Durable-Goods Monopoly with Discrete Demand ...

Durable-Goods Monopoly with Discrete Demand Mark Bagnoli; Stephen W. Salant; Joseph E. Swierzbinski The Journal of Political Economy, Vol. 97, No. 6. (Dec., 1989), pp. 1459-1478. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28198912%2997%3A6%3C1459%3ADMWDD%3E2.0.CO%3B2-0 The Journal of Political Economy is currently published by The University of Chicago Press.

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Durable-Goods Monopoly with Discrete Demand

Mark Bagnoli, Stephen W. Salant, and Joseph E. Swierzbinski I'nzrler~ztsof .Ifzchzgan

We analyze a dynamic game betlveen consumers and the sole seller of a durable good. Unlike previous analyses. we assume that there exists a finite collection o f buyers rather than a continuum. None of the main conclusions of the literature on durable-goods monopoly survives this change in assumption. Cease's corljecture that a durable-goods monopolist cannot earn supracompetitive profits in the continuous-time limit. Bulolv's proposition that renting a durable is always Inore profitable than selling it, and Stokey's proposition that preco~nrnittingto a tirne path of prices is always optimal are all false \\hen the set of buyers is finite. Thus the assumption of a continuurn of consumers-so innocuous and useful a simplification in other contexts-has proved rnisleadir~gin the context of durable-goods monopoly.

I.

Introduction

While models o f strategic interactions a m o n g firms a b o u n d , models with bu!,ers as acti1.e players a r e comparatively r a r e . I n his pioneering analysis o f o n e such g a m e , Coase (1972) cotljectured that a durablegoods monopolist would b e u n a b l e t o e x e r t m o n o p o l \ polver since strategic t~~lyer-s-anticipiiting t h a t t h e price lvould d r o p in t h e "t~vinkling o f all eyew-would r e f u s e to bu!. as l o n g as t h e price r e m a i n e d abo1.e t h e competitive level. TVe would like to thank (witllout implicating) Ken Binmore. 1)avid Levine, Sherwin Rosen, J e a n .I irole. Hal Val-ian. Robel-t W'ilson, and two anon\nioil\ referees for their comments on an earlier dr-aft. [ J o u m l rf Poiriani Eronorni. 198'2. \ u l 97. no 61

0 1989 h\ T h e L:n~verstt\of Ch~cnqo A11 r l g h ~ srere! verl 0022-3808.89 9iOIi-O00?SOI i O



1460

J O U R N A L O F POLITICAL L C O N O ~ ~ Y

Recent papers by Stokey (1981) a n d Kahn (1086) have formally modeled durable-goods nlonopoly. Under the assumptiorls that Inarg i r d cost is constarlt and the 111011opolistis unable to com~llitcredibly in the to future behavior, these papers confirm C;oase's cor~Jectu~-e corltilluous-time limit ~ v h e r ethe discount kictor :ipproaclles o n e . ' klore recently, Gul, Sonllenscllein, a n d il'ilson (1986) have reforrrlulated the durable-goods nlollopoly problem as a d!.na~llic game. Under the same cost a n d demand assunlptions, they shelved that for disc0~111tkictors near one, every subgame-perfect equilibriuln of the game satisfies Coase's cor~jectureabout monopoly profits. T h e literature or1 durable-goods monopoly also contains two other important propositions. Stokey (1979) and Sobel and Takahashi (1C)83) sho~ved that a durable-goods monopolist can i11cre:ise his ~ x o f i t sif he can precorn~nitto ii time path of prices. Bulo\v (1982) sho~vedthat, even if no such precornrnitnlellt is possible, ii durablegoods ~llorlopolistcan increase his profits by renting the durable illstead of selling i t . Until nolv, the durable-goods ~llorlopolyliterature has confined itself to dellland curves with a continuulrl of nonatonlic buyers. assumption, of course, corresponds to n o naturally occurrillg o r laboratory economy; but ill other contexts, it has proved an innocuous sirrlplificiition. il'e anal>ze durable-goods nlorlopoly ~ v h e nthe set of buyers is f i n~t~-albeit possibly \.el-y large. '1-his as sump ti or^ is both realistic and fiithf'~11to (:ease's original assumption regarding the nurnber of 11~1!.ers.As ive shall demonstrate, none of the three conclusions o f t h e durable-goods literature sut.vi\,es this single change it1 :issutrlption. '1'0 isolate the source ofthis major re~.ersiili l l conclusions, Ive retain the 0tht.1.stiindarti :issunlptions. .l'he seller is a s s u ~ n e dto have a co11stant margilliil cost of productioll. Each buyer is assunled to ha1.e a n inelastic derrl:illcl fill- a single unit of the nlonopolist's product and to be cllaracte~.ir.edby a reservation price that represents his nlaximunl \villinglless to pay for the unit. T h e market for the durable good is anal!.red as :I dynamic game. 'I'he strategies of the nlonopolist and each bu!,er a r e assunled t o form a subgame-perfect equilibrium. T o facilitate comparison \\.it11 C;ul et al. (1986), I\.e assume that the ~nonopolistcall observe the reser1.atio11 price of each buyer. As we

'

H \ the c o n t i n , ~ o u s - t i ~ n111llit. e n e tliearl the situatio~ii l l \\hicll the colnrlion tirne ~ n t e r \ a lbetween the seller-'s s ~ t c c e \ \ i \ eof.fe~-sshl-inks to rero. If the tlrne hot-iron is assumed to he finite, the11 the ~lunlher-of offer-\ that the seller is dllo\ved to mahe nlu.;t n. the al\o t ~ eexpanded to fill the un(.harlgeci t i ~ l l ehor17on. In this G t l ~ ~ i t i o altliougll d ~ s c o u n ft actor per-raining to ari\ gi\en lerlgtll of time is trnc hariged, tlie d i \ c o u ~ l tfactorper-taining to the time inter\-a1 bet\veen successive offers ,~ppr-oaclleso n e . For- details. 5ee n . 12.

