Ambiguity aversion in the long run: “To disagree, we must also agree”

Accepted Manuscript Ambiguity aversion in the long run: “To disagree, we must also agree”

Aloisio Araujo, Pietro da Silva, José Heleno Faro

PII: DOI: Reference:

S0022-0531(16)30019-9 http://dx.doi.org/10.1016/j.jet.2016.04.008 YJETH 4558

To appear in:

Journal of Economic Theory

Received date: Revised date: Accepted date:

14 April 2015 27 January 2016 10 April 2016

Please cite this article in press as: Araujo, A., et al. Ambiguity aversion in the long run: “To disagree, we must also agree”. J. Econ. Theory (2016), http://dx.doi.org/10.1016/j.jet.2016.04.008

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Ambiguity Aversion in the Long Run: ”To Disagree, We Must Also Agree”∗ Aloisio Araujoa,b , Pietro da Silvac , and Jos´e Heleno Farod a

b c

IMPA, Estrada Dona Castorina 110, 22.460-320, Rio de Janeiro, Brazil

EPGE/FGV, Praia de Botafogo, 190, 22253-900, Rio de Janeiro, Brazil

Universidade Federal de Sergipe, Av. Marechal Rondon, 49100-000, S˜ ao Crist´ ov˜ ao/SE, Brazil

d

Insper, Rua Quat´ a 300, Vila Ol´ımpia 04546-042, S˜ ao Paulo, Brazil

January 27, 2016

Abstract We consider an economy populated by smooth ambiguity-averse agents with complete markets of securities contingent to economic scenarios, where bankruptcy is permitted but there is a penalty for it. We show that if agents’ posterior belief reductions given by their “average probabilistic beliefs” do not become homogeneous then an equilibrium does not exist. It is worth noting that our main result does not imply any convergence of ambiguity perception or even the attitudes towards it. In this way, complete markets with default and punishment allows for ambiguity aversion in the long run, and the agents can disagree on their ambiguity perception but they must agree on their expected beliefs. Keywords: Ambiguity aversion, bankruptcy, complete markets, convergence of beliefs, punishments, smooth ambiguity model. JEL Classification: D53, D81, D84.

1

Introduction

When markets are complete, one necessary condition for existence of equilibrium, in a stochastic framework with expected utility agents, is that beliefs must be locally equivalent, i.e., all agents’ beliefs assign null probability over the same finite-time events (see Harrison and Kreps (1979)). In general, equilibrium existence in infinite horizon economies is not precluded by the lack of equivalence ∗ We wish to thank V. Filipe Martins-da-Rocha, Marciano Siniscalchi (the editor) and three referees for useful suggestions and comments. We also thank the participants of seminars at IMPA and the 34◦ Meeting of the Brazilian Econometric Society for their comments. J.H. Faro is also grateful for financial support from the CNPq-Brazil (Project No. 310837/20138). Corresponding author. Tel.: +55-11-45-04-24-22; fax: +55-45-04-09-23-90; e-mail adress: [email protected] (J. H. Faro).

1

as we can see from the conditions provided by Bewley (1972).1 On the other hand, Araujo and Sandroni (1999) have showed that if bankruptcy is permitted, with a penalty for it, then equivalence of beliefs is a necessary condition for equilibrium existence. In turn, according to Blackwell and Dubins (1962), such equivalence implies convergence of posteriors. In other words, when agents may become bankrupt but there is a penalty for doing so, equilibrium existence implies convergence to homogeneous expectations in the long run. An important assumption made in afore cited work is that all agents obey the expected utility hypothesis. Under this perspective, while the agents may associate subjective likelihoods to events instead of objective probabilities, the beliefs should be given by subjective probabilities. However, from the past decade the possibility that agents may not hold a single belief on scenarios has widely recognized in macroeconomics and finance. This perspective has been a classical issue in decision theory, and it goes back to the seminal work of Ellsberg (1961) proposing the notion of ambiguity by showing that may be no single probabilistic belief on the states of nature that rationalizes the pattern of behavior revealed in the well-known Ellsberg Paradox. To model individual preferences with a negative attitude towards ambiguity, we assume the smooth ambiguity aversion model proposed by Klibanoff et al. (2005), hereafter KMM. They provided an axiomatic foundation (see also Seo (2009)) for a preference representation that allows for a separation of the perception of ambiguity and attitudes towards it. As in many applications, we assume that the smooth representation is determined by a triple (u, φ, μ). The attitudes towards risk is captured by a concave von Neumann-Morgenstern utility index u : R+ → R that satisfies also some mild classical conditions. The attitudes towards ambiguity are described by the properties of φ : R → R, which we assume concave too. The second-order belief μ is defined over measurable sets of first-order probabilities and its support captures the perceived ambiguity. For a given random payoff x : Ω → R, we assume in our application that the smooth representation takes the following form    φ v (x (ω)) P (dω) μ (dP ) ≡ Eμ [φ (EP [v (x)])] , Δ(Ω)

