Expectations-Based Reference-Dependent Life-Cycle Consumption · 2017-03-15 · behavior and...

Review of Economic Studies (2016) 01, 1–49 0034-6527/16/00000001$02.00c© 2016 The Review of Economic Studies Limited

Expectations-Based Reference-Dependent Life-Cycle Consumption

MICHAELA PAGEL1

Columbia Business SchoolE-mail: [email protected]

First version received April 2014; final version accepted December 2016 (Eds.)

This study incorporates a recent preference specification of expectations-based loss aversion, which has beenapplied broadly in microeconomics, into a classic macro model to offer a unified explanation for three empiricalobservations about life-cycle consumption. First, loss aversion explains excess smoothness and sensitivity–thatis, the empirical observation that consumption responds to income shocks with a lag. Intuitively, such laggedresponses allow the agent to delay painful losses in consumption until his expectations have adjusted. Second,the preferences generate a hump-shaped consumption profile. Early in life, consumption is low due to a first-order precautionary-savings motive. However, as uncertainty resolves over time, this motive is dominated bytime-inconsistent overconsumption that eventually leads to declining consumption toward the end of life. Third,consumption drops at retirement. Prior to retirement, the agent wants to overconsume his uncertain income before hisexpectations catch up. Post-retirement, however, income is no longer uncertain, and overconsumption is associatedwith a sure loss in future consumption. As an empirical contribution, I structurally estimate the preference parametersusing life-cycle consumption data. My estimates match those obtained in experiments and other micro studies, andgenerate the degree of excess smoothness observed in macro consumption data.

JEL Codes: E03, D03, D91, D14.

1. INTRODUCTION

Economists have been devoting considerable attention to understand threeï¿œmajor empiricalobservations about life-cycle consumption: excess smoothness and sensitivity in consumption, a hump-shaped consumption profile, and a drop in consumption at retirement.1 This study offers a new,unified, and parsimonious explanation for these facts based on expectations-based reference-dependentpreferences, developed by Koszegi and Rabin (2006, 2007, 2009) to discipline and broadly apply theinsights of prospect theory. These preferences formalize the idea that changes in expectations aboutconsumption generate instantaneous utility and that losses in expectations about consumption hurt morethan gains please. While these preferences have been shown to explain empirical evidence in variousmicro domains, this study evaluates the preferences in a classic macro domain. The intuition behind myresults reflects the micro evidence that these preferences were developed to explain, and may provide newfoundations for key ideas in the macro consumption literature. Moreover, the preferences generate newbehavior and welfare predictions.

Expectations-based reference-dependent preferences can be briefly described as follows. In eachperiod, the agent’s instantaneous utility has two components: First, “consumption utility” depends on hisconsumption and corresponds to the standard model of utility. Second, “gain-loss utility” is determined byhis consumption relative to a reference point and corresponds to the prospect-theory model of utility. The

1. I AM INDEBTED TO ADAM SZEIDL, MATTHEW RABIN, AND BOTOND KOSZEGI FOR THEIR EXTENSIVEADVICE AND SUPPORT. I THANK NICK BARBERIS, ULRIKE MALMENDIER, TED O’DONOGHUE, YURIYGORODNICHENKO, JOSH SCHWARTZSTEIN, BRETT GREEN, STEFANO DELLAVIGNA, DAVID LAIBSON, MARTINLETTAU, PER KRUSELL, HANNO LUSTIG, AND SEMINAR OR CONFERENCE PARTICIPANTS AT UC BERKELEY(AND HAAS), YALE SOM, CORNELL UNIVERSITY, THE FEDERAL RESERVE BANK OF ST. LOUIS, STANFORD, IIESSTOCKHOLM, EIEF, UC IRVINE, SCIENCES PO, DEPAUL UNIVERSITY IN CHICAGO, UC SAN DIEGO, TILBURGUNIVERSITY, HUNTSMAN SCHOOL OF BUSINESS, AND UNIVERSITY OF KONSTANZ FOR THEIR HELPFUL COMMENTS AND SUGGESTIONS. I ALSO THANK THE EDITOR, DIMITRI VAYANOS, AND THREE ANONYMOUS REFEREESWHOSE COMMENTS IMPROVED THE PAPER CONSIDERABLY. ALL ERRORS REMAIN MY OWN.

1. Refer to Attanasio and Weber (2010) for a comprehensive survey of the life-cycle consumption literature.1

2 REVIEW OF ECONOMIC STUDIES

agent’s reference point is determined by his previous expectations about both his present and his futureconsumption. The agent experiences “contemporaneous” gain-loss utility when he compares his actualpresent consumption with his expectations about present consumption. In this comparison, he experiencesa sensation of gain or loss relative to each previously expected consumption outcome. Additionally, theagent experiences “prospective” gain-loss utility when he compares his updated expectations about futureconsumption with his previous expectations–experiencing gain-loss utility over what he has learned aboutfuture consumption. Thus, gain-loss utility can be interpreted as utility over good and bad news aboutconsumption.

In this paper, I analyze an agent with such news-utility preferences in a life-cycle consumptionmodel. The agent lives for a finite number of periods. At the beginning of each period, he observesthe realization of a permanent and transitory income shock, and then decides how much to consumeand save. Initially, I assume that the agent’s consumption utility is an exponential-utility function. Thisassumption produces a closed-form solution, which provides a precise understanding of the intuitionbehind the preferences’ implications. I then show that all results hold when I assume a power-utilityfunction instead.2

As the key contribution, I show that the preferences generate excess smoothness and sensitivityin consumption: the empirical observations that consumption initially underresponds to income shocksand then adjusts with a delay. Empirical evidence for these inherently related consumption phenomenais first provided by Flavin (1981), Deaton (1986), and Campbell and Deaton (1989) and later surveyedby Jappelli and Pistaferri (2010). Such consumption responses are puzzling from the standard modelperspective (in which consumption fully adjusts immediately), but they can be explained by expectations-based loss aversion. A simple intuition is that, in the event of an adverse income shock, unexpected lossesin present consumption are more painful than expected losses in the future. Therefore, the agent delaysunexpected losses in consumption until his expectations have adjusted in the future. The agent considerslosses in present consumption as more painful than losses in future consumption because the latter dependon future income shocks and are thus more uncertain.

Beyond resolving these puzzles, the preferences generate a hump-shaped life-cycle consumptionprofile, characterized by increasing consumption at the beginning of life, but decreasing consumptiontoward its end. Hump-shaped profiles are empirically documented by Fernandez-Villaverde and Krueger(2007) and Gourinchas and Parker (2002). In my model, this hump results from the interaction of twofeatures of the preferences: a first-order precautionary-savings motive and a time inconsistency.

First, the preferences motivate precautionary savings because loss aversion causes expected gainsand losses in consumption utility to be painful. However, any gains and losses in income cause smallergains and losses in consumption utility on the high and flat part of the concave utility curve–generatingan additional motive to save. This savings motive depends on concavity and is a first-order consideration,in contrast to the precautionary-savings motive under standard preferences. Second, the preferencesgenerate time-inconsistent overconsumption. The agent behaves inconsistently because he takes hispresent expectations as given when increasing current consumption, but he considers future expectationswhen increasing future consumption. However, when the future arrives, he will take future expectationsas given and will consider only the pleasure of increasing consumption above expectations–rather thanincreasing consumption and expectations. Thus, the agent overconsumes in the future relative to hispresent optimal plan. In summary, the precautionary-savings motive keeps consumption low at thebeginning of life. However, the need for precautionary savings decreases when uncertainty resolvesover time and is (at some point) dominated by the time inconsistency causing overconsumption. Thisoverconsumption forces the agent to choose a declining-consumption path toward the end of his life.

2. I assume a standard model environment, as proposed by Carroll (2001) and Gourinchas and Parker (2002), but myresults are robust to many alternative environmental assumptions, such as different income profiles, different savings devices withborrowing constraints, endogenous labor supply, mortality risk, an endogenous retirement date, different pension designs, andincome risk after retirement.

PAGEL REFERENCE-DEPENDENT LIFE-CYCLE CONSUMPTION 3

Finally, the preferences predict a drop in consumption at retirement, an empirical observationdebated by Battistin et al. (2009) among others.3 During retirement, income uncertainty is absentin a standard life-cycle model–which ends time-inconsistent overconsumption. The agent stopsoverconsuming because he allocates certain retirement income instead of uncertain labor income.Certainty implies that overconsumption today will yield a sure loss in future consumption, and this sureloss would hurt more than today’s overconsumption would give pleasure (because the agent is loss averse).Thus, the agent suddenly controls his time-inconsistent desire to overconsume and his consumption dropsat retirement. This result is robust to assuming small uncertainty, such as inflation or pension risk, anddiscrete uncertainty, such as health shocks.

Beyond introducing the dynamic version of news-utility preferences, Koszegi and Rabin (2009)analyze their implications for consumption and savings in a two-period, two-outcome model and obtainboth the precautionary-savings and overconsumption results. Moreover, Koszegi and Rabin (2009) findthat the news-utility agent consumes windfall gains, but delays windfall losses. This result depends onunexpected windfall gains and losses in addition to an initially certain consumption path. Thus, it differsfrom my excess-smoothness finding, explained by a differential response of present and future newsutility due to adjusting expectations.4 Beyond my three main implications, I show that the preferencesgenerate several new and testable predictions. For instance, excess smoothness is prevalent for temporaryincome shocks only if the agent does not face additional permanent shocks, and the degree of excesssmoothness decreases in income uncertainty. I argue that these new predictions can help empiricallydistinguish between news utility and other theoretical explanations, and that news utility provides a robustexplanation independent of other common assumptions, such as power utility, hump-shaped incomeprofiles, or borrowing constraints.

However, this study’s main contribution is that the intuitions behind my results connect robustmicro evidence on reference dependence to compelling concepts in the macro consumption literature.For instance, loss aversion is an experimentally robust risk preference, which explains importantbehavioral phenomena (such as the endowment effect), and macro phenomena (such as stock marketnon-participation and the equity-premium puzzle).5 The explanation of the equity-premium puzzle relateswell to the explanation of the excess-smoothness puzzle, as loss aversion smooths consumption relativeto movements in either asset prices or permanent income. To explain the other life-cycle evidence, thepreferences intuitively unify precautionary savings (studied extensively in the standard consumptionliterature) and time-inconsistent overconsumption (reminiscent of hyperbolic discounting). Moreover,I draw potentially important welfare conclusions: while excess smoothness increases welfare, both thehump-shaped consumption profile and the drop in consumption at retirement decrease welfare.

To quantitatively evaluate the model, I structurally estimate the preference parameters. I follow thetwo-stage method-of-simulated-moments approach of Gourinchas and Parker (2002) and use pseudo-panel data from the Consumer Expenditure Survey (CEX). I can identify all preference parametersbecause each parameter generates specific variation in consumption growth over the life cycle. Then,I compare my estimates to those in the experimental literature, eliciting attitudes toward small and largewealth gambles and intertemporal consumption tradeoffs. I then show that my estimates are in line with

3. A series of studies find that consumption drops at retirement even when considering work-related expenses (Banks et al.(1998), Bernheim et al. (2001), Haider and Stephens (2007), and Schwerdt (2005)), and my data also display such a drop. Moreover,Ameriks et al. (2007) and Hurd and Rohwedder (2003) provide evidence that the drop in consumption is expected. However, Aguiarand Hurst (2005) strongly question the idea that there is a drop in consumption at retirement after properly controlling for healthshocks, work-related expenditures, and home production.

4. To elaborate on the result about windfall gains and losses, I explore an extension that assumes that the agent receiveslarge income shocks every couple periods, while receiving only small or discrete income shocks during intercession periods. Smallor discrete uncertainty allows the agent to make a credible plan to overconsume less and implies that he will consume entire smallgains and delay entire small losses.

5. The endowment effect refers to the phenomenon that people are less willing to give up items once they own them.


not only the micro literature, but also the empirical evidence for excess smoothness and sensitivity inaggregate macro data.

The rest of the paper is organized as follows. After a literature review, I explain the modelenvironment, preferences, and equilibrium in Section 2.1. Then, I derive the model predictions in closedform assuming exponential utility in Section 2.2. After introducing the power-utility model, I structurallyestimate the preference parameters in Section 2.3. Finally, Section ?? concludes.

1.1. LITERATURE REVIEW

Expectations-based reference-dependent preferences are developed by Koszegi and Rabin (2009),following the static version of Koszegi and Rabin (2006, 2007), which explains experimental and othermicro evidence in many contexts. Heidhues and Koszegi (2008, 2014), Herweg and Mierendorff (2012),and Herweg et al. (2010) explore the implications for consumer pricing and principal-agent contracts.Among others, Sprenger (2015) provides direct evidence for the implications of stochastic referencepoints, Abeler et al. (2011) for labor supply, Gill and Prowse (2012) for real-effort tournaments, Mengand Weng (2016) for the disposition effect, and Ericson and Fuster (2011) for the endowment effect (notconfirmed by Heffetz and List (2014)). Crawford and Meng (2011) provide suggestive evidence for laborsupply, Pope and Schweitzer (2011) for golf players’ performance, and Sydnor (2010) for deductiblechoice. These studies consider the static preferences, but all experiments using monetary payoffs supportthe notion that agents are loss averse with respect to news about future consumption.

Excess smoothness in consumption cannot be generated with a time-separable utility function(Ludvigson and Michaelides (2001) among others). The most common additional assumption is ofborrowing constraints; however, the presence of borrowing constraints could not ultimately be empiricallyconfirmed.6 Theoretically, the agent expects borrowing constraints and ensures that they are not bindingfor most income paths in the standard model (Deaton (1991) among others). However, Deaton (1991)highlights the possibility to construct a model with impatient agents and particular income processes inwhich borrowing constraints bind frequently. Moreover, borrowing constraints are binding more oftenin models featuring a time-inconsistency problem (Angeletos et al. (2001) and Laibson et al. (2012)).In these models, sophisticated hyperbolic-discounting preferences imply that the agent restricts hisconsumption opportunities with illiquid savings (against which he can borrow up to some constraint).To the extent that the agent’s borrowing constraint binds or his liquid asset holdings bunch at zero, hisconsumption is excessively smooth.7 Demand for commitment is also generated by temptation-disutilitypreferences, as specified in Gul and Pesendorfer (2004) and analyzed by Bucciol (2012) in a life-cyclecontext.

Alternative preference-based explanations for excess smoothness also exist. Caballero (1995)assumes that agents consume near-rationally, while Fuhrer (2000) and Michaelides (2002) assumeinternal multiplicative habit formation. However, Michaelides (2002) finds that habit formation generatesunreasonably high wealth accumulation when calibrated to match the empirical degree of excesssmoothness. In addition, Flavin and Nakagawa (2008) consider non-durable and housing consumption,

6. Refer to Jappelli and Pistaferri (2010) for a comprehensive survey. For instance, unlike Krueger and Perri (2011), Shea(1995) finds that consumption is more excessively sensitive to expected income decreases than increases. This finding is inconsistentwith borrowing constraints, but consistent with a static model of loss aversion, as in Bowman et al. (1999). Recently, Olafsson andPagel (2016) show that less than three percent of individuals have less than one day of spending left in liquidity or cash before theirpaydays, using a very accurate panel dataset of individual spending, income, balances, and credit limits from a financial aggregationapp.

7. Laibson et al. (2012) show that the hyperbolic-discounting agent exhibits a high marginal propensity to consume out oftransitory income shocks, whereas the permanent income model predicts this propensity to consume close to zero. A high marginalpropensity to consume out of transitory income shocks is prevalent for all preference specifications that feature a time-inconsistencyproblem, if the agent has access to illiquid assets (as he tries to make wealth inaccessible to his future selves). Excess sensitivitymay refer to a high marginal propensity to consume out of transitory income shocks or to a delayed response to past income shocks(interpreted as a response to expectedly high income).


which is characterized by adjustment costs. As the utility function depends non-separably on the twogoods, non-durable consumption is excessively smooth and sensitive. Building on Grossman and Laroque(1990), Chetty and Szeidl (2010) also analyze two consumption goods, but assume separability, implyingthat consumption is excessively smooth and sensitive with respect to the durable good only. Elaboratingon Caballero (1995) and Gabaix and Laibson (2002), Reis (2006) assumes that agents face costs whenprocessing information, and, thus, optimally decide to update their consumption plans sporadically.Furthermore, Tutoni (2010) assumes that consumers are rationally inattentive (following Sims (2003))and Attanasio and Pavoni (2011) show that excessively smooth consumption results from incompleteconsumption insurance related to a moral hazard problem.

Several studies show that the standard and hyperbolic agents’ consumption profiles are hump shapedunder the assumptions of power utility, sufficient impatience, and a hump-shaped income profile (Carroll(1997), Gourinchas and Parker (2002), and Laibson et al. (2012)). Other studies that generate a hump-shaped consumption profile are Caliendo and Huang (2008) with overconfidence, Attanasio (1999) withfamily size effects, Deaton (1991) with borrowing constraints, Feigenbaum (2008) and Hansen andImrohoroglu (2008) with mortality risk, Bullard and Feigenbaum (2007) and Heckman (1974) withconsumption-leisure choice, and Fernandez-Villaverde and Krueger (2007) with consumer durables.Caliendo and Huang (2007) and Park and Feigenbaum (2015) assume that the agent has a shorter planninghorizon than his true horizon. In addition, Park and Feigenbaum (2015) shows that short-term planningcan generate the hump in a well-calibrated general-equilibrium model. Last, Caliendo and Huang (2011)show that a hump-shaped profile and an anticipated drop in consumption can be generated by assumingimplementation costs of savings.

2. THE LIFE-CYCLE CONSUMPTION MODEL

First, I define a generalized life-cycle model environment to formally introduce the preferences and theequilibrium concept.

2.1. Model environment, preferences, and equilibrium

The model environment.. The agent lives for T discrete periods indexed by t ∈ 1, ...,T. Atthe beginning of each period, a vector of random shocks St , distributed according to FSt , is realized; theshocks are independent of each other and over time. The realization of the vector of random shocks St isdenoted by st . The vector Zt represents the model’s exogenous state variables and evolves according tothe following law of motion

Zt = f Z(Zt−1,St). (2.1)After observing the realization of the vector of random shocks st and the vector of state variables Zt , theagent decides how much to consume, Ct . The model’s endogenous state variable is cash-on-hand Xt+1and is determined by the following budget constraint

Xt+1 = f X (Xt −Ct ,Zt ,St+1). (2.2)

All variables indexed by t are realized in period t. Because the agent’s preferences are defined over bothconsumption and expectations or probabilistic “beliefs,” I explicitly define his beliefs about each periodt + τ variable from the perspective of any prior period t as follows.

Definition 1. Let It = Xt ,Zt ,st denote the agent’s information set in some period t ≤ t + τ . Then, theagent’s beliefs about any model variable Vt+τ conditional on period t information is F t

Vt+τ(v) = Pr(Vt+τ <

v|It), and F t+τ

Vt+τis degenerate.


Throughout this study, I assume rational expectations: the agent’s beliefs about any of the modelvariables equals their objective probabilities as determined by the economic environment.

Expectations-based reference-dependent preferences.. Having outlined the model environment,I now introduce the agent’s preferences. To facilitate the exposition, I first explain the static model ofexpectations-based reference dependence (as specified in Koszegi and Rabin (2006, 2007)), and thenintroduce the dynamic model of Koszegi and Rabin (2009). Koszegi and Rabin (2006, 2007) assume thatthe agent’s utility function consists of two components: First, the agent experiences consumption utilityu(c), which corresponds to the standard model of utility and is determined solely by consumption c.Second, he experiences gain-loss utility µ(u(c)− u(r)). The gain-loss function µ(·) corresponds to theprospect-theory model of utility determined by consumption c relative to the reference point r. The gain-loss function µ(·) is piecewise linear with slope η and a coefficient of loss aversion λ , that is, µ(x) = ηxfor x > 0 and µ(x) = ηλx for x≤ 0. The slope η > 0 weights the gain-loss utility component relative tothe consumption utility component, and the coefficient of loss aversion λ > 1 implies that losses outweighgains. Koszegi and Rabin (2006, 2007) allow for stochastic consumption, distributed according to Fc, anda stochastic reference point, distributed according to Fr. Then, the agent experiences gain-loss utility byevaluating each possible consumption outcome relative to all other possible outcomesˆ

∞

−∞

η

ˆ c

−∞

(u(c)−u(r))dFr(r)+ηλ

ˆ∞

c(u(c)−u(r))dFr(r)

dFc(c). (2.3)

As a simple example, I assume that the agent’s consumption outcome is deterministic, c= c and Fc(c)= 1.In this case, the outer integral can be ignored and the two inner integrals (in the curly brackets) are easyto interpret. For all reference outcomes r less than the actual outcome, c > r, the agent experiences asensation of gain utility, given by the linear difference in consumption utilities u(c)−u(r) and weightedby the probability of the reference outcome Fr(r). For all reference outcomes r larger than the actualoutcome, c < r, the agent experiences a sensation of loss disutility, given by the linear difference inconsumption utilities u(c)− u(r) and weighted by the probability of that reference outcome Fr(r). Inturn, if the consumption outcome is stochastic, c ∼ Fc, the agent experiences the described gain-lossutility for each possible realization of the consumption outcome c, weighted by its probability Fc(c). Inaddition, Koszegi and Rabin (2009) make the central assumption that the distribution of the referencepoint, Fr, equals the agent’s rational beliefs about consumption c.

