22nd international symposium on transportation and traﬃc

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ScienceDirectTransportation Research Procedia 00 (2016) 000–000

www.elsevier.com/locate/procedia

22nd International Symposium on Transportation and Traffic Theory

First-Best Dynamic Assignment of Commuterswith Endogenous Heterogeneities in a Corridor Network

Minoru Osawa a, Haoran Fu b,∗, Takashi Akamatsu b,∗

aSchool of Engineering, Tohoku University, Aramaki Aoba 6–6, Aoba-ku, Sendai, Miyagi, 980–8579, JapanbGraduate School of Information Sciences, Tohoku University, Aramaki Aoba 6–6, Aoba-ku, Sendai, Miyagi, 980–8579, Japan

Abstract

We study a parsimonious theory that synthesizes short-term traffic demand management (TDM) policies with long-term endoge-nous heterogeneities of demand. In a corridor network with multiple discrete bottlenecks, we study a model of system optimalassignment that integrates the short-term problem (departure time choice with tolling) and the long-term problem (job and residen-tial location choice). For the short-term departure-time-choice equilibrium, under mild assumptions on schedule delay function,we derive analytical solutions under a first-best TDM scheme. Investigating properties of long-term equilibria, we found that theoverall equilibrium pattern exhibits remarkable spatio-temporal sorting properties. It is further shown that a lack of integration ofthe short- and the long-term policy results in excessive investments for long-term road construction.c⃝ 2016 The Authors. Elsevier B.V. All rights reserved.Peer review under responsibility of the scientific committee of the 22nd International Symposium on Transportation and TrafficTheory.

Keywords: departure time choice, job-location choice, corridor problem, user heterogeneity, dynamic system optimal assignment

1. Introduction

1.1. Background and purpose

Alleviating traffic congestion in the morning peak hours is one of the most significant challenges in transporta-tion science. Parsimonious and rich in insights, the standard departure-time choice equilibrium model with a singlebottleneck (Vickrey, 1969; Hendrickson and Kocur, 1981; Arnott et al., 1990) and its extensions have long been theworkhorse for tackling the problem, yielding a large body of theoretical studies (e.g. Smith, 1984; Daganzo, 1985;Newell, 1987; Lindsey, 2004; van den Berg and Verhoef, 2011; Liu and Nie, 2011). Even though researches have suc-cessfully provided deep implications for the short-term TDM policies, when it comes to the long term, there is roomfor further investigation. In the long term, what is given in the short term is endogenous. Origin–destination demandpatterns might vary over time because of residential location choice of commuters; commuters can also switch theirjobs, thereby affecting value-of-time (VOT) distribution over commuters. Such long-term endogenous variations in

∗ Corresponding author.E-mail address: [email protected] (Haoran Fu), [email protected] (Takashi Akamatsu)

2214-241X c⃝ 2016 The Authors. Elsevier B.V. All rights reserved.Peer review under responsibility of the scientific committee of the 22nd International Symposium on Transportation and Traffic Theory.

2 M. Osawa, H. Fu, and T. Akamatsu / Transportation Research Procedia 00 (2016) 000–000

demand inevitably affect long-term effectiveness of short-term TDM policies. Extending the short-term framework toincorporate long-term demand dynamics thus seems to be a valuable direction of study.

The purpose of the present paper is to introduce a parsimonious theory that synthesizes long-term endogenousheterogeneities of traffic demands with the short-term TDM policy. To this end, keeping the model as compact aspossible, we extend the standard bottleneck model to allow each commuter to choose his/her own residential locationand job in the long-run, so that endogenous commuter heterogeneities emerge. Also, our analysis is conducted in acorridor network with multiple bottlenecks a la Akamatsu et al. (2015) to allow rich spatial and temporal dynamics.

We first prove that, under the short-term first-best policy, any integrated equilibrium for the whole problem isPareto-efficient. We also show that there is an equivalent optimization formulation that is a highly structured linearprogramming problem, whose structure yields a natural decomposition into the two components of the short- andthe long-term. We further show that the short-term component (departure time choice equilibrium) is analyticallysolvable under mild assumptions on the schedule delay function.1

Interestingly, equilibrium job–location–departure-time choice patterns are shown to exhibit striking sorting reg-ularities in both the short- and the long-term. In the short-term equilibrium traffic flow, we observe a job- andlocation-based temporal sorting. Specifically, (i) commuters who work for jobs with higher VOT arrive at the CBD attimes closer to their desired arrival time, and (ii) commuters who reside at distant locations have strictly longer arrivaltime window. In the long-term equilibrium patterns, we observe a job-based spatial sorting, where commuters whowork for jobs with higher VOT choose to live closer to the CBD.

As a major policy implication, we show that, without integration of the short- and the long-term TDM policy,the total investment for bottleneck capacity is overestimated; without consideration of the long-term decision ofcommuters, the road manager results in excessive extension of road capacity.

1.2. Related literature

To date, only Arnott (1998), Gubins and Verhoef (2014), and Takayama and Kuwahara (2016) have developeddynamic bottleneck models that include location choice of commuters. Our study extends the scopes of these studiesin two ways. First, without sacrificing tractability, we generalize the single bottleneck setup to a corridor with multiplebottlenecks. The setting allows us to examine richer spatial dynamics in both the short and the long term. Second, weallow endogenous job choices that determine commuter heterogeneities (regarding their VOT), which is novel.

For the short-term component, our study formulates a dynamic system optimal (DSO) assignment which is achievedas the equilibrium under a short-term TDM scheme that eliminates the bottleneck queues. A DSO assignment prob-lem with similar network setting (a freeway corridor with multiple discrete on- and off-ramps) is studied by Shen andZhang (2009). Tian et al. (2012) is also in related direction, albeit the model is formulated via a continuum approx-imation a la Arnott and de Palma (2011). The prevalent problem is that analytical solutions are hard to obtain evenwhen one assume piecewise linear schedule delay and homogeneous commuters. We contribute to this literature byproviding analytical solutions (i.e., equilibrium traffic flow, equilibrium commuting cost) under mild assumptions onthe schedule delay function, as well as allowing commuter heterogeneity.

2. Model

Consider a freeway corridor that connects I residential locations to a central business district (CBD), as illustratedby Fig.1. The residential locations are indexed sequentially from the CBD. We denote the set of locations by I ≡{1, 2, . . . , I}. There is a single bottleneck with capacity µi just downstream of the on-ramp from the location i ∈ I,which we call the “bottleneck i”. Dynamic queueing at the bottlenecks are modeled by the standard point queue modelalong with first-in-first-out; at each bottleneck, a queue is formed vertically when the inflow exceeds the capacity.Each residential location i ∈ I is endowed with Ai unit of land. At the CBD, there are J job opportunities. The labordemand and the VOT for each job j ∈ J ≡ {1, 2, . . . , J} are given by L j and α j, respectively.

1 The result contrasts to previous studies which usually assume piecewise-linear schedule delay. This generalization becomes possible partlybecause we consider dynamic system optimal assignment. Nonetheless, it is noted that the basic strategy presented in this paper is effective andcan be used to tackle dynamic user equilibrium assignment in a corridor network (Fu et al., 2016).

M. Osawa, H. Fu, and T. Akamatsu / Transportation Research Procedia 00 (2016) 000–000 3

Fig. 1: The corridor network

We assume that the tradable network permits (TNP) scheme is implemented in the corridor network.

Assumption 2.1. The TNP scheme (Akamatsu et al., 2006; Akamatsu, 2007; Wada and Akamatsu, 2013) is imple-mented by the road manager as the short-term TDM policy.

