Peter P. Wakker (& Arthur Attema, Han Bleichrodt, Kirsten Rohde)

1. Time-Tradeoff Sequences for Easy Measurements and Analyses of Hyperbolic Discounting and Dynamic Inconsistency in Intertemporal Choice Make yellow comments invisible. ALT-View-O Peter P. Wakker (& Arthur Attema, Han Bleichrodt, Kirsten Rohde)

2. 2 For (5:700) ~ (t1:900) etc. We need a … volunteer.

3. 3 Topic: Intertemporal choice. (Many similarities with decision under uncertainty). (t1:1, …, tm:m): stream of outcomes, yielding outcome€j at timepoint tj, j=1,…,m, and nothing else (= €0) at all other times. [0, ): time axis; +: outcome set. m, tj, j variable. cf acts from DUU.

4. 4 Discounted utility: Evaluate streams of outcomes through (t1:1, …, tm:m)  j=1m(tj)U(j) with U(0) = 0, (0) = 1, everything continuous, and  strictly decreasing (impatience).Major empirical violation: time separability … Most common, traditional, special case: Constant discounting:j=1ntjU(j)with 0 <   1. That is, (t) = t(Samuelson 1937). Implication: "stationarity." If the first thing a baby says is not mom, or dad, or milk, but "I think, so I exist," then that would suggest to me that it is an intelligent baby. This was the 2nd paper by Samuelson and, indeed, it suggested that this is an intelligent guy.

5. 5 Imagine (t1 : 1, .…, tm : m)  (s1 : 1, …, sn : n). That is, j=1mtjU(j) i=1nsiU(i). Imagine: Common delay of all outcomes.  (t1+:1, …, tm+:m) ? (s1+:1, …, sn+:n).  j=1mtj+U(j) ?i=1nsi+U(i). j=1mtjU(j) ?i=1nsiU(i). j=1mtjU(j) ?i=1nsiU(i).   What is preference "?" ? = ! Preference is not affected.

6. 6 Stationarity: A preference is maintained if all outcomes are delayed by the same amount of time. We saw: holds if constant discounting. Well known: "iff." Also called constant impatience. A normative argument for stationarity:

7. €100 immedi-ately €X in 1 month Choice 2. €100 immedi-ately Wait 6 months. Then choose: €X in 1 month > : majority preference in choice 5. Choice 3. Wait 6 months. €100 immediately €X in 1 month Wait 6 months. Choice 5. Choice 4. 7 €100 immedi-ately Wait 6 months. Then choose: €X in 1 month Choice 1. €X: value to make you indifferent Announce decision beforehand. Are commit-ted to it! in choice 1; please write it down on a piece of paper. choice 3  choice 4 choice 5: trivial (framing) €100 in 6 months choice 2  choice 3/4/5: time consistency (rational?) > €X in 7 months choice 1  choice 2: uniformity of time choice 1  choice 3/4/5: stationarity

8. 8 Empirical finding: decreasing impatience (as in choice 5 before). Violation of stationarity. Deviations from constant discounting have been developed. So-called hyperbolic discounting. They discount: the near future stronger relative to the present, but the far future less strongly.

9.  with 0 <   1. 9 Example 1. Quasi-hyperbolic discounting. Evaluate streams of outcomes through (0:1,t2:2 …,tm:m)  U(1) + j=2ntjU(j) That is: (t) = 1 = t if t = 0. (t) = t < t if t > 0. "Constant discounting plus a present-effect." Pragmatic&popular (Laibson), but not refined.

10. 10 Example 2.Generalized hyberbolic discounting (Prelec & Loewenstein 1992). (t) = (1+ht)–r/h for 0  h <  (limit h=0: constant discounting). Comprises several popular special cases: h=1: Harvey (1986). h=r: Mazur (1987), Harvey (1995) as proportional discounting.

11. Strictly prefers the late "large" payment, paying  €1 to exchange. €900 (large) €700 (small) €900 (large) €900 (large) €700 (small) €700 (small) time axis 4 months 9 months 5 months 0 months 5 months 9 months   Here we offer to change back large for small (nontrivial move). Here we offer to exchange small for large, charging €1. 11 Example [arbitrage out of nonstationarity]. Agent owns soon "small" payment. Not stationary; preference change if moving forward by 5 months: Arbitrage: we can pump €1. Arbitrage: we gained €1, leaving agent in original situation less €1.

12. 12 For possibility of arbitrage, deviations from stationarity are important. Prelec (2004): "Decreasing impatience provides a natural criterion for assessing whether a set of time preferences represents a more or less severe departure from the stationarity axiom. The criterion is associated with a simple normative diagnostic—the selection of inefficient (dominated) outcomes in two-stage decision problems."

13. Prelec (2004): is an index of nonstationarity. (ln((t)))´´ (ln((t)))´ 13 Hard to observe (?): Have to measure discounting function . To do so, also have to estimate utility U. Then need computer to calculate …

14. 14 We introduce time-tradeoff sequences. Easily get Prelec's measure, and graph thereof. Can be determined by only pencil and paper and eyeballing of data.

