Job Search 2. Equilibrium Search Model

Job Search2. Equilibrium Search Model

Bruno Van der Linden

Université catholique de Louvain

November 23, 2010

ESL (UCL) 1 / 33

Road-map

1 Motivation

2 An equilibrium Search model with wage posting

3 Estimating the equilibrium search model

4 Note on the literature

5 References

ESL (UCL) 2 / 33

Motivation

Motivation1. The inadequacies of the basic job-search model

A. The distribution of wages is left unexplainedWhy would wages for homogeneous workers be distributed (“purewage dispersion”)?

♦ If the labor market was competitive,identical workers in identical jobs cannot durably get different wages.

♦ Under imperfect information: Diamond (1971)’s critiqueAssumption: firms post wages (i.e. set take-it-or-leave-it wage offersthat they commit to pay) and job-seekers search randomly

Equilibrium: A distribution H(w) such that every wage posted (withpositive probability) earns the same profit and no other wage earnsgreater profit.

ESL (UCL) 3 / 33

Motivation

The inadequacies of the basic job-search modelDiamond’s critique or Diamond’s paradox

Start from the basic model in continuous time that leads to thereservation wage x such that:

x = z +λ

r + q

∫ +∞

x(w − x)dH(w) (6)

Any wage offer w ≥ x is accepted;

⇒ no incentive for firms to pay w > x ;

⇒ From Eq. (6), x = z;

⇒ Diamond’s critique or Diamond’s paradox:Degenerate equilibrium distribution H(w) i.e. 1! wage (z);Firms can take the entire surplus (= marg. product - x).

ESL (UCL) 4 / 33

Motivation

The inadequacies of the basic job-search modelEmpirical issue

B. Empirical difficultiesIn the data, only information about accepted wages above thereservation wage;The distribution of wage offers cannot be identified nonparametrically;⇒ H(w) is identified under parametric assumptions: H(·) has tobe recoverable, i.e. can be identified from observations of thetruncated distribution (of accepted wages).Example: The family of log-normal distributions.However, is this parameterization appropriate?

ESL (UCL) 5 / 33

Motivation

Motivation

C. Empirical evidence“Mincer log wage equations”

log Earningsi = β0,i + β1 · Schoolingi + β2 · EXPi + β3 · EXP2i

+ β4 ·Gender&Racei + β5 · Sectori + β6 · Locationi

explain no more than 30% of differences in compensation bycharacteristics linked to labour productivity (education, experience(‘EXP’ above),...)⇒ 70% due to unobserved ability differences ...... or by differences in firms’ pay policy. In favour of the latterexplanation:Matched employer-employee data: Inter-industry differences incompensations. Two explanations are equally important:

1 differences in workers characteristics2 differences in firms fixed effects

(see Abowd and Kramarz, 1999)ESL (UCL) 6 / 33

An equilibrium Search model with wage posting

2. An equilibrium Search model with wage posting

On p. 127, Cahuc and Zylberberg (2004) starts presenting a rathergeneral framework.

Since the framework p. 127 -131 is unnecessarily complicate,I develop a simpler setting inspired by Mortensen (2003) andRogerson, Shimer and Wright (2005), henceforth RSW.(Without proof) I explain where and how the formulas are changedwhen the more general setting of Cahuc and Zylberberg (2004) isused.I do not expect that student understand the mathematicaldevelopments in this more general setting.

Seminal paper: Burdett and Mortensen (1998).

ESL (UCL) 7 / 33


Assumptions

A1 Rational forward-looking and homogeneous risk-neutral agentswho only care about their income. All unemployed are entitled to aflat UB, b, with no time limit. No taxes. z ≡ b − c.

A2 Job search intensity is fixed. In [t , t + dt ], each employer contactsa finite number of job-seekers at random.

A3 Job-seekers choose to reject or accept a job offer, if any. Anaccepted wage remains constant all along the employment spell.Rejected offers (i) cannot be recalled, (ii) lead to no sanction.

