Inter-temporal choices with temporal reference dependence

Research in Economics 73 (2019) 107–122

Contents lists available at ScienceDirect

Research in Economics journal homepage: www.elsevier.com/locate/rie

Research Paper

Inter-temporal choices with temporal reference dependence Hyeon Park Department of Economics and Finance, Manhattan College, 4513 Manhattan College Pkwy, Riverdale, NY 10471, United States

a r t i c l e

i n f o

Article history: Received 15 January 2018 Accepted 26 January 2019 Available online 30 January 2019 JEL classification: D83 D91 D14 Keywords: Loss aversion Reference dependent preference Belief updating Intertemporal choice

a b s t r a c t I develop an intertemporal choice model for rational deviators whose preferences depend not only on their actual consumption but also on comparison to their beliefs about the optimal consumption. The standard decision maker is loss averse with respect to this beliefdependent reference point. When psychologically weighted loss aversion is low, a decision maker deviates from the standard intertemporal choice behavior and over-consumption, as well as the alternative possibility of under-consumption can be rationalized. When the decision maker has time-varying degrees of loss aversion, he re-optimizes the consumption plan through adjusted beliefs as subsequent selves realize that past decision for the present period is no longer optimal. In the dynamic model, I solve for consistent intertemporal optimization rules by which a dynamic deviator should meet rational intertemporal consistency at each point in time. Finally, I demonstrate that the dynamic reference dependent model can solve a puzzling feature in lifecycle consumption data. © 2019 University of Venice. Published by Elsevier Ltd. All rights reserved.

1. Introduction In this paper, I develop a dynamic model for individuals who exhibit reference-dependent preference and loss aversion ˝ following the framework of Koszegi and Rabin (20 06,20 09). Reference dependence of utility has been widely confirmed in lab experiments: it helps understand why standard economic theory cannot explain such findings as the failure of the independence axiom in expected utility. As Rabin (20 0 0) demonstrates, reference dependence is an important factor in explaining people’s attitude toward risk. It also explains a variety of field data (major league: Humphreys and Zhou (2015); employment contract: Herweg and Schmidt (2014); tax evasion: Bernasconi et al. (2014); labor supply: Farber (2008), Fehr and Goette (2007); housing market: Genesove and Mayer (2001); finance: Barberis et al. (2006), Karlsson et al. (2009); insurance: Sydnor (2010); poverty: Gunther and Maier (2014)). The choice of utility for a reference point results from the recent development in reference-dependent preference models. A key issue in these models is what determines the reference point of a model, because as Pesendorfer (2006) expressed, the reference point can be anything and may be arbitrarily selected by researchers. Quite often, the reference point is assumed to be the current status, such as current consumption, position, or endowment. Providing one way to solve this problem, ˝ Koszegi and Rabin (2006) propose a model of reference-dependent preferences where the reference point is the individual’s ˝ rational expectation formed in the recent past about the relevant outcomes. Koszegi and Rabin provide a solution concept, by which the reference point is endogenously determined as a function of the decision maker’s beliefs on the available ˝ strategies combined with his planned action for each strategy. In another work, Koszegi and Rabin (2009) propose a dynamic reference-dependent model where they specifically introduce the equilibrium concept by which the decision maker should

E-mail address: [email protected] https://doi.org/10.1016/j.rie.2019.01.004 1090-9443/© 2019 University of Venice. Published by Elsevier Ltd. All rights reserved.

108

H. Park / Research in Economics 73 (2019) 107–122

meet rational consistency: the decision maker chooses the plan for state-contingent behavior that is consistent in each period with his future behavior, and that maximizes his expected reference dependent utility going forward. Based on this solution concept, I develop an intertemporal choice model with reference-dependent preference. I specifically examine the model’s dynamic implication for an individual who is in a specific intertemporal wealth position with respect to lifecycle income and saving/debt position. Given any information about the future income stream, the decision maker forms rational beliefs on what should be the ex ante optimal consumption among all the available consumptionsaving strategies over time. It depends on its consistency whether or not this plan is actually from the expectation that the individual is sure to follow ex post. This is because the decision maker only forms expectations that he will be fulfilling ex post. In other words, a decision maker’s intertemporal choice must be credible. A reference-dependent decision maker derives utility from comparison to the reference status, i.e. the utility from the ex ante optimal consumption: it may be a gain to the reference point or a loss to it. A loss is assumed to be more important to the decision maker than a gain of the same magnitude, a property known as loss aversion. In a two-period consumptionsavings model, the decision maker feels a “contemporaneous gain” utility if his current consumption is more than the ex ante optimal solution. As a result, his consumption will be lowered next period and this yields a “prospective loss” utility relative to the reference point. If the contemporaneous gain utility is greater than the prospective loss utility, he then chooses not to follow the ex ante optimal committed plan but to deviate for more consumption. However, if the prospective loss utility is greater than the contemporaneous gain utility, then he sticks to the ex ante optimal consumption. The decision maker’s greater concern for the loss (high loss aversion) deters him from over-consuming. This analysis can be applied the other way around: another decision maker may have a “contemporaneous loss” utility if she consumes less and saves more than the ex ante optimal solution. As a result, her consumption is elevated for the next period and this gives her a “prospective gain” utility. If her contemporaneous loss feeling is greater than the prospective gain utility, then she does not reduce her consumption but follows the committed plan. If, however, her prospective gain utility is more than the contemporaneous loss utility, then she is willing to reduce her consumption and save more. In this case, the high loss aversion of the decision maker deters her from under-consuming. By both cases, it is clear that the weight on prospective/contemporaneous loss utility relative to gain is crucial. When the decision maker’s loss aversion is high, the agent does not deviate from the ex ante optimal solution, which is equivalent to the outcome of the standard model. However, if the decision maker cares more about the gain due to low loss aversion, then deviation is desirable. The alternative consumption plans which fulfill the agent’s intention of either over-consumption or under-consumption must satisfy the consistency condition so that the ideal consumption should maximize the ex post utility among all the feasible and consistent strategies to consider.1 Constructing a multi-period dynamic model of intertemporal choice with this preference involves how to construct the multi-period reference points from overlapping layers of belief formation, as well as how to define an economic model that conveys the idea of “intended consumption” described above. These issues occur because the framework of the model is absolutely dependent on description, and admits many different formats for a model. To solve the first issue, I restrict the dimension of commodity space to one so that the agent of an economy has only the consumption space from which both the consumption utility and the gain–loss utility are derived. For the second issue, I focus mainly on the consistent consumption plans of two typical types of decision makers, assuming the agent is either an over-consumer or a natural born saver, but not both, if the agent’s loss aversion is low. The agent in the model is assumed to keep one’s natural type of spending, although one can change the degree of loss aversion over time within the same type of spending. Next I construct intertemporal models of decision making, first in two periods, then three periods, and finally T periods, based on the approach of developing the consistent consumption strategy for any committed plans. Unlike in the two-period model, in three or more period models, a decision maker can have several alternative intended plans. For example, a deviator intends to consume more than the ex ante optimal for the first two consecutive periods, and as a result has to accept a very low consumption amount for the last period. Alternatively, he may want to consume more only in the first period. The two intended plans yield two different solutions, and the agent should select the most desirable strategy between them. In a later section, I compute the utility for the best consumption plan that a rational deviator employs because it gives him the highest utility among all the available consistent consumption profiles. In fact, it turns out that a plan that keeps over- or under-consumption up to the middle point of the planning horizon gives the highest utility for a deviator (Section 2). This property implies that the consumption smoothing rule is beneficial even to the deviators. Based on this preliminary work, I analyze the model of a dynamic decision maker for the case where the decision maker may or may not change his mind over time. When the agent’s preference (loss aversion, in particular) stays constant, then because his expectation is met, there are no gain–loss utilities in the subsequent periods. However, if the agent’s preference does change, then depending on how the new reference points are formed, there are different types of gain–loss utilities following his intended plan in subsequent periods. I provide optimality conditions, as well as closed form solutions, for each of the alternative paths that the decision maker may choose. Lastly I propose a self-corrective sub-period perfect reference point (SPRP) in the dynamic model and explore the choice dynamics when the decision maker changes his mind over time due to time-varying degrees of loss aversion. The decision maker rebalances his consumption based on the adjusted belief regarding the ex ante optimal consumption through the

1

˝ This is what Koszegi and Rabin (2009) called preferred personal equilibrium (PPE).

H. Park / Research in Economics 73 (2019) 107–122

109

SPRP for his remaining life. Because the SPRP is constructed based on current wealth position, whenever the decision maker changes his mind, he solves a new maximization problem relative to the updated reference point. The resulting consumption profile reflects his current financial situation. In this paper I demonstrate that age-related loss aversion which varies over time can solve a puzzling feature in lifecycle consumption data. 1.1. Other related literature ˝ Although there are few directly related literature to this paper other than Koszegi and Rabin (20 06,20 09), there have been some other works on the development of the preference itself, as well as ample research related to the lifecycle model, which is one application of the model in this paper. The model generates a featured phenomenon of consumption data known as consumption hump. Because this is not a major part of the paper, the literature for this will not be included, but interested readers may look for it in Park and Feigenbaum (2017). ˝ Regarding the former, prior to Koszegi and Rabin (2006), several earlier studies have developed reference dependent models with non-constant and/or non-status quo,2 reference points. Shalev (20 0 0) introduces a Nash equilibrium solution concept that endogenizes reference points. Sugden (2003) even allows the reference points to be stochastic, proposing a theory of choice under uncertainty in which preferences are reference-dependent. In fact, Sugden’s work generalizes the subjective expected utility (SEU) theory into a reference dependent model based on regret theory, compared to the disappointment ˝ theory to which Koszegi and Rabin (20 06,20 09) belong. De Giorgi and Post (2011) investigate the former (state-dependent reference points), while Park (2016) demonstrates the difference between the two with the introduction of loss-tolerance. Based on a two-period intertemporal model, Park shows a greater deviation with the state-independent model than the state-dependent when there is a positive state-dependence. Apesteguia and Ballester (2009) provide an axiomatic characterization of reference dependence when the reference point belongs to the choice set. Similarly, by use of intuitive axioms, Bleichrodt (2009) presents a new class of preference foundations that extend reference-dependent expected utility to the case of incompleteness which is caused by reference dependence as the reference point is one of the available strategies. Beyond Sugden (2003) who developed reference-dependent subjective expected utility with a fixed reference point under complete preference, Bleichrodt allows incompleteness to build the expected utility. Moreover, Bleichrodt’s (2009) model extends to the case with shifting reference points, which element is related to the current paper, under reasonable conditions that are intuitive and easily tested as well. The extension is significant since the often noted deviation from expected utility is mainly explained by the shifting reference points. Regarding reference points related with risk, Matthey (2005) provides a model of reference dependence where the risk itself is included in the reference status. Matthey argues that people see unknown risks differently from the risks that are common because the common risk is anticipated while the unknown risk may turn out to be a surprise. Therefore, when such risk is included, the evaluation of contrast utility changes, known as “risk inclusion effect” compared to the usual “endowment effect.” The work by Krähmer and Stone (2013) analyzes uncertainty aversion based on fear of regret. Unlike ˝ Koszegi and Rabin, loss aversion is generated within their model from the decision maker’s ex post beliefs when the agent evaluates the original decision. Information is revealed when a compound lottery is selected, which alters the decision maker’s ex post assessment of his choice. This idea was also discussed in a similar work by Halevy and Felkamp (2005). 2. The static model 2.1. Rational deviators Consider a rational decision maker (DM hereafter) who has a reference dependent preference (RDP hereafter) and makes a two-period (t = 0, 1 ) intertemporal consumption decision. Given a nonnegative income stream {y0 , y1 }, the DM chooses the optimal consumption {c0 , c1 } by maximizing his lifetime utility which is discounted by β per period. Assume that the ˝ agent can borrow or lend freely at the market interest rate 1 + r = R. Following Koszegi and Rabin (20 06,20 09), I posit the reference dependent utility by u(c|c∗ ) = u(c ) + ημ(c|c∗ ), where η is the weight of gain–loss utility relative to consumption utility and c∗ is the “ex ante optimal consumption,” which is solved from the maximization problem of a rational DM given the income stream. The gain–loss utility μ(c|c∗ ) is then defined with



