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Decreasing relative impatience
Journal of Economic Psychology 30 (2009) 831–839
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Journal of Economic Psychology journal homepage: www.elsevier.com/locate/joep
Decreasing relative impatience Kirsten I.M. Rohde * Erasmus School of Economics, H13-27, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands
a r t i c l e
i n f o
Article history: Received 3 October 2008 Received in revised form 6 July 2009 Accepted 13 August 2009 Available online 2 September 2009 JEL classification: D90 PsycINFO classification: 3920
a b s t r a c t Prelec (2004) showed that the Arrow–Pratt degree of convexity of the logarithm of the discount function can serve as a measure of decreasing impatience and of the corresponding time-inconsistency. In decision under risk and uncertainty the convexity of the utility function itself, not of its logarithm, has empirical meaning in terms of risk attitude. This paper introduces decreasing relative impatience, which is directly related to the degree of convexity of the discount function. By giving empirical meaning to the convexity of the discount function this paper clarifies the relations between several concepts in intertemporal choice. This paper also introduces a concept of spread seeking. Spread seeking and decreasing relative impatience are likely to have distinct underlying psychological motives, but are shown to be equivalent under discounted utility. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Intertemporal choice Decreasing impatience Decreasing relative impatience Hyperbolic discounting Consumer attitudes and behaviour
1. Introduction Imagine a decision maker who is indifferent between receiving a small reward soon and a larger reward at a later point in time. Under decreasing impatience, his preference will shift to the larger and later reward, when the receipt of both rewards is equally delayed. Thus, under decreasing impatience a difference in timing is weighted less the further it lies in the future. Constant impatience, i.e. stationarity, predicts the preference of the decision maker to be unchanged after a common delay of the smaller sooner and the larger later reward. Stationarity corresponds to constant discounting as introduced by Samuelson (1937). Under assumptions commonly made in the literature constant discounting is equivalent to time-consistency. Then, decreasing and increasing impatience imply time-inconsistency. There is an increasing interest in adjusting traditional economic models to accommodate time inconsistent preferences (Akerlof, 2002; Harris & Laibson, 2001; Krusell & Smith, 2003; Laibson, 1997; Luttmer & Mariotti, 2003; O’Donoghue & Rabin, 1999; Rubinstein, 2003, 2006; Thaler & Benartzi, 2004). Decreasing impatience means that a difference in timing is weighted less the further in the future it occurs. The weight per unit of time attached to an infinitely small difference in timing is given by minus the first derivative of the discount function. Thus, this weight becoming smaller the further in the future, corresponds to an increasing first derivative of the discount function, which corresponds to the discount function being convex. This suggests that individuals can be compared in terms of their degrees of decreasing impatience by comparing the degrees of convexity of their discount functions. Prelec (1989, 2004) formalized the concept of decreasing impatience. He showed that the degree of decreasing impatience, and the corresponding time-inconsistency, is captured by the degree of convexity of the logarithm of the discount * Tel.: +31 10 408 9548. E-mail address: [email protected] 0167-4870/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.joep.2009.08.008
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function, rather than the degree of convexity of the discount function itself. The latter is quite surprising considering the above reasoning. The key in understanding why Prelec needed the convexity of the logarithm of the discount function is that decreasing impatience concerns the rates of change of the discount function and not absolute changes, as this paper will show. We introduce the notion of decreasing relative impatience which concerns absolute changes of the discount function and can be captured by the convexity of the discount function itself, rather than its logarithm. The concept of decreasing relative impatience considers sequences of two outcomes, unlike the concept of decreasing impatience, which considers single outcomes. Consider an investment project where a decision maker has to incur costs today to receive a reward in the future. This investment project is a sequence of two outcomes: ‘cost now’ and ‘reward later’. Impatience implies that the decision maker likes a speed-up of the receipt of the reward. Imagine that a decision maker is willing to incur some extra costs today to speed-up the receipt of the reward by a particular amount of time. Decreasing relative impatience holds if the decision maker is less willing to incur extra costs to speed-up the receipt of the reward by a given amount of time, the further in the future the receipt of the reward. Thus, like decreasing impatience, decreasing relative impatience means that a time difference becomes less important the further it lies in the future. We will also introduce a notion of spread seeking and relate it to decreasing relative impatience. Consider a decision maker who receives a particular benefit twice, once soon and once later. If the DM is spread seeking then he will like to speedup the receipt of the first benefit and delay the receipt of the second one by a common amount of time, thereby increasing the spread of the two benefits. Consider, for instance, a person with a chronic illness who is entitled to two treatments that both give the same temporary relief from symptoms. The more spread seeking this person is, the more he will prefer to speed-up the first treatment and delay the second one by a common amount of time. We will show how the concepts of decreasing impatience, decreasing relative impatience, and spread seeking relate to each other. Interestingly, both decreasing relative impatience and spread seeking can be captured by the convexity of the discount function. Moreover, the degree of decreasing relative impatience equals the degree of decreasing impatience plus the discount rate. 2. Decreasing impatience Consider preferences < over streams of outcomes ðt0 : x0 ; t1 : x1 ; . . .Þ 2 ðT XÞn ; where X ¼ Rm is the set of outcomes and T ¼ Rþ is the set of time points. Time point 0 is today. The decision maker (DM) receives outcome xi at time t i and a reference outcome 0 2 X at all other time points. Preferences over outcomes are derived from preferences over outcomes today, i.e. for b; c 2 X we have b
X
uðtÞUðxt Þ;
t
where u is a general, twice continuously differentiable and strictly decreasing discount function and U is a continuous and increasing utility function with Uð0Þ ¼ 0: We will often consider another DM with preferences < ; who satisfies all the aforementioned assumptions, with discount function u and utility function U : We start by listing the main concepts that are used in intertemporal choice, so that we can show the links between them later on. The discount factor at date t is uðtÞ: The discount rate at date t is the rate of decline of the discount function at date t, i.e.