Ω

where v (x (ω)) = u (x (ω)) if x (ω) ≥ 0 and, otherwise, the agent receives a penalty for each unit of payoff that she is short, which is captured by a constant M > 0 so that v (x (ω)) = −M x (ω) if x (ω) < 0. From the second-order belief we can derive the  so called reduction or weighted (expected) belief P, which is defined by P (A) = Δ(Ω) P (A) μ (dP ) = Eμ [P (A)], for any event A ⊂ Ω. This concept is the key element for our main result as we will comment below. One possible interpretation for this model, as discussed by Ju and Miao (2012, p.564), says that this is a model of robustness in which the agents are 1 See Riedel (2003) for a class of two agents economies with von Neumann-Morgenstern utilities with heterogeneous but locally equivalent beliefs. This is a special case of additively separable preferences in which the equilibrium existence result provided by Dana (1993) holds.

2

concerned about model misspecification or statistical ambiguity and, thus, seek robust decision-making a la Hansen and Sargent (2001). The support of subjective prior μ describes plausible economic models and the agent is ambiguous about which is the right model specification. The agent is averse to model uncertainty and, thus, evaluates plausible models through a concave mapping φ. We model an economy populated by a finite number I of smooth ambiguityaverse agents with complete markets of securities contingent to economic scenarios. Following the Araujo and Sandroni (1999), we allow for bankruptcy but there is a penalty for it. Our main result shows that if agents’ posterior  I reductions Pi i=1 are not equivalent then an equilibrium does not exist. It is worth noting that our main result does not imply any convergence of ambiguity perception or even the attitudes towards it. Accordingly, in the long run any agent should predict the expected belief as the same as others even they disagree in their perceived ambiguity concerning the true distribution. From this perspective, our result concerning an infinite horizon economy has the same flavor of the conclusion provided by Rigotti et al. (2008) showing that, for a finite economy with smooth agents and without aggregate uncertainty, absence of purely speculative trade means that the weighted belief have to coincide.2 An important consequence of our main result is that all speculative trade associated with weighted belief heterogeneity will disappear. Following the bankruptcy rule proposed by Araujo and Sandroni (1999), our analysis based on smooth ambiguity averse individuals as above gives that if some agent i defaults one unit of consumption good at period t contingent to an event E, then perceived  the discounted

 penalty

by this individual at period 0 is Eμi φi β t P (E) M i ≤ φi β t Pi (E) M i , where β t Pi (E) M i captures the average expected discounted penalty for the agent i. Now, if the expected beliefs of two agents, i and j, don’t eventually converge, then it is possible to find an event E with arbitrary small expected probability Pi (E) but with expected probability Pj (E) close to 1. Our main result assumes a condition called unambiguous convergence to null events (Definition 3), which guarantee that all probabilities in the of the support second-order belief μi agree that the probability of E is arbitrary small. In the same way, all probabilities in the support of the second-order belief μj agree that E has a probability close to 1. We note that the price at period 0 of the bet on E is strictly positive because of agent’s j beliefs while the cost in terms of utility for agent i to default contingent to E is arbitrarily small. As a result, agent i chooses to short-sell an arbitrarily large amount of the claim contingent to the event E while agent j is willing to trade it with agent i. Since in any equilibrium with penalties there is no default, posterior expected beliefs that don’t eventually converge and equilibrium with penalties are not compatible. To conclude, we provide an equilibrium existence result for an economy a 2 Rigotti et al. (2008) showed that their proposed notion of subjective beliefs (across constant acts) contain all of the information needed to predict the presence or absence of purely speculative trade. In the case of smooth ambiguity averse agents, the set of subjective beliefs contains only the prior given by the reduction P.

