Learning in Crowded Markets

Transcription

1 Learning in Crowded Markets Péter Kondor London School of Economics Adam Zawadowski Central European University September 8, 28 Abstract We analyze the welfare effects of learning and competition in a novel entry-game with heterogeneous agents and endogenous information acquisition. Each investor decides whether to enter a new market but is uncertain about her competitive advantage relative to other investors, i.e. her type. Investors can learn about their types using flexible but costly signals. The outcome is socially inefficient; over- or under-entry depends on market characteristics. While increasing competition results in more learning, it does not change the overall entry. Hence, incremental learning is wasteful and welfare-decreasing. We embed the game into an explicit capital reallocation model connecting welfare effects and allocative inefficiency to the features of the production technology and potential shocks in the new market. For comments and suggestions, we are grateful to Patrick Bolton, Thomas Chemmanur, Johannes Hörner, Victoria Vanasco, Marianne Andries, Ming Yang, Martin Oehmke, Gábor Lippner and seminar participants at Boston University, Central European University, ESSET 24 (Gerzensee), Brigham Young University, GLMM 25 (Boston), Paul Woolley Conference 25 (LSE), University of Vienna, 6th Annual Financial Market Liquidity Conference (Budapest), American Finance Association 26 Annual Meeting (San Fransisco), Adam Smith Workshop 26 (Oxford), University of Maryland, University of Naples, University of Copenhagen, 4nation cup workshop (Copenhagen), Federal Reserve Bank of New York, Toulouse School of Economics, University of Zurich. We thank Pellumb Reshidi and Dániel Szabó for their excellent research assistance. The support of the European Research Council Starting Grant # is acknowledged. p.kondor@lse.ac.uk, zawadowskia@ceu.edu,

2 Introduction When an investor a venture capitalist, a hedge fund manager, a real estate developer or an innovative firm decides to enter a new market segment, she will naturally gather information on her advantage relative to her future competitors. How does this learning affect the efficiency of the aggregate capital reallocation process, and how does it affect welfare? We analyze the welfare effects of learning and competition in a novel framework. We formulate an entry-game with a continuum of heterogeneous agents and endogenous information acquisition. Each agent (investor hereon) considers whether to enter to the new market. An entrant s type represents her competitive advantage relative to others. That is, the pay-off of the entrant depends differentially on the mass of entrants with better and worse types. Initially, investors are uncertain about their type. They construct an optimal signal subject to the cost of learning. Based on this signal, they decide whether to enter. We show that the outcome of this game is socially inefficient. We characterize when there is over- and under-entry. We also show that increasing competition between investors induces them to learn more, shifting the distribution of entrants towards better types. However, competition does not change overall entry, thus it neither alleviates nor aggravates the inefficiency of the outcome. Thus, incremental learning due to fiercer competition turns out to be wasteful and welfare-decreasing. We embed the game into an explicit capital reallocation model connecting the welfare effects and allocative inefficiency to the features of the production technology and potential shocks in the new market. We study extensions where some investors can find better opportunities regardless of the type of other entrants, and where investors are heterogeneous in their cost of learning. In our game, investors type is distributed over the unit interval. Each investor s pay-off upon entry is a linear function of the mass of entrants with better types, and the mass of entrants with worse types. Each investor constructs an optimal signal structure to learn about her type. Building on Sims (998, 23), the cost of any given signal structure is proportional to the implied reduction

3 of entropy. Given the equilibrium strategies, an investor s posterior on her type implies a posterior on the mass of better and worse entrants. Hence, each investor learns about her competitive advantage on the given market relative to other entrants. Our formalization results in a parsimonious structure. In equilibrium, the optimal information acquisition and entry strategies are reduced to a single function, mapping investor s possible types into a probability of entry. For example, if the investor were to decide to enter with a given probability independently of her type, we represent this choice with a constant function. This strategy does not require learning. In contrast, if the investor were to decide to enter if and only if her type is better than a given threshold, this can be represented by a step function. However, for this, her signal has to be sufficiently precise to know with certainty whether her type is above this threshold. This strategy turns out to be very costly under our specification. In equilibrium, investors typically choose an interior strategy represented by a smooth, monotonic function, implying higher entry probability for better types. Our main focus is the welfare and efficiency effects of learning and competition. We argue that the interesting parametrization of the pay-off function is when an entrant s revenue is decreased by an additional better entrant, and when the effect of an additional average entrant (if she is better or worse with equal probability) is also negative. We model increased competition by increasing the mass of investors who have the opportunity to enter. Unless the mass is so small that the entry decision is trivial, increasing competition does not improve social efficiency of total entry. Instead, the entry stabilizes at an inefficiently low or high level. To understand this result, note that investors adjust their entry decisions along two main dimensions as competition increases. First, the marginal benefit of knowing your type more precisely before entering is increasing in competition because there are more investors with a better type. Thus investors choose entry strategies that are more contingent on their types. We refer to this as the rat race effect. Second, with more investors present, crowding becomes a bigger concern. This means each investor is less likely to enter on average, we refer to this 2

