2013-14, spring term

1. EC930 Theory ofIndustrial OrganisationProduct Differentiation II – Address Models and Switching CostsGame Theory II – Mixed Strategies 2013-14, spring term

2. Outline: Address models: product positioning where consumers have “ideal” specifications appealing to the average customer’s tastes differentiation to relax price competition Vertical Differentiation: high quality is not necessarily what we’d choose to offer differentiation to relax price competition again… “natural oligopoly” even without entry barriers Switching Costs: Barriers to switching suppliers without differentiation. barriers to switching can raise prices…but “captive customers” can be so valuable that competition for them drives down profits in the long term! Game theory review – mixed strategy equilibrium Readings: Lecture notes 2, 3 Problem set 4 Game Theory Appendix Cabral, ch 12; Tirolech 7

3. Multiple Choice A lack of consumer switching among products is a good indication of the existence of market power among suppliers. Product differentiation relaxes price competition. Switching costs are a type of product differentiation. Private markets will tend to result in excessive product differentiation.

4. Address Models Assume a continuum of different consumers, each of whom has a different “ideal” product design (sweetness of a drink, location of an outlet…). 0 x y 1 preference for sweetness For simplicity, each consumes one unit of the good. Distance of consumer x from 0 is d = x; distance of consumer x from 1 is d = 1-x The “distance” represents how far away the specification of the product is from the ideal. (or, it can represent physical distance). Let t = “disutility” of travel (or it can represent a true travel cost) Ux(purchase) = V – td – P, where P is the price paid for the good V V A B

5. Surplus V Surplus of consuming from firm A P -t d* 1 0 Location of consumer If consumer located at 0 (so this is his preferred specification), then consuming from a firm supplying at 0 yields surplus V with no “disutility” from distance. If consumer located at x, then obtains V – tx. If price P is charged, surplus shifts down by P (for the single unit purchased). Consumers located to right of d* do not purchase: V – td* - P = 0  d* = [V-P]/t

6. Possible Market Configurations Local monopoly; some consumers do not purchase. Local monopoly; all consumers purchase Competition; all consumers purchase and all gain positive surplus.

7. Market Boundaries if all prices same; value V the same for all consumers and customers uniformly distributed along line A C B When we add competitors, we see that C faces competition from both A and B, whereas A and B face only a single direct competitor. Is this realistic? For some products where there are reasonable “extremes” in tastes and product specifications then probably yes. (Sweetness) For others, this may be less reasonable. Consider flights that differ only in the time of day they depart. For a 12 hour “day”, where are the end points?

8. For some products, where there are no clear “end points” in the market, a circle may be a better representation. The 5PM flight may compete, for example, against the next morning’s 9AM flight… B 2PM 9AM A C 5PM Price changes affect firms “next door” directly, but even price changes several points removed from a product may have an “knock on” effect.

9. Price cut at 2PM flight will have a direct effect on 5PM flight, so this may result in a price cut at 5PM…but 5PM price cut also reduces revenue from customers between 5 and 7PM…so the price cut may not match the 2PM cut…and D may cut even less. B 2PM 9PM A C D 5PM 7PM

10. Firms may choose not only their prices, but also their product specification, though. In other words, if we are to assume product differentiation is possible, we should assume that firms can choose their product design. What if, in the linear “Hotelling” model, firms charge the same price but may choose location. What location will they choose? Start with the same location as before – the ends of the line… ‘ 1 0 A’ A B’ B A’’ Equilibrium must be that the firms both locate just at the centre of the market… so there is no equilibrium product differentiation at all in the absence of price competition! Do firms (or politicians) always try to appeal just to the “median consumer/voter”?

11. With no differentiation, the two firms would make zero profits if they compete on prices. So if firms can both choose location and price, do we see equilibrium differentiation? Yes. Intuitively, if the products are differentiated, there is a smaller incentive to cut price since not all the consumers will flock to the firm that has the lower price. Hence, the market stealing incentives to cut price are lower with product differentiation, which leads to higher equilibrium price and so higher profits. We will show that for fixed location, with only choosing price, firms do better as the degree of potential differentiation increases. The full two stage decision problem of (1) choosing product design/location (2) setting price will be solved later in lecture (since it was an old exam question), but also as part of the problem set since it is very messy and so needs a second look!