A4’ On-the-job search. No threshold wage above which searching onthe job no longer pays. λu = λe = λ designate the equalexogenous arrival rates of job offers respectively for theunemployed and the employed. Ass.: 0 < λ < +∞.

ESL (UCL) 8 / 33


Assumptions continued

A5’ An endogenous distribution of wage offers on an a prioriunknown support. Firms choose the wage offer and commit topay that wage. To currently employed workers, firms send wageoffers ignoring their current wage. The worker’s current employerdoes not respond to the outside offer.

A6 Constant exogenous job destruction rate, 0 < q < +∞.A7 Constant exogenous discount rate, r .A8 A large given number of firms and workers. Formally, a continuum

of worker and a continuum of firms, each of unitary mass.⇒ Stationary environment.

Job-search behaviour of the unemployed:

x = z + (λu − λe)

∫H(ξ)

r + q + λeH(ξ)dξ︸︷︷︸

0

(20)

ESL (UCL) 9 / 33


♦ No employer will offer a wage below x .♦ Why would an employer offer more than x? Intuition:Posting a higher wage affects the flow of employees (i) quitting the firmbecause they face a better offer and (ii) accepting offers by the firm.For a given firm, profit maximization might be attained indifferentlythrough

high wages and many employees orlow wages and few employees.

♦ The question: In equilibrium, is H(w) degenerate?

The law of motion of the unemployment rate u :

u ≡ du/dt = q(1− u)− λu

since all offers will be ≥ x . In steady state u = 0. Hence,

u =q

q + λ

ESL (UCL) 10 / 33


Equilibrium flows on the labour market

For any w , The fraction of those employed at a wage w or less, i.e.the wage distribution function G(w), should not be confused with thedistribution of job offers H(w).

By the law of large numbers, the flow into the set of workers earning wor less is

λ · H(w) · u

The flow out of the same set is

G(w)(1− u)[q + λ(1− H(w))]

Taking u = qq+λ into account, the equality between these two flows

allows to relate G(w) to H(w) in the following way:

G(w) =q · H(w)

q + λ · H(w)with H(w) ≡ 1− H(w)

ESL (UCL) 11 / 33


Expected discounted profitThe cdf H(·) being given

The expected discounted profit from posting wage w ≥ x , Π(w | H(·)),is the product of two terms:• p(w) the probability that the wage offer w is accepted and• J(w) the expected value of hiring a worker whose real product isy > z (y constant!) at wage w .

p(w) = u + (1− u)G(w) =q

q + λ · H(w)

The higher the wage, the bigger the acceptance rate of a wage offer w(through λ · H(w)).

rJ(w) = y − w +(q + λ · H(w)

)(−J(w))

The higher the wage paid, the bigger the retention rate λ · H(w) andthe lower the instantaneous profit.

ESL (UCL) 12 / 33


Expected discounted profitThe cdf H(·) being given

The expected profit of offering a wage w is then given by:

Π(w | H(·)) =q

q + λ · H(w)· y − w

r + q + λ · H(w)

a higher wage increases Π through the acceptance and theretention rates (denominators)a higher wage lowers Π via the instantaneous profit y − w

Notice that for any w , the R.H.S. is strictly increasing in H(w).

ESL (UCL) 13 / 33


Wage posting

Clearly no firm posts w > y nor w < x = z.Each employer makes his decision in a noncooperative context inwhich the other employers’ wage policies are taken as given. So,for any given cdf H(·) and hence G(·):

w = arg maxω≥x

{Π(ω | H(·))}

= arg maxω≥x

{q

q + λ · H(ω)· y − ω

r + q + λ · H(ω)

}

ESL (UCL) 14 / 33


Equilibrium

We are not searching for a wage w given a cdf H(·).We are searching for a cdf H(·) and its support.Equilibrium requires that any posted wage yields the same profit,which is at least as large as profit from any other wage.More formally, any equilibrium wage offer w ≥ x = z in the support ofH must yield the same profit, say π:

π =q

q + λ · H(w)· y − w

r + q + λ · H(w)∀w in the support

An equilibrium solution can be described by a triple (x ,H, π) such thatx = z, π defined above, and H is such that

Π(w | H) = π for all w on the support of HΠ(w | H) ≤ π otherwise

ESL (UCL) 15 / 33


Equilibrium

Let w ≥ z and w ≤ y denote the infimum and the supremum of thesupport of H.