4 as the crowding effect. We show that these two effects cancel out in the aggregate. Thus aggregate entry remains constant and efficiency does not improve due to increased competition. Nevertheless, as competition increases, welfare decreases. The key insight is that the rat race effect increases with competition, thus investors choose to learn more. While ceteris paribus more learning can alleviate inefficient over-entry, more learning due to increased competition does not change the amount of entry. In equilibrium, learning induced by competition simply makes better type investors more likely and worse type less likely to enter. However, as the best entrants always make the best available investments, only the total mass of entrants matters for welfare, not their distribution. Thus the higher learning cost implied by more competition is socially wasteful. To gain more economic insights on the determinants of over- or under-entry, in the second part of the paper we build an explicit capital reallocation model. With this model in hand, we map the reduced form parameters of our game into economic fundamentals. We use the metaphor of two identical islands with heterogeneous pastures in each. One island is saturated with cows while the other one is empty. We represent capital reallocation by investors buying cows from resident farmers and carrying them from the earlier to the latter island. If an investor has the best type among the entrants, she is able to locate the worst pasture in the former island and the best pasture in the latter island thereby exploiting the highest surplus opportunity. We allow for idiosyncratic (e.g. a leak on the boat carrying the cow) and aggregate (e.g. a storm between the islands) shocks to the reallocation process. With the help of this model, we show that over- or under-entry depends on the technological characteristics of the new investment opportunity and the potential shocks, and not on the level of competition. In particular, we find that markets are more prone to over-investment when: () idiosyncratic liquidity shocks are less likely, (2) aggregate shocks are more likely or more severe, and (3) learning is harder. Intuitively, a market with more investors provides easier exit opportunities for those hit by an idiosyncratic liquidity shock. This is not internalized by market participants leading to under-entry of investors. Also over-entry is more likely for investments with more likely and more 3

5 severe aggregate shocks, since investors do not internalize the fire sale externalities. Finally, markets with higher cost of learning are more prone to over-investment because investors have a harder time conditioning on their relative type to avoid over-entry and thus the outcome resembles the tragedy of commons. We analyze two extensions. First, we show that when better types find socially more valuable deals in the new market, then more competition often leads to an allocation that is too efficient compared to the planner s solution. This is due to over-learning and thus it decreases welfare. Second, we also extend our model to the case in which there is heterogeneity across investors: some are more sophisticated than others. Keeping the mass of all investors fixed but increasing the share of sophisticated investors might also decrease welfare. Initially, increasing the fraction of sophisticated investors increases welfare since it raises the average sophistication of investors and this can alleviate over-entry. However, further increasing the fraction of sophisticated investors beyond a certain threshold, less sophisticated investors are afraid of being ripped off and exit the market. Once less sophisticated investors exit, sophisticated investors engage in a vicious rat race of learning which leads to decreasing welfare. Identifying the most sophisticated investors as high-frequency traders connects this result to the policy debate on the social benefit of ultra-high frequency trading. Our main contribution is to study the welfare effects of optimal learning in an entry game with uncertain competitive advantage. Our paper is connected to various branches of literature. First, there is a growing literature on the welfare effects of endogenous information acquisition, e.g. Myatt and Wallace (22) and Colombo, Femminis, and Pavan (24). While this literature focuses on a common-value learning, we analyze an environment when agents learn about their relative advantage compared to the other entrants. Second, from a methodological view point we rely on the rational inattention approach pioneered by Sims (998, 23). We follow the branch of the literature which allows for fully flexible informa- See, for example, Securities and Commission (2). 4

6 tion acquisition as Matějka and McKay (25), but restrict ourselves to binary actions similarly to Woodford (28), Yang (25a,b). 2 Third, there are numerous papers showing excessive investment in learning or effort. There is a literature on the social value of private learning: e.g. in Hirshleifer (97), private information can be detrimental as it changes ex ante incentives for insurance, in Glode, Green, and Lowery (22), learning effects ex-post trading opportunities. More generally, socially inefficient effort choice has also been emphasized in very different settings: e.g. Tullock (967), Krueger (974), and Loury (979). Our focus is different compared to the above papers: we are interested in how learning affects capital allocation and welfare. Fourth, there is a literature analyzing entry/exit in the presence of externalities induced by other investors. Stein (29) introduces a simple model of crowded markets but leaves the effect of learning in such models for future research. Abreu and Brunnermeier (23) and Moinas and Pouget (23) show that the inability to learn about one s relative position versus that of other investors is a key ingredient in sustaining excessive investment in bubbles. This highlights our contribution in adding learning to a model of crowded markets with potential over-entry. The rest of the paper is structured as follows. In Section 2, we present our model. In Section 3, we present the optimal choice of entry and learning and analyze its implications for aggregate entry, market efficiency, the median entrant and welfare. In Section 4, we give a structural microfoundation for the model and analyze its economic implications. In Section 5, we analyze extensions of the payoff function and also allow for heterogeneity in investor sophistication. Section 6 concludes. All proofs are relegated to Appendix A. Further extensions are explored in online Appendices B and C. In the latter we analyze Gaussian signals instead of fully flexible learning. 2 The other succssful approach is to allowing for continuous actions, but restricting signals to be Gaussian. See Maćkowiak and Wiederholt (29), Hellwig and Veldkamp (29) and Kacperczyk, Nieuwerburgh, and Veldkamp (26) for intriguing models using this approach. 5

