Diversion Ratio Based Merger Analysis: Avoiding Systematic Assessment Bias

Transcription

1 Diversion Ratio Based Merger Analysis: Avoiding Systematic Assessment Bias Kai-Uwe Kűhn University of Michigan 1 Introduction In many cases merger analysis heavily relies on the analysis of so-called "diversion ratios". Diversion ratios capture customer movements from a firm to others (as well as the pool of non-puchasing consumers). The empirical diversion ratio of firm A to firm B measures what proportion of the customers firm A looses substitutets into purchasing the porduct of firm B. Intuitively diversion ratios capture information about substitution between different product offerings. For this reason they are used in simple arguments of closeness of competition, in upward pricing pressure (UPP) analysis, the calculation of "indicative price increases" (IPR), as well as merger simulation by calibration on diversion ratios. In particular, the use of diversion ratios in UPP is seen as a simpler alternative to data intensive demand estimation. However, it is well known that the diversion ratios needed for a UPP test (and the IPR and merger simulation analyses that are directly related to it) are diversions caused by price increases or decreases of one of the merging firms. In principle the determination of the correct diversion ratios is subject to the same identification issues as those for price elasticities in demand estimation. Unfortunately, this problem is often ignored and UPP analysis often proceeds as if the measured diversions can be taken as unbiased estimates of the theoretically correct diversions caused by price changes needed for UPP analysis. In this paper, we systematically analyze under what circumstances measured diversion ratios correctly proxy for the theoretically correct diversion ratios used in UPP analysis. We first show that under logit demand functions a wide range of shocks to individual demand will lead to measured diversion ratios generating the theoretically correct diversion ratios. This result relies on the property of "independence of irrelevant alternatives" that is satisfied by logit demand functions. We then show that even in this benchmark case there is a systematic bias when using measured diversion ratios when from one period to the next the prices and quality offer of a rival change. We show under what circumstances diversion ratios are systematically biased up or down. Similarly, we show that, generically, diversion ratios will be systematically biased when preferences change. We show that these changes may go in different directions depending on the relative quality and price choices of the different competitors.we develop 1

2 a test for detecting whether systematic biases are present in measured diversion ratios that will lead to poor approximation of the relevant diversion ratios for UPP and IPR analysis. We then discuss to what extent the analysis gets modified when the underlying choice model does not satisfy independence of irrelevant alternatives. We develop some examples that show that measured diversions will generally not correctly approximate the theoretically relevant diversions of UPP analysis. Most importantly random changes in preferences will tend to confound systematic effects of relative price changes when there is heterogeneity in customer responses to prices and quality. This is particularly important in settings in which there is a merger between a higher quality and a lower quality competitor. We use this analysis to explain why the frequent practice of comparing the measured diversion ratios with market shares as a benchmark for closeness of competition makes very little economic sense. We develop a systematic method for analysis in practical antitrust settings in which we distinguish between UPP analysis as a filter for detecting concerns and as a means of proving effects. In this context we show that it never makes sense to use measured diversion ratios for merger simulation. 2 The distinction between Theoretical and Measured Diversion Ratios Let D i (p i, p i ) be the demand of firm i when it sets price p i and other competing firms are setting prices described by price vector p i. We have n firms and the outside option for every consumer 0, which will be chosen by D 0 (0, p 0 ) consumers. Consider firm i increasing its price p i. Then a first order approximation to its demand change would be Di(pi,p i) p i, where p i is the price change adopted by firm i when every other firm keeps its price constant. The price change in p i will increase the demand for any firm j i by approximately D j(p j,p j) p i. So the percentage of demand that moves from firm i to firm j after a price increase by firm i can simply be written as: δ i j = D i(p i,p i) p i = D j(p j,p j) p i D i(p i,p i), D j(p j,p j) which we will call the "theoretical diversion ratio". When we look at switching behavior of consumers it is immediately obvious that most of the diversions between firms that we are measuring are not diversions generated by prices. For example, even when prices and qualities of products do not change we see consumers switching between products. So there must be some underlying preference changes that generate such observed moves. However, in many markets the aggregate demand is fairly stable, so that the distribution of preferences in the population remains the same. We can model such switches by assuming that some proportion of the population, 2

