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Partial ordering of risky choices : anchoring, preference for flexibility and applications to asset pricing Sagi, Jacob S. 2000

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Partial Ordering of Risky Choices: Anchoring, Preference For Flexibility And Applications To Asset Pricing By Jacob S. Sagi 1991 B.Sc. Physics, University of Toronto 1995 Ph.D. Physics, University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In THE FACULTY OF GRADUATE STUDIES FINANCE DIVISION, FACULTY OF COMMERCE We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A May 2000 © Jacob S. Sagi In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Finance Division, Faculty of Commerce The University of British Columbia 2053 Main Mall Vancouver, Canada V6T 1Z2 Date: Abstract This dissertation describes two theories of risky choice based on a normatively ax-iomatized partial order. The first theory is an atemporal alternative to von Neumann and Morgenstern's Expected Utility Theory that accommodates the status quo bias, vi-olations of Independence and preference reversals. The second theory is an extension of the Inter-temporal von Neumann-Morgenstern theory of Kreps and Porteus (1978) that features a normatively deduced preference for flexibility. A substantial part of the thesis is devoted to examining equilibrium implications of the inter-temporal theory. In partic-ular, a multi-agent multi-period Bayesian rational expectations equilibrium is shown to exist under certain conditions. Implications to asset pricing are then investigated with an explicit parameterization of the model. ii Table of Contents Abstract i i Table of Contents i i i List of Figures v 1 Introduction 1 1.1 Additional Literature Review 4 1.1.1 Static Choice Under Risk 5 1.1.2 Non-Expected Utility and Dynamic Choice Under Risk 6 1.1.3 Utility for Flexibility 8 1.1.4 Alternative Asset Pricing Theories 9 2 Anchored Preference Relations: a theory of the status quo bias 10 2.1 Theoretical Foundations 23 2.1.1 Anchored Preference Relations 30 2.2 Further Discussion 35 2.2.1 Other Anomalies: Preference Reversals and Imprecise Certainty Equivalents 35 2.2.2 Indeterminate Anchor 37 2.2.3 Relation to semiorders 39 2.3 Appendix 40 3 Inter-temporal Flexibi l i ty Preference 49 iii 3.1 Introduction 50 3.1.1 Related literature on utility for flexibility 57 3.1.2 Induced Preferences 58 3.2 Theory 60 3.2.1 Formulation of the Choice Problem and Agents'Preferences . . . 60 3.2.2 Time Consistency 67 3.3 Discussion 70 3.3.1 Inter-temporal Flexibility Preferences and Induced Utility Functions 70 3.3.2 First-degree Flexibility Dominance And State Contingent Plans . 75 3.3.3 Relation to Additive Representations and Second-degree Dominance 78 3.4 Proofs 82 3.5 Appendix 85 4 Application to Asset Pricing 91 4.1 A General Equilibrium Model 93 4.2 A Specific Example 98 4.2.1 Further Discussion 100 4.3 Asset Pricing 103 4.3.1 Constant Drift and Diffusion Coefficients 110 4.3.2 Mean Reversion in G and C I l l 4.3.3 Simulations 114 4.4 Proofs 140 4.5 Derivation of State Prices 149 Bibliography 154 iv List of Figures 2.1 EUT indifference surfaces 13 2.2 Anchored 'better-than' sets are parallel wedges. Note that p >-p r, r yNP q and q yNP p 14 2.3 Neither p nor q can be said to 'Pareto-dominate' the other in the Kahne-man and Tversky (1979) example 15 2.4 (a) Indifference surfaces when the anchor is at $2000, and (b) when the anchor is at $1000 16 2.5 The Allais Paradox, anchoring and fanning of indifference surfaces. The shaded picture shows the 'Pareto-dominance' wedge. The 'kink' in indif-ference curves shows up near the low outcome lottery 18 2.6 General Anchored Preferences. The 'better-than' set at the anchor is con-strained to coincide with the 'Pareto-dominance' wedge. Bottom: Better-than sets must contain the wedge 19 3.1 A Dynamic Choice Problem with incomplete preferences 66 4.1 Comparative statics of the riskless rate and equity premium with respect to To and X when the coefficient of relative risk aversion does not vary through time (o~G = —o~c) 116 4.2 Comparative statics of the riskless rate and equity premium with respect to \o~c\ and X when the coefficient of relative risk aversion does not vary through time (<rG = -crc) 117 v 4.3 Comparative statics of the riskless rate and equity premium with respect to \JF and X when the coefficient of relative risk aversion does not vary through time (crG = -crc) 118 4.4 Comparative statics with respect to TQ. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, rT(X). Bottom: the equity premium on the market portfolio, rrM(xy • oM(x) 120 4.5 Comparative statics with respect to r0. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealth-dividend ratio, f ( X ) . Bottom: the normalized differential consumption coupon, d(X) = D/C 121 4.6 Comparative statics with respect to y,c. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instan-taneous riskless rate, rT(X). Bottom: the equity premium on the market portfolio, aM { X ) ' • aM(X) 122 4.7 Comparative statics with respect to \ic. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealth-dividend ratio, f ( X ) . Bottom: the normalized differential consumption coupon, d{X) = D/C 123 4.8 Comparative statics with respect to \o~c\. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instan-taneous riskless rate, rT(X). Bottom: the equity premium on the market portfolio, CJM{X)' • CTM(X) 124 vi 4.9 Comparative statics with respect to \crc\. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealth-dividend ratio, f{X). Bottom: the normalized differential consumption coupon, d(X) = D/C ' 125 4.10 Comparative statics with respect to KG . The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instan-taneous riskless rate, rt(X). Bottom: the equity premium on the market portfolio, aM{X)' • aM(X) 126 4.11 Comparative statics with respect to K ° . The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealth-dividend ratio, f(X). Bottom: the normalized differential consumption coupon, d{X) = D/C 127 4.12 Comparative statics with respect to \xg. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instan-taneous riskless rate, rt(X). Bottom: the equity premium on the market portfolio, crM(X)' • (TM{X) 128 4.13 Comparative statics with respect to ug. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealth-dividend ratio, f(X). Bottom: the normalized differential consumption coupon, d(X) = D/C 129 4.14 Comparative statics with respect to |<rG|. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instan-taneous riskless rate, rt(X). Bottom: the equity premium on the market portfolio, (TM{X)' • <rM(X) 130 vii 4.15 Comparative statics with respect to |cr G | . The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealth-dividend ratio, f(X). Bottom: the normalized differential consumption coupon, d(X) = D/C 131 4.16 Comparative statics with respect to p. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, rt(X). Bottom: the equity premium on the market portfolio, aM^xy . rjM^X) 132 4.17 Comparative statics with respect to p. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealth-dividend ratio, f(X). Bottom: the normalized differential consumption coupon, d{X) = D/C 133 4.18 Instantaneous riskfree rate in the example. Top: plot versus X. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Bottom: unconditional probability distribution 137 4.19 Market risk premium. Top: plot versus X. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Bottom: unconditional probability distribution 138 4.20 Perpetual bond price and price-dividend ratio of aggregate production. Top: plot versus X. The squares represent the 5th, 50th and 95th per-centile values of the state variable, X. Bottom: plot of derivatives versus X 139 viii Acknowledgements I would like to thank my dissertation committee and examiners: Ken MacCrimmon, who sparked my interest in decision theory and encouraged me even when my faith in this research wavered; Ron Giammarino, who contributed his typically unique and thoughtful insight; Tan Wang, who made extremely valuable comments on the theoretical development; Larry Epstein, who selflessly invested a great deal of time reading the thesis and provided many pages of detailed comments in his role as my external examiner; Yoram Halevy, who kindly agreed to act as a University Examiner; Lorenzo Garlappi, who graciously agreed to read and comment on a more awkward and preliminary version of the thesis. I also benefited from interactions with Ed Granirer, Priscilla Greenwood, Ulrich Haussmann, Burton Hollifield, Philip Loewen, Florin Sabac, Bob Sugden, Peter Wakker, and seminar participants at Queen's University, Duke University, Carnegie Mellon Uni-versity, University of British Columbia, UC Berkeley, U C L A and University of Washing-ton in Seattle. Most of all, I would like to thank my friend, mentor and colleague, Alan Kraus, who supervised this research. If it were not for Alan's interest in me and his support, I would have long tired of pursuing another academic career. ix Chapter 1 Introduction This dissertation develops a collection of original results in microeconomic theory and then applies some of them to asset pricing. The thesis can be viewed as three separate essays, but there is a strong theme throughout. Chapters 2 and 3 attempt to make a case for the virtues of a strict partial order in modeling choice behaviour1 with suitable structure as the basis for a theory of risky choice - both static and inter-temporal. Chapter 4 applies the general inter-temporal theory introduced in chapter 3 to a classical asset pricing equilibrium. Central to the thesis is an axiomatization and representation theorem for an incom-plete binary relation (a strict partial order) over atemporal risky prospects, This relation is termed an incomplete reference based preference relation. The axioms con-stitute a relaxation of the standard von Neumann and Morgenstern (1944) axioms. In particular, the partial order implies that there exists a set of von Neumann-Morgenstern utility functions such that one distribution of payoffs, q, is strictly preferred to another, p, (i.e., q yp p) if and only if the expected utility of q exceeds that of p with respect to all members of the set2. A similar ordering, but in a setting of uncertainty, has been proposed by Levi (1980) and Seidenfeld (1988), and axiomatized by Seidenfeld, Schervish and Kadane (1995). Chapter 2 focuses on applying the strict partial order in an atemporal setting. By * A strict partial order, >-, is an irreflexive and transitive binary relation wi th an intransitive reflexive complement, y. 2 T h e 'P' refers to the fact that the ordering is similar to that induced by the notion of Pareto dominance wi th respect to a set of ut i l i ty functions 1 Chapter 1. Introduction 2 adding additional axioms so as to complete yp, an Anchored Preference Relation (APR) is derived. APRs are intended to model the well-documented status quo bias and endow-ment effect (Tversky and Kahneman (1986, 1991)). In addition, APRs are among the class of non-Expected Utility preferences over risky choice that accommodate instances of preference reversal and violations of the Independence Axiom (such as the Allais paradox). Essentially, a prospect is subjectively valued based on the least scaled-utility achievable in relation to the status quo and with respect to a set of utility functions. As such, APRs can be loosely seen as dual to Gilboa and Schmeidler's (1989) maximin utility over uncertain prospects. In Chapter 3, the theory of Kreps and Porteus (1978) is extended by imposing yp instead of the usual von Neumann-Morgenstern preferences over inter-temporal lotteries3. The inter-temporal sequence of partial orders is interpreted in terms of uncertainty over future preferences. In addition to the usual recursive structure, it is shown that the time consistency condition over the inter-temporal partial orders leads to a 'preference for flexibility'. The ensuing theory of Inter-temporal Flexibility Preferences does not make use of the usual axioms over choice sets. Moreover, as in Kreps (1979), one need not refer to a state space of 'tastes' to derive a representation. In contrast with Nehring (1999), the agent's uncertainty over future preferences can be Knightian, making no reference to subjective probabilities. Several interesting properties of Inter-temporal Flexibility Preferences are examined. First, the theory is 'closed' under partial optimization under more general conditions than is Inter-temporal von Neumann-Morgenstern Utility. What this means is that the indirect or induced representation derived by optimizing over a subset of choices (e.g., unobservable actions) still obeys the basic axioms. The conditions under which this 3 It should be emphasized that the inter-temporal theory of Chapter 3 is not an inter-temporal theory of anchoring in the status quo. The relation with the partial ordering in Chapter 2 is purely technical. Chapter 1. Introduction 3 is true are far less stringent for Inter-temporal Flexibility Preferences than for Inter-temporal von Neumann-Morgenstern Preferences. Operationally this is important since our models of agents are generally abstractions of more complicated preferences. As Kreps and Porteus (1979) point out4, a detailed theory from the class of Inter-temporal von Neumann-Moregenstern preferences will not reduce to a simpler model of the same class (except under unrealistic assumptions about the detailed preferences). By contrast, models in the class of Inter-temporal Flexibility Preferences that are closed under such reduction abound. Chapter 3 also identifies concepts for flexibility preference that are analogous to those of risk. In particular, notions of First Degree Flexibility Dominance (FFD) and Second Degree Flexibility Dominance (SFD) are explored. FFD is a normative criterion that prevents the existence of manipulation or 'free-lunches'. SFD, on the other hand, is concerned with hedging. Chapter 4 constructs an example of a general equilibrium with fully informed agents (in the sense of DeMarzo and Skiadas (1998, 1999)) in a multi-period and multi-agent consumption based one-good exchange economy. In such a model, changing tastes cor-respond to uncertainty over risk aversion and inter-temporal substitution parameters. Because agents have Knightian uncertainty over future preferences, trading in this econ-omy continues even though a complete set of contingent claims is available over every future observable event. This is in contrast with Arrow (1964) and Debreu (1959), and yet there is neither arbitrage nor inconsistent preferences as in Donaldson and Selden (1981). Components of agents' future preferences that are correlated appear in the filtration of observable events and are therefore priced. This is analogous to aggregate shifts in risk aversion or 'herding'. In fact, a condition of equilibrium is that components of future preferences that are uncorrelated wash out and do not enter the pricing kernel. The latter "Also, see Kelsey and Milne (1997, 1999) and Machina (1984). Chapter 1. Introduction 4 property is the only equilibrium criterion on the 'un-modeled' process governing agents' changing tastes. This suggests that so long as agents do indeed have Inter-temporal Flex-ibility Preferences, a detailed specification of the nature of their subjective uncertainty is not necessary for the full characterization of an equilibrium. As mentioned above, a certain degree of agents' subjective uncertainty over future preferences does appear in the macro-economic filtration of states (e.g., correlated changes in risk aversion) and is priced. The source of uncertainty in such an economy is there-fore larger than in the traditional consumption based models. This is modeled in more detail in a two-factor asset pricing model based on the equilibrium derived earlier. One factor in the model corresponds to aggregate per-capita consumption while the other factor corresponds to aggregate risk aversion. The latter summarizes the degree of cor-related uncertainty over future preferences in the economy. Because the pricing kernel arises from the aggregation of agents with non-standard preferences, additional terms enter the expressions for the market price of risk and instantaneous risk-free rate. These terms correspond to the aggregate wealth (not to be confused with the aggregate con-sumption) and the price of a unit perpetual bond. Explicit model solutions entail the integration of coupled non-linear partial differential equations. This is done numerically and asymptotically for various parameterizations. Among the novel findings is that the equity premium generated by the model can be large and stems, largely, from uncertainty over future preferences. 1.1 Additional Literature Review The dissertation is related to four main branches of economic theory: static choice under risk, dynamic choice under risk, utility for flexibility and non-standard asset pricing. A brief discussion of the literature in these areas follows. Chapter 1. Introduction 5 1.1.1 Static Choice Under Risk The static EUT of von Neumann and Morgenstern (vNM (1944)) is based on three assumptions5. The first requires an agent's preference relation to be complete and tran-sitive over final outcome distributions. The second requires that 'at-least-as-good-as' sets are closed, in a topological sense. The final assumption assumes that adding the same 'noise' to two distributions does not change their ordering. The latter assumption implies the existence of a linear representation (in probabilities). Decision theorists have proposed a large number of alternative models to accommo-date the impressive body of empirical literature documenting violations of E U T 6 . For choice under risk (where probabilities or subjective beliefs are assumed known), alterna-tive theories can be classified into three major groups, each corresponding to relaxing a different axiom. Theories that give up transitivity among binary sets of choices (e.g. Fishburn (1983, 1988), Bell (1982), Loomes and Sugden (1982) and Sugden (1993)) feature either cyclic preferences, failure of stochastic dominance (Quiggin (1990)) or dependence on irrelevant alternatives (Quiggin (1994)). Theories that depend on features other than final outcome distributions have been proposed by Segal (1990) and Grant, Kajii and Polak (1998). These, however, lack either normative appeal or tractability7. Finally, theories based on semi-orders (see Luce (1956), Vincke (1980) and Nakamura (1988)), as opposed to completeness, violate dominance. 5 A s shown by Anscomb and Aumann (1963), Savage's (1954) Subjective E U T is derivable from the theory of v N M combined with some axioms imposing "event independence". 6Notable citations from the empirical literature on E U T violations include Allais (1953), Camerer (1989), Chechile and Cooke (1997), Dubourg and Jones-Lee (1994), Ellsberg (1963), Goldstein and Ein-horn (1987), Hey and Orme (1994), Kahneman and Tversky (1979), Lichtenstein and Slovic (1973), Mac-Crimmon and Larsson (1979), MacCrimmon and Smith (1986), MacCrimmon, Stanbury and Wehrung (1980), Tversky and Kahneman (1986). 7 A theory that drops the reduction hypothesis must define preferences over every conceivable multi-stage tree that produces the same final outcome distribution. This can be questioned on normative grounds if the time between the resolution of each stage is small. Chapter 1. Introduction 6 Non-archimedean theories (see Fishburn (1988)), in which 'at-least-as-good-as' sets are not closed, have been largely ignored by economists. This is likely due to lack of tractability. On the other hand, the majority of theoretical effort has been focused on relaxing the assumption of translation invariance (Strong Independence). It is well known that complete, transitive and continuous preferences can always be represented by a continuous cardinal utility function (see Machina (1982)). Theories that drop translation invariance are therefore described as 'non-linear'. Some notable examples are Chew and MacCrimmon (1979), Chew (1983), Gul (1991), the rank-dependent models summarized in Wakker (1996) and the rank- and sign-dependent models axiomatized in Luce (1991, 1997), Luce and Fishburn (1991, 1995) and Wakker and Tversky (1993, 1995). Most of these models meet basic normative criteria, such as acyclicity and stochastic dominance. The latter theories, in particular, can theoretically accommodate most of the observed EUT anomalies, including (to a limited degree) reference effects. Empirical research, however, has been less than conclusive about the superiority of non-linear models (see Camerer (1989), Chechile and Cooke (1997), and Hey and Orme (1994)). The last two references suggest that preferences are inherently unstable. 1.1.2 Non-Expected Ut i l i t y and Dynamic Choice Under Risk The classic inter-temporal theory of choice consistent with the Expected Utility Hypoth-esis is that of Kreps and Porteus (1978). Extensions include Skiadas (1997, 1998). Most realistic economic choice problems that involve risk also involve a temporal component. It is curious that many theories of static choice that, at face value, de-scribe a single-period model, indirectly make reference to dynamic principles in order to achieve normative appeal. The 'Dutch-Book' (or 'money pump') argument is, per-haps, the most familiar example. The literature seeking to directly impose restrictions on preferences through dynamic 'rationality' reaches back almost as far as the literature Chapter 1. Introduction 7 on EUT violations8. The general conclusion is that any set of preferences which can be represented by some cardinal continuous utility function and that accommodates viola-tions of translation invariance (i.e. any non-linear utility theory) must violate one of the following principles9 1. Reduction - choice/probability trees with identical reward distribution are neces-sarily deemed identical. 2. Consistency of Contingent Plans - any strategy to which a player is willing to commit is subgame perfect . 3. Consequentialism - there is no 'memory' for risk consumption (i.e. chance branches that are not realized do not affect preferences). Although Machina (1989) makes a forceful argument against the generic normative stature of the third principle, such arguments become quite tenuous when the possible outcomes are monetary and stages in the choice/probability trees are to be resolved over arbitrarily long or short periods. Relinquishing the first principle is tantamount to giving up the reduction axiom in static EUT. Giving up the second principle can create difficulties if contingent contracts are available, as agents will be indifferent to signing contracts which they will later renounce10. Related theoretical work includes Chew and Epstein (1990), Cubbit (1996), Epstein and Zin (1989), Hammond (1989) Kelsey and Milne (1997, 1999), Segal (1997) and Sarin and Wakker (1998). 8 For an excellent review, see Machina (1989) and Sarin and Wakker (1998) 9 For a somewhat different 'breakdown' of temporal consistency, see Halevy (2000) and references therein. 1 0 Thus agents will be willing to 'make book' against themselves (Green (1987)). Chapter 1. Introduction 8 1.1.3 Utility for Flexibility A branch of research in decision theory, apparently separate from work on non-expected utility theory, focuses on the benefits or utility that a decision maker may derive from having flexibility. Specifically, an agent reveals a preference for flexibility when the certainty (cash) equivalent of a future choice set (also called an opportunity set) is greater than the certainty equivalent of any constituent element. In particular, an agent will be unwilling to relinquish the opportunity set for its maximal or most desired element. The classic theoretical reference11 on the subject is Kreps (1979). Kreps considers preferences over the set of possible opportunity sets. Axiomatically, he derives a theory of changing tastes and subjective probabilities over possible future preferences. Because an agent knows that his or her preferences may change with non-zero probability, the value of a set is partly associated with how well the set accommodates contingencies of changing tastes. Kreps' major contribution consists of demonstrating that a preference for flexibility is intimately connected with an endogenous state space that he identifies with unforeseen contingencies (Kreps (1991)). The approach, however, is not normative in that there is no connection between what the agent might actually prefer in future periods and how she values a choice set. Moreover, the endogenous state space Kreps deduces is not necessarily unique. In the last few years, Kreps' initial approach has been modified and abstracted by Bossert, Pattanaik and Xu (1994), Pattanaik and Xu (1998), Puppe (1995, 1996), Nehring and Puppe (1996, 1999), Bossert (1997) and Nehring (1999). It has also been associated with theories of unforeseen contingencies (Kreps (1992) and Dekel, Lipman and Rusti-chini (1999)). One of the important conclusions of this literature is that preference for flexibility implies either a discontinuous or partial induced order on singleton opportunity 1 1 Other historical references can be found in Kreps (1979) as well. Chapter 1. Introduction 9 sets. In other words, a complete, transitive and continuous utility theory, whether linear, as in EUT, or not, implies no preference for flexibility. Dekel, Lipman and Rustichini (1999) show that forcing the agent to exhibit a preference for flexibility by aggregating over a set of expected utility functionals provides a minimal representation of the agent's endogenous state space. A trend in the literature is to axiomatize preference for flexibility directly over oppor-tunity sets. Although the notion of changing tastes motivates some approaches (Kreps (1979) and Nehring (1999)), to my knowledge inter-temporal consistency has not figured formally in the various models12. 1.1.4 Alternat ive Asset P r i c ing Theories The general dissatisfaction of researchers with the empirical performance of consumption and additive-utility based asset pricing models (e.g., Lucas (1978)) has led researchers to seek alternative models. The most successful consumption based models of asset pricing are likely those of habit formation (Sundaresan (1989), Constantinides (1990) and Detemple and Zapatero (1991))13. Non-expected utility type models include Epstein and Zin (1989), Chew and Epstein (1990), Duffie and Epstein (1992) and Epstein and Wang (1994). The asset-pricing model developed in Chapter 4 aggregates agents' non-expected util-ity preferences over consumption, and accommodates both the large equity premium and relatively low risk-free spot rate. Although this requires the addition of another priced factor (aggregate relative risk aversion), parameters relevant to this factor can be directly estimated from long-term bond prices. 