Industrial Organization Lecture 7: Product Differentiation

Transcription

1 Industrial Organization Lecture 7: Product Differentiation Nicolas Schutz Nicolas Schutz Product Differentiation 1 / 57

2 Introduction We now finally drop the assumption that firms offer homogeneous products. Differentiated products are similar, but not identical (they are close, but not perfect, substitutes). There are two major models of differentiation: 1 Horizontal differentiation: If all products were the same price, consumers disagree on which product is most preferred. E.g. films, cars, clothes, books, cereals, icecream flavors, Starbucks (by geographic location),... 2 Vertical differentiation: If all products were the same price, all consumers agree on the preference ranking of products, but differ in their willingness to pay for the top ranked versus lesser ranked products. E.g. computer memory / processors, airline tickets, different quantities,... Nicolas Schutz Product Differentiation 2 / 57

3 Introduction Another way to think about product differentiation: The hedonic approach (or characteristics approach). Think of a product as being a bundle of different attributes. For example, a car is bundle of a certain amount of horsepower, a color, weight, size, etc. Products have multiple attributes to them, and some attributes are vertical attributes (eg. horsepower) while other attributes are horizontal attributes (eg. color). In general, we think most products are differentiated in both vertical and horizontal dimensions. Nicolas Schutz Product Differentiation 3 / 57

4 Spatial Models of Product Differentiation This is a model of price competition among differentiated firms, i.e., Bertrand w/ differentiated products. The main idea comes from Hotelling (1929), referred to as Hotelling s Linear City : Consider a (uni-dimensional) city that is one mile long. Formally, the city is the segment [0, 1]. There is a mass M > 0 of consumers uniformly distributed on [0, 1]. (This means, that for every 0 a b 1, there are M(b a) consumers located between a and b.) There are two firms (say, two coffee shops). Firm l is located at point x = 0. Firm r is located at point x = 1. Consumers have unit demand for the product, i.e., a given consumer either buys one unit from firm l, or buys one unit from firm r, or buys nothing. A consumer that buys nothing receives a utility flow of 0. Nicolas Schutz Product Differentiation 4 / 57

5 Spatial Models A consumer that buys from firm i {l, r} receives u = v }{{} gross utility of consumption t distance to firm i } {{ } disutility of walking p i. t, the transport cost parameter measures the extent to which consumers dislike walking. To summarize: Fix a consumer location x [0, 1]. The utility of a consumer located at point x is given by: v tx p l if consumer buys from l, u(x) = v t(1 x) p r if consumer buys from r, 0 if consumer does not buy. Nicolas Schutz Product Differentiation 5 / 57

6 Spatial Models What kind of differentiation does this model capture? It s horizontal product differentiation: Consider the consumer located at x = 0, and assume prices are the same (p l = p r = p). That consumer s net utility from consuming the left-end good is v p, whereas its utility from consuming the right-end one is v p t. For the consumer located at x = 1, utility from consuming the left-end (resp. right-end) product is v p t (resp. v p). Consumers disagree on their preferred product. Nicolas Schutz Product Differentiation 6 / 57

7 Spatial Models What kind of differentiation does this model capture? It s horizontal product differentiation: Consider the consumer located at x = 0, and assume prices are the same (p l = p r = p). That consumer s net utility from consuming the left-end good is v p, whereas its utility from consuming the right-end one is v p t. For the consumer located at x = 1, utility from consuming the left-end (resp. right-end) product is v p t (resp. v p). Consumers disagree on their preferred product. Consumer x maximizes utility: max(v p l tx, v p r t(1 x), 0). The consumer doesn t always go to the closest firm: it depends on relative prices p l vs. p r. Similarly, the consumer doesn t always go to the cheapest firm: it depends on relative distances x vs. 1 x. Nicolas Schutz Product Differentiation 6 / 57

8 Spatial Models Suppose prices are sufficiently low, so that all consumers buy the good. Then there is a marginal consumer sitting at x for whom v p l tx = v p r t(1 x ), i.e., p l + tx = p r + t(1 x ). x = t (p r p l ). Nicolas Schutz Product Differentiation 7 / 57

9 Spatial Models Suppose prices are sufficiently low, so that all consumers buy the good. Then there is a marginal consumer sitting at x for whom v p l tx = v p r t(1 x ), i.e., p l + tx = p r + t(1 x ). x = t (p r p l ). All people to the left of x will strictly prefer going to the left seller. All people to the right of x will strictly prefer going to the right seller. Nicolas Schutz Product Differentiation 7 / 57

