Download Job Search Mechanism and Individual Behaviour

Job Search Mechanism
and Individual Behaviour
Massimo Giannini
Università del Molise∗
Preliminary Draft
Abstract
This paper is a first attempt at modelling job search mechanism at
individual level by a determinstic-stochastic approach widely used in the
analysis of epidemic models. Unlike standard literature on job search, each
single unit, firm or worker, is take into consideration; aggregate dynamics
comes directly from the micro-structure of the economy. We show that
the unemployment as well as vacancy rate converge in the long run to an
ergodic distribution whose average value lies on the Beverdige curve.
∗ Facoltà
di Economia, Palazzo Orlando, Isernia. E-mail: [email protected]
1
1
Introduction
Like most other aggregate functions in the macroeconomist’s tool kit, the matching function is a black box; we have good intuition about its existence and properties but only some tentative ideas about its microfoundations. Yet, those tentative ideas have not been rigorously tested....Further work needs to elaborate a
number of issues. The search for microfoundations needs to continue and rigorous tests of plausible alternatives done..... The most popular microeconomic
models, such as the urn-ball game, do not perform as well empirically.1
Job search is an intrinsically individual-level issue; it sums up the behaviour
of two elementary units of an economy: firm and worker. What we observe in the
job market data is the consequence of such elementary processes; unfortunately
we have no a clear idea whether the aggregate level is just the mirror, on a larger
scale, of the microcosm or only a fuzzy and distorted representation. In trying
to challenge the question, we need to model the microcosm in a ”fair” way,
i.e. sufficiently simple to be analytically tractable and, meanwhile, sufficiently
complex to cope with the goal.
The number of variables affecting the single match between firm and worker
is high and well reviewed in the Petrongolo and Pissarides, 2000, survey. Under
this point of view, our request of a ”fair” description of the microcosm could
be frustrated; nevertheless it is worth the effort. This paper presents a first,
and very preliminary, attempt at providing a simple but reasonably descriptive analytical body for modelling job search at the individual level which is
coherent with a simple macroeconomic model. It relies heavily on recent developments of mathematical and statistical research applied to the investigation of
the dynamics of epidemic processes.
The simplest way of looking at the job search mechanism is the one described by the Pissarides Mortensen approach (PM hereafter): at every time
a firm randomly ”contacts” a worker, the match is closed and the unemployed
changes its status to employed (hence she is removed from the unemployment
population), the firm fills the vacancy and it can be removed from the population of vacant firm. Meanwhile, new unemployed are born and new vacancy
are opened as the result of an exogenous separation between an existing firmworker unit. The matching rate, as well as the separation one, can depend both
on worker features, notably human capital, and firm characteristics as well as
on local spillovers.
The PM approach relies on an aggregation of the micro-economic behaviours
and focusses on the steady state solution. By doing so, it does not allow to investigate indifferently the macro and the micro point of view of the question.
Our goal is to provide an alternative approach to the problem based on a probabilistic law that allows us to analyze both sides of the problem in way that they
are coherent to each other. The two-fold result allows us to investigate both
on the variables relevant for the single unit history, as described by empirical
analysis on microdata, and on the economy as a whole, undertaking in this case
1 Petrongolo
and Pissarides, 2000.
2
empirical analysis based on aggregate data. Finally the analysis should not be
limited to the steady state but on the transition path as well.
The paper is organized in three logical steps: the first one is devoted to the
individual analysis where ”optimal” tranistion rates are obtained according to
a rational behaviour of firm and worker. The second one shows how obtaining
macro-dynamical equations by averaging over these individual rates. Finally
the probabilistic model shows how we can obtain the same aggregate laws by
analyzing micro-dynamics. In the last part, we show how the micro-model allows
to take into consideration some spillovers effect which can not be analyzed by the
aggregate analysis; under this point of view, it stresses the greater explorative
power of empirical analysis based on microdata rather than on a aggregate
account.
2
The job search approach
In this section we briefly recall the standard job search approach a la Pissarides
and Mortensen; this helps us to stress where our approach overlaps with the
former and where not. As it is well known, the crucial assumption of the job
search mechanism is the ”matching function” which provides an average measure of the instantaneous rate at which an unemployed worker and a vacant
firm match one each other. The matching rate is increasing in the number of
unemployed workers U and vacant firms V ; by assuming CRS on the function,
one can shift from U and V to the respective rate u and v and usually the
matching function is described as m( uv , 1).