qðtÞ ¼
u0 ðtÞ : uðtÞ
We will need the following observation, which can easily be proven, later in the paper. Observation 2.1
qðtÞ ¼ ½ln uðtÞ0 : Impatience means that for every outcome b 0 and all time points s < t we have ðs : bÞ ðt : bÞ: Impatience corresponds to the discount function being strictly decreasing. Constant impatience or stationarity holds if ðs : cÞ ðt : bÞ implies ðs þ s : cÞ ðt þ s : bÞ: A DM with preferences < has decreasing impatience if for all s < t and s > 0, b c 0, and ðs : cÞ ðt : bÞ imply ðs þ s : cÞ^ðt þ s : bÞ. Increasing impatience holds if the implied preference is always the reverse. The interpretation of decreasing impatience is that when indifference holds between a smaller but sooner reward and a larger but later reward, adding a constant delay to both rewards will induce a, possibly strict, preference for the larger but later reward. Thus, the further in the future, the less the timing difference t s matters, and the more preferences will be driven by the outcomes. Preferences < exhibit more decreasing impatience than < if for all s < t; s; r > 0, b c 0, b c 0, we have that ðs; cÞ ðt; bÞ, ðs þ r; cÞ ðt þ s; bÞ, and ðs; c Þ ðt; b Þ imply ðs þ r; c Þ^ ðt þ s; b Þ. Prelec (2004) showed that the degree of decreasing impatience can be measured by the Arrow–Pratt degree of convexity of the logarithm of the discount function. Thus,
DIðtÞ ¼
½ln uðtÞ00 ½ln uðtÞ0
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serves as a measure of decreasing impatience. The next observation can easily be proven. Observation 2.2. The degree of decreasing impatience is the rate of change of the discount rate, i.e.
DIðtÞ ¼ ½ln qðtÞ0 : Thus, the degree of decreasing impatience is measured by the rate of change of the discount rate. It cannot be measured by the absolute decline in the discount rate, i.e. it cannot be measured by the first derivative of the discount rate. Nevertheless, a declining discount rate is equivalent to decreasing impatience as the next theorem shows. The first derivative of the discount rate can therefore identify decreasing impatience, but not measure its degree. Theorem 2.3. A declining discount rate is equivalent to decreasing impatience.
3. Decreasing relative impatience This section introduces the concepts of relative impatience and decreasing relative impatience, which we will show to be closely related to impatience and decreasing impatience. We will show that the degree of decreasing relative impatience can be captured by the degree of convexity of the discount function, rather than the degree of convexity of its logarithm. The main difference between the concepts introduced in this section and the concept of decreasing impatience as introduced by Prelec (1989, 2004), is that the latter concerns the receipt of a single outcome, whereas the former concern a sequence of two outcomes. We will often think of this sequence of two outcomes as an investment project, which entails costs today in return for a reward in the future. Definition 3.1. Preferences < satisfy relative impatience if
ð0 : b; s : cÞ ð0 : b; t : cÞ whenever s < t and c 0. Relative impatience means that in every investment project which entails costs today and a reward in the future, a DM dislikes delaying the future reward. We name it relative impatience because the delay of the receipt of c from s to t should be considered relative to the payment or receipt of b today. Under discounted utility, which we assume throughout, relative impatience is equivalent to impatience. Definition 3.2. Preferences < satisfy decreasing relative impatience if for all outcomes a; b; c with a b and c 0, all time points s < t, and r > 0,
ð0 : a; s : cÞ ð0 : b; t : cÞ implies ð0 : a; s þ r : cÞ^ð0 : b; t þ r : cÞ: Preferences < satisfy increasing relative impatience if the implied preference is reversed. Decreasing relative impatience can be interpreted as follows. The indifference ð0 : a; s : cÞ ð0 : b; t : cÞ means that to speed-up the receipt of c from t to s, the DM is willing to pay b a. If the receipt of c is delayed to t þ r, then the DM is no longer willing to pay b a to speed-up the receipt of c by t s units of time. Thus, a delay (of t s units of time) becomes less important the further it lies in the future. Intuitively, decreasing relative impatience, thus, encompasses a similar phenomenon as decreasing impatience. In the next section we will see that formally, these two concepts are not equivalent, though. First, we have to define how two decision makers can be compared according to their degree of decreasing relative impatience. The following definition does so in a similar manner as the definition for comparing decision makers according to their degree of decreasing impatience. Definition 3.3. Preferences < satisfy more decreasing relative impatience than < if for all outcomes a; b; c; a ; b ; c with a b, c 0, a b , and c 0, all time points s < t, all r > 0, and all s
ð0 : a; s : cÞ ð0 : b; t : cÞ and ð0 : a; s þ r : cÞ ð0 : b; t þ s : cÞ and ð0 : a ; s : c Þ ð0 : b ; t : c Þ imply ð0 : a ; s þ r : c Þ^ ð0 : b ; t þ s : c Þ: The interpretation of comparative decreasing relative impatience is as follows. Imagine a decision maker with preferences < who is indifferent between (1) speeding up the receipt of c from t to s by paying a instead of b; and (2) not speeding up the receipt of c. Imagine he is also indifferent between (1) speeding up the receipt of c from t þ s to s þ r by paying a instead of b and (2) not speeding up the reward. Then, if another decision maker with more decreasing relative impatience is indifferent between (1) speeding up the receipt of c from t to s by paying a instead of b and (2) not speeding up, this other decision maker prefers not to speed-up the receipt of c from t þ s to s þ r by paying a instead of b . He would, however, be indifferent if the receipt of c would instead be speed up from t þ s to s þ r for a particular r 6 r. In fact, when ð0 : a; s : cÞ ð0 : b; t : cÞ and ð0 : a; s þ r : cÞ ð0 : b; t þ s : cÞ, the timing difference s r serves as a measure of the degree of decreasing relative impatience.
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Until now we only considered a decision maker in a static context. The definition of decreasing relative impatience considers what happens to a preference between two investment projects as specified, when the receipt of the rewards is pushed back in time. If we assume that, as time evolves, decision makers reset the clock at zero every time they make a decision, then decreasing relative impatience also has a dynamic interpretation. Suppose that, on the first of January, a decision maker can choose between two investment projects. In one project, he will receive a reward c on the first of December of the same year. He would have to pay b as soon as he commits to that project. In the other project, he would receive the same reward c, but earlier, namely on the first of November, in return for a larger price at the time of commitment: jaj > jbj. Suppose that on the first of January the decision maker has a slight preference for receiving the reward on the first of December: ð0 : a; s þ r : cÞ ð0 : b; t þ r : cÞ, where s þ r and t þ r are the durations until the first of November and the first of December, respectively. Consider which investment project he would prefer on the first of February. Note that the investment projects are specified in such a way that the receipt of the reward is on fixed dates, whereas the investment of either a or b can be done at any date before the first of November. Decreasing relative impatience implies that, as the decision maker is almost indifferent between the two projects on the first of January, he is likely to prefer the project which delivers the reward on the first of November in return for a payment of a on the first of February: ð0 : a; s : cÞ ð0 : b; t : cÞ. Thus, he switches his preference between the two investment projects as time elapses from the first of January to the first of February.