12. Fixed location, price competition We will assume transportation cost/disutility is the square of distance (for computational reasons: there are problems deriving an equilibrium for linear transportation costs) Consumers uniformly distributed along the line with mass one; the locations of the two firms are as before at the ends of the line; all customers purchase from one firm or the other. 1 0 1 B A We need to know, for any prices PA and PB, where the market divides so that we can compute the demand going to each firm as a function of prices.

13. Let x be the consumer who is just indifferent between the two firms, where x is the distance from 0: UxA = UxB V-x2t-PA = V-(1-x)2t-PB 2tx = PB – PA + t x* = So that if the prices are the same, the indifferent consumer is half-way between A and B, regardless of transport cost (as long as it is small enough that entire market still covered). If B raises its price, the market boundary shifts right (and B’s market share falls). U = V-(1-x)2t-PB U=V-x2t-PA …and V-tx*2-PA ≥0 must hold for all consumers to purchase in equilibrium. x*

14. QA(PA, PB) = = x*= QB(PA, PB) = = 1-x*= )  FOC: c = 0  And similarly, So that, solving simultaneously, we have This means that when t=0 (so that there is no scope for product differentiation) we have standard Bertrand result.

15. Substituting back into quantities, we have market shares of ½ for each firm and Profits of: ½( t+c-c) = ½ t. Hence, the greater the potential for differentiation (t), the greater are profits when firms locate at the ends of the line. (think of advertising here…) Product differentiation with price competition means that firms earn positive profits in equilibrium. Note that if both firms located at the same spot, profits are zero as well…so no incentive to move together in this case. In fact, the equilibrium in a two-stage decision problem is that the firms choose maximum differentiation.

16. Does this maximise social surplus? No. Note that prices are a pure transfer between consumers and firms if the entire market served so that the social planner does not care about pricing, per se. On the other hand, the transport cost is a pure loss to consumers and firms – so the social planner would choose locations to minimise transport costs. To minimise transport costs, we would locate firms at ¼, ¾. Why? No customer is farther away than ¼ of the line. Since transport costs are quadratic, we wish to keep travel for all consumers as small as possible: large distances quickly increase total transport cost spending. Hence, the market solution is for “too much” differentiation because firms care about price level whereas the social planner doesn’t.

17. Example (Exam, 2008) – this is a very hard question!! A version of this is on problem set for this week! 1 b a 0 x MC = 0 t = (x-y)2 Reservation price = V a. Locations fixed; choose price. What is the Nash equilibrium in prices and resulting demands, (assuming the full market is served)? This is solved in the problem set, question 2a. The marginal consumer is the one for whom PA + t(x-a)2 = PB + t(1-b-x)2. Solving this for x’s location, we have x* = (a+1-b)/2 + (pb–pa)/(2t(1-a-b)). If a and b are located at 0 and 1, this simplifies to the expression we had earlier, x* = ½ + (pb-pa)/2t. Everyone left of x* consumes from a, and similarly to the right for b. This yields demands DA = x* and DB = 1-x*. The resulting profits are PRA = x*(pA), PR = (1-x*)(pB).

18. It is messy to solve for prices, but one can obtain: P*A = c+t-(2/3)ta – (4/3)tb – (1/3)ta2 + (1/3)tb2 P*B = c+t-(4/3)ta – (2/3)tb + (1/3)ta2 – (1/3)tb2 Notice that both of these are c + t(stuff). This means that as long as the markup is positive, increasing the scope for differentiation increases price. b. Substituting these into profits yields (also after messy calculations): PRA = (1/18)t(1-a-b)(b-a-3)2 PRB = (1/18)t(1-a-b)(b-a+3)2 And if one differentiates the first profit expression with respect to a and the second with respect to b, one obtains a negative expression for each. This means that both firms would like to minimise the distance from the end of the line. Hence, maximal differentiation is the result in this case, since it allows for relaxation in pricing.

