Labor Queues
Michael Sattinger
Department of Economics
University at Albany
Albany, NY 12222, USA
Email [email protected]
Phone: (518) 442-4761
Keywords: Queueing, Search, Unemployment
April 2, 2005
Abstract
This paper applies queueing theory to derive the equilibrium of a
labor market with frictions. Queueing arises when workers can get jobs
by waiting at a firm for a position to open. Queueing theory provides
expressions for the expected numbers of vacancies, searching workers
and queueing workers. As the ratio of workers to firms increases,
unemployed workers shift from searching for vacancies to waiting in
queues.
1
Introduction
This paper applies queueing theory to the analysis of labor markets. Queueing in a labor market occurs when a worker can get a job by waiting at a firm
for a position to become available. Allowing queues benefits firms because
labor queues reduce firm vacancies. Unemployed workers can benefit from
choosing to queue by reducing the expected time until employment. Queueing theory provides labor economics with an analytic tool to calculate the
equilibrium of a labor market with frictions.
1
A.K. Erlang developed queueing theory to analyze the operation of telephone exchanges. Queueing theory continues to be an important subject of
operations research and management science.1 In a simple queueing model,
customers arrive randomly at a single server, and wait in queue and for their
order to be processed. For particular assumptions regarding arrival, queue
discipline and service times, queueing theory calculates the proportions of
time the system is in each state, characterized by the number of customers
in the queue. Queueing models can also be developed in a market context to
analyze equilibrium and demand for services.2
Queueing theory can be used to model a labor market with frictions as
follows.3 Workers leave firms at a fixed rate. Firms can allow workers to
enter a queue to take the next available job at the firm. If there are workers
in the queue, a firm can avoid vacancies. Otherwise, with no workers in the
queue, a firm would experience vacancies when workers leave. Unemployed
workers search among firms either for a vacancy or for a queue that is not
too long. With fairly straightforward assumptions, it is possible to obtain
analytical expressions for the expected numbers of firm vacancies, searching
workers and queueing workers for a given aggregate ratio of workers to firms.
Search theory provides a commonly accepted explanation for the operation of a labor market with frictions.4 On the basis of an arbitrary matching
function, search theory generates a Beveridge Curve relating unemployment
to vacancies. Queueing theory does not displace search theory. It can be incorporated into a model of labor market behavior together with search theory.
A labor market model with both queueing and searching can generate the
Beveridge Curve without assuming a matching function.
Additionally, queueing theory modifies the results of a labor market with
only search behavior. At low ratios of workers to firms, the labor market
1
See Cox and Smith, 1961; Gnedenko and Kovalenko, 1989; and Gross and Harris,
1985, for standard treatments of queueing theory. Basic treatment of queueing theory
can also be found in texts on probability and stochastic processes such as Feller, 1957;
Karlin, 1966; Karlin and Taylor, 1981; Saaty, 1961; and Taylor and Karlin, 1984. Hassin
and Haviv, 2003, present recent developments in queueing theory including applications
of game theory.
2
See Allon and Federgruen, 2004; Knudsen, 1972; Leeman, 1964; Naor, 1969; Sattinger,
2002; and Stahl, 1987, for applications of queueing in market contexts.
3
Shimer, 2001, has applied elements of queueing theory in a model of coordination
frictions in assignment.
4
Pissarides, 2000, Mortensen and Pissarides, 1999, and Rogerson, Shimer and Wright,
2004, provide surveys of search theory.
2
operates in the same manner whether or not queueing occurs, since vacancies
would be common and queues would be rare. At high ratios of workers
per firm (and with high unemployment), most unemployed workers would
be waiting in queues, and vacancies would be substantially reduced by the
presence of queues. Most workers would be searching for queues to join rather
than vacancies.
The next section develops the queueing theory of labor market equilibrium. This includes determination of the proportion of time firms have vacancies and queues, the maximum queue length, the proportion of worker
times spent employed, searching and queueing, the firm arrival rate given
the ratio of workers to firms, and the wage rate. Section 3 describes the
operation of the labor market. In particular, the operation changes qualitatively as the ratio of workers to firms increases, shifting from searching to
queueing. Section 4 compares labor markets with and without queueing in a
model compatible with both. Section 5 summarizes conclusions and describes
how queueing workers would appear in the Current Population Survey.
2
2.1
Model
Overview and Definitions
Production at a firm is proportional to employment up to a maximum n,
the same for all firms. A firm employing j workers has n − j vacancies.
Unemployed workers may contact the firm to see if there is a vacancy and
can also decide to enter a queue to wait for a job if the queue is not too long.