4. Convexity of the discount function The degree of decreasing relative impatience is captured by the degree of convexity of the discount function, as is shown in the following theorem. Theorem 4.1. Preferences < satisfy more decreasing relative impatience than < if and only if
u00 ðtÞ u00 ðtÞ P 0 0 u ðtÞ u ðtÞ
for all t P 0. Thus,
DRIðtÞ ¼
u00 ðtÞ u0 ðtÞ
is a measure of decreasing relative impatience. DRI will be non-negative in the case of decreasing relative impatience and negative otherwise. As the following theorem shows, the degree of decreasing relative impatience at any point in time is the sum of the degree of decreasing impatience and the discount rate at that point in time: decreasing relative impatience = decreasing impatience + discount rate. Theorem 4.2. For all t
DRIðtÞ ¼ DIðtÞ þ qðtÞ: Prelec (1989) showed this result before, but for completeness we repeat it here. Theorem 4.2 shows that decreasing impatience implies decreasing relative impatience and that increasing relative impatience implies increasing impatience. Corollary 4.3. Decreasing impatience implies decreasing relative impatience:
DIðtÞ P 0 ) DRIðtÞ P 0: Corollary 4.4. Increasing relative impatience implies increasing impatience:
DRIðtÞ 6 0 ) DIðtÞ 6 0: The constant discount function uðtÞ ¼ dt satisfies decreasing relative impatience, but constant impatience. Constant discounting is equivalent to a constant degree of decreasing relative impatience, which equals the discount rate. Theorem 4.5. Preferences satisfy constant discounting uðtÞ ¼ dt with 0 < d < 1 if and only if the degree of decreasing relative impatience is constant and equal to ln d: Table 1 gives the degree of decreasing relative impatience for other discount functions that are used in the literature. These degrees are derived in the Appendix.
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Table 1 DRI for various discount functions. Quasi-hyperbolic (Phelps & Pollak, 1968) uðtÞ ¼ bdt for t > 0
DRIðtÞ ¼ ln d
Hyperbolic (Loewenstein & Prelec, 1992)
uðtÞ ¼ ð1 þ atÞb=a
DRIðtÞ ¼ ða þ bÞ=ð1 þ atÞ
CADI (Bleichrodt et al., 2009) ct
uðtÞ ¼ kere uðtÞ ¼ kert ct uðtÞ ¼ kere
DRIðtÞ ¼ c þ crect DRIðtÞ ¼ r DRIðtÞ ¼ c crect
CRDI (Bleichrodt et al., 2009) 1d
uðtÞ ¼ kert uðtÞ ¼ ktr 1d uðtÞ ¼ kert
DRIðtÞ ¼ d=t ð1 dÞrt d DRIðtÞ ¼ ð1 þ rÞ=t DRIðtÞ ¼ d=t þ ð1 dÞrt d
5. Spread seeking This section shows that decreasing relative impatience relates to a concept which we call spread seeking. An individual who is spread seeking prefers to increase the spread, over time, of pleasant events. Assume that a DM receives a particular benefit at two different dates. He is spread seeking if he prefers to speed-up the early receipt and delay the late receipt by a common amount of time. Definition 5.1. Preferences < satisfy spread seeking if for all c 0 and all t > s P 0,
s > 0 with s 6 s
ðs s : c; t þ s : cÞ<ðs : c; t : cÞ: Preferences < satisfy spread aversion if the preference is the reverse. One decision maker may be more spread seeking than another. The following definition formalizes this point. Definition 5.2. Preferences < satisfy more spread seeking than < if for all c 0, c 0, and all t > s P 0, s > 0 and all r 2 R with s r P 0
ðs r : c; t þ s : cÞ ðs : c; t : cÞ implies
ðs r : c ; t þ s : c Þ< ðs : c ; t : c Þ: It follows that when ðs r : c; t þ s : cÞ ðs : c; t : cÞ for c 0, the length of time s r serves as a measure of spread seeking. As the next theorem shows, the degree of convexity of the discount function captures the degree of spread seeking. Interestingly, spread seeking and decreasing relative impatience are equivalent concepts under discounted utility. Theorem 5.3. < is more spread seeking than < if and only if
u00 ðtÞ u00 ðtÞ P 0 u0 ðtÞ u ðtÞ
for all t P 0. It is remarkable that two at first sight distinct concepts with possibly distinct underlying psychological motives, like decreasing relative impatience and spread seeking, turn out to be equivalent under discounted utility. As soon as these concepts are considered outside the discounted utility setting, though, they are likely no longer to be equivalent. 6. Discussion This paper introduced the concept of decreasing relative impatience, which is related, but not equivalent, to decreasing impatience as introduced by Prelec (1989, 2004). The main difference between the two concepts is that the latter concerns receipts of single outcomes, whereas the former concerns sequences of two outcomes. A convenient way to operationalize the concept of decreasing relative impatience, is to think in terms of investment projects that require a payment today in return for a future reward. It will be interesting to see how the concepts of decreasing relative impatience, decreasing impatience, and spread seeking are empirically related in experiments. It is conceivable that in experiments, these three concepts turn out to capture distinct psychological motives underlying intertemporal choice.
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Two papers in the literature test a concept which is directly related to decreasing relative impatience and spread seeking. Onay and Öncüler (2007) defined a decision maker to be timing risk averse if he prefers receiving an outcome with a sure delay to receiving the outcome with an uncertain delay with expected delay equal to the sure delay. Under expected discounted utility a decision maker is timing risk averse if and only if his discount function is concave. Then, timing risk aversion coincides with increasing relative impatience and spread aversion. Onay and Öncüler found evidence in favor of timing risk aversion. They do, however, not attribute their findings to concave utility but to non-linear probability weighting. Chesson and Viscusi (2003) also found evidence in favor of timing risk aversion. A test whether subjects satisfy increasing relative impatience will be a test of convexity of the discount function that is not confounded by probability weighting. Such a test will indicate whether the results of Onay and Öncüler and Chesson and Viscusi are only driven by probability weighting or also by concavity of the discount function. Acknowledgements Kirsten Rohde’s research was made possible through a VENI grant from the Netherlands Organization for Scientific Research (NWO). The author would like to thank Drazen Prelec and Peter Wakker for helpful comments. Appendix A Proof of Theorem 2.3 The result follows from DIðtÞ ¼ ½ln qðtÞ0 and from increasingness of ln.