19. In this case, where everyone consumes anyway and all consumers obtain value V from consumption of the single unit, the only thing price does is to draw away some of this value to allocate it for the firm. A social planner who maximises total surplus should not care what the division of surplus between customers and firms is, so long as all customers are served. Hence, a social planner who values total surplus would just locate the firms so as to minimise the distance travelled. This places the firms at ¼, ¾. Hence, the private market generates too much differentiation compared to the social optimum. This is because the firms wish to achieve higher prices (since they only care about their share of surplus), but the social planner does not care about prices. If the social planner only cared about firms, the optimal solution would be the market solution.

20. Some applications 1. Entry deterrence by “crowding the spectrum” – RTE breakfast cereal case x xxxxxxxx ) ( But what if new entrant imitated exactly? Wouldn’t best response be to re-locate since targeted entry forces down price of many products in the line? This would be the case if exit costs were small…and crowding the spectrum would not work as an entry deterrence strategy. 2. Merger analysis Merging top and bottom slots into a single firm has little price effect. Merging neighbouring slots has large price effect. Think of airlines…It depends on their slot ownership whether price effect will be large or small.

21. Vertical Product Differentiation Suppose that consumers all prefer one good over the other at the same price. NB: “horizontal” differentiation always is the case where some consumers prefer one, and some prefer another good at the same price. Hence, we could imagine that these two types of product differentiation could be incorporated into the same model: Good 2 of higher quality since it offers more surplus to all consumers at the same price; But it can also be offered in a variety of specifications, measured along horizontal axis. Surplus Good 2’s surplus: V2– td-P V2>V1 0 1 Good 1’s surplus: V1-td-P

22. For a representative consumer model, we can have: • With In the linear and representative consumer models of quality differences, all consumers care the same amount about quality: all consumers are identical from this point of view. Hence, demand for the high quality good simply dominates demand for the lower quality good (graphically, it is shifted out). But what if consumers have different values for quality?

23. For consumer i and product j, then, we obtain the following utility: Uij= V-Pj + θiqj Let the preference for quality, θ, be distributed uniformly over the population on interval [a, b], where b>a>0. Quality of product j is qj Two products, j= 1,2, are offered with q1 > q2. Both produced a marginal cost c. Hence, the consumer who is just indifferent about which she buys is: U1 = V – P1 + θq1 = V – P2 + θq2 = U2 θ*= It follows that all consumers who have a higher preference for quality than θ* prefer to consume the high quality good, good 1, and the rest prefer the low quality good, good 2: D1 = D2=

24. D1 = D2= Hence, we maximise profit of firm 1 = (P1-c)(D1) = (P1-c)[b- w.r.t. P1 And similarly for firm 2. Call the denominator of demand Δq:   And similarly, where Δq= Which both slope upwards in prices, as in other differentiated product Bertrand models. Solving simultaneously, we obtain: ,

25. Conclusions: Firms “value price”. In other words: We have equilibrium prices above marginal cost for b>2a (so for “enough” preference for quality). Furthermore, the equilibrium price for the high quality good is larger than that of the low quality good (and the profit ends up being higher for the high quality good). Both goods have positive demand, with some consumers purchasing the high quality good for a large price and some opting for the low quality but cheap good. Hence, consumers move to the good that yields better “value”, given their preferences, and firms price for “value”.

26. Conclusions: Firms differentiate to relax price competition It is also the case that the high quality good does not find it optimal to attempt to price the lower quality good completely out of the market. Why? The low quality firm would be willing to decrease price to cost to save the final sale(s). For the same cost level, this means that the high quality firm would earn nothing picking up the final sales and would lose on the inframarginal customers. This can’t be best. Note, too, that the low quality firm would not choose to have the same high quality as this would generate zero profits (no differentiation). Hence, the market segments naturally.

27. Conclusions: “Natural Oligopoly” Finally, there is a natural limit to the number of firms that enter such a market in equilibrium. Why? Notice that firms do not earn profit symmetrically, as they did in the representative consumer model: lower quality firms earn less (at equal cost). This means that entry “between” other firms’ quality level will end up not paying for a lower quality firm at some point: first, because it earns little even if the higher quality firm “gives it space”; second, the closer its quality to the higher level, the more tempted the high quality firm will be to cut its price – and all customers prefer the high quality brand for the same price. So…we can’t locate “too close” to the higher quality brand.

