Competitive Search with Repeated Moral Hazard · IntroductionModelSteady StateAggregate RiskPolicy...

Introduction Model Steady State Aggregate Risk Policy

Competitive Search with Repeated Moral Hazard

Fei ZhouHKUST

HKUST Seminar

Nov 28 2019

Zhou Repeated Moral Hazard 1/44


Introduction

Search friction is central in understanding key labor market moments

Moral hazard is the major reason for imperfect risk sharing in wage contract

Search Friction Meets Moral Hazard

What are the effects on employment and wage dynamics?

◦ need to solve optimal long-term contract in GE framework with aggregate risk

◦ need to measure underlying information frictions

This paper: quantify the effects of moral hazard in a labor search model



This Paper

1 Build a continuous-time search model where effort is imperfectly observed

◦ extend optimal dynamic contract a la Sannikov (2008) with aggregate risk

◦ block-recursive equilibrium a la Menzio and Shi (2011) in dynamic model

◦ remain tractable despite all the complications

2 Measure moral hazard by matching wage residual volatility in PSID

◦ larger vol. in unobservable idiosyncratic shock −→ more severe moral hazard

◦ indirect infer: moment in model simulated data −→ moment in PSID data

◦ target moment: std. of wage residual



Results

1 Unemployment volatility increases by 2-6 times with moral hazard

◦ higher productivity relaxes incentive constraint for firm

◦ firms’ profit elasticity w.r.t productivity is much larger

2 Endogenous counter-cyclical cross-sectional wage dispersion

◦ in good times, more relaxed incentive constraint, less wage dispersion

◦ Storesletten, Telmer, and Yaron (2004)

3 Endogenous wage scarring effects after job displacement

◦ wage increase gradually with tenure, upon separation, rebuild from scratch

◦ (Jacobson et al., 1993; Couch and Placzek, 2010; Barnette and Michaud, 2011)



Policy Implication

With moral hazard,

1 An increase of UI benefit significantly worsens employment rate

◦ harder to incentivize worker: the degree a firm can punish a worker is limited

◦ firms’ profit and incentive to post vacancy decline

2 Imposing minimum wage is much more likely to affect labor market

◦ without moral hazard, wage depends on average labor productivity

◦ with moral hazard, wage depends on history of labor productivity



Literature

Model: combine competitive search model with the dynamic contract model

◦ contract: Spear and Srivastava (1987), Sannikov (2008), Grochulski and Zhang(2019)

◦ search: Moen (1997), Menzio and Shi (2011)

◦ unemployment volatility: Shimer (2005), Hagedorn and Manovskii (2008), Hall(2005), Moen and Rosen (2011)

◦ wage dispersion: Storesletten, Telmer, and Yaron (2004)

◦ wage scarring: Jacobson et al. (1993); Couch and Placzek (2010); Barnette andMichaud (2011)



Part II: Model



Environment

Preference

E0

[∫ ∞0

e−rt (u (Ct)− φ (At)) dt]

Technology: a firm-worker pair produces output according to

dYt = zAtdt+ σdBt

◦ Yt: accumulated output

◦ Bt: unobserved idiosyncratic shock

◦ z: aggregate productivity

◦ σ: parameterizes information frictions



Frictional Labor Market and Directed Search

Workers and firms are subject to search and matching frictions

Directed search: sub-markets are indexed by (W0, θ)

◦ W0: life time utility of employed worker

◦ θ: market tightness

reflect ratio of vacancy measure v to unemployment worker measure u

determine job finding rate p(θ) and filling rate q(θ)

◦ firms and workers choose to visit the optimal sub-markets available

Next, we spell out firms’ problem and workers’ problem



Operating Firms: Optimal Contract Problem

Moral hazard: effort is unobservable, workers tend to shirk

Compensation needs to vary with output to provide incentive

◦ but, contract may depend on history of output, an infinite dimension object

Optimal contract remains tractable

◦ promised utility Wt summarizes relevant history

◦ profit function characterized by simple ODE



Operating Firms: Contracting problem

maxA,C

E[∫ τ

0

e−rt (dYt − Ctdt)]

◦ A, C: functions of entire history of output

◦ τ : expected duration, due to separation shock, with arrival intensity λ

1 Promise keeping condition

E[∫ τ

0

e−rt (u (Ct)− φ (At)) dt+ e−rτWu

]=W0

◦ W0: initial promised utility, lifetime utility of active submarket

◦ Wu: outside option, lower bound for continuation value

◦ both will be endogenously determined in equilibrium

2 Incentive compatibility condition: no desire to deviate from A



Employed Worker: Promised Utility

A plan {A, C} implies the following promised utility form time t

Wt = E[r

∫ τ

t

e−r(s−t) [u (Cs)− φ (As)] ds+ e−r(τ−t)Wu

]

Dynamics of promised utility follow

dWt = r [Wt − u (Ct) + φ (At)] dt+ r κ(Wt) (dYt − zAtdt)︸︷︷︸σdBt

+ . . .

