Monopolistic competition: Linear city model

Hotelling’s linear city model was developed by Harold Hotelling in his article “Stability in Competition”, in 1929. In this model he introduced the notions of locational equilibrium in a duopoly in which two firms have to choose their location taking into consideration consumers’ distribution and transportation costs. Hotelling’s model has been source of inspiration for a great amount of fruitful literature which does not only constrain to industrial organization theory but also to other sciences, such as politics, as some of its conclusion can be directly applied to them.

Initially the model was developed as a game in which firms first chose a location and after a selling price for their products. In order to set their business in the best location to maximize profits, the firms will have to evaluate three key variables: competitors’ location, customers’ distribution and transportation costs.

However, this model includes two different approaches, the first one being a static model consisting of a single stage were firms choose their location and prices simultaneously, and a dynamic model in which price is set after determining the location. The model is based on a linear city that consists of only a single straight street. For a better comprehension, Hotelling’s model is sometimes explained by using the example of a beach were two ice cream stands are trying to decide the best location.

Static model

In a beach going from west to east, of size [0,1] where consumers are distributed evenly, two identical ice cream stands (A and B) with a marginal cost of production, c > 0, try to determine their best location. A is located at a distance a from point 0, while B is located at a distance b from point 1.

It is indifferent for customers whether to go to one or the other stand, as long as their utility is maximized. For this reason, we must assume that both ice cream stands offer the exact same ice creams, and therefore consumers’ utility will be given only by the price of the ice cream and the distance to the stand. Despite differences in prices, the stand with the lowest prices will not necessary attract all the demand since consumers will also consider the distance to the stand.

If both stands’ prices were equal, the differential factor would be how close consumers are to each stand. All consumers located to the left of a would go to stand A, and all consumers located to the right of 1-b would go to stand B. The remaining consumers, located between both stands would go to whichever is the closest. In this first model,  is the exact middle of that beach, so consumers to its left would go to stand A while consumers to its right would go to stand B.

Two conditions are necessary for profits to be positive and maximized in both stands:

-Selling prices must be higher than marginal costs

–

This second condition implies that both stands can’t be located at the same point exactly in the middle of the beach. If this were the case it would be indifferent for the customers to go to either one. Each stand would decrease its prices in order to attract customers, and thus they would enter into what is referred to as a price war. If they had different marginal costs, the stand with the highest marginal cost would be in a clear disadvantage and would end up exiting the market, as the other stand would be able to push the prices further down, and so attract all customers. Depending on how prices are set it could lead to a Bertrand’s solution, in which the prices of both stands are equal to their marginal costs, thus achieving zero profits.

We can conclude by saying that in this model the key factor for product differentiation is location. Each stand will therefore set prices that will be higher than their marginal costs, and will choose a location other than the middle of the beach.

Dynamic model

In this case, the model has two stages. During the first stage, the stands chose their location and on the second stage prices are set.  With regards to the previous model there are two additional assumptions:

Both stands have equal marginal production cost.

Total cost for consumers depends both on the price of the goods and the distance to its location (t).

The point of division between the areas served by these two stands is determined by the condition that at a certain point it is indifferent for consumers between consuming in one stand or the other. Equating and equalising we get 

By solving, the demand function for each stand will be: 

By using backwards induction we can find the subgame perfect Nash equilibrium where both firms maximize their profits.

Stage 2: having already determined a location the stands will now have to set the price that will report the highest possible profits. This will lead us once again to the static model result, in which each stand sets its own price.

Stage 1: the stands anticipate and choose their location. Profits will be higher the less distance there is to the extremes, therefore maximum differentiation between stands will be given when A locates at 0 and the B at 1.

The conclusions that can be drawn from this model are two opposite effects. On one hand, there is an incentive for both stands to locate at the centre of the beach in order to increase their market share by reaching out to the greatest amount of customers, in what is known as the demand effect. And, on the other hand, there is an incentive for both stands to locate at opposite extremes in what is considered to be the strategic effect. While the first effect will reduce differentiation between the stands, the second one will increase it.

In politics we can observe how candidates follows this demand effect, as they tend to come from a specific ideology but tend to convey towards a moderate ideology with the objective of attracting as many voters as possible.