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 U N I V E R S I T Y 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 axiomatized partial order. The first theory is an atemporal alternative to von Neumann and Morgenstern's Expected Utility Theory that accommodates the status quo bias, violations 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 particular, 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 ii Table of Contents iii 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 A n c h o r e d Preference Relations: a theory of the status quo bias 10 2.1 Theoretical Foundations 23 2.1.1 30 2.2 Further Discussion 2.2.1 2.3 3 Anchored Preference Relations 35 Other Anomalies: Preference Reversals and Imprecise Certainty Equivalents 35 2.2.2 Indeterminate Anchor 37 2.2.3 Relation to semiorders 39 Appendix 40 Inter-temporal Flexibility Preference iii 49 3.1 3.2 3.3 Introduction 50 3.1.1 Related literature on utility for flexibility 57 3.1.2 Induced Preferences 58 Theory 60 3.2.1 Formulation of the Choice Problem and Agents'Preferences 3.2.2 Time Consistency ... 60 67 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 . 3.3.3 Relation to Additive Representations and Second-degree Dominance 78 75 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 4.3 Further Discussion 100 Asset Pricing 103 4.3.1 Constant Drift and Diffusion Coefficients 110 4.3.2 Mean Reversion in G and C Ill 4.3.3 Simulations 114 4.4 Proofs 140 4.5 Derivation of State Prices 149 Bibliography 154 iv List of Figures 2.1 E U T indifference surfaces 2.2 13 Anchored 'better-than' sets are parallel wedges. Note that p >- r, r p q and q y NP y NP p 14 2.3 Neither p nor q can be said to 'Pareto-dominate' the other in the Kahneman 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 indifference curves shows up near the low outcome lottery 18 2.6 General Anchored Preferences. The 'better-than' set at the anchor is constrained to coincide with the 'Pareto-dominance' wedge. Bottom: Betterthan sets must contain the wedge 19 3.1 A Dynamic Choice Problem with incomplete preferences 4.1 66 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~ = —o~ ) G 116 c 4.2 Comparative statics of the riskless rate and equity premium with respect to \o~ \ and X when the coefficient of relative risk aversion does not vary c through time (<r = -cr ) G 117 c 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 (cr = -cr ) G 4.4 118 c 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, rr (xy M 4.5 r (X). T Bottom: the equity premium on the market portfolio, • o (x) 120 M Comparative statics with respect to r . 0 The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the wealthdividend ratio, f ( X ) . Bottom: the normalized differential consumption coupon, d(X) = D/C 4.6 121 Comparative statics with respect to y, . The squares represent the 5th, c 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, portfolio, a M 4.7 r (X). T Bottom: the equity premium on the market { X ) ' • a (X) 122 M Comparative statics with respect to \i . The squares represent the 5th, c 50th and 95th percentile values of the state variable, X. Top: the wealthdividend ratio, f ( X ) . Bottom: the normalized differential consumption coupon, d{X) = D/C 4.8 123 Comparative statics with respect to \o~ \. The squares represent the 5th, c 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, r (X). portfolio, M CJ {X)' M T Bottom: the equity premium on the market 124 • CT (X) vi 4.9 Comparative statics with respect to \cr \. The squares represent the 5th, c 50th and 95th percentile values of the state variable, X. Top: the wealthdividend ratio, f{X). Bottom: the normalized differential consumption coupon, d(X) = D/C ' 125 4.10 Comparative statics with respect to K . The squares represent the 5th, G 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, r (X). Bottom: the equity premium on the market t portfolio, a {X)' • a (X) M 126 M 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 wealthdividend ratio, f(X). Bottom: the normalized differential consumption coupon, d{X) = D/C 127 4.12 Comparative statics with respect to \x . The squares represent the 5th, g 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, r (X). Bottom: the equity premium on the market t portfolio, cr (X)' • (T {X) M 128 M 4.13 Comparative statics with respect to u . The squares represent the 5th, g 50th and 95th percentile values of the state variable, X. Top: the wealthdividend ratio, f(X). Bottom: the normalized differential consumption coupon, d(X) = D/C 129 4.14 Comparative statics with respect to |<r |. The squares represent the 5th, G 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, r (X). Bottom: the equity premium on the market t portfolio, (T {X)' • <r (X) M 130 M vii 4.15 Comparative statics with respect to |cr |. The squares represent the 5th, G 50th and 95th percentile values of the state variable, X. Top: the wealthdividend 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, r (X). Bottom: the equity premium on the market portfolio, t M^ y a x . 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 wealthdividend 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 percentile 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 University, University of British Columbia, UC Berkeley, U C L A and University of Washington 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 behaviour with suitable 1 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 incomplete binary relation (a strict partial order) over atemporal risky prospects, This relation is termed an incomplete reference based preference relation. The axioms constitute 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 y p p) if and only if the expected utility of q exceeds that of p with respect to all members of the set . A similar ordering, but in a setting of uncertainty, has been 2 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 w i t h an intransitive reflexive complement, y. T h e 'P' refers to the fact that the ordering is similar to that induced by the notion of Pareto dominance w i t h respect to a set of utility functions 2 1 2 Chapter 1. Introduction adding additional axioms so as to complete y , p an Anchored Preference Relation (APR) is derived. APRs are intended to model the well-documented status quo bias and endowment 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 y p instead of the usual von Neumann-Morgenstern preferences over inter-temporal lotteries . 3 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 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. 3 3 Chapter 1. Introduction is true are far less stringent for Inter-temporal Flexibility Preferences than for Intertemporal 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 out , a detailed theory from the class of Inter-temporal 4 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. F F D 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 correspond to uncertainty over risk aversion and inter-temporal substitution parameters. Because agents have Knightian uncertainty over future preferences, trading in this economy 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). 4 Chapter 1. Introduction 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 Flexibility 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 therefore 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 correlated 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 consumption) 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. 5 Chapter 1. Introduction 1.1.1 Static Choice U n d e r R i s k The static E U T of von Neumann and Morgenstern (vNM (1944)) is based on three assumptions . The first requires an agent's preference relation to be complete and tran5 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 accommodate the impressive body of empirical literature documenting violations of E U T . For 6 choice under risk (where probabilities or subjective beliefs are assumed known), alternative 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 tractability . Finally, theories based 7 on semi-orders (see Luce (1956), Vincke (1980) and Nakamura (1988)), as opposed to completeness, violate dominance. 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". Notable 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 Einhorn (1987), Hey and Orme (1994), Kahneman and Tversky (1979), Lichtenstein and Slovic (1973), MacCrimmon and Larsson (1979), MacCrimmon and Smith (1986), MacCrimmon, Stanbury and Wehrung (1980), Tversky and Kahneman (1986). A theory that drops the reduction hypothesis must define preferences over every conceivable multistage 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. 5 6 7 6 Chapter 1. Introduction 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 E U T 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 N o n - E x p e c t e d U t i l i t y and D y n a m i c Choice U n d e r R i s k The classic inter-temporal theory of choice consistent with the Expected Utility Hypothesis 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, perhaps, the most familiar example. The literature seeking to directly impose restrictions on preferences through dynamic 'rationality' reaches back almost as far as the literature 7 Chapter 1. Introduction on E U T violations . The general conclusion is that any set of preferences which can be 8 represented by some cardinal continuous utility function and that accommodates violations of translation invariance (i.e. any non-linear utility theory) must violate one of the following principles 9 1. Reduction - choice/probability trees with identical reward distribution are necessarily 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 renounce . 10 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). For an excellent review, see Machina (1989) and Sarin and Wakker (1998) For a somewhat different 'breakdown' of temporal consistency, see Halevy (2000) and references therein. T h u s agents will be willing to 'make book' against themselves (Green (1987)). 8 9 10 8 Chapter 1. Introduction 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 reference on the subject is Kreps (1979). Kreps considers preferences 11 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 Rustichini (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 11 Other historical references can be found in Kreps (1979) as well. 9 Chapter 1. Introduction 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 opportunity 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 models . 12 1.1.4 A l t e r n a t i v e Asset P r i c i n g 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)) . Non-expected utility type models include Epstein 13 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 utility 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. T h e models are generally two-stage, where the first stage consists of choosing a 'menu' while the second selects an object from the menu. ' Schroder and Skiadas (2000) show that these are isomorphic to recursive utility models. 12 13 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= p 1= \ $1500 A = I L $1000 P l 0= p H \ n and B $2000 j ( 0.5 = q L \ $1000 0 = qj 0.5 = q H $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 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 v N M axioms lead to the existence of an expected utility representation, where p is preferred to q if and only if E [U] — E [U] > 0. U is the vNM P q utility function, and E denotes expectations taken over the probability distribution, q. 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. A n example is , 1 E [U] - E [U] > inf ue{u } E [U] - E [U] p y q <^ inf P e q e e (2.2) a where p >~ q, should be interpreted as "p is preferred to q when the anchor is at e", and e {U } is a set of utility functions. Note especially that this representation reduces to that a of vNM if {U } is a singleton set. The representation in Eqn. (2.2) is a special instance a 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 E [U] — E [U] > 0, must hold for each member of the utility function set, P e {U }, in order for p to be preferred to the anchor, e, itself . The latter can be thought 2 a of as a requirement for 'overcoming' the anchor. It is this special relationship between lotteries and an anchor that characterizes APRs. In particular, all A P R s induce a pair of binary relations between lotteries, {V , y }. p py p q NP E [U]-E [U] inf ue{u } > 0 (2.3) E [U]-E [U] inf ue{u ] < 0 (2.4) p q a qy NP p p q a The class of representations derived is more general than the example. Equivalently, E [U] - E [U] > 0 must hold for every function in the smallest set * that generates U through linear combinations. In other words, * is the smallest set such that U is in its convex hull. x 2 P a e a 11 12 Chapter 2. Anchored Preference Relations: a theory of the status quo bias q >- NP p can be interpreted as p cannot strictly overcome an anchor at g", while p >- q u p can be interpreted as "p can strictly overcome an anchor at q." Intuitively, y p Pareto dominating relation in the sense that p y p is a 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 > q for any e anchor, e (and in particular, whenever e = q). >- , on the other hand, corresponds to, NP possibly, non-Pareto dominance in the sense that p y q if and only if p is preferred NP to q with respect to at least one of the utility functions in the utility function set. It is possible that s y r and r y NP NP 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 y NP is not transitive. On the other hand, if s can 'overcome' an anchor at r (i.e., s y p r), 5 is preferred to r regardless the location of the anchor, and choice behavior is context independent. In particular, s >- r => s y p NP r. The stronger relation, y , is transitive but incomplete, p 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> T 2 = (^($1000),^($1500), V ($2000)) = (0, ^ 2) 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 y p $1500 y p $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, E [ipi] — E [ipi] — ipj • (q — p) = q p 13 Chapter 2. Anchored Preference Relations: a theory of the status quo bias H Figure 2.1: E U T indifference surfaces. > 0, and E [ip2] ~ E [ip2\ = 4>2 • (p — q) = \ > 0. | p q 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 y NP qy q and p, so knowledge of the anchor is required to determine which lottery is preferred. NP 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 elements: {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 natural bijection which exists between the three outcome lottery space and the unit simplex S 2 = {(Pi>P2) G 1Z \pi,p2 > 0,pi +p < 1}- Figure 2.1 illustrates the idea. The horizon2 2 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), 14 Chapter 2. Anchored Preference Relations: a theory of the status quo bias L Figure 2.2: Anchored 'better-than' sets are parallel wedges. Note that p )~ r, r y and q y p. p NP q NP 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, y p and >~ , derived by weakening the v N M axioms, are NP illustrated in Figure 2.2. In contrast with EUT, 'better-than' sets with respect to the Pareto dominance relation, y , p are parallel wedges ; these correspond to the intersection 3 of the two v N M '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 y p r, as well as anchor dependent preference, r y latter example demonstrates that y NP NP q and q y NP p. The 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 underlie the final choice. The bias assignment can be purely descriptive or axiomatic. One 3 T h e 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 influence 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 $1500 $1000 $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) . Consider, instead, a representation that is a more general Anchored 4 Preference Relation, py q $ [E [U] inf ue{u } u 4=^ e P -E [U] e > a $ (^E [U} inf ue{u } -E [U] u q e (2.5) a where Q : 71 —> R is 0 at 0 and is increasing for every U G {U }. Each element of the u a utility function set, U, is associated with a, possibly, different In the Allais example, by setting r™ + 5o,ooo ( T T a / {U (w)}={———^——, \ 100,000 ' a 1 v n 100,000 ) , r _ a . u ' ^ > and {$ (x)\ ™ +50,000 J ^ a K n f , 4 - {x ,x\ ' 4 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 E U T 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. 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. 4 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 L I A llowed Not A llowed Figure 2.6: General Anchored Preferences. The 'better-than' set at the anchor is constrained to coincide with the 'Pareto-dominance' wedge. Bottom: Better-than sets must contain the wedge. 20 Chapter 2. Anchored Preference Relations: a theory of the status quo bias 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, {>- ,y }p NP In the next section I axiomatically set up the necessary and sufficient conditions for the following representation theorem (2.6) where ^ is a set of bounded continuous functions on the set of final outcomes The 5 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 NP (2.7) 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) 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. 