ifdp · November 30, 1994

Loss Aversion in a Consumption/Savings Model

Abstract

Psychological evidence indicates that a person's well-being depends not only on his current consumption of goods, but on a reference level determined by his past consumption. According to Kahneman and Tversky's (1979) prospect theory, people care much more about losses relative to their reference points than about gains, are risk-averse over gains, and risk-loving over losses. We define these characteristics as loss aversion. We incorporate an extended form of loss aversion into a simple two-period savings model. Our main conclusion is that, when there is sufficient income uncertainty, a person resists lowering consumption in response to bad news about future income, and this resistance is greater than the resistance to increasing consumption in response to good news. We discuss some recent empirical research that confirms this predicted asymmetry in behavior, which seems inconsistent with other models of consumption.

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 492 December 1994

LOSS AVERSION IN A CONSUMPTION/SAVINGS MODEL

David Bowman, Deborah Minehart, and Matthew Rabin

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.

Abstract

Psychological evidence indicates that a person’s well-being depends not only on his current consumption of goods, but on a reference level determined by his past consumption. According to Kahneman and Tversky’s (1979) prospect theory, people care much more about losses relative to their reference points than about gains, are riskaverse over gains, and risk-loving over losses. We define these characteristics as loss aversion. We incorporate an extended form of loss aversion into a simple two-period savings model. Our main conclusion is that, when there is sufficient income uncertainty, a person resists lowering consumption in response to bad news about future income, and this resistance is greater than the resistance to increasing consumption in response to good news. We discuss some recent empirical research that confirms this predicted

asymmetry in behavior, which seems inconsistent with other models of consumption.

Loss Aversion in a Consumption/Savings Model

David Bowman, Deborah Minehart, Matthew Rabin!

1. Introduction

Psychological evidence indicates that a person’s well-being depends not only on his current consumption of goods, but also on how his current consumption compares to his past consumption. 2 As documented by Kahneman and Tversky (1979,1991,1992), underlying many instances of such reference-dependent preferences is a fundamental asymmetry in how increases and decreases in consumption are evaluated: People care much more about losses relative to their reference point than about gains. Moreover, while people are risk averse in gains, they are risk loving in losses. We refer to these combined characteristics as loss aversion.3

In this paper, we incorporate an extended form of loss aversion into a two-period consumption/savings model where a consumer faces uncertainty over his second-period income. To capture the idea that preferences depend on past consumption, we follow the

recent literature on habit persistence+ by assuming that a person’s second-period

International Finance Division of the Board of Governors of the Federal Reserve System, Boston University, and the University of California at Berkeley respectively. This paper represents the views of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or other members of its staff. We thank Danny Kahneman and Richard Thaler for helpful conversations, and especially Eddie Dekel-Tabak and Halsey Rogers for helpful comments. We are also grateful to Berkeley’s Institute for Business and Economic Research for funding of research assistance. Rabin thanks the National Science Foundation for financial support, through research grant SES- 9210323.

For an excellent survey of the psychological evidence along these lines, see Kahneman, Knetsch, and Thaler (1991). Other recent papers considering effects of the status quo on behavior are Samuelson and Zeckhauser (1988) and Thaler (1980). For empirical evidence regarding habit persistence, see Ferson and Constantinides (1991).

We are abusing terminology here; Kahneman and Tversky (1979) coined the term "loss

version" to mean the basic gain-loss asymmetry alone.

See Ryder and Heal (1973), and Becker and Murphy (1988), Boyer (1983), Constantinides

reference point is a weighted average of his initial reference point and his firstperiod consumption.

Our main conclusion is that, when there is sufficient uncertainty, a person resists consuming below his reference point in the first period even when his expected average per-period income is below his reference point. This resistance to changing consumption is asymmetric. Formally, though a consumer will never consume |elow his reference level when he anticipates an average per-period income above his reference level, he might consume above his reference level when he anticipates an average perperiod income below his reference level. This asymmetry is due largely to the riskloving attitude towards losses, which most clearly differentiates our model from both the classical utility function and recent theoretical literature on habit persistence.

Our model conforms with the general intuition--posited at least as far back as Duesenberry (1952)--that people resist decreasing their standard of living in response to bad news about income. Our results also conform with recent empirical research by Shea (1993a, 1993b), who provides evidence of asymmetric behavior consistent with our model. Using information on union contracts to construct a measure of expected wage growth for each household in his sample, Shea (1993a) finds that the response of consumption to predictable declines in wages is greater than to predictable increases. He also shows that, while Hall’s (1978) test of the Permanent Income Hypothesis is rejected for predicted wage declines, it cannot be rejected for predicted wage increases. Shea (1993b) tests for and finds the same asymmetry in aggregate U.S. data. These findings appear to be robust, and are inconsistent with other standard explanations of apparent violations of the Permanent Income Hypothesis, such as

liquidity constraints or Campbell and Mankiw’s (1989) rule-of-thumb behavior.

(1990), Sundaresan (1989), and Detemple and Zapatero (1991).

Another alternative hypothesis that has received attention recently is myopia. Because

he associates it with Campbell and Mankiw’s rule-of-thumb behavior, Shea (1992.a,1993b) 2

‘Shea’s empirical findings also help differentiate our model from the recent literature on habit. persistence. | While this literature posits reference-dependent behavior, they assume that utility is concave both in consumption and in the reference point, thus omitting loss aversion. In these models, as in ours, a person who receives bad news about future income may be slow to adjust consumption downward, because breaking with habit is painful. But, in these models, he should be equally slow to adjust consumption upward in response to good news. The asymmetry predicted by our model relies on the person’s risk-loving attitude towards losses, resistance to lowering consumption in response to bad news means that a person is willing to risk an even more dramatic future drop in consumption so as to avoid consuming below his reference point today.

Because loss aversion encapsulates important psychological facts about how preferences and attitudes towards risk are influenced by reference points, we feel it deserves significant attention by mainstream economists. Beyond our results on savings, therefore, we see this paper as an attempt to grapple with some more general issues involved in incorporating loss aversion into formal economic analysis. Our research indicates, for instance, that loss aversion may have little predictive power in formal models of dynamic decision-making unless ancillary assumptions are made about how a person’s welfare is affected by changes in his reference point. We assume that preferences exhibit acclimation: Fixing a person’s consumption, he is happier if his reference point matches his consumption than if it is either higher or lower. As we show in Section 3, this assumption guarantees that if a person is certain that his average

income will be just sufficient to consume each period at his reference level, he will do

argues that myopia does not explain his evidence of asymmetry. More generally, it is hard to imagine that any natural definition of myopia would cause the asymmetry found by Shea (1993a,1993b). Myopia could, however, magnify the asymmetry; Benartzi and Thaler (1992) in fact argue that loss aversion and myopia together explain the phenomenon of underinvestment in risky assets. 3

so rather than vary his consumption across periods.

In developing our formal model, we have made some compromises with behavioral evidence. For instance, while we have adopted loss aversion from Kahneman and Tversky (1979,1992), we have ignored other aspects of their more general prospect theory. They demonstrate that people are subject to framing effects--a person may behave differently if a logically equivalent decision is framed to him in two different ways.© ‘They also show that the decision weights people use in evaluating the riskiness of their choices may differ from the true probabilities. We ignore these and other issues because we feel they are simply not as tractable nor as generalizable as loss aversion, and, in taking the pragmatic strategy of incorporating new assumptions one at a time, loss aversion seems an ideal candidate to start with. Moreover, it does not appear that adding other behavioral assumptions would substantially change the qualitative differences between our model and the standard life-cycle model of savings.

We begin in Section 2 by presenting Kahneman and Tversky’s (1979) original formulation of loss aversion, our extensions of loss aversion, and two simple propositions comparing loss aversion to the standard economic model of risk aversion. In Section 3, we present our savings model and our formal results. We conclude in Section 4 by discussing possible extensions of our savings model, and briefly considering other

economic applications of loss aversion. 2. Loss Aversion and Acclimation

In this section, we review the basic model of preferences developed by Kahneman

and Tversky (1979) and outline our extensions. To formally incorporate the idea of

6Thaler’s (1980,1985,1990) notion of mental accounting is somewhat related to the idea of framing effects, and Thaler (1990) demonstrates that mental accounting may play an important role in savings behavior, implying that people treat different scurces of income differently even when standard theory predicts that they are interchangible.

reference dependence, we assume that a person’s utility is a function of both his level of consumption, c, and his reference level for consumption, r. We assume for simplicity that utility is separable in the reference level and the deviation of actual consumption

from the reference level:

U(r,c) = wir) + v(c-r),

where w(-) is the "reference utility," v(-) is the "gain-loss utility," and U(-,-) is overall utility. We shall assume that the function is defined for all non-negative values of r and c. For convenience, we shall also assume that both components of the

utility function are continuous and, except when c = 1, are twice differentiable.

2.1 Loss Aversion

We begin with the assumptions on the utility function incorporated into Kahneman and Tversky’s (1979,1992) prospect theory. Because their analysis concerned the case where the reference point is fixed, tastes are determined entirely by v(-). We normalize v(-) to be equal to zero if c = r, so that v(0) = O and U(r,r) = w(7). Kahneman and

Tversky posit the following three characteristics of v(-):

Assumptions V1-V3: V1. v(x) is strictly increasing in x. V2. v(x) is strictly concave for x > 0 and strictly convex for x < 0. V3. If y > x > 0, then v(y) + v(-y) < v(x) + v(-x) < 0.

Assumption V1 says that "more is better"--v(-) is an increasing function, which implies that U(-,-) is imcreasing in c. Assumption V2 implies that people are risk

averse in situations involving a sure gain, but also implies that people are risk loving

in situations involving a sure loss. Assumption V3 is that the marginal utility of a loss is strictly greater than the marginal utility of a comparable gain.

The experimental evidence (see Tversky and Kahneman (1992)) indicates that the ratio of loss aversion--the marginal utility of losses divided by the marginal utility of gains--is often approximately 2. That is, a loss of one unit is two times more unpleasant than a gain of one unit is pleasant. Moreover, evidence indicates that the relative distaste for losses exists even for very small changes, and the algebraic examples Tversky and Kahneman (1991,1992) estimate have a "kink" such that the ratio of loss aversion is strictly greater than 1 for even infinitesimal gains and losses. We

formalize this idea in Assumption V4:

Assumption V4: Limit, 6 v’(-x)/v’(x) =L > 1.

Assumptions V1-V4 can be graphically represented by the following classical

diagram:

Standard theory predicts that people are very close to risk neutral for small bets, and will accept any slightly-better-than-fair bet if it is small enough (Arrow (1974), Pratt (1964), and Samuelson (1961)). Because of Assumption V4, loss aversion predicts the opposite result--that people will reject any _ slightly-better-than-fair bet if it is small enough.