In the TNP sfcheme, a permit allows its holder to pass through a pre-specified bottleneck during a pre-specifiedtime period, and a perfectly competitive trading market for the network permits is launched. The road managerlimits the number of permits to be either equal to or less than the capacity of the bottleneck. Then, the queue ateach bottleneck is completely eliminated so that there is no queueing delay. This scheme internalizes congestionexternalities and is mathematically equivalent to an optimal dynamic road pricing scheme that completely eliminatesbottleneck congestion; in effect, equilibrium values of the permits coincide with optimal congestion toll—without anyknowledge of users’ preference. For details of the scheme, see Wada and Akamatsu (2013).

There are Q ex-ante identical workers in the network that commute to the CBD. Each commuter makes threechoices to maximize his/her own utility: residential location i ∈ I and job j ∈ J in the long term and the arrival timet ∈ T at the CBD in the short term, where T denotes a sufficiently long arrival time window.2 Commuters’ utility isassumed to be homogeneous and quasilinear, i.e., the utility function is linear with respect to the numeraire.3

At every morning, each commuter choose his/hers arrival time t at the CBD to minimize his/her own travel cost.4

The commuting cost of a worker who resides at location i ∈ I with job j ∈ J (henceforth “an (i, j)-commuter”) isexpressed in terms of arrival time t ∈ T at the CBD:

Ci, j(t) ≡ ci, j(t) +∑m≤i

pm(t) ∀i ∈ I, j ∈ J , t ∈ T , (1)

where ci, j(t) is the (i, j, t)-specific monetary commuting cost, pi(t) is the price of a permit at the bottleneck i with CBDarrival time t. Note that the second term is the total amount of the permit costs a commuter from location i has to payto arrive at t at the CBD. Also note that because we assume the TNP scheme, the queueing delay cost is absent.

We assume that ci, j(t) is a sum of schedule delay cost and free-flow travel cost.

Assumption 2.2. ci, j(t) is expressed as ci, j(t) ≡ α j {s(t) + di}, where α j is the VOT for job j, s(t) is the job-independentschedule delay function measured in time unit, and di is the free-flow travel time from residential location i to the CBD(d1 < d2 < · · · < dI). s(t) is continuous, strictly quasiconvex, and strictly minimized at a unique desired arrival timet0 that is common to all commuters. The job-indices { j} are in the descending order of the VOT: α1 > α2 > · · · > αJ .

Even though VOTs are differentiated with respect to job, all groups share exactly the same schedule delay preference.We will discuss relaxation of the apparently restrictive assumptions at the end of Section 4.

In the long term, in addition to their arrival time at the CBD, commuters can choose their jobs and locations tomaximize their own utility. For simplicity, we further assume that land consumption of each consumer is unity. Then,

2 For the time index, we take the arrival time t at the CBD rather than the absolute departure time at each bottleneck to simplify the analysis. Fordiscussions on the merits of using the arrival time t at the CBD (i.e., “time-based Lagrangian” coordinate system that focuses on each commuter)rather than departure time at the bottlenecks (i.e., “Eulerian” absolute coordinate system), see Akamatsu et al. (2015).

3 For the standard definition of quasilinear preference, see, for example, Mas-Colell et al. (1995) (Definition 3.B.7). In effect, the indirect utilityfunction is expressed by (2).

4 Since we assume quasilinear preference, the cost minimization behavior is consistent with long-term utility maximization discussed below.


the maximum number of commuters that can reside at location i coincide with the total land endowment Ai. Let ri

be the market land rent at i and w j the market wage of job j which are endogenously determined in the markets. Weassume that the land and job markets are perfectly competitive. Then, an (i, j)-commuter’s indirect utility is expressedas the following function of arrival time t ∈ T :

vi, j(t) ≡ w j − ri −Ci, j(t) ∀i ∈ I, j ∈ J , t ∈ T . (2)

Commuters choose i, j, and t to maximize their own utility: maxi, j, t vi, j(t) = maxi, j{w j − ri −mint Ci, j(t)}.Let qi, j(t) be the arrival-flow rate at the CBD of (i, j)-commuters at t. At an equilibrium, there is no incentive for

any commuter to switch his/hers choice of (i, j, t) unilaterally; that is, the following condition should hold true:5V − vi, j(t) = 0 if qi, j(t) > 0V − vi, j(t) ≥ 0 if qi, j(t) = 0

∀i ∈ I, j ∈ J , t ∈ T , (3)

where V is the equilibrium utility level. Also, the following three demand–supply equilibrium condition at eachmarket should be satisfied:6

(tradable permit market)

µi −∑

m≥i∑

j qm, j(t) = 0 if pi(t) > 0µi −

∑m≥i

∑j qm, j(t) ≥ 0 if pi(t) = 0

∀i ∈ I, t ∈ T , (4)

(land market)

Ai −∑

j

∫T qi, j(t)dt = 0 if ri > 0

Ai −∑

j

∫T qi, j(t)dt ≥ 0 if ri = 0

∀i ∈ I, (5)

(labor market)

L j −∑

i

∫T qi, j(t)dt = 0 if w j > 0

L j −∑

i

∫T qi, j(t)dt ≤ 0 if w j = 0

∀ j ∈ J . (6)

Lastly, the number of commuters should not change:∑i∈I

∑j∈J

∫T

qi, j(t) dt = Q. (7)

An integrated equilibrium, which is a long-run equilibrium that respects short-run effects, is defined as follows:

Definition 2.1. An integrated equilibrium is a collection of variables (V∗, q∗ ≡ {q∗i, j(t)}, p∗ ≡ {p∗i (t)}, r ≡ {r∗i },w ≡{w∗j}) that satisfies (3), (4), (5), (6), and (7).7

In the following, we decompose the model into two interdependent components of the short- and the long-term.

3. Equivalent optimization problems

First, by formulating an equivalent optimization problem (EOP) for our integrated equilibrium problem, the presentsection shows that any integrated equilibrium is efficient. Then, we decompose the EOP into two natural components:the short- and the long-term. According to the decomposition, the short- and the long-term equilibria are defined.

5 One may think that the formulation is too simplified because in reality the choice of t and those of i and j are the problems with very differenttime scales. Actually, in the following, we do decompose the integrated equilibrium problem into two interdependent components of the short- andthe long-run and analyze them sequentially.

6 To keep the model compact, we assume that the wage is dependent solely on demandsupply condition and abstract from the effects fromproductivity of firms or other realistic determinants of the wage. The job–location choice modelling in our formulation corresponds to the celebratedHerbert–Stevens (H–S) model (Herbert and Stevens, 1960; Fujita, 1989) in the urban economics literature, as per revisited by Wheaton (1974). TheH–S model has long provided rich insights into the internal structure of cities; applications include the seminal work of Fujita and Ogawa (1982).

7 Observe that our integrated equilibrium is formulated as an equilibrium under the TNP scheme. If, instead, one assumes a top-down congestion-tolling scheme, the equilibrium conditions should include the queueing delays and their dynamics. The value of optimal tolling should be alsodetermined through a mathematical programming with equilibrium constraints (MPEC).


3.1. Efficiency of integrated equilibria

We expect that our equilibrium problem is equivalent to a socially optimal (or efficient) allocation problem sincethere are no externalities: the land and labor markets are perfectly competitive, whereas congestion externalities dueto queueing at the bottlenecks are internalized by the short-term TDM policy (i.e., TNP scheme). In fact, we have thefollowing proposition and its corollary, which confirm the intuition.

