Online Companion to Pricing Services Subject to Congestion: Charge Per-Use Fees or Sell Subscriptions?

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1 Online Companion to Pricing Serices Subject to Congestion: Charge Per-Use Fees or Sell Subscriptions? Gérard P. Cachon Pnina Feldman Operations and Information Management, The Wharton School, Uniersity of Pennsylania, Philadelphia, Pennsylania 904, USA Operations and Information Technology Management, Haas School of Business, Uniersity of California, Berkeley, CA 9470, USA June 7, 008; reised July 4, 009; May, 00; August 0, 00 In this online companion we introduce propositions that are not explicitly included in the main text of the paper but are useful for deriing some of the intuitions and results that we claim in the main text without proof. We also analyze the mixed strategy equilibrium. Due to space limitations, seeral results are stated without proof. These proofs are aailable from the authors. Proofs from the Exogenous Capacity Model For the comparison between per-use and subscription, it is useful to introduce a third pricing scheme the two-part tari. A two-part tari combines a per-use fee with a subscription rate. As with per-use pricing, a consumer with a alue V for a serice opportunity is indi erent between seeking serice or not when V = p + ww (): Therefore, the arrial rate to the rm is F (); where is the unique solution to = p + ww (F ()). Reenue from per-use fees accrues at rate R p (); just as in the per-use scheme, assuming all consumers are subscribers. As in the subscription case, the optimal subscription makes all consumers indi erent between purchasing the subscription or not. Hence, subscription reenues are R s (): Total reenue from the two-part tari is then expressed in terms of the threshold for the subscription customer who is indi erent between paying the per-use fee for serice or not (the t subscript is used to signify notation associated with the two-part tari ): R t () = R p () + R s () = F () E [V jv ] ww ( F ()) Unlike with subscription pricing, and just like with per-use pricing, the threshold is a decision ariable for the rm. The following proposition establishes that there is a unique optimal threshold for the rm. The proof is similar to that of Theorem in the main text and is omitted.

2 Proposition The rm s reenue function with a two-part tari, R t (); is quasi-concae. Hence, there exists a unique optimal threshold, t = arg max R t (); where t is the unique solution to t = ww ( F ( t )) + w F ( t )W 0 ( F ( t )) () R t () is equilialent to social welfare: it is composed of the bene t from the serice minus the waiting costs, but does not include the prices charged by the rm. By charging an appropriate per-use price, the system achiees the social optimal congestion: the optimal threshold alue, t, is such that customers take into account their own waiting cost (ww (F ( t ))) and the externality they impose on others from joining (wf ( t )W 0 (F ( t ))). In addition, by charging a subscription price, the rm is able to extract the entire consumer rent. The following proposition establishes the relation between the threshold alues of the three pricing schemes. Proposition The resulting two-part tari threshold aluation, aboe which a customer requests serice, lies in between the threshold aluation in the pay-per-use case and that of the subscription case, i.e., s < t < p. Proof. Denote the right-hand-side of functions s = ww p = F ( p ) f ( p ) + ww ( F ( p )) + w F ( p )W 0 ( F ( p )); () F ( s ) and () by g p (), g s () and g t (), respectiely. g s () < g t () < g p () 8. Since the LHS is strictly increasing in and g x () is decreasing in 8x = fs; t; pg, the result follows. The following statement generalizes Theorem 3 in the main text to include results about the two-part tari. The proof is similar to that of Theorem 3 and is therefore omitted. Proposition 3 The following limits hold: (i) lim!0 R s = lim!0 R t = ( ) = and lim!0 R p = ( ) =4. (ii) lim! R x = 0; x fs; p; tg. Proposition 3 implies that subscription pricing approaches two-part tari reenues when the potential utilization is low. In addition, we show numerically, that per-use pricing approaches two-part tari for a high potential utilization: Coering the range [0; ], we obsere that as!, lim! R s =R t < and lim! R p =R t =. This implies that at high alues of potential utilization, the two-part tari is well approximated by per-use and at low potential utilizations, it is well approximated by subscription pricing. Note that because Rx 0 () < 0 and lim! R x = 0, per-use approaches the two-part tari when the rm barely makes a reenue in any scheme.

