The Two Faces of Information

Transcription

1 The Two Faces of Information Gaetano Gaballo Banque de France, PSE and CEPR Guillermo Ordoñez University of Pennsylvania and NBER October 30, 2017 Abstract Information is a double-edged sword. On the one hand, it increases investment efficiency when allocating resources. On the other hand, it decreases their trading liquidity value. We capture these two faces of information in a standard macro-finance setting in which agents have to choose an allocation of resources knowing that they may need to trade some of those resources at a latest stage. When agents acquire information to improve reallocation they do not internalize its role in raising their price volatility. We show that information (even free and infinitely precise) may be undesirable from a social perspective for empirically plausible values of risk aversion and trading needs. Also because of the information externality there are multiple equilibria, some of which may be inefficient, either with too much information and excessive volatility or with too little information and inefficient investment. 1 Introduction Information improves the allocation of resources and the efficiency of investment decisions. The positive roles of information when choosing how much to invest in a project, which projects to undertake or how to allocate resources to an endeavor are well-understood and constitute the backbones of a large literature discussing the incentives to acquire information, its costs and its aggregation. Preliminary. Please do not cite without permission. We thank Willie Fuchs, Alexandre Kohlhas and seminar participants at the 2017 Barcelona Summer Forum and the 2017 SED Meetings for comments. The usual waiver of liability applies. 1

2 Even though less acknowledged, information is also critical for the provision of liquidity and insurance. According to the Financial Accounting Standard Board, an asset is liquid when it is readily convertible to cash with minimal or no change in value. As highlighted by Dang et al. (2014), information induces fluctuations in the transaction value of assets and then it may be detrimental to the provision of liquidity and insurance in financial markets. This paper puts on equal footing these dual roles of information in a standard macro-finance setting with investment and trade. We study the problem of atomistic agents that have an endowment of capital and have to decide their production scale by committing on labor choices. We assume that a fraction of an agent s capital proves completely useless for the agent after committing to the scale of production, but it can sell it in financial markets to other agents and in exchange buy others agents capital. In choosing the scale of production, an agent can acquire an information technology. Under this technology the agent receives a signal about the productivity of all capital in the economy, this is about the productivity of the own capital as well as the productivity of capital of all other agents. Focusing on information as a technology (or language) to process and interpret all signals in the economy is a departure from the traditional use of information as signals, and it is critical to understand our efficiency-volatility trade-off. When information is a language to decode signals instead of signals that are free to decode, agents that choose to learn about their own capital cannot help but also learn about others capital. Acquiring the information technology then affects the price of others capital and creates an externality. In other words, while access to the information technology puts a single investor in a better position to choose her labor and scale of production it also contributes to increase the volatility of prices in capital markets. The implication is that investors are more efficient in making scale decision but also more uncertain about their production when many other investors have acquired the information technology. This information externality can be severe enough such that a benevolent planner would rather dispense form agents having access to the information technology, 2

3 even if the information technology is free and provides infinitely precise signals. The intuition behind this surprising result is that, while labor choices provide insurance against foreseen circumstances, transacting capital in the market provides insurance against unforeseen circumstances. In particular, the selling price of capital will be determined by the market belief about the productivity of capital, while the buying price of a composite of others capital is known and given just by the average productivity of capital in the economy. If other agents believe that the own capital is unproductive, then the agent will not be able to exchange her own capital for many of the composite capital. In contrast of the other agents think that the own capital is very productive, then the agent will be able to transact it for a large amount of the composite capital. In sum, when the market is informed about an agent s capital productivity, the value of her production becomes volatile. When the information technology is free and perfect, agents face volatility in consumption even when choosing labor optimally in response to such information. This reallocation possibility that information provides, however, is more valuable when the agent is very risk averse. In contrast, when there is no information at all, the agent only faces the uncertainty that comes from the fraction of the capital that is not traded and that the agent cannot react upon by choosing labor optimally. The impossibility to reallocate resources is less painful when the agent trades a large fraction of its capital. This implies that information may be socially undesirable when risk aversion is relatively low compared with the needs to interact in capital markets. We show that indeed these conditions are satisfied under empirically plausible parameter configurations. When information acquisition is costly, this externality produces a complementarity in information acquisition; it is optimal to acquire information if and only if others acquire information, which leads to multiple equilibria. When an agent expects to trade a large fraction of its own capital, it becomes less concerned about the productivity of the own capital that will be used in production and more concerned about the price at which that capital will be sold. On the one hand, when no other agent acquires information, the price of the own capital simply reflects its expected fundamental value, without volatility, discouraging the agent to spend re- 3

