Changing Tastes and Effective Consistency

Transcription

1 Changing Tastes Effective Consistency Larry Selden Columbia University University of Pennsylvania Xiao Wei University of Pennsylvania October 8, 0 Abstract In a simple single commodity certainty setting with changing tastes, a consumer s consumption plan can be obtained using naive or sophisticated choice. Although in general these solutions diverge, they agree in the special case where the changing tastes are represented by discounted logarithmic utilities. We show more generally that this result does not require preferences to be logarithmic, additively separable, or homothetic. Suffi cient conditions are provided for when naive sophisticated choice agree the common plan can be rationalized by a non-changing tastes utility. Because the solution generated by this utility is not revised over time, the plan is referred to as being effectively consistent. The optimal consumption plan for a popular form of quasi-hyperbolic discounted utility is shown to be rationalized by a non-changing tastes discounted utility which has a very different pattern of intertemporal discount rates. We also consider the implications of effective consistency for equilibrium interest rates in a stard representative agent term structure model. JEL Codes: D0, D, D50, D90.

2 Introduction In the years following publication of Strotz s 956) classic paper, the question of changing tastes was widely studied. Recently interest has been rekindled by behavioral studies showing that certain changing tastes models actually do a better job of predicting decisions than non-changing tastes models. The changing tastes decision problem can most simply be framed in a three period certainty setting with a single consumption good c t t =,, 3) in each period t. Assume preferences in period one are defined over c, c, c 3 ) triples represented by U ) in period two over c, c 3 ) pairs represented by U ), where for a fixed c = c, U ) c, c, c 3 ) U ) c, c 3 ) differ by more than an increasing monotonic transform. To determine an optimal plan, the consumer can follow naive choice by using U ) to make the period one consumption decision then in period two given remaining resources, use U ) to make the allocation between c c 3. Alternatively, she could follow sophisticated choice solve the problem recursively using U ) to make the allocation between c c 3 conditional on c then use U ) to select c. In general, there is no reason to suppose that the resulting naive sophisticated consumption plans should agree, so the consumer confronts the problem of which process to follow? 3 However, as Pollak 968) observed, there is no conflict in the very special case where the changing tastes U ) U ) both take the form of additively separable logarithmic utility with arbitrary discounting). Although the consumer changes her plans with the passage of time, the naive sophisticated plans always agree. Donaldson Selden 98) showed that for these preferences, the common consumption plan can be rationalized by a non-changing tastes utility Û. We refer to We thank Bob Pollak, John Donaldson, Herakles Polemarchakis especially Yakar Kannai for insightful comments suggestions. We thank the Sol Snider Research Center Wharton for support. See, for example, Pollak 968), Phelps Pollak 968), Peleg Yaari 973), Blackorby, et al. 973, 978), Hammond 976). As in each of these papers except the last, when referring to changing tastes we only consider the case of exogenous taste changes. The notion of endogenous taste changes typically associated with habit formation models is not considered. See Yaari 977) for a philosophical discussion of the differences between these models. See, for example, Ainslie 99), Laibson 997) Frederick, et al. 00). See Mulligan 996) for a critique. 3 Given that U ) U ) disagree over the ranking of c, c 3 ) pairs, it was argued by Phelps Pollak 968) Peleg Yaari 973) that one should think of the problem as being equivalent to a game between two divergent individuals, myself today myself tomorrow. Harris Laibson 003) assume sophisticated quasi-hyperbolic consumers. Caplin Leahy 006) argue that the sophisticated approach is preferable to the game theoretic models.

3 this common plan as being effectively consistent, since when obtained by maximizing Û, rather than U ) U ), the plan will not be revised over time. 4 This result seems to contradict the general view that there exists an intertemporal utility which rationalizes sophisticated choice only if preferences take the strongly recursive form U ) c, c, c 3 ) = U c, U ) c, c 3 ) ). 5 In this paper we show that the existence of an effectively consistent plan does not require preferences to be logarithmic, additively separable or homothetic. Suffi cient conditions are given for when naive sophisticated choice agree the common plan can be rationalized by a non-changing tastes Û. for constructing Û from the assumed changing tastes U ) U ). Specific formulas are derived Concrete nonadditive examples are provided which are of particular interest due to the widely held aversion to assuming that intertemporal utility is additively separable. 6 The intuition for why this enlarged class of preferences results in effectively consistent plans seems to be closely related to the optimal consumption plans generated by U ) being partially independent of prices. We demonstrate that for a popular form of the quasi-hyperbolic discounted utility model of changing tastes, 7 the optimal consumption plan is effectively consistent. Although the resulting quasi-hyperbolic nor exponential in form. Û is an additively separable discounted utility, it is neither Û is shown to discount the current period more heavily than the exponential case but not as strongly as quasi-hyperbolic utility. The assumption of changing tastes poses a number of complications for equilibrium analysis especially for equilibria associated with sophisticated dem. 8 Assuming a 4 In the special case where preferences take the quasi-hyperbolic discounted form, it has been noted that in a consumption-savings setting, the resulting optimal savings function may be "observationally equivalent" to the savings function obtained from exponential discounted utility Laibson 997; Morris Postewaite 997; Barro 999). This notion should not be confused with effective consistency where the optimal savings function corresponding to changing tastes is identical to that obtained from the non-changing tastes Û. 5 See Remark 4 below. 6 Indeed there is a long tradition in economic theory challenging time additive intertemporal utility Fisher 930; Hicks 965). Lucas Stokey 984) have argued that the only justification for studying this class of preferences is its analytic tractability. Koopmans 960) provided an alternative approach with his recursive preference structure. Finally, the extensive endogenous taste change) habit formation literature often justifies its modelling on the a priori unattractiveness of additively separable preferences. 7 See, for instance, [9] [0]. 8 Herings Rohde 006) propose specific modifications of the classic general equilibrium Pareto Optimality notions to accommodate changing tastes. 3