Suppose workers in a queue are hired in the order that they enter the queue
(i.e., the queue discipline is First In First Out). Define the balk level q as the
maximum length of a queue, so that an unemployed worker would decline
to enter a queue of length q or longer (q will be determined endogenously
in Section 2.3). Let α be the arrival rate of unemployed workers at a firm
looking for vacancies or a queue shorter than q. Let γ be the rate at which
employed workers depart jobs. Assume γ is the same for all workers and
firms. Let p−i be the (long run) proportion of time a firm has i vacancies,
i = 1, n, and let pi be the proportion of time a firm has i unemployed workers
queueing for a job, i = 1, q. Let p0 be the proportion of time a firm has exactly
n workers. The proportions pi , i = −n, ...q, are determined by the arrival
and departure rates α and γ and are derived in Section 2.2. At higher arrival
3
rates α, firms are less likely to have vacancies and more unemployed workers
are in queues.
Workers optimally choose the balk level q by comparing the expected time
until they get a job if there are q −1 workers in queue ahead of them with the
expected time if they continue searching. Let VE , VS and VQ be the expected
asset values for a worker entering the states of employment, searching and
queueing, respectively. These asset values are derived in Section 2.6. An
individual firm takes the asset value of searching, VS , as given. An individual
firm can choose the wage rate it pays but then the arrival rate at the firm
will adjust to a level such that the value of searching at the firm is the same
as searching in the rest of the labor market, given by VS , with balk level q
adjusting. The firm optimally chooses the wage subject to the constraint on
the arrival rate. It is then possible to determine the symmetric equilibrium
of a labor market consisting of identical workers and firms.
2.2
Steady-State Firm Proportions of Time in Each
State
Queueing theory provides formulae for the long run proportions of time a
firm is in each possible state. Let −i be the firm’s state when there are
i vacancies, i = 1, n, let i be the state when there are i workers in the
queue, and let 0 be the state when the firm has n workers with no workers
in the queue. Following standard procedures in queueing theory, proportions
of time in each state can be found from transition rates into and out of
particular states. Beginning with the state of no workers (state −n), the
average transition rates for a firm into and out of the state must be equal:
γp1−n = αp−n
(1)
The average transition rate of firms into state −n equals the departure rate
γ times the proportion of time the firm is in state 1 − n.
The average transition rate of workers out of state −n equals the arrival
rate of workers α times the proportion of time the firm is in state −n. The
average transition rate into state 1 − n arises from worker departures when
the firm is in state 2−n and arrivals when the firm is in state −n. The average
transition rate out of state 1 − n arises from worker arrivals or departures.
4
Setting the flows equal yields
2γp2−n + αp−n = αp1−n + γp1−n
(2)
Similarly, the average transition rates into and out of state j − n, with n − j
vacancies, are
(j + 1)γpj+1−n + αpj−1−n = αpj−n + jγpj−n
(3)
The firm moves into state j − n from state j + 1 − n when any of the j + 1
workers depart, or from state j − 1 − n when a worker arrives. The firm
moves out of state j − n when a worker arrives or when any of the j workers
departs. In state −1, with one vacancy, the average transition rates are
nγp0 + αp−2 = αp−1 + (n − 1)γp−1
(4)
When the firm has workers in the queue, n workers are employed. Then
the average transition rates in states 0 and 1 are
nγp1 + αp−1 = αp0 + nγp0
(5)
nγp2 + αp0 = αp1 + nγp1
(6)
nγpj+1 + αpj−1 = αpj + nγpj
(7)
αpq−1 = nγpq
(8)
In state j > 0,
and in state q
The equations 1 through 8 can be solved for the proportions pi , i = −n, q.
First, the proportions of time that the firm has vacancies can be expressed
in terms of p−n . From 1,
α
p1−n = p−n
(9)
γ
Combining 1 with 2 yields
2γp2−n + αp−n + γp1−n = αp1−n + γp1−n + αp−n
5
(10)
Canceling out terms yields
p2−n
1α
1
=
p1−n =
2γ
2
µ ¶2
α
p−n
γ
(11)
Similarly, combining equations for successive numbers of vacancies yields
µ ¶j+1
α
1 α
1
pj+1−n =
pj−n =
p−n
(12)
j+1γ
(j + 1)! γ
From 4
1α
1
p−1 =
p0 =
nγ
n!
µ ¶n
α
p−n
γ
(13)
In the second step, the proportions of time with workers in the queue can
be expressed in terms of pq . Beginning with 8,
γ
pq−1 = n pq
α
Then at successively lower numbers of workers in the queue,
³ γ ´q−j
γ
pj = n pj+1 = nq−j
pq
α
α
At j = 1,
q
(14)
(15)
³ γ ´q
pq
(16)
α
Now solve 9 through 16 to yield the proportions of time in terms of p0 ,
the proportion of time with no vacancies or workers in the queue.
p0 = n
n! ³ γ ´j
p0 , j = 1, n
(n − j)! α
µ ¶j
α
−j
= n
p0 , j = 1, q
γ
p−j =
pj
The sum of all the proportions of time must add up to 1, so that
à n
µ ¶j !−1
q
X n! ³ γ ´j
X
α
p0 =
+1+
n−j
(n − j)! α
γ
j=1
j=1
6
(17)
(18)
(19)
An immediate consequence of 17 through 19 is that it is possible to calculate
the proportions of time that the firm has vacancies or workers in the queue.