10 Spatial Models Suppose prices are sufficiently low, so that all consumers buy the good. Then there is a marginal consumer sitting at x for whom v p l tx = v p r t(1 x ), i.e., p l + tx = p r + t(1 x ). x = t (p r p l ). All people to the left of x will strictly prefer going to the left seller. All people to the right of x will strictly prefer going to the right seller. So demands are given by q l = Mx and q r = M(1 x ): ( 1 q l = M ) 2t (p r p l ), ( 1 q r = M ) 2t (p l p r ). These are demand functions for horizontally differentiated products. (Keep in mind that these demand functions have been derived under the assumption that everybody purchases. When we solve for equilibrium prices, we will need to check whether that condition holds.) Nicolas Schutz Product Differentiation 7 / 57

11 Spatial Models Points to note: If p l > p r, then q l 0 (unlike homogeneous goods). Demand is downward sloping in a firm s own-price, and upward sloping in their competitors price. Instead of a city + physical cost of walking, we could think about the location of my tastes and various products in product space. i.e., although people may differ in their preferences, they may still be willing to consume a product that is not closest to their ideal, as long as the price is low enough. We can also solve this model when the firms are located at points other than the ends. We can also have more than one dimension. Remark: Discrete choice models, Multinomial logit model. Nicolas Schutz Product Differentiation 8 / 57

12 Spatial Models Elasticities: We calculate own-price demand elasticities just like usual. But now the cross-price demand elasticities are also interesting. We can look at the percentage change in demand for firm l due to a one percent increase in price by firm r: ε l,r = q l p r p r q l Since products are imperfect substitutes in this framework, the cross-price elasticities will be positive. A one percent increase in price by firm r will lead to an x percent increase in demand for firm l. Of course, own-price elasticities are negative. Nicolas Schutz Product Differentiation 9 / 57

13 Spatial Models Go back to the linear-city example, but suppose there are three firms: Firm 1 located at x = 0. Firm 2 located at x = 1 5 Firm 3 located at x = 1. (Note firms 1 and 2 are much closer to each other than firms 2 and 3.) Nicolas Schutz Product Differentiation 10 / 57

14 Spatial Models To illustrate the point here, let s compute the demand functions: Assuming firm 2 has positive demand, no consumer is choosing between firms 1 and 3. Intuitively, this will be the case when the three prices are not too different. What is the location of the consumer who s indifferent between firms 1 and 2? x 1,2 = t (p 2 p 1 ) What is the location of the consumer who s indifferent between firms 2 and 3? x 2,3 = t (p 3 p 2 ) Nicolas Schutz Product Differentiation 11 / 57

15 Spatial Models Everyone to the left of x 1,2 goes to firm 1, so q 1 = Mx 1,2, i.e., ( 1 q 1 = M t p ) 2t p 2. Everyone between x 1,2 and x 2,3 goes to firm 2, so q 2 = M(x 2,3 x ), i.e., 1,2 ( 1 q 2 = M t p 1 1 t p ) 2t p 3. All consumers located to the right of x 2,3 go to firm 3, so q 3 = M(1 x 2,3 ), i.e., ( 2 q 3 = M t p 2 1 ) 2t p 3. Nicolas Schutz Product Differentiation 12 / 57

16 Spatial Models Everyone to the left of x 1,2 goes to firm 1, so q 1 = Mx 1,2, i.e., ( 1 q 1 = M t p ) 2t p 2. Everyone between x 1,2 and x 2,3 goes to firm 2, so q 2 = M(x 2,3 x ), i.e., 1,2 ( 1 q 2 = M t p 1 1 t p ) 2t p 3. All consumers located to the right of x 2,3 go to firm 3, so q 3 = M(1 x 2,3 ), i.e., ( 2 q 3 = M t p 2 1 ) 2t p 3. Demand for each firm is decreasing in its own price and increasing in the price of competitors. Demand for firm 1 does not depend on price at firm 3. Firm 2 is the only firm whose demand depends on everyone s prices. Nicolas Schutz Product Differentiation 12 / 57

17 Spatial Models Suppose that every firm sets a price of $5 and suppose that travel cost is $1 per mile (for example). Then we find: q 1 = 100, q 2 = 500, q 3 = 400 And we can compute a table of own and cross-price elasticities at these prices. Nicolas Schutz Product Differentiation 13 / 57

18 Spatial Models q 1 /q 1 q 2 /q 2 q 3 /q 3 p 1 /p p 2 /p p 3 /p The bottom line: because firm 1 is much closer to firm 2 (compared to how close firm 3 is to firm 2), demand at firm 1 is much more sensitive to p 2 than is the case for firm 3. And, the own price elasticity for firm 1 is the highest of all firms. Nicolas Schutz Product Differentiation 14 / 57

19 Spatial Models q 1 /q 1 q 2 /q 2 q 3 /q 3 p 1 /p p 2 /p p 3 /p The bottom line: because firm 1 is much closer to firm 2 (compared to how close firm 3 is to firm 2), demand at firm 1 is much more sensitive to p 2 than is the case for firm 3. And, the own price elasticity for firm 1 is the highest of all firms. This is the general result that firms with products in crowded regions of product space will face flatter residual demand curves, which means they have less market power, which will give rise to lower markups and profits in equilibrium (relative to firms in uncrowded parts of product space). Nicolas Schutz Product Differentiation 14 / 57

