Unemployment volatility puzzle and the specifications of the matching model

Hernán Ruffo* Master Thesis CEMFI 20 June 2008

Abstract This paper analyzes alternative calibration strategies for the search and matching model of the labor market, allowing for endogenous separation rate. Two extensions to the canonical model were considered: heterogeneity in new jobs and two shocks to productivity. The results are that, when compared to US business cycle stylized facts, traditional calibration proposed by Shimer (2005) generates too low volatility of unemployment, finding and separation rates in the three specifications of the model. On the other hand, alternative calibration implemented by Hagedorn and Manovskii (2005) succeeds at reproducing the observed fluctuations when original assumptions apply and produces excessive volatility when the extensions to the model are considered. This suggests the posibility of evaluating novel calibration strategies that could eventually improve the performance of the model in other dimensions than business cycle, such as the response of unemployment to policies or the elasticity of wages of new jobs.

* I want to dedicate this paper to Laura and Mateo, for being my constant support. I also want to thank the encouragement of my classmates. I am grateful to Stéphane Bonhomme and Josep Pijoan-Mas for their enthusiastic and useful advice. I want to thank all the professors that assisted to the presentation of this work and the useful comments of Javier Suarez and Manuel Arellano. Last but by no means least, I want to thank Claudio Michelacci for his excellent work as supervisor of this Master Thesis.

1 Introduction The matching models have failed in many aspects to replicate some of the most important features of labor market dynamics over the business cycle. In particular, Shimer (2005) pointed out that the basic matching model could not reproduce the volatility of unemployment and vacancies observed in the US. Other studies have been able to produce higher volatility of labor market variables, but at the cost of introducing fixed wages, as in Shimer (2004) or Hall (2005). Alternativelly, Hagedorn and Manovskii (2005) introduced a novel calibration that was successful at matching the data, but also generated some controversial implications. In particular, this alternative calibration is based on a very high instantaneous utility in unemployment state and a very low bargaining power of the workers, implying that workers are almost indifferent between being unemployed or employed. In other words, for the model to produce a high elasticity of unemployment the value of the job should be very low as a fraction of productivity, so that any change in productivity level implies a big percental change in its value, which is perceived mainly by the firm, given the low bargaining power of workers. These competing specifications generated some controversy and motivated works as the ones by Hornstein, Krusell and Violante (2005) and Mortensen and Nagypál (2005), which compare them in the context of the basic matching model. The importance of the problem justifies this discussion. In fact, the Shimer’s work follows the business cycle research project, and the identification of this puzzle restricts the application of the matching model for macroeconomic analysis. This paper analyzes these calibrations in the broader context of the Mortensen and Pissarides (1994) model. This is a natural extension to assess unemployment volatility, given that it endogenizes the separation rate, which accounts for about 1/3 of the total unemployment volatility (Pissarides, 2007). Additionally, we will test whether these calibrations are consistent with the observed evolution of separation rate. In particular, one could think that a higher volatility of the finding rate would generate a lower elasticity of separation rate. To make this argument clear, we have to analyze the cyclical properties of surplus value of a job. This element determines finding and separation rates: a higher value of new jobs encourages vacancy creation, while separations occur when surplus value is below zero, in which case both employer and employee agree to destroy the match. In this context, a shock in productivity level generates an increase in the surplus of a job, what enhances job

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creation and reduces separation rate. Thus, finding rate increases, improving the value of being unemployed, given that better jobs can be found more easily. Given that unemployment state constitutes an outside option, a higher unemployment value implies a reduction in surplus. On the whole, an expansion implies a higher surplus of the job, which is partially offset by the rise in unemployment value. This counteracting force turns out to be determinant for the assessment of volatility. The cyclical evolution of unemployment value, on the other hand, is determined by changes in the finding rate and in the value of being employed. A higher finding rate elasticity, if not offset by value effect, can compensate productivity impact over surplus of existing jobs, making separation rate too stable over the business cycle. In Hagedorn and Manovskii calibration, the finding probability effect is strong, while the value effect is weak, given that any improvement in surplus is gained by the firms, who have the bargaining power. Which of the two effects dominates is a quantitative matter that will be studied in simulations. The Mortensen-Pissarides framework offers as well the opportunity of introducing some natural extensions. In particular, we depart from the original model that assumes that new jobs productivity is the highest value of the support of idiosyncratic productivity distribution and we will evaluate calibration performance in the context of heterogeneity in productivity of newly created jobs. This offers an additional source of volatility, as will be explained. Additionally, we will analyze the introduction of another shock to productivity. Traditionally, the literature has considered that cyclical shocks were driven by changes in aggregate component of productivity. This paper presents another view, in which reallocative shocks coexist with aggregate shocks, determining the evolution of mean productivity level. Actually, the process of reallocation of resources between low and high productivity firms is also governed by the dispersion of idiosyncratic productivity, which is subject to shocks. When dispersion is increased, both job creation and job destruction are encouraged. Davis, Haltiwanger and Schuh (1998) and Caballero and Hammour (1996) have studied the evolution of these two variables and their dynamics over business cycle. Michelacci and Balakrishnan (2001) analyzes the relative importance of aggregate and reallocative shocks for unemployment in OECD coutries. In this paper, we followed a similar route that proved to be empirically relevant: almost one third of finding rate variance and one half of productivity changes is triggered by reallocative shocks.

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The first finding of this work is that, while the traditional calibration proposed by Shimer is unable to reproduce cyclical volatility, alternative one as in Hagedorn and Manovskii succeeds in matching the volatility over the business cycle of both finding and separation rates. The main reason for this is the lack of responsiveness of unemployment value to changes in labor market conditions: low bargaining power counteracts effectively the elasticity of finding rate. In fact, the latter specification base on the fact that unemployed are almost as well as employed; in such a case, expansions or recessions hardly change the welfare of unemployed and no offsetting effect arise. This is a strong and controversial implication of this calibration for which we generate additional evidence. In addition, Hagedorn and Manovskii approach is counterfactual in that it requires a too low elasticity of wages of newly created jobs to generate its outcome, while in the data this elasticity is around one (Pissarides, 2007). When extensions to the Mortensen-Pissarides model are considered alternative calibration generates an excess volatility in all labor market variables. Thus, this approach offers a framework for generating and evaluating novel calibration strategies that can help to release the tension between matching business cycle facts and generating reasonable outcomes in other dimensions of the labor market. In section 2 I will present the stylized facts of the labor market. Then I will introduce the Mortensen-Pissarides model in a general way, with the purpose of nesting the extensions that will be presented. Next, the calibration and simulation sections will be divided in three parts, first analyzing the model without idiosyncratic productivity, then introducing this component, and finally with the consideration of multiple shocks to productivity. I choose this route to firstly present the original context of Shimer (2005) and Hagedorn and Manovskii (2005) work, replicating their results. Finally, section 6 concludes.

2 Stylized facts I follow Shimer by presenting the evolution of labor market variables along the business cycle. Table 1 shows some statistics for unemployment, vacancies, job finding rate, separation rate and productivity.1 This table is constructed borrowing from the business cycle analysis. That is to say variables are quarterly data, in logs as

1

Data is described in detail in Annex B.

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deviations from their trend, captured by the Hodrick-Prescott filter with a parameter equal to 10000.2 The facts that have been discussed in the literature are: • The volatility of unemployment (u) is very high: ten times the volatility of productivity. • Similarly, vacancies (v) and market tightness (v/u) volatility is also very high: ten times or and times the one of productivity. • A well known and intuitive fact: unemployment and vacancies are inversely correlated, with a -0.9 coefficient. • Market

tightness,

vacancies

and

finding

rate

are

procyclical

while

unemployment is countercyclical, but all rates have a mild correlation coefficient, that is around 0.4. • All the variables have strong autocorrelation. These are the frequent observations, at least since Shimer (2005). We have to add that separation rate (s) is 3-4 times more volatile than productivity, it has a negative correlation with finding rate and it is also coutercyclical. Beveridge curve seems to hold (because of the strong inverse correlation of u and v), and variables seem to have strong elasticity to cycle shocks. Nevertheless, a somewhat puzzling issue is the relative low correlation with productivity. This suggests some independent driving forces in addition to aggregate productivity. A second observation that will be tested in the simulations is the decomposition of unemployment volatility. The literature has focused during years in separation rate and job destruction to explain unemployment evolution and volatility. In fact, the work of Davis, Haltiwanger and Schuh (1998), based in manufacturing, implied that most of the action in the labor market come from job destruction instead of job creation. Nevertheless, recent work of Shimer (2005b) based on a wider definition of the economy (non-farm) and accounting for time aggregation biases in estimations of finding and separation rates, shows that finding seems to be more relevant in the explanation of unemployment volatility. To formally assess this issue I will apply a decomposition of the variance that is based on the Taylor approximation of the steady state unemployment variance. Consider the unemployment level in steady state, u =

s s+ f

, which depends only

in separation (s) and finding (f) rates. As shown in Figure 1, this variable has very 2

Shimer (2005) and Reiter and Costain (2006) justifies this detrending procedure arguing that it avoids demographic effects as well as correlation of unemployment trend with NBER-identifyed business cycle.

