A Structural Model of US Aggregate Job Flows Fabrice Collard

Patrick F`eve

CNRS–GREMAQ and IDEI

Universit´e de Toulouse I GREMAQ and IDEI

Fran¸cois Langot∗

Corinne Perraudin

Cepremap

SAMOS (Universit´e de Paris I)

GAINS (Universit´e du Maine)

Publi´e dans le Journal of Applied Econometrics, 17(3), 197–223, 2002.

Abstract This paper contributes to the analysis of jobs flows dynamics through the explicit modelling of job creations and job destructions. We propose a simple matching model extended for endogenous separation and tractable heterogeneity. The parameters of the model are estimated using a simulation–based estimation method. We then test the ability of trade externalities, generated by the matching process, to (i) propagate reallocation and aggregate disturbances in the whole labor market and (ii) generate the observed distribution of aggregate job flows. The results clearly indicate that the model is able to match the dynamics of US aggregate job flows. Keywords: matching process, stochastic heterogeneity, nonlinear dynamics, simulation based estimation. JEL Classification: C51, E24, E32

Address: F. Langot, Cepremap, 142 Rue du Chevaleret, 75013 Paris, France. We would like to thank A. d’Autume, J.P. Benassy, R. Boucekkine, P. Cahuc, R. Cooper, S. Gregoir, P.Y. H´enin, T. Kollintzas, F. Kramarz, E. Lehmann, O. Licandro, R. Marimon, F. Portier, J. Rust and two anomymous referees for helpful comments on the previous version of the paper. We are particularly thankful to Phillip Schmidt–Dengler for pointing out some errors in an earlier version of the text. The data were kindly provided by J. Haltiwanger. Previous versions of this paper have also benefited from discussions during presentations at GMM (Cepremap), MAD (U. Paris 1), CREST (INSEE), IOBE (Athens), SEL (Louvain–la–Neuve) and IDEI (Toulouse) seminars, and T2M 98 (Marseille), SED 98 (Philadelphia), EEA 98 (Berlin), Macrodynamic Workshop 98 (Vigo, Spain), SCE 99 (Boston) conferences. The traditional disclaimers apply. ∗

1

Introduction In most industrial countries the aggregate dynamics of labor market can be characterized by the following stylized facts1 : (i) vacancies and unemployment are negatively correlated (the so–called Beveridge curve), (ii) aggregate job flows are large within the business cycle, (iii) the destruction rate is more volatile than the creation rate, (iv) destruction and creation rates are negatively correlated. Two additional stylized facts have also been documented from time series analysis: (v) employment adjustments display significant non–linearities (See, e.g., Burgess [1992]) and (vi) the distribution of gross job flows (more particularly destructions) exhibits significant asymmetries. In particular, Davis, Haltiwanger and Schuh [1996] pointed out that “job destruction rises dramatically during recessions, whereas job creation initially declines by a relatively modest amount.” (p.31). The recent econometric literature has provided with candidate time series models that should be able to mimic these stylized facts, in particular threshold autoregressive models (see Tong [1983]). Nevertheless, the aim of this paper is to propose an economic model that accounts and explains these asymmetries. Many recent papers focus on structural models that attempt to shed light on the main forces that can generate these empirical regularities. The most popular framework is the matching model, developed by Mortensen and Pissarides [1994].2 Nevertheless, as underlined by Cole and Rogerson [1999], “although matching models have been used rather extensively, there has been little systematic work to evaluate their quantitative properties with regard to labor-market flows over the business cycle” (p.934). A counterexample is provided by Burgess [1992] who shows that the matching model can account for non–linearities in aggregate employment dynamics (stylized fact (v)). Nevertheless, in matching models, aggregate employment dynamics is generated by the dynamics of gross job flows — i.e. job creations and job destructions. An important issue of this type of models is therefore to account for the stylized facts summarizing the distribution of aggregate job flows. The purpose of this paper is precisely to propose a model that can account for the whole set of stylized facts on aggregate job flows. Beyond the “calibration– simulation” exercise (see e.g. Cole and Rogerson [1999] or Mertz [1999]), we undertake an explicit estimation of the structural parameters and formal statistical tests of the model. At a theoretical level, a novel feature of our analysis is to extend the simple matching 1

See Davis and Haltiwanger [2000] for a recent survey of the large literature that documents the dynamics of job flows in different countries. 2 See also Mortensen and Pissarides [2000] and Hall [2000] for a large survey of the theoretical literature of matching model applied to the labor market.