h
Proof of Theorem 4.1 Assume that < satisfies more decreasing relative impatience than <. Then for all time points s < t, all r > 0, and all s we can find outcomes a; b; c; a ; b ; c with c 0 and c 0 such that
ð0 : a; s : cÞ ð0 : b; t : cÞ and ð0 : a; s þ r : cÞ ð0 : b; t þ s : cÞ and ð0 : a ; s : c Þ ð0 : b ; t : c Þ imply ð0 : a ; s þ r : c Þ^ ð0 : b ; t þ s : c Þ: The indifference ð0 : a; s : cÞ ð0 : b; t : cÞ is equivalent to
uð0ÞUðaÞ þ uðsÞUðcÞ ¼ uð0ÞUðbÞ þ uðtÞUðcÞ; which is equivalent to
½uðsÞ uðtÞUðcÞ ¼ uð0Þ½UðbÞ UðaÞ: Thus, we have,
½uðsÞ uðtÞUðcÞ ¼ uð0Þ½UðbÞ UðaÞ; ½uðs þ rÞ uðt þ sÞUðcÞ ¼ uð0Þ½UðbÞ UðaÞ; ½u ðsÞ u ðtÞU ðc Þ ¼ u ð0Þ½U ðb Þ U ða Þ; and ½u ðs þ rÞ u ðt þ sÞU ðc Þ 6 u ð0Þ½U ðb Þ U ða Þ: Combining these we have
½uðsÞ uðtÞUðcÞ ¼ ½uðs þ rÞ uðt þ sÞUðcÞ; and ½u ðsÞ u ðtÞU ðc Þ P ½u ðs þ rÞ u ðt þ sÞU ðc Þ: The assumption c 0 implies that UðcÞ > 0. Therefore, we have
uðsÞ uðtÞ ¼ uðs þ rÞ uðt þ sÞ; and u ðsÞ u ðtÞ P u ðs þ rÞ u ðt þ sÞ: Thus, u is a convex transformation of u. It follows that u is more convex that u. From Pratt (1964) it then follows that
u00 ðtÞ u00 ðtÞ P 0 ; u0 ðtÞ u ðtÞ
for all t P 0. The converse can be proven similarly. h Proof of Theorem 4.2 First observe that
d u0 ðtÞ lnuðtÞ ¼ : dt uðtÞ
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It follows that 2
d
dt
2
½u0 ðtÞ2 þ uðtÞu00 ðtÞ
lnuðtÞ ¼
½uðtÞ2
:
Thus,
DIðtÞ ¼
½ln uðtÞ00 ½u0 ðtÞ2 uðtÞu00 ðtÞ u0 ðtÞ u00 ðtÞ ¼ ¼ ¼ qðtÞ þ DRIðtÞ: uðtÞu0 ðtÞ uðtÞ u0 ðtÞ ½ln uðtÞ0
The result follows. h Proof of Corollary 4.3 By impatience and the fact that u > 0, the discount rate is always non-negative. Then, from DRIðtÞ ¼ DIðtÞ þ qðtÞ it follows that DRIðtÞ P 0 whenever DIðtÞ P 0. h Proof of Corollary 4.4 From a similar argument as in the proof of Corollary 4.3 we know that DIðtÞ 6 0 whenever DRIðtÞ 6 0.
h
Proof of Theorem 4.5 It can easily be verified that constant discounting uðtÞ ¼ dt implies a constant degree of decreasing relative impatience equal to ln d. Now assume that the degree of decreasing relative impatience is constant and equals ln d for some d > 0 with d < 1. From Pratt (1964) we then know that uðtÞ ¼ et ln d . Thus, uðtÞ ¼ et ln d ¼ dt . h Proof of Table 1 For quasi-hyperbolic discounting (Phelps & Pollak, 1968) we have
uðtÞ ¼ bdt ¼ bet ln d ; u0 ðtÞ ¼ b ln det ln d ; and u00 ðtÞ ¼ bðln dÞ2 et ln d : It follows that
DRI ¼ ln d: For hyperbolic discounting (Loewenstein & Prelec, 1992) we have
uðtÞ ¼ ð1 þ atÞb=a ; u0 ðtÞ ¼ bð1 þ atÞb=a1 ; u00 ðtÞ ¼ bða þ bÞð1 þ atÞb=a2 : It follows that
DRI ¼ ða þ bÞ=ð1 þ atÞ: For CADI discounting (Bleichrodt, Rohde, & Wakker, 2009) we have three cases. First we have ct
uðtÞ ¼ kere ; ct
u0 ðtÞ ¼ kere ðcrect Þ; ct
ct
u00 ðtÞ ¼ kere ðc2 rect Þ þ kere ðcrect Þ2 : It follows that
DRI ¼ c þ crect : Second,
uðtÞ ¼ kert ; which yields
DRI ¼ r:
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Finally, ct
uðtÞ ¼ kere yields
DRI ¼ c crect : For CRDI discounting (Bleichrodt et al., 2009; Ebert & Prelec, 2007) we also have three cases. First we have 1d
uðtÞ ¼ kert ; 1d
u0 ðtÞ ¼ kert ðð1 dÞrtd Þ; 1d
1d
u00 ðtÞ ¼ kert ðdð1 dÞrtd1 Þ þ kert ðð1 dÞrtd Þ2 : It follows that
DRI ¼ d=t ð1 dÞrt d : Second,
uðtÞ ¼ ktr ; u0 ðtÞ ¼ rktr1 ; u00 ðtÞ ¼ rðr þ 1Þkt r2 : It follows that
DRI ¼ ð1 þ rÞ=t: Finally, for 1d
uðtÞ ¼ kert ; it follows that
DRI ¼ d=t þ ð1 dÞrt d :
Proof of Theorem 5.3 Assume that < is more spread seeking than <. Let c 0, c 0, t > s P 0,
s > 0, r 2 R with s r P 0 and
ðs r : c; t þ s : cÞ ðs : c; t : cÞ: Then
ðs r : c ; t þ s : c Þ< ðs : c ; t : c Þ: By discounted utility these are equivalent to
uðs rÞ þ uðt þ sÞ ¼ uðsÞ þ uðtÞ; and
u ðs rÞ þ u ðt þ sÞ P u ðsÞ þ u ðtÞ: It follows that
uðs rÞ uðsÞ ¼ uðtÞ uðt þ sÞ: and
u ðs rÞ u ðsÞ P u ðtÞ u ðt þ sÞ: Thus, u is a convex transformation of u. It follows that u is more convex that u. From Pratt (1964) it then follows that
u00 ðtÞ u00 ðtÞ P 0 ; 0 u ðtÞ u ðtÞ
for all t P 0.
