Hyperbolic Discounting and Self-Control:An Experimental Analysis

Jess BenhabibNYU

Alberto BisinNYU

Andrew SchotterNYU

June 2004∗

1 Introduction

A vast literature in experimental psychology has documented various behavioral regulari-ties that cast doubts on exponential discounting. The most important of such anomalies,called ”reversal of preferences,” has been interpreted to suggest that agents have a prefer-ence for present consumption not consistent with exponential discounting. Psychologists(e.g., Herrnstein, 1961, de Villiers-Herrnstein, 1976, Ainsle-Herrnstein, 1981; see alsoAinsle, 1992, 2001) and, most recently, behavioral economists (e.g., Elster, 1979, Laib-son, 1997, O’Donoghue-Rabin, 1999) have noted that the evidence is consistent with adeclining rate of time preference, and have consequently suggested various specificationsof discounting with this property, notably hyperbolic discounting and quasi-hyperbolicdiscounting.1

Such specifications of discounting introduce a fundamental paradigm change in eco-nomic theory: preferences with hyperbolic (or quasi-hyperbolic) discounting, unlike thosewith exponential discounting, lack time-consistency (see Strotz, 1954, for an early dis-cussion of such issues). When his preferences are time-inconsistent, an agent’s preferenceordering changes over time. Dynamic choice problems are therefore not determined by

∗Thanks to Colin Camerer, Antonio Rangel, Aldo Rustichini, Giorgio Topa, and especially to ArielRubinstein. Kyle Hyndman exceptional work as RA is also gratefully acknowledged.

1Of course, a declining rate of time preference is not the only possible explanation of such anomaliesof time preference. Rubinstein, 2000, shows how reversal of preferences might be induced by a specificform of procedural rationality. Also, most of the documented anomalies are consistent in principle withpreferences over sets of actions, under standard rationality assumptions; see Gul-Pesendorfer, 2001.Finally, various specifications of psychological models of strategic interactions between multiple selvesat each time period may rationalize such anomalies; see Thaler-Shefrin, 1981, for an early example ineconomics.

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the solution of a simple maximization problem, and require the agent to form expec-tations regarding his own decisions in the future. Moreover, when preference orderingschange over time, the scope for normative statements and welfare analysis is of coursegreatly limited.

While experimental psychologists have collected an impressive amount of data ontime preference in support of declining discount rates, most of this data is not with-out problems. Often experiments have been conducted with hypothetical rewards, orwith ”points” redeemable at the end of the experiments (thereby eliminating any ratio-nale for time preference); the design of the experiments is seldom immune to issues ofstrategic manipulability, or of framing effects. Most importantly, rarely have the databeen analyzed with proper econometric instruments. To our knowledge the hypothe-sis of hyperbolic discounting has never been tested statistically against the alternativeof exponential discounting. (See Frederick-Loewenstein-O’Donoghue, 2002, for a com-prehensive survey of the empirical literature on time preference.) Formal statisticalprocedures are necessary to identify behavioral regularities with noisy data; this is im-portant in the context of experimental data on discounting which, the literature hasrepeatedly observed, are particularly noisy (Kirby-Hernnstein, 1995, Kirby, 1997; seealso Frederick-Loewenstein-O’Donoghue, 2002).

In this paper we elicit preferences for discounting via experimental techniques. Wethen estimate a general specification of discounting which nests exponential and hyper-bolic discounting, as well as various forms of present bias in discounting. We call presentbias the psychological phenomenon whereby subjects associate a discrete cost to anyfuture, as opposed to present, reward. The classic example of present bias is then quasi-hyperbolic discounting, as adopted e.g., by Laibson, 1997, and O’Donoghue-Rabin, 1999,in which the cost associated with future rewards is variable, that is proportional to thereward. The specification of present bias we estimate in this paper nests quasi-hyperbolicdiscounting as well as another specification in which a fixed rather than a variable costis associated to future rewards. Our methodology allows us to statistically test the ex-ponential against the hyperbolic specifications. It also allows us to identify present biasin preferences for discounting and to test the fixed versus variable cost specification.Distinguishing empirically between fixed and variable costs has important implications.If the present bias takes the form of a variable cost, that is it contains a quasi-hyperboliccomponent, then it affects all intertemporal choices; if instead the present bias takes theform of a fixed cost its effects tend to vanish for large rewards.

In accordance with the results of Kirby, 1997, we find that exponential discountingis rejected by the data. A hyperbolic specification fits our discounting data better; anddiscount rates decline with delay. If we impose no present bias in the specification,however, estimated instantaneous discount rates appear implausibly high, implying formost agents yearly discount rates of the order of 500 percent. The specification withpresent bias therefore fares better on this and on other dimensions. Regarding the formof the present bias, the data do not seem to support a quasi-hyperbolic specification: a

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positive variable cost is not present for most of our subjects and, when it is, it is verysmall. The data strongly favor a specification with a present bias in the form of a fixedcost, significantly different from 0 and estimated as of the order of $4 on average acrossagents.

Finally we report some evidence about framing. The evidence is rather inconclusive:eliciting discount rates by means of different questions produces different point estimates,but for most subjects these estimates are not statistically distinct, that is, confidenceintervals do overlap.

2 Discounting Curves

Consider evaluating a dollar amount, say y, t periods from now: t is the delay at which yis received. An arbitrary discount function D(y, t) is such that the value of y with delayt is:

yD(y, t).

Note that we allow the discount factor D(y, t) to depend on delay t as well as on theamount to be discounted y. The associated subjective discount rate at y is:

∂∂t

D(y, t)

D(y, t)

The function D(y, t) represents exponential discounting if

D(y, t) = exp{−rt}, r > 0; (1)

and it represents hyperbolic discounting if

D(y, t) =1

1 + rt, r > 0 (2)

Both exponential and hyperbolic discounting are independent of the amount to bediscounted, y. But, in contrast to exponential discounting, preferences that displayhyperbolic discounting induce declining subjective interest rates. In particular, the sub-jective interest rate associated with exponential discounting is −r, a constant, while thesubjective interest rate associated with hyperbolic discounting is −r

1+rt, and hence it is

declining in the delay t. Consequently, preferences displaying hyperbolic discountinggive rise to the phenomenon of preference reversals, see Figure 1.