5 21 Chapter 2. Anchored Preference Relations: a theory of the status quo bias includes that of v N M 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 completeness conditions on a set of binary relations, {V }> where e indexes all lotteries. In e addition, one must require that q y p q <£4> q y- NP p and that q y- p Ve e q >- p. p These requirements imply that if q y p for any e, then it must be that q y p. In partice q ular, the representation in Eqn. (2.5) follows by assuming the convexity of upper contour sets and a form of translation invariance . The representation in Eqn. (2.2) follows if, 6 in addition, one requires scale invariance . To be sure, none of convexity, translation 7 invariance or scale invariance have the normative appeal of the weakened vNM axioms that define {>- , y }. p The representations in Eqn. (2.2) and Eqn. (2.5) can therefore NP 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 willingness to buy and willingness to sell, led early investigators to postulate reference effects (Chechile and Cooke (1997), MacCrimmon and Smith (1991), Goldstein and Einhorn (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 dependence. 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 Translation invariance means that for every anchor, e, p ~ 9 P + £, ~ e + £ <J + £ whenever p + £, e 4- £, and q + £ are lotteries. Scale invariance means that for every anchor, e, a G (0,1], and lottery, r, it is the case that 6 e 7 p> q e ap + (l- a)r >- +(i- )r ae a 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 preferences 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 E U T have largely ignored this literature. As Cubbit (1996) notes, the reason may have to do with a bias for relaxing the v N M 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 A P R ) 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 {y ,y } p NP as in Eqn. 2.6. Anchored Preference Relations are then defined and representations derived under different 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, {>- , >~ }, to similar constructions in the literature. p NP 23 Chapter 2. Anchored Preference Relations: a theory of the status quo bias 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, {y }, directly, I focus initially on the a more general, yet incomplete, binary preference relations, {>- , y }. p These define the NP 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, {y , p y } such that for every p,q € NP exactly one of the following holds (with the associated interpretation): (i) p y q (ii) qy p p (When anchored at p, q is strictly preferred to p.) (iii) q~ p (When anchored at p, there is indifference between p and q.) NP p The relation, ~ P >z NP (When anchored at p, p is strictly preferred to q.) p is defined by the failure of (i) and (ii) above to hold. q is taken to denote anchored weak-preference (i.e. p y similar interpretation for q y NP p q or q ~ p Further, p), with a 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 V{X), 24 Chapter 2. Anchored Preference Relations: a theory of the status quo bias structure can be imposed on incomplete reference based preference relations by adopting a suitable form of negative transitivity. Definition 2.2 {>~ ,>~ } is a frail order on V(X) if and only if {y ,y } P NP p incomplete reference based preference relation on V{X) qy p => r y p p or q y p NP NP and for every p,q,r 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 y p is asymmetric and transitive. The symmetric complement of >- is not necessarily transitive, however, which means that >- is a strict p p partial order. B y comparison, >z , the complement of ^ NP p by Definition 1, is not even assumed to be a true order relation but merely a suborder . This is analogous to saying 8 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 {y , p y } NP 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 >- (Duggan (1999)). p is an E V(X), 25 Chapter 2. Anchored Preference Relations: a theory of the status quo bias 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 B P and W = {q\p y p p q}. In other words, B N P P = {q\q >r p} P 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, W . Further, define W NP p p = {q\p y q} and B p p p = {q\q y p}. NP W p represents all the lotteries which are no better than p, viewed with an anchor at the alternative; a complementary definition applies to B on B p and p p . The next axiom forces a particular structure Wf . p A x i o m 2.2 (Continuity) B = Closure{q\q y p p} p = {p} if {q\q y p] ^ 0 p otherwise and W" p = Closure{q\p y = {p} N P q} if {q\p y N P q] ^ 0 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 q —> q and n p —> p are Weak-* convergent sequences with q ~ n n p p for every n, then q ~ n These rather technical conditions essentially stipulate that B P in M(X,T,x,T^)B p and W . p p and W^ p p p. are closed Further, anchored indifference surfaces reside on the boundaries of It is straight forward to show that the continuity axiom implies y_ is a p 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 >~ and >~ . NP p A x i o m 2.3 (Independence or Linearity) Given p, q, r G V(X) and a G (0,1], py N P q implies ap + (1 — a)r y aq + (1 — a)r N P and qy p p implies aq + (1 — a)r y p ap + (1 — a)r. The intuition behind this axiom is similar to that behind the conventional Independence A x i o m 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 will 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 representation theorem for {y , p y } NP N P p =$> q >~ p. The following p is one of the main results of this chapter. T h e o r e m 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, 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>. O n 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), qy N P p qy p, then p where U is a real, continuous and bounded function on X defined up to an affine transformation. Further, P 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. Lemma 2.1 Linearity of Anchored Indifference For any p,q,r G V(X) and a G [0,1], q ~ p aq + (1 — a)r ~ p ap + (1 — a)r p => 28 Chapter 2. Anchored Preference Relations: a theory of the status quo bias L e m m a 2.2 Transitivity of y p For any p,q,r E V{X), q y p p and r >z q =*> r y P p p. L e m m a 2.3 No Direct 'Money Pumps' qy p p qy p and in particular, q y NP p ; p =>• q y NP p. L e m m a 2.4 Convexity of'at-least-as-good-as'Sets q,r E B p => aq + (1 - a)r € B V«G [0,1]. P If in addition either q y p p or r y p p then aq + (1 — a)r y p p Va E (0,1) (denoted as strict convexity). L e m m a 2.5 Nesting of 'at-least-as-good-as' Sets qEB ^ p B P C B P L e m m a 2.6 Strict nesting of 'at-least-as-good-as' Sets For any p,q,r E V(X), if q E B p and r E B p and either q y p p or r y p q then r y p p. Lemma 2.1 extends A x i o m 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 i n 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 >z is a partial order. This is P 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 y p p, or in other words, p would have been exchanged for r and there is no cycling. Lemma 2.4 says that the set B P indifference surface bounding B P is convex, whereas Lemma 2.1 demonstrates that the 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 i n the space of signed measures, A4(X, T,x,R)Specifically, define B p = {\(q-p)\ qt P,Xen , , eV(X)} p + q P P r o p o s i t i o n 2.1 Every anchored 'at-least-as-good-as' set, B , P intersection ofV(X) (2.8) can be written as the and the translated set, p + B . p V{X) n ( + B ) p P The purpose of the following lemma is to establish that B p L e m m a 2.7 B p is weak-* closed and convex. is a closed convex cone. 30 Chapter 2. Anchored Preference Relations: a theory of the status quo bias The discussion so far has focused on properties of the anchored sets B and by P implication, W . In particular, no mention has been made of the structure of B NP P and W . The following result completes the description of {y ,y } P p canonical cone, NP P P in terms of the B. P Proposition 2.2 w = v{x) n (p - B) P p Theorem 2.1 essentially derives from observing that y and y p by the canonical cone, B . P Because it is closed and convex, B NP P some set of supporting hyperplanes. Since B P are characterized itself can be described by C A4(X, E x , 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 2.1.1 B. P Anchored Preference Relations Theorem 2.1 derives by relaxing the v N M axioms. A lottery at p will be traded for lotteries in B and any lottery in W P W P P \ W P will always be traded for p. The ambiguous region P (or equivalently B P P \ B ), P is what characterizes A P R s . Violations of the v N M axioms will 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 assumptions that { V , y } p NP normatively 'bound' violations of E U T . Unfortunately, y p 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 y , p however, that allows one to postulate ref- erence dependence for lotteries that are incomparable through y . p In this subsection I 31 Chapter 2. Anchored Preference Relations: a theory of the status quo bias formally define Anchored Preference Relations and provide necessary and sufficient conditions for several representations. The idea is to define a set of complete and continuous preferences, { V } , each of which has an upper contour set (at-least-as-good-as set) that a agrees with B at e. In other words, y P e is a complete representation of preference when an anchor (e.g., the status quo) is known to be at e. Definition 2.3 An A n c h o r e d Preference R e l a t i o n is a set of binary relations, {y } a over V(X) such that (i) y e {y } & e e V(X) (ii) y € {y } =>• y (iii) There exists a reference based preference relation, {y & a e a is a complete, weak and continuous order. e p ,y } obeying NP Axioms 2.1-2.3 such that for any p,q E V(X) qy p q qy p e ^ qy N P p for all y E {y } ^ qy p p e a The symmetric complement of y E e not the case that q y e p or p y e {y } a is defined, as usual, as q ~ e p 4^ it is 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. T h e 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 y q p and p y p q. The second requirement formally establishes the connection between overcoming the anchor and the Pareto dominating relation, P r o p o s i t i o n 2.3 Fix {y } a y. p and its associated reference based preference relation, {y p Chapter 2. Anchored Preference Relations: a theory of the status quo bias (i) 32 For every y E {>-o} there exists a real valued bounded and continuous e function, H (-), overV(X) such that for any p,q G V{X) e qy p H (q) > H (p) e (ii) e e If H (-) has a subgradient, tp , at p G V{X) then ty is, up to an affine e g g transformation, a convex combination of members of . For any p, q E V(X), (iii) q~ v q~ p p P H {q) = H (p) P 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 >- and its representation. e 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 y e {y }, P,q,r G V(X) and a G [0,1], e a p y q and r y q e ap + (1 — a)r y q e e A x i o m 2.5 Translation Invariance For any y £ {y }, p,q eV{X) e a P~ q e for any£ G M(X,T, ,Tl), x p+ £ ~e+£q+ £ 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 y E e {y }, a p,q,r E V(X) py q and a E [0,1], otp + (1 - a)r y (!_ ) e ae+ 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. . Both translation invariance and scale invariance, however, 9 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). Translation 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. 9 Chapter 2. Anchored Preference Relations: a theory of the status quo bias 34 Proposition 2.4 //{>~ } is an Anchored Preference Relation, then a (i) Axiom 2.4 implies that >- can be represented by e H (p) = inf $f(E [ip] - E [ip}) e p e 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 y can be represented by e H {p) = inf ^(E [tp] e p - E [ip]) e 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 y e H (p) = inf (E [ib] - E [ip]) e p e can be represented by where K 7 C lin(#) where $ is the set of utility functions provided in Theorem 2.1, lin(^) 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 E q n . (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 U (x) > U (y), where x, y and t are bundles of goods and U (-) is a reference-dependent t t utility function. t 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 2.2 35 Further Discussion 2.2.1 Other Anomalies: Preference Reversals and Imprecise Certainty Equivalents 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 . 10 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 See Chechile and Cooke(1997), Goldstein and Einhorn (1987), Loomes and Sugden (1982), Lichtenstein and Slovic (1973), MacCrimmon and Smith (1991), Tversky and Kahneman (1986). 10 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) y p A, happens to coincide with the expected value of A. Thus c(A) < c(B), consistent with the empirical findings. O n 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 2.2.2 37 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 i n 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 will 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, y , p 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: M = {peF\FC F 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, >- can be interpreted as a Pareto-dominating ordering p 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 38 Chapter 2. Anchored Preference Relations: a theory of the status quo bias 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, (2.9) 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 W ^ {p} and B p p p S F = ^ {p}, then M. F The requirement that there.exists p € V{X) such that W p p ^ {p} and B p ^ {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 c o ( ^ ) . Alternatively, an anchor assignment can be seen as analogous to an assignment of a security index i n the sense of Levi (1980). It is important to stress that only revealed choice can be interpreted as the optimization 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 . 39 Chapter 2. Anchored Preference Relations: a theory of the status quo bias 2.2.3 Relation to semiorders Related to frail orders are semiorders, first introduced by Luce (1956) to model intransitive 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. A s mentioned earlier, >~ is a strict partial order which is a weaker notion than that of a p semiorder. Consequently, similar to semiorders, the symmetric complement of y p 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 y q <^> p J {dq u 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, ip = u+a\ l for some u. Moreover, assume that if u + a E ^ then u — a E $ as well. Theorem 2.1 1 1 can then be written as qy p p 4=> Ju(dq-dp)>E(p,q) (2.11) where T,(p,q) = max| J a (dp — dq) \ . There are some important differences between the l 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 E q n . 2.10 does 40 Chapter 2. Anchored Preference Relations: a theory of the status quo bias 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' i n that violations of dominance are not necessary. First degree stochastic dominance is guaranteed so long as every utility function i n ^ is increasing , and yet the reflexive complement of y is 11 p intransitive. 2.3 Appendix Proof of Lemma 2.1. B y assumption, q e B . If B p — {p} or if a 6 {0,1}, then the proof is done. Otherwise, P by A x i o m 2.2, q must be the limit point for some sequence {q } such that q y p. n n p It is possible to construct the sequence {s } such that s = aq + (1 — a)r. Clearly, n n {s } —> aq+(l -a)r. Moreover, by A x i o m 2.3, s y p n n for any n. Since B ^_ y a aq + (1 - a)r E B ^_ y. a Using the definitions of B ^_ y p p+ aq + (1 — a)r ~ p n A similar argument shows that aq -f (1 - a)r G p p+ =4> s e ap + (1 B ^_ p p+ a)r is a closed set, the sequence converges inside it to give p p+ ap+(l-a)r n — a W^ ^_ y. p a and W^ ^_ y and using A x i o m 2.1 then implies that p p a 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. 