κ(Wt): control sensitivity of compensation to output

◦ measure exposure to shocks, implied by plan {A, C}

How to make sure the plan is incentive compatible?



Employed Worker: Incentive Compatibility

Given the dynamics of promised utility

dWt = r [Wt − u (Ct) + φ (At)] dt+ rκ(Wt) (dYt − z At dt) + . . .

Worker want to choose A∗t such that

A∗t = arg minAt

{rφ(At)− rκ(Wt)zAt

}

If a plan is incentive compatible, κ(Wt) needs to satisfy

κ(Wt) =φ′(At)

z

◦ imposing restriction on possible plans {A, C}



Recursive Formulation: HJB Equation

Firm’s expected profit permits the following recursive formulation

(r + λ)F (W ) = maxA,C

zA(W )− C(W ) → current profit

drift of PU ← + F ′ (W ) [r (W − u (C) + φ (A))− (Wu −W )λ]

volatility of PU ← +F ′′ (W )

2r2(φ′ (A)

z

)2

σ2

A simple ODE to compute, lead to policy function A(W ) and C(W )



Allocation

Dynamics of Wt

dWt = r

[Wt − u (C(Wt)) + φ (A(Wt))

]dt+ r

φ′(A(Wt))

zdBt + . . .



Profit

Profit function is lower than first best, and humped-shaped

Remain to be determined: W0 and Wu



Vacant Firm

Unmatched firms can pay k to post vacancies in markets indexed by (W0, θ)

Free entry condition guarantees zero profit

post vacancy cost← k = q(θ)F (W0) → expected profit of vacancy

Active sub-markets (W0, θ) are characterized by

θ = q−1

[F (W0)

k

]

Firms are indifferent among active sub-markets, but workers are not



Unemployed Worker

Unemployed worker home production b

Choose best active sub-markets to search for job

rWu = maxW0

ru (b) + p(θ)(W0 −Wu)

subject to

θ = q−1

[F (W0)

k

]

Trade-off: job finding rate p(θ) versus expected utility W0

In equilibrium, workers’ choice determines W ?0 and Wu

General equilibrium graphic illustration



Part III: Parameterization



Functional form

Momentary utility function

u(C) =C1−η − 1

1− η , φ(A) = χ(A1+γ +A

)

Matching functionM(u, v) = ξuαv1−α

◦ job filling rate: q (θ) = ξθ−α

◦ job finding rate: p (θ) = ξθ1−α



Aggregate Statistics

Unemployment rate is determined by

u =1− λ

1− λ+ p(θ)←− du = λ(1− u)dt− p(θ)udt

Output is determined by

Output = (1− u)∫W

zA(W ) dG(W )

◦ G(W ): distribution of promised utility



Parameterization at steady state

Parameter Description Baseline Target

Externally Calibrated Parameter

r discount rate 0.01 4% annual return

λ separation rate 0.10 job duration 2.5 yrs

ξ matching efficiency 1.35 monthly job finding rate 0.45

α matching elast. 0.72 Shimer (2005)

η risk aversion 0.50 Sannikov (2008)

γ inverse of Frisch elast. 1 middle of various estimation

Internally Calibrated Parameter

z aggregate TFP 1 normalization

b home production 0.20 replacement ratio 20%

k vacancy posting cost 0.03 market tightness θ = 1

χ effort disutility 0.13 aggregate output equals 1

σ idiosyn. volatility 2.40 own estimation



Estimation of σ

Idea: identify wage movement due to idiosyncratic risks unobserved by firms

Strategy: choose σ to match wage residual in micro data

Execution: indirect inference, regression with model generated data

Data: Panel Study in Income Dynamics (PSID)

◦ longitude: 1994-2017 −→ 14 years unbalanced panel data

◦ observations: 14,137 individuals, 58,026 person-year observations



Recovering Volatility of Wage Residual

1 OLS: Mincer regression on labor income or performance pay

logwagei,j,t = βedu edui + βexpr L(expri,t) + βtenu L(tenui,j,t)

+ βt Dt + βind Dind + βocc Docc + ei,j,t

◦ residual decomposition: cross-sectional ηi,j , longitudinal εi,j,t

ei,j,t = ηi,j + εi,j,t

2 Fixed effect regression: individual idiosyncratic risk

logwagei,j,t = βedu edui + βexpr L(expri,t) + βtenu L(tenui,j,t)