7 2 A model of learning and investing in crowded markets In this part we describe our setup. We first present the payoff function, then introduce the flexible learning technology and define the real outcomes. 2. Payoffs Consider an entry game with a continuum (mass M) of investors, each with a type θ uniformly distributed over [, ]. Each investor can decide to take an action: whether to enter the market or not. θ characterizes the investor s ability to identify better investment opportunities in this new market than others. Lower θ implies a better type. The utility gain (or loss, if negative) from entry is given by u(θ) = β b(θ) + α a(θ) κ θ () where α and β are constant parameters. b (θ) denotes the equilibrium mass of entrants with a type better than θ. a(θ) denotes the equilibrium mass of entrants whose type is worse than θ. We show in the microfoundation in Section 4 that it is natural to assume that that β > α. First, this implies that, β + α >, which is without loss of generality since it is simply consistent with the interpretation that a lower θ represents a better type. Second, it follows that β α >, such that the median entrant imposes a negative externality on others, that is, the market is prone to getting crowded from a social point of view. It also follows that β > while α could be positive or negative. When κ >, better investors have an absolute advantage, that is, better types derive more utility from entering regardless of the entry decision of others. We discuss this case in Section 5., otherwise we analyze the simpler case of κ =. As we specify below, investors do not know their type, but can gather information about it through a costly learning process. 6

8 In this section and Section 3 we work with the reduced form payoff (). In Section 4 we embed the reduced form game into an explicit model of capital reallocation. 3 This provides further insights regarding the economic interpretation of the parameters α, β and κ. 2.2 Learning cost based on entropy Before entry, investors can engage in costly learning about their type. Observe that if H ( ) is any intuitive measure of uncertainty then H (θ) H (θ s), the reduction of uncertainty after observing signal s, is a measure of learning induced by signal s. Following Sims (998), we measure uncertainty by specifying H ( ) as the Shannon-entropy of a random variable. 4 Therefore, we specify the cost of learning a signal s as being proportional to the induced reduction in entropy of θ : H (θ) H (θ s). This quantity is often called the mutual information in θ and s. As Sims (998) argues, the advantage of such a specification is that it both allows for flexible information acquisition and can be derived based on information theory. Note that the payoff () for a given θ in our model is linear in entry. Woodford (28) derives the optimal signal structure and entry decision rule for such problems which we restate in the lemma below. Lemma. Optimal signal choice. The optimal signal structure is binary: investors choose to receive signal s = with probability m(θ) and s = with probability m(θ), given their type θ. The optimal entry decision conditional on the signal is: enter if s =, stay out if s =. Thus, similar to Yang (25a) the conditional probability of entry m(θ), or equivalently, the conditional probability of getting a signal, is the only choice variable. The intuition for the binary signal structure is that the only reason investors want to learn about θ is to be able to make a binary decision of whether or not to enter. Given the linearity of the problem, the cheapest signal to 3 In an earlier version of this paper, Kondor and Zawadowski (26), we provide microfoundations in various other contexts, including production with local spill-overs, consumption with externalities, and academic publications. The critical feature of all microfoundations is that each investor s pay-off is lower if better types also enter, while worse entrants can either help or hurt. 4 The entropy of a discrete variable is defined as x probability P (x), see MacKay (23). P (x) log P (x), where the random variable takes on the value x with 7

9 implement the optimal entry strategy is also binary, it simply tells the investor whether or not to enter. We now write the cost of learning, defined by the reduction in entropy, in case of a binary information structure. Denote the amount of learning L using the mutual information in signal s defined in Lemma and in θ, L(m) H (θ) H (θ s) = H (s) H (s θ) = (2) ( [ ] [ ]) ( [ ] [ ]) p log + ( p) log m(θ) log + ( m(θ)) log dθ p p m(θ) m(θ) where the first equation is a property of Shannon-entropy. p denotes the unconditional probability of entry and is defined by: p = m( θ)d θ (3) The expression for learning (2) can be understood in the following way. There is no learning if the signal is uninformative about the state, that is, if it prompts the investor to enter with probability p unconditional on its type θ. Indeed, it is easy to check that when m (θ) is constant at p then L (m) =. Thus, learning depends on how much information the signal contains about the state. Intuitively, the steeper m(θ) becomes in θ (keeping average entry p constant), the more the investor is differentiating its entry decision according to its type and the higher the entropy reduction, thus the higher the learning cost. The highest cost is achieved when m (θ) is a step function. Note that L is bounded from above but might generate infinite marginal cost of learning. Our measure of the cost of learning induced by a signal defined in Lemma is µ L (m) where µ is an exogenous marginal cost parameter. We assume that investors have to decide about the amount of information acquisition ex ante without any knowledge about the action of others. We interpret this as the cost of building an information gathering and evaluation machine which includes the costs of gathering and optimally evaluating the right data. 8

