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Wall Street and Silicon Valley: A Delicate Interaction George-Marios Angeletos Guido Lorenzoni MIT and NBER MIT, Federal Reserve Bank of Chicago NBER Alessandro Pavan Northwestern University September 23, 2007 Abstract Financial markets look at data on aggregate investment for clues about underlying pro…tability. At the same time, …rms’investment depends on expected equity prices. This generates a two-way feedback between …nancial market prices and investment. In this paper we study the positive and normative implications of this interaction during episodes of intense technological change, when information about new investment opportunities is highly dispersed. Because high aggregate investment is “good news” for pro…tability, asset prices increase with aggregate investment. Because …rms’ incentives to invest in turn increase with asset prices, an endogenous complementarity emerges in investment decisions— a complementarity that is due purely to informational reasons. We show that this complementarity dampens the impact of fundamentals (shifts in underlying pro…tability) and ampli…es the impact of noise (correlated errors in individual assessments of pro…tability). We next show that these e¤ects are symptoms of ine¢ ciency: equilibrium investment reacts too little to fundamentals and too much to noise. We …nally discuss policies that improve e¢ ciency without requiring any informational advantage on the government’ side. s Keywords: heterogeneous information, complementarity, volatility, ine¢ ciency, beauty contests. We thank Olivier Blanchard, Hyun Song Shin, Rob Townsend, Jaume Ventura, Iván Werning and seminar participants at MIT, the Federal Reserve Board, the 2007 IESE Conference on Complementarities and Information (Barcelona), the 2007 Minnesota Workshop in Macroeconomic Theory, and the 2007 NBER Summer Institute for useful comments. Angeletos and Pavan thank the NSF for …nancial support. Email addresses: angelet@mit.edu; glorenzo@mit.edu; alepavan@northwestern.edu. 1 1 Introduction Financial markets follow closely the release of macroeconomic and sectoral data, looking for signals about underlying economic fundamentals. In particular, high current levels of activity tend to forecast high future proﬁtability, leading to an increase in asset prices. At the same time, ﬁnancial prices aﬀect the real economy by changing the incentive to invest for individual ﬁrms. For a startup company, higher asset prices raise the value of a potential IPO and facilitate ﬁnancing from venture capitalists. For ﬁrms already quoted on the stock market, higher asset prices raise the value of equity issues and the market valuation of further investments within the ﬁrm. Both directions of causation—from real activity to ﬁnancial prices and from ﬁnancial prices to real investment—have been widely explored in existing theoretical and empirical work. The literature on the impact of macroeconomic news on asset prices goes back to Chen, Roll and Ross (1986) and Cutler, Poterba, and Summers (1989); the literature on the impact of asset prices on real investment goes back to Brainard and Tobin (1968), Tobin (1969), Bosworth (1975), Abel and Blanchard (1986), Barro (1990), and Morck, Shleifer and Vishny (1990). In this paper, we document novel positive and normative implications stemming from the interaction of these two channels when agents do not share the same information. We ﬁrst show that this interaction generates a feedback mechanism between the real and the ﬁnancial sector of the economy: high investment drives up aggregate activity; ﬁnancial markets interpret this as a positive signal about future proﬁtability; asset prices increase; this adds fuel to the initial increase in investment. We next show that this mechanism can exacerbate non-fundamental movements in real investment and asset prices, and can distort allocative eﬃciency. This mechanism seems particularly relevant in periods of intense technological change, when information regarding the viability and proﬁtability of new technologies is widely dispersed across the economy. Preview. We conduct our exercise within a neoclassical economy in which allocations would be ﬁrst-best eﬃcient if all agents had the same information. A large number of “entrepreneurs” gets the option to invest in a new technology. They have dispersed information about the proﬁtability of this technology and may sell their capital in a competitive ﬁnancial market before uncertainty is realized. The “traders” who participate in the ﬁnancial market are also imperfectly informed, but they observe aggregate investment, which provides a summary statistic of the information dispersed among the entrepreneurs. In this environment, movements in real investment and asset prices are driven by two types of shocks: “fundamental shocks,” reﬂecting actual changes in the long-run proﬁtability of investment, and “expectational shocks,” reﬂecting correlated mistakes in individual assessments of this proﬁtability. The positive contribution of the paper is to study how the interaction between real and ﬁnancial activity aﬀects the transmission of these shocks in equilibrium. Because high aggregate investment 1 is “good news” for proﬁtability, asset prices increase with aggregate investment. As a result, an endogenous complementarity emerges in investment decisions. An entrepreneur anticipates that the price at which he might sell his capital will be higher the higher the aggregate level of investment. He is thus more willing to invest when he expects others to invest more. In equilibrium, this complementarity induces entrepreneurs to rely more on common sources of information regarding proﬁtability, and less on idiosyncratic sources of information. This is because common sources of information are relatively better predictors of other entrepreneurs’ investment choices, and hence of future ﬁnancial prices. For the same reason, the entrepreneurs’ choices become more anchored to the common prior, and hence less sensitive to changes in the underlying fundamentals. It follows that the feedback between the real and the ﬁnancial sector of the economy ampliﬁes the impact of common expectational shocks while also dampening the impact of fundamental shocks. The normative contribution of the paper is to study whether the reaction of the economy to diﬀerent shocks is optimal from a social perspective. The mere fact that entrepreneurs care about the ﬁnancial market’s valuation of their investment does not, on its own, imply any ineﬃciency. Indeed, as long as all agents share the same information, equilibrium asset prices coincide with the common expectation of proﬁtability; whether entrepreneurs try to forecast fundamentals or asset prices is then completely irrelevant for eﬃciency. This is not the case, however, when information is dispersed. The sensitivity of asset prices to aggregate investment induces a wedge between private and social returns to investment: while the fundamental valuation of the investment made by a given entrepreneur is independent of the investments made by other entrepreneurs (i.e., there are no production externalities or spillovers), the market valuation is not. By implication, the complementarity that emerges in equilibrium due to the dispersion of information is not warranted from a social perspective. It then follows that the positive eﬀects documented above are also symptoms of ineﬃciency: equilibrium investment reacts too little to fundamental shocks and too much to expectational shocks. We conclude by examining policies that improve eﬃciency without requiring the government to have any informational advantage vis-a-vis the market. We ﬁrst consider interventions “during the fact,” while information remains dispersed. In particular, we consider a tax on ﬁnancial trades or other policies aimed at stabilizing asset prices. By moderating the reaction of asset prices to aggregate investment, these policies dampen the equilibrium impact of non-fundamental shocks, which improves eﬃciency. In so doing, however, these policies also dampen the equilibrium impact of fundamental shocks, which was ineﬃciently low to start with. It follows that these policies can raise welfare, but never achieve full eﬃciency. We next consider interventions “after the fact,” when uncertainty has been resolved. Building on results from Angeletos and Pavan (2007b), we show that full eﬃciency can be achieved by 2 introducing a tax on capital holdings that is contingent on both realized aggregate investment and realized proﬁtability. Although real and ﬁnancial decisions are sunk by the time these taxes are collected, the anticipation of these contingencies aﬀects the incentives entrepreneurs and traders face “during the fact.” By appropriately designing these contingencies, the government can induce agents to respond eﬃciently to diﬀerent sources of information, even if it can not directly monitor these sources of information. Discussion. The US experience in the second half of the 90’s has renewed interest in investment and asset-price booms driven by apparent euphoria regarding new technologies (e.g., the Internet), and on the optimal policy response to these episodes. A common view in policy discussions is that entrepreneurs and corporate managers are driven by noise traders and other irrational forces in ﬁnancial markets, or are irrational themselves. Elements of this view are formalized in Shiller (2000), Cecchetti et al (2000), Bernanke and Gertler (2001), and Dupor (2005). Similar concerns are currently raised for the investment boom in China. The presumption that the government can detect “irrational exuberance” then leads to the result sought—that it should intervene. While we share the view that expectational errors may play an important role in these episodes, we also recognize that these errors may originate from noise in information rather than irrationality. Furthermore, we doubt the government’s ability to assess fundamentals better than the market as a whole. Our approach is thus diﬀerent. On the normative side, we identify an informational externality that can justify intervention even by a policy maker with no superior information. At the same time, on the positive side, we show that the interaction between real and ﬁnancial activity can amplify the impact of noise and that this ampliﬁcation is stronger when information is more dispersed. This helps explain, without any departure from rationality, why periods of intense technological change, like the 90’s, may feature signiﬁcant non-fundamental volatility. Because the source of both ampliﬁcation and ineﬃciency in our model rests on the property that investment is largely driven by expectations about others’ choices rather than about fundamentals, our results are reminiscent of Keynes’ famous beauty-contest metaphor: “...professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself ﬁnds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors...” Keynes (1936, p.156). Implicit in Keynes’ argument appears to be a normative judgement that something goes wrong when investment is driven by higher-order expectations. However, Keynes does not explain why 3 this might be the case. More recently, Allen, Morris and Shin (2005), Bacchetta and Wincoop (2005) and Cespa and Vives (2007) have revisited rational-expectations models of asset pricing and have shown that, at least in certain cases, a mechanism similar to the one articulated by Keynes increases the impact of the common prior and of common noise on equilibrium prices. However, these papers abstract from the real sector of the economy. They also do not address whether the positive eﬀects they document are symptoms of allocative ineﬃciency. To the best of our knowledge, our paper is the ﬁrst to provide a complete micro-foundation for beauty-contest-like ineﬃciencies in the interaction between real and ﬁnancial activity. Other related literature. By focusing on the two-way feedback between real and ﬁnancial decisions as a potential explanation of “bubbly” episodes, the paper also relates to two other lines of work. One line studies rational bubbles in economies with ﬁnancial frictions or asset shortages (e.g., Ventura, 2003; Caballero, 2006; Caballero, Farhi and Hammour, 2006). The mechanisms studied in these papers also generate signiﬁcant non-fundamental movements, but they are unrelated to information. The second and more closely related line studies speculative ﬂuctuations in prices and investment due to heterogeneous priors regarding proﬁtability (e.g., Scheinkman and Xiong, 2003; Gilchrist, Himmelberg, and Huberman, 2005; Panageas, 2005). In these papers, investment and prices are largely driven by expectational shocks regarding others’ valuations. This is similar to the role of higher-order expectations in our paper. However, in these papers asset prices continue to reﬂect the social value of investment, ensuring that no ineﬃciency emerges.1 In our paper, instead, the impact of higher-order expectations is also the source of ineﬃciency. The paper also relates to the growing macroeconomic literature on heterogeneous information and strategic complementarities (e.g., Amato and Shin, 2006; Angeletos and Pavan, 2007a,b; Baeriswyl and Conrand, 2007; Hellwig, 2005; Hellwig and Veldkamp, 2007; Lorenzoni, 2006, 2007; Mackowiak and Wiederholt, 2006; Woodford, 2002). However, unlike the complementarities considered in these papers, which originate in monopolistic price competition, production or demand spillovers, or other payoﬀ externalities, the complementarity documented here is due to an informational externality: it emerges only when information is dispersed and only because aggregate activity is then a signal of the underlying fundamentals. This speciﬁc source of complementarity is the key to both the positive and the normative results of our paper.2 Also related are Subrahmanyam and Titman (2001), Goldstein and Guembel (2003), and Ozde1 In Gilchrist, Himmelberg, and Huberman (2005), ineﬃciency can arise due to the monopoly power of the owners of the ﬁrms issuing speculative stocks. 