# Advance Selling, Competition, and Brand Substitutability

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1 Advance Selling, Competition, and Brand Substitutability Oksana Loginova October 27, 2016 Abstract This paper studies the impact of competition on the benefits of advance selling. I construct a two-period price-setting game with heterogeneous consumers and two firms that produce different brands. Some consumers prefer one brand, others prefer the other brand. Consumers derive common value from their preferred brand, but they differ in how strongly they dislike their less preferred brand. I consider the situation in which one firm can offer consumers the opportunity to pre-order its product in advance of the regular selling season. I calculate the benefits of advance selling when this firm faces competition from the other firm in the regular selling season and when it does not. I show that competition enhances the benefits of advance selling when in the advance selling season consumers are uncertain about which brand they will prefer. Comparative statics analysis with respect to brand substitutability reveals some interesting results. For example, I find that in the competitive setting the firm has greater incentives to advance sell when the brands are more substitutable, while the reverse holds in the monopolistic setting. Key words: advance selling, price competition, strategic consumers, valuation uncertainty, consumer heterogeneity, substitutability of brands. JEL codes: C72, D42, D43, L12, L13, M31. Department of Economics, University of Missouri, 118 Professional Bldg, Columbia, MO 65211, USA; Phone:

7 equilibrium profits A A = B monopolistic setting competitive setting d Figure 1: Equilibrium profits in the no advance selling benchmark Proposition 2 (No advance selling benchmark, monopolistic setting). In the monopolistic setting of the benchmark we have: (i) When d v/3, firm A charges p A 2 profit is Π A = v d. = v d and all consumers buy. Firm A s equilibrium (ii) When d (v/3, v], firm A charges p A 2 = (v + d)/2. All type A consumers buy, and type B consumers with x (v d)/2 buy. Firm A s equilibrium profit is Π A = v 4 + v2 8d + d 8. All proofs are relegated to the Appendix. In contrast to the competitive setting, in the monopolistic setting firm A s profit decreases in d. This is because the differentiation effect of d that price competition subsides as the brands become less substitutable is turned off in the monopolistic setting. Figure 1 depicts the relationships between the firms equilibrium profits in the two settings of the benchmark and the brand substitutability parameter d. 3 We will use the results of Propositions 1 and 2 in the next three sections, when calculating the relative benefits of advance selling firm A s profit in model I (II, III) firm A s profit in the benchmark competition 3 Recall that d [0, v]. In Figure 1, I normalized v to 1, so the range for d becomes [0, 1]. 7

8 and firm A s profit in model I (II, III) firm A s profit in the benchmark monopoly in the competitive and the monopolistic settings, respectively. When the first ratio is larger (smaller) than the second, we say that competition enhances (diminishes) the benefits of advance selling. 5 Model I In model I consumers are initially uncertain about which brand they will prefer in the regular selling season. Thus, in period 1 they are heterogenous with regard to x, but not the brand. 5.1 Competitive Setting Below is the timeline for the competitive setting. 1. At the beginning of period 1, each consumer learns her x. 2. Firm A sets a pre-order price for its product, p A Consumers decide whether to pre-order the product from firm A. 4. At the beginning of period 2, each consumer learns her preferred brand. 5. Firm A and firm B simultaneously and independently set their regular selling season prices p A 2 and pb Those consumers who did not pre-order in period 1 decide which brand to purchase, if any. Obviously, consumers with high values of x (strong brand preferences) have more incentives to wait until the regular selling season to learn their preferred brand than those with low values of x (weak brand preferences). Hence, suppose that consumers with x (ˆx, d] are left on the market in period 2. The next lemma establishes the firms Nash equilibrium prices in period 2 as functions of the threshold ˆx. Lemma 1 (Nash equilibrium prices in period 2 given ˆx). Suppose that consumers with x (ˆx, d] are on the market in period 2. Then the firms Nash equilibrium prices are: (i) When d v/2, p A 2 (ˆx) = p B 2 (ˆx) = { d + 2 dˆx ˆx 2, if ˆx d 2 2d, if ˆx ( d 2, d). (ii) When d (v/2, v], { d + 2 dˆx ˆx p A 2 (ˆx) = p B 2, if ˆx d 2 2 (ˆx) = 2vd v 2 ( 2 ) v, if ˆx d 2 2vd v 2 2, d. 8

