Strategic choices of quality, differentiation and pricing in financial services

Journal of Banking & Finance 25 (2001) 1447±1473 www.elsevier.com/locate/econbase

Strategic choices of quality, di€erentiation and pricing in ®nancial services q Sandeep Mahajan a, Richard J. Sweeney

b,*

a

b

The World Bank, Washington, DC 20433, USA The McDonough School of Business, Georgetown University, Room 323 Old North, Washington, DC 20057, USA Received 12 April 1999; accepted 2 May 2000

Abstract Two ®nancial services ®rms (FSFs) produce information on future returns from risky assets, incorporate this information in a report and sell the reports to investors. The FSFs make strategic choices of the quality, di€erentiation and prices of their reports. Their optimal strategic choices of quality and di€erentiation divide the market for the report into three segments. Each FSF has a monopoly in its own segment, and the two compete head-to-head in a duopoly segment. The sizes of the monopoly segments increase with increases in quality and di€erentiation. An increase in di€erentiation reduces the size of the duopoly market, while an increase in quality has an ambiguous e€ect on the duopoly market: for high enough equilibrium di€erentiation, the size of the duopoly market decreases with an increase in quality. The non-cooperative FSFs pursue niche strategies, each focusing on a di€erent related set of risks, in order to coordinate the contents of their reports to achieve the equilibrium degree of product di€erentiation. Recent years have seen a worldwide wave of ®nancial-®rm mergers and acquisitions. This paperÕs model suggests that in equilibrium FSFs di€erentiate their products and develop pro®table niches; in principle, however, suciently strong economies of scope could lead to a small number of FSFs with little di€erentiation that dominate all ®nancial services markets. Ó 2001 Elsevier Science B.V. All rights reserved.

q The results and interpretations of this paper are authorsÕ alone, and should not be attributed to the World Bank, its Board of Governors, its management, or any of its members. * Corresponding author. Tel.: +1-202-687-3742; fax: +1-202-687-4031. E-mail address: [email protected] (R.J. Sweeney).

0378-4266/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 2 6 6 ( 0 0 ) 0 0 1 4 4 - 8

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JEL classi®cation: G20; D83; D43 Keywords: Financial services ®rms; Quality and di€erentiation; Strategic decisions; Niche strategies; Segmented markets

1. Introduction In ®rmsÕ strategic interactions, key choices are product quality and product di€erentiation. Quality and di€erentiation choices a€ect competition in the industry, and thus each ®rmÕs pricing±output strategy. Standard models of strategic competition frequently analyze ®rms' pricing±output decisions, but not their choices of quality and di€erentiation. Understanding the ®nancial services industry requires analysis of strategic interactions over quality and di€erentiation. Past literature does not consider strategic choices of product quality and di€erentiation by ®nancial services ®rms (FSFs). This paper helps ®ll this gap with a duopoly model that endogenizes FSFsÕ choices of quality, di€erentiation and pricing±output strategies. The paper focuses on FSFs that produce and sell information to investors on future returns from risky assets. Past literature on ®rms that sell ®nancial information assumes these ®rms are endowed with a certain quantity of information. In this paperÕs duopoly model, both ®rms face increasing costs in producing ®nancial information and di€erentiating their products. The paper assumes that risky assets in the economy are subject to a large number of risks. The FSFs inform customers in advance on the values of the realizations of a relatively small number of these risks. Each ®rm packages its information in a report that it sells, possibly with some price discrimination, to investors who cannot resell the report. The e€ect of the reportsÕ information on market prices is assumed small relative to the noise in the market; thus, the informativeness of prices can be ignored in formulating investor demands for the reports. For given prices of the reports, the investors maximize expected utility: the quantities of the reports they demand depend on the reportsÕ quality and di€erentiation. The results are striking. The FSFs endogenously decide to produce di€erentiated products. Each FSF has a monopoly niche market where it does not compete directly with the other. In its niche, it sells information to investors who buy information from both ®rms. The ®rms may compete head-on in a third market segment, where investors buy information services from one ®rm but not both. The monopoly markets are high pro®t; the duopoly market is low pro®t. The market segments exist because of heterogeneity of investorsÕ attitudes toward risk. At the given price of a particular FSF's report, the reportÕs bene®ts have a greater dollar value the more funds the investor expects to have at

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risk. Investors fall into three categories based on their risk aversion. Those with high enough risk aversion expect to invest a small dollar amount in risky assets and do not buy a report from either FSF: the dollar price of the report is too high relative to the dollar value of the bene®ts that the report is expected to provide. At the other extreme, investors with low risk aversion expect to put a large dollar amount at risk: because the two FSFs' reports are di€erentiated, with information on somewhat di€erent risks, these investors buy both reports. Other investors with intermediate risk aversion may buy a report from one but not both FSFs: these investors expect to have a large enough dollar amount at risk to justify buying one report, but view the second reportÕs extra information as not cost-e€ective. FSFi Õs report contains qi discrete pieces of information. Ex ante, each piece of information in a given report has the same value as any other in terms of revealing information about coming rates of return and risk. Further, the same information from one FSF is as good as from the other. The quality of the report from FSFi is then naturally measured by qi . As investors will pay a higher price for a report with greater quality, and each ®rm produces more information at increasing cost, FSFi endogenously chooses the quality of its report. In equilibrium, the ®rms always choose to di€erentiate their products: this product di€erentiation is necessary to give each FSF its high-pro®t monopoly niche. Di€erentiation arises endogenously even if ®rms face the same demand and cost conditions and have no competitive advantage in producing and marketing particular types of information: under identical demand and cost conditions, ®rms would identify the same number of risks in their reports, but the set of risks each ®rm identi®es would have an incomplete overlap. In order to coordinate the amount of overlap in their reports and thus achieve optimal di€erentiation, each non-cooperative ®rm announces an explicit niche, a related set of risks on which it focuses. These niches are signals to investors that promised di€erentiation will be achieved, and are signals to the other FSF on which set of risks to avoid. Each FSF respects the otherÕs niche, so that both can achieve the di€erentiation they promise buyers. Any competitive advantage the ®rms have in producing di€erent pieces of information reinforces the decision to di€erentiate by reducing di€erentiationÕs marginal cost. Competitive advantages also guide the directions in which the ®rms differentiate, thus reinforcing each ®rmÕs declared niche and allowing the FSFs to coordinate to achieve optimal di€erentiation. Section 2 ties this paper both to the literature on sales of information by FSFs, and also to the general literature on product di€erentiation. Section 3 describes the economy where investors and FSFs interact. Sections 4 and 5 derive the paperÕs results. The FSFs optimally choose product quality, di€erentiation and pricing±output strategies. Equilibrium choices of product quality and di€erentiation are shown as explicit functions of model parameters. The FSFs' choices respond (mostly) unambiguously to parameter changes, such as

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the number of assets in the economy, investor heterogeneity, the marginal cost of improving the quality of ®nancial products, and the marginal cost of increasing product di€erentiation. Section 5 also includes a discussion of how the non-cooperative FSFs can adopt niches to coordinate their behavior and achieve the di€erentiation they promise the customers. Section 6 summarizes results and o€ers some conclusions. Niche strategies are to be expected and may well be successful. The main threat to a niche strategy is economies of scope: niche players might be overwhelmed if large FSFs enjoy economies of scope that give them lower costs for most products in most circumstances. The evidence on such economies is mixed. 2. Relationship to the literature A small literature discusses decision making by ®rms that sell ®nancial information to investors: the seller is often a monopolist and does not produce the information, and usually faces no strategic issue of product di€erentiation. A much larger and older literature considers non-®nancial ®rmsÕ strategic decisions on product di€erentiation. Admati and P¯eiderer (1986, 1990) study a monopolist selling ®nancial information to investors. Their monopolist ®rm is endowed with a given quantity of information. Admati and P¯eiderer (1986) endogenize the equilibrium quality and quantity of information the ®rm sells from its endowment. They show that the ®rm might sell a noisy version of its endowed information, to reduce the dilution of the informationÕs value from its incorporation in informative asset prices. Admati and P¯eiderer (1990) compare the monopolistÕs choice of selling the endowed information directly to investors as opposed to selling it ``indirectly'' through the creation of a fund that makes portfolio choices as a function of the endowed information. They show that indirect sale allows the ®rm greater control over investorsÕ reactions to the information, but may not allow the ®rm to extract as much surplus. The ®rmÕs optimal choice depends on the informativeness of equilibrium asset prices. Allen (1990) studies the credibility problem of a monopolist who sells information and ®nds that the problem need not rule out the existence of markets for ®nancial information. He has the ®rm announce a pricing rule and a portfolio rule that describe the price of information and the ®rmÕs portfolio allocation across assets corresponding to each possible set of information with which the ®rm may be endowed. He shows that in these circumstances, there exists a set of portfolios and information prices that ensure that the ®rm correctly reveals its information to buyers. Fishman and Hagerty (1995) consider price competition between two ®rms selling ®nancial information from endowment. A ®rm has an incentive to sell

