Macroeconomics 2Lecture 8 - Labor markets: The search and matching model

Zsofia L. Barany

Sciences Po

2014 March

Last week

I an overview of some labor market facts

I a brief reminder of dynamic programmingI we covered the search model

I unemployed worker gets job offers sequentiallyI the offers arrive at exogenous rate aI from exogenous, stationary wage distribution F (w)I the unemployed has to decide whether to take a job offer or notI we concluded that the optimal decision is a reservation wage

rule, such that W (ξ) = UI ξ solves

ξ =r

r + a(1− F (ξ))b +

a

r + a(1− F (ξ))

∫ A

ξ

wdF (w)

→ depends on r , a, b and the distribution F (w)

The one-sided search model is a partial equilibrium model

I where does the arrival rate, a, come from?I the arrival rate of job offers should depend on the number of

vacancies firms createI this should be the result of a profit maximizing decision

I where does the wage distribution, F (w), come from?wages are endogenous, different views of what determinesthem:

I wage posting: firms post wages and commit to itI choose wages to maximize profitsI competitive search modelsI efficiency wages

I wage bargaining: wage determined by the meeting worker-firmpair after contact was made

The Rothschild and Diamond paradoxes

in the one-sided search model of last week – exogenous,non-degenerate wage distribution, F (w)→ where does this come from?

one option: this is the result of profit maximizing wage posting byfirms

paradox: a non-degenerate wage distribution cannot be the resultof profit maximizing behavior by firms

The Rothschild paradox

I somebody must be offering each wage in the support of thedistribution

I critique: it is difficult to rationalize the distribution F (w) as aresult from profit maximizing choices of firms

I workers search sequentially

I they discover the firm’s wage offer when they meet

I worker has to accept/reject before he can sample another firm

I worker’s optimal decision: accept any wage s.t.W (w) ≥ U ⇔ w ≥ ξ

I firm’s objective: hire workers at the lowest wage possible

I profit maximization then leads to a common wage offer w = ξ

The Diamond paradox

I Diamond goes even further (chronologically he was first)

I suppose that all firms offer jobs at wage, w

I now suppose that a single firm deviates, and offers w − εI the unemployed who gets the offer w − ε will accept it

the reservation wage in this case is

ξ =r

r + a(1− F (ξ))b +

a

r + a(1− F (ξ))

∫ A

ξwdF (w)

=rb + aw

r + a≤ w with = if w = b

I the firm has the power to set wages⇒ extracts all the surplus from job creation: w = b

I if there are out-of-pocket search costs, the market breaksdown

Resolving the Diamond paradox

in several situation the Diamond paradox disappearsI competitive search models

I let worker have more than one offerI give the worker more ex ante information

I efficiency wage modelsworkers have incentives to work harder when wages are harder

I models with wage distributionI workers search on the job → the worker can have at one time

two wage offers (current wage and new offer)I assume that workers have different search costsI allow for ex post wage heterogeneity

Second generation models

Two-sided matching

to close the one-sided search model, to derive a proper equilibriumof the economy, we need to

I make the arrival rate, a, endogenous→ specify decision of firms whether or not to offer jobs

I ensure that employment is not an absorbing state→ exogenous job destruction, at rate λinterpretation: negative shocks arrive to existing matches,that destroy the match→ the worker becomes unemployed, the job is destroyed

I the wage is the result of bargaining between the matchedworker and firmassume for now: single wage offer, which satisfies w ≥ ξ→ all job offers are accepted, outflow rate fromunemployment is a

Value of employed and unemployed

I the value of an unemployed worker is (as before)

rU = b + a(W − U)

I using that all jobs pay the same wage rate, w

I → nobody has an incentive to quit or to search for anotherjob while employed

I ⇒ the value of an employed worker is

rW = w − λ(W − U)

Combining the above two and rearranging:

W − U =w − b

r + a + λ

The matching functionkey assumption: the aggregate flow satisfies and aggregatematching function

I black box: gives the outcome of the matching process as afunction of the inputs into the search process

I m = m(u, v) – matchesI assumptions on m(·, ·)

I continuous, differentiableI positive first partial derivativesI negative second partial derivativesI CRS

I uncoordinated random search implies the following matchingfunction

m = v(

1− e−kuv

), where k > 0

I the empirical literature found that a Cobb-Douglas functionmatches the data well

m = Auηv1−η

The matching functionjob matching is pairwise ⇒

m = au = qv

I arrival rate of workers to vacant jobs

q =m(u, v)

v= m

(uv, 1)≡ m(θ−1, 1) ≡ q(θ)

I q is a decreasing function of θ: q′(θ) < 0I the elasticity of q wrt θ is ∂q

∂θθq ≡ −η ∈ (−1, 0)

I arrival rate of jobs to workers

a =m(u, v)

u= m

(1,

v

u

)= a(θ) = θq(θ)

