Sell-Side Information Production in Financial Markets

Zhaohui Chen and William J. Wilhelm, Jr.

UNDER REVISION

Abstract

We study conditions that sustain production of nonexcludable information for direct sale in thepresence of a trading opportunity. We use the Kyle (1985) framework to demonstrate that informationproduced for direct sale uniquely heightens competition in the financial market and thus promotes pricee!ciency. We then derive conditions under which firms that benefit from price e!ciency will pay agentsto produce this externality rather than trade on their private information. The model sheds light onwhy investment banks bundle sell-side analyst research with other services but are able to commandadvisory fees from corporate clients on a standalone basis.

Keywords : Financial Intermediation, Investment Banking, Information Production, Selling Informa-tion

JEL classification: L22, G24, G14, D82

1 Introduction

First and foremost, investment banks produce information. Until recently, they did not carry substantialrisk on their balance sheets [Morrison and Wilhelm (2007b)]. They simply brokered access to the capitalmarkets. If there is social value in the services they provide, presumably, it rests on their unique contributionto investment e!ciency via more informative securities prices. But this argument begs two questions thatwe address in this paper. First, how could information produced by investment banks be unique? Investorsdevote similar resources to research on the “buy-side” of financial markets and the rise of institutionalinvesting has strengthened incentives to do so by enabling greater internalization of the benefits. Doinvestment banks have a unique technology for information production relative to hedge funds and largemutual funds?

Second, unlike buy-side agents that trade on their (private) information, until recently, investmentbanks generally sold rather than traded on their information. But given the di!culty of contracting overinformation, what attracts talented people to the sell-side of the market? Alternatively, what sustainsthe seemingly rich fee structure for advisory services when the information on which it rests is largelynonexcludable?

We study these questions assuming that a fixed number of agents can opt either to sell or trade ontheir private information, that the sell-side production technology is not unique, and that exclusive sale ofinformation is impossible. We demonstrate that, although it need not be unique, information produced fromthe sell-side uniquely promotes financial price e!ciency by increasing competitive pressure on the remainingprivately informed buy-side agents. This positive externality follows directly from the nonexcludability ofinformation and does not appear in the body of literature stemming from Admati and Pfleiderer (1986).

Similar in spirit to this literature, we show that when sell-side agents are unable to internalize thiscontribution to price e!ciency, it will not be realized. However, assuming only that price e!ciency improvesfirms’ investment decisions, we derive conditions under which an endogenously determined fee contractbetween a firm and sell-side agents will induce the latter to forego trading on their private information infavor of disseminating it among buy-side agents. This result provides a rationale for the existence of directsale of information on a standalone basis to agents who pay for indirect benefits from its production.

Our results are obtained in a setting similar to that studied by Kyle (1985). At no loss of generality, thisapproach enables direct comparison with much of the existing work on the barriers to direct sale of privateinformation in the presence of a trading opportunity.1 We assume that information-producing agents maychoose either to use information for their own trading purposes or to sell it to buy-side agents who willthen trade on the purchased information as well as their own private information. The nonexcludabilityassumption implies that these sell-side agents cannot credibly commit to sell their information to a subsetof buy-side agents.2 However, they can credibly commit to not trade on the information they sell. Theseassumptions are intended to capture the essence of industry attempts to separate proprietary trading fromresearch and advisory functions and the limited capacity for contracting over information.3

1Similar results arise in a common-value auction framework. Results are available upon request.2Sell-side firms, such as investment banks and brokerage firms, produce research for both wholesale (institutional investor

and corporate) clients and retail clients who typically acquire it bundled with other investment banking or brokerage services.Buy-side firms, like mutual and pension funds, produce their own research in addition to that acquired from sell-side producersand bundle the research with asset management services for both retail and institutional investors. Cheng, Liu and Qian (2003)estimate that 71% of research is produced by buy-side firms, 24% by sell-side firms, and the remaining 5% by specialized firms.

3It would be naive to believe that proprietary trading functions are completely walled-o! from research and advisoryfunctions but investment banks expend resources to limit their sharing information and su!er considerable criticism whengross violations of market and client expectations to this e!ect are violated. In fact, it is not unusual for boutique banks thatspecialize in advisory functions to actively promote their freedom from conflicts of interest. Biais and Germain (2002) analyzethe optimal contracting problem when a monopolistic informed agent can both engage in an indirect sale of information to a

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The inability to commit to exclusive sale of sell-side information implies that it will be widely dissem-inated in equilibrium. This threat heightens competition among buy-side traders. Information producedfrom the sell-side of the market is therefore more rapidly impounded in financial market prices than wouldbe the same information produced from the buy-side and thus reserved strictly for private trading. Inother words, sell-side information production yields a positive externality in the form of more rapid priceadjustment.

Similar to previous work, sell-side production is di!cult to sustain in the presence of a trading oppor-tunity. When sell-side information-production capacity is not unique and all agents are risk neutral, it isimpossible to sustain sell-side production without an external subsidy. This result holds regardless of thelevel of bargaining power sell-side agents maintain relative to buy-side agents. We then examine conditionsunder which the competitive tension associated with sell-side information production broadens the range ofconditions under which it can be sustained. A complementary (symmetric) sell-side production technology,this distinct competitive pressure can sustain, without subsidy, a small number of sell-side agents. Withasymmetric information production, as one might imagine developing in the context of a banking rela-tionship, we derive conditions under which at least one sell-side agent is sustainable and show that lowerprecision signals among buy-side agents sustain more sell-side producers of complemetary signals.4

To this point in the analysis the sustainability of sell-side information production rests strictly on thesell-side agent’s capacity for extracting rents from buy-side agents. In turn, buy-side agents profit fromtheir own private information and that purchased from sell-side agents strictly at the expense of liquiditytraders. In other words, we do not consider investment e!cency gains that should arise from more rapidfinancial price adjustment.

We conclude the analysis by examining the connection between real e"ects of price e!ciency and theircapacity for sustaining sell-side information production. We focus on corporate control and restructuringevents where investment sensitivity to financial price information is likely to be strongest.5 This is alsothe setting in which it is perhaps least clear how payment (in the form of advisory fees) for nonexcludableinformation is sustainable, especially at the richly rewarded level observed in practice. Bank clients generallypay advisory fees upon completion as a percentage of deal value. In principle, firms could shop a bank’sadvice in search of a more favorable fee contract. And yet, increasingly the advisory function is beingcarried out by specialized (boutique) firms that have virtually no means for bundling the information theyproduce with goods or services over which they have stronger property rights. We derive an endogenouslydetermined fee contract that connects the sustainability of pure advisory functions to financial marketcharacteristics including the level of competition among buy-side agents and financial market liquidity.

2 Relation to the Existing Literature

A substantial body of research examines the implications of weak property rights for investment-bankingpractices and both organizational and industry structure. Tufano (1989), Bhattacharya and Nanda (2000),and Persons and Warther (1997) study incentives for and time patterns in financial innovation given therelative ease with which the new products and services are expropriated. Pichler and Wilhelm (2001) arguethat the unique structure and long existence of underwriting syndicates reflects banks’ ability to freee-ride on one another’s investment in information and client relationships. Benveniste et. al. (2002, 2003)

principal while also conducting proprietary trades. Germain (2005) extends the analysis to the case of competitive informationsellers.

4Admati and Pfleiderer (1987) examine how correlation in signals bears on the concentration of information ownership. Inour model, variation in correlation among signals influences the seller’s capacity for rent extraction from buy-side agents.

5See Luo (2005) for empirical evidence that firms weigh market reactions when deciding whether to carry through tocompletion an announced merger.

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argue that industry-level pricing patterns reflect investment banks coordinating the sharing of informationproduction costs among industry members. Morrison and Wilhelm (2004, 2007a,b) identify the industry’slong commitment to partnership organizations as a reflection of the primacy of human capital over whichfirms maintained weak property rights. Anand and Galetovic (AG) (2000, 2006) focus on the connectionbetween sustaining investment in expropriable assets and the industrial organization of investment bankingand other professional services.

AG (2000, pp. 358-9) argue that “incentives to gather information can be preserved simply by en-dogenous adjustment of the intermediary market structure.” While we agree with this argument and viewour contribution in very much the same spirit, we extend this perspective on investment banking by con-necting incentives for sell-side information production directly with its presumed contribution to economice!ciency. Namely, securities prices in capital markets guide investment. If information produced for saledoes not promote price e!ciency, and thereby more e!cient investment, then there is little to recommendthe considerable resources committed to the investment-banking sector. Anand and Galetovic take thisconnection as given. By contrast, we study how the connection between sell-side information and financialprice e!ciency determines the level of resources devoted to sell-side information production and why firmsare willing to pay for such information to be disseminated among buy-side agents.

We follow Kyle (1985) in modeling how information is impounded in securities through securities markettrading and Admati and Pfleiderer (AP) [(1986), (1987), (1988), (1990)] in studying incentives to sellinformation in the presence of a trading opportunity. The central message of Admati and Pfleiderer’s worklies in the relative di!culty of sustaining the production of information for sale when there is a tradingopportunity. Simply put, when compensation for such information derives strictly from trading profits, arisk-neutral, monopolistic owner of private information should benefit at least as much from direct tradingas from selling the information to someone who then becomes a competitor in its exploitation. By sellinginformation, one e"ectively gives up control over how profitably it will be exploited.

