Parallel Accommodating Conduct: Evaluating the Performance of the CPPI Index

Transcription

1 Parallel Accommodating Conduct: Evaluating the Performance of the CPPI Index Marc Ivaldi Vicente Lagos Preliminary version, please do not quote without permission Abstract The Coordinate Price Pressure Index (CPPI) measures the incentives of two competitors to engage on a specific kind of Parallel Accommodating Conduct (PAC). Specifically, it measures the incentives of a given firm to initiate a unilateral percentage price increase, with the expectation that at least one of its competitors will follow it. Using a large set of simulated economies, we measure the accuracy of the index in terms of predicting the impact of a merger on firms incentives to engage on PAC. In addition, we propose two alternative indexes that incorporate the positive pricing externalities existent between substitute brands produced by the merged firm. The results suggest that the original index displays a good performance when predicting the direction of the change on firm s incentives to engage in PAC, but only in mergers with low values of the diversion ratios between the merged firm brands. However, the percentage of cases with Error Type I is not negligible. In addition, it is shown that both alternative indexes dramatically outperform the original one in terms of predicting mergers with a significant anticompetitive impact. 1

2 1. Introduction The aim of this paper is to evaluate the performance of the Coordinate Price Pressure Index (CPPI) introduced by Moresi et al. (2011), and to propose and evaluate an alternative version of this index that incorporates more information. The CPPI measures the incentives of two competitors to engage on a specific kind of tacit coordination strategy to increase prices. Specifically, a leader firm increases its price by a certain percentage, expecting that its competitor will observe this change and will match exactly the same percentage price increase. This specific conduct is considered as a form of Parallel Accommodating Conduct (PAC). As explained by Harrington (2013), the PAC could lead firms to reach a supra-competitive outcome. Nevertheless, the conduct requires some kind of retaliation or deterrence in order to be successfully implemented by firms. The game considered by Moresi et al. (2011) is in line with this argumentation. The CPPI index is derived from a simple model of repeated interaction between two firms, which explicitly considers monitoring and retaliation. The game is as follows. (i) On a certain period, a leading firm increases its price by a given percentage. (ii) In the subsequent period a follower firm observes the price increase and decides whether to match it or not. (iii) When matching occurs, then the price increase becomes permanent. While when there is no matching, the leading firm reverses to its initial price level, with the promise of not initiating any further attempt to engage on PAC. We constructed a set of 50,000 simulated economies. The demand is derived from a model of discrete choice with random coefficients. The supply side is composed by a set of heterogeneous firms that offer differentiated products and compete in prices. The initial level of prices is obtained by computing the Bertrand-Nash equilibrium. On each economy, we considered the engagement on PAC by two firms (let s say and ), and for each firm, we simulated the percentage price increase that would maximize the present value of the firm expected profits. Then the pre-merger Actual Coordinate Price Pressure (ACPP) was computed as the minimum of these two percentage price increases. Thus, this measure can be seen as the lower bound of the supra-competitive prices that two firms could reach through PAC. 2

3 The next step was to simulate a merger between one of these firms (let s say ), the acquiring firm, and a third firm (let s say ), the acquired or target firm. Under this new scenario, we recomputed the ACPP. The impact of the merger on firms incentives to engage on PAC was measured as the variation of the ACPP induced by the acquisition. Then the ACPP variation is used as the benchmark to measure the performance of the variation predicted by the CPPI index. Note that we use a modified version of the CPPI index. The baseline index proposed by Moresi et al. (2011), considers the percentage price increase that leaves to the leader firm indifferent between increasing and not increasing its price (just-profitable variation). While the index used by us considers the percentage price increase that maximizes the present value of its expected profits (profit-maximizing variation). However, as already stated in the Section II.C.3 of Moresi et al. (2011), in practice this is translated just into a small difference with respect to the baseline index: the index built under the profit-maximization assumption, is equal to one half of the baseline one. The accuracy of the index is measured in two situations: (i) its ability to correctly predict the sign of the change ( or ), and (ii) its ability to identify mergers that generate a significant anticompetitive impact ( or ). Then we measured the percentage of cases where the index would lead us to incur on Error Type I and Error Type II. Recall that an Error Type I denotes a case for which the index erroneously identifies a merger as anticompetitive, when it is not. While Error Type II, refers to the case of the index failing to identify an anticompetitive merger. First, regarding the direction of the change, the results suggest that the CPPI displays a good performance for mergers involving brands with low diversion ratios between them. However, the percentage of cases with Error Type I is not negligible. Second, the results suggest that the CPPI index displays a poor performance when predicting mergers that generate a significant increase of firms incentives to engage on PAC. While the percentage of cases with Error Type I is almost zero, the percentage of cases where the index incurs on Error Type II is considerably high. The reason is that the index consistently underestimates the magnitudes of the actual ACPP variations. We believe that the cause of this problem, is that the index omits important information regarding the strategic interaction between the brands produced by the merging 3