1 2 The models are generally two-stage, where the first stage consists of choosing a 'menu' while the second selects an object from the menu. ' 1 3Schroder and Skiadas (2000) show that these are isomorphic to recursive utility models. Chapter 2 Anchored Preference Relations: a theory of the status quo bias Consider an experiment conducted by Kahneman and Tversky (1979) on two separate groups of subjects. Given an initial endowment of $1000, one group was told to choose between a certain gain of $500 and a fifty-fifty chance at gaining $1000. The second group was given an initial endowment of $2000 and asked to choose between losing $500 for certain and a fifty-fifty chance at losing $1000. Both groups were effectively asked to choose between the final wealth lotteries, / 0 = pL 1 = P l 0 = pH \ n ( 0.5 = qL 0 = qj 0.5 = qH A = I and B \ $1000 $1500 $2000 j \ $1000 $1500 $2000 where the probabilities are specified above the payoffs. Refer to A and B as the probability vectors, p and q, respectively. In the study, most people from the first group chose the first lottery, p, while most of those from the second group selected q. The theory of von Neumann and Morgenstern (1944) (vNM) based on lotteries over final wealth levels predicts the same choice for both groups regardless of how the situation is framed. Some decision theorists argue, on the other hand, that the a choice problem seen in terms of gain will elicit different actions than when seen in terms of a loss, even if the final outcomes are indistinguishable. In other words, the two groups are identical except that they are 'biased' towards different choices because each group 'anchors' with a different status quo. Another example of status quo anchoring is the endowment effect of Tversky and Kahneman (1991) in which subjects are asked to choose between two 10 Chapter 2. Anchored Preference Relations: a theory of the status quo bias 11 prospects, one of which is their endowment; subjects consistently prefer the prospects with which they' are endowed to the other alternative. In this chapter, I axiomatically derive a representation that rationalizes the choice behavior described above. The vNM axioms lead to the existence of an expected utility representation, where p is preferred to q if and only if EP[U] — Eq[U] > 0. U is the vNM utility function, and Eq denotes expectations taken over the probability distribution, q. The type of representation I axiomatically derive here is somewhat richer and implicitly assumes the existence of an anchor at some lottery, e - the status quo or some other suitable reference point. An example is 1, p ye q <^  inf EP[U] - Ee[U] > inf ue{ua} Eq[U] - Ee[U] (2.2) where p >~e q, should be interpreted as "p is preferred to q when the anchor is at e", and {Ua} is a set of utility functions. Note especially that this representation reduces to that of vNM if {Ua} is a singleton set. The representation in Eqn. (2.2) is a special instance of, what is termed here, an Anchored Preference Relation, or for short, an APR. To compare p to the anchor itself, q in Eqn. (2.2) is set equal to e. In this comparison, the inequality EP[U] — Ee[U] > 0, must hold for each member of the utility function set, {Ua}, in order for p to be preferred to the anchor, e, itself2. The latter can be thought of as a requirement for 'overcoming' the anchor. It is this special relationship between lotteries and an anchor that characterizes APRs. In particular, all APRs induce a pair of binary relations between lotteries, {Vp, yNP}. p yp q inf ue{ua} q yNP p inf ue{ua] Ep[U]-Eq[U] Ep[U]-Eq[U] > 0 < 0 (2.3) (2.4) xThe class of representations derived is more general than the example. 2Equivalently, EP[U] - Ee[U] > 0 must hold for every function in the smallest set * that generates Ua through linear combinations. In other words, * is the smallest set such that Ua is in its convex hull. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 12 q >-NP p can be interpreted as up cannot strictly overcome an anchor at g", while p >-p q can be interpreted as "p can strictly overcome an anchor at q." Intuitively, yp is a Pareto dominating relation in the sense that p yp q if and only if p is preferred to q with respect to every utility function in the utility function set. In this case, p >e q for any anchor, e (and in particular, whenever e = q). >-NP, on the other hand, corresponds to, possibly, non-Pareto dominance in the sense that p yNP q if and only if p is preferred to q with respect to at least one of the utility functions in the utility function set. It is possible that s yNP r and r yNP s, if 5 is preferred to r when s is the anchor, and r is preferred to 5 when r is the anchor. In such a case preference is context dependent in the sense that one needs to know the location of the anchor to completely determine choice behavior. The anchor dependence stems from the fact that yNP is not transitive. On the other hand, if s can 'overcome' an anchor at r (i.e., s yp r), 5 is preferred to r regardless the location of the anchor, and choice behavior is context independent. In particular, s >-p r => s yNP r. The stronger relation, yp, is transitive but incomplete, thus 'preference regardless of anchor' is a strict partial order. The Kahneman and Tversky situation can occur whenever bias, or alternatively, the location of the anchor, matters. To make this clearer, suppose that the set of utility functions contains two utility 'vectors', ipi and ip2, where = (^($1000), Vi($1500), Vi($2000)) = (0, | 6) V>2T = (^($1000),^($1500), V2($2000)) = (0, ^ 2) Notice that taken individually in the sense of vNM, ipi describes a risk loving individual and ip2 describes one that is risk averse. Because both ipi and ip2 are increasing in wealth, $2000 yp $1500 yp $1000. Thus it does not matter where the anchor resides when one wishes to order sure monetary outcomes: more is always better than less regardless of anchor. For the lotteries in Eqn. (2.1), the other hand, Eq[ipi] — Ep[ipi] — ipj • (q — p) = Chapter 2. Anchored Preference Relations: a theory of the status quo bias 13 H Figure 2.1: EUT indifference surfaces. | > 0, and Ep[ip2] ~ Eq[ip2\ = 4>2 • (p — q) = \ > 0. So with respect to ipi, q is the preferred choice, while p is preferred with respect to ^2- It is never the case, however, that both utility 'vectors' yield a positive product with p — q (or q — p). Thus p yNP q and q yNP p, so knowledge of the anchor is required to determine which lottery is preferred. If the anchor is with the status quo, then the group endowed with $1000 has an anchor at $1000, while the group endowed with $2000 anchors with $2000. it is simple to check from Eqn. (2.2) that p >~$iooo q while q >-$2ooo V ~ the choice behavior exhibited in Kahneman and Tversky (1979). To reflect on these issues in more detail, consider an outcome set of three ele-ments: {L,I,H}. The outcomes L, I and H correspond to low, intermediate and high value outcomes, respectively. A particularly useful representation for the lottery space on these outcomes is the Marschak-Machina Triangle. This is based on a natu-ral bijection which exists between the three outcome lottery space and the unit simplex S2 = {(Pi>P2) G 1Z2\pi,p2 > 0,pi +p 2 < 1}- Figure 2.1 illustrates the idea. The horizon-tal axis corresponds to the probability of obtaining a low outcome, while the vertical axis corresponds to the probability of a high outcome. In Expected Utility Theory (EUT), Chapter 2. Anchored Preference Relations: a theory of the status quo bias 14 L Figure 2.2: Anchored 'better-than' sets are parallel wedges. Note that p )~p r, r yNP q and q yNP p. the indifference surfaces bounding the 'better-than' sets are parallel lines (Figure 2.1). The vector normal to these parallel lines corresponds to the unique (up to an affine transformation) utility function. The preference relations, yp and >~NP, derived by weakening the vNM axioms, are illustrated in Figure 2.2. In contrast with EUT, 'better-than' sets with respect to the Pareto dominance relation, yp, are parallel wedges3; these correspond to the intersection of the two vNM 'better-than' sets associated with each of the basis utility functions in the utility function set, {ipi,^}- The diagram shows instances of anchor independent preference, p yp r, as well as anchor dependent preference, r yNP q and q yNP p. The latter example demonstrates that yNP is not a transitive relation. In the Kahneman and Tversky example discussed above, the lotteries, p and q, are situated on the boundary of the simplex. As seen in in Figure 2.3, neither lottery sits in the 'Pareto dominance' wedge of the other. Rather than interpret this situation as indifference or incomparability, it is assumed that some context-dependent bias will un-derlie the final choice. The bias assignment can be purely descriptive or axiomatic. One 3 The wedges are truncated wherever they exit the simplex. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 15 $1000 Figure 2.3: Neither p nor q can be said to 'Pareto-dominate' the other in the Kahneman and Tversky (1979) example. axiomatic approach results in the representation of Eqn. (2.2). Regardless, the wedge implied by the utility function set places bounds on instances where context can influ-ence decision-making. Assuming the representation in Eqn. (2.2), Figures 2.4a and 2.4b demonstrate the context dependent indifference surfaces when the anchor is at the high and low outcomes, respectively (i.e., the anchor is, in turn, with each one of the status quos of the two groups in the Kahneman and Tversky experiment). The most widely quoted anomalies under the v N M assumptions are probably those associated with violations of the Independence Axiom. Most noteworthy among these is the famous example of Allais (1953) (for a comprehensive list of references, see Machina (1987)). Consider a three-element outcome set. Let the high, medium and low outcomes correspond to prizes of $1 million, $ | million and $0, respectively. If p awards $ | million with certainty, and q awards $1 million with some high probability, say 0.9, and $0 otherwise, most people choose p over q. This is known as the certainty effect. Now, let the certain $0 lottery be denoted by z, and create the compounded lotteries p' = ap+(l — a)z and q' = aq + (1 — a)z, where a is small. Empirical evidence suggests that over half of those who prefer p over q also prefer q' over p', in clear violation of the vNM theory. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 16 $2000 $1500 $1000 $1500 $1000 Figure 2.4: (a) Indifference surfaces when the anchor is at $2000, and (b) when the anchor is at $1000. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 17 Kahnemann and Tversky (1979) argue that people are drawn to certainty even when tempted by risky yet potentially more lucrative payoffs, but are alternatively drawn to higher payoffs when the probabilities of winning are low. Assuming that the typical subject is anchored at her current wealth level, (the $0 prize), this behavior too can be reconciled in terms of an Anchored Preference Relation, albeit, not the representation in Eqn. (2.2)4. Consider, instead, a representation that is a more general Anchored Preference Relation, p ye q 4=^  inf ue{ua} $u [EP[U] -Ee[U] > inf ue{ua} $u(^Eq[U} -Ee[U] (2.5) where Qu : 71 —> R is 0 at 0 and is increasing for every U G {Ua}. Each element of the utility function set, U, is associated with a, possibly, different In the Allais example, by setting ( T T a / r™ + 5o,ooo 100,000 ) , r _ a . u f 4 , {Ua(w)}={———^——, ' ^ > and {$a(x)\ - {x 4 ,x\ 1 v n \ 100,000 ' ™ +50,000 J ^ K n x ' s Eqn. (2.5) implies that p >-$o q whereas q' >-$o p'• This is shown in Figure 2.5 along with the induced indifference surfaces for the chosen representation. A related phenomenon is the so-called 'Fanning Out' of indifference surfaces (For a review and references, see Machina (1987)). Empirically, indifference surfaces are seen to be nearly linear and fan out from somewhere behind the intermediate outcome vertex in the Marschak-Machina Triangle. A common interpretation is that people become more risk averse when considering lotteries with high probabilities of a good outcome. This too can be seen in Figure 2.5. There are other types of behaviors anomalous under EUT that are consistent with APRs. These are discussed in a later section. In general, Eqn. (2.5) gives rise to a representation with indifference surfaces as in Figure 2.6. The upper contour set is constrained to be a wedge only at the anchor. 4 A s will be discussed later, the representation in Eqn. (2.2) is scale invariant about the anchor. In the example, the Allais phenomenon can still occur if the anchor is located somewhere other than $0. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 18 $1,000,000 $500,000 Figure 2.5: The Allais Paradox, anchoring and fanning of indifference surfaces. The shaded picture shows the 'Pareto-dominance' wedge. The 'kink' in indifference curves shows up near the low outcome lottery. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 19 H I L A llowed Not A llowed Figure 2.6: General Anchored Preferences. The 'better-than' set at the anchor is con-strained to coincide with the 'Pareto-dominance' wedge. Bottom: Better-than sets must contain the wedge. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 20 Elsewhere, the indifference surfaces may have curvature, so long as all upper contour sets 'contain' the wedge at each point (see Figure 2.6). As the anchor shifts, the level sets are translated. Due to the presence of wedge-like indifference surfaces, the representation in Eqn. (2.5) is not everywhere Frechet differentiable, and is therefore not included in the class of general utility representations discussed by Machina (1982). When the outcome set contains more than three elements, the structure of APRs is more complicated, but the characterization by a utility function set remains intact. As mentioned earlier, central to APRs is the relationship between an anchor and any other prospect. This binary relationship is captured by the incomplete ordering, {>-p,yNP}-In the next section I axiomatically set up the necessary and sufficient conditions for the following representation theorem where ^ is a set of bounded continuous functions on the set of final outcomes5 The expression states that q is strictly preferred to p regardless of the location of the anchor if and only if the expected utility of q is greater or equal to that of p for every utility function in the utility set, ty. In particular, q can overcome the anchor at p. The analogous representation for the weaker case where, "given an anchor at q, q is strictly preferred to p", is Thus all that is required for potentially choosing the anchor over another alternative is that the expected utility of the prospect with the anchor exceeds that of the alternative for at least one of the utility functions in the utility set. The representation in Eqn. (2.6) 5 Elements of ^ are unique up to an affine transformation. The set can be made unique without loss of generality by defining all functions to have minimum and maximum of 0 and 1, respectively, over the (assumed) compact metric space of payoffs. In such a case, \I/ is closed. (2.6) NP (2.7) Chapter 2. Anchored Preference Relations: a theory of the status quo bias 21 includes that of vNM as a special case and thus may be seen as a weakening of their axioms. The set of Anchored Preference Relations is derived by requiring continuity and com-pleteness conditions on a set of binary relations, {Ve}> where e indexes all lotteries. In addition, one must require that q yq p <£4> q y-NP p and that q y-e p Ve q >-p p. These requirements imply that if q ye p for any e, then it must be that q yq p. In partic-ular, the representation in Eqn. (2.5) follows by assuming the convexity of upper contour sets and a form of translation invariance6. The representation in Eqn. (2.2) follows if, in addition, one requires scale invariance7. To be sure, none of convexity, translation invariance or scale invariance have the normative appeal of the weakened vNM axioms that define {>-p, yNP}. The representations in Eqn. (2.2) and Eqn. (2.5) can therefore be seen as descriptive, albeit axiomatic, examples of APRs. As a general notion, reference based preferences are not new. Empirical evidence, in the form of preference reversals, framing, intransitivity and differences between will-ingness to buy and willingness to sell, led early investigators to postulate reference ef-fects (Chechile and Cooke (1997), MacCrimmon and Smith (1991), Goldstein and Ein-horn (1987), Tversky and Kahneman (1986), Loomes and Sugden (1982), MacCrimmon, Stanbury and Wehrung (1980) and Lichtenstein and Slovic (1973) to mention but a few references). The rank and sign dependent theories axiomatized in Luce (1991, 1997), Luce and Fishburn (1991, 1995) and Wakker and Tversky (1993, 1995) also have reference depen-dence. These models of choice assign a value function to prospects using non-additive probability weights and a 'utility' function defined over changes from some reference 6Translation invariance means that for every anchor, e, p ~ e 9 P + £, ~ e + £ <J + £ whenever p + £, e 4- £, and q + £ are lotteries. 7Scale invariance means that for every anchor, e, a G (0,1], and lottery, r, it is the case that p>eq ap + (l- a)r >-ae+(i-a)r aq + (1 - a)r. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 22 point. As in APRs, the nature of the reference point is not normatively elucidated, but in practice it is usually taken to be the status quo. There has been discussion of strict partial orderings in the social choice and decision theory literature analogous to the theory given here. The classic reference in Aumann (1962). For example, Levi (1980) and Seidenfeld et. al. (1995) motivate a similar theory for Anscomb-Aumann (1964) 'horse-lotteries'. More recent work on incomplete prefer-ences over risky prospects includes Ok (2000), Dubra and Ok (2000), and Baucells and Shapley (1998). The existing literature focuses on representing incomplete preferences. My purpose here is to focus on the intimate relationship between the status quo bias (i.e., anchoring) and the incomplete preference it induces. Unfortunately, economic theorists seeking alternatives to EUT have largely ignored this literature. As Cubbit (1996) notes, the reason may have to do with a bias for relaxing the vNM Independence Axiom versus giving up the assumption of a complete ordering. Finally, note that the form of this representation in Eqn. (2.2) is similar to the maxmin preferences of Gilboa and Schmeidler (1989) derived in a Savage world of uncertainty. In their representation the min is over a set of beliefs, whereas here it is over a set of utility functions. In this sense, the representation is dual to theirs. The important difference (aside from the fact that Eqn. (2.2) is not the most general form of a APR) is the reference dependence of APRs. This chapter is organized as follows: Section 2.1 presents axioms that are necessary and sufficient to prove the existence of a representation for {yp,yNP} as in Eqn. 2.6. Anchored Preference Relations are then defined and representations derived under differ-ent assumptions. Section 2.2 discusses other anomalies and their 'reconciliation' under APRs, implications of the theory when the location of the anchor is not known, and the relation of the ordering, {>-p, >~NP}, to similar constructions in the literature. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 23 2.1 Theoretical Foundations Let X be a set of distinct outcomes and assume that X is a compact metric space. Denote the Borel u-algebra of X by E ^ . The space of lotteries, V(X), is defined to be the space of probability measures on (X, T,x)-Instead of axiomatizing the set of relations, {ya}, directly, I focus initially on the more general, yet incomplete, binary preference relations, {>-p, yNP}. These define the relationship between a pair of lotteries when one of them serves as the anchor (as opposed to comparing two lotteries when the anchor is located elsewhere). The relationship between the anchor itself to other lotteries is at the heart of the notion of anchoring. It is therefore sensible to focus on a binary relation that assumes that one of the lotteries being compared is the anchor. Definition 2.1 An incomplete reference based preference relation on V(X) is a collection of two irreflexive binary relations, {yp, yNP} such that for every p,q € V{X), exactly one of the following holds (with the associated interpretation): (i) p yNP q (When anchored at p, p is strictly preferred to q.) (ii) q yp p (When anchored at p, q is strictly preferred to p.) (iii) q ~p p (When anchored at p, there is indifference between p and q.) The relation, ~ p is defined by the failure of (i) and (ii) above to hold. Further, P >zNP q is taken to denote anchored weak-preference (i.e. p yNP q or q ~p p), with a similar interpretation for q yp p. Usually, preference relations are defined to be weak orders (asymmetric and negatively transitive). Asymmetry requires that if p y q, then it is not the case that q y p. Negative transitivity requires that if q y p, then either r >- p or q y r, for any prospect r . Among other things, a weak ordering implies transitivity, which precludes cycling. A n analogous Chapter 2. Anchored Preference Relations: a theory of the status quo bias 24 structure can be imposed on incomplete reference based preference relations by adopting a suitable form of negative transitivity. Definition 2.2 {>~P,>~NP} is a frail order on V(X) if and only if {yp,yNP} is an incomplete reference based preference relation on V{X) and for every p,q,r E V(X), q yp p => r yp p or q yNP r (Negative Transitivity). A n intuitive special case of this definition corresponds to replacing r with p. In this case, Negative Transitivity guarantees that if q is preferred to p when there is an anchor at p, then q must also be preferred to p when the anchor lies with q. In other words, if q is deemed better than p when there is a natural disposition or bias towards p, then surely q will be preferred to p when there is a bias towards q. It is easy to check that Negative Transitivity also ensures that yp is asymmetric and transitive. The symmetric complement of >- p is not necessarily transitive, however, which means that >-p is a strict partial order. B y comparison, >zNP, the complement of ^ p by Definition 1, is not even assumed to be a true order relation but merely a suborder 8. This is analogous to saying that rejecting a null hypothesis (with which one is naturally anchored) requires stronger conditions than accepting it. To begin the axiomatic development, a reference based preference relation is assumed to exist on V(X). Axiom 2.1 {yp, yNP} is a frail reference based preference relation on V{X). The most important sets induced by a preference relation on a space of lotteries are 'at-least-as-good-as' and 'no-better-than' sets - the upper and lower contour sets, 8 > : P turns out to be a compatible extension of >-p (Duggan (1999)). Chapter 2. Anchored Preference Relations: a theory of the status quo bias 25 respectively. In the context of A P R s , each of these must be further divided into two qualitatively different types of sets due to the reference dependence. There is a distinction between the set of lotteries preferred to, say, q assuming an anchor at q, and the set of lotteries preferred to q, assuming the anchor is with the alternative to q. Similarly, one has to define two separate 'no-better-than' sets. More precisely, define BP = {q\q >rP p} and Wpp = {q\p y N P q}. In other words, BP contains all lotteries which are at least as good as p, viewed with an anchor at p. A similar statement holds for the 'no-better-than' set, WpNP. Further, define Wp = {q\p yp q} and B p p = {q\q yNPp}. Wp represents all the lotteries which are no better than p, viewed with an anchor at the alternative; a complementary definition applies to B p p . The next axiom forces a particular structure on Bp and Wfp. A x i o m 2.2 (Continuity) Bp = Closure{q\q yp p} if {q\q yp p] ^ 0 = {p} otherwise and W"p = Closure{q\p y N P q} if {q\p y N P q] ^  0 = {p} otherwise where closed sets are defined with respect to the Weak-* topology of the linear space of real valued signed measures on (X,T,x), namely A4(X, T,x, TV). Further, if qn —> q and pn —> p are Weak-* convergent sequences with qn ~ p pn for every n, then q ~ p p. These rather technical conditions essentially stipulate that BP and W^p are closed in M(X,T,x,T^)- Further, anchored indifference surfaces reside on the boundaries of Bp and Wpp. It is straight forward to show that the continuity axiom implies y_p is a Chapter 2. Anchored Preference Relations: a theory of the status quo bias 26 partial order (reflexive, anti-symmetric and transitive). The last condition of the axiom is needed when X is not finite. The next axiom imposes mixture invariance on >~ N P and >~p. A x i o m 2.3 (Independence or Linearity) Given p, q, r G V(X) and a G (0,1], The intuition behind this axiom is similar to that behind the conventional Indepen-dence Axiom of v N M . If, given an anchoring produced bias, one lottery is preferred to another, then identical mixtures of the two lotteries with a third wil l not result in a preference reversal so long as the direction of the bias remains unchanged. From Definition 2.1, it should be clear that Axioms 2.1-2.3 are equivalent to the v N M axioms upon the addition of the requirement that q y N P p =$> q >~p p. The following representation theorem for {yp, yNP} is one of the main results of this chapter. Theorem 2.1 Axioms 2.1-2.3 are necessary and sufficient for the following to hold, where * is a set of real, continuous functions on X whose elements are defined up to an affine (positive linear) transformation. Further, p y N P q implies ap + (1 — a)r y N P aq + (1 — a)r and q yp p implies aq + (1 — a)r yp ap + (1 — a)r. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 27 and p >-NP q inf J ip(dq - dp) < 0 * is a utility function set and the theorem states that q overcomes an anchor at p if and only if the expected utility of q is greater or equal to that of p for every utility function in the utility set, \I>. On the other hand, given an anchor at p, p is preferred to q if the expected utility of p is higher than that of q for any utility function in the utility set. E U T is recovered by insisting on bias independence of preferences: Corollary to Theorem 2.1: Given Axioms 2.1-2.3, if for every p,q G V(X), q y N P p q yp p, then where U is a real, continuous and bounded function on X defined up to an affine trans-formation. Further, Although details of all proofs are relegated to the appendix, a deeper insight into A P R s can be gained by listing and discussing several key results which are instrumental in the derivation of Theorem 2.1. The following lemmas, for example, guarantee some important and intuitive results. P Lemma 2.1 Linearity of Anchored Indifference For any p,q,r G V(X) and a G [0,1], q ~p p => aq + (1 — a)r ~p ap + (1 — a)r Chapter 2. Anchored Preference Relations: a theory of the status quo bias 28 Lemma 2.2 Transitivity of yp For any p,q,r E V{X), q yp p and r >zP q =*> r yp p. Lemma 2.3 No Direct 'Money Pumps' q yp p q yNP p; and in particular, q yp p =>• q yNP p. Lemma 2.4 Convexity of'at-least-as-good-as'Sets q,r E Bp => aq + (1 - a)r € BP V«G [0,1]. If in addition either q yp p or r yp p then aq + (1 — a)r yp p Va E (0,1) (denoted as strict convexity). Lemma 2.5 Nesting of 'at-least-as-good-as' Sets qEBp^ BP C BP Lemma 2.6 Strict nesting of 'at-least-as-good-as' Sets For any p,q,r E V(X), if q E Bp and r E Bp and either q yp p or r yp q then r yp p. Lemma 2.1 extends Axiom 2.3 to include linearity on anchored indifference surfaces. Lemma 2.2 expresses the intuition that anything which is at least as good as a strict improvement on an anchored prospect is also itself a strict improvement on the anchored prospect. Lemma 2.3 states that, if anchored at p, it seems worthwhile to trade p for q, Chapter 2. Anchored Preference Relations: a theory of the status quo bias 29 then if q is acquired and serves as the new anchor, p should no longer seem attractive. In other words, it is not possible to create an exchange cycle between two lotteries when one anchors in the status quo. The Lemma 2.4 demonstrates that, anchored at p, any combination of prospects which are individually just as good as p, will also be just as good as p. This is an empirically falsifiable statement. Lemma 2.5 and Lemma 2.6 guarantee that 'at-least-as-good-as' sets are properly nested, thus >zP is a partial order. This is important if an agent anchors in the status quo in a two stage trade: suppose p —> q —> r (i.e. the anchor p is exchanged for q, and the new anchor, q is then exchanged for r). What is shown in the last two lemmas is that r yp p, or in other words, p would have been exchanged for r and there is no cycling. Lemma 2.4 says that the set BP is convex, whereas Lemma 2.1 demonstrates that the indifference surface bounding BP is spanned by rays originating from the anchor, p. The next proposition shows that all 'at-least-as-good-as' sets are similar in that they all derive from some canonical closed convex cone in the space of signed measures, A4(X, T,x,R)-Specifically, define Proposit ion 2.1 Every anchored 'at-least-as-good-as' set, BP, can be written as the The purpose of the following lemma is to establish that Bp is a closed convex cone. Bp = {\(q-p)\ qtpP,Xen+, q,PeV(X)} (2.8) intersection ofV(X) and the translated set, p + Bp. V{X) n (P + Bp) Lemma 2.7 Bp is weak-* closed and convex. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 30 The discussion so far has focused on properties of the anchored sets BP and by implication, WPNP. In particular, no mention has been made of the structure of B P P and WP. The following result completes the description of {yp,yNP} in terms of the canonical cone, BP. Proposition 2.2 wp = v{x) n (p - BP) Theorem 2.1 essentially derives from observing that yp and yNP are characterized by the canonical cone, BP. Because it is closed and convex, BP itself can be described by some set of supporting hyperplanes. Since BP C A4(X, Ex , Tl), one can find supporting hyperplanes that are continuous and bounded functions on X. In other words, specifying a utility function set is equivalent to specifying BP. 2.1.1 Anchored Preference Relations Theorem 2.1 derives by relaxing the v N M axioms. A lottery at p wil l be traded for lotteries in BP and any lottery in WP wil l always be traded for p. The ambiguous region WPP \ WP (or equivalently BPP \ BP), is what characterizes A P R s . Violations of the v N M axioms wil l occur between lotteries that sit in each other's ambiguous region. It is by deriving the existence of this ambiguous region from a parsimonious set of assump-tions that { V p , yNP} normatively 'bound' violations of E U T . Unfortunately, yp is also incomplete and there is no way to completely determine choice behavior solely through knowledge of the utility function set, unless ^ is a singleton. It is precisely the incompleteness of yp, however, that allows one to postulate ref-erence dependence for lotteries that are incomparable through yp. In this subsection I Chapter 2. Anchored Preference Relations: a theory of the status quo bias 31 formally define Anchored Preference Relations and provide necessary and sufficient con-ditions for several representations. The idea is to define a set of complete and continuous preferences, { V a } , each of which has an upper contour set (at-least-as-good-as set) that agrees with BP at e. In other words, ye is a complete representation of preference when an anchor (e.g., the status quo) is known to be at e. Definition 2.3 An Anchored Preference Relat ion is a set of binary relations, {ya} over V(X) such that (i) y&e {ya} & e e V(X) (ii) ye€ {ya} =>• ye is a complete, weak and continuous order. (iii) There exists a reference based preference relation, {yp ,yNP} obeying Axioms 2.1-2.3 such that for any p,q E V(X) qyqp ^ q y N P p qyep for all yeE {ya} ^ qyp p The symmetric complement of yeE {ya} is defined, as usual, as q ~e p 4^ it is not the case that q ye p or p ye q. Condition (i) simply asserts that one can only anchor in something tangible (i.e., a prospect over which one has preferences). Condition (ii) is standard. Condition (iii), in particular, is what relates an anchored preference relation to the partial order axiomatized earlier. The first part formally identifies y N P with a preference for the anchor (which is a weaker type of preference). In particular, it is possible that q yq p and p yp q. The second requirement formally establishes the connection between overcoming the anchor and the Pareto dominating relation, yp. Proposi t ion 2.3 Fix {ya} and its associated reference based preference relation, {yp Chapter 2. Anchored Preference Relations: a theory of the status quo bias 32 (i) For every yeE {>-o} there exists a real valued bounded and continuous function, He(-), overV(X) such that for any p,q G V{X) qyep He(q) > He(p) (ii) If He(-) has a subgradient, tpg, at p G V{X) then tyg is, up to an affine transformation, a convex combination of members of . (iii) For any p, q E V(X), q ~P v q ~p p HP{q) = Hp(p) Part (iii) of the proposition states that the indifference surface at the anchor is a truncated pointed cone with the point coinciding with the anchor (e.g., Figure 2.6). Save for Proposition 2.3, not much more can be said about any given >-e and its representation. In other words, although the indifference surface at the anchor is fully characterized by the utility function set, the indifference surfaces elsewhere are not. The next few axioms progressively place more structure on the representation. A x i o m 2.4 Convexity For any yee {ya}, P,q,r G V(X) and a G [0,1], p ye q and r ye q ap + (1 — a)r ye q A x i o m 2.5 Translation Invariance For any ye£ {ya}, p,q eV{X) P~eq p + £ ~ e + £ q + £ for any£ G M(X,T,x,Tl), such that p + £,q + £ , e + £ G V{X) Chapter 2. Anchored Preference Relations: a theory of the status quo bias 33 A x i o m 2.6 Scale Invariance For any yeE {ya}, p,q,r E V(X) and a E [0,1], pyeq otp + (1 - a)r yae+(!_a)r aq+(l- a)r Convexity implies that, fixing the anchor, if two lotteries are strictly preferred to a third, then a coin toss that is guaranteed to award one of the preferred lotteries will also be preferred to the third. This has a normative flavor, but with strictly convex upper contour sets, it also implies a possible affinity for gambling. In other words, there may be two lotteries that are individually inferior to a third, but a coin toss over the inferior prospects is strictly preferred to the third lottery. Green (1987) argues that only quasiconvex utilities are immune to Dutch Books. Translation invariance and scale invariance do not carry the same or any normative weight as does convexity. 9. Both translation invariance and scale invariance, however, are implied by the v N M axioms. Note, also that Allais type phenomenon are not allowed by scale invariance when the common consequence is the anchor (e.g., recall the example in the introduction). 9Translation invariance assumes that comparison between lotteries only depends on how each lottery deviates from the anchor. While translation invariance may seem vaguely normative, it is not clear what normative principles scale invariance conveys, save for elegance in representation. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 34 Proposit ion 2.4 //{>~a} is an Anchored Preference Relation, then (i) Axiom 2.4 implies that >-e can be represented by He(p) = inf $f(Ep[ip] - Ee[ip}) where f C 7 C lin(tf), and for any e and tp, is an increasing real valued function such that <&f(0) = 0. (ii) Axiom 2.4 & 2.5 imply that ye can be represented by He{p) = inf ^(Ep[tp] - Ee[ip]) where f C 7 C lin(tf), and for any ib, $^(-) is an increasing real valued function. (iii) Axiom 2.4 -2.6 imply that ye can be represented by He(p) = inf (Ep[ib] - Ee[ip]) where K 7 C lin(#) where $ is the set of utility functions provided in Theorem 2.1, l in(^) is the set of all positive linear combinations of elements of ^ (i.e., the cone generated by ty). Part (ii) of the proposition is the representation promised in Eqn. (2.5), while the more restricted representation in Eqn. (2.2) requires the assumption of scale invariance. Tversky and Kahneman (1991) describe a deterministic utility theory over bundles of goods that incorporates reference effects. Their theory posits the existence of a family of reference-dependent utility functions. Specifically, x is preferred to y from t if and only if Ut(x) > Ut(y), where x, y and t are bundles of goods and Ut(-) is a reference-dependent utility function. Anchored Preferences Relations are thus an axiomatic extension of Tversky and Kahneman's reference dependent utility to risky prospects. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 35 2.2 Further Discussion 2.2.1 Other Anomalies: Preference Reversals and Imprecise Certainty Equiv-Preference reversals correspond to a class of E U T violations that show inconsistency between valuation versus choice. For example, one is asked to choose between the lotteries and then asked to price each of them. Subjects often choose a lottery whose elicited value is less than the lottery foregone (e.g., choose A but price B higher). There have been two types of explanations (see Machina (1987) for an entertaining account and a more comprehensive list of references): economists label this behavior as a violation of transitivity; psychologists, on the other hand, claim a separate mechanism for choice versus valuation. The effect has been very well documented and persists even under experimental conditions that try to enhance motivation for careful choice (see Slovic and Lichtenstein (1983) and references therein). Some effort to bridge this gap has gone into describing the difference between the two actions as based on hidden transaction costs, anchoring or imprecise determination of certainty equivalents 1 0. i Theories that succeed in explaining this anomaly have been axiomatized by Fishburn (1983) and formulated in terms of 'Regret Theory' by Bell (1982) and Loomes and Sugden (1982). These theories are intransitive and allow for an inversion in preferences between the certainty equivalents of lotteries and the lotteries themselves: p >~ q, p ~ c(p), q ~ c(q), but c(q) y- c(p), where c(.) is a certainty equivalent amount of money. Since the certainty equivalents in these theories are unique, the preference reversal is truly 1 0See Chechile and Cooke(1997), Goldstein and Einhorn (1987), Loomes and Sugden (1982), Lichten-stein and Slovic (1973), MacCrimmon and Smith (1991), Tversky and Kahneman (1986). alents Chapter 2. Anchored Preference Relations: a theory of the status quo bias 36 the result of potential intransitivity (note that this does not necessarily imply a money pump, as argued by Loomes and Sugden (1982)). Luce et. al . (1993) develop a theory that can also account for preference reversals using certainty equivalents that depend on some exogenously specified reference levels. In their theory, the reference level is different depending on whether a subject is asked to compare selling prices or the lotteries themselves. To understand this phenomenon in terms of anchoring one key assumption is required. Namely, when eliciting a certainty equivalent, assume that the agent is anchored in the prospect that is to be valued. For a concrete example, consider the representation in Eqn. (2.2) with a utility function set <3> = {x, Inx} containing functions defined over the monetary outcomes of a lottery. When asked to value A, the agent anchors in A and it is easy to see that the least amount of cash, c(A), such that c(A) yp A, happens to coincide with the expected value of A. Thus c(A) < c(B), consistent with the empirical findings. On the other hand, when comparing A to B, the anchor matters. Assume that the status quo is associated with the guaranteed, but least desirable outcome of $1. Assuming an anchor at $1 implies a preference of A over B according to Eqn. (2.2). The source of the reversal is related to the difference between a willingness to buy versus a willingness to sell. In the above example, and assuming anchoring in the status quo, The agent would sell A for no less than $3.67 (assuming an anchor at A), but given $3.67 (and an anchor at $3.67), would part with no more than $3.46 to obtain A. Such a disparity, termed an imprecise certainty equivalent, occurs naturally in the representations of Proposition 2.4 and is empirically persistent (e.g., MacCrimmon and Smith (1991)). Chapter 2. Anchored Preference Relations: a theory of the status quo bias 37 2.2.2 Indeterminate Anchor Anchored Preference Relations are suited to deal with choice problems where an anchor is easily identifiable (e.g., the status quo). There are instances where this is not the case. In framing phenomena (e.g., Tversky and Kahneman (1986)), choice alternatives may be described in a variety of ways that do not distort the final outcome but affect the point of view of the subject. A n anchor may not always be unambiguously associated with a frame. A theory of anchoring when the anchor is not the present endowment is beyond the scope of this chapter. Although by choosing a suitable anchor in each choice problem one wil l likely be able to 'explain anything', a more optimistic point of view on the issue would be that A P R s shift the major descriptive components of the theory from the axioms to a subordinate theory of what can serve as an anchor and when. A n interesting question is whether something useful can still be said without direct reference to a theory of how one comes to be anchored. Equivalently, one can ask to what degree knowledge of the strict partial order, yp, which is insensitive to context, is useful in describing behavior. The problem is analogous to that of social choice where the population of agents corresponds to the set of utility functions, and one wishes to make normative statements about social welfare. Given a closed choice set, F C V(X), an agent conforming to Axioms 2.1-2.3 will choose an element of Mp, the maximal subset of F: MF = {peF\FC W^} I.e, Mp is the set of elements of F that are not dominated by other elements of F regardless of anchor. If the utility function set is seen to describe the agent as being a collection of many subagents, >-p can be interpreted as a Pareto-dominating ordering and elements of Mp are those that are not pareto-dominated by other elements of F. What I show next is that if the agent is allowed to use mixing strategies in choosing Chapter 2. Anchored Preference Relations: a theory of the status quo bias 38 prospects, then any model which 'picks' an anchor is equivalent to one that maximizes over F some convex combination of utility functions in To explore this idea define, where F C V(X) is closed and co(Vl') denotes the closure of the convex hull of ^. SF is a set of lotteries in F that maximize some convex mixture of utility functions from The following theorem gives the desired result. Theorem 2.2 Given a choice set, F C V(X), SF Q Mp. Furthermore, if F is convex and weak-* closed, and there exists p € V(X) such that Wpp ^ {p} and Bp ^ {p}, then SF = MF. The requirement that there.exists p € V{X) such that Wpp ^ {p} and Bp ^ {p} ensures that the ordering is not trivial (i.e., there is a prospect that is strictly preferred to, or is strictly inferior to another prospect). This result is none other than the Second Welfare Theorem of classical microeconomics. If an agent is allowed to use mixing strategies in making choices (e.g. throw a coin before selecting among prospects) then, effectively, any choice set is convex and thus SF = Mp- Assigning an anchor is therefore equivalent to assigning some member of co(\I/) to the choice problem. The implication is that in a static one-period model A P R s reduce to E U T but where a context dependent v N M index is chosen from co(^) . Alternatively, an anchor assignment can be seen as analogous to an assignment of a security index in the sense of Levi (1980). It is important to stress that only revealed choice can be interpreted as the optimiza-tion of some element of co(^). If, for example, an agent is asked to provide a ranking, as opposed to make a single choice, then the ranking may not correspond to E U T . (2.9) Chapter 2. Anchored Preference Relations: a theory of the status quo bias 39 2.2.3 Relation to semiorders Related to frail orders are semiorders, first introduced by Luce (1956) to model intransi-tive indifference relations. Semiorders capture the notion that indifference curves are 'fat' in the sense that agents will always be indifferent to slight changes in their endowments. A sequence of slight changes, however, can accumulate to significant differences between the initial and final endowments, over which an agent may no longer be indifferent. As mentioned earlier, >~p is a strict partial order which is a weaker notion than that of a semiorder. Consequently, similar to semiorders, the symmetric complement of yp is also not transitive. Vincke (1980) axiomatized a linear utility theory of semiorders over mixture spaces in the spirit of Herstein and Milnor (1953). The resulting representation theorem essentially states that qyp <^> Ju{dq - dp) > a(p) (2.10) where u is a utility function over X and a is a non-negative real valued threshold function over V(X). Nakamura (1988) obtains similar results. Theorem 2.1 can be interpreted in an analogous but not fully equivalent manner under certain circumstances. To see this, consider a utility function set, consisting of elements which can be written as, ipl = u+a\ for some u. Moreover, assume that if u + a1 E ^  then u — a1 E $ as well. Theorem 2.1 can then be written as qypp 4=> Ju(dq-dp)>E(p,q) (2.11) where T,(p,q) = max| J al(dp — dq) \ . There are some important differences between the i above and Vincke's representation. First, the threshold term, E(p, q) depends on both lotteries whereas in Vincke's model only the 'incumbent' lottery, p appears. Related to this is the fact that Theorem 1 guarantees strong independence whereas Eqn. 2.10 does Chapter 2. Anchored Preference Relations: a theory of the status quo bias 40 not. Specifically, if r >- p, then it is not the case that ar + (1 — a)p >- p for every a € (0,1]. It is easy to see that Vincke's model inherently violates a strong form of dominance - that is in fact what it is intended to do for small changes in endowment. The order implied by Axioms 2.1-2.3 is arguably more 'normative' in that violations of dominance are not necessary. First degree stochastic dominance is guaranteed so long as every utility function in ^  is increasing 1 1, and yet the reflexive complement of yp is intransitive. 2.3 Appendix Proof of Lemma 2.1. B y assumption, q e Bp. If BP — {p} or if a 6 {0,1}, then the proof is done. Otherwise, by Axiom 2.2, q must be the limit point for some sequence {qn} such that qn yp p. It is possible to construct the sequence {sn} such that sn = aqn + (1 — a)r. Clearly, {sn} —> aq+(l -a)r. Moreover, by Axiom 2.3, sn yp ap+(l-a)r =4> sn e Bpp+^_a)r for any n. Since Bpp+^_ay is a closed set, the sequence converges inside it to give aq + (1 - a)r E Bpp+^_ay. A similar argument shows that aq -f (1 - a)r G W^p^_ay. Using the definitions of Bpp+^_ay and W^pp^_ay and using Axiom 2.1 then implies that aq + (1 — a)r ~p ap + (1 — a)r. Proof of Lemma 2.2: Use negative transitivity. Proof of Lemma 2.3: n T h e same is true for Second Degree Stochastic Dominance as long as every function in *P is concave. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 41 The special case q yp p =>• q y N P p is proven using the Negative Transitivity as-sumption from Axiom 2.1 setting r = p. To prove the general case, which may include indifference, invert the Negative Transitivity property. Proof of Lemma 2.4. First notice that if any one of r or q coincides with p then the theorem is a straight ap-plication of Lemma 1 or Axiom 2.3. In any other case, Axiom 2.2 implies that there exists a sequence of points, {rn} — • r, such that rn >-p p. Thus by Axiom 2.3, (1 — a)rn + ap = yp p for every a E [0,1). Also, from Lemma 1 and Axiom 2.3, write q E BP =>• (1 - a)rn + aq = q™ E BPn. Lemma 2 implies that g™ E BP. Now, clearly g™ —> qa. Since BP is closed, one must conclude that qa E Bp. This takes care of the first portion. Notice that if any one of r and q is strictly preferred to p, then a slight modification of the above argument yields qa yp p. For example, if r yp p, then rn = r and g™ = qa for every n; the implied strict preference in Lemma 2 finishes the job. This takes care of the strict convexity portion. Proof of Lemma 2.5: To prove that q E BP Bp C BP, it is sufficient to show that q E Bp and r E BP r E Bp. If BP — {p} then q = p and the proof is done. Otherwise, Axiom 2.2 ensures the existence of some q y p p. The strict convexity property then implies that aq + (1 — a)q yp p for any a E (0,1]. Now, r E BP along with Axiom 2.3 and Lemma 1 imply that aq + (1 — a)r E Bp-+^_a^q. Lemma 2 can now be used to give aq + (1 — a)r E Bp for any a E (0,1]. In particular, this is true for any sequence of a's in (0,1). Picking an = 2~n and rn = (1 — an)r + anq gives {rn} —> r, and therefore r E Bp. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 42 Proof of Lemma 2.6: The case q yp p is treated by Lemma 2. Assume, therefore that q ~ p p and r yp q. Now, assume to the contrary that r ~p p. Since r ^ p (otherwise there would be a contradiction with Lemma 3), by Axiom 2.2 there exists a sequence {rn} —> r such that p y N P rn for every n. There has to be some element of this sequence, say f, which is strictly preferred to q. If not, then {rn} must converge in W^p, which means r G W^p', a contradiction. Thus by Lemma 5, f G BP, which violates the hypothesis about the sequence. The conclusion is that r ~p p cannot hold. Since r G BP it must be that P r y p. Proof of Proposi t ion 2.1: To begin, suppose that q G Bp. Clearly q G V(X), and so it remains to be shown that q = p + X(q' - p') for some A > 0 and q',p' G V(X) such that q' G Bp,. This is trivially accomplished by choosing A = 1, q' — q and p' = p. This demonstrates that BP Cp(X)n(p + Bp). To show the equivalence the other way, fix q G V(X) f l (p + Bp). This means that q = p + \[q' - p1) for some for some A > 0 and q',p' G V(X) such that q' G Bp. There are now two possibilities, A < 1 and A > 1. Consider first the case where A < 1. Since q' G Bp, Axiom 2.3 and Lemma 1 guarantee that z = \q' + (1 — X)p' G Bp,. Now construct the two lotteries, r = ^q + \p' and r' = \q + \z. Axiom 2.3 and Lemma 1 can be used again to show that r' G Bp. Note that r can also be written as r — \z + | p so that if one assumes, q 0 BP then, once again using Axiom 2.3 and Lemma 1 one deduces the contradiction, r' 0 Bp. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 43 If A > 1 then construct the lottery q = (1 — a)p + aq = p + a\(q' — p') with a > 0 sufficiently small so that a\ < 1. Using an argument identical to the one above, one concludes that q € BP. Clearly, if q ^ Bp then using Axiom 2.3 and Lemma 1 one again deduces a contradiction, q Bp. One therefore has that V(X) D (p + Bp) C BP. Proof of Lemma 2.7: Suppose that /Ui,/x 2 £ Bp. This means that px = \\{q\ — pi) and p2 = M{<12 — Pi) such that qi <E BPx and q2 G Bp2. Axiom 2.3 and Lemma 1 guarantee that aq\ + (1 — a)q2 € Bppi+(1_a)q2 and api + (1 - a)q2 € Bppi+{1_a)p2 for any a € (0,1). The nesting property (Lemma 5) gives aqi + (1 — a)g 2 € S P D 1 + ( 1 _ a ) P 2 - In turn, the definition of Bp can be used to claim that A (a(qi - Pi) + (1 - a)(5a - P2)) € B (2.12) for every positive A. This is true, in particular, for A = a\\ + (1 — a )A 2 , where a € (0,1). Choosing a = ^ (and noting that this forces a to lie in (0,1)) one can write the left hand side of Eq. (2.12) as A (a(qx - pi) + (1 - a)(q2 - p2)) = aAi(gi - pi) + (1 - a )A 2 (g 2 - P2) = a/ / i + (1 - a)u2 (2-13) This, of course, gives the desired result that any convex combination of p\ and p2 is in Bp; thus Bp is convex. To prove Bp is closed, assume the contrary. Then there is a sequence, {£ n } — • £ with £ n e S p for every n, yet £ ^ . B p . Because X is a compact metric space, V(X) is weak-* compact. It is possible, although tedious, to use the compactness of V(X) and Chapter 2. Anchored Preference Relations: a theory of the status quo bias 44 the hypothesis to demonstrate the existence of sequences {qn} —> q and {pn} —> p such that for each n, one has qn,pn G V{X), qn—pn G Bp, but q — p 0 Bp. It therefore stands o NP 0 NP o NP that q eWp , where Wp = WpNP \ Bp. Axiom 2.3 implies that \q + \pn EWiP+iPri-Note that qn G - B p implies that \qn + \p G I?f , i . The idea is to combine \q + \pn and \qn + ^p, sitting on opposite sides of indifference surfaces to obtain a lottery which is on the indifference surface. This can always be done because Bp is closed and convex, o NP and B^U Wm = V{X) for every m G P{X). There must therefore exist a number, an G (0,1), such that f1 1 A M \ f ^ 1 \ P 1 1 ani^q + ^Pnj +{l-an)l-qn + -pj ^ -p + -pn In the above equation, the left hand side converges to | g + \p and the right hand side (as well as the anchor) converges to p. Axiom 2.2 can now be used to conclude that \q + \p ~p p which contradicts the hypothesis that p >p q. The contradiction implies that Bp is closed. Proof of Proposi t ion 2.2: The proof closely follows that of Proposition 1. Proof of Theorem 2.1: B y Proposition 1, given q,p G V(X), q G Bp if and only if q - p G Bp. Since Bp is weak-* closed and convex, Theorem 2 and its corollaries from Phelps (1964) guarantees that Bp is the intersection of half spaces E{ C M{X, S) where the Ei's are defined by p E Ei -^ => f ipi dp > ^ for some ^ ^ 0 6 C(X), a continuous and bounded function Chapter 2. Anchored Preference Relations: a theory of the status quo bias 45 on X, and a{ G R. Moreover, since Bp is a cone p G Et Xp G E{ for all positive A. This means that a; = 0 for every i. One can therefore write Note that multiplying any of the ^ ' s by a positive constant does not change the result. Moreover, since p and q are probability measures, adding a constant to any of the ^ ' s does not affect the conclusion either. In other words, the ipi's are defined up to an affine transformation. Finally, let ^ be the set of all separating hyperplanes, ipi. It should be clear that in the quotient set of functions induced by affine transformations ^ is closed in the sup topology of C(X). To prove the second part of the theorem, first assume that q ~p p for some p,q G V(X). Note that p G W q 7 V P . If p = q then / ip (dq - dp) = 0 and the proof is done. Since qn—p^L Bp, min J ipi (dqn — dp) = pn < 0 for every n. Since the ^ ' s are bounded i and continuous (affine equivalence implies that one can assume ||^;|| = 1 in the strong metric of C(X)), the sequence {pn} has a subsequence that converges in [—2,0]. Since q G Bp {pn} — • 0. This implies that min J ipi (dq — dp) = 0, as required. i Suppose now that min / ipi (dq—dp) — 0. There must therefore be some ip ^ 0 G C(X) i such that J ipi (dq — dp) = 0. Set iniip(x) = a and suppose V ^ o ) = a. Now define qn = q(l — ^) + ^8x0, where 5Xo is a point mass' at xQ. Notice that for every n , qn G V(X) and J ip(dqn — dp) = £ J(a — ip)dq = an < 0 (if ip is constant then it is affine equivalent to 0, which is ruled out by hypothesis). Thus qn EWp for every n, meaning that it must converge in Wpp. Since qn —> q G Bp it must be that q ~p p, which ends the The last part of the theorem follows trivially from Axiom 2.1. This proves sufficiency of Axioms 2.1-2.3. The necessity part is a straight forward exercise in checking that Otherwise, Axiom 2.2 guarantees that there is a sequence, {qn} o NP q such that qn GWP proof. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 46 Axioms 2.2 and 2.3 hold. Proof of Corollary to Theorem 2.1: If ^ is a singleton, it should be clear that Theorem 1 reduces to Expected Utility Theory with ip € * the utility function, and that yNP implies yp. Suppose, on the other hand, that yNP implies This means that the sets Bp f l Wp = V(X) for every p. Clearly that can only hold if Bp is a half-space, which can only be if ^ is a singleton. Proof of Proposition 2.3: Part (i): This follows from Debreu (1954). Part (ii): Suppose that ipg is a gradient vector to He at p and that < tpg,p >= c. Part (iii) of Definition 3 implies that p + Bp C {q \ < ipg, q >= c}. Since Bp is the cone produced by the intersection of all the half-spaces, {q \ < ip,q >= c, ip £ this can only be true if ipg € H u l l ^ up to some affine transformation. Part (iii): Chapter 2. Anchored Preference Relations: a theory of the status quo bias 47 Recall that q ~F p <-> it is not the case that q y-p p or p yNP q} and that q ~p p <-> it is not the case that q ye p or p ye q. Using part (iii) with the anchor at p gives the desired result. Proof of Proposition 2.4: Part (i): Since He(-) is continuous and its upper and lower contour sets are compact (by compact-ness of V{X)), it follows that —He(-) satisfies the conditions in Holmes (1975), Lemma 141. The result follows immediately by setting §t{x - EMA) = sup He(q) 96{g I Eq{iP]=x} Since Proposition 3 guarantees that p + Bp C {q\He(q) = He(p)}, one need only consider ip G ^ as part of the program. Finally, setting He(e) — 0, it follows from part (iii) of Proposition 3 that $^(0) = 0. Part (ii) & (iii): These follow directly by demanding that the representation be translation and scale invariant, respectively. Proof of Theorem 2.2: The first part of the theorem is trivial. Chapter 2. Anchored Preference Relations: a theory of the status quo bias 48 To prove the second part, assume that c E MF. Denote by M°(X, E x , R), the subspace of M(X,T,x,R) containing measures, p with J dp = 0. Clearly, both Bp and F - c are subsets of M°(X, 7£). Moreover, assuming 73p ^ [p] for some p, it can be shown that Bp has a non-trivial interior relative to the space, M°(X, Ex,R)- Now, when Wjf p ^ {p} for some p, Bp is a pointed cone and cE MF only if lnt(Bp) n (F - c) = 0. Since F is weak-* closed and convex, and Int(£? p) is convex, the Hahn-Banach Theorem implies the existence of a supporting hyperplane which separates Int(f?p) and (F — c). Corollary 1 of Theorem 2 from Phelps (1964), then implies the existence of a separating weak-* supporting functional, ipc. Since 0 E Bp D (F — c), ipc is a supporting functional for Bp at 0. Thus, ip E$, and c E SF. Chapter 3 Inter-temporal Flexibility Preference In a seminal paper, Kreps (1979) showed that a preference for flexibility implies that an agent acts as if she possesses an endogenous state space. For example, if a menu from which the agent will later consume, {a, b}, is preferred to both the menu {a} and the menu {b}, Kreps demonstrates, under parsimonious assumptions on the preference over choice sets, that the agent has a utility representation that suggests an endogenous state space of possible future tastes. Specifically, Kreps derives a representation for preference over subsets of a finite space of prospects, X, with the following structure: x and y are menus (subsets of X), S is an index set derived endogenously, and for each s e S, U(s,d) is a function over X. The existence of the index set, S (i.e., the set of utility functions), is interpreted as an endogenous state space of tastes. This space, unfortunately, is not unique and the formulation is not normative. In particular, the theory does not rule out the possibility that the agent simply prefers choice sets with higher cardinality regardless of their content, thus no connection is made to eventual choice from menus or to future preference over the constituents of menus. Moreover, there are many alternative representations specifying different index sets. These, in general, take the form of a utility for a set, x: x y y / max U ( ^—J d£x s,d) > y^ max U(s,d) U(x) = ii(max U(l,d), max U(2,d), ... max U(S,d)) 49 Chapter 3. Inter-temporal Flexibility Preference 50 where u is increasing in all its arguments (it can be viewed as an aggregator of future utilities). Dekel, Lipman and Rustichini (1999) show that the state space can be pinned down by expanding it to lotteries over X and insisting that the aggregated C/(s,d)'s be expected utility functional. They also do not present any normative structure relating the set of utility functions to actual eventual choice. Moreover, it isn't clear why allowable functions ought to be expected utility. This chapter takes a different approach to utility for flexibility. Note that if the menu {a, b}, is preferred to both {a} and {b}, then it is not possible, based on the latter infor-mation alone, to order the possible future consumption choices a and b. In other words, a preference for flexibility entails a partial ordering of future prospects. The main contribu-tion of this chapter is the demonstration that by combining partial orderings and a time consistency condition, a preference for flexibility emerges in a normative manner. The theory is essentially a weakening of the Kreps and Porteus (1979) structure for recursive von Neumann-Morgenstern inter-temporal utility. This allows for a re-examination of the induced utility problem (Kreps and Porteus (1979), Machina (1984)). A further con-tribution of this chapter is the demonstration that the theory axiomatized here is closed under 'hidden optimization' (i.e., induced utility) to a far greater extent than expected utility. A final contribution in this chapter is the identification of dominance principles analogous to stochastic dominance, but with respect to flexibility. 3.1 Introduction Suppose that at any date, economic agents know how they feel about immediate payoffs but are unsure of their future feelings regarding future risky payoffs. Moreover, assume that the agents can place 'bounds' on the extent to which their feelings may later change but are uncertain, in the Knightian sense, of their precise future tastes. In other words, Chapter 3. Inter-temporal Flexibility Preference 51 the agent knows the set of future tastes but cannot necessarily assign probabilities to potential future preferences. In this chapter, a set of normative rules is proposed to which such agents should conform in order for their choices to be deemed economically rational in a dynamic sense. The set of rules, or axioms, makes use of the partial order, ( V p , yNP}, derived earlier1 from Axioms 2.1-2.3. Presented here is a theory of inter-temporal choice derived along the axiomatic lines of Kreps and Porteus (1978), but which takes as its primitives a much richer set of preferences. Kreps and Porteus (1978) assume that at each date, the agent's actions are consistent with the von Neumann and Morgenstern (1944) axioms of Expected Utility Theory (EUT) over inter-temporal lotteries. The latter are defined as gambles whose outcomes are an immediate consumption bundle and a future decision tree. Formally, the agent described in this chapter is unsure of how she will order inter-temporal lotteries in the future. In other words, as seen at date s < t, her preferences at date t are incomplete. Here, therefore, the assumption of von Neumann-Morgenstern preferences at each date is weakened to Axioms 2.1-2.3 of the last chapter2. The resulting sequence of partial orderings, {Vf , y?p}, is associated with Knightian uncertainty over future tastes. Theorem 2.1 implies that a 'basis' set of tastes at date t can be represented by a set of utility functions, tyt. If the choice set is convex, as it is in the case of inter-temporal lotteries3, Theorem 2.2 guarantees that revealed choice at date t corresponds to maximizing the expected utility of some element of co(^t), the closed convex hull of tyf Thus the description of choice behavior reduces to the assignment, at each date, of weights corresponding to that date's utility function from co(^t) (and consequently, that date's optimization program). Although the axioms are the same, it must be stressed that the interpretation is very different. In particular, this chapter is not concerned with anchoring phenomenon. 2Although the axioms used are the same, their interpretation is revised to better fit the context of changing tastes. 3Agents can always 'convexify' a choice set composed of lotteries by randomizing. Chapter 3. Inter-temporal Flexibility Preference 52 The choice problem is similar to one found in the context of social choice where the only information available ex-ante is the future utility function set representing the individuals in society. In other words, a Pareto frontier can be identified, but it is unclear how prospects on the frontier will be ranked. The Second Welfare Theorem states that revealed choice at each date is tantamount to having a social planner optimize some linear combination of individuals' utility functions. The weights assignment at date t corresponding to the convex combinations of utility functions from \&t, must be unknown before date t or else the agent essentially knows how date t prospects will be ranked ex-ante, contradicting the assumption that she cannot completely order future choices4. It is worth stressing that the uncertainty over future weights assignments is key in relating the concept of changing or uncertain future tastes with that of an incomplete ordering. Thus one can regard an inter-temporal sequence of partial orderings as a representation of Knightian uncertainty over future tastes. In the last chapter, the incomplete order, {yp, >-NP}, was introduced as a basis for anchored preferences. Here, the same foundation is used but in the context of changing tastes. Kreps and Porteus (1978) relate utility indices at different dates through a time-consistency condition. With the addition of a time-consistency condition to relate the partial order, {>-f, >-?p}, with that of the subsequent date, the theory presented here leads naturally to a preference for flexibility. Due to the latter property, the resulting theory of choice is termed an Inter-temporal Flexibility Preference. Since the approach essentially weakens the Kreps and Porteus (1978) axioms, it should be viewed as normative. In particular, if there is no uncertainty over future tastes, Inter-temporal Flexibility Preference reduces to the Inter-temporal von Neumann-Morgenstern Utility 4 A n incomplete preference relation cannot fully describe truly static (i.e., one-shot) choice problems since some decision must be made and revealed choice therefore forces a comparison among prospects. When future choice is considered ex-ante, however, there is nothing to force the comparison. In other words, inter-temporal choice is a much more natural setting in which to consider incomplete orderings. Chapter 3. Inter-temporal Flexibility Preference 53 theory of Kreps and Porteus (1978). In addition to a utility for flexibility, the formulation can represent non-linear induced preferences over future prospects. By contrast, Kreps and Porteus (1979) note that induced preferences (e.g., a preference for future wealth induced by an optimization over consumption) can not, in general, be represented by an Inter-temporal von Neumann-Morgenstern utility function. To better understand the more general development to come, consider a simple ex-ample of an agent that lives in a three-date world where decisions can be made at each date, t = 0,1, 2 , but consumption can only take place at the last date, t = 2. At date 1 the agent' must choose a menu (decision tree) from which she can select consumption at date 2. Alternative menus can contain combinations of the following: an apple (a), an orange (o) or a mango (m)5. Assume that at dates t < 2 the agent does not know which fruit she will like best at date 2. Assume, further, that the agent does not know at date 0 which menu she will prefer at date 1. If choice behavior at each date satisfies some basic normative axioms (to be presented later), then a characterization must specify a utility function set at each date. The utility function set at date 2 can contain three types of functions that differ on which fruit is ranked first. Since there are 7 menus that can be formed from combinations of a, o and m, the utility function set at date 1 can contain 7 types of functions, defined over menus, that differ on which of the seven menus is ranked first. To relate the feasible rankings at date 1 to those at date 2 in a normatively meaningful manner, one has to impose time-consistency constraints. For example, suppose that the agent knows that her date 2 utility function set contains two functions: one that ranks the apple as best and one that ranks the orange as best. If at date 1 the agent's utility function set contains only functions that rank the singleton menu {m} as best, then regardless the realization of tastes at date 1, future consumption 5 For simplicity of exposition, the example employs deterministic outcomes rather than lotteries. This is done without loss of generality. Explicitly considering lotteries would only serve to unnecessarily complicate the example. Chapter 3. Inter-temporal Flexibility Preference 54 choices made at date 1 will, in general, be inconsistent with the possible preferences at date 2. The agent will choose ex-ante a menu that she knows will be inferior ex-post. To avoid such inconsistencies, it seems reasonable to require that any menu that the agent will surely prefer at date 2 will also be preferred ex-ante regardless of tastes at date 1. Furthermore, if the agent will surely (i.e., regardless of changing tastes) prefer some menu, x, over another, say y at date 1, then there must be a realization of tastes at date 2 that justifies this ex-ante preference6. These consistency requirements guarantee that the Pareto frontier at date t, as seen at the earlier date s < t, is consistent with the realized Pareto frontier at date t. Everything else being equal, the agent will never choose a consumption menu ex-ante that she knows will be inferior ex-post. In particular, the Kreps-Porteus paradigm is recovered whenever utility function sets are singletons. A consequence of time consistency is that for any two menus, x, x', it must be that the menu xU x' weakly dominates x at date l 7 . In particular, this requires that for any member, U\, of the date 1 utility function set max U^JV^Uxix) (3.1) The time consistency condition can also be used to derive an upper bound for any date 1 utility function. If Bp(x) denotes the set of all possible outcomes (fruits) that weakly Pareto dominate the contents of the menu x, then clearly, assuming Bp(x) is not empty 6 This means that under some realization of tastes at date 2, something from the menu x will dominate anything on the menu y. A discussion of a stronger time consistency condition is found in an appendix. The stronger condition requires that if the agent will surely (i.e., regardless of changing tastes) prefer some singleton menu, x, over another singleton, say y, at date 1, then she will also surely (i.e., regardless of changing tastes) prefer the item in x over the item in y at date 2. The stronger condition leads to the interesting result that the set of possible tastes must shrink weakly in time. Unfortunately the stronger condition also generally implies a potential for manipulation. 7Although this result may seem obvious, it is important to stress that in the absence of the time consistency condition, there is nothing to relate x and x U i ' . These menus would be seen as primitives independent of their future role as menus from which further choice will be undertaken. Chapter 3. Inter-temporal Flexibility Preference 55 U,(x)< mm lM{j} ) " (3-2) jeBy{x) If the utility function set at date 2 contains only a single utility function, then the maximal element in x weakly dominates all elements of x. The two bounds in Eqs. (3.1) and (3.2) converge (that is essentially the Kreps and Porteus (1978) result). If, on the other hand, the utility function set at date 2 contains two or more functions, it is possible for-no single outcome in x to weakly dominate x. For example, suppose that at date 2, the mango Pareto dominates the apple and the orange but that the apple and orange do not dominate each other. This means that Bp({a,o}) — {m}. If it is the case that max Ui({j}) < Ui({m}), then, with respect to the utility function or 'taste' U\, the je{a,o} menu containing the apple and orange can be strictly preferred to the singleton menus containing one of the apple or the orange. I.e., it is possible that max CM{j}) < LMz) < £M{m}) j€x Whenever the utility of a menu exceeds that of its maximal element, an agent is said to possess a utility or preference for flexibility. What is important to understand is that it is possible for the decision maker to exhibit such a preference for flexibility because future preferences are incomplete yet tied together by a time consistency condition. In investigating the structure of representations satisfying the axioms in this chapter, it is useful and natural to define the concepts of First and Second Degree Flexibility Dominance. Suppose that the date 2 utility function set for the agent in the earlier example has two utility functions. One strictly prefers mangos to all other fruit and the other is indifferent to the consumption of any quantity of mangos (including zero). A date 1 utility function that exhibits First Degree Flexibility Dominance (FFD) will assign higher utility to menus with more, rather than less, mangos (holding apples and Chapter 3. Inter-temporal Flexibility Preference 56 oranges constant). In other words, an FFD utility function aggregates all possible future preferences. This property is important if one is to rule out manipulation. For example, suppose that the agent's revealed date 1 choice is rationalized through the maximization of a date 1 utility function that is insensitive to the number of mangos in a menu. The agent will therefore freely sign a contract in date 1 that limits the number of mangos she can consume in the next period. If a preference for mangos is realized at date 2, then the agent will pay to break the contract. Note that the general theory does not require a premium for flexibility to exist. Likewise, von Neumann and Morgenstern do not require their agents to have monotonically increasing utility functions. The absence of the latter property can lead to 'free lunches' in a multi-agent society. By contrast, each date t preference which exhibits FFD also adheres to the axioms postulated by Kreps (1979) and is not vulnerable to being exploited for arbitrage. If one insists that all realized tastes exhibit FFD then the agent's preferences are immune to manipulation. Second Degree Flexibility Dominance (SFD), on the other hand, requires a form of 'flexibility hedging' as well as a marginally diminishing utility for flexibility (as required in Nehring's (1999) theory). SFD requires that the added utility derived from adding a mango to the menu, {a, o] is less than or equal to the added utility derived from adding the mango to either {a} or {o}. If a date 1 utility function, say Ui, exhibited SFD then ^({a.o.m}) - C/i({a,o}) < C/i({a,m}) - £A({a}) (3.3) To see the connection to hedging, assume that the agent must choose at date 1 between a lottery, / , that awards {a,m} or {a, o} with equal probabilities, and a lottery, g, that awards {a, o, m} or {a} with equal probabilities. The agent chooses by maximizing the expected utility of some (randomly chosen) utility function in the convex hull of \I>i. If the realized utility function happens to be Ui, then Eqn. (3.3) implies that the agent will reject g in favor of / . The agent will pay a premium to move the mango from the larger Chapter 3. Inter-temporal Flexibility Preference 57 menu to the smaller one in the outcomes awarded by g. In other words, the agent prefers to hedge by reducing the dispersion between the menus. 3.1.1 Related literature on utility for flexibility The benefit or utility that a decision-maker may derive from having flexibility is the sub-ject of research for a literature that is apparently separate from that on inter-temporal utility theory. The classic theoretical reference8 on preference for flexibility is Kreps (1979). Kreps considers preferences over the set of possible opportunity sets. Axiomati-cally, he derives a theory of changing tastes. Because an agent knows that her preferences may change with non-zero probability, the value of a set is partly associated with how well the set accommodates contingencies of changing tastes. In contrast to theory presented here, however, Kreps' main purpose is to obtain a representation where the agent has specific probabilities about changes in future tastes, rather than Knightian uncertainty. In the last few years, the concept of preference for flexibility has been modified and abstracted by Bossert, Pattanaik and Xu (1994), Pattanaik and Xu (1998), Puppe (1995, 1996), Nehring and Puppe (1996, 1999), Bossert (1997) and Nehring (1999). An impor-tant finding of this literature is that preference for flexibility is intimately related to discontinuous and/or partial orderings over singleton opportunity sets. If one wishes to derive a preference ordering over opportunity sets that exhibits a preference for flexibility from a more primitive ordering on individual prospects, then the inducing ordering must be either discontinuous or incomplete9. If one wishes to retain continuity, the implication is that a normative theory of changing tastes must arise from primitives that partially order the set of future prospects. 8Other historical references can be found in Kreps (1979) as well. 9 In particular, this refers to Expected Utility Theory as well as most of its alternatives (which relax the Strong Independence Axiom). Chapter 3. Inter-temporal Flexibility Preference 58 The most prevalent approach in the literature is to axiomatize preference for flexibil-ity directly over opportunity sets of prospects as opposed to the more basic individual prospects. Moreover, although the notion of dynamic or changing tastes motivates some approaches (Kreps (1979) and Nehring (1999)), there is no truly inter-temporal theory of flexibility. The models in the literature are generally two-stage, where the first stage consists of choosing a 'menu' and, the second, to selecting an object from the menu. Kreps (1979, 1989) alludes to, yet does not fully develop, a multi-period theory. Although the agents in this chapter generally have a 'utility for flexibility', the rep-resentation of inter-temporal preference need not involve a positive linear weighting of future tastes as in Kreps (1979) and Nehring (1999). In other words, the representation does not require a probabilistically sophisticated approach to the uncertainty surround-ing future tastes. This is a desirable feature, since it allows one to model Knightian uncertainty over future tastes (i.e., without reference to subjective probability). Lastly it should be mentioned that the idea of incomplete preferences in the context of inter-temporal choice has been explored by Bewley (1986 ,1987), although he does not relate this to a preference for flexibility. 3.1.2 Induced Preferences A n interesting property of Inter-temporal Flexibility Preferences is that the theory is 'closed' under partial optimization under more general conditions than is Inter-temporal von Neumann-Morgenstern Utility. What this means is that the indirect or induced rep-resentation derived by optimizing over a subset of choices (e.g., unobservable actions) still obeys the basic axioms. The conditions under which this is true are far less strin-gent for Inter-temporal Flexibility Preferences than for Inter-temporal von Neumann-Moregenstern Preferences. Operationally this is important since our models of agents are generally abstractions of more complicated preferences. As Kreps and Porteus (1979) Chapter 3. Inter-temporal Flexibility Preference 59 point out 1 0 , a detailed theory from the class of Inter-temporal von Neumann-Moregenstern preferences wil l not reduce to a simpler model of the same class (except under unreal-istic assumptions about the detailed preferences). B y contrast, models in the class of Inter-temporal Flexibility Preferences that are closed under such reduction abound. In this chapter, no specific reference is made to the stochastic process governing the changing of tastes. The reason for the omission is that the conditions required for a general equilibrium will place necessary constraints on admissible processes of changing tastes. A n equilibrium model is presented in the next chapter. One of the conclusions there is that, to some degree, the specifics of the stochastic process governing the changes in taste that are uncorrelated across individuals are unimportant. The contribution in this chapter can be seen as a normative inter-temporal extension of Kreps' (1979, 1992) theory of utility for flexibility and unforeseen contingencies. The approach is also attractive for reasons that are not related to Knightian uncertainty over future preferences. Even if agents can, to a first order approximation, anticipate future tastes, another difficulty with the Kreps and Porteus (1978) approach is the assumption that at each date, the agent's actions are consistent with the von Neumann-Morgenstern axioms of Expected Uti l i ty Theory (EUT) . Aside from the induced utility problem men-tioned earlier, there is now an impressive body of empirical literature documenting vi-olations of the Expected Uti l i ty Hypothesis 1 1 . Although' one can argue, as does Levi (1997), that empirical evidence against E U T is not sufficient to reject it as normative theory, the view taken here is that economic rationality requires less normative structure than implied by the E U T axioms. As one line of approach, one can require, as do most 1 0 Also , see Kelsey and Milne (1997, 1999) and Machina (1984). "Notable citations from the empirical literature on E U T violations include Allais (1953), Camerer (1989), Goldstein and Einhorn (1987), Hey and Orme (1994), Kahneman and Tversky (1979), Licht-enstein and Slovic (1973), MacCrimmon and Larsson (1979), MacCrimmon, Stanbury and Wehrung (1980), Tversky and Kahneman (1986). Chapter 3. Inter-temporal Flexibility Preference 60 theoretical departures from EUT, that at each date the agent's preferences are complete, transitive and continuous (but do not necessarily obey the Strong Independence Axiom), which implies a representation by a continuous cardinal utility function (see Machina (1982)). However, one encounters a more serious problem of how to impose dynamic consistency without violating consequentialism or the reduction principle. Other related theoretical work includes Chew and Epstein (1990), Cubbit (1996), Kelsey and Milne (1997) and Segal (1997), Epstein and Zin (1989) and Sarin and Wakker (1998). The theory presented here may be roughly interpreted as a normative inter-temporal theory of random preferences. This approach has empirical justification (see, for example, Hey and Orme (1994)). The rest of the chapter is organized as follows. Section 3.2 introduces the basic axioms and concepts, and derives the main results. Section 3.3 discusses the results and their relationship to the induced utility problem of Kreps and Porteus (1979), as well as the theories of Kreps (1979) and Nehring (1999). Section 3.3 also introduces the concepts of First and Second Degree Flexibility Dominance. 3.2 Theory 3.2.1 Formulation of the Choice Problem and Agents' Preferences Following the framework suggested by Kreps and Porteus (1978), consider an arbitrary finite sequence of dates, t G 1,... ,T , where at each date an agent must choose an action, du from a current opportunity set, xt. The action, dt\ is a probability measure over outcomes. Each outcome takes the form of a pair, (zt,xt+\), where zt G Zt is a bundle of goods in the compact metric space, Z t , representing the goods available for consumption at date t. xt+i is a future opportunity set. Specifically, dt is an element of Dt, the set of all probability measures over the Borel sets of Zt x Xt+\. In turn, Xt+x, Chapter 3. Inter-temporal Flexibility Preference 61 representing all possible t + l opportunity sets viewed from date t, is the set of all closed subsets in Dt+i endowed with the Hausdorff metric. Since ZT is metrizable and compact, and assuming XT+I = {0}, DT is metrizable and compact in the Weak* topology. Kuratowski (1950 - cf. §42) proves that XT, the set of all closed subsets in DT, is also a compact metric space. Thus Z T - I X XT is compact, meaning that DT-\ is metrizable and compact in the Weak* topology. Clearly, this can be continued recursively to t = 0, when the agent must choose a distribution, do from a closed subset, x0 of Do- A choice problem at date t is simply some element of X T , say xt. Definition 3.1 A dynamic choice problem at date t induced by the sets, Z T , t £ 0,. . . ,T, is any element, xt of X T . An action at date t is any element, dt of Dt. An agent faced with a dynamic choice problem must select an action, dt, from xt C Dt consistent with some ordering over Dt. The choice behavior of the agent at date t can thus be summarized by a preference relation, yt, over Dt. In contrast to Kreps and Porteus (1978), who assume that >~4 is complete, negatively transitive, continuous and invariant under mixture (the von-Neumann and Morgenstern axioms), Axioms 2.1-2.3 impose a weaker normative structure on the agent's choice problem. These are quoted here with an interpretation appropriate to the context of changing tastes. A x i o m 3.1 Given a history, yt £ ZQ X . . . x Z T - \ , and any dt,d't £ Dt, exactly one of the following binary relations holds (with interpretations in parenthesis): (d't is strictly preferred to dt for at least one realization of taste at date t.) (dt is strictly preferred to d't for any realization of taste at date t.) (dt is at least as good as d't for any realization of taste at date t, but is no better than d't for at least one realization of taste.) (i) < yNytp dt (ii) dt yp d't (hi) dt~pd> Chapter 3. Inter-temporal Flexibility Preference 62 Moreover, for any ct € Dt, dt yPt d't => ct >Pt d't or dt >-^p ct (Negative Transitivity). The reference to 'tastes' and their realization is purely interpretive at this stage. As in Chapter 2, dt hPt d't denotes that either dt >-p d't or dt ~p d't. This corresponds to a notion of weak Pareto dominance with respect to tastes. A x i o m 3.2 (Continuity) Given the history, yt £ Z0 x . . . x Z T - \ , and any dt G Dt, at each date t Bp(dt) = K e A K hi dt} = Closure{d't\d't yp dt} if {d't\d't yp dt} ? 