Linear City Model

The linear city model (Hotelling model) is a model of spatial market differentiation with monopolistic competition, which demonstrates consumer preferences for specific brands of goods and their location, first proposed by G. Hotelling in 1929 ^[1] .

Content

Fixed Location Option

Hotelling model with non-fixed location

In 1929, in the article “Stability in Competition” ^[2], G. Hotelling proposed a model for the location of enterprises, taking for simplification two firms located on a line representing a uniformly distributed consumer market, where consumers also live, on a straight line of length L (L> 0).

Assumptions

The model has a number of assumptions ^[1] :

firms do not choose prices that are equal to fixed marginal costs and are similar in both companies
the company can change its location on a straight line
the final price of the goods for the consumer is equal to the selling price of the company plus the cost of transporting the goods (fixed per unit distance)
changing position on a straight line, firms change the volume of sales, that is, revenue and profit
firms A and B at the initial moment of time are located arbitrarily on a straight line.

Price game

The price of goods for the consumer increases in proportion to the distance from the company. Consumers buy goods from company A, being closer to it. When the consumer price when buying in company A is compared with the consumer price when buying in company B, then an indifferent consumer arises (he does not care where to buy), as well as the interface between the firms of the entire market. In the case of linear transport costs, the division between the markets will take place in the middle of the segment AB ^[1] .

This situation is not in equilibrium, as firm A can move directly to the interface of the market and get the whole part of the market on the left plus half of the new segment AB. Firm B will do likewise. Displacements are completed only when both firms are in the middle of the market, where any further movement will reduce the firm's profit. In this case, both firms will be located in one place - in the middle of the segment. Manufacturers strive to produce their products as identical as possible, which is the principle of minimal differentiation (Hotelling's law). Placing firms in one place explains the process of agglomeration and concentration of trade ^[1] .

Fixed Location Option

Fixed Location Hotelling Model

The linear city model is considered as a general product differentiation model, assuming that the distance between sellers reflects the difference in consumer characteristics of the goods of the two manufacturers. Transport costs are considered as the loss of utility of the consumer who prefers the first product, but is forced to use the second (the amount of discount necessary for the buyer who prefers the first product to choose the second). Thus, the transport tariff becomes a reflection of the degree of brand loyalty, the growth of the transport tariff - an increase in brand loyalty. The Hotelling model allows us to conclude on the impact of changes in brand loyalty on the position of sellers: increasing brand loyalty reduces price competition and strengthens the foundations of monopoly power ^[3] .

Assumption

The model has a number of assumptions ^[3] :

buyers are evenly spaced
customer preferences are identical
maximum willingness to pay for goods is - y
transport costs per unit of goods are t for the distance between two sellers (equal to 1)
transportation costs include explicit and implicit costs.
the price that the first company can assign to the product is limited by the maximum willingness to pay for the product - y, the transportation cost rate t, price competition from the second company.

Price game

The price for the goods depends, on the one hand, on the maximum willingness to pay for the goods, on the other hand, on the remoteness of the buyer from the seller. The further the buyer is located from the seller, the lower the net price that the seller can receive. For the first seller, the dependence of his net price on the location of the buyer is described by the formula ^[3] :

$p_{1}=y-tx$ ${\ displaystyle p_ {1} = y-tx}$ and $p_{2}=y-t(1-x)$ ${\ displaystyle p_ {2} = yt (1-x)}$ ,

where x is the location of the buyer, belongs to the interval [0; one].

Remoteness reduces competition between firms, as the buyer at a point $X_{1}$ ${\ displaystyle X_ {1}}$ ready to purchase goods at a price $P_{1}(X_{1})$ ${\ displaystyle P_ {1} (X_ {1})}$ at the first company and only at a price $P_{2}(X_{1})$ ${\ displaystyle P_ {2} (X_ {1})}$ the second company. This differentiation of sellers creates an area of pure monopoly power, where buyers are not ready to purchase goods from a second seller. If firms charge equal prices $(y-0,5t)$ ${\ displaystyle (y-0,5t)}$ then they will divide the market in half. Raising transport tariffs will lead to the creation of monopoly zones of firms. A sufficiently significant increase in transportation costs will lead to the emergence of zones where consumers are so far from sellers that firms cannot expect to receive any price, transactions will not occur ^[3] .

Hence, the main consequence of the model : in order to increase the profitability of firms, it is beneficial for traders to make it as difficult as possible to move buyers as much as possible ^[4] .

Nash Equilibrium

G. Hotelling received the Nash local equilibrium formula ^[5] :

$p_{1}^{h}=L+(a-b)/3$ ${\ displaystyle p_ {1} ^ {h} = L + (ab) / 3}$ and $p_{2}^{h}=L+(b-a)/3$ ${\ displaystyle p_ {2} ^ {h} = L + (ba) / 3}$ ,

$q_{1}^{h}=0.5(L+(a-b)/3)$ ${\ displaystyle q_ {1} ^ {h} = 0.5 (L + (ab) / 3)}$ and $q_{2}^{h}=0.5(L-(a-b)/3)$ ${\ displaystyle q_ {2} ^ {h} = 0.5 (L- (ab) / 3)}$ ,

where a is the distance of location of firm 1 from the start of the reference point, b is the distance of location of firm 2 from point L, $p^{h}$ ${\ displaystyle p ^ {h}}$ - the equilibrium price level of the company 1 and 2, $q^{h}$ ${\ displaystyle q ^ {h}}$ - the volume of equilibrium output.

