Consumption Smoothing and Precautionary Saving under Recursive Preferences

. 

Consumption Smoothing and Precautionary 

Saving under Recursive Preferences 

AJ A. Bostian 

Christoph Heinzel 

FOODSECURE Working paper no. 44 

May 2016 

INTERDISCIPLINARY RESEARCH PROJECT 

TO EXPLORE THE FUTURE OF GLOBAL 

FOOD AND NUTRITION SECURITY

Consumption Smoothing and Precautionary Saving 

under Recursive Preferences 

AJ A. Bostian 

Christoph Heinzel ∗ 

This Version: November 30, 2015 

Abstract 

Intertemporal choices simultaneously activate discounting, risk aversion, and intertemporal 

substitution. Future risk stimulates, more specially, higher-order aspects 

of preference. While a vast empirical literature has studied discounting, risk preferences, 

and basic consumption smoothing, empirical knowledge of higher-order preferences 

is still scarce. Based on a two-period consumption/saving model, we investigate 

the interaction of risk and time preferences in intertemporal decisions with future risk. 

We show that the main carriers of saving variation in intertemporal decisions under 

risk, according to the model, are intertemporal preferences. Risk preferences only 

play a minor role. The predictions under Expected Utility (EU) resemble those of the 

intertemporal-substitution component of recursive utility, not the risk component. Our 

simulations also show that second- and third-derivative effects are the most essential 

features of preferences in the decisions in question. Effects already from the fourth 

order on have essentially no impact. While the risk effects under EU are stronger than 

under recursive preferences, the few-relevance result regarding third- and higher-order 

risk effects persists. For a deepened understanding of preferences underlying intertemporal 

choice, correctly identifying intertemporal preference seems to be the single most 

critical aspect. The quantitative differences in the predictions for EU and recursive 

preferences may allow to empirically discriminate between the preference concepts. 

Keywords: Intertemporal choice, prudence, precautionary saving, recursive preferences, 

higher-order risk 

JEL classification: D91, C90, D81 

∗ Bostian: Fulbright Scholar, University of Tampere, Finland. Heinzel: INRA, UMR1302 SMART, F- 

35000Rennes, France. Severalcolleagueshaveprovidedvaluablecommentsonanearlierversionofthispaper: 

Alain Carpentier, Glenn Harrison, Olivier l’Haridon, Kaisa Kotakorpi, Béatrice Rey, Harris Schlesinger, as 

wellasparticipantsattheCEAR/MRICWorkshoponBehavioralInsuranceIV,theWorkshoponUncertainty 

and Public Decision-Making at Paris-Nanterre, the 2014 FUR conference, the 2015 meetings of the Finnish 

and the French Economic Associations, and seminars in Munich and Rennes. The usual disclaimer applies. 

The research leading to these results has received funding from the European Union’s Seventh Framework 

Programme FP7/2007-2011 under Grant Agreement no. 290693 FOODSECURE. 

1

1 Introduction 

Intertemporal choices under risk activate three preference facets. Discounting addresses the 

relativeutilityvaluationofcurrentversusfutureprospects, withimpatienceimputinggreater 

value to current payoffs. Risk aversion addresses the desire to avoid risky situations and 

seek out safe ones, with the compensation for taking on risk coming from a risk premium. 

Intertemporal substitution describes the willingness to postpone current consumption for 

futureconsumption. Futureriskinsuchdecisionsmakes, morespecially, higher-orderaspects 

ofpreferencesoperative. Thus,apositivethirdutilityderivative,orprudence(Kimball1990), 

induces precautionary saving, defined as the additional saving that arises when a zero-mean 

risk is added to a future aspect. Systematically, Eeckhoudt and Schlesinger (2008) derive 

the conditions on Expected Utility (EU) for saving increases in response to higher-order 

changes of income or return risk. EU, however, is often criticized for confounding risk and 

intertemporal preferences in temporal choices. 1 Conceding this criticism, Kimball and Weil 

(2009) study prudence using recursive Selden (1978)/Kreps and Porteus (1978) preferences 

for the case of income risk, showing how utility derivatives up to the third in both the risk 

and the time domain are involved in the intertemporal risk response. 

While a vast empirical literature has studied discounting, risk preferences, and basic 

consumption smoothing, 2 empirical knowledge of higher-order preferences is still scarce. 3 In 

fact,individualpreferencesofhigherorderhaveneverbeentackledempiricallyinaframework 

that involves the risk and time dimensions simultaneously. The importance of the different 

1 Typical findings in empirical macroeconomics, starting with Epstein and Zin (1991), suggest that risk 

aversion and the elasticity of intertemporal substitution are different ideas. Recent experiments back this 

result (Coble and Lusk 2010, Andreoni and Sprenger 2012a, Abdellaoui et al. 2013, Miao and Zhong 2014). 

2 A main focus of experimental preference elicitation has been on (static) risk attitudes. Intertemporal 

experimental studies usually concentrate on discounting or EU, precluding a distinct role for consumption 

smoothing (e.g., Andersen et al. 2008, Dohmen et al. 2010, Tanaka et al. 2010). A notable exception that 

treats aspects of consumption smoothing is Andreoni and Sprenger (2012a,b). 

3 An early macroeconomic literature studied prudence under EU with diverse methodologies and datasets, 

results were inconclusive (Kuehlwein 1991, Dynan 1993, Merrigan and Normandin 1996, Eisenhauer 2000, 

Ventura and Eisenhauer 2006, Lee and Sawada 2007). All experiments on higher-order preferences have 

focused on static risk preferences thus far (e.g., Deck and Schlesinger 2010, 2014, Ebert and Wiesen 2011, 

2014, Noussair et al. 2014). 

2

preference elements for the overall decision is most relevant for directing empirical research 

effort, yet by and large unknown. 

In this paper, we investigate the interaction of risk and time preferences in intertemporal 

decisionsthatinvolvefuturerisk, alsoofhigherorder. Webaseouranalysisonthetwo-period 

consumption/saving model, which has long served as the canonical theoretical framework for 

studying saving behavior in the face of risk (e.g., Drèze and Modigliani 1966, 1972, Leland 

1968, Sandmo 1970, Rothschild and Stiglitz 1971, Selden 1979, Kimball 1990). 4 The model 

analyzes the consumption/saving decision of an agent who receives income in each period 

and earns a future return on saving, under risk added to some future aspect. Our reference 

model uses a recursive preference specification, allowing to deal explicitly with the distinct 

contributions of risk and intertemporal preferences as well as discounting to overall utility. 

Its most popular specification, following Epstein and Zin (1989, 1991) (EZ), however, is 

not particularly suited for our analysis, because the implied preference-intensity coefficients, 

including prudence, are constant. For example, it cannot represent the stylized empirical 

finding that relative risk aversion exhibits an increasing shape, as experiment based on static 

lottery-choice tasks often show (e.g., Holt and Laury 2002), whereas macroeconomic data 

imply a decreasing shape of the relative resistance to intertemporal substitution (e.g., Meyer 

and Meyer 2005). A modeling that is open to increased empirical accuracy is important for 

our numerical appreciation of the distinct role of the different preference elements, especially 

as regards higher-order risk effects. We focus, therefore, on more flexible Kreps and Porteus 

(KP) preferences, of which EU and the EZ specification are special cases. For the same 

reason, our simulations rely, moreover, on an operationalization using expo-power utility, 

which nests power and exponential utility as particular specifications (Saha 1993, Xie 2000). 

For a first set of theoretical results, we adapt the Kimball and Weil analysis to our case 

with KP preferences, separate per-period incomes, and potential return risk. Based on this 

4 More recent extensions refer to the relation to social discounting (Gollier 2002), the impact of liquidity 

constraints (Carroll and Kimball 2005), hedging demand under return predictability (Gollier 2008), and 

bivariate frameworks (Flodén 2006, Gollier 2010, Nocetti and Smith 2011). 

3

framework, we generalize then the Eeckhoudt and Schlesinger conditions for saving increases 

under risk changes to recursive preferences. Our predictions mirror results from the EU case 

(e.g., Gollier 2001, Eeckhoudt and Schlesinger 2008) only on a first look. For, they show 

how both risk and intertemporal preferences are involved in reactions to any change in a 

future element carrying risk, from the first order on. The direction of the saving response to 

an N th -order risk increase is determined by the (N +1) th utility derivative. 5 However, the 

(N +1) th derivative of KP utility is not a trivial quantity, containing derivatives of all lower 

orders from both the risk-aversion and intertemporal-substitution components. 

In a next step, we quantify the contributions from the different preference elements given 

typical parameter values, as far as available. Based on a simulated empirical implementation 

of the model, we show that, of the three components of utility in the model, intertemporal 

preference captures the most variation in decisions. Risk aversion does not seem to matter 

very much, even when large risks are present. Hence, correctly identifying intertemporal 

preference seems to be the single most critical aspect of inference. This result is relevant also 

under EU: qualitatively, the EU predictions resemble those of the intertemporal-substitution 

component of recursive utility as opposed to the risk-aversion component. 

In a final step, we study the size of predicted saving increases in response to higher-order 

risk changes. The empirical relevance of higher-order preferences has often been questioned. 

Deck and Schlesinger (2014), for example, document principle difficulties to econometrically 

identify risk preferences of the fifth and sixth order from static lottery choices in an experiment. 

For the intertemporal case, our simulations show this few-relevance result already 

from the fourth order on. This result is robust to all checks we perform, including increasing 

lottery stakes up to the triple of monthly income and sensitivities in all preference parameters. 

We also find that effect sizes fall importantly with a person’s income level. Under EU, 

precautionary saving is predicted higher than with recursive utility. This stylized fact could 

potentially be used as to discriminate between the two theories also at the individual level. 

5 An N th -order risk increase corresponds to a ceteris paribus increase in magnitude of the N th moment 

of a lottery (Ekern 1980). 

4

We introduce our reference model with recursive preferences in Section 2. Section 3 providesthetheoreticalcomparative-staticsresultsunderKPpreferences. 

InSection4,westudy 

the relative importance of risk and intertemporal preferences in the overall saving decision 

under risk in calibrations of the personal saving equilibrium. Section 5 assesses numerically 

higher-order risk effects in our given intertemporal framework. Section 6 concludes. 

2 Two-Period Consumption/Saving Model 

Consider an agent receiving exogenous income y t in each of two periods t = 1,2. The firstperiod 

income is split between consumption c 1 and saving s 1 for the second period. Any 

amount saved earns gross return R 2 (net return r 2 ) in the second period. The agent exhausts 

all resources in the second period through consumption c 2 . Risk may enter through either 

y 2 or R 2 . For convenience, we denote risky variables a tilde (e.g., ỹ 2 ). 

The KP maximization problem is 

max 

s 1 

U (c 1 ,˜c 2 ) = u(c 1 )+βu(CE(˜c 2 )) s.t. 

⎧ 

⎪⎨ 

⎪⎩ 

y 1 = c 1 +s 1 

(1) 

ỹ 2 +s 1 ˜R2 = ˜c 2 

Inproblem(1), uistheagent’sper-periodfelicityfunction, β isthediscountfactorassociated 

with the pure rate of time preference, and CE(˜c 2 ) is the certainty equivalent of the ˜c 2 risk: 

CE(˜c 2 ) ≡ ψ −1 (E 1 [ψ(˜c 2 )]) 

This function serves to rank consumption paths according to the risk preference ψ. Thus, 

ψ addresses risk attitudes explicitly, while u captures the preference for consumption now 

versus later. As a result, ψ is a von Neumann-Morgenstern utility function, while u is not. 

However, EU arises if ψ = u, making EU a testable hypothesis from the KP framework. 6 

6 The increased flexibility of KP comes with a restriction on its constituent functions. Namely, strictly 

increasing and concave shapes for u and ψ – the natural extension of the EU requirements – are insufficient 

5

This decision setting is similar to the next-to-last period of a finite-horizon EZ utility 

function. The infinite-horizon EZ specification uses two power utility functions u(c) = c ρu 

and ψ(c) = c ρ ψ (ρu ,ρ ψ ∈ (0,1)), to give the recursion 

U t = 

[ 

ρu 

(1− ¯β)c t + ¯β [ ( ρ 

E 1 U 

ψ 

)] 1 ] 1 

ρ 

ρu 

ψ 

t+1 

(2) 

In this case, EIS is simply 1/1−ρ u , and RRA merely 1−ρ ψ . Importantly, risk aversion and 

intertemporalsubstitutionaredisentangled–eacheffectdependssolelyonitsownparameter. 