Admati and Pfleiderer examine the problem under a range of assumptions regarding risk toleranceand how information is sold. When a signal is sold directly (so that its use cannot be controlled), AP(1986) show that it may be optimal to add noise to the signal. Doing so influences the behavior of tradersfavorably with respect to trading profits and thus the seller’s profits. AP (1990) considers the alternative ofindirect sale of information. An indirect sale requires a buyer to pay for a response to private informationthat is taken on their behalf by the seller. For example, a mutual fund manager e"ectively bundles privateinformation in the selection of an asset portfolio and sells shares in the profit yielded by the trading strategy.The information seller (fund manager) thereby captures a larger share of the private benefits from the saleof by controlling free riding and thus more profitably coordinating its exploitation.

In contrast to the perfectly competitive noisy rational expectations equilibrium framework in AP (1986,1990), AP (1988) study how risk tolerance influences information selling and trading strategies when theseller can trade strategically on his information. In a standard Kyle (1985) setting, risk neutral agentswill not sell information.6 Risk aversion provides incentive to sell information but at the cost of sacrificingmonopoly power over its use. Admati and Pfleiderer (1988) study how this tradeo" can be tilted in favorof risk sharing via indirect sale of information.

The problem with indirect sale of information is that it prevents heterogeneous, privately-informedtraders from purchasing individual signals for optimal combination with their own private signals. The

6We obtain a similar result but also provide conditions under which unique sell-side information-production capacityprovides incentive for one or more agents to refrain from trading on their information in favor of selling it to others who will.Fishman and Hagerty (1995) show that for risk-neutral, competitive information producers, selling information undermines theseller’s monopoly power over her private information, but it also commits her to trading more aggressively on the information.This causes her competitors to trade less aggressively and the e!ect can outweigh the cost of sacrificing monopoly power overthe information.

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problem is particularly severe when the seller’s information and buyers’ information is complementary (AP,1987). AP (1990) provides a more general analysis of the relative merits of direct and indirect sale ofinformation under the assumption that the seller does not trade. They show (among other things) that itis never desirable to add noise to a signal when it is sold indirectly.7 Rather, the information seller benefitsmore by adjusting the unit price of the fund rather than by introducing noise.8

Our model is most closely related to AP (1988) in the sense that information is sold and/or traded onin a Kyle setting but we focus strictly on the risk-neutral case and endow information buyers with theirown private signal. Although we do not permit sell-side agents to trade on their information, endowingbuy-side agents with private information provides a channel through which sell-side agents can extracttrading profits (gained at the expense of liquidity traders).

By ruling out indirect sale of information of this sort (or otherwise permitting control of its use) drawsinto sharp focus a previously unrecognized externality from the production of information for sale. Namely,the inability to commit to exclusive sale creates a competitive tension that causes such information to bemore rapidly impounded in financial market prices than would be private information produced strictly fortrading purposes. We believe that limiting the seller’s control over the use of information better reflectsthe nature of information production in advisory functions. Moreover, the fact that such information isgenerally tailored to complement a corporate client’s transactional goals, suggests a relatively high cost tobundling schemes of the sort considered by Admati and Pfleiderer. By identifying scope for direct sale ofinformation on a standalone basis to agents who pay a fee for indirect benefits from its production, we o"era rationale for the presence of investment banks (or similar information intermediaries) that rests on theprovision of public goods.

3 The Basic Model

We derive an analytical baseline in an economy with a liquid financial market. The model comprises fourdates, three periods and one risky asset. The risky asset’s payo" is realized at date 3 (the end of thethird period) and is denoted ! + V , where V is a known constant and ! is a random variable. The priordistribution of ! at date 0 is N(0,"2

!). By convention, we define v! = 1"2

!as the random variable’s precision

and without loss of generality assume that v! = 1. The discount rate across periods is normalized to bezero.

3.1 Agents and Information

There are three types of risk-neutral agents in the economy: information-producing agents, market makers,and liquidity traders. There are N information-producing agents and for the sake of simplicity we assumethat N is exogenously determined. Between date 1 and date 2, each information-producing agent receivesone signal about the asset value. The signal of agent i, i = 1, 2, ..., N , takes the form:

si = ! + #i (1)

where #i ! N(0,"2i ), ! and #i are independent for any i, and #i is independent across information-producing

agents. The last assumption captures the feature that each agent has a unique perspective about the asset7It is never optimal to add noise to a signal in our model.8Although less directly related to our work, Brennan and Chordia (1993) show that charging investors brokerage commis-

sions is a way for investors to share risk with risk-neutral brokers. Vishny (1985) studies a brokerage firm’s incentive to sellinformation in order to increase market liquidity when doing so increases potential trading commissions. Cespa (2008) consid-ers how stock prices react to private information depending on whether an analyst or an insider controls its flow. Veldkamp(2006a,b) studies the relation between asset prices and pricing of information, especially as it relates to mass media.

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value.9 Notation is simplified by denoting vi " 1"2

iand we further assume that all signals have the same

quality so that vi = vj = v for any i and j. Finally, we assume that each agent’s cost of receiving thesignal is zero.10

Market makers set the trading price in the financial market at date 2. Liquidity traders enter thefinancial market at date 2 with risky asset demand z that is normally distributed with mean zero andvariance "2

z.

3.2 Sequence of Events

At date 0, each information-producing agent decides whether to trade on his private information (becomea buy-side trader) or to sell his private information to one or more other information producers.11 Afterthe information-producing agents specialize, there are m information sellers and n " N # m traders eachof whom observes all other agents’ decisions (m and n are determined endogenously). Information sellerscannot trade in the financial market. Buy-side traders cannot sell information directly, either their own(buy-side) information or any (sell-side) information bought from information sellers. They can only profitfrom their own information and any they acquire by trading in the financial market. This assumptione"ectively provides for the existence of a market for direct sale of information.

At date 1, before they receive signals about the asset value, information sellers make o"ers to tradersin the market for information. Information seller j (j = 1, 2, ...,m) makes take-it-or-leave-it o"ers for hissell-side information to a set of traders, Fj at prices p(Fj). p(Fj) is a vector whose elements are pi

j, i $ Fj .

pij is the o"er price that information seller j demands from trader i for the information.

Trader i (i = 1, 2, ..., n) receives o"ers from a set of information sellers, Si, at a vector of o"ering pricesp(Si), whose elements are pi

j, j $ Si. Trader i does not observe any o"ers made to other traders, nordoes he observe the (information) buying decisions of other traders.12 However, trader i does form beliefsabout other o"ers conditional on the o"ers that he receives and conditions whether to accept an informationseller’s o"er on these beliefs. If he accepts the o"er from seller j, the trader pays pi

j for the information.In exchange for this payment, seller j reports his signal to trader i. If trader i declines seller j!s o"er, thereis no payment and seller j does not report his signal to trader i. The set of sellers from whom trader ichooses to buy information is Ai, which is a subset of Si. In the basic model, there are no sell-side agencyproblems. We discuss the consequences of agency problems in the conclusion.

After date 1, each information-producing agent receives a signal about the value of the asset andinformation sellers report their signals to the buy-side informed traders who have paid for them.

At date 2, buy-side traders take positions in the financial market conditional on their information andtheir beliefs about other traders’ information and strategies. The trading mechanism is similar to that inKyle (1985). Each trader submits a market order, xi and the market makers clear the market by supplyingliquidity at a price, P2, conditioning on the total order flow y. Traders do not observe current prices orquantities traded by other informed traders or by liquidity traders. Market makers do not receive anyprivate information, nor do they observe individual quantities traded by the informed traders and liquidity

9We consider a more general information structure later.10Obviously, this abstracts from the decision whether to produce information. We could introduce another date, t = !1, at

which a large number of agents can decide whether to invest in a costly information production capacity understanding thatthey will have the option to exploit their information in one of two ways (selling or trading). Once the production cost is sunkand the number of players, N, is determined, the game modelled here proceeds. The boundary condition that pins down N isidentical to that derived in Chen and Wilhelm (2008).

11For technical completeness, we assume the presence of a coordination device that tells each information-producing agentwhich path to follow. In equilibrium, each agent finds it optimal to be guided by the coordination device.

12See Hart and Tirole (1990) and McAfee and Schwartz (1994) for more on multilateral contracting in a privately observablesetting.

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traders, but they do observe the total order flow, y, from all market participants. Moreover, market makersdo not observe the number of sell-side agents, m, or to whom each sell-side agent sells information. Theycan, however, infer m and the outcome of information sales in equilibirum.

Finally, at date 3, the security value is revealed.

3.3 Definition of Equilibrium

The equilibrium concept is Perfect Bayesian Equilibrium (PBE). An equilibrium comprises the followingcomponents:

(i) Each information-producing agent’s choice of whether to specialize on the sell-side or the buy-side.(ii) The number of sellers, m".(iii) For seller j, the set of buy-side traders to whom information is o"ered, Fj , and the o"er prices

p(Fj).(iv) For buy-side trader i, the set of sellers from whom information can be acquired, Ai.(v) The market order, xi, submitted to market makers by trader i conditional on his information.(vi) The market makers’ pricing conditional on total order flow, y.(vii) The information sellers’ and traders’ beliefs about o"ers made by other sellers and market makers’

beliefs about the number of sellers and the traders’ information structure.In an equilibrium, conditional on other agents’ equilibrium strategy, (a) each information seller (trader)

finds it optimal to be a seller (trader); (b) based on his information and beliefs about other traders’information, trader i chooses the set of sellers, Ai, to buy information from to maximize his profit (tradingprofit minus the cost of buying information); (c) seller j chooses the set of traders, Fj , and prices p(Fj)to maximize his total profit (investment-banking profit plus profit from selling information); (d) based onhis information and beliefs about other traders’ information, each trader submits an order to maximize hisexpected trading profit; (e) each market maker sets the trading price conditional on the total order flowto maximize his expected payo"; and (f) all beliefs are consistent with the equilibrium strategies of all theagents in the model.