3

la Araujo and Sandroni (1999) with smooth ambiguity averse agents, which is a natural question given our main result as discussed above. Also, even admitting that the question at issue is not about market selection, we find as a scholion of our existence result that all agents in this economy will survive. More precisely, we show that there is a personal positive lower bound for the consumption in equilibrium for each agent, which is uniform over all paths.3 Let us highlight that we do not study the question about the survival of ambiguity averse agents in the presence of expected utility maximizers with correct beliefs, which is briefly discussed in the next section on the related literature. Thus, in principle, any statement on the incorrectness of beliefs doesn’t make sense because we do not assume the existence of some true probability law governing the economy. On the other hand, if we additionally assume the presence of some expected utility agent with a ”correct probability law” then this belief will work as an anchor for all the other agents’ expected beliefs. It does not mean that people with incorrect beliefs or expected beliefs are priced out of the market, just that there no equilibrium by the same reasons as discussed in the previous paragraph. The paper is organized as following: bellow we present a discussion about related works in the literature; section 2 states the framework; market features and the equilibrium definition are in section 3; section 4 is devoted to present the main result; our equilibrium existence result is in section 5; section 6 concludes this paper; proofs are left in the appendix. Related Literature Our present work seeks to contribute to the literature on ambiguity aversion and its interplay with the theory of complete markets with default and punishment in the lines proposed by Araujo and Sandroni (1999). To the best of our knowledge, this is the first attempt to study the evolution of ambiguous beliefs as a necessary condition for the equilibrium existence in a complete market economy a la Araujo and Sandroni populated by smooth ambiguity averse agents. We may view our analysis as a attempt to carry out a robustness check of the Araujo and Sandroni result, but only in the context of the KMM smooth ambiguity model. Another interesting class of preference featuring ambiguity aversion is the multiple-prior model of Gilboa and Schmeidler (1989), aka maxmin expected utility (MEU) model. KMM showed that the MEU model can be viewed as the limiting case of the smooth ambiguity model with infinite ambiguity aversion.4 Two important aspects of the smooth ambiguity model are tractability and the fact that this model admits a clear-cut comparative statics analysis. Comparative statics of ambiguity attitudes can be analyzed using the function φ only, holding the perception of ambiguity fixed. In contrast, the comparative statics analysis is not evident for the MEU model since multiple-priors sets characterizing MEU preferences may characterize ambiguity perception as well as ambiguity attitudes. On the other hand, indifference 3 See

our Lemma 9 discussion is based on the behavioral foundation of ambiguity and ambiguity attitude as provided by Ghirardato and Marinacci (2002). 4 This

4

curves in the KMM’s model are smooth under regularity conditions, rather than kinked as in the case of the MEU model. Also, considering that an important evidence for ambiguity aversion is given by the tendency to equalize the demands for the bets that pay off in ambiguous states, Ahn et al. (2014) found that kinked specifications can explain this tendency while the smooth specification cannot. In this way, it seems worth to investigate in the future the Araujo and Sandroni’s model under a different pattern of ambiguity aversion behavior as, for instance, modeled in the MEU model. In another direction, Carvajal and Riascos (2008) analyzed the consequences of dropping completeness of financial markets in the Araujo and Sadroni’s model with expected utility agents and found that if markets are sufficiently incomplete then equilibrium with trade allowing for beliefs disagreement on null events generically exists. Since Alchian (1950) and Friedman (1953), a related problem to convergence of expectations aims to understand whether agents who do not predict future events as accurately as others are driven out of the market. A positive answer to this issue leads to the so called market selection hypothesis by stating that markets select traders with more accurate beliefs. A confirmation of the market selection hypothesis has been viewed as the key argument supporting the strong uniformity on the agents’ beliefs that is imposed by the rational expectations hypothesis. In general equilibrium models with complete markets, Sandroni (2000) and Blume and Easley (2006) have showed under some regularity conditions that the market selection hypothesis holds true. One of the main message in these works is that for equally patience investors only those with true belief or those whose forecasts merge with the true likelihoods survive. Blume and Easley (2006) also have showed that for competitive economies with incomplete markets the market selection hypothesis may fails. In another direction, Massari (2014) studies a general equilibrium model with a continuum of traders a la Aumann (1965) and concludes, contrary to results for economies with only finitely many traders, that risk attitudes matters for survival and that markets might select traders that do not hold accurate beliefs, but asymptotic equilibrium prices reflect accurate beliefs as claimed by the Friedman’s conjecture. More recently, Condie (2008), da Silva (2011), and Guerdjikova and Sciubba (2015) analyzed survival of ambiguity averse agents in the presence of expected utility maximizers with correct beliefs.5 They follow the same economic framework as in Sandroni (2000) or Blume and Easley (2006) where bankruptcy is not allowed. Condie (2008) studies the case of maxmin expected utility agents as axiomatized by Gilboa and Schmeidler (1989). Condie concludes that, in general, ambiguity averse agents have no importance in the long run under the presence of aggregate risk. Da Silva (2011) considers the same framework but with more general case of agents with variational preferences (introduced by Maccheroni et al. (2006)) and states that survival of ambiguity averse agents depends on the relative “weight” assigned to the true probability model and the 5 Easley and Yang (2014) studied the problem of survival by considering the class of loss -averse decision makers in the presence of Epstein and Zin ’s (1989) preferences who are not loss-averse.

5

level of aggregate risk. Guerdjikova and Sciubba (2015) deal with smooth model agents like us but assume the presence of an objective ambiguity captured by a finite set of probability measures, uniformly perceived by all ambiguity averse agents in the market. They find special conditions on the ambiguity attitudes (including decreasing absolute ambiguity aversion) and aggregate risk in which an smooth ambiguity averse agent may survive and also determine prices in the limit. But for the case of increasing or constant absolute ambiguity aversion, smooth ambiguity averse agents only survive under no-aggregate risk and always such agents have no impact in the limit prices.

