Uncertain Consumer Tastes and Two-Part Tariff

Transcription

1 NO.E Uncertain Conumer Tate and Two-Part Tariff Hao Wang * CCER, Peking Univerity NO. E Augut 2007

2 Uncertain Conumer Tate and Two-Part Tariff Hao Wang * CCER, Peking Univerity NO. E Augut 2007 * I am grateful to Edward Green and Jame Peck for very helful comment. All error are mine. Correonding addre: Room 623, China Center for Economic Reearch, Peking Univerity, Beijing, , China; Telehone: ; Fax: ; hwang@ccer.edu.cn 1

3 Uncertain Conumer Tate and Two-Part Tariff Abtract: A ervice rovider ell to homogenou rik-avere conumer through a two-art tariff. The conumer have uncertain tate toward the ervice. They ubcribe the ervice before the uncertainty reolve. In contrat with the common view that a monoolit otimal two-art tariff for homogenou conumer hould entail a uage rate equaling to the firm marginal roduction cot, I how that when conumer have uncertain tate, the ervice rovider otimal two-art tariff entail a uage rate that i greater than the marginal cot. Keyword: Uncertain tate, Rik averion, Service ricing, Two-Part Tariff JEL Code: D4, D8 2

4 1. Introduction Conumer choice or deciion often deend on ome ituational variable that are beyond the control of the conumer (Belk, 1975). A long a we cannot identify all the ituational variable that affect a conumer tate, we may have to view the conumer tate a an uncertain one. The economic ignificance of uncertain tate or reference i well recognized in the literature (McFadden, 2001). Comared to the microeconometric tudie in thi area, much le attention i allocated to the analytical modeling of economic mechanim that uncertain tate affect the behavior of firm and conumer. Chang (1993) invetigate the trategic ue of flexible manufacturing technology a an entry-deterring mechanim when conumer tate fluctuate robabilitically. It i hown that for relevant arameter value, an incumbent firm hold exce flexibility in manufacturing in order to deter entry into the market. Walh (1995) develo model of an exected-utility-maximizing conumer faced with the roblem of urchaing multile unit for multile future conumtion occaion. The conumer chooe from two conumtion alternative. The utilitie of the alternative to the conumer deend on the tate of nature. Walh how that the conumer may wih to urchae an aortment of alternative rather than multile unit of the ame alternative. Walh conider the individual uncertaintie in conumer tate, while Chang focue on the aggregate uncertainty. The fact that individual conumer have uncertain tate may influence the ricing trategie of firm. The iue i articularly intereting in market where (i) trade contract are igned before the uncertainty in tate reolve, and (ii) the roduct in conideration are non-torable. Indeed, if tranaction occur after the uncertainty reolve, the firm in the 3

5 market necearily face (ex ot) heterogeneou conumer. Thi i a tyical aymmetric information or econd-degree rice dicrimination iue, which ha been extenively dicued in the literature. If the roduct are torable, conumer may reell unued roduct or ave them for future conumtion. Hence the conumer do not take much rik although they have to trade with firm before knowing their tate in future eriod. What really matter to the firm in thi cae might be the aggregate demand of the conumer. If the individual uncertaintie are indeendent, which mean that the aggregate demand i rather table, the imact of uncertain conumer tate on firm ricing trategie would not be ignificant. Many ervice indutrie have the two feature mentioned above. In economic literature, a deciion maker i aid to face a rik if hi well-being deend on outcome that will occur with robabilitie. When conumer tate toward a ervice rely on ome ituational variable, the conumer may face a rik if they have to ubcribe the ervice for future uage. For examle, a conumer extract a large benefit from a iece of bread only when he haen to be hungry. If the conumer i rovided with a certain amount of bread each day, he may value the bread differently in different day. Hence the conumer erceive rik from ubcribing uch a tream of bread. In the Arrow-Pratt theory of rik, a conumer attitude toward rik i ortrayed by a Bernoulli utility function (the terminology follow the textbook of Ma-Colell et al., 1995, age 184). Unlike a utility function that i defined on conumtion bundle and rereent a conumer ordinal reference, a Bernoulli utility function i often defined on a numeraire good like money and rereent a conumer cardinal reference over different quantitie of the good (but the von Neumann-Morgentern exected utility can be viewed a ordinal on a ace of money 4

