Optimal Delegation, Unawareness, and Financial Intermediation

Optimal Delegation, Unawareness, and Financial Intermediation Sarah Auster⇤and Nicola Pavoni† April 3, 2017 Abstract We study the delegation problem between an investor and a financial intermediary. The intermediary has private information about the state of the world that determines the return of the investment. Moreover, he has superior awareness of the available investment opportunities and decides whether to reveal some of them to the investor. We show that the intermediary generally has incentives to make the investor aware of investment opportunities at the extremes, e.g. very risky and very safe projects, while leaving the investor unaware of intermediate investment options. We study how the extent to which the intermediary reveals available investment opportunities to the investor depends on the investor’s initial awareness and the degree of competition between intermediaries in the market. JEL Codes: D82, D83, G24. 1 Introduction One of the many striking features of the recent financial crisis was the extreme exposure of investors to risk. Both investment and commercial banks had been selling excessively risky assets to investors, sometimes hiding some of the asset characteristics (e.g., Gerardi et al., 2008). At the same time, despite the impressive amount of new financial instruments and ⇤Department of Decision Sciences, Bocconi University: [email protected] †Department of Economics, Bocconi University: [email protected] 1 the rapidly changing financial world, since the 1950s a large fraction (of approx. 33% in the US) of investment demand has remained on ’safe’ assets (e.g., Gordon et al. (2008) and Garcia (2012)). Most of financial investments are intermediated by professionals. Financial intermediaries are non-neutral brokers and, as such, direct, influence, and distort the demand of assets in the economy. For instance, investment bankers commonly underwrite transactions of newly issued securities, whereby they raise investment capital from investors on behalf of corporations and governments both for equity and debt capital. Such practices give rise to conflicts of interest, sometimes leading to investments that are not necessarily in the best interest of the client. At the same time, investors di↵er widely in their financial literacy. They not only face limits in their ability to assess the profitability of particular investments, but often also have limited awareness of the available investment opportunities and must therefore rely on professional advice. This paper studies the implications of such limitations by incorporating unawareness into the canonical delegation problem. Specifically, we consider the problem of an investor (the principal, she) who needs to choose how to invest her savings and delegates the task of picking the right project to a financial intermediary (the agent, he). The intermediary has private information about the payo↵s of each investment opportunity and the investor’s problem is to determine a set of projects from which the intermediary chooses upon observing the state of the world (see for example Alonso and Matouschek, 2008). We then depart from the traditional framework by considering a situation where the intermediary not only has better information on what the best investment choice is, but also on the actual set of available investment opportunities. This second dimension of asymmetry is captured by the assumption that the investor is only partially aware of the feasible investment projects.Before the delegation stage the intermediary has then the possibility to expand the investor’s awareness by revealing additional investment opportunities. We are interested in the im2 plications of the investor’s unawareness on the interaction with the intermediary. Will the intermediary expand the investor’s awareness? If so, which projects will the intermediary disclose/not disclose? What are the properties of the investment projects eventually realized? We address these questions in an environment with a continuum of states and a continuum of investment projects, some of which the investor is aware of. The intermediary’s and investor’s preferences are represented by quadratic loss functions, with di↵ering bliss points for the two agents. We view the bliss point to be the investment opportunity which generates the best combination between risk, illiquidity, and return as a function of the state. The divergence between the investor’s and intermediary’s bliss point can be interpreted as banks being less risk averse, having limited liability, having di↵erent liquidity needs, etc. When deciding on the set of projects from which the intermediary can select, the investor then faces the usual tradeo↵between granting flexibility to the intermediary so he can react on his private information and precluding the intermediary from choosing projects he is biased towards too often. We show that the consideration of unawareness has important implications for delegation and investment. In the benchmark case of full awareness, under certain regularity conditions on the distribution of states, the optimal delegation set for the investor is an interval. E↵ectively, the investor imposes a cap: if, for instance, the intermediary is less risk averse than the investor, the investor places an upper bound on the riskiness of projects the intermediary can select. In our environment, where