10 Conveniently, standard results in information theory imply that the entropy of a random variable is proportional to the average number of bits needed to optimally convey its realization. Hence, the parameter µ can be interpreted as the cost of building a marginally larger information gathering and evaluating machine or writing a longer code Competitive and social solution We define the competitive solution of the game as a Nash equilibrium: strategy profiles m i (θ) : [, ] [, ] for all i [, ], such that investor i s strategy is a best response to all other agents strategy. We restrict our attention to looking for a symmetric Nash equilibrium in which all agents choose the same m(θ) function, thus the i subscript can be dropped and every investor solves: max m(θ) (m(θ) u(θ) µ L(m)) dθ, (4) where u is the utility gain of entrants defined by (). In the competitive solution, each investor takes u as given. We contrast the above competitive solution with that of a social planner who can choose the amount of learning and entry for all investors. This gives us a benchmark against which we can evaluate learning and entry decisions in the competitive equilibrium. The main difference between the competitive solution and the social planner s one is that the social planner takes into account the externalities that investors exert on each other. The social planner chooses the symmetric function m s (θ) to maximize max m s(θ) (m s (θ) u(θ) µ L(m s )) dθ. (5) The social planner takes into account that u also depends on the choice function m s of all investors. 5 An alternative would be to think of capacity as limited and µ being the Lagrange multiplier of the capacity constraint. Instead, our modelling choice captures the idea that in most relevant contexts learning capacity can be expanded, even if for a cost. That is, the investor can decide to use more complex code to evaluate data, hire new staff or spend more time with the analysis before entry. 9

11 2.4 Definition of efficiency and welfare In this part, we define our main economic objects of interest. Wherever ambiguous, we denote the counterpart of each object in the social solution with a subscript s. Note first, that under symmetric strategies, the mass of lower types entering (better than investor θ) becomes: θ b(θ) = M m( θ)d θ (6) mass of higher types entering (worse than investor θ): a(θ) = M θ m( θ)d θ (7) thus M p = b(θ) + a(θ) is the aggregate entry of investors. Accordingly, our first measure for efficiency is the difference between aggregate entry in the competitive and social solutions. For instance, we refer to M p > M p s as over-entry. The expected revenue for an investor, i.e., her expected pay-off before accounting for the cost of learning is: R m(θ) u(θ) dθ (8) Aggregate revenue M R weights the probability of entry of a given type with the pay-off to that type. In Section 4, we show formally that this measure quantifies allocative efficiency in a microfounded capital allocation model. Therefore, we use the difference between aggregate revenue in the competitive and social solutions as our second efficiency measure. Whenever aggregate revenue is smaller under the competitive solution, we say that the solution is inefficient. The difference between efficiency and welfare in our model is that welfare takes into account the cost of learning. In particular, a given investor derives V R µ L (9)

12 value by following strategy m(θ). This is what the investors maximize. Using the same notation, the overall welfare in the whole economy can be computed as W M V. () Finally, we are also interested in the type distribution of entrants. One simple measure of this object is the median entrant type θ /2 defined implicitly as θ/2 M m(θ)dθ = M p 2. 3 Model Solution In this section we present our main results. We allow for any β and α in () satisfying our parameter restrictions but restrict κ to. We analyze the κ > case in Section 5.. First, we analyze the no information and full information benchmark. Second, we formulate and solve the investors and the planner s problem. Third, we analyze over- and under entry. Finally, we analyze welfare and social efficiency measured by aggregate revenue. 3. Full and no information benchmark To better understand the optimal strategies and aggregate entry, we first look at the extreme cases of µ (full information) and µ (no information). Lemma 2. Entry under full and no information. For full information (µ = ), the competitive functions m(θ) is a step function, m(θ) µ= = θ, () Mβ resulting in ( ) M p µ= = min β, M (2)

13 aggregate entry. In the social planner s optimum, aggregate entry is ( ) M p s µ= = min β α, M. (3) Any symmetric strategy implying entry with probability p s µ= is a solution to the planner s problem. For no information (µ ), the competitive and social planner s entry functions m(θ) are constant at m(θ) µ = min ( ) ( 2 M(β α), and m s (θ) µ = min M(β α) ),, respectively. Therefore, the aggregate entry is ( ) 2 M p µ = min β α, M (4) ( ) M p s µ = min β α, M. (5) Aggregate revenue, M R and welfare, W, are under the competitive solution, but positive under the planner s solution. Under full information, each investor enters if only if her pay-off is non-negative. The last entrant earns zero revenue. Whether this implies under- or over-entry compared to the social planner s choice depends on the sign of α. There is competitive under-entry (over-entry) if α > (α < ), since investors with higher θ do not take into account the positive (negative) effect of their entry that accrues to entrants with lower θ. Note that under full information without externalities α =, competitive entry choice is socially optimal. Under no information, each investor enters with the same probability, p µ, at which she earns zero expected revenue. As they do not learn, this also implies zero welfare. This probability implies over-entry under any parameter values: investors enter twice as often than they should. The intuition is analogous to the tragedy of commons. Each investor fails to internalize that her own entry reduces the benefit of entry for everyone else in expectation, given that β > α. 2

