Optimal labor market policy with search frictions and risk-averse workers

Labour Economics 35 (2015) 93–107

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Labour Economics journal homepage: www.elsevier.com/locate/labeco

Optimal labor market policy with search frictions and risk-averse workers☆ Jean-Baptiste Michau Ecole Polytechnique, France

H I G H L I G H T S • • • •

I characterize the optimal insurance policy in a search model of the labor market. General equilibrium effects are key to the provision of insurance. At the optimum, layoff taxes are typically higher than hiring subsidies. Moral hazard does not change the results, unless workers have low bargaining power.

a r t i c l e

i n f o

Article history: Received 25 March 2014 Received in revised form 18 April 2015 Accepted 27 April 2015 Available online 5 May 2015 JEL classification: D60 E62 H21 J38 J65 Keywords: Employment protection Hiring subsidies Optimal rate of unemployment Unemployment insurance

a b s t r a c t This paper characterizes the optimal policy within a dynamic search model of the labor market with risk-averse workers. In a first-best allocation of resources, unemployment benefits should provide perfect insurance against the unemployment risk, layoff taxes are necessary to induce employers to internalize the social cost of dismissing an employee but should not be too high in order to allow a desirable reallocation of workers from low to high productivity jobs, hiring subsidies are needed to partially offset the adverse impact of layoff taxes on job creation, and both unemployment benefits and hiring subsidies should be almost exclusively financed from layoff taxes. I obtain an optimal rate of unemployment which is, in general, different from the output maximizing rate of unemployment. When workers have some bargaining power, which prevents the provision of full insurance, it is optimal to reduce the rate of job creation below the output maximizing level in order to lower wages and increase the level of unemployment benefits. Thus, layoff taxes should typically exceed hiring subsidies which generates enough surplus to finance at least some of the unemployment benefits. The inclusion of moral hazard does not change this conclusion, unless workers have low bargaining power. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The design of labor market institutions is among the key determinants of the economic success or failure of a nation. There is nevertheless no consensus among economists about the optimal design of such institutions and, in many industrialized countries, the subject remains at the center of considerable controversies among policy makers. In particular, there appears to be a fundamental trade-off between the ☆ I am grateful to the editor, to two anonymous referees, and to Pierre Cahuc, Pierre Chaigneau, Pieter Gautier, Philipp Kircher, Etienne Lehmann, Alan Manning, Barbara Petrongolo, Steve Pischke, Christopher Pissarides, Fabien Postel-Vinay, Johannes Spinnewijn, Jean Tirole, and Bruno Van der Linden, and to seminar participants at the LSE, EEA-ESEM 2009 (Barcelona, Spain) and the Centre for Structural Econometrics of the University of Bristol (UK) for their useful comments and suggestions. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.labeco.2015.04.003 0927-5371/© 2015 Elsevier B.V. All rights reserved.

demand for insurance of risk-averse workers and the macroeconomic efficiency of the labor market which should allocate workers to the jobs where they are going to be most productive. Hence, a typical concern is that government interventions aimed at improving insurance, such as the provision of unemployment benefits or employment protection, might have adverse consequences for aggregate production. Search frictions are a major source of the trade-off between insurance and production since they generate some unemployment and they prevent an immediate reallocation of workers from low to high productivity jobs. 1 A macroeconomic framework is required to analyze this trade-off since search frictions induce non-trivial general equilibrium effects on job creation and job destruction, which are key to the 1 The other major source of the trade-off is moral hazard which will be allowed for in the last section of this paper.

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J.-B. Michau / Labour Economics 35 (2015) 93–107

reallocation process of workers. Furthermore, wages could be affected by macroeconomic variables such as the expected length of an unemployment spell. These general equilibrium effects imply that different labor market policy instruments do interact among each other. Hence, these instruments jointly influence the provision of insurance and the efficiency of production. They therefore need to be analyzed jointly. A search model à la Mortensen and Pissarides (1994) with riskaverse workers captures the trade-off between insurance and production as well as the aforementioned general equilibrium effects and allows for a joint analysis of the different policy instruments. In this paper, I therefore rely on such a framework to determine the main characteristics of an optimal labor market policy. Employment protection takes the form of layoff taxes. The government can also provide hiring subsidies in order to encourage job creation. The generosity of unemployment insurance is determined by the level of unemployment benefits. Payroll taxes can be used to raise revenue. If they happen to take negative values, payroll taxes can also be seen as employment subsidies. Importantly, it is assumed throughout, as in most of the literature on the topic, that the government is the sole provider of unemployment insurance.2 I begin by deriving the optimal allocation of resources chosen by a planner who wants to maximize the welfare of workers subject to matching frictions and to a resource constraint. In this ideal setup, full insurance is provided and aggregate output, net of recruitment costs, is maximized. It turns out that this first-best allocation can be implemented in a decentralized economy when workers are wage takers. To obtain an efficient rate of job destruction, layoff taxes should induce firms to internalize the social costs and benefits of dismissing a worker. The costs consist of the unemployment benefits that will need to be paid and of the forgone payroll taxes; while the benefits correspond to the value of a desirable reallocation of the worker from a low to a high productivity job. Hiring subsidies are needed to partially offset the negative impact of layoff taxes on job creation. Finally, and perhaps surprisingly, both unemployment benefits and hiring subsidies are almost entirely financed from layoff taxes. Importantly, my analysis naturally defines a welfare maximizing optimal rate of unemployment. If full insurance cannot be provided, then this optimal rate of unemployment is generically different from the output maximizing rate of unemployment commonly emphasized in the search and matching literature. I then turn to the characterization of the optimal policy when workers have some bargaining power, which prevents the provision of full insurance. Relying on numerical simulations, I show that the planner typically chooses to set layoff taxes higher than hiring subsidies such as to discourage the entry of firms with a vacant position. This reduces market tightness and, hence, wages which, by relaxing the resource constraint, makes it possible to increase the level of unemployment benefits. I then allow for moral hazard. This generates the opposite possibility that insurance may be too high, in which case the planner wants to increase market tightness. However, the simulations reveal that an insufficient provision of insurance remains the main concern whenever workers have substantial bargaining power. Thus, moral hazard does not seem to be the most important feature of the fundamental tradeoff between the provision of insurance and the level of aggregate production. General equilibrium effects on wages and on job creation and job destruction seem to be at least as important.

1.1. Related literature This paper is related to the extensive economic literature on the optimal design of labor market institutions. The main strand of this literature focuses on the provision of unemployment insurance. In their seminal work, Shavell and Weiss (1979) and Hopenhayn and Nicolini (1997) focused on a single unemployment spell and derived the optimal time profile of unemployment benefits when moral hazard introduces a trade-off between the provision of insurance and the provision of incentives to search. By contrast, Baily (1974b) and Chetty (2006) focused on the level of benefits, rather than their time profile, in a framework that allows for multiple spells. Importantly, these contributions assume that unemployment benefits are exclusively financed from payroll taxes and abstract from general equilibrium effects. The literature on employment protection is mostly positive, rather than normative. The crux of the academic debate is about the impact of layoff taxes on the level of employment; with the underlying presumption that layoff taxes are desirable if they decrease the number of jobless workers. Bentolila and Bertola (1990) showed, in a partial equilibrium context, that firing costs have a larger impact on job destruction than on job creation and should therefore be beneficial for employment. This conclusion was challenged by the general equilibrium analysis with employment lotteries of Hopenhayn and Rogerson (1993). Ljungqvist (2002) showed that, in search models à la Mortensen–Pissarides, layoff costs increase employment if initial wages are negotiated before a match is formed, while the opposite is true if bargaining only occurs after the match is formed. Importantly, these contributions either assume that workers are risk-neutral or that financial markets are complete. Hence, they do not generate any trade-off between insurance and production efficiency and cannot give sensible measures of the welfare implications of layoff taxes. These analyses are therefore hardly informative about the optimal level of employment protection. While most papers ignore the interaction between different policy instruments, there are two important exceptions which are closely related to this work. First, Mortensen and Pissarides (2003)3 analyze labor market policies in a dynamic search model with risk-neutral workers. Since there is no need for insurance, the best that the government can do is to maximize output net of recruitment costs. If the Hosios (1990) condition holds, i.e. the bargaining power of workers is equal to the elasticity of the matching function, then it is optimal for the government not to intervene; while, if it does not hold, policy parameters should only be used to correct for the resulting search externalities. An important insight is that the introduction of unemployment benefits has a positive impact on wages and, therefore, increases job destruction. This should be offset by higher layoff taxes. Hiring subsidies should also be increased such as to leave the rate of job creation unchanged. However, with risk-neutral workers, there is no trade-off between insurance and production. 4 The second closely related paper is Blanchard and Tirole (2008) which proposes a joint derivation of optimal unemployment insurance and employment protection in a static context with risk-averse workers. In their benchmark model, they show that unemployment benefits should be entirely financed from layoff taxes, rather than payroll taxes, in order to induce firms to internalize the social cost of unemployment.5 However, their static framework does not have a job creation margin and therefore ignores the adverse effect of layoff taxes on job creation. In fact, as we shall see, in a dynamic context the 3

See also Mortensen and Pissarides (1999) and Pissarides (2000, chapter 9). Interestingly, Schuster (2010) extends the framework of Mortensen and Pissarides (2003) by adding a job acceptance margin. He shows that, in general, the implementation of the optimal policy requires additional policy instruments. 5 This policy, often referred to as “experience rating”, was originally proposed by Feldstein (1976). Other related contributions on the topic, and mostly in favor of such policy, include Topel and Welch (1980), Topel (1983), Wang and Williamson (2002), Cahuc and Malherbet (2004), Mongrain and Roberts (2005), Cahuc and Zylberberg (2008) and L'Haridon and Malherbet (2009). 4

2 The implicit contract literature has argued that risk-neutral firms should be expected to provide unemployment benefits to risk-averse workers; see, for instance, Baily (1974a) or Azariadis (1975). However, in reality, such contracts remain the exception rather than the rule. Thus, although somewhat ad-hoc, the assumption that the private market does not provide insurance seems reasonable and has the merit of making the analysis transparent. This assumption has nevertheless been relaxed in the optimal policy analyses of Chetty and Saez (2010) and Fella and Tyson (2011).

