partial equilibrium search theory - gerzensee equilibrium search theory jean-marc robin sciences po...

Partial Equilibrium Search Theory

Jean-Marc Robin

Sciences Po and UCL

30th July 2017

JM Robin (Sciences Po and UCL) Partial Equilibrium Search Theory 30th July 2017 1 / 31

Road map

1 The prototypical, stationary search model2 On-the-job search and layoffs3 Non-stationary search


1. The prototypical, stationary search model


Assumptions

Time is continuous and individuals live for ever.In a small time interval [t, t +�], unemployed individuals receive apayment b�.

I Parameter b is the opportunity flow-cost of employment (includingunemployment insurance, search costs, etc.).

When unemployed, the probability of receiving an offer in [t, t +�] is�0�, irrespective of the time already spent waiting for an offer.Such a point process is called a Poisson process.Employment lasts for ever.The distribution of wage offers is F and F is continuous and differentiable.The time-discount rate is ⇢ > 0.


Poisson process

Spell durations are independent: if 0, t1 and t2 are three point realizations(three subsequent job offers), then t1 and t2 are independent.A spell duration has an exponential distribution with parameter �0:

I the density is: f0(t) = �0 exp (��0t)I the cdf is: F0(t) = 1 � exp (��0t))I the expectation is:

R10 t�0e��0ddt =

1

�0

Important property: At any time t, the elapsed time durationseparating t from the last point realization, say ⌧0, and the residual timeduration separating t from the next point realization, say ⌧1, areindependent and have the same marginal distribution: an exponentialdistribution with parameter �0.


Employment value function

Let W (w) be the value of a job offering a wage w per unit of time �.First calculation. Assuming that employment spells last for ever,

W (w) =

Z 1

0we

�⇢t

dt =w

⇢

Second calculation. Remark that value functions do not depend oncalendar time because of the stationary environment: the value tomorrowis equal to the value today.Bellman’s optimality principle yields

W (w) =w�

1 + ⇢�+

1

1 + ⇢�W (w)

, (1 + ⇢�)W (w) = w�+ W (w)

, W (w) =w

⇢


Unemployment value function 1

Let V0 be the value of unemployment.Unemployed workers accept any wage offer w such that W (w) � V0.Bellman’s principle yields

V0 =b�

1 + ⇢�+

�0�

1 + ⇢�E

w⇠F

max {W (w),V0}+1 � �0�

1 + ⇢�V0

, ⇢V0 = b + �0Ew⇠F

max {W (w)� V0, 0}

Option-value equation:

⇢V0|{z}annuity

= b|{z}revenue flow

+ �0Ew⇠F

[max {W (w),V0}� V0]| {z }option value of job offer

and⇢W = w|{z}no option


Unemployment value function 2

W (w) > V0 , w > ⇢V0 (reservation wage)Hence,

Ew⇠F

max {W (w)� V0, 0} = Ew⇠F

max

⇢w

⇢� V0, 0

�

=

Zw

⇢V0

✓w

⇢� V0

◆dF (w)

(integration by part) = �✓

w

⇢� V0

◆F (w)

�w

⇢V0

+1

⇢

Zw

⇢V0

F (w)dw

=1

⇢

Zw

⇢V0

F (w)dw

where F (w) ⌘ 1 � F (w).


Reservation wage policy

The optimal strategy when unemployed is to accept any job offering awage w such that

W (w) � V0 , w � � ⌘ ⇢V0

The reservation wage � is the solution to the equation:

� = b +�0

⇢

Zw

�F (w)dw

Remarks on computation:I The integral usually has no closed form. A numerical procedure is required

to approximate the integral (quadrature, Simpson, etc.), available insoftwares like R, Gauss or Matlab.

I The equation can then be solved for �, also numerically, using aNewton-type algorithm.


Data

Labor force surveys usually observe workers continuously during a timeinterval [t0, t1] and gather retrospective information at t0 so that, forunemployed workers at t0, it is possible to record:

the elapsed unemployment duration at t0: ⌧0

the residual unemployment duration after t0: ⌧1

the accepted wage w at t0 + ⌧1 if the worker leaves unemployment by theend of the recording period t1 � t0


Likelihood

The duration of unemployment has an exponential distribution withparameter �0F (�) (instantaneous probability of receiving an offer timesthe probability that it is acceptable).The Poisson property implies that ⌧0 and ⌧1 are independent andexponentially distributed.The density of (⌧0, ⌧1,w) is therefore equal to

` (⌧0, ⌧1,w) = �0F (�) exp

��0F (�)⌧0

�

⇥�0F (�) exp

��0F (�)⌧1

�· f (w)

F (�)

�z ⇥

exp

��0F (�)⌧1

�⇤1�z

here z = 1 if ⌧1 < t1 � t0, z = 0 if ⌧1 = t1 � t0 (the residualunemployment duration is censored).The distribution of wage offers is identified only conditional on w > �(with density f (w)

F (�)).