+ βt Dt + βind Dind + βocc Docc + βi,j Di,j + εi,j,t



Estimation

Dependent: log( Real toal labor incomeHour )

Method: OLS OLS FE FE FE FE

Controls: (1) (2) (3) (4) (5) (6)

i.worker

i.job

i.year

i.industry

i.occupation

i.y# i.ind# i.occ

Observations 58026 57932 54303 54200 37613 37462

Std of residual .528 .513 .321 .315 .247 .241

Conservative benchmark: match Std of residual = 0.241, σ = 2.4

Alternative calibration: match Std of residual = 0.315, σ = 5.5

Robustness check: performance pay



Additional Evidence

transport

public

professional

entertain

service

financial

retail

manufacture

construct

mining

agriculture

Industry with higher individual risk −→ use performance pay more often



Part IV: Steady State Analysis

Exercise 1: unemployment rate

Exercise 2: wage dispersion

Exercise 3: individual wage dynamic



Exercise 1: Elasticity of Unemployment w.r.t aggregate productivity z

0.9 0.95 1 1.05 1.1

6%

7%

8%

9%

Unemployment is more responsive to aggregate productivity with moral hazard



Incentive Constraint

With a larger aggregate labor productivity, incentive constraint is relaxed

Output becomes more informative about worker’s effort A

dYt = zAtdt+ σdBt

◦ for a target A, shirking is easier to be detected with a large z

Relaxation of incentive constraint will increase firms’ profit

z κ(Wt) ≥ φ′(A)



Amplification Through Incentive

-0.5

0

0.5

1

1.5

-0.5

0

0.5

1

1.5

z ↑ −→ standard efficiency gain


zA(W )− C(W ) → Without moral hazard

+ F′(W ) [r (W − u (C) + φ (A))− (Wu −W )λ]

with moral hazard ← +F ′′ (W )

2︸︷︷︸<0

r2

(φ′ (A)

z

)2

σ2




-0.5

0

0.5

1

1.5

-0.5

0

0.5

1

1.5

z ↑ −→ standard efficiency gain + additional incentive gain


zA(W )− C(W ) → Without moral hazard

+ F′(W ) [r (W − u (C) + φ (A))− (Wu −W )λ]

with moral hazard ← +F ′′ (W )

2︸︷︷︸<0

r2

(φ′ (A)

z

)2

σ2




0.9 0.95 1 1.05 1.1

6%

7%

8%

9%

More responsive profit function −→ More responsive vacancy posting



Exercise 2: Wage Dispersion and Aggregate Productivity

0.9 0.95 1 1.05 1.11

1.2

1.4

1.6

1.8

z ↓ −→ higher wage dispersion measured by std [log CA]



Wage Dispersion and Aggregate Productivity

With a lower z, moral hazard becomes more severe

Promised utility Wt subject to greater exposure to dBt, more dispersed

dWt = [r (Wt − u (Ct) + φ (At))− (Wu −Wt)λx] dt+ rφ′ (At)

zσdBt

Wage distribution inherits the properties of Wt



Exercise III: Wage Scarring Effects after Job Loss

Average yearly wage loss upon displacement: 40%

dWt = [r (Wt − u (Ct) + φ (At))− (Wu −Wt)λx]︸︷︷︸in general>0

dt+ rφ′ (At)

zσdBt

Previous empirical findings: wage drops 15% - 40% after job displacement

(Jacobson et al., 1993; Couch and Placzek, 2010; Barnette and Michaud, 2011)



Part V: Dynamic Model Analysis



Extension with Aggregate Risk

Quantify insights from comparative statics by adding stochastic zt

dYt = ztAtdt+ σdBt

◦ zt is perfectly observable

Challenge: potentially much more state variables are required

dut = λ(1− ut)dt− p(θt)utdt

Solution: block recursive equilibrium Menzio and Shi (2011)

◦ only need to keep zt as additional state variable



Dynamic Model

Productivity shock zt ∈ {zH , zL} with switching intensity %(zH) and %(zL)

Operating firms’ recursive problem

(r + λ+ %(z))F (W, z) = maxA,C,ι

zA(W, z)− C(W, z)

+ F ′ (W, z) [r (W − u (C(W, z)) + φ (A(W, z)))− ι(W, z)%(z)− ...]

