The Negative Value of Public Information in the Glosten-Milgrom Model Romans Pancs⇤ March 2014

Abstract In the standard market-microstructure model of Glosten and Milgrom (1985), public information can have negative social value. Equivalently, an increase in informational asymmetry can raise the total surplus from trade. Keywords: market microstructure, negative social value of public information, GlostenMilgrom JEL classifications: G14, D02, D40

1

Introduction

The seminal market-microstructure model of Glosten and Milgrom (1985) is designed to explain the positive bid-ask spread in financial markets. A market-maker, whose information is inferior to that of traders, must ask for an asset more than he bids for it, just to break even in the face of adverse selection. Potential gains from trade are forgone because some traders are excluded from trade by the positive bid-ask spread. ⇤

Department of Economics, University of Rochester, [email protected]

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I study the gains from trade, or surplus, in a simple version of the Glosten-Milgrom model, with a trader and a market-maker trading an asset of uncertain value for money. The surplus is defined as the sum of the trader’s and the market-maker’s payoffs. The paper’s contribution is to show that public information can have negative social value. That is, one can construct a signal about the asset’s value such that the commitment to publicize this signal reduces the surplus. The paper’s finding is significant because it suggests that promoting transparency can reduce the surplus even in this canonical model of financial markets. The observation that public information can have a negative social value is familiar to the economic theorist. Even though in decision problems information is always valuable (Blackwell, 1951, 1953), in games and markets, it need not be. Public information can eliminate insurance opportunities and thus have negative social value at a competitive equilibrium (Hirshleifer, 1971). In a beauty-contest model, public information can have negative social value by serving as a coordination device that motivates each player to match the actions of others, at the expense of matching the state (Morris and Shin, 2002). In a macroeconomic model, publicly releasing information about productivity can reduce welfare by reducing the informational efficiency of the price system (Amador and Weill, 2010). The goal of the present paper is to alert the reader to the ubiquity of the negative value of public information by identifying it in the workhorse market-microstructure model, which does not share the defining features of the models listed above and which has not been specifically designed to exhibit the negative value of public information. In the model, public information can have a negative value regardless of the magnitude of the informational asymmetry between the market maker and the trader. The intuition is easiest to see when the informational asymmetry is small, and so the bid-ask spread is small. In this case, if the bid-ask spread is

> 0, the forgone surplus is of order

an analogy with public finance is appropriate; a small tax wedge of order

2

2

. Here,

causes a deadweight loss

. Thus, the forgone surplus is convex in the bid-ask spread. If the release of a 2

public signal either slightly raises the bid-ask spread or slightly lowers it so that the expected bid-ask spread equals the initial bid-ask spread , then, by Jensen’s inequality, the expected forgone surplus is greater than the forgone surplus at the initial bid-ask spread , without the signal. Hence, this signal’s social value is negative.

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Model

The model is a “static” version of the model of Glosten and Milgrom (1985). The possibility of the negative value of public information can be shown to be inherited by the dynamic version but is more transparent in the originating static model. Environment A trader and a market-maker exchange a unit of an asset for money, after which the asset pays its holder a random amount v 2 {0, 1}. The commonly known probability assigned to v = 1 is

2 (0, 1).

Players The trader is informed with probability ↵ 2 (0, 1) and uninformed with the complementary probability 1

↵. The informed trader knows v and values the asset at v. The uninformed

trader does not know v and values the asset at v + u, where his private valuation u is distributed on ( u¯, u¯) according to a smooth c.d.f. G, where u¯ is positive and possibly infinite. The associated p.d.f. g is positive and bounded, with a bounded derivative. Assume also that g is symmetric around zero (i.e., g (u) = g ( u) and G ( u) = 1

G (u)), implying

that u has mean zero. The trader’s privately observed type is whether he is informed, the value of v if he is informed, and the value of u if he is uninformed. The market-maker observes neither v nor the trader’s type.