4 sources to acquire the information technology. On the other hand, when other agents acquire information, the capital price will reflect its fundamental value, introducing price volatility and increasing the individual incentives to acquire the information technology to better forecast the selling price of her own capital. This multiplicity has welfare ramifications. Under certain conditions it is socially optimal for agents to produce information and invest efficiently. It may exist, however, an equilibrium without information acquisition and inefficient investment. Similarly, there are conditions under which it is optimal for agents not to produce information, improving liquidity instead of investments, but there may exist an equilibrium with information acquisition and insufficient liquidity. These possible results are relevant in understanding fluctuations and inefficiencies in financial markets. The first possibility captures inefficient booms of liquid assets with poor asset origination quality (as it seems to have happened previous to the recent U.S. financial crisis), while the second captures the inefficiency created by a dry-up of liquid assets when the system turns into excessive information acquisition (as it seems to have happened at the wake of the crisis). Our work relates to the large branch of banking literature that explores the role of information in rationalizing the existence and organization of financial intermediaries. Such literature either focuses on the beneficial role of information for reallocating resources and improving the quality and profitability of assets (such as the seminal papers of Leland and Pyle (1977), Campbel and Kracaw (1980), Bester (1985) and Diamond (1984 and 1991)), or in the detrimental role of information for the value of liabilities (more recently discussed by Gorton and Pennacchi (1990) and Dang et al. (2014)). This paper analyzes these two effects in a unified macro setting and studies the potential multiplicity and inefficiencies arising in such a unified setting. While Goldstein and Yang (2015) and Dow, Goldstein, and Guembel (2017) study the feedback effects between investment and trading prices that generate strategic complementarities, we study the complementarity in information acquisition generated by the needs to trade, and its interaction with the welfare gains from investment reallocation. 4

5 Our work also relates to the more general discussion about complementarity on information acquisition and aggregation. In particular, Morris and Shin (2002) and Angeletos and Pavan (2007) study the social value of public information and its efficient use in the presence of externalities that are hardwired into payoffs. While their focus is on the role of public signals, ours is on the possibility of multiplicity when there is an information acquisition choice. While Colombo, Femminis, and Pavan (2014) also study information acquisition in the presence of strategic complementarities, we highlight the relevance of multiple equilibria. Gaballo (2016) shows that private information about the future can inefficiently increase volatility of market outcomes when agents learn from a public asset price. We also emphasize the detrimental effect of information in increasing volatility, but our mechanism does not rely on the endogeneity of signals and it also arises with perfect information. Finally, Barlevy and Veronesi (2000) study the multiplicity of equilibrium that arise because of information acquisition in financial markets, but their complementarity arises from the endogenous precision of information and not from the impact of precision on prices. Allen, Morris, and Shin (2006) and Hellwig and Veldkamp (2009) also study a setting in which the price of an asset depends on information aggregation in the economy. They do not consider, however, the role of trading on information acquisition. We differ from all these papers on the modeling of information as a technology, under which information, even when free and perfect, may be socially undesirable. The welfare implications of information in financial markets have been explored by Kurlat and Veldkamp (2015) as well. While they assume CARA preferences (standard in the finance literature) and explore a trade-off between the risk and return of different asset combinations, we assume CRRA preferences (more standard in the macroeconomics literature) and explore a trade-off between investment efficiency and liquidity provision. As with the previous papers, their paper is also based on the modeling of information as signals. Finally, our work relates to the more recent literature that highlights the relevance of studying origination and trading of assets in a single setting. Vanasco (2017) shows that information acquisition at origination deepens asymmetric information and may lead to a freeze in trading of assets, with a collapse of liquidity. In our setting the issue 5

6 does not come from asymmetric information in decentralized secondary markets, but rather by the aggregation of information in centralized secondary markets. Caramp (2017) also studies the role of liquidity on the incentives to originate poor quality assets, highlighting the reverse direction of liquidity affecting the quality of assets. While his focus is not on information, his paper is also an example of the close links between investments incentives and trading. The next section presents the model and Section 3 discusses multiplicity of equilibria, the information acquisition externality behind its existence and its welfare implications. Section 4 concludes. In the Appendix we show an extension to risk aversion and to the role of higher-order beliefs. 2 Model There is a single period with a continuum of island of mass one indexed by i (0, 1). Each island is inhabited by a continuum of agents of mass one. An agent j (0, 1) living in island i has utility function [ ] C 1 i,j U i,j E i,j 1 L i,j, (1) where C i,j and L i,j are consumption and labor specific to agent (i, j) respectively, is a constant relative risk-aversion parameter and E i,j [ ] is an expectation operator conditional on the information set Ω i,j of agent (i, j). Each agent is endowed with one unit of raw capital denoted by x i,j = 1. The timing of the period can be divided in three stages. In the first stage, each agent has two choices: whether or not to become informed about the productivity of raw material and the labor he commits to work. In the second stage, raw capital can be transformed into intermediate capital according to a linear production function y(x i,j ) = (θ + θ i )x i,j, where θ i N (0, 1) is an i.i.d. productivity shock and θ is a deterministic productivity component, assumed large enough to guarantee that pro- 6