28. Other Models Where Price Differences Persist Search models: Consumers do not know ex ante the characteristics or prices of all products on the market. Learning these characteristics/prices requires costly search: the more the consumer invests in search, the more informed before acquisition. Hence, even if a product is lower priced, a consumer may not purchase it because s/he has not invested in search to find it.  lower priced product may not steal all business

29. Switching Costs: Even if products are identical ex ante, they differ ex post (ie after purchase) because there is a cost to switching supplier. Examples: Changing mortgage supplier with a fee for terminating the mortgage. Learning a new software package with the old and different techniques in mind. Auxiliary products acquired to support a product make you more likely to stick with the same product type as switchingrequires purchase of new auxiliaries whereas staying doesn’t ).  Lower priced product may not steal all business because consumers unwilling to pay the switching cost.

30. Switching Cost Model: 2 firms, A and B, producing (homogeneous) good at marginal cost c. N consumers, each with reservation price (maximum willingness to pay), R, for good. Period 1: Both firms simultaneously announce prices Each consumer purchases at most one unit from one or the other firm. Period 2: Consumers purchase a second time. Those who stay pay that firm’s price. Those who switch pay new firm’s price plus switching cost, s. Note: We know that if the switching cost is zero, the equilibrium in both periods would be to price at marginal cost. Consumers allocate to the two firms (say, half and half). A monopolist would charge R

31. Switching Cost Model: Solving backwards with s>0: Let aN consumers purchase from firm A in period 1. (1-a)N consumers purchase from B in period 1. Say firm B charges R for the good. What is A’s best response? If A charges R, it earns profit = aN(R-c) If A charges less, all new customers benefit by at most R-s from the switch. Say A drops its price and picks up the entire market: profit = N(R-s-c) (actually, would need to drop price a tiny bit more than this to pick up all customers… so profit is “arbitrarily close” to this) N(R-s-c) > aN(R-c) if R-C-s>aR-ac  (R-c)(1-a)>s

32. Pure Strategy Equilibrium: “Collusive outcome” When PB is close to R, then A can charge R as well. When PB is below R-s, however, A loses all its customers unless it drops its price to PB-s. PB = PA- s PA PA = PB R PA= PB-s R-s s>(1-a)(R-c), a = ½ c c R R-s PB 0

33. No Pure Strategy Equilibrium When PB is close to R, A is tempted to steal all of B’s customers by undercutting. As PB falls, the revenue earned with these new customers falls and so A prefers to charge a higher price. PB = PA - s PA PA = PB R PA = PB -s R-s c s<(1-a)(R-c) a = ½ c R R-s PB 0

34. Equilibrium: First, note that price will be above marginal cost since if a firm charges anything equal to or less than c+s, cutting price can never be the best response since this will always yield zero profit at best. Instead, charging c+2s and obtaining the maximum from the existing customers would always yield better profits: Assuming that a=1/2, PR = (c+2s-c)(1/2) = s>0 So we can already conclude that price exceeds marginal cost, despite homogeneity ex ante and price competition. Second, in equilibrium, it can never be the case that one firm allows the other to steal all market share (since profits would be zero). Hence, we know that in the second period the firms split the market. We can also say that if a>1/2 (so one firm has a larger market share), then this firm will tend to have a greater market share in the second half… More formally…

35. Mixed Strategy Equilibrium • Example: Matching Pennies Head 2 Tail Head Tail 1, -1 -1, 1 There is no sure way to win for either of the players…and no pure strategy Nash Equilibrium 1 -1, 1 1, -1 A reasonable way to play is to randomize between H and T with equal probability. The expected payoff is zero.

