Chapter I: Labour Market Theory - uni-muenchen.de · 2012-08-19 · Chapter I: Labour Market Theory Section 4: Directed search models Timing of the game: 1. Firms decide simultaneously

Chapter I: Labour Market Theory Section 4: Directed search models

Chapter I: Labour Market Theory

Section 4: Directed search models

Literature:

Pierre Cahuc and Andre Zylberberg: Labour Economics

Chapter 9: Section 6

Burdett, Shi and Wright (2001), Pricing and matching with frictions,

Journal of Political Economy, 109(5), pp. 1060 - 1085.

Delacroix and Shi (2006), Directed search on-the-job and the wage ladder,

International Economic Review, 47(2), pp. 651 - 699.

Kircher (2009), Efficiency and simultaneous search,

Journal of Political Economy, 117(5), pp. 861- 913.

Prof. Dr. Christian Holzner Page 213


Definition: Directed search models

Firms post wages (we assume that they can commit to their posted wages) andworkers observe the posted wages and decide whether they want to apply or not.

Firms compete for workers by posting higher wages, i.e., a higher wage makes it morelikely that a worker applies.

Firms can only employ one worker, i.e., there is only one available vacancy per firm.

Commitment to posted wages allows firms to post wages that are above the reserva-tion wage. This solves the Diamond Paradox.

Research question:

What wages are offered in equilibrium?



Framework:

- One period model.

- Unemployment income (value of leisure) z = 0.

- Productivity of a firm-worker pair equals y = 1.

- Number of firms (vacancies) V .

- Denote the wage that firm i posts by wi.

- Number of unemployed workers U .

- Workers are risk neutral, i.e., their utility equals their wage.

- Denote the probability that a worker j applies to firm i by aj,i.



Timing of the game:

1. Firms decide simultaneously on the wage they want to post.

2. Workers observe all posted wages and decide simultaneously where to apply.

3. Firms with more than one application pick one candidate at random and employthe worker at the posted wage. Firms with only one candidate employ the workerat the posted wage and firms without any application remain vacant.



4.1 Microfoundation - Burdett, Shi and Wright (2001)

4.1.1 Two firm - two worker case

Utility of a worker:

If worker 1 applies to firm A, he gets employed at the wage wA with probability1 − a2 + a2/2, i.e.,

- if worker 2 does not apply to firm A (prob. 1−a2), worker 1 gets the job for sure

- if worker 2 applies to firm A (prob. a2), worker 1 is chosen with probability 1/2.

This implies the following utility, if worker 1 applies to firm A,

u1,A = (1 − a2 + a2/2) wA. (1)

Similarly, if worker 1 applies to firm B,

u1,B = (a2 + (1 − a2) /2) wB.



Stage 2: Application decision of a worker

The worker applies to the firm that offers the highest utility, i.e.,

u1 = max [u1,A, u1,B]

Depending on the wages offered, i.e., wA and wB, and on the application probabilityof worker 2, i.e., a2, worker 1 applies to firm A with the following probability, i.e.,

a1 =

0 if a2 > a (wA, wB)1 if a2 < a (wA, wB)[0, 1] if a2 = a (wA, wB)

where a (wA, wB) is defined such that the worker is indifferent between applying tofirm A or B, i.e.,

u1,A = u1,B ⇐⇒ a (wA, wB) =2wA − wB

wA + wB. (2)

Note: The worker does not necessarily apply to the firm that offers the highest wage,since rationing needs to be taken into account.



Equilibrium at stage 2:

If firm A posts a low wage compared to firm B, i.e.,

wA <wB

2=⇒ (a1, a2) = (0, 0) (3)

both workers apply to firm B, since a (wA, wB) < 0.

If firm A posts a high wage compared to firm B, i.e.,

wA > 2wB =⇒ (a1, a2) = (1, 1) (4)

both workers apply to firm A, since a (wA, wB) > 1.



Equilibrium at stage 2 (continued)

If firm A and firm B post similar wages, i.e.,

wA ∈[wB

2, 2wB

]

. (5)

There exist two pure-strategy equilibria

(a1, a2) = (1, 0) , (6)

(a1, a2) = (0, 1) , (7)

and one mixed-strategy equilibrium

a1 = a2 = a (wA, wB) . (8)



Stage 1: Wage posting

Firms post wages in order to maximize their profits, i.e.,

πA = maxwA

[1 − (1 − a1) (1 − a2)] [1 − wA]

πB = maxwB

[1 − a1a2] [1 − wB]

where a1 = a1 (wA, wB) and a2 = a2 (wA, wB) depend on the posted wages ascharacterized in stage 2.

