# Complementarities in information acquisition with short-term trades

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7 Theoretical Economics (007) Complementarities in information acquisition 447 Prices 1 Periods p ++ t +1 p + t π 0 π 0 + t +1 p t 1 π 0 + π + t +1 π 0 π 0 + π t t +1 p t π 0 + π t +1 p t +1 FIGURE 1. The evolution of the price. The true state is θ = 0 and the price is near one. (The case θ = θ 1, arbitrarily large and with a low belief, is symmetric.) For a description, see the text. The values of the prices p t +, pt and the probabilities π t are common knowledge and can be computed before the game begins. In general, π t +1 = α+β t +1 may depend on the history, hence the notation π + t +1 and π t +1. Movements along the branches are random and determined by the probabilities π o,π o + π t, etc. probabilities of a purchase and a sale in period, conditional on the good state θ = 1, are π 0 + π and π 0 (and vice-versa if θ = 0), the value of holding one unit of the asset in period 1, u (ν), is the sum of the price p 1 and the expected capital gain in each of the two states θ = 1 and θ = 0, multiplied by their probabilities ν and 1 ν: u (ν) = p 1 + ν (π 0 + π )(p + (p 1,π ) p 1 ) + π 0 (p (p 1,π ) p 1 ) + (1 ν) π 0 (p + (p 1,π ) p 1 ) + (π 0 + π )(p (p 1,π ) p 1 ). Because the price p 1 satisfies the martingale property, we can write the same equation for the market-maker, who replaces ν and u by p 1. That equation is subtracted from (4) and we have u (ν) p 1 = π (p 1,π )(ν p 1 ). (5) The absolute value of this expression represents the value of the optimal trade (buy one unit if u (ν) p 1 > 0, and sell one unit otherwise). After the agent acquires information, his belief is either ν = 1, in which case he buys at p + 1, or ν = 0, in which case he sells at p 1. The probability of his learning that θ = 1 is equal to µ, the public belief at the beginning of period 1. The value of acquiring the information at the beginning of period 1 is therefore V (π 1 ;µ,π +,π ) = µ(1 p + 1 )π+ (p + 1,π+ ) + (1 µ)p 1 π (p 1,π ). (6) A distinction is made here between π + and π, which is the value of π after a purchase or a sale in period 1, in order to have a general expression for the next section, but can (4)

8 448 Christophe Chamley Theoretical Economics (007) be ignored here. The key effect appears immediately. The value of information at the beginning of period 1 is an increasing function of the variability of the price in the next period following each of the two possible prices at the end of period 1. That variability is measured by the spread in period multiplied by the probability difference between the events of a buy and a sale, at the high or the low price, in that period. (See Figure 1.).3 Strategic complementarity of investment information within a period The value of information in (6) depends on the ask and the bid in that period, p + 1 and p 1, which are functions of π 1 = α + β 1 in (). The investment strategy β 1 is public information in a rational expectation equilibrium and is used by the market-maker to set the bid and the ask. When the investment is higher, the market-maker raises the ask p + 1 and lowers the bid p 1 in equation (). This is the Grossman and Stiglitz (1980) effect. It reduces the value of information. Now consider the impact on the variability of the price p as measured by the terms π + (p + 1,π ) and π (p 1,π ). Assume for the discussion that the initial confidence is strong, and that µ = p 0 is near 1, as in Figure 1. (The case of µ near 0 is symmetric.) A higher ask p + 1 in period 1 entails a higher confidence in the belief and therefore a lower impact of a transaction in the next period on p : the spread (p + 1,π+ ) is reduced. (Recall that π + is exogenous in this section.) But if a sale takes place in period 1, the lower confidence at the beginning of period generates more variability of p and a larger spread, which increases the value of information. When µ = p 0 is near 1 the two effects are not symmetric: an increase of confidence when confidence is already high reduces the value of information by a small amount. But news that reduces the confidence has an impact on the value of information that will be shown to dominate the combination of the first effect and the Grossman Stiglitz effect of the previous paragraph. Omitting the arguments π + and π, which are fixed, and using (6), (3), and some manipulations, we find V (π 1 ;µ) = κ(π π 1 ) π 0 + π p + 1 π 0 + π (1 p + 1 ) π 0 + π 1 µ π 0 + π p 1 π 0 + π (1 p 1 ) π 0 + π(1 µ), 3 with κ = (π ) (π + π 0 )µ (1 µ) (π 0 ). If µ is near 1 or near 0, we can make the approximation V (π 1 ;µ) π 0 3 A 1 π 1 π 0, + π 1 where A > 0 is a function of the parameters of the model that is independent of π 1. The right-hand side is positive if π 0 ( 1/3 1) < π 1. Recall that we must have π 0 + π 1 1. If π 0 < 1, the interval (π0 ( 1/3 1), 1 π 0 ] is well defined. Since the previous expression is an approximation when µ is near 0 or 1, we have the following result.

