Transparency of Information and Coordination in Economies with Investment Complementarities

ISSUES IN MACROECONOMIC POLICY DESIGN

By GEORGE-MARIOS ANGELETOS AND ALESSANDRO PAVAN*

Economies with production externalities, de- spect to public information, which increases the mand spillovers, incomplete financial markets, volatility generated by common noise in market and Keynesian frictions are only a few exam- expectations. Moreover, when information is ples of those in which macroeconomic comple- heterogeneous, an increase in the precision of mentarities play a prominent role. Within this public information may have the perverse effect class of economies, how does the precision of of increasing aggregate volatility, by increasing publicly provided and privately collected infor- the sensitivity of economic activity to common mation affect equilibrium allocations and social noise. On the contrary, an increase in the pre- welfare? And what is the optimal transparency cision of private information necessarily re- in the information conveyed, for example, by duces aggregate volatility. Nevertheless, we economic statistics, policy announcements, or show that, as long as there is no value to lotter- news in the media? To answer these questions, ies, welfare unambiguously increases with an we consider a simple real economy where the increase in either the relative or the absolute individual return to investment is increasing in precision of public information. Hence, policies the aggregate level of investment and where that either disseminate more precise informa- market participants have heterogenous expecta- tion about economic fundamentals or reduce the tions about the underlying economic fundamen- heterogeneous interpretation of economic statis- tals (the exogenous productivity). We interpret tics and policy measures necessarily boost wel- an increase in the transparency of public infor- fare. On the contrary, an increase in the mation either as a reduction in the level of precision of private information may reduce common uncertainty for a given level of idio- welfare by increasing the heterogeneity of ex- syncratic uncertainty (that is, an increase in the pectations and thereby obstructing coordination absolute precision of public information) or as a in the market, in which case policies that dis- reduction in the heterogeneity of expectations courage the private collection of private infor- across market participants for a given level of mation may increase welfare. overall uncertainty (that is, an increase in the Morris and Shin (2002) argued that, in envi- relative precision of public information). ronments with strategic complementarities and We first consider an environment where heterogeneous information, more precise public complementarities are weak, so that the equilib- information can reduce social welfare, whereas rium is unique no matter the structure of infor- more precise private information is always ben- mation. As in Stephen Morris and Hyun Song eficial. We find rather the opposite. The differ- Shin (2002), complementarities increase the ence in the results is due to an important sensitivity of equilibrium allocations with re- distinction between the environments in the two models. Morris and Shin (2002) consider a kind of “beauty contest,” where the payoff of a * Angeletos: Department of Economics, Massachusetts player decreases with the distance between his Institute of Technology, 50 Memorial Drive, E52-251, Cambridge, MA 02142, and NBER (e-mail: angelet@ own action and the action of others, but where mit.edu); Pavan: Department of Economics, Northwestern this distance is irrelevant from a social perspec- University, 2001 Sheridan Road, Andersen Hall Room tive. It follows that the complementarity is 3239, Evanston, IL 60208-2600 (e-mail: alepavan@ present only at the private level, and hence the northwestern.edu). We are grateful to Narayana Kocherla- kota for the invitation to prepare this paper. For useful attempt of the agents to align their actions is comments, we thank Daron Acemoglu, Ricardo Caballero, socially wasteful. In this case, more transparent Christian Hellwig, Bengt Holmstrom, and Ivan Werning. public information facilitates more effective 91 92 AEA PAPERS AND PROCEEDINGS MAY 2004 coordination, which is valued by the market but policy traps, where the optimal policy and not by society. In contrast, we consider envi- market outcomes are dictated largely by arbi- ronments where the complementarity is present trary self-fulfilling market expectations. In at the social level so that effective market co- the present paper, we do not consider active ordination is socially valuable, as is likely to be policy intervention. Nevertheless, a similar trap the case in economies with production and de- emerges regarding the information dissemi- mand spillovers, network externalities, or in- nated by government agencies and central complete financial markets. As shown in bankers: The optimal transparency depends Angeletos and Pavan (2003), market partici- on the aggressiveness or leniency of market pants use public information to align their in- expectations. vestment choices, but not enough as compared We conclude that, in the class of environ- to what is socially optimal, for they do not ments considered in this paper, noise in public internalize the positive externality of their in- information may be socially desirable only vestment on the return to others. As a conse- when there is a high risk that more transparency quence, more transparent public information, by will introduce coordination failures.2 Other- permitting more effective coordination in the wise, the timely and frequent provision of pub- market, necessarily increases welfare, despite lic information seems warranted from a social the fact that it may lead to higher volatility.1 perspective, even if that may lead to an increase In light of these results, we finally consider in volatility. the possibility that complementarities are strong enough that multiple equilibria emerge for cer- I. Weak Complementarities tain structures of information, in which case more effective coordination in the market need Preferences and Technologies.—The econ- not always be socially beneficial. Indeed, there omy is populated by a (measure 1) continuum of is a critical threshold for the transparency of agents, indexed by i and uniformly distributed public information above which multiple equi- over the [0, 1] interval. Agents are risk-neutral librium levels of investment are possible. with utility Above this threshold, the desirability of more ϭ Ϫ 1 2 effective market coordination and thus the wel- (1) ui Aki ki . fare effect of more transparent public informa- 2 ʦ ޒ tion depend critically on which equilibrium is We interpret ki as individual investment (or 2 selected. If the market coordinates on the so- effort), A as the return to investment, and ki /2 ϭ͐1 cially desirable equilibrium, facilitating coordi- as the cost of investment. We let K 0 ki di nation is beneficial, and welfare tends to be denote the aggregate level of investment. maximized at high levels of transparency. If Like John Bryant (1983), Russell Cooper and instead the market coordinates on the undesir- Andrew John (1988), Daron Acemoglu (1993), able equilibrium, impeding coordination by in- Jess Benhabib and Roger Farmer (1994), and troducing noise in public information can be others, we introduce a complementarity by as- welfare-enhancing. suming that the individual return to invest- This final result is related to Angeletos et al. ment is increasing in the aggregate level of (2003), where it is shown that, in coordination investment: environments where a privately informed poli- cymaker is interested in fashioning market out- (2) A ϭ ͑1 Ϫ ␣͒␪ ϩ ␣K. comes, active policy intervention may lead to The random variable ␪ parameterizes the exog-