2 Morris and Shin (2002) show that more precise public information can have a detrimental eﬀect on equilibrium welfare in a game where the complementarity in individual actions reﬂects a discrepancy between private and social objectives; but they do not study the origin of this discrepancy and they assume it to be exogenous to the information structure. In our environment, instead, the discrepancy between private and social objectives, like the complementarity, originates in the two-way feedback between real and ﬁnancial decisions and crucially depends on the information structure; this has also interesting implications for the social value of information (see Section 5.3). 4 noren and Yuan (2007). These papers study a variety of feedback eﬀects between a ﬁnancial market and a real sector that features of a form of network externality. Like the papers mentioned in the previous paragraph, the complementarity in these papers is exogenous to the information structure. On the other hand, a complementarity that originates in an informational externality is featured in the currency-crises model of Goldstein, Ozdenoren and Yuan (2007); but it is of a diﬀerent kind than ours. In their model, the central bank looks at the size of attack to learn about the underlying fundamentals. The larger the attack, the worse the bank’s perception of the fundamentals. But then also the higher the bank’s willingness to abandon the peg and hence the higher the incentive for the individual speculator to attack. Layout. Section 2 introduces the baseline model. Section 3 characterizes the equilibrium and derives the positive implications of the model. Section 4 characterizes the socially eﬃcient use of information and contrasts it to the equilibrium. Section 5 discusses policy implications. Section 6 considers a number of extensions. Section 7 concludes. All proofs are in the Appendix. 2 The baseline model We consider an environment in which heterogeneously informed agents choose how much to invest in a “new technology” with uncertain returns. After investment has taken place, but before uncertainty is resolved, agents trade ﬁnancial claims on the returns of the installed capital. At this point, the observation of aggregate investment partially reveals the information that was dispersed in the population during the investment stage. Timing, actions, and information. There are four periods, t ∈ {0, 1, 2, 3}, and two types of agents: “entrepreneurs,”who ﬁrst get the option to invest in the new technology, and “traders,” who can subsequently purchase claims on the installed capital of the entrepreneurs. Each type is of measure 1/2; we index entrepreneurs by i ∈ [0, 1/2] and traders by i ∈ (1/2, 1]. At t = 0, nature draws a random variable θ from a Normal distribution with mean µ > 0 and variance 1/πθ (i.e., πθ is the precision of the prior). This random variable represents the exogenous productivity of the new technology and is unknown to all agents. At t = 1, the “real sector” of the economy operates: each entrepreneur decides how much to invest in the new technology. Let ki denote the investment of entrepreneur i. The cost of this 2 investment is ki /2 and is incurred within the period. When choosing investment, entrepreneurs have access to various signals (sources of information) that are not directly available to the traders. Some of these signals may have mostly idiosyncratic noise, while others may have mostly common noise (correlated errors). To simplify, we assume that entrepreneurs observe two signals. The ﬁrst one has only idiosyncratic noise and is given by xi = θ + ξi , where ξi is Gaussian noise, 5 independently and identically distributed across agents, independent of θ, with variance 1/πx (i.e., πx is the precision of the idiosyncratic signal). The second has only common noise and is given by y = θ + ε, where ε is Gaussian noise, common across agents, independent of θ and of {ξi }i∈[0,1/2] , with variance 1/πy (i.e., πy is the precision of the common signal). The more general case where all signals have both idiosyncratic and common errors is examined in the Supplementary Material. At t = 2, the “ﬁnancial market” opens: some entrepreneurs sell their installed capital to the traders. In particular, we assume that each entrepreneur is hit by a “liquidity shock” with probability λ ∈ (0, 1). Liquidity shocks are i.i.d. across agents, so λ is also the fraction of entrepreneurs hit by the shock. Entrepreneurs hit by the shock are forced to sell all their capital to the traders. For simplicity, entrepreneurs not hit by the shock are not allowed to trade any claims on installed capital.3 The ﬁnancial market is competitive and p denotes the price of one unit of installed capital. When the traders meet the entrepreneurs hit by liquidity shocks in the ﬁnancial market, they observe the aggregate level of investment from period 1, K = 1 0 ki di. They can then use this observation to update their beliefs about θ.4 Finally, at t = 3, θ is publicly revealed, each unit of capital gives a cash ﬂow of θ to its owner, and this cash ﬂow is consumed. Payoﬀs. All agents are risk neutral and the discount rate is zero. Payoﬀs are thus given by ui = ci1 + ci2 + ci3 , where cit denotes agent i’s consumption in period t. First, consider an entrepreneur. 2 If he is not hit by the liquidity shock his consumption stream is (ci1 , ci2 , ci3 ) = (−ki /2, 0, θki ), so 2 that his payoﬀ is ui = −ki /2 + θki . If he is hit by the shock, he sells all his capital at the price p 2 2 and his consumption stream is (ci1 , ci2 , ci3 ) = (−ki /2, pki , 0), so that his payoﬀ is ui = −ki /2 + pki . Next, consider a trader and let qi denote the units of installed capital he purchases in period 2. His consumption stream is (ci1 , ci2 , ci3 ) = (0, −pqi , θqi ), so that his payoﬀ is ui = (θ − p)qi . Remarks. The two essential ingredients of the model are the following: (i) the agents who make the initial investment decisions have dispersed private information, so that aggregate investment is a signal of the fundamental; (ii) there is some common source of “noise” that prevents aggregate investment from perfectly revealing the fundamental to all agents, so that the dispersion of information does not completely vanish by the time agents meet in the ﬁnancial market. The speciﬁc information structure we have assumed is a convenient way to capture these two properties. In particular, the role of the common signal y is to introduce correlated errors in the entrepreneurs’ assessments of proﬁtability in stage 1, thereby adding noise to the inference problem that the traders face in stage 2: in equilibrium, aggregate investment will move both with the 3 We relax this assumption in Section 6. Letting the traders observe the entire cross-sectional distribution of investments does not aﬀect the results. This is because, in equilibrium, this distribution is Normal with known variance; it then follows that the mean investment contains as much information as the entire cross-sectional distribution. 4 6 fundamental θ and with the common error ε, ensuring that aggregate investment reveals θ only imperfectly. As mentioned above, in the Supplementary Material we dispense with the common signal y and instead consider the case where entrepreneurs observe multiple private signals, all of which have both idiosyncratic and common errors. We also consider a variant that introduces unobserved common shocks to the entrepreneurs’ cost of investment as an alternative source of noise in aggregate investment. In both cases, our main positive and normative results (Corollaries 1 and 2) remain intact, highlighting that the key for our results is the existence of a common source of noise, not the speciﬁc form of it. A similar remark applies to other simplifying modeling choices. For example, we could have allowed the entrepreneurs that are not hit by a liquidity shock to participate in the ﬁnancial market; we could further have allowed all entrepreneurs to observe a noisy signal of aggregate investment, or a noisy price signal, at the time they make their investment decisions.5 What is essential for our results is only that the dispersion of information remains present both at the investment and at the trading stage. Also note the “liquidity shock” need not be taken too literally. Its presence captures the more general idea that when an agent makes an investment decision, be him a start-up entrepreneur or the manager of a public company, he cares about the market valuation of his investment at some point in the life of the project. A start-up entrepreneur may worry about the price at which he will be able to do a future IPO; a corporate manager may be concerned about the price at which the company will be able to issue new shares. In what follows, we interpret λ broadly as a measure of the sensitivity of the ﬁrms’ investment decisions to forecasts of future equity prices.6 Finally, note that there are no production spillovers and no direct payoﬀ externalities of any 2 kind: both the initial cost (−ki /2) and the eventual return on capital (θki ) are independent of the investment decisions of other agents. The strategic complementarity that will be identiﬁed in Section 3.1 originates purely in an informational externality. A benchmark with no informational frictions. Before we proceed, it is useful to examine what happens when the dispersion of information vanishes at the time of trading in the ﬁnancial market. That is, suppose that all the information that is dispersed during period 1 (namely, the signals {xi }i∈[0,1/2] and y) becomes commonly known in period 2. The fundamental θ then also becomes commonly known and the ﬁnancial market clears if and only if p = θ. It follows that the 2 expected payoﬀ of entrepreneur i in period 1 reduces to E[ui |xi , y] = E[θ|xi , y]ki − ki /2, which in 5 See Section 6 for these extensions. See Baker, Stein and Wurgler (2003) for complementary evidence that the sensitivity of corporate investment to stock prices is higher in sectors with tighter ﬁnancing constraints (which here can be interpreted as higher λ). 6 7 turn implies that equilibrium investment is given by ki = E[θ|xi , y] = πy πθ πx µ+ xi + y. πθ + πx + πy πθ + πx + πy πθ + πx + πy The key result here is that equilibrium investment is driven solely by ﬁrst-order expectations regarding the fundamental and is independent of the intensity of the entrepreneurs’ concern about ﬁnancial prices (as measured by λ). This result does not require θ to be perfectly known in period 2. Rather, it applies more generally as long as the asymmetry of information about θ vanishes in period 2.7 This case, which we henceforth refer to as the case with “no informational frictions,” provides a convenient reference point for the rest of the analysis. 3 Equilibrium Individual investment is described by a function k : R2 → R so that k(x, y) denotes the investment made by an entrepreneur with information (x, y). Aggregate investment is then a function of (θ, y): K(θ, y) = k (x, y) dΦ(x|θ), (1) where Φ(x|θ) denotes the cumulative distribution function of x given θ. Since traders observe aggregate investment and are risk neutral, the unique market-clearing price is p = E[θ|K], where the latter denotes the expectation of θ given the observed level of K.8 Since K is determined by (θ, y), p is also a function of (θ, y). We thus deﬁne an equilibrium as follows. Deﬁnition 1 A (symmetric) equilibrium is an investment strategy k(x, y) and a price function p(θ, y) that satisfy the following conditions: (i) for all (x, y), k (x, y) ∈ arg max E k (1 − λ) θk + λp (θ, y) k − k2 /2 | x, y ; (ii) for all (θ, y), p (θ, y) = E [ θ | K(θ, y) ] , where K(θ, y) = k (x, y) dΦ(x|θ). 7 To clarify this point, consider an arbitrary information structure. Let Ii,t denote the information of agent i in period t. Impose that no agent has private information about θ in period 2 so that E[θ|Ii,2 ] = E[θ|I2 ] for all i. From market clearing we then have that p = E[θ|I2 ]. From the law of iterated expectations we then have that E[p|Ii,1 ] = E[E[θ|I2 ]|Ii,1 ] = E[θ|Ii,1 ] for all i. It follows that every entrepreneur chooses ki = E[θ|Ii,1 ]. 8 Since the price is only a function of K and K is publicly observed, the price itself does not reveal any additional information. Therefore, we can omit conditioning on p. The case where p conveys additional information is examined in Section 6. 8 Condition (i) requires that the entrepreneurs’ investment strategy be individually rational, taking as given the equilibrium price function. Condition (ii) requires that the equilibrium price be consistent with rational expectations and individual rationality on the traders’ side, taking as given the strategy of the entrepreneurs. As it is often the case in the literature, tractability requires that we restrict attention to equilibria in which the price function is linear. Deﬁnition 2 A linear equilibrium is an equilibrium in which p(θ, y) is linear in (θ, y) . 3.1 Endogenous complementarity The optimality condition for the entrepreneurs’ strategy can be written as k (x, y) = E [ (1 − λ) θ + λp (θ, y) | x, y ] . (2) The linearity of p (θ, y) in (θ, y) and of E [θ|x, y] in (x, y) then guarantees that the entrepreneurs’ strategy is linear in (x, y); that is, there are coeﬃcients (β0 , β1 , β2 ) such that k (x, y) = β0 + β1 x + β2 y. (3) By implication, aggregate investment is given by K = β0 + β1 θ + β2 y = β0 + (β1 + β2 )θ + β2 ε. Observing K is thus informationally equivalent to observing a Gaussian signal z with precision πz , where z≡ K − β0 β2 =θ+ ε β1 + β2 β1 + β2 and πz ≡ β1 + β2 β2 2 πy . (4) Standard Gaussian updating then gives the expectation of θ given K as a weighted average of the prior and the signal z: E [θ|K] = πθ πz µ+ z. πθ + πz πθ + πz Because market clearing in period 2 requires p = E [θ|K] , we conclude that the equilibrium price satisﬁes p(θ, y) = γ0 + γ1 K(θ, y), where γ0 ≡ πθ πz β0 µ− πθ + πz πθ + πz β1 + β2 and These results are summarized in the following lemma. 