Simultaneous Multi-Market Sear h

Xiaodong Fan

†

Chao He

∗

‡

O tober 5, 2015

(Preliminary Draft)

Abstra t We model sear h intensity as in how many markets workers simultaneously sear h for jobs. We nd that simultaneous sear h imposes a negative externality to other sear hers, in terms of lower job nding rate. The negative externality indu es strategi omplementarity of workers, making possible multiple equilibria: a low (sear h) intensity equilibrium (LIE) and a high intensity equilibrium (HIE), the latter of whi h may generate less jobs due to the negative externality. Based on these results, a theory of dynami bifur ation is then proposed: large enough nan ial sho ks that in rease the entry ost of rms, even temporarily, an ause the labor market to permanently swit h from the LIE to the HIE. Although workers sear hes harder, the job nding rate an be smaller if the negatively externality is strong enough. If so, the beveridge urve shifts out and the re overy of the labor market is slower than if simultaneous sear h is not allowed. On the other hand, if the negative externality is weak, then the opposite is true. Therefore, labor market u tuations an be amplied or dampened by the endogenous hanges in sear h intensity of workers.

JEL odes : E24, J64, J61. ∗

We thank Chao Fu, Greg Kaplan, John Kennan, Rasmus Lentz, Guido Menzio, Ananth Seshadri,

Lones Smith, Wei Sun, and Randy Wright for many useful omments. † ARC Centre of Ex ellen e in Population Ageing Resear h (CEPAR), University of New South Wales (UNSW), Sydney, NSW 2052, Australia. Email: x.fanunsw.edu.au. ‡ S hool of E onomi s, Shanghai University of Finan e and E onomi s, he hao1776gmail. om

1

Keywords : Sear h fri tions, Unemployment, Multiple equilibria, Labor mobility.

1

Introdu tion

We model sear h intensity as in how many markets workers simultaneously sear h for jobs. We nd that simultaneous sear h imposes a negative externality to other sear hers, in terms of lower job nding rate. When these vi tims nd it harder to get a job, additional sear h intensity be omes more useful.

Thus the negative externality an indu e these

vi tims to simultaneously sear h in more markets themselves. The strategi omplementarity of workers makes possible multiple equilibria: a low (sear h) intensity equilibrium and a high intensity equilibrium. This aptures the idea that: if everyone else is sear hing harder, I had better sear h harder, too. A

ording to our theory, the equilibrium where everybody sear hes harder is not ne essarily more e ient than the equilibrium where everybody sear hes less. Whi h one is more e ient depends on the magnitude of the negative externality aused by simultaneous sear h.

We hara terize onditions under

whi h one equilibrium is more favorable to the so iety. Based on these results, a theory of dynami bifur ation is then proposed: large enough nan ial sho ks that in rease the entry ost of rms, even temporarily, an ause the labor market to permanently swit h from the low intensity equilibrium to the high intensity equilibrium.

Although workers sear hes harder, the job nding rate an be smaller if

the negatively externality is strong enough. If so, the beveridge urve shifts out and the re overy of the labor market is slower than if simultaneous sear h is not allowed.

We

believe our theory helps us understand the re ent jobless re overy. On the other hand, if the negative externality is weak, then the opposite is true.

Therefore, labor market

u tuations an be amplied or dampened by the endogenous hanges in sear h intensity of workers. Our theory is rst motivated by the fa t that when workers in rease their sear h intensity, they not only an in rease their sear h eort in a parti ular market but also an sear h wider by looking at more markets. Potentially they an look for jobs in addi2

tional lo ations, se tors and o

opations that they do not previously onsider. Another motivation of the paper is that sear h intensity in reased a lot during the Great Re ession while the Beveridge urve shifted outward, meaning that the mat hing e ien y in the labor market a tually de reased. It is hard to have workers sear h harder in a standard mat hing model, and it would be hallenging to form a theory that an simultaneously a

ount for both the in reased sear h intensity and de reased mat h e ien y. What is more interesting is that our theory is about how endogenously in reased sear h intensity a tually an be the reason for the de reased mat h e ien y. To understand the results of the model, it is key to understand how simultaneous multi-market sear h an ause negative externality.

Suppose that there are two labor

markets, A and B, both have onstant return-to-s ale (CRS) mat hing te hnology and free entry of rms.

A mat h is an oer that would be a

epted if the worker do not

have other oers. If workers an only sear h in one of the two markets then any oer generated by the mat hing fun tion would be a

epted. If we aks a worker who initially only sear hes in Market A to only sear h in Market B, then this would not ause any negative externality to workers sear hing in Market B. This is be ause free entry in Market B indu e rms to reate a little more va an y so that the market tightness is not hanged in Market B, whi h makes workers in Market B indierent. Now suppose we instead ask the worker to simultaneously sear h in both markets, what would happen? First, rms in Market B would re eive more appli ations. Se ond, there are some probability that the worker would re eive oers from both of the markets.

Suppose the worker always

a

ept oers from Market A, then sin e some of the oers extended by rms in Market B are reje ted, the equilibrium market tinghtness has to de rease so that the in reased arrival rate for rms an ompensate the in reased oer reje tion rate. But the de reased market tightness also means the job nding rate for workers in Market B must de rease. So this is the ase where all of the negative externality is imposed on workers in Market B. Suppose the worker sometimes hooses oers from Market B over oers from Market A, then by the similar logi , some of the negative externality is imposed on other workers in Market A as well.

3

It is also important to note that in standard mat hing models without simultaneous sear h, when everyone sear hes harder, the mat h e ien y always in reases, be ause basi ally we have more e ien y sear h units thus more inputs into the mat hing fun tion. Here simultaneously sear h poses a new tradeo for the so iety: when workers simultaneously sear h more markets, more oers are extended but at the same time, more oers are reje ted, as well. Whether higher sear h intensity an translate to more jobs being

reated depends on the relative magnitude of the two ee ts. One of our ontribution is to advan e a novel theory of multiple equilibria and how temporary real sho ks (spe i ally, nan ial sho ks) an permanently ause the swit h of equilibrium. In Mortensen (1999), there is also a bifur ation system. But like many other papers, the sour e of multipli ity omes from in reasing returns to s ale in produ tion (Benhabib and Farmer 1994, Farmer and Guo 1994, and Christiano and Harrison 1999). Multipli ity an also obtain due to in reasing returns to s ale in mat hing (Diamond 1982, Diamond and Fudenberg 1989 and Boldrin, Kiyotaki and Wright 1993). In Heller (1986), Roberts (1987) and Cooper and John (1988), multipli ity obtains be ause of demand externalities. More re ently, Kaplan and Menzio (2014) propose a theory based on market power externality, whi h is present be ause a rm hiring an additional worker

reates a positive external ee t on other rms, as a worker has more in ome to spend and less time to sear h for low pri es when he is employed than when he is unemployed. As Kaplan and Menzio (2014), in our theory, both produ tion and mat hing are CRS. But we have multiple equilibria be ause of the strategi omplementarity of workers' de isions on how broadly to sear h for jobs. Their theory is also onsistent with the shift of Beveridge Curve after the Great Re ession. To our knowledge, this is the rst attempt to model simultaneous sear h in a random sear h framework. Usually, sear h intensity is treated as a te hnology parameter (Pissarides, 2000), whi h in reases the mat h probability for a worker. However, an individual worker an also improve her mat h probability by sear hing more broadly. Many authors

ales 2005, study this extensive margin with dire ted sear h (Gautier and Moraga-Gonz´ Albre ht et al. 2006, Galenianos and Kir her 2008, and Kir her 2009). In these studies,

4

there are two oordination fri tions.