14 Figure : Conditional entry function in the two benchmark cases m (conditional entry) m (conditional entry) θ θ The graphs show the probability of entry conditional on the type of the agent m(θ) in the benchmark cases of full information (µ =, left panel) and no information (µ =, right panel). For the κ = there are many social solutions, we graph the one that is the solution for κ +. The solid line is the competitive choice, while the dashed line is the socially optimal choice. Parameters: β = 4, α = 2, M =, κ =. 3.2 Optimal strategies in the general problem We derive the first order condition (FOC) of problems (4) and (5) using the variation method, i.e. we look for the function m(θ) such that if we take a very small variation around that function, the value function of the investors does not change. The following Lemma restates Proposition of Yang (25a) for our model. Lemma 3. Competitive best responses. Denote the strategy function of all other investors as m(θ), then θ u(θ) = M β m( θ)d θ + M α m( θ)d θ. (6) θ The unique best response of an investor to all other investors playing m(θ) is: i) m(θ) = if and only if u(θ) e µ dθ ; ii) m(θ) = if and only if e u(θ) µ dθ ; iii) m(θ) (, ) if and only if e u(θ) down by the first-order condition: θ M β µ dθ > and e u(θ) µ dθ >, the optimal m(θ) is pinned [ ( ) ( )] m(θ) p m( θ)d θ + M α m( θ)d θ = µ log log. (7) θ m(θ) p 3

15 To solve for the equilibrium m(θ), we differentiate the FOC (7) with respect to θ. That gives an ordinary differential equation for m(θ) where the original FOC at θ = is the boundary condition. The following Proposition presents the solution of this ordinary differential equation up to a constant m() (the boundary value). Proposition. Competitive equilibrium. There is a unique threshold M > pinned down by M (α + β) µ β M +α M = e µ e µ, (8) such that a) if M M all investors enter without learning m(θ) = and this is the unique symmetric competitive equilibrium, b) if M > M, there is a competitive symmetric equilibrium in which optimal entry function is given by: m(θ) = + W ( e α+β M µ m() θ+ +log m() ( m() m() )), (9) where W denotes the upper branch of the Lambert function 6 and m() is pinned down by the boundary condition: [ ( ) ( )] m() p M α p + = µ log log. (2) m() p We obtain the following solution for the planner s problem. Proposition 2. Socially optimal entry strategies. There is a threshold M s = β α, such that a) if M M s then all investors enter m s (θ) = 6 The definition of the upper branch of Lambert function is z = W (z) e W(z) if z >. 4

16 b) if M > M s then the socially optimal entry function is constant in θ: m s (θ) = M (β α). (2) The first feature to note is that investors want to differentiate between states, but the planner does not. Under the planner s solution the entry function is constant and there is no learning. The reason for this over-learning in the competitive equilibrium is that every investor wants to know whether she is better than the other investors even if this is wasteful from the social planner s point of view. 3.3 Over- and under-entry In this subsection we analyze how aggregate entry M p changes as the mass of investors M grows. Figure 2: Competitive entry functions for different levels of competition m(θ). Entry function M m(θ).5 Scaled entry function.8 M=.25 M= M=.25 M=.6 M=2. M= θ θ Entry functions for the competitive entry (m, left panel) and scaled competitive entry function (M m, right panel) for M =.25 (dotted line), M = (solid line), and M = 2 (dashed line). Parameters: β = 4, α = 2, µ =.5, κ =. To understand the effect of M on incentives in the general solution, see Figure 2 which shows the competitive optimal entry function for different levels of M. The left panel shows the unscaled functions, while the right panel is scaled by M, thus showing the aggregate entry by type in equilibrium. For small M =.25, investors enter for sure and there is no need for learning since revenues are high. To gain intuition on how m(θ) changes as M increases from to 2, consider the effect of larger M on 5

17 the benefit of entry u (θ) for a given investor, keeping the others strategy constant. First, note that we can measure the relative incentive for entering with a lower θ by differentiating u (θ) in θ, giving M (α + β) m (θ). Therefore, keeping other investors strategies fixed, the incentive to learn more and follow a more differentiated strategy is increasing in M. Loosely speaking, this results in a steeper m (θ) as it is apparent on the right panel of Figure 2. We refer to this as the rat race effect. Second, note that the benefit of entry for the average investor is u ( ) 2 = M β 2 m( θ)d θ+m α m( θ)d θ, 2 hence, keeping others strategy constant u ( ) 2 M = α m( θ)d θ β 2 2 m( θ)d θ < (α β) p 2 < (22) where the first inequality comes from the fact that m (θ) is decreasing in equilibrium. This suggests that for the average investor with θ = 2, increasing M decreases the incentive to enter as it is apparent in the left panel of Figure 2. We refer to this as the crowding effect that relies on β α >. While in equilibrium the strategy of other investors m( θ) also changes, implying further adjustments, as Figure 2 demonstrates, the total effect is still driven by this intuition. In the next proposition we show that these two effects exactly cancel out. Proposition 3. Entry in the competitive and socially optimal solution. The competitive aggregate entry is M p = min ( M, M ). ( ) M p s = min M, β α. The aggregate entry in the social planner s solution is Just as in the planner s solution, investors in the competitive equilibrium also enter with probability one for small M and aggregate entry M p is constant when M is large. However, that constant level, M, is in general different from the social optimum M s = β α. That is, whenever M > M, increasing the number of investors does not get the competitive equilibrium any closer to the socially optimal outcome. Figure 3 illustrates this part by showing the amount of total entry M p as a function of the mass of investors M. 6

