2. Critique of Hotelling Hotelling’s “Principle of Minimum Differentiation” was flawed No pure strategy exists if firms are close together. With quadratic rather than linear transportation costs, firms optimally differentiate (DGT) Mixed strategies imply firms differentiate (Osborne and Pitchick) Firms seek differentiation to avoid unbridled price competition Intuitively, equilibrium, prices fall when firms get too close to each other; it overcomes the positive impact that a unilateral move towards the market center has on demand Hotelling and others ignored this strategic impact

3. Multiple dimensions Has been implicitly supposed that the one-dimension result of maximum differentiation carries over to multiple characteristics, so a firm that can differentiate on two (or more) dimensions (for example, color and size) will do so on both Preliminary results in two dimensions (Neven and Thisse, Tabuchi) indicate this might not be true Maximum differentiation in both dimensions is not an equilibrium Instead, strategic firms maximize differentiation in one characteristic, minimize differentiation in the other characteristic So in multiple characteristic goods, in how many dimensions should we observe product differentiation?

4. Analysis moved to multiple dimensions Is it Max-Min-…-Min or Min-…-Min? Ansari, Economides, and Steckel, 1998 Irmen and Thisse (hyperspace), 1998 Hehenkamp and Wambach (evolutionary stability ), 2010

5. Model set up Simple Hotelling type model but with several dimensions Quadratic transportation costs Represent the costs of a firm’s good not exactly matching a consumer’s preferences. Quadratic makes it symmetric, so “too much” of some characteristics is as bad as a like amount of “too little” of that characteristic Allow each characteristic to be weighted, so some characteristics matter more than others Inelastic demand

6. Preview of the results With n characteristics, all weighted equally by consumers, there are n local equilibria in which firms maximize differentiation on one characteristic, and minimize differentiation on the others When there is a dominant characteristic, there is a unique equilibrium where firms maximize differentiation on the dominant characteristic, and minimize differentiation on the others Implication is that differentiation along a single dimension is sufficient to relax price competition. With inelastic demand, profits are highest where prices are highest, which requires the elasticity of demand to be lowest, which means the lowest mass of marginal consumers Differentiation in only one dimension is sufficient to achieve this goal Use as an example the magazines Time and Newsweek

7. Basics of the model: Consumer demand Product characteristics are given in  n with n>1. A’s location is described by a vector a=(a 1, …,a n ) and likewise B’s location is give by b=(b 1,...,b n ). Consumers are uniformly distributed over a hypecube C=[0,1] n. Consumers get net utility from consuming 1 unit of the good according to S is sufficiently large that consumers always consumer 1 unit of the good They get a net utility that decreases in price and decreases in how far the characteristics of the good they consumer differ from their optimum, which is described by z=(z 1,…,z n ). t k weights the importance of the k th characteristic. More salient characteristics have the highest weight.

8. A digression on dominance Irmen-Thisse have so that n is the dominant characteristic. Weak dominance is if Strong dominances is if Strong dominance means the marginal utility loss from deviations on the dominant characteristic is larger than the sum of marginal utility losses from other deviations. Even if the salience of characteristic n is most important. They show that this condition is always satisfied at a local Nash equilibrium, so they focus on strong dominance. It is this assumption that makes looking at two dimensions equivalent to looking at n>2 dimensions

9. Firm demand The demand for variant A is given by the mass of consumers for whom the A is weakly preferred to B, that is, for whom So where is the uniform distribution. There are only the two firms, no threat of entry. They assume a two- stage game where firms first choose location and then compete on prices. They seek a sub-game perfect Nash equilibrium

10. The model with n=2 (facilitates math and intuition A consumer buying from A enjoys a utility equal to A consumer buying from B enjoys a utility equal to The marginal consumer, indifferent between A and B is thus located Solving for z 2 as a function of z 1 gives Without loss of generality we assume strong dominance of characteristic 2 so

11. Firm Demand and profit As noted, the demand for variant A is so the middle piece of A’s demand under strong dominance is over the interval

12. More on the price range We have normalized the population to 1 so This help provide some intuition about the price ranges Suppose p A is at its lower limit. Then Anticipating the result that and A gets the entire market. Like wise, if p A is at its upper limit it is easy to show so if and B gets it all. Since the firms are identical, you can expect that each gets ½ the market

13. Profit and profit maximization (check these) Given the demand above, profit is just price times quantity, so Maximizing with respect to a 1, a 1 and p A gives

14. If we do the same for firm 2, we get unique solutions for prices as a function of the location parameters: Both prices rise as the products become more similar in the dominated characteristic.

15. Likewise for the ranges of a2 and b 2 Both prices rise as the products become less similar in the dominating characteristic. Differentiating in the dominating characteristic lessens price competition, while similarity in dominated increases demand. In Hotelling the first dominated the second, so firms differentiated in the only dimension possible.

16. Location equilibrium Given the equilibrium prices the profit function for firm A is With tedious but straightforward algebra they also show that in permissible ranges for all the ( a 1, b i ), implying the firm wants a 2 as small as possible, ie, a 2 =0, and likewise, it means b 2 =1.

17. If there is no dominant characteristic With k dimensions and no strictly dominant characteristic things are a little dicey, but if all salience characteristics are equal, there are k local equilibria, each differentiating on one characteristic and not on the others. The implication of this is that what dominates can change as relative costs of differentiating, or consumer tastes, change

18. So what do we have? Is this model much different from Hotelling? In what way yes or no? Does it explain the real world better than single dimension models? In what way?