1 In independent parallel work, Christian Hellwig (2003) and Guido Lorenzoni (2003), building on Michael Wood- 2 Matthew Canzoneri (1985), Alex Cukierman and Allan ford (2003), examine monetary economies in which Meltzer (1986), Andrew Atkeson and Patrick Kehoe (2001), complementarities arise in pricing decisions. They also find Nancy Stokey (2003), and others consider how the trans- that the Morris-Shin result about the social value of public parency of policy instruments relates to the ability of the information can be reversed. However, they do not show market to detect policy deviations in Barro-Gordon envi- how the welfare effects of public information depend on ronments where the government lacks commitment. Our whether market coordination is socially desirable. approach is clearly orthogonal to that line of research. VOL. 94 NO. 2 ISSUES IN MACROECONOMIC POLICY DESIGN 93 enous return to investment (the underlying fun- ever, xi introduces idiosyncratic variation in damentals of the economy) and the coefficient market expectations about the fundamentals and ␣ Ն 0 captures the degree of complementarity. may thus be read also as heterogeneity in the Finally, social welfare is given by a utilitarian filtering and interpretation of commonly avail- ϭ͐1 ␦ aggregator, w 0 ui di. Using (1) and (2), we able information. In this sense, measures the have that level of conformity in market expectations, whereas ␴ measures the quality of available 1 information. w ϭ AK Ϫ 1 ͵ k2 di In the following, we interpret an increase in 2 i the transparency of public information either as 0 ␴ ␴ a reduction in z for given x (that is, an increase in the absolute precision of public infor- ϭ ͑1 Ϫ ␣͒␪K Ϫ ͑1 Ϫ 2␣͒ 1K2 Ϫ 1var 2 2 mation) or as an increase in ␦ for given ␴ (an increase in the relative precision of public in- ϭ͐1 Ϫ 2 4 where var 0 (ki K) di measures the formation). What we have in mind is that the cross-sectional heterogeneity in investment. transparency of public announcements, policy Note that w is concave in K for ␣ Ͻ 1⁄2 , whereas measures, and news in the media may affect it is convex for ␣ Ͼ 1⁄2 . In the latter case, either the noise in publicly available informa- lotteries would be welfare-increasing, and vol- tion or the extent to which such information is atility in K would be desirable. Since we are interpreted differently across market partici- interested in the canonical case where welfare is pants. As will become clear, our results are not decreasing in both volatility and heterogeneity, very sensitive to which of the two interpreta- we restrict ␣ ʦ [0, 1⁄2). This also suffices for the tions we adopt. equilibrium to be unique.3

