Transportation Research Part B 38 (2004) 853–867www.elsevier.com/locate/trb

Wage payoffs and distance deterrence in the journey to work

Paul Glenn a, Inge Thorsen a, Jan Ubøe b,*

a Stord/Haugesund University College, Bjørnsonsgate 45, N-5528, Haugesund, Norwayb Norwegian School of Economics and Business Administration, Helleveien 30, N-5045 Bergen, Norway

Received 13 May 2002; received in revised form 11 November 2003; accepted 24 November 2003

Abstract

In this paper we suggest a microeconomic model for how commuting flows relate to traveling distance in

a two-region system. Commuting is the preferred choice of a worker whenever he can obtain an increase in

wages greater than the cost of commuting. Our framework is based on an approach where workers apply

for jobs according to a strategy that maximizes their expected payoffs (wages minus commuting costs). We

also discuss the possibility of a systematic bias when actual traveling distances are represented by distances

between city centers, ignoring intrazonal distances.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Commuting; Deterrence functions; Wage differences

1. Introduction

It is well known from the literature that spatial interaction behavior is affected by several typesof separation between origins and potential destinations. For a discussion of spatial separationmeasures, see for instance Sen and Smith (1995). In this paper we focus on the effect of travel timeand travel cost between actors and opportunities. Time and cost aspects are represented by the(generalized) traveling distance. To keep the analysis free from disturbing elements we onlyconsider geographies with two areal units or central places. The basic mechanisms underlying theanalysis are of course also relevant in systems with more complex configurations of central places.Studies of spatial interaction and travel demand are, for instance, basic ingredients in economicassessments of investments in transportation infrastructure as well as in assessments of roadpricing schemes.

* Corresponding author. Tel.: +47-5-595-9978; fax: +47-5-595-9650.

E-mail address: jan.uboe@nhh.no (J. Ubøe).

0191-2615/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.trb.2003.11.002

854 P. Glenn et al. / Transportation Research Part B 38 (2004) 853–867

For a standard textbook presentation of travel demand models, see Ortuzar and Willumsen(1994). In this paper we consider trip generation and trip distribution aspects of journeys to work.Consider the following general expression of a spatial interaction model:

Tij ¼ fiðAiÞfjðBjÞCij ð1Þ

Here, Tij represents traffic flows from origin i to destination j, fiðAiÞ represents characteristics ofthe origin/production zone i, while fjðBjÞ represents characteristics of the destination/attractionzone j. If the relevant type of spatial interaction is commuting the production characteristics arerelated to the concentration of workers residing in zone i, while the attraction factors are relatedto the number of jobs in zone j. Cij is introduced as a friction effect, representing the disutility oftravel. This formulation subsumes a set of spatial interaction models. Most commonly applied tothe problems relevant in this paper are models belonging to the gravity modeling tradition. For adiscussion of the theoretical foundation and practical applications of gravity models, see Erlanderand Stewart (1990) or Sen and Smith (1995). For applications of gravity models in studies ofjourneys to work, see Thorsen and Gitlesen (1998, 2002) and Gitlesen and Thorsen (1999, 2000).Gravity models are also often involved when commuting flows are determined within operationalintegrated land use and transport models. For an overview of such models, see Wegener (1998). Insome large-scale models of this kind the commuting flow pattern is determined through thespecification of a multinomial logit model, see for instance Andersstig and Mattsson (1998). Sucha model formulation is, however, formally equivalent to a conventional gravity model (see Anas,1983). Hence, the analysis in this paper is relevant also for multinomial logit models.In the literature on spatial interaction problems a lot of interest has been focused on the

specification of the friction effect, i.e., the distance-deterrence function. One possible specificationis the power deterrence function, defined by Cij ¼ db