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strong comovement with observed unemployment rate. If we calculate the variance of the log of this rate, then we can compute the first order Taylor approximation: 2

2

⎛ ∂ ln(u ) ⎞ ∂ ln(u ) ∂ ln(u ) ⎛ ∂ ln(u ) ⎞ ⎟⎟ V [ f ] + V [ln(u )] ≅ ⎜ Cov[s, f ] , ⎟ V [s ] + ⎜⎜ ∂s ∂f ⎝ ∂s ⎠ ⎝ ∂f ⎠

where

derivatives are evaluated at mean levels of s and f. Applying this formula, we can say that 2/3 of the variance of unemployment can be explained by finding and separation rates with equal weights, while around 1/6 is explained by their (negative) covariance (see Table 2). The remaining proportion of the variance in steady state unemployment is related to the errors of the Taylor approximation. We can conclude that finding and separation has similar effects on unemployment volatility.3 While these are the main issues that we will consider when testing the specifications of the model, some other observations are also worth noting. In particular, many authors (Davis, Haltiwanger and Schuh, 1998, and Caballero and Hamour, 1996, among others) have stressed the role of asymmetries and dynamics in the labor market. The data used here also shows some of the characteristics: when recession begins, separation rate jumps up in around 10% on average and has a maximum in the second or third quarter after the beginning of the recession which is over 15% of its initial value. After this, separation rate tends to go down smoothly. On the other hand, finding rate decreases strongly but at half of the rate of the jump of separation rate, and converges to a 15% lower value after five quarters. When recession ends, separation rate goes down around 5% in two quarters, staying at a lower level (that seems to be a floor); finding rate recovers gently and goes up around 10% after 5 quarters. Thus, shocks are asymmetric for both variables: a negative shock generates a quick and ample response, while in the recovery and growth periods variables seem to converge smoothly to a new level. While both variables are responsive, separation rate seems to jump at a negative shock, while finding rate movements are smoother. To summarize this section: i) unemployment volatility is around ten times the one of productivity; ii) all variables are very responsive over the business cycle; iii) volatility of each finding and separation rates explains one third of unemployment 3

It is important to note that this unemployment level is logged but not filtered. In other words, the estimated contribution of separation rate includes both its cyclical component and its trend. This generates some differences when comparing this decomposition with simulated ones, in which there is no trend of separation (exception made from the two shock specification). When applying this method to filtered variables, the approximation renders too inaccurate, and the error represents most of the variance. More work should be done to generate a clear variance decomposition of cyclical unemployment.

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volatility, while their negative correlation explains another ¼; iv) asymmetric responses are present, with higher response to negative shocks and jumps in separation rate. These are the issues that will be considered when analyzing the simulation results for different calibrations.

3

The Mortensen-Pissarides model

In this section I will present the Mortensen and Pissarides (1994) (MP) model, which endogenizes the separation decision. I will introduce extensions to the model in the subsequent subsections. In the MP model there is a continuum of mass one of two kinds of agents: workers and firms. For generating a productive activity, workers must search for a job during a period of unemployment, while firms must search for a worker by posting a vacancy. Workers and firms meet at a rate given by the matching process, for which a matching function is a reduced form expression (Lagos, 2000). The matching function is assumed to be m(v, u ) = kv1−η uη . In terms of probability of

m (v , u ) = kθ 1−η , where θ=v/u is the so u m(v, u ) called market tightness. The probability of filling a vacancy is q (θ ) = = kθ −η . v finding a job, this function implies f (θ ) =

Production only takes place when a match is created. The productivity of a job is determined by an aggregate component (p) and an idiosyncratic shock (εi), which arrives with a certain probability (λ) and is drawn from a given distribution (F(.)). Then productivity of a match is y=p+σεi, where σ is the dispersion of the shock. In this context, there is a critical value of the idiosyncratic productivity level (εd) below which the job has a negative value and both parts agree to destroy it.4 We add to this model an exogenous separation rate that could be interpreted as an independent draw of idiosyncratic productivity with probability δ that takes εi to ∞. Given this setup, the value functions for the agents are as follows. When looking for a worker, firms has to pay a cost of c per period, and has a probability of filling the vacancy of q(θ), in which case the value of the firm turns to

4

We interpret as productivity what MP calls prices. For them, differences between jobs are related to changes in prices at individual level.

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be the expectation over all possible values that the idiosyncratic productivity can take for new jobs.

rV = −c + q (θ )[ J (ε u ) − V ]

(1)

When producing, the firm earns each period the output net of wages, and it faces a change in firm level productivity with probability λ and an exogenous separation risk with probability δ.

rJ (ε ) = p + σε − w(ε ) + λ ∫ [max{J (ω ), V } − J (ε )]dF (ω ) − δJ (ε )

(2)

On the other hand, workers, when unemployed, receive a flow of b that accounts for a mixture of unemployment benefits, household production and leisure. As searching for jobs, they have a probability f(θ) of finding a job. As in the firm case, the new job value is the expectation of all possible values of the job given the distribution of new jobs idiosyncratic productivity.

[

rU = b + f (θ ) W (ε u ) − U

]

(3)

When working, the worker perceives a wage and, as the firm, two sources of change in the surplus: the idiosyncratic shock and the exogenous separation shock. rW (ε ) = w(ε ) + λ ∫ [max{W (ω ), U } − W (ε )]dF (ω ) − δW (ε )

(4)

The surplus of a job is the value that the job generates for the firm and for the worker. Moreover, wages are set in a Nash bargaining process, that is to say, a proportion β of surplus is acquired by workers and the remaining by firms. W (ε ) − U J (ε ) − V S (ε ) = J (ε ) − V + W (ε ) − U = (5) = 1− β β While there is no on-the-job search, workers do have the opportunity of switching jobs with an intermediate step in unemployment, and this opportunity increases the proportion of output that they finally get. In fact, it is a direct implication of the model that unemployment value enters negatively in the determination of the surplus appraisal. In other words, unemployment is the outside option for workers, who would prefer to search for other jobs if the current one is not worthy enough. This is explicitly used in equation (4), where if W(ε)
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and would partially compensate the increase of the surplus. As will be analyzed later on, the magnitude of this compensating effect is crucial for the assessment of the cyclicality of the labor market variables. The decisions in this model are the vacancy decision (when to open a vacancy) and the separation decision (when to split the match). More formally: An equilibrium is a pair {εd,θ} that satisfies: c a) free entry condition: V = 0 ⇒ q (θ ) = (1 − β ) S (ε u ) b)

reservation productivity condition: S (ε d ) = 0

where surplus is: (r + λ + δ ) S (ε ) = p + σε − b −

εu

βcθ + λ ∫ S (ω )dF (ω ). 1− β ε d

Surplus formula comes from replacing value functions in the definition of surplus, applying the Nash bargaining solution and considering that, when free entry condition is satisfied, the third and the forth term of the right hand side stands for the unemployment value. From the inspection of the surplus formula, it is easy to see that a higher θ reduces surplus for every value of ε by increasing outside option value. This implies that higher market tightness generates higher critical value of the idiosyncratic shock (εd), from the reservation productivity condition. This positive relationship is called in the original paper the Job Destruction curve. On the other hand, the first condition implies that the cost of opening a vacancy must be equal to its benefits. In other case, more vacancies would be opened, increasing congestion in the market, the spell of the vacancy and reducing its value. This equation generates a downward sloping relationship, the Job Creation curve, between market tightness and the critical value of idiosyncratic shock: when εd is higher, surplus value lowers due to the shorter duration of job relationships and employment creation incentives are lessened. The intersection of these curves determines the equilibrium in the model (see Figure 2). Then, changes in surplus determination would imply a direct impact in the equilibrium. For example, an expansion in the aggregate component of productivity, p, raises the value of the surplus for a given level of idiosyncratic productivity, what will lead to a lower critical value for separation for a given market tightness (job destruction curve shifts to the left). Thus, an increase in p generates a higher θ, and also a lower εd.

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An increase in b is an exogenous increase in outside option value, what generates a lower surplus, with opposite consequences than the increase in p (job destruction shifts right). Lower vacancies and higher separation tend to increase unemployment, in line with the intuition of the consequences of more generous unemployment benefits. The impact of bargaining power of the workers, β, is twofold. It splits a given surplus value, but it also changes the level of the surplus: a higher β implies a higher value of the new jobs for the workers, increasing unemployment value and decreasing the surplus. Given that firms face both a lower surplus and a lower proportion of it, incentives to open vacancies are greatly reduced. Thus, a higher β generates a shift to the right of the job destruction curve and a shift down of the job creation curve. Changes in σ have more subtle consequences. Higher dispersion in idiosyncratic shocks makes that for negative values of ε productivity lowers while for positive values of ε productivity increases. In particular, a higher σ generates higher productivity values for new entrants, strengthening the incentives to open vacancies. On the other hand, if εd is negative it is clear that εd should go up, increasing separation; but also if εd is positive, the outside option value has improved and εd would tend to raise for this reason. (See Mortensen and Pissarides, 1994, for a formal derivation). All these effects are discussed extensively in the original paper. Let us consider now the effect of the introduction of the exogenous separation rate. Exogenous job destruction generates lower employment spells, which makes the value of the surplus to be lower. From this point of view, δ is equivalent to an increase in discount rate. This analogy can be furthered applied to analyze the labor hording effect. To see this, we can write more explicitly the condition (b), to be: p + σε d = b + f (θ ) β S (ε u ) − λ

σ r + λ +δ

εu

∫ [1 − F (ω )]dω

εd

where the last equality holds after integration by parts (see Mortensen and Pissarides, 1994, for details). The last term of the right hand side is related to labor hoarding: it reduces the level of εd because a low value idiosyncratic shock is not permanent, and an employer would accept temporary losses in anticipation of future improvements in job level productivity. Nevertheless, the higher the interest rate and the exogenous separation rate, the milder the labor hording effect: if the job has a positive probability of being destroyed, then it is less worthy to maintain it in bad times.