In the dynamic model of Koszegi and Rabin (2009), the utility function consists of consumptionutility, contemporaneous gain-loss utility about current consumption, and prospective gain-loss utilityabout the entire stream of future consumption. Thus, lifetime utility in each period t is

Et [∑T−tτ=0 β

τUt+τ ] = u(Ct)+n(Ct ,F t−1Ct

)+ γ ∑T−tτ=1 β

τnnn(F t,t−1Ct+τ

)+Et [∑T−tτ=1 β

τUt+τ ], (2.4)

where β ∈ [0,1) is an exponential discount factor. The first term on the right-hand side of Equation(2.4), u(Ct), corresponds to consumption utility in period t. The other terms in Equation (2.4) dependon consumption and beliefs. The second term in Equation (2.4), n(Ct ,F t−1

Ct), corresponds to gain-loss

utility over contemporaneous consumption; here, the agent compares his present consumption Ct with hisbeliefs F t−1

Ct. According to Definition 1, the agent’s beliefs F t−1

Ctcorrespond to the conditional distribution

of consumption in period t, given the information available in period t− 1. Thus, the agent experiencesgain-loss utility over “news” about contemporaneous consumption by evaluating his contemporaneousconsumption Ct relative to his previous beliefs F t−1

Ct

n(Ct ,F t−1Ct

) = η

ˆ Ct

−∞

(u(Ct)−u(c))dF t−1Ct

(c)+ηλ

ˆ∞

Ct

(u(Ct)−u(c))dF t−1Ct

(c). (2.5)


The third term in Equation (2.4), γ ∑T−tτ=1 β τnnn(F t,t−1

Ct+τ), corresponds to gain-loss utility, experienced in

period t, over the entire stream of future consumption. Prospective gain-loss utility about period t + τ

consumption depends on F t−1Ct+τ

, the beliefs with which the agent entered the period, and on F tCt+τ

, theagent’s updated beliefs about consumption in period t + τ . Importantly, the prior and updated beliefsabout consumption Ct+τ , namely F t−1

Ct+τand F t

Ct+τ, are not independent distribution functions because

both contain the uncertainty about future income shocks. Thus, there exists a joint distribution of priorand updated beliefs, denoted by F t,t−1

Ct+τ6= F t

Ct+τF t−1

Ct+τ. The agent experiences gain-loss utility over news

about future consumption by evaluating his updated beliefs about future consumption F tCt+τ

relative to hisprevious beliefs F t−1

Ct+τas follows

nnn(F t,t−1Ct+τ

) =

ˆ∞

−∞

(η

ˆ c

−∞

(u(c)−u(r))+ηλ

ˆ∞

c(u(c)−u(r)))dF t,t−1

Ct+τ(c,r). (2.6)

I thus calculate prospective gain-loss utility nnn(F t,t−1Ct+τ

) by generalizing the “outcome-wise” comparison(specified in Koszegi and Rabin (2006, 2007) and reported in Equation (2.3)) to account for thedependence of the distributions of the reference point Fr = F t−1

Ct+τand consumption Fc = F t

Ct+τ

nnn(Fc,r) =

ˆ∞

−∞

ˆ∞

−∞

µ(u(c)−u(r))dFc,r(c,r). (2.7)

If the distributions of the reference point Fr and consumption Fc are independent, Equation (2.7) reduces toEquation (2.3). However, if Fr and Fc are not independent, Equations (2.7) and (2.3) yield different values.Suppose that the distributions of the reference point Fr and consumption Fc are perfectly correlated, asif no updates of information occurred. The outcome-wise comparison, Equation (2.3), would yield anegative value because the agent experiences gain-loss disutility over his previously expected uncertainty–which seems unrealistic. In contrast, my generalized comparison, Equation (2.7), would yield zero,since the agent considers the dependence of prior and updated beliefs (capturing the future uncertaintycontained in both). Thereby, the agent separates the uncertainty that has been realized from the uncertaintythat has not been realized. Thus, I call this gain-loss formulation the “separated comparison.” Koszegi andRabin (2009) generalize the outcome-wise comparison to a “percentile-wise” ordered comparison. Theseparated and ordered comparisons are equivalent for contemporaneous gain-loss utility. However, forprospective gain-loss utility, they are qualitatively similar, though slightly different quantitatively. As alinear operator, the separated comparison is more tractable and simplifies the equilibrium-finding processbecause it preserves the outcome-wise nature of contemporaneous gain-loss utility.8

As can be seen in Equation (2.4), the agent discounts exponentially prospective gain-loss utilityby β ∈ [0,1]. Moreover, he discounts prospective gain-loss utility relative to contemporaneous gain-lossutility by a factor γ ∈ [0,1]. Thus, he puts a weight γβ τ < 1 on prospective gain-loss utility regardingconsumption in period t +τ . Because the agent experiences both contemporaneous and prospective gain-loss utility over news, the preferences can be referred to as news utility.

To describe the probabilistic structure of the agent’s beliefs about any of the model’s variables atany future date explicitly, I now define an “admissible consumption function.” I consider a consumptionfunction that depends only on this period’s cash-on-hand Xt , the vector of exogenous state variables Zt ,the realization of the vector of shocks st , and calendar time t, because the agent fully updates his beliefsin each period and because the shocks are independent over time.

Definition 2. The consumption function in any period t is admissible if it can be written as a functionCt = gt(Xt ,Zt ,st) that is strictly increasing in the realization of each shock ∂gt (Xt ,Zt ,st )

∂ st> 0. Repeated

8. Pagel (2013) describes the differences between the separated and ordered comparisons in detail.


substitution of the law of motion, Equation (2.1), and the budget constraint, Equation (2.2), yieldsCt+τ = gt+τ(Xt+τ ,Zt+τ ,St+τ) = ht

t+τ(Xt ,Zt ,st ,St+1, ...,St+τ).

Via the admissible consumption function, I can now explicitly describe the beliefs in theformulas for contemporaneous and prospective gain-loss utility, that is, Equations (2.5) and (2.6). Forcontemporaneous gain-loss utility, contemporaneous consumption Ct and beliefs F t−1

Ct(c) are described

by Ct = gt(Xt ,Zt ,st) = ht−1t (Xt−1,Zt−1,st−1,st) and F t−1

Ct(c) = Pr(ht−1

t (Xt−1,Zt−1,st−1,St)< c). Forprospective gain-loss utility, the probabilistic structure of the beliefs about prospective consumption Ct+τ

can be described by Ct+τ = htt+τ(Xt ,Zt ,st ,St+1, ...,St+τ) with the joint distribution of the agent’s prior and

updated beliefs F t,t−1Ct+τ

(c,r) determined by

F t,t−1Ct+τ

(c,r) = Pr(htt+τ(Xt−1,Zt−1,st−1,st ,St+1, ...,St+τ)< c,ht−1

t+τ(Xt−1,Zt−1,st−1,St ,St+1, ...,St+τ)< r).For certain parameter combinations, the Koszegi and Rabin (2009) preferences reduce to well-

known specifications. For η = 0 or λ = 1 and γ = 1, the preferences reduce to standard preferences(Carroll (2001), Gourinchas and Parker (2002), and Deaton (1991)). For η > 0, λ = 1, and γ < 1, thepreferences correspond to hyperbolic-discounting preferences with the hyperbolic-discount factor givenby 1+γη

1+η(Laibson (1997) and O’Donoghue and Rabin (1999)). More specifically, the hyperbolic agent’s

lifetime utility is u(Cbt )+bEt [∑

T−tτ=1 β τ u(Cb

t+τ)] where b ∈ [0,1] is the hyperbolic-discount factor.

Model equilibrium.. I define the model’s “monotone-personal” equilibrium in the spirit of thepreferred-personal equilibrium solution concept (defined by Koszegi and Rabin (2009)), but I do so withinthe outlined environment and admissible consumption function.

Definition 3. The family of admissible consumption functions Ct = gt(Xt ,Zt ,st) is a monotone-personal equilibrium for the news-utility agent if, in any contingency, Ct = gt(Xt ,Zt ,st) maximizes(2.4) subject to (2.2) and (2.1) under the assumption that all future consumption corresponds to Ct+τ =gt+τ(Xt+τ ,Zt+τ ,st+τ). In each period t, the agent takes his beliefs about consumption F t−1

Ct+τT−t

τ=0 as givenin the maximization problem.

I can obtain the monotone-personal equilibrium by simple backward induction, such that beliefsmap into behavior and vice versa. More specifically, I derive the equilibrium consumption function underthe premise that the agent enters period t, takes his beliefs as given, optimizes over consumption, andrationally expects to behave in the same manner in the future. If the consumption function obtainedby backward induction is admissible, the monotone-personal equilibrium corresponds to the preferred-personal equilibrium defined by Koszegi and Rabin (2009). For the hyperbolic-discounting agent, themonotone-personal equilibrium corresponds to the solution of Laibson (1997). I choose this equilibriumconcept as opposed to a naive solution concept (a behaviorally reasonable alternative discussed inO’Donoghue and Rabin (1999)), because it has desirable technical properties and is more commonlyassumed in the literature. In Appendix Appendix A.4, I compare the monotone-personal equilibrium toa monotone-pre-committed equilibrium that maximizes expected utility, but is not consistent with theagent’s actual behavior without commitment.

While the preferences are very helpful in explaining a range of economic phenomena, they canalso make some counterfactual predictions. For example, in some contexts, they predict that agents willchoose a dominated alternative. However, the monotone-personal equilibrium concept rules out suchbehavior. As an example, consider a two-period model in which the agent believes he will certainlyconsume c tomorrow; then, suppose he learns about a lottery (c,1− p; c+ ε, p) today, which will resolvetomorrow and first-order stochastically dominates consuming c. He then compares the expected utilityof learning about the gamble today and taking the gamble tomorrow to his expected utility of decliningthe gamble. However, even if the agent wanted to decline the gamble, he could not decline under the


monotone-personal equilibrium solution concept. The reason is that he expects to deviate and take thegamble tomorrow, since consuming c+ ε dominates consuming c once the agent takes his beliefs asgiven. While the preferences can make unappealing predictions in some circumstances, their success inexplaining a range of empirical facts suggests that it is nonetheless useful to examine their predictions forconsumption behavior.

I prove the existence and uniqueness of the monotone-personal equilibrium in an environmentassuming exponential utility and both permanent and transitory normal shocks under one parametercondition. According to this condition, the joint distribution of the vector of shocks FSt must be dispersedsufficiently, such that all equilibrium consumption functions fall into the class of admissible consumptionfunctions. For other environments, such as power utility and permanent and transitory log-normalshocks, simulations using numerical backward induction suggest that the monotone-personal equilibriumexists and is unique (more specifically, the consumption function is admissible for a vast majority ofreasonable calibrations and the equilibrium appears stable and easy to solve for). Carroll (2011) andHarris and Laibson (2002) demonstrate the existence and uniqueness of equilibria for the standard andsophisticated hyperbolic-discounting agents in similar environments. In these models, the equilibriumconsumption functions fall in the class of admissible consumption functions. In any case, the requirementof consumption function admissibility is not relevant for the model’s predictions. The predictions holdeven if the consumption functions were not admissible, which does not affect the equilibrium muchqualitatively or quantitatively (as I argue in Section 2.2.4 and Appendix 1.3.1).

2.2. Theoretical predictions about consumption

In the following subsections, I describe the closed-form solution of the exponential-utility model indetail to illustrate the model predictions both intuitively and formally. After briefly outlining themodel’s monotone-personal equilibrium, I analyze the second-to-last period’s decision problem to explainthe model’s theoretical predictions. In Section 2.2.1, I examine excess smoothness and sensitivity inconsumption. In Section 2.2.2, I discuss the hump-shaped consumption profile, and in Section 2.2.3 Ianalyze the consumption drop at retirement. Then, I discuss several new predictions in Section 2.2.4.

I begin by briefly explaining the model environment and stating the equilibrium consumptionfunction. The agent’s utility function is exponential u(C) = − 1

θe−θC, where the coefficient of risk

aversion is θ ∈ (0,∞). His additive income process, Yt = Pt−1 +SPt +ST

t , is characterized by a permanentshock SP

t ∼ N(0,σ2Pt) with realization sP

t , a transitory shock, STt ∼ N(0,σ2

Tt) with realization sTt , and his

permanent income Pt = Pt−1 + SPt . As before, I denote the agent’s cash-on-hand by Xt , consumption by

Ct , and end-of-period asset holdings by At = Xt −Ct . His budget constraint with 1+ r denoting risk-freeinterest is

Xt+1 = (Xt −Ct)(1+ r)+Yt+1⇒ At+1 = At(1+ r)+Yt+1−Ct+1. (2.8)

Proposition 1. A unique monotone-personal equilibrium exists if√

σ2Pt +(1−a(T − t))2σ2

Tt ≥ σ∗t forall t ∈ 1, ...,T. The admissible consumption function is

Ct = (1−a(T − t))(1+ r)At−1 +Pt +(1−a(T − t))sTt −a(T − t)Λt (2.9)

with a(·) denoting the annuitization function a(T − t) =∑

T−t−1j=0 (1+r) j

∑T−tj=0(1+r) j , and the function Λt is given by

Λt =1θ

log(1−a(T − t)

a(T − t)ψt + γQtη(λ − (λ −1)FSt

(st))

1+η(λ − (λ −1)FSt(st))

), (2.10)

where the distribution function of the realization of the income shocks st = sPt + (1− a(T − t))sT

t , isFSt

(st) = Pr(SPt + (1− a(T − t))ST

t < st) and Qt and ψt are constants given by Equations (A17) and


(A18) in Appendix 1.3.1.

The proof of this and all the following propositions can be found in Appendix Appendix A.3 and theoptimal consumption function is derived in Appendix 1.1.1. All the following propositions are derivedwithin this model environment and hold for any monotone-personal equilibrium, if one exists. The agent’sconsumption function, Equation (2.9), depends on his income, assets, horizon, and interest rate, with thelast two captured by the annuitization function a(·). The function Λt , Equation (2.10), varies with theincome shock realization st , but it is independent of permanent income Pt and assets At . The constantsQt and ψt represent the recursively determined discounted sum of future marginal consumption utilitiesplus contemporaneous and future marginal gain-loss utilities, explained in greater detail in Section 2.2.1.The standard and hyperbolic-discounting agents’ monotone-personal equilibria have the same structure,except that the corresponding functions Λs

t and Λbt determining consumption vary only with the agents’

horizons.

2.2.1. Excess smoothness and sensitivity in consumption. Excess smoothness and sensitivity inconsumption are two robust empirical observations that emerged from tests of the permanent incomehypothesis. This hypothesis postulates that the marginal propensity to consume out of permanentincome shocks equals one and that future consumption growth is not predictable using past income.However, many studies find that the marginal propensity to consume is less than one becauseconsumption underresponds to permanent income shocks; thus, according to Deaton (1986), consumptionis excessively smooth. Moreover, numerous studies find that past changes in income have predictivepower for future consumption growth because consumption adjusts with a delay; thus, according toFlavin (1985), consumption is excessively sensitive. Campbell and Deaton (1989) explain how theseobservations are intrinsically related; consumption underresponds to permanent income shocks, and thus,adjusts with a delay. In this spirit, I define excess smoothness and sensitivity for the exponential-utilitymodel as follows.

Definition 4. Consumption is excessively smooth if ∂Ct∂ sP

t< 1 everywhere, and is excessively sensitive if

∂∆Ct+1∂ sP

t> 0 everywhere.

This definition has an empirical counterpart given by the ordinary least squares (OLS) regressionof period t +1 consumption growth on the realizations of the permanent shock sP in periods t +1 and t.For the two OLS coefficients β1 and β2, Definition 4 implies that consumption is excessively smooth ifβ1 =

∂Ct+1∂ sP

t+1|sP

t+1=0 < 1 and excessively sensitive if β2 =∂∆Ct+1

∂ sPt|sP

t =0 > 0.

Proposition 2. The news-utility agent’s consumption is excessively smooth and sensitive.

I first explain the intuition for this result and then derive the agent’s first-order condition to describethe intuition more formally. Suppose the agent experiences an adverse income shock and compares hisdisutility of losing present versus future consumption. Because losses in future consumption also dependon future income shocks and are therefore uncertain–the agent considers losses in present consumptionas more painful. In turn, he delays unexpected losses in present consumption and plans expectedreductions in future consumption. Now, suppose the agent experiences a favorable income shock andnote that he is subject to a baseline overconsumption problem: increasing present consumption aboveexpectations is more pleasant than increasing future consumption–which also increases expectations.However, overconsuming unexpected gains increases utility less than overconsuming unexpected losses


(as the agent is loss averse).9 Thus, in the event of a favorable income shock, the agent comparesgains in present consumption to uncertain gains in future consumption and prefers to delay some ofhis consumption to buffer expected gains and losses. Thus, relative to his baseline consumption, the agentunderconsumes in the event of favorable income shocks, while he overconsumes in the event of adverseincome shocks.

To explain this result in greater detail, I describe the agent’s decision-making problem in the second-to-last period, making the following assumptions: transitory income shocks are absent, end-of-periodasset holdings and permanent income in period T − 2 are zero, AT−2 = PT−2 = 0, and the permanentincome shock is independent and identically distributed (i.i.d.) normal SP

T−1,SPT ∼ FP = N(0,σ2

P), withrealizations sP

T−1 and sPT . In period T − 1, the agent chooses how much to consume CT−1 and save

sPT−1−CT−1, according to his maximization problem

-1θET−1[e

−θCT−1+

ˆ∞

−∞

µ(e−θCT−1 − e−θc)FT−2CT−1

(c)︸︷︷︸contemporaneous gain-loss

+γβ

ˆ∞

−∞

ˆ∞

−∞

µ(e−θc− e−θr)FT−1,T−2CT

(c,r)︸︷︷︸prospective gain-loss

+βe−θCT +β

ˆ∞

−∞

µ(e−θCT − e−θc)FT−1CT

(c)︸︷︷︸expected gain-loss

]

which results in the following first-order conditione−θCT−1(1+η(λ − (λ −1)FP(sP

T−1)))︸︷︷︸contemporaneous gain-loss

= β (1+ r)e−θ((sPT−1−CT−1)(1+r)+sP

T−1)ET−1[e−θSPT

+ γe−θSPT η(λ − (λ −1)FP(sP

T−1))︸︷︷︸prospective gain-loss

+η(λ −1)ˆ

∞

SPT

(e−θSPT − e−θs)dFP(s)︸︷︷︸

expected gain-loss

].

I now explain each component of Equation (2.2.1) in detail. The left-hand side represents marginalconsumption and contemporaneous gain-loss utility in period T − 1. While marginal consumptionutility e−θCT−1 is self-explanatory, I now derive marginal gain-loss utility. When experiencingan utility gain, the agent compares his actual consumption to all worse consumption outcomes,weighted by their probabilities and multiplied by the weight of gain-loss versus consumption utilityη , − 1

θη´ CT−1−∞

(e−θCT−1 − e−θc)FT−2CT−1

(c). When experiencing an utility loss, the agent compareshis actual consumption to all better outcomes, weighted by their probabilities and multiplied bythe weight of gain-loss versus consumption utility times the coefficient of loss aversion ηλ ,− 1

θηλ´

∞

CT−1(e−θCT−1 − e−θc)FT−2

CT−1(c). Thus, the agent’s marginal contemporaneous gain-loss utility

equals e−θCT−1(ηFT−2CT−1

(CT−1) + ηλ (1 − FT−2CT−1

(CT−1))) because the agent takes his beliefs about

consumption FT−2CT−1

(CT−1) as given in the monotone-personal equilibrium. This expression can berewritten as in Equation (2.2.1) and simplified by replacing the agent’s beliefs about consumptionFT−2

CT−1(CT−1) with his beliefs about the income shock FP(sP

T−1) (because any admissible consumptionfunction is increasing in income).

9. Koszegi and Rabin (2009) find that the news-utility agent delays surprise losses in consumption but overconsumessurprise gains in a model without previously expected uncertainty. In this no-uncertainty environment, the news-utility agent makesan initial plan that corresponds to a standard agent’s plan (as he does not expect to experience any news utility). In turn, the authorsfind that the news-utility agent overconsumes surprise gains relative to the standard agent’s consumption plan. In the same setup,the agent’s consumption would be excessively smooth and sensitive for gains if they were expected or sufficiently large and notconsumed entirely. In a model in which the news-utility agent expects to receive either good or bad news, the standard agent’sconsumption plan would not be a feasible equilibrium. Consequently, the news-utility agent would plan to overconsume in the eventof both good and bad news. In such a situation, he plans to overconsume more in the event of bad news and less in the event of goodnews and, thus, effectively underconsumes the gain, as he does in my model.


The first and third term on the right-hand side of Equation (2.2.1) represent expectedmarginal consumption and gain-loss utility. Because period T consumption is given by CT =(sP

T−1 −CT−1)(1 + r) + sPT−1 + SP

T , expected marginal consumption and gain-loss utilities equal (1 +

r)e−θ((sPT−1−CT−1)(1+r)+sP

T−1) times the expected marginal utility of the income shock βET−1[e−θSPT ] plus

the expected marginal gain-loss utility of the income shock

βET−1[η(λ −1)ˆ

∞

SPT

(e−θSPT − e−θs)dFP(s)].

Note that e−θSPT can be conveniently separated from the agent’s savings because of the assumed

exponential utility function (in the presence of additive income shocks). Moreover, note that expectedgains and losses partly cancel, so that only the overweighted part of the coefficient of loss aversionλ −1 times the expected loss utility remains. The second term on the right-hand side of Equation (2.2.1)represents marginal prospective gain-loss utility over future consumption multiplied by the prospectivegain-loss discount factor γ . Here, the agent compares his expected consumption (based on his updatedbeliefs determined by the period T − 1 income shock sP

T−1) to all outcomes that would have been moreor less favorable. As the agent’s future admissible consumption is increasing in the shock realization andhe takes his beliefs as given, his marginal prospective gain-loss utility corresponds to the same weightedsum of his beliefs about the income shock FP(sP

T−1) determining his marginal contemporaneous gain-lossutility. I denote the expected marginal utility of the income shock by QT−1 = βET−1[e−θSP

T ] plus theexpected marginal gain-loss utility by ψT−1 = QT−1 +βET−1[η(λ −1)

´∞

SPT(e−θSP

T − e−θs)dFP(s)].In turn, the first-order condition determines the agent’s optimal consumption given by

CT−1 = sPT−1−

12+ r

1θ

log((1+ r)ψT−1 + γQT−1η(λ − (λ −1)FP(sP

T−1))

(1+η(λ − (λ −1)FP(sPT−1)))

). (2.11)

The agent consumes his period T − 1 permanent income, sPT−1, minus a term that is increasing in the

income shock sPT−1 for all prospective gain-loss discount factors γ ∈ [0,1] iff ψT−1 > QT−1, that is, the

expected marginal gain-loss utility of the income shock is positive. This term varies with sPT−1 because 1)

expected marginal gain-loss utility

ψT−1−QT−1 = βET−1[η(λ −1)ˆ

∞

SPT

(e−θSPT − e−θs)dFP(s)]

is positive and constant because future expectations adjust to the current savings plan and future losses areoverweighted, whereas 2) both marginal contemporaneous and prospective gain-loss utilities vary withthe realization of income η(λ − (λ − 1)FP(sP

T−1)). Thus, a positive share of future marginal utility isinelastic to the present income shock–which implies that future marginal utility is less sensitive to changesin savings than present marginal utility is. More specifically, while expected marginal gain-loss utility isconstant because the future reference point will have adjusted to the current plan, present marginal gain-loss utility is relatively high or low in the event of an adverse or favorable shock. Therefore, the agentconsumes relatively more in the event of an adverse shock and relatively less in the event of a positiveshock. As gain-loss utility is concave (since it is proportional to consumption utility), the agent wants tosmooth gain-loss utility, as he wants to smooth consumption utility over time. According to Definition4, consumption is excessively smooth, as ∂CT−1

∂ sPT−1

< 1, and excessively sensitive, as ∂∆CT∂ sP

T−1> 0. In contrast,

the standard agent’s consumption is CsT−1 = sP

T−1−1

2+r1θ

log((1+ r)QT−1), and the hyperbolic agent’sconsumption is Cb

T−1 = sPT−1−

12+r

1θ

log((1+ r)bQT−1). Because these agents’ consumption is linearlyincreasing in income, they are neither excessively smooth nor excessively sensitive.

