Time-Tradeoff Sequences for Easy Measurements and Analyses of

Hyperbolic Discounting and Dynamic Inconsistency in Intertemporal Choice

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

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We need a …

volunteer.

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For

(5:700) ~ (t1:900)

etc.

For

(5:700) ~ (t1:900)

etc.

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.

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m, tj, j variable.

cf acts from DUU.

m, tj, j variable.

cf acts from DUU.

Discounted utility: Evaluate streams of outcomes through(t1:1, …, tm:m) j=1

m (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=1

n tjU(j)with 0 < 1. That is, (t) = t (Samuelson 1937).

Implication: "stationarity."

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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.

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.

Imagine (t1 : 1, .…, tm : m) (s1 : 1, …, sn : n).

That is, j=1

m tj U(j) i=1

n si U(i).

Imagine: Common delay of all outcomes.

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(t1+:1, …, tm+:m) ? (s1+:1, …, sn+:n).

j=1m

tj+ U(j) ? i=1

n si+

U(i). j=1

m tj

U(j) ? i=1n

si U(i).

j=1m

tj U(j) ? i=1n si U(i).

What is preference "?"? = !

Preference is not affected.

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:

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choice 3 choice 4 choice 5: trivial (framing)

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€ X in 1 month

€ 100 immedi-ately

Announce decisionbeforehand. Are commit-ted to it!

€ X in 1 month

€ 100 immediately

Wait 6 months.

Wait 6 months.

€ X in 7 months

€ 100 in 6 months

Wait 6 months. Then choose:

€ X in 1 month

€ 100 immedi-ately

Wait 6 months. Then choose:

€ X in 1 month

€ 100 immedi-ately

Choice 1. Choice 2.

Choice 3.

Choice 4.Choice 5.

€ X: value to make you indifferentin choice 1;please write it down on a piece of paper.

>>

: majority preference in choice 5.

choice 1 choice 2: uniformity of time

choice 2 choice 3/4/5: time consistency (rational?)

choice 1 choice 3/4/5: stationarity

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.

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Example 1. Quasi-hyperbolic discounting.

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Evaluate streams of outcomes through(0:1, t2:2 …, tm:m) U(1) + j=2

n 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.

with 0 < 1.

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.

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Example [arbitrage out of nonstationarity]. Agent owns soon "small" payment.

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Not stationary; preference change if moving forward by 5 months:

0 months€700 (small)

4 months€900 (large)

5 months€700 (small)

9 months€900 (large)

5 months€700 (small)

Strictly prefers the late "large" payment, paying €1 to exchange.

9 months

€900 (large)

Here we offer to exchange small for large, charging €1.

Here we offer to change back large for small (nontrivial move).

Arbitrage: we can pump €1.

time axis

Arbitrage: we gained €1, leaving agent in original situation less €1.

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."

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Prelec (2004):(ln((t)))´´(ln((t)))´

is an index of nonstationarity.

Hard to observe (?):Have to measure discounting function .To do so, also have to estimate utility U.Then need computer to calculate …

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.

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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.

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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 .

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0

t

1

4/5

t1t0

Normalize: (t0) = 1; (tn) = 0.

t5

1/5

t4t3

2/5

Further:(tj) = (n–j)/n.

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t2

3/5

Say: n = 5.

This curve is the time-tradeoff curve.

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.

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(t) =ln((t)) – ln((tn))

ln((t0)) – ln((tn))

We conclude:

Degree of convexity of ln((t)) is degree of convexity of ! Can immediately be inspected from graph depicted before.

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Figure. The time-tradeoff curve of subject 7.

t0= 5

t1= 12

t2= 18

t3= 25

t4= 37

t5= 49

1.0

0.8

0.6

0.4

0.2

0.0

time in months

Explain that stationarity is the diagonal.

Explain that stationarity is the diagonal.

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1,000,800,600,400,200,00

seq3

1,00

0,80

0,60

0,40

0,20

0,00

phi

49,00

38,00

24,00

13,00

10,00

7,00

5,00

49,00

38,00

24,00

13,00

10,00

7,00

5,00nr

subject 7

subject 24

subject 38

subject 13

subject 5 subject 49

subject 10

linear = stationary

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:

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(ti+1–ti) – (ti–ti–1 )

ti(ti–ti–1 ) – ti–1(ti+1–ti)

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.

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Experiment.

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

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(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)

(5:700) ~ (t1:900) (t1:700) ~ (t2:900) (t2:700) ~ (t3:900) (t3:700) ~ (t4:900) (t4:700) ~ (t5:900)

(0:1600) ~ (t1:1900) (t1:1600) ~ (t2:1900) (t2:1600) ~ (t3:1900) (t3:1600) ~ (t4:1900) (t4:1600) ~ (t5:1900)

In addition, demographic variables such as gender, age, length, body weight, smoker?, field of study, nationality.

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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.

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

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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).

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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.

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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.

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The end.

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