William Vickrey and Transportation Economics

William Vickrey made major contributions to a wide range of fields within the economics discipline and was recognized for his work with 1996 Nobel Prize in Economic Science. Many of his contributions fall within my interests of Urban and Transportation Economics, so I thought it worthwhile to review a few of his publications. He also happens to be the only Nobel Laureate, of which I am aware, who was borne in my home province of British Columbia. My focus will be on his work in marginal cost pricing, specifically his article “Congestion Theory and Transport Investment”, which was published in American Economic Review in 1969. This is among the first discussions of congestion pricing. His focus is on measures to relieve congestion, and the efficient allocation of public investment towards this objective. He suggests that investment does not sufficiently consider the marginal cost, and may therefore misrepresent consumer surplus as a metric for investment benefit.

Vickrey was one of those great geniuses who, in an off-hand comment, makes insights that would form the basis for an entire paper by others. One such case is that of bus headway, which he argued to be optimally related to demand by a square root relationship - to be proven analytically by Gordon Newell at UC Berkeley a few years later. Vickrey assumes delay to vary by the square of total volume, with no indication of an empirical reference point. This forms a cornerstone of his theory of traffic congestion. On average, a vehicle will inflict an amount of delay on other vehicles equal to the delay experienced by itself.

The principle situation of delay considered by Vickrey is what he terms a bottleneck. This is an instance of a segment of roadway that has a lower capacity than its upstream and downstream adjacent segments. The bottleneck is typically a function of a reduction in the number of travel lanes due to an accident or geometric constraint (e.g. bridge or road narrowing). Vickrey recognizes this is an atypical form of congestion, but argues for its tractable setup for analytical analysis of costs and measures of welfare.

He considers the example of a bottleneck on a roadway with a demand for travel of N = 7,200 commuters. It is assumed that their departure from the bottleneck is distributed uniformly over a period between \(t_a = 08:00\) and \(t_b = 09:00\). The critical capacity can be defined as

\[ v_m = \frac{N}{t_b - t_a} = 120\, veh/min \label{eq:mc1} \]

If the capacity of the bottleneck is smaller than \(v_m\), then it is impossible for all the commuters to reach their destinations at their desired arrival times. Some of the commuters will arrive at their destinations close to their arrival time, but will be required to stay in the queue for a longer time. While those who push their arrival time away from the desired arrival time, will spend relatively less time in the queue, but experience a penalty through time spent at home when they desire to be at their destination (or, if not desire, lose pay).

Vickrey makes a set of assumptions about value of time (VOT), which initially confused me on the basis of disutility of time - until I realized that they described activities with positive utility parameters. He defines the value of time spent at home at \(w_h = 2\, cents/min\), time spent at work before the desired starting time at \(w_o = w_p = 1\, cent/min\), and time spent at work after the desired starting time at \(w_o = w_j = 4\, cents/min\). This means that, before the desired starting time, a person would rather be at home (greater VOT), while they would rather be at work after the desired start time. This tends to assume a wage of 2 cents per minute above the VOT for leisure (i.e. time spent at home). Time spent in the queue is valued at \(w_q = 0\, cents/min\). With these monetary conditions, a fraction of commuters, \(r = \frac{2}{3}\), will pass through the queue during queue development. The time of maximum queue size will occur at \(t_p = 08:40\), subjective to the above prices. Vickrey specifies a linear relationship for the queue size, producing a triangular shape in time and queue size, with a maximum at time \(t_p\). He suggests that a dynamic price structure be set, which eliminates the queue. The toll is set to zero at \(t_i\) and rises to a maximum, given by

\[ p_p = \frac{Nr}{v}(w_h - w_p) = \frac{4800}{v} \label{eq:mc2} \]

at \(t_p\), before falling to zero at \(t_j\). Commuters will leave their home at a time, such that they leave the location of the bottleneck at the same time they would leave the queue absent prices. This will distribute commuters throughout the time period in a more even manner, such that no queue forms and no commuter would be better off changing their departure time, in terms of VOT and congestion toll.

An interesting aspect to this result is that it has a net neutral effect from the perspective of the commuter, but produces additional revenue for the local government agency levying the toll. Revenue from the toll could be used to offset an existing flat toll or invested in long-term infrastructure improvements (e.g. transit or capacity expansion). Vickrey extends his study through empirical analysis of the results, consideration of alternative routes, and variation in the VOT. He provides some interesting commentary on the assessment of investments in new infrastructure. He argues that many of the new routes put forth at the time of publication based their benefit measures on increased speeds and flow volumes. These are not accurate measures of benefits to the user because they are frequently circuitous routes and do not decrease overall travel time. Vickrey argues for the application of congestion pricing as a policy lever in the cost-benefit analysis. The inclusion of such a measure would give a more accurate picture of the monetized improvement in travel time.

Vickrey published “The City as a Firm” in the proceedings of a conference held by the International Economic Association in 1977. The significance of this work is its early development of concepts of New Economic Geography. Vickrey identifies the importance of transportation costs and economies of scale as model inputs. The absence of transportation costs represents a futuristic notion of instantaneous movement of goods and persons. Vickrey defines economies of scale in this context as positive neighbourhood effects. This is a measure of the efficiency of proximity. If each activity could be carried out efficiently within small shops, then production could be uniformly spread throughout a region without loss of efficiency. Evidence has found neither to be the case and this is an important point in the context of autonomous vehicles (AV). They are often touted as significantly reducing transportation costs, but this argument fails on two counts. Based on the arguments of Vickrey, this argument fails to recognize the structural importance of scale economies. Such effects are wholly unaffected by the introduction of AV and economic efficiency will outweigh any desire to relocate activities to more dispersed locations. Vickrey discusses the importance of natural features in the location of our largest cities, while also recognizing that their size would be far smaller in the absence of economies of scale.

A related publication, in the context of the effect of AV adoption on city formation, is the book “Peak Car: the Future of Travel” by Professor David Metz at UCL. He provides a strong argument for hard limits on the speed of travel within an urbanized region. Metz argues that average commuting time has an equilibrium about 30 minutes, going back to the dawn of human civilization. While not a new argument, it is an important one and has been largely overlooked in the AV literature. These vehicles have the potential to introduce marginal increases in travel speed and therefore travel distance, but this 30 minute threshold will likely remain. Transportation costs will remain a factor in the formation and development of urban regions.

Vickrey assumes a perfect market in developing a model of the city as a firm but recognizes this is not an ideal assumption. Dixit and Stiglitz published their monopolistic competition model in the same year, which led to the NEG work of Krugman in the following decades (for which he scored himself a Nobel).