Time Inconsistency and Naivety of Consumer within Hotelling Competition

Transcription

1 Time Inconsisency and Naivey of Consumer wihin Hoelling Compeiion Léa Bousque Paris School of Economics March 1, 14 Absrac In his paper, we look a firms compeing à la Hoelling wih ime inconsisen consumers, boh sophisicaed and naives. The previous lieraure claims ha a monopoly offering a wo-par ariff conrac o such consumers do no exploi he ime inconsisency bias bu he naivey bias. Moreover, wih perfec compeiion, firms do no succeed anymore o exploi he naivey bias even if an allocaion inefficiency says. We adap DellaVigna and Malmendier (4) framework o sudy he impac of an imperfec compeiion beween firms offering a wo-par ariff conrac and he inroducion of ranspor coss on he exploiaion by he firms of his wo bias. These coss can be inroduced a he conrac period or a he consumpion period. A he conrac period, we observe ha he compeiion even imperfec erases he exploiaion of he naivey bias bu he inroducion of he ranspor coss conducs o a disorion beween consisen and ime inconsisen consumers. Indeed, his las consumer gives more weigh o he ranspor coss which leads o a higher marke power for firms. A he consumpion period, ranspor coss generae an heerogeneiy beween consumers, firms are no longer able o provide a perfec commimen o sophisicaed consumers. Consequenly, allocaion wih boh sophisicaed and naive are no efficien. Keywords: Two-par ariff conrac; Hoelling compeiion; ime inconsisency; quasihyperbolic discouning; naivey. JEL Classificaion Numbers: D43, L13 Campus Jourdan, 48 boulevard Jourdan 7514 Paris bousquelea@gmail.com 1

2 Inroducion An imporan and growing empirical and experimenal lieraure highlighs several biases of raionaliy in consumer decision. These resuls have an imporan impac on he ineracion beween firms and hose consumers. Especially, i is ineresing o wonder if firms are exploiing his bias or if hese bias have an impac on he social welfare; if he marke sill allows efficien allocaions. In his aricle, we focus on consumers wih ime inconsisency problem and naivey bias. Various consumpion goods involve coss and benefis delayed in ime. I is he case wih leisure goods (obacco, alcohol, gambling... ) or invesmen goods (gym, culural goods, saving... ) and facing he opporuniy o consume hese goods, consumer can have self-conrol problems due o a presen bias (ie a very srong preference for he presen). Tha behavior can lead consumer o over consume leisure goods or under consume invesmen bias. Indeed, here is a srong empaion o pospone or even never realize a aciviy wih immediae cos if one has a srong preference for he presen and he benefis are fuure. However, laer, one can regre ha choice because i is no opimal for her long erm uiliy. On he conrary, for he same reason i is easy o be emped by an aciviy or a good wih an immediae benefis even if one knows ha i will definiely cos her he nex day. Alhough consumer can have a naivey problem ie hinks she will resis o he empaion of consumpion when she will face he choice o consume (eg saying I am going o a chocolae salon bu I will no ea one ) or on he conrary make some resoluions abou her consumpion and never respec hem (eg saying This year, I will go o he gym club hree imes a week ). A way o ake ino accoun hese biases is o adap he uiliy funcion. There are wo main mehods o rea ime inconsisency in he lieraure. The firs one is he muli selves model (Sroz 1956, Peleg and Yaari 1973). The ime inconsisen agen changes her uiliy beween oday and omorrow. Today she has he uiliy u whereas omorrow she will have he uiliy v 1. The second way o include ime inconsisency bias is wih quasi-hyperbolic preferences (Phelps and Pollak 1968, Laibson 1997, Gul and Pesendorfer 1). From ha (β, δ) model, he agen has wo discoun raes, he radiional long-erm one, δ and he shor-erm one, β, which represens he presen bias. The uiliy in a period is, U = u + n βδ u =1 A ime inconsisen consumer can be disinguished beween a sophisicaed one and a naive one (O Donoghue and Rabin 1). The sophisicaed is perfecly aware of her ime consisen problem whereas he naive hinks wrongly she is ime consisen. In he wo previous models, he naive believes her shor erm discoun rae β is equal o 1 (like he consisen) or she can be parially naive and having some belief on her shor erm discoun rae ˆβ wih β < ˆβ < 1. In he firs model, he naive believes his uiliy will say u insead of v in he nex period or if she is only parially naive, believes her uiliy may be u wih a 1 The consisen agen does no change and keeps he uiliy u in he fuure.

3 probabiliy θ or v wih a probabiliy (1 θ). Several empirical or experimenal sudies have prove he exisence of ime inconsisen and naivey bias in differen indusries. Many experimenal sudies show ha he annual discoun rae decreases wih ime wih moneary rewards and explain i hanks o quasi-hyperbolic preferences (Thaler 1981, Benzion e al. 1989, Chapman 1996, Andersen e al. 8). Ohers lab or field experimens find evidence of presen bias (Augenblick e al. 13) and naivey (Acland and Levy 13, Ariely and Werenbroch ) hanks o real effor coss. Empirical sudies based on several indusries end o poin ou ha consumers have ime inconsisency and naivey problem: phone marke (Miravee 3, Grubb 9), gym aendance (DellaVigna and Malmendier 6), credi marke (Heidhues and Köszegi 1). Thus, i is ineresing o sudy he ineracion beween hese consumers and firms. DellaVigna and Malmendier (4) use he Phelps and Pollacks design ie a (β, δ) model o sudy he ineracion beween a monopoly and such consumers wih a wo-par ariff conrac. Their main resuls are monopoly explois he naivey bias of consumer bu no he ime inconsisency problem. Indeed, she offers a sophisicaed a perfec commimen device and hanks o his, she is able o consume as a consisen one. On he conrary, he conrac offered o naive consumer leads o a negaive surplus for consumer and a lower welfare due o inefficien allocaions. These resuls are robus wih perfec compeiion. Eliaz and Spiegler generalize his analysis wih Sroz framework, unspecified non-linear conrac and heerogeneiy on he naivey degree (Eliaz and Spiegler 6, Eliaz and Spiegler 8). In his paper, we will focus on ime inconsisen consumers (boh sophisicaed and naive) who consume an invesmen good. Furhermore, here are wo firms which offer a wo-par ariff conrac and compee à la Hoelling. Indeed, wih a monopoly, here is no exploiaion of he ime inconsisency problem if he consumer is sophisicaed only of he naivey bias and wih perfec compeiion, he naivey is no exploied anymore bu he inefficiency problem says. Thus, we wan o sudy he impac of he inroducion of ranspor cos firs a he conracing period and hen a he consumpion period on he firm profi. We wonder if he ime inconsisency bias sill has no impac on he firm profi and if an imperfec compeiion erases he exploiaion of naivey problem. We use a model very similar o DellaVigna and Malmendier one. There is hree periods. In he firs period, firms propose a wo-par ariff conrac and consumers accep or no. A he second period, agens choose wheher o consume or no. If hey do, hey suppor he coss of he consumpion and have he benefis a he las period. Adding some ranspor coss inroduces an heerogeneiy in ase of consumers. We can assume ha hese coss will be suppored when consumers accep he conrac a he firs period or suppor i if hey consume a he second period. Wihin his framework, we find ha he inroducion of he ranspor cos a he conrac allows firms o exploi he ime inconsisency problem. Indeed, ranspor coss represen Grubb (9) found almos he same resuls wih overconfidence of consumers. 3