3 σ, randomly redraws its preferences in each period, but that the group that redraws its preferences draws it from the same distribution as the initial population. This means the demand from this sub-population is simply a scaled demand of that of the whole population. Formally, this means that every period σd i (p i, p i ) draw their preferences again. Of the σd i (p i, p i ) consumers of i D j(p j,p j) n k=0 Dj(p k,p k ) who draw their preferences anew, a proportion, so the diversion to firm j from firm i without changes in price or quantity will be σd i (p i, p i ) D j(p j,p j), while the number of consumers leaving firm i will be given by n k=0 Dj(p k,p k ) k i σd i (p i, p i ) Dj(pj,p j) n k=0 Dj(p k,p k ). The measured diversion ratio therefore becomes: ˆδi j = D j (p j, p j ) k i D j(p j, p j ) where a "ˆ" indicates a measured number, i.e. it is the market share of firm j among all other options, excluding i. Suppose that over time period of measurement prices did not change. The measurement would still generate a diversion ratio that an analyst would use to compute a measure like upward pricing pressure. Are there any circumstances in which this would be a reasonable way to proceed. In fact, ˆδ i j is the correct measure for the theoretical diversion ratio if the choices between all options k i remain unchanged, whatever the reason is that a consumer stops purchasing good i. This implies that, if good i would disappear from the market, the relative market shares between the remaining products would remain unchanged. But it is even stronger: if price increased a little so that the customers who just decided to purchase i now switch, they switch in exactly the same proportions as the relative market shares at current prices. This is a strong restriction on preferences in the population that is generally known as "independence of irrelevant alternatives": whatever the reason product i is dropped, the choice between the remaining products is unaffected. This is a very strong restriction on demand systems in differentiated products markets and we know it is generally not satsified. The one demand system we know satisfies this condition is the logit demand system. We know that this demand system imposes very strong constraints on derivatives and crossderivatives and essentially leads to elasticities of demand being determined entirely by market share. Empirically we know that logit demand does very poorly in the sense of generating unreasonable estimated demand elasticities and alternatives like mixed logit (see BLP) and the almost ideal demand system provide flexibility that allows estimates to much better fit actually observed behavior. As soon as switches between products occur for other reasons than relative price changes, we therefore have to be concerned about the validity of using measured diversion ratios as proxies for the theoretical diversion ratio in a UPP analysis. However, the logit demand model is a very useful benchmark to show that using measured diversions can lead to highly misleading results even when the assumption of independence of irrelevant alternatives does hold. We therefore develop these fundamental issues of bias in measured diversion ratios for this 3

4 most favorable case. We then return to the issue of bias for more general demand systems in section 4. 3 Bias of measured diversion ratios when the Logit Demand System is valid 3.1 The Logit Deman system In the logit demand system the utility obtained from a good i by individual h is modelled as u ih = α h + β h q i + γ h p i + ε ih for each i where ε ih is a random utility term that follows a double exponential distribution. firm i offers a product of quality q i and at price p i. For the outside good 0 we set quality to and price to zero. The terms α h > 0, β h > 0, and γ h < 0 are preference parameters. With the distributional assumption on ε ih, individual h will buy good i with probability s ih = ( e u ih n ) 1 µ n k=0 e = e u kh u ih u µ kh µ k=0 For the standard logit model we assume that the preference parameters for all potential customers are the same so that we drop the inidividual subscript h. Then s i can be interpreted as the demand for good i (modulo a scaling parameter for the size of the population). Note that this demand system satifies independence of irrelevant alternatives. The relative probability between two choices i and j, = e uj ui µ only depends on the difference between these s i s j utilities. This means that the probability that a product j is chosen, conditional on i not being chosen, is e u j µ k i e u k µ, which is independent of the utility that could be obtained from i. This means that any change that affects only the level of utility u i will lead to the same distribution of choices for customers leaving good i - whatever the reason. So any change in the market that only affects product i will generate a correct estimate of the theoretical diversion ratios generated from an increase in the price i. Furthermore, any separation due to idiosyncratic preference changes in ε ih that do not change the distribution of ε ih in the population that changes preferences will generate measured diversion ratios that contain the correct information. Note also, that under the same assumption of changes that affect u i only one can measure diversion ratios both as incoming diversions or outgoing diversions. By the fact of independence of irrelevant alternatives it must be the case that inward and outward diversions ratios must be the same. 4