Formally, consider a bet as a distribution f over gains and losses, and say that we are scaling it by kK > O when we consider the bet where all values of f are multiplied by k. Then standard theory predicts that for every better-than-fair bet f, there exists e > O such that for all k < e, a person will accept the bet f scaled by k.

In. contrast, loss aversion predicts:

Proposition 2.1:7

Let f be any better-than-fair bet with a lower bound on losses, and with a positive probability of a loss; let g be any fair bet with a lower bound on losses; and let h(p) = pf + (1-p)g for p € (0,1). Assumptions V1-V4 imply that there exists ane > 0 and p- e (0,1] such that for all kK < © and p < p, a person will reject h(p) when scaled by k.

Locally, then, loss aversion implies a stronger form of risk aversion than standard theory.8 Globally, however, the situation is different. As is well known, the standard assumption of a concave utility implies that a person will reject all fair bets. While loss aversion implies risk aversion for both small bets and symmetric bets

of any size, the more general risk aversion predicted by the concave utility function is

TAI proofs are in the Appendix.

8Recently, papers such as Segal and Spivak (1990) consider ways to modify utility theory to get “first-order risk aversion" (where slightly-better-than-fair bets are turned down no matter how small) of the type we get, rather than the classical "second-order risk aversion." This literature, however, attains first-order risk aversion by relaxing the assumption of expected-utility maximization. By contrast, Proposition 2.1 shows that first-order risk aversion can be a consequence of loss aversion, even if one works within the expected-utility framework.

not guaranteed. In fact, empirical evidence suggests that some systematic: deviations from risk aversion do exist.? However, we do not consider these deviations to be a central issue in savings behavior, and so on this issue we conform to the standard theory. Specifically, Assumption V5 states that a person will turn down any fair bet

centered around his reference point--including asymmetric ones of any size:

Assumption VS: For all 1 = p= 0, x > O and y < O such that p-x + (1-p):y = 0,

p-v(x) + (1-p)-v(y) < 0.

Assumption V5 is equivalent to the condition that the marginal disutility of a

loss is everywhere greater than the marginal utility of a gain. Formally:

Proposition 2.2: If v(-) satisfies Assumptions V1-V4, then it satisfies Assumption V5 if and only

if for all x = O and y < O, v’(y) > v’(x).

If preferences meet Assumptions V1-V5, we say they exhibit general loss aversion.

2.2 Acclimation In the type of intertemporal setting we consider in this paper, future reference points may be affected by current choices. Individuals may realize this, and be

influenced by reference-point effects when making consumption decisions. For instance, a

9while Kahneman and Tversky do not assume that people refuse all fair bets, their discussion of violations of this assumption focuses more on the manner in which people use probabilities in making decisions (for instance, ignoring small, but non-zero, probability outcomes) than on the properties of v(-). Tversky and Kahneman (1992) estimate a parameterization of v(-) that implies that the marginal disutility of a loss is greater than the marginal utility of a gain whenever the loss is less than 860 times the size of the gain; only extremely asymmetric fair bets will be accepted. Especially because we do not in any event conceive of our results as being relevant in the case of very large losses in income, we feel that the addition of Assumption V5 does relatively little damage to the realism of our results. 8

consumer may be cautious in developing a luxurious lifestyle because he knows that doing so will make him less happy if he later becomes impoverished. We must therefore make assumptions about how a person’s welfare depends on his reference point. We have little empirical guidance on this issue, but make what we feel are sensible choices for the problem at hand.

The gain-loss utility v(-) represents the amount of pleasure or displeasure the consumer experiences from gains or losses relative to his reference point w(-), whereas w(:) corresponds to that portion of a person’s well-being that is determined by his reference point. Assuming that reference points tend eventually to adjust to levels of consumption, w(-:) can be interpreted a kind of “long-term" utility function, representing how much material pleasure a consumer gets from consuming amount r after he becomes accustomed to that consumption level. We shall make the conventional assumptions

that w(-) exhibits non-satiation and diminishing marginal utility:

Assumption W1: w(-) is increasing and weakly concave everywhere.

While our assumptions about w(-) are relatively straightforward, we must also make assumptions about the relationship between w(-) and v(-). For reasons which will be

explained shortly, we assume the following:

Assumption VW1:

We assume U(r,c) exhibits acclimation: When r > c, dU(r,c)/ar < 0, and whenr < c, aU(r,c)/ar > 0. For preferences meeting Assumptions V1-V5 and W1, this is true if and only if for all r = 0, x = 0, and y = 0, v’(x) > w’(r) > v’(y).

This assumption says that, fixing his consumption level, an individual is happier

the smaller his loss or the smaller his gain. The first part of this assumption seems

uncontroversial, but the second part may seem less natural--it says that. somebody who is consuming a lot now: will be happier:if he is accustomed to it than if he had previously been poor. We show in the next section, however, that this assumption is needed to guarantee reasonable intertemporal behavior. In particular, acclimation guarantees that a person will not purposely consume below his reference point in the present: solely to consume above his reference point in the. future. 10

On a technical level, acclimation is simply an upper envelope condition. To see this, suppose for the moment that we relax the separability assumption and set v(c - 7) = w(c) - wir) for c 2 r. Then U(r, c) = w(c) for c = r. This captures the idea that consumers adjust to gains immediately, evaluating them according to their "long-term" utility function. Then as long as U(r,c) satisfies general loss aversion, W1, and the more intuitive part of acclimation, all of our results in section 3 still obtain. Although in what follows we assume separability for expositional simplicity, more

generally the second part of acclimation can be expressed as U(r,c) = w(c) for ¢ 2 r.

ee 10Some empirical and experimental research (see, for instance, Frank and Hutchens (1990), Loewenstein and Sicherman (1991), and Loewenstein and Prelec (1992)) has found that, fixing average lifetime income, people prefer their income to be increasing over time, rather than either flat or decreasing. The assumption of acllimation is consistent with this behavior, as long as a person is not willing to consume below his current reference point in order to consume above his reference level in the future. 10

3. A Two-Period Model of Savings

We now consider the implications of general loss aversion and acclimation in a simple two-period model of a consumer’s savings decision. We assume that there are two periods remaining in a consumer’s life, and any uncertainty in his total remaining income will be resolved only in the second period. We do not allow a consumer to consume less than zero in either period, and assume that he cannot borrow against uncertain future income (which would risk default); but we assume that there are no liquidity constraints which prevent him from consuming any guaranteed second-period income in the first period. For simplicity, we assume no discounting and that savings earn no interest. We let a be the reference level in period ¢, and let C be consumption level

in period t. The consumer’s total two-period utility is given by:

U(r 64579 2€p) = [w(r)) + v(c,-r)1 +E {[w(r5) + v(Cy- 2)]}-

The consumer chooses his first-period consumption after he knows his first-period income, but with probabilistic beliefs about his second-period income. The nature of the maximization problem is clearly determined by the way in which reference points are formed. We will model reference-point formation in a manner consistent with recent literature. on habit persistance in consumption. We take the first-period reference point, ry, as exogenously determined. The second-period reference point, r,, will be

determined in part by ry and in part by the first-period consumption level, Cy:

e) = (1-a)ry + aCy, where ae [0,1].

The parameter « represents the speed at which the reference point changes in

response to recent consumption. If a = 0, then first-period consumption has no effect on

the consumer’s second-period reference level, so that utility is time-separable; if

a = 1, then the second-period reference level adjusts fully to first-period consumption.

3.1 Consumption Behavior When Income is Certain

We begin our analysis by examining the consumer’s behavior when he faces no uncertainty. We denote by c4(Y) and cy(¥) the consumer’s choice of first- and secondperiod consumption if faced with a sure total income of Y. Our results are summarized by

Theorem 1:

Theorem 1: If the constraints Cc) 2 0 and Cy = 0 are ignored, then for all « > Q, c,(¥) and c4(¥) are continuous and strictly increasing, with c,(¥) = c,(Y) =n when Y = 2r,- If the constraints cy 2 0 and Cy > 0 are imposed, then for all « > 0, c,(Y) and c4(Y) are continuous and non-decreasing, and for Y = (l-a)r,, c(¥) and c,(Y) are strictly increasing, with c(¥) = c4(¥) =" when Y = 2r,-

Theorem 1 says that--except for cases where the constraint that consumption cannot be negative in either period is binding--a consumer will increase both his first and his second period consumption in response to an increase in his lifetime income. In other words, both first and second period consumption are normal goods. It also says that he will consume below his current reference level if and only if his lifetime income will not be able to support continued consumption at or above his current reference level. We consider both these features of behavior to be natural when there is no uncertainty.

In Section 2, we introduced the assumption of acclimation (Assumption VW1), which describes a consumer’s preferences over his reference point for a fixed consumption level. We discussed how one aspect of acclimation--that a person with a high standard of living is happier if he is acclimated to that standard of living than if he is accustomed to being poor--may strike some readers as counterintuitive. In particular,

this assumption may seem to go against the intuition that people like improvements rather than staying where they are. Proposition 3.1 establishes, however, that this aspect of acclimation is in fact necessary in order to obtain the behavior implied by

Theorem 1.

Proposition 3.1:

If preferences meet Assumptions V1-V5 and W1, then Theorem 1 holds for all re 0 and all Y only if for all r = O andc > 1, w’(r) > v’(c-r). Because v(-) is concave for positive values, this last condition holds if and only if for all r = 0 and c > r, w(r) = U(e,c) > U(r,c).

Proposition 3.1 suggests an indirect empirical test for the validity of acclimation, which is hard to test directly: If the behavior implied by Theorem 1 captures reasonable intemporal behavior when a person’s lifetime income is certain, Proposition 3.1 indicates that acclimation may be a realistic restriction on preferences.

The more intuitive aspect of acclimation--that, when a person’s reference level is above his current consumption, he is happier the lower is his reference point--is sufficient but not necessary for Theorem 1 to hold. Proposition 3.2 establishes a weaker upper envelope condition on preferences which also conforms with this intuition and is

in fact necessary for Theorem 1:

Proposition 3.2: If preferences meet Assumptions V1-V5 and W1, then Theorem 1 holds for all r= 0 and all Y only if for all c = 0 andr > c, w(r) = U(c,c) > U(’,c).

3.2 Consumption Behavior When Income is Uncertain In the conventional concave-utility framework, the monotonicity obtained in

Theorern 1 extends to cases of uncertainty: If lottery g(-) first-order stochastically 13

dominates lottery h(-), then first-period consumption will be higher if faced with income stream g(-) than if faced with h(-). Example 1 shows that this feature does not hold in our model. The convexity of v(-) over losses can cause a stochastic rise in

income to yield a decline in first-period consumption.

