Matching with Frictions and Entry with Poisson Distributed Buyers and Sellers.

Transcription

1 Matchig with Frictios ad Etry with Poisso Distributed Buyers ad Sellers. Peter Norma February 4, 015 Abstract I cosider a simple directed search model with a fiite umber of buyers ad sellers. The mai iovatio is that buyers ad sellers are radomly draw from idepedet Poisso distributios. This provides a simple justificatio for the usual equilibrium selectio where oly symmetric radomizatios by buyers are cosidered, because ay equilibrium is payoff equivalet to such symmetric equilibrium. The Poisso assumptio also makes the model more tractable. A simple proof of uiqueess of equilibrium prices uses the fact that prices are strategic complemets. It also becomes tractable to hadle etry with a fiite umber of players, which is diffi cult with a fixed fiite set of sellers. Keywords: Poisso Distributio, Directed Search, Etry. JEL Classificatio Number: D43, L13. I thak Beoit Julie, Joh Kees, Ia Kig, Fei Li, Ca Tia, Roald Wolthoff, Rady Wright, semiar participats at the Chicago Fed ad participats at the 013 Midwest Macro coferece at Uiversity of Illiois at Urbaa-Champaig for commets. The usual disclaimer applies. Departmet of Ecoomics, Uiversity of North Carolia at Chapel Hill, 107 Garder Hall, CB 3305, Chapel Hill, NC ormap@ .uc.edu

2 1 Itroductio This paper cosiders a simple versio of a directed search model, where a fiite set of buyers ad sellers are draw from idepedet Poisso distributios. The most importat cosequece of this modellig strategy is that there is o loger ay loss of geerality i focusig o equilibria where all buyers follow a symmetric mixed strategy. I stadard fiite directed search models, such as Peters 000 ad Burdett et al 001 there are equilibria where idetities are used i order to improve coordiatio. To uderstad this, cosider the simplest case with two sellers ad two buyers. It is the easy to see that there is a multiplicity of cotiuatio equilibria followig ay pair of prices that are ot too far apart. Oe possibility is that buyer 1 visits seller 1 ad seller visits seller. Aother possibility is that the buyers radomize i a way so that both buyers are idifferet. All players, icludig the seller, are strictly worse off i the mixed strategy cotiuatio equilibrium. Hece buyers ca use radomized cotiuatio equilibria as credible threats to support a cotiuum of equilibrium prices. 1 I order to obtai predictive power the directed search literature has focused o equilibria where the buyers or workers i labor market applicatios follow the same mixed strategy after ay history of play. This equilibrium selectio is usually justified by a iformal argumet based o the idea that usig idetities to coordiate is diffi cult i a market with may buyers ad sellers or workers ad firms. A more formal argumet justifyig the stadard equilibrium selectio ca be foud i Blad ad Loertscher 01. I cotrast, there is o eed to make a equilibrium selectio i the framework cosidered i this paper. Ay equilibrium is payoff equivalet to a equilibrium i which all buyers use a symmetric mixed strategy after ay history of play. There are typically equilibria where, say, gree buyers are more likely to visit firm 1 ad red buyers are more likely to visit firm, but we may as well igore them as both the buyers ad the sellers ear the same payoffs i a symmetric equilibrium. This payoff equivalece makes it impossible to build equilibria built o puishig ad rewardig sellers by selectig differet types of cotiuatio equilibria depedig o which prices are posted. Hece, the uiqueess of equilibrium prices that will be discussed further below is ot cotiget o selectig the symmetric mixed strategy i every cotiuatio game as it is i Galeiaos ad Kircher, 01 ad Kim ad Camera 014. While justifyig symmetric mixed strategies is arguably the most importat cotributio of the paper, the setup with Poisso distributed players also makes the aalysis cosiderably more tractable tha stadard fiite models. There are two mai sources for the improved tractability. The first reaso is that equilibrium arrivals of buyers are distributed i accordace with a Poisso distributio which makes the seller profit maximizatio problem somewhat cleaer. However, a more importat chage is that the Poisso assumptio creates a model with what Myerso 000 refers to as evirometal equivalece: beig bor or beig of ay particular type does ot affect the beliefs over the umber or the types of what other players are preset. This property is particularly importat i that it facilitates the costructio of equilibrium etry strategies, which 1 See Burdett et al 001 for details. 1

3 otherwise is a rather diffi cult task with fiite umbers. While differig i terms of specifics, most papers i the applied directed search literature cosider what I heceforth refer to as market utility models. These models assume that all agets are small, so that the choice of ay particular seller, firm or mechaism desiger, caot affect the utility of the agets that search. This simplifies the aalysis i may ways, but is also coceptually awkward as oe caot ask what happes if a sigle actor deviates from the equilibrium path. To build i sequetial ratioality it is therefore ecessary to use various tricks, such as askig what will happe if a small measure of firms would deviate ad the take the limit as this measure approaches zero. Naturally, some papers have asked whether there are game theoretic foudatios for the market utility approach. I particular, Peters 000 ad Burdett et al 001 cosider simple fiite models that are very similar to the oe studied i this paper ad establish that sequeces of subgame perfect equilibria coverge to the market utility outcome. More recetly, Galeiaos ad Kircher 01 provide a substatial geeralizatio of Peters 000 that allows for more flexible productio ad matchig fuctios. Market utility models simplify the aalysis for two reasos. Firstly, the absece of strategic effects creates a much simpler fixed poit problem tha i the fiite versio. Secodly, if there is a etry decisio, equilibrium etry boils dow to a simple idifferece coditio. The literature has focused o the first of these issues, but largely igored the secod. 3 Naively, the problem of hadlig etry with a fiite set of agets may appear rather uiterestig. Oe may thik that idifferece ca be replaced by a coditio that says that the etry cost must be i betwee the equilibrium seller profits ad the profits i case a extra seller eters. This is ot the case. The reaso is that the profit that a firm makes whe m other firms eter is higher whe the firm eters the market uexpectedly compared to the case whe the other m firms expect the firm to eter. For this reaso, equilibria with pure etry strategies will geerally fail to exist. I this paper I cosider two variats of the simplest possible directed search model with homogeous buyers ad sellers. I both cases, buyers are draw from a Poisso distributio, but I have results for both the case with a fixed set of sellers ad with Poisso distributed sellers. The aalysis with a fixed set of sellers cotais o surprisig results. Usig argumets that are more or less idetical with those i Galeiaos ad Kircher 01 oe shows strict cocavity of the profit fuctio i the relevat rage. Also, agai reproducig a kow result for the case with a fixed set of buyers ad sellers, I show that there is a uique equilibrium. 4 While this is ot a ew result i itself, the proof is differet ad, arguably, more istructive. The argumet relies o two properties, symmetry ad prices beig strategic complemets. That is, I first show the best respose is strictly icreasig i the price posted by ay other seller that attracts buyers with See also Peters 1984, Julie et al 000, ad Lester The oe exceptio I am aware of is Geromichalos See Galeiaos ad Kircher 01 ad Kim ad Camera 014.

