labor in rbc model, stylized facts, wage posting€¦ · postel vinay and robin (2002) contraction...

BackgroundEmpirical evidence on labor market stocks and �ows

Search and Matching modelsWage postingEstimation

Postel Vinay and Robin (2002)Contraction Mappings

Labor in RBC model, stylized facts, wage posting

Pieter Gautier (Vrije Universiteit Amsterdam)

January 11, 2011 CES Ifo

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Introduction RBCIntroduction RBCLabor in the standard Real Business Cycle ModelThe goalSolving the RBC modelBusiness cycle factsIndivisible labor

Nobel prize 2011

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Realism in economic models

"Nothing is less than realism ... Details are confusing.It isonly by selection, by elimination, by emphasis, that we get atthe real meaning of things." (Georgia O�Keefe)

Economics is partly an art, partly a science.

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3 lectures macro labor

1 Labor in RBC model, stylized facts, partial search2 Wage posting, Burdett and Mortensen3 (Diamond-Mortensen-Pissarides) Matching models

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Introduction RBC

What do standard RBC models predict with respect to thelabor market?

Stylized facts of worker and job stocks and �ows

Basics of search and matching

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Labor in the standard Real Business Cycle Model

Point of departure

How far can we get in explaining business cycle withWalrasian model (no frictions)?

We need

shocks (otherwise convergence to balanced growth in Ramseymodel)

variations in employment (working time)

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Think as starting point of average person (representativeagent)

What happens if the agent experiences a positive technologyshock?

Production " .C ", I "! more capital next period

2 e¤ects on labor supply: w "

1 substitution e¤ect: work harder to exploit high wage period2 income e¤ect: use part of income gain to "buy" leisure

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2 e¤ects on saving rate:

1 substitution e¤ect: s # (Kt+1 "! rt+1 #, s #)2 income e¤ect: income ", s "

In the most simple RBC models, income and substitutione¤ects cancel.

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Y is output

Output:

Yt = K αt (AtLt )

1�α

Yt = Ct + It

Technology follows a stochastic process:

lnAt = A+ gt + eAteAt = ρAAt�1 + εA,t

We abstract from population growth9 / 102






Kt+1 = (1� δ)Kt + It

wage (w) is price for labor

interest (r) is price for capital

Both are determined at aggregate level and are paid theirmarginal product

Why not more? Reduces pro�ts

Why not less? Competition

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wtMCL

=∂Yt∂Lt

= (1� α)

�KtAtLt

�α

At

MPL

rtMCK

+ δdepr .

= α

�AtLtKt

�1�α

MPK

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Households

simpli�cation: Nt single person households:

Normalize Nt to 1 and assume no growth of population

Representative agent:

maxU = E∞

∑t=0

βtu (ct , 1� lt ) (1)

instantaneous utility of consumption, assume:

ut = ln ct + b ln(1� lt ) (2)

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The goal

exogenous variables: K0 and fεA,tg∞t=0

endogenous variables: fYtg∞t=0 , fKtg

∞t=0 , fctg

∞t=0 , fltg

∞t=0

solve endogenous variables as functions of exogenous variables

then check whether the endogenous variables display�uctuations around the non-stochastic trend that correspondto the data in terms of magnitude, persistence andco-movement with GDP.

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(1) Intertemporal substitution labor supply

labor supply today vs tomorrow

relevant for current discussion about retirement

maximization problem for 2 periods (Lagrange)

L = ln c1 + b ln(1� l1)utility in period 1

+ βt [ln c2 + b ln(1� l2)]discounted utility in period 2

+λ

24w1l1 + 11+ r

w2l2 � c1 �1

1+ rc2

budget constraint

35

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This can be extended to n periods. FOC:

Ll1 = 0

Ll2 = 0

This yields:1� lt1� lt+1

=1

β(1+ r)wt+1wt

(3)

work more today if your wage is relatively high, and or theinterest rate is high (you can save your wage) and or β is high(discount rate is low).

Because of the logarithmic utility, the elasticity of substitutionbetween leisure in the 2 periods is 1.