+F ′′ (W, z)

2r2(φ′ (A(W, z))

z

)2

σ2

+ %(z)F (W + ι(W, z), zc)

Unemployed workers’ problem

rWu (z) = maxX∈W

u (b) + p(θ) [X −Wu (z)] + %(z) [Wu (zc)−Wu (z)]



Quantitative Result I: Unemployment Volatility

Unemployment volatility is 2− 6 times larger considering moral hazard

Common difficulty in generating high employment volatility Shimer (2005)

Moments std(z) autocorr(z) std(u) autocorr(u)

Data 0.02 0.88 0.190 0.93

No moral hazard 0.02 0.88 0.016 0.73

Benchmark 0.02 0.88 0.028 0.84

High individual volatility 0.02 0.88 0.095 0.83



Quantitative Result II: Counter-Cyclical Wage Dispersion

Wage dispersion is countercyclical

◦ wage disper. is negatively correlated with the GDP

◦ wage disper. in downturn > wage disper. in upturn

Consistent with empirical findings in Storesletten, Telmer, and Yaron (2004)

Moments Baseline

corr(GDP, wage disp.) -0.45

autocorr(wage disp.) 0.79

wage disp.|zL 1.36

wage disp.|zH 1.19



Part IV: Policy Implications



Unemployment Insurance

An increase of UI improves workers’ outside option

◦ without moral hazard: workers switch to better jobs but wait longer

◦ with moral hazard: limit firms’ ability to punish workers, tighten IC constraint

0.1 0.2 0.3 0.4

5%

10%

15%



Minimum Wage

Conventional wisdom: only affects workers at bottom

Without moral hazard: no workers are affected

◦ wage equals average productivity with long-term contract

With moral hazard: all workers are affected

◦ wage depends on history of performance



Minimum Wage

Additional cost: more difficult to provide incentive

0 0.1 0.2 0.3

7.5%

8%

8.5%

0 0.1 0.2 0.3

0.96

0.97

0.98

0.99



Conclusion

Incorporate repeated moral hazard into a search framework

Measure the underlying information friciton using micro data

Quantify the effects of moral hazard on labor market dynamics

◦ higher unemployment volatility with aggregate productivity shocks

◦ larger wage dispersion in business cycle downturn

Promising to see interaction between dynamic contract theory and GE


Barnette, J., and A. Michaud (2011): “Wage scars from job loss,”Discussion paper.

Couch, K. A., and D. W. Placzek (2010): “Earnings losses of displacedworkers revisited,” American Economic Review, 100(1), 572–89.

Grochulski, B., and Y. Zhang (2019): “Termination as an incentivedevice,” .

Hagedorn, M., and I. Manovskii (2008): “The cyclical behavior ofequilibrium unemployment and vacancies revisited,” American EconomicReview, 98(4), 1692–1706.

Hall, R. E. (2005): “Employment fluctuations with equilibrium wagestickiness,” American economic review, 95(1), 50–65.

Jacobson, et al. (1993): “Earnings losses of displaced workers,” TheAmerican economic review, pp. 685–709.

Menzio, G., and S. Shi (2011): “Efficient search on the job and the businesscycle,” Journal of Political Economy, 119(3), 468–510.

Moen, E. R. (1997): “Competitive search equilibrium,” Journal of politicalEconomy, 105(2), 385–411.

Moen, E. R., and A. Rosen (2011): “Incentives in competitive searchequilibrium,” The Review of Economic Studies, 78(2), 733–761.

Sannikov, Y. (2008): “A continuous-time version of the principal-agentproblem,” The Review of Economic Studies, 75(3), 957–984.


Shimer, R. (2005): “The cyclical behavior of equilibrium unemployment andvacancies,” American economic review, 95(1), 25–49.

Spear, S. E., and S. Srivastava (1987): “On repeated moral hazard withdiscounting,” The Review of Economic Studies, 54(4), 599–617.

Storesletten, K., C. I. Telmer, and A. Yaron (2004): “Cyclicaldynamics in idiosyncratic labor market risk,” Journal of political Economy,112(3), 695–717.

Van Nieuwerburgh, S., and L. Veldkamp (2006): “Learning asymmetriesin real business cycles,” Journal of monetary Economics, 53(4), 753–772.


Equilibrium

Given F (W ), workers determine W ?0 and Wu

Given Wu, firms determine F (W )

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Estimation

Dependent: log( Real performance payHour )

Method: OLS OLS FE FE FE FE

Controls: (1) (2) (3) (4) (5) (6)

i.worker

i.job

i.year

i.industry

i.occupation

i.y# i.ind# i.occ

Observations 13302 13146 11888 11677 10947 10712

Std of residual .707 .679 .542 .516 .519 .491

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Competitive Search with Repeated Moral Hazard · IntroductionModelSteady StateAggregate RiskPolicy...

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