Game Theory Problem Set 4 1. (Hotelling°s model) Consumers are uniformly distributed along a boardwalk that is 1 mile long. Ice-cream prices are regulated, so consumers go to the nearest vendor because they dislike walking. Suppose that at the regulated prices all consumers will purchase an ice cream even if they have to walk a full mile.
Hotelling's law is an observation in economics that in many markets it is rational for producers to make their products as similar as possible. This is also referred to as the principle of minimum differentiation as well as Hotelling's linear city model. The observation was made by Harold Hotelling (1895–1973) in the article 'Stability in Competition' in Economic Journal in 1929.[1]
The opposing phenomenon is product differentiation, which is usually considered to be a business advantage if executed properly.
1Example
Example[edit]
Two pharmacies on the main street in Mrzeżyno. Possibly an effect of Hotelling's location competition.
Suppose there are two competing shops located along the length of a street running north and south, with customers spread equally along the street. Both shop owners want their shops to be where they will get most market share of customers. If both shops sell the same range of goods at the same prices then the locations of the shops are themselves the 'products'. Each customer will always choose the nearer shop as it is disadvantageous to travel to the farther.
One shop[edit]
For a single shop, the optimal location is anywhere along the length of the street. The shop owner is completely indifferent about the location of the shop since it will draw all customers to it, by default. However, from the point of view of a social welfare function that tries to minimize the distance that people need to travel, the optimal point is halfway along the length of the street.
Two shops: halfway[edit]
Hotelling's law predicts that a street with two shops will also find both shops right next to each other at the same halfway point. Each shop will serve half the market; one will draw customers from the north, the other all customers from the south.
Another example of the law in action is that of two takeaway food pushcarts, one at each end of a beach. If there is an equal distribution of rational consumers along the beach, each pushcart will get half the customers, divided by an invisible line equidistant from the carts. But, each pushcart owner will be tempted to push his cart slightly towards the other, moving the invisible line so that the owner is on the side with more than half the beach. Eventually, the pushcart operators will end up next to each other in the center of the beach.
Social optimum[edit]
It would be more socially beneficial if the shops separated themselves and moved to one quarter of the way along the street from each end — each would still draw half the customers but customers would, on average, make a shorter journey. However, neither shop would be willing to do this independently, as it would then allow the other to relocate and capture more than half the market.
Deviating assumptions[edit]
When people along the street, or along the range of possible different product positions, consume more than a minimum number of goods (i.e. have discretionary income), companies can position their products to sections where consumers exist to maximize profit; this will often mean that companies will position themselves in different sections of the street, occupying niche markets. When prices are not fixed, companies can modify their prices to compete for customers; in those cases it is in the company's best interest to differentiate themselves as far away from each other as possible so they face less competition from each other.[2]
Application[edit]
The street is a metaphor for product differentiation; in the specific case of a street, the stores differentiate themselves from each other by location. The example can be generalized to all other types of horizontal product differentiation in almost any product characteristic, such as sweetness, colour, or size. The above case where the two stores are side by side would translate into products that are identical to each other. This phenomenon is present in many markets, particularly in those considered to be primarily commodities, and results in less variety for the consumer.
An extension of the principle into other environments of rational choice such as election 'markets' can explain the common complaint that, for instance, the presidential candidates of the two largest American political parties are 'practically the same'. The candidates elected during primaries are usually established figures within their own partisan camps. Electors in the middle of the political spectrum are more likely to be swing voters, and there is a tendency for the candidates to 'rush for the middle' to appeal to this crucial bloc. The assumption is that people will choose the candidate with a closer ideology to their own, so that the most votes can be had by being directly in the center.
