value of information in competitive economies with ... of information in competitive economies with...
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Value of Information in
Competitive Economies with
Incomplete Markets
⇤
by
Piero GottardiDepartment of Economics
European University Institute
Villa San Paolo
Via della Piazzuola 43
50133 Florence, Italy
http://www.eui.eu/Personal/Gottardi/
and
Rohit RahiDepartment of Finance
and Financial Markets Group
London School of Economics
Houghton Street
London WC2A 2AE, U.K.
http://vishnu.lse.ac.uk/
November 28, 2012.
⇤We are grateful to Atsushi Kajii and Tito Pietra for extensive research discussions. We wouldalso like to thank Alex Citanna, Andy Postlewaite, Antonio Villanacci, and the referees for helpfulcomments.
Abstract
We study the value of public information in competitive economies with incom-plete asset markets. We show that generically the welfare e↵ect of a change inthe information available prior to trading can be in any direction: there existchanges in information that make all agents better o↵, and changes for whichall agents are worse o↵. In contrast, for any change in information, a Paretoimprovement is feasible, i.e. attainable by a planner facing the same informa-tional and asset market constraints as agents. In this sense, the response ofcompetitive markets to changes in information is typically not socially optimal.
Journal of Economic Literature Classification Numbers: D52, D60, D80.
Keywords: Competitive Equilibrium, Incomplete Markets, Value of Informa-tion.
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1 Introduction
The objective of this paper is to analyze the value of public information in the setupof a competitive exchange economy under uncertainty in which agents trade in assetmarkets to reallocate risk. It is well-known that when markets are complete, andagents are initially uninformed, the receipt of a public signal prior to trading impairsrisk sharing and cannot improve welfare. Indeed, if the true state of the world isrevealed before markets open, no mutually beneficial risk sharing trade is possible.This e↵ect has come to be known as the Hirshleifer e↵ect, after Hirshleifer (1971) whoproduced an early example of it. If markets are incomplete, the arrival of informationstill reduces the possibilities of trading gains from risk sharing, but a second welfaree↵ect arises. Provided some residual uncertainty remains, agents can hedge it moree↵ectively by conditioning their portfolios on the available information. We referto this as the Blackwell e↵ect, after Blackwell (1951) who compared the value ofdi↵erent information structures in single-agent decision problems.
We begin our analysis with an example of the welfare impact of the arrival ofinformation before markets open in which the Hirshleifer and Blackwell e↵ects can beclearly disentangled. We show that, depending on the nature of this information, oneor the other e↵ect dominates so that in equilibrium welfare can worsen or improve. Inthe rest of the paper we extend the analysis to a general two-period exchange economywith a single consumption good and a single round of asset trade wherein we studythe welfare e↵ects of arbitrary changes in information, starting from a situationwhere agents may have some initial information, changes that do not necessarilyentail an increase or decrease in the information available to agents prior to trading.While in our general analysis the welfare impact of a change in information cannotbe interpreted in terms of the Hirshleifer and Blackwell e↵ects, it is neverthelesspossible to provide an analogous interpretation that is helpful in understanding howcompetitive markets deal with changes in information.
For this purpose it is instructive to compare the change in agents’ welfare inequilibrium to welfare changes that can be achieved by a hypothetical planner whofaces the same constraints as agents in terms of the available assets and information.We show that for a generic economy a (strict) Pareto improvement can be attained,for any change in information, by tailoring agents’ portfolios to the new informa-tion. When we compare competitive equilibria associated with di↵erent informationstructures, an additional welfare e↵ect obtains because of adjustments in equilibriumasset prices. We find that the overall e↵ect on welfare may be positive or negative.More precisely, we show that generically, if markets are su�ciently incomplete, thereexists a change in information that makes all agents better o↵ as well as a changethat makes all agents worse o↵; indeed, there is a change in information such thatany subset of agents is better (or worse) o↵.
Thus agents’ welfare can vary in any direction, depending on the change in in-formation, in line with our findings in the example. Moreover, we can interpretthe welfare e↵ect of a change in information as having a positive component due to
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agents being able to tailor their portfolios to the available information, and a secondcomponent that may be positive or negative for any individual agent arising fromthe adjustment in equilibrium asset prices induced by the change in information.This is in the same spirit as the decomposition of the overall welfare e↵ect in theexample into the Blackwell and Hirshleifer e↵ects.1 Our analysis provides an answerto the question of how competitive markets respond to changes in information, andhow equilibrium changes in welfare compare with welfare changes that are attain-able. Our results lead to the conclusion that competitive markets typically do notdeal with changes in information in a way that is welfare-improving even though itis feasible to do so.
There is an extensive literature on the value of information in a competitive pureexchange setting. A number of papers have followed Hirshleifer’s lead in comparingcompetitive equilibrium allocations before and after the arrival of information. As-suming complete markets, Schlee (2001) derives some conditions under which moreinformation is Pareto worsening.2 In an economy with incomplete markets, Green(1981) and Hakansson et al. (1982) provide (quite restrictive) conditions under whichbetter information leads to a Pareto improvement, while Milne and Shefrin (1987)present an example showing that welfare can also decrease. In contrast to the variousexamples in the literature, the example in the present paper allows a clear under-standing of the welfare impact of the arrival of information by explicitly decomposingthe overall change in welfare into the Hirshleifer and Blackwell e↵ects.
In our general welfare analysis we utilize the analytical apparatus developedby Geanakoplos and Polemarchakis (1986), and later generalized by Citanna et al.(1998), to study the welfare e↵ects of local perturbations in competitive economieswith incomplete markets. These methods have been employed in particular by Cassand Citanna (1998) and Elul (1995) to show that generically the welfare e↵ect of theintroduction of a new asset can be in any direction. We provide an analogous resulton the e↵ect of a change in public information prior to trading.3 While there aresome parallels between these two problems, they are analytically quite distinct. Theintroduction of a new asset enlarges the set of feasible allocations, while this is not
1Unlike the Blackwell e↵ect, which we can properly speak of only for an increase in information,the first (positive) component is present for any change in information. The second component bearsa closer relationship to the classical Hirshleifer e↵ect. Indeed, Gottardi and Rahi (2001) argue thatthe Hirshleifer e↵ect is due to changes in equilibrium prices, induced by a change in information,that alter agents’ budget sets. See footnote 13 for further discussion in the context of this example.
2For the case of complete markets, an alternative to the welfare analysis of competitive equilib-ria is provided by Campbell (2004), who compares welfare at allocations associated with di↵erentinformation structures when these allocations have to be feasible and, in addition, satisfy ex-postindividual rationality constraints. He shows that any increase of information in the sense of Black-well (1951) has a negative e↵ect on agents’ welfare in the sense that each allocation associated withthe better information is ex-ante Pareto dominated by some allocation associated with the worseinformation.
3We are able to exploit di↵erential techniques by employing a smooth parametrization of changesin information, just as Cass and Citanna (1998) and Elul (1995) are able to use these techniquesby modeling the introduction of an asset in a way that avoids discontinuities.
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in general true for a change in information.4 Furthermore, the price changes thatunderpin the welfare e↵ects are di↵erent. In our single-good setting, only the currentasset prices are a↵ected by a change in information. In contrast, the results of Cassand Citanna (1998) and Elul (1995) also depend on adjustments in the future relativeprices of physical commodities that modify the set of transfers of state-contingentconsumption achievable through asset trading.5
The aforementioned techniques have also been used by Citanna and Villanacci(2000) to study the e↵ect on welfare of the information revealed by prices in anasymmetric information economy with multiple physical goods and nominal assets.In their model, there is a continuum of equilibria which can be parametrized by thefuture (state-contingent) price level. The authors compare a non-revealing equilib-rium to a nearby equilibrium of the same economy at which asset prices reveal someinformation, and show that generically there is an arbitrary change in welfare whengoing from one equilibrium to the other, with the utility of any individual agent go-ing up or down. In contrast, we consider an exogenous change in public information,and show that the welfare change is arbitrary when we move from any equilibriumof a generic economy to a nearby equilibrium associated with the new informationstructure. Moreover, in our model with a single physical good and real assets, thewelfare change can be attributed entirely to the change in information under con-sideration; it arises from agents being able to condition their portfolios on the newinformation and from the adjustment in equilibrium asset prices resulting from thechange in information. In the analysis of Citanna and Villanacci (2000), on the otherhand, welfare e↵ects cannot be traced solely to the change in information, which isendogenous, being the information revealed by asset prices. In their welfare results,an important role is played by two additional price e↵ects which do not arise in ourmodel: adjustments in relative spot commodity prices in the future, and changes inreal asset payo↵s induced by changes in the future price level. The latter e↵ect isunrelated to the change in information.
The paper is organized as follows. We describe the economy in Section 2. InSection 3 we provide an example to illustrate the welfare e↵ects of changes in infor-mation. The remainder of the paper is devoted to our formal results, on attainablewelfare changes in Section 4, and on equilibrium changes in welfare in Section 5.Proofs are collected in the Appendix.
2 The Economy
There are two periods, 0 and 1, and a single physical consumption good. The econ-omy is populated by H � 2 agents, with typical agent h 2 H (here, and elsewhere,
4We elaborate on this point in footnote 12.5In a single-commodity environment, Elul (1999) shows that it is generically possible to intro-
duce an asset in order to make all agents better o↵ (while it can never be the case that all agentsare worse o↵). The argument relies on identifying an asset whose introduction leaves the prices ofall existing assets unchanged. Thus price e↵ects play no role in this result.
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we use the same symbol for a set and its cardinality). Uncertainty, which is resolvedat date 1, is described by S states of the world.