The other specification of discounting that has been studied in psychology and eco-nomics is quasi-hyperbolic discounting :

yD(y, t) =

{y if t = 0α exp{−rt}y if t > 0

, α < 1. (3)

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Discounting is quasi-hyperbolic if it displays a present bias, the larger the bias the smallerthe parameter α.

Note that the present bias in quasi-hyperbolic discounting takes the form (the inter-pretation) of a variable cost associated to future payoffs: any payoff y is valued at mosty − (1 − α)y when received in the future. The cost (1 − α)y is variable in the sensethat it increases linearly with the amount y.2 This interpretation induces us to consideranother possible specification of discounting, in which the present bias is represented bya fixed cost b rather than by a variable cost. In this case, the amount y with delay t isvalued

yD(y, t) =

{y if t = 0exp{−rt} (y + b e−mt)− b if t > 0

(4)

The parameter m controls the distribution of costs over time. This specification reducesto

yD(y, t) =

{y if t = 0exp{−rt}y − b if t > 0

(5)

for m = ∞, where the cost b is not discounted. For m = 0, where the cost is discounted,it reduces instead to:

yD(y, t) =

{y if t = 0y exp{−rt} − (1− exp{−rt}) b if t > 0

(6)

Note that, in contrast with the all other specifications of discounting we have consid-ered, when present bias is represented by a fixed cost the discount factor D(y, t) declineswith the amount y. This is the property that we can exploit in the data to identify thefixed cost from the variable cost (quasi-hyperbolic discounting) specification of presentbias.

3 Revealed Preferences Experiments: Is Discount-

ing Hyperbolic ?

As noted in the Introduction, the experimental literature which documents preferencereversals cannot provide a statistical test of the hyperbolic against the exponential spec-ifications of discounting.

While under the assumption that ”reversals” are due to hyperbolic preferences, thedelay at which a reversal occurs contains some information about r in equation (2) or

2It is immediate to see that quasi-hyperbolic discounting implies time inconsistency, as well as re-versal of preferences. This formulation has been introduced by Phelps-Pollak, 1968, and has beenadopted by behavioral economists over the hyperbolic specification for its tractability; see Laibson,1997, O’Donaghue-Rabin, 1999.

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(3), individual discount rates cannot be estimated with the data generated by prefer-ence reversal experiments. It is then impossible to evaluate results statistically, e.g., toformally distinguish consistent empirical regularities from the effects of noise.

3.1 The Experimental Design

A total of 27 inexperienced subjects were recruited from the undergraduate populationof New York University to engage in our experiment. Each subject did the experimenton two different days. During both sessions they were asked a set of questions whoseaim was to elicit their discount rates for money using a version of the Becker-DeGroot-Marschak mechanism to be described later. The two sessions differed by the types ofquestions asked which we will also describe in more detail later in this section.3

The paper-and-pencil experiment took place at the Center for Experimental SocialScience (C.E.S.S) at New York University. When subjects arrived in the lab they wereseated at tables and separated from each other for the duration of the experiment. Theywere then given a set of instructions which were read out loud to them after they hadhad a chance to read them individually. Each experimental session lasted about 1/2an hour and subjects earned on average approximately $28 in each session. (These areundiscounted amounts; some of these earnings were paid to the subjects at later times).

In Session 1 subjects were asked to reply to a set of 30 questions of the followingform:

What amount of money, $y, if paid to you today would make you indifferentto $x paid to you in t days. [Q− present]

In the actual experiment x and t were specified so a typical question would be:

What amount of money, $y, if paid to you today would make you indifferentto $10 paid to you in 1 month.

The amounts $x varied from $10, to $20, $30, $50, to $100 while t varied from 3 daysto 1 week, 2 weeks, 1 month, 3 months, to 6 months. So for each amount we asked sixquestions involving the six different time frames; with five different amounts this totalled30 questions.

To give the subjects an incentive to answer these questions truthfully we used aversion of the Becker-DeGroot-Marschak mechanism to determine what amount wouldbe paid to the subjects and when. This mechanism was employed on one of the thirtyquestions drawn at random. For example, say that at the end of the experiment we drewthe question that asks the subject what amount, $y, he would require today to make himindifferent between that amount today and $50 to be paid to him in one month. Assume

3A set of experimental instructions are provided at the end of the paper. We modified the notationto make it consistent with the one used in the paper.

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he says y = $40. In that case we would draw a random number uniformly from theinterval [0, $50]. If the number drawn was less than the $40 indifference amount statedby the subjects then he or she would have to wait for one month at which time the $50would be paid. If the number drawn was greater than the $40 indifference amount stated,that amount would be paid immediately. To insure waiting the subject need only statean indifference amount of $50 while to insure receiving some money today the subjectsneed only state an indifference amount of $0. It is a dominant strategy to report yourtrue indifference amount in this procedure (assuming risk neutrality) and this fact wasexplained to the subjects. We had no doubt that the subjects understood the incentiveproperties of the mechanism.

In this experimental session, therefore, subjects received either money today or moneyin the future. If money today were to be paid subjects were handed a check. If futuremoney were to be paid subjects were asked to supply their mailing address and weretold that on the day promised a check would arrive at their campus mailboxes with thepromised amount. This was done to minimize any possible transaction costs involved inwaiting, i.e., when paid in the future no subject had to travel to the lab to pick up hismoney etc., it would just arrive at their door.