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Needless to say that the effect of the exogenous separation rate is indeed different than a mere higher interest rate: it increases the separation rate not only indirectly but also in a direct and evident way. In a context of aggregate shocks, the previous conditions would determine the steady state, and the equilibrium definition should be changed: For a given state variables vector x, the equilibrium is a pair {θ(x),εd(x)} such that: c a) free entry condition: V = 0 ⇒ q (θ ( x) ) = (6) (1 − β ) S (ε u , x) and productivity reservation condition: S (ε d ( x), x) = 0

b)

(7)

are satisfied, where surplus is: εu

(r + λ + δ ) S (ε , x) = p + σε − b − f (θ ( x) )β S (ε u , x ) + λ ∫ S (ω , x)dF (ω ) + Ε x [max{S (ε , x' ),0}− S (ε , x)]. (8) εd

The last term in this formula, in which Ex is the expectation conditional on x, stands for the fact that state variables for the next period are random, that this is expected by the agents and that their future value is conditioned on the present one. The application of the model with shocks is mostly done with a Markov Chain approximation with two levels of p, following MP original paper. For concreteness, I will present the simplified formulas for this case. The reservation productivity equation or job destruction curve is similar to the one previously analyzed. In the presence of aggregate shocks, when there is a probability µ that p changes its value, this formula becomes: εd

εu

* d

εd

βc * λσ p + σε = b + θ − [1 − F (ω )]dω − λσ ∫ 1− β r + λ +δ + µ ε r + λ +δ *

* d

∫ [1 − F (ω )]dω

(9)

and p + σε d = b + *

*

βc λσ θ− 1− β r + λ +δ

εd*

εu

∫ε [1 − F (ω )]dω − µS (ε ). *

d

(10)

d

where p , θ and are the values when productivity is high and where S*(εd) can be replaced by its definition. The interpretation and intuition of this formula is similar to the previous one. However, it is important to stress that under anticipated cyclical shocks, the gap between reservation productivity values is less than implied by steady state analysis. The same occurs for market tightness. In effect, the job creation condition can be rewritten as: c 1 q (θ ) = (11) 1 − β S (ε u )

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q (θ * ) =

1 c * 1 − β S (ε u )

(12)

where S(εu) is derived from

(r + λ + δ + µ )S * (ε u ) = p* + σε u − b − βθ

*

c

1− β

(r + λ + δ + µ )S (ε u ) = p + σε u − b −

+

εd

εu

* d

εd

λσ [1 − F (ω )]dω + λσ r + λ + δ + µ ε∫ r + λ +δ

βθc λσ + 1− β r + λ + δ

εu

∫ [1 − F (ω )]dω + µ S (ε ), u

∫ [1 − F (ω )]dω + µ S (ε ). ε *

u

d

These formulas are more general than the ones presented by MP in their original paper. In particular, the formulas of MP assumes that S(εd)=0, but when ε support is constrained, then the surplus in the critical value can be positive. 3.1 The model with the original assumptions The MP paper does some particular assumptions that has generated most of the work in this model: a) the distribution of the idiosyncratic shock is an uniform with support [-1 , 1 ], b) the idiosyncratic productivity of created jobs is always in the upper end of the support (εu=1). The justification for the last is that new jobs tend to be oppened with higher productivity than the mean level. In that case, then, S (ε u ) = S (1) and q (θ ) = kθ −η .

3.2 Heterogeneity in newly created jobs productivity In the MP original context, a new job is created whenever there is a meeting between a worker and a firm, and its productivity is at εu=1. We consider another approximation: after a meeting, a draw of ε is realized and conditional of its value, the job is created or not. That is, idiosyncratic productivity of new jobs is not known ex-ante, but after the meeting between workers and firms. Additionally, the value of ε is assumed to be known by both parts without additional costs and before creating the job, say, by a simple test. This changes the rather arbitrary assumption of MP model, and introduces some variance in the newly created jobs productivity (and increase the variance of wages).

Furthermore, while the MP assumption can be justified for a new firm (for which the productivity tend to be higher than the mean), it is not necessarily the case if we consider new jobs.

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This comprises a more direct relationship between the critical value of the idiosyncratic productivity (εd) and the market tightness (θ) in equilibrium. In fact, the probability of filling a vacancy is now q (θ ) = k0 [1 − F (ε d )]θ −η

(equivalently,

k = k0 [1 − F (ε d )] ). This generates an amplification of the response of tightness to aggregate conditions: when aggregate productivity drops, εd goes up, reducing the probability that meetings become jobs, what limits incentives to open vacancies. Nevertheless, this specification somewhat dubious when εd is high. In such a case, meetings are multiplied but jobs are not created. The surplus of the newly created jobs should be interpreted now as a conditional expectation, but also as the surplus of the conditional mean of εu, that is

S (ε u ) = S (Ε[ε ε > ε d ]) (see a formal derivation in the Appendix A).

3.3 The model with multiple shocks In the original MP paper, shocks to aggregate productivity (p) was the source of cyclical fluctuation. Nevertheless, they illustrate the possibility of different changes in parameters, for instance in σ. Following this vein, we will consider an extension for which mean productivity changes not only because of changes in p but also in σ. This last can be thought as reallocative shocks, which are conceptually interpreted as situations in which productivity between sectors changes and some of them become more productive while others have to face an important adjustment process. In this model, we can think of an increase in the volatility of idiosyncratic shocks as causing these same outcomes: those jobs with lower productivity (negative values of ε) would be affected, while the remaining would increase their productivity. In terms of the model, εd would go up, increasing job destruction, while productivity at εu would be higher involving higher θ and job creation. There are other admissible shocks to the model, but this work will concentrate on the shocks that affect productivity, both shocks to aggregate and reallocative components.

4 Calibration Calibration of the matching model is an open issue (Hornstein, Krusell and Violante, 2005). In fact, the evaluation of different calibrations approaches is what motivates this work. Alternative calibration strategies will be presented in this section in the

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context of the model without idiosyncratic shocks. Then, the calibration of λ and σ will be discussed, along with the problems of the model to generate the observed dispersion of wages. Finally, the calibration of the model with two shocks will be commented.

4.1 The model without idiosyncratic shocks The parameters that have to be calibrated in the context of the model without idiosyncratic shocks are: the interest rate r, the probability of arrival of a shock to aggregate productivity µ, the exogenous separation rate δ, the elasticity of matching function η, the cost of a vacancy c, the scale parameter of the matching function k, the utility flow of unemployed b and the bargaining power of workers β. Before considering the differences between specifications, I will present the less problematic parameters. The period of time that we will consider for the simulations is a month, but quarterly averages will be used to present the outcomes. Thus, all variables are expressed to be consistent with this time period. For example, if we consider an annual discount factor of 0.953 as in Shimer, then r=0.004. The probability of arrival of a shock to aggregate productivity (µ) is set to 0.023 to generate the autocorrelation of productivity similar to the data. In this context, the exogenous separation rate (δ) is set in the value of the montly separation rate of 3.4%, as in Shimer and HM. The calibration of the elasticity of matching function, η, is based on time series estimations. In general, this value can be set from a range of 0.2 (Hall, 2005) to 0.7 (Petrangolo and Pissarides, 2001). This parameter will change in each alternative calibration. Finally, θ is normalized to 0.7 following Pissarides (2007) (in Shimer is set to 1 and in HM to 1.8), given that, as discussed in Shimer (2005), the value of θ in steady state is irrelevant for the solution of the model. The remaining parameters k and c, are calibrated in equilibrium for finding rate to be 0.45 and θ to reach the normalized value. 4.1.1 Traditional calibration The traditional calibration is based on approximating the instantaneous value of unemployment state to unemployment benefit (and a plus for a value of leisure), and the bargaining power of the workers to be the efficient one, that is, β=η using Hosios (1990) result.

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Shimer justifies the choice of b=0.4 as being the upper bound of a distribution of unemployment benefit replacement ratio (mean wages are 0.993 in his framework). As said, β=η based on the efficiency result, while η is estimated as the elasticity of the matching function for US detrended data and set to 0.72. 4.1.2 Alternative calibration Alternative calibration builds on other set of observations. It uses wage equation and determines β and b to generate the observed elasticity of mean wages with respect to productivity (which they estimate in 0.47) and a profit rate of 3% (according to Basu and Fernald, 1997). HM identify profit rate to Π =

p−w . Finally, η is set to 0.5 by p

HM, as a midpoint of the commonly used values. 4.1.3 Discussion At this point, we can stress the fact that both calibrations are rather arbitrary, and by no means the ultimate calibration practice. For concreteness, traditional calibration procedure does not justify further the choice of b, which does not include formally any value of leisure or home production, and setting β=η for efficiency reasons is not beyond controversy, given that there is no reason why the model should converge to this value. The alternative calibration seeks for a source of lower flexibility of wages that can increase the elasticity of finding rate. For this reason, they set the value of beta to match the elasticity of mean wages of 0.47. But the relevant elasticity for the model is the one of newly created jobs, as stressed by Pissarides (2007). This elasticity is close to unity, as measured by Haefke, Sonntag and van Rens (2007). Given that in the model the mean and newly created jobs elasticities are very similar, HM calibration is implying a counterfactual issue in the process of calibration. Thus, the target choice seems to be misleading. Furthermore, the profit rate of 3% (that is, that 97% of the output is assigned to wages) can also be discussed: it is not clear which ratio should be used given that there is no capital in the model. In any case, we will analyze the outcomes of these calibrations and look for a more definite conclusion about the convenience of each calibration when idiosyncratic shocks are included.