4 stard representative agent equilibrium interest rate model, if the optimal consumption plan is effectively consistent then the first order conditions based on Û can be directly used to obtain the equilibrium interest rates. 9 Moreover, the corresponding equilibrium comparative statics are greatly simplified. These points are illustrated for two different changing tastes models quasi-hyperbolic preferences CES constant elasticity of substitution) preferences with changing intertemporal elasticities of substitution. The resulting equilibrium interest rates for these two cases are seen to respond quite differently to changes in the supply of bonds. The rest of the paper is organized as follows. In the next Section after introducing notation definitions, we derive our new results for the existence of effectively consistent plans the construction of Û, including extensions to the case of a finite number of periods. Section 3 considers the special case of quasi-hyperbolic discounted utilities. In Section 4, we examine the implications of effective consistency for the equilibrium term structure of interest rates. The last Section contains concluding comments. Proofs are provided in the Appendix. Effective Consistency. Preliminaries: Changing Tastes Assume a single consumption good, certainty setting in which a consumer is endowed with income or initial wealth of y which she seeks to allocate over time periods t =,, 3. 0 Let c t p t denote, respectively, consumption present value) price in period t. In period one, the consumer determines optimal c a plan for c, c 3 ). In period two, she decides whether or not to revise her planned consumption for periods two three given her remaining income y p c. Time preferences for periods one 9 It is interesting to observe that the log additive special case of Pollak plays a special role in the term structure analysis of Kocherlakota 00): "Is there any equilibrium at all...? I do not know the answer to this question for general specifications of the [period utility]. In the Appendix, though, I construct an equilibrium for log utility." Kocherlakota 00, p. 8) Kocherlakota employs the sophisticated choice solution technique without seemingly being aware that his representative agent s plan is effectively consistent for the log additive case can be rationalized by a Û. 0 The assumption of three periods is made for simplicity where appropriate, results are extended to T > 3 periods. 4

5 two are represented respectively by U ) c, c, c 3 ) : C C C 3 R ) U ) c, c 3 ) : C C 3 R, ) where C t denotes the set of possible consumption values in period t, which is a subset of) the positive orthant. Both U ) U ) are assumed to satisfy the following Property. Property The utility U is a real-valued function defined on a subset of) the positive orthant of a Euclidean space, which is C, strictly increasing in each of its arguments strictly quasiconcave. At the heart of time inconsistency is the notion of changing tastes. Definition An agent s tastes will be said to have changed if only if for any increasing transformation T, U ) c, c 3 ) T U ) c, c, c 3 )), 3) where c C denotes any fixed value of period one consumption. It is clear from this definition that whether or not preferences change is the absence or presence of a very special nesting of U ) in U ). Proposition Blackorby, et al. 973) Given preferences corresponding to U ) U ), the necessary suffi cient condition for tastes not to change in the sense of Definition is that they satisfy U ) c, c, c 3 ) = Uc, U ) c, c 3 )). 4) To define consistent choice or planning, suppose the consumer faces the following two optimization problems: P : max U ) c, c, c 3 ) S.T. y = p c + p c + c 3 5) c,c,c 3 Although here the investment element of a consumption plan is ignored, in Section 4 we modify the budget constraints to allow for consideration of the equilibrium term structure of interest rates. 5

6 P : maxu ) c, c 3 ) S.T. y p c = p c + c 3. 6) c,c 3 Let c = c, c, c 3) denote the optimal three period consumption plan for P. Applying the terminology from Machina 989) McClennen 990), the c plan is said to be resolute if only if the consumer does not modify her c, c 3) plan even if her tastes change. The naive sophisticated choice models for solving these two problems, where no assumption is being made about whether or not preferences change, are defined as follows. Definition P P are said to be solved by naive choice if P is solved for optimal c = c then optimal c c 3 are solved via P conditional on c. Definition 3 P P are said to be solved by sophisticated choice if P is solved for conditionally optimal c c ) c 3 c ) then optimal c is determined from solving P conditional on c c ) c 3 c ). The vectors c = c, c, c 3) c = c, c, c 3 ) denote respectively the solutions resulting from the naive sophisticated choice procedures. A time consistent plan is defined as follows. Definition 4 A consumption plan c, c 3) which optimizes P is said to be consistent if only if c, c 3) = c, c 3). Together Definitions 4 imply that in a certainty setting, a consumption plan will be consistent if only if tastes do not change. Proposition Blackorby, et al. 973) Assume a consumer confronts choice problems P P. Then her consumption plan will be consistent if only if U ) U ) satisfy eqn. 4). In stard intertemporal choice problems only U ) is specified since the continuation of U ) can be viewed as the period two utility U ), tastes do not change. As pointed out by Peleg Yaari 973), the sophisticated choice process need not always generate an optimal plan. This problem arises when substitution of the P solution into the P optimization results in U ) not being concave in c. Consistent with Peleg Yaari 973), footnote, it follows from Blackorby, et al. 978) that a suffi cient condition for a sophisticated solution to exist is that U ) is homothetic. 6