I a is an increasing function of θ: a′(θ) > 0I the elasticity of a wrt θ is ∂a

∂θθa ∈ (0, 1)

Market tightness

I θ = vu is a measure of market tightness

I u is a statev is the firm’s control → this drives unemployment

I however, probably both firms and workers ignore their effecton θ when they make their search choices→ search externalities

Equilibrium

Things you need to know

I derivation of the aggregate flows from individual transitions

I derivation of individual transitions from optimizing decisionsunder rational expectations about the constraints and thefuture path of variables

I monopoly rents and wage determination

I derivation of the dynamics of the stock of employment andunemployment from the aggregate flows

I the equilibrium rate of unemployment (also called the naturalrate)

Timing of decisions

t

1 costly search 2 production 3

1. firm posts vacancyI the firms post vacancies until there are rents to be madeI the value of a vacancy in equilibrium has to be zero ⇔ V = 0I new jobs produce with the best technology

2. worker arrivesI wage bargaining takes place

w acceptable to both parties ⇔ W ≥ U and J ≥ 0I job creation takes place if there is an agreementI production begins

3. idiosyncratic productivity shock arrivesI investment is irreversible

if the shock reduces the net value of the job below zero ⇔J + W < U

I job destruction

Timing of decisions

t1 costly search

2 production 3

1. firm posts vacancyI the firms post vacancies until there are rents to be madeI the value of a vacancy in equilibrium has to be zero ⇔ V = 0I new jobs produce with the best technology

2. worker arrivesI wage bargaining takes place

w acceptable to both parties ⇔ W ≥ U and J ≥ 0I job creation takes place if there is an agreementI production begins

3. idiosyncratic productivity shock arrivesI investment is irreversible

if the shock reduces the net value of the job below zero ⇔J + W < U

I job destruction

Timing of decisions

t1 costly search 2 production

3

1. firm posts vacancyI the firms post vacancies until there are rents to be madeI the value of a vacancy in equilibrium has to be zero ⇔ V = 0I new jobs produce with the best technology

2. worker arrivesI wage bargaining takes place

w acceptable to both parties ⇔ W ≥ U and J ≥ 0I job creation takes place if there is an agreementI production begins

3. idiosyncratic productivity shock arrivesI investment is irreversible

if the shock reduces the net value of the job below zero ⇔J + W < U

I job destruction

Timing of decisions

t1 costly search 2 production 3

1. firm posts vacancyI the firms post vacancies until there are rents to be madeI the value of a vacancy in equilibrium has to be zero ⇔ V = 0I new jobs produce with the best technology

2. worker arrivesI wage bargaining takes place

w acceptable to both parties ⇔ W ≥ U and J ≥ 0I job creation takes place if there is an agreementI production begins

3. idiosyncratic productivity shock arrivesI investment is irreversible

if the shock reduces the net value of the job below zero ⇔J + W < U

I job destruction

Constraints

I the distribution of productivities

I the processes that

I bring together firms and workers – the matching function→ affects unemployed people and vacant jobs

I change the productivity of a job – the idiosyncratic shocks→ affects a pair of employed person and a firm/job

Flow of people and jobs

v , vacancies

control

value: V

u, unemployment

state

value: U

1− u, employment

productivity p

value for worker: W

value for firm: J

q(θ)

a(θ)

m(u, v)

λ

jobs

job destroyed

p = 0

workers

The value of vacancies and jobs

I V – value of a vacant job

I J – value of an occupied job

I a job is an asset owned by the firm, and its valued isdetermined by arbitrage equations

rV = −pc + q(θ)(J − V )

rJ = p − w − λJ

I r – discount rate

I p – value of product, productivity → higher value for betterworkers

I pc – cost of maintaining vacancy → it is more costly to findpeople to fill skilled vacancies

I w – wage rate

Job creation

firm creates a job vacancy when there are gains from entering themarket and there is free entry⇒ zero profits, V = 0

rV = −pc + q(θ)(J − V )

V = 0⇔ J =pc

q(θ)

I J – the value of having a worker = the PV(expected profits)

I 1q(θ) – the expected duration of a vacancy

I pcq(θ) – the expected total cost of finding a worker

Job creation

the value of a filled job

rJ = p − w − λJ

J=p − w

r + λ

note: for the firm to accept a wage w ⇔ p ≥ w

combining the two equations on the value of filled jobs:

J =pc

q(θ)& J =

p − w

r + λ

p − w︸ ︷︷ ︸profit flow

− (r + λ)pc

q(θ)︸ ︷︷ ︸expected cost of finding a worker

= 0

job creation condition: generalization of the labor demandcondition, downward sloping in the θ - w space

The value of the unemployed and the employed

Human capital as an asset valued by arbitrage conditions

rU = z + θq(θ)(W − U)

rW = w + λ(U −W )

I z – income during unemployment (previously b)

I θq(θ) = a(θ) – arrival rate of jobs

I w – wage rate

Wage determination

How does the firm and employee divide the surplus?