4 positive probability. This strategic complemetarity together with symmetry immediately rules out ay asymmetric equilibrium as the best respose of the seller postig the higher price is lower, which is a cotradictio. It follows from a direct calculatio that there exists a uique symmetric equilibrium, ad combied with the o-existece of asymmetric equilibria, this is ideed the uique equilibrium of the model. Addig etry to the model I first demostrate that pure strategy equilibria geerally fail to exist i the model with a give set of potetial buyers. Presumably mixed strategy equilibria exist, but these are ot easy to characterize. I cotrast, whe sellers are distributed i accordace with a Poisso distributio a essetially uique equilibrium is remarkably simple to characterize. I equilibrium, all sellers that eter the market post the same price. The price is strictly decreasig i the etry probability, ad the equilibrium is essetially uique i that the average etry probability is determied uiquely by a zero profit coditio. I the limit as the umber of buyers ad sellers go to ifiity, the equilibrium approaches the market utility bechmark. The Model The model is a variat of Burdett et al 001 where buyers ad potetial sellers are draw from a Poisso distributio, ad where sellers face a o-trivial etry decisio. As moves are sequetial it is ot strictly speakig a Poisso game i the sese of Myerso 000, but the aalysis is very similar. 5 There are s sellers draw from a Poisso distributio with expected value m ad b buyers draw from a Poisso distributio with expected value. All buyers value the object at 1 ad all sellers value the object at some K 0 which is commo kowledge. A alterative iterpretatio is that K is the cost of eterig the marketplace. The extesive form is as follows: 1. First, ature draws s potetial sellers ad b buyers. To allow for coordiatio we also assume that each seller ad buyer is equipped with a observable characteristic. Formally, we let t T {1,..., T } be the type of a seller ad c C {1,..., C}. Type t sellers are draw from a Poisso with expected value m t ad type c buyers are draw from a Poisso with expected value c where m t T m t ad c C c are the expected umber of buyers ad sellers. Types are payoff irrelevat i the baselie model as all sellers of ay type attach value K 0 to it ad all buyers of ay type value the good at 1.. Each potetial seller decides whether or ot to eter the market. Coditioal o etry sellers post prices ad sellers do t observe how may other sellers or which types are realized whe takig their actios. A seller strategy is thus a pair e, p where e : T [0, 1] maps the type of player to a probability of etry ad p : T R + is the pricig strategy. 5 I directed search models, buyers react to the posted prices. Hece, buyers must be able to ex post distiguish betwee differet sellers. 3

5 3. Buyers observe how may sellers of each type are realized ad the posted ad decide which seller to visit. That is, a history is a vector m 1, p 1,.., m T, p T N R + T where N is the set of o-egative itegers. Hece, a strategy is a map θ : C N R + T m 1+m +...+m T +1, where we ote that the assumptio is that buyers ca distiguish betwee differet sellers of the same type. The reaso that it is the m 1 +m +...+m T +1 dimesioal simplex ad ot m 1 + m m T is that we allow the buyer to visit obody. If the buyers igore the type of the player or if there is a fixed kow set of players, which is a case that will be cosidered below a strategy ca be writte θ : [0, 1] m m If at least oe buyer visits a seller oe of the visitig buyers is chose to receive the good at the posted price. As buyers do t kow how may other buyers there are whe decidig which seller to visit the secod stage is ot a proper subgame, so subgame perfectio is formally ot applicable. However, the solutio algoritm is still just like how oe would solve for a subgame perfect equilibrium. After ay set of posted prices the buyers must optimize ad, while moves of ature are ivolved, the relevat beliefs over the umber of buyers are give by the prior. Clearly, it is possible for a firm to post a price high eough so that o buyer shows up i equilibrium, implyig that the arrival of a buyer is off the equilibrium. However, this is irrelevat as there are o moves left for the firms, implyig that a equilibrium say weak perfect Bayesia will be characterized by backwards iductio. 3 The Market Utility Bechmark Cosider a competitive versio of the model where we imagie a cotiuum of buyers ad sellers. The, i equilibrium, ay buyer must be idifferet betwee all sellers, so we let U be the utility for a buyer that visits ay active seller. Also assume that the ratio of buyers to potetial sellers is give by ρ ad that a proportio α of the potetial sellers that are active. The ratio of buyers to active sellers is the ρ α. Also, assume that a sigle seller posts price p ad let λ p, U be the Poisso arrival at this firm as a fuctio of it s posted price ad the utility of visitig ay other active seller. Note that if the Poisso arrival probability of buyers at a seller is λ ad that the seller radomizes with equal probabilities whe pickig a buyer, the the probability that a buyer is successful whe visitig this seller is 1 e λ λ. 6 It follows that the coditio that buyers are idifferet betwee visitig the firm postig price p ad ay other firm is 1 e λp,u 1 p λ p, U U. 1 6 This is a well-kow property of the Poisso distributio, but a direct calculatio is provided below i 5 i the cotext of the fiite model i order to make the paper self cotaied. 4