5 However, we will now show that any contemporaneous changes in the market in which rivals change their qualities or prices will lead to systematic bias when using the measured diversion ratios as estimators for the relevant diversion ratios in a UPP analysis. 3.2 Measured Diversion Ratios when product characteristics and prices can change between periods The measured diversion from product i to product j will be all the consumers who stop purchasing from i and start purchasing from j divided by all consumers who stop purchasing from i. We assume that such switches can come from two types of sources. First, there may be a proportion of the population that gets a new draw of ε ih, i.e. there is a part of the population that changes its preferences. However, in aggregate their choices remain the same, so that this would not affect market shares. Secondly, any firm may change its contract offerings by either changing quality or price of its product. We can now compute a first order approximation to the measured diversion from good i to good j for a general switch from period t 1 to t: ( D(pi, p i ) i j = σd(p i, p i )s j (p j, p j ) min{0, p i + D(p ) i, p i ) q i } s j(p j, p j ) q i 1 s i (p i, p i ) ( D(pi, p i ) min{0 p j + D(p ) i, p i ) q j } p j q j where the first term corresponds to all of the customers who get a new preference draw. Of these a proportion of s i stays with firm i and a proportion of s j goes to firm j. The second term in brackets are the effects of product changes of firm i that decrease the utility through increases in price or increase it through increases in quality. So this is the number of customers who suffer a suffi cient net decrease in utility to leave i. They will spread among the other offerings according to the probability of purchase conditional on not purchasing from i, namely s j /(1 s i ) for diversions from i to j. Note that the second term could be positive if the effect of the quality increase outweighed the effect of the price increase and then this effect would add nothing to outward diversion. The third effect is that of firm j changing its prices and quantities. This can only have an effect on outflows if either the price is decreased or the quality increased (or both). So this is precisely the number of customers substituting to firm j because firm j has become more attractive. Note that changes in other firms offerings have no impact on the flows from i to j because of the independence of irrelevant alternatives properties of logit demand. The total measured outward diversion can analogously be calaculated as: ( D(pi, p i ) i i = σd(p i, p i )(1 s i (p j, p j )) min{0, ( D(pi, p i ) min{0 j i p j + D(p i, p i ) q j p j q j p i + D(p ) i, p i ) q i } q i ) } 5

6 Noting that D(pi,p i) p k as th fact that si(pi,p i) the measured outward diversion ratio as: ˆδi j = = M si(pi,p i) p k, and D(p i, p i ) = Ms i (p i, p i ), as well = γs i (1 s i ) and sj(pi,p i) p j = γs i s j, we can write σs i s j + max{0, γ p i s i (1 s i ) β q i s i (1 s i )} sj 1 s i max{0, γ p j s i s j β q j s i s j } σs i (1 s i ) + max{0, γ p i s i (1 s i ) β q i s i (1 s i )} + k i max{0, γ p ks i s k β q k s i s k } This gives us directly the first result: Proposition 1 The the measured (outward) diversion ratio is an unbiased estimator for the diversion ratio caused by a unilateral price increase of firm i if and only if only firm i changed its product offering from period t 1 to period t The proof of this proposition is simple. The last term in both the numerator and the denominator are zero if the competitors have not changed prices or qualities. Then the above ratio collapses to s j /(1 s i ), which is the correct diversion ratio for diversions caused by a price increase by firm i: ˆδi j = s j σs i (1 s i ) + max{0, γ p i s i (1 s i ) β q i s i (1 s i )} min{0, γ p j s i (1 s i ) β q j s i (1 s i ) 1 s i σs i (1 s i ) + max{0, γ p i s i (1 s i ) β q i s i (1 s i )} k i min{0, γ p ks i s k β q k s i s k } Clearly the numerator and denominators of the second term are the same when there are no changes in any price or qualities of all of the other firms except firm i. If there are changes in the other prices that lead to outward diversions, the measured diversion ratio is an upward biased estimator of the true diversion ratio ˆδ i j if j changed its offerings inducing diversions and a downward biased estimator if only firms excluding j have changed their prices so as to induce an outward diversion. To see the result without further proof for the case that only j changes its offerings in such a way that there it increases diversion from i to j. Then one can write the measured diversion ratio as a + ξ (1 si) s j a + ξ for ξ > 0 if and only if j induces outward diversions from i to itself through its offerings. The first derivative of this is given by: (1 s i) s j (a + ξ) (a + ξ (1 si) s j ) (a + ξ) 2 = 1 s i s j s j a (a + ξ) 2 > 0 which proves the upward bias. In the general case one would have to prove that in a well-defined sense the improvement in the offering of j is greater than the weighted average of the improvements of all other competitors k i. Note that the case for only k j improving its offering downward biasing the measured diversion ratio as an estimator of the true diversion ratio is also obvious since the terms in j are zero. 6

