Financial Complexity and Trade

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Feb 3, 2016 - We study the effect of complexity of the asset structure on the occurence .... 2006 led to the freezing of the market for asset-backed commercial ...
Financial Complexity and Trade∗ Spyros Galanis† February 3, 2016

Abstract Can frictions generated by financial complexity be so severe that they lead to a complete shutting down of trading activity, such as the one that happened during the financial crisis of 2007? We consider a worst-case scenario, where each security is so complex that any investor can understand and trade at most one. We show that even with this extreme complexity, trading activity will not stop, as long as the indifference curves of investors are not kinked. Our result is robust to all less extreme definitions of complexity, as long as they allow investors to trade at least one security. JEL-Classifications: D70, G01, Keywords: Incomplete Markets, Financial Complexity, Financial Crises, Agreeable Bets, Agreeable Trades, No Trade, Betting, Ambiguity Aversion.

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Introduction

An implicit assumption when modelling financial markets is that each investor is able to understand and simultaneously trade any number of the available securities. In reality, however, frictions do exist. A reason is that financial innovation has led to the emergence of increasingly complicated securities, whose understanding requires time, cognitive effort and special training, which are usually in short supply. Moreover, information acquisition about past performance of complex securities may be too costly. ∗

I am grateful to Stelios Kotronis for numerous discussions on this topic. I also thank for their useful comments Hector Calvo-Pardo, Martin Cripps, Thomas Gall, Itzhak Gilboa, Luca Rigotti, Anastasios Karantounias, Alessandro Mennuni and participants at the Essex workshop on ambiguity and its implications in Finance and Macroeconomics. † Department of Economics, University of Southampton, Southampton, UK, [email protected].

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The Economic Affairs Committee (2009) reports that “It is hard for investors to evaluate complex financial instruments, because difficult risk modelling is required, and because they are often unaware of the details of the asset pool which backs financial securitisations”. C´el´erier and Vall´ee (2014) show that financial complexity is increasing over time and that issuers who target low income investors offer more complex securities than issuers who target high income investors. As a result, frictions that arise due to financial complexity do have an effect on how many securities each investor trades. Polkovnichenko (2005) reports data from the Survey of Consumer Finances, showing that many households invest significant fractions of their wealth simultaneously in well-diversified mutual funds and in un-diversified portfolios of individual stocks. Nieuwerburgh and Veldkamp (2010) derive optimal under-diversification in a framework with costly information acquisition. Carlin et al. (2013) show experimentally that complexity makes subjects less inclined to trade. The obvious negative impact of not being able to simultaneously trade any subset of the available securities is that the ability of financial markets to allocate resources efficiently is restricted. However, is it possible that the effect of complexity is so large that opportunities for trade cease to exist completely? In such a case, other market functions, such as aggregating information and providing risk-sharing opportunities, are hampered as well. Therefore, when considering introducing complex securities, these potential costs need to be compared against their benefits, such as their ability to complete inherently incomplete markets (Tufano (2004)). We study the effect of complexity of the asset structure on the occurence of trade in an environment with incomplete markets and general convex preferences, by characterizing the existence of trading in terms of the investors’ beliefs, both in the standard case of no complexity, where each investor is able to trade any subset of the available securities, and in the case of maximum complexity, where all securities are so complicated that each investor can trade at most one. Holding the set of available securities and the initial allocation fixed, if there is trade in the former environment, will there be trade in the latter? The answer depends on how investors perceive risk and uncertainty. In particular, we find that for smooth preferences which do not allow for kinked indifference curves (e.g. expected utility, smooth ambiguity, multiplier and mean-variance preferences), trading will not stop because of complexity. More importantly, because we are always able to find a trading opportunity where each investor trades at most one security, this result is robust to any less extreme definition of complexity, as long as it allows each investor to trade at least one security. However, we also find that complexity can completely shut down trade if the indifference curves of at least one investor have

