Changing tastes and e⁄ective consistency

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Changing Tastes and E?ective Consistency

Larry Selden

Xiao Wei

April 22, 2015

Abstract

In a single commodity setting with changing tastes, an individual's consumption plan can be obtained using naive or sophisticated choice. We provide two su? cient conditions for when (i) the solutions are unique and agree and (ii) the common plan is representable by a non-changing tastes utility. Because the solution is not revised over time, the plan and associated preferences are referred to as being e?ectively consistent. Afriat-style revealed preference tests are derived. The assumption of e?ective consistency can mitigate the problems of vulnerability to Dutch Books, non-existence of a competitive equilibrium and the aggregation of heterogeneous agents with changing tastes. JEL Codes: D01, D11, D50, D90.

KEYWORDS. Naive choice, sophisticated choice, e?ective consistency, revealed preference, Dutch book, competitive equilibrium and aggregation.

Almost 60 years after the publication of Strotz's (1956) classic paper, there continues to be considerable interest in the question of changing tastes.1 Following the appearance of behavioural studies showing that changing taste models can do a better job of

Corresponding author: Larry Selden: Columbia University, Uris Hall, 3022 Broadway, New York, NY, USA. Email: larry@. We are indebted to the Editor and two Referees for their many valuable comments and suggestions. We thank the participants in our session at the 2014 North American Econometric Society Summer meeting in Minneapolis as well as Bob Pollak, Herakles Polemarchakis, John Donaldson and especially Felix Kubler and Yakar Kannai for many helpful comments and discussions. Support of the Sol Snider Research Center ?Wharton is gratefully acknowledged.

1See, for example, Pollak (1968), Phelps and Pollak (1968), Peleg and Yaari (1973), Blackorby, et al. (1973, 1978) and Hammond (1976). As in each of these papers except the last, when referring to changing tastes we only consider the case of exogenous taste changes.

predicting individuals'actions,2 a number of diverse applications and theoretical extensions have appeared.3 The changing tastes optimisation problem can most simply be framed in a three period certainty setting with a single consumption good ct (t = 1; 2; 3) in each period t. Assume preferences in period one are de...ned over (c1; c2; c3) triples and represented by U (1). Preferences in period two de...ned over (c2; c3) pairs, which can depend on c1, are represented by U (2). For a ...xed c1 = c1, U (1)(c1; c2; c3) and U (2)(c2; c3jc1) di?er by more than a strictly increasing transform. To determine an optimal plan, an individual can follow naive choice by using U (1) to make the period one consumption decision and then in period two given remaining resources, use U (2) to make the allocation between c2 and c3. Alternatively, she could follow sophisticated choice and solve the problem recursively using U (2) to make the allocation between c2 and c3 conditional on c1 and then use U (1) to select c1.

In general, there is no reason to suppose that the resulting naive and sophisticated consumption plans should agree, and so the consumer confronts the problem of which process to follow.4 However, as Pollak (1968) observed, there is no conict in the very special case where the changing tastes U (1) and U (2) 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 and sophisticated plans always agree. Donaldson and Selden (1981) showed that for these preferences, the common consumption plan can be rationalised by a non-changing tastes utility Ub. We refer to this common plan as being e?ectively consistent, since when obtained by maximising Ub, rather than U (1) and U (2), the plan will not be revised over time. This result seems to contradict the general view that there exists an intertemporal utility which rationalises sophisticated choice only if preferences take the strongly recursive form U (1)(c1; c2; c3) = U c1;U (2) (c2; c3jc1) .5

In this paper we show that the existence of an e?ectively consistent plan does not require preferences to be logarithmic, additively separable or homothetic. Two mutually exclusive su? cient conditions are given for when naive and sophisticated choice are

2See, for example, Ainslie (1992), Laibson (1997) and Frederick, et al. (2002). Mulligan (1996) provides an interesting critique.

3Examples of the former include Diamond and Koszegi (2003) and of the latter Luttmer and Mariotti (2006, 2007).

4Phelps and Pollak (1968) and Peleg and Yaari (1973) argue that one should think of the problem as being equivalent to a game between two divergent individuals, myself today and myself tomorrow. Harris and Laibson (2001) assume sophisticated quasi-hyperbolic consumers. Caplin and Leahy (2006) argue that the sophisticated approach is preferable to the game theoretic models.