4 a marke power for he firm and since we assume firms raional, here is a difference in discouning hese ranspor coss beween consumers and firms. Time inconsisen consumers overweigh hese coss and give o firms a higher marke power. Also, he inroducion of compeiion, even imperfec, prevens firms o exploi he naivey bias. However, he allocaion inefficiency problem says. The inroducion of ranspor coss a he consumpion period has more impac. I induces an heerogeneiy in he consumpion probabiliy. The immediae resul which follows is he incapaciy of he firms o provide a perfec commimen for sophisicaed consumers. Then, i implies a non opimal welfare wih sophisicaed compared o consisen ones. Moreover, as he ranspor cos play a role in he consumpion probabiliy, he profi made facing a sophisicaed and a naive will no be he same. In secion 1, we presen he main resuls from DM wih a monopoly and hese resuls exended o perfec compeiion. In secion, we inroduce a compeiion la Hoelling wih ranspor coss a he conrac period. Then, we look he impac of he inroducion of ranspor coss a he consumpion period (secion 3) and finally we conclude. 1 Framework wih Monopoly and Perfec Compeiion This secion sands o presen he principal resuls for a monopoly offering a wo-par ariff conrac a ime-inconsisen consumers. This model is a simplified version of DellaVigna and Malmendier s one (DM herefore) (DellaVigna and Malmendier 6). Resuls can be easily exrapolaed o perfec compeiion (DellaVigna and Malmendier 6, Heidhues and Köszegi 1, Huck and Zhou 11). 1.1 The seing Timing and payoffs The monopoly offers a wo-par ariff conrac (L,p) in period (P ) a a represenaive consumer. This agen can accep or rejecs he conrac. If she acceps, she will pay he lump sum L in period 1 (P 1 ). In P 1, consumer faces her consumpion cos c and decide or no o consume given p deermined by he conrac she acceped in P. If she consumes, she suppors c and p and will have he consumpion benefi b a period (P ). If she does no consume, she has no more payoff. Figure 1 summarizes decisions and payoffs. 4

5 Period Period 1 Period Payoffs: -c-p b Monopoly offers (L,p) conrac Consumer acceps Payoff: Consumer rejecs Payoffs: -L Agen consumes Agen does no consume Payoffs: Figure 1: Timing of he model The consumer suppors he cos c when she consumes. This cos is unknown o boh firm and consumer when conracing a P, only he disribuion of c is common knowledge. We denoe by F is cumulaive disribuion funcion and by f is probabiliy densiy funcion 3. Consumer There is a unique represenaive consumer. This consumer can experimen boh ime inconsisency and naivey. We model, following DM, his self-conrol problem by using quasi-hyperbolic discouning preferences. Wih ime inconsisency problem, her uiliy in n P 1 is U 1 = u 1 + βδ u. We add a shor erm discoun facor β beween he presen and = he period righ afer oday o he radiional long erm discoun facor δ 4. A ime inconsisen person can be sophisicaed (fully aware abou her ime inconsisency problem) or (parially) naive. If she is naive, she has a wrong belief abou her shor erm discoun rae. She hinks she will be more paien ha she will acually be. In P, she hinks n her uiliy in P 1 will be U 1 = u 1 + ˆβδ u wih ˆβ [β, 1. When she is sophisicaed, she = has he righ belief abou her impaience ( ˆβ = β). We sudy he case wih a naive consumer bu i is easy o replace ˆβ by β o have he behavior for he sophisicaed consumer and boh ˆβ and β by one when he consumer is consisen. In his framework, he consumer in P acceps he conrac if her expeced uiliy is superior o her reserved uiliy u. This expeced uiliy depends on he lump-sum L bu also on he uiliy she expecs o have wih he consumpion. She expecs o consume in P 1 for all c < ˆβδb p, ie wih a probabiliy F ( ˆβδ p). We can already noe ha her expeced probabiliy is superior o her real probabiliy o consume. Indeed, she will acually consume if c < βδb p. The difference F ( ˆβδ p) F (βδ p) is a 3 To garanee he exisence of he conracs, DM used Assumpion ABP: There exiss a pair (M, z) R such as f(y ) Mf(y ) for all y and y, wih z < y < y and y.y >. 4 When consumer is consisen, hen β = 1 5

6 measuremen of he naivey degree. The expeced uiliy of he consumpion depends also on c and i is equal o δb p c for all c. A he end, he agen expeced uiliy in P is: βδ[ L + ˆβδb p (δb p c)df (c) (1) Here, his expeced uiliy is discouned wih he real shor-erm discoun rae β and no her belief on i since i is he uiliy she is facing and no her anicipaed one. Only her consumpion probabiliy is anicipaed and depends on her belief on he shor-erm discoun rae. Monopoly The monopoly is considered being ime consisen. She does no add he shor urn discoun rae by discouning beween oday and omorrow. She ries o maximize her profi under he paricipaion consrain of he consumer. We consider she is facing one represenaive consumer ype. Her profi includes he lump-sum if he consumer acceps he conrac and he consumpion price if she consumes wih he (real) probabiliy F (βδb p). Wihou los of generaliy, we consider he monopoly have no cos. Thus, he monopoly program can be wrien as, max δ[l + pf (βδb p) L,p s.. βδ[ L + ˆβδb p (δb p c)df (c) βδu 1. Opimal pricing I is easy o see ha he paricipaion consrain is binding since oherwise he monopoly can arise his profi by increasing he lump-sum. Then we obain he consumpion price hanks o he firs order condiion of monopoly program facing naive consumer. These condiions give us he following opimal prices, L = ˆβδb p (δb p c)df (c) u p = (1 ˆβ)δb f( ˆβδb p ) f(βδb p ) F ( ˆβδb p ) F (βδb p ) f(βδb p ) DM proves he exisence of such conrac hanks o he ABP assumpion. From hese previous prices, we can deduce properies on prices for each ype of consumer. () 6

