On-the-job Training and On-the-job
Search: Wage-Training Contracts in a
Frictional Labor market
Seung-Gyu (Andrew) Sim
University of Tokyo

December, 2014

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Introduction
Motivation
Previous Literature

The Model
Primitives
Steady State Equilibrium
Efficiency Analysis

Quantitative Analysis
Steady State
Comparative Statics

Conclusion

Seung-Gyu (Andrew) Sim (University of Tokyo)

December, 2014

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Motivation
Human Capital Accumulation (post schooling)the major contributor individual wage growth and economic growth
the first best outcome: Workers should pay for the cost of general training.
In reality,
only firms can provide general training in many cases andworkers cannot commit to staying with the training firms
This paper studies the coexistence of On-the-job Training and Search
Do productive firms provide more training?
Do firms provide the efficient level of training?
Do firms provide more training, as search friction is mitigated?

Seung-Gyu (Andrew) Sim (University of Tokyo)

1 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Illustrative Examples
period 1

period 2worker:

w1

w2

firm:

y − x − w1y + f (x) − w2

Becker (1964): Perfect Competitionx B ∈ arg max −x + f (x)w2 = y + f (x B )w1 = y − x B

π2 = 0π1 = 0Under perfect competition, the firm provides the efficient level of training,
and the worker pays the training cost through lower wage during training.

Seung-Gyu (Andrew) Sim (University of Tokyo)

2 / 22

On-the-job Training and On-the-job Search
IntroductionThe Model

Quantitative Analysis

Conclusion

Illustrative Examples
period 1

period 2

worker:

w1

w2

firm:

y − x − w1

y + f (x) − w2

Acemoglu (1997): exogenous job-turnover shockx A ∈ arg max −x + (1 − α)f (x)w2 = φ(y + f (x A ))w1 = φ(y − x A )π2 = (1 − α)(1 − φ)(y + f (x A ))p

π1 = (1 − φ)(y − x A )A

π = α(1 − φ)(y + f (x ))positive externality for subsequent poaching firms (free rider problem)

Seung-Gyu (Andrew) Sim (University of Tokyo)

3 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Previous Literature
Moen and Rosen (2004)
no on-the-job search by unskilled workers
no skilled unemployed workers
no productivity differential
Fu (2011)
incorporates the piece rate sharing rule into Burdett and Mortensen (1998)
ends up with inefficient level of training
Sanders and Taber (2012)
over-investment on job specific human capital
under-investment on general human capital

Seung-Gyu (Andrew) Sim (University of Tokyo)

4 / 22

On-the-job Training and On-the-job SearchIntroduction

The Model

Quantitative Analysis

Conclusion

Introduction
Motivation
Previous Literature

The Model
Primitives
Steady State Equilibrium
Efficiency Analysis

Quantitative Analysis
Steady State
Comparative Statics

Conclusion

Seung-Gyu (Andrew) Sim (University of Tokyo)

December, 2014

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

The Basic Model
Burdett and Mortensen (1998) with productivity differentials
a unit measure of risk neutral (lifetime income maximizing) workers
A newly born worker enters the labor market as unskilled and unemployed.
The unemployed worker gets employed at rate λ0 .
The employed worker finds another job at rate λ1 and gets laid off at rate δ.
The employed worker acquires (general) skills at rate µx through training.
All workers retire at rate ρ and they are replaced with newly born workers.
a unit measure of heterogenous firms (p ∼ H(p))
Each firm maintains one vacancy at every instant.
The recruiting firm with p posts (Eu (p), Es (p)) = ((wu (p), x(p), Est (p)), (ws (p))).
It meets an employed searcher at rate λ1 and unemployed searcher at rate λ0 .
ε-measure of noise firms
They offer only skilled wages from F̂n : [p + s, p + s] → [0, 1].
Seung-Gyu (Andrew) Sim (University of Tokyo)

5 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

The Basic Model
Unemployed Workers
retire at rate ρ, and get employed at rate λ0 .
rUi = b − ρUi + λ0R

max{z − Ui , 0}dFi (z), for each i ∈ {u, s}Skilled Employed Workers
retire at rate ρ, get laid off at rate δ, and find offers at rate λ1 .
rEs (p) = ws − ρEs (p) + δ(Us − Es (p)) + λ1