5 See Remark 2 below.

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unique and agree and the common plan can be rationalised by a non-changing tastes Ub. Speci...c formulas are derived for constructing Ub from the assumed changing tastes U (1) and U (2). One of the su? cient conditions assumes that U (1) and U (2) take the myopic separable form introduced in Kannai, et al. (2014). This form of utility implies that the consumer exhibits a strong form of two stage budgeting where the choice among commodities in a group is based on within group prices and income (the expenditure on the group becomes independent of prices of goods not in the group).6 Concrete non-additive examples of e?ectively consistent preferences are provided which are of particular interest due to the widely held aversion to assuming that intertemporal utility is additively separable.7 The second su? cient condition requires the period utilities to take a quasilinear form. For our two forms of e?ectively consistent preferences, we derive revealed preference tests in the spirit of Afriat (1967) and Varian (1983) such that observed demand-price pairs are consistent with maximising Ub.8;9

We demonstrate that for a popular form of the quasi-hyperbolic discounted utility model of changing tastes,10 the optimal consumption plan is e?ectively consistent. Although the resulting Ub is an additively separable discounted utility, the discount function is neither quasi-hyperbolic nor exponential in form. Ub is shown to discount the current period more heavily than the exponential case but not as strongly as the quasi-hyperbolic utility.

While the hypothesis that an individual's future tastes can be di?erent from those currently assumed or perceived is both intuitively plausible and in some instances, such as the case of quasi-hyperbolic discounted utility, consistent with behavioural data, it nevertheless can pose a number of challenging problems for standard economic analyses. We consider three di?erent complications and show how the assumption of e?ective consistency can mitigate these challenges. First given the changing tastes U (1) and

6See Deaton and Muellbauer (1980, ch.5) for a discussion of the weaker form of two stage budgeting considered by Strotz (1957, 1959) and Gorman (1959).

7See Fisher (1930), Hicks (1965) and Lucas and Stokey (1984). 8Since, as emphasised by Kubler (2004) and discussed in Section 4 below, only spot demands and prices (and incomes) are observed over time in the form of a single "extended" observation, our tests which require more observations would need to be performed in a laboratory setting such as in Choi, et al. (2007). 9 One cannot determine whether a consumer's preferences correspond to U (1) and U (2) or Ub based solely on observed consumption demands, since they are the same. However if the consumption optimisation problem is reformulated as a consumption-bond optimisation where there are both one and two period bonds, then it is possible to distinguish the consumer's naive consumption and bond purchases from those based on Ub (see the online Appendix I). 10See, for instance, Phelps and Pollak (1968) and Laibson (1997).

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U (2), a consumer in general is vulnerable to a Dutch book or money pump sequence of trades. However if the consumer's preferences are e?ectively consistent, then in a market setting she cannot be manipulated to impoverish herself. This result can be viewed as an alternative to the requirement in Laibson and Yariv (2007) that both U (1) and U (2) must be time separable. A second complication is that in the absence of transitive intertemporal preferences, Gabrieli and Ghosal (2013) have shown that a representative agent competitive equilibrium can fail to exist. Luttmer and Mariotti (2006, 2007) avoid this problem by assuming the intertemporal utilities U (1) and U (2) are both additively separable. Alternatively for the case where a representative agent Ub exists, one can accommodate changing tastes without having to confront the possibility of the nonexistence of a competitive equilibrium.11 Moreover given the existence of a Ub, one can often signi...cantly simplify the characterisation of the equilibrium by using the ...rst order conditions based on Ub. Third in economies where tastes do not change, well-known conditions exist such that the aggregate demands for a collection of agents can be rationalised by a well-behaved utility function or aggregator.12 It is natural to ask whether the aggregate naive or sophisticated demands of consumers exhibiting changing tastes can be rationalised by an aggregator. We provide su? cient conditions such that e?ectively consistent preferences of the individual agents can be aggregated for both the myopically separable and quasilinear cases. Moreover, we provide explicit formulas for constructing the aggregator from the changing tastes U (1) and U (2) of the individual agents.

The rest of the paper is organised as follows. In the next section, notation and some preliminary de...nitions are given. Section 2 provides a motivating example. In Section 3, we derive two su? cient conditions for e?ectively consistent plans. Section 4 gives a revealed preference test for the myopic separable utility associated with e?ectively consistent preferences. Section 5 considers quasi-hyperbolic discounted utilities. In Section 6, we consider (i) the existence of Dutch Books or money pumps, (ii) naive and sophisticated equilibria and (iii) aggregation where consumers exhibit changing tastes. The last section contains concluding comments. Selected proofs are provided in the Appendix of this paper and the remaining proofs and supplemental materials are available in an online Appendix.

11Herings and Rohde (2006) propose speci...c modi...cations of the classic general equilibrium and Pareto Optimality notions to accommodate changing tastes.