2

model for endogenous separation and (tractable) firm heterogeneity. Although it borrows from Mortensen and Pissarides [1994], our model departs from their specification with four respects. First of all, for sake of tractability, we take the wage process as an exogenous forcing variable and, following Campbell and Fisher [1996], use this external information to identify the aggregate shock. Second, we relax the key assumption of job irreversibility. In others, firing costs are not infinite, but are assumed to be flexible over the business cycle. Third, we relax the extreme assumption that newly–created jobs are the most profitable. Indeed, if we are to compare the productivity of newly– created jobs across firms, it may be profitable to create jobs in one type of activity, whereas, although endowed with the same productivity, this job may not be profitable in another activity. Fourth, we do not rely on plant specific idiosyncratic shocks. This raises the question of the level of disaggregation assumed in the model. Davis and Haltiwanger [1992] have shown that plant specific idiosyncratic shocks are important to account for highly disaggregated phenomena on reallocation process.3 They however show that, as far as time variation in overall job destruction and overall job creation are concerned, the variance of the idiosyncratic component of job flows accounts for only a small part of the overall variance of job flows (12–16 percents for overall job creations and 6–8 percents for overall job destructions). Davis and Haltiwanger [2000] also show that sectoral shocks can only explain 13–14 percents of the aggregate variation of the reallocation process. These results together with Davis and Haltiwanger’s [1999] empirical study suggest that only two types of disturbances are necessary to account for the dynamics of aggregate job flows: a positive aggregate shock, inducing simultaneously a increase of job creations and a decrease of job destructions, and a positive reallocation shock, inducing increases both in jobs creations and destructions. This last shock can be interpreted as an increase in the gap between the actual and the desired distribution of the characteristics (the location and the skill) of labor.4 Davis and Haltiwanger’s [1999] empirical results show that, conditionally to the identifying scheme, the fraction of the short term variance in the rate of growth of net job that is attributable to reallocation shock lies between 44% and 88%. We therefore consider an economy divided in two physical locations, each of them experiencing specific reallocation shocks, identified by perfectly negatively correlated technology shocks, and 3

Davis and Haltiwanger [1992] report the following stylized facts: (i) fluctuations of job reallocations across industries and regions are countercyclical, (ii) the countercyclical behavior of job reallocation reflects time variation in the magnitude of idiosyncratic plant–level employment movements, (iii) job reallocation rates among young, small and single–unit plants exhibit little or no systematic relationship in the cycle and (iv) job reallocation among older, larger and multi–unit plants exhibits pronounced countercyclical patterns of variation. 4 The reallocation shock exerts no net effect on aggregate productivity.

3

an aggregate shock — the aggregate real wage. Given this stochastic heterogeneity and that firing costs are internal to the firm, we test the ability of trade externalities, generated by the matching process, to propagate shocks in the whole labor market and generate the observed distribution of aggregate job flows. Given the structure of the model and given that the reallocation stochastic process has no observable counterpart, the structural parameters of the model are estimated using a simulation–based estimation method.5 This procedure can be easily implemented even when the likelihood function is intractable or when moments cannot be computed using direct integration methods. We use unconditional and time series moments that encompass as many features of the data as possible in order to avoid too much arbitrariness.6 Thus, the set of moments combines usual moments (sample mean, variance and correlation) and higher order moments that account for asymmetries in the data. In the lines of Gallant, Hsieh and Tauchen [1994], we further develop a diagnostic test that aims at locating potential failures of the structural model in terms of moment matching. Our results indicate that the model matches the selected moments computed using quarterly data on creation and destruction rates in the US manufacturing sector — and thus accounts for the aforementioned stylized facts. This first confirms our specification choices, and more particularly the degree of stochastic heterogeneity. Further, it is shown that the non–linearity generated by the theoretical model is sufficient to characterize the empirical distribution of the data. Hence, these results show that trade externalities provide a mechanism sufficient to match the distribution of aggregate job flows. More precisely, congestion effects limit the magnitude of the response of aggregate hirings following a positive technology shock. Conversely, aggregate congestion effects do not affect firing decisions, explaining the sharper response of firings following a negative shock. The paper proceeds as follows. Section 1 presents the model. Section 2 is devoted to the econometric methodology. Finally, the estimation and testing results are reported in section 3. This section also provides some insights on the propagation mechanisms. A last section concludes. 5