Information and Transparency.—The funda- A. Equilibrium mentals ␪ ʦ ޒ are not known at the time investment decisions are made. Furthermore, Each agent chooses ki so as to maximize ␪ ޅ agents have heterogenous beliefs about . For i[ui]. It follows that the optimal investment is simplicity, we let the common prior about ␪ be given by uniform over ޒ. We summarize the public in- ϭ ϭ ޅ ͓ ͔ ϭ ͑ Ϫ ␣͒ޅ ͓␪͔ ϩ ␣ޅ ͓ ͔ formation by a sufﬁcient statistic z such that z (3) ki i A 1 i i K . ␪ ϩ ␴ ␧ ␧ z , where is standard normal, independent of ␪ and common across agents. Similarly, Individual investment is thus increasing in the the private information of agent i is summarized expected level of the fundamentals and in the ϭ ␪ ϩ ␴ ␰ ␰ by a sufﬁcient statistic xi x i, where i expected level of aggregate investment. is standard normal, independent of ␪ and inde- Given the linearity of (3) and the normality of pendent and identically distributed across posterior beliefs about ␪, equilibrium invest- ␴ ␴ ϭ ␤ ϩ ␥ agents; z and x parametrize the precision of ment decisions are linear, so that ki xi z, public and private information, respectively. where ␤ and ␥ are constants determined in ␦ ϵ ␴Ϫ2 ␴Ϫ2 ϩ ␴Ϫ2 ␴ ϵ ␴Ϫ2 ϩ ϭ ␤␪ ϩ ␥ Letting z /( x z ) and ( x equilibrium. Then, K z, and thus, ␴Ϫ2 Ϫ1/2 ␪ z ) , the posterior belief of agent i about ޅ ␪ ϵ ޅ ␪͉ ϭ Ϫ ϭ ޅ ͓ ͔ is normal with mean i[ ] [ xi, z] (1 ki i A ␦ ϩ ␦ ␪ ϵ ␪͉ ϭ )xi z and variance Vari[ ] Var[ xi, z] ␴2 ϭ ͑ Ϫ ␣ ϩ ␣␤͓͒͑ Ϫ ␦͒ ϩ ␦ ͔ ϩ ␣␥ . Literally interpreted, the dependence of 1 1 xi z z. ޅ ␪ i[ ]onxi is the result of the observation of private signals about ␪. More generally, how- It follows that ␤ ϭ (1 Ϫ ␣)(1 Ϫ ␦)/[1 Ϫ ␣(1 Ϫ ␦)] and ␥ ϭ ␦/[1 Ϫ ␣(1 Ϫ ␦)]. Clearly, this is the unique symmetric linear (rational-expectations) 3 The equilibrium is unique if and only if ␣ Ͻ 1. The case ␣ ʦ [1⁄2 , 1) is considered in Angeletos and Pavan (2003). In that case, introducing noise in public information can be desirable to the extent that this substitutes for the 4 Note that ␦ is an increasing transformation of the absence of socially valuable lotteries. relative precision of public information. 94 AEA PAPERS AND PROCEEDINGS MAY 2004

equilibrium. Furthermore, as proved in Morris B. Welfare and Shin (2002), when best responses are linear ޅ ␪ ޅ in i[ ] and i[K], there do not exist equilibria We now consider social welfare evaluated at 5 ␪ ϭ 1 ␪2 Ϫ 1 ⍀ other than this one. equilibrium. This is given by w( ) 2 2 , where PROPOSITION 1: The equilibrium exists, is ϭ ␤ ϩ ␥ ⍀ ϭ ͑␤␴ ͒2 ϩ ͑ Ϫ ␣͒͑␥␴ ͒2 unique, and is given by ki xi z, where x 1 2 z ϭ ͑ ͉␪ ͒ ϩ ͑ Ϫ ␣͒ ͑ ͉␪͒ (4) ␤ ϭ 1 Ϫ ␦ Ϫ ␳ Var ki , z 1 2 Var K .

