Investing in trading opportunities in dynamic adverse selection Sander Heinsalu

∗

February 10, 2017

Abstract I analyse a market with asymmetric information, interdependent values, multiple trading opportunities and trade frictions. The frictions can be reduced at a cost, e.g. by increasing attention, search, lobbying or computing power. For some parameters, there is a unique equilibrium in which an increase in the gains from trade delays trade. An increase in the trading surplus may also reduce the payoff of all types. The driving force is a novel feedback loop between the endogenous trading frictions and the signalling motive. Keywords: Lemons market, frictions, signalling, asymmetric information. JEL classification: D82, D83, C72.

Markets with asymmetric information (mortgages, health insurance, used cars) may feature inefficiently few transactions. One solution used in practice is to subsidise trade, e.g. by government guarantees, purchases of troubled assets, or a lower interest rate. By increasing the gap between buyer and seller valuations, such interventions may reduce the lemons problem of Akerlof (1970) enough to restore the efficient level of trade. This insight holds for one shot markets with observable actions, but may fail in a dynamic environment with trading frictions, as the current work will show. With multiple opportunities for exchange, a higher difference between buyer and seller valuations may delay trade, reducing its volume initially. Rejecting an offer can be used to signal high quality. Greater gains from trade raise the benefit of signalling, but also increase the incentive to trade early. With exogenous (or absent) trading frictions, the signalling motive does not outweigh impatience, so subsidising trade makes it occur earlier. If trading opportunities can be increased at a cost, then ∗ Australian National University, Research School of Economics, HW Arndt Building 25a Kingsley St, Acton, ACT 2601, Australia. Email: [email protected] Website: http://www.sanderheinsalu.com/ The author thanks George Mailath, Gabriel Carroll, Yingni Guo, Mihkel Tombak, Colin Stewart, Simon Grant and the seminar audiences in UNSW, U Melbourne, Johns Hopkins, U Penn, Yale, UCSD, UT Austin and Berkeley for insightful suggestions. All errors are the author’s.

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a novel feedback effect arises between the signalling motive and the investment in the probability of trade. This feedback can strengthen the signalling motive sufficiently to overwhelm impatience. The resulting delay can be severe enough to outweigh the subsidy on trade and reduce the payoff to all types. In the model, the players are a privately informed buyer and perfectly competitive sellers.1 The buyer is one of two types: high or low valuation. The buyer knows the type, but the sellers only have a common prior belief about the type. Selling to the high-valuation buyer is more costly for the sellers, but gains from trade are positive for any seller matched with any buyer type. First, the buyer privately invests to increase the probability of being able to trade later. Second, the sellers make a price offer. Third, with the probability chosen earlier, the buyer obtains the trading opportunity, and in this case, decides whether to accept or reject the lowest price offered. There is unit demand, so acceptance ends the game. Fourth, if the buyer cannot trade or rejects, then after some delay, the sellers make another offer. Fifth, the buyer again decides to accept or reject. The probability of being able to trade in this last stage is one and does not require investment. This assumption is for simplicity; the results are robust to changing it. The solution concept is perfect Bayesian equilibrium (PBE). In a one shot interaction, all types would trade, because the gains from trade are positive for any buyer. With multiperiod trading, reservation values are endogenous and some types may delay trade. Delay is inefficient, due to discounting. For a nonempty open set of parameters, there is a unique PBE. For a nonempty open subset of parameters, raising the valuation of the high-value buyer type changes the equilibrium to both types trading with lower probability initially. The buyer reduces the investment in the chance to trade, and may switch from accepting to rejecting the first price offer. Counterintuitively, a higher expected trading surplus delays trade. Raising the valuation of the low-value buyer increases the probability of trading for both types. Raising the valuations of both types may reduce acceptance of the sellers’ initial offer and make both types worse off. Rejecting an initial offer is a stronger signal when the benefit of accepting that offer is higher. This signal changes the belief of the uninformed sellers so that they make a better offer later. On the other hand, there is a cost of rejecting the initial offer, because the buyer is impatient. In a model with no frictions or exogenous frictions, the signalling motive never outweighs the impatience. Raising the gains from trade then always shifts trade earlier. The contrast between exogenous and endogenous frictions illustrates that the results in the current paper are driven by a novel force, not just by signalling by waiting. With endogenous frictions, a larger benefit from trade leads to a feedback effect in the best responses. Raising the gains from trade for the high type buyer causes her to accept earlier, ceteris paribus. The high type buyer is more costly to sell to. In response to the high type trading earlier, the sellers increase the initial price and lower the later, because fewer costly high types remain later. This price change causes the low-valuation buyer to optimally 1 The

situation with an informed seller and competitive buyers is symmetric.

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delay acceptance. Fewer low types accepting initially raises the expected cost for the sellers, leading to an even higher initial price and a reduced later price. This motivates the low-valuation buyer to delay even more, etc. This feedback strengthens the signalling motive, so it may outweigh the impatience. The higher initial and lower later price also incentivises the high type to wait before trading. Delay is costly, because of impatience. Waiting reduces payoffs, which may outweigh the subsidy on trade. The gains from trade do not have to rise uniformly across time periods for the result to hold. A higher surplus from exchange in the initial period has the same effect as a greater surplus every period. More gains from trade in the last period has the opposite effect—it reduces the signalling motive. This is somewhat surprising, because signalling is like investment—a cost paid at the start and a benefit reaped later. Despite this, there is less signalling with a higher later surplus, because the greater later gains from trade motivate the low-value type to imitate the high-value more. The increased similarity between the actions of the types leads to belief responding less to signals, reducing the signalling motive for all. The results do not depend on having two types or valuations constant over time. Neither do sellers have to update exactly by Bayes’ rule nor be perfectly competitive. The geometry of the game (illustrated in the two type case) makes the results robust to various small perturbations. Investment in reducing frictions can be interpreted as rational inattention— the buyer decides how frequently to check advertisements, email, or the bid and ask prices on the stock market. The more costly attention is directed to these activities, the smaller the chance of missing a trading opportunity. The task of paying attention can be delegated to more or less competent workers, with the corresponding salary expenses. For example, a Deutsche Bank forex dealer mistakenly transferred 6 billion USD to a hedge fund in June 2015. The recipient returned the money the next day, but the hedge fund lost some trading opportunities due to less available capital. Investing to increase the chance to trade may take the form of preventing or fixing technical problems with phone or email, to ensure noticing offers. High frequency traders invest in computing power and fast connections to stock exchanges to be the first to trade on a profitable opportunity. The probability of a correct decision is never 0 or 1, even with large stakes, e.g. Knight Capital Group went bankrupt on 1 Aug 2012 after losing 450 million USD due to mistaken computerised asset trades. In a search and matching context, investing in reducing frictions means increasing the search intensity, resulting in fewer missed buying opportunities. The chance to trade can be increased at a cost also when a permit is required for purchasing (weapons, controlled drugs, radioactive isotopes) or operating in an industry (mining, insurance, utilities). Satisfying the legal criteria and lobbying for the license are expensive. If the buyer invests in trading opportunities, then he must expect a positive share of the surplus. Significant market power is held by e.g. a foreign government buying weaponry, a pharmaceutical company purchasing poppy extract to make painkillers, a hospital system sourcing isotopes for medical imaging. 3

Regulated industries similarly have a few large firms, this being one reason to regulate them. In the applications, the buyer has private information, with a higher valuation corresponding to a larger cost for the seller. A government buying weapons or radioactive material for aggression values them more than a country focussing on defence or civilian nuclear applications, because the chance of using the weapons is higher and nuclear blackmail is profitable. The government that allows the sale of weaponry or nuclear material suffers a political cost if the products are used for aggression. Narcotics dealers value drug precursors more than pharmaceutical firms, and the regulator permitting purchases of controlled substances is punished if these end up in the illicit drug trade. A firm planning to break environmental, antitrust or consumer protection laws in a regulated industry expects higher profit than an honest company, and thus values the operating licence more. Permitting the entry of a lawbreaking firm is worse for the regulator. Markets are dynamic—if one does not trade now, there is still a positive probability to transact after a costly delay. Regulations and political circumstances change, permitting trades that were previously prohibited. The investment in reducing frictions probabilistically leads to earlier trade. The result that subsidising trade may delay trade and reduce payoffs provides an alternative interpretation for the reluctance of US financial institutions to accept government bailouts during the 2008 financial crisis. If the institutions expected a Pareto worsening as a result of the intervention, then opposing it was natural. This interpretation is complementary to the two traditional ones: (a) refusing the bailout was a Spence signal of a strong financial position, (b) limiting executive pay was a condition of the bailout, so moral hazard led the executives to refuse the bailout that their institution needed. When Lehman Brothers went bankrupt on 15 Sept 2008, the financial markets froze and the share prices of financial firms fell. The standard interpretation is that the public believed the financial positions of the firms to be correlated. One firm failing was a signal that the others were in a bad state. The uninformed buyers of shares were afraid of getting an Akerlof ‘lemon’ and reduced their demand. The current paper provides an alternative interpretation in which the financial states of the firms are negatively correlated. One source of negative correlation is competition—if one competitor goes bankrupt or is incapable of investing in capacity expansion, then the others have more market power and profit in the future. With negative correlation, the bankruptcy of Lehman Brothers increased the expected profit (type) of its competitors. Without the feedback loop of this paper, a higher expected profit would increase trade and the competitors’ share prices, and reduce adverse selection. The result may be the opposite under the interaction of the investment in trading opportunities and signalling.

