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1 MANAGEMENT SCIENCE oi 87/mnsc.8.93ec pp. ec ec6 e-companion ONLY AVAILABLE IN ELECTRONIC FORM informs 28 INFORMS Electronic Companion Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumers by Maxim Sinitsyn, Management Science, oi 87/mnsc.8.93.

2 ec2e-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumers Aitional material an iscussion EC.. Piecewise linear eman specification To check the robustness of my results I consier an alternative specification of consumer preferences, in which the switchers are Hotelling-like consumers who are uniformly istribute on segment [; b], have linear transportation costs, an purchase the prouct from the firms that are locate at the ens of the segment. Here, b is the measure of consumer heterogeneity. The eman for the strong firm s prouct is shown in Figure EC.. Figure EC. Piecewise linear eman function of the strong firm Deman of Firm s b b p s p w Sinitsyn (28) consiere the symmetric case with the firms having the same percent of the loyal consumers an prove that in the mixe strategy equilibrium the firms will use a finite number of prices. The same argument works for the asymmetric case. Figure EC.2 shows the regions with ifferent types of Nash equilibria. The comparison with Figure 2 in the main boy of the paper reveals that the structure of the regions is the same. One ifference is that the value that separates the regions with ifferent sales behavior of the strong firm is close to for the case of piecewise linear emans whereas it is close to for the case of normal emans. Similarly, the optimal pricing strategies illustrate in Figure EC.3 reveal that the main conclusions erive from Figure 3 remain intact. The strong firm charges its sale price more often than

3 e-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumersec3 Figure EC.2 Regions with ifferent types of Nash equilibria (piecewise linear emans) A (; ).9.7 σ B (p s <; p w <) C (; p w <).3 D its regular price when it uses only two prices, the firms either use the same number of prices in the equilibrium or the strong firm uses one price more than the weak firm, the number of prices the firms use increases with a ecrease in consumer heterogeneity, an when both firms use sales the weak firm promotes more often than the strong firm. EC.2. Explanation of the shape of region A in Figure 2 The bounary of region A (Figure 2) oes not epen on the percent of loyal consumers because oes not affect the profit maximization problem of the weak firm the weak firm maximizes its profit from the switching segment of the consumers, but the size of this segment is irrelevant for the weak firm s optimal price choice. The only variables that matter are the price of the strong firm an the consumer heterogeneity. Thus, if the strong firm charges the reservation price, the optimal price of the weak firm epens only on consumer heterogeneity σ. If σ is sufficiently large, the weak firm will also charge the reservation price. If the weak firm oes not fin it profitable to unercut the rival who is charging the reservation price, neither oes the strong firm its incentives to compete for the switchers are the same as the weak firm s, an the presence of the loyal cohort puts an upwar pressure on the strong firm s price. The smallest value of σ which results in (; ) equilibrium is given by the formula π w( {; })= an is equal to σ.798. Since it oes not have loyal consumers, the weak firm will have more incentives to eviate from (; ) equilibrium than the strong firm. When σ is large, π w( {; }) is positive, since the price competition is weak. As σ eclines,

4 ec4e-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumers Figure EC.3 Equilibrium prices, probabilities, an profits for =.3 (piecewise linear emans) Prices p w, p s, p w,2 p s,2.9.7 b.3. Probabilities γ s, γ w,.9.7 b γ s,2 γ w,2.3. Profits π s π w.9.7 b.3. EC.3. Firms strategies when the weak firm has loyal consumers The major results of the paper remain unchange if I allow for the weak firm to have some loyal consumers as well. Figure EC.4 illustrates the optimal strategies of the firms when the percent of the loyal consumers of the strong firm is fixe at.3 an the percent of the loyal consumers of the weak firm is either 2 =. or 2 =. As expecte, when the weak firm has some loyal consumers, the price competition is softene. Therefore, the region, in which the firms use mixe strategies, is smaller. Comparing Figures 3 an EC.4, we see that the larger is the weak firm s loyal consumer base, the smaller is the level of consumer heterogeneity at which there exists a mixe strategy equilibrium with two prices charge by the strong firm an one price charge by the weak firm. When the amounts of loyal consumers of both firms are close to each other, the weak firm can also charge as its regular price. π w( {; }) ecreases. For the values of σ below σ, π w( {; }) <, sotheweakfirm can increase its profits by charging the price less than.