6 A represetative seller will therefore solve the problem max p p 1 e λp,u, where λ p, U is defied i 1. However, otig that 1 e x x is strictly decreasig i x we may chage variable from p to λ. That is, suppressig argumets ad solvig 1 we have that p 1 λu 1 e λ implyig that the problem for a idividual seller is 1 max λ λu 1 e λ 1 e λ max λ 1 e λ λu. The problem i is a strictly cocave problem ad by takig the first order coditio ad substitutig the coditio U 1 p 1 e λ λ which must hold i a symmetric equilibrium we have that the price must be p λ 1 e λ 1 + λ 1 e λ 1 λ e λ 1. 3 Substitutig the price i 3 back ito the objective fuctio oe fids that the maximized equilibrium profit is 1 e λ 1 + λ, which is strictly icreasig i the Poisso arrival parameter λ. To make the etry decisio o-trivial, I assume that 1 e ρ 1 + ρ < K < 1. This meas that the cost of etry is somewhere i betwee the profit sellers make if all sellers are active ad the limitig profit for the case whe sellers match with a buyer for sure. Equilibrium etry must the be such that all sellers are idifferet betwee stayig outside the market ad beig active. That is, the probability of etry, α, must solve 1 e ρ α 1 + ρ α K, 4 which has a uique solutio. We coclude that λ ρ α i order to make active ad passive sellers idifferet, ad the associated equilibrium price is give by 3 evaluated at ρ α The procedure described i this Sectio is a simple example of a market utility model with etry. Clearly, there are several somewhat ad hoc steps i the aalysis, ad the rest of the paper is cocered with the questio of whether this aalysis ca be viewed as a limit of a model with fiite umbers of sellers ad buyers. 4 The Case with a Determiistic Set of Sellers As a first step i the aalysis I cosider the case with m sellers who assig a zero reservatio value of the product. The reaso for this is twofold. Firstly, it allows me to idetify why hadlig etry is diffi cult with a kow fiite set of firms. Secodly, I show that all equilibria are payoff equivalet with a equilibrium i which all buyers follow the same mixed strategy, which is sometimes referred to as a directed search equilibrium. As a by-product, I also show that prices are strategic complemets which allows me to costruct a proof of uiqueess of equilibria which is cosiderable simpler tha existig argumets. 4.1 Buyer Payoffs with Symmetric Buyer Strategies If the umber of buyers is a Poisso with expected value ad all buyers go to seller i with probability θ i the the umber of buyers at seller i is a Poisso with expected value θ i. This is a 5

7 well kow property of the Poisso distributio, but for the coveiece of the reader I demostrate this i the appedix. Next, a Poisso setup exhibits what Myerso 000 labels evirometal equivalece. 7 That is, if the ucoditioal probability distributio over the umber of buyers at seller i is a Poisso with parameter θ i, the, from the poit of view of a idividual buyer, the probability distributio of the umber of other buyers at seller i is also a Poisso with parameter θ i. Hece, beig bor cotais o iformatio over how may other agets are realized. It follows that the probability that a buyer is successful ad acquires the good whe visitig seller i is Pr [buyig from seller i] v0 1 v + 1 e θ i θ i v } v! {{} Prob v other buyers 1 θ i [ 1 e θ i [θ i ] v e θ i θ i v! 0! v0 ] v0 e θ i v + 1! [θ i] v e θ i θ i. I write θ i p for the probability to visit firm i as a fuctio of the price vector p, which so far is restricted to be the same for all buyers. To simplify otatio slightly, I defie λ i p θ i p for each seller i. As choosig λ p λ 1 p,..., λ p such that i λ i p is a just chage i variables from choosig a mixed strategy it is irrelevat whether we view λ or θ as the strategic variable. I will use this chage of variables for the remaider of the paper. Whe cosiderig a arbitrary p ad seller i there are two possibilities. Either o buyer visits seller i i which case gettig the good at price p i is weakly worse tha visitig ay other seller. The other possibility is that the arrival rate is positive i which case the buyer must be idifferet betwee firm i ad ay other active seller, that is 1 p i 1 e λ ip λ i p must hold for each i, j such that λ i p > 0 ad λ j p > 0. 1 p j 1 e λ jp, 6 λ j p 4. The Poisso Assumptio Justifies Lack of Coordiatio I directed search models with a fiite set of agets there is a plethora of equilibria that ca be supported by buyers coordiatig o qualitatively differet cotiuatio equilibria after differet price vector posted by the sellers. Almost all of the literature, however, focus o the case where buyers play a symmetric mixed strategy after ay history of prices posted. This approach is typically justified by arguig that coordiatio is hard i large aoymous markets. 8 I cotrast, the setup with Poisso distributed buyers is a eviromet i which there is lack of coordiatio a ay equilibrium. 7 This property is also plays a role i Lester et.al. 014 where it is called idepedece. 8 See Blad ad Loertscher 01 for a more formal justificatio. 6

8 Propositio 1 Suppose that θ : C R m + m is a equilibrium cotiuatio strategy for the buyers. 9 The there exists a symmetric equilibrium cotiuatio strategy θ : R m + m such that all buyers ad sellers ear the same payoffs as whe buyers are followig θ. Proof. Let θ i1 be the probability that type 1 visits i ad θ i be the probability that type visits seller i. Sice buyers of the two types are draw idepedetly the probability that v buyers visit seller i is v e θ i1 1 θi1 k 1 e θ i θi v k k0 k! v k! e θ i1 1 + θ i v! v k0 v! k v k θic 1 θi k! v k! e θ i1 1 + θ i θic 1 + v! θ v i, 7 by use of the biomial theorem. We coclude that the umber of sellers that are either type 1 or is a Poisso distributio with parameter θ ic 1 + θ i. By iductio it follows that the probability that v buyers from C visit i is a Poisso with expected queue legth c C θ ic c. For every price vector p ad every seller i let θ i p be defied as c C θ i p θ ic p c c C. 8 c If each buyer of ay type follows θ i p the expected queue legth is θ i p c C c c C θ ic p c. Hece, the payoffs for all buyers ad sellers are uchaged, implyig that the symmetric radomizatio is also a equilibrium. Hece, either buyers or sellers ca do ay better or worse by coditioig o payoff irrelevat buyer characteristics. I will therefore assume that all buyers follow a symmetric mixed strategy i the remaider of the paper. 4.3 Seller Payoffs Propositio 1 establishes that for ay price vector p we may restrict attetio to a symmetric cotiuatio equilibrium i which the arrival rate at firm i is some λ i p. A seller ears p i provided that at least oe buyer cotacts the seller. The probability that o buyer visits is e λ ip, so a sale is cosummated with probability 1 e λ ip. The payoff fuctio for a seller, takig sequetially ratioal cotiuatio strategies by the buyers ito cosideratio, is thus o the form exactly like the market utility bechmark. π i p p i 1 e λ ip, 9 The differece with the market utility model is that the equilibrium buyer utility will be affected by prices set by idividual sellers. However, the form of 9 is evertheless makig the model more tractable tha the stadard settig with a give fiite set of buyers. 9 We have removed seller types.the argumet trivially geeralizes to the case with radom sellers with differet observables. The reaso is that we assume that buyers ca distiguish differet sellers of the same type, which implies that the type is redudat at this stage. 7