2

Notation and Framework

Consider a dynamic model with discrete time T = {0, 1, . . . }. There is a finite set of agents I = {1, . . . , I}, which have common information modeled by a filtered space (Ω, (Ft )t∈T ), where Ω := {ω0 } × t≥1 St , with ω0 the sure state occurring at the first time and St is the finite set of possible states occurring at each time t ≥ 1. A representative element of Ω will be denoted by ω = t (ω0 , ω1 , . . . ) and ω t = (ω0 , . . . , ωt ) ∈ Ωt := {ω0 } × τ =1 Sτ denotes the partial history until period t. The σ-algebra generated by (t+1)-dimensional cylinders is Ft = σ({Gt (ω); ω ∈ Ω}), where Gt (ω) := {ω t } × τ >t Sτ denotes the cylinder6 with base ω t . Let F 0 = ∪t∈T Ft be the algebra of finite-time events and F = σ(F 0 ) the σ-algebra generated by F 0 . The filtered space (Ω, (Ft )t∈T , F) represents the informational process known by agents. The process ωt might be governed by a probability Q(·|ω t−1 ) on St , which can be understood as the conditional probability given ω t−1 in the past. These probabilities generate the law Q on (Ω, F) by constructing the partials Q(ω t ) = Q(ω t−1 )Q(ωt |ω t−1 ) on Ft for each t ∈ T , and evoking the Kolmogorov’s extension theorem (see, for instance, Shiryayev (1996)). The set of all probabilities on a measurable space (Ω, F) is denoted by Δ(Ω, F), or Δ(Ω) for simplicity. If P ∈ Δ(Ω), Pt denotes its restriction to (ω t ,s) denotes the conditional one step ahead probabilFt , and P (s|ω t ) := Pt+1 Pt (ω t ) ity, also called posterior belief, from P . Note that we can consider P (·|ω t ) ∈ Δ(Gt (ω), Ft+1 ). Given two probabilities, P and Q, we say that Q is absolutely continuous with respect to P if for A ∈ F, P (A) = 0 implies Q(A) = 0, and we denote Q P . If Q P and P Q we say that P and Q are equivalents. Again, for P and Q we denote the total variation distance between P and Q by P −Q := supA∈F |P (A) − Q(A)|. The acts considered by the agents must be based on the common knowledge about the world. In this way, the possible consequences of an act at a period t 6 The

cylinder Gt (ω) can also be understood as a subtree after the node ω t .

6

will be contingent to events on Ft . The individual choice space is a subset of 

X = (xt )t∈T ; xt : Ω → R is Ft -adapted and sup |xt (ω)| < ∞ , t,ω

whose dual, which contains the prices, is  ∗

X =

(pt )t∈T ; pt : Ω → R is Ft -adapted and



 |pt (ω)| < ∞ ,

t,ω

considering the dual pairing

x, p =



xt (ω)pt (ω),

t,ω

which generates the Mackey topology τ (X, X ∗ ) on X, and also the weak topology σ(X ∗ , X) on X ∗ . It is worth noting that X can be identified with the set given by    t t ({t} × Ω ) → R; sup |x(t, ω )| < ∞ , x: t,ω

t∈T

which in turn is basically ∞ since

3

 t∈T

({t} × Ωt ) is a countable set.

Market, Agents and Optimal Choices

At period 0 agent i ∈ I can trade contingent claims for all periods, in other words, each agent i choose an asset allocation k i = (kti )t≥1 ∈ X, where kti (ω) represents the amount that i will receive (deliver in the negative case) at time t if ω t occurs. Agent i has a positive consumption at period 0 such that ci0 = ei0 − q, k i ,

(1)

∗ is the price of assets. Each agent is endowed with an initial where q ∈ X+ consumption stream ei ∈ X+ satisfying, for all i ∈ I  e < ei < ej < e¯, j

for positive constants e and e¯. For t ≥ 1, agent i’s consumption stream derived of his choice k i is cit = (eit + kti )+ , and dit = (eit + kti )− is the amount which he is short of (we also denote k0i := − q, k i  in order to get ci0 = ei0 + k0i ≥ 0). We assume the existence of a penalty for each unity of dit , which is constant and denoted by M i > 0. As discussed in our Introduction, we do not rule out negative attitudes towards ambiguity implicitly done by Araujo and Sandroni (1999). Actually, we allow for ambiguity-sensitive behavior by considering that each agent i behaves in a way inspired by the smooth ambiguity averse model as proposed by KMM. 7

Following Araujo and Sandroni (1999), each agent i is endowed with a utility index vi over R determined by ui and M i in the following way:

ui (x) if x > 0 vi (x) = . −M i x if x ≤ 0 support The second order belief of agent i is denoted by μi and the corresponding 7 perception of the agent. For each first order Supp μi captures the ambiguity

probability P ∈ Supp μi we have the discounted expected utility associated to each profile ci0 , k i given by  ∞  t i i β vi (et + kt ) . EP t=0

