Product Policy in Markets with Word-of-Mouth Communication. Technical Appendix

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1 rodut oliy in Markets with Word-of-Mouth Communiation Tehnial Appendix August 05

2 Miro-Model for Inreasing Awareness In the paper, we make the assumption that awareness is inreasing in ustomer type. I.e., those that have higher valuation for quality are more likely to know about the existene of the produt. Here, we present a model in whih we endogenize awareness as a funtion of onsumer searh. We might onsider this searh to inlude not only literal searh in stores but also the general investment in ategory awareness inluding, for example, reading ategory blogs, magazines and literature. In this way, we expet that those making a larger investment are more likely to know about the latest produts. Consider a onsumer with expeted utility for produts in the ategory given by E [Uj] > 0: for any given values of and (whih are learned upon awareness), higher types will earn higher utility. The onsumer hooses an amount of searh S () at ost C (S ()) whih gives rise to awareness: A = A (S () ;!) Awareness is a funtion of both word of mouth and onsumer searh. Sine a onsumer invests in awareness building but an only buy the produt if awareness beomes available, we an write the following as the onsumer s objetive funtion: A = arg max A A (S () ;!) E [Uj] C (S ()) First-order onditions for the optimal searh are: and (S ) E (S S S 0 Di erentiating both sides of () with respet to and re-arranging @S S S E [Uj] > 0 whih follows from the seond-order onditions and the > 0. Treating Awareness as an Expetation While our exposition in the main body of the paper treated the awareness funtion as ommon knowledge, given the linearity of the model, it is straightforward to show that without loss of generality

3 we ould also onsider awareness as being only in expetation. We would then rewrite the objetive funtion as: E [] = where we ll us the supersript to denote expetation. A 0 (;!) d () Then, essentially, everything proeeds diretly as before. First-order onditions for optimal and, respetively, are: A 0 = A 0 (;!) d (3) A 0 = A 0 () d (4) whih, ombined, yields: = (5) Substituting (5) into (3) and then di erentiating impliitly with respet = A 0! () d A 0! ( ) 3A 0 ( ) + A 0 ( ) (6) Finally, we an rewrite this in analogous form as in the > 0 () A 0! () d A 0 () d > A0! ( ) A 0 ( ) (7) The analysis of the persuasive ase proeeds in exatly the same way. 3 Informative WOM - General Type Distribution Here, we analyze a somewhat more-general version of the model studied in the main body of the paper. We allow here for a general population distribution g () whih, along with the awareness funtion A (;!) gives rise to the e etive market size m (). Spei ally, we think of A (;!) as the proportion of ustomers of type who are aware of the produt and g () as the proportion of ustomers who are of type. Thus, we an think of the weighted awareness level (i..e, that whih takes into aount di erent population sizes aross types) m () = g () A (;!) ro ts in this model are: = m () d

4 First-order onditions for optimal are: m () d + Seond-order onditions for global onavity in are: m () d m 3 m m = 0 (8) m + m 0 (9) A su ient ondition for this to hold is that m () 0. This was guaranteed in the uniformdistribution ase analyzed in the main body of the paper sine m = A > 0. Here, though: m = g () A () + g () A () Thus, we will maintain the assumption that m () 0 8 while also noting that this will depend on the joint impat of awareness hanges (for whih we ontinue to assume that A > 0) and the distribution of ustomer types. In partiualr, if population falls o more rapidly than awareness grows at higher levels of, this assumption may be violated. Similarly, rst-order onditions for are: m () d Seond-order onditions for global onavity in are: m m + m = 0 (0) m 0 8; () whih again holds given the su ient (but not neessary) assumption that m () 0 8. Combining the optimal hoies of and by substituting (0) into (8) yielding: = () This is exatly the same relationship as before. Substituting () into (8) yields the following (again, analogous) impliit equation for : m ( ) = m () d (3) Di erentiating both sides of (3) and = m! () d m! ( ) [ m ( ) + 3m ( )] (4) 3