Example 1: A consumer solves the problem max U(r, ,.¢)) + E{ U(r, Cy) }

C4 2l5 subject to Cy +o = Y; + Y5 n= 1, Ty = Cy (i.e., a = 1) Y, = 2.75, prob(Y,= 0) = 3/8, prob(Y, = 1) = 5/8 1.25r + (c-r) - .25(c-r)* ife zr

where U(r,c) = 2 1.25r + 2.5(c-r) + .S(c-r)” if c zr

In this example, the optimal choice of Cy is locally approximated by the equation cy, * 3.25 - 59°), so that increases in Yy lead to decreases in Cy: The irituition for this non-monotonicity is as follows. When the consumer chooses optimally in this example, he will choose to risk the possibility of consuming below his reference point in period 2. A rise in income gives the consumer the ability to erase some of his possible loss in the second period; since the marginal utility of consumption is rising in the loss region, the consumer may even reduce first-period consumption in order to take advantage of this new opportunity. In this example, first-period consumption is an inferior good.!1

Strict non-monotonicities as exhibited in Example 1 appear to require that preferences are highly risk loving in losses and therefore are, in a loose sense,

unlikely. Example 2 illustrates what seems, however, to be a very common outcome in our

1l]t is also possible to construct examples in which second-period consumption is an inferior good, so that a rise in expected income persuades the consumer to raise firstperiod consumption by more than the expected increase in income.

model: when expected average per-period income is near the reference level of consumption, a consumer may rigidly set c= lp and be totally unresponsive to modest

changes in expected income.

Example 2: A consumer solves the problem Max, C5 U(r .¢,) + E{ U(r, ,€4) }

subject to Cc) +o = Yy + Y>

r= 1, Ty = Cy 1.5 = Yy = 2.5 and Y2 uniformly distributed over [-.5,.5] 3r + Uc-r) - .25(c-r)* ifczr where U(r,c) = 2 3r + 4(c-r) + .25(c-n)” ifc =r The solution is: c, = 2.0625 + .5-Y, - .25-(73.0625 + yi? if Y, = 2.131 cqHn= 1 if 1.7081 =< Y; =< 2.131

Cy = 1.9375 + .5-Yy - .25-(53.0625 - y)?

if Y; = 1.7081

The results from Example 2 can be illustrated as follows:

“a a

Figure 2

The flat part of c,(¥) in Example 2 reflects the non-differentiability in the

value function v(-). But the more general fact that consumption changes little around the reference point demonstrates the key feature of our model: When there is enough uncertainty, people resist lowering consumption in response to the news that they will probably not be able to maintain the standard of living to which they are accustomed. If a person learns that his average per-period income is likely to be lower than his current reference point, he will not immediately consume below his reference level unless he is very sure that he will never recover his former income level. Theorem 2 states that as long as future per-period income exceeds the current reference level with at least probability a/(1+«), a person will consume at least at his reference level in

the first period:

Theorem 2: Suppose that P[Y/2 = ri] = a/(1+a). Then Czy if P[Y = ri] = 1.

The intuition for this result depends on precisely the two aspects of loss aversion that are most non-standard to economists--the dependence on refererice points, and risk-loving preferences in losses relative to the reference point. 12 Consumers want to avoid losses, but they are willing to risk rather large losses in the future to avoid even small losses in their current standard of living. This "risk-loving" behavior is therefore crucial in yielding the prediction that people will resist decreasing their consumption.

Theorem 2 also provides some insight into how changes in the speed at which a person’s reference point adjusts to recent consumption affects his behavior. The higher is a, the more willing is a person to decrease his current consumption in response to

bad news about future income. This is because a higher « means that lowering consumption

121m fact, our proof of Theorem 2 shows that only a weaker form of Assumption VW1 is required. So long as Assumptions V1-V5 and W1 hold, then Theorem 2 holds if w'(r) = Vale") for all c 2 r.

will more dramatically lower the future reference point; if future consumption is likely to be lower, then such a lowering of the reference point is a positive feature. Note that if a person’s reference point is completely unaffected by current changes in consumption--if « = 0--then Theorem 2 says that the person will never consume below his reference level unless his income is so low that consuming above his reference level in the first period risks default.

While Example 2 illustrates that consumption can be sticky both when expected income. is below the reference level and when it is above, there is more generally an asymmetry in our model regarding such stickiness. Theorem 2 implies that consumer behavior under loss aversion is asymmetric with respect to the reference level, because there ure examples where a consumer will consume above his reference level in response to bad news. Example 3 describes a situation in which, even though the consumer expects income in the second period to be below his reference level, he consumes above his

reference point in the first period.

Example 3: A. consumer solves the problem max 0 ey + E{U(r sCy)}

Ch subject to Cy) to = Y, + Y,

n= 1; r= Sy + Cy

1 < Yy <2

Y,, uniformly distributed over [0,.5]

2 r+ (c-r)- 25(c-n)* if czr where U(r,c) = 2 r + 1.05(c-r) + .0625(c-rn)” if czr The solution is: cy = 1052 + .6667Y, - (-.2126 + 1580Y ,)"7 if 1.6944 < Y, <2 c= 1 if 1 < Y, = 1.6944

For 1.6944 < Yy < 1.75, Yy + E(Y,) <2= ary. but Cy >1= Ty.

Example 3 can be illustrated graphically as follows:

Figure 3

<, cy)

2=T1,C1 =F]

It is this asymmetry implied by Theorem 2 and Example 3 together that we feel is

the main feature differentiating loss aversion from other models of preferences.

3.3 Consumption Behavior when Income Uncertainty is Increased : Precautionary Savings

It is a commonly held belief that people will increase savings in response to an increase in uncertainty. Leland (1968) shows that--in a conventional, time-separable, concave-utility model--this intuition is correct only when the third derivative of the utility function is positive (see also Kimball (1990a,1990b)).

In our model, an increase in uncertainty can either increase consumption or decrease it. Combining Theorems 1 and 2 show that an increase in uncertainty can decrease savings. Theorem 1 shows that if a consumer knows for sure that his future income is below his current reference level, he will consume below his reference level. Theorem 2 shows that if the consumer faces expected per-period income below his reference level, but believes that there is a high probability that his per-period

income will be above his reference level, he will consume at or above his reference

level.

Thus, when expected income is below the reference level by a small amount, then an increase in the probability that a consumer will be able to consume above his reference level each period will lead him to decrease savings. Part (i) of Proposition 3.3 illustrates this idea. Part (ii), on the other hand, suggests a set of circumstances where the classical intuition is likely to hold: If expected per-period income is above the reference level by a small amount, then an increase in uncertainty is likely to

increase savings.

Proposition 3.3: Let c,(¥,k) be the consumer’s first-period consumption when faced with probabilistic total income (.5 (1-k)Y, .5 (1+k)Y), where k ¢ [0,1]. i) There exists Y, < 2+) such that for all Y e (Y 4,274) and all & for which (1-k)-Y > rp cy (V4) > cy (¥,0). . ii) There exists kK > Qand Y > a+r; such that for all Y e (2r,,¥ ) and all k<k, c,(Y.0) > c,(Y,4);

While part (ii) of Proposition 3.3 need not generalize to non-binary symmetric distributions, the idea driving the proof is that when expected average per-period incorne exceeds your reference point, an increase in uncertainty that raises the odds that you will not be able to maintain your current consumption level will increase your savings. 13 Roughly, this holds whenever the effect of increasing the odds of being forced to consume below one’s reference level is not outweighed by behavioral changes

relating to third-derivative features of the various components of the utility function.

13Theorems 1 and 2 make clear, however, that part (i) generalizes to any symmetric distr:bution. The fact that part (i) of Proposition 3.3 holds more generally than part (i) seems to be related to the asymmetry in Theorem 2 and Example 3.

4. Discussion and Conclusion

Our exploration of the savings problem calls for a few natural extensions. We would like to extend the analysis to a multiple-period or infinite-horizon model. To do this, we would need to add discounting and positive interest rates to the model. Given our analysis, one can make an informed guess about how results would be altered under these extensions. No matter how many periods there are, for instance, it is staightforward to show that, if a = 0, an agent will never consume below his reference point unless his income absolutely forces him to. As « rises, a consumer will be more willing to consume below his reference point, just as in the two-period model.

Extending the analysis to many periods would allow us to explore some issues that cannot be completely addressed within the two-period framework. As we show in Proposition 3.3, consumers may respond to an increase in income uncertainty either by saving more or by saving less, depending on whether expected per-period income is above or below the reference level. Given that losses are so costly, it is natural to suppose that over a longer time frame consumers will plan their path of consumption in order to avoid losses. In this case, the probability of a consumer having to consume below his reference level will be greatly diminished. As in standard models of precautionary savings, the consumer’s typical response to small increases in uncertainty might therefore be to increase savings. 14

As we stated in the introduction, we feel that loss aversion can usefully be incorporated more generally into economics. Benartzi and Thaler (1992) invoke loss aversion in explaining under-investment in risky assets. And Sheffrin and Statman (1985) find that trading volume on the stock market falls when stock values fall, which they

interpret as due to loss aversion--investors are unwilling to realize the loss on their

14For discussions of precautionary savings in the standard framework, see Sibley (1975), Miller (1976), Zeldes (1989), and Caballero (1990). 20

investment and so hold on to the stocks.15 Fershtman (1993) considers the effect of loss aversion on the willingness of incumbents in an industry to fight potential entrants. We feel that a formal model along the lines developed in this paper will help researchers begin to systematically investigate the implications of loss aversion in a wider array

of economic situations.

15These examples, and indeed the savings example in this paper, all have a similar flavor to the results found in some studies in non-economic settings. McGlothlin (1956) documents the tendency of race track bettors to bet increasingly long shots as the betting day goes on, behavior that he interprets as an attempt to end the day without a loss. Thaler and Johnson (1990) discover a similar pattern in a different gambling context.

Appendix: Proofs Note: In all proofs, we denote the function v(x) by vjQ) when x < 0, and by V(X) when = 0.

Proof of Proposition 2.1: Assumption V4 implies that there exists 5 > 0, y > O such that vi%) - v'(0) > x > O for all y € (-5,0]. Let G and H denote the c.d.f.’s of g and h, and let y, = be the lowest y realized with positive probability by either f or g.