The approach is basically a steady state analysis. It provides the number
of jobs opened by rational, forward-looking, firms in equilibrium by taking into
account the equilibrium conditions arising from the inflow and outflow of workers
from the unemployment status. The main feature of the model is that, in
equilibrium, the market does not clear, as both u and v are positive; more
precisely, the equilibrium points lie on a downward sloping, convex-to-the-origin
curve, in the u, v space, known as the ”Beveridge curve” since Lord Beveridge’s
work on the empirical relationship between these two variables. The lack of full
employment is due to the search mechanism and not to wage rigidity; the profit
for a single firm is given by (terms have the usual meaning):
f (k) − w − rk −
(r + s)γ
q( uv )
(1)
A comparison with the profit in a walrasian context shows immediately that the
extra term (r+s)γ
v
q( u
) matters; this is zero only if either the search cost γ is zero or
the matching rate q( uv ) is very high , theoretically infinite; both conditions are
true in a walrasian market. Since a competitive equilibrium implies the zero
profit condition, it is straightforward to see that equation 1 can be solved with
respect to ( uv ), leading to the result that, in equilibrium, u and v are not zero.
This static framework can be analyzed in a dynamic setting (see Pissarides,
1990, ch.3); the adjustment dynamics towards the Beveridge curve is monotone;
3
off the equilibrium, both u and v are decreasing or increasing depending on
the initial conditions. Anti-clock wise adjustments are also possible in response
to productivity shocks. It is worth stressing that the long run equilibrium is
characterized by a saddle dynamics; convergence to the unique stable branch
requires a well specified initial condition on the market tightness v(0)/u(0).
In general, empirical observations on the Beveridge curve over time show a
rather complex dynamics, characterized by loops and shifts which are usually
induced by structural changes as well as business cycle (Nickell et al. 2002, Wall
and Zoega, 2002). Nevertheless, we can not exclude that a relevant part of this
odd dynamics is induced by a more complex adjustment to the steady state that
the standard job search model does not capture.
However, our aim is to provide a theoretical framework which can describe
the adjustment dynamics both in a macro and in a micro framework. Basically
we derive the aggregate dynamics from the micro structure of the economy;
under this point of view we can indifferently analyse the temporal evolution of
the variables u and v either by the aggregate equations or by the more complex
micro-structure. There are different gains in working at different levels; the
macro-equations are simpler and can be account for a large range of dynamical
behaviour, even including chaotic dynamics2 . From this point of view, suitable
modifications of the macro equations modify heavily the dynamical paths; these
modifications can span from a simple coefficient endogeneity to a more complex
form for the matching function. If so, the wide range of observed dynamics
for the UV curve could be induced not only by structural changes or business
cycle but by different parameters, initial conditions and functional form for the
matching function.
Once the macro-equations successfully replicate the observed dynamics, the
micro-structure can be fruitfully exploited to investigate individual behaviour
at a finer level than the aggregate one.
Concluding, we are going to use a dynamical framework where unemployed
workers and vacant firms match at random; in this paper we use a highly stylized
economy. Workers need work in order to gain income and firms need workers for
producing and making profits. Under this point of view, both the single worker
and the single firm behave as a two-state process switching from unemployed to
employed at random times following a Poisson process. In this paper we investigate this type of simple dynamics both at the macro and micro level, showing
that we can switch from the former to the latter without lack of coherency.
3
Single-Worker Analysis
We start by considering the individual working life. In official statistics agency,
the workforce is divided between unemployed and employed. Nevertheless this
dicotomy does not represent the total supply of labour since we have to account
for individuals temporary out of the workforce who would like to have a job.
2 We are going to show that such dynamical system involves three non linear differential
equations and it is known that such systems can generate chaotic dynamics.
