Lecture 6 Search and matching theory

Lecture 6Search and matching theory

Leszek Wincenciak, Ph.D.

University of Warsaw

Lecture 6 – Search and matching theory

2/48

Lecture outline:

Introduction

Search and matching theorySearch and matching theoryThe dynamics of unemploymentJob creation by firmsWage determinationSteady stateComparative statics


Introduction 3/48

Introduction

� Even in the absence of marked changes in overall employment,there are simultaneous processes of job creation anddestruction, reaching 20% of total employment inmanufacturing during a year

� Workers are searching for the best jobs� Firms are looking for the best workers� Searching for job or a worker and matching takes time and iscostly

� This leads to frictional unemployment


Search and matching theory 4/48

Search and matching theory





Frictional unemployment

� Firms create job openings (vacancies)� Workers search for jobs� Match of a worker and a vacancy results in a productive job� Matching is not coordinated (workers and firms dedicate timeand resources to find a suitable match)

� Probability that a firm or a worker find the partner dependson a relative number of vacant jobs and unemployed workers

� labour supply (L) = unemployed + employed� labour demand = filled jobs + vacancies




� Total number of unemployed workers – uL� Total number of vacancies – vL� Total number of matches between unemployed workers andvacant firms in each unit of time – mL

� The process of matching is summarized by a matchingfunction, which expresses the number of newly created jobs(mL) as a function of the number of unemployed workers(uL) and vacancies (vL):

mL = m(uL, vL) (1)




The matching function, assumed to be increasing in botharguments, can be thought of as similar to aggregate productionfunction. Workers and vacant jobs can be viewed as productiveinputs which produce a match, which results in a productive job.Creation of employment requires presence of both unemployedworkers and vacant jobs – m(0, 0) = m(0, vL) = m(uL, 0) = 0.Typically it is assumed that matching function exhibits a constantreturns to scale (empirical evidence seems to support thisassumption).




In case of CRS matching function, we can write:

m =m(uL, vL)

L= m(u, v). (2)

The function m(·) determines the flow of workers who find a joband who exit the unemployment pool within each time interval.Consider the case of an unemployed worker: at each moment intime, the worker will find a job with probability p = m(·)/u. Withconstant returns to scale for m(·) we may thus write:

m(u, v)

u= m

(1,

v

u

)≡ p(θ), (3)

where θ = v/u is called the labour market tightness.




The instantaneous probability p that a worker finds a job ispositively related to the tightness of the labour market which ismeasured by θ, the ratio between the number of vacancies andunemployed workers. An increase in θ, reflecting a relativeabundance of vacant jobs relative to unemployed workers, leads toan increase in p.




The average length of an unemployment spell is given by 1/p(θ),and is thus inversely related to θ. Similarly, the rate at whicha vacant job is matched to a worker may be expressed as:

m(u, v)

v= m

(1,

v

u

) u

v=

p(θ)

θ≡ q(θ), (4)

a decreasing function of the vacancy/unemployment ratio. Anincrease in θ reduces the probability that a vacancy is filled and1/q(θ) measures the average time that elapses before a vacancy isfilled.The dependence of p and q on θ captures the dual externalitybetween agents in the labour market: an increase in the number ofvacancies relative to unemployed workers increases the probabilitythat a worker finds a job (dp(·)/dv > 0), but at the same time itreduces the probability that a vacancy is filled (dq(·)/dv < 0).



The dynamics of unemployment






Changes in unemployment result from a difference between theflow of workers who lose their job and become unemployed, andthe flow of workers who find a job. The inflow into unemploymentis determined by the ’separation rate’ which we take as given forthe moment: at each moment in time a fraction s of jobs(corresponding to a fraction 1− u of the labour force) is hit bya shock that reduces the productivity of the match to zero: in thiscase the worker loses her job and returns to the pool ofunemployed, while the firm is free to open up a vacancy in order tobring employment back to its original level. Given matchdestruction rate s, jobs therefore remain productive for an averageperiod of 1/s.




Given these assumptions we can now describe the dynamics of thenumber of unemployed workers. Since L is constant,d(uL)/dt = uL and hence:

uL = s(1− u)L− p(θ)uL

⇒ u = s(1− u)− p(θ)u. (5)

The dynamics of the unemployment rate depend on the tightnessof the labour market θ: at a high value for the ratio of vacancies tounemployed workers, workers easily find a job leading to a largeflow out of unemployment.