To illustrate excess smoothness and sensitivity in greater detail, I now describe the infinite-horizonequilibrium of the exponential-utility model, which I derive in Appendix 1.1.2.


Proposition 3. For T −t→∞, a monotone-personal equilibrium exists if√

σ2Pt +( r

1+r )2σ2

Tt ≥ σ∗t , r > 0,

and β < β . The admissible consumption function is

Ct = Pt +r

1+ rsT

t + rAt−1−1

1+ rΛt (2.12)

with the function Λt given by

Λt =1θ

log(rψt + γQtη(λ − (λ −1)FSt

(st))

1+η(λ − (λ −1)FSt(st))

), (2.13)

where st = sPt + r

1+r sTt , FSt

(st) = Pr(SPt + r

1+r STt < st), and Qt and ψt are constants given by Equations

(A19) and (A20) in Appendix 1.3.3.

To explain the intuition behind the consumption function, I now explain each component of theinfinite-horizon first-order condition given by

e−θCt (1+η(λ − (λ −1)FSt(st))) = re−θrAt e−θPt (ψt + γQtη(λ − (λ −1)FSt

(st))). (2.14)

The left-hand side of Equation (2.14) represents marginal contemporaneous consumption and gain-lossutility (derived in the same fashion as for the second-to-last period). The first part of the right-hand sideof Equation (2.14) represents the marginal value of savings re−θrAt times marginal continuation utilityψt plus marginal prospective gain-loss utility γQtη(λ − (λ − 1)FSt

(st)) (both normalized by marginalpermanent income e−θPt ). The constants Qt and ψt represent the recursively determined discountedinfinite sum of future marginal consumption utilities plus contemporaneous and future marginal gain-loss utilities.

The first-order condition can be solved for the optimal consumption function (2.12) using the budgetconstraint Ct = (1+ r)At−1+Yt−At . Intuitively, Equation (2.12) states that the agent consumes his entirepermanent income Pt , plus the per-period value of the transitory income shock r

1+r sTt , plus the interest

generated by his savings rAt−1, minus the term 11+r Λt . The function Λt increases in the realization of

the income shock st for any prospective gain-loss discount factor γ if ψt > Qt –which introduces excesssmoothness and sensitivity. This can be easily seen by looking at the first-order condition (2.14) andassuming that e−θCt ≈ re−θrAt e−θPt Qt . The intuition is that expected marginal gain-loss utility containedin ψt−Qt is constant, i.e., the agent expects a mix of gains and losses as future expectations adjust, whilepresent marginal gain-loss utility is either high or low in the event of an adverse or favorable incomerealization. Thus, the news-utility agent consumes relatively more or less, because he wants to smoothgain-loss utility (since it is concave like consumption utility).

To illustrate the quantitative implications of excess smoothness and sensitivity, I run the linearregression of consumption growth on income growth ∆Ct+1 = α + β1∆Yt+1 + β2∆Yt + εt+1. For thenews-utility model, I obtain an OLS coefficient on lagged income growth of β2 ≈ 0.36 and an excess-smoothness ratio, σ(∆Ct )

σ(∆Yt )as defined in Deaton (1986), of approximately 0.78. In constrast, for the standard

agent, the marginal propensity to consume out of permanent shocks is one. Figure 1 shows the calibrationand the news-utility and standard agents’ consumption functions for realizations within 2 standarddeviations of each shock, while the other is held constant. The flatter part of the news-utility consumptionfunction generates excess smoothness and sensitivity.

2.2.2. The hump-shaped consumption profile. Fernandez-Villaverde and Krueger (2007) amongothers, show that lifetime consumption profiles are hump-shaped, even when controlling for cohort,family size, number of earners, time effects, and non-separability between consumption and leisure. Inthe following subsection, I demonstrate that the preferences generate a hump-shaped consumption profileas the net result of two competing features–an additional first-order precautionary-savings motive and an


FIGURE 1EXPONENTIAL-UTILITY CONSUMPTION FUNCTIONS

In this figure, I display the news-utility and standard agents’ consumption functions for realizations within 2standard deviations of each shock, while the other is held constant. I choose the agent’s horizon T , his retirementperiod R, the interest rate r, and the volatility of the income process in accordance with the life-cycle literature;

σPt = 5% and σTt = 7% for all t , r = 2%, A0 = 0, and P0 = 0.1. Additionally, I choose the preference parametersin line with the microeconomic literature and experimental evidence, which is explained in Section 2.3.2,

β = 0.978, θ = 2, η = 1, λ = 2.5, and γ = 0.7.

overconsumption problem exacerbated by the agent’s prospective gain-loss discount factor γ . I explainboth in the second-to-last period setting introduced in Section 2.2.1.

Precautionary savings and prospective news discounting.. Income uncertainty has a first-ordereffect on savings in the news-utility model. This “first-order precautionary-savings motive” is added tothe precautionary savings motive of the standard agent, which is a second-order motive.

Definition 5. A first-order precautionary-savings motive exists iff ∂ (sPt −Ct )

∂σPt|σPt=0 > 0.

At the same time, the agent wishes to increase his consumption and decrease his savingsbecause he has an overconsumption problem, and he discounts prospective gain-loss utility relative tocontemporaneous gain-loss utility if γ < 1. Lemma 1 formalizes these conflicting effects.

Lemma 1. In the second-to-last period example, it holds that:1. Precautionary savings: news utility introduces a first-order precautionary-savings motive.2. Overconsumption: there exists an upper bound for the discount factor on prospective versuscontemporaneous gain-loss utility γs < 1, implicitly determined by equalizing news-utility and standardconsumption growth ∆CT = ∆Cs

T , such that, iff γ < γs, the news-utility agent’s consumption growth inperiod T is lower than the standard agent’s consumption growth.

The intuition for the first part of Lemma 1 is as follows. The agent anticipates being exposedto gain-loss fluctuations in period T , which are painful in expectation because losses hurt more thangains give pleasure. In addition, the painfulness of these fluctuations is proportional to marginalconsumption utility, which decreases on a higher point of the concave utility curve (where fluctuationsin consumption generate fewer fluctuations in consumption utility). Thus, these gain-loss fluctuationsgenerate an additional incentive to increase savings. I now develop a more formal intuition for thestandard and additional precautionary-savings motive and demonstrate that expected marginal gain-loss


utility is positive, that is, ψT−1 > QT−1. As shown in Section 2.2.1, the marginal value of savings is(1+ r)e−θ((sP

T−1−CT−1)(1+r)+sPT−1)ψT−1, where ψT−1 equals the expected marginal consumption plus the

expected marginal gain-loss utility of the income shock

βET−1[e−θSPT ]+βET−1[η(λ −1)

ˆ∞

SPT

(e−θSPT − e−θs)dFP(s)]. (2.15)

The integral in Equation (2.15) reflects the expected marginal utility of all gains and losses, which partlycancel, such that only the overweighted component of the losses remains (multiplied by η(λ − 1)(·)).This integral is always positive if the utility function is concave, u′′ < 0. Thus, it captures the additionalprecautionary-savings motive, which implies that the expected marginal gain-loss utility is positiveψT−1−QT−1 > 0 (and increasing in the weight of gain-loss versus consumption utility η , the coefficientof loss aversion λ , and income uncertainty σP). This precautionary-savings motive is first order, that is,∂ (sP

T−1−CT−1)

∂σP|σP=0 > 0, because the news-utility agent is first-order risk averse. In contrast, the standard

precautionary-savings motive is captured by the expected marginal utility of the income shock QT−1 =

βET−1[e−θSPT ], which is larger than the marginal utility of the expected income shock βe−θET−1[SP

T ] ifthe utility function is prudent, u′′′ > 0, (according to Jensen’s inequality). This standard precautionary-

savings motive is second order, that is,∂ (sP

T−1−CT−1)

∂σP|σP=0 = 0, as the standard agent is second-order risk

averse.10

The intuition for the second part of Lemma 1 is as follows. If the prospective discount factor equalsone, that is γ = 1, the precautionary savings motive will always decrease period T − 1 consumptionbelow the standard agent’s consumption, via ψT−1

QT−1> 1. But, if γ < 1, the agent’s overconsumption

problem is exacerbated. The agent is subject to time-inconsistent overconsumption because the monotone-personal equilibrium assumes that the agent wakes up in any period, takes his expectations as given, andoptimizes over consumption. He thus chooses higher consumption ex post compared to the optimal levelhe would like to commit to ex ante. Ex ante, he considers increasing both his future consumption andexpectations–but, when the future arrives, he takes his expectations as given, and enjoys the pleasantsurprise of increasing consumption above expectations. He, therefore, puts a higher weight on futuremarginal consumption utility in the future versus the present. This problem is exacerbated if the discountfactor on prospective versus contemporaneous gain-loss utility is smaller than one (γ < 1), implying thatthe agent is more concerned about contemporaneous than prospective gain-loss utility. In that case, healways wishes to increase his consumption and decrease his savings.

Consequently, news utility might increase or decrease consumption relative to the standard model,depending on the net effect of two parameters 1) uncertainty about permanent income, σP > 0, increasingprecautionary savings, and 2) prospective gain-loss discounting, γ < 1.

The resulting hump-shaped consumption profile.. The additional precautionary-savings motiveand the overconsumption problem compete which causes a hump-shaped news-utility life-cycleconsumption profile. The reason is that the precautionary-savings motive increases with incomeuncertainty (as the agent’s horizon increases), while the overconsumption problem does not.

Definition 6. The agent’s consumption profile is hump shaped if consumption is increasing at thebeginning of life, ∆C1 > 0, and decreasing, ∆CT < 0, at the end of life.

10. Benartzi and Thaler (1995) and Barberis et al. (2001) show that first-order risk aversion resolves the equity premiumpuzzle (highlighting that agents must have implausibly high second-order risk aversion to generate the historical equity premiumbecause aggregate consumption is smooth compared to asset prices). The excess-smoothness puzzle highlights that aggregateconsumption is overly smooth compared to labor income, and again, first-order instead of second-order risk aversion is a necessaryingredient to resolve the puzzle. In a canonical asset-pricing model, Pagel (2015) demonstrates that news-utility preferencesconstitute an additional step toward resolving the equity-premium puzzle as they match the historical level and the variation ofthe equity premium, while simultaneously implying plausible attitudes toward small and large wealth gambles.


FIGURE 2EXPONENTIAL-UTILITY LIFE-CYCLE PROFILES

This figure displays the news-utility and standard agents’ life-cycle consumption profiles; both the averageconsumption profiles of a sample of 300 identical agents, who encounter different realizations of the permanent and

transitory income shocks sPt and sT

t , and the consumption profiles if sPt = 0 and sT

t = 0 for all t. I use the samecalibration as in Section 2.2.1 explained in the caption of Figure 1.

Proposition 4. Suppose T is large, σPt = σP and σTt = σT for all t, and β > β ; then, a γ in [γ, γ] exists,so that the news-utility agent’s lifetime consumption path is hump shaped.

The basic intuition is illustrated in Lemma 1. The relative strengths of the additional precautionary-savings motive and the overconsumption problem (exacerbated by the discount factor on prospectiveversus contemporaneous news utility γ < 1) determine whether gain-loss utility increases or decreasesthe news-utility agent’s consumption relative to the standard model. Toward the end of life, theadditional precautionary-savings motive is relatively small, and the discount factor on prospective versuscontemporaneous news utility γ < 1 is likely to decrease consumption growth. However, when the agent’shorizon increases, the precautionary-savings motive increases because uncertainty accumulates; hence,at the beginning of life, the presence of gain-loss utility is likely to reduce consumption and increaseconsumption growth unless γ is very small.

More formally, the two conditions on consumption growth, ∆Ct+1 ≶ 0, reduce to the function thatadds on to consumption, Λt ≶ 0, as the agent’s horizon T − t becomes small or large. The sign of Λtis determined by the relative values of the expected marginal gain-loss utility, ψt

Qt> 1, and the discount

factor on prospective versus contemporaneous news utility, γ < 1. If the agent’s horizon T − t is small, ψtQt

is small, such that γ < 1 is likely to make the function that adds to consumption Λt negative. However,as T − t increases, ψt

Qtincreases, such that γ < 1 loses relative importance and Λt is more likely to be

positive, as ψt increases at a first-order rate and faster than Qt .Figure 2 illustrates the news-utility and standard agents’ life-cycle consumption profiles. The figure

shows the average consumption profile of a sample of 300 identical agents, who face different realizationsof the permanent and transitory income shocks, sP

t and sTt , and the consumption profile of one agent, for

whom sPt = 0 and sT

t = 0 for all t. As can be observed in Figure 2, the news-utility agent’s consumptionprofile is hump shaped; that is, consumption increases in the beginning, but decreases toward the endof life. This hump is very robust for different parameter choices, which I discuss in Section 2.3.2. Incontrast, the standard agent’s profile is V-shaped, showing that exponential utility and a random-walkincome process do not promote the desired hump. Moreover, Figure 2 shows the profiles in the presenceof a retirement period, which induces a rather different consumption profile toward the end of life, which


I explain next.

2.2.3. Consumption during and at the beginning of retirement.

Consumption during retirement.. I now add a retirement period at the end of life to the model.More specifically, I assume that in periods t ∈ T − R, ...,T, the agent earns his permanent incomewithout uncertainty. First, I formalize a general prediction of the news-utility agent’s consumption duringretirement, that is, CT−R to CT .

Proposition 5. If uncertainty is absent, the monotone-personal equilibrium of the news-utilityagent corresponds to the standard agent’s equilibrium iff the discount factor on prospective versuscontemporaneous gain-loss utility is weakly larger than the inverse of the coefficient of loss aversionγ ≥ 1

λ. Iff γ < 1

λ, then the monotone-personal equilibrium of the news-utility agent corresponds to a

hyperbolic-discounting agent’s monotone-personal equilibrium with the hyperbolic-discount factor givenby b = 1+γηλ

1+η.

The news-utility agent is likely to follow the standard agent’s path if uncertainty is absent and if theagent’s prospective gain-loss discount factor is high enough. The basic intuition is that, when the agentdecides to increase present consumption, he also considers a certain loss in future consumption, whichis very painful. Thus, unless the agent discounts prospective gain-loss utility significantly, he decides tonot increase present consumption. More formally, suppose that the agent allocates his deterministic cash-on-hand between present consumption CT−1 and future consumption CT . Under rational expectations, hecannot fool himself; hence, he cannot experience actual gain-loss utility in equilibrium in a deterministicmodel. Accordingly, his expected utility maximization problem corresponds to the standard agent’smaximization problem (determined by setting present and future marginal consumption utilities equal,with the discount factor and interest rate e−θCT−1 = β (1+ r)e−θCT ). Suppose that the agent’s beliefsabout consumption in both periods correspond to this equilibrium path. Taking his beliefs as given, theagent deviates if the gain from consuming more exceeds the discounted loss from consuming less in thefuture, that is,

e−θCT−1(1+η)> β (1+ r)e−θCT (1+ γηλ ).

Thus, he follows the standard agent’s path iff the discount factor on prospective versus contemporaneousgain-loss utility is weakly larger than the inverse of the coefficient of loss aversion, γ ≥ 1

λ. In this case,

the pain associated with a certain loss in future consumption is larger than the pleasure gained frompresent consumption. However, if γ < 1

λ, the agent deviates and must choose a consumption path that just

meets the consistency constraint, thereby behaving as a hyperbolic-discounting agent, with a hyperbolicdiscount factor b = 1+γηλ

1+η< 1. Thus, during retirement, the implications of the agent’s prospective gain-

loss discount factor γ are simple: it must be sufficiently high to keep the news-utility agent on the standardagent’s track.

The drop in consumption at retirement.. The empirical evidence of a drop in consumption atretirement is a topic of debate in the literature. While a series of studies find that consumption drops atretirement (summarized in Attanasio and Weber (2010)), Aguiar and Hurst (2005) strongly question thisfinding, when controlling for the sudden reduction in work-related expenses, the substitution of homeproduction for market-purchased goods and services, and health shocks. In my data, I find such a drop inconsumption at retirement, even for non-work-related spending. Moreover, Schwerdt (2005) finds a drop,explicitly controlling for home production and for German retirees, who receive large public pensions,and for whom health is a complement to consumption owing to health insurance. Moreover, Amerikset al. (2007) and Hurd and Rohwedder (2003) provide evidence that the drop in consumption is expected.First, I define a drop in consumption as follows.


Definition 7. A drop in consumption at retirement occurs if consumption growth at retirement ∆CT−R isnegative and less than consumption growth after retirement ∆CT−R+1.

Proposition 6. If the prospective gain-loss discount factor is less than 1, γ < 1, and β < β , the news-utility agent’s monotone-personal consumption path is characterized by a drop at retirement.

After the beginning of retirement, the agent is less inclined to overconsume than before. The basicintuition for overconsumption in the period just before retirement is that the agent consumes housemoney–that is, labor income that he was not certain to receive. Such uncertain income wants to beconsumed before his expectations catch up, iff the prospective gain-loss discount factor is less thanone, γ < 1. In the period just before retirement, the agent finds the loss in future consumption merelyas painful as a slightly less favorable realization of his labor income–that is, the agent trades off beingsomewhere in the gain domain today versus being somewhere in the gain domain in the future. In contrast,during retirement, the agent associates a certain loss in future consumption with an increase in presentconsumption–that is, he trades off a current gain with a sure loss in the future. For example, supposethe agent’s retirement period is period T only. The agent’s first-order condition in period T − 1, absentuncertainty in period T , is given by

e−θCT−1(1+η(λ − (λ −1)FP(sPT−1))) = β (1+ r)e−θCT (1+ γη(λ − (λ −1)FP(sP

T−1))). (2.16)

In Equation (2.16), it can be seen that, iff the prospective gain-loss discount factor equals 1 (γ = 1),contemporaneous and prospective marginal gain-loss utility cancel. However, iff γ < 1, the agent reducesthe weight on future utility relative to present utility by a factor 1+γηλ

1+ηλ< 1+γη

1+η< 1. During retirement, the

news-utility agent follows the standard agent’s consumption path (if the prospective gain-loss discountfactor γ is sufficiently high), and otherwise follows a hyperbolic agent’s consumption path (with discountfactor b = 1+γηλ

1+η). Because min 1+γηλ

1+η,1 > 1+γη

1+ηiff γ < 1, the agent’s factor that reduces the weight

on future utility is necessarily lower in the period just before retirement than during retirement–implyingthat consumption drops at the beginning of retirement. The drop in consumption is thus brought about bya change in the agent’s effective time-inconsistency problem, rather than by the absence of precautionarysavings in the period just before retirement. Therefore, the other agents’ consumption paths do not showsuch a drop. Quantitatively, Figure 2 shows a noticeable drop in the news-utility consumption profile atretirement.

The assumption of no uncertainty during retirement is made in all standard life-cycle consumptionmodels, as these are abstracted from portfolio choice; thus, the drop in consumption at retirement is anecessary artifact of news-utility preferences in the standard environment. However, the drop is robustto three alternative assumptions: small income uncertainty during retirement (for instance, inflationrisk); potentially large discrete income uncertainty (for instance, health shocks); and mortality risk.11

Furthermore, if I were to observe a consumption path that is much flatter during retirement than beforeretirement and interpret this observation from the perspective of the standard model, I may conclude thatthe agent does not decumulate assets rapidly enough after retirement (compared to his asset decumulation

11. The drop in consumption is due to the fact that the agent overconsumes before retirement, but consumes efficiently afterretirement. If income uncertainty is very small, the agent can credibly plan a flat consumption level, independent of the realizationof his income shocks as the benefits of smoothing consumption perfectly are too small relative to the costs of experiencing gain-lossutility. In such a small-uncertainty situation, the agent can stick to a flat consumption level that induces less overconsumption afterretirement than before retirement, such that consumption drops. I formally explain this result about overconsumption in Section2.2.4. Moreover, discrete uncertainty after retirement induces less overconsumption than before retirement for the same reason thatno uncertainty causes less overconsumption. If uncertainty is discrete, overconsumption is associated with a discrete gain in presentconsumption and a discrete loss in future consumption. Because the discrete loss hurts more than the discrete gain, the agent maycredibly plan a consumption level that induces less overconsumption than the baseline continuous-outcome equilibrium. Finally,mortality risk does not affect the result, as the agent would not experience gain-loss utility relative to being dead. Additionally, thedrop in consumption is robust to uncertainty becoming small in the period just before retirement, even if the agent then chooses aflat consumption function as described in Section 2.2.4.


before retirement). Such a lack of asset decumulation during retirement is another commonly observedlife-cycle consumption puzzle (Hurd (1989), Disney (1996), and Bucciol (2012)), but it could also beexplained by bequest motives (Hurd (1989)) or medical expenditures shocks (Nardi et al. (2011)).

