Data Abundance and Asset Price Informativeness. On-Line Appendix

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1 Data Abundance and Asset Price Informativeness On-Line Appendix Jérôme Dugast Thierry Foucault August 30, 07 This note is the on-line appendix for Data Abundance and Asset Price Informativeness. It contains the proofs for findings mentioned in the paper but not proven there for the sake of brevity. It is not intended for publication. The on-line appendix is organized as follows. Section derives the equilibrium demand for the raw signal (given in Lemma in the paper. Section analyzes the case in which the upper bound on the mass of speculators, i.e., ᾱ, is less than. Section 3 analyzes the case in which speculators can make their decision to buy the processed signal contingent on the realization of the price at date. Section completes Proposition 3 (case. Section 5 completes Proposition by considering the case in which C p > C min (θ, α e, derives sufficient and necessary conditions for E (C r, C p / C r > 0, and shows that long run price informativeness jumps down when the market for the raw signal takes off, provided the processed signal is produced when the raw signal is not (i.e., E ( θ 8, C p < E (, C p if C p C max (θ, 0. Luxemburg school of finance. Tel: ( ; jerome.dugast@uni.lu HEC, School of management, Paris, GREGHEC, and CEPR. Tel: ( ; foucault@hec.fr

2 Section 6 shows that C max (θ, α decreases with α as claimed in the proof of Proposition 3. Section 7 analyzes the case in which speculators who only buy one signal can trade at both dates if they wish. Section 8 shows that it is optimal for speculators to trade when they receive their signal if they cannot trade more than one share overall. Equilibrium demand for the raw signal (Proof of Lemma. Let π gross,a (α = α π (α = α max { θ ( α, 0 } be the aggregate gross expected profit for speculators who receive the raw signal. We represent by α e the equilibrium value of α, the demand for the raw signal at date. Proceeding as in the market for the processed signal, we deduce that if α e > 0 then α e solves: { } θ π gross,a (α e = α e max ( αe, 0 = C r. (. As, π gross,a (α reaches its maximum for α = /, we deduce that the previous equation has no solution if C r > θ 8. In this case, for all values of α, π gross,a (α < C r and therefore there is no fee at which trades between the seller of the raw signal and buyers of this signal are mutually beneficial. Thus, in this case, α e = 0. For C r < θ, eq.(. has two solutions in (0,. As explained in the text, we select the 8 highest as the equilibrium demand since it yields the lowest fee charged by the information seller. This solution is: α e = + C r θ, (. and the corresponding equilibrium fee is therefore F e = C r /α. e Last, if C r = θ, the only 8 solution to eq.(. is α e = 0.

3 The case in which ᾱ. In equilibrium speculators demands for each type of signal (i.e., α e and α e must satisfy: 0 α e ᾱ, and 0 α e ᾱ. When ᾱ >, the right hand sides of these constraints are never binding (as α e and α e. This is the case on which we focused in the baseline version of the model. In contrast, when ᾱ, these constraints can be binding, in which case one obtains a corner solution for equilibrium demands. We discuss each case in turn below. Note that a corner solution obtains for the demand for the raw signal iff C r > θ 8 (in which case α e = 0 or C r < Ĉr (in which case α e =, where Ĉr = θ ( (min{(ᾱ 0.5, 0}.. Case. The zero profit conditions eq.(9 in the text and eq.(. in this appendix have solutions α e and α e such that 0 < α e < ᾱ and 0 < α e < ᾱ. In this case, the equilibria of the markets for the raw signal and the processed signal are interior. This case is identical to that analyzed in the paper and all results are identical to those obtained when ᾱ >.. Case. The equilibrium demand for (i the raw signal is a corner solution, that is α e = 0 (if C r > θ or 8 αe = ᾱ (if C r < Ĉr and (ii the equilibrium demand for the processed signal is interior (0 < α e < ᾱ. In this case, the cost of producing the raw signal C r has (locally no impact on α e and thus has no local effect on α. e Therefore, if C r < Ĉr or C r > θ 8 then a small change in the cost of the raw signal has no effect on asset price informativeness or trade and price patterns. For other values of C r, results are identical to that in the text. 3. Case 3. The equilibrium demand for the processed signal is a corner solution, that is α e = 0 or α e = ᾱ. Thus, α e does not depend on α e since its value is either zero or ᾱ. Therefore, a change in cost of the raw signal C r has (locally no impact on α. e Thus, it does not affect price informativeness in the long run. 3

4 3 Speculators can buy the processed signal at date. In this section, we consider the case in which speculators can buy the processed signal after observing the price of the asset at date, i.e., just before trading at date. The equilibrium of the market for the raw signal is unchanged in this case and is still given by the second part of Lemma. The main difference with the case considered in the text is that (i the demand for the processed signal at date, α, depends on p, the price of the asset at the end of the first period and (ii the seller the processed signal can charge a different fee for this signal depending on the price realized at date. We consider two subcases. In the first case (Section 3., we assume that the seller of the processed signal pays the fixed cost of information processing after observing the realization of the price at date. In the second case (Section 3., we assume that this cost is paid before observing the realization of the price at date. The break-even conditions for the information seller differ in each case. In the second case, the information seller must set its fees for the processed signal so that it covers its fixed cost of producing the signal on average across the possible realizations for the price at date. In the first case, the information seller sets its fee so that it breaks even conditional on each realization of the price at date. The two cases yield an identical equilibrium outcome when there is a demand for the processed signal for each realization of the price at date. Equilibrium outcomes (e.g., the fees for the processed signal differ when the demand for the processed signal is nil for some realization of the price at date. However, we show that the implications obtained in the baseline model (about price informativeness and price and trade patterns are preserved in each case. 3. Investment in information processing takes place after the realization of the price at date At the end of the first period, there are three possible outcomes (see Panel A of Figure 3 in the text : (i the asset price has not changed, p = p 0 ; (ii the asset price has