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Literature This paper builds on the literature on markets with adverse selection that started from Akerlof (1970), adding to it the endogenous decision errors studied by Stahl (1990); van Damme (1991) in the context of normal form complete information games. Costly investment in trading opportunities can be interpreted in terms of the rational inattention proposed in Sims (2003), the search frictions of Diamond (1982); Mortensen and Pissarides (1994) or the lobbying all-pay auction of Baye et al. (1993). Other trading frictions can have similar effects. Dynamic adverse selection (in two periods) is studied in Fuchs et al. (2016), who focus on the comparison between public and private offers without trading frictions. They find private offers to be welfare-enhancing. H¨orner and Vieille (2009) derive a result similar to Fuchs et al. with an infinite horizon and a different monopolist making an offer each period. With public offers, trade may stop forever after a rejection. The current paper assumes private offers, imposes frictions that can be reduced at a cost and considers a different question, namely the comparative statics when the gap between the valuations of the buyer and sellers increases. The work closest to the current one is Fuchs and Skrzypacz (2015). They exhibit a condition ensuring that an initial subsidy of trade followed by a tax high enough to shut the market is optimal. The present paper, using different assumptions, derives the opposite result: an initial subsidy with a later tax may freeze the market. Contrary to Fuchs and Skrzypacz (2015), there are frictions, but no static adverse selection (all types would trade in a one shot situation). The market freeze is entirely due to dynamic incentives, which may change in either direction when the gap between the buyer and seller valuations increases. Intervention in a static market with adverse selection has been studied in Philippon and Skreta (2012); Tirole (2012), in which a round of government financing of privately informed firms is followed by one shot competitive trade. Government intervention affects this later trade and the expectation of this effect influences the firms’ response to the intervention. Both Philippon and Skreta (2012) and Tirole (2012) show that intervention cannot increase welfare in their static frictionless context. The current paper shows that it is possible to structure subsidies and taxes to raise welfare in a dynamic environment with frictions. The focus is on the dynamics, which open the possibility of the novel feedback loop. Adverse selection is combined with maturity mismatch in Bolton et al. (2011); Heider et al. (2009). The current paper does not model maturity mismatch directly, but this could be one reason for the gains from trade and asymmetric information between the buyer and sellers. Dynamic competitive markets with adverse selection are studied in Janssen and Roy (2002) who show that equilibrium prices increase over time and eventually all types trade. Unlike Janssen and Roy (2002), the current paper focusses on the comparative statics of interventions, not on the price path, but the price change is similar in the two papers. An increasing price is also found in Camargo

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and Lester (2014); Chiu and Koeppl (2016) in a search context. Camargo and Lester argue that sunset provisions can improve welfare, because the expectation of a future subsidy can delay trade. The present work qualifies this finding—a current or future subsidy targeted to the high-value type may have the opposite effect to targeting the low-value buyer. Chiu and Koeppl present an argument for an increase in total surplus from delaying asset purchasing programs. The current paper shows that present and future subsidies have opposite effects for any parameter configuration. The welfare impact of a (current or future) subsidy may be positive or negative. The driving force in Chiu and Koeppl (2016) is the build-up of selling pressure over time, which is absent in the current work, just like the feedback loop of the current work is absent in Chiu and Koeppl (2016).

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Model

The players are a buyer and, in each period, two sellers, who are in Bertrand competition. The sellers are indexed by j = A, B. The buyer has a type θ ∈ {H, L}. The buyer knows the type, but the sellers only have a common prior belief µ0 = Pr(H) ∈ (0, 1). Selling to type θ costs the market cθ , normalised to cH = 1, cL = 0 w.l.o.g. Type θ buyer values the good at vθ , with vH > vL . Gains from trade are large enough for all types to be willing to trade in a one shot interaction: vL > cH . Thus there is no static adverse selection problem, allowing the paper to focus on the dynamics. The timing of actions is as follows. 1. Buyer type θ chooses the probability rθ ∈ [r, 1) of being able to accept the sellers’ initial offer, with r ∈ (0, 1). 2. Sellers j = A, B make price offers P1j ∈ R+ without observing rθ . 3. With probability rθ , the buyer observes P1j and chooses to accept or reject the lower of these. Acceptance is denoted by 1, rejection by 0. If P1A = P1B , then the tie is broken uniformly randomly. Acceptance ends the game. 4. If the buyer did not observe the prices or rejected, then the sellers j = A, B in period 2 make price offers P2j ∈ R+ , without observing rθ or P1j . Calendar time is observable, so the sellers know that the buyer rejected a previous offer. 5. The buyer observes P1j and chooses to accept or reject the lower of these. Ties are broken uniformly. The buyer cannot commit to ignoring the initial offer (rθ = 0). The interpretation is that with probability r, the buyer gets a trading opportunity even without trying (the buyer sees an advertisement or gets a call or email). Making the final prices stochastically observable to the buyer, either exogenously or endogenously (controlled by rθ or by a second investment rθ2 made 6

at the start or after the first rejection) does not change the qualitative results. Similarly, a noisy offer by the sellers has little effect. For simplicity, the final price offers are assumed freely and noiselessly observable. The first accept-reject choice of the buyer is subject to controllable error, which does affect the results. The benchmarks of noiseless observability and exogenous noise are discussed in Section 6. Each seller has only one information set, because the sellers making the final offer only know that trade did not happen earlier. Both information sets of the sellers are reached with positive probability, so all seller beliefs are determined by Bayes’ rule. The buyer knows the history, which determines the buyer’s belief. Each seller’s strategy is Ptj ∈ R+ . The best price in period t that the buyer ∗ ∗ expects in equilibrium is Pt∗ := min {PtA , PtB }. The best price encountered is Pt := min {PtA , PtB }. The buyer’s strategy is a function s : {H, L} → [r, 1) × R R {1, 0} + × {1, 0} + . This definition does not permit the buyer to condition on max {PtA , PtB }, condition the final acceptance on the initial prices P1j or break ties between sellers with other than uniform probability. These restrictions are w.l.o.g. In particular, by Mailath and Samuelson (2006) p. 330, public strategies are w.l.o.g., because the game has a product structure. A product structure means that a player’s action does not alter the informativeness of the public signal about the actions of other strategic players. A public strategy means that a player does not condition his behaviour on his own past actions at any information set. Type θ’s component of the strategy is s(θ, P1 , P2 ) = (rθ , s1 (θ, P1 ), s2 (θ, P2 )). Here, st (θ, Pt ) ∈ {1, 0} for t = 1, 2 is the buyer’s attempted choice, which for s1 (θ, P1 ) has probability rθ of mattering. The realised choice is denoted σt (θ, Pt ) ∈ [0, 1], which incorporates both possible intentional mixing and the probability of not being able to trade. Formally, σ1 (θ, P1 ) = rθ Pr(s1 (θ, P1 ) = 1) and σ2 (θ, P2 ) = Pr(s2 (θ, P2 ) = 1). The strategy that the market expects the buyer to use in equilibrium is denoted s∗ , with realised choices σt∗ . If the sellers expect σ1∗ (θ, P1 ), then the probability of H conditional on acceptance of P1 is Pr(H|1, P1 ) :=

σ1∗ (H, P1 )µ0 . σ1∗ (H, P1 )µ0 + σ1∗ (L, P1 )(1 − µ0 )

(1)

The probability that period 2 sellers put on H conditional on initial rejection and expected period 1 price P1∗ is Pr(H|0) :=

(1 − σ1∗ (H, P1∗ ))µ0 . (1 − σ1∗ (H, P1∗ ))µ0 + (1 − σ1∗ (L, P1∗ ))(1 − µ0 )

(2)

If the sellers expect type θ to accept the second offer P2 with probability σ2∗ (θ, P2 ), then Pr(H|01, P2 ) :=

σ2∗ (H, P2 ) Pr(H|0) ∗ σ2 (H, P2 ) Pr(H|0) + σ2∗ (L, P2 )(1

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− Pr(H|0))

.

(3)

The buyer pays a cost κ(r) for choosing r, with κ : [r, 1] → R twice differentiable unless stated otherwise, κ0 (r) = 0, κ00 > 0, κ0 (1) > vL and κ0 (1) > (1−δ)vH −µ0 . Accepting P1 yields vθ −P1 to type θ. There is discounting between offers: accepting P2 gives the buyer δ(vθ − P2 ), with δ ∈ (0, 1). Rejecting both P1 and P2 gives zero. The buyer accepts P2 conditional on reaching period 2 if vθ − P2 ≥ 0. The buyer accepts P1 if vθ − P1 ≥ max {0, δ(vθ − P2 )}. The total expected payoff when choosing r, assuming later choices are optimal, is πθ := r max {vθ − P1∗ , max {δ(vθ − P2∗ ), 0}}

(4)

+ (1 − r) max {δ(vθ − P2∗ ), 0} − κ(r). Denote the willingness to pay of type θ in period t by wθt , where wθ1 = (1 − δ)vθ + δP2∗ and wθ2 = vθ . Clearly wL < wH . Due to Bertrand competition, the payoff of seller A in period t from setting PtA is  1 ∗ ∗   2 [µ0 rH (PtA − 1) + (1 − µ0 )rL PtA ], PtA ≤ wLt ∧ PtA = PtB ,   ∗ ∗   PtA ≤ wLt ∧ PtA < PtB , µ0 rH (PtA − 1) + (1 − µ0 )rL PtA , 1 ∗ πA (PtA ) := 2 µ0 rH (PtA − 1), PtA ∈ (wLt , wHt ] ∧ PtA = PtB ,   µ0 r∗ (PtA − 1), PtA ∈ (wLt , wHt ] ∧ PtA < PtB ,  H   0, PtA > wHt ∨ PtA > PtB . (5) The payoff of seller B is symmetric. Based on (5), w.l.o.g. the action set of each seller can be restricted to [0, vH ]. In each period, each seller prefers to undercut the other, as long as prices are above the marginal cost conditional on acceptance. This marginal cost is Pr(H|1, P1j )cH + (1 − Pr(H|1, P1j ))cL = Pr(H|1, P1j ) in period 1 and Pr(H|01, P2j ) in period 2. It equals both the cost and the expected type conditional on acceptance.