Agent h 2 H has endowments !h0 > 0 in period 0 and !h 2 RS
++ in period 1.He has time-separable expected utility preferences with von Neumann-Morgensternutility functions uh
0 : R++ ! R for period 0 consumption and uh : R++ ! R forperiod 1 consumption. We assume that uh
0 and uh satisfy the standard smoothpreference assumptions, i.e. they belong to the set U := {� : R++ ! R |� 2 C2,�0 >0,�00 < 0, limx!0 �0(x) = 1}.
Asset markets, in which J � 2 securities are traded, open at date 0. At date 1assets pay o↵. The payo↵ of asset j in state s (in terms of the physical consumptiongood) is denoted by rjs, and the vector of asset payo↵s in state s by rs 2 RJ . Bydefault all vectors are column vectors, unless transposed. Thus r>s = (r1s . . . r
Js ). Let
R be the S ⇥ J matrix whose s’th row is r>s . We assume, without loss of generality,that R has full column rank J . Markets are complete if J = S, and incomplete ifJ < S.
Prior to trading, agents observe a public signal correlated with the state of theworld s. This signal does not directly a↵ect utility functions, endowments or assetpayo↵s. We fix a finite set of “signal realizations” ⌃, #⌃ � 2, with a typical elementof ⌃ denoted by �. The uncertainty over fundamentals, parametrized by s 2 S,and the public signal, parametrized by � 2 ⌃, can then be jointly described by aprobability measure on S⇥⌃, i.e. by the probabilities ⇡ := {⇡s�}s2S,�2⌃ 2 RS⌃
+ , where⇡s� denotes Prob(s, �). Let ⇡s|� := Prob(s|�), ⇡s := Prob(s), and ⇡� := Prob(�).6
The uncertainty over fundamentals is thus given by the marginal distribution overS, {⇡s}s2S.
We refer to the vector ⇡ as an information structure.7 More precisely, an infor-mation structure lies in the set8
⇧ :=
(⇡ 2 RS⌃
+
��� ⇡s > 0, 8s 2 S; ⇡� > 0, 8� 2 ⌃;X
s,�
⇡s� = 1
).
This specification admits a range of possible information structures. It includesinformation structures that have full support over S for every �, i.e. S⇡(�) :={s 2 S | ⇡s� > 0} = S for all �. The set of these information structures is the interiorof ⇧, given by ⇧0 := {⇡ 2 RS⌃
++ |P
s� ⇡s� = 1}. A subset of ⇧0 consists of infor-mation structures satisfying the independence condition ⇡s� = ⇡s⇡�, for all s, �, inwhich case the public signal provides no information about s; we refer to such infor-mation structures as uninformative. The set ⇧ also includes information structures
6Note that ⇡s and ⇡� are scalars.7This is a slight abuse of terminology since ⇡ represents both the uncertainty over funda-
mentals and the information of agents. Describing information in terms of the joint distribution{⇡s�}s2S,�2⌃ is clearly equivalent to using the conditional distributions {⇡s|�}s2S and the marginaldistribution {⇡�}�2⌃, with ⇡s� = ⇡s|�⇡�. But working directly with the joint distribution, capturedby the single vector ⇡, is notationally more convenient.
8The restriction on the marginal distributions over S and ⌃ is without loss of generality.
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for which the support S⇡(�) is a strict subset of S for some signal realizations. Aspecial case of the latter is one where the signal induces a partition over S.
We take as given the date 0 endowment !h0 and preferences uh
0 of all agents h 2 H,and parametrize economies by agents’ date 1 endowments and preferences, and theinformation structure. Let ! := {!h}h2H 2 ⌦ := RSH
++, and u := {uh}h2H 2 UH .An economy is then a tuple (!, u, ⇡) 2 E := ⌦ ⇥ UH ⇥ ⇧, specifying the period1 endowments and utility functions of agents and the information structure. Weformalize our notion of genericity as follows. The sets ⌦ and ⇧ are endowed with theusual (Euclidean) topology. The set U is endowed with the C2 uniform convergencetopology on compact sets, i.e. the sequence uh
n in U converges to uh if and only ifuhn, u
hn0and uh
n00converge uniformly to uh, uh0 and uh00 respectively, on any compact
subset of R++. The set of economies E is endowed with the product topology. By“generic subset of ⌦,” we mean “for an open, dense subset of ⌦,” and likewise forUH , ⇧ and E . By “generically” or “for a generic economy” we mean “for an economyin an open, dense subset of E .”
Consider an economy (!, u, ⇡) 2 E . For signal realization �, let p� 2 RJ be thevector of asset prices (date 0 consumption serves as the numeraire), and yh� 2 RJ
the portfolio of agent h. The consumption of agent h for signal � is then given bych0� := !h
0 � p� · yh� at date 0, and chs� := !hs + rs · yh� in state s at date 1. Let
y� := {yh�}h2H , y := {y�}�2⌃, c0 := {ch0�}h2H,�2⌃, and p := {p�}�2⌃. A competitiveequilibrium is defined as follows:
Definition 1 Given an economy (!, u, ⇡) 2 E, a competitive equilibrium consists ofa portfolio allocation y, and prices p, satisfying the following two conditions:
(a) Agent optimization: 8h 2 H and � 2 ⌃, yh� solves
maxx2RJ
⇣uh0
⇥!h0 � p� · x
⇤+X
s
⇡s|� uh⇥!hs + rs · x
⇤⌘. (1)
(b) Market clearing: 8� 2 ⌃, X
h
yh� = 0. (2)
We will often refer to an equilibrium (y, p) of the economy (!, u, ⇡) as a ⇡-equilibriumin order to emphasize the dependence of the equilibrium on the information struc-ture. Since portfolios uniquely determine period 1 consumption, we can write a⇡-equilibrium allocation as (c0, y).
We wish to study the welfare consequences of changes in information. Given ⇡ 2⇧, we consider the set of alternative information structures ⇧(⇡) ⇢ ⇧ that representpurely a change in information relative to ⇡ without a↵ecting the uncertainty overfundamentals. Formally, ⇧(⇡) is the set of vectors ⇡ in ⇧ satisfying the followingconditions:
a1.
P�2⌃ ⇡s� = ⇡s, for all s 2 S; and
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a2. {⇡s�}s2S is not proportional to {⇡s�}s2S, for some � 2 ⌃.
Condition a1 says that ⇡ does not alter the marginal distribution over S implied by⇡, while condition a2 says that there is an actual change in information.9 If ⇡ 2 ⇧(⇡)we say that (⇡, ⇡) is an admissible pair of information structures.
We begin with an example to illustrate the welfare e↵ects of changes in informa-tion.
3 An Example
Consider an economy with four equally likely states and two assets. The state spaceis S = {s1, s2, s3, s4}, and the asset payo↵ matrix is
R =
0
BB@
1 + � 00 �1�1 00 1 + �
1
CCA
,
for some � > 0. There are two agents. For simplicity we assume that no consumptiontakes place at date 0 and agents have no endowment in that period. Both agentshave the same quadratic utility function, u(c) = c� a
2c2, for period 1 consumption,10
where the parameter a is positive and chosen to be small enough to ensure that themarginal utility of consumption is positive in every state in equilibrium. Agents’endowments are given by the matrix
0
BB@
! � ↵ + � ! + ↵� �! � ↵� � ! + ↵ + �! + ↵� � ! � ↵ + �! + ↵ + � ! � ↵� �
1
CCA
,
the (s, h)’th element of which is the endowment in state s of agent h. The parameters↵ and � are positive, and ! > ↵+�. Notice that there is no aggregate uncertainty inthis economy. We can think of endowments as being subject to two kinds of shocks,the ↵-shock and the �-shock, which a↵ect the agents in opposite directions.
The information of agents is described by a signal � that can take values �1 or �2with equal probability; thus ⌃ = {�1, �2}. We consider an initial situation in whichagents have no information; the information structure is ⇡, which is uninformativewith ⇡s|� = 1
4 , for all s, �. We analyze the welfare e↵ect of two alternative changesin information, to either ⇡ or ⇡0; in both cases the signal conveys information about
9As an example of a “change” in information that does not satisfy a2, suppose ⇡ is uninforma-tive, ⇡s|� = ⇡s|� = ⇡s for all s,�, and ⇡� 6= ⇡� for some �. Clearly ⇡ is also uninformative.
10While for our general results we assume that utility functions lie in the set U specified inSection 2, for the purpose of the example it is convenient to use quadratic utility which is clearlynot a member of this class.
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the state of the world s, inducing a partition over S represented by the event treesin Figures 1 and 2 respectively. Thus ⇡ reveals the ↵-shock, while ⇡0 reveals the�-shock. In both cases, the two states in each cell of the partition are equally likely.Conditional probabilities are shown on each branch.
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�1
�2
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w
w
w
Asset Payo↵s !1 !2
s1
s2
s3
s4
12
12
12
12
12
12
1 + �
0
�1
0
0
�1
0
1 + �
! � ↵ + �
! � ↵� �
! + ↵� �
! + ↵ + �
! + ↵� �
! + ↵ + �
! � ↵ + �
! � ↵� �
Figure 1: Agents are worse o↵ under ⇡ (Hirshleifer dominates Blackwell)
As a benchmark it is useful to consider the first-best symmetric allocation inwhich both agents consume their ex-ante (i.e. evaluated prior to the receipt of thepublic signal) expected endowment !, in all states.11 The ex-ante expected utilityof both agents is given by
U := ! � a
2!2
= UA +a
2(↵2 + �2),
11This is the unique competitive equilibrium allocation of this economy if markets are completeand the signal is uninformative.
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Asset Payo↵s !1 !2
s1
s4
s2
s3
12
12
12
12
12
12
1 + �
0
0
�1
0
1 + �
�1
0
! � ↵ + �
! + ↵ + �
! � ↵� �
! + ↵� �
! + ↵� �
! � ↵� �
! + ↵ + �
! � ↵ + �
Figure 2: Agents are better o↵ under ⇡0 (Blackwell dominates Hirshleifer)
where UA is the expected utility of both agents in autarky. Thus the first-best utilitygain from sharing the ↵-shock is a
2↵2, while the corresponding gain from sharing the
�-shock is a2�
2.Under the assumed incomplete asset structure, agents are not able to insure fully.