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Trading Protocol The trader may request to buy a unit from, or sell a unit to, the market marker, or can refrain from trading. The market-maker posts the smallest ask A 2 [0, 1] and the largest bid B 2 [0, 1] at each of which he breaks even in expectation. Let a ⌘ A

and b ⌘

B denote

the market-maker’s markups, which capture how much the uninformed trader loses from transacting at the posted ask and bid relative to transacting at , which is his expectation of v. The break-even conditions that define A and B are

(1

↵) (1 (1

G (a)) a

↵) G ( b) b

↵ (1

A) = 0

(1)

↵ (1

) B = 0.

(2)

Thus, (1) specifies that the trader is uninformed with probability 1 buys at A with probability Pr { + u > A} = 1

↵, in which case he

G (a), and the market-maker earns a.

With the complementary probability ↵, the trader is informed, in which case he buys with probability

(i.e., if v = 1), and the market-maker loses 1

A. The zero-profit condition in

(2) is interpreted analogously. One can always find an A and a B that solve (1) and (2). Furthermore, by inspection of (1) and (2), the symmetry assumption G ( u) = 1

G (u) implies b ( ) = a (1

).

Payoffs Each trader is a risk-neutral expected-payoff maximizer. The informed trader’s payoff is v

A if he buys and B

and B

v

v if he sells. The uninformed trader’s payoff is v + u

A if he buys

u if he sells.

Surplus from Trade For a measure of the gains from trade, it is natural to consider the sum of the trader’s and the market-maker’s expected payoffs. A market-maker’s trade with the informed trader does 4

not contribute to this sum, because they both value the asset the same, at v. By contrast, a market-maker’s trade with the uninformed trader contributes to this sum amount u if the trader buys and amount

u if he sells. Hence, the analysis will focus on the gains

from trade, or surplus, defined as the expected sum of the uninformed trader’s and the market-maker’s payoffs:1 ⇥

S ⌘ E 1{u>a} u + 1{ where 1{u>a} and 1{

u>b}

⇤

u>b} ( u) =

ˆ

a

1

sg (s) ds +

ˆ

1

sg ( s) ds.

(3)

b

are the indicator functions for the events at which the uninformed

trader buys or sells, respectively, and E is the expectation operator with respect to u. By inspection of (3), the symmetry assumption g (u) = g ( u), together with a ( ) = b (1 implies the symmetry S ( ) = S (1

),

).

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The Main Result

At

= 1/2, the informational asymmetry about v between the informed trader and the

market-maker is maximal; both markups are positive and hence exclude some uninformed traders, thereby leaving some gains from trade unrealized. By contrast, at

2 {0, 1}, the

trader and the market-maker are symmetrically informed; both markups are zero, the uninformed trader always trades, and the gains from trade are maximal. From this comparison, however, it is wrong to conclude that better public information about v raises the surplus. In particular, it will be shown that (i) the surplus need not rise if a public signal about v becomes available, and (ii) the surplus need not be minimized when the informational asymmetry is maximized. Define the social value of public information as the expected change in surplus associated with the policy of publicly releasing a signal about v. The shape of S determines this value. To see how, consider a binary public signal that, with probability p, 1

The surplus is defined to be conditional on the trader being uninformed for notational parsimony.

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S 0.25 0.24 0.23 0.22 0.21 0.20 0.19 0.2

0.4

0.6

0.8

1.0

g

Figure 1: S, the gains from trade, in Example 1. induces

as a posterior probability of v = 1 and, with probability 1

0

p, induces

00

as

the posterior probability. By Bayes’s rule, the prior must equal the expected posterior: 00

. The social value of public information provided by this signal is neg-

ative if pS ( 0 ) + (1

p) S ( 00 ) < S ( ), which holds if S is locally strictly concave and if

=p

0

and

0

+ (1

00

p)

are sufficiently close to . While the value function in any decision problem is

convex (Blackwell, 1951, 1953), the surplus function S is necessarily strictly locally concave for some : Lemma 1. The surplus function S is locally strictly concave near for some parameter values, also near

= 0, near

= 1, and,

= 1/2.