DURABLE-GOOL)S MONOPOLY

146 I

sllo~vin a comp:inion papet-, I%agtloli,Salant, and Swierzbinski (1989), similar results hold u n d e r i~lco~riplete inf'or~natiotl:as long as the set of buyers is finite, perfect Bayesi:in equilibria exist in ivhich all three propositiotls about durable-goods trionopol) fiiil.' C:orlclusions dratz.11 frotri the analysis of dynarriic gannes ;ire often sensitive t o iissumptions about the finiteness of the horizon. Sonle readers may regard finite-horizon analyses rvith suspicion because of "end-effects." 111 the context of durable-goods morlopoly, for example, a finite ho~-izollallo~vsthe seller to make a credible take-it-orleave-it offer in the final period. O t h e r readers may. regard infinite., horizon analyses with suspicio11 because t h r set of subga~~ie-perfect equilibt-ia often "exparlds disco~lti~luously at infinity." A familiar example is the prisoner's dilemma, lvhich has ;I single subga~ne-perfect equilibrium for an! finite tirrie horizon but multiple equilibria (suppot-ted b!. history-dependent strategies) if the hot-izon is unbounded. Ausubel and Deneckere ( 1989) have derived analogous results f;jr the durable-goods monopoly game of Gul rt al. (15186). B\. relaxing the assurription of (;ul et a1. that strategies a r e "statiorlary." Xusubel and Denecket-e derrionstrate the existence of additiolliil equilibria that 1.iol;ite C:oase's cotljecture altliough not Bulo~tr'sand Stokey's PI-opositiorls:' Ho~vever,these additional equilibria disappear if the assumption of a n unbounded hot-iron is replaced by the assumptio~iof a lorlg but finite horizon. T o persuade the \videst audience, tve show that the three propositions of C;oasr, Stoke!,, a n d Bulow are false under pitlu~r;lss~i~liptio~l about horizon length. Indeed, ill each of o u r finite-horizon exanlples, the strategy cornbirl~itior~ that we consider forrris a subganne-perfect equi1ibriur-n not only for a finite horizon of any length but also for the unbounded horizo11. \Ve ,l\\urne that the I-eser-\ation price of each t)u\er is c-ornrnon kno~vledge,hut this is not essential. Hagnoli et al. ( l 9 X 9 ) assunle that tllc monopolist knows the initial aggr-egate t l i s t r i b ~ ~ t ~ofo nreservation pr-ices, ob\er-\es a g g r e p t e p ~ ~ r c l l a s eIn\ each period, but c c i r z ~ i i ~iclerrtif\ t the reser-\,rtlorl prices of indi\ idual t)u)er-s. I.nder this alternatl\e infor-rn;rtion structur-e. \at-iarlts ot the str-ategies sho\\,11h e r e to form a s11t)gamepertect ec~uilit)i-~uni in the t\lo-l)u\er exatllple ot Sec. I11 a n d tlre infinite-horizoil exanlple of' Sec I\' rvill for-rn a perfect Ha\e\ian equilibr- urn (suppor-ted I,\ appr-opr-iate belief\). 'I'hese eclu~lit~ria, \\hich generate the same p a \ o f t i , derlion\trate that-when the set of t)u\ers is finite-the ( : o ~ s econjecture as \\ell as the pr-opositio~lsof Bulo\\ a n d Stoke\ ar-e iri\alid nhether- the time horizor~is h n ~ t eor- inhrlrte " S o n e of the n d d i t ~ o n a le q i ~ i l i b r ~r a d e n t ~ h e dI>\ . \ t ~ \ i ~ b e2nd l De~recLereI-esult\ in p,a\offs ;IS lligll as static nlonopol\ profits Since static nlonopol\ pr-ohts ( a n be achie\ed ertller tx pr-econrmitting to ,I PI-ice path o r b\ I-enting, treither Stokep's nor Bulo\v3\ proposrtlon is contr-ad~c ted in ;in) oi .\ri\i~t)ela n d 1)eneckere's eqliilibri;~.In corrtr;~st. the \ellet- e;lrns s t r i c t l ~m o r e than st;rt~cmonopol\ pr-oflts in each of the e q r ~ ~ l i b r that ia

\ve discuss. for both h n ~ t ea n d iinbounded horizo~ls.Indeed. for- di\count factors ap-

proaching o n e , the seller- tar1 gerle~-all\I-eap the pr-oht of a perfectl\ discr-irnrnat~ng

nlonopolrst ~f the hot-won IS u r l b o u ~ l d e da n d I-eaps that pa\off in o n e of our- t\vo h n ~ t e -

Ilor-17on esample, as 5% ell.

1462

J O U K K A L OF POLITICAL E C O N O M Y

O u r finite-horizon analysis pertains to specific distributiorls of reservation prices. While these serve as counterexamples to the three propositions, the possibility remains that-for some ;is yet uncharacterized class of reservation prices-these venerated propositions may still be true. However, our- infinite-horizon results hold for a r ~ distribution j of I~uyers'reser\.ation prices. I n all such cases, for sufficiently high discount factors, there is a subgame-perfect equilibrium in ~vhichthe monopolist extracts (virtually) the rr~tirrconsumer surplus. I n this equilibrium, the present value of the monopolist's profits in the continuous-time limit approaches the value obtained by ii perfectly discriminating monopolist. Renting o r precommitting to a price path u n d e r such circumstances would be folly. Rather than losing all market polver, M-hichCoase conjectured is ine\.itable, the monopolist attains almost I p r f ' e r t market power fi)r discount E~ictorsnear one. C;oase conjectured that a durable-goods monopolist selling to a finite set of buyers could not earn supracompetitive profits in the continuous-time limit. O u r analysis not only denlorlstrates that his conjecture is false but illuminates which of his supporting arguments is f:~llacious.Chase kvas correct that in the continuous-time limit of the infinite-horizon game, the price would quickly d r o p to the competitive level. But he was wrong to think that no sales ~vouldtake place before the competitive level rvas reached. I n the equilibrium kve study, strategic buyers a r e willing to purchase at their own reservation price l~ecauseeach realizes that it would not be in the monopolist's interest to cut price if' the purchase were delayed. I n the continuous-time lirnit, to sell to a finite set of' buyers a n d approach the profit of a perfectly discriminating monopolist takes only "the twinkling of an eye." I n Section 11, we intl-oduce notation and formulate o u r model. Section 111 considers two finite-horizon examples. Section IV treats the infinite-horizon case for any finite set of reservation prices. Section V concludes the paper. 11. The Specification of the Market Model

'This section formulates the model m d introduces o u r notation and terminology. T h e r e is a finite o r countable n u ~ n b e rof' periods indexed by t = 1, . . . , T. 'The horizon, T, can be finite; otherwise T = =. Each period consists of two stages. I n the first stage, the monopolist offers to sell to any a n d all buyers at the pricep(t). I n the second stage, each buyer simultaneously decides whether to accept the monopolist's current offer o r to reject the offer a n d continue. .There are initially ,V buyers, each ofwhorn can consume either one

14~3

IIL'KABLE GOODS MONOPOLY

or zero units of the monopolist's good. A type 1 buyer who accepts the monopolist's offer in per-iod t obtains the utility

u, =

p'

-

' [I!, - p (t)].

(1

A buyer who neber accepts a n offer- obtains a utility of zero. Hence, a type 1 buyer is ne\.er lvilling to pay more than his I-eser-\.ationprice, z!,. There are at most L distinct resen ation prices, with L 5 S , and the resenation prices are indexed so that z!, > z12 > . . . > vl r 0. Forsimplicity, we follow the literature and assume that the discount factor, p E (0, l ) , is the same for all buyers and for the monopolist. Following Gill et al. (I%%), we assume that the reser\,ation price of each buyer is common knowledge (see n. 2). T h e monopolist is assumed to incur no fixed costs and to produce the good at constant marginal cost. Hence, without loss of generality, the price, p(t), and the reser\.ation prices, 711, can be considered as net of n~arginalcost. Equation ( 1 ) remains the same under this interpretation. If B (t) is the number- of buyers accepting the monopolist's offer in period t , then V, the present value of the nlonopolist's profits, is A~' i v e n by

?'he nlonopolist's goal is to maximize V. Without loss of generality, Itre can restrict attention to the case in ~vhich > 0.4 In this case, the lo~vestreservation price is greater than the monopolist's marginal cost, and the Coase conjecture predicts that the monopoly profit is equal to L\'~!L. T h e history o f t h e game prior to period t is described by a list of the monopolist's price offers in periods 1 through t - 1 and a list of'the t~uyerswho accept the monopolist's offer in each period. A pure strategy is a function specifiing a player's choice at each stage for each history of the game prior to that stage. Hence, a pure strategy for the monopolist specifies the nlonopolist's price offer in each period t as a function of the garne's history u p to t. A buyer's pure strategy specifies the choice "accept" or- "continue" in stage 2 of each period t as a function of the history u p to t and the monopolist's current price offer, p(t). -ijL

'

If ZIL = 0, then i t is never optimal for t \ p e L b u ~ e r sto b u ~at a positive price As a result, the equilibrium strategies that \ve consider for- the monopolist and other- bu,ers are also equilibrium strategies in a game with a finite number ol zer-o-I-eservation-price bu)ers added, and the nlonopolist's equilibrium profits are the same in both cases. (Once all the b u ~ e r swith positive reservation prices have purchased the good, we can assume that the monopolist sells [or not] to the zero-reservation-price bu~er-sat a zer-o price.)