5 increased p = +θ; (iii the asset price has decreased p = θ. Due to the symmetry of the model, the expected profit from trading on the processed signal is the same in the last two cases. Thus, the decision to buy the processed signal is identical whether the price has increased or decreased during the first period. This means that only two states are relevant for the analysis of the market for information at date : either (i the price has not changed in the first period (p = p 0 or (ii the price has changed (p p 0. We denote by (α nc, F nc the equilibrium of the market for information in the first case and by (α c, F c the equilibrium of this market in the second case ( superscript c stands for change while superscript nc stands for no change. Case : p p 0. We first derive the equilibrium of the market for the processed signal when p p 0, i.e., (α c, F c. As in the baseline model, (α c, F c must satisfy the following zero profit conditions when α c > 0: Zero profit for speculators: π net,c (α c, F c, θ = π c (α c, θ F c = 0. (3. Zero profit for the information seller: Πseller,c (α c, F c = α c F c C p = 0, (3. where superscript c for expected profits indicates that these profits are computed conditional on a price change in the first period. If there is no solution (α, c F c to this system of equations for which α c > 0 then the market for the processed signal is inactive when p p 0 and we set α c = 0. As p p 0, the gross expected profit (per speculator from trading on the processed signal is (see Step 3 in the proof of Proposition in the text for a derivation: π c (α c = θ( θ( α c. Thus, in aggregate, speculators net expected profit is: π net,c,a (α c, C p = α c (π c (α c F c = α c θ( θ( α c C p, where the second equality follows from eq.(3.. In equilibrium, if α c > 0, speculators net 5

6 expected profit must be equal to zero (see eq.(3.. Thus, we deduce from the previous equation that α c solves: π net,a,c (α c, C p = α c θ( θ( α c C p = 0. (3.3 As in the baseline model, if eq.(3.3 has multiple positive solutions then we retain the highest because it yields the smallest (hence most competitive fee for the information seller. If eq.(3.3 has no positive solution then the market for information for the processed signal is inactive (i.e., α c = 0 when p /. Solving for eq.(3.3, we obtain: α(θ, c C p = + ( Cp θ( θ 0 if C p > θ( θ if 0 C p θ( θ,, (3. It follows from eq.(3. that the equilibrium fee for the processed signal is F c = Cp α c α c > 0. when Case : p = p 0 = /. In this case, there is no change in the price at date. The net expected profit (per speculator from trading on the processed signal is then (see Step 3 in the proof of Proposition in the text for a derivation: π nc (α = θ ( θ α ( θ if α, θ θ ( α θ if α >, (3.5 where superscript nc for the expected profit indicates that it is computed conditional on no price change in the first period. We can then solve for the equilibrium demand for the processed signal as in the case in which p /. After some algebra, we obtain: α nc (θ, C p = + ( ( θ C θ( θ p θ + ( ( θ ( θ C θ p if C p θ θ, θ if θ θ < C θ p θ( θ 8 (3.6 0 if C p > θ( θ, 8 6

7 Thus, when p = /, it follows from eq.(3. that the equilibrium fee for the processed signal is F nc = Cp α nc when α nc > 0.. Using the expressions for α c and α nc (eq.(3. and eq.(3.6, it is easily shown that α c < α nc when C p > 0. Thus, the demand for processed information is smaller when prices have changed at date than when they have not. In equilibrium, the expected demand for the processed signal in equilibrium is: E(α e = P r(p p 0 α c + ( P r(p p 0 α nc = α α c + ( α α nc, where α e denotes the realization of the demand for the processed signal at date (i.e., α c or α nc. An increase in the demand for the raw signal increases the probability that the price changes at date since P r(p p 0 = α. As the demand for the processed signal is smaller when the asset price changes at date than when it does not (α c < α nc, we obtain the following result. Proposition 3.. The expected demand for the processed signal in equilibrium decreases with the demand for the raw signal, that is, E(αe α < 0. Thus, a decrease in the cost of the raw signal reduces the expected demand for the processed signal in equilibrium. This result is the analog of Proposition 3 in the baseline model. It is stronger in the sense that it holds for all parameter values. Figure 3. illustrates the previous proposition for specific parameter values. Asset price informativeness at date t=. We now study the effect of a reduction in the cost of producing the raw signal on the informativeness of the price at date. As in the baseline model, we define asset price informativeness at date as: E (C r, C p = E[(Ṽ p ], (3.7 where p denotes the realization of the equilibrium price at date. We obtain the following result. Proposition 3.. If C p θ( θ 8 then 7

8 .5.0 E[α e ] C r (θ=0.75, C p =0.06 Figure 3.: Expected demand for the processed signal (E(α e. When θ /, a decrease in the cost of producing the raw signal reduces asset price informativeness at date.. When θ > /, a decrease in the cost of producing the raw signal reduces asset price informativeness at date iff C p C p where C p is a threshold that belongs to [ θ( θ, ] θ( θ ( θ 8 Thus, when C p < θ( θ 8, the informativeness of the price at date can decline when C r declines as emphasized by Proposition in the paper. Figure 3. illustrates this claim for specific parameter values E * (C r,c p E * (C r,c p C r (θ=0.75, C p =0.06 Figure 3.: The informativeness of the price at date 8

9 When C p > θ( θ 8, the cost of producing the processed signal is so high that there is no fee at which it can be profitably produced (i.e., a which buyers and sellers of the processed signal can mutually trade, for all possible realizations of the price at date. Thus, the processed signal is not produced and the informativeness of the price at date is equal to that at date. The latter increases when α increases and therefore, in this case, a reduction in the cost of the raw signal raises price informativeness at date as well. Price and trade patterns. If C r θ/8, some speculators buy the raw signal (see Lemma in the text of the paper. Using Proposition in the paper, we can write the equilibrium strategy of a speculator receiving the raw signal, s, as x (s, C r = I s= I s=0 = u [I V = I V =0 ] + ( u [I ɛ= I ɛ=0 ]. (3.8 If C r > θ, no speculator buys the raw signal in equilibrium and therefore 8 x (s, C r = 0. In sum: I s= I s=0 = u [I V = I V =0 ] + ( u [I ɛ= I ɛ=0 ] if 0 C r θ/8, x (s, C r = 0 if C r > θ/8. (3.9 Similarly, using Proposition in the paper, we can write the optimal trading strategy of speculators at date as: u [I V = I V =0 ] + ( u [I p =( θ/ I p =(+θ/] x (p, u, C p = u [I V = I V =0 ] I p =/ if 0 C p θ( θ, if θ( θ < C p θ( θ 8, 0 if C p > θ( θ, 8 (3.0 where the last equality follows from the fact that there is no demand for the processed signal if C p > θ( θ 8. Now, assume C r θ 8 and C p θ( θ 8. These assumptions guarantee that some speculators trade at date (x 0 and that some speculators trade at date at least if 9

