Labor Market Dynamics in Continuous Time
An Outline of Burdett and Mortensen
Labor Supply under Uncertainty
and Some Extensions
C. Flinn
May 2001
Partial Revision January 2006
Model Assumptions and Structure
An individual can occupy the following labor market states:
1. u Unemployment - that is, without a job, actively seeking work, and not subject to
recall from a formerly held job
2. e(w) Employment at wage w. In this case the individual is attached to some firm
that pays her a wage of w.
3. d(w) Laid-off from a job that paid a wage of w. In this case the individual was
“temporarily dismissed” from a job that had paid her a wage of w.
4. o Out of the labor force (OLF). In this case the individual is not actively searching
for work, is not on layoff from a job from which she is subject to the possibility
of recall, and is simply at home “enjoying leisure.” In the Burdett and Mortensen
framework there is really no need to distinguish between u and o, since o can be
determined endogenously as a special case of u for which the optimal search intensity
choice is 0. In this case the rate of leaving the u state will be 0.
BM begin by constructing a simple stationary stochastic process that can mechanically
describe movements between these states. They go on to provide a behavioral rationale
for all the movements using a stationary search framework. We will attempt to mix the
two steps to some extent in these notes.
In terms of the most important assumptions:
• They assume that wage offers are i.i.d. draws from some fixed discrete distribution
F (wj ), where the support points of the distribution are given by 0 < w1 < . . . <
wN < ∞. Every time an individual encounters a new firm, a potential wage is drawn
from this distribution. The distribution is fully known by all individuals when they
enter the labor market (i.e., no learning).
• Any individual who searches must decide a rate of search intensity, denoted by s.
Then the rate at which the individual encounters new firms is given by λs. Searchers
may be currently employed, unemployed, or laid off.
• All employed individuals are separated from their current employers at a rate η.
This separation rate applies both to individuals who are actively employed and
those on layoff from the firm.
• All individuals actively employed at any firm are subject to a risk of layoff. Layoff
occurs at rate ζ.
• All laid off individuals are subject to a recall. Recall occurs at rate ν.
• Individuals have a contemporaneous utility function given by
U(c, l),
where c is “instantaneous” consumption and l is instantaneous leisure consumption.
• The individual has a nonlabor income flow of y > 0. There exist no capital markets,
so that instantaneous consumption is equal to instantaneous income.
• The total amount of “time” in any instant is normalized to 1. In terms of instantaneous time allocation, the individual can allocate her time between:
— Leisure
— Labor Supply
— Search
Let s be search time, and l leisure, so that labor supply is h = 1 − l − s.
• The intertemporal discount rate is ρ > 0.
Structure of the Optimization Problem
The problem is stationary, which considerably simplifies solving it. Nevertheless,
dynamic programming (DP) problems almost always must be solved numerically, and
this problem is no exception. Let us consider a generic way to approach continuous time,
discrete state space optimization problems.
Begin by setting up the Bellman equation as though we were solving a discrete time
problem. Let ε denote the length of a “time period.” The state of world is the individual’s
current labor market status. Let’s first consider the problem faced by an individual who
is currently unemployed. The Bellman equation is given by
−1
Vn (y) = max(1 + ρε) {U(y, 1 − s)ε + λsε
s
+(1 − λsε)Vn (y) + o(ε)}.
N
X
pi max[Vn (y), Ve (w; y)]
(1)
i=1
The interpretation of this expression is as follows. ε must be considered to be a relatively
small period of time, over which (approximately) nothing changes. The unemployed
individual will consume all of their income over this period, and at every instant their
income is y. In addition, since the individual does not have a job, their time is spent
either in job market search (s) or in leisure (l). Then their “instantaneous” utility level
is U(y, 1 − s), and they will attain this constant utility level for a period of time equal to
ε - so that the total payoff at the end of ε is U(y, 1 − s)ε. This payoff must be discounted
back to the beginning of the “period,” which is now, and that is the role played by the
“instantaneous” discount factor
1
.
1 + ρε
Therefore the term
1
U(y, 1 − s)ε
1 + ρε
is the discounted value of the current period return over the short interval ε.