J.-B. Michau / Labour Economics 35 (2015) 93–107

share of unemployment benefits financed from layoff taxes, net of expenditures on hiring subsidies, is determined by the job creation side of the economy. Also, and more fundamentally, a static approach entails an entirely negative view of unemployment; whereas in a dynamic setting an unemployed worker is a useful input into the matching process. In fact, a well-known result from the search and matching literature is that, to maximize output in an economy without governmental intervention, the bargaining power of workers that satisfies the Hosios condition actually maximizes the rate of job destruction! Finally, this paper is also related to a small literature on policy analyses within dynamic search models of the labor market with risk-averse workers. Cahuc and Lehmann (2000), Fredriksson and Holmlund (2001) and Lehmann and Van der Linden (2007) focus on the optimal provision of unemployment insurance under moral hazard. All three contributions pay particular attention to the general equilibrium effects of unemployment insurance. More specifically, they emphasize that an increase in benefits leads to an increase in wages which reduces the rate of job creation. Along similar lines, Krusell et al. (2010) investigate the optimal provision of unemployment insurance in a search model where the accumulation of risk-free savings is the only source of private insurance available to risk-averse workers. They show that, even in the absence of moral hazard, the adverse impact of unemployment insurance on job creation is so large that the optimal replacement ratio is close to zero. It should be mentioned that none of these papers allow for the possibility of using hiring subsidies and firing taxes to control the rates of job creation and job destruction. Acemoglu and Shimer (1999, 2000) showed, in the context of directed search with risk-averse workers, that higher unemployment benefits can improve the quality, and productivity, of job-worker matches. By contrast, in this paper, match quality is unrelated to the length of unemployment. Alvarez and Veracierto (2000, 2001) rely on calibrated search models with risk-averse workers to investigate the effects of different labor market policies. However, their approach is entirely positive and does not attempt to characterize optimal policies.6 In a closely related paper, Coles and Masters (2006) show that there is some complementarity between the provision of unemployment insurance and that of hiring subsidies. The idea is that, by boosting the job creation rate, subsidies exert a downward pressure on unemployment and, hence, on the cost of providing unemployment insurance. However, their model does not have an endogenous job destruction margin and, therefore, cannot be used to determine the optimal level of employment protection. Schaal and Taschereau-Dumouchel (2010) and Jung and Kuester (2015) are two complementary contributions which were written at the same time as this paper. Schaal and Taschereau-Dumouchel (2010) allow for unobservable heterogeneity in productivity across workers. Their analysis therefore focuses extensively on redistribution. However, they do not allow for an endogenous job destruction margin and, hence, layoff taxes are absent from their model. They also assume throughout their analysis that workers have no bargaining power. Jung and Kuester (2015) focus on the optimal labor market policy in recessions. They show numerically that, at the optimum, layoff taxes and vacancy subsidies are both strongly countercyclical while unemployment benefits are almost acyclical. As their focus is numerical, they do not provide detailed intuitions for their results in simple benchmark cases. In particular, they do not explain how, in the absence of moral hazard, the optimal policy of full insurance can be implemented in a decentralized economy when workers have some bargaining power. In my analysis, I investigate in much greater detail the steady state properties of the optimal labor market policy. Note that none of these two papers allow for private savings.

6 Ljungqvist and Sargent (2008) also investigate the interactions between unemployment insurance and employment protection in a positive analysis of the labor market, but with risk-neutral workers.

95

To begin, Section 2 offers a brief reminder of some of the key features of the Mortensen and Pissarides (1994) framework. In the following section, I derive the first-best policy, which then serves as a benchmark. Section 4 relies on numerical simulations to investigate optimal policies when workers have some bargaining power. Finally, the last section deals with the consequences of moral hazard. This paper ends with a conclusion. 2. Search model Before characterizing the optimal labor market policies, it is necessary to describe the main features of the dynamic search model on which I rely throughout this paper. The structure of the economy builds on the Mortensen and Pissarides (1994) framework. Time is continuous. Production requires that vacant jobs and unemployed workers get matched, which occurs at rate: m ¼ mðu; vÞ;

ð1Þ

where u stands for the number of unemployed and v for that of vacancies. For simplicity, the mass of workers is normalized to one, so that u also stands for the rate of unemployment. The matching function m is increasing in both arguments, exhibits decreasing marginal product to each input and satisfies constant returns to scale. Let θ denote market tightness which is defined as the ratio of vacancies to unemployment, i.e. θ = v/u. The rate at which vacant jobs meet unemployed workers is given by:  
u  mðu; vÞ 1 ¼ m ; 1 ¼ m ; 1 ¼ qðθÞ; v v θ

ð2Þ

where q(θ) is a decreasing function of θ. Similarly the rate at which unemployed workers find jobs is: mðu; vÞ ¼ mð1; θÞ ¼ θqðθÞ: u

ð3Þ

Clearly, the constant returns to scale assumption implies that market tightness is the key parameter which summarizes labor market conditions for both unemployed workers and recruiting firms. The elasticity of the matching function is defined as7: ηðθÞ ¼ −

θ dqðθÞ : qðθÞ dθ

ð4Þ

The standard Mortensen–Pissarides framework is extended by allowing for stochastic job matching. Thus, the initial match productivity x is randomly drawn from the c.d.f. G(x) with support [ψ, 1], where the maximal productivity has been normalized to 1. Newly created matches only survive if the initial productivity x is above a job acceptance threshold A. Then, as in the standard Mortensen–Pissarides framework, at Poisson rate λ, the match is hit by a productivity shock and a new productivity x ∈ [ψ, 1] is randomly drawn from the same c.d.f. G(x). The employment relationship ends if the new productivity is below a job destruction threshold R. The remaining features of the model will be given in the following section as the optimal policy is being derived. 3. First-best policy The optimal policy is derived in two steps. First, I characterize the optimal allocation of resources chosen by a benevolent social planner. 7 Note that η is the elasticity of the matching function with respect to the number of unemployed, i.e. η ¼ mu ∂m , and 1 − η the elasticity with respect to the number of vacancies, ∂u i.e. 1−η ¼ mv ∂m . ∂v

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J.-B. Michau / Labour Economics 35 (2015) 93–107

Then, I turn to its implementation in a decentralized economy with free entry of risk-neutral firms. 3.1. Optimal allocation The optimal allocation maximizes a utilitarian social welfare function subject to a resource constraint and to the search frictions that characterize the labor market. It is therefore the solution to the following problem: Z

∞

max

fθ;A;R;b;wg

subject to

e−rt ½ð1−uÞvðwÞ þ uvðz þ bÞdt

aggregate output y. It should be emphasized that while the time subscripts have been omitted for conciseness, the five control variables of the planner, θ, A, R, w, and b, and the two states variables, u and y, are all time dependent. The planner's problem is straightforward to solve using standard optimal control techniques. Throughout this paper, I investigate the optimal policy in steady state. I therefore focus on the first-order conditions in steady state, where all the variables are constant over time. The first characteristic of the optimal allocation is that workers are offered perfect insurance against the unemployment risk:

ð5Þ

0

w ¼ z þ b;



u¼ λGðRÞð1−uÞ−θqðθÞ½1−GðAÞu

ð6aÞ

Z 1 Z 1 y¼ θqðθÞu sdGðsÞ þ λð1−uÞ sdGðsÞ−λy

ð6bÞ

ð1−uÞw þ ub ¼ y−cθu

ð6cÞ



A

ð7Þ

which is a direct consequence of workers' risk aversion, i.e. of the concavity of v(.). This can be combined with the resource constraint, (6c), to give the optimal value of w and b:

R

where r stands for the planner's (or workers') discount rate, w for the income of the employed, z for the value of home production, b for the income of the unemployed, y for the aggregate output of the economy, and c for the flow cost of posting a vacancy. The instantaneous utility function of risk-averse workers is denoted by8 v(.), which is increasing and concave. The planner's objective is to maximize intertemporal social welfare, which, according to a utilitarian criterion, is composed at each instant of the instantaneous utility of u unemployed and 1 − u employed workers.9 The first constraint depicts the dynamics of unemployment, driven by the difference between the job destruction flow and the job creation flow. A match dissolves when it is hit by an idiosyncratic shock that generates a new productivity below the job destruction threshold R, which occurs at rate λG(R). This rate of job destruction applies to the mass 1 − u of existing matches. The job creation flow is equal to the rate at which unemployed workers find jobs with productivity above the job acceptance threshold, θq(θ)[1 − G(A)], multiplied by the mass u of job seekers. It should be emphasized that this first constraint captures the fact that even the social planner is subject to matching frictions. The second constraint gives the dynamics of aggregate output, y. At each instant, θq(θ)u new matches are formed and their productivity x is randomly drawn from G(x). These matches only survive if their productivity is above the job acceptance threshold A. The 1 − u existing jobs are hit at rate λ by idiosyncratic shocks which destroy their current productivity y/(1 − u) and replace it, in case of survival, by a randomly drawn number greater or equal to the threshold R. Finally, any feasible allocation must satisfy the economy's aggregate resource constraint.10 The expenses, composed of the income of the employed and of the unemployed, cannot exceed total output net of the resources allocated to recruitment, which amount to a flow cost c paid for each of the θu vacancies. The planner's control variables are market tightness θ, the job acceptance threshold A, the job destruction threshold R, the income w of the employed and the income b of the unemployed. The state variables are the unemployment rate u and

8 In the previous section v denoted the number of vacancies. However, this variable will not appear in the rest of the text (except when I define the matching function under moral hazard in the last section of the paper). I focus instead on θ and u and, where needed, v is just replaced by θu. 9 An alternative would be to maximize the weighted average between the expected utility of an employed and of an unemployed worker. Such objective function would be more appropriate in a political economy context focusing on the conflict between insiders and outsiders. However, without time discounting, this would be identical to the planner's objective of this paper. 10 Replacing the resource constraint (6c) by an intertemporal resource constraint would not change any of the results provided that the interest rate at which the planner can transfer resources across time is equal to the planner's discount rate.

w ¼ y−cθu þ zu;

ð8Þ

b ¼ y−cθu−zð1 −uÞ:

ð9Þ

Note that perfect insurance necessitates a replacement ratio smaller than one whenever the value z of home production is strictly positive. The other straightforward characteristic of the optimal allocation is that the job acceptance and the job destruction thresholds are equal to each other: A ¼ R:

ð10Þ

Indeed, whether a match should survive at any given point in time is independent of the history of the employment relationship. The productivity threshold is therefore the same whether the match is new or old. The optimal values of θ and R are implicitly determined by the following two first-order conditions: Z ½1−ηðθÞ

R¼zþ

1 R

ðs−RÞdGðsÞ rþλ

¼

ηðθÞ λ cθ− 1−ηðθÞ rþλ

c ; qðθÞ

Z

1

ðs−RÞdGðsÞ;

ð11Þ

ð12Þ

R

where η(θ) denotes the elasticity of the matching function, cf. Eq. (4). These two optimality conditions are exactly identical to the ones that would result from output maximization.11 This is not surprising as, when nothing prevents the provision of full insurance, the best that the planner can do is to maximize output. The first Eq. (11), guarantees an optimal rate of job creation. The cost of job creation consists of the flow cost of having a vacancy, c, multiplied by the expected time that has to be spent before a worker can be found, 1/q(θ). The value of a 1 newly created match is equal to ∫R(s − R)dG(s)/(r + λ). However, optimally, recruitment costs should only absorb a fraction 1 − η(θ) of this value as, otherwise, there is too much job creation and an excessive amount of resources is allocated to recruitment. Eq. (12) ensures an optimal rate of job destruction. In the static context of Blanchard and Tirole (2008), the optimal threshold is just equal to the value of home production, i.e. R = z. Making the model dynamic yields two extra terms. First, when a low productivity job is destroyed, the corresponding worker returns to unemployment with the hope of finding a new job with productivity

11 Under risk neutrality, the objective of the planner is to maximize the net present value of the flow of net output, where this flow is given by y − cθu + uz.