projects are ordered along the real line, this translates into a threshold below which the intermediary is free to choose and above which he is not allowed to take an action. Imposing the same regularity condition on the distribution of states, we characterize the equilibrium delegation set in an environment where the investor is partially unaware. Our main result shows that the intermediary makes the investor fully aware if and only if the investor is initially aware of the project at the optimal threshold under full awareness. If this is not the case, the intermediary leaves the investor unaware 3 of an interval of projects around the threshold. The intermediary thus makes the investor aware of investment opportunities at the extremes (e.g. very safe and very risky projects), while he does not disclose intermediate investment opportunities. This makes it optimal for the investor to permit projects at both extremes and thus gives the intermediary the possibility to invest in them. Next, we introduce competition among intermediaries by incorporating our baseline model into a search environment with imperfect competition. In this framework multiple intermediaries compete for investors over the set of investment opportunities they disclose: the larger this set is, the larger are the chances of attracting an investor. We show that, for intermediate degrees of competition, in equilibrium an intermediary either reveals everything or leaves investors unaware of a significant set of investment options. We thus obtain polarization in the market: some banks fully disclose all investment opportunities, others try to obtain a higher profit by concealing some of them. Finally, we discuss the policy implications of our findings. Clearly, promoting financial literacy among investors improves their welfare in our model. Interestingly though, our results show that it is not necessary to educate investors about all possible investment projects. Instead, we demonstrate that making investors aware of only one intermediate project -at the optimal threshold in the benchmark case of full awareness - is enough. Making investors aware of this project gives incentives to intermediaries to reveal all remaining projects as well. We therefore have an interesting complementarity between the regulator and the market, suggesting a surprisingly simple, yet powerful, policy intervention. The paper makes two main contributions. First, it makes a methodological contribution in that we introduce limited awareness into the model of delegation. The potential applications are much broader than the financial market. Second, the stated framework is able to generate predictions on the equilibrium portfolio di↵erentiation, in particular on the demand 4 of safe and risky assets, as a function of the nature of misalignment between the investor and the financial intermediary. Related Literature: This paper is first of all related to the literature on optimal delegation. Starting with Holmstrom (1984), who first defines the delegation problem and provides conditions for the existence of its solution, this literature, which includes Melamud and Shibano (1991), Martimort and Semenov (2006), Alonso and Matouschek (2008), Armstrong and Vickers (2010) and Amador and Bagwell (2013), studies optimal delegation problems in environments of increasing generality. None of them consider limited awareness in this framework. Furthermore, this paper is related to a small literature on contract theory and unawareness. The application of the concept of unawareness to contracting problems is still at its beginnings. In contrast to our setting, existing work considers contracting problems where contingent transfers are feasible and where the agent is unaware - either of possible actions (Von Thadden and Zhao, 2012 and 2014) or of possible states (Zhao, 2011; Filiz-Ozbay, 2012; Auster, 2013), while the principal is fully aware. This paper is also related to the literature on financial intermediation which sees banks as ’efficient’ brokers who reduce transaction and information costs. The issue of this role of financial intermediation has been studied by many authors starting from Diamond (1984). A summary of the literature can be found in Bhattacharya and Thakor (1993) and Allen and Santomero (1998). These works focus on the possibility of partially monitoring the financial intermediaries ex-post. Most of the work is on the load provision side of the banks and on the design of the optimal loan contract, whereas we concentrate on the brokerage role of banks. 2 Environment There is an investor (she) who acts as the principal and a financial intermediary (he) who acts as the agent. The intermediary has access to a set of investment projects Y=[ymin,y max], 5 the return to which depends on the state of the world. Let ⇥= [0,1] be the set of states and let F(✓) denote the cumulative distribution function on ⇥,assumedtobetwicedi↵erentiable in [0,1].1Both the investor and the intermediary have a von-Neumann-Morgenstern utility function that takes the quadratic form u(y,✓)=(y✓)2and v(y,✓)=(y(✓))2. The intermediary’s preferred policy is y=✓,whiletheinvestor’spreferredpolicyisy=✓. We assume >0, hence the intermediary has an upward bias of size .2In Appendix A we provide a micro foundation for the assumed utility functions by starting with preferences defined over the mean and variance of the investment. Assuming that the