Result 1:

Offering a wage wA < wB/2 or wB < wA/2 cannot be an equilibrium, since afirm that offers such a low wage does not attract any worker and therefore makeszero profit.

Thus, only wages in the following range are potential equilibrium wages, i.e.,

wA ∈[wB

2, 2wB

]

and wB ∈[wA

2, 2wA

]

.



Stage 1: Wage posting (continued)

If posted wages are in the range

wA ∈[wB

2, 2wB

]

and wB ∈[wA

2, 2wA

]

,

firms earn the profit

πA = [1 − wA] and πB = [1 − wB] ,

if workers play one of the two pure-strategy equilibrium.

If workers play the mixed-strategy equilibrium a1 = a2 = a (wA, wB) firms earnthe profits

πA =

[

1 −

(

1 −2wA − wB

wA + wB

)2]

[1 − wA] ,

πB =

[

1 −

(

2wA − wB

wA + wB

)2]

[1 − wB] .



Stage 1: Wage posting (continued)

Conditional on workers playing a mixed-strategy equilibrium the best responsewage function w∗

A (wB) and w∗B (wA) can be derived from ∂πA/∂wA = 0 and

∂πB/∂wB = 0, i.e.,

w∗A (wB) =

4wB + w2B

5wB + 2and w∗

B (wA) =4wA + w2

A

5wA + 2

The profit functions π∗A (wB) and π∗

B (wA) are obtained by substituting w∗A (wB)

and w∗B (wA) into firm profits, i.e.,

π∗A (wB) =

(2 − wB)2

4wB + 4and π∗

B (wA) =(2 − wA)2

4wA + 4.

As shown by Burdett, Shi and Wright (2001) the best response wage function andthe profit function have the shape as shown in Figure 4.1.



Figure 4.1: Best response wage function and the profit function



Stability of pure strategy equilibria:

If firms post wages (wA, wB) that are not symmetric, workers play pure strategies,i.e., each worker visit a different firm with probability one.

If a firm deviates and offers a different wage, workers deviate to the mixed-strategy equilibrium.

This trigger strategy of deviating to the mixed-strategy equilibrium sustains the pure-strategy equilibrium, as long as the mixed-strategy equilibrium implies lowerprofits, i.e.,

πpureA = 1 − wA > πmixed

A =(2 − wB)2

4wB + 4.



Stability of pure strategy equilibria (continued):

Thus, all wages that satisfy the following condition are equilibrium wages, i.e.,

wA < 1 −(2 − wB)2

4wB + 4

=⇒ All these pure strategy equilibria require a lot of coordination.

In large markets with many workers and firms coordination is likely to fail andworkers would play the mixed-strategy equilibrium.



Stability of mixed strategy equilibria:

If firms post wages (wA, wB) = (0, 0), workers have no incentive to search (DiamondParadox).

=⇒ A pure strategy equilibrium with (wA, wB) = (0, 0) cannot exist.

If firms post wages (wA, wB) = (1/2, 1/2), workers play the mixed applicationstrategy.

The equilibrium is self-supporting, since no profitable worker deviation exists. Wor-kers apply according to a (wA, wB) in equation (2) with probability (a1, a2) =(1/2, 1/2).



4.1.2 The case of V vacancies and U workers

We focus only on mixed-strategy equilibria.

Thus, applications are send according to the Urnball model.

The Urnball model implies:

A worker sends with probability a an application to a particular firm.

The probability that at least one worker applies to a specific vacancy equals (compareCh. I, Sec. 2, p. 25)

1 − (1 − a)U .



Acceptance probability of a worker:

Denote by Ω the probability that a worker, who applies to a specific vacancy, getsaccepted. The probability that a worker is matched with a specific vacancy is thereforeequal to aΩ, i.e., the probability that she applies to a specific vacancy time theacceptance probability.

The probability that a worker is matched to a specific vacancy must equal the pro-bability that a specific vacancy is matched to a worker, i.e.,

aΩ =1 − (1 − a)U

U.



Acceptance probability at different wage offers:

• Suppose all firms offer a wage w and one firm contemplates deviating to wd.

• Let the probability that a worker applies to the deviating firm be ad.

• The probability that a worker applies to a nondeviating firm is then given by(

1 − ad)

/ (V − 1).

• The acceptance probability at the deviating firm is then given by

Ωd =1 −

(

1 − ad)U

adU.