μ ( c |c ∗ ) =

u (c ) − u (c∗ ) λ[u(c ) − u(c∗ )]

if u(c ) − u(c∗ )  0 if u(c ) − u(c∗ ) < 0



in which λ > 1 is the coefficient of loss aversion. To build a simple RDP version of the intertemporal decision, I first define ω as the initial psychological weight on loss relative to gain utility. At the of the  beginning t = 0, the DM forms a  first period 1/γ y0 + y1 R−1 R y0 + y1 R−1 ∗ ∗ belief regarding the ex ante optimal consumption, which is {c0 , c1 } = , where φ −1 = (β RR) . φ 1 + φ −1 1 + φ −1

2 For the status quo bias, see Masatlioglu and Ok (2005): they propose a rational choice theory that allows for the presence of the bias and that incorporates the standard choice theory as a special case.

110

H. Park / Research in Economics 73 (2019) 107–122

Because the consumption gives the DM maximum utility ex ante, it is necessary to check if this is a personal equilibrium (PE) for the RDP agent; (c0 , c1 ) is a PE if

u(c0 , c1 |c0 , c1 ) ≥ u(c0 , c1 |c0 , c1 )

(1)

for all feasible (c0 , c1 ). To verify this, consider the following situation: if the DM intends to consume more than the ex

ante optimal at the first period t = 0, and as a result he ends up with consuming less than the optimal at t = 1, i.e. {c0 > c0∗ , c1 < c1∗ }, then the optimization objective of the DM is summarized by Max u(c0 , c1 |c0∗ , c1∗ ) : 1 −γ



1 −γ

∗1−γ

c c0 c0 +η − 0 1−γ 1−γ 1−γ





+β



1 −γ

1 −γ

∗1−γ

c c1 c1 + ηωλ − 1 1−γ 1−γ 1−γ



(2)

subject to

c0 + b1 = y0 c1 = y1 + Rb1

(3)

where b1 is the agent’s bond holding at t = 0 for the next period t = 1. The overall utility comes from not only the consumption utility for each period, i.e. u(c0 ) and β u(c1 ), but also the respective gain–loss utility. Because the DM intends to consume more than the reference point at the first period, he expects a contemporaneous gain utility for that period, which is [u(c0 ) − u(c0∗ )] when the weight of gain–loss utility relative to consumption utility is equal to one. As a result, his consumption will be lowered next period, and this yields a prospective loss utility relative to the ex ante optimal. This portion of utility, the prospective loss feeling relative to the contemporaneous gain feeling is given by βωλ[u(c1 ) − u(c1∗ )]. There is no gain–loss utility in the second period (t = 1 ) because the expectation is met once he conforms to the intended consumption plan. Is consuming more than the ex ante optimal at the first period worthwhile to the DM? Constructing the following derivative will show the possibility:

du −γ = (1 + η )c0 − β R(1 + ηωλ )[y0 + R(y0 − c0 )]−γ dc0

(4)

If the derivative is evaluated at the planned consumption c0∗ , then



du y0 + y1 R−1 = η (1 − ωλ ) dc0∗ 1 + φ −1

 −γ

>0

if

ωλ < 1

(5)

There is no incentive to deviate from the ex ante optimal committed consumption c0∗ if ωλ  1; this is the PE for the DM with ωλ  1 since he is willing to choose c∗ given that c∗ is his reference point. The ex post optimal consumption of this case is in fact the same as the one solved in the standard model. Thus, the standard agents are thought to be those who care more about the prospective loss utility than the contemporaneous gain utility.3 However, the planned strategy (c0∗ , c1∗ ) is not optimal, and thus not a PE for those with ωλ < 1, because deviation from the ex ante solution increases the utility ex post, i.e. u( c0 ,  c1 |c0∗ , c1∗ ) > u(c0∗ , c1∗ |c0∗ , c1∗ ) for a feasible ( c0 ,  c1 ). The DM whose concern about loss is relatively low must follow a new optimality condition, which is

(1 + η )c0−γ = Rβ (1 + ηωλ )c1−γ

(6)

The consistent consumption for the intended plan {c0 >



{c0 , c1 } =

y0 + y1 R−1 1 + φ1

R



1+ηωλ 1/γ , φ 1 +η

1+ηωλ 1+η

1/γ

c0∗ ,

c1 <

c1∗ }

y0 + y1 R−1 1 + φ1

is



1+ηωλ 1/γ

(7)

1 +η

This choice (c0 > c0∗ and c1 < c1∗ ) maximizes the over-consumer’s utility among all feasible, consistent plans to consider, and thus is a preferred personal equilibrium (PPE) as the choice satisfies u(c0 , c1 |c0 , c1 ) ≥ u(c0 , c1 |c0 , c1 ) for all (c0 , c1 ) which are PE. When the DM cares more about contemporaneous gain than about prospective loss, his present consumption can be higher than the one of his ex ante optimal committed. However, the RDP model can be illuminated in the opposite direction, which has been noticed by few researchers, if any. When a DM who is an under-consumer such as a natural born saver intends to consume less than the ex ante optimal in the first period, and as a result ends up with consuming more than the optimal in the next period, i.e. {c0 < c0∗ , c1 > c1∗ }, then her objective is described by Max u(c0 , c1 |c0∗ , c1∗ ) : 1 −γ



1 −γ

∗1−γ

c c0 c0 + ηωλ − 0 1−γ 1−γ 1−γ





+β

1 −γ



1 −γ

∗1−γ

c c1 c1 +η − 1 1−γ 1−γ 1−γ



(8)

3 Here, DM’s prospective loss feeling serves as an internal commitment device. The RDP agent would not over-consume even though he does not have any external commitment device like illiquid assets, or golden eggs as in Laibson (1997).

H. Park / Research in Economics 73 (2019) 107–122

111

subject to the budget constraint Eq. (3). At t = 0, the DM experiences a contemporaneous loss feeling relative to the reference consumption behavior, represented by ωλ[u(c0 ) − u(c0∗ )] when η = 1. Consequently, her consumption is elevated at t = 1, and this gives her a prospective gain feeling, which is β [u(c1 ) − u(c1∗ )]. Like in the over-consumer’s maximization procedure, because the expectation is met, there is no gain–loss utility in the second period. The solution to this problem is summarized by (i ){c0 , c1 } = {c0∗ , c1∗ } if ωλ  1, and

⎧ ⎪ ⎪ ⎪ ⎨

⎫ ⎪ ⎪

1+η 1/γ ⎪ −1 ⎬ 1+φ 1+ ηω λ

(ii )

1+η 1/γ ⎪ ⎪ y0 + y1 R−1 R ⎪ ⎪ ⎪

1+η 1/γ ⎪ ⎩c1 = φ 1+ηωλ ⎭ −1 1+φ 1+ηωλ y0 + y1 R−1

c0 =

if

ωλ < 1

(9)

Again, the consumption profile {c0 < c0∗ , c1 > c1∗ } is PPE for the DM who has low aversion for contemporaneous loss. When the DM cares enough about her prospective gain utility, she deviates from the ex ante optimal committed for less consumption (more saving) in the first period. 2.2. Optimal plan for more than two periods The intertemporal model of belief dependent preference for three or more periods requires acknowledging potentially many reference points on which the DM forms expectations. This topic is postponed until Section 3 where I rigorously analyze the dynamically dependent reference models. Another important issue is regarding what type of strategy gives the DM the highest total utility among all the alternative consumption plans because with longer periods there are more of these alternative strategies. For this, I first study the model with three, four, and five periods in detail to obtain an implication for the longer periods. If the DM lives for T periods, and if his degree of loss aversion stays constant, the static RDP maximization problem is given by Max u(c|c∗ ): T  t=0



βt

1 −γ

ct +ηIt 1-γ



1 −γ





∗1−γ

ct c − t 1-γ 1-γ

+ηωλ(1 − It )

1 −γ

∗1−γ

ct c − t 1-γ 1-γ



(10)

subject to

ct + bt+1 = yt + Rbt

(11)

and b0 = bT +1 = 0. The index It = {0 or 1} represents the DM’s gain–loss status for each period t. For example, if the DM has a plan of {c0 > c0∗ , c1 > c1∗ , . . . , cT −1 > cT∗ −1 , cT < cT∗ }, then the plan is described by It = 1∀t = 0, 1, . . . , T − 1, but IT = 0. It is also expected that the DM would not deviate from c∗ when his loss feeling is grave (ωλ  1) throughout his life. Likewise, if ωλ < 1, it is always profitable for the DM to deviate, in which case the DM must satisfy the following optimality condition for all t: −γ

ct

−γ

+ ηIt ct

−γ

+ ηωλ(1 − It )ct

−γ

−γ

−γ

= Rβ [ct+1 + ηIt+1 ct+1 + ηωλ(1 − It+1 )ct+1 ]

(12)

Consider an over-consumer with three periods, T = 2. Given any income stream {y0 , y1 , y2 }, the ex ante utility maximizer