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(2) Household inter-temporal FOC�s (trade-o¤ current andfuture consumption

uc (ct , 1� lt ) = βEt [(1+ rt+1)uc (ct+1, 1� lt+1)]

(3) household intra-temporal FOC�s (trade-o¤ consumption,leisure)

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uc (ct , 1� lt )wt = ul (ct , 1� lt ) (*)

Barro and King (1984): necessary condition for ct and lt to beboth procyclical is that wt is procyclical. I.e. (*) implies

1ctwt �

b(1� lt )

= 0

ct =wt (1� lt )

b

lt is strongly procyclical, ct is mildly procyclical so wt must bestrongly procyclical (while it is mildly pro cyclical)

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Solving the RBC model

Log linearize endogenous variables around the steady stategrowth path.

Solve numerically (only special case with logarithmic utilityand δ = 1) can be solved analytically.

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Business cycle facts

Detrend data with Hodrick Prescott �lter

Choose smooth trend, τt , according to

minτt

T

∑t=1

h(yt � τt )

2 + λ ((τt+1 � τt )� (τt � τt�1))2i

the residual yt � τt is the business cycle component

λ = 1600 typically for quarterly data and λ = 13000 formonthly data. λ punishes for non smooth trend.

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Performance basic RBC

Hansen and Wright (1992, FED, Minneapolis QR )US data Baseline RBC

σy 1.92 1.30σc/σy 0.45 0.31σi/σy 2.78 3.15σl/σy 0.96 0.49

Employment �uctuates more than RBC model predicts.

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Indivisible labor

Hansen (1985, JME), Rogerson (1988, JME)

Typically you can either work 8 hours a day or not (�xed costof working or production restriction)

l 2 f0, l̂gCan we still use the rep. agent framework?

Yes, assume employment lotteries that divide people betweenemployment and unemployment randomly

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Indivisible labor

Indivisibility increases the labor input response to shocksbecause an increase in Lt , linearly decreases utility(ln�1� l̂

�< 0)

while ln�1� Lt

Nt

�is decreasing at a decreasing rate.

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Indivisible labor

performance of RBC labor with indivisible labor

US data RBC RBC+indiv. labσy 1.92 1.30 1.73σl/σy 0.96 0.49 0.76If you are only interested in movement of employment andoutput over time, the RBC models work �ne.

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Critique

This is arbitrary set of variables, why not include labor markettightness and wage dispersion?

.......and stocks are just top of iceberg

Large scale gross job and worker �ows cannot be explainedwith competitive rep. agent models

Allowing for search frictions can overcome those weaknesses

Andolfatto (1996, AER) and Merz (1995) show thatintroducing search frictions in the RBC models increasesemployment �uctuations.

den Haan, Ramey and Watson (2000, AER) show thatendogenizing job destruction together with capital adjustmentcost increase ampli�cation and persistence of output shocks.

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StocksFlowsWages

Empirical evidence on labor market stocks and �ows

Stocks: (unemployment, employment, vacancies)

Flows: employment in- and out�ow, job creation, jobdestruction

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StocksFlowsWages

Stocks

Between 1951-2003, in a typical month, 5.67% of US laborforce was unemployed (available for and actively seekingwork), Shimer (2004) .

For European countries this fraction was till the 82-83recession the same and then increased to 7-10%

Ljungqvist, Sargent (1998) argue that increased turbulenceincreased unemployment in Europe because long bene�tdurations cause long term unemployment and skilldepreciation.

Huge variability: di¤erence between log unemployment and(HP, λ = 105) trend is 0.19

Persistence: quarterly autocorrelation is 0.94.

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StocksFlowsWages

Unemployment in- and out�ow

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StocksFlowsWages

De�nitions of Flows

E int : # workers who leave unemployment and enteremployment in period t

E outt :# workers who leave employment and enterunemployment in period t

S+ set of growing �rms (in terms of employment)

EMPest = # workers at employer e in sector s in period t.