The effects of variables on these proportions follow directly.
Theorem 1 The proportion of time the firm has vacancies decreases and the
proportion of time the firm has workers in the queue increases as the arrival
rate α increases, the departure rate γ decreases, or the balk level q increases.
2.3
Balk Level q
Workers are free to balk at joining a queue to get a job. If wage and arrival
rates were the same for each firm, a worker would balk at joining a queue if
the expected wait time until employed would be greater than if the worker
continued searching for a vacancy or a shorter queue. The comparison generates a condition that determines the optimal balk level for workers in a
symmetric equilibrium.
Consider first the expected wait until getting a job when there are j
workers currently in the queue and the balk level among workers is q, j < q.
The rate at which the firm experiences the loss of a worker is γn. If the
worker joins the queue, j + 1 workers must leave the firm before the worker
gets a job. The probability density function for the waiting time until j + 1
events is
γn (γnt)j e−γnt
fj [t] =
(20)
j!
The expected time until the worker gets a job, if there are already j workers
in the queue, is
j+1
(21)
γn
If there are q − 1 workers in the queue, and the worker joins the queue, the
expected wait would then be q/(γn).
Now consider the alternative that arises if the worker continues searching. Suppose the worker meets firms at the rate m, a constant that does
not depend on the number of workers searching.5 In the next section, an
aggregate condition on the labor market relates m to the firm arrival rate α.
5
As in search theory, it would be possible to incorporate a matching function relating
the rate of meetings between workers and firms to the ratio of searching workers to firms.
At this stage in the development of labor queues, the simpler assumption that m is constant
is adopted.
7
The worker gets a job immediately if the firm has a vacancy, joins the queue
if the length is less than the worker’s balk level, and continues searching if
the queue length equals or exceeds the worker’s balk level. In a symmetric
equilibrium, all workers would have the same balk level and all firms would
have the same proportions of time with vacancies and queues. The expected
proportion of time a firm is in a particular state will equal the expected proportion of all firms in that state at a particular point of time. If the balk
level for all workers and firms is q, the expected wait time until employment,
conditional on a firm having a queue to enter, is
Pq−1
j=0 pj (j + 1)/(γn)
(22)
Pq−1
j=0 pj
The expected time until employment for a searching worker, ES [q], satisfies
the implicit condition generated by the alternatives facing a searching worker:
Pq−1
Xn
Xq−1
1
j=0 pj (j + 1)/(γn)
+0
ES [q] =
p−j +
pj
+ pq ES [q] (23)
Pq−1
j=1
j=0
m
j=0 pj
Solving for Es [q] yields
1
Es [q] =
1 − pq
µ
¶
1 Xq−1
+
pj (j + 1)/(γn)
j=0
m
(24)
Now consider whether q is a balk level that could prevail for all workers
in equilibrium. If
q+1
q
≤ ES [q] ≤
(25)
γn
γn
then workers facing queue length q − 1 or less would join the queue. Additionally, workers would balk at joining a queue of length q. Then q would
be an equilibrium balk level. Suppose instead that (q + 1)/(γn) is also less
than ES [q]. Then consider q + 1 as the possible equilibrium balk level. In
recalculating the expected wait time from searching for balk level q + 1,
ES [q + 1] increases since the firm proportion of time with a queue increases.
Then (q + 1)/(γn) will also be less than the recalculated expected wait time
from searching, ES [q + 1], and the equilibrium balk level would be q + 1 or
greater. Considering sequentially higher balk levels, a balk level consistent
with symmetric equilibrium arises for the first value such that 25 holds.
8
2.4
Worker Times Spent in Each State
As a step towards the construction of asset values for workers in each state,
it is necessary to calculate the proportions of time workers spend in each
state, taking the arrival and departure rates and the balk level as given. Let
Ti be the proportion of time workers spend in state i, where the states are
E, Q and S for employed, queueing and searching, respectively. Let Sij be
the transition rate for workers from state i to state j, where the possible
states are again E, Q and S. Equating flows of workers into and out of states
generates the following three conditions.
SQE TQ + SSE TS = SES TE
SES TE = SSE TS + SSQ TS
SSQ TS = SQE TQ
(26)
(27)
(28)
In the above,
SES = γ
SSE = m
SSQ = m
Xn
j=1
Xq−1
SQE = γn
j=0
Xq
(29)
p−j
(30)
pj
(31)
j=1
pj /
Xq
j=1
jpj
(32)
In 32, SQE solves the condition that the average number of workers in a
queue (given that there is a queue) times the transition rate into employment
equals γn, the rate at which jobs become available when there are n workers
employed. Imposing the condition that TE + TS + TQ = 1 and substituting
the transition rates from 29 through 32, it is possible to solve for TE , TS and
TQ .