14~4

J O U R N A L OF POLITICAL ECONOMY

X subgarrie-perfect equilibl-iurn is a strateg!. corribin;ition such that the strategy for eacli pla>er is a s ~ q u ~ n t i best a l replj, that is, optimal at every st:ige arld for every history g i ~ . e nthe strategies of' the other players. Since we seek not to characterize the entire set of subgarneperfect equilibria b ~ i merely t to exhibit equilibria with char:icteristics ;it odds with accepted ~visdorri,we restrict attention throughout this paper to subgame-perfect equilibria in p u r e strategies..' For brevity, Ive henceforth omit explicit reference to this restriction o n the strategies of' the players.

111. Monopoly Power in the Finite-Horizon Market Game I n this section, Ive show for the finite-horizon case that the three key propositions about durable-goods rnonopoly 111;iy fail ivhen the nuniher of' buyers is finite. We consider t ~ v oexamples." 'l'he first e x a ~ n p l ehas t11o buyers (indexed 2 and 3). .Fhe secor~dhas three buyers (irldexed 1, 2, ;irld 3). Suppose that the buyer's reservation PI-icesa r e distinct and recall that T I , > 712 > 713 > 0. I n each of the t ~ v oexaniples, i t is optimal for the ~nonopolistto use the same strategy in equilihriuni. 111any period he sets the price that ~vouldbe optimal Ivhen there is no future to consider. Hence, in periods in ~vhichonly one hu!,er ( i ) rerriains, the nionopolist sets the price in each period :it i's reservation price. 'l'hat is, p(t) = i l , . 111 periods in ~vhiclltlvo bu!,ers ( i and j rvith 71, > il,) remain, the Inonopolist ~ v o ~ ichoose ld p(t) = 7 1 , if i!, > 271,and p(t) = 71, other~vise.Firlally, in periods in \vhich three buyers remain, the monopolist would choose P ( t ) = 7 1 1 if ill > r11;1~(271~, 371g) and p(t) = 712 if 2 7 1 > ~ rnax(z~,, 311:~); ' A I I \ str-ateg\ c o r n b ~ n a t ~ ot nl i , ~ tforrns a subgame-perfect ec~uilihriirmrvlien plakers a r e I-estritted to p u r e stl-ategies \vill I-ernain ,I sul~ganie-per-fecte c ~ u i l ~ b r i u ifm pla\er-\ a r e allo\\ed to pla! t)eha\~or-al( n ~ i s e d str-ategles. ) 'To see this. suppose the tontrar-\. T h e n there niilst be some s u l ~ g a n i eIn \\liich this cotlibination of piire strategies n o longer forms a Sasll equilit)riun~.For- that to t ~ etrue. some p l a ~ e r -must be able to str-ictl\ rrnpro\e his pa~of'f'l)v unrlaterall\ slvitching to a l,eha\.ioral strate#\ in that sut)garrie \\bile the other- p l a ~ e r - scontinue to pl,~\ their- purt. str-ategies. Hut for tliis de\iation to I I ~s trictly profitable, ther-e mlrst Oe some put-e strateg\ o \ e r which the mlxing occurs that \ields a higher pa)off to the d e v ~ a n than t h ~ initral r p u r e strate#\. If such a strateg\ did exist. h o \ \ e \ e r , lle could Iia\e p l a ~ e dit 100 per-tent of the ti111e pre\iousl\ (a p u r e a n d hence acirnissil)le strategk) a n d t l ~ e r e b \str-ictl\ ir~cr-easedhis pa)off' Hut this contr-adicts the assurriptron that the original strateg) cotiibirlatron \v,~s subgarne perfect In p u r e str-ategres. " T h e sul)ganie-perfect ec[uilrbria in the finite-hot-izon game car1 he quite complex, l\ them for- ar-hitr-as\ d e m a n d cur-\e\. a n d we Ila\e not sutceeded iri f ~ ~ lch,~s,icterizing IVIiile \uch 21 char-,1cteri7ation rlirght 11e e x t r e n ~ e l~~n t e r e s t i n gi,t is unnecessar-\ f o l - o u r le is all that is ~ l e e d e dt o denionstr-ate that a proposipurposes. A s ~ r ~ gtounterexample tion is false.

1465

DCRABLE GOODS ~ ~ O N O P O L E '

otherwise, he would setp(t) = 7 1 : ~ . We refer to this (optirnitl) strategy as the "niyopic" monopoly stl-ategy.' I3uyers' strategies differ in the t w o examples a n d are described its the need arises.

Consider first the simple case o f a market with two buyer-s (indexed 2 and 3) ~vithreservation prices - 0 ~> 2 7 1 : ~> 0 that lasts for tlvo periods (7' = 2). Suppose that the buyers always accept the first offer that generates nonnegati1.e utility. 'I'hat is, tor a type I buyer, accept i11 period t if and only if'p(t) 5

11,.

(3)

il'e refer to this as the "get-it-ivhile-).ou-can"strategy of a type I buyer since he ~ v o u l dseize the first opportunity for surplus as if no future opportunities might present themse11.e~.Using the get-it-while-youcar1 strategy in period t is clearly a sequential best reply for any buyer ~ v h ocan obtain only zero utility in all future periods lvhen the other players use their equilibrium stl-ategies.' iVe now show that the niyopic monopoly strategy for the seller and the get-it-lvhile-yoti-c:~~~ strategy tor each buyer- fi)rm a subgit~r~eperfect equilibrium. iVe show this first for the tivo-period horic.011, then for finite horizons of length T > 2, and finally f'or the unbounded horizo11. ?'he get-it-while-yo~1-car1strategy is clearly optimal i11 the final period of the two-pel-iod game. Sirlce iri this period the myopic monopoly strategy is optimal by construction, the monopolist chooses p(2) = zlz if' buyer 2 has not yet purchased the good and p(2) = z1:3 if only buyer 3 remains. Hence, in the priol- period neither buyer car1 ever obtain 1x)sitive utilit). by ~vaitinguntil period 2, and the get-it-ivhile~ O L I - c astrategy n is optimal f'or each bu!,er in period 1. 'lVhe11 buyers use the get-it-while-yo~1-c:111strategy in each period, it is clearly suboptimal for the ~nonopolistto choose a price in period 1 other than or 7'. Bv choosi~lgp ( l ) = 7 ~ ? , the nionopolist sells o111>,to bu\.er 2 in ' .-lltllougll the monopolist in both of' ~ L I I finite-1101-i7on examples uses the rn\oplc rnonopol\ str-ateg~.\ \ e d o not contend (and d o rot l)elie\e) that this strateg\ is par-t o f an e q u i l ~ h r ~ uin m ;11l hr~ite-hor-i7onexatrlples. ' T h e r e is a c o n t i n u ~ ~ of n i distinct suhgames beginning in the second st'tge ot each p e r ~ o d1 and indexed h \ the price. p ( t ) , that the nronopolist otter5 In thc fit-st \tage ot the period. T h e htrateg! tor each bu)er- i\ a function that specihes the l)u\er-'5 choice in period I ("accept" o r - " ~ ( ~ n t i ~ l uase "a)tune tion of'the first-stage price ( a n d , rnore generall!. as a tc111ction ol the histor! of the game). H \ 1 r r i f \ i n g the o p t ~ n ~ ~ r loift \this tunction in a g i ~ e nper-iud, \\.e a r e sirnultaneousl\ \ e r i t \ i n g the opt~nlalit\of'the b u ) r r ' \ str-ateg\ In each of the second-\t mediately to both buyers and obtains the present value 2 - 0 ~Since ~ z I : it~ ,is optirrlal for the monopolist to choose p(1) = z12 as specified by the myopic nlorlopoly strategy. T h e equilibrium present value of the monopolist's profits in the two-period game is given by