♦ What is the value of the infimum w ?As H(w) = 1, it is easily seen that the employer offering the lowestwage in the market maximizes Π(w | H(·)) by setting w = x = z.

♦ What is the value of the supremum w ?As H(w) = 0, w solves

y − wr + q

=q

q + λ· y − z

r + q + λ

from which it turns out that w is a weighted average of z and y .As long as the weight of z, namely q(r+q)

(q+λ)(r+q+λ) is not one, the propertyw > w is guaranteed since y > z.

ESL (UCL) 16 / 33


Equilibrium

♦ To characterize the distribution H(w) (and, hence, G(w)), one knowsthat in equilibrium

Π(w | H) = Π(z | H) ∀w ∈ [z,w ]

i.e.q

q + λ · H(w)· y − w

r + q + λ · H(w)=

qq + λ

· y − zr + q + λ

∀w ∈ [z,w ], the LHS increases and is continuous in H(w) while theRHS remains constant. Hence, for any wage in the support, onegets a unique H(w).

ESL (UCL) 17 / 33


Equilibrium

Some technical issues:

♦ Suppose a mass point at some w , such that z ≤ w < w .1

A firm posting w + ε would increase its revenue discretely (hiringaway any worker contacted paid w) while paying an ε more.So, a wage such as w associated to a mass point cannotmaximize profits.

♦ Can there be “gaps” in H(w)?A gap between say w and w means H(w) = H(w). Then a firmposting w could reduce its offer and hence its cost without reducing itsflow of entrants or increasing its flow of exits.Hence, w cannot be optimal.

1Hence, there is a finite number of workers paid exactly w .ESL (UCL) 18 / 33


EquilibriumIf r → 0

In this limit case, explicit expressions can be found:

w =

[1−

(q

q + λ

)2]

y +

(q

q + λ

)2

z (21)

From Π(w | H) = Π(z | H) in the support,

H(w) =q + λ

λ

[1−

√y − wy − z

](32)

From (32) one easily derives:

G(w) =qλ

[√y − zy − w

− 1]

Note: in the general case λe 6= λu, these formulas are modified asfollows: (i) λe instead of λ and (ii) x instead of z (see CZ2004)

ESL (UCL) 19 / 33


Monopsony powerThe case where r → 0

In all the previous relationships, the matching friction parameter q/λplays a key role. See the empirical part for estimates of λ/q.In particular, if frictions vanish i.e. if λ→ +∞, q/λ→ 0, then:

w → yG(w)→ 0 for all w < wu → 0

In the absence of frictions, the search equilibrium has all the propertiesof perfect competition. However, perfect competition is a limit case.As long as q/λ > 0, one has w = z < w < y . All workers are paid lessthan their marginal product.2 In this sense, they are “exploited”.

2Without proof (see Manning, 2003, p. 42), the expected wage can be written as:

E(w) =q

q + λz +

λ

q + λy (33)

ESL (UCL) 20 / 33


“It is frictions, broadly defined, that give employersmonopsony power in the labor market. The most importantsources of these frictions are:

ignorance among workers about labor marketopportunities;individual heterogeneity in preferences over jobs;mobility costs.