intermediary is less risk averse than the investor, the bias bthen captures the di↵erence between the investor’s and the intermediary’s preferred investment on the mean-variance frontier. Accordingly, we can interpret low values of yin [ymin,y max]asrelativelysafeinvestmentsandhighvaluesof yas relatively risky ones. As in the canonical delegation problem, we assume that the intermediary is informed about the state of the world ✓, while the investor is not. We rule out monetary transfers and assume that the intermediary’s participation constraint is always satisfied. The contracting problem of the investor then reduces to the decision of which projects to let the intermediary choose from.3 In contrast to the canonical setting, we assume that the investor is not aware of the whole set Ybut only of a closed subset YP✓Y. The intermediary, on the other hand, is 1For ✓= 0 and ✓= 1, this condition holds for, respectively, the right and left derivative. 2We can interpret itself as the result of a contracting problem that generates as the minimal level of conflict of interest between the principal and the agent. An alternative reading is that the intermediary and the investor have portfolios with di↵erent correlations across assets. 3Formally, the investor commits to a mechanism that specifies the project which will be implemented as a function of the intermediary’s message. Alonso and Matouschek (2008) show that this contracting problem is equivalent to delegating a set of projects D✓Yfrom which the investor can choose freely after observing the state of the world. 6 fully aware and has the possibility to make the investor aware of additional projects. After updating her awareness, the investor then makes her delegation choice. The precise timing is as follows: 1. The intermediary reveals a set of projects X✓Yand the investor updates her awareness to b Y⌘YP[X. 2. Given b Y,theinvestorchoosesadelegationsetD2D(b Y), where D(b Y) is the collection of closed subsets of b Y.4 3. The intermediary observes the state of world ✓and chooses an action from set D. 4. Payo↵s are realized. In general, optimal awareness sets ˆ Yand optimal delegation sets Dwill not be unique since di↵erent awareness sets may induce the same delegation set and di↵erent delegation sets may induce the same implemented actions for each state of the world. We will assume that if the investor is indi↵erent between two delegation sets Dand D0such that D0⇢D,shechooses the larger set D. Similarly, we assume that if the intermediary is indi↵erentbetweentwo revelation strategies that yield awareness sets b Yand b Y0such that b Y0⇢b Y, he expands the investor’s awareness to b Y. That is, we will consider the sets that yield maximal awareness and maximal discretion. Remark: It is important to point out that our model is agnostic to the question of whether or not the investor is aware of her unawareness. What matters for the investor’s expected payo↵is the set of projects she permits the intermediary to implement. The investor may be well aware of the fact that there exist other projects outside her awareness but since she cannot include such projects in the delegation set, their existence does not a↵ect her expected payo↵or optimization problem. 4As discussed in Alonso and Matouscheck, the restriction to closed sets is without loss of generality. 7 3 Equilibrium Analysis 3.1 Full Awareness We will start our analysis by describing optimal delegation under full awareness: YP=Y. For this specification, the existing literature shows that if the density function f(✓)⌘F0(✓) satisfies the regularity condition f0(✓)+f(✓)>0 for all ✓2[0,1], the optimal delegation set is an interval (Martimort and Semenov, 2006, and Alonso and Matouschek, 2008). In particular, if the bias is sufficiently small so that the expected best project for the investor E[✓]5is strictly greater than ymin, the optimal delegation set is given by [ymin,ˆy], where ˆyis such that ˆy=E[✓|✓ˆy].(1) Otherwise the optimal delegation set is the singleton {ymin}. Hence, only if E[✓]>y min, the implemented project varies with the state of the world and delegation is valuable. In that case, the intermediary chooses his preferred project y=✓for all ✓<ˆyand the project ˆyin all remaining states. To understand why the optimal delegation set takes this form, it is useful to describe the investor’s tradeo↵in more detail. We can first explain why the optimal delegation set does not have ”holes”. By adding the missing projects to the delegation set, the intermediary’s choice of projects as a function of the state becomes smoother: instead of switching from a low to a high project at some threshold state ✓0, the implemented action increase continuously with the realized state. Around ✓0the intermediary’s preferred project lies in between the low and the high project, which implies that he switches to higher actions in states below ✓0and to lower actions in states above ✓0. Since the agent is upward bias and since the cost of moving away from the bliss point is convexly increasing for the investor, the gain of moving closer to the bliss point in the states above ✓0outweighs the cost of moving 5All expectations are taken with respect to F. 8 away from the bliss point in the states below ✓0, as long as the probability weight attached to the latter states is not too large. The regularity condition on the state distribution assures that this is indeed the case. The investor’s