9 Theoretical Economics (007) Complementarities in information acquisition 449 PROPOSITION 1 (Strategic complementarity within a period). If the noise parameter π 0 is smaller than 1, there exist µ and π such that if the public belief µ at the beginning of the first period is smaller than µ or greater than 1 µ, and the probability of an informed agent in the first period, π 1, is greater than π, then the value of information is an increasing function of π 1. The upper-bound condition on noise trading is not very restrictive and is intuitive: if noise trading is large, the trades of the informed agents have little impact on the price. As the variability of the price becomes smaller, the incentive for a short-term trader to get information also becomes smaller. Under the conditions of Proposition 1, the payoff of anyone s investment in information increases in the investment made by others: there is strategic complementarity in getting information within the first period..4 Strategic complementarity between periods A higher information investment in period raises π and the variability of investment π (p 1,π ) for any p 1. This intuitive property is verified by simple algebra. The higher variability of the price in period increases the value of information at the beginning of the previous period 1 (in expression (6)). One has the following result. PROPOSITION (Strategic complementarity from a period to the previous one). The value of information investment in period 1, V (p 0,π 1,π +,π ), defined in (6), is increasing in the level of next period investment, β, and therefore increasing in π + and π. The increasing value of information may generate multiple equilibria, a trivial property at this stage in the two-period model. After the exposition of the main mechanisms, we now consider the setting with an infinite number of periods with an endogenous information investment in each of them. Proposition shows that the equilibrium levels of investment in all periods are linked. This non-trivial problem is analyzed in the next section. 3. INFINITE HORIZON The model is extended to an infinite number of periods, with a small probability of revelation of the fundamental in each period. This assumption is introduced to obtain a stationary solution. As before, θ is set randomly before the first period and is constant through time. In each period, θ is revealed with probability δ, conditional on no previous revelation. The value of δ is small, in a sense that is made more precise later. If θ is not revealed at the beginning of a period t (with probability 1 δ), trading takes place as in the simple model: a new agent is of one of the three types described in the previous section, acquires information at a fixed cost c if he can and finds it profitable, and meets a risk-neutral profit-maximizing market-maker to trade one unit of the asset or not to trade. The market-maker does not observe the type of the agent but has rational expectations and can compute the strategy of an information agent, which is based on public information.