Session 2 was identical to Session 1 except the question asked was reversed. here weasked:

What amount of money, $y, would make you indifferent between $x todayand $y t days from now. (yLarge = w) [Q− future]

Note that in this question instead of asking what amount of money they need todayto make them indifferent to a given amount of money in the future, we ask them whatamount of money if given at a specific time in the future would make them indifferentto a fixed amount today. In this question they were not allowed to state an amountlarger than some pre-determined quantity, yLarge. Here yLarge varied from $10, to $20,to $30, to $50 and finally to $100 while the time horizons varied form 3 days to 1 week,1 month, to 3 months and finally to 6 months, just as in Session 1. The $x amountsgiven them were derived from the answers to questions received in Session 1 and were theminimum of the amounts stated there. For example, one question was, ”What amountof money, $y, would make you indifferent between $14 today and $y 3 weeks from now(yLarge = 20)”. Here the subject was forced to give an answer of $20 or less but as statedabove, from our Session 1 answers we never observed an amount less than $14 beingasked for $20 in 3 weeks and since this question is simply the inverse we expected theseamounts to be sufficient. If we repeatedly saw responses equal to yLarge we would knowthat we had chosen x inappropriately. That was not the case.

In summation, we employed a within-subject design using 27 subjects and two treat-ments where the treatments varied according to the type of question asked.

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3.2 Estimates of discounting

Kirby, 1997, uses econometric methods to fit discount curves. In his main experiment,the hyperbolic discount specification fits better that the exponential, in the sense that theR2 is higher, for 19 out of 23 participants. What is missing from his analysis is a formalstatistical test of the hyperbolic specification against the exponential alternative. Thecomparison of R2 across specifications, while illustrative, cannot be considered sufficientstatistical evidence.

To this end we introduce a two-parameter class of discount factor specifications nest-ing hyperbolic and exponential discounting:

D(y, t; θ, r) = (1− (1− θ)rt)1

1−θ (7)

It is immediate to see that:

D(y, t; θ = 1, r) = exp{−rt}

D(y, t; θ = 2, r) =1

1 + rt;

and hence estimating the parameter θ from experimental data will possibly allow us todistinguish hyperbolic discounting, that is θ = 2, from exponential discounting, θ = 1.

More generally, it is straightforward to extend the two-parameter specification (7) toa four-parameter specification which also nests quasi-hyperbolic discounting and fixedcosts:

yD(y, t; θ, r, α, b) =

{y if t = 0

α (1− (1− θ)rt)1

1−θ y − b if t > 0, α < 1 (8)

Our data consists, for each subject h = 1, 2, ...., of answers to a battery of questionssuch as [Q − present] and [Q − future], for different values of x and t. We presentfirst our analysis of data regarding [Q − future]. Qualitative results are the same for[Q− present]; we discuss framing in Section 3.3.

Let yh(x, t) denote the answer given by subject h to question [Q−future] for amountx and delay t. We start by estimating (7), to statistically document declining subjectivediscount rates, and we explore Kirby’s, 1997, analysis of the failure of exponential dis-counting. We then estimate (8) to better characterize the functional form of discounting,and the present bias.

To estimate (7) we assume that yh(x, t), the data is generated by

yh(x, t) = x(1− (1− θh)rht

) 1

1−θhεh(x, t)

where the error εh(x, t) is i.i.d. with respect to subjects h and questions (x, t). Moreover,we assume εh(x, t) is lognormally distributed. Note that we allow the parameters of the

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discount curve, (θh, rh), to be indexed by the subject. We estimate individual discountcurves, independently across subjects, (θh, rh)h=1,...,25, by non-linear least squares.

Results are collected in Table 1 and are somewhat consistent with Kirby’s, 1997,conclusions.4 In fact, for 23 of the 27 agents the exponential specification, θ = 1, isrejected by the data. Nonetheless the estimates do not appear particularly appealing,essentially because the point estimates for r are extremely high, in 17 cases of the orderof thousands of percentage points. Even though when discounting is not exponential rdoes not represent the discount rate, it is still the case that one dollar with no delay isworth more than 5 in a year, for more than half of the agents in the sample at the pointestimates. The point estimates of r are in only 1 case less than 100%.5

We turn then to a second specification. In this formulation discounting is allowed tobe hyperbolic, as in the previous specification. But we include a fixed cost component tothe preference for the present, that is we estimate (8) under the restriction that α = 1.Results are reported in Table 2. The fixed cost b is estimated to be significantly differentthan 0 for all the subjects (except subject 19 for which the estimate does not converge).It is, on average, about $4 (with a minimum value of $.31 and a maximum value of$5.38). The estimates of r are also more reasonable when we include fixed costs. Forinstance, the point estimates of r are less than 100% for 15 subjects and less than 30%for 9 subjects. The estimates of θ, the curvature of the discounting function at delays tdifferent than zero, are however very imprecise when a fixed cost is added. For 5 subjectsthe hypothesis of exponential discounting is not rejected at the 95% confidence interval,and for 1 of these neither is the hypothesis of hyperbolic discounting . For 9 subjectsthe confidence interval of θ lies in the region smaller than 2, while for 16 subjects it isin the region greater than 1. In other words, the data seem to be clearly consistent witha present bias which we postulated in the form of a fixed cost, but do not have muchpower to distinguish exponential from hyperbolic discounting.

The final specification we estimate is (8), where both fixed and variable costs areallowed for, that is, where we allow for a quasi-hyperbolic component of present bias.Note that we can identify fixed versus variable costs components since we have data fordifferent amounts x.6 Results are striking, and are reported in Table 3. For 16 subjects

4Players 1, 9, 13, 18, 21, 23 and 26 gave virtually identical responses.5This is by no means only a property of our data. Similar discount rates have been generally imputed

from experimental data; see Frederick-Loewenstein-O’Donoghue, 2002, Table 1.1.6We have also estimated the different functional forms for present bias discussed in Section 2. In

particular we have estimated

D(y, t; θ, r, b, α, m) ={

y if t = 0α ((1− θ) rt)

11−θ (y + b e−mt)− b if t > 0

(9)

Results (not reported) are not significantly different from those obtained with specification (8) illustratedin Table 3. The estimates of m are large enough so that (8) appears to represent a good approximation,with one less parameter.