4
This share of labour supply is usually labelled as ”person not in the labor force
who want a job now” according to the U.S. Bureau of Labor Statistics and in
general is a form of hidden unemployment or labour reserve ( see Castillo 1998
for a detailed description). According to the US Current Population Survey, in
September 2003 the number of workers who were out of workforce not looking
for a job but that would like to work was about 4.6 million while the total
official unemployment was about 16 million; it is worth stressing that the larger
share of hidden unemployment lies between age 16 and 54, i.e. in the most
productive part of the working life. Average duration of hidden unemployment
is about 12 months in the CPS; one workerk recorded as ”person not in the
labor force who want a job now” in the earler year becames a job seeker the
following year (see Castillo 1998 again). The hidden uemployement is a sort of
voluntary temporary retirement from the job market and this choice depends
on a lot of individual characteristics not least the ”discouraged worker” effect;
according to the US BLS: Discouraged workers are those persons who say that
they want a job, were available to work, had searched for a job some time in the
previous 12 months but had stopped looking for work because they believed that
there were no jobs available for them.
In this paper we are going to take into consideration hidden unemployment
and, according, we are going to divide the total labour supply in unemployed,
employed and temporary retired, L = U + E + R. A worker randomly drawn
from the population occupies one of these three status; nevertheless the status
will be ordered hierarchically since a worker is employed only after she was a job
seeker, i.e. U −→ E. At same time, we assume that once the worker separates
from the firm she does not became immediately a job seeker but stays in the
state R before of entering again the state U ; if the transition rate from state R
to U is very high, ideally infinite, then the temporal occupancy of R is close to
zero and basically this means that a separation gives birth immediately to a job
seeker. As will be clearer later, this assumption of stages hierarchically ordered
allows us to characterize the steady state of the job market as a balancing of
flows out and in the U, E and R states; in this way the model fits the official
partition of the work force in unemployed, employed and labour reserve.
Under a probabilistic point of view, the simplest way at modelling the single
worker history over time is a ”three states process”. Worker life history is a
continuos jump from three possible states: U, E and R. Under this point of
view, each agent is characterized by an own string of U, E and R recording
her history over time; the length of time the individual stays in a given state is
a random variable and the Poisson distribution is a good aproximation of this
occupancy time.
This process can be easily described in a probabilistic way: if the worker
is unemployed but actively engaged in search at time t, she can change her
status to employed at the instantaneous rate α, and the probability that she
will be effectively employed in the time interval t+∆t, is simply α∆t3 . Likewise,
3 More precisely is α∆t + o(∆t) where o(∆t) means a function which tend to zero more
rapidly than ∆t. Intuitively this means that in a small time interval (ideally when ∆t tends
5
when the worker is employed, she has a probability β∆t to be separated during
t+∆t,finally a temporary retired worker has a probability m∆t to be a job seeker
during t+∆t. The coefficients α, β and m are the transition rates, i.e. the chance
for a transition in a unit of time and they are not constrained in the unitary
interval as they are not probabilities; they can depend both on individual and
firm characteristic, although we are going to treat them as exogenous parameters
along this section.
When the worker changes her status, she stays in the new state for a random
time τ ; as said, this waiting time is well modelled by a Poisson process. Under
this point of view, our worker is a job seeker for a given time T1U after then
she becomes employed for another random time T1E then changes again to nonseeker for a time T3R and finally come back to the status of job seeker and so
on. The dynamic of the probability to be in a given state at time t follows the
Kolmogorov forward equations (see Cox and Miller, page 172):
dP E (t)
dt
dP U (t)
dt
dP R (t)
dt
= −βP E (t) + αP U (t)
= −αP U (t) + mP R (t)
(2)
= −mP R (t) + βP E (t)
with initial condition and P U (0) = 1 − P E (0)− P R (0) given. The above equations have a prompt interpretation; they describe the evolution of the probability
over time according to a balancing of inflows and outflow. As an example, let
us focus on the first one; the probability of finding a worker in the employed
status increases with the inflows from unemployed to employed (αP U (t)) and
decreases with the separation βP E (t).
The steady state solution is :
PE
=
PU
=
PR
=
mα
βα + βm + αm
βm
βα + βm + αm
βα
βα + βm + αm
The steady state is a stable node and so the long run distribution does not
depend on initial conditions; this means that if we observe a single worker job
history over a sufficient long horizon, then the percentage of time spent in a
given state, E , U, or R is close to the theoretical distribution. The strong
convergence and the independence of initial conditions are valuable properties
for empirical analysis.
to zero) the process can undertake only a single change of state (point process).
