Homo Economicus Is Stochastically Patient

Homo Economicus is Stochastically Patient

Eric R. Ulm, Daniel Bauer, Justin Sydnor∗

July 2020

Abstract

DeJarnette et al. (2020) introduce the concept of generalized expected discounted utility (GEDU) and show that it can explain the empirical finding that people tend to be risk-averse over time lotteries but violates an intuitively appealing property they call stochastic impatience. This note extends their analysis to the classical case of an agent that maximizes expected utility of future consumption subject to a budget constraint. We show that these classical agents adhere to GEDU preferences with risk aversion over time lotteries and will often be stochastically patient. We argue that while stochastic impatience may be an intuitively appealing criterion when lottery payouts map directly to concurrent utility experiences, when agents can save, bor- row, and optimize, stochastic patience is a natural result. We present results of a simple survey experiment consistent with this view, finding that individuals exhibit stochastic impatience for small-scale monetary risks but stochastic patience for large-scale risks. JEL classification: D81; D90; C91. Keywords: Risk and Time Preferences, Stochastic Impatience, Expected Consumption Utility.

Introduction

DeJarnette et al. (2020) advance our understanding of consumers’ preferences across risk and time by analyzing time lotteries. As one of their key results, they show that under Expected Discounted Utility (EDU), individuals must be risk-seeking over time lotteries (RSTL). Yet their experimental results show that the majority of people are risk-averse over time lotteries (RATL). They show that a modiﬁcation of EDU they call Generalized EDU (GEDU) can account for this risk aversion over time lotteries. However, they highlight that a necessary sacriﬁce for overcoming RSTL is to accept that individuals violate an intuitively appealing property they call stochastic impatience,

∗Ulm: School of Economics and Finance, Victoria University of Wellington, 23 Lambton Quay Wellington, NZ, [email protected]. Bauer/Sydnor: Department of Risk and Insurance, University of Wisconsin-Madison, 975 Uni- versity Avenue, Madison, WI 53706, USA, [email protected]/[email protected].

1 HOMO ECONOMICUSIS STOCHASTICALLY PATIENT 2 the “risky counterpart to the standard notion of Impatience” that individuals “should [behave] in accordance with” (p. 621). They show that this incompatibility plagues a broad class of models, including many forms of non-expected utility and time discounting. Ultimately, they conclude that their results call into question some of the foundations of the models that are core to many applied economics insights. In the EDU framework and related models that are the focus of DeJarnette et al. (2020), lottery payouts at a moment in time map directly to utility realizations at that time. This framework is likely the relevant approach for situations where risk and time delays are over direct consumption experiences that cannot be reallocated or reoptimized over time. It may also be the right as if approximation for psychological reactions to small dollar rewards at different points in time. However, in many economic applications the dominant modeling approach is intertemporal maximization of consumption utility subject to a budget constraint so that it is probably inappropriate to consider EDU as the “standard model in economics” as DeJarnette et al. claim (Frederick, Loewen- stein, and O’donoghue, 2002; Cohen el al., 2020, and references therein). In models of intertemporal consumption utility, individuals reoptimize consumption streams in reaction to stochastic monetary realizations and this has important implications for how to think about reactions to risky lotteries over time. In this note we show that when extending the DeJarnette et al. analysis to this classical case of an agent who maximizes utility over consumption subject to an intertemporal budget constraint, there is nothing problematic about the inconsistency between risk aversion over time lotteries and stochastic impatience. We show theoretically that agents maximizing intertemporal utility of consumption will adhere to DeJarnette et al.’s Generalized EDU preferences in the context they consider, albeit with speciﬁc parameter choices. They will (thus) often display risk aversion over time lotteries. Moreover, these classical agents will – and should – violate stochastic impatience as long as they are sufﬁciently risk-averse. Stochastic patience, rather than stochastic impatience, is the natural result in this classical model of intertemporally maximized consumption utility. To provide intuition for our arguments, it is helpful to revisit one of the examples DeJarnette et al. present to showcase stochastic impatience (p. 627):

• Lottery A: (50-50) $100 today or $20 in one month.

• Lottery B: (50-50) $20 today or $100 in one month.