5 where the denominator is positive given our assumptions. We now substitute in and re-arrange to yield the following neessary and su ient ondition for the impat of an inrease in WOM on quality hoies in this > 0, m! () d m () d > m! ( ) m ( ) The following proposition establishes that our main result is a general one. roposition For general population density g (), (i) when m! < 0, optimal quality always delines in WOM; (ii) when m! > 0; quality may inrease or derease. When m! is high enough, optimal quality inreases in WOM. roof: (i) Assume that m! < 0 and quality doesn t deline in WOM. This implies that m! () d m () d m! ( ) m ( ) However, sine m! < 0, we know that m! ( ) > m! () 8 > whih, in turn implies that m! () d m! ( ) d = ( ) m! ( ) m ()! () d m! ( ( ) (5) ) Now, sine m > 0 by assumption, we know that m ( ) < m () 8 > and therefore Combined, this implies that ( ) m ( ) < m () d m () d () m ( > ( ) ) m!()d m!( ) < m()d m( ) whih ontradits the premise, proving the laim. (ii) First, note that when m! = 0, it will always be the ase that quality stritly dereases in < 0). To see this, assume that m! = The latter assumption implies that: m! () d m () d m! ( ) m ( ) However m! = 0 implies that m! () = m! 8 and thus we an rewrite the ondition as: ( ) m! m! = ( ) 4 m () d m ( )

6 However, the impliation that ( ) m ( ) m () d ontradits the assumption that m > 0. Sine this holds stritly, it implies there will exist a region in whih m! > < 0. Now we investigate what happens when m! >> 0. quality is dereasing in WOM. From (7), this implies that Assume that this is the ase and that optimal Now, x the HS as well as m! ( ). m! () d m () d m! ( < ) m ( ), m ( ) m! () d < m! ( ) m () d The idea is to keep overall awareness ( m () d) onstant but just vary the sensitivity of the slope of the awareness funtion with respet to WOM (that is, m! ). Sine the LHS is monotonially inreasing in m!, we an arbitrarily inrease it until a ontradition is found. 4 ersuasive WOM - General Type Distribution As we did above in our analysis of the informative WOM ase with a general distribution, we will simply reprove the main laims by substituting for F (;!) the funtion G (;!) whih aounts for both the proportion of informed ustomers and the distribution of types: G (;!) = g () F (;!) As noted in the main body of the paper, it is not neessary to make the assumption that G 0. In fat, the key assumption ensuring the seond-order onditions hold is that the slope is not too steeply positive. We an proeed here exatly as we did in the informative ase by simply replaing with G (;!) the ourrenes of F (;!) in our persuasive WOM analyses in the main body of the paper. ather than reviewing all elements, we ll simply walk through the ore result that optimal quality is inreasing in WOM. As above, we ll walk through the same approah to the proof as in the uniform ase. Optimal quality is given by the impliit expression ( + ) G 4 = 0 (6) where the argument to G is = + 4. The seond-order neessary onditions are: (G 4) + ( + ) 4 G 0 5

7 Sine G (0; ), this holds if G is not too steep. Next, we impliitly di erentiate (6) with respet to! and rearrange = G! ( + ) (4 F ) ( + ) 4 G To appreiate the sign of the inequality, note that (a) G! > 0 by assumption; and (b) the denominator is positive by the seond-order onditions. > 0 5 Fixed Costs We onsider rst an extension to the informative-wom model in whih quality is a xed ost and then do the same for the persuasive model. While we believe the variable ost model to be an aurate one in many ases, it is likely that some produt ategories may be better aptured in this xed ost model. 5. Informative WOM with Fixed Costs Here, the pro t funtion beomes = A (;!) d and rst- and seond-order onditions, respetively, for are as follows: A A A Similarly, rst- and seond-order onditions for are: = 0 (7) + 0 8; (8) A (;!) d A = 0 (9) A + A 0 8; (0) Substituting (9) into (7) yields: A (;!) d = 6

8 or " # = A (;!) d () whih implies that =. The following proposition in ontrast to our results in the main body of the paper highlights the important role of ost struture in determining the relationship between optimal quality and WOM. roposition For any awareness funtion, A (;!), when the ost of quality is xed, an upward shift in informative WOM leads to higher-quality produts as long as A! () > 0 for some >. roof of roposition : We di erentiate both sides of () with respet A @! A + whih, after = A! = 0 h A! () A A () d i A () Notie that the denominator is positive by (7). Moreover, from (9) we know that A = 0 allowing us to re-write () as follows: A (;!) d whih proves = A! () d A > 0 The intuition here is straightforward. Sine osts are xed, the relevant trade-o is simply the marginal ost of an inrease in quality whih is independent of A () and the marginal bene t of inreasing quality in order to attrat more ustomers. When WOM shifts the level of awareness up above the margin regardless of whether this shift is higher or lower than anywhere else this will result in an inrease in quality. On the ontrary, if the only inrease in awareness ours with respet to infra-marginal ustomers, it has no e et on the marginal deision and thus quality will not hange. 5. ersuasive WOM with Fixed Costs Our hoie of quality follows in the same way as before in that we ll announe e and then nd the optimal produed level of quality onditional on the announement and onditional on the uninformed 7