• The worker, who applies at a nondeviating firm, gets served with probability

Ω =1 −

(

1 −[(

1 − ad)

/ (V − 1)])U

U [(1 − ad) / (V − 1)].



Stage 2: Workers application decision

Workers apply to the deviating firm with the following probability

ad =

0 ifw

wd>

Ωd

Ω

∣

∣

∣

∣

ad→0

1 ifw

wd<

Ωd

Ω

∣

∣

∣

∣

ad→1

ad(

wd, w)

ifw

wd=

Ωd

Ω

∣

∣

∣

∣

ad=ad(wd,w)

where the application probability ad(

wd, w)

is such that workers are indifferent bet-ween applying to the deviating firm or any other nondeviating firm, i.e.,

wd1 −(

1 − ad(

wd, w))U

ad (wd, w) U= w

1 −(

1 −[(

1 − ad(

wd, w))

/ (V − 1)])U

U [(1 − ad (wd, w)) / (V − 1)]. (9)



Stage 1: Firms’ wage posting decision

The expected profit of a deviating firm is given by

πd(

wd, w)

=[

1 −(

1 − ad(

wd, w))U

]

[

1 − wd]

.

The first order condition implies:

[

U(

1 − ad(

wd, w))U−1

] ∂ad(

wd, w)

∂wd

[

1 − wd]

−[

1 −(

1 − ad(

wd, w))U

]

= 0

where the workers application reaction ∂ad(

wd, w)

/∂wd is given by implicitly diffe-rentiating equation (9),

∂ad(

wd, w)

∂wd=

1−

(

1−1−ad

V −1

)U

1−adV −1

wdU(1−ad)

U−1ad−

[

1−(1−ad)U

]

[ad]2 − w

V −1

−U(

1−1−adV −1

)U−11−adV −1 +

[

1−(

1−1−adV −1

)U]

[

1−adV −1

]2



Stage 1: Firms’ wage posting decision (continued)

The symmetric equilibrium requires that all firms offer the same wage, i.e., w = wd.

Symmetric mixed application strategies then imply that each firm has the same pro-bability to receive an application, i.e., a = 1/V .

These symmetry conditions imply the following equilibrium wage offer, i.e.,

w =U (V − 1) (1 − 1/V )U

V (V − 1) − (U + V (V − 1)) (1 − 1/V )U

The matching probability of a worker is then given by (compare Ch. I, Sec. 2, p. 26)

M (V, U) = V

[

1 −

(

1 −1

V

)U]

.



Stage 1: Firms’ wage posting decision (as V, U → ∞)

The wage offer in the limit, i.e., as V, U → ∞ and θ = V/U constant, is given by

w = limV →∞

(V/θ) (V − 1) (1 − 1/V )(V/θ)

V (V − 1) − ((V/θ) + V (V − 1)) (1 − 1/V )(V/θ)=

1

θ[

e1/θ − 1]

The matching probability of a firm and a worker are given by

m (θ) = 1 − e−1/θ and θm (θ) = θ[

1 − e−1/θ]



4.1.3 Alternative solution method in a large market

Workers will only apply to those firms that provide the highest utility of being unem-ployed.

All firms must offer a wage that ensures the same value of being unemployed for allunemployed.



Value of being unemployed (in a one-period model):

Vu = θ[

1 − e−1/θ]

w

Profit of a firm (in a one-period model):

π = maxw

[

1 − e−1/θ]

[1 − w]

By choosing the wage firms indirectly choose the number of workers that apply, i.e.,the market tightness θ they face.



Substituting the wage w

π = maxθ

[

[

1 − e−1/θ]

−Vu

θ

]

and differentiating with respect to the market tightness implies the following firstorder condition,

e−1/θ 1

θ2 =Vu

θ2 =

[

1 − e−1/θ]

θw,

w =1

θ[

e1/θ − 1],

and the same wage as above.

Note: This method is only correct for large markets. In small markets the value ofunemployment Vu cannot be takes as exogenous, because a change in the postedwage implies a change in the application probability of a worker and therefore achange in Vu.



Directed search in the Mortensen-Pissarides framework:

Value of being unemployed and employed, i.e.,

rVu = z + θm (θ) (Ve − Vu)

rVe = w + q (Vu − Ve)

Substitution implies

rVu =(r + q) z + θm (θ) w

r + q + θm (θ)(10)

The value of being unemployed:

• increases with the offered wage w and with the acceptance probability θm (θ),

• is the weighted average of unemployment benefits z and wages w.