6
4
Single-firm Analysis
In this section we are going to duplicate the previous point for the firm. For
sake of simplicity,we are goint to assume that firms can occupy only two states:
vacant and filled. In this case the system 2 reduces to two equations only
dP F (t)
dt
dP V (t)
dt
= −βP F (t) + αF P V (t)
= −αF P V (t) + βP F (t)
where αF is the transition rate for vacant firms. The steady state solution of
the system is:
PF
PV
PF + PV
5
αF
β + αF
β
=
β + αF
= 1
=
(3)
Individual Choice
We start by analyzing firm behaviour. Any single firm knows the probability
of being filled or vacant, according to 3. If we look at αF as the intensity of
search, the higher αF the higher the probability of being a productive firm.
But intensity is a resource wasting activity and so the single firm has to choose
αF in order to maximize the expected discounted flow of profits net of the
searching cost. Assuming that each single firm produces a fixed amount of the
only commodity in the economy whose price is the numeraire, and it uses a
simple one-to-one technology, the problem is quite simple:
Z ∞
¤γ
£
M ax
e−ρt (pF π − (1 − pF )C(α) dt
α
0
where π = y − w is the profit when the firm is filled, C(α) the search cost that
we assume quadratic, C(α) = α2 , and γ ≤ 1 is the risk aversion coefficient.
The probability pF comes from 3 under the assumption that the firm life is
sufficiently long, and that it can be approximated by the long run probability.
This simple maximization scheme brings to α∗ = y−w
β ; the higher the wage
the lower the search intensity. This means that the demand curve is negatively
sloped; in fact the average number of matches is α∗ V and for a given V the
number of matches is decreasing in w.
The worker faces a similar problem; let us suppose that the transition from
unemployed to hired worker depends on the investment in human capital or
7
search intensity which is costly. By assuming that the worker earns w when
employed, 0 when retired and −C(αe ) when seeker, the problem is then:
Z ∞
e−ρt [(pe w − pu C(αe )]γ dt
M ax
αe
0
where the subfix e stands for ”employed”. The optimal human capital is α∗e = w
β.
In this case the average number of matches, from the side of unemployed workers,
is increasing in w.
Instantaneous equilibrium requires that same number of matches comes form
both sides of the market, i.e. α∗ V = α∗e U or α∗ v = α∗e u and from this condition
yv
we obtain the equilibrium wage w = u+v
< y hence:
α∗
=
α∗e
=
y u
y 1
=
β u+v
β 1+θ
y v
y θ
=
β u+v
β 1+θ
(4a)
It is worth stressing that: α∗e = α∗ θ, with θ = v/u and depends only on θ. Moreover the average instantaneous matching function m = α∗ v = α∗e u is increasing,
concave and linearly homogenous of degree one in u and v, as in the spirit of
the MP approach.
6
Macro-Dynamic Analysis
The model is essentially dynamic in its nature and in this section we are going
to obtain the differential equations characterizing the average dynamics; given
an initial condition, the job market evolves towards the steady state represented
by a point on the Beverdige Curve.
Given the instantaneous transition rates 4a we derive the differential equations by averaging over firms and workers. For a job seeker, the average probability, per unit of time, of changing status from unemployed to hired (U → E)
is α∗e times the probability of finding an individual in the unemployment status, PU , i.e. α∗e PU , where PU = U/L. In the time interval ∆t, corresponding
to L elementary steps4 ,the average number of unemployed being employed is
α∗e PU L∆t; hence, the number of unemployed decreases at the rate ∆U/∆t = −
θ
uv
α∗e PU L = − βy 1+θ
uL = −L βy u+v
. The number of unemployed instead increases
by mR. Likewise the number of vacancy decreases at the rate ∆V /∆t = −
1
uv
uv
vL = −L βy u+v
. Clearly, the matching function βy u+v
must
α∗ PV L = − βy 1+θ
be the same both for unemployed and vacant firm since that, once a match is
closed, both variables decrease by one unit. Conversely the number of vacancies
increases, in the unit of time, when a separation occurs, i.e. β(1 − v). Finally
the number of workers out of labor force increases by β(1 − v) and decreases by
mR.