5 e-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumersec5 Figure EC.4 Equilibrium prices,probabilities, an profits for =.3 an varying loyal segments of weak firm Prices ( 2 =.) Prices ( 2 =) p w, p s, p s,2 p s,2 p w, p s, Probabilities ( 2 =.) γs, γ w, Probabilities ( 2 =) γ s, γ w, γ s,2 γ s,2 Profits ( 2 =.) Profits ( 2 =) π s π s π w π w EC.4. Firms strategies for large values of The firms optimal pricing strategies an profits for large values of are shown in Figure EC.5. As σ eclines, the single price equilibrium (; p w < ) persists until the strong firm has enough incentives to charge the sale price close to p w with some small probability an keep charging with large probability. In this equilibrium, the strong firm charges two prices while the weak firm charges one price. Contrary to the case <,inwhichregularprice was introuce with a small probability, for > it is the sale price that is introuce with a small probability. This ifference in strategies is intuitive for small values of, the strong firm can not extract large profits from its loyal cohort, therefore, in orer to fight for the switchers the sale price is charge with large probability. For large values of, the strong firm can extract large profits from its loyal cohort, thus, it charges the regular price with large probability. The weak firm s strategy, on the other han, is similar to the one for small values of. Whenσ becomes sufficiently low, the weak firm introuces its secon, regular price with small probability an most of the time charges the sale

6 ec6e-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumers Figure EC.5 Equilibrium prices, probabilities, an profits for = pw, Prices p s, p p s,2 w,2.7 σ.3. Probabilities γ s, γ w, γ s,2 γ w,2.7 σ.3. Profits π s π w.7 σ.3. price. For smaller values of σ, the structures of the equilibrium strategies for the large an small values of are similar. It is still true that when both firms use the sales, the weak firm promotes more frequently than the weak firm. As coul be observe from Figures 3 an EC.5, the well-known result that the firms profits ecline with a ecrease in consumer heterogeneity (see, for example, Anerson an e Palma (992)) oes not hol everywhere when one of the firmshasloyalconsumers.itisstillthecasethata ecrease in σ exerts ownwar pressure on the firms prices if they were to compete only for the switching cohort. However, in the region where the strong firm charges price with certainty, the presence of loyal consumers prevents the strong firm from competing aggressively for the switchers. Therefore, as long as the strong firm charges, theweakfirm coul increase its profits by charging the same price an getting more switchers as σ eclines.

7 e-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumersec7 EC.5. Changing the percent of loyal consumers Fixing an changing heterogeneity level σ, as one in the preceing section, was helpful in explaining how multiple prices appear in the equilibrium. The firms coul affect the level of consumer heterogeneity by choosing the prouct characteristics before the proucts are launche or using avertising to change the consumer perception of the prouct characteristics after the proucts are launche. The more similar are the proucts (as perceive by the consumers), the more homogeneous are the consumers in their willingness to pay the price premium. The consumer loyalty is another variable that the firms coul influence to some extent using marketing techniques such as loyalty programs. In this section, I examine what happens to the firms optimal pricing strategies an profits as percent of loyal consumers varies while the level of consumer heterogeneity σ is fixe. Figure EC.6 shows the firms optimal pricing strategies an profits for σ = an σ =.3. Figure EC.6 Equilibrium prices, probabilities, an profits for σ = an σ =.3 Firms prices for σ = Firms prices for σ =.3 p s,2 p p w,2 s,2 p p w, p w, p s,3 s, p s, Firms probabilities for σ = γ w, γ s,2 Firms profits for σ = γ s, π s π w Firms probabilities for σ =.3 γs, γ w, γ s,2 γ s,3 γ w,2 Firms profits for σ =.3 π s π w