7 ductivity is positive almost surely. 1 In the third stage, intermediate capital becomes finally productive once combined with (the already committed) labor, generating consumption goods, according to the following production function: C i,j = eˆk i,j (θ i ) c i,j L i,j, (2) where ˆk i,j (θ i ) is the quantity of intermediate capital own by agent (i, j) after the second stage and c i,j {0, c} accounts for the cost of acquiring information (c if acquiring information and 0 if not), which is expressed in terms of intermediate capital. Before going over each stage in more detail, we first discuss the information structure that can be acquired in case of becoming informed. Information Structure: Acquiring information implies receiving a set of private signals that includes a signal θ i +η i,j about the productivity of raw capital of the own island and a signal about the productivity of raw capital for each of the other islands, that is a θ h + η h for each h i. We assume that the signal that an agent acquires about his own island is unbiased and has an idiosyncratic component (there is dispersion among signals acquired by domestic agents), this is η i,j N(0, τ 1/2 ). In contrast, the signal the agent acquires about other islands is also unbiased but common to all foreign agents of that island, this is η h N(0, τ 1/2 h ). Therefore, if an agent acquires information at a cost c it = c, her information set becomes Ω i,j = {θ i + η i,j, {θ h + η h } h i }. If the agent does not acquire information (c it = 0) its information set is Ω i,j =. Now we will go through the different stages backwards. As the third stage is just deterministic production of consumption goods based on equation (2), we discuss first the potential market for raw capital and production of intermediate capital that take place in the second stage. Then we discuss labor choices during the first stage that are conditional on an information set. Finally, we show the optimal choice of 1 More precisely, such that Pr ( θ + θi < 0 ) 0. 7

8 the information set. 2.1 Second Stage: Markets for raw capital In the second stage, raw capital is transformed in intermediate capital. To introduce the role of the market and asset trading in the model, we assume that only a fraction β of the own island raw capital reveals productive for generating intermediate capital in the own island. This fraction, however, can be sold to other islands who can use it productively. Symmetrically, agents can use the proceedings of these sales to buy unproductive raw capital in other islands to use it in production of intermediate capital in the own island. This reallocation of raw capital across islands happens through a market for raw capital in the second stage, after information and labor has been chosen but before production of consumption goods takes place. As we will show, information that was produced in the first stage can become pervasive for the operation of this market, in particular increasing the volatility of raw capital prices, a negative face of information. What is the trading protocol in the raw capital market? We assume that each agent j living in island h i can buy raw capital from island i in a centralized islandspecific market of raw capital i let us denote this demand by z h,j (i) at a unit price R i, which represents an enforceable claim on future production of intermediate capital. In other words, the agent (h, j) can use raw capital from island i that is not productive in island i, and promise to repay with intermediate capital generated in island h. There is, however, a quadratic adjustment cost γ 2 z2 h,j (i) that an agent j from island h has to incur to operate with raw capital from island i. Since labor is already fixed in this stage and production of consumption goods is increasing in the amount of available intermediate capital (from equation 2), in this stage an agent (i, j) seeks to maximize the total quantity of intermediate capital to operate. After selling his own raw capita at price R i, buying foreign raw capital at a price R h from islands h i (which we denote as i) and covering the corresponding information costs and adjustment costs with intermediate capital, the volume of 8

9 intermediate capital available in island i to produce consumption goods in the third stage is ˆk i,j (θ i ) = β ( θ ) + θi + (1 β)ri + Π i,j (h)dh c i,j (3) where Π i,j (h) is the profit for agent j in island i from buying raw capital from island h, which is given by Π i,j (h) = ( θ + θh ) zi,j (h) γ 2 z2 i,j(h) R h z i,j (h), (4) Symmetrically, the profit for each agent j in island h from buying raw capital from island i is given by Π h,j (i) = ( θ + θi ) zh,j (i) γ 2 z2 h,j(i) R i z h,j (i). An agent (h, j) chooses the quantity z h,j (i) of raw capital from island i to buy in order to maximize their expected profits E h,j [Π h,j (i)]. As we have assumed that all agents j in all islands h = i receive the same signal about the productivity θ i, the demand of raw capital from island i is the same for all foreign agents (this is, z h,j (i) = z i for all agents j in island h = i), such that the total demand of raw capital of island i is z i = θ + E i (θ i ) R i. γ As that total supply of raw capital of island i is 1 β in equilibrium we have z i = 1 β. As a consequence, the equilibrium price in the capital market of the raw capital of island i is R i = θ + E i (θ i ) γ(1 β). (5) The actual profit (replacing the price R i and the demand z h,j (i) = z i = 1 β in equation 4) is then equal to Π h,j (i) = (θ i E i (θ i ))(1 β) + γ 2 (1 β)2. 9

10 Symmetrically, Π i,j (h) = (θ h E h (θ h ))(1 β) + γ 2 (1 β)2. As the expected total profits of an agent (i, j) is given by the expectation of actual profits for each raw capital of all islands i, then E i,j (Π i,j (h)) dh = Π i,j (h)dh = γ 2 (1 β)2. (6) From the law of large numbers these profits are deterministic so that their ex-ante and ex-post evaluations coincide. Substituting the price received from domestic raw capital (equation 5) and the profits from buying foreign raw capital (equation 6), the end of this stage quantity of intermediate capital for all agents j in island i (equation 3) is ˆk i,j (θ i ) = βθ i + (1 β)e i (θ i ) + θ γ 2 (1 β)2 c i,j (7) where c i,j is either c or 0 in a symmetric equilibrium. In words, an agent in an island will operate with intermediate goods that come from three sources. First, by transforming a fraction β of its own raw capital into intermediate capital with productivity θ i. Second, by selling a fraction 1 β of its own raw capital to other islands in exchange for R i of intermediate capital per unit of raw capital. Finally, by buying raw capital from other islands, the agent obtains a profit in terms of intermediate capital after producing and repaying to the other islands. As can be seen, the total intermediate capital available for an agent of island i to produce consumption is given by the productivity of a fraction β of own raw capital and the expected productivity of a fraction 1 β of foreign raw capital, after deducting the adjustment costs. Finally, note that the financial market remains well defined no matter how small is γ. Remark on a single price per island, R i : Notice that there is a single price for the raw capital of island i. The reason is that all foreign individuals have the 10