Directed search in the Mortensen-Pissarides framework:

Value of a vacancy and of employing a worker, i.e.,

rΠv = −h + m (θ) (Πe − Πv) ,

rΠe = y − w + q (Πv − Πe)

The firm posts a wage that maximizes the value of a vacancy,

rΠv = maxw

[

−h +m (θ)

r + q + m (θ)(y − w + h)

]

subject to the participation constraint (10),

rVu =(r + q) z + θm (θ) w

r + q + θm (θ).



Using the Lagrangian implies

L = maxw,θ

[

−h +m (θ) (y − w + h)

r + q + m (θ)− λ

(

rVu −(r + q) z + θm (θ) w

r + q + θm (θ)

)]

First order conditions:

1.∂L

∂w= −

m (θ)

r + q + m (θ)+ λ

θm (θ)

r + q + θm (θ)= 0

2.∂L

∂θ=

m′ (θ) (y − w + h) (r + q)

(r + q + m (θ))2+ λ

[θm (θ)]′ (w − z) (r + q)

(r + q + θm (θ))2= 0



The free entry condition implies that the value of a vacancy will be zero inequilibrium, i.e.,

Πv = 0 ⇐⇒y − w

r + q=

h

m (θ)or

(y − w + h)

r + q + m (θ)=

h

m (θ)or θm (θ) = (r + q)

θh

y − w

Substituting λ implies

m′ (θ)(y − w + h)

(r + q + m (θ))= −

1

θ

[θm (θ)]′ (w − z)

(r + q + θm (θ))

Using the free entry condition allows us to write the wage equation as follows,

m′ (θ)y − w

r + q= −

1

θ

[θm (θ)]′ (w − z)

r + q + (r + q)θh

y − w

θm′ (θ) (y − w + θh) = − [θm (θ)]′ (w − z)



Wage equation:

Using the fact that [θm (θ)]′ = m (θ) + θm′ (θ), implies

w = (1 − η (θ)) z + η (θ) (y + θh) ,

where η (θ) equals the elasticity of the matching function with respect to the unem-ployment rate u, i.e.,

η (θ) = −θm′ (θ)

m (θ)

Note: The posted wage satisfies the Hosios condition, i.e., γ = η (θ), and leadsto efficient vacancy creation (compare Ch. I, Sec. 2, pp. 100).



Intuition for the efficiency of directed search:

By posting a certain wage, firms can price the probability to receive an application.This internalizes the externalities, since workers and other firms observe the postedwages and adjust their behavior accordingly.

Unemployed workers apply to the market that maximizes their utility, i.e., a marketthat would waste resources by opening to many vacancies will not be attractivefor unemployed workers.

A market with too few vacancies would also not be attractive, since this would implya suboptimal matching probability for unemployed workers.



4.2 Directed search on-the-job - Delacroix and Shi (2006)

Idea:

Employed workers search in the submarket that maximizes their utility given theircurrent wage.

For a given wage only one wage and the associated matching probability is optimal.

Thus, employed workers climb a wage ladder step by step. This result is differentfrom random search (BM or PVR), where all workers are equally likely to receive aspecific wage.



Framework:

- All firms are equally productive, i.e., y. Firms post wages w. The set of postedwages W is observed by workers.

- With probability λ unemployed and employed workers receive an applicationopportunity.

- a (w′, w) is the probability with which an applicant that currently earns awage w applies for a job offering the wage w′. If a worker applies, he hasto pay the cost c.

- n (w) denotes the number of workers employed at wage w. The total numberof workers is normalized to unity. The unemployment rate equals u = n (z) andthe wage earnings density g (w) = n (w) / [1 − u].

- V denotes the endogenously determined number of vacancies. The fractionof vacancies offering a wage w is given by v (w).



Market tightness for a w′-submarket (posted wage w′):

Workers with potentially different wages w receive an application opportunity withprobability λ and apply with probability a (w′, w) to the w′-submarket.

The expected number of workers applying at a w′-submarket is therefore given by∑

w

λn (w) a (w′, w)

The tightness in a w′-submarket is then given by

θ (w′) =V v (w)

∑

w λn (w) a (w′, w).



Matching probabilities:

A worker’s employment probability in a w′-submarket is given by

θ (w′) m (θ (w′)) .

A firm’s hiring probability in a w′-submarket is given by

m (θ (w′)) .



Value of being unemployed:

With probability λ an unemployed worker receives an application opportunity, i.e.,

rVu = z + λ max[

∑

w′a (w′, z) E (w′, z) − c, 0

]

.