7 This result has profound consequences for how to deal with diversion ratios empirically. Suppose we have only two firms in the market and one change the price in month 1 and the other in month 2. Now suppose only aggregate diversion data is available. Such diversion ratios must be biased upwards for both parties. Unless one can disaggregate the data so that individual changes in price and quality level can be identified, the analysis cannot filter out upward biases in the diversion ratios. Those periods can generate good proxies for the diversion ratios of the firm that has changed its offering, but only upward biased poxies for the diversion ratios for competing firms. An appropriate way of approaching diversions empirically is therefore to use highly disaggregated data, and measure the diversions only at the changes of offerings for each firm. The current way of measuring diversion ratios therefore clearly overestimates the relevant diversion ratios for UPP and merger simulation analysis. 3.3 Systematic Bias with changes in Preferences and Product Introduction In the previous subsection we have shown that the typical way of using diversion ratios in pactic will typically lead to an overestimations of diversions even when the demand system has the independence of irrelevant alternatives property and there are no changes shifts in demand over time. One could still interpret the previous analysis as the typical demand estimation problem. What we need to estimate is the ratio of the cross-elasticity of demand of firm j with respect to the price of firm i and the own price elasticity of demand of product i. The problem is that the price of firm j also changes so that demand shifts, counfounding the effect of the own price. This problem that the measured diversion ratios includes the impact from uncontrolled for demand shifts more generally happens when there are changes in preferences. We will here as an example simply look at how an increase in the valuation of a given quality provision will systematically lead to a bias in the measured diversion ratio as an estimator of the true underlying diversion due to prices. Suppose that α is increased, and lest us maintain the assumption that there is some diversion simply because of random turnover of preferences (redrawing ε ih ). Note that the market share of i can be written as: s i = 1 k e u k u i µ = e βqi+γpi + k 0 e β(qi q k)+β(p i p k ) The total outward diversion from an increase in β is thus every marginal term 7

8 that leads to a outflow: s i β = β min{0, q i}e βqi+γpi + k i,0 β min{0, q i q k }e β(qi q k)+β(p i p k ) ( 1 + e βqi+γpi + ) 2 k i,0 e β(qi q k)+β(p i p k ) = k i,0 β min{0, q i q k }e β(qi q k)+β(p i p k ) ( 1 + e βqi+γpi + k i,0 e β(qi q k)+β(p i p k ) = s 2 i β min{0, q i q k }e β(qi q k)+β(p i p k ) k i,0 Note that each term in the sum can be interpreted as the outflow from firm i to firm k that is induced by the higher quality of firm k. When firm i s quality is higher, no outflow is induced. So the diversion ratio becomes: ) 2 ˆδi j = = σs i s j s 2 i β min{0, (q j q i )}e β(qi qj)+β(pi pj) σs i (1 s i ) s 2 i k i,0 β min{0, q i q k }e β(qi q k)+β(p i p k ) s j σs i β min(0, q j q i )s i 1 s i σs i s i k β min{0, q i q k } s k 1 s i We get our second result from this analysis: Proposition 2 Suppose that there is a systematic shift in greater preference for quality. If firm i is not the highest quality firm, the measured diversion ratio will be a biased estimator of the relevant diversion ratio from firm i to j caused by a price increase by firm i. In particular, the estimate will be upwards biased if firm j is the highest quality firm and the estimator will be downward biased if firm i has a higher quality level than firm j Proof. Follows very much the argument in the proof of proposition 1. We have thus shown that any change in the measured diversion ratio caused by changes in the market that are not idiosyncratic to firm i will lead to systematic bias of the measure diversion ratio as an estimator of the true diversion ratio. 4 How to detect the importance of bias in measured diversion ratios Can we distinguish situations in which measured diversions are a good approximation to the theoretically correct diversion ratios in the sense that we should expect only a small bias? One way to construct a test is to look at the diversion ratios for outbound flows and for inbound flows. We show here that these will be the same when the measured diversion ratio is an unbiased estimator but will show differences when it is not. 8