11 same expectation about θ i. This result is a combination of two assumptions. One is that the equilibrium is symmetric, so all agents either acquire information or no agent does. Two, all foreign individuals receive the same signal about θ i. These assumptions are indeed less stringent than it seems. It would be possible that all foreign individuals who acquire information receive correlated idiosyncratic signals, which are aggregated into a single price. Upon observing this price all buyers of raw material i would share E i (θ i ), reaching the same conclusion (see the Appendix for a formal proof). Remark on selling only unproductive raw capita: In our setting agents sell all their unproductive raw capital and none of their productive raw capital. On the one hand, an agent does not want to sell just a fraction of unproductive raw capital as it can be exchanged in the market for some intermediate capital. On the other hand, an agent does not want to sell any own productive raw capital because we have assumed that the agent puts the raw capital in the market before observing its price, R i. Then the selling decision is determined by E i R i = θ+e i [E i (θ i )] γ(1 β) (from equation 5). If the agent works with the own productive raw capital it obtains in expectation θ+e i (θ i ) of intermediate capital. If the agent sells an unit of raw capital it gets in expectation E i R i = θ + E i (θ i ) γ(1 β) if only supplying unproductive raw capital. The agent then clearly prefers to use own raw capital instead of selling it, as foreign agents value it less because they have to pay marginal adjustment costs γ to use foreign raw capital. 2.2 First Stage: Labor choice In the first stage, an agent (i, j) needs to choose labor to use in producing consumption goods in the last stage. For this decision, however, the agent forms an expectation of the quantity of intermediate capital available in the last stage, which was shown before to only depend on the productivity to transform raw capital into intermediate capital in the own island. Becoming informed allows agents to form those expectations more accurately and make better labor choices, the positive face of information. 11

12 Conditional on the information acquisition decision, the agent commits to work a certain amount of hours. According to the first-order condition, labor is conditional on Ω i,j. L i,j = E i,j [e (1 )(ˆk i,j (θ i )) ] 1. (8) A first observation concerns the value of. With < 1 larger expected quantities of intermediate capital induce higher labor as a substitution effect prevails: the more productive is labor (because the level of intermediate capital to operate with is higher) the agent is willing to has less leisure and consume more. With > 1 instead the opposite occurs as a wealth effect prevails: the more productive is labor, the agent produces more and prefers to cut on labor and enjoy more leisure. The second observation is that, according to equation (2), consumption increases on intermediate capital k i,j (θ i ) for a given labor choice. In the next section we study the first decision of the agents, which is whether to become informed or not. 3 Information Acquisition In order to study the decision to acquire information we first define the ex-ante utility of an individual. Given the symmetry across islands and ex-ante symmetry across individuals (previous to possibly acquiring heterogenous signals), this is also the utility of a representative agent in any island and coincides with social welfare. Formally, both the ex-ante expected utility of an agent and social welfare in the economy are given by E[U i,j ] = E [e (1 )ˆk i,j (θ i ) E i,j [e (1 )ˆk i,j (θ i ) ] 1 1 E i,j [e (1 )ˆk i,j (θ i ) ] 1 which obtains after replacing (8) and (2) into (1), where E[ ] denotes the unconditional expectation operator. Notice that in a symmetric equilibrium, equation (7) implies that ˆk i,j (θ i ) = ˆk(θ i ) 12 ],

13 for all j in island i. We decompose ˆk (θ i ) from equation (7) into a stochastic component k (θ i ) = βθ i +(1 β)e i (θ i ) and a deterministic component κ = θ γ 2 (1 β)2 c. Given the properties of the log-normal distributions (see full derivation in the Appendix 1) we have E[U i,j ] = 1 e (1 ) 2 2 ( 1 V 0(E i,j (k(θ i )))+V i,j (k(θ i ))+ 2 1 κ), (9) where V 0 ( ) represents the unconditional volatility operator and V i,j ( ) the volatility operator conditional to the information set held by agent i, j. As our analysis will be concerned with evaluating the effect of information acquisition on welfare for given β and, we can now define a welfare criterion as the following. Definition 1. For given β and social welfare is increasing whenever is increasing. ( 1 V(θ i ) sign(1 ) V 0 (E i,j (k (θ i ))) + V i,j (k (θ i )) 2 ) 1 c In practice, with < 1, utility increases with volatility; in contrast, when > 1 utility decreases with volatility. The beneficial effect of volatility is related to the findings of Lester, Pries, and Sims (2014) and Cho, Cooley, and Kim (2015). When an agent can react by adjusting labor to volatile productivity, welfare increases with volatility when risk aversion is small relative to the elasticity of labor supply (in our case the Frisch elasticity is 1). In such case the option value of adjusting labor increases total expected output to a level that compensates risk aversion. There are two components of volatilities that determine welfare in this economy. One is the volatility that comes from making mistakes once labor has been chosen, which is captured by V i,j ( ) and represents the variance of the intermediate capital available to work after fixing labor. The other is the ex-ante volatility that comes from the uncertainty about what signal will be acquired and which level of labor will be chosen. As the unconditional volatility also encodes the volatility conditional 13