2.5
Arrival Rate α
The arrival rate α is determined endogenously and does not depend on the
wage rate. Let NW be the number of workers and let NF be the number of
firms (assuming the labor force is fixed). The expected number of workers
9
at a firm is given by
FE =
Xn
j=1
(n − j)p−j +
Xq
j=0
npj
(33)
In equilibrium, the number of workers employed can be calculated as the
number of workers times the proportion of time employed or as the number
of firms times expected employment per firm. Setting the two measures of
the number of workers employed equal yields:
NW TE = NF FE
(34)
Since TE and FE are functions of α, 34 provides a condition on α (treating the
balk level q as being determined by α). As α approaches zero, employment
per firm approaches zero, the transition rate from searching to employment
approaches m (since almost all firms have vacancies), and TE approaches
m/(m + γ). As α increases, employment per firm increases, the balk level
goes up or stays the same, and TE declines. As α continues to increase, employment per firm approaches n and TE approaches zero. There will therefore
exist an arrival rate such that 34 holds, and the arrival rate will be unique.
The relation between the arrival rate α and the ratio of workers to firms can
be found by solving 34 for NW /NF as a function of α.
2.6
Worker Asset Values for Each State
Let Vi , i = E, S and Q, be the expected asset values for workers entering each
of the three states.6 Since workers in a queue become employed after some
time, VQ can be calculated from VE by discounting. Let r be the discount
rate used by workers to calculate the present value of future income. Then
if a worker enters employment t units of time later, the present value of the
asset value VE is e−rt VE , i.e. VE is discounted by a factor e−rt . If there are j
workers in line, the expected discount factor is
µ
¶j
Z ∞
γn
−rx
fj [x]e dx =
(35)
γn + r
0
6
The expected asset value while in the queue will differ from the expected asset value
upon entering a queue since the expected time until employment will decline as the worker
advances in the queue.
10
where fj [t] is from 20. Multiplying discount factors by the probabilities of
queue lengths and summing over queue lengths yields the expected discount
factor, CDF :
³
´q
α
µ
¶
j
1
−
Xq−1
γn+r
γn
γn − α
³ ´q
pj
=
(36)
CDF =
j=0
γn + r
γn − α + r 1 − α
γn
Then
VQ = CDF VE
(37)
The other relations among the asset values are
rVE = w + γ(VS − VE )
rVS = b + SSE (VE − VS ) + SSQ (VQ − VS )
(38)
(39)
where SSE and SSQ are given by 30 and 31. Then it is possible to solve for
the worker asset values VS , VE and VQ .
2.7
Firm Profits
As a further step towards determination of the equilibrium wage rate, firm
profits can be expressed as
π = (p − w)FE − rnK
(40)
where p is the price of output, w is the wage rate, FE is expected firm
employment from 33, and K is the amount of capital required per position
at the firm. The interest rate r is assumed to be the same as the worker
discount rate, although this simplification is not essential.
2.8
Wage Determination
Wage determination in a labor queueing market follows the methodology
in queueing markets for products (Sattinger, 2002) and is based on similar
methods in product and labor markets (Carlton, 1978, 1989, and Sattinger,
1990). The method assumes that a firm can attract more workers to search
at it by offering a higher wage, as in models of directed search with wage
posting.
11
Suppose that workers are able to achieve an expected value of searching
given by V S . Each firm is small relative to the labor market and takes V S as
given. Suppose that, starting from a symmetric equilibrium, a firm deviates
by offering a higher wage than other firms. If workers can influence where
they search, they would initially be more likely to search at the firm with the
higher wage. To be consistent with the previous assumption that workers get
interviews at the rate m, assume that workers can change the probabilities
that interviews occur at specific firms without changing the interview rate.
Then in response to a higher wage rate, the arrival rate at the firm would
increase, reducing the likelihood of vacancies, lengthening the expected queue
length, and reducing the value of searching at the firm. Workers could choose
an optimal balk level at the firm that could differ from balk levels at other
firms. Eventually, the arrival rate at the firm would rise to a level such
that the value of searching at the firm falls to the level in the rest of the
labor market. At that arrival rate, workers would not change further their
probabilities of getting interviews at different firms. In response to a wage
rate that deviates from the prevailing wage rate, worker behavior generates
an arrival rate such that the value of searching at the firm is the same as
searching at other firms in the labor market. Treating the arrival rate at the
firm αi as a function of the firm’s wage rate wi , the firm maximizes profits in
40 with respect to wi . The second order condition on maximizing firm profits
requires
µ 2
¶
∂ FE /∂α2i
∂ 2 αi /∂wi2
(p − wi )
−
−2<0
(41)
∂FE /∂αi
∂αi /∂wi
The symmetric equilibrium for the labor market consists of an arrival rate
satisfying 34, and a value of searching VS and wage rate w that satisfy 37
through 39 and the firm’s first order condition for maximization of profits
(with firm proportions and balk level determined as before from α).
Computationally, the relations among ratios of workers to firms, arrival
rate, value of searching and wage rate are derived in a different order but in a
manner fully consistent with the previous derivation. Set VS from 37 through
39 equal to an arbitrary constant V S to derive the wage rate as a function
of α and V S . Substituting this expression into the first order condition for
the maximization of profits, solve for V S as a function of the arrival rate.