Set © = -3/Y;- Then k < e implies that

i) E{v(k-g)} = Ny VUNG) + SG Ve(kx)IG(X)

10 WG ViEsds]-dG) + SG LG votks)ds] -dG@0)

1° fv.) +]xdG(x) +. $6 Vu(0)-xdG(x), V 20 I xdG@x) + wy xdG(x) 0 Iv. (0)+2]- xdG(%) < 0.

il) E{v(k:f} = Ny yon + Sr 0 Y g(kx)dF(x) < v’ g(0): Uy, x-dF(x) + ° x-dF(x)]

= Vg(0) . Syd O),

Set a(p) = p-v5(0)-S9xdF(x) + (-p)-9-SY, x- dG). Then a(1) > 0, since f is better-than-fair; a0) < 0. Thus, there exists pD- > 0 such that a(p ‘= 0 and a(p) < 0 for all p < D. Sop < pD- and k < e imply that E{v(kh(p)} = p-E{v(k)} + (1-p)-Efvikg)}. But this is less than a(p), which is negative, so you reject h(p) scaled by k < e. Q.E.D.

Remark on Proposition 2.1: This proposition would also be true even if the supports of f and g contained infinitely negative outcomes, so long as the expected value of losses were finite for

each of the bets.

Proof of Proposition 2.2:

Proof of "only if" direction: Assurne v(-) satisfies V1-V5.

Suppose there exists Yo < 0 such that Vv’ Yo) < vg. Then v’ (yy) < Vg(9) for all ¥1 < Yo: Therefore, the continuity of v gt ) implies that v’ (yy) =v 2 for some « > 0. Thus, by Assumption V2, v’(y) < v’(e) for all y < yy and v’(e) - v’(y) imcreases as y decreases.

Then v(y) - v’(e):y = Ny [v’ (e)-v’ (s)]ds

= 5 fv'(e)-v'(s)]ds + SY0 [v’ (e)-v’ (s)]ds Yo y and lim, _5 -@ VY) - Vi(e)-y = @ In particular, we can choose y < 0 sufficiently negative such that v(y) - v’(e)-y > 2v(e). Set p= el(e-y). Then i) D- y + (1 -p): e = 0, and ii) p-w(y) + (1-p): v(e)_ > p[v'(e)-y + 2v(e)] + (1-p): v(e) > -e-v’(e) + wep + 1), (since 0 > p-y = -(I-p):e > -e.) = v(e) - e-v’(e) + p-v(e) > 0 (by Assumption V2). Proof of "if" direction: Suppose v’(y) > v3(0) for all y < 0. (Choose any y,z such that p-y + (1-p)-z = 0. Thus, z = -p-y/(1-p). Then p-v(y) + (1-p):v(z) < P-v,(0)-y + (1-p)-v(z) (since v’(y) > vg(9)) < p-v,(0)-y - [(1-p)-p/(1-p)] -v,)-y = p-v,(0)-y - p-vg(0)-y = 0 Q.E.D.

Proof of Theorem 1:

For any given T, and Y, substituting Y- Cy for Cy and ac, + (l-a)ry for 5, we can write UE, 45¢yFp) as a function of Cy alone:

u(cy) = w(t) + v(cy-Ty) + w(ac, +(1-0)F4) + WY - (1 +a)Cy - (1-«)r4)

* * We will let cj denote the cy which maximizes u(c,), and let Cy = Y- Cy. If we impose the non-negativity constraint, we say that cy) is feasible if c, € [0,Y]; if we do not impose the non-negativity constraint, then any C, is feasible.

(1) Claim: c,(Y) = c4(Y) = Ty when Y = 2r)-

Proof: When Y = 2ry> if Cy >t then C5 < IT, and if Cy <vy then ¢, > I». If Cy > ry and Cy < rn, thenu’ (cy) = Vales Ty) + aw’ (rt) -(_ +a)v4(¥-CyF)- By VW1 this is negative, so that the consumer will decrease C,- If Cy <r and Cy > ra, then uw’ (c,) = v4 C47) + aw’ (rf) - (1 ta)va(¥-cy19)- By VW1 this is positive, SO that the

consumer will increase C,.

(2) Claim: c,(¥) and c4(Y) are continuous and strictly increasing when Y > 2ry, and both converge to r; when Y approaches 2ry from above.

. Proof: Assume Y> 2ry- We know from the above argument that at the optimum, czy and Cy = 9; this implies tr, + Cy < (Y-(1-a)r, )/( +). Note that uc;) is strictly concave in this region. If c= rp then, by VW1,

u’ (cy) = v0) + aw’ (ry) - (1+a)yv,(¥-2r) > 0. Therefore, ©; 7 4 if Y > 2r;. Now consider C, = 19; this implies c) = (Y-(1-a)r,)/( +e). Then

w (C,) =V o((¥-2rv +a)) + aw’ (@Y + (1-a)r,)/(. +@)) 7 (t +a)v/ (0). If u’(c,) is greater than zero when evaluated at this point, then ¢, = (Y-(l-a)r, /G +a) and Cy = (a¥ +(1-a)r,/G +a). As Y increases, wU’ ((Y-(-a)r/G +o)) decreases continuously by the concavity of Vv g and w, and by the assumption that V3 and w’ are continuous.

If u’ ((¥-C-a)r)/(1 +4) < 0, then there is a Cy E (7, (¥-C-a)r V/A - )h such that uw’ (C}) = 0. Thence, = Cy and cy = Y- Cy. We know Cy is a continuous function of Y since v’. and w’ are continuous. Totally differentiating the equation u’ (C,) = 0 yields 1W/(it+a) > de ,/d¥ > 0.

Therefore, c,(¥) and c4(¥) are continuous and strictly increasing, and since (Y-

(1-o)r)/ +a) r, as Y—- ary we have c,(¥) and c4(Y) — 1, as Yo ar, from above.

(3) Claim: If Y < 2r, and negative values of Cy and Cy are allowed, then c,(Y) = (Y - (1-a)r,)/(1 +e). If Y < 2r, and the constraints C12 0 and Cy = 0 are imposed, then c,(¥) = max[0,(Y-(1-«)r,)/( +a)].

Proof: From the proof of (1), we know that cj <r and 5 = 14, SO that (Y-(1-e)r,)/ +a) = C, = 7). Consider c, «€ [(Y--a)r,)/0 +a),(¥ tar,)/(2+a)I. In this region C€)-Ty = CyT * 0, and so we have

u’(C;) = vi (cy-rp) + aw’ (ac, +(1-a)r 4) - (1+a)vj (C4-F5) < 0 by VW1 and V2. Now consider the region Cie ((Y¥ +ar,)/(2 +a), ri. In this region we have Oz=c,-y > Cy - My. We can rewrite this range of possible choices for C, as (Y-2r,)/(2 +a) <c)- 0. Consider any choice of Cyr, 2 in this range, then u(r, +2) =v,(Z) + w(az+ry) + v,(¥-C. +a)z-2r;) is the utility from this choice. Compare this to the choice of cy-r, = Z (This choice implies a c, which lies in the range c, E [(Y-C-e)r,)/C +a),(Y +ar,)/(2+a))). If Cc, = r,+z is feasible, then Cy = I, +z is also feasible. The choice c, = 1, +z results in utility

u(Y-r-2) = v(¥-(1+a)2-2r4 +k) + w(az +r, -k) + v,@), where k = a[2r,-Y¥ +(2+@)z]/(1-a) > 0. Then

U(Y-15-Z) - u(r, +2) =

[v(Y-G. +a)z-2r) +k) - v(¥- +a)z-2r)] + [w(az+r,-k) - w(az+r,)I. VW1 implies vjxt+q) - vj) > w(s+q) -w(s) for any q > 0, x = -q and s = 0. Therefore U(Y-F5-Z) > u(r, +z) for any z such that (Y-2r,)/(2 +a) < z <0. This implies that for any feasible choice of Cc) «€ (Y+ar,\/(2+a), r] there is a feasible choice of c) € [(Y-(1-a)r Wa), (¥ Far, )/(2+0)] which is preferred. Therefore

ct € [(Y-(L-a)r, WL +a), (¥ tar, )/(2+a)],

ard therefore if negative values of c,and C» are allowed, c,(Y) = (Y-(1-a)r)/(L +e); if Cy < 0 and Cy < 0 are not allowed, then c,(¥) = Max[0,(Y-(1-a)r,)/(1 +)].

This establishes Claim (3); Claims (1), (2), and (3) together establish the theorem. Q.E.D.

Remark on Theorem 1: The proof above shows that c,(¥) is never steeper for Y/2 > ry than it is for Y/2 <r,. 1

Proof of Proposition 3.1: For any given TY; and Y, substituting Y - c, for Cy and ac, + (l-a)r, for Ty, we can write U(C, 1135.15) as a function of Cy alone: u(c;) = w(r,) + v(c,-r;) + w(ac, +(1-a)r)) + WY - (1+a)c, - (1-a)r,) We will let ct denote the Cy which maximizes u(c,), and let C3 =Y- cf. Suppose that there is some 7 = 0, c = 7, such that v’(c-r) = w’(r). Then by W1 for any r > randc2=c > r we have v’(c-r) > w’(1). Let r=r and take Y = 2r)- Theorem 1 implies that c)=r and Cy =r If c)=r then u’(r) = v 39) + aw’ (r) - (1 +a)ve(¥-2r). Then Limy 5, u’(r) = a-[w’ (r) - v’(0)] < 0. Because v’ and w’ are continuous, this implies that there is some x > O such that for allO < y = x,

u(r) = v,(0) + aw’(r) - (1 +a)vo(y) < 0.

u(c,) is concave in the region r < c) = (Y-(1-«)r)/(1+a). Therefore if CF lies in this region, cy = r for Y ¢€ [2r,2r+x]. This means c,(Y) is not strictly increasing in this region, which contradicts Theorem 1.

To see that C,€ [7,(Y-(1-«)r)/(1 +«)] note that by supposition r = CF = Y-r.

