Labor Queues. April2,2005

Transcription

1 Labor Queues Michael Sattinger Department of Economics University at Albany Albany, NY 12222, USA Phone: (518) Keywords: Queueing, Search, Unemployment April2,2005 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

2 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, 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

3 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 Model 2.1 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 betherateatwhich 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 p i be the proportion of time a firm has i unemployed workers queueing for a job, i =1,q.Let p 0 be the proportion of time a firm has exactly n workers. The proportions p i, 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

4 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 V E,V S and V Q 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, V S, 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 V S, 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: γp 1 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

5 Setting the flows equal yields 2γp 2 n + αp n = αp 1 n + γp 1 n (2) Similarly, the average transition rates into and out of state j n, with n j vacancies, are (j +1)γp j+1 n + αp j 1 n = αp j n + jγp j 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 nγp 0 + 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 In state j>0, andinstateq nγp 1 + αp 1 = αp 0 + nγp 0 (5) nγp 2 + αp 0 = αp 1 + nγp 1 (6) nγp j+1 + αp j 1 = αp j + nγp j (7) αp q 1 = nγp q (8) The equations 1 through 8 can be solved for the proportions p i,i= n, q. First, the proportions of time that the firm has vacancies can be expressed in terms of p n. From 1, p 1 n = α γ p n (9) Combining 1 with 2 yields 2γp 2 n + αp n + γp 1 n = αp 1 n + γp 1 n + αp n (10) 5

6 Canceling out terms yields p 2 n = 1 α 2 γ p 1 n = 1 2 µ 2 α p n (11) γ Similarly, combining equations for successive numbers of vacancies yields From 4 p j+1 n = 1 α j +1γ p j n = p 0 = 1 α n γ p 1 = 1 n! 1 (j +1)! µ α j+1 p n (12) γ µ n α p n (13) γ In the second step, the proportions of time with workers in the queue can beexpressedintermsofp q. Beginning with 8, p q 1 = n γ α p q (14) Then at successively lower numbers of workers in the queue, p j = n γ α p j+1 = n q j ³ γ α q j pq (15) At j =1, p 0 = n q ³ γ α q pq (16) Now solve 9 through 16 to yield the proportions of time in terms of p 0, the proportion of time with no vacancies or workers in the queue. p j = n! ³ γ j p0,j=1,n (17) (n j)! α p j = n j µ α γ j p 0,j=1,q (18) The sum of all the proportions of time must add up to 1, so that à nx n! ³ γ j qx µ! j 1 α p 0 = +1+ n j (19) (n j)! α γ j=1 j=1 6

7 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. Therateatwhichthefirm experiences the loss of a worker is γn. If the worker joins the queue, j +1workers must leave the firm before the worker gets a job. The probability density function for the waiting time until j +1 events is f j [t] = γn(γnt)j e γnt (20) j! Theexpectedtimeuntiltheworkergetsajob,iftherearealreadyj 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

8 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 P q 1 j=0 p j(j +1)/(γn) P q 1 j=0 p (22) j The expected time until employment for a searching worker, E S [q], satisfies the implicit condition generated by the alternatives facing a searching worker: E S [q] = 1 m +0X n p j + X P q 1 q 1 p j=0 p j(j +1)/(γn) j P j=1 j=0 q 1 j=0 p + p q E S [q] (23) j Solving for E s [q] yields E s [q] = 1 1 p q µ 1 m + X q 1 p j(j +1)/(γn) j=0 (24) Now consider whether q is a balk level that could prevail for all workers in equilibrium. If q γn E S[q] q +1 (25) 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 E S [q]. Then consider q +1 as the possible equilibrium balk level. In recalculating the expected wait time from searching for balk level q +1, E S [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, E S [q +1], and the equilibrium balk level would be q +1or greater. Considering sequentially higher balk levels, a balk level consistent with symmetric equilibrium arises for the first value such that 25 holds. 8