Rmax{z − Es (p), 0}dFs (z)Unskilled Employed Workers
retire at rate ρ, get laid off at rate δ, find offers at rate λ1 , andacquire (general) skills at rate µx.
RrEu (p) = wu − ρEu (p) + δ(Uu − Eu (p)) + λ1 max{z − Eu (p), 0}dFu (z)+µx(Est − Eu (p))

Seung-Gyu (Andrew) Sim (University of Tokyo)

6 / 22

On-the-job Training and On-the-job SearchIntroduction

The Model

Quantitative Analysis

Conclusion

The Basic Model
Operating Firms with Skilled Matchesdeliver the committed values through ...
rJs (p) = p + s − ws (p) − [ρ + δ + λ1 (1 − Fs (Es (p)))]Js (p)Operating Firms with Unskilled Matchesdeliver the committed values through ...
rJu (p) = max p − wu − c(x) − [ρ + δ + λ1 (1 − Fu (Eu (p)))]Ju (p)wu , x, Est+µx(Js (p) − Ju (p))

subject to the promise-keeping constraint on Eu (p).
F.O.C.
wst (p) = p + sc 0 (x) = µ(Est (p) − Eu (p) − Ju (p))
The promise keeping constraint determines unskilled wages.
Seung-Gyu (Andrew) Sim (University of Tokyo)

7 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

The Basic Model
Recruiting Firms
post (Eu (p), Es (p)) to maximize[λ0 us + λ1 Gs (Es )]Js (Es , p) + [λ0 uu + λ1 Gu (Eu )]Ju (Eu , p)

Seung-Gyu (Andrew) Sim (University of Tokyo)

8 / 22

On-the-job Training and On-the-job Search
Introduction

The ModelQuantitative Analysis

Conclusion

The Basic Model
Given firms’ productivity distribution H(p), a steady state equilibrium withon-the-job training and on-the-job search consists of value equations {Ui , Ei , Ji }compensation packages {(wu (p), x(p), Est (p)), (ws (p))} and steady statemeasures {Fi , Gi , ui } that jointly satisfy the following conditions.
(i) Given Fi , workers make optimal job turnover decision.
(ii) Given {Fi , Ei }, operating firms optimally deliver the committed values.
(iii) Given {Gi , ui }, recruiting firms post their contract to maximize their profit.
(iv ) {Fi , Gi , ui } are stationary and consistent with the behavior of each agents.

Seung-Gyu (Andrew) Sim (University of Tokyo)

9 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Steady State Equilibrium

Figure 1: Equilibrium Support of Wages
Seung-Gyu (Andrew) Sim (University of Tokyo)

10 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Steady State Equilibrium
Proposition 1 The optimal training intensity is characterized byc 0 (x(p))(r + ρ + δ)/µ + x(p)c 0 (x(p)) − c(x(p)) = s + δ(Us − Uu )RERE+λ1 E ts(p) [z − Est (p)]dFs (z) − λ1 Euu(p) [z − Eu (p) − Ju (p)]dFu (z)sIn particular, x(p) < x(p) for any p ∈ [p, p) if and only if
R Es

Est (p)[z − Est (p)]dFs (z) >R Eu
Eu (p)

Seung-Gyu (Andrew) Sim (University of Tokyo)[z − Eu (p) − Ju (p)]dFu (z)11 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Efficiency Benchmarks
Constrained Social Plannermaximizes the present value of the expected output flow throughout the lifeof a newly born worker in the steady state equilibrium.
Rp(r + ρ)Ss∗ (p) = p + s + δ(Us∗ − Ss∗ ) + λ1 p [Ss∗ (p 0 ) − Ss∗ (p)]dH(p 0 )(r + ρ)Su∗ (p) = p − c(x ∗ (p)) + δ(Uu∗ − Su∗ ) + µx ∗ (p)(Ss∗ (p) − Su∗ (p))
Rp+λ1 p [Su∗ (p 0 ) − Su∗ (p)]dH(p 0 )Rp(r + ρ)Ui∗ (p) = b + λ0 p [Si∗ (p 0 ) − Ui∗ (p)]dH(p 0 )chooses the training intensity such that c 0 (x ∗ (p)) = µ(Ss∗ (p) − Su∗ (p))(vsSeung-Gyu (Andrew) Sim (University of Tokyo)

c 0 (x(p)) = µ(Est (p) − Eu (p) − Ju (p)))