␥ ϭ ␦ ϩ ␳ ⍀ measures the welfare consequences of heterogeneity in individual investment and volatility ␣ Ͻ 1 ␣␦͑1 Ϫ ␦͒ in aggregate investment. Since ⁄2 , welfare ␳ ϭ decreases with both heterogeneity and volatil- Ϫ ␣͑ Ϫ ␦͒ . 1 1 ity. Furthermore, the relative weight on heterogeneity is increasing in ␣. This reﬂects the Note that ␤ ϭ 1 Ϫ ␦ and ␥ ϭ ␦ for ␣ ϭ 0. social value of coordination: The stronger the The term ␳ thus measures the excess sensi- complementarity, the more important the align- tivity of equilibrium allocations to public in- ment in individual investment decisions. ␴ ϭ ␴ ͌ Ϫ ␦ formation as compared to the case where Using (4) and substituting x / 1 ␳ ␴ ϭ ␴ ͌␦ there are no complementarities. Moreover, and z / ,weget is increasing in ␣. Stronger complementarities thus lead to a higher sensitivity of invest- ͑1 Ϫ 2␣͒ ϩ ␣2͑1 Ϫ ␦͒ ⍀ ϭ ␴2. ment to public information. This is a direct ͓1 Ϫ ␣͑1 Ϫ ␦͔͒2 implication of the fact that, in equilibrium, the public signal is a relatively better predic- It follows that ␣ ʦ [0, 1⁄2) sufﬁces for ⍀Ͼ0 tor of aggregate behavior than the private which leads to the following proposition. signal. The equilibrium levels of volatility and het- PROPOSITION 3: Welfare necessarily in- ͉␪ ϭ ␥␴ 2 ͉␪ ␦ erogeneity are Var(K ) ( z) and Var(ki , creases with either an increase in or a reduc- ϭ ␤␴ 2 ␤ ␥ ␴ z) ( x) , respectively, where and are tion in . given by (4). This leads to the following proposition. That welfare decreases with ␴ for given ␦ follows directly from the fact that an increase in PROPOSITION 2: (i) Volatility necessarily in- ␴ means an increase in both volatility and het- creases with an increase in ␦ for given ␴, and erogeneity. On the other hand, as shown in ␴ ␴ ␦ increases with a reduction in z for given x if Proposition 2, an increase in implies lower ␴ 2 Ͼ ␴ 2 Ϫ ␣ and only if z x/(1 ). (ii) Heterogeneity heterogeneity at the expense of higher volatility. falls with either an increase in ␦ or a reduction To understand why the effect of lower hetero- ␴ in z. geneity dominates, note that social welfare under a utilitarian objective coincides with the This result may suggest that transparency can expected utility of an agent. Suppose for a mo- be socially undesirable when it increases vol- ment that there were no complementarity (␣ ϭ atility. However, we will see that welfare 0), in which case ␤ ϭ 1 Ϫ ␦ and ␥ ϭ ␦.It ⍀ϭ ␤␴ 2 ϩ ␥␴ 2 ϭ Ϫ necessarily increases with an increase in ei- follows that ( x) ( z) (1 ther the relative or the absolute precision of ␦)␴2 ϩ ␦␴2 ϭ ␴2, so that the expected utility of public information. an agent depends only on ␴, and not on ␦. This result should be expected: An increase in ␦ given ␴ substitutes a higher precision in public information for a lower precision in private 5 Note that, although the two models are different, the information, without altering the overall preci- structure of the best responses is identical and so are the sion of information. When ␣ ϭ 0, individual equilibrium strategies. choices are not interdependent, and the decom- VOL. 94 NO. 2 ISSUES IN MACROECONOMIC POLICY DESIGN 95