8 ec8e-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumers For small values of, there exists a pure strategy Nash equilibrium (region C in Figure 2) since the percent of loyal consumers is relatively small, the strong firm concentrates on competing for the switching cohort. The prices of both firms are small, but the strong firm oes not have enough incentives to eviate to a higher price. When is large, again there is a pure strategy Nash equilibrium (region B in Figure 2) this time the percent of loyal consumers is so large that the strong firm concentrates only on serving its loyal cohort an charges the reservation price. The weak firm uses its optimal response to the strong firm s strategy of charging. The price of the weak firm oes not epen on the percent of loyal consumers since the weak s firm optimal response epens only on the price of the strong firm, an that price is regarless of. For intermeiate values of, mixe strategy equilibria exist (region D in Figure 2). When becomes sufficiently large, the strong firm starts charging two prices. Regular price is charge with a small probability. As increases, the strong firm increases the probability of charging until becomes large enough to generate a pure strategy equilibrium, in which is charge with certainty. As expecte, the profit of the strong firm increases with. Therearetwoeffects from an increase in on the weak firm s profit. There is a irect effect as increases, there are less switchers, an this ecreases the weak firm s profit. An inirect effect comes from the fact that an increase in the loyal consumer base gives less incentive to the strong firm to fight for the switchers an, therefore, its pricing strategy will be less aggressive. This leas to an increase in the profit oftheweakfirm. Which effect ominates epens on the values of an σ. Figure EC.7 shows, for ifferent values of σ, how affects the weak firm s profit in the regions epicte in Figure 2. The weak firm s profit function always strictly ecreases in regions A an B. This happens because in these regions, for the fixe level of σ, the optimal strategies of the firms o not epen on the percent of loyal consumers. The strong firm always charges the reservation price in these regions, an the weak firm charges its optimal response to the reservation price. Thus, the weak firm s profit per switching consumer, π w, oes not epen on, antheweakfirm s total profit, π w ( ), eclines linearly with an increase in.

9 e-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumersec9 Figure EC.7 Profits of the weak firm for ifferent values of σ σ=. σ=.3 σ= C D B C D B C D B σ= σ= σ=.3 C D B C D B C D B σ= σ= σ= C B C B A When σ is greater than approximately.3, the profit function of the weak firm strictly increases in regions C an D. These regions are characterize by the strong firm using a price lower than either as a part of a sales strategy in a mixe strategy equilibrium (region D) or as the only price in a pure strategy equilibrium (region C). The situation is less clear for the smaller values of σ (σ 6.3). The weak firm s profit function still strictly increases in region C, but it is not monotonic in region D. Figure EC.7 shows that the values of, atwhichtheweakfirm s profit starts to ecline, epen on the level of consumer heterogeneity σ, an are large for the smaller values of σ. This means that the investment in the loyal consumer base might not be an efficient mechanism for eterring entry. This argument was previously mae by Schmalensee (983), who analyze a similar case using the concept of Cournot equilibrium. Of course, that oes not mean that the strong firm shoul not invest in acquiring loyal consumers. An increase in the loyal consumer base oes increase the strong

10 ece-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumers firm s profit when the weak firm is present in the market. Therefore, if the entry eterrence is not possible, the strong firm shoul invest in loyal consumers until the point where the marginal revenue from having a larger loyal consumer base equates to the marginal cost of acquiring loyals. Using the taxonomy of strategic behavior introuce by Fuenberg an Tirole (984), since the investment makes the strong firm soft, an prices are strategic complements, the strong firm shoul behave like a fat cat, overinvesting in orer to force a less aggressive response by the weak firm. On the contrary, if the strong firm wants to eter entry, it shoul stay lean an hungry unerinvesting in loyal consumers in orer to generate a more aggressive price response that reuces the profitability of an entry by the weak firm. Thefactthattheweakfirm actually benefits from the strong firm having more loyal consumers coul have implications for the entry ecision by the firms. For example, a supermarket chain wants to open a new store an chooses between two locations: near an ol store with an establishe loyal consumer base or near a recently opene store that i not yet acquire loyal consumers. My analysis showsthatitcoulbemoreprofitable to choose a location near the store with the larger loyal consumers base as this store will be less likely to engage in aggressive price competition. The following is a summary of the effect of the percent of the loyal consumers on the firms profits for the fixe level of consumer heterogeneity σ: ) The profit of the strong firm always increases with. 2) If σ is high (σ>σ.798), the weak firm s profit always eclines with an increase in. 3) If σ takes on intermeiate values (.3 <σ<σ), the weak firm s profit strictly increases in regions C an D, where the strong firm uses prices smaller than. Theweakfirm s profit reaches the maximum value on the bounary with region B an then strictly eclines. 4) If σ is small (σ<.3), the weak firm s profit strictly increases in region C, strictly ecreases in region B, an is non-monotonic in region D. EC.6. Asymmetric switchers In this section, I introuce asymmetry into the switchers preferences an examine the effect of the average preferences of the switchers on the optimal pricing strategies an profits of the firms. The

