2012-2013 – Master 2 – Macro I

Lecture notes #9 : the Mortensen-Pissaridesmatching model

Franck Portier(based on Gilles Saint-Paul lecture notes)

franck.portier@TSE-fr.eu

Toulouse School of Economics

Version 1.214/11/2012

Changes from version 1.0 are in redChanges from version 1.1 are in purple

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Disclaimer

These are the slides I am using in class. They are notself-contained, do not always constitute original material and docontain some “cut and paste” pieces from various sources that Iam not always explicitly referring to (not on purpose but because ittakes time). Therefore, they are not intended to be used outside ofthe course or to be distributed. Thank you for signalling me typosor mistakes at franck.portier@TSE-fr.eu.

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1. A simple frameworkThe matching function

I The basic building block is the matching function, whichrelates hirings per unit of time to the two key inputs in thesearch process, unemployment and vacancies :

Ht = m(Ut ,Vt). (1)

I Here Ht = the gross hiring rate per unit of time, Ut = thenumber of unemployed workers,Vt = the number of vacantjobs.

I The matching function is similar to a production function, andwe assume it has the same properties.

I Note that in this framework, unemployment and vacancies arenot a waste : they are a productive input in the production ofnew matches.

I This defines the process for job creation. To begin with, weassume a simple process of job destruction : a fraction s of alljobs is destroyed per unit of time.

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1. A simple frameworkEvolution of unemployment

I Let L = the total labor force, Lt = employment at t. Then wecan define the hiring, unemployment, and vacancy rates inrelation to the total workforce :

ut =Ut

L=

L− Lt

L,

vt =Vt

L,

ht =Ht

L.

I Because of constant returns to scale, we can rewrite (1) as

ht = m(ut , vt).

I The evolution of the unemployment rate is

du

dt= −ht + s(1− ut)

= −m(ut , vt) + s(1− ut).4 / 37

1. A simple frameworkThe Beveridge curve

I This defines a du/dt = 0 locus in the (u, v) plane which iscalled the ”Beveridge curve”.

I Along this locus, we have

0 = −m′udu −m′vdv − sdu

=⇒ dv

du= −m′u + s

m′v< 0.

I Furthermore,

d2v

du2= −

(m′′uu + m′′uv

dvdu

)m′v − (m′u + s)(m′′uv + m′′vv

dvdu )

m′2v

∝ (−m′′uum′v ) + (m′′vvdv

du)(m′u + s) + (m′′uv (m′u + s − dv

dum′v )),

and all the terms in parentheses in the last expression are > 0,therefore

d2v

du2> 0.

I Note : ”∝” means ”proportional with the same sign”. 5 / 37

1. A simple frameworkLabor market tightness

I Convexity of the Beveridge curve : because decreasingmarginal returns to each input in the matching function.

I When I increase vacancies by one unit when vacancies arelarge, the effect on hirings is small, and only a small reductionin unemployment would maintain a balance betweenemployment outflows and inflows.

I Given constant returns, it is easier to think in terms of labormarket tightness rather than vacancies. By definition, labormarket tightness is

θ = v/u.

I The probability per unit of time of finding a job is

p = h/u =m(u, v)

u= m(1, θ) = p(θ), p′ > 0, p′′ < 0.

I The probability per unit of time of filling a vacancy is

q =m(u, v)

v= m(

1

θ, 1) = q(θ), q′ < 0.

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1. A simple frameworkAlong the Beveridge curve

I Furthermore,

p(θ) = θm(1

θ, 1) = θq(θ).

I The Beveridge curve can be reexpressed in the (u, θ) plane :

u = s(1− u)− up(θ)

= s(1− u)− uθq(θ).

I Along this curve :

dθ

du= −s + p

up′< 0;

d2θ

du2∝ −up′2 dθ

du+ (s + p)(p′ + p′′u

dθ

du) > 0.

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1. A simple frameworkBeyond the Beveridge curve

I The Beveridge curve delivers one dynamic relationshipbetween u and v (or θ). Above it vacancies are larger than insteady state, so unemployment is falling. Below it,unemployment is rising. Hence the arrows on Figure 1.