41 Chapter 2. Anchored Preference Relations: a theory of the status quo bias The special case q y p =>• q y p p is proven using the Negative Transitivity as- N P sumption from A x i o m 2.1 setting r = p. To prove the general case, which may include indifference, invert the Negative Transitivity property. P r o o f of L e m m a 2.4. First notice that if any one of r or q coincides with p then the theorem is a straight application of Lemma 1 or Axiom 2.3. In any other case, A x i o m 2.2 implies that there exists a sequence of points, {r } n (1 — a)r n + ap = 2.3, write q E B y P a p p. Thus by Axiom 2.3, Also, from Lemma 1 and Axiom + aq = q™ E B n. Lemma 2 implies that g™ E B . P n clearly g™ —> q . Since B >- n p for every a E [0,1). p =>• (1 - a)r P — • r, such that r P is closed, one must conclude that q E B . p a Now, 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 q y p a r p. For example, if r y p p, then = r and g™ = q for every n; the implied strict preference in Lemma 2 finishes the n a job. This takes care of the strict convexity portion. P r o o f of L e m m a 2.5: To prove that q E B B P B P r E B. p If B p P C B, p p. The strict convexity property then implies p p for any a E (0,1]. Now, r E B p P Lemma 1 imply that aq + (1 — a)r E B - ^_ ^ . p + aq + (1 — a)r E B p (0,1). Picking a a q along with A x i o m 2.3 and Lemma 2 can now be used to give for any a E (0,1]. In particular, this is true for any sequence of a's in = 2~ and r = (1 — a )r + a q gives {r } n n and r E — {p} then q = p and the proof is done. Otherwise, Axiom 2.2 ensures the existence of some q y that aq + (1 — a)q y it is sufficient to show that q E B P n n n n —> r, and therefore r E B. p 42 Chapter 2. Anchored Preference Relations: a theory of the status quo bias P r o o f of L e m m a 2.6: The case q y p p is treated by Lemma 2. Assume, therefore that q ~ Now, assume to the contrary that r ~ p n N P r n p q. —> r such that for every n. There has to be some element of this sequence, say f, which is strictly preferred to q. If not, then {r } n must converge in W^ , p a contradiction. Thus by Lemma 5, f G B , P sequence. p and r y p. Since r ^ p (otherwise there would be a contradiction with Lemma 3), by A x i o m 2.2 there exists a sequence {r } p y p The conclusion is that r ~ p which means r G W^ ', p which violates the hypothesis about the p cannot hold. Since r G B P it must be that P r y p. P r o o f of P r o p o s i t i o n 2.1: To begin, suppose that q G B . Clearly q G V(X), p 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 B ,. This is p trivially accomplished by choosing A = 1, q' — q and p' = p. This demonstrates that B P Cp(X)n(p + B ). p To show the equivalence the other way, fix q G V(X) f l (p + B ). q = p + \[q' - p ) for some for some A > 0 and q',p' G V(X) 1 p This means that such that q' G B . p There are now two possibilities, A < 1 and A > 1. Consider first the case where A < 1. Since q' G B , p A x i o m 2.3 and Lemma 1 guarantee that z = \q' + (1 — X)p' G B ,. p Now construct the two lotteries, r = ^q + \p' and r' = \q + \z. A x i o m 2.3 and Lemma 1 can be used again to show that r' G B . p that if one assumes, q 0 B P the contradiction, r' 0 B. p Note that r can also be written as r — \z + | p so then, once again using A x i o m 2.3 and Lemma 1 one deduces 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 € B . Clearly, if q ^ B P then using A x i o m 2.3 and Lemma 1 one again p deduces a contradiction, q B. One therefore has that V(X) D (p + B ) C B . p p P P r o o f of L e m m a 2.7: B . This means that p = \\{q\ — pi) and p = Suppose that /Ui,/x that qi <E B x and q G B . B and api + (1 - a)q € B P _ p pi+(1 2 £ p x 2 — Pi) such 2 _ p a)q2 M{<12 A x i o m 2.3 and Lemma 1 guarantee that aq\ + (1 — a)q € p 2 2 2 (Lemma 5) gives aqi + (1 — a)g € S 2 pi+{1 P for any a € (0,1). The nesting property a)p2 ( _ ) - In turn, the definition of B p D 1 + 1 a P 2 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 , where a € (0,1). 2 Choosing a = ^ (and noting that this forces a to lie in (0,1)) one can write the left hand side of E q . (2.12) as A (a(q - pi) + (1 - a)(q - p )) = a A i ( g i - pi) + (1 - a ) A ( g - P2) x 2 2 2 2 = a / / i + (1 - a)u (2-13) 2 This, of course, gives the desired result that any convex combination of p\ and p is in 2 B ; thus B p is convex. p To prove B p with £ e S n p is closed, assume the contrary. Then there is a sequence, { £ } — • £ n for every n, yet £ ^ . B . Because X is a compact metric space, V(X) is p weak-* compact. It is possible, although tedious, to use the compactness of V(X) and 44 Chapter 2. Anchored Preference Relations: a theory of the status quo bias the hypothesis to demonstrate the existence of sequences {q } —> q and {p } —> p such n that for each n, one has q ,p o NP NP n , where W p G V{X), q —p G B , but q — p 0 B . It therefore stands o NP p n 0 that q eW n = W p n NP p \ B. A x i o m 2.3 implies that \q + \p p n Note that q G - B implies that \q + \p G I ? f , i p n p n n EWi +i P Pri . The idea is to combine \q + \p n and \q + ^p, sitting on opposite sides of indifference surfaces to obtain a lottery which n is on the indifference surface. This can always be done because B o NP p and B^U W m is closed and convex, = V{X) for every m G P{X). There must therefore exist a number, a G (0,1), such that n 1 f 1 ani^q A M \f ^ 1 + ^Pnj +{l-a )l-q n \ P + -pj ^ n 1 1 -p + -p n 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. A x i o m 2.2 can now be used to conclude that \q + \p ~ p that B p p which contradicts the hypothesis that p > q. The contradiction implies p is closed. P r o o f of P r o p o s i t i o n 2.2: The proof closely follows that of Proposition 1. P r o o f of Theorem 2.1: B y Proposition 1, given q,p G V(X), q G B p if and only if q - p G B . p Since B is p weak-* closed and convex, Theorem 2 and its corollaries from Phelps (1964) guarantees that B p 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 45 Chapter 2. Anchored Preference Relations: a theory of the status quo bias on X, and a G R. Moreover, since B p { is a cone p G E Xp G E for all positive A. t { 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 V(X). Note that p G W 7 V P q p for some p,q G . If p = q then / ip (dq - dp) = 0 and the proof is done. NP o Otherwise, A x i o m 2.2 guarantees that there is a sequence, Since q —p^L B , p n {q } n q such that q n GW P min J ipi (dq — dp) = p < 0 for every n. Since the ^ ' s are bounded n n i and continuous (affine equivalence implies that one can assume ||^;|| = 1 in the strong metric of C(X)), the sequence {p } n q GB p {p } has a subsequence that converges in [—2,0]. Since — • 0. This implies that min J ipi (dq — dp) = 0, as required. n 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 q = q(l — ^) + ^8x0, where 5 n Xo is a point mass' at x . Notice that for every n , q G V(X) Q and J ip(dq — dp) = £ J(a — ip)dq = a n n < 0 (if ip is constant then it is affine equivalent to 0, which is ruled out by hypothesis). Thus q n proof. converge in W . must p p Since q —> q G B p n n EW p for every n, meaning that it it must be that q ~ p p, which ends the The last part of the theorem follows trivially from A x i o m 2.1. This proves sufficiency of Axioms 2.1-2.3. The necessity part is a straight forward exercise in checking that 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 y NP that y implies NP implies y . Suppose, on the other hand, fl W = V(X) for every p. Clearly p This means that the sets B that can only hold if B p p p 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 ip is a gradient vector to H at p and that < tp ,p >= c. Part (iii) of g Definition 3 implies that p + B e p g C {q \ < ip , q >= c}. Since B p g by the intersection of all the half-spaces, {q \ < ip,q >= c, ip £ true if ip € H u l l ^ up to some affine transformation. g Part (iii): is the cone produced this can only be 47 Chapter 2. Anchored Preference Relations: a theory of the status quo bias Recall that q ~ F p <-> it is not the case that q y- p or p y p it is not the case that q y p or p y e e NP q and that q ~ p <-> } p q. Using part (iii) with the anchor at p gives the desired result. Proof of Proposition 2.4: Part (i): Since H (-) is continuous and its upper and lower contour sets are compact (by compacte ness of V{X)), it follows that —H (-) satisfies the conditions in Holmes (1975), Lemma e 141. The result follows immediately by setting §t{x - EMA) = sup H (q) e 96{g I E {iP]=x} q Since Proposition 3 guarantees that p + B p C {q\H (q) = H (p)}, one need only consider e e ip G ^ as part of the program. Finally, setting H (e) — 0, it follows from part (iii) of e 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. 48 Chapter 2. Anchored Preference Relations: a theory of the status quo bias To prove the second part, assume that c E M . F subspace of M(X,T,x,R) Denote by M°(X, E , R), the x containing measures, p with J dp = 0. Clearly, both B p F - c are subsets of M°(X, and 7£). Moreover, assuming 73 ^ [p] for some p, it can be p shown that B has a non-trivial interior relative to the space, M°(X, Ex,R)- Now, when p W j f ^ {p} for some p, B p p is a pointed cone and cE M only if lnt(B ) n (F - c) = 0. p F Since F is weak-* closed and convex, and Int(£? ) is convex, the Hahn-Banach Theorem p implies the existence of a supporting hyperplane which separates Int(f? ) and (F — c). p Corollary 1 of Theorem 2 from Phelps (1964), then implies the existence of a separating weak-* supporting functional, ip . Since 0 E B D (F — c), ip is a supporting functional p c for B p c at 0. Thus, ip E$, and c E S . F 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: xy y / ^— J max U ( s,d) > y^max U(s,d) d£x 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. general, take the form of a utility for a set, x: U(x) = ii(max U(l,d), max U(2,d), ... max 49 U(S,d)) These, in 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 information 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 contribution 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 contribution 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, 51 Chapter 3. Inter-temporal Flexibility Preference 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 , y }, p NP derived earlier from Axioms 2.1-2.3. 1 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 chapter . The resulting sequence 2 of partial orderings, { V f , 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, ty . If the choice set is convex, as it is in the case of intert temporal lotteries , Theorem 2.2 guarantees that revealed choice at date t corresponds 3 to maximizing the expected utility of some element of co(^ ), the closed convex hull of t 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(^ ) (and consequently, that t date's optimization program). A l t h o u g h 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. Although the axioms used are the same, their interpretation is revised to better fit the context of changing tastes. Agents can always 'convexify' a choice set composed of lotteries by randomizing. 2 3 52 Chapter 3. Inter-temporal Flexibility Preference 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 \& , must be unknown before date t or else the agent essentially knows how t date t prospects will be ranked ex-ante, contradicting the assumption that she cannot completely order future choices . It is worth stressing that the uncertainty over future 4 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, {y , p >- }, was introduced as a basis for anchored NP 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 timeconsistency condition. With the addition of a time-consistency condition to relate the partial order, {>-f, >-? }, with that of the subsequent date, p 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 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. 4 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 example 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) . Assume that at dates t < 2 the agent does not know which 5 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 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. 5 54 Chapter 3. Inter-temporal Flexibility Preference 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 preference . These consistency requirements guarantee 6 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 . In particular, this requires that for any 7 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 B (x) p denotes the set of all possible outcomes (fruits) that weakly Pareto dominate the contents of the menu x, then clearly, assuming B (x) p is not empty T h i s 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. Although 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. 6 7 55 Chapter 3. Inter-temporal Flexibility Preference U,(x)< mm l M { j } ) " (3-2) jeB {x) y 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 B ({a,o}) p — {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 56 Chapter 3. Inter-temporal Flexibility Preference oranges constant). In other words, an F F D 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 F F D 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 F F D 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 subject of research for a literature that is apparently separate from that on inter-temporal utility theory. The classic theoretical reference on preference for flexibility is Kreps 8 (1979). Kreps considers preferences over the set of possible opportunity sets. Axiomatically, 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 X u (1994), Pattanaik and X u (1998), Puppe (1995, 1996), Nehring and Puppe (1996, 1999), Bossert (1997) and Nehring (1999). A n important 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 incomplete . If one wishes to retain continuity, the implication 9 is that a normative theory of changing tastes must arise from primitives that partially order the set of future prospects. Other historical references can be found in Kreps (1979) as well. I n particular, this refers to Expected Utility Theory as well as most of its alternatives (which relax the Strong Independence Axiom). 8 9 Chapter 3. Inter-temporal Flexibility Preference 58 The most prevalent approach in the literature is to axiomatize preference for flexibility 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 representation 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 surrounding 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. W h a t 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 is true are far less stringent for Inter-temporal Flexibility Preferences than for Inter-temporal von NeumannMoregenstern Preferences. Operationally this is important since our models of agents are generally abstractions of more complicated preferences. A s Kreps and Porteus (1979) 59 Chapter 3. Inter-temporal Flexibility Preference point o u t , a detailed theory from the class of Inter-temporal von Neumann-Moregenstern 10 preferences will not reduce to a simpler model of the same class (except under unrealistic 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 Utility Theory ( E U T ) . Aside from the induced utility problem mentioned earlier, there is now an impressive body of empirical literature documenting v i olations of the Expected Utility Hypothesis . Although' one can argue, as does Levi 11 (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. A s one line of approach, one can require, as do most A l s o , 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), Lichtenstein and Slovic (1973), MacCrimmon and Larsson (1979), MacCrimmon, Stanbury and Wehrung (1980), Tversky and Kahneman (1986). 10 60 Chapter 3. Inter-temporal Flexibility Preference 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 3.2.1 Theory 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, d from a current opportunity set, x . The action, d \ is a probability measure u t t over outcomes. Each outcome takes the form of a pair, (z ,x \), t t+ where z G Z is a t t bundle of goods in the compact metric space, Z , representing the goods available for t consumption at date t. x +i is a future opportunity set. Specifically, d is an element of t t D , the set of all probability measures over the Borel sets of Z x X \. t t t+ In turn, X x, t+ 61 Chapter 3. Inter-temporal Flexibility Preference representing all possible t + l opportunity sets viewed from date t, is the set of all closed subsets in Since endowed with the Hausdorff metric. D +i t is metrizable and compact, and assuming ZT 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 D -\ is metrizable and compact in the Weak* topology. Clearly, this can T be continued recursively to t = 0, when the agent must choose a distribution, do from a closed subset, x of Do- A choice problem at date t is simply some element of X , say x . 0 T t Definition 3.1 A dynamic choice problem at date t induced by the sets, Z , t £ T 0,... ,T, is any element, x of X . An action at date t is any element, d of Dt. t T t An agent faced with a dynamic choice problem must select an action, d , from x C D t t t consistent with some ordering over D . The choice behavior of the agent at date t can t thus be summarized by a preference relation, y , over D . In contrast to Kreps and t t Porteus (1978), who assume that >~ is complete, negatively transitive, continuous and 4 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, y £ ZQ X . . . x Z - \ , and any d ,d' £ D , exactly one of t T t t t the following binary relations holds (with interpretations in parenthesis): (i) <y N p yt d t (d' is strictly preferred to d for at least one realization of taste at t t date t.) (ii) (hi) d y d' p t t dt~ d> p (d is strictly preferred to d' for any realization of taste at date t.) t t (d is at least as good as d' for any realization of taste at date t, but t t is no better than d' for at least one realization of taste.) t 62 Chapter 3. Inter-temporal Flexibility Preference Moreover, for any c € D , d y P t t t d' => c > d' or d >-^ c (Negative Transitivity). P t t t p t t t t The reference to 'tastes' and their realization is purely interpretive at this stage. A s in Chapter 2, d h d' denotes that either d >- d' or d ~ P t p t t d' . This corresponds to a p t t t t notion of weak Pareto dominance with respect to tastes. A x i o m 3.2 (Continuity) Given the history, y £ Z x . . . x Z - \ , and any d G D , at t 0 T t t each date t = K e A K hi dt} = Closure{d' \d' y B (d ) p d} p t t t if {d' \d' y d} ? 