See e.g. Gouri´eroux, Monfort and Renault [1993], Smith [1993], Gouri´eroux and Monfort [1994] and Gallant and Tauchen [1996] for a general statement of these methods. 6 Concerning the choice of the moments, we follow Gallant, Rossi and Tauchen [1993] or Foster and Viswanathan [1995].

4

1

A Matching Model of the Labor Market

This section is devoted to the exposition of the behavior of individuals on the job market and to the characterization of the reallocation process governing trade on this market.

1.1

Technology and labor market arrangements

Technology We consider an economy that consists of 2 employment pools indexed by j. Each pool is composed of a large number of firms indexed by i. Each firm has access to a constant returns–to–scale production function given by: Yi,j,t = ηj,t Ni,j,t

(1)

Ni,j,t denotes the employment level involved in the productive process. Each pool is characterized by a specific implementation of technology, summarized by ηj,t which is assumed to follow a covariance stationary AR(1) process: log(ηj,t ) = ρη log(ηj,t−1 ) + ση νj,t

(2)

where νj,t is zero mean, unit variance Gaussian white noise that satisfies ν1,t = −ν2,t . This latter assumption implies that this shock will essentially shift employment from one employment pool to the other, such that it primarily exerts microeconomic effects and can be interpreted as a reallocation shock. Given this specific structure of the reallocation shocks, the stochastic dimension of the model is the same as the one found in the data. Labor market arrangements We assume that allocation of resources in the labor market is driven by a search process for a given real wage. Each firm controls its hiring policy through Vi,j,t , the number of vacancies, and its firing policy through Fi,j,t , which denotes the level of firing. An important feature of this economy is that vacancies can be opened or closed but they cannot move between employment pools. Following Pissarides [1990], trade in the labor market is a costly and uncoordinated economic activity. There is a unified labor market where workers are perfectly mobile. In each and every period, a firm can post Vi,j,t vacancies. As Bertola and Caballero [1994], we depart from the standard vacancy cost setup by letting the marginal cost of posting vacancies be an increasing function of the number of vacancies posted. 5