9 γ1 ≡ (5) πz 1 . πθ + πz β1 + β2 (6) Lemma 1 In any linear equilibrium, there are coeﬃcients (β0 , β1 , β2 , γ0 , γ1 ) such that k (x, y) = β0 + β1 x + β2 y and p (θ, y) = γ0 + γ1 K (θ, y) . Moreover, γ1 > 0 if and only if β1 + β2 > 0. The key result here is the relation between K and p. Provided that high investment is “good news” for proﬁtability (in the sense that a higher realization of K raises the traders’ expectation of θ), ﬁnancial prices increase with aggregate investment (γ1 > 0). This in turn induces strategic complementarity in investment decisions. Indeed, when the entrepreneurs are choosing a higher level of investment, they are sending a positive signal to the ﬁnancial market, thus increasing the price at t = 2. But then each entrepreneur’s willingness to invest at t = 1 is higher when he expects a higher level of investment from other entrepreneurs, which means precisely that investment choices are strategic complements. We formalize these intuitions in the next result, which follows directly from replacing condition (5) into condition (2). Lemma 2 In any linear equilibrium, the investment strategy satisﬁes k (x, y) = E[(1 − α)κ (θ) + αK (θ, y) | x, y ], where α ≡ λγ1 and κ (θ) ≡ (7) (1−λ)θ+λγ0 . 1−λγ1 Condition (7) can be interpreted as the best-response condition in the coordination game that emerges among the entrepreneurs for a given price function: it describes the optimal strategy for each individual entrepreneur as a function of his expectation of aggregate investment (the relevant summary of the strategy of other entrepreneurs), taking as given the impact of the latter on ﬁnancial prices. The coeﬃcient α then measures the degree of strategic complementarity in investment decisions: the higher α, the higher the slope of the best response of individual investment to aggregate investment, that is, the higher the incentive of entrepreneurs to align their investment choices. The function κ(θ), on the other hand, captures the impact of the fundamental on the individual return of investment for given K, normalized by 1 − α.9 A similar best-response condition characterizes the class of linear-quadratic games examined in Angeletos and Pavan (2007a), including the special case of Morris and Shin (2002). However, there are two important diﬀerences. First, while in those games the degree of strategic complementarity α is exogenously determined by the payoﬀ structure, here it is endogenously determined as an integral 9 This normalization serves two purposes. First, it identiﬁes κ(θ) with the complete-information equilibrium level of investment in the game among the entrepreneurs, for given price function. Second, it ensures that the unconditional mean of investment is given by Ek(x, y) = Eκ(θ). 10 part of the equilibrium. Second, while in those games the degree of complementarity is independent of the information structure, here it actually originates in the dispersion of information. In fact, the complementarity in our setup is solely due to the informational content of aggregate investment. How much information aggregate investment conveys about θ determines the coeﬃcient γ1 , which captures the sensitivity of prices to aggregate investment. In turn, the coeﬃcient γ1 pins down the value of α, which captures the degree of complementarity in the entrepreneurs’ investment decisions. In the absence of informational frictions (the benchmark case examined in the previous section), aggregate investment provides no information to the traders, prices are thus independent of K, and the complementarity in investment decisions is absent. When instead information is dispersed, aggregate investment becomes a signal of θ, prices respond to aggregate investment and a complementarity in investment decisions emerges. The more informative aggregate investment is about θ, the stronger the complementarity. Because the complementarity depends on the informational content of aggregate investment, which in turn depends on the entrepreneurs’ strategies, to determine the equilibrium value of α we need to solve a ﬁxed-point problem. Before doing so, we ﬁrst show how this endogenous complementarity is instrumental in understanding the incentives entrepreneurs face in using their available sources of information. Lemma 3 In any linear equilibrium, β2 πy 1 = . β1 πx 1 − α (8) Therefore, provided that β1 , β2 > 0, the sensitivity of the entrepreneurs’ equilibrium strategy to the common signal relative to the idiosyncratic signal is higher the higher the equilibrium degree of complementarity. Let us provide some intuition for this result. Consider an entrepreneur’s best response to the strategy that other entrepreneurs follow, holding ﬁxed the price function. Suppose that the other entrepreneurs’ strategy is k (x, y) = β0 + β1 x + β2 y, with β1 , β2 > 0.10 Aggregate investment is then given by K (θ, y) = β0 + β1 θ + β2 y and an agent’s best predictor of aggregate investment is E[K|x, y] = β0 + β1 E[θ|x, y] + β2 y. 10 This condition means that investment responds positively to both signals. Since an entrepreneur’s expectation of θ increases with either signal, it is quite natural to expect this condition to be satisﬁed in equilibrium. Below we will show that an equilibrium that satisﬁes this condition always exists and that this equilibrium is unique for λ small enough. For λ high enough, however, it is possible to construct equilibria in which entrepreneurs ﬁnd it optimal to react negatively to a signal because they expect others to do the same. 11 The private signal x helps predict aggregate investment only through E[θ|x, y], while the common signal y helps predict aggregate investment both through E[θ|x, y] and directly through its eﬀect on the term β2 y. Therefore, relative to how much the two signals help predict the fundamental, the common signal y is a better predictor of aggregate investment than the private signal x. But now recall that a higher α means a stronger incentive for an individual entrepreneur to align his investment choice with that of other entrepreneurs. It follows that when α is higher entrepreneurs ﬁnd it optimal to rely more heavily on the common signal y relative to the private signal x, for it is the former that best helps them align their choice with the choice of others.11 3.2 Equilibrium characterization As noted earlier, completing the equilibrium characterization requires solving a ﬁxed-point problem. On the one hand, how entrepreneurs use their available information depends on α, the endogenous complementarity induced by the response of prices to aggregate investment. On the other hand, how sensitive asset prices are to aggregate investment, and hence how strong α is, depends on how informative aggregate investment is about the fundamental, which in turn depends on how entrepreneurs use their available information in the ﬁrst place. This ﬁxed-point problem captures the essence of the two-way feedback between the real and the ﬁnancial sector in our model. Its solution is provided in the following lemma. Lemma 4 There exist functions F : R × (0, 1) × R3 → R and G : R × (0, 1) × R3 → R5 such that + + the following are true: (i) In any linear equilibrium, β2 /β1 solves β2 =F β1 β2 ; λ, πθ , πx , πy β1 (9) while (β0 , β1 , β2 , γ0 , γ1 ) = G (β2 /β1 ; λ, πθ , πx , πy ); (ii) Equation (9) has at least one solution at some β2 /β1 > πy /πx ; ¯ ¯ (iii) For any (πθ , πx , πx ), there exists a cutoﬀ λ = λ (πθ , πx , πy ) > 0 such that (9) admits a ¯ unique solution if λ < λ; (iv) There exists an open set S such that (9) admits multiple solutions if (λ, πθ , πx , πx ) ∈ S. The ﬁxed-point problem that leads to the equilibrium characterization is set up in terms of the variable b = β2 /β1 , which represents the relative sensitivity of entrepreneurial investment to the two signals. Given b, we can determine the sensitivity of the price to aggregate investment γ1 . 11 A similar property holds for the more general information structures considered in the Supplementary Material: a stronger complementarity shifts the use of information towards the signals whose errors are relatively more correlated across agents. 12 Given γ1 , we can then determine the complementarity α and then the sensitivity b of individual best responses to the two signals. These steps describe the mapping F used in Lemma 4 and provide the intuition for part (i) of the lemma: the ﬁxed points of F identify all the linear equilibria of our economy. Parts (ii)-(iv) then characterize the ﬁxed points of F , establishing than a linear equilibrium always exist and, although it is not always unique, it is unique for λ small enough. The possibility of multiple equilibria for high values of λ is interesting for several reasons. First, it illustrates the potential strength of the two-way feedback between real and ﬁnancial activity. Second, this multiplicity originates solely from an informational externality rather than from the more familiar payoﬀ eﬀects featured in coordination models of crises ` la Diamond and Dybvig (1984) a and Obstfeld (1996). Finally, this multiplicity can induce additional non-fundamental volatility in both real investment and ﬁnancial prices. However, the possibility of multiple equilibria is not central to our analysis. When there is a unique equilibrium, the key positive and normative predictions documented in Corollaries 1 and 2 below necessarily hold. When there are multiple equilibria, these predictions continue to hold for any equilibrium that satisﬁes the natural property that investment increases with both signals. For the rest of the paper we thus leave aside the possibility of multiplicity and focus on the case where the equilibrium is unique. The next proposition then summarizes some key equilibrium properties. Proposition 1 There always exists a linear equilibrium in which the following properties are true: (i) Individual investment increases with both signals (β1 , β2 > 0) and hence the equilibrium price increases with aggregate investment (γ1 > 0); (ii) The equilibrium degree of complementarity satisﬁes 0 < α < 1 and is increasing in λ; (iii) The sensitivity of investment to the common signal relative to the private is higher than πy /πx and is increasing in λ. Moreover, λ small enough suﬃces for this equilibrium to be the unique linear equilibrium. Part (i) guarantees that individual investment increases with both signals, which in turn ensures that high aggregate investment is necessarily “good news” for proﬁtability and hence that α is positive. Part (ii) further establishes that α is higher the stronger the entrepreneurs’ concern about ﬁnancial prices. Combining this with Lemma 3 then gives part (iii), which spells out the implications for the equilibrium use of information. 3.3 Impact of fundamental and expectational shocks To further appreciate the positive implications of informational frictions—and the complementarity thereof—it is useful to rewrite aggregate investment as K = β0 + (β1 + β2 ) θ + β2 ε. 13 Aggregate investment thus depends on two types of shocks: fundamental shocks, captured by θ, and expectational shocks, captured by ε. How entrepreneurs use available information aﬀects how investment respond to these shocks: the sensitivity to fundamentals is governed by the sum β1 + β2 , while the sensitivity to expectational shocks is governed by β2 . When information is dispersed, prices react positively to aggregate investment, the equilibrium degree of complementarity is positive, and hence the relative sensitivity to the common signal satisﬁes β2 /β1 > πy /πx . In contrast, when there are no informational frictions, prices do not react to aggregate investment, the equilibrium degree of complementarity is zero, and hence β2 /β1 = πy /πx . The following is then an immediate implication. Corollary 1 (Main positive prediction) The impact of expectational shocks relative to fundamental shocks is higher in the presence of informational frictions. This result is the key positive prediction of the paper: informational frictions amplify nonfundamental volatility relative to fundamental volatility; that is, they reduce the R-square of a regression of aggregate investment on expected proﬁts. Importantly, because the equilibrium α increases with λ, this ampliﬁcation eﬀect is stronger the more entrepreneurs care about asset prices.12 Corollary 1 regards the relative impact of the two shocks. The next proposition reinforces this ﬁnding by examining the absolute impact of the two shocks. ˆ ˆ Proposition 2 There exists λ > 0 such that, for all λ ∈ (0, λ], there is a unique linear equilibrium and the following comparative statics hold: (i) higher λ reduces β1 + β2 , thus dampening the impact of fundamental shocks; (ii) higher λ increases β2 , thus amplifying the impact of expectational shocks. The key intuition for these results is again the role of the complementarity for the equilibrium use of information. To see this, suppose for a moment that γ0 = 0 and γ1 = 1, meaning that p (θ, y) = K (θ, y) in all states. The entrepreneurs’ best response then reduces to k (x, y) = E [ (1 − λ) θ + λK (θ, y) | x, y ] , 12 (10) This result extends to the more general information structures considered in the Supplementary Material. Because all signals have common errors, there are multiple “expectational shocks.” We can then decompose the total variance of investment in two components: the one that is explained by θ (which deﬁnes “fundamental volatility”), and the one that is explained by the combination of all common errors (which deﬁnes “non-fundamental volatility”). We then show that the ratio of the latter to the former is higher with dispersed information (in which case α > 0) than in the frictionless benchmark (in which case α = 0). 