J.-B. Michau / Labour Economics 35 (2015) 93–107

higher than R. To make this explicit, the corresponding term of Eq. (12) can be rewritten, using Eq. (11), as: Z ηðθÞ cθ ¼ θqðθÞηðθÞ 1−ηðθÞ 2Z 1 6 ¼ θqðθÞ6 4

R

1 R

ðs−RÞdGðsÞ rþλ

ðs−RÞdGðsÞ rþλ

ð13Þ

c 7 7 : − qðθÞ5

This says that, once a job is destroyed, an unemployed worker gets matched at rate θq(θ) which generates a social value of 1 ∫R(s − R)dG(s)/(r + λ) net of the expected recruitment cost c/q(θ). In other words, the threshold R has to be sufficiently high to induce an efficient reallocation of workers from low to high productivity jobs. The second additional term to the expression for the optimal threshold R corresponds to the option value of a match. Even if current productivity is very low, keeping the match alive preserves the option of being hit by an idiosyncratic shock that restores a profitable level of productivity. The option value decreases the optimal threshold R. In steady state, the optimal allocation of resources chosen by a benevolent social planner is fully characterized by the first-order conditions (7), (10), (11), and (12) together with the constraints (6a), (6b), and (6c) with u¼y¼ 0. 

I now proceed backward and start by determining the threshold R chosen at Stage 5 by a risk-neutral employer. The asset value of a producing firm with productivity x, J(x), solves the following Bellman equation:

; 3



3.2. Implementation Having characterized the optimal allocation, I now turn to its implementation in a decentralized economy. Throughout the paper, I restrict the government to rely exclusively on the following four policy instru~ payroll taxes τ, layoff taxes F and hiring ments: unemployment benefits b, subsidies H. I choose to focus on these four as they are the most standard instruments through which governments do in practice affect labor market outcomes.12 Moreover, as we shall see in this section, they are sufficient to implement the first-best allocation of resources in a benchmark case. I assume that all workers have an equal ownership of all the firms in the economy. Thus, aggregate profits π are homogeneously distributed ~ denote the wage rate of an employed worker across workers. Let w ~ þ π his income. Similarly, the income of an unemployed and w ¼ w ~ þ π. worker is given by b ¼ b In the decentralized economy, five stages of interest can be distinguished. • Stage 1: The government chooses the level of unemployment benefits ~ payroll taxes τ, layoff taxes F and hiring subsidies H. b, • Stage 2: Entrepreneurs decide whether or not to create a firm with a vacant position. • Stage 3: Once a match occurs, the employer and employee agree on a wage rate. • Stage 4: As soon as match productivity is revealed, the employer enforces a job acceptance threshold A. • Stage 5: Each time the match is hit by a productivity shock, the employer enforces the job destruction threshold R. Importantly, the wage rate is determined before the employment relationship begins and, hence, before match productivity is revealed. It is indeed natural to assume that match productivity cannot be revealed before the employee starts working.13

12 In practice, the minimum wage also is a very important labor market policy instrument. However, any meaningful analysis of the minimum wage must allow for heterogeneity among workers, which is beyond the scope of this paper. 13 As we shall see in subsequent sections, in the absence of full insurance, the riskneutral firm is assumed to provide insurance to the risk-averse worker. It is therefore important that the wage contract is signed before match productivity is revealed.

97

~ þ τÞ þ λ r J ðxÞ ¼ x−ðw

"Z

1

# J ðsÞdGðsÞ−GðRÞF− J ðxÞ

R

;

ð14Þ

where r denotes the risk-free interest rate, which is taken to be iden~ the net wage that the worker tical to the planner's discount rate, w ~ þ τ the gross wage paid by the employer. This Bellman receives and w equation states that, for a firm, the flow return from having a filled job with productivity x is equal to the instantaneous surplus it generates to which the possibility of a change in productivity should be added. An idiosyncratic shock destroys the value of the firm at the current productivity and replaces it by either a corresponding expression, if the new productivity is above the threshold, or by the cost of layoff,14 if the match is to be destroyed. As J(x) is strictly increasing in x, the employer chooses a job destruction threshold R that is determined by: J ðRÞ ¼ −F:

ð15Þ

This says that, at the threshold, the employer is indifferent between closing down and continuing the relationship. Simple algebra15 on Eqs. (14) and (15) gives the expression for the value of R chosen by firms: ~ þ τ−r F− R¼w

λ rþλ

Z

1

ðs−RÞdGðsÞ:

ð16Þ

R

The threshold productivity is smaller than the cost of labor because of the firing tax and of the option value of continuing the match. Note that, for this to be possible, firms must be able to borrow and lend from perfect financial markets, an assumption that is maintained throughout this paper. Eq. (16) is our first implementability constraint: the decentralized job destruction condition. At Stage 4, the firm needs to enforce the job acceptance threshold A. Importantly, it is assumed that, if a match dissolves as soon as its productivity is revealed, the firm nevertheless needs to pay the layoff tax F.16 Indeed, this is the only way to ensure that the job acceptance threshold is determined by: J ðAÞ ¼ −F;

ð17Þ

and, hence, that A = R, as required to implement the first-best allocation. The planner wants to destroy all matches with productivity inferior to a given threshold, regardless of whether the matches are new or old. Hence, the layoff tax should be applied to all matches, including new ones. In other words, setting up a trial period would not be desirable in this setup. Let us now turn to the determination of the wage rate that occurs at Stage 3. The formation of a match generates a surplus that needs to be shared between the two parties. But, from Eq. (7), at the optimum ~ þ π of an employee must be equal to the income the income w ¼ w 14 Throughout this paper, it is assumed that firms are able to pay the layoff tax. Blanchard and Tirole (2008) investigate the consequences of having employers constrained by shallow pockets. See also Tirole (2010) for a deeper analysis on the topic which allows for extended liability to third parties in the context of employment protection. 15 An analytic expression for the function J(.) can be obtained by taking the difference between Eq. (14) evaluated at x and the same equation evaluated at R. This expression for J(.) can then be substituted into Eq. (14) evaluated at R. Finally, using the value of J(R) given by Eq. (15) yields Eq. (16). 16 As mentioned above, the match productivity is only revealed after the employee has started working and, hence, after an employment contract has been signed. Thus, it is administratively feasible to impose a layoff tax on newly created jobs.

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J.-B. Michau / Labour Economics 35 (2015) 93–107

~ þ π of being unemployed. This immediately equivalent z þ b ¼ z þ b leads to the following lemma: Lemma 1. A necessary condition to implement the first-best allocation is that workers are wage takers and that all the match surplus is captured by the firm. This guarantees that, as desired: ~ ~ ¼ z þ b: w

ð18Þ

The intuition for this result is straightforward. If workers have some bargaining power, they will obtain a mark-up over and above their outside option which is the income they get while unemployed. But this prevents the provision of full insurance which is a characteristic of a first-best allocation.17 Clearly, with a binding resource constraint (Eq. (6c)) and perfect insurance, the optimal values of w and b are still given by Eqs. (8) and (9), respectively. The requirement that workers have no bargaining power could be seen as an important benchmark.18 In the context of this paper, it could also be seen as part of the optimal policy to be implemented. For example, the labor market could be organized in such a way that firms and workers first meet without exchanging any information on the wage rate. Then, firms make a take-it-or-leave-it offer to workers.19 Finally, the following corollary is an immediate consequence of the above lemma: Corollary 1. The first-best allocation cannot be implemented when the Hosios condition holds, i.e. when the bargaining power of workers is equal to the elasticity of the matching function η(θ). The Hosios condition balances search externalities on both sides of the labor market such that, without government intervention, output is maximized. It is, however, inconsistent with the provision of perfect insurance. Since, in the first-best allocation, output is maximized and workers must have zero bargaining power, the optimal policy will need to correct the rates of job creation and job destruction for the failure of the Hosios condition to hold. Stage 2 is solved by assuming free entry. Vacancies keep being created by entrepreneurs until the returns from doing so reduce to zero. More formally, the value V of a vacant position solves: " rV ¼ −c þ qðθÞ H þ

Z

1 A

# J ðsÞdGðsÞ−GðAÞF−V

:

ð19Þ

This states that the return from a vacancy consists of the flow cost c of recruitment and of the value of being matched with a worker, which occurs at rate q(θ). The employer receives a hiring subsidy H as soon as an employment contract is signed, and hence before the productivity of the match is revealed. The worker then starts producing, which reveals productivity. If it is below A, then the firm lays off the worker and pays the tax F. Free entry implies: V ¼ 0:

ð20Þ

17 In their benchmark case, Blanchard and Tirole (2008) also assume that the bargaining power of workers is nil. Thus, the first-best benchmark derived in this section is a dynamic counterpart to theirs. 18 Hagedorn and Manovskii (2008) argue that workers have a bargaining power close to 0.05, which suggests that this benchmark is not necessarily implausible. 19 This is the assumption made by Schaal and Taschereau-Dumouchel (2010) throughout their analysis. Lehmann and Van der Linden (2007) proposed an alternative solution for an environment with Nash bargaining. It consists in setting a 100% marginal tax rate ~ This completely absorbs the bargaining above the full insurance level, i.e. above z þ b. power of the worker. However, there are good reasons outside the model for not setting such a confiscatory tax rate, which is why I shall consider in the rest of the paper that the non-linear labor income tax is not an instrument used by the government to influence labor market outcomes. Relaxing this assumption would require a model that allows for endogenous margins of labor supply and for heterogeneity across workers.