Stochastic impatience implies that agents prefer the larger prize earlier, so that Lottery A will be preferred. This seems to an intuitively appealing criterion when the monetary prizes map directly to utility received at that time. If I can have either utility of $100 or $20 and that utility will arise either today or in one month, preferring the higher utility earlier is intuitive. HOMO ECONOMICUSIS STOCHASTICALLY PATIENT 3

However, for an agent that is optimizing consumption utility over time, when they learn which monetary prize they are receiving, they will smooth consumption and allocate part of that prize to both time periods. Prizes received later have a lower present-discounted value than prizes received earlier and result in lower utility in both periods. That reduction in utility from the later receipt of the prize has a bigger effect when the overall level of consumption is lower. As a result, a sufﬁciently risk-averse person anticipating smoothing consumption will not prefer lotteries where low wealth positively correlates with later receipt of the wealth. Instead they prefer diversiﬁcation, where better wealth realizations correlate with more delay.1 That is, they will be stochastically patient. This intuition can be made more transparent by recasting DeJarnette et al.’s example into a naturalistic situation with larger stakes and income received over time, where consumption utility maximization is likely more intuitive. Consider the following example where a currently unem- ployed person (e.g., a new college graduate) has two potential jobs they might land with very different annual salaries and the potential for a delayed start:

• Lottery A’: (50-50) $100,000 starting right away or $50,000 with 6 months wait to start.

• Lottery B’: (50-50) $50,000 starting right away or $100,000 with 6 months wait to start.

We suspect that for most people in this formulation, Lottery B’ – the stochastically patient option – is more naturally appealing.2 To test that intuition, we conducted a simple pre-registered survey experiment on Amazon’s Mechanical Turk platform with 162 participants. We randomized participants to choose between either Lotteries A/B (DeJarnette et al.’s small-dollar choice) or Lotteries A’/B’ (our large-stakes choice), with both being hypothetical choices. Consistent with DeJarnette et al.’s conjecture that stochastic impatience is an intuitive property for their lottery, 68 percent of subjects made the stochastically impatient choice of Lottery A over Lottery B. However, in the larger-stakes option, the majority of subjects (57 percent) instead made the stochastically patient choice of Lottery B’ over Lottery A’. Overall, the results of our theoretical analysis and this supplementary experiment suggest that violations of stochastic impatience are natural in environments where people are likely to optimize in response to stochastic realizations. More generally, these results call for nuance in interpreting

1We show that the intuition for this result can be illustrated using familiar insights from financial asset pricing theory by decomposing DeJarnette et al.’s stochastic impatience lotteries into an identical background lottery and differing supplemental lotteries. 2The optimal consumption streams will diverge more between realizations in Lottery A’ than Lottery B’: Note that the two lotteries are equivalent after the first year and therefore someone who is smoothing consumption might focus on the total income differences over the first year; Lottery A’ involves a total of either $100,000 or $25,000 in income during the first year while Lottery B’ guarantees a certain level of $50,000 income during the first year. HOMO ECONOMICUSIS STOCHASTICALLY PATIENT 4 the implications of DeJarnette et al. (2020). Their analysis suggests the need to reconsider the EDU model and its close cousins. Yet there is little reason to believe these concerns are an issue for economic analysis in environments when people save and optimize, where models of intertemporal consumption utility maximization subject to budget constraints are the norm. In order to keep the discussion that follows transparent and accessible, we first describe our analysis of DeJarnette et al.’s simple lotteries in a setting with two pay-out days, simple preferences with a constant relative risk aversion/elasticity of intertemporal substitution, and their motivating example in favor of stochastic impatience (p. 627). However, we also provide results that document the general validity of our main points (with proofs relegated to an online appendix).

Utility of Intertemporally Optimized Consumption and GEDU

To illustrate our ﬁrst point that individuals optimizing intertemporal consumption utility adhere to GEDU, consider simple lotteries with monetary prizes m0 and m1 paid at time 0 and 1. As in DeJarnette et al., we assume uncertainty “is resolved as soon as one of the options is chosen” and before the realization at time zero (p. 620). This assumption is critical as gradual resolution over time “could improve planning” (see Footnote 1 in DeJarnette et al.), and particularly it ties together assumptions on preferences over risk (as often characterized by risk aversion) and time (as often characterized by elasticities of intertemporal substitution). We will return to this point in the last part of our note. Unlike in DeJarnette et al., the lotteries are offered to an individual that optimizes utility of consumption subject to the budget constraint:

c m c + 1 = w = m + 1 , 0 1 + r 0 0 1 + r where r is the objective rate earned on assets deferred to the later period. Hence, the individual’s total initial wealth is w0. The (intertemporal) utility function in consumption is:

c1−γ c1−γ U C (c , c ) = 0 + β 1 , 0 1 1 − γ 1 − γ where β is the subjective time discount factor for utility, and for simplicity we assume individuals exhibit a constant relative risk aversion (CRRA) of γ – and thus also a constant elasticity of 1 intertemporal substitution (CEIS) of γ . We assume the individual chooses c0 so as to maximize utility subject to the budget constraint. HOMO ECONOMICUSIS STOCHASTICALLY PATIENT 5