Excess smoothness and sensitivity of consumption in the period just before retirement.. Theexcess smoothness and sensitivity results derived in Section 2.2.1 abstract from retirement. I now showthat the results hold when I introduce a retirement period in the model. After all, the intuition explained inSection 2.2.1 is based on no uncertainty associated with bad news in present consumption, but uncertaintyassociated with bad news in future consumption, as future expectations adjust. Because uncertainty isabsent after retirement begins, this intuition does not apply in the period just before retirement. However,consumption is excessively smooth and sensitive in the period just before retirement iff the prospectivegain-loss discount factor is smaller than 1 (γ < 1). For all other periods, the intuition described in Section2.2.1 about the adjustability of expectations (rather than γ < 1) generates most of the excess smoothnessand sensitivity in consumption.

Corollary 1. Iff the prospective gain-loss discount factor is less than 1, γ < 1, the news-utility agent’smonotone-personal equilibrium consumption is excessively smooth and sensitive in the period just beforeretirement.

Iff the prospective gain-loss discount factor is less than 1, γ < 1, the agent cares more aboutcontemporaneous than prospective gain-loss utility, and thus, overconsumes in the presence of uncertainty(explained in the beginning of Section 2.2.2). Moreover, the agent overconsumes more when experiencinga relatively bad realization because losses are overweighted, and he can effectively reduce his sense ofloss by delaying the cut in consumption. Because the agent overconsumes relatively more in the eventof a bad shock and relatively less in the event of a good shock, consumption is excessively smooth andsensitive. More formally, Equation (2.16) shows that the agent behaves as a hyperbolic-discounting agent,weighting future consumption by a factor b ∈ 1+γηλ

1+ηλ, 1+γη

1+η< 1. Moreover, the agent’s weight on future

consumption is particularly low when his income realization is relatively bad, that is, FP(sPT−1) ≈ 0. In

turn, variation in FP(sPT−1) leads to variation in consumption growth, as in Section 2.2.1. Additionally, for

any weight of gain-loss versus consumption utility η and coefficient of loss aversion λ , consumption ismore excessively smooth and sensitive if the prospective gain-loss discount factor γ is low.

2.2.4. Discussion and new predictions about consumption. Many different explanations existfor excess smoothness and sensitivity, the hump-shaped consumption profile, and the drop in consumptionat retirement (see Section 1.1). One of the key contributions of this study is that it uses a single calibrationto provide a parsimonious and unified explanation robust to different choices of parameter values anddifferent assumptions about the model environment. After all, loss aversion and present bias have beendocumented empirically for poor, rich, as well as professional individuals (as surveyed in Angner andLoewenstein (fthc), DellaVigna (2009), and Barberis (2013)).

If an agent is somewhat less loss averse or present biased in my model, his predicted consumptionstill features excess smoothness, a hump shape, and a drop at retirement, although all of these aresomewhat smaller in magnitude. This is consistent with the empirical evidence. Although the evidenceis less strong for rich households, the literature documents that rich households’ consumption featuresexcess smoothness, a hump, and a drop at retirement. As surveyed in Jappelli and Pistaferri (2010), thereexists evidence that households without borrowing constraints have excessively smooth consumption.Gourinchas and Parker (2002) and Fernandez-Villaverde and Krueger (2007) document a hump-shapedconsumption profile throughout all income and occupation classes (after controlling for time, cohort, andfamily size effects), but find that consumption peaks later for rich households. Bernheim et al. (2001)find that the drop in consumption for the top wealth quartile is only half of the drop in consumption


for the bottom wealth quartile, but both drops are statistically significant. Nevertheless, because theseempirical observations are explained by a variety of theories, I will now highlight a number of additionalnews-utility predictions, which are new and testable comparative statics.

The first novel prediction of the news-utility model is that consumption is more excessively sensitivefor permanent shocks than for transitory shocks–when both types of shocks are present. The intuitionis that the agent consumes only the per-period value of the transitory shock, which is a small fractionof the shock, whereas he consumes the entire permanent shock. In the presence of permanent shocks,the transitory shock thus barely affects how the agent compares his actual consumption to his beliefs(which determines marginal gain-loss utility). Instead, the permanent shock almost fully determines howconsumption compares to beliefs. However, in an environment with only transitory shocks, consumptionis excessively smooth and sensitive with respect to transitory shocks because they fully determine howconsumption compares to beliefs.

To formally describe this result, I return to the agent’s consumption function for the infinite-horizonmodel, Equation (2.12). The consumption function in each period t says that the agent consumes hispermanent income Pt , the per-period value of his temporary shock r

1+r sTt , and the return of his last period’s

asset holdings rAt−1. Because he consumes the entire permanent shock but only a very small fraction,that is, r

1+r of the transitory shock–almost all the variation in consumption is due to the permanentshock (which almost fully determines F t−1

Ct(Ct), marginal gain-loss utility, and the function Λt ). This

prediction could be tested empirically by comparing the consumption responses of individuals exposedmainly to transitory shocks, such as cab drivers or waiters, with those exposed to transitory and permanentshocks, such as white-collar workers. News utility predicts that cab drivers exhibit excessively smoothconsumption in response to transitory income shocks, as shown by Crawford and Meng (2011), whilewhite-collar workers do not. This prediction stands in contrast to borrowing constraints, inattention (Reis(2006)), adjustment costs (Chetty and Szeidl (2010)), and incomplete consumption insurance (Attanasioand Pavoni (2011)).

A second testable new prediction is that the degree of excess smoothness and sensitivity is decreasingin income uncertainty. The intuition is that, when income uncertainty is small, small variation in incomecauses a relatively large amount of gain-loss utility and a corresponding response in consumption (becauseeven small variation in income causes large variation in how consumption compares to beliefs). Becausethe agent’s beliefs change more rapidly relative to the change in the income shock, the consumptionfunction is flatter. This prediction could actually result in a perfectly flat consumption function over somerange, if income uncertainty was very small (as also highlighted by Heidhues and Koszegi (2008)).

To formally describe the implications of flat consumption, I return to the two-period one-shockmodel of Section 2.2.1. Suppose that the realization of the income shock, sP

T−1, increases; then, holdingconsumption CT−1 constant, the marginal value of savings declines and the agent’s first-order conditionimplies that present consumption should increase. However, the value of the cumulative distributionof the income shock FP(sP

T−1) also increases, and marginal gain-loss utility is lower, such that presentconsumption should decrease. Now suppose that the income shock increases marginally but FP(sP

T−1)increases sharply because σP is small. In this case, the lower marginal gain-loss utility that decreasesconsumption dominates the decrease in the marginal value of savings (which increases consumption). Inturn, the first-order condition predicts decreasing consumption over some range in the neighborhoodof zero, where the distribution function FP increases most sharply (if FP is bell shaped). However,a decreasing consumption function cannot be an equilibrium because the agent would unnecessarilyexperience gain-loss utility over the decreasing part of the consumption function (which decreasesexpected utility, as losses are overweighted). Thus, the agent could increase his expected utility bychoosing a flat consumption function instead of a decreasing consumption function to avoid gain-lossutility in the flat region.

In such a situation, he does not respond to shocks at all–that is, his consumption is perfectlyexcessively smooth and sensitive, as in the presence of borrowing constraints or adjustment costs


to consumption. Moreover, the agent may choose a credible consumption plan with a flat section,which induces less overconsumption than the original plan.12 That the degree of excess smoothnessand sensitivity is decreasing in income uncertainty clearly differentiates news utility from borrowingconstraints, inattention, and adjustment costs. To empirically test this prediction, one could check whetherhouseholds with little income uncertainty, such as tenure-track academics, seem to have more stableconsumption and suffer less overconsumption.

A third testable new prediction is that consumption is more excessively smooth and sensitive whenthe agent’s horizon increases. The intuition relates to the fact that the agent takes his present expectationsas given, while his future expectations are adjustable and depend on future uncertainty. When there aremany future periods, the future income shocks are well diversified and the agent can diversify across timebecause he is first-order risk averse. Thus, he is more inclined to reduce his present gain-loss utility viaexcessively smooth present consumption.

Formally, the gain-loss induced variation in the optimal consumption function increases for tworeasons: first, the variation in the function Λt increases and thus the marginal propensity to consume out ofpermanent shocks decreases when the precautionary-savings motive increases; second, the annuitizationfactor a(T − t) increases in the agent’s horizon T − t. While both effects increase excess smoothness,only the first effect increases excess sensitivity, such that consumption is relatively less excessivelysensitive when the agent’s horizon increases. These predictions can be tested empirically by comparingthe excess smoothness and sensitivity in consumption of young and old households (as in Section 2.3.2). Ayoung household’s consumption should be much more excessively smooth and slightly more excessivelysensitive relative to that of an old household. Such empirical evidence can distinguish between newsutility, borrowing constraints, inattention, adjustment costs, and consumption insurance.

Empirically, there is abundant field and laboratory evidence for time-inconsistent overconsumption,preference reversals, and demand for commitment devices (as surveyed in DellaVigna (2009) andFrederick et al. (2002)). In addition, theoretically, the hyperbolic-discounting model of Laibson et al.(2012) is very successful in explaining life-cycle consumption. While hyperbolic-discounting preferencesdo not generate excess smoothness and sensitivity in consumption per se, they induce borrowingconstraints to bind much more often. However, the news-utility time inconsistency is observationallydistinguishable from hyperbolic-discounting preferences.

First, news utility introduces an additional precautionary-savings motive that is absent in thehyperbolic-discounting model. Second, because of this precautionary-savings motive, the news-utilityagent does not have a universal desire to pre-commit to the standard agent’s consumption path, in contrastto hyperbolic-discounting preferences. Third, news utility predicts that the agent’s degree of present biasis reference dependent and lower in less favorable times. Intuitively, increasing consumption in bad timesis a best response. Fourth, the news-utility agent’s degree of present bias depends on the uncertainty hefaces (as in Section 2.2.3 and Footnote 12). Absent uncertainty, the agent’s present bias is absent, as longas the prospective gain-loss discount factor is weakly larger than the inverse of the coefficient of lossaversion, γ ≥ 1

λ. In presence of small or discrete uncertainty, the agent’s degree of present bias is less

than that in presence of large and dispersed uncertainty. All these predictions could be tested empiricallyor in a laboratory with on-the-spot consumption, as in Pagel and Zeppenfeld (2014).

2.3. Quantitative predictions about consumption

In the following subsection, I assess whether the model’s quantitative predictions match the empiricalevidence. To demonstrate that all predictions hold in model environments that are commonly assumed

12. Suppose the agent chooses a flat consumption level for realizations of the income shock sPT−1 in some range s and s.

Then, he chooses s where the original consumption function just stops decreasing, which corresponds to the lowest possible levelof the flat section of consumption CT−1. Moreover, the agent’s consistency constraint for not increasing consumption beyond CT−1for any realization of the income shock sP

T−1 ∈ [s,s] always holds (Appendix 1.3.1).


in the life-cycle consumption literature, I present the numerical implications of a power-utility model,that is, u(C) = C1−θ

1−θwith θ being the coefficient of constant relative risk aversion.13 In Section 2.3.1,

I first outline the power-utility model. In Section 2.3.2, I structurally estimate the power-utility model’sparameters, compare my estimates with those in the microeconomic literature, and show that my estimatesgenerate the degree of excess sensitivity and smoothness found in aggregate data.

2.3.1. The power-utility model.

The income processes and model environment.. I follow Carroll (1997) and Gourinchas andParker (2002), who specify income Yt as log-normal and characterized by deterministic permanent incomegrowth Gt , permanent shocks NP

t , and transitory shocks NTt , which allow for a low probability p of

unemployment or illnessYt = PtNT

t = Pt−1GtNPt NT

t

NTt =

esT

t with probability 1− p and sTt ∼ N(µT ,σ

2T )

0 with probability p

NP

t = esPt sP

t ∼ N(µP,σ2P).

The life-cycle literature suggests fairly tight ranges for the parameters of the log-normal income process,which are approximately µT = µP = 0, σT = σP = 0.1, and p = 0.01. Gt typically implies a hump-shapedincome profile. Nevertheless, I initially assume that income is flat, that is, Gt = 1 for all t, to highlight themodel predictions in an environment that does not simply generate a hump-shaped consumption profilevia a hump-shaped income profile. For the preference parameters, I initially use the same calibration as inSection 2.2.1. In addition to the standard and hyperbolic agents, I show results for internal, multiplicativehabit-formation preferences, as assumed in Michaelides (2002), and temptation-disutility preferences, asdeveloped by Gul and Pesendorfer (2004) and assumed in Bucciol (2012). The utility specifications can befound in Appendix Appendix B.1. For the habit-formation agent, I follow Michaelides (2002) and choosea habit-forming parameter h= 0.45, which matches the excess-smoothness evidence. The tempted agent’sadditional preference parameter τ = λ td

1+λ td = 0.1 is chosen following the estimate of Bucciol (2012).

Comparison of life-cycle consumption profiles.. Figure 3 contrasts the five agents’ consumptionpaths with the empirical consumption and income profiles. I show only part of the habit-formationagent’s consumption profile because he engages in extremely high wealth accumulation, owing to his higheffective risk aversion (even if I choose a lower value for the habit-forming parameter h, the value fittingthe excess-sensitivity evidence), confirming the findings of Michaelides (2002). Hyperbolic-discountingpreferences push the consumption profile upward at the beginning and downward at the end of life.Temptation disutility causes severe overconsumption at the beginning of life, which then reduces whenconsumption opportunities are depleted. Standard, hyperbolic-discounting, news-utility, and temptation-disutility preferences all generate a hump-shaped consumption profile. The consumption path of thestandard and hyperbolic agents is increasing at the beginning of life because power utility makesthem unwilling to borrow; however, these agents are also sufficiently impatient, such that consumptioneventually decreases. Because power utility eliminates the possibility of negative or zero consumptionand because of the small possibility of zero income in all future periods, the agents will never find itoptimal to borrow.

Moreover, power utility implies prudence, such that all agents have a standard precautionary-savingsmotive. However, this motive is weak because the standard agent’s consumption begins to decrease ratherearly in life, criticized by Attanasio (1999) among others. Therefore, at first glance, the news-utility

13. The power-utility model cannot be solved analytically, but it can be solved by numerical backward induction (Gourinchasand Parker (2002) or Carroll (2001) among others). The numerical solution is illustrated in greater detail in Appendix AppendixB.3.


FIGURE 3LIFE-CYCLE PROFILES AND CEX CONSUMPTION AND INCOME DATA

This figure contrasts the five agents’ consumption paths with the average CEX consumption and income data. Theparameter values are µT = µP = 0, σT = σP = 0.1, p = 0.01, and Gt = 1 for all t. The preference parameters areβ = 0.97, r = 1%, θ = 2, η = 1, λ = 3, γ = 0.8, the hyperbolic discounting parameter is 0.8, the habit-forming

parameter is h = 0.45, and the temptation-disutility parameter is τ = 0.1. The unit of consumption and income is thelog of 1984 dollars controlling for cohort, family size, and time effects.

agent’s hump looks closer to the empirical consumption profile than the other agents’ humps, with slowlyincreasing consumption at the beginning of life and decreasing consumption shortly before retirement.Additionally, the news-utility hump is more robust than the standard and hyperbolic humps to alternativeassumptions about the discount factor, interest rate, and income profile. For instance, in a model withonly transitory shocks and no unemployment, the standard agent’s consumption is basically flat and thehyperbolic agent’s consumption is decreasing throughout because the precautionary-savings motive isvery weak; in contrast, the news-utility model generates a hump-shaped consumption profile. Obviously,in partial equilibrium, matching the hump shape is trivial because the preference and environmentalparameters are jointly calibrated; thus, I confirm that in a simple real-business-cycle model, news-utilitygenerates realistic moments with the preference parameters assumed in this study.

Moreover, Figure 3 shows a drop in consumption at retirement for both the news-utility consumptionprofile and the CEX consumption data. Thus, I conclude that the news-utility agent’s lifetime consumptionprofile looks very similar to the average consumption profile from CEX data, which I explain in greaterdetail in the next subsection.

2.3.2. Structural estimation.

Data, estimation procedure, identification, and results.. I use data from the CEX for 1980to 2002. Because the CEX does not survey households consecutively, I generate a pseudo panel thataverages household consumption and income at each age. To control for cohort, family size, and timeeffects, I employ average cohort techniques, which are explained in detail in Appendix 2.3.2. Figure 3shows the average empirical income and average empirical consumption profiles. Then, I structurallyestimate the news-utility parameters using the methods-of-simulated-moments procedure (Gourinchasand Parker (2002) among others). The procedure has two stages: In the first stage, I estimate the structuralparameters governing the environment using standard techniques, and my results are perfectly in linewith the literature. In the second stage, I estimate the preference parameters by matching the simulatedand empirical average life-cycle consumption profiles. The estimation procedure and identification of allpreference parameters is described in greater detail in Appendix 2.3.2.


TABLE 1

FIRST-STAGE PARAMETER ESTIMATION RESULTS

µP σP µT σT p Gt r P0A0P0

R T

estimate 0 0.19 0 0.15 0.0031 eYt+1−Yt 3.1% 1 0.0096 11 54

SECOND-STAGE PARAMETER ESTIMATION RESULTS

news-utility model standard model

β θ η λ γ β θ

estimate 0.96 1.11 1.16 3.11 0.46 0.91 2.12standard error (0.004) (0.227) (0.111) (0.682) (0.082) (0.013) (0.305)

χ(·) 72.4 135.5

This table displays all first- and second-stage parameter estimates as well as their standard errors. Theoveridentification test’s critical value at 5% is 66.3.

All first- and second-stage structural parameter estimates are shown in Table 1. In the first stage, Iestimate the environment parameters µP =−0.002, µT =−0.0031, σP = 0.18, σT = 0.16, and p= 0.0031in line with the literature. The mean of Moody’s municipal bond index is r = 3.1%. Moreover, becauseGourinchas and Parker (2002) choose 25 years as the beginning of life, I choose the retirement periodR = 11 years and the working life span T = 54 years, in accordance with the average US retirement ageaccording to the OECD, and the average US life expectancy according to the UN. At age 25, I estimatethe mean ratio of liquid wealth to income as 0.0096, assuming that initial permanent income equals one,that is P0 = 1.

In the second stage, I estimate the preference parameters discussed below with standard errorsobtained by numerically estimating the gradient of the moment function at its optimum. The preferenceparameters are estimated tightly, and I marginally reject the overidentification test, as do Gourinchasand Parker (2002). Finally, I obtain suggestive evidence for one of the new prediction generated by newsutility; the excess-smoothness ratio in the CEX data increases from 0.68 at age 25 to 0.82 at the beginningof retirement.

Discussion of the estimated preference parameters.. I now argue that my initial calibration andestimates are in line with the micro literature, generate reasonable attitudes toward small and large wealthgambles, and match the empirical evidence for excess smoothness and sensitivity in aggregate data. Irefer to the literature for the standard preference parameter estimates of the exponential discount factorβ ≈ 1 and the coefficient of risk aversion θ ≈ 1 but discuss the news-utility parameter estimates, η , λ ,and γ , in greater detail. In particular, I demonstrate that my estimates are consistent with the existingmicro evidence on risk and time preferences. In Table 3 in Appendix Appendix B.8, I illustrate the riskpreferences over gambles with various stakes of the news-utility and standard agents. In particular, Icalculate the required gain G for a range of losses L to make each agent indifferent between accepting orrejecting a 50-50 win G or lose L gamble at a wealth level of $300,000, in accordance with Rabin (2001)and Chetty and Szeidl (2007).

First, I aim to demonstrate that my estimates match risk attitudes toward gambles about immediateconsumption, which are determined solely by the weight of gain-loss versus consumption utility η

and by the coefficient of loss aversion λ (because it can be reasonably assumed that utility overimmediate consumption is linear). η(λ − 1) ≈ 2 implies that the equivalent Kahneman and Tversky


TABLE 2

EXCESS-SMOOTHNESS AND SENSITIVITY REGRESSION RESULTS

Model news-utility habit standard hyperbolic temptedβ1 β2 β h

1 β h2 β s

1 β s2 β b

1 β b2 β td

1 β td2

coefficient 0.66 0.26 0.92 0.09 0.94 0.01 0.95 0.01 1.02 0.00t-statistic 84.2 42.9 119 12.2 108 0.72 108 0.63 103 0.35e-s ratio 0.72 0.92 0.95 0.96 1.03

The table displays the aggregate regression results of 200 individuals with 200 simulated data points.

(1979) coefficient of loss aversion is around 2. The reason is that the news-utility agent experiencesconsumption and news utility, whereas classical prospect theory effectively consists of news utility only,and consumption utility works in favor of any small-scale gamble. In Table 3, it can be seen that thenews-utility agent’s contemporaneous gain-loss utility generates reasonable attitudes toward small andlarge gambles over immediate consumption. Moreover, the parameters are consistent with the laboratoryevidence on loss aversion over immediate consumption, that is, the endowment effect literature.14 Incontrast, since I assume linear utility over immediate consumption, the standard agent is risk neutral.

Second, I elicit the agents’ risk attitudes by assuming that each of them is presented with a gambleafter all consumption in the current period has taken place. The news-utility agent will experience onlyprospective gain-loss utility over the gamble’s outcome, which is determined by the prospective gain-loss discount factor γ . γ < 1 implies attitudes toward intertemporal consumption tradeoffs similar to ahyperbolic-discounting parameter of the same value. Empirical estimates for the hyperbolic-discountingparameter model typically range around 0.7 (see Angeletos et al. (2001) among others). Thus, theexperimental and field evidence on individuals’ attitudes toward intertemporal consumption trade-offsdictates a prospective gain-loss discount factor around 0.7 (γ ≈ 0.7), when the exponential discount factoris around one (β ≈ 1). This is roughly in line with my estimate, that is, as the one in a life-cycle contextby Laibson et al. (2012), on the lower end. Table 3 shows that the news-utility agent exhibits reasonablerisk attitudes for small, medium, and large stakes, whereas the standard agent does so for very large stakesonly.

