Consumer Theory with Endowment Effect

Transcription

1 Consumer Theory with Endowment Effect Marek Hudík Abstract The paper incorporates the endowment effect into the traditional consumer theory. Following Bernardelli (1938, 195), it is shown that when preferences depend on the endowment, the law of diminishing marginal utility can reintroduced into the theory. The notion of loss aversion is newly formalized and the older definition of Kahneman and Tversky (1991) is criticized. Finally, it is argued that as a result of the existence of the endowment effect the traditional analysis need not be reected; it can be thought of as a special case of a more general approach. Keywords: consumer theory, endowment effect, loss aversion, diminishing marginal utility JEL: D01, D03, D11

2 The goal of this paper is to incorporate the influence of endowment into the consumer theory. It focuses on three points: first, it shows that the explicit account of endowment enables the law of diminishing marginal utility (LDMU) to be formulated while preserving ordinalism; it is also shown that the standard assumption of convexity of preferences in fact involves several notions that must be treated separately. Second, loss aversion is newly defined and the older definition of Kahneman and Tversky (1991) is criticized. Third, it is argued that endowment effect leaves the traditional analysis of consumer behaviour almost untouched. The paper proceeds as follows: The part 1 reviews relevant literature; Part briefly presents definitions, while part 3 introduces the assumptions of the theory. Part 4 discusses properties of the utility function and the relation of the traditional theory and the suggested reformulation thereof. It is shown that the former is a generalization of the latter. Part 5 concludes. 1. Literature Present paper draws upon two branches of literature; first of them deals with ordinalism and marginal utility following the Hicks-Allen reformulation of consumer theory. This includes Phelps Brown s (1934) insight that the change in utility induced by a change of consumption bundle is not the same thing as the utility of a change of consumption bundle; Samuelson (1938) then showed under what conditions are the two notions equivalent. In two important papers, Bernardelli (1938, 195) introduced utility function, which depends on initial position of a consumer; in this way he attempted to rehabilitate the concept of marginal utility, while preserving ordinalism. In this he failed due to some awkwardness in formalization (see the criticisms of Samuleson (1939) and Lancester (1953, 1954) and Bernardelli s (1939, 1954) replies). Second branch of literature is explicitly concerned with endowment effect and loss aversion. There seem to be only one attempt, namely that by Kahneman and Tversky (1991), to incorporate these biases into the standard consumer theory. Present paper differs from theirs in considering solely the effect of endowment and not of any reference point, independent of the consumer s endowment; it is also more concerned with the consequences for the traditional theory.. Definitions Since our approach is conceptually very similar to the traditional one, this section may be brief. We call the pair (e; x), where e is the initial position (endowment) and x is the end position, an action; we denote the set of all actions relative to e as Ae ; on each Ae we define preference [1]

3 relation satisfying standard assumptions of completeness, transitivity and continuity (note that comparability of actions across different endowments is not assumed). The set of equivalent actions (E) includes all actions between which the consumer is indifferent; we may represent E with the action curve; E is, of course, equivalent to what is usually called indifference set, however, it is convenient to reserve this term only for the set of actions that are indifferent to status quo. We denote the indifference set relative to e as Ie ; as we shall see, indifference set plays a special role, because virtually all assumptions of the theory are about the properties of this set. Finally, we define better set to e as B = {( e ; x) : ( e ; x) ( e ; e )} and worse set to e as W {( e ; x) : ( e ; e ) ( e ; x) } e e = ; similarly, we define the set of better actions to Ek as Bk = {( e ; x) : ( e ; x) ( e ; x );( e ; x ) E } set of worse actions to Ek as Wk = {( e ; x) : ( e ; x ) ( e ; x);( e ; x ) E } and the. 3. Assumptions This section can be divided in two parts: first four assumptions are to replace the standard assumption of convexity of preferences; the fifth assumption incorporates loss aversion into the theory. [CB] Strict convexity of better set: for any two actions ( e ; x ),( e ; x ) I e and k (0,1), ( e ; kx ) + ( e ;(1 k) x ) ( e ; x ). The motivation behind this assumption is that the greater amount of commodity a consumer gives up the progressively greater compensation he or she requires in order to remain on the indifference curve, since he or she has to sacrifice more and more important needs. [CBA] Strict convexity of better actions set: for any two actions ( e ; x ),( e ; x ) E and k (0,1), ( e ; kx ) + ( e ;(1 k) x ) ( e ; x ). Although formally similar to [CA], [CBA] has different interpretation than the former. Here the motivation is that consumer prefers more balanced consumption to less balanced one. The next two assumptions are to incorporate LDMU. The law has two implications for the shape of indifference curves: first, given the endowment, for giving up a unit of a commodity the consumer has to be compensated more to remain on the indifference curve, than he is willing to sacrifice for obtaining an additional unit. Second, the greater a stock of a commodity, the less is the consumer willing to sacrifice (must be []