Suppose instead that the finite horizon is lorlger than two periods. We have already noted that it is optimal for a buyer to use the get-itwhile-you-can strategy in the current period if the buyer can expect to obtain at most zero utility in any future period. Moreover, when > 2u:(, no buyer can obtain positive utility in any period in which the monopolist uses the myopic monopoly strategj-. Finally, it is straightforlvard to extend the analysis in the previous paragraph to show that the myopic monopoly strategy is optimal for the monopolist in period t when every buyer uses the get-it-while-you-car1 strategy from period t onward. By working backward from the final period, these three "facts" are easily combined to prove that the use of the get-it-while-you-can strategy by each buyer and the myopic monopoly strategy by the monopolist corlstitutes a subgame-perfect equilibrium of the two-buyer, Tperiod game for all T > 2. Since the monopolist uses the myopic strategy when t = T, the get-it-while-you-can strategy is optimal for each buyer in period T - 1 as well as period T. Hence, the myopic monopoly strategy is optimal for the monopolist in period T - 1, the get-it-while-you-can strategy is optimal for each buyer in period T 2, and so on. Moreover, for any T > 2 , the equilibrium present value for- the morlopolist is given by equation (4). T h e Coase conjecture predtcts that in the absence of precommitment or renting, the monopolist in the T-period, two-buyer game with reservati~nprices as described above will obtain a present value . (1979) proved that if the mo~lopolistcan precomequal to 2 - 0 ~Stokey nlit to a price path in a T-period game, then the optimal choice is to charge the static monopoly price in every period and earn a present value equal to the static nlonopoly profits. In our two-buyer example with zlz, > 2 ~ ' the ~ . static monopoly price is u2 and so the optimal precommitment strategy is p(1) = p(2) = . . . = p(T) = ~ 2 which , yields a present value of vz,since it is optimal for buyer 2 to pur-chase the good in the first period. Bulow (1982) observes that a monopolist can obtain (at most) the static monopoly profits by renting rather than sellirlg the good." Hence, for the T-period, two-buyer game ~vhereu2 "

Bulow acsumes a stationary rental demand

curie

for the durabie good. .I o capture

DURABLE GOODS MONOPOLY

14~7

> 2zl3, the monopolist who rents obtains at most z12 Since U Z , + P713 > u2 > 229, o u r analysis of the two-buyer case demonstrates that Coase, Bulolv, and to key"' a r e each incorrect for any discount factor and any finite time horizon (7' 2 2)." 'The get-it-while-you-can strategy for each buyer and the myopic monopoly strategy for the monopolist constitute a subgame-perfect equilibrium for o u r two-buyer- exatnple fhr a 9 finite horizon, T , and any discount factor- between zero and one. Moreover, as p + 1 and/or T + x, the difference between the equilibrium present value obtained by the nlonopolist and the static monopoly profits is strictly positive and bounded away from zero. Since static monopoly profits in turn exceed the present \,slue predicted by the Coase conjecture, the propositions of Cease, Bulow, and Stokey will also be invalid in any limiting case of our- two-buyer-, finite-horizon example in which the discount factor is allolved to approach one, the horizon T is allowed to approach infinity, o r both.12 this assumption when each of a finite number of buyers has unit demands, assume that the rental dernand curve is generated b) the service demands of the various agents. Assume that agent 1 would be willing to par as much as I., to enjo) the services of the durable for a single period arid that r, is independent of tirrie. \Ve refer to r, as willingness to rent. Gill et al. assume, on the other hand, that agent 2 would be willing to pa\ as much as 11, to p z ~ r c h a sthe ~ durable outright and rhat this "willingness to purchase" is irldeperlderit of time. It is possible to reconcile these two stationarity assumptions even if the seller/renter's horizon is finite, provided the service stream of the durable is independent of this horizon. If we assume perfect durability, then rl, = r , / ( l - P). Hence, to convert the inverse demand curve genel-ated by the initial set of burers to the inverse rental demand curve, sin~pl)rnultiply each vertical distance by 1 - P. Let V, be the maximal net revenue obtained b\ the monopolist in a single-period garnc; i.e., I.5 equals the static monopoly prohts. Clearly, the maxirnal net rental income per period is Vs(l - p). Let 1',, denote the maximal net discounted wealth of a monopolist renter. Then if the renter can operate over infinite time, 1,' = I.,. If instcad the renter can operate for only ?'periods, then 1', = 1',(1 - P ' ) < 1.'(.. Hence, as asserted in the text, a monopolist can obtain at rnost the static monopol) profits by renting rather than selling the good. l o It ma) seem paradoxical that the rnonopoliqt could lo% profits b) precomrnitting. After all, his profits would not change if he sirnplr precornrnitted to the strategb that is optirnal for him in the uncommitted equilibrium. Stokey's propo5ition. however, concerns precommitment to a tzmppath of prices. This form of precommitment is injurious because the morlopolist would relinquish his abilit\ to corlditiorl hi? pricing on observed bu)er behavior. ' I JVhen 1 ' 2 < 20:~.~t is straightforwarti to show that the subgame-perfect equilihriurn strategy for the monopolist is to offer a price p( 1 ) = 71.~ in the first period and that both bubers 1 and 2 accept this offer. T h e static monopol) price when rl, < 218.~is alsop = arid the static monopoly profits are ~ Z JHence, . ~ . the profits obtained from the optirnal precornrnitment strategy, the prohts obtained by the optirnal rental strateg), and the profits predicted by the (:ease conjecture all coincide with the equilibrium prohts in this case. ' q I . e t S be the tirne interval between the seller's successive offers, p be the discount factor pertaining to rhat interval, and p be a hxed parameter expre5sing the in5tantancous rate of time preference of the bukers and the monopolist. 1-hen. P satisfies P = exp( - p6). Let r be the length of the "tinle" horizon of the game. Note that both 6 and r

~ l . ~ ,

1468

JOUKNAI. OF POLITICAI. ECONOMY

To conclude our discussion of the two-buyer game, we note that the myopic monopoly strategy for the seller and the get-it-while-you-can strategy for each buyer also form a subgame-perfect equilibrium if the horizon is unbounded. This follows since, with I!., > 279, the myopic monopoly strategy is the same as the Pacman strategy defined in discount the next section, and corollary 1 there implies that-for factors sufficiently close to one-this strategy conlbination forms a subgame-perfect equilibrium in the infinite-horizon Hence, the samp two-buyer example provides a counterexample to the propositions of (:ease, Stokey, and Bulow not only for any finite horizon but also for the infinite horizon. In the two-buyer game above, equation (4) implies that as the discount factor approaches one, the present value obtained by the monopolist approaches the value obtained by a perfectly discriminating monopolist. It might be supposed that the failure of'the Coase con-jecture and of Stokey's and Bulow's predictions depends on this characteristic of our counterexample. 'The follotvirlg three-buyer example shows that such a supposition would be incorrect.'