To obtain our two-period formulation, we can set t = 1, U t+1 to c 2 , and note that ψ −1 is a 

positive monotone transformation. We can translate between discount factors with β = ¯β 

1−¯β . 

To be clear, we consider more flexible functions for u and ψ than these isoelastic ones, but 

this special case underscores the fact that our model also reaches a canonical specification 

in RU. 

The Euler condition of (1) determines how intertemporal balance is achieved. For the 

basic intuition, consider the EU case. This Euler condition is 7 

E 1 

[ 

β u′ (˜c 2 ) 

u ′ (c 1 ) ˜R 2 

] 

= 1 (3) 

TheEulerconditionillustratesthe“pricing”interpretationofoptimalsaving: theequilibrium 

occurswhentheexpecteddiscountednetreturntosavingisnil. The“discount”isdetermined 

by the stochastic discount factor (SDF): 

β u′ (˜c 2 ) 

u ′ (c 1 ) 

(4) 

for a well-behaved maximization problem (Gollier 2001). Concavity of ψ −1 in s is also required, which occurs 

if its absolute risk tolerance (inverse of absolute risk aversion) is concave. These assumptions are implicit 

throughout. 

7 If the agent faces borrowing or lending constraints, the Euler condition may not hold with equality. Our 

proofs, which are based on comparative static exercises at the saving optimum, do require equality, but this 

is not a hugely restrictive assumption as the exercises hold (albeit trivially) at a boundary. Empirically, 

however, this may be of importance. 

6

The SDF contains all of the behavioral implications of the model, particularly with regard 

to consumption-smoothing and precautionary motives for saving. Under EU, the smoothing 

motive is captured by the Arrow-Pratt coefficient − u′′ (c)c 

, which is the inverse of the EIS. 

u ′ (c) 

Importantly, the Arrow-Pratt coefficient does not enter into the intertemporal risk response. 

Thisprecautionaryor“prudent”dependsinsteadonthecoefficientofrelativeprudence(RP): 

RP(c) ≡ − u′′′ (c)c 

u ′′ (c) 

(Kimball 1990). 

The intertemporal component of condition (3) can be highlighted by grouping contemporaneous 

terms (Carroll and Kimball 2005, Kimball and Weil 2009): 

u ′ (c 1 ) = E 1 

[ 

βu ′ (˜c 2 ) ˜R 2 

] 

(5) 

Under this reading, a “personal intertemporal equilibrium” within the agent occurs when the 

marginal utility from foregoing consumption in period 1 (i.e., saving a marginal amount) 

is equal to the expected discounted marginal utility from consuming instead in period 2. 

Because saving is sourced from period-1 resources, the left-hand side of (5) can be thought of 

as the intrapersonal supply of saving, and the right-hand side as the intrapersonal demand. 8 

This supply-and-demand interpretation forms the basis of our calibration exercises. 

EU has the unfortunate side effect of functionally linking intertemporal and risk responses. 

To illustrate, consider the simple but widespread example of power utility: u(c) = 

c 1−ρu 

1−ρ u 

. In this case, EIS is 1/ρ u and RP is 1+ρ u , which is quite restrictive. 9 The KP Euler 

8 Pursuing this interpretation further, we could invoke the metaphor of “two selves” – a present and a 

future one – whose Nash bargaining determines the equilibrium saving amount. For example, Andersen et al. 

(2008) refer to a dual-selves model (e.g., Benhabib and Bisin 2005, Fudenberg and Levine 2006) when arguing 

that responses to immediately rewarded risk tasks are probably temptation-driven, while the responses to 

tasks with certain but delayed payments are probably self-controlled. We are agnostic regarding the mental 

mechanisms that underpin these internal tradeoffs. We simply note that the equilibrium condition does 

break cleanly into current and future incentives. 

9 Non-additive EU formulations also contain an explicit link between risk aversion and intertemporal 

substitution (Bommier 2007, Andersen et al. 2011). The isomorphization in these cases is mediated by 

“correlation aversion,” or the desire to avoid highly similar consumption paths. Fully separating risk aversion 

from intertemporal substitution ultimately requires non-EU choice axioms. 

7

condition involves an explicit interaction of risk and intertemporal preferences: 

[ ] 

E 1 β u′ (CE(˜c 2 ))CE ′ (˜c 2 ) 

= 1 (6) 

u ′ (c 1 ) 

where 

CE ′ (˜c 2 ) ≡ dCE(˜c [ ] 

2) ψ ′ (˜c 2 ) 

= E 1 

ds 1 ψ ′ (CE(˜c 2 )) ˜R 2 

(7) 

is a “risk-preference adjusted return to saving” (Selden 1979). The additional terms in the 

KP stochastic discount factor reflect the manner in which the certainty-equivalent ranking 

of ˜c 2 changes with s 1 . Collecting the ψ ′ terms in condition (6) together gives 

u ′ (c 1 ) 

u ′ (CE(˜c 2 )) = βCE′ (˜c 2 ) (8) 

Thetermontheleftisthemarginalrateofsubstitutionbetweencurrentandfuturecertaintyequivalent 

consumption. The term on the right is a discounted expected return to saving, 

where the discount in this case only arises from risk factors. Intuitively, this equation means 

that the agent saves until the intertemporal utility tradeoff from saving equals the riskadjusted 

return to saving. This explicit tension between risk and intertemporal forces is not 

present in EU. 

Like EU, the consumption-smoothing motive under KP is determined solely by u. However, 

the KP precautionary motive is more nuanced. Kimball and Weil (2009) show that 

RP U is a mixture of risk and intertemporal effects: 

( 

RP U (c) = RRA ψ (c) 1+ ε ) 

ψ(c) 

RRIS u (c) 

(9) 

RRA ψ is the coefficient of relative risk aversion associated with ψ, RP ψ its coefficient of 

relative prudence, and ε ψ = RP ψ − RRA ψ its elasticity of absolute risk tolerance. RRIS u 

is the relative resistance to intertemporal substitution (inverse of EIS) associated with u. 

Thus, KP prudence requires the derivatives of ψ up to degree 3, and the derivatives of u 

8

up to degree 2. To determine the rate of change in prudence, an additional degree in each 

domain would need to be evaluated. 10 

3 Comparative Statics 

Togain somedirectional intuitionon howchangingincentivescan impact saving, weexamine 

theoretical comparative statics on the consumption/saving model (1) in non-risk and risk 

elements in turn. The comparative statics on non-risk elements adapt and extend the results 

in Kimball and Weil to our RU framework. 11 The comparative statics on risk extend the 

higher-order risk effects on saving in Eeckhoudt and Schlesinger (2008) to RU. Like Eeckhoudt 

and Schlesinger, we analyze income and return risk separately. The precautionary 

effect differs markedly in these two cases: it is possible to steer return-risk exposure directly 

via saving, but income risk must be borne. Remark 1 summarizes this principal difference. 12 

Remark 1 Denote period-2 consumption under income risk as ˜c y 2 ≡ ỹ 2 +s 1 R 2 , and under 

return risk as ˜c R 2 ≡ y 2 +s 1 ˜R2 . Let m n (˜z) denote the n th central moment of a random variable 

˜z. Under income risk, saving only alters the first moment (mean) of ˜c 2 : 

m 1 (˜c y 2) = E 1 (ỹ 2 )+s 1 R 2 , m n (˜c y 2) = m n (ỹ 2 ) for n > 2 

(10a) 

Under return risk, saving magnifies all moments of ˜c 2 by the amount s n 1: 

m 1(˜c ) ) 

R 

2 = y2 +s 1 E 1 

(˜R2 

, m n(˜c ) ) 

R 

2 = s 

n 

1 ·m 

(˜R2 n 

for n > 2 

(10b) 

In the latter case, risk can be entirely avoided by saving nothing. 

10 UnderKP,agloballynecessaryandsufficientconditionforprudenceisthatψ exhibitsdecreasingabsolute 

riskaversion(DARA)(Gollier2001). Agloballysufficientconditionisthatψ ′ isconvexanduismoreconcave 

than ψ. 

11 Notably, Kimball and Weil focus on wealth and future income risk and do not consider comparative 

statics related to the return on saving. 

12 Proofs in this section that do not appear in the text can be found in Appendix A. 

9

3.1 Non-Risk Elements 

We first consider comparative statics on the incentives y 1 , y 2 , y = y 1 = y 2 , and R 2 . The 

optimality condition associated with problem (1) written as a function of s 1 can be stated 

as 

−u ′ (c 1 )+βu ′ (CE(˜c 2 ))CE ′ (˜c 2 ) = 0 

(11a) 

Given our prior assumptions on u and ψ, the second-order condition 

u ′′ (c 1 )+β 

[ 

] 

u ′′ (CE(˜c 2 ))[CE ′ (˜c 2 )] 2 +u ′ (CE(˜c 2 ))CE ′′ (˜c 2 ) < 0 (11b) 

ensures that condition (11a) provides a unique solution for s 1 . 13 Optimal saving is positive 

as long as the expected net returns are positive, which we assume to occur. Letting F(s 1 ;Θ) 

denote the expression on the left-hand side of equation (11a) with parameter vector Θ ≡ 

(y 1 ,ỹ 2 , ˜R 2 

) 

, thechangeinoptimalsavingduetoaparticularincentiveθ ∈ Θfollowsfromthe 

implicit function theorem: 

of ds∗ 1 

dθ 

fully determines its sign. 

ds ∗ 1 

dθ = − ∂F(s 1;Θ)/∂θ 

∂F(s 1 ;Θ)/∂s 1 

∣ ∣∣s1 

=s ∗ 1. Given condition (11b), the numerator 

The following proposition provides these sign conditions for incentive changes. Proposition 

1 extends results in Gollier to KP preferences. 14 

Proposition 1 Marginal increases in the incentives y 1 , ỹ 2 , y = y 1 = ỹ 2 , and ˜R 2 result in 

the following directional changes in saving s ∗ 1: 

[ ] ds 

∗ 

sgn 1 

dy 

[ 1 

] ds 

∗ 

sgn 1 

dỹ 

[ 2 

] ds 

∗ 

sgn 1 

dy 

= sgn[−u ′′ (c 1 )] > 0 (12a) 

= sgn [ u ′′ (CE(˜c 2 ))CE y2 (˜c 2 )CE ′ (˜c 2 )+u ′ (CE(˜c 2 ))CE ′ y 2 

(˜c 2 ) ] < 0 (12b) 

= sgn [ −u ′′ (c 1 )+β ( u ′′ (CE(˜c 2 ))CE y2 (˜c 2 )CE ′ (˜c 2 )+u ′( CE(˜c 2 )CE ′ y 2 

(˜c 2 ) ))] 0 

(12c) 

13 Kimball and Weil (2009: Appendix A) show that CE ′′ (˜c 2 ) < 0 under the above assumptions on ψ. 

14 In the following, CE y2 (˜c 2 ) and CE ′ y 2 

(˜c 2 ), respectively, denote the parameter derivatives of CE(˜c 2 ) and 

CE ′ (˜c 2 ) with respect to y 2 at the optimum. 

10

Saving decreases if 

• ψ exhibits DARA and u′′ (CE(˜c 2 )) 

u ′ (CE(˜c 2 )) 2 is weakly increasing; or 

• ψ exhibits DARA and decreasing relative prudence, ARIS u ≥ ARA ψ , and RP u ≤ RP ψ . 

[ ] ds 

∗ 

sgn 1 

d ˜R 

= sgn [ u ′′ (CE(˜c 2 ))s 1 CE y2 (˜c 2 )CE ′ (˜c 2 )+u ′ (CE(˜c 2 )) [ s 1 CE y ′ 2 

(˜c 2 )+CE y2 (˜c 2 ) ]] 0 

2 

Saving decreases if −s 1 

u ′′ (CE(˜c 2 )) 

u ′ (CE(˜c 2 )) CE′ (˜c 2 ) > 1. 

(12d) 

Proposition 1 shows that the direction of the saving response to y 1 increases depends 

on the concavity of the intertemporal preferences u, but that the properties of both u and 

risk preferences ψ are important if risky future income varies. The signs of the two effects 

reflect a typical preference for smooth consumption: if current income rises, smoothing (12a) 

involves sending some funds into the future through the saving mechanism. If future income 

rises, smoothing (12b) involves foregoing some saving for current use. 