One easily ﬁnds the standard result that utility in initial wealth is also of CRRA form:

n γ−1 o (1 + r)1−γ [(1 + r) β] γ + β WI 1−γ U (w0) = 1−γ w0 . (1) n − 1 o (1 − γ) 1 + [(1 + r) β] γ (1 + r) | {z } c˜

Consider now a simple time lottery p that pays mt at a random time t ∈ {0, 1}. Hence, with (1), the individual’s preference order over such lotteries is represented by: " # m 1−γ m 1−γ V (p) = (t = 0)c ˜ m1−γ + (t = 1)c ˜ 1 = c˜ t . (2) P 0 P 1 + r E (1 + r)t

1−γ 1 This is the GEDU representation (8) in DeJarnette et al. when setting φ(x) =c ˜ x , β = 1+r , and u(x) = x. Hence, individuals that maximize consumption subject to the intertemporal budget constraint exhibit GEDU preferences over time lotteries with rather specific choices of the corresponding parameters. In particular, agents are not necessarily RSTL as in the DaJarnette et al.’s EDU setting. This result is not driven by the specific assumptions – it is true generally that optimizing utility of consumption is equivalent to a GEDU model. Suppose we allow for the existence of a util- C ity function for intertemporal consumption U (c0, c1, . . . , cT ). If the individual instead receives M monetary endowments of m0, m1, . . . , mT , the monetary utility function U is defined by:

M C U (m0, m1, . . . , mT ) = max {U (c0, c1, . . . , cT )} c0,c1,...,cT

subject to the budget constraint f(c0, c1, . . . , cT ) = f(m0, m1, . . . , mT ), where f reﬂects the capi-

tal market rules. The initial wealth is then deﬁned as the value of w0 such that:

f(c0, c1, . . . , cT ) = f(m0, m1, . . . , mT ) = f(w0, 0,..., 0).

Assume f(w0, 0,..., 0) is increasing in w0. Then the capital market rule can be scaled, so that

without loss of generality we can assume f(w0, 0,..., 0) = w0.

C Proposition 1. All utility functions U (c0, c1, . . . , cT ) and capital market rules:

f(m0, m1, . . . , mT ) = w0 HOMO ECONOMICUSIS STOCHASTICALLY PATIENT 6

M produce monetary utility functions U (m0, m1, . . . , mT ) of the type:

M U (m0, m1, . . . , mT ) = φ (f(m0, m1, . . . , mT )) ,

W and therefore correspond to a unique utility function in initial wealth U (w0) = φ(w0).

It is important to note that the utility function here can be general, and particularly does not need to satisfy additive separability. While Proposition 1 is quite general, it reduces to GEDU under the usual capital market rule.

Corollary 1. Assume the capital market rule:

f(m0, m1, . . . , mT ) = D(0) m0 + D(1) m1 + ... + D(T ) mT , where D(t) are (objective) discount factors. Then the value of a lottery p with (stochastic) monetary payments M0,M1,...,MT is:

" T !# M X V (p) = Ep U (M0,M1,...,MT ) = Ep [φ(w0)] = Ep φ D(t) Mt (3) t=0

This again is equivalent to the GEDU representation in Equation (8) in DeJarnette et al. or Equation (4) in Dillenberger, Gottlieb, and Ortoleva (2018) when u(x) = x. And this equivalence originates from the form of the capital market rule rather than the assumption on preferences. A key result in DeJarnette et al. is that if the function φ in the GEDU representation is a concave transformation of the natural logarithm, agents will be risk-averse over time lotteries (RATL). Hence, we can conclude that agents maximizing intertemporal consumption utility subject to the standard budget constraint from Corollary 1 will be RATL as long as the utility function exhibits sufficient curvature – which in the simple setting from above corresponds to a CRRA parameter above one or a CEIS below one. However, DeJarnette et al.’s Proposition 4 then implies that these agents will violate Stochastic Impatience, which they flag as a key downside when relying on this preference specification.