ij, and another is the exponential deterrencefunction, Cij ¼ expð�bdijÞ. Here, dij is the physical distance between origin i and destination j,while b is the distance-deterrence parameter. The exponential distance-deterrence function followsdirectly as a result of a standard derivation of the gravity model from an optimizing framework.One such theoretical basis for the gravity model is the entropy-maximizing procedure, that wasintroduced by Wilson (1967). As pointed out by Ortuzar and Willumsen (1994), among others,however, reasonable reformulations of the optimizing problem result in a power function speci-fication. It is also well known in the literature on spatial choice behavior that traditional gravitymodels can be derived from stochastic utility maximization (see for example Anas, 1983). Thisderivation results in an exponential distance-deterrence function if (indirect) utility is linearlyrelated to distance, while a power function results if the (indirect) utility function is linearly relatedto the natural logarithm of distance. In general, the literature offers no strong theoretical argu-ments in favor of one particular specification. For example, Nijkamp and Reggiani (1992) claimthat the choice of a deterrence function is essentially a pragmatic one. Still, they refer to empiricalevidence of how the two alternative specifications have proved to be appropriate for differentconfigurations of spatial structure leading to the interaction.One approach to achieve an appropriate specification of the distance-deterrence function is to

let the data decide through a Box–Cox transformation of the distance variable. Fik and Mulligan(1998) concluded that the appropriateness of the functional form should be critically examined.Based on migration data they found that Box–Cox transformations significantly improved thegoodness of fit and changed the qualitative evaluation of their gravity-related model formulation.

P. Glenn et al. / Transportation Research Part B 38 (2004) 853–867 855

Similar results were found in Gitlesen and Thorsen (1999) in a study of commuting flows. Thorsenet al. (1999) argue in favor of a logistic specification. Their argumentation is based on theintroduction of models defined in terms of extreme states of the system. In our point of viewThorsen et al. (1999) set up a common framework for the discussion, comparison, and visuali-zation of any kind of model within the field, and the same set of ideas will be used here to visualizethe responsiveness to distance of microeconomic models for commuting.This paper is based on the fact that the functional form reflects aggregation aspects in

addition to the outcome of individual optimization behavior. To be more precise we consider asituation where workers evaluate the commuting cost against prospects of achieving higherwages in more remote job destinations. Decisions are interdependent, however, as workerscompete for attractive jobs. We consider a scenario with a multitude of different worker cat-egories (professions), and we account for the possibility that each profession is faced withseveral types of jobs. Based on this rather general framework a commuting flow pattern resultsfrom a Nash equilibrium solution, and we demonstrate that this pattern corresponds to aglobally concave distance-deterrence function. In addition, however, we account for possiblemeasurement errors resulting when the measured distances between zonal centers diverge fromreal interzonal distances of journeys to work; jobs and workers are not completely spatiallyconcentrated in the zonal center. Due to the existence of such measurement errors we argue infavor of a geometric correction of distances. This correction gives the otherwise globally con-cave function a logistic profile.A short survey of the extreme state approach is given in Section 2 of this paper. In Section 3 we

study the determination of commuting flows through a game where wage differences are evaluatedagainst commuting costs. In Section 4 we restrict the discussion to the cases where each individualworker is left with two or three job alternatives. In Section 5 we extend the discussion in Section 4to the case where the labor force is divided into a number of different categories. Systematic biasdue to measurement errors and a geometrical correction procedure for this are considered inSection 6. We consider the aggregate effect of labor market conditions and geometrical correctionsin Section 7. Finally, in Section 8 we offer some concluding remarks.

2. Extreme states and distance deterrence

We now give a short survey of the discussion in Thorsen et al. (1999). Consider a frameworkwith two towns. Let L1, L2 denote the number of workers residing in towns 1 and 2, respectively,and let E1, E2 denote the corresponding number of jobs in each town. All workers are assumed tohave a job, so L1 þ L2 ¼ E1 þ E2. The trip distribution matrix T is then defined as T ¼ fTijg2i;j¼1where

Tij ¼ Traffic flow; i:e:; the number of people commuting from town i to town j ð2Þ

Thorsen et al. (1999) focus on two extreme situations:

1. When commuting in the system is determined by random choice only, the expected trip distri-bution matrix can be expressed as follows:

856 P. Glenn et al. / Transportation Research Part B 38 (2004) 853–867

Trandom ¼

L1E1E1 þ E2

L1E2E1 þ E2

L2E1E1 þ E2

L2E2E1 þ E2

2664

3775 ð3Þ

2. If, on the other hand, we consider a situation where the total traveling cost is as low as possible,we get � �

Tminimal cost ¼ min½L1;E1 L1 �min½L1;E2L2 �min½L2;E1 min½L2;E2

ð4Þ

The basic idea in Thorsen et al. (1999) is then to write any trip-distribution matrix as a convexcombination of the two extremes, i.e.,