14 so that the degree of complementarity now coincides with λ. One can then easily show that the unique solution to (10) is k (x, y) = β0 + β1 x + β2 y, with β0 = πθ µ, πθ + πx (1 − λ) + πy β1 = πx (1 − λ) , πθ + πx (1 − λ) + πy and β2 = πy . πθ + πx (1 − λ) + πy It is then immediate that a higher λ increases the sensitivity to the prior (captured by β0 ) and the sensitivity to the common signal (captured by β2 ), while it decreases the sensitivity to the private signal (captured by β1 ). We have already given the intuition for the result that a stronger complementarity ampliﬁes the reliance on the common signal and dampens the reliance on the private signal. That it also increases the reliance on the prior is for exactly the same reason as for the common signal: the prior is a relative good predictor of others’ investment choices. However, note that the average return on investment coincides with the mean of θ, irrespective of λ. This is because the average price must equal the mean of θ, for otherwise the traders would make on average non-zero proﬁts, which would be a contradiction. But then the average investment must also be equal to the average of θ, that is, β0 + (β1 + β2 ) µ must equal µ. It is then immediate that, because a higher λ increases β0 , it also reduces the sum β1 + β2 . In simple words, investment is less sensitive to changes in fundamentals simply because the complementarity strengthens the anchoring eﬀect of the prior. These intuitions would be exact if γ0 = 0 and γ1 = 1, or more generally if these coeﬃcients were exogenous to λ. In our model, the price is an increasing function of aggregate investment, but the coeﬃcients γ0 and γ1 depend on λ. This explains why these intuitions are incomplete and why the absolute eﬀects documented in Proposition 2 hold only for a subset of the parameter space. However, the prediction regarding the ampliﬁcation of the relative impact of non-fundamental shocks (Corollary 1) holds true more generally. 4 Constrained eﬃciency The analysis so far has focused on the positive properties of the equilibrium. We now study its normative properties by examining whether there is an allocation that, given the underlying information structure, leads to higher welfare. The question of interest here is whether society can do better, relative to equilibrium, by having the agents use their available information in a diﬀerent way—not whether society can do better by giving the agents more information. We thus adopt the same constrained eﬃciency concept as in Angeletos and Pavan (2007a,b): we consider the allocation that maximizes ex-ante welfare subject to the sole constraint that the choice of each agent must depend only on the information available to that agent. In other words, we let the planner dictate how agents use their available 15 information, but we do not let the planner transfer information from one agent to another. In so doing, we momentarily disregard incentive constraints; later on we will identify tax systems that implement the eﬃcient allocation as an equilibrium. Note that the payments in the ﬁnancial market represent pure transfers between the entrepreneurs and the traders and therefore do not aﬀect ex-ante utility.13 We can thus focus on the investment strategy and deﬁne the eﬃcient allocation as follows. Deﬁnition 3 The eﬃcient allocation is a strategy k(x, y) that maximizes ex-ante utility 1 2 Eu = with K (θ, y) = (1 − λ) θk (x, y) − 1 k (x, y)2 dΦ(x|θ) + 1 [θλK (θ, y)] dΨ(θ, y) 2 2 (11) k (x, y) dΦ(x|θ). Condition (11) gives ex-ante utility for an arbitrary strategy. The ﬁrst term in square brackets is the payoﬀ of an entrepreneur with information (x, y); the second term in square brackets is the payoﬀ of a trader when aggregate investment is K(θ, y); ﬁnally Ψ denotes the cumulative distribution function of the joint distribution of (θ, y). Note that the transfer of capital from the entrepreneurs that are hit by the liquidity shock to the traders does not aﬀect the return to investment. It follows that (11) can be rewritten compactly as14 Eu = 1 E[V (k (x, y) , θ) ] = 1 E[E[V (k (x, y) , θ) |x, y]], 2 2 where V (k, θ) ≡ θk − 1 k2 . From the society’s viewpoint, λ is irrelevant and it is as if the en2 trepreneurs’ payoﬀs are V (k, θ). It is then obvious that a strategy k(x, y) is eﬃcient if and only if, for almost all x and y, k(x, y) maximizes E[V (k, θ)|x, y]. The following result is then immediate. Proposition 3 The eﬃcient investment strategy is given by k (x, y) = E [θ|x, y] = δ0 µ + δ1 x + δ1 y, where δ0 ≡ πθ , πθ + πx + πy δ1 ≡ πx , πθ + πx + πy δ2 ≡ πy . πθ + πx + πy 13 By ex-ante utility, we mean before the realization of any random variable, including those that determine whether an agent will be an entrepreneur or a trader. However, note that, because utility is transferable, any strategy k(x, y) that maximizes ex-ante utility also maximizes any weighted average of the expected utility of an entrepreneur and of a trader. By the same token, any strategy k(x, y) that improves upon the equilibrium in terms of ex-ante utility can yield a Pareto improvement. It suﬃces, for example, to let the entrepreneurs and the traders continue to trade at a price p(θ, y) = E(θ|K(θ, y)), where K(θ, y) = k (x, y) dΦ(x|θ). 14 It suﬃces to substitute the expression for K (θ, y) in (11). 16 Note that the eﬃcient strategy would have coincided with the equilibrium strategy if it were not for informational frictions. It follows that our earlier positive results admit a normative interpretation.15 Corollary 2 (Main normative prediction) In the presence of informational frictions, the impact of expectational shocks relative to fundamental shocks is ineﬃciently high. 5 Policy implications Having identiﬁed a potential source of ineﬃciency, we now analyze the eﬀect of diﬀerent policies. First, we consider interventions “during the fact,” by which we mean interventions in the ﬁnancial market at t = 2, when uncertainty about θ has not been resolved yet. Next, we consider policies “after the fact,” by which we mean policies contingent on information that becomes public at t = 3, after uncertainty about θ has been resolved. In both cases, we impose that the government has no informational advantage vis-a-vis the private sector, which is our preferred benchmark for policy analysis. At the end of this section, however, we also consider situations where the government can directly aﬀect the information available to the agents. 5.1 Interventions “during the fact”: price stabilization We start by considering policies aimed at reducing asset-price volatility. In particular, suppose the government imposes a proportional tax τ on ﬁnancial trades (purchases of capital) at date 2. This tax can depend on the price, which is public information. For simplicity, it takes the following linear form: τ (p) = τ0 + τ1 p, (12) where (τ0 , τ1 ) are scalars. Tax revenues are rebated as a lump sum. The equilibrium price in the ﬁnancial market is now given by p = E [θ|K] − (τ0 + τ1 p) ; equivalently, p= 1 1 (E [θ|K] − τ0 ) = (γ0 + γ1 K − τ0 ) , 1 + τ1 1 + τ1 (13) where γ0 and γ1 are given, as before, by (6). When the tax is pro-cyclical (i.e., τ1 > 0), its eﬀect is to dampen the response of asset prices to the traders’ expectation of θ, and thereby their response to the news contained in aggregate investment. In equilibrium, this tends to reduce the degree of complementarity in investment decisions. To see this more clearly, note that the degree 15 Corollary 2 presumes that the equilibrium is unique. When there are multiple equilibria, the result holds for any equilibrium in which β1 , β2 > 0. Since the eﬃcient allocation satisﬁes β1 , β2 > 0, this also ensures that no equilibrium is eﬃcient. 17 Welfare −0.012 −0.014 −0.016 −0.018 −0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.21 1 0.205 β 0.2 0.195 0.19 −0.1 0.5 β 2 0.45 0.4 0.35 0.3 −0.1 τ 1 Figure 1: The impact of price-stabilization policies. of complementarity is now given by α= λγ1 . 1 + τ1 If γ1 were exogenous, it would be immediate that α decreases with τ1 . But if α falls, we know from Lemma 3 that β2 /β1 , the relative weight on common sources of information, must also fall. This in turn means that aggregate investment becomes more informative about θ, so that γ1 increases, counteracting the direct eﬀect of τ1 on α. However, one can show that, at least as long as the equilibrium is unique, the direct eﬀect dominates, guaranteeing that the equilibrium degree of complementarity decreases with τ1 .16 We conclude that a higher τ1 , by reducing the degree of complementarity, necessarily reduces the relative impact of expectational shocks. However, by reducing the overall sensitivity of prices to all sources of variation in investment, a higher τ1 also reduces the impact of fundamental shocks. As argued in the previous section, in the absence of policy intervention, investment is excessively sensitive to expectational shocks and insuﬃciently sensitive to fundamental shocks. It follows that the welfare consequences of the tax are ambiguous: while reducing the impact of expectational shocks improves eﬃciency, reducing the impact of fundamental shocks has the opposite eﬀect. These intuitions are illustrated in Figure 1 where for each value of τ1 , the value of τ0 is chosen 16 This follows from an argument similar to the one that establishes that α is monotonic in λ. 18 optimally to maximize welfare. The top panel depicts the diﬀerence in welfare under the stabilization policy considered here and under the constrained eﬃcient allocation; the bottom panels depict the sensitivity to expectational shocks β2 and to fundamental shocks (β1 + β2 ).17 The ﬁgure is drawn for a baseline set of parameters: πθ = πx = πy = 1 and λ = 0.5. However, its qualitative features are robust across a wide set of parametrizations. In particular, we have randomly drawn 10,000 parameter vectors (λ, πθ , πx , πy ) from (0, 1) × R3 . For each such vector, we have found + that the optimal τ1 is positive and it induces a lower β2 and a lower β1 + β2 as compared to the equilibrium without policy, reﬂecting the trade oﬀ discussed above. While these numerical results, which span the entire parameter space, reveal that the optimal policy always involves a strictly positive degree of price stabilization (i.e., τ1 > 0), we have not been able to establish this result formally. However, it is easy to show that full price stabilization (i.e., τ1 → ∞) is never optimal. In this limit, prices cease to react to aggregate investment, the strategic complementarity disappears, and equilibrium investment reduces to k(x, y) = (1 − λ) E[θ|x, y]. By implication, the relative sensitivity of investment to expectational shocks β2 /(β1 + β2 ) is at its eﬃcient level, but its overall sensitivity to the fundamental is λ times lower than at the eﬃcient level. At this point, a marginal increase in the relative sensitivity implies only a second-order welfare loss, while a marginal increase in the overall sensitivity implies a ﬁrst-order welfare gain. It follows that it is never optimal to fully stabilize the price. Proposition 4 A tax that stabilizes prices can increase welfare; however, a tax that completely eliminates price volatility is never optimal. 5.2 Interventions “after the fact”: corrective taxation Suppose now that the government imposes a proportional tax τ on asset holdings in period 3. The tax is now paid by the entrepreneurs not hit by the liquidity shock and by the traders who acquired capital in period 2. The advantage of introducing a tax in period 3 is that the tax rate τ can now be made contingent on all information which is publicly available in that period, including K and θ. We focus on linear tax schemes of the form τ (θ, K) = τ0 + τ1 θ + τ2 K, (14) where (τ0 , τ1 , τ2 ) are scalars; tax revenues are again rebated in a lump-sum fashion. The following result shows that these simple tax schemes can implement the constrained eﬃcient allocation. 17 Note that τ0 aﬀects the unconditional average of k (x, y), but has no eﬀect on the sensitivity of investment to the signals x and y, i.e., on β1 and β2 . We henceforth concentrate on τ1 . 19 Proposition 5 There exists a unique linear tax scheme that implements the eﬃcient allocation as an equilibrium. The optimal tax satisﬁes τ0 < 0, τ1 < 0 and τ2 > 0. The intuition behind this result is that the government can control the degree of strategic complementarity perceived by the agents by appropriately designing the contingency of the marginal tax rate τ on aggregate investment: the higher the elasticity τ2 of the marginal tax rate with respect to K, the lower the degree of complementarity in investment choices α and the lower the sensitivity of equilibrium investment to common noise relative to idiosyncratic noise. This eﬀect is analogous to that of the stabilization policies discussed above. However, the government now has an extra instrument available: the elasticity τ1 of the tax to the realized fundamental. Through τ2 the government can thus induce the optimal relative sensitivity to expectational shocks β2 /(β1 + β2 ) while, at the same time, adjust τ1 to obtain the optimal absolute sensitivities to each shock.18 Although this result does not require any informational advantage on the government’s side, it assumes that the government observes perfectly the fundamental θ and the agents’ capital holdings at the time taxes are collected. However, the result easily extends to situations where these quantities are observed with measurement error. In particular, suppose that in stage 3 the government ˜ only observes θ = θ + ǫ and si = si + ηi for each i, where si is the capital holding of agent i, ˜ while ǫ and ηi are measurement errors, possibly correlated with one another, but independent of ˜ θ and of the agents’ information in period 2. Then let K be the cross-sectional mean of si (i.e., ˜ the government’s measure of aggregate investment) and consider a proportional tax on si of the ˜ ˜ K) = τ0 + τ1 θ + τ2 K. It is then easy to check that there continues to exist a ˜ ˜ ˜ following form: τ (θ, unique set of coeﬃcients (τ0 , τ1 , τ2 ) that implement the eﬃcient allocation as an equilibrium and that these coeﬃcients continue to satisfy τ1 < 0 < τ2 . To recap, the key insight here is that the government can use the contingency of the tax rate on public signals of θ and K that will be revealed at stage 3 to achieve eﬃciency in the decentralized use of information during stage 1. Although this information becomes available only after all investment decisions are sunk, by promising a speciﬁc policy response to this information the government is able to manipulate how entrepreneurs use their available sources of information when making their investment decisions, even if it cannot directly monitor these sources of information.19 18 The optimal τ0 is then chosen to induce the optimal level of unconditional average investment. Similar tax schemes implement the eﬃcient investment strategy in all the extensions considered in Section 6. 