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kinks, as it is the case with maxmin expected utility, for example. To illustrate our approach, we describe the asset structure using a partition P of state space S, which describes the uncertainty about tomorrow. Each available security pays 1 if a particular event in P occurs and 0 otherwise. Hence, the finest partition, which contains all states in S, generates a complete markets structure, consisting of all Arrow-Debreu securities. If P is coarser, it generates an incomplete markets structure. The safe asset is generated by having a unit of each available security. In the absence of complexity, we show in Theorem 1 that there is trade (i.e. the initial allocation is not constrained Pareto optimal) if and only if the investors do not agree on the probability they assign to all events in P, according to one of their subjective beliefs. The subjective beliefs, which are revealed by market behavior and are defined by Rigotti et al. (2008) (RSS), depend on the initial endowment and are not necessarily part of the utility representation of preferences. In the presence of maximum complexity, we show in Theorem 2 that there is trade if and only if an investor’s maximum subjective belief about an event in P is lower than another investor’s minimum subjective belief. To provide a graphical illustration, consider Figure 1, where partition P of state space S (with at least three states) contains three elements, E1 , E2 and E3 . The triangle represents the probability simplex, so that each point represents a probability on {E1 , E2 , E3 }. The economy has two investors and an incomplete markets structure with three securities, where security k = 1, 2, 3 pays 1 if Ek occurs and 0 otherwise.1 Investor i’s subjective beliefs, restricted to partition P, are given by the rhombus, whereas investor j’s are given by the circle. Theorem 1 specifies that these sets are disjoint if and only if there is trade in the absence of complexity. Because the rhombus and the circle are disjoint, there is trade in the absence of complexity. The discontinuous lines denote i’s maximum and minimum subjective beliefs about E1 , E2 and E3 , whereas the shaded area containing the rhombus consists of all probabilities that agree with these bounds. Similarly, the shaded area containing the circle consists of all probabilities that agree with j’s maximum and minimum subjective beliefs about E1 , E2 and E3 . Theorem 2 specifies that the shaded areas are disjoint if and only if there is trade in the presence of maximum complexity. Because they intersect, there is no trade in the presence of maximum complexity. It is straightforward that the latter type of disagreement of subjective beliefs implies the former, hence trade in the presence of maximum complexity implies trade in the absence of complexity. As shown in Figure 1, however, the reverse direction 1

If S consists of three states, sk , k = 1, 2, 3, and Ek = sk , then we are in a complete markets structure.

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E1

E2

E3

Figure 1: Trade occurs only in the absence of complexity does not hold, which means that as complexity increases, trade may eventually stop completely. Nevertheless, the two definitions of disagreement are equivalent if each investor’s set of subjective beliefs consists of a single probability measure, thus excluding kinked indifference curves. This observation allows us to obtain a dichotomy of models with convex preferences (including most models with ambiguity aversion), in terms of whether complexity impedes trade, using the results on subjective beliefs provided by RSS. In particular, trade cannot stop due to complexity for preference models that generate unique subjective beliefs and therefore do not allow for kinked indifference curves. Examples are subjective expected utility, the smooth variational preferences of Maccheroni et al. (2006) (including, as special cases, the mean-variance preferences of Markowitz (1952) and Tobin (1958) and the multiplier preferences of Hansen and Sargent (2001)), the smooth ambiguity of Klibanoff et al. (2005) and the second-order expected utility of Ergin and Gul (2009). The models where subjective beliefs given an allocation are not necessarily unique and therefore complexity can impede trade are the choquet expected utility (CEU) with convex capacity of Schmeidler (1989), the maxmin expected utility (MEU) of Gilboa and Schmeidler (1989), the non-smooth variational preferences of Maccheroni et al. (2006), the confidence preferences of Chateauneuf and Faro (2009) and the uncertainty averse preferences of Cerreia-Vioglio et al. (2011). For instance, in the MEU model

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with a full insurance allocation, each investor’s subjective beliefs coincide with his (non unique) priors and his indifference curve has a kink at his endowment. As shown in Figure 1, in that case complexity completely shuts down trade. This dichotomy provides behavioral implications. Suppose that we observe trade in an environment with a less extreme definition of complexity, allowing investors to trade up to k > 1 securities. If, by increasing the complexity of the asset structure (by decreasing k), we eventually observe that trading stops, it must be that the investors’ preferences are from the second set. Alternatively, if we know the preference model, we have a test of complexity. Brunnermeier and Oehmke (2010) argue that a definition of complexity is not straightforward: “while CDOs are viewed by most as highly complex, equity shares of financial institutions, whose payoff structures are even more complicated, are often seen as less complex”. Part of the difficulty is that a security may be considered simple during normal times, where past prices provide a good indication of future payoffs, but extremely complex during a crisis, where, for example, an investor needs to calculate how the hundreds of thousands of mortgages included in the CDO will perform. Suppose that financial crises generate a lot of uncertainty (Caballero and Simsek (2013), Brunnermeier and Sannikov (2014)) and that investors have MEU preferences. If we know that during the crisis the investors’ priors do not change significantly but we nevertheless observe a market freeze, this is an indication that the investors suddenly realize that the asset structure of this market is complex. To provide an example, Acharya et al. (2009) describe how a series of events that was triggered by an unexpected decrease of the US house prices in the first quarter of 2006 led to the freezing of the market for asset-backed commercial paper in 2007, right after BNP Paribas announced that it was suspending redemptions from its structured investment vehicles, which were trading these types of securities. We can interpret the decline of US house prices as an event that created uncertainty about fundamentals, thus generating MEU preferences. However, these preferences were not sufficient for shutting down trade. This happened one year later, exactly when the suspension of redemptions informed everyone that asset-backed commercial paper was no longer easy to price and value. That this second event triggered an immediate suspension of trade can be explained by investors realising that these securities have a complex payoff structure and therefore cannot simultaneously trade any subset of them. Brunnermeier and Oehmke (2010) also argue that identifying complex securities is important in establishing whether they should be regulated subject to an FDAtype approval system, or limit who is able to invest in them. A case in support of policy intervention is depicted in Figure 1. Because of complexity, there is no trade,