It is straightforward to check that there is a ⇡-equilibrium in which the two assetshave the same (nonzero) price. Agent 1 buys an amount 2+�
1+(1+�)2 ↵ of asset 1 andsells an equal amount of asset 2. Agent 2 trades the same amounts in the oppositedirection. The ex-ante expected utility of both agents is given by
U⇡ := U � a
2�↵2 � a
2�2,
where
� :=�2
2[1 + (1 + �)2].
By going long in one asset and short in the other, agents can transfer consumptionbetween the events {s1, s2} and {s3, s4}. Thus they are able to partially insure
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the ↵-shock. The proportion of the potential welfare gain from insuring this shock,a2↵
2, that is not realized in equilibrium is given by �. In other words, a2�↵
2 isthe magnitude of the unrealized welfare gain from trading the ↵-shock. Notice that� 2 (0, 12) and is strictly increasing in �. As � tends to zero, risk sharing relative tothe ↵-shock approaches the first-best, i.e. full insurance. On the other hand, agentsare not able to insure the �-shock at all, since that would require them to trade theassets in one direction in the event {s1, s2} and the opposite direction in the event{s3, s4}. Since the equilibrium relative price of the assets is 1, it is not optimal toinsure the �-shock in one of these events at the cost of exacerbating it in the other.
Given the information structure ⇡, markets are complete conditional on �.12
There is a unique equilibrium in which the consumption of each agent, conditionalon �, is riskless. The consumption of agent 1 is ! � ↵ and ! + ↵ in �1 and �2respectively, while for agent 2 it is the other way around. The ex-ante expectedutility of both agents is given by
U⇡ := U � a
2↵2.
Since ⇡ reveals the ↵-shock, this risk cannot be shared in equilibrium; hence theutility loss a
2↵2 relative to the first-best. On the other hand, agents can trade the
assets in opposite directions in the events {s1, s2} and {s3, s4}, which allows them tofully insure the �-shock. The equilibrium welfare gain from changing the informationstructure from ⇡ to ⇡ is
U⇡ � U⇡ = �a
2(1��)↵2 +
a
2�2.
The Hirshleifer e↵ect is captured by the term a2(1��)↵2, which is equal to the welfare
loss, relative to the ⇡-equilibrium, arising from agents’ inability under ⇡ to share the↵-shock.13 The second term a
2�2 is the magnitude of the Blackwell e↵ect, the welfare
gain from being able to trade the �-shock under ⇡, which was not possible under⇡. The Hirshleifer e↵ect is increasing in ↵ and decreasing in �, while the Blackwell
12In this example, the release of information leads to an increase in the asset span (conditionalon �), and is thus analogous to the introduction of new assets. This is not the case in general,however. For example, for information structures in ⇧0 (which is a generic subset of the set of allinformation structures ⇧), i.e. those for which the conditional distribution over S has full supportfor every realization of the signal, the asset span conditional on the signal is the same for everyrealization and equal to the asset span with no information.
13As mentioned in footnote 1, the Hirshleifer e↵ect can be interpreted in terms of changes inequilibrium prices induced by the change in information. In the present example, agents face twobudget constraints, one for each value of �. Under ⇡, these constraints are identical, and hencereduce to a single budget constraint which allows agents to transfer wealth between the events{s1, s2} and {s3, s4}. The change in information from ⇡ to ⇡ causes equilibrium prices to diverge inthe two events, resulting in two distinct budget constraints. Agents cannot transfer wealth betweenthese events without violating one of the budget constraints. In Hirshleifer’s original two-stateexample, each of the two events consists of a single state; thus in each event the budget set facedby an agent is trivial, containing only the agent’s endowment in that event.
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e↵ect is increasing in �. For su�ciently large ↵ and/or su�ciently small � and �,the Hirshleifer e↵ect dominates the Blackwell e↵ect, and both agents are worse o↵.
Next consider the information structure ⇡0. As with ⇡, markets are completeconditional on �. Again there is a unique equilibrium, with riskless consumption forboth agents conditional on �. Ex-ante expected utility is given by
U⇡0 := U � a
2�2.
Since ⇡0 reveals the �-shock, this risk cannot be traded in equilibrium; hence theutility loss a
2�2 compared to the first-best. On the other hand, the ↵-shock can be
smoothed out completely. The equilibrium welfare gain from changing the informa-tion structure from ⇡ to ⇡0 is
U⇡0 � U⇡ =a
2�↵2.
It consists of only one term, which is equal to the unrealized welfare gain from tradingthe ↵-shock under ⇡. This is the Blackwell e↵ect. The Hirshleifer e↵ect is zero inthis case. While agents are unable to insure the �-shock under ⇡0, due to this shockbeing revealed prior to trading, it was not insurable even under the uninformativeinformation structure ⇡, because of the structure of asset payo↵s. Clearly both agentsare better o↵ under ⇡0 for all values of ↵, � and �.
Let us assume that the parameters ↵, � and � are such that the Hirshleifere↵ect dominates the Blackwell e↵ect under ⇡. Comparing competitive equilibriumallocations before and after the release of information, we see that agents are ex-anteworse o↵ under ⇡, and ex-ante better o↵ under ⇡0, relative to the ⇡-equilibrium. Wenow consider what a hypothetical planner is able to achieve, when he is restricted bythe same feasibility condition as competitive markets, i.e.
Ph y
h� = 0 for all �, and
has access to the same information. We show that, for both ⇡ and ⇡0, the plannercan attain an allocation that is Pareto improving ex-post (i.e. conditional on bothvalues of �), and hence also ex-ante.14
Under ⇡, the planner can improve ex-post upon the ⇡-equilibrium by assigningto each agent the sum of his ⇡-equilibrium and ⇡-equilibrium portfolios, for bothvalues of �. He is thus able to exploit the Blackwell e↵ect (in this case, insuring the�-shock) just as competitive markets can under ⇡, while also insuring the ↵-shockto the extent that competitive markets can under ⇡. Under ⇡0, the competitiveequilibrium is itself an ex-post (not just ex-ante) Pareto improvement relative to the⇡-equilibrium. Clearly the ⇡0-equilibrium allocation is feasible for the planner as well(though he cannot improve upon it since it is ex-post e�cient).
14The first-best allocation in which both agents have a riskless consumption of ! is a feasibleallocation under both ⇡ and ⇡0, and is an ex-ante Pareto improvement relative to the ⇡-equilibrium.However, it is not in general an ex-post Pareto improvement. The ex-post expected utility of bothagents at this allocation is U for both values of �. One can verify that, under ⇡, the expectedutility of agent 1 conditional on �2 is higher than U , for su�ciently small �. Similarly, under ⇡0,the expected utility of agent 1 conditional on �1 is higher than U , for � su�ciently small relativeto �.
12
To summarize, when markets are incomplete, the release of information expandsthe set of achievable utility levels via the Blackwell e↵ect. The planner can takeadvantage of this and so can competitive markets. At the same time, the release ofinformation can impair risk sharing due to the Hirshleifer e↵ect, by precluding thesharing of risks that have already been resolved. The planner is not subject to thise↵ect since the initial equilibrium allocation remains feasible even after the arrivalof information. The example suggests that while equilibrium welfare e↵ects can goin either direction, depending on the nature of the information being released, theplanner can achieve an ex-post (hence also ex-ante) Pareto improvement for anyincrease in information.
In the remainder of the paper we extend the analysis to a general economy whereinwe consider the e↵ects of arbitrary changes in information. In the case of non-ordered information structures, welfare changes cannot be interpreted in terms of theBlackwell and Hirshleifer e↵ects. Nevertheless, there is a sense in which the findingsof the example remain valid. In Theorem 1 we establish that, for any change ininformation, an ex-post welfare improvement can generically be attained by tailoringagents’ portfolios to the new information structure. An additional welfare e↵ect arisesin equilibrium because of the adjustment in asset prices induced by the change ininformation; we show in Theorem 2 that this may have a positive or negative e↵ecton welfare. More precisely we show that, for a generic economy, there is a change ininformation that makes all agents (ex-ante) better o↵ as well as a change that makesall agents worse o↵. Identifying the set of information changes that lead to a welfareimprovement (or worsening) is not possible, however, at the level of generality of thisresult; clearly such a characterization can be obtained only under strong restrictionson the environment as in the example. We combine Theorems 1 and 2 in Corollary 1to argue that the conclusions drawn from the example, summarized at the end of theprevious paragraph, have generic validity, i.e. competitive markets typically do notrespond to changes in information in a way that is welfare-improving even though itis feasible to do so.
4 Attainable Changes in Welfare
In this section we consider allocations attainable by a hypothetical planner whois subject to the same asset market and informational constraints as agents. Wesay that an allocation (c0, y) :=
�ch0�, y
h�
h2H,�2⌃ is feasible if
Ph c
h0� =
Ph !
h0 andP
h yh� = 0, for all � 2 ⌃.
Note that a feasible allocation is defined as an allocation contingent on �, andis therefore trivially measurable with respect to the information available to agents.Moreover, since all signals have the same set of possible realizations ⌃, allocationsthat are measurable with respect to ⇡ are the same as those that are measurablewith respect to ⇡. Thus a change in information does not a↵ect the set of feasibleallocations. However, it does a↵ect the set of attainable utility levels: the value to
13
an agent of making his consumption conditional on � depends on the informationconveyed by it. For example, if ⇡ is uninformative and ⇡ induces a partition over S,under ⇡ nothing is gained by (nontrivially) conditioning on �, but under ⇡ this istypically not true for values of � associated with distinct cells of the partition. Evenif ⇡ does not induce a partition, it is in general beneficial to condition on � as longas the conditional distribution over S varies with �.