19 These intuitions, and the implementation result in Proposition 5, build on Angeletos and Pavan (2007b). This paper considers optimal policy within a rich, but abstract, class of economies with dispersed information on correlated values. See also Lorenzoni (2007) for monetary policy in a business-cycle model with dispersed information on underlying productivity. 20 5.3 Optimal release of information We now turn to policies that aﬀect the information available to the agents. This seems relevant given the role of the government in collecting (and releasing) macroeconomic data. To capture this role, suppose that in stage 2 traders can only observe average investment with noise, that is, they observe ˜ K = K + η, where η is aggregate measurement error, which is a random variable, independent of all other shocks, with mean zero and variance 1/πη .20 Suppose now that the government can aﬀect the precision πη of the macroeconomic data available to ﬁnancial traders. By changing πη the government determines ˜ the weight that traders assign to K when estimating future proﬁtability. This is another channel by which the government is able to aﬀect the degree of strategic complementarity in investment decisions. Indeed, the choice of πη is formally equivalent to the choice of τ1 in the setup with a tax on ﬁnancial transactions (Section 5.1). For each value πη of the precision of the signal about aggregate activity, there is a value τ1 of the tax elasticity that induces the same equilibrium strategy, and vice ˜ versa. To see this, note that in any linear equilibrium, the observation of K = β0 +(β1 +β2 )θ+β2 ε+η is informationally equivalent to the observation of a signal z≡ ˜ K − β0 β2 1 =θ+ ε+ η, β1 + β2 β1 + β2 β1 + β2 with precision πz = β2 β1 + β2 2 −1 πy + 1 β1 + β2 2 −1 πη −1 . The equilibrium price is then given by p(θ, y, η) = γ0 + γ1 [K(θ, y) + η], with γ0 and γ1 given by (6), and hence the degree of strategic complementarity remains equal to α ≡ λγ1 , as in the baseline model. By changing the value of πη , the government can then directly manipulate γ1 and thus the degree of strategic complementarity perceived by the entrepreneurs.21 We conclude that the choice of πη is subject to the same trade-oﬀs emphasized for the choice of τ1 : decreasing πη reduces the relative response of investment to expectational shocks, but it also reduces its response to fundamental shocks. The results of Section 5.1 then imply that an intermediate degree of release of macroeconomic data may be optimal even when the cost of collecting such data is zero.22 20 The equilibrium characterization for this case is a straightforward extension of the baseline case. See the Supplementary Material for the proof of this claim. 22 Note, however, that this holds only as long as the equilibrium is ineﬃcient. If, instead, the policies considered in Section 5.2 are in place, thus guaranteeing that the equilibrium is eﬃcient, then a higher πη is always welfare 21 21 Finally, one could consider policies which aﬀect directly the agents’ information regarding the fundamental θ. In particular, the government can collect some information about θ in period 1 and decide whether to disclose this information to the entrepreneurs, or to both the entrepreneurs and the traders. In the ﬁrst case, the policy corresponds to an increase in the precision of the signal y in the baseline model. Although entrepreneurs have a more precise estimate of the fundamental, this information is not shared with the traders. Therefore, this policy could exacerbate the asymmetry of information and could magnify the feed-back eﬀects between investment and asset prices, with possible negative consequences on social welfare. In the second case, instead, the policy corresponds to an increase in the precision of the common prior in the baseline model. This policy is socially beneﬁcial for two reasons: ﬁrst, it improves the quality of the information available to the entrepreneurs and hence it permits them to better align their decisions to the fundamental. Second, it reduces the reliance of ﬁnancial markets on the endogenous signal K in their estimate of the fundamental. This second eﬀect tends to reduce the degree of strategic complementarity in investment decisions, and hence also the discrepancy between equilibrium and eﬃcient allocations. Both eﬀects then contribute to higher welfare. 6 Extensions Our analysis has identiﬁed a mechanism through which the dispersion of information induces complementarity in real investment choices, ampliﬁcation of non-fundamental disturbances, and ineﬃciency of market outcomes, all at once. In the baseline model, we have made a number of assumptions to illustrate this mechanism in the simplest possible way. In particular, we have assumed that the traders’ demand for installed capital is perfectly elastic, that entrepreneurs who are not hit by the liquidity shock do not trade in the ﬁnancial market, and that the traders’ valuation of the asset coincides with that of the entrepreneurs. In this section, we relax each of these assumptions. We ﬁrst extend the model to allow for the traders’ demand for capital to be downward sloping. This extension is interesting because it introduces a potential source of strategic substitutability in the entrepreneurs’ investment decisions: when aggregate investment is higher, the supply of installed capital in the ﬁnancial market is also higher, putting a downward pressure on asset prices and lowers the ex-ante incentive to invest. In a second extension, we allow entrepreneurs not hit by the liquidity shock to participate in the ﬁnancial market. This extension is interesting for two reasons: ﬁrst, it allows for some of the entrepreneurs’ information to be aggregated in the ﬁnancial market; second, it introduces a non-trivial allocative role for prices. improving. 22 Although some interesting diﬀerences arise, the key positive and normative predictions of the paper (Corollaries 1 and 2) remain valid in both extensions: as long as the dispersion of information does not completely vanish in the ﬁnancial market, the signaling eﬀect of aggregate investment continues to be the source of ampliﬁcation and ineﬃciency in the response of the equilibrium to common sources of noise. Finally, we consider a variant that introduces shocks to the ﬁnancial-market valuation of the installed capital. This variant brings the paper closer to the recent literature on “mispricing” and “bubbly” asset prices. It also helps clarify that the details of the information structure we assumed in the baseline model are not essential: any source of common noise in the information that aggregate investment conveys about the underlying fundamentals opens the door to ampliﬁcation and ineﬃciency. 6.1 The supply-side eﬀect of capital: a source of strategic substitutability We modify the benchmark model as follows. The net payoﬀ of trader i, who buys qi units of capital at the price p, is now given by ui = (θ − p) qi − 1 2 q , 2φ i (15) where φ is a positive scalar. The diﬀerence with the benchmark model is the presence of the last term in (15), which represents a transaction cost associated to the purchase of qi units of capital. A convex transaction cost ensures a ﬁnite price elasticity for the traders’ demand, which is now given by q(p, K) = φ (E [θ|K] − p). The parameter φ captures the price elasticity of this demand function and our benchmark model corresponds to the special case where the demand is inﬁnitely elastic, i.e. φ → ∞.23 As in the benchmark model, in any linear equilibrium the traders’ expectation of θ is given by E[θ|K] = γ0 + γ1 K, for some coeﬃcients γ0 and γ1 . However, unlike in the benchmark model, the equilibrium price does not coincide with E[θ|K]. Market clearing now requires that q(p, K) = λK, so the equilibrium price is p = E[θ|K] − λ K = γ0 + γ1 − φ λ φ K. (16) It follows that aggregate investment has two opposing eﬀects on the price of installed capital, p. On the one hand, it raises the traders’ expectation of θ, thereby pushing the price up. On the other hand, it raises the supply of capital, thereby pulling the price down. The strength of these two eﬀects determines whether investment choices are strategic complements or substitutes. 23 A more familiar way of introducing a ﬁnitely elastic demand is to assume risk aversion. The alternative we use here captures the same key positive and normative properties—namely, demands are ﬁnitely elastic and individual payoﬀs are concave in own portfolio positions—but has the advantage of keeping the analysis tractable by making the elasticity of demands invariant to the level of uncertainty. 23 Proposition 6 (i) In any linear equilibrium, the investment strategy satisﬁes k (x, y) = E [ (1 − α) κ (θ) + αK (θ, y) | x, y ] , with α ≡ λγ1 − λ2 /φ and κ (θ) ≡ (17) (1−λ)θ+λγ0 1−λγ1 +λ2 /φ . (ii) λ small enough suﬃces for the equilibrium to be unique, for investment to increase with θ, and for γ1 to be positive. The degree of complementarity α is now the sum of two terms. The ﬁrst term λγ1 captures the, by now familiar, informational eﬀect of investment on asset prices documented in the benchmark model. The second term, −λ2 /φ, captures the simple supply-side eﬀect that emerges once the demand for the asset is ﬁnitely elastic. If either the information contained in aggregate investment is suﬃciently poor (low γ1 ) or the price elasticity of demand is suﬃciently low (low φ), investment choices become strategic substitutes (α < 0). However, the question of interest here is not whether investment choices are strategic complements or substitutes, but how the positive and normative properties of the equilibrium are aﬀected by the dispersion of information. In this respect, the implications that emerge in this extension are essentially the same as in the benchmark model. First, consider the positive properties of the equilibrium. Lemma 3 immediately extends to the modiﬁed model: equilibrium investment satisﬁes k(x, y) = β0 + β1 x + β2 y with β2 πy 1 = . β1 πx 1 − α (18) Provided that investment increases with both signals, then aggregate investment is necessarily good news for θ (i.e., γ1 > 0), in which case Proposition 6 implies that α > −λ2 /φ. In contrast, when there are no informational frictions, the equilibrium price is simply p = θ − (λ2 /φ)K, so that α = −λ2 /φ. It follows that the dispersion of information increases the value of α and hence ampliﬁes the impact of common expectational shocks relative to that of fundamental shocks, even if α happens to be negative. We conclude that Corollary 1, which summarizes the key positive predictions of the model, continues to hold. Next, consider the normative properties. Because of the convexity of the transaction costs, it is necessary for eﬃciency that all traders take the same position in the ﬁnancial market: qi = λK for all i ∈ (1/2, 1]. Ex-ante utility then takes the form Eu = = 1 2 (1 − λ) θk(x, y) − 1 k(x, y)2 dΦ(x|θ) + 2 1 k2 1 E − + θk − (λK)2 2 2 2φ 24 1 2 θλK(θ, y) − 2 1 2φ [λK(θ, y)] dΨ(θ, y) (19) and the eﬃcient investment strategy is the function k(x, y) that maximizes (19). Proposition 7 The eﬃcient investment strategy is the unique linear solution to k (x, y) = E [(1 − α∗ ) κ∗ (θ) + α∗ K (θ, y) | x, y ] , where α∗ ≡ −λ2 /φ < 0, κ∗ (θ) ≡ θ/ 1 + λ2 /φ , and K(θ, y) = (20) k(x, y)dΦ(x|θ). To understand this result, note that the social return to investment is given by (1 − λ) θ + λ (θ − λK/φ) = θ − λ2 K/φ. The new term, relative to the benchmark model, is −λ2 K/φ and it reﬂects the cost associated with transferring λK units of the asset from the entrepreneurs to the traders. If information were complete, eﬃciency would require that each agent equates his marginal cost of investing to the social return to investment, which would give k = θ − λ2 K/φ.24 The analogue under incomplete information is that each agent equates the marginal cost to the expected social return: k (x, y) = E θ − λ2 /φ K (θ, y) x, y . (21) Rearranging this condition gives (20). The key ﬁnding here is that the introduction of downward sloping demands has a symmetric eﬀect on the private and social returns to investment. This is simply because the negative pecuniary externality caused by the higher supply of capital perfectly reﬂects the social cost associated with having traders absorb this additional capital. As a result, it is only the informational eﬀect that generates a discrepancy between the equilibrium and the eﬃcient allocation. As in the benchmark model, this discrepancy manifests itself in the response of equilibrium to expectational and fundamental shocks. Indeed, while equilibrium investment satisﬁes (18), eﬃcient ∗ ∗ ∗ investment satisﬁes k(x, y) = β0 + β1 x + β2 y with ∗ β2 πy 1 ∗ = π 1 − α∗ . β1 x Because in any equilibrium in which β1 , β2 > 0 the complementarity satisﬁes α∗ < α < 1, the relative sensitivity of the equilibrium strategy to common noise is ineﬃciently high. We conclude that Corollary 2, which summarizes the key normative predictions of the model, continues to hold.25 24 Note that under full information the optimal level of investment would be equal to κ∗ (θ). As common in competitive environments, there are other forms of pecuniary externalities that could induce strategic substitutability in the entrepreneurs’ investment decisions, even with a perfectly elastic demand for capital. For example, suppose that, in order to complete their investment, entrepreneurs need to purchase certain inputs whose aggregate supply is imperfectly elastic (e.g., labor or land). Higher aggregate investment then implies higher aggregate demand for these inputs, and hence higher input prices and lower entrepreneurial returns, once again inducing strategic substitutability in the entrepreneurs’ investment