2

Equilibrium

The equilibrium concept is perfect Bayesian equilibrium, hereafter simply called equilibrium. Definition 1. A buyer strategy s∗ and sellers’ price offers Ptj∗ , t = 1, 2, j = A, B are an equilibrium if (a) vθ − P2 > 0 ⇒ s∗2 (θ, P2 ) = 1 for any P2 , <

0

(b) vθ − P1 > max {δ(vθ − P2∗ ), 0} ⇒ s∗1 (θ, P1 ) = 1 for any P1 , 0

<

(c) rθ∗ maximises (4), (d) Ptj∗ maximises (5).

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Due to vH > vL , if L prefers to accept Pt , then H also, and strictly. Conditional on some type accepting, reducing the price weakly reduces the sellers’ cost, as well as increases demand. The sellers make zero profit in any equilibrium, by the standard Bertrand reasoning. The incentive to cut price is further strengthened by the falling cost. The zero-profit price must be in [cL , cH ] = [0, 1], so restricting price offers to [0, 1] is w.l.o.g. The assumption vL > 1 implies Pt∗ < vL for t = 1, 2, so the buyer’s final choice is s∗2 (θ, P2∗ ) = 1 in any equilibrium, even after P1 6= P1∗ . The final price is then P2∗ = Pr(H|0), and the continuation payoff after rejecting P1 is δvθ − δP2∗ . The equilibrium price in period 1 satisfies P1∗ = Pr(H|1, P1∗ ), where Pr(H|1, P1∗ ) is given in (1). There may be multiple initial prices s.t. P1∗ = Pr(H|1, P1∗ ), which makes multiple equilibria possible. If rθ∗ > r, then s1 (θ, P1∗ ) = 1, because the otherwise the cost κ(rθ∗ ) could be reduced, keeping the probability of initial trade the same. In other words, the buyer does not both invest in being able to trade and refuse to trade, unless the price is unexpectedly high (off the equilibrium path). The equilibrium rθ∗ equals r iff the incentive constraint (IC) (1 − δ)vθ − P1∗ + δP2∗ ≤ 0

(6)

holds. Otherwise, rθ∗ ∈ (r, 1) and the first order condition (FOC) (1 − δ)vθ − P1∗ + δP2∗ = κ0 (rθ∗ )

(7)

holds. Due to κ0 (r) = 0, at r the FOC ensures (6). The buyer payoff (4) is concave, so the FOC is sufficient for a maximum. For θ = H, L, the left hand side (LHS) of (7) is the marginal benefit (MB) of raising rθ and the right hand side (RHS) is the marginal cost (MC). The MB can be further decomposed into the impatience motive (1 − δ)(vθ − P1∗ ) and the signalling motive δ(P1∗ − P2∗ ). The impatience motive describes the gain from trading early, instead of later. The signalling motive captures the benefit of waiting. This benefit is a better price later. Proposition 1. In any equilibrium, s∗2 (θ, P2∗ ) = 1, P2∗ = Pr(H|0), s∗1 (L, P1∗ ) ∈ L −µ0 +δµ0 ∗ ∗ {0, 1}, s∗1 (H, P1∗ ) > 0, and if r < µ0(1−δ)v [(1−δ)vL −µ0 ]+δµ0 , then s1 (H, P1 ) = 1. Proof. Due to κ0 (1) > (1 − δ)vH − µ0 , we have rθ∗ < 1 and demand positive in period 2. By the Bertrand undercutting argument, P2∗ = Pr(H|0) ∈ [0, 1] for any P1 . From Pr(H|0) < vθ , we get s∗2 (θ, P2∗ ) = 1 for any P1 . ∗ ∗ Due to vH > vL and (7), we have ∀x s∗t (H, x) ≥ s∗t (L, x) and rH ≥ rL . From ∗ ∗ ∗ ∗ ∗ ∗ ∀x s1 (H, x) ≥ s1 (L, x) and rH ≥ rL , we get σ1 (H, P1 ) ≥ σ1 (L, P1 ) If s∗1 (θ, P1 ) = 0, then πj = 0 for both sellers in period 1. If s∗1 (θ, P1∗ ) < 1, then rθ∗ = r. If P2∗ = µ0 and rθ∗ = r, then seller A deviating to P1A = (1 − δ)vL + δµ0 −  < P1B = P1∗ for  ∈ (0, (1 − δ)(vL − µ0 )) ensures s1 (θ, P1A ) = 1 ∗ ∗ and πA = µ0 rH [(1 − δ)vL + δP2∗ −  − 1] + (1 − µ0 )rL [(1 − δ)vL + δP2∗ − ] = r[(1 − δ)vL + δµ0 −  − µ0 ] > 0. Therefore, in no equilibrium is s∗1 (θ, P1∗ ) = 0 for both types. 9

∗ ∗ Suppose s∗1 (L, P1∗ ) ∈ (0, 1), then by (7), s∗1 (H, P1∗ ) = 1 and rL = r < rH . ∗ ∗ ∗ ∗ Buyer L is indifferent between s1 (L, P1 ) = 0, 1 iff P1 = (1 − δ)vL + δP2 . Seller ∗ ∗ ∗ ∗ ∗ (P1∗ − 1) + (1 − µ0 )rL s1 (L, profit is πj = 12 [µ0 rH   P1 )P1 ], zero in equilibrium.

Deviating to P1A = P1∗ −  < P1B for  ∈ ∗ µ0 rH (P1∗

∗ µ0 )rL (P1∗

0,

∗ ∗ [1−s∗ 1 (L,P1 )](1−µ0 )rP1 ∗ +(1−µ )r ∗ µ0 rH 0 L

yields

πA = −  − 1) + (1 − − ), an increased profit. Thus in no equilibrium is s∗1 (L, P1∗ ) ∈ (0, 1). Suppose s∗1 (H, P1∗ ) ∈ (0, 1), then s∗1 (L, P1∗ ) = 0 and rθ∗ = r. By (1) and ∗ µ0 [1−rs∗ 1 (H,P1 )] P1∗ = Pr(H|1, P1∗ ), we have P1∗ = 1. By (2), we have P2∗ = µ0 [1−rs ∗ (H,P ∗ )]+1−µ . 0 1 1 Buyer H is indifferent between s∗1 (H, P1∗ ) = 0, 1 iff P1∗ = (1 − δ)vH + δP2∗ , i.e. H −1+δµ0 s∗1 (H, 1) = µ(1−δ)v . We have s∗1 (H, 1) ∈ [0, 1] iff both (1 − δ)vH ≥ 1 − δµ0 0 r(1−δ)(vH −1) 1−r . Deviating to P1A = (1 − δ)vL + δP2∗ −  < P1B = and (1 − δ)vH ≤ 1 − δµ0 1−µ 0r P1∗ for some  > 0 leads to s1 (θ, P1A ) = 1 and πA = r[(1 − δ)vL + 1 − (1 − δ)vH −  − µ0 ]. Deviation is unprofitable for any  iff

(1 − δ)vH ≥ 1 − µ0 + (1 − δ)vL .

(8)

Due to 1 − µ0 + (1 − δ)vL > 1 − δµ0 , (8) tightens (1 − δ)vH ≥ 1 − δµ0 above. 1−r Consistency of (8) and s∗1 (H, 1) ≤ 1 holds iff (1 − δ)vL ≤ µ0 − µ0 δ 1−µ , which 0r occurs iff r ≥

(1−δ)vL −µ0 +δµ0 µ0 [(1−δ)vL −µ0 ]+δµ0 .

The probability of noticing offers without paying attention is likely to be small in practice, so the conditions for s∗1 (H, 1) ∈ (0, 1) in Prop. 1 are unlikely L −µ0 +δµ0 ∗ to be satisfied. Assume r < µ0(1−δ)v [(1−δ)vL −µ0 ]+δµ0 henceforth, so s1 (H, P1 ) = 1. ∗ Call equilibria in which s1 (L, P1 ) = 1 class 1 equilibria. Class 0 equilibria have s1 (L, P1∗ ) = 0. Substituting st (θ, Pt∗ ) = 1 into (1) and (3), prices in class 1 are P1∗ :=

∗ µ0 rH , ∗ ∗ rH µ0 + rL (1 − µ0 )

P2∗ :=

∗ )µ0 (1 − rH ∗ )(1 − µ ) . ∗ (1 − rH )µ0 + (1 − rL 0

(9)

Substituting s1 (L, P1∗ ) = 0 and s1 (H, P1∗ ) = s2 (H, P2∗ ) = 1 into (1) and (3), prices in class 0 are P1∗ := 1,

P2∗ :=

∗ )µ0 (1 − rH . ∗ (1 − rH )µ0 + 1 − µ0

(10)

The next result characterises class 0 and 1 equilibria in terms of FOCs and ICs. This simplifies subsequent proofs and numerical calculations. ∗ ∗ Proposition 2. (a) The strategy rH ≥ rL = r, s∗1 (H, P1∗ ) > 0 = s∗1 (L, P1∗ ), ∗ ∗ ∗ s2 (θ, P2 ) = 1 and prices Ptj , t = 1, 2, j = A, B form an equilibrium (of class 0) iff Ptj∗ satisfies (10), s∗ (H) satisfies (7) and

(1 − δ)vL + δP2∗ ≤ holds for the sellers. 10

∗ µ rH 0

∗ rH µ0 + r(1 − µ0 )

(11)

∗ ∗ (b) The strategy rH > rL > r, s∗1 (θ, P1∗ ) = s∗2 (θ, P2∗ ) = 1 and prices Ptj∗ , t = 1, 2, j = A, B form an equilibrium (of class 1) iff Ptj∗ satisfies (9) and s∗ (θ) satisfies (7) for θ = H, L.