18 Figure 3: Real outcomes as a function of investor competition M p Aggregate entry M R Aggregate revenue μ=.2. μ=.2 μ=.5.2 μ=.5 μ=5.5 μ=5 social social M M θ /2 Median entrant type W Welfare μ=.2 μ=.5.3 μ=.2 μ=.5.5 μ=5 social.2 μ=5 social M Aggregate entry, revenue, type of the median entrant and welfare as a function of the mass M of investors allowed to invest. The thin solid line is the social optimum, the thick lines are the competitive outcomes for three different values of µ: µ =.2 (dotted line), µ =.5 (solid line), µ = 5 (dashed line). Parameters: β = 4, α = 2, κ =. M For the intuition, recall that changing M changes the optimal strategy m (θ) for every investor through the rat race effect and the crowding effect. As the rat race effect primarily affects the slope of the entry function, as opposed to its level, it has little influence on M p. In contrast, due to the crowding effect the average entry, p, decreases. In equilibrium, the decrease in p is exactly proportional to the increase in M, keeping M p constant. This can be observed on Figure 2: even though the entry functions look very different in case of M = and M = 2, the areas under the scaled entry functions, i.e. aggregate entry, are the same. 7 7 Allowing investors to flexibly choose their information structure is crucial in generating the result of constant entry as the mass of investors increases. With flexible learning, the investors can optimally devise their information to exactly counter the increase in the mass of investors and thus enter at a constant aggregate rate. When learning is constrained, this is not 7

19 In Proposition 3 we showed that whether there is under- or over-entry is independent of the mass of investors M for M > M. In the following Proposition 4 we show that the level of under- or over-entry is determined by the parameters of the model. Proposition 4. Comparative statics of crowding. If there is a sufficient mass M > max[ M, M s ] of investors, the relative amount of competitive aggregate entry to social aggregate entry (crowding) M M s is. increasing in µ, the marginal cost of information, 2. decreasing in α, 3. increasing in β if α >, decreasing in β if α < and does not depend on β if α =. More costly information leads to more crowding, because the game is closer to a tragedy of commons problem as explained in Section 3.2. To understand the effects of β and α for fixed µ, consider their effect on the difference of social and private incentives in the exterme benchmarks presented in Lemma 2. In the full information benchmark, M M s µ = β α. As α measures the social benefit of low-type entrants which investors β fails to internalize, this ratio is decreasing in α. In contrast, for fixed α, larger β alleviate the effect of this externality. The reason is that the effect of β on total entry is proportional to the original level. For example, if investors suffer more by better entrants (larger β), total entry decreases both under the planner s and the competitive solutions. However, when α >, this decrease in entry is smaller in the competitive solution, because α > implies under-entry. Thus, the ratio M Ms µ = β α β goes up, leading to less severe under-entry. In contrast, in the no-learning benchmark, crowding is M M s µ = 2, a ratio unaffected by the parameters. Therefore, intuitively, the effects driving the full information benchmark are carrying over for intermediate values of µ. necessarily the case: we demonstrate this in Online Appendix C in which investors can only buy Gaussian signals about their type subject to the same entropy cost as before. 8

20 In Section 4 we build an explicit capital allocation model to better understand the determinants of α and β and to analyze the effect of the underlying externalities on over- and under entry. In particular, in Proposition 8, we examine the comparative statics in more detail. 3.4 Decoupling of welfare and social efficiency Now we turn to the question of welfare as more and more investors are present. We show that the presence of some investors (M < M) unambiguously increases welfare in the competitive equilibrium. Note that in the small M case, welfare does not depend on µ since no resources are spent on learning. The total mass of investors is small in this range, hence they do not try to beat each other by learning about their relative type. Instead, all decide to enter without putting resources in learning. As investors mass is marginally increasing, there is more entry, which increases aggregate revenue. Thus entry, revenue and welfare go hand-in-hand. The above insight that larger M means (at least weakly) higher welfare and more revenue to investors remains true in case of the social planner s optimum since no learning is chosen in that case. In the competitive equilibrium, raising M above M leads to decoupling of welfare and revenue: while aggregate revenue stays constant (even though at a suboptimal level), welfare decreases, see Figure 3. The reason is that as the amount of investors in the market grows, they start worrying about crowding and thus their relative type θ, inducing them to learn about it. A rat race ensues with increasing amounts invested in learning and reduced welfare. Thus an increasing mass of investors leads to a drop in welfare not because of crowding (the total amount of investors entering is constant) but because of increased spending on learning. Proposition 5. Welfare. If M > M, aggregate entry and revenue of investors stays constant as we increase M. However, welfare becomes decoupled from revenue, and converges to zero from above as M : W ( M) > lim W (M) = (23) M 9