0 p t t = {d } t t otherwise t and {d' eD \d' t t y dy p t = Closure{d' \d ^ t t d' } if {d' \d y% d' } ± 0 p p t t t = {d } t t otherwise t where closed sets are defined with respect to the topology of weak convergence. Further, if d ^ —> dt and c™ —> c are weakly convergent sequences in D with d" ~ 1 t n, then d ~ P t t p c" for every Q . A x i o m 3.3 (Independence) Given d , d' , c E D and a e (0,1], and the consumption history, y & ZQ X . . . x Z - \ , t t t t t dt y T d' implies ad + (1 — a)ct >~ otd' + (1 — a)c P P t t t t t t and d t >-y P t d' implies ad + (1 t t — a)ct >~^ ad' + (1 P t — a)c t 63 Chapter 3. Inter-temporal Flexibility Preference A x i o m 3.2 is a technical condition on the 'at-least-as-good-as' and 'no-better-than' sets of y P t while A x i o m 3.3 extends the Independence axiom to the non-standard ordering defined by the pair ( vt> yt )y y Restating Theorem 2.1: P T h e o r e m 3.0 Axioms 3.1-3.3 are necessary and sufficient for the following representation: for every p,q € D and history, y € ZQ X . . . x Z - \ , qy p inf {E [U }t t P T t q q~ p p « t inf {E [U \- sup (E [U }- q q E [U \) > 0 (3.4a) E [U ]) = 0 (3.4b) E [U }) > 0 (3.4c) p t p p where ^ is a subset of C(Z x X i), T t+ t t t t the real-valued, continuous and bounded functions on Z x Xt+\, and E [-\ denotes an expectation taken over the distribution q. The elements T of ^ q are defined up to an affine (positive linear) transformation. has the interpretation of a utility function set. The agent can be of many 'minds' v L t regarding future preference, as if she represents a group of many ordinary von NeumannMorgenstern utility maximizers instead of just one. Thus the representation defining q y- P t 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 p y t 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. 64 Chapter 3. Inter-temporal Flexibility Preference If \I/f contains more than one element, it is possible to have p q and q >-y P >-y F t p. This corresponds to a situation in which q is preferred to p with respect to some utility function i n ^ 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, y . p t If the agent is forced to choose, however, she will pick one of p or q and may not be indifferent between t h e m . This renders the 12 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 y = (y , yt- ) where y 6 ZQ X . . . x Z -i, y € ZQ x . .. x Z _ i and t € Z x .. . x y -t s S Z -\T s If d s t >-y t P t t s s d' and d' >-^ d then contingent on the future path, P t t t y , as seen from date s, the agent cannot be certain of her date t choice between d and t t d'tFinally, from here on it is implicitly assumed that all choice sets, x , are convex. t A x i o m 3.5 (Convex Choice Sets) All choice sets, x , are convex. t 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. 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. 12 65 Chapter 3. Inter-temporal Flexibility Preference T h e o r e m 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, y G ZQ X . . . x Z - \ , t T there is an associated set, ty'f, containing basis 'utility' functions over Z x X +\. Each Ut G ^'f is bounded and continuous in both arguments. T (ii) t The agent's revealed choice at date t corresponds to the maximization over the choice set, x , of the expected value of some uncertain element of y^ C c o ^ ^ ' ) 1 t where \&f C y™ and c o ( * f ) is the closure o / H u l l * f . The situation is analogous to that of aggregate decision making for a society of individuals 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 will 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 66 Chapter 3. Inter-temporal Flexibility Preference —I 1 d ,, d ,, d ,,...} 1 Choice Set: x, = { 2 3 Choice Set: x,+y = { cf,+i, c / , Utility Function Set: 4*, w .cft+i,...} y Utility Function Set: ¥ , + ; Bias assignment ( y ' z ) Bias assignment 7 v choose if = (p = Vi, (z, x, ); t J-p='/2 ,(z', x',+;)) +1 Resolution of uncertainty Next period's choice set Uncertainty resolution => draw. 0, x,+i) • Consume z to give next period's Utility Function Set: ^' Z>) Figure 3.1: A Dynamic Choice Problem with incomplete preferences presented i n the next chapter . In addition, it is important to emphasize that Theorem 13 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 d G x is degenerate, then it will be denoted henceforth as (z ,x i). t t t If an oppor- t+ tunity set, x , consists of only a single choice, say d , then it is denoted x = {dt}. Such t t t singleton sets are of primary importance in the formulation of the theory. A sketch of the setup is in Figure 3.1. T h e 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. 13 67 Chapter 3. Inter-temporal Flexibility Preference 3.2.2 T i m e Consistency Utility function sets in consecutive periods are related through the following. A x i o m 3.6 (Time Consistency) For any t, z £ Z , f,g& X i, t max E [U i] def d and consumption history y if t+ t) > max E [U ] deg d t+ for every U t+l € y^{ then (zj) z) t+l > P t (z,g). Recall that 3^+Y^ is the set containing all possible linear combinations of utility functions in ^ ^ { ^ that may serve as a bias assignment (including ^t+i^ itself) . Suppose, 14 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 A x i o m 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 A x i o m 3.2, it is easy to prove that part of A x i o m 3.6 also applies to the case of weak dominance . The converse of the 15 weak form with respect to singleton menus then implies L e m m a 3.1 For any t, z € Z / , <? E X i, and g = {d' } where d i,d' <E A + i , and (z,f) y^ t) t+1 t+ and consumption history y , if f = {d i} t+ t (z,g) then d i >-g p t+1 t+ t+ 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 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 y and z only. Note that the stochastic process associated with bias assignment may depend on the date t + 1 choice set. 14 t 15 t i.e., if max Ed [ip] > max Ed[ip] for every ip € X+Y^) d€f d€g t n e n ( >/) h z P J t (z,9) 68 Chapter 3. Inter-temporal Flexibility Preference future action d preferred to d' t + 1 t+l to d' . The lemma implies that it must be that d \ will also be t+l t+ 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 A x i o m 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. A x i o m 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 A x i o m 3.6 is equivalent to the Temporal Consistency A x i o m in Kreps and Porteus (1978), and the theory reduces to their recursive utility formulation. A n interesting consequence of A x i o m 3.6 is that the utility of a choice set at date t, say x , depends only on the set of maximal utilities attainable at date t + 1 . Formally, t let a be an index set for elements of 3^+i^ and define the mapping wf : X •->• 72.^'+' ' 1 1 t through {w?{x)) = maxE \ib ) (3.5) a a where ip £ 3 ^ + 1 ^ a a n a - t{ ) w t d represents a (possibly) uncountably infinite vector of x maximal cardinal utilities for the set x. Denote wf{x) > wf'^x') whenever (wt (x)) > t a (wt'(a;')) for every a , and there is some a for which the inequality is strict. Likewise, a denote w^ix) w^ix') whenever {wf {x)) , a > (^'(x')),* for every a. P r o p o s i t i o n 3.1 Fix f / f € tyf. Then for any history y € Z$ x . . . x Z -\, z & Z and t t t x€ X , t+1 U?(z,x) = u? (z,wj*i*\x)) t where w^ (x) z) > w^[ \x') =• uf(z,w^\x)) u?(z,w%{ (x)) z z) > uf{z, ^\x')). W > uT{z,w^\x')) (3.6) and w%{\x) » 69 Chapter 3. Inter-temporal Flexibility Preference 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') > \J ( {z,x) for any x,x' x l Kreps' other condition, namely that f / f (z, x U x') = U? {z, x) l Ut (z,x"Ux), t £ X +\. t => U^(z, x"UxU may not hold, since u\ is not necessarily strictly increasing in l x') = wf^\x). This, however, does not imply that there is no preference for flexibility, as the next theorem demonstrates. T h e o r e m 3.2 Fix Uf £ * f . Then for any history y £ Z 1 t 0 x x Z _ i , z £ Z and t t x £ Xt+i, such that (3.7) B (x)^f] [ (d)^(D p B yttZ) d€x it is the case that max [If(z,{d}) < U?{z,x) < min U {z,{d'}) yt t (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 every element of x. 16 which, at date t + 1 , dominates The difference between the upper and lower bounds in E q . (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 and, Proposition 3.1 implies 17 that, E q . (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 16 Recall that ^(d), defined in Axiom 3.2, is the set of t + 1-prospects that Pareto dominate d. B (x) is therefore the set of t + 1-prospects that dominate every member of x. I f ty^i^ is a singleton, the set of elements that Pareto dominate everything in x has a non-empty intersection with x. p 17 70 Chapter 3. Inter-temporal Flexibility Preference 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 ^[+{ ^ is not a singleton. z 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 If ^{ Inter-temporal F l e x i b i l i t y Preferences a n d Induced U t i l i t y Functions z) = {U^{ } is a singleton, Proposition 3.1 implies that Uf (z,x) = z) l uf (z, m&xEd[Ut+{ ^]) z for any menu, x S X i and any Uf 6 yf . l t+ u\ is continuous 1 l 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+i \ z for d € A + i , is a linear functional on A + i , the indifference surfaces corresponding to U^{ ^ are linear. Thus indifference surfaces on z D i induced by singleton menus at date t are also linear for each Uf (z, {•}) € yf*. l t+ If Vp|+i^ is n o ta singleton, however, there is nothing that guarantees that a given date t utility function restricted to singleton menus, for example Uf^z, {•}) where Uf G yf\ l 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 U?(z, and U^{ Zt] 2 = uf (z, E [U^{ Zt) d , then in general \ E [U^{ Zt) ]) 2 d for any Uf <E yf . In particular it should be clear t h a t , t 1 U?(z,{d }) 18 = U?(z,{d' })^ t+1 U?(z, {d }) = U?(z, {ad t+1 t+1 t+1 + (1 - «K }) +1 I f Ed[Ut+{^ '} = Ed'^Jt+i^ \ f ° both i = 1,2 then mixture invariance holds. That is a direct result of Axiom 3.3. 18 1 l r 71 Chapter 3. Inter-temporal Flexibility Preference where ad i + (1 — a)d' t+ t+1 with a E (0,1) corresponds to a probabilistic mixture of the distributions d i and d' . Thus [/^(z, {•}), does not necessarily have linear indifference t+ t+1 surfaces in D +\. t In fact, there is a compelling argument for allowing derived utility functions over singleton menus (i.e., U (z, {•})) to have convex lower contour sets in D \- In a followyt t+ 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 i n D +\- To explore this in t 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 Z = A x B for each date, t, and that at t t t each date the agent has the opportunity to optimize consumption by selecting, a E A , t t from a closed set, Af (bt)- Here, y E Z x . . . x Z -\, can be thought of as a history of t 0 t optimization and endowment (a and b , respectively). Note that the feasible set for a t t t can depend on the current endowment, b . A^ (b ) itself can be seen either as a set of l t t t controls or simply a set of consumption choices that can be integrated out of the choice problem. A n y direct utility function corresponding to a date t bias assignment has the form derived in Proposition 3.1: uf (ztMlfixt+i)) =< (a., h, ™ & (at A ) ) (x i)) t + 72 Chapter 3. Inter-temporal Flexibility Preference After selecting a vector, a from a closed feasible set, Af'ibt), the indirect or induced t utility is vf{b wti \x ),x ) = bt) u t+1 t+l max uf (a u b, w^ t M ) (x t + 1 )) where vj^{ ^ *(x x) = wj.+f ' ^ (x i) is the vector of maximal date t+1 utilities as in bt at bt t+ t+ Eqn. (3.5) and a* corresponds to the optimized vector of 'actions'. t Consider the set of all date t induced utility functions j>r = {vr{bt,w%i *(x ),x ) i uf e yt bt) t+1 Note that yf 1 t+1 depends on y = (yt-i, (a-t-i> ^t-i))- Since a^_ is a function of the realized t 1 date t opportunity set, x , this means that yf in general depends on x . Thus if yf itself 1 1 t t represents some Inter-temporal Flexibility Preference , a necessary condition is that yf 19 1 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: (J 4*W= Mgmzxuf(a b ,w ^> \x)\ { u (3.9) ht t T h e o r e m 3.3 Fix an Inter-temporal Flexibility Preference described by the sequence of feasible bias assignments, yf , 1 and Z — A x B . Af(b ) is where y G Z x . . . x Z -\ t 0 t t t t t a continuous mapping (with respect to the Hausdorff Metric) from B to closed subsets t of A . t Then the sequence of induced bias assignments, yf , 1 describes an Inter-temporal Flexibility Preference if and only if for every t and y G Z x . . . x Z -\ t y{yt-i,at-iM-l)) 0 t A_ ^(yt-i,a' _ ,bt-i)) t ^ 1 ^ 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 b ,... , b -i19 0 t 73 Chapter 3. Inter-temporal Flexibility Preference whenever a _i,a' _ t t 1 GA ^ 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 represent 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 preserved after optimization up to an affine transformation. This negative result is contained in Proposition 8 of Kreps and Porteus (1979). Utility 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 Utility 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^' }, is composed of real-valued and continuous functions (indexed co by a) over the closed and bounded interval Co + C C 7Z. The date 0 direct utility (bias x assignment) over U ey Q (co,Ci) and a transfer strategy, a G [0,Co], must take on the form, C/ (a;(co,ci)) = U (CQ - a, tuj°(ci + a)) = i z ( c - a, {£?[t/ °(ci + a)]}) a,c 0 0 0 0 0 1 The induced utility is therefore Vo(co, ci) = max u (co - a, {E[U?' °(ci c 0 + 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[) 74 Chapter 3. Inter-temporal Flexibility Preference 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 U °, then to be consistent with Proposition 3.1, it must x be that Vo(c ,aci + (1 — a)c\) = Vo(co,Ci) . Thus the reason why Inter-temporal von 20 0 Neuman-Moregenstern Preferences cannot generally represent induced utility is that they do not allow for the utility aggregator (i.e., V ) to be convex under mixture. 0 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 specifications. 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)}, xeC x Some examples follow: i) 3^i° is the set of continuous functions on Co + C\. ii) y ° is the set of continuous and strictly increasing functions on Co + C\. x iii) 3^i° is the set of continuous, strictly increasing and concave functions on Co + C i . iv) ^ i is a family of twice differentiable concave functions {U \ RA = — %(x) 0 ~ 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 " ^ , a p > OV/3} 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"' {c) up to an affine transformation for every a € [0,a]. This is generally true only for constant absolute risk aversion ( C A R A ) and linear utility. 20 ca 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 thereafter . 21 As mentioned earlier, the bounds in E q . (3.8) always converge when there is a single utility function in the utility set. its maximal element. In such a case the utility of a choice set is that of 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 E q . (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 ( F S D and S S D , 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 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. 2 I 76 Chapter 3. Inter-temporal Flexibility Preference 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, U - If U does not assign a premium to flexibility then there is a potential t t for regret and/or manipulation. In particular, the agent will 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 will seem inferior relative to an alternative that was unnecessarily removed from the choice set ( at least with respect to some date t+1 of the opportunity set will be regretted ex-post. bias assignment). The reduction Even if unable to forecast changing tastes, a 'rational' agent will 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 theory should, in general, require a non-zero premium for flexibility much the same way that a sensible expected utility theory requires monotonically increasing utility functions. Note that one can also make 'free-lunch' profits from inter-temporal von NeumannMorgenstern agents who possess preferences that violate F S D . 