Given a ⇡-equilibrium, if no welfare improvement is feasible under the new infor-mation structure ⇡, i.e. when agents evaluate consumption using the beliefs deter-mined by ⇡, we say that the ⇡-equilibrium is ⇡-e�cient. As in the example in theprevious section, this gives rise to two welfare notions according to whether agentsevaluate allocations ex-ante or ex-post, i.e. before or after the observation of thesignal. Formally:
Definition 2 A ⇡-equilibrium (y, p) is ex-ante ⇡-e�cient if there does not exist afeasible allocation (c0, y) that ex-ante Pareto dominates the ⇡-equilibrium allocation,given the information structure ⇡, i.e.
X
s,�
⇡s�
⇣uh0
⇥ch0�⇤+ uh
⇥!hs + rs · yh�
⇤� uh
0
⇥!h0 � p� · yh�
⇤� uh
⇥!hs + rs · yh�
⇤⌘� 0,
for all h 2 H, where at least one of these inequalities is strict.
Definition 3 A ⇡-equilibrium (y, p) is ex-post ⇡-e�cient if there does not exist afeasible allocation (c0, y) that ex-post Pareto dominates the ⇡-equilibrium allocation,given the information structure ⇡, i.e.
X
s
⇡s�
⇣uh0
⇥ch0�⇤+ uh
⇥!hs + rs · yh�
⇤� uh
0
⇥!h0 � p� · yh�
⇤� uh
⇥!hs + rs · yh�
⇤⌘� 0,
for all h 2 H and � 2 ⌃, where at least one of these inequalities is strict.
Notice that both the allocations (c0, y) and (c0, y) are evaluated at the same odds,given by ⇡.15
It is well-known (Diamond (1967)) that competitive equilibria of a two-period,single-good economy with incomplete markets are (ex-post) constrained Pareto e�-cient. Thus, in our terminology, ⇡-equilibria are ex-post ⇡-e�cient. This conditioncan be characterized in terms of the equality of agents’ marginal rates of substitutionbetween assets and period 0 consumption, for every �, i.e.
Ps ⇡s� u
h0⇥!hs + rs · yh�
⇤rjs
uh00⇥!h0 � p� · yh�
⇤ =
Ps ⇡s� u
h0⇥!hs + rs · yh�
⇤rjs
uh0
0⇥!h0 � p� · yh�
⇤ , 8h, h 2 H; j 2 J ; � 2 ⌃.
(3)
15It would not be sensible to evaluate (c0, y) at ⇡, and (c0, y) at ⇡. Doing so can lead to thepossibility of a “Pareto improvement” with no change in the allocation (unless the allocation underconsideration is �-invariant for any given s).
14
This means that, if there is no change in information, an ex-post welfare-improvingreallocation of portfolios cannot be found. In this sense, competitive markets makee�cient use of the available information. In contrast, we show that if markets areincomplete ⇡-equilibria are generically ex-post ⇡-ine�cient, for every ⇡ 2 ⇧(⇡). Inother words, while a competitive equilibrium makes e�cient use of the availableinformation, a Pareto improvement can typically be achieved for any change in in-formation.
We begin by verifying that incompleteness is necessary for this result. Whenmarkets are complete (and the initial information structure has full support), at acompetitive equilibrium there are no further gains from trade, whatever the changein agents’ information is. Hence no portfolio reallocation can be found that allowsan improvement in agents’ welfare.
Lemma 1 Suppose markets are complete and (⇡, ⇡) is an admissible pair of infor-mation structures with ⇡ 2 ⇧0. Then every ⇡-equilibrium is ex-post ⇡-e�cient.
Proof :
If markets are complete and ⇡ 2 ⇧0, condition (3) for the ⇡-e�ciency of a ⇡-equilibrium (y, p) is equivalent to the equality of agents’ marginal rates of intertem-poral substitution for every �:
uh0⇥!hs + rs · yh�
⇤
uh00⇥!h0 � p� · yh�
⇤ =uh
0⇥!hs + rs · yh�
⇤
uh0
0⇥!h0 � p� · yh�
⇤ , 8h, h 2 H; s 2 S⇡(�); � 2 ⌃. (4)
Since S⇡(�) ⇢ S⇡(�) = S for all �, condition (4), and hence also (3), must in fact holdwhen ⇡ is replaced by ⇡. Therefore, the ⇡-equilibrium (y, p) is ex-post ⇡-e�cient,and this is true for all ⇡ 2 ⇧(⇡). 2
When markets are incomplete, condition (3) still holds at a ⇡-equilibrium, butdoes not imply (4). In fact for a generic economy competitive equilibria are Paretoine�cient, so that (4) does not hold (this is a standard result; see, for example, Magilland Quinzii (1996)). We now show that, in addition, a ⇡-equilibrium is genericallyex-post ⇡-ine�cient, i.e. condition (3) is violated when ⇡ is replaced by ⇡, indeed forany ⇡ 2 ⇧(⇡). Intuitively, when we change ⇡ to ⇡, (3) continues to hold only underrestrictive conditions on agents’ marginal utilities for state-contingent consumption.These conditions are not satisfied for a generic economy. We prove the result underthe assumption that the asset payo↵ matrix R is in general position, i.e. every J ⇥Jsubmatrix of R is nonsingular.16
Theorem 1 Suppose markets are incomplete and R is in general position. Supposefurther that (⇡, ⇡) is an admissible pair of information structures such that at leastone element of the pair (⇡, ⇡) lies in ⇧0. Then there is a generic subset of ⌦ suchthat every ⇡-equilibrium is ex-post ⇡-ine�cient.
16General position is a generic property of the set of S ⇥ J matrices, for S � J .
15
Theorem 1 allows for arbitrary changes in information,17 including those thatentail an increase or decrease in the support of the information structure.18 Inparticular, it implies that generically an increase in information generates additionalgains from trade, so that agents would trade again if asset markets were to reopenafter the increase in information. This idea is developed more fully in Gottardi andRahi (2012), which provides several extensions of Theorem 1 in the specific contextof retrading after the arrival of information. However, Theorem 1 su�ces for ourpresent purpose, which is to contrast welfare changes in equilibrium to those thatcan be achieved by a planner.
5 Equilibrium Changes in Welfare
We have shown that, for any ⇡-equilibrium, and for any change in information from⇡ to ⇡, generically there is a feasible allocation that is ex-post Pareto improving.While this is no longer true if we restrict attention to feasible allocations that arealso ⇡-equilibrium allocations, we show in this section that generically there existsan information structure ⇡ and a corresponding ⇡-equilibrium such that all agentsare better o↵ ex-ante. But there also exists a ⇡0 and a corresponding ⇡0-equilibriumsuch that all agents are worse o↵. Indeed, the e↵ect of a change in information onequilibrium welfare can be in any direction. Unlike the welfare changes consideredin the previous section, the change in equilibrium welfare is attributable not only tothe e↵ect of the change in information on the available gains from trade, but also tothe e↵ect of this change on equilibrium asset prices. The proof of this result exploitsthe general framework laid out in Citanna et al. (1998), and exposited more fully inVillanacci et al. (2002), Chapter 15, to analyze the e↵ect on equilibrium welfare ofperturbing a parameter of the economy.
Given a ⇡-equilibrium (y, p), and a ⇡-equilibrium (y, p), let W (y, p, ⇡; y, p) bethe vector of di↵erences in the ex-ante expected utilities of agents between the twoequilibria. This vector is given by(X
s,�
⇡s�
⇣uh0
⇥!h0 � p� · yh�
⇤+ uh
⇥!hs + rs · yh�
⇤� uh
0
⇥!h0 � p� · yh�
⇤� uh
⇥!hs + rs · yh�
⇤⌘)
h2H
We consider local changes in (y, p, ⇡) in a neighborhood of (y, p, ⇡). An ex-ante Paretoimprovement is attained in equilibrium if W (y, p, ⇡; y, p) is nonnegative and nonzero.We show in fact that for a generic economy, for any vector in RH , there exists a ⇡and a corresponding equilibrium (y, p) such that the change in equilibrium welfaregiven by W (y, p, ⇡; y, p) is equal to the specified vector. In other words, it is possible
17The assumption that either ⇡ or ⇡ has full support is just a matter of convenience. The resultholds even if neither has full support, provided a2 and the market incompleteness condition applyfor the relevant subset of S.
18Notice that an ex-post Pareto improvement can typically be attained even if ⇡ resolves all theuncertainty in the economy, since there is still room for intertemporal reallocation of consumption.
16
to choose ⇡ in order to generate a local change in welfare in any direction. Our resultrequires that markets be su�ciently incomplete, and that the heterogeneity of agentsnot be too large.
Theorem 2 Suppose S � 2JH, J � H � 1, and R is in general position. Then,for a generic subset of E , for any ⇡-equilibrium (y, p), there exists a local change ininformation d⇡ satisfying (⇡ + d⇡) 2 ⇧(⇡), and a corresponding local change in theequilibrium (dy, dp), such that W (y + dy, p+ dp, ⇡ + d⇡; y, p) is any desired nonzerovector in RH .
We can now summarize the sense in which Theorems 1 and 2 generalize theexample in Section 3.
Corollary 1 Suppose S � 2JH, J � H�1, and R is in general position. Then, forany (u, ⇡) in a generic subset of UH ⇥⇧, there is a generic subset of ⌦ such that forany ⇡-equilibrium, there exist ⇡ and ⇡0 in ⇧(⇡) for which the following statementshold:
i. There is a ⇡-equilibrium at which all agents are worse o↵ ex-ante.
ii. There is a ⇡0-equilibrium at which all agents are better o↵ ex-ante.
iii. The ⇡-equilibrium is ex-post ⇡-ine�cient as well as ex-post ⇡0-ine�cient.