9 In this lemma ˆx takes values from the interval [0, d). When ˆx = d (no consumers are left on the market in period 2), we will assume that the firms prices are p A 2 (d) = pb 2 (d) = min{2d, v}, obtained by substituting ˆx = d into the expressions for p A 2 (ˆx) and pb 2 (ˆx). Also, note that the special case ˆx = 0 implies p A 2 (0) = pb 2 (0) = d, consistent with the results of Proposition 1. After the announcement of p A 1, consumers form their expectations about the firms regular selling season prices that p A 2 and pb 2 will be the same and equal some price p 2. If a consumer pre-orders the product from firm A, her payoff will be v p A 1 if brand A turns out to be her preferred brand, and v x p A 1 otherwise. Thus, the consumer s expected payoff is v x 2 pa 1. If the consumer waits until period 2 to purchase her preferred brand (provided that p 2 v), she receives the payoff of v p 2. Therefore, the consumer will optimally pre-order the product from firm A if and only if v x 2 pa 1 v p 2, x 2(p 2 p A 1 ). Lemma 2 (Optimal consumer behavior given p A 1 and p 2). Suppose that firm A charges p A 1 in period 1 and that consumers expect the firms prices in period 2 to be equal p 2. Then consumers with x ˆx will pre-order the product from firm A, while those with x (ˆx, d] will wait until period 2 to purchase their preferred brand, where { 0, if p A ˆx = 1 > p 2 min{d, 2(p 2 p A 1 )}, if pa 1 p 2. For a given value of p A 1, a rational expectations equilibrium consists of prices pa 2 and pb 2, consumer expectations p 2, and a threshold ˆx such that: (i) p A 2 and pb 2 are the Nash equilibrium prices given ˆx (Lemma 1); (ii) ˆx describes the optimal consumer behavior given p 2 (Lemma 2); and (iii) consumer expectations are fulfilled (p A 2 = pb 2 = p 2). The statement of the next lemma, Lemma 5 (Rational expectations equilibrium given p A 1 ), is relegated to the Appendix because of its length. It gives the equilibrium values of ˆx and p 2 as functions of p A 1. As it turns out, both are (weakly) decreasing functions. The intuition is as follows. As p A 1 increases, the number of consumers who pre-order the product from firm A decreases. Because lower values of ˆx lead to more intense price competition in period 2, p 2 decreases. That a higher pre-order price leads to lower regular selling season prices is an interesting result driven by the information structure assumed in model I. 9

10 To complete the equilibrium analysis, it is left to calculate the optimal pre-order price p A 1 that maximizes firm A s expected profit Π A (p A 1 ) = p A 1 ˆx(p A 1 ) d where the functions ˆx(p A 1 ) and p 2(p A 1 ) are from Lemma 5. + p 2 (p A 1 ) d ˆx(pA 1 ), (1) 2d Proposition 3 (Model I, competitive setting). In the competitive setting of model I, the equilibrium prices, profits, and consumer behavior are given by: (i) When d v/2, firm A charges p A 1 = 3d/2 inducing all consumers to pre-order. In period 2 firm A and firm B charge p 2 = 2d. The firms equilibrium profits are Π A = 3d 2 and Π B = 0. (ii) When d (v/2, v], firm A charges p A 1 = 3v/4 inducing consumers with x v/2 to preorder. In period 2 firm A and firm B charge p 2 = v. The remaining consumers purchase their preferred brand. The firms equilibrium profits are Π A = v 2 + v2 8d and Π B = v 2 v2 4d. When the level of brand substitutability is high (d v/2), price competition in period 2 is intense (part (i) of Lemma 1). Low second-period prices imply firm A s pre-order price must be low to generate positive demand. When both p A 1 and p 2 are low, firm A s profit is maximized when the total number of consumers who purchase its brand, ˆx d + d ˆx 2d, is maximized. In equilibrium firm A charges p A 1 = 3d/2, which induces all consumers to pre-order: ˆx = d. When the level of brand substitutability is low (d > v/2), price competition in period 2 is less intense (part (ii) of Lemma 1). Maximizing (1) yields an interior solution in the sense that firm A s equilibrium pre-order price attracts only a fraction of consumers: ˆx < d. Recall that in the competitive setting of the no advance selling benchmark (Proposition 1) the firms profits are increasing functions of d. Here, firm A s profit first increases, then decreases in d. Firm B s profit increases in d. Figure 2 depicts the firms equilibrium profits as functions of d. With Propositions 1 and 3 in hand, we can calculate the relative benefits of advance selling when firm A competes with firm B in the regular selling season: firm A s profit in model I firm A s profit in the benchmark = competition { 3, if d v 2 if d ( v v d + v2 4d 2, 2, v]. The ratio is strictly above one for all values of d: firm A always benefits from advance selling in the competitive setting of model I. Also note that the ratio (weakly) decreases in d. That is, the relative benefits of advance selling decrease as the two brands become less substitutable. 10 (2)