position of information between private and would permit him to predict more accurately the public is irrelevant. When instead ␣ Ͼ 0, public aggregate level of investment. In other words, information becomes more important than pri- the private collection of information may create ␴ vate information in predicting the return to in- a negative externality, implying that a lower x vestment, so that a substitution from private to need not be beneficial from a social viewpoint. public information raises the expected utility of We can thus have a market failure in the amount an agent. In other words, an increase in ␦ raises of private information collected by individual welfare because it permits the agents to second- agents, in which case government intervention guess each others’ actions better and therefore that discourages the collection of private infor- facilitates more coordination in the market. mation may actually increase welfare. We next consider the comparative statics of w Finally, our result that more transparency in ␴ ␴ with respect to x and z, which combine the public information increases welfare even effects of ␦ and ␴. To this aim, we first rewrite though it may also increase volatility, contrasts ⍀ as with the result of Morris and Shin (2002). As we anticipated in the Introduction, the differ- ␴2␴2͓͑ Ϫ ␣͒␴2 ϩ ͑ Ϫ ␣͒2␴2͔ x z 1 2 x 1 z ence is due to the social value of coordination. ⍀ ϭ . ͓␴2 ϩ ͑1 Ϫ ␣͒␴2͔2 In our model, the complementarity is equally x z present at the private and the social level.6 In It follows that the next proposition holds. Morris and Shin (2002), instead, there is an additional externality that perfectly offsets the ␴ complementarity at the social level. To see this, PROPOSITION 4: (i) A reduction in z nec- ␴ let L ϭ (␪ Ϫ A)K and suppose individual pay- essarily increases welfare. (ii) A reduction in x 1 1 offs are u ϭ Ak Ϫ k2 ϩ L, in which case decreases welfare if and only if ␣ Ͼ and i i 2 i 3 ϭ͐ ϭ͐␪ Ϫ 1 2 welfare reduces to w uidi ( ki 2 ki )di. ͑1 Ϫ ␣͒2 This is the analogue of Morris and Shin (2002) ␴2 Ͼ ␴2 in our setting. L is the externality that renders x ␣ Ϫ z . 3 1 the social (gross) return to investment independent of K, thus removing the complementa- Hence, more precise public information neces- rity at the social level. It follows that the sarily increases welfare. This is because a re- coordination motive is not warranted from a ␴ ␴ duction in z for given x implies both better social perspective, in which case stronger coordination (higher ␦) and lower uncertainty complementarity or more transparent public in- (lower ␴). On the contrary, more precise private formation may decrease welfare by exacerbat- information has an ambiguous effect on wel- ing this motive. If instead coordination is ␴ fare. This is because a reduction in x for given socially valuable, as is probably the case in ␴ ␴ z means lower uncertainty (lower )atthe economies with production or demand spill- expense of lower coordination (lower ␦). If the overs, network externalities, or incomplete fi- complementarity is sufficiently weak, so that nancial markets, then more transparency is most there is little value to coordination, or if the likely to boost welfare. relative precision of public information is very low, so that volatility is low compared to het- II. Strong Complementarities erogeneity, the benefit of lower uncertainty out- weighs the cost of lower coordination. In this section, we consider environments in ␴ Otherwise, a reduction in x reduces welfare. which the complementarity is sufficiently ␴ One can interpret x as the amount of infor- strong to induce multiple equilibria for some mation collected privately by market partici- information structures. To capture this possibil- pants. Given the precision of the information ity, we now let and the strategies of the other agents, an individual decision-maker always values more pre- A ϭ ␪ ϩ 1͓K Ն r͔ ␴ cise information. Hence, a lower x is always beneficial from a private perspective. On the other hand, an agent may prefer other agents to 6 See also Angeletos and Pavan (2003) for further have less precise private information, since this discussion. 96 AEA PAPERS AND PROCEEDINGS MAY 2004