The amount of job creation can then be determined by plugging Eq. (20) into Eq. (19) and by using the value of J(s) deduced from Eqs. (14) and (15). This, together with A = R, gives: Z

1 R

ðs−RÞdGðsÞ −F ¼

rþλ

c −H: qðθÞ

ð21Þ

The left hand side is the expected value of a new match to a firm (before productivity is revealed); while the right hand side corresponds to the expected cost of recruiting a worker. Eq. (21) is our second implementability condition: the decentralized job creation condition. At Stage 1, the government needs to choose the optimal policy. The corresponding implementability condition is the usual government budget constraint: ~ þ uθqðθÞH: ð1−uÞτ þ uθqðθÞGðAÞF þ ð1−uÞλGðRÞ F ¼ ub

ð22Þ

Revenues consist of payroll taxes paid by employed workers and of layoff taxes that apply whenever new or old matches dissolve; while the expenses are the payment of benefits to the unemployed and of hiring subsidies to the flow of newly created matches. Aggregate profits in the economy are given by: Z π ¼ ð1−uÞ

1

~ ðs−w−τ ÞdGðsÞ

R

−ð1−uÞλGðRÞF

1−GðRÞ

−uθc þ uθqðθÞH−uθqðθÞGðRÞF: ð23Þ

The first term corresponds to the profits realized by producing firms, the second to the cost of laying off a worker following an adverse productivity shock, the third to the recruitment cost (which is paid by a mass uθ of vacant firms) and the last two terms correspond to the hiring subsidy received upon recruitment net of the cost of laying off newly recruited workers who turn out to be insufficiently productive. Note that, in steady state, i.e. when u¼ 0 and ẏ = 0, Eqs. (6a) and (6b) jointly 1 imply that y = (1 − u)∫ sdG(s)/[1 − G(R)]. Using this expression toR gether with the government budget constraint (Eq. (22)) implies that profits are equal to: 

~ ~ π ¼ y−ð1−uÞw−u b−cθu:

ð24Þ

This corresponds to the resource constraint, which by Walras' law had to be satisfied. It is straightforward to solve for the optimal policy by matching the implementability conditions of the decentralized economy to the equations that characterize the first-best allocation of resources. More specifically, H and F must be chosen such that Eq. (21) reduces to Eq. (11) and Eqs. (16) to (12). This gives: Z F−H ¼ ηðθÞ ~ þ τ− rF ¼ b

1 R

ðs−RÞdGðsÞ rþλ

ηðθÞ cθ; 1−ηðθÞ

;

ð25Þ ð26Þ

where θ and R are jointly determined by Eqs. (11) and (12). These are key equations characterizing the optimal policy in the benchmark model. They ensure that the rate of job creation and job destruction prevailing in the decentralized economy coincide with the planner's optimum. These conditions have a potentially insightful interpretation. Let us start with the implementation of the optimal level of job creation, Eq. (25). Eq. (11) implies that, under free entry, firms should only capture a fraction 1 − η(θ) of the match surplus; otherwise, entry is too

J.-B. Michau / Labour Economics 35 (2015) 93–107

high and too many resources are allocated to recruitment. However, employers have all the bargaining power and this must be offset by setting a firing tax that exceeds the hiring subsidy such as to absorb a fraction η(θ) of the match surplus which reduces job creation to an efficient level. Let us now turn to the interpretation of the equation implementing the optimal level of job destruction, Eq. (26). As can be seen from Eq. (16), a layoff tax only affects the threshold R if firms discount the future, i.e. if r N 0. Indeed, any match will eventually be destroyed and, hence, by not laying off its worker now, the firm is only postponing the payment of the tax. Thus the relevant cost imposed by the layoff tax is rF, rather than just F. A firm that dismisses its worker imposes a double externality on the financing of unemployment insurance. First, the worker will qualify for benefits and, second, he will no longer contribute to its funding by paying payroll taxes. The layoff tax should therefore be sufficiently high to ensure that employers internalize these effects. This is the main message of Blanchard and Tirole (2008).20 The additional insight that is obtained by extending the analysis to a dynamic context is that there is also a social benefit from laying off a worker: it allows a desirable reallocation of this worker from a low to a high productivity job. This is captured by the third term of Eq. (26) which was given an intuitive interpretation when the optimal allocation was derived, cf. Eq. (13). This effect reduces the net social cost of dismissal and, hence, the level of the optimal layoff tax. Interestingly, the option value of keeping the match alive is properly taken into account by firms and therefore does not affect the size of the optimal layoff tax. Note that Eqs. (25) and (26) imply that the optimal job creation (Eq. (11)) and job destruction (Eq. (12)) conditions are satisfied. As workers are wage takers, we have Eq. (18) which implies that the full insurance condition (7) is met. Finally, Eq. (24) implies that the resource constraint (Eq. (6c)) is also satisfied. We have therefore fully characterized the policy that implements the optimal allocation in the decentralized economy. The fact that we did not need to specify the level τ of payroll taxes implies that τ and π are not separately identified. Let us rely on the government budget constraint (Eq. (22)) to derive the relationship between payroll taxes and profits that must hold at the ~ þ π and that, in steady state, the job optimum. Using the fact that b ¼ b destruction flow is equal to the job creation flow, (1 − u)λG(R) = uθq(θ)[1 − G(A)], we obtain: ð1−uÞτ þ uπ ¼ ub−uθqðθÞð F−H Þ:

ð27Þ

Note that, by Eq. (25), the right hand side of this expression is a function of b, θ, and R only. It is therefore fully determined by the optimal allocation of resources. Hence, τ and π are not determined, only (1 − u)τ + uπ ~ þ π for the unemployed can either be imis. Indeed, a high income b ¼ b ~ which require high payroll plemented through generous benefits b, taxes, or through high profits π. An important insight from this analysis is that the job destruction side of the economy determines the level of layoff taxes, F; while the job creation side determines the difference between layoff taxes and hiring subsidies, F − H. Note that this result is fundamentally due to the implementability conditions, Eqs. (16) and (21), and will therefore remain true in all extensions of the benchmark model. An important implication, which follows from Eq. (27), is that the share of income ub of the unemployed that is financed from the revenue raised from layoff taxes net of the expenditures on hiring subsidies is essentially determined from the job creation side of the economy, a margin that is absent from Blanchard and Tirole (2008). 20 In fact, in Blanchard and Tirole (2008) payroll taxes do not appear as they should optimally be set equal to zero. However, Cahuc and Zylberberg (2008), who propose a generalization to the case where the government needs to raise taxes on income in order to redistribute wealth across heterogeneous individuals, did explicitly have them affecting the level of layoff taxes.

99

Further insights on the financing of the income of the unemployed can be gained by replacing F − H in Eq. (27) by its value from Eq. (25), which, after some straightforward rearrangement using Eq. (11), yields: 2

2Z

6 6 6 ð1−uÞτ þ uπ ¼ u6 4b−θqðθÞ4

1 R

33

ðs−RÞdGðsÞ rþλ

−

7 c 7 77 : 5 qðθÞ 5

ð28Þ

The flow b of income of the unemployed constitutes the social cost of having an unemployed worker. The second term represents the corresponding social benefit. Indeed, at rate θq(θ), an unemployed finds a job which generates a social value equal to the expected profits from production net of the recruitment costs. Since the optimal rate of unemployment should ensure that the social benefit from joblessness is not too distant from its social cost, we expect the two terms of the main bracket of Eq. (28) to be close to each other. In fact, with time discounting, we expect the first term to be slightly larger than the second one since the benefit will only be realized in the future. This intuition is formally confirmed by rewriting expression (Eq. (28)) as: ð1−uÞτ þ uπ ¼

h y i r uð1−uÞ −R : rþλ 1−u

ð29Þ

The derivation is provided in Appendix A. Hence, without time discounting, i.e. when r = 0, the left hand side of Eq. (27) is equal to zero, which implies that both the income of the unemployed and the hiring subsidies should be financed, exclusively, from layoff taxes. The optimal policy can now be fully characterized. Proposition 1. When workers are wage takers, the first-best allocation can be implemented in a decentralized economy with any given level π ~ of profits by choosing the values of the policy instruments H, F, τ, and b ~ that jointly satisfy Eqs. (25), (26), (29) and b ¼ b−π with b given by Eq. (9). The four policy instruments under investigation are sufficient to implement the first-best allocation as each one of them takes care of ~ determine the proan important margin. The unemployment benefits b vision of insurance, the hiring subsidy H controls the job creation margin, the layoff tax F controls the job destruction margin, and the payroll tax τ balances the government budget constraint (for a given level of profits). Knowing that the first-best allocation is implementable, we can derive the equilibrium rate of unemployment by setting u¼ 0 in Eq. (6a) determining the dynamics of unemployment. This yields the familiar expression: 

u¼

λGðRÞ ; λGðRÞ þ θqðθÞ½1−GðRÞ

ð30Þ

where θ and R are jointly determined by Eqs. (11) and (12). This equation nevertheless has an interesting new interpretation in this framework. Whereas, with risk-neutral workers, this is the output maximizing rate of unemployment21; here, given the microfoundations laid in terms of risk-averse workers, this is the optimal rate of unemployment. Not only can unemployment be too low from an output maximization perspective, it can also be too low from a welfare point of view, which is conceptually very different. Here, the output maximizing and optimal rates of unemployment coincide. However, this only occurs because full insurance is provided at the optimum. For instance, assume that there is a fixed non-insurable utility cost B N 0 of being unemployed. Thus, in the planner's problem (Eq. (5)), 21 This is often referred to as the “efficient rate of unemployment” in the search and matching literature with risk-neutral workers.