Proposition 3.1. An integrated equilibrium with any utility level V is obtained as a solution of the following linearprogramming problem (LP):

[Int] minq≥0

z(q) ≡∑i∈I

∑j∈J

∫T

ci, j(t) qi, j(t) dt, (8)

s.t.∑m≥i

∑j∈J

qm, j(t) ≤ µi (pi(t)) ∀i ∈ I, t ∈ T , (9)

∑j∈J

∫T

qi, j(t) dt ≤ Ai (ri) ∀i ∈ I, (10)

∑i∈I

∫T

qi, j(t) ≥ L j (w j) ∀ j ∈ J , (11)

with∑

i Ai =∑

j L j = Q, where the price variables p, r, and w are determined as the Lagrange multipliers for theconstraints (9), (10), and (11), respectively.

Proof. Comparing the first-order conditions with the equilibrium conditions, one shows the assertion.

Corollary 3.1. An integrated equilibrium (Definition 2.1) achieves a socially optimal (efficient) allocation in the sensethat it minimizes the total social transportation cost in the network.

Note that, since at any equilibrium the total permit costs paid by the commuters coincides with the total revenueof the road manager from the TNP system, the permit costs cancels out if one consider the total social cost; one canactually regard z(q∗) as the total social cost incurred by an (integrated equilibrium) assignment q∗. We thus concludeany integrated equilibrium under the TNP scheme is regarded as a social optimal assignment.

3.2. Decomposition into the short- and the long-term components

We study the properties of integrated equilibria, i.e., those of solutions for the EOP [Int]. The EOP admits a naturaldecomposition into the short- and the long-term components. To see this, first define an intermediate variable Qi, j by

Qi, j ≡∫T

qi, j(t) dt ∀i ∈ I, j ∈ J . (12)

Then, Q = {Qi, j} is interpreted as a job–location choice pattern of commuters, which reflects the long-term decisionof commuters. Observe that z(q) is decomposed as z(q) = z1(q) + z2(Q), where

z1(q) ≡∑i∈I

∑j∈J

∫T{α js(t)} qi, j(t) dt and z2(Q) ≡

∑i∈I

∑j∈J{α jdi}Qi, j. (13)

Also, (10) and (11) can be rewritten by Q. In effect, [Int] is decomposed into the long- and short-term components.8

8 The decomposition corresponds to the so-called Benders decomposition.


3.2.1. Short-term problemThe short-term problem for a given Q is formulated as the following LP:

[ST–Primal] minq≥0

z1(q) s.t. (9) and (12), (14)

where Q constrains q via (12). Its associated dual is

[ST–Dual] maxp≥0,ρ

z1(ρ, p) ≡∑i∈I

∑j∈Jρi, j Qi, j −

∑i∈Iµi

∫T

pi(t) dt, (15)

s.t. α js(t) +∑m≤i

pm(t) ≥ ρi, j (qi, j(t)) ∀i ∈ I, j ∈ J , t ∈ T , (16)

where ρ ≡ {ρi, j} and p = {pi(t)} are the Lagrange multipliers associated to the constraints (9) and (12), respectively.Via strong duality, it follows that the optimal values of [ST–Dual] and [ST–Primal] coincide:

z1(Q) ≡ z1(ρ∗, p∗ | Q) = z1(q∗ | Q), (17)

where the superscript “∗” denotes that the associated variable is evaluated at the optimal solution; (ρ∗, p∗) and q∗ arethe dual and primal optimal points, respectively. It is also used to express equilibrium variables, as in Definition 2.1.

The first-order optimality conditions for [ST–(Primal /Dual)] areρi, j = α js(t) +∑

m≤i pm(t) if qi, j(t) > 0ρi, j ≤ α js(t) +

∑m≤i pm(t) if qi, j(t) = 0

∀i ∈ I, j ∈ J , t ∈ T (18)

along with (4) and (12). Observe that the condition (18) is interpreted as a condition for optimal arrival-time choicebehavior of commuters under fixed Q (i.e., a fixed job-location choice pattern) if we interpret {pi(t)} as permit prices.In this context, ρi, j is regarded as the equilibrium commuting cost of (i, j)-commuters, whereas

∑m≤i pm(t) is the total

permit cost a commuter from the residential location i has to pay to arrive at the CBD at time t ∈ T . Conversely, weshall regard the problems [ST–(Primal /Dual)] as EOPs for the short-term (arrival-time-choice) equilibrium problem.We define a short-term equilibrium as follows.

Definition 3.1 (Short-term equilibrium). A short-term equilibrium is a collection of variables (q∗, ρ∗, p∗) that satisfies(4), (12), and (18).

Note that the right hand sides of (18) differs from Ci, j(t) defined by (1) in a sense that the time-independent fixedcommuting cost α j di is absent. It is because α j di is irrelevant for commuters’ choices if we fix the pattern of Q.Instead, {α jdi} appears in z2(Q) defined by (13), the objective function of the long-term component discussed below.

3.2.2. Long-term problemUsing the (dual) optimal value function z1(Q), the long-term problem is formulated as follows, where we note that

(10’) and (11’) are reformulations of (10) and (11) in terms of Q, respectively:

[LT] minQ≥0

z1(Q) + z2(Q), (19)

s.t.∑j∈J

Qi, j ≤ Ai (ri) ∀i ∈ I, (10’)∑i∈I

Qi, j ≥ L j (w j) ∀ j ∈ J , . (11’)

Applying the envelope theorem to [ST–Dual], we see that ρ∗ satisfies ∇z1(Q) = ρ∗(Q). Note that ρ∗i, j is interpretedas a function of Q (i.e., the dual solution for [ST–Dual] for given Q). The first-order optimality conditions for [LT] arew j − ri − ρ∗i, j(Q) − α jdi = 0 if Qi, j > 0

w j − ri − ρ∗i, j(Q) − α jdi ≤ 0 if Qi, j = 0∀i ∈ I, j ∈ J , (20)


Fig. 2: T , t−(T ), t+(T ), and s(T ), in relation to s(t).

and

(land market)

Ai −∑

j Qi, j = 0 if ri > 0Ai −

∑j Qi, j ≥ 0 if ri = 0

∀i ∈ I, (5’)

(labor market)

L j −∑

i Qi, j = 0 if w j > 0L j −

∑i Qi, j ≤ 0 if w j = 0

∀ j ∈ J . (6’)

Here, {ri} and {w j} are the Lagrange multipliers associated to (10’) and (11’), respectively.As in [ST–(Primal /Dual)], we interpret (20) as the condition for optimal job–location choice behavior of com-

muters, where {ri}, {w j}, and {ρ∗i, j(Q) + α jdi} are each interpreted as land rent, wage, and commuting cost. Theconditions (5’) and (6’) express market clearing. For instance, the condition (5’) determines land rent {ri} via demand–supply balance. It states that, if the total land demand

∑j Qi, j at a location i ∈ I is smaller than its land supply Ai,

i.e., if there is excess supply, ri should be zero; if ri is positive, then demand equals supply. In a similar manner, (6’)determines wage {w j}. Analogous to the short-term equilibrium, we define a long-term equilibrium as follows.

Definition 3.2 (Long-term equilibrium). A long-term equilibrium is a collection of variables (Q∗, r∗,w∗) that satisfies(20), (5’), and (6’).

In [LT], the short-run (dual) optimal value function z1(Q) represents the feedback effect from the short-run compo-nent. Even though z1(Q) is, conceptually, a function of Q, we do not yet have analytical formula of it. Thus, usuallywe cannot solve [LT] independently of [ST–(Primal /Dual)] and an iteration between [ST–(Primal /Dual)] and [LT]is required. Fortunately, Section 4 shows that [ST–(Primal /Dual)] is analytically solvable; i.e., the short-term flowpattern q∗(Q), as well as the associated Lagrange multipliers ρ∗(Q) and p∗(Q), are determined as functions of Q. Usingthe result, Section 5 demonstrates that z1(Q) is also analytically evaluated in terms of Q.