4 parties. Indeed, the index does not consider any of the diversion rations between the acquiring and the target firms products. When the acquiring firm decides to increase its prices, part of the loss of sales from one brand goes to its other brand, and vice versa. Thus, the higher the diversion ratios between the brands produced by the acquiring and target firms, the lower the cost of initiating a PAC. Therefore, not considering this information, would lead us to underestimate the impact of the merger. Two alternative indexes are proposed. The first one requires the same set of information as the original one, plus the diversion ratio from the target firm to the acquiring firm. The second one instead, requires a much richer set of information, including: the diversion ratio from the target to the acquiring firm and vice versa, the pre-merger price, margin and own-price elasticity of the target firm. We tested the performance of these alternative indexes against the original one. The results suggest that the original index still outperform the two alternative ones in terms of predicting the sign of the change. Nevertheless, the two alternative indexes dramatically outperform the original one in terms of predicting mergers with a significant anticompetitive impact. Indeed, the percentage of cases with Error Type II decreases substantially, while the percentage of cases with Error Type I is just moderately increased. Thus, considering the interaction between the merged firm brands significantly increases the effectiveness of the index in this regard. In addition, the results show that the alternative index that uses more information, on average outperform the one with less information. The rest of the paper is organized as follows. On Section 2 we briefly review the CPPI index proposed by Moresi et al. (2011). On Section 3 we explain the simulations approach. On Section 4 the main results are shown. Finally, on Section 5 we present our main conclusions and policy recommendations. 4

5 2. The CPPI index 2.1 Pre-merger case We closely follow the definition proposed by Moresi et al. (2011). The PAC strategy consists on a game where two (or more) competitors engage on a joint price increase without the need of explicit communication among them. A leader firm increases its price by a certain percentage with the expectation that its competitors will accommodate and follow a similar strategy. The game is defined as follows. (i) On period, Firm will raise its price by a percentage equal to for at least two periods. (ii) On period, Firm observes the price increase and decides whether to match it or not, by increasing its price by the exact same percentage. (iii) On period there are two possible results. If Firm decides to match the price increase, then the change becomes permanent for both firms. If Firm decides not to match the price increase, then Firm will return to its initial price level and will not make any additional attempt to engage on PAC anymore. Moresi et al. (2011) propose an index that captures a firm s incentives to initiate a unilateral price increase (assuming that its competitor will match it). However, we chose to study a variation of the original index. Instead of using the maximum percentage price increase that a firm would be willing to initiate (just-profitable variation), we use the percentage price increase that maximizes the expected profits from initiating a PAC (profit-maximizing variation). In practice, it is equal to one half of the original index 1. The percentage price increase that Firm is willing to initiate is given by; With: 1 The mathematical derivations of both indexes are presented on the Technical Appendix of Moresi et al. (2011). 5

6 With being equal to the original index proposed by Moresi et al. (2011). The pre-merger market shares of Firms and are given by and, respectively. The term is the percentage margin charged by Firm, and are the own-price elasticities of firms and respectively and is the inter-temporal discount rate assumed to be equal for every firm in the market. The term is the diversion ratio from Firm to Firm The cost/benefit trade-off faced by Firm when initiating a price increase is captured by the term. The numerator represents the size of the demand that Firm will capture once its competitor decides to match the price increase. While the denominator contains the size of the demand lost by Firm when initiating a unilateral price increase. The term measures potential deviations of Firm with respect to the Bertrand-Nash equilibrium in prices. In equilibrium, it has to be the case that and. Therefore, when Firm is already pricing above the equilibrium, its incentives to initiate a price increase are reduced ( ). The percentage price increase that Firm procedure and it mirrors equation (1). is willing to initiate is obtained by an identical Finally, the pre-merger CPPI index is given by; { } With being equal to the original (or baseline) index proposed by Moresi et al. (2011). Notice that as pointed out by Moresi et al. (2011), the percentage price increase that a firm is willing to follow should be always higher that the percentage price increase that a firm is willing to initiate. Thus, the index captures the lower bound of the range of percentage price increases that two firms and could sustain through PAC. 6