The marginal increase in y = y 1 = ỹ 2 in Conditions (12c) considers a homogeneous 

income shock to the already-balanced consumption path of an agent. The effect of this 

shock on saving is ambiguous, because it hinges on the relative strengths of the two effects 

in (12a) and (12b). 

Thedirectionofthesavingresponsetoanincreasethereturn ˜R 2 dependsontwoopposing 

effects. First, higher future wealth permits higher consumption (less saving) today due to 

consumptionsmoothing; thiswealtheffectiscapturedbythenegativetermsin(12d). Higher 

future returns also increase the incentive to substitute current for future consumption by 

reducing the opportunity cost of future consumption (lowering the implicit price of saving); 

this substitution effect toward future consumption is captured by the positive term. Saving 

decreases if the wealth effect overcomes the substitution effect. A coefficient of partial 

resistance to intertemporal substitution exceeding one is sufficient for saving to decrease. 15 

15 Following Menezes and Hanson (1970) and Chiu et al. (2012), we refer to a quantity of the form 

− f(N) (x+t)t 

f (N−1) (x+t) 

with t > t for all t, where the lower bound t > 0 of the range of t is chosen such that the 

11

3.2 Precautionary Saving under Risk Changes 

Like Eeckhoudt and Schlesinger, we structure changes in risk using N th -order stochastic 

dominance (NSD) as well as “increases in N th -degree risk” (Ekern 1980). The latter kind of 

increases require that the moments in two lotteries up to N −1 be identical, and that one 

stochastically dominates the other at the N th order. Several well-known “increases in risk” 

fall under this latter definition. A mean-preserving spread (Rothschild and Stiglitz 1970) is 

an increase of second degree, an increase in downside risk (Menezes et al. 1980) is of third 

degree, and an increase in outer risk (Menezes and Wang 2005) is of fourth degree. 

Lemma 1 links NSD to higher-order risk preferences (Eeckhoudt and Schlesinger 2008, 

Heinzel 2013). It is essential for the proofs that follow. 16 

Lemma 1 (NSD Equivalence) For any utility function f(t) with sgn [ f (n) (t) ] = (−1) n+1 

for all n = 1,2,...,N, and two random variables ˜z a and ˜z b with identical real bounded 

supports, where ˜z a dominates ˜z b via NSD, the expected marginal utility premium Ef ′ (˜z b )− 

Ef ′ (˜z a ) is non-negative. 

If (−1) N f (N) (.) ≤ 0, the agent is said to be “N th -degree risk averse.” This definition encompasses, 

for example, the standard EU concepts of risk aversion (n = 2), prudence (n = 3), 

temperance (n = 4), and edginess (n = 5). 

ToextendtheEeckhoudtandSchlesingerconditionsunderincomerisktoKPpreferences, 

let the future return R 2 be certain, and suppose there are two future income lotteries ỹ 2,a 

and ỹ 2,b . Let s ∗ y 2,a 

and s ∗ y 2,b 

be the solutions to the optimality condition (11a) in each case. 

Moreover, let ˜z in the NSD Equivalence correspond to ỹ 2 , and f(t) correspond to u(CE(t)). 

Then, Lemma 1 implies that 

s ∗ y 2,b 

≥ s ∗ y 2,a 

⇔ d 

ds 1 

u(CE(˜c 2,b )) ≥ d 

ds 1 

u(CE(˜c 2,a )) (13) 

probability Pr(x+t < 0) = 0, as a measure of “partial N th -degree risk aversion.” In our case, x = y 2 . The 

term “partial resistance to intertemporal substitution” arises in analogy. 

16 Where unambiguous, we use the notation f (n) (t) ≡ ∂n f(t) 

∂t 

for an n th derivative. 

n 

12

[ ] 

if sgn d n+1 u(CE(˜c 2 )) 

= (−1) n for all n = 1,2,...,N. The following proposition details this 

ds n+1 

1 

result. It generalizes a standard result for mean-preserving spreads (Kimball and Weil 2009) 

to the N th order. 

Proposition 2 Let R 2 be certain, and let s ∗ y 2,a 


be the optimal saving choices of 

(11a) under income lotteries ỹ 2,a and ỹ 2,b . The following statements are equivalent: 

1. s ∗ y 2,b 

≥ s ∗ y 2,a 

, if sgn [ u (n) (.) ] = sgn [ CE (n) (˜c 2 ) ] = (−1) n−1 for all n = 1,2,...,N +1. 

2. ỹ 2,a dominates ỹ 2,b via NSD. 

The sign conditions on the u(.) and CE(.) derivatives in Proposition 2 are sufficient for the 

successive derivatives of u(CE(˜c 2 )) to similarly alternate in sign for all n = 1,2,...,N +1 

(Appendix A). 

Applying the definition of an N th -degree risk increase allows us to limit the conditions 

in Proposition 2 to just the sign for n = N +1. 

Corollary 1 Let s ∗ y 2,a 


be the optimal choices from Proposition 2. The following 

statements are equivalent: 

1. s ∗ y 2,b 

≥ s ∗ y 2,a 

, if sgn [ u (N+1) (.) ] = sgn [ CE (N+1) (˜c 2 ) ] = (−1) N . 

2. ỹ 2,b represents an N th -degree increase in risk over ỹ 2,a . 

With regard to the signs of u (n) (CE(˜c 2 )), Proposition 2 and Corollary 1 are analogous to 

Eeckhoudt and Schlesinger’s conditions under EU. The main difference here is that the argument 

contains the certainty-equivalent ranking of ˜c 2 , rather than ˜c 2 itself. Proposition 2 and 

Corollary 1 underscore the simultaneous importance of intertemporal and risk preferences 

for the saving response. This interplay is actually quite intricate relative to EU, because the 

uniform-signing requirement for successive CE(.) derivatives involves a non-obvious dependence 

on ψ derivatives. While the positivity of CE ′ (.) flows naturally from the properties 

of ψ, the negativity of just the next derivative CE ′′ (.) is exceptionally tedious to establish 

13

(Kimball and Weil 2009). We thus refrain from a general sign proof for the n th derivative 

of CE(.), but show in Appendix A that such a proof would depend upon all ψ derivatives 

up to n th order. As n increases, these conditions become more and more restrictive, and are 

thus probably less likely to hold in reality. Indeed, our calibrations suggest that most of the 

practical relevance of the proposition is found in first- and second-order increases in risk, so 

we do not appear to be omitting critical behavior by foregoing these higher orders. 

The conditions under which an increase in return risk leads to higher saving are more 

involved under KP, because the agent can directly limit his or her risk exposure through 

the saving choice (see Remark 1). Prior results on risk increases under RU include Selden 

(1979), Langlais(1995)undermean-preservingspreads, andWeil(1990)forgeneralisoelastic 

preferences. We extend these results to an arbitrary increase in N th -degree risk under KP. 17 

Proposition 3 Let y 2 be certain, and let s ∗ R 2,a 

and s ∗ R 2,b 

be the optimal saving choices of 

(11a) under return lotteries ˜R 2,a and ˜R 2,b . The following statements are equivalent: 

1. s ∗ R 2,b 

≥ s ∗ R 2,a 

, if sgn [ u (n) (.) ] = sgn [ CE y n 

2 

(.) ] = (−1) n+1 CE y 

n+1 

2 

and −s 1 R 2 

for all n = 1,2,...,N. 

(c 2 ) 

CE y n 

2 

(c 2 ) 

≥ n 

2. ˜R2,a dominates ˜R 2,b via NSD. 

Applying the definition of an N th -degree risk increase again simplifies this proposition. 

Corollary 2 Let s ∗ R 2,a 

and s ∗ R 2,b 

be the optimal choices from Proposition 3. The following 

statements are equivalent: 

1. s ∗ R 2,b 

≥ s ∗ R 2,a 

, if sgn [ u (n+1) (.) ] [ ] 

= sgn CE y 

n+1 (.) = (−1) n CE y 

n+1 

2 

and −s 1 R 2 

2 

for all n = 1,2,...,N. 

(c 2 ) 

CE y n 

2 

(c 2 ) 

≥ n 

2. ˜R2,b represents an N th -degree increase in risk over ˜R 2,a . 

The EU analog to the first condition requires the coefficient of partial (N +1) th -degree risk 

aversiontoexceedN. Here, thiscoefficientinvolvesthecertaintyequivalentinsteadofutility. 

17 In what follows, CE y n 

2 

(.) stands for the n th derivative of CE(.) at the optimum with respect to y 2 . 

14

AswithEU,itismoredifficulttoestablishunambiguousdirectionalchangesunderreturn 

risk. Proposition 3 and Corollary 2 show that intertemporal and risk preferences are again 

simultaneously active under return risk, though the channel is different from income risk. To 

illustrate, consider an increase in second-order risk. When the risk is on income, Collorary 

1 shows that saving rises for sure if the agent is prudent in the KP sense. However, when 

the risk is on the return, two competing effects arise. These can be seen by examining 

h(R 2 ) ≡ − ∂u(CE(c 2)) 

∂s 1 

, constructed so that its n th derivative determines the response to an 

n th -order increase in risk (see proof of Proposition 3). Using the shorthand CE for CE(c 2 ), 

the second derivative of h is: 

h ′′ (R 2 ) = u ′′′ (CE)CE 3 y 2 

s 1 R 2 +u ′′ (CE)CE y2 s 1 

[ 

3CE y 2 

2 

s 1 R 2 +2CE y2 

] 

+u ′ (CE)s 1 

[ 

CE y 3 

2 

s 1 R 2 +2CE y 2 

2 

] 

At the optimum s ∗ 1 where CE y2 > 0, the first term is positive if the agent is prudent in 

u. This aspect contributes positively to the precautionary motive. However, a negative 

substitution effect occurs in the second and third terms, and the former must dominate the 

latter for saving to rise. Under the standard signs for u derivatives, the net effect is actually 

(c 2 ) 

CE y 

n+1 

2 

positive if −R 2 s 1 CE y n(c 2 

≥ n for n = 1,2. 

) 2 

3.3 Comparing Risk Effects under EU and RU 

The preference conditions above show the different roles played by risk and intertemporal 

preferences in RU versus EU, mainly with regard to signs. Another interesting difference 

arises with regard to the strength of the saving responses: the precautionary response in 

EU consistently tends to exceed that under RU. The following proposition shows that this 

result, originally due to KW for an actuarially-neutral risk added to future income, holds 

similarly for increases in risk and under return risk. 

Proposition 4 Consider an EU agent and a RU agent with similar preferences (i.e., u eu = 

15

u = ψ), who both save more (less) in response to any risk increase in NSD. In view of a 

deterioration in the risk on period-two income or, alternatively, on the interest rate, the EU 

agent’s saving increases (decreases) at least as much as the RU agent’s. 

The proof in Appendix A relies on the precautionary-premium concepts associated with 

higher-order risk increases defined in Liu (2014) and Heinzel (2015a,b). The latter concepts 

elaborate on the equivalent precautionary premia developed by Kimball (1990) and KW. 

According to Liu and Heinzel, the precautionary premia θ y 2 

eu and θ y 2 

ru for increases in income 

risk arise, respectively, from 

E 1 f ′ (ỹ 2,a −θ y 2 

eu +s 1 R 2 ) = E 1 f ′ (ỹ 2,b +s 1 R 2 ) 

(14a) 

f ′ (CE(ỹ 2,a −θ y 2 

ru +s 1 R 2 ))CE y2 (ỹ 2,a −θ y 2 

ru +s 1 R 2 ) = f ′ (CE(ỹ 2,b +s 1 R 2 ))CE y2 (ỹ 2,b +s 1 R 2 ) 

(14b) 

The precautionary premia θ r 2 

eu and θ r 2 

ru related to return-risk increases follow from 

[ ( 

E 1 f ′ y 2 +s ∗a ˜R 

) ] ( 

1 2,a −θ r 2 

eu ˜R2,a = E 1 

[f ′ y 2 +s ∗b ˜R 

) ] 

1 2,b ˜R2,b 

(15a) 

f ′ (CE(y 2 +s ∗a 

1 ˜R 2,a −θ r 2 

ru))CE ′ (y 2 +s ∗a 

1 ˜R 2,a −θ r 2 

ru) = f ′ (CE(y 2 +s ∗b 

1 ˜R 2,b ))CE ′ (y 2 +s ∗b 

1 ˜R 2,b ) 

(15b) 

Following the approach of Briys et al. (1989) to endogenous risk premia, Equations (15) 

compare future marginal utility evaluated at optimal saving s ∗i 

1 , i = a,b, under return ˜R 2,a 

and future marginal utility under the more risky return ˜R 2,b . The reference to the optimal 

saving levels accounts for the different levels of risk exposure in the two risk situations, which 

are immediately co-determined with saving choices. Proposition 4 holds, obviously, similarly 

for Ekern risk increases. 