9 ustomers buying if they re above the uto given the announement. Formally, we solve the following: " # Max F (;!) d + ( F (;!)) d e First-order onditions for quality and prie are, respetively: " # F (;!) d + ( F (;!)) d e F F + e F = 0 (3) = 0 (4) e After applying the equilibrium ondition that = e and re-arranging, we nd that the latter redues to: whih substitute into (3) to yield = ; F = = 4 where it is obvious that optimal quality is inreasing in exogenous WOM. 6 Untargeted Informative Advertising We ll speify the analogous model that spei ed in the main body of the paper exept that the rm an t target its ads to spei types. We ll again assume onsumers di er in terms of their reeptiveness, or attentiveness, to rm ommuniations suh that higher types are more likely to beome aware as a funtion of these rm ations: A () = p () V; p () > 0 (5) where V is the rm s hosen level of overall ommuniation omes at a onvex ost V. The objetive funtion is: = V First-order onditions for in this model are: V p () d + p () d V V p = 0 (6) We retain the p () term for onsisteny with the main model. It has no e et on the results. 8

10 Unlike the targeted advertising ase, when advertising is untargeted, even exogenously-endowed advertising has no impat on optimal quality hoies. This an be seen by the fat that V drops out of this solution. First-order onditions for optimal advertising are: V = p () d Substituting this into (6) yields p () d! + p () dp = 0 where, as before drops out of the model and, thus, doesn t impat optimal quality hoies. 7 ersuasive Advertising Model In this ontext, we assume the rm an send advertising messages targeted to type that are redible in the sense that onsumers viewing them know the exat level of quality being produed. The ost of advertising to a proportion V () of type is V (). As with the model of informative advertising, we also allow for the possibility that onsumers will di er as to the likelihood that they attend to or, equivalently, reall the ads: p (). The rm announes a quality level e and selets a prie ; a true quality level and ad level V () [0; ]. We will de ne f : V () > 0g as the set of ustomer types that reeive advertising. Informed ustomers (those reeiving advertising messages) will buy if h i while uninformed ustomers buy, in equilibrium, if E e >. In equilibrium, of ourse, E [] = e =. Lemma 3 No pure strategy equilibrium exists in whih V () > 0 for type suh that p () <. roof of Lemma 3: Assume suh an equilibrium exists. Then, some will not see advertising messages in equilibrium and will, thus, need to base their purhase deisions on the relative magnitude of e and. However, given this, the rm will always deviate to V () = 0. This is an intuitive result in any BE. It says that if there s a hane that a ustomer won t see an ad in equilibrium, then the rm will never send an ad in a pure strategy equilibrium. Using similar reasoning, we an also eliminate advertising at any intensity other than 0 or. Lemma 4 No pure strategy equilibrium exists for any value of V () other than 0 or. 9

11 roof of Lemma 4: Assume suh an equilibrium exists. Then, some will not see advertising messages in equilibrium and will, thus, need to base their purhase deisions on the relative magnitude of e and. However, given this, the rm will always deviate to V () = 0. Now that we ve established that we an fous our attention on p () = and V () f0; g, we show that we an restrit our fous to the rm s equilibrium advertising strategy onsisting of a single onneted region of types. Lemma 5 For any prie and quality ; the following is a pro t-maximizing equilibrium advertising strategy: where = E (j ; V = 0) = 0. 8 < V ; () = : 0 otherwise and = + <. O -path beliefs over quality to support this inlude roof of Lemma 5: First, we ll establish that this is an equilibrium and then demonstrate that any other equilibrium that may exist would have inferior pro ts. To see that the rm has no stritly pro table deviation for a given, rst notie that a deviation to 0 = 0 would be notied by the informed ustomers but not by the uninformed ustomers. Thus, pro ts would be: 0 = sine the rm would not need to pay for quality prodution or advertising. This deviation is not pro table i : () + whih aptures the equilibrium de nition of. This will tehnially be true for all, however there is no bene t to advertising above and thus the onstraint holds stritly. Sine the rm would never advertise to onsumers that will not buy, we need to hek three possible alternative strategies: (i) advertising to all ustomers above the uto : = ; (ii) not advertising to a measure of low-value ustomers: > ; and (iii) a set of disonneted regions: 90 ; suh that V () = for < 0, V () = for > 0 and V () = 0 for 0. As for (i), for a given, 0