4.2 The model with idiosyncratic shocks

14

The MP model introduces two additional parameters, λ and σ. In the original paper, these parameters are set to match mean and volatility of job creation of the manufacturing sector (Pissarides, 2007, and Cole and Rogerson, 1999, use similar targets). Nevertheless, as our aim is to test volatility of both finding and separation rates, we follow an alternative route. The rate of arrival of the idiosyncratic productivity shock (λ) is set to match the persistence of firm level productivity. According to Baily, Hulten and Campbell (1992), the productivity within a firm has an autocorrelation of 30% after 5 years, which implies a monthly autocorrelation of 98%. This allows us to set λ=0.02.5 The most natural target to match σ, the parameter of dispersion of idiosyncratic component of productivity, is wage or productivity dispersion. According to Heathcote, Storesletten and Violante (2003) using PSID from 1967 to 1995, the standard deviation of log wages is around 0.3. But not all the dispersion in the data is relevant, given that in this model we have homogenous firms and workers. Only the unexplained proportion of the variance of wages, once controlling for firm and workers observables must be targeted. An alternative would be to consider the proportion of the volatility of wages which is driven by differences in productivity between firms, once some characteristics of the firm are considered.6 While the empirical counterpart of the idiosyncratic component of productivity of a match is rather unclear, any discussion in this context is useless: the model generates a volatility of 0.02, which represents 0.5% of the variance of wages. In other words, by no means this specification can match the target of volatility of wages, whatever the definition that one uses. Hornstein, Krusell and Violante (2007) also found that the matching model can only generate an unsuitable low dispersion of wages. According to their measure of dispersion, which is the ratio of the mean wage to the minimum wage, the model is 20 times off the target. This is an important 5

It is worth noting that wages are also very persistent. Heathcote, Storesletten and Violante (2003) using PSID estimated an autocorrelation of 0.95 for the persistent component of wage residuals, after controlling for experience and worker fixed effects, and net of transitory shocks. This number is in line with the ones used to calibrate, given that the measure is for yearly wages. 6 We considered some approximations for the value of wage dispersion that the model should match. A first approach is to target the dispersion of wages due to differences in productivity for unobservable components of worker and firms. According to Abowd, Karamarz, Margolis and Troske (1999) the firm effects account for 30-40% of the total variance of wages, while Faggio, Salvanes and Van Reenen (2007) assess that the between firm and within industry component is around 30% of the productivity variance. Thus, following this definition, the total variance accounted by the model should be around 10% of the variance of wages. The other approach would be to consider the unexplained proportion of the variance of a wage regression in which characteristics of workers and firms are taken into account. The R2 of the wages regressions are around 62% (Abowd, Karamarz, Margolis and Troske, 2000), what leaves the remaining proportion to be explained by the model.

15

drawback, which we can consider as a basic puzzle, and solving it can be considered a priority. We will come back to this issue later on. We tried alternative targets for σ, one of which is the elasticity of newly created jobs wages with respect to productivity. But targeting this value to unity implies that mean wages are also elastic, what makes the HM calibration impossible. As Pissarides (2007) pointed out, the model cannot generate different elasticities for new job wages and for mean wages. To reconcile the gap between observed and simulated dispersion, we considered an extension of the model, namely, normal independent draws for newly created jobs. The intuition is that, in this context, newly created jobs wages replicate the dispersion of existing jobs. The model doubles the dispersion of wages, but it is still far off from its objective. We conclude that a deeper modification to the model must be done for it to account for observed wage dispersion, and we simulated the model for a wide range of values of σ. Fortunately, the model converges to the same outcomes when σ is high. Finally, the exogenous separation rate (δ) is calibrated in equilibrium to target the monthly mean separation rate. This rate includes the sum of three job destruction processes. The first source of destruction is the change in aggregate productivity, which causes that some jobs are render unfeasible. This implies a jump in separation when aggregate productivity changes. The remaining jobs face two idiosyncratic shocks: a first one with a probability of arrival of δ that takes productivity to a negative value, such that the job must be destroyed; a second one, the arrival of an idiosyncratic shock as described in MP for which there is some positive probability that the productivity becomes insufficient to maintain the job, in which case it is closed. The value of δ is set in equilibrium so that the sum of the three components adds to the observed monthly separation rate of 3.4%. In the context of idiosyncratic shocks, traditional and alternative calibrations base on the same principles as before. For concreteness, we shall clarify that b value in traditional calibration will be 42% of the newly created jobs wages7, and for HM calibration we consider the same targets for mean values of the variables: elasticity of mean wages is set to 0.47 and mean profit rate is 3%.

7

The justification for this choice is that it generates the same incentives than in the Shimer model, given that what is relevant for the unemployed is the wage of the new job that he can acquire. In any case, results are unchanged if we consider mean wages instead of new wages

16

Furthermore, the aggregate productivity shock is simulated as in the previous section, and amplitude of the shocks is modified for each value of σ to match the

volatility of filtered productivity. On the whole, for a given value of σ, the calibration procedure is as follows: a) the Shimer calibration is solved in steady state for the parameters b, c and k, while δ is calibrated in equilibrium (out of steady state); b) the HM calibration is solved in steady state for values of b, c and k, while β and δ are found in equilibrium (out of steady state).

4.3 The model with multiple shocks In the extension of the model with multiple shocks to productivity, calibration procedure is the same, in the sense that we borrow from parameter values from the previous section, as an approximation of equilibrium calibration (steady state parameters are the same by definition, and out of steady state parameter targets are also roughly matched). The shocks that will feed the model are Markov Chain approximations of AR(1) processes for p and σ. Details will be provided in the simulation section.

5 Simulations and results As commented above, σ is not calibrated; we calibrate and simulate the model for each value of σ. The shocks that will be analyzed in the simulations are only driven by changes in p, the aggregate component of productivity, which takes two values and a probability µ of changing state (equations (9)-(12) will be used). Only in section 5.3 reallocative shocks (changes in σ) will be introduced and explained in more detail. The simulation involves timing assumptions. In this case, we follow MP closely. The timing is as follows: 1. Aggregate productivity shocks arrive, and θ and εd jump. 2. New wages are calculated, given the new level of productivity and the distribution of jobs. Mean wages are: wt = β ( pt + σ ε t ) + βcθ t + (1 − β )b where ε is the mean level of idiosyncratic productivity given the distribution of jobs.

17

3. Before the arrival of subsequent shocks, the jobs with idiosyncratic productivity below εd are computed as separations. With two levels of productivity, this implies that if there is a mass of workers between εd* and εd (denoted as ndh) and a bad shock arrives, all these jobs are considered separations. Formally, D1t = ntdh {pt = pl } , where D1t stands for the amount of jobs destroyed at impact given a change in productivity, pl is the low value of productivity and {.} is an indicator function. In other words, the jump in separation rate is computed first. 4. The exogenous separation probability is realized. That is, a constant proportion of jobs is destroyed in addition to the previous ones. Formally,

[

]

D2t = δ N t − ntdh {pt = pl } . 5. The idiosyncratic shock is realized. The jobs with low productivity are destroyed. That is to say, the last component of separation is computed:

[

]

D3t = λF (ε dt )(1 − δ ) N t − ntdh {pt = pl } . 6. Unemployed searchers find a match according to the matching function. Job creation is, then, Ct = f (θ t )(1 − N t ) .

7. Separation and finding rates are computed: finding rate in the period is

ft = f (θ t ) = components

st =

dh t

n

Ct ; separation rate is computed as the sum of its three 1 − Nt in

relation

to

the

total

amount

of

workers:

{pt = pl }+ δ [N t − nt {pt = pl }] − λF (ε dt )(1 − δ )[N t − nt {pt = pl }] . Nt

8. These flows determine the stock of workers in the next period. Formally, the total employed are: Nt +1 = N t − Dt + Ct , where Dt = D1t + D2t + D3t , while the unemployed are ut +1 = 1 − N t +1 . The distribution of workers are driven by the

following formulas: ⎧nu t +1 = ft (1 − N t ) + (1 − δ )(1 − λ )ntu ⎪ dl dl dh , ⎨nt +1 = (1 − δ )(1 − λ )nt + (1 − δ )λ [1 − F (ε dl )] N t − nt {p = pl} ⎪ dh dh dh ⎩nt +1 = (1 − δ )(1 − λ )nt + (1 − δ )λ [F (ε dl ) − F (ε dh )] N t − nt {p = pl}

[

[

]

]

where each nt is a mass of workers (nu is the mass of workers at εu, ndl is the mass of matches with idiosyncratic productivity between εu and εd, and finally ndh is the mass of matches with productivity between εd and εd*), and the distribution within ndl or ndh is a uniform.

18

All the simulations are done in monthly terms. Then quarterly averages are calculated. Output tables are computed filtering the simulated series of separation, finding, unemployment, vacancies and productivity. Unemployment variance decomposition is computed (as in the data) without filtering the variables.

5.1 Simulations of the model without idiosyncratic shocks

With no idiosyncratic shocks, we are back in the Pissarides (1985) framework, with exogenous separation rate, the original context of Shimer and HM work. I will present the results for this calibration for both Shimer and HM calibrations and comment about the intuition about this result. Shimer’s calibration implies a low volatility of unemployment, around twenty times lower than the observed one (see Table 3). This is based on a low volatility of market tightness, of about the one in productivity. A low elasticity of matching function translates this into a really low volatility of finding rate. On the whole, labor market variables show a low elasticity to shocks on productivity. The cyclicality is like the observed one but with a much higher correlation coefficient. The model generates the right correlation between unemployment and vacancies. HM calibration is much more successful in matching the business cycle facts (see Table 4). Volatility in unemployment and market tightness in this case is near the one observed. Finding rate volatility is much higher than the data, what can be expected, given that in this model all the volatility of unemployment is due to finding rate only. As well as in the Shimer case, negative correlation of vacancies and unemployment is as observed, but the correlation of labor market variables to productivity is also too strong. Thus, the results of both specifications are clearly different. The literature has stressed that the reason for the lack of responsiveness of labor market variables was the effect of wage increments in the booms. While increases in productivity generate additional incentives for creating jobs, the subsequent increase in wages undermines the proffits, and compensates the job creation. For that reason, Shimer (2005) stress the need of some wage rigidity in the model for increasing unemployment volatility. HM alternative calibration is also benefiting from low responsive wages: their calibration target is an elasticity of wages of 0.47. Additionally, setting proffits to 3% implies that an increase in productivity along with the low elasticity of wages generates a very high proportional increase in job value for the firm, what makes job creation boosted. 19

5.2 Simulations of the model with idiosyncratic shocks Given that σ is not calibrated, the results for idiosyncratic shocks implies simulating the model for different levels of σ. The chosen range of this parameter goes from zero to 500, but the outputs rapidly converge for σ>2.8 We present the results for the model with the original assumptions (uniform distribution and εu in the highest level of support) and for the model with independent draws for newly crated jobs. 5.2.1 Original assumptions Figure 3 shows the standard deviation of filtered finding and separation rates for the two specifications. For reference, observed values of these statistics are 0.118 and 0.072, respectively. For all levels of σ, HM calibration generates more volatility in finding rate, which is always above 0.07, while for Shimer calibration is always below 0.01. In other words, the difference is as striking as without idiosyncratic shocks. It is worth noting that for low values of σ, a given change in p implies a higher proportional shock to the surplus of the newly created jobs when σ is low. In words, if dispersion of idiosyncratic shocks is higher, the productivity of newly created jobs will be determined in a grater proportion by its idiosyncratic component, rather than by the aggregate productivity. This explains why finding rate volatility tends to go down with higher σ. Separation rate volatility is clearly non monotonic as a function of σ. It is zero for low levels of σ, because in this case restrictions on εd are active (εd both for high and low levels of productivity is -1). In other terms, when dispersion of firm level productivity is low (and support is constrained), the difference between newly created jobs and the least productive job is insignificant. Thus no job is destroyed for endogenous reasons, but only because of exogenous separation rate. For σ around 0.05, the critical value is not constrained. When this occurs, separation rate volatility jumps up, because at low levels of σ, elasticity of εd is highest. This is because a change in aggregate productivity implies a higher

8

It is worth saying that the case with σ=0 implies also λ=0, given that it is the case of no idiosyncratic shocks; thus, a discrete jump is observed for low values of σ.