20 Spatial Models Go back to the two-firm model. Define a game: Players: Firms 1 (the left firm) and 2 (the right firm). Actions: p 1 and p 2. Payoffs: Profits, π i = (p i c)q i. We look for the symmetric Nash equilibria of this game. Set M = 1. This is without loss of generality. To begin with, assume there exists a symmetric Nash equilibrium in which the market is covered (i.e., all consumers consume), and the marginal consumer receives a strictly positive net utility. Then, profits are given by: ( 1 π i (p 1, p 2 ) = (p i c) 2 + p j p ) i 2t Nicolas Schutz Product Differentiation 15 / 57

21 Spatial Models Notice that the demand functions are much smoother than under Bertrand competition w/ homogenous products. This implies that we can take the FOC to find firm i s best response: FOC: π i p i = 0 (p i c) 1 2t p j p i = 0 2t R i (p j ) = t 2 + c + p j 2 Best responses are upward-sloping, i.e., prices are strategic complements. This is the case in most models of price competition w/ differentiated products. If my rival increases its price, then I want to increase my price as well. Nicolas Schutz Product Differentiation 16 / 57

22 Spatial Models Solve for two equations w/ two unknowns: p 1 = p 2 = c + t Notice that the only Nash equilibrium in which the market is covered and the marginal consumer receives a > 0 utility is symmetric. Check that the marginal consumer receives positive surplus: Since p 1 = p 2, the marginal consumer is located at x = 1/2. His net utility is v t 2 p 1 = v 3 2 t c. Nicolas Schutz Product Differentiation 17 / 57

23 Spatial Models Conclusion: There exists a Nash equilibrium w/ covered market and positive surplus for all consumers iff v > 3 2 t + c. Firms equilibrium profits are given by π i = t 2. Nicolas Schutz Product Differentiation 18 / 57

24 Spatial Models Conclusion: There exists a Nash equilibrium w/ covered market and positive surplus for all consumers iff v > 3 2 t + c. Firms equilibrium profits are given by π i = t 2. Now, assume there exists a symmetric Nash equilibrium where market is covered, and the marginal consumer receives exactly 0 surplus. Denote by p the price set by both firms in this equilibrium. We need to be careful when computing firms best responses. Consider firm 1, and assume that it sets a price in the neighborhood of p, while firm 2 sets p : { π 1 (p 1, p (p1 c) ( 1 ) = 2 + ) p p 1 2t if p 1 p (p 1 c) v p 1 t if p 1 p Firm 1 s profit function is kinked at price p. Nicolas Schutz Product Differentiation 18 / 57

25 Spatial Models Let us look for conditions under which p is a local maximum for function π 1 (., p ): Assume p 1 < p. Then, π 1 = p 1 c + 1 p 1 2t 2 + p p 1 2t For p to be a local maximum, the limit of this derivative as p 1 p has to be non-negative: p c 1 2t 2 Nicolas Schutz Product Differentiation 19 / 57

26 Spatial Models Now assume p 1 > p. Then, π 1 = p 1 c + v p 1 p 1 t t For p to be a local maximum, the limit of this derivative as p 1 p + has to be non-positive. Remember that our equilibrium candidate is symmetric, and that the market is covered. Therefore, firm 1 s demand if it sets p is equal to 1/2. This yields the following inequality: p c + 1 t 2 0 Notice that function π 1 (., p ) is strictly concave for p 1 < p, and strictly concave for p 1 > p. Of course, this does not mean that this function is globally concave, but this implies that any local maximum is also a global maximum. Nicolas Schutz Product Differentiation 20 / 57

27 Spatial Models At the end of the day, it is a best response for firm 1 to set p when firm 2 sets p iff: t 2 p c t Remember that we are looking for a symmetric equilibrium where the marginal consumer (located at x = 1/2) gets 0 surplus. v p t/2 = 0, i.e., p = v t/2. Plugging this in the above inequalities, we conclude that such an equilibrium exists iff: c + t v c t In this case, both firms earn π i = 1 2 (v t 2 c). Nicolas Schutz Product Differentiation 21 / 57

28 Spatial Models As a last step, let us look for equilibria in which the market is not covered. In this case, firm i s demand is independent of firm j s price: q i = v p i t. Profit maximization implies: p i c t + v p i t = 0 i.e., p i = v+c 2. In this case, each firm supplies (v c)/(2t) consumers. Check that the market is not covered: v c 2t < 1 2 v < c + t If the above inequality holds, there is a unique Nash equilibrium with a noncovered market, and both firms earn (v c)2 4t. Nicolas Schutz Product Differentiation 22 / 57