T ¼ Trandomð1� DÞ þ Tminimal costD ð5Þ

D measures the level of deterrence from the random choice case, and the basic hypothesis is thatD ¼ DðcÞ where c represents a measure of generalized distance, including both time and moneycosts of traveling. As argued in Thorsen et al. (1999), we expect that commuting will be randomwhen the distance is very short, i.e., there is no deterrence and D 0. When the traveling distanceis very long, however, we expect that action is taken so as to minimize the traveling cost in thesystem, i.e., there is full deterrence and D 1. To allow for some friction in the system, weconsider a marginal level of interaction, a. The distance d0 signifies the distance where the cost issuch that DðcÞ is marginally close to no deterrence, and d1 signifies the distance at which DðcÞ ismarginally close to full deterrence. The conditions above can then be expressed as follows:

1. d P d1 ) DðcÞP 1� a2. d 6 d0 ) DðcÞ6 a

In Fig. 1, we show a distance-deterrence function corresponding to the discussion above. InFig. 1, d0 ¼ 10 km and d1 ¼ 60 km. In the regions where d06 d and d 6 d1, the function fallswithin a ¼ 5% of its extreme values. Fig. 1 is based on a linear cost function; c ¼ 80d.The main purpose of this paper is to provide a theoretical discussion of the functional form of

such distance-deterrence functions. In particular, we will demonstrate that the basic properties

Fig. 1. A distance-deterrence function.

P. Glenn et al. / Transportation Research Part B 38 (2004) 853–867 857

mentioned above can be argued to be the result of a framework where the level of commuting isdetermined from game theory together with a geometric correction procedure.

3. A game theoretical approach to commuting

We now consider a situation where there are N1 different types of jobs in town 1, each of whichhave E1j, j ¼ 1; . . . ;N1 employment positions, with wages W1j, j ¼ 1; . . . ;N1. Correspondinglythere are N2 different types of jobs in town 2, with E2j, j ¼ 1; . . . ;N2 employment positions, andwages W2j, j ¼ 1; . . . ;N2. The workers in each town apply for the positions with the purpose ofmaximizing their individual wages net of commuting costs. We assume that the jobs are sortedwith the N1 jobs in town 1 first followed by the N2 jobs in town 2. The game is played as follows:

1. The workers in each town hand in a set of applications for the jobs in the system. Each workerapplies for one and only one job.

2. If there are more applicants than jobs, the individuals getting the jobs are determined by ran-dom choice. We hence assume that all workers are equally qualified for the jobs, and invoke thelaw of large number to compute the distribution between towns. These jobs and the corre-sponding numbers of workers are then removed.

3. When the number of applications is less than the number of employment positions, all theapplicants get hired, and the corresponding number of employment opportunities is removedfrom the game.

4. The game is then repeated until all positions are resolved.

Since E1 þ E2 ¼ L1 þ L2, at least one type of job will be resolved in each round, so the maximumduration of the game will be N1 þ N2 rounds. The point of importance here is that workers ap-pointed to jobs in the opposite town must adjust their wages with respect to the generalizedtraveling cost, c. If LðiÞ

j denotes the number of workers residing in town i appointed to job j, thenthe average outcome of the game can be computed as follows:

Average outcome for town 1: O1 ¼XN1j¼1

Lð1Þj W1j þ

XN2j¼1

Lð1ÞjþN1ðW2j � cÞ

Average outcome for town 2: O2 ¼XN1j¼1

Lð2Þj ðW1j � cÞ þ

XN2j¼1

Lð2ÞjþN1W2j

ð6Þ

To simplify the notation, we define N ¼ N1 þ N2, and let W ij represent net wages for workers in

town i and job category j:

W ð1Þj ¼

W1j if 16 j6N1ðW2j � cÞ if N1 < j6N

W ð2Þj ¼

ðW1j � cÞ if 16 j6N1W2j if N1 < j6N

ð7Þ

858 P. Glenn et al. / Transportation Research Part B 38 (2004) 853–867

With this notation we get O1 ¼PN

j¼1 Lð1Þj W ð1Þ

j and O2 ¼PN

j¼1 Lð2Þj W ð2Þ

j . Note that all mixed strat-egies for this game can be defined from a choice probability P defined on the set of all types ofjobs, i.e., a function defining the probability that a worker will apply for each type of job giventhat his choice is restricted to a particular subset of the jobs.The game defined above has L1 þ L2 individual players. Each player in town 1 gets an average

outcome equal to the average outcome for town 1 divided by L1, with a corresponding result fortown 2. Since L1 and L2 are fixed, we can assume without loss of generality that the players try tomaximize the outcomes in (7). All the players within the same town are by assumption identical. Ifwe make the additional assumption that each such player has a unique best strategy, and that allthe players use this strategy, we may essentially consider this as a two player game; all the workersin town 1 apply the same choice probability, P , and all the workers in town 2 apply the samechoice probability Q.It is well known from the literature that game theoretical approaches are commonly used in

studies of travel demand. This, however, primarily applies to traffic assignment problems, whichare problems of how traffic is distributed between different route alternatives connecting twogeographical points. Such studies are based on the concept of user equilibrium, which states thatno trip maker can reduce the travel cost by changing to another route. This concept refers back toWardrop (1952). For an overview of the later development of assignment models based on gametheoretical concepts, see for instance Ortuzar and Willumsen (1994) or Hensher and Button(2000). Those books also address the four stage model of travel demand in general.

4. 2-job and 3-job games

To simplify the discussion further, we will restrict ourselves to some special cases. The simplestsuch case occurs when there is only one job category in each town, i.e., N1 ¼ N2 ¼ 1. We refer tothis situation as a 2-job game. In this case we let p denote the fraction of workers in town 1applying for the job in town l, and q denotes the fraction of workers in town 2 applying for the jobin town l. The outcome of the game in this case is fairly obvious, but we include some detailsmerely as an illustration of the technical complexity of the game.

O1½p;q ¼

pL1e11pL1þ qL2

W ð1Þ1 þ L1�

pL1E11pL1þ qL2

� W ð1Þ2 if pL1þ qL2PE11

L1�ð1� pÞL1E21

ð1� pÞL1þð1� qÞL2

� W ð1Þ1 þ ð1� pÞL1E21

ð1� pÞL1þð1� qÞL2W ð1Þ2 if pL1þ qL26E11

8>>><>>>:

ð8Þ

From (8) we get

oO1op

¼

qL1L2E11ðpL1 þ qL2Þ2

ðW ð1Þ1 � W ð1Þ

2 Þ if pL1 þ qL2 PE11

ð1� qÞL1L2E11ðð1� pÞL þ ð1� qÞL Þ2

ðW ð1Þ1 � W ð1Þ

2 Þ if pL1 þ qL26E11

8>>><>>>:

ð9Þ

1 2

Fig. 2. A distance-deterrence function for a 3-job game.

P. Glenn et al. / Transportation Research Part B 38 (2004) 853–867 859

Similar expressions hold for O2. From (9) it follows that O1 is monotone in p, and similarly that O2is monotone in q. Hence if W ð1Þ

1 6¼ W ð1Þ2 and W ð2Þ

1 6¼ W ð2Þ2 , then there is a unique Nash equilibrium

characterized by a strategy where a player always applies for the job with the best wages. IfW ð1Þ1 ¼ W ð1Þ

2 , then player 1 is indifferent between all the strategies, and may just as well only applyfor job l. If W ð2Þ

1 ¼ W ð2Þ2 , then player 2 is indifferent between all the strategies, and may just as well

only apply for job 2.Now consider the case where there are two different job categories in town 1 and one job

category in town 2, i.e., N1 ¼ 2 and N2 ¼ 1. The outcome of the game is now quite different. Theplayers have to balance the possibility of getting the best job against the danger of falling into theworst category. In this case the game is too complex to admit a straightforward analysis, so wewill resort to a numerical approach. One basic principle remains from the 2-job game, however:the players never apply for the worst category.EXAMPLE 4.1In this example we set L1 ¼ 4000, L2 ¼ 6000, E11 ¼ 4000, E12 ¼ 2500 and E21 ¼ 3500. The wages

are W11 ¼ 29; 000 ($/year), W12 ¼ 22; 000 ($/year), and W21 ¼ 30; 000 ($/year). From the numericalsimulations, we see that the players now continuously change their strategy. The resulting dis-tance-deterrence function is shown in Fig. 2.In this case the non-trivial choice probabilities are the ones for job 1 and job 3 on the first level.