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however in the absence of complexity there would be gains from trade, as the two sets of subjective beliefs are disjoint. The policy intervention could take the form of regulation that forces the issuers to make securities simple, or of not allowing investors to trade securities that may be viewed as complex during crises. Our paper is related to Billot et al. (2000, 2002), who characterize trading in terms of disjoint sets of priors in a complete markets setting with a full insurance allocation, using the MEU and CEU models. RSS generalize these results for all convex preferences, encompassing many models with ambiguity averse preferences. Ghirardato and Siniscalchi (2015) analyze the case of non-convex preferences. Additionally, RSS characterize trading from any initial allocation at a (not necessarily full insurance) allocation. We generalize this approach, by providing with Theorem 1 a similar characterization in an incomplete markets setting. In the MEU model with two investors and a full insurance allocation, Kajii and Ui (2006) characterize trading in the case of a partition with two elements, which in our terminology is an incomplete markets structure with only one security and maximum complexity. In the special case where each investor’s set of priors is the core of a convex capacity, they also characterize trade in an economy with more than one security and effectively show that trade in the absence of complexity is equivalent to trade in the presence of maximum complexity. Dominiak et al. (2012) extend this result for the CEU model with not necessarily convex capacities. We improve on these results with Theorem 2, by providing a characterization for any finite number of investors with general convex preferences and any initial allocation, as well as for partitions with more than two elements, hence for asset structures with more than one security. The only condition that we impose is that of no arbitrage, which has clear economic content. More importantly, our characterization applies irrespective of whether trade in the absence of complexity is equivalent to trade in the presence of maximum complexity, which is crucial in separating the models in terms of whether they satisfy this property. Our definition of complexity relates to how investors perceive the payoff structure. Alternatively, Caballero and Simsek (2013) use ambiguity and the notion of complexity about the structure of cross exposures of banks to explain market freezes. Rigotti and Shannon (2005, 2012) show that generic determinancy is a robust feature of general equilibrium models with ambiguity averse preferences, because kinks are relatively rare, whereas robust indeterminacies arise naturally in the incomplete preferences model of Knightian uncertainty of Bewley (1986), where kinks are prevalent. This model is used also by Easley and O’Hara (2010) to explain no trade.2 2

Although our results apply only to complete convex preferences, it would be straightforward to extend them in the incomplete preference model of Bewley (1986). In this model each indifference curve has a kink

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Our main difference from these papers is that we use complexity, and the subsequent “enlargement” of beliefs, in order to explain no trade in an environment where there are actually gains from trade. This difference has policy implications, because we suggest that if complexity was lifted then there would be gains from trade, whereas the aforementioned papers suggest that uncertainty has destroyed all gains from trade. Moreover, our mechanism can help explain the BNP Paribas story, where the market froze not because of the initial event that created uncertainty (and hence ambiguity or incompleteness) about house prices, but due to the further enlargement of beliefs, caused by the investors’ realization that the payoff structure was too difficult to understand. The paper is organized as follows. Section 2 introduces the model, the subjective beliefs and the two notions of common beliefs. Section 3 characterizes trading in the absence of complexity, whereas Section 4 characterizes trading in the presence of maximum complexity. Section 5 concludes.

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Set up

Consider a set I of investors with typical element i and a single consumption good. There are two periods, 0 and 1. Uncertainty about period 1 is represented by a finite set of payoff relevant states S, with typical element s. An event E is a nonempty subset of S. The set of consequences is R, interpreted as monetary payoffs. The set of acts is F = RS with the natural topology. An act f is strictly positive if f (s) > 0 for all P p(s)f (s) be the expectation of f given measure p ∈ ∆S. s ∈ S. Let Ep f = s∈S

We assume throughout that each preference relation %i on F satisfies the following standard axioms. Axiom 1. (Preference). %i is complete and transitive. Axiom 2. (Continuity). For all f ∈ F, the sets {g ∈ F : g %i f } and {g ∈ F : f %i g} are closed. Axiom 3. (Strict Convexity). For all f 6= g and α ∈ (0, 1), if f %i g, then αf + (1 − α)g i g. Axiom 4. (Strong Monotonicity). For all f 6= g, if f ≥ g, then f i g. at the endowment, hence maximum complexity would shut down trade.