11 equilibrium profits A monopolistic setting A competitive setting B d 5.2 Monopolistic Setting Figure 2: Equilibrium profits in model I Now we consider the monopolistic setting. The results of the equilibrium analysis are summarized in the proposition below. Proposition 4 (Model I, monopolistic setting). In the monopolistic setting of model I, the equilibrium prices, profits and consumer behavior are given by: (i) When d v/3, firm A sets a high pre-order price so that no consumer pre-orders. In period 2 firm A charges p A 2 = v d. All consumers buy. Firm A s equilibrium profit is Π A = v d. (ii) When d (v/3, v/2], firm A charges a high pre-order price so that no consumer preorders. In period 2 firm A charges p A 2 = (v + d)/2. All type A consumers buy, and type B consumers with x (v d)/2 buy. Firm A s equilibrium profit is Π A = v 4 + v2 8d + d 8. (iii) When d (v/2, v], firm A charges p A 1 = 3v/4 inducing consumers with x v/2 to preorder. In period 2 firm A charges p A 2 = v. The remaining type A consumers buy. Firm A s equilibrium profit is Π A = v 2 + v2 8d. 11

12 When the level of brand substitutability is high (d v/2), the monopolist sets a high preorder price so that all consumers wait until the regular selling season. This pricing strategy is payoff equivalent to not implementing advance selling. Inability to set the regular selling season price in advance hurts the monopolist. If only firm A could commit to p A 2 = v! It then could charge pa 1 = v d/2, inducing all consumers (even the one with x = d) to pre-order. Firm A would earn Π A = v d 2, a higher profit than that in parts (i) and (ii) of Proposition 4. When the level of brand substitutability is low (d > v/2), the monopolist sells in both periods. Observe that firm A sets the same prices and earns the same profit as in the competitive setting of model I. The only difference between part (ii) of Proposition 3 and part (iii) of Proposition 4 is that in the competitive setting type B consumers with x > v/2 are served by firm B. Figure 2 depicts firm A s equilibrium profit as a function of d. It is straightforward to show that when ( (1 + ] 12)v d, v, 11 2 firm A s profit is actually lower in the monopolistic setting than in the competitive setting. Counterintuitive at first, this result is driven by firm A s inability to commit to a particular regular selling season price in advance. With Propositions 2 and 4 in hand, we can calculate the relative benefits of advance selling when firm A is a monopolist: firm A s profit in model I firm A s profit in the benchmark = monopoly { 1, if d v 2 v/2+v 2 /(8d) v/4+v 2 /(8d)+d/8, if d ( v 2, v]. The ratio equals one when d v/2: firm A finds it optimal not to implement advance selling when the level of brand substitutability is high (parts (i) and (ii) of Proposition 4). We can now answer the main question of the paper, Does competition enhance the benefits of advance selling? I plot the benefit ratios (2) and (3) in Figure 3. We see that in model I competition always enhances the benefits of advance selling. This is because in model I advance selling attracts consumers with weak brand preferences (low x), relaxing price competition in the regular selling season. (3) 6 Model II In model II, in period 1 a consumer knows her preferred brand, but is uncertain about her disutility from consuming the alternative. We first analyze the competitive setting. Type B consumers have more incentives to wait until period 2 than type A consumers. In Lemma 3 I calculate the firms Nash equilibrium prices when the market in period 2 consists of the fraction 1 β of type B consumers (i.e., all type A consumers and the fraction β of 12