'

The Thrpr-B1qpr E Y ( I ? ~ I ~ ~ P Consider a three-buyer, T-period game in which the buyers have distinct reservation prices satisfying the inequalities

A three-buyer denland curve satisfying the inequalities in equation (3) can be constructed from our two-buyer example (where z12 > 2 ~ 3 ) by adding a new "top" reservation price ( I ! ' ) "close to" the original 112. It is verified in the Appendix that the follolving strategies constitute a subgame-perfect equilibrium of the three-buyer, T-period game. T h e n~onopolistuses the myopic monopoly strategy in every period. Buyers 2 and 3 use the get-it-while-you-can strategy (defined in eq. [3]) in every period. Buyer 1 uses the get-it-while-you-can strategy in are measured in units of calendar time (e.g.. d a ~ s )Let . T denote the maxirnum number of offers the seller can rnake during the time her-iron of length T. rhcn T satishes 7'= r/6. A continuous-time limit of the discrete-per~od market game occurs when the interval length, 6, is allowed to appl-oach zero. Note that even in the case in which the time horizon I' is held constant, allowing 6 tu approach zero cauces 7' to appr-oach intinitv as well as causing p to approach one. I '3 Indeed, in the two-bu)er case the nibopic monopol> strategb for the seller and the get-it-while-!ou-can ctrategb for each buver form a subgame-perfect equilibrium in the inhriite-horizon game for c ~ r rdiccourit ~ factor between zero and one. This can he verihed using the argurnerlt presented at the end of the Appendix for the threr-buyer

case.

We wish to thank an anon\nious referee for encouraging us to discuss the three-

buyer, hnite-horizon example.

''

14%

DURABLE-GOODS MONOP0I.Y

the final period and in any period in which buyers 1 and 3 are the only remaining buyers who have not yet purchased the good. Finally, in periods in which only buyers 1 and 2 or all three buyers remain and t < T, buyer 1 uses the strategy accept in period t if and onlv if

p(t) 5 p*

=

(1

-

P)zl,

+

Pz12.

(6)

Since 271p > max(ill,379), the monopolist sets a pricep(1) = u2 in the first period of the three-buyer game, which buyers 1 and 2 accept. I n the second period (if T 2 2), the nlorlopolist sets the price p(2) = z ~ : ~ , which buyer 3 accepts. Hence, the equilibrium present value obtained by the monopolist in the three-buyer game with T 2 2 is given by

Note that even as the discount factor approaches one, the monopolist's present value does not approach the value obtained by a perfectly discriminating monopolist, hloreover, buyer 1 retains the surplus i l , - 7Ip. Nevertheless, for any 7' 2 2 and any discount tactor, the equilibrium in the three-buyer game contradicts the propositiorls of Coase, Bulow, and Stokey. For three-buyer games in which the reservation prices satisfy the inequalities in equation (3),the static monopoly price is p = 712 and the static monopoly profit is 271~. The Coase corljecture predicts that the monopolist will obtain a present value of 379. Since ~ ~3zlg for P > 0, Bulolv's rental strategy and Stokey's 2 - r ~+~P71:{ > 2 7 > preconlnlitmerlt strategy do not produce higher present values than the equilibrium strategy without preconlmitmerlt or renting; nor is the Cease conjecture correct. For the same reasons as in the two-buyer example, the three-buyer example also provides a counterexample to the propositiorls of C:oase, Stokey, and Bulow in any limit of finite-horizon games in which the discourlt factor is allowed to approach one, the horizon, T, is allowed to approach infinity, or both. Finally, it is also verified in the Xppendix that the strategies discussed above constitute a subgame-perfect equilibrium in the three-buyer, infinite-horizon game with 0 < P < 1 and reservation prices satisfying equation (3). Hence, the samr threebuyer example provides a courlterexanlple to the propositions of Chase, Stokey, and Bulow not only for any finite-horizon game and any limit of finite-horizon games but also for the infinite-horizon game. ii It should he noted that ~ n r ~ l t i p lequilibria e niav arise ~ I I C I I the hol.i7o11 i5 unbounded. For example, in the three-hu)er case with a sufhcientl) high discount factor. there is a second equilibrium, wliich we discuss in Scc. I\'. I n this equilibrium, the present value o f the monopolist's prohts approaches the value obtained b\ a perfcctl\ discriminating nionopolist.

IOL'RNAI. OF '470 IV. Monopoly Power in the Infinite-Horizon Market Game: The Pacman Theorem

POI.ITICA1. E C O N O M Y

This section demorlstrates that for an a r b i t r a ~ ,finite collection of buyers, there exists a subgame-perfect equilibrium of the infinitehorizon market game that violates the claims of Coase, Bulow, and Stokey as long as the discourlt factor is sufficiently close to one. hloreover, the monopolist's profits in this equilibrium approach the surplus achievable by a perfectly discrinlirlating monopolist as the discourlt factor approaches one. In the equilibrium that we shall consider, each buyer uses the get-itwhile-you-can strategy defined in equation ( 3 ) . T h e monopolist's strategy is equally straightforward. Let 'c~,,,,,(t)be the highest reservation price belonging to some buyer who has not yet purchased the good at the beginning of period t. 'The monopolist sets the price in period t as

A monopolist who uses the strategy specified in equation (8) sets the price in each period to the highest remaining reservation price and drops this price if and only if all buyers with the highest reservation price have purchased the good. We refer to this pricing strategy as the "Pacman strategy" since the monopolist attempts to eat his way down the demand curve. If the morlopolist plays the Pacnlarl strategy, then no buyer can achieve positive utility. Hence, fbr a buyer of type I , a sequential best reply to the Pacman strategy requires that-if an informatior1 set with p(t) < ul is ever reached-the buyer accepts the n~onopolist'soffer. For in this case the consumer would gain strictly positive surplus from accepting the offer and would anticipate zero surplus from continued play against the Pacnlan strategy. If p(t) = 711, then either choice, "accept" o r "continue," is optimal for type 1 buyers. MTefocus o n the get-it-while-you-can strategy, which assigns the choice "accept" in this case. Finally, if an information set with p(t) > zr( is ever reached, declining the offer is a sequential best reply since the consumer can get a strictly larger surplus (zero) just by declining all future offers. 'These observations are formalized in lemma 1. LEMMA1. 'The get-it-while-you-can strategy is a sequential best reply for a buyer playing against the Pacman strategy. It remains to determine corlditions under which the Pacman strategy is a sequential best reply for the nlorlopolist when all buyers use the get-it-while-you-can strategy. Lemma 2 simplifies the analysis by demonstrating that we can restrict attention to strategies for the mo-

DURABLE-GOODS MONOPOLY

'47l

rlopolist that set the price in each period to the reservation price of some remaining buyer. LEMMA 2 . If all buyers use the get-it-while-you-can strategy, then in each period t there exists some I such that settirlgp(t) = 711 is a sequential best reply for the monopolist where at least one buyer who has not yet purchased the good at the beginning of period t has the reservation price 7f1. Proof. From the definition of zl,,,;,,(t) and the get-it-while-you-can strategy, the monopolist has no incentive to choose p(t) > u,,,;,,(t). So buyer would accept such an offer, and for discount factors strictly less than one, the delay decreases the present value of' the monopolist's profits. Reindex by I = 1, . . . , L(t) the reservation prices of buyers who have not yet purchased the good at the begirlnirlg of' period t. Suppose that it were optimal for the nlonopolist to choose p(t) such that 71,< p(t) < I zl,,,,,(t) for some 1 5 L 2 ( t ) - 1. By choosing instead p(t) + E with E > 0 and p(t) + E < zll, the morlopolist can strictly increase his profits without changing the behavior of any buyer. Hence, the original choice could not have been optimal. T h e same argument also shows that p(t) < ul,(,, is not optimal. Q.E.D. T h e followirlg notation will be useful. For Id = 1, let A = ~ ~ 1For 2 .I, 2 2 , define A as

T h e symbol A denotes the minimum distance between reservation price levels. This distance will be used as a lower bound on the mo~lopolist'smarginal loss from deviating from the Pacman strategy. Denote the initial number of buyers at each reservation price level by the vector n = ( n , , . . . , nl.), where nl is the initial number of buyers with reservation price ul. Let n2 = (0, n ~ ,., . . , n l ~ ) T . h e vector n2 represents the same distribution of buyers as n except that buyers with the highest reservation price have been removed. More generally, let n l = (0, . . . , 0 , nl, . . . , n,_) be the same distribution as n but with the highest 1 - 1 le\,els removed (I 5 L). T h e final lemma presents accounting identities that will simplify the proof of o u r main theorem. For a given set of reservation prices and initial buyers, let V*(n) be the present value of the monopolist's profits when all buyers use the get-it-while-you-can strategy and the monopolist uses the Pacman strategy. Let V(p, n ) denote the present value of the monopolist's profits when all buyers use the get-it-while-you-can strategy while the monopolist chooses an initial price p and uses the

'47'

J O U R S A L OF POLITICAI. E C O N O M Y

Pacman strategy thereafter. T h e following fornlulas follow immediatelv from these definitions. LEMMA3.