The players never apply for job 2, and on the second level there are at most two jobs left, so theanalysis for the 2-job game applies. We mention a few typical values: At d ¼ 20 the players intown 1 apply p1 ¼ 0:58 and p3 ¼ 0:42, while the players in town 2 use q1 ¼ 0:48 and q3 ¼ 0:52. Atd ¼ 40 km the players in town 1 apply p1 ¼ 0:67 and p3 ¼ 0:33, while the players in town 2 useq1 ¼ 0:46 and q3 ¼ 0:54. The corresponding trip distribution matrices are as follows:

T20 ¼2774 1226

3726 2274

� T40 ¼

2978 1022

3522 2478

� ð10Þ

and, as can be expected, we observe an increase in internal commuting.

5. The N-job game with multiple categories

To proceed one step further, we consider a case where there are M different job categories, andwhere the same job categories are found in both towns. Each worker is only qualified for work in

860 P. Glenn et al. / Transportation Research Part B 38 (2004) 853–867

one job category, but jobs offered for a specific category are not homogeneous. Each category(profession) consists of workers with specific qualifications, but as a group they are faced with joboffers of varying tasks, responsibilities, and wages.Hence the labor force L1 in town 1 can be divided into categories L11; L12; . . . ; L1M , where each

category is qualified for its corresponding job category. The corresponding categories in town 2are denoted L21;L22; . . . ; L2M . We will assume that the workers L1k;L2k in category k are qualifiedfor the jobs E1jk, k ¼ 1; . . . ;N1k and E2jk, k ¼ 1; . . . ;N2k only. All the workers within each categorycompete for the jobs according to the game in the previous section. By a slight adjustment ofnotation, we define E1k ¼

PN1jj¼1 E1jk and E2k ¼

PN2jj¼1 E2jk, i.e., the total number of employment

opportunities in each town and category.In the numerical simulations, we only consider two special cases. In the first case we assume

that within each category, there is one and only one job option in each town. Hence N1k ¼ N2k ¼ 1for all k, and E1k ¼ E11k, E2k ¼ E21k denote the number of jobs available for category k. In thesecond case we assume N1k ¼ 2, N2k ¼ 1 for each k, in which case E11k, E12k and E21k denote thenumber of available jobs for category k. Then E1k ¼ E11k þ E12k denotes the total number ofavailable jobs in town 1 in category k, and E1k ¼ E21k denotes the total number of available jobs intown 2 in category k.When a population is divided into categories as indicated above, the matrices Trandom and

Tminimal cost must be computed from the expressions

Trandom ¼

Xk

L1kE1kE1k þ E2k

Xk

L1kE2kE1k þ E2kX

k

L2kE1kE1k þ E2k

Xk

L2kE2kE1k þ E2k

26664

37775 ð11Þ

and

Tminimum cost ¼P

k minfL1k;E1kg L1 �P

k minfL1k;E1kgL2 �

Pk minfL2k;E2kg

Pk minfL2k;E2kg

� �ð12Þ

Note that the matrices defined by (11) and (12) will in general be different from the correspondingexpressions given by (3) and (4).For a thorough discussion of aggregation and the modeling issues it poses with respect to

gravity models, see Ubøe (2001). Here we will need to compute distance-deterrence functions foraggregated systems, and we refer to the following result from Ubøe (2001).

Theorem (Ubøe (2001)). Consider an aggregated system of M categories, where each of the cate-gories k has a distance-deterrence function Dk, k ¼ l; . . . ;M . If we let Ek ¼ E1k þ E2k, k ¼ l; . . . ;Mand Lk ¼ L1k þ L2k, k ¼ l; . . . ;M denote the total number of employment opportunities/workers ineach category k in the whole system, then the distance deterrence D for the aggregated system can befound as follows:

D½c ¼PM

K¼1 LkminE1kEk

� L2kLk; E2kEk

� L1kLk

h i� Dk½cPM

K¼1 LkminE1kEk

� L2kLk; E2kEk

� L1kLk

h i ð13Þ

Fig. 3. Distance-deterrence functions for 2-job and 3-job games with 500 different categories.

