Abstract
In this paper, it is shown that integrated tariffs can be used to extract the consumer’s surplus when there are a lot of connections supplied so that the law of large numbers applies in the estimation of the consumer’s willingness to pay. The time validity limitations of tickets are explained by a nonlinear pricing approach. Links between optimal pricing in local public transport and network characteristics are highlighted.
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Notes
While quantity (number of journeys of a given type) can vary only by discrete amounts, for the sake of simplicity, it will be considered continuous.
It is assumed that the quantity demanded assumes a finite value at (marginal) price p=0.
Armstrong [1], by assuming that the error made by the monopolist in estimating the mean total rent is itself an increasing function of the coefficient of variation, calculates a convergence of profits to the level corresponding to the first degree price discrimination at the rate of \(\frac{1}{{\sqrt[3]{n}}}\). Note also that \(\frac{{{\sigma ^{{\text{*}}}_{i} } \mathord{\left/ {\vphantom {{\sigma ^{{\text{*}}}_{i} } {{\sqrt n }}}} \right. \kern-\nulldelimiterspace} {{\sqrt n }}}}{{n\mu ^{{\text{*}}}_{i} }}\) is, by construction, an overestimation of the true coefficient of variation of the total consumer’s surplus.
V can be interpreted as the agent’s salary, while T is a fixed amount of time to be used either for travelling or for working. For the sake of simplicity, leisure is disregarded.
While this characteristic might go unnoticed in everyday life, it becomes apparent when, e.g., visiting another city, one must plan an itinerary that meets the validity time of an hourly ticket. On the other hand, if the demand for public transport encompasses the break-even point with, e.g., a daily card so that the latter is chosen, one feels freer to wander and improvise the route of the visit.
In practice, the two groups might differ in many dimensions (e.g., age, profession, etc.). What matters here is that they differ in the opportunity cost of time.
See Katz [8] for a review.
For consumption above the efficient level, the (marginal) demand price is still positive for a while, even if lower than the marginal cost.
For example, the monopolist could sell bundles of two units of a good at the price of 4, and bundles of three units at the price of 7. A consumer who buys the largest bundle pays a quantity premium, as the average price is larger than 2, but he might prefer this solution to the alternative of buying two small bundles, paying 8, and having one unit of the good in excess.
It is assumed that the two types have the same utility function and differ only in the opportunity cost of time. Results do not change if one introduces differences in the utility function that involve a multiplicative term that shifts the willingness to pay of one group: for a review, see Armstrong [1].
On this topic, see Mirman and Sibley [10] for a review.
For a more detailed treatment of this problem, see the working paper [9] version of this study.
See Section 2.
On this point, see also Fang and Norman [4], who refer to an excludable public good. The network considered in this paper is in fact a collective good, with specificities due to the fact that consumption is optional and costly in terms of time.
References
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Acknowledgements
Financial support by MIUR (grant No. 2003131910-002, year 2003) is gratefully acknowledged. I wish also to thank two anonymous referees of this review; the usual disclaimer applies.