 2

y +y R−1 +y R−2

2 for the three-period consumption is {c0∗ , c1∗ , c2∗ } = {c0∗ , φR c0∗ , φR c0∗ } with c0∗ = 0 1+1φ −1 +φ −2 . The consumption maximizes the consumption utility, as well as the ex ante expected gain/loss utility, which is zero. Thus, this is the ex ante optimal strategy. To see if this strategy is a consistent plan for the DM, construct the following consumption plan. At t = 0, the DM considers a consumption scheme {c0 , c1 , c2 } that satisfies his intention of over-consumption for the first two periods; {c0 > c0∗ , c1 > c1∗ , c2 < c2∗ }. Then, the strategic situation of the DM is given by the gain–loss utility index It = {1, 1, 0} in Eq. (10),4 subject to Eq. (11) with b0 = b3 = 0. The total utility comes from the consumption utility for each period, u(c0 ), β u(c1 ), and β 2 u(c2 ), plus the relevant gain–loss utility to the consumption choice. Because the DM consumes more than his reference point at the first two periods, he has the contemporaneous gain utility for the first period (which is [u(c0 ) − u(c0∗ )]) and the prospective gain utility (β [u(c1 ) − u(c1∗ )]) for the second period when η = 1. But, the DM feels a prospective loss for the third period (β 2 ωλ[u(c2 ) − u(c2∗ )]) because of his expected consumption, which is lower than the reference point. There will be no gain–loss utility in the second and third period because the DM has formed adapted beliefs based on the reference points for the subsequent periods. To examine the consistency of the ex ante optimal choice, consider the following derivatives for possible deviation. Since c2 = [y2 + Ry1 + R2 (y0 − c0 ) − Rc1 ], the derivatives w.r.t. the other two variables (c0 , c1 ) are

du −γ = (1 + η )c0 − β 2 R2 (1 + ηωλ )[y2 + Ry1 + R2 (y0 − c0 ) − Rc1 ]−γ dc0 4

u(c0 , c1 , c2 |c0∗ , c1∗ , c2∗ ) =

1−γ

c0 1 −γ

+η



1−γ

c0 1 −γ

−

∗1−γ

c0 1 −γ



+β



1−γ

c1 1 −γ

+η



1−γ

c1 1 −γ

−

∗1−γ

c1 1 −γ



+ β2



1−γ

c2 1 −γ

+ ηωλ

(13) 

1−γ

c2 1 −γ

−

∗1−γ

c2 1 −γ

 .

112

H. Park / Research in Economics 73 (2019) 107–122 Table 1 Utility comparison with three-period model. Plan ∗

c : A: B: C: D: E:

c0 = c0∗ , c1 = c1∗ , c0 > c0∗ , c1 > c1∗ , c0 > c0∗ , c1 < c1∗ , c0 = c0∗ , c1 > c1∗ , c0 > c0∗ , c1 = c1∗ , c0 > c0∗ , c1 < c1∗ ,

c2 = c2∗ c2 < c2∗ c2 < c2∗ c2 < c2∗ c2 < c2∗ c2 = c2∗

u(c)

u(G/L)

Total U

Rank

29.4461 29.4421 29.4419 29.3483 29.4430 29.3418

0 0.0115 0.0119 –0.0766 0.0088 0.0823

29.4461 29.4536 29.4538 29.2717 29.4518 29.2595

4 2 1 5 3 6

du −γ = β [(1 + η )c1 − β R(1 + ηωλ )[y2 + Ry1 + R2 (y0 − c0 ) − Rc1 ]−γ dc1

(14)

If these are evaluated at the ex ante optimal consumption {c0∗ , c1∗ , c2∗ },

du ∗−γ = (1 + η )c0 − β 2 R2 (1 + ηωλ )[y2 + Ry1 + R2 (y0 − c0∗ ) − Rc1∗ ]−γ dc0∗

(15)

y0 + y1 R−1 + y2 R−2 = η (1 − ωλ ) 1 + φ −1 + φ −2

(16)



−γ

>0

if

ωλ < 1

du ∗−γ = β (1 + η )c1 − β 2 R(1 + ηωλ )[y2 + Ry1 + R2 (y0 − c0∗ ) − Rc1∗ ]−γ dc1∗

 = η (1 − ωλ )R

−1

y0 + y1 R−1 + y2 R−2 1 + φ −1 + φ −2

 −γ > 0 if

(17)

ωλ < 1

(18)

Each of the derivatives are obtained from the fact that since c2 (c0∗ , c1∗ ) = c2∗ , the resource constraint satisfies c2∗ = [y2 + Ry1 + R2 (y0 − c0∗ ) − Rc1∗ ]. When ωλ  1, there is no incentive to deviate from the ex ante optimal committed c0∗ and c1∗ . For the agent who is concerned about the loss feeling regarding the future, consuming more than the committed would not pay off. However, this does not apply to the agent who has a low weight on loss (ωλ < 1). It always pays off to choose another strategy and the alternative choice should be consistent to his intention. Proposition 1. When the DM has a high weight (ωλ  1) on prospective loss, he would not deviate from the ex ante optimal committed plan. If, however, his on loss is low,  then the consistent consumption plan for the DM who intends to consume  weight    2  1/γ y + y1 R−1 + y2 R−2 1+ηωλ R R ∗ ∗ ∗ {c0 > c0 , c1 > c1 , c2 < c2 } is c0 , c0 φ , c0 φ μ with c0 = 0 , and μ = . 1+η 1 + φ −1 + φ −2 μ ηωλ ∗ ∗ ∗ Because 1+1+ η < 1 when ωλ < 1, it is true that c0 > c0 and c1 > c1 , but c2 < c2 . When the DM cares more about the gain feelings near future than about remote loss feelings, his early consumption can be higher. It is also possible that the DM with a low weight may intend to overconsume for the first period, but not the second period; the DM intends to  

consume {c0 > c0∗ , c1 < c1∗ , c2 < c2∗ }. Then the consistent consumption for this case is

 

 2

c0 , c0 φR μ, c0 φR

μ

where

y0 +y1 R−1 +y2 R−2 . Likewise, if the DM with ωλ < 1 intends a consumption strategy in which he clings to one of the 1+φ −1 μ+φ −2 μ ex ante optimal consumption such as c1 = c1∗ , then the consistent consumption for the plan {c0 > c0∗ , c1 = c1∗ , c2 < c2∗ } is c0 =



  y +y R−1 +y2 R−2  2  y0 +y1 R−1 +y2 R−2 − φ1 0 1+1φ −1 +φ −2 c0 , c1∗ , c0 φR μ where c0 = . Because the second-period consumption is fixed at 1 + φ −2 μ

the ex ante optimal c1∗ , the other two consumption levels are adjusted accordingly. When the DM sticks to the intended plan for the subsequent periods, the ex post lifetime total utility from the three scenarios has the following order: {c0 > c0∗ , c1 < c1∗ , c2 < c2∗ }  {c0 > c0∗ , c1 > c1∗ , c2 < c2∗ }  {c0 > c0∗ , c1 = c1∗ , c2 < c2∗ }. 2.2.1. Utility comparison I compare the overall utility from alternative consumption plans and determine the order of the total ex post utility among the plans. The key assumption for the comparison is that the DM executes the intended plan without changing his mind over time, although the case of changing mind will be analyzed in Section 3. In the three-period model, there are five possible consumption plans for an over-consumer. Once the consistent consumption strategies for these five alternatives are solved, it is possible to rank the total utility like Table 1. Although the table shows the result from a specific income stream

H. Park / Research in Economics 73 (2019) 107–122

113

Table 2 Utility comparison with five-period model. Plan c0 c0 c0 c0

> > > >

u(c) c0∗ , c0∗ , c0∗ , c0∗ ,

c1 c1 c1 c1

> > > <

c1∗ , c1∗ , c1∗ , c1∗ ,

c2 c2 c2 c2

> > < <

c2∗ , c2∗ , c2∗ , c2∗ ,

c3 c3 c3 c3

> < < <

c3∗ , c3∗ , c3∗ , c3∗ ,

c4 c4 c4 c4

< < < <

c4∗ c4∗ c4∗ c4∗

46.6852 46.6562 46.6491 46.6707

u(G/L)

Total U

Rank

0.1258 0.2024 0.2178 0.1570

46.8110 46.8586 46.8669 46.8277

4 2 1 3

{y0 = 1, y1 = 1, y2 = 1} and a set of parameter values {R = 1.05, β = 0.98, γ = 0.9, ωλ = 0.8}, I observe this ranking is robust with all the parameter values I have simulated given any alternative income profile.5 In terms of total utility, the best plan (B) does not provide a highest consumption utility. In fact, the highest consumption utility comes from plan D, which gives the DM the smoothest consumption bundle among all the deviation strategies, except for the ex ante optimal solution. This implies that the DM has the highest consumption utility if he follows the most efficient plan. However, because of his gain–loss feeling, this is suboptimal to his preferred choice. Plans C and E have negative gain–loss utilities, implying these choices are not fulfilling his preference at all, as well as giving the DM the two lowest consumption utilities. Once the DM decides to deviate, retaining any part of the original plan would not be profitable. Thus, it is proper to analytically compare only those plans that do not have any fixed consumption point (c∗ ). This leaves us the two main plans: {A: c0 > c0∗ , c1 > c1∗ , c2 < c2∗ } and {B: c0 > c0∗ , c1 < c1∗ , c2 < c2∗ }. The DM will consider whether he should consume more for both periods and takes the burden of deep loss at the last period (A), or mitigate the burden by over-consuming only in the first period (B). Which is better for a deviator? Although the table shows a direct answer to 1/γ this question, it is also intuitive. Assume Rβ = 1. Then because (Rβ ) = 1, it follows that c0 = c1 in the plan A. Thus, if ωλ < 1, then c0 = c1 > c2 . If ωλ  1, then no deviation occurs and c0 = c1 = c2 . In plan B, because c1 = c2 , it follows that c0 > c1 = c2 if ωλ < 1. Solving the model, it is easy to see that c0B > c0A and c2B > c2A for the deviator, and plan B is more likely to give him a higher utility. The utility difference between the two plans is (1 + η )[u(c0B ) − u(c0A )] + β [(1 + ηωλ )u(c1B ) − (1 + η )u(c1A ) − (1 − ωλ )u(c1∗ )] + β 2 (1 + ηωλ )[u(c2B ) − u(c2A )]. How about the four-period model? By deriving consistent consumption for each of the alternative plans for the DM with ωλ < 1, I obtain the following utility order6 :{c0 > c0∗ , c1 > c1∗ , c2 < c2∗ , c3 < c3∗ }  {c0 > c0∗ , c1 < c1∗ , c2 < c2∗ , c3 < c3∗ }  {c0 > c0∗ , c1 > c1∗ , c2 > c2∗ , c3 < c3∗ }  {c0∗ , c1∗ , c2∗ , c3∗ }. The ranking is preserved for a general class of parameter values. Finally, with the five-period model, I have the result for {β = 0.95, R = 1.035, γ = 0.9, η = 1, ωλ = 0.4} in Table 2. Using the index It , the order is described in this fashion: {1, 1, 0, 0, 0}{1, 1, 1, 0, 0}{1, 0, 0, 0, 0}{1, 1, 1, 1, 0}(Standard). As shown in the tables, the overall utility culminates ex post when the DM keeps the over-consumption behavior up to half of the entire periods for the general class of parameters. Thus, if the DM has a consumption plan for S periods, shifting from high consumption (c > c∗ ) to low consumption (c < c∗ ) at the midpoint of the planning horizon (which is S/2) gives him the highest ex post utility. For example, if S = 4, then {1, 1, 0, 0} is the best plan. If S is an odd number, then over-consumption up to one period less than the midpoint gives the best result. Although I register the utility comparison only for the over-consumer here, this rule also applies to the case of a saver, who achieves a highest utility when she keeps the under-consumption behavior until the midpoint of the planning horizon. 3. The dynamic model 3.1. Dynamic decision makers In the static model, there are no gain–loss utilities in the subsequent periods because the DM will follow his plan in those periods and this forms a new expectation that determines his reference point on the periods. However, the DM of the next period might not follow the rule he has intended in the previous period. This section analyzes what would happen to the consistency condition under this scenario. 3.1.1. Reference points in the dynamic model I will call the agent of each period in terms of decision time such as DM(0), DM(1), and DM(2). The analysis of the first scenario in the three period model (Section 2.2) is for DM(0) (ω0 λ < 1) who is at period zero and has a great pleasure of consuming more during the first two periods, but is aware that his consumption will be lower in the third period. Thus, DM(0) expects that he would consume in the second period the planned amount he set, which is more than the ex ante optimal (c1 > c1∗ ), and he would not have any gain or loss feeling at the period because his expectation of high consumption is already met. Likewise, DM(0) expects that during the third period, he would not feel any loss regarding the low consumption because of his already lowered reference point. At t = 1, if the preference of DM(1) stays the same as that of his old self, deviating in the subsequent periods is not desirable. However, DM(1) could change his mind and plan 5 The plan of alternating consumption {c0 > c0∗ , c1 < c1∗ , c2 > c2∗ } is excluded because it not only gives the lowest total utility, but also does not clearly characterize the type of an over-consumer. 6 The plans with fixed consumption profiles are excluded by the same argument.