P. Glenn et al. / Transportation Research Part B 38 (2004) 853–867 861

In systems with a small number of categories, the distance-deterrence function can have almostany shape according to the characteristics of each category. As the system is refined into a largenumber of small categories, however, it seems reasonable to conjecture that the local anomalieswill be significantly reduced. Fig. 3a shows a numerical simulation of the distance-deterrencefunction in a 2-job system where the workers are subdivided into 500 different professions. In thisframework, commuting is computed on the basis of wage differences between the two towns. InFig. 3a we have used a construction where the difference in wages decreases with the size of thecategories (see the theoretical discussion below).The resulting picture in Fig. 3a is a globally-concave function. To examine this further, we

repeated the experiment for a 3-job game. In this case the game comes out with a variety ofdifferent curves and patterns depending on the relative size of the various groups. When thesesystems are aggregated, however, all the peculiarities are wiped out, and the result is again aglobally-concave function. Fig. 3b below shows the result of an aggregation of 500 categories.The curves in Fig. 3 have been constructed as follows: We first sampled random numbers

Lk ¼ Ek, k ¼ 1; . . . ; 500 from a uniform distribution on the interval [100, 10,000]. Then each Lk

was subdivided into L1k, L2k by random choice, and correspondingly, we split Ek into E11k, E12k andE21k. To simplify the programming, we always placed 2 job categories in town 1. Finally the wagesW11k, W12k, and W21k were drawn from a uniform distribution on the interval [30,000(1–100/Lk),33,000].

5.1. Theoretical discussion

To try to explain these results from a theoretical point of view, we consider the 2-job game. Welet DWk ¼ jW1k � W2kj, k ¼ 1; . . . ;M , i.e., the difference in wages between the two towns in categoryk. For this game the distance-deterrence functions are given by the simple expression

Dk½c ¼ vDWk 6 c ¼1 if DWk 6 c0 otherwise

ð14Þ

Hence from (13), we get

D½c ¼PM

K¼1 LkminE1kEk

� L2kLk; E2kEk

� L1kLk

h i� vDWk 6 cPM

K¼1 LkminE1kEk

� L2kLk; E2kEk

� L1kLk

h i ð15Þ

862 P. Glenn et al. / Transportation Research Part B 38 (2004) 853–867

If we consider a distance where c ¼ DWk, the curve jumps upwards with a jump of relative size

Rk ¼ LkminE1kEk

� L2kLk

;E2kEk

� L1kLk

� �ð16Þ

To explain what is happening with the curve, we fix a distance d0 and consider all the job cate-gories where c06DWk 6 c0 þ Dc0. If Dc0 is not too small, there will be many job categories withinthis range of wage differences. What controls the size of the jumps is mainly the first term in (16),

i.e. the size of the total population, Lk. Consider the second term in (16), i.e., minE1kEk

� L2kLk; E2kEk

� L1kLk

h i.

Note that the largest value of this term is obtained whenever

E1k ¼ E2k ¼ L1k ¼ L2k ¼ 0:5 � Ek

If E1k 6¼ 0:5 � Ek, however, the largest value is obtained whenever L1k ¼ E1k. Hence, the moresymmetry, the more impact the term has on the final curve. This term thus adjusts for an unevenspread in the fraction of employment opportunities in a given job category between each town inrelationship to the fraction of workers in the same job category. If we assume that the two sto-chastic terms in (16) are independent of each other, i.e., that the proportion of local employmentin each job category is largely independent of the population size, Lk, then the effect of theadjustments will average out. This being so, the size of the jump in the interval ½c0; c0 þ Dc0 will beproportional to

Xk:c0 6DWk 6 c0þDc0

Lk X

k:c0 6DWk 6 c0þDc0

Expectation½Lk ð17Þ

invoking the law of large numbers. What happens with the average number of workers in thesejob categories when c0 increases and Dc0 is fixed? We think it is reasonable to assume that

• The number of job categories with c06DWk 6 c0 þ Dc0 goes down.

For example, there will be fewer job categories where the wage differences between the twotowns are in the interval between $10,000 and $10,500 than in the interval between $1000 and$1500.

• The expected population, Expectation[Lk], goes down.

