Bellman Equations and Sequential Auctions

Value of Unemployment

20 Let V_u,t denote the worker’s lifetime value of being unemployed at time t, and V_e,t(w, y) the value of being employed in a match of quality y with a contract that specifies a current wage w. Then:

21 where the expectation is taken over realizations of aggregate TFP z_t+1, new match quality y and associated wage contract w_t+1.

22 Because firms have all the bargaining power, they extract all the rents from unemployed workers by making them indifferent between working or remaining unemployed. Therefore, the value of unemployment is time-invariant:

Value of Employment

23 In this section, we focus on the simple case where b is small enough to ensure that no match ever breaks up endogenously, all separations are exogenous and occur with probability δ; we will later relax this assumption. The value of employment V_e,t(w, y) at the beginning of period t equals the wage plus the discounted expected continuation value, which comprises three terms. With probability δ the match separates and the worker joins unemployment and receives a value βV_u; otherwise, with probability 1 – sϕ(θ_t) he receives no outside offer and continues with value βV_e,t+1(w_t+1, y) according to the wage w_t+1 in the current contract, while with probability sϕ(θ_t) he receives an outside offer from a firm, and an auction takes place. We now analyze the auction.

24 Let {w^*_τ(y)}^∞_τ=t+1 denote the continuation (state-contingent) contract which delivers to the worker the maximum value V_t+1(y) = V_e,t+1 (w^*_t+1(y), y) that the firm is willing to promise to deliver to the worker at the beginning of time t + 1, after the TFP realization z_t+1 is observed but before production takes place in that period. By promising this continuation contract, the firm breaks even, namely its profits equal the value of continued search (of the vacancy). The auction between an employer and a poacher takes place at the end of period t, after exogenous separations have unfolded but before observing the TFP realization z_t+1. Therefore, is the firm y’s willingness to pay for the worker in the time-t auction, the maximum time-t expected value that the firm is willing to promise to the worker from t + 1 on. Now consider a firm currently employing a worker at match level y and promising any continuation contract {w_τ(y)}^∞_τ=t+1, and confronting an outside offer made to its employee by a firm with match y’. The second-price auction has one of three mutually exclusive outcomes:

1. , in which case the incumbent employer needs to do nothing to retain the worker, the offer is irrelevant, and the promised value at time t, in case of no separation, is the one delivered with no outside offer and no renegotiation, ;
2. , in which case the incumbent employer retains the worker by renegotiating the offer, for a raise to ;
3. , in which case the worker is poached with an offer worth .

25 If the current continuation contract specifies {w^*_τ(y)}^∞_τ=t+1 the firm is already breaking even, thus will not match any outside offers. In that case, either no threatening outside offers arrive, the promised continuation value is V_t+1(y) ex post, and in expectation, or the worker is poached, at value , which is the second price in the auction. Either way, the continuation value of the worker who holds a contract {w^*_τ}^∞_τ=t+1 and survives an exogenous separation is , which is the maximum the firm can deliver. Under the maintained assumption that the value of leisure b is low enough never to trigger an endogenous separation (i.e., ) by backward induction, the maximum value that the firm is willing to deliver to the worker at the beginning of time t must solve:

26 In words, the continuation value is already maximized, and the maximum current wage that the firm can pay at time t without making a loss is the entire flow output z_t y. Substituting forward, using the L.I.E. and Transversality, and replacing V_u from Equation 1,

27 Evaluating at t + 1, taking expectations at time t, and using again the L.I.E., the firm’s willingness to pay in the auction equals

28 where

29 is a known function of z_t, and z_t only because of the Markov property. So the willingness to pay is affine in y, and the firm with the higher y wins the auction. We draw the main conclusion of this subsection: the equilibrium, if it exists, must be Rank Preserving, and the direction of EE reallocation is efficient, always from less to more productive matches.

Value of a Vacancy

30 By the time a firm and a worker who have met on the search market must decide whether or not to consummate the match, they know the quality of the potential match, y’. The firm’s willingness to pay is the maximum profits that the firm can make because, when giving this to the worker, the firm breaks even. The worker’s outside option is known, too: it is βV_u for an unemployed worker, and the willingness to pay of the current employer for a worker employed in an existing match y. The firm earns the difference between its own willingness to pay and the worker’s outside option. For an unemployed job applicant this is, if positive,

31 and for a job applicant initially employed in a match of quality y it is

32 again, if positive. Let L_t(y) denote the measure of employment at matches with productivity in [y, y], a non-normalized c.d.f. of employment on the job ladder (such that ), with domain . The value of a vacancy V_v,t then solves the Bellman equation:

33 where the expectation is taken over z_t+1| z_t and

34 is the probability that a randomly drawn job applicant in period t is unemployed.

Employment (Distribution) Dynamics

35 We now describe the dynamics of L_t(y). Due to the rank-preserving (RP) property of equilibrium, this measure increases with hires from unemployment that draw a match quality below y and decrease with separations to unemployment and with quits to better matches: [5]

36 Differentiating both sides and letting γ(y) = Γʹ(y), the measure of employment ℓ_t(y) = L^ʹ_t(y) at match y follows

37 which nets out flows in and out of the match both from/into unemployment and from/into other matches. The law of motion of the unemployment rate is familiar:

Free Entry and Equilibrium

38 The free entry condition is V_v,t = 0 for all t. As mentioned, for now we restrict attention to the simple case where b is small enough to ensure that no match ever breaks up endogenously, namely . Then Equation 3 writes as

39 where the last term:

40 is the expected return to the firm from a contact with an employed job applicant, a key object that we will discuss in detail. On the LHS of Equation 8 are vacancy costs times the expected duration of a vacancy, on the RHS the average of the expected discounted profits from hiring an unemployed and an employed job applicant, weighted by the respective shares of the two types of job applicants in the pool of job searchers. Unemployed hires are homogeneous, while employed hires are distributed according to the measure dL_t(y)/(1 – u_t) of match quality y in their current jobs, which determines their bargaining power in wage negotiations. Given the predetermined (at time t) distribution of employment L_t(·), and the resulting unemployment rate , this equation uniquely pins down equilibrium tightness θ_t.

41 Given initial conditions z₀ ∈ ℝ₊ and , a Rational Expectations Equilibrium is a stochastic process for job market tightness θ_t solving the free entry condition 8, given: P(u) in Equation 4, and the dynamics of L_t(·) in Equation 5.