7 Proposiion a (DM prices properies) 1. For consisen consumer, he wo-par ariff is se radiionally. The consumpion price equals o marginal cos (here ) and he lump-sum allows o cach all he consumer surplus.. For he sophisicaed consumer, he consumpion price is below he marginal cos and he lump-sum is higher han for he consisen o compensae. I implies a perfec commimen for he sophisicaed; he consumpion probabiliy is he same han he consisen. 3. For he naive consumer, he consumpion price is also se below he marginal price bu he lump-sum is even higher since she overesimaes her consumpion. Proof. See DM (6) For a consisen consumer, since (1 ˆβ) = and F ( ˆβδb p ) F (βδb p ) =, he monopoly ses he marginal price a he marginal cos (here ) and he lump-sum is equal o L = δb (δb c)df (c) u. Facing he sophisicaed consumer, monopoly offers p = (1 ˆβ)δb since F ( ˆβδb p ) F (βδb p ) = and f( ˆβδb p ) = 1. Thus, he consumpion price allows a perfec commimen f(βδb p ) o he sophisicaed since F (βδb p ) = F (δb) ie he same probabiliy o consume han for he consisen. Moreover, L can be rewrien as L = δb (δb c)df (c) p F (δb) u so he monopoly is recovering he loss induced by a below marginal cos price in he lump-sum. For a naive consumer, he monopoly has wo reasons o offer a low consumpion price. The firs one is he same ha for he sophisicaed. The second one is explicaed by consumer overesimaing her consumpion probabiliy. Since, she believes she will consume more ofen han she would acually do, he monopoly can arise he lump-sum by recovering he price weighed by he belief on he consumpion probabiliy (which is higher han he real one) and decrease he marginal price o arac naive people. The naive consumer looses because of lower benefis she will acually do wih her (lower) real consumpion probabiliy. The reurn of invesmen represened by he consumpion is no enough o compensae her higher lumpsum. 1.3 Profi and welfare Profi The monopoly makes profi hanks o he opimal lump-sum which binds he paricipaion consrain and hanks o he opimal consumpion price weighed by he real consumpion probabiliy. π = δ[l + p F (βδb p ). Subsiuing L by L, [ βδb p π = δ (δb c)df (c) u + ˆβδb p βδb p (δb p c)f (c) (3) The firs par of equaion (3) corresponds o he social surplus generaed by he ineracion beween he monopoly and he consumer. This par represens he expeced uiliy of he consumpion wihou he consumpion price. Indeed, his price is payed by he consumer bu received by he monopoly so a he end, he has no consequence on he social surplus. The second par of (3) corresponds o he addiional ficive surplus he naive consumer 7

8 beliefs she will make wih her consumpion and he monopoly manages o capure. Since, his consumpion does no happen and so he ineracion, he consumpion price is aken ino accoun in his surplus. This ficive surplus does no exis when he monopoly faces consisen or sophisicaed consumer. Consumer surplus and social welfare As we saw wih he expression (1), he uiliy of he consumer in P facing he opimal conrac proposed by he monopoly is, U = βδ[ L + ˆβδb p (δb p c)df (c) Since L = ˆβδb p (δb p c)df (c) u, for all he consumer U =. Wih his resul, he social welfare W arises wih he profi. In ha case, one conclusion would be ha he social welfare arises wih he naivey degree. However, his measure does no consider he fac ha he consumer has wrong belief on her fuure shor erm discoun rae. Also, we can wonder if here is a loss of surplus beween he sophisicaed consumer who uses a shor-erm discoun rae and a consisen one. One manner o deal wih hese problem is o consider for all consumers heir long erm uiliy (O Donoghue and Rabin 1). In his case, he shor-erm discoun rae is no aken ino accoun anymore since we suppose ha in he long erm uiliy funcion, here is no more presen bias so β = 1 as he belief on he shor-erm discoun rae. We are only ineresed in he real consumer surplus and no her anicipaed one. Wih his assumpion, he long-erm consumer surplus is, CS = δ[ L + βδb p Subsiuing L and rewrien, equaion (7) gives us, [ CS i = δ ˆβδb p βδb p (δb p c)df (c) (4) (δb c p )df (c) + u Or, [ βδb p CS i = δ (δb c)df (c) π i Wih i, he consumer ype and π i, he monopoly profi facing he ype i. The social surplus is he profi added o he consumer surplus, [ βδb p W i = CS i + π i = δ (δb c)df (c) (5) 8

9 Proposiion b (DM profi and welfare properies) 1. For he consisen and he sophisicaed consumer, profi, consumer surplus and welfare are he same because of he perfec commimen. Time inconsisency does no change he welfare and is sharing.. For naive consumer, he welfare is lower because of he non-opimaliy of p and is sharing beween he monopoly and he consumer is differen. Consumer surplus is negaive due o an overesimaion of he consumpion. This ficive surplus leads o addiional profi for he monopoly. Proof. See DM (6) 1.4 Perfec compeiion Thanks o he resuls wih he monopoly cases and he reservaion uiliy, we can exrapolae o perfec compeiion case. Proposiion C (Perfec compeiion) 1. The consumpion price is he same as monopoly case for all he consumer ypes. Perfec compeiion decreases he lump-sum o se he profi equals o. The lump-sum is equal o for consisen consumer and superior o for he inconsisen one.. Perfec compeiion does no change he social welfare bu is sharing. 3. For naive consumer, here remains an inefficiency effec of he naivey bu is impac is lower. The loss in consumer surplus due o monopoly power becomes larger as naivey increases (for fixed β). The compeiion has an impac hrough he reservaion uiliy u. In he expression () of he opimal prices, in one hand he marginal price does no depend on u, i resuls ha he consumpion price is he same in boh perfec compeiion case and monopoly one. In he oher hand, u decreases he lump-sum L. u is deermined as o equae profis o in he case of perfec compeiion. The social welfare, in expression (5) does no depend on u ie i is no affeced by he monopoly power. However, is sharing will change since boh profi and consumer surplus depends on u. Moreover, for naive consumer, he degree of u does no affec he efficiency problem since he consumpion probabiliy says he same. The welfare is sill affeced by his problem. Moreover, he naivey degree arises he monopoly power on he consumer surplus. Her losses arises wih he naivey degree. We sudied he case wih he monopoly case and he perfec compeiion case. However, i is ineresing o sudy he imperfec compeiion case, especially Hoelling case. Indeed, he ranspor cos, wheher supored a he momen consumers choose heir conrac or a he momen hey have o choose o consume, can affec he welfare and is sharing differenly. We sudy his impac in he nex wo secions. 9