K. D'Aspermont, J. Ya. Gabzhevich, J.-F. Thyss in 1979 was supplemented by the restriction of the existing Nash equilibrium ^[6] :

$(L+(a-b)/3)^{2}>=4/3L(a+2b)$ ${\ displaystyle (L + (ab) / 3) ^ {2}> = 4 / 3L (a + 2b)}$ and $(L+(b-a)/3)^{2}>=4/3L(b+2a)$ ${\ displaystyle (L + (ba) / 3) ^ {2}> = 4 / 3L (b + 2a)}$ ,

that is, equilibrium can be achieved if firms are not close to each other.

When firms are located too close to each other, but not at the same point, they begin to bring down prices, lowering prices, without leading to the establishment of an equilibrium state ^[4] .

Variant with a quadratic increase in transportation costs

The equilibrium problem was solved by K. D'Aspermont, J. Ya. Gabzhevich, J.-F. Tiss , when they proposed using quadratic transport cost functions instead of linear ones, at which price equilibrium always exists ^[6] :

$p_{1}=y-tx^{2}$ ${\ displaystyle p_ {1} = y-tx ^ {2}}$ and $p_{2}=y-t(1-x)^{2}$ ${\ displaystyle p_ {2} = yt (1-x) ^ {2}}$ .

The utility function of the consumer located at point x:

$U_{x}=-p_{A}-t(x-a)^{2}$ ${\ displaystyle U_ {x} = - p_ {A} -t (xa) ^ {2}}$ if he buys from firm A

$U_{x}=-p_{B}-t(x-(L-b)^{2})$ ${\ displaystyle U_ {x} = - p_ {B} -t (x- (Lb) ^ {2})}$ if he buys from firm B.

Price game

Using the quadratic function of transport costs, we consider a two-period game, where in the first period the placement of firms is decided, and in the second, pricing. We find the perfect balance in the subgames (the outcome of the game in which there is a Nash equilibrium in each of the subgames of the original game). In the first period, we maximize the profit function of firms by a for firm A and by b for firm B, fulfilling the condition $\delta \pi _{A}/\delta a<0$ ${\ displaystyle \ delta \ pi _ {A} / \ delta a <0}$ firm A chooses $a=0$ ${\ displaystyle a = 0}$ , and firm B chooses the location at point L. Using the quadratic functions of transport costs leads to the fact that firms will choose the maximum differentiation of brands, in which profits grow with an increase in the degree of differentiation ^[4] .

Notes

↑ ¹ ² ³ ⁴ Limonov L.E. Regional economics and spatial development . - M.: Yurayt Publishing House, 2015. - T. 1. - S. 82-83. - ISBN 978-5-9916-4444-0 .
↑ Hotelling H. Stability in competition // The Economic Journal. - March 1929. - Vol. 153, No. 39 . - P. 41-57. Archived February 21, 2016.
↑ ¹ ² ³ ⁴ Avdasheva S.B., Rozanova N.M. Theory of organization of industrial markets . - M .: Publishing house Magister, 1998. - 312 p. - ISBN 5-89317-082-2 .
↑ ¹ ² ³ Shai Oz. Organization of industry markets. Theory and its application. - M .: HSE Publishing House, 2014 .-- S. 171-187. - 503 s. - ISBN 978-5-7598-0555-7 .
↑ Iskakov M.B., Iskakov A.B. Complete solution of the Hotelling problem: the concept of equilibrium in safe strategies for the game of pricing // Journal of the New Economic Association. - 2012. - T. 13 , No. 1 .
↑ ¹ ² d'Aspremont S., Gabszewicz JJ, Thisse J.-F. On Hotelling's “Stability in Competition” // Econometrica. - 1979. - Vol. 47, No. 5 . - P. 10-33.

[:1-1] ¹ ² ³ ⁴ Limonov L.E. Regional economics and spatial development . - M.: Yurayt Publishing House, 2015. - T. 1. - S. 82-83. - ISBN 978-5-9916-4444-0 .

[:2-2] Hotelling H. Stability in competition // The Economic Journal. - March 1929. - Vol. 153, No. 39 . - P. 41-57. Archived February 21, 2016.

[:3-3] ¹ ² ³ ⁴ Avdasheva S.B., Rozanova N.M. Theory of organization of industrial markets . - M .: Publishing house Magister, 1998. - 312 p. - ISBN 5-89317-082-2 .

[:4-4] ¹ ² ³ Shai Oz. Organization of industry markets. Theory and its application. - M .: HSE Publishing House, 2014 .-- S. 171-187. - 503 s. - ISBN 978-5-7598-0555-7 .

[:5-5] Iskakov M.B., Iskakov A.B. Complete solution of the Hotelling problem: the concept of equilibrium in safe strategies for the game of pricing // Journal of the New Economic Association. - 2012. - T. 13 , No. 1 .

[:6-6] ¹ ² d'Aspremont S., Gabszewicz JJ, Thisse J.-F. On Hotelling's “Stability in Competition” // Econometrica. - 1979. - Vol. 47, No. 5 . - P. 10-33.

Linear City Model

Content

Fixed Location Option

Assumptions

Price game

Fixed Location Option

Assumption

Price game

Nash Equilibrium

Variant with a quadratic increase in transportation costs

Price game

Notes

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