Match Surplus and the Labor Wedge

42 The expected return from an unemployed hire is the (expected, capitalized) difference between Marginal Product of Labor (MPL) and value of leisure. Due to risk neutrality, the value of leisure also equals the Marginal Rate of Substitution (MRS) of consumption for leisure. In the Business Cycle accounting literature (Chari, Kehoe and McGrattan [2007]), the ratio between the MRS and the MPL is the “labor wedge.” Measured in the data through the lens of a neoclassical growth model with balanced growth preferences, this ratio is procyclical (the implicit “tax” rate on labor, equal to one minus this ratio, is countercyclical) and plays a key role in amplifying business cycle fluctuations. Estimated New-Keynesian models (Smets and Wouters [2007]) define the “wage markup” as the ratio between the real wage and the MRS, and find that changes in this mark-up are key to explain inflation and output dynamics. Lacking a mechanism to generate endogenous changes in the wage mark-up, they attribute them to shocks, that they estimate to be procyclical. Galí [2011] calls for a theory of an endogenous wage mark-up. In the business cycle search literature, which typically abstracts from OJS, this wedge corresponds to the firm’s surplus, which compensates for hiring costs. This surplus is procyclical, as long as TFP is persistent, making the returns to hiring unemployed workers, hence labor market tightness, procyclical. In the absence of OJS (setting s = 0) our model reduces to the stochastic Nash Bargaining search model of Shimer [2005] where firms have all the bargaining power. In that class of models, Hagedorn and Manovskii [2008] argue that this surplus is small, relative to mean output, providing a rationale for the observed high volatility of unemployment. We now show that OJS substantially changes the terms of this debate.

The Mismatch Wedge

43 Our model contains an additional, novel wedge, and related transmission mechanism of aggregate shocks to job creation, which is not present in any strands of the literature, and which greatly reduces the importance of the size of the unemployed surplus. When posting vacancies, firms also mind the expected return Ω_t from an employed hire, defined in Equation 9. This is independent of the MRS, hence of the surplus, and depends entirely on the distribution of employment L_t(·), which is a slow-moving aggregate state variable. Ω_t can be viewed as an index of misallocation relative to the frictionless limit, where workers sample jobs at unbounded rate and thus are always in the best possible match. We call Ω_t the “Mismatch Wedge,” because it measures the expected marginal productivity effect of moving one worker between jobs.

44 This wedge introduces an additional, time-varying component to labor demand, with a complex cyclical pattern. To illustrate, it is useful to observe that this wedge is larger the worse the normalized employment distribution on the ladder L_t(y)/(1 – u_t) in a first-order stochastic dominance sense: just integrate Equation 9 by parts and observe that is decreasing

45 At a cyclical peak, workers have had time and opportunities to climb the ladder, so poaching employees from other firms is both difficult and expensive, and the returns to hiring employed workers are low. After a recession, as the unemployed regain employment, they restart from random rungs on the match quality ladder, which are worse than the employment distribution at the cyclical peak. Hence, early in a recovery many recent hires are easily “poachable.” Moving cheap unemployed job applicants into low-quality jobs makes them only slightly more expensive, and still quite profitable to hire. As time goes by, and unemployment declines, employment reallocation up the ladder through job-to-job quits accumulates, employed workers become more and more expensive to hire, ultimately putting pressure on wages, until we are back to a cyclical peak. Misallocation and the resulting mismatch wedge imply a procyclical wage mark-up, or countercyclical labor “tax,” but only as long as employment is still misallocated and vulnerable to poaching. [6]

46 In the US economy, the transition probability from job to job is fairly small, similar to the separation probability into unemployment, and both are an order of magnitude smaller than the transition probability from unemployment to employment. Therefore, movements in the employment distribution up the job ladder are slow. An important implication is that, in our model, job market tightness, thus the unemployment rate, have significant transitional dynamics. This stands in contrast to the canonical model with only unemployed job search, where job market tightness is a jump variable, the unemployment rate converges very quickly to its new steady state, and both track the current state of TFP essentially one for one.

47 Before concluding, we show another important property of the mismatch wedge Ω_t. Since L_t(y) is strictly increasing in at least part of its domain, then and the mismatch wedge is lower than the unconditional average match quality:

48 where in the second equality we used, again, integration by parts. Intuitively, employed workers must be compensated for giving up their current job. Furthermore,

49 where the last inequality follows from the assumption that b is small enough to make all matches acceptable, namely for all y. We conclude that an employed hire is always less profitable in expectation than an unemployed hire, a property that will play an important role in shaping aggregate equilibrium dynamics.

Calibration

64 To calibrate model parameters we proceed in two steps. First, we choose the values of all parameters except the aggregate TFP process so that the steady state equilibrium matches a few key statistics for the US economy. Second, given this set of parameter values, we calibrate the TFP process so that the stochastic simulation of the model generates an empirically accurate persistence of innovations to and unconditional volatility of Average Labor Productivity (ALP). This second step calls for some discussion.

65 Job ladder reallocation makes the Solow residual in the model endogenous and different than the underlying exogenous TFP process. In principle, we could estimate an “empirically plausible” TFP process outside of the model and verify what the model then implies for ALP dynamics. We prefer to match the latter by construction, because our emphasis is on job finding probability and unemployment, not on ALP, so we are interested in the model’s predictions for the former. In fact, we are mostly interested in the model’s potential to amplify and to propagate aggregate shocks. Thus, in principle, we could just study the volatility and persistence of job finding probability relative to those of the TFP driving force. Because the model is non linear, however, outcomes are not invariant to the scale and persistence of the shocks, hence we discipline both using the closest empirical analogue to TFP, namely ALP.

66 We calibrate the model parameters and compute equilibrium at a monthly frequency. We start with preferences. We set the discount factor β = 0.95^1/12, and the value of leisure b = 0 so that no existing job is ever destroyed endogenously. We will later explore the case of endogenous job destruction. Importantly, once we allow for OJS the amplification properties of the model are much less dependent on the value of b. This value determines the returns to hire unemployed job applicants, while the returns from hiring employed job applicants depend on their current wages, which may have been renegotiated multiple times and thus no longer retain any memory of the opportunity cost b. So OJS allows to sidestep the debate on the opportunity cost of time that originated from Hagedorn and Manovskii [2008].