That is, that jobs with large wage differences are generally more specialized, consisting of asmaller number of workers.Under these assumptions, the relative increase in D will decrease with increasing c, giving rise to

a globally concave profile. This argument applies to the 2-job game. For the N -job game theanalysis is much more difficult. Some parts of the previous discussion applies, however. Formula(13) can be used in this case as well. Hence much of the behavior is controlled by the coefficient Rk,defined by (16). For the N -job case the particular shape of each Dk, k ¼ 1; . . . ;M is too compli-cated to admit a detailed analysis. From the numerical simulations shown in Fig. 3b, however, it

P. Glenn et al. / Transportation Research Part B 38 (2004) 853–867 863

seems reasonable to expect that the same principle holds for this case as well, i.e., that distance-deterrence curves for aggregated systems can be expected to be globally concave.

5.2. Some remarks

Ubøe (2001) considers aggregated systems of gravity models with an exponential-deterrencefunction, and obtains a similar result: the deterrence functions are concave at moderate and largedistances and close to linear at short distances. Notice that the behavior of each category iscompletely different from the approach in the present paper. In Ubøe (2001) the categories behaveaccording to a random utility maximization. In the present paper the behavior is determined froma fixed rule determined by the game. The deterrence functions for each subcategory of workers arecompletely different in the two cases. Nevertheless we get exactly the same kind of response in theaggregate system.

6. Geometric corrections

Our model takes into account the expenses for commuting between towns, and we have usedthe distance between the centers in each town to calculate the traveling costs. This is a reasonableapproach as long as all internal traveling distances are small. In general, however, the distancesbetween urban centers will have to be adjusted with respect to intrazonal traveling distances.Traveling distance in itself is not crucial; the important factor is the actual difference in travelingdistance between the alternative job locations.Consider the framework in Fig. 4. The first town is separated from the second one by a distance

d between the two centers. The actual difference in traveling distance is, in general, different fromd. To analyze this further we start out with a number of simplifying assumptions. We assume thatmovement is unrestricted, i.e., that any pair of positions within the two towns can be joined by alink with length equal to the euclidean distance between the two positions. We also assume thatjob and residential sites are uniformly distributed within two circles, each of which has radiusr. We let ITD¼ ITD(r) denote the average internal traveling distance. Correspondingly welet ATD¼ATD(r; d) denote the average total commuting distance from the one town to theother. If we are to compare the average difference in traveling expenses between intrazonal andinterzonal commuting, we must consider the average difference in traveling distances,GrðdÞ ¼ ATDðr; dÞ � ITDðrÞ. The graph of this function in the case r ¼ 10 km is shown in Fig. 6(thick line). From the graph we may conclude that the effect of intrazonal traveling is quite sig-nificant. It is easy to verify that limd!1 d � GrðdÞ ¼ ITDðrÞ. Hence the long range correction isequal to the average intrazonal traveling distance.

Fig. 4. Euclidean distances.

Fig. 5. Restricted geometries.

864 P. Glenn et al. / Transportation Research Part B 38 (2004) 853–867

One argument against the construction above is that motion is usually restricted to movementalong fixed roads, and that euclidean distance cannot be obtained. To determine the effect of thiswe carried out the same kind of construction for the cases shown in Fig. 5.In both of these cases travel is restricted to movement along the straight lines in each figure. The

resulting differences in the average traveling distance are shown in Fig. 6 together with the resultsfrom the euclidean case. In Fig. 6 the geometry from the left-hand side of Fig. 5 is represented bythe dotted line, while the geometry from the right-hand side is depicted by the dashed line. Thestraight line corresponds to the case where internal distances are ignored, and the thick linecorresponds to the euclidean correction.Ignoring the part of the curve near d ¼ 20, it is surprising to see that the incorporation of just a

few interconnecting lines pulls the solution strongly in the direction of the euclidean case. Thebehavior near d ¼ 20 is very artificial, and is due to collapsing geometries when the two circlestouch each other. This part of the curve has no practical significance. If two towns are situated ina position such as this, one would expect a number of small roads connecting the areas. Thiswould pull the solution strongly in the direction of the euclidean case.We do not know of studies of trip distribution or trip generation problems that explicitly

introduce a correction procedure related to the measurement of distances. Thorsen and Gitlesen(1998) considered the performance of gravity models to explain commuting flows. One result isthat model performance improves significantly when a parameter is introduced in the distance-deterrence function that represents an additive constant term attached to the diagonal elements ofthe trip-distribution matrix. One possible interpretation for this is that aggregation problems andmeasurement errors are involved in the division of the study area into discrete zones and asimplified representation of the road network. The need for corrections depends on the nature ofthe geography that is considered. If employment and population are scattered over a wide areawithin a zone, while the distances between the corresponding centroids are relatively short, then alarge gain can be expected to result from a correction procedure. As De la Barra (1998) points out

Fig. 6. Geometric corrections.