9 We will do this here only for the example of the systematic changes in preferences that we have discussed before. But the basic argument holds more generally. Let us first construct the inward diversion ratio, i.e. the ratio of inflows from j to the total inflows for i. With general turnover as modelled previously the inflow from j will be σs j s i and the more general inflow from all other competitors will be σ(1 s i )s i. What about the inflows because α is increased. The inflow from j to i due to a higher quality of i would then be s 2 jβ max{0, (q i q j )}e β(qj qi)+β(pj pi) and the corresponding sum over all k i for the total inflow. We therefore get an inflow measured diversion ratio of: ˆδI j i = = σs i s j + s 2 j β max{0, (q i q j )}e β(qj qi)+β(pj pi) σs i (1 s i ) + k i s2 k β max{0, (q i q k )}e β(q k q i)+β(p k p i) s j σs i + s i β max{0, (q i q j )} 1 s i σs i + s i k i s k max{0, q i q k } We can now state directly our third result: Proposition 3 Generically, the inward diversion ratio ˆδ I j i is equal to the outward diversion ratio ˆδ i j if and only if the neither ratio is biased. Proof. Note that when preferences change, all of the terms in (q i q k ) are zero in ˆδ I k i if and only if q i q j. But a term that is zero for ˆδ I k ithe opposite holds for ˆδ i k. This means that whenever there are any differences in qualities the terms appearing due to the increase in the valuation of quality are completely distinct in the two diversion ratios. This means that generically they will have different values. Intutively a shift in demand psuhes demand in one direction, away from lower quality firms and towards higher quality firms. This means that firms that have high outflows tend not to have high inflows and vice versa. The issue is slightly complicated because a firm that is of medium quality may suffer outflows to a higher quality firm, but a lower quality firm may be so much worse, that the inflows are much bigger than the outflows. Without further information it is therefore not possible to determine in which direction the diversion ratios are biased. However, when UPP analysis is performed as a means of proving price effects, a conservative analysis should be performed. In that case a pragmatic approach would be to use the lower of the two diversion ratios, which will lead to a certain degree of under estimation of the diversion ratios assuming that a model with independence of irrelevant alternatives is seen as an appropriate benchmark model. We will discuss further below what the appropriate practical implications are in order to avoid upward biases in diversion ratios and their effects on the estmation of price effects of merger transactions. 9

10 5 Dropping the Assumption of Independence of Irrelevant Alternatives While the above provides a nice conceptual benchmark to explain the estimation bias that can arise from using measured diversion ratios to estimate the inputs for UPP and IPR analysis as well as diversions base merger simulation, the approach does generate some conceptual problems in itself. The most important is that under the assumption of a logit demand system we do not have to estimate diversion ratios from measured diversions, but can simply use the market shares (when measured including the share of the consumers not purchasing). If current market shares of firms i and j are s i and s j it follows from the functional form that the best estate for the diversion ratio is s j /(1 s i ). So none of the measurement problems would be an issue. However, we know that the logit demand function is not a good approximation for true demand systems. To generate reasonable own and cross-price elasticities in demand estimation one cannot use demand systemats with the property of independence of irrelevant alternatives. This complicates a proper implementation of UPP and IPR approaches to estimating the price effects of a merger even further. Note that under these circumstances there is no sensible interpretation of diversion ratios not being the same as market shares. First, note that the often used argument that diversion ratios being smaller than market shares indicates that firms are not so close competitors is obviously false under all assumptions on hte properties of demand. When the demand function is not logit, there is in the first place no reason to expect that diversion ratios equal the market shares and these then are not a meaningful benchmark. When logit demand is the correct demand function, it is also clear that the interpretation of closeness of competition is definitely wrong, because as we just showed this would simply indicate a bias in the measured diversion ratios, not an indication of clsoeness of competition. [Give an example here, to be worked out: suppose that buyers with higher willingness to pay sort more towards the higher quality product. A price incrase therefore will lead to more of the marginal consumers with higher willingness to pay to leave to the higher quality product than when all a proportion of customers get a preference shock ε ih, which will lead to switching of a smaller proportion of consumers to the high quality firm. The measured diversion ratios may then not be an unbiased measure of the true diversion ratios related to price. The problem is that the test for inward and outward diversions will also not be valid. You need that the general turnover is a fixed proportion of the true diversion ratio for that trick to work. Still need to work something out on this. 10

11 6 Policy Conclusions: 1. To minimize biases, analyze the data at a highly disaggregated level and only calculating outward and inward diversions for a period in which only the contract conditions for the firm changes for whom the diversion ratio is calculated or where there are no changes in contract conditions. 2. Calclate the difference between inward and outward measured diversion ratios 3. As a conservative approach calculate the diversions on the basis of the minimum of the inward and outward diversion ratios 4. If demand estimation can identify the elasticities of demand, do demand estimation and calculate the correct theoretical diversion ratios on that basis 7 Further stuff to Do Develop some further implications for IPR and merger simulation. 11