The vacancy cost, ψ(Vi,j,t , Vj,t ) : R+ × R+ −→ R+ , is thus an increasing and convex function, and satisfies the additional assumption ψ(0, .) = 0. The convexity assumption is necessary to determine vacancies for an individual firm which has access to a constant return-to-scale production function. Indeed, this convexity assumption prevents firms with high productivity from posting an infinite number of vacancies. Moreover, it is assumed that the cost of advertising a vacancy is lower when the total number of vacancies in the employment pool is large. This latter assumption simply reflects increases in the returns of information when the size increases. A firm cannot transfer its vacancies from an employment pool to the other. Further, we assume that, in each pool j, there exists a constant returns to scale matching function linking the number of hirings Hj,t to the number of vacancies Vj,t and unemployed workers Ut in the aggregate economy: Hj,t = h(Ut , Vj,t ) for j = 1, 2 h(.) is an increasing and concave function with respect to both arguments. It further satisfies: h(0, .) = h(., 0) = 0. In each and every period, vacant jobs and unemployed workers are matched and move from trade to productive activity. But some existing jobs disappear, therefore generating persistent unemployment. The size of the population is normalized to one, such that Ut = 1−Nt denotes the unemployment rate. Job vacancies and unemployed workers matched at time t are randomly selected. The rate at which job vacancies are filled is given by qj,t = Hj,t /Vj,t . This transition rate depends on the relative number of traders. Thus, there are two types of trade externalities. First, as Vj,t increases, the probability of rationing firms increases. This trade externality amounts to a congestion externality. Second, as Ut increases, the probability of rationing firms decreases: there thus exists some positive trade externality between traders.7 In each firm and in every period t, the number of employment outflows has two components. On the one hand, there are “exogenous” separations, given by the product of the separation rate, s, with the current number of employees Ni,j,t . On the other hand, firms adopt an active firing policy, denoted by Fi,j,t . At the firm level, firing costs depend on the firing level Fi,j,t . These firing costs generalize the simple case exposed in Pissarides [1986]. Firing costs are defined by the function: φ(Fi,j,t , Nj,t ) : R+ × R+ −→ R+ , strictly increasing and convex. We impose φ(0) = 0 and lim φ(Fi,j,t , Nj,t ) = ∞. Given that all firms of the pool Fi,j,t →(1−s)Nj,t

7

Unemployed workers find jobs more easily as the number of vacancies is high relatively to the number of workers involved in the matching process; symmetrically, the greater the number of searching workers is, the easier it is for a firm to fill up a vacancy.

6

have the same employment level at the beginning of the period (Ni,j,t = Nj,t ∀i), this condition simply amounts to assume that reorganization costs increase as firms fire. This assumption aims to account for the following fact: higher–tenure workers may have accumulated job-specific capital, such that firing them generates higher costs (see Ruhm [1991]). It follows that ∂φ(Fi,j,t , Nj,t )/∂Nj,t < 0. 8 In pool j, employment evolves according to the following law of motion: Nj,t+1 = (1 − s)Nj,t + Hj,t − Fj,t

for j = 1, 2

(3)

Thus, productive employment at time t + 1 is hired at time t, implying some labor hoarding phenomenon. Therefore, the number of unemployed is an adjustment variable. However, the aggregate job availability only determines the rate at which jobs vacancies are filled. This externality causes a stochastic rationing on the individual hiring policy. It is worth noting at this point that when a firm hires, it has to go on the labor market in order to find a worker. In this case it faces aggregate trade externalities. Conversely, when it fires, it does not face aggregate trade externalities.

1.2

Individual decision rules

The problem of the firm i of the employment pool j is to decide the number of vacancies Vi,j,t and firings Fi,j,t so as to maximize the expected discounted sum of profit flows: (∞ ) X max ∞ Et (1 + r)−τ [Πi,j,t+τ − ψ(Vi,j,t+τ , Vj,t+τ ) − φ(Fi,j,t+τ , Nj,t+τ )] {Vi,j,t+τ ,Fi,j,t+τ }τ =0

subject to

τ =0

  Ni,j,t+1 = (1 − s)Ni,j,t + qj,t Vi,j,t − Fi,j,t Vi,j,t > 0  Fi,j,t > 0

(Xi,j,t ) (λi,j,t ) (µi,j,t )

where (1 + r)−1 denotes the firm’s discount factor, r ∈ (0, 1). λi,j,t and µi,j,t are the Lagrange multipliers associated to the positivity constraints on, respectively, vacancies and firings. Xi,j,t can be interpreted as the marginal valuation of employment. Πi,j,t is the operating profit flow of the firm i at time t in pool j, given the real wage wi,j,t : Πi,j,t = ηj,t Ni,j,t − wi,j,t Ni,j,t Note that instantaneous operating profits are linearly homogeneous in employment. The first–order conditions for a firm i shows that firm posts vacancies up to the point 8

From a technical point of view, this assumption guarantees that firing will never rise up to the point where employment becomes zero.