Fig. 7. A model for distance deterrence.

P. Glenn et al. / Transportation Research Part B 38 (2004) 853–867 865

the relevant kind of spatial aggregation problems might be a source of considerable stochasticvariation in modeling spatial interaction, reducing the potential for significant conclusions.

7. Composition of distance deterrence and geometric corrections

From the arguments in Section 6, it seems reasonable to model the distance deterrence from aglobally concave function depending on the actual traveling distance x. As an example of this, wesuggest using the simple expression

DðxÞ ¼ 1� e�cx ð18Þ

where we, in the spirit of Section 2, set c ¼ 1

d1lnð1aÞ. In applications of this theory, however, we

usually do not refer to actual traveling distances as d but rather refer to the difference between twocity centers as d. As argued in Section 6, we quite often expect to find a systematic bias between dand the average difference in traveling distance between intrazonal and interzonal traveling. Fromthis point of view, we suggest using the euclidean correction v ¼ GrðdÞ. Here, r is the radius of thetown. In the spirit of Section 2 again, we put r ¼ d0. This gives us

Dd0;d1ðdÞ ¼ DðxÞ ¼ 1� e�cGd0ðdÞ with c ¼ 1

Gd0ðd1Þln

1

a

� ð19Þ

Assuming a constant relationship between physical and generalized distance, we let physicaldistance appear in the deterrence function in this section. A plot of this function using theparameter values a ¼ 5%, d0 ¼ 10 and d1 ¼ 60 is shown in Fig. 7.Thorsen et al. (1999) suggested that one could model DðdÞ using a logistic function. This is the

function shown in Fig. 1. As we can see from Fig. 7, the resulting graph is not very different.Nevertheless, it is our opinion that the composite structure above is more satisfying from atheoretical point of view.

8. Concluding remarks

In this paper we have discussed some theoretical aspects of distance-deterrence functions inmodeling commuting to work. We started out by defining a game where each worker applies for a

866 P. Glenn et al. / Transportation Research Part B 38 (2004) 853–867

job in one of two towns in the geography. The simplest case occurs when there is only one kind ofjob in each town. Commuting flows are the result of a unique Nash equilibrium determined bywage differentials and the distance between the towns. A more complex situation arises in the casewhere there are many different job alternatives for each type of worker (profession). We base ourdiscussion on the assumption that the individual workers are qualified for one and only one jobcategory.We demonstrate that commuting flows in such situations result not only from wage dif-

ferentials and distances, but also from a spatial mismatch between the types of jobs and thecategories of workers. This represents a potential problem when distance-deterrence parame-ters are estimated from aggregated data on commuting flows. We then carry out simulationexperiments based on the specification of the game. Finally, we discuss analytical results forthe resulting distance-deterrence function in the case with a very large number of workercategories (professions). Under quite general conditions we are able to demonstrate that thedistance-deterrence function is globally concave when observed on a scale that is not toosmall.The globally concave distance-deterrence function does not account for possible measurement

errors resulting from a practice where distances between zones are measured relative to the zonalcenters, while the jobs and the residents are in general more evenly scattered over the region. Weargue that this calls for a geometric correction of distances, and demonstrate that this correctiontypically should correspond to the average intrazonal traveling distance. Through this geometriccorrection of the otherwise globally concave function we end up with a distance-deterrencefunction with a logistic profile.It is of course possible to extend the game-theoretical analysis considerably in many directions

from the simple framework that is offered in this paper. The main purpose of this paper is todemonstrate that it is possible to derive a profile for the relationship between distances andcommuting flows from a game-theoretical specification of a labor market equilibrium. This profilecan next be compared to the alternative distance-deterrence functions that appear in the literature.Considering, for example, the experiences with Box–Cox transformations that are mentioned inthe introduction, we find it surprising that so little attention has been directed towards purelytheoretical aspects of the distance-deterrence relationship in the literature on spatial interactionproblems.

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