Firm and Worker Dynamics in a rictionalF Labor...

Firm and Worker Dynamics

in a Frictional Labor Market

Adrien Bilal - Niklas Engbom - Simon Mongey - Gianluca Violante

December 12th, 2019

Labor reallocation

Reallocation of labor in the economy is key to understand:

dynamics of aggregate productivity

adjustment after sectoral/occupational shocks

impact of labor and product market policies

1. Jobs move across rms: job turnover (JD = 4%)

2. Workers move across jobs: worker turnover (SEP = 10%)

Two approaches to the analysis of labor reallocation

Labor reallocation

Two approaches

1. Firm dynamics with competitive labor market

Span of control or taste for variety: DRS

Lucas (1978), Hopenhayn (1992), Melitz (2003)

Firm size distribution and job turnover: determined

Worker ows: indeterminate

2. Worker dynamics with frictional labor market

Job ladder: OJS with 1w-1f / CRS

Postel-Vinay & Robin (2002), Burdett & Mortensen (1998)

Worker ows: determined

Firm size distribution indeterminate/pinned down by frictions

Two approaches

1. Firm dynamics with competitive labor market

Span of control or taste for variety: DRS

Lucas (1978), Hopenhayn (1992), Melitz (2003)

Firm size distribution and job turnover: determined

Worker ows: indeterminate

2. Worker dynamics with frictional labor market

Job ladder: OJS with 1w-1f / CRS

Postel-Vinay & Robin (2002), Burdett & Mortensen (1998)

Worker ows: determined

Firm size distribution indeterminate/pinned down by frictions

This paper

A theory of joint rm & worker dynamics in a frictional LM

Hopenhayn-type rms: DRS, enter, receive shocks, hire/re, exit

PVR-type workers: search at random o and on the job

Challenge: value sharing btw workers and rm with DRS + OJS

1. Propose a contractual environment s.t.

Allocation characterized by joint value of rm + all employees

Parsimonious: joint value depends only on productivity + size

Privately ecient

Endogenous job ladder in marginal surplus

2. Characterization of rm dynamics, job and worker ows

3. Parameterization + Questions

This paper

Privately ecient

This paper

Privately ecient

Questions for an environment with DRS + OJS

1. What is the misallocation cost of labor market frictions?

Can't do this counterfactual with CRS, nor w/o OJS

2. Which rm characteristics determine its rank on the job ladder?

Speak to new micro data on poaching ows by rm type

3. Why the failure of the job ladder in Great Recession?

Conjecture: related to the sharp drop in rm entry

Environment

Demographics and preferences

Continuous time

Unit continuum of ∞-lived workers

Ex-ante homogeneous

Risk neutral with discount rate ρ

Supply inelastically one unit of labor

Endogenous wage payment w as employed

Exogenous ow value of leisure b as unemployed

Technology

Endogenous mass m of rms produce the nal good (numeraire)

Production function:

y(z, n)

z is idiosyncratic productivity:

dzt = µdt+ σdWt

n is the number of workers

Properties: yz > 0, yn > 0, ynn ≤ 0, yzn ≥ 0

Workers recruited by posting vacancies v at cost c(v;n)

Firm entry cost c0 and scrap value ϑ

Random search in a single labor market

Unemployed: meet vacancies at rate λU

Employed:

Search with relative intensity ξ → λE = ξλU

Lose job endogenously, or exogenously at rate δ

Vacancies v: meet workers at rate λF

CRS matching function (v, s) → θ = v/s

s = u + ξ(1− u) is the mass of eective job seekers

φ = u/s is the share of unemployed job seekers

Desired representation

Allocative decisions (entry, exit, vacancy, mobility) are obtainedfrom the joint value Ω of the rm (owner of the technology) andits incumbent workers

Appealing properties of such representation:

1. All decisions within `coalitions' are privately ecient

2. Parsimonious state space: (z, n)

3. Endogenous job ladder in marginal value Ωn (or Sn = Ωn − U)

With complete information & contracts: Coase Theorem

What if we impose plausible restrictions on contracts?