67 Next, we move to transition probabilities between employment and unemployment. Since for now all separations into unemployment are exogenous, we set δ equal to the average monthly transition probability from employment into unemployment (EU). Since all new matches are acceptable to the unemployed, we set the job contact probability in steady state equilibrium ϕ(θ) equal to the average monthly transition probability from unemployment into employment (UE). We estimate these probabilities from unemployment duration stocks (Shimer [2012]) in the monthly CPS, respectively the number of workers who report being unemployed for 5 weeks or less divided by employment a month before (EU), which averages 2.4%, and one minus the ratio between the number of workers who report being unemployed for more than 5 weeks and unemployment a month before (UE), which averages 41%. The implied steady-state unemployment rate is u = 0.024/(0.024 + 0.41) = 0.055. [9]

68 Given these parameter values, the efficiency of OJS s is identified by the pace of EE reallocation. To show how, we first find the stationary employment distribution ℓ(y) = Lʹ(y), which solves the following ordinary linear differential equation:

69 The solution can be found in closed form:

70 Now we can write the average EE transition probability in steady state equilibrium:

71 where in the first line we integrate by parts, in the second line we replace the expressions for the steady state unemployment rate and employment distribution from Equation 15, in the third line we change variable to quantiles . The last expression shows that the steady state EE probability does not directly depend on the specific match quality distribution Γ: when given the opportunity, workers move up the job ladder, no matter how steep it is, at a speed that depends only on s. [10] Given values of ϕ(θ) and δ hence u, we solve for the value of s that equates the last expression to the average monthly transition probability from job to job, which is about 2% in the monthly CPS after its 1994 survey re-design that introduced Dependent Coding.

72 Next, we calibrate meeting frictions. The free entry condition determines the vacancy filling probability. To map it into the the job finding probability from unemployment, which is the main object of interest in the stochastic simulation, we need to specify a functional form for the matching function. We cannot reject empirically the hypothesis of constant returns to scale in matching. Hence, we assume a Cobb-Douglas form ϕ(θ) = min〈Φθ^α, 1〉. We estimate the value of α by regressing, after HP-filtering with smoothing parameter 8.1 × 10⁶ (see Shimer [2012]), the log of the monthly job finding probability on the log of vacancies and on the log of total worker search effort u_t + (1 – δ)s(1 – u_t), where the values of δ and s were calibrated before. For the job finding probability we use the unemployment-duration based measure described above, which has standard deviation (in log deviations from HP trend) equal to 0.147 over the post-war period. [11] For vacancies we use the monthly Composite Help-Wanted Index of Barnichon [2010], updated by the author to cover 1955–2016, and very close to JOLTS vacancies since its 2001 inception. For u_t we use the civilian unemployment rate from the monthly CPS, 1948–2018. We filter each series separately using the longest time span available for each. We then run regressions with filtered monthly data on the time period where the series overlap, 1955–2016. [12] For our preferred calibration of s targeting a 2% EE transition probability and δ targeting a 2.4% EU separation probability, and the resulting search pool u_t + (1 – δ)s(1 – u_t), the estimated regression coefficient of the log UE job finding probability on log tightness is . [13] Crucially, as we will soon show, when we estimate a standard matching function ignoring OJS, i.e., when we identify the search pool with just unemployment u_t, we obtain a lower elasticity . The reason is simple: this standard method incorporates a term equal to α times the log of relative search effort by the unemployed vs the employed u_t/[u_t + (1 – δ)s(1 – u_t)] into the residual, which is then negatively correlated with ln(v_t /u_t), creating a downward bias in the estimated elasticity .

73 In the data, the job finding probability is always less than one at a monthly frequency. To guarantee that this is also true of the vacancy filling probability in the model, Φθ^α–1_t we choose the scale of vacancies so that job market tightness θ_t always exceeds the maximum job finding probability observed in the data, the model counterpart of which is Φθ^α_t. Indeed, min_t θ_t > max_tΦθ^α_t guarantees Φθ^α–1_t < 1 for all t.

74 Once both probabilities are less than one at all times, the vacancy filling probability can be written as . Using Equations 15, 4 and in the free entry condition 8, pins down the steady state equilibrium level of job market tightness θ.

75 Normalizing average steady-state TFP to z = 1 and average match quality to μ = 1, this free entry condition simplifies to:

76 To use this equation, and to solve for expected hiring cost (vacancy cost times vacancy duration) on the LHS, we only need to compute the value of the steady-state mismatch wedge Ω. Its expression

77 shows that, given the calibrated values of transition rates δ, s, ϕ(θ), Ω is uniquely determined by the sampling distribution Γ. We assume it to be Pareto with lower bound y and parameter λ > 1. Changing variable in Equation 18 to , and using the normalization we obtain

78 a direct mapping from values of λ to steady-state equilibrium mismatch wedge, thus hiring costs. As explained below, we do not specify a value of λ, but will experiment with different values.

79 Given ϕ(θ) = 0.41. the estimated value of , the value of Ω = Ω(λ) and of the parameters, we can use free entry condition 17 to calibrate the composite parameter . Without specifying the scale of vacancies, we cannot separately identify κ from Φ, nor do we need to. The free entry condition equates the returns to a contact with a job applicant, which depends on other parameter values and shock realizations, with . Therefore, knowing the value of ψ allows computation of the job finding probability ϕ(θ_t) at each point in time. On the other hand, as illustrated in the Appendix, the steady-state elasticity of the job finding probability ϕ(θ) to aggregate TFP is independent of the values of the parameters that make up the multiplicative constant ψ. Hence, we expect the stochastic properties (volatility, persistence, impulse response) of the job finding probability ϕ(θ_t) to be fairly insensitive to the value of ψ.

80 Therefore, the value of λ matters for shock amplification and propagation not through the scale of total returns to hiring and corresponding hiring costs ψ, but only through the relative returns to hire unemployed and employed workers Ω(λ) which receive time-varying weights in the free entry condition. Given the exogenous returns to contact a jobless worker, less dispersed match outcomes (higher value of λ) reduce the mismatch wedge and the importance of poaching. In the limit, as λ → ∞, all matches are the same, as in Robin [2011], the worker captures all rents from any outside offer, and the free entry condition looks exactly like in the DMP model without OJS; the only difference is the procyclical congestion that employed workers impose on the unemployed in the matching process and that dampens aggregate volatility.

81 As a final step, we are interested in assessing the size of hiring costs in steady state, a non-targeted moment, as a share of output. These costs equal κv = κθ[u + (1 – δ)s(1 – u)] per period. As mentioned, we cannot separately identify κ and the scale of vacancies v. But these hiring costs equal, by free entry, the total number of hires, namely the stock of searchers u + (1 – δ)s(1 – u) times the contact probability ϕ(θ), multiplied by the expected return from each contact with a randomly drawn job applicant. All of these objects are pinned down by the steady state calibration, therefore we can also compute hiring costs κv, without having to disentangle its two components.