The proof, which is in the Appendix, is not an immediate consequence of Theorems1 and 2. For a given economy (!, u, ⇡) in the generic subset for which Theorem 2applies, statements (i) and (ii) of the corollary hold. However, the generic subset ofendowments for which statement (iii) holds, given by Theorem 1, may not contain!. An additional argument is required to establish the result.
Corollary 1 tells us, in particular, that while there is (almost) always a change ininformation that makes all agents worse o↵ in equilibrium, for the same change ininformation there is a feasible allocation at which all agents are better o↵ (even ex-post). Thus, while competitive markets make e�cient use of the available information(in the sense of being constrained e�cient for a given information structure), theytypically do not deal with changes in information in a welfare-enhancing mannereven though it is feasible to do so.
17
A Appendix
In our analysis we use the following shorthand notation for matrices. Given an indexset N with typical element n, and a collection {zn}n2N of vectors or matrices, wedenote by diagn2N [zn] the (block) diagonal matrix with typical entry zn, where nvaries across all elements of N . For a given vector or matrix z, diagn2N [z] is thenthe diagonal matrix with the term z repeated #N times. In similar fashion, we write[. . . zn . . .n2N ] to denote the row block with typical element zn, and analogously forcolumn blocks. We drop reference to the index set if it is obvious from the context:for example diagh2H is shortened to diagh, and [. . . zs . . .s2S,s 6=s1 ] to [. . . zs . . .s 6=s1 ]. Weuse the same symbol 0 for the zero scalar and the zero matrix; in the latter case weoccasionally indicate the dimension in order to clarify the argument. We denote byIN the N ⇥N identity matrix. A “⇤” stands for any term whose value is immaterialto the analysis. We will sometimes need to order the set S (and similarly the sets ⌃and H) as {s1, s2, . . .}, s1 being the first state, and so on.
We begin by providing a characterization of competitive equilibria. The first-order conditions for the utility-maximization program (1) are:
X
s
⇡s�⇣uh0⇥!h
s + rs · yh�⇤rs � uh
00⇥!h0 � p� · yh�
⇤p�⌘= 0, 8h 2 H; � 2 ⌃. (5)
The tuple (y, p) is a competitive equilibrium if and only if it satisfies equations (2)and (5). Let ⇡� := {⇡s�}s2S, and define
f�(y�, p�,!, u, ⇡�) :=
X
s
⇡s�⇣uh0⇥!h
s + rs · yh�⇤rs � uh
00⇥!h0 � p� · yh�
⇤p�⌘, 8h 2 H,
g�(y�) :=X
h
yh�.
Then the equations that characterize a competitive equilibrium, (2) and (5), can bewritten as
F�(y�, p�,!, u, ⇡�) :=
✓f�(y�, p�,!, u, ⇡�)
g�(y�)
◆= 0, 8� 2 ⌃, (6)
or more compactly as
F (y, p,!, u, ⇡) :=
✓f(y, p,!, u, ⇡)
g(y)
◆= 0, (7)
where f := {f�}�2⌃, and g := {g�}�2⌃. Here the endogenous variables are (y, p),while (!, u, ⇡) are parameters.19
19For ease of notation, we will often drop parameters that remain fixed and play no role inthe argument at hand. For example, in the proof of Theorem 2, we write F (y, p, ⇡) instead ofF (y, p,!, u, ⇡).
18
Suppose ⇡ 2 ⇧0. Consider the equilibrium system (6) for a given value of �.These are JH + J equations, equal to the number of unknowns (y, p). The Jacobianof this equation system can be written as follows:
Dy� ,p� ,!F� =
✓Dy� ,p�f� D!f�Dy� ,p�g� 0
◆
,
withD!f� = diagh
h. . . ⇡s� u
h00⇥!hs + rs · yh�
⇤rs . . .s
i
andDy� ,p�g� = (. . . IJ . . .h 0).
The matrix D!f� has full row rank since R has full column rank and the term⇡s�uh00[!h
s + rs · yh�] is nonzero, for all h, s,�. Clearly, Dy� ,p�g� has full row rank aswell (indeed, Dy�g� has full row rank). By a standard argument, for a generic subsetof endowments ⌦R�, the number of equilibria conditional on � (zeros of F�) is finite,and positive. Hence, for endowments in the generic subset ⌦R := \�2⌃⌦R�, thenumber of equilibria (zeros of F ) is finite as well.
Since we exploit the transversality theorem repeatedly in the proofs below, it isuseful to summarize the argument at the outset. Consider a function : Rn ⇥ E ! Rm,m > n. For e 2 E , let e be the function (·, e). The argument involves identifyingsuch a function , such that the desired result can be formulated as �1
e (0) = ?, forevery e in a generic subset of E . We show that the Jacobian Dx,e has full row rankat all zeros (x, e) of , i.e. is transverse to zero. By the transversality theorem,there is then a dense subset E of E such that, for each e 2 E , e : Rn ! Rm istransverse to zero. It follows that �1
e (0) = ?. In other words, the equation system e(x) = 0 has no solution since the number of (locally) independent equations mexceeds the number of unknowns n. A standard argument (see, for example, Citannaet al. (1998)) establishes that the set E is open, and hence a generic subset of E .
In order to economize on notation, we use the shorthand uh0�0:= uh
00⇥!h0 � p� · yh�
⇤,
uhs�0:= uh0⇥!h
s + rs · yh�⇤, and similarly for the second derivatives, uh
0�00and uh
s�00.
Proof of Theorem 1:
A ⇡-equilibrium (y, p) is ex-post ⇡-e�cient if and only if the marginal rates of sub-stitution between assets and period 0 consumption, evaluated at ⇡, are equal acrossagents, for every �, i.e. condition (3) holds with ⇡ replaced by ⇡. Letting
µhh� (y�, p�) :=
uh0�0
uh0�
0 ,
we can rewrite this condition as follows:X
s
⇡s�
⇣uhs�
0 � µhh� uh
s�
0⌘rjs = 0, 8h, h 2 H; j 2 J ; � 2 ⌃. (8)
19
Fix a � for which {⇡s�}s2S is not proportional to {⇡s�}s2S. We will show that aPareto improvement can be achieved conditional on every � with this property.
We assume that ⇡ 2 ⇧0 in the following argument, outlining in the last paragraphof the proof how it extends to the case where ⇡ does not have full support. Let
R⇤ :=⇣diags[⇡s�]R diags[⇡s�]R
⌘
.
We claim that rank(R⇤) � J +1. If S⇡(�) = S, this is immediate from the followingresult in Geanakoplos and Mas-Colell (1989) (Step 4, page 31):
Fact Suppose markets are incomplete, and R is in general position. Considernonzero scalars ✓s, ✓0s, s 2 S, such that {✓s}s2S is not proportional to {✓0s}s2S. Then,diags[✓s]R and diags[✓
0s]R do not have the same column span.
On the other hand, if S⇡(�) ( S, R⇤ is row-equivalent to
R⇤ :=
diags2S⇡(�)[⇡s�]R1
�� diags2S⇡(�)[⇡s�]R1———————————————————
0�� diags/2S⇡(�)[⇡s�]R2
!
,
where R1 consists of the rows of R corresponding to the states in S⇡(�), while R2
consists of the remaining rows. If we delete the rows of R⇤ corresponding to theredundant rows of its upper left block, and also delete the rows corresponding to theredundant rows of its lower right block, we are left with a block-triangular matrixwhose diagonal blocks have full row rank, and hence whose rank is equal to thesum of the ranks of the diagonal blocks. It follows that for the full matrix R⇤,rank(R⇤) � rank(R1) + rank(R2), which is at least J + 1 due to the general positionof R. Consequently, rank(R⇤) = rank(R⇤) � J + 1.
Let rj 2 RS be the payo↵ of asset j. Since rank(R⇤) � J + 1, we can pick j suchthat diags[⇡s�] rj lies outside the column span of diags[⇡s�]R, and hence the matrix
✓. . . ⇡s� rjs . . .s. . . ⇡s� rs . . .s
◆(9)
has full row rank J + 1.We will show that, for a generic subset of ⌦, there is no solution to the equation
system
1(y�, p�,!; ⇡�, ⇡�) :=
F�(y�, p�,!; ⇡�)P
s ⇡s�
⇣uh1s�
0 � µh1h2� uh2
s�0⌘rjs
!= 0,
i.e. the ex-post ⇡-e�ciency condition (8) is violated. The Jacobian of 1 is
Dy� ,p� ,! 1 =
0
BB@
⇤�� diagh
⇥. . . ⇡s� uh
s�00rs . . .s
⇤—————————————————Dy� ,p�g�
�� 0—————————————————
⇤�� ⇥. . . ⇡s� uh1
s�00rjs . . .s
⇤⇤
1
CCA
,
20
which is row-equivalent to
0
BBBB@
⇤�� . . . ⇡s� uh1
s�00rjs . . .s
�� ⇤———————————————————————————
⇤�� . . . ⇡s� uh1
s�00rs . . .s
�� 0———————————————————————————
⇤�� 0
�� diagh 6=h1
⇥. . . ⇡s� uh
s�00rs . . .s
⇤———————————————————————————Dy� ,p�g�
�� 0�� 0
1
CCCCA
.
This matrix is in turn column-equivalent to
0
BBB@
⇤��� ⇤
���✓
. . . ⇡s� rjs . . .s
. . . ⇡s� rs . . .s
◆
————————————————————————————⇤
�� diagh 6=h1
⇥. . . ⇡s� uh
s�00rs . . .s
⇤ �� 0————————————————————————————Dy� ,p�g�
�� 0�� 0
1
CCCA
.
(10)
This is a block-triangular matrix. The top right block is the same as (9) which hasfull row rank. The other two diagonal blocks of (10) have full row rank as well, bythe properties of ⇡-equilibria established at the beginning of the Appendix. Hencethe matrix (10) has full row rank.