0 = {dt} otherwise and {d't eDt\d't yp dty = Closure{d't\dt ^ p d't} if {d't\dt y%p d't} ± 0 = {dt} otherwise where closed sets are defined with respect to the topology of weak convergence. Further, if d1^ —> dt and c™ —> ct are weakly convergent sequences in Dt with d" ~ p c" for every n, then dt ~ P Q . A x i o m 3.3 (Independence) Given dt, d't, ct E Dt and a e (0,1], and the consumption history, yt & ZQ X . . . x Z T - \ , dt yPt d't implies adt + (1 — a)ct >~Pt otd't + (1 — a)ct and dt >-ytP d't implies adt + (1 — a)ct >~^P ad't + (1 — a)ct Chapter 3. Inter-temporal Flexibility Preference 63 Axiom 3.2 is a technical condition on the 'at-least-as-good-as' and 'no-better-than' sets of yPt while Axiom 3.3 extends the Independence axiom to the non-standard ordering defined by the pair (yvt>yytP)- Restating Theorem 2.1: Theorem 3.0 Axioms 3.1-3.3 are necessary and sufficient for the following representa-tion: for every p,q € Dt and history, yt € ZQ X . . . x Z T - \ , Ep[Ut\) > 0 (3.4a) Ep[Ut]) = 0 (3.4b) Ep[Ut}) > 0 (3.4c) where ^ is a subset of C(ZT x Xt+i), the real-valued, continuous and bounded functions on ZT x Xt+\, and Eq[-\ denotes an expectation taken over the distribution q. The elements of are defined up to an affine (positive linear) transformation. ^ v t L has the interpretation of a utility function set. The agent can be of many 'minds' regarding future preference, as if she represents a group of many ordinary von Neumann-Morgenstern utility maximizers instead of just one. Thus the representation defining q y-Pt p has the desired interpretation of a Pareto dominating type of preference in which all the utility functions in tyf agree on the relative ranking of q and p. B y contrast, y y t p can be interpreted as a non-Paretian type of preference where some, but not necessarily all, of the utility functions agree on a ranking. The different utility functions in tyf can be interpreted as a fundamental or basis set of 'tastes' for the decision-maker in a manner which is to be made clearer shortly. Choice between prospects is unambiguous only if one prospect Pareto dominates the other. Note that if every utility function set is a singleton, the Kreps and Porteus (1978) assumptions are recovered. qyPtp inf {Eq[Ut}-q~pp inf {Eq[Ut\-« sup (Eq[Ut}-Chapter 3. Inter-temporal Flexibility Preference 64 If \I/f contains more than one element, it is possible to have p >-yP q and q >-ytF p. This corresponds to a situation in which q is preferred to p with respect to some utility function in ^ f , while p is preferred to q with respect to another member of \&f" (i.e., neither prospect Pareto dominates the other). In such a case, the prospects, p and q are incomparable through the Pareto ordering, ypt. If the agent is forced to choose, however, she will pick one of p or q and may not be indifferent between them 1 2 . This renders the partial ordering somewhat irrelevant in the case of static choice. However, it is important to emphasize that in an inter-temporal context the concept of a partial ordering remains non-trivial for it corresponds to the inability of the agent to anticipate future choice (even though current decisions are well defined). The following axiom formalizes this and ties the notion of changing tastes to the partial ordering defined earlier A x i o m 3.4 (Changing Tastes) Assume s < t and yt = (ys, yt-s) where yt 6 ZQ X . . . x Zt-i, ys € ZQ x . .. x Z s _i and ys-t € ZS x .. . x ZT-\- If dt >-ytP d't and d't >-^P dt then contingent on the future path, yt, as seen from date s, the agent cannot be certain of her date t choice between dt and d't-Finally, from here on it is implicitly assumed that all choice sets, xt, are convex. A x i o m 3.5 (Convex Choice Sets) All choice sets, xt, are convex. This is a natural supposition since the agent can convexify any choice set by mixing (i.e., throwing dice to decide among prospects), observed choice consistent with Axioms 3.1-3.5 will satisfy the following theorem. 1 2 I n this case, one cannot be sure whether the agent will pick p or <j, but if one could run a repeated experiment, the probability of picking p would not necessarily be the same as that of picking q. Chapter 3. Inter-temporal Flexibility Preference 65 Theorem 3.1 If Axioms 3.1-3.5 apply to a dynamic choice problem then, (i) For each time period, t, and given a consumption path, yt G ZQ X . . . x Z T - \ , there is an associated set, ty'f, containing basis 'utility' functions over ZT x Xt+\. Each Ut G ^'f is bounded and continuous in both arguments. (ii) The agent's revealed choice at date t corresponds to the maximization over the choice set, xt, of the expected value of some uncertain element of y^1 C c o ^ ^ ' ) where \&f C y™ and co (* f ) is the closure o / H u l l * f . The situation is analogous to that of aggregate decision making for a society of indi-viduals with different utility functions. In fact, a variant of the Second Welfare Theorem applies in this case (see Theorem 2.2). The theorem states that, as long as the choice set is convex, revealed choice can always be rationalized by maximizing a positive linear combination of the utility functions in the utility function set. The condition, C yf\ ensures that all the 'basis' tastes are possible at date t (otherwise, a smaller basis would apply). A key implication is that the utility function to be maximized at date t is not known with certainty prior to date t, otherwise future preference ordering would essentially be complete. To preserve the assumption of incompleteness, viewed from before date t, the utility function that wil l be used at date t must be "assigned" randomly by the agent at date t. The 'assignment' is, of course, not something that the agent chooses - it simply happens. In reference to this process the term bias assignment is used throughout. In particular, the agent facing the choice problem may know nothing about her mechanism determining the bias assignment and thus may not possess a probability distribution for future tastes. It is at this point especially that I must ask for the reader's forbearance in deferring a discussion of the bias assignment. This is done with the belief that only certain processes will be admissible in a general equilibrium, an example of which will be Chapter 3. Inter-temporal Flexibility Preference 66 1 Choice Set: x, = { d1,, d2,, d3,,...} Utility Function Set: 4*,y Bias assignment v choose if t = (p = Vi, (z, x,+1); J-p='/2 ,(z', x',+;)) —I Choice Set: x,+y = { cf,+i, c / , w .cft+i,...} Utility Function Set: ¥ ,+; ( y ' z ) Bias assignment 7 Resolution of uncertainty Uncertainty resolution => draw. 0, x,+i) • Consume z to give next period's Utility Function Set: ^'Z > ) Next period's choice set Figure 3.1: A Dynamic Choice Problem with incomplete preferences presented in the next chapter 1 3. In addition, it is important to emphasize that Theorem 3.1 refers to revealed preference. In other words, the agent is acting as if some uncertain utility function is assigned at each date. If dt G xt is degenerate, then it will be denoted henceforth as (zt,xt+i). If an oppor-tunity set, xt, consists of only a single choice, say dt, then it is denoted xt = {dt}. Such singleton sets are of primary importance in the formulation of the theory. A sketch of the setup is in Figure 3.1. 1 3 The equilibrium demonstrates that the part of an agent's preferences about which she is uncertain is independent of observable economic variables. Thus it is not necessarily important to address the question of 'how' the bias assignment comes to be. Chapter 3. Inter-temporal Flexibility Preference 67 3.2.2 Time Consistency Util i ty function sets in consecutive periods are related through the following. A x i o m 3.6 (Time Consistency) For any t, z £ Zt, f,g& Xt+i, and consumption history yt) if max Ed[Ut+i] > max Ed[Ut+l] for every Ut+l € y^{z) then (zj) >Pt (z,g). def deg Recall that 3^+Y^ is the set containing all possible linear combinations of utility func-tions in ^ ^ { ^ that may serve as a bias assignment (including ^t+i^ i tself) 1 4 . Suppose, that regardless of the utility function maximized at date t + 1 (i.e, regardless of tastes), an element from the menu / will be strictly preferred to any element from the menu g. According to Axiom 3.6, at date t the agent foresees this and, to avoid certain regret, strictly prefers the opportunity set / to g. In particular, this holds true when / and g are singleton sets. Using the assumed continuity of preferences from Axiom 3.2, it is easy to prove that part of Axiom 3.6 also applies to the case of weak dominance 1 5 . The converse of the weak form with respect to singleton menus then implies Lemma 3.1 For any t, z € Zt) / , <? E Xt+i, and consumption history yt, if f = {dt+i} and g = {d't+1} where dt+i,d't+1 <E A + i , and (z,f) y^p (z,g) then dt+i >-g P 2 ) d't+1. The lemma requires that any action taken at date t will not be regretted with certainty at date t+1. More specifically, suppose that the agent must choose a current consumption and future action combination at date t. Suppose further, that conditional on the current consumption choice and realization of a bias assignment, the agent prefers at date t the 1 4 Note that 3^t+Y^  does not depend on the date t + 1 choice set endowed at date t. It is the set of all possible bias assignments conditional on yt and zt only. Note that the stochastic process associated with bias assignment may depend on the date t + 1 choice set. 1 5 i .e. , if max Ed [ip] > max Ed[ip] for every ip € X+Y^) t n e n (z>/) hP (z,9) d€f d€g J t Chapter 3. Inter-temporal Flexibility Preference 68 future action d t + 1 to d't+l. The lemma implies that it must be that dt+\ will also be preferred to d't+l under some future realization of taste. I.e., any current preference for an action must be justifiable ex-ante through the possible realization of some future taste. Note that Axiom 3.6 does not rule out regret per se. A n agent who cannot perfectly predict future preferences is virtually guaranteed to experience regret some of the time. Axiom 3.6 simply rules out situations when regret is certain and foreseeable. Further, if the utility function set at date t contains only a single element, then Axiom 3.6 is equivalent to the Temporal Consistency Axiom in Kreps and Porteus (1978), and the theory reduces to their recursive utility formulation. A n interesting consequence of Axiom 3.6 is that the utility of a choice set at date t, say xt, depends only on the set of maximal utilities attainable at date t + 1 . Formally, let a be an index set for elements of 3^+i^ and define the mapping wf1 : Xt •->• 72.^ '+1' ' through {w?{x))a = maxEd\iba) (3.5) where ipa £ 3^+1^ a n a - wtt{x) represents a (possibly) uncountably infinite vector of maximal cardinal utilities for the set x. Denote wf{x) > wf'^x') whenever (wtt(x))a > (wt'(a;'))a for every a, and there is some a for which the inequality is strict. Likewise, denote w^ix) w^ix') whenever {wf,{x))a > (^ '(x ')) ,* for every a. Proposit ion 3.1 Fix f /f € tyf. Then for any history yt € Z$ x . . . x Zt-\, z & Zt and x € Xt+1, U?(z,x) = u?t(z,wj*i*\x)) (3.6) where w^z)(x) > w^[z\x') u?(z,w%{z)(x)) > uT{z,w^\x')) and w%{\x) » =• uf(z,w^\x)) > uf{z,W^\x')). Chapter 3. Inter-temporal Flexibility Preference 69 The properties of Uf* outlined in Proposition 3.1 are similar to those possessed by the utility functions denned in Kreps (1979) when agents have a preference for flexibility. Indeed, Proposition 3.1 implies that Uf^z,x U x') > \Jx(l{z,x) for any x,x' £ Xt+\. Kreps' other condition, namely that f / f (z, x U x') = U?l{z, x) => U^(z, x"UxU x') = Utt(z,x"Ux), may not hold, since u\l is not necessarily strictly increasing in wf^\x). This, however, does not imply that there is no preference for flexibility, as the next theorem demonstrates. Theorem 3.2 Fix Uf1 £ * f . Then for any history yt £ Z0 x x £ Xt+i, such that Bp(x)^f]B[yttZ)(d)^(D d€x it is the case that max [If(z,{d}) < U?{z,x) < min Utyt{z,{d'}) (3.8) Theorem 3.2 states that the utility of an opportunity set, x £ Xt+i, as seen at date t, is bounded between the utility of the maximal element in x and the lowest possible utility that can be derived from a singleton opportunity set 1 6 which, at date t + 1 , dominates every element of x. The difference between the upper and lower bounds in Eq. (3.8) corresponds to a maximum flexibility premium, associated with the choice set, x, over the utility that can be derived from any one of its constituent elements. Note that if ^[+1^ contains only one function the two bounds converge 1 7 and, Proposition 3.1 implies that, Eq. (3.8) reduces to the time consistent recursive inter-temporal utility introduced by Kreps and Porteus (1978). Indeed, it is only where there is some uncertainty about 1 6 Recal l that ^(d), defined in Axiom 3.2, is the set of t + 1-prospects that Pareto dominate d. Bp(x) is therefore the set of t + 1-prospects that dominate every member of x. 1 7 I f ty^i^ is a singleton, the set of elements that Pareto dominate everything in x has a non-empty intersection with x. x Z t _ i , z £ Zt and (3.7) Chapter 3. Inter-temporal Flexibility Preference 70 future endowments, beliefs or utilities that an intuitively rational basis exists for having a non-zero premium for flexibility. In the present context this corresponds to a situation where ^[+{z^ is not a singleton. Theorem 3.2 motivates characterizing any agent facing a dynamic choice problem, and who acts in accordance with Axioms 3.1-3.6, as having an Inter-temporal Flexibility Preference. 3.3 Discussion 3.3.1 Inter-temporal Flexibi l i ty Preferences and Induced U t i l i t y Functions If ^{z) = {U^{z)} is a singleton, Proposition 3.1 implies that Ufl(z,x) = uf (z, m&xEd[Ut+{z^]) for any menu, x S Xt+i and any Ufl 6 yf1. u\l is continuous in both arguments and strictly increasing in the second. In particular, if x is taken to be a singleton menu, all date t utility functions are a monotonic transformation of date t + 1 expected utility. Since Ed\Ul+iz\ for d € A+i, is a linear functional on A+i, the indifference surfaces corresponding to U^{z^ are linear. Thus indifference surfaces on Dt+i induced by singleton menus at date t are also linear for each Ufl(z, {•}) € yf*. If Vp|+i^ is n o t a singleton, however, there is nothing that guarantees that a given date t utility function restricted to singleton menus, for example Uf^z, {•}) where Ufl G yf\ is itself a linear functional over A+i- To see this, consider in Proposition 3.1 that if there are two functions in 3 ^ ° , say U^{Zt) 1 and U^{Zt] 2 , then in general U?(z, = uf (z, Ed[U^{Zt) \ Ed[U^{Zt) 2]) for any Uft <E yf1. In particular it should be clear that 1 8 , U?(z,{dt+1}) = U?(z,{d't+1})^ U?(z, {dt+1}) = U?(z, {adt+1 + (1 - «K+1}) 1 8 I f Ed[Ut+{^1'} = Ed'^Jt+i^ l\ f ° r both i = 1,2 then mixture invariance holds. That is a direct result of Axiom 3.3. Chapter 3. Inter-temporal Flexibility Preference 71 where adt+i + (1 — a)d't+1 with a E (0,1) corresponds to a probabilistic mixture of the distributions dt+i and d't+1. Thus [/^(z, {•}), does not necessarily have linear indifference surfaces in Dt+\. In fact, there is a compelling argument for allowing derived utility functions over singleton menus (i.e., Uyt(z, {•})) to have convex lower contour sets in Dt+\- In a follow-up paper, Kreps and Porteus (1979) demonstrate that preference for wealth induced through a preference for inter-temporal consumption implies an induced inter-temporal utility function, Uf* *(z, {•}), with convex lower contour sets in Dt+\- To explore this in the current context, consider an agent who at date t has some preference over the set of current payoff-opportunity set pairs but that this is induced through an optimization over a set of 'actions', from a more primitive utility for consumption. The relevant question is whether the induced or optimized utility can be modeled as Inter-temporal Flexibility Preference, given that the more primitive preference relation is assumed to be an Inter-temporal Flexibility Preference. Specifically, assume the agent faces a dynamic choice problem and has Inter-temporal Flexibility Preference. Assume further that Zt = At x Bt for each date, t, and that at each date the agent has the opportunity to optimize consumption by selecting, at E At, from a closed set, Af (bt)- Here, yt E Z0 x . . . x Zt-\, can be thought of as a history of optimization and endowment (at and bt, respectively). Note that the feasible set for at can depend on the current endowment, bt. A^tl(bt) itself can be seen either as a set of controls or simply a set of consumption choices that can be integrated out of the choice problem. Any direct utility function corresponding to a date t bias assignment has the form derived in Proposition 3.1: uf (ztMlfixt+i)) = < (a., h, ™ & ( a t A ) ) ( x t + i ) ) Chapter 3. Inter-temporal Flexibility Preference 72 After selecting a vector, at from a closed feasible set, Af'ibt), the indirect or induced utility is vf{buwtibt)\xt+1),xt+l) = max uf (au bt, w ^ M ) ( x t + 1 ) ) where vj^{bt^ *(xt+x) = wj.+fat'bt^ (xt+i) is the vector of maximal date t+1 utilities as in Eqn. (3.5) and a*t corresponds to the optimized vector of 'actions'. Consider the set of all date t induced utility functions j>r = {vr{bt,w%ibt)*(xt+1),xt+1) i uf e yt Note that yf1 depends on yt = (yt-i, (a-t-i> ^t-i))- Since a^_1 is a function of the realized date t opportunity set, xt, this means that yf1 in general depends on xt. Thus if yf1 itself represents some Inter-temporal Flexibility Preference 1 9, a necessary condition is that yf1 be independent of the optimization history. The next theorem demonstrates that this is also a sufficient condition. Before stating it, however, it is useful to define the set of optimal actions: 4 * W = (J Mgmzxuf(aubt,w{^>ht\x)\ (3.9) Theorem 3.3 Fix an Inter-temporal Flexibility Preference described by the sequence of feasible bias assignments, yf1, where yt G Z0 x . . . x Zt-\ and Zt — At x Bt. Af(bt) is a continuous mapping (with respect to the Hausdorff Metric) from Bt to closed subsets of At. Then the sequence of induced bias assignments, yf1, describes an Inter-temporal Flexibility Preference if and only if for every t and yt G Z0 x . . . x Zt-\ y{yt-i,at-iM-l)) A_ ^(yt-i,a't_1,bt-i)) ^ ^ 1 9 I t is easy to show that one can always find an Inter-temporal Flexibility Preference that rationalizes induced preferences by enlarging the set of potential bias assignments. Bias assignment in such cases will necessarily depend on the endowed opportunity set and not simply the history b0,... , bt-i-Chapter 3. Inter-temporal Flexibility Preference 73 whenever at_i,a't_1 G A ^ 1 , 6 ' ^ *. The equality, = denotes that corresponding elements in each the two sets are identical up to an affine transformation. According to Theorem 3.3, a theory with a singleton utility function set can only rep-resent the indirect preferences derived from optimal choice of, for example, consumption, under the very restrictive condition that the form of the utility function is essentially pre-served after optimization up to an affine transformation. This negative result is contained in Proposition 8 of Kreps and Porteus (1979). Uti l i ty for wealth is derived from utility for consumption, which in turn depends on many other factors. Since induced inter-temporal preferences are not expected, in general, to be Inter-temporal von Neumann-Morgenstern, Kreps and Porteus (1979) note with disappointment that an inter-temporal theory based on the Expected Uti l i ty Hypothesis may, at best, furnish a crude approximation. For an example, consider a two period dynamic choice problem where at date 0 the agent selects among combinations of current endowment plus future stochastic dividend distribution, (co, c\). It is assumed that realizations of endowments Co and C j are elements of the bounded and closed intervals, CQ,C\ C TZ, respectively. Moreover, the agent can transfer some or all of her endowment for later consumption. Assume the date 1 utility function set, yi° = {U^'co}, is composed of real-valued and continuous functions (indexed by a) over the closed and bounded interval Co + Cx C 7Z. The date 0 direct utility (bias assignment) over ( co ,C i ) and a transfer strategy, a G [0,Co], must take on the form, UQ ey0 C/ 0(a;(co,ci)) = U0(CQ - a, tuj°(ci + a)) = iz 0 (c 0 - a, {£?[t/ 1 a , c°(ci + a)]}) The induced utility is therefore Vo(co, ci) = max u0(co - a, {E[U?'c°(ci + a)}}) ae[0,co] Note that due to the 'hidden' optimization over transfer of consumption, Vo is generally convex under mixture of its second argument. In other words, given (CQ,CI) ^ {co,c[) Chapter 3. Inter-temporal Flexibility Preference 74 with Vo(co,ci) = VotccCi), it must be that Vo(co,aci + (1 — a)c[) < Vb(co,c"i), where ac\ + (1 — a)c'j is a probabilistic mixture of the distributions, C\ and c[. If there is only one utility function in y%°, say Ux°, then to be consistent with Proposition 3.1, it must be that Vo(c0,aci + (1 — a)c\) = Vo(co ,Ci) 2 0 . Thus the reason why Inter-temporal von Neuman-Moregenstern Preferences cannot generally represent induced utility is that they do not allow for the utility aggregator (i.e., V0) to be convex under mixture. The restriction implied by Eqn. (3.10) on Inter-temporal Flexibility Preferences, on the other hand, is not as severe and can be accommodated by various appealing specifi-cations. In the two period example, there are many parameterizations of 3 i^° = {Ui'00} that satisfy Theorem 3.3. A sufficient requirement is that for every a G Co, {Ur°{x + a)} = {Ur\x)}, xeCx Some examples follow: i) 3 i^° is the set of continuous functions on Co + C\. ii) yx° is the set of continuous and strictly increasing functions on Co + C\. iii) 3 i^° is the set of continuous, strictly increasing and concave functions on Co + C i . iv) ^ i 0 is a family of twice differentiable concave functions {U \ RA = — %(x) ~ f(x + a),a € TZ} for any fixed positive real valued function, / . v) 3^1° is generated by an arbitrary set of Constant Absolute Risk Aversion ( C A R A ) functions through positive linear combination: {U | U(x) = - E / j O / j e " ^ , ap > OV/3} 2 0 According to Theorem 3.3, if a*(co, •) can take on any value between, say, 0 and a, then it is necessary and sufficient that f/j°(c + a) = U"'ca{c) up to an affine transformation for every a € [0,a]. This is generally true only for constant absolute risk aversion (CARA) and linear utility. Chapter 3. Inter-temporal Flexibility Preference 75 Thus the class of utility function sets that can be associated with induced utility is rich. 3.3.2 First-degree Flexibility Dominance And State Contingent Plans In the dynamic programming approach to inter-temporal (von Neumann-Morgenstern) utility maximization, an agent assigns the utility of the most desirable choice to the choice set itself. Moreover, a choice element can be seen as a state contingent plan. In other words, contingent on the state of nature, the conventional approach to inter-temporal choice does not assign a premium for flexibility to a choice set over and above the value associated with the best (as viewed from the current period) of the set's elements. It is precisely this property of indifference between the choice set and a state contingent plan which implies that inter-temporal von Neumann-Morgenstern agents will trade, if allowed, to achieve their state contingent plan at date zero and will not trade thereafter2 1. As mentioned earlier, the bounds in Eq. (3.8) always converge when there is a single utility function in the utility set. In such a case the utility of a choice set is that of its maximal element. W i t h non-singleton utility function sets, however, agents with Inter-temporal Flexibility Preferences cannot forecast future tastes and, to reflect the Knightian uncertainty, may assign a premium to flexibility whenever Eq . (3.8) makes such a premium possible. The theory outlined in Section 3.2 is silent on the normative value and normative form of a premium for flexibility. It is useful to view these issues concerning flexibility as analogous to the criteria of First- and Second-degree Stochastic Dominance (FSD and SSD, respectively) for risky choice. To understand the analogy, recall first that the axioms of von Neumann and Morgenstern are also silent on the attitude of agents towards wealth and risk (i.e., the signs of the first and second derivatives 2 I T h e sufficient requirements are that state contingent claims are available for purchase at date zero and that the inter-temporal price equilibrium is characterized by Bayesian rational expectations. Chapter 3. Inter-temporal Flexibility Preference 76 of the utility function). F S D is commonly assumed as normative for the simple reason that anyone violating such a principle would give up 'something for nothing' (or less than nothing!). In this subsection a direct analogy is made for flexibility. SSD, on the other hand, is assumed to account for the economically relevant and near universal behavior of hedging. In the next subsection, an appropriate parallel is demonstrated for flexibility as well. W i t h respect to the first point, consider an agent whose date t bias assignment is a utility function, Ut- If Ut does not assign a premium to flexibility then there is a potential for regret and/or manipulation. In particular, the agent wil l be indifferent to trading a current opportunity set (menu) for one which contains a single element (the maximal one as seen from date t). If tastes can change, it is possible that the ex-ante maximal element wil l seem inferior relative to an alternative that was unnecessarily removed from the choice set ( at least with respect to some date t+1 bias assignment). The reduction of the opportunity set will be regretted ex-post. Even if unable to forecast changing tastes, a 'rational' agent wil l at least notice that the consequences of a binding contract limiting future choice lead to regret. Worse yet is the possibility that a market maker can make a 'free-lunch' profit by costlessly inducing agents to bind themselves to some contingent plan which ignores the possibility of changing tastes. The 'bound' agents will later willingly pay a penalty to escape from the contract. Thus, from both a normative and arguably descriptive point of view, the imposition of zero premium for flexibility seems unappealing. The discussion suggests that an acceptable Inter-temporal Flexibility Preference the-ory should, in general, require a non-zero premium for flexibility much the same way that a sensible expected utility theory requires monotonically increasing utility func-tions. Note that one can also make 'free-lunch' profits from inter-temporal von Neumann-Morgenstern agents who possess preferences that violate F S D . Chapter 3. Inter-temporal Flexibility Preference 77 A premium for flexibility ensures that, generally, possession of a choice set is strictly preferred to a commitment to any single member of the set. Agents do not, therefore, commit to contingent plans in the conventional sense, since this implies that they must be indifferent between the specific plan and the whole choice set. Moreover, the flexibility premium associated with the choice set reflects the degree to which the agent is conscious of her changing preferences. In that sense, the agent does account for the possibility of changing tastes even when these cannot be anticipated probabilistically as in the additive representation of Kreps (1979) or Nehring (1999). A final but important implication of this framework is that trading wil l , in general, occur in every period even when state contingent claims are available for purchase and the market is complete 2 2. This is in contrast with the models of Arrow (1964) and Debreu (1959). The discussion thus far motivates a more formal analogy between utility for flexibility and First-degree Stochastic Dominance. Definition 3.2 A utility function, Uf G yf, is said meet the criterion of First-degree Flexibility Dominance (FFD) for a given z G Zt iff its representation in Proposition 3.1 is strictly increasing in w^{z\ In other words, Uf respects F F D if and only if for every closed 2 3 set, xt+i G Xt+i, it is the case that w^{z){xt+1) > w{^[z)(x't+l) =» Uf (z,xt+i) > Uf{z,x't+1). To understand the intuition behind the definition, assume that Uf(z,x) is not strictly increasing in wt+i\x)- A t date t the agent is therefore indifferent to changes in the opportunity set 2 2 The use of the term 'complete market' requires elaboration when agents have changing tastes. A complete market in this context means one where a standard inter-temporal von Neumann-Morgenstern agent can completely hedge her future consumption. The possible existence of such a market is discussed later. 