The rst is the urn-ball fri tion, some va an ies

re eive no appli ation while others re eive more than one. In this paper, if we interpret a mat h for a va an y as re eiving an appli ation, then some va an ies re eive no appli ation but other at most re eive one appli ation. In addition, some workers re eive no oer while others re eive more than one. This is the same as in this paper. But aside from using a random sear h framework, our paper is dierent in that we assume workers have dierential taste over jobs from dierent markets.

This is the reason why when

workers in our model sear h harder by rea hing the other market, they pollute the pool of appli ants in that market. We would argue that it is only natural for workers to do so. A worker would always sear h in the market he likes rst, and then start sear hing in the other market(s) if ne essary. Thus the in remental sear hes would be assigned a lower priority if she re eives oers from both markets. This paper also ontributes to our understanding of how the re overy of the labor market is still slow after the nan ial sho ks fade away. planations for jobless re overy:

There exists three main ex-

mismat h, nan ial sho ks and wage rigidity (real or

nominal). The mismat h hypothesis says that idle workers are seeking employment in se tors, o

upations, industries, or lo ations dierent from those where the available jobs are. Sahin and et al. (2014) nd that mismat h, a ross industries and three-digit o

upations, explains at most one-third of the total observed in rease in the unemployment rate. Lazear and Spletzer (2012) nd that although mismat h in reased during the re ession, it retreated at the same rate. Regarding nan ial sho ks, Chen et al. (2012) and Jermann and Quadrini (2011) study how nan ial sho ks ae t ma roe onomi variables through traditional hannels of fri tion, whereas Mona elli et al. (2011) model how nan ial onditions ae t rms' bargaining position with workers. However, these studies

annot explain the small job nding rate, thus the jobless re overy after nan ial onditions improve. The third andidate explanation is about wage rigidity. Shimer (2012)

e and Uribe (2012) study how real wage rigidities ause jobless re overies. S hmitt-Groh´ argue nominal rigidity and la k-of- onden e sho ks help explain the jobless re overy. Bils et al. (2014) propose a model in whi h if wages of mat hed workers are stu k too

5

high in a re ession, then rms will require more eort, lowering the value of additional labor and redu ing new hiring. These theories an a

ommodate many previous US experien e whereas miss out on the unemployment after the Great Re ession, espe ially the breakdown in the Beveridge urve. What we have shown in this paper is that relative to the standard sear h and mat hing model, our theory an generate slower re overy of unemployment rate. Of ourse the omparison with the above papers is in omplete sin e the quantitative feature of the model is yet to be done. The rest of the paper is organized as follows. Se tion 2 introdu e the model setup, Se tion 3 study the steady state property of the model and dis uss onditions for multiple equilibria, Se tion 4 looks at the dynami s and Se tion 5 on ludes.

2

The Stati Model

In this se tion we present a stati model that illustrates how workers an hoose whether to simultaneous sear h in two markets.

2.1

Environment

Consider a stati enviroment where a given measure of unemployed workers are looking



for jobs. Type

2.

Spe i ally, assume there are two types of unemployed workers Type The measure of ea h type is normalized to

namely Market and

1

y˜i (yi ≥ y˜i )

and Market

2.

ui .

i

and Market

j,

and

There are also two markets,

Workers' utility is linear. Type

if employed in Market

1

i

workers produ es

respe tively (j

6= i).

yi

Workers

an either hoose to only sear h for their favorite market, or to sear h simultaneouly in both markets with additional ost

c

if employed in Market

j.

cu .

In addition, they need to pay an additional ost

Su h ost an represent a dire t ost of relo ation (in luding

the realized apital loss when selling a house, whi h is very important during the fall of housing pri es), or preferen e over markets. We say Market of Type

i

workers.

We would fo us on the spe ial ase where

i

is the favorite market

yi = y˜i

and

c = 0,

we

introdu e these notations to make future dis ussions easier. So the essential dieren e

6

between markets is the additional sear h ost if a worker wants to sear h in two markets simultaneously. Sin e workers might sear h in both markets, the measure of appli ants in market

ni ,

whi h we denote by Market are

i.

might be greater than

ui .

Let

vi

be the measure of va an ies in

In ea h market, given the measure of va an ies and appli ants,

m (v, n)

v

and

n,

there

of su

essful oers extended to appli ants in a market. An appli ant (rm)

in a market at most would re eive (extend) one oer in that market. Assume in reasing in both argument, on ave, onstant-return-to-s ale and Let the market tightness be

θ = v/n


m (v, n) /v = m 1, nv = M (θ)

of

i,

in reasing fun tion of

θ.

m (v, n) ≤ min {v, n}.

for extending an oer and every appli ant has a han e

On the rms' side, the ost of reating a va an y is Noti e

is

for a market, then every rm has a han e of


m (v, n) /n = nv m (v, n) /v = nv m 1, nv = θM (θ)

rms' prot to zero.

m (v, n)

M (θ)

for re eiving an oer in the market.

cv

in either market. Free entry drives

is a de reasing fun tion of

Of ourse, both

θM (θ)

and

M (θ)

θ,

whereas

θM (θ)

is an

must be positive but smaller

than one. If a worker has no oer at all, she would re eive the unemployed benet, is smaller than workers get

γ

y.

yu ,

whi h

We onsider two ways of determining the wage, one is to assume

fra tion of the total output; the other is to assume Nash bargainning with

bargainning power

ρ

of workers.

In the bargaining, we assume that the ost

c

is not

ontra table. In other words, bargaining happens after workers have paid the ost a workers re eives only one oer, she would a

ept it. both markets, then it matters whether oer in her favorite market. If

c = 0,

c

is positive. If

c.

