Liquidity and Asset Prices in Rational Expectations Equilibrium with Ambiguous Information

Liquidity and Asset Prices in Rational Expectations Equilibrium with Ambiguous Information Han Ozsoylev SBS, University of Oxford Jan Werner University of Minnesota September 006, revised March 007 Abstract: The quality of information in financial asset markets is often hard to estimate. This paper analyzes information transmission in asset markets when agents treat information of unknown quality as ambiguous. We study the effects of information ambiguity on asset prices, trading volume, and market liquidity in noisy rational expectations equilibrium. We consider a market with risk-averse informed investors, risk-neutral competitive arbitrageurs, and noisy supply of the risky asset, first studied in Vives 1995a,b) with unambiguous information. Ambiguity increases sensitivity of asset prices to information signals and to changes in asset supply. Further, it may make markets less liquid. Preliminary Draft. 1

1. Introduction Information in financial markets is plentiful. There are earnings reports, announcements of macroeconomic indices, political news, expert opinions, and many others. Investors use various pieces of information to update their expectations about asset returns. In standard models of asset markets, agents update their prior probabilistic beliefs about asset returns in Bayesian fashion upon observing an information signal drawn from a precisely known distribution. The quality of some information signals in the markets may be difficult to judge. Investors may not have a single probability belief about the information signal. The situation of insufficient knowledge of probability distribution is, of course, reminiscent of the famous Ellsberg Paradox where agents have to choose between bets based on draws from an urn with a specified mix of balls of different colors and an urn with an unspecified mix. Many agents choose bets with known odds over the bets with the same stakes but unknown odds. A decision criterion which - unlike the standard expected utility - is compatible with this pattern of preferences is the maxmin or multiple-prior) expected utility. Under the maxmin expected utility, an agent has a set of probability beliefs priors) instead of a single one, and evaluates an action, such as taking a bet, according to the minimum expected utility over the set of priors. Such behavior is often referred to as ambiguity aversion, for it indicates the dislike of uncertainty with unknown or ambiguous odds. Axiomatic foundations of maxmin expected utility are due to Gilboa and Schmeidler 1987). The effects of ambiguous uncertainty and ambiguity aversion in financial markets have been studied in in finance and economic over the past two decades. Dow and Werlang 199) were the first to point out that portfolio choice of an agent facing ambiguous uncertainty may display an inertia in that the agent may choose not to trade the asset for a range of prices. Mukerji and Tallon 003) showed that ambiguity may impede risk sharing in financial markets. Other models of asset markets with ambiguity are Cao, Wang and Zhang 005) who focus on asset pricing, and Easley and O Hara 005, 006) who focus on market liquidity. The recent paper by Epstein and Schneider 006) studies asset prices in dynamic markets with ambiguous information signals, but without information transmission. This paper analyzes information transmission in asset markets when the quality of some information signals is unknown and agents treat information of unknown quality as ambiguous. The questions we ask are how does ambiguity of information affect the process of information