2 3Since only convex choice sets are considered here, this criterion is stronger than it has to be. Chapter 3. Inter-temporal Flexibility Preference 78 which make it strictly worse from the point of view of some possible date t+1 utility function(s). This situation invites the type of manipulation discussed earlier. A natural question to ask is why not simply enlarge the state space to include 'tastes'. A rational agent would then be willing to commit to state-contingent plans in the enlarged space and the theory would essentially correspond to state-dependent recursive utility (e.g., Skiadas (1997, 1998) and Epstein and Zin (1989)). There are several rebuttals to this. First, recall that the bias assignment is a result of Axioms 3.1-3.5 which do not make any explicit reference to 'states'. The Axioms only imply that the agent acts as if there are states associated with bias assignment. Moreover, as in the usual literature on preference for flexibility, the endogenous state space of possible future tastes is derived endogenously from preferences. More crucially, these 'states' are not truly observable even by the agent herself, since revealed choice is the only clue to the realized bias assignment. Given that the bias assignment states are not fully observable, the agent cannot condition on them directly, thus making 'fully' contingent plans rather vacuous. What has been so far modeled here and derived axiomatically is not the filtration of states, as in Skiadas (1998), but the rationalization of choice behavior, satisfying some normative rules, by a cardinal representation. What will be shown in the next chapter is that specific reference to the internal 'states' and their evolution, from an operational standpoint, is largely unnecessary in the exercise to fully characterize a market equilibrium. 3.3.3 Relation to Additive Representations and Second-degree Dominance It should be clear from Definition 3.2 that any utility function that meets the First-degree Flexibility Dominance criterion also satisfies the conditions specified by Kreps (1979) for a cardinal representation of 'utility for flexibility'. A s Kreps notes, however, a continuum of outcomes and possible 'tastes', as in the theory presented here, suggests Chapter 3. Inter-temporal Flexibility Preference 79 that there is no guarantee of a general additive representation 2 4, never mind an additive representation of the form, Ur(z,x) = u?{z, Y, Aamax Ed[U?+1]) Xa > 0, U?+1 € } f t 2 ) (3.11) a where a indexes elements of y^{z\ Note, in particular, that the second argument of uf(z,x) reduces to a linear combination of date t + 1 utility functions when a; is a sin-gleton. In particular, such a representation assumes linear induced indifference surfaces for Uft(z, {•}) over Dt+i and is, in general, not a good proxy for an induced utility (see Section 3.3.1). Another important paper on utility for flexibility is that of Nehring (1999). Nehring derives an additive representation of preference over choice sets by demanding that a utility function over opportunity sets, U, possess a property which he terms indirect stochastic dominance. In particular, this implies that, given arbitrary opportunity sets, U(x Uz)- U{x) >U(xUyUz)- U(x U y) (3.12) In Nehring's world, the additional utility for flexibility achieved by adding the set z to x U y never exceeds the utility for flexibility gained when z is added to x. The utility for flexibility is thus 'diminishing marginally'. Another way to state this is in terms of lotteries over opportunity sets. Let / be a lottery which awards the agent x\Jy or xU z, each with | probability. This is to be compared with the lottery g which awards the agent x U y U z or x, again, with probability \ for each case. According to Nehring's principle and Eq. (3.12) the utility of / is at least as great as that of g: l-U(x Uz) + ^U(x Uy)> l-U{x U y U z) + l-U{x) (3.13) 2 4 A general additive representation would imply that Uft{z,x) = uf' (z,^gmax vf(d)) where uf (d) is not necessarily a linear functional over Dt+\ (i.e., an expected utility). Chapter 3. Inter-temporal Flexibility Preference 80 In other words, an agent always benefits from hedging and is willing pay to move z from the larger set to the smaller set in g. A n alternative way to think of Nehring's indirect stochastic dominance is as an expres-sion of Second-degree Stochastic Dominance for flexibility. The premium for flexibility is assigned in such a way as to induce hedging against future changes in tastes. Such re-strictions are descriptive as opposed to normative, similar to imposing risk aversion (i.e., marginally diminishing utility) on von Neumann-Morgenstern agents. As with SSD, there is no way to 'manipulate' an agent who demonstrates a strict preference for g in the example, and thus violates indirect stochastic dominance. This argument prompts the following definition, Definition 3.3 A utility function, Uf G co(\Pf), is said meet the criterion of Second-degree Flexibi l i ty Dominance (SFD) for a given z G Zt if and only if for arbitrary Xx,X2,X3 G Xt+X Uf{z, xx U x2) - Uf{z, x2) > Uf(z, XlUx2U x3) - Uf{z, x2 U x3) (3.14) I end the chapter with some examples. Consider the date t+1 utility function set ^(V^) _ ^jj\,{yt,zt) rj2,(yt,zt)y and suppose that y(*izt) = {Ux {yt'zt) | Ux{vt'zt) = XU1' {vt'zt) + (1 - X)U2' {yt'zt) | A G [0,1]} Thus possible bias assignments (i.e., elements of J^t+i'2^) include all convex combinations of the basis functions, U1, and U2, up to an affine transformation. The agent wil l choose at date t + 1 by optimizing with respect to some Ux^yt'Zt\ and A is only 'revealed' 2 5 at date t + 1. Now suppose that for some date t utility function, Uf G yf Utp(zt,xt+1)= r max Jzt,s, Edt+1[( e - "d -p) /P[ / i . (*«.*) + (1 _ e - « ( i - p ) / p ) [ / 2 . (vt,«)) ]) d s JO d&xt+i \ \ / / 2 5 Recal l , again, that the agent only behaves as if a A is revealed at date t + 1. Chapter 3. Inter-temporal Flexibility Preference 81 where <p is increasing in its third argument and p G (0,1). Note that Ut(zt,xt+i) satisfies Proposition 3.1. Uf(zt, xt+i) also satisfies F F D 2 6 . It is also easy to show that t/f (z t , £ t +i) satisfies S F D . The representation conforms to a probabilistic interpretation (additive representa-tion) of changing tastes only if cp is separable in its last two arguments (i.e., (p(z,s,a) = p(z,s)a(z,a)). In such a case, the agent acts as if she has expected utility over fu-ture outcomes based on subjective probability over future tastes. For example, setting tp(z,s,a) = f(z)e~sa gives Ufizt, H + i } ) = f(z)Edt+1[pU^ + (1 - p)U2' <»•*>] so, at least with respect to singleton sets, the agent makes choices at date t as if she were at date t + 1 (although the bias assignment may change at date t + 1, p at date t may not be the same as A at date t + 1 ) . If, on the other hand, <p is not separable, there is no probabilistic interpretation in terms of 'weighted' tastes. The representation can, in such a case, be interpreted as reflecting the agent's Knightian uncertainty with respect to future tastes. Now consider a somewhat simpler example in which and assume that for some Uf* G yfl, U?(zt,xt+1) = $?(z, max ^ ( E ^ J E / 1 - <*•*>]) + max <p2{Edt+1 [U2' ^ } ) ) where <p\ and <pi are strictly increasing real-valued univariate functions and <f>f is strictly increasing in its second argument. It is clear that C/ t y t G yfl satisfies F F D . On the other 2 6 T o prove this, note that Ux^yt'z'^ is continuous in A (in the sup topology). Thus if max Edt+1[Ux <~VuZ'^} - max E d f + 1 [C/A ( y , ' 2 f ) ] = eA > 0, for some A, then there is a neighborhood of A, say N(X), over which this equality remains true with ex > e for some e > 0 and every A' € N(X). This is sufficient to prove that Uf- is F F D . Chapter 3. Inter-temporal Flexibility Preference 82 hand, whether it satisfies S F D depends entirely on the curvature of $ f . If $ f is convex, it is possible that Uf G yf will violate S F D . Consider the following scenario: dt+i M^iW1'{vt'zt)]) <p2(Edt+l[U2>^]) di 0 2 d2 1 1 d3 2 0 and set x = {di}, y = {d2} and z = {d 3 }. If <&f(z, a) = af(z), then Uf (z,xUy)- Uf (z,y) = f(z) = Uf (z,x U y U z) - Uf (z,y U z) and t / f G 3 f^ satisfies S F D . If, on the other hand, $f(z ,a) = a 2 / (^ ) , then Uf (z, x U y) - t / f (z, y) = 5/(z) < 7/(z) = C/f (z, x U y U z) - C/f (z, y U z) thus violating S F D . Note that $ j ' (z ,a) acts as an inter-temporal aggregator to separate inter-temporal substitution from risk aversion (see Kreps and Porteus (1978) or Epstein and Zin (1989)). If $( ' (z ,a) is convex in its second argument then the agent tends to prefer resolving uncertainty earlier rather than later. If such preferences are deemed desirable, then Nehring's (1999) S F D representation is not appropriate. It is important to realize that if there is only one utility function in the date t + 1 utility function set, all date t utility functions are S F D . Thus predictions of violations of this property are unique to Inter-temporal Flexibility Preferences. 3.4 Proofs Proof of Theorem 3.1: Part (i) is simply a consequence of Theorem 2.1. Part (ii) follows from Axiom 3.4 and 3.5 and Theorem 2.2. Chapter 3. Inter-temporal Flexibility Preference 83 • Proof of Lemma 3.1: The weak form of Axiom 3.6 implies that if max Ed[ip] > max Ed[ip} for every %p 6 then (z,g) >zP (z,f). In particular, fixing / = {dt+1} and g = {d't+1} implies that not (z,g) >pyt (z,f) not EM > Ed[iP] E yfrf) Axiom 3.1 implies that not (z,g) >zP (z,f) (z,f) y^p (z,g). To derive the implication, note that, according to the above, there is some ip e ^t+iz)) such that Ed[ip] > Ed,[ip}. B y Theorem 3.0 this is true if and only if d y^ypz) d'. • Proof of Proposition 3.1: Suppose w[v!{z\x) = w*j.+{z\x') and x ^ x'. The weak form of Axiom 3.6 implies that (z,x) yp (z,x') and (z,x') >zPt (z,x). B y Eq. (3.4) it must be that Uf{z,x) = Uf(z,x') for every Uf 6 * f . Thus Uf(z,x) = uf (z, w^{z\x)). Clearly, Ax iom 3.6 and Eq. (3.4) also guarantee the stated properties of u. • Proof of Theorem 3.2: Chapter 3. Inter-temporal Flexibility Preference 84 Note that the result easily holds when x is a singleton. One can therefore assume that x is not a singleton. Consider first the lower bound in Eq . (3.8), and set d* = argmax Uyt(z, {d}). The weak form of Axiom 3.6 implies that (z,x) >Pt (z,{d*}) and thus, by Eq . (3.4) U?(z, x) > U?(z, {d*}) = U^'z\d*). To obtain the upper bound, consider d & Bp(x), and note that Axiom 3.6 implies that (z,d) tp (z,x). By Theorem 3.0, it must be that U{z,x) < u[yt'z){d). Since this holds for every d € Bp(x), the theorem obtains. • Proof of Theorem 3.3: To prove sufficiency, first note that under the hypothesis, the optimization history does not matter. Without loss of generality, and to simplify the notation, refer to the history, bo,- • • ,h-i as yt. Also, denote the set of possible date t opportunity sets by Xf. The hypothesis implies that all functions in yfl are continuous and bounded on Bt x Xp+l. That is enough to satisfy Axioms 3.1 - 3.3. Axioms 3.4 - 3.5 are also satisfied by hypothesis. To prove that Axiom 3.6 is satisfied, one only needs to show that vf(bt, w^{bt^ *(xt+i), xt+i) does not depend on the last argument explicitly. To this end, let a*(bt, xt+i) = argmax uf [at,bt, w^iat'bt)(xt+i) Now consider two sets, xt+\ and x't+1 such that w^{a'bt\xt+i) = wl+{a'bt\x't+1) for some a G AVt_\. Since y^ i s assumed invariant, this must remain true for any a G Ayt_*i- In Chapter 3. Inter-temporal Flexibility Preference 85 particular, vf {btAXtt] =uf (a'(bt, xt+l), bu l i , ^ * ^ 0 * Wi)) = uf ( a * ( 6 t , x t + 1 ) , b t ,^ ( + r* ( b " " + l ) ' b t ) ( ^ + i ) ) << (a*{bux't+il fc, w%fM+l)M\x^S) = vf(bt,w^)*(x't+1)1x't+1) (3.15) The above also works when reversing xt+iandx't+l. The conclusion is that vf(bt, u>l+{bt^ *(xt+i), xt+i) does not depend on the last argument explicitly. It's easy to show that the other properties in Proposition 3.1 are obeyed, thus Axiom 3.6 is satisfied. 3.5 Appendix This appendix explores a stronger form of time consistency. Specifically, assume that utility function sets in consecutive periods are related through the following. A x i o m 3.7 (Time Consistency) For any t, z £ Zt; /, g € Xt+i, and consumption history yt, i) If f = {dt+i} and g = {d't+1} where dt+i,d't+l £ Dt+\, and (z,f) yPt {z,g) then it) If max Ed[Ut+i] > max Ed[Ut+i} for every Ut+1 e y[v+{z) then (z,f) yp (z,g). The first part of the axiom requires that, holding z constant, the strict partial order on singleton opportunity sets induced by yPt be commensurate with the date t + 1 strict partial order ypyt zy More specifically, suppose that the agent must choose a current consumption and future action combination at date t. Suppose further, that conditional on the current consumption choice, the agent surely (i.e., regardless of realized date t tastes) prefers at date t the future action dt+i to d't+1. The first part of Axiom 3.7 implies Chapter 3. Inter-temporal Flexibility Preference 86 that in the above situation dt+\ will also be preferred (regardless of changing tastes) to d't+l even if the actual choice between the actions can be deferred to date t + 1 . I.e., If it is not the case that dt+\ Pareto dominates d't+1 in the future, then it is not the case that {dt+i} Pareto dominates {d't+1} at any earlier time. This suggests that Pareto dominating sets can only shrink weakly through time. That intuition wil l be shortly confirmed. It is possible to argue that there is no strong normative motivation for including part (i) of the axiom. Indeed, an agent cannot be manipulated based only on the fact that she violates part (i). The second part of the axiom is as before. Using the assumed continuity of preferences, it is easy to prove that the conditions of Axiom 3.7 apply to the case of weak dominance 2 7. If the utility function set at date t contains only a single element, then Axiom 3.7 is equivalent to the Temporal Consistency Axiom in Kreps and Porteus (1978), and the theory reduces to their recursive utility formulation. Theorem 3.4 Suppose that every U^1'^ € ^Vt'z^ is Frechet dijferentiable28 and that at date t + 1 there exist prospects, dt+\, d't+1 € Dt+\ such that dt+i y-pyt z^ d't+1. Then there exists a continuous linear mapping from ^ Vt onto ^ j+i^ • Proof of Theorem 3.4: The proof makes use of the intuition that every neighborhood of Dt+i contains a scaled down copy of Dt+\- Since the relevant date t utility functions are linear at small scales they must support the same strict partial ordering as the utility functions at date t + 1 . 2 7 i .e. , (z,f) yP (z,g) = 4 - dt+i >zf , d't+1 and, if max Ed[ip] > max Ed[tp] for every ip e c o ( # ^ z ) ) then (z , / ) hPt (z,g) 2 8Unfortunately, we are unable to provide a version of Theorem 3.4 without assuming Frechet differ-entiability (except when ^ [^{^ has finitely many elements). Chapter 3. Inter-temporal Flexibility Preference 87 Consider any point, d G Dt+\. Since Dt+x is a bounded subset of a separable normed linear space (the space of signed measures on Zt+\ x Xt+i), Holmes (1975) §15 Lemma 2 guarantees that relative Weak-* neighborhoods of d can be defined in terms of some norm, || • ||. Moreover, the norm, || • || is dominated by the strong norm, Ms = sup p(Ei) (3.16) {Ei} . where {Ei} is a partition of Dt+\-B y hypothesis, every U^'^ G &yt'^ is Frechet differentiable. Since, in addition, all such are assumed to form a closed set, there must be a sufficiently small neigh-borhood of d, say N(d), where the U^^'s can be approximated by a linear functional, Vd\j[yuz). Clearly, VdU^yt'z) G *|+{ z ) . Consider now the set, BN{d) = {X(q~p)\ q y p p,XeTZ+, q,peN{d)} (3.17) If it can be shown that Bjv(d) = Br)t+l then assuming that Bol+i is not empty, Axiom 3.7 and the techniques used in proving Theorem 2.1 can be used to establish that cd(Vd) = cd(^[y^), where the equivalence of the sets is determined up to an affine transformation. Bpt+1 is not empty by hypothesis (i.e. the partial ordering is non-empty). To prove the proposition we must therefore establish that Br>t+l Q Bpj(dy Assume without loss of generality that N(d) = {d' | ||d — d'|| < e > 0. Consider £ S BDt+1. B y definition, £ = X(q — p) for some positive A and p, q, G A+i - Because Dt+i is a mixture space and < 1, d' = (1 — t/2)d+ (e/2)p e N(d). Also, by the same reasoning, d! + (e/2)(q — p) G N(d), and consequently, (e/2)(q — p) G Naturally, this also implies that £ G BN^dy • Theorem 3.4 essentially states that, as long as the strict partial ordering, ypyt zy is not empty, utility function sets do not increase in size with time. A n appealing interpretation Chapter 3. Inter-temporal Flexibility Preference 88 is that as time moves forward, an agent's preferences can only become more precise in the sense that tastes vary less. Another interpretation is that the agent can more accurately anticipate changes in taste. Whether utility sets shrink and in what manner, in general, depends on the evolution of the consumption path, yt. Unfortunately there are special instances in which members of a non-singleton utility function set never possess a premium for flexibility (i.e., the bounds in Eq. (3.8) converge for every opportunity set). A bias assignment corresponding to such utility functions must be ruled out if only F F D utility functions are to be considered. Example 1: Consider an agent with Inter-temporal Flexibility Preference where at some period, t, the utility function set ^t+i^ contains only two utility functions, ipi and ip2. Suppose that there is some Uf E $ f such that for one of i = 1 or 2, and where (p : Zt x 7Z i—• 1Z is continuous and monotonically increasing in its second argument. One interpretation of (3.18) is that the individual acts at date t as if she is certain that her preferences at date t + 1 will correspond to ipi. In other words, the individual is myopic about changes in future tastes. U^1'^ has linear indifference surfaces as in the inter-temporal von Neumann-Morgenstern theory of Kreps and Porteus (1978). (</> plays the role of an aggregator function separating inter-temporal substitution from risk aversion as in Kreps and Porteus (1978)). In this special case, the lower bound in Theorem 3.2 binds: Lemma 3.2 Suppose that Uf(z, {d}) — <p(z, Ed[ipi\) for one of i = 1 or 2. Then u?(z,{d}) = <K z,Ed[ipi\) (3.18) Uf(z, x) = max <p ( z,Ed[ipi\) (3.19) for any x € Xt+\ and B (x) non-empty. Chapter 3. Inter-temporal Flexibility Preference 89 Proof of Lemma 3.2: To prove the lemma, it is sufficient to demonstrate that min Ed[ipi] = max Ed[ipi] for d£Bp(x) d^x either i = 1 or i = 2 (i.e. the bounds in Theorem 3.2 are equal). To this end, given x E Xt+i, define Cx{x) = argmax Ed[tpi] C2{x) = argmax Ed[rp2] (3.20) d€x In the case that x is such that Ci(x)nC2(x) ^ 0, any common element to both C\(x) and C2(x) is clearly in Bp(x), and thus the lemma is proved. To address the other situation, consider p E Bp(x), and choose some d\ E Cx(x) as well as some d2 E C2(x). Now define d\ = Xdi + (1 - \)p for A E [0,1]. Since p yp+1 dx, it must be that dx >zP+1 dx for all A. More importantly, since Cx(x) D C2(x) = 0, £^[^2] < Ed2[4>2] < Ep[ip2]. B y the continuity of yj2, there must be some A E [0,1) such that ^ [ ^ 2 ] = ^2 [^2]- Moreover, for such a A we must have that dx E Bp(x) (since Ed2[yj\] < Edl[ipi] < Ed\[tjji\). The conclusion is that min Ed[ip2} < Ed^[tp2] = Ed2[ip2], which proves the lemma for ip2-d€Bp(x) 1 The corresponding argument for ipx follows a similar line. • Note that the requirement of a non-empty Bp(x) is not particularly stringent. This wil l always be the case so long as there is some 'most-coveted' outcome, dt+\-The example demonstrates that, in some cases, insisting on utility functions that respect F F D requires that zero probability be assigned to the realization of certain bias assignments (e.g., Uf in Eq. (3.18)). This is an unpleasant situation since without functions such as the ones in Eqn. (3.18) it is difficult to build up a reasonable basis set that would satisfy part (i) of Axiom 3.7. Put simply, in the absence of the functions in Chapter 3. Inter-temporal Flexibility Preference 90 Eqn. (3.18) the utility function set at date t would have to be infinite. Such a drastic change in the cardinality of sets, necessitated by 'normative' considerations makes Axiom 3.7 unattractive. Chapter 4 Appl icat ion to Asset Pr ic ing To explore Inter-temporal Flexibility Preferences further and to establish the practi-cal relevance of the formulation, a multi-period multi-agent general equilibrium model with Bayesian-rational beliefs about prices is presented. The model is similar to stan-dard general equilibrium formulations of consumption and investment except that agents have F F D Inter-temporal Flexibility Preferences. A set of contingent claims is traded at each date over a set of observable states. Observable states arise from two sources: an exogenous aggregate dividend process and correlated changes in tastes across agents. To understand the latter, recall that a pricing kernel essentially depends on the aggre-gated risk tolerances of agents in the economy. One contribution of this chapter is the demonstration that when agents have changing tastes, equilibrium can exist only if the aggregate risk tolerance is a function of the current state (i.e., the state price of an observable state must be a function of current and past observables only). This leads to the result that changes in taste in any given observable state should be decomposed into idiosyncratic and systematic components. The idiosyncratic component must wash out upon aggregation of risk tolerance while the systematic component is priced. This is akin to an incomplete market where agents possess some internal states that are not contractible. The internal states need not be assigned subjective probabilities, so while observable states are associated with risk (homogeneous probabilities) the internal states can be associated with (possibly Knightian) uncertainty. Because agents have a utility for flexibility they will not, in general, commit to an 91 Chapter 4. Application to Asset Pricing 92 observable-state contingent plan. Instead, they wil l continue to trade contingent claims even in a complete markets 1 subsequent to the initial trading of endowments. Thus trad-ing does not stop with the introduction of a complete set of contingent claims. Another important contribution distinct from standard models is that the state prices depend on the common factors in agents' changing tastes. This implies that, in general, additional factors must be introduced into the standard pricing kernel. These additional factors represent the uncertain aggregate risk aversion and inter-temporal discount factor. Due to the presence of these additional factors, the model can exhibit a greater degree of volatility than could be explained by variations in aggregate consumption alone. In par-ticular, if agents have changing tastes that cannot be perfectly foreseen, a model with a non-degenerate pricing kernel can result even if aggregate consumption is deterministic. After deriving these general results, the analysis specializes to an economy where agents are uncertain about future parameterization of their risk aversion. The resulting pricing kernel is analyzed by approximating it in continuous time. The pricing kernel has two correlated stochastic factors: aggregate consumption and relative risk aversion. Because the relative risk aversion is time varying, macro-economic variables such as the market price of risk and the riskless rate have non-trivial evolution. Numerical analysis demonstrates that the price equilibrium can exhibit a highly right-skewed equity premium distribution with large mean but small median. The riskless rate can be low even while equity premia are high, features interesting dynamics and can exhibit regimes or cycles even though the exogenous aggregate consumption process is not cyclic. Finally, it is shown that in certain cases the additional factor parameters (those describing the dynamics of the aggregate risk aversion) can be directly estimated from long-term coupon bond prices. Thus, in principle, all but one of the model parameters can be estimated lA complete market in this context is one in which a standard von Neumann-Morgenstern agent would be able to perfectly hedge inter-temporal consumption. Chapter 4. Application to Asset Pricing 93 directly. 4.1 A General Equi l ib r ium M o d e l This section applies last chapter's theory to a multi-agent model and presents an example of a 'complete market' general equilibrium with Bayesian-rational expectations under broad assumptions regarding agents' bias assignments. The assumed economy is one of pure exchange, consisting of N agents with Inter-temporal Flexibility Preferences. There are T periods and one perishable consumption good that is produced according to some exogenous process (all of the good produced at date t must be consumed at date t). Just prior to the beginning of period t the date t macro state 2 of the economy is revealed and the aggregate amount of consumption for that period becomes known to all agents. Simultaneously, each agent's bias assignment (inter-temporal utility function) becomes known, but to that agent only. Following the revelation of the state and the bias assignment, each agent begins period t with a set of claims over current consumption and a portfolio of claims to future (state contingent) amounts of the consumption good. The total value of each agent's endowment of claims (in terms of current consumption) forms that agent's budget. Following this, a market opens for trading current and future claims. B y trading, agents can revise their holdings subject to the budget constraint and choose a desired amount of consumption for period t, and a set of state contingent claims for consumption in the remaining periods. After trade, agents consume their share of current production of the consumption good. Denote the budget set for agent i at date t by x\. The primitive of choice for agent i at date t is a pair, (c\, x\+1) G x\, corresponding to consumption at date t and a portfolio of state contingent claims for future periods. The latter, xlt+1, is a random budget set for 2 The word 'state' is henceforth used to denote an observable macro-state of the economy and should not be confused with the bias assignment of each agent. Chapter 4. Application to Asset Pricing 94 the next period. The utility function of agent i at the beginning of date £ is a member of the set ^ t = yt and is parameterized by two indices, (gl,Ri) G [g,g] x [R, R], corre-sponding loosely to risk aversion and inter-temporal substitution, respectively. Thus each agent 'draws' her utility function from the same set of possible bias assignments3 and this set is stationary. As discussed in section 3.2, U^9t'R^ g \&t is defined over (c j , .T t + i ) - a deterministic pair of consumption and date t + 1 budget set. The sequence of events at date t as seen by agent i is as follows: Beginning of period t Middle of period t End of period t Choice Consumption Resolution of Uncertainty Choose ( c j , £ j + 1 ) G x\ Uncertainty resolved by maximizing expected utility Consume c\ x t+ l ^t+l) max . Et[U^itRi)] (ct> t^+i)^xt (glRi).