On Highway Fuel and Time Expenditures

User cost is one of the major considerations when designing a new highway. The vertical alignment of the highway (i.e., the determination of the pavement level along the highway's trajectory), is one of the design oriented factors which affects this cost. In this paper we investigate the influence of the slopes along the highway, which are determined as part of the vertical alignment, on the main parts of the user cost, namely, fuel cost and travel time value. We show that if, and only if, we evaluate each time unit at not less than a value we derive for it by a revealed preference method (from the driver's speed choice), then the sum of the fuel and time costs is a convex function of the slope. We also show that this sum is a convex function of the (traffic) flow. However, if we take the fuel cost alone, we do not get a convex function for either of these cases. For instance, the fuel cost as a function of the slope fails to be convex due to a drop in its derivative at the “critical slope” where the vehicle starts losing speed. Also we note that the same sum is not necessarily a convex function of both variables (slope and flow) together. Finally, we use the convexity of the fuel and time costs to show that the vertical alignment problem is convex too.