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we estimate a value of α greater than one. For the remaining 11 subjects, α is estimatedsignificantly smaller than 1 (that is, we reject α ≥ 1) only in 5 cases. Finally, the pointestimates of α are never smaller than .92, and the lower bound of the 95% interval neverlower than .84. These estimates stand in sharp contrast to the much lower (around.6) imputed value of α obtained in a consumption-saving model by Laibson-Repetto-Tobacman (2004). The estimates we obtain for the fixed cost b in this specification arenot much varied from those obtained in the previous specification, and still around $4 onaverage across agents. The estimates for the curvature of discounting, θ, are still quiteimprecise but seem now to favor, at the margin, the exponential specification: for 15subjects the hypothesis of exponential discounting is not rejected at the 95% confidenceinterval, and for 11 of these neither is the hypothesis of hyperbolic discounting; but forall 15 subjects exponential discounting is not rejected at the 97% level, while for 12 ofthese the hypothesis of hyperbolic discounting is rejected at the 97%.

We conclude that the data do not seem to significantly support the quasi-hyperbolicspecification, while they do support a fixed cost specification. In fact, if the fixed costis correctly estimated at about $4, it would appear to be negligible in most economicapplications of interest. Of course strong caution in interpreting our results is necessary,because our estimates are derived from a sample which do not include amounts greaterthan $100.

3.3 Framing

Do results depend on how the question is posed? To address the issue of framing, weestimate the same specifications of discounting with the data obtained from question[Q − present], and compare the results with those just discussed, that is, with theestimates with the data from [Q − future].7 The results are reported in Table 4, 5,6 for our three specification of the discounting curves, respectively.8 To identify andmeasure framing effects we proceed as follows: for each of the three specifications ofdiscounting, i) we produce estimates of the parameters with data from questions of theform [Q−future] and, in addition, of the form [Q−present], ii) we construct confidenceintervals on parameters’ estimates, and iii) we check, parameter by parameter, if theconfidence intervals obtained with the data generated by different questions overlap.

It is not easy to find clear statistical evidence for framing in our data. The firstspecification (equation 7), without fixed costs and without a quasi-hyperbolic component,has 2 parameters. For 14 out of the 25 subjects for which we could obtain estimates,

7For a survey of framing effects in discounting, see Frederick-Loewenstein-O’Donoghue, 2002; seealso Frederick, 2003.

8It should be noted that when answering this question: subjects 1, 9, 16, 18 misunderstood thequestion (for instance, they claimed to be willing to accept an amount y in the present to avoid waiting30 days for $15, but to require an amount y′ > y in the present to avoid waiting 60 days for the same$15.; subjects 3, 17, 19 and 23 essentially did not discount

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the confidence intervals for both parameters overlap; and for 9 of them the confidenceintervals of one of the parameter overlap. For only one subject the estimates obtainedwith the different question are statistically distinct. The second specification (equation 8with α = 1), with fixed costs but no quasi-hyperbolic component, has 3 parameters. Weobtain 3 overlapping confidence intervals for 7 out of 24 subjects, 2 overlapping intervalsfor 9 subjects, 1 overlapping interval for 7 subjects, and finally distinct estimates for only1 subject. The results for the third specification (equation 8), which has 4 parameters,are as follows: 4 overlapping confidence intervals for 6 out of 17 subjects, 3 overlappingintervals for 5 subjects, 2 overlapping intervals for 3 subjects, 1 overlapping interval for2 subjects, and finally distinct estimates for only 1 subject.

4 Conclusions

This paper provides the first experimental study of discounting in which data are an-alyzed with formal statistical methods and the various hypothesis regarding the spec-ification of discounting preferences adopted in the behavioral economics literature aretested. We find clear experimental evidence against exponential discounting. The datafavors a specification of discounting which contains a present bias in the form of a fixedcost, and no quasi-hyperbolic component. The curvature of discounting (exponential vs.hyperbolic), in the fixed cost specification, is not precisely estimated with our data, andis consistent for most subjects with exponential discounting. This finding implies, whenextrapolated outside the sample, that present bias vanishes with large rewards. Experi-ments with relatively large rewards are needed to confirm the fixed cost representationof present bias that we identify in the experimental data.

We hope to soon extend our experiments in two directions. First, to explore howpresent bias behaves as rewards become large, we want to conduct experiments in coun-tries where wealth and income are low. Second, to better understand discount curvesand interest rates under alternative framing, we want to conduct experiments wheresubjects are paid a lump sum, but are then expected to choose between repaying a sumimmediately, or a different sum at a different point in the future.

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References

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Ainsle G. and R.J. Herrnstein (1981): ‘Preference Reversal and Delayed Reinforcement,’Animal Learning and Behavior, 9, 476-82.

Ainsle, G. and N. Haslam (1992): ‘Hyperbolic Discounting,’ in G. Lowenstein and J.Elster (eds.), Choice Over Time, Russell Sage Foundation.

de Villiers, P.A. and R.J. Herrnstein (1976): ‘Towards a Law of Response Strength,’Psychological Bulletin, 83, 1131-53.

Elster, J. (1979): Ulysses and the Sirens: Studies in Rationality and Irrationality,Cambridge University Press, Cambridge.

Frederick, S., 2003, ‘Measuring Intergenerational Time Preference: Are Future LivesValued Less?,’ mimeo, MIT.

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Gul, F. and W. Pesendorfer (2001): ‘Temptation and Self-Control,’ Econometrica, 69,1403-35.

Herrnstein, R.J. (1961): ‘Relative and Absolute Strengths of Response as a Functionof Frequency of Reinforcement,’ Journal of the Experimental Analysis of Behavior,4, 267-72.

Herrnstein R.J. (1997): The Matching Law: Papers in Psychology and Economics, H.Rachlin and D. Laibson (Eds.), Sage Publishers.

Kirby, K.N. (1997): ‘Bidding on the Future: Evidence Against Normative Discountingof Delayed Rewards,’ Journal of Experimental Psychology: General, 126, 54-70.

Kirby, K.N. and R.J. Herrnstein (1995): ‘Preference Reversals Due to Myopic Discount-ing of Delayed Reward,’ Psychological Science, 6, 83-9.

Laibson, D. (1997): ‘Golden Eggs and Hyperbolic Discounting,’ Quarterly Journal ofEconomics, CXII, 443-77.