20

proportional change in job productivity when σ is low than when σ is high, what necessarily implies a higher change in critical value of idiosyncratic shock.9 For low levels of σ but with unconstrained values of εd, volatility in separation rate is driven by both jumps effects and changes in separation probabilities between levels of productivity. When σ is higher, εd tends to converge to a level near one, both for εd and εd*. In this case, separation rates are not so different for different values of aggregate productivity, but the jump effect is still present, given that the mass of workers between the two critical points is high. Figure 4 shows a diagram in which volatility of separation rate is plotted in the x-axis, and volatility of finding is drawn in the y-axis. For HM calibration, low σ implies highest finding volatility and zero separation volatility; for the highest levels of σ, volatility converges to a low value of separation volatility (around 0.01) and a volatility of finding of around 0.1. The shape of this plot is determined by the ups and downs of the separation rate volatility. In the Shimer case, though hidden, the shape is the same as in the HM case, and convergence is to a point with very low volatility for both variables. For reference, observed volatility values are drawn as a single point in this diagram. It can be seen that only HM calibration is near the facts for low levels of σ (between 0.08 and 0.1). As can be seen from Figure 5, correlation between finding and separation is around the ones observed for low levels of σ, just when separation rate begins to be endongenous. Finally, Figure 6 shows the unemployment volatility. As previous graphs suggested, the highest volatility of unemployment is for low σ (in fact for the lowest σ), while for high levels it converges to a low value. Given that the model should provide volatility for both variables, finding and separation, as in the data, the model is closest to observed values for σ between 0.05 and 0.1. For these values, both finding and separation are volatile and unemployment has the highest standard deviation. In the case of HM unemployment volatility is around 0.16 while for Shimer calibration it is well far off the target, around 0.04. To get more details about the properties of both calibrations, we computed the same statistics than for the stylized facts table (see Table 5). We fixed the σ value to a low level of 0.1. Shimer calibration output shows a low volatility of unemployment, finding and separation rates (ten, twelve and three times lower than the facts, respectively), while correlation between unemployment and vacancies seems very 9

In the case of HM it can be seen that there are two points in which separation rate volatility tends to increase. This is because in the first case only εd is above -1 (but close to it), while in the second they are both above this level.

21

weak (-0.25 while in the facts it is around -0.9). Correlation between separation and finding is around the observed one. For HM calibrations things are much more in line with the facts: volatilities of unemployment, finding and separation rates are around the ones in Table 1, with differences lower than 25% relative to the observed ones. Negative correlation between vacancies and unemployment is much stronger (-0.6), while correlation between separation and finding is almost the one observed (-0.52). Nevertheless, the correlation between labor productivity and labor market variables is far away from observed ones: it is around 1, while in the data it is around 0.4. The only exception is the separation rate, for which this correlation is -0.54. The variance decomposition of unemployment shows that for the HM case, inflow and outflow effects are as in the data, while correlation effect is near the one observed. In the Shimer calibration, the only significant source of volatility in unemployment is the inflow effect, that is, volatility of separation rate (particularly, jumps in separation). This is the first, and comforting, contribution of these simulations: HM calibration not only generates higher volatility in unemployment in the Pissarides (1985) context, but also in the broader MP model, and it does so by matching both finding and separation volatility, and generating a decomposition of unemployment variance as in the data. Up to this point, it is clear that HM calibration is far better for matching the labor market in business cycle, for any specification of idiosyncratic productivity. But, why is this so? Particularly, higher finding elasticity implies that in bad aggregate conditions, workers would find it very difficult to get a job. If this is the case, outside option value would be lower and surplus fall would be compensated by this effect. Thus, if we assume that elastic finding rates generates higher differences between unemployment values for high or low productivity levels, then elastic finding rates should be encompassed by inelastic separation rates. In other words, if workers find it unlikely to find a job when labor market conditions are bad, they will accept a lower wage because of their lower outside option; in this case, then, some jobs with low productivity will be acceptable for workers and firms. But this is not the case in HM calibration. The low bargaining power of workers (β) and high utility flow when unemployed (b) of this calibration implies that unemployed are not affected by their probability of finding a job. In fact, the difference between being unemployed or being employed in this specification is minimal: workers are almost indifferent between both states. Then, unemployment value does not fluctuate. Its volatility is one fifth of the volatility of productivity; this

22

is an inexistent standard deviation of 0.4%. This means that welfare of unemployed is not affected by recessions. While the previous is a questionable implication of the model, it is not counterfactual in the sense of the observable facts that we have discussed before. Nevertheless, this calibration does have counterfactuals in other dimensions. First of all, the elasticity of wages of newly created jobs is too low, around a half of the observed one as measured in Haefke, Sonntag and van Rens (2007). Moreover, the elasticity of the present value of wages of the newly created jobs is around 0.3, a very low value for any elasticity of this kind. Additionally, HM calibration implies that the workers share of output would be around 97%. It is difficult to appraise this number within a model without capital, but in any case this is a dubious proportion. Finally, as stressed by Costain and Reiter (2006), the response of unemployment to changes in unemployment insurance benefits is too strong in HM calibration, much more than observed (HM discuss this point as a problem of changes in taxes). To sum up, while HM specification of the model is quite successful in generating volatility of labor market variables in the business cycle, it is directly counterfactual or dubious in other dimensions, such as wage level and volatility, welfare of unemployed and unemployment responses to policy changes. 5.2.2 Heterogeneity in newly created jobs productivity In a context where productivity of newly crated jobs is randomly assigned, the results are somewhat different. The observation that HM calibration generates more volatility than traditional calibration is still true for this specification, but in this case volatility is excessive. The first thing that is worth noting is that this specification does generate a higher dispersion of wages, but in any case far off the observed values (dispersion is 1% of total observed one). For σ close to zero, outcomes are similar to the ones in previous section: volatility of separation is null, while finding standard deviation is around 0.2 for HM and 0.02 for Shimer (see Figure 7). The differences arise when σ tend to grow. In this specification, a higher σ produces an increase in volatility of finding rate for both calibrations, particularly notable for HM case. The reason for this is that when σ increases, the reservation productivity value (εd) tends to grow, what implies that, at a given meeting rate, the probability of creating jobs are reduced. Thus, productivity impact on εd also affect

23

finding rate in this context. In other words, the incentives to create jobs in good times rather than in bad times are twofold: surplus is higher and probability that meetings are effectively converted in jobs is also stronger. This is why finding rate gap between high and low states have widened so much, being almost null for low productivity levels, and almost one in good times. The shape of separation rate volatility is also similar to the previous case, but for other reasons. While in this specification there is no restriction to the value of εd, for σ near zero the value of εd is negative and the cummulative probability of this value is almost zero for both levels of p. When σ augment, εd tends to be close to zero, the mode of the distribution, what implies a huge difference in probability between the lower and the higher value of p. Moreover, the mass of workers that are displaced when productivity drops is maximum when εd is close to zero (in fact, when εd*<0 and εd >0). For higher levels of σ, the probability of separation is high for both productivity levels, and the difference between them is not significant. On the whole, HM calibration generates too much unemployment volatility with this specification, while Shimer calibration is still far off observed values (see Figure 8). The higher values of σ not only generates excessive volatility but also corresponds to a rather unnatural environment in which offers are made with an exacerbated frequency but only a few (say 5%) of them are converted into jobs. Thus, lower values of σ are more interesting. Tables 9 and 10 presents the results for σ=0.035 for both calibrations. In the Shimer case, results do not present much improvement. On the other hand, HM case shows a higher volatility than observed for all variables: standard deviation of finding rate is almost 0.5, while for separation rate this is over 0.6. Unemployment volatility more than doubles the observed one. Additionally, correlations in this specification is counterfactual: is very weak for the relationships between u and v, and between s and f, while is too strong for all variables with respect to p, with the exception of s.

5.3 Simulations of the model with multiple shocks Shimer (2005) and Hagedorn and Manovskii (2005) had identifyed business cycles with changes in p, aggregate component of productivity. This is a natural interpretation in their context. In general, the literature has built on this consideration, even in the more general context of MP model.