Estimation - First approach

Treat reservation wage � as a parameter.Maximize the likelihood of a sample {(⌧0i

, ⌧1i

,wi

, zi

) , i = 1, ...,N} subjectto w

i

� �, 8i .The reservation wage � is estimated by the minimal accepted job offer:b� = min{w

i

| z

i

= 1}.I It is a superconsistent estimator: N

⇣b��

⌘L! non Gaussian distribution.

Estimate �0 and the parameters of F by maximising the log likelihoodconditional on � = b� (b� is like a constant as it is superconsistent);

estimate b as bb = b�� b�0

⇢

Rw

b�bF (w)dw .

One can cluster the sample by individual type.For a complete treatment see Christensen and Kiefer (1991)


Estimation - Second approach

Same but do ML subject to reservation wage equation (gets you sd for b).Add conditioning variables (determining b,F or �0) if necessary.


2. On-the-job search and layoffs

Mortensen and Neumann (1989)I also deal with wage offer distribution F that may exhibit mass points orbe discrete (important for the sequel).


Cumulative Distribution Functions

Let F : R ! [0, 1] be the CDF of a random variable X .It is cadlag. “Continue à droite, limite à gauche.” Right continuous withleft limits.It has at most countably many discontinuities {x1, x2, ...} andPr{X = x

i

} = F (xi

)� F (xi

�) ⌘ �F (xi

).

It has a left-derivative F

0(x) = lim"#0F (x)�F (x�")

" at almost every point.

F

c

(t) =R

t

�1 F

0(x) dx is the absolutely continuous part of F andF

s

(t) = F (t)� F

c

(t) =P

x2(0,t] �F (x) is the singular part:

F (t) = F (0) +

Zt

0F

0(x) dx +X

x2(0,t]

�F (x)


Lebesgue-Stieltjes measure and integral

Lebesgue-Stieltjes measure. This is the Borel measure µ on R such that1 µ(a, b] = F (b)� F (a)2 µ[a, b] = F (b)� F (a�)3 µ[a, b) = F (b�)� F (a�)4 µ(a, b) = F (b�)� F (a)5 µ{a} = F (a)� F (a�) = �F (a).

Stieltjes integral. For any Borel-measurable real-valued function g andany Borel set B (union of intervals), define

RB

g dF to beRB

g dµ, with theconvention that

Rb

a

g dF =R(a,b] g dF .


Stieltjes integral

For non pathological CDFs on [0,+1),Z

b

a

g dF =

Zb

a

g(x)F 0(x) dx +X

x2(a,b]

g(x)�F (x)

Integration by part. Given two non-decreasing functions (or of boundedvariation on any finite integral), which are both cadlag, then

F (b)G (b)� F (a)G (a) =

Zb

a

F (x�)dG (x) +

Zb

a

G (x)dF (x)

=

Zb

a

F (x�)dG (x) +

Zb

a

G (x�)dF (x) +X

x2(a,b]

�F (x)�G (x)

I See e.g. Fleming, Harrington (2005), Counting Processes and Survival

Analysis, John Wiley & Sons, Inc.


Assumptions

When employed, job offers accrue at constant rate �1. (It is likely that�1 < �0.)Job are exogenously destroyed at constant rate �.Employees at wage w accept an alternative job paid x if onlyW (x) > W (w).


Unemployment values

The unemployment value function proceeds from the same definition as before,

⇢V0 = b + �0Ew⇠F

max {W (w)� V0, 0}

= b +

Zw

�[W (w)� V0] dF (w)


Employment value

The employment value is

W (w) =w�

1 + ⇢�+

�1�

1 + ⇢�E

x⇠F

max {W (x),W (w)}

+��

1 + ⇢�V0 +

1 � �� 1�

1 + ⇢�W (w)

that is

⇢W (w) = w + �1Ex⇠F

max {W (x)� W (w), 0}+ � [V0 � W (w)]

= w + �1

Zw

w

[W (x)� W (w)] dF (x) + � [V0 � W (w)]


W is continuous for all F

For all real z , x and y ,

|max{z , x}� max{z , y}| |x � y | .