Each agent i is also endowed with a function φi which captures her ambiguity attitudes. To

summarize, the way in which each agent

i ranks alternatives ci ≡ ci0 , k i ∈ R+ × X is determined by the couple ui , φi , μi , ei , M i that generates the functional ⎧ ⎛ ⎡ ⎤⎞⎫ ∞ ⎬ ⎨  β t vi (eit + kti )⎦⎠ . (2) V i (ci ) ≡ V i (ci0 , k i ) = Eμi φi ⎝EP ⎣ ⎭ ⎩ t≥0

From the second-order belief μi we can derive the so called reduction or expected belief Pi of the agent i: Pi (A) := Eμi [P (A)] , for any event A ⊂ Ω. Also, we assume that the utility index ui and the ambiguity index φi are twice continuously differentiable with ui (0) = 0, ui , φi > 0, ui < 0, φi ≤ 0 and lim+ ui (x) = ∞. x→0

Definition 1 A competitive equilibrium with penalties is described by a profile of allocations and price ((¯ ci0 , k¯i )i∈I , q¯) such that each agent i optimizes and markets clear, i.e.   ci = ei , i



i

k i = 0.

i

has

μi



 μi is the smaller closed set (w.r.t. the vague topology) in Δ whose complement null measure.

7 Supp

8

4

Ambiguity in the Long Run

We show in this section our main result which says that a necessary condition for existence of equilibrium is the equivalence between the reductions or expected beliefs of every individual. Moreover, by Blackwell and Dubins (1962), the expected beliefs must eventually become homogeneous. Araujo and Sandroni (1999) assumed that individuals are risk averse, and further features on attitude toward risk plays no role. In our case, the level of risk aversion is still not important and the type of non-positive attitude toward ambiguity also does not matter. Before presenting our result about global equivalence of the expected beliefs, we need to introduce an important regularity condition over the agents perceived ambiguity. Definition 2 We say that the agent i obeys the unambiguous convergence to null events property if for any sequence of events {At }t≥1 , where At ∈ Ft , ∀t ≥ 1, if lim Pi (At ) = 0, t→∞

then lim P (At ) = 0,



uniformly for P ∈ Supp μi .

t→∞

As usual, uniform convergence is a kind of strong condition on the mode of convergence. In our case, given an agent i, by imposing the unambiguous convergence to null events property we mean that if the expected belief Pi says that a sequence of events converge to a miracle then all first-order priors describing the perceived ambiguity must agree with this convergence and also it must be uniform over all priors in Supp μi . We note that this condition is automatically satisfied when we assume finite supports. Theorem 3 Suppose that all agents in the economy satisfies the unambiguous convergence to null events property. Let ((¯ ci0 , k¯i )i∈I , q¯) be an equilibrium with i j penalties, then P P ∀i, j ∈ I. Moreover, lim Pi (·|ω t ) − Pj (·|ω t ) = 0, Pi − a.s., for all i, j ∈ I.

t→∞

5

Existence of Equilibrium

In the previous section we have considered an economy a la Araujo and Sandroni (1999) but with smooth ambiguity averse agents holding second-order beliefs that satisfy the ”unambiguous convergence to null events” condition. Our main result says that a necessary condition for the existence of a competitive equilibrium without default is that agents’ expected beliefs must be globally equivalent. It is clear that in order to get that this result is economically meaningful we need to give reasonable conditions in which the equilibrium exists. 9

The following condition proposed by Sandroni (1994) will be used in our next result about the existence of a competitive equilibrium. Definition 4 We say that a set of probabilities P displays strong compatibility condition if there is a constant K > 0 such that P1 (A) ≤ KP2 (A), ∀A ∈ F for any P1 , P2 ∈ P. We note that this condition can be rewritten by saying that for all P1 , P2 ∈ P sup

A∈F

P1 (A) < +∞. P2 (A)