7 2.2 Post-merger variation The idea is to measure the change on Firms and incentives to engage on PAC, after the acquisition of a third Firm. Thus, assuming that Firm (the acquiring firm) merges with Firm (the target firm), we re-build the post-merger index considering the exact same set of assumptions as in Moresi et al. (2011). These are: The is measured in relation to the pre-merger price level We abstract from any unilateral effects The merged Firm would raise the price of all its products by the same percentage The post-merger sales volume of the merger Firm, is going to be equal to the sum of the pre-merger sales of the merging parties The diversion ratio from Firm to the merged Firm will be equal to the sum of the pre-merger diversion ratios and. The diversion ratio from the merged Firm to Firm, will be equal to the share of lost sales from the merged Firm that goes to Firm. We approximate it by the following expression: The product produced by Firm has the same price and margin than the product produced by Firm. After the acquisition, the merged Firm will face the same elasticity, price and margin for both products. Then the post-merger percentage price increase that the merged Firm given by; is willing to initiate is While the percentage price increase that the outsider Firm given by; is willing to initiate post-merger is 7

8 Note that both formulas are derived from our interpretation of Moresi et al. (2011), since they do not present an explicit equation for the post-merger CPPI index on the paper. Finally, the impact of the merger on firms incentives to engage on PAC, is given by: { } { } 3. Simulations We simulated 50,000 economies, with 10,000 consumers and 5 single-brand firms on each of them. It is assumed that consumer preferences behave according to a model of discrete choice demand with random coefficients. In addition, we assume that firms offer products with differentiated characteristics or attributes, including a continuous one and a discrete one. Firms have heterogeneous and constant marginal costs of production and compete in prices. Under the absence of collusion or PAC, on each period prices are determined by the static Bertrand- Nash equilibrium. Using this approach, it is ensured that the simulated economies exhibit a much more realistic pattern of own and cross-price elasticities (See Nevo (2000)). In addition, having both continuous and discrete quality attributes gives more generality to the model and allows us to capture a wider range of preferences (See Grigolon and Verboven (2013)). On each economy we simulated a PAC strategy among Firm 1 (Firm or acquiring firm) and Firm 3 (Firm or outsider firm). The actual percentage price increases (as opposite to the predicted ones presented in the previous section) initiated by firms involved on PAC, are computed by maximizing the sum of firms present and future stream of expected payoffs, assuming that the competitor will follow the price increase. Therefore, we observe two percentage price increases: the one potentially initiated by Firm and the one potentially initiated by Firm. Then the Actual Coordinate Price Pressure (ACPP) is defined as the minimum of these two values. In other words, the ACPP represents the actual (instead of the 8

9 predicted one by the CPPI) lower bound of supra-competitive prices that firms and could reach through PAC. As a next step, we simulated the impact of a merger between Firm 1 and Firm 2 (Firm or target firm) on firms incentives to engage on PAC. Specifically, we recomputed the percentage price increase that Firm would be willing to initiate after acquiring Firm (which we call merged firm ), assuming that it applies the same percentage increase to both brands. At the same time, we recomputed the percentage price increase that Firm would be willing to initiate, assuming that it will be followed by the two brands of the merged firm. The post-merger ACPP is then defined as the minimum of these two corrected percentage price increases, and the impact of the merger is measured as the change on the ACPP. Thus, a positive ACPP variation represents an increase on the lower bound of prices that two firms could reach through PAC, and it could be considered as anticompetitive. However, an additional adjustment was made to the simulated price increase initiated by the acquiring firm post-merger. Since we are evaluating the impact of the merger with respect to the pre-merger level of prices, we need to adjust for the potential presence of unilateral effects. In order to do it, we compute the percentage price increase that the merged firm would be willing to unilaterally initiate, even if there are not competitors willing to follow it. Thus, the post-merger price increase initiated by the acquiring firm and motivated only by PAC, is obtained as follows: PAC Percentage Price Increase initiated by the merged (or acquiring) firm = Percentage price increase initiated by the merged firm and followed by a third competitor - Percentage price increase that the merger firm would unilaterally initiate In other words: Coordinated Effects = Overall Effect - Unilateral Effects 9