Desired representation

Allocative decisions (entry, exit, vacancy, mobility) are obtainedfrom the joint value Ω of the rm (owner of the technology) andits incumbent workers

Appealing properties of such representation:

1. All decisions within `coalitions' are privately ecient

2. Parsimonious state space: (z, n)

3. Endogenous job ladder in marginal value Ωn (or Sn = Ωn − U)

With complete information & contracts: Coase Theorem

What if we impose plausible restrictions on contracts?

Contractual environment I

Contract: a constant wage in exchange for labor services

[A-LC] Two-sided limited commitment

Quits: Workers can always quit the rm

Layos / Exit: Firms can always re workers / exit

[A-MC] Self-enforcing labor contracts

Contracts can be renegotiated only under mutual consent

Both parties agree to renegotiate when one has a credible threat

Credible threat: when one party is better o dissolving the match

Contractual environment I

Contract: a constant wage in exchange for labor services

[A-LC] Two-sided limited commitment

Quits: Workers can always quit the rm

Layos / Exit: Firms can always re workers / exit

[A-MC] Self-enforcing labor contracts

Contracts can be renegotiated only under mutual consent

Both parties agree to renegotiate when one has a credible threat

Credible threat: when one party is better o dissolving the match

Contractual environment II

Negotiation protocol between rm and workers

[A-EN] External negotiation between rm and jobseekers

Meeting with unemployed: rm makes TOL oer

Meeting with employed: sequential auction with TOL oers

[A-IN] Internal negotiation between rm and its incumbents

Zero sum game

With additional assumptions: can determine wage payments

Zero sum game

Contractual environment III

DRS + OJS: prot maximizing vacancies are privately inecient

1. Fire and swap incentives, even if hiring is not protable

CRS + OJS: absent because hiring is always protable

DRS w/o OJS: absent because all workers paid same wage

2. Over-hiring incentives to push down MPL and keep wages low

CRS + OJS: absent

DRS w/o OJS: present, but tractable (Elsby-Michaels, 2013)

DRS + OJS: Private eciency yields tractability

[A-VP] Ecient vacancy posting

After rm chooses vacancies, incumbents make TOL counteroer

CRS + OJS: absent

Joint value representation

The maximized joint value

ρΩ(z, n) =

maxv≥0

y (z, n)− c (v;n)︸︷︷︸net output ow

+(vλFφ− δn

)·[Ωn(z, n)− U

]︸︷︷︸

net hires from unemployment

+ vλF (1− φ) ·Ωn(z,n)

[Ωn(z, n)− Ω′n

]dG(Ω′n)

︸︷︷︸hires from rms ranked lower in the job ladder

− nλE

Ωn(z,n)

[Ωn(z, n)− Ωn(z, n)

]dF(Ω′n)

︸︷︷︸quits to other rms ranked higher in the job ladder

+ µ(z)Ωz(z, n) +σ2(z)

2Ωzz(z, n)︸︷︷︸

productivity shocks

ρΩ(z, n) =

maxv≥0

+(vλFφ− δn

)·[Ωn(z, n)− U

]︸︷︷︸

+ vλF (1− φ) ·Ωn(z,n)

]dG(Ω′n)

− nλE

Ωn(z,n)

]dF(Ω′n)

+ µ(z)Ωz(z, n) +σ2(z)

2Ωzz(z, n)︸︷︷︸

productivity shocks

ρΩ(z, n) =

maxv≥0

+(vλFφ− δn

)·[Ωn(z, n)− U

]︸︷︷︸

+ vλF (1− φ) ·Ωn(z,n)

]dG(Ω′n)

− nλE

Ωn(z,n)

]dF(Ω′n)

+ µ(z)Ωz(z, n) +σ2(z)

2Ωzz(z, n)︸︷︷︸

productivity shocks

ρΩ(z, n) =

maxv≥0

+(vλFφ− δn

)·[Ωn(z, n)− U

]︸︷︷︸

+ vλF (1− φ) ·Ωn(z,n)

]dG(Ω′n)