Under the strong compatibility condition, the next result finds that there exist a profile of default penalties that ensures the existence of equilibrium with penalties.8 Concerning the interpretation of default penalties, Dubey et al. (2005) claims that it might be interpreted in terms of some extra-economic punishments (prison or a self blame recognized generating utility loss). On the other hand, Zame (1993) come up with an interpretation of default penalties based on the notion of a non-modeled economic punishment given, for instance, by reputation loss or exclusion from credit markets. Proposition 5 Suppose that the set {Pi }i∈I displays the strong compatibility condition, then there exists a profile (Mi )i∈I of penalties such that there exists a competitive equilibrium with penalties. Let us examine some examples fitting the conditions of Proposition 5. First, consider the case generated by a finite set St = {s1 , s2 } for every t ≥ 1. Assume also that there are two probability measures P and Q over Ω generated by i.i.d. trials that assign p(s1 ) and q(s1 ) to state s1 , respectively, satisfying 0 < p(s1 ) < q(s1 ) < 1. Also, we assume that the economy is populated by two agents with second order beliefs μ1 and μ2 such that Supp(μ1 ) = Supp(μ2 ) = {P, Q} and μ1 (P ) = m1 < m2 = μ2 (P ). We note that for each partial history ω t = (ω0 , . . . , ωt ) t t m2 s=0 p(ωs ) + (1 − m2 ) s=0 q(ωs ) P2 (ω t ) = t t P1 (ω t ) m1 s=0 p(ωs ) + (1 − m1 ) s=0 q(ωs ) $ % t t t s=0 q(ωs ) + m2 s=0 p(ωs ) − s=0 q(ωs ) $ % = t t t s=0 q(ωs ) + m1 s=0 p(ωs ) − s=0 q(ωs ) $ % t p(ωs ) 1 + m2 s=0 q(ωs ) − 1 $ % = t p(ωs ) 1 + m1 s=0 q(ωs ) − 1 8 In the proof of Proposition 5 we provide an explicit lower bound for each individual default penalty.

10

2x We have that f (x) = 1+m 1+m1 x with x ∈ [−1, ∞) is a increasing and continuos 2 2 f (−1) ≤ f (x) ≤ limx→∞ f (x) = m function such that 1−m 1−m1 = m1 , therefore ' $ % & t p(ωs ) P2 (ω t ) 1−m2 m2 s=0 q(ωs ) − 1 ∈ 1−m1 , m1 . Since this happens in every Ft we P1 (ω t ) = f

get9 sup

A∈F

Pi (A) Pi (A) = sup <∞ Pj (A) A∈F 0 Pj (A)

for i, j ∈ {1, 2}. Now, in another example, assume that P and Q are two arbitrary probabilities over Ω and define [P, Q] := {αP + (1 − α)Q : α ∈ [0, 1]}. Also, assume 1 that I = {1, 2} and Pi (A) = 0 fi (α)Pα dα, where Pα = αP + (1 − α)Q and 1 fi : [0, 1] → [0, ∞) is a continuous function satisfying 0 fi (α)dα = 1. Thus, the 1 ≤ ff12 (x) support of μi is given by the convex set [P, Q]. Provide that K (x) ≤ K for some K > 0, the reductions P1 and P2 are strongly compatible.

6

Conclusion

Araujo and Sandroni (1999) showed that in an open-ended complete market economy with subjective expected utility agents a la Savage (1954) and the possibility of costly bankruptcy, there are some compatibility conditions required for existence of equilibrium. These necessary compatibility conditions imply that posteriors subjective beliefs must converge and so, eventually agents must hold near homogeneous beliefs. Our main contribution is to extend this result to the case of smooth ambiguity-averse agents a la KMM. Analogous compatibility conditions must hold for an equilibrium with penalties to exist and so agents’ posteriors of the expected beliefs must converge in equilibrium. However, this does not imply any convergence of ambiguity perception or attitudes towards it. Therefore, some disagreement may persist.

Appendix Before the proof of our Theorem 3, we need the following two lemmas: Lemma 6 In every equilibrium with penalties, agent i’s first order conditions (3) β t ui (cit (ω t ))Eμi {φi (EP [Ui (ci )])P (ω t )} = λ(i)qt (ω t ) ( are satisfied, where Ui (ci ) := t β t ui (cit ), ω t ∈ Ωt and λ(i) > 0 is agent i’s Lagrange multiplier. Proof. In an equilibrium with penalties there is no default and by the assumption given in the Inada conditions the equilibrium allocations must be positive. 9 In

fact, it was proved for events {ω t }, butan event in Ft is a finite disjunct union of

unitary events and if m <

an bn

< M , then m <

a n n n bn

11

< M.

So, since we have interior solutions of maximization problems we get first order conditions (see, for instance Luenberger (1969) section 9.3, Theorem 1). Lemma 7 Let P, Q ∈ Δ, if P is not absolutely continuous with respect to Q then there exists a sequence At ∈ Ft such that Q(At ) → 0 and P (At ) > δ > 0 ∀t ∈ T . Proof. By hypothesis there exists A ∈ F such that Q(A) = 0 but P (A) > δ for some δ ∈ (0, 1). Since F = σ(∪t Ft ) and ∪t Ft is an algebra, by the Carath´eodory Extension Theorem (see Shiryayev (1996)) Q(A) = inf{Q(B); A ⊂ B ∈ ∪t Ft }. So, for each n ∈ N, ∃Atn ∈ Ftn such that A ⊂ Atn and Q(Atn ) < n1 , with tn < tn+1 because Ft ⊂ Ft+1 . On the other hand P (Atn ) ≥ P (A) > δ. For t ∈ N∩(tn , tn+1 ) choose At = Atn . Theorem 3: Proof. Assume that Pi is not absolutely continuous with respect to Pj for some i, j ∈ I. Thus, by Lemma 7, there exists a sequence of events At ∈ Ft and δ > 0 such that limt Pj (At ) = 0 but Pi (At ) ≥ δ for all t ≥ 1. By the first order condition as obtained in our Lemma 6 we get * )   