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2.1

Subjective beliefs

Based on Yaari (1969), RSS define investor i’s subjective beliefs at an act f to be the set of all (normalized) supporting hyperplanes of f , π i (f ) = {p ∈ ∆S : Ep g ≥ Ep f for all g %i f }. For convex preferences, π i (f ) is nonempty, convex and compact. RSS define two other sets of beliefs. The first contains the subjective beliefs revealed by unwillingness to trade at f , πui (f ) = {p ∈ ∆S : f %i g for all g such that Ep g = Ep f }. We can interpret these beliefs as the set of (normalized) Arrow-Debreu prices such that the investor would optimally not be willing to trade his endowment f . For the second set, let E i (f ) denote the collection of all compact, convex sets P ⊆ ∆S such that if Ep g > Ep f for all p ∈ P , then g + (1 − )f i f for sufficiently small . Then, the subjective beliefs revealed by willingness to trade at f are defined to be T i i (f ) = πw E (f ). RSS show that for convex preferences and strictly positive acts f , i (f ). π i (f ) = πui (f ) = πw

2.2

Common beliefs

In a complete markets setting, RSS show that allocation {ei }i∈I ∈ RSI is Pareto T i i π (e ) 6= ∅. We provide two optimal if and only there is a common belief, so that i∈I

successively weaker notions of a common belief, given a collection of events P ⊆ 2S \ ∅. Let i πP (ei ) = {q ∈ ∆S : q(E) = p(E) for all E ∈ P, for some p ∈ π i (ei )}

be the set of probability measures that agree on all events in P, according to one of investor i’s subjective beliefs at ei ∈ RS . We say that there is a P-common belief if the investors agree on each event E ∈ P, according to one of their subjective beliefs. T i i Definition 1. There is a P-common belief at {ei }i∈I if πP (e ) 6= ∅. i∈I

A common belief implies a P-common belief but the reverse is not true. However, for a large enough collection P (for example containing all states s ∈ S), a P-common belief implies a common belief. In Section 3, we show that there is no P-common belief if and only if the allocation is not constrained Pareto optimal relative to P, so that there is trade in the absence of complexity.

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Given an event E and a set π i (ei ) of subjective beliefs, let pi (E) = min p(E) p∈π i (ei )

and

pi (E)

= max p(E) be i’s minimum and maximum subjective belief about E, p∈π i (ei )

respectively. A weaker notion of a common belief is provided if we let π iP (ei ) = {q ∈ ∆S : pi (E) ≤ q(E) ≤ pi (E) for all E ∈ P} be the set of probability measures that are within i’s minimum and maximum subjective beliefs at ei ∈ RS , for each event E ∈ P.3 Definition 2. There is a P-common belief at {ei }i∈I if

T i∈I

π iP (ei ) 6= ∅.

If there is a P-common belief then there is a P-common belief but the reverse is not true, even if P contains all possible events. However, if π i (ei ) is a singleton for each i, then the two notions are equivalent. In Section 4, we show that there is no P-common belief if and only if there is trade in the presence of maximum complexity.

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Absence of complexity

Consider a standard incomplete markets setting.4 In period 0, there is no uncertainty and the traders buy or short sell any combination of the available securities. These securities pay in period 1, where there is uncertainty, described by state space S. The available securities are described by a collection P of nonempty events of S. In particular, for every event E ∈ P, there exists a security fE that pays 1 if E occurs and 0 otherwise. We denote its price by qE and the price vector by q = {qE }E∈P . i } P i Let y i = {yE E∈P ∈ R denote i’s portfolio of securities. Portfolio y and prices P P i f , where ei = {ei , ei } ∈ R × RS is i’s i q )1 + ei + yE q generate act (ei0 − yE E E 0 1 1 E∈P E∈P P i endowment and 1 is an S × 1 vector of 1’s. Investor i pays ( yE qE )1 in period 0 to E∈P P i i > 0 (y i < 0) then i has obtain this portfolio and receives yE fE in period 1. If yE E E∈P

positive (negative) net demand for security fE . The main simplifying assumption we make is that there is no discounting. This is consistent with other papers (e.g. Billot et al. (2000), RSS), which provide a (complete markets) two period model with no discounting.5 In particular, we assume that investor i with preference relation %i on RS weakly prefers consumption vector x = {x0 , x1 } ∈ Note that π iP (f ) is a closed and convex polytope, as it is bounded and the intersection of half spaces. See, for example, the financial market described in Chapter 2 of Magill and Quinzii (1996). 5 In these models there is no consumption in period 0. Here we allow for consumption in period 0 because this is when investors buy or short sell securities. 3

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R×RS over consumption vector z = {z0 , z1 } ∈ R×RS if and only if x0 1+x1 %i z0 1+z1 . In other words, irrespective of what x1 is, in period 0 he is indifferent between getting x0 in period 0 and x0 in every state of period 1. We assume that P is a partition of S. An economy is a tuple h{%i }i∈I , {ei }i∈I , Pi, where {ei }i∈I = R(S+1)I is the allocation of endowments and P specifies the available securities. A portfolio allocation is a vector y = {y i }i∈I . Definition 3. Given economy h{%i }i∈I , {ei }i∈I , Pi, an equilibrium {y, q} consists of portfolio allocation y and prices q such that we have • Investor optimization: For each i ∈ I, for all x ∈ RP , (ei0 −

X E∈P

i yE qE )1 + ei1 +

X

i yE fE %i (ei0 −

E∈P

• Market clearing: For each E ∈ P,

X

xE qE )1 + ei1 +

E∈P

P i∈I

X

xE fE .

