Panos Kouvelis Olin School of Business Washington University

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1 Quality-Based Cometition, Profitability, and Variable Costs Chester Chambers Co Shool of Business Dallas, TX Panos Kouvelis Olin Shool of Business Washington University John Semle Co Shool of Business Dallas, TX 7575

2 Aendi Euivalene of Utility Seifiations Consider i the utility funtion U, : θ θ where θ has the umulative distribution funtion F θ orresonding to a uniform distribution on [ε,] versus ii the lassial utility funtion U,,α α where α has the unknown umulative distribution F α t The infinitesimal ε is introdued to avoid division by 0, although taking the limit ε 0 oses no diffiulties in our subseuent analysis The two utility seifiations are euivalent if and only if the roortion of the oulation referring one rodut over another in our model euals the roortion of the oulation having idential referenes in the lassial model Straightforward algebra shows that these roortions are indeed eual if and only if Pr α t Pr θ / t owever, α θ t < / t ε t / ε Fα t Pr α : α t Pr θ : θ / t Fθ / t ε 0 t > / ε Differentiating eah side with reset to t imlies the euivalent density for the taste arameter in the lassial model is f t α ε t for t / ε and 0 elsewhere In the seial ase where ε 0, the euivalent density for the taste arameter in the lassial model is f α t / t on [, Proosition Condition The terms,, and are onsidered fied given Suose < / Then me must have / < / < / see Figure a and s rofit is π min /, It is easy to show that s rofit is stritly inreasing on this interval Thus /, whih is Condition Condition By Condition, we must have / / / For suh that / > see Figure, s rofit is π, whih is stritly inreasing Therefore, will always resond with a rie suh that / or euivalently Best Resonse for Beause must satisfy roosition, we have π / π and

3 If <, the funtion π is stritly inreasing and thus this rie is simly large and finite if one takes θ [ε,] for ε > 0 and small If, then s rofit is, regardless of s rie rovided it is feasible see Figure Thus selets any rie satisfying, / If >, then π is stritly dereasing and will set the lowest ossible feasible rie, whih is, / In summary, s best resonse is R, /, / < > Proosition Given 0 < < let hose a rie, and hose his best resonse Thus, the total size of the served market is and π Sine > has ositive market share and rofits as long as > This is easily verified by diret substitution The following lemma is needed to simlify the analysis of s best resonse It is based on straightforward algebra and therefore offered without roof emma If satisfies a market overing rie, then Condition of roosition imlies Condition If >, then Condition imlies Condition Best resonse for Market is overed ase On the interval, overs the market and the resulting rofit funtion is π Assuming the market is overed imliitly reuires ; if this is not the ase, the market annot be overed and one roeeds diretly to the analysis of the unovered market For the overed market ase, we have π / The euation π / 0 has two roots owever, only one of these roots is less than, and therefore it is the only relevant resonse The root is Beause of s rie floor,, one an show by diret alulation that i is an inreasing funtion of, ii, and iii Condition of 3

4 roosition holds observe that we must still show s rie resonse forms a feasible air so that the initial rofit reresentation for is valid If the root additionally satisfies, then, must be a feasible air of ries by emma In this ase, we may make the stronger statement that is a global imum for all < This follows from the fat that the funtion is onave on the etended interval < and vanishes at most one on this interval Thus for feasible ries satisfying >, we observe The latter ineuality ensures that is a global imum for π if If >, then the derivative of π is ositive on the interval and so π is inreasing on this interval The imum and thus s best resonse ours at This resonse forms a feasible air beause a if Condition of roosition is satisfied for then it is satisfied for and b Condition and Condition are idential if Market not overed ase If has a ositive market share and >, the market is not overed Observe that > an only our if > otherwise the market is already overed by see Figure b and Figure Additionally, we may assume has ried above his rie floor,,, For satisfying, s rofit funtion is given by π, and π / If the derivative vanishes at an interior oint >, then it is a global imum on the interval sine π is stritly onave on this interval Again, there are two ossible roots for π / 0 owever, in this ase only one root is ositive: It an be shown we omit the tedious algebra that, satisfies Condition of roosition rovided Note that the latter ondition is guaranteed by s rie floor

5 Moreover, it an be shown the root is an inreasing funtion of on the interval,, and further satisfies If, then emma imlies Condition must hold as well Thus, is a feasible air of ries whenever This solution is also a global imum for π for reasons elained net We observe first that the derivative for π annot vanish on both of the intervals and > For suose π / 0 for, then we have for all feasible ries Observe that the two rofit eressions agree at the rossover oint owever, the atual onave rofit funtion that alies on the interval, π, is bounded above by the stritly dereasing onave funtion This rohibits π / from vanishing on the interval > if it has already vanished for Thus if >, π must be inreasing on [, ], inreasing on [, ], and then dereasing thereafter This makes a global imum If the solution satisfies, then the rofit funtion is dereasing on the interval and the imum on this interval ours at the endoint, whih is feasible sine > see Figure It an be shown that < rovided hooses above his rie floor Conseuently, the best resonse for is if < R if > otherwise Theorem : Prie Euilibrium Eistene of a rie euilibrium We observe that the best resonse urves always interset This an be shown in three stes, whose details are left to the reader Ste : The minimum oint on s resonse urve has oordinates,, This oint ours at the bottom of a vertial line segment ositioned at see Figure 3 Ste If the minimum resonse on s urve is, then <, whih imlies R < Ste 3 If 5