We focus on pure strategy equilibria in which trading strategies in the date 2 financial market arelinear. The equilibrium is solved by backward induction beginning with the trading game comprising theinformed traders’ strategies and the market makers’ pricing rule at date 2. Then we analyze date 1trading in the market for information. Finally, we characterize the date 0 specialization decisions of theinformation-producing agents and the equilibrium number of information sellers.

4 A Benchmark Equilibrium

The benchmark equilibrium analysis sheds light on two results that will be central to our arguments. First,we illustrate that buy-side and sell-side information have distinct e"ects on financial market prices. This isa result of externalities arising in competition among buy-side traders in the financial market. Second, weestablish a baseline for further analysis of the information-producing agent’s decision whether to specializeon the buy-side or the sell-side of the market for information.

To avoid obscuring these results, all proofs and much of the analytical machinery is relegated to theappendix. It is useful at this point to highlight one of the results proved in the appendix. At date 1, tradingin the market for information is a multilateral contracting game with privately observable contracts. Underpassive beliefs, the following lemma holds.13

13If out-of-equilibrium beliefs are not restricted, there may be a plethora of PBE. See Bolton and Dewatripont (2005) fora summary of the literature. McAfee and Schwartz (1994) provide an extensive discussion of reasonable out-of-equilibrium

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Lemma 1 Equilibrium with passive beliefs is characterized by each agent who specializes on the sell-sideof the information market selling his information to every agent who specializes on the buy-side of theinformation market.

Any information seller has incentive to deviate from an equilibrium in which he sells to a subset ofbuy-side traders because privately observable contracts prevent credible commitment to not doing so.14This is a strong conclusion that imposes boundaries on the analysis. It is consistent with the relativedi!culty of establishing and enforcing formal property rights over information and, as such, it enablesus to examine conditions under which sell-side information production can occur absent any extra-legalenforcement arrangements.

4.1 Trading in the Financial Market

At date 2, liquidity traders, market makers, and n buy-side informed traders participate in the financialmarket. Lemma 1 implies that trader i’s information set is Fi = {si, s1, ..., sm}; each informed traderhas his own signal, si, and m signals purchased from information sellers. Conditional on this information,trader i (i = 1, 2, ..., n) submits the market order xi that maximizes his expected profit. Market makersobserve only the total order flow, y =

!ni=1 xi + z, and with information extracted from that observation,

they establish the market clearing prices, P2(y). Thus, trader i’s problem is

maxxi

E[xi(V + ! # P2(n"

k=1

xk + z))|Fi]. (2)

Each trader takes the pricing rule P2(·) and other informed traders’ strategies as given and exploits hisinformation advantage by accounting for the impact of his trading decision on the date-2 price.

Most of our results are not directly dependent on strategic behavior among market makers. We needonly to assume that market makers price according to

P2(y) = V + $y, (3)

where $, the slope of the price impact function, is a positive number. Thus, as is true in the standard Kyle(1985) model, the market-clearing price is linear in total order flow. This formulation includes the case inwhich the market makers are perfectly competitive:

P2(y) = E[!|y] + V. (4)

Although $ and P2(y) are endongenously determined throughout the following analysis, it is worth notingthat the decisions of information producing agents are not dependent on a particular value of $ as long asit is positive. We demonstrate this point in proposition ?? below.

We conjecture that buy-side informed trader i!s trading strategy takes the symmetric form:

xi = %si + a(m"

j=1

sj), (5)

beliefs for sharpening predictions about equilibrium outcomes, the most common of which is their passive beliefs restriction.We adopt this convention and note that it implies that upon receiving an out-of-equilibrium o!er from a seller, trader i believesthat all other o!ers remain equilibrium o!ers.

14See McAfee and Schwartz (1994) for analysis of supplier commitments to competing downstream firms under unobserv-ability in a more general setting. We also assume a zero marginal cost in selling information. Given recent advances ininformation technology, this is not a particularly strong assumption.

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where % and a measure how aggressively a buy-side agent trades on his own information and on sell-sideinformation, respectively (this assumption is without loss of generality, since the unique linear equilibriumis symmetric as shown in Proposition ??). We can rewrite trader i’ strategy as

xi = %si + &sp, (6)

where sp = 1m

!mj=1 sj and & = am. Note that sp is the su!cient statistic for !, given all sell-side

signals. We show in the appendix that this strategy is indeed the unique linear equilibrium and that it ischaracterized by Proposition ??:

Proposition 2 (i) There exists a unique linear equilibrium for date 2 trading, in which a fund manager’strading strategy is given by (5), where

a =2v

$(n + 1)[2(1 + mv) + (n + 1)v](7)

% =v

$[2(1 + mv) + (n + 1)v], (8)

and the market makers’ pricing rule is given by (3).(ii) In equilibrium, the fund manager’s expected trading profit is

' =14$

[4v(1 + 4m + 2n + n2 + (1 + 2m + n)2v)

(1 + n)2[2 + (1 + 2m + n)v]2]. (9)

(iii) The equilibrium asset price is

P2 = V +2nv

(n + 1)[2(1 + mv) + (n + 1)v]

m"

j=1

sj +v

[2(1 + mv) + (n + 1)v]

n"

i=1

si + $z. (10)

(iv) If market makers are perfectly competitive so that (4) holds, then:

a =2"zv%

nD(11)

% ="z(n + 1)v%

nD(12)

$ =%

nD

"z(n + 1)[2(1 + mv) + (n + 1)v], (13)

where D " 4mv + 4(mv)2 + 4nv2 + (n + 1)2(v + v2).

Proposition ?? calls attention to the fact that (buy-side) informed traders exploit buy-side and sell-sideinformation di"erently and thus these two sources of information have distinct e"ects on the market price.This is true in spite of the fact that buy-side and sell-side information are of the same quality. Corollary 3outlines their e"ects on trading behavior and market prices:

Corollary 3 (i) na# = 2n

n+1 & 1 implies that buy-side traders, as a group, trade more aggressively on sell-sideinformation than on their own information.

(ii) a# = 2

n+1 ' 1 implies that individual buy-side traders trade less aggressively on sell-side informationthan on their own information.

(iii) Ceteris paribus, sell-side information generates more trading volume than does buy-side information.(iv) Ceteris paribus, sell-side information has greater price impact than does buy-side information.

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Buy-side traders individually trade less aggressively on sell-side information than on their own (buy-side)information because they recognize and fully internalize the private consequences of common knowledgeof sell-side signals. But common knowledge of sell-side signals promotes competition in the aggregate.By definition, a buy-side trader is a monopolist in his private buy-side signal. If no agent opted forthe sell-side of the market, then every agent’s signal would be exploited monopolistically. Recognizing thedi!culty in contracting for exclusive sale of information, an agent who sells his private information heightenscompetition around its exploitation relative to what would have existed had he remained a monopolistictrader in his signal.

It is worth noting that a# is decreasing in n while na

# is increasing in n. This means that the distinct pricee"ects of buy-side and sell-side information are more pronounced when there are more buy-side traders inthe economy. Finally, because the market makers’ pricing rule is linear in total order flow, a greater changein order flow implies a greater change in price. Therefore, sell-side information is more influential in thesense that it generates a larger price impact.15

Note that the inability to sustain arms-length, exclusive-use contracts over information is a double-edged sword. On the one hand, weak property rights undermine incentives to produce sell-side information.But if ample incentive is provided to attract agents to the sell-side of the market, direct, non-exclusive salepromotes competition and leads to their information being more rapidly impounded in financial marketprices.

At this stage of the analysis, redistribution of trading profits is the only channel for sustaining sell-sideinformation production. As we noted in the introduction, if rapid impoundment of information in financialmarket prices has positive real investment e"ects, however, operating firms may have incentive to promotesell-side information production by sharing the surplus. Before we analyze this potentially complementarychannel, we conclude the benchmark analysis by characterizing the limits of redistribution as a means ofsustaining sell-side information production.

5 Conditions that Support Sell-Side Information Production

5.1 Specialization in the Information Market

The final stage of the benchmark equilibrium analysis characterizes the specialization decision facinginformation-producing agents. In equilibrium m" information-producing agents operate from the sell-sideof the information market. The remaining information-producing agents buy information from the sell-sidespecialists and trade in the financial market conditional on both their own private information and thatacquired from the sell-side.

The equilibrium definition implies that m" is an equilibrium composition if and only if

'b(m") & 's(m" + 1) if m" ' N # 1, and (14)'s(m") & 'b(m" # 1) if m" > 0. (15)

15This is formally demonstrated in the appendix. Also see Chen (2004). With a fixed number of information-producingagents in the economy, increasing the number of sell-side agents, m, has o!setting e!ects on the revealed uncertainty inshare prices. Corollary 3 shows that sell-side information is impounded in financial markets prices more aggressively thanis buy-side information. On the other hand, increasing the number of sell-side agents reduces the number of, and thereforecompetition among, buy-side agents in the financial market. It can be shown, however, that when N goes to infinity whilethe total information in the economy ! = Nv is fixed, increasing the number of sell-side analysts unambiguously improvesinformational e"ciency in the financial market. If all buy-side traders could commit to trading on sell-side information withthe same intensity with which they trade on their own private information (that is, maintain a such that na equals theequilibrium !) they would earn greater trading profits, ceteris paribus.

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The buy-side agent’s incentive-compatibility condition, (14), implies a smaller profit if he unilaterally defectsto the sell-side. Similarly, the sell-side incentive-compatibility condition, (15), prevents a sell-side agentunilaterally defecting to the buy side. Proposition 4 establishes that sell-side information production cannotbe sustained in equilibrium under the conditions set out thus far.

Proposition 4 In the absence of an external subsidy for sell-side information production, all informationis produced from the buy-side of the information market in equilbrium.