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Contents lists available at ScienceDirect
Journal of Economic Psychology journal homepage: www.elsevier.com/locate/joep
Decreasing relative impatience Kirsten I.M. Rohde * Erasmus School of Economics, H13-27, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands
a r t i c l e
i n f o
Article history: Received 3 October 2008 Received in revised form 6 July 2009 Accepted 13 August 2009 Available online 2 September 2009 JEL classification: D90 PsycINFO classification: 3920
a b s t r a c t Prelec (2004) showed that the Arrow–Pratt degree of convexity of the logarithm of the discount function can serve as a measure of decreasing impatience and of the corresponding time-inconsistency. In decision under risk and uncertainty the convexity of the utility function itself, not of its logarithm, has empirical meaning in terms of risk attitude. This paper introduces decreasing relative impatience, which is directly related to the degree of convexity of the discount function. By giving empirical meaning to the convexity of the discount function this paper clarifies the relations between several concepts in intertemporal choice. This paper also introduces a concept of spread seeking. Spread seeking and decreasing relative impatience are likely to have distinct underlying psychological motives, but are shown to be equivalent under discounted utility. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Intertemporal choice Decreasing impatience Decreasing relative impatience Hyperbolic discounting Consumer attitudes and behaviour
1. Introduction Imagine a decision maker who is indifferent between receiving a small reward soon and a larger reward at a later point in time. Under decreasing impatience, his preference will shift to the larger and later reward, when the receipt of both rewards is equally delayed. Thus, under decreasing impatience a difference in timing is weighted less the further it lies in the future. Constant impatience, i.e. stationarity, predicts the preference of the decision maker to be unchanged after a common delay of the smaller sooner and the larger later reward. Stationarity corresponds to constant discounting as introduced by Samuelson (1937). Under assumptions commonly made in the literature constant discounting is equivalent to time-consistency. Then, decreasing and increasing impatience imply time-inconsistency. There is an increasing interest in adjusting traditional economic models to accommodate time inconsistent preferences (Akerlof, 2002; Harris & Laibson, 2001; Krusell & Smith, 2003; Laibson, 1997; Luttmer & Mariotti, 2003; O’Donoghue & Rabin, 1999; Rubinstein, 2003, 2006; Thaler & Benartzi, 2004). Decreasing impatience means that a difference in timing is weighted less the further in the future it occurs. The weight per unit of time attached to an infinitely small difference in timing is given by minus the first derivative of the discount function. Thus, this weight becoming smaller the further in the future, corresponds to an increasing first derivative of the discount function, which corresponds to the discount function being convex. This suggests that individuals can be compared in terms of their degrees of decreasing impatience by comparing the degrees of convexity of their discount functions. Prelec (1989, 2004) formalized the concept of decreasing impatience. He showed that the degree of decreasing impatience, and the corresponding time-inconsistency, is captured by the degree of convexity of the logarithm of the discount * Tel.: +31 10 408 9548. E-mail address: [email protected] 0167-4870/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.joep.2009.08.008
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function, rather than the degree of convexity of the discount function itself. The latter is quite surprising considering the above reasoning. The key in understanding why Prelec needed the convexity of the logarithm of the discount function is that decreasing impatience concerns the rates of change of the discount function and not absolute changes, as this paper will show. We introduce the notion of decreasing relative impatience which concerns absolute changes of the discount function and can be captured by the convexity of the discount function itself, rather than its logarithm. The concept of decreasing relative impatience considers sequences of two outcomes, unlike the concept of decreasing impatience, which considers single outcomes. Consider an investment project where a decision maker has to incur costs today to receive a reward in the future. This investment project is a sequence of two outcomes: ‘cost now’ and ‘reward later’. Impatience implies that the decision maker likes a speed-up of the receipt of the reward. Imagine that a decision maker is willing to incur some extra costs today to speed-up the receipt of the reward by a particular amount of time. Decreasing relative impatience holds if the decision maker is less willing to incur extra costs to speed-up the receipt of the reward by a given amount of time, the further in the future the receipt of the reward. Thus, like decreasing impatience, decreasing relative impatience means that a time difference becomes less important the further it lies in the future. We will also introduce a notion of spread seeking and relate it to decreasing relative impatience. Consider a decision maker who receives a particular benefit twice, once soon and once later. If the DM is spread seeking then he will like to speedup the receipt of the first benefit and delay the receipt of the second one by a common amount of time, thereby increasing the spread of the two benefits. Consider, for instance, a person with a chronic illness who is entitled to two treatments that both give the same temporary relief from symptoms. The more spread seeking this person is, the more he will prefer to speed-up the first treatment and delay the second one by a common amount of time. We will show how the concepts of decreasing impatience, decreasing relative impatience, and spread seeking relate to each other. Interestingly, both decreasing relative impatience and spread seeking can be captured by the convexity of the discount function. Moreover, the degree of decreasing relative impatience equals the degree of decreasing impatience plus the discount rate. 2. Decreasing impatience Consider preferences < over streams of outcomes ðt0 : x0 ; t1 : x1 ; . . .Þ 2 ðT XÞn ; where X ¼ Rm is the set of outcomes and T ¼ Rþ is the set of time points. Time point 0 is today. The decision maker (DM) receives outcome xi at time t i and a reference outcome 0 2 X at all other time points. Preferences over outcomes are derived from preferences over outcomes today, i.e. for b; c 2 X we have b
X
uðtÞUðxt Þ;
t
where u is a general, twice continuously differentiable and strictly decreasing discount function and U is a continuous and increasing utility function with Uð0Þ ¼ 0: We will often consider another DM with preferences < ; who satisfies all the aforementioned assumptions, with discount function u and utility function U : We start by listing the main concepts that are used in intertemporal choice, so that we can show the links between them later on. The discount factor at date t is uðtÞ: The discount rate at date t is the rate of decline of the discount function at date t, i.e.