4 For computational purposes we normalize the number of elementary units to the size of
the economy.
8
Summing up, we obtain the following dynamical equations (we assume the
normality condition y = 1):
du
dt
dv
dt
dr
dt
1 uv
+ mr
β u+v
1 uv
= −
+ β(1 − v)
β u+v
= −
(5)
= β(1 − v) − mr
By imposing the steady state condition dv
dt = 0 in the second equation we
obtain the long run relationship between vacancy and unemployemnt, i.e. the
Beveridge curve.
v∗ = −
∗
∗ 2
1 u − (1 − u )β −
2
q
(u∗ − β 2 )2 + β 2 u∗ (2u∗ + 2β 2 + β 2 u∗ )
β2
It must be stressed that the determinant of the Jacobian matrix of eqs. 5 is
zero and there are two complex roots with a negative real part while the third is
zero. In such a situation, all solutions converge to a three-dimensional manifold
with the consequence that the steady state, as well as the transition path,
depends on initial conditions (see for example Kamien and Schwartz page 347).
The Beveridge curve is the projection of such a manifold in the bidimensional
plane u,v. Convergence to the steady state can be both clockwise and anticlockwise depending on parameters and initial conditions; finally the evolution
of variables over time follows a stable oscillatory dynamics. Figure 1 shows both
the Beveridge curve and some transitional paths for some initial conditions.
Figure 2 shows the unemployment transition path:
This sort of ”path-dependency” of the system is interesting in the comparison of the working of the job market and equilibrium unemployment across
countries: long run differences are induced not only by a different institutional
setting or search mechanism but by different initial conditions as well.
In the above scheme the parameter m is exogenous; however the US Labour
Statistics survey underlines that individual decisions about the staying in R on
in U depends on individual expectations about the probability of being hired
once the individuals becomes a seeker. Such a probability depends on job market conditions and in general on the tightness of the labour market. The ”discouraged worker” effect is expected to be as higher as unemployment is and
vacancies are low. Even in absence of a clear microfoundation of such an effect,
we have carried out the simulations of figure 1 when we allow m to depend on
job market tightness, in the idea that workers comes out from R at a higher
rate if unemployment is low and/or vacancies are high, i.e. m = g(θ). Figure 3
show results when g(θ) is driven by a concave function:
The comparison of figure 3 with figure 1 does not show relevant changes
in the results; there is still convergence to the Beverdige curve and tranistional
9
Figure 1:
Figure 2:
10
Figure 3:
dynamics are not dramatically different from the case when m is exogenous. We
have performed other simulations, here not reported for brevity, involving more
complex g(θ) but with similar results. In any case working with an exogenous
m rate seems a reasonable assumption that does not modify remarkably the
results.
7
Micro-Dynamic Analysis
In the previous section we have obtained the dynamical equations describing
the behaviour of the averages (macro-variables) over time. In this section we
show the microeconomic behaviour driving the macro one. In other words, we
are going to set-up a microeconomic world whose averages behave exactly as
the equations obtained in the previous section; if both ways of obtaining the
average dynamics produce same results then we conclude that the macro and
micro are coherent to each other. In case of a positive result, we can indifferently
investigate both the macro-world and the life history of a single agent, depending
on our research goal.
Probabilistic approach at modelling our problem involves, theoretically, the
analysis of the Kolmogorov equation for the entire economy, as we did for a
single agent. Unfortunately the dependence of the transition rates on u and v
makes very hard or impossible managing it. This type of problems are common
to other disciplines making use of a probabilistic approach to systems charac-
11
terized by a large number of states. Recently epidemiologists working on the
spreading of diseas across individuals have used a dynamic Monte Carlo algorithm which can be fruitfully applied both to the numerical solutions of the
probabilistic models used to forecast the evolution over time of diseases and to
the estimation of individual transition rates; a good description of these methods is in Gilks et al., 1996. In particular we are going to follow the Monte Carlo
algorithm presented by Aiello et al., 2000 and 2001. For a technical and detailed
description of the algorithm and its probabilistic features the reader should refer to the original papers; here we present the main steps of the methodology
applied to our problem of job search.