11 Theoretical Economics (007) Complementarities in information acquisition 451 The expression Ṽ represents the expected payoff of information investment from short-term trading and is similar to the expression (6) in the simple model, while L represents the payoff from long-term trading (for an agent who waits for the revelation of the fundamental). The value of V depends on π + t +1 and π t +1 only through the function Ṽ and one can show, as in Proposition, that Ṽ is increasing in these two variables. It follows immediately that information investment in period t + 1 increases the payoff of information investment at the beginning of period t. COROLLARY 1 (Strategic complementarity between periods). The expected value of information investment in any period t, V (p t 1,π t,π + t +1,π t +1 ), is an increasing function of π + t +1 = α + β + t +1 and π t +1 = α + β t Symmetry The model exhibits a symmetry between the high and the low values of the price p t 1 with respect to the middle price 1. This symmetry is expressed by the following property of the value function: V (p t 1,π t,π + t +1,π t +1 ) = V (1 p t 1,π t,π t +1,π+ t +1 ). (9) The analysis focuses on values of the price above The impact of the belief from history on the value of information In the present setting, the uncertainty about the fundamental is measured by its variance p t 1 (1 p t 1 ). When p t 1 is greater than 1, this uncertainty is a decreasing function of p t 1. There is a positive relation between uncertainty and the value of infor- mation. This intuitive property is formalized in the next result, which is proved in the Appendix. Throughout the paper, increasing (decreasing) means strictly increasing (decreasing). LEMMA 1. For any probabilities (π t,π + t +1,π t +1 ) [α,α + β]3, the value of information V (p t 1,π t,π + t +1,π t +1 ) defined in (7) is decreasing in p t 1 if p t 1 > ˆp (increasing if p t 1 < 1 ˆp ), where ˆp is defined by min β [0,β] p t ( ˆp,α + β) = 1. Because of the interactions between periods, an equilibrium strategy is defined as a sequence of investment probabilities. We identify such a strategy β t /β by the probability π t = α + β t of an informed agent in period t. Any such sequence beginning in period t must depend on the history up to period t. It is reasonable to assume that this dependence is only on the public belief at the beginning of period t, which is equal to p t 1. (Other previous prices could matter as coordination devices for an equilibrium, but such an assumption would be artificial.) We consider therefore the following class of equilibria. DEFINITION 1 (Equilibrium). An equilibrium strategy is defined by a sequence of measurable functions {B t (p)} t 1 from (0, 1) to [0,β] such that for any p (0, 1),

12 45 Christophe Chamley Theoretical Economics (007) if B t (p) = 0, then V (p, B t (p),π + t +1,π t +1 ) c if 0 < B t (p) < 1, then V (p, B t (p),π + t +1,π t +1 ) = c if B t (p) = β, then V (p, B t (p),π + t +1,π t +1 ) c, with π + t +1 = α + B t +1 p + (p, B t (p)), π t +1 = α + B t +1 p (p, B t (p)). An equilibrium strategy B t in period t depends on the strategy B t +1 in the next period. In general, the structure of equilibria is complex. We focus on stationary strategies where the function B t does not depend on t ; this class is sufficiently rich, and, under imperfect information (Section 5), the unique equilibrium is a stationary strategy. 4. STATIONARY EQUILIBRIUM STRATEGIES We have seen in Lemma 1 that the information payoff is decreasing in p if p is high, and increasing if p is low. It is therefore natural to consider strategies in which an information agent invests if and only if the price is in some interval (p, p ). DEFINITION (Trigger strategy). A (stationary) trigger strategy is defined by an investment interval 7 (p, p ) such that B(p) = 1 if p (p, p ), and B(p) = 0 if p / (p, p ). Since there is a symmetry between the high and the low value of p (equation (9)), we focus on the determination of the upper-end of the investment interval, p. The value of information in period t depends on the level of information investment in period t + 1, which depends on the price p t. Assume that p t 1 is near or at p. If the information agent learns that θ = 1, he buys at the ask p t + which is above p and, by definition of the stationary strategy, there is no information investment in the next period. If he sells at the bid, the price p t is in the investment interval and β t +1 = β. We are thus led to introduce zero-one expectations such that π + t +1 = α and π t +1 = α + β. Under zero-one expectations, in the period that follows a transaction at the ask (bid), no information agent (any information agent) invests. Using (7), omitting the time subscript with p t 1 p, and recalling π = α+β, the payoff of investment under zero-one expectations is equal to W (p,β) = V (p,π,α,α + β) = (1 δ)w (p,β) + δl(p,π), (10) with W (p,β) = Ṽ (p,α + β,α,α + β), defined in (8). The expression W (p,β) defines the payoff of information with pure short-term trade and zero-one expectations as a function of the last transaction price p and the information investment β in the current period. We first analyze the properties of this function. We later show that the component from long-term trade, δl(p,π), can be neglected if δ is sufficiently small. The next result shows that under some assumptions, the payoff from pure short-term trade generates strategic complementarities. 7 The interval (p, p ) is open, but one could include boundaries without altering the equilibrium since the price is at one of the boundaries with zero probability.