I To complete the model we need another relationship betweenu and θ. This will come from labor demand.

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1. A simple frameworkThe Beveridge curve

v

uFigure 1 – The Beveridge curve

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1. A simple frameworkLabor demand

I We assume there is a single homogeneous good. Once aworker finds a job, he produces a constant flow of this goodequal to y per unit of time. He is paid a fixed wage w . Thereis a fixed real interest rate equal to r . Let Jt be the value ofthe firm at t. Since the job is destroyed with flow probabilitys, the asset valuation equation for J is

rJ = y − w + J − sJ. (2)

I The only non explosive solution is

J =y − w

r + s.

I To recruit workers, firms must post vacancies. Posting avacancy costs c per unit of time. Let Vv be the value of avacancy. Its asset valuation equation is

rVv = −c + q(θ)(J − Vv ) + Vv .

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1. A simple frameworkLabor demand (continued)

I There is free entry in posting vacancies. Therefore,

Vv = 0.

I Thus we get

J =c

q(θ).

I Note that the expected duration of a vacancy is 1/q(θ),therefore this tells us that the value of a job is equal to theaverage recruiting cost per job.

I This determines the equilibrium value of θ, which is constantand equal to

θ = q−1[c(r + s)

y − w

].

I Figure 2 shows the adjustment dynamics.

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1. A simple frameworkEquilibrium

θ

uFigure 2 – Adjustment dynamics under fixed wages

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1. A simple frameworkComparative statics

I θ goes up, and u falls, if the profitability of a job goes up, i.e.if r goes down, y goes up,w goes down.

I θ goes up if the cost of a vacancy falls.

I All these changes do not affect the BC. Thus the economymoves along the BC. (Figure 3)

I A rise in s shifts both the labor demand curve and the BCthrough the discounting and mechanical effects of jobdestruction. (Figure 4)

I Assume shocks to y alternate : this suggests that businesscycles induce counter-clockwise loops around the Beveridgecurve.

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1. A simple frameworkComparative statics

θ

uFigure 3 – Impact of an increase in labor demand

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1. A simple frameworkComparative statics

θ

uFigure 4 – Impact of an increase in the job destruction rate s

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2. Endogenous wagesBargaining

I An elegant way to endogenize wages is to assume thatincumbent employees and employers bargain over the surpluscreated by the sunk recruiting costs.

I Bargaining is individual between each worker and the firm. Itis easiest to assume that 1 firm = 1 job.

I Let Ve be the value of being employed, Vu be the value ofbeing unemployed.

I The asset valuation equations for Ve and Vu are (assuming nounemployment benefit)

rVe = w + s(Vu − Ve) + Ve ; (3)

rVu = θq(θ)(Ve − Vu) + Vu. (4)

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2. Endogenous wagesBargaining (continued)

I It is convenient to use the net surplus of the match, i.e.

W = J + Ve − Vu.

I Consolidating (2),(3), and (4) we get

rW = y − θq(θ)ϕW − sW + W . (5)

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2. Endogenous wagesBargaining (continued)

I At each date wages are set so as to maximize the joint logNash product :

ln[(J − Vv )1−ϕ (Ve − Vu)ϕ

].

III Bargaining takes place at the firm-worker level, taking asgiven aggregate conditions Vu and Vv .

I This is because the bargaining sets the wage within th match,not for all the economy

I Recall that J = y−wr+s

I Recall that rVe = w + s(Vu − Ve) + Ve so that thenon-explosive solution is Ve = w+sVu

r+s

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2. Endogenous wagesBargaining (continued)

I For any increase in wages ∆w we have (all else equal)∆Ve = 1

r+s ∆w and ∆J = − 1r+s ∆w = −∆Ve .

I Therefore the FOC is :

1− ϕJ − Vv

=ϕ

Ve − Vu.

I Since Vv = 0, this is equivalent to

Ve = Vu +ϕ

1− ϕJ. (6)

I That is :

Value of being employed = Outside option + rent.

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2. Endogenous wagesBargaining (continued)

I Since J = cq(θ) , the rent is proportional to the total recruiting

cost that has been spent.