82 This completes the steady state calibration. To introduce aggregate shocks and simulate the stochastic equilibrium of the model, we specify the TFP process as an AR(1) in logs:

83 In the spirit of the steady state normalization z = 1, we set average log TFP ζ = 0, and we calibrate the parameters ρ, σ so that the model’s Average Labor Productivity (ALP) , where , matches time series properties of ALP in the data. depends on the worker allocation on the job ladder, i.e., by match quality, a slow-moving state variable, hence it has interesting dynamics. To measure ALP in the data, following Shimer [2005] for comparison, we use quarterly Real Output Per Person in the Nonfarm Business Sector (BLS series PRS85006163) in 1947:I–2017:IV, HP filter its log with parameter 100,000 (which corresponds to 8.1 × 10⁶ monthly) and trim the first and last two years of data. We assume that the filtered series follows an AR(1) at quarterly frequency and estimate the first order serial correlation (0.88) and standard deviation (0.093) of innovations. These imply an unconditional standard deviation which almost exactly equals the actual value 0.0197, providing support to the AR(1) assumption. In each model, we choose values of the monthly log TFP parameters ρ and σ to target both persistence of innovations (0.88) to and unconditional volatility (0.0197) of filtered log ALP using simulated log ALP aggregated to quarterly frequency.

84 Given the AR(1) specification for TFP with Gaussian innovations, we can derive a closed-form expression for . In the Appendix we show

85 The AR(1) for ln z_t can be approximated by a finite Markov chain using Tauchen’s method, and then can be pre-computed from Equation 19 for all values of z in its finite support.

86 To compute equilibrium dynamics, we discretize the Pareto distribution of match quality on a 500-point support . Given the value of λ > 1 we choose to be the 99.9% percentile of the underlying Pareto distribution, and y to guarantee that the resulting mean of the discrete distribution equals μ = 1. We then specify z_t as a 500-point discrete Markov chain approximation to the AR(1) process described above.

87 In order to understand the role of OJS, besides the benchmark model just described, we compare it to a model without OJS. We restrict s = 0 to eliminate OJS altogether, and do not target the EE probability, so we also re-estimate the elasticity of the matching function by assuming that the pool of search equals only unemployment, and obtain, as mentioned, . We also recalibrate the TFP process parameters ρ and σ and the value of hiring costs Ψ, but do not expect the latter to make a material difference. Note that the model with no OJS has no endogenous productivity component, and ALP = TFP, so in this case ρ and σ can be directly set to the ALP empirical counterparts.

88 All parameter values for each version of the model are gathered in the upper panel of Table 1. The values of β = 0.995, b = 0 and δ = 0.024 are maintained in all versions of the model. The value of the parameter λ, which is varied exogenously across our different simulations exercises, is highlighted in gray in the table.

Quantitative Results

89 We study the unconditional second moments of unemployment, job-finding probability, ALP and other statistics from a stochastic simulation of the model, and their impulse response functions to a one-period negative TFP shock.

90 Table 1 illustrates calibration and results. The first three columns, OJS I-III, refer to three versions of the model with OJS, where the value of the Pareto slope λ of the match quality distribution is allowed to vary. The only material difference this slope makes to the rest of the calibration is in the implied hiring costs: a more dispersed match quality distribution (lower λ), given the mean, generates a higher value of climbing the job ladder, hence a larger surplus appropriated by firms that hire unemployed workers, which must then translate into larger hiring costs to match the same, observed job finding probability (JFP) from unemployment of 41% per month. The other parameter values are essentially unchanged. [14]

91 Given this similarity, we compare this calibration of the OJS model to a version where we shut down OJS (s = 0) and recalibrate all parameters (except λ whose value is immaterial without OJS). The results are in the NO OJS column. The comparison reveals that the model with OJS features very modest amplification of aggregate TFP shocks, with the log job finding probability varying between a quarter and a third as much as log ALP, as opposed to about ten times in the data, and yet this is even lower without OJS. This difference is entirely due to the impact of OJS on the estimated elasticity of the matching function , which is raised from 0.32 to 0.5. To corroborate this claim, in column OJS-IV we fix the OJS calibration of column OJS-I and just replace the value of the matching function elasticity with the lower value we estimate under NO OJS in the last column. Amplification reverts to a level below the model without any OJS. This is intuitive: once we allow for procyclical employment to create congestion for the unemployed job seeker, vacancies must be more effective in order to generate the variation in the job finding probability from unemployment that we observe in the data and that we use for this estimation. In turn, a higher estimated elasticity implies that vacancies create less mutual congestion, making the volume of new job creation more responsive to its returns, therefore to aggregate shocks. The other two effects of OJS on the calibration, the implied change in hiring costs and moments of the aggregate impulse, explain the remaining, negligible difference in amplification.

Table 1

Calibration and quantitative results: baseline model

Model ⇒		OJS			NO OJS	Data
Parameters calibrated externally	I	II	III	IV
discount factor β	0.995	0.995	0.995	0.995	0.995
flow value of leisure b	0	0	0	0	0
exogenous separation prob. δ	0.024	0.024	0.024	0.024	0.024
match inequality λ	1.1	3	5	1.1	–
Parameters calibrated internally
OJS efficiency s	0.176	0.176	0.176	0.176	0
persistence of log TFP innov’s ρ	0.947	0.953	0.954	0.947	0.955
volatility of log TFP innov’s σ	0.0067	0.0066	0.0066	0.0067	0.0065
matching function elasticity α	0.50	0.50	0.50	0.32	0.32
Targeted moments
average unemployment rate	0.055	0.055	0.055	0.055	0.055	0.057
average UE prob.	0.410	0.410	0.410	0.410	0.410	0.410
average EU prob.	0.024	0.024	0.024	0.024	0.024	0.024
average EE prob.	0.020	0.020	0.020	0.020	0	0.020
std(Quarterly log ALP)	0.0198	0.0198	0.0198	0.0193	0.0192	0.0197
persistence of Quarterly log ALP innov’s	0.884	0.884	0.888	0.877	0.886	0.885
Untargeted moments
std(log unempl. rate)	0.0058	0.0048	0.0043	0.0028	0.0032	0.214
std(log UE prob.)	0.0064	0.0054	0.0050	0.0031	0.0036	0.147
std(log EU prob.)	0	0	0	0	0	0.112
elasticity of UE prob w.r.t. TFP	0.332	0.271	0.244	0.160	0.178
std(log UE prob.)/std(log ALP)	0.317	0.268	0.246	0.156	0.178
elasticity of UE prob. to vacancies/unempl.	0.37	0.38	0.38	0.27	0.32	0.32
correlation(vacancies,unempl.)	–0.89	–0.85	–0.83	–0.90	–0.85	–0.86

Calibration and quantitative results: baseline model

92 The OJS results in the first three columns also show that greater dispersion in match quality (a lower λ) helps volatility. This is because, as explained earlier, dispersion raises the future expected returns from climbing the job ladder through OJS, more so when the level of TFP is high, because TFP multiplies match quality. Firms appropriate these returns, so that a positive shock raises the returns to hiring both employed and especially unemployed job applicants, more so when match draws have a thicker tight tail. In addition, as firms post more vacancies, they raise the meeting probability for employed job searchers, further raising the future returns to OJS, hence to hiring, which has a multiplier effect.