If the maximum expected surplus E (w′, z) that an unemployed applicant canget given the set of posted wages W , i.e.,

E (w′, z) = maxw′∈W

θ (w′) m (θ (w′)) [Ve (w′) − Vu]

is higher than the cost c of an application, then the unemployed worker sends anapplication to the submarkets that maximize the expected gain from searching.



Value of being employed:

With probability λ an employed worker receives an application opportunity, i.e.,

rVe (w) = w + λ max[

∑

w′a (w′, w) E (w′, w) − c, 0

]

+ q (Vu − Ve (w)) .

If the maximum expected surplus E (w′, w) that an employed applicant can get, i.e.,

E (w′, w) = maxw′∈W

θ (w′) m (θ (w′)) [Ve (w′) − Ve (w)]

is higher than the cost c of an application, then the employed worker send an appli-cation to the submarkets that maximize the expected gain from searching.

The value of employment Ve (w) increases with the wage w, since E (w′, w)includes the same set of posted wages W to which a worker can apply.



Workers’ application decision:

Workers apply to wi-submarket with the following probability

a(

wi, w)

=

0 if θ(

wi)

m(

θ(

wi)) [

Ve

(

wi)

− Ve (w)]

< E(

w−i, w)

1 if θ(

wi)

m(

θ(

wi)) [

Ve

(

wi)

− Ve (w)]

> E(

w−i, w)

(0, 1) if θ(

wi)

m(

θ(

wi)) [

Ve

(

wi)

− Ve (w)]

= E(

w−i, w)

where E(

w−i, w)

equals the expected surplus that an applicant can get in all sub-markets except wi, i.e.,

E(

w−i, w)

= maxw′∈W\wi

θ (w′) m (θ (w′)) [Ve (w′) − Ve (w)] .



Value of a vacancy:

rΠv (w) = −h + m (θ (w)) [Πe (w) − Πv (w)]

+ (1 − m (θ (w)))[

Πv − Πv (w)]

whereΠv = max

wΠv (w)

Thus, if a vacancy is not matched this period, which happens with probability(1 − m (θ (w))), the firm will post the wage next period that maximizes the value ofa vacancy.

Free entry:

Free entry implies that Πv = Πv (w) = 0 for all equilibrium wage offers w ∈ W .



Value of employing a worker at the wage w:

rΠe (w) = y − w + (q + ρ (w))[

Πv − Πe (w)]

where ρ (w) denotes the probability that a worker employed at the wage w quits andworks for another employer, i.e.,

ρ (w) = λ∑

w′

a (w′, w) θ (w′) m (θ (w′)) .

The quitting probability ρ (w) depends on the worker’s application decision a (w′, w)and the respective employment probabilities θ (w′) m (θ (w′)).(Comparable to λs [1 − F (w)] in the BM-model).



A recruiting firm’s optimal offer:

The firm offers the wage that maximizes the gain from employing a worker subjectto the constraint that the worker is willing to apply for a job at the offered wage, i.e.,

maxwi

m(

θ(

wi)) [

Πe

(

wi)

− Πv

]

subject to

θ(

wi)

m(

θ(

wi)) [

Ve

(

wi)

− Ve (w)]

≥ E(

w−i, w)



Separation of applicants by their current wage:

The worker’s indifference curve is given by

θ(

wi)

m(

θ(

wi)) [

Ve

(

wi)

− Ve (w)]

= E(

w−i, w)

.

The indifference curve constitutes a negative relationship between the employmentprobability θ

(

wi)

m(

θ(

wi))

and the wage wi, since the value of employment Ve

(

wi)

is strictly increasing in the wage wi, i.e.,

d[

θ(

wi)

m(

θ(

wi))]

dwi= −

θ(

wi)

m(

θ(

wi))

V ′e

(

wi)

Ve (wi) − Ve (w).



Single crossing property implies separation:

An applicant with a higher current wage, i.e., w1 > w2, cares more about a higherwage than about the employment probability, i.e.,

−d

[

θ(

wi)

m(

θ(

wi))]

dwi

∣

∣

∣

∣

∣

w=w1

> −d

[

θ(

wi)

m(

θ(

wi))]

dwi

∣

∣

∣

∣

∣

w=w2

.

This single crossing property implies the following result:

Each equilibrium wage attracts at most one type of applicant, i.e., workers with

different current wages apply for different wages.



q

indifference curve

for worker wj (wj>wi)

direction of higher

expected utility

A indifference curve

for worker wi

B

direction of

higher expected

surplus for firm iso-profit curve

w



The equilibrium must be a wage ladder:

posted wages

w1 w2 wi-1 wi wi+1 wM

….. ….

w0=b w1 w2 wi-1 wi wi+1 wM

employed wages

The wage wi therefore denotes the wage paid at step i, with w0 = z and wi > wi−1.