114

H. Park / Research in Economics 73 (2019) 107–122

a new strategy for the remaining periods. In this case, the initial plan by DM(0) may not be consistent with the intention of subsequent DMs. To construct an optimization problem for DM(1), the reference point of DM(1) must be defined. Would the reference point be the consumption utility that DM(0) set for the period to fulfill his taste, which is u(c1 )? Or would it be the optimal consumption utility solved from the original problem for the period, u(c1∗ )? To explore this, assume first that DM(0) has the usual high weight on loss (ω0 λ  1). Then c1 = c1∗ and there is no difference between the two reference points, regardless of the weights of subsequent DMs. Thus, deviating from the planned path is not a consistent choice for DM(1) if he also has a high loss aversion (ω1 λ  1). But if DM(1) has a low weight (ω1 λ < 1) and intends to consume more, then the consistent choice for DM(1) is a new consumption solved from t = 1, c1 (1) which should be more than c1∗ . The bracket in the new consumption denotes the choice by DM(1). Proposition 2. When DM(0) has a preference of ω0 λ  1 and made up a plan of {c0∗ , c1∗ , c2∗ }, if DM(1) has a preference of ω1 λ  1, then it does not pay off for him to deviate from DM(0)’s plan. However,   if DM(1) has  a preference of ω 1 λ < 1, then it pays off for him to deviate and the consistent consumption is c1 (1 ) = 



1+ηω1 λ 1+η

1/γ

1 1+ φ

μ 1+ φ1

R

φ 

y y y0 + 11 + 22 R

R  2  =

1+ 1 1+ φ φ



1 1+ φ μ 1+ φ1



c1∗ , with μ1 =

. Thus c1 (1 ) > c1∗ .

Second, assume that the initial DM(0)’s weight on loss is low (ω0 λ < 1) so that he deviates with a plan of {c0 > c0∗ , c1 > c1∗ , c2 < c2∗ }. Because the ex ante optimal consumption for t = 1 is not optimal any longer and the planned consistent consumption by DM(0) for the period is not equal to c1∗ , the two specifications of the reference point are different; u(c1 ) = u(c1∗ ). To analyze further, assume that the consistent consumption choice by DM(0) for t = 1 is c1 . Then, with respect to the loss utility, the second reference point defines the utility loss of the new plan, which deviates from the original plan by u(c1 (1 )) − u(c1∗ ), while by the first, the loss feeling arises from not fulfilling the adjusted plan recently revised by DM(0), i.e. u(c1 (1 )) − u(c1 ). Although in the next section I define a novel concept of reference points that can be applied to any subsequent period in the dynamic model, here I explore the implication of dynamic decision making based on both notions of the reference point.7 First, let the reference point be defined by the second notion u(c1∗ ). Consider an optimization problem of DM(1), at the beginning of t = 1, who has cash on hand (current labor income plus the financial wealth accrued from the last period) and is supposed to consume c1 , which is more than the ex ante optimal initially set for the period. The agent may or may not desire to keep high consumption for this period. DM(1) may regret his over-consumption during the first period and would go back to the optimal path by reducing his consumption when he has a reference point of {c0∗ , c1∗ , c2∗ }. Thus, DM(1) realizes that he would have a prospective loss utility if he ends up consuming much less than the optimal for the next period as a result of consuming more this period. The situation for DM(1) who has gain–loss utility by deviating from the original plan c∗ is described by Max u1 (c1 , c2 |c1∗ , c2∗ ) :



1 −γ

1 −γ

∗1−γ

c c1 c1 +η − 1 1−γ 1−γ 1−γ





+β



1 −γ

1 −γ

∗1−γ

c c2 c2 + ηω1 λ − 2 1−γ 1−γ 1−γ



(19)

subject to

c1 + b2 = y1 + Rb1

(20)

c2 = y2 + Rb2 Observe that the new utility is indexed by the time of decision making, t = 1. The cash on hand is defined by x1 = y1 + Rb1 = y1 + R(y0 − c0 ). Because c0 > c0∗ , it is true that x1 < x∗1 incorporating the over-consumption at the first period. The preference of DM(1) is represented by ω1 λ. When DM(1) consumes more than the ex ante optimal at t = 1, he has a contemporaneous gain utility for the period, but experiences a prospective loss for the next period. To see if it is profitable for DM(1) to deviate from the preset plan, rewrite c2 = [y2 + Ry1 + R2 b1 − Rc1 ]. Then the derivative, w.r.t. c1 (1) is

du1 −γ = (1 + η )c1 − β R(1 + ηω1 λ )[y2 + Ry1 + R2 b1 − Rc1 ]−γ dc1 (1 ) b1 = y0 − c0 =

2 1+ηω0 λ 1/γ

y0 φ1 + y0 φ1

1+η

−

y1 R1

1 2 1+ηω0 λ 1/γ

1 + φ1 + φ

−

(21)

y2 R2

(22)

1+η

The bond demand is results from DM(0)’s plan of {c0 > c0∗ , c1 > c1∗ , c2 < c2∗ } from his taste ω0 λ < 1. If the derivative is evaluated at the planned consumption c1 = φR c0 ,

du1 = dc1

 R −γ φ

c

0



(1+η ) 1 −



1+ηω1 λ 1+ηω0 λ



 0 if > 0 if

ω1 λ  ω0 λ ω1 λ < ω0 λ < 1



(23)

7 ˝ Regarding this, the proposal by Koszegi and Rabin (2009) vaguely suggests “the most recent past expectation,” which is more likely to indicate the latter. But I find the first one is more appealing in this application.

H. Park / Research in Economics 73 (2019) 107–122

115

The first condition says that deviation for any amount larger than the planned c1 is not optimal if DM(1) cares about the future as much as DM(0), who has deviated for over-consumption at t = 0. By the second, it pays off to consume more than the planned if DM(1) has a lower weight. As long as DM(1) has a lower weight than DM(0)’s, deviation from the revised plan set by DM(0), i.e. c1 for an even larger consumption is possible. Since ω0 λ < 1, if DM(1) cares about his future (ω1 λ > 1), it is always better for him to choose a consumption strategy other than the one set by DM(0). Also, even though DM(1) might not care for his future as much (ω1 λ < 1), as long as his weight is higher than DM(0)’s, it is still optimal for him not to follow DM(0)’s plan of over-consumption. Solving the maximization problem yields a consistent consumption plan for each case:



c1 (1 ) = c1 (0 )

1 + φ1

c1 (1 ) = c1 (0 )

1 + φ1 1

1+ φ

c1 (1 ) = c1 (0 )

1 + φ1

1+ηω0 λ 1/γ 1+η

1+ηω1 λ 1/γ

1

1+ φ

if

ω0 λ < 1  ω1 λ

(24)

1+η

1+ηω0 λ 1/γ 1+η

1+ηω1 λ 1/γ

< c1 (0 )

if

ω0 λ < ω1 λ < 1

(25)

> c1 (0 )

if

ω1 λ < ω0 λ < 1

(26)

1+η

 1 /γ 1 1+ηω0 λ

1+ φ

< c1 (0 )

1+η

1+ηω1 λ 1/γ 1+η

Now I look for the optimal strategy with an alternative scenario: DM(0), with a taste of ω0 λ < 1, has a three-period plan by which he initially consumes more than the predetermined and less for the later two periods, {c0 > c0∗ , c1 < c1∗ , c2 < c2∗ }. The derivative evaluated at the consumption point set by DM(0) is



du1 R  = ( 1 − ω1 λ )η c dc1 φ 0

−γ  1 + η  0 >0 1 + ηω0 λ

if if

 ω1 λ  1 ω1 λ < 1

(27)

Unlike the first scenario, the sign of this derivative does not depend on ω0 λ, implying that only the attitude of DM(1) matters. Proposition 3. When DM(0) with ω0 λ < 1 has a plan {c0 > c0∗ , c1 < c1∗ , c2 < c2∗ }, DM(1) should not increase his consumption from the preset amount of c1 = φR



1+ηω0 λ 1+η

1/γ

c0 if he cares about future (ω1 λ  1). If, however, DM(1) also has a low weight

on loss, then it pays off for DM(1) to deviate from c1 by consuming more. Thus, c1 = c1 if ω1 λ  1 and c1 =

ω1 λ < 1.

