Lecture 2: Frictional unemployment

II. Efficiency

Can unemployment be efficient

• In efficiency wage models and I/O models, unemployment is inefficient– We want to reduce the insider bargaining

power– We want to make contracts complete to

eliminate rents

• In matching models, unemployment is a productive input into the creation of new jobs

Too much or too little unemployment?

• To have a low unemployment rate, vacancies must be high

• But vacancies consume resources we can have too many vacancies

• On the other hand, the unemployed are also costly because they do not produce we can have too much unemployment

Congestion externalities

• An increase in search exerts positive externalities on the other side of the market– Unemployment reduces the length of

vacancies; vacancies reduce the length of unemployment

• Because m() is concave, negative externality on the same side of the market– More unemployment reduces job finding rate

Appropriability

• As in the I/O model, a surplus arises from turnover costs

• Part of the surplus is appropriated by the worker the firm only appropriates a fraction of the surplus created by a vacancy

• Symmetrical problem on the worker’s side on evaluating the value of being unemployed: only a fraction of the value of future jobs will be appropriated

What to do next?

• Have a more complex wage formation model, borrowed from the dynamic I/O model

• Derive equilibrium conditions

• Set up a dynamic welfare maximization problem

• Compare its optimality conditions to the equilibrium conditions

The wage bargaining problem

)(1

)ln()1()ln(max

)()(

))((

)(

Vue

Vuew

uueu

eeue

VJVV

VJVV

tJtsJwyrJ

VVVqbrV

VVVswrV

How is θ determined?

substitutecan we;)()1(1

)(

)1(;

]0[

ismatch theof surplusnet totalThe

q

cJW

WWqsWbyrW

WJWVV

VVVJW

ue

Vue

The evolution of θ:

1

)(')()1(

)(

)()1(

)(2

cq

q

csrby

q

csr

The welfare problem

tu

usvumeecvuyH

usvumu

dtecvuy

tttrt

trt

ttt

tttt

rttt

vt

at )1( job a of valuemarginal

))1(),(())1((

)1(),( s.t.

))1((max0

The F.O.C:

job a of value

social marginal for theequation eAsset valu

)(

),(0

tttt

rttt

t

u

msyr

eru

Hv

vumc

v

H

Comparing equilibrium and optimum (b=0)

WWqsWyrW

W)q- (c

)(

:job a of value theofEvolution

)(1

:conditioncreation Job

mEquilibriu

tttt

t

u

msyr

v

vumc

:job a of value theofEvolution

),(

:conditioncreation Job

:Optimum

The congestion externality

• The benefit of a vacancy:– The SP looks at its marginal effect on job creation– The firm looks at the average probability of filling the

vacancy

• Future jobs:– The match looks at the opportunity cost of foregone

future other jobs, driven by average job finding probability

– The SP looks at the marginal effect of an extra unemployed on job creation

The appropriability problem

• The benefit of a vacancy:– The SP looks at the whole value of the jobs λ– The firm only expects a fraction 1-φ of W

• Future foregone jobs:– The SP values their whole social value λ– The match considers that it loses only the

worker’s share φ W

When is the equilibrium efficient?

• Congestion externalities private transition rates > marginal effects on hirings too much unemployment and vacancies

• Appropriability problem firms and workers only appropriate a fraction of their job search effort not enough unemployment and vacancies

• If these two effects cancelled each other, equilibrium would be efficient

Assume eqm θ is efficient:

sign) opposite of bracketsin terms(

1 and

.)1(

1)1(

)(

)( e,Furthermor

.)1()1(

)(

)1()1(

vu

vuv

vv

v

u

msr

Wu

myWrs

u

mWyWrs

WWvmv

m

vm

c

The Hosios Condition:

1

v

u

Inefficient unemployment

• Because of congestion externalities and appropriability problems, the equilibrium is generally inefficient

• Unemployment is inefficiently low iff

u

What is going on?

• Little incentive to look for another job as I appropriate little of the surplus

• But unemployment is useful in creating new jobs:– Vacancy costs would be reduced

• This effect is sizable as ηu is large

What is the market failure?

• Congestion externalities could be avoided if there was a price for entering the job search pool

• Appropriability could be avoided if wages could be contracted ex-ante

• Moen (1995) develops a « Club Theory » approach of matching and shows the outcome is efficient

Example 1: The Pond

)

:Optimum

/)(

:mEquilibriu

fishing ofcost

fish; of value

caughtfish ofquantity )(

fishermen; of #

vf'(Xc

XXvfc

c

v

Xf

X

Introduce N clubs

cXvf

cNxvfxcvNxfNx

N

cv

A

AxAfx

)('

)('))(1

max(

:members ofutility total themaximize and size their controlThey

pond theof /1 controls clubA

fishing ofcost fish; of value

size of areain

caughtfish ofquantity )/( fishermen; of #

Example 2: The hold-up problem

1)(')(max

investors across same theis Wage

wagespostingby compete Investors :2 Case

1)(')1 toupinvest I

)( gets Workers

cake on the bargainingpost -Ex :1 Case

itexploit worker toa need I

)( size of cake a create toinvest I

ICIwIC

IC-(

IC

ICI

Competitive search equilibrium:

• Firms and workers must join club to match

• Each club characterized by– Its local tightness θ– Its local wage w(θ)

• In CSE:– Workers choose their club optimally– Each nonempty club yields zero value to firms

The firms’ participation decision

)(

)()(

)()()(0

)()(

q

srcyw

JqcV

Jsr

wyJ

v

The worker’s choice of a club

0)(

)('

))()((

)()('

0)(':FOC

)(

)(

)()(

))()()(()(

))(()()(

)(max

u

u

uu

ueu

uee

uu

Vsw

w

qrq

qqr

V

sr

Vsw

qr

qV

VVqrV

VVswrV

VV