3 The next proposition establishes the potential utilization alues under which per-use and subscription dominate, for di erent alues of the parameter > 0. Proposition 4 When approaches zero, subscription dominates pay-per-use when < p and ice ersa. When =, subscription dominates pay-per-use when < and ice ersa. Proof. (i)! 0: Pay-per-use. Rearranging we get: p = p + + p =! p ; (3) (4) If! 0; there are two roots that sole the aboe. The larger of the two is the maximum. This implies Substituting, it follows that Subscription. Rearranging we get: R p ( p ;! 0) = p = if if > : s = s ( 4 if ( ) if > + s s ; (5) = (6) When! 0; there are two roots that sole the aboe. The larger of the two is the maximum. This implies that in this case, we hae s 0 if = if > : It thus follows that R s ( s ;! 0) = ( if if > Comparing the two reenue functions, we get that lim!0 e = p. The corresponding reenue functions are then gien by R s = R p = 4. (ii) = : Rearranging (5) and substituting = we get: s + ( ) s = 0. The releant root suggests s!. As p > s (Proposition ), we must hae p! as well. Next, we 3

4 conjecture that e () = and erify that this is indeed the case. Substituting = in (4) and in p (6), we get = s and = ; respectiely. Note that = and s = p = sole p both equations. This solution implies that R s = R p and thus satis es e () =. Proofs from the Endogenous Capacity Model Like with a xed capacity, with a two-part tari the rm s reenue is R t (); where is the alue of the consumer who, at a serice opportunity, is indi erent between seeking serice or not. By adding the cost of capacity, the rm s pro t function is t (; ) = F () E [V jv ] w F () c: As with per-use pricing, t (; ) is concae in for a xed and t () = p () is the optimal capacity. The pro t function can then be written as t () = t (; t ()) = F () E [V jv ] c! p F () The next proposition characterizes the rm s optimal policy. The proposition holds for any increasing generalized failure rate (IGFR) distribution. Proposition 5 If > c, there exists an upper bound, such that for eery there exists a unique interior optimal threshold leel, t = arg max t (); which is implicitly de ned by the smaller of two possible solutions to: The optimal capacity is t = t = p + c: (7) F (t ) w t ( t c) ; (8) As with xed capacity, the two-part tari is the optimal pricing strategy. For any gien serice rate leel, the indi erent consumer has a threshold alue which satis es: e = ww F ( e ). If the rm chooses a serice rate t, the social optimal threshold alue, t ; is gien by equation (7) and is equialent to e + c. To guarantee customers join according to the social optimal threshold, the rm can choose a per-use fee p t = c. Conditional on choosing p t, a subscribed customer s expected utility equals k t = R t df () F (t ) ( e + c). Setting the subscription rate to k t enables the rm to extract the remaining welfare and his expected pro t rate results in the optimal social welfare. 4

5 Proposition 6 Compared to the two-part tari, the following hold: (i) In pay-per-use, the aluation threshold for requesting serice is higher ( p > t ) and there is lower inestment in capacity ( p < t ); and (ii) If V U[0; ] subscription results in a lower aluation threshold ( s = t c) and in a higher inestment in capacity ( s > t ). Proof. (i) Pay-per-use. compare the thresholds gien in (7) and F p = ( p ) f ( p ) + q + c: (9) F (p ) Let g t () = p cw= F () + c and g p () = F ()=f() + p cw= F () + c. Since F ()=f() > 0, g p () > g t (), 8. Thus, it must be that t < p. the optimal serice rate function () is the same for per-use and two-part tari. Because 0 () < 0, it follows that ( t ) > ( p ). (ii) Subscription. Assume t = s + c holds and check that the FOC gien in (7) are satis ed. For the uniform case, condition (7) becomes t = p =( t ) + c: Substituting t = s + c, we get s s s = F ( s ) cf( s ) = ( s c), (0) when applied for the uniform distribution. For the uniform case, we get ( s ) = w= s +( s )= and ( t ) = w t = ( t c) = w= s + wc= s; where the last equality follows from substituting t = s + c. Now, ( s ) > ( t ) i < s( s ) : () (7) implies that for the FOC to hold, we must hae: = p F (t )( t c) or = F ( t )( t c). () follows, because s < t and because for the uniform distribution, s = t c. Let u x (c) = F ( x )= x ; x fs; t; pg be the actual utilization rates under each pricing scheme. The next proposition compares between the actual utilization rates. Proposition 7 The following inequalities hold: (i) u t (c) > u p (c) 8c > 0 (ii) if V U[0; ], u s (c) > u t (c) 8c > 0. Proof. (i) Theorem 6 of the main text and Proposition 5 imply u x = + p w=c F ( x ) x ft; pg:the result follows for eery IFR distribution, because t < p (Proposition 6). (ii) Let V U[0; ]. We hae: and u t = u s = w s ( s ) + () w t ( t c) = w ( s + c) t s ( s c) = w + ; (3) s ( s c) 5