7 . Definition Motivating Example Based on the definitions in the prior Subsection, if the consumption plan is consistent, then c = c. However, it follows from the changing tastes examples in Pollak 968) Donaldson Selden 98), where U ) U ) are log additive but with different discount functions, that the consistency of a consumption plan is suffi cient but not necessary for i) c = c ii) the existence of a non-changing tastes utility which rationalizes the common plan. 3 Consider the following definition. Definition 5 Given U ), U )), a plan is said to be eff ectively consistent if it corresponds to c can be rationalized by a non-changing tastes Property. Otherwise, the plan is effectively inconsistent. 4 Û, which satisfies The reason for referring to c as being effectively consistent is that if a non-changing tastes Û exists, the agent will never revise her period one plan based on Û in period two. Remark Although we don t require c = c in this definition, we conjecture that if sophisticated choice can be rationalized, we always have c = c. Indeed for each of the suffi cient conditions derived below as well as all of the Examples considered, this equivalence will be seen to hold. If the plan is consistent, c can be rationalized by U ) hence the plan is also effectively consistent. If the plan is inconsistent, we may still be able to rationalize c implying that the plan is effectively consistent as illustrated below. The following example shows that the existence of a Û agreement of naive sophisticated plans holds for more general forms of utility than ordinally) additively separable logarithmic utilities. In addition to the period one two utilities not needing to take the additive logarithmic form, the period one utility need not be separable or homothetic. 3 It should be noted that although Pollak 968) realized the consistency of a consumption plan is not necessary for c = c, he never discussed the existence of a non-changing tastes utility which rationalizes the common plan. 4 It should be noted that our notion of effectively consistent plans is very different from Hammond s 976) concept of essentially consistent preferences. Hammond in effect argues that consistency essential consistency are almost equivalent when he states "It seems that an essential inconsistency is almost certain to occur unless... the dynamic utility function... [is]... fully consistent." Hammond 976, p. 7). 7

8 Example Assume the following period one two utilities U ) c, c 3 ) = c δ δ where both satisfy Property. order conditions for P that U ) c, c, c 3 ) = c c ) 4 + c c 3, 7) c δ 3, δ >, δ 0, 8) δ Considering naive choice first, it follows from the first c = c = y p c c + 4 p c ) 3 c = p p, 9) implying that c c 3 via the budget constraint) are nonlinear in income. Solving P, one obtains c = y ) ) c p p 3 = +δ + p 3 ) +δ p y ) p +δ p + p 3 ), 0) which are linear in income. inconsistent. Therefore, we have c c the consumption plan is Next to apply the sophisticated choice strategy, solve P resulting in the period two conditional dems c c ) = y p c ) p +δ p + p 3 c 3 c ) = ) p +δ y p c ). ) ) p +δ p + p 3 Maximizing U ) c, c c ), c 3 c )) = with respect to c yields y p c ) c ) p p + p 3 +δ 4 + ) p +δ y p c ) c ) p +δ p + p 3 ) c = y p, c = y ) ) c p p 3 = +δ + p 3 ) +δ p y ) p +δ p + p 3 ). 3) Thus, even though the consumption plan is inconsistent, we always have c = c. Moreover, it can be verified that the common naive sophisticated dem functions 8

9 can be rationalized by the following non-changing tastes three period utility function 5 which satisfies Property. ) Û c, c, c 3 ) = c c δ + c δ δ 3, 4).3 Suffi cient Conditions Building on Example, we next show when in the presence of changing tastes c c agree can be rationalized by a non-changing tastes Û. prove convenient to introduce the following definition. First, however, it will Definition 6 For any homothetic utility function U, denote by L U the monotone increasing transformation of U which results in L U U being homogeneous of degree. Proposition 3 Given the decision problems defined by P P, assume where f g satisfy Property g > 0. U ) c, c, c 3 ) = f g c ) c, g c ) c 3 ), 5) be rationalized by Û if only if U ) is homothetic. Then c = c, the common plan can Û is given by Û c, c, c 3 ) = g c ) u ) c, c 3 ), 6) where u ) = L U ) U ) Û satisfies Property. At first glance it may appear that U ) in Proposition 3 does not include the ordinally equivalent additive logarithmic Cobb-Douglas utilities assumed in Pollak 968) Donaldson Selden 98), respectively, as special cases. However if one defines α α g c ) = c +α 3 f x, y) = x α y α 3, then it follows from Proposition 3 that If we further assume that U ) c, c, c 3 ) = c α c α c α ) U ) c, c 3 ) = c β c β 3 3, 8) where β + β 3 =, implying that u ) = U ), then Û is given by Û c, c, c 3 ) = c α α +α 3 c β c β 3 3, 9) 5 A simpler example which illustrates that U ) need not be logarithmic or homothetic for the consumption plan to be effectively consistency can be constructed for the case where U ) c, c, c 3 ) = exp c ) + ln c c 3 U ) is homothetic. 9

10 which is ordinally equivalent to the Û given in Donaldson Selden 98). It should be noted that in Proposition 3, when assuming U ) is homothetic, it is not necessary to also assume additive separability as demonstrated by the following Example. Example Assume the following period one two utilities U ) c, c 3 ) = U ) c, c, c 3 ) = ln c + ln c + ln c 3, 0) ) c δ + c δ β δ 3 β c β 3, δ, β >, δ, β 0, ) β where both satisfy Property. It can be easily seen that L U ) U ) = βu )) β, implying Therefore from Proposition 3, u ) c, c 3 ) = c ) δ + c δ β δ 3 Û c, c, c 3 ) = ln c c β ln ) δ + c δ β δ 3 ) + c β β 3. ) ) + c β 3. 3) Remark Assuming U ) takes the form 0) U ) takes the CES form 8), one has the very natural changing tastes interpretation that the consumer s intertemporal elasticity of substitution between c c 3 changes between periods one two. 6 equilibrium interest rate implications of this form of changing tastes will be analyzed in Subsection 4., Example 6, compared with those of the quasi-hyperbolic case. We next provide another suffi cient condition for the existence of an effectively consistent plan, which demonstrates that the homotheticity of U ) in Proposition 3 is not required if U ) takes a different form than 5). Proposition 4 Given the decision problems defined by P P, assume The U ) c, c, c 3 ) = f ) c ) + g ) c ) + c 3, 4) 6 It is well-known that for the CES utility 8), the elasticity of substitution equals +δ. The log additive form 0) can be viewed as the special CES case where δ 0, with an elasticity of substitution of. Thus by varying δ away from 0 either in the negative or positive direction we can control whether the consumer s elasticity of substitution increases or decreases over time. 0