10 Note that it is assumed that competitors that are not involved on the PAC strategy do not react and keep their prices at the Bertrand-Nash level. In addition, we are restricting the unilateral effects to be a percentage price increase equally applied to all the brands produced by the merged firm. However, the post-merger level of prices (the Bertrand-Nash equilibrium) does not necessarily satisfy this condition. We will try to relax these assumptions in the robustness checks section of a future version of this paper. For more details regarding the maximization problem, please refer to the Appendix. The Table 3.1 summarizes the main descriptive statistics of the set of simulated economies. There are already two interesting results that can be taken from this table. First, on average the CPPI index significantly underestimates the actual impact of the merger. Second, as predicted by Moresi et al. (2011), the merger can actually reduce firms incentives to engage on PAC (a negative ACPP variation). Indeed, the ACPP change is negative on 13.57% of the sample. Table Summary statistics Variable Mean Std. Dev. Min. Max. Own-price elasticity Firm Own-price elasticity Firm Diversion Ratio (Firm to Firm ) Diversion Ratio (Firm to Firm ) HHI pre-merger Predicted HHI variation ACPP pre-merger ACPP variation CPPI pre-merger CPPI variation* Number of observations 46,093 *It only considers observations with a CPPI variation higher than

11 4. Results 4.1 The index significantly underestimates the ACPP variation Graph 4.1 contains a set of scatter plots displaying the relationship between the value predicted by the index ( ( ) and the actual variation on firms incentives to engage on PAC ). The sample was classified in four groups, according to the actual value of the diversion ratio from the acquired Firm to the acquiring Firm. The upper left panel displays the scatter plot of the observations under the 25% percentile, while the upper right one displays the plot of the observations between the 25% and 50% percentiles, and so on. It is clear from a visual examination of the graph, that the index have a better predictive power for those acquisitions with lower diversion ratios between the merging parties. Graph 4.1 Predicted ( variation on firms incentives ) and actual ( to engage on PAC p25 - p50 p50 - p75 p75 - p p0 - p CPPI (Predicted var.) ACPP (Actual var.) Fitted values Graphs by Diversion Ratio (Firm C to Firm A) Percentiles The obvious explanation for the existence of this asymmetry, it is that the index omits important information regarding the diversion ratios between the brands produced by the merged firm. Indeed, when the acquiring Firm is evaluating to initiate a post-merger PAC with Firm, it has to consider the cost of unilaterally initiating the price increase. However, the higher the diversion ratios between the merging parties brands, the lower the cost of initiating the PAC, and thus the 11

12 higher the impact of the merger on the acquiring firm s incentives to initiate such a conduct. Therefore, for higher values of the merging parties diversion ratios, the index will tend to underestimate the real impact of the merger and to be considerably less accurate. Graph 4.2 displays the empirical distribution of the ratio between the predicted variation of the percentage price increase that the acquiring firm is willing to initiate ( ), and its actual variation. In other words, it shows the percentage of the actual variation that is explained by the index, and it is denoted by. As is can be seen from the graph, for higher values of the diversion ratio from the acquired Firm to the acquiring Firm, the distribution is centered around 0 (zero). This fact has two implications. First, the index consistently underestimates the actual price variation. And second, for higher values of this diversion ratio, the index predicts the wrong direction (or sign) of the change on a high percentage of the sample (almost half of it). Graph Histogram of p0 - p25 p25 - p50 Percent p50 - p75 p75 - p rs Graphs by Diversion Ratio (Firm C to Firm A) Percentiles Result 1 For higher values of the diversion ratios between the brands produced by the merged firm, the index tends to significantly underestimate the actual impact of the merger of firms incentives to engage on PAC and to be significantly less accurate when predicting the direction of the change. 12