JCst = ∑eεS+ ∆EMPest (job creation)JDst = ∑eεS� ∆EMPest (job destruction)JRst = JCst + JDst (job reallocation)

EXCst = JRst � jNETst j (excess job reallocation)

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StocksFlowsWages

Labor market �ows

Shimer (2007)

For period between 1967-2004, monthly UE �ow is 39% andmonthly EU �ow is 2%,implying, mean duration:1/0.39 = 2.56 months and st.st.u = EU

EU+UE = 4.88%

Davis and Haltiwanger HBL 3B, p.2756,yearly �ows as fraction ofemployment

Country E in JC E out JD JC/E in JD/E out

USA 18.4 9.0 18.7 9.3 49.1 49.7Denmark 28.5 12.0 28.0 11.5 42.1 41.1Netherlands 16.3 7.3 15.7 8.3 44.8 52.9Germany 4.4 3.7

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StocksFlowsWages

Labor market �ows

Job reallocation is mainly driven by idiosyncratic shocks

There is still debate on whether JR is countercyclical

var(JD)>var(JC )

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StocksFlowsWages

Wages

Pissarides (2007), wages of workers who are newly hired in tare as cyclical as y while existing wages are half as cyclical.

Rich (Mincer type) wage equations typically only explain30-40% of variance in log wages

I.e. similar workers earn di¤erent wages (law of 1 price breaksdown)

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Dynamic programming in a simple partial job search modelThe reservation wageContinuous timeDuration analysis and search modelsComparative statics

Requirements

We want model based on sound micro foundations that jointlydetermines equilibrium employment in- and out�ow, thestocks of unemployment, vacancies and equilibrium wages?

Is the resulting outcome e¢ cient?

Is there a potential role for minimum wages and/orunemployment bene�ts?

For simplicity we ignore capital and savings and focus on thelabor market.

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With micro-foundations we mean here: individual optimizationin search environment

Dynamic programming is a useful tool to characterize thisoptimization process

(i) partial model, (ii) wage dispersion and then bargainingmodels and (iii) cyclicality of wages and unemployment

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Dynamic programming in a simple partial job search model

�rst discrete time then continuous time

workers know the wage distribution F (w)

jobs, once accepted last forever, no on the job search

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Value of employment

When to accept a job o¤er?

Wage income,

E∞

∑t=0

βtwt

0 < β < 1 is the discount factor

wt is the wage in period t

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Value of employment

worker receives one (i.i.d.) wage o¤er from a cdf F (w) perperiod

accept or reject?

Rejection!unemployedAcceptance!job foreverAssume stationary environment

value functions for working (VE (w)):

VE (w) = w∞

∑t=0

βt =w

(1� β)

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Value of unemployment

unemployment (VU )

VU = b+ βE [max [VU ,VE ]]

b are bene�ts plus (dis)utility of leisureWhat is the value of a wage o¤er, O(w)?Explore recursive relationBellman equation:

O(w) = max [VU ,VE (w)] = (1)

max�b+ βEO,

w(1� β)

�.

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Existence. uniqueness of reservation wage

Existence of reservation wage can be shown as follows

b+ βEO does not vary with ww

(1�β)is monotonically increasing in w .

There is a unique point where both lines cross.

worker is indi¤erent between working or remainingunemployed.

This is reservation wage, wr .

worker accepts all w � wr and rejects the rest.

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Reservation wage

derivation wr :

VE (wr ) � VUwr

(1� β)= b+ βEO,

wr = (1� β)b+ β(1� β)EO (2)

Next, eliminate EO by substituting in VU = VE (wr ) � wr(1�β)

in (1):

O(w) =

(wr

(1�β)for w < wr

w(1�β)

for w � wr.

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Reservation wage

substitution in (2) gives the following mapping:

wr = T (wr ) = (1� β)b+ βZ ∞

0max [w ,wr ] dF (w) ((3))

T is a contraction mapping (see appendix notes)Contraction mappings have following properties:there is a unique solution to wr = T (wr )can be found by repeated iteration.I.e. w1r = T

�w0r�, ...w ir = T

�w i�1r

�.

In the limit w ir � T�w i�1r

�! 0.

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Reservation wage

wr and VU are expressed in many ways in the literature:

VU = b+ βZ ∞

wrVE (w)dF (w)

(1� β)VU = b+ β (4)Z ∞

wr[VE (w)� VU ] dF (w)

wr = b+ (5)β

1� β

Z ∞

wr[w � wr ] dF (w)

(4) has a clear economic intuition.current income, b plus his expected discounted wealthimprovement in the next period.