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information rather than only trade on it. They show that selling information allows the ®rm credibly to commit to trade more aggressively, inducing other informed traders to trade less aggressively, and thus allowing the information seller to capture a larger share of trading pro®ts. This paperÕs analysis of competition has similarities to the literatures on spatial or horizontal di€erentiation, beginning with Hotelling (1929), and vertical di€erentiation or di€erentiation by quality, beginning with Shaked and Sutton (1982). In these literatures, ®rms soften competition by di€erentiating themselves horizontally or vertically, and then price their products. The model developed here di€ers in important ways from these earlier models of di€erentiation. First, though the location decision in earlier di€erentiation models is typically onedimensional, this paperÕs model has two-dimensional optimization±choices of quality and di€erentiation. Second, the demand functions for the ®rmsÕ products, typically exogenous in spatial di€erentiation models, are derived endogenously here. Third, in the current analysis, at a given price, the exact same output produced by di€erent ®rms would give the same utility to consumers. In earlier models of product di€erentiation, at a given price, otherwise identical products of di€erent ®rms give di€erent levels of utilities to consumers. Fourth, earlier models of di€erentiation restrict consumers to buying from one or none of the ®rms. Here, investors may buy from one, both or neither ®rm. In earlier models, sellers bene®t from product di€erentiation because it reduces price competition among them. Here, ®rms bene®t because di€erentiation expands the two high-pro®t monopoly markets, largely at the expense of shrinking the low-pro®t duopoly market where they compete directly. 3. The economy This section presents the risky consumption/investment environment where consumers maximize expected utility and FSFs make strategic decisions on quality, di€erentiation and pricing±output. Sections 4 and 5 derive the paperÕs results. Assume an economy with a continuum of population, where agents are heterogeneous in their risk aversion. Each consumerÕs index is a point in the continuity between 0 and K. Each agent's coecient of absolute risk aversion corresponds to a point in the continuous uniform interval (a0 ; . . . ; aK ); or each point in the interval 0; . . . ; K has a one-to-one mapping with one point in the interval a0 ; . . . ; aK . Each agent is born with an initial wealth endowment Wk ; lives for two days (days 1 and 2), and derives utility only from day-2 consumption, C. There are N risky assets and one risk-free asset. Investors make portfolio decisions on day 1 and consume on day 2. The risk-free asset gives the rate of return r from day 1 to day 2. The value of r is known to all on day 1 before information on risky-asset returns is produced

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and sold. Without loss of generality r ˆ 0. The rate of return on the risky asset j, unknown until day 2, is represented by a random variable, Rj whose realization on the morning of day 2 is Rj . (The superscript * emphasizes that a symbol represents a random variable.) Rj is generated as Rj ˆ

n X iˆ1

ei;j ;

j ˆ 1; . . . ; N :

…1†

The realized value of the risk-variable ei;j on day 2 is denoted ei;j . n is the total number of ei;j s per asset, equal for all assets for convenience. Prior to buying information from the FSFs, investors have only their common beliefs about the distributional properties of the ei;j s in Eq. (1). Each believes the ei;j s are multinormal with (a) E…ei;j † ˆ l; i ˆ 1; . . . ; n; j ˆ 1; . . . ; N ; (b) cov…ei;j ; eg;h † ˆ 0, unless i ˆ g and j ˆ h, (c) var…ei;j † ˆ r2 ; i ˆ 1; . . . ; n; j ˆ 1; . . . ; N : This simple structure is tractable and aids in presenting results; it could be substantially generalized without changing qualitative results. The two FSFs produce and sell information on day-2 rates of return from risky assets. The ®nancial services sector exists in this model because information production is costly, and because the FSFs have a comparative advantage over individual investors in producing information on future returns from risky assets. FSF1 produces q1 units of information, with a cost function speci®ed below, incorporates this information in a report, and sells the report to investors; and similarly for FSF2 . Each unit of information in the report perfectly identi®es one realization ei;j . Thus, ql measures the quality of FSFl Õs report ± the higher ql , the higher the quality. Assume that the FSFs truthfully reveal their information. 1 In the morning of day 1, each agent faces the choice of buying a report from each FSF, or not. She buys a maximum of one report from each FSF, the decision depending on parameters speci®ed below. In the evening of day 1, each investor allocates her portfolio using the information she then has; her information depends on the number of reports she bought in the morning, the quality of the reports and their di€erentiation. The results of the investor's investment are realized on day 2, after which all investors consume. For tractability, the analysis assumes that asset prices are not informative enough to a€ect investorsÕ expectations about distributions of rates of return. This assumption is the limit of the case where the percentage of investors who buy the report is small and the e€ects of their portfolio decisions on asset prices 1

Allen (1990) discusses in detail the incentive problem between a monopolist information seller and the buyers of information. In the present model, incentive problems would likely be resolved in a repeated game setting, where FSFs would have incentives to build reputations for honesty.

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are negligible relative to noise in those prices. (See, for example, Grossman and Stiglitz, 1980; Hellwig, 1980; Diamond and Verrecchia, 1981; Admati and P¯eiderer, 1986; Mahajan, 1996). Admati and P¯eiderer (1986) study the case where a monopolist FSF takes account of the e€ect of its information sales on asset prices and thus on the demand it faces. A two-®rm ®nancial services sector. There are two ®nancial-services ®rms; results are easily generalized to a ®nancial services sector with more FSFs. The two FSFs are pre-existing; incumbent±entrant issues are not examined. For convenience, government regulations prevent entry of other ®rms. Each FSF produces one ®nancial product, its report. The paper neglects issues of the FSFsÕ economies of scope, to preserve tractability. Each FSF may have a competitive advantage in producing and marketing particular types of information, or may have natural niches; as shown below, competitive advantage is not necessary to ensure that reports are di€erentiated in equilibrium. The FSFsÕ decisions can be thought of in two steps. Subject to the demand function for their products, derived in Section 4, the two FSFs ®rst choose the quality and di€erentiation of their products. They then decide on their pricing± output strategies in light of the competition they face from each other. It is costly for a given FSF to increase the quality of its product by ®nding information on a larger number of ei;j s, and to di€erentiate its product by ®nding information on di€erent ei;j s. Di€erentiation depends inversely on the number of common ei;j s identi®ed in both reports. The increasing marginal cost of product di€erentiation depends inversely on the strength of the FSF's competitive advantage, if any. In deciding how much to di€erentiate its product, each FSF weighs the marginal cost of increasing di€erentiation against the marginal bene®ts of expanding its market niche. 4. Demand for the FSFs' products This section discusses the investorÕs choice problem, shows how the problem is solved recursively, and derives some results for the competitive structure of the ®nancial services industry. The following section then discusses the strategic choice problem facing each ®rm, and characterizes equilibrium in the industry, including how niche strategies coordinate the risks identi®ed to minimize costs and give optimal product di€erentiation. Due to the structure of risk in (a)±(c) above, the value of the reports to the consumer can be summarized by the amount of information in each report, q1 and q2 , and by the di€erentiation across the reports, or how much q1 overlaps with q2 . This overlap is measured below by the variable h, the heterogeneity of the reports. This simple structure allows analysis of the consumerÕs choice problem, at a given set of prices for the reports, in terms of the variables q1 , q2 and h.