transmission in markets, and how does ambiguous information affect asset prices and trading in equilibrium. Information transmission in financial markets has been much studied when information is of precisely known quality. Models of competitive markets with asymmetric information that is partially revealed by asset prices have been developed by Grossman and Stiglitz 1980), Hellwig 1980), Diamond and Verecchia 1980), and Admati 1985). Models of strategic trading under asymmetric information take their origin in the work of Kyle 1985) and Glosten and Milgrom 1985). We consider a market with risk-averse informed investors, risk-neutral uninformed arbitrageurs and random supply of a single risky asset, first studied in Vives 1995a,b) with unambiguous information. Prior to the arrival of information, all investors have ambiguous beliefs about probability distribution of the asset payoff. Ambiguous beliefs are described by a set of probability distributions of the payoff. Informed agents receive a private information signal about the payoff. The signal reveals the payoff only partially so that the payoff remains uncertain, but it removes the ambiguity of informed investors beliefs. They know precisely the conditional distribution of the payoff. Uninformed arbitrageurs do not observe the signal and their beliefs about payoff distribution remain ambiguous. They extract information from prices. Our main focus is on the process of information transmission through prices in the presence of ambiguity. Our description of ambiguous information is different from the one seen in Epstein and Schneider 006). In Epstein and Schneider 006) investors have unambiguous beliefs prior to the arrival of information. Their posterior beliefs after receiving information signals become ambiguous. Investors in our model have ambiguous prior beliefs. The ambiguity disappears for investors who observe the information signal but persists for uninformed investors. The model has CARA-normal specification. Informed investors utility of wealth has CARA form. All random variables are normally distributed. The set of multiple prior beliefs about the distribution of the asset payoff consists of normal distributions with means and variances from some bounded intervals. Informed investors and arbitrageurs behave as competitive price-takers. Arbitrageurs, who have linear utility of wealth, take arbitrary positive or negative positions in the asset if the price equals the minimum expected payoff over their set of beliefs or, respectively, the maximum expected payoff. If the price is between the maximum and the minimum expected values, they do not trade. Arbitrageurs are uniformed and extract information from market prices. We derive a rational expectations equilibrium in closed-form. The equilibrium price function is 3

piecewise linear. It is partially revealing the signal as long as the noise in supply is non-degenerate i.e., non-deterministic.) If there is no noise, the equilibrium is fully revealing. We study the effects of ambiguous information on asset prices, trading volume, and market liquidity. We find that ambiguous information increases sensitivity of prices to information signals and to supply shocks when compared to a benchmark equilibrium in markets with no ambiguity. Since price sensitivities can be used as proxies for volatility, we conclude that ambiguous information increases volatility of prices. We obtain interesting results pertaining to liquidity of the market. As long as there is ambiguity about the mean of the information signal, there is range of values of the information signal and the asset supply such that the arbitrageurs do not trade. We identify this range as the regime of market illiquidity. The regime of illiquidity exists if an only if there is ambiguity about the mean. If the ambiguity is only about the variance but not about the mean, then the market is always liquid although, as we demonstrate, the equilibrium trade of the arbitrageur gets smaller as the ambiguity gets larger. If there is ambiguity about the mean, sensitivity of equilibrium prices is higher when the market is illiquid than when it is liquid.. Asset Markets with Ambiguous Information There are two assets: a single risky asset, and a risk-free bond. The payoff of the risky asset is described by random variable ṽ. The bond has deterministic payoff equal to one. Asset are traded in a market with risk-averse informed agents, risk-neutral uninformed arbitrageurs, and noise traders. The price of the risky asset is denoted by p. The price of the bond is normalized to 1. There is a single informed agent, called the speculator, whose utility of end-of period wealth w has the CARA form e ρw, where ρ > 0 is the Arrow-Pratt measure of absolute risk aversion. The arbitrageur has linear utility of end-of-period wealth equal to w. These two agents should be thought of as representative agents for large numbers of two types of traders. Initial wealth is w s for the speculator and w a for the arbitrageur. The payoff of the risky asset is the sum of two random variables, ṽ = θ + ω. 1) The distribution of θ is ambiguous while that of ω is unambiguously known. The informed 4