13 benefits of advance selling monopoly competition d Figure 3: Benefits of advance selling in model I type B consumers have pre-ordered the product from firm A) and when it consists of all type B consumers and the fraction 1 α of type A consumers (i.e., the fraction α of type A consumers have pre-ordered the product from firm A). Lemma 3 (Nash equilibrium prices in period 2). Suppose the market in period 2 consists of the fraction 1 β of type B consumers, then the firms Nash equilibrium prices are: p A 2 = d 3 and p B 2 = 2d 3. Consumers with x d/3 will purchase brand A and those with x (d/3, d] will purchase brand B. If the market consists of all type B consumers and the fraction 1 α of type A consumers, then p A d(3 2α) 2 = and p B d(3 α) 2 =. 3 3 Type B consumers with x αd/3 and type A consumers will purchase brand A, while type B consumers with x (αd/3, d] will purchase brand B. Where we are solving for Nash equilibrium, β takes values from the interval [0, 1). It follows from the first part of the lemma that the Nash equilibrium prices are independent of β, so we will assume that when β = 1 (no consumers are left on the market in period 2), p A 2 = d/3 and p B 2 = 2d/3. For a given price p A 1 that firm A might be charging, Lemma 4 derives the firms equilibrium prices in period 2 and the optimal consumer purchasing behavior. 13

14 Lemma 4 (Rational expectations equilibrium given p A 1 ). Suppose that firm A charges pa 1 period 1. Under the rational expectations equilibrium we have: in (a) If p A 1 d/9, all consumers pre-order; in period 2 the firms charge pa 2 = d/3 and pb 2 = 2d/3. (b) If p A 1 (d/9, d/3], only type A consumers pre-order; in period 2 p A 2 = d 3 and p B 2 = 2d 3, type B consumers with x d/3 purchase brand A, while those with x (d/3, d] brand B. (c) If p A 1 (d/3, d], the fraction 3(d pa1 α = ) 2d of type A consumers pre-order; in period 2 and p A 2 = p B 2 = d(3 2α) 3 d(3 α) 3 = p A 1 = d + pa 1 2 type B consumers with x αd/3 and the remaining type A consumers purchase brand A, while type B consumers with x (αd/3, d] brand B., (d) If p A 1 > d, no consumer pre-orders; in period 2 p A 2 = p B 2 = d, each consumer purchases her preferred brand. It turns out that firm A maximizes its profit when it sets a high pre-order price so that all consumers wait until period 2 (part (d) of the above lemma). Proposition 5 (Model II, competitive setting). In the competitive setting of model II, firm A sets a high pre-order price so that no consumer pre-orders. In period 2 firm A and firm B charge p A 2 = pb 2 = d. Each consumer purchases her preferred brand. The firms equilibrium profits are Π A = Π B = d 2. Why is it the case that any p A 1 that generates positive demand (parts (a), (b) and (c) of Lemma 4) is inferior to not implementing advance selling (charging a high pre-order price so that no consumer pre-orders)? In model II, type A consumers have more incentives to preorder the product than type B consumers, so there will be more type B consumers than type A consumers in period 2. This intensifies price competition. Take, for example, p A 1 (d/9, d/3] 14