Excess smoothness and excess sensitivity in aggregate data.. Finally, I now demonstrate that myestimates are not only consistent with those found in the micro literature, but that they also generate thedegree of excess smoothness found in macro data. I simulate 200 consumption and income data pointsfor 200 agents, aggregate their consumption and income, and run the regression

∆log(Ct+1) = α +β1∆log(Yt+1)+β2∆log(Yt)+ εt+1

following Campbell and Deaton (1989).15 The results are shown in Table 2. In the news-utility model,I obtain a regression coefficient β2 ≈ 0.26 and the excess smoothness ratio, σ(∆log(Ct ))

σ(∆log(Yt ))as defined in

14. To illustrate, I take a concrete example from Kahneman et al. (1990), in which the authors distribute an item (mugs orpens) to half of their subjects and ask those who received the item about their willingness to accept (WTA) and those who did notreceive it about their willingness to pay (WTP), if they traded the item. The median WTA is $5.25, whereas the median WTP is$2.75. Accordingly, I infer (1+η)u(mug) = (1+ηλ )2.25 and (1+ηλ )u(mug) = (1+η)5.25, which implies λ ≈ 3 when η ≈ 1.I obtain a similar result for the pen experiment.

15. The regression results are similar for different assumptions about the number of data points and citizens as well as theirwealth levels and time horizons. I simulate data for each citizen at normalized wealth level A0

P0= 1 and age 40.


Deaton (1986), is 0.72.16 On the other hand, in the standard model, I obtain β s2 ≈ 0 and 1.17 Regressing

consumption growth on lagged labor income growth in aggregate data, I obtain a regression estimatefor β2 of approximately 0.23 and an excess-smoothness ratio of approximately 0.68.18 Not surprisingly,temptation disutility does not generate excess smoothness or sensitivity, while habit formation generatessome. However, habit formation has unrealistic implications for the life-cycle consumption profile, asexplored in Section 2.3.1. I conclude that the estimates obtained from CEX data simultaneously matchthe degree of excess smoothness and sensitivity found in aggregate data.

3. CONCLUSION6

This study demonstrates that expectations-based reference-dependent preferences explain not only microevidence (such as the endowment effect or cab-driver labor supply), but also three major life-cycleconsumption phenomena.

Excess smoothness and sensitivity in consumption, two widely analyzed macro consumptionpuzzles, are explained by loss aversion, a robust risk preference analyzed in experimental research and apopular explanation for the equity premium puzzle. Intuitively, the agent wants to allow his expectations-based reference point to adjust prior to adjusting consumption. Moreover, a hump-shaped consumptionprofile and a drop in consumption at retirement are both explained by the interplay of news-utility riskand time preferences. The hump shape results from the net effect of two preference features. First,the agent’s consumption path is steeper at the beginning of his life because loss aversion generates anadditional precautionary-savings motive, which increases more rapidly than the standard precautionary-savings motive in the agent’s horizon. Second, the agent’s consumption path declines toward the endof his life because the expectations-based reference point introduces time-inconsistent overconsumption.However, once the agent retires, overconsumption is associated with a certain loss in future consumption.Thus, the agent can suddenly refrain from overconsumption, and his consumption drops at retirement.

I explore the intuition behind these results by solving an exponential-utility model in closed form.Moreover, assuming power utility, as standard in the literature, I structurally estimate the preferenceparameters, obtain estimates that are in line with the micro evidence, and generate the degree of excesssmoothness found in aggregate data.

Finally, I draw potentially important welfare conclusions. My results suggest that excess smoothnessis an optimal response, in contrast to the welfare implications of borrowing constraints, the potentiallymost popular alternative explanation for excess smoothness. Instead, the hump-shaped life-cycleconsumption profile and the consumption drop at retirement decrease welfare, which is more in linewith alternative explanations.

16. As explained in Ludvigson and Michaelides (2001), the transitory shock introduces a spurious negative correlationbetween consumption growth ∆Ct+1 and income growth ∆Yt that will be aggregated away if enough individuals are simulated.As an alternative to aggregating individual observations, I can omit the transitory shock realizations.

17. All consumption adjustment occurs after a single period because the agent’s preferences are characterized by full beliefupdating. However, the variation in consumption adjusts via end-of-period asset holdings and is thereby spread out over the entirefuture. Empirically, Fuhrer (2000) and Reis (2006) find that consumption peaks one year after the shock and that the consumptionresponse dies out briefly after the first year.

18. I follow Ludvigson and Michaelides (2001) and use NIPA deflated total, nondurable, or services consumption and totaldisposable labor income for 1947 to 2011.


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APPENDIX A. DERIVATIONS AND PROOFS

Appendix A.1. Derivation of the exponential-utility model

1.1.1. The finite-horizon model. The “news-utility” agent’s lifetime utility in each period t = 0, ...,T is

u(Ct)+n(Ct ,F t−1Ct

)+ γ ∑T−tτ=1 β



τUt+τ ]

with β ∈ [0,1], η ∈ (0,∞), λ ∈ (1,∞), and γ ∈ [0,1]. Also suppose σPt = σP and σTt = σT for all t (for simplicity of the exposition).I can solve the exponential-utility model through backward induction. The text contains a simple derivation of the second-to-lastperiod. In the following, I outline the model’s solution for period T − i in which the agent chooses how much to consume CT−i andhow much to invest in the risk-free asset AT−i. I guess and verify the model’s consumption function

CT−i =(1+ r)i

f (i)(1+ r)AT−i−1 +PT−i−1 + sT−i−

f (i−1)f (i)

ΛT−i

with sT−i = sPT−i +(1− f (i−1)

f (i) )sTT−i and f (i) = ∑

ij=0(1+ r) j = (1+ r)i 1+r−( 1

1+r )i

r (I define a(i) = f (i−1)f (i) in the text) and

ΛT−i =1θ

log((1+ r)i

f (i−1)ψT−i + γQT−iη(λ − (λ −1)FS(st))

1+η(λ − (λ −1)FS(st)))

with FS(st) = Pr(SPt +(1− f (i−1)

f (i) )STt < st). Then, the budget constraint AT−i = (1+ r)AT−i−1 +YT−i−CT−i determines end-of-

period asset holdings

AT−i =f (i−1)

f (i)(1+ r)AT−i−1 +

f (i−1)f (i)

sTT−i +

f (i−1)f (i)

ΛT−i.

ΛT−i is a function independent of AT−i−1 and PT−i−1 but dependent on sPT−i and sT

T−i. In the last period the agent consumeseverything such that ΛT = 0. As a first step to verify the solution guess, I simply plug the asset-holdings function into each futureconsumption function. For instance, CT−i+1 is given by

CT−i+1 =(1+ r)i−1

f (i−1)(1+ r)AT−i +PT−i + sT−i+1−

f (i−2)f (i−1)

ΛT−i+1

=(1+ r)i

f (i)(1+ r)AT−i−1 + sT−i +

(1+ r)i

f (i)ΛT−i +PT−i−1 + sT−i+1−

f (i−2)f (i−1)

ΛT−i+1.

Then, I sum the expectation of the discounted consumption function utilities from period T − i to T :

βET−i−1[∑iτ=0 β

τ u(CT−i+τ )] = u(PT−i−1 +(1+ r)i

f (i)(1+ r)AT−i−1)QT−i−1

=− 1θ

exp−θ(PT−i−1 +(1+ r)i

f (i)(1+ r)AT−i−1)QT−i−1,

with QT−i−1 given by

QT−i−1 = βET−i−1[exp−θ(sT−i−f (i−1)

f (i)ΛT−i)+ exp−θ(sT−i +

(1+ r)i

f (i)ΛT−i)QT−i]

QT−i−1 is a constant if ΛT−i depends only on sPT−i and sT

T−i. The consumption function and its sum allows me to write down theagent’s continuation utility in period T − i−1 as follows

u(PT−i−1 +(1+ r)i

f (i)(1+ r)AT−i−1)ψT−i−1 =−

1θ

exp−θ(PT−i−1 +(1+ r)i

f (i)(1+ r)AT−i−1)ψT−i−1

with ψT−i−1 given by

ψT−i−1 = βET−i−1[exp−θ(sT−i−f (i−1)

f (i)ΛT−i)+ω(exp−θ(sT−i−

f (i−1)f (i)

ΛT−i))

+γQT−iω(exp−θ(sT−i +(1+ r)i

f (i)ΛT−i))+ exp−θ(sT−i +

(1+ r)i

f (i)ΛT−i)]ψT−i

and ω(x) for any random variable X ∼ FX , where the realization is denoted by x, is

ω(x) = η

ˆ x

−∞

(x− y)dFX (y)+ηλ

ˆ∞

x(x− y)dFX (y).


The above expression for ψT−i−1 can be easily inferred from the agent’s utility function. The first component in ψT−i−1 correspondsto the expectation of consumption utility in period T − i, the second to contemporaneous gain-loss in period T − i, the third toprospective gain-loss in period T − i that depends on the sum of future consumption utilities QT−i, and the last to the agent’scontinuation value. Moreover, for any random variable Y ∼ FY = FX note that

ˆ∞

−∞

ω(g(x))dFX (x) =ˆ

∞

−∞

ηˆ x

−∞

(g(x)−g(y))︸︷︷︸<0 if g′(·)<0

dFY (y)+ηλ

ˆ∞

x(g(x)−g(y))︸︷︷︸>0 if g′(·)<0

dFY (y)dFX (x)> 0

ˆ∞

−∞

ηˆ x

−∞

(g(x)−g(y))︸︷︷︸<0 if g′(·)<0

dFY (y)+η

ˆ∞

x(g(x)−g(y))︸︷︷︸>0 if g′(·)<0

dFY (y)+η(λ −1)ˆ

∞

x(g(x)−g(y))︸︷︷︸>0 if g′(·)<0

dFY (y)dFX (x)> 0

=

ˆ∞

−∞

η(λ −1)ˆ

∞

x(g(x)−g(y))︸︷︷︸>0 if g′(·)<0

dFY (y)dFX (x)> 0

if λ > 1, η > 0, and g′(·)< 0. The above implies ψT−i−1 > QT−i−1 necessarily if θ > 0 such that u(·) is concave. I now turn to theagent’s maximization problem in period T − i given by

u(CT−i)+n(CT−i,FT−i−1CT−i

)+ γ ∑iτ=1 β

τnnn(FT−i,T−i−1CT−i+τ

)+u(PT−i +(1+ r)i

f (i−1)AT−i)ψT−i.

I want to find the agent’s first-order condition. I begin by explaining the first derivative of contemporaneous gain-loss utilityn(CT−i,FT−i−1

CT−i). The agent takes his beliefs about period T − i consumption FT−i−1

CT−ias given such that

∂n(CT−i,FT−i−1CT−i

)

∂CT−i=

∂ (η´ CT−i−∞

(u(CT−i)−u(c))dFT−i−1CT−i

(c))+ηλ´

∞

CT−i(u(CT−i)−u(c))dFT−i−1

CT−i(c))

∂CT−i

= u′(CT−i)(ηFT−i−1CT−i

(CT−i)+ηλ (1−FT−i−1CT−i

(CT−i))) = u′(CT−i)(ηFS(sT−i)+ηλ (1−FS(sT−i)))

= u′(CT−i)η(λ − (λ −1)FS(sT−i))

the last step results from the guessed consumption function and the assumption that admissible consumption functions are increasingin both shocks. The first derivative of the agent’s prospective gain-loss utility ∑

iτ=1 β τnnn(FT−i,T−i−1

CT−i+τ) over the entire stream of

future consumption utilities u(PT−i +(1+r)i

f (i−1)AT−i)QT−i can be inferred similarly. Recall that QT−i is a constant under the guessedconsumption function; thus, the agent only experiences gain-loss utility over the realized uncertainty in period T − i, i.e.,

∂ ∑∞τ=1 β τnnn(FT−i−1,T−i

CT−i+τ)

∂AT−i= ∑

∞

τ=1 βτ ∂

∂AT−i

ˆ∞

−∞

ˆ∞

−∞

µ(u(c)−u(r))dFT−i−1,T−iCT−i+τ

(c,r)

=∂

∂AT−i

ˆ∞

−∞

µ(u(PT−i +(1+ r)i

f (i−1)AT−i)QT−i−u(x)QT−i)dFT−i−1

PT−i+(1+r)if (i−1) AT−i

(x)

=(1+ r)i

f (i−1)exp−θ(PT−i +

(1+ r)i

f (i−1)AT−i)QT−iη(λ − (λ −1)FS(sT−i))

and again, FS(sT−i) = Pr(SPT−i +

(1+r)i

f (i) STT−i < sT−i) results from the solution guess for AT−i times (1+r)i

f (i−1) and the fact that future

consumption is increasing in both shocks (because end-of-period asset holdings AT−i are always increasing in sPT−i and sT

T−i as∂ΛT−i∂ sP

T−i> 0 and ∂ΛT−i

∂ sTT−i

> 0). The derivative of the agent’s continuation utility with respect to AT−i is simply

(1+ r)i

f (i−1)exp−θ

(1+ r)i

f (i−1)AT−iψT−i.

In turn, in any period T − i the news-utility agent’s first-order condition (normalized by PT−i) is

exp−θ((1+ r)AT−i−1 + sTT−i−AT−i)︸︷︷︸

=−θ(CT−i−PT−i) budget constraint

(1+η(λ − (λ −1)FS(sT−i)))

=(1+ r)i

f (i−1)exp−θ

(1+ r)i

f (i−1)AT−i(ψT−i + γQT−iη(λ − (λ −1)FS(sT−i)).


I can rewrite he first-order condition to obtain the optimal consumption and end-of-period asset holdings functions and the functionΛT−i

ΛT−i =1θ

log((1+ r)i

f (i−1)ψT−i + γQT−iη(λ − (λ −1)FS(sT−i))

1+η(λ − (λ −1)FS(sT−i)))

and verify the guessed consumption function.

1.1.2. The infinite-horizon model. Suppose σPt = σP and σTt = σT and T, i→∞. I use a simple guess and verify procedureto find the infinite-horizon recursive equilibrium; alternatively, I can obtain the solution by simple backward induction taking T andi to infinity. I guess and verify that the infinite-horizon model consumption and asset-holdings functions are

Ct = Yt + rAt−1−1

1+ rsTt −

11+ r

Λt = Pt−1 + st + rAt−1−1

1+ rΛt = Pt +

r1+ r

sTt + rAt−1−

11+ r

Λt

where st = sPt + r

1+r sTt with the budget constraint At = (1+ r)At−1 +Yt −Ct determining end-of-period asset holdings

At = At−1 +1

1+ rsTt +

11+ r

Λt

and the function Λt being i.i.d. in the infinite-horizon model and given by

Λt =1θ

log(rψ + γQη(λ − (λ −1)FS(st))

1+η(λ − (λ −1)FS(st))).

The function Λt results from the first-order condition derived analogously to the finite horizon model. First, marginal consumptionand contemporaneous gain-loss utility, i.e.,

exp−θ(1+ r)At−1−θsTt +θAt︸︷︷︸

=−θ(Ct−Pt ) (budget constraint)

(1+η(λ − (λ −1)FS(st))),

is equalized with the agent’s marginal continuation utility captured by ψ (the infinite sum of future consumption and gain-lossutility) and Q (the infinite sum of future consumption utility) with the latter multiplied by marginal prospective gain-loss utility

= rexp−θrAt(ψ + γQη(λ − (λ −1)FS(st))).

To understand the first-order condition, take the guessed consumption function and note that consumption in period t + 1 isCt+1 = Pt + st+1 + rAt − 1

1+r Λt+1 and thus depends on the end-of-period asset holdings At . Similarly, consumption in period t +2can be iterated back to At and Pt , i.e.,

Ct+2 = Pt+1 + st+2 + r(At +1

1+ rsTt+1 +

11+ r

Λt+1)−1

1+ rΛt+2 = Pt + st+1 + rAt +

r1+ r

Λt+1 + st+2−1

1+ rΛt+2.

These two equations for Ct+1 and Ct+2 help to understand the continuation utility terms in the first-order condition. Let Q be theinfinite discounted sum of marginal consumption utility normalized by rAt and Pt , which is determined by Ct+1, Ct+2, and so forthusing the following recursion

Q = βEt [exp−θ(st+1−1

1+ rΛt+1)+ exp−θ(st+1 +

r1+ r

Λt+1)Q].

More specifically the agent’s infinite discounted sum of marginal future consumption utilities normalized by Pt is (as in the first-order condition)

rexp−θrAtQ with Q =βEt [exp−θ(st+1− 1

1+r Λt+1)]1−βEt [exp−θ(st+1 +

r1+r Λt+1)]

in turn, this term gets multiplied by marginal prospective gain-loss utility γη(λ − (λ − 1)FS(st)). Now, I move on to the secondcontinuation utility term that is the infinite discounted sum of marginal consumption and gain-loss utility ψ , which is determinedanalogously to Q, but simply includes gain-loss in addition to consumption utility

ψ =βEt [exp−θ(st+1− 1

1+r Λt+1)+ω(exp−θ(st+1− 11+r Λt+1))+ γQω(exp−θ(st+1 +

r1+r Λt+1))]

1−βEt [exp−θ(st+1 +r

1+r Λt+1)].

Thus, the agent’s marginal continuation value of end-of-period asset holdings (normalized by Pt ) is determined by rexp−θrAtψ .Solving the first-order condition for optimal end-of-period asset holdings At yields

At = At−1 +1

1+ rsTt +

11+ r

1θ

log(rψ + γQη(λ − (λ −1)FS(st))

1+η(λ − (λ −1)FS(st)))︸︷︷︸

=Λt


which verifies the expression for At and Λt above. In turn, consumption can be verified by the budget constraint

Ct = Yt + rAt−1−1

1+ rsTt −

11+ r

Λt = Pt−1 + st + rAt−1−1

1+ rΛt .

Appendix A.2. The other agent’s exponential-utility consumption functions

By the same arguments as for the derivation of the news-utility model, the “standard” agent’s consumption function in period T − iis

AsT−i =

f (i−1)f (i)

(1+ r)AsT−i−1 +

f (i−1)f (i)

sTT−i +

f (i−1)f (i)

ΛsT−i

CsT−i =

(1+ r)i

f (i)(1+ r)As

T−i−1 +PT−i−1 + sT−i−f (i−1)

f (i)Λ

sT−i

ΛsT−i =

1θ

log((1+ r)i

f (i−1)Qs

T−i)

QsT−i−1 = βET−i−1[exp−θ(sT−i−

f (i−1)f (i)

ΛsT−i)+βexp−θ(sT−i +

(1+ r)i

f (i)Λ

sT−i)Qs

T−i].

The “hyperbolic-discounting” agent’s consumption in period T − i is

AbT−i =

f (i−1)f (i)

(1+ r)AbT−i−1 +

f (i−1)f (i)

sTT−i +

f (i−1)f (i)

ΛbT−i

CbT−i =

(1+ r)i

f (i)(1+ r)Ab

T−i−1 +PT−i−1 + sT−i−f (i−1)

f (i)Λ

bT−i

ΛbT−i =

1θ

log((1+ r)i

f (i−1)bQb

T−i)

QbT−i−1 = βET−i−1[exp−θ(sT−i−

f (i−1)f (i)

ΛbT−i)+βexp−θ(sT−i +

(1+ r)i

f (i)Λ

bT−i)Qb

T−i].

Appendix A.3. Proofs of Section 2.2:

1.3.1. Proof of Proposition 1. ψt and Qt are constants recursively determined by

Qt = βEt [exp−θ(st+1−a(T − t−1)Λt+1)+ exp−θ(st+1 +(1−a(T − t−1))Λt+1)Qt+1] (A17)

ψt = βEt [exp−θ(st+1−a(T − t−1)Λt+1)+ω(exp−θ(st+1−a(T − t−1)Λt+1)) (A18)

+γQt+1ω(exp−θ(st+1 +(1−a(T − t−1))Λt+1))+ exp−θ(st+1 +(1−a(T − t−1))Λt+1)ψt+1]

with

ω(x) = η

ˆ x

−∞

(x− y)dFX (y)+ηλ

ˆ∞

x(x− y)dFX (y).

If the consumption function derived in Appendix 1.1.1 belongs to the class of admissible consumption functions, then theequilibrium exists and is unique as the equilibrium solution is obtained by maximizing the agent’s objective function in any periodT − i, which is concave as its derivative is monotonically decreasing in CT−i. Moreover, there is a finite period T that uniquelydetermines the equilibrium. The derivative of the agent’s maximization problem (as derived in Appendix 1.1.1) is given by

exp−θ(CT−i)(1+η(λ − (λ −1)FST−i(sT−i)))

− (1+ r)i

f (i−1)exp−θ

(1+ r)i

f (i−1)((1+ r)AT−i−1 +YT−i−CT−i)exp−θPT−i(ψT−i + γQT−iη(λ − (λ −1)FST−i

(sT−i)).

Note that, exp−θPT−i(ψT−i + γQT−iη(λ − (λ − 1)FST−i(sT−i)) is constant in CT−i and positive, which can be easily seen in

Equations A17 and A18. As an illustrative example, in period T −1 (given that QT = ψT = 0)

QT−1 = βET−1[exp−θ st+1] and ψT−1 = βET−1[exp−θ st+1+ω(exp−θ st+1)]


and for any random variable X it holds that

E[ω(exp−θX)] = E[η(λ −1)ˆ

∞

X(e−θX − e−θy)dFX (y)]> 0 as e−θX > e−θy for all y > X .

In turn, it has to be shown that the second derivative of the agent’s maximization problem

−θexp−θ(CT−i)(1+η(λ − (λ −1)FS(sT−i)))

−θ(1+ r)i

f (i−1)(1+ r)i

f (i−1)exp−θ

(1+ r)i

f (i−1)((1+ r)AT−i−1 +YT−i−CT−i)positive const. < 0

which is always true. In turn, this derivative has a unique point at which it equals zero, which is the global maximum as themaximization problem is concave. σ∗T−i is implicitly defined by the two admissible consumption function restrictions ∂CT−i

∂ sPT−i

> 0

and ∂CT−i∂ sT

T−i> 0 for any i as

CT−i =(1+ r)i

f (i)(1+ r)AT−i−1 +PT−i−1 + sT−i−a(i)ΛT−i

the restrictions are equivalent to ∂a(i)ΛT−i∂ sP

T−i< 1 and ∂a(i)ΛT−i

∂ sTT−i

< 1− a(i) as ∂ΛT−i∂ sP

T−i,

∂ΛT−i∂ sT

T−i> 0 (since ψT−i > γQT−i (as u(·) is

concave)). Recall that a(i) = 1− (1+r)i

f (i) = f (i−1)f (i) . Then, σ∗T−i (note that, t = T − i) is implicitly defined by the two restrictions

∂a(i)ΛT−i

∂ sPT−i

=a(i)θ

(ψT−i−γQT−i)η fST−i(sT−i)(λ−1)

1+η(λ−(λ−1)FST−i(sT−i))

ψT−i + γQT−iη(λ − (λ −1)FST−i(sT−i))

< 1

and

∂a(i)ΛT−i

∂ sTT−i

=a(i)θ




< 1−a(i).