This result is analogous to Admati and Pfleiderer’s (1988) result for the risk-neutral case but with afew subtle wrinkles that will be important for the remainder of our analysis. To gain intuition for thedistinguishing characteristics, note that an agent may opt for the sell-side of the market for one of tworeasons. First, the agent’s information might generate greater trading profits when widely disseminatedthan when exploited monopolistically. Second, the sell-side agent’s ability to extract large rents frombuy-side agents could make selling information attractive relative to trading on the information directly.

The first possibility is ruled out by competition among buy-side traders who acquire the sell-side agent’ssignal. Corollary 3 shows that buy-side agents trade more aggressively on a sell-side signal (with intensityna) than on buy-side signals (with intensity %). As a result, a sell-side signal is more rapidly impounded inthe price than would be the case if buy-side traders could coordinate to trade less aggressively on sell-sidesignals and thus limits the price at which sell-side signals can be sold.

The second possibility is ruled out because the information seller’s information is assumed to be asubstitute for the information buyer’s signal (albeit an imperfect one). Thus the marginal sell-side agent’sinformation has a relatively small marginal benefit to buy-side agents and the rents that sell-side agents canextract from buy-side agents are limited despite their having all of the bargaining power. In equilibrium,buy-side agents pay only the marginal benefit associated with the last sell-side agent’s signal. This paymentis decreasing in both the number of information sellers and in the degree of correlation among informationsignals. In the extreme case where si = !, signals are perfect substitutes for one another and buy-sideagents will not pay a positive price for sell-side signals.

The idea also can be understood in terms of the Prisoner’s dilemma. In much of the existing literature,the seller of information is the only source of private information. If our buy-side agents were similarlyuninformed, their payo"s in equilibrium would be zero regardless of whether they bought information ordid not because the seller is assumed to have all of the bargaining power in the relationship. But then thereis nothing for sell-side agents to extract from buy-side traders and thus there is no incentive for an informedagent to sell rather than trade on private information. Thus, the fact that buy-side agents are informed inour model provides a channel for rent extraction by information sellers that sets our analysis apart fromprevious work and suggests means of sustaining sell-side information production in equilibrium. In theremainder of this section we explore how unique sell-side production capacity might be used to exploit thecompetitive tension that gives rise to the results in Corollary 3.

5.2 A Simple Example with Symmetric Information

Thus far we have assumed that there is no unique information production capacity on the sell-side ofthe market. Now we examine information structures that can sustain sell-side production. We beginby assuming a unique (or complementary) sell-side information production technology. This is not anarbitrary assumption. Investment banks and other financial market intermediaries deal repeatedly withsome counterparties and to a greater or lesser degree in their various intermediary capacities.16 It is notonly plausible but, we believe, likely that such interaction yields insight that cannot be gained through

16See Morrison and Wilhelm (2007a, ch. 8) and Asker and Ljungqvist (2010).

11

buy-side information production. Moreover, quasi-rents stemming from reputational barriers to entry willmitigate forces that prevent sell-side agents from capturing a fair return on their investments in information.

We develop intuition with the special case of two information producing agents whose signals havecorrelated errors:

si = ! + (i, i = 1, 2, (16)

with V ar((i) = 1v and corr((1, (2) = ). Complementarity arises when the errors are negatively correlated

so that combining the signals yields a more precise estimate of ! than either signal alone. In the extremecase where ) = #1, obtaining both signals yields an estimate of ! with infinite precision in contrast toprecision 1 + v for one signal.

If both agents produce information from the buy-side of the market each expects trading profit equalto

'b(0) =14$

4v(1 + v)(2 + 3v + ))2

. (17)

Note that expected trading profits approach zero with the precision (v) of information signals.By contrast, if one agent opts to produce information from the sell-side of the market, his expected

profit from selling information is

's(1) =14$

(1 # ))v(1 + v)(1 + ) + 2v)

. (18)

If ) = #1, the profit from selling information is positive ( 14$ ) even as signal precision goes to zero. The

following proposition provides the condition under which one agent will opt to produce information fromthe sell-side of the market:

Proposition 5 If ) ' #v, an equilibrium exists in which one agent will opt to produce information fromthe sell-side of the market without a subsidy.

Two forces drive this result. First, in the special case where there are only two agents, their totaltrading profit is maximized when one trades monopolistically with their private information. Second, thesell-side information producer maintains considerable bargaining power because failure to acquire sell-sideinformation sharply diminishes expected trading profits. Buy-side traders are thus relatively insensitive tothe price set for sell-side information. This enables the sell-side producer to capture a large fraction of theexpected trading profits. In other words, although the sell-side producer sacrifices monopoly power over theinformation he produces, this sacrifice is more than compensated by rents captured by, but extracted from,the buy-side trader. This result generalizes to cases where there are more than two agents. Although sell-sideagents lose monopoly power over the information they produce, su!cient bargaining power over buy-sideproducers yields a net expected private benefit to sell-side information production. Client relationships andthe reputations on which they rest provide a natural barrier to entry and access to unique information thatreinforce such bargaining power.

5.3 Asymmetric Information Production with Large N

Because it can be shown that the lower bound on ) in the information structure assumed in the precedingsection (corr((i, (j) = ), for any i (= j) is # 1

N#1 , the simple symmetric information structure sustainsunsubsidized sell-side information production only when the number of information-producing agents issmall. In this section we explore the limits of unsubsidized sell-side production when agents produceinformation asymmetrically. We do so by imposing the following information structure on agent signals:

si = ! + (i + *i. (19)

12

The noise terms, (i and *i have precision v and v% , respectively, and *i and *j are independent for anyi (= j. To keep things simple, we assume that agent 1 has unique, complementary information-productioncapacity in the sense that (1 = #(i, for any i (= 1. The following proposition shows that with su!cientlystrong complementarity, there is no equilibrium in which all agents produce information from the buy-sideof the market. Further, we establish a condition under which only agent 1 produces information from thesell-side of the market.

Proposition 6 (i) When v is low, there is no equilibrium in linear strategies with passive beliefs in whichall agents produce information from the buy-side of the market.

(ii) When v is su!ciently low and v% is su!ciently high, there is an equilibrium in which only agent 1produces information from the sell-side of the market.

The first condition arises because the marginal value of agent 1’s complementary signal is high whensignals have low precision. In this case, if all agents opt to produce information from the buy-side, monop-olistic trading on low precision signals yields modest trading profits. If agent 1 opts to produce informationfrom the sell-side, his complementary signal, when joined with others’ signals yields substantially largertrading profits in spite of the fact that buy-side agents trade aggressively on the more precise informationnow at their disposal. Agent 1 has an incentive to produce from the sell-side if he can command a su!cientlylarge share of buy-side trading profits.

The incentive to produce information from the sell-side extends beyond a single agent when v is low.Because agent 1 commands a large fraction of the trading profits, other buy-side agents have little to losefrom opting to produce sell-side information. Moreover, once buy-side agent i acquires agent 1’s signal, (iis eliminated. The remaining noise in agent i’s information arises from *i and *1. If another agent, 2, alsoopts for sell-side production then his information sold to the remaining buy-side agents further increasesthe precision of information on which they trade in the financial market. Fewer buy-side agents trading onmore precise information yields additional profits in which agent 2 can then share. Because these profitsstem from reducing noise associated with *i and *1, they persist as v goes to zero. Thus agent 2 hasincentive to produce sell-side information if v% is not too high. This implies that multiple agents may optfor the sell-side of the market in equilibrium (if one exists).17

In summary, the results in this section rea!rm the idea that direct sale of information is feasibleonly under a relatively narrow range of conditions even when taking account of the unique competitivee"ect generated by sell-side information production. Investment research is perhaps the most visible andcontroversial example of the industry’s information production because it appears to flourish only when itis subsidized by or bundled with other products or services. From our perspective, this is not surprising.With the possible exception of their having somewhat greater access to senior managers of the firms theycover, it is not obvious that research analysts have unique production capacity relative to that of theirinstitutional peers on the buy-side of the market.18 If such information yields only modest trading profits,there is less scope for sell-side agents extracting rent from noise traders via the fee they charge buy-sideagents. Moreover, this suggests reconsideration of whether analyst research warrants costs arising fromeither internal conflicts of interest or the Global Settlement’s demand that banks pay independent researchfirms for research to be distributed along with their own internal research.19

17Exploring equilibria with multiple sell-side agents is complicated by increasing returns to scale potentially arising withcomplementarity among signals. As a consequence, prices for sell-side information may be indeterminant. The proof ofproposition 6 illustrates that even in the simple case where there are only two sell-side agents, there are a continuum ofequilibrium prices for their information.

18See Lim (2001) and Chen and Matsumoto (2006) for evidence that analysts gain or sustain access via favorable forecasts.19A substantial academic literature identifies systematic biases in analyst research [See Hong and Kubik (2003)] and recent

investigations led by the New York Attorney General’s o"ce and the Securities and Exchange Commission produced evidence

13

6 Real Investment E!ects and Sell-Side Information Production

We have identified a potentially positive function for investment banks: the externality associated with sell-side information production uniquely promotes financial price e!ciency. But thus far it appears that thescope for its realization is quite limited. This conclusion does not rest comfortably with the longstandingcentrality of investment banks to capital market-oriented economies much less recent claims that thissector of the economy has grown too large and that banker compensation is excessive. However, we havenot examined whether a connection between price e!ciency and investment e!ciency opens new channelsfor sustaining sell-side information production. As a consequence, the sustainability of sell-side informationproduction has depended entirely on the sell-side agent’s capacity for extracting trading rents from buy-sidetraders. Buy-side traders profit from their own and sell-side private information at the expense of liquiditytraders. In other words, there are no e!ciency gains sustaining sell-side information production but ratheran exogenous wealth transfer mechanism running from liquidity traders to sell-side information producersthrough privately-informed, buy-side traders.