If

But if she re eives oers from

c > 0,

then she would a

ept the

then she is indierent between the two oers so

we assume she a

ept either oer with half the han e. In addition, be ause we assume sear hing in their favorite market in ur no ost, workers always sear h in their favorite market. The a tive de isions workers must hoose is whether to simultaneously sear h in the other market. The net benet of simultaneously sear h in the non-favorite market for a Type

i

worker is given by

7

Γi = −cu + θj M (θj ) [1 − θi M (θi )] (w˜i − c) , where

i 6= j

and

w˜i

is the wage for a type

i

(1)

worker employed in market

j.

This

aptures the idea that an oer from the non-favorite market is only useful if no oer from the favorite market is re eived, whi h happens with

1 − θi M (θi )

would simultaneously sear h in both markets if and only if

probability. A worker

Γi > 0 .

Next onsider the

wage determination. The fra tional wage is straightforward. The wage in a bargaining must maximize the following Nash produ t:

max (wˆ − yu )ρ (ˆ y − w) ˆ 1−ρ ,

(2)

w ˆ

where

yˆ ould

be

yi

or

y˜i

and

wˆ ould

be

wi

or

w˜i .

So we have

wˆ = ρˆ y + (1 − ρ) yu . A

ording to the above wage equation, we have two pure strategy equilibria:

yˆ − wˆ = (1 − ρ) (ˆ y − y u ).

(3)

We onsider

workers only sear h in their favorite market (FJE) and

sear h in both markets (AJE). The free entry ondition for rms are then

cv = M (θi ) (yi − wi ) , in F JE;   uj ui (yi − wi ) + [1 − θj M (θj )] (˜ yj − w˜j ) , in AJE. cv = M (θi ) ui + uj ui + uj

(4) (5)

From equation (5), we an see that sear hing in the non-favorite market imposes two kinds of externalities. The rst is that with some probability workers reje t the oers from the non-favorite market. This happens when they re eive oers from their favorite market at the same time. The se ond is that their produ tivity is lower, whi h gives rms lower prot per mat hed worker. Both types of negative externality is dire tly imposed on the rms but are entirely borned by the workers in the other market, due to the free entry ondition of rms. Te hni ally, sin e the va an y ost is the same in both equilibria,

8

we must have

θA ,

M(θ)

to be higher in the AJE. This means the market tightness in AJE,

θF .

would be smaller than that in the FJE,

Intuitively, this means that rms must

fa e a higher probability of extending the oers so as to ompensate the above two kinds of externality. In our formula of (5) we assume workers from the other market get mat hed with the same probability. This assumption is somewhat tri ky if

y˜ < y ,

be ause why would

the rms extend oers to these unprodu tive workers with the same probability? So we would fo us on the ases where

y˜ = y .

The job nding rate in ea h ase are then:

λF = θi M (θi ) , in F JE;

(6)

λA = θi M (θi ) + [1 − θi M (θi )] θj M (θj ) , in AJE.

(7)

Sin e the mat hing fun tion

m (v, n)

is CRS, therefore the job nding rate in AJE is

also determined by the market tightness, whi h means the adjusted mat hing fun tion in AJE is also onstant return to s ale. Spe i ally, the number of mat hes in AJE would double if

2.2

u

and

v

double.

Symmetri Equilibria with

c = 0 , ui = uj ,

and

y = y˜

We only onsider symmetri equilibria where the same type of workers hoose the same pure strategy.

As mentioned before, there are two possible equilibria: equilibrium in

whi h workers only sear h in their favorite market (FJE) and equilibrium in whi h workers simultaneously sear h in both markets (AJE). A FJE means given the wages, the two market tightness

θi

and

θj

that satisfy (4) also make both

means given the wages, the two market tightness and use

Γj θF

θi

and

θj

Γi

and

Γj

negative. An AJE

that satisfy (5) also make

positive. Sin e the two markets are symmetri , it is obviously that and

θA

θi = θj .

Γi

We

to denote the andidate market tightness in FJE and AJE, respe tively.

From the two free entry onditions, they must satisfy

9

M (θF ) = M (θA )



1 1 y˜ − yu + [1 − θA M (θA )] 2 2 y − yu



=

cv . (1 − ρ) (y − yu )

(8)

Note that the job nding rate an be writtn as:

λF = 1 − [1 − θF M (θF )] , in F JE;

(9)

λA = 1 − [1 − θA M (θA )]2 , in AJE.

(10)

In the se ond equation, a worker an nd a job in a market with

θA M (θA ) probability 1

therefore he annot nd a job if he re eives no oer from neither of the markets .

θF > θA ,

multiple equilibria exist, (8) implies

If

whi h means in AJE the probability of

nding jobs in the favorite market is lowered be ause

θA M (θA ) < θF M (θF ).

However,

whether the total han e of nding a job, whi h in lude the han e of nding jobs in non-favorite market is not so obvious. If

θA

is su iently lower than

θF ,

then

λF > λA .

Whether this is true would depend on the mat hing fun tion and the relative size of and

y.

In addition, we need

Note that the

Γ (θ)

Γ (θA ) > 0 and Γ (θF ) < 0 for ea h

y˜

of the equilibria to exists.

an be written as

) (  2 1 1 [ρ˜ y + (1 − ρ) yu − c] , + Γ(θ) = −cu + − θM (θ) − 2 4

(11)

cu / [ρ˜ y + (1 − ρ) yu − c] ∈ (0, 1/4] then there exist two utos θ and θ¯, with θ ≤ θ¯,
  ¯ = 0. In this ase, Γ (θ) ≥ 0 if θ ∈ θ, θ¯ , and Γ (θ) < 0, otherwise. su h that Γ (θ) = Γ θ
¯ and θF > θ¯. If both θA and Therefore multiple equilibria exists if and only if θA ∈ θ, θ
θF are in the θ, θ¯ , then only AJE is feasible. If both market tightness are not in that If

region, then only FJE is feasible. If

cu / [ρ˜ y + (1 − ρ) yu − c] > 1/4,

then only FJE is feasible.

Of ourse, in the spe ial ase where

y˜ = y ,

everything works, as well.

Now the two types of workers only dier in one thing: Type 1 Note that if we have

n ¯

markets and workers simultaneously sear h in

n

i

workers has no ost

of them, then the above

formula for job nding rate would still be valid if we hange the power of the bra ket to

10

n.

of sear hing in Market

j

simultaneously.

i

but need to pay ost

cu

if (additionally) she sear hes in Market

Now if a worker re eives two oers ea h from a market, she would

randomly hoose one. The

Γ

fun tion is the same as (11), though now

y˜ = y .

The net

benet of additionally sear h in another market is the same as before, although even a worker re eives two oers ea h from a market, she sometimes hoose the other market. This is be ause the benet is the same: get the added han e of getting a job. Next think about rms' arrival rate. We now have the following:

cv = M (θF ) = M (θA ) (1 − ρ) (y − yu )



1 θA M (θA ) + [1 − θA M (θA )] 2



(12)

In the AJE, if a rm extends an oer to a worker who has another oer, the worker a

ept the job with one half probability; the worker would a

ept for sure if she does not have another oer. Interestingly, (12) and (8) are mathemati ally the same when

y = y˜.