100

J.-B. Michau / Labour Economics 35 (2015) 93–107

social welfare at each instant is given by (1 − u)v(w) + u[v(z + b) − B] (rather than (1 − u)v(w) + uv(z + b)). As the unemployed's marginal utility of consumption is not affected ~ The welfare loss from unem~ ¼ z þ b. by B, it remains optimal to set w ployment is therefore not insurable. It can be shown that the planner nevertheless finds it optimal to mitigate the problem by reducing the rate of job destruction below its output maximizing level such as to reduce the rate of unemployment. Thus, in that case, the optimal rate of unemployment is below the output maximizing rate of unemployment.

When the employment contract is signed, i.e. before match productivity is revealed, the expected utility W i of the worker is equal to: Z

1

Wi ¼

W i ðxÞdGðxÞ þ UGðRi Þ;

where the expected utility of an unemployed, U, and of a worker in a match of productivity x, Wi(x), are implicitly determined by:   rU ¼ vðz þ bÞ þ θqðθÞ W−U ;

4. Workers with bargaining power Under risk aversion, it is desirable to suppress any fluctuations in income between employment and unemployment spells. Hence, the implementation of a first-best allocation requires workers to have zero bargaining power, as stated in Lemma 1. However, it could be objected that workers fundamentally do have some bargaining power and that this cannot be influenced by government policy. Thus, when solving for the optimal policy, the expression for the wage rate resulting from the bargaining process should be added as an extra constraint to the planner's problem. An important limitation of the analysis of this section, which is shared with much of the literature on the topic, is that the model does not allow for private savings.22 When workers have some bargaining power, their income fluctuates over time which should induce them to accumulate some precautionary savings in order to avoid sharp drops in consumption when unemployed.23

~ i ðxÞ þ π Þ þ λ rW i ðxÞ ¼ vðw

"Z

1

Before setting up the planner's problem, it is necessary to solve the bargaining problem between the worker and the firm. When a match is formed, the firm and the worker bargain over an ~ ðxÞgx∈½R;1 , and entire wage schedule, as a function of productivity, fw on a minimal productivity R below which the match is destroyed.24 Indeed, as the firm is risk-neutral and the worker risk-averse, it is quite natural that they initially bargain on a state-contingent contract which allows the risk-neutral firm to commit to absorb some of the future productivity risk facing the match. This employment contract is determined by Nash bargaining and it is signed before the employee starts working and, hence, before the initial productivity of the match is revealed. If an agreement is not reached, the employer does not receive the hiring subsidy but does not have to pay the layoff tax. Thus, the employment contract is determined by:

max

 β  1−β J i þ H−V ; o W i −U

ð31Þ

~ i ðxÞgx∈ R ;1 ;Ri fw ½  i

where W i denotes the expected utility of an employed worker in match i, U the expected utility of an unemployed worker and J i the expected profitability of match i.

# W i ðsÞdGðsÞ þ GðRi ÞU−W i ðxÞ

;

ð34Þ

where, as before, v(.) stands for the instantaneous utility of consumption and the income b of the unemployed is composed of unemploy~ and of profits π. Similarly, the expected profitability J ment benefits b of the match to the firm before its productivity is revealed is given by: Z Ji ¼

1 Ri

J i ðxÞdGðxÞ−FGðRi Þ;

i

ð35Þ

where the value of a producing firm with productivity x, Ji(x), is determined by:

~ i ðxÞ þ τÞ þ λ r J i ðxÞ ¼ x−ðw

4.1. The planner's problem

ð33Þ

Ri

"Z

n o ~ ðxÞgx∈½R;1 ; R ¼ arg n fw

ð32Þ

Ri

1 Ri

# J i ðsÞdGðsÞ−GðRi ÞF−J i ðxÞ

:

ð36Þ

The subscript i in the bargaining problem (Eq. (31)) and in the value of employment to a worker, Eqs. (32) and (34), or to a firm, Eqs. (35) and (36), is used to stress that the wage rates and the threshold productivity bargained in match i do not affect the values of the outside options, i.e. the values of U or V.25 The implicit contract literature suggests that risk-neutral firms might be willing to provide their risk-averse workers with insurance against unemployment. However, note that, in the absence of savings, a severance payment cannot be used as an insurance device. If, instead, firms are allowed to provide unemployment insurance to their former employees until they receive a job offer, then they will choose to provide perfect insurance. In that situation, as in the benchmark case of Section 3, the government can implement the first-best allocation of resources and the corresponding policy can easily be characterized analytically. However, this case is neither theoretically interesting nor empirically relevant. Hence (as mentioned in footnote 2), I rule out transfers from a firm to a worker after their work relationship has ended. As shown in Appendix B, the wage schedule and the job destruction threshold that solve Eq. (31) are jointly determined by the following three equations. First, the wage rate is independent of productivity: ~ ðxÞ ¼ w ~ for all x ∈ ½R; 1: w

ð37Þ

22

Indeed, most of the papers mentioned in the introduction on the optimal provision of unemployment insurance within matching models of the labor market also abstract from private savings. 23 A possible justification of the no-savings assumption is that, in practice, many employees hardly accumulate any savings. However, a key question is whether these workers rationally choose not to accumulate any savings or whether there is something that prevents them from doing so (such as a lack of information, exclusion from the banking sector or myopia). 24 For simplicity, I do not distinguish the job acceptance threshold from the job destruction threshold. Given the structure of the economy, they should trivially be equal to each other.

Thus, the risk-neutral firm absorbs all the productivity risk. The income ~ þ π of an employed worker is determined by: w¼w vðwÞ−vðz þ bÞ r þ λGðRÞ þ θqðθÞ½1−GðRÞ β c ; ¼ v0 ðwÞ 1−GðRÞ 1−β qðθÞ

ð38Þ

25 Importantly, the value of employment W that appears in Eq. (33) is not indexed by i since this value is not affected by the employment contract that is bargained in match i.

J.-B. Michau / Labour Economics 35 (2015) 93–107

and the job destruction threshold solves: Z 1 λ ðs−RÞdGðsÞ rþλ R r þ λGðRÞ vðwÞ−vðz þ bÞ : − r þ λGðRÞ þ θqðθÞ½1−GðRÞ v0 ðwÞ

~ þ τ−r F− R¼w

Table 1 Exogenous parameter values.

ð39Þ

Note that the last term of this job destruction condition would not appear in the absence of commitment, cf. Eq. (16). This shows that firms use both margins to provide insurance to risk-averse workers: they pay a constant wage and they lower the job destruction threshold. Relying on the free-entry condition, Appendix C shows that the decentralized job creation condition is: 2Z 6 ð1−βÞ6 4

1 R

ðs−RÞdGðsÞ rþλ

3 7 c þ H− F 7 5 ¼ qðθÞ :

ð40Þ

The optimal policy can then be derived by adding the wage Eq. (38) as a constraint to the original problem. Thus, the planner should maximize Eq. (5) with respect to θ, R, b, and w subject to Eqs. (6a), (6b), (6c), and (38).26 Note that, as in the previous section, the level π of profits does not affect the allocation of resources. In other words, the optimal allocation can be implemented in a decentralized economy for any given level of profits. The three remaining implementability constraints, Eqs. (39), (40), and (22), can be left out since they jointly determine, for a given level of profits, the values of F, H, and τ which do not appear elsewhere in the planner's problem. This shows that the choice of policy instruments is not a constraint on the optimal allocation of resources. Indeed, the only additional constraint to the planner's problem from Section 3 is the wage Eq. (38), that captures the fact that workers do have bargaining power. It turns out that, unlike in the case without bargaining power, the first-order conditions to the planner's problem are cumbersome and hardly interpretable. Hence, I first perform a reasonable calibration of the model. I then rely on numerical simulations of the optimal policy for different values of the bargaining power of workers to provide a number of key qualitative insights. 4.2. Calibration Empirical studies have provided some support for a constant elasticity of the matching rate with respect to the unemployment rate (Petrongolo and Pissarides, 2001). Let η be this fixed elasticity. We must therefore have: qðθÞ ¼ q0 θ−η ;

ð41Þ

where the two parameters, q0 and η, need to be calibrated. Following Mortensen and Pissarides (2003), the distribution of idiosyncratic shocks is assumed to be uniform on [ψ, 1]; hence its c.d.f. is: GðxÞ ¼

x−ψ : 1−ψ

ð42Þ

Finally, I use a standard constant relative risk aversion (CRRA) instantaneous utility function with CRRA coefficient ϕ: vðxÞ ¼

x1−ϕ −1 : 1−ϕ

ð43Þ

I calibrate my model to the US economy assuming that, currently, ~ the government only intervenes to provide unemployment benefits b

26

In Eqs. (6a) and (6b), A should be set equal to R.

101

r

ϕ

η

z

c

λ

q0

ψ

0.004

3

0.5

0.392

0.431

0.120

0.679

0.563

which are entirely financed by payroll taxes τ. I perform a monthly calibration. I set r = 0.004, which implies a yearly interest rate of 4.8%. Workers are characterized by a coefficient ϕ of relative risk aversion equal to 3. I take the elasticity η of the matching function to be equal to 0.5, which is in the mid-range of the empirical estimates reported by Petrongolo and Pissarides (2001). The calibration is performed assuming that workers and firms have equal bargaining power, i.e. β = 0.5. Hall and Milgrom (2008) argue that the income-equivalent of being unemployed is equal to 71% of the average match productivity, a third of which consists of unemployment benefits. I therefore impose ~ ¼ 0:71y=ð1−uÞ together with 2b ~ ¼ z, where y/(1 − u) is the averzþb age match productivity. To balance the government budget constraint, ~ ð1−uÞ. The flow cost c of the payroll taxes τ must be set equal to bu= posting a vacancy is calibrated such that the equilibrium market tightness θ is equal to 0.72, consistently with the empirical evidence reported by Pissarides (2009). The lower bound ψ of the distribution of idiosyncratic shocks is calibrated such that the coefficienthp offfiffiffivariation i of the distribution of match productivity, given by ð1−RÞ= 3ð1 þ RÞ , is equal to 0.119, i.e. such that R = 0.658, consistently with the findings of Hagedorn and Manovskii (2013). Finally, the scale parameter q0 of the matching function and the rate λ of occurrence of idiosyncratic shocks are jointly calibrated such that the monthly job finding rate is 0.45 while the unemployment rate is 5.5%, consistently with the empirical evidence provided by Shimer (2012). The parameter values implied by this calibration are all displayed in Table 1. 4.3. Simulation Throughout the simulations, profits are normalized to zero. Thus, the income b of the unemployed exclusively consists of unemployment benefits, which are either financed from payroll taxes or from the revenue raised from layoff taxes net of the spending on hiring subsidies. The simulation results are reported in Table 2. As, in this section, I want to investigate the impact of the bargaining power β of workers on the optimal policy, I report the solution to the planner's problem for four different values of β (but for the same underlying calibration of the model). The initial case, β = 0, corresponds to the first-best benchmark of Section 3 and, hence, to the output maximizing allocation. The welfare loss is computed as the proportional decline in consumption in the first-best case necessary to reach the new steady state level of welfare. For example, when β = 0.5, welfare in steady state is equal to what it would be in the first-best allocation, β = 0, with consumption decreased by 0.49%. In steady state, the gross job flow is given by uθq(θ)[1 − G(R)] or, equivalently, by (1 − u)λG(R). Finally, the last row reports the share of income of the unemployed financed by payroll taxes. It can easily be checked that, when the Hosios condition holds, i.e. when β = η = 0.5, output maximization requires F = H, such as to leave the rate of job creation undistorted.27 This would characterize the welfare maximizing policy if workers were risk-neutral. However, as can be seen from Table 2, such is not the case with risk-averse workers. Thus, when workers have some bargaining power, there is a trade-off between output maximization and insurance provision. More precisely, the planner sets layoff taxes higher than hiring subsidies in

27 This can easily be seen by comparing the decentralized job creation condition (Eq. (40)) to the first-best job creation condition (Eq. (11)).