The view that employers have some market power can hardybe controversial: it is undoubtedly true that a wage cut of acent does not cause all existing workers to instantaneouslyleave the employer.” (Manning, 2003, p. 360)

Empirical issue: Sensitivity of quits and recruits to the wage?Special issue of the Journal of Labor Economics of April 2010.Conclusion of the introductory paper: This sensitivity is quite small,“suggesting significant levels of market power for employers, especiallyas they are interpreted a long-run elasticities.” (p. 207)No consensus however: See van den Berg and van Vuuren (2010).

ESL (UCL) 21 / 33


Exercises

Exercise1. Wen λe 6= λu (both being exogenous), Eq. (33) becomes

w =

[1−

(q

q + λe

)2]

y +

(q

q + λe

)2

z

Use this formula to retrieve the Diamond (1971) paradox.

2. Wen λe = λu (both being exogenous), show that in equilibriumG(w) statistically dominates H(w) (distribution G dominatesdistribution H stochastically (at first order) if, for any argument w,H(w) ≥ G(w)). Interpret !Hint: compute successively G(w)− H(w) to look at w = w and abovethis bound exploit (after showing) that H(w)−G(w)

H(w)G(w)= λ

q .

ESL (UCL) 22 / 33


Exercise

Exercise3. Take r → 0, λe = λu, and look at the impact of a small increase inz = b − c.

3.1 case with exogenous λ: (a) Show that the support of theequilibrium wage distribution shifts upwards and shrinks (so thatthe extent of wage inequalities is smaller). (b) What is the effecton H(w) within the support? (Interpret!)

3.2 case with endogenous λ: Look at the slide where λ will beendogeneized and check the direction of change in λ (Interpret!).Then, without computing the derivatives (as they lead toambiguous analytical results), check intuitively that the propertiesfound in 3.1 are far from obvious now.

ESL (UCL) 23 / 33


Limitations and extensions

1. Posting a constant wage and committing not to revise it isprobably too simple. Extensions:

If an employee receives a better wage offer, Postel-Vinay and Robin(2002) introduces the possibility of a counter-offer by the currentemployer. Cahuc, Postel-Vinay and Robin (2006) introduces wagebargaining: “unemployed workers negotiate with a single employerin a conventional way, but when an employed worker receives anoutside job offer, a three-player bargaining process is startedbetween the worker, her/his initial employer, and the employer whomade the outside offer.”Often firms do not post a single wage, but post contracts where thewage paid can vary with an employee’s tenure (Burdett and Coles,2003, and Stevens, 2004). Summary on p. 980 of RSW.

2. No job-to-job transitions with wage cuts; relaxed by Postel-Vinayand Robin (2002) and Carrillo-Tudela and Smith (2009).

ESL (UCL) 24 / 33


Limitations and extensions

3. Model in steady state only; relaxed by Moscarini and Postel-Vinay(2009).

4. Multiple applications - Multiple job offers:In continuous time models, the probability that a job-seeker getsmore than 1 offer during a small interval of time is negligible.In reality, job-seekers can simultaneously apply to more than onjob.What does this change? The possibility that the job-seekers hasmore than one offer in hand⇒ more competition between firms.

When time consists of a single non infinitesimal period during whichan unemployed can receive several job offers, Mortensen (2003)shows that the wage posting game implies “pure wage dispersion”in equilibrium (p. 16 - 20). ⇒ Diamond’s critique disappears.Extended by Albrecht, Gautier and Vroman (2006) and Gautier andWolthoff (2009).

ESL (UCL) 25 / 33


Extension: Endogeneizing λ

Up to now, both q and λ are exogenous parameters.The latter can be endogeneized (Mortensen 2003, p. 41):Let κ > 0 denote the constant cost of contacting a worker.Since the expected profit per worker contacted verifies in equilibrium:

Π(z | H) = Π(w | H) ∀w ∈ [z,w ]

The aggregate λ consistent with optimality should verify:

qq + λ

· y − zr + q + λ

= κ

The higher κ, the lower λ conditional on all other parameters.

