Salop model – Wikipedia

The two best known models of space competition are the Hotelling model of the straight road and the Steven Salop model of the circular road. In the two cases, the model is also used to explain the position of the goods sold in the product space. This is on the other hand the main purpose of the Salop model. The n companies on the circular road become the nars of products sold. The Salop model is also an example of oligopolio with differentiated products.

Salop takes the case of a circular road l length meters long. There are in stores distributed regularly along the road. The distance between a shop and the nearby shops is therefore l/n.

The good sold is homogeneous and the cost is of the unit. Consumers are distributed evenly along the road by reason of a consumer per meter. Each consumer acquires a unit of good. The choice of the shop depends on the sale price and the transport cost that is of the meter t $. It is therefore a case of oligopolio with differentiated assets since the transport costs make the goods sold by the shops different.

Let’s take the case of the shop A (see graphic). Its competitors are shop B on its right and shop C on its left.
If the prices are identical, to have half of the consumers who are between A and B and half of those found between A and C. in the general case, the distribution of consumers depends on the price set by the shop.

Both x the distance between A and a consumer who is between A and B. This consumer indifferently chooses the shop A or B when:

${Displaystyle P+TX = {bar {p}}+T ({Frac {L} {N}}-x)}$

Where

${displaystyle p}$

It is the price of the property sold by the representative shop A,

${displaystyle {bar {p}}}$

that of the other shops (B in this case) e

${DisplayStyle vert p- {bar {p}} vert leq t {frac {l} {n}}}$

. You get:

${Displaystyle x = {Frac {1} {2t}} [{bar {p}}-P+{frac {tl} {n}}]}}$

The customers of the representative shop come from the right and left. The question will be 2x and the profit:

${displaystyle Pi ={frac {1}{t}}[{bar {p}}-p+{frac {tL}{n}}](p-c)}$

The shop sets the price that maximizes its profit. The first order condition is:

${displaystyle {frac {partial Pi }{partial p}}={frac {1}{t}}({bar {p}}-p)+{frac {L}{n}}-{frac {1}{t}}(p-c)=0}$

All shops have the same costs and the same questions. The prices will then be the same. By applying this result to Nash’s balance, you get:

${Displaystyle {bar {p}} = C+{frac {tl} {n}}}$

An increase in transport costs or a decrease in the number of stores lead to an increase in the balance price. In the past, consumers had to go on foot and then there were numerous stores of food.

The number of long -term stores depends on the amount of fixed costs (

${displaystyle c_{o}}$

). In the long term the profit is null:

${Displaystyle Pi = {frac {tl^{2}} {n^{2}}}-C_ {O} = 0}$

And then the number of shops will be:

${displaystyle n=L{sqrt {frac {t}{c_{o}}}}}$

Fixed costs limit the number of companies. There will be more hairdressers than dentists. In addition, if the transport costs increase, the number of stores will increase.

The long -term price will be:

${displaystyle {bar {p}}=c+{sqrt {tc_{o}}}}$

An increase in transport costs or fixed costs lead to an increase in long -term price.

Suppose we are looking for the number of stores that minimize consumer transport costs. When the prices are the same (Nash’s balance), the cost of transport of consumers who go to the same shop is:

${displaystyle 2int _{o}^{L/2n}xdx={frac {L^{2}}{4n^{2}}}}$

The total cost for the purchase of goods in the stores will therefore be:

${Displaystyle n [C_ {O}+C {Frac {L} {N}}+{Frac {TL^{2}} {4n^{2}}}]}}]$

The minimum cost is obtained when:

${displaystyle n={frac {L}{2}}{sqrt {frac {t}{c_{o}}}}}$

We therefore have half of the number of long -term stores in case of oligopoly. According to the criterion of total cost, there are too many shops or too many long -term products.

• S.C. Salop, « Monopolistic competition with outside goods », The Bell Journal of Economics, vol. 10, 1979, pp. 141–156