6 where the rst equality follows from (8), the second equality follows because s = t c (Proposition 6) and the third equality from substituting in (0). Comparing () and (3), the result follows. Propositions 9 and 8 nd conditions for the waiting cost w > 0 and the capacity cost c under which the rm is better-o selling subscriptions or charging per-use fees. Proposition 8 Assume F () is IFR and s > p for all c. Let ec be the maximum c such that p (c) 0 and p (c) s (c). Then either there exists a unique capacity cost threshold ec, below which it is better to use subscription and aboe which using per use pricing results in higher pro ts or subscription is always better than pay-per-use, conditional on obtaining positie pro ts. Proof. At c = 0, we hae s = 0 and p = F ( p )=f( p ). These lead to s ( s ; c = 0) = E[V ] and to p ( p ; c = 0) = F ( p ) p, respectiely. Because s ( s ; c = 0) = t ( t ; c = 0), we must hae p ( p ; c = 0) s ( s ; c = 0). Let M s (c) s ( s ; c) and M p (c) p ( p ; c). By the Enelope Theorem, Ms(c) 0 = s < 0 and Mp(c) 0 = p < 0. From s > p, we get that jms(c)j 0 > Mp(c) 0 8c. From this and the fact that s ( s ; c = 0) > p ( p ; c = 0), the result follows. Because it might be that s ( s ; ec) = p ( p ; ec) < 0, and we are only interested in the non-negatie range of the pro t functions, subscription might be better than per-use in the entire releant range. Proposition 9 Assume that V U[0; ] and w > 0. Let ew be the maximum w such that p (w) 0 and p (w) s (w). If > + p c, then either there exists a unique threshold ew such that for w < ew it is better to sell subscriptions and for w > ew, using pay-per-use results in higher pro ts, or subscription is always better than pay-per-use, conditional on obtaining positie pro ts. Otherwise, if c < < + p c, pricing on the basis of pay-per-use is always better than selling subscriptions. Proof. Obsere rst, that at w! 0, we hae s! 0 and p = F ( p )=f( p ) + c = ( + c) = (when the second inequality follows from substituting for the uniform distribution). These lead to s ( s ; w! 0) = R 0 df () c = (E[V ] c) and to p ( p ; w! 0) = F ( p ) ( p c) = ( c) =4, respectiely. Comparing between the pro t functions, we get that s ( s ; w! 0) p ( p ; w! 0) i > + p c. Rewriting equation (9) for the uniform distribution, we get ( p ) ( p =) = c + p cw: Similarly, from equation (7) we get: s p s = c= = p cw. Let M s (w) s ( s ; w) = and M p (w) p ( p ; w) =. Substituting p and s, we hae M p (w) = p (p c) q cw p and Ms (w) = s qcw s c c s. By Note that the condition s > p holds for the uniform distribution (Proposition 6) and is shown to hold numerically for the Weibull distribution (with a wide range of parameter combinations) as well (the numerical study is aailable from the authors). 6

7 the Enelope Theorem, we rst nd that both pro t functions are decreasing in w: Mp(w) 0 = p q w c ( p =) and Ms(w) 0 = w c s q c = w c t ; where the second equality follows from Proposition 6. Thus, both pro t functions are decreasing in w. Applying some algebra along with the fact that t < p, we get that M 0 s(w) < M 0 p(w) < 0 8w. Combining this with the condition at w = 0, the three cases follow. 3 Proofs from the Heterogeneous Model Proposition 0 When approaches zero, selling subscriptions only to heay users dominates payper-use when < = p + 0 and ice ersa. When =, subscription dominates pay-per-use when < and ice ersa. Proof. (i)! 0: High-usage subscription. Rearranging we get: sh sh = + 0 sh ; (4) sh = (5) When! 0; there are two roots that sole the aboe. The larger of the two is the maximum. This implies that in this case, we hae 8 < s = : It thus follows that R s ( s ;! 0) = 0 if = if > = + 0 : 0 ( + 0 (+ 0 ) if = + 0 if > = + 0 Comparing this with the per-use reenue function (Proof of proposition 4), we get that lim!0 e = = p + 0. The corresponding reenue functions are then gien by R h = R p = 4. (ii) = : Rearranging (4) and substituting = we get: +0 sh sh = 0. The releant root suggests sh!. As p > sh, we must hae p! as well. Next, we conjecture that e ; 0 = and erify that this is indeed the case. Substituting = in (4) and in (5), we get p p = +0 and sh + 0 sh = ; respectiely. Note that = and sh = p = sole both equations. This solution implies that R h = R p and thus satis es e () =. Proposition max f l ( sl ; ) ; h ( sh ; )g s ( s ) 8. 7