6 lotterie ). In a market where eole make conumtion deciion after an uncertain ituational variable reolve, one way to cature the role of conumer attitude toward rik might be defining a Bernoulli utility function on the ex ot atifaction level that the conumer achieve from the gaming. Indeed, all the rik that the conumer care about i the uncertainty in the ex ot atifaction level. Service indutrie often have relatively high etu cot but low marginal cot, i.e., increaing return to cale. Such a cot tructure imlie that comared to a two-art tariff, linear ricing in ervice trade i tyically le efficient ince it lead to more deadweight lo. On the other hand, two-art tariff are often feaible in ervice market becaue it i relatively eay to revent reelling. Standard ervice ricing theorie, e.g., Oi (1971), often ugget that when conumer are homogenou and firm caacitie are non-binding, the otimal two-art tariff hould entail uage rate equaling to the firm marginal roduction cot. Such two-art tariff aear attractive becaue the deadweight lo i avoided. However, the reult i baed on the aumtion that conumer tate are contant and anticiatable. It might not hold when conumer have uncertain tate. Thi aer conider a imle model where a ervice rovider directly ell to a large number of homogenou rik-avere conumer. The conumer have uncertain tate toward the ervice, or equivalently, they attain uncertain benefit from a certain amount of the ervice. In the game, the ervice rovider move firt by offering a ervice lan rereented by a two-art tariff. The conumer then chooe whether to ubcribe the ervice before the uncertainty in their tate reolve. Finally the uncertainty reolve and the conumer chooe conumtion bundle to maximize their ex ot well-being. In the model, a Bernoulli 5

7 utility function i defined on the conumer ex ot benefit from the conumtion. Hence the ex ot benefit, which deend on the tate of nature, lay the role of money lottery in the Arrow-Pratt theory of rik. The model how that if the conumer are rik-avere, the ervice rovider otimal two-art tariff generally entail a uage rate that i greater than the rovider marginal roduction cot. The rovider chooe uch a uage rate becaue it reduce the rik faced by the conumer. The rovider face the tradeoff between inuring the conumer and avoiding the deadweight lo when it chooe the otimal uage rate. The ret of the aer will roceed a follow. Section 2 reent a monooly model of a ervice market. I firt conider the cae where conumer have contant and anticiatable tate, then the cae where conumer have uncertain tate. The comarion of the two cae demontrate how the uncertainty in conumer tate affect the ervice rovider ricing trategy. An examle that illutrate the main oint of the model i given at the end of Section 2. Section 3 conclude the aer. 2. A Model A ervice rovider ell to a large number of homogenou conumer. The rovider roduction cot i rereented by Cq ( ) = F+ cq, where F 0 i the fixed cot, c 0 i the marginal cot, and q 0 i the quantity of the ervice. The rovider offer a ervice lan rereented by a two-art tariff ( T, ), where T i a lum um fee and i the rate of uage charge. The conumer alo ue a numeraire good that rereent their other conumtion. The quantity of the numeraire i denoted a m. I firt conider the cae where the homogenou conumer have contant and 6

8 anticiatable tate toward the ervice. Suoe that a conumer obtain benefit of bq ( ) from conumtion bundle ( qm, ). Note that function bq ( ) + m i a houehold roduction function : conumer ue conumtion bundle to roduce atifaction meaured by the numeraire good. The houehold roduction function can alo be viewed a a utility function when the conumer have anticiatable tate. Let b'( q ) > 0, b"( q ) < 0, which mean that + m the conumer obtain oitive but diminihing marginal benefit from uing the ervice. Alo aume that b'(0) > c and b'( ) = 0 for imlicity. Finally, each conumer ha an endowment of I > 0 in term of the numeraire good. The timing of the game i a follow. Firt, the rovider announce a ervice lan rereented by a two-art tariff ( T, ); Second, the conumer decide whether to accet the lan. If the conumer reject the lan, the game i over. Otherwie the conumer ay the lum um fee T and the game goe to the next tage; Third, the conumer decide how much of the ervice to ue. Note that the game can be eaily extended to a game where conumer ubcribe the ervice for multile eriod, a long a the conumer demand for the ervice ha zero intertemorary elaticity of ubtitution. I will olve the game backward and find the ervice rovider otimal two-art tariff. Note that the rovider would never offer a ervice lan with a uage rate b'(0), becaue that would totally revent the conumer from uing the ervice. The outcome of the econd and third tage of the game i characterized by following lemma. Lemma 1: Given the rovider two-art tariff (T, ) with < b'(0), define conumtion bundle ( q, m ) by 7