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Investors' optimization and the market demand for the reports. The kth investor, k 2 …0; . . . ; K†, has the utility function Vk …Wk †, where Wk is investor k's day-2 wealth. The Vk function is strictly increasing, strictly concave, and everywhere twice di€erentiable ± Vk0 > 0 and Vk00 < 0. For analytical tractability, assume that Vk is the negative exponential utility function Vk …Wk † ˆ

exp … ak Wk †;

Wk ˆ Ck ; k 2 …0; . . . ; K†;

…2†

where ak is investor k's absolute risk aversion and Ck her day-2 consumption. Because Vk is strictly increasing, the investor consumes all of her day-2 wealth, Wk ˆ Ck . Each investor maximizes the expected value of her utility function, Eq. (2), in period 1. Consider the investorÕs maximization problem (the subscript k is suppressed). The investor is endowed with Xj units of risky asset j, j ˆ 1; . . . ; N , and B units of the riskless asset. Pj is the current price of asset j; use the riskless asset asP the numeraire and set its price to unity. The investorÕs initial wealth, W ,  The investor faces the choice of buying a report from equals Njˆ1 Pj Xj ‡ B. one, both or none of the FSFs at price PRi per report for i ˆ 1; 2. The investor's budget constraint is N X

Pj Xj ‡ B ‡ PR1 Q1 ‡ PR2 Q2 ˆ

jˆ1

N X Pj Xj ‡ B ˆ W ;

Qi ˆ ‰0; 1Š; i ˆ 1; 2;

jˆ1

…3† where Qi is the number of reports, one or none, bought from FSFi at PRi per unit, in the morning of day 1, with i ˆ 1; 2. Xj is the number of units of the jth risky asset the investor holds in the evening of day 1. Given the portfolio …X1 ; . . . ; XN ; B† the investor holds at the end of day 1, her day-2 wealth will be W ˆ

N X jˆ1

…1 ‡ Rj †Pj Xj ‡ B;

…4†

where Rj is the random day-2 rate of return from risky asset j. Combining Eqs. (3) and (4) gives W ˆ W ‡

N X Rj Pj Xj

PR1 Q1

PR2 Q2 :

…5†

jˆ1

Subject to Eq. (5), the investor maximizes the expected value of Eq. (2) with respect to the Qi in the morning of day 1 and with respect to the Xj in the evening. The investorÕs choice of Qi in the morning a€ects her optimal choice of Xj in the evening. The problem is solved recursively, ®rst for the Xj conditional on the information the investor has in the evening, and then for Qi using the computed functional values of the Xj .

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Making use of Eq. (5), from Eq. (2) the investor's objective function in the evening is ( " !#) N X Max  EEv Rj Pj Xj ‡ W PR1 Q1 PR2 Q2 exp a ; Xj jˆ1 Qi ˆ ‰0; 1Š; i ˆ 1; 2;

…6†

by choice of Xj , where EEv is the expectations operator conditioned on the information set IEv available in the evening. IEv , therefore, depends on the choice of Qi in the morning of day 1. For a normally distributed random variable e, it is known that Eee ˆ exp ‰Ee

var…e†=2Š:

Using this rule for the random variables Ri;j s in Eq. (6), or the ei;j s that might be substituted in, rewrite Eq. (6) as maximizing Eq. (7), with respect to Xj " ! N X Max 0  Xj ……Rj jIEv † Pj † exp a W PR1 Q1 PR2 Q2 ‡ Xj jˆ1 #! N a2 X X 2 Var…Rj † ; Qi ˆ ‰0; 1Š; i ˆ 1; 2: …7† ‡ 2 jˆ1 j From Eq. (7), the investor's optimal choice of Xj is Xj ˆ E…Rj jIEv †=‰aPj Var…Rj jIEv †Š;

j ˆ 1; . . . ; N ;

…8†

the familiar result for models with negative exponential utility functions (for example, Grossman and Stiglitz, 1980). The investor's demand for the risky asset j depends directly on the assetÕs expected rate of return, inversely on both the assetÕs price and the conditional variance of its returns, and is independent of the investorÕs endowed wealth. To choose which report(s) to buy in the morning of day 1, the investor inserts the functional values of Xj , given in Eqs. (7) and (8). Taking expectations based on information available in the morning of day 1, the investorÕs problem is ( "   Max EM exp a W PR1 Q1 PR2 Q2 Qi #) N X 2   …E…Rj jIEv †† =…2a Var…Rj jIEv †† ; Qi ˆ ‰0; 1Š; i ˆ 1; 2; …9† ‡ jˆ1

where the subscript M on the expectations operator stands for morning. The investor buys a report from one, both or neither of the FSFs, depending on which of the three choices gives her the highest expected utility. If the re-

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portsÕ prices …PR1 ; PR2 † are small enough, and the range of absolute risk aversion coecients, aK , has a large enough maximum (aK ) and small enough minimum (a0 ), then three classes of investors exist: (1) investors with high enough risk aversion buy neither report; (2) investors with low enough risk aversion buy both reports; (3) investors with intermediate risk aversion may buy one but not both reports. This is formalized in Theorems 1 and 2 in Appendix A. The following assumes the range aK to a0 is suciently large. To sharpen the analysis, assume the two FSFs are identical in the following two ways. First, it costs each the same amount to identify a given number of ei;j s, i.e., both face identical increasing cost functions for increasing the quality of the report. The cost function is c1 ˆ c…q1 †; c0 > 0, where the cheapest risks to identify are chosen ®rst. Second, both face identical increasing costs for increasing the di€erentiation of the report, in the sense of holding constant the number of ei;j s identi®ed, but identifying a set of them that has less overlap with the other ®rmÕs set. The cost function for di€erentiation is Gi ˆ G…hje1 ; e2 †, where ei is the set of the ei;j s chosen by ®rm i. Now, let qdi and hdi be the equilibrium quality and heterogeneity of the report o€ered by ®rm i fi ˆ 1; 2g in equilibrium, where the superscript d is for the duopoly equilibrium. Then, by symmetry of their optimization problem for product quality and di€erentiation, qd1 ˆ qd2 ˆ qd ; hd1 ˆ hd2 ˆ hd . 2 Because the two FSFs o€er reports with the same quality and di€erentiation, the prices of d d reports in the duopoly market are equal, PR1 ˆ PR2 ˆ PRd , and the prices of m m reports in the monopoly market are equal, PR1 ˆ PR2 ˆ PRm . But prices may di€er between the monopoly and duopoly markets because of price discrimination; that is, PRm and PRd may di€er. In equilibrium, the FSFs choose the same values of product quality, qd , that is, identify the same number of ei;j s in their reports, but do not necessarily identify the same ei;j s in their reports, which allows for di€erentiation. Let hd measure the number of ei;j s that are common to the two FSFsÕ reports. Scale hd such that for hd ˆ 1 no ei;j is identi®ed in the reports of both FSFs; for hd ˆ 0:5, all ei;j s identi®ed in the report of one FSF are identi®ed in the report of the other. With these assumed demand and cost conditions, each FSF has a monopoly segment but also competes head-to-head in a duopoly segment. Theorems 1 and 2 in Appendix A show that, when the ratio 2hqd /Nn is small enough, as required for prices to be uninformative, there exist absolute risk aversion coecients abuy and abuy2 , (i) abuy ˆ …N =2PRd † ln ‰…Nn†=…Nn qd †Š; (ii) abuy2 ˆ …N =…2…2PRm PRd †† ln‰…Nn qd †=…Nn 2hd qd †Š.

2

Tedious but straightforward complications arise when q1 6ˆ q2 and the report prices di€er to re¯ect this quality di€erence. Theorems 1 and 2 in Appendix A provide examples of the complications.