− nλE

Ωn(z,n)

]dF(Ω′n)

+ µ(z)Ωz(z, n) +σ2(z)

2Ωzz(z, n)︸︷︷︸

productivity shocks

ρΩ(z, n) =

maxv≥0

+(vλFφ− δn

)·[Ωn(z, n)− U

]︸︷︷︸

+ vλF (1− φ) ·Ωn(z,n)

]dG(Ω′n)

− nλE

Ωn(z,n)

]dF(Ω′n)

+ µ(z)Ωz(z, n) +σ2(z)

2Ωzz(z, n)︸︷︷︸

productivity shocks

ρΩ(z, n) = maxv≥0

+(vλFφ− δn

)·[Ωn(z, n)− U

]︸︷︷︸

net hires from U

+ vλF (1− φ) ·Ωn(z,n)

Ω′n

)︸︷︷︸

hires from below in job ladder

+(vλFφ− δn

)·[Ωn(z, n)− U

]︸︷︷︸

net hires from U

+ vλF (1− φ) ·Ωn(z,n)

Ω′n

)︸︷︷︸

+ µ(z)Ωz(z, n) +σ2(z)

2Ωzz(z, n)︸︷︷︸

productivity shocks

+(vλFφ− δn

)·[Ωn(z, n)− U

]︸︷︷︸

net hires from U

+ vλF (1− φ) ·Ωn(z,n)

Ω′n

)︸︷︷︸

+ µ(z)Ωz(z, n) +σ2(z)

2Ωzz(z, n)︸︷︷︸

productivity shocks

subject to:

Ωn(z, n) ≥≥≥ U︸︷︷︸endogenous separations if ===

; Ω(z, n) ≥≥≥ nU + ϑ︸︷︷︸exit if ===

; E[Ω(z, n0)

]≤≤≤ n0U + c0︸︷︷︸

entry if ===

Static example

Static Example

Equilibrium

and limiting economies

Equilibrium

Stationary equilibrium

(i) Ω(z, n) satises the HJB equation and its boundary conditions

(ii) The vacancy policy v(z, n) satises the rst order condition:

cv(v(·)) = λF (θ)

[φ (Ωn(z, n)− U) + (1− φ)

Ωn(z,n)

(Ωn(z, n)− Ω′n

)dG(Ω′n)

(iii) Law of motion for rm-level employment satises:

dt(z, n) = λF (θ)v(z, n)

[φ+(1−φ)G(Ωn(z, n))

]−n[δ+λE(θ)F (Ωn(z, n))

](iv) Free-entry determines m0

(v) F (Ωn) and G(Ωn) are stationary and solve KFE(dndt , m0

)(vi) u, v, θ are consistent with vacancy policy and distributions

Two limiting economies

1. If y(z, n) and c(v, n) are linear in n =⇒ PVR (2012)

ρΩ(z) = maxv≥0

y(z) − c(v) +(vλFφ− δ

)[Ω(z) − U

+ vλF (1 − φ)

[Ω(z) − Ω′

]dG(Ω′)

Exogenous job ladder in productivity z

2. If matching eciency →∞ =⇒ Hopenhayn (1992)

Non degenerate rm size distribution

No dispersion in marginal product of labor across rms

Perfect correlation between size and productivity

Zero unemployment

Two limiting economies

1. If y(z, n) and c(v, n) are linear in n =⇒ PVR (2012)

ρΩ(z) = maxv≥0

y(z) − c(v) +(vλFφ− δ

)[Ω(z) − U

+ vλF (1 − φ)

[Ω(z) − Ω′

]dG(Ω′)