Proof. See Appendix A. Figure 1 illustrates Lemma 1 by plotting S for the case described in Example 1: Example 1. Let ↵ = 1/3 and G (u) = u + 1/2 for any u 2 ( 1/2, 1/2). From (1) and (2), the markups are

a ( ) = 1{

< 13 }

+ 1{

1 1 3

}

2 6

and

b ( ) = a (1

).

Substituting these markups into (3) gives the surplus:

S ( ) = 1{

2 < 13 }

5 8

2

+ 1{ 1  3

1+2  23 }

2

2

+ 1{

8

2 > 23 }

5 (1 8

)2

,

which combines three concave parabolas and hence is locally concave almost everywhere. Lemma 1 and the discussion in the paragraph preceding it imply Theorem 1: Theorem 1. There exists a prior belief

and a signal about v such that the social value of

public information is negative.

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A Heuristic Argument

The intuition for Theorem 1 will be conveyed by a heuristic argument for Lemma 1. The plan is to understand the local concavity of S first at Near

2 {0, 1} and then at

= 1/2.

= 0, approximate the markups a and b by their Taylor expansions. Each markup

is of order . The (probability) measure of the excluded uninformed traders (those with private valuations in ( b, a)) is of the order of the markups, and hence of order . The surplus lost due to markups is of the order of the product of a markup and of the measure of the excluded traders. Hence, the lost surplus is of order

2

, and so is convex in . The

realized surplus, S, equals the zero-markup surplus (which is E [|u|] and independent of ) less the lost surplus (which is convex in ) and hence is strictly concave in . Because S is symmetric, the above argument also implies that S is strictly concave near The strict concavity of S at A( )

= 1.

= 1/2 is subtler and prevails when (i) the bid-ask spread

B ( ) ⌘ a ( ) + b ( ) varies little in , and (ii) the lost surplus is convex in each

markup. When both (i) and (ii) hold, a small deviation of

from 1/2 increases one markup

by approximately as much as it decreases the other one, thereby increasing the lost surplus, as Figure 2 illustrates for Example 1. The idea is that, when the same. Once

= 1/2, both markups are

deviates from 1/2, the markups diverge, and the higher markup excludes 7

|u|g(u)

R1 b

R3 R4

R2

a

0

½ "

𝛾 Figure 2: For the parameters in Example 1, the fourth (southeastern) quadrant plots the markup a as a function of . The first (northeastern) quadrant translates this markup (which is also the private valuation of the marginal excluded buyer) into |a| g (a), the surplus lost by excluding the marginal buyers, whose private valuations equal the markup (i.e., u = a). The area R3 + R4 is the surplus lost by excluding the marginal and infra-marginal (those with positive private valuations smaller than the markup, u 2 (0, a)) buyers when = 1/2. Analogously, the third (southwestern) quadrant depicts the markup b as a function of , and the second (northwestern) quadrant translates this markup into the surplus lost by excluding marginal sellers. The area R2 is the surplus lost from excluding marginal and infra-marginal sellers when = 1/2. As rises from 1/2 to 2/3, a falls, b rises, and the bid-ask spread a + b remains unchanged. As a result, the surplus lost due to a falls by R4 , whereas the surplus lost due to b rises by R1 . Because |u| g (u) is increasing in u, the surplus reduction R1 R4 is positive. Graphically, R1 R4 is twice the area of the shaded triangle in the first quadrant.

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the uninformed traders with higher private valuations than the valuations of the traders “released” by the lower markup. Beyond Example 1, the conditions for the concavity of S near from the Taylor expansion of S. Using b ( ) = a (1

= 1/2 can be gleaned

) and differentiating S once gives

S 0 (1/2) = 0; differentiating S twice gives ✓ ◆ 1 S = 2 00

2 (a0 )

where a, a0 , and a00 are all evaluated at S is2, 3

✓ ◆ 1 S( )=S 2

0 2

(a )



2

@ (ag (a)) @a

2a00 ag (a) ,

= 1/2. The second-order Taylor approximation of

@ a00 ag (a) (ag (a)) + @a (a0 )2

✓

1 2

◆2

+O

3

.