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Reservation wage

Finally, use integration by parts to rewrite

Z w̄

wr[w � wr ] dF (w) = [w � wr ] jw̄wr �1�

Z w̄

wrF (w)dw

[w̄ � wr ] � 1�Z w̄

wrF (w)dw

=Z w̄

wr[1� F (w)] dw (6)

wr = b+β

1� β

Z ∞

wr[1� F (w)] dw

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From discrete to continuous time

allow for the possibility that the worker receives more o¤ers

most conveniently in continuous time

Suppose that the length of a period is ∆discount β = 1

1+∆r .

let l (n,∆) be the probability of n o¤ers in this periodlet G (w , n,∆) be the distribution of accepted o¤ers.

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we can generalize (4) as

r∆1+ r∆

VU = b∆+1

1+ r∆

∞

∑n=1

l (n,∆) (7)Z ∞

wr[VE (w)� VU ] dG (w , n,∆)

�rst multiply both sides by 1+ r∆ then divide by ∆.make ∆ arbitrarily smallprobability to receive 2 or more job o¤ers! 0arrival of o¤ers is Poisson process with parameter λ :

l(1,∆) = λ∆+ o (∆) and∞

∑n=2

l (n,∆) = o (∆)

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P(N = n) = λn exp(�λ)n!

o (∆) (pronounce small "o" delta) means lim∆!0o(∆)

∆ ! 0let ∆ ! 0 in (7),gives continuous time Bellman equation forunemployment:

rVU = b+ λZ ∞

wr[VE (w)� VU ] dF (w)

(1� β)VE (w) = w

lim∆!0

�∆r

1+ ∆r

�VE (w) = lim

∆!0w∆

rVE (w) = w

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Continuous time Bellman equations

Value of working:

VE (w) = wr and

VU � wrr . Then,

wr = b+ λZ ∞

wr

hwr� wrr

idF (w)

rVU = b+λ

r

Z ∞

wr[w � wr ] dF (w) (8)

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Duration analysis and search models

probability to exit unemployment is called the hazard rate

equals the probability that w � wr which equals

h = λ (1� F (wr ))

expected duration of unemployment is:

D =1h=

1λ (1� F (wr ))

(9)

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stationary environment! h does not depend on the elapsedduration of unemployment

.in the stationary environment that we consider, the durationof unemployment has an exponential distribution withparameter h

structurally estimate λ.

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Changes in b

Suppose we increase unemployment bene�ts b

makes workers more choosy and increases the reservation wage

consequently reduces the probability that a given draw fromF (w) is acceptable.

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Changes in b

From (8) and using Leibniz�s rule, we get:

rVU = wr = b+λ

r

Z ∞

wr[1� F (w)] dw (10)

dwrdb

= 1� λ

r[1� F (wr )]

dwrdb

dwrdb

=1�

1+ λr [1� F (wr )]

� 2 [0, 1]Further, since dF (wr )dwr

> 0 and dDdF (wr )

> 0, dDdwr > 0 anddDdb > 0

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Changes in job o¤er arrival rate

From (9), direct e¤ect of changes in λ on D is negative.

also positive e¤ect on duration, higher λ makes workerschoosier

How large is the total e¤ect?

Average duration decreases if h0(λ) > 0.

h0(λ) = (1� F (wr ))� λdF (wr )dwr

dwrdλ

(11)

= (1� F (wr ))� λf (wr )dwrdλ

.

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From (10) we get:

dwrdλ

=1r

Z ∞

wr[1� F (w)] dw

�λ

r[1� F (w)] dwr

dλ

dwrdλ

=

R ∞wr[1� F (w)] dwr + λF (wr )

> 0

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substitute in (11):

h0(λ) = (1� F (wr ))� λf (wr )

R ∞wr[1� F (w)] dwr + λF (wr )

.

For most distributions, h0(λ) > 0

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Diamond paradoxAlbrecht Axell modelBurdett MortensenThe model

Wage posting

Before we considered partial model and treated F (w) as given

Now we want to endogenize F (w)

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Diamond Paradox

Assume identical �rms and workers.

Workers set reservation wages according to

wr = b+λ

r

Z ∞

wr[w � wr ] dF (w)

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Diamond Paradox

No �rm has incentive to o¤er w > wrNash equilibrium strategy for all �rms is to o¤er wr = b.