The model we are investigating follows these steps:
1. There is an equal number of firms and workers, say L, .and the technology
is 1:1, so full employment is theoretically possible. At the beginning,
workers are divided in three sub-populations: employed M0 , unemployed
looking for a job (seekers hereafter) U0 and unemployed not looking for a
job, R0 . Firms are divided in vacant V0 and filled M0 .
2. Workers and entrepreneurs are completely separated and there is not flow
from one to other; we can assume that this partition of the population is
based on different abilities, managerial skills, business risk aversion and
so on. The population is stationary.
3. When an unemployed worker meets successfully a vacant firm they become
a productive unit labelled by the status M . Matched units can separate
at the exogenous rate β. When separated, workers are temporary out of
labor force and labelled R (retired); they come out from this state as a
new seeker at the exogenous rate m. When separated, the firm becomes
V . By doing so, workers have three possible stages hierarchically ordered,
i.e. U → M → R whilst the firm has only two V → M.
4. As in MP, firms and workers do accept the match hence the latter comes
indifferently either from the supply or demand side of the job market; we
are interested in counting the number of matches and separations over
time, and updating the set {U, M, R, V } consequently, and it does not
matter who is ”driving” the match. There is a strict duality between
workers and firms spaces.
5. Transition rates from U to M and from V to M are given by eqs 4a and
from M to R and M to V by the exogenous separation rate β. Finally the
exogenous parameter m is the transition rate from R to U.
The DMC algorithm allows us to investigate the individual history over
time. As stressed at point 4, it does not matter who is leading the matching
process; what is important is that when a match occurs in the economy, the
sub-populations are adequately updated. Since we are interested in analyzing
the search mechanism on the side of the individual worker, we are going to focus
on this population. Workers are divided into three sub-populations randomly
12
allocated in a one-dimensional vector; workers are hence represented as a random
collection of indexed individuals U, U, M, R, M, U... and so on. The dual vector
representing firms is V, M, M, V, V, M.... and so on.
1. At each elementary time step l one individual is randomly drawn from
the population; let us suppose an unemployed worker. This agent could
change her status from unemployed to matched according to α∗e but from
a probabilistic point of view there can be other agents in the economy
whose probability of changing the status is higher than the one characterizing our individual. So in deciding whether effectively the drawn individual will undertake a status change, we have to compare her transition
rate with the higher one in the population, i.e. pU = α∗e /WMax where
WMax = sup P, .P = {α∗e , β, m} The method suggested in Aiello et al.,
2000, is to compare pU against a random number r in the closed unitary
interval taken from a uniform distribution; the uniform assumption models a situation where every agent has the same a-priori probability of a
jump; this means that when pU < r the real situation is less informative
than the one characterized by a total ignorance about individual transition
rates. Hence, if pU > r the drawn individual changes her status otherwise
she stays in the original one. When the individual changes the status the
sub-populations U, V, M, R are consequently updated.
2. The waiting time between two consecutive jumps follows a Poisson process
at rate r = α∗e U + βM + mR.
3. After the delay time, a new draw is performed and the simulation restarts.
The result of this Markov Chain5 is the convergence of the sub-populations to
a long run probability measure whose first moment is given by the deterministic
equations 5. In the following section we show some simulations.
8
Results
We have performed 20 replications of the Monte Carlo method described in
the previous section; we have chosen 20 runs because it is enough to obtain a
smooth curve to be compared to the deterministic solution. Figure 2 shows the
adjustment path obtained by the average equations, rotated in the v, u plane,
and the average of 20 simulations obtained by the micro-model. As the reader
can see, the fit is quite good.
By doing so, we can describe the adjustment path to the Beveridge curve
indifferently by the macro-equations and the microeconomic model; both analytical approach provides same results. Under this point of view, the macro
model is well founded in the microstructure. The probabilistic law characterizeing the micro-behaviour is working properly: it does provide same dynamic
path of the deterministic rules driving the first moment of the distributions.
5 The dual DMC is the one where we analyze firms instead of workers but the two chains
lead obviously to same results.