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Proposition 1. All investors with risk aversion less than or equal to abuy prefer to buy 1 over 0 reports. All investors with risk aversion less than or equal to abuy2 prefer to buy 2 reports over 1 report. Investors with risk aversion less than or equal to abuy2 buy both reports, investors with risk aversion higher than abuy2 but less than or equal to abuy , buy 1 and only 1 report, and investors with risk aversion higher than abuy buy 0 reports. This proposition follows from Theorems 1 and 2 in Appendix A. The demand conditions FSFs face in their monopoly markets and the duopoly market are: Proposition 2. In two segments, each firm has monopoly power over sales to investors whose risk aversion is less than or equal to abuy2 . The demand function faced by each firm in its monopoly segment is D ˆ K…abuy2 a0 †=…aK a0 †. In the third segment, the firms are direct competitors; the demand function is D ˆ K…abuy abuy2 †=…aK a0 †. Firms' equilibrium choices of quality and differentiation determine the sizes of the three markets. An increase in equilibrium differentiation increases the sizes of the two monopoly markets equally, and simultaneously decreases the size of the duopoly market. The decrease in the size of the duopoly market equals the aggregate increase in the sizes of the monopoly markets. An increase in equilibrium quality increases the size of each firm's monopoly market equally, but has an ambiguous effect on the size of the duopoly market. This proposition follows from de®nitions of abuy and abuy2 in (i) and (ii), and the market-segment demand functions. 5. Equilibrium in the ®nancial services sector For the demanderÕs decisions, analysis in terms of q and h is sucient. From the modelÕs simple structure, it does not matter which risks the ®rm includes in its set of q risks. The FSFs, however, choose the q risks to minimize costs. Further, they must act to ensure that their non-cooperative choices of risks in their qÕs achieve the desired degree of heterogeneity. This section ®rst analyzes the FSFsÕ choice of q and h. It then analyzes how the FSFs adopt niche strategies to guide their choices of q1 and q2 , both to control costs and to achieve their optimal heterogeneity, in other words, how choices of qi map into the ei;j s ®rms identify. 5.1. Optimal choices of quality and di€erentiation In strategically choosing quality and di€erentiation, the FSFs face two steps:

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Step 1. FSFs simultaneously choose the optimal quality and di€erentiation of their reports, qd and hd . As Section 4 shows, these choices give three market segments. In two market segments, each FSF has monopoly power. In a third segment, they directly compete as duopolists. Step 2. The FSFs simultaneously choose the optimal price, PRd , of their reports in the duopoly market, and also choose PRm , the optimal price each sets in its monopoly market. Duopoly-market interaction is assumed to be Bertrand. From the demand functions found in Section 4, choice of prices implies quantities sold. Step 2 optimization problem is solved ®rst to ®nd the FSFs' optimal pricingoutput choices in the monopoly and duopoly markets as functions of product quality and di€erentiation (Step 1 choice variables) and model parameters. These optimal functional values are used to formulate the reduced-form optimization problem in Step 1, which is then solved to give the FSFs' optimal product quality and di€erentiation as functions of model parameters. These optimal values of quality and di€erentiation are then used in Step 2 optimal functional solutions to give optimal monopoly and duopoly prices/quantities as functions of model parameters. 5.1.1. Pricing decisions: Conditional on quality and di€erentiation The monopoly market. Assume the two FSFs successfully price discriminate between the duopoly and monopoly markets. Let the FSFs face the same constant marginal cost, C, of producing, distributing, and in general supporting the report. As Section 4 shows, in its monopoly market, ®rm 1 faces the demand function DR1 ˆ K…abuy2 a0 †=…aK a0 †. Its monopoly-market pro®t function, using this demand function and the expression for abuy2 in (ii), is   3 2 Nn qd p1 K…PR1 C† 4 N ln Nn 2hd qd …10† ˆ a0 5 : …aK a0 † Max PR1 2…2PR1 PRd † m Optimizing Eq. (10) with respect to PR1 gives ®rm 1Õs optimal price, PR1 , and m quantity, Q1 ,  

 1=2 1 m …CN =2a0 † ln Nn qd PR1 ˆ ‡ C=2; …11a† Nn 2hd qd 2 h i 1=2 2 Dm qd †=…Nn 2hd qd †Šg a0 K =…aK a0 †; R1 ˆ f…K Na0 =2C† ln‰…Nn

…11b† where the superscript m is for equilibrium choice in the monopoly market. Because the two FSFs face identical cost and demand functions, in equilibrium m m m m ˆ PR2 ˆ PRm > 0, and Dm PR1 R1 ˆ DR2 ˆ DR > 0. The total number of reports m sold in the two monopoly markets is 2DR . From Eq. (11b),

S. Mahajan, R.J. Sweeney / Journal of Banking & Finance 25 (2001) 1447±1473 d oDm R =oq > 0;

1459

d oDm R =oh > 0;

an increase in either quality or di€erentiation causes an increase in the monopoly-market quantity demanded. Using the demand function each ®rm faces, the equilibrium functional value of the monopoly-market price is PRm ˆ

NK ln …… Nn 4‰…aK

qd †=… Nn 2hd qd †† C ‡ : 2 a0 †Dm ‡ a0 K Š

…12†

The duopoly market. Assume duopoly-market interaction is Bertrand ± the FSFs simultaneously choose duopoly-market prices. Total duopoly-market demand is Dduo abuy2 †=…aK a0 †. Let each ®rm face half the duoR ˆ K…abuy poly-market demand (this assumption is not crucial). Using de®nitions (i) and (ii) and the duopoly-market demand function, ®rm-1Õs objective function is   3 2  Nn qd Nn ln ln d d d p1 …PR1 C†NK 4 Nn q Nn 2h q 5; …13† ˆ 4…aK a0 † PR1 2PRm PR1 Max PR1 where C is the constant marginal cost of physically producing, distributing, and in general supporting its reports. Since the two FSFs simultaneously choose their report prices, a Bertrand± Nash equilibrium results. The only equilibrium report price for both FSFs is d d PR1 ˆ PR2 ˆ PRd ˆ C: price equals the common, constant marginal cost (Bertrand, 1883), PRm > PRd ˆ C. This condition implies each ®rm makes zero duopoly-market pro®t in equilibrium. In equilibrium, each FSF sells DdR units of its report in the duopoly market. From the duopoly-market demand function and de®nitions (i) and (ii),   3 2  d ln NnNnqd ln NnNn2hqd qd NK 4 5: …14† Dd ˆ 4…aK a0 † C 2P m C Total duopoly-market sales are 2DdR . The FSFs' optimization in the monopoly and duopoly markets therefore gives: Proposition 3. (1) Equilibrium sales in each monopoly market increase in quality or differentiation. (2) An increase in differentiation decreases equilibrium sales in the duopoly market, and the aggregate decrease in duopoly-market sales equals the aggregate increase in monopoly-market sales by both firms. (3) An increase in quality has an ambiguous effect on duopoly-market sales. These results are clear from Eqs. (11a), (11b) and (14).

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5.1.2. Choices of optimal quality and di€erentiation In Step 1, both FSFs simultaneously choose optimal product quality and di€erentiation, with the knowledge of how choices of qd and hd a€ect monopoly-market price and output in Step 2, from Eqs. (11b) and (12). Each also knows that quality and di€erentiation do not a€ect equilibrium duopolymarket pro®ts, because these are always zero. In Step 1, then, ®rm 1 optimizes the function P1 1 ‰ NK ln …… Nn q2 †=… Nn q1 h1 q2 h2 †† C ŠDm R …q1 ; h1 ; q2 ; h2 † ˆ 2‰…aK a0 †Dm Maxq1 ; h1 2 …q ; h ; q ; h † ‡ a KŠ 1 1 2 2 0 R c…q1 †

G…h1 †;