choices. However, such pecuniary externalities do not, on their own, cause discrepancies between private and social returns. Indeed, it is easy to construct variants of 25 25 6.2 Information aggregation through prices The analysis so far has imposed that the entrepreneurs who are not hit by the liquidity shock can not access the ﬁnancial market. Apart from being unrealistic, this assumption rules out the possibility that the price in the ﬁnancial market aggregates, at least partly, the information that is dispersed among the entrepreneurs. To address this possibility, in this section we extend the analysis by allowing entrepreneurs not hit by the liquidity shock to participate in the ﬁnancial market. To guarantee downward sloping demands, we assume that traders and entrepreneurs alike incur a transaction cost for trading in the ﬁnancial market of the same type as in the previous section.26 Thus, the payoﬀ of an entrepreneur i who is not hit by a liquidity shock, has invested ki units in the ﬁrst period, and trades qi units in the second period, is given by 1 2 1 2 q + θ (ki + qi ) , ui = − ki − pqi − 2 2φ i while the payoﬀ of a trader i is given by (15), as in the previous section. Because the observation of K in the second period perfectly reveals θ to every entrepreneur,27 their demand for the asset in the second period reduces to qE = φ (θ − p). The demand of the traders, on the other hand, is given by qT = φ (E [θ|K, p] − p). Note that traders now form their expectation of θ based on K and on the information revealed by the equilibrium price p.28 Because the aggregate demand for the asset is 1 2 (1 − λ) qE + 1 qT and the aggregate supply is 1 λK, market 2 2 clearing implies p= 1 2−λ E [θ|K, p] + 1−λ 2−λ θ − 1 φ(2−λ) λK. It follows that the joint observation of K and p perfectly reveals θ to the traders as well. The 1 asymmetry of information thus vanishes and the equilibrium price satisﬁes p = θ − φ(2−λ) λK. As in the previous section, this is just the social return to investment, adjusted for the fact that the total capital of the entrepreneurs hit by the liquidity shock (λK/2) is now equally distributed among the traders and the entrepreneurs not hit by the liquidity shock. Because the equilibrium price coincides with the social return to investment it follows that the equilibrium is eﬃcient. This result is no diﬀerent from what we established for the frictionless benchmark at the end the model that capture such sources of strategic substitutability while retaining the property that the informational eﬀect of aggregate investment is the sole source of ampliﬁcation and ineﬃciency, as in the example analyzed here. 26 We assume that the entrepreneurs hit by the liquidity shock do not pay the transaction cost for the units of the asset that they have to sell in the second period; this simpliﬁcation has no impact on the results. 27 This presumes that entrepreneurs use their private information when deciding how much to invest (i.e. β1 = 0), which is indeed true in equilibrium. 28 In the benchmark model, as well as in the extension examined in the previous section, we did not condition on the information revealed by the equilibrium price simply because all agents trading voluntary in the ﬁnancial market had symmetric information. 26 of Section 2: if the dispersion of information vanishes at the time of trade in the ﬁnancial market, equilibrium investment is driven merely by ﬁrst-order expectations of θ and is eﬃcient. However, this result hinges on the equilibrium price perfectly revealing θ. To make this clear, in the subsequent analysis we introduce an additional source of noise, which prevents prices from being perfectly revealing. Assume that the cost of trading for the entrepreneurs is subject to a shock ω, that is revealed to them at the time they trade but which is not observed by the traders. In particular, the payoﬀ of an entrepreneur not hit by the liquidity shock is now given by 1 2 1 2 ui = − ki − pqi − ωqi − q + θ (ki + qi ) , 2 2φ i where ω is assumed to be independent of all other random variables, with E[ω] = 0 and V ar[ω] = 2 −1 σω ≡ π ω . In what follows, we look at linear rational expectations equilibria; we continue to denote the investment strategy by k (x, y) and we denote by p (θ, y, ω) the equilibrium price. Because the observation of aggregate investment in the second period continues to reveal θ to the entrepreneurs (but not to the traders), asset demands can be written as qE = φ (θ − ω − p) for the entrepreneurs and qT = φ (E [θ|K, p] − p) for the traders. Market clearing then implies that the equilibrium price is p= 1 2−λ E [θ|K, p] + 1−λ 2−λ (θ − ω) − 1 φ(2−λ) λK. (22) Once again, the price is a weighted average of the traders’ and of the entrepreneurs’ valuation of the asset, net of trading costs. However, because the shock ω is not known to the traders, the price no longer perfectly reveals θ, ensuring that the informational eﬀect of K on the traders’ expectation of the fundamental reemerges. This eﬀect is captured by the ﬁrst term in the right-hand-side of (22). At the same time, the supply-side eﬀect of K is also present and is captured by the last term in (22). While the supply-side eﬀect induces strategic substitutability, the informational eﬀect induces complementarity. In any rational expectations equilibrium in which the price is linear in (θ, y, ω), E [θ|K, p] is the projection of θ on (K, p) and, by (22), the price p can be expressed as a linear combination of (K, θ, ω). It follows that, for any linear equilibrium, there exist coeﬃcients (γ0 , γ1 , γ2 , γ3 ) such that29 E [θ|K, p] = γ0 + γ1 K + γ2 θ + γ3 ω. (23) 29 Note that γ1 , which captures the eﬀect of K on the traders’ expectation of θ, now combines the information that is directly revealed to the traders by the observation of K with the information that is revealed to them through the observation of the equilibrium price. 27 Using (23), (22), and the fact that the private return to investment is the expectation of (1 − λ) θ + λp, we reach the following characterization result. Proposition 8 (i) In any linear equilibrium, the investment strategy satisﬁes k (x, y) = E[(1 − α) κ(θ) + αK (θ, y) | x, y], where α = λ 2−λ γ1 − λ2 φ(2−λ) and κ(θ) = λγ0 +[2(1−λ)+λγ2 ]θ 2−λ−λγ1 +λ2 /φ . (ii) λ small enough suﬃces for the equilibrium to be unique, for investment to increase with θ, and for γ1 to be positive. As in the previous section, α combines an informational eﬀect (captured by λ 2−λ γ1 ) with a λ2 supply-side eﬀect (captured by − φ(2−λ) ). The supply-side eﬀect always contributes to strategic substitutability, while the informational eﬀect contributes to strategic complementarity if and only if high investment is good news for θ (i.e. γ1 > 0). Once again, the overall eﬀect is ambiguous, but the role of informational frictions remains the same as before: Corollary 1 continues to hold. We now turn to the characterization of the eﬃcient allocation for this economy. The eﬃciency concept we use is the same as in the preceding sections; however, now we need to allow the planner to mimic the information aggregation that the market achieves through prices. We thus proceed as follows. First, we deﬁne an allocation as a collection of strategies k(x, y), qE (x, y, K, p, ω) and qT (K, p), along with a shadow-price function p (θ, y, ω) with the following interpretation: in the ﬁrst period, an entrepreneur with signals (x, y) invests k (x, y); in the second period, all agents observe the realizations of aggregate investment K = K (θ, y) and the shadow price p = p (θ, y, ω) ; the amount of capital held by an entrepreneur not hit by a liquidity shock (in addition to the one chosen at t = 1) is then given by qE (x, y, K, p, ω), while the amount of capital held by a trader is given by qT (K, p). Next, we say that the allocation is feasible if and only if, for all (θ, y, ω) , λK (θ, y) = (1 − λ) qE (x, y, K(θ, y), ω, p(θ, y, ω))dΦ (x|θ) + qT (K(θ, y), p(θ, y, ω)). (24) As with equilibrium, this constraint plays two roles: ﬁrst, it guarantees that the second-period resource constraint is not violated; second, it deﬁnes the technology that is used to generate the endogenous public signal (equivalently, the extent to which information can be aggregated through the shadow price). Finally, for any given k(x, y), qE (x, y, K, p, ω) and qT (K, p), ex ante utility can be computed 28 as Eu = W (k, qE , qT ), where W (k, qE , qT ) ≡ 1 2 1 {− k(x, y)2 + λp(θ, y, ω)k(x, y) + (1 − λ) θk(x, y) + 2 + (1 − λ) R θ − ω, q E (x, y, K(θ, y), ω, p(θ, y, ω) }dΦ (x|θ) dΨ (θ, y, ω) 1 + R θ, q T (K(θ, y), p(θ, y, ω)) }dΨ (θ, y, ω) , 2 where R (v, q) ≡ vq − q 2 / (2φ) and where K(θ, y) = k(x, y)dΦ (x|θ) . We then deﬁne an eﬃcient allocation as follows. Deﬁnition 4 An eﬃcient allocation is a collection of strategies k(x, y), qE (x, y, K, p, ω) and qT (K, p), along with a shadow price function p(θ, y, ω), that jointly maximize ex-ante utility, Eu = W (k, qE , qT ), subject to the feasibility constraint (24). Because utility is transferable, the shadow price does not aﬀect payoﬀs directly; its sole function is to provide an endogenous public signal upon which the allocation of the asset in the period 2 can be conditioned. The next lemma then characterizes the eﬃcient allocation of the asset. Lemma 5 The eﬃcient allocation in the second period satisﬁes ∗ qE = λK φω − 2−λ 2−λ and ∗ qT = λK (1 − λ) φω + . 2−λ 2−λ (25) To understand this result, suppose for a moment that information were complete in the second period. For any given K, eﬃciency in the second period would require that all entrepreneurs hold the same qE and that (qT , qE ) maximize θqT − 1 2 q 2φ T + (1 − λ) θqE − ωqE − 1 2 q 2φ E subject to (1 − λ) qE + qT = λK. Clearly, the solution to this problem is (25). In our environment, information is incomplete but the same allocation can be induced through the following shadowprice and demand functions: p(θ, y, ω) = − φ(1−λ)ω , qE (x, y, K, p, ω) = 2−λ λK 2−λ − λK 2−λ − φω 2−λ , and qT (K, p) = p.30 We now characterize the eﬃcient investment decisions. Using Lemma 5, ex ante utility reduces to 1 1 1 Eu = E − k2 + θk − (λK)2 2 2 2φ(2 − λ) + (1 − λ) φσ 2 . 2 (2 − λ) ω (26) 30 Note that the proposed shadow price is also the unique market-clearing price given the proposed demand functions. The eﬃcient trades can thus be implemented by inducing these demand functions through an appropriately designed tax system and then letting the agents trade in the market. 29 Except for two minor diﬀerences—the smaller weight on (λK)2 , which adjusts the cost associated with absorbing the ﬁxed supply λK in the second period for the fact that now this quantity is split across a larger pool of agents, and the last term in (26), which captures how the volatility of ω aﬀects the allocation of capital across entrepreneurs and traders in the second period—exante utility has the same structure as in (19) in the previous section. The following result is then immediate. Proposition 9 The eﬃcient investment strategy is the unique linear solution to k (x, y) = E [(1 − α∗ ) κ∗ (θ) + α∗ K (θ, y) | x, y] , 2 λ where α∗ ≡ − φ(2−λ) , κ∗ (θ) ≡ 1 1−α∗ θ, and K(θ, y) = (27) k(x, y)dΦ(x|θ). Comparing the eﬃcient strategy with the equilibrium one, we have that, once again, as long as investment increases with both signals, so that high investment is good news for proﬁtability, then α remains higher than α∗ , in which case the key normative prediction of the paper, as summarized by Corollary 2, continues to hold. 6.3 Financial-market shocks In the speciﬁcations considered so far, entrepreneurs and traders share the same valuation for the installed capital. We now develop a variant of the model in which entrepreneurs and traders have diﬀerent valuations. In this variant, additional non-fundamental volatility originates from correlated errors in the entrepreneurs’ expectations about the traders’ valuations; once again, our mechanism ampliﬁes the impact of these errors. This variant thus helps connect our model to the recent work on speculative trading ` la Harrison and Kreps (1978).31 a We consider the following modiﬁcation of the baseline model. The traders’ utility in period t = 3 is given by (θ + ω) ki , where ω is a random variable, independent of θ and of any other exogenous random variable in the economy, Normally distributed with mean zero and variance 2 σω . This random variable is a private-value component in the traders’ valuation. It can originate from the hedging motive of the traders, from a diﬀerent discount factor, or from heterogeneous valuations ` la Harrison and Kreps (1978). For our purposes, what matters is that the presence of a ω in the traders’ utility is taken as given by the social planner; that is, the planner respects the preference orderings revealed by the agents’ trading decisions. We thus choose a neutral label for ω and simply call it a “ﬁnancial market shock.” We also modify the entrepreneurs’ information set, to allow for information regarding ω to aﬀect investment decisions. In particular, the entrepreneurs observe a common signal w = ω+ζ, where ζ is 31 See Scheinkman and Xiong (2003), Gilchrist, Himmelberg, and Huberman (2005), and Panageas (2005). 