9 4.4 Equilibrium While there are formally o proper subgames we ca defie a equilibrium i aalogy with subgame perfectio: Defiitio 1 The pair λ, p with λ : [0, 1] m { x R + i1 x i } ad p [0, 1] m is a equilibrium if: 1. The idifferece coditio 6 holds each p ad each i, j such that λ i p > 0 ad λ j p > 0. Moreover λ i p 0 holds if there exists j i such that 1 p i 1 p j 1 e λ jp λ j p.. p i arg max p i p i 1 e λ i p i,p i. Coditio 1 says that all buyers must follow a commo sequetially ratioal rule to select which seller to visit after every coceivable price vector. This geerates a sequetially ratioal expected arrival fuctio, ad the secod coditio says that each seller must maximize profits, takig prices by other sellers ad the sequetially ratioal demad as give. 4.5 Equilibrium Characterizatio Cocavity of Seller s Profit Fuctio The typical approach to characterize equilibria i directed search models is to focus o symmetric equilibria. This allows for a chage i variable from price to the Poisso arrival rate whe solvig the seller profit maximizatio problem, followig exactly the same logic as the derivatio of the payoff fuctio i the market utility bechmark. As I allow for asymmetric pricig strategies this approach does o loger work. Prices are therefore kept as the strategic variables for the firms. To simplify otatio I defie g x 1 e x, 10 x ad rewrtie the idifferece coditios 6 as 1 p i g λ i p 1 p j g λ j p Oe should ote that λ i p 0 for p i that are too close to 1. Specifically, if p i p i p i defied as the uique solutio to 1 p i 1 p j g λ j p i 0, 1 the λ i p 0. This creates a o-cocavity of the seller profit fuctio that is discussed i further detail by Galeiaos ad Kircher 01. Because of the simple structure of payoffs i this paper, 8

10 the o-cocavity does ot create ay aalytical diffi culties as it occurs i a rage where o firm likes to be. For each firm i there are m 1 idifferece coditios ad, as all remaiig idifferece coditios are satisfied provided that a buyer is idifferet betwee i ad ay other seller we may simply igore all other coditios. It is easy to use the implicit fuctio theorem to check that λ i is cotiuously twice differetiable i all argumets explicit expressio for the derivatives are give i 15 ad 16 so we may differetiate the profit fuctio i 9 with respect to p i to obtai the first ad secod derivative π i p π i p p i 1 e λ ip + p i e λ ip λ i p 13 e λ ip λ i p [ ] p i e λ λi p ip + p i e λ ip λ i p. Give that all sellers are active which is the case i equilibrium we have that the derivatives of λ 1 p,..., λ m p are implicitly defied by the m 1 idifferece coditios i 11. Differetiatig the idetities i 11 with respect to p i we get m 1 distict equalities p i g λ i p + 1 p i g λ i p λ i p 1 p j λ j p. 14 Rearragig ad summig over the the coditios i 14 we express the partial derivative of the expected umber of arrivig buyers with respect to p i as λ i p ad that the cross derivatives are give by g λ i p 1 p i g λ i p gλ i p. 15 λ j p g λ i p 1 p i g λ i p λ i p k i gλ k p g λ k p gλ k p k1 g λ k p 16 It is easy to check that λ ip < 0 ad λ jp > 0 for every j i. This is also easy to uderstad, as a icrease i the price charged by seller i reduces the expected payoff for a buyer who visits seller i. To restore idifferece the probability that a buyer is served must icrease, ad to keep idifferece across all other sellers the probability that a buyer is served must all move i the same directio. Take together, the oly possibility is that λ i p decreases ad λ j p icreases for every j i as p i is icreased ad all other prices are held fix. It ivolves more work to evaluate the secod derivative of the profit fuctio. To do so it turs out that the followig fact is very helpful. Lemma 1 For ay x > 0 we have that g x < g x g x < [g x]. 9

11 For completeess, a direct calculatio that establishes Lemma 1 is i the appedix, but it is easy to check the result by simply plottig the fuctios. Usig the two iequalities i Lemma 1 it is easy to show that λ i p is strictly cocave i p i, which i tur implies that the payoff fuctio is globally cocave i p i i the rage where λ i p 0. This should be of o surprise give Lemma 3 i Galeiaos ad Kircher 01, but sice the matchig fuctio is ot exactly the stadard ur-ball matchig, a proof is required. Propositio λ i p < 0 for every seller i ad every p [0, 1] m such that all firms are visited p i with positive probability. Hece, the profit fuctio for firm i is globally strictly cocave i p i o [0, p i p i ]. Obviously, the o-cocavity is ot relevat if all sellers use pure strategies. I that case we ca simply rule out p i > p i p i as attractig o buyers is the worst possible outcome. Ideed, usig the symmetry of the model we ca show that ay equilibrium must be i pure strategies, implyig that the o-cocavity at the upper ed is irrelevat for the aalysis: Propositio 3 There ca be o mixed strategy equilibrium Proof. For each player i let p 1 be the lower boud o the support of the mixed strategy used by player i i equilibrium. Also, relabel the players so that p 1 p... p m. First cosider seller 1 ad ote that p 1 arg max π i p 1, p 1 p 1 m df p j where the cumulative distributio F p j is the possibly mixed strategy played by j. Obviously p 1 < p 1 p 1 for ay p 1 i the support of the possibly mixed strategy equilibrium. As 1 sums of strictly cocave fuctios are strictly cocave; each π 1 p 1, p 1 is strictly cocave o [0, if p 1 p 1 ] > p 1, ad 3 π 1 p 1, p 1 > π 1 p 1, p 1 0 if p 1 p 1 p 1. Take together this implies that m p 1 π i p 1, p 1 j df p j > p 1 π i p 1, p 1 m j df p j for ay p 1 p 1 so player 1 must play a pure strategy. For iductio, assume that players 1,..., m play pure strategies. The, p i < p i p i for ay p i i the support of the equilibrium. Usig the same argumet as above this implies that player i must play a pure strategy. By iductio, all players must play pure strategies. j 4.5. Symmetric Equilibria The usual approach to derive a symmetric equilibrium is to chage the strategic variable from p i to λ i ad ote that if all other sellers post the same price, the it must be that the arrival for ay j i. That is, the idifferece coditio 11 implies that p i ca be probability is λ i m 1 expressed i terms of λ i ad p, the price charged by all other sellers as λ i p i 1 1 p 1 e λ i 10 1 e λ i m 1 m 1 λ i. 17