93 In Figure 1 we illustrate the impulse response of the (log of the) job finding probability from unemployment to a 1% decline in log TFP that lasts only one period, in calibrations OJS-I (λ = 1.1), OJS-III (λ = 5) and NO OJS. [15] The larger amplification caused by OJS through the estimated elasticity of the matching function is clear. As is well known, the canonical model with no OJS has no transitional dynamics in job market tightness or transition rates: the job finding probability returns immediately to its steady state value. In the OJS case, however, we observe some propagation, which is, to the best of our knowledge, a novel result. In this case, the job finding probability overshoots the steady state, and then slowly declines back. We identify the three opposing forces through which OJS affects amplification and propagation.

Figure 1

Response of Job Finding Probability (JFP) to one-period negative TFP shock

Response of Job Finding Probability (JFP) to one-period negative TFP shock

94 One the one hand, there are two composition effects, one between employment and unemployment and one within employment, which accelerate convergence. First, after a negative TFP shock, the temporarily higher unemployment and lower employment improve the quality of the jobseekers’ pool from the viewpoint of firms, that earn higher rents from hiring the unemployed. Second, within employment, the temporary decline in the contact probability slows down reallocation on the job ladder and leads to a temporary deterioration of the employment distribution, or to an increase in misallocation, which makes employed workers more “poachable.” Both composition effects raise the returns to job creation, accelerating the recovery. Neither composition effect exists without OJS. On the other hand, OJS also generates a congestion effect: as job postings recover and unemployment declines, the employed still create some congestion, making it harder for the unemployed to find jobs. In other words, the pool of job applicants improves, both across employment states and within employment, but locating the better sub-pool (the unemployed) becomes increasingly hard. In this calibration, the first effect dominates on impact, explaining the overshooting, because, after one period of slow job creation, firms benefit from a return to the initial level of TFP, but the composition of job search, both across employment states and within employment, is better than the initial one. The second effect is smaller, and unfolds more gradually. Comparison of the two panels of Figure 1 further shows that the first composition effect, which causes the job finding probability to overshoot, is more potent when match quality is less dispersed (Panel (b), λ = 5). This is intuitive: with lower dispersion in match quality, competition between employers is more intense, the mismatch wedge is small, and the returns to hiring an employed worker are low. Thus, the shift of the jobseekers’ pool towards more unemployed workers following a negative TFP shock is more beneficial to firms.

95 In Figure 2 we illustrate the Impulse Response Function of the (log of the) job finding probability from unemployment to a 1% negative innovation in log TFP, which then mean-reverts at the same rate ρ, the value estimated in the NO OJS model, in the same calibrations as Figure 1. The amplification and propagation of the shock now differs qualitatively from the case analyzed in Figure 1. The impact response is larger with OJS because of the higher estimated matching function elasticity. The distribution of employment of the ladder is a slow-moving variable, which deteriorates as the job contact probability remains lower for a while and can no longer offset exogenous separations. This rising misallocation of employment feeds back, through the composition and congestion effects described above, on job creation itself. With lower match inequality and less scope for reallocation (Panel (b), λ = 5), convergence is faster, because the distribution of employment moves less, but the impact is slightly stronger, because less muted by composition effects. In the limit as λ → ∞ all matches are identical and the model reduces to a special case of Robin [2011] with identical workers, which then features high response on impact, but modest propagation and overall volatility.

Figure 2

Response of Job Finding Probability (JFP, solid) to mean-reverting negative Total Factor Productivity shock (TFP, dashed)

Response of Job Finding Probability (JFP, solid) to mean-reverting negative Total Factor Productivity shock (TFP, dashed)

Endogenous Separations to Unemployment

96 Finally, we turn to the version of the model with endogenous job destruction analyzed in the fifth section, which is our preferred specification. This version features additional parameters. A conventional choice in the literature, motivated by empirical evidence on the age profile of firm-level productivity and worker separation rates, formalizes idiosyncratic match-specific shocks as a persistent process, typically an AR(1) in logs with mean zero. In our model, this process alone cannot generate our target 2.4% monthly probability of separation to unemployment, unless we assume a very high value of leisure b, close to mean productivity (normalized to one), thus a tiny match surplus. The reason is that, for plausible size of its innovations, the AR(1) process concentrates much of the mass of its ergodic distribution near its unit mean; therefore, the probability at each point in time that a match hits the equilibrium separation threshold is very small, unless the value of leisure b is very close to the unit mean productivity. In that case, though, the model with NO OJS is already known to generate large amplification, while our goal is to study the effects of OJS in a more plausible environment. Accordingly, we set the value of leisure b at 0.75. We assume that exogenous separation shocks occur with probability δ = 1% in all models. Because this exogenous component is acyclical, however, and the endogenous separation threshold is far from the mass of the match distribution, this formalization cannot reconcile average level and cyclical volatility of the separation rate. To resolve this tension, we introduce an infrequent, “large” idiosyncratic shock, which reduces match productivity by a certain percentage, but does not necessarily cause a separation. This third component makes endogenous separations more likely in recessions, when aggregate TFP drags the productivity distribution closer to the threshold. Accordingly, we choose the parameters of the “small” AR(1) idiosyncratic shock (persistence and standard deviation) and those of the “large” shock (probability of arrival and proportional size) to match the average level of the EU separation probability and the unconditional volatility of its HP filtered log, which equals 0.112. To avoid introducing additional parameters, we calibrate the sampling distribution Γ to equal the ergodic distribution of the match quality process.

97 The algorithms to calibrate the model’s parameters from steady state equilibrium and to compute a stochastic equilibrium are only slightly more complex than in the exogenous separation case. Given parameter values, the acceptance cutoff function ŷ(·) can be computed beforehand by solving Equation 10 by value function iteration. Steady state equations for the employment distribution now depend on the stationary level of TFP z, which determines the feasible matching set through the cutoff ŷ(z). Because of idiosyncratic shocks, endogenous separations exist also in steady state. Thus, the EU transition probability in steady state equilibrium contains the endogenous object which depends on unobserved parameters s, b, Γ, π etc. So we have to loop over the values of these parameters.