ESL (UCL) 26 / 33

Estimating the equilibrium search model

3. Estimating the equilibrium search model

Note: Results below come from structural estimations of the abovestationary model. Implementation requires that many measurementissues be addressed.

To what extent does the equilibrium search model with wage postingprovide a good fit of the distribution of wages?

The equilibrium density function of wages paid in steady state isincreasing and convex in the wage if r → 0 (easily checked whenr → 0).This is at odds with observed distributions of wages ( DENSITIES ).Introducing heterogeneities in y helps to reconcile the propertiesof the model with the data. See Bontemps, Robin and van denBerg (2000).

ESL (UCL) 27 / 33


Source: Jolivet, Postel-Vinay and Robin (2006) BACK

ESL (UCL) 28 / 33


According to Hornstein, Krusell and Violante (2007), properlycalibrated equilibrium search models generate too low wagedifferentials among ex-ante similar workers in the U.S.

Extending the framework to on-the-job (general) learning bydoing, Burdett, Carrillo-Tudela and Coles (2009) conclude that theequilibrium search model can generate reasonable wagedifferentials.

ESL (UCL) 29 / 33


Estimation of the “index of search frictions”

“Index of search frictions” λe/q (Ridder and van den Berg, 2003):Over a given length d of the employment spell, the expected number ofoffers equals λe · d .Then, the unconditional expectation of the number of offers equalsλe · E [d ] = λe/q in a simple setting where jobs end only because of alayoff (at constant rate q).If λe/q → +∞, degenerate distribution (competitive case).

Estimations by Ridder and van den Berg (2003), Table 3:λe/q from ≈ 5 (France) to ≈ 20 (U.S.).Estimations by Christensen, Lentz, Mortensen, Neumann andWerwatz (2005), Table 2: λe/q ≈ 2 for Denmark.

ESL (UCL) 30 / 33


Estimation of the “index of search frictions”

To improve the fit of the model more recent work extends the model byconsidering heterogenous workers and firmsendogeneizing search effort and recruiting effortsintroducing more complex labor contracts

Then, λe/q can be quite different according to the occupation. ForFrance, variation between 1 and 6.4 according to the sector and theoccupation in Table III of Cahuc, Postel-Vinay and Robin (2006).

ESL (UCL) 31 / 33


Other estimations of the “index of search frictions”

The partial job-search model is also used to estimate an “index ofsearch frictions”:

BEL DNK ESP FRA UK GER ITA NLD PRT USA1.2 0.5 0.5 2.0 0.5 1.1 0.4 1.2 0.3 1.2

Table: Index of search friction λe/(q + λ2), where λ2 = rate of layoffimmediately followed by a job offer. Source: Table 2 of Jolivet, Postel-Vinayand Robin (2006).

NoteThe European data are taken from the European CommunityHousehold Panel survey (ECHP 1994-7), and the U.S. data arefrom the Panel Study of Income Dynamics (PSID 1993-6).With Current Population Survey data for the US, Jolivet (2009)estimates an average index of frictions of 0.5!

ESL (UCL) 32 / 33

Note on the literature

An island metaphor has often been used in search theory along withthe assumption of perfect competition (since Lucas and Prescott,1974).

According to this metaphor,3 workers are “searching across islands, oneach of which there are many firms with a constant returns to scaletechnology using only labor. The productivity of a randomly selectedisland is distributed according to” the cdf H(w). There is on-the-jobsearch. “If the labor market on each island is competitive, a worker onan island with productivity w is paid w . This trivially makes theequilibrium wage distribution the same as the productivity distribution",H.

Under this reinterpretation, the partial-equilibrium job-search modelwith on-the-job search (see Section 1) is embedded into an equilibriumcontext if workers can fill any occupation and productivity isheterogeneous but exogenously given.

3Quotations are from the survey of RSW, p. 967.ESL (UCL) 33 / 33

References

Abowd, J. and F. Kramarz (1999) The analysis of labor marketsusing matched employer-employee data.In O. Ashenfelter and D. Card, editors, Handbook of LaborEconomics, volume 3B. North-Holland.