13
Figure 4:
14
9
Local Effects and Individual Behaviour
Once built a suitable analytical tool that allows us to obtain average dynamics
from the micro-structure, it can be used in a very flexibile way to modify the
individual behaviour. In this section we show how we can account for local
interactions in the search mechanism. It is not hard to assume that matching can depends not only on macro-variables (the so-called field effect) such as
total unemployment and vacancy in the economy but on ”local” variables as
well. Local interactions, or neighborhood effects, are often invoked in the human capital literature for explaining ”club convergence” and inter-generational
inequality persistence. We are going to assume a rather simple local effect; individual matching rates depends both on aggregate variables and on the number
of workers individual in a neighbor of the unemployed individual. The higher
the number of working agent in a surrounding of the unemployed worker, the
higher the matching rate. We can think to a sort of positive spillover or complementarity effect. Let us define p as the probability that an unemployed worker
becomes working due to the presence of a worker in the neighbor and so (1 −p)n
is the probability, per unit of time, for a unemployed not to become working
if she has n working neighbors hence (1 − (1 − p)n ) is the probability, per unit
of time, for a unemployed to be hired if she has n working neighbors.;When
n is large the local term tends to 1 whilst when n = 0 the local term is zero
(no-spillover); an increase in n makes higher the transition rate from U to M .
The individual transitions rates modify in:
µ
¶
1 θ
U → M : WU = Γ
+ Λ(1 − (1 − p)n )
β 1+θ
where Γ + Λ = 1 in order to balance macro and local effects (see again Aiello et
al. 2000).
The effect of this neighbor effect is summarized in figure 3; this shows the
adjustment path without local variables and the one risulting from the introduction of the spillover term.
The new path is righward shifted w.r.t. the baseline; this means that the
Beveridge curve, representing the steady state of the path, can shift in the UV
plane because of externalities effect in the individual behaviour, a novel result
respect to the current literature. Moreover, for a given Γ and Λ, such a sifht is
as higher as n is lower. By varying n, the beveridge curve undertake either shift
in the plane according to the empirical findings.
It is worth stressing that this type of local interaction can not be analyzed by
the standard search model nor by the aggregate dynamical equations; under this
point of view, this analytical tool represents an improvement over the standard
tecnique.
15
Figure 5:
10
Conclusions
This first attempt at modelling job search at individual level has interesting potentialities that the macro-dynamics can not capture. The debate on investment
in human capital and the effects on job search and quitting probabilities can
only be faced by analyzing the single worker; in particular, neighborhood effects
can have important effects on the dynamics of the job market. The approach
we are developing fits particularly well in coping with these spillover effects.
Individual transition rates can depend on the social and economic neighbor of
the individual and in general there is heterogeneity in such rates; this brings to
multiple equilibria and to dynamical path which are quite different by the ones
we can describe by the aggregate equations. In a more mature stage of our work
we will exploit such interesting opportunity.
References
[1] Aiello O.E., Haas V.J., da Silva M.A., Caliri A., 2000, Solution
of deterministic-stochastic epidemic models by dynamical Monte Carlo
method, Physica A, 282, 546-558
[2] Aiello O.E., M.A.A. da Silva, 2003, New approach to Dynamical
Monte Carlo Methods: application to an Epidemic Model, unpublished
manuscript.
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[3] Bailey, N.T., 1964, The elements of Stochastic Processes with applications
to the natural sciences, John Wiley & Sons, London
[4] Castillo, 1998, Monthly Labor Review, US Bureau of Labor Statistics.
[5] Cox, D.R, Miller H.D., 1965, The Theory of Stochastic Processes, London:
Champman and Hall
[6] Gilks W.R., Richardson S., Spiegelhalter C.J. (Eds.) 1996, Markov Chain
Monte Carlo in practice, London: Chapman and Hall
[7] Nickell S., Nunziata L., Ochel W., Quintini G., 2002, The Beveridge Curve,
Unemployment and Wages in the OECD from the 1960s to the 1990s. Centre for Economic Performance, LSE.
[8] O’Neill P., 1996, Strong approximations for some open population epidemic
models, Journal of Applied Probability, 33, 448-457
[9] Petrongolo B., Pissarides C.A., 2000, Looking into the black box: a survey of the matching function, Center for Economic Performance discussion
paper .
[10] Wall, H.J., Zoega G., 2002, The British Beveridge Curve: A Tale of Ten
Regions. w.p. Federal Reserve Bank of St.Louis 2001-007B.
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