-* (gt+1,Rl+1) It is assumed that agent i's utility from the consumption-opportunity pair, (ct,xt+i), is defined recursively by u f ^ \ c t , x ^ r l ) = ( u max X ^ (g', R', Et+1[U^R'\ct+1,xt+2)}) dg' dR (ct+i,xt+2)ext+i \ ' \ [g,g}x[EA (4.1) 3It is not assumed that every agent 'draws' a utility function subject to the same stochastic laws. At this point no restrictions are placed on the specific stochastic process for agents' changing tastes. The goal is to investigate the conditions under which such a process will result in a general equilibrium. Chapter 4. Application to Asset Pricing 95 where (g\, R\) G [g,g] x [R, R], u^giR^ : TZ x TZ i-> TZ is continuous and strictly increasing in both arguments, A j 5 t , i ^ : [g,g] x [R, R] x TZ t-^ TZ is continuous in its arguments and increasing in the third, and (c\+i,xt+2) denotes a random distribution of date t + 1 consumption-opportunity pairs. The function u^gi,#«)(-, •) aggregates current consumption and future utility. Since future utility depends on possible bias assignments, an agent with utility functions exhibiting F F D must place a positive weight on the maximum utility that can be obtained by each of the possible bias assignments. The integral in Eq. (4.1) is such a weighting scheme with Xt " ' acting as a weighting function. If Xt " ' is linear in its third argument, the agent can be said to possess a subjective probability distribution over future tastes. If X\ " 1 is non-linear in its third argument, the agent's uncertainty over future tastes is Knightian since it cannot be represented by a probability distribution. Note that these assumptions guarantee that all agents are F F D and thus no manipula-tion is allowed. Using Eq. (4.1), the utility of a consumption stream, ( Q , { ( Q + I , . . . , Cy)}) as seen by agent i from a date t bias assignment (associated with the index (gl, R\)), can be written as U[t9lM)(ct,{(ct+1,...,cT)}) = ct, J [g',R',U^iR')(ct+1,{(ct+2,...,cT)})) dg> dR> V l9,gME,R] J (4.2) Eq . (4.2) is the joint utility derived from some current consumption level, ct, and a singleton future consumption choice set, { ( Q + I , . . . , CT)}-As mentioned previously, just prior to the beginning of each period uncertainty is partially resolved with respect to current and future availability of the consumption good, as well as preferences of agents in the economy (i.e. the bias assignment). Let Q be a finite set of 'macro' states, is a collection of all states reflecting different values of Chapter 4. Application to Asset Pricing 96 observable economy-wide variables, such as the total amount of consumption good, and prices. Assume some common information structure which is an increasing sequence of partitions of fl, T = [T\,... ,TT). It is taken for granted that T\ — {fl} and that TT is the set of all singleton subsets of fl. A n event at date t is an element of Tt. W i t h each final state, to G fl, there is an unconditional probability, 7T(LU) and agents agree on these probabilities. The unconditional probability of an event at date t, at G Tt is simply 7r(a t) = J2u>£at 7 r ( ' J ) - For t < s and any two events, at G Tt and as G Ts such that as C at, Bayes' Rule gives the probability of the event as conditional on the occurrence of at as -n{as\at) = 7r(a s)/7r(a t). For t < s and any two events, at G Tt and as G Ts, we define (f>(as\at) to be the date t price of a contingent claim that awards one unit of the consumption good in the event as at date s, conditional on the date-t realization of the event at. The market is 'complete' in the sense that agents can trade using a full set of contingent claims. This means that all risk but not all uncertainty can be hedged, since agents have changing tastes. Agents whose tastes do not change, however, can completely hedge their future consumption by purchasing a set of contingent claims at the first session of trade. A Bayesian-rational expectations equilibrium is characterized by a price system which is common knowledge at all dates with, where (f>(as) denotes the date-1 price of a claim contingent on the event as. Before trading begins at date t, agent i has c ' _ 1 ( a s ) units of the state contingent claim that pays out in the event as (s > t). These are traded between the agents when markets subsequently open to achieve the new equilibrium allocations c\{as). Equilibrium is achieved when at each date every agent trades to maximize her current period utility (4.3) Chapter 4. Application to Asset Pricing 97 function and the following market clearing conditions are met: N J24(as)^C(as) V t < s (4.4) i=i where C(at) is the economy's production of the consumption good in the event at. To demonstrate a Bayesian-rational expectations equilibrium one needs to show that Eq. (4.3) holds and that agents can anticipate the price system correctly when consuming and investing. Since Eq. (4.3) is essentially a no-arbitrage condition, it arises naturally whenever each agent maximizes a strictly increasing utility function at every period. To guarantee that agents can anticipate prices, it must be that the set of states, fl, are exhaustive, or in other words, that the Pareto optimal price equilibria that can be achieved at date t are known at earlier dates and are adapted to the observable event partition. In particular, the last condition requires that at each date, t, the price system, {(f)(as\at)} takes into account the macro-economic impact of the current as well as all possible future bias assignments, {(glT,RlT)}, r > t, among agents. This does not mean that the price for each contingent claim, (f>(as\at), is independent of {(gl,Rl)}, but that each possible realization of the {{gl, -Rj)}'s in aggregate corresponds to some price system, {(p(as\at)}. It should be stressed that the observable or macro states in fl need not uniquely specify all of the possible bias assignments. It is only necessary that the equilibrium prices resulting from different bias assignments be incorporated in f l Essentially this requires that any changes in taste that do not wash "out upon aggregation be priced. The point is restated in the following Lemma (proved in the appendix). Lemma 4.1 Assume that at every date each agent, i £ 1,... ,N, trades in a full set of contingent claims (with respect to the partition sequence, T) and consumes to maximize a utility function as in Eq. (4.1). Assume further that the date t utility function assign-ment, {(glt, R])}, for any t is a random variable (not necessarily adapted to the partition sequence, T). Then there exists a Bayesian-rational expectations equilibrium if and only Chapter 4. Application to Asset Pricing 98 if at each date t the set of equilibrium prices generated by all realizations of {(gl, R\)} is {4>(as\at)}. 4.2 A Specific Example The investigation of general necessary and sufficient requirements on {(gl, PJj.)} in Lemma 4.1 is left to be pursued elsewhere. Instead, consider a specific example to demonstrate that a Bayesian-rational expectations equilibrium can indeed be implemented by mildly restricting agents' bias assignments. Assume that any utility function optimized by agent i at date t < T — 1 is a member of the set tyt and is parametrized by (gl, G R2+: t / , ( » ! ' R i , (Q,z 1 + 1 ) = - ^ + 9t rfiJZ) f r x ( 9 l R l ) tg,t R / j Et+l[UliiR\ct+l,xt+2)\) dg' dR! (4.5) where XT+\ is empty, and the functions \ \ 9 ' R ^ and tp[9'R^ are chosen as follows A|^V, R>, y) = - e~3/9e'R/R H_g>y) _ I < < o (4.6) gR 9 -y-R 9 The function ipi9'^ is a re-scaling of aggregated utilities and the bias assignment, R, parameterizes inter-temporal substitution. A t date T it is assumed that each agent's utility function takes the form in Eqn. (4.5) with ip^'R^ = 0. The choice of utility function sets is reminiscent of the C A R A (constant absolute risk aversion) class of functions. Since the bias assignment at different time periods can include any level of risk aversion, g, the agent can potentially exhibit a robust range of behavior. d9'R)(y) = - — (4-7) Chapter 4. Application to Asset Pricing 99 Under these assumptions, it is shown in Appendix 4.5 that at any date t, the state prices are recursively given by K7r(at+i\at)' (C(at+i) - at+1{at+l)w{at+1)) (G(t + 1) - G(t)) + qt+1(at+l)(R{t + 1) - R(tj) - R(t + 1) (4.8) where wt(at) = (C(at) + ]P (f)(at+1\at)wt+1(at+1)) (4.9) a-t + iCat 1 £ ^ m k l ) ( 4 1 0 ) 9t(Qt) _ <f>(at+i\at)qt+i(at+i) 1 ^ ^ a ^ a ^ _ a t ^ a t ott+1(at+1) a(at) 1 N W - G W g f l (4.13) vjt(at) is aggregate wealth calculated as the risk adjusted present value of the current and future consumption coupon streams. a t 1 ^ can be interpreted as the value of a perpetual unit coupon bond, and is the price of a perpetual bond that pays a stochastic coupon equal to a ^ a ^ at each date. G(t) and R(t) are, respectively, measures of the aggregate risk aversion and aggregate discount rate across investors. G(t) and R(t) are the only quantities in Eqn. (4.8) that directly depend on the realization of investors' bias assignment. The state price system in Eq . (4.8) fully accounts for the distribution of bias assign-ments and resulting Pareto optimal allocation of resources if and only if at each date t, Chapter 4. Application to Asset Pricing 100 G(t) = G(at) and R(t) = R(at). This leads to the following constraints on an equilibrium bias assignment Proposit ion 4.1 Assume that the economy is composed of agents with preferences as In other words, the aggregate risk tolerance and discount rate must depend only on the path of observable macro-economic variables (the filtration Tt). In particular, the impact of uncorrelated private states corresponding to individuals' changing tastes must wash out upon aggregation. Correlated private states may give rise to macro-economic or priced states, and these must already be part of the filtration, T. For example, if the economy consists of many agents (i.e., N,C(at) —> oo) but the per-capita aggregate consumption good stays finite in all events, each agent's bias assignment may be broken into a systematic component and an additional independent and identically distributed idiosyncratic random term that is not necessarily adapted to the partition T. This randomness washes out in (4.14) leaving the price system unchanged. 4.2.1 Further Discussion The rational expectations equilibrium presented above is similar to one with a repre-sentative consumer having state-dependent inter-temporal von Neumann-Morgenstern preferences. The state-dependence in the utility function of the representative consumer described in Eqns. (4.5)-(4.7) and that agent i's bias assignment at date t is (g\, R\) € R2+. Then a Bayesian-rational expectations equilibrium exists if and only if and (4.14) (4.15) Chapter 4. Application to Asset Pricing 101 corresponds to the coherent changes in the tastes of agents. Thus the representative agent can be said to have changing tastes but, in contrast with the agents in the above economy, can attach precise probabilities to states in which tastes change. In addition, the states corresponding to different tastes are in T and can thus be hedged completely using contingent claims. To the representative agent, utility for flexibility is irrelevant. This is because trading in a complete market allows the representative agent to hedge every relevant contingency. Although one can analyze a representative agent economy for an agent with Inter-temporal Flexibility Preferences, it is far from clear that such an economy can be more generally derived through aggregation as it can with recursive preferences (see Dumas, Uppal and Wang (1998)). The agents in the economy of the previous subsection possess Inter-temporal Flex-ibility Preferences. In particular, they assign a flexibility premium to choice sets and therefore do not make contingent plans. Contingent claims are bought partly for their investment potential rather than only for the purpose of setting future consumption. This implies that equilibrium trading will occur whenever markets open and trading volumes will therefore exceed that implied by a world with complete markets and von Neumann-Morgenstern agents. Donaldson and Selden (1981) have a general equilibrium model where agents possess non-Expected Uti l i ty preferences. In their model markets also re-open after initial trading. A shortcoming of their model, however, is the inherent time-inconsistency of their agents. For example, agents are indifferent between contracts that restrict their future consumption opportunities and their current opportunity set but subsequently prefer to deviate from an 'optimal plan'. As discussed earlier, this in-troduces an arbitrage opportunity. B y contrast, the model here admits no possibility of manipulation when binding contracts are introduced into the market. Note that another important implication of the model is that prices can move with-out any change in consumption. Consider, for example, Eq . (4.8) at date T — 1. The Chapter 4. Application to Asset Pricing 102 state prices, <p(aT\a,T-i), depend on the current state, aT_x, only through the coherence factors G(ar-i) and R(a,T-i), the aggregate supply of consumption good, C ( a r - i ) and C(O,T\CLT-I), and the conditional probability, 7r(ar |ax- i ) . Suppose that two date T — 1 events, ar-i and are distinguishable only by the different coherence factors (i.e., C(ax-i) = C(bx-i), and ^{a^ar-i) = ^(a-rjfrr-i)). In a world without changing tastes, the state-prices would be the same: (f*(aT\a,T-i) = 0(a^|6r_i). In a world with chang-ing tastes and where, say, G(CLT-\) ^ G7(6r-i), the prices would be different. In other words, the source of uncertainty regarding future prices can come purely from changes in tastes. A n extreme example is one where the supply of the aggregate consumption good is deterministic, but agents' tastes with respect to inter-temporal substitution are random. This observation suggests a possible equilibrium explanation for the excess volatility seen in financial markets relative to what could be reasonably expected within standard consumption-based models. Coherent but random changes in consumer tastes add additional uncertainty and therefore volatility. A n interesting question is whether agents can somehow conspire to reduce the residual uncertainty left due to changing tastes. The answer is, in general, no. Insurance markets for flexibility will not be sustainable because there is no observable state to tie to the changing of tastes (all the observable states are already in the filtration, T). There is therefore a moral hazard problem for anyone trying to sell insurance for 'flexibility' - it would be analogous to selling insurance on mood swings. Finally, note that the equilibrium conditions in Proposition 4.1 on the uncertain component of agents' changing tastes (i.e., the components that wash out of the pricing kernel) are very weak when there are a large number of agents in the economy 4. In par-ticular, if an agent could actually observe her own bias assignment, it is not unreasonable 4 I f there were few agents, say two, then the conditions would be quite restrictive and agents would be able to infer their personal tastes from the current state alone. In a two-agent economy, in fact, there, in general, can be no idiosyncratic (unpriced) component to the bias assignment. Chapter 4. Application to Asset Pricing 103 to assume that it is only weakly correlated with that of the aggregate (i.e., the agent's idiosyncratic component of changing tastes is large). This suggests that to a large ex-tent, a detailed model of the stochastic nature of the bias assignment is unnecessary. To characterize an equilibrium, one need only model the correlated components (i.e., R(at) and G(at))- The result also justifies Chapter 3's relative silence on the mechanics of the bias assignment - it simply is not important. What is important is that such a process exists and, that upon aggregation over all agents, obeys the equilibrium conditions. 4.3 Asset Pr ic ing In this section additional assumptions are imposed in order to derive some concrete economic implications from the model. Under the assumption of a rational expectations equilibrium and assuming only uncertainty over agents' risk aversion parameter, the pricing kernel becomes 4>(at+1\at) = prc(at+1\at) x exp (-(C(at+1) - a(at+1)W(at+1)) (G(at+1) - G(at)) - G(at){C(at+1) - C(at))) (4.16) Note that a(at)W(at) is a term which can be viewed as an economic coupon value 5. To facilitate the analysis, approximate Eqn. (4.16) in continuous time 6 and assume T —• oo. To do so we must re-interpret Ct as a consumption stream. Assume that the underlying dynamics in the economy are governed by 2-dimensional Brownian motion. 5 To justify the identification of a(at)W(at) as an economic coupon, consider that if the per period interest, rtdt and a(at) are constant, a = (1 -f- a( 1+ r.rf i))~ 1 giving a = (^*dt) • Similarly, if C(at) is constant and T —> oo, then W — C^^dt^ • The product, a(at)W(at), is therefore equal to the coupon, value C. 6 A formal derivation of a price system from an economy of agents with continuous time Inter-temporal Flexibility preferences is a technical, rather than conceptual, matter that will not be addressed here. Chapter 4. Application to Asset Pricing 104 Denote the vector of 2 independent Brownian motion variables as W t . 7r(at+dt\at) is therefore the probability associated with a multi-variate normal distribution with variance dt. The evolution of aggregate consumption, Ct, and aggregate risk aversion, Gt, are parameterized as follows ^ = p?dt + af'.dWt (4.17) ^=fj?dt + tr°,-dWt (4.18) W(at) and a(at) have continuous time analogues that require W(at) ==> dtWt and a(at) ==£' at/dt. Thus the economic coupon, a(at)W(at), remains finite in the limit. Define the differential consumption coupon (DCC) as, Dt = Ct-atWt (4.19) Its evolution is dDt = ifidt + erf • dWt (4.20) where p^ and cr® are determined via Ito's Lemma, p? is not relevant as it does not appear explicitly in any expressions for macro-economic variables, erf, however, does appear in the continuous-time limit of the pricing kernel. It is given by a? = Ct(dcD)o-f + Gt{dGD)<rf (4.21) The pricing kernel can be approximated in continuous time as <t>(at+dt\at) = 7T(at+dt\at) exp ^ - ( - * _ _ _ * _ + T t ) d t _ af<. m \ ( 4 2 2 ) Chapter 4. Application to Asset Pricing 105 where it is assumed that p = e T0 and =DtGt*? + GtCt<r? (4.23) CTM' • £ T M rt =r0 + DtGtpf + Gter?' • <xf + GtCtpc - (4.24) This can be seen as the starting point of the analysis. It was shown that the pricing kernel can be derived through aggregation of non-standard yet normative preferences, but one can alternatively view the expression as resulting from a representative consumer with state dependent preferences. erf1 and rt are, respectively, the vector of volatilities associated with the market price of risk and the instantaneous risk free rate. The risk premium for an asset with instantaneous return mean and standard deviation of pt and o~t, respectively, is given by p - r t = erf • ert (4.25) In particular, the risk premium on the market portfolio (defined to have volatility erf1) is simply erf4' • erf4. The DCC plays an important role in the dynamics of the riskfree rate and the market price of risk. Its evolution, in turn, is determined by that of the economic coupon, atWt. In continuous time, these can be expressed as /°° „Mi M C°° Cse-f^^^+r^du-^<'dW-ds}^E;[j^ Cse-hs^ds\ (4.26) 1 r00 „MI „ M , poo ± = Zt=Et[ e-MZ^+r^du-K<'"'-dW»ds] = E;[ e'^T-duds) (4.27) at Jt Jt where E% denotes a risk-adjusted expected value with respect to the information available at time t. The economic coupon is therefore the price ratio of a bond that pays Chapter 4. Application to Asset Pricing 106 the aggregate endowment each period to a bond that pays one unit (per unit time) of the consumption good each period. Equivalently, Wt/zt is a weighted average of all possible future coupons. Either way, Wt/zt is an economic variable that is sensitive to the term structure of interest rates and the expected stream of future consumption. The evolution of Wt and zt is given in the following lemma. Lemma 4.2 + Ctpf - CtGt {Dt*? • erf + Ctaf • o f ) Gtpf - Gl (Dtorf • erf + Ctcrf • erf) dcW dGW + 2 C2af'-erfd2cW + G2taf'-afd2cW + 2CtGtaf'-afdcdGW - r0 + DtGtpf + Gt (Ct(dcDt)crf + Gt(dGDt)cr?) • erf + GtCtpf - ( A G ^ f + G ^ f ) - W=-Ct (4.28) and Ctpf - CtGt {Dterf • erf 4- Cterf • erf) + Gtpf - G\ (Dterf • erf + Ctaf • erf) dcz dGz +-, C2crf • crfd2cz + G\crf • of'b%z + 2CtGter? • crfdcdGz rQ + DtGtpf + Gt (Ct(dcDt)erf' + Gt(dGDt)erf) • erf + GtCtpf -(DtGto-f + GtCto-f) z = -l (4.29) These equations are non-linear and coupled due to the presence of Dt in both rt and the risk adjusted drift. To simplify the analysis, assume that the drifts and diffusion coefficients of the state variables in (4.17) are functions of Xt = GtCt only. Xt represents Chapter 4. Application to Asset Pricing 107 the aggregate relative risk aversion of the representative consumer. The model here is distinct from standard models largely by virtue of the fact that the aggregate relative risk aversion is time varying. This allows one to write zt = zt(Xt) and Wt = Ctf(Xt) Dt = Ctd(Xt) = Ct(l- 4 | T ) ( 4 ' 3 ° ) v zyy^t)J The coupled non-linear set of partial differential equations above reduce to a coupled set of non-linear ordinary differential equations for z(Xt) and f(Xt). A(X)fxx + B'(X, /, z)fx + Cf(X, f, z, fx, zx)f = -1 (4.31) A(X)zxx + BZ(X, f, z)zx - r(X, f, z, fx,zx)z = -1 (4.32) where A(X)=^-{af(X) + af{X))2 (4.33) Bf(X,f,z)= X(jj°(X) + p°(X)) + X<rf(X)'-<Tf(X)-x*(i-t)<T?ixy.(o-?(x) + v?(x))-X(X - 1) <rf{X)' • (af(X) + <rf(X)) (4.34) B'(X, f, z) = X(p°(X) + tf(X)) + X a?(X)' • <rf(X) -X\\ - t) a°{X)' • (vf(X) + crf(X)) -X**?(Xy-(*?(X) + <T?(X)) (4.35) Chapter 4. Application to Asset Pricing 108 C'(X, / , z, fx, zx) =tf(X) - X(af(X)' • <T?{X) + (1 - *-) <rf{X)' • vf(X)) -r(XJ,z,fx,zx) (4-36) (X, / , z, fx,zx) = r 0 + X((1 - L)p°(X) + tf(XJ) + X(l - S-) crf(X)' • *f'X) <rf(Xy-(*?(X) + <rf{XJ)-X* _X2 zfx ~ fzx _ G , v V („G( v ^ A „c(v,\ _ ^ ( ( i - l ) cTf(X) + a?(X)) z2 (4.37) In addition, assume there are growth conditions at X = 0 (a situation synonymous with risk neutrality) on the exogenous drift and diffusion components: ljmXtf(X) = 0 (4.38) x—o \lmXp?(X) = 0 (4.39) l i m o f p O , l i m o f p O < oo (4.41) x^o 1 v ' X->0 Naturally, the growth condition in Eqn. (4.40) must be verified after locally solving for f(X) and z(X). If /xf (0) is defined, z(0) and /(0) can be calculated from Eqns. (4.31)-(4.32) to be ( r o - / x f (0 ) ) / (0 )= l (4.42) If lim\fit(X)\ = oo then /(0) may diverge. To integrate Eqns. (4.31)-(4.32) and X—*0 derive an equilibrium one needs two boundary conditions for each equation. For example, if / / f (0) is finite, Eqns. (4.42) provide half of the requirement. The other half may be obtainable from the derivatives of z(X) and f(X) at X = 0. In other cases, it Chapter 4. Application to Asset Pricing 109 is necessary to deduce an asymptotic form for z(X) and f(X) as X —> 0. This is useful both for pinning down boundary conditions for numerical integration as well as for analytic purposes. In particular, one can analyze the economy 'exactly' when it is near the asymptotic limit. To pursue this, rewrite Eqns. (4.26) and (4.27): f(Xt) =Et .It +r(Xu,f,zJx,zx) du X • /' \Xu)-<rC(Xu))'-dWuds , \ ;°° -^(^(Xu)'2^(Xu)+r(XJ,zJx,zx))du-Jts^(Xuy-dWl z{Xt) =Et[ / e V / ds (4.43) (4.44) Let Xu = Qu(Xt,u-t) (4.45) Under the growth conditions assumed, one can express fx and zx asymptotically as, e - ( s - t ) r 0 £ t fxxZQ d_ t dQ eIts(^(Qu)-k^u(oy^(o)) du+lt° «c(o)'.<iw„ x ( r(QU) f(Qu), z(Qu),fx(Qu),zx(Qu)) + pCu{Qu) - \<r°(Qu)' • o£(Q u ) ) dxQu du + f dxQu (dQ(Qud(Qu))af(0) + crf(0)-dQcrf(Qu)y-dWu (4.46) Chapter 4. Application to Asset Pricing 110 and zx ~ x^o e-{s-t)roE J dQ r ^ u ' fxiQu), zx(Qu)) dxQu du + [dxQu ( dQ(Qud(Qu))trG(0) + o f (0))' • dW /oo fs d e-(.-t)roEt J —r(Qu,f(Qu),z(Qu)Jx(Qu),zx(Qu))dxQudu\ds (4.47) where only the (possibly) dominating terms are left as the limit X —* 0 is approached (i.e., non-dominant terms are set at their limit). To proceed, further assumptions over the functional form of the exogenous drift and diffusion terms must be made. 4.3.1 Constant Drift and Diffusion Coefficients If Gt and Ct follow geometric Brownian motion, Q(X,u- t) = CUGU = X e ( ^ + M c - | ( ^ ' ^ - f < T G ' - ^ ) ) ( u - t ) + ( < T c + ^ ) ' w u _ t ( 4 4 g ) Where Wu_t = / " d W „ . In this case /(0) = z(0) = ^ , and thus d(0) is well defined. Note that pc must be constrained to be less than ro. Lemma 4.3 Under the assumptions of constant drift and diffusion coefficients, Eqn. (4.47) becomes, ' l ro—lJ. ^ n pG + <7CL • aG ) + pc ro (ro - (n° + pc + o~Cl (4.49) and Eqn. (4.46) is Chapter 4. Application to Asset Pricing 111 ( l - ^ ) (pG + 2*G' • <rG) + pG + *G< • cr c (4.50) c » where the following parameter restrictions apply: r0>pG + pc + c r c ' • a G r0>p° + 2pc + 2crCl • a G + a C / • a c 4.3.2 Mean Reversion in G and C Here it is assumed that crc and o~G are constant, but one or both of the state variables, C and G, mean revert. In particular, one is interested in a situation in which the consumption process is not necessarily stationary in level but risk aversion "keeps up" with consumption in such a way as to ensure that the aggregate relative risk aversion, Xt = GtCt, mean reverts. In other words, risk tolerance increases with consumption. Here, as opposed to the case in the last subsection, the riskless rate and the risk premium are stationary. In this case /(0), /A'(0) and zx(0) may diverge but we can still calculate their asymptotic form. Specifically, set pc(X) =pc -KCln(X) pG(X) = pG - KG\n{X) (4.51) this implies d\nXt = G))dt+{: <7° + CT Chapter 4. Application to Asset Pricing 112 and In QU(X, u-t) = E-(Kc+-G)(-t) in Xt + [pc + \iG --{<TGi • crG + *G> • *G)) = + [CTG + *G)' • / e ^ C + * G ^ d W v 2V / K + K V ' Jt (4.53) Lemma 4.4 Under the assumptions in Eqn. (4.51), with KG > 0 and K° > 0, ><*> „~. ^ ^ ( ^ ) ^ r( /o (4.54) where T(z) = J0°° qz xe ldt is the Euler Gamma function, and c \{„c+„ay.{ac + „ a ) _ £ _ ( 4 . 5 5 ) r> — (,,C i ,,G 1 „Gi „G ,^ „Ci „ C , - C i rTG\ K ° ( K G + K G ) S ' {KC + K~ KG2 {CTC + ( T C ) I . { ( T C + ( T G ) _ J ^ ( 4 . 5 6 ) C s { a C + v a y . { a C + ^ ) _ ^ _ ( 4 . 