In this section we introduce real e"ects arising from price e!ciency to show how sell-side information’sunique contribution to price e!ciency can contribute to its sustainability.20 Simply put, if sell-side infor-mation uniquely promotes financial market price e!ciency which, in turn, improve corporate investmentdecisions, firms may have incentive to supplement sell-side information production beyond that sustainablethrough direct (but nonexclusive) sale to buy-side agents.

We formalize this idea by considering a firm that conditions an investment decision on its second-periodstock price. We define the optimal number of sell-side agents, m"", as the number that the firm wouldchoose if it bore no cost of sustaining sell-side production.21 We focus on the interesting case where m"" > 0and assume that q" = P2

F , where q" is a firm’s optimal investment and F measures the responsiveness ofinvestment to the firm’s second-period stock price.

We model the fee contract by assuming that the firm makes a take-it-or-leave-it o"er to potential sell-side agents before they specialize at date 0. Agents who accept the o"er produce information from thesell-side and receive a fee, 'I , from the firm. Agents who reject the o"er may opt to produce from eitherthe buy-side or the sell-side. Finally, we assume for simplicity that market makers observe which agentsaccepted the firm’s contract. This implies that they set the market price conditional on the number ofsell-side agents.

that some biased research was linked to conflicts of interest within the organizations producing the research. The $1.5 billionGlobal Settlement reached in 2003 required sell-side firms to pay $460 million for independent research over five years, and todistribute independent research reports together with their own reports. See the Securities and Exchange Commission pressrelease at http://www.sec.gov/news/press/2003-54.htm for details.

20See Qi, Goldstein, and Jiang (2007) for recent evidence and Baker, Stein, and Wurgler (2003) for a review of the literatureon the relation between asset prices and corporate investment. A straightforward application of Leland’s (1992) model canbe used to study the connection between price e"ciency and sell-side information production. In our setting, increasingthe number of sell-side agents has o!setting e!ects on the revealed uncertainty in share prices. Having more sell-side agentsimproves informational e"ciency because their information is impounded more aggressively into the stock price than is buy-sideinformation. On the other hand, if there is a fixed number of information-producing agents, there is an o!setting (competition)e!ect that arises as competition among buy-side agents diminishes with their number. The relative magnitude of these toe!ects determines whether increasing the number of sell-side agents has a positive net e!ect financial market price e"ciency.As the number of information-producing agents, N , grows, ambiguity diminishes. It can be shown that when N goes toinfinity while the total information in the economy is fixed, increasing the number of sell-side agents unambiguously improvesthe informational e"ciency of financial market prices.

21A finite number of sell-side agents implies that m!! exists.

14

Under these assumptions, the firm optimizes by setting the fee and the number of agents to whom itwill be o"ered,22

maxm,&I

E[q"(P2)P2 # f(q"(P2))] # m'I . (20)

subject to

'b(m) & 's(m + 1) if m ' N # 1, and (21)'s(m) + 'I & 'b(m # 1) if m > 0. (22)

The constraints (21) and (22) are identical to (14) and (15) except that now sell-side agents receivethe fee 'I . Proposition 4 implies that (21) is satisfied and that (22) is binding – sell-side agents require asubsidy. Therefore the binding constraint is

'I(m) = 'b(m # 1) # 's(m) (23)

Substituting this into (20), the firm chooses m to maximize its net profit:

maxm

E[q"(P2)P2 # f(q"(P2))] # m'I(m). (24)

Thus the firm trades o" its expected profit from production (ignoring fees paid to sell-side agents),which is increasing in financial market price e!ciency, against the price of e!ciency as measured by theproduct of the fee paid to sell-side agents and the number to whom it is paid. The fee itself is a functionof the relative profits expected by sell-side and buy-side agents. Proposition 7 states the conditions thatfavor contracting for m"" sell-side agents (an explicit condition is derived in the appendix).

Proposition 7 The firm will o"er a fee contract that attracts m"" sell-side agents when its investmentdecision is su!ciently sensitive to its stock price (when F is small) and/or noise trader participation issu!ciently low (when "z is small).

Proposition 7 reflects the fact that there are two sources of surplus from which agents can benefit if theyopt for sell-side information production: i) trading profits extracted from noise traders and ii) fees paid bythe firm. When noise trader participation is low, trading profits from private information are small. Thishas two e"ects. On the one hand, it means that agents who opt for the sell-side of the market can expectto extract only a modest price from buy-side agents to whom they sell their information. However, theopportunity cost of opting for the sell-side also is low. Thus a relatively modest fee o"er from the firm willbe su!cient to attract agents to the sell-side of the market.

The e"ect of greater investment sensitivity to the firm’s stock price is straightforward. When investmentis highly sensitive to the firm’s stock price, greater price e!ciency is worth more to the firm and thus it hasincentive to o"er a higher fee to attract sell-side agents. The combination of these two e"ects (specifically,the product of F and "z) drives the firm’s behavior. When F"z is su!ciently small, the first term in(24) so dominates the second term that the firm o"ers a fee contract that attracts m"" sell-side agents. Ingeneral, one would expect firms to opt for less e!cient share prices and therefore underinvest in sell-sideinformation production (relative to m"").

Corporate control and restructuring events often arise in response to changes in product market condi-tions or changes in production technology when investment decisions plausibly are most sensitive to financial

22Note that the symmetric fee structure is without loss of generality given the symmetric nature of the equilibrium at bothdate 1 and date 2. The reason is that if the fees for sell-side agent j and k are di!erent, i.e., "j

I > "kI , then the firm can do

better by lower j"s fee to "kI . Agent j would have no incentive to deviate to be a buy-side agent because agent k has no such

incentive and agent j and k are symmetric in that they have the same incentive compatibility condition.

15

market prices. Thus Proposition 7 suggests that the investment bank’s advisory function has potential forattracting fees su!cient to sustain sell-side information production beyond what buy-side agents wouldsupport. Moreover, control and restructuring transactions generally are su!ciently large to tax financialmarket liquidity. In our model, financial market liquidity is reflected in the level of noise trader participa-tion measured by "z. Once again, Proposition 7 indicates that a relative dearth of liquidity will increasefirms’ incentive to supplement sell-side information production. Finally, recall from Corollary 3 that sell-side information’s unique contribution to price e!ciency stems from the fact that it heightens competitionamong buy-side agents.23

In practice, buy-side competition in corporate acquisitions often is limited by both capital constraintsand the presence of asymmetrically informed strategic (as opposed to financial) buyers.24 Upon signingconfidentiality and standstill agreements, the seller’s bank invites prospective bidders to submit preliminaryindicative bids. Conditional on the indicative bids, a subset of bidders are selected for subsequent roundsof bidding or negotiations. Our analysis suggests that competition among bidders with such limited partic-ipation can be increased through dissemination of sell-side information. In practice, bidders are providedaccess to a common data room and management presentations. In principle, the seller could reveal thisinformation directly but because direct representation by the seller would lack credibility, we believe thatsuch information is most accurately interpreted as the bank’s work product. The seller has further incentiveto compensate a bank to disseminate such information because asymmetrically informed bidders will havelittle incentive to pay for nonexcludable sell-side information that might weaken their position relative tothe seller or other bidders.

7 Robustness

The preceding discussion suggests that the externality arising from sell-side information production couldprovide a foundation for fee-based corporate advisory services. However, it is worth considering more care-fully whether the results are simply an artifact the highly stylized model from which they are derived.Perhaps most importantly, we have not considered whether our results survive in a repeated game settingwhere, given the nonexcludability assumption, sell-side producers would have incentive to undercut oneanother. In this setting, Anand and Galetovic (2000, 2006) show that investment in nonexcludable assetsof the sort that underlie investment-banking and other professional services requires, in equilibrium, “soft”price competition that gives rise to an oligopolistic industry structure. Anand and Galetovic (2000) em-phasize the necessity of the information producer having a local monopoly at the transaction level. Asa consequence, prices (fees) are determined not by competition among information producers but ratherthrough bilateral bargaining between the buyer and seller.

Our simple contract setting can be interpreted as a reduced form representation of this feature of theirequilibrium. We could formally embed our analysis in a repeated game setting but doing so would not changeour conclusions nor would it shed further light on industry structure. In turn, our focus on the feedbackmechanism involving the number of sell-side agents and the magnitude of the sell-side externality providesa foundation for Anand and Galetovic’s analysis of investment-banking industry structure by linking thesize of the industry (reflected in the number of sell-sdie agents) to financial market fundamentals.

INSERT23See Bulow and Klemperer (1996) on the value of attracting additional bidders in an auction setting.24Andrade, Mitchell, and Sta!ord characterize the prototypical corporate acquisition during the 1990s as involving only one

bidder. Boone and Mulherin (2007) suggest competition is stronger when one examines closely the private bidding process butthe actual number of bidders in their sample averages less than two in auction transactions. In any event, bona fide biddersare few in such extraordinary circumstances, especially if making a competitive bid depends on private information to whichbuy-side agents might not have ready access.

16

8 Conclusion

In this paper, we develop a model for studying conditions that sustain production of nonexcludable in-formation for direct sale in the presence of a trading opportunity. Consistent with the existing literature,we show that when sell-side production capacity is not unique, it cannot be sustained in the absence ofan external subsidy. But we extend this vein of analysis by identifying a previously unrecognized benefitof sell-side information production – information produced from the sell-side heightens competition in thefinancial market and thus promotes price e!ciency. We then derive conditions under which firms thatbenefit su!ciently from price e!ciency will pay agents to produce information for sale rather than for theirown trading purposes. The model sheds light on why sell-side analyst research is di!cult to sustain unlessit is bundled with other services over which banks maintain stronger property rights while investment bankadvisory functions command fees from corporate clients.