E∈E

i = 0. yE

An allocation {xi }i∈I = R(S+1)I specifies a consumption xi0 ∈ R at period 0 and consumption xi1 ∈ RS in period 1, for each investor. Let e ∈ R+ ×RS++ be the aggregate endowment. We next define feasibility and P-constrained Pareto optimality. P i x = e. It is P-feasible if it is Definition 4. An allocation {xi }i∈I is feasible if i∈I P i yE fE , for each i ∈ I. feasible and there exists y i ∈ RP with xi1 = ei1 + E∈P

Definition 5. An allocation {xi }i∈I is P-constrained Pareto optimal if it is P-feasible and there does not exist another P-feasible allocation {z i }i∈I such that z0i 1 + z1i %i xi0 1 + xi1 for all i ∈ I and z0i 1 + z1i i xi0 1 + xi1 for some i ∈ I. We say that an allocation {ei }i∈I ∈ R(S+1)I is interior if ei is strictly positive for each i ∈ I. There is trade in the absence of complexity if and only if the interior allocation {ei }i∈I ∈ R(S+1)I is not P-constrained Pareto optimal. The following theorem shows that there is trade if and only if the investors agree on the probability they assign to each event in P, according to one of their subjective beliefs. Theorem 1. There is trade in the absence of complexity if and only if there is no P-common prior. Proof. Consider economy h{%i }i∈I , {ei }i∈I , Pi and let π i = π i (ei0 1 + ei1 ) be i’s subi = π i (ei 1 + ei ). Note first that an equilibrium jective beliefs at ei . Similarly, let πP 1 P 0

always exists for this one-good economy with an interior allocation (see Theorem 10.5 in Magill and Quinzii (1996) and subsequent remark). For each equilibrium {y, q 0 }, we

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q0 PE 0 qF

, for each E ∈ P. This implies that there exists P probability measure xq ∈ ∆S such that xq (s) = qE for each E ∈ P. normalise prices so that qE =

F ∈P

s∈E

Suppose that {ei }i∈I is P-constrained Pareto optimal. We first show that at any P P i i ≤ equilibrium {y, q} (with normalised prices), we have p(E)zE zE qE for some E∈P

E∈P

i } P p ∈ π i , for all i ∈ I and all {zE E∈P ∈ R . i > p(E)zE

i q for all zE E E∈P E∈P P i f − ∈ RP . This implies that Ep (ei1 + zE E

Suppose that at some equilibrium {y, q} we have

P

P

i } p ∈ π i , for some i ∈ I and {zE E∈P E∈P P i i , there exists k ∈ ( zE qE )1) > Ep ei1 for all p ∈ π i . From the definition of πw E∈P P i P i (0, 1) such that k(ei1 + zE fE − ( zE qE )1) + (1 − k)ei1 i ei1 , which implies ei1 + E∈P E∈P P i P i kzE fE − ( kzE qE )1 i ei1 . Because we are in equilibrium {y, q}, we have ei1 + E∈P E∈P P i P P i P i i q )1 %i ei + kzE qE )1 i ei1 and for j 6= i we kzE fE − ( yE yE fE − ( E 1 E∈P E∈P E∈P E∈P P j P j yE qE )1 %j ej1 . Because the allocation generated by {y, q} is yE fE − ( have ej1 + E∈P

E∈P

P-feasible, {ei }i∈I is not constrained Pareto optimal. Because P provides a partition of S, we can construct state space S P = {sE1 , sE2 , . . . , sEn } where Ek ∈ P, k = 1, . . . , n and n is the number of elements in P. Consider the following mapping from ∆S to ∆S P . Each p ∈ ∆S maps to pP ∈ ∆S P , where i ∈ ∆S P accordingly. Note pP (sEk ) = p(Ek ), k = 1, . . . , n. Each set π i is mapped to πP i . Map x ∈ ∆S, which represents the that because π i is closed and convex, so is πP q

prices of the securities, to qP ∈ ∆S P . P i i } For each {zE E∈P ∈ R , let zsE

P i ≤ i , k = 1, . . . , n. Then, p(E)zE = zE k k E∈P P P P i i . zE qE for some p ∈ π i implies that pP (s)zsi ≤ qP (s)zsi for some pP ∈ πP P P E∈P s∈S s∈S P P This implies that min pP (s)zsi ≤ qP (s)zsi . Because any {zsi }s∈S P ∈ RP can i pP ∈πP s∈S P