24

Nevertheless, in the MP model additional shocks to mean productivity can be considered: changes in σ, as showed before, generate an increase in mean productivity. This is achieved by a particular mechanism: causing separations in the low productive jobs and stimulating job creation in the highest productive jobs. We intend to highlight the importance of widening the literature view of identifying a cycle to changes in p only, and considering that a reallocative process driven by an amplification in productivity dispersion can coexist within a cycle, explaining part of the observed mean labor productivity movements. The objective of this subsection is to measure the relative importance of this shock, and then to analyze how it modifies the outcomes of both calibrations. With the purpose of measuring the importance of this kind of shocks we estimate a VAR using three variables (finding rate, separation rate and mean productivity) and then recovering structural innovations. We identify structural shocks imposing sign restrictions to impulse response functions, following Uhlig (1999) and Canova and Nicolo (2002) method. The assumption is that a shock to p generates on impact an increase in finding rate and in productivity, as well as a reduction in separation rate, while shocks to σ lead to an increase in the three variables on impact. In other words, reallocation shock is identified using the fact that it increases both separation and finding rates. (Details on variables and on the method used are provided in Annex B and C respectively). The Figure 9 shows the results in terms of impulse response functions. These functions present reasonable shapes: all shocks have no long term impact; separation rate reponds more in the short run and the effects exhaust after 5 periods; finding rate is equally responsive but with more long lasting effects (even after 5 periods, the responses are significant); shocks to aggregate component is significant but concentrated in the short run, while for reallocation component it losses significancy sooner. The responses to unidentifyed shock are insignificant for the three variables. It is important to stress that these results are generated by imposing a restriction only in the first period of the impulse response function. After identifying these shocks, it is relevant to assess the variance decomposition of the variables. From Figure 10 we can conclude that the reallocative shocks explain a 15 of the separation rate volatility, almost 40 of the finding rate volatility and a half of productivity changes. Thus, it seems relevant to include these shocks in the model. To this end we construct simulated processes for p and σ that base on the shocks recovered from the SVAR. In fact, they are AR1 processes that match the historical

25

decomposition of observed productivity. These simulations are described in more detail in the Annex C. The model in the context of these two shocks should be solved using equations (6)-(8).10 One way of doing so is applying the contraction theorem and using value function iterations. As an approximation to this solution, the steady state equilibrium values for the different levels of p and σ are used. It is important to note that steady state analysis generates a wither response of equilibrium variables because there is no anticifation of future change in state variables (results are as if shocks were permanent). The model is not solved for every possible value of state variables p and σ, but only for the combination of five values of p and five values of σ. These levels are established using a Markov Chain approximation of the AR1 processes. Then, twenty five points for the functions εd(p,σ) and θ(p,σ) are known. The remaining ones are interpolated using these points. The simulation runs as follows: beginning in a particular period (1:1963) in which separation and finding rates are similar to the ones matched (3.4% and 45%), we use the estimated values of p and σ of the following period to solve for εd(p,σ) and θ(p,σ). That is, εdt and θt are the interpolations using the known points of the functions. From these equilibrium values, separation and finding rates are computed, and unemployment and employment distribution for the next period are calculated. We follow this procedure until the last observation that we have, which is 1:2007. As a result, we can compare the evolution of the simulation with the observed one. In particular, we follow filtered mean productivity, separation rate, finding rate and filtered unemployment. Results are shown in Figures 11 for Shimer calibration and Figures 12 for HM case. It is worth noting that productivity measures depend on equilibrium values of εd and θ as much as on p and σ, what results in different productivities for each calibration. Thus, it is an important verification that mean productivity of the model have the same properties (in terms of autocorrelation and variance) as the observed one. For both cases, this is satisfied, what can be seen both in the graphs and in the Tables 11 and 12. Furthermore, generally speaking, observed and simulated productivity values are very similar for most of the periods.11 10

We use for this section the assumption of original Mortensen-Pissarides model, namely, uniform distribution with support [-1,1] and homogeneous productivity of newly created jobs in the highest value of the support. 11 For HM calibration, processes of p and σ imply that 40% of the variance of aggregate productivity is explained by changes in σ and the remaining by changes in p, in line with variance decomposition of SVAR.

26

Traditional calibration is clearly misleading when compared to actual evolution: finding rate presents a low responsiveness, while separation rate volatility is extreme, driven by huge jumps. On the whole, unemployment evolution is clearly different from observed (see Figures 11). On the other hand, HM calibration is very much in line with observed values (see Figures 12). Particularly, finding rate, while more volatile, has an evolution comparable to the data, as well as unemployment that is very close to the actual evolution. Separation rate, however, is too jumpy. In any case, Table 12 shows that HM calibration generates too much volatility in this specification, but with some important similarities to the stylized facts: volatility of finding is much higher than for separation, correlation between unemployment and vacancies is -0.9, and between finding and separation is -0.5, while cyclicality is in the correct direction (while still too strong). To conclude, the model with two shocks has the advantage of generating an additional source of volatility to the variables that proves relevant in empirical approximation and promising in simulations. In particular, given that HM calibration generates too much volatility, this specification of the model can provide a framework for improving the performance of other possible calibrations, particularly those in line with other facts than volatility, namely wage elasticity and unemployment value. In other words, given that HM calibration is counterfactual with some observations, it is important to analyze alternative strategies. The model with multiple shocks can provide a more realistic framework for testing them and reconcile the tension between matching labor market business cycle or wage elasticity and responses to unemployment benefits.

6 Concluding remarks This work analyzes alternative calibration strategies applied to the matching model. These approaches have been previously compared in the context of exogenous separation rate. A main contribution of this paper is to evaluate them in a broader and more flexible context, alternatively with endogenous separation rate, heterogeneity in newly created jobs and multiple shocks to productivity. The first conclusion is that allowing for endogenous separation within the standard model does not change the previous results of the literature: traditional calibration is unable to generate cyclical volatility, while alternative one produces the observed cyclicality of labor market variables. The finding is that this last

27

specification is succesful at creating the observed volatility and correlation of both finding and separation rates. Moreover, both variables account in the simulations for the same proportion of unemployment variance as observed. This can be seen as a comforting result for the use of matching model in macroeconomic analysis. Nevertheless, as pointed out above, this calibrations fails at generating reasonable outcomes in other dimensions, such as wage elasticity of new jobs or unemployment responses to policy shocks, what encourages further research. To this end, this paper offers some promising options. It presents two specifications of the model in which alternative calibration procedure proved to generate an excess volatility in labor market outcomes. When considering heterogeneity to idiosyncratic productivity of newly created jobs and when introducing two shocks to productivity, Hagedorn and Manovskii’s calibration results in a volatility that almost doubles observed one. Particularly interesting is this last specification. The literature has traditionally considered cyclical shocks as changes in aggregate component of productivity. Nevertheless, shocks to reallocative component of productivity can also be subject to shocks. If both sources of volatility coexist, then there are motives for an increase in both separation and finding within the business cycle, enhancing the reallocation of resources from low productivity to high productivity firms. The speed of this process is an additional source of cyclicality. This framework could be a more powerful tool if capital is added to the model. In this case, some of the calibration targets (such as profit rate or wage share) would become clearer and additional volatility in the labor market, namely, the price of capital, could be addressed, allowing for novel calibration testing. Moreover, business cycle analysis can benefit by studying additional processes that are present within expansions: job creation and job destruction are both incentivated, and expansions entails adjustment costs up to some extent. The posible line of research suggested by the results of this paper should be qualified, though, in the sense that prior work should be done to make it applicable. Firstly, the wage dispersion puzzle should be solved. As has been seen in the calibration section, the model is unable to generate the dispersion in wages or in productivity that is observed in micro level data. To solve this issue is a priority, given that any analysis that depends on a rather arbitrary level of dispersion of idiosyncratic shocks is only a conjecture. In other words, if the model is able to replicate the wage dispersion of the data, this would fix the uncalibrated parameter (σ) and would allow to test the calibrations on more conclusive grounds. Secondly, a

28

deeper and broader review of evidence of reallocative component should be done, to justify on solid base the relevance of the two shock approach. In addition, it would be very important to test wether the different specifications of the model can reproduce business cycle facts of other economies rather than the US. This would entail a new approach to the calibration analysis, given the differences between countries in important quantitative aspects, such as the generosity of unemployment benefits or the finding and separation mean rates, among others.

29

Annex A Derivation of the value of a newly created job in the model with an independent draw of idiosyncratic productivity for new matches The derivation is for the case of two values of p, and expectations are taken with respect to ε conditional on that surplus is positive. S (ε u ) = S (ε u ) =

⎡ βθc λσ 1 Ε ⎢ p + σε − b − + r + λ + δ + µ ⎣⎢ 1− β r + λ + δ

∞

∫ [1 − F (ω )]dω + µ S (ε )ε > ε ε *

u

d

⎛ 1 ⎜ p + σΕ[ε ε > ε ] − b − βθc + λσ d r + λ + δ + µ ⎜⎝ 1− β r + λ + δ

S (ε u ) = S (Ε[ε ε > ε d ])

d

⎤ ⎥ ⎦⎥

∫ [1 − F (ω )]dω + µΕε [S (ε )ε > ε ε ∞

*

d

d

S (ε u ) = S (ε u ) defining ε u ≡ Ε[ε ε > ε d ]

Annex B In this annex I describe the data briefly. Labor market variables were constructed by Robert Shimer. For additional details, please see Shimer (2007) and his webpage http://robert.shimer.googlepages.com/flows. The data from June 1967 and December 1975 were tabulated by Joe Ritter and made available by Hoyt Bleakley. The finding and separation series are quarterly averages of monthly transition probabilities, corrected for time-aggregation.

The job finding probability is

constructed from unemployment and short term unemployment. The separation probability is constructed from employment, unemployment, and the job finding probability. Employment, unemployment, and short term unemployment data are constructed by the BLS from the CPS and seasonally adjusted. Short term unemployment data are adjusted for the 1994 CPS redesign. Seasonally adjusted unemployment is constructed by the BLS from the Current Population Survey (CPS). Average labor productivity p is seasonally adjusted real average output per person in the non-farm business sector, constructed by the Bureau of Labor Statistics (BLS) from the National Income and Product Accounts and the Current Employment Statistics.

