ENDOGENOUS SCHEDULING PREFERENCES AND CONGESTION

• Date: 01 May 2017
• Author: Kenneth Small,Mogens Fosgerau
• Published date: 01 May 2017
• DOI: http://doi.org/10.1111/iere.12228

INTERNATIONAL ECONOMIC REVIEW

Vol. 58, No. 2, May 2017

ENDOGENOUS SCHEDULING PREFERENCES AND CONGESTION∗

BYMOGENS FOSGERAU AND KENNETH SMALL1

Technical University of Denmark,Denmark; University of California, Irvine and Resources for

the Future, U.S.A.

We consider the timing of activities through a dynamic model of commuting with congestion, in which workers

care solely about leisure and consumption. Implicit preferences for the timing of the commute form endogenously

due to temporal agglomeration economies. Equilibrium exists uniquely and is indistinguishable from that of a

generalized version of the classical Vickrey bottleneck model, based on exogenous trip-timing preferences, but

optimal policies differ: the Vickrey model will misstate the benefits of a capacity increase, it will underpredict

the benefits of congestion pricing, and pricing may make people better off even without considering the use of

revenues.

1. INTRODUCTION

The scheduling of people’s activities determines many economic actions and investment

needs. In particular, the tendency toward wanting to do similar things at the same time results

in many significant expenses, for example, large sports stadia, concert halls, and convention

centers. This synchronization, along with a desire for spacious residential surroundings, also

requires expensive peak-capacity communications links and traffic arteries in order to enable

people to communicate or congregate simultaneously.

Yet, standard economic tools do not deal well with these phenomena because they typically

involve some sort of increasing returns to scale for activities at a given location, as elegantly

explained by Starrett (1974) and Krugman (1991). These and other authors have developed the

consequences of such spatial “agglomeration economies,” showing how they produce product

differentiation (involving scale economies for a given product variety), pecuniary externalities

(by which one firm’s competitive action or one consumer’s preferences affect others’ ability to

reap scale economies), and spatial concentration.

Similarly, the advantages of concentrating many individuals’ and firms’ activities in time seem

likely to create new phenomena. In particular, congestion in transportation results because the

demands for moving people and goods are agglomerated in space and time. Thus, understanding

it requires being able to model explicitly how those demands and the congestion resulting from

them are simultaneously determined.

Of the two types of agglomeration, the spatial type is much better understood (see, e.g.,

Rosenthal and Strange, 2004). Researchers have learned much about the strength of spatially

agglomerating forces such as labor market pooling, knowledge transmission, and building of

trust—each at a variety of geographical levels including regions, metropolitan areas, urban

subcenters, and even small industrial districts.2In some cases, explicit models can be solved to

∗Manuscript received March 2013; revised September 2015.

1We are grateful for comments by Richard Arnott, Clifford Winston, participants in several conferences, and

anonymous referees. All responsibility remains with the authors. Mogens Fosgerau has received support from the

Danish Strategic Research Council. Please address correspondence to: Mogens Fosgerau, Technical University of

Denmark, Produktionstorvet 426 Vest, 2800 Kongens Lyngby, Denmark. Phone: +4545256521. E-mail: mfos@dtu.dk.

2See, for example, Chinitz (1961) and Scott (1988) on central business districts; Anas and Kim (1996) and Helsley

and Sullivan (1991) on urban subcenters; Glaeser and Gottlieb (2009) on metropolitan areas; and Krugman (1991) on

regions.

585

C

(2017) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

586 FOSGERAU AND SMALL

explain complex equilibrium spatial patterns, such as those studied by Fujita and Ogawa (1982)

and Lucas and Rossi-Hansberg (2002) concerning the internal structure of metropolitan areas.

Most of this work is concerned with productivity at the workplace. Some, such as Glaeser et al.

(2001), also consider the value of a location to consumers.

Temporal agglomeration, by contrast, has elicited a much sparser literature. Henderson (1981)

shows that if workers are more productive when large numbers are at work simultaneously and

wages reflect those productivity differences, then workers are induced to find an equilibrium that

produces temporal clustering and hence traffic congestion. Wilson (1988) provides supporting

empirical evidence.3Vovsha and Bradley (2004) show empirically that the timing of work trips

involves preferences related both to the workplace and to the home: an example of the latter is

an aversion to departing from home too early in the day or returning too late. Thus, in addition

to the strong support for workplace agglomeration based on productivity, there is some evidence

that people also care about the timing of their activities at home.

However, there has been only limited success with modeling the simultaneous formation of

these scheduling preferences and of congestion. The problem is difficult because it is inherently

dynamic as well as nonlinear. As an example of the difficulties encountered, Henderson (1981)

derives an equilibrium pattern from his model of workplace productivity, but he is forced to

assume that travel times are determined in a way that allows for overtaking of earlier vehicles by

ones departing later. Small and Chu (2003) derive an equilibrium congestion pattern in a dense

downtown street network, but are forced to make a different unrealistic assumption, namely,

that travel time is determined solely by traffic density at the end of the trip.

The most successful theoretical models of equilibrium temporal aggregation rely instead

on exogenous scheduling preferences. Vickrey (1969) and many successors such as Arnott

et al. (1990, 1993) assume that each worker has a predetermined preferred work arrival time

and suffers disutility from deviating from that time. These papers describe congestion as a

deterministic queue behind a bottleneck, and this description has enabled them to shed light on

numerous questions including the effects of heterogeneous users, parallel and serial routes, and

various pricing and investment strategies. For useful reviews, see Arnott et al. (1998) or Small

and Verhoef (2007).

This article returns to the problem of understanding the origin of scheduling preferences.

We address agglomeration in time not only in the workplace but also at another location, here

described as “home,” where nonwork activities (“leisure”) take place. The result is a firmer

microfoundation for the demand for travel, based on a few simple technological relationships

along with the assumption that people choose schedules to maximize their combined utility

of work and leisure. We provide conditions under which equilibrium exists and explore its

properties.

We do so by making strong simplifying assumptions about the nature of the agglomerative

forces and of the travel network connecting the locations where they occur. First, we ignore

heterogeneity in order to highlight the role of endogenous preferences. Thus, we rule out

certain empirically observed phenomena, such as people commuting completely outside the

normal peak hours and occupations requiring multiple shifts.4

Second, we assume that worker productivity increases with the number of people simulta-

neously at work and that utility from leisure similarly increases with the number of people

3Arnott (2007) reviews these papers and further applications, while adding his own innovation (still within a static

framework) by allowing aggregate labor supplied to be affected by congestion tolls via a reduction in the net wage.

Gutierrez-i Puigarnau and Van Ommeren (2012), on the other hand, present evidence suggesting that the relationship

may be weak empirically.

4Some explorations of heterogeneity have occurred using exogenous scheduling preferences. Vickrey (1969) allows

commuters to have different preferred arrival times, as does Newell (1987). Other papers allow for heterogeneity in the

parameters of users’ travel-cost function (Hendrickson and Kocur, 1981; Cohen, 1987; Lindsey, 2004; van den Berg and

Verhoef, 2011). Heterogeneity clearly leads to interesting results and greater realism, at the cost of a rapid increase in

model complexity; we hope our model can be extended in this way in the future.