Albrecht, J., P. Gautier and S. Vroman (2006) “Equilibrium directedsearch with multiple applications”.Review of Economic Studies, 73:869–891.

Bontemps, C., J.-M. Robin and G. van den Berg (2000)“Equilibrium search with continuous productivity dispersion:Theory and nonparametric estimation”.International Economic Review, 41(2):305–358.

Burdett, K., C. Carrillo-Tudela and M. Coles (2009) “Human capitalaccumulation and labour market equilibrium”.4215, Institute for the Study of Labor (IZA).

Burdett, K. and M. Coles (2003) “Equilibrium wage-tenurecontracts”.

ESL (UCL) 33 / 33

References

Econometrica, 71:1377–1404.

Burdett, K. and D. Mortensen (1998) “Wage differentials, employersize, and unemployment”.International Economic Review, 39:257–273.

Cahuc, P., F. Postel-Vinay and J.-M. Robin (2006) “Wagebargaining with on-the-job search: Theory and evidence”.Econometrica, 74:323–364.

Cahuc, P. and A. Zylberberg (2004) Labor economics.MIT Press, Cambridge.

Carrillo-Tudela, C. and E. Smith (2009) “Wage dispersion andwage dynamics within and across firms”.Working Paper 4031, Institute for the Study of Labor (IZA).

Christensen, B., R. Lentz, D. Mortensen, G. Neumann andA. Werwatz (2005) “On-the-job search and the wage distribution”.Journal of Labor economics, 23:31–58.

ESL (UCL) 33 / 33

References

Diamond, P. (1971) “A model of price adjustment”.Journal of Economic Theory, 3:156–168.

Gautier, P. and R. Wolthoff (2009) “Simulataneous search withheterogeneous firms and ex post competition”.Labour Economics, 16:311–319.

Hornstein, A., P. Krusell and G. Violante (2007) “Frictional wagedispersion in search models: A quantitative assessment”.Working Paper 13674, NBER.

Jolivet, G. (2009) “A longitudinal analysis of search frictions andmatching in the U.S. labor market”.Labour Economics, 16:121–134.

Jolivet, G., F. Postel-Vinay and J.-M. Robin (2006) “The empiricalcontent of the job search model: Labor mobility and wagedistributions in Europe and the US”.European Economic Review, 50:877—907.

ESL (UCL) 33 / 33

References

Lucas, R. and E. Prescott (1974) “Equilibrium search andunemployment”.Journal of Economic Theory, 4(2):103–124.

Manning, A. (2003) Monopsony in motion: Imperfect competition inlabour markets.Princeton University Press.

Mortensen, D. (2003) Wage dispersionn: Why are similar workerspaid differently?MIT Press.

Moscarini, G. and F. Postel-Vinay (2009) “Non-stationary searchequilibrium”.http://www.econ.yale.edu/faculty1/moscarini.htm.

Postel-Vinay, F. and J.-M. Robin (2002) “Equilibrium wagedispersion with worker and employer heterogeneity”.Econometrica, 70:2295–2350.

ESL (UCL) 33 / 33

References

Ridder, G. and G. van den Berg (2003) “Measuring labor marketfrictions: A cross-country comparison”.Journal of the European Economic Association, 1:224–244.

Rogerson, R., R. Shimer and R. Wright (2005) “Search-theoreticmodels of the labor market: A survey”.Journal of Economic Literature, 43:959–988.

Stevens, M. (2004) “Wage-tenure contracts in a frictional labourmarket: Firms’ strategies for recruitment and retention”.Review of Economic Studies, 71:535–551.

van den Berg, G. and A. van Vuuren (2010) “The effect of searchfrictions on wages”.Labour Economics, 17:875–885.

ESL (UCL) 33 / 33

Documents

Job Search 2. Equilibrium Search Model