Laibson, D., A. Repetto, and J. Tobacman (2004): ‘Estimating Discount Functionsfrom Lifecycle Consumption Choices,’ Harvard University.

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O’Donaghue, T. and M. Rabin (1999): ‘Doing It Now or Later,’ American EconomicReview, 89, 103-24.

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Figure 1: Reversal of Preferences

[Figure 1 here]

Consider the following rewards: $x at time t and $x′ > x at time t′ = t+d. An agentwith hyperbolic discounting, at time 0, would evaluate the first as

x

1 + rt

and the second asx′

1 + rt′

If t′ − t = d > x′ − x, then at t = 0 any agent with hyperbolic discounting wouldprefer the smaller immediate amount x to the larger delayed amount x′. Note thoughthat, as the joint delay t is increased, the agent’s preference for the earlier reward willdecline, until the agent will in fact ”reverse his preferences” and prefer x′. This is thephenomenon of preference reversal, represented in the figure by crossing discount curves.It is straightforward to show that preference reversals never happen with exponentialpreferences: in fact, in this case if at t = 0 the earlier reward if preferred, that is,if x > x′e−rd, then the earlier reward is preferred for any joint delay t > 0, as thenxe−rt > x′e−rt′ = x′e−r(t+d) for any t.

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Table 1: Question [Q− future]; Specification with No Present Bias

Person θ se(θ) r se(r)1 3.88 0.61 33.64 14.502 -2.00 4.40 0.50 0.373 2.85 1.52 4.16 2.694 2.26 0.48 4.99 1.345 1.96 0.83 2.88 1.056 2.65 0.41 7.24 1.787 4.67 3.55 2.66 2.298 4.25 0.86 14.22 6.199 3.88 0.61 33.64 14.5010 3.15 1.16 18.36 15.4111 16.62 5.27 93.63 125.1212 4.14 0.84 16.42 7.4513 3.88 0.61 33.64 14.5014 3.55 0.92 13.06 6.9315 2.46 0.56 6.37 2.1216 2.33 0.91 4.08 1.8017 1.88 2.72 1.20 0.7318 3.88 0.61 33.64 14.5019 4.61 2.42 10.45 10.9320 4.86 1.20 13.95 7.4721 3.92 0.61 33.81 14.4422 2.62 0.49 7.03 2.0323 3.88 0.61 33.64 14.5024 6.53 1.77 66.44 60.5525 3.62 0.60 26.63 11.1926 3.88 0.61 33.64 14.5027 3.91 0.91 20.25 11.14

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Table 2: Question [Q− future]; Specification with Fixed Cost

person θ se(θ) r se(r) b se(b)1 6.27 0.75 14.27 4.30 5.38 0.962 -22.74 16.10 0.08 0.04 0.44 0.153 -4.94 3.72 0.28 0.11 2.38 0.444 -1.02 1.14 0.73 0.22 4.46 0.835 -1.39 1.30 0.56 0.14 2.26 0.476 1.81 0.52 2.07 0.42 3.51 0.697 -6.10 24.75 0.12 0.09 1.33 0.238 2.69 1.24 1.53 0.48 4.81 0.719 6.27 0.75 14.27 4.30 5.38 0.9610 3.45 1.31 2.37 0.93 4.68 1.0011 -0.90 1.41 0.66 0.20 2.17 0.6512 5.46 1.11 3.00 0.85 4.51 0.5913 6.27 0.75 14.27 4.30 5.38 0.9614 4.77 0.69 3.74 0.79 3.40 0.5115 -1.24 2.22 0.62 0.30 5.31 1.1416 -0.13 0.83 0.78 0.15 2.45 0.4217 -4.95 5.83 0.23 0.10 0.52 0.3118 6.27 0.75 14.27 4.30 5.38 0.961920 30.78 18.44 309.07 979.21 3.30 1.4321 6.28 0.75 14.44 4.37 5.32 0.9622 1.92 0.62 2.01 0.46 3.78 0.7423 6.27 0.75 14.27 4.30 5.38 0.9624 41.34 23.05 9370.38 45832.09 4.16 1.0725 4.35 0.55 6.49 1.56 5.37 0.8326 6.27 0.75 14.27 4.30 5.38 0.9627 8.47 1.47 9.80 3.92 4.42 0.89

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Table 3: Question [Q − future]; Specification with Fixed Cost and Quasi-HyperbolicComponent

person θ se(θ) r se(r) b se(b) α se(α)1 1.24 0.80 1.36 0.35 4.67 0.77 1.21 0.032 -8.13 9.66 0.13 0.05 0.73 0.17 0.98 0.013 -0.92 2.19 0.48 0.15 3.51 0.47 0.94 0.014 -1.28 1.38 0.68 0.24 4.22 1.02 1.01 0.035 -0.62 1.31 0.66 0.18 2.62 0.58 0.98 0.026 1.70 0.64 1.97 0.52 3.42 0.80 1.01 0.037 21.74 24.80 0.41 0.48 1.68 0.28 0.98 0.018 -0.51 1.28 0.68 0.18 3.41 0.60 1.09 0.029 1.24 0.80 1.36 0.35 4.67 0.77 1.21 0.0310 4.39 1.70 3.43 2.28 4.94 1.16 0.97 0.0511 -3.21 2.39 0.42 0.17 1.27 0.69 1.06 0.0312 5.60 1.52 3.15 1.55 4.53 0.66 1.00 0.0313 1.24 0.80 1.36 0.35 4.67 0.77 1.21 0.0314 6.21 0.78 8.90 4.12 3.87 0.52 0.92 0.0415 -0.06 2.06 0.83 0.43 6.17 1.52 0.96 0.0416 0.21 0.92 0.85 0.19 2.66 0.52 0.99 0.0217 -2.66 5.84 0.27 0.13 0.72 0.39 0.99 0.0118 1.24 0.80 1.36 0.35 4.67 0.77 1.21 0.0319 -11.59 19.25 0.15 0.21 6.17 1.43 0.93 0.0320 -48.75 0.04 0.00 3.04 1.28 1.20 0.0421 1.21 0.80 1.35 0.35 4.62 0.77 1.21 0.0322 4.57 0.75 1.74 0.50 3.54 0.85 1.02 0.0323 1.24 0.80 1.36 0.35 4.67 0.77 1.21 0.0324 41.33 23.02 17944.30 4.16 1.07 0.97 0.1325 3.95 0.79 5.11 2.20 5.29 0.89 1.02 0.0526 1.24 0.80 1.36 0.35 4.67 0.77 1.21 0.0327 -0.02 1.44 0.74 0.23 3.79 0.70 1.16 0.03