…15†

subject to Dm R1 P 0, where G…h1 † is ®rm 1's cost of increasing di€erentiation, and c…q1 † its cost of improving the quality of the report; G0 …h1 † > 0; c0 …q1 † > 0; c00 …q1 † > 0. The function G re¯ects the strength of ®rm 1Õs competitive advantage; the stronger its competitive advantage, the cheaper to di€erentiate its product. The ®rst of the three terms in Eq. (15) is ®rm 1's monopoly-market pro®t m function in Eq. (10), the product of PR1 in Eq. (12) and quantity in Eq. (11b). Call this ®rst term in Eq. (15) the bene®t function. Eq. (15) is optimized simultaneously, subject to constraint Dm R1 P 0, to determine the equilibrium values of h1 and q1 . To ®nd the Kuhn±Tucker (K±T) conditions for this problem, combine constraint Dm R1 P 0 with Eq. (15) to form the Lagrangian function Z ˆ p1 ‡ k1 Dm R …q1 ; h1 ; q2 ; h2 †; where k1 is a Lagrange multiplier. Assuming that the p1 function is concave and continuous, the K±T conditions are oZ op1 oDm …q1 ; h1 ; q2 ; h2 † ˆ ‡ k1 R ˆ 0; oq1 oq1 oq1

…16†

oZ op1 oDm …q1 ; h1 ; q2 ; h2 † ˆ ‡ k1 R ˆ 0; oh1 oh1 oh1

…17†

oZ ˆ Dm R …q1 ; h1 ; q2 ; h2 † P 0; ok1

k1 P 0; k1

oZ ˆ 0: ok1

…18†

The K±T conditions in Eqs. (16)±(18) are solved simultaneously to ®nd hd and qd , the equilibrium choices of ®rm 1, and k1 . From Eqs. (16)±(18), Proposition 4. The extreme case of zero differentiation between the products of the two FSFs is not a feasible equilibrium.

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Zero di€erentiation implies h ˆ 05, from the de®nition of h. From the monopoly-market demand function in Eq. (11b), h ˆ 0:5 implies Dm R < 0, which is ruled out by Eq. (18). Suppose further that equilibrium monopoly-market sales are positive. Then Dm R1 > 0, and from Eq. (18), k1 ˆ 0. Using this in Eqs. (16) and (17), in equilibrium m oP1 oP1 opm 1 oDR ˆ ‡ oq1 oq1 oDR1 oq1

c0 …qd † ˆ 0;

…19†

m oP1 oP1 opm 1 oDR ˆ ‡ oh1 oh1 oDR1 oh1

G0 …hd † ˆ 0;

…20†

where pm 1 is ®rm 1Õs bene®t function, its pro®t function in the monopoly market (the ®rst term in Eq. (15)). From the envelope theorem, opm 1 =oDR1 ˆ 0; because ®rm 1 maximizes with respect to DR1 (equivalently PR1 ) in Step 2. Therefore, Eqs. (19) and (20) can be rewritten as oP1 opm ˆ 1 oq1 oq1 ˆ

4…Nn

oP1 opm ˆ 1 oh1 oh1 ˆ

4…Nn

c0 …qd † NKDmR …qd ; hd †hd d d 2qd hd †‰…aK a0 †Dm R …q ; h † ‡ a0 KŠ

c0 …qd † ˆ 0;

…21†

G0 …hd † ˆ 0:

…22†

c0 …qd † d d d NKDm R …q ; h †q d d 2qd hd †‰…aK a0 †Dm R …q ; h † ‡ a0 KŠ

By symmetry of the problem for the two FSFs, in equilibrium q1 ˆ q2 ˆ qd and h1 ˆ h2 ˆ hd : these equalities are imposed in the ®rst-order conditions above. Eqs. (21) and (22) are solved simultaneously to ®nd the equilibrium values qd and hd . Inserting the functional value of Dm R from Eq. (11b) into the equilibrium conditions (21) and (22), p
 NKhd … Na0 ln …… Nn qd †=… Nn 2qd hd †† †=2C a0 …aK a0 † oP1 p ˆ oq1 4…Nn 2qd hd † Na0 ln …… Nn qd †=… Nn 2qd hd ††=2C c0 …qd † ˆ 0;

…23†

p  NKhd … Na0 ln …… Nn qd †=… Nn 2qd hd †† †=2C a0 oP1 p ˆ oh1 4…Nn 2qd hd †…aK a0 † Na0 ln …… Nn qd †=… Nn 2qd hd ††=2C G0 …hd † ˆ 0;

…24† d

From Eq. (23), for given values of h , the marginal bene®t function of qd , MB(qd ), is upward sloping. From Eq. (24), for given values of qd , the marginal

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bene®t function of hd , MB(hd ), is upward sloping. Assume that both the upward-sloping marginal cost functions c0 (qd ) and G0 (hd ) are steeper than their respective marginal bene®t functions. Then, from the equilibrium conditions (23) and (24), Proposition 5. The equilibrium values of product quality and differentiation decrease with: (1) an increase in N, the number of risky assets in the economy; (2) an increase in n, the number of risk variables per asset; (3) an increase in C, the marginal cost of physically producing, distributing, and in general supporting the reports; (4) an increase in investor heterogeneity, measured by (aK a0 †=K; (5) an increase in c0 , the marginal cost of increasing product quality; (6) an increase in G0 , the marginal cost of increasing product differentiation. From Proposition 6 and Eqs. (11a), (11b) and (14), Proposition 6. The equilibrium quantity of reports sold in the monopoly market decreases with increases in parameters (1)±(6) listed in Proposition 5. An increase in any of these six parameters has an ambiguous effect on the quantity of reports sold in the duopoly market. 5.2. Achieving optimal heterogeneity: The role of niche strategies The modelÕs simple risk structure allows analysis in terms of q1 , q2 and h. The consumerÕs choice problem naturally focuses on these three variables. For tractability, this focus on q1 , q2 and h continues in analysis of the ®rmsÕ strategic choices of quality and di€erentiation. Analytical results are found for quality and di€erentiation, but not for the actual risks that the FSFs identify in their reports. Turn now to how the FSFs map their optimum choices of quality and di€erentiation into the actual risks ei;j their reports identify. Firms can calculate the optimal qd and can thus be assured of achieving their optimum quality. To be concrete, if qd ˆ 20, each ®rm chooses 20 ei;j s. Achieving their optimal hd is less straightforward. If hd implies an overlap of 10 to achieve optimum di€erentiation, then the non-cooperative FSFs must develop a mechanism for choosing ei;j s and e2;j s that reliably results in identifying 10 common and 10 separate risks. This choice problem arises in starkest terms in the limiting case where ®rms face identical demand and cost conditions, but arises even if ®rms have competitive advantages on the cost side in identifying particular risks or particular types of risks. Assume that each ®rm orders the risks from cheapest to most expensive to identify, and that the ®rms face identical, increasing-cost conditions: they have the same orderings, the same increasing cost functions, and the same optimal qd , hd . Each ®rm chooses to identify the same 10 risks with lowest costs. Each aims at identifying 10 more risks that do not overlap. But this optimal heter-

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ogeneity can be obtained in many ways, and the heterogeneity of one ®rmÕs product depends in part on the other ®rmÕs choices. Firm 1 may choose its second set of 10 risk factors with an eye towards achieving optimal heterogeneity, but ®rm 2 may end up choosing the same 10; both achieve no heterogeneity rather than the jointly optimal degree. A non-cooperative way to avoid fortuitous overlap is required. One possibility is to assume that ®rms have competitive advantages that impose unique, cost-minimizing sets of non-overlapping risks for each ®rmÕs second set of 10, but this is ad hoc and need not hold in general. Niche strategies provide a mechanism that reliably results in the non-cooperative FSFs achieving the jointly optimal heterogeneity. In this case, where neither ®rm has a competitive advantage, each may stake out an explicit niche, a group of related risks, in which it specializes. Because neither has a competitive advantage, niche selection is wholly arbitrary. But once each selects a niche, the other knows which risks to avoid in order to generate the jointly optimal heterogeneity. When one ®rm declares its niche, this information does not alter either ®rmÕs strategic problem or solution to the strategic problem. Rather, explicit niches provide a coordination mechanism and ensure that the qd , hd solution can be achieved, and hence strengthen each ®rmÕs commitment to the solution. Some niches are more expensive to service than others. If many niches exist with the same low but increasing costs for heterogeneity, then it does not matter which niches the identical ®rms pick. Both have incentives to respect otherÕs niche to achieve their optimal heterogeneity; each loses by failure to respect niches. (One may imagine that one niche is so much lower cost than any other niche that the ®rms battle intensely over who gets this niche; there appears to be no reason to view this as the general case.) FSFs have a strong incentive to follow niche strategies, in the sense of having only partial overlap in the information they provide. To ensure that they achieve their optimal heterogeneity, FSFs have a strong incentive to specify their niches, signals that allow them to avoid unintentionally producing suboptimal heterogeneity. The consumer is indi€erent to the niches chosen, as long as the promised quality and di€erentiation of the ®nancial information are achieved. The niches that the FSFs choose and advertise are not oriented toward the demand side as a marketing tool, nor are they cost reducing devices in the case of identical ®rms. Rather, the niche strategies are devices that allow non-cooperative ®rms to coordinate their activities. Cost-side competitive advantages make it cheaper, but by no means straightforward, for FSFsÕ to achieve optimum di€erentiation: ®rms still have to make a conscious and coordinated e€ort to achieve di€erentiation. FSFs with competitive advantages are naturally biased towards selecting risks ei;j s that are low cost for them to identify; there is built-in heterogeneity. But this built-in heterogeneity is optimal only by chance; otherwise, the FSFs face the