Until the 1960s, merger advice and similar services were e"ectively bundled with underwriting services.25Underwriting services rested on distribution capacity over which banks maintained relatively strong propertyrights. Stable and relatively exclusive client relationships enabled banks to command underwriting feessu!cient to cover the costs of advisory services. As banking relationships weakened, this loose linkagebetween the provision of and compensation for advisory services unravelled.26 But what changed to enablebanks to begin charging advisory fees? Our model suggests that the subsequent sharp increase in capitalmarket activity in general and a long period of sustained merger, buyout, and restructuring activity providedthe conditions necessary to sustain a fee-based advisory function.

Finally, we suggest that primitive communications technology enabled greater (temporary) exclusivityfor producers of sell-side information by preventing simultaneous, low-cost, and widespread distribution.Advances in information and communications technology surely have contributed to greater nonexclud-ability of sell-side information and, from our perspective, amplified the positive externality that we haveidentified. But the same technological advances lowered execution costs for buy-side activity [Morrison andWilhelm (2007a,b)]. Coupling the e"ects of technological advance suggested by our model with those ofbank deregulation and historically low costs of capital may help us better understand the shifting balance ofpower toward buy-side activity within investment banks at a time when fee-based advisory services remainlucrative.

25Morrison and Wilhelm (2007a).26Eccles and Crane (1988) developed the idea of loose linkage between bank services and compensation. See Morrison and

Wilhelm (2007b) for evidence on the high degree of relationship exclusivity prior to 1970 and Asker and Ljungqvist (2010) forpost-1970 evidence of declining exclusivity.

17

Appendix

Proof of Lemma 1: Suppose instead that there existed an equilibrium in which sell-side agent j didnot sell to buy-side agent i. Because having agent j’s information can never reduce agent i’s profit, thisalternative implies that agent i values agent j’s information at zero. Otherwise agent j would gain bymaking an o"er to i at a positive price and i would accept the o"er. This contradicts the equilibriumdefinition.

Under passive beliefs, if i values j’s information at zero, i’s trading strategy xi must be the same withsj as without sj given the uniqueness of the buy-side agent’s problem max

xiE[xi(V + ! # (V + $y))|Fi],

where Fi is i’s equilibrium information set. The first-order conditions for xi thus imply that

xi(Fi) =E[!|Fi] # $

!k $=i

E[xk|Fi]

2$

=E[!|Fi, sj] # $

!k $=i

E[xk|Fi, sj]

2$= xi(Fi, sj)

for any realization of signals in Fi and sj. E[!|Fi] and E[!|Fi, sj ] are linear in agent i’s signals. The linearequilibrium implies that E[xk|Fi], and E[xk|Fi, sj ] are also linear in i’s signals. Matching coe!cients yieldsa set of linear equations on the model parameter values. For model parameter values outside the set definedby these equations, xi(Fi) (= xi(Fi, sj), which is a contradiction. The set of parameter values defined bythe set of equations is of measure zero in the Euclidean space. Therefore, Lemma 1 holds generically.

Proof of Proposition ??: Substituting equations (3) and (6) into buy-side agent i’s objective function(2) and simplifying yields:

maxxi

xi{E[!|Fi] # &$(n # 1)sp # %$(n # 1)E[!|Fi] # $xi}, (25)

where E[!|Fi] = mvsp+vsi

1+mv+v by Bayes rules. The first-order condition for this problem is

x"i =

[1 # %$(n # 1)]E[!|Fi] # &$(n # 1)sp

2$

=12$

{[[1 # %$(n # 1)]mv

1 + mv + v# &$(n # 1)]sp + [1 # %$(n # 1)]

v

1 + mv + vsi}. (26)

Because xi = &sp + %si in symmetric equilibrium:

& =12$

[(1 # %$(n # 1))mv

1 + mv + v# &$(n # 1)], (27)

% =12$

(1 # %$(n # 1))v

1 + mv + v. (28)

Solving equations (27) and (28) yields & and %.A general linear trading strategy is defined as xi = %isi +

!j%Ai aj

i sj. Equations (27) and (28) thusbecome

2$%i =v

1 + mv + v(1 # $

"

j $=i

%j), (29)

2$ali =

v

1 + mv + v(1 # $

"

j $=i

%j) # $"

alj

j $=i

, )l such that sl $ Ai. (30)

18

Substracting v1+mv+v $%i from both sides of equation (29) and rearranging terms yields

%i =1

(2 # v1+mv+v )$

v

1 + mv + v(1 # $

"

j

%j).

Therefore, %i = %j = %, for any i and j. Similarly, substracting $ali from both sides of equation (30)

and rearranging terms yields ali = 1

$ [ v1+mv+v (1 # $(n # 1)%) # $

!al

jj

], or ali = al

j = al for any i and j.

Substituting al and % into equation (30), yields al = 1(n+1)$

v1+mv+v (1 # $(n # 1)%). That is, al = aq = a

for any l and q. Thus the symmetry assumption is without loss of generality. Since &, % are uniquelydetermined, the uniqueness of the linear equilibrium follows.

Proof of Corollary 3: Proofs of Part (i), (ii), and (iii) follow directly. For part (iv), the volumegenerated by sell-side agent j and by buy-side agent i are naE[|sj|] and %E[|si|], respectively. Because% > na, as shown in part (i), and si and sj are identically distributed, naE[|sj|] > %E[|si|].

Proof of Proposition 4: The proof consists of the following propositions.We first examine the buy-side agent’s demand for sell-side information. More specifically, given the set,

Si, of sell-side agents who o"er signals for sale, how buy-side agent i should determine the subset, Ai, ofsell-side agents from whom to buy signals.

Upon choosing Ai, buy-side agent i!s trading strategy is the solution to:

maxxi

E[xi(V + ! # (V + $y))|Fi], (31)

where Fi = {si, sj , )j $ Ai}. Buy-side agent i!s information now includes both his own, si, and that whichhe acquired, sj, j $ Ai. Trader i solves problem (31) with the belief that everybody else will play theequilibrium strategy: every other buy-side agent buys signals from all sell-side agents and trade accordingto the strategy specified in Proposition ??. We therefore have the following proposition.

Proposition 8 (i) Conditional on other buy-side agents’ equilibrium strategies, buy-side agent i’s optimaltrading profit, 'i(Ai), only depends on, l " l(Ai), the number of sell-side agents in Ai.

(ii) 'i(l) is strictly increasing and concave in l.

Proof : Conditional on all other buy-side agents buying from all sell-side agents and trading as specifiedin Proposition ??, buy-side agent i’s objective is

maxxi

xi(E[!|Fi] # &$(n # 1)E[sp|Fi] # %$(n # 1)E[!|Fi] # $xi) (32)

where & and % are as in Proposition ??. Also

E[sp|Fi] =1m

("

j%Ai

sj +"

k/%Ai

E[sk|Fi]) (33)

=l

msl +

m # l

mE[!|Fi]. (34)

Substituting (34) into the objective function and taking the first-order condition yields buy-side agent i’soptimal trading strategy

x"i (A

i) =(1 # %$(n # 1) # &$(n # 1)m#l

m )E[!|Fi] # &$(n # 1) lmsl

2$. (35)

19

The expected profit under the optimal trading strategy is

'i(Ai) =14$

E[[1 # %$(n # 1) # &$(n # 1)m # l

m]E[!|Fi] # &$(n # 1)

l

msl)2] (36)

Substituting E[!|Fi] = lvsl+vsi1+v+lv into equations (35) and (36), and simplifying by using E[(E[!|Fi])2] =

v+lv1+lv+v , E[(E[!|Fi])sl] = 1, and E[s2

l ] = 1 + 1lv yields

'i(Ai) = 'i(l) =14$

{(1 # $%(n # 1) # $&(n # 1)m # l

m)2

v + vl

1 + v + vl+ [&$(n # 1)

l

m]2

1 + vl

vl

#2(1 # $%(n # 1) # $&(n # 1)m # l

m)&$(n # 1)

l

m}, (37)

which is part (i).Part (ii) follows from:

d'i(l)dl

=16(1 + v + mv)2

(1 + n)2(2 + (1 + 2m + n)v)2(1 + v + lv)2> 0,

d2'i(l)dl2

= # 32v(1 + v + mv)2

(1 + n)2(2 + (1 + 2m + n)v)2(1 + v + lv)3< 0.

!The concavity of 'i(l) implies that the classical marginal analysis yields trader i!s optimal set of sell-side

agents from whom to acquire signals as summarized in the following proposition:

Proposition 9 Given the prices of the sell-side agent’ information, p(Si), buy-side agent i chooses to buyfrom the l" cheapest sellers, and l" is determined by

'i(l") # 'i(l" # 1) & pil! if l" > 0, and

'i(l" + 1) # 'i(l") ' pil!+1 if l" < l(Si), (38)

where pil! (pi

l!+1) denotes the l"th (l" + 1th) lowest price in p(Si).

Proof : Buy-side agent i’s problem is

'b(Si) " maxAi&Si

'i(l(Ai)) #"

j%Ai

pij. (39)

Recall that pij is sell-side agent j’s o"er price to fund manager i. The first step in solving (39) determines

the optimal Ai for a fixed l. Because buy-side agent i’s profit depends only on the number of sell-sideagents he buys from, not their identities, if buy-side agent i buys from l sellers, he buys from the sellers withthe lowest prices. In the second step we determine the optimal l. On the one hand, the marginal benefitof sell-side agent’s information is decreasing, as shown by Proposition 8. On the other hand, the marginalcost of information is increasing because as l increases, more expensive sell-side agents will be included inAi. Thus, (38) is a necessary and su!cient condition for the unique optimum for problem (39).!