Here, the normal pdf of any random variable X is denoted by fX . Increasing σPt and σTt unambiguously decreases fSt(st) and

thereby ∂a(i)ΛT−i∂ sP

T−iand ∂a(i)ΛT−i

∂ sTT−i

(additionally, FSt(st) decreases which increases η(λ − (λ −1)FSt

(st)) (but η(λ − (λ −1)FSt(st)) ∈

[η ,ηλ ] is bounded anyway) and ψt and Qt are independent of σPt and σTt ). Thus, there exists a condition σ2Pt +( (1+r)i

f (i) )2σ2Tt ≥ σ∗t

which ensures that an admissible consumption function exists for any t that uniquely determines the equilibrium (given theadmissible consumption functions in each future period until the final period).

As a side note, if uncertainty is small then the consumption function may be decreasing over some range, i.e., ∂CT−i∂ sP

T−i< 0

or ∂CT−i∂ sT

T−i< 0. I now show that the agent would pick a consumption function that is instead of decreasing flat and thus weakly

increasing in the shock realizations, i.e., ∂CT−i∂ sP

T−i≥ 0 or ∂CT−i

∂ sTT−i≥ 0. To discuss this result in a simple framework, I return to the two-

period, one-shock model. Suppose that the absolute level of the shock increases; then, holding CT−1 constant, the marginal value ofsavings declines and the agent’s first-order condition implies that consumption should increase. However, FP(sP

T−1) also increases,and marginal gain-loss utility is lower, such that the agent’s optimal consumption should decrease. Suppose that sP

T−1 increasesmarginally but FP(sP

T−1) increases sharply, which could occur if FP is a very narrow distribution. In this case, the lower marginalgain-loss utility that decreases consumption dominates such that the first-order condition predicts decreasing consumption oversome range in the neighborhood of zero where FP increases most sharply if FP is bell shaped. However, a decreasing consumptionfunction cannot be an equilibrium because the agent would unnecessarily experience gain-loss utility over the decreasing partof consumption, which decreases expected utility unnecessarily. In the decreasing-consumption function region, the agent couldchoose a flat consumption function instead. In the following I show that the agent may choose a credible consumption plan with aflat section. Suppose the agent chooses a flat consumption level for realizations of sP

T−1 in sP and sP. Then, sP is chosen where theoriginal consumption function just stops decreasing, which corresponds to the lowest possible level of the flat section of consumptionCT−1. In that is then determined by

u′(CT−1) = (1+ r)u′((sP−CT−1)(1+ r)+ sP)ψT−1 + γQT−1η(λ − (λ −1)FP(sP))

1+η(λ − (λ −1)FP(sP))

in which case s is determined by

u′(CT−1) = (1+ r)u′((sP−CT−1)(1+ r)+ sP)ψT−1 + γQT−1η(λ − (λ −1)FP(sP))

1+η(λ − (λ −1)FP(sP)).


The agent’s consistency constraint for not increasing consumption beyond CT−1 for any sPT−1 ∈ [sP,sP] is given by

u′(CT−1)< (1+ r)u′((sPT−1−CT−1)(1+ r)+ sP

T−1)ψT−1 + γQT−1η(λ − (λ −1)FP(sP

T−1))

1+η(λ − (λ −1)FP(sP))

and always holds as can be easily inferred. This result can be easily generalized to any horizon, thus, if the consumption function isdecreasing over some range, the agent can credibly replace the decreasing part with a flat section as described above.

1.3.2. Proof of Proposition 2. Please refer to the derivation of the exponential-utility model Section Appendix A.1 for adetailed derivation of ΛT−i. According to Definition 4 consumption is excessively smooth if ∂Ct

∂ sPt< 1 and excessively sensitive if

∂∆Ct+1∂ sP

t> 0. Consumption growth is

∆CT−i = sT−i−a(i)ΛT−i +ΛT−i−1

so that ∂CT−i∂ sP

T−i< 1 iff ∂ΛT−i

∂ sPT−i

> 0 and ∂∆CT−i∂ sP

T−i−1> 0 iff ∂ΛT−i−1

∂ sPT−i−1

> 0. Since ψT−i > γQT−i (as can be seen in the previous proof or the

derivation Appendix A.1), ∂ΛT−i∂ sP

T−i> 0, i.e.

∂ΛT−i

∂ sPT−i

=1θ




> 0.

1.3.3. Proof of Proposition 3. Qt and ψt are given by Equations (A17) and (A18) with a(T − t−1) = 11+r ; and if σPt = σP

and σTt = σT , thenψ and Q are constants determined by

Q =βEt [exp−θ(st+1− 1


r1+r Λt+1)]

(A19)

ψ =βEt [exp−θ(st+1− 1

1+r Λt+1)+ω(exp−θ(st+1− 11+r Λt+1))+ γQω(exp−θ(st+1 +

r1+r Λt+1))]

1−βEt [exp−θ(st+1 +r

1+r Λt+1)]. (A20)

If the consumption function derived in Section 1.1.2 belongs to the class of admissible consumption functions, then an equilibriumexists if βEt [exp−θ(st+1 +

r1+r Λt+1)] < 1 (as β can be chosen such that β < β ) and r > 0 such that Qt and ψt converge and

Λt is well defined. Please refer to Section 1.1.2 for the guess-and-verify derivation of the consumption function and to the proof ofProposition 1.3.1 for a demonstration that the sufficient condition for an optimum always holds. σ∗t is implicitly defined by the twoadmissible consumption function restrictions ∂Ct

∂ sPt> 0 and ∂Ct

∂ sTt> 0 as

Ct = Pt +r

1+ rsTt + rAt−1−

11+ r

Λt

Λt =1θ

log(rψt + γQt η(λ − (λ −1)FSt

(st))

1+η(λ − (λ −1)FSt(st))

).

the restrictions are equivalent to∂

11+r Λt

∂ sPt

< 1 and∂

11+r Λt

∂ sTt

< r1+r as ∂Λt

∂ sPt, ∂Λt

∂ sTt> 0 (since ψt > γQt ). Then, σ∗t is implicitly defined by

the two restrictions

∂1

1+r Λt

∂ sPt

=1

1+ r1θ

(ψt−γQt )η fSt(st )(λ−1)

(1+η(λ−(λ−1)FSt(st )))

ψt + γQt η(λ − (λ −1)FSt(st))

< 1

and

∂1

1+r Λt

∂ sTt

=1

1+ r1θ

(ψt−γQt )η fSt(st )(λ−1)

(1+η(λ−(λ−1)FSt(st )))

ψt + γQt η(λ − (λ −1)FSt(st))

<r

1+ r.

Here, the normal pdf of any random variable X is denoted by fX . As in the previous proof, increasing σPt and σTt unambiguously

decreases fSt(st) and thereby

∂1

1+r Λt

∂ sPt

and∂

11+r Λt

∂ sTt

. Thus, there exists a condition σ2Pt + ( r

1+r )2σ2

Tt ≥ σ∗t which ensures that anadmissible consumption function exists that determines the equilibrium.

1.3.4. Proof of Lemma 1. I start with the first part of the lemma, the precautionary-savings motive. In the second-to-lastperiod of the simple model outlined in the text, the first-order condition is given by

u′(CT−1)+u′(CT−1)η(λ − (λ −1)FP(sPT−1))


= (1+ r)u′((sPT−1−CT−1)(1+ r)+ sP

T−1)γ βET−1[u′(SPT )]︸︷︷︸

QT−1

η(λ − (λ −1)FP(sPT−1))

+(1+ r)u′((sPT−1−CT−1)(1+ r)+ sP

T−1)βET−1[u′(SPT )+η(λ −1)

ˆ∞

SPT

(u′(SPT )−u′(y))dFP(y)]︸︷︷︸

ψT−1

.

From Section Appendix A.1 I know that ψT−1 > QT−1 because for any two random variables X ∼ FX and Y ∼ FY with FX = FY inequilibrium ˆ

∞

−∞

ηˆ x

−∞

(g(x)−g(y))︸︷︷︸<0 if g′(·)<0

dFY (y)+ηλ

ˆ∞

x(g(x)−g(y))︸︷︷︸>0 if g′(·)<0

dFY (y)dFX (x)> 0

if λ > 1, η > 0, and g′(·) < 0. Thus, βET−1[η(λ − 1)´

∞

SPT(u′(SP

T )− u′(y))dFP(y)] > 0 if u′′(·) > 0 the agent is risk averse or

u(·) is concave. Moreover, it can be seen that∂βET−1[η(λ−1)

´∞

SPT(u′(SP

T )−u′(y))dFP(y)]

∂η> 0 and

∂βET−1 [η(λ−1)´

∞

SPT(u′(SP

T )−u′(y))dFP(y)]

∂λ> 0.

Then, for any value of savings AT−1 = sPT−1−CT−1 the right hand side of the first-order condition is increased by the presence of

expected gain-loss disutility if σP > 0, whereas, if σP = 0, then ψT−1 = QT−1. The increase of the agent’s marginal value of savingsby the presence of expected gain-loss disutility depends on σP > 0, but does not go to zero as σP → 0; therefore, the additional

precautionary savings motive is first-order∂ (sP

T−1−CT−1)

∂σP|σP=0 > 0 as can be easily shown for any normally distribution random

variable X ∼ FX = N(µ,σ2) with F01 denoting a standard normal distribution function

E[η(λ −1)ˆ

∞

X(u′(X)−u′(y))dFX (y)]

= e−θ µ

ˆ∞

−∞

(η

ˆ z

−∞

(e−θσz− e−θσε )dF01(ε)+ηλ

ˆ∞

z(e−θσz− e−θσε )dF01(ε))dF01(z) with z,ε ∼ F01 = N(0,1)

= e−θ µη(λ −1)

ˆ∞

z(e−θσz− e−θσε )dF01(ε))dF01(z)

= e−θ µη(λ −1)

ˆ∞

−∞

(1−F01(z))e−θσz− e12 θ 2σ2

F01(−θσ − z)dF01(z)

∂ (·)∂σ|σ=0 = e−θ µ

η(λ −1)ˆ

∞

−∞

−θz(1−F01(z))e−θσz−θσe12 θ 2σ2

F01(−θσ − z)+θe12 θ 2σ2

F01(−θσ − z)dF01(z)|σ=0

= e−θ µθη(λ −1)

ˆ∞

−∞

−z+ zF01(z)+F01(−z)dF01(z)≈ e−θ µθη(λ −1)0.7832 > 0.

Koszegi and Rabin (2009) prove this result more generally: let F be the cumulative distribution function of a nondeterministicrandom variable y with risk scaled by σ , gain-loss utility isˆ ˆ

µ(u(σy)−u(σy′))dF(y′)dF(y) =−12

η(λ −1)ˆ ˆ

(u(σmax(y,y′))−u(σmin(y,y′)))dF(y′)dF(y)

and marginal gain-loss is therefore

12

η(λ −1)ˆ ˆ

(u′(σmax(y,y′))−u′(σmin(y,y′)))dF(y′)dF(y)

and the derivative of this expression with respect to σ evaluated at σ = 0 is

12

η(λ −1)ˆ ˆ

|y− y′|dF(y′)dF(y)> 0 if u′′(·)< 0.

Thus, news-utility introduces a first-order precautionary-savings motive.

In the second part of the lemma the implications for consumption can be immediately seen by comparing the agents’ first-order conditions. The standard agent’s first-order condition in period T −1 is given by

u′(CT−1) = Ru′((sPT−1−CT−1)R+ sP

T−1)QT−1.


The difference to the news-utility model can be seen easily: First, ψT−1QT−1

> 1 implies that

ψT−1QT−1

+ γη(λ − (λ −1)FP(sPT−1))

1+η(λ − (λ −1)FP(sPT−1))

> 1

for γ high enough such that the news-utility agent consumes less than the standard agent if he does not discount prospective gain-lossutility very highly.

1.3.5. Proof of Proposition 4. The agent optimally chooses consumption and asset holdings in periods T − i = 1, ...,T forany horizon T . I define a hump-shaped consumption profile as characterized by increasing consumption and asset holdings in thebeginning of life C1 <C2 and decreasing consumption in the end of life CT <CT−1 “on average.” The first characteristic requiresC1 <C2 which implies that

(1+ r)T−1

f (T −1)(1+ r)A0 +P0−

f (T −2)f (T −1)

Λ1 <(1+ r)T−2

f (T −2)(1+ r)A1 +P0−

f (T −3)f (T −2)

Λ2

so that Λ1 >f (T−3)f (T−2)Λ2 and since f (T−3)

f (T−2) < 1 this holds always if Λ1 > 0 and (as T becomes large) Λ1 ≈ Λ2 if an equilibrium exists.Note that, if λ > 1 and η > 0, then ψT−i > QT−i, ψT−i > ψT−i+1 and QT−i > QT−i+1 and ψT−i−QT−i > ψT−i+1−QT−i+1 forall i and ψT−i

QT−iis increasing in i and approaches its limit ψ

Q as i and T become large. γ is then defined by the requirement E0[Λ1]> 0for which a sufficient condition is (because FS1

(s1) ∈ [0,1])

(1+ r)T−1

f (T −2)

ψ1 + γQ1η(λ − (λ −1)FS1(s1))

1+η(λ − (λ −1)FS1(s1))

=r

1− ( 11+r )

T−1Q1

ψ1Q1

+ γη(λ − (λ −1)FS1(s1))

1+η(λ − (λ −1)FS1(s1))

> 1.

For T → ∞ and σPt = σP and σTt = σT for all t, the condition boils down to

rψ + γQη(λ − (λ −1)FS(st))

1+η(λ − (λ −1)FS(st))> 1⇒ γ =

1rQ (1+η(λ − (λ −1)FS(st)))− ψ

Q

η(λ − (λ −1)FS(st)).

The second characteristic requires CT <CT−1 on average, which implies that

(1+ r)AT−1 <1+ r2+ r

(1+ r)AT−2−1

2+ rΛT−1

and is equivalent to ΛT−1 < 0 as ∆CT = sT +ΛT−1 =ΛT−1 < 0. Thus, we can define γ by ET−2[ΛT−1]< 0 and again use a sufficientcondition

1θ

log((1+ r)ψT−1 + γQT−1η(λ − (λ −1)FS(sT−1))

1+η(λ − (λ −1)FS(sT−1)))< 0⇒ (1+ r)QT−1

ψT−1QT−1

+ γη(λ − (λ −1)FS(sT−1))

1+η(λ − (λ −1)FS(sT−1))< 1.

Thus, we have two sufficient conditions that have to simultaneously hold

rQψ

Q + γη(λ − (λ −1)FS(st))

1+η(λ − (λ −1)FS(st))> 1 and (1+ r)QT−1

ψT−1QT−1

+ γη(λ − (λ −1)FS(sT−1))

1+η(λ − (λ −1)FS(sT−1))< 1.

⇒1

rQ (1+η(λ − (λ −1)FS(st)))− ψ

Q

η(λ − (λ −1)FS(st))<

(1+η(λ − (λ −1)FS(sT−1)))− ψT−1QT−1

η(λ − (λ −1)FS(sT−1))

or γ =

1(1+r)QT−1

(1+η(λ − (λ −1)FSt(st)))− ψT−1

QT−1

η(λ − (λ −1)FSt(st))

and γ =

1rQt

(1+η(λ − (λ −1)FST−1(st)))− ψt

Qt

η(λ − (λ −1)FST−1(st))

.

Intuitively, these conditions hold because for β (1 + r) ≈ 1 the standard agent’s consumption growth is flat in the absence ofuncertainty and increasing in the presence of uncertainty such that rQ > (1+ r)QT−1. In addition, ψ

Q >ψT−1QT−1

then implies thatγ > γ . More formally, rQ > (1+ r)QT−1 holds

rβEt [exp−θ(st+1− 1


r1+r Λt+1)]

> (1+ r)βET−1[exp−θ(sT )]

rβ︸︷︷︸>0

Et [exp−θ(st+1−1

1+ rΛt+1)]︸︷︷︸

>0

> ET−1[exp−θ(sT )]︸︷︷︸>0

(1−βEt [exp−θ(st+1 +r

1+ rΛt+1)]︸︷︷︸

>0 but→0

)


as an equilibrium for T large only exists iff r > 0 and βEt [exp−θ(st+1 +r

1+r Λt+1)] < 1 (otherwise Λt would be undefinedand Qt would be an exploding sum). In turn, because st+1 has a zero mean and Et [Λt+1] > 0 can be chosen as close to zero aswished (as this condition ultimately defines γ), Et [exp−θ(st+1 +

r1+r Λt+1)]> 1 and β > β ensures the inequality holds. In turn,

ψT−1QT−1

< ψ

Q ⇒ ψT−1Q < ψQT−1 (using the law of iterated expectations)

βEt [(exp−θ(sT )exp−θ(st+1−1

1+ rΛt+1)+ exp−θ(st+1−

11+ r

Λt+1)ω(exp−θ(sT )))]

< βEt [exp−θ(sT )exp−θ(st+1−1

1+ rΛt+1)+ exp−θ(sT )ω(exp−θ(st+1−

11+ r

Λt+1))

+exp−θ(sT )γQω(exp−θ(st+1 +r

1+ rΛt+1)))]

⇒ Et [exp−θ(st+1−1

1+ rΛt+1)︸︷︷︸

<Et [exp−θ(sT )]

ω(exp−θ(sT )))]

< Et [exp−θ(sT )ω(exp−θ(st+1−1

1+ rΛt+1))︸︷︷︸

>0

+exp−θ(sT )γQω(exp−θ(st+1 +r

1+ rΛt+1))︸︷︷︸

>ω(exp−θ(sT ))

)]

if γQ> 1 because Et−1[Λt ]> 0 can be chosen close to zero via γ , 0 <∂

11+r Λt

∂ sPt

,∂

11+r Λt

∂ sTt

,∂

r1+r Λt

∂ sPt

,∂

r1+r Λt

∂ sTt

< 1, and E[ω(exp−θX)]>0 and increasing in the variance of X (as shown by Koszegi and Rabin (2009) for any random variable and in the proof of Lemma1). γQ > 1 can be shown in the same fashion as rQ > (1+ r)QT−1 and is always true as β can be chosen accordingly. Thus, a γ in[γ, γ] exists as γ < γ .

1.3.6. Proof of Proposition 5. In the deterministic setting, sPt = sT

t = 0 for all t such that the news-utility agent will notexperience actual news utility in a subgame-perfect equilibrium because he cannot fool himself, and, thus, ψt = Qt for all t.Consequently, the expected-utility maximizing path corresponds to the standard agent’s one which is determined in any periodT − i by the following first-order condition

exp−θ(1+ r)AT−i−1 +θAT−i=(1+ r)i

f (i−1)exp−θ

(1+ r)i

f (i−1)AT−iQs

T−i.

If the agent believes he follows the above path then the consistency constraint (increasing consumption is not preferred) has to hold

exp−θ(1+ r)AT−i−1 +θAT−i(1+η)<(1+ r)i

f (i−1)exp−θ

(1+ r)i

f (i−1)AT−iQs

T−i(1+ γηλ ).

Thus, if η < γηλ ⇒ γ > 1λ

the agent follows the expected-utility maximizing path. Whereas for γ ≤ 1λ

news-utility consumption ischaracterized by equality of the consistency constraint, because the agent will choose the lowest consumption level that just satisfiesit. Then, the first-order condition becomes equivalent to a βδ−agent’s first-order condition with b = 1+γηλ

1+η< 1.

1.3.7. Proof of Proposition 6. I say that the news-utility agent’s consumption path features a drop in consumption atretirement, if consumption growth after the start of retirement is negative and smaller than it is at the start of retirement, i.e.,∆CT−R is negative and smaller than ∆CT−R+1. The proof can be best illustrated using the hyperbolic-discounting model, after thestart of retirement the news-utility agent’s consumption growth follows the standard model or the hyperbolic-discounting model.Thus, the news-utility agent’s implied hyperbolic-discount factor after retirement is bR ∈ 1+γηλ

1+η,1, which is larger than the

news-utility agent’s implied hyperbolic-discount factor before retirement. In period T −R−1 the weight on future marginal valueversus current marginal consumption is between 1+γηλ

1+ηλ, 1+γη

1+η and since 1+γηλ

1+ηλ< 1+γη

1+η< 1+γηλ

1+η< 1 the hyperbolic-discount

factor implied by at-retirement consumption growth ∆CT−R is necessarily lower than the hyperbolic-discount factor implied by post-retirement consumption growth ∆CT−R+1. Thus, consumption growth at retirement will necessarily be less than consumption growthafter retirement. Moreover, if consumption growth after retirement is approximately zero or negative, as β < β , then consumptiongrowth at retirement will be negative.