29 Spatial Models Summary: 1 If v > 3t/2 + c, then the market is covered and all consumers get positive surplus thanks to competition. Firms profits, t/2, are increasing in transport costs. 2 If v < c + t, then the market is not covered: we say that firms are local monopolies. Firms profits, (v c)2 4t, are decreasing in transport costs. 3 Intermediate case, when c + t v c + 3t/2. Firms are competing against each other (in the sense that firm 1 s price constrains firm 2 s price), the market is covered, but the marginal consumer is left with zero surplus. Firms profits, 1 2 (v t 2 c), is decreasing in transport costs. In any case, both firms have some market power, and are therefore able to charge positive markup, due to product differentiation. Nicolas Schutz Product Differentiation 23 / 57

30 Spatial Models From the firms point of view, an increase in t makes it harder to reach consumers that are far away: When t is not too large, this is good, as it softens competition (case 1). But if t becomes too large, firms have to lower their prices in order not to lose their customers, which is bad (cases 2 and 3). Nicolas Schutz Product Differentiation 24 / 57

31 Spatial Models and Product Positioning Now assume that the two firms are able to choose their locations on the Hotelling segment. In other words, firms are also competing on the product space. From now on, let s assume that transportation costs are quadratic: If a consumer chooses a product located at distance x from him, it pays transport cost τx 2. [I ll tell you why we make this assumption at the end of the lecture] Start with a monopoly benchmark: Assume firm 1 is the only firm on the Hotelling segment? Where should the monopolist locate himself? Nicolas Schutz Product Differentiation 25 / 57

32 Spatial Models and Product Positioning Now assume that the two firms are able to choose their locations on the Hotelling segment. In other words, firms are also competing on the product space. From now on, let s assume that transportation costs are quadratic: If a consumer chooses a product located at distance x from him, it pays transport cost τx 2. [I ll tell you why we make this assumption at the end of the lecture] Start with a monopoly benchmark: Assume firm 1 is the only firm on the Hotelling segment? Where should the monopolist locate himself? Of course, at the middle of the segment, in order to attract as many consumers as possible, without having to charge a low price. Nicolas Schutz Product Differentiation 25 / 57

33 Spatial Models and Product Positioning Now, consider this problem with two firms, A and B. Timing: 1 First decide where to locate: (a, b) where a, b [0, 1]. 2 Then decide on prices: (p A, p B ). We look for the subgame-perfect equilibria of this two-stage game. Assume that v is sufficiently large, so that the market is always covered at the equilibrium of the second stage. Nicolas Schutz Product Differentiation 26 / 57

34 Spatial Models and Product Positioning Now, consider this problem with two firms, A and B. Timing: 1 First decide where to locate: (a, b) where a, b [0, 1]. 2 Then decide on prices: (p A, p B ). We look for the subgame-perfect equilibria of this two-stage game. Assume that v is sufficiently large, so that the market is always covered at the equilibrium of the second stage. Start with stage 2, and assume a < b. Utility of consumer located at point x: { v pa τ(x a) U x = 2 if she buys from A v p B τ(x b) 2 if she buys from B The marginal consumer is at ˆx where v p A τ(ˆx a) 2 = v p B τ(ˆx b) 2 ˆx = p B p A 2τ[b a] + a + b 2 Nicolas Schutz Product Differentiation 26 / 57

35 Spatial Models and Product Positioning So we can find demand for A and B: q A = ˆx and q B = 1 ˆx: Compute firms profits as a function of p A and p B. Assume zero marginal cost: ( a + b π A (p A, p B, a, b) = p A 2 ( π B (p A, p B, a, b) = p B 1 a + b 2 + p ) B p A 2τ(b a) + p A p B 2τ(b a) ) Set first order conditions equal to zero. equations in p A, p B : Solve the corresponding system of p A (a, b) = τ(b a) a + b (a + b) p B (a, b) = τ(b a) 3 Nicolas Schutz Product Differentiation 27 / 57

36 Spatial Models and Product Positioning Now, go back to stage 1. Define Π A (a, b) the profit of firm A when firms choose locations a and b in period 1, and play the corresponding Nash equilibrium in period 2: + Similar definition for Π B (a, b). Differentiate Π A wrt a: Π A (a, b) π A (p A (a, b), p B (a, b), a, b) Π A a = π A p A p A a + π A p B p B a + π A a Can you say something about the first term on the RHS? Nicolas Schutz Product Differentiation 28 / 57