problem then reduces to finding the optimal upper and lower bound of the delegation interval. Since the intermediary is upward biased it is never optimal to reduce the intermediary’s flexibility from below, so the optimal lower bound is ymin. The optimal upper bound is determined by condition (1). Notice that, conditional on the state being greater than y,theinvestor’sexpectedpreferredprojectisE[✓|✓>y]. Condition (1) says that the optimal threshold ˆyis such that the project the intermediary implements in all states above the threshold is exactly the investor’s expected preferred project in those states. The assumption f0(✓)+f(✓)>0 assures that there is only one such project. We will adopt the condition f0(✓)+f(✓)>0 throughout the analysis. Furthermore, we will assume that in each state of the world both the investor’s and the intermediary’s preferred project is available.6 Assumption 1. f0(✓)+f(✓)>0for all ✓2(0,1). Assumption 2. ymin <and ymax >1. 3.2 Partial Awareness: Main Result Our main result shows that, maintaining the regularity condition on the state distribution, it is strictly optimal for the intermediary to leave the investor partially unaware if and only if the investor is initially unaware of the project at the optimal threshold under full awareness, ˆy. In that case, the intermediary optimally reveals projects at the extremes but leaves the investor unaware of intermediate projects. Theorem 3.1. Let Assumptions 1 and 2 be satisfied. 6The purpose of the latter assumption is to reduce the number of cases we need to distinguish. 9 D* y ` ymax ymin Y `* D* YP y D* y ` ymax ymin Y `* YP y Figure 3: Optimal delegation set D⇤(ˆ Y) that the optimal value of is strictly positive. As can be verified, the unconstrained solution ⇤is increasing in the size of the bias . That is, the larger the divergence between the investor’s and the intermediary’s preferred investment is, the more investment projects the intermediary wants to hide from the investor. The solution ⇤is implemented whenever the investor’s initial awareness does not constrain the intermediary in his choice of the gap. If, however, the investor is aware of some project in the interval (ˆy⇤,ˆy+ ⇤), the intermediary’s optimal strategy is to simply choose the largest feasible gap, as shown in Figure 3. The following example illustrates the findings for the uniform distribution. Example: Suppose f(✓)=1. The optimal threshold in the benchmark case of full awareness is then given by ˆy=12and the interior solution for the optimal awareness set, characterized by condition (2), is ⇤=2p21.If2p21¯ (YP), the intermediary leaves the investor unaware of all projects in the interval 12p2,1+2p22. The resulting equilibrium awareness and delegation sets are then given by b Y⇤=[ymin,12p2] [[1 + 2⇣p22⌘,y max], D⇤(b Y⇤)=[ymin,12p2] [{1+2⇣p22⌘}. 16 If 2p21¯ (YP)is not satisfied, the intermediary is restricted by the investor’s initial awareness and therefore leaves her unaware of all projects in the smaller interval ⇣12¯ (b YP),12+¯ (b YP)⌘. 4 Competition The previous section characterized the equilibrium in an environment where the intermediary is a monopolist and, by implication, fully determines the investor’s awareness. In reality, investors can seek consult from multiple financial professionals, possibly to expand their choice between di↵erent investment options. To capture the interaction between multiple intermediaries, we adopt a simple model of imperfect competition that has been recently proposed by Lester et al. (2017) and is based on the work of Burdett and Judd (1983). Considering a model of imperfect competition accounts for some important features of the financial market, especially over-the-counter trading. It further allows us to consider the e↵ect of the degree of competition on the awareness of investors and the composition of financial products traded in the market. Environment: Following the approach of Lester et al. (2017), we assume that there are two intermediaries and a unit measure of investors.7Intermediaries have no capacity constraints and can therefore contract with many investors. There is a friction in that investors do not necessarily have access to both intermediaries. In particular, a fraction of investors is matched with one intermediaries, while the remaining investors are matched with both. As in Lester et al. (2017), we refer to investors that have access to only one intermediary as captive. Whether an investor is captive or non-captive is not observable to the intermediaries. Instead, from the viewpoint of an intermediary, conditional on meeting a particular investor, the investor is non-captive with some probability, denoted by ⇡. The parameter ⇡ can then be viewed as a measure of competitiveness in the market: if ⇡=0wearebackin 7As Lester et al. (2017) show, the restriction to two intermediaries can be easily relaxed. 