Proof. Period 2 is a standard Bertrand game after any play in period 1, so s∗2 (θ, P2∗ ) = 1 and P2∗ is given by (3). Raising price from P1∗ results in zero demand and profit. Reducing price from P1∗ at which L rejects to P1 = (1 − δ)vL + δP2∗ makes both types accept ∗ and yields profit πj = µ0 rH [(1 − δ)vL + δP2∗ − 1] + (1 − µ0 )r[(1 − δ)vL + δP2∗ ] to the deviating seller j. The deviation profit πj is nonpositive iff (11) holds. The IC (6) for L is implied by (11). Buyer type H finds s1 (H, P1∗ ) ≥ 0 optimal given Pt∗ iff (7) holds for H, because (1 − δ)vH + δP2∗ ≥ P1∗ iff the FOC (7) holds for H. Prop. 1 shows ∗ the impossibility of s∗1 (H, P1∗ ) = 0. A strategy with rL = r, s∗1 (L, P1∗ ) = 0 is ∗ ∗ optimal for L given P1 , P2 iff (6) holds for L. ∗ ∗ = r, s∗1 (H, P1∗ ) > 0 = s∗1 (L, P1∗ ), s∗2 (θ, P2∗ ) = 1, ≥ rL If the sellers expect rH then Bertrand competition drives prices to the levels in (10). In class 1, buyer type θ finds s∗1 (θ, P1∗ ) = 1 optimal given Pt∗ iff (7) holds, because in this case, (1 − δ)vθ + δP2∗ ≥ P1∗ . Of course, rθ∗ is optimal iff (7) holds. ∗ ∗ If the sellers expect rH ≥ rL ≥ r, s∗1 (θ, P1∗ ) = s∗2 (θ, P2∗ ) = 1, then Bertrand competition drives prices to (9). Raising prices from (9) results in zero demand and profit. The sellers have no incentive to reduce prices from (9), because both H and L already accept, so the cost per unit sold remains the same. Decreasing prices reduces the profit per unit below zero. Demand weakly increases, reducing the total profit. ∗ ∗ ∗ = r to rH ≥ r. As rH → 1 or The seller IC (11) generalises (8) from rH r → 0 or µ0 → 1, (11) converges to (6). The intuition why the seller IC is sufficient for the buyer IC is that the buyer IC requires the L type buyer not to deviate to accepting the equilibrium P1∗ , but the seller IC requires L not to accept a price reduced to P1 < P1∗ . The price reduction is such that it is profitable for a seller if L accepts it.

3

An example and interpretations

This section presents a numerical example of the comparative static of interest. The subsequent sections show that the result arises in the general model introduced in Section 1, as well as in various extensions of it. p 1 (1 − r)2 − (r − r)2 . This cost Let δ = µ0 = 0.5, r = 0.1 and κ(r) = − 50 function is the bottom right quarter of an ellipse. For vHo = 1.3, vLo = 1.05, there exist two class 1 equilibria and no class 0. The buyer’s strategy in one ∗ ∗ ∗ equilibrium features rHo ≈ 0.994, rLo ≈ 0.93 and in the other rˆHo ≈ 0.997, ∗ ∗ ∗ rˆLo ≈ 0.988. In both, s1o (θ, P1∗ ) = 1. Prices are P1o ≈ 0.517, P2o ≈ 0.079 and ∗ ∗ ≈ 0.199. The payoffs of the buyer types are πHo ≈ 0.784, Pˆ1o ≈ 0.502, Pˆ2o πLo ≈ 0.537 in one equilibrium and π ˆHo ≈ 0.798, π ˆLo ≈ 0.549 in the other.

11

Increasing the valuations to vHn = 1.6, vLn = 1.1, there exists one class 0 ∗ ∗ equilibrium and no class 1. The buyer’s strategy has rHn ≈ 0.307, rLn = r, ∗ ∗ ∗ ∗ s1n (H, P1 ) = 1, s1n (L, P1 ) = 0. Prices are P1n = 1, P2n ≈ 0.409. The payoffs are πHn ≈ 0.614, πLn ≈ 0.345, lower for both types than before the rise in valuations. For an open set of parameters around those in this example, the counterintuitive comparative static result holds. Interpretation 1. The buyer is a potential consumer of health insurance, with type being the future usage of medical services. A frequent user values insurance more and is more costly to insure. The sellers are insurance companies. The cost is attention, with rθ being the probability of noticing one’s need for (different) coverage. The valuation is the utility difference between a (new) policy and the default option. The default may be no insurance or the old policy. The valuations rise, because the government increases the subsidy for cover, or mandates (more) coverage and penalises noncompliers, which reduces the utility from the default option. These default options vary between H, L, because their past optimal choices that led to the defaults differed. So the subsidy or penalty affects the types differently. The same effective valuation change occurs when the government covers more of the insurance companies’ cost—just re-normalise payoffs so that costs are again 0, 1 and the valuations rise. The cost subsidy may be a fixed amount per person, covering a greater fraction of a cheaper claim. Or expenses above a certain level may be refunded, which pays part of a higher, but not a lower cost. The example shows that even if a certain level of funding for the intervention is free, it is sometimes optimal not to intervene, or at least not to the full extent of the funds. Actual interventions should satisfy stronger requirements than showing that the parameters are not close to those in this section. There should be a high enough probability that the types on average benefit sufficiently to cover the cost of the intervention. Interventions cannot usually target only one type, because types are unobservable. The effect of an intervention on the valuations of the types is likely unequal. Which type’s valuation increases more depends on the specific environment and intervention. The consequences of interventions may be opposite to those desired and are difficult to predict based on observable data. Interpretation 2. The buyer is an informed trader of a firm’s shares. The sellers are the uninformed traders selling for liquidity motives. A type H buyer is one who expects the share price to rise significantly, but L anticipates a small rise. Selling to H imposes a greater opportunity cost, because the seller gives up a greater future gain from holding the asset. A firm’s future profit and share price are negatively correlated with its competitor’s, because a successful firm steals business from its rivals. When a firm goes bankrupt, an informed trader values the shares of its competitor more, so both vθ rise. A stronger competitor is better able to expand into the now vacant niche, so its expected profit rises more, i.e. vH rises more than vL . The increase in vθ may delay trade and reduce payoffs, as the numerical example above illustrates. A possible instance is Lehman Brothers going bankrupt, which increased traders’ valuations for its 12

competitors’ shares. The higher valuations may have led to a market freeze, which reduced the profit of most market participants.

4

Comparative statics

This section derives the main result of the paper in the model introduced in Section 1. In a nonempty open region of parameters, as the prior probability ∗ ∗ of H or the valuations rise, the equilibrium (set) moves from rH > rL > r and ∗ ∗ ∗ ∗ ∗ s1 (θ, P1 ) = 1 to rH ≥ rL = r and s1 (H, P1 ) = 1, s1 (L, P1 ) = 0. The payoffs also decline for all types. Define fθ (rL ) for θ = H, L as the rH ∈ [0, 1] that solves (1 − δ)vθ −

(1 − rH )µ0 rH µ0 +δ = κ0 (rθ ), rH µ0 + rL (1 − µ0 ) (1 − rH )µ0 + (1 − rL )(1 − µ0 ) (12)

which is the FOC (7) for θ with Pt∗ determined by (9). For some rL , there is ∗ ∗ no solution when θ = L, so fL (rL ) is undefined. Prop. 2 shows that (rL , rH ) in a class 1 equilibrium is an intersection of fH , fL in (rL , rH )-space satisfying ∗ rθ ≥ r. Prop. 2 also shows that a class 0 equilibrium has rH ≥ r defined by fH (0). Solving fH (rL ) = fH (rL ) = 0, is not possible in general. When a closed form exists, it is too complicated to be useful. For example, when κ is quadratic, the equilibrium is the solution to a cubic equation with coefficients long expressions in the parameters. In the special case δ = 1, all equilibria feature rθ∗ = r, P1∗ = P2∗ = µ0 , s∗1 (H, P1∗ ) = s∗1 (L, P1∗ ) ∈ [0, 1], regardless of the other parameters. There is no investment in early acceptance, because δ = 1 makes the buyer indifferent between accepting in period 1 and 2. The indifference allows any s∗1 (θ, P1∗ ) ∈ [0, 1]. If s∗1 (H, P1∗ ) 6= s∗1 (L, P1∗ ), then P1∗ 6= P2∗ , in which case both types would try to buy in the period with the minimal price, leading to s∗1 (H, P1∗ ) = s∗1 (L, P1∗ ). If vH − vL & 0 and δ < 1, then by (12), the unique equilibrium has κ0 (rθ∗ ) = (1 − δ)(vL − µ0 ) and s∗1 (θ, P1∗ ) = 1. The functions fθ are characterised in o n the following lemma and illustrated in 0 0 Fig. 1. For Lemma 3, define rL0 := min rL : (1 − δ)vL + δ 1−rLµ(1−µ = κ (r ) . L 0) Lemma 3. Both fθ are continuous functions into [0, 1], with fH strictly increas0 ing. The domain of fH is [0, 1]. If ∀rL (1 − δ)vL − µ0 +rLµ(1−µ < κ0 (rL ), then 0) fL (rLh) ∈ [0, 1) for all rL ∈ [0, rL0 ]. Otherwise, fL is defined and onto [0, 1] on i  0 −1  µ0 both 0, (κ0 )−1 (1 − δ)vL − µ0 +r(1−µ and (κ ) ((1 − δ)v − µ ) , r . L 0 L0 ) 0 Proof. By definition, fθ (rL ) ⊆ [0, 1] ∀rL . To show that fθ is a function, use (a) for a fixed rL , κ0 (rθ ) weakly increases in rH , and (b) the LHS of (12) strictly decreases in rH . For any rL , there is thus at most one rH at which (12) holds. The LHS and RHS of (12) are jointly continuous in (rL , rH ), so both fθ are continuous. 13