10 p = /. Using eq.(3.9 and eq.(3.0, we obtain: θ( α, e if C p > θ( θ, Cov(x, x = θ ( θα, e if C p θ( θ. Thus, as implied by Corollary in the baseline model, we obtain that Cov(x, x declines when α e increases, i.e., when the cost of the raw signal decreases. Moreover, when C p θ( θ, one obtains exactly the same expression for the covariance as that in Corollary in the baseline model. Thus, as in this case, Cov(x, x becomes negative if θ < / and C r is small enough. Using the expression for the equilibrium price at date (Proposition in the paper and eq.(3.0, we also obtain Cov(r, x = E [( p ] x = 0, if C p > θ( θ, θ(θ α, e if C p θ( θ (as in the paper, where r = p p 0 is the return from date 0 to date. Thus, when C p θ( θ (so that some speculators buy the processed signal whether the price at date has changed or not, we obtain the same expression for Cov(r, x as that in Corollary 5 in the paper. Otherwise Cov(r, x is zero. Hence, Corollary 5 holds true even if speculators can make their decision to buy the processed signal contingent on the price realized at date. Last consider Cov(x, r where r = (p p is the return from date to date. Calculations yield: Cov(x, r = θ( α e αnc, when C ( θ p > θ( θ, ( θ θ( α e (+( θ(αnc ( θ, when C p θ( θ ( θ. Observe that this is the same expression as that obtained in the baseline model (see eq.(7, with α nc replacing α e. The intuition is the following. In equilibrium, the innovation in r = (p p is orthogonal to dealers information set set at t = since 0

11 equilibrium prices follow a martingale. If the order flow at date reveals the raw signal then x is known to dealers at date and therefore r = p p is orthogonal to x. This means that if p / then r is orthogonal to x. Thus, the covariance between x and r is only driven by the cases in which p = /, in which case, the demand for the processed signal is α nc. This explains why only α nc affects Cov(x, r. As α nc does not depend on α e, we obtain that Cov(x, r decreases when α e increases, i.e., when C r decreases, for the same reason as in the baseline model. Proofs for Section 3. Proof of Proposition 3.. When C p > θ( θ 8, the demand for the processed signal is nil whether the price changes at date or not. Thus, E[α e ] = 0 in this case. When θ( θ 8 C p > θ( θ, the demand for the processed signal is nil if the price changes at date and strictly positive otherwise. Thus, in this case, E[α e ] = ( α α nc, which is clearly decreasing with α. Finally, when C p θ( θ, we have, using the expressions for the equilibrium demand for the processed signal: E[α] e = ( α + ( ( θ θ( θ C p + α + ( C p, θ( θ which decreases with α since ( Cp /. θ( θ Proof of Proposition 3.. First, observe that E (C r, C p is inversely and linearly related to E[(Ṽ pe ]. It is easily shown that E[(Ṽ p ] = E[p ( p ]. To obtain the proposition, we can therefore analyze how E[p ( p ] varies with C r. We first compute E[p ( p ]. If α nc, we obtain (e.g., using Figure 3, in the

12 paper: [ E[p ( p ] = α ( θα c ] + ( ( θ( + θ αc [ + ( α ( α nc ( α nc + ( θαnc + θ ] ( θ = α [ θ + ( θθ( α] c + [ ( ( α ( θ θ ( θ = α [ θ + ( θθ( α] c + ( ( α θ θ αnc = α ( θ + α ( θθ( α c + ( ( α θ θ αnc = ( θ + α π c (α c + ( α π nc (α nc. α nc ] + ( α ( θ Similarly, if α c, we obtain: [ E[p ( p ] = α ( θα c ] + ( ( θ( + θ αc [ + ( α ( θ(α nc ( α nc + ( θ( αnc + = α ( θ + α π c (α c + ( α ( θ + ( α = α ( θ + α π c (α c + ( α ( θ + ( α = α ( θ + α π c (α c [ ( θ(α nc + αnc [ θ θ θ ] ( α nc + ( α ( θ + ( α θ( θ ( αnc ( θ = ( θ + α π c (α c + ( α π nc (α nc θ ] ( θ ( θ θ ] ( θ Thus, in all cases: E[p ( p ] α = πc (α c π nc (α nc. (3. Thus, a reduction in C r (i.e., an increase in α increases the informativeness of the price at date (i.e., reduces E[p ( p ] if and only if π c (α c < π nc (α nc. Now we analyze

13 when this is the case and when it is not. For this, note that: π c (α c = C p /α c if α c > 0 and π c (0 = θ( θ if α c = 0. Similarly, π nc (α nc = C p /α nc if α nc > 0 and π c (0 = θ/ if α nc = 0 (see eq.(3.5. If C p > θ( θ 8, α nc = α c = 0. In this case, using eq.(3., we deduce from the previous remarks that: which is decreasing with α. E[p ( p ] α = θ < 0, If θ( θ < C p θ( θ 8 we have α c = 0 and α nc > 0. In this case, In this case, using eq.(3., we obtain that: E[p ( p ] α = θ( θ C p /α nc. (3. Thus, the pricing error increases with α if and only if α nc (θ, C p C p θ( θ. (3.3 Using eq.(3.6, we deduce that α nc (θ, C p increases when C p decreases. And using eq.(3., we deduce that the derivative of the pricing error with respect to α increases when C p decreases. We also observe that when C p θ( θ ( θ, αnc and therefore eq.(3.3 is satisfied. Moreover, when C p = θ( θ 8, α nc = θ. Thus, in this case, eq.(3.3 is satisfied if and only if θ < /. Combining these two observations, we deduce that when θ( θ < C p θ( θ, there is a C 8 p [ θ( θ, ] θ( θ ( θ 8 such that asset price informativeness decreases when C r decreases if (i θ < / or if (ii θ > / and C p < C p. If C p θ( θ, using eq.(3., we obtain that: E[p ( p ] α = C p ( α c > 0, (3. α nc where the inequality follows from the fact that α c α nc when C p θ( θ. 3

