In traditional economic models, consumers display preference given the constraints of a product characteristic space.
Hence, the chocolate with nuts is a constraint of its product characteristic space.
On the other hand, consumers in location models display preference for both the utility gained from a particular brand's characteristics as well as its geographic location; these two factors form an enhanced “product characteristic space.” Consumers are now willing to sacrifice pleasure from products for a closer geographic location, and vice versa.
For example, consumers realize high costs for products that are located far from their spatial point (e.g. transportation costs, time, etc.)
[1] He represented this notion through a line of fixed length.
In Hotelling's Location Model, firms do not exercise variations in product characteristics; firms compete and price their products in only one dimension, geographic location.
Therefore, traditional usage of this model should be used for consumers who perceive products to be perfect substitutes or as a foundation for modern location models.
Assume that the line in Hotelling's location model is actually a street with fixed length.
All consumers are identical, except they are uniformly located at two equal quadrants
Consumers face a transportation/time cost for reaching a firm, denoted by
Given the assumptions of the Hotelling model, consumers will choose either firm as long as the combined price
For example, if both firms sell the product at the same price
is the price of the product including the cost of transportation.
, the halfway point between the two firms, will be indifferent between the two product locations.
Assume that the line in Hotelling's location model is actually a street with fixed length.
All consumers are identical, except they are uniformly located in four quadrants
Consumers face an equal transportation/time cost for reaching a firm, denoted by
, the halfway point of the street where each firm has the same number of customers.
This result is known as Hotelling's law, however it was invalidated in 1979 by d'Aspremont, J. Jaskold Gabszewicz and J.-F.
[2] Consider that quick (short run) price adjustment and slow (long run) location adjustment is modelled as a repeated two-stage game, where in the first stage firms will make an incremental relocation and in the second period, having observed each other's new locations, they will simultaneously choose prices.
d'Aspremont et al. (1979) prove that when firms are sufficiently close together (but not located in the same place) no Nash equilibrium price pair (in pure strategies) exists for the second stage subgame (because there is an incentive to undercut the rival firm's price and gain the entire market).
For example, when firms are equidistant from the centre of the street, no equilibrium price pair exists for locations 1/4 or closer than 1/4 of the length of the street from the centre.
The non-existence of a Cournot equilibrium precludes the ending of the game, and so it is not repeated.
[3] Similar to the previous spatial representations, the circle model examines consumer preference with regards to geographic location.
However, Salop introduces two significant factors: 1) firms are located around a circle with no end-points, and 2) it allows the consumer to choose a second, heterogeneous good.
The model will occur for one time period, in which only one product is purchased.
In this example, the consumer wants to purchase their ideal variation of Product A.
They are willing to purchase the product, given that it is within the constraint of their utility, transportation/distance costs, and price.
Now suppose the consumer also has the option to purchase an outside, undifferentiated Product B.
The consumer surplus gained from Product B is denoted by