qðtÞ ¼
u0 ðtÞ : uðtÞ
We will need the following observation, which can easily be proven, later in the paper. Observation 2.1
qðtÞ ¼ ½ln uðtÞ0 : Impatience means that for every outcome b 0 and all time points s < t we have ðs : bÞ ðt : bÞ: Impatience corresponds to the discount function being strictly decreasing. Constant impatience or stationarity holds if ðs : cÞ ðt : bÞ implies ðs þ s : cÞ ðt þ s : bÞ: A DM with preferences < has decreasing impatience if for all s < t and s > 0, b c 0, and ðs : cÞ ðt : bÞ imply ðs þ s : cÞ^ðt þ s : bÞ. Increasing impatience holds if the implied preference is always the reverse. The interpretation of decreasing impatience is that when indifference holds between a smaller but sooner reward and a larger but later reward, adding a constant delay to both rewards will induce a, possibly strict, preference for the larger but later reward. Thus, the further in the future, the less the timing difference t s matters, and the more preferences will be driven by the outcomes. Preferences < exhibit more decreasing impatience than < if for all s < t; s; r > 0, b c 0, b c 0, we have that ðs; cÞ ðt; bÞ, ðs þ r; cÞ ðt þ s; bÞ, and ðs; c Þ ðt; b Þ imply ðs þ r; c Þ^ ðt þ s; b Þ. Prelec (2004) showed that the degree of decreasing impatience can be measured by the Arrow–Pratt degree of convexity of the logarithm of the discount function. Thus,
DIðtÞ ¼
½ln uðtÞ00 ½ln uðtÞ0
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serves as a measure of decreasing impatience. The next observation can easily be proven. Observation 2.2. The degree of decreasing impatience is the rate of change of the discount rate, i.e.
DIðtÞ ¼ ½ln qðtÞ0 : Thus, the degree of decreasing impatience is measured by the rate of change of the discount rate. It cannot be measured by the absolute decline in the discount rate, i.e. it cannot be measured by the first derivative of the discount rate. Nevertheless, a declining discount rate is equivalent to decreasing impatience as the next theorem shows. The first derivative of the discount rate can therefore identify decreasing impatience, but not measure its degree. Theorem 2.3. A declining discount rate is equivalent to decreasing impatience.
3. Decreasing relative impatience This section introduces the concepts of relative impatience and decreasing relative impatience, which we will show to be closely related to impatience and decreasing impatience. We will show that the degree of decreasing relative impatience can be captured by the degree of convexity of the discount function, rather than the degree of convexity of its logarithm. The main difference between the concepts introduced in this section and the concept of decreasing impatience as introduced by Prelec (1989, 2004), is that the latter concerns the receipt of a single outcome, whereas the former concern a sequence of two outcomes. We will often think of this sequence of two outcomes as an investment project, which entails costs today in return for a reward in the future. Definition 3.1. Preferences < satisfy relative impatience if
ð0 : b; s : cÞ ð0 : b; t : cÞ whenever s < t and c 0. Relative impatience means that in every investment project which entails costs today and a reward in the future, a DM dislikes delaying the future reward. We name it relative impatience because the delay of the receipt of c from s to t should be considered relative to the payment or receipt of b today. Under discounted utility, which we assume throughout, relative impatience is equivalent to impatience. Definition 3.2. Preferences < satisfy decreasing relative impatience if for all outcomes a; b; c with a b and c 0, all time points s < t, and r > 0,
ð0 : a; s : cÞ ð0 : b; t : cÞ implies ð0 : a; s þ r : cÞ^ð0 : b; t þ r : cÞ: Preferences < satisfy increasing relative impatience if the implied preference is reversed. Decreasing relative impatience can be interpreted as follows. The indifference ð0 : a; s : cÞ ð0 : b; t : cÞ means that to speed-up the receipt of c from t to s, the DM is willing to pay b a. If the receipt of c is delayed to t þ r, then the DM is no longer willing to pay b a to speed-up the receipt of c by t s units of time. Thus, a delay (of t s units of time) becomes less important the further it lies in the future. Intuitively, decreasing relative impatience, thus, encompasses a similar phenomenon as decreasing impatience. In the next section we will see that formally, these two concepts are not equivalent, though. First, we have to define how two decision makers can be compared according to their degree of decreasing relative impatience. The following definition does so in a similar manner as the definition for comparing decision makers according to their degree of decreasing impatience. Definition 3.3. Preferences < satisfy more decreasing relative impatience than < if for all outcomes a; b; c; a ; b ; c with a b, c 0, a b , and c 0, all time points s < t, all r > 0, and all s
ð0 : a; s : cÞ ð0 : b; t : cÞ and ð0 : a; s þ r : cÞ ð0 : b; t þ s : cÞ and ð0 : a ; s : c Þ ð0 : b ; t : c Þ imply ð0 : a ; s þ r : c Þ^ ð0 : b ; t þ s : c Þ: The interpretation of comparative decreasing relative impatience is as follows. Imagine a decision maker with preferences < who is indifferent between (1) speeding up the receipt of c from t to s by paying a instead of b; and (2) not speeding up the receipt of c. Imagine he is also indifferent between (1) speeding up the receipt of c from t þ s to s þ r by paying a instead of b and (2) not speeding up the reward. Then, if another decision maker with more decreasing relative impatience is indifferent between (1) speeding up the receipt of c from t to s by paying a instead of b and (2) not speeding up, this other decision maker prefers not to speed-up the receipt of c from t þ s to s þ r by paying a instead of b . He would, however, be indifferent if the receipt of c would instead be speed up from t þ s to s þ r for a particular r 6 r. In fact, when ð0 : a; s : cÞ ð0 : b; t : cÞ and ð0 : a; s þ r : cÞ ð0 : b; t þ s : cÞ, the timing difference s r serves as a measure of the degree of decreasing relative impatience.
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Until now we only considered a decision maker in a static context. The definition of decreasing relative impatience considers what happens to a preference between two investment projects as specified, when the receipt of the rewards is pushed back in time. If we assume that, as time evolves, decision makers reset the clock at zero every time they make a decision, then decreasing relative impatience also has a dynamic interpretation. Suppose that, on the first of January, a decision maker can choose between two investment projects. In one project, he will receive a reward c on the first of December of the same year. He would have to pay b as soon as he commits to that project. In the other project, he would receive the same reward c, but earlier, namely on the first of November, in return for a larger price at the time of commitment: jaj > jbj. Suppose that on the first of January the decision maker has a slight preference for receiving the reward on the first of December: ð0 : a; s þ r : cÞ ð0 : b; t þ r : cÞ, where s þ r and t þ r are the durations until the first of November and the first of December, respectively. Consider which investment project he would prefer on the first of February. Note that the investment projects are specified in such a way that the receipt of the reward is on fixed dates, whereas the investment of either a or b can be done at any date before the first of November. Decreasing relative impatience implies that, as the decision maker is almost indifferent between the two projects on the first of January, he is likely to prefer the project which delivers the reward on the first of November in return for a payment of a on the first of February: ð0 : a; s : cÞ ð0 : b; t : cÞ. Thus, he switches his preference between the two investment projects as time elapses from the first of January to the first of February.