13 Theoretical Economics (007) Complementarities in information acquisition 453 W higher β W (p,β ) information cost, c W (p,0) investment strategy p L p p H p FIGURE. Continuum of constant equilibria. The function W (p, β) is the ex ante value of information for a last transaction price equal to p, a level of investment this period equal to β [0,β] and investment equal to β (0) if the price this period goes up (down). PROPOSITION 3. For given α and β with α < β/3, there exists p such that for any p (p, 1), the value of information with pure short-term trading and zero-one expectations, W (p,β), defined in (10), is decreasing in p and increasing in β. The first part of the result is proved as Lemma 1. The second part, which is proved in the Appendix, holds only if the exogenous level of information α is not too large relative to the range of values of the endogenous information, β. Such an assumption is not surprising: a higher value of α generates a higher ask and a lower bid by the rational market-maker, which reduces the payoff of information investment. The properties of the function W (p,β) in Proposition 3 are illustrated in Figure, where we can replace W by W. From Proposition 3 and since for given β, W (p,β) is decreasing to 0 when p tends to 1, if c is not too high, the equation W (p, 0) = c has a unique solution p L such that p L > p, where p is defined in Proposition 3. In this case, there is another solution p H such that W ( p H,β) = c as represented in Figure (where we can substitute p L for p L and p H for p H ). If the probability of revelation δ is sufficiently small, then the function W with pure short-term trading approximates arbitrarily closely the payoff of information W and Figure applies to the graph of the function W (p,β). This is the meaning of the next result. PROPOSITION 4. Assuming α < β/3, there exists c such that if c < c, then there is δ such that if δ < δ, for any β [0,β], the equation W (φ(β),β) = c has a unique solution

14 454 Christophe Chamley Theoretical Economics (007) φ(β) [ 1, 1]. Furthermore, W (p,β) > c if p [ 1,φ(β)) < c if p (φ(β), 1), W (p,β) > 0 if p [p L, p H ] with W (p L, 0) = W (p H,β) = c. β Choose a value p (p L, p H ) as represented in the figure, and a value p (1 p H, 1 p L ) (in the low range that is not represented). The next result shows that the trigger strategy (p, p ) defines an equilibrium. PROPOSITION 5. Under the assumptions of Proposition 4, there is a continuum of stationary equilibrium strategies: any pair {p, p } such that p [p L, p H ] and p [1 p H, 1 p L ], where {p L, p H } is defined in Proposition 4, defines a trigger strategy that is an equilibrium strategy. The sufficient conditions for the existence of a continuum of equilibrium strategies are simple: the cost of information should be smaller than some value, traders should sufficiently care about the short-term profits, and the occurrence of exogenously informed agents should not be too high compared to that of the traders for whom information is endogenous. 4.1 Remarks The case of high information cost When c is sufficiently large (but not too large), the solution p L of W ( p L, 0) = c may be smaller than p and Propositions 3 and 4 may not apply. In this case, there may be strategic substitutability and a unique equilibrium strategy. If p is greater than some value p, there is no information acquisition. If the price decreases from p, information investment increases gradually, possibly up to its maximum β : the strategy for an information agent is to randomize with an increasing probability to acquire information as the price decreases. (If the price becomes lower than 1, investment in information decreases.) The detailed analysis of this case is not the main focus in this paper and is left aside Heterogeneous costs of information We have assumed for simplicity that all agents have the same cost of information, but the equilibrium properties are robust when there is some cost heterogeneity. Suppose that an information agent can acquire information at the fixed cost c, which is an increasing function c(β) for β [0,β], with c(0) > 0. The distribution of costs is represented by a density function over β on the interval [0,β]. An information agent is now characterized by a random draw from this distribution. If there is strategic complementarity, Proposition 4 holds with a minor alteration: p L and p H are defined by W (p L, 0) = c(0) and W (p H,β) = c(β). If there is strategic substitution, the equilibrium strategy becomes deterministic with a threshold value c. The observed investment in information is still random because of the random determination of the agent s cost.