 1 1 EP ui ci χAt qt , t χAt = Eμi φi EP Ui ci β λi 1 ≥ φi (Ui (¯ e)) ui (¯ e) Pi (At ) λi 1 ≥ φi (Ui (¯ e)) ui (¯ e) δ =: η > 0. λi Since the unambiguous convergence to null events property holds for all agents, we obtain that if limt Pj (At ) = 0 then for any n ∈ N there is t ≥ 1 such that

P (At ) < n1 for all P ∈ Supp μj . So, let t1 ≥ 1 such that for any P ∈ Supp μj we get K , P (At1 ) < Mj + uj (¯ e) $ $ % % where K := uj c¯j0 + η − uj c¯j0 > 0. Define + k j by cj and + ktj = k¯tj if t = t1 , and + ktj1 = k¯tj1 − + cj0 := c¯j0 + η, + and note that

χAt1 , β t1

* - ) , , χ At k j = c¯j0 + η − ej0 + q, + k j − qt1 , t11 ≤ 0, + cj0 − ej0 + q, + β

12

% $ increasing + cj0 if we need to. Now, we can verify that + cj0 , + k j is a better choice than (¯ cj , k¯j ) for the agent j. Actually, 0

. $ V j (+ c j ) = E μj φ j E P . $ = E μj φ j E P . $ ≥ E μj φ j E P . $ > E μj φ j E P

& &

β t vj (ˆ cjt )

'%/

cjt ) + (vj (ˆ cj0 ) − vj (¯ cj0 )) + β t1 (vj (ˆ cjt1 ) − vj (¯ cjt1 )) β t vj (¯ '%/ & β t vj (¯ cjt ) + K − χAt1 β t1 (β −t1 Mj + uj (¯ e)) & ' %/ β t vj (¯ cjt ) + K − P (At1 )(Mj + uj (¯ e))

'%/

cj ). > V j (¯ For the second part, the proof is an immediate consequence of the Blackwell and Dubins (1962)’s Theorem. Next lemmas provide important results that will be used in the proof of our Proposition 5 about equilibrium existence. Lemma 8 If W : X → R is an increasing and concave function, then for each p¯ ∈ X ∗ , the mapping ψ (r) := max {W (c) : ¯ p, c ≤ r, c ≥ 0} is an increasing and concave real function. Proof. If r < r , then {W (c) : ¯ p, c ≤ r, c ≥ 0} ⊂ {W (c) : ¯ p, c ≤ r , c ≥ 0}, and so ψ(r) ≤ ψ(r ). If r, r > 0 and α ∈ [0, 1], by concavity of W W (αc + (1 − α)c ) ≥ αW (c) + (1 − α)W (c ), therefore p, c ≤ r, ¯ p, c  ≤ r and c, c ≥ 0} max {W (αc + (1 − α)c ) : ¯ p, c ≤ r, ¯ p, c  ≤ r and c, c ≥ 0} ≥ max {αW (c) + (1 − α)W (c ) : ¯ = max {α(c) : ¯ p, c ≤ r, c ≥ 0} + max {(1 − α)W (c ) : ¯ p, c  ≤ r , c ≥ 0} = αψ(r) + (1 − α)ψ(r ). On the other hand ψ(αr + (1 − α)r ) = max {W (αc + (1 − α)c ) : ¯ p, αc + (1 − α)c  ≤ αr + (1 − α)r and c, c ≥ 0}  p, c ≤ r, ¯ p, c  ≤ r and c, c ≥ 0} . ≥ max {W (αc + (1 − α)c ) : ¯

Lemma 9 If {Pi }i∈I displays strong compatibility condition and (ci )i∈I is an equilibrium, then ∃li > 0 such that cit ≥ li for each i ∈ I and all t ∈ T . 13

( ( Proof. Since e < i ei = i ci , for each node ω t there exists a j ∈ I such that cjt (ω t ) > e. For a fixed i ∈ I , from the first order conditions (3) uj (cjt (ω t ))Eμj {φj (EP [Uj (cj )]P (ω t ))} λj , = λi ui (cit (ω t ))Eμi {φi (EP [Uj (ci )]P (ω t ))} so

uj (e) φj (Uj (e))KPi (ω t ) λj ≥ ,  (U (¯  i t t φ e ))P (ω ) λi ui (ct (ω )) i i i λi uj (e)φj (Uj (e))K and cit (ω t ) ≥  . λj φi (Ui (¯ $e))  %/ λi uj (e)φj (Uj (e))K minj=i e, u−1  i λj φi (Ui (¯ e))

$

λi uj (e)φj (Uj (e))K λj φi (Ui (¯ e))

%

therefore ui (cit (ω t )) ≤

u−1 i

Finally, put li = does not depend on the history ω.

and notice this constant

.