12 Elimiatig p i from the objective fuctio i 9 we obtai the reduced form optimizatio problem i λ i give by max λ i 1 e λ i 1 p λ i 1 e λ i m 1 λ i m Oe ca solve 18 ad the impose the equilibrium coditio λ i m. The result is the exact same cadidate equilibrium price as what I derive below. However, the symmetric equilibrium is as easily derived by directly maximizig 9 over p i. By strict cocavity ad the fact that corers ca be ruled out ay equilibrium must satisfy the first order coditio If p j p for all j it follows that λ i p m ad λ i p 0 1 e λ ip + p i e λ ip λ i p. 19 g m 1 p i g m g m g m g m g m ad substitutig ito 19 ad rearragig gives p 1 e m 1 + m 1 e [ ] 1 e ρ 1 + ρ m 1 + [1 e ρ ] m 1 e m m 1 p i 1 e m 1+ m m 1 ρ e ρ 1 m 1 m, 0 as, m ad m ρ. The reader should ote that Propositio guaratees that p p,..., p is a symmetric equilibrium. Additioally, usig the fact that corers ca be ruled out, 19 is ecessary, so there ca be o other equilibria. Hece, sice p is the uique price such that 19 holds whe p j p for all j we ca coclude that there exists a uique symmetric equilibrium i the model where each seller posts price p Prices are Strategic Complemets A price p i 0 ca be ruled out as the profit is 0 ad that a slightly higher price must attract buyers with positive probability. A price p i such that λ i p i, p i 0 ca be ruled out as there must be some sellers attractig customers, implyig that mimickig such a seller will ear a positive profit uless all sellers charge 0 i which case a slightly higher price will ear a profit.. As the seller profit fuctio is strictly cocave i the relevat rage, the uique best respose to ay p 1 [0, 1] m 1, deoted β i p i, must satisfy the first order coditio. 0 π i p i, β i p i 1 e λ ip i,β i p i + β i p i e λ ip i,β i p i λ i p i, β i p i Differetiatig the first order coditio for optimality with respect to p k p i we have that π i p i, β i p i β i p i p + π i p i, β i p i 0, 3 i 11

13 where π i p i,β i p i p i sig the effect o p i from chagig p k p i we eed to be able to sig < 0 because the profit fuctio is strictly cocave i p i. Hece, i order to π i p i, β i p i e λ ip 1 β i p i λ i p i, β i p i +β i p i e λ ip i,β i p i λ i p i, β i p i. λi p i, β i p i 4 The first term i 4 simplifies as we oly eed to evaluate at a best respose p i β i p, so by usig the first order coditio we have that e λ ip i,β i p i 1 β i p i λ i p i, β i p i λi p i, β i p i 5 e λ ip i,β i p i 1 1 e λ ip i,β i p i e λ ip i,β i p i λ i p i, β i p i λ i p i, β i p i > 6 0. Hece, the first term i 4 ca be siged with o effort at all. Ufortuately, the same is ot true for the secod term i 4.Differetiatig 15 with respect to p k oe obtais [ ] λ i p 1 [g i λ i p] gλ i pg λ i p 1 p i [g i λ i p] gλ i p g λ i p d g + λ j p 1 p i g, λ i p dp k gλ i p λ i p 7 where oe ca show that the first term is egative, whereas the secod term is a rather complicated object that is hard to sig for geeral choices of p i. However, i the appedix we establish that the terms i 7 are small eough so that they are domiated by the direct effect o the arrival rates from a chage i p k, implyig that π i p i,β i p i is strictly positive. The critical step i this argumet is to use the buyer idifferece coditios to show that the secod term i 7 is ot too large i absolute value. That is, I show that: Lemma For ay p such that λ k p > 0 we have that d dp k gλ i p g λ i p gλ i p > 1 λ i p The key steps i the argumet uses Lemma 1 ad the relatio betwee the first derivatives ad cross-derivatives i 16. Combiig Lemma, which holds regardless of whether prices are cosistet with equilibrium or ot, with the optimality coditios we ca prove that prices are strategic complemets. Propositio 4 Pick ay p such that λ k p > 0 ad p i β i p i is a optimal respose to p i. The π i p i,β i p i > 0, implyig that β i p i π i p i,β i p i p i π i p i,β i p i p i > 0. 1

14 The proof ca be foud i the appedix. Usig symmetry ad the fact that prices are strategic complemets we ca rule out ay asymmetric equilibria. The reasoig is simple. If seller i posts a lower price tha seller j the seller i faces a distributio of competig prices that is idetical to the oe for seller j except that p j is replaced with p i i the best respose problem for firm j. As p i < p j ad the best reply is icreasig i the price of ay competitor it follows that firm j should optimally post a lower price, which is a cotradictio. It has already bee show by a direct calculatio that there ca be oly oe symmetric equilibrium, so this proves uiqueess of of equilibria: Propositio 5 There exists o asymmetric equilibrium i the model. Hece, the uique equilibrium is the oe i which every seller posts the price i 1. Proof. Suppose that there exists a equilibrium p i which p i > p j. Let p i,,j deote all prices but p i ad p j ad write the equilibrium coditios for i ad j as p i β i p i,j, p j p j β j p i,j, p i. But, the best respose fuctios β i ad β j are idetical so we may drop the idex ad write β β i β j ad ote that from the fudametal theorem of calculus p j β p i,j, p i p j β p i,j, p p i j + p j a cotradictio. β p i,j, p j p p i i + p j p j 8 β p i,j, p j > p i, 9 p j 5 Etry 5.1 Failure of Existece of Pure Strategy Etry Equilibria Naively, it would seem that addig a etry stage to the model should be easy. After all, we have that the uique equilibrium profit of a seller whe m sellers compete is 1 e m 1 + m Π m, 1 e [ ] 1 e m 1 e ρ 1 + ρ 30 m 1 + m as m, ad m ρ. Clearly, 1 e ρ 1 + ρ is icreasig i ρ. It takes some work, but oe ca check that Π m, is strictly decreasig i m for fiite m, as oe would expect. Hece, it seems that a equilibrium etry profile would be characterized by some m such that Π m + 1, K Π m,, 31 13