In a stationary model like this one in which the state space is discrete, all random
events are described by separate Poisson processes. By the structure of such point process
models, in any small interval of time (like ε), at most one of these events can occur.
When the individual is unemployed, and is supplying a positive amount of time to search
activities (i.e., s > 0), then at most one event can occur in a short period of time and
that is for her to encounter a potential employer. The “instantaneous” probability of this
event is given by λsε, and by the laws of probability the probability of nothing occurring
must be equal to 1 − λsε.
Given that a new firm is encountered, a draw from the distribution F is made. The
individual can choose to reject the offer, in which case the value of the encounter was
equal to the value of continuing in the unemployment state. Conversely, if the offer is
attractive enough the individual can accept it, in which case she enters the employment
state at the accepted wage w. Note that if no potential employer is encountered, the value
of the state remains the same, which is Vn (y). This is the simplicity of a stationary model
- because no changes in the environment take place, an individual using stationary policy
rules will continue to make the same choices whenever the state variables describing the
environment remain the same. At the “end” of the period of length ε, if the individual
receives no acceptable offer of employment we know that she will continue to make the
same decisions (regarding s in this case) as she did at the beginning of the period.
Finally, note that the term o(ε) is defined as
o(ε)
.
ε→0 ε
lim
Whenever ε is finite, there is in principle a small probability that two or more events
could occur with time ε. In this sense, our expression for [1] is an approximation and is
only strictly accurate as we shrink ε to 0.
We can now rewrite [1] in a slightly more useful way. Mulitplying through by (1 + ρε),
we get
(1 + ρε)Vn (y) = U(y, 1 −
s∗u (y))ε
+
λs∗u (y)ε
N
X
pi max[Vn (y), Ve (w; y)]
i=1
+(1 − λs∗u (y)ε)Vn (y) + o(ε),
where we let s∗u (y) denote the solution to the optimization problem for an unemployed
individual with nonlabor income equal to y. Since max[a, b] = a + max[0, b − a], we can
rewrite this expression as
(1 + ρε)Vu (y) = U(y, 1 −
s∗u (y))ε
+
λs∗u (y)ε
i=1
+Vu (y) + o(ε)
⇒ ρVu (y) = U(y, 1 −
N
X
s∗u (y))
+
λs∗u (y)
= U(y, 1 − s∗u (y)) + λs∗u (y)
N
X
i=1
N
X
i=1
pi max[0, Ve (w; y) − Vu (y)]
o(ε)
ε→0 ε
pi max[0, Ve (w; y) − Vu (y)] + lim
pi max[0, Ve (w; y) − Vu (y)].
Given knowledge of the function Ve (w, y), this expression in practice is used to simultaneously solve for the optimal search intensity in this state, s∗u (y), and the value of the
unemployment state to an individual of “type” y.
The value of being in the employment state characterized by (w, y) is only slightly
more complicated to write down. Using the “ε look ahead” formulation of the DP problem, we write
Ve (w; y) = max(1 + ρε)−1 {U(y + w(1 − s − l), l)ε + λsε
s,l
N
X
pi max[Ve (w̃, y), Ve (w; y)]
i=1
+ηεVu (y) + ζεVd (w, y) + (1 − λsε − ηε − ζε)Ve (w, y) + o(ε)},
where we now have allowed for the possibility that the individual be permanently separated from his job (at rate η) or laid off (at rate ζ). The term w̃ is the alternative
wage offer received given that the individual encounters another potential employer. The
choice of whether to accept a new job is extremely simple in this case, since the only
characteristic of a job that yields utility is the wage rate. Therefore, the individual will
change employers whenever she encounters a higher wage offer. Rearranging terms and
passing to the limit, we have
(ρ + η + ζ)Ve (w; y) = U (y + w(1 − s∗e (w; y) − l∗ (w; y)), l∗ (w; y))
N
X
pi max[0, Ve (w̃; y) − Ve (w, y)] + ηVn (y) + ζVd (w; y).
+λs∗e (w; y)
i=1
This equation includes the optimal choice of leisure consumption and search intensity
given the wage w and nonlabor income flow y. Because the individual only leaves her
current employer for others paying a higher wage, we can rewrite the equation as
(ρ + η + ζ)Ve (w; y) = U(y + w(1 − s∗ (w; y) − l∗ (w; y)), l∗ (w; y))
X
pi [Ve (wi ; y) − Ve (w, y)] + ηVu (y) + ζVd (w; y).