9 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 T i 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 S ij 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. In the above, S QE T Q + S SE T S = S ES T E (26) S ES T E = S SE T S + S SQ T S (27) S SQ T S = S QE T Q (28) S ES = γ (29) S SE = m X n p j (30) j=1 S SQ = m X q 1 p j (31) j=0 S QE = γn X q p j/ X q jp j (32) j=1 j=1 In 32, S QE 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 T E + T S + T Q =1and substituting the transition rates from 29 through 32, it is possible to solve for T E,T S and T Q. 2.5 Arrival Rate α The arrival rate α is determined endogenously and does not depend on the wage rate. Let N W be the number of workers and let N F be the number of firms (assuming the labor force is fixed). The expected number of workers 9

10 at a firm is given by F E = X n (n j)p j + X q np j (33) j=1 j=0 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: N W T E = N F F E (34) Since T E and F E 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 T E approaches m/(m + γ). As α increases, employment per firm increases, the balk level goes up or stays the same, and T E declines. As α continues to increase, employment per firm approaches n and T E 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 andtheratioofworkerstofirms can be found by solving 34 for N W /N F as a function of α. 2.6 Worker Asset Values for Each State Let V i,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, V Q can be calculated from V E 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 V E is e rt V E, i.e. V E is discounted by a factor e rt. If there are j workers in line, the expected discount factor is Z 0 f j [x]e rx dx = µ j γn (35) γn + r 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

11 where f j [t] is from 20. Multiplying discount factors by the probabilities of queue lengths and summing over queue lengths yields the expected discount factor, C DF : Then C DF = X q 1 j=0 p j ³ µ j γn = γn α 1 γn + r γn α + r 1 The other relations among the asset values are q α γn+r ³ α γn q (36) V Q = C DF V E (37) rv E = w + γ(v S V E ) (38) rv S = b + S SE (V E V S )+S SQ (V Q V S ) (39) where S SE and S SQ are given by 30 and 31. Then it is possible to solve for the worker asset values V S,V E and V Q. 2.7 Firm Profits As a further step towards determination of the equilibrium wage rate, firm profitscanbeexpressedas π =(p w)f E rnk (40) where p is the price of output, w is the wage rate, F E 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 offeringahigherwage,asinmodelsofdirectedsearchwithwage posting. 11

12 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 firmwouldrisetoalevelsuch 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 firmisthesameas searching at other firms in the labor market. Treating the arrival rate at the firm α i asafunctionofthefirm s wage rate w i, the firm maximizes profits in 40 with respect to w i. The second order condition on maximizing firm profits requires µ 2 F E / α 2 i (p w i ) 2 α i / wi 2 2 < 0 (41) F E / α i α i / w i The symmetric equilibrium for the labor market consists of an arrival rate satisfying 34, and a value of searching V S 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 V S 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 V S, thewageratecanbefoundfromthepreviouslyderivedwagerate as a function of V S and the arrival rate, since V S willbethesameasv S in 12

13 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

14 Balk Level Arrival Rate 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

15 Arrival Rate Workers per Firm 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, F E, is an increasing function of the arrival rate α, and the proportion of worker time employed, T E, 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 N W /N F. 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

16 Proportions Searching and Queueing Queueing Searching Workers per Firm 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 searchingforvacantpositionsasinstandardsearchtheory. Atahighratio 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

17 Vacancy Rate Unemployment Rate 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

18 assumptions as for earlier figures. The vacancy rate is given by 1 F E /n, and the unemployment rate (combining searching and queueing) is given by 1 T E. ThecurvesharesthegeneralshapewiththeBeveridgeCurvederived 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 V S 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 V E 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

19 Wage Rate Indifference 0.7 Isoprofit 0.6 Locus of Tangencies Arrival Rate Figure 5: Wage Determination where V E is determined from 38 with V S = V S. 4 Consequences of Queueing 4.1 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

20 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, N f ], where S is the number of searching workers not in queues, and N f 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, N f ] 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 N f ) is to reduce the firm arrival rate at low ratios of workers to firms (when S/N f 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 firmsinqueueingmodelcanbedeterminedasfollows. Settingthe arrival rate α equal to m[s, N f ], it is possible to solve for S, the number of searching workers. The proportion of time spent searching, T S in Section 2.5, is a function of the arrival rate α. Setting S equal to T S N w, it is possible to solve for the ratio of workers to firms, N w /N f, 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

