2. Quick summary One-dimensional vertical (quality) differentiation model is extended to two dimensions Use to analyze product and price competition Two stage game 1.Firms first compete on product position 2.Then they compete on price Firms do not tend towards maximal differentiation When firms have equal opportunities, they end up with max-min differentiation, just like in Irmen and Thisse

3. Choice of product features Need to consider consumer preferences In quality differentiation, all consumers agree that more of quality is better. The vary in their willingness to pay for quality Need to consider strategic responses of competitors Two forces determine the locational equilibrium (both vertical and horizontal differentiation models) A demand force that draws firms close together A strategic force that causes them to differentiate Early research on vertical differentiation maximal differentiation shows maximal quality differentiation when demand is inelastic, but when one option is to not buy, quality differs exist, but not maximal

4. Model: market and firms Goods have more than one type of quality product quality service quality Two firms, indexed by 1 and 2 choose one product to market the product has two characteristics, x and y, defined as a point, (x i,y i ) where these define the two types of quality, and are both bounded by (allowably different) maximum and minimum values production has constant marginal cost, set equal to zero regardless of product position (does not depend on quality characteristics), and no fixed cost

5. Model: consumers Consumers prefer more of each quality to less, and prefer a low price to a higher price Consumer reservation price, R, is sufficiently high that every consumer in the market buys one unit from either firm 1 or firm 2 Consumer valuation is defined by so consumers care about absolute rather than relative prices The parameters  1 and  2 are uniformly distributed over the population. We restrict the range to [0,1], where the scale is different for (x,y) if one characteristic is more important than the other.

6. Characteristic competition Asymmetric characteristics competition – each firm has a relative advantage in one quality dimension (figure a), for example “ease of use” and “power” (Apple versus Windows) Dominated characteristics competition – one firm has a relative advantage on both characteristics (Windows 10 vs Windows 7) For both types of competition, the relative positions of the products are described by the ratio This ratio equals the tangent of the angle between the horizontal axis and a line from the origin perpendicular to a line joining This angle is called the angle of competition

7. Indifference space Panel a shows the angle of competition with asymmetric characteristics (firm 1 dominates in x, firm 2 dominates in y ) Consumers buy the good that maximizes their utility – there is a set of consumers indifferent between the two goods where as shown in panel b when prices are equal, in which case the indifference line lies along the angle of competition Differences in prices shifts the indifference line up or down. More skewed dominance in increases or decreases the angle of competition

8. Indifference space Panel a shows the angle of competition with dominant characteristics competition (firm 2 has the superior product) When prices are equal the indifference line is outside the characteristic space – everyone buys from firm 2. Hence, for the indifference equation to be in the product space we need and we graph as before Relatively greater dominance by firm 2 in y increases the angle of competition. Relatively greater dominance in x decreases the angle of competition

9. Price Equilibrium Sequential game: firms first choose quality, then choose price, with subgame perfect criterion An equilibrium consists of product choice for firms 1 and 2 such that neither would choose a different product unilaterally, for whatever prices equilibrium that follows, using backward induction Profit function for firm i, i=(1,2) is A noncooperative Nash equilibrium is a pair of prices such that

10. Asymmetric characteristics equilibrium The indifference line is positively sloped with angle With product positions fixed the indifference line is shifted up or down by changes in Look at the problem from firm 1’s perspective (so p 2 is fixed). We see in the next figure there are four boundaries

11. is the lowest price at which no consumers buy from firm 1 These are the upper and lower bounds of the price the firm will charge given The two remaining key prices are those where the indifference line passes through (0,0), at price, and when it passes through (1,1), at price

12. Price equations Replacing in the boundary equations with the boundary point coordinates gives where, for example, in (3) we set These are all increasing in, and, if it is in the equation, increasing in The greater firm 1’s relative advantage over 2, the higher the price it can charge to generate a similar demand