11 where f ) g ) satisfy Property. rationalized by Û if only if Then c = c, the common plan can be U ) c, c 3 ) = g ) c ) + c 3, 5) where g ) satisfies Property. Û is given by Û c, c, c 3 ) = f ) c ) + g ) c ) + c 3 6) satisfies Property. 7 Remark 3 Although the suffi cient conditions for effective consistency provided by Propositions 3 4 are mutually exclusive 8, both require period one dem to be rationalizable by at least two different utility functions. In general this is necessary for the plan to be effectively consistent but not consistent. That is, given U ) a resolute) optimal consumption vector c, c, c 3), one needs to find another vector c, c, c 3 ) with the same period one dem function but different period two three dems that is generated by a Û. Since there is total freedom to choose the form of the c dem function with the optimal c 3 being determined from the budget constraint), one might think that it would not be diffi cult to find such a dem function utility. However for a Û to exist, the requirement that the Slutsky matrix of the new dem system be symmetric seems quite diffi cult to satisfy. Thus it remains an open question whether in the presence of changing tastes, the U ) forms in Propositions 3 4 together are also necessary for the existence of an effectively consistent plan. Given that from Propositions 3 4, neither additive separability nor homotheticity of U ) U ) are necessary for the existence of effective consistency, what then is required? The following Proposition provides a partial answer to this question for the case where U ) is homothetic. Proposition 5 Given the decision problems P P, if U ) is homothetic, then the sophisticated solution c can be rationalized only if it satisfies c 3 c p = c c. 7) 7 It should be noted that it is not necessary for U ) to be quasilinear in c 3. If U ) is quasilinear in c, then the optimal plan is effectively consistent when U ) is quasilinear in c. 8 To see this, note that if U ) c, c 3 ) is quasilinear in c 3 as in Proposition 4, it cannot also be homothetic as assumed in Proposition 3 otherwise by Euler s Theorem, U ) c, c 3 ) would be linear in both c c 3, violating the Property assumption of strict quasiconcavity).

12 As can be seen from the proof, although much simpler in form, 7) is fully equivalent to the stard Slutsky symmetry condition required for the existence of a utility function rationalizing sophisticated choice when U ) is homothetic). We can consider two cases satisfying 7). The first is where U ) is nested in U ). In this instance since the optimal plan is consistent, the Slutsky symmetry condition is satisfied implying that 7) holds. Because demonstrating the latter implication directly is not obvious, we sketch out the argument in Appendix D.) The second case where 7) holds is when the sophisticated period one dem is independent of period two three prices c p = c = 0. 8) This condition corresponds to the stard definition of a myopic plan Kurz 987, p. 579). 9 Definition 7 Given the pair U ), U )) the budget constraint y = 3 p t c t, 9) t= optimal period one consumption, c, c or c, is said to be myopic if only if it is independent of p. By directly applying a result from Kannai, Selden Wei 0), it is possible to characterize the necessary suffi cient restrictions on U ) for period one resolute consumption to be myopic in the sense of Definition 7. Proposition 6 Assume the decision problems are defined by P P. myopic if only if U ) takes the following form Then c is where f g satisfy Property g > 0. 0 U ) c, c, c 3 ) = f g c ) c, g c ) c 3 ), 30) 9 Although Strotz 956) Hammond 976), among others, use the terms myopic naive planning interchangeably, we distinguish these notions using Definition 7. 0 In order to simplify the statement of this result, we assume without loss of generality that g > 0. However it should be noted that for suffi ciency, f, g satisfy Property g > 0 cannot guarantee that U ) satisfies Property, which is always assumed in this paper, since the strict quasiconcavity of f g cannot ensure the strict quasiconcavity of U ). For necessity if U ) satisfies Property g > 0, we can show that this implies f, g satisfy Property. It should be emphasized that when g < 0, one can always reverse the sign of g the signs of the arguments in f such that the form of U ) remains the same f, g > 0 satisfy Property.

13 Figure : It follows immediately from Propositions 3 6 that the plan will be effectively consistent if resolute naive) period one dem is myopic in the sense of Definition 7 period two preferences are homothetic. Moreover, in this case, since u ) is homogeneous of degree one, eqn. 6) can be written as Ûc, c, c 3 ) = g c ) u ) c, c 3 ) = u ) gc )c, gc )c 3 ), 3) implying that the optimal period one consumption solved from Û is also myopic. The geometric intuition gleamed from c being myopic is illustrated in Figure. Assume the conditions in Proposition 3 hold. Consider the two unshaded budget planes characterized by the same y p, but different prices p. The budget lines AB CD are drawn, respectively, on the upper lower planes. Given that U ) takes the form of 5), it follows from Proposition 6 that c = c is myopic implying that c is independent of p. Thus U ) determines a vertical shaded plane corresponding to c = c, which intersects the two budget planes. utility U ) defines a set of indifference curves on the vertical plane. The period two Tangent points on AB CD correspond to the naive solutions for the two budget planes. It can be also easily verified that when U ) takes the quasilinear form, the optimal naive period one dem is not myopic. If the plan is consistent, then U ) generates the same set of indifference curves as U ) on each budget plane. Otherwise, U ) produces another set of indifference curves, where the different tangent On 3