Then the value of searching can be obtained as a function of the arrival rate.
Given VS , the wage rate can be found from the previously derived wage rate
as a function of V S and the arrival rate, since VS will be the same as V S in
12
a symmetric equilibrium. Finally, the relation between the arrival rate and
the ratio of workers to firms can be found from 34. These computational
procedures differ from the theoretical derivation by starting with the arrival
rate and solving for the remaining variables, instead of solving for the arrival
rate as functions of other variables. The computational procedures are necessary because firm proportions of time and other variables are complicated
functions of α that cannot be explicitly solved for α.
3
Vacancies, Unemployment, Queueing and
Search
This section demonstrates the underlying relations derived in labor queues.
A fundamental result of search theory is that there will be an inverse relation
between firm vacancies and worker unemployment. This relation is generated
by the matching function, which yields the transition rates for unemployed
workers and firms with vacancies as a function of the ratio of vacancies to
unemployed (market tightness). Analogously, the results in the previous section on labor queues generate an inverse relation between firm vacancies and
worker unemployment as a consequence of the proportion of time a firm has
vacancies or queues shorter than the balk level. With labor queues, however,
unemployment consists of workers either searching or waiting in queues. Labor market conditions (as reflected in the firm arrival rate α) determine the
mix of unemployed workers between those searching and queueing. As α increases, the labor market passes from circumstances where most unemployed
workers are searching to circumstances where most are in queues.
3.1
Balk Level Versus Arrival Rate
From Theorem 1, the proportions of time spent in each state in 17, 18 and 19
depend on the balk level. In turn, the balk level can be found by comparison
of expected times until employment. Figure 1 shows the calculation of the
balk level for particular parametric assumptions.7 Nevertheless, it exhibits
general features of the relation between the balk level and the arrival rate.
Since the balk level only takes integer values, it is a non-decreasing step
function of the arrival rate. At low arrival rates, firms have many vacancies
7
The figure assumes n = 20, p = 1, b = .2, γ = .1, r = .05, m = 1, and K = 2.
13
Balk Level
14
12
10
8
6
4
2
Arrival Rate
0.5
1
1.5
2
2.5
Figure 1: Balk Level Versus Arrival Rate
on average and the expected time until employment depends mostly on the
rate at which searching workers meet with firms. Then the balk level will be
constant over a wide range at low arrival rates, as shown in the figure.8 As
the arrival rate increases, vacancies at firms decline, the expected wait until
employment for searching workers increases, and the optimal balk level goes
up. At high arrival rates, firms generally have queues and few vacancies, and
searching workers must find a queue to join in order to get a job. The balk
level then goes up rapidly as vacancies disappear.
The rapid increase in the balk level at higher arrival rates can also be
understood in terms of formal queueing theory. Consider a queueing process
with arrival rate α, service rate σ at a single server, and no balking. As the
ratio α/σ (the erlang or traffic intensity) increases and approaches one, the
expected queue length increases indefinitely. In labor queues, as the ratio
α/(γn) increases, the expected queue length also increases. With reduced
alternatives to joining a queue as α/(γn) increases, workers increase the
8
If instead the meeting rate depended on the ratio of vacancies to searching workers, as
in the standard matching function for search models, the expected time until employment
would approach zero at lower arrival rates and the balk level would also approach zero.
14
Arrival Rate
2.5
2
1.5
1
0.5
Workers per Firm
5
10
15
20
25
Figure 2: Arrival Rate Versus Workers per Firm
length of the queue they are willing to join.
3.2
Arrival Rate Versus Workers per Firm
From Theorem 1, employment per firm, FE , is an increasing function of
the arrival rate α, and the proportion of worker time employed, TE , is a
decreasing function (with the balk level q optimal at each α). Then the
equilibrium arrival rate satisfying 34 will be a non-decreasing function of the
ratio of workers per firm, given by NW /NF . Figure 2 shows the relationship
for the same parametric assumptions as Figure 1 and exhibits some general
features.
At low ratios of workers per firm, most firms have vacancies because of
the low arrival rate, worker expected time to employment is constant, and
the balk level is also constant. When vacancies decline at higher ratios of
workers per firm, workers raise their balk level as described in the previous
section. At the arrival rate at which a balk level increases, additional workers
(from more workers per firm) are absorbed in queues with no change in the
arrival rate. As a consequence, the arrival rate exhibits horizontal sections at
levels where the balk level increases. As the ratio of workers per firm reaches
15
Proportions Searching and Queueing
0.2
Queueing
0.175
0.15
0.125
0.1
Searching
0.075
0.05
0.025
Workers per Firm
5
10
15
20
25
30
Figure 3: Proportions Searching and Queueing
high levels, vacancies decline to zero, and additional workers are absorbed
in increasingly lengthy queues without raising the arrival rate further. As a
consequence, the arrival rate approaches a maximal level.