16

4 Calibrations of the Personal Equilibrium 

In this and the next section, we simulate an empirical implementation of the above model. 

We start in this section by investigating the relative contributions of risk and intertemporal 

preferences to the overall saving decision and compare the results under KP preferences to 

the EU case. To visualize the effects, we calibrate some median agent’s “intrapersonal saving 

market,” which arises when grouping Euler condition (6) by its current and future elements. 

Starting from a set of baseline values for the preference parameters and economic incentives, 

we consider how the agent’s saving response under risk depends on the different preferences. 

To operationalize the model, we first select functional forms for u and ψ. Our choices 

of ψ and u are motivated by a desire to accommodate the varying shapes of risk aversion 

and the elasticity of intertemporal substitution observed in the empirical literature. In fact, 

while for relative risk aversion an increasing shape is a typical finding from experimental 

lottery-choice tasks with substantial incentives (e.g., Holt and Laury 2002), macroeconomic 

data often exhibit a decreasing shape of the relative resistance to intertemporal substitution 

(e.g., Meyer and Meyer 2005). EZ preferences as in specification (2) are not well suited to 

reflect these findings, because the related RRA ψ and RRIS u coefficients are constant. To 

accommodate potentially non-constant shapes, we use utility functions of the “expo-power” 

form (Saha 1993, Xie 2000). We set ψ to 

ψ(c) = 1 [ ( )] 

c 1−ρ ψ 

1−exp −α ψ · 

α ψ 1−ρ ψ 

(16a) 

Its coefficients of absolute and relative risk aversion are 18 

ARA ψ (c) = α ψ c −ρ ψ 

+ρ ψ c −1 , RRA ψ (c) = α ψ c 1−ρ ψ 

+ρ ψ 

18 The coefficients of absolute and relative prudence follow easily from these coefficients: 

AP ψ (c) = ARA ψ (c)− ARA′ ψ (c) 

ARA ψ (c) , RP ψ(c) = RRA ψ (c)− ARA′ ψ (c) 

ARA ψ (c) c 

These coefficients additionally imply that the elasticity of absolute risk tolerance is ε ψ (c) = −ρ ψ 

α ψ c+c ρ ψ 

α ψ c+ρ ψ c ρ ψ. 

17

and nest several important special cases. When ρ ψ = 0, the utility function is exponential 

(CARA). As α ψ → 0, the utility function is power (CRRA), and log if ρ ψ → 1 in addition. 

Relative risk aversion is increasing when α ψ > 0 and ρ ψ < 1, and decreasing when α ψ > 0 

and ρ ψ > 1. Risk neutrality occurs when ρ ψ = 0 and α ψ → 0. The expo-power form has 

thus, in particular, the ability to capture increasing, decreasing, and constant RRA. 19 

We set u to a different expo-power function 

u(c) = 1 ( )] 

c 

[1−exp 

1−ρu 

−α u · 

α u 1−ρ u 

(16b) 

andnotethatthesameanalysisofriskaversioninψ appliestotheresistanceofintertemporal 

substitution in u. That is, when α u > 0, the elasticity of intertemporal substitution increases 

if ρ u > 1, and decreases if ρ u < 1. 

We present theoretical comparative statics of optimal saving in the parameters of the 

expo-power functions for both KP preferences and EU in Appendix B. The effects in the 

α parameters depend all in particular on whether the related ρ is smaller or larger than 

unity and, thus, on whether the related relative preference intensity coefficient is increasing 

or decreasing. The bounds delimiting the effects in the ρ parameters are endogenous to the 

optimal choices. The effects in the parameters of the ψ function of KP preferences claim for 

uniform signing the presence of a preference for early risk resolution. The gain in accuracy 

by adopting a more flexible functional form and, eventually, recursive preferences comes thus 

at the cost of richer conditions on preferences for uniform effects. Persuasive procedures to 

obtainvalidestimatesoftheinvolvedparametersseemallthemoreimportant. Topotentially 

help to guide such empirical research, we proceed here with simulations giving an idea of the 

relative importance of the difference preference elements that are represented. 

19 Excluding pathological parametrizations, the expo-power function has derivatives with alternating sign, 

concave absolute risk tolerance, and decreasing absolute risk aversion (except in the edge case of ρ ψ = 0). 

The underlying KP optimization problem is thus well-behaved. 

18

Our baseline preference parameters are 

α ψ = 0.03 ρ ψ = 0.7 α u = 0.05 ρ u = 1.5 β = 1 

The ψ parameters are approximately equal to those from Holt and Laury. The ρ u parameter 

is in the neighborhood of empirical macroeconomic estimates, and DRRIS (increasing elasticity)isaddedthroughaslightlypositiveα 

u . Wesetthediscountfactorβ to1toaccentuate 

the effects of risk aversion and intertemporal substitution. Its impact upon saving demand 

is always multiplicative, so shifts arising from this channel can be easily intuited. 

The stakes in our calibration are oriented towards small gains an agent could encounter 

in daily life, which would typically induce a short-term saving response. “Short-term” and 

“small” are only meaningful relative to field incentives, so we fix those first. We set the 

baseline background income to y = $1,000, which is on the order of monthly income for the 

first and second quintiles of the US population. We abstract from other saving opportunities 

than the ones we present in the scenarios to the agent. This is, for example, the case if other 

returns are so small that they generate only relatively minor effects over such horizons. Of 

course, theirimpactscanbeeasilyintuitedbyanalogytoR 2 . Weeliminateotherbackground 

risks for a similar reason: its effects can be intuited by examining the relatively large risks 

associated with the lotteries we consider. 

Given these background incentives, we center scenario incomes at y 1 = y 2 = $100, or 

at 10% of background income. We center the baseline net return r 2 in a scenario at 20%. 

The baseline income lotteries involve 50-50 chances of a $50 deviation in y 2 , and the return 

lotteries 50-50 chances of a 15 percentage-point deviation in r 2 . All of the results we present 

are sensitive to the scaling of the involved payments. We consider a few different scalings in 

each case to illustrate these effects. The scalings typically involve the baseline (1x scaling) 

and the double of the baseline (2x scaling), but sometimes go up to five times (5x scaling) 

or thirty times the baseline (30x scaling). 

19

4.1 Preference Effects under KP Preferences 

We first examine how the preference components of U in the recursive case as in specification 

(1) affect the saving decision. We approach the question by perturbing the parameters in 

the risk and intertemporal domains, and observing the resulting shifts in the intrapersonal 

equilibrium. We concentrate here on scenarios with income risk, the findings for return risk 

are the same. 20 

Risk Preferences 

Recall that ψ under KP preferences captures risk effects by ranking future consumption 

streams according to their certainty equivalents. Because it deals exclusively with future 

values, ψ only affects saving demand. Figure 1 plots the intrapersonal saving market for ψ 

perturbations in 2x scalings of the baseline income lottery. The boundaries of the choice set 

are marked by vertical lines. We focus on interior equilibria in this paper. We consider fairly 

large changes in risk aversion: α ψ ranges from 0.01 to 1, and ρ ψ from 0.3 to 2. 

4 x 10−5 1x Income Lottery 

D α ψ 

=0.01 


D ρ ψ 

=0.3 

3.8 

3.6 

D α ψ 

=0.1 

D α ψ 

=1 

S 

3.8 

3.6 

D ρ ψ 

=0.7 

D ρ ψ 

=2 

S 

Marginal Utility 

3.4 

3.2 

3 


3.4 

3.2 

3 

2.8 

2.8 

2.6 

2.6 

2.4 

−50 0 50 100 150 

Saving 

2.4 

−50 0 50 100 150 


Figure 1: Comparative statics for α ψ (left) and ρ ψ (right) at $100 scale. 

Ceteris paribus, increasesinα ψ increasesavingdemand. Thisoccursbecauseriskaversion 

strengthens the precautionary motive (see the coefficient of prudence (9)). The fact that the 

20 We present the corresponding graphs with return risk in Appendix C. 

20

slope of RRA ψ changes sign with ρ ψ makes its impact more peculiar. Recall that RRA ψ 

increases when ρ ψ < 1 and decreases when ρ ψ > 1; this causes saving demand to fall in the 

former case but rise in the latter. Overall, large variations in risk preferences generate only 

small effects on saving demand. 

Intertemporal Preferences 

Consumption-smoothing preferences are captured by u, which appears in saving supply and 

saving demand, respectively. Thus, changing α u or ρ u will shift both supply and demand. 

We consider changes that are sized similarly to those of ψ: α u ranges from 0.01 to 1, and ρ u 

from 0.75 to 1.5. 

Figures2showsthatsupplyanddemandaremuchmoreresponsivetotheseintertemporal 

parameters than they are to risk parameters. Additionally, the ρ u effects are much more 

substantial than the α u effects. Perturbing u generates opposing effects on supply and 

demand, yetthenetresultisintuitive. Namely, savingfalls astheresistancetointertemporal 

substitution rises, because there is a reduced desire for consumption smoothing. However, 

the magnitude of these net effects is not very large. 


4 x 1x Income Lottery 

10−5 

3.5 

S α u 

=0.01 

D 

S α u 

=0.1 

D 

S α u 

=1 

D 


2.5 x 1x Income Lottery 

10−3 

2 

1.5 

1 

S ρ u 

=0.75 

D 

S ρ u 

=1.1 

D 

S ρ u 

=1.5 

D 

3 

0.5 

2.5 

−50 0 50 100 150 


0 

−50 0 50 100 150 


Figure 2: Comparative statics for α u (left) and ρ u (right) at $100 scale. 

21

4.2 Preference Effects under Expected Utility 

Utility curvature under EU is commonly referred to as “risk aversion,” which has a clear 

meaning in a static risk environment. It is a murkier term in an intertemporal context 

due to the two potential domains of operation: does it primarily reflect a desire to avoid 

future risk, a desire to substitute intertemporally, or some mixture of the two? We address 

its interpretation in intertemporal choice here by comparing the preference effects between 

KP preferences and EU. The EU case requires equating risk aversion and resistance to 

intertemporalsubstitution. Wedothisbysettingψ ≡ u, sothattheKPoptimalitycondition 

collapses to its EU analog. 

Figure 3 shows the saving market for income scenarios as the expo-power parameters 

α EU and ρ EU are varied. Supply and demand shift substantially. These movements do not 

qualitatively resemble at all the analogous graphs for ψ under KP preferences (Figure 1), but 

bear a marked resemblance to those of u (Figure 2). This confirms that EU “risk aversion” 

(a second-derivative effect) primarily conveys a desire to smooth consumption. The same 

observationariseswhenexaminingreturnscenarios(Figure25inAppendixDascomparedto 

Figures 19–20 in Appendix C). We return to the comparison of effects under KP preferences 

and EU in Section 5.2. 



10−5 

3.5 

S α EU 

=0.01 

D 

S α EU 

=0.1 

D 

S α EU 

=1 

D 



10−3 

2 

1.5 

1 

S ρ EU 

=0.75 

D 

S ρ EU 

=1.1 

D 

S ρ EU 

=1.5 

D 

3 

0.5 

2.5 

−50 0 50 100 150 


0 

−50 0 50 100 150 


Figure 3: Comparative statics for α EU (left) and ρ EU (right) at $100 scale. 

22

5 Higher-Order Risk Effects 

Our theoretical results show how higher moments of lotteries activate higher-order aspects 

of utility under KP preferences. As is well known, a particular moment activates a different 

aspect of utility depending on whether a choice is static or dynamic (e.g., Eeckhoudt and 

Schlesinger 2006, 2008). For example, the second-moment mean-preserving spread (MPS) 

activates risk aversion in a static context, but the precautionary motive (prudence) in an 

intertemporalcontext. Thethird-momentincreaseindownsiderisk(IDR)activatesdownside 

risk aversion in a static context, and temperance in an intertemporal context. 

The choices we consider are rooted in the dynamic interpretation, in which an N th -order 

risk activates (N+1) th -order aspects of intertemporal utility. Because risk and time are both 

present in the modeled decision, an agent’s risk aversion and intertemporal substitution are 

activated simultaneously. Corollary 1 provides some guidance on the response of saving to 

increases in risk in this more complex case. Namely, if we observe a sequence of decisions 

involving increases in N th -order risk, the direction of the incremental change in saving is 

predictable. The interaction of risk aversion and intertemporal substitution in this process 

is complex, however. 