14 3. Investment in information processing takes place before the realization of the price at date Now we consider the case in which the seller and the buyers of the processed signal contract before observing the realization of the price at date but the buyers can make their decision to eventually buy the signal contingent on the realization of the price at date. This corresponds to the case in which (i information processing must start early at date for the processed signal to be delivered in time at date and (ii the seller offers a schedule (F c,f nc, i.e., a fee for the processed signal that is contingent on the realization of the price. A speculator can then decide to buy the processed signal only when (i the price has changed at date, (ii only when the price has not changed or (iii in both cases. Denote R nc = α nc F nc and R c = α c F c. An equilibrium of the market for the processed signal is a vector (α nc, α c, F nc, F nc such that: Zero profit for speculators if p p 0 : π net,c (α c, F c, θ = π c (α c, θ F c = 0. (3.5 Zero profit for speculators if p = p 0 : π net,c (α c, F c, θ = π c (α c, θ F c = 0. (3.6 Zero profit for the information seller: Π seller (α nc, α c, F c, F nc = P r(p p 0 R c +( P r(p p 0 R nc C p = α R c +( α R nc C p = 0. (3.7 The only difference with the case analyzed in Section 3. is the zero profit condition for the seller of the processed signal. This condition accounts for the fact that the information seller must pay the fixed cost of information processing before knowing whether p p 0 (which happens with probability P r(p p 0 = α or p = p 0 (which happens with probability ( α. Note that R nc = α nc F nc and R c = α c F c are the revenues of the seller of the processed signal conditional on the two possible outcomes at the end of period : (i no change (nc in the price of the asset or (i a change (c in the price of the asset. These revenues are

15 bounded by the largest possible value for the aggregate expected profits from trading on the processed signal conditional on each possible outcome, that is: R nc θ( θ, and R c 8 θ( θ Thus, the largest possible expected profit for the seller of the processed signal is Π(α ( α θ( θ 8 +α θ( θ We deduce that a necessary condition for the seller of the processed signal to pay the fixed cost of producing this signal is that: C p Π(α. (3.8 Henceforth we assume that this condition is satisfied. Otherwise the processed signal is not produced and α c = α nc = 0. Observe that there might be multiple pairs (R c, R nc for the zero expected profit condition for the seller of the processed signal (in eq.(3., which leads to the possibility of multiple equilibria in the market for the processed signal. When this happens, we we select the equilibrium that minimize the variance of revenues for the information seller, i.e., (R nc R c since the seller s revenue is binomial random variable. Indeed, this would be the strictly preferred outcome for the information seller if it is risk averse. We can rewrite the zero profit condition (3.7: R nc R c = C p R c α (3.9 We then consider two cases. The first case arises when C p θ( θ. In this case, the variance of revenues is minimized for R c = C p and is equal to zero, so that R nc = C p as well. This outcome is possible because C p < θ( θ, the largest possible value for R c. The zero profit condition (3. can thus be replaced by two conditions: α c F c = C p, 5

16 and α nc F nc = C p. The analysis is then identical to the case analyzed in Section 3.. In particular, as in Section 3., we obtain that the equilibrium demand for the processed signal when p p 0 and when p = p 0 are respectively: α c (θ, C p = + ( C p θ( θ (3.0 and α nc (θ, C p = + ( ( θ θ( θ C p (3. The second case arises when θ( θ 8 C p > θ( θ. In this case, the constraint that R c θ( θ/ is binding. Inspection of eq.(3.9 shows that the equilibrium such that the variance of revenues for the information seller is minimal is such that R c is set at its maximal possible value, i.e., θ( θ, which is therefore the aggregate expected trading profits obtained by speculators buying the processed signal when the price changes at date. This implies that in this case the demand for the processed signal must be: α c =, (3. and therefore F c = θ( θ/. Moreover, using eq.(3.9, we deduce that R nc must be such that R nc = α nc F nc = θ( θ + C p θ( θ. α Let denote the R.H.S of this equation by R(α, C p (i.e., R(α, C p = θ( θ + Cp θ( θ α. We can then derive the equilibrium demand for the processed signal when p = p 0 as we did in Section 3., except that the role of C p is now played by R(α, C p. We deduce that the demand the processed signal conditional on no change in the price of the asset 6

17 at date is: α nc (α, θ, C p = + ( ( θ R(α θ( θ, C p θ + ( ( θ ( θ θ R(α, C p if θ( θ < R(α, C p θ θ, θ if θ θ < R(α θ, C p θ( θ 8 (3.3 In sum, the only difference with the case studied in Section 3, is the case where θ( θ 8 C p > θ( θ. There are two differences relative to what we obtain in 3. First, even if the price changes at date, there is a demand for the processed signal when C p > θ( θ while this demand was nil in the case considered in Section 3. The reason is that conditional on the realization of this state (a change in price, the information seller sells the processed signal at a price that is too low to recover its fixed cost of producing the signal. As this cost has been sunk, selling the signal is optimal anyway but, in this state, the fee charged for the signal is less than the cost of producing it. This implies that when p = p 0, the seller of the processed signal sells it at a price that exceeds his fixed cost of producing the processed signal. This price cannot be profitably undercut because the decision to produce the processed signal (and pay the associated fixed cost must be made before observing the realization of the price of the asset at date. Second, α nc is negatively related to α. Yet, as in Section 3., we still have α nc > α c. Moreover, the expected demand for information is equal to E(α e = α +( α α nc (α. As in Section 3., the expected demand for the processed signal decreases with the demand for the raw signal since E(α e = α + ( α α nc (α and α nc (α decreases with α. Asset price informativeness at date t=. As in Section 3, we analyze the effect of a reduction of C r (i.e., an increase in α on the informativeness of prices at t =. Proposition 3.3. If C p > Π(α, a decrease in the cost of producing the raw signal improves asset price informativeness at date. If C p Π(α then there is a C p (α [ θ( θ, Π(α ] such that a decrease in the cost of producing the raw signal reduces asset 7