15 where K is the the seller valuatio of the good or the cost of etry. The problem with this is that if m sellers eter ad a additioal seller cosiders a deviatio, the relevat optimizatio problem is to solve max p i 1 e λ ip i,p mm, 3 p i where p m m is the price profile with m sellers postig the price i 1 for the equilibrium with m sellers, that is I cotrast, we have that 1 e m 1 + m p m m 1 e [ ],..., 1 e m 1 + m m e [ ] m m 1 + m }{{ } m idetical coordiates 33 Π m + 1, max p i 1 e λ ip i,p mm+1, 34 p i where p m m + 1 is the price profile with m sellers postig the price i 1 for the equilibrium with m sellers, that is p m m e m m+1 1 e m m+1 [ [ ],..., ] 1 e m e m+1 m m+1 }{{ } m idetical coordiates. 35 It is easy to check ad it is also a cosequece of Propositio 4 that the equilibrium price with m+1 sellers is strictly lower tha the equilibrium price with m sellers. Hece, max pi p i 1 e λ i p i,p mm > Π m + 1,. If i additio Π m + 1, < K < max p i 1 e λ ip i,p mm, 36 p i the it is impossible to have a pure strategy equilibrium i the etry stage. A similar issue arises also i models without etry, but where differet sellers or firms take differet actios targetig differet types. For example, i Galeiaos ad Kircher 009 firms specialize as high wage or low wage firms. They use a market utility framework with a cotiuum of firms, implyig that oe of the equilibrium coditios is a idifferece coditio betwee postig a high ad a low wage. I a fiite versio the same problem arises as with etry. I equilibrium, firms kow that there are, say, m high wage firm, so that high wage firms would be strictly worse off tha low wage firms if oe additioal firm switched to a high wage. However, to check whether the wage postigs are cosistet with equilibrium it must be that a firm that would uexpectedly switch to a high wage would be worse off, ad for the same reasos as above, the profit is higher for a firm that switches uexpectedly tha if the other firms ca react. Hece, there will ot be pure strategy equilibria with fiite umbers i these type models either. 14

16 5. Equilibrium Aalysis with Radomly Draw Sellers I ow cosider the full model as specified i Sectio, i which buyers ad sellers are distributed i accordace with idepedet Poisso distributios. Each seller is ow also goig to attach a strictly positive valuatio K to the object, which will be set i a way so as to guaratee o-trivial etry decisios. Cosider first a symmetric etry strategy where each potetial etrat eters with probability α. Coditioal o potetial k etrats the probability distributio is a biomial with parameters a ad k implyig that the probability that v sellers eter is a Poisso with expected umber of etrats αm. This observatio follows from the same derivatio as i Sectio A.1 i the appedix Symmetric Seller Strategies are Without Loss of Geerality It was already show i the cotext of a fixed set of sellers that it is without loss of geerality to restrict attetio to equilibria where all buyers follow the same mixed cotiuatio strategy. This argumet trivially exteds to this settig as buyers are assumed to be able to distiguish sellers based o prices posted. Cosequetly, we will assume a symmetric cotiuatio strategy for the buyers ad the cotiuatio game from this poit o is o the same form as with a kow set of sellers. Next, we will ask whether etry ca be type depedet ad to what extet it matters. Assume that sellers are distiguished by color, where, for simplicity, they ca be either gree or blue. Let gree sellers be draw from a Poisso with expectatio m g ad blue sellers be draw from a Poisso with expectatio m b m m g. Furthermore, suppose that gree ad blue sellers radomize with probabilities α g ad α b, which implies that the probability that v sellers eter is v e αgmg α g m g k e α bm b α b m b v k k! v k! k0 e αgmg+α bm b v! e αgmg+α bm b v! v k0 v! k! v k! α gm g k α b m b v k α g m g + α b m b v, 37 by use of the biomial theorem. This shows that the probability of v players eterig is a Poisso with expected queue legth α g m g + α b m b. Hece, if all active sellers post the same price I will show that this must be the case i the proof of Propositio 6 below, the it is irrelevat whether differet groups choose the same radomizatio as oly the expected umber of etrats matter. That is, all radomizatios α g, α b satisfyig α g m g + α b m m g αm are equivalet i terms of the icetives to eter. I geeral, recall that t T {1,..., T } deotes the payoff-irrelevat type of a seller where m t is the expected umber of type t sellers. As T is arbitrary it is possible to make the probability of two idetical cloes arbitrary small. The result is that these types ca be used to decide who eters ad who does ot, but that payoffs caot be affected by use of them. Propositio 6 I ay equilibrium there exists a uique equilibrium price p such that p t p i ay equilibrium. Moreover, if a 1,..., a T is a equilibrium etry profile there exists α such that 15

17 α t α for each t such that all buyers ad sellers ear the same payoffs as i the equilibrium with asymmmetric etry. Proof. Let p p 1,..., p T ad α α 1,..., α T be a equilibrium. We eed to itroduce otatio for the Poisso arrivals of buyers coditioal o etry vector k k 1,..., k T ad to so we write λ p, p, k λ p, p, k 1,.., k T, which deotes the expected queue legth at a seller of ay type who posts p give that for each t T there are k t other sellers who posts p t. The queue legth is uiquely pied dow by the idifferece coditios i 11 give ay p, p, k. The expected profit for the seller postig p is thus π p, p, α... e α 1 m 1 α 1 m 1 k 1 k 1! k 1 0 k T 0... e α T m T α T m T k T k T! p 1 e λp,p,k K. From Propositio we kow that each p 1 e λp,p,k is strictly cocave i p o some rage [0, p p ], but has a o-cocave rage where o buyers visit the firm. However, all active sellers face the same optimizatio problem with the same set of maximizers. Let p mi be the smallest maximizer. Obviously, a seller postig p mi must be visited with strictly positive probability regardless of which sellers eter, so p mi is i the rage where the all profit fuctios i the support of the expected profit fuctio is strictly cocave, implyig that 38 is strictly cocave i a rage aroud p mi. It follows that p mi must be the uique global maximum. It follows that there is a uique price p p, α that is a best respose, which implies that i ay equilibrium there must be a uique price p such that p t p for each t. Sice all types post the same price oly the total umber of etrats matter. This is distributed Poisso with parameter t T α t m t implyig that there is a symmetric radomizatio probability α t T α t m t t T m t that is cosistet with equilibrium Solvig for a Symmetric Equilibrium Sice ay equilibrium is payoff equivalet to a symmetric equilibrium I ow simplify the profit fuctio by expressig it i terms of the ow price p i, the commo equilibrium price p, ad the commo etry probability α as π p i, p, α e αm p i + e αm αm k p i 1 e λ kp i,p K, 40 k! k1 where λ k p i, p deotes the buyer queue at a seller postig p i give that k sellers post p. Notice that e αm is the probability that o other seller eters i case the seller attracts a buyer for sure. A equilibrium price must be symmetric ad satisfy the first order coditio e αm e αm αm k + 1 e λ kp,p + p e λ kp,p λ k p, p 0 41 k! k1 16