speculator observes realization θ of random variable θ. Observation θ resolves partially, but not fully, the riskiness of the payoff of the risky asset. It removes though the ambiguity about the payoff distribution. The arbitrageur does not observe θ. She is uninformed about θ and has ambiguous beliefs about the distribution of θ. The ambiguity of beliefs about θ is described by a set P of probability distributions. Each probability distribution in P is assumed normal and independent of ω. More specific, we take P to be the set of normal distributions with mean lying in an interval [µ, µ] and variance in an interval [σ, σ ]. The set P could, for instance, be described as a relative entropy neighborhood of a normal distribution see Cao, Wang, and Zhang 005)). We use µ = µ µ and σ = σ σ to measure the magnitudes of ambiguity about the mean and the variance. Random variable ω is known to have normal distribution with mean m and variance τ. We assume that the supply of the risky asset is random. Random asset supply serves as an additional source of uncertainty, other than information signal, and prevents asset prices from fully revealing agents information. Randomness in L can be thought as resulting from trade by noise traders. We assume that L is normally distributed, independent of θ and ω, with mean zero and variance σ L. Assuming zero mean of asset supply L is inessential see a remark at the end of Section 3.) The speculator s random wealth resulting from purchasing x shares of the risky security at price p is w = w s +ṽ p)x. Upon observing realization θ of θ her information is I s = { θ = θ}. Her portfolio choice is described by max E[ e ρws+ṽ p)x) I s ]. ) x Since the conditional distribution of ṽ on I s is normal, maximization problem ) simplifies to max ρ w s + x E [ṽ p) I s ]) 1 ) x ρ x var[ṽ I s ]. 3) The solution to 3) is the speculator s risky asset demand, and it takes the form x s I s, p) = E [ṽ I s] p ρ var[ṽ I s ]. 4) The ambiguity about the distribution of θ is reflected in the arbitrageur s choice of portfolio. Arbitrageur s preferences are represented by maxmin expected utility with linear utility function and the set of priors P. These preferences are motivated by the famous Ellsberg Paradox which 5

most clearly exemplifies the impact of ambiguous information on agent s decision. Maxmin expected utility prescribes that the agent considers the worst-case distribution when making a decision. It has been extensively studied in decision theory, and an axiomatization has been given by Gilboa and Schmeidler 1989). The arbitrageur s maxmin expected utility of random wealth resulting from purchasing x shares of risky security is min E P[w a + ṽ p)x I a ], 5) P P where E P denotes expectation under belief P P, and I a is the arbitrageur s information. Her portfolio choice, given information I a, is thus max min E P [w a + ṽ p)x I a ], 6) x P P The set of solutions x a I a, p) to this maximization problem is 0 if min P P E P [ṽ I a ] < p < max P P E P [ṽ I a ] x a I a, p) = [0, + ) if p = min P P E P [ṽ I a ] 7), 0] if p = max P P E P [ṽ I a ] The arbitrageur s portfolio choice problem has no solution for other values of price p. Arbitrageur s demand shows an inertia that is typical to maxmin expected utilities, as first pointed out by Dow and Werlang 199). For a range of prices, the agent chooses not to trade the asset at all. 3. Rational Expectations Equilibrium The definition of rational expectations equilibrium in our model is standard see Grossman and Stiglitz 1980), Hellwig 1980), Diamond and Verecchia 1980), and Admati 1985), among many others.) Rational expectations equilibrium consists of an equilibrium price function Pθ, L) and equilibrium demand functions X s θ, L) and X a θ, L) such that, for p = Pθ, L), it holds X s θ, L) = x s Is, p), X a θ, L) x a Ia, p) 8) X a θ, L) = L X s θ, L), 9) 6