9 * b'( q ) = and m = I T q. (1) If bq ( ) b(0) T+ q, the conumer accet the ervice lan and chooe conumtion bundle ( q, m ); Otherwie the conumer reject the lan. Proof: Given tariff (T, ), a conumer obtain benefit of b(0) + I if he reject the lan. The conumer benefit from taking the lan i given by following maximization roblem: Max q 0, m bq ( ) + m, (2).t, T + q+ m I, (3) Solving the roblem yield conumtion bundle ( q, m ) recribed in (1). Obviouly, if bq ( ) m b(0) I + +, i.e., bq ( ) b(0) T+ q, (4) the conumer refer acceting the lan, which mean ( q, m ) i hi otimal conumtion bundle. Otherwie the conumer refer rejecting the lan. Q.E.D. The rovider doe not want to rooe a lan that care the conumer away and reult in zero rofit. A long a F and c are not too large, the rovider rofit-maximization roblem, a hown below, ha a non-degenerated olution. Max T F + ( c) q (5) T,.t., bq ( ) b(0) T+ q, (6) b'( q) =, (7) The model can be viewed a a ecial cae of Oi (1971) Dineyland model with homogenou conumer (age 80). We have following rooition, which tate that the rovider otimal two-art tariff entail a uage rate equaling to the rovider marginal 8

10 roduction cot. The roof i eay if one notice that the contraint (6) mut be binding at the otimal two-art tariff. Detail are omitted. Prooition 1 (Oi, 1971): With contant and anticiatable conumer tate, the ervice rovider otimal two-art tariff ( T, ) i c ), where ( bq ( ) b(0) cq, * q i given by * b'( q ) = c. Now uoe that the conumer have uncertain but indeendent tate toward the ervice. Secifically, a rereentative conumer tate deend on the tate of nature, which i a random variable with cumulative ditribution function G(.) on [, ] where <. In tate, the conumer ha ex ot houehold roduction function of bq (, ) + m. Note that the houehold roduction function i definitely not a Bernoulli utility function. Hence the quai-linearity of the function doe not imly that the conumer i rik-neutral. I aume bq (, q ) > 0, bqq (, q ) < 0, b (, q ) > 0 and bq (, q) = bq (, q) > 0 for all [, ] and q 0. Hence both the conumer benefit and marginal benefit from uing the ervice increae with variable. To imlify the exoition, I aume that bq (,0) = b> c for all [, ]. Hence the conumer would not ue the ervice at all if the uage rate i higher than b. We view the ex ot benefit of the conumer bq (, ) + m a a money lottery againt the tate of nature. Therefore we can define a Bernoulli utility function ux ( ), where x bq (, ) + m and u'( x ) > 0 for all x R. The rik-averion of the conumer imlie that u"( x ) < 0. The timing of the game with uncertain conumer tate i a follow. Firt, the ervice rovider announce a two-art tariff ( T, ); Second, the conumer decide whether to 9

11 accet the lan. If the conumer reject the lan, the game i over. Otherwie the conumer ay the lum um T and the game goe to the next tage; Third, the uncertainty in tate reolve and the conumer decide how much of the ervice to ue. Given the rovider ervice lan (T, ) with < b, if a conumer accet the lan, he chooe (q, m) to olve following roblem after the tate of nature realize. Max q 0, m bq (, ) + m, (8).t., T + q+ m I, (9) Conditional on the ervice lan (T, ) being acceted by the conumer, the otimal conumtion bundle in tate, denoted a ( q(, T, ), m(, T, )), atifie bq (, q(, T, )) = and mt (,, ) = I T qt (,, ). (10) To imlify the exoition, I denote the conumtion bundle ( q(, T, ), m(, T, )) a ( q ( ), m ( )) hereafter. A conumer ex ot benefit in tate i thu bq (, ()) + m (). The main reult of the model i reented in following rooition. Prooition 2: When the rik-avere conumer have uncertain tate toward the ervice, the ervice rovider otimal two-art tariff ( T, ) entail a uage rate * > c. Proof: At the firt tage of the game, the ervice rovider rofit-maximization roblem i Max T F + ( c ) q ( ) dg ( ), (11) T,.t., ubq ( (, ( )) + m ( )) dg ( ) ub ( (,0) + IdG ) ( ). (12) The conumtion bundle ( q ( ), m ( )) i given by (10). Since there i no direct contraint on T or, the roblem mut have an interior olution. It can be tranformed into following otimization roblem: 10