a b The LHS of (12) strictly increases in rL , but κ0 (rH ) is constant. If rL < rL , a a b a then, fixing rH = fH (rL ), the LHS is higher at rL than at rL , but the RHS-s b are equal. The RHS increases in rH , so for the RHS at rL to equal the LHS, it a b is necessary that fH (rL ) < fH (rL ). The assumption κ0 (1) > (1 − δ)vH − µ0 applied to (12) gives fH (1) < 1, so fH is defined at rL = 1. Extend the definition of κ to [0, 1] by setting κ(r) = 0 ∀r ∈ [0, r]. Then fH (0) ≥ 0, because setting rH = 0, rL > 0 in (12), the LHS becomes (1 − δ)vH + δ µ0 +(1−rµL0 )(1−µ0 ) , which is positive and cannot equal the RHS κ0 (0) = 0. If rθ → 0 with limrθ →0 rrHL = ηH , then (12) becomes µ0 + δµ0 = 0. This is satisfiable iff (1 − δ)vH ≥ 1 − δµ0 , in (1 − δ)vH − ηH µη0H+1−µ 0 H +δµ0 ](1−µ0 ) which case ηH = [(1−δ)v µ0 [1−δµ0 −(1−δ)vH ] . Thus fH (0) = 0 iff (1 − δ)vH ≥ 1 − δµ0 , a b otherwise fH (0) > 0. If fH (rL ), fH (rL ) ∈ [0, 1], then fH (rL ) ∈ (0, 1) for all a b rL ∈ (rL , rL ), because fH is strictly increasing. Denote by rL0 a solution of fL (rL ) = 0. Suppose rL0 > 0 and set rH = 0 in 0 (12) with θ = L to get (1 − δ)vL + δ 1−rL0µ(1−µ = κ0 (rL0 ). As rL0 → 0, we get 0) (1 − δ)vL + δµ0 > κ0 (0). At rL0 = 1, we get (1 − δ)vL + δ < κ0 (1). By the Mean Value Theorem, there exists rL0 ∈ fL−1 ({0}) satisfying     µ0 0 −1 0 −1 rL0 ∈ (κ ) (1 − δ)vL + δ , (κ ) ((1 − δ)vL + δ) . 1 − r(1 − µ0 )

Using rH = rL in (12), there is exactly one nonzero intersection rθ45 of fθ and the 45 degree line. The equation fθ (rL ) = rL is κ0 (rθ45 ) = (1 − δ)(v  θ − µ0 ). µ0 By continuity of fL , for rL ∈ [rL45 , (κ0 )−1 (1 − δ)vL + δ 1−r(1−µ ], we have 0)

fL (rL ) ∈ [0, rL45 ]. H )µ0 = 0, Suppose fL (0) > 0, then (12) becomes (1 − δ)vL − 1 + δ (1−r(1−r H )µ0 +1−µ0 L −1+δµ0 yielding rH = fL (0) = (1−δ)v . This is consistent with fL (0) > 0 iff µ0 (1−δ)vL (1 − δ)vL > 1 − δµ0 . Suppose fL (0) = 0 with limrθ →0 rrHL = ηL ∈ [0, ∞]. µ0 Then (12) becomes (1 − δ)vL − ηL µη0L+1−µ + δµ0 = 0. This is satisfiable iff 0 L +δµ0 ](1−µ0 ) (1 − δ)vL ≤ 1 − δµ0 , in which case ηL = [(1−δ)v µ0 [1−δµ0 −(1−δ)vL ] . Set rH = 1 in (12) with θ = L to obtain the rL at which fL (rL ) = 1. This yields µ0 = κ0 (rL ). (13) (1 − δ)vL − µ0 + rL (1 − µ0 )

µ0 If (1 − δ)vL ≥ µ0 +r(1−µ , then by (1 − δ)vL − µ0 < κ0 (1) and the Mean 0) Value  Theorem ∃rL1 s.t. fL (rL1 ) = 1. Based on (13), this rL1 is between

(κ0 )−1 (1 − δ)vL −

µ0 µ0 +r(1−µ0 )

and (κ0 )−1 ((1 − δ)vL − µ0 ). Denote the solu-

tion set of (13) by {rL1 }. In terms of primitives, @rL1 , fL (rL1 ) = 1 is µ0 (1 − δ)vL − < κ0 (rL ) (14) µ0 + rL (1 − µ0 )     µ0 ∀rL ∈ (κ0 )−1 (1 − δ)vL − , (κ0 )−1 ((1 − δ)vL − µ0 ) . µ0 + r(1 − µ0 ) 14

Combine the fL (rL ) = 1 result with those for fL (rL ) = 0 and fL (0). If (14) holds, then fL (rL ) ∈ [0, 1) for all rL ∈ [0, rL0 ]. If (14) fails, then the continuity of fL implies that fL is onto [0, 1] and defined on [0, min {rL1 }] ∪ [max {rL1 } , rL0 ].

0.8

0.8

0.6

0.6 rH

1.0

rH

1.0

0.4

0.4

0.2

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

rL

0.0 0.0

0.2

0.4

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0.8

1.0

rL

Figure 1: The effect of the prior p on the equilibrium set. Blue curve: fH , orange: fL . Parameters κ(r) = − 15 (1 − r)2 − (r − r)2 , δ = µ0 = 0.5, r = 0.1, left panel: vHo = 1.2, vLo = 1.1, right panel: vHn = 1.8, vLn = 1.2. The next lemma characterises the fθ functions further. It is used to prove the existence and uniqueness results later. Lemma 4. For θ = H, L, fθ (r) > r. Further, fH (r) > fL (r) iff κ0 (fL (r)) < (1 − δ)(vH − vL ). Proof. Suppose fθ (r) ≤ r, then (7) becomes (1 − δ)vθ − P1 + δP2 ≤ 0, in which rH ≤ rL and prices are determined by (9). With rH ≤ rL , (9) gives P1 ≤ µ0 ≤ P2 . This implies (1 − δ)vθ − P1 + δP2 ≥ (1 − δ)(vθ − µ0 ) > 0, a contradiction. (1−rH )µ0 H µ0 At r, (12) with θ = L becomes (1−δ)vL − rH µ0r+r(1−µ +δ (1−rH )µ = 0) 0 +(1−r)(1−µ0 ) √

−b± d 0. The solutions are rH = 2µ0 (1−δ)(v , where −b = (1 − δ)vL − 1 + δµ0 − L −1) 2r(1 − µ0 )(1 − δ)vL + r(1 − µ0 )(1 − δ) and d = δ 2 (µ0 + r − µ0 r − vL )2 + (vL − 1 + r − µ0 r)2 − 2δ[−(1 + µ20 )r(1 − r) + vL (vL − 1) − µ0 (vL − 1 − 2r(1 − r))]. There is exactly one solution in [0, 1], because Lemma 3 showed that fθ are functions. This solution is fL (r). Substituting rH = fL (r) into the LHS of (12) with θ = H gives (1 − δ)(vH − vL ). The RHS κ0 (fL (r)) is smaller than (1 − δ)(vH − vL ) iff fH (r) > fL (r). Raising rH increases the RHS and reduces the LHS, restoring equality.

15

The next proposition provides existence conditions for the class 0 equilibrium, using the assumption (1 − δ)(vL − µ0 ) µ0 [(1 − δ)vL + δ − µ0 ] µ0 (1 − vL + δvL ) ∨ (1 − δ)vL ∈ (µ0 , 1) ∧ r < . (1 − µ0 )(1 − δ)vL

(1 − δ)vL < µ0 ∧ r <

∗ A lower bound on the equilibrium rH will be shown to be s 1 d −b sol , rH := + 2[µ0 (1 − δ)(vL − 1)] 2 [µ0 (1 − δ)(vL − 1)]2

(15)

(16)

where −b = vL − 1 − vL r(1 − µ0 ) − δ[r(1 − µ0 )(vL − 1) + vL − µ0 ] and d = δ 2 [r(1 − µ0 )(vL − 1) + vL − µ0 ]2 + [vL − 1 + r(1 − µ0 )]2 − 2δ[(vL − 1)(vL − µ0 ) + 2 r2 (1 − µ0 )2 (vL − 1)vL + r[−1 + µ20 (vL − 2) + 2(vL − 1)vL + 3µ0 + vL µ0 − 2vL µ0 ]]. The comparative static results rely on Prop. 5 and Prop. 6, which provides existence conditions for class 1 equilibria. The comparative static of interest involves moving between class 0 and 1 equilibria. Proposition 5. A class 0 equilibrium exists iff (15) and (1 − δ)(vH − vL ) − r(1−µ0 ) sol sol ≥ κ0 (rH ) hold, where rH is defined in (16). A class 0 equilibr sol µ0 +r(1−µ0 ) H

rium exists only if (1 − δ)vL ≤

µ0 µ0 +r(1−µ0 )

(1−r)µ0 ≥ 1. and (1 − δ)vH + δ (1−r)µ 0 +1−µ0

Proof. By Prop. 2, a class 0 equilibrium exists iff the FOC (7) holds for θ = H, rL = 0, rH ≥ r, and the seller IC (11) is satisfied. The LHS of (11) decreases in rH and the RHS increases, so (11) holds iff rH is above a cutoff. By the Mean Value Theorem, (7) has a solution satisfying (11) iff solving (11) with ‘=’ instead of ‘≤’ and substituting into (7) results in (7) holding with ‘≥’. sol The solution of the quadratic (11) with ‘=’ is rH , defined in (16). It satisfies sol rH ∈ (r, 1) iff (15) holds, as can be verified by checking the discriminant and sol < 1. simplifying the inequalities r < rH sol Substituting rH into (7) with ‘≥’, a class 0 equilibrium exists iff (15) and r sol µ

0 sol (1 − δ)(vH − vL ) + rsol µ0H+r(1−µ − 1 ≥ κ0 (rH ) hold. 0) H µ0 At rH = 1, (11) is (1 − δ)vL ≤ µ0 +r(1−µ0 ) . This with ‘<’ is necessary for ∗ class 0 existence, because rH < 1. At rH = 1, (7) has ‘<’ instead of ‘=’. By the Mean Value Theorem, (7) has a solution rH ≥ r iff at rH = r, (7) has ‘≥’ instead (1−r)µ0 of ‘=’. Thus (1 − δ)vH + δ (1−r)µ ≥ 1 is necessary for class 0 existence. 0 +1−µ0   ∗ 0 −1 Based on (12), rH ∈ (κ ) ((1 − δ)vH − 1) , (κ0 )−1 ((1 − δ)vH − 1 + δµ0 ) .