4 compensated) for obtaining (sacrificing) an additional unit thereof. These ideas are defined formally as follows: [DMU1] Diminishing marginal utility (see Appendix for n commodities case): for any endowment ( e 1, e ) and any two actions ( e ; e 1 + h1, x), ( e ; e 1 h1, x ) I e, x e > e x (see Fig. 1). [DMU] Diminishing marginal utility (see Appendix for n commodities case): let ( e 1, e ) and ( e 1, e ), where e 1 < e, be two endowment points; (i) for any two actions ( e ; e 1 h1, x) I e and ( e ; e1 h1, x ) I e, x > x (see Fig. ). (ii) for any two actions ( e ; e 1 + h1, x ) I e and ( e ; e 1 + h1, x ) I e, x > x (see Fig. ). [DMU] is similar to Bernardelli s (1938, 195) definition of LDMU; his definition, however, required that consumer is able to compare actions across different endowment points. x x e x e Ie h1 h1 Fig. 1 x1 x x x x x e Ie e Ie h1 h1 Fig. h1 h1 x1 [3]

5 [LA] Loss aversion (see Appendix for n-commodities case): let ( e 1, e ) and ( e 1, e ), where e 1 < e, be two endowment points and let ( e ; e 1, x) I e and ( e ; e 1, x ) I e be two actions; then x e < e x (see Fig. 3). x x e x e e e 1 e 1 Ie Ie x1 Fig. 3 [LA] is motivated by the experiments reported in Kahneman and Tversky (1991) and Kahneman et al. (1991). However, present definition of loss aversion differs from that of Kahneman and Tversky (1991), which runs as follows: [LAKT] Let ( e 1, e ) and ( e 1, e ), where e 1 < e, be two endowment points and let x = ( e 1, x ) and x = ( x 1, x ), where x 1 e and x < x, be two commodity bundles. Then ( e ; x ) ( e ; x ) implies ( e ; x ) ( e ; x ). Although the two definitions are different, there is a common motivation behind them: loosely speaking, both of them attempt to incorporate the idea that a unit of a good gained is valued differently from a unit lost. This may seem similar to [DMU1]; the difference is that in the case of [LA] and [LAKT] the unit in question is used to satisfy the same want and not a different one as in the case of [DMU1]. I will now criticize [LAKT] because it is inconsistent with the original motivation to account for the influence of consumer s endowment. The reasoning behind the definition is lucidly explained by the authors (Tversky and Kahneman 1991:1048); if we translate their reasoning into the language of the present paper, it goes as follows: let as assume that ( e ; x ) = (, e ;, x ) + ( e, ; e, ) ( e ; x ) = ( i, e ; i, x ) + ( e, i; e, i ), i i 1i 1i, 1 1 [4]