30 common noise, independent of any other exogenous random variable in the economy, with variance 2 σζ . The signal w is observed by the entrepreneurs but not by the traders; as in the baseline model, this is a shortcut for introducing correlated errors in the entrepreneurs’ expectations regarding the ﬁnancial-market shock. Finally, to focus on common expectational shocks about ω rather than about θ, we remove the common signal y : the entrepreneurs observe only private signals about θ, xi = θ + ξi , where ξi is idiosyncratic noise as in the baseline model. In this environment, the asset price in period two is given by p = E [θ|K, ω] + ω. It follows that equilibrium investment choices depend not only on the entrepreneurs’ expectations of θ, but also on their expectations of ω : there exist coeﬃcients (β0 , β1 , β2 ) such that individual investment is given by k (x, w) = β0 + β1 x + β2 w and, by implication, aggregate investment is given by K (θ, w) = β0 +β1 θ +β2 w. Following similar steps as in the baseline model, leads to the following result. Proposition 10 (i) In any equilibrium, there exist a scalar α > 0 and a function κ(θ, ω) such that k (x, w) = E [ (1 − α) κ (θ, ω) + αK (θ, w) | x, w] . (ii) λ small enough suﬃces for the equilibrium to be unique and for investment to increase with both θ and w. (iii) The eﬃcient investment satisﬁes k (x, w) = E [ θ + λω | x, w] . (iv) In any equilibrium in which investment increases with both θ and w, investment underreacts to θ and overreacts to w. In this economy, entrepreneurs pay too much attention to their signals regarding shocks in the ﬁnancial market. The reason is essentially the same as in the benchmark model. When traders interpret high investment as good news for θ, ﬁnancial prices increase with aggregate investment. Because the noise in the entrepreneurs’ signals about the ﬁnancial market shock ω is correlated, these signals are relatively better predictors of aggregate investment than the signals about θ. By implication, entrepreneurs’ investment decisions are oversensitive to information about ﬁnancial market shocks relative to information about their fundamental valuation θ. Through this channel, an increase in investment that was purely driven by expectations regarding ﬁnancial market shock is ampliﬁed. 31 Absent informational frictions (i.e., if θ were known at the time of ﬁnancial trade)), the response of investment to θ and ω would be eﬃcient. Since ω can be interpreted as the diﬀerence between the traders’ and the entrepreneurs’ fundamental valuations of the asset, this case is reminiscent of the eﬃciency results obtained in richer models of “bubbles” based on heterogeneous priors; in particular, Panageas (2006) derives a similar eﬃciency result for a model that introduces heterogeneous valuations in a q-theory model of investment. The interesting novelty here is that ineﬃciency arises once we introduce dispersed information. Traders are then uncertain whether high investment is driven by good fundamentals or by the entrepreneurs’ expectations of speculative valuations. This uncertainty opens the door to our feedback eﬀect between ﬁnancial prices and investment, creating ineﬃciency in the response of investment to diﬀerent sources of information. 6.4 Other extensions An important function of stock prices is to guide corporate investment choices by revealing valuable information that is dispersed in the marketplace and not directly available to corporate managers (e.g., Dow and Gorton, 1997; Subrahmanyam and Titman, 1999; Chen, Goldstein and Jiang, 2007). This eﬀect is absent in the preceding analysis, because the entrepreneurs’ investment choices are made before the opening of the ﬁnancial market. However, we can easily incorporate such an eﬀect by letting the entrepreneurs make an additional investment in stage two, after observing the price in the ﬁnancial market.32 Provided that the dispersion of information does not vanish, the source of complementarity and ineﬃciency we have documented remains. Interestingly, though, an additional information externality emerges: if all agents were to increase their reliance on idiosyncratic sources of information, then the information contained in prices would be more precise, which in turn would improve the eﬃciency of the investment decisions that follow the observation of these prices. Clearly, this informational externality only reinforces the conclusion that agents rely too much to common sources of information, and hence that non-fundamental volatility is ineﬃciently high. Throughout the preceding extensions, we have maintained the assumption that traders cannot directly invest in the new technology during the ﬁrst period. Clearly, our results do not hinge on this assumption. For example, consider the benchmark model and suppose that each trader 2 j chooses ﬁrst-period real investment kj at cost kj /2 and then trades an additional qj units in the second-period ﬁnancial market. Neither the equilibrium price in the ﬁnancial market nor the entrepreneurs’ choices in the ﬁrst period are aﬀected; all that happens is that aggregate investment now includes the investment of the traders, which is simply given by kT = Eθ, which does not aﬀect the information structure in the second period. More generally, one could drop the distinction 32 Alternatively, we could introduce a ﬁnancial market in stage 1 or let entrepreneurs observe a noisy signal of K instead of a noisy price signal. 32 between entrepreneurs and traders altogether and simply talk about diﬀerentially informed agents who ﬁrst make real investment decisions and then trade ﬁnancial claims on the installed capital. Next, consider the assumption that a fraction λ of the entrepreneurs is hit by a liquidity shock and is forced to sell their capital in the ﬁnancial market; this was a modeling device that ensured that the private return to ﬁrst-period investment depends on (anticipated) second-period ﬁnancial prices while ensuring tractability. If one were to drop the assumption of risk neutrality, or assume that the second-period transaction costs depend on gross positions, or introduce short-sale constraints in the ﬁnancial market, then the proﬁts an agent could do in the ﬁnancial market would depend on how much capital he enters the market with; this in turn would ensure that private returns to ﬁrst-period investment depend on expectations of future ﬁnancial prices, even in the absence of liquidity shocks.33 Finally, consider the assumption that proﬁtability is perfectly correlated across entrepreneurs. Clearly, what is essential is only that there is a common component about which agents have dispersed information. For example, we could let the productivity of the new technology for en˜ trepreneur i be θi = θ +vi , where θ is the common component and vi is an idiosyncratic component; ˜ we could then also let the entrepreneurs’ signals be θi plus noise instead of θ plus noise. Alternatively, we could introduce common and idiosyncratic shocks to the entrepreneurs’ cost of investment during period 1. In this case, unobservable common shocks to the cost of investment would also act as a source of noise in the information that aggregate investment conveys about θ, essentially playing the same role as the correlated errors in the entrepreneurs’ signals about θ.34 7 Conclusion This paper examined the interaction between real and ﬁnancial decisions in an economy in which information about underlying proﬁtability is dispersed. By conveying a positive signal about profitability, higher aggregate investment stimulates higher asset prices, which in turn raise the incentives to invest. This creates an endogenous complementarity, making investment decisions sensitive to higher-order expectations. In turn, this can dampen the impact of fundamental shocks and amplify the impact of common expectational shocks. Importantly, all these eﬀects are symptoms of ineﬃciency. These eﬀects are likely to be stronger during periods of intense technological change, when the dispersion of information about the potential of the new technologies is particularly high. Our analysis therefore predicts that such periods come hand-in-hand with episodes of high non-fundamental 33 Note, however, that these extensions may feature additional deviations from the ﬁrst best (e.g., short-sale constraints), which may introduce novel eﬀects in addition to the ones we have considered. 34 Such an extension is studied in the Supplementary Material. 33 volatility and comovement in investment and asset prices. At some level, this seems consistent with the recent experiences surrounding the internet revolution or the explosion of investment opportunities in China. What looks like irrational exuberance may actually be the ampliﬁed, but rational, response to noise in information. While both explanations open the door to policy intervention, the one suggested by our theory is not based on any presumption of “intelligence superiority” on the government’s side. Our mechanism also represents a likely source of non-fundamental volatility and ineﬃciency over the business cycle. Indeed, information regarding aggregate supply and demand conditions seems to be widely dispersed in the population, which explains the ﬁnancial markets’ anxiety preceding the release of key macroeconomic statistics. Extending the analysis to richer business-cycle frameworks is an important direction for future research. 34 Appendix: Proofs omitted in the main text Proof of Lemma 3. The derivations of β1 and β2 are in the proof of the Lemma 4. Rearranging (30), gives β2 = 1 (1 − λ + λγ1 β1 ) δ2 . 1 − λγ1 Using (29), α = λγ1 and δ2 /δ1 = πy /πx , then gives the result. Proof of Lemma 4. The proof proceeds in several steps. We start by proving part (i). We continue with some auxiliary results regarding the function F which are used in the last steps. We conclude by establishing parts (ii), (iii) and (iv). Throughout, to simplify notation, we suppress the dependence of F and G on (λ, πθ , πx , πy ) and let π ≡ πθ + πx + πy , δ0 ≡ πθ /π, δ1 ≡ πx /π, and δ2 ≡ πy /π. Part (i). Substituting K(θ, y) = β0 + β1 θ + β2 y into (7) and using E [θ|x, y] = δ0 µ + δ1 x + δ2 y gives k (x, y) = (1 − λ) E [θ|x, y] + λγ0 + λγ1 (β0 + β1 E [θ|x, y] + β2 y) = (1 − λ + λγ1 β1 ) E [θ|x, y] + λγ0 + λγ1 β0 + λγ1 β2 y = [(1 − λ + λγ1 β1 ) δ0 µ + λγ0 + λγ1 β0 ] + + [(1 − λ + λγ1 β1 ) δ1 ] x + [(1 − λ + λγ1 β1 ) δ2 + λγ1 β2 ] y Because in equilibrium the above must coincide with β0 + β1 x + β2 y for all x and y, the following conditions must hold β0 = (1 − λ + λγ1 β1 ) δ0 µ + λγ0 + λγ1 β0 , (28) β1 = (1 − λ + λγ1 β1 ) δ1 , (29) β2 = (1 − λ + λγ1 β1 ) δ2 + λγ1 β2 . (30) It is immediate that any equilibrium must satisfy β1 = 0. Then let b ≡ β2 /β1 . From (4) and (6), γ1 β1 = h (b) ≡ δ2 (1 + b) δ0 b2 + δ2 (1 + b)2 (31) while from (29) and (30), b= δ2 λγ1 β1 b + . δ1 (1 − λ + λγ1 β1 ) δ1 35 (32) Substituting (31) into (32) gives b = F (b), where F (b) ≡ δ2 δ1 1+ λ (1 + b) b (1 − λ) (δ0 + δ2 ) b2 + (2 − λ)δ2 b + δ2 . (33) Note that the domain of F is the set of all b ∈ R such that 1 − λ + λγ1 β1 = 0. Using (31), the latter is given by B ≡ {b ∈ R : (1 − λ)(δ0 + δ2 )b2 + (2 − λ)δ2 b + δ2 = 0}. It follows that, in any linear equilibrium, b is necessarily a ﬁxed point of F , while the coeﬃcients (β0 , β1 , β2 , γ0 , γ1 ) are given by the following conditions: β1 = [1 − λ + λh (b)]δ1 (34) β2 = bβ1 = b[1 − λ + λh (b)]δ1 (35) γ1 = γ1 β1 h (b) = β1 [1 − λ + λh (b)]δ1 β0 = (1 − λ + λh (b)) δ0 µ + γ0 = (36) λδ0 µ δ0 + δ2 1 + (37) 1 2 b δ0 µ − γ1 β0 δ0 + δ2 (1 + 1 )2 b (38) Conditions (34)-(38) uniquely deﬁne the function G. Auxiliary results. Let g(b) ≡ (1 − λ)(δ0 + δ2 )b2 + (2 − λ)δ2 b + δ2 ; the domain of F is B = {b ∈ R : g(b) = 0} and its complement is Bc = {b ∈ R : g(b) = 0}. Note that the discriminant of g(b) is (2−λ)δ2 ∆ ≡ (δ2 λ)2 − 4δ0 δ2 (1 − λ). If ∆ < 0, then Bc = ∅; if ∆ = 0, then Bc = − 2(1−λ)(δ0 +δ2 ) ; ﬁnally, if ∆ > 0, then Bc = √ √ (2−λ)δ2 (2−λ)δ2 − 2(1−λ)(δ++δ∆) , − 2(1−λ)(δ−+δ∆) . Because there are values for (δ0 , δ2 , λ) that 0 2 0 2 make ∆ negative, zero, or positive, all three cases are possible in general. However, because ∆ is continuous in λ and ∆ = −4δ0 δ2 < 0 when λ = 0, λ small enough suﬃces for Bc = ∅. Moreover, because g(b) ≥ δ2 > 0 for any b ≥ 0, R+ ⊂ B always. The function F is continuously diﬀerentiable over its entire domain, with F ′ (b) = λ δ2 φ1 (b) δ1 g (b)2 where φ1 (b) ≡ [δ2 − (1 − λ) δ0 ]b2 + 2δ2 b + δ2 . Moreover, lim F (b) = lim F (b) = F∞ ≡ b→−∞ b→+∞ δ2 δ1 and F (−1) = F (0) = δ2 /δ1 < F (δ2 /δ1 ) . 36 1+ λ (1 − λ) (δ0 + δ2 ) > δ2 , δ1 Consider the case δ2 = (1 − λ) δ0 . Then φ1 (b) = 0 admits a unique solution at b = −1/2. Because ∆ < 0, the function F is deﬁned over the entire real line, it is decreasing for b < −1/2 and increasing for b > −1/2. Next, consider the alternative case, δ2 = (1 − λ) δ0 . Then φ1 (b) = 0 admits exactly two solutions, at b = b1 and at b = b2 , where b1 ≡ −δ2 − (1 − λ)δ0 δ2 −δ2 + (1 − λ)δ0 δ2 and b2 ≡ . δ2 − (1 − λ) δ0 δ2 − (1 − λ) δ0 The function F then reaches a local maximum at b1 and a local minimum at b2 . Part (ii). By the preceding results we have that F is continuous over R+ , with F (δ2 /δ1 ) > δ2 /δ1 and limb→∞ F (b) < ∞. It follows that the equation F (b) = b admits at least one solution at b > δ2 /δ1 = πy /πx . Part (iii). Fix any (δ1 , δ2 ) ∈ (0, 1)2 . If λ is such that δ2 = (1 − λ) δ0 , where δ0 = 1 − (δ1 + δ2 ), ¯ then let F ≡ F (−1/2) and F ≡ F∞ . If, instead, λ is such that δ2 = (1 − λ) δ0 , then let F ≡ ¯ ¯ min{F∞ , F (b2 )} and F ≡ max{F∞ , F (b1 )}. It is easy to check that both F and F converge to δ2 /δ1 as λ → 0. Since F is continuous over its entire domain, B, and λ small enough suﬃces for ¯ B = R, we have that λ small enough also suﬃces for F to be bounded in [F , F ]. But then F ˆ ˆ converges uniformly to δ2 /δ1 as λ → 0. It follows that for any ε > 0 there exists λ = λ (ε) > 0 such ˆ that, whenever λ < λ, B = R and F has no ﬁxed point outside the interval [δ2 /δ1 − ε, δ2 /δ1 + ε]. Now note that the function F ′ (b; λ) is continuous at (δ2 /δ1 , 0) with F ′ (δ2 /δ1 ; 0) = 0. It follows ˜ ˜ that, for any η ∈ (0, 1), there exist ε = ε (η) > 0 and λ = λ (η) such that −1 < −η < F ′ (b; λ) < ˜ ˜ ˜ η < 1 for all b ∈ [δ2 /δ1 − ε, δ2 /δ1 + ε] and all λ ∈ [0, λ]. ˜ ˜ Combining the aforementioned results with the continuity of F , we have that there exist ε > 0 ¯ ¯ ¯ and λ > 0 such that, for any λ ∈ [0, λ], the following are true: for any b ∈ [δ2 /δ1 − ε, δ2 /δ1 + ε], / ¯ ¯ F (b) = b; for b ∈ [δ2 /δ1 − ε, δ2 /δ1 + ε], F is continuous and diﬀerentiable in b, with F ′ (b) < 1. It ¯ ¯ ¯ follows that, for λ < λ, F has at most one ﬁxed point. Together with the fact that F necessarily has at least one ﬁxed point (from part (ii)), this proves part (iii). Part (iv). It is easy to check that (δ1 , δ2 , λ) = (.2, .1, .75) implies that B = R (so that F is continuous over the entire real line) and F (b2 ) < b2 < 0. These properties, together with the properties that F (0) > 0 and limb→−∞ F (b) > 0 > −∞, ensure that, in addition to a ﬁxed point in (δ2 /δ1 , +∞), F