From equation (5) we can immediately derive the steady staterelationship between the unemployment rate and θ:

u =s

s+ p(θ). (6)

Since p′(·) > 0, the properties of the matching function determinea negative relation between θ and u.To obtain job creation and destruction rates, we may divide theflows into and out of employment by the total number of employedworkers (1− u)L. The rate of destruction is simply equal to s,while the rate of job creation is given by p(θ)[u/(1− u)].




u = 0

uu0

θ0

θ

θ0

v = θ0u

u = 0

uu0

v0

v

Figure 1. Dynamics of the unemployment rate




The steady-state relationship (6) is illustrated graphically in theleft panel of Figure 1: to each value of θ corresponds a uniquevalue for the unemployment rate. Moreover, the same properties ofm(·) ensure that this curve is convex. For points above or belowu = 0, the unemployment rate tends to move towards thestationary relationship: keeping θ constant at θ0, a value u > u0causes an increase in the flow out of unemployment and a decreaseof the flow into unemployment, bringing u back to u0.




Moreover, given u and θ, the number of vacancies is uniquelydetermined by v = θu, where v denotes the number of vacanciesas a proportion of the labour force. The picture on the right handside of the figure shows the curve u = 0 in (v, u) space. This locusis known as the Beveridge curve, and identifies the level ofvacancies v0 that corresponds to the pair (θ0, u0) of the left handpanel of the Figure 1.It is important to note that variations in the labour markettightness are associated with a movement along the curve u = 0,while changes in the separation rate s or the efficiency of thematching process (captured by the properties of the matchingfunction) correspond to movements of the curve u = 0.




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Job

vaca

ncy

rate

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Unemployment rate

Jan 2001 − Jul 2008

Aug 2008 − Jul 2009

Aug 2009 − Jul 2016

Figure 2. Beveridge curve for USA, 2001-2016 (BLS)




Equation (6) gives a first steady state relationship between u andθ. To find the actual equilibrium values, we need to specifya second relationship between these variables. This secondrelationship can be derived from the behavior of firms and workerson the labour market.



Job creation by firms






The crucial decision of firms concerns the supply of vacancies onthe labour market. The decision of a firm whether to createa vacancy depends on the expected future profits over the entiretime horizon of the firm, which we assume to be infinite. Formally,each individual firm solves an intertemporal optimization problemtaking as given the aggregate labour market conditions which aresummarized by θ, the labour market tightness. Individual firmstherefore disregard the effect of their decisions on θ, andconsequently on the matching rates p(θ) and q(θ). To simplify theanalysis, we assume that each firm can offer at most one job. Ifthe job is filled, the firm receives a constant flow of output equal toy. Moreover, it pays a wage w to the worker and it takes this wageas given. The determination of this wage will be described later on.




On the contrary, if the job is not filled, the firm incurs a flow costc, which reflects the time and resources invested in the search forsuitable workers. Firms therefore find it attractive to createa vacancy as long as its value, measured in terms of expectedprofits, is non-negative; in the opposite case, the firm will not findit attractive to offer a vacancy and will exit the labour market. Thevalue that a firm attributes to a vacancy (denoted by V ) and toa filled job (J) can be expressed using the asset equations. Givena constant real interest rate r, we can express these values as:

rV (t) = −c+ q(θ(t))(J(t)− V (t)) + V (t), (7)

rJ(t) = (y −w(t)) + s(V (t)− J(t)) + J(t). (8)




(7) and (8) are explicit functions of time. The flow return ofa vacancy is equal to a negative cost component (−c), plus thecapital gain in case the job is filled with a worker (J − V ), whichoccurs with probability q(θ), plus the change in the value of thevacancy itself (V ). Similarly, (8) defines the flow return of a filledjob as the value of the flow output minus the wage (y − w), plusthe capital loss (V − J) in case the job is destroyed, which occurswith probability s, plus the change in the value of the job (J).




Subtracting (7) from (8) yields the following expression for thedifference in value between a filled job and a vacancy:

r(J(t)− V (t)) =(y − w(t) + c)

− [s+ q(θ(t))](J(t) − V (t))

+ (J(t)− V (t)). (9)




Now, if we focus on steady state equilibria we can imposeV = J = 0 in equations (7) and (8). Moreover, we assume freeentry of firms and as a result V = 0 : new firms continue to offervacant jobs until the value of the marginal vacancy is reduced tozero. Substituting V = 0 in (7) and (8) and combining theresulting expressions for J , we get:

J = c/q(θ)J = (y − w)/(r + s)

}⇒ y −w = (r + s)

c

q(θ). (10)

Equation (7) gives us the first expression for J . According to thiscondition the equilibrium value of a filled job is equal to theexpected costs of a vacancy, that is the flow cost of a vacancy ctimes the average duration of a vacancy 1/q(θ).