17 the monopoly case; if ⇡= 1 intermediaries engage in Bertrand competition. To simplify matters we will first assume that investors are initially unaware of all projects. Upon meeting an investor, intermediaries disclose a set of investment projects, as before. If an investor meets with two intermediaries, her updated awareness set is the union of the projects that are revealed by either of the two intermediaries. The investor then needs to decide to which of the intermediaries to delegate her investment. In principle, after updating her awareness, the investor is indi↵erent between both intermediaries. To make competition matter, we assume here that the investor chooses the intermediary that reveals more investment opportunities. More specifically, if the set revealed by the first intermediary is a strict subset of the set revealed by the second, the investor chooses the second, and viceversa. If both intermediaries choose the same awareness set or if the awareness sets cannot be ordered, the investor chooses either intermediary with equal probability. Once an investor selects an intermediary, the interactions unfolds exactly as in the monopoly case. For a given investor, the timing can be summarized as follows: 1. The investor privately observes whether she meets one or two intermediaries. 2. Each intermediary reveals a set of projects and the investor updates her awareness to the union of both sets. 3. The investor chooses an intermediary and a delegation set. 4. The selected intermediary observes the state of world and chooses an action from the delegation set. 5. Payo↵s are realized. The assumption that an investor delegates to the intermediary that reveals more investment opportunities directly implies that intermediaries optimally choose awareness sets of the form [ymin,ˆy][[ˆy+,y max],0. Intermediaries, therefore, compete over awareness gaps, parameterized by : a smaller value of increases an intermediaries chances of 18 being chosen by the investor he meets. Remark: An alternative assumption that gives rise to the same property is the assumption that the set of projects delegated to a particular intermediary has to be a subset of the set of projects the intermediary reveals. This can be interpreted as the intermediary o↵ering a set of projects to which he has access and the investor restricting the intermediary’s choice to a subset of those. Payo↵s: An intermediary’s expected payo↵is the product of the probability of being chosen by the investor and his conditional expected payo↵. Using the intermediary’s optimal policy y⇤(✓;) as defined in the previous section, his conditional payo↵as a function of  is defined by: U()⌘1 2Zˆy ˆy (ˆy✓)2f(✓)d✓1 2Z1 ˆy (ˆy+✓)2f(✓)d✓. From the analysis in Section 3 we know that U() is strictly concave and attains its maximum at ⇤, as determined by (2). The probability that an intermediary is selected by the investor depends on the strategy of the other intermediary. Letting H() denote the probability that the other intermediary chooses an awareness gap smaller than , the probability that the investor delegates to the competitor is given by the product of the conditional probability that the investor meets the other intermediary and H(). Letting ¯ Udenote the outside option of both intermediaries, an intermediary’s expected payo↵is then given by (1 ⇡H())U() + ⇡H()¯ U. (3) We assume that intermediation can be profitable, that is: ¯ U<U(⇤). Equilibrium: A symmetric equilibrium in this environment is a (cumulative) distribution function H⇤() such that intermediaries are indi↵erent between all elements in the 19 support of H⇤and weakly prefer those over any other values of . Since U() is strictly decreasing for >⇤, the support of H⇤has to be a subset of [0,⇤]. Standard arguments show that H⇤is continuous. The intermediaries’ strategy thus has no mass point, except possibly at = 0. We are now ready to describe the equilibrium distribution H⇤in this environment. While the appendix provides a complete characterization of H⇤,wefocushereonsomekeyfeatures of the equilibrium. Proposition 4.1. Let Assumptions 1 and 2 be satisfied. In the environment with imperfect competition, there exists an equilibrium, characterized by H⇤, with the following properties: •if ⇡⇡, the support of H⇤is [0,⇤]for some 00; •if ⇡<⇡<⇡, the support of H⇤is {0}[[0,⇤]for some 0>0; •if ⇡⇡, the support of H⇤is {0}; where 0<⇡⇡1. Proof. See Appendix B.5. Proposition 4.1 shows that there are three parameter regions to be distinguished. If the degree of competition is sufficiently small, then intermediaries choose awareness gaps parameterized by values of in the interval [0,⇤]. As we show in the proof of Proposition 4.1, the size of this interval is strictly increasing in the competition parameter ⇡.Consequently, the more competition there is, the smaller is the minimal awareness gap. At the threshold ⇡,thelowerboundoftheinterval, 0, is zero. When ⇡increases further we observe polarization: there is a strictly positive probability that intermediaries disclose everything (= 0), while otherwise they leave investors unaware of a significant part of the available investment opportunities. As ⇡increases further, the probability that intermediaries leave investors unaware of certain projects becomes smaller, up to the point where this probability is zero, which happens at the second threshold ⇡.Forallvaluesof⇡greater than this 20 D* p £ p p < p <p p £ p D 1 H* Figure 4: Equilibrium distribution H⇤under imperfect competition threshold, intermediaries then fully reveal. Figure 4 depicts the equilibrium distribution for the di↵erent regimes of ⇡. We thus find that competition promotes unawareness. In fact we can show that as ⇡ increases the equilibrium distribution is shifted towards smaller values of . Corollary 4.2. Let Assumption 1 and 2 be satisfies and let 0⇡<⇡ 01. In the environment with imperfect competition, the equilibrium distribution H⇤under ⇡first-order stochastically dominates the one under ⇡0. Proof. See