Since an information seller makes take-it-or-leave-it o"ers to the sell-side agents, he extracts the entiremarginal surplus of his information from the buy-side agents. In equilibrium, the marginal benefit of aseller’s information to a sell-side agent is '(m) # '(m # 1) (symmetry enables dropping the subscript i).Therefore, a sell-side agent’s o"ering price is:

pij = p = '(m) # '(m # 1), )i, j. (40)

20

Proof: a sell-side agent will not deviate from this strategy because o"ering a lower price to any buy-sideagent would lead to its acceptance (by Proposition 9) and thus diminish the seller’s profit. Moreover,o"ering his information at a higher price to any sell-side agent only leads to the o"er being rejected (byProposition 9) and thus once again diminishes the sell-side agent’s profit.!

Because by Lemma 1, all the buy-side agent buy infomration from a sell-side agent in equilibrium. Aninformation seller’s equilibrium profit from selling information is 'r " np. Proposition 10 characterizesthe equilibrium in the market for information:

Proposition 10 In the market for sell-side agents’ information,(i) an information sellers makes o"ers to sell his information to all buy-side agent, and the o"er price

isp =

14$

[16v(1 + v + mv)

(1 + n)2(1 + mv)[2 + (1 + m + n)v]2]; (41)

(ii) all buy-side agent accept the o"er;(iii) equilibrium profits for a buy-side agent and a sell–side agent are

'b =14$

{4v[(n + 1)2 + [4m2 + (1 + n)2 + m(1 + n(6 + n))]v + m(1 + 2m + n)2v2](1 + n)2(1 + mv)[2 + (1 + m + n)v]2

} (42)

and's =

14$

[16nv(1 + v + mv)

(1 + n)2(1 + mv)[2 + (1 + m + n)v]2] + 'I(m,n), (43)

respectively.

Proof: Simplifying (40) using 37, yields part (i). Part (ii) is true by Proposition 9. Part (iii) followsby simplifying 'b = '(m) # mp and 's = np + 'I(m,n).!

Proposition 10 characterizes the equilibrium payo"s of a buy-side agent and a sell–side agent as functionsof m and n. Because n = N # m, 'b and 's are functions of m only.

Finally, we prove Proposition 4 by showing that (15) is impossible to satisfy for any m" > 0. Thatis, 's(m") < 'b(m" # 1) for all m" & 1. The following three equations are derived using Mathematica.Simplification yields

's(m) # 'b(m # 1) =

# 4v(1 + N # m)4

[4(m # 1)2

1 + (m # 1)v+

(N # m # 1)(1 + N # m)(5 + 2N + N2 # 2(3 + N)m + m2)(2 + (N + m # 1)v)2

+(N + m # 1)(5 + 2N + N2 # 2(3 + N)m + m2)

2 + (N + m # 1)v# 4(N # m # 1)(1 + N # m)(N # m)

(2 + (N + m # 1)v)2

+4(N # m)m

1 + mv# 4(1 + N + m)(N # m)

2 + (N + m # 1)v].

21

Thus, 's(m) # 'b(m # 1) and the term in square brackets have opposite signs. Combine the terms insquare brackets into a single fraction. Because the denominator is positive, we need only show that thenumerator is positive. The numerator is (1 + N # m)2 times:

B(N,m) " 4(1 + N # m)2 + 4[5 + N3 + N2(m # 1) # K(m + 1)(5m # 3) + m(m(7 + 3m) # 7)]v

+ [13 + 24N # 6N2 + N4 + 8(3 + N + N2 + N3)m # 2(45 + N(3N # 8))m2 # 8(2N # 5)m3

+ 13m4]v2 + [#8(N # 1)2N + 2(21 + N(4 + N(#6 + N(N + 4))))m + 4(1 + N(#7 + N(7 + N)))m2

# 8(10 + (#4 + N)N)m3 # 4(#7 + N)m4 + 6m5]v3 + [#(#1 + N)4

+ (#1 + N)(N + 1)(3 + (#12 + N)N)m + (23 + N(#8 + N(#12 + N(12 + N))))m2

+ 2(#3 + N(#14 + 11N))m3 # (23 + 2(#10 + N)N)m4 + 9m5 + m6]v4

+ (#1 + m)m(#1 + N + m)2(1 + N + m)2v5. (44)

We show that B > 0 in three steps: B is convex in N, B is increasing in N , and B is positive. First

+2B

+N2= 4{2 + 2(#1 + 3N + m)v + [#3 + 3N2 + 12Nm + (4 # 3m)m]v2

+2[4 + 3N2m + m(#3 + (7 # 2m)m) + 3N(#2 + m(2 + m))]v3

+[#3 + m # m2(6 + (#11 + m)m) + 6N(1 + 3(#1 + m)m) + 3N2(#1 + m + m2)]v4

+(#1 + m)m(#1 + 3(N + m)2)v5}. (45)

The coe!cients of v, v2, v3, and v5 are non-negative for m & 1 and N & m + 1. The coe!cient of v4 isincreasing in N. So, for a given m, the smallest possible value of the coe!cient of v4 can be achieved bythe smallest N, which is m + 1. Substituting N = m + 1 into the coe!cient, the coe!cient of v4 becomes

2m[#7 + m2(19 + m)], (46)

which is positive for m & 1. Thus, the coe!cient of v4 is positive and thus '2B'N2 > 0.

Because 'B'N is increasing in N for a given m, the smallest value for 'B

'N is achieved by the smallest valuefor N. Again, the smallest possible N is m + 1. At N = m + 1, 'B

'N is

+B

+N|N=m+1 = 16v(1 + v)(1 + v + mv)(1 + m(1 + (#1 + m)v)(1 + v + 2mv) > 0. (47)

Thus, 'B'N > 0.

Finally, because 'B'N > 0, the smallest B can be achieved by the smallest N. Substituting N = m + 1

into B yieldsB|N=m+1 = 16v(1 + mv)2(1 + v + mv)[2 + (#1 + m)m(1 + v + mv)], (48)

which is positive. Thus, for a given m & 1, B is positive. Hence, 's(m) # 'b(m # 1) is negative for1 ' m ' N # 1.

Finally, we establish that in equilibrium all agents choose to be traders by checking that m" = 0 satisfiesboth (14) and (15).

Proof of Proposition 5: Suppose, without loss of generality, in equilibrium agent 1 opts for thesell-side of the market and agent 2 remains on the buy-side. If agent 1 deviates so that both agent 1 and2 produce information from the buy-side of the market, agent i’s trading strategy is

maxxi

E[xi(V + ! # (V + $y))|Fi],

22

The optimal solution is xi = E[!|Fi]#$E[xj]2$ . Assuming xi = %isi for i = 1 and 2, we get

xi =12$

{E[!|si] # $%jE[sj|si]}

=12$

[v # $%j(v + ))]1

1 + vsi.

The second equation follows because E[!|si] = vsi1+v and E[sj |si] = v+(

1+v . The conditional expectation ofjoint normal distribution is given by the following well-known result from Anderson (2003).

Theorem 11 If####

X1

X2

#### ! N(µ,!), where µ =####

µ1

µ2

#### and ! =####

!11 !12

!21 !22

#### , then X1|X2 = a ! N(µ,!),

where µ = µ1 + !12!#122 (a # µ2), and ! = !11 # !12!#1

22 !21.

We thus have two equations, %i = 12$ [v # $%j(v + ))] 1

1+v , for i = 1 and 2. Solving them, we get%i = %j = v

$(2+3v+() . Each agent’s trading profit is

'b(0) =14$

E{[v # $%j(v + ))]1

1 + vsi}2

=14$

{[v # $%j(v + ))]1

1 + v}2(1 +

1v)

=14$

4v(1 + v)(2 + 3v + ))2

.

If agent 1 does not deviate, he sells his information to agent 2, who is the trader. The first-ordercondition for the buy-side agent’s trading strategy is

x2 =12$

E[!|F2]

=12$

(s1 + s2)v1 + 2v + )

.

His trading profit is '(1) = 14$ ( v

1+2v+( )2V ar[s1 + s2] = 14$

2v1+2v+( . However, if he does not buy information

from agent 1, his trading startegy is x2 = 12$E[!|s2] = 1

2$v

1+vs2. His trading profit is 'b(1) = 14$

v1+v . Since

the sell-side agent makes a take-it-or-leave-it o"er, the price for his information, which is also his profit, isthus

's(1) = p = '(1) # 'b(1)

=14$

(1 # ))v(1 + v)(1 + ) + 2v)

.

Therefore, agent 1 would not have incentive to deviate if 's(1) & 'b(0), which is equivalent to

(1 # ))v(1 + v)(1 + ) + 2v)

& 4v(1 + v)(2 + 3v + ))2

.

That is,(2 + 3v + ))2(1 # )) # 4(1 + v)2(1 + ) + 2v) & 0. (49)

23

The right-hand-side is a 3rd order polynomial of ), with only one real root, #v. Since the right-hand-side derivative with respect to ) is negative at ) = #v, we can conclude that if ) ' #v, (49) is true, andagent 1 has no incentive to deviate. As for (buy-side) agent 2, if he deviates while agent 1 does not, bothagents become earn zero profit from the sell-side of the market. So agent 2 will not deviate.