7

where the marginal value of a vacancy equals its marginal cost: −ψ ′ (Vi,j,t , Vj,t ) + Xi,j,t qj,t + λi,j,t = 0

(4)

This optimality condition is equivalent to the standard “free entry” condition in the labor market when firms can have only one job — filled or unfilled. Concerning firings, the optimality condition states that firms fire up to the point where the marginal value of employment is equal to the marginal cost of firings: −φ′ (Fi,j,t , Nj,t ) − Xi,j,t + µi,j,t = 0

(5)

We also have the two additional conditions: λi,j,t Vi,j,t = 0

(6)

µi,j,t Fi,j,t = 0

(7)

From the first order conditions (4)–(5) together with (6) and (7), for any given value of Xi,j,t , we get the following property. Proposition 1 A firm will not fire when it has positive vacancies. Conversely, a firm will not post any vacancy when it fires. Proof: See appendix A. According to equations (4)–(5) and proposition 1, a firm posts vacancies and does not fire when the expected marginal value of employment is greater than the marginal cost of posting no vacancy — i.e. ψ ′ (0, .)/qj,t . Further, a firm does not post vacancies and fires when the expected marginal value of employment is negative and less than minus the marginal cost of firing, when firings are zero — i.e. −φ′ (0, .). There exists a third regime, characterized by a marginal value lying between the two last values, in which firms are totally inactive: they neither fire nor post vacancies.

1.3

Equilibrium decision rules

In order to compute the equilibrium on the labor market, we specify first the wage process in the lines of Campbell and Fisher [1996]. As workers are perfectly mobile, the wage is the same across employment pools. This exogenous process can be interpreted as the reservation wage of the workers. If workers have the same preference, the dynamics of the wage allows us to identify the aggregate disturbance in our economy.

8

For sake of tractability, we assume that it is modelled as an exogenous process given by log(wt ) = ρw log(wt−1 ) + σw εt where |ρw | < 1 and εt is a zero mean, unit variance Gaussian white noise. The symmetric equilibrium in employment pool j is defined by a set of functions {Vj (.), Fj (.), Nj (.)} which depends on the information set It .9 Thus, Vj,t = Vj (It ), Fj,t = Fj (It ) and Nj,t+1 = Nj (It ) solve: ψ ′ (Vi,j,t , Vj,t ) = qj,t Xi,j,t

if

Xi,j,t > ψ ′ (0, .)/qj,t

(8)

Vi,j,t = Fi,j,t = 0

if

(9)

−φ′ (Fi,j,t , Nj,t ) = Xi,j,t Z 1 Vj,t = Vi,j,t di 0 Z 1 Fj,t = Fi,j,t di

if

−φ′ (0) 6 Xi,j,t 6 ψ ′ (0, .)/qj,t

Xi,j,t < −φ′ (0, .)

(10) (11) (12)

0

As our economy displays constant returns to scale, the marginal value of employment corresponds to its average value and satisfies the following property. Proposition 2 At a symmetric equilibrium, marginal value of employment is exogenous in each employment pool. Proof: See appendix A. We thus get:10 1 Et [ηj,t+1 − wt+1 + (1 − s)Xj,t+1 ] 1+r = (1 − s)Nj,t + qj,t Vj,t − Fj,t

Xj,t = Nj,t+1

(13) (14)

Equation (8) gives the level of hiring, while equation (10) furnishes the firing level both in terms of Xj,t . As aforementioned, the model displays three regimes. The shift from one regime to another is driven by the shocks, as implied by proposition 2. Marginal value of employment results from changes in specific technology (the reallocation shock) and in real wage (the aggregate shock). Thus when a negative shock occurs in a given employment pool, the marginal valuation of employment decreases, and can reach a level lower than the bound that renders a firing policy worth to implement. In this situation, the equalization of the marginal value of employment to the marginal cost 9 10

We denote It = {Nt , wt , {ηj,t }j=1,2 }. See appendix A for the complete resolution of the model

9

of firings implies that firings shift up. In the case of a positive shock, the increase in the marginal value of employment leads firms to post vacancies, which are determined by the equalization of the marginal value of employment to marginal cost of posting vacancies. In the third regime, firms will neither hire nor fire. Employment will decrease at a constant rate, determined by the exogenous quit rate. As shown in proposition 1, hirings and firings cannot coexist at the individual level. Nevertheless, as we consider a 2–employment pools economy, hirings and firings will coexist at the aggregate level, because each pool experiences a specific history of productivity shocks. Heterogeneity is thus a necessary condition to study aggregate creation and destruction rates.