Then

u(c, = Y-(l-«)r)/(1+a«)) = w(r) + Vg((¥-2r)/(1 +0) + w((aY +(1-«)7r)/(1+a)). Compare this to uc, = (Y-(l-a)r)/(1+a) + z), where z > 0:

u(Y-(1-a)r)/(1+a)+z) = w(r) +

V_(((Y-2r)/(1 +a)) +z) + w(((aeY +(1-«)r)/(1+a))+az) + vcd +a)z). Let X = u((Y-(1-«)r)/(1+a))+z) - u(Y-(1-«)r)/(1 +«)) = [V ((CY-2)/(1 +0) +2) - Va((¥-2r/( +a))] + vcd +a)z) + [w(((a¥ +(1-«)r)/(1 +a))+az) - w((«Y +(1-a)r)/(1 +a))]. Then X = [v (2) - V(9)] + vj +e)z) + [w(((aY +(1-a)r)/(1+a))+az) - w((aY¥ +(1-«)r)/(1+a))]. =v (z) + vjCC +e)z) + [W(((aY +(1-a)r)/(1 +a))+az) - w((aY¥ +(1-«)r)/(1 +a))]. By supposition, c,(2r) = cy(2r) = r. Therefore when Y = 2r we must have u(r+k) < u(r) for any z > 0. u(r+z) - u(r) = Vg(2) + w(rtaz) + vjCC +a)z) - w(r) = Vo(2) + VCC. +@)z) + [w(r+az) - wr]. Therefore, u(r+z) - u(r) < O for all r= 0, z > O, and Y = 27 implies X < 0. Therefore u(((Y-(1-«)r)/(1 +a))+z) < u((Y-(1-«)r)/(1+«a)) for any z > 0, which implies r < Cy < (Y- (1-a)r)/(1 +). This proves that theorem 1 and V1-V5, W1 imply w’(r) > v’(c-r) for any r = 0 andc

will more dramatically lower the future reference point; if future consumption is likely to be lower, then such a lowering of the reference point is a positive feature. Note that if a person’s reference point is completely unaffected by current changes in consuniption--if « = 0--then Theorem 2 says that the person will never consume below his refererce level unless his income is so low that consuming above his reference level in the first period risks default.

While Example 2 illustrates that consumption can be sticky both when expected income is below the reference level and when it is above, there is more generally an asymmetry in our model regarding such stickiness. Theorem 2 implies that consumer behavior under loss aversion is asymmetric with respect to the reference level, because there are examples where a consumer will consume above his reference level in response to bad news. Example 3 describes a situation in which, even though the consumer expects income in the second period to be below his reference level, he consumes above his

reference point in the first period.

Example 3: A consumer solves the problem MAX cy) + E{U(ry,C>)} subject to Cy + Cy = Y, + Y, n= 1; ry = Sy + SC} 1s Yy <2 Y, uniformly distributed over [0,.5] r+ (c-r) - 25(c-r)* if c2zr

where U(r,c) = 2 r + 1.05(c-r) + .0625(c-r)" if cr The solution is: ¢, = 1052 + .6667Y, - (-.2126 + .15g0y ,)/? if 1.6944 < Y, <2 C= 1 if 1< Yy = 1.6944

For 1.6944 < Y, < 1.75, Y; + E(Y,) <2= 2r but Cc, > 1 = Ty:

Example 3 can be illustrated graphically as follows:

Figure 3

<,¢v)

QTL CLST]

It is this asymmetry implied by Theorem 2 and Example 3 together that we feel is

the main feature differentiating loss aversion from other models of preferences.

3.3 Consumption Behavior when Income Uncertainty is Increased : Precautionary Savings

income will be above his reference level, he will consume at or above his reference

level.

Thus, when expected income is below the reference level by a small amount, then an increase in the probability that a consumer will be able to consume above his reference level each period will lead him to decrease savings. Part (i) of Proposition 3.3 illustrates this idea. Part (ii), on the other hand, suggests a set of circumstances where: the classical intuition is likely to hold: If expected per-period income is above the reference level by a small amount, then an increase in uncertainty is likely to

increase savings.

Proposition 3.3: Let c, (Yk) be the consumer’s first-period consumption when faced with probabilistic total income (.5 (1-k)Y, .5 (1+k)Y), where k e€ [0,1]. i) There exists Y, < 2+; such that for all Y ¢ (Y4,2r)) and all k for which (1-k):-Y > rh cy) > cy(¥,0). . ii) There exists kK > Oand Y > ary such that for all Y ¢ (2r,.¥ ) and all k<k, c,(Y¥.0) > ¢,(¥,4);

While part (ii) of Proposition 3.3 need not generalize to non-binary symmetric distributions, the idea driving the proof is that when expected average per-period income exceeds your reference point, an increase in uncertainty that raises the odds that you will not be able to maintain your current consumption level will increase your savings. !3 Roughly, this holds whenever the effect of increasing the odds of being forced to consume below one’s reference level is not outweighed by behavioral changes

relating to third-derivative features of the various components of the utility function.

13Theorems 1 and 2 make clear, however, that part (i) generalizes to any symmetric distribution. The fact that part (i) of Proposition 3.3 holds more generally than part (13) seems to be related to the asymmetry in Theorem 2 and Example 3.

4. Discussion and Conclusion

Our exploration of the savings problem calls for a few natural extensions. We would like to extend the analysis to a multiple-period or infinite-horizon mocel. To do this, we would need to add discounting and positive interest rates to the model. Given our analysis, one can make an informed guess about how results would be altered under these extensions. No matter how many periods there are, for instance, it is staightforward to show that, if « = 0, an agent will never consume below his reference point unless his income absolutely forces him to. As a rises, a consumer will be more willing to consume below his reference point, just as in the two-period model.

As we stated in the introduction, we feel that loss aversion can usefully be incorporated more generally into economics. Benartzi and Thaler (1992) invoke loss aversion in explaining under-investment in risky assets. And Sheffrin and Statrran (1985) find that trading volume on the stock market falls when stock values fall, which they

interpret as due to loss aversion--investors are unwilling to realize the loss on their

14For discussions of precautionary savings in the standard framework, see Sibley (1975), Miller (1976), Zeldes (1989), and Caballero (1990). 20

of economic situations.

a 15These examples, and indeed the savings example in this paper, all have a similar flavor to the results found in some studies in non-economic settings. McGlothlin (1956) documents the tendency of race track bettors to bet increasingly long shots as the betting day goes on, behavior that he interprets as an attempt to end the day without a loss. Thaler and Johnson (1990) discover a similar pattern in a different gambling

conte xt. 21

Appendix: Proofs Note: In all proofs, we denote the function v(x) by v&) when x = 0, and by V(x) when x 2 0.

Proof of Proposition 2.1: Assumption V4 implies that there exists 5 > 0, y > 0 such that vi) - v,(0) > y > 0 for all y € (-8,0]. Let G and H denote the c.d.f.’s of g and h, and let y; = be the lowest y realized with positive probability by either f or g. Set « = -3/Y;. Then k < ec implies that

i) E{v(k-g)} = Ny vkx)dG(K) + Sq Vy(kx)dGOX)

Ny [19 vj &s)ds] -dG(x) + 54 Lg Vg(ks)ds] -dG(x)

ee + 59 V(0)-xdG(x), v! a" Sy XxdG(x) + Wyi 0 _ xdG(x) 0 Wve (0)+9]- oe < 0.

ii) E{v(k-f} = Ny yah + Sa oY g(kx)dF(x) < v’ (9) UY, x-dF(x) + "° o x dF(x)]

= v0) . Jy 4F QO).

Set a(p) = p-vi(0)-J9xdF(x) + (1-p)-a-JY, x-dG(0). Then a(1) > 0, since f is better-than-fair; a(O) < 0. Thus, there exists pD- > 0 such that ap ) = 0 and a(p) < 0 for all p < D. Sop < D. and k < e imply that E{v(kh(p)} = p-E{v(f)} + (1-p)- Efv(kg)}. But this is less than a(p), which is negative, so you reject h(p) scaled by k < e. Q.E.D.

Remark on Proposition 2.1: This proposition would also be true even if the supports of f and g contained infinitely negative outcomes, so long as the expected value of losses were finite for

each of the bets.

Proof of Proposition 2.2: Proof of "only if" direction:

Assume v(-) satisfies V1-V5.

Suppose there exists y) < 0 such that Vv’ Yo) = v'(0). Then v’(y,) < vg(0) for all ¥, < Yo: Therefore, the continuity of Val") implies that v’ (y;) < Vale) for some ce > 0. Thus, by Assumption V2, v’(y) < v‘(e) for all y < yy and v’(e) - v’(y) increases as y decreases.

Then wy) - v'(e)-y = JY [v’()-v )lds

= 59 [v(e)-v/(s)]ds + S%0 [v’ (e)-v’ (s)]ds Yo y and lim, 5 v(y) - v’(e):y = ®.

In particular, we can choose y < 0 sufficiently negative such that v(y) - v’(e)-y > 2v(e). _. . Set p = e/(e-y). Then i) p-y + (1-p)-e = 0, and ii) p-v(y) + (1-p)-v(e) . > p-[v'(e)-y + 2v(e)] + (1-p)-w(e) > -e-v'(e) + wep + 1, (since 0 > p-y = -(I-p):e > -e.) = v'(e) - e-v/(e) + p-v(e) > 0 (by Assumption V2). Proof of "if" direction: Suppose v’(y) > vg(0) for all y < 0. (Choose any y,z such that p-y + (1-p):z = 0. Thus, z = -p-y/(1-p). Then p-v(y) + (1-p)-v(z) < p-v/(0)-y + (I-p):v(z) (since v’(y) > v0) < p-v,(0)-y - {(1-p)-p/C-p)]-v,)-y = p-v,(0)-y - P-v,(0)-y = 0 Q.E.D.

Proof of Theorem 1: For any given T, and Y, substituting Y- Cy for Cy and ac, + (1-o)ry for r, we

can write UC, 45¢yF9) as a function of Cy alone: u(c;) = w(t) + v(c4-Ty) + w(ac, +(1-a)r 4) + vV(Y - (it+a)c, - (1-«)r;)

* * We will let c; denote the cy which maximizes uc,), and let Cy = Y - Cy. If we impose the non-negativity constraint, we say that Cc, is feasible if c, « [0,Y]; if we do not impose the non-negativity constraint, then any C; is feasible.

(1) Claim: c,(¥) = c4(¥) = when Y = 2ry-

Proof: When Y = 2), ife, > Ty then c, < [, and ifc, < Ty then c., > [- If Cy > ry and ) <1 then u’ (C,) = Voley Ty) + aw’ (f5) - (1+a)vj(¥-C4-Fy). Bv VW1 this is negative, sO that the consumer will decrease Cy. If Cy <r and ) > My) then u’ (cy) = v4 C47) + aw’ (I) - (1 +a)vg(Y-CyF9)- By VW1 this is positive, sO that the

consumer will increase Cy.

(2) Claim: c,(Y) and c(Y) are continuous and strictly increasing when Y > 2ry> and both converge to 7, when Y approaches 2r, from above.