If we start in a FJE, now if a worker de ides to additionally sear h in the other market, this poses two kinds of externalities: rst, by in reasing her han e of reje ting oers in her own market, she dis ourages va an y reation and lowers the market tightness in her own market. Workers of the same type would nd it harder to nd jobs in their favorite market, making them want to sear h in their non-favorite market, as well. Se ond, she also lowers the han e of the other type of workers for nding jobs in their favorite market. Making them want to sear h in their non-favorite market as well. Surprisingly, from the two kinds of setup, we arrive at the same theoreti al onditions for equilibrium.

2.3

Symmetri Equilibria with

For now we assume

c>0

ui = uj .

Interpretation. For rm i. Compare ase 1 (Symmetri Equilibria with Equilibria with

c > 0) with ase 2 (Symmetri

c = 0 and y = y˜), the probability of getting a su

essful mat h is de reased

11

by

i

1 2

·

1 2

· θA M (θA )



be ause now this half of appli ations who would have hosen rm

whi h is their favorite jobs now hoose randomly if they also get mat hes in the other

market.

1 · 12 2



Symmetri ally, the probability of getting a su

essful mat h is

· θA M (θA )

hosen rm

j



i

by

as well for the reason that the other half of appli ations who would have

whi h is their favorite jobs now hoose randomly if they also get mat hes in

this market where rm for rm

in reased

i is in.

So they an el ea h other out and the mat hing probability

is the same in ase 1 or 2.

For worker, look at equation 1, the

Γi

fun tion. Compare ase 1 with ase 2. The

expe ted return from sending appli ation to the other market remains exa tly the same, be ause if she gets two mat hes, she will get a job regardless of ase 1 or 2, and the return from working in either market is the same sin e

c=0

and

y = y˜.

Therefore we have the same equilibrium onditions for these two ases.

When we

say the two setups give the same equilibrium onditions, an additinal assumption is that

u1 = u2 .

Now suppose

u1 = γu2 ,

then even

y = y˜ and c > 0

so that workers prefer their

own market, then in AJE

 1 θ2 M (θ2 ) cv = (1 − ρ) (y − yu ) M (θ1 ) 1 − 1+γ   γ cv = (1 − ρ) (y − yu ) M (θ2 ) 1 − θ1 M (θ1 ) 1+γ 

This is be ause when 1, whereas there are

γ 6= 1,

γ/ (1 + γ)

then there are

1/ (1 + γ)

Type

2

workers in Market

Type 1 workers in Market 2. For these workers, if they

re eive oers from their favorite market, they would surely take that instead of those from their non-favorite market. Of ourse, when

γ 6= 1,

then in prin iple we ould have one

type of workers sear h only in one market and the other type sear h for both markets. In the above expressions, we have assumed that both type of workers sear h in both markets. On the other hand, if the workers have no preferen e over markets, and when re eiving

12

both oer a

ept ea h of them with half of the han e, then



1 cv = (1 − ρ) (y − yu ) M (θi ) 1 − θj M (θj ) 2

where

(i, j)

(1, 2)

ould be

or

(2, 1).



This expression does not depend on

γ.

Be ause,

if every worker applies in both market, then for a worker having re eived an oer from market

i,

with onditional probability

θj M (θj )

he also has an oer from Market

j,

and

he a

pet either of the oer with half of the han e. In summary, in the symmetri ase, of the workers who re eives oers from both markets, they a

ept the oer from the other market with half of the han e, so

1 θM 2 j

(θj )

of the oers would be wasted; whereas in the asymmtri ase, of the workers who re eives oers from both markets, exa tly those from the other market a

ept oers from that market, so

1 θM 1+γ 2

(θ2 )

of the oers in Market 1 and

2 would be wasted. When

2.4

γ = 1, the

γ θM 1+γ 1

(θ1 )

symmetri ase is the same as the asymmetri ase.

Simulation

m (v, u) = m0 v α u1−α , example how

θ

and

θ¯

hold others xed, see how

cv

hange ae t the equilibrium, for

hange and when multiple equilibria exist: we an plot the job

nding rate as unemployment rate before. Plot similar graphs. similar, use (4) and (10) to get

θF

and

these ondition hold, then al ulate

3 3.1

of the oers in Market

λ.

θF ,

then he k if

ρ ∈ (0, 1).

Γ (θF ) < 0

Also plot a graph of how

θ

and if

and

The logi is

Γ (θA ) ≥ 0.

θ¯ hange

with

If

cv .

The Dynami Model Environment

We onsider a dis rete-time, innite-horizon e onomy. workers is normalized to one.

Let

vit

and

uit

13

The measure of ea h type of

be the measure of va an y in market

i

and that of the unemployed Type

i

workers, in period

t.

Spe i ally, Type

i

have zero disutility from working in market i, but have a positive xed disutility,

workers

c,

from

working in the other market. We further assume that su h disutility is unobservable to employers. Workers' utility is linear in periodi in ome minus the disutility from work. An employed worker produ es

2y

every period as long as the employment ontinues. In

ea h period, an employment relationship is destroyed by nature with probability

δt

after

urrent produ tion. The output of ea h employed worker is divided half-half between this worker and her employer. On the rms' side, the periodi ost of reating a va an y is

cvt

for rms in market i. This ost ould vary over time, say, be ause of hanging nan ing

ost.

3.2

Agents' Problem

3.2.1 Workers' Problem For an employed worker, there is no a tive a tion needed. Let Type employed in market

where

i Ut+1

j

be

i

workers' value of

W ij :


ij
i Wtij = y − 1 {i 6= j} c + βδt Et Ut+1 + β (1 − δt ) Et Wt+1 , is the value fun tion of an unemployed Type

i

(13)

worker. Note that we assume

2

an equal split of the total output between the rm and the worker.

There are two hoi es in the a tion spa e of an unemployed Type for job in market

i only,

i

worker: applying

3

or applying for jobs in both markets . To ondense notation, let

Hti (1) and Hti (2) be his in remental value of hoosing the above two a tions, respe tively.  i Uti = yu + max Hti (1) , Hti (2) + βEt Ut+1

(14)

2 The Nash-bargaining determination of the wage, as in the stati model, will be one with private information, whi h is more ompli ated. For the sake of simpli ity we adopt this equal-splitting wage setting and leave the Nash-bargaining wage determination as the robustness he k.

3 We are interested in the symmetri equilibrium where identi al agents take identi al a tions. If an

unemployed type-i worker applies for market

j 6= i

job only, then applying for market

a protable deviation.