The wage ladder implies the following simplification:

For a firm:

rΠv

(

wi)

= −h + m(

θ(

wi)) [

Πe

(

wi)

− Πv

]

,

rΠe

(

wi)

= y − wi +(

q + λθ(

wi+1)

m(

θ(

wi+1))) [

Πv − Πe

(

wi)]

For a worker, if E(

wi+1, wi)

≥ c, i.e.,

rVu = z + λθ(

w1)

m(

θ(

w1)) [

Ve

(

w1)

− Vu

]

− c,

rVe

(

wi)

= wi + λθ(

wi+1)

m(

θ(

wi+1)) [

Ve

(

wi+1)

− Ve

(

wi)]

− c

+q(

Vu − Ve

(

wi))

.

For submarket M it must be true that E(

wi+1, wi)

< c, i.e.,

rVe

(

wM)

= wM + q(

Vu − Ve

(

wM))

.



The optimal wage offer:

maxwi,θi

m(

θi) [

Πe

(

wi)

− Πv

]

subject toθim

(

θi) [

Ve

(

wi)

− Ve

(

wi−1)]

= E(

wi, wi−1)

Implies

[

1 − η(

θi)] [

Ve

(

wi)

− Ve

(

wi−1)]

= η(

θi)

Πe

(

wi)

(11)

θim(

θi) [

Ve

(

wi)

− Ve

(

wi−1)]

= E(

wi, wi−1)

(12)

where η(

θi)

= −θim′(

θi)

/m(

θi)

.



Feasibility condition:

The free entry condition Πv = 0 implies the following feasibility condition forwages wi and market tightness θ

(

wi)

combinations, i.e.,

h

m (θ (wi))=

y − wi

r + q + λθ (wi+1) m (θ (wi+1))(13)

This condition (that resembles the vacancy creation condition in the MP-model)determines for which wages it is profitable to be posted in equilibrium.



Search incentive:

Given the search cost c, workers will search for a better paid job paying wage wi, iff

E(

wi, wi−1)

≥ c.

Using the FOCs (11) and (12)

θim(

θi) η

(

θi)

1 − η(

θi)Πe

(

wi)

= E(

wi, wi−1)

together with the feasibility condition (13) that the market tightness has to satisfythe following condition to ensure that workers are willing to search, i.e.,

η(

θi)

1 − η(

θi)θi ≥

c

h.



The highest wage:

Since search cost are positive, i.e., c > 0,

and firms are only willing to pay wages below the marginal product, i.e., wi < y,

the maximum wage wM is so high that no worker is willing to search given that wage,i.e.,

θMm(

θM) η

(

θM)

1 − η(

θM)

y − wM

r + q= E

(

wM+1, wM)

< c.

The maximum wage therefore satisfies

wM > y −r + q

θMm(

θM)

1 − η(

θM)

η(

θM) c.



Given wM the worker has the following value of being employed,

rVe

(

wM)

=wM + qVu

r + q.

The feasibility condition determines the market tightness,

h

m(

θM) =

y − wM

r + q.

This determines the maximum expected surplus E(

wM , wM−1)

at the wagewM−1,

θMm(

θM) η

(

θM)

1 − η(

θM)

y − wM

r + q= E

(

wM , wM−1)

and the value of being employed at the wage wM−1,

Ve

(

wM−1)

= Ve

(

wM)

−E

(

wM , wM−1)

θMm(

θM)



This allows us to determine the wage wM−1 using the Belmann-Equation for anemployed worker,

rVe

(

wM−1)

= wM−1 + λθ(

wM)

m(

θ(

wM)) [

Ve

(

wM)

− Ve

(

wM−1)]

− c

+q(

Vu − Ve

(

wM−1))

.

The feasibility condition determines the market tightness θ(

wM−1)

h

m (θ (wM−1))=

y − wM−1

r + q + λθ (wM) m (θ (wM))

and so on and so on...

The equilibrium wage ladder has to be consistent with the maximum expectedsurplus of an unemployed worker, i.e., E

(

w1, z)

.

Delacroix and Shi (2006) show under which conditions equilibrium expectations areconsistent with the wage ladder described above.



Equilibrium characterization:

The matching rate of firms increases along the wage ladder, i.e.,

m(

θ(

wi))

> m(

θ(

wi−1))

,

since firms that pay higher wages can only make equal profits, if they have a highermatching rate.