for almost every realizations θ and L of θ and L. Condition 8) expresses optimality of agents portfolio demands. Condition 9) is the market clearing condition. Information set I s of the speculator and Ia of the arbitrageur reflect rational expectations, that is, they result from observations of private signals and equilibrium prices. We describe these information sets next. The speculator observes realization θ of θ. Equilibrium price could reveal extra information about realization of liquidity supply L, but such information would be irrelevant for the speculator. This is so because the probability distribution of risky part ω of the asset payoff is independent of the supply L. Consequently, the information set I s is equal to { θ = θ}. We write I s = {θ} for short. The arbitrageur does not observe θ and extracts information about θ from equilibrium price. Her information set I a is {P θ, L) = p}. Let us consider information revealed to the arbitrageur by the order flow against her, instead of the equilibrium price. If the observed order flow is f and asset price is p, that information is described by the set { L x s Is, p) = f}. Conditions 8) and 9) imply that in equilibrium, that is, if p = Pθ, L) and f = L X s θ, L), information revealed by order flow is the same as information revealed by price. We will use this observation when deriving an equilibrium and verify it again at the end of this section for the derived equilibrium. The equivalence between information revealed in equilibrium by order flow and information revealed by price has been pointed out by Vives 1995b) see also Romer 1993)). In the absence of ambiguity, this equivalence and the fact that the arbitrageur has linear utility permit a different interpretation of the model. Instead of competitive market with price-taking speculator and arbitrageur, one could imagine the arbitrageur acting as a market maker who sets asset price using zero-expected-profit rule and executes orders submitted by the speculator and noise traders. The arbitrageur sets price equal to the expected payoff conditional on information revealed by total order flow. This zero-expected-profit condition could be justified by Bertrand competition among many risk-neutral market makers. Speculator s orders are price-dependent limit orders determined by competitive demand. Such market structure resembles Kyle s 1985) auction, an important difference being that speculator s order is competitive instead of monopolistic. Such alternative interpretation of the model is also possible in the presence of ambiguity as will be explained at the end of this section. We proceed now to derive a rational expectations equilibrium. The demand function of the 7

speculator at information I s = {θ} can be obtained from 4), x s I s,p) = m + θ p ρτ. 10) The information revealed by the order flow L x s I s, p) when its observed value is f is { L m + θ p ρτ = f} This information set can be written as {ρτ L θ = a}, where a = m p + ρτ f is a parameter known to the arbitrageur. Thus the content of information revealed by the order flow against the arbitrageur is the same as the content of observing random variable ρτ L θ. This random variable has ambiguous distribution out of a set of multiple normal distributions. Taking our above discussion into account, we can write I a = {ρτ L θ}. 11) Conditional expectation of asset payoff ṽ on Ia under a probability distribution P from the set of multiple priors P can be obtained using the Projection Theorem. We have E P [ṽ Ia ] = E P[ṽ ρτ L θ] ) cov P θ, ρτ L θ = m + E P [ θ] + var P ρτ L θ ) = m + µ P [ ρτ L θ E P ρτ L θ ]) σ P σ P + σ L ρ τ 4 ρτ L + µ P θ ). 1) where µ P and σ P denote the mean and the variance of distribution P of θ. Equation 1) allows us to find the maximum and the minimum expected asset payoffs over the set of priors P needed for the arbitrageur s asset demand 7). Market clearing condition 9) is then used to determine equilibrium price function and equilibrium asset demands. These calculations have been relegated to the Appendix. Our main result is Theorem 1 There exists a unique rational expectations equilibrium with price function given by σ m + µ + σ +ρ τ 4 σ θ ρτ L µ ), if θ ρτ L µ L Pθ, L) = m + θ ρτ L, if µ < θ ρτ L < µ 13) σ m + µ + θ ρτ L µ), if θ ρτ L µ, σ +ρ τ 4 σl 8

speculator s demand given by X s θ, L) = ρτ σ L σ +ρ τ 4 σ L θ µ) + σ σ +ρ τ 4 σ L L, if θ ρτ L µ L, if µ < θ ρτ L < µ ρτ σ L σ +ρ τ 4 σ L θ µ) + σ σ +ρ τ 4 σ L L, if θ ρτ L µ, and arbitrageur s demand given by ρτ σ L σ +ρ τ 4 σ θ ρτ L µ ), if θ ρτ L µ L X a θ, L) = 0, if µ < θ ρτ L < µ ρτ σl θ ρτ L µ). if θ ρτ L µ. σ +ρ τ 4 σ L Equilibrium price function 13) depends on θ and L only through the value of ρτ L θ, and is a strictly increasing function thereof. This implies that information revealed by equilibrium price function 13) is the same as observing a realization of ρτ L θ, which in turn is the same as observing order flow L x s Is, p) against the arbitrageur. This verifies again that 11) holds and shows that the price and the demand functions of Theorem 1 are indeed a rational expectations equilibrium. The equilibrium of Theorem 1 is partially revealing. Information signal θ is only partially revealed to the uninformed arbitrageur. If asset supply is deterministic, that is, if σ L 14) 15) = 0, then the equilibrium becomes fully revealing, for then the equilibrium price function reveals θ. Price function 13) is piecewise linear. It is linear only if µ = 0 or σ L ambiguity about the mean of signal θ, or if the asset supply is deterministic. = 0, that is, if there is no 4. Asset Prices Equilibrium price function P can take one of three values as indicated be 13). If θ ρτ L µ, then the equilibrium price equals the minimum over the set of priors of expected payoff conditional on information Ia. In this case equilibrium order flow against the arbitrageur is positive, see 15). Similarly, the equilibrium price equals maximum expected payoff conditional on I a when the order flow is negative. The price lies between the maximum and the minimum values when the order flow is zero. Therefore equilibrium prices satisfy multiple-prior version of the zero-expected-profit condition. A useful benchmark for further analysis of rational expectations equilibrium with ambiguity is the case of no ambiguity, that is, when agents have unique prior probability distribution of θ, 9