+λs∗ (w; y)
{i:wi >w}
Given solutions for Vu (y) and Vd (w; y), this equation simultaneously determines the functions Ve (w; y) and the optimal choices s∗e (w; y) and l∗ (w; y).
Finally, the value of being laid-off at a wage w is given by
−1
Vd (w; y) = max(1 + ρε) {U(y, 1 − s)ε + λsε
s
N
X
pi max[Ve (w̃, y), Vd (w; y)]
i=1
+νεVe (w; y) + ηεVu (y) + (1 − λsε − νε − ηε)Vd (w, y) + o(ε)},
with a “limiting” representation of
(ρ + ν + η)Vd (w; y) = U(y, 1 − s∗d (w; y)) + λs∗d (w; y)
+νVe (w; y) + ηVu (y).
X
{i:Ve (wi ;y)>Vd (w;y)}
pi [Ve (w̃i , y) − Vd (w; y)]
This completes the specification of the model. All behavior and equilibrium outcomes are
determined by Vu (y), Ve (w; y), and Vd (w; y). For a given y, there are 2N + 1 equations to
solve for. For any given set of solutions, the optimal choices of the individual in terms of
s∗ and l∗ must be consistent with them.
Transition Rates between States
We now consider the transition rates between states, which with a stationary framework is relatively straightforward to do. To begin, consider the rate of leaving the state
of being employed at a wage w. Thus we are interested in the transition rate out of the
state (e, w).
An individual can leave this state three distinct ways.
1. She can be dismissed, which occurs at rate η.
2. She can be laid off, which occurs at rate ν.
3. She can find a better job. The rate at which she meets other potential employers
is given by
λs∗ (w; y).
Given that she meets another potential employer, she will leave if and only if she
receives a better job offer. The probability of this event is
X
F̃ (w) =
pi .
{i:wi >w}
Therefore the exit rate out of this job into another job spell is
λs∗ (w; y)F̃ (w).
The “total hazard rate” out of the state (e, w) is simply the sum of these three
“destination specific” hazards, or
h(e, w) = η + ν + λs∗ (w; y)F̃ (w).
Note that this hazard rate is partially determined by model assumptions and partially
determined by behavior - due to the endogeneity of s∗ and the choice rule in terms of
which wage offers to accept.
We will now consider the hazard function out of the layoff state (d, w), which will
give us a chance to discuss the behavioral rule used by individuals in the layoff state. We
provided a verbal argument as to the nature of the turnover decision used by individuals
currently employed at a job paying a wage w. In this case, any job paying a wage w̃ > w
would be accepted. The situation is different for individuals currently on layoff from a job
that paid a wage of w. This state is in fact some kind of combination of the employment
state and the unemployment state. Clearly, when the recall rate ν is equal to 0, the value
of being in this state is exactly the same as is the value of being unemployed (convince
yourself of this fact). By the same token, when the rate of recall is extremely high the
value of this state can become arbitrarily close to that of someone currently employed at
the same wage.
When ν assumes intermediate values, however, clearly the value of this state is strictly
bounded by Vu (y) < Vd (w; y) < Ve (w, y), where it is presumed that the wage w belongs
to the set of wages that are acceptable. Clearly, whenever the individual is in the layoff
state from a job that pays w, if she were to encounter a new potential employer that paid
the same wage, she would accept the job. Moreover, depending on the parameter values
characterizing the environment, she may well be willing to accept wages lower than w
in compensation for having an “active” job. Let wd∗ (w, y) denote the lowest wage she
would accept given layoff from a job paying w and nonlabor income flow y. We can now
compute the exit rate out of the state (d, w). The individual can leave this state for the
following reasons:
1. She can be permanently dismissed from the job, which occurs at rate η.
2. She can be recalled to her job, which occurs at rate ζ.
3. She can find a “better” job, though in this case it need not offer a higher wage.
The rate at which this occurs is
λs∗d (w; y)F̃ (wd∗ (w; y)).
Thus, the rate at which individuals exit the state (d, w) is equal to
h(d, w) = η + ζ + λs∗d (w; y)F̃ (wd∗ (w; y)).