12 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Efficiency Benchmarks
Proposition 2 The training intensity in the social planner’s problem
is characterized byc 0 (x ∗ (p))(r + ρ + δ)/µ + x ∗ (p)c 0 (x ∗ (p)) − c(x ∗ (p)) = s + δ(Us∗ − Uu∗ )In particular, dx ∗ /dp = 0, dx ∗ /dλ1 = 0, and dx ∗ /dλ0 > 0

Seung-Gyu (Andrew) Sim (University of Tokyo)

13 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Efficiency Benchmarks
In the market equilibrium,c 0 (x(p))(r + ρ + δ)/µ + x(p)c 0 (x(p)) − c(x(p)) = s + δ(Us − Uu )
+λ1

R Es

Est (p)[z − Est (p)]dFs (z) − λ1

R Eu
Eu (p)[z − Eu (p) − Ju (p)]dFu (z)In the social planner’s problem,c 0 (x ∗ (p))(r + ρ + δ)/µ + x ∗ (p)c 0 (x ∗ (p)) − c(x ∗ (p)) = s + δ(Us∗ − Uu∗ )Seung-Gyu (Andrew) Sim (University of Tokyo)

14 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Efficiency Benchmarks

Figure 2: Training Intensity

Seung-Gyu (Andrew) Sim (University of Tokyo)

15 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Introduction
Motivation
Previous Literature

The Model
Primitives
Steady State Equilibrium
Efficiency Analysis

Quantitative Analysis
Steady State
Comparative Statics

Conclusion

Seung-Gyu (Andrew) Sim (University of Tokyo)

December, 2014

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Baseline Simulation
Table 1: Parameter Values[p, p] = [0.75, 1.75]
η = 1.0s = 0.25
γ = 2.0r = 0.012
ρ = 0.008
δ = 0.064
λ0 = 1.35
λ1 = 0.45

the productivity support
the shape parameter of H(p)productivity improvement through training
cost function parameterinterest rate
retirement rate
separation rate
job finding rate by unemployed workers
job finding rate by employed workers

Productivity Distribution: H(p) =1 − (p/p)η
1 − (p/p)η

cost function: c(x) = x γ

Seung-Gyu (Andrew) Sim (University of Tokyo)

16 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Baseline Simulation

Figure 3: The Baseline Simulation Result I
Seung-Gyu (Andrew) Sim (University of Tokyo)

17 / 22

On-the-job Training and On-the-job Search
IntroductionThe Model

Quantitative Analysis

Conclusion

Baseline Simulation

Seung-Gyu (Andrew) Sim (University of Tokyo)

18 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Comparative Statics

Figure 5: Training Intensity

Seung-Gyu (Andrew) Sim (University of Tokyo)

19 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Comparative Statics

Figure 6: Training Intensity
Seung-Gyu (Andrew) Sim (University of Tokyo)

20 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

IntroductionMotivation
Previous Literature

The Model
Primitives
Steady State Equilibrium
Efficiency Analysis

Quantitative Analysis
Steady State
Comparative Statics

Conclusion

Seung-Gyu (Andrew) Sim (University of Tokyo)

December, 2014

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Conclusion
Human Capital Accumulation (post schooling)
the major contributor individual wage growth and economic growth
the first best outcome: Workers should pay for the cost of general training.
In reality,
only firms can provide general training in many cases andworkers cannot commit to staying with the training firms
This paper studies the coexistence of On-the-job Training and Search
Hump-shaped training intensity
over-intensified general training
Mitigating search friction intensifies training but improves net output.

Seung-Gyu (Andrew) Sim (University of Tokyo)

21 / 22

On-the-job Training and On-the-job Search
Introduction

The Model

Quantitative Analysis

Conclusion

Thanks for listening!
Seung-Gyu (Andrew) Sim
University of Tokyosgsim@e.u-tokyo.ac.jp

December, 2014