Wage distribution becomes degenerate at b.

But then workers have no incentive to participate and themarket collapses.

In reality we do observe that "identical" workers earn di¤erentwages (R2 < 0.4 in "rich" wage equation)

How do we get out of the Diamond paradox?

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Albrecht Axell model

Two types of workers

fraction µ has high value of leisure and consequently highreservation wage

fraction (1� µ) has low value of leisure (and low reservationwage)

Firms face trade-o¤ between o¤ering high and low wage

This results in a mixed strategy equilibrium

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K di¤erent worker types with K di¤erent reservation wagesmakes �rms o¤er in equilibrium K wages, each of themcorresponding to a particular reservation wage.

Problem: an ε search cost will prevent the workers with thehighest value of non-employment from searching because theexpected pay-o¤ from search is �ε for them.

For the remaining K � 1 searchers there is a new group ofworkers with the highest value of non-employment for whomthe same argument can be made and so on and so forth.

Then we are back in the Diamond paradox.

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Alternatives to solve the Diamond paradox

! continuous wage distribution:

Burdett and Mortensen (discussed below). On the job search

Multiple applications

Directed search

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Burdett Mortensen model

Assumptions

ex ante identical workers and jobs

unemployed workers accept all job o¤ers that exceed theirreservation wage

employed workers continue searching on the job and accept allwages that exceed their current wage

worker strategy: �nd reservation wage given F (w)

�rm strategy: wage o¤er F (w)

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There exists a unique non-generate steady state wagedistribution to the wage posting game that the �rms play

mixed strategy equilibrium balances two forces:

1 high wage attracts more employed workers and reduces theprobability that workers at your �rm leave

2 low wage generate higher pro�ts if you have a worker.

There is equal pro�t (as is required for mixed strategy) over arange of wages

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continuous time

measure of identical workers m, measure of identical �rms isnormalized to 1

Workers can be in 2 states:

(0) unemployed or (1) employed with associated (Poisson) jobo¤er arrival rates: λ0 and λ1.

search is random, i.e. each o¤er is a random draw from: F (w)

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Bellman equation for unemployed worker

rVU = b+ λ0

�Z ∞

RVE (z)� VU

�dF (z)

Accept if VE (z) > VU .

Bellman equation for employed worker who currently earns w .

rVE (w) = w � δ [VE (w)� VU ] (2)

+λ1

�Z ∞

w[VE (z)� VE (w)] dF (z)

�

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Then from (1) and (2) and using

rVE (R) = rVU

b+ λ0

�Z ∞

RVE (z)� VU

�dF (z) (3)

= R + λ1

�Z ∞

R[VE (z)� VE (R)] dF (z)

�R � b = (λ0 � λ1)

�Z ∞

R[VE (z)� VU ] dF (z)

�

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use integration by parts to write:Z ∞

R[VE (z)� VU ] dF (z) = [VE (z)� VU ] j∞R �1�

Z ∞

RV0E (z)F (z)dz

=Z ∞

RV0E (z) [1� F (z)] dz

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complete di¤erentiation of (2) wrt w yields:

V 0E (w) =1

r + δ+ λ1 (1� F (w))Substitute in (3) gives:

R � b = (λ0 � λ1)Z ∞

R

1� F (z)r + δ+ λ1 (1� F (w))

dz

Typically: δ >> r . So we let r ! 0 and let κ0 =λ0δ and

κ1 =λ1δ :

R � b = (κ0 � κ1)Z ∞

R

1� F (z)1+ κ1 (1� F (w))

dz (4)

This is the crucial workers equation66 / 102






Steady state �ows!unemployment rate.Let 1 = e + u be labor force

unemployment in�ow is unemployment out�ow:

δ (1� u) = λ0 [1� F (R)] u

u =1

1+ κ0 [1� F (R)]

implies the following relation between u and e = 1� u

u1� u =

δ

λ0� 1

κ0(5)

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Let E (w) be the number of employed at jobs that pay lessthan w .

E (w) = (1� u)G (w)where G (w) is the distribution of accepted wages (not thesame as o¤er distribution, F (w))

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Steady state equation for E (w)

λ0 (F (w)� F (R)) u = (δ+ λ1 [1� F (w)])E (w) (6)

in�ow from unemployment is out�ow (due to job destructionor into better job).