Stochastic Patience when Optimizing Consumption Over Time

To illustrate stochastic impatience for agents that optimize intertemporal consumption, consider the stochastic impatience lotteries from DeJarnette et al. (2020, p. 627):

• Lottery A: (50-50) $100 today or $20 in one month. HOMO ECONOMICUSIS STOCHASTICALLY PATIENT 7

• Lottery B: (50-50) $20 today or $100 in one month.

Stochastic impatience implies that agents prefer the larger prize earlier, so that Lottery A will be preferred. But is this true for agents that maximize utility of intertemporal consumption? For simplicity of presentation, ﬁx the parameters of our simple CRRA model as β = 0.98, r = 5%, and γ = 2. $20 Lottery A is between w0 = $100 or w0 = 1.05 = $19.05, which, when optimizing consumption, results in a 50-50 lottery between the two consumption streams:

• A1: c0 = $50.86 and c1 = $51.59, and

• A2: c0 = $9.69 and c1 = $9.83.

$100 Lottery B, in turn, is between w0 = $20 or w0 = 1.05 = $95.24, and the equations above imply a 50-50 lottery between the two consumption streams:

• B1: c0 = $10.17 and c1 = $10.32, and

• B2: c0 = $48.44 and c1 = $49.14.

In this light, Lottery A is riskier in consumption than Lottery B, and individuals with high risk aversion will prefer B to A – and thus be stochastically patient rather than impatient. Indeed, for

γ = 2, E[UA] = −0.121 and E[UA] = −0.117 and Lottery B is prefered to Lottery A. Again, this result is not due to the particular choices/parameters and persists in far more general settings. Indeed, a general condition for an agent to exhibit RATL and stochastic patience in terms of the utility function in initial wealth from Proposition 1 is given in the following.

Proposition 2. Adopt the same assumptions and notations as in Proposition 1. Then the individual will be RATL and stochastically patient if and only if:

∂2 − ∂w2 φ 0 w > 1. (4) ∂ φ 0 ∂w0

Thus, the condition requires a lower bound of one for an expression akin to the relative risk aversion parameter of the utility function in initial wealth – although conditions in terms of the C object φ that derives from primitive assumptions on U (c0, c1, . . . , cT ) may be difﬁcult to process. However, it turns out that the link to relative risk aversion persists when considering additively separable utility.

C C C C Corollary 2. Assume U (c0, c1, . . . , cT ) = U0 (c0) + U1 (c1) + ... + UT (cT ) (additive separability) with continuous, monotonically increasing, and concave component utility functions. For HOMO ECONOMICUSIS STOCHASTICALLY PATIENT 8

P C pt = D(t) c(t) denote Ut (pt) = Ut (ct), where D(t) are (objective) discount factors. Then the individual will be RATL and stochastically patient if and only if:

p0 p1 pT w0 = p0 + p1 + ... + pT > + + ... + , RRA0 RRA1 RRAT

∂2 P − 2 Ut where RRA = ∂pt p is the relative risk aversion coefficient of U P (p ). t ∂ U P t t t ∂pt t In particular, a sufficient condition for RATL and stochastic patience is if the relative risk aversion coefficients of the (additive) component utility function uniformly exceed one. To obtain an intuition of why risk-averse, consumption-optimizing individuals are stochastically patient, it is helpful to follow DeJarnette et al.’s description of the intuition behind stochastic impatience (p. 621):

“Note that the choice above can be rephrased as: Given a base lottery paying $20 either in a day or a month, which prize do we want to increase by $80, the earlier or the later one? To the extent that receiving earlier payments is better, the individual should prefer the ﬁrst option, in accordance with Stochastic Impatience.”

In other words, the individual has a base lottery (Lottery C: (50-50) $20 today or $20 in one month) and wishes to add to it one of two lotteries:

• Lottery D: (50-50) $80 today or $0 in one month.

• Lottery E: (50-50) $0 today or $80 in one month.