We have shown that the Jacobian Dy� ,p� ,! 1 has full row rank, at every zero of 1. Thus 1 is transverse to zero, and �1
1! (0) = ?, for every ! in a generic subsetof ⌦.
In this proof we assumed that ⇡ 2 ⇧0. If not, then ⇡ 2 ⇧0, due to the as-sumption in the theorem that at least one of these information structures has fullsupport. Then rank(R⇤) � J+1 by the same argument as above, swapping the rolesof ⇡ and ⇡. Dropping any assets that are redundant with respect to the states inS⇡(�), let y� and p� be the portfolios and prices of the remaining assets. The aboveargument shows that, for a generic subset of ⌦, there is no solution to the equationsystem 1(y�, p�,!; ⇡�, ⇡
�) = 0. In particular, Dy� ,p� ,!F� has full row rank, andconsequently so does Dy� ,p� ,! 1. This establishes the result. 2
Proof of Theorem 2:
We fix an economy e = (!, u, ⇡) 2 ⌦⇥UH ⇥⇧0, and consider a ⇡-equilibrium (y, p).Thus F (y, p, ⇡) = 0. We consider (y, p, ⇡) in a neighborhood of (y, p, ⇡) such that(y, p) is a ⇡-equilibrium, i.e. F (y, p, ⇡) = 0.
Let
�(y, p, ⇡; y, p, ⇡) :=
0
@W (y, p, ⇡; y, p)
F (y, p, ⇡)�P� ⇡s� � ⇡s
s2S
1
A
.
Notice that �(y, p, ⇡; y, p, ⇡) = 0. As we vary (y, p, ⇡) in a neighborhood of (y, p, ⇡)we move locally along the equilibrium manifold. The theorem is established by show-ing that the Jacobian of � with respect to (y, p, ⇡) has full row rank at (y, p, ⇡) =
21
(y, p, ⇡), i.e. Dy,p,⇡�(y, p, ⇡; y, p, ⇡) has full row rank. This full rank property im-plies that there is a local change (dy, dp, d⇡) such that d�(y, p, ⇡; y, p, ⇡) is any de-sired vector. In particular, (dy, dp, d⇡) can be chosen so that dW (y, p, ⇡; y, p) :=W (y + dy, p+ dp, ⇡ + d⇡; y, p) is any desired nonzero vector in RH , dF (y, p, ⇡) :=F (y + dy, p+ dp, ⇡ + d⇡) = 0, and
P� d⇡s� = 0 for all s 2 S. The condition dF = 0
ensures that (y + dy, p + dp) is a (⇡ + d⇡)-equilibrium. The conditionP
� d⇡s� = 0implies that (⇡ + d⇡) satisfies condition a1 in the definition of the set ⇧(⇡). Con-dition a2 in this definition must also be satisfied, for otherwise there is no changein the equilibrium allocation and consequently no change in welfare. Therefore,(⇡ + d⇡) 2 ⇧(⇡).
The matrix Dy,p,⇡�(y, p, ⇡; y, p, ⇡) is given by
0
BBBBBBBBBBBBBBBB@
0
�����
...
. . .� ⇡�uh0�0yh�
>. . .�
...h
����� 0
———————————————————————————————————–
⇤
����� ⇤
����� diag�
0
BB@
...
. . .⇣uhs�0rs � uh
0�0p�⌘. . .s
...h
1
CCA
———————————————————————————————————–diag�[. . . IJ . . .h]
�� 0�� 0
———————————————————————————————————–0
�� 0�� . . . IS . . .�
1
CCCCCCCCCCCCCCCCA
.
The top left block is zero because of the first-order condition (5). This matrix isrow/column-equivalent to
0
BBBBBBB@
...
. . . uh0�0yh�
>. . .�
...h
����� 0
����� 0
—————————————————————0
�� diag�[. . . IJ . . .h]�� 0
—————————————————————⇤
�� ⇤�� Q(y)
1
CCCCCCCA
,
(11)
where
Q(y) :=
0
BBBBB@
diag�
0
BB@
...
. . .⇣uhs�0rs � uh
0�0p�⌘
. . .s...h
1
CCA
—————————————————. . . IS . . .�
1
CCCCCA
.
(12)
Later in the Appendix we show that, for every competitive equilibrium of a genericeconomy, under the dimensionality restrictions in the statement of the theorem, the
22
portfolios and marginal utilities of agents satisfy the following conditions:20
C1. The vectors {yh�}h2H,h 6=h are linearly independent, for all h, h 2 H and � 2 ⌃.
C2. The vectors (. . . uh10�
0. . .�) and (. . . uh2
0�0. . .�) are not collinear.21
C3. Q(y) has full row rank.
Each of these conditions holds for a di↵erent generic subset of E (see Lemmas A.1, A.3and A.5). In the remainder of the proof, we assume that the economy e consideredin the foregoing analysis is in the intersection of these three generic subsets. Thuse is a generic economy for which conditions C1, C2 and C3 hold. We will provethat, for such an economy, (11) has full row rank, and hence so does the JacobianDy,p,⇡�(y, p, ⇡; y, p, ⇡).22
Since (11) is a block-triangular matrix, it su�ces to show full row rank of thethree diagonal blocks. The middle block clearly has full row rank. The bottom rightblock Q(y) has full row rank due to C3.
It remains to show that the top left block of (11) has full row rank. The lastH � 1 rows of this block are linearly independent by condition C1. We claim thatthe first row is not in the span of the last H � 1 rows. Suppose not. Then the firstrow [. . . uh1
0�0yh1�>. . .�] can be written as a unique linear combination of the remaining
rows {[. . . uh0�0yh�
>. . .�]}h 6=h1 , i.e. there exist unique coe�cients {�h}h 6=h1 such that
yh1� =
X
h 6=h1
�huh0�0
uh10�
0 yh�, 8� 2 ⌃.
By the market-clearing condition, this linear combination must in fact be yh1� =
�P
h 6=h1yh�, for all �. It follows that the ratio uh1
0�0/uh2
0�0is �-invariant, violating con-
dition C2. 2
In the remainder of the Appendix we establish conditions C1, C2 and C3, thatwere used in the proof of Theorem 2, for a generic subset of E , and provide a proofof Corollary 1.
Lemma A.1 Suppose J � H � 1 and ⇡ 2 ⇧0. Then, for a generic subset ⌦Y of ⌦,condition C1 is satisfied at any ⇡-equilibrium.
20The upper bound on the number of agents relative to the number of assets, J � H � 1, isclearly necessary for condition C1, which in turn is used in the proofs of conditions C2 and C3. Inaddition, C2 and C3 rely on a property that agents’ marginal utilities for assets vary su�cientlywith respect to ⇡, across both s and � (see Lemma A.2); a lower bound on the number of states,S � J(H + 1) + 1, is needed to establish this property. Finally, the proof of condition C3 requiresa tighter lower bound on S, given by S � 2JH.
21The choice of agents h1 and h2 is just a matter of convenience.22Notice that, in order to establish the full rank property of D�, we do not apply a transversality
argument to the map �, i.e. we do not perturb parameters of �. Instead, we use transversality toshow the generic validity of conditions C1, C2 and C3.
23
Proof :
Without loss of generality we can take h = h1. Consider the equation system
2(y�, p�, 2,!) :=
0
@F�(y�, p�,!)
[ . . . yh� . . .h 6=h1 ] 2
>2 2 � 1
1
A = 0,
where 2 2 RH�1, with typical element 2h, h 2 H, h 6= h1. Thus we have appendedJ + 1 equations and H � 1 unknowns to the equilibrium system F� = 0. Under thedimensionality condition J � H� 1, the number of equations exceeds the number ofunknowns, for given !. We will show that, for a generic subset of endowments, thissystem has no solution, and hence the vectors {yh�}h 6=h1 are linearly independent atany equilibrium.
The Jacobian of 2 is
Dy� ,p� , 2,! 2 =
0
BBBB@
Dy�f��� Dp�f�
�� 0�� D!f�
——————————————————————–. . . IJ . . .h
�� 0�� 0
�� 0——————————————————————–0 [. . . 2hIJ . . .h 6=h1 ]
�� 0�� ⇤
�� 0——————————————————————–
0�� 0
�� 2 >2
�� 0
1
CCCCA
,
which is column-equivalent to
0
BBBB@
0�� ⇤
�� D!f�————————————————————————–
⇤
�����
IJ
�� . . . IJ . . .h 6=h1———————————0�� . . . 2hIJ . . .h 6=h1
!0
����� 0
————————————————————————–2 >
2
�� 0�� 0
1
CCCCA
.
This is a block-triangular matrix. At any zero of 2, 2 6= 0 (since >2 2 = 1), and
hence the bottom left and middle blocks have full row rank. Furthermore, D!f� alsohas full row rank. It follows that the JacobianDy� ,p� , 2,! 2 has full row rank, at everyzero of 2. Thus 2 is transverse to zero, and �1
2! (0) = ? for all ! 2 ⌦Y �, a genericsubset of ⌦. The set ⌦Y in the statement of the lemma is given by ⌦Y := \�2⌃⌦Y �.2
In order to establish conditionsC2 andC3, we will need to independently perturbthe marginal utilities of agents for di↵erent signal realizations �. This cannot beachieved via endowment perturbations, since endowments do not vary with respect to�. Instead, we perturb the information structure ⇡. More precisely, at any ⇡ 2 ⇧0, weconsider derivatives with respect to the probabilities over S for two signal realizations�1 and �2, i.e. with respect to ⇡1 := {⇡s�1}s2S and ⇡2 := {⇡s�2}s2S, along directionsthat leave ⇡�1 and ⇡�2 unchanged. In particular this ensures that the perturbedprobabilities remain in ⇧0.