77 Chapter 3. Inter-temporal Flexibility Preference 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 will, in general, occur in every period even when state contingent claims are available for purchase and the market is complete . This is in contrast with the models of Arrow (1964) and Debreu 22 (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 Z iff its representation in Proposition t 3.1 is strictly increasing in w^{ \ z In other words, Uf respects F F D if and only if for every closed the case that w^{ {x ) z) t+1 > w ^[ (x' ) { z) t+l =» Uf (z,x i) t+ t+i\ )x set, x +i G X +i, it is t > Uf{z,x' ). the intuition behind the definition, assume that Uf(z,x) w 23 t+1 t To understand is not strictly increasing i n A t date t the agent is therefore indifferent to changes i n the opportunity set T h e 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. Since only convex choice sets are considered here, this criterion is stronger than it has to be. 22 23 Chapter 3. Inter-temporal Flexibility Preference which make it strictly worse from the point of view of some possible date t+1 78 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. W h a t 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 Firstdegree 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 79 Chapter 3. Inter-temporal Flexibility Preference that there is no guarantee of a general additive representation , never mind an additive 24 representation of the form, Ur(z,x) = u?{z, Y, Aamax E [U? ]) d X > 0, U? +1 a +1 € }ft 2 ) (3.11) a where a indexes elements of y^{ \ Note, in particular, that the second argument of z 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 Uf (z, {•}) over D +i and is, in general, not a good proxy for an induced utility (see t t 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 E q . (3.12) the utility of / is at least as great as that of g: -U(x l 2 4 Uz) + ^U(x Uy)> -U{x l U y U z) + A general additive representation would imply that Uf {z,x) t = uf' (z,^ max is not necessarily a linear functional over D +\ (i.e., an expected utility). t -U{x) l g (3.13) vf(d)) where uf (d) 80 Chapter 3. Inter-temporal Flexibility Preference 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 expression 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 restrictions 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 Seconddegree F l e x i b i l i t y Dominance (SFD) for a given z G Z if and only if for arbitrary t X G Xx,X2,X3 t+X Uf{z, x U x ) - Uf{z, x ) > Uf(z, Ux U x 2 2 Xl x ) - Uf{z, x U x ) 2 3 2 3 (3.14) I end the chapter with some examples. Consider the date t+1 utility function set ^(V^) _ ^jj\,{y ,z ) t t rj2,(yt,zt)y and suppose that y(*i zt) = {U ' |U ' x {yt zt) = XU ' ' x{vt zt) 1 {vt zt) + (1 - X)U ' ' 2 | A G [0,1]} {yt zt) Thus possible bias assignments (i.e., elements of J^t+i' ^) include all convex combinations 2 of the basis functions, U and U 1, 2, up to an affine transformation. The agent will choose at date t + 1 by optimizing with respect to some U ^ ' \ x 'revealed' U (z ,x )= p t t t+1 25 and A is only at date t + 1. Now suppose that for some date t utility function, Uf G yf r JO 25 yt Zt max Jzt,s, d&x i t+ \ E [( dt+1 e - " d - p ) / P [ / i . (*«.*) + (1 _ - « ( i - p ) / p ) [ / 2 . (vt,«)) e \ Recall, again, that the agent only behaves as if a A is revealed at date t + 1. ]) / / d s 81 Chapter 3. Inter-temporal Flexibility Preference where <p is increasing in its third argument and p G (0,1). Note that Ut(z ,x i) t satisfies t+ Proposition 3.1. Uf(z , xt+i) also satisfies F F D . It is also easy to show that t/f (z , £ + i ) 2 6 t t t satisfies S F D . The representation conforms to a probabilistic interpretation (additive representation) 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~ a gives s Ufizt, H + i } ) = f(z)E [pU^ + (1 - p)U ' <»•*>] 2 dt+1 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. T h e 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 yf , l U?(zt,x ) t+1 = $?(z, max ^ ( E ^ J E / 1 - <*•*>]) + max <p {E 2 [U ' ^ } ) ) 2 dt+1 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/ yt t G yf l satisfies F F D . O n the other T o prove this, note that U ^ ' '^ is continuous in A (in the sup topology). Thus if max E [U ~ '^} - max E [C/ ' ] = e > 0, for some A, then there is a neighborhood 2 6 x x dt+1 < yt z VuZ A ( y , 2 f ) A d f + 1 of A, say N(X), over which this equality remains true with e This is sufficient to prove that Uf- is F F D . x > e for some e > 0 and every A' € N(X). 82 Chapter 3. Inter-temporal Flexibility Preference 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: <p (E [U >^]) 2 dt+i M^iW ' ' ]) 1 {vt zt) 2 dt+l di 0 2 d 1 1 2 0 2 d 3 and set x = {di}, y = {d } and z = { d } . If <&f(z, a) = af(z), then 2 Uf (z,xUy)- 3 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 / ( ^ ) , then 2 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 i n 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 i n 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 A x i o m 3.4 and 3.5 and Theorem 2.2. 83 Chapter 3. Inter-temporal Flexibility Preference • Proof of Lemma 3.1: The weak form of A x i o m 3.6 implies that i f max E [ip] > max Ed[ip} for every %p 6 d then (z,g) >z (z,f). In particular, fixing / = {d } and g = {d' } implies P t+1 t+1 that not (z,g) > p yt (z,f) not EM > Ed[iP] A x i o m 3.1 implies that not (z,g) >z (z,f) (z,f) P E y^ p implication, note that, according to the above, there is some E [ip] d yfrf) (z,g). ip > Ed,[ip}. B y Theorem 3.0 this is true if and only if d y^ e p y z) To derive the ^t+i ) z) such that d'. • Proof of Proposition 3.1: Suppose w[ !{ \x) = w*j.+{ \x') and x ^ x'. The weak form of A x i o m 3.6 implies that v (z,x) y p z (z,x') and (z,x') >z z P t (z,x). B y E q . (3.4) it must be that Uf{z,x) for every Uf 6 * f . Thus Uf(z,x) = uf (z, w^{ \x)). z = Uf(z,x') Clearly, A x i o m 3.6 and E q . (3.4) also guarantee the stated properties of u. • Proof of Theorem 3.2: 84 Chapter 3. Inter-temporal Flexibility Preference Note that the result easily holds when x is a singleton. that x is not a singleton. One can therefore assume Consider first the lower bound in E q . (3.8), and set d* = argmax U (z, {d}). The weak form of A x i o m 3.6 implies that (z,x) > yt P thus, by E q . (3.4) U?(z, x) > U?(z, {d*}) = z and note that A x i o m 3.6 implies p p (z,{d*}) and U^' \d*). To obtain the upper bound, consider d & B (x), that (z,d) t t (z,x). B y Theorem 3.0, it must be that U{z,x) < u[ ' {d). Since this yt z) holds for every d € B (x), p 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 Xf. as y . t Also, denote the set of possible date t opportunity sets by The hypothesis implies that all functions in yf l Bt x X . are continuous and bounded on That is enough to satisfy Axioms 3.1 - 3.3. Axioms 3.4 - 3.5 are also p +l satisfied by hypothesis. To prove that A x i o m 3.6 is satisfied, one only needs to show that vf(b , w^{ ^ *(x i), x +i) does not depend on the last argument explicitly. To this end, bt t t+ t let a*(b , x i) t t+ = argmax uf [a ,b , Now consider two sets, x +\ and x' t t a G A _\. Vt w^i ' (x i) at bt) t t+ such that w^{ ' \x +i) a bt t+1 t = wl+{ ' \x' ) a bt t+1 for some Since y^ i assumed invariant, this must remain true for any a G A _*i- In s yt 85 Chapter 3. Inter-temporal Flexibility Preference particular, vf {b AXt =uf (a'(bt, x ), t] t = uf ( a * ( 6 , x t+l ),b ,^ r* "" ( t t + 1 t ( b + l ) + ' b t ) u ( ^ i ) ) << (a*{b x' il fc, + u t+ l i , ^ * ^ * Wi)) 0 b w%f \x^S) M+l)M = vf(b ,w^ *(x' ) x' ) (3.15) ) t t+1 1 t+1 The above also works when reversing x iandx' . t+ vf(b , u>l+{ ^ *(x i), x i) does not depend on the last argument explicitly. It's easy to bt t t+ The conclusion is that t+l t+ show that the other properties in Proposition 3.1 are obeyed, thus A x i o m 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 £ Z /, g € X +i, t; i) If f = {d i} and consumption history y , t t and g = {d' } where d i,d' t+ t+1 it) If max E [U +i] > max E [U i} d t d t+ t+ £ D +\, and (z,f) y P t+l for every U t e y[ +{ then (z,f) y v t+1 z) p t {z,g) then (z,g). The first part of the axiom requires that, holding z constant, the strict partial order on singleton opportunity sets induced by y P partial order y y p yt z t be commensurate with the date t + 1 strict 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 d +i to d' . The first part of A x i o m 3.7 implies t t+1 86 Chapter 3. Inter-temporal Flexibility Preference that in the above situation d \ will also be preferred (regardless of changing tastes) to t+ 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 d \ Pareto dominates d' t+ t+1 in the future, then it is not the case that {d i} Pareto dominates {d' } at any earlier time. This suggests that Pareto dominating t+ t+1 sets can only shrink weakly through time. That intuition will 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 A x i o m 3.7 apply to the case of weak dominance . If the utility function set at date t 27 contains only a single element, then Axiom 3.7 is equivalent to the Temporal Consistency A x i o m i n Kreps and Porteus (1978), and the theory reduces to their recursive utility formulation. Theorem 3.4 Suppose that every U^ '^ € ^ ' ^ 1 Vt z date t + 1 there exist prospects, d +\, d' t is Frechet dijferentiable 28 € D +\ such that d i y- ^ d' . Then there p t+1 exists a continuous linear mapping from ^ Vt t t+ and that at yt z t+1 onto ^ j + i ^ • Proof of Theorem 3.4: The proof makes use of the intuition that every neighborhood of D t+i contains a scaled down copy of D +\- Since the relevant date t utility functions are linear at small t scales they must support the same strict partial ordering as the utility functions at date t+1. 27 i.e., (z,f) y P (z,g) =4- d +i >zf , d' t and, if max E [ip] > max E [tp] for every ip e c o ( # ^ ) z ) t+1 d d then ( z , / ) h (z,g) P t Unfortunately, we are unable to provide a version of Theorem 3.4 without assuming Frechet differentiability (except when ^ [ ^ { ^ has finitely many elements). 28 87 Chapter 3. Inter-temporal Flexibility Preference Consider any point, d G D +\. Since D x is a bounded subset of a separable normed t t+ linear space (the space of signed measures on Z +\ x X +i), t t 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) {Ei} (3.16) . where {Ei} is a partition of D +\t B y hypothesis, every U^'^ such G & '^ is Frechet differentiable. Since, in addition, all yt are assumed to form a closed set, there must be a sufficiently small neigh- borhood of d, say N(d), where the U^^'s V \j[ . Clearly, V U^ ' yuz) G *|+{ . Consider now the set, yt z) d z) d B = {X(q~p)\ N{d) can be approximated by a linear functional, qy p p,XeTZ , If it can be shown that Bjv(d) = Br) + t+l q,peN{d)} (3.17) then assuming that Bo is not empty, l+i A x i o m 3.7 and the techniques used in proving Theorem 2.1 can be used to establish that cd(V ) = cd(^[ ^), y d transformation. Bp t+1 where the equivalence of the sets is determined up to an affine 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( y d Assume without loss of generality that N(d) = {d' | ||d — d'|| < e > 0. Consider £ S B . Dt+1 B y definition, £ = X(q — p) for some positive A and p, q, G A + i - Because D +i is a mixture space and < 1, d' = (1 — t/2)d+ (e/2)p e N(d). Also, by the same t reasoning, d! + (e/2)(q — p) G N(d), and consequently, (e/2)(q — p) G Naturally, this also implies that £ G B ^ y N d • Theorem 3.4 essentially states that, as long as the strict partial ordering, y y is not p yt z empty, utility function sets do not increase in size with time. A n appealing interpretation 88 Chapter 3. Inter-temporal Flexibility Preference 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, y . t 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 E q . (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 ip . Suppose 2 that there is some Uf E $ f such that z,E [ipi\) u?(z,{d}) = <K (3.18) d for one of i = 1 or 2, and where (p : Z x 7Z i—• 1Z is continuous and monotonically t 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: L e m m a 3.2 Suppose that Uf(z, {d}) — <p(z, Ed[ipi\) for one of i = 1 or 2. Then Uf(z, x) = max <pz,E ( [ipi\) d for any x € X +\ and B (x) non-empty. t (3.19) 89 Chapter 3. Inter-temporal Flexibility Preference P r o o f of L e m m a 3.2: To prove the lemma, it is sufficient to demonstrate that min E [ipi] = max E [ipi] for d d d£B (x) ^ p d x either i = 1 or i = 2 (i.e. the bounds in Theorem 3.2 are equal). To this end, given x E X i, t+ define Cx{x) = argmax E [tpi] d C {x) = argmax E [rp ] 2 d (3.20) 2 d€x ^ 0, any common element to both C\(x) and In the case that x is such that Ci(x)nC (x) 2 C (x) 2 is clearly in B (x), p consider p E B (x), p and thus the lemma is proved. To address the other situation, and choose some d\ E C (x) as well as some d E C (x). Now define x d\ = Xdi + (1 - \)p for A E [0,1]. Since p y 2 d , it must be that d p all A. More importantly, since C (x) D C (x) x 2 2 +1 x >z = 0, £^[^2] < E [4>2] < E [ip ]. d2 d for P x p 2 +1 x B y the continuity of yj , there must be some A E [0,1) such that ^ [ ^ 2 ] = ^ 2 [ ^ 2 ] - Moreover, 2 for such a A we must have that d E B (x) x conclusion is that min (since Ed [yj\] < E [ipi] p 2 Ed[ip } < E ^[tp ] = E [ip ], 2 d d€B (x) p 2 d2 2 dl < E \[tjji\). d The which proves the lemma for ip 2 1 The corresponding argument for ip follows a similar line. x • Note that the requirement of a non-empty B (x) p is not particularly stringent. This will always be the case so long as there is some 'most-coveted' outcome, d +\t 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 E q . (3.18)). This is an unpleasant situation since without functions such as the ones in E q n . (3.18) it is difficult to build up a reasonable basis set that would satisfy part (i) of A x i o m 3.7. P u t 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 A p p l i c a t i o n to Asset P r i c i n g To explore Inter-temporal Flexibility Preferences further and to establish the practical 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 standard 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 aggregated 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 will continue to trade contingent claims even in a complete markets subsequent to the initial trading of endowments. Thus trad1 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 particular, 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 A complete market in this context is one in which a standard von Neumann-Morgenstern agent would be able to perfectly hedge inter-temporal consumption. l 93 Chapter 4. Application to Asset Pricing directly. 4.1 A General E q u i l i b r i u m 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 of the economy is 2 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, x , is a random budget set for l t+1 T h e 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. 2 94 Chapter 4. Application to Asset Pricing the next period. The utility function of agent i at the beginning of date £ is a member of the set ^ t = y t 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 assignments and 3 this set is stationary. As discussed in section 3.2, U^ ' ^ g \& is defined over ( c j , . T i ) 9t R t t+ - 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 E n d of period t Choice Consumption Resolution of Uncertainty Uncertainty resolved Choose ( c j , £ j ) G x\ +1 by maximizing expected utility max Consume c\ x (glRi).-* Et[U^ ] . itRi) ( t>^t+i)^ t c t+l ^t+l) (gt ,Rl ) +1 +1 x It is assumed that agent i's utility from the consumption-opportunity pair, (c ,x i), t t+ is defined recursively by u f ^ \ c u t , x ^ r l ) = ( max X ^ (c +i,x 2)ex +i t \ t+ t (g', R', E [U^ '\c ,x )}) dg' dR R t+1 \ t+1 t+2 ' [g,g}x[EA (4.1) It 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. 