4. Analysis of the short-term equilibrium

This section analyzes the properties of the problems [ST–(Primal /Dual)], which are equivalent to the short-termarrival-time-choice equilibrium under TNP (Definition 3.1), or to a dynamic system optimal assignment (in the senseof Corollary 3.1). We first review known results for two special cases of the problem: (i) the single-bottleneck case(i.e., I = 1) (Akamatsu et al., 2016), and (ii) the homogeneous commuter case (i.e., J = 1) (Yodoshi and Akamatsu,2008; Fu, 2015; Fu et al., 2016). We show that analytical solutions for general setting (i.e., I > 1 and J > 1) isnaturally obtained by combining the properties of (i) and (ii).

To simplify the presentation throughout this section, we introduce some notations. First, we define t−, t+, ands based on the common schedule delay function s(t). t−(T ) is defined as the unique solution for the equation s(t) =s(t+T ) with t ≤ t0, T ≥ 0 (see Fig.2). Using t−, we define t+ and s by t+(T ) ≡ t−(T )+T and s(T ) ≡ s(t−(T )) = s(t+(T )).

Next, we define {α j} and {µi} by α j ≡ α j − α j+1 and µi ≡ µi − µi+1, respectively, where αJ+1 ≡ 0, µI+1 ≡ 0.In addition, t−i, j and t+i, j denote the earliest and latest arrival time of (i, j)-commuters, respectively. We denote the

arrival-time window Ti, j of (i, j)-commuters and its length Ti, j by Ti, j ≡ [t−i, j, t+i, j] and Ti, j ≡ |Ti, j| = t+i, j − t−i, j. Note that

Ti, j is the convex hull of the support of (i, j)-specific arrival flow. Finally, we denote T0 ≡ {t0}.


(a) Equilibrium arrival flow rates at the CBD. (b) Equilibrium permit prices and commuting costs.

Fig. 3: Equilibrium solutions for [ST–I1]

4.1. Special case (i): Analytical solutions for a single bottleneck problem with heterogeneous commuters

The single-bottleneck case (I = 1) is well-studied in the discipline; for this case, [ST–Primal] reduces to:9

[ST–I1] minq≥0

∑j∈J

∫Tα js(t) q1, j(t) dt, (21)

s.t.∑j∈J

q1, j(t) ≤ µ1 (p1(t)) ∀t ∈ T , (22)∫T

q1, j(t) dt = L j (ρ1, j) ∀ j ∈ J . (23)

Note that, as we abstract from location choice decision of commuters in this case, we interpret L j = Q1, j.Akamatsu et al. (2016) showed that the problem can be treated as an instance of mass transportation problem

(Rachev and Ruschendorf, 1998). In particular, the coefficients {α js(t)} is shown to satisfy Monge property.10 Mongeproperty of {α js(t)} yields that the equilibrium is unique, and commuters of higher VOT α j arrive at times closer tothe unique desired arrival-time t0 in equilibrium.11 In concrete terms, the arrival time windows exhibit the followingjob-based temporal sorting property:

Lemma 4.1 (Job-based sorting properties in [ST–I1]). Consider an equilibrium solution for [ST–I1]. Under Assump-tion 2.2, the arrival-time windows {T1, j} satisfy {t0} ⊂ T1, j ⊂ T1, j+1 for all j ∈ J \ {J}, or equivalently

t−1,J < · · · < t−1,2 < t−1,1 < t0 < t+1,1 < t+1,2 < · · · < t+1,J . (24)

The equilibrium values of t±i, j are determined by t−1, j = t−(T1, j) and t+1, j = t+(T1, j), where T1, j =∑

n≤ jLn/µ1.

The above sorting regularity yields the following analytical solution for [ST–I1].12

9 The EOP [ST–I1] is the one proposed by Iryo and Yoshii (2007). In light of this, our model [ST–Primal] is a corridor generalization of it.10 An array {Mi, j} is said to satisfy the (strict) Monge property if Mi, j +Mi+1, j+1 > Mi, j+1 +Mi+1, j for all i, j. For reviews on Monge property and

its applications, see for example Bein et al. (1995) and Burkard (2007).11 Some previous studies also discuss temporal sorting properties in departure-time choice equilibrium problems with user heterogeneity (Newell,

1987; Arnott et al., 1992, 1994; Liu et al., 2015). In contrast to the results presented in this section, the previous studies assume piecewise linearschedule delay to obtain concrete results on sorting phenomena.12 Due to space limitation, proofs are provided online at: https://sites.google.com/site/minoosawa/


Proposition 4.1. The solution for [ST–I1] (q∗, ρ∗, p∗) is obtained as follows:

• arrival flow rates: q∗1, j(t) =

µ1 if t ∈ T1, j \ T1, j−1

0 otherwise∀ j ∈ J , (25)

• commuting cost: ρ∗1, j =∑n≥ j

α j s(T1, j) ∀ j ∈ J , (26)

• permit price: p∗1(t) =

ρ∗1, j − α js(t) if t ∈ T1, j \ T1, j−1

0 otherwise∀ j ∈ J . (27)

Fig.3 illustrates Lemma 4.1 and Proposition 4.1 for the case of J = 2. Fig.3a depicts equilibrium flow rates, whichalways coincide with the bottleneck capacity µ1 during the peak periods. Note that the length of arrival time windowsare simply determined via exogenous constants {L j} by T1,1 =

∣∣∣T1,1∣∣∣ = L1/µ1 and T1,2 =

∣∣∣T1,2∣∣∣ = (L1 + L2)/µ1 (Lemma

4.1). From (25), it follows that the area of the red region equals L1, whereas the total area of the blue regions L2. Alsonote that L1 + L2 = Q. Using {T1, j}, the actual values of {t±1, j} is determined via the functions t± illustrated in Fig.2.

On the other hand, Fig.3b depicts permit price p∗i (t) at the bottleneck as a function of arrival time (in relation toequilibrium commuting costs {ρ∗1, j}). It is noted that the black curve is closely related to the so-called isocost curve(Cohen, 1987; Lindsey, 2004). Yet, there is a subtle but significant difference in that usual isocost curves expresstime-dependent queueing delay—instead of permit prices. Nonetheless, we shall call it the isocost curve (in terms ofmoney), as the role it plays for our analysis is quite similar to usual ones.

Fig.3b also graphically illustrates how to recursively construct the isocost curve (or graph of time-dependent permitprices). We first define the following time-varying “job-specific willingness to pay” for permit costs conditional on agiven level of the commuting cost ρ:

P j(t | ρ) ≡ ρ − α js(t). ∀ j ∈ J (28)

Then, in equilibrium it must be that p∗1(t) = max j{P j(t | ρ∗1, j)}. As we already know the arrival time windows T1,1and T1,2, we know that P2(t−1,2 | ρ∗1,2) = P2(t+1,2 | ρ∗1,2) = 0. It yields that ρ∗1,2 = α2 s(T1,2), so that the equilibriumcurve P2(t | ρ∗1,2) in Fig.3b is determined. Then, since t±1,1 are the crossing times of P2(t | ρ∗1,2) and P1(t | ρ∗1,1), theequilibrium cost ρ∗1,1 for job j = 1 is determined by the equation P1(t+1,1 | ρ∗1,1) = P2(t+1,1 | ρ∗1,2). One can observe thatno commuter can improve his/her commuting cost by changing the arrival time; e.g., no commuter with job j = 2 canafford to buy permit to arrive at t ∈ T1,1 since p∗1(t) = P1(t | ρ∗1,1) ≥ P2(t | ρ∗1,2) for t ∈ T1,1.