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problem of adjusting their chosen ei;j s to achieve optimal heterogeneity. The problem returns: how are the FSFs to ensure they achieve the heterogeneity they both desire? As above, the device of declaring a niche, and if necessary a list of mainstream common risks, serves to coordinate the risk selections of non-cooperative FSFs. 6. Summary and conclusions This paper is the ®rst to study the duopolistsÕ strategic choices of quality, di€erentiation and pricing-output in the ®nancial services industry. In previous work, Admati and P¯eiderer (1986, 1990) and Allen (1990) study monopoly markets for ®nancial information. Fishman and Hagerty (1995) study pricing strategies for ®nancial services in a duopoly framework, but do not consider issues of product quality and di€erentiation. These papers also assume information sellers are endowed with a certain quantity of information, rather than producing it at increasing marginal costs. In contrast, the competitors in the present paper face increasing costs in the quality and di€erentiation of their products. This paperÕs duopoly ®nancial service ®rms (FSFs) strategically choose the optimal quality and di€erentiation of their products as well as their optimal pricing-output strategies. The duopolistsÕ optimal quality and di€erentiation are solved for as explicit functions of the modelÕs parameters. The model provides rich results on strategies on quality and di€erentiation. In equilibrium, both quality and di€erentiation decrease with an increase in: (1) the number of risky assets in the economy; (2) the number of factors that contributes to the riskiness of each risky asset; (3) the marginal cost of physically producing, distributing and in general supporting the ®nancial products; (4) the heterogeneity of investors, measured here by their absolute risk aversion; (5) the marginal cost of improving quality; and (6) the marginal cost of increasing the di€erentiation of their ®nancial products. The duopolistsÕ optimal strategic choices of quality and di€erentiation result in market segmentation. Some investors who are quite risk averse, expect to put only a small dollar value of funds at risk, and do not ®nd it worthwhile to buy information from either FSF. At the other extreme are the economyÕs least risk averse investors. Their unconditional expectation, before buying information, is that they will have a relatively large exposure to risky assets; therefore, they have a relatively high demand for the FSFsÕ information products, and buy information from both FSFs. Thus, each FSF has monopoly power in selling to these investors, and each sets a monopoly price. Another market segment consists of those investors, with intermediate risk aversion, who ®nd it worthwhile to buy information from one, but only one, FSF; the two FSFs compete head-to-head in this market.

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In equilibrium, the ®rms always choose to di€erentiate their products, even if they face the same demand and cost conditions and have no competitive advantage in producing and marketing particular types of information. Any competitive advantage the ®rms have in producing di€erent pieces of information reinforces the decision to di€erentiate by reducing the marginal cost of increasing di€erentiation. In order to coordinate the amount of overlap in their reports and thus achieve optimal di€erentiation, each non-cooperative ®rm announces an explicit niche, a related set of risks that it focuses on. These niches are signals to investors that promised di€erentiation will be achieved, and are signals to the other FSF on which area to avoid. Each FSF respects the otherÕs niche, so that both can achieve the di€erentiation they promise buyers. Any competitive advantage the ®rms have will help guide the directions in which the ®rms di€erentiate, thus reinforcing each ®rmÕs declared niche and helping the FSFs to coordinate to ensure they achieve optimal di€erentiation. An increase in di€erentiation of the FSFsÕ products increases the size of each monopoly market, and simultaneously reduces the size of the duopoly market. The decrease in the duopoly marketÕs size equals the aggregate increase in the sizes of the monopoly markets. An increase in the quality of the FSFs' product increases the size of each monopoly market. This increase in quality has an ambiguous e€ect on the size of the duopoly market. An increase in any one of the six exogenous factors mentioned above causes a decrease in both quality and di€erentiation, and thus in sales in the monopoly markets. The e€ects of the six factors on sales in the duopoly market are ambiguous; for low enough levels of equilibrium product di€erentiation, sales in the duopoly market fall with an increase in the six factors. Recent years have seen a worldwide wave of ®nancial-service-®rm mergers and acquisitions, and increased competition in ®nancial services in emerging market economies (EMEs). Some observers fear that the increased M and A activity will result in the survival of only very large FSFs. This paperÕs model suggests that equilibrium will see FSFs develop pro®table niches and o€er partial product di€erentiation, where investors pay for incremental information they cannot get elsewhere. Contrariwise, ®rms that attempt to compete across the market may fail. Similarly, some EME policy makers fear that developed-country FSFs will decimate local FSFs. This paperÕs model suggests that local FSFs may well survive by ®nding niches in the identi®cation of local risks, leaving identi®cation of international risks to developed-country ®rms. This paperÕs results depend crucially on the assumption that economies of scope are not dominant. If large ®rms that compete across the ®nancial-services market have lower costs in most areas because of economies of scope, then the outlook for niche ®rms is bleak. The evidence on economies of scope is mixed. Both those FSFs aiming to compete across the market and those FSFs

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focussing on niche strategies may, in e€ect, be making large bets on simple hunches about economies of scope. This paper models only two FSFs, but the analysis is easily generalized to any ®xed number of ®rms. Entrant±incumbent issues are not considered here, but would be an interesting application of this paperÕs approach. This paper assumes the duopolists have no concerns about entry by other ®rms±government restrictions on entry were assumed. Generalizing the analysis to allow free entry would permit consideration of strategies to deter entry, or if it is infeasible or suboptimal to deter entry, strategies for the period before entry occurs.

Acknowledgements For helpful comments, thanks are due to an anonymous referee. This paper builds on parts of MahajanÕs dissertation, ``Liberalization and recent developments in ®nancial services sectors in emerging market economies,'' Georgetown University, 1996. For comments on earlier versions, thanks are due to Gwen Eudey and Matt Canzoneri. The McDonough Business School provided summer support for Sweeney. Georgetown UniversityÕs Capital Markets Research Center provided summer research assistance. Part of the work on this paper was done at the Gothenburg School of Economics, Sweden.