13 4.2 Predicting the sign of the ACPP change To study the performance of the index when predicting the direction of the change, the sample was classified in two groups: cases with and cases with.. Then we measured the percentage of cases where the index would lead us to incur on Error Type I and Error Type II. Recall that Error Type I refers to the case when the index erroneously classify a merger as potentially anticompetitive, when it is not. While Error Type II, corresponds to the case when the index fails to detect an anticompetitive merger. Table 4.1 summarizes the results. It can be seen that in terms of Error Type II cases, the index displays a better performance for mergers with a low value of the diversion ratio Type I is still significant., however the number of cases with Error Table Accuracy when predicting the sign of the change Diversion Ratio (Firm / Firm ) Percentiles Freq. Type-I Error Freq. Type-II Error 25% 2, % 9, % 50% 2, % 9, % 75% 1, % 10, % 100% % 11, % Total 6, % 39, % Result 2 When predicting the sign of the ACPP change, the index displays a better performance for mergers with low values for the of the diversion ratios between the brands produced by the merged firm. However, the probability of incurring on an Error Type I is not negligible. 4.3 Identifying mergers that generates a significant increase on ACPP To study the performance of the index when predicting a significant increase on firms incentives to engage on PAC, the sample was classified in two groups: cases with and cases with. Then as before, we measured the percentage of cases where the index would lead us to incur on Error Type I and Error Type II. Table 4.2 summarizes the results. First, the percentage of cases with Error Type I is really low (0.41% for the whole sample) and stable across the sample. Second, there is a positive relationship between and the percentage of cases that incur on Error Type II. Nevertheless, the overall predictive power of the index for 13

14 detecting anticompetitive cases is quite poor. The index incurs on Type II Error on 75.08% of the cases, and almost on 100% of the cases for mergers with high values of the diversion ratio from the acquired Firm to the acquiring Firm. Table 4.2 Accuracy when predicting a significant variation of the ACPP Diversion Ratio (Firm / Firm ) Percentiles Freq. Type-I Error Freq. Type-II Error 25% 8, % 3, % 50% 9, % 2, % 75% 9, % 1, % 100% 8, % 2, % Total 36, % 9, % Result 3 The port-merger variation of the CPPI index displays a poor performance when detecting mergers that generate a significant anticompetitive impact ( engage on PAC. on firms incentives to 4.4 Incorporating the strategic interactions between the merged firm brands In order to overcome this drawback of the original CPPI index, we propose an alternative version that incorporates the strategic interaction between the brands produced by the acquiring and acquired firms, respectively. When the merged firm is evaluating to initiate a unilateral percentage price increase, its decision has to incorporate the fact that some of the sales lost by one brand are captured by the other brand and vice versa. This effect generates a positive externality between the merged firm brands and it could increase the acquiring firm incentives to initiate a PAC. However, the fact that the merged firm is generating a loss of sales for two brands instead of only one (as in the pre-merger case), it could increase the cost of initiating such a conduct in the first place. Therefore, the merger could also decrease the acquiring firm incentives to engage on PAC. The direction of the overall impact is going to depend on which of these effects predominates. 14

15 Two alternative indexes are proposed: 1. The first one uses the same information than the original index, plus the value of the diversion ratio from the acquired Firm to the acquiring Firm ( ). We denote it by. For its construction we kept the same set of assumptions from section The second one uses much more information. In addition to the information required for the construction of the original index, it also requires: the diversion ratios between the two brands produced by the merged firm, the pre-merger own price elasticity of the acquired firm, the pre-merger margin and price of the product produced by the acquired firm. We denote it by. For its construction we relaxed the assumption than the pre-merger prices, margins, diversion rations and own-price elasticities are the same for the brans produced by the merged firm. For further details about the derivation of both indexes, please refer to the Appendix at the end of this document. Table 4.3 displays the performance of the three indexes (the original one plus the two alternative ones) when predicting the direction of the ACPP change. The two alternative indexes display a better performance in terms of Error Type II, however the percentage of cases with Error Type I is considerably higher. Therefore, there is no evidence that the alternative indexes have a better performance in this regard. Table Accuracy when predicting the sign of the change Diversion Ratio (Firm / Firm ) Percentiles Freq. Type-I Error Freq. Type-II Error 25% 2, % 39.43% 41.16% 9, % 8.87% 7.57% 50% 2, % 42.26% 37.16% 9, % 9.45% 9.02% 75% 1, % 71.51% 58.67% 10, % 7.76% 9.75% 100% % 88.34% 80.76% 11, % 5.09% 7.26% Total 6, % 49.16% 45.18% 39, % 7.66% 8.39% Result 4 In terms of predicting the direction of the ACPP change, the two alternative indexes do not display a better performance than the original one. 15