4.2. Special case (ii): Analytical solutions for a corridor problem with homogeneous commuters

Next, we consider another special case where commuters are identical with respect to their jobs (i.e., J = 1) butthe network structure is the original corridor in Fig.1. This homogeneous case also admits analytical solutions andexhibits another temporal sorting regularities in equilibrium (Fu, 2015; Fu et al., 2016).

We set J = 1, α1 = 1 and [ST–Primal] reduces to the following problem:

[ST–J1] minq≥0

∑i∈I

∫T

s(t) qi(t) dt, (29)

s.t.∑m≥i

qm(t) ≤ µi (pi(t)) ∀i ∈ I, t ∈ T , (30)∫T

qi(t) dt = Ai (ρi) ∀i ∈ I, (31)

where we omit job subscript j = 1 and denote qi(t) ≡ qi,1(t), ρi ≡ ρi,1 for expositional simplicity. Note that, as weabstract from job choice of commuters, we interpret Ai = Qi,1 in this case.

First, two concepts of false bottleneck and reduced network are introduced to simplify the presentation.

Definition 4.1 (False bottleneck). A false bottleneck is a bottleneck i whose permit price is always zero in equilibrium.


Definition 4.2 (Reduced network). A reduced network is a network with no false bottlenecks.

In a corridor network, a false bottleneck does not affect essential properties of the solution. In equilibrium, passingsuch a bottleneck at any time does not affect commuting cost for any commuter since permit price always equals zero.It causes some irrelevant freeness in analytical solution. To avoid such unnecessary complications in presentation, weshall consider a reduced corridor network. Fortunately, we have a simple criterion to detect false bottlenecks. First,define the normalized demand Di, or the demand/supply ratio, at each bottleneck i, by Di ≡

∑m≥i Am/µi. Note that the

numerator of Di is the total upstream demand which is to pass through the bottleneck i. Using the normalized demandDi, we have the following lemma that provides the means to detect a false bottleneck.

Lemma 4.2 (Detection of a false bottleneck). For the problem [ST–J1], a bottleneck i ∈ I is a false bottleneck if andonly if Dm ≥ Di for some m < i.

From any setting of {Ai} and {µi}, one can obtain a reduced corridor network by sequentially removing false bottle-necks. We thus justify proceeding with a reduced corridor network assumption.

The following lemma and proposition summarizes the analytical results for [ST–J1] in the reduced corridor network.First, arrival time windows Ti ≡ Ti,1 satisfy the following location-based temporal sorting property.

Lemma 4.3 (Location-based sorting properties in [ST–J1]). Consider a equilibrium solution for [ST–J1]. UnderAssumption 2.2, the arrival-time windows {Ti} satisfy {t0} ⊂ Ti ⊂ Ti+1 for all i ∈ I \ {I}, or equivalently

t−I < · · · < t−2 < t−1 < t0 < t+1 < t+2 < · · · < t+I . (32)

The equilibrium values of t±i are determined according to Ti ≡ |Ti| as t−i = t− (Ti) and t+i = t+ (Ti), where Ti = Ai/µi.

In other words, commuters with a smaller location index always arrive at the CBD nearer to the desired arrival time.On the basis of Lemma 4.3, we obtain the following proposition which summarize the equilibrium solution in a

reduced corridor network with J = 1.

Proposition 4.2. In a reduced corridor network, the solution for [ST–J1] (q∗, ρ∗, p∗) is obtained as:

• arrival flow rates: q∗i (t) =

µi if t ∈ Ti

0 otherwise∀i ∈ I (33)

• commuting cost: ρ∗i = s(Ti) ∀i ∈ I (34)

• permit price: p∗i (t) =

ρ∗i − s(t) if t ∈ Ti \ Ti−1

ρ∗i − s(t) −∑m≤i−1 p∗m(t) if t ∈ Ti−1

0 otherwise∀i ∈ I. (35)

Fig.4 illustrates the analytical solution in this proposition for the case of I = 3. Fig.4a depicts equilibrium arrivalflow rates at the CBD. One can reconstruct disaggregated, i-specific arrival flow rates q∗i (t) from the figure sinceq∗i (t) = µi for all t ∈ Ti and zero otherwise. Because Ti = Ai/µi, the areas of the gray regions equal {Ai}, where thedarker color corresponds to that of the closer residential location.

Fig.4b depicts the time-varying permit cost∑

m≤i p∗m(t) that a commuter from residential location i has to pay toarrive the CBD at t ∈ T (in relation to {ρ∗i }). The top curve corresponds to i = 3, the middle i = 2, and the bottomi = 1. As in [ST–I1], the curves can be interpreted as isocost curves for commuters from each residential location. Ateach location, the isocost curve is the (homogeneous) willingness-to-pay function P(t | ρi) of its residents, subject toa given level ρi of commuting cost at location i:

P(t | ρi) ≡ ρi − s(t). ∀i ∈ I (36)

It is interesting to compare (36) with (28), where commuters are heterogeneous. In equilibrium, it must be that

P(t | ρ∗i ) = ρ∗i − s(t) =∑m≤i

p∗m(t) ∀t ∈ Ti, i ∈ I, (37)

which can be readily verified from Fig.4b.The false-bottleneck condition in Lemma 4.2 is understood using Fig.4b. For instance, if D1 > D2, it follows that

T1 > T2. It in turn implies that ρ∗1 > ρ∗2, but this contradicts to the fact that p∗i (t) ≥ 0 for all t ∈ Ti; when D1 = D2, it

implies ρ∗1 = ρ∗2, but then the bottleneck 2 becomes irrelevant for analyzing the properties of the equilibrium.


(a) Aggregated arrival flow rates at the CBD. (b) Equilibrium permit prices and commuting costs.

Fig. 4: Equilibrium solutions for [ST–J1]

4.3. Analytical solutions for the general case

Equipped with intuitions from the two special cases, we turn our attention to the general case of [ST–(Primal /Dual)].In essence, the general case is understood as a combination of the two special cases presented in the above.

For a given pattern of Q, we introduce the location-specific job set Ji, which is the set of jobs that have employeesresiding at location i ∈ I, by Ji ≡ { j ∈ J | Qi, j > 0} for all i ∈ I. Its maximal and minimal elements are defined byj−(i) ≡ minJi and j+(i) ≡ maxJi.

Analogous to [ST–J1], we consider a reduced network for the general setting to simplify the presentation. In doingso, we define a false bottleneck for the problem [ST–(Primal /Dual)] as follows.

Definition 4.3 (False bottleneck in [ST–(Primal /Dual)]). A false bottleneck in [ST–(Primal /Dual)] is a bottleneck isuch that p∗i (t) = 0 for all t ∈ supp[{qi, j(t)}] for some job index j× ∈ Ji in equilibrium.

Suppose that a bottleneck i is a false bottleneck in the sense of Definition 4.3 and that j = j×. Then, any (i, j)-commutercan pass through the bottleneck for free of charge, thereby causing unnecessary degeneracy in the equilibrium flowrates; we will discuss this point later using Fig.6. To screen out false bottlenecks, we define the normalized demandfor the problem [ST–(Primal /Dual)] by Di, j ≡

∑l≥ j αi,l

∑k≥i

∑n≤l Qk,n/µi, where αi, j = α j for j ∈ Ji \ { j+(i)} and

αi, j+(i) = α j+(i). Analogous to Lemma 4.2, we conclude as follows:

Lemma 4.4 (Detection of a false bottleneck in [ST–(Primal /Dual)]). For the problem [ST–(Primal /Dual)], a bottle-neck i ∈ I is a false bottleneck if and only if Dm, j ≥ Di, j for some m < i and some j ∈ Ji.

We can always construct a reduced corridor using Lemma 4.4; we thus assume a reduced corridor network in thefollowing presentation: Di, j < Di+1, j for all i ∈ I \ {I} and j ∈ Ji.