98 The simulation of a stochastic equilibrium proceeds as before. We draw a path of {z_t}T_t=0. Then, given chosen initial conditions L₀, thus u₀, we use the free entry condition 11 to find the value of θ₀, and finally use this value in Equations 12, 13 and 14 to update L₁, thus u₁, and so on for every t ≥ 1.

Table 2

Calibration and quantitative results: endogenous separations model

Model ⇒	OJS	NO OJS	Data
Parameters calibrated externally
discount factor β	0.995	0.995
flow value of leisure b	0.75	0.75
exogenous separation prob. δ	0.010	0.010
persistence of log match quality ρy	0.99	0.99
volatility of log match quality innov’s σy	0.002	0.002
Parameters calibrated internally
OJS efficiency s	0.086	0
persistence of log TFP innov’s ρ	0.962	0.974
volatility of log TFP innov’s σ	0.007	0.007
prob. of large idiosyncratic shock	0.074	0.045
size of large idiosyncratic shock	0.124	0.152
elasticity of contact prob. to tightness α	0.450	0.322
Targeted moments
average unemployment rate	0.055	0.055	0.057
average UE prob.	0.414	0.416	0.410
average EU prob.	0.024	0.024	0.024
std(log EU prob.)	0.112	0.112	0.112
average EE prob.	0.020	0	0.020
std(Quarterly log ALP)	0.0197	0.0196	0.0197
persistence of Quarterly log ALP innov’s	0.886	0.888	0.885
Untargeted moments
std(log unempl. rate)	0.113	0.117	0.214
std(log UE prob.)	0.035	0.040	0.147
elasticity of EU prob w.r.t. TFP	–3.63	–2.80
elasticity of UE prob w.r.t. TFP	1.42	1.42
std(log UE prob.)/std(log ALP)	1.74	1.97
elasticity of UE prob. to tightness	0.499	0.345	0.49
elasticity of UE prob. to vacancies/unempl.	0.249	0.345	0.32
elasticity of contact prob. to vacancies/unempl.	0.223	0.323
correlation(vacancies/unempl.)	–0.36	0.276	–0.86

Calibration and quantitative results: endogenous separations model

99 Table 2 reports the quantitative results, which now include statistics on the job contact probability, which differs from the job finding probability because some unemployed job applicants are rejected. As well known in the business cycle search literature, endogenous separations greatly amplify the volatility of unemployment through the contribution of countercyclical EU probability. This, however, comes at a dual cost.

100 First, the volatility of the UE exit probability from unemployment remains a valid empirical target, that the model still needs to explain. Improvement in explaining the volatility of unemployment through job destruction does not resolve the issue of the volatility of job creation. Endogenous separations help also in this respect, because not all new matches are acceptable, especially in recessions, a powerful lever on firms’ incentives to post vacancies. While still far from the empirical target, the UE probability volatility is greatly increased in this model, relative to the previous version with exogenous separations.

101 Second, waves of layoffs during recessions raise the pool of workers available for hire, stimulating vacancy creation and even turning the Beveridge curve upward-sloping (Mortensen and Nagypal [2007]). For this reason, the business cycle search literature has not pursued this endogenous separation avenue, which was central to the seminal steady state analysis by Mortensen and Pissarides [1994]. Fujita and Ramey [2012] show that OJS can help remedy this undesirable byproduct of endogenous separations, while bringing the model closer to the empirical evidence that layoffs do spike in every recession. Intuitively, when search continues on the job, unemployed and employed job searchers are closer substitutes in the eyes of the firm, hence vacancy creation responds less to the sheer size of unemployment, than in the baseline model with unemployed search alone. Consequently, the large inflow of unemployed made available by a wave of endogenous separations does not, per se, stimulate vacancy postings as much.

102 We find the same result: the Beveridge curve slopes up without OJS, and back down (although not as much as in the data) with it. Fujita and Ramey [2012] make their case introducing OJS in a business cycle version of Mortensen and Pissarides [1994], where wages are determined by Nash Bargaining and all new matches, whether with unemployed and already employed workers, start at the top of the ladder, and deteriorate stochastically over time. As a consequence, all outside offers are accepted, and the allocation of employment on the job ladder is irrelevant to the job creation decision. It has in fact been long thought that tracking the distribution of employment would make the problem of business cycles with random OJS intractable—Menzio and Shi [2011], for example, make this case forcefully. In Moscarini and Postel-Vinay [2013] we showed that equilibrium remains tractable when Rank-Preserving, namely when workers always prefer a more productive match, and that this is the case under commitment by firms to employment contracts without renegotiation. Even more tractable is the case studied here, where we allow firms to renegotiate wage contracts following outside offers. The employment distribution on a job ladder, albeit possibly infinitely-dimensional, is a predetermined state variable that, under the sequential auctions protocol, does not need to be forecasted, but is backward-looking, thus can be computed recursively along the way.

103 Figure 3 reports the impulse responses of the accession rate from and separation rate to unemployment to the usual 1% TFP shock, returning to its long-run mean at the rate estimated in the NO OJS model. The effects of OJS are, once again, visible, but the main novelty is the size of the response in both models.

Figure 3

Responses of Job Finding Probability (JFP) and Job Loss Probability (JLP) to mean-reverting negative Total Factor Productivity shock (TFP): endogenous Job Destruction model

Responses of Job Finding Probability (JFP) and Job Loss Probability (JLP) to mean-reverting negative Total Factor Productivity shock (TFP): endogenous Job Destruction model

104 We conclude that endogenous separations both to unemployment and to other jobs are very promising joint ingredients of a successful quantitative business cycle model of the US labor market.

Screening Costs

106 In this first extension, we introduce an additional source of hiring cost, which we interpret as a screening cost. As before, the firm pays a vacancy posting cost κ to advertise the position, receive applications, and observe the employment status of the applicant. But now, in order to hire a job applicant and to observe the match quality draw y’ as well as the existing match quality y of an employed applicant with their current employer, the firm must also pay an additional cost . This parameter captures all investment into hiring, such as interviewing, screening, and training, that the firm has to make after receiving job applications. As pointed out by Pissarides [2009], unlike advertising/vacancy costs, this part of the hiring costs are unaffected by market tightness and congestion, a powerful general equilibrium force that tends to offset aggregate shocks. In an estimated equilibrium search model, Christiano, Eichenbaum and Trabandt [2016] exploit this insight to resolve the tension highlighted by Shimer [2005].