2

Test of the Model

This section describes the econometric method we used to estimate the deep parameters of the model. We first present the specifications for cost and hiring functions. We then introduce the simulated method of moments and discuss the choice of the moment conditions. Finally, we present various practical issues related to the estimation method.

2.1

Specification

In order to estimate the model, the matching function, the cost of posting vacancies and the firing costs must be specified first. As far as the hiring process is concerned, rather usual and simple functional forms were adopted. In the lines of the standard literature on matching process (see Blanchard and Diamond [1989]), the matching technology is assumed to be represented by a Cobb–Douglas function: α 1−α Ut for j = 1, 2 Hj,t = HVj,t

where α ∈ (0, 1) and H > 0. The costs of posting vacancies are given by: ψ(Vi,j,t , Vj,t ) =

2 ω Vi,j,t for j = 1, 2 2 Vj,t

It is worth noting that at the individual level, the marginal cost of posting vacancies is zero when Vi,j,t = 0. This modelling has the advantage to yield the same reduced form as the standard linear vacancy cost in the symmetric equilibrium. In order to preserve homogeneity at the individual level, firing costs are assumed to be more important when the number of firings in a firm is large relative to the net employment in the employment pool, (1 − s)Nj,t . Therefore, the firing cost function 10

depends on an external effect and is given by: 2 Fi,j,t φ(Fi,j,t , Nj,t ) = ϕ for j = 1, 2 (1 − s)Nj,t − Fi,j,t

This specification implies that in symmetric equilibrium the marginal cost of firings will be zero whenever the firings are zero. The third regime — inaction — therefore degenerates to a unique point and its probability of occurrence is reduced to zero. This particular form for the adjustment cost has the attractive feature that it allows for sudden and large changes in firings while warranting positive employment as, at symmetric equilibrium, firings approach (1 − s)Nj,t by above the cost of firings tends to infinity. Given these specifications, the realization of shocks generates a particular value at Xj,t , implying the following decision rules: • When Xj,t > 0, individual firms choose to hire new workers, Vj,t > 0. Then, given the optimality condition on vacancies, the hiring function in equilibrium can be written: 1

α

α

1−α H(Xj,t , N1,t , N1,t ) = H 1−α ω α−1 Xj,t (1 − N1,t − N2,t ), j = 1, 2

(15)

The individual hiring decisions are thus non–linear functions of the latent variable Xj,t , whose influence is controlled by the parameter of the matching function, α. It further shows how the state of the labor market — the unemployment rate (Ut = 1 − N1,t − N2,t )— affects individual decision rules. Finally, given a value 1

α

for Xj,t and Ut , the composite parameter H 1−α ω α−1 sets the level of hirings.

• when Xj,t = 0, it is optimal for firms to be inactive, Vj,t = Fj,t = 0 — i.e. the level of employment decreases at the quit rate, s. • when Xj,t < 0, firms adopt an active firing policy, Fj,t > 0. In this case, firms decide actively to fire workers. Given the optimality condition on firings, the equilibrium firing function is given by: "  −1/2 # ϕ − Xj,t F (Xj,t , Nj,t ) = (1 − s)Nj,t 1 − (16) ϕ At symmetric equilibrium, the level of firing in each employment pool j = 1, 2 is a fraction of net employment, (1−s)Nj,t . This fraction depends negatively on the marginal value of employment Xj,t . Then, the role of the firing cost parameter, ϕ, is, as in standard quadratic adjustment costs function, to smooth the firing 11

decision. The overall job destruction in each employment pool is then defined as the sum of endogenous firings and exogenous quits: Fj,t + sNj,t . It follows that the value of s partly determines the mean of the job destruction rate in the economy.