37 Spatial Models and Product Positioning Now, go back to stage 1. Define Π A (a, b) the profit of firm A when firms choose locations a and b in period 1, and play the corresponding Nash equilibrium in period 2: + Similar definition for Π B (a, b). Differentiate Π A wrt a: Π A (a, b) π A (p A (a, b), p B (a, b), a, b) Π A a = π A p A p A a + π A p B p B a + π A a Can you say something about the first term on the RHS? Firm A knows that it will set its price optimally tomorrow, given a and b, and given that firm b is playing the Nash equilibrium strategy. π A p A = 0. Nicolas Schutz Product Differentiation 28 / 57

38 Spatial Models and Product Positioning Therefore, Π A a = π A p B p B a + π A a ( qa = p A (a, b) = p A (a, b) p B p B a + q A a ) ( 2 a 3(b a) p B(a, b) p A (a, b) 2τ(b a) 2 = p A(a, b) ( 2 + b 3a) < 0 a, b 6(b a) ) So, if firm B positions its product at point b > 0, then Π A (0, b) > Π A (a, b) for all a (0, b]. By symmetry between the two firms, we also have that Π B / b > 0 for all b > a. If firm A chooses a < 1, then, Π B (a, 1) > Π B (a, b) for all b [a, 1). Nicolas Schutz Product Differentiation 29 / 57

39 Spatial Models and Product Positioning Again, by symmetry, this also implies that, Π A (1, b) > Π A (a, b) b a < 1 Π B (a, 0) > Π B (a, b) 0 < b a Best response of firm A when firm B locates at point b [0, 1]: { 1 if ΠA (1, b) Π R A (b) = A (0, b) 0 otherwise + Similar best-response function for firm B. By working out the calculations, you can show that R a (b) = 0 if b > 1/2, and R a (b) = 1 otherwise. Nicolas Schutz Product Differentiation 30 / 57

40 Spatial Models and Product Positioning But we don t really need to do this: Given the above best responses, we know that there is no Nash equilibrium in which 0 < a < 1 or 0 < b < 1. We are left with four equilibrium candidates: (0, 0), (0, 1), (1, 0) and (1, 1). Of course, if both firms locate at point 0 (or 1), products are undifferentiated, and the Bertrand result applies: Π i (0, 0) = Π i (1, 1) = 0. We conclude that the only equilibria are (0, 1) and (1, 0). This is the principle of maximum differentiation: firms choose to differentiate their products as much as they can. Nicolas Schutz Product Differentiation 31 / 57

41 Spatial Models and Product Positioning Some robustness considerations: It is possible to show that this model has no subgame-perfect equilibria when transport costs are linear (tx). The shape of transport costs seems to have an impact on the equilibrium degree of product differentiation. Nicolas Schutz Product Differentiation 32 / 57

42 Spatial Models and Product Positioning Nevertheless, the tradeoff faced by the firms when they decide how to position their products is robust to other types of transport costs: Assume firm B sets b > 1/2. Then, if a < 1/2, whatever the transport costs, Π A a = π A p B p B a + π A a Moving your product toward the center (i.e., positioning your product closer to the average consumer s taste) creates two effects: A direct effect: being close to the center enables the firm to attract more consumers (or to sell to as many consumers, but with a higher price). This is term. the π A a A strategic effect: if you move closer to your rival, he will react by cutting its price in order to protect its market shares. This is the π A p B p B term. a Nicolas Schutz Product Differentiation 33 / 57

43 Spatial Models and Product Positioning Of course, when there is only one firm, the strategic effect does not exist, and the monopoly always wants to locate at the middle of the segment. When there are two firms, the comparison b/w the strategic effect and the direct effect depends on the shape of the transport cost. Nicolas Schutz Product Differentiation 34 / 57

44 A Model of Vertical Differentiation Two firms, A and B, sell a product of quality a and b, respectively. Assume 0 a, b 1. There is a mass 1 of consumers, whose marginal utility for quality x is uniformly distributed on the segment [0, 1]. Consumer x s preferences: { v + xa pa if he chooses product A U x = v + xb p B if he chooses product B This is indeed a model of vertical differentiation: For instance, if a < b and p A = p B, then all consumers derive a higher utility from consuming product B. We want to solve the following two-period game: 1 Firms choose their qualities a and b simultaneously. 2 Firms set their prices p A and p B. Nicolas Schutz Product Differentiation 35 / 57

45 A Model of Vertical Differentiation For simplicity, assume: Firms have the same constant unit costs, which we normalize to zero. v is sufficiently large, i.e., full market coverage in equilibrium. No investment costs to improve quality. As usual, solve by backward induction, and start with period 2. 0 a < b 1. Clearly, if p B p A, then all consumers will choose product B. Assume p B > p A, and look for the marginal consumer, ˆx: v + aˆx p A = v + bˆx p B ˆx = p B p A b a Assume All consumers with x ˆx consume good A, all consumers with x > ˆx buy product B. [Draw consumers utilities] q A = ˆx, q B = 1 ˆx. Nicolas Schutz Product Differentiation 36 / 57