Note that we consider jobs that pay w or less, so if anemployed worker leaves a "< w job" and enters a job thatpays w , E (w) remains the same.

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Finding G(w)

get a relation between the o¤er distribution and the wagedistribution which is useful for empirical work

G (w) =E (w)1� u = (7)

λ0 (F (w)� F (R)) u(δ+ λ1 [1� F (w)]) (1� u)

=δ (F (w)� F (R))δ+ λ1 [1� F (w)]

=(F (w)� F (R))1+ κ1 [1� F (w)]

1st step follows from (6) , 2nd step from (5)

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Firm behavior

Let l(w jR,F ) be the measure of workers available per �rmthat o¤ers a wage w :

l(w jR,F ) = limε!0

G (w)� G (w � ε)

F (w)� F (w � ε)(1� u)

which is the steady state number of workers earning w(G (w)� G (w � ε)(1� u)) divided by the number of �rmso¤ering w (F (w)� F (w � ε))

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Expected pro�ts

π (p,w) = l(w jR,F ) (p � w) . (8)

In appendix of class notes I show:

l(w jR,F ) = κ0 (1+ κ1) / (1+ κ0)

(1+ κ1 [1� F (w)]) (1+ κ1 [1� F (w�)])

Note that the trade-o¤ is between l(w jR,F ) (increasing inw) and (p � w) (decreasing in w)

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Equilibrium wage dispersion

Equilibrium is the set {F (w),G (w), u,Rg that satis�es thesteady state conditions, given λ

F (w) is the only thing we have not solved for.

How does F (w) look?

No �rm will o¤er a wage below R (because workers will simplyreject it)

Can F (w) be degenerate (all �rms pay same wage)? No!

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Continuous wage distribution

Proof by contradiction.

Could all �rms o¤er the competitive wage? No!

Then a deviant could do better by o¤ering the reservationwage and make positive pro�ts!

This suggest mixed strategy equilibrium where F (w) iscontinuous and non-degenerate.

In mixed strategy equilibrium, expected pro�ts of all �rmsmust be equal, including the �rm that pays the lowest possiblewage, w = R.

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Wage o¤er distribution

From appendix:

l (w jR,F ) = κ0(1+ κ1) (1+ κ0)

π (p,w) = π (p,w) =κ0 (p � R)

(1+ κ0) (1+ κ1)

=κ0 (1+ κ1) (p � w)

�1

1+κ0

��

(1+ κ1 [1� F (w)])� (1+ κ1 [1� F (w)])

� .

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This implicitly determines F (w).

F (w) =�1+ κ1

κ1

� "1�

�p � wp � R

�1/2#. ((9))

Substitute (9) in (4) yields

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Wage o¤er distribution

R � b

=(κ0 � κ1)

κ1

Z w̄

R

κ1 [1� F (x)]1+ κ1 (1� F (w))

dx

=(κ0 � κ1)

κ1

Z w̄

R

"1�

�1

1+ κ1

��p � xp � R

��1/2#dx

=(κ0 � κ1)

κ1

"w̄ � R + 2 (p � R)

1+ κ1

�p � w̄p � R

�1/2

� 1#

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Next, use the fact that F (w̄) = 1 in (9) to get

p � w̄ = (p � R)(1+ κ1)

2

implying that

R =

"(1+ κ1)

2 b+ (κ0 � κ1) κ1p

(1+ κ1)2 + (κ0 � κ1) κ1

#

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Derivation labor supply

l (w jR,F ) = limε!0

G (w)� G (w � ε)

F (w)� F (w � ε)(1� u)

(#workers earning w/# �rms o¤ering w)

limε!0

(F (w )�F (R ))1+κ1 [1�F (w )] �

(F (w�ε)�F (R ))1+κ1 [1�F (w�ε)]

F (w)� F (w � ε)(1� u) =

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= limε!0

(F (w)� F (R)) (1+ κ1 [1� F (w � ε)])� (F (w � ε)� F (R)) (1+ κ1 [1� F (w)])(F (w)� F (w � ε)) (1+ κ1 [1� F (w)])

(1+ κ1 [1� F (w � ε)])�κ0 [1� F (R)]

