Search Models of Unemployment
 Another labor market friction apart from wage rigidity is “search”
 Search is required on both sides of the labor market – it takes time for
an individual who wants to work to find a suitable job and it takes
time for a firm to fill a vacancy.
 The idea is that unemployed workers and job openings do not
instantaneously find each other
o There are frictions in the labor market
o There is a search process
 Source of frictions:
o Heterogeneity of workers and firms
o Uncertainty about the skills, quality of jobs
o Imperfect information: invest resources in locating firm/worker
takes time for worker to find an ‘acceptable job and similarly
for an employer to find an ‘acceptable’ worker
o Mobility costs
 Unemployment is a stock variable that adjusts slowly as a result of
this job finding process
o Workers that would be willing to work if they found the ‘right
job’
o But they will keep turning down job offers that are not good
enough
o Unemployment is an equilibrium phenomenon: all workers and
firms are maximizing utility/profits
o No need to have wage rigidities: search frictions are sufficient
 The model can also be used to study the vacancy rate, the
participation rate, the employment/population ratio, etc.
One period search model of unemployment (DMP)
 Peter Diamond-Dale Mortensen-Christopher Pissarides (DMP) won
the nobel prize in 2010 for their analysis of markets with search
frictions.
Set up:
 One period
 Many consumers and firms
 N consumers, who are all potential workers, so N is the working age
population
 The number of firms is endogenous, to be determined.
 Each of the N consumers can choose to work outside the market
(household production) or to search for market work
 Let Q denote the quantity of consumers who decide to search for
work, so N-Q is the number of consumers who choose household
production. Q is interpreted as the labor force and N-Q is those
working age people not in the labor force.
 P(Q) is a supply curve for workers who choose to search for market
work. Thus it represents the expected payoff to search for market
work that would induce Q consumers to search
P(Q) is upward sloping: as the expected payoff from searching is higher, this
induces more consumers to forgo household production to search for market
work.
 In order to produce, a firm must post a vacancy to hopefully match
with a worker
 Recruiting is costly – the firm incurs cost k (in units of consumption
goods) to post a vacancy
 Firms that do not post a vacancy are inactive and cannot produce. Let
A denote the number of active firms, which is the number that choose
to post vacancies.
 At the beginning of the period, there will be Q consumers searching
for work and A firms posting vacancies
 Matching workers with firms is a time-consuming and cost process,
characterized by a matching function, M:
M = em(Q,A)
where e denotes matching efficiency and m(.)is a function
Note: with larger e, more matches occur given the numbers of
consumers Q and firms searching A
- Matching efficiency, e, can increase in practice due to better
information (eg. Efficient search technologies such as internet
advertising, or because the skills that consumers have are better
matched to the skills that firm want.)
- The matching function has constant returns to scale
Em(xQ,xA)=xem(Q,A)
- m(0,A)=m(Q,0)=0
If there are no consumers searching for work or no firms searching
for workers, then there are no matches
- The number of matches M increases when either Q or A increase
- Marginal products are diminishing, in that the increase in matches
obtained for a one-unit increase in Q decreases as Q increases, and
similarly for A.
 Each consumer chooses between household production and searching
for work
 If they choose to search for work, then he or she finds a match with a
firm with probability
pc = M/Q = em(Q,A)/Q=em(1,A/Q)=em(1,j)
with the constant returns to scale property of the matching function.
Therefore, the probability of finding work for a consumer depends
only on j=A/Q which is the ratio of firms searching for workers
relative to consumers searching for work. This is a measure of market
tightness.
 If the consumer search for work and is matched, he/she receives wage
w.
 If the consumer searches and is not matched, then he/she is
unemployed and received an unemployment insurance benefit b.
This occurs with probability 1-pc=1-em(1,j)
 Recall that P(Q) is the supply curve for workers who choose to search
for market work. In equilibrium, P(Q) must be equal to the expected
payoff a consumer receives from searching
P(Q) = pcw + (1- pc)b = b+(w-b)em(1,j)
The market wage, UI benefit, labor market tightness determine the
expected payoff to searching for work for a consumer. Then, given
this expected payoff, the supply curve for searching consumers
determine the labor force.
The Supply Side of the Labour Market
The market wage, t
and labour market
determine the expe
to searching for wo
consumer. Then, giv
expected payoff, th
curve for searching
determines the labo
 A worker in competitive equilibrium model observes the market wage
Junjiedecides
Liu – Econ
305
then
how
much labor to sell on the market at that wage
 In the DMP model, a would be worker takes into account not just the
market wage, but his or her chances of finding work and the UI
benefit if his or her job search fails
 On the firms side, firms choose to bear the cost k of posting a vacancy
and have a probiality of pf=M/A of finding a worker, since the ratio of
total matches to the number of firms searching determines the chances
of achieving a successful match.
 Then, from the matching function, we have
pf = M/A = em(Q,A)/A = em(Q/A,1) = em(1/j,1)
 Given a successful match with a worker, the firm and worker produce
output z, so the profit the firm receives from the match is z-w
 Firms will enter the labor market, post vacancies, until the expected
net payoff from doing so is zero pf(z-w)-k=0
 In equilibrium, k must be equal to the expected payoff for the firm
from posting the vacancy
em(1/j,1) = k/(z-w)
The Demand Side of the Labour Market
Firms post vacancies up
the point where the
probability for a firm o
matching with a worke
equal to the ratio of th
of posting a vacancy to
profit the firm receives
successful match.