10 Hoelling Wih Transpor cos a he Conrac Period In his secion, we sudy he Hoelling compeiion case wih ranspor coss a he conrac period. Firms may offer producs or services wih differen characerisics or qualiies. The ranspor cos represens he difference beween he consumer mos preferable conrac and he conracs offered by firms..1 Framework In order o sudy he impac of he ranspor coss on pricing and welfare, we keep a framework closed o he monopoly case of DM. The iming of choices and payoffs say he same, he conrac is sill a wo-par ariff and he cos and benefis of he consumpion have he same properies. However, now, he consumer has o suppor a cos a he momen she chooses her conrac. Moreover, we have a represenaive segmen of demand [,1 in which consumers are uniformly disribued. We have wo symeric firms (A, B) locaed on each par of he segmen: A in and B in 1. The consumer suppors a ranspor cos depending on her siuaion on he segmen demand ie her disance o he firms. We suppose his cos is linear, (x) = x wih he marginal cos and x he disance beween he consumer and he firm. Consumers and demand Consumer chooses o consume a firm 1 or firm and her uiliy depends on he conrac offered by firm i. The uiliy of he consumer locaed in x and who has chosen he firm A is U A x = βδ[ L A + And her uiliy if she chooses he firm B is U B (1 x) = βδ[ L B + ˆβδb pa ˆβδb pb (δb p A c)df (c) x (δb p B c)df (c) (1 x) These uiliies are he same as DM case minus he cos of localizaion for he consumer. The demand o he firm i is deermined by he marginal consumer x who is indifferen beween consuming a he firm A and a he firm B ie U A x = U B (1 x). I follows D A = x = U A U B + 1 D B =1 x = U B U A + 1 1

11 Firms Firms are in sraegic ineracion and maximize heir profi regarding he consumer uiliy bu also he oher firm behavior hrough he demand funcion. The firm i program is max π i = δ[l i + p i F (βδb p i )D i (6) L i,p i Wih D i = U i U i + 1 or, D i = βδ[l i L i + ˆβδb p i (δb p i c)df (c) ˆβδb p i (δb p i c)df (c) + 1 (7). Opimal conracs The firs order condiion o he program (6) considering he demand (7) gives he opimal conrac offered by i p i = (1 ˆβ)δb f( ˆβδb p i ) f(βδb p i ) F ( ˆβδb p i ) F (βδb p i ) f(βδb p i ) Or, L i = βδ + F (δβb p i ) [ (1 ˆβ)δb f( ˆβδb p i ) f(βδb p i ) F ( ˆβδb p i ) F (βδb p i ) f(βδb p i ) p i = (1 ˆβ)δb f( ˆβδb p i ) f(βδb p i ) F ( ˆβδb p i ) F (βδb p i ) f(βδb p i ) L i = βδ F (δβb p i )p As firms are symmeric, he same conrac is offered by boh firm A and B. Moreover, he marginal consumer is locaed in he middle of he segmen demand, x = 1 x = D A = D B = 1. By symmery, we sudy only he firm i. Proposiion 1 (Pricing wih ranspor cos a conrac period) 1. The consumpion price is he same han in DM case wih he same properies. This price is equal o zero wih consisen consumer and negaive (below marginal cos) for sophisicaed and naive one.. The lump-sum is se such as i allows o recover he loss due o a negaive marginal price and firms can make some profi hanks o her marke power hrough he ranspor coss. 3. The marke power of he firm arises wih he degree of ime inconsisency of he consumer. The marginal price does no depend on and has he same properies han in DM case. In paricular, he proof of he exisence of he conrac is he same han in DM case. Alhough his price sill allows perfec commimen o sophisicaed consumer, ie he consumpion probabiliy is he same for sophisicaed consumer han for consisen one. However, ranspor coss have an impac on he lump-sum. The lump-sum is composed of 11 (8)

12 discouned ranspor coss and he marginal price weighed by he consumpion probabiliy. The ranspor coss reflec he firms power. This resuls is very closed o Armsrong and Vickers one sudying wo-par ariffs in duopoly compeiion 5. However, since he marke power is capured a he consumpion period and he ranspor coss suppored a he conrac period (a period before), hese coss is evaluaed by consumers wih ime-inconsisency, ie divided by δ and β. I follows ha he marke power is more imporan for he firms when he degree of ime inconsisency arises (when β decreases). Neverheless, since hese coss are already payed a he consumpion period, he marke power does no depend on he naivey. Furhermore, firms recover only he consumpion price wih he real consumpion price. In conras o DM model, hey are no longer able o capure he consumer ficive surplus..3 Profi and welfare Profi The firm i makes profi hanks o he lump-sum and he marginal price weighed by he real consumpion probabiliy for he served porion of he segmen demand. This profi a equilibrium is, π i = δ[l i + p i F (βδb p i )D i Firms are symmeric, i follows ha demand is equal o 1 for each firm. Moreover, we see wih expression (8) ha he lump-sum is a linear funcion of he marginal price. I follows, πi = β Again, we find a resul very closed o Armsrong and Vickers (1). They claimed he sum of all he firms profis in he indusry is equal o. The difference here is he weighed by he shor-run discoun rae. Indeed, he firms and an inconsisen consumer do no evaluae he ranspor cos wih he same way. Firms use only δ whereas consumers use βδ. Consumer pu more weigh on his ranspor cos han firms and in his way give more marke power o he firms. As we claim in proposiion 1 (4), firms can no longer exploi he consumer naivey wih her conrac. Consumer surplus and social welfare Since he uiliy of he consumer depends o her ranspor cos, each consumer has a differen uiliy. Thus we calculae he oal surplus 6, CS = ˆx (sc 1 x)dx + 1 ˆx (sc (1 x))dx (9) Wih x = 1/, he marginal consumer and sc i = δ[ L i + βδb p i (δb p i c)df (c) he surplus wihou ranspor coss of all he consumer who urn o he firm 1. Since firms are 5 (Armsrong and Vickers 1, Armsrong and Vickers 1) 6 Here, since he consumers are uniformly disribued on he demand segmen [,1, he oal surplus is equal o he average surplus. 1

13 symmeric, we have sc 1 = sc. We remind ha we ake ino accoun he long-erm surplus of he consumer, ie wih he real probabiliy of consumpion and discouned only wih he long-erm discoun rae. By replacing L and rewrie (9), Or, I follows he oal welfare, [ βδb p i CS = δ [ βδb p i CS = δ [ βδb p i W = CS i + π i = δ (4 + β) (δb c)df (c) 4β (δb c)df (c) 4 π i (δb c)df (c) 4 This oal welfare is he same as DM framework minus he impac of he ranspor coss. Since, hese coss do no depend on he consumer ime inconsisency or naivey and marginal prices are he same, we have he same resuls on he welfare han for DM. The ineresing hing is he sharing of his welfare. Proposiion (Profi and welfare wih ranspor cos a conrac period) 1. Time inconsisency increases he profi and decreases he consumer surplus bu no change he oal welfare.. Naivey has no impac on he profi bu since he marginal price is no opimal he degree of naivey decreases he consumer surplus and so he social welfare. In his secion, we saw ha he inroducion of ranspor coss suppored by consumers a he conrac period leads o a disorion in he firm profi beween he consisen and he sophisicaed consumer. Indeed, he ime inconsisency and he difference beween he evaluaion of ranspor cos beween firms and consumers lead o a greaer power marke for he firm. The inroducion of compeiion even imperfec removes he exploiaion of naivey by he firm bu no he inefficiency problem since he consumpion price says non opimal. 3 Hoelling Wih Transpor cos a he Consumpion Period 3.1 Framework In his secion, we sudy he case in which he ranspor coss is suppored by consumers when hey consume. In his framework, we assume firms even compeing, may exploi he naivey bias since he ranspor cos will impac he probabiliy consumpion. 13