with mean µ and variance σ. Rational expectations equilibrium in the absence of ambiguity is derived by setting µ = µ = µ and σ = σ = σ in Theorem 1. We obtain Corollary If µ = 0 and σ = 0 so that there is no ambiguity, then the unique rational expectations equilibrium has linear price function given by P na θ, L) = m + µ + and demand functions given by σ σ + ρ σ L τ4 θ ρτ L µ ), 16) Xs na ρτ θ, L) = σ L σ θ µ) + L σ + ρ τ 4 σl σ + ρ τ 4 σl X na a θ, L) = ρτ σl θ ρτ L µ ). σ + ρ τ 4 σl The equilibrium of Corollary is a modification of the one obtained by Vives 1995, Proposition 1). This equilibrium is partially revealing as long as the asset supply is non-deterministic. Otherwise, the equilibrium is fully revealing. Conditions for partial and full revelation are the same without ambiguity as they are with ambiguity. Equilibrium price 16) equals the expected payoff conditional on arbitrageur s information I a. Expression 16) can be decomposed in two parts: the first part, m + µ, is the unconditional expected payoff and the remaining part is the information premium for arbitrageur s information. The premium is proportional to the order flow against the arbitrageur. It is positive when the order flow is positive, and negative a discount) when the order flow is negative. The magnitude of the information premium increases with an increase of variance of signal θ, that is, with a decrease in the signal s precision. Equilibrium price in the presence of ambiguity 13) has similar structure. If the order flow against the arbitrageur is positive, equilibrium price is the sum of minimum expected payoff and information premium. The information premium is positive and proportional to the order flow. If the order flow is negative, the price is the sum of maximum expected payoff and negative information premium i.e., discount). The discount is proportional to the order flow. Information premium per unit of order flow is greater when there is ambiguity than when there is no ambiguity. Comparing Corollary with Theorem 1 reveals that, if there is ambiguity about the variance but not about the mean of the information signal, so that µ = 0, then the equilibrium takes 10

exactly the same form as the equilibrium under no ambiguity, but with variance of the signal set at the upper bound σ. Loosely speaking, the market behaves as if the variance were unambiguously known as the upper bound. An important characteristic of equilibrium asset prices is their sensitivity to changes in information and asset supply. These are measured by Pθ,L) and Pθ,L), where we took the absolute θ L value of the latter to assure a positive number. Since information signal θ directly affects the asset payoff, price sensitivity to information signal can be considered as fundamental. This is in contrast to the asset supply L, which is uncorrelated with the asset payoff. Price sensitivities are often used in empirical literature as measures of volatility of prices. and For equilibrium price function 13), the price sensitivities are P θ, L) = θ Pθ, L) L = 1, if µ < θ ρτ L < µ, σ, otherwise. σ +ρ τ 4 σl ρτ, if µ < θ ρτ L < µ ρτ σ, otherwise. σ +ρ τ 4 σl For price function 16) of the benchmark equilibrium, price sensitivities are It follows from 17-0) that Pθ, L) θ for every θ and L. P na θ θ, L) = σ σ + ρ τ 4 σl P na θ, L) ρτ σ L = σ + ρ τ 4 σl P na θ, L), and θ 17) 18) 19) 0) Pθ, L) L P na θ, L) L 1) Proposition 3 The sensitivity of equilibrium asset prices to information and to supply shocks is greater in the presence of ambiguous information than under no ambiguity. Proposition 3 says that ambiguity about information increases volatility of prices in rational expectations equilibrium. Except for the region of values of θ and L such that µ < θ ρτ L < µ, the magnitude of the increase gets higher as the ambiguity about the variance increases. 11

