### Do 60 improvements each saving 1 minute equal 1 improvement saving 60 minutes

This is the third in a continuing series asking deep questions about the nature of transportation analysis. Previous episodes include:

1. Why do commute distances and times rise with income

So,

3. Can small units of time be given the same value of time as larger units of time. In other words, do 60 improvements each saving a traveler 1 minute equal 1 improvement saving a traveler 60 minutes? Similarly, does 1 improvement saving a 1000 travelers 1 minute equal the value of time of a single traveler of 1000 minutes. These are different problems, one is intra-traveler and one is inter-traveler, but related.

Several issues arise.

A. Is value of time linear or non-linear? To this we must conclude the value of time is surely non-linear. I am much more agitated waiting 3 minutes at a red light than 2, and I begin to suspect the light is broken. Studies of ramp meters show a similar phenomena, as in our paper Weighting Waiting:

Evaluating Perception of In-Vehicle Travel Time Under Moving and Stopped Conditions .

B. How do we apply this in a benefit-cost analysis? If we break one project into 60 smaller projects, each with a smaller value of travel time saved, and then we added the gains, we would get a different result than the what obtains with a single large project. For analytical convenience, we would like our analyses to be additive, not sub-additive, otherwise arbitrarily dividing the project changes the result. In particular many smaller projects will produce an undercount that is quite significant, and result in a much lower benefit than if the projects were bundled.

As a practical matter, every Benefit/Cost Analysis I have seen assumes a single value of time, rather than assuming non-linear value of time. (Alert me if you have a counter-example).

On the other hand, mode choice analyses do however weight different components of travel time differently, especially transit time (i.e. in-vehicle time is less onerous than waiting time). The implicit value of time for travelers does depend on the type of time (though generally not the amount of time). Using the log-sum of the mode choice model as a measure of benefit would implicitly account for this.