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Table 4: Question [Q− present]; Specification with No Present Bias

Person θ se(θ) r se(r)12 3.20 10.24 0.10 0.053 -80.90 156.93 0.00 0.054 3.90 0.98 7.10 2.915 6.90 2.49 1.00 0.276 7.40 0.53 31.10 6.247 -4.40 8.24 0.30 0.198 16.80 3.34 16.50 9.119 48.70 24.78 9.10 14.0510 14.60 4.59 7.90 5.5411 2.70 0.55 6.30 1.9012 11.50 1.85 20.30 8.8613 3.00 0.83 12.10 6.6114 9.10 3.29 3.10 1.6915 1.30 3.58 0.70 0.3516 24.00 5.77 18.80 14.0017 334.40 4.72E+08 0.00 0.0018 177.70 433.96 2000.00 44794.411920 7.70 1.13 4.40 1.0521 -0.60 5.32 0.30 0.1622 2.70 4.58 0.80 0.5123 334.40 4.72E+08 0.00 0.0024 2.00 0.65 6.70 3.1025 73.20 45.04 1.20 1.4826 -0.80 1.66 0.60 0.1827 10.40 2.00 48.90 31.27

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Table 5: Question [Q− present]; Specification with Fixed Cost

person θ se(θ) r se(r) b se(b)12 4.06 12.35 0.14 0.06 -0.02 0.163 crazy4 5.58 1.29 2.55 0.76 2.80 0.715 4.33 1.57 0.62 0.11 0.22 0.226 8.83 0.69 24.47 5.35 1.34 0.617 -17.04 24.08 0.09 0.06 0.69 0.378 18.31 3.07 11.76 5.76 0.55 0.699 161.48 626.71 0.25 1.81 1.99 0.8910 2.27 4.45 0.40 0.16 1.91 0.4411 1.39 0.61 1.64 0.34 3.28 0.7412 16.75 2.43 115.38 72.81 -0.86 0.8113 1.27 0.71 1.75 0.45 3.31 1.0114 17.51 6.44 0.95 0.48 1.66 0.4115 -42.61 60.27 0.05 0.06 1.11 0.4916 21.57 3.57 2.41 0.83 1.46 0.3317 8.17 0.00 0.00 0.00 0.0018 -192.27 164.16 -6.12 21.76 2.14 0.501920 7.48 1.19 3.36 0.87 0.26 0.5721 0.12 17.10 0.14 0.09 0.41 0.2922 4.28 2.70 0.84 0.31 -0.74 0.5823 29.37 0.00 0.00 0.00 0.0024 2.28 0.57 2.49 0.52 1.95 0.7825 57.46 17.25 2.56 2.11 -0.31 0.3726 -41.53 0.05 0.00 0.99 0.4627 8.92 1.15 30.11 11.27 -0.78 1.03

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Table 6: Question [Q − present]; Specification with Fixed Cost and Quasi-HyperbolicComponent

person θ se(θ) r se(r) b se(b)1 α se(α)2 10.80 12.29 0.20 0.09 0.18 0.203 0.99 0.014 8.01 1.27 7.72 4.47 3.64 0.695 4.52 1.87 0.63 0.14 0.24 0.27 0.91 0.046 1.00 0.017 -3.45 17.53 0.15 0.11 1.15 0.488 0.98 0.019 163.32 865.87 0.26 4.36 2.00 0.9610 7.37 4.04 0.78 0.38 2.69 0.51 1.00 0.0511 1.87 0.67 1.99 0.49 4.24 0.88 0.97 0.0112 1.94 2.90 0.68 0.30 -0.89 0.64 0.96 0.0213 2.47 0.57 3.05 0.75 5.91 0.91 1.24 0.0214 0.89 0.0215 -12.57 13.95 0.12 0.08 2.09 0.6116 25.68 4.69 8.04 13.09 1.56 0.34 0.97 0.0117 9.62 0.00 0.00 0.00 0.00 0.97 0.0418 1.00 0.001920 9.57 1.17 10.40 6.33 0.92 0.5221 11.29 12.75 0.28 0.18 1.02 0.34 0.92 0.0422 5.38 2.84 1.09 0.52 -0.12 0.68 0.98 0.0123 9.62 0.00 0.00 0.00 0.00 0.97 0.0224 3.57 0.48 4.84 1.11 3.61 0.68 1.00 0.0025 0.90 0.0226 -16.31 10.87 0.12 0.07 2.05 0.6927 0.97 0.01

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Instructions for the Experiment

Personal Information

Name: _____________________________ e-mail address: _____________________________ Mailing Address (For next six months): _____________________________ Social Security Number: ______________________________

Telephone Number: _____________________________

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Instructions This is an experiment in decision-making. Various research foundations have provided funds for this experiment and depending on the decisions you make you will be able to earn money that will be paid to you when you leave. Your Decision Task: We will be interested in learning your response to three types of questions. In one we ask what amount of money, if paid to you at some future time, would make you indifferent between having a fixed amount of money today and that amount of money in the future. For example, say that we offered you $2 today and asked you to tell us what is the smallest amount of money that, if paid to you in 7 days, would make you indifferent between $2 today. (By indifferent we mean that you do not care in any way which one of the alternatives you ultimately get.) If your answer was $y then that would mean that you would not care whether we gave you $2 today or you had to wait 7 days to get $y. We will call the $y your future-indifference amount since it is the amount that would make you indifferent between $2 today and that amount next week. Let’s call this type of question the future indifference-amount question since it asks you to state an amount of money which if paid to you in the future would make you indifferent to a given amount of money today. The next type of question is just the opposite. It starts by positing that you have a certain amount of money at some time in the future, say $10 in two weeks, and then asks you how much money you would ask for today to make you indifferent to it. Let’s call this the present indifference-amount question since it asks you to state an amount of money that, if paid to you today, would make you indifferent to a given amount of money paid to you at some fixed time in the future. Finally, we will ask a third type of question called a time-indifference question. In this question we give you two amounts of money, say $10 today and $15 sometime in the future. Here we ask you to tell us the number of days you would be willing to wait to make you indifferent between $10 today and $15 in that number of days. Note that here we specify the two amounts to be paid either today or in the future, and then ask you to state an amount of time between them that makes you indifferent. The way we will elicit any of these three indifference amounts depends on the type of problem at hand. Let’s start with the future indifference-amount question. The Future Indifference-Amount Question