30

]⎞⎟⎟ ⎠

Alternative labor productivity measure y is seasonally adjusted real average output per hour worked in the non-farm business sector, constructed by the Bureau of Labor Statistics (BLS).

Annex C The identification scheme is based on analyzing the signs of Impulse Response functions, in particular, the contemporaneous effect of a structural shock to some particular variables. This approach, following Uhlig (1999) and Canova and Nicolo (2002), somewhat less demanding in terms of assumptions than other identification procedures: only some elements of the matrix (the ones that are relevant for the exercise) are restricted. Concretely, this method identifies the aggregate and reallocative shocks in which we are interested in, but the third shock is not identified. A three lag VAR is estimated using filtered separation and finding rates, as well as productivity defined as output per hour worked (we tested the robustness of results to other definition of variables, namely, transition probabilities and productivity as output per person). A Bayesian approach was preferred, using a diffuse prior.

BVAR allows for a conceptually simple draw of error bands of

statistics of interest such as impulse responses, that is, of impulse responses that are admitted by the sign identification procedure. With this method we estimate B0, B1, B2 and B3 matrixes of coefficients, ut errors and Σ (variance-covariance matrix of errors). Following, A(L) matrixes of orthogonal Wold (MA) representation of the VAR are found. There are three fundamental innovations, mutually independent and with normalized variance. To identify these shocks a matrix Ao such that AoAo’=Σ is required. With this matrix, a vector or fundamental innovations can be recovered knowing that vt = Ao ut. This identification can be done using Cholesky factorization of Σ; from structural relationships between innovations and one-step ahead prediction errors; imposing long term restrictions to responses to shocks (as in Blanchard and Quah, 1989); or by imposing some sign restrictions to the impulse response function. We implemented this last approach. The sign identification procedure is as follows. From a random matrix (drawn from a normal distribution), a R=QZ decomposition is applied, where Q is an orthogonal matrix. Additionally, one particular orthogonal MA representation (A(L)) is picked from the set of all the candidates estimated by the Bayesian VAR

31

procedure. Then, responses are computed as Â(L)=A(L)Q. This procedure is repeated and responses are selected only if they satisfy the sign restrictions. In this exercise, the sign restrictions were the following: f and y respond positively to v1 (aggregate shocks) while s responds negatively; responses to v2 (reallocative shocks) are all positive. Given the randomness of R, Q and A(L) matrixes, the exercise reflects the uncertainty about coefficients of VAR and about the possible decompositions. A natural way of presenting the results is to show the distribution (median, and 15 and 85 percentiles) of all the valid collected Impulse Response Functions. On the other hand, following Uhlig, the variance decomposition is estimated from the median impulse responses. Also, historical decomposition of productivity was calculated, to estimate the evolution of productivity due to v1 or v2, which we can denote as y1 and y2. I then estimated an autorregresive process for both components. Autocorrelation of both series were very similar and around 0.86. Then I reproduced the evolution of productivity simulating processes for p and σ from v1 or v2 innovations. To this end, I used the following formulas:

pˆ t = p +

p ~ pt p +σε

σˆ t = σ +

σ ~ σt , p +σε

where ~ variables are the processes estimated through structural innovations using: ~ pt = ρ~ pt −1 + v1t , σ~ = ρσ~ + v t

t −1

2t

where p t and σ t are steady state values (1 and 0.08 respectively), and where pˆ t and σˆ t are the estimated values of p and σ used for simulations.

32

References Abowd, J.M., F. Kramarz, D.N. Margolis, and K.R. Troske (1998): "The Relative Importance of Employer and Employeee Effects on Compensation: a Comparison of France and the United States." Papers 1998-10, Paris I - Laboratoire de Microeconomie Appliquee. Baily, M., Hulten, C. and Campbell, D. (1992): “Productivity Dynamics in Manufacturing Plants.” Brookings Papers on Economic Activity: Microeconomics, 187– 249. Basu, S., and J. G. Fernald (1997): “Returns to Scale in U.S. Production: Estimates and Implications.” Journal of Political Economy, 105(2), pp. 249–283. Canova, Fabio and Gianni de Nicolo (2002): "Monetary disturbances matter for business fluctuations in the G-7." Journal of Monetary Economics, Elsevier, vol. 49(6), pp 1131-1159, September. Caballero, Ricardo J. and Hammour, Mohamad L. (1996): “On the Timing and Efficiency of Creative Destruction," The Quarterly Journal of Economics, MIT Press, vol. 111(3), August. Costain, James S. and Michael Reiter (2006): “Business Cycles, Unemployment Insurance, and the Calibration of the Matching Models”, mimeo, Universitat Pompeu Fabra. Davis, Steven J., John C. Haltiwanger and Scott Schuh (1998): Job Creation and Destruction. Cambridge, MIT Press, 1998. Elsby, Michael W. L. (2008): “Marginal Jobs, Heterogeneous Firms and Unemployment Flows”, NBER Working Paper, 13777, Feb. 2008. Faggio, Giulia, K.G. Salvanes, and J.M. Van Reenen (2007): "The Evolution of Inequality in Productivity and Wages: Panel Data Evidence". NBER Working Paper No. W13351. Haefke, C., M. Sonntag and T. van Rens (2007): “Wage Rigidity and Job Creation”, mimeo. Hagedorn, Marcus and Iourii Manovskii (2005): “The Cyclical Behavior of Equilibrium Unemployment and Vacancies Revisited”, mimeo, University of Pennsylvania. Hall, Robert (2005): “Employment Fluctuations with Equilibrium Wage Stickiness”, American Economic Review, 95 (1), March 2005. Hornstein,

Andreas,

Per

Krusell

and

Giovanni

L.

Violante

(2005):

“Unemployment and Vacancy Fluctuations in the Matching Model: Inspecting the Mechanism”, Federal Reserve Bank of Richmond Economic Quarterly, 91 (3).

33

Hornstein, Andreas, Per Krusell and Giovanni L. Violante (2007): "Frictional Wage Dispersion in Search Models: A Quantitative Assessment," NBER Working Papers 13674, National Bureau of Economic Research. Hosios, Arthur J. (1990): “On the Efficiency of Matching and Related Models of Search and Unemployment.” Review of Economic Studies, 1990, 57(2), pp. 279–98. Michelacci, C. and R. Balakrishnan (2001): “Unemployment Dynamics across OECD Countries.” European Economic Review, 45, 135-165 Mortensen, Dale T. and Christopher A. Pissarides (1994): “Job Creation and Job Destruction in the Theory of Unemployment”, Review of Economic Studies, 61 (3). Mortensen, Dale T. and Eva Nagypál (2005): “More on Unemployment and Vacancy Fluctuations”, Review of Economic Dynamics, 10. Petrongolo, Barbara and Pissarides, Christopher A. (2001): “Looking into the Black Box: A Survey of the Matching Function.” Journal of Economic Literature, 2001, 39(2), pp. 390–431. Pissarides, Christopher A. (1985): “Short-Run Equilibrium Dynamics of Unemployment, Vacancies, and Real Wages”, American Economic Review, 75. Pissarides, Christopher A. (2007): “The Unemployment Volatility Puzzle: Is Wage Stickiness the Answer?” Mimeo, London School of Economics, 2007. Shimer, Robert (2004): “The Consequences of Rigid Wages in Search Models”, Journal of the European Economic Association (Papers and Procedings), 2. Shimer, Robert (2005): “The Cyclical Behavior of Equilibrium Unemployment and Vacancy”, The American Economic Review, 95(1). Shimer, Robert (2005b): “Reassessing the Ins and Outs of Unemployment.” University of Chicago, mimeo. Uhlig, Harald (1999):

“What are the effects of monetary policy on output?

Results from an agnostic identification procedure.” CEPR Discussion Papers 2137.

34

Tables Table 1: Summary Statistics, quarterly US data, 1951 to 2003 Standard Deviation Quarterly Autocorrelation u v Correlations v/u f p

u

v

v/u

f

p

s

0.190 0.936 1.000

0.202 0.940 -0.894 1.000

0.382 0.941 -0.971 0.975 1.000

0.118 0.908 -0.949 0.897 0.948 1.000

0.020 0.878 -0.408 0.364 0.369 0.396 1.000

0.072 0.694 0.658

-0.527 -0.443

Shimer (2005a). u: (unemployment rate) seasonally adjusted constructed by BLS from CPS. v: (vacancies) seasonally adjusted help-wanted advertising index constructed by Conference Board. f: (job finding rate) constructed by Shimer from employment, unemployment rate and duration of BLS source. p: (average labor productivity seasonally adjusted real average output per person in the non-farm business sector constructed by the BLS. s: separation rate constructed by Shimer (2005b). Variables reported in logs as deviations from an HP trend with smoothing parameter 10e+5.

Table 2: Unemployment decomposition Unemployment volatility Inflows effect Outflows effect Covariance effect Approximation error

0.049 37.2% 38.4% 14.5% 9.9%

Decomposition using steady state unemployment value and approximating variance decomposition of the effect of s and f using taylor approximation

Table 3: Simulated model with Shimer type calibration. σ = 0. Standard Deviation Quarterly Autocorrelation u v Correlations v/u f p

u

v

v/u

f

p

s

0.008 0.894 1.000

0.024 0.807 -0.900 1.000

0.032 0.853 -0.943 0.994 1.000

0.009 0.853 -0.943 0.994 1.000 1.000

0.020 0.853 -0.943 0.994 1.000 1.000 1.000

0.000 . . . . . .

Average quarterly rates of monthly simulations. Variables reported in logs as deviations from an HP trend with smoothing parameter 10e+5

Table 4: Simulated model with HM type calibration. σ = 0. Standard Deviation Quarterly Autocorrelation u v Correlations v/u f p

u

v

v/u

f

p

s

0.153 0.895 1.000

0.204 0.722 -0.821 1.000

0.341 0.851 -0.940 0.967 1.000

0.171 0.852 -0.941 0.966 1.000 1.000

0.020 0.853 -0.941 0.964 0.999 1.000 1.000

0.000 . . . . . .