We first show that, at each bottleneck in the corridor network, job-based temporal sorting property is present asit is in the single bottleneck case [ST–I1] (Lemma 4.1). Assuming a reduced corridor, we show that the arrival timewindows Ti, j exhibit the following sorting property.

Lemma 4.5 (Job–location-based sorting property in [ST–(Primal /Dual)]). Consider a equilibrium solution for [ST–(Primal /Dual)]. Under Assumption 2.2, the arrival-time windows {Ti, j} satisfy {t0} ⊂ Ti, j ⊂ Ti, j+1 for all j ∈Ji \ { j+(i)} and for all i ∈ I, or equivalently

t−i, j+1 < t−i, j < t0 < t+i, j < t+i, j+1 ∀ j ∈ Ji \ { j+(i)}, i ∈ I (38)

The equilibrium values of {t±i, j} are determined according to Ti, j as t−i, j = t−(Ti, j) and t+i, j = t+(Ti, j), where Ti, j isobtained as a function of Qi ≡ {Qi, j | j ∈ J} such that Ti, j(Qi) =

∑n≤ j Qi,n/µi.

Observe that the length of the arrival time window Ti, j(Qi) is interpreted as a natural generalization of its counterpartfor [ST–I1] and [ST–J1]; if we set I = 1 or J = 1, it reduces to T1, j in Lemma 4.1 or Ti in Lemma 4.3, respectively. Itis also noted that for j > j−(i) and j > j+(i), we define Ti, j = 0 and Ti, j = Ti, j+(i), respectively.


(a) Aggregate arrival flow rates at the CBD. (b) Equilibrium permit prices and commuting costs.

Fig. 5: Equilibrium solutions for [ST–(Primal /Dual)]

The following proposition summarizes the analytical solution for [ST–(Primal /Dual)]. Note that Ti ≡ ∪ jTi, j.

Proposition 4.3. In a reduced corridor network, the solution for [ST–(Primal /Dual)] (q∗, ρ∗, p∗) is obtained as:

• arrival flow rates: q∗i, j(t) =

µi if t ∈ Ti, j \ Ti, j−1

0 otherwise∀ j ∈ Ji, i ∈ I (39)

• commuting cost: ρ∗i, j(Qi) =∑n≥ j

αi,n s(Ti,n(Qi)) ∀ j ∈ Ji, i ∈ I (40)

• permit price: p∗i (t) =

ρ∗i, j − α js(t) if t ∈ (Ti, j \ Ti, j−1) \ Ti−1

ρ∗i, j − α js(t) −∑m≤i−1 p∗m(t) if t ∈ (Ti, j \ Ti, j−1) ∩ Ti−1

0 otherwise

∀ j ∈ Ji, i ∈ I. (41)

Fig.5 illustrates Lemma 4.5 and Proposition 4.3 for the case of I = 2, J = 2. The figure can be understood as acombination of Fig.3 and Fig.4. Note that for simplicity we assume Q > 0 here, so that J1 = J2 = J = {1, 2}. InFig.5, from Lemma 4.5, we see that

T1,1 =∣∣∣T1,1

∣∣∣ = Q1,1/µ1, T1,2 =∣∣∣T1,2

∣∣∣ = (Q1,1 + Q1,2)/µ1 = A1/µ1, (42)

T2,1 =∣∣∣T2,1

∣∣∣ = Q2,1/µ2 = Q2,1/µ2, T2,2 =∣∣∣T2,2

∣∣∣ = (Q2,1 + Q2,2)/µ2 = A2/µ2. (43)

Once {Ti, j} is determined, the arrival time windows {Ti, j} are determined via the functions t− and t+ illustrated by Fig.2Fig.5a illustrates equilibrium arrival flow rates at the CBD. The location-specific disaggregated arrival flow rates

q∗i (t) ≡ ∑j q∗i, j(t) always equals µi in its support, in coincidence with the homogeneous problem [ST–J1]. At each

location i, the arrival time windows Ti, j is sorted according to Lemma 4.5, analogous to the single bottleneck case[ST–I1]. In the figure, the area of red regions equals L1, whereas the total area of blue regions L2. One can also find{Ai}, {Qi, j} as areas in Fig.5a.

Fig.5b illustrates equilibrium isocost curves for [ST–(Primal /Dual)] in relation to equilibrium permit prices andcommuting costs. We immediately notice that Fig.5b is a combination of Fig.3b and Fig.4b. The isocost curve at eachbottleneck is constructed as in [ST–I1]. Specifically, the isocost curve at the bottleneck i is∑

m≤i

p∗m(t) = maxj

{P j(t | ρ∗i, j)

}∀t ∈ Ti, j ∈ Ji, i ∈ I, (44)

where P j(t | ρi, j) ≡ ρi, j −α js(t) is the (i, j)-specific willingness-to-pay for permit costs; special cases are given by (28)and (36). Then, as in [ST–J1], the isocost curves are vertically sorted according to the location index, as long as weconsider a reduced corridor network. Observe that the isocost curve at i ∈ I depends only on Qi = {Qi, j | j ∈ J}.


Fig. 6: Example of a false bottleneck in [ST–(Primal /Dual)].

Fig.6 illustrates a situation where a false bottleneck in Definition 4.3 appear. In the figure, the bottleneck i = 2is a false bottleneck, where p∗2(t) = 0 for T2,1. Under such situation, the equilibrium arrival flow rates for commuterwith job 1 (i.e., q∗1,1(t) and q∗2,1(t)) are not uniquely determined in T2,1, since equilibrium costs are the same for bothlocation 1 and 2 during the time window. In contrast to the homogeneous case [ST–J1], however, such a situation doesnot necessarily mean that the equilibrium costs are not affected by the existence of the bottleneck. Even though thebottleneck 2 is irrelevant for commuters with job 1, it does affect commuters with job 2 since p∗2(t) > 0 for T2,2 \ T2,1.As Fig.6 illustrates, a false bottleneck appears as a result of some “overlaps” between isocost curves for consecutivebottlenecks; Lemma 4.4 allows one to detect and sort out such situations.

Since our analysis is conducted under specific assumptions on ci, j(t), it would be beneficial to discuss robustnessof the obtained short-run results against relaxations of our simplifying assumptions (made in Section 2) before pro-ceeding to the analysis of the long-run equilibria. The most important point is that s(t) is job-independent. It saysthat all groups share exactly the same schedule delay preference. Though seemingly restrictive, we can generalize ouranalytical results to “single-crossing” cases, provided that the desired arrival time is unique. Consider a case wheres(t) is differentiated by job as s j(t). With job-dependent schedule delay functions {s j(t)}, define the willingness-to-payfunctions by P j(t | ρ j) ≡ ρ j − s j(t). A single-crossing case is where for any given pair of jobs j, k ∈ J , for any givenlevels of ρ j and ρk, the equation P j(t | ρ j) = Pk(t | ρk) has at most a single solution in each of t < t0 and t > t0. It isnoted that the single-crossing property is actually general, as majority of previous studies were conducted under theproperty. To simplify the presentation, however, we employ job-independent s(t).

5. Analysis of the long-term equilibrium and comparison with a “naıve” long-term equilibrium model

Having established the analytical formula (40) of the short-term equilibrium commuting cost function ρ∗(Q), weare now in a position to analyze the properties of the long-term problem.

5.1. Properties of the long-term equilibrium

5.1.1. Qualitative propertyThe properties of ρ∗(Q), which in turn determine those of long-term equilibria, are summarized as follows:13

Lemma 5.1 (Properties of ρ∗(Q)). The commuting cost function ρ∗(Q) satisfy the following properties:

(a) ρ∗(Q) is separable with respect to location. That is, ρ∗i ≡ {ρ∗i, j | j ∈ J} depends on only Qi = {Qi, j | j ∈ J}.(b1) Each ρ∗i (Qi) is strictly monotone.