46 A Model of Vertical Differentiation Profit functions: π A (p A, p B, a, b) = p A p B p A b a ( ) p π B (p A, p B, a, b) = p B 1 B p A b a Take FOCs, solve the 2x2 system, get equilibrium prices: And profits: Now, it is clear that, in equilibrium: p A (a, b) = b a 3 p B (a, b) = 2 3 (b a) Π A (a, b) = b a 9 Π B (a, b) = 4 9 (b a) Firms always choose extreme quality levels (0 or 1). A firm never chooses the same quality level as its rival. We conclude that there are two subgame-perfect equilibria: a = 0, b = 1, and a = 1, b = 0. The principle of maximum differentiation also applies in this vertical differentiation model. Nicolas Schutz Product Differentiation 37 / 57

47 Non-Spatial Models Consider a two-firm industry, and assume the demand-side of the economy can be described by a representative consumer with the following preferences: U(q 0, q 1, q 2 ) = q q i ( q i ) γ i=1 i=1 2 q 2 i ( 2 i=1 q i) 2 2 and budget constraint q i=1 p iq i I. As in the perfect competition lecture, q 0 represents the rest of the economy, while q i, i = 1, 2 is consumption of firm i s product. Clearly, the budget constraint is binding at the optimum, so we can plug q 0 = I 2 i=1 p iq i in the consumer s utility. Take the FOCs: i=1 1 p i 2 j=1 q j γ (2q i 2 q j ) = 0 j=1 Nicolas Schutz Product Differentiation 38 / 57

48 Non-Spatial Models Solving the 2x2 system, we get: q i = 1 2 (1 p i γ(p i p)), where p p 1+p 2 2 denotes the average price. Notice that, when p j is too large, q j becomes negative. Of course, the true demand is not negative, so this just means that the consumer does not want to purchase q j at all. This happens when 1 p j γ p j p i 2 0, i.e., when p j 1 + γ 2 p i 1 + γ 2 When this is the case, the consumer maximizes U(q 0, q 1, q 2 ) subject to the budget constraint, and to constraint q j = 0. Again, take the FOC: 1 p i q i q i 1 + γ = 0, i.e., q i = 1 + γ 2 + γ (1 p i). Nicolas Schutz Product Differentiation 39 / 57

49 Non-Spatial Models But again, if p i 1, then the above expression for q i is negative, and the consumer does not consume firm i s product. We can now write the demand faced by firm i as a function of prices: q i (p 1, p 2 ) = 0 if p i > 1 or p i > 1+ γ 2 p j 1+ γ 2 1+γ 2+γ (1 p i) if p i 1 and p i < 1+ γ 2 p j 1+ γ (1 p i γ(p i p)) otherwise Nicolas Schutz Product Differentiation 40 / 57

50 Non-Spatial Models Comments: γ 0 is the substitutability parameter. If γ = 0, q i = 1 2 (1 p i). In this case, the two products are unrelated (i.e., they are not substitutes), and each firm is a monopoly in its submarket. When γ, products become perfect substitutes / homogenous. Indeed, the demand function converges to 0 if p i > 1 or p i > p j q i (p 1, p 2 ) = 1 p i if p i 1 and p i < p j 1 p i 2 if p i = p j Nicolas Schutz Product Differentiation 41 / 57

51 Non-Spatial Models Differences wrt Hotelling demands: In the Hotelling model, firms are either competing, or local monopolies. When firms are competing (in Hotelling), the market is covered, and the total demand does not depend on prices. When firms are local monopolies, the price of a firm does not affect its rival s demand, and the total demand is decreasing in prices. In our non-spatial model of product differentiation, we don t have this discontinuity b/w local monopolies and competition: The total demand is always decreasing in prices, and firms are always competing against each other. Nicolas Schutz Product Differentiation 42 / 57

52 Non-Spatial Models Assume firms set their prices simultaneously, and let us look for the Nash equilibria of this game. Firms profits (normalize costs to zero for simplicity): First-order condition: 1 π i (p 1, p 2 ) = p i 2 (1 p i γ p i p j ) 2 p i (1 + γ 2 ) + 1 p i γ p i p j 2 = 0 This yields the best-response function: R i (p j ) = 2+γp j 2(2+γ). Since these best responses are upward-sloping, prices are strategic complements. Solving for the 2x2 system, we get equilibrium prices, quantities and profits: p i = 2 4+γ q i = 2+γ 8+2γ π i = 2+γ (4+γ) 2 Nicolas Schutz Product Differentiation 43 / 57

53 Non-Spatial Models Notice that dp i dγ = 2 (4+γ) 2. Similarly, dπ i dγ = γ (4+γ) 3. An increase in the degree of product differentiation (i.e., a decrease in γ, the substitutability parameter) raises equilibrium prices and profits. Nicolas Schutz Product Differentiation 44 / 57