5 7 ) C 2 with the restriction that A must be positive. Further, Chapter 4. Application to Asset Pricing 113 z{X) 1 KG2f0\lnX\"^ f°° _ r a ^ - i r ° ° x^o r0 E ^ r l ^ + « W ( o , i ) - - z o l l n X f ^ T ^ -v + a + /W(0,1) i - - C+KG dvdq with and _ yF + pG - \{<JCI • crc + (7°' • *G) 0 1 ~ KC + KG (4.58) (4.59) (4.60) 2{KP + KG) Af(0,1) is a standard normal variate and the expectation in Eqn. (4.97) is taken with respect to its distribution. As long as A < KC + KG , which is true under most reasonable parameterizations of the consumption process, the integral in the expression for z(X) has a well defined limit, although there is no explicit closed form solution for the last expression. It is straight forward, however, to calculate it on a digital computer. Note that for any positive value of KC, as long as KP > 0, the growth condition in Eqn. (4.40) holds. It is arguable that this, in fact, covers all cases of economic interest. A situation where KC > 0 and KG — 0 would correspond to a consumption process that mean reverts according to level of risk aversion. In other words, the level of aggregate risk aversion Granger causes the production level of the aggregate consumption good. Intuition would suggest that a causal relationship, if present, should be in the other direction. If K° = 0, a different asymptotic result holds: Chapter 4. Application to Asset Pricing 114 Lemma 4.5 Under the assumptions in Eqn. (4.51), with KG > 0 and KC = 0, CaA'+B' z(X) ~ 1 + , / 1 + ^ ) r ( ^ ) (4.61) and 2 jjpe^'^~G' / ro / i ^ " \ ro / x G where ^ ^ • - i ^ ^ . ^ ) ( 4 6 3 ) S ' ^ ( ( r c + < T G ) ' . ( a c + < T G ) (4.64) C" =B' + 4T<TC' • (<^C + « - G ) (4-65) 4.3.3 Simulations The asymptotic expressions allow one to uniquely specify the boundary conditions when numerically integrating the ODE's, and in turn, characterize the equilibrium7. For the rest of the chapter, I consider the mean reverting case where KC = 0. This corresponds to an economy where the aggregate dividend process is non-stationary, while the aggregate relative risk aversion, Xt, mean reverts. To understand the difference between this model and standard consumption based models, note that if o~G = —ac (i.e., the representative agent acts as if she has constant relative risk aversion), Xt reverts to its median value, Xm, and, once there, remains 7 I f the intergration must start at some finite value of X (e.g., in the mean reverting case), one can obtain further accuracy by solving the ODE's using a series expansion. This is made easier by the fact that at small values of X the ODE's decouple. Chapter 4. Application to Asset Pricing 115 constant. In this case, the economy possesses constant macro-variables (superscripted by with rs solving rs x2 / rs \2 Using the benchmark values of r 0 = 5.0%, \fi = 1.5% and | rx c | = 3.0%, Figures ?? plot the riskless rate, rs, and equity premium on the market portfolio, es, versus the coefficient of relative risk aversion. Although the plots show the values of macro variables at different levels of the relative risk aversion, the model is static and X must be fixed exogenously at Xm. As in other standard consumption based models with constant aggregate relative risk aversion, the steady state model above features a riskless rate, rs(Xm), that is nearly linear in ro and increasing in pc. Also, es(Xm) is independent of ro and pc, and is negligible for realistic values of |<T c | unless the the constant relative risk aversion, Xm is huge8. This also implies that the impact of |er c | on rs is negligible for Xm ~ 0(1) or smaller. To compare the static model above with the more general case of changing aggregate relative risk aversion, consider the following 'benchmark': 8 This is the equity premium puzzle. Related to this is the risk-free rate puzzle: if Xm is large, rs will also be large for reasonable values of pF. Chapter 4. Application to Asset Pricing 116 Figure 4.1: Comparative statics of the riskless rate and equity premium with respect to ro and X when the coefficient of relative risk aversion does not vary through time (aG = -*c) Figure 4.2: Comparative statics of the riskless rate and equity premium with respect to \ac\ and X when the coefficient of relative risk aversion does not vary through time (aG = -<r c). Figure 4.3: Comparative statics of the riskless rate and equity premium with respect to \ £ and X when the coefficient of relative risk aversion does not vary through time (aG = -ac). Chapter 4. Application to Asset Pricing 119 Parameter Benchmark Value ro 5.0% P° 1.5% \ * c \ 3.0% 0.20 y° -0.04 \CTG\ 47.17% -0.53 Under this parameterization the aggregate risk tolerance changes rapidly compared to the consumption process, but reverts in the long-run to roughly twice the value of the dividend (Xm ~ 0.5). Figures 4.3-4.9 show graphs of the riskless rate, equity premium on the market portfolio, the wealth-dividend ratio and DCC. Each figure demonstrates the comparative statics with respect to a model parameter. On each curve the unconditional 5th, 50th and 95th percentile values of the relative risk aversion, Xt, are denoted by boxes (recall that Xt mean reverts about the median). At the bottom of the figure a table indicates the mean values for the riskless rate and equity premium. In contrast with the case in which Xt does not vary, Figures 4.3-4.5 demonstrate that for |crG| ^ \o~c\, r(Xt) is non-linear in ro, strongly decreasing in a° and very sensitive to |cr c|. The equity premium, e(Xt) = o-M(Xt)' • crM(Xt), has a tail distribution that is highly sensitive to ro and fic (which shows up in the radically different mean equity premia). Note that r(Xt) and e(Xt) tend to be affected in opposite ways by changes in the model parameters. The tail distribution of r(Xt) is sensitive to changes in all of the model parameters, whereas the median riskless rate is sensitive only to the drift parameters: Chapter 4. Application to Asset Pricing 120 Figure 4.4: Comparative statics with respect to r0. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, rt(X). Bottom: the equity premium on the market portfolio, o~M(X)' • crM{X). Chapter 4. Application to Asset Pricing 121 Figure 4.5: Comparative statics with respect to r 0 . The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealth-dividend ratio, f{X). Bottom: the normalized differential consumption coupon, d(X) = D/C. Chapter 4. Application to Asset Pricing 122 Figure 4.6: Comparative statics with respect to pc. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, rt(X). Bottom: the equity premium on the market portfolio, crM(X)' • crM(X). Chapter 4. Application to Asset Pricing 123 He mean r mean e 0.014 0.034 0.052 0.015 0.028 0.066 0.016 0.021 0.084 Figure 4.7: Comparative statics with respect to pc. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealth-dividend ratio, f{X). Bottom: the normalized differential consumption coupon, d(X) = D/C. Chapter 4. Application to Asset Pricing 124 Figure 4.8: Comparative statics with respect to \crc\. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, rt(X). Bottom: the equity premium on the market portfolio, aM(X)' • crM(X). Chapter 4. Application to Asset Pricing 125 -0.8 mean r mean e 0.02 0.031 0.063 0.03 0.028 0.066 0.04 0.024 0.071 Figure 4.9: Comparative statics with respect to |<T c | . The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealth-dividend ratio, f(X). Bottom: the normalized differential consumption coupon, d(X) = D/C. Chapter 4. Application to Asset Pricing 126 Figure 4.10: Comparative statics with respect to KG. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, rt(X). Bottom: the equity premium on the market portfolio, *M(X)' • crM(X). Chapter 4. Application to Asset Pricing 127 -0.S5 K mean r mean e 0.15 0.035 0.046 0.2 0.028 0.066 0.25 0.022 0.084 Figure 4.11: Comparative statics with respect to KG. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealth-dividend ratio, f(X). Bottom: the normalized differential consumption coupon, d(X) = D/C. Chapter 4. Application to Asset Pricing 128 Figure 4.12: Comparative statics with respect to pg. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, rt(X). Bottom: the equity premium on the market portfolio, crM(X)' • crM{X). Chapter 4. Application to Asset Pricing 129 f(X) 55 1 , , , , , , • i , 1 , 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 X -0.8 Ho mean r mean e -0.03 0.024 0.073 -0.04 0.028 0.066 -0.05 0.032 0.060 Figure 4.13: Comparative statics with respect to p9. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealth-dividend ratio, f(X). Bottom: the normalized differential consumption coupon, d(X) = D/C. Chapter 4. Application to Asset Pricing 130 Figure 4.14: Comparative statics with respect to |crG | . The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, rt(X). Bottom: the equity premium on the market portfolio, crM(X)' • crM(X). Chapter 4. Application to Asset Pricing 131 f ( X ) 7 5 70 55 50-1 , , , , . . . . . • 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 X -0.6 i 1 1 1 1 1 1 1 1 1 ' v > 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -0.8 o0 mean r mean e 0.2 0.031 0.063 0.2225 0.028 0.066 0.25 0.025 0.069 Figure 4.15: Comparative statics with respect to \crG\. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealth-dividend ratio, f{X). Bottom: the normalized differential consumption coupon, d(X) = D/C. Chapter 4. Application to Asset Pricing 132 Figure 4.16: Comparative statics with respect to p. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, rt(X). Bottom: the equity premium on the market portfolio, crM(X)' • crM(X). Chapter 4. Application to Asset Pricing 133 -0.8 p mean r mean e -0.4 0.029 0.065 -0.53 0.028 0.066 -0.66 0.027 0.067 Figure 4.17: Comparative statics with respect to p. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealth-dividend ratio, f(X). Bottom: the normalized differential consumption coupon, d(X) = D/C. Chapter 4. Application to Asset Pricing 134 r0,/j,c,nG and the rate of mean reversion, KG. The instantaneous riskfree rate tends to decrease towards the median from extreme values of the relative risk aversion, Xt. Agents are less eager to borrow as their absolute risk tolerance matches their consumption level (or vice versa). Agents borrow at low relative risk aversion so that they can invest in risky assets. This is accompanied by a relatively high levels of the short-term interest rates and wealth-dividend ratio, f{Xt). At higher levels of relative risk aversion there is the familiar trade-off between the need to smooth consumption due to the higher emphasis (proportional to Xt) on the dividend growth rate and the need for cautionary savings due to the higher emphasis (proportional to X2) on the volatility of the market portfolio. Eventually, as Xt becomes disproportionately large the need to save prevails and the riskless rate plummets. Note the lower wealth-dividend ratio when the relative risk aversion, Xt, is high. Also note that the inability to move aggregate consumption across periods can lead to negative interest rates (although the parameters chosen for the example tend to hide this fact). The equity premium is universally increasing in Xt. This is because it depends on X2d2(Xt), which is increasing in Xt. A t the median relative risk aversion, the market risk premium is low (between 2 and 3 percent),. The unconditional mean market risk premium, however, depends strongly on the right tail distribution. In particular, as the distribution of possible Xt spreads, the unconditional mean market risk premium dramat-ically increases. Whereas in standard models the market risk premium is constant, here it has a highly right-skewed distribution. Times of high returns are rare and spectacular. In other words, to realize high returns on the market, one has to invest over sufficiently long horizons so as to "be in the market" during the short periods of abnormally high returns. Finally the extreme right-skewness wil l lead to the familiar implied volatility smirks in option prices. Another feature of the model that can be gleaned from the graphs is the general Chapter 4. Application to Asset Pricing 135 presence of negative correlation in returns. The long-term negative correlation is an artifact of mean-reversion in Xt. To see the negative auto-correlations in the shorter horizons, consider that the instantaneous auto-correlation of excess returns on the market portfolio is given by Mtv \J4. Et[d{<rM(xty • <rM(xt)} *M{xty • dwt] Since crM(Xt) « Xtd{Xt)crG, the auto-correlation can be written using Ito's Lemma as pM(Xt) « 2<rG' • crGX~(Xt d{Xt)) This last expression is typically negative, as can be seen in the graphs. Note, in particu-lar, that in the absence of time-variations in the aggregate relative risk aversion, pM(Xm) is zero. Although the presence of short-term auto-correlation of the model is in disagree-ment with empirical findings (i.e., the presence of momentum effects in equity returns), it is consistent with the intuition that a positive movement in asset prices generaly results in a lower equity premium. Specific Example To continue, it is useful to specialize to a particular example. Consider an economy parameterized by Parameter Benchmark Value 4.7% 1.5% \ac\ 1.75% KG 0.15 -0.02 \aG\ 51.5% P ~ |cr«| | ^ ' | -0.63 Chapter 4. Application to Asset Pricing 136 The risk-free rate and market risk premium have distributions characterized by mean and standard deviation of Parameter Unconditional Mean Unconditional Standard Deviation e(X) 8.45% 18.8% r(X) 1.71% 0.63% Figures 4.10-4.12 describe the macro-economic variables. In particular, Figure 4.12 plots the long-term coupon bond price, z(Xt) as well as the derivatives of the wealth-dividend ratio and long-term coupon bond prices. The latter are useful in characterizing the volatility for the respective variables. The riskfree rate (a real rate) typically fluctuates between 1.5 and 2.5 percent. As with the previous analysis, notice that the market risk premium is typically low but the unconditional mean is quite large. This is due to the relatively slow mean reversion of the aggregate relative risk aversion (decay time of about 7 years). I complete the chapter and the dissertation with an analysis of correlations between macro-variables in the economy and testable predictions of the model based on the ex-ample above. • Long-term coupon bond prices (z(X ( ))are roughly linear in aggregate relative risk aversion, X. Thus aG, KG and fiG can be estimated from returns on long-bonds. In particular, the model predicts that long-term coupon bonds are mean reverting, and thus exhibit negative auto-correlation in returns. • Evolution of the term structure is regime dependent. When long-term coupon bond prices decline towards their median value short-term interest rates tend to Chapter 4. Application to Asset Pricing 137 Figure 4.18: Instantaneous riskfree rate in the example. Top: plot versus X. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Bottom: unconditional probability distribution. Chapter 4. Application to Asset Pricing 138 Figure 4.19: Market risk premium. Top: plot versus X. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Bottom: unconditional probability distribution. Chapter 4. Application to Asset Pricing 139 Figure 4.20: Perpetual bond price and price-dividend ratio of aggregate production. Top: plot versus X. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Bottom: plot of derivatives versus X. Chapter 4. Application to Asset Pricing 140 decrease, thus the term structure steepens. When long-term coupon bond prices increase towards their median value, short-term interest rates usually decrease, thus the term structure shifts down. During times of extreme volatility, however, short-term interest rates increase as the economy returns to its long-run median; in this case, the term structure flattens since long-term bonds still increase in price towards their median. • Long-term coupon bond prices are negatively correlated with expected returns. • Short-term real interest rates are not log-normally distributed. Rather, the shape of their distribution is more consistent with a mixture of two log-normal distributions (due to the 'double hump'). • the return volatilities of long-term coupon bonds and the economy's 'wealth-dividend' ratio is decreasing with increasing expected returns on risky assets. The volatility of long-term bond returns is highest when bond prices are above their median value. • Expected returns on the market portfolio are consistent with a highly right-skewed distribution (e.g., log-normal). 4.4 Proofs Proof of Lemma 4.1: First note that at date t, investor i maximizes a derived utility of form, V®™ (c<(a 4 ) ,^(a t + 1 )) (4.66) where at+\ C at (4.67) a s C a t + l s>t+l Chapter 4. Application to Asset Pricing 141 This is simply a consequence of optimizing under a budget constraint. The maxi-mization program is as follows: max { V ir(at+1\at)Vti9t'Rt) (cl(at), Wi(at+1)) -,>t U t + l C a t A \ 4(at)+ c%as)4>{as\at) - Wi(at) V as(Zat s>t (4.68) J where A is a Lagrange multiplier. The Lagrange multiplier can be eliminated from the first order conditions for c*(at+i) and c\{at) to give 4>(at+1\at) = ir(at+i\at) £ V t { 9 ^ {c\{at),Wi{at+l)) d *tr(.9l,R\) (J, d~cVt Vr^'icHat)^^)) (4.69) Similarly, using the fact that the events, {as} for s > t partition the state space, the first order condition for c\{as) for s > t + 1 can be written as , , ( , u , , , £ V f (c1(q t ) ,^(a f + 1 )) (f){as\at) = TT{at+l\aT)4>{as\at+i) d - — fVt ( q ( a t ) , ^ ( a i + i ) ) Substituting from Eq. (4.69) and rearranging, the last equation becomes ,/ i x <t>(as\at) (p{as\at+1) = — — (p(at+1\at) These equations can be used to inductively derive Eq. (4.3). (4.70) (4.71) • Proof of Lemma 4.2: Recall the Feynman-Kac formula: f(t,s:Yt) = Et[e-l'°^dMYs)} {At-q(Yt))f = d3f (4.72) Chapter 4. Application to Asset Pricing 142 where Yt denotes the vector of exogenous state variables (Gt and Ct), and At is the second order partial-differential operator associated with diffusion: At = J2 + E ^ T ^ ^ A (4-73) If the expected value is risk adjusted, the Girsanov theorem instructs one to shift the drift terms: p( —» p%, as follows A T = pi - e4 • °f (4.74) where 0 t is a matrix whose ijth element is the jth diffusion component of the ith state variable. Applying this to the state variables, Ct and Gt gives, A t = [ C t " ? ) - ( C t ( T ? ' ) • (DtGt<r? + GtCt*c) (4.75) Ctpc-CtGt(Dtac'.af + Cto-c'-ac) (4.76) Gttf - Gl (Dt*r • <rf + Ctaf • a?) Thus denoting A\ as the risk adjusted diffusion operator, the Feynman-Kac formula becomes, f(t,s,Yt) = E;[e-X^dup(Ys)} * (A*t-q(Yt))f = dJ (4.77) To derive the dynamics for Wt and zt, consider a general expression, /oo e-K^dup(Ys)ds} (4.78) Chapter 4. Application to Asset Pricing 143 Applying Al — q(Yt) to both sides of the equation and assuming that the integral is absolutely convergent, we get /CO (A* - q{Yt))El[e~q{Yu)dup{Ys)]ds /O O dsE*t[e~ S° q{Yu)dup{Y s)]ds =E;[e-ir^y{Yoo)}-p(Yt) = -p(Yt) (4.79) where in the last step we assumed the standard transversality or growth condition on p(Yt) so as to ensure that its expected value grows at a sufficiently small rate. The evolution equations for Wt and zt are {A*t-rt)Wt = -Ct (4.80) (A*t-rt)zt = - \ (4.81) Writing this out explicitly gives the desired result. • Proof of Lemma 4.3: zx(0) = - J\-^roEt dxQu(X,u- t)(d(0)pG + pc + d{0) ac' • o-^du ds roUo - {n° + PG + o-c' (4.82) Chapter 4. Application to Asset Pricing 144 / x ( 0 ) = _ J"eHs-t){ro-f + \°C'<rC)+S;<>Cl<™« x J dxQu(X, u-t) [d{Q)pG + pc + d(0) CTC' • o-G) du + Et dxQu(X, u-t) ( d(0)*G + ac)'-dW. ds using Et ei:°c'^dxQu{x,u-t) = Lc+»a+CTC'-ac+2<Ta'-<TC)(u-t)+±CTC'-<TC(s-t) t < u < Et ehs°c'dW»dxQu(X,u-t)dWv = crGEt eftS<rC'<™»dxQu(X,U-t) du (4.83) t < U < s (4.84) In addition, c _ L „a . a c fx(0) = ( l - ^ ) ( p G + 2aG'.crG)+pG + (r ( r 0 - pc) ( r 0 - (pG + 2\ic + 2crc' • crG + ac> • <rc)) (4.85) • Proof of Lemma 4.51: Begin by analyzing f(X) as X —> 0. Assuming Eqn. (4.40) holds (which will be Chapter 4. Application to Asset Pricing 145 verified soon), f(X)xZEt ~ / e A'->0 r0-^c + \<TC'<JC ) (s-t)-ft3 K° ln(Q„) du+J* <rc'dW. 0 0 -(r0-nC + \*C'-*C ){s-t) ds \ l x C + ^ G l [ ( T C , . ( T C + r T G l . ( T G ) £ - £ r ^ l - e x p ( - ( K C + K G ) ( s - t ) ) ^ lnX 3 _ K c j - e x p ( _ ( K c + K G ) ( u _ „ ) j ( < T C ' + < T g ) ' . d w „ d « + < r c ' d w u ds (4.86) The expectation is calculated as, - K C jt* / t " e x p ( - ( K C + / c G ) ( U - i ; , ) ( /+ / ) ' • * ,+ J ' T S ac'dWu ds = \ ( ° C ' * C + { < * C + ° G ) ' { ° C + ° G ) ^ ^ - 2 ^ ^ + ^ ) ^ ^ J ( a_ t) X 1 - C X P ( - ( K C + K G ) ( S - £ ) ) / C , r r , G , « C 2 5. i ^ , ( f f c + f f G ) ^ _ ( f f c + f f G y . ( a C + a 0 ) _ ^ , l - e x p ( - 2 ( K G + K G ) ( s - t ) ) g 4 ( K C + K < : ; ) 3 (4.87) g i v i n g x->o -A(s-t)+B 1^—exp( — ( K C +« G ) ( s -4 )> )+C ' ^1-exp (-2(/tG+KG)(s-t)) j -TC^U ( l -exp(-( K c + K G )(s-t))) lnX (4.88) with Chapter 4. Application to Asset Pricing 146 A= (r0-pc) + (pc + p G - ^ ~Gi _ G , 1 C / i _C7 / T G \ _ _ K _ _ _ 2 2 J KC + KG ,C2 l (<T G + a - G ) ' > c + <x G). G . 2 (4.89) D — / C , G 1 G / _ G , 1 C7 „ C , „ . C / ^ . G \ _ _ ^ 1 _ _ B=(P -if ° + f -o- +* cr ) ^  + r G ) 2 ( < T G + 0 . G ) / . ( o . G + 0 . a ) ft C 2 ( /C C + ftG)3 (4.90) KC2 ,C , _ G y / _ C , „ G \ K C = ( ^ + 0 ' - ( » ° + O i 5 ? ^ 5 F ( « 1 ) Now making the change of variables, y = e~(K C + K G)( s~*)| InX\, v 1 x^o J0 \\lnX\J {Kc + nG)y v ; The diverging part can now be isolated in closed form: eB+C roo i c f(X) ~ g — / y«c+*a e*c+*aydy X^\KC + K G ) X ^ U \ l n X | ^ t ^ J o > B + C SKC + K g ^ ^ f ( A rc A V KC I V K C i K G {KC + KG)X^^\\nX\^; V K K + K (4.93) where T(z) = JQ°° qz~1e~tdt is the Euler Gamma function, and A must be positive. For any postive value of KC, as long as KG > 0, the growth condition in Eqn. (4.40) holds. From Eqn. (4.93) we can also deduce the behavior of fx(X) as X —> 0: ^ H ^ ) " ^ ^ ) <«•*> 0 (KC+ KG)X^a~+1\lnX\^u K K ' K + K ' Chapter 4. Application to Asset Pricing 147 To derive the asymptotic behavior of Z(X) for X —» 0 express its derivative from Eqn. (4.47) up to leading divergent terms: Z X ^ XZOkG[°°e~(s~t)r°Et [ dAdQ(QUd(Qu)HQu))du To leading order this is zx(X) ds (4.95) X ^ O X{KC + KG) e-(°-t>°Et e - { K C + K G ) { u - t ) \ \ n Q u \ l ~ ^ a Q ' f ^ du ds (4.96) Making the variable changes, v = e - (« c +« G ) («-*) | \ n X \ and q = e-(-c+«G)(^-t)| l n x | , for u and s, respectively, and using the functional form for Qu derived in Eqn. (4.53) gives zx[X) ~ f j X^°X\\nX\L+^^{KC + Kg)*JO | lnX| rllnAI - + « ( i - r ^ ) + / 3 ( i - n ^ ) ^ ( ° - 1 ) ) - - + - ( 1 - n n ^ ) + / 3 ( 1 - n ^ W 1 ) In X I dvdq (4.97) where a = PC + pG - l(*C' • * G + CTG' • <TG) KC + K G (4.98) and (4.99) 2{KG + KG) Af(0,1) is a standard normal variate and the expectation in Eqn. (4.97) is taken with respect to its distribution. As long as A < KF + K g , which is true under most reasonable Chapter 4. Application to Asset Pricing 148 parameterizations of the consumption process, the integral in Eqn. (4.97) has a well defined limit: zx{X) nG2r0fo x ^ ° X | l n X | 1 + ^ ^ ( f t c + KG \ 3 J0 Jq X E ^ p ( - « + « W ( o , i ) -v + a + BM{0X 1 - -dwdg (4.100) There is no explicit closed form solution for the last expression. It is straight forward, however, to calculate it on a digital computer. A n expression for z(X) for X ~ 0 is readily integrated (asymptotically) to yield the desired expression. • Proof of Lemma 4.5: Because pc(X) is constant, Eqns (4.42) imply that z(0) = ^ and /(0) = ^z^p- To derive the asymptotic behavior of Z(X) for X —> 0 express its derivative from Eqn. (4.47) up to leading divergent terms: zx(X) ~ Kg [ e-t'-^Et f dxQudQ(Qud{Qu)HQu Jt Jt v du ds (4.101) since d(0) is finite, this can be expressed to leading terms as zx(X) K G p C ^ ,c\ /O O PS e - ( s - t ) r 0 E t J Q u x^o X(r0 - p' j J t Setting y = e-K < 3( u-*) and calculating the expectation ,c e-K°(u-t)duds (4.102) MX) x - o X(r0 - pG) O O fl - ( s - t ) r o o ! / l n ( X ) + / l ' ( l - i / ) + B ' ( l - y 2 ) x (y ln(X) + A ' ( l -y) + 2B'(1 - y2)) dy ds (4.103) Chapter 4. Application to Asset Pricing 149 where C + G _ l ( ( T C , . a C + (TG, ,aG\ A' =- ^—^ (4-104) B' =^G(O-c + vG)' • (<xc + <rG) (4.105) proceeding as in the previous lemma leads to * W , - o K o ( n _ £wZxiw* (' + £ ) r(^> <«•«*» Integrating over X asymptotically gives Eqn. (4.106). The result for f(X) is derived in the same way. • 4.5 Derivation of State Prices The argument is inductive. We conjecture that at date t the investor maximizes a derived utility as in Eq. (4.66), with each event at+i G Ft+i giving a utility of , M D p-9i{ty\{at) r{gi{t),Ri{t)) ft VM)M» (cl(at),Wl(at+1)) = --9i(t) ^ , * p-OLt+i(.at+i)gi{t)wi(at+i) - P t + l ( a t + 1 ) e - ^ a ^ R ^ - — — (4.107) at+1{at+i)gi{t) where Wi(at+i) is as in Eqn. (4.67). Given Eqns. (4.5)-(4.7) and the fact that (pr = 0, xT+i = 0, it can be readily verified, by setting 4>(as\at) = 0 for t > T, that Eqn. (4.107) holds for dates T and T — 1. Specifically, pr+i = 0, PT(O-T) = 1, QT(O-T) = 1 and O;T(O-T) = 1-The optimization program facing the consumer is given by Eq. (4.68). Eq. (4.69) can Chapter 4. Application to Asset Pricing 150 be written as (f)(at+i\at) = 7 r ( a t + i | a t ) p t + 1 ( a t + i ) e x p ^ ( t )c*(a t ) - at+1{at+i)gi{t)wi(at+i) - qt+i(at+i)Ri{t)^ (4.108) Multiplying each side by a t + 1 ( a ^ and summing the last equation over all the date t + 1 events in the at partition leads to E (f>(at+i\at) -gi(t)cHat) -a*+i(at+i) ^ ^pt+l(at+1)e-^a^R'V ( \ } Tr{at+l\at) r exp -at+i{at+1)gl(t)wi{at+1) at+1{at+i) \ I (4.109) The optimized utility from Eq. (4.68) in the event at is therefore, The left hand side is not an explicit function of Wi(at). To fix this, note that Eq. (4.108) can be manipulated to give C*(Q0 _ qt+i{a,t+i)R{t) _ j = 1 l n ( <t>(<h+i\<h) s ( 4 n i ) Ut+i{at+\) at+i(at+i)gi(t) % t + gi(t)at+i(at+i) ^{at+\\at) Pt+i{at+i) Wi{at+\) in the last expression can be written out as in Eq. (4.67), afterwhich multiplying both sides of the above equation by 4>{at+i\at) and using the relationship, (j)(at+i\at)4>(as\at+1) = (f>(as\at) when as C at (from Eq. (4.3)) gives <^K^ _ ^ g t + 1 (a t + ^K + 1 | a t ) _ / t { ) < K M + £ c* ( a^( a > t )) = a t + i ( a t + 1 ) 5 i ( t) a t + i ( a t + 1 ) V a ^ + i / s>t + l 0(a t + i |at) l n / <t>(at+i\at) x (4112) gi(t)at+i(at+i) v 7r (a t + i | a t )p t + i ( a t +i ) ' Chapter 4. Application to Asset Pricing 151 After summing this last expression over all date-t + 1 events in the at partition one can once more refer to Eq. (4.67), but this time as applied to the date t budget. The result yields t/ x 4>{at+i\at) Ri(t) v - ^ Qt+i(<k+i)<l>{at+i\<k) if \ . t, \ cAat) I . 7 - c TTT y. 7 - x ui {at) + ci{at) = at+1{at+i) gl{t) . at+1{at+l) <H+lCat - r \ T- / v o t + iCoi ^ v 1 y- (j>(at+i\(k) l n / <t>(<h+i\<k) x (4 113) 9i(t) ^-tn a t +i(a t + i) {ir(at+i\at)pt+i(at+1)J The above can be solved for c\{at) in terms of Wi(at): „ <n \,n (r. U W K f t ( a t ) - - l ) lm>(at)) , . Ci(at) = a i ( a# i f l t + TTT T r r — (4.114) 9i{t) 9i{t) with 1 _ {•> , 0(at+i|at) (1+ Y (4.115) qt{at) = </>(at+ilQt)fr+i(Qt+i) + 1 ^ Substituting this back into the expression for the maximum date t+1 utility, Eq. (4.110), results in at (at )gi (t)wl- (at) [ / ; ( * W A ( 0 ) ( a t i ( a t ) ) = _ ( a t ) c - f t W t e ( « , ) - i ) ! _ ^ ^ _ ( 4. i i8) ost{at)gi{t) Note that pt{at), qt(at) and at (at) do not depend on the bias assignment at date t (i.e., (gi(t), Ri(t)), and are therefore universal across investors, r^rry can be interpreted as the price of a bond that pays a unit coupon at each date. is the price of a bond that pays a stochastic coupon equal to the value of r^rry at each date. The date-(t — 1) Chapter 4. Application to Asset Pricing 152 derived utility conditional on event at can be calculated from Eq. (4.5): ~ - f l i ( t - l ) c J - 1 ( a t _ i ) r(9i(t-l),Ri(t-l) f t - l 9iit ~ 1) e-Ri(t-l) / f°° e-g'/gi(t-l)-R'/Ri(t-l) / r°° e-g' 9i{t-l)-R' Ri(t-V / , \ , A (4.119) This simplifies to e - 9 i ( * - l ) c , ' - 1 ( a t - l ) - 1) p t ( f l t ) e - "^Wf t - i ) e (4.120) at{at)gi{t - 1) which is of the form conjectured in Eq. (4.107). It is now possible to return directly analyze the state prices. Specifically, summing Eqns. (4.111) and (4.114) over all investors and using the market clearing conditions, Eqns. (4.4), yields C(at) - qt+1{at+l) V - at+1(at+1)w(at+l) = ln(——^* + 1 ^ f ) r) J ] —777 (4.121) N Rlt) N 1 C(at) = at(at)w(at) + (qt(at) - 1) £ _ l n ( A ( a t ) ) £ — (4.122) i=i ^ ' i=i ^ ' One can now solve for p t + 1 ( a t + i ) by using the date t + 1 version of Eqn. (4.122) and substitute the result in Eqn. (4.121): ^ ftW v 7 r ( a i + 1 | a t ) y ^ & ( ; = 1 * f ( C ( a t + 1 ) - a t + 1 ( a i + 1 ) W ( a f + 1 ) - (qt+l(at+1) - 1) £ ^±ii) (4.123) Chapter 4. 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