Appendix A. Proofs of propositions in the text Theorem 1. Let superscript I represent an informed investor who owns C reports, either 1 and 2, and superscript U an uninformed investor who owns K reports, fewer than the informed investor, either 0 or 1, at the time of portfolio allocation. Thus, C > K; C ˆ ‰1; 2Š and K ˆ ‰0; 1Š; when K ˆ 0; C ˆ 1 or 2, and when K ˆ 1; C ˆ 2. Given a choice between buying C or K number of reports, an investor chooses to buy C reports over K reports if and only if the expected utility from doing so is at least as great, or, because expected utility is a negative number, Evk …W I †=Evk …W U † 6 1. But " #N =2 Var…Rj jC report…s†† Evk …W I † ak F ˆe Evk …W U † Var…Rj jK report…s†† " #N =2 Var…Rj jC report…s†† ac F 6 …>†e ˆ 1; …A:1† Var…Rj jK report…s†† where W I is the day-2 wealth of the investor if she chooses to buy C reports, WU her day-2 wealth if she chooses to buy K reports, N the number of risky assets,

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Var…Rj jC† and Var…Rj K† are conditional variances for any asset j and are the same across all assets j and investors k, and V a negative exponential utility function for all investors. F is the cost of (C K) reports, that is, buying an extra one or two reports. Therefore, Evk …W I †=Evk …W U † 6 1 if and only if ak F ‡ …N =2† ln‰Var…Rj jC†=Var…Rj jK†Š 6 0, where a0 6 ak P aK , F > 0 and ln‰Var…Rj jC†=Var…Rj jK†Š < 0. ac is the absolute risk aversion of an investor who is indifferent between buying C and K reports. Any investor with absolute risk aversion less than ac has higher expected utility if she buys C rather than K reports. Proof. From Eq. (9) in the paper, the expected utility of an investor in the morning of day 1 is ( " EM exp a W PR1 Q1 PR2 Q2 ‡

N X jˆ1

#)

!, 2

…E…Rj jIEv ††

…2aVar…Rj jIEv ††

;

Qi ˆ ‰0; 1Š; i ˆ 1; 2: …A:2†

Investor buying zero reports. For an investor who buys 0 reports, Q1 ˆ Q2 ˆ 0; E…Rj jIEv † ˆ nl, and Var…Rj jIEv † ˆ nr2 . For such an investor, Eq. (A.2) can be written as #) ( " !, N X 2 2  …nl† …A:3† EM exp a W ‡ …2anr † : jˆ1

Investor buying one report. Assume that information is perfectly divisible, and that the FSFs spread qd equally across all risky assets such that qd =N ˆ y d , where yd is the number of units of information produced per asset. For an P risky d investor who buys one report at price PRi , E…Rj jIEv † ˆ yiˆ1 ei;j ‡ …n y d †l, and Var…Rj jIEv † ˆ …n y d †r2 . For such an investor, Eq. (A.2) can be written as EM

8 < :

2

exp 4

0

a@W

yd N X X d PRi ‡ ei;j ‡ …n jˆ1

!2 ,

y d †l

iˆ1

…2a…n

139 = y d †r2 A5 : ;

…A:4† Let yd X iˆ1

ei;j ˆ Xj ;

j ˆ 1; . . . ; N :

The Xj are joint normally distributed with Var…Xj jIM † ˆ y d r2 . Then, Eq. (A.4) can be written as

E…Xj jIM † ˆ y d l

and

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Z

1 …2p†N …y d r2 †N =2 N X ‰Xj jˆ1

1

(

1

y d lŠ 2y d r2

" exp

a W

PR ‡

N X ‰Xj ‡ …n jˆ1

#) 2

2a…n

dX1 ; . . . ; dXN :

Eq. (A.5) can be rewritten as " !# 2 N X …nl† 1 exp a W PR ‡ N d 2 N =2 2 2anr …2p† …y r † jˆ1 " #! Z 1 2  N X ‰n‰Xj Š  exp dX1 ; . . . ; dXN : d …n d †r2 † 2I y 1 jˆ1 Let Z ˆ Then dZj

r n X : y d …n y d †r2 j

r n ˆ dXj : y d …n y d †r2

Using Eqs. (A.7) and (A.8) in Eq. (A.6) gives " !# N =2 2 N X …nl† n yd 1 exp a W PR ‡ N 2 2anr n …2p† jˆ1 " #! Z 1 N X Zj2  exp dZ1 ; . . . ; dZN : 2 1 jˆ1

y d †lŠ2 y d †r2

!

…A:5†

…A:6†

…A:7†

…A:8†

…A:9†

From Eq. (A.9) Zj  N…0; 1†. Therefore, the last term in Eq. (A.9) equals 1. Eq. (A.9) can now be rewritten as " !# N =2 2 N X …nl† n yd exp a W PR ‡ : …A:10† 2anr2 Nn jˆ1 Since by assumption y d ˆ qd =N , Eq. (A.10) can be rewritten as " !# N =2 2 N X …nl† Nn qd exp a W PR ‡ : 2anr2 Nn jˆ1

…A:11†

Investor buying two reports. Using a derivation similar to the one used above for the one report case, the expected utility in the morning, prior to buying the reports, of the investor who buys two reports is

S. Mahajan, R.J. Sweeney / Journal of Banking & Finance 25 (2001) 1447±1473

" exp

a W

2PR ‡

2 N X …nl† jˆ1

!#

2anr2

Nn

2hd qd Nn

1469

N =2 ;

…A:12†

where hd 2 …0:5; 1† and is the equilibrium PN heterogeneity chosen by each ®rm. The variance of the portfolio, R …R ˆ jˆ1 Rj †, when the investor buys 0, 1 or 2 reports is Nnr2 , …Nn qd †r2 and …Nn 2qd †r2 . Therefore, using Eqs. (A.3), (A.11) and (A.12), for any investor k…k ˆ 1; . . . ; K† with absolute risk aversion ak , the ratio of her utility if she buys C reports …Evk …W2U †† to her expected utility if she buys K reports …Evk …W2U †† is " #N =2 Var…Rj jC report…s†† Evk …W I † ak F ˆe Var…Rj jK report…s†† Evk …W U † " #N =2 Var…Rj jC report…s†† ac F 6 …>†e ˆ 1; …A:13† Var…Rj jK report…s†† where ac is the absolute risk aversion of the investor who is indi€erent between m m buying C reports and K reports. F ˆ PRd if C ˆ 1; K ˆ 0; F ˆ …PR1 ‡ PR2 † if m m d d C ˆ 2; K ˆ 0; F ˆ …PR1 ‡ PR2 PR † if C ˆ 2; K ˆ 1; and PRi is the duopolym market price and PRi is the monopoly price for i But this is the same as Eq. (A.1). The expected utility of the investor k from buying C reports will be greater than (or equal to) her expected utility from buying K reports if and only if Ev…W2I †=Ev…W2U † 6 1 (since the utility function in model considered has a negative sign before it); i.e., if and only if Ee

ak W2I

Ee

ak W2U

6 1:

But from Eq. (A.13), this is true if and only if ak 6 ac . Thus, all investors with absolute risk aversion less than or equal to ac will prefer to buy C reports over buying K reports, and all investors with risk aversion greater than ac will prefer to buy K reports over buying C reports. Hence proved.  Theorem 2. From Theorem 1, the number of reports that different investors buy depends on their risk aversion. (i) C ˆ 1; K ˆ 0 (the informed investor buys one report, the uninformed 0): An investor who buys a report from only one FSF, FSFi (i ˆ 1; 2), in the duopoly market, faces the conditional variance of Rj of …Nn qi †r2 ; she faces the unconditional Nnr2 if she does not buy any report. From Eq. (A.2), the absolute risk aversion of the investor indi€erent between buying a report from ®rm i …i ˆ 1; 2† and not buying any report is abuy…i† , d abuy…i† ˆ …N =2PRi † ln‰…Nn†=…Nn

qi †Š;

a0 6 abuy…i† 6 aK ; i ˆ f1; 2g:

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All investors with absolute risk aversion less than or equal to abuy…i† ˆ abuy d prefer to buy ®rm iÕs report fi ˆ …1; 2†g, which o€ers quality qi at price PRi , rather than no report. All investors with absolute risk aversion greater abuy…i† ˆ abuy prefer to buy no report rather than ®rm iÕs. (ii) C ˆ 2; K ˆ 1 (the informed investor buys two reports, the uniformed one): let h1 2 …0:5; 1† be a measure of the product di€erentiation (heterogeneity) that ®rm 1 chooses, conditional on the fact that ®rm 2 chooses h2 . If in equilibrium h1 ˆ h2 ˆ 0:5 and q1 ˆ q2 , then the FSFs have no product di€erentiation and produce completely homogenous products. If in equilibrium h1 ˆ h2 ˆ 1, then the FSFs have complete di€erentiation and produce completely heterogeneous products. With this de®nition, an investor who buys reports from both ®rms faces the conditional variance of Rj of (Nn q1 h1 q2 h2 †r2 , but faces the conditional variance of Rj of …Nn q2 †r2 with only ®rm 2Õs report. From Eq. (A.1), m m ‡ PR2 abuy…1†j…2† ˆ …N =2…PR1

d PR2 †† ln‰…Nn

q2 †=…Nn

h1 q1

h2 q2 †Š;

a0 6 abuy…1†j…2† 6 aK ; where abuy…1†j…2† is the absolute risk aversion of the investor indi€erent between buying both ®rmÕs report and buying only ®rm 2Õs report. All investors with absolute risk aversion less than or equal to abuy…1†j…2† prefer to buy reports from both ®rms, rather than only ®rm 2Õs report. All investors with absolute risk aversion greater than abuy…1†j…2† but less than or equal to abuy…2† buy only ®rm 2Õs report rather than both ®rmsÕ. Similarly all investors with absolute risk aversion less than or equal to abuy…2†j…1† , where m m ‡ PR2 abuy…2†j…1† ˆ …N =2…PR1

d PR2 †† ln‰…Nn

q1 †=…Nn

h1 q1

h2 q2 †Š;

a0 6 abuy…2†j…1† 6 aK ; buy both reports, rather than only ®rm 1Õs report. In equilibrium, abuy…1†j…2† ˆ abuy…2†j…1† ˆ abuy…2j1† . (iii) C ˆ 2; K ˆ 0 (the informed investor buys two reports, the uninformed 0): the conditional variance of Rj faced by an investor who buys both reports is …Nn q1 h1 q2 h2 †r2 ; the variance of Rj conditional on no reports is Nnr2 . From Eq. (A.2), m m abuy…2j0† ˆ ‰N =2…PR1 ‡ PR2 †Š ln‰…Nn†=…Nn

h1 q1

h2 q2 †Š;

a0 6 abuy…2j0† 6 aK ; where abuy…2j0† is the absolute risk aversion of the investor who is indi€erent between buying the two reports and not buying any report. All investors with absolute risk aversion less than or equal to abuy…2j0† , when given the alternatives of buying both reports or not buying any report choose to buy both. All in-

S. Mahajan, R.J. Sweeney / Journal of Banking & Finance 25 (2001) 1447±1473

1471

vestors with absolute risk aversion greater than abuy…2j0† buy neither report rather than buying both. (iv) Monopoly-market demand conditions; existence of the duopoly market: m m let qd1 ˆ qd2 ˆ qd ; hd1 ˆ hd2 ˆ hd , and thus PR1 ˆ PR2 ˆ PRm in the monopoly d d d markets, and PR1 ˆ PR2 ˆ PR in the duopoly market. Let PRd < PRm from price discrimination. Then, from above abuy…2j0† ˆ …N =4PRm † ln ‰…Nn†=…Nn abuy…2j1† ˆ …N =2…2PRm

2hd qd †Š;

PRd †† ln ‰…Nn

a0 6 abuy…2j0† 6 aK ;

qd †=…Nn

abuy  abuy…1j0† ˆ …N =2PRd † ln ‰…Nn†=…Nn

qd †Š;

2hd qd †Š;

a0 6 abuy…1†j…2† 6 aK ;

a0 6 abuy…1j0† 6 aK :

Monopoly-market demand. Any investor k with ak 6 abuy2 buys both reports, where abuy2 ˆ min…abuy…2j1† ; abuy…2j0† ). If 2hq=Nn is not implausibly large, then abuy…2j1†† < abuy…2j0†† when N =…2…2PRm <

PRd †† ln‰…Nn

q†=…Nn

…N =4PRm † ln‰…Nn†=…Nn

2hq†Š

2hq†Š

or PRd ln ‰…Nn†=…Nn

2hq†Š < 2PRm ln‰…Nn†=…Nn

q†Š:

Since in equilibrium, PRd ˆ C and PRm > C, a sucient condition for abuy…2j1† < abuy…2j0† is Nn=…Nn

2hq† < ‰Nn=…Nn 2

…Nn

q† < Nn=…Nn 2

2

q†Š ;

2hq†;

…Nn† ‡ q2

2Nnq < …Nn†

q < 2Nn…1

h†;

2

2Nnhq;

or 2hq=Nn < 4h…1

h†:

This condition holds for small 2hq=Nn, where 2hq=Nn is the two reportsÕ ratio of risks revealed to total risks. Because the model requires uninformative prices, small 2hq=Nn is assumed. Nevertheless, in principle, the demand curve is kinked: As  2hq=Nn () 4h…1 h†; then abuy2 ˆ abuy…2j1† < abuy…2j0† ; abuy2 ˆ abuy…2j1†  ˆ abuy…2j0† ; abuy2 ˆ abuy…2j0† < abuy…2j1† : Duopoly-market existence. A duopoly market exists for small 2hq=Nn. Existence requires that abuy > abuy2 . If monopoly-market risk aversion is abuy2 ˆ abuy…2j1† , then existence requires

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S. Mahajan, R.J. Sweeney / Journal of Banking & Finance 25 (2001) 1447±1473

abuy =abuy2 > 1; …Nn

……2PRm

P d †=P d † f ln‰…Nn†=…Nn

q†Š= ln‰…Nn

q†=

2hq†Šg > 1;

or PRd ln ‰…Nn†=…Nn

2hq†Š < 2PRm ln‰…Nn†=…Nn

q†Š:

But this condition is same as the monopoly market condition abuy2 ˆ abuy…2j1† . Therefore, existence of the duopoly market too requires 2qh=Nn < 4h…1 h†. At extreme heterogeneity of h ˆ 0:99, the condition is satis®ed if 2hq is less than 3.96% of the risk factors, Nn. Nevertheless, for suciently large 2hq=Nn, the duopoly market does not exist: all report-buyers buy two reports. Hence proved.  Proposition 5. The equilibrium values of product quality and differentiation decrease with: (1) an increase in N, the number of risky assets in the economy; (2) an increase in n, the number of risk variables per asset; (3) an increase in C, the marginal cost of physically producing, distributing, and in general supporting the reports; (4) an increase in investor heterogeneity, measured by …aK a0 †=K. (5) An increase in c0 , the marginal cost of increasing product quality; (6) an increase in G0 , the marginal cost of increasing product differentiation. Proof. From Eq. (24), hd ˆ f …qd ; N ; n; C; …aK a0 †=K; G0 †; f 1 > 0; f 2 < 0; f 3 < 0; f 4 < 0; f 5 < 0 and f 6 < 0 where f is a function and superscripts indicate the partial derivatives with respect to the corresponding parameters; G0 is the marginal cost of increasing hd . Use this functional value f to replace hd in Eq. (23). Eq. (23) now gives the following implicit solution: qd ˆ gd …N ; n; C; …aK a0 †=K; G0 ; c0 †, where gd is the optimal equilibrium functional form of g; gd1 < 0; gd2 < 0; gd3 < 0; gd4 < 0; gd5 < 0 and gd6 < 0; c0 is the marginal cost of qd units of quality. Use qd ˆ gd …N ; n; C; …aK a0 †=K; G0 ; c0 † to replace qd in Eq. (24) to get hd ˆ f d …N ; n; C; …aK a0 †=K; G0 ; c0 †, where fd is the equilibrium functional form of f; f d1 < 0; f d2 < 0; f d3 < 0; f d4 < 0; f d5 < 0 and f d6 < 0. Hence proved.  Proposition 6. The equilibrium quantity of reports sold in the monopoly market decreases with increases in parameters (1)±(6) listed in Proposition 6. An increase in any of these six parameters has an ambiguous effect on the quantity of reports sold in the duopoly market. Proof. From Eqs. (11a) and (11b), the equilibrium monopoly output increases in qd and hd and, for given qd and hd , decreases in N, n, C, and …aK a0 †=K. But Proposition 6 concludes that qd and hd decrease in N, n, C, …aK a0 †=K, G0 or

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c0 . Therefore, the equilibrium quantity in the monopoly market decreases in N, n, C, …aK a0 †=K, G0 and c0 . From Proposition 3, the six parameters have ambiguous e€ects on equilibrium duopoly output. 

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