Empirical evidence on higher-order preferences has been fairly inconclusive thus far (cf. 

footnote 3). Experimental studies, which have focused on static risk, find downside risk 

aversion in large parts of their samples, have ambiguous conclusions about fourth-order 

preferences, and do not detect significant expressions of any fifth- or sixth-order preferences. 

In this section, we study the size of higher-order risk effects as predicted by our calibration 

of the two-period consumption/saving model. 

23

5.1 KP Preferences 

Income Risk 

Figure 4 plots the intrapersonal saving market for MPS of the ỹ 2 lottery in 1x and 5x scalings 

of the baseline income lottery. Fixing the mean at $100 and the probabilities at 50-50, we 

vary the standard deviation from $0 to $100 by adding and subtracting two equally-sized 

amounts in the two lottery outcomes. 21 We then scale the MPS lotteries from 1x to 5x. 

3.8 

3.6 


D sd=0 

D sd=50 

D sd=100 

S 

3.5 x 10−5 5x Income Lottery 

3 

D sd=0 

D sd=250 

D sd=500 

S 


3.4 

3.2 

3 


2.5 

2 

2.8 

2.6 

1.5 

2.4 

−50 0 50 100 150 


1 

−100 0 100 200 300 400 500 600 


Figure 4: Comparative statics for MPS of y 2 lottery at $100 and $500 scales. 

The graphs support Corollary 1. As second-order income risk rises, the KP precautionary 

motive leads to higher saving. For the baseline parameters, the actual variation in saving is 

smallevenwhentherisksarelarge: atthe5xscalingwherethelotteryrepresents50%offield 

income, a standard deviation of $500 generates only a few dollars more saving than complete 

certainty does. Saving appears, thus, to be mostly driven by consumption smoothing rather 

than prudence. 

Figures 5–6 show that the risk reaction is sensitive to the level of the risk-preference 

parameters. Rising the latter, for example, via an increase of α ψ to (high) 0.5 ceteris 

paribus (Figure 5), provides for a stronger reaction to risk. By contrast, a variation of 

21 We stop increasing the variance once one of the lottery outcomes reaches zero, ensuring that the agent 

cannot get worse off by the lottery. Otherwise loss-aversion preferences, for example, might become relevant. 

24

the consumption-smoothing preference, like in Figure 6, shifts the saving rate but has virtually 

no influence on the risk response. In the example, a lower resistance to intertemporal 

substitution by reducing ρ u to 0.75 only leads to clearly higher saving rates as compared to 

Figure 4. 


10−5 

3.8 

3.6 

D Var=0 

D Var=2500 

D Var=10000 

S 


10−5 

3.5 

D Var=0 

D Var=62500 

D Var=250000 

S 


3.4 

3.2 

3 


3 

2.5 

2 

2.8 

2.6 

1.5 

2.4 

−50 0 50 100 150 


1 

−100 0 100 200 300 400 500 600 


Figure 5: Comparative statics for MPS of y 2 lottery at $100 and $500 scales with α ψ = 0.5. 


10−3 

2.2 

2.1 

D Var=0 

D Var=2500 

D Var=10000 

S 


10−3 

1.8 

D Var=0 

D Var=62500 

D Var=250000 

S 


2 

1.9 

1.8 


1.6 

1.4 

1.2 

1.7 

1.6 

1 

1.5 

−50 0 50 100 150 


0.8 

−100 0 100 200 300 400 500 600 


Figure 6: Comparative statics for MPS of y 2 lottery at $100 and $500 scales with ρ u = 0.75. 

Figure 7 shows the saving-market predictions for stakes levels of $200 and $500 for a 

higher background income of $3,000. Diluting the importance of lotteries relative to overall 

resources leads (compare, e.g., to Figure 4) to less saliency: increasing y makes saving 

variation exceptionally small for large variations in risk. 

25


10−6 

6.8 

6.6 

D Var=0 

D Var=10000 

D Var=40000 

S 


10−6 

6.2 

6 

5.8 

D Var=0 

D Var=62500 

D Var=250000 

S 


6.4 

6.2 

6 


5.6 

5.4 

5.2 

5.8 

5 

4.8 

5.6 

4.6 

5.4 

−50 0 50 100 150 200 250 


−100 0 100 200 300 400 500 600 


Figure 7: Comparative statics for MPS of y 2 lottery at $200 and $500 scales for y = $3,000. 

Figures 8–9 illustrate the saving market for an IDR of the baseline lottery. To construct 

the IDR, we preserve the mean and variance of the initial lottery while simultaneously 

subtracting two amounts from both outcomes, and transferring probability mass from the 

lower outcome to the upper. 22 The lower outcome shifts left to a greater extent than the 

upper, and this shift combined with the probability transfer gives the lower outcome the 

appearance of a left-tail outlier. This process generates an IDR by increasing the left skew 

of the lottery. We vary the standardized skew from 0 in the baseline to -4.69, 23 and scale 

the IDR lotteries from 1x to 30x. 

Although it is not visually apparent, the prediction of Corollary 1 is supported. However, 

increasing third-order risk has essentially no impact on saving. This is not simply a side 

effect of small stakes: even when the lottery incentives are 200% higher than the background 

incentives, there is no meaningful response to an increase in downside risk. This result is 

also robust with respect to the variations of the baseline parameters we considered for MPS 

above. Only if α ψ is raised to (high) 0.5 and for $500 scales and higher a risk response 

becomes visible (Figure 9). 

22 Of course, the initial lottery needs to have some positive variance. The varied ỹ 2 posits a 50-50 chance 

to receive $80 or $120. 

23 Again, the bound is selected to preclude losses from a lottery. 

26


3.4 

D Skew=0 

D Skew=−2.67 

D Skew=−4.69 

S 


3 

D Skew=0 

D Skew=−2.67 

D Skew=−4.69 

S 

3.2 

2.5 


3 


2 

1.5 

2.8 

1 

2.6 

0.5 

2.4 

−50 0 50 100 150 


0 

−500 0 500 1000 1500 2000 2500 3000 3500 


Figure 8: Comparative statics for IDR of (varied) y 2 lottery at $100 and $3,000 scales. 


3 

D Skew=0 

D Skew=−2.67 

D Skew=−4.69 

S 


3 

D Skew=0 

D Skew=−2.67 

D Skew=−4.69 

S 


2.5 

2 


2.5 

2 

1.5 

1.5 

1 

1 

−100 0 100 200 300 400 500 600 


0.5 

−200 0 200 400 600 800 1000 1200 


Figure 9: Comparative statics for IDR of (varied) y 2 lottery at $500 and $1,000 scales with 

α ψ = 0.5. 

The ability to make inferences about fourth-order intertemporal preferences from variations 

in third-order risk seems thus strongly limited. Indeed, these are effects that are 

marginal to third-order preferences, which were already shown to be quite small. This result 

for the intertemporal case is even more limiting than Deck and Schlesinger’s (2014) finding 

regarding the non-detectability of fifth- and sixth-order risk effects from static experimental 

choices. 

27

Return Risk 

The case with return risk confirms our basic results for income risk, with a few interesting 

twists. We focus here on these points of dissimilarity. 24 Figure 10 shows the saving market 

equilibrium for MPS of return risk. We consider lotteries that involve 15, 80, and 120 

percentage-point deviations from r 2 = 20%. 25 In contrast to MPS of income risk (Figure 4), 

a MPS of return risk does not induce a positive precautionary effect. This occurs because 

the substitution effect of the risk change outweighs the precautionary effect under this calibration. 

The shifts in saving demand do not substantially vary with scaling, and are much 

larger than under income risk (especially for small scalings). 

4 x 1x Interest Lottery 

10−5 

3.8 

3.6 

D Var=0 

D Var=6400 

D Var=14400 

S 

3.5 x 10−5 5x Interest Lottery 

3 

D Var=0 

D Var=6400 

D Var=14400 

S 

3.4 

2.5 


3.2 

3 


2 

2.8 

1.5 

2.6 

2.4 

1 

2.2 

−50 0 50 100 150 


0.5 

−100 0 100 200 300 400 500 600 


Figure 10: Comparative statics for MPS of r lottery at $100 and $500 scales. 

Figure11indicates, moreover, thatunderreturnrisktheriskresponseincreasesbothwith 

the level of the risk preferences and with decreasing resistance to intertemporal substitution, 

the latter due to the expanded substitution effect. 

24 For the interested reader, an identical set of saving-market equilibria that is fully analogous to income 

risk is provided in Appendix C. 

25 Capping the downside return at -120% ensures that the agent is just as well off from a lottery, because 

the worst-case scenario is that the agent saves all of y 1 and loses it. 

28


10−5 

4.5 

4 

D Var=0 

D Var=6400 

D Var=14400 

S 

2.3 x 1x Interest Lottery 

10−3 

2.2 

2.1 

D Var=0 

D Var=6400 

D Var=14400 

S 


3.5 

3 

2.5 


2 

1.9 

1.8 

1.7 

2 

1.6 

1.5 

1.5 

1 

−50 0 50 100 150 


1.4 

−50 0 50 100 150 


Figure 11: Comparative statics for MPS of r lottery at $100 scales with α ψ = 0.5 (left) and 

ρ u = 0.75 (right). 

An IDR for return risk is constructed in a manner similar to income risk, by lowering 

the probability of the low outcome while shifting both outcomes left. Figure 12 present left 

skews up to about -7. While this skew is stronger than the highest considered under income 

risk (Figures 8–9), the effects on saving demand are similarly minute. Numerics confirm that 

the IDR also affects saving demand negatively, for reasons analogous to the MPS analysis. 

Moreover, like MPS, the magnitude of the effect is relatively constant across scalings. Even 

strong increases in the level of risk preferences do not lead to a sizable risk response. 


3 

D Skew=0 

D Skew=−4.69 

D Skew=−9.85 

S 


3 

D Skew=0 

D Skew=−4.69 

D Skew=−9.85 

S 


2.5 

2 


2.5 

2 

1.5 

1.5 

1 

1 

−100 0 100 200 300 400 500 600 


0.5 

−200 0 200 400 600 800 1000 1200 


Figure 12: Comparative statics for IDR of r lottery at $500 and $1,000 scales with α ψ = 0.5. 

29

5.2 Comparison with Expected Utility 

Figures 13–16 reconstruct the ỹ 2 MPS of Figures 4–7 under EU. The precautionary response 

of saving to an increase in income risk is clearly stronger here than under KP. 

3.8 

3.6 


D sd=0 

D sd=50 

D sd=100 

S 


3 

D sd=0 

D sd=250 

D sd=500 

S 


3.4 

3.2 

3 


2.5 

2 

2.8 

2.6 

1.5 

2.4 

−50 0 50 100 150 


1 

−100 0 100 200 300 400 500 600 


Figure 13: Comparative statics under EU for MPS of y 2 lottery at $100 and $500 scales. 

Figure 14 and 15 show the saving responses under varied preference parameters similar 

to Figures 5 and 6 above. In contrast to an increase of α ψ under KP (Figure 5), the increase 

of α EU to 0.5 yields a slight decrease in the saving rates and only a minute increase in the 

risk response. The two effects are in line with the increases in resistance to intertemporal 

substitution and prudence captured jointly by the EU parameters. In fact, apart from the 

typical higher risk response under EU, this graph corresponds closely to the respective KP 

graph with α u = 0.5 (Figure 18 in Appendix C). An analogous finding holds for a decrease 

of ρ EU to 0.75 as compared to Figure 18 in Appendix C (with, conversely, ρ ψ increased to 

1.5) and Figure 6. As in the KP case, the risk response is depressed for higher income levels 

(Figure 16). Moreover, also under EU similar results are evident for return risk (comparing 

Figure 26 in Appendix D to Figure 10, as well as Figure 27 in Appendix D to Figure 11 and 

Figure 21 in Appendix C). 

30

3.8 


D Var=0 

D Var=2500 

D Var=10000 

S 

3.5 


D Var=0 

D Var=62500 

D Var=250000 

S 

3.6 

3 


3.4 

3.2 


2.5 

3 

2 

2.8 

1.5 

2.6 

−50 0 50 100 150 


1 

−100 0 100 200 300 400 500 600 


Figure 14: Comparative statics under EU for MPS of y 2 lottery at $100 and $500 scales with 

α EU = 0.5. 