21 The welfare in the social planner s optimum for M > M s is constant: W s (M) = 2 (β α). (24) Note, unlike in the no learning benchmark, welfare is strictly positive under competitive solution for any parameter values. This shows that learning is useful under the competitive solution. In fact, learning is a substitute for coordination. When learning is prohibitively expensive, each investor enters with a probability that is too large. When investors learn, they can partially condition their entry on their type allowing for smaller aggregate entry. Indeed, as Proposition 4 states, entry decreases when learning is less costly. The resulting coordination is inefficient, because social and private incentives for entry do not coincide. Still, it is always better than no learning at all. In contrast, the planner does not have to learn to coordinate as it can choose all investors actions. For further intuition, it is instructive to see how increasing the cost of learning µ affects welfare. This is illustrated on the last panel of Figure 3. Interestingly, the sign of the effect depends on the level of competition, M. When competition is fierce (high M), the dominating effect is that investors burn more resources on gathering information which decreases welfare. However, there is also a range of M when higher cost of learning increases welfare. The reason is that for low cost of learning, there is under-entry for α >. As the cost of learning increases, entry gets closer to the efficient level, which increases welfare. When M is not that high, implying relatively modest aggregate cost of learning, this effect dominates the increase in learning cost. Interestingly, this implies that if the policymaker could influence transparency of a competitive market (change µ), he should make markets with lots of investors more transparent, while markets with fewer investors and high α might benefit from less transparency. 2

22 3.5 Median entrant As the lower left panel on Figure 3 illustrates, increasing competition implies that the median type entering is lower (better). The right panel of Figure 2 gives the intuition why this is so. As more learning implies a steeper scaled entry function M m (θ), better types enter with higher probability, while worse types enter with lower probability. Therefore, in the aggregate, the median entrants have lower types. Regardless of whether there is too much or too little entry, the median entrant always has a lower θ (has a better competitive advantage) than in the social optimum. However, in our baseline model, there is no social benefit of this improved type distribution. 8 In fact better types entering just destroys welfare since more information is necessary for this to happen and learning is costly. In Section 4.3, we present a dynamic interpretation of our model under which smaller θ /2 corresponds to faster reallocation of capital. Under that interpretation, our model predicts inefficiently too fast capital reallocation. Contrary to full-information benchmark, the median entrant is not even with a large amount of investors. That is, θ /2 converges to a positive constant as M. The intuition is that, given individual rationality, the cost of aggregate learning is bounded from above by the total revenue. If only the very best were to enter, that would necessitate large amounts of individual learning. However, as aggregate learning is bounded from above, individual learning has to converge to zero when M grows without bound. We state the formal result in the next proposition. Proposition 6. Median entrant. The median entrant in the social planner s optimum is θ /2 = 2. If the mass of investors is small M M, in the decentralized solution θ /2 = 2. If the mass of investors is large M > M, the median type is lower θ /2 < 2. As the number of investors increases, 8 In Section 5. we introduce a variation of the model in which better types entering is valuable and show that this does not change any of the major insights. 2

23 in the limit it converges to: lim θ /2 = M + e M (α+β) 2µ (25) In the following Proposition 7, we show how the type distribution depends on the parameters of the game when there is a large mass of investors. Proposition 7. Comparative statics of the median entrant. In the limit of a large mass of investors (M ), the median entrant θ /2 is. increasing in µ, 2. decreasing in α, 3. increasing in β if α >, decreasing in β if α < and does not depend on β if α =. When M, competition is fierce and only the highest types earn a positive pay-off. If the cost to learn is smaller (smaller µ) then investors learn more, higher types avoid to enter, decreasing the median type. The effect of α and β is more subtle. From expression (25), the median type depends inversely on the product of the private benefits of learning, (α + β), and the level of aggregate entry, M. When private benefits are larger, there is more learning, higher types avoid entering, decreasing the median type. However, consistent with the discussion of Proposition 4, when α >, larger β also implies smaller aggregate entry, M. For α >, this second effect dominates, leading to a larger median-type as β increases. We further discuss how the equilibrium speed depends on deep parameters of our micro-foundation in Section Microfoundation: Capital reallocation To be able to better understand the economic forces and the comparative statics we present a microfoundation of our model. 22