14

i

job only will be

where

 ii  i Hti (1) = βθit M (θit ) Et Wt+1 − Ut+1 n h io  ii  i6=j i i Hti (2) = −cu + β θit M (θit ) Et Wt+1 − Ut+1 + θjt M (θjt ) [1 − θit M (θit )] Et Wt+1 − Ut+1 where

i6=j Wt+1

means

ij Wt+1

if

j 6= i.

Let

Γit = Hti (2) − Hti (1),

so that

Hti (1) > Hti (2)

if

and only if

h i i6=j i Γit = −cu + βθjt M (θjt ) [1 − θit M (θit )] Et Wt+1 − Ut+1 <0 And of ourse we need to guarantee that

Hti (1) > 0.

employed only sear h in their favorite market in period

(15)

In equilibria where the un-

t,

we need

Γit < 0;

whereas in

equilibria where the unemployed sear h in both markets in period t, we must have is su iently large so that Type

i

agents, upon re-

eiving oers from both markets, will a

ept the oer from market

i.

Lastly, suppose

Note we have used the fa t that

at the beginning of period

t

c

Γit > 0.

a previously employed worker fa e a separation sho k, she

is in the unemployed pool this period, so an only get

yu

for onsumption. Even if she

immediately nd a mat h, she will start working in the next period. It is possible for a newly employed worker to lose job and get ba k to the unemployed pool immediately. But this worker gets to produ e at least one period after she a

epts the oer.

3.2.2 Firms' Value The rm de ides how many va an ies to reate by omparing the ost and the benet of opening a va an y. The ost of opening a va an y is given by the disutility ost

cv .

The

benet of opening a va an y is given by the produ t of the probability of lling a va an y, and the present dis ounted value of the prots generated by an additional employee,

Jt .

Sin e the rm operates a onstant return to s ale te hnology, the value of an additional employee to the rm is independent of the number of workers employed by the rms and, hen e, the rm's problem is linear. The value of a lled va an y to a rm is given by

15

Jt = y + βEt (1 − δt ) Jt+1

(16)

3.2.3 Free Entry and Law of Motion of Unemployment Depending on whether unemployed workers sear h in their non-favorite market, we have (a)

Favorite-Job Strategy (FJS),

market; and (b)

in whi h they only apply for jobs in their favorite

Any-Job Strategy (AJS), in whi h they apply for jobs in both markets.

In every period, free entry requires

cvt = βEt (Jt+1 ) M (θit ) , in F JS   ujt uit + [1 − θjt M (θjt )] , in AJS. cvt = βEt (Jt+1 ) M (θit ) uit + ujt uit + ujt Note in FJS, the probability of forming a employment relationship is

M(θit ).

(17) (18)

But if

every unemployed worker sear hes for both markets, then the above expression says an oer is a

epted if this is the appli ant's favorite market, or if she prefers the other market but re eives no oer from that market. Similarly, the law of motion for unemployment rate for Type

4

i

given workers dierent sear h strategies are as follows:

ui,t+1 = uit + (1 − uit ) δit − uit θit M (θit ) , in F JS

(19)

ui,t+1 = uit + (1 − uit ) δit − uit {θit M(θit ) + θjt M (θjt ) [1 − θit M (θit )]} , in AJS

(20)

Steady State Equilibria

We only onsider symmetri equilibrium in the sense that all of the unemployed workers take the same strategy. Depending on whether unemployed workers sear h in their nonfavorite market, we have (a)

Favorite-Job Equilibrium (FJE),

apply for jobs in their favorite market; and (b)

Any-Job Equilibrium (AJE), in whi h

they apply for jobs in both markets. In steady state,

16

in whi h they only

cvt , δt

are onstant over time, so are

Jt , θit , Wtii , W ij

and

Uti , where i 6= j

and

i, j = 1 or 2.

will render dierent solutions of market tightness, we use

θF

and

θA

Sin e the two types of equilibrium

θ, from rms' free entry ondition (17),

to denote the market tightness in the two types of equilibria. In steady

state, from (16), and (13), we an write

y 1 − β (1 − δ) y − 1 {i 6= j} c + βδU i = 1 − β (1 − δ)

J= W ij

(21)

(22)

From (22) we have

W i6=j − U i =

4.1

y−c 1−β − Ui 1 − β (1 − δ) 1 − β (1 − δ)

(23)

Favorite-Job Equilibrium

In FJE, every unemployed worker prefers sear hing in favorite market only so every oer extended by rms will be a

epted. Thus the steady state the value fun tion for unemployed and the free entry ondition for rms an be written as

UF = yu − cu + βθF M (θF ) WFii + β [1 − θF M (θF )] UF

(24)

cv = βM (θF ) J.

(25)

Of ourse in order to guarantee that workers do not want to deviate we need

0.

The expression of

with

θF M (θF )

denition of

U i , (24), says that the unemployed will either be an employed worker

probability and remains unemployed with

W ij

Γ (θF ) <

(when

j 6= i)

1 − θF M (θF )

probability. The

is ompli ated in the sense we are al ulating the value

of an o-equilibrium obje t. As dened in (22) and (24),

W ij

(when

j 6= i)

looks like

a one-time deviation, in the sense that although this person works in her non-favorite market, as soon as she loses this job, she would apply for jobs in her favorite market from

17

4

then on . From (25), and be ause uniquely pinned down by

M (θ)

cv /βJ .

is monotoni , we know that the market tightness is We an use (22), (13), and (14) to solve for

W ij

and

U i:

1 {i 6= j} c y [1 − β + βθF M (θF )] + βδ (yu − cu ) − (1 − β) [1 + βδ − β (1 − θF M (θF ))] 1 − β (1 − δ) (yu − cu ) (1 − β + βδ) + βθF M (θF ) y UF = (1 − β) [1 + βδ − β (1 − θF M (θF ))]

WFij =

(26)

(27)

Lastly, to pin down unemployment rate, the inow (those lose their jobs in a market) and outow (those nd a job in the same market) of the unemployed pool will be the same. So we must have

uF = The number of va an y is just

δ δ + θF M (θF )

vF = θF uF .

(28)

Average duration of unemployment is

[1 − θF M (θF )] /θF M (θF ).

4.2

Any-Job Equilibrium

In the AJE steady state, (21) and (22) are the same. But the expression for be ause now

Hti (1) < Hti (2).

UA is dierent

Thus in steady state

UA = yu − 2cu + βθA M (θA ) W ii + βθA M (θA ) [1 − θA M (θA )] W i6=j + β [1 − θA M (θA )]2 UA (29) Now the probability an oer is a

epted is no longer one. If a worker re eives an oer from her favorite market, she a

epts for sure. But if a worker re eives an oer from her 4 Another possible setup is to assume all-time deviation, so that a person working in her non-favorite market will always hoose to work in her non-favorite market. This ontradi ts the requirement that it is optimal for an unemployed to sear h only in her favorite market. Noti e in the next subse tion, where ij (when j 6= i) is no longer an o-equilibrium obje t. agents apply for both markets in equilibrium, W

18

non-favorite market, she a

epts only if she doesn't have an oer in her favorite market, whi h happens with probability

1 −θM (θ).