15 (part (b) of Lemma 4). Such a price induces type A consumers to pre-order. In period 2 the firms compete for type B consumers. To attract customers, firm A has to lower its price. Firm B responds by lowering its price, too. As a result, p A 2 = d/3 and pb 2 = 2d/3. (If all type A consumers and all type B consumers were present in period 2, the equilibrium prices would be much higher, p A 2 = pb 2 = d.) Intense price competition in period 2, in turn, puts pressure on firm A s pre-order price. Indeed, if firm A pursues a strategy of selling to some consumers in period 1, p A 1 cannot be higher than p A 2. Thus, it becomes very costly for firm A to generate positive demand in period 1. Firm A is better off charging a high pre-order price so that no consumer pre-orders. The same result that firm A finds it optimal not to implement advance selling also arises in the monopolistic setting. Proposition 6 (Model II, monopolistic setting). In the monopolistic setting of model II, firm A sets a high pre-order price so that no consumer pre-orders. In period 2, firm A s equilibrium price and profit are the same as in the monopolistic setting of the no advance selling benchmark (Proposition 2). The proof of Proposition 6 (in the Appendix) is rather involved despite its simple conclusion. Again, firm A s inability to commit not to lower its price in period 2 for type B consumers hurts the monopolist. If firm A could credibly commit to p A 2 = v, then it would charge pa 1 = v d/2, inducing both types of consumers to pre-order. Firm A s profit of Π A = v d 2 would be higher than that in Proposition 2. To sum up, in both competitive and monopolistic settings of model II firm A finds it optimal not to implement advance selling. This happens because, in model II, type B consumers have more incentives to wait until the regular selling season than type A consumers, which implies firm A s price in period 2 cannot be high. To encourage consumers who prefer brand A to preorder, the firm must set its pre-order price below this low regular selling season price. Generating positive demand in period 1 turns out to be too costly. 7 Model III In model III, in period 1 a consumer does not know which brand she will prefer, nor her disutility from consuming the alternative. In contrast to the previous two models, there is no heterogeneity among consumers in the advance selling season, implying all consumers either pre-order or all wait until the regular selling season. This makes the analysis of model III fairly simple. Let us start with the competitive setting. It is easy to show that in equilibrium firm A sets its advance selling price to p A 1 = 3d/4, inducing all consumers to pre-order. In period 2, firm A 15

16 equilibrium profits A monopolistic setting A competitive setting B d Figure 4: Equilibrium profits in model III and firm B charge p A 2 = pb 2 = d.4 The resulting profits are and Π A = 3d 4 Π B = 0. I plot these profits, as well as firm A s profit in the monopolistic setting (see Proposition 7 below) in Figure 4. The relative benefits of advance selling when firm A competes with firm B in the regular selling season equal firm A s profit in model III firm A s profit in the benchmark = 3d/4 competition d/2 = 3 2. (4) Since the ratio is above one for all values of d, firm A always benefits from advance selling. Now consider the monopolistic setting. By charging a high pre-order price firm A can achieve the equilibrium profit under the no advance selling benchmark (Proposition 2). The other option is to induce all consumers to pre-order. 4 Suppose the fraction γ [0, 1) of consumers have pre-ordered the product. Hence, the market in period 2 consists of the fraction 1 γ of consumers. The firms Nash equilibrium prices are p A 2 = p B 2 = d. These prices are independent of γ, so we will assume that when γ = 1 (no consumers are left on the market in period 2), p A 2 = p B 2 = d. 16