14 Indeed, consumers have o choose one of he firms conrac in P and she anicipaes she consumes only if δ ˆβb > c + p i + x ie δ ˆβb p i x > c, i follows he believed consumpion probabiliy F (δ ˆβb p i x). The real one is F (δβb p i x). Thus each consumer has a differen consumpion probabiliy. In his secion, we make more resricive assumpions on he effor cos densiy funcion. Assumpion f(c) is posiive, decreasing in c and deermined on [, +. A way o inerpre his assumpion is saying ha each consumer has a differen ase ie cos for he invesmen good consumpion capured by he ranspor cos bu some hazards can affec her cos when hey consume. 7 The uiliy of he consumer locaed in x is U x = δβ[ L i + δ ˆβb pi x (δb p i x c)df (c) The ranspor cos play a role boh in he consumpion probabiliy and in he evaluaion of he consumpion benefis. The marginal consumer x who deermine he demand o he firm is characerized by U i = U i ie δβ[ L i + Or, δ ˆβb pi x (δb p i x c)df (c) = δβ[ L i + δ ˆβb p i (1 x) (δb p i (1 x) c)df (c) L i L i = δ ˆβb pi x (δb p i x c)df (c) δ ˆβb p i (1 x) (δb p i (1 x) c)df (c) (1) We can no deermine x bu we will use he expression (1) o deermine he opimal conrac and we know ha x = 1 since firms are symmeric. Facing a consumer x, he firm i realize he profi π i = δ[l i +p i F (δβb p i x). However, since he consumpion probabiliy is differen for each consumer due o he differen localizaion of consumer, he oal profi is Wih D i = x π i = δ[l i + Di (p i F (δβb p i x))dxd i (11) 7 We can ake he example of spor. Suppose a consumer wan o make some exercises, he can choose beween firm A which is a rugby club and B which is an indoor soccer club. She suppors a ranspor cos o consume rugby or soccer depending on her ase. This cos is common knowledge and we suppose ha ase do no vary wih ime and is differen for each consumer. However, facing he decision wheher o consume, she can face some exra unusual coss: if she is sick, if i is raining, if she has an imporan work o do for he nex day... 14

15 To simplify fuure equaions, we denoe : X i = δβb p i x ˆXi = δ ˆβb p i x Xĩ = δβb p i x ˆXĩ = δ ˆβb p i x Thus, he firm i program is Wih x such as, max π i = δ[l i + p i (F (X i ))dx x L i,p i 3. Opimal conracs L i L i = δb(1 ˆβ)(F ( ˆXĩ) F ( ˆX ( ĩ) )) + ˆXĩ ˆX ( ĩ) F (c)dc Using he parial derivaives of he demand wih respec o he wo prices and he symmery assumpion, we obain he parial derivaives of he profi wih respec o he prices 8 Π [ i x = δ [L i + p i Π i = δ (F (δβb p i) F (Xĩ)) x F (X i )dx + x[1 + p i F (Xĩ) [ p i f( δb(1 ˆβ) ˆXĩ) F (δβb p i ) F (Xĩ) + F (X i)dx F ( ˆXĩ) F (δβb p i ) F (Xĩ) The firs order condiion gives us he opimal prices [ p i = δb(1 ˆβ) f( ˆXĩ) F (δβb p i ) F (Xĩ) + F (X i)dx F ( ˆXĩ) F (δβb p i ) F (Xĩ) ( L i = p i F (X i )dx + 1 ) [ F (X ĩ ) + δb(1 ˆβ)f( ˆXĩ) + F ( ˆXĩ) Proposiion 3 (Exisence) There exis a conrac (L, p ) which is a profi maximizing conrac. Proof. The second order condiions are very difficul o verify. To ensure ha his conrac is a maximum, we can insead confine p in a defined inerval and show ha if p is inferior han his inerval, he profi is increasing in p and if i is superior, he profi is decreasing in p. Thus by coninuiy of he profi funcion, here exiss a price p which represens a local maximum beween he boundary of he inerval. This inerval [M, M is such ha, 1 + δb(1 ˆβ)f() M = M f() M = 1 Assumpion There exis M such ha for all y < y, f(y ) < Mf(y) wih M > 1. 8 see calculaion in annex A1 A A3 15

16 The calculaion of his inerval is in annex. In his framework, since he consumpion probabiliy is no he same for all he consumer served by he firm, i is difficul for he firm o exer her marke power. Also, he marginal price akes ino accoun ha difference in he consumpion probabiliy. I induces ha he marginal price is differen han in DM framework and i is no eviden ha we will find he same properies. To beer undersand he differen effecs of ime inconsisency and naivey bias, we propose o sudy separaely he differen conracs proposed for each ype of consumer: he consisen one, he sophisicaed one and he naive one. Consisen consumer Facing consisen consumers, firms se he conrac, [ p c = F (δb p c x)dx F (δb p c /) F (δb p c) F (δb p c /) ( L c = p c F (δb p c x)dx + 1 F (δb p c ) ) + [F (δb p c ) Wih his expression, i is difficul o deermine he sign of he marginal price and o inerpre i. However, we can focus on he lump-sum. Is composiion is similar han previous cases. Indeed, he firs par of his lump-sum sands o compensae he marginal price and he second par ensures ha firms exer heir marke power. However, he probabiliy of consumpion is no he same for each consumer. The closes one does no have any ranspor cos so he has he highes consumpion probabiliy. On he oher side, he marginal consumer suppors he highes ranspor cos so he has he lower consumpion probabiliy. Wih he expression (1), we see ha he marke power of firms depending on ranspor cos is weighed by he marginal consumpion probabiliy ie firms have only a marke power for he lowes consumpion probabiliy. Noneheless, in he firs par of he lump-sum expression, he marginal price is compensaed hanks o a probabiliy included beween he marginal one and he closes one. Proof. F (δb p c x)dx > F (δb p c ) so, And, F (δb p c x)dx < F (δb p c ) so, F (δb p c x)dx + 1 F (δb p c ) > F (δb p c ) F (δb p c x)dx + 1 F (δb p c ) < F (δb p c) (1) 16