2.2

Estimation method

As previously stated, we want to estimate and test our model using US aggregate job flows data. This enables to detect the dimensions along which our simple structural model is capable to mimic some features of the US labor market. More specifically, we would like to assess the ability of the model to match a set of moment restrictions and locate the potential failures of our specification choices. But, the statistical evaluation of the model raises some specific problems and standard methods are inoperative for various reasons in our framework. First of all, the dynamics of employment in each pool is characterized by an aggregate externality, as stated in the previous section. Second, both creation and destruction levels are a non–linear policy functions of an unobserved random variable (see the non–linear state–space form (17)). Third, the series are aggregated over employment pools. Fourth, gross job creation and destruction rate are computed deflating the levels by the average of begin and end of period stocks. This normalization is performed for consistency sake with aggregate data.11 All this leads to complicated reduced forms for aggregate creation and destruction rates, which makes the likelihood function intractable and/or makes the moments not calculable using direct integration methods. This implies that neither maximum likelihood nor GMM can be implemented in this framework. We therefore turn to a simulated based method. The basic idea of the Simulated Method of Moments (SMM hereafter) is to replace the computation of analytic moments by simulations — i.e. moments are obtained through Monte–Carlo integration (see Ingram and Lee [1991], Duffie and Singleton [1993]). The SMM has the advantage of being easily implementable even when the structural model is complicated. Let Yt = {ct , dt }′ denote the vector of the variables of interest, the model can be rewritten in the following non–linear state–space form   H +H Yt

11

 =  

P

1/2(

P

1,t

2 j=1

Nj,t +

P

2,t 2 j=1

P

Nj,t−1 )

F1,t +sN1,t +F2,t +sN2,t 1/2( 2j=1 Nj,t + 2j=1 Nj,t−1 )

  ≡ Y(Xt ) 

Simulated data are built in accordance with Davis and al.’s definition (see 3.1 for further details)

.

12

where the state variable Xt = {H1,t , H2,t , F1,t , F2,t , N1,t , N2,t , X1,t , X2,t , log(η1,t ), log(η2,t ), log(wt )} evolves as

                 



1

α

α

1−α (1 − N1,t − N2,t ) 1[X1,t >0] H 1−α ω α−1 X1,t

α 1  α 1−α  α−1 X 1−α (1 − N 1 ω H ) 1,t − N2,t [X >0] 2,t 2,t  −1/2      1,t H1,t  1[X1,t 0 (µi,j,t = 0). Then, the optimality condition on vacancies implies ψ ′ (Vi,j,t , Vj,t ) >0 Xi,j,t = qj,t whereas the optimality condition on firings implies Xi,j,t = −φ′ (Fi,j,t , Nj,t ) < 0 These two conditions are mutually inconsistent. We thus have: • either Vi,j,t > 0 and Fi,j,t = 0, which implies: λi,j,t = 0 and µi,j,t =

ψ ′ (Vi,j,t , Vj,t ) + φ′ (0, .) > 0, qj,t

• either Vj,t = 0 and Fj,t > 0, which implies: λi,j,t = φ′ (Fi,j,t , Nj,t )qj,t + ψ ′ (0, .) > 0 and µi,j,t = 0, • or Vi,j,t = 0 and Fi,j,t = 0, which implies: λi,j,t = ψ ′ (0, .) − (µi,j,t − φ′ (0, .)) qj,t This condition is satisfied as long as λi,j,t > 0 and µi,j,t > 0 — i.e. as long as 0 6 µi,j,t < ψ ′ (0, .)/qj,t + φ′ (0, .)  Proposition 2: At a symmetric equilibrium, the Euler equation takes the form Xj,t = βEt [ηj,t+1 − wt+1 + (1 − s)Xj,t+1 ] where β = 1/(1 + r). Then, iterating forward, one gets: ) (K X Xj,t = β lim (β(1 − s))k Et [ηj,t+1+k − wt+1+k ] + (β(1 − s))K Et Xj,t+K K→+∞

k=0

The transversality condition associated to the firms’ program implies that lim (β(1 − s))K Et Xj,t+K = 0