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J.-B. Michau / Labour Economics 35 (2015) 93–107

Table 2 Optimal policy under Nash bargaining. β

0

0.25

0.5

0.75

θ R u (%) y w b τ F H F−H Welfare loss (%) Gross job flow (1 − u)τ/ub (%)

0.906 0.638 3.716 0.789 0.789 0.396 0.00022 1.461 0.856 0.605 0 0.0199 1.42

0.784 0.633 3.662 0.786 0.792 0.303 0.00021 1.259 0.764 0.495 0.12 0.0185 1.78

0.531 0.613 3.043 0.782 0.792 0.233 0.00022 1.040 0.582 0.458 0.49 0.0133 3.01

0.254 0.563 0.000 0.781 0.781 0.182 0.0000 0.804 0.320 0.484 0.92 0.0000 .

order to reduce entry and, hence, market tightness. This decreases wages,28 which by relaxing the resource constraint, allows an increase in the income of the unemployed. In a nutshell, the worker's bargaining power introduces a discrepancy between the income w of the employed and the income z + b of the unemployed, which is detrimental to the provision of insurance. The planner responds by reducing market tightness such as to reduce this discrepancy, which enhances the provision of insurance. Thus, when β is low, F is higher than H in order to compensate for the failure of the Hosios condition to hold (as discussed in Section 3). As β increases, this becomes a smaller concern, but insufficient insurance becomes a bigger one. The planner then wants to decrease market tightness, which becomes the main reason why F exceeds H. Note that F is so much higher than H that it generates sufficient surplus to finance almost entirely the unemployment benefits. This is true even though, for all values of β, the magnitude of F only amounts to less than two months of wage payments. This is more than sufficient to pay for the unemployment benefits given that either β is low and the expected length of unemployment is short or β is high and the replacement ratio is low. The reservation threshold R declines with bargaining power in order to compensate for the imperfect provision of insurance. Indeed, at the margin, a decrease in R reduces the rate of unemployment. But, this fall in R comes at the cost of a more sclerotic labor market characterized by a lower reallocation of workers from low to high productivity jobs, as shown by the lower gross job flow. A very high bargaining power of workers results in such a low level of unemployment benefits that the optimum is characterized by a corner solution where no job is ever destroyed, i.e. G(R) = 0.29 This eliminates unemployment, which completely prevents the reallocation of workers across jobs.30 Importantly, the downward adjustment in θ and R, which enhances the provision of insurance, hinders the reallocation of workers from low to high productivity jobs, which reduces aggregate output. This is the essence of the trade-off between insurance and production. Also, it should be emphasized that a moderate amount of private savings is likely to reduce, but certainly not to eliminate, the demand for insurance. Thus, a trade-off would remain, albeit of a smaller magnitude, and the key qualitative insights about the optimal policy would presumably remain unaltered.

28 This can be seen from expression (Eq. (38)) for the wage rate while recalling that q(θ) is a decreasing function of θ and θq(θ) an increasing function of θ. 29 As the model is degenerate when G(R) = 0, the allocation reported in Table 2 when β = 0.75 was obtained by taking the limit as R tends to ψ. 30 This extreme feature would not occur if the lower bound ψ of the productivity distribution was closer to zero. Alternatively, following Pissarides (2007), it would be possible to introduce exogenous job destruction shocks in addition to the productivity shocks. This would guarantee that the job destruction flow never disappears.

The wages and the job destruction threshold could be determined by directed search, rather than by Nash bargaining. In such an environment, competitive market makers jointly choose the wage schedule, the threshold and the length of queues, equal to 1/θq(θ), such as to maximize the expected utility of an unemployed worker subject to a free entry condition for firms; or more formally: max rU subject to V ¼ 0: ffw~ ðxÞgx∈½R;1 ;R;θg

ð44Þ

This yields exactly the same equations as Eqs. (37), (38), and (39) with β replaced by η. Thus, in Table 2, directed search corresponds to the case where β = η = 0.5. As implied by Corollary 1, directed search and the associated Hosios condition fail to implement a first-best allocation of resources in an economy with risk-averse workers as they fail to entail a sufficient provision of insurance. Finally, to investigate the usefulness of firing taxes and hiring subsidies, Table 3 compares the allocations implemented by three different policies when β = 0.5. The first column corresponds to the current US policy, as described in the calibration section, the second to the optimal policy when the government is restricted to rely on unemployment benefits and payroll taxes only, while the last column corresponds to the fully optimal allocation which is reproduced from Table 2 (when β = 0.5). Comparing the last two columns is informative about the usefulness of firing taxes and hiring subsidies. Switching from the current U.S. policy to the restricted optimal policy raises steady state welfare by 0.1%, while implementing the fully optimal policy raises welfare by 0.37%. Note that, in the absence of firing taxes and hiring subsidies, the government has a very limited ability to affect the provision of insurance. Indeed, the gap between the income of the employed and of the unemployed is fully determined by the outcome of Nash bargaining, given by Eq. (38), and by the resource constraint, given by Eq. (6c). It follows that ~ alone have a very limited ability to affect the unemployment benefits b provision of insurance. Under the restricted policy, the only way to enhance the provision of insurance, relative to the current US benchmark, is to reduce the equilibrium rate of unemployment. To achieve this, the government reduces the provision of unemployment benefits. This decreases the level of payroll taxes which lowers the job destruction threshold. This reduces unemployment, as desired, but it also induces matches to survive for longer which stimulates entry. Market tightness therefore rises, which strengthens the bargaining power of workers thereby increasing the gap between the income of the employed and of the unemployed. Thus, the reduction in unemployment comes as the expense of a fall in the income of the unemployed. By contrast, with both firing taxes and hiring subsidies, the government can enhance the provision of insurance through both a reduction in R, which reduces unemployment, and a reduction in θ, which

Table 3 Restricted vs. full optimum under Nash bargaining when β = 0.5.

θ R u (%) y w b τ F H F−H π Welfare gain (%) Gross gain (%) (1 − u)τ/ub (%)

US benchmark

Restricted optimum

Full optimum

0.720 0.685 5.500 0.784 0.799 0.199 0.01142 0 0 0 0.0026 0 0.0248 98.69

1.033 0.626 2.854 0.790 0.796 0.149 0.00429 0 0 0 0.0029 0.10 0.0168 98.03

0.531 0.613 3.043 0.782 0.792 0.233 0.00022 1.040 0.582 0.458 0 0.37 0.0133 3.01

J.-B. Michau / Labour Economics 35 (2015) 93–107

weakens the bargaining power of the worker. The former is implemented by setting a sufficiently high firing tax F, while the latter requires a hiring subsidy H that is smaller in magnitude than the firing tax. These results illustrate that, in a search model of the labor market, the provision of insurance must rely on general equilibrium effects, which is why firing taxes and hiring subsidies are such valuable instruments. 5. Moral hazard So far, we have seen that, when workers have some bargaining power, the planner is always seeking to improve the provision of insurance. However, reducing the level of insurance might be a virtue if it increases the search intensity of unemployed workers. Indeed, concerns about the moral hazard effects of unemployment insurance have been at the heart of the literature on the topic. Hence, this section characterizes the optimal policy when job search monitoring is not available and, hence, when the unemployed freely choose their search intensity. 5.1. Determination of search intensity Let s denote the average search intensity of the unemployed. Vacant jobs and unemployed workers now get matched at rate31: m ¼ mðsu; vÞ;

ð45Þ

where the matching function satisfies the same properties as before. Vacancies become filled at rate:
s  mðsu; vÞ ¼ m ; 1 ¼ qðθ; sÞ; v θ

si mðsu; vÞ si ¼ θqðθ; sÞ: u s s

5.2. The planner's problem n o ~ ðxÞgx∈½R;1 ; R is still determined by The employment contract fw Nash bargaining as specified in Eq. (31) with the value of unemployment now given by:   rU ¼ vðz þ bÞ−σ ðsÞ þ θqðθ; sÞ W−U

ð51Þ

where s is determined by Eq. (50). Proceeding as before (cf. Appendix B), it can easily be established that the wage rate is still independent of pro~ ðxÞ ¼ w ~ for all x ∈ [R, 1], that this wage rate is determined ductivity, i.e. w by: vðwÞ−vðz þ bÞ þ σ ðsÞ r þ λGðRÞ þ θqðθ; sÞ½1−GðRÞ β c ; ð52Þ ¼ v0 ðwÞ 1−GðRÞ 1−β qðθ; sÞ and that the job destruction threshold solves: Z 1 λ ðx−RÞdGðxÞ rþλ R r þ λGðRÞ vðwÞ−vðz þ bÞ þ σ ðsÞ ; − r þ λGðRÞ þ θqðθ; sÞ½1−GðRÞ v0 ðwÞ

~ þ τ−r F− R¼w

ð53Þ

~ ¼ w−π. where w From the value of employment to a worker, given by Eqs. (32) and (34), and the value of unemployment, given by Eq. (51), it can be shown that the first-order condition for search intensity can be written as: sσ 0 ðsÞ ¼