24
We denote by ⇡�i the value of ⇡�i that remains fixed in such a perturbation (withderivatives being evaluated at ⇡, this value is equal to
Ps ⇡s�i). As a first step we
show in Lemma A.2 below that, for a generic subset of endowments, the Jacobian of
Z(y�1 , p�1 , y�2 , p�2 , ⇡1, ⇡2,!) :=
0
BB@
f�1(y�1 , p�1 , ⇡1,!)P
s ⇡s�1 � ⇡�1f�2(y�2 , p�2 , ⇡
2,!)Ps ⇡s�2 � ⇡�2
1
CCA
with respect to ⇡1 and ⇡2 has full row rank at any equilibrium. Lemma A.2 is invokedin all the lemmas that follow.23
Lemma A.2 Suppose S � J(H + 1) + 1, and R is in general position. Let ⇡ 2 ⇧0
be an information structure withP
s ⇡s�i = ⇡�i, i = 1, 2. Then, for a generic subset⌦Z of ⌦, at any ⇡-equilibrium, D⇡1,⇡2Z has full row rank.
Proof :
For � = �1, �2, we have
f� =X
s
⇡s� uh0⇥!h
s + rs · yh�⇤rs � ⇡� u
h00⇥!h0 � p� · yh�
⇤p�, h 2 H.
Therefore,
D⇡1,⇡2Z = diag�2{�1,�2}
0
BBBB@
.... . . uh
s�0rs . . .s...h
———————. . . 1 . . .s
1
CCCCA
.
Let
�� :=
0
B@
.... . . uh
s�0rs . . .sJH+1
...h
1
CA
and 1
> := (1 1 . . . 1)1⇥(JH+1). We claim that, for a generic subset of ⌦, at a ⇡-equilibrium, the square matrix
�� :=
��
——–1
>
!
has full rank, i.e. for 3 2 RJH+1, there is no solution to
23The assumptions in the statement of Lemma A.2 are implied by those in the statements ofLemmas A.3, A.4 and A.5.
25
3(y�, p�, 3,!) :=
0
BB@
F�(y�, p�,!)��(y�, p�,!) 3
1
> 3
>3 3 � 1
1
CCA = 0.
The Jacobian, Dy� ,p� , 3,! 3, is row-equivalent to
0
BBBBB@
⇤���� ⇤
���� D!
✓f��� 3
◆
—————————————————
0
����1
>
>3
���� 0
—————————————————Dy� ,p�g�
�� 0�� 0
1
CCCCCA
.
(13)
We wish to show that this matrix has full row rank at any zero of 3. As we haveseen already, Dy� ,p�g� has full row rank. Also, 3 is orthogonal to 1 and nonzero(since >
3 3 = 1). Hence, due to the block-triangular structure of (13), it su�ces toshow that the upper right block, given by
D!
✓f��� 3
◆=
0
@diagh
⇥. . . ⇡s� uh
s�00rs . . .s
⇤———————————————————diagh
h�. . . 3s uh
s�00rs . . .sJH+1
�(0 . . . 0)
i
1
A
,
has full row rank. This matrix is row-equivalent to a block-diagonal matrix, withblocks indexed by h. The h’th block is
. . . ⇡s� uh
s�00rs . . .sJH+1
�� . . . ⇡s� uhs�00rs . . .s>JH+1
——————————————————————–. . . 3s uh
s�00rs . . .sJH+1
�� 0
!
.
(14)
This matrix is block-triangular as well, and its upper right block has full row ranksince it has at least J columns and R is in general position. It remains to showthat the lower left block of (14) has full row rank. Let S be the subset of statesfor which 3s 6= 0. This is a nonempty subset at any zero of 3.24 Then we haveP
s2S 3suhs�0rs = 0. Since R is in general position, and uh
s�0is nonzero for all s, we
must have #S � J + 1. Full row rank of the lower left block of (14) now followsfrom the general position of R.
We have shown that the Jacobian Dy� ,p� , 3,! 3 has full row rank, at every zeroof 3. Thus 3 is transverse to zero, and �1
3! (0) = ? for all ! 2 ⌦Z�, a genericsubset of ⌦. This establishes that �� has full row rank. It follows that D⇡1,⇡2Z hasfull row rank for the generic subset of endowments ⌦Z := ⌦Z�1\ ⌦Z�2 . 2
Lemma A.3 Suppose S � J(H + 1) + 1, J � H � 1, and R is in general position.Then, for a generic subset of ⌦⇥ ⇧, condition C2 is satisfied at any equilibrium.
24S may depend on the zero of 3 we are considering, but this does not a↵ect our argument.
26
Proof :
Let ⇡ 2 ⇧0. We restrict endowments to lie in the generic subset ⌦Y \ ⌦Z of ⌦ forwhich condition C1 holds and D⇡1,⇡2Z has full row rank (see Lemmas A.1 and A.2),and show that, for a generic subset of ⇧0, and hence of ⇧, there is no solution to
4(y�1 , p�1 , y�2 , p�2 , ⇡1, ⇡2) :=
0
BBBB@
F�1(y�1 , p�1 , ⇡1)
F�2(y�2 , p�2 , ⇡2)P
s ⇡s�1 � ⇡�1Ps ⇡s�2 � ⇡�2
uh10�1
0uh20�2
0 � uh10�2
0uh20�1
0
1
CCCCA= 0.
The Jacobian, Dy�1 ,p�1 ,y�2 ,p�2 ,⇡1,⇡2 4, is row/column-equivalent to
0
BBBBB@
⇤�� D⇡1,⇡2Z
———————————————————————————Dy�1
g�1 0 0��
0 Dy�2g�2 0
��� 0
⇤ ⇤ Dp�1 , p�2
huh10�1
0uh20�2
0 � uh10�2
0uh20�1
0i ���
1
CCCCCA
.
(15)
Furthermore,
Dp�1
huh10�1
0uh20�2
0 � uh10�2
0uh20�1
0i= uh1
0�2
0uh20�1
00yh2�1
� uh10�1
00uh20�2
0yh1�1,
which is nonzero since yh1� is not collinear with yh2
� by condition C1. Since Dy�g� hasfull row rank for all �, it follows that the lower left block of (15) has full row rank.Since D⇡1,⇡2Z also has full row rank, the Jacobian Dy�1 ,p�1 ,y�2 ,p�2 ,⇡
1,⇡2 4 has full rowrank. Thus 4 is transverse to zero, and �1
4⇡ (0) = ? for all ⇡ in a generic subset of⇧0. 2
In order to establish condition C3, we will employ finite-dimensional perturba-tions of agents’ period 1 utility functions. This procedure requires the followingresult.
Lemma A.4 Suppose S � J(H + 1) + 1, and R is in general position. Then, for ageneric subset ⌦C⇥⇧C of ⌦⇥⇧, at any equilibrium, chs� 6= chs� for all (s, �) 6= (s, �),and for all h 2 H.
Proof :
Consider first the case where s 6= s and � = �. Without loss of generality, we cantake s = s1 and s = s2, and prove the result for the first agent, h1, for a given �. Wewill show that, for a generic subset of ⌦, there is no solution to
5(y�, p�,!) :=
✓F�(y�, p�,!)ch1s1�
� ch1s2�
◆= 0.
27
The Jacobian, Dy� ,p� ,! 5, is row-equivalent to
0
B@⇤
���� D!
✓f�
ch1s1�
� ch1s2�
◆
———————————————Dy� ,p�g�
�� 0
1
CA
,
and
D!
✓f�
ch1s1�
� ch1s2�
◆=
diagh[. . . ⇡s� u
hs�00rs . . .s]———————————–
1 � 1 01⇥(S�2)
�� 0
!
.
(16)
Note that the zero block in the lower right of (16) corresponds to all agents otherthan h1. The matrix (16) is row-equivalent to
0
BBB@
⇡s1� uh1s1�
00rs1 ⇡s2� u
h1s2�
00rs2 [. . . ⇡s� uh1
s�00rs . . .s>2]
���� 01 �1 0
��——————————————————————————————————–
0�� diagh 6=h1
[. . . ⇡s� uhs�00rs . . .s]
1
CCCA
.
(17)The lower right submatrix of (17) has full row rank since R has full column rank.The (1, 3) block of the upper left submatrix of (17) also has full row rank due to thegeneral position of R (note that the assumed dimensionality condition implies thatS � J+2). Consequently, the whole matrix (17) has full row rank, and hence so does(16). Furthermore, due to the full row rank of Dy� ,p�g�, the Jacobian Dy� ,p� ,! 5 hasfull row rank as well. Thus 5 is transverse to 0, and �1
5! (0) = ? for all ! 2 ⌦1�, ageneric subset of ⌦. Let ⌦1 := \�2⌃⌦1�.
Now consider the case where � 6= �, while s may or may not be equal to s.Without loss of generality, we can take � = �1, � = �2, and s = s1, and prove theresult for the first agent, h1. Let ⇡ 2 ⇧0. We restrict endowments to the genericsubset ⌦Z for which D⇡1,⇡2Z has full row rank (see Lemma A.2), and show that, fora generic subset of ⇧0, and hence of ⇧, there is no solution to
6(y�1 , p�1 , y�2 , p�2 , ⇡1, ⇡2) :=
0
BBBB@
F�1(y�1 , p�1 , ⇡1)
F�2(y�2 , p�2 , ⇡2)P
s ⇡s�1 � ⇡�1Ps ⇡s�2 � ⇡�2ch1s1�1
� ch1s�2
1
CCCCA= 0.
28
The Jacobian, Dy�1 ,p�1 ,y�2 ,p�2 ,⇡1,⇡2 6, is row-equivalent to
0
BBBB@
⇤�� D⇡1,⇡2Z
————————————————————————0
B@Dy�1 ,p�1
✓ch1s1�1
g�1
◆ ���� ⇤———————————————–
0�� Dy�2 ,p�2
g�2
1
CA
����� 0
1
CCCCA
.