17 Proposition 7 (Model III, monopolistic setting). In the monopolistic setting of model III, the equilibrium prices, profits and consumer behavior are given by: (i) When d v/3, firm A sets a high pre-order price so that no consumer pre-orders. In period 2 firm A charges p A 2 = v d. All consumers buy. Firm A s equilibrium profit is Π A = v d. (ii) When d (v/3, 2v/(3 + 8)], firm A charges a high pre-order price so that no consumer pre-orders. In period 2 firm A charges p A 2 = (v + d)/2. All type A consumers and type B consumers with x (v d)/2 buy. Firm A s equilibrium profit is Π A = v 4 + v2 8d + d 8. (iii) When d (2v/(3+ 8), v], firm A charges p A 1 = v v2 /(8d) d/8 inducing all consumers to pre-order. In period 2, firm A charges p A 2 = (v + d)/2. Firm A s equilibrium profit is Π A = v v2 8d d 8. This proposition implies that when the level of brand substitutability is high (d 2v/(3 + 8)), the monopolist finds it optimal not to implement advance selling. When the level of brand substitutability is low (d > 2v/(3 + 8)), the monopolist induces all consumers to pre-order. Recall that the same result that firm A s incentives to advance sell decrease with the level of brand substitutability was obtained in the monopolistic setting of model I. Finally, we calculate the relative benefits of advance selling when firm A is a monopolist firm A s profit in model III firm A s profit in the benchmark = monopoly { 1, if d 2v 3+ 8 v v 2 /(8d) d/8 v/4+v 2 /(8d)+d/8, if d ( 2v 3+ 8, v ] (5) and compare them with the relative benefits under the competitive setting (4). As we see from Figure 5, competition enhances the benefits of advance selling, to a greater extent when the level of brand substitutability is high (d is low). 8 Concluding Remarks In this paper I looked at the effect of competition on the advantages of advance selling driven by consumer valuation uncertainty. I assumed two firms that produce different brands. Some consumers prefer one brand, and others prefer the other brand. Consumers derive common value from their preferred brand, but they differ in how strongly they dislike their less preferred brand. I considered the situation in which one firm can offer consumers the opportunity to pre-order its product in advance of the regular selling season. I calculated the benefits of advance selling when this firm faces competition from the other firm in the regular selling season and when it does not, under three demand specifications. In model I consumers are initially uncertain 17

19 Π B = p/2. Then p is the highest price such that no firm has incentives to undercut it by ɛ (0, min{p, d}]. Suppose firm A deviates to p ɛ. It will attract type B consumers with x ɛ, so the firm s profit will go up by At the same time, firm A will lose (p ɛ) ɛ 2d. ɛ 1 2 on its existing customers due to the price decrease. Such deviation is not profitable if (p ɛ) ɛ 2d ɛ1 2, p ɛ d. The highest p that satisfies this inequality for all ɛ (0, min{p, d}] is p = d. Proof of Proposition 2: Any p A 2 [v d, v] will induce type A consumers and type B consumers with x v p A 2 buy the product. Firm A faces the following optimization problem: ( 1 max p A p A 2 [v d,v] v ) pa 2. 2d Differentiating w.r.t. p A 2 and setting the derivative to zero yields v 2pA 2 2d = 0, implying that when d v/3 the firm should charge p A 2 Otherwise, the firm should charge p A 2 = v + d 2, earning Π A = v 4 + v2 8d + d 8. = v d and sell to all consumers. Proof of Lemma 1: Suppose we found a symmetric Nash equilibrium in which p A 2 = pb 2 = p d (p d will be confirmed later), implying that type A consumers buy from firm A and type B consumers from firm B. Since only consumers with x (ˆx, d] are on the market in period 2, a firm (say, firm A) should decrease its price by at least least ˆx to attract any of type B consumers. Then p is the highest price such that a deviation to p ˆx ɛ, ɛ (0, d ˆx] is unprofitable. Suppose firm A deviates. It will attract type B consumers with x (ˆx, ˆx + ɛ], so the firm s profit will go up by (p ˆx ɛ) ɛ 2d. 19