14 the other h if the consumer follows sophisticated choice, then on each vertical c plane there exist tangent points on the respective budget lines. Given that c equals c, it also is myopic, U ) determines the same shaded vertical plane corresponding to c = c = c independent of how the budget plane shifts with changing p. Given that there exists a Û which rationalizes sophisticated or naive) choice, Û generates the same set of indifference curves as U ) on each c plane selects the same c = c = c vertical plane as U ). 3 Propositions 3 4 can be easily generalized to the case with T > 3) periods. 4 For Proposition 4, one can simply assume that U i) i =,,..., T ) is additively separable quasilinear in c T. Then the optimal plan is effectively consistent Û is also quasilinear in c T. is summarized in the following Proposition. The extension of Proposition 3 is a little more complicated Proposition 7 Assume the following utility functions in periods to T U ) c, c,..., c T ) = f ) g c ) c, g c ) c 3,..., g c ) c T ), 3) ) U i) c i,..., c T ) = f i) ci αi) c i+, c αi) i c i+,..., c αi) i c T i =,..., T ), 33) where f i) i =,..., T ) g satisfy Property, g > 0, α i) > 0 i =,..., T ) U T ) is homothetic. The budget constraint is given by T p t c t = y. 34) t= Then naive sophisticated choice always coincide the common plan is rationalized by the intertemporal utility Û which satisfies Property.5 The method for constructing Û can be found in the proof of this Proposition.) points on AB CD correspond to the resolute solution for the two budget constraints. 3 It should be noted that if U ) U ) are quasilinear in c 3, then the optimal c = c ) does not stay on the same vertical c plane when shifting the budget plane with changing p since c c ) is not myopic. 4 In the following Proposition, we assume the natural extension of the three period optimization setup 5)-6) to T periods assume that Property is satisfied by the utility functions in each period. 5 Although g c ) in most of our examples has the form g c ) = c α α > 0), it is allowed to take a more general form such as g c ) = exp exp c )) as can be seen by rewriting U ) in Example 4 below. 4

15 Remark 4 Blackorby, et al. 973) state that "an intertemporal utility function which generates the dem functions of a sophisticated society exists if only if the society preferences are strongly recursive with a consistent representation" Blackorby, et al. 973, Theorem 6, p. 47). 6 The suffi ciency part is not surprising since if preferences are strongly recursive i.e., satisfy eqn. 4) in Proposition above) with a consistent representation, then sophisticated choice can be rationalized by a Û = U ). On the other h, the necessity part seems to suggest that when sophisticated choice can be rationalized, preferences must be consistent. But this is contradicted by Propositions 3, 4 7 Examples. Based on the results in this Subsection, we have the following conjecture. Conjecture Naive sophisticated) choice can be rationalized by an intertemporal utility function Û N Û S ) if only if the naive sophisticated solutions coincide. In this case, both solutions can be rationalized by the same utility function Û = Û N = Û S. 7 3 Quasi-hyperbolic Discounted Utility In this Section, we discuss the implications of effective consistency for the quasihyperbolic discounted utility model first introduced by Phelps Pollak 968). Assume the period one utility function takes the following form U ) c, c, c 3 ) = 3 D t) u c t ), 35) t= 6 A similar assertion can be also found in Blackorby, et al. 978, Theorem 0.5), where they introduce additional assumptions but seem to reach essentially the same conclusion. 7 Donaldson Selden 98) prove that if U ) U ) are homothetic the distribution of income between periods one two is price aggregate income independent, then the naive solution the sophisticated solutions can be generated by Û N Û S, respectively. They comment in their Remark 3 that this conclusion doesn t ensure that the naive solution the sophisticated solution will give the same dem behavior. To the contrary, Conjecture states that one can never find a Û N that isn t also a Û S vice versa. 5

16 where D t) is the discount function the ratio D t) /D t ) is the discount factor. 8 The period two utility U ) exhibits the same discount pattern U ) c, c 3 ) = 3 D t ) u c t ). 36) t= Following Strotz 956), the plan is consistent if only if the discount function is exponential D t) = γ t. However, empirical studies suggest that the decision making behavior of individuals is not compatible with exponential discounted utility, as they tend to overweight the current time period relative to future periods. 9 This has led to the development of quasi-hyperbolic discounted utilities which in the simple T = 3 case take the following form U ) c, c, c 3 ) = uc ) + β 3 γ t u c t ), 37) t= U ) c, c 3 ) = uc ) + u c 3 ), 38) where 0 < β, γ. The discount function Dt) = if t = Dt) = t if t > the discount factor between periods one two is between periods two three γ). Clearly eqn. 37) converges to the exponential discounted form when β =. When β, U ) cannot be nested in U ), implying that 37) 38) exhibit changing tastes. The economic implications of the quasi-hyperbolic discounted form have been studied extensively Laibson 997 Diamond Koszegi 003). In the following Example, we consider the case where quasi-hyperbolic utility results in the optimal plan being effectively consistent. Example 3 Assume the following period one two quasi-hyperbolic discounted utilities U ) c, c, c 3 ) = ln c + ln c + ln c 3, 39) U ) c, c 3 ) = ln c + ln c 3, 40) 8 As is stard, the discount rate ρ t t > ) is defined by D t) = ) t ) t ρ t =. + ρt D t) 9 See, for instance, the extensive survey of Frederick, Loewenstein O Donoghue 00). 6