Let me formalize the agent’s consumption growth at and after retirement. After retirement the news-utility agent’sconsumption growth is ∆CT−R+1 =CT−R+1−CT−R =−a(R−1)ΛT−R+1 +ΛT−R and will correspond to a hyperbolic-discountingagent’s consumption with b∈ 1+ηγλ

1+η,1 such that the agent’s continuation utilities in period T−R and T−R+1 (which determine

ΛT−R and ΛT−R+1) correspond to

QT−R = ψT−R = QbT−R and QT−R+1 = ψT−R+1 = Qb

T−R+1


such that

ΛT−R =1θ

log((1+ r)R

f (R−1)bQb

T−R) and ΛT−R+1 =1θ

log((1+ r)R−1

f (R−2)bQb

T−R+1)

thus if β < β then consumption growth after retirement will be approximately zero (if b = 1 and the news-utility agent followsthe standard agent’s path and (1+ r)β ≈ 1) or negative if b < 1 (if the news-utility agent follows a hyperbolic-discounting path or(1+ r)β < 0)). Consumption growth at retirement is ∆CT−R = CT−R−CT−R−1 = −a(R)ΛT−R +ΛT−R−1. ΛT−R will correspondto a hyperbolic-discounting agent’s value with b ∈ 1+ηγλ

1+η,1 as above. But, ΛT−R−1 will correspond to a hyperbolic-discounting

agent’s value with b ∈ 1+γηλ

1+ηλ, 1+γη

1+η and it can be easily seen that 1+γηλ

1+ηλ< 1+γη

1+η< 1+γηλ

1+η< 1. Thus, from above

ΛT−R =1θ

log((1+ r)R

f (R−1)bQb

T−R)

and if the news-utility agent would continue this hyperbolic path implied by the past retirement b ∈ 1+ηγλ

1+η,1 then

ΛbT−R−1 =

1θ

log((1+ r)R+1

f (R)bQb

T−R−1)

whereas in fact his ΛT−R−1 is given by

ΛT−R−1 =1θ

log((1+ r)R+1

f (R)ψT−R−1 + γQT−R−1η(λ − (λ −1)FS(sT−R−1))

1+η(λ − (λ −1)FS(sT−R−1)))

with ψT−R−1 = QT−R−1 = QbT−R−1 because there is no uncertainty from period T −R on. As can be easily seen iff γ < 1 then

ΛT−R−1 < ΛbT−R−1 because

QbT−R−1 + γQb

T−R−1η(λ − (λ −1)FS(sT−R−1))

1+η(λ − (λ −1)FS(sT−R−1))< Qb

T−R−1

for instance, if F(·) = 0.5 then 1+γ12 η(1+λ )

1+ 12 η(1+λ )

< 1 iff γ < 1. Thus, news-utility consumption growth is smaller at retirement than after

retirement.

1.3.8. Proof of Corollary 1. After retirement the news-utility agent’s consumption from period T −R on will correspond toa hyperbolic-discounting agent’s consumption with b ∈ 1+ηγλ

1+η,1 such that the agent’s continuation utilities correspond to

QT−R−1 = ψT−R−1 = QbT−R−1

thus ΛT−R−1 is given by

ΛT−R−1 =1θ

log((1+ r)R+1

f (R)ψT−R−1 + γQT−R−1η(λ − (λ −1)FS(sT−R−1))

1+η(λ − (λ −1)FS(sT−R−1)))

as can be easily seen iff γ < 1 then ∂ΛT−R−1∂ sP

T−R−1> 0 and consumption is excessively smooth and sensitive in the period just before

retirement.

1.3.9. Proofs of the new predictions about consumption (Section 2.2.4). The new predictions can be easily inferred fromthe above considerations.

1 Consumption is more excessively sensitive for permanent than for transitory shocks in an environment with permanentshocks. In an environment with transitory shocks only, however, news-utility consumption is excessively sensitive withrespect to transitory shocks. Λt varies more with the permanent shock than with the transitory shock, because the agent isconsuming only the per-period value r

1+r sTt of the period t transitory shock, such that F t−1

Ct(Ct) varies little with sT

t . However,in the absence of permanent shocks, Λt would vary with FST

t(sT

t ) which fully determines F t−1Ct

(Ct) even though consumptionitself will increase only by the per-period value r

1+r sTt of the transitory shock. Thus, consumption is excessively sensitive

for transitory shocks when permanent shocks are absent. With permanent shocks, however, consumption is excessivelysensitive for transitory shocks only when the horizon is very short or the permanent shock has very little variance such that

the transitory shock actually moves FST−i(sT−i) despite the fact that it is discounted by (1+r)i

f (i) .2 The degree of excess smoothness and sensitivity is decreasing in the amount of economic uncertainty σPt . If σPt is small, the

agent’s beliefs change more quickly relative to the change in the realization of the shock; hence, the consumption functionis more flat for realizations around zero. The consumption function CT−i is less increasing in the realizations of the shocks


sPT−i if ∂ΛT−i

∂ sPT−i

is relatively high. As can be seen easily, ∂ΛT−i∂ sP

T−iis increasing in f P

T−i(sPT−i) which is high if f P

T−i(sPT−i) is very

high at sPT−i = 0 which happens if f P

T−i(sPT−i) is a very narrow distribution, i.e., σPT−i is small.

3 Consumption is more excessively smooth and sensitive when the agent’s horizon increases, because the marginal propensityto consume out of permanent shocks declines when the additional precautionary-savings motive accumulates. ψT−i

QT−iis

increasing in i and approaches a constant ψ

Q when T → ∞ and i→ ∞. Then, the variation in ΛT−i is increasing in i. Andsince consumption growth ∆CT−i is determined by −a(i)ΛT−i +ΛT−i−1 on average the larger variation in ΛT−i translatesinto a higher coefficient in the OLS regression. This can be seen by looking at ∂ΛT−i

∂ sPT−i

, i.e.,

∂a(i)ΛT−i

∂ sPT−i

=a(i)θ

(ψT−i−QT−i)η fST−i(sT−i)(λ−1)


ψT−i +QT−iη(λ − (λ −1)FST−i(sT−i))

> 0.

As a(i) = f (i−1)f (i) is increasing in i because f (i) = ∑

ij=0(1+ r) j and thus (a(i))2

1−a(i) is increasing in i and approaching a constant

and ψT−iQT−i

is increasing in i and approaching a constant it can be easily seen that ∂a(i)ΛT−i∂ sP

T−iis increasing in i which means that

consumption becomes more excessively smooth as the agent’s horizon increases. However, because consumption growth isgiven by ∆CT−i = sT−i−a(i)ΛT−i +ΛT−i−1 for excess sensitivity it only matters that ψT−i−1

QT−i−1is increased which increases

the variation in ΛT−i−1 while it does not matter that a(i) is increased. Thus, consumption becomes more excessively sensitiveas the agent’s horizon increases but not as much more as excessively smooth.

Appendix A.4. Comparison to the agent’s pre-committed equilibrium

In order to assess the preferences’ welfare implications, I briefly explain the consumption implications of the monotone-pre-committed equilibrium. The monotone-personal equilibrium maximizes the agent’s utility in each period t when he takes his beliefsas given. However, if the agent could pre-commit to his consumption in each possible contingency, he would choose a differentconsumption path. I define this path as the model’s “monotone-pre-committed” equilibrium in the spirit of the choice-acclimatingequilibrium concept, as defined by Koszegi and Rabin (2007), but within the outlined environment and admissible consumptionfunction as follows.

Definition 8. The family of admissible consumption functions, Ct = gt(Xt ,Zt ,st) for each period t, is a monotone-pre-committedequilibrium for the news-utility agent, if, in any contingency, Ct = gt(Xt ,Zt ,st) maximizes (2.4) subject to (2.2) and (2.1) underthe assumption that all future consumption corresponds to Ct+τ = gt+τ (Xt+τ ,Zt+τ ,st+τ ). In each period t, the agent’s maximizationproblem determines both his beliefs F t−1

Ct+τT−t

τ=0 and consumption Ct+τT−tτ=0.

I derive the equilibrium consumption function under the premise that the agent can pre-commit to an optimal, history-dependent consumption path for each possible future contingency and thus jointly optimizes over consumption and beliefs. Thisequilibrium is not time consistent because the agent would deviate if he were to take his beliefs as given and optimize overconsumption alone. In the next proposition, I formalize how the consumption implications differ in the monotone-pre-committedequilibrium if one exists. Then, I explain beliefs-based present bias in detail and show how it differs from hyperbolic discounting.

Proposition 7. Comparison to the monotone-pre-committed equilibrium.1. If income uncertainty σPt > 0 for any period t, then the monotone-pre-committed consumption path does not correspond

to the monotone-personal equilibrium consumption path.2. The news-utility agent’s monotone-pre-committed consumption is excessively smooth and sensitive.3. News-utility preferences introduce a first-order precautionary-savings motive in the monotone-pre-committed equilibrium,

monotone-pre-committed consumption is lower CcT−1 < CT−1, and the gap increases in the event of good income realizations

∂ (CcT−1−CT−1)

∂ sPT−1

> 0.

5. The news-utility agent’s monotone-pre-committed consumption path is not necessarily characterized by a hump-shapedconsumption profile and consumption does not drop at retirement.

The proof can be found below. Suppose that the agent can pre-commit to an optimal, history-dependent consumptionpath for each possible future contingency. Then, the agent’s marginal gain-loss utility is no longer solely composed of thesensation of increasing consumption in one particular contingency; additionally, the agent considers that he will experience fewersensations of gains and more feelings of loss in all other contingencies. Thus, marginal gain-loss utility has a second component,−u′(CT−1)(η(1−F t−1

Ct(Ct))+ηλF t−1

Ct(Ct)), which is negative such that the pre-committed agent consumes less. Moreover, this

negative component dominates if the realization is above the median, i.e., F t−1Ct

(Ct) > 0.5. Thus, in the event of good income


realizations, pre-committed marginal gain-loss utility is negative. In contrast, non-pre-committed marginal gain-loss utility is alwayspositive because the agent enjoys the sensation of increasing consumption in any contingency. Therefore, the degree of present biasis reference dependent and less strong in the event of bad income realizations, when increasing consumption is the optimal responseeven on the pre-committed path. Moreover, this negative component implies additional variation in marginal gain-loss utility suchthat pre-committed consumption is more excessively smooth and sensitive. From the above consideration it can be easily inferredthat the optimal pre-committed consumption function in the exponential-utility model is thus given by

ΛcT−i =

1θ

log((1+ r)i

f (i−1)

ψcT−i + γQc

T−iη(λ −1)(1−2FST−i(sT−i))

1+η(λ −1)(1−2FST−i(sT−i))

)

with

QcT−i−1 = ET−i−1[βexp−θ(sT−i−

f (i−1)f (i)

ΛcT−i)+βexp−θ(sT−i +

(1+ r)i

f (i)Λ

cT−i)Qc

T−i]

and

ψcT−i−1 = βET−i−1[exp−θ(sT−i−

f (i−1)f (i)

ΛcT−i)+ω(exp(−θ(sT−i−

f (i−1)f (i)

ΛcT−i))

+γQcT−iω(exp−θ(sT−i +

(1+ r)i

f (i)Λ

cT−i))+

1θ

exp−θ(sT−i +(1+ r)i

f (i)Λ

cT−i)ψc

T−i.

Proof of Proposition 7. In the following, I assume that the following parameter condition (which ensures that the agent’smaximization problem is globally concave) holds η(λ − 1) < 1. All of the following proofs are direct applications of the priorproofs for the monotone-personal equilibrium just using Λc

T−i instead of ΛT−i. Thus, I make the exposition somewhat shorter.

1 The personal and pre-committed consumption functions are different in each period as can be seen in Section Appendix A.1.But, if there’s no uncertainty and γ > 1

λthen the personal and pre-committed consumption functions both correspond to the

standard agent’s consumption function as shown in the proof of Proposition 5.2 Please refer to the derivation of the exponential-utility pre-committed model in Section Appendix A.1 for a detailed

derivation of ΛcT−i. According to Definition 4 consumption is excessively smooth if

∂Cct+1

∂ sPt+1

< 1 and excessively sensitive

if∂∆Cc

t+1∂ sP

t> 0. Consumption growth is

∆CcT−i = sP

T−i +(1−a(i))sTT−i−a(i)Λc

T−i +ΛcT−i−1

so that∂Cc

T−i∂ sP

T−i< 1 iff

∂ΛcT−i

∂ sPT−i

> 0 and∂∆Cc

T−i∂ sP

T−i−1> 0 iff

∂ΛcT−i−1

∂ sPT−i−1

> 0. Since ψcT−i > γQc

T−i it can be easily seen that∂Λc

T−i∂ sP

T−i> 0,

i.e.

∂ΛcT−i

∂ sPT−i

=1θ

(ψT−i−γQT−i)η(λ−1)2 fST−i(sT−i)

1+η(λ−1)(1−2FST−i(sT−i))

ψcT−i + γQc

T−iη(λ −1)(1−2FST−i(sT−i))

> 0.

Thus, optimal pre-committed consumption is excessively smooth and sensitive.3 The first-order condition of the second-to-last period in the exemplified model of the text is

u′(CcT−1) = (1+ r)u′((sP

T−1−CcT−1)(1+ r)+ sP

T−1)ψT−1 + γQT−1η(λ −1)(1−2FP(sP

T−1))

1+η(λ −1)(1−2FP(sPT−1)))

.

By the exact same argument as above ψT−1 > QT−1 such that news utility introduces a first-order precautionary-savingsmotive in the pre-committed equilibrium. Compare the above first-order condition with the one for personal-monotoneconsumption CT−1, i.e.,

u′(CT−1) = (1+ r)u′((sPT−1−CT−1)(1+ r)+ sP

T−1)ψT−1 + γQT−1η(λ − (λ −1)FP(sP

T−1))

1+η(λ − (λ −1)FP(sPT−1))

.

Because η(λ − (λ − 1)FP(sPT−1)) > η(λ − 1)(1 − 2FP(sP

T−1)) for all sPT−1 and ψT−1 > γQT−1 monotone personal

consumption is higher CT−1 >CcT−1 than pre-committed consumption. Moreover, the difference η(λ − (λ −1)FP(sP

T−1))−η(λ −1)(1−2FP(sP

T−1)) = η(1−FP(sPT−1))+ηλFP(sP

T−1) is increasing in sPT−1 such that the difference in consumption

CT−1−CcT−1 is increasing in sP

T−1.4 Consider the pre-committed first-order conditions before and after retirement. After retirement the news-utility agent’s

consumption from period T −R on will correspond to the standard agent’s one, i.e.,

QT−R−1 = ψT−R−1 = QsT−R−1


thus ΛT−R−1 is given by

ΛT−R−1 =1θ

log((1+ r)R+1

f (R)ψT−R−1 + γQT−R−1η(λ −1)(1−2FS(sT−R−1))

1+η(λ −1)(1−2FS(sT−R−1)))

it can be easily seen that∂ΛT−R−1

∂γ< 0 only if FS(sT−R−1)< 0.5.

Thus, γ < 1 does not necessarily increase or decrease ΛT−R−1 and thereby consumption growth at retirement is notnecessarily negative and smaller than consumption growth after retirement. There is no systematic underweighting ofmarginal utility before or after retirement, and a drop in consumption at retirement for γ < 1 does not occur. The sameargument that γ < 1 does not necessarily lead to a reduction in consumption growth even in the end of life when ψT−i andQT−i are small implies that the pre-committed consumption path is not necessarily hump shaped.

APPENDIX B. DERIVATION AND ESTIMATION OF THE POWER-UTILITY MODEL

Appendix B.1. Summary of utility functions under consideration

I briefly summarize the lifetime utility of all preference specifications that I consider. I define the “news-utility” agent’s lifetimeutility in each period t = 0, ...,T as

u(Ct)+n(Ct ,F t−1Ct

)+ γ ∑T−tτ=1 β



τUt+τ ]

with β ∈ [0,1], η ∈ (0,∞), λ ∈ (1,∞), and γ ∈ [0,1]. Additionally, I first consider standard preferences as analyzed by Carroll(2001), Gourinchas and Parker (2002), and Deaton (1991) among many others. The “standard” agent’s lifetime utility is given by

u(Cst )+Et [∑

T−tτ=1 β

τ u(Cst+τ )].

Second, I consider βδ− or hyperbolic-discounting preferences as developed by Laibson (1997). The “βδ−” or “hyperbolic-discounting” agent’s lifetime utility is given by

u(Cbt )+bEt [∑

T−tτ=1 β

τ u(Cbt+τ )]

with b ∈ [0,1] corresponding to the βδ−agent’s β .Third, I consider internal, multiplicative habit-formation preferences as assumed in Michaelides (2002). The “habit-forming”

agent’s lifetime utility is given by

u(Cht ,Ht)+Et [∑

T−tτ=1 β

τ u(Cht+τ ,Ht+τ )]

with u(Ct ,Ht) =( Ct

Hht)1−θ

1−θand Ht = Ht−1 +ϑ(Ct−1−Ht−1). I assume ϑ = 1 such that Ht =Ct−1with h ∈ [0,1].

Fourth, I consider temptation-disutility preferences as developed by Gul and Pesendorfer (2004) following the specificationof Bucciol (2012). The “tempted” agent’s lifetime utility is given by

u(Ctdt )−λ

td(u(Ctdt )−u(Ctd

t ))+Et [∑T−tτ=1 β

τ (u(Ctdt+τ )−λ

td(u(Ctdt+τ )−u(Ctd

t+τ )))]

with Ctdt being the most tempting alternative consumption level and λ td ∈ [0,∞).

In the following, I outline the numerical derivation of the model with a power-utility specification u(Ct) =C1−θ

t1−θ

. I start withthe standard model to then explain the news-utility model in detail and finish with hyperbolic-discounting, habit-formation, andtemptation-disutility preferences. In turn, I will give details on the estimation procedure.

Appendix B.2. The standard model

The standard agent’s maximization problem in any period t = T − i is

maxu(CT−i)+∑iτ=1 β

τ ET−i[u(CT−i+τ )]

subject to Xt = (Xt−1−Ct−1)R+Yt and Yt = Pt−1Gt esPt esT

t = Pt esTt .

I can normalize the maximization problem by P1−θ

T−i and then becomes in normalized terms (Xt = Pt xt for instance)

maxu(cT−i)+∑iτ=1 β

τ ET−i[∏τ

j=1(GT−i+ jesPT−i+ j )1−θ u(cT−i+τ )]


subject to xt = (xt−1− ct−1)R

Gt esPt+ yt and yt = esT

t .

I can solve the model by numerical backward induction as done by Gourinchas and Parker (2002) and Carroll (2001). The standardagent’s first-order condition is

u′(cT−i) = Φ′T−i = βRET−i[

∂cT−i+t

∂xT−i+1(GT−i+1esP

T−i+1 )−θ u′(cT−i+1)+(1− ∂cT−i+1

∂xT−i+1)(GT−i+1esP

T−i+1 )−θΦ′T−i+1]

which shows his continuation value Φ′T−i where it can be seen that

P−θ

T−iΦ′T−i = ET−i[∑

iτ=1 β

τ ∂CT−i+τ

∂AT−iu′(CT−i+τ )]

= ET−i[β∂XT−i+1

∂AT−i

∂CT−i+1

∂XT−i+1u′(CT−i+1)+∑

i−1τ=1 β

τ+1 ∂XT−i+1

∂AT−i

∂CT−i+1+τ

∂XT−i+1u′(CT−i+1+τ )]

= βET−i[∂XT−i+1

∂AT−i

∂CT−i+1

∂XT−i+1u′(CT−i+1)+∑

i−1τ=1 β

τ ∂XT−i+1

∂AT−i

∂XT−i+2

∂XT−i+1

∂XT−i+1

∂XT−i+2

∂CT−i+1+τ

∂XT−i+1u′(CT−i+1+τ )]

= βET−i[R∂CT−i+1

∂XT−i+1u′(CT−i+1)+∑

i−1τ=1 β

τ ∂XT−i+1

∂AT−i

∂XT−i+2

∂XT−i+1

∂CT−i+1+τ

∂XT−i+2u′(CT−i+1+τ )]


∂XT−i+1u′(CT−i+1)+∑

i−1τ=1 β

τ R∂XT−i+2

∂XT−i+1

∂CT−i+1+τ

∂XT−i+2u′(CT−i+1+τ )]


∂XT−i+1u′(CT−i+1)+R

∂AT−i+1

∂XT−i+1ET−i+1[∑

i−1τ=1 β

τ ∂CT−i+1+τ

∂AT−i+1u′(CT−i+1+τ )]︸︷︷︸

P−θ

T−i+1Φ′T−i+1

]

= βRET−i[∂u(CT−i+1)

∂XT−i+1+(1− ∂CT−i+1

∂XT−i+1)P−θ

T−i+1Φ′T−i+1].

Φ′T−i = βRET−i[

∂cT−i+1

∂xT−i+1(

1PT−i

)−θ u′(PT−i+1cT−i+1)+(1− ∂cT−i+1

∂xT−i+1)(GT−i+1eSP

T−i+1 )−θΦ′T−i+1]

= βRET−i[∂cT−i+1

∂xT−i+1(GT−i+1eSP

T−i+1 )−θ u′(cT−i+1)+(1− ∂cT−i+1

∂xT−i+1)(GT−i+1eSP

T−i+1 )−θΦ′T−i+1]

Φ′T−i is a function of savings aT−i thus I can solve for each optimal consumption level c∗T−i on a grid of savings aT−i as

c∗T−i = (Φ′T−i−1)

− 1θ = ( f Φ

′(aT−i))

− 1θ to then find each optimal level of consumption for each value of the normalized cash-

on-hand grid xT−i by interpolation. This endogenous-grid method has been developed by Carroll (2001). Alternatively, I could usethe Euler equation instead of the agent’s continuation value but this solution illustrates the upcoming solution of the news-utilitymodel of which it is a simple case.

Appendix B.3. The news-utility model

The monotone-personal equilibrium in the second-to-last period. Before starting with the fully-fledged problem,I outline the second-to-last period for the case of power utility. In the second-to-last period the agent allocates his cash-on-handXT−1 between contemporaneous consumption CT−1 and future consumption CT , knowing that in the last period he will consumewhatever he saved in addition to last period’s income shock CT = XT = (XT−1−CT−1)R+YT . According to the monotone-personalequilibrium solution concept, in period T −1 the agent takes the beliefs about contemporaneous and future consumption he enteredthe period with FT−2

CT−1,FT−2

CT as given and maximizes

u(CT−1)+n(CT−1,FT−2CT−1

)+ γβnnn(FT−1,T−2CT

)+βET−1[u(CT )+n(CT ,FT−1CT

)]

which can be rewritten as

u(CT−1)+η

ˆ CT−1

−∞

(u(CT−1)−u(c))dFT−2CT−1

(c)+ηλ

ˆ∞

CT−1

(u(CT−1)−u(c))dFT−2CT−1

(c)

+γβ

ˆ∞

−∞

ˆ∞

−∞

(u(c)−u(r))dFT−1,T−2CT

(c,r)+βET−1[u(CT )+η(λ −1)ˆ

∞

CT

(u(CT )−u(c))dFT−1CT

(c)].