4. Convexity of the discount function The degree of decreasing relative impatience is captured by the degree of convexity of the discount function, as is shown in the following theorem. Theorem 4.1. Preferences < satisfy more decreasing relative impatience than < if and only if
u00 ðtÞ u00 ðtÞ P 0 0 u ðtÞ u ðtÞ
for all t P 0. Thus,
DRIðtÞ ¼
u00 ðtÞ u0 ðtÞ
is a measure of decreasing relative impatience. DRI will be non-negative in the case of decreasing relative impatience and negative otherwise. As the following theorem shows, the degree of decreasing relative impatience at any point in time is the sum of the degree of decreasing impatience and the discount rate at that point in time: decreasing relative impatience = decreasing impatience + discount rate. Theorem 4.2. For all t
DRIðtÞ ¼ DIðtÞ þ qðtÞ: Prelec (1989) showed this result before, but for completeness we repeat it here. Theorem 4.2 shows that decreasing impatience implies decreasing relative impatience and that increasing relative impatience implies increasing impatience. Corollary 4.3. Decreasing impatience implies decreasing relative impatience:
DIðtÞ P 0 ) DRIðtÞ P 0: Corollary 4.4. Increasing relative impatience implies increasing impatience:
DRIðtÞ 6 0 ) DIðtÞ 6 0: The constant discount function uðtÞ ¼ dt satisfies decreasing relative impatience, but constant impatience. Constant discounting is equivalent to a constant degree of decreasing relative impatience, which equals the discount rate. Theorem 4.5. Preferences satisfy constant discounting uðtÞ ¼ dt with 0 < d < 1 if and only if the degree of decreasing relative impatience is constant and equal to ln d: Table 1 gives the degree of decreasing relative impatience for other discount functions that are used in the literature. These degrees are derived in the Appendix.
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Table 1 DRI for various discount functions. Quasi-hyperbolic (Phelps & Pollak, 1968) uðtÞ ¼ bdt for t > 0
DRIðtÞ ¼ ln d
Hyperbolic (Loewenstein & Prelec, 1992)
uðtÞ ¼ ð1 þ atÞb=a
DRIðtÞ ¼ ða þ bÞ=ð1 þ atÞ
CADI (Bleichrodt et al., 2009) ct
uðtÞ ¼ kere uðtÞ ¼ kert ct uðtÞ ¼ kere
DRIðtÞ ¼ c þ crect DRIðtÞ ¼ r DRIðtÞ ¼ c crect
CRDI (Bleichrodt et al., 2009) 1d
uðtÞ ¼ kert uðtÞ ¼ ktr 1d uðtÞ ¼ kert
DRIðtÞ ¼ d=t ð1 dÞrt d DRIðtÞ ¼ ð1 þ rÞ=t DRIðtÞ ¼ d=t þ ð1 dÞrt d
5. Spread seeking This section shows that decreasing relative impatience relates to a concept which we call spread seeking. An individual who is spread seeking prefers to increase the spread, over time, of pleasant events. Assume that a DM receives a particular benefit at two different dates. He is spread seeking if he prefers to speed-up the early receipt and delay the late receipt by a common amount of time. Definition 5.1. Preferences < satisfy spread seeking if for all c 0 and all t > s P 0,
s > 0 with s 6 s
ðs s : c; t þ s : cÞ<ðs : c; t : cÞ: Preferences < satisfy spread aversion if the preference is the reverse. One decision maker may be more spread seeking than another. The following definition formalizes this point. Definition 5.2. Preferences < satisfy more spread seeking than < if for all c 0, c 0, and all t > s P 0, s > 0 and all r 2 R with s r P 0
ðs r : c; t þ s : cÞ ðs : c; t : cÞ implies
ðs r : c ; t þ s : c Þ< ðs : c ; t : c Þ: It follows that when ðs r : c; t þ s : cÞ ðs : c; t : cÞ for c 0, the length of time s r serves as a measure of spread seeking. As the next theorem shows, the degree of convexity of the discount function captures the degree of spread seeking. Interestingly, spread seeking and decreasing relative impatience are equivalent concepts under discounted utility. Theorem 5.3. < is more spread seeking than < if and only if
u00 ðtÞ u00 ðtÞ P 0 u0 ðtÞ u ðtÞ
for all t P 0. It is remarkable that two at first sight distinct concepts with possibly distinct underlying psychological motives, like decreasing relative impatience and spread seeking, turn out to be equivalent under discounted utility. As soon as these concepts are considered outside the discounted utility setting, though, they are likely no longer to be equivalent. 6. Discussion This paper introduced the concept of decreasing relative impatience, which is related, but not equivalent, to decreasing impatience as introduced by Prelec (1989, 2004). The main difference between the two concepts is that the latter concerns receipts of single outcomes, whereas the former concerns sequences of two outcomes. A convenient way to operationalize the concept of decreasing relative impatience, is to think in terms of investment projects that require a payment today in return for a future reward. It will be interesting to see how the concepts of decreasing relative impatience, decreasing impatience, and spread seeking are empirically related in experiments. It is conceivable that in experiments, these three concepts turn out to capture distinct psychological motives underlying intertemporal choice.
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Two papers in the literature test a concept which is directly related to decreasing relative impatience and spread seeking. Onay and Öncüler (2007) defined a decision maker to be timing risk averse if he prefers receiving an outcome with a sure delay to receiving the outcome with an uncertain delay with expected delay equal to the sure delay. Under expected discounted utility a decision maker is timing risk averse if and only if his discount function is concave. Then, timing risk aversion coincides with increasing relative impatience and spread aversion. Onay and Öncüler found evidence in favor of timing risk aversion. They do, however, not attribute their findings to concave utility but to non-linear probability weighting. Chesson and Viscusi (2003) also found evidence in favor of timing risk aversion. A test whether subjects satisfy increasing relative impatience will be a test of convexity of the discount function that is not confounded by probability weighting. Such a test will indicate whether the results of Onay and Öncüler and Chesson and Viscusi are only driven by probability weighting or also by concavity of the discount function. Acknowledgements Kirsten Rohde’s research was made possible through a VENI grant from the Netherlands Organization for Scientific Research (NWO). The author would like to thank Drazen Prelec and Peter Wakker for helpful comments. Appendix A Proof of Theorem 2.3 The result follows from DIðtÞ ¼ ½ln qðtÞ0 and from increasingness of ln.