18 price informativeness at date if and only if C p < C p (α. Qualitatively, we find the same result as in Section 3.: when the cost of producing processed information is high, a decrease in the cost of producing the raw signal improves asset price informativeness at date, and conversely when the cost of producing processed information is low. Price and trade patterns. As speculators are either active in both cases ( c or nc or not at all, a speculator s trade at t = is as follows, 0 if C p > Π(α, x (p, u, C p = u [I V = I V =0 ] + ( u [I p =( θ/ I p =(+θ/] if 0 C p Π(α. Therefore, proceeding as in Section 3., we obtain: Cov(x, r = Cov(r, x = E 0, if C p > Π(α, Cov(x, x = θ ( θα, e if C p Π(α. 0, if C p > Π(α, θ( α e αnc (α ( θ [( p ] x = θ( α e (+( θ(αnc (α ( θ 0, if C p > Π(α, θ(θ α, e if C p Π(α., if θ( θ < C p Π(α and R(α, C p > θ( θ ( θ, θ( α e (+( θ(αnc, if C ( θ p θ( θ. (3., if θ( θ < C p Π(α and R(α, C p θ( θ ( θ, The expression of covariances Cov(x, x and Cov(r, x are as in the paper. The only difference is the threshold Π(α that replaces C max. Compared to Section 3., we lose the intermediate cases which correspond to the situation where α c = 0 and α nc > 0, when θ( θ 8 C p > θ( θ. Cov(x, r admits more cases than in the paper or in Section 3.. Notice that in the 8

19 second and third cases, α also enters the expression via α nc (α, which is not the case in the fourth case, and in Section 3. Yet, in all cases, Cov(x, r decreases with α, as in Section 3.. Proofs for Section 3. Proof of Proposition 3.3. When C p > Π(α, we have α nc = α c = 0 then informativeness increases with α as in Proposition 3., when C p > θ( θ 8. When C p > θ( θ Using equation (3., we can calculate the derivative of the pricing error with respect to α as follows E[p ( p ] α = πc (/ π nc (α nc = θ( θ α nc θ( θ + C p θ( θ α, We can first notice that this derivative is decreasing function of C p since α nc declines with C p and π nc (α nc declines with α nc. Second, the derivative can be rewritten as θ( θ ( α nc α nc C p θ( θ. α If θ θ R(α θ, C p θ( θ, we have α nc 8, then the derivative of the pricing error is negative. It follows that price informativeness increases with α. When R(α, C p < θ θ, θ αnc >. Consider the case where R(α, C p converges to θ( θ, or equivalently C p converges to θ( θ from above. The derivative converges towards θ( θ 8 ( > 0. α nc Since E[p ( p ]/ α is a decreasing function of C p, it follows that there is a threshold C p (α such that price informativeness decreases with α, i.e., E[p ( p ]/ α > 0, if and only if C p < C p (α. When C p θ( θ, we are in the same case as in Proposition 3.. Asset price informativeness decreases with α. 9

20 Complement to the proof of Proposition 3. Let ᾱ (θ = result. ( θ +θ ( θ[( θ( θ ] and C p (θ = (θ( θ ( θ ( θ(( θ Lemma.. For θ < / and C min (θ, α C p < C max (θ, α, αe α θ <, (ii α > ᾱ (θ, and (iii C p > C p (θ. Proof. When θ < /, we know from Corollary that αe α α e (α < ˆα (θ = ( θ( θ ( θ( θ θ. We first prove the following > 0 if and only if (i > 0 if and only if Using the expression of α e in Lemma in the text, we obtain that α e (α < ˆα (θ if and only if α max (θ, α ( + C p < ˆα (θ. (. C max (θ, α That is, if and only if C p C max (θ, α < ˆα (θ. (. α max (θ, α For this inequality to be verified, a necessary condition is that the right hand side is positive. That is, we must have: ˆα (θ > α max (θ, α = (θ α + ( (= ( θ α θ θ ( θ( (θ α. ( + (( θ( θ α (.3 Observe that α max (θ, α decreases with α. Moreover, α max (θ, 0 = ( θ/ and α max (θ, = /. We have: α max (θ, 0 ˆα (θ = θ θ ( θ( θ ( θ ( θ = θ ( ( θ θ ( θ( θ ( θ + = ( θ( θ + = ( θ ( > 0, ( θ θ θ 0

21 for θ /. Moreover, α max (θ, ˆα (θ = θ ( θ θ = θ ( θ ( θ θ = = ( θ + ( θ( θ. = θ ( θ + θ ( θ θ θ( θ ( θ( θ Therefore α max (θ, ˆα (θ > 0 iff θ >. Consequently if θ > ˆα (θ > 0 for all α and Condition (.3 cannot hold true. Hence, αe α true if θ >. then α max (θ, α > 0 cannot hold If θ <, there exists ᾱ (θ [0, ] such that α max (θ, ᾱ (θ = ˆα (θ and for all α > ᾱ (θ, α max (θ, α < ˆα (θ, since α max (θ, α is decreasing with α. Solving the equation α max (θ, ᾱ (θ = ˆα (θ for ᾱ (θ, we obtain: ᾱ (θ = ( [ θ θ ( θ θ ] + θ ( θ[( θ( θ ]. (. We deduce that for θ <, ᾱ (θ increases with θ. Moreover, ᾱ (0 = 3 and Thus, if θ < ᾱ ( = + ( ( [ ( + ] =. and α > ᾱ (θ, Condition (. can be satisfied. This condition is equivalent to: C p > C p (θ, (.5 where C p (θ C max(θ, α α max (θ, α (αmax (θ, α ˆα (θˆα (θ

22 We have C max (θ, α α max (θ, α = θ [ ( θ + ( θ θ α ]. Moreover: α max (θ, α ˆα (θ = = (θ α + ( ( θ θ θ ( θ θ α θ ( θ θ ( [ ( θ + ( ( θ θ θ θ α ] Thus: C p (θ = θ( θ θ ˆα (θ ( θ θ = θ( θ θ θ ( = θ( θ ( θ( θ ( θ (( θ( θ. θ This achieves the proof of Lemma..Q.E.D Finally, we observe in equilibrium, the condition α e (C r > ᾱ (θ is equivalent to: C r < θ ( max ( ᾱ (θ, 0 = C r (θ. 5 Sufficient and necessary conditions for E (C r, C p / C r > 0. In Proposition, we have shown that if C p < C min (θ, α e then a decrease in the cost of producing the raw signal, C r, reduce the informativeness of the price at date (i.e., E (C r, C p / C r > 0. We now derive sufficient and necessary conditions for this result to hold when C min (θ, α e < C p < C max (θ, α e (Case A. We also analyze the case C p > C max (θ, α e at the end of this section (Case B. Overall the conditions derived in Cases A and B complete the proof of Proposition and yields Figure 9 in the paper. In case C, we show that long run price informativeness jumps down when the market for the raw signal takes off provided the processed signal is produced when the raw signal is not, i.e., E ( θ, C 8 p < E (, C p if C p C m ax(θ, 0. This supports the claim in the two