107 The firm would not pay the advertising cost κ to post the vacancy in the first place if it was not then willing to pay the screening/training cost to hire some of the job applicants it hopes to come in contact with, at least unemployed job applicants who are homogeneous and most profitable. This property follows from our timing of events: vacancy posting and matching occur in the same period, i.e., under the same information set. Employed job applicants are also homogeneous ex ante (i.e., before paying the screening cost), but less profitable. Therefore, it is possible that, for low enough levels of aggregate TFP and the mismatch wedge, firms may be unwilling to pay the screening cost to hire an employed job applicant. In order to avoid a complete shutdown of the economy, should this situation arise, we assume that the screening cost and the value of leisure are still small enough that firms are always willing to post vacancies, because they will encounter sufficiently often some unemployed job applicants, whom they will then be willing to always hire. We also maintain the assumption that the value of leisure is small enough that firms are willing to match with any unemployed job applicant, conditional on screening/training. The free entry condition now writes as: [16]

108 Such a screening/training cost thus allows us to reduce the size of the expected returns to job creation. Accordingly, the economy achieves substantial amplification. The intuition is simple: small average returns are more sensitive to a given aggregate impulse. To leverage this mechanism, Hagedorn and Manovskii [2008] raise the value of leisure b closer to average output (here equal to one) and thus reduce total match surplus. We do it through additional hiring costs, paid only once, right after meeting. In this sense, our mechanism is formally (albeit not conceptually) similar to Hall and Milgrom [2008]’s cost for the firm of continuing wage negotiations, which is never paid by the firm in equilibrium, but raises the wage and shrinks profits. One advantage of our approach, in terms of computation and understanding of the mechanism, is that a high value of b would make some jobs infeasible and lead to endogenous separations.

109 Table 3 repeats the exercises of Table 1. In Table 3, we keep dispersion in match quality constant at λ = 1.1, but vary the magnitude of the screening cost exogenously such that screening costs account for a share of 0%, 50%, and 90% of total hiring costs in OJS columns I, II, and III, respectively (column OJS-I which has a screening cost share of 0% is only a replica of the same column in Table 1, for comparison). Note that the more important are hiring costs, the more cyclically volatile is the job contact probability, hence reallocation on the ladder. Given the target volatility of ALP, this requires a much less persistent TFP impulse. Finally, the NO-OJS column shows results from a version of the model where we shut down OJS by setting s = 0 and set the screening cost share to 90% of total hiring costs.

Table 3

Calibration and quantitative results: screening cost model

Model ⇒		OJS			NO OJS	Data
Parameters calibrated externally	I	II	III	I	II
discount factor β	0.995	0.995	0.995	0.995	0.995
flow value of leisure b	0	0	0	0	0
exogenous separation prob. δ	0.024	0.024	0.024	0.024	0.024
match inequality λ	1.1	1.1	1.1	–	–
share of screening in hiring costs	0	0.5	0.9	0.5	0.9
Parameters calibrated internally
OJS efficiency s	0.176	0.176	0.176	0	0
persistence of log TFP innov’s ρ	0.947	0.941	0.908	0.955	0.955
volatility of log TFP innov’s σ	0.0067	0.0067	0.0067	0.0065	0.0065
matching function elasticity α	0.50	0.50	0.50	0.32	0.32
Targeted moments
average unemployment rate	0.055	0.055	0.053	0.055	0.054	0.057
average UE prob.	0.410	0.410	0.430	0.410	0.420	0.410
average EU prob.	0.024	0.024	0.024	0.024	0.024	0.024
average EE prob.	0.020	0.020	0.020	0	0	0.020
std(Quarterly log ALP)	0.0198	0.0198	0.0198	0.0197	0.0197	0.0197
persistence of Quarterly log ALP innov’s	0.884	0.885	0.884	0.886	0.886	0.885
Untargeted moments
std(log unempl. rate)	0.0058	0.0101	0.0285	0.0064	0.0312	0.214
std(log UE prob.)	0.0064	0.0113	0.0323	0.0071	0.0346	0.147
std(log EU prob.)	0	0	0	0	0	0.112
elasticity of UE prob w.r.t. TFP	0.332	0.609	2.04	0.355	1.71
std(log UE prob.)/std(log ALP)	0.317	0.559	1.61	0.355	1.72
elasticity of UE prob. to vacancies/unempl.	0.37	0.37	0.38	0.32	0.32	0.32
correlation(vacancies,unempl.)	–0.89	–0.88	–0.83	–0.85	–0.85	–0.86

Calibration and quantitative results: screening cost model

110 Figure 4 echoes Figure 2 for the new screening costs calibrations. The larger amplification of a given TFP shock guaranteed by screening costs makes the effects of reallocation on the cyclical job ladder even more clearly visible.

Figure 4

Response of Job Finding Probability (JFP, solid) to mean-reverting negative Total Factor Productivity shock (TFP, dashed): screening cost model

Response of Job Finding Probability (JFP, solid) to mean-reverting negative Total Factor Productivity shock (TFP, dashed): screening cost model

Worker Heterogeneity

111 In this second extension, we revert to the case of zero screening cost but assume that each worker has individual-specific, time-varying human capital h_t evolving stochastically over a finite grid 0 = h₁ <…< h_K = 1 according to a state-dependent Markov process, similar to Ljungqvist and Sargent [1998]. Specifically, the human capital of an employed worker changes from h to h′ with probability π_e(h, h′) that of an unemployed worker or a laid off worker (on impact of the layoff) with probability π_u(h, h′). Besides having an important tradition in the unemployment literature, this extension allows us to control the relative appeal of hiring unemployed and employed job applicants, thus to fine tune of the composition effects that we highlighted. Intuitively, employed job applicants, while more expensive to hire, are also on average more productive.

112 Upon being hired (either from unemployment or from employment), a worker draws an idiosyncratic match quality “potential” , y ~ Γ. Match output is then:

113 where h ∈ [0, 1] is the worker’s human capital and z is current aggregate TFP. The interpretation is that all matches have the same baseline productivity y, and an idiosyncratic supplementary productivity potential y – y. How much of that added productivity potential is realized depends on the worker’s human capital. Unemployed workers produce b, regardless of their human capital. We assume that b and y are such that it is always marginally profitable to hire a worker with human capital h = 0, even in the worst TFP state.