24 4. Microfoundation without shocks In the next part we present the core of such a model in the context of capital reallocation. A richer model is presented in Section 4.2. There are two islands A and B indexed by i {A, B}. Each island is divided into infinitesimal farms owned by a farmer. On both islands, the farmer and his land are both indexed by ϑ uniformly distributed over [, ]. The quality of farms is heterogenous, lower ϑ means a better farm. Namely, on each island if there is a cow on farm ϑ, it produces γ δ ϑ of the consumption good, where δ < γ. Initially, on island A, each farmer ϑ [, k A, ] has a cow, while the rest of the farmers do not have any. On island B there are no cows. Farmers cannot move and transfer cows across islands, however, there is a mass M of investors uniformly distributed over types denoted by θ [, ] who can. Each can take a single cow from island A and take it to island B. Of those who decide to do so, the better types (lower θ) will be able to pick the cows of the worse quality (higher ϑ) non-empty farms on A not yet picked by better investors and take them to the best quality (lowest ϑ) empty farms on island B. This is consistent with the interpretation that θ measures investors ability, as better investors are matched with higher surplus deals. Investors have a large endowment of the consumption good which they can use to pay the farmers for the cows. For simplicity, we assume that each farmer contacted by an investor engages in Nash bargaining over the surplus from transferring the cows. As a result, investors end up with ξ (, ] share of the surplus. At this abstract level, cows stand for capital, and island B represents a profitable investment opportunity. In fact, our analysis might apply in various contexts. One interpretation that fits well is a real-economy decision in which global firms decide whether to enter a new market segment. In another interpretation, venture capitalists decide which start-up to support knowing that they will make a profit only if they identify the start-up which can produce the best product in that future market segment. Alternatively, we can interpret our variables in a financial context where hedge funds and other sophisticated investors are deciding whether to be active in a novel trading strategy like trading the CDS-bond basis. In this context, relative advantage might mean that they can identify a 23

25 given profitable strategy earlier than others. For this specific case, we explicitly consider a dynamic interpretation of our static model in Section 4.3. Finally, our results might also shed light on welfare effects of high-frequency trading where participants can make profit only if they can be the fastest traders among those who decide to enter. In this latter case, relative type is relative speed. 9 We show the following result. Lemma 4. Microfoundation of reduced form parameters. Choosing k A, = ξ δ, the expected payoff of investor θ from transporting capital (given that investor θ can enter at time t) simplifies to () with α = and β = 2δ ξ. This results in both our parametric assumptions being met as β α = α + β = 2δ ξ >. Note that both our parameter assumptions, β α > and α + β >, are met because δ > which measures the extent of decreasing return to scale. That is, entering with a better type is beneficial because there are more profitable investment opportunities available when not many better investors have entered. This shows that in this context our parameter restrictions are natural even in the absence of any externalities. While in this simple setup α = κ =, in the next section we introduce shocks leading to α. In Section 5. we analyze the case in which the technology of transferring the cows depends on the investors type, leading to κ >. To simplify the exposition we set ξ =. Setting ξ < does not change any of our main insights and the analysis is relegated to Appendix B. 9 The interpretation of the size of µ also depends on the economic context. In the real economy context, it might measure the uncertainty about the most valued product characteristics in the given market segment. In the financial context, µ might depend on the extent prices are informative about the mass of early entrants into a strategy. Examples for a trade with low µ would be that of twin stocks or on-the-run-off-the-run bonds: a large price gap implies that the strategy is not yet very crowded. Another example is merger arbitrage, where the price offered by the bidder is known. On the other hand, with high µ, it is very hard for the investor to determine whether to enter, e.g. because there is no clear price signal whether the trade is still profitable. Examples for such trades include: emerging markets, carry trade, January effect. In the language of Stein (29), high µ represents unanchored strategies. 24

26 4.2 Introducing shocks in the microfoundation In the above microfoundation, on each island if there is a cow on farm ϑ, it produces γ δ ϑ of the consumption good. In the present extension, we allow δ to be state and island-contingent and denote it by δ i. For simplicity we assume that investors have full bargaining power (ξ = ). To explore the effect of various direct externalities in our problem, we consider the possibility that the reallocation of cows (capital) is subject to shocks. There are two time periods t =,. At t = the investor decides whether to transfer a cow. The transfer is concluded at t =, by which time two types of shocks could hit the investor or the island. First, there could be a leakage on the boat transporting the cow. The transport is successful only with probability ν, with probability ν, the investor is hit by an idiosyncratic ( liquidity ) shock and has to go back to island A and sell the cow there in random order at t =. Second, a hurricane might devastate island B. With ex ante probability η, even if the capital transfer is successful, upon arriving on island B, there is an aggregate shock ( crisis ) and all investors have to sell their cows in a fire sale. In case of a fire sale the relative advantage of investors does not matter, they sell their cows on island B randomly, i.e. the best investors are not matched with the best farmers. On island A, we always have δ A = δ. However, on island B, δ B = δ only if there is no crisis but δ B = δ c + δ in case of the aggregate shock, where δ c >. In case of both shocks, the cows are sold to the best available farms. Note that unless there is an idiosyncratic shock, investors buy cows from the farmers at its marginal product on island A and sell capital at its marginal product on island B. Whenever this activity is profitable, it also decreases the difference between the marginal return on cows across the two islands, that is, it increases market efficiency. Idiosyncratic shock complicates this picture only to the extent that it introduces some redistribution among investors; an element which washes out by aggregation. Therefore, we can still interpret the aggregate revenue of investors as a measure of market efficiency. ν For technical reasons we assume ν < η δc 2δ, which ensures that not more than k A, cows are transported from island A. 25