17 Sophisicaed consumer Facing sophisicaed consumers, firms se he conrac, [ p f(xĩ) s = δb(1 β) F (δβb p s) F (Xĩ) + F (X i)dx F (Xĩ) F (δβb p s) F (Xĩ) ( L s = p s F (X i )dx + 1 ) [ F (X ĩ ) + δb(1 β)f(xĩ) + F (Xĩ) For his consumer, we can reduce he exisence inerval and can discuss according o he new boundary of his inerval he sign of he marginal price. We keep he lower boundary M bu we can reduce he higher boundary M 9. We find ha, p s < 1 δb(1 β) Thanks o his boundary, we can claim ha he opimal marginal price is negaive if his boundary is negaive. One condiion is, This condiion can also be wrien as β < δb δb < δb(1 β) Even if we can no find condiions where he profi is posiive, we see ha wih a sufficienly high ime inconsisency degree or sufficienly low ranspor coss, he marginal price is sure o be negaive. Inuiively, his resul is consisen wih he previous resuls. Indeed, he ime inconsisency degree ends o decrease he marginal price o offer a he consumer a commimen whereas he ranspor coss offer o firms a marke power. The analysis of he lump sum is really closed o he consisen one. The marginal price is weighed by a probabiliy beween he closes consumer one and he marginal consumer one. The marke power is exered hanks o ranspor cos weighed by he marginal consumer probabiliy bu in his case also wih δb(1 β)f(xĩ) which represens he effec of he ime inconsisency degree on he marke power of he firm 1. 9 see calculaion in annex A5 1 I is no obvious ha ime-inconsisency will arise he marke power since i has an impac on he consumpion probabiliy bu also on he price which has an impac on he probabiliy. 17

18 Proposiion 4 Wih Hoelling compeiion, firms facing sophisicaed consumer offer a wo-par ariff conrac wih he following properies 1. The marginal price p s decreases wih β ie when he ime inconsisency degree increases 11.. Perfec commimen for sophisicaed hanks o he marginal price is no allowed anymore. Proof. To prove (1), we used he parial derivaive of profi wih respec o he marginal price. Π i = δ [ (F (δβb p i) F (Xĩ)) p i f(xĩ) δb(1 β) F (δβb p i ) F (Xĩ) + F (X i)dx F (Xĩ) F (δβb p i ) F (Xĩ) If we consider he profi as a funcion of β and p(β) and we know ha firs order condiions are verified hen ( Πi ) (β, p i (β)) + (β) β β. ( Πi ) (β, p i (β)) = Now, we know ha he second order condiions are respeced ie Π i < in p p c. Then he i sign of ( Π i β ) gives us he sign of ie he variaion of he marginal price wih respec o β β. We have, Π i = δ [ p i β [δb(f(δβb p i) f(xĩ)) (δb) (1 β)f (Xĩ) + δb (F (δβb p i) F (Xĩ)) The hird erm is posiive because F is increasing. The second erm is posiive oo because f <. For he las erm, we have f(δβb p i ) f(xĩ) < because f is decreasing so he sign of his erm depends on he sign of he marginal price. If he price is posiive, hen i is posiive. If he marginal price is negaive, we show in annex A5 ha if we assume f is linear, we find ha his las erm is also posiive. We can conclude ha >. β This resuls is again very closed o previous ones. The inuiion is he ime inconsisency reduces he consumpion probabiliy so firms have incenives o reduces he marginal prices o mainain he consumpion probabiliy. Especially wih a wo-par ariffs where she can compensae he marginal price in he lump-sum. Perfec commimen happens if he consumpion probabiliy of he sophisicaed consumer is he same han consumpion probabiliy of he consisen one ie if F (δβb p s x) = F (δb p c x) or, given he monoony of F, if δβb p s = δb p c. Thus, he condiion for he perfec commimen is, f(x i )dx = f(xĩ) This condiion is oo paricular o be validaed in he general case. We can conclude here is no perfec commimen. 11 A leas when p s is posiive or negaive and f(c) linear 18

19 The inuiion behind his comes from he fac ha each consumer has a differen consumpion probabiliy and he firm can no offer a perfec commimen o each consumer. Naive consumer Facing naive consumers, firms se he conrac, [ p i = δb(1 ˆβ) f( ˆXĩ) F (δβb p i ) F (Xĩ) + F (X i)dx F ( ˆXĩ) F (δβb p i ) F (Xĩ) ( L i = p i F (X i )dx + 1 ) [ F (X ĩ ) + δb(1 ˆβ)f( ˆXĩ) + F ( ˆXĩ) As wih consisen and sophisicaed consumer, i is hard o claim he sign of he marginal price. However, we observe ha if he lump-sum is composed by he same wo pars. The second par is differen han for he consisen and he sophisicaed. indeed, he ranspor cos is now weighed by he belief on he consumpion probabiliy and no he real one. On he conrary of he previous case wih ranspor coss a he conrac period, he naivey is now impacing he marke power of he firms. By sudying he variaion of he marginal price regarding he ime inconsisency degree and naivey degree, we can highligh some resuls. Proposiion 5 Wih Hoelling compeiion, firms facing naive consumers offer a wo-par ariff conrac wih he following properies 1. The marginal price p s increases wih ˆβ (wih consan β) ie when he naivey degree increases.. Time inconsisency has an ambiguous impac on he marginal price. If he opimal price is posiive, hen he ime inconsisency degree decreases he marginal price however if his price is negaive, he ime inconsisency degree increases his price. Proof. To prove his proposiion, we use he same echnique as previous ie o prove (1), we use, Π i ˆβ = δ [ δbf ( ˆXĩ) This equaion is he same sign as he parial derivaive of he marginal price wih respec o he belief on he shor erm discoun rae ˆβ (wih a consan β). Since, f is decreasing, i is easy o saw ha he expression (44) is posiive and conclude ha he naivey degree arises he marginal price. To prove (), we have, Π i = δ [ β δb p i [(f(δβb p i) f(xĩ)) + (F (δβb p i) F (Xĩ)) The second erm is posiive bu he firs erm depends on he sign of p n. If he marginal price is posiive, he firs erm is posiive and so he equaion. I follows ha he ime inconsisency degree decreases he marginal price. However, if p n is negaive, we show in 19

