Endogenous Scheduling Preferences and Congestion

Endogenous Scheduling Preferences and Congestion


M Fosgerau, DTU Transport / CTS / ENS, DK; K Small, UC Irvine, US


We analyse capacity expansion and optimal tolling in a bottleneck in a case where scheduling preferences are endogenous.


The Vickrey (1969) bottleneck model is a much used work horse for the analysis of congestion. Vickrey's model regards travellers' preferences regarding the timing of trips as exogenous. This paper provides a more fundamental view of travellers who derive utility just from consumption and leisure. Agglomeration economies at home and at work lead to scheduling preferences forming endogenously in equilibrium and a queue that looks like Vickrey (1969) would assume. However, the policy implications change.

Call, for ease of exposition, a "Vickrey" someone who has adopted the Vickrey (1969) model, even when the present model with endogenous scheduling preferences is true.

A Vickrey, observing untolled equilibrium and taking scheduling preferences as exogenous, would underinvest in capacity, underestimate the marginal external cost of congestion, fail to identify the optimal time varying toll and underestimate its benefits. In contrast to the case of exogenous scheduling preferences, travellers gain from optimal time varying tolling, even if revenues are not returned.

Consider a continuum of N homogenous workers. Each worker must start his day at home and end his day at work. Utility is a strictly concave and strictly increasing function of effective leisure produced at home and output produced at work, U(H,W), with the interpretation that work output is exchanged for consumption at a constant price normalized to one.

Time is measured as an interval [0,?¶]. Transit between home and work occurs through a one-way bottleneck with a capacity of ?Õ workers per time unit. Capacity is assumed large enough that all workers can pass through the bottleneck during a day and still have time left over: ?Õ?¶>N.

Workers depart from home at the time-dependent rate ?Ï(t). At any time t, the numbers of workers at home, traveling and at work are N_{H}(t), N_{T}(t), and N_{W}(t).

We now describe agglomeration. Worker productivity (aggregate output per worker per unit time) is assumed positively related to the number of workers through agglomeration parameter ?Î_{W}>0. Specifically, a worker who is at work from time t produces output W(a)=ç_{a}^{?¶}(((N_{W}(s))/N))^{?Î_{W}}ds.

Thus a worker's productivity per unit time is equal to zero when a single atomistic worker is alone at work, and is equal to 1 when everybody is at work simultaneously. This normalization means that we exclude an effect on maximal productivity at work from changing the total number of workers. The speed at which productivity rises with number of workers is governed by ?Î_{W}; for example when ?Î_{W}=1, productivity is linear in N_{W}. In the limit ?Î_{W}«0, productivity is constant at value 1 so long as there is some positive mass of workers present.

We similarly assume that leisure is produced in a social context, with effective leisure for a worker at home until time t equal to H(t)=₀^{t}(((N_{H}(s))/N))^{?Î_{H}}ds.

While we treat this as occurring at home, it can be interpreted as social activities that involve travel that, because it is uncongested, is omitted from the model. It can also be interpreted as incorporating home production subject to agglomeration economies; for example, household member may engage in activities together. In this way we capture the other all-important driving force, besides work agglomeration, that is needed to explain why people will undertake additional congestion costs in order to be at home during preferred times.


Association for European Transport