Exogenous job ladder in productivity z

2. If matching eciency →∞ =⇒ Hopenhayn (1992)

Non degenerate rm size distribution

No dispersion in marginal product of labor across rms

Perfect correlation between size and productivity

Zero unemployment

Characterization

Firm dynamics and job reallocation in (n, z) space

dn = 0

Job Creation

Job Destruction

Ω(z, n) = nU + ϑ

dn = 0

Job Creation

Job Destruction

Ω(z, n) = nU + ϑ

dn = 0

Job Creation

EntryJob Destruction

Ω(z, n) = nU + ϑ

dn = 0

Worker reallocation in (n, z) space

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Ω(z, n) = nU + ϑ

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Fire & JD

Ω(z, n) = nU + ϑ

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Fire & JD

Ω(z, n) = nU + ϑ

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Fire & JD

Ω(z, n) = nU + ϑ

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Fire & JD

Ω(z, n) = nU + ϑ

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Fire & JD

Ω(z, n) = nU + ϑ

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Fire & JD

Ω(z, n) = nU + ϑ

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Fire & JD

Ω(z, n) = nU + ϑ

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Fire & JD

Ω(z, n) = nU + ϑ

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Fire & JD

Ω(z, n) = nU + ϑ

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Fire & JD

Ω(z, n) = nU + ϑ

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Fire & JD

Ω(z, n) = nU + ϑ

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Fire & JD

Ω(z, n) = nU + ϑ

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Fire & JD

Ω(z, n) = nU + ϑ

Ωn(z, n) = U

dn = 0dn = 0

Hire & JC

Hire & JD

Fire & JD

Ω(z, n) = nU + ϑ

Composition of worker ows

n= λF v

∗(Ωn)

φ︸︷︷︸UE

+ (1− φ)G(

)︸︷︷︸

− δ︸︷︷︸

+ λEF(

)︸︷︷︸

Gross ows

Ω = nU + ϑ

n∗E(z)

dnn = 0 Ωn = U

n∗S(z)

qv∗(z, n)φ

qv∗(z, n)(1− φ)G(Ωn)

Net poaching

Composition of worker ows

n= λF v

∗(Ωn)

φ︸︷︷︸UE

+ (1− φ)G(

)︸︷︷︸

− δ︸︷︷︸

+ λEF(

)︸︷︷︸

Gross ows

Ω = nU + ϑ

n∗E(z)

dnn = 0 Ωn = U

n∗S(z)

qv∗(z, n)φ

qv∗(z, n)(1− φ)G(Ωn)

Net poaching

Quantitative Analysis

Parameters - SMM Estimation

Productivity distribution

- Dispersion of rm-level TFP

Firm dynamics

- Average rm size

- Employment share of large rms

- Exit rate

Job reallocation

- Job reallocation rate (JC + JD)

- Job creation by entrants

Worker reallocation

- UE rate, EU rate, EE rate

Parameters Local Global

Firm and employment distributions by size and age

Group A. Firms B. Employment

Model Data Model Data

A. By rm size

0-9 0.792 0.761 0.116 0.10710-49 0.169 0.195 0.135 0.16750-249 0.032 0.036 0.133 0.153250-999 0.005 0.006 0.096 0.1111000-4999 0.001 0.002 0.116 0.1285000+ 0.000 0.001 0.404 0.335

B. By rm age

1-5 0.350 0.242 0.058 0.0876-10 0.218 0.176 0.072 0.07911+ 0.432 0.499 0.871 0.813

Data: Census BDS

Job and worker reallocation by size and age

Job reallocation Worker reallocation

Creation Destruction Hire Separate

Model Data Model Data Model Data Model Data

A. By rm size

0-19 0.045 0.049 0.038 0.038 0.116 0.122 0.109 0.11020-49 0.029 0.033 0.029 0.029 0.104 0.106 0.105 0.10250-249 0.031 0.031 0.031 0.027 0.105 0.102 0.105 0.097250-499 0.032 0.031 0.033 0.025 0.105 0.101 0.107 0.096500+ 0.029 0.029 0.031 0.024 0.103 0.087 0.106 0.084

A. By rm age

0-1 0.161 0.161 0.024 0.038 0.233 0.212 0.096 0.1402-3 0.049 0.049 0.030 0.051 0.121 0.150 0.102 0.1444-5 0.042 0.040 0.032 0.044 0.115 0.134 0.104 0.1306-10 0.035 0.035 0.032 0.037 0.109 0.125 0.106 0.12111+ 0.028 0.026 0.033 0.024 0.102 0.086 0.107 0.084

Data: Census BDS and J2J (quarterly rates) Vacancies

Questions

Life-cycle dynamics of rm distribution over (z, n)

Misallocation costs of labor market frictions

Who poaches from whom?