This exercise can be repeated for all of the states of the model. The results are
summarized in the following table.
State
Destination
Hazard Rate
u
(e, w)
(d, w)
−
−
−
λs∗u (y)F̃ (wu∗ (y))
η + ν + λs∗ (w; y)F̃ (w)
η + ζ + λs∗d (w; y)F̃ (wd∗ (w; y))
u
(e, w)
(e, w)
u
(d, w)
(e, w̃)
(d, w)
u
(d, w̃)
(e, w̃)
λs∗u (y)p(w); w ≥ wu∗ (y)
0; w < wu∗ (y)
η
ν
λs∗e (w; y)p(w̃); w̃ > w
0; w̃ < w
η
0
∗
λsd (w; y)p(w̃); w̃ ≥ wd∗ (w; y)
0; w̃ < wd∗ (w; y)
Burdett and Mortensen are able to derive a few comparative statics results from this
relatively general model. Albrecht, Axel, and ? are able to derive many more using an
insightful method of differentiating policy functions in stationary DP problems. For our
present purposes, we will simply describe a few of their results.
One particularly interesting comparison is between the labor market dynamics of the
same individual (i.e., y constant) when she is employed at a job paying w and when she
is laid off from such a job. When she is employed, she has an opportunity cost of search
that is not only composed of the value of foregone leisure, but also includes foregone
consumption. This is the case because when employed she has the option of selling her
time for increased consumption at the rate w. When she is laid off, the opportunity cost
of search is lower and therefore she will engage in more. Therefore s∗e (w; y) < s∗d (w; y).
In addition, we can establish that wd∗ (w; y) ≤ w, so that the rate of exit into another
job when the individual is on layoff from a job paying w is higher than when the individual
is employed at a job paying w since
λs∗d (w; y)F̃ (wd∗ (w; y)) > λs∗e (w; y)F̃ (w+ ),
where w+ denotes the next higher wage than w. In terms of the total rate of exit from the
states (e, w) and (d, w), this would be ambiguous because of the existence of the different
rates ν and ζ in the two “total” hazard functions. If it so happened that ν = ζ, then the
total hazard out of layoff at w would be greater than the total hazard out of employment
at w.
Heterogeneity
Though it has been awkward at times, we have tried to consistently carry through
the argument y in the previous discussion. The reason is that y plays a prominent role
in standard neoclassical analyses of labor supply behavior (typically in a static context),
and because it is reasonable to expect that the is a large degree of variation in y in
virtually any large population. It is the heterogeneity in y that is most important for our
discussion.
A key characteristic of stationary point process models is that the duration between
events follows an exponential distribution. In our previous discussion, all hazard rates
were constant, i.e., no matter how long the individual has occupied a given state, the
rate of leaving is the same. In this case, we say that the dynamics of the model are
independent of the history and only depend on the contemporaneous values of the state
variables. It is well known that the only distribution that has associated hazard functions
that are history independent is the (negative) exponential. If a random variable X has a
negative exponential distribution, then
FX (x) = 1 − exp(−αx), x ≥ 0
fX (x) = α exp(−αx), x ≥ 0
hX (x) = α, x ≥ 0,
for some α > 0. Then the parameter α is the hazard rate associated with the distribution,
and the mean of the random variable X is equal to 1/α.
If one finds that duration data are not adequately characterized by a negative exponential distribution, does it necessarily follow that the framework developed by Burdett
and Mortensen (and others) is a serious misspecification of the labor market behavior of
a population of individuals? The answer to this question depends on the extent to which
we have conditioned on individual differences in the population, as well as the nature of
the departure from the negative exponential distribution.
To begin, let’s consider the conditioning argument. For the purposes of argument, let
us assume that there are two types of agents in the population in terms of the characteristic Y. Rich individuals have a value of Y = y2 , while poor individuals have a nonlabor
income flow of y1 , with y2 >> y1 . Denote the proportion of the population that is rich
by π.
If an individual’s type is perfectly observable by us, and there are only two types,
than it may be perfectly feasible to conduct essentially two separate analyses of labor
market behavior within each subpopulation. Within each subpopulation, all durations
would follow a negative exponential distribution and departures from this distributional
form would be evidence against the model (for the particular subpopulation in question).