14 on the choice of labor, when the risk aversion is very large, most of the effect is determined already by the volatility of consumption given labor, and not much by the ex-ante volatility in terms of the specific labor used. Next we study the components of these two volatility components and how they depend on the precisions of signals. This discussion will be relevant when discussing information acquisition choices. 3.1 The pieces of volatility We discuss here the properties of first and second order moments involved in our analysis. We first compute the expectation of an agent (i, j) about the stochastic component of the quantity of intermediate capital available to combine with labor, E i,j (k(θ i )) = βe i,j (θ i ) + (1 β)e i,j [E i (θ i )], which depends on the individual expectation of domestic productivity, θ i, and of the market expectation of domestic productivity that determines the market compensation for the own raw capital in the market, E i (θ i ). The individual expectation of θ i is E i,j (θ i ) = τ i,j 1 + τ i,j (θ i + η i,j ), with τ i,j = τ if the agent acquires information and τ i,j = 0 otherwise. Similarly, the market expectation of θ i is E i [θ i ] = τ i 1 + τ i (θ i + η i ), with τ i = τ h if foreign agent acquire information and τ i = 0 otherwise. Finally, the individual expectation of the market expectation is E i,j [E i (θ i )] = τ i 1 + τ i τ i,j 1 + τ i,j (θ i + η i,j ), which implies that E i,j [E i (θ i )] is different from the prior, which is equal to zero, if 14

15 and only if both domestic and foreign agents are informed. Now we can compute the unconditional variance of the expected k(θ i ), which captures the consumption volatility that can be reacted upon by choosing labor optimally, ) ) τ i τi,j V 0 (E i,j (k (θ i ))) = V 0 ((β + (1 β) (θ i + η i,j ) = 1 + τ i 1 + τ i,j ( ) 2 τ i τ i,j = β + (1 β), (10) 1 + τ i 1 + τ i,j which is increasing in both τ i,j and τ i. The domestic precision increases the ex-ante volatility of labor allocations that respond to more precise (and then more extreme) signals. The foreign precision increases the volatility of market expectations, and then the volatility of consumption conditional on a specific labor choice. We can also note that, ceteris paribus, the higher the needs for trading (the lower is β), the weaker is the reaction of intermediate capital to the realized fundamental θ i. If the agent believes that a large fraction of intermediate capital will be generated with own raw capital the volatility of labor choices will be highly responsive to the fundamental θ i. If not, the volatility of labor will be more responsive to what the agent expects of the market expectations of θ i and less responsive to what the own agent s expectation about the own productivity. We compute now the volatility of the individual forecast errors about k (θ i ), which captures the consumption volatility that cannot be reacted upon by choosing labor optimally, τ i V i,j (k (θ i )) = V 0 ((β + (1 β) 1 + τ i ( τ i = β + (1 β) 1 + τ i ) ( θ i τ ) i,j (θ i + η i,j ) + (1 β) 1 + τ i,j ) τ i η i = 1 + τ i ) τ i,j + (1 β) 2 τ i (1 + τ i ) 2 (11) which is decreasing in τ i,j but increasing in τ i. While the foreign precision still increases the volatility of market prices and consumption conditional on a labor choice, domestic precision tends to decrease the forecasting errors about market 15

16 expectations, therefore about market prices. The unconditional volatility of V 0 (k (θ i )) can also be decomposed into the variance that comes from productivity and the variance that comes from noise. We can rewrite the unconditional variance as ( ) 2 τ i V 0 (k (θ i )) β + (1 β) 1 + τ i }{{} V 0 (k(θ i )) θ + (1 β) 2 τ i (1 + τ i ) 2 }{{} V 0 (k(θ i )) η = β2 + τ i 1 + τ i. While V 0 (k (θ i )) θ measures the volatility due to θ i, V 0 (k (θ i )) η the volatility due to the presence of idiosyncratic noise into the price. V 0 (k (θ i )) θ into V 0 (k (θ i )) θ = V 0 (E i,j (k (θ i ))) θ + V i,j (k (θ i )) θ. We can further decompose To see this notice that V 0 (E i,j (k (θ i ))) θ = V 0 (k (θ i )) θ V (E i (θ i )) θ and V i,j (k (θ i )) θ = V 0 (k (θ i )) θ V i,j (θ i ) θ, where V (E i (θ i )) θ = which by the law of total variance implies τ i,j 1 + τ i,j and V i,j (θ i ) θ = V 0 (θ i ) θ = V (E i (θ i )) θ + V i,j (θ i ) θ = τ i,j, This decomposition is relevant to highlight that domestic precision, τ i,j does not change the unconditional variance of θ i, just the weight of its two components. Notice that in a setting where the price is determined by an average market expectation, everything will equally follow fixing V 0 (k (θ i )) η = 0, which only depends on the market information acquisition choice. Notice also that the unconditional volatility of the amount of intermediate capital is increasing in τ i and β, meaning that smaller fluctuations are associated with high reliance on an uninformed market. 16