Here we will elicit your indifference amount as follows: First we will offer you an amount for consumption today, say $x. We will then ask you to tell us what amount, say $y, would make you indifferent between $x today and $y paid to you in t days. $y can be any amount between $0 and some larger amount $yLARGE that will change depending on the problem. After you tell us $y we will use a random devise, (to be described later) and randomly choose an amount of money between $0 and $yLARGE . If the random amount we choose is below the $y you stated as your indifference amount, we will pay

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you the $x today. If it is above the $y, say it is $z with $z > $y, we will deliver $z to you t days from now. You will not have to do anything to get your money in the future; it will be delivered to you by mail on the day promised. We will ask you 30 such questions each with different amounts and different future times. Example To illustrate how we will run this part of the experiment, say that we offer you $4 today and ask you to tell us what amount of money, if paid to you in 7 days, would make you indifferent between the $4 today and that amount in 7 days later. If you said $4.51 (you can make your indifference amount denominated in pennies) then we would randomly choose an amount between $0 and $yLARGE, say $25 (in 1-cent intervals), with each amount being chosen with equal probability so that there is an equal chance of choosing $0 as there is of choosing $7.11, as there is of choosing $20.20, and there is of choosing $24.44 etc.). If the random amount is $8, which is greater than the $4.51 you stated, we will arrange to pay you the $8 in exactly 7 days by mailing it to you so that it arrives exactly one week from today without you doing anything to get it. If the random amount is below $4.51, then we will pay you the $4 today and nothing next week. (If your indifference amount is more than $yLARGE, $25 in this example, then simply state it as $25. This will insure you that you will receive the $4 today which is what you would want if your indifference amount were truly above $25.) It can be demonstrated (and will be below) that given the procedures we are using it is best for you, in terms of maximizing your payoff in this experiment, to report your indifference amount truthfully since doing anything else would reduce your welfare. So it pays to report your indifference amount truthfully. Why it Pays to Report Truthful Indifference Amounts To prove to you that you should reveal your truthful valuations, let us work through one example. Say that you are asked to tell us what amount, $y given to you in one year from today would make you indifferent to $20 today. Let us just assume, for the sake of argument, that you would require $22 to be given to you in one year in order to make you indifferent to having $20 today. Hence, your truthful value is $22. The question is, Should you tell us $22 when we ask you? Let us say that $yLARGE = $50 for this question.

To see why the answer is yes, let us say that you are thinking of not telling us the truth and reporting that it would only take $21 to make you indifferent, i.e., you state an amount lower than your truthful amount. Then, for all random amounts below $21 both reports, the truthful one of $22 and the one of $21, will yield the same results; you will be paid the $20 today. However, for random amounts between $21 and $22, say $21.5, you will be given the value of the random number one year from now. However, if your truthful number is $22 then this is bad for you since you would rather have $20 today rather than $21.50 in one year since $22 is your true indifference amount. Hence it never pays to say less than your truthful value.

What about saying more? Maybe you should say $24.5 instead of $22? In this case, for all random values of $24.5 or more both reports, $22 or $24.5 yield the same result, you will be paid that amount in one year and this is good since it is an amount above your stated value. But for values between $22 and $24.5 (i.e. for values above your

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true one but below your stated one), you will be paid $20 today since those random numbers are below your false report of $24.5. However, this is hurting you since, given your truthful value is $22 for all those random values between $22 and $24.5 you would rather wait and get the money next year yet, due to your false report, you are given the money today. Hence, reporting a too high value will not help you either. (Note that if you reported an amount of $50 you would always be paid $20 today no matter what random value is determined.) IN SHORT, TELL THE TRUTH.

The Present Indifference-Amount Question

For the present indifference-amount type of question, we follow an identical procedure but modified slightly to meet the new circumstance. Here we will ask you to state that amount of money, $y that, if paid to you today, would make you indifferent between that amount and $x paid to you t days from now. So here we offer you an amount in the future and ask you for an amount of money today that makes you indifferent. It is the opposite question.

Once you have responded to this question, we will employ the same random device and again choose an amount at random. However, this random device will choose a number between $0 and $x, the amount we offered to pay you in t days. We will then compare the random amount to your stated indifference amount for money today. If the random amount is less then your stated indifference amount, then we will wait and pay you the $x t days from now. If the random amount is greater than your stated indifference amount, then we will pay you the random amount realized today. Again, it can be demonstrated in an exact manner as done above, that your best action is to state your truthful indifference amount when engaging in a present indifference amount type question. Why it Pays to Report Truthful Indifference Amounts To prove to you that you should reveal your truthful valuations, let us work through one example. Say that we offer you an amount, say $25 one year from now which we are willing to pay you. However, here we ask you what amount, say $y, if given to you today, would make you indifferent to $25 in one year. (This is just the reverse of the previous question). Let us just assume, for the sake of argument, that you would require $22 to be given to you today in order to make you indifferent to having $25 in one year. Hence, your truthful value is $22. The question is should you tell us $22 when we ask you?