23 first paragraphs on page 3 in Section 6.. Case A. Consider the case in which C min (θ, α e < C p < C max (θ, α e. In this case, 0 < α e (Proposition. Using eq.(3 and eq.(5, we have: E (C r, C p = θ ( π (α e, α e θ( θ α e ( α e, (5. where we omit the arguments of functions α e and α e to simplify notations. As α e > 0, in equilibrium, α e π e = C p (see eq.(8. Thus, we can rewrite eq.(5. as: E (C r, C p = θ ( Cp α e θ( θ α e ( α e, (5. that: Using the fact that C r affects α e only through its effect on α e, we deduce from eq.(5. ( E (C r, C p α e ( α e = C r C r α e ( Cp (α e θ( θ α e + θ( θ ( α e, (5.3 As αe C r 0, we deduce that the sign of E (C r,c p C r function: G(α e, α e αe α e ( Cp (α e θ( θ is opposite to the sign of the following α e + θ( θ ( α e. (5. To determine the sign of G(α, e α, e we first compute αe. Using eq.(5, we obtain: α e αe α e = [α e π (α e,αe ] α e [α e π (α e,αe ] α e = α e π (α e,αe α e α e π (α e,αe + π α e (α, e α. e Moreover, as 0 < α e, we deduce from Proposition that: π (α e, α e = θ { [ ( (θ α e θ + ( θ θ ] } α e α e. This implies that π (α e, α e α e = θ [ ( θ + ( θ ] α e, θ 3

24 π (α e, α e α e = θ [ ( θ + ( θ θ ] α e. Therefore, αe α e = α e α e [ θ + ( ( θ [ θ + ( ( θ θ θ α e ] α e ] C p θ. α e The denominator of this expression is equal to [αe π (α e,αe ], which is strictly positive α e (see the discussion that precedes Lemma in the paper or Figure 5. Hence, we deduce that G(α e, α e < 0 iff: α e [ ( θ + ( θ ( θ( α e [ α e ] ( α e C p θ θ ( ( θ + ( θ θ (α ( e θαe, α e C p θ ] > 0 α e After some algebra, one can show that this condition is equivalent to: Υ(C r, C p, θ > 0, where Υ(C r, C p, θ θ ( ( Cp θ θ αe ( α e + C p α α α e e ( θ e θ. (5.5 In sum, G(α e, α e < 0 iff Υ(C r, C p, θ > 0. Thus, when C min (θ, α e < C p < C max (θ, α e, E (C r,c p C r > 0 iff Υ(C r, C p, θ > 0. In other words, when C min (θ, α e < C p < C max (θ, α e, a decrease in the cost of the raw signal lowers the informativeness of the price at date if and only if Υ(C r, C p, θ > 0. Case B. Last consider the case in which C max (θ, α e < C p. In this case, no speculator buys the processed signal (α e = 0. Thus, E (C r, C p = E (C r, C p (see eq.(5 in the proof of Corollary 3. That is, long run price informativeness is equal to short run price informativeness because there is no information production after date. In this case, a decrease in C r raises price informativeness simply because it raises the demand for the

25 raw signal and thereby price informativeness at date. Case C. Figure 5. plots for each pair (C p, θ the sign of E ( θ 8, C p E (C r > θ 8, C p. The grey (blue area are the pairs for which this difference is negative (resp., positive. The red curve plots C max (θ, 0 = θ( θ 8. For a given θ, if C p is below this curve then the demand for the process signal is strictly positive when there is no market for the raw signal (α = 0. The graphic shows that for all pairs (C p, θ such this condition is satisfied (C p < C max (θ, 0 then E ( θ 8, C p < E (C r > θ 8, C p. This proves that long run price informativeness jumps down when the market for the raw signal takes off provided the processed signal is produced when the raw signal is not (i.e., E ( θ 8, C p < E (, C p if C p C max (θ, E (C r =θ/8-e (C r >θ/ Cp C p = θ (-θ θ Figure 5. 6 Proof that C max decreases with α when θ >. Case. For θ > /, we can write C max as C max (θ, α = θ ᾱ(θ, α ( (θ α, Note that Figure 5. covers all possible cases for the parameter values since C r is fixed here. Thus, a graphical proof is sufficient. 5

26 which is the product of two decreasing and positive functions of α. Thus, in this case, C max decreases with α. Case. For θ < /, we use the fact that for a given α, the aggregate speculators profit at t =, that is α π (α, α can be written as α π (α, α = C max(θ, α α max (θ, α (αmax (θ, α α α ( α α = C max (θ, α α max (θ, α α max (θ, α We know that for α = ˆα (θ, the expected profit π (α, ˆα (θ does not depend on α. Therefore the aggregate profit taken in ˆα (θ is also independent of α and happen to be equal to C p (θ calculated previously. Indeed, ˆα (θ π (α, ˆα (θ = C max (θ, α ( ˆα (θ ˆα (θ α max (θ, α α max (θ, α = C p (θ > 0. (6. Notice that the the function h(x = x( x reaches a maximum for x =, increases for x < and decreases for x >. In previous section, we have shown that when θ < /, Cp (θ > 0, α max (θ, α decreases with α, and α max (θ, α > ˆα (θ if θ >. Therefore, the ratio r(θ, α = ˆα (θ α max (θ,α increases with α, and is always less than. Hence r(θ, α ( r(θ, α increases with α. We deduce that C max (θ, α must decrease with α since C max (θ, α r(θ, α ( r(θ, α is strictly positive and does not depend on α (see eq.(6.. 7 Speculators can trade at both dates even if they buy only one signal. In this section we consider the case in which speculators who buy only one signal can trade at both dates if they wish. We follow the same steps as in the baseline model. In Section 7., we derive speculators equilibrium trading strategies (Proposition 7. and their expected profits (Proposition 7., holding (α, α constant. We then show 6