3 95 Chapter 4. Application to Asset Pricing where (g\, R\) G [g,g] x [R, R], u^ i ^ : TZ x TZ i-> TZ is continuous and strictly increasing g R 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,x +2) + denotes a random distribution of date t t+1 consumption-opportunity pairs. The function u^ i,#«)(-, •) aggregates current consumption g 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 E q . (4.1) is such a weighting scheme with X " ' acting as a weighting function. If X " ' t t 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 manipulation is allowed. Using E q . (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 (c ,{(c ,...,c )}) [ 9lM) t t t+1 ct, V T J = [g',R',U^i ' (c ,{(c ,...,c )})) R ) t+1 t+2 T l9,gME,R] dg> dR> (4.2) J E q . (4.2) is the joint utility derived from some current consumption level, c , and a t 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 96 Chapter 4. Application to Asset Pricing 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\,... It is taken for granted that T\ — {fl} and that TT ,TT). is the set of all singleton subsets of fl. A n event at date t is an element of Tt. each final state, to G fl, there is an unconditional probability, 7T(LU) With and agents agree on these probabilities. The unconditional probability of an event at date t, a G T is simply t 7r(a ) = J2u>£a ( ' ) 7r t t J For t < s and any two events, a G T t t and a t G T s s such that a C a , Bayes' Rule gives the probability of the event a conditional on the occurrence s t s of a as -n{a \a ) = 7r(a )/7r(a ). t s t s t For t < s and any two events, a G Tt and a G T , we define (f>(a \a ) to be the date t t s s s t price of a contingent claim that awards one unit of the consumption good in the event a s at date s, conditional on the date-t realization of the event a . The market is 'complete' t 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, (4.3) where (f>(a ) denotes the date-1 price of a claim contingent on the event a . s s Before trading begins at date t, agent i has c ' ( a ) units of the state contingent _1 s claim that pays out in the event a (s > t). These are traded between the agents when s markets subsequently open to achieve the new equilibrium allocations c\{a ). Equilibrium s is achieved when at each date every agent trades to maximize her current period utility 97 Chapter 4. Application to Asset Pricing function and the following market clearing conditions are met: N J24(a )^C(a ) s V t < s (4.4) s i=i where C(a ) is the economy's production of the consumption good in the event a . t t 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 E q . (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)(a \a )} takes into account the macro-economic impact of the current as well s t as all possible future bias assignments, {(g ,R )}, l l T T r > t, among agents. This does not mean that the price for each contingent claim, (f>(a \a ), is independent of s t {(gl,Rl)}, but that each possible realization of the {{gl, -Rj)}'s in aggregate corresponds to some price system, {(p(a \a )}. It should be stressed that the observable or macro states in fl s t 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). L e m m a 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 assignment, {(g , R])}, for any t is a random variable (not necessarily adapted to the partition l t sequence, T). Then there exists a Bayesian-rational expectations equilibrium if and only 98 Chapter 4. Application to Asset Pricing if at each date t the set of equilibrium prices generated by all realizations of {(gl, R\)} is {4>(a \a )}. s 4.2 t A Specific E x a m p l e 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 ty and is parametrized by (gl, G R : 2 t t/, » ' (Q,z ( ! Ri, 1+1 rfiJZ) ) = - f r ^ 9t + + x t, (9lRl) g t E [Ulii \c ,x )\) R R / j t+l t+l t+2 dg' dR! (4.5) where XT+\ is empty, and the functions \ \ ' ^ and tp[ ' ^ 9 A|^V, R>, y) = - ~ ' e d ' (y) = 9 R) R 9 _> 3/9e R/R — are chosen as follows R H g y) gR _I<<o 9 (4.6) (4-7) -y-R 9 The function ipi '^ is a re-scaling of aggregated utilities and the bias assignment, R, 9 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^' ^ = 0. The choice of utility function R 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. 99 Chapter 4. Application to Asset Pricing Under these assumptions, it is shown in Appendix 4.5 that at any date t, the state prices are recursively given by 7r(a i\at)' K t+ (C(at+i) - a {a )w{a )) t+1 t+l (G(t + 1) - G(t)) t+1 + q (a )(R{t t+1 + 1) - R(tj) t+l - R(t + 1) (4.8) where w (a ) = (C(a ) + t t ]P t (f)(a \a )w (a )) t+1 t t+1 (4.9) t+1 a-t + iCat 1 £ <f>(at i\a )qt+i(at+i) 9t(Qt) _ a ^ a ^ _ ^ m k l ) + a t ^ a t ot (a ) t t+1 t+1 ( 1 ^ 4 1 0 ) ^ a(a ) t 1 N W-GWgfl (4.13) vj (a ) is aggregate wealth calculated as the risk adjusted present value of the current and t t future consumption coupon streams. unit coupon bond, and 1 a t ^ can be interpreted as the value of a perpetual is the price of a perpetual bond that pays a stochastic coupon equal to ^ ^ at each date. G(t) and R(t) are, respectively, measures of the a a 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 E q . (4.8) fully accounts for the distribution of bias assignments and resulting Pareto optimal allocation of resources if and only if at each date t, 100 Chapter 4. Application to Asset Pricing G(t) = G(a ) and R(t) = R(a ). This leads to the following constraints on an equilibrium t t bias assignment P r o p o s i t i o n 4.1 Assume that the economy is composed of agents with preferences as described in Eqns. (4.5)-(4.7) and that agent i's bias assignment at date t is (g\, R\) € R . 2 + Then a Bayesian-rational expectations equilibrium exists if and only if (4.14) and (4.15) 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(a ) t —> oo) but the per-capita aggregate consumption good stays finite i n 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 i n (4.14) leaving the price system unchanged. 4.2.1 Further Discussion The rational expectations equilibrium presented above is similar to one with a representative consumer having state-dependent inter-temporal von Neumann-Morgenstern preferences. The state-dependence in the utility function of the representative consumer 101 Chapter 4. Application to Asset Pricing 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 Flexibility 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 Utility 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'. A s discussed earlier, this introduces 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 without any change in consumption. Consider, for example, E q . (4.8) at date T — 1. The 102 Chapter 4. Application to Asset Pricing state prices, <p(aT\a,T-i), depend on the current state, a _ , T factors G(ar-i) and R(a,T-i), x only through the coherence the aggregate supply of consumption good, C ( a r - i ) and C(O,T\CLT-I), and the conditional probability, 7 r ( a r | a x - i ) . Suppose that two date events, ar-i and C(ax-i) T — 1 are distinguishable only by the different coherence factors (i.e., = 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 changing 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 . In par4 ticular, if an agent could actually observe her own bias assignment, it is not unreasonable 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. 4 103 Chapter 4. Application to Asset Pricing 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 extent, 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(a ) t and The result also justifies Chapter 3's relative silence on the mechanics of the G(a ))t 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 P r i c i n g 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>(a \a ) t+1 t = prc(a \a ) t+1 exp (-(C(a ) x t - a(a )W(a )) t+1 t+1 (G(a ) t+1 - G(a )) t+1 - G(a ){C(a ) t t - t+1 C(a ))) t (4.16) Note that a(a )W(a ) t t is a term which can be viewed as an economic coupon value . 5 To facilitate the analysis, approximate Eqn. (4.16) in continuous time and assume 6 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. T o justify the identification of a(a )W(a ) as an economic coupon, consider that if the per period interest, r dt and a(a ) are constant, a = (1 -f- ( + .rf ))~ giving a = (^*dt) • Similarly, if C(a ) is constant and T —> oo, then W — ^^ ^ • The product, a(at)W(a ), is therefore equal to the coupon, value C. 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. 5 t t 1 t t a C 1 r i t dt t 6 104 Chapter 4. Application to Asset Pricing Denote the vector of 2 independent Brownian motion variables as W . 7r(a \a ) t t+dt is t therefore the probability associated with a multi-variate normal distribution with variance dt. The evolution of aggregate consumption, C , and aggregate risk aversion, G , are t t parameterized as follows ^ = p?dt + af'.dW ^= j?dt + tr° -dW t a(a ) t (4.18) , f W(a ) (4.17) t t and a(a ) have continuous time analogues that require W(a ) ==> dtW and t ==£' at/dt. t Thus the economic coupon, a(a )W(a ), remains finite in the limit. t t t Define the differential consumption coupon ( D C C ) as, D = Ct-a Wt t (4.19) t Its evolution is dD = ifidt + erf • dW t (4.20) t where p^ and cr® are determined v i a 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? = C (d D)o-f t c + G {d D)<rf t (4.21) G The pricing kernel can be approximated in continuous time as <t>(a \a ) t+dt t = 7T(a t\a ) t+d t exp ^ - ( - * _ _ _ * _ + T t ) d t _ f<. a m \ ( 4 2 2 ) 105 Chapter 4. Application to Asset Pricing where it is assumed that p = e =D G *? t and T0 + G C <r? t t (4.23) t • £T CT ' M r =r + D G pf t 0 t t + G er?' • <xf + G C p c t t t M - (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. erf and r are, respectively, the vector of volatilities 1 t 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 p and t o~ , respectively, is given by t p-r t = erf • er (4.25) t In particular, the risk premium on the market portfolio (defined to have volatility erf ) 1 is simply erf ' • erf . 4 4 The D C C 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, a W . t t In continuous time, these can be expressed as „Mi °° / C°° M C e-f^^^ ^ -^<' -ds}^E;[j^ +r du C e-h ^ds\ dW s s 1 ± t a r 00 = =E [ Zt t „MI t , e-M ^ ^ -K '"'- »ds] Z J „ M +r du < s dW = E;[ poo e'^ - ds) T Jt du (4.26) (4.27) 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 106 Chapter 4. Application to Asset Pricing the aggregate endowment each period to a bond that pays one unit (per unit time) of the consumption good each period. Equivalently, W /z is a weighted average of all possible t t future coupons. Either way, W /z is an economic variable that is sensitive to the term t t structure of interest rates and the expected stream of future consumption. The evolution of W and z is given in the following lemma. t t L e m m a 4.2 C pf • of) dW - C G {D *? • erf + C af t t t t c t + G pf - Gl (Dtorf • erf + Ctcrf • erf) d W G t C af'-erfd W 2 + + G af'-afd W 2 2 c 2 G (C (d D )crf t t c + 2C G af'-afdcd W 2 t c t + G (d Dt)cr?) t t t - G r + D G pf 0 t + t • erf + GtCtpf - ( A G ^ f + G ^ f ) - W=-C G t (4.28) and Ctpf - C Gt {D erf • erf 4- C erf • erf) d z t G pf + t t - G\ (Dterf • erf + C af t 2 G 2 c G (C (d Dt)erf' t • erf) d z • crfd z + G\crf • of'b%z + 2C G er? +-, C crf t c t c t + Gt(d D )erf) G t t • erf + G C pf t t r + D G pf • crfdcd z Q G - t t (DtGto-f + GtCto-f) + z = -l (4.29) These equations are non-linear and coupled due to the presence of D in both r and t t 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 X = G C only. X represents t t t t 107 Chapter 4. Application to Asset Pricing 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 z = z (X ) t t W = and t C f(Xt) t t 4zyy^t) |T) D = C d(Xt) = C (lt t t v ( J 4 ' 3 ° ) The coupled non-linear set of partial differential equations above reduce to a coupled set of non-linear ordinary differential equations for z(X ) t A(X)f and f(X ). t + B'(X, /, z)f + C (X, f, z, fx, z )f = - 1 f xx x A(X)z xx + B (X, f, z)z x Z x - r(X, f, z, f ,z )z x x = -1 (4.31) (4.32) where A(X)=^-{af(X) + af{X)) 2 Bf(X,f,z)= X(jj°(X) + p°(X)) + x*(i-t < ? xy.(o-?(x) ) X(X T i - 1) (4.33) X<rf(X)'-<Tf(X)- + v?(x))- <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) 108 Chapter 4. Application to Asset Pricing C'(X, / , z, fx, z ) =tf(X) x - X(af(X)' • <T?{X) + (1 - *-) <rf{X)' • vf(X)) r(XJ,z,fx,zx) (X, / , z, fx,z ) zfx ~ fzx z _2 X 2 _ G , (4-36) = r + X((1 - x 0 v V („G ( v ^ <rf(Xy-(*?(X) L)p°(X) + tf(XJ) + „c ,\ _ ^ ( ( + <rf{XJ)-X* A - (v i - l ) X(l --) S crf(X)' • cTf(X) + *f'X) a?(X)) (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) \lmXp?(X) =0 (4.39) l i m o f p O < oo (4.41) x—o limofpO, 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 -/xf(0))/(0)=l (4.42) o 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. be obtainable from the derivatives of z(X) and f(X) The other half may at X = 0. In other cases, it 109 Chapter 4. Application to Asset Pricing 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(X ) du +r(X ,f,zJ ,zx) .It =E t u x X t • /' , \ z{X ) t \X )-<rC(X ))'-dW u u ;°° / =Et[ (4.43) uds -^(^ ' ^ +r(XJ,zJ ,z ))du-J ^(X y-dW (Xu) (Xu) s 2 e V x x t u / l (4.44) ds Let X = u (4.45) Q (X ,u-t) u t Under the growth conditions assumed, one can express fx and zx asymptotically as, xZ e fx - ( s - t ) r 0 £ du+l ° «c(o)'.<iw„ I (^(Qu)-k^u(oy^(o)) s t e t t x Q d_ t dQ ( r(Q f(Q ), z(Q ),fx(Qu),z (Qu)) U) u u x + f dxQu (d (Q d(Q ))af(0) Q u u + +p {Qu) - \<r°(Q )' • o £ ( Q C u u crf(0)-d crf(Q )y-dW Q u u (4.46) u )) dQ x u 110 Chapter 4. Application to Asset Pricing and z x - -t)ro ~ x^o e {s J E dQ ^ r [dxQu ( d (Q d(Q ))tr (0) u u f / J -(.-t e fxiQu), )roEt du + d Qu x(Qu)) z x d s oo ' + o f (0))' • dW G Q u — (Q ,f(Q ),z(Q )J (Q ),z (Q ))dxQudu\ds (4.47) r u u u x u x u 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 D r i f t and Diffusion Coefficients If Gt and Ct follow geometric Brownian motion, Q(X,uWhere W _ u t t) = C G U U = (^+M -|(^'^-f<T c X e G '-^))(u-t)+(<T c + ^)'w _ u t ( 4 4 g ) z(0) = ^ , and thus d(0) is well = / " d W „ . In this case /(0) = defined. Note that p must be constrained to be less than ro. c L e m m a 4.3 Under the assumptions of constant drift and diffusion coefficients, Eqn. (4.47) becomes, 'l ^ ro—lJ. ro (ro and Eqn. (4.46) is p G n + <7 - (n° + p CL c •a + o~ G Cl ) + p c (4.49) 111 Chapter 4. Application to Asset Pricing (p (l - ^ ) + * < • crc + 2* ' • <r ) + p G G G G G c (4.50) » where the following parameter restrictions apply: +p + cr ' • a r >p G r >p° 0 4.3.2 c c 0 + 2p + 2cr • a c G Cl + a G C / • ac Mean Reversion in G and C Here it is assumed that cr and o~ are constant, but one or both of the state variables, c C and G, mean revert. G 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, X = G C , mean reverts. In other words, risk tolerance increases with consumption. t t t 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 p (X) =p p (X) = p c G -K ln(X) c C G - (4.51) K \n{X) G this implies d\nX = t ))dt+{ G : <7° + CT 112 Chapter 4. Application to Asset Pricing and In Q (X, U u-t) -{<T Gi 2 = -(K +- )(-t) c in X G E • cr G + [p + \i c t + * > • * )) = G G / V - G + [CT + * )' • / ' V K + K e^ G G C + * ^dW G J t (4.53) L e m m a 4.4 Under the assumptions in Eqn. (4.51), with K > 0 and K° > 0, G ><*> „~. ^ ^ ( ^ ) ^ r( /o (4.54) where T(z) = J °° q e dt is the Euler Gamma function, and z x l 0 c \ „c „ay. c { r> — (,,C i + {a ,,G „Gi 1 „ + „G ,^ a „Ci ) _ £ _ „ C , - C i G\ K rT (K C {CT + ( T C I . ) _K J ( T C + ( T G ) + C s { a C + v a y . G 5 5 ) ° + K G ) S G2 { ' {K C . ( 4 { a C + ^ ) with the restriction that A must be positive. Further, _ ^ ( . 4 5 6 ) K~ C 2 ^ _ ( 4 . 5 7 ) v 113 Chapter 4. Application to Asset Pricing z{X) 1 K f \lnX\"^ f°° G2 0 _ra^-i r °° x^o r 0 ^ r l ^ + « W ( o , i ) E - -v + a + /W(0,1) i - - C+ G K dvdq (4.58) - z o l l n X f ^ T ^ with _ yF + p G 0 1 - \{<J CI ~ K C • cr + (7°' • c + K G *) G (4.59) and 2{KP + K) G (4.60) 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 < K + K , which is true under most reasonable parameterizations of C G 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 K , as long as KP > 0, the growth condition in C Eqn. (4.40) holds. It is arguable that this, in fact, covers all cases of economic interest. A situation where K > 0 and K — 0 would correspond to a consumption process that C G 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: 114 Chapter 4. Application to Asset Pricing L e m m a 4.5 Under the assumptions in Eqn. (4.51), with K > 0 and K = 0, G z(X) ~ 1 + C A'+B' a , / C ^ ) 1 + r ( ^ ) (4.61) and 2 jjpe^'^~ ' / G ro /i^" \ ro / x G where ^ ^ • S ' ^ ( ( - r c i + < ^ T G ^ )'.(a . c + ^ ) ( 4 6 C" = B ' + 4T<T ' • (<^ + « - ) C 4.3.3 C ) (4.64) T ) G < 3 (4-65) G Simulations The asymptotic expressions allow one to uniquely specify the boundary conditions when numerically integrating the ODE's, and in turn, characterize the equilibrium . For the 7 rest of the chapter, I consider the mean reverting case where K = 0. This corresponds to C an economy where the aggregate dividend process is non-stationary, while the aggregate relative risk aversion, X , mean reverts. t To understand the difference between this model and standard consumption based models, note that if o~ = —a (i.e., the representative agent acts as if she has constant G c relative risk aversion), X reverts to its median value, X , and, once there, remains t m 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. 