Lemma 10 Under the same assumptions of the previous lemma, we get qt (ω t ) ≤ Li β t Pi (ω t ), where Li is a positive constant. Proof. By the first order conditions and using the Lemma 9 bound we get 1 t  i t β ui (ct (ω ))Eμi {φi (EP [Ui (ci )])P (ω t )} λ(i) 1 t  ≤ β ui (li )Eμi {φi (EP [Ui (li )])P (ω t )} λ(i) 1 t  β ui (li )φi (Ui (li ))Pi (ω t ) = λ(i)

qt (ω t ) =

then we take Li =

1   λ(i) ui (li )φi (Ui (li )).

Before the next Lemma, note that X can be viewed as a l∞ space. Since x ∈ X is Ft -adapted it is a function on the nodes t N = {(t, ω t )/t ∈ {0, 1, 2, . . . }, ω t ∈ Ωt } = ∪∞ t=0 ({t} × Ω )

which is an enumerable set. Lemma 11 The utility function    V˜ i (c) = φi Ui (c(ω))P (dω) μi (dP ) Δ(Ω)

Ω

is Mackey continuous in X. Proof. Remember V˜ i is a concave function, so −V˜ i is convex. The Mackey continuity follows from Proposition 2 section 3.3.2 in Aubin (1982), because −V˜ i ≤ 0. Proposition 5: Proof. If we consider the utilities V˜i (c) = Eμi {φi (EP [Ui (c)])} , 14

I ∗ by Lemma 11, Bewley (1972) gives that there exists ((¯ ci )i∈I , p¯) ∈ X+ ×X+ such that:  (¯ ci − ei ) = 0; i

and,

. / c¯i = arg max V˜ (ci ) : p¯(ci − ei ) ≤ 0, ci ≥ 0 .

Define k¯i by k¯0i := 0, k¯ti := c¯it − e¯it , t ≥ 1 and q¯ by q¯0 := 0, q¯t := p10 p¯t t ≥ 1. We claim that ((¯ ci , k¯i )i∈I , q¯) is equilibrium with penalty for suitable i ¯i c , k ) is in the constraint (1) by ¯ p, c¯i − ei  = 0 we (Mi )i∈I . Note that (¯ ( ¯i ( since i i i i i ¯ get c¯0 − e0 + ¯ q , k  = 0, furthermore i k = i (¯ c − e ). Now, consider an arbitrary (ci , k i ) satisfying the constraint (1). As cit − dit = eit + kti , multiplying both sides by p¯t and summing over t > 0 we get     p¯t cit − p¯t dit = p¯t eit + p¯0 q¯t kti , t>0

t>0

t>0

t>0

by summing over all nodes and rearranging we get p, ei  + ¯ p, di .

¯ p, ci  = ¯ Denote by r¯ = ¯ p, ei  and

. / p, ci  ≤ r . ψi (r) = max V˜i (ci ) : ci ∈ X+ and ¯

Since ψi is concave (see Lemma 8) if r < r ψi (r) − ψi (r ) ≥ −D+ ψi (r)(r − r) where D+ ψi (r) denotes the derivative from the right of ψi at r. By the first order conditions (Lemma 6) we get that   

 β t ui (cit (ω t ))Eμi φi EP Ui (ci ) P (ω t ) = λi p¯t (ω t ), ∀ω t ∈ Ωt , so, by Lemma 10 we get qt (ω t ) ≤ Li β t Pi (ω t ), where the constant incorporates p0 . Therefore, by this bound and the concavity of φi and ψi . $ & . $ & '%/ '%/ β t ui (¯ β t (ui (ci ) − Mi dit ) Vi (¯ ci ) − Vi (ci ) = Eμi φi EP ci ) − E μi φ i E P . $ & . $ & '%/ '%/ ci ) − E μi φi E P ≥ E μi φi E P β t ui (¯ β t ui (ci ) . $ & & '/ '% β t ui (ci ) EP β t Mi dit + Eμi φi EP & ' ≥ψi (¯ r) − ψi (¯ r + ¯ p, di ) + φi (Ui (¯ e))Mi EPi β t dit & ' ≥ −D+ ψi (¯ r) ¯ p, di  + φi (Ui (¯ e))Mi EPi β t dit & ' β t dit , ≥ (φi (Ui (¯ e))Mi − D+ ψi (¯ r)Li ) EPi thus, if Mi ≥

D+ ψi (¯ r )Li φi (Ui (¯ e)) ,

((¯ ci , k¯i )i∈I , q¯) is an equilibrium with penalties. 15

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