Considered in isolation, Lottery D is preferable to Lottery E, irrespective of the preference speci- ﬁcation as long as the individual is impatient. However, this lottery is not considered in isolation, but will be added to an already existing correlated asset – namely the background lottery C. As can be seen in the usual discussions of the CAPM and Portfolio Theory, an asset that is inferior in isolation can become the preferred asset when added to an already existing portfolio. That is the case here. To elaborate, in the original base lottery, $20 later is the “Bad” outcome for the background lottery C. Lottery E is positively correlated with the “Bad” outcome and therefore offsets some of its pain. For this reason, Lottery E is more valuable than one might expect if it is considered in isolation. This is analogous to the situation with put options, which are correlated with bad stock outcomes and are therefore “overvalued” relative to what one would expect if they were considered in isolation from the context of the broader portfolio. If an individual is sufﬁciently risk averse, Lottery E will even become the preferred option and be valued higher than Lottery D. HOMO ECONOMICUSIS STOCHASTICALLY PATIENT 9

Survey Experiment on Stochastic Impatience

The prior section highlights that stochastic impatience, while an intuitively attractive property for static utility ﬂows, is not a natural property for monetary risk in situations where individuals dynamically optimize consumption ﬂows. Thus, we hypothesize that stochastic impatience may be natural for many people when presented with the small-dollar lottery that DeJarnette et al. use as their motivating example. For these small dollar amounts, it seems reasonable that people would think about the lottery payouts equating directly to utility experiences at the moment of receipt rather than engaging in consumption smoothing and re-optimization over these amounts. How- ever, we also hypothesize that lotteries over higher stakes that more naturally generate thoughts of consumption smoothing behavior would make people more stochastically patient. We tested these hypotheses using a very simple preregistered hypothetical-choice survey experiment on the Amazon Mechanical Turk platform.3 Participants were paid $1 to make one (hypothetical) lottery choice. We targeted 160 participants for the experiment and ended up having 162 complete the survey before the platform cut off enrollment. Half of the participants were randomly selected to receive the lottery from DeJarnette et al. (2020) discussed above, which we operationalized as follows:

“Which would you prefer between these two options where your payments would depend on the ﬂip of a fair coin (50% chance heads and 50% chance tails)?

• Option A: If the coin lands heads you get $100 today and if it lands tails you get $20 in a month. • Option B: If the coin lands heads you get $20 today and if it lands tails you get $100 in a month.”

The other half of participants were randomly assigned to receive our large-stakes analogue to this lottery put into the context of a job situation. Speciﬁcally we asked:

“Please imagine a situation where you are out of work and looking for a job. Suppose there are two possible jobs you might land that are similar and equally appealing work, but pay very differently: one would pay you $50,000 per year and one would pay you $100,000 per year. When you get a job you might get to start right away or you might have to wait six months before the job and your paychecks start. Which would you prefer between these two options where the combination of your job and start time would depend on the ﬂip of a fair coin (50% heads, 50% tails)?

3The preregistration is available at https://aspredicted.org/blind2.php. HOMO ECONOMICUSIS STOCHASTICALLY PATIENT 10

Figure 1: Outcomes of the Survey Experiment on Stochastic Impatience

• Option A: If the coin lands heads you get the $100,000 per year job and it starts right away. If the coin lands tails you instead get the $50,000 per year job and have to wait 6 months to start. • Option B: If the coin lands heads you get the $50,000 per year job and it starts right away. If the coin lands tails you instead get the $100,000 per year job and have to wait 6 months to start.”

In both of these examples Option A is the stochastically impatient option. To ensure that results did not depend on the exact ordering of the presentation of the outcomes in these options, for both questions we randomized the order of the higher and lower outcomes within each of the options and also the order of whether the stochastically impatient option was listed ﬁrst (as Option A) or second (as Option B). The results of the survey experiment are shows in Figure 1. Consistent with the claims in HOMO ECONOMICUSIS STOCHASTICALLY PATIENT 11

DeJarnette et al. (2020), only 32 percent of people made the stochastically patient choice when presented with their small-dollar lottery used to motivate stochatic impatience. However, the majority (57 percent) chose the stochastically patient option in the larger-stakes job context question. This 25 percentage point difference is highly statistically signiﬁcant (95% CI 0.10 - 0.40).