The employment rate of workers decreases with the wage ladder, i.e.,

θ(

wi)

m(

θ(

wi))

< θ(

wi−1)

m(

θ(

wi−1))

,

since workers at low wages prefer to get a higher wage with a relatively high proba-bility, while workers at high wages are willing to take a low employment probabilityinto account in order to be able to receive even higher wages.

The wage gains between two adjacent steps decrease as the worker climbs the wageladder, i.e.,

wi − wi−1 > wi+1 − wi.

The feasibility constraint restricts the wage gains that are possible at higher wages.



Unemployment and wage distributions in equilibrium:

Given the market tightness θ1 in the first submarket, steady state unempoymentis given by

u =q

q + λθ1m(

θ1).

The steady state number of workers employed at the wage wi is given byequating in- and outflow, i.e.,

λuθ1m(

θ1)

=[

q + λθ2m(

θ2)]

n(

w1)

for i = 1

and

λn(

wi−1)

θim(

θi)

=[

q + λθi+1m(

θi+1)]

n(

wi)

for all i ∈ 2, ..., M

The wage earnings distribution is given by g(

wi)

= n(

wi)

/ [1 − u].



Extensions:

Shi (2009):

Shi extends the model to on-the-job search with wage-tenure contracts like in Burdettand Coles (2003). The existence is simpler to establish, since firms offer a continuumof contracts in equilibrium, which makes it easier to consider the payoff of a deviatingfirm.

Menzio and Shi (2010):

Show that directed on-the-job search models can be characterised quite easily due tothe Block-Recursive structure of the equilibrium. Block Recursive Equilibria have theproperty that the worker’s decision (where to apply) does not depend on aggregatevariables like unemployment, wage earnings distribution, etc.. To see this note, thatthe worker’s decisio depends on the market tightness, but not on unemployment ornumber of workers earning a certain wage.



4.3 Multiple Applications - and directed search

Idea:

Workers send more than one application.

Depending on the wage mechanism, there might be ex-ante or ex-post wage disper-sion.



4.3.1 Multiple Applications - Wage commitment

Galenianos and Kircher (2009) and Kircher (2009):

If workers send multiple applications, firms offer different wage.

Firms commit to their posted wages and do not compete ex-post, if the workerreceives another offer.

Firm entry and the number of applications send by workers is efficient, if firms cancontact all workers.



Framework:

Suppose all worker applies to both firms.

Denote the weakly highest wage by w2 ≥ w1.

Ex-ante wage dispersion:

1. Like in directed on-the-job search workers prefer to send

• their first application to a low wage that is associated with a high employmentprobability,

• their second application to a high wage that gives them a higher income incase they receive two offers.

2. Firms offer different wages, since offering the same wage would require on averagea higher wage to be able to attract workers. To see this note that offering twodifferent wages provides a higher utility to workers despite a lower average wage.



Wage w

Effective

Queue

Length µ(w)

w

IC2

IC1

IP2

IP1

w2w1



Value of being unemployed (in a one-period model):

Vu (w1, w2) = p (w2) w2 + [1 − p (w2)] p (w1) w1 − 2c

where p (w) denotes the probability to get an offer in the w-submarket, i.e.,

p (w) = θ[

1 − e−1/θ]

In the high wage market the worker’s matching probability is determined by themarket tightness in the w2-submarket, i.e.,

θ2 =v2

u.

In the low wage market the worker’s matching probability depends on the effectivenumber of workers in w1-submarket, i.e.,

θ1 =v1

u [1 − p (w2)].



Profit of a firm (in a one-period model):

π = maxw

[

1 − e−1/θ]

[1 − w]

Both firms in w1-, and w2-submarket must make equal profits.

Firms make equal profits, since high wage firms have a higher matching probabilitythan low wage firms, i.e., θ2 < θ1.



Efficiency:

By posting a certain wage, firms can price the probability to receive an application.This internalizes the externalities, since workers and other firms observe the postedwages and adjust their behavior accordingly.

Workers apply to the market that maximizes their utility, i.e., their first applicationis send to the market with the highest utility, i.e., p (w1) w1 > p (w2) w2, and thesecond application goes to the market that maximizes the expected increase inutility, i.e., sending the second application to the w1-submarket would give lessutility, i.e.,

p (w2) w2 + [1 − p (w2)] p (w1) w1 > 2p (w1) w1,

p (w2) [w2 − p (w1) w1] > p (w1) w1.