The second condition for J can be derived from (8): the value ofa filled job is equal to the value of the constant profit flow y − w.These flow returns are discounted at rate r+ s to account for bothimpatience and the risk that the match breaks down. Equatingthese two expressions yields the final solution (10), which gives themarginal condition for employment in a steady state equilibrium:the marginal productivity of the worker (y) needs to compensatethe firm for the wage w paid to the worker and for the flow cost ofopening a vacancy.The latter is equal to the product of the discount rate r + s andthe expected costs of a vacancy c/q(θ).




This last term is just like an adjustment cost for the firm’semployment level. It introduces a wedge between the marginalproductivity of labour and the wage rate. However, in this modelthe size of the adjustment cost is endogenous and depends on theaggregate conditions on the labour market. In equilibrium, the sizeof the adjustment costs depend on the unemployment rate and onthe number of vacancies, which are summarized at the aggregatelevel by the value of θ. If, for example, the value of output minuswages (y − w) increases, then vacancy creation will becomeprofitable (V > 0) and more firms will offer jobs. As a result, θ willincrease, leading to a reduction in the matching rate for firms andan increase in the average cost of a vacancy and both these effectstend to bring the value of a vacancy back to zero.




Finally, notice that equation (10) still contains the wage rate w.This is an endogenous variable. Hence the ’job creation condition’(10) is not yet the steady state condition which together with (6)would allow us to solve for the equilibrium values of u and θ. Tocomplete the model we need to analyze the process of wagedetermination, to which we now turn.



Wage determination

Wage determination



Wage determination

Wage determination

The process of wage determination that we adopt here is based onthe fact that the successful creation of a match generatesa surplus. That is, the value of a pair of agents that have agreedto match (the value of a filled job and an employed worker) islarger than the value of these agents before the match (the valueof a vacancy and an unemployed worker). This surplus has thenature of a monopolistic rent and needs to be shared between thefirm and the worker during the wage negotiations. Here we shallassume that wages are negotiated at a decentralized level betweeneach individual worker and her employer. Since workers and firmsare identical, all jobs will therefore pay the same wage.



Wage determination

Wage determination

Let E and U denote the value that a worker attributes toemployment and unemployment, respectively. The joint value ofa match (given by the value of a filled job for the firm and thevalue of employment for the worker) can then be expressed asJ +E, while the joint value in case the match opportunity is notexploited (given by the value of a vacancy for a firm and the valueof unemployment for a worker) is equal to V + U . The totalsurplus of the match is thus equal to the sum of the firm’s surplus,J − V , and the worker’s surplus, E − U :

(J + E)− (V + U) ≡ (J − V ) + (E − U). (11)



Wage determination

Wage determination

The match surplus is divided between the firm and the workerthrough a wage bargaining process. We take their relativebargaining strength to be exogenously given. Formally, we adoptthe assumption of Nash bargaining. This assumption is common inmodels of bilateral negotiations. It implies that the bargained wagemaximizes a geometric average of the surplus of the firm and theworker, each weighted by a measure of their relative bargainingstrength (Nash maximand). In our case the assumption of Nashbargaining gives rise to the following optimization problem:

maxw

(J − V )1−β(E − U)β, (12)

where 0 ≤ β ≤ 1 denotes the relative bargaining strength of theworker.



Wage determination

Wage determination

Given that the objective function is a Cobb-Douglas one, we canimmediately express the solution (the first order conditions) of theproblem as:

E −U =β

1− β(J − V ) ⇒ E −U = β[(J − V ) + (E −U)]. (13)

The surplus that the worker appropriates in the wage negotiations(E−U) is thus equal to a fraction β of the total surplus of the job.



Wage determination

Similar to what is done for V and J in (7) and (8), we can express thevalues E and U using the relevant asset equations (reintroducing thedependence on time t):

rE(t) = w(t) + s(U(t)− E(t)) + E(t) (14)

rU(t) = z + p(θ)(E(t) − U(t)) + U(t). (15)

For the worker the flow return on employment is equal to the wage plusthe loss in value if the worker and the firm separate, which occurs withprobability s, plus any change in the value of E itself; the return onunemployment is given by the imputed value of the time that a workerdoes not spend working, denoted by z, plus the gain if she finds a job andthe change in the value of U . Parameter z includes the value of leisureand/or the value of alternative sources of income including possibleunemployment benefits. It is assumed to be exogenous and fixed.



Wage determination

Restricting attention to steady state equilibria, so that E = U = 0,we can derive the surplus of the worker E − U directly from (14)and (15).