16 456 Christophe Chamley Theoretical Economics (007) Market-makers have perfect information as befits their role. (A noisy observation on their part would probably not change the results.) The other parameters of the model are the same as in Section 4 without observation noise, and are assumed to satisfy the assumptions in Proposition 4. If an information agent, after observing his signal s t, does not invest in information, he stays out of the market and does not trade because of the bid ask spread, as in the case with no observation noise. If he invests at the cost c, he learns the exact value of the fundamental θ, and trades whatever the equilibrium prices. A strategy is now a measurable function of the signal s t and the level of the investment β t of any other information agent who may be called to the market in the same period. The strategy β t is rationally anticipated by the market-maker. Since he knows the price p t 1, he can compute the distribution of private signals and using the probability of facing an informed agent, he sets the bid and ask to maximize his expected profit from trade, as in the previous sections. We do not restrict the strategy to be a trigger strategy, but the payoff with a trigger strategy is a useful tool. In a trigger strategy, an information agent invests if his signal is in some interval (ŝ, ŝ ). We focus on the behavior of agents near the value ŝ, which is shown to be in the interval (p L, p H ). Without loss of generality, we may assume that ŝ = 1 ŝ, and the trigger strategy is defined by ŝ. For σ sufficiently small, the level of investment is equal to ˆp + σ s β(p, ŝ ) = β min max σ, 0, 1. (11) In the model with perfect information, we used the payoff function W (p,β) with the zero-one expectations that in the period after a price rise (at the ask), there is no information investment, whereas after a transaction at the bid, investment is at the maximum β. A similar function plays an important role here. An information agent with signal s has a subjective probability of state θ = 1 equal to µ(s ) and a density function φ(p s ) on the last transaction price p. Assuming that he has zero-one expectations about the next period investment, and that the market-maker anticipates the strategy ŝ, by extension of (6), the payoff of information is equal to the function J σ (s, ŝ ) = (1 δ) J σ (s, ŝ ) + δk σ (s, ŝ ), (1) with J σ (s, ŝ ) = µ(s ) K σ (s, ŝ ) = s +σ s σ s +σ s σ (1 p + )π + (p +,π + )φ(p s ) d p + (1 µ(s )) s +σ s σ p π (p,π )φ(p s ) d p µ(s )(1 p + ) + (1 µ(s ))p φ(p s ) d p, assumption is that the prior of an information agent has a support that includes an open interval that includes the interval [1 p H, p H ].