Lr(-o1, n)

~

+

~[ ? 1 2 7 ' (

nlzll

+

\'(I!/,

= 1 =

+

~ n3zll ~

+

..

+ ? t 1 7 ~ c+ pl'"(nl+ ,)I

n 2 ) for all 1 2 3 .

(12)

Lenlma 3 shows how the present value of the profits obtained by setting a n initial price and thereafter using the Pacnlarl strategy can be tiecornposed into the value obtained b) selling to the bu),ers with the highest reservation price a n d the value obtained by selling to the other buyers. Define S(n) by L

~ ( n =)

1

riiu1.

(13)

I= I

I n equation (13), S(n) represents the area under the initial demand curve and also the surplus available to a perfectly discri~ninatingtnonopolist. T h e term ( 1 - P)S(n)turns out to be an upper bound o n the monopolist's gain from deviating from the Pacman strategy. ?'he nlairl result of the section is a corollary of' the follo~vingtheorem. THEOREM 1. P ( L C T ~ C (all I I .buyers - ~ ~ use the get-it-rvhile-you-car1 strategy, then for all values of the discount factor such that 1

A lr(i12,n ) . Subtracting equation (1 1 ) f r o ~ t ei quation (10) produces the following result:

T h e first inequalit) follows from the definition of A and the definition of the Pacmar~strategy, Ivhich, together with equation (13), implies that S(n) > S(n.,) > I'*(nl,).T h e second inequalit!. follows from equation (14). It remains to s h o ~ vthat charging 7 1 in ~ period 1 and playing Pacman thereafter is in turn superior to charging any lower price in the first period and then playing Pacrnan. 'l'hat is, I r ( z l 2 , n) > I ' ( t l i , n). Since the induction hypothesis implies that it is optimal to use the Pacman strategy once the buyers with the highest r-eservation price

l474

JOURNAL. OF P O L I T I C A L E C O N O M Y

have been eliminated, I r * ( n y ) 2 I.'(vi, n 2 ) for all 1 2 3. Hence, equations (1 l ) and (12) imply that Lr(zl2,n) > L7(z1/,n ) for all l > 2. Since I ' ( - ~ Jn~) ,> \ ' ( z J ~n, ) > I'(vl, n) for 1 > 2, Pacman is a sequential best reply to get-it-while-you-can. Q.E.D. COROLI.ARY 1. T h e use of the Pacrnan strategy by the monopolist and the get-it-while-you-can strategy by each buyer constitutes a subgame-perfect equilibrium of the infinite-horizon market game for all discount factors satisfying the inequality in equation (14). T h e intuition underlying theorem 1 car1 be understood by considering the nionopolist's decision to set an initial price equal to 71, rather than to drop the price immediately to zlg. T h e benefits and costs of this decision are summarized in equation ( l 3 ) , which can be rewritten in the form

.The term r ~ ~ ( zu2) ~ on ~ the left-hand side of equation (16) is the marginal cost incurred by the monopolist when the price is prematurely lobvered to vy. 'The symbol A denotes a lower bound on this cost that is independent of the discount factor. In equation (16), (1 - P ) I r * ( n n )is the monopolist's marginal gain from dropping the price to 712. It represents the additional interest obtained by advancing the receipt of the payrnents from all inframarginal buyers by one period. Since V*(n2) is bounded above by the available surplus, the additional interest approaches zero as the discount factor approaches one. At each distinct reservation price level, the marginal cost of dropping the price before every buyer with that reservation price has purchased the good has a strictly positive lorzer bound that is independent o f t h e discount factor. On the other hand, the marginal gain from droppirlg the price can always be made smaller than the marginal cost by choosing a discount factor sufficiently close to one. Hence, for discount factors sufficiently close to one, the Pacman strategy is a sequential best reply when all buyers use the get-it-while-youcan strategy. From the definition of the Pacman strategy, we obtain the following equation for the present value of the monopolist's profits in the subgame-perfect equilibrium referred to in corollary 1 :

A conlparisorl of equations (13) arid (17) indicates that as P + 1, Ir*(n) -, S ( n ) . 'The present value of the monopolist's profits approaches the surplus achievable by a perfectly discrinlinatirlg monopolist as the discount factor approaches one. For any demand curve

DURABLE GOODS MONOPOLY

I473

with two o r more distirlct reservation prices, this surplus is larger than the present value achievable by Stokey's preconlnlitmerlt strategy. Bulo~r'srental strategy, o r the present value predicted by the Coase conjecture. Hence, the propositiorls of Coase, Stokey, and Bulow will generally be incorrect for the infinite-horizon game when the discount factor is sufficiently close to one.

V.

Concluding Remarks

I n this paper, we have examined three classic propositions of durablegoods monopoly theory when there is a finite number of buyers. I n each of the examples in Section 111, the strategy combination that we corlsidered forms a subgame-perfect equilibrium not only for the unbounded horizon but also for finite horizons of any length. For these examples, the seller earns strictly more than static monopoly profits for any discount factor and any assumption about horizon length. T h u s all three classic propositions are false if the standard assunlption of a c o n t i ~ l u u mof buyers is replaced by the more rlatural assumption that the set of buyers is finite. We have also shown that for a n arbitrarily large, finite collectiorl of bu!.ers and a n infinite horizon, there exists a subgame-perfect equilibrium in which a durable-goods monopolist can earn approximately the profits of a perfectly discrimirlatirlg monopolist for discourlt factors near one. This result differs strikingly from the case of a continuum of nonatomic buyers considered by the previous literature. If the monopolist is allowed to use the history-dependent strategies introduced by Xusubel a n d Deneckere (1989), then in an infinitehorizon game with a continuum of buyers, h e can obtain a present value that is bounded from above by the static monopoly profits. If these a n d similar strategies are ruled out, then C;ul et al. (1986) have shown that the monopolist loses virtuall!. all monopoly power in the remaining subgame-perfect equilibria as the discount factor approaches one.16 O u r finite-horizon examples also show that the classic propositions of Chase, Bulow, and Stokey cannot be rescued by any refinement of '"11 addition to assuming stationarity, GuI et al, impose a technical restriction o n the n~onopolist'sstrategy: they assume that the rnonopolist cannot condition his price offers o n sets of measure zero. Thiq assumption facilitates the anal)sis wit11 a continuum of buvers a n d is adopted as well by Ausubel a n d Ileneckere. (;ul et al. discuss this restriction o n p . 170 of their article. While the) show that the restriction doe5 exclude certain equilibr-ia, it remains unknown whether an) of t h r equilibria excluded by this technical assuniptioll conflicts with the thrust of their paper-that the Coase conjecture is valid with a continuum of b u l e r s . It is likewise unknown if any of the excluded equilibria violates Rulow's o r Stokey's propositions. Hence, it is concel\.able that the dornain In which o u r clairns a r e valid is even wider than we contend.

