The value of reliability Fosgerau, Mogens and Karlström, Anders_MPRA_PAPER_5733

Munich Personal RePEc Archive The value of reliability Fosgerau, Mogens and Karlström, Anders Technical University of Denmark, Royal Institute of Technology, Sweden 13 November 2007 Online at MPRA Paper No. 5733, posted 14 Nov 2007 06:00 UTC The value of reliability ∗ Mogens Fosgerau Technical University of Denmark 2800 Kgs. Lyngby, Denmark ∗We are grateful to Ken Small, John Bates and Steffen Andersen for comments. We also thank Rene´ Ring from the Danish Roads Directorate for supplying data. Mogens Fosgerau has received financial support from the Danish Social Science Research Council. The value of reliability Abstract We derive the value of reliability in the scheduling of an activity of ran- dom duration, such as travel under congested conditions. We show that the minimal expected cost is linear in the mean and standard deviation of du- ration, regardless of the form of the standardized distribution of durations. This insight provides a unification of the scheduling model and models that include the standard deviation of duration directly as an argument in the cost or utility function. The results generalize approximately to the case where the mean and standard deviation of duration depend on the starting time. Empirical illustration is provided. KEYWORDS: Welfare; Random duration; Time; Scheduling; Reliability; Vari- ability JEL codes: D01; D81 1 Introduction In this paper we consider the value of reliability for an agent who wishes to under- take an activity of random duration and must decide when to initiate the activity, knowing only the distribution of its duration. We are concerned with the value of changes to the distribution of the duration. The value of a change in the mean duration is just the value of time, which is a concept with a long history in eco- nomics (Becker, 1965; Beesley, 1965; Johnson, 1966; DeSerpa, 1971) and there is a large literature on its measurement.1 The concept of the value of reliability of duration, i.e. the value of a change in the standard deviation of duration, is less well established but not much less important. We incorporate reliability by building on the model of Small (1982), who con- sidered the scheduling of commuter work trips when the commuters have schedul- ing costs as well as time and monetary costs.2 We formulate the scheduling costs as an opportunity cost of starting early and a greater cost of finishing late rela- tive to some fixed deadline. In contrast to earlier contributions (e.g., Noland and Small, 1995), we are able to derive the optimal expected cost for a general distri- bution of durations. We obtain the simple result that the optimal head start as well as the optimal expected cost depend linearly on the mean and standard deviation of the distribution of durations, provided the standardized distribution of durations ...................................................continued.................................................