20 he annex A5 ha his expression can no be posiive, i follows ha in ha case he ime inconsisency degree decreases he marginal price. 3.3 Profi and welfare Profi A he equilibrium, by replacing he opimal lump-sum in he expression (11) of he profis, we obain, π i = δ [ p 1 i F (X ĩ ) + [δb(1 ˆβ)f( ˆXĩ) + F ( ˆXĩ) Or by replacing he opimal marginal price, π i = δ [ (δb(1 ˆβ)f( ˆXĩ) + F ( ˆXĩ))(1 + F (Xĩ) (F (δβb p i ) F (X ĩ ))) F (X ĩ ) F (X i)dx F (δβb p i ) F (X ĩ ) To improve he undersanding of he profi funcion and he impac of he localizaion, we can compare he profi firms realize wih a consumer locaed in x o he profi wih x. Wih he marginal consumer, he profi is π x = δ[l i + p i (F (δβb p i x)) ie, [ ( F (Xĩ) π x = δ p i ) (F (X i ))dx + [δb(1 ˆβ)f( ˆXĩ) + F ( ˆXĩ) And wih any consumer x, π x = δ[l i + p i (F (δβb p i x)) ie, [ ( F (Xĩ) π x = δ p i ) (F (X i ))dx Xi + Xĩ p i df (c) + [δb(1 ˆβ)f( ˆXĩ) + F ( ˆXĩ) (13) In boh expressions, hrough he firs erm, he marginal price has a negaive effec on he profi if i is posiive and a posiive one if i is negaive since F (X ĩ ) (F (X i))dx <. The negaive effec of his price (if i is posiive) decreasing he consumpion probabiliy ( p x i (F (X i))dx < ) is higher han he profi made hanks o his price wih he marginal consumer (p F (Xĩ) i ). However, we see hanks o expression (13) ha since a consumer x has a higher consumpion probabiliy o consume han a marginal one, firm receive p wih a highes probabiliy so his price is a posiive source of profi if i is posiive. Generally, he marginal price has 3 differen impacs on he profi. Firs, when he marginal price increases, he consumpion probabiliy decreases bu i increase he profi made when here is consumpion and finally i can decrease he demand a he firm giving he sraegic ineracion beween firms. Consumer surplus As we did for he firm profi, i is more ineresing o sudy he surplus of any consumer locaed in x and he surplus of he marginal consumer.

21 The marginal consumer surplus is, SC x = [ Xĩ δ (δb c)df (c) + p i (δb(1 ˆβ)f( ˆXĩ) + F ( ˆXĩ)) And he surplus of one consumer x is, Or, [ X i SC x = δ (δb x c)df (c) + p i (δb(1 ˆβ)f( ˆXĩ) + F ( ˆXĩ)) SC x = δ Xi I follows he social welfare for one consumer x, W x = SC x + π x = F (X i )dx p i F (X ĩ ) F (X i )dx p Xi i F (X ĩ ) p i df (c) Xĩ (δb x c)df (c) π x δβb p i x (δb x c)df (c) Thus he consumer surplus is he oal surplus generaed by he ineracion beween his consumer and one firm minus he firm profi. The social welfare depends on he consumpion probabiliy rough he marginal price and since he price are differen for each consumer ype and does no allow he perfec commimen, he profi, he consumer surplus and so he social welfare of hese hree ypes are differen. Proposiion 6 The inroducion of Hoelling compeiion and ranspor cos a he period of consumpion generaes, 1. Because of he heerogeneiy in he consumpion probabiliies, a perfec commimen is impossible for he sophisicaed. Tha implies differen profi, consumer surplus and social welfare beween a consisen and a sophisicaed consumer. We can conjecure ha he consumpion probabiliy is lower wih he sophisicaed and so he social welfare.. Because of wrong beliefs on consumpion probabiliy, profi, consumer surplus and social welfare are differen han for sophisicaed. Again, we can conjecure ha he social welfare is no opimal since he price and so he consumpion probabiliy are no opimal for he naive consumer. 1

22 4 Conclusion We inroduce imperfec compeiion and ranspor coss in DM model. These coss allows us o inroduce an heerogeneiy on he ase of consumers. I can be inroduced a he conrac period ie consumers suppor i one ime by choosing beween he wo firms he bes conrac for hem or a he consumpion period ie consumers suppor i only if hey consume. When ranspor coss are inroduced a he conrac period, hey do no impac he consumpion. The marginal price proposed o each ype of consumers is he same han DM framework. The lump-sum sands o recover loses due o negaive marginal price in case of ime inconsisen consumer and o exer a marke power due o ranspor coss. Thereby, firms are sill able o offer o sophisicaed consumers a perfec commimen, heir consumpion probabiliy says he same han consisen consumers one. However, ime inconsisen overweigh he ranspor coss wih heir shor erm discoun run which gives more power marke o firms. In ha sense, because of he difference in evaluaion of hese coss beween he firm and hese consumers, firms can exploi he ime inconsisency bias. Thus, firm profi increases (and consumer surplus decreases) when he degree of ime inconsisency increases. Facing naive consumers, we find ha he inroducion of even imperfec compeiion is enough o preven he exploiaion of he naivey bias by firms. However as perfec compeiion, he problem of allocaion inefficiency is no solve which implies ha naivey bias reduces he social welfare, even if i has no impac on he profi. When ranspor coss are inroduced a he consumpion period, hey impac he consumpion probabiliy. The consumer heerogeneiy sands boh in he difference beween consumpion probabiliies and in he difference beween consumpion uiliy. This mainly implies ha firms are no longer able o provide a perfec commimen o sophisicaed consumers. Thereby, we can conjecure he social welfare is lower wih sophisicaed consumers han consisen ones. Moreover, wihin his framework, we observe ha he ime inconsisency degree decreases he marginal price whereas he naivey degree increases i. The inuiion is ha firms have incenives o decrease he price o arac ime inconsisen consumers while he naivey ends o give hem more marke power a he consumpion period. In conclusion, we can claim ha wihin his framework boh ime inconsisency bias and naivey one can have impac on consumer surplus and social welfare. This resul reminds he significance of he ime inconsisency bias. Indeed, sophisicaed people are ofen considered as raional since hey ry o commi hemselves in order o ac like a consisen. Furhermore, he lieraure we sudied in he firs secion poin ou ha ime inconsisency, wih a wopar ariff conrac, has no impac on consumer surplus and social welfare. Thereby, one can conclude ha considering his bias is no imporan in fron of he naivey one.

23 References Acland, Dan and Mahew Levy, Naiveé, Projecion Bias, and Habi Formaion in Gym Aendance, 13. Andersen, Seffen, Glenn W Harrison, Moren I Lau, and E Elisabe Rusröm, Eliciing risk and ime preferences, Economerica, 8, 76 (3), Ariely, Dan and Klaus Werenbroch, Procrasinaion, deadlines, and performance: Self-conrol by precommimen, Psychological Science,, 13 (3), Armsrong, Mark and John Vickers, Compeiive price discriminaion, RAND Journal of Economics, 1, pp and, Compeiive Non-Linear Pricing and Bundling, The Review of Economic Sudies, 1, 77 (1), 3 6. Augenblick, Ned, Muriel Niederle, and Charles Sprenger, Working over ime: Dynamic inconsisency in real effor asks, Technical Repor, Naional Bureau of Economic Research 13. Benzion, Uri, Amnon Rapopor, and Joseph Yagil, Discoun raes inferred from decisions: An experimenal sudy, Managemen science, 1989, 35 (3), Chapman, Grechen B, Temporal discouning and uiliy for healh and money., Journal of Experimenal Psychology: Learning, Memory, and Cogniion, 1996, (3), 771. DellaVigna, Sefano and Ulrike Malmendier, Conrac Design and Self-Conrol Theory and Evidence, The Quaerly Journal of Economics, 4, 119 (), and, Paying No To Go o he Gym, The American Economic Review, 6, 96 (3), Eliaz, Kfir and Ran Spiegler, Conracing Wih Diversely Naive Agens, Review of Economic Sudies, 6, 73 (3), and, Consumer opimism and price discriminaion, Theoreical Economics, 8, 3 (4), Grubb, Michael D, Selling o overconfiden consumers, The American Economic Review, 9, pp Gul, Faruk and Wolfgang Pesendorfer, Tempaion and self-conrol, Economerica, 1, 69 (6), Heidhues, Paul and Boond Köszegi, Exploiing Naïvee abou Self-Conrol in he Credi Marke, The American Economic Review, 1, 1 (5), Huck, Seffen and Jidong Zhou, Consumer Behavioural Biases in Compeiion: A Survey, 11. 3