5

See footnote . Thus we have the steady state

free entry ondition for rms:

cv = βM (θA )



 1 1 + [1 − θA M (θA )] J 2 2

We an use (22), (13), and (29) to solve for

UA =

the

1−u

UA :

(yu − 2cu ) (1 − β (1 − δ)) + βθM (θ) [y + (1 − θM (θ)) (y − c)]   (1 − β) 1 + βδ − β (1 − θM (θ))2

The expression of does not hold.

(30)

W ij

is the same as (26). Now we an he k if

H i (1) > 0

Next think about the ow of unemployed pool of type

employed Type

i

outow of unemployed type

workers,

i

δ

i

(31)

and (15)

workers.

Of

fra tion will lose their jobs. Thus the inow and

workers will be equalized whi h gives us an expression of

the unemployment rate:

uA =

δ δ + θA M (θA ) [2 − θA M (θA )]

The number of appli ations in ea h market is

vA = 2θA uA .

nA = 2uA ,

Average duration of unemployment is

(32)

so the number of va an y is

{1 − θA M (θA ) [2 − θA M (θA )]} /θA M (θA ) [2 − θA M

The fra tion of type 2 workers in market 1 is:

1 − θA M (θA ) 1 + [1 − θA M (θA )] 5 Here we have used the assumption that re eiving an oer from one market is independent of re eiving an oer in another market. So onditioning on re eiving an oer in non-favorite market, the probability of not re eiving an oer from favorite market is

1 − θM (θ).

Further noti e sin e the measure of two types

are the same, so onditioning on extending an oer, half of the han e it is extended to Type 1 workers.

19

4.3

Comparing FJE and AJE

4.3.1 Dire t Comparison Sin e

1 − θM (θ) < 1, by omparing (24) and (30),

θF > θA .

we know

M (θF ) < M (θA ).

Therefore

This is intuitive be ause in the AJE some mat hes are wasted so the rms must

be ompensated by a lower tightness. Again, from (25) and (30), we know

M (θA ) [2 − θA M (θA )] = 2

cv = 2M (θF ) βJ

(33)

Therefore from (28) and (32), we have

uA = if and only if

2θA ≤ θF .

δ δ ≥ uF = δ + 2θA M (θF ) δ + θF M (θF )

Similarly, if

2θA ≤ θF ,

then there is fewer number of va an y in

the AJE,

vA = 2θA uA =

δ ≤ vF = θF uF = + M (θF )

δ 2θA

δ + M (θF )

δ θF

Job nding rate is also lower in the AJE,

θA M (θA ) [2 − θA M (θA )] ≤ θF M (θF ) So when will lling rate Red:

2θA ≤ θF

happen? The following graphs is the two urves of va an y

M (θ) and Blue: M (θ) [1 − θM (θ) /2].

both urves to be equal to onstant in AJE (Blue urve) is a fa tor of de reasing in

θ.

Therefore, when

more likely to have

2θA ≤ θF .

cv /βJ .

Noti e given same

[1 − θM (θ) /2]

cv /βJ

The free entry onditions requires

θ,

the va an y lling rate

of that in FJE, whi h is

1

when

θ=0

is small, the two urves will be further apart and

Figure 1 show an example where

20

θF > 2θA .

4.3.2 Beveridge Curve The idea is that

u

and

v

are fun tions of

θ,

whi h is determined by the two free entry

onditions. I exogenously hange theta (we an do that by hanging

cv

or

J

as well), and

plot the implied Beveridge urve. In general there ould be two rossings. However, when unemployment rate is not too big, the urve in AJE is above that in FJE.

4.4

Existen e and Multiple Equilibria

FJE and AJE require the following two onditions, respe tively:

h i i6=j i ΓF (θF ) = −cu + βθF M (θF ) [1 − θF M (θF )] WF (θF ) − UF (θF ) < 0 h i ΓA (θA ) = −cu + βθA M (θA ) [1 − θA M (θA )] WAi6=j (θA ) − UAi (θA ) > 0

(34) (35)

When both of the above two onditions hold, we have multiple equilibria.

Lemma 1: ΓF (θ) and ΓA (θ) either ross the horizontal line simultaneously, or neither of them rosses it. Proof: After some algebra, we an write

ΓA (θ) = where

λ = θM (θ)

1 − β + βδ + βλ ΓF (θ) . 1 + βδ − β (1 − λ)2

is the job nding rate. Q.E.D.

So we an draw the two graphs with dierent parameter value as in Figure 6 and Figure 7. We are now in the position of showing the existen e of multiple equilibria under ertain

ondition. For the sake of simpli ity, we make two assumptions. First of all, we argue that without loss of generality we may assume the appli ation ost to be zero, Essentially the appli ation ost plays a very similar role as the ost model.

c

cu = 0 .

in the dynami

This is dierent from the stati model where a positive appli aiton ost

21

cu

is

ne essary to generate multiple equilibria. In the stati model without appli ation ost, if AJE exists, then it is the only equilibrium. If the AJE exists, it means that onditional on a su

essful mat h, the benet of applying the non-favorite market whi h is the wage outweights its ost c. Both the benet and the ost are paid only after the mat h is formed in the non-favorite market, and more importantly both of them are independent of the market tightness

θ.

Therefore, for any worker, submitting another appli ation to the

non-favorable market has a positive expe ted return regardless of the market tightness. Therefore if AJE exists, then AJS is always a dominant strategy, and the AJE will be the unique equilibrium. Thus, there is either the FJE or the AJE but not both if the appli ation ost

cu

is zero.

However, this is not the ase in the dynami model with no appli ation ost. Now both the benet and the ost of applying the non-favorite market depend on the market tightness, as shown in equations (23), (27) and (31). If the AJE exists under the market tightness

  θA WAi6=j (θA ) − UAi (θA ) > 0 , it does not mean that WFi6=j (θF ) −UFi (θF ) > 0.6

The intuition is that in the dynami game, the market tightness ae ts the probabilities of nding a job in both the favorite market and the non-favorite market. In summary, in the dynami model, even if the appli ation ost

cu

is zero, it is still possible there exist

multiple equilibria. For the sake of tra kability, we assume

cu = 0.7

A se ond assumption is that we assume the mat hing fun tion takes the form of

M (θ) = κθ−σ

and the elasti ity of the mat hing fun tion w.r.t. the market tightness is

no greater than

0.5.

This is onsistent with many empiri al ndings in the literature.