17 where 0 < β, γ <. Given that 39) 40) exhibit changing tastes, it is straightforward to show that the resolute naive consumption plans for periods two three diverge. However since U ) U ) satisfy the conditions in Proposition 3 for a plan to be effectively consistent, naive sophisticated choice agree. As a result, the common solution can be rationalized by a discounted additive logarithmic Û. note that applying Proposition 3 + g c ) = c u ) c, c 3 ) = c c 3 To see this, ) +, 4) implying which is ordinally equivalent to Û c, c, c 3 ) = c + c c 3 ) +, 4) Û c, c, c 3 ) = ln c + Remark 5 The existence of + + ln c + + ) ln c 3. 43) + Û in this example seems to present a paradox. On the one h the representation 43) does not exhibit changing tastes hence its resulting dems are consistent, but on the other h the form of the discount function is not exponential. The source of this paradox is the fact that one can either maintain the same pattern of discounting over time or maintain the same absolute discount functions in subsequent time periods. 30 To illustrate this distinction, note that the quasi-hyperbolic utilities U ) U ), 39) 40), preserve exactly the same discounting pattern between periods. relative to future periods. form In this case, the consumer always overweights the current period Alternatively were the period two utility instead to take the U ) c, c 3 ) = ln c + ln c 3, 44) then the same period two three discount functions would apply for U ) U ). Since U ) would then be nested in U ), the optimal plan would be consistent. However in this case, the consumer only overweights the current period relative to future periods in period one not in subsequent periods. When there exists a functions carryover from period to period since one can take U ) = Û U ) c, c, c 3 ) = ln c + Û, the discount + + ln c + + ) ln c 3 45) + 30 See Rasmusen 008) for a thoughtful discussion of the general distinction between maintaining the same discount pattern or absolute discount functions. 7

18 U ) to be the continuation of Û U ) c, c 3 ) = + + ln c + + ) ln c 3, 46) + implying that the consumer necessarily changes her discounting pattern with the passage of time. It is demonstrated in Appendix H, Part I, that if U ) U ) in Example 3 take the more general non-log) CES quasi-hyperbolic discounted form, although U ) homothetic, U ) does not satisfy the conditions in Proposition 3 c there exists no Û. is c We next compare the behavior of discount functions over time corresponding to the quasi-hyperbolic rationalized Û utility models. discussion to T periods. But first, we generalize the above Proposition 8 Assume in period one the following T period quasi-hyperbolic discounted utility U ) c, c,..., c T ) = ln c + β T γ t ln c t, 47) where 0 < β, γ in each future period utility preserves the same discounting pattern as U ). Then c,..., c T ) = c,..., c ) the common solution can be rationalized by t= Û c, c,..., c T ) = ln c + where the discount function α t for t =,..., T is given by T T α t ln c t, 48) t= α t = i= t T i β t i= j= γj t + β T i ) j= γj < γ t. 49) For the exponential case U ) c, c,..., c T ) = T γ t ln c t, 50) t= it follows from 47) that the quasi-hyperbolic discount function for each t =,...T, will always be smaller than the exponential discount function γ t so long as β <. Eqn. 8

19 Discounted Value Exponential Discount Rationalized Discount Quasi hyperbolic Discount Time a) Figure : 49) implies that for each period t {,..., T }, the Û discount function α t will also be less than the exponential discount function. But when t, the relationship between α t t is ambiguous in general. In Figures, we plot the value of the discount functions for the exponential, quasihyperbolic Û cases where γ = 0.95 β = 0.6. The utility Û smooths the discount functions for the quasi-hyperbolic model with the value of the Û discount functions being higher in the earlier years smaller in the later years. Next consider limit cases. If β, the discount functions for the exponential, quasi-hyperbolic rationalized discounted utilities all converge to the exponential curve. If γ, the value of the discount function for the exponential discounting model would be reflected in Figure by a horizontal line at. Moreover, the value of the discount function for the quasi-hyperbolic model would always be β except for period one implying in Figure, a sharp drop followed by a flat segment. In this case, the smoothed time pattern of the discount function for Û given by eqn. 49) would simplify to α t = β t t T i) i=. 5) t β + β T i)) i= 9

20 4 Equilibrium Bond Returns: Changing Tastes In this Section, the implications of effective consistency for the term structure of interest rates are examined using a simple representative agent equilibrium model. We follow the general set-up in Kocherlakota 00) assume, without loss of generality, that there are three time periods General Setting Assume a representative agent economy where U ) c, c, c 3 ) U ) c, c 3 ) exhibit changing tastes. The utilities satisfy Property. In period one, let y, b b 3 denote, respectively, income or wealth), units of a one period bond that pays off one unit of consumption at the beginning of period two units of a two period bond paying off one unit of consumption at the beginning of period three but can be retraded in period two. Assume period one consumption is the numeraire income is in units of c. Let q q 3 denote respectively the prices at the beginning of period one of the one two period bonds. Denote by + r, + r + f, respectively, the one period, two period implied forward gross interest rates, 3 where q = + r q 3 = + r ) = + r ) + f ). 5) It follows that c = q b + r ) = b 53) c 3 = q 3 b 3 + r ) = b 3. 54) In period two, the one period bond matures. Let b 3 q 3 denote, respectively, the period two dem price for the two period bond with one period of remaining maturity. The corresponding period two spot rate r for the bond can be derived from q 3 = + r. 55) 3 Although our setting differs from that of Kocherlakota 00) in not introducing a commitment asset, the impact of such an asset is discussed below in footnote For interest rates, the pre-subscript ) indicates that the bond implied forward contract) is held beginning in period one two). The post subscript indicates the number of periods for which the interest rate applies. The use here of f to denote the forward interest rate should not be confused with the use of f in Propositions