. Proof: Assume Y > 2r;- We know from the above argument that at the optimum, cB and Cy > ly; this implies m1 =o < (Y-(1-a)r, Wi +e). Note that u(c;) is strictly concave in this region. If c, = > then, by VW1,

u’ (cy) = v4 (0) + aw’ (T4) - (1+a)vi(¥-2r4) > 0. Therefore, C, > ry if Y > 2ry- Now consider Cc, = [5 this implies c, = (Y-(1-a)r,)/( +2). Then

u’ (cy) =V o((¥-2r yO +a)) + aw’ ((aY + (1-a)r)/G +a)) 7 {t +a)v/ (0). If u’(c,) is greater than zero when evaluated at this point, then c, = (Y-(1-a)r, /G +a) and C) = (a¥ +(1-a)r,)/G +a). As Y increases, U’ ((Y-(1-a)r)/(1 +2) decreases continuously by the concavity of Vv g and w, and by the assumption that Vg and w’ are continuous.

If u’ ((¥-C-a)ry)/ +0) < 0, then there is a Cy ra (r, (Y-C-a)r 0 +99] such that w (c,) = 0. Thenc, = Cy and Cy = Y - Cy: We know Cy is a continuous function of Y since v’ and w’ are continuous. Totally differentiating the equation u’ (C,) = 0 yields Wdta) > de, /d¥ > 0.

Therefore, cy) and c4(Y) are continuous and strictly increasing, and since (Y-

(1-a)r, /. +a) > r, as Y- ary we have c,(¥) and c4(Y) — 1; as Yo ar, from above.

(3) Claim: If Y < 2r, and negative values of Cy and C» are allowed, then c,(Y) = (Y - (1-a)r, /(1 +e). If Y < 2ry and the constraints C= 0 and Cy = 0 are imposed, then c,(¥) = max([0,(Y-(1-«)r,)/( +«)].

Proof: From the proof of (1), we know that Cj <r) and C5 = Ty, SO that (Y-(l-a)r,)/Q +a) 5 C) = 7). Consider cy é [(Y-(1-a)r +a), (¥ tar, )/(2+0)]. In this region ¢,-ly = CT = 0, and so we have

u’(c,) = Vi (ey-7y) + aw’ (ac, +(1-«)ry) - (1+a)vj (Cy) < 0 by VW1 and V2. Now consider the region cy) « ((Y +ar,)/(2 +a), rl. In this region we have Ozc,-% > Cy - My. We can rewrite this range of possible choices for cy as (Y-2r,)/(2 +e) <c)- 7 5 0. Consider any choice of Cyr) =z in this range, then u(r, +Z) =v, (Z) + w(az+r}) + v(¥-C. +a)z-2r4) is the utility from this choice. Compare this to the choice of Cory = 2 (This choice implies a Cy which lies in the range Cy E [(Y-(-e)r )/ +a),(¥ +ar,)/(2+a))). If Cc, = +z is feasible, then Cy = I, +Z is also feasible. The choice c, = t,+Z results in utility

u(Y-r-Z) = v(¥-(1 +a)z-2r, +k) + w(az +r) -k) + v,@), where k = a{2r,-¥ +(2+a)z]/(1-«) > 0. Then

u(Y-T-Z) - u(r, +Z) =

[v,(v- +a)z-2r, +k) - v(¥-. +@)z-2r7 )] + [w(az+r,-k) - w(az+r;)I. VW 1 implies vj(x+q@) - vj) > w(s+q) -w(s) for any q > 0, x = -q and s 2 O. Therefore u(Y-14-Z) > u(r, +z) for any z such that (Y-2r,)/@ +a) < z <0. This implies that for any feasible choice of cy) «€ ((Y +or,)/i(2+a), ry] there is a feasible choice of Cc, é [(¥-(L-a)r (1 +a). tor, )i(2+@)] which is preferred. Therefore

ct E [(¥-(1-a)r (+a), (¥ tar, (2 +a),

and therefore if negative values of c,and Cy are allowed, ¢,(Y) = (Y-(1-a)r )/( +e); if Cy < 0 and C5 < 0 are not allowed, then c,(¥) = Max(0,(¥-(1-«)r,)/( +a)].

This establishes Claim (3); Claims (1), (2), and (3) together establish the theorem. Q.E.D.

Remark on Theorem 1: The proof above shows that c,(¥) is never steeper for Y/2 > r, than it is for Y/2 <r 1

Proof of Proposition 3.1: For any given ry and Y, substituting Y - Cy for c, and acy + (1-a)r, for r5, we can write UC] .13¢9,Fp) as a function of Cy alone: u(c;) = w(r,) + v(C4-Ty) + Ww(ac, +(1-«)r;) + WY - (l+a)c, - (1-a)r;) We will let cy denote the Cy which maximizes ucc,), and let ch =Y- cj. Suppose that there is some 7 = 0, c = 7, such that v’(c-r) = w’(r). Then by W1 for any r > randc2c > r we have v’(c-r) > w’(r). Let moar and take Y 2: 2r)- Theorem 1 implies that c, =r and Cy =r If cj =r then u’(r) = v ‘(O) + aw’(r - (1 to)vi(¥-27). Then Limy .» r u’(r) = a:[w’(r) - v’(0)] < 0. Because v’ and w’ are continuous, this implies that there is some x > O such that for allO < y= x, u(r) = vg(9) + aw’(r) - (1t+a)v’(y) < 0. ucc;) is concave in the region r = cy) = (Y-(1-a)n)/(1+a). Therefore if cy lies in this region, cy = r for Y e [2r,2r+x]. This means c,(Y) is not strictly increasing in this region, which contradicts Theorem 1. To see that Cc, « [r,(Y-(1-a«)r)/(1+«)] note that by supposition r = CF = Y-r. Then u(c, = Y-(l-a)r)/(1+a)) = w(r) + Vo((¥-21)/(1 +0) + w((aY +(1-«)r)/(1+a)). Compare this to uc, = (Y-(1-«)r)/(1+a) + z), where z > 0: u(Y-(1-«)r)/(1+«)+z) = wi) + v_((CY-2r)/(1+a))+z) + w(((aY +(1-a)r)/(1 +a))+az) + vj. +a)z). Let X = u((Y-(1-«)r)/(1+a))+z) - u(Y-(1-«)r)/(i +@)) = Lv ((¥-2r)/(1 +a))+Z) - Vv g(¥-2r/(1 +a))] + sige! +a)z) + [w(((aY +(1-a)r)/(1+a))+az) - w((aY +(1-«)r)/(1 +«))]. Then X = [v (2) -v g] + vj CC +a)z) + [w(((aY +(1-«)r)/(1+«))+az) - w((aY +(1-«)r)/(1+«))]. = Vg(2) + vcd +«)Z) + [w(((aY +(1-«)r)/(1+a))+az) - w((a¥ +(1-a)r)/(1 +«))]. By supposition, c, (27) = cy (2r) = r. Therefore when Y = 2r we must have u(r+k) < u(’) for any z > 0. u(r+z) - u(r) = V gi) + w(r+az) + vcd +a)z) - w(r) = v_(z) + vjCC. +a)2z) + [w(r+az) - w(n)]. Therefore, u(r+z) - u(r) < 0 for all r= 0, z > 0, and Y = 2r implies X < 0. Therefore u(((Y-(1-@)r)/(1+a))+z) < u((Y-(1-«)r)/(1+«)) for any z > 0, which implies r = C) < (Y- (1-a)r)/(1 +a). This proves that theorem 1 and V1-V5, W1 imply w’(r) > v‘(c-r) for any r = 0 and c

> r. Given that w and v, are strictly concave, this can be true if and only if U(c,c) > U(r,c) for any rt = Q and c > I. Q.E.D.

Proof of Proposition 3.2: For any given iT and Y, substituting Y - Cy for Cy and acy + (1-o)ry for Ty, we can write UC, 14 3€2-7y) as a function of Cy alone: -

u(c,) = wiry) + veyrp) + w(ac, +(1-a)ry) + VY - (L+a)e, - (1-0)ry)

* * * We will let c, denote the cy which maximizes u(c,), and let Cy = Y-Cy. Consider the case in which negative values of Cy and Cy are allowed, so. that

theorem 1 implies that c,(¥) and c4(Y) are continuous and strictly increasing for all Y.

* Consider Y < 2ry- By supposition c; <1) and Cy = Ty. We can show that

(Y-(1-a)r,)/G +a) <= cy < Ty. If Cy < (Y-(1-o)r,)/G +a) then C, > [5, SO that u’ (Cy) = v4 (C4-Fy) + aw’ (ac, +(1-2)r 4) (1 +a)v,(¥-(1-a)ry (1 +a)e4) > 0 by assumption V5 and the proof of Proposition 3.1. Over the range (Y-(1-o)r,)/C +a) = es uc;) = vyey tp + w’ (ac, +(1-a)r4) + v4 (¥-(L-a)ry-C. +a)cy) In this region uc,) need be neither concave or convex. Suppose c; is in the interior

of this region. For this to be so uc,) must be concave in some neighborhood of cy. At

ok such an optimum we must have u’ (c,) = 0, Totally differentiating this equation with *

respect to c, and Y implies that dc,/4¥ = [(1-t avs (Y-(1-0)ry-(1-+a)eq V0 () < 0, which contradicts the claim that c,(Y) is strictly increasing. Therefore there can be

no interior optimum, which implies that either Cj = 1%, or cj = (Y-(1-a)r,)/( +a). Since

* ns also violates the claim that c,(Y¥) is strictly increasing, we must have

ch = (¥-(-a)ry/(1 +e). | Ci = (Y-(1-«)r, )/ +a) implies that u((Y-(1-a)r,)/( +a)) 2 uCc,) for any C). In particular we must have X = u((¥-(1-«)r,)/( +a)) - u(r) > OifY < 2ry- Writing out the terms of X we have: X = vy((¥-2r)/0 +a)) + w(z) - w(z + k) - v((¥-2r)/ +a)) -k) > 0,

where k = a(2r,-Y)/( +a) > 0, andz = 1, - k. Since

[vy(0) - vO] > [vy((Y-2r,)/(. +0) - vy((¥-2r (+a) - W)]), X > 0 for any T= QOand Y < 2r, implies -v,Ck) > w(z + k) - w(z) for any k > 0, z20. This last inequality implies U(c,c) > U(r,c) for any c = 0 andr > c. Q.E.D.

Proof of Theorem 2:

We prove the proposition for lotteries that are represented by probability measures of the form m = fa + d, where A is Lebesgue measure on R, f: R—> R is a measurable function, and d is a measure which has support consisting of a countable collection of point masses with no point of accumulation.

Recall the consumer’s problem: (P) mar, Ole 5t2.¢y) S.t. ct Cy = Y= Y, + Y>; where Y, is a fixed amount representing first-period income, and Y> is a lottery over second-period income.