7 115 Chapter 4. Application to Asset Pricing constant. In this case, the economy possesses constant macro-variables (superscripted by with r solving s s x 2 r Using the benchmark values of r / r \2 s = 5.0%, \fi = 1.5% and |rx | = 3.0%, Figures ?? c 0 plot the riskless rate, r , and equity premium on the market portfolio, e , versus the s s 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 X. m As in other standard consumption based models with constant aggregate relative risk aversion, the steady state model above features a riskless rate, r (X ), s m linear in ro and increasing in p . Also, e (X ) c s m that is nearly is independent of ro and p , c and is negligible for realistic values of |<T | unless the the constant relative risk aversion, X c m huge . This also implies that the impact of |er | on r 8 c s is negligible for X is ~ 0(1) or m smaller. To compare the static model above with the more general case of changing aggregate relative risk aversion, consider the following 'benchmark': T h i s is the equity premium puzzle. Related to this is the risk-free rate puzzle: if X will also be large for reasonable values of pF. 8 is large, r s m 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 (a = -* ) G c Figure 4.2: Comparative statics of the riskless rate and equity premium with respect to \a \ and X when the coefficient of relative risk aversion does not vary through time (a = -<r ). c G 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 (a = -a ). G c 119 Chapter 4. Application to Asset Pricing Parameter Benchmark Value ro 5.0% P° 1.5% \* \ 3.0% c 0.20 y° -0.04 \CT \ 47.17% G -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 (X m ~ 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, X , are denoted by t boxes (recall that X mean reverts about the median). At the bottom of the figure a t table indicates the mean values for the riskless rate and equity premium. In contrast with the case in which X does not vary, Figures 4.3-4.5 demonstrate that t for |cr | ^ \o~ \, r(X ) is non-linear in ro, strongly decreasing in a° and very sensitive G c t to |cr |. The equity premium, e(X ) = o- (X )' c M t is highly sensitive to ro and fi c t • cr (X ), M t has a tail distribution that (which shows up in the radically different mean equity premia). Note that r(X ) and e(X ) tend to be affected in opposite ways by changes in the t t model parameters. The tail distribution of r(X ) is sensitive to changes in all of the model t parameters, whereas the median riskless rate is sensitive only to the drift parameters: 120 Chapter 4. Application to Asset Pricing Figure 4.4: Comparative statics with respect to r . The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, r (X). Bottom: the equity premium on the market portfolio, o~ (X)' • cr {X). 0 M t M 121 Chapter 4. Application to Asset Pricing Figure 4.5: Comparative statics with respect to r . 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. 0 122 Chapter 4. Application to Asset Pricing Figure 4.6: 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, r (X). Bottom: the equity premium on the market portfolio, cr (X)' • cr (X). c M t M 123 Chapter 4. Application to Asset Pricing He 0.014 0.015 0.016 mean r mean e 0.034 0.028 0.021 0.052 0.066 0.084 Figure 4.7: 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. c 124 Chapter 4. Application to Asset Pricing Figure 4.8: Comparative statics with respect to \cr \. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, r (X). Bottom: the equity premium on the market portfolio, a (X)' • cr (X). c M t M 125 Chapter 4. Application to Asset Pricing -0.8 0.02 0.03 0.04 mean r mean e 0.031 0.028 0.024 0.063 0.066 0.071 Figure 4.9: Comparative statics with respect to | < T | . T h e 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. c 126 Chapter 4. Application to Asset Pricing Figure 4.10: Comparative statics with respect to K . The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, r (X). Bottom: the equity premium on the market portfolio, * (X)' • cr (X). G M t M 127 Chapter 4. Application to Asset Pricing -0.S5 K 0.15 0.2 0.25 mean r 0.035 0.028 0.022 mean e 0.046 0.066 0.084 Figure 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. G 128 Chapter 4. Application to Asset Pricing Figure 4.12: 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, r (X). Bottom: the equity premium on the market portfolio, cr (X)' • cr {X). g M t M 129 Chapter 4. Application to Asset Pricing f(X) 55 1 , , , , 0 0.2 0.4 0.6 0.8 , , 1 1.2 •i 1.4 , 1.6 1 , 1.8 2 X -0.8 Ho -0.03 -0.04 -0.05 mean r mean e 0.024 0.028 0.032 0.073 0.066 0.060 Figure 4.13: 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. 9 130 Chapter 4. Application to Asset Pricing Figure 4.14: Comparative statics with respect to |cr |. The squares represent the 5th, 50th and 95th percentile values of the state variable, X. Top: the instantaneous riskless rate, r (X). Bottom: the equity premium on the market portfolio, cr (X)' • cr (X). G M t M 131 Chapter 4. Application to Asset Pricing f(X) 7 5 70 55 , 50-1 0 -0.6 v> 0.2 i 1 0 0.2 , 0.4 1 0.4 , , 0.6 . . 1 0.8 X 1 1 1 0.6 0.8 1 . 1.2 1 1 1.2 . 1.4 1.4 . 1.6 • 2 1.8 1 1 ' 1.6 1.8 2 -0.8 o mean r mean e 0.2 0.2225 0.25 0.031 0.028 0.025 0.063 0.066 0.069 0 Figure 4.15: Comparative statics with respect to \cr \. 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. G 132 Chapter 4. Application to Asset Pricing 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, r (X). Bottom: the equity premium on the market portfolio, cr (X)' • cr (X). M t M 133 Chapter 4. Application to Asset Pricing -0.8 p -0.4 -0.53 -0.66 mean r mean e 0.029 0.028 0.027 0.065 0.066 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. 134 Chapter 4. Application to Asset Pricing r ,/j, ,n c G 0 and the rate of mean reversion, K . The instantaneous riskfree rate tends to G decrease towards the median from extreme values of the relative risk aversion, X . t 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{X ). t A t 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 X ) on the dividend growth rate and the need for cautionary t savings due to the higher emphasis (proportional to X ) 2 portfolio. Eventually, as X t on the volatility of the market becomes disproportionately large the need to save prevails and the riskless rate plummets. Note the lower wealth-dividend ratio when the relative risk aversion, X , t 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 X . t X d (X ), 2 2 t which is increasing in X . t This is because it depends on 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 X spreads, the unconditional mean market risk premium dramatt 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 will 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 135 Chapter 4. Application to Asset Pricing presence of negative correlation in returns. artifact of mean-reversion in X . The long-term negative correlation is an To see the negative auto-correlations in the shorter t horizons, consider that the instantaneous auto-correlation of excess returns on the market portfolio is given by Mtv Since cr (X ) M t E [d{<r (x y M \J4. t « X d{X )cr , • <rM(xt)} * {x y M t • dw ] t the auto-correlation can be written using Ito's Lemma as G t t t p (X ) M t « 2<r ' • cr X~(X G G t d{X )) t This last expression is typically negative, as can be seen in the graphs. Note, in particular, that in the absence of time-variations in the aggregate relative risk aversion, p (X ) M m is zero. Although the presence of short-term auto-correlation of the model is in disagreement 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 E x a m p l e To continue, it is useful to specialize to a particular example. parameterized by Parameter Benchmark Value 4.7% 1.5% \a \ 1.75% K 0.15 c G -0.02 \a \ G P ~ |cr«| | ^ ' | 51.5% -0.63 Consider an economy 136 Chapter 4. Application to Asset Pricing 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(X ) t 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 example above. • Long-term coupon bond prices (z(X ))are roughly linear in aggregate relative risk ( aversion, X. Thus a , G K G and fi G 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. W h e n 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. 140 Chapter 4. Application to Asset Pricing 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 P r o o f of L e m m a 4.1: First note that at date t, investor i maximizes a derived utility of form, V®™ (c<(a ),^(a )) 4 t+1 (4.66) where a +\ C a t t (4.67) a Cat+l s>t+l s 141 Chapter 4. Application to Asset Pricing This is simply a consequence of optimizing under a budget constraint. The maximization program is as follows: max { V ir(a \a )V ' (cl(a ), Wi(a )) i9t Rt) t+1 t t t t+1 - Ut+lCat ,>t \ A 4( t)+ c a a t J a (Zat s>t V (4.68) % s)4>{ s\at) - Wi(a ) a s where A is a Lagrange multiplier. The Lagrange multiplier can be eliminated from the first order conditions for c*(a +i) and c\{a ) to give t 4>(a \a ) = t+1 t t ir(a i\a ) t+ t £V { 9 t ^ {c\{a ), {a )) t Wi t+l (4.69) d *tr(.9l,R\) (J, V d~c t Vr^'icHat)^^)) Similarly, using the fact that the events, {a } for s > t partition the state space, the first s order condition for c\{a ) for s > t + 1 can be written as s , , , , , , £ V f (c1(q ),^(a )) (f){a \a ) = TT{a \a )4>{a \a i) — fVt (q(a ),^(a i)) ( s u t t+l T s t+ t f+1 t i + (4.70) d Substituting from E q . (4.69) and rearranging, the last equation becomes ,/ i x <t>(a \at) (p{a \a ) = — — (p(at \a ) (4.71) s s t+1 +1 t These equations can be used to inductively derive E q . (4.3). • P r o o f of L e m m a 4.2: Recall the Feynman-Kac formula: f(t,s Y ) : t = E [e-l'°^ MYs)} d t {A -q(Y ))f t t = df 3 (4.72) 142 Chapter 4. Application to Asset Pricing where Y denotes the vector of exogenous state variables (G and C ), and A is the t t t t 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 • °f A T = pi - e 4 where 0 is a matrix whose ij (4.74) element is the j th diffusion component of the i th t th state variable. Applying this to the state variables, C and G gives, t At = [ C " t ) - ? ( C t ( T ? ' t ) • (D G <r? + G C * ) C p -C G (D a '.af c t G tf t t t - Gl t t + c t (D *r t (4.75) c t t C o- '-a ) c c t (4.76) • <rf + C af • a?) t Thus denoting A\ as the risk adjusted diffusion operator, the Feynman-Kac formula becomes, f(t,s,Y ) t = E;[e-X^ p(Ys)} * du (A* -q(Y ))f t t = dJ (4.77) To derive the dynamics for W and z , consider a general expression, t t oo / e-K^ p(Y )ds} du s (4.78) 143 Chapter 4. Application to Asset Pricing Applying Al — q(Y ) to both sides of the equation and assuming that the integral is t absolutely convergent, we get CO (A* - / q{Y ))El[e~ p{Y )]ds q{Yu)du t s O O d E* [e~ S° s / p{Y )]ds q{Yu)du t s =E;[e-ir^y )}-p(Y ) {Yoo t = -p(Y ) (4.79) t where in the last step we assumed the standard transversality or growth condition on p(Y ) so as to ensure that its expected value grows at a sufficiently small rate. t The evolution equations for W and z are t t {A* -r )W t t = -C t (A* -r )z t t (4.80) t t =- \ (4.81) Writing this out explicitly gives the desired result. • P r o o f of L e m m a 4.3: z (0) = - J\-^ E ro x t d Q (X,ux u t)(d(0)p G + d{0) a ' • o-^du ds +p c c roUo - {n° + P + o- ' G c (4.82) 144 Chapter 4. Application to Asset Pricing / ( 0 ) = _ J" Hs-t){ro-f x + \° '<r )+S;<> <™« C e J E t d Q (X, x Cl u-t) u d Qu(X, u-t) C [d{Q)p + p + d(0) CT ' • o- ) du + G ( d(0)* + c C G a )'-dW. ds G x x c using i:° '^ c Et x,u-t) L +» +CT '-a +2<T '-<T )(u-t)+±CT '-<T (s-t) c = a C c a C C C t < u < dxQu{ e (4.83) h ° ' »d Q (X,u-t)dW s E t e c dW x u G f <r '<™»d Q (X,U-t) S cr E = v t du C e t x u t < U< s (4.84) In addition, ( l - ^ ) ( p fx(0) = c + 2a '.cr ) p G G G G + _ + L r„a . c a ( (4.85) ( r - p ) ( r - (p + 2\i + 2cr ' • cr + a > • <r )) c 0 G c c G c c 0 • P r o o f of L e m m a 4.51: Begin by analyzing f(X) as X —> 0. Assuming E q n . (4.40) holds (which will be 145 Chapter 4. Application to Asset Pricing verified soon), r -^ xZ ) (s-t)-f + \<T '<J c f(X) E C 0 C ° ln(Q„) du+J* <r 'dW. 3 t c ds K t 0 -(r -nC 0 + \*C'-*C 0 ~ / £ - £ r ^ l - e x p ( - ( K + K ) ( s - t ) ) ^ lnX ){s-t) C G e A'->0 \ C ^Gl l x + 3 _ c K [ ( T C,. j - e ( T x C (_( p G . + r T l K c + ( T K G G ) ( u '.dw„ _„)j (<T '+<T ) C ) g d « + < r c ' d w ds u (4.86) The expectation is calculated as, - \ ( ° 1 - C X P ( - ( K K C C + j* / " e x p ( - ( K C t ' * K G t C + { < * ) ( S - £ ) ) 5. C G U + ° C G ) ' { ° C / i ^ , (/+/)'•*,+ +/c )( -i;,) + ° f f c + f f G ) , ^ _ ( r f 4 ( K C G + + f f G y . , G ( ds = u , r c f , l-exp(-2(K g S T ) ^ ^ - 2 ^ ^ + ^ ) ^ ^ G C ( a 'dW c J ' a C J _) (a t X « C 2 + a 0 ) _ ^ + K )(s-t)) G K < : ; ) (4.87) 3 giving -A(s-t)+B ^1—exp( — ( K + « ) ( s - 4 ) > )+C' ^1-exp (-2(/t + )(s-t)) j C G G G K x->o -TC^U with (l-exp(-( + c K G K ) ( s - t ) ) ) lnX (4.88) 146 Chapter 4. Application to Asset Pricing A= (r -p ) + (p c + p c 0 -^ 2 G _ G , ~Gi 1 C/ i _C7 / G \ _ _ T 2 JK _ _ _ K + K C G ,C2 l(<T D —/ C , + a - G G ) ' > G 1 G/ _G ( G < T + 0 . G G .G /. ) ( o .G -o- +* + 0 .a (4.89) 2 , 1 C7 „ C , „ . C / ° +f -if B=(P + <x ). c ^.G\__^1__ cr ) ^ + r G ) 2 C2 ft ) (/C + (4.90) ft ) C G 3 C2 K ,C C = , _Gy / _ C , „G\ ^ ( 0 ' - ( » ° +O + Now making the change of variables, y = e~( v 1 x^o J i K 5 KC+KG ^ 5 ? («1) F )( ~*)| InX\, s \\lnX\J 0 {K + n )y c G v ; The diverging part can now be isolated in closed form: B+C roo e f(X) ~ g ^\K X + K C G ) X ^ U — / \ l n X | ^ t ^ > B + J c SK + C C c V A r G c e* +* dy a c ay o {K + K )X^^\\nX\^ C i y« +* ; V K C g I K K ^ ^ f A ( V C i G K K K + K (4.93) where T(z) = J °° q ~ e~ dt z 1 t Q is the Euler G a m m a function, and A must be positive. For any postive value of K , as long as K > 0, the growth condition i n Eqn. (4.40) holds. C G From Eqn. (4.93) we can also deduce the behavior of fx(X) as X —> 0: ^ 0 H ^ ) " ^ ^ ) <«•*> (K + K )X^ ~ \lnX\^ C G a +1 u K K ' K + K ' 147 Chapter 4. Application to Asset Pricing 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 ^ ZO [°° ~ ~ ° kG e (s t)r [ Et X ds AdQ(Q (Qu)HQu))du d d (4.95) U To leading order this is zx(X) X^O X{K C e- -(°-t>°Et e + K) G { K C + K G ) { u - \\nQ \ ~^ Q'f^ t ) l a u du ds (4.96) Making the variable changes, v = - ( « + « ) ( « - * ) | \ c \ and q = e-(- +« c G e n X G )(^-t)| x | , l n for u and s, respectively, and using the functional form for Q derived in Eqn. (4.53) u gives |lnX| zx[X) ^°X\\nX\ ^^{K X rllnAI f j ~ L+ + K )*JO C g -+«(i-r^)+/3(i-n^)^(°- )) 1 dvdq - - + - ( - n ^ ) + / ( In - nX I^ W ) 1 3 1 1 (4.97) n where P +p C a = G - •* l(* ' C K + K C G + CT ' • <T ) G G (4.98) G and 2{K + K ) G (4.99) G Af(0,1) is a standard normal variate and the expectation in Eqn. (4.97) is taken with respect to its distribution. A s long as A < KF + K , which is true under most reasonable g 148 Chapter 4. Application to Asset Pricing parameterizations of the consumption process, the integral i n Eqn. (4.97) has a well defined limit: n r fo G2 zx{X) 0 x ^°X|lnX| E 1 + ^^(ft + c K G\3 J X J 0 q 1-- ^p(-«+«W(o,i) dwdg -v + a + BM{0X (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. • P r o o f of L e m m a 4.5: Because p (X) is constant, Eqns (4.42) imply that z(0) = ^ and / ( 0 ) = ^z^p- To c derive the asymptotic behavior of Z(X) for X —> 0 express its derivative from Eqn. (4.47) up to leading divergent terms: ~ z (X) x f dxQud (Qud{Q )HQ e-t'-^Et K [ g Q Jt Jt u (4.101) duu ds v since d(0) is finite, this can be expressed to leading terms as K zx(X) e K<3 p ^ 0 J e -(s-t)r 0 E t J Q -K°(u-t) d u e du (4.102) s t ( -*) and calculating the expectation u ,c MX) PS O O C X(r - p',c\j / x^o Setting y = - G O O -(s-t)ro x-o fl o ! / l n ( X ) + / l ' ( l - i / ) + B ' ( l - y 2 ) x X(r - p ) G 0 (y ln(X) + A ' ( l -y) + 2B'(1 - y )) dy ds 2 (4.103) 149 Chapter 4. Application to Asset Pricing where C G + _ l ( A' =- ( T C , . C a G, + , G\ (T a ^—^ B' =^G(O- (- ) 4 + v )' • (<x + <r ) c G c 104 (4.105) G proceeding as in the previous lemma leads to * W ,-o o K ( n _ £wZ w* (' + £ ) (^> <«•«*» r xi 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 E q . (4.66), with each event a +i G F +i giving a utility of t , M D {gi{t),Ri{t)) M)M» r V t p-9i{ty\{at) ft (cl(a ), (a )) t Wl t+1 = -9i(t) ^ - P t + l (a , * p-OLt+i(.a i)gi{t)wi(at+i) t+ )e-^ ^ ^ a t + 1 - — — R (4.107) a {a i)gi{t) t+1 where Wi(a i) t+ x i T+ t+ is as in Eqn. (4.67). Given Eqns. (4.5)-(4.7) and the fact that (pr = 0, = 0, it can be readily verified, by setting 4>(a \a ) = 0 for t > T, that Eqn. (4.107) s t holds for dates T and T — 1. Specifically, pr+i = 0, O;T(O-T) = PT(O-T) = 1, QT(O-T) = 1 and 1- The optimization program facing the consumer is given by E q . (4.68). E q . (4.69) can 150 Chapter 4. Application to Asset Pricing be written as (f)(a i\a ) t+ = t 7r(a i|a )p t + t t + 1 ( a i ) e x p ^ ( t ) c * ( a ) - a {a i)gi{t)wi(a i) t + t t+1 t+ - t+ q i(a i)Ri{t)^ t+ t+ (4.108) Multiplying each side by a t + 1 ( ^ and summing the last equation over all the date t + 1 a events in the a partition leads to t (f>(a +i\at) - (t)cHat) t E - gi a*+i( t+i) a ^ } ^p (a )e-^ ^ 'V a Tr{a \a ) t+l t+l R t+1 t r a {a i) t+1 exp t+ ( -a i{a )g (t)w {a ) t+ \ t+1 l i t+1 \ I (4.109) The optimized utility from E q . (4.68) in the event a is therefore, t The left hand side is not an explicit function of Wi(a ). To fix this, note that E q . (4.108) t can be manipulated to give *( 0 _ qt i{a,t+i)R{t) Ut+i{at+\) a i(a i)gi(t) C _ Q + t+ j % t 1 = l n gi(t)a i(a i) + t+ t+ t+ ( <t>(<h+i\<h) s ^{a +\\a ) Pt+i{a i) t t ( 4 n i ) t+ Wi{a \) in the last expression can be written out as in E q . (4.67), afterwhich multiplying t+ both sides of the above equation by 4>{a i\a ) and using the relationship, t+ (j)(a i\a )4>(a \a ) t+ t s t + t + 1 = (f>(a \a ) when a C a (from E q . (4.3)) gives t+1 <^K^ a