µ0 A corollary of Prop. 5 is that if vH is above a cutoff, (1 − δ)vL ≤ µ0 +r(1−µ 0) sol and (15) holds, then a class 0 equilibrium exists. Neither rH nor (15) depends µ0 on vH . If r → 0 or µ0 → 1, then (15) implies (1 − δ)vL ≤ µ0 +r(1−µ , and both 0) together reduce to (1 − δ)vL < 1. If (1 − δ)vL ≥ 1, then a class 1 equilibrium exists, as Prop. 6 below shows.

16

µ0 Proposition 6. A class 1 equilibrium exists if (1 − δ)vL ≥ µ0 +r(1−µ or vH ∈ 0)   κ0 (fL (r)) ∗ vL , vL + 1−δ . If (14) holds, then there exist κ0∗ (1) > 0 and vH (κ0 (1)) ∈   0 0 ∗ s.t. if κ0 (1) > κ0∗ (1) and vH > vH (κ0 (1)), then a class 1 equilibvL , κ (1)+µ 1−δ rium does not exist.

Proof. By Lemma 3, if (14) fails, then on [(κ0 )−1 ((1 − δ)vL − µ0 ) , rL0 ], fL is continuous and onto. In this case, fL intersects the continuous strictly increasing fH on [(κ0 )−1 ((1 − δ)vL − µ0 ) , rL0 ]. The intersection is a class 1 equilibrium. 0 L (r)) Lemmas 4 and 3 imply that if (14) holds and vH ∈ [vlo , vL + κ (f1−δ ), then fH (r) ≤ fL (r), fH (rL0 ) > fH (0) ≥ 0 = fL (rL0 ) and both fθ are defined and continuous on [0, rL0 ]. By the Mean Value Theorem, the fθ intersect on [0, rL0 ]. The intersection is a class 1 equilibrium. If (14) holds, then define fLmax := maxrL ∈[0,1] fL (rL ). From −P1∗ + δP2∗ ∈ [−1, δ] and (12), get κ0 (rH ) ∈ [(1 − δ)vH − 1, (1 − δ)vH + δ] for any rL , therefore 0 0 , rH = fH (rL ) ∈ [(κ0 )−1 ((1 − δ)vH − 1), (κ0 )−1 ((1 − δ)vH + δ)]. As vH → κ (1)+µ 1−δ 0 −1 max the bound minrL ∈[0,1] [fH (rL ) − fL (rL )] ≥ (κ ) ((1 − δ)vH − 1) − fL becomes minrL ∈[0,1] [fH (rL ) − fL (rL )] ≥ (κ0 )−1 (κ0 (1) + µ0 − 1) − fLmax . As κ0∗ (1) → ∞, we get (κ0 )−1 (κ0 (1) + µ0 − 1) → 1 for any κ0 with κ0 (1) > κ0∗ (1). Then fH (rL ) > fLmax for any rL and κ0 (1) ≥ κ0∗ (1), so there is no class 1 equilibrium. By continuity of (κ0 )−1 ((1 − δ)vH − 1), there exist κ0∗ (1) > 0 and 0

0 ∗ ∗ (κ0 (1)) ∈ vL , κ (1)+µ vH (κ0 (1)), then a s.t. if κ0 (1) > κ0∗ (1) and vH > vH 1−δ class 1 equilibrium does not exist. ∗ Due to a class 1 equilibrium existing when vH → vL , we have vH (κ0 (1)) > vL . 0 0 ∗ The assumption vH (κ0 (1)) < κ (1)+µ is κ0 (1) > (1 − δ)vH − µ0 from Section 1; 1−δ 0 it becomes vacuously true as κ (1) → ∞.

The main comparative static follows from Theorem 7 below. The theorem establishes sufficient conditions for uniqueness of class 0 and class 1 equilibria. The unique equilibrium may be of class 0 at higher vL , vH and of class 1 at lower. This situation is depicted in Fig. 1, where in the right panel, valuations are higher than in the left panel and there is a unique class 1 equilibrium in the left panel and a unique class 0 in the right panel. Theorem 7. There exists ∆v > 0 s.t. when vH − vL < ∆v, a unique class 1 equilibrium exists. There exists M > ∆v s.t. when (14) holds and vH −vL > M , a unique class 0 equilibrium exists. 0) sol Proof. The class 0 necessary condition (1−δ)(vH −vL )− rsol µr(1−µ ≥ κ0 (rH ) 0 +r(1−µ0 ) H

0) in Prop. 5 is violated when vH − vL < µ0r(1−µ +r(1−µ0 ) is below a positive cutoff. At ∗ ∗ vH = vL , the unique equilibrium has rH = rL = (κ0 )−1 ((1 − δ)(vL − µ0 )). Because fθ vary continuously in vθ , there is an open set of vH around vH = vL for which the intersection of fH , fL is unique. As vH → vL , this intersection 0 −1 converges to r ((1 H = rL = (κ )  − δ)(vL − µ0 )). By Prop. 6, the intersection

exists if vH ∈ vL , vL +

κ0 (fL (r)) 1−δ

.

17

There is at most one class 0 equilibrium. If (14) and (1 − δ)(vH − vL ) − r(1−µ0 ) 0 sol sol µ +r(1−µ ) ≥ κ (rH ) from Prop. 5 hold, then there exists a class 0 equirH 0 0 librium. The inequality is satisfied for vH above a positive cutoff. If vH > ∗ vH (κ0 (1)), κ0 (1) > κ0∗ (1) as in Prop. 6, then there does not exist a class 1 equilibrium. As Theorem 7 shows, increasing the valuations of the buyer types from vHo − vLo < ∆v to vHn − vLn > M , keeping r low and (1 − δ)vL < 1, moves the equilibrium from class 1 to class 0. The trading probability σ1 (L, P1∗ ) of L falls ∗ to zero in period 1. If the increase in vH is not too large, then σ1 (H, P1∗ ) = rH also decreases. The equilibrium is on the strictly increasing fH before and after the rise in vθ , but fH itself rises in vH . If the shift in fH is not too large relative to the move along fH , then σ1 (H, P1∗ ) decreases. Section 3 uses a cost function with κ0 (1) = ∞, so the requirement κ0 (1) > 0 κ∗ (1) is satisfied. In the real world, probability 1 of most events, e.g. being able to trade, is unattainable for any cost. There is no guarantee against force majeure that may prevent a transaction, so κ0 (1) = ∞ is reasonable. Section 3 shows that the uniqueness of a class 0 equilibrium does not require a particularly high vH , either on an absolute scale or relative to vL . Due to the continuity of fθ in vθ , there is an open set of parameters around those in the example in which the comparative statics are similar to the example. This means σ1 (θ, P1∗ ) decreases in vθ , as do the equilibrium payoffs. The feedback loop that drives the counterintuitive comparative static is explained next, using the FOC (7). Increasing vθ raises the impatience motive (1−δ)(vθ −P1∗ ) for type θ. Then θ invests more in being able to accept early, i.e. rθ∗ increases. There is greater incentive to accept given the chance, so σ1∗ (θ, P1 ) rises, for any P1 . The sellers revise their expectation of σ1∗ (θ, P1 ) upward and change price accordingly. A higher σ1∗ (H, P1 ) increases P1∗ and decreases P2∗ , but a higher σ1∗ (L, P1 ) has the opposite effects. The signalling motive δ(P1∗ −P2∗ ) rises in σ1∗ (H, P1∗ ), but falls in σ1∗ (L, P1∗ ). A greater signalling motive reduces σ1∗ (θ, P1∗ ) for both θ. Now the lower σ1∗ (L, P1∗ ) increases the signalling motive more, which reduces σ1∗ (L, P1∗ ) more, etc. This positive feedback can cause a large change in the equilibrium in response to a small variation of the parameters. The payoff (4) of type θ after imposing the FOC (12) for θ is πθ1 = δvθ1 − δ

(1 − rH1 )µ0 + rθ1 κ0 (rθ1 ) − κ(rθ1 ) (1 − rH1 )µ0 + (1 − rL1 )(1 − µ0 )

in a class 1 equilibrium. For L in a class 0 equilibrium, the payoff is πL0 = δvL0 − δP2∗ . For H in class 0, the payoff is πH0 = δvH0 − δ

(1 − rH0 )µ0 + rH0 κ0 (rH0 ) − κ(rH0 ). (1 − rH0 )µ0 + 1 − µ0

By the standard strategy-stealing argument, πHk > πLk for k = 0, 1. In the limit as vH1 → vL1 , both types’ class 1 payoffs become δvL1 − δµ0 + 18