ð46Þ

θqðθ; sÞ½1−GðRÞ ½vðwÞ−vðz þ bÞ þ σ ðsÞ : r þ λGðRÞ þ θqðθ; sÞ½1−GðRÞ

ð54Þ

Using the wage Eq. (52), this expression can be simplified to:

where market tightness remains defined as the ratio of vacancies to unemployment, i.e. θ = v/u. Unemployed worker i who searches with intensity si finds a job at rate: ~ðθ; s; si Þ ¼ q

103

ð47Þ

sσ 0 ðsÞ ¼

β cθv0 ðwÞ: 1−β

ð55Þ

The planner's problem is the same as in the previous section with s as a new control variable and either Eq. (54) or (55) as an additional constraint.32 5.3. Calibration

The expected utility U of unemployed worker i is implicitly determined by:   ~ðθ; s; si Þ W−U ; rU ¼ max vðz þ bÞ−σ ðsi Þ þ q si

ð48Þ

where σ(.) denotes an increasing and convex cost of search, with 1 ∫ R W ðxÞdG

þ UGðRÞ is the expected value of a new σ′(0) = 0, and W ¼ job to a worker before match productivity is revealed. The first-order condition for search intensity is: −σ 0 ðsi Þ þ

 ~ðθ; s; si Þ  ∂q W−U ¼ 0: ∂si

ð49Þ

The convex search cost is assumed to be given by a power function: σ ðsÞ ¼ k

sγþ1 ; γþ1

ð56Þ

where k and γ are positive parameters. I follow the same procedure as in the previous section to calibrate the model. In addition, I choose k such that s is normalized to 1 and γ such that the elasticity of the unemployment duration with respect to the benefit level is equal to 0.5, consistently with the extensive empirical evidence surveyed by Krueger and Meyer (2002). The parameter values resulting from this calibration are displayed in Table 4. 5.4. Simulation

Hence, using the symmetry which prevails in equilibrium, i.e. si = s, the search intensity of unemployed workers is implicitly determined by:   sσ 0 ðsÞ ¼ θqðθ; sÞ W−U :

31

ð50Þ

The intensity of job advertising made by vacant firms is exogenously set to 1 as, even if endogenously determined, it would not be affected by any policy parameters; cf. Pissarides (2000, chapter 5.3).

The simulation results are reported in Table 5, where profits have again been normalized to zero. The welfare of workers is maximized for β = 0.4245. To see why there is such a welfare maximizing value of β ∈ (0, 1), note that, when the bargaining power of workers is very 32 The other changes are that search intensity should be included in the matching function, i.e. q(θ) should be replaced by q(θ, s), and the search cost σ(s) should be subtracted from the objective function for a mass u of unemployed workers, i.e. the last term of the objective should be u[v(z + b) − σ(s)] instead of uv(z + b).

104

J.-B. Michau / Labour Economics 35 (2015) 93–107

Table 4 Exogenous parameter values with moral hazard. r

ϕ

η

z

c

λ

q0

ψ

k

γ

0.004

3

0.5

0.392

0.866

0.058

0.971

0.374

1.308

0.640

Table 5 Optimal policy under Nash bargaining with moral hazard. β

0.125

0.25

0.4249

0.5

0.75

θ R u (%) y w b s τ F H F−H Welfare loss (%) Gross job flow (%) (1 − u)τ/ub (%)

1.385 0.603 3.947 0.770 0.741 0.283 0.501 0.0000 2.885 2.536 0.349 1.96 0.0203 0.09

1.203 0.628 4.122 0.780 0.759 0.248 0.739 −0.0002 2.378 2.102 0.275 0.52 0.0224 −1.80

0.904 0.636 4.362 0.782 0.773 0.207 0.974 0.0003 1.857 1.637 0.220 0 0.0231 2.92

0.776 0.635 4.473 0.781 0.777 0.192 1.057 0.0007 1.651 1.452 0.200 0.08 0.0230 8.34

0.367 0.606 4.826 0.764 0.780 0.145 1.300 0.0035 0.935 0.820 0.114 1.76 0.0204 47.23

low, the provision of insurance is too high which results in excessively small incentives to search while unemployed. Conversely, a high bargaining power of workers results in an excessively small provision of insurance. Thus, β = 0.4245 optimally balances the trade-off between the provision of insurance and the provision of incentives to search while unemployed. The corresponding allocation is the one that the planner would choose to implement if he could freely set the wage rate.33 It can therefore be seen as the optimal allocation under moral hazard. The (consumption equivalent) welfare loss that is reported in Table 5 is computed relative to that welfare maximizing benchmark. When workers have a smaller bargaining power, β b 0.4245, search intensity is excessively low, which is partially offset by the planner choosing a higher market tightness than in the benchmark. Indeed, a higher market tightness reduces the provision of insurance,34 which boosts the returns to search. Conversely, when β N 0.4245, search intensity is higher than in the optimal allocation. The previous intuitions, without moral hazard, dominate again and the planner decreases market tightness in order to enhance the provision of insurance. Thus, for high values of β, the introduction of moral hazard does not modify the qualitative conclusions of the previous section about the key characteristics of an optimal policy. When the Hosios condition holds, i.e. when β = 0.5, output (net of search costs) is again maximized when F = H. The fact that, with risk aversion, the planner chooses to set layoff taxes higher than hiring subsidies confirms that he seeks to reduce market tightness below the output maximizing level in order to enhance the provision of insurance. Interestingly, when β = 0.4245, the gross job flow is close to being maximized. Indeed, when β N 0.4245, market tightness is decreased which reduces the job creation flow; when β b 0.4245, the low search intensity of unemployed workers also depresses the job creation flow. In sum, the simulation results have revealed that the force pushing for more insurance, i.e. risk aversion, is only dominated by the force pushing for less insurance, i.e. moral hazard, for β b 0.4245. It follows that moral hazard, and the resulting over provision of insurance, is only a dominant concern for rather low values of β. 33 Indeed, allowing β to be a control variable of the planner is equivalent to not imposing the equation for the wage rate that results from Nash bargaining, Eq. (52), as a constraint to the planner's problem. 34 This can be seen from expression (Eq. (52)) for the wage rate while recalling that q(θ, s) is a decreasing function of θ and θq(θ, s) an increasing function of θ.

If the coefficient of relative risk aversion is set equal to 1, instead of 3, then the welfare maximizing bargaining power of workers becomes equal to 0.4730. It is not surprising that a lower degree of risk aversion makes insurance a smaller concern and, hence, moral hazard a bigger one. If workers were risk-neutral, then the optimal bargaining power would be given by the Hosios condition and would therefore be equal to 0.5. This would implement the first-best allocation of a planner who could directly control the search intensity of the unemployed. Hence, under risk-aversion, when β N 0.5, reducing bargaining power is good for both insurance and incentives. The elasticity of unemployment duration with respect to the benefit level has been set equal to 0.5. However, Landais (forthcoming) finds a value as low as 0.3. In that case (and with a coefficient of relative risk aversion equal to 3), the welfare maximizing bargaining power becomes equal to 0.3628. A smaller elasticity reduces the severity of the moral hazard problem, which reduces the bargaining power above which the planner wants to increase the provision of insurance. 6. Conclusion In this paper, I have investigated optimal policies in a dynamic search model of the labor market with risk-averse workers. More precisely, I have focused on the joint derivation of the optimal level of unemployment benefits, layoff taxes, hiring subsidies, and payroll taxes. I began by abstracting from moral hazard in order to focus on the general equilibrium effects of the different policy instruments. I have shown that the first-best allocation of resources can be implemented in a decentralized economy when workers are wage takers. In this situation, full insurance is provided and output is maximized. Layoff taxes are higher than hiring subsidies in order to prevent an excessive entry of vacancies induced by the absence of bargaining power of workers. The gap between layoff taxes and hiring subsidies generates a budgetary surplus which is sufficiently large to finance nearly all the unemployment benefits. The analysis being properly microfounded with risk-averse workers, it naturally defines an optimal rate of unemployment which only coincides with the output maximizing rate of unemployment when full insurance can be provided, i.e. when there is no trade-off between the provision of insurance and the maximization of production. When workers have some bargaining power, the planner wants to reduce wages in order to relax the resource constraint and raise the income of the unemployed. In particular, this is achieved by reducing market tightness which lowers wages, as desired, but also hinders the reallocation of workers from low to high productivity jobs. Introducing moral hazard adds a counteracting force to the model. When workers have a very low bargaining power, it is typically desirable to increase market tightness and to boost wages in order to enhance the incentive to search while unemployed. However, when workers have more substantial bargaining power, under-provision of insurance, rather than moral hazard, remains the primary concern of the planner. By emphasizing the liquidity effect of unemployment insurance, Chetty (2008) has already argued that the issue of moral hazard might have been over-emphasized in the literature. The present paper adds to this by showing that general equilibrium effects on job creation, job destruction and wages might be at least as important for the determination of the optimal labor market policy.35 There are essentially two reasons which could justify setting layoff taxes higher than hiring subsidies; in which case the difference between the two could cover at least some of the costs of providing unemployment benefits. First, to compensate for the failure of the Hosios condition to hold; or, in other words, to reduce entry in order to save on recruitment costs when the bargaining power of workers is lower 35 Krusell et al. (2010) reach a similar conclusion, even though they abstract from the possibility of using hiring subsidies and firing taxes to affect the rates of job creation and job destruction.