h
Proof of Theorem 4.1 Assume that < satisfies more decreasing relative impatience than <. Then for all time points s < t, all r > 0, and all s we can find outcomes a; b; c; a ; b ; c with c 0 and c 0 such that
ð0 : a; s : cÞ ð0 : b; t : cÞ and ð0 : a; s þ r : cÞ ð0 : b; t þ s : cÞ and ð0 : a ; s : c Þ ð0 : b ; t : c Þ imply ð0 : a ; s þ r : c Þ^ ð0 : b ; t þ s : c Þ: The indifference ð0 : a; s : cÞ ð0 : b; t : cÞ is equivalent to
uð0ÞUðaÞ þ uðsÞUðcÞ ¼ uð0ÞUðbÞ þ uðtÞUðcÞ; which is equivalent to
½uðsÞ uðtÞUðcÞ ¼ uð0Þ½UðbÞ UðaÞ: Thus, we have,
½uðsÞ uðtÞUðcÞ ¼ uð0Þ½UðbÞ UðaÞ; ½uðs þ rÞ uðt þ sÞUðcÞ ¼ uð0Þ½UðbÞ UðaÞ; ½u ðsÞ u ðtÞU ðc Þ ¼ u ð0Þ½U ðb Þ U ða Þ; and ½u ðs þ rÞ u ðt þ sÞU ðc Þ 6 u ð0Þ½U ðb Þ U ða Þ: Combining these we have
½uðsÞ uðtÞUðcÞ ¼ ½uðs þ rÞ uðt þ sÞUðcÞ; and ½u ðsÞ u ðtÞU ðc Þ P ½u ðs þ rÞ u ðt þ sÞU ðc Þ: The assumption c 0 implies that UðcÞ > 0. Therefore, we have
uðsÞ uðtÞ ¼ uðs þ rÞ uðt þ sÞ; and u ðsÞ u ðtÞ P u ðs þ rÞ u ðt þ sÞ: Thus, u is a convex transformation of u. It follows that u is more convex that u. From Pratt (1964) it then follows that
u00 ðtÞ u00 ðtÞ P 0 ; u0 ðtÞ u ðtÞ
for all t P 0. The converse can be proven similarly. h Proof of Theorem 4.2 First observe that
d u0 ðtÞ lnuðtÞ ¼ : dt uðtÞ
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It follows that 2
d
dt
2
½u0 ðtÞ2 þ uðtÞu00 ðtÞ
lnuðtÞ ¼
½uðtÞ2
:
Thus,
DIðtÞ ¼
½ln uðtÞ00 ½u0 ðtÞ2 uðtÞu00 ðtÞ u0 ðtÞ u00 ðtÞ ¼ ¼ ¼ qðtÞ þ DRIðtÞ: uðtÞu0 ðtÞ uðtÞ u0 ðtÞ ½ln uðtÞ0
The result follows. h Proof of Corollary 4.3 By impatience and the fact that u > 0, the discount rate is always non-negative. Then, from DRIðtÞ ¼ DIðtÞ þ qðtÞ it follows that DRIðtÞ P 0 whenever DIðtÞ P 0. h Proof of Corollary 4.4 From a similar argument as in the proof of Corollary 4.3 we know that DIðtÞ 6 0 whenever DRIðtÞ 6 0.
h
Proof of Theorem 4.5 It can easily be verified that constant discounting uðtÞ ¼ dt implies a constant degree of decreasing relative impatience equal to ln d. Now assume that the degree of decreasing relative impatience is constant and equals ln d for some d > 0 with d < 1. From Pratt (1964) we then know that uðtÞ ¼ et ln d . Thus, uðtÞ ¼ et ln d ¼ dt . h Proof of Table 1 For quasi-hyperbolic discounting (Phelps & Pollak, 1968) we have
uðtÞ ¼ bdt ¼ bet ln d ; u0 ðtÞ ¼ b ln det ln d ; and u00 ðtÞ ¼ bðln dÞ2 et ln d : It follows that
DRI ¼ ln d: For hyperbolic discounting (Loewenstein & Prelec, 1992) we have
uðtÞ ¼ ð1 þ atÞb=a ; u0 ðtÞ ¼ bð1 þ atÞb=a1 ; u00 ðtÞ ¼ bða þ bÞð1 þ atÞb=a2 : It follows that
DRI ¼ ða þ bÞ=ð1 þ atÞ: For CADI discounting (Bleichrodt, Rohde, & Wakker, 2009) we have three cases. First we have ct
uðtÞ ¼ kere ; ct
u0 ðtÞ ¼ kere ðcrect Þ; ct
ct
u00 ðtÞ ¼ kere ðc2 rect Þ þ kere ðcrect Þ2 : It follows that
DRI ¼ c þ crect : Second,
uðtÞ ¼ kert ; which yields
DRI ¼ r:
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Finally, ct
uðtÞ ¼ kere yields
DRI ¼ c crect : For CRDI discounting (Bleichrodt et al., 2009; Ebert & Prelec, 2007) we also have three cases. First we have 1d
uðtÞ ¼ kert ; 1d
u0 ðtÞ ¼ kert ðð1 dÞrtd Þ; 1d
1d
u00 ðtÞ ¼ kert ðdð1 dÞrtd1 Þ þ kert ðð1 dÞrtd Þ2 : It follows that
DRI ¼ d=t ð1 dÞrt d : Second,
uðtÞ ¼ ktr ; u0 ðtÞ ¼ rktr1 ; u00 ðtÞ ¼ rðr þ 1Þkt r2 : It follows that
DRI ¼ ð1 þ rÞ=t: Finally, for 1d
uðtÞ ¼ kert ; it follows that
DRI ¼ d=t þ ð1 dÞrt d :
Proof of Theorem 5.3 Assume that < is more spread seeking than <. Let c 0, c 0, t > s P 0,
s > 0, r 2 R with s r P 0 and
ðs r : c; t þ s : cÞ ðs : c; t : cÞ: Then
ðs r : c ; t þ s : c Þ< ðs : c ; t : c Þ: By discounted utility these are equivalent to
uðs rÞ þ uðt þ sÞ ¼ uðsÞ þ uðtÞ; and
u ðs rÞ þ u ðt þ sÞ P u ðsÞ þ u ðtÞ: It follows that
uðs rÞ uðsÞ ¼ uðtÞ uðt þ sÞ: and
u ðs rÞ u ðsÞ P u ðtÞ u ðt þ sÞ: Thus, u is a convex transformation of u. It follows that u is more convex that u. From Pratt (1964) it then follows that
u00 ðtÞ u00 ðtÞ P 0 ; 0 u ðtÞ u ðtÞ
for all t P 0.
h
K.I.M. Rohde / Journal of Economic Psychology 30 (2009) 831–839
839
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