15. 15 t0,t1,…,tn is a time-tradeoff (TTO) sequence if there exist outcomes    such that (t0:) ~ (t1:) (t1:) ~ (t2:) . . . (tn–1:) ~ (tn:) Claim: a normalized graph of ln() can immediately be inferred from TTO sequences. In particular, Prelec's convexity index can immediately be inferred.

16. Demonstration. (ti–1:) ~ (ti:) implies (ti–1)U() = (ti)U(), (ti–1)/(ti) = U()/U(). So, (ti–1)/(ti) is the same for all i. ln((ti–1)) – ln((ti)) is the same for all i. t0,t1,…,tn are equally spaced in ln((t)) units. We next show how this leads to the normalized graph of ln((t)), through what we call time-tradoff (TTO) curves, denoted . 16

17. 1  3/5 2/5 1/5 0 t4 t2 t3 t 17 Further:(tj) = (n–j)/n. Normalize:(t0) = 1; (tn) = 0. Say: n = 5. 4/5 This curve  is the time-tradeoff curve. t1 t0 t5

18. 18 ti's are equally spaced in terms of . That's how we drew . ti's are also equally spaced in terms of ln((t)). So,  must be a linear (for mathematicians: affine) transformation of ln((t)). So,  = b + aln((t)) for a > 0, b.

19. ln((t)) – ln((tn)) (t) = ln((t0)) – ln((tn)) 19 We conclude: Degree of convexity of ln((t)) is degree of convexity of ! Can immediately be inspected from graph depicted before.

20. 1.0 0.8 0.6 0.4 0.2 0.0 t1= 12 t2= 18 t3= 25 t4= 37 t5= 49 t0= 5 time in months Figure. The time-tradeoff curve of subject 7.  20 Explain that stationarity is the diagonal.

21. 21 linear = stationary subject 10 subject 13 subject 5 subject 49 subject 7 subject 38 subject 24

22. (ti+1–ti) – (ti–ti–1 ) ti(ti–ti–1 ) – ti–1(ti+1–ti) 22 Quantitative measures of convexity can be devised, such as area between curve and stationarity-diagonal. Rohde (2005) introduced an alternative measure of decreasing impatience, more suited for generalized hyperbolic discounting, the hyperbolic factor:

23. 23 Theorem. Generalized hyperbolic discounting ((t) = (1+ht)–r/h) holds iff hyperbolic factor is always h.  Thus, we can immediately test special cases of hyperbolic such as Harvey's (1986) special case of h=1.

24. 24 Experiment. N=55 subjects, students from Maastricht & Rotterdam.Received flat payment €10. All interviewed individually. Training questions prior to experimental. We measured:

25. 25 (0:700) ~ (t1:900) (t1:700) ~ (t2:900) (t2:700) ~ (t3:900) (t3:700) ~ (t4:900) (t4:700) ~ (t5:900) (0:2800) ~ (t1:3300) (t1:2800) ~ (t2:3300) (t2:2800) ~ (t3:3300) (t3:2800) ~ (t4:3300) (t4:2800) ~ (t5:3300) (0:1600) ~ (t1:1900) (t1:1600) ~ (t2:1900) (t2:1600) ~ (t3:1900) (t3:1600) ~ (t4:1900) (t4:1600) ~ (t5:1900) (5:700) ~ (t1:900) (t1:700) ~ (t2:900) (t2:700) ~ (t3:900) (t3:700) ~ (t4:900) (t4:700) ~ (t5:900)

26. 26 In addition, demographic variables such as gender, age, length, body weight, smoker?, field of study, nationality. Also we measured T for (5:700, 11:700) ~ (1:700, T:700). Then (5)r + (11)r = (1)r + (T)r identifies the power of , allowing an entire "two-stage" measurement of time attitude.

27. 27 Median data: median TTO curves p. 20 in pdf file. median di curves on p. 21 in pdf files.

28. 28 Literature study: often increasing impatience. Decreasing impatience is not universal: Airoldi, Read, & Frederick (2005) Frederick (1999) Read, Airoldi, & Loewe (2005) Read, Frederick, Orsel, & Rahman (2005) Rubinstein (2003) Sayman & Onculer (2005).

29. 29 Our data also very clearly rejected generalized hyperbolic discounting (hyperbolic factor not constant and often not even defined). In general, men were more impatient but less deviating from stationarity, but not significantly so.

30. 30 Conclusions. - Nonstationarity (arbitrage-proneness) easily measurable, quantifiable, and visualizable through time-tradeoff sequences. Using only pencil and paper! - Impatience not universally decreasing.In our data first increasing, then constant. - Quasi-hyperbolic discounting & generalized hyperbolic discounting performed badly. Problem: these families are entirely focused on decreasing impatience. We need new discounting families, with more flexibility regarding increasing impatience.

31. 31 The end.