The Mechanics and Identification of Search Technology in the Canonical Random Search Model. Rasmus Lentz March, 2017

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Introduction

The worker mobility patterns in the canonical on-the-job random search model do not depend on the cardinal characteristics of the model. Consequently, the search technology can be identified without use of for example wages. This is in contrast with mobility models such as directed search or endogenous search intensity models, where mobility patterns depend on the magnitude of the economic gains to mobility. This note describes how search technology parameters in the exogenous random search model are identified off readily available aggregate labor market data.

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Model

Time is continuous. There is a mass of workers normalized at unity. Workers can be either employed or unemployed, and they meet employment opportunities at Poisson rates λ1 and λ0 , respectively. Each worker can rank the employment opportunities according to an offer distribution. A currently employed worker makes a job-to-job transition if offered a job at a higher ranked firm. Workers need not agree on the ranking and so may face different offer distributions. If a worker accepts a job, it maintains its ranking relative to the worker’s offer distribution throughout the duration of the match. Characterize a job by its rank ρ in the worker’s offer distribution. By definition ρ is uniformly distributed over the unit interval. Denote by R ∈ [0, 1] the reservation rank of an unemployed worker and assume it is identical across workers. Jobs are destroyed at rate δ. Burdett and Mortensen (1998) and Postel-Vinay and Robin (2002) are special cases of the above where workers face identical offer distributions and R = 0.

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Steady state worker mobility

Denote by hue the unemployment hazard which is immediately given by hue = λ0 (1 − R). The layoff hazard is similarly straightforward, heu = δ. The following determines the steady state rate at which an employed worker makes a job-to-job transition, hee . Denote by Mn the mass of employed workers that have P received n meetings during the past employment spell. Let the ∞ mass of unemployed workers be given by M0 . By definition n=0 Mn = 1. In steady state we have that for n ≥ 2, Mn (δ + λ1 ) = Mn−1 λ1 and M1 (δ + λ1 ) = M0 λ0 (1 − R), where M0 = δ/(δ + λ0 (1 − R)). The distribution of number of meetings in the employed population is mn = Mn /(1 − M0 ) which by the above steady state conditions is, mn =

δγ n , n ≥ 1, λ1

where γ = λ1 / (δ + λ1 ). A worker who is currently employed at rank ρ moves to another job at rate λ1 (1 − ρ). The employed job-to-job hazard rate is, Z 1  Z 1 ∞ X δλn−1 1 ρ − R n−2 n−1 1 hee = λ1 ρ (1 − ρ)dρ + (n − 1) ρ (1 − ρ)dρ (δ + λ1 )n R 1−R R 1−R n=1 "∞ # ∞ ∞ X γn X X δ γ n Rn+1 γn = − −R , 1 − R n=1 n + 1 n=1 (n + 1) n n n=1 1

A bit of algebra yields, 

hee

1 − γR =δ ln γ (1 − R)



1 − γR 1−γ



 −1 .

(1)

This result also features in Hornstein et al. (2011).

3

Calibration

The US Current Population Survey (CPS) allows observation of aggregate labor flows between employment and unemployment as well between jobs. At annual frequency, the conventional estimate of the U.S job finding rate is hue = 5.2, which implies an average unemployment duration of 2.3 months. As argued in Fujita and Moscarini (2017) this includes a significant number of recalls and should therefore be considered an upper bound estimate for the model in this note. Shimer (2012) estimates the layoff rate to an annual frequency of heu =0.24. Following Rogerson and Shimer (2011) I take a low end estimate of job-to-job transition rate from the CPS at annualized frequency of hee = 0.26. Finally, survey results in Hall and Mueller (2017) show that unemployed workers reject somewhere between 10% and 30% of offers. I pick the midpoint of this range and set R = 0.2. With these values, one obtains, R = 0.2 δ = heu = 0.24 hue = 6.5 λ0 = 1−R λ1 = 1.35. Hence, for the United States labor market λ0 /λ1 = 4.8. That is, in the exogenous search model, unemployed search is almost 5 times as efficient as employed search. Note, the above calibration is in some ways extreme. In particular, it implies that hue /hee = 20, which is high. As such, it should be viewed as an upper bound on the unemployed search efficiency relative to employed efficiency. Also, note that the reservation level, R, has a modest impact on the calibrated level of λ0 /λ1 . In addition, note that a carefully estimated endogenous search model such as Bagger and Lentz (2017) finds that employed search is estimated to be more efficient than unemployed search. Faberman et al. (2017) find the same result based on direct survey evidence.

References Bagger, Jesper and Rasmus Lentz (2017). An empirical model of wage dispersion with sorting. Forthcoming in The Review of Economic Studies. Burdett, Kenneth and Dale T. Mortensen (1998). Wage differentials, employer size, and unemployment. International Economic Review 39, no. 2: 257–273. Faberman, R. Jason, Andreas I. Mueller, Ayşegül Şahin, and Giorgio Topa (2017). Job search behavior among the employed and non-employed. Working Paper . Fujita, Shigeru and Giuseppe Moscarini (2017). Recall and unemployment. Working paper . Hall, Robert E. and Andreas I. Mueller (2017). Wage dispersion and search behavior: The importance of non-wage job values. Working paper . Hornstein, Andreas, Per Krusell, and Giovanni L. Violante (2011). Frictional wage dispersion in search models: A quantitative assessment. American Economic Review 101: 2873–2898. Postel-Vinay, Fabien and Jean-Marc Robin (2002). Equilibrium wage dispersion with worker and employer heterogeneity. Econometrica 70, no. 6: 2295–2350. Rogerson, Richard and Robert Shimer (2011). Search in macroeconomic models of the labor market. In Handbook of Labor Economics, Volume 4a. Elsevier Science B.V. Shimer, Robert (2012). Reassessing the ins and outs of unemployment. Review of Economic Dynamics 15, no. 2: 127–148.

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