6 the minimum resonse on s urve is, then this imlies < One an then show < <, whih imlies R < Sine R is an inreasing funtion satisfying R as, it must ross the infinite vertial segment of s resonse urve see Figure 3 at some oint above, Observe that a simle erturbation argument imlies the eistene of a similar intersetion for the ase where θ [ε,], rovided ε is suffiiently small The rie euilibrium The only otential euilibrium solution ours when In this ase, the rofit funtion beomes π Conseuently, aears indifferent to any feasible rie sine they all result in an idential rofit of this indifferene disaears for θ [ε,] with ε > 0 and small owever, must still hoose his rie arefully so that aets the rie and has no inentive to hange This is indeed the ase if is set so that is s otimal resonse Analysis of the first term in s resonse urve shows that if < then the value of whih drives to is / / One an readily hek that as defined above and form a feasible air of ries this ensures that our original rofit reresentations are valid These are the euilibrium ries when < Analysis of the seond term in s resonse urve shows that if >, the aroriate seletion is α / α / α where α One an hek that as defined above and form a feasible air of ries These are the euilibrium ries when > Finally, if, then there are an infinity of rie euilibrium solutions One an alulate a value for using either of the reeding formulas, although any rie seleted between these two values will also suffie This ours beause the finite vertial segment in s best resonse urve erfetly oinides with the infinite vertial segment in s resonse urve see Figure 3 We note that this situation does not our when the rie sensitivity arameter satisfies θ [ε,] ε > 0 The solution with the lowest rie for is the limiting rie as ε 0 Theorem : Theorem The result follows from diret substitution into euation using the results from Theorem 3: Sine < by definition, the market is neessarily unovered unless, whih is therefore the only situation to assume For notational onveniene, 6

7 7 reall The roof roeeds by showing that s market share for the unovered market as see 0 eeeds s market share for the entire overed market see 9 Sine rofit margins are also larger on the unovered market side, it follows that the rofit for as see 0 will dominate all rofits for 9 with Aording to 0, the market share for as is im / / Sine >, Beause / is log-onave, so is Thus for any y we must have y y Beause is onve, for any y we must have y y, where the differene uotient at y is interreted as Thus for y, and z we have z z, where the last ineuality follows from the onveity of It follows that for any z, z z The last term is reisely s market share in 9 for z in the interval z This demonstrates that the market share for as eeeds that for all where 9

8 alies This imlies the imum rofit ours over the region market region whose rofit is determined by 0, ie, the unovered We introdue three lemmas that will hel with the roof of Theorem emma Suose / is onve and log-onave Then the ratio funtion / r is non-dereasing Proof Observe that r is ontinuous on 0, ] with r 0 0 / and [ ] [ 0, r / Moreover, r is differentiable on and therefore it is non-dereasing on [ 0, ] if and only if its derivative is nonnegative on 0, The latter is euivalent, after suitable algebrai maniulations, to the ineuality ondition We will now show the latter ineuality is true Observe that we may write h with h onve and log-onave og-onavity of h imlies h / h is a dereasing funtion of Therefore, beginning with the left hand side of the revious ineuality ondition h h h h h h But h is onve, so the final term in urly brakets satisfies { h h } { h h } h h h h h emma 3 Suose f is nonnegative, differentiable, and stritly onave on [ a, b] Suose g is differentiable, non-inreasing, and ositive on [ a, b] et the imum of f on [ a, b] our at the oint f, and let a global imum of f g on [ a, b] our at the oint fg Then fg f Proof For, b], f < 0 After alying the rodut rule and the various sign f onditions stated in the theorem, it follows that < 0 the result fg on this interval as well This roves emma Suose n and d are nonnegative, ontinuous funtions on [ a, b] that are 8

9 n also differentiable on a, b Assume n is non-dereasing If the ratio d is ontinuous n on [ a, b], non-dereasing on [ a, b], and differentiable on a, b, then α d is ontinuous and non-dereasing on [ a, b] for any α > 0 n n Proof Continuity of on [ a, b] is lear Sine α d d is non-dereasing and differentiable on a, b, n d n d 0 on a, b, and so n n [ α d ] n d 0 on a, b, whih imlies α d is non-dereasing on [ a, b] Theorem Part a By lemma, r is non-dereasing on [ 0, ] Thus r / is non-dereasing with a removable singularity at 0 The following hain of non-dereasing non-d for short funtions is imlied: r / non-d emma non-d non-d emma non-d Conseuently, / non-d - / g is seen to be ositive and non-inreasing If we define f art a of Theorem follows immediately from emma 3, then / For art b of the theorem, observe that an uer bound on s rofit in is π π If leafrogs, the same ineuality alies where reresents the high uality osition Sine <, <, s best rofit from leafrogging is bounded K K 9

10 above by K Part b follows by omaring the lower bound on s rofit ensured by art a and the uer bound on rofit for leafrogging K For art, observe that the rofit for is bounded above by the eression K[0, ] If leafrogs, then the same bound alies for the otimal osition that takes below : K, where is s urrent osition It is immediately lear that [0, ] K K [0, ] K We will be done if we an show that [0, ] K K[0, ] [ 0, ] The ondition stated in art lays an essential role Consider an arbitrary ost funtion Χ Construt the assoiated funtion λ v λ, λ defined for 0 λ and 0 < Then λ λ λ v λ, λ λ λ λ 0 [ ] [ ] The last ineuality follows beause / is assumed to be non-dereasing, and so the v λ, braketed term must be nonnegative, too Beause 0, it follows that λ λ K [ 0, ] Ma λ Ma K λ [0,] [0, ] [0,] λ λ Sine Χ and Χ, we must have Χ, and onseuently the same result alies for The imum rofit an obtain by leafrogging is therefore bounded above by K K[0, ] Part now follows by insisting that the uer bound on s rofit for leafrogging is no better than the uer bound on s rofit as reviously established in art a for staying at 0