12See the classic papers of Gorman (1953) and Chipman (1974) as well as the discussion of more contemporary work in Chipman (2006) and Chiappori and Ekeland (2011).

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1. Preliminaries: Changing Tastes

Assume a single consumption good, certainty setting in which a consumer is endowed with income or initial wealth of y1 which she seeks to allocate over time periods t = 1; 2; 3.13 Let ct and pt denote, respectively, consumption in period t and the present value price in period one of consumption in period t. Preferences for periods one and two are represented respectively by

U (1)(c1; c2; c3) : C1 C2 C3 ! R

and14

U (2)( c2; c3j c1) : C2 C3 ! R; 8c1 2 C1;

where Ct denotes the set of possible consumption values in period t, which is (a subset of) R+. Both U (1) and U (2) are assumed to satisfy the following property throughout this paper.

Property 1. The utility U is (i) a real-valued function de...ned on (a subset of) the positive orthant of a Euclidean space, (ii) C2, (iii) strictly increasing in each of its arguments and (iv) strictly quasiconcave.

At the heart of time inconsistency is the notion of changing tastes.

Definition 1. A consumer's tastes will be said to have changed if and only there exists a c1 2 C1 such that for every strictly increasing transformation T

U (2)( c2; c3j c1) 6= T (U (1)(c1; c2; c3)):

It is clear from this de...nition that whether or not preferences change is the absence or presence of a very special nesting of U (2) in U (1).

Proposition 1. (Blackorby, et al., 1973) Given preferences corresponding to U (1)(c1; c2; c3) and U (2)( c2; c3j c1), the necessary and su? cient condition for tastes not to change in the sense of De...nition 1 is that for any given c1 2 C1, there exists a strictly increasing transformation T such that

U (1)(c1; c2; c3) = T U (2)( c2; c3j c1) :

(1)

13The assumption of three periods is made for simplicity. The general T period case is discussed in

the online Appendix G. 14 Since U (2) can depend on c1 as a ...xed parameter, we use U (2)( c2; c3j c1) for the general case. For

situations where U (2) is independent of c1, U (2) (c2; c3) is used.

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To de...ne consistent choice or planning, suppose the consumer faces the following two optimisation problems:15

P1 : max U (1)(c1; c2; c3) S:T: y1 p1c1 + p2c2 + p3c3

(2)

c1;c2;c3

and

P2 : max U (2)( c2; c3j c1) S:T: y1 p1c1 p2c2 + p3c3:

(3)

c2;c3

Let c = (c1; c2; c3) denote the optimal three period consumption plan for P1. Applying terminology from Machina (1989) and McClennen (1990), the c plan is said to be

resolute if and only if the consumer does not modify her (c2; c3) plan even if her tastes change.

The naive and sophisticated choice models for solving these two problems, where

no assumption is being made about whether or not preferences change, are de...ned as

follows.

Definition 2. P1 and P2 are said to be solved by naive choice if P1 is solved for optimal c1 = c1 and then optimal c2 and c3 are solved via P2 conditional on c1.

Definition 3. P1 and P2 are said to be solved by sophisticated choice if P2 is solved for conditionally optimal c2 (c1) and c3 (c1) and then optimal c1 is determined from solving P1 conditional on c2 (c1) and c3 (c1).

The vectors c = (c1; c2; c3) and c = (c1 ; c2 ; c3 ) denote respectively the solutions resulting from the naive and sophisticated choice procedures.16 Given the assumptions on U (1) and U (2), whereas c will be unique c need not be (see Blackorby, et al., 1973, p. 245). A time consistent plan is de...ned as follows.

Definition 4. A consumption plan (c2; c3) which optimises P1 is said to be consistent if and only if (c2; c3) = (c2; c3) for any (p1; p2; p3; y1).

Together De...nitions 1 and 4 imply that in a certainty setting, a consumption plan will be consistent if and only if the U (1) and U (2) used to solve P1 and P2 are equivalent up to an increasing transform.

15Although here the investment element of a consumption plan is ignored, in Subsection 6.2 we modify the budget constraints to allow for the investment in one and two period bonds.

16As pointed out by Peleg and Yaari (1973), the sophisticated choice process need not always generate an optimal plan. This problem arises when substitution of the P2 solution into the P1 optimisation results in U (1) not being concave in c1. Consistent with Peleg and Yaari (1973, fn.1), it follows from Blackorby, et al. (1978) that a su? cient condition for a sophisticated solution to exist is that U (2) is homothetic.

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