Competitive Location Models - Location Science - page 357

An extension is the utility function
￿ UD1b: u ij D - p j - td ij
Maximizing such a utility is equivalent to minimizing the full price of the good,
i.e., the mill price plus the transportation costs. Hotelling's own contribution falls
into this category, and so do the papers by Serra and ReVelle ( 1999 ) and Pelegrín
et al. ( 2006 ). Note that the utility UD1a is a special case of the utility UD1b with
zero prices (or prices that are equal at all existing facilities).
Consider now the utility function
￿ UD1: u ij D R i - p j - td ij ,
where R i denotes the reservation price customer i assigns to one unit of the good in
question, an upper bound customers are prepared to pay for one unit of the good.
Given that, the utility is an expression of the amount of money that the customer
“saved,” i.e., the amount that he was prepared to, but did not have to, spend on a
unit of the product. Some authors refer to R i as the valuation of the product, other
refer to it as income, while still others think of it as the budget. In all cases, R i - p j -
td ij is an expression of the money that was available for the purpose, but did not have
to be paid for the product. It is apparent that the utility functions UD1a and UD1b
are special cases of the function UD1: Given equal reservation prices R i D R k , i ¤ k ,
maximizing the utility UD1 reduces to UD1b, which, in turn, reduces to UD1a for
fixed and equal prices p j . One important feature of the utility function UD1 is that,
in case the utility u ij is nonpositive, it allows customer i to refrain from making any
purchases.
Finally, there exists a variety of other deterministic utility functions used by some
authors. Among them is Lane ( 1980 ), who uses a Cobb-Douglas-style function
that expresses the utility as the product of three components: a measure of a
characteristic raised to a power, another measure of the facility raised to some power,
and the available income of the individual. Neven ( 1987 ) frames his discussion in
the context of brand positioning, and his utility function is the difference between a
(very high) reservation price, and the price plus the squared of the customer-facility
distance (which, in this context, is actually the difference between the customer's
ideal point and the actual feature of the product). Finally, Kohlberg ( 1983 )usesa
utility function that includes the sum of travel time and waiting time, a utility that
is important in the context of facilities that feature congestion, such as health-care
facilities.
￿ UD1c: u ij D R i - p j - td ij - W i ,
where W i denotes the waiting time. One pertinent example in the context of health
services is found in Marianov et al. ( 2008 ).
Another utility function incorporates not only distances, which are present in
all spatial models—after all, they are what makes a model “spatial”—but also
the “attractiveness” of the facilities. As already briefly alluded to above, this one-
dimensional measure attempts to capture differences between facilities the way they
are perceived by customers: floor space as a proxy for selection (even though the