17 Theoretical Economics (007) Complementarities in information acquisition 457 where p + = p + (p,α + β(p, ŝ )), p = p (p,α + β(p, ŝ )), π + = α, π = α + β, and β(p, ŝ ) is given by (11). The functions J σ (s, ŝ ) and K σ (s, ŝ ) and are continuous and have continuous partial derivatives. Using the uniform distributions of p t 1 and the signal s, the previous expressions can be rewritten J σ (s, ŝ ) = s 1 σ s +σ s σ (1 p + )α (p +,α) d p K σ (s, ŝ ) = s 1 s +σ (1 p + ) d p + (1 s ) 1 σ σ s σ + (1 s ) 1 s +σ p (α + β) (p,α + β)d p σ s σ s +σ s σ 5.1 Vanishingly small observation noise p d p. We have to consider the case ŝ = s in an equilibrium. After some elementary manipulations, 10 we find lim σ 0 J σ(s, s ) = 1 β β 0 (13) W (s,β)d β, (14) where W (s,β) is defined in (10). Using the differentiability of the Bayesian functions p + and p on [0, 1], d J σ (s, s ) lim = 1 β W (s,β) d β. σ 0 d s β s 0 These equations show that for σ arbitrarily small, the function J σ (s, s ) is approximated by an average of the functions W (s,β). The next result follows from the properties of W (p,β) in Propositions 4 and 5. LEMMA. Under the assumptions of Proposition 4, there exists ˆσ such that if σ < ˆσ, the equation J σ (s, s ) = c has a unique solution s on the interval [p L, 1]. Furthermore, s (p L, p H ), J σ (s, s ) < c for s > s, and if σ 0 then s S which is defined by 1 β β 10 Using p = s + σ(β/β 1) from (11), we have 0 W (S,β)d β = c. with J σ (s, s ) = s β β 0 (1 p + )α (p +,α) d β + 1 s β β 0 p (α + β) (p,α + β)d β, p + = p s + + σ ββ 1,β, p = p s + σ ββ 1,β. Recall that with perfect information, the short-term payoff of information (with δ 0) is given in (10) and (8): W (p,β) = p(1 p + (p,π))α (p +,α) + (1 p)p (p,π)(α + β) (p,α + β), with π = α + β. Equation (14) follows with the expression of J σ in (1) and K σ in (13).

18 458 Christophe Chamley Theoretical Economics (007) J (s,s ) W (p,β ) W (p,0) c s 4 s 3 s s 1 s J (s,s ) J (s,s ) s FIGURE 3. The value of information J (s, s ) at the threshold s of a trigger strategy, and iterated dominance. J (s, s ) is the value of information for an agent with signal s when other agents invest only if their signals are greater than s. For a discussion, see the proof of Proposition 6 in the Appendix. The function J σ (s, s ) replaces the function W (p,β) that was used with perfect information. It is represented in Figure 3, which illustrates the following iterative dominance argument. In the first step, there is a value of the private signal s 1 such that if s > s 1, the agent is sufficiently confident that the state is good and the value of information is below the cost c even if all other agents invest. Investment is dominated, which implies that J (s 1, s 1 ) < c. When agents with a signal higher than s 1 do not invest in period T, T arbitrary, the value of investment in period T 1 is bounded above by the value under zero-one expectations, which is J (s, s 1 ). By continuity of J, J (s, s 1 ) < c on an interval (s, s 1 ]. Hence, investment is dominated in period T 1 for s < s if it is dominated in period T for s > s 1. The argument is used iteratively for the following result, which is proved in the Appendix. PROPOSITION 6. Assume c and δ such that Proposition 4 holds, and α > 0. There exists σ such that if σ < σ, (i) investment is iteratively dominated for any s > s where s (p L, p H ) is defined by J σ (s, s ) = c. If σ 0, then s S such that 1 β β 0 W (S,β)d β = c (ii) there exists a value ĉ c (defined in Proposition 3), such that if c < ĉ and σ < σ, then for s (1 s, s ), not to invest in information is iteratively dominated. In this