11 e-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumersec switchers preferences are istribute normally with mean an variance σ 2. shows the average preferences for the firms proucts among the switchers. If <, the switchers on average prefer the weak firm s prouct. In orer to examine how affects the optimal pricing strategies, I fix to be.3, σ to be, anlet vary from to.. The equilibrium prices, probabilities, an profits are shown in Figure EC.8. Figure EC.8 Effect of the switchers average preferences on the equilibrium prices, probabilities, an profits for =.3, σ= p w,2 Prices p s, p s,2 p w,.2 Probabilities γ w, γ s, γ s,2 γ w,2.2 Profits π s π w.2 When is negative an large in absolute terms, the weak firm has a large avantage within the switching cohort. Therefore, it is able to charge the reservation value. If the strong firm chooses to compete for the switchers, its price woul have to be so small that the strong firm is better off targeting only its loyal consumers an also charging. As increases, the incentives for the strong firm to compete for the switchers also increase. When 4, itbecomesprofitable for

12 ec2e-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumers the strong firm to target the switchers an it starts using two prices: the regular price that extracts maximum profits from the loyal consumers an the sale price, esigne to compete for the switchers. Atthesametime, theweakfirm also starts using two prices. As the weak firm s avantage among the switching cohort iminishes, its pricing strategy becomes more aggressive. Its sale price an, eventually, its regular price go own with most of the probability place on the sale price. The sale price of the strong firm stays practically constant until reaches approximately.3, atwhichpoint both firms are charging one price. The strong firm s profit stays approximately constant in the region of sales. In this region, an improvement in the strong firm s position among the switchers has very little effect on the strong firm s profit. This fact is ue to the weak firm competing more aggressively an charging lower prices when increases. On the other han, more aggressive pricing strategies by the weak firm lea to a ecrease in its profit. These features of the profit functions are robust to the changes in the parameters of the eman function Figure EC.9 shows the profit functions of the two firms for ifferent values of an σ ( takes values,, an; σ takes values,.3, an5). For each graph in Figure EC.9, in the region between the two vertical lines the strong firm charges multiple prices, i.e. uses the strategy of sales. In the sales region, the weak firm s profit strictly eclines with an increase in, whilethestrongfirm s profit stays approximately constant. To the left of the sales region, there exists a pure strategy equilibrium with both firms charging. Inthat region, is negative an large in absolute terms, so the strong firm chooses not to compete for the switchers. When is sufficiently low, the strong firm s profit iscloseto an the weak firm s profit is close to. To the right of the sales region there exists a pure strategy equilibrium with the weak firm charging a price strictly less than. In this region, an increase in leas to an increase in the profit of the strong firm an a ecrease in the profit oftheweakfirm. Theanalysisofhowtheprofits of both firms epen on changes in an in the region of sales highlights an important ifference between the two groups of consumers. If the strong firm wants to eter entry, it shoul o so by gaining an avantage among switchers, not by increasing