(κ0 )−1 ((1 − δ)(vL1 − µ0 )) (1−δ)(vL1 −µ0 )−κ (κ0 )−1 ((1 − δ)(vL1 − µ0 )) . The payoff comparison between the class 1 and 0 equilibria is given in the next theorem. Together with the conditions in Theorem 7, the assumptions in Theorem 8 ensure that increasing the valuations changes the equilibrium from a unique class 1 to a unique class 0 with lower payoffs for both types. 0 s.t. if vL1 − µ0 > vH0 − vL0 , Theorem 8. There exist  ∆2 v > 0 and r∗ >  δ(vH0 −vL1 +µ0 ) 0 (1 − δ)(vH0 − vL0 ) > κ (1−δ)(vL1 −µ0 −vH0 +vL0 ) , vH1 − vL1 < ∆2 v and r < r∗ , then πθ1 > πθ0 for θ = H, L. 0 Proof. The is positive and strictly increasing in r, because R r function rκ (r)−κ(r) Rr κ(r) = 0 κ0 (x)dx < κ0 (r) 0 dx = rκ0 (r) and [rκ0 (r) − κ(r)]0 = rκ00 (r) > 0. Geometrically, rκ0 (r) − κ(r) subtracts the area under the κ0 curve from the area of the rectangle with sides [0, r] and [0, κ0 (r)]. Switching the r-axis and κ0 (r)R κ0 (r) 0 −1 axis, rκ0 (r) − κ(r) is also the area 0 (κ ) (z)dz under the (κ0 )−1 curve on 0 [0, κ (r)]. ∗ than in the proof of Prop. 5 is obtained A smaller upper bound on rH0 ∗ by subtracting the seller IC (11) from the FOC (7), resulting in κ0 (rH0 ) ≤ r(1−µ0 ) ∗ (1 − δ)(vH0 − vL0 ) − r∗ µ0 +r(1−µ0 ) . As r → 0, the bound becomes rH0 ≤ H0 (κ0 )−1 ((1 − δ)(vH0 − vL0 )). This is used in the first inequality below. H0 )µ0 limvH1 →vL1 , r→0 πH1 − πH0 = δ(vL1 − vH0 ) − δ(µ0 − (1−r(1−r )+ H0 )µ0 +1−µ0
(1 − δ)(vL1 − µ0 )(κ0 )−1 ((1 − δ)(vL1 − µ0 )) − κ (κ0 )−1 ((1 − δ)(vL1 − µ0 )) − rH0 κ0 (rH0 ) + κ(rH0 ) R (1−δ)(v −µ ) (1−rH0 )µ20 +µ0 (1−µ0 )−(1−rH0 )µ0 + κ0 (rH0 ) L1 0 (κ0 )−1 (z) dz = δ(vL1 − vH0 ) − δ (1−rH0 )µ0 +1−µ0 R (1−δ)(vL1 −µ0 ) 0 −1 > δ(vL1 − vH0 ) − δµ0 + (1−δ)(vH0 (κ ) (z) dz −vL0 ) > −δ(vH0 −vL1 +µ0 )+(1−δ)(vL1 −µ0 −vH0 +vL0 )(κ0 )−1 ((1 − δ)(vH0 − vL0 )).

δ(vH0 −vL1 +µ0 ) , If vL1 − µ0 > vH0 − vL0 and (1 − δ)(vH0 − vL0 ) > κ0 (1−δ)(v L1 −µ0 −vH0 +vL0 ) then limvH1 →vL1 , r→0 πH1 > πH0 . If πH1 ≥ πH0 , then πL1 ≥ πL0 , because πH1 − πH0 − (πL1 − πL0 ) = δvH1 − ∗ ∗ ∗ δP2,1 +rH1 κ0 (rH1 )−κ(rH1 )−δvL1 +δP2,1 −rL1 κ0 (rL1 )+κ(rL1 )−δvH0 +δP2,0 − 0 ∗ rH0 κ (rH0 ) + κ(rH0 ) + δvL0 − δP2,0 = δ(vH1 − vL1 − vH0 + vL0 ) + rH1 κ0 (rH1 ) − κ(rH1 ) − rL1 κ0 (rL1 ) + κ(rL1 ) − rH0 κ0 (rH0 ) + κ(rH0 ) < 0. This is negative, because rHk ≥ rLk and rκ0 (r) − κ(r) > 0 is strictly increasing. Proving πH1 > πH0 thus suffices for πL1 > πL0 . By the joint continuity of πθk in (vLk , vHk , r), if πθ1 > πθ0 at vH1 = vL1 , r = 0, then there exist ∆2 v > 0 and r∗ > 0 s.t. vH1 − vL1 < ∆2 v and r < r∗ together imply πθ1 > πθ0 .

∗ In the proof of Theorem 8, the upper bound on rH0 could be replaced with ∗ 0 −1 the bound rH0 ≤ (κ ) ((1 − δ)vH0 − 1 + δµ0 ) from the proof of Prop. 5. Then the requirement r < r∗ may be omitted. This is not done above, because (κ0 )−1 ((1 − δ)vH0 − 1 + δµ0 ) is larger than (κ0 )−1 ((1 − δ)(vH0 − vL0 )) for all r ∈ [0, 1), given the necessary condition (1 − δ)vL < 1 from Prop. 5.

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The results are robust to small perturbations of the fθ functions. One such perturbation is changing the updating from Bayes’ rule to a function uniformly close to it. Another such small change occurs when the sellers have slightly positive market power. The buyer’s payoff changes continuously in the prices, which are continuous in the competitiveness of the sellers, so the comparative statics still hold. The reason for the robustness can be seen in Fig. 1: when fθ are perturbed slightly, generically their intersection still exists and is unique and close to its original location. The next section discusses other robustness checks formally and in greater detail.

5 5.1

Extensions Valuations vary over time

The first extension is to valuations that change over time. Instead of valuing the good at vθ at the time of both offers, the buyer’s benefit from the first offer is vθ1 and from the second offer vθ2 . The FOC becomes vθ1 −δvθ2 −P1∗ +δP2∗ = κ0 (rθ ). Similarly, (1 − δ)vθ is replaced with vθ1 − δvθ2 in the buyer and seller IC-s. The proofs in the preceding sections are unchanged when time subscripts are added to valuations. Based on the new FOC and IC-s, the comparative statics in vθ1 are the same as in the time-invariant vθ previously, but the comparative statics in vθ2 are the opposite. The counterintuitive results are driven by the change in the first period valuation, meaning that subsidising trade early and taxing it later may delay trade and reduce the payoffs of all types. This is the opposite of the result of Fuchs and Skrzypacz (2015) that to encourage trade, there should be an early subsidy and a later tax.

5.2

Uniformly distributed types

The buyer’s type θ is uniformly distributed on [0, 1], and equals a seller’s cost of serving the buyer. The buyer’s valuation is v(θ) = a + bθ, with a ≥ 1, b > 0. The rest of the primitives of the model are the same as in Section 1. In period 2, all buyer types accept. The type who is indifferent between accepting and rejecting in period 1 is denoted θ∗ (P1 ) and satisfies v(θ∗ ) − P1 − δv(θ∗ ) + δP2∗ = 0 if interior. All types above θ∗ strictly prefer to accept. If θ∗ (P1 ) = 0, then v(0) − P1 − δv(0) + δP2∗ ≥ 0. If θ∗ (P1 ) = 1, then v(1) − P1 − δv(1) + δP2∗ ≤ 0, but the sellers would deviate to lower price if no type accepts P1∗ . There is at most one indifferent type, because v 0 (θ) > 0. The prices depend on the r∗ that the sellers expect and on θ∗ given P1 . The sellers in period 2 set R θ∗ (P1∗ ) 0

θdθ +

P2 = R θ∗ (P ∗ ) 1 0

dθ +

R1

θ[1 − r∗ (θ)]dθ

θ ∗ (P1∗ ) R1 [1 θ ∗ (P1∗ )

20

− r∗ (θ)]dθ

.

(17)

The cost a period 1 seller expects if setting P1 below the competitor’s price is R1 θr∗ (θ)dθ θ ∗ (P ) . (18) E[θ|P1 ] = R 1 1 r∗ (θ)dθ θ ∗ (P1 ) Seller A in period 1 maximises  1 ∗   2 [P1A − E[θ|P1 ]], θ (P1A ) < 1 ∧ P1A = P1B , πA (P1A ) := P1A − E[θ|P1 ], θ∗ (P1A ) < 1 ∧ P1A < P1B ,   0 θ∗ (P1A ) ≥ 1.

(19)

Seller B’s payoff is symmetric. An equilibrium price in period 1 satisfies P1∗ = E[θ|P1∗ ]; there may be multiple such prices. The FOC and the buyer IC are (7) and (6) as in the benchmark model, with only the prices changed. The seller IC now becomes a continuum of constraints, one for each P1 < P1∗ . These constraints hold iff the most profitable deviation yields nonpositive profit: maxP1 ≤P1∗ Pr(θ ≥ θ∗ (P1 ))(P1 − E[θ|P1 ]) ≤ 0. Subtracting the FOCs of different types yields (1 − δ)[v(θ1 ) − v(θ2 )] = κ0 (r∗ (θ1 )) − κ0 (r∗ (θ2 )), which allows the function r∗ (·) to be reduced to a scalar in (17) and (18). Take θ2 = θ∗ , then the scalar is v(θ∗ ), because r∗ (θ) = r and κ0 (r) = 0 for all θ ≤ θ∗ , but r∗ (θ) = (κ0 )−1 ((1 − δ)[v(θ) − v(θ∗ )]) for all θ > θ∗ . This simplifies numerical solving. There is generally no closed form solution for r∗ (θ). Next, the counterintuitive comparative static is demonstrated numerically. p At vo (θ) = 1.17 + 0.4θ, δ = 0.6, r = 0.1, κ(r) = −0.01 (1 − r)2 − (r − r)2 , the ∗ ∗ ∗ ) = 0, P1o ≈ 0.523 and P2o ≈ 0.096. All types try unique equilibrium has θo∗ (P1o ∗ to accept P1o . The equilibrium payoff is approximately linear in θ between 0.65 at θ = 0 and 1.05 at θ = 1. Raising the valuations to vn (θ) = 1.18 + 0.6θ, there ∗ ∗ ∗ is a unique equilibrium with θn∗ (P1n ) ≈ 0.743, P1n ≈ 0.884 and P2n ≈ 0.39. The payoff is approximately linear between 0.47 at θ = 0 and 0.85 at θ = 1. It is not surprising that the comparative static results continue to hold with a continuum of types, because the driving forces of signalling and investment in trading opportunities are present whenever there are at least two types. The feedback loop in this section operates similarly to the two type case.