Ն Ն ␴ Ͼ ␴ where 1[K r] equals 1 if K r and 0 alently z ˆ z, then F is monotonic in x for all otherwise; r ʦ (0, 1) represents the critical size z, and thus the equilibrium threshold x*is of aggregate investment necessary for the unique. F is instead nonmonotonic whenever ␦ Ͼ ␦ˆ ␴ Ͻ ␴ complementarity to pay off. Without serious , or equivalently z ˆ z. In this case, let Ϫ1 loss of generality, we let r ϭ 1⁄2 . For tractabil- z៮, zគ ϭ 1⁄2 Ϯ {m Ϫ⌽ (m)␴(1 Ϫ ␦)/␦ Ϫ 1⁄2}, Ϫ1 ity, we also let the cost of investment be linear, where m Ͼ 1⁄2 solves ⌽Ј(⌽ (m)) ϭ ␴(1 Ϫ so that ␦)/␦, and note that 0 Ͻ zគ Ͻ 1⁄2 Ͻ z៮ Ͻ 1. For z ԫ (zគ, z៮), (5) has a unique solution x*. But for z ʦ ϭ Ϫ គ ៮ Ͻ ui Aki ki (z, z), (5) admits three solutions, x*low x* Ͻ x* . The two extreme solutions rep- ʦ medium high and constrain ki [0, 1]. resent stable equilibria; the intermediate one Let ␪ ϵ 0 and ␪ ϵ 1. If ␪ ʦ [␪, ␪] were represents an unstable equilibrium, which we ϭ ϭ common knowledge, both ki 1 and ki 0 for disregard. all i would be an equilibrium; the former coincides with the first best, whereas the latter PROPOSITION 5: (i) If ␦ Ͻ ␦ˆ (equivalently, ␴ Ͼ ␴ ԫ គ ៮ represents a coordination failure. With hetero- z ˆ z), or z (z, z), there exists a unique ϭ Ն geneous information, the possibility of multiple threshold equilibrium: ki 1 if and only if xi equilibria depends on the transparency of public x*(z), where x*(z) is the unique solution to information, as we show next. F(x*; z, ␦, ␴) ϭ 0. (ii) If instead ␦ Ͼ ␦ˆ (equiv- ␴ Ͻ ␴ ʦ គ ៮ alently, z ˆ z) and z (z, z), there exist A. Equilibrium exactly two stable threshold equilibria: In the ϭ Ն first, ki 1 if and only if xi x*low(z); in the ϭ ϭ Ն An agent finds it optimal to invest ki 1if second, ki 1 if and only if xi x*high(z), where ޅ Ն ϭ i[A] 1, and ki 0 otherwise. We restrict x*low(z) and x*high(z) are the lowest and highest attention to equilibria with monotonic strate- solutions to F(x*; z, ␦, ␴) ϭ 0. gies, in which case for every z there exists x*(z) ϭ Ͼ ϭ ␦ ␴ such that ki 1ifxi x*(z) and ki 0 A high level of transparency (high or low z) otherwise. Aggregate investment is thus in- may thus lead to multiple equilibria. When this ␪ ␪ ϭ⌽ ␪ Ϫ creasing in and is given by K( , z) ([ is the case, the second equilibrium (x*high)is ␴ ⌽ x*(z)]/ x), where is the standard normal cu- characterized by less aggregate investment than mulative distribution function. This implies that the first (x*low). K Ն r (ϭ1⁄2) if and only if ␪ Ն ␪*(z), where ␪ ϭ ޅ ͉ ϭ ޅ ␪͉ *(z) x*(z), and therefore [A xi, z] [ xi, B. Welfare ϩ ␪ Ն ␪ ͉ z] Pr( * xi, z), or equivalently The probability that a coordination failure is ޅ͓ ͉ ͔ ϭ ͑ Ϫ ␦͒ ϩ ␦ A xi , z 1 xi z possible depends on the transparency of information. Indeed, ͑1 Ϫ ␦͒x ϩ ␦z Ϫ ␪* ϩ ⌽ i ͩ ␴ ͪ. z៮ Ϫ ␪ zគ Ϫ ␪ ͓ ʦ ͑គ ៮͉͒␪͔ ϭ ⌽ Ϫ ⌽ Pr z z, z ͩ ␴ ͪ ͩ ␴ ͪ ϭ Ն ϭ z z It follows that ki 1ifx x*, and ki 0 otherwise, where x* solves ޅ[A͉x*, z] ϭ 1. increases with z៮, decreases with zគ, and decreases with ␪ ϭ ␴ ␪ ʦ គ ៮ Ѩគ Ѩ␦ Ͻ Ͻ Ѩ៮ Ѩ␦ Combining this with * x*, we conclude that z for (z, z). Moreover, z/ 0 z/ . the equilibrium threshold x* must solve ␦ Ͼ ␦ˆ ␴ Ͻ PROPOSITION 6: If (equivalently, z ͑ ␦ ␴͒ ϵ ␦͑ Ϫ ͒ ␴ ␪ ʦ គ ៮ (5) F x*; z, , z x* ˆ z) and (z, z), the probability of multiple equilibrium levels of investment increases with Ϫ ␴⌽Ϫ1͓ Ϫ ͑ Ϫ ␦͒ Ϫ ␦ ͔ ϭ ␦ ␴ ␴ 1 1 x* z 0. either a higher for given or a lower z for ␴ given x. It is easy to check that the above always admits a solution. Let ␦ˆ ϵ ͌2␲␴/(1 ϩ ͌2␲␴) An increase in the level of transparency may ␴ and let ˆ z be the unique positive solution to thus decrease welfare by increasing the proba- ␴ 2͌ ␲ ϭ ␴ 2͌␴Ϫ2 ϩ ␴Ϫ2 ␦ Ͻ ␦ˆ z 2 x x z .If , or equiv- bility that the “bad” equilibrium is played. VOL. 94 NO. 2 ISSUES IN MACROECONOMIC POLICY DESIGN 97