Firms post vacancies up to the point where the probability for a firm of
Junjie Liuwith
– Econ
305 is equal to the ratio of the cost of posting a vacancy
matching
a worker
to the profit the firm receives from a successful match.
 When a firm is matched with a worker, together they produce output z
 The firm and worker need to come to an agreement concerning the
wage w that the worker is to receive
 We use Nash bargaining theory to determine how a matched firm and
worker split the total surplus.
Nash bargaining
 Worker’s surplus = w-b (wage minus UI benefit)
 Firm’s surplus = z-w (profit)
 Total surplus = z-b
 a=worker’s share of total surplus (the bargaining power of the worker)
 Then the worker and firm agree to a contract such that the work’s
surplus is a fraction a of total surplus, ie. w-b=a(z-b)
 As a result, w = az+(1-a)b
Substitute w in the labor supply and labor demand equations
The above two equations determine two endogenous variables, Q and j.
In equilibrium, the unemployment rate, the vacancy rate, and the level of
aggregate output are
Equilibrium in the DMP M
In panel (b), the ratio of the cost of posting a vacancy to the firm’s surplus
from a successful match determines labor market tightness. Then in panel (a)
labor market tightness determines the size of the labor force.
In panel (b), t
posting a vac
surplus from
determines la
tightness. The
labour marke
determines th
force.
Junjie Liu – Econ 305
Case study 1: An increase in the UI benefit b
 An increase in b reduces the total surplus from a match, z-b (shifts the
curve up in panel a)
 The wage w increases, as the alternative to working becomes more
tempting for a searching consumer
 Posting vacancies become less attractive for firms, so labor market
tightness falls (panel b)
 For consumers, searching for work becomes more attractive, as the
wage is higher. But searching for work is also less attractive, as the
chances of finding a job are lower (j is lower)
 As a result, Q may rise or fall given these two opposing effects
 Y is ambiguous, u rises, v falls.
Case study 2: An increase in productivity z
 An increase in z causes the total surplus from a match z-b to increase
(panel a)
 The wage w increases as the worker gets a larger slice of the pie
 As profits are higher, posting vacancies become more attractive for
firms, so labor market tightness, j, rises (panel b)
 For consumers, searching for work becomes more attractive, as the
wage is higher and the chances of finding work are better.
 Q rises, u falls, v rises, Y rises.
Case study 3: A decrease in the matching efficiency, e
 There is no change in total surplus, or in the wage
 Because firms find it more difficult to find the right workers, fewer
firms post vacancies and j falls
 For consumers, searching is less attractive – the wage is the same but
the chances of finding a job are lower, so Q falls
 u rises, but vacancy rates states the same, and Y falls
 The Beveridge curve shifts to the left
 These predictions are consistent with the observations from the 20082009 recession and the recovery from the recession
- a decrease in matching efficiency implies a rightward shift of
the Beveridge curve, a decrease in the labor force, higher
unemployment and lower aggregate output
The Beveridge Cu
The
obs
rate
Dec
200
clea
soli
Dec
201
to h
200
The points in the scatter plot are observations on the unemployment
rate and the vacancy rate for December 2000 through Decemeber
2007,
and those
Junjie
Liu –points
Econexhibit
305 a clear Beveridge curve relation. A
solid line joins observations from December 2007 through March
2012. The Beveridge curve appears to have shifted in about December
2009.
What can be the cause of decrease in matching efficiency?
- worse information about vacancies and searching workers
 not a very plausible explanation for this recession
- mismatch between skills firm needed and skills workers possess
 possibly if there are sectoral shocks, and workers leaving
declining industries do not have skills required by growing
industries
- mismatch between locations of vacancies posted and where
workers search
 quite relevant for financial crises: many workers are not
moving to different geographical areas for work
A Keynesian DMP model
 There is a type of market failure associated with the setting of
wages and prices
 Wages and prices are sticky (not fully flexible)
 Instead of wages being determined by Nash bargaining, w is
exogenous. It can be determined by
o “animal spirits”. The relative bargaining power of
workers and firms are in some sense random. Firms may
decide to drive a hard bargaining with workers and this is
contagious, making the market wage relatively low, or
firms in a similar contagious manner decide to go easy on
workers and pay them a high wage
o Alternatively, the market wage may be sticky and
determined from history.
 Market wages may be too low or too high, relative to
what is socially optimal
Given w, j, Q, we have
There could be two equilbria:
 In one equilibrium, the market wage is high, Q is low, j is low, u is
high, v is low, aggregate output is low and labor participation is low
 In the other equilibrium, the market wage is low, Q is high, j is high,
the unemployment rate is low, the vacancy rate is high, aggregate
output is high and labor force participation is high.
 Either equilibrium could arise and be self fulfilling.
A comparison of equilibria with high and low wages, with w1>w2. In
the high wage equilibrium, labor market tightness is low and the labor
force is low, though it is possible for the labor force to be high in the
high wage equilibrium.
 The Keynesian DMP model gives an alternative explanation for
the Beveridge curve relationship
 If business cycles are driven by animal spirits, with fluctuation
wages, then when wages are high the unemployment rate will
be high and the vacancy rate low, and when wages are low the
unemployment rate will be low with high vacancy rate.
 This is the Beveridge relationship.