In real life[edit]
This phenomenon can be observed in real life, not just in commodity businesses like bars, restaurants, and gas stations, but even in large, branded chains:[3]
McDonald's and Burger King[4]
McDonald's and Jollibee in the Philippines
Alfamart and Indomaret in Indonesia
Target and Walmart
Lowe's and Home Depot
CVS and Walgreens[5]
Whole Foods and Trader Joe's
See also[edit]
References[edit]
^Hotelling, Harold (1929), 'Stability in Competition'(PDF), Economic Journal, 39 (153): 41–57, doi:10.2307/2224214
^D'Aspremont, C.; Gabszewicz, J. Jaskold; Thisse, J.-F. (1979). 'On Hotelling's 'Stability in Competition''. Econometrica. doi:10.2307/1911955.
^Why Do Certain Retail Stores Cluster Together?
^Why are McDonald’s and Burger King usually located near each other? Fast food location game theory
^Nation's biggest pharmacies sidle right up to each other
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A location (spatial) model refers to any monopolistic competition model in economics that demonstrates consumer preference for particular brands of goods and their locations. Examples of location models include Hotelling’s Location Model, Salop’s Circle Model, and hybrid variations.
2Hotelling's Location Model
2.1An example of fixed firms
2.2An example of firm relocation
3Salop’s Circle Model
3.1An example of a second good
Traditional vs. location models[edit]
In traditional economic models, consumers display preference given the constraints of a product characteristic space. Consumers perceive certain brands with common characteristics to be close substitutes, and differentiate these products from their unique characteristics. For example, there are many brands of chocolate with nuts and others without them. 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.) and also for products that deviate from their ideal features. Firms have greater market power when they satisfy the consumer’s demand for products at closer distance or preferred products.
Hotelling's Location Model[edit]
In 1929, Hotelling developed a location model that demonstrates the relationship between location and pricing behavior of firms.[1] He represented this notion through a line of fixed length. Assuming all consumers are identical (except for location) and consumers are evenly dispersed along the line, both the firms and consumer respond to changes in demand and the economic environment.
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.
An example of fixed firms[edit]
Assumptions[edit]
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 and , which is divided in the center by point . Consumers face a transportation/time cost for reaching a firm, denoted by ; 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.
Advanced analysis[edit]
Given the assumptions of the Hotelling model, consumers will choose either firm as long as the combined price and transportation cost of the product is less than the competitive firm.
For example, if both firms sell the product at the same price , consumers in quadrants and will pick the firm closest to them. The price realized by the consumer is
, where is the price of the product including the cost of transportation.
As long as for Firm x is greater than Firm y, consumers will travel to Firm y to purchase their product; this minimizes . Only the consumers who live at point , the halfway point between the two firms, will be indifferent between the two product locations.
An example of firm relocation[edit]
Assumptions[edit]
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 , , , and ; the halfway point between the endpoints is point . Consumers face an equal transportation/time cost for reaching a firm, denoted by ; 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.
Analysis[edit]
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 , the halfway point of the street where each firm has the same number of customers. This result is known as Hotelling's law.
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 gained from Firm X significantly increase, while Firm Y incurs a significant loss.
Salop’s Circle Model[edit]
One of the most famous variations of Hotelling’s location model is Salop’s circle model.[2] 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.
An example of a second good[edit]
Assumptions[edit]
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.
Analysis[edit]
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 for a particular product at distance is represented in the following equation:
Where is the utility from a superior brand, denotes the rate at which an inferior brand lowers the utility from the superior brand, is the location of the superior brand, and is the location of the consumer. The distance between the brand and the consumer is thereby given in .
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 which is given by:
, where the difference is between the utility of a product at location and the 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 .
Therefore, for a given amount of money, the consumer will purchase the superior variation of Product A over Product B as long as
, 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
, where the difference between the surplus of the superior variation of Product A and the surplus gained from Product B is positive.
See also[edit]
References[edit]
^Hotelling, Harold (1929), 'Stability in Competition', Economic Journal, 39 (153): 41–57, doi:10.2307/2224214
^Salop, Steven C. (1979), 'Monopolistic competition with outside goods', The Bell Journal of Economics, 10 (1): 141–156, JSTOR3003323
External links[edit]
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