Appendix B.6. Summing up, in the proposed environment the disclosure of available investment opportunities is an instrument to compete for costumers. An interesting question is how the extent to which investors are left unaware of certain investment opportunities varies over the business cycle. In our framework the state of the economy might be captured by the profitability of investments: when the economy is doing well, financial market investments yield particularly high returns, some of which are appropriated by the financial intermediaries. In our model we thus interpret good times as an upward shift of U() relative to U.Wefindthat 21 as the gap between U() and Uincreases, the equilibrium distribution H⇤shifts towards smaller values of , illustrated in Figure 5. That is, when the gains from intermediation increase, investors become aware of more and more investment opportunities. Intuitively, when the value of attracting an investor becomes larger, competition for investors increases and this results in smaller awareness gaps. Viceversa, when times are bad and gains from intermediation are small, intermediaries worry less about loosing investors to competitors and hence behave more predatory. Our model therefore predicts that in bad times we will observe more banks taking advantage of costumers by hiding certain investment opportunities than in good times. D* Good Times Bad Times D 1 H* Figure 5: Equilibrium awareness in good and bad times 4.1 Heterogenous Investors Thus far we have assumed that investors are equally sophisticated, in particular that they are all completely unaware.8In reality, investors vary widely in their financial literacy, which gives rise to the question of how intermediaries optimally act when investors di↵er in their 8The extension to the case where investors are homogenous but aware of some projects is straight forward. In this case we find the largest feasible value of and, if it turns out to be smaller than ⇤, repeat the derivation of the equilibrium above, replacing ⇤with that value. 22 awareness. Will the presence of more sophisticated investors be beneficial or detrimental to the welfare of the less sophisticated ones? To address this question in the simplest fashion, we assume that there are two types of investors, sophisticated ones that are aware of all investment opportunities and naive ones that are aware of none. We denote the fraction of naive investors by µand assume that each investor is privately informed about her type. Upon meeting an investor, an intermediary is then confronted with two unknowns. He does not know whether the investor has access to the second intermediary and he does not know whether the investor is sophisticated or naive. If the investor is sophisticated, she delegates the optimal interval under full awareness, [0,ˆy], no matter how much the intermediary reveals. Nevertheless, she still rewards an intermediary for disclosure by choosing the one that reveals more. If the investor is naive, everything remains as above. Intermediaries then compete over awareness gaps, parametrized by . An intermediary’s expected payo↵as a function of is now given by (1 ⇡H())[µU() + (1 µ)U(0)] + ⇡H()¯ U. (4) Following the steps of the equilibrium construction above, we can characterize the equilibrium for this environment and obtain the same properties as described in Proposition 4.1.9Of course, the equilibrium distribution H⇤will now depend on µ. The following proposition shows that, provided intermediaries can make positive profits with sophisticated investors, an increase in the fraction of sophisticates leads intermediaries to disclose more investment opportunities in equilibrium. Proposition 4.3. Let Assumption 1 and 2 be satisfies. Assume U<U(0) and let 0 µ<µ 01. In the environment with heterogenous investors, the equilibrium distribution H⇤ under µis first-order stochastically dominated by the distribution under µ0. 9For details see the proof of Proposition 4.3. 23 Proof. See Appendix B.7. The result in Proposition 4.3 is intuitive. Whenever an intermediary meets a sophisticated investor, revealing additional investment opportunities does not a↵ect the delegation set the investor determines but increases the probability with which the investor delegates to him. The assumption U(0) >¯ Uimplies that intermediaries make strictly positive profits with sophisticated investors, thus, conditional on meeting a sophisticated investor, it is optimal to fully reveal. By implication, the larger the probability an intermediary attaches to the event of meeting a sophisticated investor is, the more attractive revealing additional projects becomes. The presence of sophisticated investors in the market consequently leads to more disclosure and thereby benefits the naive types. Suppose now that intermediaries can make positive profits with naive investors but not with sophisticated ones. If intermediaries can reject sophisticated investors, the equilibrium is as if they did not exist and so the previous analysis applies. There are, however, situations where it is reasonable to assume that intermediaries cannot avoid negative profits with certain type of investors. For example, advising and setting up a contract may imply certain opportunity costs. If the investor’s type is initially unknown and if the expected profits with sophisticated investors do not compensate the opportunity costs, intermediaries make losses with such