27 in Section 7. that an increase in the demand for the raw signal can exert a positive or negative externality on investors trading on the processed signal, as in the baseline model, and we write conditions characterizing equilibrium demands for each type of signal. Using these conditions, in Section 7.3, we show numerically that a decrease in the cost of the raw signal can lead to a decrease in long run price informativeness, as in the baseline model. In Section, 7., we show that the implications of the model for price and trade patterns are preserved in this extension. 7. Equilibrium strategies and profits It is straightforward to show that, at date t =, the equilibrium is the same as in the baseline model. In particular, speculators who only buy the raw signal behaves as described in Proposition in the paper while speculators who only buy the processed signal do not trade. Indeed, the latter expects to trade at E(p = p 0, which is just their valuation for the asset given that they have no information at date. Thus, as in the baseline model, there are two possible outcomes at date : (i either the price has changed relative to date 0 (with probability α or (ii it has not changed (with probability ( α. If the price has changed at date, it fully reflects the information in the raw signal. In this case, speculators who only purchase the raw signal cannot profitably trade at date since they have no informational advantage relative to the dealers. Thus, the equilibrium at date is identical to that in the baseline model if the price changes at date. Hence, in this case, speculators trading on the raw signal gets a zero expected profit at date while speculators who trade on the processed signal obtain the same expected profit as in the baseline model, i.e., π c (α = θ( θ max[ α, 0]. If the price does not change at date, speculators who only purchase the raw signal might find optimal to trade again at date (see below. We denote by π,nc (α, α, the expected profit of a raw signal speculator in this case. Hence, the ex-ante expected trading profit of trading on the raw signal is: π (α, α = π (α + ( α π,nc (α, α (7. 7

28 with π (α = θ max[ α, 0]. Similarly, we denote by π,nc (α, α, the expected profit of trading on the processed signal when the price has not changed at t =. The total expected profit of trading on the processed signal is therefore: π (α, α = α π c (α + ( α π,nc (α, α. (7. Lemma 7.. If π,nc (α, α > 0 then no speculator buys both signals. Proof. The expected profit of a speculator who buys the raw and the processed signals net of the fees charged by information sellers is: π (α + π (α, α F r F p = π (α F r < π (α, α F r, where the second equality follows from the fact that, in equilibrium, the fee charged for the processed signal is such that π (α, α = F p and the last inequality from eq.(7. and the condition π,nc (α, α > 0. This inequality implies that buying only the raw signal dominates buying both signals when π,nc (α, α > 0. Thus, in the rest of this section, we call speculators who only trade on the raw signal as raw information speculators and those who only trade on the processed signal as processed information speculators. The next proposition describes the equilibrium trading strategies of both types of speculators at date when the price has no changed at date. An equilibrium of the market at date when the price has not changed at date is a triplet (x r(s, x (s, u, p (f such that (i p (f is equal to the expected payoff of the asset given f, (ii each speculator receiving the raw signal maximizes his expected profit by trading x r(s shares given other speculators trading strategies and the market maker s pricing policy (p (f and (iii each speculator receiving the processed signal maximizes his expected profit by trading x p(s, u shares given other speculators trading strategies and and the market maker s pricing policy (p (f. Let denote k = min [α /α, ], α = α + α, and β = α k α. 8

29 Proposition 7.. At t =, when the price has not changed at t =,. The speculators who receive the raw signal buy one share when s = and sell one share when s = 0, i.e., x r( = and x r(0 =.. The speculators who receive the processed signal buy one share when (s, u = (,, sell one share when (s, u = (0,, sell k shares when (s, u = (, 0, and buy k shares when (s, u = (0, 0. That is, x p(, =, x p(0, =,x p(, 0 = k, and x p(, 0 = k. 3. The equilibrium price is: p (f = θφ(f α + θφ(f + β + θφ(f β θ φ(f α + θφ(f + α + θφ(f + β + θφ(f β. (7.3 In line with intuition, when the price has not changed at date, speculators who receive the raw signal trade in the same way at dates and since their informational advantage relative to dealers is identical at both dates. When u =, speculators who receive the processed signal trade as in the baseline case. In contrast, when u = 0, they act differently. Indeed, they trade even when u = 0 and p = p 0 while they do not in this situation in the baseline case. For instance, when s = and u = 0, the process information speculators expect the raw information speculators to buy the asset at date. These buys tend to push the price above processed information speculators estimate of the payoff of the asset given that u = 0 (i.e., /. Thus, when s = and u = 0, they sell the asset. Given the aggregate size of their trade, k, the order flow from all speculators (those trading on the processed signal and those trading on the raw signal is β = α kα. The trade size of processed information speculators in equilibrium (k is such that β 0 so that their sells never more than offset raw information speculators buys when s = and u = 0. Intuitively doing so cannot be optimal in equilibrium because the expected price for the asset at date would then be less than the estimate of the asset by the processed information speculators. Intuitions are symmetric when when s = 0 and u = 0 9

30 From Propositions 7., we deduce the expected profit of the speculators (on their trade at date t = when the price has not changed at date t =. Proposition 7.. Speculators expected profits on their trade at date, conditional on no change in the price at date t =, are as followed. Γ : α + α, and α α π,nc (α, α = π,nc (α, α = θ ( θ (α + α (7.. Γ : α + α, and α < α π,nc (α, α = θ [ α θ ] θ α π,nc (α, α = θ ( (θ α θ (θ α }{{} 0 (7.5 ( Γ 3 : < α + α, and α α π,nc (α, α = π,nc (α, α = θ( θ ( θ ( α α (7.7. Γ : < α + α, α < α, and α < π,nc (α, α = 5. Γ 5 : α, and α π,nc (α, α = θ( θ θ ( α (7.8 θ( θ ( θ ( + ( θα ( θα (7.9 π,nc (α, α = 0 (7.0 π,nc (α, α = θ( θ( α (7. 30