1 1+ φ



1 1+ φ 1+ηω1 λ 1+η

1/γ c1 if

It is clear that c1 > c1 when ω1 λ < 1. The proposition implies that when DM(0) with a low weight consumes a lot in the first period not caring about the future, DM(1), who wishes to correct this, should accept the low consumption amount set by his precedent self. However, this does not apply when DM(1) also has a low weight. If I turn to the second specification of the reference point: the reference point is the consumption utility from the solution that DM(0) solves for the period to fulfill his taste of ω0 λ < 1. Let {c } ≡ {c0 , c1 , c2 } be the solution to the consumption schedule of DM(0) who intends to consume more for the first two periods: {c0 > c0∗ , c1 > c1∗ , c2 < c2∗ }. DM(1) may or may not keep the high consumption for the second period. Should he regret his over-consumption in the previous period, he can reduce his consumption. Since the reference point is now u(c1 ), DM(1) realizes that he would have a contemporaneous loss utility if he consumes less than this solution, but would have a prospective gain utility for the next period. Then DM(1), who has gain–loss utility by deviating from the predetermined consumption {c } for an alternative plan {c1 < c1 , c2 > c2 }, maximizes u1 (c1 , c2 |c1 , c2 ): 1 −γ



1 −γ

 1 −γ

c1 c1 c + ηω1 λ − 1 1−γ 1−γ 1−γ





+β

1 −γ



1 −γ

 1 −γ

c2 c2 c +η − 2 1−γ 1−γ 1−γ



(28)

subject to Eq. (20). This may be called “reverting saver’s optimization problem.” DM(1) inherits the financial wealth, positive or negative, from period zero. To see if it is profitable to deviate from the high consumption that DM(0) set, I look for the derivative w.r.t. c1 :

du1 −γ = (1 + ηω1 λ )c1 − β R(1 + η )[y2 + R(y1 + Rb1 − c1 )]−γ dc1

(29)

Evaluating this at the planned consumption c1 = Rφ −1 c0 gives

 R −γ  φ

c

0

 1+η  0 i f (1+ηω1 λ ) − (1+η ) 1+ηω0 λ > 0 if

1+ηω1 λ 1+η 1+ηω1 λ 1 +η

 >

1+η 1+ηω0 λ 1+η 1+ηω0 λ

 (30)

116

H. Park / Research in Economics 73 (2019) 107–122

The first condition demonstrates that because ω0 λ < 1, deviation from c1 for a smaller consumption is not desirable if DM(1) has a preference of ω1 λ  1.8 The condition is also satisfied, even with ω1 λ > 1, as long as it is not too high. Therefore, deviation for a lower amount is optimal only if ω1 λ 1, which is what the second condition says. If DM(1) does not have a strong attachment to DM(0)’s plan then he can reduce the consumption for the period. If, however, DM(1) cares strongly about DM(0)’s intention, then he would not deviate from the plan. It is possible for DM(1) to lower his consumption in vision of a remote gain only if DM(1) overcomes the strong attachment to DM(0). The new consistent consumption for this saver is



c1 (1 ) = c1 (0 )

1 + φ1 1

1+ φ

1+ηω0 λ 1/γ 



1+η



(31)

1 /γ 1+η 1+ηω1 λ

Thus, c1 (1 ) < c1 (0 ) if ω1 λ 1 > ω0 λ and c1 (1 )  c1 (0 ) if ω1 λ  1. The new consumption c1 (1 ) is smaller than c1 (0 ) only if ω1 λ is much greater than one. As an exercise, I extend this analysis for T − period optimization problem where there are (T + 1) selves of DM(0), DM(1),..., DM(T). To generalize, let the initial DM set a T – period plan, and the reference point {c0∗ , c1∗ , . . . , cT∗ } be known to all the subsequent DMs. Their consistent consumption plans are revised based on this reference status each time. Also, assume that all DMs up to period τ − 1 have high loss aversion, ω0 λ  1, ω1 λ  1, . . . , ωτ −1 λ  1, and thus {c0∗ , c1∗ , . . . , cτ∗ −1 } has been chosen. If at t = τ , DM(τ ) has a low loss aversion ωτ λ < 1, then it pays off for DM(τ ) to choose an alternative consumption path and he will start a new plan {cτ , cτ +1 , . . . , cT } from time τ . Proposition 4. If there is a deviation by DM(τ ), who has a preference of ωτ λ < 1 and sets an alternative consumption ct (τ ) for t = τ , τ + 1, . . . , T , then the deviation by DM(τ + 1 ) from cτ +1 for a larger consumption would not pay off if ωτ +1 λ  ωτ λ. His choice is cτ +1 at most. If, however, ωτ +1 λ < ωτ λ < 1, then it is possible for DM(τ + 1 ) to deviate from cτ +1 for an even larger amount, cτ+1 > cτ +1 . As long as DM(τ + 1) has a lower weight than DM(τ ), deviation from the plan of DM(τ ), i.e. c1 is optimal. Since ωτ λ < 1, it always pays off for DM(τ + 1) to choose a consumption strategy which is lower than the intended plan of DM(τ ), if DM(τ + 1) cares more about the future (ωτ +1 λ > 1 ). Even though DM(τ + 1) does not care about his future as much (ωτ +1 λ < 1 ), it still pays off for him to abandon DM(τ )’s plan and reduce current consumption as long as his weight is higher than DM(τ ). The next proposition describes the alternative dynamics when DM(τ ) intends more consumption at the cost of the very next period: (cτ > cτ∗ , cτ +1 < cτ∗ +1 ). Proposition 5. If there is a deviation by DM(τ ), who has a preference of ωτ λ < 1 and sets a consumption ct (τ ) for t = τ , τ + 1, . . . , T , then DM(τ + 1 ) would not deviate from cτ +1 < cτ∗ +1 if ωτ +1 λ  ωτ λ. If however, ωτ +1 λ < ωτ λ < 1, then DM(τ + 1 ) would deviate from c for a larger consumption cτ+1 > cτ +1 . When DM(τ ) overconsumes for the period from his taste (ωτ λ < 1), and has a lower subsequent consumption as a result, then DM(τ + 1) should not increase his consumption from the predetermined cτ +1 if he cares about the future more than the previous DM (ωτ +1 λ  ωτ λ ). If, however, DM(τ + 1) also has a lower weight than DM(τ ), then it is better for DM(τ + 1) to choose more consumption. Thus, cτ +1 = cτ +1 if ωτ +1 λ  ωτ λ and cτ +1 = cτ+1 > cτ +1 if ωτ +1 λ < ωτ λ < 1. The proposition implies that when DM(τ ) with ωτ λ < 1 consumes more than the optimal for the current period without caring about the future, DM(τ + 1), who wants to correct this, should accept the low consumption point set by his precedent self. However, this would not apply when DM(τ + 1) has a loss aversion lower than the precedent self. 3.1.2. Saver’s dynamic optimization Consider the optimization procedure of an initial saver (ω0 λ < 1) who plans to consume less for the first two periods, {c0 < c0∗ , c1 < c1∗ , c2 > c2∗ }. The optimal solution to the initial saver’s problem is {c0 , (Rβ )1/γ c0 , (Rβ )2/γ ν 0 c0 }, where

ν0 =



1+η 1+ηω0 λ

1/γ

and c0 =

y0 +y1 R−1 +y2 R−2 . 1+φ −1 +φ −2 ν0

Contrary to DM(0)’s intention, DM(1) may want to deviate from the low con-

sumption for t = 1 and re-plan to increase consumption. Following the analysis in Section 3.1.1, let his reference point be the DM(0)’s solution. Since the reference point is now u(c1 ), DM(1) realizes that he would have a contemporaneous gain utility if he consumes more than this, but would have a prospective loss for the next period as a result of consuming more this period. Then the utility status for DM(1) when he has gain–loss utility by deviating for {c1 > c1 , c2 < c2 } is described by Max u1 (c1 , c2 |c1 , c2 ) : 1 −γ



1 −γ

 1 −γ

c1 c1 c +η − 1 1−γ 1−γ 1−γ





+β

The derivative is evaluated at c1 :

 R −γ  φ

c0

( 1 +η ) -



1 −γ

1 −γ

 1 −γ

c2 c2 c + ηω1 λ − 2 1−γ 1−γ 1−γ

(1+ηω1 λ )(1+ηω0 λ ) 1+η



 0 if

> 0 if

1+η 1+ηω1 λ 1+η 1+ηω1 λ

 >

1+ηω0 λ 1 +η 1 +ηω0 λ 1 +η



(32)

 (33)

8 To compare the two weights in a consumption plan, the weights should be related to the same reference point. Here this is not satisfied. Because of this, I indirectly explain the loss aversion to the new reference point in terms of the initial reference point.

H. Park / Research in Economics 73 (2019) 107–122

117

By the second condition, the deviation for a larger consumption amount is always profitable if he does not care as much 1+ηω0 λ η about the next period (ω1 λ < 1). The condition follows from 1+1+ because ω0 λ < 1. If DM(1)’s attachment ηω λ > 1 > 1+η 1

to the plan of low consumption is not too strong, he can deviate and replan. Now turn to the other case. If DM(0) is a saver (ω0 λ < 1) and his plan is to save only in the first period, {c0 < c0∗ , c1 > c1∗ , c2 > c2∗ }, then the consistent strategy is {c0 , (Rβ )1/γ ν 0 c0 , (Rβ )2/γ ν 0 c0 }, where c0 =

y0 +y1 R−1 +y2 R−2 . 1+φ −1 ν0 +φ −2 ν0

Instead of enjoying high consumption, if DM(1) intends to save again

for t = 1, then DM(1)’s utility position for {c1 < c1 , c2 > c2 } is described by Max u1 (c1 , c2 |c1 , c2 ): 1 −γ



1 −γ

 1 −γ

c c1 c1 + ηω1 λ − 1 1−γ 1−γ 1−γ The deviation condition is

du1 = −η (1 − ω1 λ ) dc1







+β

1 −γ



1 −γ

 1 −γ

c c2 c2 +η − 2 1−γ 1−γ 1−γ



  1 + η 1/γ −γ   0 if ω1 λ  1 > 0 if ω1 λ > 1 φ 1 + ηω0 λ R  c0

(34)

(35)