Proof of Proposition 6: For part i), suppose there is an equilibrium in which all N = n + 1 agentschoose to be traders. Suppose the agents’ trading strategies are linear, i.e., xi = %isi. For agent i, thefirst-order condition of his strategy is

xi =12$

{E[!|si] # $"

j $=i

%jE[sj |si]} (50)

We know from theorem 11, E[!|si] = 11+"+)si, and E[sj|si] = 1#"

1+"+)s1 if either i or j is 1. Here for theease of notation, " = 1

v , and , = 1v"

. Further, E[sj |si] = 1+"1+"+)si, for i, j (= 1. Therfore, for agent 1,

%1 =12$

(1

1 + " + ,# $

"

j $=1

%j1 # "

1 + " + ,) (51)

For agent i (= 1,

%i =12$

(1

1 + " + ,# $

"

j $=1,i

%j1 + "

1 + " + ,) # $%1

1 # "

1 + " + ,) (52)

From this equation, we can conclude that %i = %j = % for any i (= 1 and j (= 1 using the method as inthe proof of Proposition ??. We then solve for the unique % and %1 using equations (51) and (52). Becausethe solution is too complicated, we do not report it here. (We use Mathematica to derive the equationsand the progam is available upon request.) The trading profit for agent 1 is thus

'1b(0) = $E[x2

1]= $%2

1(1 + " + ,)

=vv%(v + v% + vv%)(v% + 2nv% + v(2 + v%))2

[(2 + n)v2% + 2vv%(3 + n + 2v% + 3nv%) + v2(2 + v%)(2 + (2 + n)v%)]2$

.

The first equation follows from by using (50) to calculate agent 1’s trading profit. The second equationfollows because E[s2

1] = 1 + " + ,. The third equality follows by substituting in %1.Now suppose agent 1 deviates to the sell-side. By the same logic as in Lemma 1, each of the n buy-

side agents would buy from the the sell-side agent in equilibrium. By the same steps as solving for thebenchmark equilibrium, that is, 1) we first solve for all buy-side agents’ unique linear strategy, which issymmetric; 2) then calculate a buy-side agent’s total trading profit, '(1); 3) then calculate his profit if hedoes not buy sell-side information, 'b(1); finally 4) the price for sell-side information is thus p = '(1)#'b(1),and the sell-side agent’s profit is '1

s(1) = np, we can solve for the unqiue equilibrium profit for sell-sideagent 1,

'1s(1) =

4nv%(1 + 2v%)(v + 2v%)(v + v% + nv%)2

(1 + n)2%(v + v% + vv%)[2v + (3 + n)(1 + v)v% + 4(1 + n)v2% ]2$

Agent 1 has incentive to deviate to the sell-side of the market if and only if

'1s(1) # '1

b(0) > 0. (53)

24

Butlimv'0

'1s(1) # '1

b(0) =8v%(n + 2nv%)

(3 + n + 4(1 + n)v%)2$> 0,

therefore, as v approaches 0, all agents opting for the buy-side of the market cannot be an equilibrium inwhich they all follow linear trading strategies and the out-of-equilibrium belief is passive. We thus haveproved part (i).

As for part (ii), we first check that if all agents but agent 1 become buy-side agents, none will deviateto the sell-side. Suppose without loss of generality that agent 2 deviates to the sell-side. That is inequilibrium there are n # 1 buy-side agents and 2 sell-side agents (1 and 2). We will show that agent 2’sprofit from selling information, '2

s(2) is no more than his equlibrium trading profit 'b(1). By the samelogic as in Lemma 1, in equilibrium all buy-side agents buy information from all sell-side agents. We firstcalculate each buy-side agent’s equilibrium strategy and trading profit, '(2). Second we calculate agent2’s profit from selling information '2

s(2). Notice that now, agent 1 and 2 have complementary signals, inthat the sum of the marginal contributions of their signals to a buy-side agent’s profit may be larger thantheir individual marginal contributions.27 As a result, the prices for their information may not be uniquelydetermined.28 However, we know that the price for an agent’s information should be no more than itsmarginal contribution to a buy-side agent’s trading profit. Otherwise he would not buy the information.That is, the price for agent 2’s information satisfies

p2 ' '(2) # '(analyst 1)

where '(agent 1) is a buy-side agent’s trading profit from buying from agent 1 only. Therefore, '2s(2) '

(n # 1)['(2) # '(agent1)]. Some tedious algebra yields that

limv"'(

'b(1) # '2s(2) & lim

v"'(['b(1) # (n # 1)['(2) # '(agent 1)]] =

v

(1 + n)2(1 + v)$> 0. (54)

That is, agent 2 would not have incentive to deviate to be a sell-side agent if v% is large. On the otherhand, if (53) holds, agent 1 would not deviate as shown in the proof of part (i). Some algebra yields that

limv'0

limv"'(

'1s(1) # '1

b(0) =n

(1 + n)2$> 0. (55)

(54) and (55) imply that there exists a v > 0 such that for any 0 < v < v, if v% > v%(v), where v%(v)may depends on v, both 'b(1) # '2

s(2) > 0 and '1s(1) # '1

b(0) > 0. In other words, if v is small and v% islarge, neither agent 1 nor agent 2 has incentive to deviate. We thus have proved part (ii).

The proof of Proposition ??: It is su!cient to show in two economies, denoted by 1 and 2, that if!r0 > !r1, then W0 > W1.

Because !r0 > !r1 and P 02 and P 1

2 are normally distributed, there exists a , " N(0,!r0 # !r1) suchthat P 0

2 and P 12 + , have the same distribution and , and P 1

2 are independent. Thus

W0 = E[P 02 q"(P 0

2 ) # f(q(P 02 ))] = E[(P 1

2 + ,)q"(P 12 + ,) # f(q"(P 1

2 + ,))]. (56)

But(P 1

2 + ,)q"(P 12 + ,) # f(q"(P 1

2 + ,)) & (P 12 + ,)q"(P 1

2 ) # f(q"(P 12 )), (57)

27That is, "(2) ! "(analyst 1) + "(2) ! "(analyst 2) > "(2) ! "(0).28More specifically, the prices for both signals can be any pair such that p1 + p2 = "(2)! "(0), p1 " "(2)! "(agent 2), and

p2 " "(2) ! "(agent 1).

25

because q"(P 12 + ,) uniquely maximizes (P 1

2 + ,)q # f(q). Furthermore, the inequality in (57) is strict for, (= 0. Taking expectation of both sides yields

E[(P 12 + ,)q"(P 1

2 + ,) # f(q"(P 12 + ,))] > E[(P 1

2 + ,)q"(P 12 ) # f(q"(P 1

2 ))]= E[E[(P 1

2 + ,)q"(P 12 ) # f(q"(P 1

2 ))|P 12 ]]

= E[P 12 q"(P 1

2 ) # f(q"(P 12 ))]. (58)

The last equality follows because E[,|P 12 ] = 0. Combining (56) and (58) yields W0 > W1.

Proof of Proposition 7: The firm’s production profit is

12F

E[P 22 ] =

12F

[V 2 + "2(P2)]

=1

2F(V 2 + !r)

The first equality follows because E[P2] = V. !r = V ar[$y] = $2E[y2]. The second equality follows fromthe fact that E[y] = E[

!nk=1 xk + z] = 0.

The market maker sets the price according to (4). But,

E[!|y] =1

n(*+#)y

1 + ( **+# )2 1

mv + n#2

(n(*+#))2v + "2z

(n(*+#))2

, (59)

which implies

$ =1

n(*+#)

1 + ( **+# )2 1

mv + n#2

(n(*+#))2v + "2z

(n(*+#))2

. (60)

Together with equations (27) and (28) in the proof of Proposition ?? , we get

a =2"zv%

nD(61)

% ="z(n + 1)v%

nD(62)

$ =%

nD

"z(n + 1)[2(1 + mv) + (n + 1)v], (63)

where D " 4mv + 4(mv)2 + 4nv2 + (n + 1)2(v + v2). Substituting a, %, and $ into !rand simplifying, weget

!r(m) =(m # N)v[m3v # m(N # 1)2v + m2(2 + (N # 3)v) # (1 + N)2(2 + v + Nv)]

(1 # m + N)2[2 + (1 + m + N)v]2

Notice that !r(m) does not depend on "z.From the proof of Proposition ??, we know that

'b(m) =14$

{4v[(n + 1)2 + [4m2 + (1 + n)2 + m(1 + n(6 + n))]v + m(1 + 2m + n)2v2](1 + n)2(1 + mv)(2 + (1 + m + n)v)2

} (64)

and's(m) =

14$

(16nv(1 + v + mv)

(1 + n)2(1 + mv)(2 + (1 + m + n)v)2) + 'I(m,n). (65)

26

Therefore, substituting (64) and (65) into (23) we get

'I(m) =14$

"(m)

where "(m) does not depend on "z. The investment banking fee is thus negligible if "z is small. When noisetraders are scarce there is little opportunity for either buy- or sell-side agents to extract trading profits.Trading profits are then negligible relative to the firm’s benefit from more informative pricing, which isindependent of "z. Thus the firm is willing to choose m"" if and only if

12F

[!r(m"") # !r(m)] & 14$

["(m"")m"" # "(m)m], ) m (= m"", and 1 ' m ' N # 1

4$2F

& maxm$=m!!

["(m"")m"" # "(m)m][!r(m"") # !r(m)]

;

1F"z

&max

m$=m!!

[!(m!!)m!!#!(m)m]["r(m!!)#"r(m)]

2)

n!!D(n!!+1)[2(1+m!!v)+(n!!+1)v]

.

(In case there are multiple m"", m (= m"" should be understood as m not equal to any one of those m"").Because the feasible set of m is finite, max

m$=m!!

[!(m!!)m!!#!(m)m]["r(m!!)#"r(m)] always exists. Thus we have proved the

proposition.

27

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