21 It follows that c 3 = q 3 b 3 + r ) = b 3. 56) Let b b 3 denote the exogenous supply of bonds maturing at the beginning of period two three, respectively. 33 We can now convert the individual s decision problems P P, 5) - 6), into the representative agent s decision problems which will be used to determine equilibrium prices Q : max U ) c, c, c 3 ) S.T. y + q b + q 3 b 3 = c + q b + q 3 b 3, 57) c,b,b 3 Q : S.T. c = b, 58) S.T. c 3 = b 3 59) max c,b 3 U ) c, c 3 ) S.T. W = c + q 3 b 3, 60) S.T. c 3 = b 3, 6) where in 60) period two income or wealth) W equals the period two value of bonds b + q 3 b 3 bought in period one, but valued in period two). The budget constraint for Q is a natural extension of that used in P. The budget constraint implied by P can be written as y + q b + q 3 b 3 c = c + q 3 c 3 + r, 6) where the left h side is the unconsumed wealth at the end of period one the right h side is the present value of future consumption. However this constraint cannot be used for Q, since in the presence of changing tastes the price of the two period bond in period two q 3 can diverge from that based on the period one implied forward rate, i.e., q 3 q 3 /q. the bonds in period two based on q 3 in the Q constraint. This requires us instead to use the present value of Given the fixed initial wealth y supply of bonds b b 3, one can solve for the equilibrium prices q, q 3 q 3 assuming the representative agent follows either naive or sophisticated choice. We next define the corresponding equilibrium concepts 33 Here we assume that b, b 3 0. Such an assumption is not atypical. It could for instance be associated with the debt being issued by a government which is outside the model see Parlour, et al. 0 the literature cited therein). Moreover, the assumption of nonzero supplies of bonds could be dropped if we were to allow for endowments in the form of income in periods two three. This would change none of the conclusions, only making the notation more complicated.

22 following Herings Rohde 006). For a naive equilibrium, the representative agent first solves Q in period one determining q q 3 then in period two, solves Q determining q Definition 8 An equilibrium c, b, b 3, c, b 3, q, q 3, q 3 ) is a naive equilibrium if only if the equilibrium prices are solved from Q Q, respectively, by setting the period one resolute dems equal to the endowments y, b, b 3 ) the period two naive dems equal to b, b 3 ). To determine a sophisticated equilibrium, the representative agent first uses the first order condition corresponding to Q to find a relation between c b 3. In contrast to the naive equilibrium analysis, sophisticated equilibrium prices cannot be determined until both Q Q are solved. To solve Q, one follows three steps. First, the relationship between the period one two vectors b, b 3 ) = c, b 3 ) the period two first order conditions are used to obtain a relationship between b b 3. Second, this relationship is combined with the Q budget constraint to express b b 3 as functions of c. Third, since U ) can now be expressed as a function of just c, it is maximized to find optimal c. The resulting optimal sophisticated dem c is then used to find optimal c = b c 3 = b 3 = b 3. The resulting equilibrium is defined as follows. 35 Definition 9 An equilibrium c, b = c, b 3 = b 3, q, q 3, q 3 ) is a sophisticated equilibrium if only if the equilibrium prices are determined by solving Q Q setting the sophisticated dems c, c, c 3 ) equal to the endowments y, b, b 3 ). For a sophisticated equilibrium, the relation f = r is automatically embedded in the equilibrium without it, one cannot in general find a unique sophisticated 34 The notion of a naive equilibrium is essentially the same as that of a temporary equilibrium Grmont 008; Balasko 003). Strictly speaking, the naive equilibrium is a sequence of period one period two temporary equilibria. Alternatively, a sophisticated equilibrium is sometimes referred to as an intertemporal equilibrium. 35 This process is illustrated in Appendix H, Part II. It should be noted that although the representative agent sophisticated optimization problems in Laibson 997) may appear different from our Q Q, they are actually equivalent. Laibson allows for income y t in each period. But if we define y = y + y y + r r ), then the representative agent faces the equivalent budget constraint as in Q, y = c + c c + r + 3 +, similarly for Q r )+ f ).

23 equilibrium. 36 II. 37 See the non-log) CES quasi-hyperbolic Example in Appendix H, Part Remark 6 Given the existence of a nonchanging tastes Û which rationalizes the common naïve sophisticated plan, it is natural to wonder whether in a representative agent economy with identical consumers the common Û can be used for welfare analysis. In Pollak 976), the author considers a simple changing tastes multiperiod representative agent setting characterized by preferences in each period being represented by a simple linear habit formation utility. He shows that the representative consumer s short-run dem in each period depends on the prior period s consumption history. Under certain conditions, there exists a long-run utility that generates the same consumption in every period. In this set-up, the short-run utilities are analogous to our utility functions U ) U ) the long-run utility function is analogous to our Û. Pollak argues that because the long-run utility may order consumption streams differently from the short-run utilities, even though a long-run utility function exists, it cannot serve as a general welfare criterion. The long-run utility function, when it exists, does not reflect "long-run preferences," but is merely an indicator of long-run behavior Pollak 976, p. 97). It would seem that the same logic applies to the use of our nonchanging tastes Û. When a plan is effectively consistent but not consistent), one can use Û to rationalize sophisticated choice. However, Û will be different from U ) U ). Therefore, in general, Û will not reflect the preferences in each period hence it is not suitable for welfare analysis. 4. Effectively Consistent Preferences As discussed in the prior Subsection when the representative agent exhibits changing tastes corresponding to U ), U )), the resulting naive sophisticated equilibria, 36 Herings Rohde already embed this assumption in their definition of a sophisticated equilibrium Herings Rohde 006, p. 600). Kocherlakota 00) argues that q q 3 = q 3 f = r ) must hold as a result of dynamic arbitrage assuming continuous trading. 37 As noted in footnote 9 in Section, Kocherlakota leaves as an open question whether a sophisticated) equilibrium exists for period utilities other than the log case. It follows from Appendix H, Part II, that in our setting without a commitment asset a unique sophisticated equilibrium also exists when the period utilities take the more general CES form. 3