investors. As long as these losses are compensated by the profits with other investors, intermediaries may still find it worthwhile to enter the market. In our framework this situation is captured by the specification U(0) <¯ U<µU(⇤)+ (1 µ)U(0) and the assumption that intermediaries cannot reject any delegation sets. In contrast to the previous case, intermediaries are then no longer interested in attracting sophisticated investors but would rather have them go to competitors. This brings about a new interesting equilibrium property. Proposition 4.4. Let Assumption 1 and 2 be satisfies. Assume U()<U<µU(⇤)+ (1 µ)U(0) and let 0µ<µ 01. In the environment with heterogenous investors, 24 the equilibrium distribution H⇤under µfirst-order stochastically dominates the distribution under µ0. Proof. See Appendix B.7. Proposition 4.4 shows that a larger share of sophisticated investors leads to more unawareness among naive investors. Given U(0) <U, revealing all investment opportunities cannot be optimal for intermediaries. Indeed, regardless of how large ⇡is, there is no ’full awareness’ equilibrium. Instead, intermediaries randomize across an interval of values of  bounded away from zero. The losses they make with sophisticates can thereby be compensated with the profits they make with naive investors. The larger the share of sophisticated investors is, the larger this compensation has to be. Hence, as µdecreases, the equilibrium distribution shifts towards higher values of . The presence of sophisticated investors in the market thus reduces the awareness and, by implication, the welfare of naive investors. The feature that there is unawareness in equilibrium - no matter how intense competition is - with cross-subsidization across naive and sophisticated investors is reminiscent of the shrouding equilibrium in Gabaix and Laibson (2006). In their work, firms hide costly addons, which in equilibrium will be purchased by naive customers only. 5 Policy Implications and Conclusion The paper has implications for bank regulation and brokerage practices. Given that small investors are those more likely to have limited awareness, our results show that banks may have incentives to eliminate investment opportunities with intermediate levels of risk so as to induce investments in risky assets. Of course educating investors about available investment options, thereby expanding YP,benefitsinvestorsinourenvironment. Inreality,however, promoting full awareness in that way might not always be feasible or might be very expensive. 25 B.3 Proof of Lemma 3.4 Consider delegation set Dand suppose max D<max b Y.Lety=maxDand consider project y>max Dsuch that y2b Y.Lett=y+y 2denote the state at which the intermediary is indi↵erent between the two projects. When t<1, the change in the investor’s payo↵when including project yis then given by11 Z1 t (y✓+)2f(✓)d✓+Z1 t (y✓+)2f(✓)d✓, =Z1 t [(yy)(y+y)2(yy)(✓)] f(✓)d✓, =2(yy)S(t). This change is weakly positive if S(t)0, i.e. if tˆy. The condition tˆyis equivalent to y+y 2ˆy,yˆyˆyy. Since y>ˆy, this condition can only be satisfied if y<ˆy. Given this, the inequality is always satisfied when yˆy.Itisalsosatisfiedwheny>ˆy, provided that d(y, ˆy)d(y,y). By a perfectly symmetric argument 2(yy)S(t) is strictly negative if yˆy>ˆyy,whichis satisfied if and only if d(y, ˆy)>d(y, ˆy). B.4 Proof of Proposition 3.5 Let the intermediary’s payo↵as a function of be defined by U() := Zˆy ˆy (ˆy✓)2f(✓)d✓Z1 ˆy (ˆy+✓)2f(✓)d✓ The first and second derivative of U() are @U() @=2 Zˆy ˆy [ˆy✓]f(✓)d✓2Z1 ˆy [ˆy+✓]f(✓)d✓,(6) @2U() @2=2[1 F(ˆy)] <0(7) 11When t1 including yhas no e↵ect on the investor’s payo↵since the intermediary never chooses y. 32 U() is strictly concave in and hence has a unique solution on [0,¯ (b Y)]. Recalling that ˆy=E[✓|✓ˆy], the first derivative is strictly positive when evaluated at = 0: @U() @=0 =2[1F(ˆy)]>0. The derivative is strictly positive, which implies that whenever ¯ (YP)>0, the optimal value of is strictly positive. The interior solution of the intermediary’s optimization problem, ⇤, is characterized by the first-order condition that equalizes the expression (6) to zero. After rearranging terms ⇤solves: (1 F(ˆy⇤)) (E[✓|✓ˆy⇤](ˆy⇤)) = 2(1 F(ˆy)). B.5 Proof of Proposition 4.1 Suppose first that both intermediaries choose = 0 in equilibrium and consider the deviation of one intermediary. Since = 0 maximizes an investor’s payo↵, any deviating o↵er will only be accepted if the investor is captive. The best deviating o↵er is thus characterized by ⇤. This deviation is not profitable if ⇡¯ U+(1⇡)U(⇤)✓11 2⇡◆U(0) + 1 2⇡¯ U. (8) Letting ⇡be the value of ⇡at which (8) holds as equality, it is easy to verify that (8) holds for all ⇡⇡. Suppose now there is a positive probability with which intermediaries choose a non-zero gap. Since Hcannot have a mass point at >0, it follows that there is an interval of values of across which intermediaries are indi↵erent. Di↵erentiation of (3) yields the following first order conditions: (1 ⇡H())U0() = ⇡H0()(U()¯ U)for>0; (1 ⇡H())U0()⇡H0()(U()¯ U)for=0; 33 Let ˆ H(·) be the solution of the di↵erential equation defined by the first order condition with border condition ˆ H(⇤)=1. Wehave ˆ H() = U()[⇡¯ U+(1⇡)U(⇤)] ⇡[U()¯ U]. Suppose ˆ H(0) 0 and consider the following candidate equilibrium distribution: H⇤() = 8 > < > : 0if0 U()[⇡¯ U+(1⇡)U(⇤)] ⇡[U()¯ U]if 2(min,⇤) 1if⇤ with 0such that U(0)=⇡¯ U+(1⇡)U(⇤). Any value of strictly