6 6.1

Benchmarks: No frictions or exogenous frictions Two types and frictionless trading

Suppose the buyer always has the opportunity to trade, without paying any cost. Type θ accepts the first offer P1 if vθ − P1 > δvθ − δP2∗ and rejects it if the reverse inequality holds. If L accepts, then H strictly prefers to accept, because vH > vL .

21

If the sellers expect both types to accept the first offer, then P1 = µ0 and P2 is off-path and undefined. If the sellers expect both types to reject the first offer, then P2 = µ0 and P1 is undefined. Impose the refinement that a deviation to accepting the undefined price comes from the type for whom this deviation is more profitable. This sets P2 = 0 when the sellers expect both types to accept and P1 = 1 when they expect both to reject. These prices are obtained as limits when H accepts and L mixes with probability of acceptance approaching 1, or when L rejects and H mixes with acceptance probability going to 0. This refinement respects the abovementioned property of best responses that if L accepts, then H strictly prefers to accept. The equilibrium where both types accept exists iff (1 − δ)vL ≥ µ0 . Both rejecting is never an equilibrium, because a seller will reduce price to make positive profit. The equilibrium where H accepts and L rejects the first offer exists iff vH − 1 ≥ δvH , vL − 1 ≤ δvL and the seller IC µ0 [(1 − δ)vL − 1] + (1 − ∗ = r = 1 and µ0 )(1 − δ)vL ≤ 0 hold. This IC is a special case of (11) when rH ∗ P2 = 0. As before, the buyer IC is implied by the seller IC, which reduces to (1 − δ)vL ≤ µ0 . Mixed equilibria may also exist, but I focus on pure. The comparative statics are intuitive: raising the gains from trade leads to earlier trading. Increasing vL may create the equilibrium where both types accept and destroy the equilibrium where H accepts and L rejects. This is clear from the inequalities in the preceding paragraph. Increasing vH may create the equilibrium where H accepts and L rejects.

6.2

Two types and exogenous frictions

Now suppose there are exogenous frictions, so that the buyer has the option to trade with probability r ∈ (0, 1). The conditions for acceptance and rejection of the first offer are the same as without frictions; only the prices change. The second period price P2 is now always defined and in (0, 1). When both types are expected to reject, P1 is undefined. If the sellers expect both types to accept, then P1∗ = P2∗ = µ0 . With δ < 1, it is not possible that both types reject, because the sellers would cut price. The equilibrium where both accept exists iff vL ≥ µ0 , which holds by assumption. When P1 ≤ P2 , there is no reason to delay acceptance. If the sellers expect H to (1−r)µ0 accept and L to reject, then P1∗ = 1 and P2∗ = (1−r)µ . This equilibrium 0 +1−µ0 exists iff (1 − δ)vL ≤ 1 − δP2∗ ≤ (1 − δ)vH and the seller IC (1 − δ)vL + δP2∗ ≤ µ0 hold. The seller IC implies the L type IC. Mixed equilibria may also exist. Increasing both vθ does not affect the all-accept equilibrium, but may create or destroy the equilibrium where H accepts and L rejects the initial offer. This is the intuitive direction of comparative statics, and it is unchanged when the exogenous probability r of noticing offers is type-dependent: rL , rH ∈ (0, 1).

6.3

Continuum of types with exogenous frictions

Buyer types θ ∈ [0, 1] are distributed according to the atomless prior cdf F . The R1 prior expected type is θ¯ := 0 θdF (θ). The valuation of buyer type θ is v(θ), 22

with v : [0, 1] → (1, ∞) a strictly increasing function.   Each buyer type θ chooses the probability σ(θ, P1 ) ∈ 0, σ 1 (θ) ⊆ [0, 1] of accepting the first price offer P1 . Assume σ 1 (θ) > 0 ∀θ. For simplicity, the choice of accepting P2 is frictionless. If σ 1 (θ) = 1 for all θ, then the choice of accepting P1 is also frictionless. The bound σ 1 (θ) on the choice probability defines the frictions. Due to v(0) > 1, all types accept the last offer of the market conditional on reaching that point in the game. A best response of buyer type θ to the first offer P1 of the sellers is σ(θ, P1 ) = 1 if v(θ) − P1 ≥ δv(θ) − δP2∗ . If the reverse inequality holds, then σ(θ, P1 ) = 0 is optimal. Clearly, θ1 ≤θ2 implies  P −δP ∗

σ(θ1 , P1 ) ≤ σ(θ2 , P1 ) for any P1 . At most one type, θ∗ (P1 ) := v −1 11−δ 2 , is indifferent between the first and the second offer, so at most one type mixes. Due to the atomless prior, the action of one type is irrelevant for equilibrium. A strategy σ(θ, P1 ) can be replaced with θ∗ (P1 ) w.l.o.g. The probability the sellers put on H conditional on acceptance in period 1 and 2 is, respectively, R1 1 ∗ zσ (z)dF (z) ∗ Pr(H|1, θ ) = Rθ 1 , (20) σ 1 (z)dF (z) θ∗ R θ∗ R1 zdF (z) + θ∗ z[1 − σ 1 (z)]dF (z) ∗ 0 Pr(H|01, θ ) = R θ∗ . R1 dF (z) + θ∗ [1 − σ 1 (z)]dF (z) 0

In equilibrium, P1 = Pr(H|1, θ∗ (P1 )) and P2 = Pr(H|01, θ∗ (P1∗ )). If both prices are defined, then P1 > θ¯ > P2 and both prices increase in θ∗ (·). The price increase is strict if θ∗ rises over a region with a positive mass of types. If the sellers expect all types to reject the first offer, then P1 is off-path and undefined. But then a seller will deviate to a lower price that causes some buyer types to accept, making positive profit. If the sellers expect all types to accept with certainty (σ 1 (θ) = 1 = σ ∗ (θ, P1∗ ) for all θ), then P2 is undefined. Impose the refinement that the off-path P2 is the limit of Pr(H|01, θ∗ (P1∗ )) when θ∗ (P1∗ ) → 0, i.e. P2 = 0. This is consistent with θ1 ≤ θ2 ⇒ ∀P1 σ(θ1 , P1 ) ≤ σ(θ2 , P1 ) above. Define the function γ : [0, 1] → R as γ(θ) := (1 − δ)v(θ) − Pr(H|1, θ) + δ Pr(H|01, θ). All types accepting is an equilibrium iff γ(0) ≥ 0. If σ 1 (θ) is constant in θ, then γ(0) ≥ 0 reduces to (1 − δ)v(0) ≥ (1 − δ)θ, which holds by assumption. The inequality v(0) ≥ θ is also sufficient for the all-accept equilibrium when σ 1 is decreasing, because then P1 ≤ θ ≤ P1 . For other σ 1 , there is a cutoff δ ∗ ∈ (0, 1] s.t. the all-accept equilibrium exists for δ ≤ δ ∗ . Interior zeros of γ, i.e. θ ∈ (0, 1) s.t. γ(θ) = 0, are equilibria iff the seller IC holds. The seller IC states that the profit from cutting price to P1 < P1∗ is nonpositive. Equivalently, reducing the indifferent type to θ˜ := θ∗ (P1 ) < θ∗ (P1∗ ) ˜ + δP ∗ . Formally, is not profitable. The type θ˜ is indifferent iff P1 = (1 − δ)v(θ) 2

23

the seller IC is ˜ + δP ∗ ] max [(1 − δ)v(θ) 2

˜ ∗ (P ∗ ) θ≤θ 1

Z

1 1

Z

1

σ (θ)dF (θ) − θ˜

θσ 1 (θ)dF (θ) ≤ 0,

θ˜

∗ ∗ ˜ ˜ equivalently maxθ≤θ ˜ ∗ (P ∗ ) (1 − δ)v(θ) − Pr(H|1, θ) + δ Pr(H|01, θ (P1 )) ≤ 0. The 1 ∗ ∗ LHS of this inequality increases in θ (P1 ). Assume a positive mass of types in any open subinterval of [0, 1], then the LHS of the seller IC strictly increases ∗ ∗ in θ∗ (P1∗ ). Consider two candidate equilibria a, b, with θa∗ (P1a ) < θb∗ (P1b ). ∗ ˜ − Pr(H|1, θ) ˜ + Then the seller IC is violated at θb∗ (P1b ), because (1 − δ)v(θ) ∗ ∗ δ Pr(H|01, θb∗ (P1b )) > γ(θa∗ (P1a )) ≥ 0. Only the candidate equilibrium with the ∗ ∗ lowest θ (P1 ) can be an equilibrium. Raising v increases γ(0) and may thus create the all-accept equilibrium and destroy an interior equilibrium. If there is an interior equilibrium both before and after increasing v, then γ(0) < 0 in both cases. Then γ must cross 0 from below at the interior equilibrium. Raising v increases γ, moving this zero of γ left, i.e. to a lower θ∗ (P1∗ ), making more types accept. In summary, the unique equilibrium (whether all-accept or interior) has the intuitive comparative static that higher gains from trade lead to earlier trade.

7

Conclusion

Markets with adverse selection and multiple opportunities to trade feature an incentive to signal by delay. This signalling may manifest as seemingly counterintuitive behaviour. Subsidising trade early and taxing it later may delay trade. Similarly, raising the gains from trade in all periods may delay trade. These findings run counter to the previous literature, which uses different assumptions, in particular exogenous probability of being able to trade. Usually this probability is one. The importance of considering controllable trading frictions, e.g. rational inattention, is illustrated by the significant change in the play of the game when there is a small cost of gaining the option to trade. Generalisations to more than two periods and types are left for future research, as is the addition of moral hazard (investing in increasing one’s valuation). An important modification of the model to study is the case where the uninformed party of the transaction is a monopolist. This would connect the current work to the large literature on the Coase conjecture. Another avenue of extension is to endogenise the valuations of the buyers and sellers by embedding the model in a search and matching framework. There has been recent interest in search markets with adverse selection, but to the author’s knowledge, no search model has considered the agents being able to control the matching intensity under adverse selection.

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