Since D⇡1,⇡2Z has full row rank, it su�ces to show that the lower left block of theabove matrix has full row rank. This block is itself block-triangular, and Dy�2 ,p�2
g�2has full row rank. The other diagonal term, given by
Dy�1 ,p�1
✓ch1s1�1
g�1
◆=
r>s1
�� 01⇥J(H�1)
�� 0————————————IJ
�� . . . IJ . . .h 6=h1
�� 0
!
,
also has full row rank since rs1 is nonzero by the general position of R.It follows that the Jacobian Dy�1 ,p�1 ,y�2 ,p�2 ,⇡
1,⇡2 6 has full row rank, at every zeroof 6. Thus 6 is transverse to zero, and �1
6⇡ (0) = ? for all ⇡ 2 ⇧C , a genericsubset of ⇧0. The set ⌦C in the statement of the lemma is given by ⌦C := ⌦1 \⌦Z .2
The space of utility functions U is infinite-dimensional. For the genericity ar-guments that we use in order to establish condition C3, it su�ces to consider afinite-dimensional submanifold of U . This submanifold consists of linear pertur-bations of the von Neumann-Morgenstern utility index uh of each agent h in theneighborhood of consumption in each state (s, �), for a given equilibrium. This is astandard construction (see, for example, Citanna et al. (1998)).25
Consider an economy (!, u, ⇡) 2 ⌦C⇥ UH⇥⇧C , and a corresponding equilibriumwith period 1 consumption allocation {chs�}h2H,s2S,�2⌃. By Lemma A.4, the consump-tion level of agent h, chs�, is distinct across (s, �), for all h. Therefore, we can findopen intervals Bh
s�, Bhs� such that chs� 2 Bh
s� ( Bhs� ⇢ R++, where the intervals Bh
s�
are disjoint across (s, �). Define C2 functions ⇢hs� : R++ ! [0, 1] such that ⇢hs� = 1on Bh
s� and ⇢hs� = 0 on the complement of Bhs�.
26 Now consider the class of utilityfunctions uh parametrized by ⌫h 2 RS⌃:
uh(c, ⌫h) := uh(c) +X
s,�
⇢hs�(c) ⌫hs�(c� chs�).
It can be verified that, for ⌫h su�ciently small in norm, uh 2 U . We have
uh0(c, ⌫h) = uh0(c) +X
s,�
⇢hs�0(c) ⌫hs�(c� chs�) +
X
s,�
⇢hs�(c) ⌫hs�,
25Unlike Citanna et al. (1998), we perturb the gradient of the utility functions instead of theirHessian. Also, we have state-independent separable utility so additional care has to be exercised inperturbing utilities in di↵erent states.
26The existence of such a “bump” function is well-known. See Guillemin and Pollack (1974),chapter 1.
29
so thatD⌫hs�
uh0(chs�, ⌫h) = 1. (18)
Let ⌫� := {⌫hs�}h2H,s2S, and ⌫ := {⌫�}�2⌃. In order to show that condition C3 holds,we will perturb agents’ period 1 utility functions via perturbations of ⌫.
Lemma A.5 Suppose S � 2JH, J � H � 1, and R is in general position. Then,for a generic subset of E , condition C3 is satisfied at any equilibrium.
Proof :
Let ⇤� be the JH ⇥ S matrix defined by
⇤� :=
0
BB@
...
. . .⇣uhs�0rs � uh
0�0p�⌘
. . .s...h
1
CCA
.
We first establish that
⇤ :=
✓⇤�⇤�
◆
has full row rank, for all � 6= �. Without loss of generality we can take � = �1 and� = �2. Let ⇤ be the (square) submatrix of ⇤ consisting of the first 2JH columnsof ⇤. Let 7 2 R2JH . We will show that, for a generic subset of UH ⇥⇧, there is nosolution to
7(y�1 , p�1 , y�2 , p�2 , 7, ⌫�1 , ⌫�2 , ⇡1, ⇡2) :=
0
BBBBBB@
F�1(y�1 , p�1 , ⌫�1 , ⇡1)
F�2(y�2 , p�2 , ⌫�2 , ⇡2)P
s ⇡s�1 � ⇡�1Ps ⇡s�2 � ⇡�2
⇤(y�1 , p�1 , y�2 , p�2 , ⌫�1 , ⌫�2) 7
>7 7 � 1
1
CCCCCCA= 0.
We restrict ourselves to the generic subset of endowments and information structures(⌦R \ ⌦C) ⇥ ⇧C for which Lemma A.4 applies, so that we can parametrize agents’period 1 utility functions by the vector ⌫, and for which the number of equilibria isfinite. Recall that ⌦C is a subset of ⌦Z , for which D⇡1,⇡2Z has full row rank (seeLemma A.2).
The Jacobian, Dy�1 ,p�1 ,y�2 ,p�2 , 7,⌫�1 ,⌫�2 ,⇡1,⇡2 7, is row-equivalent to
0
BBBBBB@
*�� ⇤
�� D⇡1,⇡2Z———————————————————————————
*�� D⌫�1 ,⌫�2
(⇤ 7)�� 0
———————————————————————————Dy�1 ,p�1
g�1 0 0�� ��
0 Dy�2 ,p�2g�2 0
�� 0�� 0
0 0 >7
�� ��
1
CCCCCCA
.
(19)
30
We wish to show that this matrix has full row rank at any zero of 7. Since D⇡1,⇡2Z,Dy�1 ,p�1
g�1 and Dy�2 ,p�2g�2 have full row rank, and 7 6= 0, it su�ces to show that
the middle block of (19) has full row rank. Using (18), we see that this block isblock-diagonal with respect to h 2 H and � 2 {�1, �2}, with typical diagonal termgiven by [. . . 7srs . . .s2JH ]. This diagonal term has full row rank by the sameargument that we used for the bottom left block of (14). It follows that the JacobianDy�1 ,p�1 ,y�2 ,p�2 , 7,⌫�1 ,⌫�2 ,⇡
1,⇡2 7 has full row rank, at every zero of 7. Thus 7 istransverse to zero, and �1
7u⇡(0) = ? for all (u, ⇡) in a generic subset of UH ⇥ ⇧C .Now that we have established that ⇤ has full row rank, consider the matrix Q,
defined in (12). We have
Q =
diag�[⇤�]
——————–. . . IS . . .�
!
.
The upper submatrix of Q has full row rank due to the full row rank of ⇤, and clearlythe lower submatrix of Q has full row rank as well. If Q does not have full row rank,there exist vectors a� 2 RJH , � 2 ⌃, and b 2 RS, not all of which are zero, such thata>�⇤� + b = 0, for all �. Moreover, since ⇤� has full row rank, b 6= 0. It follows thatthe row spaces of {⇤�}�2⌃ have a nontrivial intersection. But this contradicts thefull row rank property of ⇤.
The utility perturbations in this proof apply only to the particular equilibriumunder consideration. However, we can repeat the same construction for each equi-librium, and take the intersection of the generic subsets for which the Jacobian of 7 has full row rank. This intersection is itself a generic subset since the number ofequilibria is finite (recall that endowments are restricted to ⌦R). 2
Proof of Corollary 1:
Let E⇤ be the generic subset of economies for which Theorem 2 applies. An inspectionof the proofs of Lemmas A.1–A.5 reveals that E⇤ can be written as ⌦⇤ ⇥ ⇥, where⌦⇤ is a generic subset of ⌦, and ⇥ is a generic subset of UH ⇥ ⇧ (in particular, wedo not perturb endowments jointly with either utilities or probabilities in any of thetransversality arguments in these proofs). Moreover, ⌦⇤ is a subset of ⌦R, the set ofregular economies (this condition was imposed in the proof of Lemma A.5).
Fix a (u, ⇡) in ⇥, and consider an !⇤ in ⌦⇤. Since the economy (!⇤, u, ⇡) isregular, it has a finite number of equilibria, which we index by i = 1, . . . , n. ByTheorem 2, for the i’th ⇡-equilibrium, there exist ⇡i and ⇡0
i in ⇧(⇡), such thatW ⌧ 0 at a ⇡i-equilibrium and W � 0 at a ⇡0
i-equilibrium (the information struc-tures ⇡i and ⇡0
i are in a neighborhood of ⇡, and the corresponding equilibria arein a neighborhood of the i’th ⇡-equilibrium). Using the regularity of (!⇤, u, ⇡)once again, there exists ✏i > 0, such that the preceding statement applies for all! 2 B✏i(!
⇤) := {! 2 ⌦⇤ s.t. |! � !⇤| < ✏i}, for the same choice of ⇡i and ⇡0i.
By Theorem 1, there is a generic subset ⌦i of ⌦ such that every ⇡-equilibriumis ex-post ⇡i-ine�cient, and a generic subset ⌦0
i of ⌦ such that every ⇡-equilibriumis ex-post ⇡0
i-ine�cient. Therefore, statements (i)–(iii) of the corollary apply for all
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! 2 O!⇤ := \ni=1[B✏i(!
⇤) \ ⌦i \ ⌦0i]. Indeed, they apply for all ! 2 ⌦ := [!⇤2⌦⇤ O!⇤ .
We claim that ⌦ is a generic subset of ⌦. Let ✏ := mini ✏i. Then, clearly, the setO!⇤ is an open, dense subset of B✏(!⇤). The set ⌦, being the union of these opensubsets of ⌦⇤, is itself an open subset of ⌦⇤. Moreover, ⌦ is dense in ⌦⇤, since for all!⇤ 2 ⌦⇤, there is a corresponding set O!⇤ , the closure of which contains !⇤. Thus ⌦is a generic subset of ⌦⇤, and hence also of ⌦. 2
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