramsey economy. We simulate a one percent negative aggregate productivity shock.
31
4
1
2 0 % Deviation
% Deviation
0 −2 −4 −6 −8
−1 −2 Job finding rate
−3 Unemployment
−10
Separation rate −4
Vacancies
−12 0
Q5
Q10
Q15
Q20
Q25
Q30
Hiring rate 0
Q5
Q10
Q15
Q20
Q25
Q30
0.2 Lump−sum tax
−0.2
% Deviation
% Deviation
Layoff tax
3
0
−0.4 −0.6
Output
Unemp. Benefits
2 1 0
Consumption
−0.8
Employment
−1
−1 0
Q5
Q10
Q15
Q20
Q25
Q30
0
Q5
Q10
Q15
Q20
Q25
Q30
Figure 3: Impulse response functions - Second best allocation. We simulate a one percent negative aggregate productivity shock.
32
0.7 Benchmark 0.6
2nd best allocation
0.5
% Welfare loss
(Ψ) 0.4
0.3
0.2
0.1
0 0
0.1
0.2
0.3
0.4
0.5
0.6
Wage rigidity (1−γ)
0.7
0.8
Figure 4: Welfare losses with respect to wage rigidity.
33
0.9
1