16 Table 4.4 compares instead the performance of the indexes in terms of predicting mergers that generate a significant anticompetitive effect. It can be seen that the two alternative indexes incur on a higher percentage of Type I Error, however, this increase seems to be moderate. Regarding the Type II Error, both indexes dramatically outperform the original one. As expected, from the two alternative indexes, the one that uses the most information is the most accurate one. Table Accuracy when predicting a significant variation of the ACPP Diversion Ratio (Firm / Firm ) Percentiles Freq. Type-I Error Freq. Type-II Error 25% 8, % 2.77% 2.87% 3, % 48.79% 31.82% 50% 9, % 1.89% 1.27% 2, % 57.20% 36.14% 75% 9, % 3.82% 2.28% 1, % 52.40% 37.37% 100% 8, % 12.63% 6.5% 2, % 26.95% 25.14% Total 36, % 5.24% 3.19% 9, % 45.44% 31.99% Result 5 The two alternative versions of the CPPI index significantly outperform the original index in terms of identifying mergers that generate a significant anticompetitive impact ((, while moderately increasing the occurrence of Error Type I. And as expected, the alternative index which uses more information is the one with the best performance. 5 Conclusions We tested the accuracy of the CPPI index in a simulated environment, considering a system of non-linear demands and a supply side composed by heterogeneous firms that compete in prices. The results suggest that the index displays a poor performance when predicting significant changes on firm s incentives to engage on PAC. There are two potential explanations for this result. First, the index was derived from a model with linear demands. Thus, it is expected that its accuracy will be reduced on a model based on non-linear demands. Second, the index does not consider the strategic interactions between the merged firm brands. Indeed, the cost of initiating a PAC is reduced by the fact that some of the lost sales by a given brand go 16

17 to the other brands. Therefore, not considering this positive externality could lead us to inaccurate predictions. We showed that incorporating the strategic interactions between the merged firm brands substantially increase the accuracy of the index. The number of Error Type II cases decreases dramatically, while the number of Error Type I cases increases just moderately. Therefore, these results highlight the importance of considering these interactions when building a model or an index which attempts to predict the coordinated effects of a merger. 6 References Foncel, Jérôme, Marc Ivaldi and Aleksandra Khimich (2013), Assessing the accuracy of merger guidelines' screening tools, Preliminary version July 8, Grigolon, Laura and Frank Verboven (2013), Nested logit or random coefficients logit? A comparison of alternative discrete choice models of product differentiation, The Review of Economics and Statistics, forthcoming. Harrington, Joseph (2013), Evaluating Mergers for Coordinated Effects and the Role of Parallel Accommodating Conduct, 78 Antitrust Law Journal No. 3 (2013) Moresi, Serge, David Reitman, Steven C. Salop and Yianis Sarafidis (), Gauging Parallel Accommodating Conduct Concerns with the CPPI Nevo, Aviv (2000), A Practitioner's Guide to Estimation of Random-Coefficients Logit Models of Demand, Journal of Economics & Management Strategy, Volume 9, Number 4, Winter 2000, 513_

18 7 Appendix 7.1 Alternative CPPI variation 1: { } { } 1. With having the same formulas introduced on Section The post-merger percentage price increase initiated by the acquiring firm is given by: With; 3. While the post-merger price increase initiated by the outsider firm is given by: 18

19 7.2 Alternative CPPI variation 2: { } { } 1. With having the same formulas introduced on Section The post-merger percentage price increase initiated by the acquiring firm is given by: With; And; 3. While the post-merger price increase initiated by the outsider firm is given by: With: 19