13 Lemma 5.1 can also be generalized to single-crossing cases. In effect, the results of Section 5 are robust against relaxation of assumptions onschedule delay preference.


(a) {Ci, j} (b) Ri (c) {S i, j}

Fig. 7: Schematic illustration of the relation (49) at bottleneck i with three job classes j = 1, 2, 3.

(b2) Each ρ∗i (Qi) is differentiable and ∇ρ∗i (Qi) is positive, symmetric, and positive definite, if s(t) is differentiable.(c1) ρ∗i, j(Q) is monotonically increasing in i.(c2) ρ∗i, j(Q) is monotonically decreasing in j.

Combining the definition of the long-term equilibrium (Definition 3.2) and Lemma 5.1 (b2), we show:

Proposition 5.1. The long-term equilibrium is obtained as a solution for the following convex optimization problem.

[LT’] minQ≥0

z1(Q) + z2(Q) s.t. (10’) and (11’), (45)

where z1(Q) ≡∑i∈I

∮0→Qi

ρ∗i (ω) dω. (46)

Since strict monotonicity of ρ∗(Q) (Lemma 5.1 (b1)) implies strict convexity of z1(Q) (and hence z1(Q) + z2(Q)),we have the following corollary regarding uniqueness of the equilibrium.

Corollary 5.1. The solution for [LT’], or the long-term equilibrium, is unique.

The above problem can also be regarded as a part of integrated equilibrium problem which includes feedbacksfrom the short-term component. As the short-term equilibrium is uniquely determined in a reduced corridor, we cansequentially solve [LT’] and [ST–Primal] to obtain integrated equilibrium pattern (Definition 2.1).

5.1.2. Equivalence and graphical interpretationsAs [LT’] in the above and [LT] in Section 3 should be equivalent to each other by construction, it must be that

z1(Q) = z1(Q). In other words, recalling the definition of z1(Q) in (17), we must have the following identity inshort-term equilibrium for any given values of Q:

z1(Q) =∑i∈I

∮0→Qi

ρ∗i (ω) dω =∑i∈I

(ρ∗i (Qi) · Qi − µi

∫T

p∗i (t | Q) dt)= z1(Q) (47)

where {p∗(t | Q)} is permit price at the short-term equilibrium. Actually, we conclude as follows.

Lemma 5.2. z1(Q) = z1(Q) holds true. Furthermore, the value is explicitly computed as

z1(Q) = z1(Q) =∑i∈Iµi

∑j∈J

S i, j(Qi) where S i, j(Qi) ≡ α j

( ∫ t−(Ti, j−1(Qi))

t−(Ti, j(Qi))s(t) dt +

∫ t+(Ti, j(Qi))

t+(Ti, j−1(Qi))s(t) dt

). (48)

Observe that z1(Q) is a weighted sum of integrals of job-specific schedule delay functions.The last equality of (47) has intuitive graphical interpretation; recalling that Qi, j = µi · (Ti, j − Ti, j−1) in equilibrium

(Proposition 4.3), (47) implies the following relation which is satisfied at each bottleneck i ∈ I:

µi

∑j∈J

Ci, j − µi · Ri = µi

∑j∈J

S i, j where Ci, j ≡ ρ∗i, j · (Ti, j − Ti, j−1) and Ri ≡∫T

∑m≤i

pm(t) dt. (49)


(a) Equilibrium pattern Q∗ for the integrated model. (b) Equilibrium pattern QN for the naıve model.

Fig. 8: Equilibrium patterns of Q for the integrated and naıve models. Gray regions are positive elements, whereas black markers { j−(i)} and { j+(i)}.

Notice that µi · Ri is the total permit cost paid by commuters residing at residential location i. The above equality thusstates that, in equilibrium, the total commuting cost net of the total permit cost coincides with the total schedule delaycost. In other words, it represents the strong duality between [ST–Primal] and [ST–Dual]. The relation is schematicallyillustrated by Fig.7. The isocost curve approach in the literature, which determine the boundary in Fig.7, is thusinterpreted to have its basis on the primal–dual relation of the short-term equivalent optimization problem.

5.1.3. Properties of equilibrium job–location choice patternSolving [LT’], we obtain the equilibrium job–location choice pattern Q∗ ≡ {Q∗i, j}, or travel demand for the short-

term problem. Analogous to Ji, we define job-specific location sets by I j ={i ∈ I | Qi, j > 0

}for all j ∈ J . We show

the following regularity in equilibrium job-location choice pattern Q∗:

Proposition 5.2. j−(i) = minJi, j+(i) = maxJi, i−( j) = minI j, and i+( j) = maxI j satisfy j−(i) ≤ j−(i + 1)j+(i) ≤ j+(i + 1)

∀i ∈ I \ {I} and

i−( j) ≤ i−( j + 1)i+( j) ≤ i+( j + 1)

∀ j ∈ J \ {J}. (50)

Proposition 5.2 states that commuters with larger job index (i.e., lower VOT α j) are located at residential locationsdistant from the CBD, and vice versa. Or, we have a spatial sorting property in Q∗. An example of equilibrium patternQ∗ is illustrated by Fig.8a, in which one can observe both of the sorting property (50).14

5.1.4. Properties of equilibrium wage and land rentIn the long-term equilibrium, by (20), the following equality should hold true in the support of Q∗:

w∗j − r∗i − ρ∗i, j(Qi) − α j di = 0 ∀ j ∈ Ji, i ∈ I. (51)

Recall that ρi, j(Qi) is monotonically decreasing in the job index j and monotonically increasing in the location indexi. Then, (51) yields the following proposition.

Proposition 5.3. w∗j is monotonically decreasing in j and r∗i is monotonically decreasing in i.

Without loss of generality, suppose that α j is a j-independent constant. We can interpret this case as the continuouslimit of the job set where there are large number of jobs. In this case, {L} should be understood as the probabilitydistribution of VOT {α}. We show the following proposition.

Proposition 5.4. Assume that α j is a j-independent constant. Then, {w∗j} is strictly concave on Ji at every residentiallocation i ∈ I, in the sense that the second-order difference is negative. Overall, {w∗j} is piecewise concave in j ∈ J .

14 For Fig.8 and Fig.9, exogenous parameters and functions are set as follows: I = 10, J = 20, Ai = 18 + 4i and µi = 11 − i for all i ∈ I, L j = 20and α j = 2.1 − 0.1 j for all j ∈ J , and s(t) = max {−0.2t, 0.4t} for all t ∈ T .


(a) Comparison of land rents {r∗i } and {rNi }. (b) Comparison of wages {w∗j} and {wN

j }.

Fig. 9: Comparison of wages and land rents.

5.2. The “naıve” model and its analytical solutions

To investigate the role of interactions between the long- and the short-term components, we consider a special casein which we completely ignore interactions between the two time scales. We call it the “naıve” model.

Definition 5.1 (Naıve model). The long-term component of the naıve model is defined by

[LT–N] minQ≥0

z2(Q) s.t. (10’) and (11’), (52)

along with∑

i∈I Ai =∑

j∈J L j = Q. In the model, we first solve [LT–N] to obtain the long-term solution QN. Then,we solve [ST–Primal] to determine the short-term dynamic equilibrium flow pattern.

The long-term component of the naıve model, [LT–N], is obtained by ignoring the feedback from the short-termcomponent, i.e., the first term z1(Q) of the problem [LT’]. Even though we sequentially solve the long-term and theshort-term problems also in the naıve model, there is no consideration of the short-term effect in the first step; eventhough both components each seek to achieve efficient allocation, the two components are completely separated inthe naıve model. We note that the naıve model is equivalent to the Herbert–Stevens model (as revisited by Wheaton,1974), which is an important model in theoretical literature in urban land use forecasting.