17 This analysis reveals an externality of own information acquisition onto other agents labor choices that is channeled through the market and affects the unconditional variance of intermediate capital Remark 2. A price externality emerges in information acquisition as where V 0 (k (θ i )) τ i = 1 β2 (1 + τ i ) 2 > 0, V 0 (k (θ i )) θ τ i = 2 (1 β) (β + τ i) (1 + τ i ) 3 > 0 V 0 (k (θ i )) η τ i = (1 β)2 (1 τ i ) (1 + τ i ) 3 > 0 iff τ < 1 In words, the market price becomes more volatile with a more informed market, and in particular, the increase in unconditional volatility of k (θ i ) is mainly driven by the comovement of market prices with fundamental productivities. 4 Multiple equilibria with costly information In determining whether to acquire information or not, the only relevant consideration is about the expectation about domestic productivity. When individuals buy raw capital from other islands they pay their expected productivity of those units of raw capital, therefore there is no gain from having superior information when buying. In contrast, when individuals sell raw capital to other islands, foreign agents pay the market expected productivity of the domestic raw capital. Then the agent would like to forecast the domestic raw capital price. In other words, when choosing information acquisition, an individual only cares about learning about its own productivity, either because it works with his own raw capital (a fraction β) or because it has to sell raw capital (a fraction 1 β) to individuals from other islands at a price determined by the market expectations 17

18 about the own raw capital. Acquiring information has three effects. One, it improves allocations on the part of the own capital that can be used for own production. Two, it improves the forecasting of the price foreign individuals will pay for the own raw capital. Three, it provides information about how much to pay for raw capital from other islands. Even though this last effect does not generate any individual benefit of information because individuals pay the expected value of the raw capital when buying, it does generate an externality on the labor and informational choices of individuals in other islands. Individual information about domestic raw capital productivity affects the forecast error of labor decisions, but the weight on those forecasts errors depends on the decision of other individuals. We discuss how this weight may induce multiple equilibrium in this setting. As we focus on symmetric equilibria, denote by V(θ i ) τi,j τ i the individual ex-ante expected utility for an agent j in island i from acquiring information with precision τ i,j {0, τ} when foreign individuals acquire information of precision τ i {0, τ}. when An equilibrium in which all agents pay the cost of acquiring information exists V(θ i ) τ τ V(θ i ) 0 τ, (12) whereas an equilibrium where nobody acquires information exists when V(θ i ) 0 0 V(θ i ) τ 0. (13) Because of the aforementioned externality in information acquisition, there are parameter configurations under which these two conditions are simultaneously satisfied when information is costly. Substituting equations (10) and (11) in the Definition 1 18

19 for welfare, V(θ i ) can be rewritten as τ i ( ( V(θ i ) = sign(1 ) β + (1 β) 1 + τ i ) 2 ( 1 τ i,j 1 + τ i,j τ i,j ) ) + (1 β) 2 τ i (1 + τ i ) c i. (14) Note immediately that, for any, β and τ i an agent will always have incentive to acquire costless (i.e. fixing c = 0) information. When < 1 an increase in τ i,j increases (1+τ i,j /)/ (1 + τ i,j ) and then welfare. In contrast, when > 1 an increase in τ i,j reduces (1 + τ i,j /)/ (1 + τ i,j ), which also increases welfare as the sign of 1 is negative. When information is costly, however, the incentives to acquire information depends on whether others acquire information. After some algebraic manipulations we can rewrite (12) as and (13) as c ( β + (1 β) 2 (1 )2 c β 2 τ h 1 + τ h ) 2 (1 ) 2 τ 1 + τ + 1 2γ 2 τ 1 + τ, (15) τ h 1 + τ h. (16) The first inequality characterizes an upper bound to the cost of information for acquisition to be a rational choice. The second characterizes a lower bound to the cost for not acquiring information being optimal. If τ i = τ h (this is, foreign agents acquire information), then V 0 (k (θ i )) is high, so the value of information is high. In this case, the condition of information acquisition is more likely to be fulfilled, which justifies that all agents are informed and then τ i = τ h. Intuitively, when there are many informed investors in the market the variance of the market price of the raw capital is large and there are more incentives to forecast it in order to choose labor more precisely. In contrast, if τ i = 0 (this is, foreign agents do not acquire information), then V 0 (k (θ i )) is low, so the value of information is relatively low. In this case, the 19