One market would waste resources by opening to many vacancies and not providinga higher expected income.



4.3.2 Multiple Applications - Ex-post Bertrand competition

Gautier and Holzner (2011):

Workers and firms are linked via applications. The resulting network provides thebasis for matches to be formed.

The wage mechanism (wage commitment or ex-post competition) will result in adifferent network clearing.

The wage mechanism will also determine the application behavior of workers andtherefore network formation.



Network clearing:

Wage commitment does generally not lead to efficient network clearing.

Consider the following 3by3 example:

- There are 3 workers and 3 firms.

- One firm offers a high wage and two firms offer a low wage.

- Each worker sends 2 applications.

- The first application is send to a low wage firm and the second application to thehigh wage firm.



Inefficient network clearing under wage commitment

Inefficient matching Efficient matching

1 2 3

H L L

1 2 3

H L L

• Solid lines are linkes (applications)

• Dashed lines are matches



Inefficient network clearing under wage commitment

1. If all 3 workers apply to the same 2 firms, 2 matches occur. This is the maximummatching possible given the network.

2. If 3 workers apply to 3 firms, wage commitment sometimes results in 2 sometimesin 3 matches depending on which worker is chosen by the high-wage firm

• If the high-wage firm offers the job to one of the workers who are linked tothe low-wage firm with two applicants, i.e., to worker 2 or 3, the number ofmatches is equal to the maximum number of matches (3).

• If the high-wage firm offers the job to the worker linked to the low-wage firmwith only one applicant, i.e. to worker 1, there will be only two matches, sincethe low-wage firm with only one applicant will remain unmatched.



Efficiency with ex-post Bertrand competition

To see why ex post Bertrand competition always leads to 3 matches, if three vacanciesare collectively linked to three workers, we show below that you get a contradictionif this does not hold.

• Suppose a worker and a firm remain unmatched.

• This implies that the unmatched worker receives her reservation value equal tozero. The firm that is linked to the unmatched worker must pay a wage equal tothe reservation value to its matched worker, since any higher wage would not beprofit maximizing.



• The unmatched firm is willing to pay a wage equal to the marginal product. Thus,the worker that is linked to the unmatched firm but hired by another firm mustbe paid a wage equal to the marginal product, since any lower wage would beoutbid by the unmatched firm.

• Thus, one of the three firms pays the reservation wage, one the marginal productand one remains unmatched.

• The unmatched worker cannot be linked directly to the unmatched firm, sinceboth parties would then form a match.

• Thus, the unmatched worker can only be linked to both matched firms. This,however, implies that both matched firms must pay a wage equal to the reservationvalue.

• This cannot be the case as we argued above.



Efficient network clearing under ex post Bertrand competition

Contradiction Efficient matching

w = 1 w = 0 Not matched

w = 1 Not matched w = 0

w = 0w = 0w = 0

w = 0w = 0w = 0

Firms

Workers

• Solid lines are linkes (applications)

• Dashed lines are matches



Network network formation:

Kircher (2009) shows that in a model with directed search and wage commitmentdifferent market tightness θ1 > θ2 leads to efficient firm entry, since firms can pricetheir probability to be matched and these prices are observed by all other agents.

Does wage commitment with different wages lead to efficient networkformation?

• Directed search with different wage offers implies that some firms get syste-matically more offers than others.

• This leads to less balanced networks and therefore to less matches.

• Efficiency requires that all vacancies should be contacted by workers withequal probability, i.e., θ1 = θ2.



Efficient network formation under random search:

Assume: Network clearing is efficient (i.e., maximum matching).

• ai is the probability that a worker sends one of her two applications to vacancy i.

• The expected maximum number of matches is,

M =

3∑

i=1

(

1 − (1 − ai)3)

, with

3∑

i=1

ai = 2,

⇒ Since(

1 − (1 − ai)3)

is concave in ai, all vacancies should have same proba-

bility to receive an application, ai = 2/3.



Comparing wage commitment with ex-post Bertrand competition:

Wage commitment:

- Leads to efficient firm entry and efficent number of applications send by workers.

- Leads to inefficient network formation and network clearing.

Ex-post-Bertrand competition:

- Leads to more efficient network formation and to efficient network clearing.

- Will most likely not leads to efficient firm entry and efficient number of applicationssend by workers.

=⇒ No directed search model can be efficient on all margins.


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Chapter I: Labour Market Theory - uni-muenchen.de · 2012-08-19 · Chapter I: Labour Market Theory Section 4: Directed search models Timing of the game: 1. Firms decide simultaneously