Moreover, for all couple of rv X ,Y , |EX � EY | E |X � Y | . Hence

|Ew

max{W (w),W (x)}� Ew

max{W (w),W (y)}| E

w

|max{W (w),W (x)}� max{W (w),W (y)}| E

w

|W (x)� W (y)| = |W (x)� W (y)|

Bellman equation

(⇢+ �1 + �)W (w) = w + �1Ex⇠F

max {W (x),W (w)}+ �V0

implies that(⇢+ �) |W (x)� W (y)| |x � y |

and so W is Lipschitz-continuous.


W is increasing for all F

For all k and x < y ,

max{k ,W (x)}� max{k ,W (y)} � min{W (x)� W (y), 0}.

Then,

(⇢+ �1 + �) [W (x)� W (y)] � x � y + �1 min{W (x)� W (y), 0}.

If W (x)� W (y) 0 then

(⇢+ �) [W (x)� W (y)] � x � y > 0,

yielding a contradiction. Hence W (x) > W (y).


Integration by part

Because W is continuous,Unemployment value:

Zw

�[W (w)� V0] dF (w) =

Zw

�F (w)dW (w)

Employment value:Z

w

w

[W (x)� W (w)] dF (x) =

Zw

w

F (x)dW (x)


W is left-differentiable

This function is differentiable (because W is continuous – like acontinuous cdf) and its left-derivative (like a pdf) is equal to�F (w�)W 0(w), where W

0(w) is the left-derivative of W (w)

IndeedZ

w

w

�Z

w

w�"=

Z

(w ,w ]�Z

(w�",w ]=

Z

(w�",w ]

On the interval (w � ",w) and small " the value of F (x) is close toF (w�) = Pr{wage > w} assuming that the jump points of F are notdense.


W is left-differentiable

Using the Bellman equation for W (x), the left-derivative of W is obtainedas

W

0(x) =1

⇢+ � + �1F (x�)> 0

(Left- and right-derivatives differ at points of discontinuity of F .)Then,

Zw

w

F (x)dW (x) =

Zw

w

F (x)W 0(x)dx =

Zw

w

F (x)

⇢+ � + �1F (x)dx

(The discontinuities don’t matter inside the Riemann integral becausethey are at most countably many.)


Reservation wage

Unemployment value:

⇢V0 = b + �0

Zw

�F (w)dW (w) = b + �0

Zw

�

F (w)

⇢+ � + �1F (w)dw

Employment value:

(⇢+ �)W (w) = w + �V0 + �1

Zw

w

F (x)

⇢+ � + �1F (x)dx

Reservation wage:

W (�) = V0 , � = b + (�0 � �1)

Zw

�

F (w)dw

⇢+ � + �1F (w)

I Note that �0 > �1 ) � > b. Because of low wages, b is often estimatednegative.


Non-stationary search

Van den Berg (1990)


Assumptions

b now depends on unemployment duration (or exit rates, ...)For simplicity, we rule out on-the-job search and layoffs, so that theemployment value continues to be W (w) = w

⇢ .


Unemployment value

Let V0(t) denote the value of unemployment for unemployment durationt.Bellman equation:

V0(t) =b(t)�

1 + ⇢�+

�0�

1 + ⇢�Emax {W (w),V0(t +�)}+ 1 � �0�

1 + ⇢�V0(t +�)

Hence,

⇢V0(t) = b(t) + �0Ew⇠F

max {W (w)� V0(t +�), 0}+ V0(t+�)�V0(t)�

= b(t) + �0

Zw

⇢V0(t)

✓w

⇢� V0(t)

◆dF (w) + V0(t+�)�V0(t)

�

= b(t) +�0

⇢

Zw

⇢V0(t)F (w)dw + V0(t+�)�V0(t)

�

This is an integral-differential equation that can be solved backwardassuming stationarity for t > T (V0(t) = V0(T ), 8t > T ).


Reservation wage

�(t) = ⇢V0(t)

If b(t) is decreasing then so is the reservation wage. Workers are lesspicky as they approach the end of the insurance period.


References

Christensen, B. J. and N. M. Kiefer (1991): “The Exact LikelihoodFunction for an Empirical Job Search Model,” Econometric Theory, 7,464–486.

Mortensen, D. T. and G. R. Neumann (1989): “Estimating structuralmodels of unemployment and job duration,” in Proceedings of the Third

International Symposium in Economic Theory and Econometrics, ed. byW. A. Barnett, E. R. Berndt, and H. White, Cambridge University Press,Series: International Symposia in Economic Theory and Econometrics (No.3), chap. 15, 335–354.

Van den Berg, G. J. (1990): “Nonstationarity in Job Search Theory,”Review of Economic Studies, 57, 255–277.


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