J.-B. Michau / Labour Economics 35 (2015) 93–107

than the elasticity of the matching function. Second, in order to reduce wages, by reducing market tightness, when the provision of insurance is insufficient. Importantly, as the bargaining power of workers increases, the first reason becomes less relevant while the second becomes more important. This is why layoff taxes exceed hiring subsidies in all realistic calibrations of the model and for any bargaining power of workers. An important limitation of my analysis is that the model does not allow for on-the-job search. To understand how this would change the results, it is useful to consider the benchmark case where all employed workers are searching for jobs and where they do so as efficiently as the unemployed. In that case, the optimal rate of unemployment chosen by the planner is trivially equal to zero. Indeed, in such an environment, unemployment would be a pure waste of time. This optimal allocation would be implemented by a prohibitive cost of laying off a worker, which would induce firms to set their job destruction threshold equal to zero. However, on-the-job search can only meaningfully be analyzed with an endogenous search intensity. In that case, it is only the dispersion in labor income across employed workers that can generate on-the-job search. Note that my model implies that all workers are paid the same wage, which is consistent with the absence of on-the-job search. Characterizing the optimal labor market policy with an endogenous search intensity of employed job seekers when wages are affected by match productivity is an interesting issue that is left for further research. A natural question to ask in such a setup is whether the government should redistribute across employed workers through a progressive income tax. On the one hand, this would reduce wage fluctuations but, on the other hand, this would also discourage poorly matched workers from searching on-the-job. Michau (2013) relies on a simple model of the wage ladder to investigate this question and finds that low wage workers should receive little insurance in order to be induced to search on the job for a better match. However, by assuming a fixed market tightness and a fixed rate of job destruction, the analysis abstracts from the general equilibrium effects that have been at the heart of this paper. Some other important issues are left for further research. First, an accurate empirical knowledge of the main determinants of wages, at the macroeconomic level, is key for the optimal design of labor market policies. 36 Knowing, quantitatively, how wages are affected by market tightness or by the different policy instruments is obviously essential if the planner wants to increase the provision of insurance at the smallest cost in terms of output. The precise specification of wages also crucially affects the implementability constraints. For instance, if layoff taxes and hiring subsidies are passed on to workers through adjustment in wages, then they have a much smaller effect on the job creation and job destruction decisions of firms. Throughout this paper, I have only considered time invariant policy instruments. In fact, in a dynamic context, it would be interesting to allow the level of unemployment benefits to be affected by the length of unemployment and that of layoff taxes and hiring subsidies to depend on the age of the match, among other things. Also, in the proposed model, the length of unemployment does not directly matter, only its rate does. 37 This could be relaxed by assuming that the level of human capital depreciates during an unemployment spell38 or, more simply, by assuming that workers have a preference for shorter spells even if this is associated with a higher probability of being unemployed. The length of unemployment being decreasing in market tightness, the resulting optimal policy would presumably advocate for a smaller reduction in the rate of job creation. 36 Blanchflower and Oswald (1994) provide extensive evidence of the negative impact of unemployment on wages. However, their work does not control for the number of vacancies and, hence, cannot identify the impact of market tightness on wages. 37 The length of unemployment nevertheless has an impact on the speed of the reallocation of workers from low to high productivity jobs. 38 See the related analyses of Pavoni (2009) and Shimer and Werning (2006) who determine the optimal unemployment insurance policy with human capital depreciation.

105

Appendix A. Payroll tax in first-best policy Before deriving Eq. (29), it is necessary to rewrite the expression for the optimal value of b given by Eq. (9). b ¼ y−cθu−z"ð1−uÞ # Z 1 ηðθÞ λ ¼ y−cθu− R− ðs−RÞdGðsÞ ð1−uÞ cθ þ 1−ηðθÞ rþλ R i i r h y ηðθÞ λGðRÞ h y ¼ ð1−uÞ −R þ cθð1−uÞ þ ð1−uÞ −R −cθu r þ λ 1−u 1−ηðθÞ r þ λ 1−u 2Z 1 3 ð s−R ÞdG ð s Þ h i 6 R r y c 7 7: ¼ ð1−uÞ −R þ θqðθÞ6 − 4 r þ λ 1−u qðθÞ5 rþλ

The second line was derived by using the optimal job destruction condition (Eq. (12)) to get rid of z. Note that, combining Eqs. (6a) and (6b) while imposing the steady state conditions u¼ 0 and ẏ = 0, implies that the steady state level of output can be expressed as 1 y = (1 − u)∫ sdG(s)/[1 − G(R)]. To obtain the third line, and to get R rid of the integral, I have used that expression for the steady state level of output and then rearranged the terms. Finally, to get the last line, I have used Eq. (13) to rewrite the second term of the third line and used the fact that, in steady state, λG(R)(1 − u) = θq(θ)[1 − G(R)]u to rewrite the third term of the third line, which was then sim1 plified using y = (1 − u)∫RsdG(s)/[1 − G(R)]. Substituting this expression into Eq. (28) yields Eq. (29). 

Appendix B. Solving the Nash bargaining problem Before solving the bargaining problem, we need to find an expression for W i and J i as a function of the wage rates and of the job destruction threshold. Taking the difference between the expression for Wi(x), as given by Eq. (34), evaluated at productivity s and the same expression at productivity x yields:

W i ðsÞ ¼

~ i ðsÞ þ πÞ−vðw ~ i ðxÞ þ π Þ vðw þ W i ðxÞ: rþλ

Substituting this expression into the value of employment to a worker, given by Eq. (34), gives: W i ðxÞ ¼

1 rþ " λGðRi Þ

# Z 1 λ ~ i ðxÞ þ π Þ− ~ i ðxÞ þ πÞ−vðw ~ i ðsÞ þ πÞdGðsÞ þ λGðRi ÞU  vðw ½vðw r þ λ Ri " # Z 1 1 r þ λGðRi Þ λ ~ i ðx Þ þ π Þ þ ~ i ðsÞ þ πÞdGðsÞ þ λGðRi ÞU : vðw vðw ¼ r þ λGðRi Þ rþλ r þ λ Ri

Thus, W i defined by Eq. (32) is equal to: Wi ¼

" # Z 1 rþλ 1 ~ i ðxÞ þ πÞdGðxÞ þ GðRi ÞU : vðw r þ λGðRi Þ r þ λ Ri

This expression implies that: ~ i ðxÞ þ π Þg ðxÞdx ∂W i v0 ðw ; ¼ ~ i ðxÞ r þ λGðRi Þ ∂w

ðB1Þ

and: " # Z 1 ðr þ λÞg ðRi Þ r þ λGðRi Þ ∂W i λ ~ i ðRi Þ þ π Þ þ ~ i ðxÞ þ π ÞdGðxÞ−rU : v w ¼− ð v ð w rþλ r þ λ Ri ∂Ri ½r þ λGðRi Þ2

ðB2Þ

106

J.-B. Michau / Labour Economics 35 (2015) 93–107

Similarly, using Eq. (36), the value of employment to a firm can be written as: J i ð xÞ ¼

1 r þ λGðRi Þ "

JþH ¼

~ i ðxÞ þ τ Þ−  x−ðw

λ rþλ

Z

1 Ri

" # Z 1 rþλ 1 ~ Ji ¼ ½x−ðwi ðxÞ þ τÞdGðxÞ−GðRi ÞF : r þ λGðRi Þ r þ λ Ri

;

ðB3Þ

It follows that: ∂J i g ðxÞdx ; ¼− ~ i ðxÞ r þ λGðRi Þ ∂w

ðB4Þ

and: ∂J i ðr þ λÞg ðRi Þ ¼− ∂Ri ½r þ λGðRi Þ2 " # Z 1 r þ λGðRi Þ λ ~ i ðRi Þ þ τÞÞ þ ~ i ðxÞ þ τ ÞdGðxÞ þ r F : ðR−ðw ½x−ðw  rþλ r þ λ Ri

ðB5Þ ~ i ðxÞ and the threshold Ri are The first-order conditions for the wage w obtained by differentiating the logarithm of the Nash product in Eq. (31). This yields: β ∂W i 1−β ¼ ~ i ðxÞ J i þ H−V W i −U ∂w

−

c : qðθÞ

ðB9Þ

# ~ i ðxÞÞ−ðs−w ~ i ðsÞÞdGðsÞ−λGðRi ÞF ½ðx−w

which implies that J i defined by Eq. (35) is equal to:

∂J i ~ i ðxÞ ∂w

Substituting Eqs. (B1), (B4), (B8), and (B9) into Eq. (B6) yields Eq. (38) which implicitly determines the wage rate. Note that the derivative (Eq. (B2)) can be simplified by using the fact that, from Eqs. (33) and (B8), we have: vðwÞ−rU ¼ ½r þ λGðRÞ

for all x ∈ ½Ri ; 1;

ðB6Þ

vðwÞ−vðz þ bÞ : r þ λGðRÞ þ θqðθÞ½1−GðRÞ

ðB10Þ

Substituting V = 0, J þ H ¼ c=qðθÞ , Eq. (B8), the derivative of the worker's welfare (Eq. (B2)) simplified with Eq. (B10) and the derivative of the firm's expected profits (Eq. (B5)) into the first-order condition for the threshold (Eq. (B7)) gives an expression for the equilibrium threshold which can be simplified using Eq. (38) to give Eq. (39). Appendix C. The decentralized job creation condition when workers have some bargaining power From Eq. (B3) together with Eq. (37), we know that: " # Z 1 rþλ 1 1−GðRÞ ~ ðR−w−τ Þ−GðRÞF : J¼ ðx−RÞdGðxÞ þ r þ λGðRÞ r þ λ R rþλ Substituting the expression for the equilibrium value of the threshold Eq. (39) yields: Z

! J¼

1 R

ðs−RÞdGðsÞ rþλ

− F−

1−GðRÞ vðwÞ−vðz þ bÞ : r þ λGðRÞ þ θqðθÞ½1−GðRÞ v0 ðwÞ

Finally, using the expression for the equilibrium wage (Eq. (38)), we obtain:

and: β ∂W i 1−β ¼ W i −U ∂Ri J i þ H−V

−

! ∂J i : ∂Ri

Z ðB7Þ

Now that the first-order conditions and the corresponding derivatives have been derived, we can drop the subscript i and use the fact ~ ðxÞ and Ri = R. ~ i ðxÞ ¼ w that in equilibrium, from symmetry, w Substituting the derivatives (B1) and (B4) into the first-order condition for the wage rate (B6) immediately reveals that the wage rate is independent of productivity, as stated by Eq. (37). Note that, with a fixed wage, we have W(x) = W for all x ∈ [R, 1]. Thus, the expression for the value of employment, given by Eq. (34), simplifies to: rW ¼ vðwÞ þ λGðRÞ½U−W ; ~ þ π. Also, the expected utility W of a newly matched where w ¼ w worker given by Eq. (32) can be written as: W ¼ ½1−GðRÞW þ GðRÞU:

These two equations together with the value of unemployment given by Eq. (33) jointly imply that: W−U ¼ ½1−GðRÞ

From the value of a vacancy (Eq. (19)) together with the free-entry condition V = 0, we have:

vðwÞ−vðz þ bÞ : r þ λGðRÞ þ θqðθÞ½1−GðRÞ

ðB8Þ

J¼

1 R

ðs−RÞdGðsÞ rþλ

− F−

β c : 1−β qðθÞ

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