13. Characteristic i dominance, i=x,y If α  45 o we have and which is characteristics x dominance, while if then 45 o  α we have characteristic y dominance and α  45 o characteristic x dominance 45 o  α, characteristic y dominance 11 11 22 22

14. Characteristic x dominance Three regions In demand for firm 1 increases at an increasing rate In demand for firm 1 increases at an constant rate In demand for firm 1 increases at an increasing rate

15. To see the rate of change Suppose we are in. As price falls from to the area of demand grows at an increasing rate We can easily see that z 1 is given by and demand by

16. Total demand under characteristic x dominance It is easy to see that Note D 2 =1-D 1 and we have three regions that correspond to the regions in the above graph. In region 1 D 1 is convex, in region 2 it is linear and in region 3 it is convex. Firm 2’s demand follows the opposite pattern

17. Equilibrium prices Start in region 2. Total profit is price times quantity, so Maximizing this with respect to p x gives a unique solution if in region 2 so but that is only satisfied if (the condition for characteristic x dominance which we started with). A similar condition holds that puts firm 2 in the same region. A similar analysis under characteristic y dominance results in so whichever characteristics is dominant in the asymmetric dominance situation we end up in the middle region

18. Dominated characteristic competition In this case one firm (2) has higher quality in both characteristics. With characteristic x dominance we have With the same type of tortured logic they show that there are price equilibrium in all three regions. This is different from the asymmetric characteristic competition. There is a similar picture if y is dominant, but the ordering of prices becomes because the blue lines are flatter. R2 R1 R3 11 22

19. Product equilibrium strategy Product positions are dependent on the equilibrium prices Process 1.Establish which regions need to be considered 2.Calculate the firms profit function within each relevant region 3.Find the FOC for profit along with the demand restrictions that define the region indicate the maximum profit location in a region 4.Compare the maximum profit in each region to find the optimal location choice

20. Location decision under asymmetric dominance Need only look in region 2 for each firm. From the demand for each firm, and the equilibrium prices we find Now look at table 1

22. Implications about produce choice Maximum quality in both directions yields the highest profit, therefore both firms want to choose that location Both can’t be there, and it is not clear which will, so we can arbitrarily pick one Given that one firm is in the supreme position of highest quality in both characteristics, it is the strategy of the lower quality firm that determines how much differentiation there is One solution is max-min (panel a on next page), in which both firms have maximum quality in one characteristic, one has maximum quality in the first characteristic, and the other has minimal quality in the second characteristic Other solutions are max-max, and max partial

23. Max-min Max-max

24. Max-partial

25. What is interesting about these three pictures? If the range in quality in one direction is approximately equal to that of the other, the result is max-min As the ratio of the ranges grows, we move to max-max But as it gets very large, we get max-partial. Moreover, as the range of x quality relative to the range of y quality gets bigger and bigger, the partial differentiation gets further away from max-max (but not necessarily closer to max-min So why? (they do a lousy job of explaining why these results make sense)

26. Both firms want to have maximum quality. When quality dimensions are approximately equal, but because of strategic force to avoid price competition, only one locates there. The other firm, to maintain demand as much as possible keeps quality high in one dimension, but differentiates in the other dimension to reduces price competition. When quality are very different maintaining high quality in the lesser dimension does not bring enough demand to counter the price competition effect. So the second firm relinquishes demand (by lowering the quality on the second dimension) but gains in price. But the strategic tradeoff doesn’t always dominate. If the ranges of quality get sufficiently large, relative quality would get too low (they would lose too much demand for any diminishment of the price competition to compensate, so chooses an intermediate level in the second quality dimension.

27. So why is this important? Depending on the range of quality differences, we go, as the range of x relative to the range of y increases, from max-min to max-max to max- partial in differentiation along the two quality dimensions This is different from everything we’ve seen before, because within one model we are seeing all possibilities Also shows that the way quality differentiates and how consumers value different quality components are important in determining the equilibrium (different consumer values would show up as different scales, increasing or decreasing the ranges of x and y ) Helps explain the wide variety of quality we see among differentiated products