54 Non-Spatial Models Notice that dp i dγ = 2 (4+γ) 2. Similarly, dπ i dγ = γ (4+γ) 3. An increase in the degree of product differentiation (i.e., a decrease in γ, the substitutability parameter) raises equilibrium prices and profits. Intuitively, as γ decreases, the demand faced by each firm becomes less elastic: firms can therefore increase their prices without losing too many consumers. This leads to higher equilibrium prices and profits. Nicolas Schutz Product Differentiation 44 / 57

55 Non-Spatial Models Now, we would like to compare these prices and quantities to the equilibrium prices and quantities under Cournot competition. To do so, we need to find the inverse demand functions for the two products. Remember that firms demands are given by: q 1 = 1 ( 1 p 1 γ p 1 p ) q 2 = 1 ( 1 p 2 γ p 2 p ) Subtract these two equations: q 1 q 2 = 1 2 (1 + γ)(p 2 p 1 ) Plug this into firm 1 s demand function: 2q 1 = 1 p 1 γ 1 + γ (q 2 q 1 ) Rearrange terms: p 1 = γ 1 + γ q 1 γ 1 + γ q 2 Nicolas Schutz Product Differentiation 45 / 57

56 Non-Spatial Models p i = γ 1 + γ q i γ 1 + γ q j Because products are substitutes, an increase in firm j s quantity decreases firm i s price. But, because products are imperfect substitutes, p i / q i < p i / p j. Sanity checks: If γ = 0, then p i = 1 2q i. This is indeed the inverse demand curve of a monopoly. Conversely, if γ, then p i 1 q i q j, which is indeed the inverse demand curve in a Cournot duopoly with homogenous products. Nicolas Schutz Product Differentiation 46 / 57

57 Non-Spatial Models Now that we have derived the inverse demand functions, we can solve for the Nash equilibrium of the quantity-setting game. Profits: FOC: ( π i (q i, q j ) = ( γ 1 + γ q i γ 1 + γ q j γ 1 + γ q i γ ) 1 + γ q j ) q i 2 + γ 1 + γ q i = 0 which yields the best-response functions: R i (q j ) = 1 + γ ( 1 γ ) 2(2 + γ) 1 + γ q j This is Cournot competition, so quantities are strategic substitutes. Solving for the system of equations, we get: q i = 1+γ 4+3γ p i = 2+γ 4+3γ π i = (1+γ)(2+γ) (4+3γ) 2 Nicolas Schutz Product Differentiation 47 / 57

58 Non-Spatial Models Now, let s compare the Bertrand and the Cournot outcome: p C p B = q C q B = 2+γ 4+3γ 2 4+γ = 1+γ 4+3γ 2+γ π C π B = (2+γ)(1+γ) (4+3γ) 2 2+γ (4+γ) 2 = γ 2 (4+γ)(4+3γ) > 0 γ 8+2γ = 2 (8+2γ)(4+3γ) < 0 γ 3 (2+γ) > 0 (4+γ) 2 (4+3γ) 2 Under Cournot competition, prices and profits are higher, and quantities are lower. Remarks: We already knew this result when products were homogenous (i.e., when γ ). Now, we see that this result is much more general. When γ = 0, p C = p B, q C = q B and π C = π B. Intuition? When γ = 0, each firm is a monopoly in its submarket, and we know that it does not matter whether the monopoly sets prices or quantities. Nicolas Schutz Product Differentiation 48 / 57

59 Non-Spatial Models Now, let us try to understand why the Cournot outcome is less competitive than the Bertrand outcome. Rewrite firm i s inverse demand function as p i = 1 aq i bq j. Under Cournot competition, each firm expects its rival to hold its output level constant. If firm i expands its output unilaterally, both p i and p j decrease, but firm i only cares about the impact on p i. The impact on firm i s profit from a unilateral output expansion can be written as follows: π C i q i = p i + p i q i q i = p i aq i Nicolas Schutz Product Differentiation 49 / 57

60 Non-Spatial Models Under Bertrand competition, each firm expects its rival to holds its price constant (and not its output level). If a firm expands its output (by lowering its price), its rival reacts by decreasing its quantity. π B i q i = p i + p i q i q i = p i (a + b q j q i )q i > p i aq i > πc i q i This means that firm i has more incentives to increase its output / behave aggressively under Bertrand competition. Nicolas Schutz Product Differentiation 50 / 57

61 Bertrand with Sequential Moves Now assume that firm 1 is a Stackelberg leader, whereas firm 2 is the follower. Assume also that firms compete in prices. As usual, start with period 2. We know that, if firm 1 sets p 1 in period 1, then firm 2 reacts by setting p 2 = R 2 (q 1 ) = 2+γp 1 4+2γ. In period 1, firm 1 anticipates that firm 2 will play its best responses. Therefore, firm 1 maximizes Π 1 (p 1 ) = π 1 (p 1, R 2 (p 1 )). Take the FOC: dπ 1 dp 1 = π 1 p 1 + dr 2 dp 1 π 1 p 2 = 0 Nicolas Schutz Product Differentiation 51 / 57