5. Market Liquidity In the equilibrium of Theorem 1, there is a region of values of θ and L such that the arbitrageur does not trade, i.e. has zero asset demand. This region of values is non-void only if µ > 0. If µ = 0, so that there is no ambiguity about the mean of θ, then the arbitrageur participates in trading for almost every value of θ and L. Then his equilibrium demand shows no inertia despite ambiguity about the variance i.e. σ > 0). Further, as the ambiguity about the variance increases i.e. σ, or equivalently σ ), the equilibrium asset demand of the arbitrageur goes to zero. In other words, when ambiguity about variance is large, the arbitrageur participation in trade is small. If µ > 0 so that there is ambiguity about the mean of θ, then the region values of θ and L where the arbitrageur does not trade is non-void. When the arbitrageur does not trade, the speculator trades so as to match the supply of noise traders. On the other hand, when the arbitrageur participates in trading, then asset price is determined by the arbitrageur s expectation of the asset payoff and the arbitrageur matches the order flow of the speculator and noise traders. Thus the arbitrageur can be thought of as providing liquidity to the market. The set of θ and L where she does not participate in trading will be called the regime of market illiquidity. The set is characterized by µ < θ ρτ L < µ. The likelihood of market illiquidity can be measured by the probability of the event that θ ρτ L lies betwen µ and µ under a true distribution of signal θ which can be taken as any distribution from the set of normal priors P. We show in the Appendix that the likelihood of market illiquidity increases when the magnitude of ambiguity µ gets larger, but decreases when either one of variances τ or σl, or speculator s risk aversion ρ get larger. These observations are reminiscent of the results of Cao, Wang and Zhang 005) and Easley and O Hara 005, 006) who find that market illiquidity increases with the level of ambiguity. An inspection of equations 17) and 18) reveals that price sensitivities in a rational expectations equilibrium take two different values in the regime of illiquidity and outside of it. Proposition 4 Assume that µ > 0. The sensitivity of equilibrium asset prices to information and to supply shocks is higher when the market is illiquid than when it is liquid. As already mentioned, price sensitivities are used in empirical literature as measures of volatility of prices. Proposition 4 is consistent with this literature which finds negative corre- 1

lation between liquidity and volatility see, e.g., Tinic 197), Stoll 1978a, 000), Pastor and Stambaugh 003)). Negative relation between liquidity and volatility has also been established in the market microstructure theory see Amihud and Mendelson 1980), Copeland and Galai 1983), Admati and Pfleiderer 1988), Foster and Viswanathan 1990)). However, liquidity in the market microstructure theory is measured by the bid-ask spreads. The market is liquid if the bid-ask spread is low. Equilibrium price sensitivities 17) and 18) change in a discontinuous way between two values as the market switches between the regimes of illiquidity and liquidity. The magnitudes of the jumps in price sensitivities to information and supply shocks are ρ τ 4 σl, and σ + ρ τ 4 σl ρ 3 τ 6 σl, σ + ρ τ 4 σl respectively. These magnitudes increase as the upper bound σ of ambiguity about the variance becomes smaller, or the speculator becomes more risk averse i.e. ρ increases), or the variance σ L of asset supply increases. 5. Conclusion We presented a model of competitive asset markets where uninformed traders consider the information signals received by informed traders as ambiguous. The effects of ambiguity on information transmission in the markets and on equilibrium asset prices and trading were analyzed. A noisy rational expectations equilibrium was characterized in closed-form in a CARA-gaussian setting with risk averse informed speculators and uniformed arbitrageurs who have maxmin multiple-prior expected utility. The equilibrium price function is in general non-linear. We found that the presence of ambiguous information increases sensitivity of equilibrium prices to information signal and asset supply. If there is ambiguity about the mean of the signal, then arbitrageurs choose not to participate in trading in equilibrium for a non-zero measure set of information and supply realizations. Thus market illiquidity arises in equilibrium. 13