The estimated mat hing fun tion elasti ity w.r.t. va an ies is typi ally no greater than 0.5 (e.g., 0.18 in

?, 0.28 in ?, 0.42 in ?, 0.5 in ?, 0.3-0.5 in ?)8

Proposition 1: ing fun tion is

(Existen e of multiple equilibria) In the ase of

M (θ) = κθ−σ

where

σ ≤ 0.5,

and

1 κ



˜

1 − β2 rc



cu = 0 ,

< 1,

if the mat h-

and the following

6 In fa t, with the absense of appli ation ost c , the ondition for the existen e of multiple equilibria u

is

WAi6=j (θA ) − UAi (θA ) > 0

WFi6=j (θF ) − UFi (θF ) < 0. appli ation ost cu is positive but

and

7 As shown in Figure (??), if

equilibria as well.

not too large, there may exist multiple

8 Some other study estimates a mu h higher elasti ity, for example 0.7 in

22

?.

ondition holds,

β˜ 1 − rc 2

1 κ

!"

β˜ (rc )σ κ

1 #− 1−σ

"

β˜ (rc )σ < rv < κ

then both the FJE and the AJE are Nash Equilibria, given are:

β˜ =

1−β(1−δ) ; β

rc =

y−yu c

− 1 ; rv =

y

1 #− 1−σ

(36)

high enough. The notions

cv . y

Proof: Appendix (A.1).

Dis ussion: We an see that when are larger. When

ΓA

and

ΓF

c

c

is small, the range of

ΓA

and

ΓF

positive

θ.

The ase for appli ation ost

θ

that makes the

is big, sin e agents only want to work in their favorite markets, both

are negative for any

large range of

θ

cu

is similar. When appli ation ost is small, we have a

that makes applying for the other market protable. When appli ation

ost is high, both

ΓA

and

ΓF

are negative for all possible

θ.

So when will we have multiple equilibria? The answer lies in Figure 4. Noti e for the same

cv /βJ ,

Take

cu = 0.2

the market tightness are dierent in the two types of equilibria: in Figure 5 for example. If

multiple equilibria, sin e now

θA ∈ (0.04, 0.3)

and

θF > 0.3,

and

ΓF (θF )

cv /βJ

is big (small), both

θA

and

θF

cu = 0.2,

by looking at

are small (big), so that both

are negative, resulting the only equilibrium: FJE; similarly when

not too big or too small, it is possible that

ΓF (θF )

then we have

ΓA (θA ) > 0 and ΓF (θF ) < 0 (noti e both ΓA (θ) and ΓF (θ)

ross the horizontal axis at 0.04 and 0.3). In this example, given Figure 1, when

θF > θA .

θA , θF ∈ (0.04, 0.3)

so that both

cv /βJ

ΓA (θA )

are positive, resulting the only equilibrium being AJE. Of ourse, when

enough (cu

= 0.4

in Figure 5), we only an have FJE; vi e versa for

In Figure 8, we show the and AJE requires

ΓA > 0 ,

ΓF

and

ΓA

as fun tion of

cv .

cu

cv

cu

is

and

is big

being small.

Sin e FJE requires

ΓF < 0

we also plot the steady state unemployment rate of ea h type

of equilibrium if that equilibrium exists. We an see that FJE exist if whereas AJE exists if

ΓA (θA )

is in the middle range. In the region where

cv

is small or big,

ΓF < 0

and

ΓA > 0 ,

we have multiple equilibria. But it is not always true that the unemployment rate in AJE

23

is always higher than FJE.

5

Dynami s

In this se tion, we would like to see if we an onstru t a dynami equilibrium that looks like the jobless re overy in Figure 1. period sho k to the separation rate

δ

We onsider two kinds of sho ks: a one-

that exogenously in reases the unemployment rate,

cv .

and a one-period nan ial sho k to the rm entry ost,

The standard model in

this se tion is when we for e agents to sear h their favorite market only. Note that in the standard model, whether we have nan ial sho k or not only matters slightly for the height of the peak of the unemployment in Period 3 but does not ae t how it behaves afterward.

In the urrent model, if there is no nan ial sho k, then it looks

exa tly the same as in the standard model. Be ause the one-period unexpe ted sho k to

δ

i.e.

only hanges unemployment rate in Period 3, but not the in entives of the workers,

ΓF (θF )

is un hanged. And sin e market tightness is un hanged, unemployment rate

qui kly de reases as in the standard model. However, if there is nan ial sho k in the

urrent model, things are dierent.

First, suppose everyone else only sear hes in his

favorite market at present and in the future, think about the expe ted net benet of sear h additionally to the non-favorite market for an individual:

h i i6=j i ΓF (θF,2 ) = −cu + βθF,2 M (θF,2 ) [1 − θF,2 M (θF,2 )] E2 WF (θF,3 ) − UF (θF,3 ) Noti e for

θF,j

θF,2

and

and

θF,3

cv2 > cv3 = c∗v .

are both determined by (17), with

j ≥ 3, cvj = c∗v

Further, sin e

is equal to its original value so we know that the value in the

expe tation sign is equal to its steady state value. If state value then we must have

ΓF < 0 .

But now

θF,2

is also the same as its steady

θF,2 < θF .

This means if

θF,2 M (θF,2 ) [1 − θF,2 M (θF,2 )] > θF M (θF ) [1 − θF M (θF )] ,

24

then we ould have

ΓF (θF,2 ) > 0,

whi h means any individual have in entive to deviate

from the FJS strategy, applying for favorite market only. This means applying for their favorite markets is no longer a dynami equilibrium. equilibrium and then verify if in this equilibrium

We then assume agents hange

ΓA (θA,2 )

is indeed positive. Figure 9

shows an example of the swit h of equilibrium after the one-period sho ks of

δ

and

cv

in period 2. The blue urve is the standard one market sear h and mat hing model and the green urve is ours. Now from Period 3 on, if the agents an oordinate and forbid sear hing in the non-favorite market, the e onomy an swit h ba k to the FJE and the unemployment rate an de rease as the standard model. However, we would like to argue that on e the AJE is already the equilibrium played by the workers, oordination is not a trivial matter. As in our steady state analysis, it is not always true that AJE would always have lower job nding rate than FJE. From equation (33), we need

2θA ≤ θF

to

guarantee the job nding rate in our dynami AJE is smaller than that in our dynami FJE, whi h is the same as in the standard model.