12 Max LT (, λ, ) = T F + ( c ) q ( ) dg ( ) λ T,, 0 λ [ ubq ( (, ( )) + m ( )) ub ( (,0) + I)] dg ( ), (13) where LT (, λ, ) i the Lagrangian function of the original roblem. The otimal two-art tariff mut atify the firt order condition, a hown below. LT (,, λ) = [1 '( (, ( )) ( ))] ( ) 0 T λu b q + m dg =, and (14) LT (,, λ) q ( ) = [ q ( ) + ( c) ] dg ( ) q () m () λ u'( b(, q( )) + m( ))[ bq (, q( )) + ] dg( ) q () = [() q + ( c) ] dg () λ u'((, bq ()) + m ())() qdg () q () = q ()(1 λu'((, bq ()) + m ())) dg () + ( c) dg () = 0. (15) Note that 1 λu'( b(, q( )) + m( )) i trictly increaing with reect to becaue λ u'( b(, q( )) + m( )) = λu"((, b q()) + m()) b (, q()) < 0. (16) 0 Hence equation (14) imlie that there exit (, ) uch that 1 '( (, ( )) + ( )) < 0, λu b q m 0 < 1 '( (, ( )) + ( )) = 0, λ u b q m 0 = Meanwhile, from bq (, q( )) = of (10), we have and 1 λ u'( b(, q( )) + m( )) 0 > 0. (17) < q () 1 = < 0 b (, q()) qq and q () bq (, q()) = > 0. (18) b (, q()) qq Hence the quantity demanded increae with the tate of nature. We have q ()[1 λ u'((, bq ()) + m ())] dg () 11

13 = 0 q ()[1 λ u'((, bq ()) + m ())] dg () + q ()[1 λ v'((, bq ()) + m ())] dg () > q ( ) [1 λu'( bq (, ( )) + m ( ))] dg ( ) + q ( ) [1 λu'( bq (, ( )) + m ( ))] dg ( ) 0 = q ( ) [1 λu'( bq (, ( )) + m ( ))] dg ( ) = 0. (19) The inequality i obtained by uing (17) and (18). From (19) and (15), we have q () ( c) dg( ) < 0. Since q () < 0, we have * > c. Q.E.D. Prooition 2 ugget that when conumer tate are uncertain, the uage rate in the ervice rovider otimal two-art tariff i greater than the rovider marginal roduction cot. Comared to a lan with a uage rate equaling to the marginal cot, the otimal lan reduce the rik faced by the conumer. The rovider balance between inuring the conumer and avoiding the deadweight lo caued by a high uage rice. Note that the ervice rovider doe not take much rik under the otimal tariff becaue the individual uncertaintie are indeendent and thu the aggregate tate i almot certain. The conumer obtain zero exected urlue and all the benefit from the trade goe to the monoolitic ervice rovider. The formulation of the rovider rofit-maximization roblem alo ugget that with the otimal tariff, the market outcome i allocative efficient ex ante, though it i normally inefficient ex ot. The model can alo be ued to dicu the cae where conumer are not rik-avere. From the roof of Prooition 2, we can eaily obtain following reult. Corollary 1: If the conumer are rik-neutral, the ervice rovider otimal two-art tariff ( T, ) entail * = c. 12