24 if they exist, will be different. effectively consistent. This conclusion also holds when the resulting plan is However if one uses Û instead of U ) U ), the naive sophisticated dems will be the same resulting in the same set of equilibrium prices or interest rates r, r, r ). Moreover, the sophisticated equilibrium prices can be obtained directly from the first order conditions based on Û. Proposition 9 Assume the representative agent s decision problems are characterized by Q Q the resulting consumption plan is effectively consistent, implying the existence of a Û. Then given the exogenous supplies y, b, b 3 ), there exists a unique sophisticated equilibrium, where q = Û c, c, c 3 ) c c,c,c 3 )=y,b,b 3) / Û c, c, c 3 ) c, 63) c,c,c 3 )=y,b,b 3) q 3 = Û c, c, c 3 ) c 3 c,c,c 3 )=y,b,b 3) / Û c, c, c 3 ) c c,c,c 3 )=y,b,b 3) 64) q 3 = q 3 = Û c, c, c 3 ) q c 3 c,c 3 )=b,b 3) / Û c, c, c 3 ) c. 65) c,c 3 )=b,b 3) As the following demonstrates, this result overcomes the problem that in general there may be no way to characterize sophisticated equilibrium prices analytically when tastes change. Example 4 Assume the representative agent s decision problems are characterized by Q Q, where U ) U ) are given by both utilities satisfy Property. U ) c, c, c 3 ) = 3 exp c ) + ln c + ln c 3, 66) U ) c, c 3 ) = c c 3, 67) Given that U ) cannot be nested in U ), the representative agent has changing tastes. To characterize the sophisticated equilibrium, consider deriving the representative agent s sophisticated dems. that c satisfies It can be verified c + exp c = y + q b + q 3 b 3. 68) 4

25 Because this equation does not admit derivation of closed form expressions for dems, it is not possible to equate dems with endowments solve for equilibrium prices. However, using Proposition 3 to derive Û Û c, c, c 3 ) = exp exp c )) c c 3, 69) Proposition 9 can be applied to obtain the following very simple expressions for sophisticated equilibrium prices = q = Û / Û + r c c c,c,c 3 )=y,b,b 3) c,c,c 3 )=y,b,b 3) + r ) = q 3 = Û / Û c 3 c c,c,c 3 )=y,b,b 3) c,c,c 3 )=y,b,b 3) = exp y b, 70) = exp y b 3 7) + r = q 3 = + r + r ) = b b 3. 7) Next we use Proposition 9 to derive the sophisticated equilibrium term structure analyze its comparative statics for two different models of changing tastes. Example 5 Assume the representative agent s decision problems are characterized by Q Q, with quasi-hyperbolic preferences defined by U ) c, c, c 3 ) = ln c + ln c + ln c 3 73) U ) c, c 3 ) = ln c + ln c 3, 74) where 0 < β, γ <. It directly follows from Proposition 3 that 38 Û c, c, c 3 ) = ln c ln c + + ) ln c 3. 75) + 38 It is common in applications of the quasi-hyperbolic discounted utility model to introduce a commitment asset, which is illiquid or non-tradeable Laibson 997; Kocherlakota 00). In the context of this three period example, suppose the two period bond is replaced by a two period commitment asset which must be held until maturity at the beginning of period three. Because the asset cannot be retraded, the resolute plan becomes the naive solution. However, it is clear that the sophisticated solution does not change continues to be rationalized by the same Û, 75). It should be noted that if one modifies P P to allow for the existence of a commitment asset, the Proposition 3 4 conclusion that there exists a Û which rationalizes sophisticated choice continues to hold although naive sophisticated choice no longer agree. 5

26 Applying Proposition 9, the equilibrium interest rates are given by 39 r = + ) b + ) y, r = To guarantee that r, f = r > 0, we require b y > ) b 3 + ) y f = r = b 3 b. 76) b 3 b >. 77) Moreover, whether the yield curve is upward sloping, constant or downward sloping can be determined from r r b y b 3 Defining the slope of the yield curve by we have k β, γ) γ k β, γ) β + + ). 78) kβ, γ) = def r r, 79) 0 b 3y b + + γ) + ) 80) + γ + ) + ) β)γ + ) + γ). 8) 0 b 3y b If one assumes that the period endowments grow at a constant rate α > 0, i.e., b 3 = + α) b = + α) y, 8) it follows from 77) that all the interest rates are positive. Noticing that + + ) b = y b 3 83) + = + β >, 84) γ it follows from 78) that the yield curve is always upward sloping, r > r. Moreover, we have b 3 y b = > + k β, γ) + γ) + ) β < 0 85) 39 Although our setting is similar to that of Kocherlakota 00), his inclusion of both a commitment asset a two period bond complicates undertaking an equilibrium term structure analysis as in this Example. 6

27 b 3 y b = > + γ + ) + ) β)γ + ) + γ) k β, γ) γ implying that the yield curve becomes flatter with increasing β or γ. 40 < 0, 86) Building on our observation in Remark that changing tastes can alternatively be modeled by assuming that the representative agent s intertemporal elasticity of substitution changes over time, we next derive the corresponding equilibrium term structure investigate its associated comparative statics. Example 6 Assume the representative agent s decision problems are characterized by Q Q with U ) U ) defined by 4 U ) c, c, c 3 ) = ln c + ln c + ln c 3 87) U ) c, c 3 ) = c δ δ It follows from Proposition 3 that c δ 3, δ >. 88) δ Û c, c, c 3 ) = ln c δ ln ) c δ + c δ 3. 89) Applying Proposition 9, the equilibrium interest rates are given by ) ) δ ) ) b b + δ b 3 b b b r =, r = f = r = y y b3 b ) +δ. 40 As a simple numerical example, assume an endowment growth rate + α =.05 preference parameters take on the typical values β = 0.6 γ = 0.99 Laibson 997, p. 450). Then the equilibrium interest rates are given by r = 0.38 r = 0.544, which seem unreasonably high for the moderate growth rate of.5% in aggregate consumption between the periods. More generally, this problem of very high interest rates occurs for the quasi-hyperbolic discounted CES utility even when δ 0 see Appendix H) no Û exists. This diffi culty can be overcome by assuming either β is suffi ciently close to or a different changing tastes model such as the one introduced in Example 6 below. 4 It should be noted that if δ in eqn. 88) goes to 0, this Example converges to the special case of Example 5 where β = γ =. 90) 7