i(a s _^ ) t 5i gt+1 t s t (a ^K |a ) _ / t+ (t) +1 a i(a t + t ) t + 1 t { V ) < K M + £ a c ^ * ^( + (a i a>t) )= / s>t + l 0(a i|at) gi(t)a i(a i) t+ t+ t+ l n v / <t>(at+i\at) x 7r(a i|a )p i(a +i)' t+ t t+ t (4112) 151 Chapter 4. Application to Asset Pricing After summing this last expression over all date-t + 1 events in the a partition one t can once more refer to E q . (4.67), but this time as applied to the date t budget. The result yields t/ x A t) c a Ri(t) v - ^ Qt+i(<k+i)<l>{at+i\<k) i \ . t, T T T y. 7x ui {at) + c {a ) = g {t) . a {a ) 4>{a i\a ) 7c a {a i) I. t+ t+1 <H+lCa f i t+ -r \ t t l T- / t+1 v o iCoi y^-t 1 9i(t) ^ t+ n t+l v (j>(at+i\(k) a +i(a i) t \ t l n / <t>(<h+i\<k) x ir(at+i\at)pt+i(a ) { t+ (4 113) J t+1 The above can be solved for c\{at) in terms of Wi(a ): t „ <n \,n (r. U W K f t ( a t ) - - l ) Ci(at) = a i ( a # i f l t + lm>(at)) TTT 9i{t) with _ {•> , 1 (1+ qt{a ) t Trr— . (4.114) 9i{t) 0(at+i|at) Y (4.115) </>( t+il t)fr+i( t+i) a = , Q Q + 1 ^ Substituting this back into the expression for the maximum date t+1 utility, E q . (4.110), results in at (at )gi (t)w - (at) - f t W t e ( « , ) - i ) ! _ ^ ^ _ l [/;(*WA(0) ( a t i ( a t ) ) = _ ( a t ) c ( 4.ii8) ost{a )gi{t) t Note that p {a ), qt(a ) and at (at) do not depend on the bias assignment at date t t t 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) 152 Chapter 4. Application to Asset Pricing derived utility conditional on event a can be calculated from E q . (4.5): t ~-fli(t-l)cJ r(9i(t-l),Ri(t-l) - 1 (a _i) t f t - l 9iit ~ 1) -Ri(t-l) // e f°° r°° -g' e -g'/gi(t-l)-R'/Ri(t-l) 9i{t-l)-R' Ri(t-V e / , \ , A (4.119) This simplifies to e -9i(*-l)c,' (at-l) - 1 - 1) p ( )e-"^Wft-i) t (4.120) e flt a {a )gi{t - 1) t t which is of the form conjectured in E q . (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 V C(a ) - q {a ) t t+1 t+l - a (a )w(a ) t+1 t+1 t+l = ln(——^* ^ ) +1 f r) J ] —777 (4.121) Rlt) N C(a ) = a (a )w(a ) + (q (a ) - 1) £ t t t t t One can now solve for p _ ln( (a )) £ t A i=i ^ t + 1 1 — N t ' (4.122) i=i ^ ' ( a i ) by using the date t + 1 version of Eqn. (4.122) and t + substitute the result in Eqn. (4.121): ^ ; = 1 * f (C(a ftW t + 1 ) - a t + 1 (a i + 1 ) (a W v f + 1 7r(a ) - (q (a ) t+l t+1 |a ) ^ & ( y i + 1 t - 1) £ ^±ii) (4.123) 153 Chapter 4. Application to Asset Pricing Now define the aggregate risk tolerance and aggregate discount rate via. N — = (4.124) = ( i=l 5 i 4 - 1 2 5 ) w Eqn. (4.123) becomes, ]n(^° l° J = -G(t)(C(a 7r(a +i|at) t + 1 t ) v t + 1 ) - C(a,)) - t (C(a ) m a t + 1 K iM«m)) + + 1) - G(t)) + ft+i^+i) + D ~ *(*)) - + 1) (4.126) This is Eq. (4.8). • Bibliography [1] Allais, M . , Le Comportement de l'homme rationnel devant le risque: Critique des postulats et axiomes de l'ecole americaine, Econometrica 21 (1953), 503-46. [2] Aumann, R . J . , Utility Theory without the Completeness A x i o m , Econometrica 30 (1962), 445-462. [3] Anscombe, F . , and R. Aumann, A definition of subjective probability. Annals of Mathematical Statistics 34 (1963), 199-205. [4] Arrow, K . , , The Role of Securities in the Optimal Allocation of Risk-Bearing. Rev. Econ. Stud. 31 (1964), 91-96. [5] Baucells, M . and L.S. Shapley, Multiperson Utility, UCLA Economics Working Paper (1998). [6] Bell, D . E . , Regret in Decision Making Under Uncertainty, Operations Research 30 (1982), 961-981. [7] Bewley, T . F . , Knightian Decision Theory: Part I, Cowles Foundation Discussion PaperNo. 807, Yale University, (1986). [8] Bewley, T . F . , Knightian Decision Theory: Part II, Cowles Foundation Discussion PaperNo. 835, Yale University, (1987). [9] Bossert, W . , Opportunity Sets and Individual Well-Being, Social Choice and Welfare 14 (1997), 97-112. [10] Bossert, W . , P . K . Pattanaik, and Y . X u , Ranking Opportunity Sets: A n Axiomatic Approach, J. Economic Theory 63 (1994), 326-345. [11] Camerer, C . F . , A n Experimental Test of Several Generalized Utility Theories, Journal of Risk and Uncertainty 2 (1989), 61-104. [12] Chechile, R . A . and D . J . Cooke. A n Experimental Test of a General Class of Utility Models: Evidence for Context Dependency, Journal of Risk and Uncertainty 14 (1997), 75-93. [13] Chew, S.Ff. and L . G . Epstein, Nonexpected Utility Preferences in a Temporal Framework with an Application to Consumption-Savings Behaviour, J. Economic Theory 50 (1990), 54-81. 154 Bibliography 155 [14] Chew, S.H. and K . R . MacCrimmon, Alpha-Nu Choice Theory: A Generalization of Expected Utility, UBC Working Paper No. 669 (1979). [15] Chew, S.H., A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox, Econometrica 51 (1983), 1065-1092. [16] Campbell, J . Y . and J . H . Cochrane, Explaining the Poor Performance Consumption-Based Asset Pricing Models, U. Chicago Working Paper (1999). of [17] Cubitt, R . P . , Rational Dynamic Choice and Expected Utility Theory, Oxford Econ. Papers 48 (1996), 1-19. [18] Constantinides, G . M . , Habit Formation: A Resolution of the Equity Premium Puzzle, J. Political Economy 98 (1990), 519-543. [19] Dekel, E , B . Lipman and A . Rustichini, A Unique Subjective State Space for U n foreseen Contingencies, Northwestern University Working Paper (1999). [20] Demarzo, P. and C . Skiadas, O n the Uniqueness of Fully Informative Rational E x pectations Equilibria, Economic Theory 13 (1999), 1-24. [21] Demarzo, P. and C. Skiadas, Aggregation, Determinacy, and Informational Efficiency for a Class of Economies with Asymmetric Information, J. Economic Theory 80 (1998), 123-152. [22] Donaldson, J . B . and L . Selden, Arrow-Debreu Preferences and the Reopening of Contingent Markets, Economics Letters 8 (1981), 209-216. [23] Dubourg, W . R . , M . W . Jones-Lee and G . Loomes, Imprecise References and the W T P - W T A Disparity, Journal of Risk and Uncertainty 9 (1994), 115-133. [24] Dubra, J . and E . Ok, Theory of Risk and Incomplete Preferences, NYU Economics Working Paper (2000). [25] Debreu, G . , Theory of Value. (1959) New York: Wiley. [26] Detemple, J . B . and F . Zapatero, Asset Prices in an Exchange Economy with Habit Formation, Econometrica 59 (1991), 1633-1657. [27] Duffie, D . and L . G . Epstein, Asset Pricing with Stochastic Differential Utility, Review of Financial Studies 5 (1992), 411-436. [28] Duggan, J . , A General Extension Theorem for Binary Relations, J. Economic Theory 86 (1999), 1-16. Bibliography 156 [29] Dumas, B . , R . Uppal and T . Wang, Efficient Intertermporal Allocations with Recursive Utility, NBER Working Paper No. 231 (1998). [30] Ellsberg, D . , Risk, ambiguity and the Savage axioms. Quarterly Journal of Economics 75 (1963), 643-669. [31] Epstein, L . G . and T . Wang, Intertemporal Asset Pricing under Knightian Uncertainty, Econometrica 62 (1994), 283-322. [32] Epstein, L . G . and S.E. Zin, Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework, Econometrica 57 (1989), 937-969. [33] Fishburn, C P . , Nontransitive Measurable Utility, Journal of Mathematical Psychology 26 (1983), 31-67. [34] Fishburn, C P . , "Nonlinear Preference and Utility Theory," Johns Hopkins University Press, Baltimore, (1988). [35] Gilboa, I and D . Schmeidler, M a x m i n Expected Utility with Non-Unique Prior, J. Math. Econ. 18 (1989), 141-153. [36] Goldstein, W . M . and H . J . Einhorn, Expression Theory and the Preference Reversal Phenomena, Psychological Review 94 (1987), 236-254. [37] Grant, S., A . K a j i i , and B . Polak, Intrinsic Preference for Information, J. Economic Theory 83 (1998), 233-259. [38] Green, J . , 'Making Book Against Oneself,' The Independence A x i o m and Non-linear Utility Theory, Quart. J. Econ. 102 (1987), 785-796. [39] G u l , F . , A Theory of Disappointment Aversion, Econometrica 59 (1991), 667-686. [40] Hammond, P . J . , Consistent Plans, Consequentialism, and Expected Utility, Econometrica 57 (1989), 1445-1449. [41] Halevy, Y . , The Possibility of Speculative Trade Between Dynamically Consistent Agents, UBC Working Paper (2000). [42] He, H . and J . Wang, Differential Information and Dynamic Behavior of Stock Trading Volume, Review of Financial Studies 8 (1995), 919-972. [43] Hey, J . D . and C . Orme, Investigating Generalizations of Expected Utility Theory Using Experimental Data, Econometrica 62 (1994), 1291-1326. [44] Herstein, I . N . and J . Milnor, A n Axiomatic Approach to Measurable Utility, Econometrica 21 (1953), 291-297. Bibliography 157 [45] Holmes, R . B . , Geometric functional analysis and its applications. (1975) New York: Springer-Verlag. [46] Kahneman, D . and A . Tversky, Prospect Theory: A n Analysis of Decision Under Risk, Econometrica 47 (1979), 263-291. [47] D . Kelsey and F . Milne, Induced Preferences, Dynamic Consistency and Dutch Books, Economica 47 (1997), 471-481. [48] D . Kelsey and F . Milne, Induced Preferences, Nonadditive Beliefs, and Multiple Priors, International Economic Review 40 (1999), 455-477. [49] Kreps, D . M . , and E . L . Porteus, Temporal Resolution of Uncertainty and Dynamic Choice Theory. Econometrica 46 (1978), 185-200. [50] Kreps, D . M . , and E . L . Porteus, Temporal von Neumann-Morgenstern and Induced Preferences. Journal of Economic Theory 20 (1979), 81-109. [51] Kreps, D . M . , A Representation Theorem for "Preferences for Flexibility". Econometrica 47 (1979), 565-577. [52] Kreps, D . M . , Static Choice in the Presence of Unforeseen Contingencies. In Economic Analysis of Markets and Games, ed. B y P. Dasgupta, et. al. (1992) Cambridge: M I T Press, pp. 565-577. [53] Kuratowski, C , Topologie, Vol. 2. (1950) Warsaw: Polska Akademia. [54] Levi, I., The Enterprise of Knowledge. (1980) Cambridge: M I T Press. [55] Lichtenstein, S. and P. Slovic, Reversals of Preference Between Bids amd Choices in Gambling Decisions, Journal of Experimental Psychology 101 (1973), 16-20. [56] Loomes, G . and R. Sugden, Regret Theory: A n Alternative Theory of Rational Choice Under Uncertainty, Economic Journalbi 92 (1982), 805-824. [57] Lucas, R . E . , Asset Prices i n an Exchange Economy, Econometrica 46 (1978), 14291445. [58] Luce, R . D . , Semiorders and a Theory of Utility Discrimination, Econometrica 24 (1956), 178-191. [59] Luce, R . D . , Associative Joint Receipts, Math. Soc. Sci. 34 (1997), 51-74. [60] Luce, R . D . and P . C . Fishburn, Rank- and Sign-Dependent Linear Utility Models for Finite First-Order Gambles, J. Risk and Uncertainty 4 (1991),29-59. [61] Luce, R . D . and P . C . Fishburn, A Note on Deriving Rank-Dependent Utility Using Additive Joint Receipt, J. Risk and Uncertainty 11 (1995),5-16. Bibliography 158 [62] R . D . Luce, B . A . Mellers and S. Chang, Is Choice the Correct Primitive? O n Using Certainty Equivalents and Reference Levels to Predict Choices Among Gambles, Journal of Risk and Uncertainty 6 (1993), 115-143. [63] MacCrimmon, K . R . and M . R . Smith, Imprecise Equivalences: Preference Reversals in Money and Probability, Working Paper #1211, University of British Columbia, Vancouver (1986). [64] MacCrimmon, K . R . and S. Larsson, Utility Theory: Axioms Versus "Paradoxes", In M . Allais and O. Hagen (ed.), "Expected Utility Hypothesis and the Allais Paradox", Dordrecht, Holland: Reidel (1979), 333-409. [65] MacCrimmon, K . R . , W . T . Stanbury and D . A . Wehrung, Real Money Lotteries: A Study of Ideal Risk, Context Effects and Simple Processes, In T . S . Wallsten (ed.), "Cognitive Process i n Choice and Decision Behavior", Erlbaum, Hillsdale, N J (1980), 155-177. [66] Machina, M . J . , Choice Under Uncertainty: Problems Solved and Unsolved, Journal of Economic Perspectives 1 (1987), 121-154. [67] Machina, M . J . , Temporal Risk and the Nature of Induced Preferences, J. Economic Theory 33 (1984), 199-231. [68] Machina, M . J . , Expected Utility Analysis Without the Independence Axiom, Econometrica 50 (1982), 277-323. [69] Machina, M . J . , Dynamic Consistency and Non-Expected Utility Models of Choice Under Uncertainty, J. Econ. Lit. 27 (1989), 1622-1668. [70] Nakamura, Y . , Expected Utility with an Interval Order, J. Math. Psych. 32 (1988), 298-312. [71] Nehring, K . , Preference for Flexibility in a Savage Framework, Econometrica 67 (1999), 101-119. [72] Nehring, K . and C . Puppe, O n the Multi-Preference Approach to Evaluating O p portunities, Social Choice and Welfare 16 (1999), 41-63. [73] Nehring, K . and C . Puppe, Continuous Extensions of an Order Set to the Power Set, J. Economic Theory 68 (1996), 456-479. [74] Ok, E . , Utility Representation of an Incomplete Preference Relation, NYU Economics Working Paper (2000). [75] Pattanaik, and Y . X u , O n Preference and Freedom, Theory and Decision 44 (1998), 173-198. Bibliography 159 [76] Phelps, R . R . , Weak-* Support Points of Convex Sets in E*, Israel Journal of Mathematics 2 (1964), 177-82. [77] Puppe, C , Freedom of Choice and Rational Decision, Social Choice and Welfare 12 (1996) , 137-153. [78] Puppe, C , A n Axiomatic Approach to "Preference for Freedom of Choice", J. Economic Theory 68 (1996), 174-199. [79] Quiggin, J . , Stochastic Dominance i n Regret Theory, Rev. Econ. Stud. 57 (1990), 503-511. [80] Quiggin, J . , Regret Theory W i t h General Choice Sets, Journal of Risk and Uncertainty 8 (1994), 53-65. [81] Sarin, R. and P.P. Wakker, Dynamic Choice and Non-Expected Utility, J. Risk and Uncertainty 17 (1998), 87-119. [82] Savage, L . J . , "The Foundations of Statistics", Wiley, New York, (1953): [83] Schroder, M . and C . Skiadas, A n Isomorphism between Asset Pricing Models with and without Linear Habit Formation, Working Paper No. 247, Kellogg Graduate School of Management, Department of Finance. (2000). [84] Segal, U . , Two-Stage Lotteries Without the Reduction A x i o m , Econometrica 58 (1990), 349-377. [85] Segal, U . , Dynamic Consistency and Reference Points, J. Economic Theory 72 (1997) , 208-219. [86] Seidenfeld, T . , Decision Theory Without "Independence" or Without "Ordering": What is the Difference?, Economics and Philosophy 4 (1988), 267-290. [87] Seidenfeld, T . , M . J . Schervish and J . B . Kadane, A Representation of Partially Ordered Preferences, Ann. Stat. 23 (1995), 2168-2217. [88] Skiadas, O , Recursive Utility and Preferences for Information, Economic Theory 12 (1998) , 293-312. [89] Skiadas, C , Conditioning and Aggregation of Preferences, Econometrica 65 (1997), 347-367. [90] Sugden, R . , A n Axiomatic Foundation for Regret Theory, Journal of Economic Theory 60 (1993), 159-180. [91] Sundaresan, S . M . , Intertemporally Dependent Preferences and the Volatility of Consumption and Wealth, Review of Financial Studies 2 (1989), 73-88. Bibliography 160 [92] Tversky, A . and D . Kahneman, Rational Choice and the Framing of Decisions, Journal of Business 59 (1986), S251-S278. [93] Tversky, A . and D . Kahneman, Loss Aversion in Riskless Choice: A Reference Dependent Model, Qtrly. J. Econ. (1991), 1049-1061. [94] Vincke, P., Linear Utility Functions on Semiordered Spaces, Econometrica48 (1980), 771-775. [95] von Neumann, J . and O. Morgenstern, "Theory of Games and Economic Behavior", Princeton University Press, Princeton, N J , (1944). [96] Wakker, P., The Sure-thing Principle and the Comonotonic Sure-thing Priniciple: A n Axiomatic Analysis, J. Math. Econ. 25 (1996), 213-227. [97] Wakker, P. and A . Tversky, A n Axiomatization of Cumulative Prospect Theory, J. Risk and Uncertainty 7 (1993),147-176. [98] Wakker, P. and A . Tversky, Risk Attitudes and Decision Weights, Econometrica 63 (1995), 1255-1280. [99] Wang, J . , A Model of Competitive Stock Trading Volume, J. Political Economy 102 (1994), 127-168.
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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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Title | Partial ordering of risky choices : anchoring, preference for flexibility and applications to asset pricing |
Creator |
Sagi, Jacob S. |
Date Issued | 2000 |
Description | This dissertation describes two theories of risky choice based on a normatively axiomatized partial order. The first theory is an atemporal alternative to von Neumann and Morgenstern's Expected Utility Theory that accommodates the status quo bias, violations 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 particular, 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. |
Extent | 6932226 bytes |
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Risk management -- Mathematical models Equilibrium (Economics) -- Mathematical models Capital assets pricing model |
Genre |
Thesis/Dissertation |
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Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-07-23 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0089874 |
URI | http://hdl.handle.net/2429/11161 |
Degree |
Doctor of Philosophy - PhD |
Program |
Business Administration - Finance |
Affiliation |
Business, Sauder School of Finance, Division of |
Degree Grantor | University of British Columbia |
Graduation Date | 2000-11 |
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UBCV |
Scholarly Level | Graduate |
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