s∈S P

i } P be constructed from some {zE E∈P ∈ R , probability measure qP cannot be separated i , hence q i from closed and convex set πP P ∈ πP . This is true for all i ∈ I, hence T i qP ∈ πP 6= ∅. i∈I T i Conversely, suppose that πP 6= ∅ but {ei }i∈I is not P-constrained Pareto optii∈I

mal. Because any allocation resulting from some equilibrium {y, q} is P-constrained P i Pareto optimal (Theorem 12.3 in Magill and Quinzii (1996)), we have ei1 + yE fE − E∈P P i P P j j ( yE qE )1 %i ei1 for all i ∈ I and ej1 + yE fE − ( yE qE )1 j ej1 for some E∈P

E∈E

E∈P

j ∈ I. From the definition of πui and Axiom 4, we have that for all i ∈ I, for all p ∈ π i , P i P i Ep (ei1 + yE fE − yE qE ) ≥ Ep ei1 , with strict inequality for j. Rearranging we have E∈P

E∈P

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P i P i that, for all i ∈ I and all p ∈ π i , yE p(E) ≥ yE qE , with strict inequality for j. E∈P E∈P T i Let p ∈ πP 6= ∅. Then, there exists pi ∈ π i , for each i ∈ I, such that p(E) = pi (E) i∈I P P i P P i for each E ∈ P. By adding over all i we have p(E) yE > qE yE . BeE∈P i∈I E∈P i∈I P i cause yE = 0 for each E ∈ P, we have 0 > 0, a contradiction. Hence, {ei }i∈I is i∈I

P-constrained Pareto optimal.

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Maximum complexity

Consider now a more restrictive setting, where the set of available securities is still the same and determined by partition P, however each investor is allowed to buy or sell at most one security, instead of any linear combination of securities, as in the standard incomplete markets setting of the previous section. We say that a trade is a tuple {f i }i∈I , where f i ∈ RS and

P

f i = 0. It is a bet on

i∈I

P if for each i ∈ I there exist E i ∈ P and ai , bi ∈ R, ai 6= bi , such that f i (s) = ai if s ∈ E i and f i (s) = bi otherwise. To compare it with the setting of the previous section, bet {f i }i∈I specifies that it is as if i buys yE i units of security fE i (that pays 1 if E i occurs and 0 otherwise) at price qE i , such that ai = yE i − yE i qE i and bi = −yE i qE i . i Let {ei }i∈I ∈ RSI ++ be the initial allocation, where each e is strictly positive. We

assume that ei = ei0 1 + ei1 , where {ei0 , ei1 } is i’s allocation in Section 3. A trade {f i }i∈I is agreeable if ei + f i %i ei , for each i ∈ I, and ej + f j j ej , for some j ∈ I. An agreeable bet on P is an agreeable trade which is a bet on P. We make the assumption that |I| ≥ |P| + 2. Although we do not employ the notion of an equilibrium, we impose a no arbitrage condition. Definition 6. Tuple {f i }i∈K is an arbitrage trade if {−f i − α}i∈K is a trade for some α > 0 and ei + f i i ei for all i ∈ K ⊆ I. It is an arbitrage bet if {−f i − α}i∈K is also a bet on P. The condition that {−f i − α}i∈K is a trade for some α > 0 simply specifies that P

− f i (s) > 0 for all s ∈ S, hence taking the opposite side of {f i }i∈K yields a positive

i∈K

payoff always. The second condition specifies that each i ∈ K would accept taking f i . Hence, the existence of an arbitrage trade means that an investor can profit without incurring any risk or uncertainty.

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We assume that either there are no arbitrage trades, or the economy is large and there are no arbitrage bets. An economy is large if for each investor i with preference relation %i , there are at least n−1 investors j with identical preference relation %i =%j , where n is the number of the partition cells of P. Hence, our no arbitrage condition implies that that either (at least) some investors are smart enough to understand an arbitrage opportunity that involves more than one security, or the economy is large and there are no arbitrage bets. Proposition 1 in the Appendix identifies the consequences, in terms of beliefs, of no arbitrage. We say that there is trade in the presence of maximum complexity if there is an agreeable bet on P. Note that an agreeable trade exists if and only if {ei }i∈I is not Pareto optimal. Moreover, if an agreeable bet on P exists, then {ei }i∈I is not P-constrained Pareto optimal. This is intuitive, because the incomplete markets setting of the previous section allows investors to buy and sell any subset of available securities, whereas the current setting restricts each investor to trade at most one security. However, the more interesting question is the reverse. If {ei }i∈I is not Pconstrained Pareto optimal, does there exist an agreeable bet on P? In other words, if we observe trade in the absence of complexity, would we also observe trade in the presence of maximum complexity? The following theorem answers this question, by characterizing the existence of trade in the presence of maximum complexity in terms of a P-common prior. Theorem 2. There is trade in the presence of maximum complexity if and only if there is no P-common prior. T i i T i [p (E), pi (E)] = ∅ for some Proof. Suppose that π P (e ) = ∅. We first show that i∈I i∈I T i [p (E), pi (E)] 6= ∅. E ∈ P. Suppose not, so that for each E ∈ P, [q(E), q(E)] = i∈I