18 Usig symmetry, we have that λ k p, p λ k p, p g 1 p g for each k 1 ad that k k p we ca solve for the cadidate equilibrium price i closed form, p α e αm + e αm αm k k1 k! e αm + e αm αm k k1 k! 1 e 1 e 1+ 1 e 1 e 1 e 1 + ] [1 e 1 + k k + 1, Give ay α [0, 1] we have that p α is uiquely defied, so we ca immediately coclude that there is a uique cadidate equilibrium price p α for every etry probability α. However, i order to show that there is a uique equilibrium price give ay cost of sellig K we eed to establish that p α is mootoe i α. Let G k ad let H : R + R + be defied as H k { 1 e 1 e 1+ 1 for k 0 1 e for k 0, 44 1 for 0 k < 1 ] 1 + for k 1 [1 e To demostrate that p α is mootoe i α we show that. Lemma 3 G is strictly decreasig ad H is icreasig o R + ad strictly icreasig at ay k 1. The proof is relegated to the appedix. Usig the mootoicity of G ad H we are i a positio where we ca demostrate that the price is strictly mootoe i the etry probability, as oe would expect. Lemma 4 p defied i 43 is strictly decreasig i α. Proof. Suppose that a < a. The the Poisso with probability mass desity e α m α m k k! is first order stochastically domiated by the Poisso with probability mass desity e α m α m k k!. Sice G k is strictly decreasig ad H k is icreasig also whe restricted to itegers it follows that E α G k e α m + > e α m + e α m αm k k1 k! e α m αm k k1 k! 17 1 e 1 e E α G k 45 46

19 ad E α H k e α m + e α m + e α m αm k k1 k! e α m αm k k1 E α H k, which implies that p a > p a. k! 1 e 1 e 1 e 1 e [ 1 e 1 + ] k + 1 [ ] 1 e 1 + k + 1 Corollary 1 For ay K > 0 there is a uique etry probability α K that is cosistet with a equilibrium i symmetric etry strategies. Moreover if [ e m + ] e m m 1 k e e m + e m m k k1 k! k1 k! 1 e 1 e 1+ ] [1 e < K < 1, 48 the α K 0, 1 ad every seller that eters is idifferet betwee sellig ad o sellig. The proof is immediate ad left to the reader. If sellers have differet characteristics to coditio o oe ca costruct equilibria with differet types eterig with differet probabilities, but the average probability of etry is uiquely defied, so the equilibrium is essetially uique Covergece Towards the Market Utility Bechmark With some further otatioal abuse I write α K,, m for the uique average radomizatio probability whe the cost of sellig is K ad there are buyers ad m potetial sellers. While p α K,, m i 43 is a determiistic umber that ca be thought of as a expectatio ad ot a radom variable it turs out that it is useful to ote that G k ad H k coverges i probability i order to derive the determiistic limit of p α K,, m as, m ad m ρ. k That is, we use the fact that the umber of active sellers, m, coverges i probability to α K,, m i order to derive the limitig price ad expected equilibrium profit. More formally, Chebyshev s iequality implies that Pr [ k α K,, m m > δm] σ α K,, m δ m δ m holds for ay m, because σ α K,, m m is the variace of the umber of active sellers k as α K,, m m is both the first ad the secod momet of the Poisso geerated whe the expected umber of potetial sellers is m ad every seller eters with probability α K,, m.it follows that if m ρ we have that [ k Pr α K,, m ρ > δ ] ρ 18 α K,, m δ m

20 as m. More geerally, if, m ad m ρ ad α K,, m α takig a subsequece if ecessary we have that k α coverges i probability to ρ. Hece, 1 e ρ α p α K,, m 1 1 ρ e α 1 + ρ α 1 e ρ α 1 e ρ 1 e ρ α α 1 e ρ α 1+ ρ α ρ α e ρ α 1, 51 as, m ad m ρ ad α K,, m α. Also ote that as k α coverges i probability to ρ we have that e k coverges i probability to e ρ α implyig that the equilibrium expected profit π p α K,, m, p α K,, m, α K,, m 1 e ρ α 1 + ρ α K 5 alog ay coverget subsequece. As 1 e x 1 + x is strictly icreasig there is a uique solutio α to 1 e ρ α 1 + ρ α K implyig that is a α K,, m coverget sequece. Hece, the right had side of 5 is the limitig equilibrium profit. Also otice that the right had side of 5 is idetical with the profit 4 i the market utility model ad that the limitig price i 51 correspods to the equilibrium price 3 for the equilibrium active buyer to seller ratio. We coclude that the market utility bechmark ca ideed be see as the limit a large fiite model i which both buyers ad sellers are distributed i accordace with a Poisso. 6 Cocludig Remarks The results i this paper suggests that drawig players radomly from a Poisso distributio ca be a useful compromise betwee a market utility model ad a stadard fiite directed search model. I the simplest possible setup I demostrated that there is o loger ay eed to make a equilibrium selectio ad that a otherwise utractable etry problem ca be hadled without too much diffi culty. The added tractability should be helpful also i richer models where buyers or sellers have payoff relevat types, ad or i models where cotracts are more elaborate tha postig a sigle price, but I do t explore this i the curret paper. Refereces [1] Blad, J. ad S. Loertscher 01, Mootoicity, No-Participatio, ad Directed Search Equilibria, mimeo, Uiversity of Melboure. [] Burdett, K. Shi S., ad R. Wright, 001. "Pricig ad Matchig with Frictios," Joural of Political Ecoomy, 1095, October 001, [3] Galeiaos, M. ad P. Kircher, 009 Directed Search with Multiple Job Applicatios, Joural of Ecoomic Theory, 144, 009, [4] Galeiaos, M. ad P. Kircher, 01 O the Game-theoretic Foudatios of Competitive Search Equilibrium, Iteratioal Ecoomic Review, 531,