The problem [LT–N] admits a unique analytical solution QN, thanks to Monge property of the coefficient array{α jdi} of z2(Q).15 It is noted that the analytical solution QN corresponds to the so-called northwest corner rule.

Lemma 5.3 (Analytical solution for [LT–N]). Define the cumulative distributions for {Ai}, {L j}, and {QNi, j} by Ai ≡∑

m≤i Am, L j ≡∑

n≤ j Ln, and QNi, j ≡

∑k≤i

∑l≤i QN

i, j, respectively. Then, QNi, j = min

{Ai, L j

}.16

Fig.8b illustrates the analytical solution QN in marginal form. Observe that the commuters with higher VOT areprioritized and sequentially assigned to residential locations closer to the CBD. Equilibrium land rent and wage areshown to satisfy similar properties as Proposition 5.3.

5.3. Comparisons and policy implications

5.3.1. Equilibrium job–location choice patternsDefine location-specific job setJN

i and job-specific location set INj for the naıve model byJN

i ≡{j ∈ J | QN

i, j > 0}

and INj ≡

{i ∈ I | QN

i, j > 0}, respectively. Then, we have the following regularity:

15 Since {α j} is decreasing in j and {di} is increasing in i, one can easily verify that α jdi + α j+1di+1 > α jdi+1 + α j+1di.16 In marginal form, we have JN

i ≡ { j ∈ J | Qi, j > 0} = { j ∈ J | Ai−1 < L j < Ai} and QNi, j = min{Ai, L j} −max{Ai−1, L j−1} for all i ∈ I, j ∈ JN

i .


Proposition 5.5. The location-specific job set for the naıve model is contained by that of integrated model: JNi ⊂ Ji

for all i ∈ I. Similarly, INj ⊂ I j holds true.

Fig.8 illustrates the proposition. Even though the total number of commuters at location i always equals to Ai, theintegrated equilibrium has a greater job variety at every location than the equilibrium in the naıve model. In otherwords, the extra term z1(Q) acts as—due to its strict convexity—a “regularization” which prefers interior solutions.

5.3.2. Equilibrium land rents and wagesIn equilibrium of [LT–N], we have wN

j − rNi − α jdi = 0 for all i ∈ I, j ∈ JN

i where {wNj } and {rN

i } are theequilibrium wage and land rent of the naıve model, respectively. Compared with (51), this condition misses theshort-term equilibrium cost ρ∗i, j(Q). It leads to some “bias” in the price variables in the naıve model:

Proposition 5.6. Let rNI = r∗I = 0 for normalization. Then, rN

i < r∗i for all i ∈ I \ {I} and wNj < w∗j for all j ∈ J .

That is, equilibrium land rent and wage are underestimated in the naıve model. Fig.9 illustrates the above Proposition5.6, as well as Proposition 5.3 and Proposition 5.4. First, Fig.9a illustrates the land rent curves. One readily observethe relation rN

i < r∗i except for i = I. By Henry George Theorem, the optimal land investment at residential locationi coincides with the total land rent at i. Recall that the total number of commuters at location i is fixed to Ai, so thatthe land investment also follows a similar inequality (rN

i · Ai ≤ r∗i · Ai). The proposition thus implies that if we employ[LT–N] to determine the long-term land investment policy, it results in a insufficient development of residential areas.

On the other hand, Fig.9b compares wages. Observe that wNj < w∗j . Analogous to land development, if one consider

industry zoning at the CBD, then the total wage of each job j is the measure to determine which industry to lure. Therelation wN

j · L j < w∗j · L j then implies that high VOT industries are undervalued in the naıve model.

5.3.3. Comparison of the total revenues of the road managerThe total equilibrium revenue R(Q) of the road manager from the TNP scheme is of interest for considering long-

term road capacity investments. R(Q) is the total of the equilibrium permit revenues Ri(Q) at every bottleneck i ∈ I:

R(Q) ≡∑i∈I

Ri(Qi) where Ri(Qi) ≡ µi ·∫Ti

pi(t) dt. (53)

The analytical solution QN (Lemma 5.3) and the relation JNi ⊂ Ji (Proposition 5.5) yields the following proposition:

Proposition 5.7. The following properties are satisfied in equilibrium: (a) R1(QN) ≥ R1(Q∗), and (b) R(QN) ≥ R(Q∗).

In words, the total permit revenue in the network and the revenue at the first bottleneck are overestimated bythe naıve model. The property can be understood using intuitions from Section 4. Recall that commuters with higherVOT are concentrated around the CBD in the naıve model (see Fig.8b). This causes higher permit prices at bottleneckswhich are close to the CBD. This in turn shifts all the isocost curves for the upstream bottlenecks upward, resulting inlarger commuting cost as well as a greater TNP revenue on the side of the road manager.

Proposition 5.7 has an important policy implication for long-term road construction decisions. The TNP schemeis known to satisfy a self-financing property (Akamatsu, 2007; Wada and Akamatsu, 2013); that is, the total TNPrevenue of the road manager at a bottleneck i ∈ I coincides with the optimal investment on the capacity of thebottleneck i, so long as the cost function for road construction is homogeneous of degree one. If we employ theself-financing property of the TNP scheme to determine the amount of investment on bottleneck capacity expansion,the optimal value of the total investment also coincides with R(Q). Then, Proposition 5.7 implies that the naıve modeloverestimates the total required investment on capacity expansion. Without consideration of the short-term dynamicsin the long-term component, one ends up in excessive construction of roads. A naıve approach to separately analyzethe short- and long-term problems thus falls short. The result should be regarded as an illustrative example of definiteneed to synthesize the short-term TDM policy with the long-term evolution of travel demand patterns.

6. Concluding remarks

Using a minimalist model, this study theoretically demonstrated the fundamental need to synthesize the short-termTDM policy with the long-term demand dynamics and, conversely, to include the effect of short-term traffic dynamics


in the long-term analysis. In doing so, our analysis involved many choices and compromises which call for furtherfollow-up studies. We discuss three important directions below.

First, we assumed not only that the long-term preference of commuters is quasilinear but also that land consumptionis fixed to unity. Introducing elastic demand for land would be a natural next step. Mathematically, this correspondsto introducing another entropy-like strictly convex term into the long-term problem [LT’] in addition to the short-termobjective z1(Q). Even though we expect that basic intuitions of Section 5.3.2 and, in particular, Section 5.3.3 are keptunchanged, this is a claim that we should verify.

Second, job-choice in our model is a simplified one. The labor demands are inelastic. As a result, the onlydeterminant of the wage in our model is commuting cost. Introducing profit-maximizing firms along with otherappropriate assumptions such as productivity will provide richer implications (cf. the single bottleneck model ofTakayama, 2015).

Third, as the short-term problem, we focused solely on a DSO assignment achieved as the equilibrium underan optimal short-term TDM policy. Comparison with the dynamic user equilibrium (DUE) assignment without anytolling (Friesz and Mookherjee, 2006; Ma et al., 2015; Akamatsu et al., 2015) is also a fascinating direction, sinceanalytical solutions are unavailable for DUE assignment even in a corridor network—in contrast to the clear-cut resultsfor the DSO assignment presented in Section 4. However, due to the complicated dynamics of the queueing delays,this would involve much more efforts.

Acknowledgments. We are grateful for the constructive comments and criticisms made by the three anonymous re-viewers. This research is partially supported by JSPS KAKENHI (15H04053, 14J02901).

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Omitted proofs are available online at: https://sites.google.com/site/minoosawa/.

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