3.3
Proportions of Workers Searching and Queueing
With queues, the labor market is qualitatively different at low and high
ratios of workers per firm. At a low ratio of workers per firm, firms often
have vacancies and seldom have queues. Unemployed workers are essentially
searching for vacant positions as in standard search theory. At a high ratio
of workers to firms, the arrival rate is high and firms rarely have vacancies.
Workers, to get a job, must find a firm with a queue shorter than their
balk level. Most unemployed workers are then in queues rather than actively
searching. Proportion of time spent searching remains substantially constant,
however, because workers must continue searching when they meet firms with
queues at the balk level.
Figure 3 shows proportions of workers searching and queueing as functions
of the ratio of workers to firms. Both proportions exhibit discontinuities
in slope when balk levels change. Proportion of time queueing starts at
16
Vacancy Rate
0.2
0.175
0.15
0.125
0.1
0.075
0.05
0.025
Unemployment Rate
0.125
0.15
0.175
0.2
0.225
0.25
Figure 4: Beveridge Curve
negligible levels for low ratios of workers per firm and increases rapidly at
high levels when vacancies disappear. The proportion of workers searching
varies over a small range when workers per firm increase, as workers shift
from searching for vacancies to searching for queues shorter than their balk
levels.
3.4
Unemployment Versus Vacancies
A fundamental accomplishment of search theory is the generation of the
relation between unemployment and vacancies observed by Beveridge. This
relation is constructed by deriving the transition rates between employment
and unemployment and between vacant and filled jobs from the matching
function. Queueing provides an alternative relation between unemployment
and vacancy rates generated by the proportions of times a firm has vacancies
or queues. Unemployment in a queueing model of the labor market is defined
here as including searching as well as queueing workers.9
Figure 4 shows the Beveridge Curve generated by queueing using the same
9
Bureau of Labor Statistics procedures may not classify queueing workers as unemployed. See Section 5.
17
assumptions as for earlier figures. The vacancy rate is given by 1 − FE /n,
and the unemployment rate (combining searching and queueing) is given by
1 − TE . The curve shares the general shape with the Beveridge Curve derived
from standard search theory but shows discontinuous slopes where balk levels
change.
3.5
Wage Determination
In the wage determination process, firms maximize their profits taking the
asset value for searching workers, V S , as given. This results in an optimal
wage that is determined at a point of tangency between the worker’s indifference curve for V S and the highest firm isoprofit curve. These curves are
shown in Figure 5, using the parameter values and functional forms for the
previous figures. The indifference curve is constructed by finding the combination of wage and arrival rate at the firm that would generate the market
value of V S . The isoprofit curve for the firm is constructed as combinations
of w and α that yield a constant profit rate. As the ratio of workers per firm
increases, the labor market value of VS decreases, shifting the indifference
curve downwardward, and firm profit increases. The locus of tangencies in
Figure 5 shows the points of tangency between the indifference and isoprofit
curves as workers per firm increases. When the arrival rate reaches 2, the
balk level rises from 2 to 3 using the parameter values for this figure. The increase in balk level reduces expected vacancies at firms, worsening the labor
market condition of workers.
In considering arrival rate and wage combinations that a firm would offer
searching workers when the asset value of search is V S , it is necessary to
consider the balk level that workers would choose at the firm. This balk level
at a firm that deviates from the symmetric equilibrium could be different
from the balk level at other firms.10 A worker searching at a firm joins a
queue whenever the expected asset level from doing so exceeds the value of
continuing to search. If there are j workers in the queue at a firm, the worker
would then join the queue if the expected discounted value of employment
at the firm equals or exceeds the value of searching elsewhere:
µ
¶q
γn
VE ≥ V S
(42)
γn + r
10
In Section 2.2, the balk level is calculated for a symmetric equilibrium in which all
firms pay the same wage rate and have the same arrival rate for workers.
18
Wage Rate
1
0.9
0.8
0.7
Indifference
Isoprofit
0.6
Locus of Tangencies
Arrival Rate
1.2
1.4
1.6
1.8
2
2.2
2.4
Figure 5: Wage Determination
where VE is determined from 38 with VS = V S .
4
4.1
Consequences of Queueing
Comparable Basis
An evaluation of labor queues requires that models with and without queues
be placed on a comparable basis, so that the only difference is the presence
or absence of queues. One immediate difference between the labor queues
model introduced in Section 2 and the standard search and matching model
(Pissarides, 2000; Mortensen and Pissarides, 1999) is the use of a matching
function. Although the labor queueing model can be constructed without
a matching function, there is no inconsistency and a matching function can
be incorporated to facilitate comparison. A second difference is that in the
standard search and matching model, a firm is coextensive with a single job.