admits at least one ﬁxed point in (−∞, b2 ) and one in (b2 , 0). Indeed, in this example F admits exactly three ﬁxed point, which are “strict” in the sense that F (b)−b changes sign around them. Because F is continuous in (b, δ1 , δ2 , λ) in an open neighborhood of (δ1 , δ2 , λ) = (.2, .1, .75) , there necessarily exists an open set S ⊂ (0, 1)3 such that F admits three ﬁxed points whenever (δ1 , δ2 , λ) ∈ S. Proof of Proposition 1. From Lemma 4, there always equilibrium in which b > πy /πx . Pick 37 the equilibrium that corresponds to the highest solution to F (b; λ) = b and let b(λ) ∈ (πy /πx , +∞) denote this solution. Part (i) follows from conditions (34), (35) and (36) observing that b > 0 suﬃces for h (b) > 0 and hence for β1 , β2 , γ1 > 0. For part (ii), note that α > 0 follows from γ1 > 0; that α < 1 follows from (8) along with β2 /β1 > 0; ﬁnally, that α increases with λ follows from Lemma 3 and part (iii), which we prove next. First, note that ∂F (b; λ) δ2 b (1 + b) δ2 (1 + 2b) + (δ0 + δ2 ) b2 = , ∂λ δ1 g (b)2 so that b > 0 suﬃces for ∂F (b; λ) /∂λ > 0. Next, note that the function F (b; λ) − b is continuous in b over (πy /πx , +∞) and satisﬁes limb→+∞ {F (b; λ) − b} = −∞. Since b(λ) ∈ (πy /πx , +∞) is the highest solution to F (b; λ) − b = 0, it is then necessarily the case that F (b; λ) − b < 0 for any b > b(λ). Part (iii), then follows from this property together with the fact that F increases with λ. ¯ Proof of Proposition 2. Take any λ < λ. Let b (λ) denote the unique ﬁxed point to F (b; λ) = b and denote by β0 (λ) , β1 (λ) , β2 (λ) , γ0 (λ) and γ1 (λ) the corresponding equilibrium coeﬃcients, as given by (34)-(38). Note that all these functions are continuous. Part (i). Using conditions (34)–(38), the sensitivity of investment to the realization of θ is given by β1 (λ) + β2 (λ) = W (λ) (δ1 + δ2 ) , where W (λ) ≡ w (b (λ) , λ) , with w (b, λ) ≡ (1 + b) (1 − λ + λh (b)) δ1δ1 2 and h (b) deﬁned as in +δ (31). We can compute b′ (λ) and b′′ (λ) applying the Implicit Function Theorem to F (b, λ) − b. We can then use this to compute W ′ (λ) and W ′′ (λ) . After some tedious algebra (which is available upon request), we ﬁnd that W ′ (0) = 0 and W ′′ (0) = − (δ 2δ0 δ1 δ2 < 0. Together with the fact ˆ ¯ that b (0) = δ2 /δ1 and hence W (0) = 1, this ensures that there exists λ ∈ (0, λ] such that, for all ˆ λ ∈ (0, λ), W (λ) < W (0) = 1 and W ′ (λ) < 0; that is, β1 + β2 is lower than δ1 + δ2 , its value in 1 (δ1 +δ2 )+δ2 ) 2 the frictionless benchmark, and is decreasing in λ. ′ Part (ii). From condition (34), we have that β1 (λ) = [1 − λ + λh (b (λ))]δ1 and hence β1 (λ) = δ1 [−1 + h (b (λ)) + λh′ (b (λ) b′ (λ))] . Since b (0) = δ2 /δ1 and h (δ2 /δ1 ) = that ′ β1 (0) δ1 (δ1 +δ2 ) δ1 (δ1 +δ2 )+δ2 < 1, we have ′ = δ1 [−1 + h (δ2 /δ1 )] < 0, which together with the result from part (i) that β1 (0) + ′ ′ ′ β2 (0) = 0 gives β2 (0) > 0. The result then follows from the local continuity of β2 (λ) in λ. Proof of Proposition 4. The ﬁrst claim is proved by the numerical example in the main text. Thus consider the second claim. Given any linear strategy k(x, y) = β0 + β1 x + β2 y, ex-ante utility 38 is given by 1 2Eu = E − k (x, y)2 + θk (x, y) 2 1 2 −1 1 1 2 −1 = − β0 + β0 (1 − β1 − β2 ) µ − β1 πx − (β1 + β2 )2 πθ 2 2 2 1 2 −1 1 2 − β2 πy + (β1 + β2 ) σθ + (β1 + β2 ) 1 − (β1 + β2 ) µ2 . 2 2 (39) Now suppose prices are fully stabilized at p = p. Substituting p(θ, y) = p into the entrepreneurs’ ¯ ¯ best response (2) gives the following coeﬃcients for the equilibrium investment strategy: β0 = (1 − λ) δ0 µ + λ¯, p β1 = (1 − λ) δ1 , and β2 = (1 − λ) δ2 . (40) Note that p aﬀects only the ﬁrst two terms in (40) through its eﬀect on β0 . Hence, the maximal ¯ welfare that can be achieved with full price stabilization is obtained by choosing p so that β0 = ¯ 1 − (1 − λ)(δ1 + δ2 ). Next, note that for any a ∈ (0, 1) and any b ∈ R, there exists a policy τ (p) = τ0 + τ1 p that induces an equilibrium in which the investment strategy is given by35 β0 = b β1 = (1 − λ)δ1 1 − aδ1 and β2 = (1 − λ + aβ1 )δ2 . 1−a (41) To see this, suppose that, given (τ0 , τ1 ), the entrepreneurs follow the linear strategy deﬁned in (41). Then E[θ|K] = γ0 + γ1 K, where (γ0 , γ1 ) are obtained from (41) using the formulas given in (6). The market clearing price is then equal to p= 1 (γ0 + γ1 K − τ0 ) 1 + τ1 (42) Replacing (42) and K(θ, y) = β0 + β1 θ + β2 y into (2), we then have that the best response for each ˜ ˜ ˜ entrepreneur consists in following the strategy k (x, y) = β0 + β1 x + β2 y given by λ(γ0 − τ0 ) + (1 − λ) δ0 µ + α [β0 + β1 δ0 µ] ˜ 1 + τ1 = (1 − λ) δ1 + αβ1 δ1 ˜ ˜ β0 = ˜ β1 ˜ β2 = (1 − λ) δ2 + α [β1 δ2 + β2 ] ˜ ˜ ˜ where α = λγ1 /(1 + τ1 ). It is then immediate that there exists a (τ0 , τ1 ) such that β0 = β0 , β1 = β1 ˜ 35 Equivalently, for any a ∈ (0, 1) and any κ0 ∈ R, there exists a policy (τ0 , τ1 ) that sustains an equilibrium in which the investment strategy satisﬁes k(x, y) = E[(1 − a)˜ (θ) + aK(θ, y) | x, y] with κ(θ) ≡ κ ˜ 39 1−λ 1−a θ + κ0 . ˜ ˜ and β2 = β2 (it suﬃces to choose τ1 so that α = a and then adjust τ0 so that β0 = b). ˜ Now let b0 ≡ 1 − (1 − λ)(δ1 + δ2 ) and for any a ∈ [0, 1) let (1 − λ) δ1 ˜ β1 (a) ≡ , 1 − aδ1 (1 − λ + aβ1 )δ2 ˜ β2 (a) ≡ , 1−a and 2 1 ˜ 1 ˜ 1 −1 ˜ ˜ ˜ β1 (a) πx − β1 (a) + β2 (a) W (a) ≡ − b2 + b0 1 − β1 (a) − β2 (a) µ − 0 2 2 2 2 1 ˜ −1 −1 ˜ ˜ − β2 (a) πy + β1 (a) + β2 (a) πθ 2 1 ˜ ˜ ˜ ˜ + β1 (a) + β2 (a) 1 − β1 (a) + β2 (a) µ2 2 2 −1 πθ Note that welfare under full price stabilization is given by W (0), whereas welfare under any policy (τ0 , τ1 ) that implements a linear strategy as in (41) with a ∈ (0, 1) and b = b0 is given by W (a). Next note that W is continuously diﬀerentiable over [0, 1). To prove the second claim in the proposition it thus suﬃces to show that ˜ ˜ dW ∂W dβ1 ∂W dβ1 = + >0 ˜ ˜ da ∂ β1 da ∂ β2 da at a = 0. First note that ∂W ˜ ∂ β1 −1 −1 ˜ −1 ˜ ˜ ˜ ˜ = −b0 µ − β1 πx − β1 + β2 πθ + πθ + 1 − β1 + β2 µ2 ∂W ∂β2 −1 −1 ˜ −1 ˜ ˜ ˜ ˜ = −b0 µ − β2 πy − β1 + β2 πθ + πθ + 1 − β1 + β2 µ2 ˜ ˜ Using β1 (0) = (1 − λ) δ1 , β2 (0) = (1 − λ) δ2 and b0 ≡ 1 − (1 − λ)(δ1 + δ2 ), we thus have that ∂W ˜ ∂ β1 = ˜ ˜ β1 =β1 (0) ∂W ˜ ∂ β2 ˜ ˜ β2 =β2 (0) −1 −1 = −b0 µ + [1 − (1 − λ) (δ1 + δ2 )] µ2 + λπθ = λπθ > 0. ˜ ˜ Because β1 and β2 are both increasing in a, it follows that dW (0)/da is positive, which establishes the result. Proof of Proposition 5. Rewrite the tax rule as τ (θ, K) = −φ0 − (φ1 − 1)θ − φ2 K, where φ0 ≡ −τ0 , φ1 ≡ 1−τ1 , and φ2 ≡ −τ2 . Suppose that all other entrepreneurs follow the eﬃcient 40 strategy k(x, y) = E [θ|x, y]. The equilibrium price is then given by p(θ, y) = E [θ − τ (θ, K(θ, y)) | K(θ, y)] = φ0 + φ1 E [θ | K(θ, y)] + φ2 K(θ, y) = φ0 + φ1 γ0 + (φ1 γ1 + φ2 ) K(θ, y) where we have used E [θ | K(θ, y)] = γ0 + γ1 K(θ, y), with (γ0 , γ1 ) determined by substituting (β0 , β1 , β2 ) = (δ0 µ, δ1 , δ2 ) into (6). The best response for each individual entrepreneur is then to follow the strategy k(x, y) = E [(1 − λ) (θ − τ ) + λp | x, y] (43) = E [(1 − λ) (φ0 + φ1 θ + φ2 K) + λ (φ0 + φ1 γ0 + (φ1 γ1 + φ2 ) K) | x, y] = (1 − λ) φ0 + λ (φ0 + φ1 γ0 ) + (1 − λ) φ1 E [θ|x, y] + [(1 − λ) φ2 + λ (φ1 γ1 + φ2 )] E [K|x, y] For the tax τ (θ, K) to implement the eﬃcient allocation, it is thus necessary and suﬃcient that the strategy in (43) coincides with k(x, y) = E [θ|x, y] , which is possible if and only if (1 − λ) φ2 + λ (φ1 γ1 + φ2 ) = 0, (1 − λ) φ1 = 1, and (1 − λ) φ0 + λ (φ0 + φ1 γ0 ) = 0. Equivalently, τ0 = −φ0 = λ γ0 , 1−λ τ1 = 1 − φ1 = − λ , 1−λ and τ2 = −φ2 = λ γ1 . 1−λ The result then follows from the fact that γ0 < 0 < γ1 when the entrepreneurs follow the eﬃcient allocation. Proof of Proposition 6. For part (i), it suﬃces to substitute the price as in (16) into the entrepreneurs’ best response (2). Thus consider part (ii). Substituting (4) into (6) gives γ1 = πz 1 (β1 + β2 ) πy (β1 + β2 ) δ2 = 2 = 2 . 2 πθ + πz β1 + β2 β2 πθ + (β1 + β2 ) πy β2 δ0 + (β1 + β2 )2 δ2 1 )δ In the limit, as λ → 0, we have that β0 → δ0 , β1 → δ1 , β2 → δ2 , and hence γ1 → δ2 δ (δ1 +δ+δ 2)2 δ > 0. 1 2 2 0 +(δ1 ˆ ˆ By continuity, then, there exists λ > 0 such that, for all λ ∈ (0, λ), (β1 + β2 ) > 0, i.e. investment increases with θ, γ1 > 0, i.e. the traders’ expectation of θ increases with K, and α = λ(γ1 −λ/φ) > 0, i.e. entrepreneurs perceive a complementarity in their investment decisions. Proof of Proposition 7. Let V (k, K, θ) ≡ − 1 k2 + θk − 2 λ2 2 2φ K . From (19), Eu = 1 EV (k, K, θ). 2 The result then follows from Proposition 3 in Angeletos and Pavan (2007a), noting that κ∗ (θ) ≡ 41 arg maxK V (K, K, θ) = 1 1+λ2 /φ θ Proof of Proposition 8. and α∗ ≡ 1 − Vkk +2VkK +VKK Vkk = VKK = −λ2 /φ. From (2), in any equilibrium in which p is linear in (θ, y, ω), there are coeﬃcients (β0 , β1 , β2 ) such that k(x, y) = β0 + β1 x + β2 y. From (22) and (23), the equilibrium price is then p(θ, y, ω) = P (K(θ, y), θ, ω) ≡ η0 + η1 K(θ, y) + η2 θ + η3 ω. (44) for some (η0 , η1 , η2 , η3 ). Now consider the optimality of the traders’ strategies. As in the benchmark model, the information that K(θ, y) reveals about θ is the same as that of a signal z≡ whose precision is πz ≡ β1 +β2 β2 2 K(θ, y) − β0 β2 =θ+ ε β1 + β2 β1 + β2 πy ,while the information that p(θ, y, ω) reveals about θ given K(θ, y) is the same as that of a signal s= whose precision is πs = η2 η3 2 1 η3 [p(θ, y, ω) − η0 − η1 K(θ, y)] = θ + ω η2 η2 πω . A trader who observes K and p thus believes that θ is normally distributed with mean πθ πz πs µθ + z+ s πθ + πz + πs πθ + πz + πs πθ + πz + πs = γ0 + γ1 K(θ, y) + γ2 θ + γ3 ω E [θ | K(θ, y), p(θ, y, ω)] = where πθ πz β0 µθ − πθ + πz + πs πθ + πz + πs β1 + β2 1 πz γ1 = πθ + πz + πs β1 + β2 πs γ2 = πθ + πz + πs πs η3 γ3 = . πθ + πz + πs η2 γ0 = 42 (45) (46) (47) (48) Combining (22) with (44) we then have that η0 = η1 = η2 = η3 = γ0 2−λ 1 λ γ1 − 2−λ φ 1 (γ2 + 1 − λ) 2−λ 1 (γ3 − 1 + λ) . 2−λ (49) (50) (51) (52) Lastly, consider the optimality of the entrepreneurs’ investment strategies. From condition (2), the strategy k (x, y) = β0 + β1 x + β2 is individually rational if and only if (β0, β1 , β2 ) satisfy β0 +β1 x+β2 y = (1 − λ) E [θ|x, y]+λE [p(θ, y, ω)|x, y] . That is, (β0, β1 , β2 ) must satisfy the following: β0 = [1 − λ + λη1 β1 + λη2 ]δ0 µθ + λη0 + λη1 β0 (53) β1 = (1 − λ + λη1 β1 + λη2 ) δ1 (54) β2 = (1 − λ + λη1 β1 + λη2 ) δ2 + λη1 β2 (55) A linear equilibrium is a thus a solution to (45)-(55). The existence of a linear equilibrium and its uniqueness for λ small enough can be established following steps similar to those in the benchmark model. Here we prove that λ small enough suﬃces for γ1 > 0, and even for α > 0. Substituting πz ≡ β1 +β2 β2 2 πy and πs = γ1 = = = η2 η3 2 πω = πω into (46) gives πz 1 πθ + πz + πs β1 + β2 (β1 + β2 ) πy 2 2 β2 πθ + (β1 + β2 )2 πy + β2 πω (β1 + β2 ) δ2 2 β2 δ0 + (β1 + β2 )2 δ2 + 2 β2 πω πθ +πy +πx . In the limit, as λ → 0, we have that β0 → δ0 , β1 → δ1 , β2 → δ2 , and hence γ1 → (δ1 + δ1 ) δ2 2 δ2 δ0 + (δ1 + δ1 )2 δ2 + 2 δ2 πω πθ +πy +πx > 0. ˆ ˆ By continuity, then, there exists λ > 0 such that, for all λ ∈ (0, λ), (β1 + β2 ) > 0, γ1 > 0 and α= λ 2−λ γ1 − λ φ > 0. 43 Proof of Proposition 9. Let λ2 1 K 2. V (k, K, θ) ≡ θk − k2 − 2 2φ(2 − λ) The result then follows for the same argument as in the proof of Proposition 7. Proof of Proposition 10. Part (i). In any equilibrium in which the price p (θ, ω, w) is linear in (θ, ω, w), there are coeﬃcients (β0 , β1 , β2 ) such that the investment strategy can be written as k (x, w) = β0 + β1 x + β2 w, implying that aggregate investment satisﬁes K(θ, w) = β0 + β1 θ + β2 ω + β2 ζ. For the traders, who know ω but do not know either ζ or θ, observing K is then equivalent to observing a Gaussian signal z with precision πz , where z≡ β2 K − β0 − β2 ω =θ+ ζ β1 β1 and πz ≡ β1 β2 2 πζ , −2 with πζ ≡ σζ . It follows that the equilibrium price satisﬁes p (θ, ω, w) = E[θ|K, ω] + ω = γ0 + γ1 K (θ, w) + (1 − γ1 β2 ) ω, where γ0 = πθ πz β0 πz µ− and γ1 = = πθ + πz πθ + πz β1 β1 (πθ + πz ) πζ β1 β2 β1 2 (56) . (57) πθ + πζ Substituting (56) into the entrepreneurs’ best response gives k(x, w) = (1 − λ) E [θ|x, w] + λE [p (θ, ω, w) |x, w] = (1 − λ) E [θ|x, w] + λγ0 + λγ1 E [K (θ, w) |x, w] + λ (1 − γ1 β2 ) E [ω|x, w] which can be rewritten as in part (i) of the proposition by letting α ≡ λγ1 and κ (θ, ω) ≡ (1 − λ) θ + λγ0 + λ (1 − γ1 β2 ) ω . 1 − λγ1 Finally, that α > 0 is shown in the next part. Part (ii). Substituting K (θ, w) = β0 + β1 θ + β2 w into (58) gives k(x, w) = λ (γ0 + γ1 β0 ) + (1 − λ + λγ1 β1 ) E [θ|x, w] + λγ1 β2 w + λ (1 − γ1 β2 ) E [ω|x, w] 44 (58) Using the facts that E [θ|x, w] = E [θ|x] = δ0 + δ1 x and E [ω|x, w] = E [ω|w] = ηw, where δ0 ≡ −2 −2 −2 −2 −2 −2 −2 −2 −2 σθ /(σθ + σx )µ, δ1 ≡ σx /(σθ + σx ), and η ≡ σζ /(σω + σζ ), the above reduces to k (x, w) = λ (γ0 + γ1 β0 ) + (1 − λ + λγ1 β1 ) δ0 + (1 − λ + λγ1 β1 ) δ1 x +λ [η + (1 − η) γ1 β2 ] w For this strategy to coincide with k (x, w) = β0 + β1 x + β2 w, it is necessary and suﬃcient that the coeﬃcients (β0 , β1 , β2 ) solve the following system: β0 = λ (γ0 + γ1 β0 ) + (1 − λ + λγ1 β1 ) δ0 (59) β1 = (1 − λ + λγ1 β1 ) δ1 , (60) β2 = λ [η + (1 − η) γ1 β2 ] . (61) By (57), γ1 β1 = πζ β2 β1 2 ∈ (0, 1) , (62) πθ + πζ which together with (60) guarantees that β1 ∈ (0, δ1 ). From (60) and (61) we then get β2 =λ β1 β2 β1 η 2 πθ + πζ β2 β1 δ1 (1 − λ) + (1 − η) πζ 2 πθ + πζ β2 β1 + λπζ or equivalently β2 =F λβ1 where F (b; λ) ≡ η δ1 1+λ β2 ;λ λβ1 λ2 πθ b2 + πω b (1 − λ) λ2 πθ b2 + πζ . It is then easy to show that, for λ small enough, F has a unique ﬁxed point and this ﬁxed point is in a neighborhood of β2 η = . λβ1 δ1 Along with the fact that β1 > 0 always, this guarantees that β2 > 0 for λ small enough. Part (iii). The social planner’s problem can be set up as in the baseline model, giving the optimality condition stated in part (iii) of the proposition. Part (iv). From part (iii) the eﬃcient strategy is given by ∗ ∗ ∗ k (x, w) = β0 + β1 x + β2 w 45 with ∗ β0 = δ0 , ∗ β1 = δ1 and ∗ β2 = λη. ∗ We have already shown, in the proof of part (ii), that β1 < δ1 = β1 , which means that investment underreacts to θ. Next, note that β1 > 0 implies γ1 > 0. 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