1. Introduction
It is a well-known empirical observation that transport rates normally taper with
increasing haulage distance, as described in Fig 1., thereby producing a generally
concave relationship between transport costs and haulage distance. At the same time it
is also observed that in many cases, transport rates fall with respect to the haulage
quantity for any given distance, as described by Fig.6, thereby producing a convex
relationship between transport rates and the quantity shipped. These observations are
normally termed economies of distance and economies of scale in transportation,
respectively. For many transportation models such observations are simply taken as
assumed and any analysis proceeds on the basis of these assumptions. However,
exactly why transport rates should behave in this way is actually not at all clear from
an analytical point of view. The problem appears initially when we try to relate the
observed structure of distance-transport costs to the costs of moving vehicles or
vessels, as part of the overall problem of determining the relationship between the
optimum size of a vehicle or vessel and variations in haulage distance. Under
conditions where observed transport costs taper with haulage distance, there is no
formal analytical proof that the optimum size of a vehicle or vessel (from hereon
referred to as ‘vehicle-vessel’), increases with haulage distance (Thorburn 1960;
Jansson and Shneerson 1982; 1987). Although we normally assume this fact, as we
will see shortly, the existing heuristic attempts at accounting for this can be shown to
be analytically indeterminate. The reason for this is that while the outcome of the
vehicle-vessel size optimisation problem depends on the behaviour of distance-
transport costs, the behaviour of distance-transport costs can also be shown to depend
on the outcomes of the vehicle size optimisation problem (Bacon 1984, 1992; 1993;
McCann 1993, 1998; McCann and Fingleton 1996). The result of this is not only that
a formal proof of the relationship between optimal vehicle-vessel size and haulage
distance has not previously been provided, but also that no theoretical explanation as
to why transport costs are concave with respect to haulage distance has yet been given.
The aim of this paper is to provide a general proof to both of the above
problems. In order to do this, the derivation of the structure of transport costs and
haulage distance will be discussed within the overall framework of shipment size
optimisation theory. Our purpose here is not to question the validity of standard
shipment or vehicle-vessel size optimisation theory. Rather, we show here that both of
these questions, namely that of the optimum vehicle-vessel size, and that of the non-
linear structure of transport costs with respect to haulage distance, are in fact the same
problem. Moreover, they can be solved within the existing framework by unifying
them into a single consistent theoretical approach, which avoids the anomalies
inherent in previous analyses. This approach allows us to prove the very general
conditions under which the optimum size of a vehicle-vessel does increase with the
haulage distance or the haulage weight. Furthermore, it also allows us to prove that
transport rates are almost always concave with haulage distance irrespective the
number of vehicle-vessel choices we have or the returns to scale that they exhibit. The
reason is that the observed concave structure of distance-transport costs, is itself
endogenously generated by the derivation of the optimum size of a shipment or
vehicle-vessel.