The deviation for less consumption is profitable if DM(1) cares more about the future consumption than about current consumption. If DM(1) has low attachment to the previous plan of high consumption, then he can reduce the consumption to save even more. However, if DM(1) prefers DM(0)’s intention of high consumption, then he would not deviate from the plan. 3.2. The sub-period perfect reference point In the dynamic model, where the subsequent reference points are changing over time, there is no universal way to describe the DM’s behavior. Since there can be many reference points to consider at any time of decision making, the consistent consumption profiles can be numerous based on each of the alternative reference points. Here I introduce a novel way to solve this problem and propose the following reference point that can be applied to any period of the dynamic model. Assume that there is no uncertainty and the deterministic income stream is known. ∗ , . . . , c∗ } Definition. Given any wealth position at time t, a sub-period perfect plan is (i) the ex ante optimal solution {ct∗ , ct+1 T to the time-t maximization problem of the DM whose preference follows RDP and (ii) whose preference is not changing over the planning horizon of T − t + 1 periods. At any time t, given any debt or savings inherited from the previous period, the sub-period perfect plan gives the DM a clear notion of what the ex ante optimal consumption should be for the remaining periods. I postulate that this plan serves as the time t-contingent reference point (sub-period perfect reference point: SPRP) to which the DM will refer. The merit of this definition is that the current asset/liability of the DM directly affects the optimal consumption stream for the remaining life so that the plan itself is self-adjusting over the subsequent periods.9 SPRP should be a proper reference point in the dynamic model in which the DM may change his degree of loss aversion and revise the strategy following his new taste. Whenever the DM changes his mind, the reference point is updated, and the updating reflects his current financial situation. If there is no deviation at t − 1, then the new reference point for time t is the same as DM(t − 1 ) set for the period because DM(t) does not have any extra debt or saving other than what was planned from the last period. Likewise, if there is no deviation at all up to time t − 1, then the reference point is exactly what the initial DM set for the period. Because of this, the initial sub-period perfect plan, c∗ (0 ) = {c0∗ , c1∗ , . . . , cT∗ } may be called the super plan. Any plan other  , . . . ,c }. than c∗ (t) is called an alternative plan c (t ) = {ct , ct+1 T Proposition 6. If DM(t) has a preference of ωt λ  1, then there is no deviation from the optimal plan at t. If DM(t) has ωt λ < 1, then it pays off for the DM(t) to choose an alternative consumption path c (t) other than the sub-period perfect plan c∗ (t). If DM(t) is an over-consumer, then he consumes more at t, ct > ct∗ . Given the SPRP(t), the consumption {c (t)} is a PE if u(c (t)|c (t))  u(c(t)|c(t)) for all feasible c(t). There is no incentive to deviate from the ex ante optimal committed {c∗ (t)} if ωt λ  1. However, the planned strategy {c∗ (t)} is not consistent for those withωt λ < 1, because deviation from ct∗ increases the utility ex post, i.e. u( c (t )|c∗ (t )) > u(c∗ (t )|c∗ (t )) for a feasible  c (t ). The DM(t) whose concern about loss is relatively low must meet the consistency condition. The consistent consumption path for any intended plan {Iτ }tT is given by {c  (t)}, which solves the new maximization problem from time t. This choice {c  (t)} maximizes DM(t)’s utility among all feasible, consistent plans to consider, and thus is PPE as the choice satisfies u(c  (t)|c  (t))  u(c (t)|c (t)) for all c (t) which are PE. Proposition 7. Assume that all DMs up to period τ have high loss aversion, ω0 λ  1, ω1 λ  1, . . . , ωτ −1 λ  1, and thus {c0∗ , c1∗ , . . . , cτ∗ −1 } has been chosen based on the reference point of the sub-period perfect plan up to τ − 1. If at τ , DM(τ ) has low loss aversion ωτ λ < 1, then it pays off for the DM(τ ) to choose an alternative consumption path to fulfill his taste, which is 9 For example, if there was over-consumption by the previous DM, leaving a great debt, then the new plan by the subsequent DM automatically incorporates this debt. By SPRP, the new reference point has been adjusted accordingly to this debt.

118

H. Park / Research in Economics 73 (2019) 107–122

T t ∗ ∗ 10 c (τ ) = arg max t= τ β {u (ct ) + ηIt [u (ct ) − u (ct )] + ηωτ λ(1 − It )[u (ct ) − u (ct )]}, where the reference point is the sub-period perfect plan at time τ . 3.3. Dynamic lifecycle model Being allowed to change his mind over time, DM(t) decides either to follow the consumption rule determined by his earlier self or start a new plan from his current taste. Once the DM decides to change his mind, he re-optimizes for the remaining periods. The reference point where the DM’s new belief is formed is now the ex ante optimal consumption to the maximization problem over the remaining periods, which reflects that the DM has inherited a financial wealth (positive or negative) from the previous consumption. Because each DM rebalances his consistent consumption as he re-optimizes according to his new belief on the ex ante optimal plan for the rest of life, the actual consumption profile is the envelope of the initial consumption points set by each DM(t). 3.3.1. Dynamic loss aversion and consumption hump Consider a simple model where the DM with a dynamic loss aversion parameter ωt , lives for four periods and optimizes for t = 0, 1, 2, 3. If, at any time t, DM(t) changes his mind, he will re-plan for the remaining life according to his new taste. Also, the reference point is the SPRP defined in Section 3.2. Assume that the agent intends to deviate from the ex ante optimal path for more consumption for the first half of the life, lowering his consumption in the second half as suggested in Section 2.2.1 (utility comparison). At t = 0, the agent maximizes his lifecycle utility u0 (c0 , c1 , c2 , c3 |c∗ (0)): 3 



β

t

t=0

1 −γ

ct + ηIt 1−γ



1 −γ

∗1−γ

ct c − t 1−γ 1−γ





1 −γ

∗1−γ

ct c + ηωλ(1 − It ) − t 1−γ 1−γ



(36)

subject to Eq. (11) with b0 = b4 = 0, and It = {1, 1, 0, 0}. It is clear that deviation from the optimal path (c0∗ , c1∗ , c2∗ , c3∗ ) is not profitable if ω0 λ  1. But if ω0 λ < 1, then deviation is optimal, and the consistent consumption is {c0 , (R/φ )c0 , (R/φ )2 μ0 c0 ,  1/γ y + y1 R−1 + y2 R−2 + y3 R−3 1+ηω0 λ (R/φ )3 μ0 c0 }, where c0 = 0 and μ = . Note that μ0 < 1 (deviators) corresponds to 0 1+ η 1 + φ −1 + φ −2 μ0 + φ −3 μ0 ω0 λ < 1, while μ0 = 1 for the non-deviators. In fact, μt is the parameter that represents the main property of gain–loss utility, and can be called the “relative loss aversion parameter.” For the DM with μ0 < 1, it is also obvious that c0 > c0∗ , c1 > c1∗ but c2 < c2∗ , c3 < c3∗ . Thus, he consumes c0 at the first period and intends to consume c1 in the next period. At t = 1, if DM(1) changes his mind and decides to deviate from this plan, then he must re-optimize for the remaining periods according to his current taste ω1 , given the temporal financial position. A negative position (debt), which may be linked to the over-consumption of the first period, comes from an extra setback beyond the scheduled bond holding. At t = 1, the re-optimization problem of DM(1) for the remaining periods t = 1, 2, 3 is Max u1 (c1 , c2 , c3 |c∗ (1)): 3 



β

t

t=1

1 −γ

ct +ηIt 1−γ



1 −γ

∗1−γ

ct c − t 1-γ 1 -γ





1 −γ

∗1−γ

c c +ηω1 λ(1 − It ) t − t 1-γ 1 -γ



(37)

subject to Eq. (11) with b1 = y0 − c0 , and It = {1, 0, 0}. The budget constraint shows  that b1 is given at time t= 1 by the consumption behavior at t = 0. The solution for the deviator is {c1 , c2 , c3 } = c1 , (R/φ )μ1 c1 , (R/φ )2 μ1 c1 , with





1/γ y1 + y2 R−1 + y3 R−2 + Rb1 1+ηω1 λ and μ1 = . This solution is the consistent consumption profile for t = {1, 2, 3}, 1+η −1 −2 1 + φ μ1 + φ μ1 starting from t = 1. The new solution c1 by DM(1) is different from what DM(0) calculated last period. To emphasize the difference, name this c11 denoting the new consistent consumption starting from t = 1. Likewise, c21 is the consistent consumption for the next period, i.e. t = 2, planned at t = 1. Similarly, at t = 2, DM(2) with ω2 λ < 1 re-plans for the rey +y R−1 +Rb2 maining two periods, yielding the consistent consumption {c22 , c32 } = {c2 , (R/φ )μ2 c2 } in which c2 = 2 3 −1 and 1 + φ μ2

c1 =

μ2 =



1+ηω2 λ 1+η

1/γ

. Altogether, in accordance with each DM(t)’s preference ωt λ, the consistent consumption profile for the

entire four periods {c00 , c11 , c22 , c32 } is



y

y

y

y0 + R11 + R22 + R33

y

1+ φ1 + φ12 μ0 + φ13 μ0 where μt =



1+ηωt λ 1+η

1/γ

,

y

y1 + R21 + R32 +Rb1 1+ φ1 μ1 + φ12 μ1



y

,

y2 + R31 +Rb2 1+ φ1 μ2

,

R

φ

μ2 c22

(38)

. If the preference of the DM μt changes over time, it is possible to obtain the featured consumption

hump even with a flat (or monotonic) income profile. This is important because the property of a hump-shaped consumption profile is not expected in the canonical model with far-sighted rational agents in the frictionless world. The consumption hump can be obtained with a model of time-inconsistent preferences, but even with this, it usually requires certain restrictions on income.11 Can the hump be obtained with the four-period model? A hump comes with either {c0 < c1 < c2 > c3 } or 10 11

Here ωτ is fixed and does not change over time within the planning horizon of DM(τ ). For example, a hump-shaped income profile can generate a consumption hump in certain models. See Park and Feigenbaum (2017).

H. Park / Research in Economics 73 (2019) 107–122

119

{c0 < c1 > c2 > c3 } depending on the peak location c1 or c2 . In either case, the hump is obtained when c0 < c1 and c2 > c3 . Solving recursively for the bond demands b1 and b2 , and substituting into Eq. (38) pins down the entire consumption. The resulting consistent consumption {c0 , c1 , c2 , c3 } ≡ {c00 , c11 , c22 , c32 } is

⎛

c0 =

⎞

h0

0 1+ φ + μ + μ0  φ2 φ3  (h1 + Ry0 ) φ1 + μφ 20 + μφ 30

⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟   c1 =  ⎜ ⎟ μ1 μ1 μ0 μ0 1 ⎜ ⎟ 1 + φ + φ2 1 + φ + φ2 + φ3 ⎜ ⎟   

 ⎜ μ1 μ1 ⎟ 0 ⎜ ⎟ h2 + Ry1 + R2 y0 1 + μφ0 + μ + φ2 φ2 φ3 ⎟ ⎜   ⎟ ⎜c2 =

 μ0 ⎟ ⎜ 1 0 1 + μφ2 1 + μφ1 + μ 1 + φ1 + μ ⎜ φ2 φ2 + φ3 ⎟ ⎝ ⎠ R c 3 = μ2 c 2 φ 1

(39)

 where ht = 3τ =t Ryττ−t represents “human wealth” at time t. When the DM is not deviating, the consumption profile is obtained by letting μt = 1. To characterize the consumption property out of this simple version of the dynamic model, I compare this with the consumption profile from the standard lifecycle model in which the DM has no gain–loss utility. The optimization in the standard model predicts that the marginal utility of consumption between any two periods conforms to the Euler rule, and thus the consumption profile over every period exhibits monotonic movement. Because of the monotonicity, the consumption profile is either increasing, decreasing, or constant over the entire life. The consumption profile h0 of the standard agent is characterized by {c0 , (R/φ )1 c0 , (R/φ )2 c0 , (R/φ )3 c0 }, where c0 = , with the human 1+φ −1 +φ −2 +φ −3 3 wealth h0 = t=0 . Note that this is equal to {c0∗ , c1∗ , c2∗ , c3∗ }. The monotonicity of the consumption profile is preserved across any choice of income processes. For the RDP agent this monotonicity holds for the special case of μt = 1. In fact, this is the situation where the DM does not deviate from the ex ante optimal path because of high loss aversion. Therefore, by letting μt = 1 for t = 0, 1, 2, 3, the RDP consumption Eq. (39) return to the {c0∗ , c1∗ , c2∗ , c3∗ }. To explore whether the dynamic consistent consumption can generate the featured consumption hump, I further assume that the gross interest rate is positive (1 + r = R > 0) and the risk aversion parameter is positive (γ > 0). Proposition 8. Given R and β , if the preference of a dynamic agent, {μ0 , μ1 , μ2 }, satisfies that Rφ

−1







1 + φ −1 μ1 + φ −2 μ1 <

1 + φ −1 μ0 + φ −2 μ0 , then the consistent consumption is initially increasing, i.e. c0 < c1 , regardless of the income stream

 

of the agent. The upper bound of μ1 for a hump is μ0 φR

+

R−φ

1+φ −1

.