welfare along the low-investment equilibrium. In general, the welfare effects of transparency are neither monotonic nor homogeneous across ␪. However, the effects tend to be small for sufficiently low or sufficiently high values of ␪. We thus choose to concentrate on intermediate fundamentals. ␴ ϭ Since z 0 restores common knowledge for ␴ Ͼ any given x 0, and since the “good” equilibrium under common knowledge coincides with the first best, it is trivial that welfare is ␴ ϭ maximized at z 0. Therefore, provided that there is little risk of a coordination failure, full transparency is desirable. If instead the “bad” equilibrium is played whenever multiple equilibria are possible, welfare tends to decrease ␴ ␴ with a reduction in z beyond ˆ z, in which case welfare is maximized at an intermediate level of transparency. The patterns revealed by the examples of Figure 1 appear to be robust across a large number of simulations. Also, simulations of the welfare effect of ␦ give similar results and are omitted due to space limitations. We thus conclude that the following must hold.

PROPOSITION 7: If the socially preferable equilibrium is selected with high probability whenever multiple equilibria are possible, welfare is robustly maximal at ␦ ϭ 1 for given ␴, ␴ ϭ ␴ and necessarily maximal at z 0 for given x. If instead the worse equilibrium is selected with high probability, welfare is robustly maximized ␦ Յ ␦ˆ ␴ Ն ␴ at , or z ˆ z.

III. Concluding Remarks

This paper has examined the welfare effects of public and private information in an economy with investment complementarities. If the complementarity is weak so that multiple equilibria are never possible, more transparency in ␴ public information increases welfare, despite FIGURE 1. THE EFFECT OF z ON WELFARE the fact that it also increases volatility. This is because more transparency facilitates more effective coordination in the market, which Unfortunately, it is impossible to characterize is socially valuable given that the complemen- welfare analytically. We thus resort to numeri- tarity is present at the aggregate level. On the cal simulations. other hand, if the complementarity is strong so ␴ ␪ The effect of z on w( ) is illustrated in that multiple equilibria are possible for high Figure 1 for various values of ␪. The solid lines levels of transparency, more precise public in- represent welfare along the high-investment formation facilitates more effective market equilibrium, whereas the dashed lines represent coordination on either equilibrium. In that case, 98 AEA PAPERS AND PROCEEDINGS MAY 2004

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