21 [5] Galeiaos, M. P. Kircher ad G. Virag 011, Market power ad effi ciecy i a search model, mimeo, Lodo School of Ecoomics. [6] Geromichalos, A. 01 Directed search ad optimal productio, Joural of Ecoomic Theory, 476, Pages [7] Julie, B., J. Kees ad I. Kig, 000, Biddig for labor, Review of Ecoomic Dyamics, 34, [8] Kircher, P., 009, Effi ciecy of simultaeous search, Joural of Political Ecoomy, 1175, [9] Kim, J. ad G. Camera 014. Uiqueess of equilibrium i directed search models, Joural of Ecoomic Theory 151, [10] Lester, B., 011 "Iformatio ad Prices with Capacity Costraits," America Ecoomic Review, 1014, [11] Lester, B, Visschersz L., ad R. Wolthoff, Meetig Techologies ad Optimal Tradig Mechaisms i Competitive Search Markets, mimeo, September 014. [1] McAfee P. 1993, Mechaism desig by competig sellers, Ecoometrica 616, [13] Moe, E. R Competitive Search Equilibrium. Joural of Political Ecoomy, 105, [14] Myerso, Roger B Large Poisso Games, Joural of Ecoomic Theory 94 1: doi: /jeth [15] Peters, M Bertrad Equilibrium with Capacity Costraits ad Restricted Mobility., Ecoometrica, 55: [16] Peters, M. 000, Limits of Exact Equilibria for Capacity Costraied Sellers with Costly Search, Joural of Ecoomic Theory, 95, [17] Shimer, R., 005, The assigmet of workers to jobs i a ecoomy with coordiatio frictios, Joural of Political Ecoomy, 1135, A Appedix A.1 A Remider about a Poisso Distributio Property Claim If every buyer visits seller i with probability θ i ad if buyers are distributed i accordace with a Poisso with parameter, the the umber of buyers visitig seller i is a Poisso distributio with parameter θ i. 0

22 Proof. Coditioal o there beig b buyers the umber of buyers at seller i follows a biomial with parameters θ i ad b, so that Pr [v buyers at i b buyers] The ucoditioal probability of v buyers arrivig at seller i is thus Pr [v buyers at i] e b b! b! v! b v! θv i [1 θ i ] b v e [θ i ] v v! bv e v! θ i v r0 a Poisso distributio with parameter θ i. b! v! b v! θv i [1 θ i ] b v A1 1 r! [ 1 θ i] r e θi θ i v, v! bv 1 b v! [ 1 θ i] b v A A. Proof of Lemma 1 Proof. The first iequality holds as [ g x ] g x g x [ 1 e x ] [ ] [ ] 1 + x 1 e x 1 e x 1 + x + x x x x 3 1 [ 1 e x x x 1 e x ] 1 e x 1 + x + x 1 x 4 [ 1 e x 1 + x 1 e x 1 e x 1 + x + x 1 [ 1 e x x x 1 e x 1 + x + x 1 [ 1 e x ] x 4 1 e x 1 + x + x 1 x 4 [ 1 e x 1 + x + x 1 [ 1 e x x 4 1 e x 1 + x + x 1 [ x 4 1 e x 1 + x + x e x x 1 [ 1 e x x 4 1 e x 1 + x + x 1 [ x 4 1 e x 1 + x + x e x x 1 x 4 [ 1 e x 1 e x 1 + x + x + e x x ] 1x [ 1 e x ] 4 1 e 1 x + x + x 1 e x ] 1 e 1 x + x + x 1 e x 1 + x 1 + e x] + 1 [ 1 e x 1 + x e ] x 4 ] + 1x 4 [ 1 e x 1 + x e ] x x ] [ + 1x 4 1 e 1 x + x + x ] ] x x e e x x x x ] ] 1

23 1 [ x 4 1 e x 1 + x + x x x [e e x x 1 e x ]] 1x 4 e x x 1 ] x 4 1 e 1 x + x + x [1 e x + x + x 1 e x x 1 e x }{{} 4 <0 ad the secod follows from x x [ g x ] [ 1 e g x g x ] [ ] [ ] 1 + x 1 e x 1 e x 1 + x + x x x x x 3 [ 1 e x x x 1 e x ] 1 e x 1 + x + x for every x > 0. < 0, [ 1 e x x x 1 e x 1 + x + e x x 1 e x 1 + x e [ x x x 4 e 1 e x 1 + x ] e x x 1 e x x x 1 + x e [ x 1 e x 1 + x ] 1 e x 1 + x e e x x 3 e x x 3 [ x e x x 3 [ x 1 e x x x e 1 e x 1 + x e 1 e x 1 e x 1 + x ] e x x 3 [ x 1 + e x 1 + x ] > 0 x x x x ] x x ]

24 A.3 Proof of Propositio Proof. Differetiatig 15 we obtai λ i p p i g λ i p 1 p i g λ i p + 1 [g λ i p] g λ i p g λ i p 1 p i [g λ i p] g λ i p d g + λ j p 1 p i. g λ i p dp i λ i p A3 By substitutig λ ip gλ i p 1 p i g λ i p we write this as λ i p p i g λ i p 1 p i g λ i p + 1 [g λ i p] g λ i p g λ i p 1 p i [g λ i p] }{{} <0 g λ i p d + 1 p i g λ i p dp i, A4 g λ i p 1 p i g λ i p }{{} <0 where we used the first part of Lemma 1 to sig the secod term. Notig that 0 < which implies that 0 < < positive we get a upper boud o the secod derivative give by λ i p p i < < 1, < 1 ad that the secod term i A4 is [ ] g λ i p 1 p i g λ i p } [g λ i p] g λ i p g λ i p [g λ i p] {{ } g λ i p d + 1 p i g λ i p dp i >0, A5 3

25 where we have usedthe secod part of Lemma 1 to sig the bracketed expressio. Clearly, the first term i A5 is egative. Moreover, d g λ j p d 1 dp i dp i + gλ i p g λ i p Combiig A6 ad A5 establishes the result. [ [g λ i p] gλ i pg λ i p λ i p m [g λ i p] [ m ] ] gλ i p [] g λ j p λ j p g λ i p [] [ m ] <0 <0 <0 {}}{{}}{{}}{ [g λ i p] g λ i p g λ i p λ i p g λ j p [g λ i p] [ m ] <0 {}}{{}}{{}}{ g λ i p [] g λ j p g λ j p λ j p g λ i p [g + λ j p] ] > 0. [ m <0 >0 A6 4