DM(t)’s preference μt is directly (one to one) related to ωt λ given other parameter values. If the DM is not deviating (μ0 = μ1 = μ2 = 1), the condition returns to 1 < (R/φ ), which in turn returns β R > 1. This is the sufficient condition for any strictly increasing consumption in the standard model. Therefore, when the DM deviates, increasing property of consumption can be obtained without the stronger condition β R > 1. To clearly see the meaning of the condition, first look at the condition with β = 1/R:

1 + R−1 μ1 + R−2 μ1 < 1 + R−1 μ0 + R−2 μ0

(40)

This condition implies μ1 < μ0 . Whenever the loss aversion of DM(1) is less than the one of DM(0), the consumption is increasing because μt is positively related to the loss aversion parameter ωt . When β R > 1, the restriction on the loss aversion parameter can be relaxed due to the time preference. However, when β R < 1, the condition requires μ1 < μ0 to insure the above property.12 Proposition 9. Given R and β , if the preference of the RDP dynamic agent, {μ0 , μ1 , μ2 }, satisfies (R/φ )μ2 < 1, then the consistent consumption is decreasing in the later periods, c2 > c3 , regardless of the income stream. The preference of DM(3) does not affect this condition because his consumption is residual. Again with β = 1/R, the condition returns μ2 < 1. If β = 1/R, then the time preference and the loss aversion jointly operate to yield the decreasing property. This result contrasts to the standard model; when the DM is not deviating so that it produces the same result as in the standard model, the time preference is the sole determinant of the inequality, R/φ < 1. However, with the RDP model, With the static model, the hump is obtained if ωλ <1 for a DM who has intention of a humped-shaped consumption profile {c0 c1∗ , c2


2 

η 1/γ y0 +y1 R−1 +y2 R−2 and ν = 1+1+ . This profile produces , with c0 = deviation is optimal (ωλ < 1), the consistent consumption is c0 , c0 φR ν, c0 φR ηωλ 1+φ −1 ν +φ −2 the consumption hump even in the simplest environment, such as β = 1 and R = 1, while the standard model produces a flat consumption profile. The  y +y +y ν y +y +y  ( 0 1 2 ) y0 +y1 +y2 0 1 2 . Since ν >1 when ωλ <1 for all values of the risk aversion parameter γ >0, , , RDP consumption under this condition is 1 +ν + 1 1+ν +1 1+ν +1 it is clear that c0 < c1 > c2 . This result is important because this profile is not dependent on any assumption about either income, time preference, or the magnitude of interest rate. Only the loss aversion matters to produce a hump. 12

120

H. Park / Research in Economics 73 (2019) 107–122

Fig. 1. The consumption is obtained with a flat income stream for a dynamic overconsumer (A) and a dynamic saver (B). The parameters are η = 1, λ = 2, R = 1.035 for both. For (A), ωt = {0.45, 0.4, 0.35}, β = 0.98, γ = 0.9. For (B), ωt = {0.35, 0.45, 0.5}, β = 0.94, γ = 0.5.

it is not a necessary condition anymore. Even with β R > 1, it is possible to get the result if μ2 is sufficiently small. Finally, combining both conditions (Propositions 8 and 9) yields the consumption hump. How about the saver’s hump? In a four-period model, the consumption profile is

⎛

c0 =

⎞

h0

⎜ ⎟ 1 + φ1 + φν02 + φν03   ⎜ ⎟ ⎜ ⎟ μ0 μ0 1 (h1 + Ry0 ) φ + φ 2 + φ 3 ⎜ ⎟ ⎜ ⎟   c1 =  ⎜ ⎟ ⎜ ⎟ 1 + νφ1 + φν12 1 + φ1 + φν02 + φν03 ⎜   ⎟

 ⎜ μ1 μ1 ⎟ 0 ⎜ ⎟ h2 + Ry1 + R2 y0 1 + μφ0 + μ φ2 φ2 + φ3 ⎟ ⎜    ⎟ ⎜c2 =

 ⎜ 1 + νφ2 1 + νφ1 + φν12 1 + φ1 + φν02 + φν03 ⎟ ⎜ ⎟ ⎝ ⎠ R c 3 = ν2 c 2 φ where the relative loss aversion parameter is νs =



1+η 1+ηωs λ

1/γ

(41)

 1.

Therefore, with {ν0 , ν1 , ν2 }, the condition for increasing part for a hump is







1 + φ −1 ν1 + φ −2 ν1 < Rφ −1 1 + φ −1 ν0 + φ −2 ν0



(42)

The condition for decreasing consumption is (R/φ )ν2 < 1, together with R/φ < 1, because ν s 1. Although this condition looks to be the mirror image to the over-consumer’s, its interpretation is the opposite. For example, the increasing condition is satisfied ν1 < ν0 whenever R/φ  1. Since ν s (relative loss aversion of a saver) is inversely related to ωs , the consumption is increasing for the saver when the loss aversion of the subsequent DM is greater than the precedent one. In this case, the DM can reduce his savings by consuming more than before. Likewise, the decreasing property is obtained when the saver reduces his intention to save. Fig. 1 shows the consumption hump for a dynamic DM who deviates from the optimal plan to consume more (A) and save more (B). 4. Conclusion Reference-dependent preferences help explain many phenomena: excessive aversion to small risk, the reluctance to sell a house at a loss, the equity premium puzzle, insurance against small risk, and target earnings in labor supply decision. Yet, there are few works, if any, which demonstrate how these types of preferences can help explain consumption dynamics in lifecycle data. Based on this motivation, I develop a dynamic model of belief dependent preferences and examine its implication when the reference status is the consumer’s belief on the optimal consumption for current and future periods. To dissolve the key issue of the determination of reference points in these models, I take the solution concept of the rational ˝ expectation equilibrium by Koszegi and Rabin (2006). The main agents in the paper are those who have low loss aversion with respect to the ex ante optimal consumption strategy. Throughout this paper, I look for the consistent intertemporal optimization rule for a decision maker when he has a certain weight on loss relative to gain. Anyone who cares about the future is considered to have a high weight since he would not over-consume early in life at the cost of severe low future consumption. Likewise, anyone who cares more of the current pleasure of extra consumption is thought to have a low weight because this agent has tolerance of the pain from a prospective loss. The same argument applies to the saver’s case as well. In any of these, the choice of consumption strategy should meet the rational intertemporal consistency. This paper contributes to the literature on intertemporal choice, lifecycle models and consumption dynamics by demonstrating that the reference dependent preferences can explain some noted features of consumption dynamics in lifecycle data

H. Park / Research in Economics 73 (2019) 107–122

121

like the consumption hump, as well as by providing a multi-layered understanding of dynamic decision making. I prove the consumption hump for an over-consumer and for a saver in the dynamic model. In fact, the hump is closely related to loss aversion, combined with the time preference, regardless of income structure. I also find the best consumption scheme for deviators, providing closed form solutions when they have alternative reference points. Furthermore, I show how to dissolve the problem of many reference points in the dynamic model by introducing the novel concept of the sub-period perfect reference points. Finally, throughout this paper, I illuminate the existence of savers in an economy and the indirect relationship between consumption and income through age. Acknowledgments I would like to thank John Duffy, James Feigenbaum, Botond Koszegi, Andreas Blume, Liz Vesterlund, Daniele Coen-Pirani, Sourav Bhattacharya, Luca Rigotti and Attila Ambrus for helpful comments and discussions. Appendix for proofs Proposition 1. The maximization problem of the DM with ωλ < 1, who for the first two   plans to overconsume periods {c0 > c0∗ , c1 > c1∗ , c2 < c2∗ } is max u(c|c∗ ) =

 ∗1−γ c ωηλ 12−γ −

1 −γ c2 1−γ

 }. The optimality condition requires

yields {c0 , c1 , c2 } = {c0 , (Rβ )

1/γ

c 0 , ( Rβ )

2/γ

1 −γ c0 1−γ

1 −γ c0 1−γ

+η

∗1−γ c0 1−γ

−

c

1 −γ

1 −γ c1 1−γ

+ β{ 11−γ + η

∗1−γ c1 1−γ

−

c

1 −γ

} + β 2 { 12−γ −

γ γ (1+η )c− (1+η )c− 0 1 −γ = Rβ and γ = Rβ . From the budget constraint, this (1+η )c1 (1+ωηλ )c− 2

μc0 }, where c0 = y0 +

y1 R1

+

y2 R2



2



/ 1 + φ1 + φ1 μ .

Proposition 2. By the same logic as in DM(0), if DM(1) has ω1 λ  1, he would not deviate from the optimal soy y1 + R2 + Rb1 lution to the maximization problem that starts from t=1: Max u1 (c|c∗ ). The solution is c1 = and c2 = −1

−1 −2  1 +y1φ y2  y1 y2 y0 φ +φ − R1 + R2 y0 + R1 + R2 = . From this, (R/φ )c1 . Because DM(0) consumes c0∗ , leaving b1 = y0 − c0∗ = y0 − −1 −2 −1 1+φ +φ 1+φ +φ −2

 y1 y2 R y2 y0 + R1 + R2 φ y1 + R +Rb1  = c1∗ . Therefore, c1 = c1∗ . The consistent consumption is not differc1 is obtained by c1 = =

−1 1+φ 1 + φ −1 + φ −2 ent from the one DM(0) planned for the period. When ω1 λ < 1, he would deviate  because deviation increases his utility:

du1 (c|c∗ ) = dc1∗

ηω1 λ )β R φ (1+φ R

−1





)

[1+φ −1 μ1 ](

(1 + η )

RR ∗ φ c0



R ∗ φ c0

−γ



−γ



3. The c2 1−γ

Rβ .

b1 = y0 − c0 =

 

y Ry0 +y1 + R2

1+φ −1



1+ηω1 λ 1+η



1/γ

φ −1 



=

maximization

 ∗1−γ

 1 −γ c βηω1 λ 12−γ − Because



R 1+φ −1 +φ −2

−γ

+β

1 −γ c2 1−γ

1+ηω0 λ 1+η

1+φ −1



1/γ

problem

subject y0 φ −1



to

1+ηω0 λ 1+η

1+φ −1

+φ −2

1+ηω0 λ 1+η



1/γ

1+ηω0 1+η

+φ −2



c2 1/γ



1+ηω0 λ 1+η

of

DM(1)

=

y2 + R(y1 + Rb1 − c1 ).