21 unemployment, so S would equal the unemployment rate times the number of workers, N w. 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/N f 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/N f,moresearching 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,thearrivalratewillbelowerwith queueing than without queueing. Firms will have fewer vacancies (because of the presence of queues) but the vacancies will be filled less quickly because ofthelowerarrivalratewithqueues. 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, N f ]=3S.5 N f.5,n=20,γ=.1, r=.05, K =2and b =0. 21

22 Arrival Rate Without Queueing 2 With Queueing Workers per Firm Figure 6: Arrival Rate With and Without Queueing 22

23 Wage With Queueing Without Queueing Workers per Firm Figure 7: Wages With and Without Queueing for a given arrival rate when there are queues, firms choose lower wage rates. Second, foragivenratioofworkersperfirm (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 wagesthaniftherewerenoqueues. 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

24 Profits Without Queueing With Queueing 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 With Queueing Without Queueing Arrival Rate Workers per Firm Wage Rate Unemployment Rate Vacancy Rate With a high level of capital costs, the zero profit ratioofworkersperfirm for a labor market with queueing will be greater than for a market without queueing. 24

25 Production per Firm With At Most One Queueing Without Queueing 18 With Endogenous Queueing Workers per Firm 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 ). 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

26 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

27 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. References [1] Allon, G. and Federgruen, A.: Competition in Service Industries, unpublished paper, 2004, Columbia University. [2] Carlton, D. W.: Market Behavior with Demand Uncertainty and Price Inflexibility. American Economic Review, Vol. 68, No. 4, 1978, pp [3] Carlton, D. W.: The Theory and the Facts of How Markets Clear: Is Industrial Organization Valuable for Understanding Macroeconomics? Chapter 15, , in: Handbook of Industrial Organization, Vol. I, edited by R. Schmalensee and R. Willig. Amsterdam: North Holland, [4] Cox, D. R. and Smith, W. L.: Queues. New York: John Wiley, [5] Diamond, Peter, Wage Determination and Efficiency in Search Equilibrium, Review of Economic Studies, XLIX,

28 [6] Feller, W.: An Introduction to Probability Theory and its Applications, Volume I, second edition. New York: John Wiley, [7] Hassin, R. and Haviv, M.: To Queue or Not to Queue; Equilibrium Behavior in Queueing Systems. Boston: Kluwer [8] Hosios, Arthur J., On the Efficiency of Matching and Related Models of Search and Unemployment, Review of Economic Studies, Vol. 57, [9] Karlin, S.: A First Course in Stochastic Processes. New York: Academic Press, [10] Karlin, S. and Taylor, H. M.: A Second Course in Stochastic Processes. New York: Academic Press, [11] Knudsen, Niels Chr. Individual and Social Optimization in a Multiserver Queue with a General Cost-Benefit Structure. Econometrica Vol. 40, No. 3, May 1972, pp [12] Leeman, W.A.: The Reduction of Queues Through the Use of Price. Operations Research Vol. 12, 1964, pp [13] Moldovanu, B. and Kittsteiner, T.: Priority Auctions and Queue Disciplines that Depend on Processing Time, forthcoming, Management Science. [14] Mortensen, D., and Pissarides, C.: New Developments in Models of Search in the Labor market. In Handbook of Labor Economics, Vol. 3B, edited by Orley C. Ashenfelter and David Card, pp , North- Holland, Amsterdam, 1999, pp [15] Naor, P.: The Regulation of Queue Size by Levying Tolls. Econometrica Vol. 37, No. 1, January 1969, pp [16] Pissarides, Christopher, Equilibrium Unemployment Theory, second edition. Cambridge: MIT Press. [17] Saaty, T. L.: Elements of Queuing Theory. New York: McGraw-Hill,

29 [18] Sattinger, M.: Unemployment, the Market for Interviews, and Wage Determination. Journal of Political Economy Vol. 98, No. 2, April 1990, pp [19] : A Queuing Model of the Market for Access to Trading Partners. International Economic Review Vol. 43, 2002, pp [20] Shimer, R.: The Assignment of Workers to Jobs in an Economy with Coordination Frictions. NBER Working Paper 8501 (2001). [21] Stahl II, D. O.: Queue-Rationing and Price Dynamics. Scandinavian Journal of Economics, Vol. 89, No. 4, 1987, pp [22] Taylor, H. M. and Karlin, S.: An Introduction to Stochastic Modeling. Orlando: Academic Press