31 6. Γ 6 : α + α >, and α > π,nc (α, α = π,nc (α, α = 0 (7. Figure 7. shows the case that obtains (denoted Γ j for j {,..., 6} for each possible pair ((α, α. Proposition 7. shows that in case Γ 6, all speculators obtain zero expected profits. This case cannot arise when the costs of producing the signals are strictly positive. Thus we ignore it in the rest of the analysis. Similarly, in case Γ 5, speculators trading on the raw signal obtain a zero expected profit. Again this case cannot arise when the raw signal is costly to produce. Thus, we ignore it as well. When α α (cases Γ and Γ 3, the expected profits of speculators on their trades at date are the same whether they buy the processed or the raw signal, when the price has not changed at date (i.e., π,nc (α, α = π,nc (α, α. The reason is that in this case, β = 0. Thus, the aggregate demand from speculators trading on the raw signal is perfectly offset by that of speculators trading on the processed signal when u = 0. Thus, when u = 0, all speculators trade at price / and make zero expected profits. When u =, they trade as the speculators who receive the processed signal and therefore obtain the same expected profit. 3

32 Γ 6 α Γ 3 Γ Γ Γ 5 Γ α Figure 7.: Cases in Proposition 7. for each pair (α, α. 7. Equilibrium in the market for information Let π agg (α, α = α π (α, α denote the aggregate profits of speculators who buy the raw signal and π agg (α, α = α π (α, α denote the aggregate profits of those who buy the processed signal. (α e (C r, C p, α e (C r, C p such that, An interior equilibrium in the market for information is a pair π agg (α e (C r, C p, α e (C r, C p = C r, (7.3 π agg (α e (C r, C p, α e (C r, C p = C p, (7. and, π agg α (α e (C r, C p, α e (C r, C p < 0, π agg α (α e (C r, C p, α e (C r, C p < 0. (7.5 Condition (7.5 guarantees that we select as an equilibrium the pair (α e (C r, C p, α e (C r, C p with the largest demand for each signal when there are multiple solutions to the system 3

33 of equation eq.(7.3 and eq.(7.. Demand externalities. As in the paper, we can assess how an exogenous increase in the demand for the raw signal, α, affects the demand for the processed signal α, around an equilibrium, by computing the best response function, α (α, implicitly given by π agg (α, α = C p, around (α e (C r, C p, α e (C r, C p. This function is (locally increasing if and only if π α (α e, α e < 0. Using eq.(7. and the definition of π agg (α, α, we obtain: π = π α (α c π,nc (α, α + ( α π,nc. (7.6 α We obtain a similar the same expression for π α in the baseline model (see eq.(5, except for the last term (( α π,nc α. As explained in the baseline model, the first term (π c (α π,nc (α, α can be positive or negative, which implies that an increase in the demand for the raw signal can increase or reduce the value of trading on the processed signal. Eq.(7.6 shows that this key mechanism for our results still operates in the case in which the speculators can trade on the raw signal at date. In the baseline model, however, an increase in the demand for the raw signal affects the expected profit of speculators receiving the processed signal only through its effect on the likelihood α that the price at date reveals the raw signal s. When speculators receiving the raw signal can also trade at date, an increase in their mass also directly affects the expected profit of trading on the processed signal when there is no change in the price at date. Indeed, as shown in the previous section (Proposition 7., in this case, speculators with the raw signal trade at date and this affects the expected profit of speculators who receive the processed signal. This effect is captured by the last term in eq.(7.6 ( π,nc α. The sign of this effect can be positive or negative. The reason is identical to the reason for which π c (α π,nc (α, α can be positive or negative. Indeed, when the raw 33

34 information speculators signal is correct (u =, trading by these speculators increases competition for the speculators trading on the processed signal and thereby reduces their profits. In contrast, when their signal is incorrect (u = 0, trading by the raw speculators at date open new profit opportunities for the speculators trading on the processed signal since they can trade against the noise introduced by raw signal speculators in the price at date. This is exactly the same mechanism as that present in the baseline model, which is already captured by the first term in eq.(7.6 ((π c (α π,nc (α, α. Thus, as in the baseline model, an increase in the demand for the raw signal can either be (locally (i a positive externality for those buying the processed signal (it increases their expected profit or (ii a negative externality. To see this, consider two cases. First, consider the subcase of case Γ 3, in Proposition 7., where α. In this case, π c (α = 0 and therefore, using eq.(7. and eq.(7.7, we have: π agg (α, α = θ( θ ( θ α ( α ( α α. It follows that π agg / α < 0 and thus α / α < 0. Alternatively, consider case Γ. We deduce from eq.(7. and (7.9 that in this case, θ( θ (α, α = α θ( θα ( α + ( α ( θ α ( + ( θα ( θα. π agg The partial derivative with respect to α can be positive for some values of α. Another difference with the baseline case is that the demand for the processed signals (α affects the expected profit of speculators who receive the raw signal. Indeed, when they trade at date, their expected profit is affected by the demand of speculators who receive the processed signal (see Proposition 7. and therefore their total expected profit depends on this demand as well. In all cases, considered in Proposition 7., the expected profit for speculators trading on the raw signal at date decreases with the demand for the processed signal. We deduce that: π agg α < 0. (7.7 3

35 Solving for the equilibrium. Equations (7.3and (7.5 show that when speculators can trade on the raw signal at date, the demand for the raw signal and the demand for the processed signal are simultaneously determined in equilibrium because (i the expected profit from trading on the processed signal depends on the demand for the raw signal, α (as in the baseline model and (ii the expected profit from trading on the raw signal depends on the demand for the processed signal, α. In contrast, in the baseline model, the expected profit from trading on the raw signal does not depend on the demand for the processed signal. As a result, in the baseline model, one can solve for equilibrium demands in closed-form by first solving for the demand for the raw signal and then the demand for the processed signal at the equilibrium point obtained for the demand for the raw signal. The simultaneous determination of the demands for both signals in the extension considered here precludes the use of this approach and, for this reason, no analytical solutions can be obtained for equilibrium demands. However one can solve numerically for the system of equations (7.3 and (7.5 to find the equilibrium. Panel A in Figure 7. provides an example. In this example, when C r declines, the demand for the raw signal (blue line increases and the demand for the processed signal (red line increases. Thus, as in the baseline model, this is a case in which an increase in the demand for the raw signal exerts a negative externality on the demand for the processed signal. 7.3 Asset price informativeness. Remember that as asset price informativeness at date t is defined as, E t = E[(V p t ] = E[p t( p t ]. At date, the equilibrium is as in the baseline model and we therefore obtain the same expression for E (eq.(53 in the baseline model. For date, we obtain the following result. We obtain the following result. 35