14 Proof: If the conumer are rik-neutral, we have u'( x ) being a contant for all x R. From (14) we have 1 λu'( b(, q( )) + m( )) = 0. Inerting it to (15), we immediately have * = c. Q.E.D. Moreover, one can how that if the conumer are rik-eeking, the otimal uage rate hould be le than the marginal cot. The roof of the concluion i arallel to that of Prooition 2. Rik-eeking conumer refer a relatively lower uage rate becaue it increae the variation in their ex ot atifaction level. The reult alo ugget that whether the uage rate i greater than the rovider marginal cot doe not deend on the houehold roduction function. To give a brief illutration of the key idea of the model, I will dicu a tylized examle a follow. Examle: A monooly telehone comany ha zero marginal cot. Conumer are homogenou and trictly rik-avere. In each eriod, a conumer need to make either one call or none, deending on the tate of nature. Each cae occur with robability of 0.5. The value of a call i $10 to the conumer. It i imortant to realize that the market i not like a health care or auto reair market, in which conumer benefit from the ervice only after they uffer a lo. In the current tory, conumer need to ue the ervice jut becaue they find themelve in certain ituation. If the monooly comany ue linear ricing, it aarently hould charge $10 for each call. The conumer take no rik ince they obtain zero ex ot benefit from the ervice in either tate of nature. The comany earn (gro) exected rofit of $5 from each conumer. 13

15 On the other hand, if the comany offer a flat fee of T>0 dollar er eriod, the conumer ha equal chance to loe T dollar (no call) and gain 10 T dollar (one call) of benefit in each eriod. The uncertain ex ot benefit ugget that the conumer take ome rik in thi ervice lan. Since the exected ex ot benefit i 1 1 T + (10 T ) = 5 T dollar, the 2 2 otimal flat fee T mut be trictly le than 5 otherwie the rik-avere conumer would not take the lan. Hence the flat fee lan generate le exected rofit than the linear rice lan, which mean a flat fee lan cannot be otimal although the comany ha zero marginal cot. More generally, uoe the comany offer a two-art tariff (T, ). A conumer who ha igned u the lan obtain ex ot benefit of either T dollar (no call) or 10 T dollar (one call), each with robability of 0.5. The conumer exected ex ot benefit i 1 1 T + (10 T ) = 5 T dollar. Becaue the conumer are rik-avere, they are willing to ay le than 5 T dollar for the ervice lan a long a T 10 T. 2 To make the lan attractive to the conumer, the comany two-art tariff mut atify 5 T 0 (or T + 5), with the equality holding if and only if T = 10 T, i.e., 2 2 = 10. Note that with uage rice = 10, the lum um fee T mut be zero. Since the comany exected rofit from each conumer i exactly T + dollar, a two-art tariff i 2 otimal if and only if it degenerate to a linear rice cheme with = Concluding remark Thi aer conider a ervice market where a rovider ell to homogenou rik-avere conumer through a two-art tariff. The conumer tate toward the ervice deend on an exogenou ituational variable. It i hown that the ervice rovider rofit-maximizing 14

16 two-art tariff entail a uage rate that i greater than the rovider marginal roduction cot. Comared to a two-art tariff with a uage rate equaling to the marginal cot, the otimal ricing cheme allow the rik-avere conumer to take le rik. The ervice rovider balance between inuring the conumer and avoiding the deadweight lo when it chooe an otimal uage rate. The finding may hel to undertand why it i common for uage rate to be greater than firm marginal cot in ervice indutrie. It may a well hel to jutify the linear rice oberved in ublic utilitie, video rental, toll road, banking, and other indutrie where conumer do not ubcribe the ervice in advance. Indeed, linear ricing in indutrie with increaing return to cale may not be a inefficient a reviou theorie recommend when conumer have uncertain tate. Reference Belk, Ruell W., Situational Variable and Conumer Behavior, Journal of Conumer Reearch, Vol. 2, No. 3, Chang, Myong-Hun, Flexible Manufacturing, Uncertain Conumer Tate, and Strategic Entry Deterrence, Journal of Indutrial Economic, Vol. 41, No. 1, Ma-Colell, Andreu, Michael D. Whinton and Jerry R. Green, Microeconomic Theory, Oxford. McFadden, Daniel, Economic Choice, American Economic Review, Vol. 91, No. 3, Oi, Walter Y., A Dineyland Dilemma: Two-art Tariff for a Mickey Moue 15

17 monooly, Quarterly Journal of Economic, Vol. 85, No. 1, Walh, John W., Flexibility in Conumer Purchaing for Uncertain Future Tate, Marketing Science, Vol. 14, No. 2,