I the worker does not get his marginal productbecause there are sunk costs (the search is costly, long)→ matches enjoy some local monopoly rents due to thefrictions

I wages are determined by a bargainusually by Nash bargaining

I net rent: J + W − (U + V ) ≥ 0 → is the room for bargaining→ given this range, where do they fix wages?

I worker rent: W − U

I firm rent: J − V

I contract: initial wage and continuation wageboth depend on job productivity and on outside conditions

Wage bargain

I J – firm’s reward from agreement

I V – firm’s payoffs when there is no agreement

I W – worker’s rewards from agreement

I U – worker’s payoff when there is no agreement

assumption: wage rate maximizes the Nash product

(W − U)β(J − V )1−β where β ∈ (0, 1)

Solution:W − U = β(J + W − V − U)

β share of the net (monopoly) rents go to the worker, which implies

w = (1− β)z + βp(1 + cθ)

Wage bargain

this is a weighted average of the income (flow value) inside andoutside the firm:

w = (1− β)z + βp(1 + cθ)

Wage bargain

this is a weighted average of the income (flow value) inside andoutside the firm:

w = (1− β)z + βp + βpcθ

= (1− β)

(z +

β

1− βpcθ

)+ βp

Wage bargain

this is a weighted average of the income (flow value) inside andoutside the firm:

w = (1− β)z + β(p + cθ)

= (1− β)

(z +

β

1− βpcθ

)+ βp

= (1− β)

(z +

β

1− βpc

q(θ)q(θ)θ

)+ βp

Wage bargain

this is a weighted average of the income (flow value) inside andoutside the firm:

w = (1− β)z + β(p + cθ)

= (1− β)

(z +

β

1− βpcθ

)+ βp

= (1− β)

(z +

β

1− βpc

q(θ)q(θ)θ

)+ βp

= (1− β)

(z +

β

1− β(J − V )q(θ)θ

)+ βp

Wage bargain

this is a weighted average of the income (flow value) inside andoutside the firm:

w = (1− β)z + β(p + cθ)

= (1− β)

(z +

β

1− βpcθ

)+ βp

= (1− β)

(z +

β

1− βpc

q(θ)q(θ)θ

)+ βp

= (1− β)

(z +

β

1− β(J − V )q(θ)θ

)+ βp

= (1− β) (z + (W − U)q(θ)θ) + βp

Wage bargain

this is a weighted average of the income (flow value) inside andoutside the firm:

w = (1− β)z + β(p + cθ)

= (1− β)

(z +

β

1− βpcθ

)+ βp

= (1− β)

(z +

β

1− βpc

q(θ)q(θ)θ

)+ βp

= (1− β)

(z +

β

1− β(J − V )q(θ)θ

)+ βp

= (1− β)(z + (W − U)q(θ)θ)+βp

= (1− β)rU+βp

Equilibrium wages and market tightness

θ

wwage curve

job creation

θ∗

w∗

wage curve – bargained wage as a function of θ

w = (1− β)z + βp(1 + cθ)

job creation curve – zero profit from vacancies, optimal θ as afunction of w

p − w − (r + λ)pc

q(θ)= 0

Equilibrium

in this economy the equilibrium is a path for

the controls:

I wages – from the wage equation: J ≥ 0,W ≥ U

I job vacancies (or tightness) – from job creation V = 0

the state:

I employment – condition for the evolution ofunemployment

u = λ(1− u)−m(u, v) = λ− (λ+ θq(θ))u

given θ∗, this has a unique stable solution, the natural rate ofunemployment

u =λ

λ+ θ∗q(θ∗)

the above is the Beveridge curve

Equilibrium tightness and unemployment

u

θ

job creation, θ = 0θ∗

Beveridge curve, u = 0

I θ is a control variable, its equilibrium is independent of uI u is a state variable, and it is stableI the θ = 0 line is the saddle pathI θ jumps to its equilibrium value, and the economy moves

along the saddle path to the steady state

Equilibrium vacancies and unemployment

u

v

JC, v = θ∗u

BC, u = 0

I v is a control variableI u is a state variable, and it is stableI the v = θ∗u line is the saddle pathI v jumps on the saddle path, and the economy moves along it

to the steady state

The Beveridge curve in the US

Source:https://sites.google.com/site/robertshimer/

the Beveridge curve shifted out in the great recession in the US

What caused the rightward shift of the BC?

I data: fewer hirings per job openings compared to historicaldata→ decline in the matching efficiency?

I composition of labor demand changed:away from high labor turnover industries (construction)towards low labor turnover industries (engineering, medicalcare→ might seem like a decline in matching efficiency

I skill mismatch

I extended unemployment insurance benefits