+y0 φ −2

1/γ

1+ηω0 λ 1+η 1/γ  λ

the consistent consumption is c1 (1 ) = c1 (0 ) c1 (0 ) if ω1 λ < 1.



y y y0 φ −1 +y0 φ −2 − 11 − 22

− (1 + ηω1 λ )β R[y2 + R(y1 +R

− φR c0∗ )]−γ = (1 + η ) φR c0∗

R

1+ηω0 λ 1+η

+φ −2





1/γ



to

maximize

y y − 11 − 22 R

1+ηω0 λ 1+η

 R 1/γ  = c0 φ 1 1+ φ

1+φ −1



is

1/γ

1+ηω0 λ 1+η

R

,

it

1/γ

is

1+φ −1



=

1 −γ c1 1−γ

 +η

optimality

condition

true

c1 (1 ) =

1 1+ φ

= c1 (0 ) when  1+ηω1 λ 1/γ 1+η

The

u1

( c ( 1 )|c ∗ )

1+ηω1 λ 1+η

that

1/γ = c1 (0 )

1 −γ c1 1−γ

1+φ −1

− (1 +

−

∗1−γ c1 1−γ

=

 +

(1+η )c1−γ = (1+ηω1 λ )c2−γ

is

y y1 + R2 +Rb1

1+φ −1

1 1+ φ



−γ

y y1 + R2 +Rb1 1+φ −1 μ1

R ∗ [η (1 − ω1 λ )]>0 if ω1 λ <1. The new consumption is c1 = φ c0     y y y R −1 1 y2 y0 + 11 + 22 φ (1+φ ) y0 + R1 + R2 R  ∗  ∗ R R . Because c1 > = φ 1+φ −1 +φ −2 = c1 , it is true that c1 > c1 . [1+φ −1 ](1+φ −1 +φ −2 ) )

∗ 1+φ −1 − RR φ c0

y y y0 + 11 + 22 R R 1+φ −1 +φ −2

Proposition



1+ηω1 λ 1+η

ω1 λ  1. However, c1 (1 ) = c1 (0 )



1+ηω1 λ 1+η

1/γ =

1/γ . Therefore, 1 1+ φ

1+φ −1



1+ηω1 λ 1+η

1/γ >

Proposition 4. Suppose that DM(τ + 1 ) has a higher weight than the previous DM such that ωτ +1 λ  ωτ λ, but consumes cτ+1 , which is larger than c, and as a result, cτ+s < cτ∗ +s , s = 1, 2, . . . , T − s − τ . Then this consumption is not consistent with respect to his belief on his reference point: because the reference point for the subsequent periods is u(cτ∗ +s ), he has loss utility −[u(cτ∗ +s ) − u(cτ+s )], and if his loss aversion is greater than his previous DM, then this loss utility should not be bigger than −[u(cτ∗ +s ) − u(cτ +s )] for each of the subsequent periods. Therefore, cτ+s  cτ +s . If however, ωτ +1 λ < ωτ λ < 1, then because his loss aversion is lower than before, cτ +1 is no more a consistent solution. Thus he deviates.

122

H. Park / Research in Economics 73 (2019) 107–122

Proposition 5. Suppose DM(τ + 1 ) with ωτ +1 λ  ωτ λ consumes cτ+1 , which is larger than cτ +1 , and as a result, cτ+s < cτ +s < cτ∗ +s , s = 1, 2, . . . , T − s − τ . Then this consumption must be a consistent consumption profile with respect to his belief on his reference point. Since the reference point for the subsequent periods is u(cτ∗ +s ), he definitely has loss utility −[u(cτ∗ +s ) − u(cτ+s )], which is bigger than −[u(cτ∗ +s ) − u(cτ +s )] for each of the subsequent periods. Since his loss aversion is greater than his previous DM, cτ+s is not the consistent consumption. If, however, ωτ +1 λ < ωτ λ < 1, then because his loss aversion is lower than before, cτ +1 is no more a consistent solution. He deviates for more. Proposition 6. Suppose DM(t) with ωt λ  1 deviates for more consumption than the sub-period perfect plan and consumes ct which is higher than ct∗ . Then by doing this, the DM incurs prospective losses of −[u(ct∗ ) − u(ct )] . Comparing his expected loss utility of zero if he follows the sub-period perfect plan, it is clear that the plan is not fulfilling the DM’s high loss aversion. Thus he would not deviate for more. Likewise deviation for less produces a current loss of −[u(ct∗ ) − u(ct )] < 0 and thus this can’t be an optimal choice to the DM either. If however, ωt λ < 1, then deviation is justified because the losses −[ηωt λ[u(ct ) − u(ct∗ )]] are not bigger than the expected gains η[u(ct ) − u(ct∗ )] from the alternative consumption he chooses. That is, η[u(ct ) − u(ct∗ )] − ηωt λ[u(ct ) − u(ct∗ ) > 0. Proposition 7. Since the path {c0∗ , c1∗ , . . . , cτ∗ −1 } has been chosen, this path leaves the DM with no extra debts or savings for the period τ other than the amount that is necessary for the optimal path for the remaining periods. Let this be the optimal bond demand at t = τ , b∗τ . Since the sub-period perfect plan at time t is defined by the optimization rule based on the current financial wealth at period τ , this wealth is equal to the optimal bond demand for time τ . Thus, if the new plan starts from period τ , then the reference point must be from the sub-period perfect plan at time τ , which is cτ∗ . The proof of deviation by DM(τ ) for more or less consumption follows the same logic as in the previous proposition. Proposition



μ

8. Notice μ

R φ1 + φ 20 + φ 30

that

Rh0 = h1 + Ry0 .

The

first two

consumption

amounts

are

c0 =



h0

μ

μ ,

1 + φ1 + φ 20 + φ 30

c1 =

 , each of which does not depend on the income stream (ht ). It is straightforward to get c0 < c1 if the condition c0  μ μ 1 + φ1 + φ 21 is satisfied. The upper bound of μ1 is obtained directly from the condition.

Proposition 9. It is clear that R

φ

h2 + Ry1 + R2 y0



μ

μ

1 + φ0 + φ 20



μ1 μ + φ 31 φ2



c3    and c3 = does not depend on ht , because c2 =  c2 μ μ μ μ μ 1 + φ2 1 + φ1 + φ 21 1 + φ1 + φ 20 + φ 30

μ2 c2 . Therefore, it is straightforward that c2 > c3 , if the condition is satisfied.

References Apesteguia, J., Ballester, M.A., 2009. A theory of reference-dependent behavior. Econ. Theory 40, 427–455. Barberis, N., Huang, M., Thaler, R., 2006. Individual preferences, monetary gambles, and stock market participation: a case of narrow framing. Am. Econ. Rev. 96, 1069–1090. Bernasconi, M., Corazzin, L., Seri, R., 2014. Reference dependent preferences, hedonic adaptation and tax evasion: does the tax burden matter? J. Econ. Psycho. 40, 103–108. Bleichrodt, H., 2009. Reference-dependent expected utility with incomplete preferences. J. Math. Psychol. 53, 287–293. De Giorgi, E., Post, T., 2011. Loss aversion with a state-dependent reference point. Manage. Sci. 57, 1094–1110. Farber, H., 2008. Reference-dependent preferences and labor supply: the case of new york city taxi drivers. Am. Econ. Rev. 98, 1069–1082. Fehr, E., Goette, L., 2007. Do workers work more if wages are high? Evidence from a randomized field experiment. Am. Econ. Rev. 97, 298–327. Genesove, D., Mayer, C., 2001. Loss aversion and seller behavior: evidence from the housing market. Q. J. Econ. 116, 1233–1260. Gunther, I., Maier, J.K., 2014. Poverty, vulnerability, and reference-dependent utility. Rev. Income Wealth 60, 155–181. Halevy, Y., Felkamp, V., 2005. A Bayesian approach to uncertainty aversion. Rev. Econ. Stud. 72, 449–466. Herweg, F., Schmidt, K., 2014. Loss aversion and inefficient renegotiation. Rev. Econ. Studies 82, 297–322. Humphreys, B., Zhou, L., 2015. Reference-dependent preferences, team relocations, and major league expansion. J. Econ. Behav. Org. 109, 10–25. Karlsson, N., Loewenstein, G., Seppi, D., 2009. The ostrich effect: selective attention to information. J. Risk Uncertainty 38, 95–115. ˝ Koszegi, B., Rabin, M., 2006. A model of reference-dependent preferences. Q. J. Econ. 121, 1133–1165. ˝ Koszegi, B., Rabin, M., 2009. Reference-dependent consumption plans. Am. Econ. Rev. 99, 909–936. Krähmer, D., Stone, R., 2013. Anticipated regret as an explanation of uncertainty. Econ. Theory 52, 709–728. Laibson, D., 1997. Golden eggs and hyperbolic discounting. Q. J. Econ. 112, 443–477. Masatlioglu, Y., Ok, E.A., 2005. Rational choice with status quo bias. J. Econ. Theory 121, 1–29. Matthey, A., 2005. Getting Used to Risk: Reference Dependence and Risk Inclusion. Working Paper. Park, H., 2016. Loss aversion and consumption plan under stochastic reference points. B. E. J. Theor. Econ. 16, 303–336. Park, H., Feigenbaum, J., 2017. Bounded rationality, lifecycle consumption, and social security. J. Econ. Behav. Org. 146, 65–105. Pesendorfer, W., 2006. Behavioral economics comes of age: a review essay on ‘advances in behavioral economics. J. Econ. Lit. 44, 712–721. Rabin, M., 20 0 0. Risk aversion and expected-utility theory: a calibration theorem. Econometrica 68, 1281–1292. Shalev, J., 20 0 0. Loss aversion equilibrium. Int. J. Game Theory 29, 269–287. Sugden, R., 2003. Reference-dependent subjective expected utility. J. Econ. Theory 111, 172–191. Sydnor, J., 2010. (Over)insuring modest risks. Am. Econ. J. Appl. Econ. 2, 177–199.