1476

JOUKNAI. OF POI.ITICAL ECOSOMY

t h e subgame-perfect equilibrium concept. T h i s follows since in t h e two-buyer, 7 - p e r i o d e x a m p l e discussed in Sectiotl 111, o n e c a n show by working backward fro111 t h e final period that t h e equilibrium payoff o f t h e monopolist is t h e s a m e i n every subgatne-perfect equilibriurrl. L V e conclude, t h e r e f o r e , that equilibrium behavior I\-hen t h e collection o f bu!,ers is finite may d i f t e r markedly f r o m t h e c o r r e s p o n d i n g behavior w h e n t h e r e is a c o n t i n u u m o f nonatornic buyers. T h e r e is n o d i s p l ~ t i n gt h e fact t h a t , b o t h in t h e "naturally o c c u r r i n g world" a n d in laboratory settings, t h e set of b u y e r s is finite. T h e artifice o f assuming a c o n t i n u u m o f nonatornic buyers tnay nonetheless b e a useful a p proxirnation if it facilitates analysis ulithout distorting results. It is t o r this I-easo11 t h a t t h e a p p r o x i m a t i o n has p r o v e d s o valuable in o t h ~ 7 contexts. H o r i e v e r , it is plainly nlisleading in t h e analysis o f durablegoods nlonopoly.

Appendix In this .-\ppendix \re verif.!-Sol. an) finite horicon or for the infinite horion-that for all 0 < p < 1, the follo\ring strategies coristitute a subganieperfect equilibl-iuni of' the three-11uyer. game in \vhich the b u ~ e r ' sresenation prices satisfr, the inequalities in ( 3 ) .111every period, the nionopolisr uses the tn!opic rnonopolr str-ategy that Ira5 defined in Section 111. Buyers 2 anrl 3 use strateg in e\el.y period. Huyer 1 uses the get-itthe get-it-\vliile-vo~~-c~~~i ivhile-)ou-can strateg) in the final period and in any periotl 1 < T in \\hich 11~r)er-s1 and Y are the onl! 1,emainirig bu!er-s \rho have not !et purchased the good 11) the beginning of period 1 . Finall\, in periorls in I\ hich buyers 1 arid 2 01- all three buyers remain ancl / < ?', bu\er 1 uses the strategy described in equation (6). \Ye need to \erify that, given the strategies of the other ~~l:t)ers, these strategies are optimitl for each pla)er- for- evel-1 possi1,le collection of remaining hulers at the beginning of e\er! period 1 (see also 11. 8). 'That is, the strategies ;Ire sequential best replies. In the course of' anal! ~ i n gthe t\\o-hu>erexample in Section 111, \ve ha\,e alread) \erifiecl that these strategies are sequential best replies in all periods in 12.hic.h onl\ one ~.ernainingbu)er has not e t pul-chased the good. h1ol.eover. a sut~gamethat begins in per-iod 1 with oril? 11uyer.s 1 and Y or bu)ers 2 and 3 rerriaining is the salrie as the t\\,o-l~uye~. garne in Section I11 except fhr- a change in the indexing of the l~iryers.Hence, the an:il>sis in Section 111 also demonstr:ites that the strategies are sequential best replies for thew SLI~Igarnes. Recall also that \c-e have pre\iously obser-\ed that the get-it-while-youcan strateg) is optimal fol- each hu)er in the final periorl of'the garne. Rloreover-, b constl-uction the m\opic monopol\ strateg! i? optirnal in the final periocl. Hence, these strategies are seq~~ential best replie? in the final period of the garrie. Recall that using the get-it-\\hile-)ou-can strateg) in period 1 is a sequential best reply fi~ra n i b11)el. who (.;In obtain o n 1 zero utilitt in all futur-e periods \\-lien the other player-s use their eq~~ilibr-iuni str-ategies.

'477

DURABLE-C;OOI)S MONOPOLY

Now considel. a suhgarrie I~egirinirigin period 1 in ~ v h i c honl) b ~ i > e r 1s anrl 2 remain and 1 < T. Bii)er 2 can ne\,er-obtain p o s i t i ~ e~itilit)in fiiture periods of this siil~gamegiven the strategies of the o t h e r player-s; hence, the get-it~vhile-yo~i-car1 s t r a t e g is optinla1 for- hirn. I t is clearl! optirri:il f o r bu!er 1 to accept any price less t h a n o r eqiizil to ? l a since bii)er 2 \vill accept siich a pr-ice a n d b ~ i ) e r1 ~voiilrlget ~ e r oiitilit) iri t h e eq~iilibriiimof the fhllo~vingone. 711 - p by accepting IILI\-ersubgame. For- :in price p > ?)?, I ~ i i > e1~obtains irn;nediatel anrl P ( 7 j 1 - T I ? )t) delii! irig acceptance o n e per-iod. (If' I ~ ~ i y e1r dela!s, then the rrionopolist, follo~vingthe myopic nioriopol! strateg), ~vill , \ve have a l r e ; ~ dseen, it \vill be offer a price 71.2 i r i t h e next period, ~ v h i c h ;IS optinla1 for I ~ i i > e1~to . accept.) T h e value obtained ,t accepting inllnerliatel! is greater than o r equal to the valiie obtained 1, delay as long as p 5 ,!I* = (1 - P)7l1 P 7 j 2 . Hence, b i i > e ~1's . proposerl strategy is optimzil. If the rnonopolist in this subg;~rrielo\vered his price belo~v712, \,olurrie ~ v o ~ i l r l not increase, so such ;I deviation ~ v o u l dbe less profital~le.As f o r 111-ice increases, the best of t h e higher- prices i s p * = (1 - P)7'1 + P71a. But deviating to that price is suboptim:il since the rrloriopolist ~ v o u l d then e a r n p * in the (the ecl~iilib~.i~im profit in the continu;~tiorione-hii!erciirrent period a n d subgame) in t h e next, which together prodiice a pr-esent valiie less than 271.?, the present \,slue o l ~ t a i n e diisirig the rri>opic rrionopol! strateg) ( [ l - P]711 + P271., < 27'.,). Hence, t h e myopic rnoriopoly str;~teg\is optinial for- the monopolist in periods in ~ v h i c honl! buyers 1 a n d 2 remain. Finall), consirler the thr-ee-11u)er subgames beginning in per-iod 1 with 1 < T. (;i\en t h e ecl~iilil~r-i~irri strategies, b~i)er-s2 a n d :4 cannot obtairi positi\,e ~itilit)in f ~ i t i i r eperiods of' the three-bii!er subgarrie. Hence, the get-it-~vhileo u - c a n strateg! is optimal f o r therri. I t is optimal for- 11ii)er 1 to accept :in) price less than o r equal to since b ~ i > e2r ~voulclaccept such a 111-icea n d i r nthe follo~vingone- o r I~u!er 1 \vo~ildget ~ e r osiirplus i r i t h e e q ~ i i l i l ~ r i ~of t~vo-11u)ersu11g:irries. As in t h e pr-e\,io~ist~vo-buyerciise, if I~iiyer1 fails to p > iri the ciirrent period, then t h e rrionopolist ~villofftr- a accept . a price . price 7'2 i r i the ilext period, ~ v h i c h11u)el. 1 \vill accept. ~ e r i c e , - f o ran! - pricep > buyer 1 o11t;lins 7 ' 1 - I!, 13: accepting irrimediatel! a n d P ( 7 ' ] - I ) ? ) by del:i)ing acceptance o n e per-iod. T h e \.slue ol~tainerlI]> accepting irrirrierliatel! is greater than o r equal to the value obtainerl hy rlela) ;IS long :is p 5 p*. Hence, bii!er 1's strategy is optimal. Il'hen the rnonopolist uses the rnyopic monopol! s t r a t e g in the threem \,:ilue 2?1,2 + buyer subgame ~ v i t ht < T, he receives the e q ~ i i l i b r i ~ ipresent P I ~ . ~If'. the rrionopolist \\.ere to de\,i;~tef r o m this str-ateg) b) lo~veringthe price, then the only sensible choice ~ v o ~ i lbe d ?I.(. But that ~vo~ilcl result in a smaller pr.ofit since 371:~ < 2712 + PI^.^. If' the rnonopolist deviaterl 1, raising his price, then t h e oril! sensible choice \voiilrl he ,!I* = (1 - P ) ? l I + PI)?, ~vhich o n 1 11~i)er 1 ~voulcl accept. T o g e t h e r ~ v i t hthe eqiiilibri~irn profits in the this de\iation p~.orlucesthe present value (1 follo~vingt \ v o - I ~ i i > esiihgame, ~. P ) 7 ' l + P2712 if 1 = T - 1 a n d the present \,:ilue (1 - P)ill P2?1? + if 1 < T - 1. Both these \ d u e s a r e less than the present \,aliie ol~tairiedh! using the rri)opic rrionopoly s t r a t e g ; hence, such ;I deviation ~ v o u l dbe suboptimal. T h e r e f o r e , t h e myopic ~ r i o n o p o l \strateg! is optimal for the monopolist in t h r e e - b u ) e r subgarries. T h e analysis abo\ e is Lalid f o r all 0 < P < 1 . Hence, for- all discoiint f:icto~.s I~et~veeri zero a n d o n e , the strategies disciissed in Section 111 constitute a sul~garrie-perfecteq~iilibr-iurno f t h e thr-ee-11~i)er.T-period garrie ~ v h e nthe initial reservation prices satisf? equ:itiori ( 5 ) .