13 e-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumersec3 Figure EC.9 Effect of the switchers preferences on the profits of the strong (soli line) an the weak (otte line) firms for ifferent values of an σ =; σ= sales =; σ= sales =; σ= sales =; σ=.3 sales =; σ=.3 sales =; σ=.3 =; σ=5 =; σ=5 =; σ=5 its loyal consumer base. A better position among switchers woul force the weak firm to price more aggressively, thus, ecreasing the weak firm s profit. A larger loyal consumer base, on the other han, leas to a less aggressive pricing of the strong firm which, actually, can increase the weak firm s profit. If the strong firm wants to increase its profits while allowing the weak firm to operate in the market, it shoul o so by increasing its loyal consumer base, not by improving its position among switchers. If the strong firm invests in gaining a better position among switchers, the weak firm will respon by pricing more aggressively, which keeps the strong firm s profit approximately constant. The following is a summary of the effect of the average preferences of the switchers on the firms profits for the fixe level of consumer heterogeneity σ an the percent of loyal consumers : ) The switchers average preferences o not affect the strong firm s profits in the region of sales.

14 ec4e-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumers An increase in the switchers average preferences for the strong firm s prouct increases the strong firm s profit in the regions where a pure strategy equilibrium exists. 2) An increase in the switchers average preferences for the strong firm s prouct ecreases the profit oftheweakfirm. EC.7. Asymmetric costs In this section, I examine the effect of the cost asymmetry on the firms optimal pricing strategies an profits. Recall that the marginal cost of both firms was normalize to zero. Now, I keep the marginal cost of the strong firm, c s, constant at zero an allow the marginal cost of the weak firm,,tovary. can be positive or negative. When is negative, the marginal cost of the weak firm is smaller than the marginal cost of the strong firm. Figure EC. illustrates the epenence of the equilibrium prices, probabilities, an profits on the value of. Figure EC. Effect of on the equilibrium prices, probabilities, an profits for =.3, σ= Prices p s,2 p w, p w,2 p s, γ w, Probabilities γ s, γ s,2 γ w,2 Profits π w π s In a simple moel of prouct ifferentiation without loyal consumers, an increase in the marginal

15 e-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumersec5 cost of the weak firm leas to a ecrease in the weak firm s profit an an increase in the strong firm s profit (see, for example, Tirole (998), pg. 282, 3). As Figure EC. shows, the weak firm s profit always ecreases with an increase in, but the strong firm s profit increases with an increase in only in the regions where a pure strategy equilibrium exists. In the region where sales are present, the profit of the strong firm stays approximately constant. These features of the profit functions are robust to the changes in the parameters of the eman Figure EC. shows the profit functions of the two firms for ifferent values of an σ ( takes values,, an; σ takes values,.3, an5). The flat segments of the strong firm s profits occur in the regions that involve sales. Figure EC. Effect of on the profits of the strong (soli line) an the weak (otte line) firms for ifferent values of an σ =; σ= =; σ= =; σ= sales sales sales =; σ=.3 sales =; σ=.3 sales =; σ=.3 =; σ=5 =; σ=5 =; σ=5 The following is a summary of the effect of the marginal cost of the weak firm on the firms profits for the fixe level of consumer heterogeneity σ an the percent of loyal consumers :

16 ec6e-companion to Maxim Sinitsyn: Technical Note: Price Promotions in Asymmetric Duopolies with Heterogeneous Consumers ) The marginal cost of the weak firm oes not affect the strong firm s profits in the region of sales. An increase in the marginal cost of the weak firm increases the strong firm s profit inthe regions where a pure strategy equilibrium exists. 2) An increase in the marginal cost of the weak firm ecreases its profits. References [] Anerson, S. P. an A. e Palma, "The Logit as a Moel of Prouct Differentiation," Oxfor Economic Papers, 992, 44(), [2] Fuenberg, D. an J. Tirole, "The Fat Cat Effect, the Puppy Dog Ploy an the Lean an Hungry Look," American Economic Review, Papers an Proceeings, 74, [3] Schmalensee, Richar, Avertising an Entry Deterrence: An Exploratory Moel, The Journal of Political Economics, 983, 9(4), [4] Sinitsyn, Maxim, "Price Dispersion in Duopolies with Heterogeneous Consumers," 28, International Journal of Inustrial Organization, oi:.6/j.ijinorg [5] Tirole, Jean, The Theory of Inustrial Organization. Cambrige, Mass.: MIT Press, 999.