Resolution of Uncertainty and Recursive Preferences

Our basic results, like much of the analysis in DeJarnette et al. (2020), use the simple time- separable model where risk aversion (RA) and the elasticity of intertemporal substitution (EIS) are linked. It is natural to wonder how the insights change if we consider preferences that allow for a separation between the two concepts, such as Epstein and Zinn (1989) preferences. However, in the situation that DeJarnette et al. (2020) study, and we follow, where uncertainty gets revealed immediately after the lottery choice is made, there is no meaningful way to separate the two concepts. To see this, consider Epstein and Zinn (1989) preferences that allow for a separation between the two concepts, which in the current setting take the form (see also Equation (9) in DeJarnette et al.):

1 n 1−ρ o 1−ρ C 1−ρ 1−α 1−α U (c0, c1) = (1 − β) c0 + β E0 c1 .

Now since the uncertainty in c0 gets revealed simultaneously with the uncertainty in c1, the ex-

pectation is conditional on knowing c1 (both are measurable with respect to the ﬁltration at the

onset F0+) so that α cancels and the only relevant parameters is ρ – which is associated with the speciﬁcation of EIS rather than RA in Epstein and Zinn (1989). While the assumption of immediate resolution of uncertainty is obviously unrealistic in many situations, we agree with DeJarnette et al. (2020) that this is a useful abstraction, especially for analyzing stochastic patience. If uncertainty gets revealed over time, the situation changes as “planning motives” become relevant (DeJarnette et al., 2020, p. 620). More precisely, in environments where individuals optimize consumption utility dynamically, adjustments in consumption

will depend on the nature of the considered lotteries. Will the realization at time zero (M0) imply

anything about the distribution of later payoffs (M1, say)? And are investments available to the agent that correlate with the payoffs of the lotteries? The analysis will depend on the particularities of the modeled situation, as for instance within dynamic models of choices over an individual’s life cycle given exposure to various risks (Cocco, Gomes, and Maenhout, 2005; Gourinchas and Parker, 2002, among many others). HOMO ECONOMICUSIS STOCHASTICALLY PATIENT 12

Conclusion

The primary take-way from our analysis is that the relevance of the insights in DeJarnette et al. (2020) depends on the economic setting and models being considered. Their work substantially en- hances our understanding of preferences across risk and time for direct utility experiences that are often modeled in the expected discounted utility framework. However, their results do not have the same immediate implications for analysis and modeling of consumption-utility maximization subject to a budget constraint, which is the workhorse framework for a signiﬁcant portions of applied economics. The results of our survey suggest that individuals’ actual attitudes toward monetary risk over time can vary between these two frameworks depending on stakes and context. Improv- ing our understanding in which context the two frameworks tend to be better approximations of individuals’ actual thought processes may be a fruitful direction for future research.

References

Cocco, Joao F., Francisco J. Gomes, and Pascal J. Maenhout (2005). “Consumption and portfolio choice over the life cycle.” Review of Financial Studies, 18: 491-533.

Cohen, Jonathan D., Keith Ericson, David Laibson, and John Miles White (2020). “Measuring time preferences.” Journal of Economic Literature, 58: 299-347.

DeJarnette, Patrick, David Dillenberger, Daniel Gottlieb, and Pietro Ortoleva (2020). “Time lotteries and stochastic impatience.” Econometrica, 88: 619-656.

Dillenberger, David, Daniel Gottlieb, and Pietro Ortoleva (2018). “Stochastic Impatience and the separation of Time and Risk Preferences” PIER Working Paper No. 18-020 (available on SSRN).

Epstein, Larry G., and Stanley E. Zinn (1989). “Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework.” Econometrica, 57: 937-969.

Frederick, Shane, George Loewenstein, and Ted O’donoghue (2002). “Time discounting and time preference: A critical review.” Journal of Economic Literature, 40: 351-401.

Gourinchas, Pierre-Olivier, and Jonathan A. Parker (2002). “Consumption over the life cycle.” Econometrica, 70: 47-89. Online Appendix to “Homo Economicus is Stochastically Patient”

Eric R. Ulm, Daniel Bauer, Justin Sydnor1

July 2020

This online appendix collects the proofs for the paper “Homo Economicus is Stochastically Pa- tient.”