The job moves between the two states of filled and vacant, generating a welldefined vacancy. Despite the almost universal adoption of this simplifying
assumption, it is unnecessary for the construction of the basic relationships
19
in a search and matching model. Section 2 describes methods that can be
applied to determine the probabilities that firm with up to n workers has 0
through n vacancies. In the standard model, an unemployed worker searches
only among jobs that are vacant. The number of matches then depends on
the numbers of unemployed workers and vacant jobs. When firms are greater
than a single job, it is necessary to specify how the arrival rate varies with
the number of vacancies at the firm. However, to be consistent with the
determination of state probabilities in Section 2, the arrival rate must be
independent of the number of vacancies or number in queue. The resolution
adopted here assumes searching workers seek jobs or queues at all firms, not
just firms with vacancies.
Specifically, assume the number of interviews between workers and firms
is given by m[S, Nf ], where S is the number of searching workers not in
queues, and Nf is the number of firms.11 An interview results in employment
if the firm has a vacancy. If there are queues and the queue length is less
than the balk level, the interview could result in a transition of the worker
from searching to queueing. As in the standard literature, assume m[S, Nf ]
is a continuous, concave function of its arguments and is homogeneous of
degree one. The same matching function will be used for the model with and
without queueing.
In Section 2, each searching worker was assumed to get a constant number
of interviews per unit of time. The effect of using a more general matching
function (e.g., a Cobb-Douglas function of both S and Nf ) is to reduce the
firm arrival rate at low ratios of workers to firms (when S/Nf is low) and
raise the firm arrival rate at high ratios. Use of a matching function then
has predictable effects on the relations generated by labor queueing. With
a matching function, the relation between the arrival rate and the ratio of
workers to firms in queueing model can be determined as follows. Setting the
arrival rate α equal to m[S, Nf ], it is possible to solve for S, the number of
searching workers. The proportion of time spent searching, TS in Section 2.5,
is a function of the arrival rate α. Setting S equal to TS Nw , it is possible to
solve for the ratio of workers to firms, Nw /Nf , as a function of α. In this way,
a matching function can be incorporated into a queueing model, generating
all the results of the previous section.
When queueing does not occur, workers move between employment and
11
This implicitly assumes that firms are passive in the matching process, except for their
influence on the arrival rate through the wage rate.
20
unemployment, so S would equal the unemployment rate times the number
of workers, Nw . The value of searching can be generated in the standard
manner using asset values for the states of employed and unemployed. The
same wage determination mechanism is assumed to operate with and without
queueing. Firms choose a wage rate that maximizes their profits, with arrival
rates adjusting to yield the same value of searching at the firm as at other
firms.
4.2
Arrival Rates
Figure 6 shows a fundamental difference between labor markets with and
without queues.12 At low ratios of workers per firm, labor markets operate
substantially the same whether or not queues are allowed, because searching
workers can easily find firms with vacancies and would not choose to enter
a queue. At higher ratios of workers per firm, the two models diverge. In a
labor market without queues, more workers per firm generate more searching
workers, and the arrival rate increases indefinitely as S/Nf increases. In a
model with queues, searching workers have the option of joining queues,
which reduces the number of searching workers and the arrival rate. As
vacancies disappear in response to higher ratios of S/Nf , more searching
workers choose to join queues. The queueing mechanism places an upper
limit on the arrival rate since the rate at which workers fill vacancies or
join queues must equal the rate at which workers separate from firms. As a
result, at high ratios of workers per firm, the arrival rate will be lower with
queueing than without queueing. Firms will have fewer vacancies (because
of the presence of queues) but the vacancies will be filled less quickly because
of the lower arrival rate with queues.
4.3
Wage and Profit Rates
The effects of queueing on the arrival rate, as shown in Figure 6, have significant consequences for wage setting. Wages with and without queueing are
shown in Figure 7. The wage rate for a given ratio of workers per firm is
determined from two factors. First, for a given arrival rate, queues reduce
the proportion of time that a firm has vacancies. With lower vacancy rates
12
Figures 6 through 9 and Table 1 assume m[S, Nf ] = 3S .5 Nf.5 , n = 20, γ = .1, r = .05,
K = 2 and b = 0.
21
Arrival Rate
5
4
Without Queueing
3
2
With Queueing
1
5
10
15
20
25
30
35
Workers per Firm
Figure 6: Arrival Rate With and Without Queueing
22
Wage
1
0.8
With Queueing
0.6
0.4
Without Queueing
0.2
Workers per Firm
5
10
15
20
25
30
35
Figure 7: Wages With and Without Queueing
for a given arrival rate when there are queues, firms choose lower wage rates.
Second, for a given ratio of workers per firm (beyond some level), the arrival rate will be lower when queues are present, as shown in Figure 6. At
relatively low ratios of workers per firm, the second factor does not operate.
Then the first factor dominates, so that the wage rate is lower with queueing
as shown in Figure 7 for ratios of workers to firms below about 19. At higher
ratios, the arrival rate is lower for queueing, leading firms to choose higher
wages than if there were no queues.
As a result of these two factors, the wage rate is less sensitive to the ratio
of workers per firm when workers have the option of joining queues than
when there is no queueing.