Because there are no arbitrage opportunities, Proposition 1 in the Appendix implies P P P P q(E). Let a > 0 such that a q(E)+(1−a) q(E) = 1. For that q(E) ≤ 1 ≤ E∈P E∈P E∈P E∈P P each E ∈ P, let p(E) = aq(E) + (1 − a)q(E). We then have p(E) = 1, so that there E∈P

exists p0 ∈ ∆S such that p0 (E) = p(E) for all E ∈ P. Moreover, p0 (E) ∈ [q(E), q(E)] T i i for all E ∈ P, contradicting that π P (e ) = ∅. i∈I T We therefore have that, for some i, j ∈ I and E ∗ ∈ P, [pi (E ∗ ), pi (E ∗ )] [pj (E ∗ ), pj (E ∗ )] = ∅. Suppose without loss of generality that pi (E ∗ ) > pj (E ∗ ). Let c be such that pi (E ∗ ) > c > pj (E ∗ ). Define act f such that f (s) = a if s ∈ E ∗ and f (s) = b otherwise, where a = 1 − c and b = −c. Note that a − b > 0. We then have that Ep f = p(E ∗ )(1 − c) − (1 − p(E ∗ ))c > 0 for all p ∈ π iP (ei ) and Ep (−f ) = p(E ∗ )(−1 + c) + (1 − p(E ∗ ))c > 0 for all p ∈ π jP (ej ).

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Let PI be a partition of the set I \ {i, j} of investors, with |P| elements. Because we have assumed that |I| ≥ |P| + 2, each element of the partition is nonempty. Denote an element of that partition as IE , where E ∈ P. Construct tuple {g k }k∈I such that g i (E) = f (E) −  for all E ∈ P, where  > 0 and g j = −f . For any other investor P k g = 0. k ∈ I \ {i, j}, set g k (E) = |IE | if k ∈ IE and g k (E) = 0 otherwise. Note that k∈I

For small enough  > 0 we have Ep g i > 0 for all p ∈ π iP (ei ) and Ep g j > 0 for all i (ei ), there exist small enough λi , λj > 0 such that p ∈ π jP (ej ). From the definition of πw

λi (ei + g i ) + (1 − λi )ei = ei + λi g i i ei and ej + λj g j j ej . By taking 0 < λ < λi , λj , we have that ek + λg k k ek for all k ∈ I, where the claim is true for k 6= i, j because P k λg = 0, {λg k }k∈I is an agreeable bet on P. of Axiom 4. Because k∈I

Conversely, suppose there is an agreeable bet {f0i }i∈I on P. This means that P i∈I

f0i = 0 and each f0i is of the form f0i (s) = ai if s ∈ E i ∈ P, f0i (s) = bi otherwise,

where ai 6= bi . Moreover, f0i + ei %i ei for all i ∈ I, f0j + ej j ej for some j ∈ I. Using Axioms 2, 4 and the fact that each ei is a strictly positive act, we can find an agreeable bet {f i }i∈I (by reducing f0j at all states and distributing the difference to all other investors) where f i (s) = ai if s ∈ E i ∈ P, f i (s) = bi otherwise, where ai 6= bi and f i + ei i ei for all i ∈ I. From the definition of π i (ei ), we have that Ep (ei + f i ) > Ep ei P i for all p ∈ π i (ei ), all i ∈ I, and f = 0. i∈I

We next show that Ep f i > 0 for all p ∈ π iP (ei ) and all i ∈ I. Fix i ∈ I and let E i = E. Note that pi (E)ai + pi (E c )bi > 0 and pi (E)ai + pi (E c )bi > 0, because 1 − pi (E) = pi (E c ) and 1 − pi (E) = pi (E c ). Take p ∈ π iP (ei ). If p(E) = p(E) or p(E) = p(E) then we are done. Suppose p(E) < p(E) < p(E). Then, there exist q1 , q2 ∈ π i (ei ) such that 21 q1 (E) + 12 q2 (E) = p(E), which implies that 12 q1 (E c ) + 21 q2 (E c ) = p(E c ). We also have that q1 (E)ai + q1 (E c )bi > 0 and q2 (E)ai + q2 (E c )bi > 0. Multiplying with