Average quarterly rates of monthly simulations. Variables reported in logs as deviations from an HP trend with smoothing parameter 10e+5

35

Table 5: Simulated model with Shimer type calibration. σ = 0.1 Standard Deviation Quarterly Autocorrelation u v Correlations v/u f p

u

v

v/u

f

p

s

0.023 0.586 1.000

0.018 0.650 -0.249 1.000

0.032 0.853 -0.848 0.725 1.000

0.009 0.853 -0.848 0.724 1.000 1.000

0.020 0.851 -0.848 0.724 1.000 1.000 1.000

0.024 0.080 0.607 -0.071 -0.469 -0.470 -0.473

Average quarterly rates of monthly simulations. Variables reported in logs as deviations from an HP trend with smoothing parameter 10e+5

Table 6: Simulated model with HM type calibration. σ = 0.1 Standard Deviation Quarterly Autocorrelation u v Correlations v/u f p

u

v

v/u

f

p

s

0.148 0.770 1.000

0.086 0.656 -0.622 1.000

0.212 0.852 -0.947 0.841 1.000

0.106 0.852 -0.948 0.839 1.000 1.000

0.020 0.843 -0.944 0.843 0.998 0.999 1.000

0.089 0.121 0.537 -0.362 -0.511 -0.517 -0.536

Average quarterly rates of monthly simulations. Variables reported in logs as deviations from an HP trend with smoothing parameter 10e+5

Table 7: Unemployment decomposition Shimer type calibration. σ = 0.1. Unemployment volatility Inflows effect Outflows effect Covariance effect Approximation error

100.0% 71.1% 10.5% 12.7% 5.7%

Decomposition using steady state unemployment value and approximating variance decomposition of the effect of s and f using taylor approximation

Table 8: Unemployment decomposition HM type calibration. σ = 0.1. Unemployment volatility Inflows effect Outflows effect Covariance effect Approximation error

100.0% 36.9% 39.0% 17.6% 6.5%

Decomposition using steady state unemployment value and approximating variance decomposition of the effect of s and f using taylor approximation

36

Table 9: Simulated model. Shimer type calibration. Normal dist. σ = 0.035 Standard Deviation Quarterly Autocorrelation u v Correlations v/u f p

u

v

v/u

f

p

s

0.011 0.894 1.000

0.027 0.783 -0.850 1.000

0.037 0.853 -0.921 0.988 1.000

0.011 0.853 -0.921 0.988 1.000 1.000

0.023 0.853 -0.921 0.988 1.000 1.000 1.000

0.001 0.091 0.533 -0.197 -0.302 -0.301 -0.301

Average quarterly rates of monthly simulations. Variables reported in logs as deviations from an HP trend with smoothing parameter 10e+5

Table 10:

Simulated model with HM type calibration. σ = 0.035 Standard Deviation Quarterly Autocorrelation u v Correlations v/u f p

u

v

v/u

f

p

s

0.492 0.853 1.000

0.332 0.393 -0.151 1.000

0.634 0.860 -0.851 0.647 1.000

0.487 0.861 -0.852 0.645 1.000 1.000

0.020 0.861 -0.852 0.639 0.997 0.998 1.000

0.690 -0.039 0.217 0.146 -0.087 -0.081 -0.056

Average quarterly rates of monthly simulations. Variables reported in logs as deviations from an HP trend with smoothing parameter 10e+5

Table 11 Simulated model with Shimer type calibration. Shocks to p and σ. Standard Deviation Quarterly Autocorrelation u v Correlations v/u f p

u

v

v/u

f

p

s

0.046 0.849 1.000

0.178 0.826 -0.800 1.000

0.063 0.849 -1.000 0.800 1.000

0.018 0.849 -1.000 0.800 1.000 1.000

0.020 0.850 -0.920 0.660 0.920 0.920 1.000

0.201 0.335 -0.800 1.000 0.800 0.800 0.660

Average quarterly rates of monthly simulations. Variables reported in logs as deviations from an HP trend with smoothing parameter 10e+5. Steady state p=1; σ=0.08.

Table 12 Simulated model with HM type calibration. Shocks to p and σ. Standard Deviation Quarterly Autocorrelation u v Correlations v/u f p

u

v

v/u

f

p

s

0.311 0.888 1.000

0.360 0.792 -0.924 1.000

0.613 0.886 -1.000 0.924 1.000

0.308 0.887 -1.000 0.924 1.000 1.000

0.020 0.922 -0.863 0.772 0.864 0.864 1.000

0.129 0.564 0.488 -0.398 -0.478 -0.481 -0.322

Average quarterly rates of monthly simulations. Variables reported in logs as deviations from an HP trend with smoothing parameter 10e+5. Steady state p=1; σ=0.08.

37

Figures Figure 1: Unemployment: observed vs. steady state 14% 12% 10% 8% 6% 4% 2% u obs

u steady state

Note: Observed unemployment: Seasonally adjusted unemployement serie. BLS. Steady state unemployment constructed from finding and separation rates: u=s/(f+s). Rates are from Shimer (2005). Details in the text.

Figure 2: Determination of θ and εd θ JD

JC

εd

38

2005

2002

1999

1996

1993

1990

1987

1984

1981

1978

1975

1972

1969

1966

1963

1960

1957

1954

1951

1948

0%

Figure 3: Volatility of rates for diff. values of sigma 0.25 SH-Finding HM-Finding SH-Separ. HM-Separ.

Std of filtered rates

0.2

0.15

0.1

0.05

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

sigma

Figure 4 Volatility of rates for sigma = 0 to 500 0.25 SH HM Facts

0.2

std(f)

0.15

0.1

0.05

0

0

0.02

0.04

0.06 std(s)

0.08

0.1

0.12

Figure 5 Correl of rates for diff. values of sigma 0 SH HM Facts

-0.1

cov(f,s)

-0.2

-0.3

-0.4

-0.5

-0.6

-0.7

0

0.1

0.2

0.3

0.4 0.5 sigma

39

0.6

0.7

0.8

0.9

Figure 6: Volatility of unemployment for diff. values of sigma 0.2 0.18

SH HM Facts

0.16

Std of unemployment

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0

0.1

0.2

0.3

0.4 0.5 sigma

0.6

0.7

0.8

0.9

Figure 7: Volatility of rates for diff. values of sigma

1.2

Std of filtered rates

1 SH-Finding HM-Finding SH-Separ. HM-Separ.

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1 sigma

1.2

1.4

1.6

1.8

Figure 8: Volatility of unemployment for diff. values of sigma 0.8 SH HM Facts

0.7

Std of unemployment

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1 sigma

40

1.2

1.4

1.6

1.8

Figure 9: Impulse response functions -3 IRF: Aggregate to SeparationIRF: Aggregate to FindingIRF: xAggregate to Productivity 10 6 0 0.04 4

-0.02

0.02

2

-0.04

0

-2

5

10

15

0

20

5

10

15

20

5

10

15

20

IRF: Reallocation to Separation IRF: Reallocation to Finding IRF: Reallocation to Productivity x 10 0.03 0.03 4 0.02 0.02 0.01 2 0.01 0 0 0 -0.01 5 10 15 20 5 10 15 20 5 10 15 20 -3

IRF: Other to Separation

IRF: Other to Finding

-3 IRF: Other to Productivity x 10 5

0.02

0.01 0

0

-0.01

-0.02 5

10

15

0 -5

20

5

10

15

20

5

10

15

Figure 10: Variance decomposition Variance decomposition Separation 0.9

Variance decomposition Finding 0.8

Agg shock Reall shock Other shock

0.8

Agg shock Reall shock Other shock

0.7

0.7 0.6 0.5 variance %

0.5 0.4

0.2

0.2

0.1

0.1 0

0.4 0.3

0.3

0

2

4

6

8

10 time

12

14

16

18

0

20

0

2

4

6

8

Variance decomposition Productivity 0.7 Agg shock Reall shock Other shock

0.6

0.5

variance %

variance %

0.6

0.4

0.3

0.2

0.1

0

0

2

4

6

8

10 time

41

12

14

16

18

20

10 time

12

14

16

18

20

20

Figure 11: Time series of observed and simulated values. Two shocks to productivity. Shimer calibration. Productivity

Filtered Unemployment

0.05

0.6

SH Simulated Observed

0.04

0.4

0.02

0.3

0.01

0.2

0

0.1

-0.01

0

-0.02

-0.1

-0.03

-0.2

-0.04 -0.05

SH Simulated Observed

0.5

0.03

-0.3 0

20

40

60

80

100

120

140

160

180

time Finding Rate 0.65

0

20

40

60

80 100 time Separation Rate

120

140

160

180

0.09

SH Simulated Observed

0.6

-0.4

SH Simulated Observed

0.08

0.55

0.07

0.5

0.06

0.45

0.05

0.4

0.04

0.35

0.03

0

20

40

60

80

100

120

140

160

180

time

0.02

0

20

40

60

80

100 time

42

120

140

160

180

Figure 12: Time series of observed and simulated values. Two shocks to productivity. Shimer calibration. Productivity

Filtered Unemployment

0.05

0.5 HM Simulated Observed

0.04

HM Simulated Observed

0.4

0.03

0.3

0.02

0.2

0.01 0.1 0 0 -0.01 -0.1

-0.02

-0.2

-0.03

-0.3

-0.04 -0.05

0

20

40

60

80

100

120

140

160

180

-0.4

0

20

40

60

80

100

time

time

Finding Rate

Separation Rate

120

140

160

180

0.05

0.8

HM Simulated Observed

HM Simulated Observed

0.7

0.045

0.6 0.04 0.5 0.035 0.4 0.03 0.3 0.025

0.2

0.1

0

20

40

60

80

100

120

140

160

180

0.02

0

20

40

60

80

100 time

time

43

120

140

160

180