Profits, shown in Figure 8, are inversely related to wage rates. From
Figures 7 and 8, queueing favors firms and disfavors workers when the ratio
of workers to firms is relatively low (below about 19 in the case used to
generate the figures). At relatively high ratios of workers per firm, queueing
favors workers and disfavors firms.
However, the ratio of workers to firms is not exogenously determined in
the long run. Assuming firm entry and exit in response to profits and losses,
23
Profits
15
10
Without Queueing
With Queueing
5
0
10
15
20
25
30
35
Workers per Firm
Figure 8: Firm Profits With and Without Queueing
the long run profit level will be zero. In the example used for the figures,
this occurs at a lower ratio of workers per firm with queueing compared to
without queueing.13 Table 1 shows the long run (zero profit) equilibrium
with and without queueing.
Table 1: Long Run Equilibria
Arrival Rate
Workers per Firm
Wage Rate
Unemployment Rate
Vacancy Rate
With Queueing Without Queueing
1.55
1.95
15.58
17.08
0.868
0.880
0.0279
0.0248
0.2429
0.1672
13
With a high level of capital costs, the zero profit ratio of workers per firm for a labor
market with queueing will be greater than for a market without queueing.
24
Production per Firm
20
With At Most One Queueing
Without Queueing
19
With Endogenous Queueing
18
17
16
15
Workers per Firm
16
18
20
22
24
Figure 9: Production Levels With and Without Queueing
4.4
Efficiency
Both searching and queueing can generate externalities that interfere with the
efficient operation of a labor market. When individuals decide to enter a labor
market and search for a job, they raise the match rate for firms but reduce
the match rate for other workers. The net externalities could be positive
or negative and have been analyzed in the literature on search congestion
(Diamond, 1982; Hosios, 1990; Sattinger, 1990). Hosios and Sattinger show
that if a particular wage level prevails, net externalities are absent and search
is efficient.
Externalities also arise from queueing (Hassin and Haviv, 2003, pp. 1819). An individual who enters a queue lengthens the time it takes individuals
entering later to reach the head of the queue. This externality is not considered by an individual entering a queue but could be balanced by the positive
externalities generated for firms at which the individual queues. Whether
queueing is efficient can be determined from the economy’s aggregate level
25
of production. The results do not unambiguously favor queueing. In Figure
9, with balk levels and queue lengths determined as in Section 2.3, production at each ratio of workers per firm is greater without queueing. Allowing
individuals to form queues would not be efficient because of the externalities generated by queueing. However, some queueing may still be optimal.
Allowing at most one individual to queue at a firm generates a greater level
of production than with no queueing, as shown in the figure. The optimal
level of queueing at each ratio of workers per firm could be greater than
one, generating a still higher level of aggregate production. The increase in
production by restricting the queue length indicates that beyond some level,
queueing generates negative net externalities.
5
Conclusions
This paper has applied methods of queueing theory to the analysis of a labor
market with frictions. Queueing theory provides expressions for the equilibrium numbers of vacancies, searching workers and queueing workers in terms
of the arrival rate. As the ratio of workers to firms and the arrival rate increase, unemployed workers shift from searching to queueing and the balk
level increases. The results generate a Beveridge Curve without the assumption of a matching function. Queueing theory also carries implications for
the dynamic adjustment of labor markets with frictions. Changes in unemployment take place mostly through the number of workers in queues rather
than through the number of workers actively searching. In a comparison with
the standard search and matching model, queueing places limits on the numbers of unemployed workers who aresearching. As a result, at higher ratios
of workers per firm, the arrival rate is lower when there is queueing. Then
the wage rate is less sensitive to the ratio of workers per firm when there is
queueing. Because of queueing externalities, a labor market with queueing
is less efficient than a labor market without queueing in the example worked
out here. However, when queue length is limited to one, a labor market with
queueing is more efficient.
Queueing theory suggests a number of questions for the analysis of labor
queues. There are alternative formulations of a labor market with queues
depending on specification of queue discipline and whether the queues are
observable (Hassin and Haviv, 2003). In labor markets with heterogeneous
workers, sequence of employment of workers in queue may be determined by
26
priority auctions rather than First In First Out (Moldovanu and Kittsteiner,
2004; Hassin and Haviv, Chapter 4).
It is also necessary to examine the empirical evidence of queueing behavior
in labor markets. This is somewhat complicated by the absence of direct
measures in U.S. labor force statistics. In the Current Population Survey, a
person who is waiting to start a job is not separately classified as in a queue.
Instead, the individual could be classified as unemployed or out of the labor
force depending on whether he or she meets the criteria for unemployment
(primarily having actively looked for a job within the last four weeks). Before
1994, a person waiting to start a new job within 30 days would be treated as
unemployed without regard to their search activity. A consequence of these
practices is that if workers queue for jobs, they would be counted as out of the
labor force. At times of high unemployment, the amount of unemployment
would be understated by the numbers of workers in queues. Dynamic changes
in the numbers of workers in queues would appear as movements of workers
into or out of the labor force. It is not yet clear which labor force data would
be used to establish whether workers queue for jobs.
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