6 ( e ; x ) = (, e ;, x ) + ( e, ; e, ) ( e ; x ) = ( i, e ; i, x ) + ( e, i; e, i ) ; i i 1i 1i, 1 1 Then ( e ; x ) ( e ; x ) ( i, e ; i, x ) + ( e, i; e, i) ( i, e ; i, x ) + ( e, i; e, i ) (1) and ( e ; x ) ( e ; x ) ( i, e ; i, x ) + ( e, i; e, i) ( i, e ; i, x ) + ( e, i; e, i ). () Cancelling the equal terms in (1) and () and setting ( e 1, i; e 1, i) = ( e 1, i; e 1, i ) = 0, [LAKT] says that ( e 1, i; e 1, i) ( e 1, i; e 1, i ), which indeed captures the idea of loss aversion. The problem is however in cancelling the equal terms, which presupposes that equivalent increase of a commodity is valued equivalently, regardless what the amounts of other commodities are. This assumption is somewhat curious in a theory that is motivated by the idea that endowment matters. 4. Comparison to the traditional analysis For each endowment e, a utility function u : A R, which is unique to positive monotone transformations, may be constructed. Since the endowment is fixed, utility function is, as in the standard analysis, function of the end point only. Elementary analysis of consumer behaviour thus remains untouched, with the following three caveats: first, it must be explicitly stated that the consumer is endowed only with money income; second, movement along action and indifference curves has no meaning and thus marginal rate of substitution cannot have any meaning either, except for in the points of endowment; third, changes in endowment cannot be treated independently from the changes in preferences. The traditional analysis is thus not reected it can be thought of as a special case of a more general approach. 5. Conclusion We attempted to incorporate the endowment effect into the standard theory. Incidentally, it turned out that when preferences depend on the endowment, LDMU can be reintroduced back to the consumer theory. Further the idea of loss aversion was newly formalized and the older definition of Kahneman and Tversky was criticized. Finally, it was argued that as a result of the existence of the endowment effect the traditional theory need not be reected; it can be incorporated into a more comprehensive approach. e e [5]

7 REFERENCES Bernardelli, Harro A Rehabilitation of the Classical Theory of Marginal Utility. Economica 19: Bernardelli, Harro A Reply to Mr. Samuelson's Note. Economica 6: Bernardelli, Harro Comment on Mr. Lancaster's "Refutation". Economica 1:40-4. Bernardelli, Harro The End of the Marginal Utility Theory?. Economica 5:19-1. Brown, E. H. Phelps Notes on the Determinateness of the Utility Function. The Review of Economic Studies : Kahneman, Daniel, Jack L. Knetsch, and Richard H. Thaler Anomalies: The Endowment Effect, Loss Aversion, and Status Quo Bias. The Journal of Economic Perspectives 5: Lancaster, Kelvin A Refutation of Mr. Bernardelli. Economica 0:59-6. Lancaster, Kelvin Reoinder to Mr. Bernardelli. Economica 1:4-43. Samuelson, Paul A The End of Marginal Utility: A Note on Dr. Bernardelli's Article. Economica 6: Samuelson, Paul A The Numerical Representation of Ordered Classifications and the Concept of Utility. The Review of Economic Studies 6: Tversky, Amos, and Daniel Kahneman Loss Aversion in Riskless Choice: A Reference-Dependent Model. The Quarterly Journal of Economics 106: Appendix The assumptions [DMU1], [DMU] and [LA] are generalized for the case of n commodities as follows: [DMU1] Diminishing marginal utility: for any endowment ( e 1,..., e n ) and any two actions ( e ; e,..., e, e + h, e,..., e, x, e,..., e ), 1 i 1 i i i n e 1 i 1 i i i n ( e ; e,..., e, e h, e,..., e, x, e,..., e ) I, where i, = 1,..., n and i, x e > e x. [6]

8 [DMU] Diminishing marginal utility: let ( e 1,..., e n ) and ( e 1,..., e i 1, e i, e i+ 1,..., e n), where e i < e i, be two endowment points; (i) for any two actions ( e ; e,..., e, e h, e,..., e, x, e,..., e ) I and 1 i 1 i i i n e 1 i 1 i i i n e ( e ; e,..., e, e h, e,..., e, x, e,..., e ) I, where i, = 1,..., n and i, x > x. (ii) for any two actions ( e ; e 1,..., e i 1, e i + hi, e i + 1,..., e 1, x, e + 1,..., e n) I e and ( e ; e,..., e, e + h, e,..., e, x, e,..., e ) I, where i, = 1,..., n and i, x > x. 1 i 1 i i i n e [LA] Loss aversion: let ( e 1,..., e n ) and ( e 1,..., e i 1, e i, e i+ 1,..., e n), where e i < e i, be two endowment points and let ( e ; e,..., e, x, e,..., e ) I and n e n e ( e ; e,..., e, x, e,..., e ) I, for i = 1,..., n, be (n 1) actions; then for all i = 1,..., n, x e < e x. [7]