1+ κ0 [1� F (R)]

�

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= limε!0

F (w) + κ1F (w)� κ1F (w)F (w � ε)� F (w � ε)�κ1F (w � ε) + κ1F (w � ε)F (w)� (F (R)) (κ1 [F (w)� F (w � ε)])

(F (w)� F (w � ε)) (1+ κ1 [1� F (w)])(1+ κ1 [1� F (w � ε)])�

κ0 [1� F (R)]1+ κ0 [1� F (R)]

�

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= limε!0

(F (w)� F (w � ε))(1+ κ1 � κ1 (F (R)))

�κ0 [1�F (R )]1+κ0 [1�F (R )]

�(F (w)� F (w � ε)) (1+ κ1 [1� F (w)])

(1+ κ1 [1� F (w � ε)])

= limε!0

[1+ κ1 (1� F (R))]�

κ0 [1�F (R )]1+κ0 [1�F (R )]

�(1+ κ1 [1� F (w)]) (1+ κ1 [1� F (w � ε)])

In equilibrium (i) wage distribution is continuous and (ii) no �rmo¤ers a wage below R. This implies F (R) = 0 :

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l (w jR,F ) = κ0 (1+ κ1)

(1+ κ1 [1� F (w)])(1+ κ1 [1� F (w�)])

�1

1+ κ0

�

and at the minimum wage, F (w) = 0 and

l (w jR,F ) = κ0(1+ κ1) (1+ κ0)

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Duration Analysis

Estimation

5 unknowns: {λ0,λ1, δ, p,R}

data longitudinal labor force panel surveys

create individual labor market histories

su¢ cient data: for each individual: unemployment spell or jobspell with wage and the state after the job (new job orunemployment)

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Duration Analysis

Duration analysis

Let unemployment duration be a random variable T withF (t) =

R t0 f (s)ds = Pr(T � t).

The probability that the length of an unemployment spell is atleast t is S(t) = 1� F (t) = Pr (T � t).The probability to leave unemployment conditional on beingunemployed for a period of length t is the hazard rate,

h(t) =f (t)S(t)

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Duration Analysis

Duration analysis

Further,

h(t) =�d lnS(t)

dt

you see one you see them all.

Once you know either h you know S and vice versa.

in a stationary environment, h does not depend on the elapsedduration of unemployment.

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Duration Analysis

Then,

h(t) = h�d lnS(t)

dt= h

lnS(t) = k � htS(t) = K exp (�ht)S(0) = 1 implies K = 1

S(t) = exp (�ht)

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Duration Analysis

Estimation

distributions of duration, wage, exit destination areendogenous and depend on the parameters of the model

Duration of job: exponential with parameterδ+ λ1 (1� F (w))wages of �rst job after unemployment are random draws fromF (w)

wages of random sample of employed workers are randomdraws from G (w)

In practice: κ0, κ1, p are estimated by ML, e.g. Van den Bergen Ridder (1998)

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Duration Analysis

Shape of g(w)

g(w) and f (w) are increasing while empirical wagedistributions are not

Van den Berg and Ridder (1998)!segmented (by occupation,age, etc.) labor market

Assume mixtures of G and F and log normal.

Heterogeneity within markets, Bontemps et al. (1997), i.e. pfollows a log normal or Pareto distribution

Note search frictions are still important because productivitydi¤erences between �rms can by itself not explain wagedi¤erences (in competitive environment the law of 1 priceholds and unproductive �rms are driven out of the market).

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Matching counter o¤ers


Postel Vinay and Robin (2002), Econometrica

Firms make take-it-or-leave-it o¤ers to workers

�rms can make better o¤ers to the more productive workers

long term contracts

No Diamond paradox because of on-the-job-search andBertrand competition

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After job-to-job movement, initial wage change can benegative, why?

Both workers and �rms are heterogeneous

worker e¤ect is independent of �rm and friction e¤ect (strongassumption)

�rm and friction e¤ect are dependent : more productive �rmshave more market power and loose less workers throughBertrand competition

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40% of wage variance due to worker characteristics for highskilled workers

0% of wage variance due to worker characteristics for lowskilled workers

Abowd, Kramarz and Margolis (Econometrica, 1999) �nd thatthe worker e¤ect is more than 50% for all types

They assume no frictions

Take 2 workers A and B who work at same type of �rm. Aearns more and has made 4 job-to-job transitions while Bmade only 2. AKM assign all wage di¤erences to workercharacteristics while PR take into account that with eachmovement, Bertrand competition pushes up the wage.