K→+∞

30

such that the marginal value of employment reduces to Xj,t = β

∞ X k=0

(β(1 − s))k Et [ηj,t+1+k − wt+1+k ]

We have to take expectations of both ηj,t+k and wt+k which are both log–normally distributed AR(1) processes. Let’s consider a generic AR(1) process zt , log(zt ) = ρz log(zt−1 ) + σz εz,t where εz,t ; N (0, 1). The k steps ahead value of z is thus given by log(zt+k ) = ρkz log(zt ) + σz

k−1 X

ρℓz εz,t+k−ℓ

ℓ=0

Then the expected value of zt+k is "

Et zt+k = Et exp ρkz log(zt ) + σz "

k−1 X

ρℓz εz,t+k−ℓ

ℓ=0

!#

k−1 X
k = exp ρz log(zt ) Et exp σz ρℓz εzt +k−ℓ ℓ=0

!#

using standard results for log–normal distribution, one gets Et zt+k

k σ 2 X 2(k−ℓ) ρ = exp ρkz log(zt ) + z 2 ℓ=1 z

!

(18)

Therefore, the conditional expectation of the specific technology shock is given by ! k ση2 X 2(k−ℓ) k ρ (19) Et ηj,t+k = exp ρη log(ηj,t ) + 2 ℓ=1 η

and the expected real wage writes as Et wt+k

k σw2 X 2(k−ℓ) k = exp ρw log(wt ) + ρ 2 ℓ=1 w

!

Plugging (19) and (20) in the Euler equation, one gets Xj,t = β

∞ X k=0

     log(ηj,t ) + ξη,k − exp ρk+1 (β(1 − s))k exp ρk+1 η w log(wt ) + ξw,k

where ξη,k and ξw,k are given by ξη,k =

2(k+1) ση2 1 − ρ2(k+1) σw2 1 − ρw η and ξ = w,k 2 1 − ρ2η 2 1 − ρ2w

The marginal value of employment therefore depends only on exogenous shocks. 31

(20)

B

Computation of Impulse Response Functions

The computation of the IRF is conducted in the lines of Koop et al. [1996]. In case of an arbitrary shock of magnitude δ, given state variables at time 0, an IRF for variable x ∈ {c, d} at horizon h is defined as: Ix (h, δ, 0) = E[xt+h |δ; I0 ] − E0 [xt+h |I0 ] where I0 is the information set. Conditional expectations, involved in IRF computations, are calculated using Monte-Carlo integration. Then IRF are obtained as follows: Step 1: Given an initial value for employment, n0 , we draw the shocks from independent normal distributions for a fixed horizon H (summarized by νt and εt ). We randomly draw H × R values for the 2–dimensional innovations. Step 2: We compute R realizations for horizon h = 0, . . . , H of creation and destruction rates using the draws from Step 1. They are denoted crh (n0 , νt , εt ) and drh (n0 , νt , εt ), for h = 0, . . . , H and r = 1, . . . , R. Step 3: We compute R realizations of aggregate creation and destruction rates using the same draws plus one additional arbitrary shock δ in a given employment pool. They are denoted crh (n0 , δ, νt , εt ) and drh (n0 , δ, νt , εt ), for h = 1, . . . , H and r = 0, . . . , R. Step 4: We form the averages for each simulated data: R

x¯R h,0

1X r x (n0 , νt , εt ) , h = 0, . . . , H = R r=1 h R

x¯R h,δ,0

1X r x (n0 , δ, νt , εt ) , h = 0, . . . , H = R r=1 h

where x(.) = {c(.), d(.)}. Step 5: We compute the IRF as the difference between the two averages of the creation and the destruction rate: Ix (h, δ, 0)R = x¯R ¯R h,δ,0 − x h,0

h = 0, . . . , H

Thus for R large, we have: lim Ix (h, δ, 0)R = Ix (h, δ, 0) h = 0, . . . , H

R→∞

32

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