114 Similar extensions of the basic search model have been applied elsewhere in the literature, in different contexts and with different purposes: Ljungqvist and Sargent [1998], Kehoe, Midrigan and Pastorino [2019], and, closer to this model, by Walentin and Westermark [2018] in a model featuring OJS and Sequential Auction bargaining. In the context of our model, heterogeneity in human capital affects the firms’ relative returns to contacting an unemployed vs an employed jobseeker, generically to the detriment of the former, who have lower human capital in equilibrium. Specifically, a TFP shock now perturbs the distribution of human capital amongst employed and unemployed workers, which in turn impacts the returns to vacancy posting, and future hiring. Because the distribution of human capital is another slow-moving state variable, the effect of TFP shocks to future hiring rates is further propagated by that transmission channel.

115 The firm’s willingness to pay for a given match now depends on the worker’s human capital as well as on match quality. Following the same reasoning as in the fourth section:

116 where the t subscript on the expectation operator is now shorthand for conditioning on current TFP z_t and worker’s current human capital h_t. Note that for any pair (y, y’):

117 We next turn to the dynamics of the employment distribution. Let L_t(y, h) denote the measure of workers with human capital h employed in matches with quality within [y, y]. Further let u_t (h) denote the measure of unemployed workers with human capital h. Then:

118 The latter law of motion can be written more compactly in matrix form. Introducing the vector notation and and, for x = e, u, defining Π_x as the K × K matrix whose (i, j) entry is π_x(h_i, h_j):

119 The (non-normalized) distribution of human capital amongst employed worker thus evolves following:

120 Finally, the (non-normalized) distribution of human capital amongst unemployed worker evolves following:

121 or, in matrix form:

122 Note that the total measure of unemployed workers, obeys a familiar law of motion. Summing the last equation over h and remembering that for all h′ and x = e, u, we obtain u_t+1 = [1 – ϕ(θ_t)]u_t + δ(1 – u_t). Similarly, the total measure of workers employed in matches with quality y or less, , follows .

123 In steady state, the laws of motion derived above boil down to:

124 which implies

125 so that the steady-state distribution of human capital amongst unemployed workers is the vector u in the null space of the matrix in curly brackets whose elements sum to . Once u is known, the joint distribution of match quality and human capital amongst employed workers is given by:

126 We finally adjust the free-entry condition to this extension of our model. Let:

127 Note that the expected returns, per unit of match quality, from hiring an unemployed and an employed worker, (resp.) and , are functions not only of current TFP z_t, but also of the current distribution of human capital in, respectively, the populations of unemployed and employed workers. Dependence on those latter state variables is subsumed into the t subscript. The free entry condition writes as:

128 where the mismatch wedge, Ω_t, is defined as in the basic model. Note that, when computing the expected return of a random contact with a worker (the RHS of the free-entry equation above), the firm must take expectations over the sampling distribution of match qualities, as before, but also over the distribution of worker human capital. The latter is reflected in the variables and , which involve the current distributions of human capital amongst employed and unemployed workers, respectively, and evolve only as fast as the learning-by-doing process will allow.

129 The functions and , can be expressed in terms of the matrix notation introduced above, using the assumption that the process of individual human capital is independent of the aggregate TFP process. Defining the function Z:ℝ ↦ ℝ^K as:

130 where is the K × 1 vector of human capital values, we have that:

131 To explore the quantitative implications of this extension, we parameterize learning-by-doing using a slightly amended version of Ljungqvist and Sargent [1998]:

132 In words: in employment, skills appreciate by one notch with probability π_e each period; in unemployment, skills depreciate by one notch with probability π_u each period.

133 The results reported in Table 4 were produced using K = 11 equally spaced human capital levels, π_e = 0.042, and π_u = 0.28. Those numbers imply that it takes on average 20 years of continuous employment for a worker’s human capital to rise from the lowest to the highest level, and three years to fall from the highest to the lowest level. Even under this calibration, when match quality dispersion is set to λ = 1.1 a contact with the average employed worker is only 7% more valuable to employers than a contact with the average unemployed. Those relative returns drop quickly as the match quality distribution becomes more concentrated (an employed contact is only worth 8% of an unemployed contact when λ = 5), as the average return of a poaching firm becomes very small when all matches are close in quality. [17]

Table 4

Calibration and quantitative results: Human capital model

Model ⇒		OJS		NO OJS	Data
Parameters calibrated externally	I	II	III
discount factor β	0.995	0.995	0.995	0.995
flow value of leisure b	0	0	0	0
exogenous separation prob.	0.024	0.024	0.024	0.024
match inequality λ	1.1	3	5	–
Parameters calibrated internally
OJS efficiency s	0.176	0.176	0.176	0
persistence of log TFP innov’s ρ	0.947	0.953	0.953	0.955
volatility of log TFP innov’s σ	0.0067	0.0066	0.0066	0.0066
matching function elasticity α	0.50	0.50	0.50	0.32
Targeted moments
average unemployment rate	0.055	0.055	0.055	0.055	0.057
average UE prob.	0.410	0.410	0.410	0.410	0.410
average EU prob.	0.024	0.024	0.024	0.024	0.024
average EE prob.	0.020	0.020	0.020	0	0.020
std(Quarterly log ALP)	0.0197	0.0197	0.0198	0.0197	0.0197
persistence of Quarterly log ALP innov’s	0.885	0.885	0.885	0.885	0.885
Untargeted moments
relative returns to contact, empl./unempl.	1.07	0.22	0.08	–
std(log unempl. rate)	0.0060	0.0048	0.0043	0.0033	0.214
std(log UE prob.)	0.0067	0.0054	0.0050	0.0036	0.147
std(log EU prob.)	0	0	0	0	0.112
elasticity of UE prob w.r.t. TFP	0.346	0.273	0.244	0.180
std(log UE prob.)/std(log ALP)	0.330	0.270	0.245	0.179
elasticity of UE prob. to vacancies/unempl.	0.37	0.38	0.38	0.32	0.32
correlation(vacancies,unempl.)	–0.90	–0.85	–0.83	–0.86	–0.86

Calibration and quantitative results: Human capital model

134 Figure 5 once again echoes Figure 2. While differences between the results of the two models are not immediately evident from a visual comparison of these two figures, human capital dynamics do add to the propagation of TFP shocks. For example, in the λ = 1.1 case, the half-life of the JFP is 26.8 months in the human capital model, compared to only 14.9 months in the baseline case. [18]

Figure 5

Response of Job Finding Probability (JFP, solid) to mean-reverting negative Total Factor Productivity shock (TFP, dashed): Human capital model

Response of Job Finding Probability (JFP, solid) to mean-reverting negative Total Factor Productivity shock (TFP, dashed): Human capital model