20 At the same time, firm A will lose (ˆx + ɛ) d ˆx 2d on its existing customers. This deviation is unprofitable if for ɛ (0, d ˆx] (p ˆx ɛ) ɛ ˆx (ˆx + ɛ)d 2d 2d, ɛ 2 ɛ(p d) + dˆx ˆx 2 0. The left-hand side attains its minimum at ɛ = min{d ˆx, (p d)/2}. When ɛ = d ˆx, the above inequality becomes p 2d. When ɛ = (p d)/2, it becomes (p d) 2 4(dˆx ˆx 2 ), As long as p v, we have p d + 2 dˆx ˆx 2. p = { d + 2 dˆx ˆx 2, if ˆx d 2 2d, if ˆx ( d 2, d]. When d > v/2 and ˆx > d/2 2vd v 2 /2, p = v. Lemma 5 (Rational expectations equilibrium given p A 1 ). Suppose that firm A charges pa 1 period 1 and let g(p A 1 ) 1 ( ) 10d 2p A d dp A1 4(pA1 )2. in Under the rational expectations equilibrium we have: (i) When d v/2, (a) p A 1 3d/2 induces all consumers to pre-order; in period 2 the firms charge p 2 = 2d; (b) p A 1 (3d/2, 7d/4] induces consumers with x 2(2d pa 1 ) to pre-order; in period 2 the firms charge p 2 = 2d and the remaining consumers purchase their preferred brand; (c) p A 1 (7d/4, (3 + 17)d/4] induces consumers with x g(p A 1 ) to pre-order; in period 2 the firms charge p 2 = p A 1 +g(pa 1 )/2 and the remaining consumers purchase their preferred brand; (d) p A 1 > (3 + 17)d/4 induces all consumers to wait until period 2; in period 2 the firms charge p 2 = d and each consumer purchases her preferred brand. (ii) When d (v/2, v], 20

21 (a) p A 1 v d/2 induces all consumers to pre-order; in period 2 the firms charge p 2 = v; (b) p A 1 (v d/2, v (d 2vd v 2 )/4] induces consumers with x 2(v p A 1 ) to pre-order; in period 2 the firms charge p 2 = v and the remaining consumers purchase their preferred brand; (c) p A 1 (v (d 2vd v 2 )/4, (3 + 17)d/4] induces consumers with x g(p A 1 ) to pre-order; in period 2 the firms charge p 2 = p A 1 + g(pa 1 )/2 and the remaining consumers purchase their preferred brand; (d) p A 1 > (3 + 17)d/4 induces all consumers to wait until period 2; in period 2 the firms charge p 2 = d and each consumer purchases her preferred brand. Proof of Lemma 5: We first consider the case d v/2 that corresponds to part (i) of the lemma. (a) Suppose p A 1 3d/2. It follows from Lemma 1 that when ˆx = d, p 2 = 2d. It is left to check that under these prices all consumers will pre-order the product. Indeed, by Lemma 2 as 2(2d p A 1 ) d holds for pa 1 3d/2. ˆx = min{d, 2(p 2 p A 1 )} = d, (b) Suppose p A 1 (3d/2, 7d/4]. It follows from Lemma 1 that when ˆx = 2(2d pa 1 ), which decreases from d to d/2 as p A 1 increases from 3d/2 to 7d/4, p 2 = 2d. We need to check that it is optimal for consumers with x 2(2d p A 1 ) to pre-order the product. By Lemma 2 ˆx = min{d, 2(p 2 p A 1 )} = 2(2d p A 1 ), as 2(2d p A 1 ) < d holds for pa 1 (3d/2, 7d/4]. (c) Suppose p A 1 equations (7d/4, (3 + 17)d/4]. In this case ˆx and p 2 are found from the system of { p2 = d + 2 dˆx ˆx 2 ˆx = 2(p 2 p A 1 ). Substituting p 2 = p A 1 + ˆx/2 into the first equation allows us to solve for ˆx: ˆx = 1 ( ) 10d 2p A d dp A1 4(pA1 )2. As p A 1 increases from 7d/4 to (3+ 17)d/4, ˆx decreases from d/2 to ( 17 1)d/(2 17) and p 2 = p A 1 + ˆx/2 decreases from 2d to (4 + 17)d/ 17. (d) Suppose p A 1 > (3 + 17)d/4. The above system of equations does not have a solution. In this case ˆx = 0, implying p 2 = d (Lemma 1). We need to check that under these prices no consumer will pre-order the product. Indeed, by Lemma 2 ˆx = 0, as p A 1 > p 2 = d holds for p A 1 > (3 + 17)d/4. 21