62 Bertrand with Sequential Moves Let us evaluate dπ 1 /dp 1 when p 1 = p B (the Bertrand equilibrium price). By definition of the Bertrand equilibrium, R 2 (p B ) = p B. dπ 1 dp 1 (p B ) = π 1 p 1 (p B, p B ) + dr 2 dp 1 (p B ) π 1 p 2 (p B, p B ) = 0 + R 2 (p B)p B q 1 p 2 (p B, p B ) > 0 This means that the Stackelberg leader can increase its profit by charging a price above p B. Nicolas Schutz Product Differentiation 52 / 57

63 Bertrand with Sequential Moves Intuition: Start from p 1 = p B, and assume firm 1 increases its price. This creates several effects: A direct effect on firm 1 s profit ( π 1 / p 1 ), which can be decomposed as: A price effect (q1 ), And a quantity effect (p1 q 1 / p 1 ) p B is chosen so that these two effect exactly cancel out, i.e., q 1 + p 1 q 1 p 1 = 0. A reaction effect: with a sequential timing, firm 2 reacts to firm 1 s price increase (R 2 (p B)p B q 1 p 2 (p B, p B )). Since prices are strategic complements under Bertrand competition, R 2 (p B) > 0, i.e., firm 1 benefits from the reaction effect. This implies that firm 1 charges a higher price than in the simultaneous moves game: p L > p B. Nicolas Schutz Product Differentiation 53 / 57

64 Bertrand with Sequential Moves What about firm 2 s price? Nicolas Schutz Product Differentiation 54 / 57

65 Bertrand with Sequential Moves What about firm 2 s price? Remember that prices are strategic complements: R (.) > 0. 2 Therefore, p F = R 2 (p L ) > R 2 (p B ) = p B. What about profits? Nicolas Schutz Product Differentiation 54 / 57

66 Bertrand with Sequential Moves What about firm 2 s price? Remember that prices are strategic complements: R (.) > 0. 2 Therefore, p F = R 2 (p L ) > R 2 (p B ) = p B. What about profits? Let s look at the leader s profit first: The leader could replicate the simultaneous moves outcome, by setting p B in stage 1... but he does not choose to do so. By revealed profitability, the leader is better off under sequential moves: π L > π B. Now, look at the follower: π B = π 2 (p B, p B ) < π 2 (p L, p B ) since p L > p B < π 2 (p L, p F ) by revealed profitability < π F In words, the follower makes more profits than in the simultaneous moves game, since its rival charges a higher price. Nicolas Schutz Product Differentiation 54 / 57

67 Bertrand with Sequential Moves To summarize when firms compete in prices with a Stackelberg timing, the outcome is less competitive, and firms make more profits than in the simultaneous moves game. By contrast, remember that, under Cournot competition with a Stackelberg timing, the outcome is more competitive, and π L > π C > π B. Intuition: Under price competition, prices are strategic complements. Firm 1 uses its Stackelberg leadership as a commitment to set a higher price. This is good for both firms, as it softens competition. Under Cournot competition, quantities are strategic substitutes. Firm 1 uses its leader position as a commitment to increase its quantity. This intensifies competition, makes the leader better off, and the follower worse off. Nicolas Schutz Product Differentiation 55 / 57

68 Bertrand with Sequential Moves Now, let us compare the leader / follower s profits. completely solve the model. Remember firm 1 s FOC: To do so, we need to This yields and dπ 1 dp 1 = π 1 p 1 + dr 2 dp 1 π 1 p 2 = 0 p F = 4 + 3γ p L = 8 + 8γ + γ 2 1 2(1 + γ) γ 8 + 8γ + γ 2 You can check that p L is indeed greater than p F. Plug these prices into profit functions: π L = (4+3γ) 2 8(2+γ)(8+γ(8+γ)) π F = (16+5γ(4+γ))2 16(2+γ)(8+γ(8+γ)) 2 Nicolas Schutz Product Differentiation 56 / 57

69 Bertrand with Sequential Moves Compare these two profits: γ 3 (8 + 7γ) π L π F = 16(2 + γ)(8 + γ(8 + γ)) < 0 2 The follower makes more profits than the leader! Remember that we had the opposite result under Stackelberg in quantities. Intuition: R 2 (p 1) = γ 4+2γ < 1: the slope of firm 2 s best response is smaller than 1. When the leader increases its prices from p B to p L, the follower reacts by increasing its price from p B to p F < p L. Obviously, firms prefer facing a less aggressive competitor, so π L < π F. Nicolas Schutz Product Differentiation 57 / 57