∗ ∗ ∗ Proof of Proposition 1. We ﬁrst show that if f(m0, m1, . . . , mT ) = f(m0, m1, . . . , mT ), then M M ∗ ∗ ∗ U (m0, m1, . . . , mT ) = U (m0, m1, . . . , mT ). We have: M C U (m0, m1, . . . , mT ) = max U (c0, c1, . . . , cT ) c0,c1,...,cT subject to the constraint f(c0, c1, . . . , cT ) = f(m0, m1, . . . , mT ). Since f(m0, m1, . . . , mT ) = ∗ ∗ ∗ f(m0, m1, . . . , mT ), this implies:

M C U (m0, m1, . . . , mT ) = max U (c0, c1, . . . , cT ) c0,c1,...,cT

∗ ∗ ∗ subject to the constraint f(c0, c1, . . . , cT ) = f(m0, m1, . . . , mT ), which is exactly the deﬁnition of M ∗ ∗ ∗ U (m0, m1, . . . , mT ). This then implies that:

M M M U (m0, m1, . . . , mT ) = U (w0, 0,..., 0) = U (f(m0, m1, . . . , mT ), 0,..., 0).

The proof thus follows by setting φ(x) = U M (x, 0,..., 0).

Proof of Corollary 1. Follows immediately when inserting the corresponding capital market rule into Proposition 1.

Proof of Proposition 2. Point 3 of Proposition 2 in DeJarnette et al. (page 634) states:

The relation ∼ is RATL if and only if φ is a concave transformation of ln. 1Ulm: School of Economics and Finance, Victoria University of Wellington, 23 Lambton Quay Wellington, NZ, [email protected]. Bauer/Sydnor: Department of Risk and Insurance, University of Wisconsin-Madison, 975 Uni- versity Avenue, Madison, WI 53706, USA, [email protected]/[email protected]. Online Appendix to “Homo Economicus is Stochastically Patient” 2

∂2f That is, φ = f (x) = f (ln{w0}), where ∂x2 < 0. Now: ∂φ 1 ∂f ∂2φ 1 ∂2f 1 ∂f 1 ∂2f 1 ∂φ = and 2 = 2 2 − 2 = 2 2 − , ∂w0 w0 ∂x ∂w0 w0 ∂x w0 ∂x w0 ∂x w0 ∂w0 so that: ∂2φ 1 ∂φ 1 ∂2f ∂2f 2 + = 2 2 < 0 ⇐⇒ 2 < 0, ∂w0 w0 ∂w0 w0 ∂x ∂x which given that φ is increasing is equivalent to:

∂2 − ∂w2 φ 0 w > 1. ∂ φ 0 ∂w0

The proof of Corollary 2 involves several steps so we break it downs into several lemmas, imposing the assumptions of Corollary 2.

P C Lemma 1. Deﬁning the utility function U (p0, p1, . . . , pT ) = U (c0, c1, . . . , cT ) with the capital PT market rule w0 = t=0 pt, we obtain:

∂φ ∂U P = ∂w0 ∂pt

∗ ∗ ∗ for all t, where the partial derivative is evaluated at the optimal values p0, p1, . . . , pT .

Proof. The ﬁrst-order conditions are:

∂U P ∂U P ∂U P = = ... = ∂p0 ∂p1 ∂pT

∗ ∗ ∗ at the optimal values p0, p1, . . . , pT . Hence, at the optimum:

T P ∗ ∂φ X ∂U ∂pt = . ∂w ∂p ∂w 0 t=0 t 0

The first partial derivative in the sum is equal to the constant ∂U P from the first-order conditions, ∂p0 so: P ∗ P ∗ P ∂ pt P P ∂φ ∂U X ∂p ∂U t ∂U ∂w0 ∂U = t = = = . ∂w ∂p ∂w ∂p ∂w ∂p ∂w ∂p 0 0 t 0 0 0 0 0 0 Again, from the first-order conditions ∂φ = ∂U P implies ∂φ = ∂U P at the optimum for all t. ∂w0 ∂p0 ∂w0 ∂pt Online Appendix to “Homo Economicus is Stochastically Patient” 3

P Lemma 2. If the utility function U (p0, p1, ..., pT ) is time separable, the derivative of the ﬁrst period optimal consumption with respect to wealth is given by:

∂p∗ 1 0 = T −1 ∂w0 ∂2U P P h ∂2U P i ∂p2 ∂p2 0 t=0 t

∗ ∗ ∗ at the optimal values p0, p1, . . . , pT .