10−3 

2.2 

2.1 

D Var=0 

D Var=2500 

D Var=10000 

S 


10−3 

1.8 

D Var=0 

D Var=62500 

D Var=250000 

S 


2 

1.9 

1.8 


1.6 

1.4 

1.2 

1.7 

1.6 

1 

1.5 

−50 0 50 100 150 


0.8 

−100 0 100 200 300 400 500 600 


Figure 15: Comparative statics under EU for MPS of y 2 lottery at $100 and $500 scales with 

ρ EU = 0.75. 

31


10−6 

6.8 

6.6 

D Var=0 

D Var=10000 

D Var=40000 

S 


10−6 

6.2 

6 

5.8 

D Var=0 

D Var=62500 

D Var=250000 

S 


6.4 

6.2 

6 


5.6 

5.4 

5.2 

5.8 

5 

4.8 

5.6 

4.6 

5.4 

−50 0 50 100 150 200 250 


−100 0 100 200 300 400 500 600 


Figure 16: Comparative statics under EU for MPS of y 2 lottery at $200 and $500 scales for 

y = $3,000. 

Figure 17 reconstructs the IDR of y 2 risk from Figure 8. The effects of increasing risk 

are stronger in EU than under KP, but the differences are visible only at scalings of 10x or 

more (i.e., on the scale of y). Variations of the baseline preference parameters as for the 

MPS analysis do not lead to sizable changes in the risk response (Figure 24 in Appendix 

D). Under return risk, no change in the saving response to a third-order increase in risk is 

detectable visually (Figures 28–29 in Appendix D). This finding persists even when the left 

skew increases to nearly -10. 

32


3.4 

D Skew=0 

D Skew=−2.67 

D Skew=−4.69 

S 


3 

D Skew=0 

D Skew=−2.67 

D Skew=−4.69 

S 

3.2 

2.5 


3 


2 

1.5 

2.8 

1 

2.6 

0.5 

2.4 

−50 0 50 100 150 


0 

−500 0 500 1000 1500 2000 2500 3000 3500 


Figure 17: Comparative statics under EU for IDR of (varied) y 2 lottery at $100 and $3000 

scales. 

We thus find two meaningful implications of assuming that agents act according to EU. 

First, utility curvature under EU primarily conveys information about intertemporal preferences 

and not risk responses. Second, for levels of risk preferences that seem a priori 

reasonable, EU implies a slightly stronger precautionary-saving motive. This latter observation 

suggests that the amount of precautionary saving could be used as a discriminant 

between the two models. 

Because decisions under KP and EU both seem to be driven primarily by intertemporal 

substitution, one could question whether any predictive difference exists between the two 

preference assumptions. In comparing the KP and EU calibrations, we see that equilibrium 

saving is often noticeably different under KP than under EU, especially when lottery stakes 

are more substantial. Thus, like the empirical macroeconomics literature, we are inclined to 

view the KP/EU choice as a relevant specification issue, even at the agent level. 

33

6 Conclusion 

Empirical knowledge of preferences underlying intertemporal decisions under risk, notably 

of higher order, is still few detailed. While a main focus of empirical research, especially 

experiments,hasbeenon(static)riskpreferences,includinghigher-order,andondiscounting, 

consumption smoothing has thus far almost exclusively been studied using revealed or selfdeclared 

data. 26 The theoretical literature shows the potential importance of higher-order 

preferences also in temporal contexts, but we are not aware of any empirical study that 

measures such preferences at the individual level. A number of confounding factors seem 

to hamper the coordination between existing pieces of knowledge. Many studies still adopt 

a convenient power-utility specification under EU. However, studies that focus on the risk 

domain and studies that examine the intertemporal domain have found shapes of relative 

preference-intensity coefficients that vary, potentially in a converse way per domain. Power 

utility, however, precludes to account for the one or the other variation. Varying shapes are 

alsoparticularlyrelevanttojudgetheimportanceofhigher-ordereffects. Inmacroeconomics, 

the distinct contributions of risk and time preferences to intertemporal decisions under risk 

are well-established (e.g., Epstein and Zin 1991), but the standard modeling of recursive 

preferences similarly forgoes any account of varying shapes of such preference coefficients. A 

more flexible modeling of EU, where shapes can vary, in turn, does not allow to disentangle 

preferences along the risk and time dimensions. 

We adopt the more flexible framework of KP preferences to study the distinct roles of 

risk and time preferences, also of higher order, in intertemporal decisions under risk. The 

decision framework we focus on is the canonical two-period consumption/saving model of 

the theoretical precautionary-saving literature. 27 On theoretical level, we show how risk and 

intertemporal preferences are consistently jointly underlying saving variations in response 

to any change in a future element carrying risk. Especially, we extend the Eeckhoudt and 

26 An exception is the Andreoni and Sprenger (2012a) experiment. Their setup, however, does not allow 

to observe a subject’s saving decision explicitly (Bostian et al. 2015). 

27 To our knowledge, none of the previous applied or experimental literature has used a KP specification. 

34

Schlesinger (2008) conditions on EU for saving increases under higher-order risk increases to 

the case of KP preferences. 

For quantitative results, we simulate an empirical implementation of the model using, as 

far as available, typical parameter values. We show that the main carriers of saving variation 

in intertemporal decisions under risk, according to the model, are intertemporal preferences. 

Risk preferences only play a minor role. This finding is significant because it underlines the 

importance of a detailed and specific knowledge of intertemporal preferences for all studies 

that involve predictions of temporal behaviors. Even if EU is found correct for a subject 

or a sample, results from static risk-preference elicitation cannot simply be adopted in the 

intertemporal domain. 28 

In a second step, we study the importance of higher-order risk effects in an intertemporal 

context, that is, the extent to which saving reacts to second- and higher-order increases of 

future risk. The simulations show that second- and third-derivative effects are the most 

essential features of preferences in the decision in question. Effects already from the fourth 

order on have essentially no impact. This result is reminiscent of Deck and Schlesinger’s 

(2014) finding that risk preferences of the fifth and sixth order are not identifiable from 

static lottery choices. Our result for an intertemporal context is stronger as here already 

responses to third-order risk changes are predicted not to be identifiable. 

For EU, the risk effects are stronger than under KP. This difference may allow to empirically 

discriminate between the preference concepts. However, the few-relevance result 

regarding third- and higher-order risk effects from the KP case is persistent also here. 

Our theoretical and numerical results do not substitute for empirical verification. Given 

the clear modeling of risk increases, the preference conditions for saving increases in response 

to risk changes are, in principle, amenable to empirical testing. For example, for a suitable 

selection of incentives and a sufficient number of observations preference intensities could 

28 Such adoption has been common practice in parts of the literature (e.g., Andersen et al. 2008, Johansson- 

Stenman 2010, Schechter 2007). 

35

e measured based on experiments that implement the modeled decision. 29 Such empirical 

research might reveal the relevance of other preference aspects, such as loss aversion or 

hyperbolic discounting, in decisions of this kind. Such evidence should, in turn, be used to 

enrich the theoretical predictions developed in this paper. 

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40

Appendix 

A Proofs for Section 3 

Proof of Remark 1: 

The statements derive by applying the definition of an n th central moment. 

Proof of Proposition 1: 

The expressions in equations (12a)–(12d) arise from straightforward calculation of all derivatives 

involved. The sign conditions and the qualifications in condition (12c) are derived from 

Kimball and Weil (2009: Appendix A). For the expression in (12d) note that 

∂CE(˜c 2 ) 

∂ ˜R 2 

= s 1 CE y2 (˜c 2 ) and ∂CE′ (˜c 2 ) 

∂ ˜R 2 

= s 1 CE ′ y 2 

(˜c 2 )+CE y2 (˜c 2 ) 


To prove Proposition 2 it remains to show that, if the successive derivatives of u(.) and 

[ ] 

CE(.) alternate in sign starting with a positive first, then, also sgn d n+1 u(CE(˜c 2 )) 

= (−1) n 

ds n+1 

for all n = 1,2,...,N, as required for condition (13) to hold. The proof will rely on Lemma 

2 below, which uses the following definition of the class of N-increasing concave functions 

(e.g., Denuit and Eeckhoudt 2010): 

Definition 1 The class U D K−icv 

of all differentiable N-increasing concave functions f, with 

D ⊆ R, is defined as 

U D K−icv := { f : D → R ∣ ∣(−1) k+1 f k (x) ≥ 0 for k = 1,2,...,K } . 

In Definition 1, the term ‘increasing’ relates to the non-negative sign of the first derivative, 

the term ‘concave’ refers to the non-positive second derivative of the functions in this class. 

41

K represents the number of times the functions are differentiable. 

Lemma 2 Let the two univariate real-valued functions F,G ∈ UK−icv D . Then, also H(x) ≡ 

F(G(x)) ∈ U D K−icv . 

Proof. The result follows by straightforward calculation applying the chain and product 

rules of differentiation. Note that in any derivative, deriving any term in one function makes 

the new term have the opposite sign of the original. 

Proposition 2 derives then as follows. 

Proof. Lemma 2 holds in particular for F ≡ u(.), G ≡ CE(.), and K = N + 1. As a 

consequence, also the successive derivatives of the composite function H(x) ≡ u(CE(x)) 

alternate in sign starting with a positive first. 

For illustrative purposes, we indicate the derivatives of period-two utility under KP 

preferences. We use the notation dn CE(˜c 2 ) 

ds n 1 

≡ CE (n) (˜c 2 ) (extended from equation (7)) and the 

shorthands u ≡ u(CE(˜c 2 )) and CE ≡ CE(˜c 2 ). The first four derivatives read: 

d 

ds u(CE(˜c 2)) = u ′ CE ′ 

d 2 

ds 2u(CE(˜c 2)) = u ′′ CE ′2 +u ′ CE ′′ 

d 3 

ds 3u(CE(˜c 2)) = u ′′′ CE ′3 +3u ′′ CE ′ CE ′′ +u ′ CE ′′′ 

d 4 

ds 4u(CE(˜c 2)) = u ′′′′ CE ′4 +6u ′′′ CE ′2 CE ′′ +3u ′′ CE ′′2 +4u ′′ CE ′ CE ′′′ +u ′ CE ′′′′ 

Based on Faà di Bruno’s formula, the n th derivative of u(CE(˜c 2 )) can be stated as: 

d n 

ds nu(CE(˜c 2)) = ∑ n! n 

Π n j=1 k j! u∑ j=1 k j 

(CE(˜c 2 ))·Π n j=1 

( ) CE (j) kj 

(˜c 2 ) 

(17) 

j! 

the sum being over all n-tupels of non-negative integers (k 1 ,...,k n ) satisfying ∑ n 

i=1 i·k i = n. 

42


To prove Proposition 3, the following lemma is helpful. 

Lemma 3 Consider the two functions f ≡ F ′ and g ≡ G ′ , with F and G defined as in 

Lemma2, andI(x) withx ∈ D ⊆ Rbeingthe identity. Then, h(x) ≡ −f(g(I(x)))g(I(x))I(x) ∈ 

UN−icv D , if − g(m) (I(x)) 

I(x) ≥ m for m = 

g (m−1) 1,2,...,N.30 

(I(x)) 

Proof. The lemma follows, similar to Lemma 2, by straightforward calculation of the 

successive derivatives of h(x), noting that I ′ (x) = 1 and I (j) (x) = 0 for all j > 1 and factoring 

out such that terms of the form [ g (m) (I(x))I(x)+g (m−1) (I(x)) ] arise in brackets for 

m = 1,2,...,N. The successive h(x) derivatives alternate in sign as indicated in the lemma 

if sgn [ g (m) (I(x))I(x)+g (m−1) (I(x)) ] = (−1) m . 

Proposition 3 derives then as follows. 

Proof. Condition (13) holds analogously for any sure second-period income y 2 and risky 

return ˜R 2 = ˜R 2,i , i = a,b, if h(R 2 ) ≡ − ∂u(CE(c 2)) 

∂s 1 

∈ UN−icv D . Proposition 3, then, follows 

from Lemma 3 by setting f ≡ u ′ (.), g ≡ CE ′ (.), and I(.) ≡ R 2 . Regarding the notation 

in the level conditions on CE(c 2 ), note that for c 2 = y 2 + R 2 s 1 and s 1 = s ∗ 1, it holds that 

∂CE(c 2 ) 

∂s 1 

= R 2 CE y2 and ∂CE y 

2 

n−1 

∂R 2 

(c 2 ) 

= s 1 CE y n 

2 

for n = 1,...,N. 