+

7j2,

+

~

~

?

l

,

4

147~

jOCRNAL O F POLITICAL. ECONOMY

n o t e that the process of \ e r ~ f Ing \ the subgame perfect~onof a set of strategles 111\ol\escorrlparlng the \slue ol~talnedfrom a urlllateral dellatlon to the \slue obtalned frorn the proposed e q ~ u l ~ b r ~strategles urn For o u r three-hu\ e r game, the l a l u e obtained frorrl a dellatlon t \ p ~ c a l l \cons~sterlof a o n e - p e r ~ o d pa\ off a n d (poss~bl\ ) the pa\ off obtalnecl In the eclull~br~unl of solrle contlnuatlon subgame hloreoler, the e c j u l l ~ b r ~ istrategles ~ln that we describe take at most t\\o periods to execute Hence, for a n \ subgame u ~ t hfour o r more paboffs used to \ e r ~ f \the optlperlods to go, the e q u ~ l ~ b r ~ ucontlnuatlon rn rnal~t\of the e q u ~ l ~ h r ~cuhron~ c e In s the first perlod of the subgarne a r e ohtalrled b\ pla\ing out the \arlous contlnuat~onsubgarnes unconstmlned b\ the f i n ~ t e h o r l ~ o nanrl u ~ t h o u using t the h n a l - p e r ~ o dstrategles (\\hlch can d ~ f f e rfrorn the statlonnrb strategles used In all earller perlocis) 1h ~ r sthe salrle cornparlsons that \ e r ~ f \the o p t ~ m a l ~of t \ the proposed strategles In a n \ s ~ t b g a ~ nu e~ t h four o r more p e r ~ o d sto g o also l n i p l ~that these strategles constitute a subgallie-perfect e q ~ u l ~ h r ~ for u r nthe ~ n f i n ~ t e - h o r galrle e the ~ ~ o n We c o n c l ~ ~ dthat t e q u l l ~ b r ~ ufor ~n strategle5 speclhccl abobe also constitute a s~rl~garne-perfet the three-hu\er garrle u ~ t ha n ~ n f i n ~ horlron, te 0 < P < 1, and reserlatlon prices s a t ~ s f t l n gequatlon (5) " References

Ai~suhel,La~vrence,a n d Deneckere, Ra)rnond. "Reputation in Bargaining a n d Durable Goods Monopol! ." Econorn~tricc~ 57 (Ma! 1989): 5 1 1-3 1. Bagnoli, hlark; Salant, Stephen W.; and S~vierzbi~iski. Joseph E. "Price Discrilrlination \,ia Irlterternporal Self-Selectior~."hlanuscript. Ann Arbor: Uni\,. Michigan, 1989. Bulo\v, Jereln) I. "Durable-Goods hlonopoIists.",J.P.E. 90 (April 1982): 31432. Coase, Ronalcl F I . "Dural~ilit! a n d hfonopol\." J. Luzc) urld Econ. 15 (April 1972): 143-49. Gul, Faruk; Sonnenschein, Hugo; a n d Wilson, Robert. "Fo~rndationsof D!narrlic hlonopol! a n d the C:oase C;or!jecture." J. Econ. Tfleo~y 39 (.June 1986): 155-90. Kahn, Charles hf. "The Durable C;oorls hlonopolist a n d Consistenc ~vith Increasing C:osts." Economelricu 54 (hlarch 1986): 275-94. Sohel, Joel, a n d Takahashi, Ichiro. "A Multistage hlodel of Bargaining." Re?). Econ. .Studie., 50 (Jul) 1983): 41 1-26. Stoke), Nanc! L. "Interternporal Price Discrimination." 4 J . E . 93 (August 1979): 355-7 1. . "Rational Expectations a n d Durable Goocls Pricing." Bell J. Econ. 12 (Spring 198 1): 1 12-28.

1;

I n a n v period f in which either buyers 1 and 2 o r all three buyers have not \ct purchased the good, buyer 1 i11 out- three-buyer game uses the get-it-rvhile-you-can strategy if' t = 7' and the st1-ategy described in eq. (6) it / < 7.. Since there is no "final game, bu?er 1 a1wa)s uses the strategy in eq. (6) in any

period" in the infir~ite-horizor~ perlod o f the infinite-horizon game in lvhich eithei- only buyer-s 1 and 2 or all three

buyers renlai11.

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You have printed the following article: Durable-Goods Monopoly with Discrete Demand Mark Bagnoli; Stephen W. Salant; Joseph E. Swierzbinski The Journal of Political Economy, Vol. 97, No. 6. (Dec., 1989), pp. 1459-1478. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28198912%2997%3A6%3C1459%3ADMWDD%3E2.0.CO%3B2-0

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References Reputation in Bargaining and Durable Goods Monopoly Lawrence M. Ausubel; Raymond J. Deneckere Econometrica, Vol. 57, No. 3. (May, 1989), pp. 511-531. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198905%2957%3A3%3C511%3ARIBADG%3E2.0.CO%3B2-1

The Durable Goods Monopolist and Consistency with Increasing Costs Charles Kahn Econometrica, Vol. 54, No. 2. (Mar., 1986), pp. 275-294. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198603%2954%3A2%3C275%3ATDGMAC%3E2.0.CO%3B2-P

A Multistage Model of Bargaining Joel Sobel; Ichiro Takahashi The Review of Economic Studies, Vol. 50, No. 3. (Jul., 1983), pp. 411-426. Stable URL: http://links.jstor.org/sici?sici=0034-6527%28198307%2950%3A3%3C411%3AAMMOB%3E2.0.CO%3B2-D