The proof has two parts. First we show the theorem for binomial bets in which Y, takes on only two values, Yy and Y,- Then we show that bets of this type can be combined, to yield the result for all Y in the class described above.

Part I of proof:

Suppose Y,, is such that P[Y,, =Yqy) =p, PLY, =Y,] = 1-p, where (Y yt Ypp/22r, and p= («/(1+«)). For binomial bets Y>, this is equivalent to the hypothesis of the proposition. We show that c) = 1. To prove this, we consider two cases:

Case 1 of Part I:

(Y, +Y,)/2 =1)- Substituting Yy + Y,5- cy for Cy and (1-e)r, + Cy for Ty, we can rewrite (P) as a function of c,:

(P2) max UC,) = w(r,) + v(c)-r,) + w((1-«)r ytec;) +

p: vV(Y, +Y yo (-)r yt(1+«)c,)) + (1-p)- vW(Y, +Y pdr jt +e)ec,)). Then Cc, < i” implies, by “Assumption VW1, that aU(c - (1-p)(1 +«)v/ mes +¥7° ~((1-e)r, + asoe,) is greater than zero. Thus maximization in Cy implies c Case 2 of Part I:

(Y,+Y¥p2 = ry > (Y,+Y,)/2

1771:

ce, < min{r, (CY, +Y, -(-~)r,)/. +9}: pe TY erty) tw! (yt +904) - pO ta)vi(Y, +Y¥pp-((1-)ty +1 +5)e4) -(1-p)(1 teaver y +Y¥, -(l-«)r, + t+a)c,)) > 0 by assumption VW1. li) T= C= (Y, +Y, -C-«)r /1 +«) Let UCc,) =X) + Xo, where x, = (-p) +)v(C,-1,) +C-p)v(Y, + ¥y -(-odry +1 +094) X» = [1-(1-p)(1 to)]v (Cy) +w((l-)ry +c 1) + PV (CY + ¥ pe (Cl-ory +C1 +2)e4)) Then if oq < ty gee = (L+odpodvj (cpr) tow! (Loar toe) - Ut edpvg(¥y +¥q (dry ++ ey)

> [a +o)p-o]vy (Cy -tp)10 +o)p-eolv (Yt ¥ y(n +1 +«)C,)) = 0 since p = («/(1+«)). So X» is maximized in this region at C) =).

Bel = Cp M1 +9)v4 -7)-C-Pd Foy (XY, FY -(C-ry +0 +4)

2 aX = (1-p)(1 +o)vi(C-Ty) + (1-p)(1 +0)*vi(¥ FY, -(l-o)r + +@)c,)) > 0 dc, dc 1

Since Xy is convex in this region, it has exactly two local maxima, at cq =r and at C1) = (Y, + YY) -C-o)r /(1 +) respectively. But by V2

x (¥, +; -C-«)r)/. +) = (1-p)( +e)v, (CY) +¥y - 2r,)/(1+)) < (1-p)v,(Y, + Y; -2r,) = X,(,)

Thus xy is strictly maximized over the region at C= Ty. Since xy and X» are both

maximized at c, = r,, U(c,) is strictly maximized over [Y, + Y, -(1-«)r,)/(1+«), r,] atc 1 1 1 1° °L 1 1 1 Case 1 and Case 2 imply that (P2) is maximized at c) =r.

Part II of the Proof: Case 1 of Part II: Suppose Y., is a countable sum of point masses with no accumulation point:

i=N, j= N. Y, =P ps6, + ¥ q:6 2421 TH a1 FY;

Pi 4, > 0, x; = 0, yj < 0; i>i’,j|>yj' 2X > Xi, 5 Yj < Yjei N., N_eNvu {o}.

Claim: Because P[Y = 2r,] = a/(1 + a), we can rewrite Y, in the form

i=M_ 1=M°5 jJ=Q_ Y, = Dioé~ +P iJa5 +¥ 5.8, 2 21 1X isi) Yi Ga 12;

1> *M }7J2 *4> 4 2, > Zu, Vi <5 y,. Proof of Claim: The proof of this is tedious, but straightforward. Match successively lower outcomes with successively higher outcomes in the proportion of 1/c

z z

to 1. Because the support of Y> has no point of accumulation, this countable process

eventually exhausts all of the negative weight of Y>.

= ~ j=Q_

_ a + Ia) Pp fa/(l + O3z + 1/1 + “35 J +y _| S85,

Each term in brackets [] is a binomial bet (the oz are trivially so) satisfying the J

i Then Y, = a

hypothesis of Proposition 3.2. (P2) max Uc,) 1

M Uc.) = z {a + La) Bj} bw(r,) + v(cy-t)) + w(t) +

(UA +) +¥ (C9) + +e)ey)) + oll bevy, + ¥ -(1-«)r, +0 +3)e,))] +E 8 lwp + v(cy-1,) + w(t) + WY, +2-((1-«)ry +(1 +«)e,))].

From Part 1, we know that each Square bracketed term in (P2) takes on a large value at c=, than at Cc < Ty. But this implies that Ur) > UC,) for Cc) < Ty; and hence that (P2) is maximized at c) =0). Case 2 of Part II:

Suppose Y, has a density with respect to Lebesgue measure in the sense that there is a measurable function h such that PLY, = x} = fs h(x)dx. Further suppose

va y h(x)dx = «/(1+«). Let H(a) = 5 “h(x)dx be the cumulative distribution function 11 for h. Define g(a) = inf H'[(«+1)H((2r,-7 )-oH(@)]. Then (2r,-¥) q(a) a hcx)dx = hx)dx. (%) a @r¥)) Then q is increasing and hence measurable.

Claim: For measurable f:R —> R, aS f(q(x)h(x)dx = fs f(x)h(x)dx . [o,q(a)] Proof of Claim: The statement holds for simple functions f (those taking a finite number

of values), since we can then break the integrals down into the sum of integrals over regions [a;,b;] so that f is constant on both [a;,b;] and [q(b;),q(a;)]. On_ these regions the equalities follow from (%). Approximate f by simple functions fy 0 =< fi, < f, so that f. ne) — f(x) for all x e R. Then we have:

ass f(qx)ndx = S f (x)n(x)dx. (%%)

[0,g(a )] Since q is increasing, f I) is a simple function defined on (-,0]. Clearly 0 =

f (4) = f(q(x)), and f,(qx)) — f(q(x)) for all x e€ (-»,0]. Applying Lebesgue’s theorem of monotone convergence to both sides of the equation (%%), we are done. Finally,using the claim, we can rewrite the consumer’s problem:

(P2) max (i+0f twee) + v(cy- ty) + W(t) + WL +a)v(¥, +x-c)-1y) Cy ~00 + al(1+a)v(Y 1 + q(x)-C 1-5) a@x)dx

+ Jc) [w(r)) + v(c,- Ty) + w(t») + V(Y¥ | +X-Cy- 5 )Ih(x)dx.

In the first integral, Y; + q(x) = 2r, while in the second integral, Y, + x2 2ry. So each term in brackets [] represents the utility from a binomial lottery on income satisfying the hypothesis of Proposition 3.2, and hence is larger at csr, than at any C, < ry: That is, (P2) is maximized at cy) = Ty.

Q.E.D.

Proof of Proposition 3.3:

Part (i) is a straightforward implication of Theorems 1 and 2. By the proof of Theorem 1, we know that c,(¥,0) < ry if Y < 2-1). But Theorem 2 says that if we choose Y, such that Y,(1+k) = 2-ry, then c,(Y,4) =r) for all Y = Y,, as long as (1-k)Y = ry:

Proof of part (ii): The consumer will maximize with respect to Cy Us= W(a-c, + (1-a) +7) + v(cy-r}) + .5:v(Y(1+k) - (l+a)-c, - (1-o)-r;) + 5-V(¥(1-k) - (1 +a)*Cy - (1-a)-7}).

By VWI, we know that we can choose Y > a+r; sufficiently close to ar; such that, for all Y e« (2-r,,¥ ),

(1 Fav, (0) < Vi((¥-27 +a)) + a-w' ((a- ¥+(1-a)-7r))/(1 +e). . As in Theorem 1, this implies that c,(Y,0) = (Y-(1-a)-1,)/(. +a). By VW1, we can take k sufficieritly small such that, for all k € [0,k ),

((1 +a)/2)- (vi C2KY) + Vg(2k¥)) > v5(9) + a-w’ (})-

Claim: c,(Y,k) = ((1-k) ¥-(1-a) 74 )/ +a) < (Y-(1-«)-7))/(. +a) = C1 (Y,0).

Proof of claim: Note first that, since Y > 2r 2 provided k = 1/2. a) Suppose that (C1-k)Y-(1-a) +7 /(1. +a) < Cy < (1 +k)Y-(1-a) +r )/(1 +0). Then aU/ac, = vey") + a-W’ (ro) - (C1 +4)/2): [vi (1-K)¥-(1 +a) +c, -(1-a)ry) + VC +K)¥-C1 +a), -(l-a)r, J] < V0) + a-w'(r;) - ((1+a)/2)- (v5 (-2kY) + Vg(2k¥)) < 0. (The second-to-last inequality follows from the fact that, for Cy in this region, alvi(cy-r) + aw’ (r,)\/ac, < 0, [(1-4)Y-(1 +a)°C,-(1-a)ry] e (-2kY,0) and [1 +k)Y-(1 +a)-C4-(l-a)r;] € (0,2kY).)

y we have (1+k)Y > 2r)- Thefore, cer by Theorem

b) Suppose that Cc, > [A +k)Y - (l-e)r VC +a). Then aU/ac, = Vo(CqFy) + aw’ (ry) - +a)/2)- [vi (C1-k)Y-(1 +a)-c,-(1-a)r}) + vid +k)Y-(1 +a)c,-(1-a)r, J], which, by VWI, is negative. (a) and (b) together establish the claim, which proves Proposition 3.2.

Q.E.D.

REFERENCES

Arrow, K. J. (1974), Essays in the Theory of Risk Bearing, Amsterdam: North Holland.

Becker, G., and Murphy, K. M. (1988), "A Theory of Rational Addiction," Journal of Political Economy 96, 675-700.

Benartzi, S. and Thaler, R. (1992), "Myopic Loss Aversion and the Equity Premium Puzzle," manuscript, Johnson Graduate School of Management, Cornell University, October.

Boyer, M. (1983), "Rational Demand and Expenditure Patterns Under Habit Formation," Journal of Economic Theory 31, 27-53.

Caballero, R. (1990), "Consumption Puzzles and Precautionary Savings," Journal of Monetary Economics 25, 113-136.