Proof. Income risk. Define, similar to Liu (2014), the risk premium π y 2 

ψ 

deterioration from ỹ 2,a to ỹ 2,b , where ỹ 2,a dominates ỹ 2,b via NSD, as 

associated with a 

Eψ(ỹ 2,b +s 1 R 2 ) = Eψ(ỹ 2,a −π y 2 

ψ +s 1R 2 ) 

(18a) 

30 Contrary to the definitions of functions f and g, function h(.) does not analogously constitute the first 

derivative of function H(.) of Lemma 2. 

43

Similar to the case with compensating premia in Kimball and Weil, it holds, when ψ exhibits 

DARA, for all δ ≥ 0 that: 

Eψ(ỹ 2,b +s 1 R 2 +δ) ≥ Eψ(ỹ 2,a −π y 2 

ψ +s 1R 2 +δ) 

so that, 

Eψ ′ (ỹ 2,b +s 1 R 2 ) ≥ Eψ ′ (ỹ 2,a −π y 2 

ψ +s 1R 2 ) 

(18b) 

Comparing (18b) to the defining equation (14a) for θ y 2 

eu specified for f = ψ, shows that, 

thanks to ψ ′′ ≤ 0, θ y 2 

eu ≥ π y 2 

ψ . To compare the precautionary premia θy 2 

ru and θ y 2 

eu note that, 

starting from (14b) defining θ y 2 

ru, it holds that: 

u ′ (CE(ỹ 2,a −θ y 2 

ru +s 1 R 2 ))CE y2 (ỹ 2,a −θ y 2 

ru +s 1 R 2 ) = u ′ (CE(ỹ 2,b +s 1 R 2 ))CE y2 (ỹ 2,b +s 1 R 2 ) 

= u ′ (CE(ỹ 2,b +s 1 R 2 )) E 1ψ ′ (ỹ 2,a −π y 2 

ψ +s 1R 2 ) 

ψ ′ (CE y2 (ỹ 2,b +s 1 R 2 )) 

≤ u ′ (CE(ỹ 2,b +s 1 R 2 )) E 1ψ ′ (ỹ 2,a −θ y 2 

eu +s 1 R 2 ) 

ψ ′ (CE y2 (ỹ 2,b +s 1 R 2 )) 

= E 1 u ′ (ỹ 2,a −θ y 2 

eu +s 1 R 2 ) 

where the second line follows from the definition of CE and (18a), the inequality in the third 

line follows from θ y 2 

eu ≥ π y 2 

ψ , and the last line is due to the identity u eu = u = ψ under EU. 

As a consequence, θ y 2 

eu ≥ θ y 2 

ru, establishing the income-risk case in Proposition 4. 

Return risk. The proof is analogous as under income risk, starting only now, following 

Heinzel (2015a,b), from the defining equation of the risk premium π r 2 

ψ 

deterioration from ˜R 2,a to ˜R 2,b , where ˜R 2,a dominates ˜R 2,b via NSD, 

associated with a 

E 1 ψ(y 2 +s ∗a 

1 ˜R 2,a −π r 2 

ψ ) = E 1ψ(y 2 +s ∗b 

1 ˜R 2,b ) 

and using equations (15) defining θ r 2 

eu and θ r 2 

ru. 

44

B Comparative Statics in Preference Parameters 

We consider the parameter comparative statics in the KP and EU versions of the model 

operationalized with expo-power utility. The parameters specific to the KP version are 

Θ KP = (α ψ ,ρ ψ ,α u ,ρ u ), where α ψ ,ρ ψ determine risk preferences and α u ,ρ u intertemporal 

preferences. The parameters specific to EU are Θ EU = (α EU ,ρ EU ). We concentrate on the 

ranges α,ρ > 0 with ρ ≠ 1 across all cases. 

B.1 KP Preferences Under Expo-Power Utility 

For the recursive KP specification, the following proposition arises: 

Proposition 5 Let s ∗ denote the optimal solution to equation (11a), Ω ⊂ R + be the sample 

space of random variable ˜c 2 with c min 

2 ≡ min(Ω) and c max 

2 ≡ max(Ω), and ¯ρ 1,2 (x) ≡ 1 + 

( √ ) 

1 

x ∓ 1 

− 1 for x as below. A marginal increase in a preference parameter changes 

2 4 xln(x) 

optimal saving ceteris paribus as follows. 

[ ] [ [ ] 

ds 

∗ ∂ψ ′ (˜c 2 ) 

[ 

] 

sgn = sgn E 1 

˜R2 +E 1 ψ ′ (˜c 2 )˜R 2 

]CE αψ (˜c 2 )(ARA(CE(˜c 2 ))−ARIS(CE(˜c 2 ))) 

dα ψ ∂α ψ 

[ ] ds 

∗ 

sgn = sgn 

dρ ψ 

≷ 0 for ρ ψ ≷ 1, if ARA(CE(˜c 2 )) ≥ ARIS(CE(˜c 2 )) . 

[ [ ] ∂ψ ′ (˜c 2 ) 

[ 

E 1 

˜R2 +E 1 

∂ρ ψ 

> 0 for ρ ψ < ¯ρ ψ1 (c min 

2 ) ∨ ρ ψ > ¯ρ ψ2 (c max 

2 ) and, if sgn 

(19a) 

] 

ψ ′ (˜c 2 )˜R 2 

]CE ρψ (˜c 2 )(ARA(CE(˜c 2 ))−ARIS(CE(˜c 2 ))) 

[ ] ds 

∗ 

< 0, then (19b) 

dρ ψ 

ρ ψ ∈ (¯ρ 1 (c min 

2 ), ¯ρ 2 (c max 

2 ) ) , both given that ARA(CE(˜c 2 )) ≥ ARIS(CE(˜c 2 )). 

[ ] [ 

] 

ds 

∗ 

sgn = sgn − ∂u′ (c 1 ) 

+β · ∂u′ (CE(˜c 2 )) 

·CE ′ (˜c 2 ) 0 (19c) 

dα u ∂α u ∂α u 

c 

( ) 

1 

depending on 

CE(˜c 2 ) 1 if ρ u > (

[ ] ds 

∗ 

sgn 

dρ u 

[ 

= sgn 

⎧ 

⎪⎨ 

⎪⎩ 

− ∂u′ (c 1 ) 

∂ρ u 

+β · ∂u′ (CE(˜c 2 )) 

∂ρ u 

·CE ′ (˜c 2 ) 

> 0 , if c 1 ≶ CE(˜c 2 ) ∧ ρ v ≷ 1+CE(˜c 2 )lnCE(˜c 2 ) 

= 0 , if c 1 = CE(˜c 2 ) 

. 

< 0 , if c 1 ≶ CE(˜c 2 ) ∧ ρ v ≶ 1+c 1 lnc 1 

] 

(19d) 

Note that the bounds on the ρ parameter of expo-power utility in conditions (19b) and 

(19d), as well as similarly in (19j) below, are endogenous to the levels of the optimal choices. 

The ARA(CE(˜c 2 )) ≥ ARIS(CE(˜c 2 )) restriction in conditions (19a) and (19b) claims the 

presence of a preference for the early resolution of risk (e.g., Epstein et al. 2014). 

Proof. For conditions (19a), note that, given ARA(CE(˜c 2 )) ≥ ARIS(CE(˜c 2 )), the sign of 

[ ] 

ds ∗ 

dα ψ 

dependsonE ∂ψ ′ (˜c 2 ) 

1 

˜R2 

∂α ψ 

andCE αψ (˜c 2 ) ≡ ∂CE(˜c 2) 

∂α ψ 

= ∂ψ−1 (Eψ(˜c 2 )) 

∂α ψ 

+ψ −1′ ∂ψ(˜c 

(Eψ(˜c 2 ))E 2 ) 

1 ∂α ψ 

, 

as all other terms are positive. Denote f a function of the expo-power form (16a) or (16b). 

Then, generally, for 

( ) −α 

f ′ (x) = x −ρ exp 

1−ρ x1−ρ 

(19e) 

it holds, for x > 0 and because f ′ (.) > 0, that 

∂f ′ (x) 

∂α = − x1−ρ 

1−ρ f′ (x) ≷ 0 for ρ ≷ 1 

(19f) 

Moreover,notethatinCE αψ (˜c 2 ), ∂ψ−1 (Eψ(˜c 2 )) 

∂α ψ 

= 1 

[ 

Eψ(˜c 2 ) 

+ ln(1−α ψEψ(˜c 2 )) 

α ψ 1−α ψ Eψ(˜c 2 ) 

α ψ 

](ψ −1 (Eψ(˜c 2 ))) ρ ψ 

≷ 

0forρ ψ ≷ 1, because, whileα ψ andψ −1 (Eψ(˜c 2 )) > 0, theexpressioninthesquarebracketsis 

( )] 

≷ 0forρ ψ ≷ 1. Toderivethelatterresult,notethatforEψ(˜c 2 ) = 1 −αψ 

α ψ 

[1−Eexp 

1−ρ ψ˜c 1−ρ ψ 

2 

( ) 

−αψ 

(cf. equation (16a)) and z ≡ Eexp 

1−ρ ψ˜c 1−ρ ψ 

2 ≷ 1 for ρ ψ ≷ 1. Therefore, it holds for the 

term in square brackets that 

1−z 

α ψ z + ln(z) 

α ψ 

0 ⇔ 1 z 1−ln(z) 

46

The latter relation holds with equality if z = 1, and it is d 1 

= − 1 ≷ − 1 = d [1−ln(z)] 

dz z z 2 z dz 

[ ] 

for z ≷ 1 (given that α ψ ,z > 0). Due to ψ −1′ (Eψ(˜c 2 )) > 0, thus, E ∂ψ ′ (˜c 2 ) 

1 

˜R2 

∂α ψ 

and 

CE αψ (˜c 2 ) ≷ 0 for ρ ψ ≷ 1. Conditions (19a) follow. 

For conditions (19b), note that, given ARA(CE(˜c 2 )) ≥ ARIS(CE(˜c 2 )), the sign of ds∗ 

dρ 

[ ] 

ψ 

depends on E ∂ψ ′ (˜c 2 ) 

1 

˜R2 

∂ρ ψ 

and CE ρψ (˜c 2 ) ≡ ∂CE(˜c 2) 

∂ρ ψ 

= ∂ψ−1 (Eψ(˜c 2 )) 

∂ρ ψ 

+ψ −1′ ∂ψ(˜c 

(Eψ(˜c 2 ))E 2 ) 

1 ∂ρ ψ 

, as 

all other terms are positive. Generally, for f ′ as in (19e) it holds, for x > 0 and because 

f ′ (.) > 0, that 

∂f ′ (x) 

∂ρ 

where ¯ρ 1 (x) ≡ 1+x 

[ ( 

= f ′ x 

(x) 

(1−ρ) −ln(x) 1+ x )] 

2 1−ρ 

( 

1 

2 − √ 

1 

− 1 

4 xln(x) 

⎧ 

⎪⎨ 

⎪⎩ 

) 

and ¯ρ 2 (x) ≡ 1+x 

> 0 , if ρ < ¯ρ 1 (x) ∨ ρ > ¯ρ 2 (x) 

= 0 , if ρ = ¯ρ 1 (x) ∨ ρ = ¯ρ 2 (x) 

< 0 , if ρ ∈ (¯ρ 1 (x), ¯ρ 2 (x)) 

( 

1 

2 + √ 

1 

− 1 

4 xln(x) 

(19g) 

) 

arethesolutions 

to the following second-order polynomial, equivalent to the expression in square brackets in 

(19g), 

ln(x)(1−ρ) 2 +xln(x)(1−ρ)−x = 0 

Because f ′ (.) > 0, the sign of ∂f′ (x) 

∂ρ 

in (19g). Note, moreover, that in CE ρψ (˜c 2 ), 

depends on the sign of the expression in square brackets 

∂ψ −1 (Eψ(˜c 2 )) 

∂ρ ψ 

[ 

1 

= − 

(1−ρ ψ ) 2 

( 1−ρψ 

ln 

α ψ 

( ( ))) ] 1−ρψ 

ln Eexp ˜c 1−ρ ψ 

2 +1 ψ −1 (Eψ(˜c 2 )) > 0 

α ψ 

because, while (1 − ρ ψ ) −2 and ψ −1 (Eψ(˜c 2 )) > 0, the term in square brackets in the latter 

( ) 

expression is negative due to Eexp −α ψ˜c 1−ρ ψ 

2 < exp(α ψ exp(−1)). Due to (19g), then 

E 1 

[ ∂ψ ′ (˜c 2 ) 

