Location model

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 ${\displaystyle a\,}$ and ${\displaystyle b\,}$, which is divided in the center by point ${\displaystyle o\,}$. Consumers face a transportation/time cost for reaching a firm, denoted by ${\displaystyle c\,}$; they have no preferences for the firms.

There are two firms in this scenario, Firm x and Firm y; each one is located at a different end of the street, is fixed in location and sells an identical product.

Given the assumptions of the Hotelling model, consumers will choose either firm as long as the combined price ${\displaystyle P\,}$ and transportation cost ${\displaystyle c\,}$ of the product is less than the competitive firm.

For example, if both firms sell the product at the same price ${\displaystyle P\,}$, consumers in quadrants ${\displaystyle a\,}$ and ${\displaystyle b\,}$ will pick the firm closest to them. The price realized by the consumer is

${\displaystyle P+c=P1\,}$, where ${\displaystyle P1\,}$ is the price of the product including the cost of transportation.

As long as ${\displaystyle c\,}$ for Firm x is greater than Firm y, consumers will travel to Firm y to purchase their product; this minimizes ${\displaystyle P1\,}$. Only the consumers who live at point ${\displaystyle o\,}$, 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 ${\displaystyle a\,}$, ${\displaystyle b\,}$, ${\displaystyle c\,}$, and ${\displaystyle d\,}$; the halfway point between the endpoints is point ${\displaystyle o\,}$. Consumers face an equal transportation/time cost for reaching a firm, denoted by ${\displaystyle c\,}$; they have no preferences for the firms.

There are two firms in this scenario, Firm x and Firm y; each one is located at a different end of the street, is able to relocate at no cost, and sells an identical product.

In this example, Firm x and Firm y will maximize their profit by increasing their consumer pool. Firm x will move slightly toward Firm y, in order to gain Firm y’s customers. In response, Firm y will move slightly toward Firm x to re-establish its loss, and increase the pool from its competitor. The cycle repeats until both firms are at point ${\displaystyle o\,}$, 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. Thisse.[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 chose 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. Thus, although both firms at the halfway point itself is an equilibrium, there is no tendency for firms to agglomerate here.

If only Firm x can relocate without costs and Firm y is fixed, Firm x will move to the side of Firm y where the consumer pool is maximized. Consequently, the profits for Firm X significantly increase, while the profits for Firm Y significantly decrease.

Assume that the consumers are equidistant from one another around the circle. The model will occur for one time period, in which only one product is purchased. The consumer will have a choice of purchasing variations of Product A (a differentiated product) or Product B (an outside good; undifferentiated product).

There are two firms also located equidistant around the circle. Each firm offers a variation of Product A, and an outside firm offers a good, Product B.

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.

The utility ${\displaystyle u\,}$ for a particular product at distance ${\displaystyle d\,}$ is represented in the following equation:

${\displaystyle U(d,d_{1})=u-r|d-d_{1}|\,}$

Where ${\displaystyle u\,}$ is the utility from a superior brand, ${\displaystyle r\,}$ denotes the rate at which an inferior brand lowers the utility from the superior brand, ${\displaystyle d\,}$ is the location of the superior brand, and ${\displaystyle d_{1}\,}$ is the location of the consumer. The distance between the brand and the consumer is thereby given in ${\displaystyle |d-d_{1}|\,}$.

The consumer’s primary goal is to maximize consumer surplus, i.e. purchase the product that best satisfies any combination of price and quality. Although the consumer may receive more pleasure from their superior brand, the inferior brand may maximize the surplus ${\displaystyle CS\,}$ which is given by:

${\displaystyle U(d,d_{1})-P=CS\,}$, where the difference is between the utility of a product at location ${\displaystyle d\,}$ and the price ${\displaystyle P\,}$.

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 ${\displaystyle u^{*}\,}$.

Therefore, for a given amount of money, the consumer will purchase the superior variation of Product A over Product B as long as

${\displaystyle U(d,d_{1})-P\geq u^{*}\,}$, where the consumer surplus from the superior variation of Product A is greater than the consumer surplus gained from Product B.

Alternatively, the consumer only purchases the superior variation of product A as long as

${\displaystyle u-u^{*}-r|d-d_{1}|-P\geq 0\,}$, where the difference between the surplus of the superior variation of Product A and the surplus gained from Product B is positive.