I Equation (6) determines the wage despite that the wage doesnot explicitly appear in it.

I We now need to derive a relationship between θ and u in thismore complicated model.

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2. Endogenous wagesBargaining (continued)

I Note that (6) implies that this net surplus is shared inproportion (ϕ, 1− ϕ), that is

Ve − Vu = ϕW ,

J − Vv = J = (1− ϕ)W .

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2. Endogenous wagesThe real wage

rW = y − θq(θ)ϕW − sW + W . (5)

I The term in θq(θ)ϕW is the opportunity cost to the worker ofbeing employed in this match instead of being looking foranother job which would yield a net value ϕW to the workerand have an arrival rate θq(θ).

I Last, W can be expressed as a function of θ, since

c

q(θ)= J = W (1− ϕ).

I We get a dynamic equation for θ :

(r + s)c

(1− ϕ)q(θ)= y − c

(1− ϕ)q(θ)2q′(θ)θ − θϕc

1− ϕ. (7)

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2. Endogenous wagesDynamics

I The θ = 0 schedule is such that θ is constant and solves

(r + s)c

(1− ϕ)q(θ)= y − θϕc

1− ϕ.

I Since q′ < 0, (7) has unstable dynamics locally around θ = 0.

I As θ is a non-predetermined variable, we pick the onlynon-explosive solution, i.e. the saddle-path which is horizontal(Figure 5).

I The adjustment dynamics are qualitatively the same as in thecase where w is fixed.

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2. Endogenous wagesDynamics

θ

uFigure 5 – Saddle path stability under dynamic wage bargaining 24 / 37

3. Endogenous job destructionProductivity shocks

I Another direction in which we may want to enrich the modelis by endogenizing job destruction.

I For this we assume the firm has idiosyncratic productivityshocks.

I Productivity at any date is y = σε, where ε is distributed over[εl , εu].

I Newly created jobs have productivity εuI With arrival rate λ per unit of time, ε is then redrawn with a

c.d.f. F (),F ′ = f , over [εl , εu].

I The endogenous job destruction margin is determined by athreshold εd such that the job is destroyed if ε < εd .

I Note : it may be that εd ≤ εu in which case the job is neverdestroyed).

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3. Endogenous job destructionJob destruction rate

I The job destruction rate is now

s = λF (εd).

I The negotiated wage now generally depends on the currentvalue of ε :

w = w(ε).

I We need to rewrite the asset valuation equation, in steadystate, for J :

rJ(ε) = σε−w(ε)+λ

∫ εu

εd

(J(x)−J(ε))f (x)dx+λF (εd)(0−J(ε)).

(8)

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3. Endogenous job destructionBargaining

I Similarly, the value of the worker is given by

rVe(ε) = w(ε)+λ

∫ εu

εd

(Ve(x)−Ve(ε))f (x)dx +λF (εd)(Vu−Ve(ε)).

(9)

I Finally, the value of being unemployed obeys

rVu = θq(θ)(Ve(εu)− Vu). (10)

I LetW (ε) = J(ε) + Ve(ε)− Vu.

I The Nash bargaining solution implies that

J(ε) = (1− ϕ)W (ε); (11)

Ve(ε) = Vu + ϕW (ε).

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3. Endogenous job destructionThe real wage

I Using (8), (9) and (10) we get the new version of (5) :

rW (ε) = σε−θq(θ)ϕW (εu)+λ

∫ εu

εd

(W (x)−W (ε))f (x)dx−λF (εd)W (ε).

I We note that the integral W =∫ εuεd

W (x)f (x)dx is a constantwhich is independent of the current value of ε.

I Therefore,

W (ε) =σ

r + λε+

λW

r + λ− θq(θ)ϕW (εu)

r + λ. (12)

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3. Endogenous job destructionSurplus from a match

I Integrating, we get the actual value of W

W =

∫ εu

εd

W (x)f (x)dx

=

∫ εu

εd

σx − θq(θ)ϕW (εu) + λW

r + λf (x)dx

=σ

r + λ

∫ εu

εd

xf (x)dx − (1− F (εd))θq(θ)ϕW (εu)

r + λ

+(1− F (εd))λ

r + λW

=⇒ W =σ∫ εuεd

xf (x)dx − (1− F (εd))θq(θ)ϕW (εu)

r + λF (εd).