20 condition of information acquisition is less likely to be fulfilled, which justifies that no agent is informed and τ i = 0. Intuitively, when there are few informed investors in the market the variance of the market price of the raw capital is small and there are less incentives to forecast it in order to choose labor more precisely. Notice at this point that the individual incentives to acquire information relate to the choice of only τ and not τ h. The reason is that the effect of information precision about other islands vanishes when paying the expected productivity for foreign raw capital and when taking expectations about a continuum of foreign islands (recall that the profits from buying raw capital is deterministic and independent of τ h ). Then the individual incentives to acquire information takes τ i as given, but in equilibrium determines τ i when choosing τ ij, as acquiring domestic signals also imply acquiring foreign signals. Finally, parameters may be such that there is no equilibrium. This is the case when the lower bound on c from equation (16) is higher than the upper bound on c from equation (16). If c is between these bounds, then there is non-existence. If everybody acquires information the benefit of information is too low to justify the cost, while if nobody acquires information the benefit is high enough to justify the cost. The intuition for non-existence is reminiscent of Grossman and Stiglitz (1980). If nobody acquires information there are gains from information acquisition as a buyer, as each agent is atomistic and the price is determined by uninformed agents. This gain is captured by the term 1 τ h 2γ 1+τ h in equation (16). In contrast, when other agents acquire information, such information gets revealed in the price, and then this term does not appear in equation (15). Interestingly, since the gains from information doe snot come only from trading as in Grossman and Stiglitz (1980), there are several combinations of parameters that make this non-existence problem moot, as agents still may want to acquire information for additional allocation reasons. 20

21 4.1 Comparative Statics Here we discuss how this range of multiplicity in terms of the cost of acquiring information depends on the parameters of the model and its implications. Figure 1 shows the range of multiplicity for different trading needs, which are captured by β. Quite naturally, the higher is the cost of information, c for a given β, the lower are the incentives to acquire information. Fixing c, however, when β is high there are a lot of incentives to acquire information about the own raw capital as the agents work with it directly and information allows for better labor decisions. When β is low there is more dependence on market expectations of the raw capital productivity. This reduces the incentives to acquire information but also increases the range of multiplicity as externalities become more prevalent. Intuitively, when a large fraction of the raw capital has to be sold, information is relevant to forecast the selling price. How difficult it is to forecast the selling price, however, depends on the information acquisition choices of potential buyers. When buyers acquire information, prices become more volatile and the private incentives to acquire information are large. When buyers do not acquire information, prices are less volatile and the private incentives to acquire information are small. Figure 2 shows the range of multiplicity for different levels of relative risk aversion. When = 1 (log utility case), no precision of information affects welfare. The nonexistence problem arises for relatively low levels of information costs and there is no information acquisition at higher cost levels. When < 1, and in particular as approaches zero (almost risk neutrality) there are very large incentives to acquire information, not because the individuals want to prevent forecast errors but instead because they want to take advantage of choosing labor optimally and increasing total output. On the other extreme, when > 1, and in particular as relative risk aversion increases the incentives to acquire information also become larger but mostly because agents try to avoid forecasting errors once labor is fixed. When risk aversion is high the range of multiplicity also widens because the gains to avoid these forecast errors depend on the information choices of the potential buyers of raw capital. Again the multiplicity range widens for parameter configurations for which the information 21

22 Multiplicity and Trading Needs Figure 1: Multiplicity and Trading Needs No Exist c 0.06 No acquisition Mult. Full acquisition gains arise from reducing the forecast errors of variance generated by the operation of markets. Figure 3 shows that the range of multiplicity is increasing in the precision of foreign signals. As such precision increases it is more difficult for individuals to forecast the market price of raw capital, increasing then the incentives to acquire information to avoid forecast errors. As it is clear from equations (15) and (16), this result depends on the the upper bound increasing with τ h at a faster rate than the lower bound. Figure 4 shows that the range of multiplicity is also increasing in the precision of domestic signals. Not only information becomes more valuable when signals are more precise for a given informational cost, but also it becomes relative more valuable when the market also acquires information (from equation (15) the relative domestic precision multiplies a larger constant than in equation (16). When the precision of domestic signals get very small compared to the precision of foreign signals, the standard non-existence problem arises. The following proposition summarizes this discussion. 17 / 25 22

23 Multiplicity and Risk Aversion Figure 2: Multiplicity and Risk Aversion c 0.06 No acquisition Mult Full acquisition No Exist picture multiplicity.nb Multiplicity and Foreign Precision Figure 3: Multiplicity and Foreign Precision 20 / c No acquisition Mult Full acquisition Proposition 3. Multiplicity 0.08 is more prevalent (the range of informational costs c 18 /

24 Multiplicity and Domestic Precision Figure 4: Multiplicity and Domestic Precision 0.10 c No acquisition Mult No Exist i,j 1+ i,j Full acquisition that admits multiple equilibrium is larger) when individuals are very risk averse (high ), when the trading needs are high (low β) and when the precision of both domestic and foreign information is high (high τ and τ h ). 19 / 25 5 The social costs of costless information We have characterized equilibria and the possibility of multiplicity generated by the presence of externalities for different combinations of information costs, c and precision of domestic and foreign signals, τ and τ h. In short, when individuals acquire information they do not internalize their effect on the volatility of market prices that affect the incentives of other individuals to acquire information. This calls for attention to the analysis of social welfare and the optimal information acquisition of a social planner. From the previous analysis it is clear that, when information is free (this is, c = 0), information acquisition (this is τ i,j = τ and τ i = τ h for all agents and islands) is always an equilibrium, and except in the knife-edge cases of β = 0 or = 1, this is 24