(4)

Thus, S is approximately quadratic, and is strictly concave if the bracketed term in (4) is positive, for which it suffices that • a00 (1/2) = 0, implying a00 (1/2) + b00 (1/2) = 0 (by a ( ) = b (1 with a0 (1/2) + b0 (1/2) = 0 (again, by a ( ) = b (1

)), which combined

)) means that, to the second

order, the bid-ask spread, a ( ) + b ( ), is constant in the neighborhood of

= 1/2;

• @ (ug (u)) /@u|u=a(1/2)=b(1/2) > 0, meaning that, as one divorces the markups by raising a a little and lowering b by the same amount, more surplus is lost from the uninformed buyers newly excluded by the higher a than is gained from the uninformed sellers newly included by the lower b.

A

Appendix: Proof of Lemma 1

To establish that S is locally concave at S 00 ( ) = 2 3

a0 ( )2 g (a ( ))

= 0, differentiate (3) twice to obtain:

b0 ( )2 g ( b ( ))

a ( ) g (a ( )) a00 ( )

b00 ( ) b ( ) g ( b ( )) .

The “big-O” notation uses O 3 for the terms of order 3 and higher. 2 2 The term (a0 ) [@ (ag (a)) /@a]h( 1/2) in (4) i is twice the area of the shaded triangle in the first

quadrant in Figure 2. The term a00 ag (a) / (a0 )

2

2

1/2) is not in the figure because, in Example 1,

(

a00 = 0.

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Using a (0) = b (0) = 0 and g, g 0 < 1, S 00 (0) =

g (0) a0 (0)2 + b0 (0)2

(A.1)

provided |a00 (0)| < 1 and |b00 (0)| < 1, as will be ascertained. Moreover, if a0 (0) > 0 and b0 (0) > 0, then (A.1) implies S 00 (0) < 0. To establish the requisite conditions on a0 (0), b0 (0), a00 (0), and b00 (0), implicitly differentiate (1) with respect to , twice, to obtain: ↵ (1 2 a) ↵ + (1 ↵) (1 G (a) aG0 (a)) 2↵ (1 + a0 ) (1 ↵) (a0 )2 (2G0 (a) + aG00 (a)) = , ↵ + (1 ↵) (1 G (a) aG0 (a))

a0 = a00

where the argument of a, a0 , and a00 has been suppressed. Substituting a0 into a00 , and evaluating both at 2 {0, 1} gives: a0 (0) =

2↵ 1 ↵

and

a0 (1) =

2↵ 1+↵

and a00 (0) =

4↵ (1 + ↵ (1 4g (0))) (1 ↵)2

and

a00 (1) =

4 (1

↵) ↵ (1 + ↵ (1 (1 + ↵)3

4g (0)))

.

whence a0 (0) > 0 and |a00 (0)| < 1, as conjectured, and, by b0 (0) = a0 (1) and b00 (0) = a00 (1), b0 (0) > 0 and |b00 (0)| < 1, also as conjectured. Hence, by (A.1), S 00 (0) < 0. Furthermore, by the symmetry of S, S 00 (1) < 0. The possibility of S 00 (1/2) < 0 is established in Example 1.

References Amador, Manuel and Pierre-Olivier Weill, “Learning from Prices: Public Communication and Welfare,” Journal of Political Economy, 2010, 118 (5), 866–907. Blackwell, David, “Comparisons of Experiments,” Proceedings of the Second Berkeley Symposium in Mathematical Statistics, 1951, pp. 93–102. , “Equivalent Comparisons of Experiments,” Annals of Mathematical Statistics, 1953, 24 (2), 265–272.

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Glosten, Lawrence R. and Paul R. Milgrom, “Bid, ask and transaction prices in a specialist market with heterogeneously informed traders,” Journal of Financial Economics, 1985, 14, 71–100. Hirshleifer, Jack, “The Private and Social Value of Information and the Reward to Inventive Activity,” The American Economic Review, 1971, 61 (4), 561–574. Morris, Stephen and Hyun Song Shin, “Social Value of Public Information,” The American Economic Review, 2002, 92 (5), 1521–1534.

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