24 Laibson, David, Golden eggs and hyperbolic discouning, The Quarerly Journal of Economics, 1997, 11 (), Miravee, Eugenio J, Choosing he wrong calling plan? Ignorance and learning, The American Economic Review, 3, 93 (1), O Donoghue, Ted and Mahew Rabin, Choice and Procrasinaion, The Quaerly Journal of Economics, 1, 116 (1), Peleg, Bezalel and Menahem E Yaari, On he exisence of a consisen course of acion when ases are changing, The Review of Economic Sudies, 1973, 4 (3), Phelps, Edmund and Rober Pollak, On Second-Bes Naional Saving Game- Equilibrium Growh, Review of Economic Sudies, 1968, 35 (), Sroz, Rober, Myopia and Inconsisency in Dynamic Uiliy Maximizaion, Review of Economic Sudies, 1956, 3, Thaler, Richard, Some empirical evidence on dynamic inconsisency, Economics Leers, 1981, 8 (3),

25 Annex A1-Marginal consumer Characerisics of he marginal consumer The consumer x is indifferen beween he firm i and he firm i if Ux ĩ = U i. Or if: x δβ[ L i + δ ˆβb pi x (δb p i x c)df (c) = δβ[ L i + δ ˆβb p i (1 x) We denoe ˆXĩ = δ ˆβb p i x e ˆX = δ ˆβb ĩ p i (1 x). So, L i L i = ˆXĩ These inegrals can be rewrien as, (δb p i x c)df (c) ˆX ĩ (δb p i (1 x) c)df (c) (δb p i + x c)df (c) ˆXĩ By symmery, ˆXĩ (δb p i x c)df (c) = [(δb p i x c)f (c) ˆXĩ + = δb(1 ˆβ)F ( ˆXĩ) + ˆXĩ F (c)dc F (c)dc ˆX ĩ (δb p i + x c)df (c) = δb(1 ˆβ)F ( ˆXĩ) + Thus, he marginal consumer is such ha, ˆX ĩ F (c)dc ˆXĩ L i L i = δb(1 ˆβ)(F ( ˆXĩ) F ( ˆX )) + ĩ F (c)dc (14) We can no formally explici he expression of his consumer however we can use he expression (14) o find he demand variaion wih respec o he prices p i and L i hanks o derivaives. Since, firms are symmeric and we assume here are localized in each par of he demand segmen, he marginal consumer is siuaed in he middle of his segmen, ie x = 1. Derivaives of he marginal consumer wih respec o he prices To calculae he opimal conracs, i would be necessary o know he derivaive of he demand (ie he marginal consumer) wih respec o he prices. (We only sudy he case of he firm i, resuls for he oher firm are simply found by symmery). Derivaive of he demand funcion wih respec o he lump-sum ˆX ĩ 5

26 (L i L i ) = ˆXĩ (δb(1 L ˆβ)(F ( ˆXĩ F ( ˆX ))) + ĩ F (c)dc) i ˆX ĩ 1 = δ(1 ˆβ)[ ˆXĩ f( ˆXĩ) ˆX ĩ f( ˆX ĩ ) + ˆXĩ ˆX ĩ F (c)dc (15) We have ˆXĩ And, = x e ˆX ĩ ˆXĩ ˆX ĩ = x F (c)dc = [ [P (c) ˆXĩ L ˆX i ĩ = [P ( ˆXĩ) P ( ˆX ĩ ) = ˆXĩ F ( ˆXĩ) ˆX ĩ F ( ˆX ĩ ) = x (F ( ˆXĩ) + F ( ˆX ĩ )) Wih P (c) : primiive of F (c). Thus, he expression (15) can be rewrien, So, x 1 = δb(1 ˆβ)[ (f( L ˆXĩ) + f( ˆX x ĩ )) (F ( i L ˆXĩ) + F ( ˆX )) ĩ i 1 = x [ δ(1 L ˆβ)(f( ˆXĩ) + f( ˆX )) + F ( ˆXĩ) ĩ + F ( ˆX ) ĩ i x 1 = [δb(1 ˆβ)(f( ˆXĩ) + f( ˆX )) + F ( ˆXĩ) ĩ + F ( ˆX ) ĩ Or, since firms are symmeric, x 1 = [ δb(1 ˆβ)f( ˆXĩ) + F ( ˆXĩ)) (16) Derivaive of he demand funcion wih respec o he consumpion price 6

27 (L i L i ) = ˆXĩ (δb(1 p ˆβ)(F ( ˆXĩ F ( ˆX ))) + ĩ F (c)dc) i ˆX ĩ = δ(1 ˆXĩ ˆβ)[ f( p ˆXĩ) ˆX ĩ f( i p ˆX ) + ĩ i ˆXĩ ˆX ĩ F (c)dc (17) We have ˆXĩ And = 1 x ˆXĩ ˆX ĩ e ˆX ĩ = x F (c)dc = [ [P (c) ˆXĩ p ˆX i ĩ = [P ( ˆXĩ) P ( ˆX ĩ ) = ˆXĩ F ( ˆXĩ) ˆX ĩ F ( ˆX ĩ ) = (1 x )(F ( p ˆXĩ) + F ( ˆX ĩ )) F ( ˆXĩ) i Thus, he expression (17) can be wrien as, = δb(1 ˆβ)[ x (f( ˆXĩ) + f( ˆX ĩ )) f( ˆXĩ) x (F ( ˆXĩ) + F ( ˆX ĩ ) F ( ˆXĩ)) δb(1 ˆβ)f( ˆXĩ)) F ( ˆXĩ)) = x (δb(1 ˆβ)(f( ˆXĩ) + f( ˆX ĩ )) + F ( ˆXĩ) + F ( ˆX ĩ )) So, x δb(1 = ˆβ)f( ˆXĩ)) F ( ˆXĩ)) [δb(1 ˆβ)(f( ˆXĩ) + f( ˆX )) + F ( ˆXĩ) ĩ + F ( ˆX ) ĩ Or, since firms are symmeric, x = 1 Finally, we can noice ha x = δb(1 ˆβ)f( ˆXĩ)) F ( x 1 Or, = δb(1 ˆβ)f( x ˆXĩ)) F. ( ˆXĩ). ˆXĩ). x. 7