We might term this type of analysis a fully conditional one, in the sense that within each
subpopulation all agents are considered to be homogeneous.
More commonly it is the case that an individual’s type is not perfectly observable.
While the analyst may have some information regarding an individual’s sources and
amounts of nonlabor income, survey methodologists have shown that such information
is often highly unreliable. For purposes of this example we will assume that we have no
information regarding wealth, though the analysis is easily extended to allow for us to
have access to imperfect measures of this characteristic.
Consider the exit rate from unemployment for individuals in the two wealth groups.
For a rich individual the hazard rate is
hu (y2 ) = λs∗u (y2 )F̃ (wu∗ (y2 )),
while for a poor individual
hu (y1 ) = λs∗u (y1 )F̃ (wu∗ (y1 )).
All members of the cohort are assumed to enter unemployment at the same time.
The hazard function for the aggregate is often referred to as the “marginal” or “empirical” hazard - by this we signify that it is an unconditional hazard function, since it
is not possible to explicitly condition on values that are unobservable. We derive this
hazard function as follows. The hazard function associated with the random variable X
is defined as
fX (x)
hX (x) =
.
F̃X (x)
In the case of a negative exponential, we know that
fX (x) = hX exp(−hX x).
The marginal density of unemployment durations is given by
fT (t) = fT |Y =y1 (t) × (1 − π) + fT |Y =y2 (t) × π
= (1 − π)hu (y1 ) exp(−hu (y1 )t) + πhu (y2 ) exp(−hu (y2 )t).
Given that we have access to the marginal density, it is straightforward to define the
marginal survivor function, which is given by
Z
F̃T (t) = fT (u)du.
t
Then the marginal hazard rate is
hu (t) =
fT (t)
.
F̃T (t)
It is not difficult to show that hu (t) has the property that dhu (t)/dt < 0.1 Moreover, this
is true no matter what the form of the “mixing distribution,” which in our case is the
1
The definitie reference on this topic is Barlow and Prsochan, Statistical Theory of Reliability and
Life Testing: Probability Models (1975, reprinted 1981). See the sections on mixtures of hazards.
distribution of Y. This negative duration dependence arises solely from the composition
effect.
The accompanying figure illustrates this phenomenon. In it, we have calculated the
marginal hazard rate for a situation in which the population is evenly divided (i.e., π = .5)
between two subpopulations. In one, the constant hazard rate is equal to .5, and for the
other it is equal to .1. At the initiation of the spell, there has been no selection, so
that the hazard rate is equal to the average of the two (.3). However, as time goes on
the individuals with high hazard rates are increasingly underrepresented in the group of
individuals who have not left the state. The marginal hazard has a relatively simple form
in this discrete heterogeneity case. The marginal density is
fT (t) = πα1 exp(−α1 t) + (1 − π)α2 exp(−α2 t).
From this, the survivor function is
F̃T (t) = π exp(−α1 t) + (1 − π) exp(−α2 t),
so that the empirical hazard is
hT (t) =
πα1 exp(−α1 t) + (1 − π)α2 exp(−α2 t)
π exp(−α1 t) + (1 − π) exp(−α2 t).
You can show that for any π ∈ (0, 1), α1 > 0, α2 > 0, then h0T (t) < 0, t > 0. Asymptotically, the group of individuals who have not left the state will be comprised only of those
with low hazard rates. As we can see from the Figure, the marginal hazard is indeed
asymptoting to the value .1.
As a result, the “trick” of allowing for unobserved heterogeniety (although in this case
the source of the heterogeneity, y, is potentially observable) is often used to simplify the
solution of continuous time, DP problems when one wants to get around the implication
of no duration dependence. Note however there are strong implications of the form of
duration dependence when constant hazards are mixed. It is not difficult to show that
any mixed constant hazard model yields a non-increasing hazard function - this is true
no matter what the form of the mixing distribution. Thus the mixtures technique is fine
so long as one observes nonpositive duration dependence of the process everywhere on
R+ . If there are any subsets of R+ for which h0u (t) > 0, this is not consistent with mixed
constant hazards, and so you have to go back to the drawing board.