E − U =w − z

r + s+ p(θ). (16)

According to (16) the surplus of a worker depends positively on thedifference between the flow return during employment andunemployment (w − z) and negatively on the separation rate s andon θ: an increase in the ratio of vacancies to unemployed workersincreases the exit rate out of unemployment and reduces theaverage length of an unemployment spell.



Wage determination

Using (16) and noting that in steady state equilibrium

J − V = J =y −w

r + s,

we can solve the expression for the outcome of the wagenegotiations given by (13) as:

w − z

r + s+ p(θ)=

β

1− β

y − w

r + s.

Rearranging terms, and using (10), we obtain the followingequivalent expressions for the wage:

w − z = β[(y + cθ − w) + (w − z)] (17)

⇒ w = z + β(y + cθ − z). (18)



Wage determination

From equation (17) it follows that the flow value of the worker’ssurplus, i.e. the difference between the wage (w) and alternativeincome z, is a fraction β of the total flow surplus. The term(y − w + cθ) represents the flow surplus of the firm, where cθdenotes the expected cost savings if the firm fills a job. If weeliminate the wage payments in (17) we obtain the flow value ofthe total surplus of a filled job (y + cθ − z), which is equal to thesum of the value of output and the cost saving of the firm minusthe alternative costs of the worker. Finally, equation (18) expressesthe wage as the sum of the alternative income and the fraction ofthe surplus that accrues to the worker.



Wage determination

It can easily be verified that the only influence of aggregate labourmarket conditions on the wage occur via θ, the ratio of vacanciesto unemployed workers. The unemployment rate u does not haveany independent effect on wages. The explanation is that wagesare negotiated after a firm and a worker meet. In this situation thematch surplus depends on θ, as we saw above. This variabledetermines the average duration of a vacancy, and hence theexpected costs for the firm if it would continue to search. Thedetermination of the equilibrium wage completes the description ofthe steady state equilibrium.



Steady state

Steady state



Steady state

Steady state

The steady state equilibrium can be summarized by equations (6),(10) and (18) which we shall refer to as BC (Beveridge curve),JC (job creation condition) and W (wage equation):

u =s

s+ p(θ)(BC) (19)

w = y − (r + s)c

q(θ)(JC) (20)

w = (1− β)z + β(y + cθ) (W ) (21)



Steady state

For a given value of θ, the wage is independent of theunemployment rate. The system can therefore be solved recursivelyfor the endogenous variables u, θ and w. Using the definition for θwe can then solve for v. The last two equations jointly determinethe equilibrium wage w and the ratio of vacancies/unemployed θ,as is shown in the left panel of Figure 5. Given θ, we can thendetermine the unemployment rate u, and consequently also v,which equate the flows into and out of unemployment (the righthand panel of the figure).



Steady state

W

JC

θθ0

w0

w

θ0

JC +W

BC

uu0

v0

v

Figure 3. Equilibrium of the labour market with frictional unemployment



Comparative statics

Comparative statics



Comparative statics

Comparative statics

This dual representation facilitates the static comparative analysis,which is intended to analyze the effect of changes in theparameters on the steady state equilibrium.

Scenario 1: Assume, that we observe an increase in unemploymentbenefits, a component of z, or an increase in the relativebargaining strength of workers β.



Comparative statics

Comparative statics, z ↑ or β ↑

W

W ′

JC

θθ0θ1

w0

w1

w

θ0 θ1

JC +W

JC +W ′

BC

uu0 u1

v0v1

v

Figure 4. The effects of an increase in z or β



Comparative statics

Interpretation

As a result of an increase in unemployment benefits (captured byz) or an increase in the bargaining power of the workers (capturedby β) the wage curve defined by (21) shifts upwards. This causesan increase in the wage and a reduction in the labour markettightness, θ. This reduction, along the Beveridge Curve (BC), isaccompanied by an increase in u and a reduction in v.

Scenario 2: Consider now a ’reallocative’ shock, i.e. the increase inthe separation rate s.



Comparative statics

Comparative statics, s ↑

W

JC

JC′θθ1 θ0

w1

w0

w

θ0 θ1

JC +W

JC +W ′

BC

BC′

uu0 u1

?

v

Figure 5. The effects of an adverse reallocative shock



Comparative statics

In the case of a reallocative shock we observe an inward shift ofJC along W schedule. This results in a joint decrease of the wageand the labour market tightness θ, as in the case of the aggregateshock. At the same time, however, the Beveridge Curve BC shiftsto the right. Hence, while the unemployment rate increasesunambiguously, it is in general not possible to determine the effecton the rate of vacancies. In reality, however, v appears to beprocyclical and this suggests that an increase of the separation ratewill reduce the rate of vacancies.

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Lecture 6 Search and matching theory