∂ρ ψ 

˜R2 

] 

if E 1 

[ ∂ψ ′ (˜c 2 ) 

∂ρ ψ 

˜R2 

] 

> 0 , if ρ ψ < ¯ρ ψ1 (c min 

2 ) ∨ ρ ψ > ¯ρ ψ2 (c max 

2 ) 

< 0 , then ρ ψ ∈ (¯ρ 1 (c min 

2 ), ¯ρ 2 (c max 

2 ) ) 

47

For ρ ψ < ¯ρ ψ1 (c min 

2 ) or ρ ψ > ¯ρ ψ2 (c max 

2 ), with ∂ψ−1 (Eψ(˜c 2 )) 

∂ρ ψ 

> 0, also ∂CE(˜c 2) 

∂ρ 

[ ] [ ] [ ] 

ψ 

ds 

sgn 

∗ 

dρ ψ 

> 0. IfE ∂ψ ′ (˜c 2 ) 

ds 

1 

˜R2 

∂ρ ψ 

< 0,thenCE ρψ (˜c 2 )and,thus,sgn 

∗ 

dρ ψ 

< 0,if 

−1. 

and, thus, 

∂ψ −1 (Eψ(˜c 2 ))/∂ρ ψ 

ψ −1′ (Eψ(˜c 2 ))E 1 

∂ψ(˜c 2 ) 

∂ρ ψ 

< 

Forconditions(19c), notethat, whileβ andCE ′ (˜c 2 ) > 0, thesignsof ∂u′ (c 1 ) 

∂α u 

and ∂u′ (CE(˜c 2 )) 

∂α u 

depend on ρ u as in (19f). Rewrite the sum to be signed in (19c) using formula (19f) for u ′ (.): 

c 1−ρu 

1 

1−ρ u 

u ′ (c 1 )−β CE(˜c 2) 1−ρu 

1−ρ u 

u ′ (CE(˜c 2 ))CE ′ (˜c 2 ) 0 

For ρ u > 1, this is equivalent to 

c 1−ρu 

1 u ′ (c 1 )−βCE(˜c 2 ) 1−ρu u ′ (CE(˜c 2 ))CE ′ (˜c 2 ) 0 

⇔ β u′ (CE(˜c 2 )) 

CE ′ (˜c 

u ′ 2 ) 

(c 1 ) 

( ) 1−ρu c1 

(19h) 

CE(˜c 2 ) 

Note that the expression on the left-hand side of (19h) is identical to the left-hand side of 

the Euler equation of the optimal saving problem under SKP preferences and is, thus, (for 

an interior choice) equal to one. The respective condition in (19c) follows. For ρ v > 1, the 

reasoning is analogous. 

Forconditions(19d), notethat, iftheEulerequationoftheoptimalsavingproblemunder 

KP preferences holds with equality, then, 

sgn 

[ ds 

∗ 

dρ u 

] 

0 ⇔ CE(˜c 2)−(lnCE(˜c 2 ))(1−ρ u +CE(˜c 2 )) 

c 1 −(lnc 1 )(1−ρ u +c 1 ) 

1 

where the latter condition holds with equality if and only if c 1 = CE(˜c 2 ). For c 1 ≠ CE(˜c 2 ), 

the sign depends on whether the numerator and denominator move uniformly in the one or 

the other direction. Conditions (19d) arise. 

48

B.2 Expected Utility Under Expo-Power Utility 

The EU analog of Proposition 5 is: 

Proposition 6 Let s ∗ denote the optimal solution to equation (11a). A marginal increase 

in a preference parameter changes optimal saving ceteris paribus as follows. 

[ ] [ [ ]] 

ds 

∗ 

sgn = sgn − ∂u′ (c 1 ) ∂v ′ (˜c 2 ) 

+β ·E 1 

˜R2 

dα EU ∂α EU ∂α EU 

0 

[ ) ] 

1−ρEU (˜c2 v ′ (˜c 2 ) 

depending on βE 1 

c 1 u ′ (c 1 ) ˜R 2 

[ ] [ [ ]] 

ds 

∗ 

sgn = sgn − ∂u′ (c 1 ) ∂v ′ (˜c 2 ) 

+β ·E 1 

˜R2 0 

dρ EU ∂ρ EU ∂ρ EU 

( ) 

 

[ v ′ (˜c 2 ) 

depending on βE 1 

u ′ (c 1 ) · ˜c ] 

2 −(1−ρ EU )ln˜c 2 

· 

c 1 −(1−ρ EU )lnc ˜R 2 

1 

1 if ρ EU > (

C Additional Figures under KP Preferences 

C.1 Income Scenarios 

Higher-Order Risk Effects 


10−5 

3.4 

3.2 

D Var=0 

D Var=10000 

D Var=40000 

S 


D Var=0 

D Var=10000 

D Var=40000 

S 


3 

2.8 

2.6 


3 

2.5 

2.4 

2.2 

2 

−50 0 50 100 150 200 250 


2 

−50 0 50 100 150 200 250 


Figure 18: Comparative statics for MPS of y 2 lottery at $200 scale with α u = 0.5 (left) and 

ρ ψ = 1.5 (right). 

C.2 Return Scenarios 

Preference Effects 

4 x 10−5 1x Interest Lottery 

D α ψ 

=0.01 


D ρ ψ 

=0.3 

3.8 

3.6 

D α ψ 

=0.1 

D α ψ 

=1 

S 

3.8 

3.6 

D ρ ψ 

=0.7 

D ρ ψ 

=2 

S 


3.4 

3.2 

3 


3.4 

3.2 

3 

2.8 

2.8 

2.6 

2.6 

2.4 

−50 0 50 100 150 


2.4 

−50 0 50 100 150 


Figure 19: Comparative statics for α ψ (left) and ρ ψ (right) at $100 scale. 

50



10−5 

3.5 

S α u 

=0.01 

D 

S α u 

=0.1 

D 

S α u 

=1 

D 



10−3 

2 

1.5 

1 

S ρ u 

=0.75 

D 

S ρ u 

=1.1 

D 

S ρ u 

=1.5 

D 

3 

0.5 

2.5 

−50 0 50 100 150 


0 

−50 0 50 100 150 


Figure 20: Comparative statics for α u (left) and ρ u (right) at $100 scale. 



3 

D Var=0 

D Var=6400 

D Var=14400 

S 

1.8 


D Var=0 

D Var=6400 

D Var=14400 

S 

2.5 

1.6 


2 

1.5 


1.4 

1.2 

1 

1 

0.5 

0.8 

0 

−100 0 100 200 300 400 500 600 


0.6 

−100 0 100 200 300 400 500 600 


Figure 21: Comparative statics for MPS of r lottery at $500 scale with α ψ = 0.5 (left) and 

ρ u = 0.75 (right). 

51


6.8 

6.6 

6.4 

D Var=0 

D Var=6400 

D Var=14400 

S 

6.5 


6 

D Var=0 

D Var=6400 

D Var=14400 

S 


6.2 

6 


5.5 

5 

5.8 

5.6 

4.5 

5.4 

4 

5.2 

−50 0 50 100 150 200 250 


3.5 

−100 0 100 200 300 400 500 600 


Figure 22: Comparative statics for MPS of r lottery at $200 and $500 scales for y = $3,000. 


3.4 

D Skew=0 

D Skew=−4.69 

D Skew=−9.85 

S 


3 

D Skew=0 

D Skew=−4.69 

D Skew=−9.85 

S 

3.2 

2.5 


3 


2 

1.5 

2.8 

1 

2.6 

0.5 

2.4 

−50 0 50 100 150 


0 

−500 0 500 1000 1500 2000 2500 3000 3500 


Figure 23: Comparative statics for IDR of r lottery at $100 and $3,000 scales. 

52

D Additional Figures under Expected Utility 

D.1 Income Scenarios 


3.5 


D Skew=0 

D Skew=−2.67 

D Skew=−4.69 

S 

3.5 


D Skew=0 

D Skew=−2.67 

D Skew=−4.69 

S 

3 

3 


2.5 


2.5 

2 

2 

1.5 

1.5 

1 

1 

−100 0 100 200 300 400 500 600 


0.5 

−200 0 200 400 600 800 1000 1200 


Figure 24: Comparative statics under EU for IDR of (varied) r lottery at $500 and $1,000 

scales with α EU = 0.5. 

D.2 Return Scenarios 

Preference Effects 



10−5 

3.5 

S α EU 

=0.01 

D 

S α EU 

=0.1 

D 

S α EU 

=1 

D 



10−3 

2 

1.5 

1 

S ρ EU 

=0.75 

D 

S ρ EU 

=1.1 

D 

S ρ EU 

=1.5 

D 

3 

0.5 

2.5 

−50 0 50 100 150 


0 

−50 0 50 100 150 


Figure 25: Comparative statics for α eu (left) and ρ eu (right) at $100 scale. 

53



10−5 

3.8 

3.6 

D Var=0 

D Var=6400 

D Var=14400 

S 


10−5 

3 

D Var=0 

D Var=6400 

D Var=14400 

S 

3.4 

2.5 


3.2 

3 

2.8 


2 

2.6 

1.5 

2.4 

1 

2.2 

2 

−50 0 50 100 150 


0.5 

−100 0 100 200 300 400 500 600 


Figure 26: Comparative statics under EU for MPS of r lottery at $100 and $500 scales. 


10−5 

4 

3.8 

D Var=0 

D Var=6400 

D Var=14400 

S 


10−3 

2.2 

2.1 

D Var=0 

D Var=6400 

D Var=14400 

S 

3.6 

2 


3.4 

3.2 

3 


1.9 

1.8 

2.8 

1.7 

2.6 

1.6 

2.4 

1.5 

2.2 

−50 0 50 100 150 


1.4 

−50 0 50 100 150 


Figure 27: Comparative statics under EU for MPS of r lottery at $100 scale with α EU = 0.5 

(left) and ρ EU = 0.75 (right). 

54


3.4 

D Skew=0 

D Skew=−4.69 

D Skew=−9.85 

S 


3 

D Skew=0 

D Skew=−4.69 

D Skew=−9.85 

S 

3.2 

2.5 


3 


2 

1.5 

2.8 

1 

2.6 

0.5 

2.4 

−50 0 50 100 150 


0 

−500 0 500 1000 1500 2000 2500 3000 3500 



scales. 

3.5 


D Skew=0 

D Skew=−4.69 

D Skew=−9.85 

S 

3.5 


D Skew=0 

D Skew=−4.69 

D Skew=−9.85 

S 

3 

3 


2.5 


2.5 

2 

2 

1.5 

1.5 

1 

1 

−100 0 100 200 300 400 500 600 


0.5 

−200 0 200 400 600 800 1000 1200 



scales with α EU = 0.5. 

55

The FOODSECURE project in a nutshell 

Title 

Funding scheme 

Type of project 

Project Coordinator 

Scientific Coordinator 

Duration 

FOODSECURE – Exploring the future of global food and nutrition security 

7th framework program, theme Socioeconomic sciences and the humanities 

Large-scale collaborative research project 

Hans van Meijl (LEI Wageningen UR) 

Joachim von Braun (ZEF, Center for Development Research, University of Bonn) 

2012 - 2017 (60 months) 

Short description 

In the future, excessively high food prices may frequently reoccur, with severe 

impact on the poor and vulnerable. Given the long lead time of the social 

and technological solutions for a more stable food system, a long-term policy 

framework on global food and nutrition security is urgently needed. 

The general objective of the FOODSECURE project is to design effective and 

sustainable strategies for assessing and addressing the challenges of food and 

nutrition security. 

FOODSECURE provides a set of analytical instruments to experiment, analyse, 

and coordinate the effects of short and long term policies related to achieving 

food security. 

FOODSECURE impact lies in the knowledge base to support EU policy makers 

and other stakeholders in the design of consistent, coherent, long-term policy 

strategies for improving food and nutrition security. 

EU Contribution 

Research team 

€ 8 million 

19 partners from 13 countries 

FOODSECURE project office 

LEI Wageningen UR (University & Research centre) 

Alexanderveld 5 

The Hague, Netherlands 

T +31 (0) 70 3358370 

F +31 (0) 70 3358196 

E foodsecure@wur.nl 

I www.foodscecure.eu 

This project is funded by the European Union 

under the 7th Research Framework Programme 

(theme SSH) Grant agreement no. 290693

Consumption Smoothing and Precautionary Saving under Recursive Preferences

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