19 Theoretical Economics (007) Complementarities in information acquisition 459 case the strategy to invest if and only s (1 s, s ) is a SREE. (It is the only strategy to survive the iterated elimination of dominated strategies.) The proposition introduces two minor assumptions: the restriction α > 0 ensures that the variation of the price after a transaction has a strictly positive lower bound. This lower bound ensures that zero-one expectations apply to an agent near a threshold value s : if he buys (sells), he is sure that for σ sufficiently small, all information agents in the next period will have a signal strictly higher (lower) than s and by definition of s will not invest (will invest) in information. The restriction on the cost c in Part (ii) is used to ensure that information investment is dominant if the price is near 1. This restriction can be removed if we assume that agents use a trigger strategy in some period. 6. CONCLUSION We began by showing that the combination of short-term trades and endogenous information generates strategic complementarities, and that these complementarities may be sufficiently strong to generate a continuum of equilibria when agents have common knowledge of the last transaction price. Each equilibrium defines regions of prices with sharply different levels of investment. When the price crosses a threshold, the level of investment is discontinuous because of the strategic complementarity. Under small imperfect information, the continuum is reduced to a singleton, but there is no contradiction between the results without observation noise. The important property is the discontinuity of the information investment. The multiplicity of equilibria is not robust to a perturbation with observation noise, but the discontinuity in behavior is robust. Take an initial price near near 1 and a fundamental equal to 0. Eventually, the price must decrease (since it converges to the truth). When agents observe perfectly the history, the price enters a region with multiple equilibrium strategies with zero or maximum investment. When the observation of history is subject to a noise, however small, Proposition 6 states that the level of investment grows from zero to its maximum when the price crosses the interval [S ˆσ,S + ˆσ], which is arbitrarily small. There is a linear relation in the model between the probability of a trade and the level of endogenous information. Hence, in the equilibrium there is a positive relation between the volume of trade and the information that is generated by the market. Information frenzy is equivalent to trade frenzy. The mechanism presented here is robust for other models of micro-structure, as verified numerically in the Appendix for standard model with a continuum of agents who place simultaneous limit orders in the same period. A next step is to consider small random changes of the fundamental. 11 One may anticipate that the present results will be extended and that in an equilibrium, there will be random switches between two regimes with sharply different levels of trade and information. 11 David (1997) analyzes the learning process in a financial market when the state switches randomly between discrete values and agents have exogenous private information.

20 460 Christophe Chamley Theoretical Economics (007) APPENDIX A.1 Proofs PROOF OF LEMMA 1. The value function V is defined in (7), which is repeated here: V = (1 δ)ṽ + δ p t 1 (1 p + t ) + (1 p t 1)p t. We omit the time subscripts when there is no ambiguity. In the second term of this expression, p t + and pt are given by the Bayesian equations (), and simple algebra shows that this term is a decreasing function of p t 1 if p t 1 > 1. Focusing now on the first term, from (8), Ṽ (p t 1,π) = p t 1 (1 p + t )π+ t +1 (p + t,π+ t +1 ) + (1 p t 1)p t π t +1 (p t,π t +1 ). The prices p t + and pt are increasing in p t 1. Since pt > 1 under the assumption in the lemma, the spreads in period t + 1, (p t +,π+ t +1 ) and (p t,π t +1 ), are decreasing in p t 1. A small exercise shows that p t 1 (1 p t + ) is decreasing in p t 1 if p t 1 > 1 and that (1 p t 1 )pt is decreasing in p t 1 if pt > 1 p t 1, which is satisfied by the assumption of the lemma (pt > 1 > 1 p t 1). The last part of the lemma follows from the symmetry in (9). PROOF OF PROPOSITION 3. The first part of the proposition was proved in Lemma. Recall the definition of W (p,β) in (10): W (p,β) = (1 δ)w (p,π) + δl(p,π) with π = α + β. We first prove that the short-term component of the information value exhibits strategic complementarity ( W (p,π)/ π > 0), and then show that the long-term component can be ignored if δ is sufficiently small. The short-term component is given in (8) for any period t and is equal to p t + Ṽ t = p (1 p t + ) (π + t +1 )(π+ t +1 + π0 ) t 1 D 1 (p t +,π+ t +1 )D (p t +,π+ t +1 ) + (1 p t 1 ) (p t ) (1 pt )(π t +1 )(π t +1 + π0 ) D 1 (pt,π t +1 )D (pt,π t +1 ). Using () and setting π = α + β, π + t +1 = α, and π t +1 = α + β because of the definition of W (p,β), replace p t 1 by p, and omit the time subscript because there is no ambiguity: W (p,π) = p (1 p) (π 0 ) (p,π), (15) with (p,π) = (π 0 + π)α(α + π 0 ) D1 3 (p,π)d 1(p +,α)d (p +,α) + (π 0 + π)(α + β)(α + β + π 0 ) D 3 (p,π)d 1(p,α + β)d (p,α + β), (16) where D 1 (p,π) = π 0 + πp and D = π 0 + π(1 p) are the denominators in ().

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