Proof. We derive the ﬁrst-order condition when increasing w0 by dw0. Using a Taylor expansion (Lagrange form):

P P 2 P 2 P 3 P ∂U ∂U ∂ U ∗ X ∂ U ∗ 1 X ∂ U ∗ ∗ = + 2 dp0+ dpj + dpj dpk. ∂p0 ∗ ∗ ∗ ∗ ∂p0 ∗ ∂p0 ∗ ∂p0∂pj ∗ 2 ∂p0∂pj∂pk p0+dp0,...,pT +dpT p p j>0 p j,k ξj,k

The cross derivatives vanish for time-separable U P so that:

P P 2 P ∂U ∂U ∂ U ∗ ∗ 2 = + 2 dp0 + O [dp0] . ∂p0 ∗ ∗ ∗ ∗ ∂p0 ∗ ∂p0 ∗ p0+dp0,...,pT +dpT p p

Similarly for t ∈ {0, 1,...,T }:

P P 2 P ∂U ∂U ∂ U ∗ ∗ 2 = + 2 dpt + O [dpt ] , ∂pt ∗ ∗ ∗ ∗ ∂pt ∗ ∂pt ∗ p0+dp0,...,pT +dpT p p

where from the ﬁrst-order conditions we obtain:

P P ∂U ∂U = . ∂p0 ∗ ∗ ∗ ∗ ∂pt ∗ ∗ ∗ ∗ p0+dp0,...,pT +dpT p0+dp0,...,pT +dpT

Therefore:

P 2 P P 2 P ∂U ∂ U ∗ ∗ 2 ∂U ∂ U ∗ ∗ 2 + 2 dp0 + O [dp0] = + 2 dpt + O [dpt ] , ∂p0 p∗ ∂p0 p∗ ∂pt p∗ ∂pt p∗

with the ﬁrst-order conditions at p∗ implies:

2 P 2 P ∂ U ∗ ∗ 2 ∂ U ∗ ∗ 2 2 dp0 + O [dp0] = 2 dpt + O [dpt ] , ∂p0 p∗ ∂pt p∗

or: ∂2U P ∂p2 ∗ 0 p∗ ∗ ∗ ∗ 2 dpt = dp0 + O [max{dp0, dpt }] . ∂2U P 2 ∂pt p∗ Online Appendix to “Homo Economicus is Stochastically Patient” 4

This means:

−1 2 P " 2 P # X ∗ ∂ U X ∂ U ∗ X ∗ 2 dw0 = dpt = dp0 + O [max{dpt }] , ∂p2 ∂p2 t t 0 p∗ t t p∗ t or: dp∗ 1 P O ([max {dp∗}]2) 0 = + t t t . dw −1 dw 0 ∂2U P P ∂2U P 0 ∂p2 t ∂p2 0 p∗ t p∗ The claim follows by taking limits.

P Lemma 3. If the utility function U (p0, p1, ..., pn) is time separable, then:

∂2φ 1 = 2 T −1 ∂w0 P h ∂2U P i ∂p2 t=0 t

∗ ∗ ∗ at the optimal values p0, p1, . . . , pT .

Proof. Differentiating the result of Lemma 1 for t = 0 with respect to w0, we ﬁnd:

2 2 P ∗ 2 P ∗ 2 P ∗ ∂ φ ∂ U ∂p0 ∂ U ∂p1 ∂ U ∂pT 2 = 2 + + ... + . ∂w0 ∂p0 ∂w0 ∂p0∂p1 ∂w0 ∂p0∂pT ∂w0

The cross derivatives vanish for time-separable U P so that:

2 2 P ∗ ∂ φ ∂ U ∂p0 2 = 2 . ∂w0 ∂p0 ∂w0

Substituting in the result from Lemma 2 then yields the claim.

Proof of Corollary 2. Starting from the result from Proposition 2, we ﬁnd:

∂φ w > − ∂w0 . 0 ∂2φ 2 ∂w0

Substituting in the results from Lemmas 1 and 3, we obtain:

P P P T 2 P −1 T ∂U T ∂U ∂U X ∂ U X ∂p X ∂p w > − = − 0 = − t , 0 2 ∂2U P ∂2U P ∂p0 ∂pt 2 2 t=0 t=0 ∂pt t=0 ∂pt where the last equality follows from the ﬁrst-order condition. Multiplying each term by pt, we Online Appendix to “Homo Economicus is Stochastically Patient” 5 have: T T X pt X pt w > = . 0 ∂2UP 2 RRAt t=0 ∂pt t=0 − ∂UP pt ∂pt