36. Let p, 0<p<1, be player 1’s probability to play H, and q, 0<q<1, be player 2’s prob. to play H. • Player 1’s expected payoff for playing H is: • U1 = 1 . q + -1 . (1-q). • Player 1’s payoff for T is U1 = -1 . q + 1 . (1-q). U1 1 And similarly, we could draw a graph For player 2 with U2 = -1p +1(1-p) for playing H U2 = 1p - 1(1-p) for player T T 1 q 0 H 1/2 -1

37. We can write best response correspondences in terms of mixed strategy: • p = B1(q) = 0 if q < ½ (play T) B2(p) = 1 if p < ½ • p = B1(q) = [0,1] if q = ½ B2(p) = [0,1] if p = ½ • p = B1(q) = 1 if q > ½ (play H) B2(p) = 0 if p > ½ q 1 ½ B2(p) The unique Nash Equilibrium is (p, q) = (½, ½). B1(q) 0 ½ 1 p

38. Example of Nash equilibrium (2) • “Battle of the sexes” • 3 Nash equilibria • 2 in pure strategies: (football, football), (opera, opera) • 1 in mixed strategies • Each player randomises such that the other player is indifferent between their own component strategies • Chris: (2/3 football, 1/3 opera); Pat: (1/3 football, 2/3 opera)

39. Returning to the Switching Cost game: Prices are continuous in our framework, so we don’t have a simple 2x2 matrix. In order to solve for the mixed strategy equilibrium, we would have to compute the support for prices (the range over which prices could be charged), assume that a probability density exists over these prices, and integrate over all prices to obtain the equilibrium. Support for prices where a = ½ : A does not undercut prices below: PB-s-c = (PB+s-c) ½  PB = 3s+c and similarly for PAWe won’t compute the mixed strategy equilibrium for continuous prices… Assume, instead, that prices can take two values: 3s+c or R:

40. Firm B R 3s+c R (R-c)/2; (R-c)/2 0, 3s Firm A 3s+c 3s, 0 3s/2; 3s/2 So that if R>6s+c we have two pure strategy Nash Equilibria (NW, SE corners) – so we know that a mixed strategy equilibrium exists… The mixed strategy equilibrium is, then: a probability q such that where q is B’s probability of playing R, 1-q is the probability of playing 3s+c and q = p, where p is the equivalent probability for A. For 2s+c<R<6s+c, we can also look for the mixed strategy equilibrium…but we won’t do it in lecture.

41. Firm B R 3s+c R (R-c)/2; (R-c)/2 0, 3s Firm A 3s+c 3s, 0 3s/2; 3s/2 First period: Notice (1) that profits in period 2 are positive, (2) that whatever market share was obtained in period 1 is preserved in period 2. Hence, market share is valuable because it is maintained in equilibrium and also generates profit. Assume that customers are “myopic” so that they purchase in period 1 from the Low-priced firm. This means that firms compete for market share – by cutting price – in period 1.

42. Hence, we have a pattern of first period low prices to entice consumers to a firm followed by second period “fleecing” prices. How low would firms be willing to drive prices in the first period? …potentially down so low that the total discounted profit over the two periods is zero. Does this hurt consumers? First period gains may offset second period losses to CS. Hence, we cannot say whether overall this is better or worse for consumers than a simple Bertrand game.

43. Other Considerations: • “Growing markets”: • New customers arriving in period 2  If firms cannot price discriminate • then they must trade off profits on old “locked in” customers against losing • market share from new customers. • “Price Discrimination” • On the other hand, a firm able to price discriminate can essentially always • price low to new customers and price high to old customers, as before. • “Subsidised switching” • As new customers are valuable, firms should be willing to subsidise switching • and hence eliminate switching costs. For example a firm can introduce • a completely compatible new product so as to avoid any learning costs. This • reduces the first period intensity of competition.

44. References: Chen, Y. (1997) “Paying Customers to Switch”, Journal of Economics and Management Strategy 6, 877-897. Garcia-Marinoso, G., (2001) “Technological Incompatibility, Endogenous Switching Costs, and Lock-In”, Journal of Industrial Economics Padilla, A. J., (1992)“Mixed Pricing in Oligopoly with Consumer Switching Costs”, International Journal of Industrial Organization 10(3), 393-411. Klemperer, P., (1987) “Markets With Consumer Switching Costs”, Quarterly Journal of Economics Shaked, A. and Sutton, J (1983) “Natural Oligopolies” Econometrica51(5), 1469-1484.