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3. Endogenous job destructionXX

II Substituting into (12), we get

W (ε) =σε

r + λ+

λσ

(r + λ)(r + λF (εd))I (εd)− θq(θ)ϕW (εu)

r + λF (εd),

(13)where

I (εd) =

∫ εu

εd

xf (x)dx .

I The model is closed by deriving a job creation condition and ajob destruction condition.

I Each condition gives us a relationship between εd , the jobdestruction margin, and θ, the labor market tightnessparameter.

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3. Endogenous job destructionThe job destruction condition

I The job destruction condition is

J(εd) = 0.

I Therefore :

W (εd) = 0

=⇒ Ve(εd) = Vu.

I In this class of models, the separation decision is jointlyprivately efficient.

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3. Endogenous job destructionThe job destruction condition (continued)

I We can then rewrite :

W (ε) =σ

r + λ(ε− εd). (14)

I This is because we know from (13) that W () is linear with aslope equal to σ

r+λ , while it must satisfy W (εd) = 0.

I Substituting into (13) we get

0 =σεdr + λ

+λσ

(r + λ)(r + λF (εd))I (εd)

− θq(θ)ϕ

r + λF (εd)

σ

r + λ(εu − εd)

⇐⇒ εd(r + λF (εd)) + λI (εd)− θq(θ)ϕ(εu − εd) = 0.

I This defines a relationship between εd and θ.

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3. Endogenous job destructionThe job destruction condition (continued)

I Differentiating, we get

[r + θq(θ)ϕ+ λF (εd)] dεd −[ϕ(εu − εd)

d

dθθq(θ)

]dθ = 0.

I Since ddθθq(θ) > 0, both terms in brackets are positive.

I Hence this defines a positive relationship between θ and εd .

I When the labor market is tighter, the opportunity cost of workis larger for the worker, because he could find a job starting atthe highest productivity level more quickly if he wereunemployed. Therefore the productivity threshold below whichit is efficient to destroy the job is higher–jobs are destroyedmore often.

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3. Endogenous job destructionThe job destruction condition (continued)

Other interesting aspects :

I An increase in ϕ increases εd : this is because the workerexpects to get more out of future jobs, thus reducing thevalue of staying in the current job.

I An increase in λ reduces εd : as shocks are more frequent, theoption value of waiting until a new shock arrives instead offiring the worker right now goes up ; hence I fire lessfrequently. This option value is the term inλ∫ εuεd

(J(x)− J(ε))f (x)dx in (8). It is always positive becauseI can always decide to fire the worker later, while if I fire himright now I will have to pay again the hiring cost to re-startmy business.

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3. Endogenous job destructionThe job creation condition

I The job creation condition is

J(εu) =c

q(θ).

I Using (14) and (11), this is equivalent to

(1− ϕ)σ

r + λ(εu − εd) =

c

q(θ).

I Since q′ < 0, this defines a negative relationship between εdand θ.

I When the labor market is tighter, it takes more time to recruitpeople. This makes it more costly to get rid of incumbentworkers. Therefore the productivity threshold where they arefired is lowered.

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3. Endogenous job destructionThe job creation condition (continued)

The other comparative statics are similar to those in the basicmodel :

I An increase in ϕ reduces θ : firms get a lower share of thesurplus out of each job and post fewer vacancies.

I An increase in r reduces θ, since the cost of funds for payingthe vacancy cost goes up.

I An increase in λ reduces θ : this is because the first shockreduces productivity (since initial productivity is εu), so I wantthis shock to occur as late as possible.

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3. Endogenous job destructionEquilibrium

I Equilibrium is determined by the intersection of the JC andJD schedules

θ

Job destructionJob destruction

Job creationJob creation

εdd

Figure 6 – Equilibrium determination in the Mortensen‐Pissarides matching model

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