8 Proof. s () = MF () (E[V jv ] ) c MF () + w=, l (; ) = M lf () (E [V jv ] ) c F ( l ) + w= l and h (; ) = M hf () (E [V jv ] ) = c Mh F () = + w=. (i) comparing s () and l (; ), l ( sl ; ) s ( s ) 8 because l. (ii) h ( sh ; ) s ( s ) 8: note h ( sh ; ) =@ > 0 (Enelope Theorem) and that h ( sh ; ) = s ( s ). Corollary Let ec 0 be the cost for which max f l ( sl ; ; ec 0 ) ; h ( sh ; ; ec 0 )g = p ( p ; ec 0 ) : Then, ec 0 ec 8. Proof. The result follows because max f l ( sl ; ) ; h ( sh ; )g s ( s ) 8 (Proposition ) and the three pro t functions are decreasing with c: 4 Analysis of Mixed Strategy Equilibria We extend the analysis of the two capacity scenarios to allow for mixed strategy equilibria. Under pure strategies, the monopolist charges a subscription price so that all customers buy. Alternatiely, the monopolist can charge a subscription price k () such that a fraction of the customers purchase and do not ( [0; ]). We show that a mixed strategy equilibrium in which the rm charges k () and a fraction < of the customers subscribes is sustainable when capacity is xed and the serice rate is low compared to the potential arrial rate (so that congestion matters). In this case, the rm can extract higher reenues by charging a high subscription fee so that only a fraction of the consumers buy. We also show that in all other cases the unique equilibrium is in pure strategies. We assume throughout the section that W () = = ( ) and that V U[0; ]. 4. Fixed Capacity Gien that a fraction of the customers purchased a subscription, they will request serice i s () = ww F ( s ()) (6) s is increasing in : As less customers subscribe, the customer requests serice more often. A customer expects that a subscription generates the following net alue per serice opportunity: F ( s ()) (E [V jv s ()] s ()) : Gien that serice opportunities arise at rate, the rm can choose which k () k to set (where k is the subscription price charged in a pure strategy equilibrium, i.e. when = ). Each subscription price chosen corresponds to a unique fraction of customers,, buying a subscription: k () = F ( s ()) (E [V jv s ()] s ()) : The rm s 8

9 resulting reenue can be expressed in terms of the fraction : R s (; s ()) = k () M = F ( s ()) (E [V jv s ()] s ()) ; (7) where s () is gien by (6). The rm can control the fraction of customers subscribing by changing the subscription price, thereby also controlling congestion. Proposition In the xed capacity model there exists a unique equilibrium in which the rm charges a subscription price k () and a fraction of the customers purchase. Moreoer, if, then = is always optimal the resulting equilibrium is in pure strategies. Howeer, if > then the resulting equilibrium is in mixed strategies customers purchase a subscription with probability = =. Proposition implies that the equilibrium found in the exogenous capacity model ( 3 in the main text) represents a lower bound on the rm s reenues. If we allow for mixed strategy equilibria, gien that >, the rm can do better by charging a higher subscription price and haing a fraction = = of all customers purchase. Thus, in the xed capacity case, subscription can do een better relatie to per-use. If ; the rm s optimal strategy is to charge a subscription fee such that all customers purchase. (The proof of Proposition is analogeous to the proof of Theorem 6 in the main text and is therefore omitted.) 4. Endogenous Capacity Proposition 3 demonstrates that when capacity choice is endogenous, the unique equilibrium is in pure strategies. That is, the optimal subscription price for the rm is the one found in 4 in the main text, which results in the entire customer base purchasing. Proposition 3 In the endogenous capacity case, if E [V ] > c and s, then = is always optimal the resulting equilibrium is in pure strategies. Proof. Proposition states that if < ; then () <. In this case, the indi erent customer threshold satis es s () = p w=. If >, then () =, in which case s is implicitly de ned by s = w= F ( s ) = w= ( ( s ) =). Gien this optimal behaior for a xed serice In proposition 3 we assume that E [V ] > c and s, which implies that the rm can make positie pro ts. If positie pro t cannot be made, then the best the rm can do is to set s = 0, in which case nobody buys. 9

10 rate, we want to nd the optimal serice rate, s. The rm s pro t function is gien by: s () = = F (s ) (E [V jv s ()] s ()) c( s ()) if < F ( s ) (E [V jv s ] s ) c if ( q w c if < q w ( s) c if where the last equality follows from the M=M= and uniform assumptions. It is easy to erify that s is continuous and that it is strictly conex in the < domain. From s, it follows that w= and s (w=) < 0. If, the optimal serice rate satis es s = F ( s ) + w= s ( 4 in the main text). Proided that a positie pro t can be made (i.e. if E [V ] > c and s ), s if and only if s p w=. This always holds, because s = w= F ( s ) = p w= when = and it is decreasing in. Continuity, s ( s ) > 0 and the fact that s imply that the optimal s inoles the rm selling subscriptions to the entire customer base ( = is optimal) and it is chosen to be high enough so that s >. 0