1 2

and adding the two inequalities we have that Ep f i = p(E)ai + p(E c )bi > 0. Finally, we show that Ep f i > 0 for all p ∈ π iP (ei ) and all i ∈ I implies that there is no P-common prior. Construct state space S P = {sE1 , . . . , sEK }, where Ek ∈ P, k ∈ K. Let |S P | = n. For each i and π iP (ei ) ⊆ ∆S, define π iS P (ei ) ⊆ ∆S P such that if p ∈ π iP (ei ) then pS P ∈ π iS P (ei ), where pS P (sE ) = p(E), for each E ∈ P. T Because each π iP (ei ) is a polytope, so is each π iS P (ei ). Moreover, π iS P (ei ) = ∅ if and i∈I T i i only if π P (e ) = ∅. The main Proposition in Samet (1998) states that there exist i∈I P i P fS1P , . . . , fSnP ∈ RS , such that Ep fSi P > 0 for each p ∈ π iS P (ei ) and fS P = 0, if and i∈I T i only if π S P (ei ) = ∅. If we set {f i }i∈I such that f i (s) = fSi P (sE ) if s ∈ E, then we i∈I P T have that Ep f i > 0 for all p ∈ π iP (ei ) and f i = 0 if and only if π iP (ei ) = ∅. Hence, i∈I

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i∈I

there is no P-common prior.

Combining Theorems 1 and 2, we can conclude that if no P-common prior implies no P-common prior, then trade in the absence of complexity implies trade in the presence of maximum complexity. As explained in Section 1, this is not true in general, however a sufficient condition (satisfied by several models, such as subjective expected utility, multiplier, mean-variance and smooth ambiguity preferences) is that each investor has a unique subjective belief at the initial allocation, hence his indifference curve has no kinks.

5

Concluding remarks

We examine trading in two separate environments. The first is the standard incomplete markets setting, where each investor can trade any subset of securities. The second depicts an extreme case of an asset structure, where each security is so complex that an investor can trade at most one. We first show that for a wide variety of preference models (e.g. expected utility, multiplier and mean-variance preferences, smooth ambiguity), maximum complexity does not destroy all trading opportunities. More importantly, the same conclusion would hold even if we were to provide a more detailed but (necessarily) weaker definition of complexity (e.g. generating complexity endogenously through cognitive costs or computational complexity), as long as each investor can trade at least one security. In other words, our result for these preferences that complexity does not impede trade is powerful, because it is independent of the particular details of how a complex security is defined. For the second set of preference models (e.g. maxmin expected utility, certain classes of variational and uncertainty averse preferences), we find that maximum complexity does impede trade. This means that a policy which simplifies the securities can have a positive welfare effect. An interesting next step would be to examine these welfare effects using a more detailed but weaker definition of complexity. This is relegated to future research.

A

Appendix

In the setting of Section 4, the following proposition identifies the implications of no arbitrage in the presence of maximum complexity, where each investor is allowed to trade at most one security.

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Proposition 1. In the setting of Section 4, suppose that for each E ∈ P, [q(E), q(E)] = T i [p (E), pi (E)] 6= ∅. Suppose that either there are no arbitrage trades, or there are no i∈I P P arbitrage bets and the economy is large. Then, q(E) ≤ 1 ≤ q(E). E∈P

Proof. Suppose that for each E ∈ P, [q(E), q(E)] =

E∈P

T

[pi (E), pi (E)] 6= ∅ and

i∈I

P

q(E)
pi (E) = q(E), we have that Ep (fE,a + ei ) = a − p(E) + ei > Ep ei = ei for all i , there exists small enough k ∈ (0, 1) such that p ∈ π i (ei ). From the definition of πw

k(fE,a + ei ) + (1 − k)ei i ei . Therefore, investor i would strictly prefer to get act kfE,a which pays ka for all s ∈ / E and ka − k if s ∈ E. Moreover, this is also true for any 0 < k0 < k. By repeating the same procedure for each E ∈ P, we can create a tuple {fE,kE aE }E∈P of acts. If the economy is not large, there is the possibility that for the same investor i we have pi (E) = q(E), pi (E 0 ) = q(E 0 ), where E, E 0 ∈ P, and this can be true for more than two partition cells of P. Then, the same arguments show that investor i would strictly prefer to get act kE fE,aE + kE 0 fE 0 ,aE0 . However, in this case i receives a bet that provides different payoffs in E, E 0 and S \ E ∪ E 0 , hence it is a trade and not a bet on a partition cell of P. By setting k = min kE , the new tuple is {kfE,aE }E∈P , where each investor i for E∈P

which q(E) = pi (E) would strictly prefer taking kfE,aE . Taking the opposite side of all P aE ). This is strictly positive of these acts yields an act that pays at all states k(1 − E∈P P because each aE is strictly bigger but arbitrarily close to q(E), where q(E) < 1. E∈P

Hence, we have created an arbitrage opportunity. We can create a similar arbitrage P opportunity if q(E) > 1, with acts that pay −kaE for all s ∈ / E and k − kaE if s ∈ E, i∈I

where q(E) > aE , for small enough k > 0.

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