20 7.3 Simulations baseline setting We almost replicated the baseline simulations setting used by Foncel, Ivaldi and Khimich (2013), there are a few differences marked with a (*) in the table below. Parameter Baseline Setting Number of firms is fixed to 5 for all the economies. Each firm produces only one product. Number of consumers is set to 10,000 for all the economies. Number of simulations for computing the expected market shares is fixed to 10,000 for all the economies. (*) It is constant within each economy, but it varies across economies with uniform distribution [ ]. For a given economy varies among consumers with exponential distribution. The parameter is distributed uniformly [ ] across economies. (*) It is constant within each economy, but it varies across economies with uniform distribution [ ]. (*) It is constant within each economy, but it varies across economies with uniform distribution [ ]., For a given economy both vary among consumers with normal distributions [ ] and [ ], respectively. The parameters and are distributed uniformly [ ] across economies., They are both drawn from an extreme value distribution [ ], where the scale parameter is equal to 0.5. (*) For each economy where and are distributed normally with [ ]. For each economy where are distributed normally with [ ]. For each economy is drawn from a normal distribution [ ]. For a given economy varies among consumers with normal distribution [ ]. The parameter is distributed uniformly [ ] across economies. For each economy is drawn from a normal distribution [ ]., Both are fixed for each economy, but they vary across economies with the same uniform distribution [ ]. It is fixed on each economy and common for all firms. It is either equal to zero (when marginal costs are assumed to be constant), or vary across economies with uniform distribution [ ] (when marginal costs are increasing). 20

21 7.4 Simulations of the actual and The actual pre-merger, for, is computed as follows: { ( ) } While the post-merger, for, are obtained with the following equations: { ( ) ( ) ( ) } { ( ) ( ) } { ( ) } 21

22 7.5 Derivation of the corrected post-merger CPPI index We propose a modified version of the CPPI index that takes into consideration the strategic interactions between the brands produced by the merged firm. For the construction of this index we closely follow the methodology proposed by Moresi et al. (2011) Acquiring firm (Firm A) 1. At period the acquiring firm increases the prices of its two brands by percent. It incurs on a loss of profits equal to the difference between: (i) the lower volume of sales generated by the price increase, times the margin charged for the two brands, and (ii) the higher price charged on its remaining sales. This is given by the following expression: [ ] [ ] [ ] [ ] We define and, and the previous formula becomes: [ ] [ ] 2. Assuming that from period the price increase is followed by Firm, then the acquiring firm gets profits equal to the difference between: (i) the higher price charged on its overall sales, and (ii) the lower volume of sales generated by the price increase, times the margin charged for each of its two brands. Thus the per-period gain from PAC is given by: 22

23 [ ] [ ] 3. Assuming that the price increase followed by Firm becomes permanent, then the merged firm will choose in order to maximize the present value of its expected payoffs. Therefore, the optimal is found by maximizing the following expression: { } And is given by; With; 4. However, we need to make an additional adjustment to this formula. Provided that the PAC incentives are evaluated at the pre-merger prices, the merging parties should have an incentive to initiate an unilateral percentage price increase, regardless the fact that competitor follows the price increase or not. Therefore, since the considers the strategic interactions between the merged firm brands, it also consider this unilateral price increase. In order to clean our index from this effect, we propose the following corrected formula:. 23

24 The unilateral price increase can be found by maximizing the following expression: { } And it is given by; Thus, the post-merger percentage price increase that the acquiring firm is willing to initiate, assuming that competitor will follow the same conduct, is given by: Finally, considering the same set of assumptions than in section 2.2 2, the previous formula is simplified to: 2 The assumptions are: 1. The prices, margins and own-price elasticities of the brands produced by the merged firm are the same (, and ). 2. The diversion ratios between the brands produced by the merged firm are identical ( ). 24

25 7.3.2 Outsider firm (Firm B) 1. At period the outsider firm increases it price by percent. It incurs on a loss of profits equal to the difference between: (i) the lower volume of sales generated by the price increase, times the margin charged for its brand, and (ii) the higher price charged on its remaining sales. This is given by the following expression: [ ] [ ] 2. Assuming that from period the price increase is followed by the merged Firm, then the outsider firm gets profits equal to the difference between: (i) the higher price charged on its overall sales, and (ii) the lower volume of sales generated by the price increase, times the margin charged for its brand. Thus the per-period gain from PAC is given by: [ ] 3. Assuming that the price increase followed by the merged firm Firm becomes permanent, then the outsider firm will choose in order to maximize the present value of its expected payoffs. Therefore, the optimal is found by maximizing the following expression: { } And is given by; 25

26 With: Finally, considering the same set of assumptions than in section 2.2, the previous formula is simplified to: Alternative variations of the CPPI index 1. Finally, and considering the previous formulas, the corrected versions of the postmerger variation of CPPI indexes are given by: { } { } And; { } { } 26