smaller than 0 cannot be optimal as it yields the same trading probability but a lower conditional payo↵ than 0. The equilibrium exists if indeed ˆ H(0) 0, or equivalently U(0) ⇡¯ U+(1⇡)U(⇤)(9) Letting ⇡denote the value of ⇡at which (9) is satisfied with equality, the characterized equilibrium exists if ⇡⇡. Finally, consider an equilibrium where H⇤has a mass point at = 0. Notice first that the intermediaries’ strategy cannot have a mass point at zero and at the same time positive density arbitrarily close to zero, since = 0 yields a strictly higher payo↵than any close to zero. The support of H⇤() thus has to have a gap. More precisely, suppose the support of H⇤() is given by {0}[[0,⇤], where 0is such that, given H⇤() = ˆ H(),80, the intermediary is indi↵erent between o↵ering 0and = 0. That is, ⇣1⇡ˆ H(0)⌘U(0)+⇡ˆ H(0)¯ U=✓1⇡1 2ˆ H(0)◆U(0) + ⇡1 2ˆ H(0)¯ U. (10) The left hand side is equal to ⇡¯ U+(1⇡)U(⇤)andthusconstantin 0, while the right hand side is strictly decreasing in 0.At 0= 0 the left hand side is strictly larger than the 34 right hand side when condition (9) is violated and at 0= ⇤the left hand side is strictly smaller than the right hand side when condition (12) is violated. Hence, whenever neither of the above equilibria exists, condition (10) has a unique solution in (0,⇤). We then have H⇤() = 8 > > > > < > > > > : 0if<0 ˆ H(0)if2[0,0] U()[⇡¯ U+(1⇡)U(⇤)] ⇡[U()¯ U]if 2(0,⇤) 1if⇤ where 0is defined by (10). B.6 Proof of Proposition 4.2 We can first show that ˆ H() shifts upwards as ⇡increases. We have indeed: @ˆ H() @⇡ =U(⇤)U() ⇡2[U()¯ U]>0 (11) For the case ⇡⇡, this immediately yields the stated property. Consider then the case ⇡<⇡<⇡.Giventheaboveproperty,itremainstoshowthatthemasspointat=0is greater when ⇡is greater. A sufficient condition is that the lower bound of the randomization interval, 0as determined by (10) is increasing in ⇡. Solving (10) for U(0) we obtain U(0)= 1 2(⇡U+(1⇡)U(⇤))(U(0) + U)U(0)U ⇡U+(1⇡)U(⇤)1 2(U+U(0)) The first derivative with respect to ⇡is given by @U(0) @⇡ = 1 4(U(⇤)U)(U(0) U)2 (⇡U+(1⇡)U(⇤)1 2(U+U(0)))2>0 Since Ustrictly increases in on [0,⇤], it follows that 0strictly increases in ⇡.Together with (11) this implies that the statement of Proposition 4.2 is satisfied. Finally, for ⇡⇡, a marginal change in ⇡has no e↵ect. 35 B.7 Proof of Proposition 4.3 The construction of the equilibrium is analogous to the one in Section B.5. The equilibrium in which both intermediaries choose = 0 exists if a deviation to =⇤is not profitable. This requires ⇡¯ U+(1⇡)[µU(⇤)+(1µ)U(0)] ✓11 2⇡◆U(0) + 1 2⇡¯ U. (12) When the above condition is not satisfied intermediaries randomize across di↵erent values of . We can first derive the function ˆ H(). Di↵erentiating the intermediaries’ expected payo↵in (4) yields (1 ⇡H())µU0() = ⇡H0()(µU() + (1 µ)U(0) ¯ U)for>0; (1 ⇡H())µU0()⇡H0()(µU() + (1 µ)U(0) ¯ U)for=0. With the condition ˆ H(⇤)=1weobtain ˆ H() = µU()[⇡¯ U+(1⇡)µU(⇤)⇡(1 µ)U(0)] ⇡[µU() + (1 µ)U(0) ¯ U]. If ˆ H(0) 0, or equivalently (1 ⇡)µ(U(⇤)U(0)) ⇡(U(0) ¯ U),(13) there exists an equilibrium with H⇤() = 8 > < > : 0ifmin ˆ H() if 2(min,⇤) 1if⇤ (14) where min is such that µU(min)=[⇡¯ U+(1⇡)µU(⇤)⇡(1 µ)U(0)]. Finally, we consider the case where the equilibrium distribution H⇤with a mass point at = 0. As before, let 0be defined as the value of such that, given H⇤() = ˆ H(),8 36 0, an intermediary is indi↵erentbetween 0and = 0. That is, ⇣1⇡ˆ H(0)⌘[µU(0)+(1µ)U(0)] + ⇡ˆ H(0)¯ U=✓1⇡1 2ˆ H(0)◆U(0) + ⇡1 2ˆ H(0)¯ U.(15) By the same argument as in Section B.5, condition (15) has a unique solution in (0,⇤)if and only if neither of the above equilibria exists. Just notice that the left hand side of (15) is equal to ⇡¯ U+(1⇡)[µU(⇤)+(1µ)U(0)]. We then have H⇤() = 8 > > > > < > > > > : 0if<0 ˆ H(0)if2[0,0] ˆ H() if 2(0,⇤) 1if⇤ (16) with 0defined by (15). Consider now an increase in µ.Forvaluesofµsufficiently close to zero condition (12) is always satisfied, so a marginal change of µhas no e↵ect on the equilibrium distribution. As µincreases condition (12) may be violated and we enter the region where the equilibrium is described by (16). Notice first that 0, as determined by (15) is decreasing in µ.This follows from the fact that the left hand side is shifted upwards as µis increases, whereas the right hand side does not depend on µ. The probability that an intermediary chooses =0 in equilibrium is thus decreasing in µ. Notice further that di↵erentiating ˆ H() with respect to µwe obtain: @ˆ H() @µ=(1 ⇡)(U(⇤)U())(U(0) ¯ U) ⇡[µU() + (1 µ)U(0) ¯ U]2(17) The above term is strictly negative for all <⇤, which implies that the function ˆ H() shifts downwards as µincreases. Together with the property that the probability with which intermediaries choose = 0 decreases as µincreases this implies that, within the considered parameter region, a higher value of µresults in an equilibrium distribution function that first order stochastically dominates the distribution associated to a lower value of µ. 37 Finally, when µincreases further, we may enter the parameter region where the equilibrium is characterized by (14). The fact that ˆ H() shifts downwards when µincreases here directly validates the statement of Proposition 4.3. B.8 Proof of Proposition 4.4 Under the assumption U(0) <¯ U, condition (13) is always satisfied, so the equilibrium distribution H⇤is given by (14). We then just need to show that ˆ Hshifts upwards as µ increases. This follows directly from (17). 38