6

Con lusion

In the urrent version of the paper, we analyze the unemployed workers' hoi e of how broadly to sear h for jobs, i.e. sear h in their favorite market only or not. The strategi

omplementarity makes the multiple equilibria possible. We study the steady state properties of this model and nd that sear hing harder sometimes improves job nding rate, but sometimes it does not. Then we ompare our model with the standard model and nd that nan ial sho ks, whi h in reases the rms' entry ost even only temporarily,

an potentially swit h the labor market equilibrium type permanently. With the swit h of equilibrium, we an generate slower re overy of the labor market as well as the shift of Beveridge Curve. In the future study it is worthwhile to quantify whether the sho ks in reality is enough to ause the swit h of equilibrium and that the resulted new AJE indeed has smaller job nding rate.

25

Figure 1: Va an y Filling Rates

26

Figure 2: Beveridge Curve

27

Figure 3: Gamma Fun tions with Changing C

28

Figure 4: Gamma Fun tions with Changing Cu

29

Figure 5: Unemployment Rate with Multiple Equilibria

30

Appendix A

Proofs

A.1

Proof of Proposition 1

Lemma 1 establishes that

ΓF (θ) = 0

and

ΓA (θ) = 0

have same roots. Condition (34)

an be rewritten as



y−c y u − cu − 1 − β (1 − δ) 1 − β (1 − δ) + βλ  βλy − [1 − β (1 − δ)] [1 − β (1 − δ) + βλ]

ΓF (θ) = −cu + βλ (1 − λ)

where

λ = θM (θ)

is the job nding rate.

We examine the spe ial ase where

ΓF (λ (θ)) =

When

λ = 0

βλ (1 − λ), λ∗ ∈ (0, 1)

or

cu = 0 .

Then the above fun tion be omes

βλ (1 − λ) {(y − yu − c) [1 − β (1 − δ)] − βλc} [1 − β (1 − δ)] [1 − β (1 − δ) + βλ]

1, ΓF (λ (θ)) = 0.

For any

λ ∈ (0, 1),

the rst part of

ΓF (λ (θ)),

is always positive, while the se ond part is a de reasing fun tion of su h that

ΓF (λ∗ (θ)) = 0, λ∗ =

rosses

Dene

whi h yields the unique solution,

1 − β (1 − δ) y − yu − c · β c

There is a reasonable region in the parameter spa e su h that

λ ∈ (0, 1), ΓF (λ (θ))

λ.

ΓF (λ∗ (θ)) = 0

λ∗ ∈ (0, 1).

And for

only on e, and it is from above. Therefore,

the su ient and ne essary ondition for both of (34) and (35) to hold is

λA < λ∗ < λF ,

or

θA < θ∗ < θF

where

λ = θM (θ) .

(A.1)

Now we provide a su ient ondition for the above ondition to hold. To save some notation, dene

β˜ =

1−β(1−δ) and β

31

rc =

y−yu c

−1, then the above equation

be omes

˜ c. λ∗ = βr Given the mat hing fun tion,

M (θ) = κθ−σ , we an solve θ∗ su h that ΓF (λ (θ∗ )) = 0,

θ∗ =

We an then solve

θF

rv =

1 ! 1−σ

˜ v βr κ

!− σ1

cv . y

The su ient and ne essary ondition for

˜ v βr κ

!− σ1

>

θF > θ∗ ˜ c βr κ

or

"

β˜ rv < (rc )σ κ The

θA

.

from (25) and (21),

θF =

where

˜ c βr κ

is

1 ! 1−σ

1 #− 1−σ

(A.2)

is determined by the following equation,

κθA−σ



  1 1 1−σ ˜ v = 0. + 1 − κθA − βr 2 2

(A.3)

Dene

F (θ) = κθ The derivative of

F (θ)

w.r.t.

θ

−σ



  1 1 1−σ ˜ v − βr + 1 − κθ 2 2

is,

1 F ′ (θ) = −κσθ−σ−1 − κ2 (1 − 2σ) θ−2σ 2 When

σ ≤ 0.5, F ′ (θ) < 0, ∀θ > 0.

Sin e

limθ→0 F (θ) > 0

32

and

limθ→∞ F (θ) < 0,

the

su ient and ne essary ondition for

˜ c βr κ

κ

θA < θ∗

σ !− 1−σ

F (θA ) > F (θ∗ ),

β˜ 1 − rc 2

whi h is

˜ c βr κ

˜ v>κ βr

is

σ !− 1−σ

!

˜ v<0 − βr

β˜ 1 − rc 2

or

1 rv > κ As long as

1 κ



˜

1 − β2 rc



< 1,

β˜ 1 − rc 2

!"

or

β˜ (rc )σ κ

!

1 #− 1−σ

(A.4)

onditions (A.2) and (A.4) would generate a non-empty

feasible parameter set. If

y

is su iently high, then the parti ipation ondition will satisfy in both the FJE

and AJE. Note that de rease with

y

when

A.2

y.

rc

in reases while

rv

y . In ondition (A.2) both sides   β˜ 1 1 − r de reases with y . Therefore κ 2 c

de reases with

Same for ondition (A.4). The

in reases, it does not ne essarily shrink the feasible parameter set.

Proof of a More General Proposition

Lemma 1 establishes that

ΓF (θ) = 0

and

ΓA (θ) = 0

have same roots. Condition (34)

an be rewritten as



y u − cu y−c − 1 − β (1 − δ) 1 − β (1 − δ) + βλ  βλy − [1 − β (1 − δ)] [1 − β (1 − δ) + βλ]   (y − c) − (yu − cu ) βλc = −cu + βλ (1 − λ) − 1 − β (1 − δ) + βλ [1 − β (1 − δ)] [1 − β (1 − δ) + βλ]

ΓF (λ (θ)) = −cu + βλ (1 − λ)

where

λ = θM (θ)

When

λ=0

is the job nding rate.

or

λ = 1, ΓF (λ (θ)) = −cu .

Therefore

horizontal axis twi e or none. That is, the equation none. If it has two roots, root,

λ∗∗ ,

ΓF (λ (θ))

ΓF (λ (θ))

in general rosses the

ΓF (λ (θ)) = 0 has either two roots or

rosses the horizontal axis from below at the smaller

and from above at the larger root,

λ∗ .

33

In FJE,

M (θF ) =

cv [1 − β (1 − δ)] βy

(A.5)

In AJE,

  cv 1 [1 − β (1 − δ)] M (θA ) 1 − θA M (θA ) = 2 βy

(A.6)

We want to show that there exists a mat hing fun tion, su h that the inverse job nding rate,

θ = λ−1 (θ),

satises (A.5), (A.6) and the following,

θA (λA ) < θ∗ (λ∗ ) < θF (λF )

The roots of

ΓF (λ (θ))

(A.7)

are solutions of the following equation,

cu (y − c) − (yu − cu ) βλc = − βλ (1 − λ) 1 − β (1 − δ) + βλ [1 − β (1 − δ)] [1 − β (1 − δ) + βλ] The right hand side (RHS) is a de reasing fun tion of

34

λ,

while the LHS is

(A.8)