12 there is no reason to advertise to all onsumers sine, by de nition, advertising to ensures that those onsumers ; who do not see messages will believe that quality is. Any additional advertising thus omes at a ost but o ers no bene t. (ii) There are two ways to think about this. Either we might shift up suh that stays onstant but both and inrease by the same amount. Alternatively, we might x while allowing to inrease. The latter will not be an equilibrium in that there would not be a su ient measure of informed ustomers to support the quality laim of : eall the ondition is: " () + + " " whih is a ontradition. The former just shifting up the advertising interval is also not an equilibrium. While there is no advertising deviation given and, the rm will always ut quality. To see this, onsider that we ll now assume that > + " and = + + ". If we assume that types ; will not buy, the non-deviation ondition would be: " () + + sine the " s anel out of the advertising segment. Again, this is a ontradition given the de nition of : Now, onsider what happens when we assume that types ; will buy. Note that the purhase deisions by uninformed ustomers (; ) are based on the fat that they know that, in equilibrium, types ; are buying. This is rational sine the latter have utility stritly lower than they do. However, this is not true of the uninformed ustomers ;. The rm will always have the opportunity to ut quality, maintain sales to the informed ustomers and do stritly better. To see this, imagine a ut in quality to 0 = +". " This is the minimum quality required for all informed ustomers (given the advertising strategy) to ontinue to purhase. Sine types ; would ontinue to buy, total demand remains onstant. At the same time, the rm saves 0 in marginal osts of prodution. Thus, this is not an equilibrium. To evaluate alternative advertising strategies of the form (iii), we assume that the measure of informed ustomers is the same as in the lemma. Changes in the measure are overed in the other onditions. Anything less would not be su ient to prevent the rm from deviating to 0 = 0 and

13 V () = 0 for all. To be formal, we ll de ne a set of J ordered pairs n j ; jo j=;:::;j whih apture the lower- and upper-limits of the advertising regions suh that D JX j= j j = Note that, at J =, the problem region ollapses to that stated in the Lemma. We further assume, using the same reasoning as in our proof of (ii) above, that =. First, we show that there s no deviation to 0 = 0 and V () = 0. The non-deviation ondition is: D D sine, one the informed ustomers observe the quality deviation, they won t buy but the uninformed ustomers will. Notie, though that by simple substitution, this redues to the original equation whih de ned. Thus, when equilibrium exists, it will have equal pro ts to our foal (onneted) equilibrium. It is important to reiterate that while the strategy presented in Lemma 5 is not neessarily unique indeed, there may be in nite suh equilibria the pro ts are all the same. Intuitively, the rm needs to advertise to su ient ustomers in order to be able to ommit redibly to the uninformed that they an expet quality to be high enough. Thus, we restrit our fous in our analysis to the onneted equilibria. It is important as well to note that not all disonneted sets of advertised-to onsumers will represent an equilibrium. Intuitively, our equilibrium works beause higher types are willing to buy as long as lower types have inspeted the produt and are willing to buy. roposed equilibria with too muh advertising weight on higher types will not hold. Consider, for example, the following advertising strategy for given and : = ; = + " = + + "; =

14 It is straightforward to hek that the measure of informed ustomers is the same as our onneted equilibrium: X j j j= = " + = " Consider a deviation to 0 = whih ensures that all the high type informed buyers ontinue to buy at the lower quality. We would expet suh a deviation to destroy the proposed equilibrium beause for this relatively large drop in quality, the rm saves a great deal on prodution osts but only loses a measure " of ustomers sine all onsumers ; buy based on the rm s announement only, aording to the equilibrium: () " Note that " 0 "!0! 0 sine 0 "!0!. Sine + a ontradition and, thus, the equilibrium annot hold. " " (7) 0 "!0! K > 0, we have We stress that the key impliation of this analysis is simply that, in these disonneted equilibria, there must be su ient advertising weight on the lower-type buyers. 3