Challenge to Hotelling s Principle of Minimum

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3 Challenge to Hotelling s Principle of Minimum Differentiation Two conclusions 1. There is no equilibrium when sellers are too close i.e., Hotelling is wrong 2. Under a slightly modified version, get maximum differentiation

4 Starts with same Hotelling model One line of length l, two sellers, A and B, of an identical product, no production costs, each located at respective distances a0 and b0 from the end points of the line Customers uniformly located along the line, each consumers 1 unit of the good (inelastic demand) Customer buys from the seller who quotes the lowest mill price plus transportation costs, where per unit transportation costs are c, p 1 is the mill price of A and p 2 is the mill price of B Gives a 2-person game between A and B where the prices are the strategies and the payoffs are the profit functions

5 Firm A Firm A gets none of the market if p 1 -p 2 >c(l-a-b), that is, if its price is more than the cost of transporting the unit of good between both sellers. It gets all the market if p 2 -p 1 >c(l-a-b), that is, if B charges so much more it is cheaper for people to buy from A and transport it to them. Firm A gets part of the market if the price difference is in-between. Firm B has a symmetric relationship to the market

6 Firm A s profit function for a given p 2. There are 2 discontinuities where a group of buyers are indifferent between the two sellers

7 Nash-Cournot Equilibrium A pair of prices, (p 1 *, p 2 *) such that p 1 * is the best reply against p 2 * and p 2 * is the best reply against p 1 *. They derive the proposition

8 Proof outline If a+b=l means they are located at the same exact point, have a Bertrand problem where they are competing over quantity, and so we have a Bertrand equilibrium is unique at p 1 *=p 2 *=0. So assume a+b<l. First, show any equilibrium requires p 1 *-p 2 * <c(l-a-b) by contradicting other relationships If p 1 *-p 2 * >c(l-a-b) then one seller makes zero profit, and can gain by lowering its price to equal the other. But then we don t have the inequality. If p 1 *-p 2 * =c(l-a-b) then if p 1 *=0 then A s profit is zero, and would gain by charging a price 0< p 1 *<p 2 *+c(l-a-b). If p 1 *>0 then either A gets whole market so B will lower its price to gain a share, or A has part of the market so can gain share by lowering its price.

9 Proof outline (continued) So we have p 1 *-p 2 * <c(l-a-b) which means (p 1 *-p 2 *) maximize the respective profit functions But they show that 1 / a>0 and 2 / b>0 too, which moves both towards the center. But if a=b (at the center) as a consequence of this incentive, conditions (1) and (2) cannot hold, that is 2 a b 2 2 l l l a b lb l lb 3 for condition (1), and likewise a similar conclusion for condition (2). So a Cournot equilibrium is no longer a reference point.

10 Their modified version Replace linear transportation costs with quadratic transportation cost, so for any distance x the cost of transportation is cx. Get demand function and symmetric for firm B (note type, second set is for q 2 )

11 Interpretation of conditions Firm A gets entire market if l a b p2 p1 ( l a) 2 c( l a b) 2 p p 2 c( l a b)( l a b b) c( l a b) p2 p1 2 c( l a b) c( l a b) 2 c( l a b) b 2 p2 p1 c( l a b) 2 c( l a b) b where the first term is the cost of transporting the entire distance between the two firms and the second term is the MC wrt b of transporting the far side of B 2

12 Firm B get the entire market (firm A gets 0) if l a b p2 p1 ( a) 2 c( l a b) 2 2 p1 p2 c( l a b) 2 c( l a b) a where again the first term is the cost of transporting the entire distance between the two firms and the second term is the MC wrt a of transporting the far side of A If the price difference is in-between these values they each get part of the market.

13 Unique equilibrium The profit functions ensure a unique Nash-Cournot equilibrium when a and b are fixed with prices Moreover, with these prices 1 / a<0 and 2 / b<0 so firms gain an advantage by moving away from each other. Doing so increases their market power without diminishing market share. Footnote 5 characterizes this as a sequential game where first location and then price is chosen. Then equilibrium locations are the extremes

14 SO.. Why do we see four gas stations on every corner?