25 a unique equilibrium. This is because individuals have incentives to have access to signals when they are costless. How about the social planner? First, assume that the social planner can manage the two precisions τ and τ h independently. Then the planner s problem is max V(θ i) {τ,τ h } The following proposition characterizes the solution. Proposition 4. τ (τ h ) for any, β and τ h and τ h (τ) with 0 < < 1 τ h (τ) = 0 with > 1 for any β < 1 and τ. Proof. Postponed to Appendix D. The social planner s problem have a bang-bang solution. The planner would always like agents to have perfect information about domestic and foreign raw capital (this is τ = and τ h = ) when < 1 and perfect information about domestic raw capital but no information about foreign raw capital (this is τ = and τ h = 0) when > 1. Intuitively the planner would like agents to have information to choose labor optimally but would like to avoid them having information that generates volatility in the market after labor has been chosen. This way the planner would take advantage of the positive face of information avoiding the negative face of information. Intuitively, even when information is costless the planner would optimally like agents to have information that can be exploited in terms of choosing labor optimally, but not information to what agents cannot ex-post react and only introduces volatility in consumption. This result naturally applies when risk aversion is such that individuals do not like volatility, this is when > 1, which is also the level of relative risk aversion that the literature finds empirically most plausible. 25

26 Base don this benchmark, we turn back to our main setting. First, the planner cannot discriminate between domestic and foreign signals and can either provide the information technology or not, such that giving information to agents about domestic raw capital implies the planner also have to given them information about foreign raw capital. For simplicity, we assume symmetry on signals precision, τ = τ h. Therefore the planner chooses between providing the free signals to agents (τ P = τ) or not (τ P = 0). Hence the planner s problem becomes max V(θ i ) τ P ={0,τ} The next proposition characterizes the solution. Proposition 5. The planner wants to provide to agents free signals with a precision τ about domestic and foreign raw capital if and only if where α = (1 1 β2 ). Proof. Postponed to Appendix E. τ > 0 for < 1 τ > α β 1 α for > 1. Notice first that, as in the previous benchmark in which the planner could choose the signals precision independently, there is a large difference in the incentives to provide free information if < 1 and if > 1. When < 1 welfare increases in volatility, and regardless of other parameters, the planner would rather provide to agents as much information as possible, even of low precision, to introduce volatility. This is consistent with the unconstrained planner s problem of choosing the precision of signals separately. In contrast, when > 1 the unconstrained planner would like to provide to agents domestic signals but not foreign signals. Then the constraint that forces the planner to provide either both signals of a given precision or no signal introduces a trade-off for the planner when > 1. The proposition shows that the planner is more likely to allow agents to obtain signals of a given precision as the 26

27 relative risk aversion increases above 1. Indeed notice that whenever < 1 the β 2 planner would only provide the signals if τ >, which means that in such region the planner prefers to discard free signals and do not give them to agents to operate even when those signals represent perfect information. The trade-off manifests itself in the following way. When > 1, the planner wants agents to have domestic signals to choose labor optimally, however this comes together with providing agents foreign signals that increase the volatility of transactions once labor has been chosen. This last effect, however introduces a reinforcing incentive to provide agents with domestic signals so they can more effectively forecast these errors. This is why information becomes more likely when is larger. This can be easily seen in the proposition. The parameter α declines with given β, while the threshold of information precision increases with α. Then an increase in decreases the information precision required for the planner to provide free signals of a given precision to agents. Similarly, for a given level of risk aversion, when β is large the first effect dominates and the trade-off biases towards acquiring information, as the forecast errors from trading become less relevant. This is why information becomes more likely when β is larger, which can also be easily seen in the proposition. The parameter α declines with β given, while the threshold of information precision increases with α and decreases with β. Then an increase in β leads to a decline in the information precision required for the planner to provide free signals of a given precision to agents. In other words, the planner chooses not to provide free signals to agents when risk aversion is relatively low ( is low) and the need for trading is relatively large (β is low), as the planner is more worried to reduce the volatility created in transactions after labor has been decided than improving the labor choice. Figure 5 shows the conditions for the acquisition of costless information by the social planner for different β and configurations. Recall that with costless signals the only sustainable equilibrium is one with information acquisition for all. This outcome is only socially optimal for all when β = 1. As long as there are trading needs, there is some range of risk aversion for 27

28 Figure 5: Optimal Information with Costless Signals τ P = τ iff τ > π τ P = τ τ P = 0 τ P = τ β which the planner would rather not provide information to agents but they would instead acquire the information in equilibrium. In other words, when there are trading needs there is a region of risk aversion in which the equilibrium with information is not socially optimal. Also notice that, when < 1 and trading needs are large (this is β 0), equation (16) implies that no information acquisition is sustained for an small informational cost c relative to γ. However, when < 1 and information is costless (or very small by continuity), the social planner would like to provide such information to agents, regardless of β. This implies that, when trading needs are large, there is a region of risk aversion in which the equilibrium with no information is not socially optimal. This discussion leads to the following corollary Corollary 6. Both excessive information and insufficient information can be sustained in equilibrium. 28