To gain intuition for the model’s predictions, I explain the derivation of the first-order condition

u′(CT−1)(1+η(λ − (λ −1)FT−2CT−1

(CT−1))) = γβRET−1[u′(CT )]η(λ − (λ −1)FT−2AT−1

(AT−1))

+βRET−1[u′(CT )+η(λ −1)ˆ

∞

CT

(u′(CT )−u′(c))dFT−1CT

(c)].

The first two terms in the first-order condition represent marginal consumption utility and gain-loss utility over contemporaneousconsumption in period T − 1. As the agent takes his beliefs FT−2

CT−1,FT−2

CT as given in the optimization, I apply Leibniz’s rule for

differentiation under the integral sign. This results in marginal gain-loss utility being the sum of states that would have promisedless consumption FT−2

CT−1(CT−1), weighted by η , or more consumption 1−FT−2

CT−1(CT−1), weighted by ηλ ,

∂n(CT−1,FT−2CT−1

)

∂CT−1= u′(CT−1)η(λ − (λ −1)FT−2

CT−1(CT−1)).

Note that, if contemporaneous consumption is increasing in the realization of cash-on-hand then I can simplify FT−2CT−1

(CT−1) =

FT−2XT−1

(XT−1). Returning to the maximization problem the third term represents prospective gain-loss utility over future consumptionCT experienced in T −1. As before, marginal gain-loss utility is given by the weighted sum of states γβRET−1[u′(CT )]η(λ − (λ −1)FT−2

AT−1(AT−1)). Note that FT−2

CT(c) is defined as the probability Pr(CT < c|IT−2) and

Pr(CT < c|IT−2) = Pr(AT−1R+YT < c|IT−2) = Pr(AT−1 <c−YT

R|IT−2).

Thus, if savings and therefore future consumption are increasing in the realization of cash-on-hand, then I can simplifyFT−2

AT−1(AT−1) = FT−2

XT−1(XT−1).

The last term in the maximization problem represents consumption and gain-loss utility over future consumption CT in thelast period T , i.e., the first derivative of the agent’s continuation value with respect to consumption or the marginal value of savings.Expected marginal gain-loss utility η(λ − 1)

´∞

CT(u′(CT )− u′(c))dFT−1

CT(c) is positive for any concave utility function such that

Ψ′T−1 = βRET−1[u′(CT )+η(λ − 1)

´∞

CT(u′(CT )− u′(c))dFT−1

CT(c)] > βRET−1[u′(CT )] = Φ

′T−1. As expected marginal gain-loss

disutility is positive, increasing in σY , absent if σY = 0, and increases the marginal value of savings, I say that news-utility introducesan “additional precautionary-savings motive.” The first-order condition can now be rewritten as

u′(CT−1) =Ψ′T−1 + γΦ

′T−1η(λ − (λ −1)FT−2

XT−1(XT−1))

1+η(λ − (λ −1)FT−2XT−1

(XT−1)).

Beyond the additional precautionary-savings motive Ψ′T−1 > Φ

′T−1 implies that an increase in FT−2

XT−1(XT−1) decreases

Ψ′T−1

Φ′T−1

+ γη(λ − (λ −1)FT−2XT−1

(XT−1))

1+η(λ − (λ −1)FT−2XT−1

(XT−1)),

i.e., the terms in the first-order condition vary with the income realization XT−1 so that consumption is excessively smooth andsensitive.

The news-utility monotone-personal equilibrium path in all prior periods. The news-utility agent’s maximizationproblem in any period T − i is given by

u(CT−i)+n(CT−i,FT−i−1CT−i

)+ γ ∑iτ=1 β

τnnn(FT−i,T−i−1CT−i+τ

)+∑iτ=1 β

τ ET−i[U(CT−i+τ )].

Again, I can normalize maximization problem by P1−θ

T−i as all terms are proportional to consumption utility u(·). In normalizedterms, the news-utility agent’s first-order condition in any period T − i is given by

u′(cT−i) =Ψ′T−i + γΦ

′T−iη(λ − (λ −1)FT−i−1

cT−i(cT−i))

1+η(λ − (λ −1)FT−i−1aT−i (aT−i))

I solve for each optimal value of c∗T−i for a grid of savings aT−i, as Ψ′T−i and Φ

′T−i are functions of aT−i until I find a fixed point

of c∗T−i, aT−i, FT−i−1aT−i

(aT−i), and FT−i−1cT−i

(cT−i). I can infer the latter two from the observation that each cT−i +aT−i = xT−i has acertain probability given the value of savings aT−i−1 I am currently iterating on. However, this probability varies with the realizationof permanent income GT−iesP

T−i ; thus, I cannot fully normalize the problem but have to find the right consumption grid for each


value of GT−iesPT−i rather than just one. The first-order condition can be slightly modified as follows

u′(GT−iesPT−i cT−i) =

(GT−iesPT−i )−θ Ψ

′T−i + γ(GT−iesP

T−i )−θ Φ′T−iη(λ − (λ −1)FT−i−1

cT−i(cT−i))

1+η(λ − (λ −1)FT−i−1aT−i (aT−i))

to find each corresponding grid value. Note that, the resulting two-dimensional grid for cT−i will be the normalized grid foreach realization of sT

t and sPt , because I multiply both sides of the first-order conditions with (GT−iesP

T−i )−θ . Thus, the agent’sconsumption utility continuation value is

Φ′T−i−1 = βRET−i−1[

∂cT−i

∂xT−i(GT−ieSP

T−i )−θ u′(cT−i)+(1− ∂cT−i

∂xT−i)(GT−ieSP

T−i )−θΦ′T−i].

The agent’s news-utility continuation value is given by

P−θ

T−i−1Ψ′T−i−1 = βRET−i−1[

dCT−i

dXT−iu′(CT−i)+η(λ −1)CT−i<CT−i−1

T−i

ˆ(

dCT−i

dXT−iu′(CT−i)− x)dFT−i−1

dCT−idXT−i

u′(CT−i)(x)

+γη(λ −1)AT−i<AT−i−1T−i

ˆ(

dAT−i

dXT−iP−θ

T−iΦ′T−i− x)dFT−i−1

dAT−idXT−i

P−θT−iΦ

′T−i

(x)+(1− dCT−i

dXT−i)P−θ

T−iΨ′T−i]

(here, CT−i<CT−i−1T−i

´means the integral over the loss domain) or in normalized terms

Ψ′T−i−1 = βRET−i−1[

dcT−i

dxT−iu′(cT−i)(GT−iesP

T−i )−θ

+η(λ −1)CT−i<CT−i−1T−i

ˆ(

dcT−i

dxT−iu′(cT−i)(GT−ieSP

T−i )−θ − x)dFT−i−1dcT−idxT−i

u′(cT−i)(GT−ieSPT−i )−θ

(x)

+γη(λ −1)AT−i<AT−i−1T−i

ˆ(

daT−i

dxT−iΦ′T−i(GT−ieSP

T−i )−θ − x)dFT−i−1daT−idxT−i

Φ′T−i(GT−ie

SPT−i )−θ

(x)+(1− dcT−i

dxT−i)(GT−ieSP

T−i )−θΨ′T−i].

Appendix B.4. The hyperbolic-discounting model

I consider an agent with hyperbolic-discounting preferences with the hyperbolic-discounting parameter denoted by γ . The agent’smaximization problem in any period T − i is

maxu(CT−i)+ γ ∑iτ=1 β

τ ET−i[u(CT−i+τ )].

I can normalize the maximization problem by P1−θ

T−i as for the standard agent. In turn, I can solve the model by numerical backwardinduction (as Laibson et al. (2012)) and the first-order condition is

u′(cT−i) = γΦ′T−i = γβRET−i[

∂cT−i+τ

∂xT−i+1(GT−i+1esP

T−i+1 )−θ u′(cT−i+1)+(1− ∂cT−i+1

∂xT−i+1)(GT−i+1esP

T−i+1 )−θΦ′T−i+1].

Appendix B.5. The habit-formation model

Consider an agent with internal, multiplicative habit formation preferences u(Ct ,Ht) =( Ct

Hht)1−θ

1−θwith Ht = Ht−1 +ϑ(Ct−1−Ht−1)

and ϑ ∈ [0,1] (Michaelides (2002)). Assume ϑ = 1 such that Ht = Ct−1. For illustration, in the second-to-last period hismaximization problem is

u(CT−1,HT−1)+βET−1[u(R(XT−1−CT−1)+YT ,HT )]

=(

CT−1Hh

T−1)1−θ

1−θ+βET−1[

11−θ

(R(XT−1−CT−1)+YT

HhT

)1−θ ]

which can be normalized by P(1−θ)(1−h)T−1 (then CT−1 = PT−1cT−1 for instance), and the maximization problem becomes

P(1−θ)(1−h)T−1 (

cT−1hh

T−1)1−θ

1−θ+βP(1−θ)(1−h)

T−1 ET−1[1

1−θ(GT esP

T )(1−θ)(1−h)(

(xT−1− cT−1)R

GT esPT+ yT

hhT

)1−θ ]


which results in the following first-order condition

c−θ

T−1 = h−θh+hT−1 βET−1[(GT esP

T )−h−θ(1−h)(hT )−h(R+h

cT

hT)(

cT

hhT)−θ ] = h−θh+h

T−1 Φ′T−1

with Φ′T−1 being a function of savings xT−1− cT−1 and habit hT . The first-order condition can be solved robustly by iterating on

a grid of savings aT−1 with c∗T−1 = (h−θh+hT−1 Φ

′T−1)

1−θ = (h−θh+h

T−1 f Φ′ (aT−1,hT ))1−θ and hT = c∗T−1

1

GT esPT

until a fixed point of

consumption and habit has been found. The normalized habit-forming agent’s first-order condition in any period T − i is given by

c−θ

T−i = h−θh+hT−i Φ

′T−i = h−θh+h

T−i βET−i[(GT−i+1eSPT−i+1 )−h−θ(1−h)h−h

T−i+1(RdcT−i+1

dxT−i+1+h

cT−i+1

hT−i+1)(

cT−i+1

hhT−i+1

)−θ

+(1− dcT−i+1

dxT−i+1)R(GT−i+1eSP

T−i+1 )−h−θ(1−h)Φ′T−i+2].

Appendix B.6. The temptation-disutility model

Consider an agent with temptation-disutility preferences as developed by Gul and Pesendorfer (2004) following the specification ofBucciol (2012). The “tempted” agent’s lifetime utility is given by

u(Ct)−λtd(u(Ct)−u(Ct))+Et [∑

T−tτ=1 β

τ (u(Ct+τ )−λtd(u(Ct+τ )−u(Ct+τ )))]

with Ct being the most tempting alternative consumption level and λ td ∈ [0,∞). Note that, in a life-cycle model context the mosttempting alternative is to consume the entire cash-on-hand. However, not more as borrowing could be infinitely painful with powerutility and a chance of zero income in all future periods. For illustration, in the second-to-last period the agent’s maximizationproblem is

u(CT−1)−λtd(u(XT−1)−u(CT−1))+βET−1[u(R(XT−1−CT−1)+YT )]

which can be normalized by P(1−θ)T−1 (then CT = PT cT for instance) and the maximization problem becomes

(PT−1)1−θ (u(cT−1)−λ

td(u(xT−1)−u(cT−1)))+(PT−1)1−θ

βET−1[(GT esPT )1−θ u(

R

GT esPT(xT−1− cT−1)+ yT )]

which results in the following first-order condition

u′(cT−1) =1

1+λ td βET−1[(GT esPT )−θ Ru′(

R

GT esPT(xT−1− cT−1)+ yT )]

with Φ′T−1 being a function of savings xT−1− cT−1. The first-order condition can be solved very robustly by iterating on a grid of

savings aT−1 assuming c∗T−1 = (Φ′T−1)

− 1θ = ( f Φ′ (aT−1))

− 1θ . The normalized agent’s first-order condition in any period T − i is

given by

c−θ

T−i =1

1+λ td βET−i[(GT−i+1esPT−i+1 )−θ R

dcT−i+1

dxT−i+1u′(cT−i+1)]

+(1− dcT−i+1

dxT−i+1)(GT−i+1eSP

T−i+1 )−θΦ′T−1].

Appendix B.7. Details of the estimation procedure

I use data from the Consumer Expenditure Survey (CEX) for 1980 to 2002 (from the NBER).19 The CEX is conducted bythe Bureau of Labor Statistics and surveys a large sample of the US population to collect data on consumption expenditures,demographics, income, and assets. As Harris and Sabelhaus (2001) suggest, consumption expenditures consist of food, tobacco,alcohol, amusement, clothing, personal care, housing, house operations (such as furniture and housesupplies), personal business,transportation such as autos and gas, recreational activities such as books and recreational sports, and charity expenditures.Alternately, I could consider non-durable or discretionary consumption only. Income consists of wages, business income, farmincome, rents, dividends, interest, pension, social security, supplemental security, unemployment benefits, worker’s compensation,public assistance, foodstamps, and scholarships. The data is deflated to 1984 dollars.

19. This data set extraction effort is initiated by John Sabelhaus and continued by Ed Harris, both of the Congressional BudgetOffice. The data set links the four quarterly interviews for each respondent household and collapses all the spending, income, andwealth categories into a consistent set of categories across all years under consideration.


Because the CEX does not survey households consecutively, I generate a pseudo panel that averages each household’sconsumption and income at each age. I only consider non-student households that meet the BLS complete income reporterrequirement and complete all four quarterly interviews. Furthermore, I only consider households older than 25 years; retired afterage 68 (the average retirement age in the US according to the OECD); and younger than 78 (the average life expectancy in the USaccording to the UN list).

To control for cohort, family size, and time effects, I employ average cohort techniques (see, e.g., Verbeek (2007), Attanasio(1998), and Deaton (1985)). More precisely, as I lack access to the micro consumption data for each household i at each age a,I pool all observations and estimate log(Ci,a) = ξ0 +αa + γc + fs +X

′i,aβ ia + εi,a. Here, ξ0 is a constant, αa is a full set of age

dummies, γc is a full set of cohort dummies, and fs is a full set of family size and number of earners dummies. Essentially, thesesets of dummies allow me to consider the sample means of my repeated cross-section C∗c,a = E[log(Ci,a)|c,a] and X∗c,a = E[X∗i,a|c,a]for each cohort c at age a. Using the sample means brings about an errors-in-variables problem, which, however, does not appearto make a difference in practice as the sample size of each cohort-age cell is large. Because age αa and cohort γc effects are notseparately identifiable from time effects, I proxy time effects by including the regional unemployment rate as an additional variablein X∗i,a beyond a dummy for retirement following Gourinchas and Parker (2002). As an alternative to a full set of dummies, Iuse fifth-order polynomials in order to obtain a smooth consumption profile. After running the pooled regression, I back out theconsumption data uncontaminated by cohort, time, family size, and number of earners effects and construct the average empiricallife-cycle profile by averaging the data across households at each age. The same exercise is done for income. Figure 3 displays theaverage empirical income and average empirical consumption profiles.

I structurally estimate the news-utility parameters using a methods-of-simulated-moments procedure following Gourinchasand Parker (2002), Laibson et al. (2012), and Bucciol (2012). The procedure has two stages. In the first stage, I estimate the structuralparameters governing the environment Ξ= (µP,σP,µT ,σT , p,G,r,a0,R,T ) using standard techniques and obtain results perfectly inline with the literature. Given the first-stage estimates Ξ, in the second stage, I estimate the preference parameters θθθ = (η ,λ ,γ,β ,θ)

by matching the simulated and empirical average life-cycle consumption profiles. The empirical life-cycle consumption profile isthe average consumption at each age a∈ [1,T ] across all household observations i. More precisely, it is lnCa =

1na

∑nai=1 ln(Ci,a) with

ln(Ci,a) being the household i’s log consumption at age a of which na are observed. The theoretical population analogue to lnCa isdenoted by lnCa(θ ,Ξ) and the simulated approximation is denoted by lnCa(θθθ ,Ξ). Moreover, I define g(θθθ ,Ξ) = lnCa(θθθ ,Ξ)− lnCaand g(θθθ ,Ξ) = lnCa(θθθ ,Ξ)− lnCa. In turn, if the estimated preference and environment parameter vectors θθθ 0 and Ξ0 are the truevectors, the procedure’s moment conditions imply that E[g(θ0,Ξ0)] = E[lnCa(θ ,Ξ)− lnCa] = 0. In turn, let W denote a positivedefinite weighting matrix then q(θθθ ,Ξ) = g(θθθ ,Ξ)W−1g(θθθ ,Ξ)′ is the weighted sum of squared deviations of the simulated fromtheir corresponding empirical moments. I assume that W is a robust weighting matrix rather than the optimal weighting matrixto avoid small-sample bias. More precisely, I assume that W corresponds to the inverse of the variance-covariance matrix of eachpoint of the average log consumption at age a lnCa, which I denote by Ω−1

g and consistently estimate from the sample data. Takingthe first-stage estimates Ξ as given, I minimize the weighted sum of squared deviations of the simulated from their correspondingempirical moments q(θθθ , Ξ) with respect to the vector of preference parameters θθθ to obtain θθθ the consistent estimator of θθθ that isasymptotically normally distributed with standard errors

Ωθ = (G′θWGθ )

−1G′θW [Ωg +Ω

sg +GΞΩΞG

′Ξ]WGθ (G

′θWGθ )

−1.

Here, Gθ and GΞ denote the derivatives of the moment functions ∂g(θθθ 0 ,Ξ0)∂θθθ

and ∂g(θθθ 0,Ξ0)∂Ξ

, Ωg denotes the variance-covariancematrix of the second-stage moments as above that corresponds to E[g(θθθ 0,Ξ0)g(θθθ 0,Ξ0)

′], and Ωsg = na

nsΩg denotes the sample

correction with ns being the number of simulated observations at each age a. I can estimate ΩΞ directly and consistently fromsample data. For the minimization, I employ a Nelder-Mead algorithm. For the standard errors, I numerically estimate the gradientof the moment function at its optimum. If I omit the first-stage correction, the expression is Ωθ = (G

′θ

Ω−1g Gθ )

−1. Alternatively, Ican obtain the standard errors by bootstrapping to account for simulation uncertainty, which yields similar values. Finally, I can testfor overidentification by comparing the minimized weighted sum of squared deviations of the simulated from their correspondingempirical moments g(θ , Ξ) to a chi-squared distribution with T −5 degrees of freedom.

Theoretically, the functional form of the function determining optimal consumption Λt , or the agent’s first-order conditionin the power-utility model, implies that news utility introduces such specific variation in consumption growth that all preferenceparameters are identified in the finite-horizon model because the Jacobian has full rank.20 Roughly speaking, the shape of theconsumption profile identifies the exponential discounting parameter β and the coefficient of risk aversion θ . Because consumptiontracks income too closely and peaks too early in the standard model, the weight of gain-loss versus consumption utility being largerthan zero η > 0 and the coefficient of loss aversion being larger than one λ > 1 can be identified. Finally, the drop in consumptionat retirement identifies the prospective gain-loss discount factor γ < 1. More precisely, as explained in Appendix Appendix B.3, the

agent’s consumption is determined by the following first-order condition u′(cT−i) =Ψ′T−i+γΦ

′T−i(ηFT−i−1

cT−i(cT−i)+ηλ (1−FT−i−1

cT−i(cT−i)))

1+ηFT−i−1aT−i (aT−i)+ηλ (1−FT−i−1

aT−i (aT−i))

20. Numerically, I confirm this result in a Monte Carlo simulation and estimation exercise. Moreover, because previousstudies cannot separately identify the weight of gain-loss versus consumption utility η and the coefficient of loss aversion λ , Iconfirm that I obtain similar estimates when I assume η = 1 and only estimate the other parameters.


TABLE 3RISK ATTITUDES OVER SMALL AND LARGE WEALTH GAMBLES

standard news-utilityLoss (L) contemp. prospective

10 10 20 18200 200 401 365

1000 1000 2020 18425000 5097 10526 9306850000 52414 194870 96633

100000 110900 2955800 205350

For each loss L, the table’s entries show the required gain G to make each agent indifferent betweenaccepting and rejecting a 50-50 gamble win G or lose L at a wealth level of 300,000 and a permanent

income of 100,000 (power-utility model).

of which I observe the inverse and log average of all households. Φ′T−i represents future marginal consumption utility, as in

the standard model, and is determined by the exponential discounting parameter β and the coefficient of risk aversion θ , whichcan be separately identified in a finite-horizon model. Ψ

′T−i represents future marginal consumption and news utility and is

thus determined by something akin of weight of gain-loss versus consumption utility times the coefficient of loss aversionη(λ − 1). ηFT−i−1

aT−i(aT−i) +ηλ (1−FT−i−1

aT−i(aT−i)) and ηFT−i−1

cT−i(cT−i) +ηλ (1−FT−i−1

cT−i(cT−i)) represents the weighted sum

of the cumulative distribution function of savings, aT−i, and consumption, cT−i, of which merely the average determined byη0.5(1 + λ ) is observed. Thus, I have two equations in two unknowns and can separately identify weight of gain-loss versusconsumption utility η and the coefficient of loss aversion λ . Finally, the prospective gain-loss discount factor γ enters the first-ordercondition distinctly from all other parameters.

Appendix B.8. Risk attitudes over small and large stakes

First, I suppose the agent is offered a gamble about immediate consumption in period t after that period’s uncertainty has resolvedand that period’s original consumption has taken place. I assume that utility over immediate consumption is linear. Then, the agentis indifferent between accepting or rejecting a 50-50 win G or lose L gamble if 0.5G− 0.5L+ 0.5ηG− 0.5ηλL = 0. Second, Isuppose the agent is offered a monetary gamble or wealth bet that concerns future consumption. Suppose T → ∞. I assume that hisinitial wealth level is At = 300,000 and Pt = 100,000. Let f Ψ(A) and f Φ(A) be the agent’s continuation value as a function of theagent’s savings At . Then, the agent is indifferent between accepting or rejecting a 50-50 win G or lose L gamble if

0.5γη( f Φ(At +G)− f Φ(At))+0.5γηλ ( f Φ(At −L)− f Φ(At))+0.5 f Ψ(At +G)+0.5 f Ψ(At −L) = f Ψ(At).

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