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Complete metric spaceCauchy sequenceContractionBlackwell�s su¢ cient conditionsMonotonicity and contraction

Contraction Mappings

Loose description of the contraction mapping theorem

The contraction mapping theorem is de�ned in a metric space

Metric Space is a set X and a metric, d (some distancemeasure).

For our purposes we can just consider the set of boundedcontinuous functions C [0,w ] mapping the interval [0,w ] intoR

and we use the distance metric:d (x , y) = max jx(w)� y(w)j

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Complete metric space

This metric space is complete because

all convergent sequences (Cauchy sequences) converge to apoint that belongs to the metric space.

Sequences for which d (am , an)! 0 as m, n! ∞ are calledCauchy sequences.

To illustrate this, we consider a sequence that converges in Xand one that does not.

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Cauchy sequence

Illustration 1 Cauchy sequence:

consider the sequence: t1+t in C [0, 1]

This sequence converges to 1

x(1) = 12 , x(2) =

23 , ...x(9) =

910 , ...x (∞) = 1.

Formally, t1+t converges because jx(w)� y(w)j ! 0 as

x , y ! ∞.

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Non Cauchy sequence

Illustration 2 non Cauchy sequence

consider the sequence: 1, 1+ 12 , 1+

12 +

13 , ... which does not

converge.

Let x = 2y .

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limx ,y!∞jx(w)� y(w)j = j�1+

12+13+ ...

1y, ...

12y

��1+

12+13+, ...

1y

�j

=1

y + 1+

1y + 2

, ...12y

� 12y+12y, ...+

12y

= y�12y

�=12.

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Contraction mapping

The reason we focus on complete metric spaces is that we donot want our space to have missing limits.

In a complete metric space, the condition d(x(w), y(w))! 0is su¢ cient to ensure convergence.

We claimed before that (3) (wr = T (wr )) was a contractionmapping. What do we mean by this?

The operator T is a contraction if it reduces the distancebetween subsequent iterations.

If we can establish that T (wr ) is a contraction mapping thenwr = T (wr ) converges to a unique �xed point and we haveproved existence of a search equilibrium.

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Blackwell�s su¢ cient conditions

To show that T (wr ) is a contraction mapping we useBlackwell�s su¢ cient conditions.

1 Monotonicity. x � y implies T (x) > T (y).2 Discounting. Let c be a constant then,T (x + c) � T (x) + βc for some β 2 [0, 1i .

Then T is a contraction mapping with modulus β.

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Search problem

we work in the complete metric space (X , d) with X being theset of bounded continuous functions C [0,w ] mapping theinterval [0,w ] into R and d (x , y) = jx(w)� y(w)j.to proof existence and uniqueness it is su¢ cient to establishthat T (wr ) is a contraction mapping:

wr = T (wr ) = (1� β)b+

β�Z w

0max [w ,wr ] dF (w).

The mapping T maps functions z in [0,w ] into functionsT (z) in [0,w ] . If T meets Blackwell�s su¢ cient conditions, itis a contraction.

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Monotonicity and discounting

Let f > g . Then Tf > Tg sinceZ ∞

0max [w , f ] dF (w) �

Z ∞

0max [w , g ] dF (w)

Next, we show that T "contracts" (discounts). I.e.T (f + c) � Tf + γc

T (f + c)

= (1� β)b+ βZ w

0max [w , f + c ] dF (w)

� (1� β)b+ βZ w

0max [w , f ] dF (w) + γc

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Contraction mapping

Therefore, T is a contraction mapping

According to the contraction mapping theorem, wr = T (wr )has a unique solution in C [0,w ] which is approached in thelimit as n! ∞ by T n

�v0�= vn

where v0 is any starting point in C [0,w ] and T n�v0�is the

nth iteration starting from wr = vn.

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labor in rbc model, stylized facts, wage posting€¦ · postel vinay and robin (2002) contraction...

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