EE+ − EE−

n= λF

v∗(Ωn)

n(1− φ)G

)− λEF

Net poaching by size and age

Net poaching by size and age: model and data

Net poaching by labor productivity & growth rate

Growth

Three key facts about the Great Recession

Fact I: Firm entry dropped by 30%

Fact II: EE rate declined by 30%

Fact III: Net poaching of high-wage rms fell → failure of job ladder

Conjecture: causal link from I ⇒ II & III

Need a model with rm dynamics + OJS

What explains the failure of the job ladder?

Shock to the discount rate: ρ ↑ (proxy for nancial shock)

Path for the shock calibrated to match dynamics of unemployment

Transmission mechanism:

Firm continuation value ↓ ⇒ rm entry ↓

Vacancy creation at the top of job ladder ↓

Fewer workers move up the job ladder

IRFs for a discount factor shock

Why do we care about the failure of the job ladder?

Yt = TFPt ×Nαt , TFPt =

)αdGt (n, z)

Slower labor reallocation ⇒ persistent decline in TFP

Conclusion

Framework to study rm and worker dynamics with frictions

Low-dimensional state space (z, n): tractable

Can be applied to a number of macro/labor/trade questions

`MIT' shock → aggregate uncertainty

Next step:

Document rm-level wage dynamics from admin data

Model wage determination protocol that matches those facts

Thanks!

Static example: hires from unemployment

Technology y(z, n) = z · nα, n = 1

Firm has met unemployed worker → TOL oer b [A-EN]

Let w∗ < w1 be the renegotiated wage of the incumbent

Credible threat: z − b > z2α − w1 − b [A-LC+A-MC]

Hire with ren. : z2α − w∗ − b > z − w∗ [A-IN]

Hire w/o ren. : z2α − w1 − b > z − w1

Firm hires if and only if the coalition is better o:

z · (2α − 1) > b ⇒ Ωn > U

Even if hiring not protable, rm may gain through wage cut:

[A-VP] ensures vacancy posted only if hiring protable

Back to Equilibrium

Static example: hires from employment

Firm z attempts to poach a worker in rm z′ (both n = 1)

Let w∗∗ < w1 the wage oer to the external worker

Credible threat: z − w∗∗ > z2α − w1 − w∗∗

Hire with ren. : z2α − w∗ − w∗∗ > z − w∗

Hire w/o ren. : z2α − w1 − w∗∗ > z − w1

Firm hires if and only if coalition better o:

z · (2α − 1) > w∗∗ = z′ · (1α − 0) = z′

Back to Equilibrium

z · (2α − 1) > w∗∗

= z′ · (1α − 0) = z′

Back to Equilibrium

z · (2α − 1) > w∗∗ = z′ · (1α − 0) = z′

Back to Equilibrium

Static example: separation after drop in z

Consider a rm with productivity z and two incumbents paidw2, w1, with w2 > w1

Technology y(z, n) = z · nα

Assume z > w1, so the rst worker is never under threat

Credible threat: z − w1 > z2α − w1 − w2

Separation : z − w1 > z2α − w1 − b

Firm res worker 1 if and only if the coalition is better o:

z · (2α − 1) < b

Back to Equilibrium

Static example: separation after drop in z

Consider a rm with productivity z and two incumbents paidw2, w1, with w2 > w1

Technology y(z, n) = z · nα

Assume z > w1, so the rst worker is never under threat

Credible threat: z − w1 > z2α − w1 − w2

Separation : z − w1 > z2α − w1 − b

Firm res worker 1 if and only if the coalition is better o:

z · (2α − 1) < b

Back to Equilibrium

Static example: vacancy creation

Now rm must pay vacancy cost ccc to meet unemployed worker

Conditional on paying ccc, all decisions unchanged

Threat to post vacancy and swap is credible if w1 > b+ ccc

But this vacancy would be jointly inecient

Worker makes counteroer: no vacancy + wage cut to w∗ = b+ c

Firm indierent, worker better o, threat removed

Firm posts vacancy and hires if and only if:

z · (2α − 1)︸︷︷︸marginal hiring gain

> b+ c︸︷︷︸marginal hiring cost

Back to Equilibrium

Parameters

Parameter Value Target

ρ Discount rate 0.004 5% annual real interest rateβ Elast. of matches w.r.t. v 0.5 Petrongolo and Pissarides (2001)ϑ Firm scrap value 1/ρ Normalizationκ Shifter in vacancy cost 100 Normalization

Parameter Value

µ Mean of productivity shocks -0.004σ Std. of productivity shocks 0.054ζ Shape of entry distribution 2.955ce Entry cost 379n0 Size of entrants 2.058α Curvature in production 0.587γ Curvature of vacancy cost 6.023A Matching eciency 0.157ξ Search eciency of employed 0.142δ Exogenous separation rate 0.017b Flow value of leisure 0.295

Target Model Data

Exit rate (unweighted) 0.084 0.076Std. dev. of log TFP 0.484 0.500Std. dev. of log TFP (age 1-5) 0.362 0.400Average rm size 22 22JC rate at age 1 0.237 0.244Employment share 500+ 0.551 0.518Std. dev. emp. growth 0.337 0.420UE rate 0.243 0.242EE rate / UE rate 0.068 0.076EU rate 0.018 0.016Leisure to output 0.367 0.400

Back to SMM

Parameters

Parameter Value Target

ρ Discount rate 0.004 5% annual real interest rateβ Elast. of matches w.r.t. v 0.5 Petrongolo and Pissarides (2001)ϑ Firm scrap value 1/ρ Normalizationκ Shifter in vacancy cost 100 Normalization

Parameter Value

µ Mean of productivity shocks -0.004σ Std. of productivity shocks 0.054ζ Shape of entry distribution 2.955ce Entry cost 379n0 Size of entrants 2.058α Curvature in production 0.587γ Curvature of vacancy cost 6.023A Matching eciency 0.157ξ Search eciency of employed 0.142δ Exogenous separation rate 0.017b Flow value of leisure 0.295

Target Model Data

Exit rate (unweighted) 0.084 0.076Std. dev. of log TFP 0.484 0.500Std. dev. of log TFP (age 1-5) 0.362 0.400Average rm size 22 22JC rate at age 1 0.237 0.244Employment share 500+ 0.551 0.518Std. dev. emp. growth 0.337 0.420UE rate 0.243 0.242EE rate / UE rate 0.068 0.076EU rate 0.018 0.016Leisure to output 0.367 0.400

Back to SMM

Are the targeted moments informative?

-10 -8 -6 -4 -2

0.2 0.4 0.6 0.8

5 10 15 20

0.02 0.04 0.06 0.08

0.1 0.2 0.3 0.4

0.01 0.02 0.03

0.2 0.4 0.6 0.8

0.1 0.2 0.3

2 4 6 8

Back to SMM

Global identication: a graphical approach

-8 -6 -410-3

0.10.20.30.4

0.1 0.2 0.3 0.4

0.10.20.30.4

2 4 6 8 10 12

0.10.20.30.4

0.02 0.04 0.06 0.08

0.10.20.30.4

0.1 0.15 0.2 0.25

0.10.20.30.4

0.01 0.02

0.10.20.30.4

0.5 0.6 0.7 0.8

0.10.20.30.4

1 2 310-3

0.10.20.30.4

0.05 0.1 0.15 0.2 0.25

0.10.20.30.4

2 4 6 8

0.10.20.30.4

1 2 3 4 5

0.10.20.30.4

Minimum Distance plotted in each dimension of the parameter space

Back to SMM

Vacancy rates and vacancy yields

Missing: endogenous recruiting intensity

Back to JR Table

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