Appendix Proof of Theorem 1. Using 1) we find min P P E P [ṽ I a] and max P P E P [ṽ I a]. Depending on θ and L, they are If θ ρτ L < µ, then If µ θ ρτ L µ, then If θ ρτ L > µ, then min P [ṽ Ia ] = m + µ + σ θ ρτ L µ ), P P σ + ρ τ 4 σl ) max P [ṽ Ia ] = m + µ + σ θ ρτ L µ ). P P σ + ρ τ 4 σl 3) min P P E P [ṽ I a] = m + µ + max P P E P [ṽ I a] = m + µ + min E P [ṽ Ia ] = m + µ + σ P P σ + ρ τ 4 σl max P P E P [ṽ I a] = m + µ + σ θ ρτ L µ ), σ + ρ τ 4 σl 4) σ θ ρτ L µ ). σ + ρ τ 4 σl 5) σ σ + ρ τ 4 σ L θ ρτ L µ ), 6) θ ρτ L µ ). 7) Using )-7) we find the demand x s Is, p) when the price p is either min P P E P [ṽ Ia ] or max P P E P [ṽ Ia ]. If p = m + µ + σ σ +ρ τ 4 σl θ ρτ L µ ), then x s I s, p) = ρτ σl σ θ µ) + L, σ + ρ τ 4 σl σ + ρ τ 4 σl σ If p = m + µ + σ +ρ τ 4 σ θ ρτ L µ ), then L x s I s, p) = ρτ σl θ µ) + σ + ρ τ 4 σl σ L, σ + ρ τ 4 σl 14

σ If p = m + µ + θ ρτ L µ), then σ +ρ τ 4 σl x s I s, p) = ρτ σl θ µ) + σ + ρ τ 4 σl σ L, σ + ρ τ 4 σl If p = m + µ + σ σ +ρ τ 4 σl θ ρτ L µ), then x s I s, p) = ρτ σl σ θ µ) + L, σ + ρ τ 4 σl σ + ρ τ 4 σl σ From these we find that if p = m+µ+ σ +ρ τ 4 σ θ ρτ L µ ) σ or p = m+ µ+ θ ρτ L µ), L σ +ρ τ 4 σl total order flow L x s Is, p) lies in the arbitrageur demand set x aia, p) at p. Market clearing condition 9) also holds with demand x a Ia, p) at zero for some prices between these maximal and minimal values. These give the rational expectations equilibrium of 13), 14) and 15). Likelihood of market illiquidity Section 5): true distribution of θ with mean µ and variance σ is Pr µ < θ ) ρτ L < µ = 1 πvar[ θ ρτ L] = 1 erf = 1 erf The probability of market illiquidity under the µ µ µ E[ θ ρτ L] var[ θ ρτ L] exp erf µ + µ µ σ + ρ τ 4 σ L ) ) erf u E[ θ ρτ L] ) var[ θ ρτ L] du µ E[ θ ρτ L] var[ θ ρτ L] )) µ µ σ + ρ τ 4 σl ) where erf is the Gauss error function erfx) = π x 0 exp u )du. Using d erfx) = dx π exp x ), one can show that Prµ < θ ρτ L < µ) is increasing in µ and decreasing in τ, σl and ρ. 15

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