22 Proof of part (ii) of the lemma is very similar to that of part (i). Proof of Proposition 3: We first consider the case d v/2 that corresponds to part (i) of the proposition. (a) Suppose p A 1 3d/2. Then (1) is simply Π A (p A 1 ) = p A 1. The profit is maximized at p A 1 = 3d/2, ΠA (3d/2) = 3d/2. (b) Suppose p A 1 (3d/2, 7d/4]. Then (1) becomes Π A (p A 1 ) = p A 1 2(2d p A 1 ) d + 2d d 2(2d pa 1 ) 2d = 2(pA 1 )2 d + 6p A 1 3d. The function attains its maximum at p A 1 = 3d/2, ΠA (3d/2) = 3d/2. (c) Suppose p A 1 (7d/4, (3 + 17)d/4]. Then (1) becomes Π A (p A 1 ) = p A 1 g(p A 1 ) d ( + p A 1 + g(pa 1 ) ) d g(p A 1 ). 2 2d Numerical calculations show that for any value of p A 1 (7d/4, (3+ 17)d/4], Π A (p A 1 ) < 3d/2. (d) Suppose p A 1 > (3 + 17)d/4. Then (1) is simply This is less than 3d/2. Π A (p A 1 ) = d 2. It follows that when d v/2, firm A should set its pre-order price to 3d/2, which induces all consumers to pre-order. Firm A s equilibrium profit is 3d/2 and firm B s profit is zero. Next, we consider the case d (v/2, v] that corresponds to part (ii) of the proposition. (a) Suppose p A 1 v d/2. Then (1) is simply Π A (p A 1 ) = p A 1. The profit is maximized at p A 1 = v d/2, ΠA (v d/2) = v d/2. (b) Suppose p A 1 (v d/2, v (d 2vd v 2 )/4]. Then (1) becomes Π A (p A 1 ) = p A 1 2(v p A 1 ) d + v d 2(v pa 1 ) 2d = 2(pA 1 )2 d + 3v d pa 1 + v(d 2v). 2d The function attains its maximum at p A 1 greater than v d/2. = 3v/4, ΠA (3v/4) = v/2 + v 2 /(8d). This is 22

23 (c) Suppose p A 1 (v (d 2vd v 2 )/4, (3 + 17)d/4]. Then (1) becomes Π A (p A 1 ) = p A 1 g(p A 1 ) d ( + p A 1 + g(pa 1 ) ) d g(p A 1 ). 2 2d Numerical calculations show that for any value of p A 1 (v (d 2vd v 2 )/4, (3 + 17)d/4], Π A (p A 1 ) < v/2 + v2 /(8d). (d) Suppose p A 1 > (3 + 17)d/4. Then (1) is simply This is less than v/2 + v 2 /(8d). Π A (p A 1 ) = d 2. It follows that when d (v/2, v], firm A should set its pre-order price to 3v/4. This price induces consumers with x v/2 to pre-order. In period 2 firm A and firm B charge p 2 = v. The remaining consumers purchase their preferred brand. The firms equilibrium profits are Π A = v/2 + v 2 /(8d) and Π B = v/2 v 2 /(4d). Proof of Proposition 4: The proof is completed in four steps. 1. We start with period 2 and assume that consumers with x (ˆx, d] are left on the market. Firm A can charge p A 2 = v and sell only to type A consumers. In this case it will earn v d ˆx 2d. If the firm charges p A 2 = v d, it will sell to all consumers and earn (v d) d ˆx d. Finally, the firm can charge p A 2 [v d, v ˆx] and sell to all type A consumers and type B consumers with x (ˆx, v p A 2 ], earning ( d ˆx p A 2 2d + v pa 2 ˆx ). 2d The above function is maximized at p A 2 = { v+d 2ˆx 2, if ˆx 3d v 2 v d, if ˆx ( 3d v 2, d ]. It can be shown that p A 2 = v maximizes firm A s profit in period 2 when d (v/2, v] and ˆx > (d 2vd v 2 )/2; p A 2 = v d is optimal when d v/3 or when d (v/3, v/2] and ˆx ((3d v)/2, d]. Otherwise, the firm is better off charging p A 2 = v+d 2ˆx 2. 23

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