Campbell, J. and Mankiw, N. G. (1989), "Consumption, Income and _ Interest Rates: Reinterpreting the Time Series Evidence," NBER Macroeconomics Annual, 185-250.

Constantinides, G. M. (1990), "Habit Formation: A Resolution of the Equity Premium Puzzle," Journal of Political Economy 98, 519-5430.

Detemple, J. B. and Zapatero, F. (1991), "Asset Prices in an Exchange Economy With Habit Formation," Econometrica 59, November, 1633-1657.

Duesenberry, J. S. (1952), Income Saving and the Theory of Consumer Behavior, Cambridge, MA: Harvard University Press.

Fershtman, C’. (1993), "On the Value of Incumbency: Managerial Reference Point and Loss Aversion," Discussion Paper No. 1020, Center for Mathematical Studies in Economics and Management Science, Northwestern University, January.

Ferson, W. and Constantinides, G. (1991), "Habit Persistance and Durability in Aggregate Consumption; Empirical Tests," Journal of Financial Economics 29, October, 199-240. Frank, R., and Hutchens, R. (1990), "Feeling Better versus Feeling Good: A Life-Cycle Theory of Wages," Working Paper, Department of Economics, Cornell University.

Hall, R. (197'8), "Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence," Journal of Political Economy 86, 971-987.

Kahneman, D., Knetch, J., and Thaler, R. (1991), "Anomalies: The Endowment Effect, Loss Aversion, ancl Status Quo Bias," Journal of Economic Perspectives 5, Winter, 193-206.

Kahneman, D., and Tversky, A. (1979), "Prospect Theory: An Analysis of Decision Under Risk," Econometrica 47, March, 263-291.

Kimball, M. S. (1990a), "Precautionary Saving in the Small and in the Large," Econometrica 58, 53-73.

Kimball, M. S. (1990b), "Precautionary Saving and the Marginal Propensity to Consume," NBER Working Paper #3403, July.

Leland, H. (1968), "Saving and Uncertainty: The Precautionary Demand for Saving," Quarterly Journal of Economics 82, 465-473.

Loewenstein, G. and Prelec, D. (1992), "Anomalies in Intertemporal Choice: Evidence and an Interpretation," Quarterly Journal of Economics 107, 573-598.

Loewenstein, G. and Sicherman, N. (1991), "Do Workers Prefer Increasing Waye Profiles?"

McGlothlin, W. H. (1956), "Stability of Choices among Uncertain Alternatives," American Journal of Psychology 69, 604-615.

Miller, B. (1976), "The Effect on Optimal Consumption of Increased Uncertainty in Labor Income in the Multiperiod Case," Journal of Economic Theory 13, 154-167.

Pratt, J. (1964), "Risk Aversion in the Small and the Large," Econometrica 22, 122-136.

Ryder, H.E. and Heal, G.M. (1973), "Optimal Growth with Intertemporally Dependent Preferences," Review of Economic Studies 40, 1-33.

Samuelson, P. (1961), "The St. Petersberg Paradox as a Divergent Double Limit," International Economic Review 1, 31-37.

Samuelson, W. and Zeckhauser, R. (1988), "Status Quo Bias in Decision Making," Journal of Risk and Uncertainty 1, 7-59.

Segal, U. and Spivak, A. (1990), "First Order Versus Second Order Risk Aversion," Journal of Economic Theory 51, 111-125.

Shea, J. (1993a), "Union Contracts and the Life Cycle-Permanent Income Hypothesis," forthcoming, American Economic Review.

Shea, J. (1993b), "Myopia, Liquidity Constraints, and Aggregate Consumption: A Simple Test," forthcoming, Journal of Money, Credit and Banking.

Sheffrin, H. and Statman, M. (1985), "The Disposition to Sell Winners Too Early and Ride Losers Too Long," Journal of Finance, 777-790.

Sibley, D. (1975), "Permanent and Transitory Income Effects in a Model of Optimal Consumption with Wage Income Uncertainty," Journal of Economic Theory 11, 68-82.

Sundaresan, S.M. (1989), "Intertemporally Dependent Preferences and the Volatility of Consumption and Wealth," The Review of Financial Studies 2, 73-89.

Thaler, R. (1980), "Toward a Positive Theory of Consumer Choice," Journal of Economic Behavior and Organization 1, 39-60.

Thaler, R. (1985), "Mental Accounting and Consumer Choice," Marketing Science 4, 199- 214, Summer. 34

Thaler, R. (1990), "Anomalies: Saving, Fungibility, and Mental Accounts," Journal of Economic Perspectives 4, Winter, 193-205.

Thaler, R., and Johnson, E. (1990), "Gambling with the House Money and Trying to Break Even: The Effects of Prior Outcomes on Risky Choice," Management Science 36, 643-660.

Tversky, A. and Kahneman, D. (1991), "Loss Aversion in Riskless Choice: A Reference- Dependent Model," Quarterly Journal of Economics, November, 1039-1061.

Tversky, A. and Kahneman, D. (1992), "Advances in Prospect Theory: Cumulative Representation of Uncertainty," Journal of Risk and Uncertainty 5, November, 297-323.

Zeldes, S. (1989), "Optimal Consumption with Stochastic Income," Quarterly Journal of Economics 104, May, 275-298.

IFDP Number

492

491

490

489

488

487

486

485

484

483

482

481

480

International Finance Discussion Papers

Titles

1994

Loss Aversion in a Consumption/Savings Model

Terms-of-Trade Uncertainty and Economic Growth:

Are Risk Indicators Significant in Growth Regressions

Politics, Economics, and Investment: Explaining Plant and Equipment Spending by U.S. Direct Investors in Argentina, Brazil, and Mexico

On The Dynamic Properties of Asymmetric Models of Real GNP

A distributed block approach to solving near-block-diagonal systems with an application to a large macroeconometric model

Conditional and Structural Error Correction Models

Bank Positions and Forecasts of Exchange Rate Movements

Technological Progress and Endogenous Capital Depreciation: Evidence from the U.S. and Japan

Are Banks Market Timers or Market Makers? Explaining Foreign Exchange Trading Profits

Constant Returns and Small Markups in U.S. Manufacturing

The Real Exchange Rate and Fiscal Policy During the Gold Standard Period: Evidence from the United States and Great Britain

The Debt Crisis: Lessons of the 1980s for the 1990s

Who Will Join EMU? Impact of the Maastricht

-Convergence Criteria on Economic Policy Choice

and Performance

Author(s)

David Bowman Deborah Minehart Matthew Rabin

Enrique G. Mendoza Guy V.G. Stevens

Allan D. Brunner

Jon Faust Ralph Tryon Neil R. Ericsson

Michael P. Leahy Robert Dekle

John Ammer Allan D. Brunner

Susanto Basu John G. Fernald

Graciela L. Kaminsky Michael Klein

Graciela L. Kaminsky Alfredo Pereira

R. Sean Craig

Please address requests for copies to International Finance Discussion Papers, Divisior. of

International Finance, Sto Washington, D.C. 20551.

p 24, Board of Governors of the Federal Reserve System,

IFDP Number

479

478

477

476

475

474

473

472 471

470

469

468

467

466

International Finance Discussion Papers

Titles

1994 Determinants of the 1991-93 Japanese Recession: Evidence from a Structural Model of the Japanese Economy

On Risk, Rational Expectations, and Efficient Asset Markets

Finance and Growth: A Synthesis and Interpretation of the Evidence

Trade Barriers and Trade Flows Across Countries and Industries

The Constancy of Illusions or the Illusion of Constancies: Income and Price Elasticities for U.S. Imports, 1890-1992

The Dollar as an Official Reserve Currency under EMU

Inflation Targeting in the 1990s: The Experiences of New Zealand, Canada, and the United Kingdom

International Capital Mobility in the 1990s

The Effect of Changes in Reserve Requirements on Investment and GNP

International Economic Implications of the End of the Soviet Union

International Dimension of European Monetary Union:

Implications For The Dollar

European Monetary Arrangements: Implications for the Dollar, Exchange Rate Variability and Credibility

Fiscal Policy Coordination and Flexibility Under European Monetary Union: Implications for Macroeconomic Stabilization

The Federal Funds Rate and the Implementation

of Monetary Policy: Estimating the Federal Reserve’s Reaction Function

Author(s)

Allan D. Brunner Steven B. Kamin

Guy V.G. Stevens Dara Akbarian Alexander Galetovic Jong-Wha Lee Phillip Swagel

Jaime Marquez

Michael P. Leahy John Ammer Richard T. Freeman Maurice Obstfeld

Prakash Loungani Mark Rush

William L. Helkie David H. Howard Jaime Marquez

Karen H. Johnson

Hali J. Edison Linda S. Kole

Jay H. Bryson

Allan D. Brunner

IFDP Number

465

464 463

462

International Finance Discussion Papers

Titles 1994

Understanding the Empirical Literature on Purchasing Power Parity: The Post-Bretton Woods Era

Inflation, Inflation Risk, and Stock Returns

Are Apparent Productive Spillovers a Figment of Specification Error?

When do long-run identifying restrictions give reliable results?

Author(s)

Hali J. Edison Joseph E. Gagnon William R. Melick John Ammer

Susanto Basu

' John S. Fernald

Jon Faust Eric M. Leeper

Cite this document

APA

David Bowman, Deborah Minehart, & and Matthew Rabin (1994). Loss Aversion in a Consumption/Savings Model (IFDP 1994-492). Board of Governors of the Federal Reserve System, International Finance Discussion Papers. https://whenthefedspeaks.com/doc/ifdp_1994-492

BibTeX

@techreport{wtfs_ifdp_1994_492,
  author = {David Bowman and Deborah Minehart and and Matthew Rabin},
  title = {Loss Aversion in a Consumption/Savings Model},
  type = {International Finance Discussion Papers},
  number = {1994-492},
  institution = {Board of Governors of the Federal Reserve System},
  year = {1994},
  url = {https://whenthefedspeaks.com/doc/ifdp_1994-492},
  abstract = {Psychological evidence indicates that a person's well-being depends not only on his current consumption of goods, but on a reference level determined by his past consumption. According to Kahneman and Tversky's (1979) prospect theory, people care much more about losses relative to their reference points than about gains, are risk-averse over gains, and risk-loving over losses. We define these characteristics as loss aversion. We incorporate an extended form of loss aversion into a simple two-period savings model. Our main conclusion is that, when there is sufficient income uncertainty, a person resists lowering consumption in response to bad news about future income, and this resistance is greater than the resistance to increasing consumption in response to good news. We discuss some recent empirical research that confirms this predicted asymmetry in behavior, which seems inconsistent with other models of consumption.},
}