To see why the answer is yes, let us say that you are thinking of not telling us the truth and reporting that it would only take $21 to make you indifferent, i.e., you state an amount lower than your truthful amount. Then, for all random amounts below 21 both reports, the truthful one of $22 and the one of $21, will yield the same results; you will be paid the $25 in one year since under both reports the random variable was realized at a value less than your reported amount. However, for random amounts between $21 and $22, say $21.5, you will you will be given $21.5 today, since the random variable realized itself at a value greater than your reported value of $21. But this value is less than your truthful value of $22 and therefore you will be given $21.5 when if fact you would prefer to wait one year and get $25. The same is true for all values between your lower report and your truthful value so saying less is not beneficial.

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What about saying more than your truthful amount? Maybe you should say $24.5 instead of $22? In this case, for all random values of $24.5 or more both reports, $22 or $24.5 yield the same result, you will be paid whatever the random amount is today. But if the random amount is between $22 and $24.5 since it is below your reported amount, you will have to wait one year to receive the $25. However, this is not good for you since you would be willing to receive any amount above $22 today rather than waiting and you just rejected such an amount by reporting $24.5. Hence it does not pay to report an amount above your truthful amount. IN SHORT, TELL THE TRUTH.

The Time-Indifference Question

Here we will elicit your time indifference amount as follows: First we will offer you two amounts $x and $y where $y is greater than $x. The amount $x we are willing to give to you today while the amount $y we will be willing to give to you at some future time to be determined by you. After stating these amounts we will then ask you to tell us that future time, say t days, that would make you indifferent between having $x today and $y at that point in the future. Your answer can be any number of days between 0 and tLARGE that is some large number of days which may vary with the problem at hand. We will call this your indifference time.

Once you have responded to this question, we will employ a random device similar to the one used above but here we will randomly choose a time, t between 0 and tLARGE , say 90 days from now with equal probability. Again, by equal probability we mean that 1 day ahead has the same probability of being chosen as does 35 days as does 67 days as does 90 days. They are all equally likely. We will then compare the random time selected to your stated time. If the random time is less than your stated time, we will give you the $y at that time instead of the one you stated. If the random time is greater than the one you stated, we will give you $x today. Again, it can be demonstrated in an exact manner as done above, that your best action is to state your truthful indifference amount when engaging in a present indifference amount type question. Why it Pays to Report Truthful Indifference Amounts To prove to you that you should reveal your truthful indifference time, let us work through one example. Say that we offer you two amounts say $5 and $25 and ask you to choose a time, t that is such that you would be indifferent to waiting t days and receiving $25 as opposed to receiving $5 today. Let tLARGE = 90 days here. Let us just assume, for the sake of argument, that you would be indifferent between waiting 60 days and receiving $25 as opposed to receiving $5 today; t = 60 is your indifference time The question is should you tell us t = 60 when we ask you?

To see why the answer is yes, let us say that you are thinking of not telling us the truth and reporting that it would only take 50 to make you indifferent, i.e., you state an time lower than your truthful indifference time. Then, for all random amounts below 50 both reports, the truthful one of 60 and the stated one of 50, will yield the same results; you will be paid the $25 at that time. (For example, if the random device chose 43 days, then you would get the $25 in 43 days.) However, for random times between 50 and 60 days, say 54 days, you will receive the $5 today. But this is not good for you since if you had stated your truthful 60 days then you would receive $25 in 54 days which is better

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than receiving $5 today if your true indifference time is 60 days. The same is true for all values between your lower report and your truthful value so saying less is not beneficial.

What about stating an indifference time greater than your true one. For example, maybe you should say 65 days? In this case, for all random values of 65 or more both reports, 60 and 65 days yield the same result; you will be paid $5 today. But if the random time is between 60 and 65, say 63 days, since it is below your reported time, you will be paid $25 in 63 days. However, this is not good for you since you would rather receive $5 today than wait as long as 63 days to get $25. Remember your indifference time is 60 days. The same is true for all values between 60 and 65 days. Hence it does not pay to report a time above your truthful indifference time. IN SHORT, TELL THE TRUTH.

Procedures: The experiment will work as follows:

Once a week for the next three weeks we will pass out a questionnaire listing 30 different questions. In the first week the questions will be of the present indifference-amount type, in the second week they will be of the future indifference-amount type, and in the third week they will be of the time-indifference type. When you are done filling in your answers, we will bring one four-sided die (with numbers 0, 1,2,3) and one ten-sided die (with numbers 0, 1, 2, ….,9) to your table and toss each one in sequence. The two numbers showing on the die will indicate which of the 30 questions will be relevant for your payoff in the experiment. For example, if the first die lands on 1 and the second on 7, we will choose question 17 and pay you according to the results of that question. So, while you answer 30 questions, only 1 of them will be relevant for your payoff. You will not know which question it is, however, until after you have answered all the questions.

When your payoff-relevant question has been determined we will determine your payoff using the procedures explained above. That is we will bring a number of die to your table and spin them. These will be how we determine the random numbers discussed above. For example, in the future-indifference amount type question, if $yLARGE = $50, we will bring one 6-sided die (with numbers 01,2,3,4,5) and three ten-sided die with numbers (0, 1, 2, …., 9) and throw them in sequence. These four die, when thrown independently, generate all random numbers between $0 and $50 in 1-cent intervals with equal probability. The random number determined this way will be compared to your stated indifference amount (or stated indifference time). If the problem is a present indifference amount problem or a time indifference amount problem, we will use the appropriate die to generate the correct random

Your final earnings in the experiment will be equal to the money you earn on the question drawn plus a $5 show-up fee. If you are due money today as a result of your answers to the questions, you will be paid it today. If you are due money in the future, as stated before, we will have that money mailed to you so that it will arrive on the exact day specified. (We may hand deliver it to your dorm in you are living in a dorm). Are there any questions?

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Present Indifference-Amount Questions

Questions 1: What amount of money, $Y, if paid to you today would make you indifferent to $10 paid to you in 3 days. State Indifference Amount ______________. Questions 2: What amount of money, $Y, if paid to you today would make you indifferent to $10 paid to you in 1 week. State Indifference Amount ______________. Questions 3: What amount of money, $Y, if paid to you today would make you indifferent to $10 paid to you in 2 weeks. State Indifference Amount ______________. Questions 4: What amount of money, $Y, if paid to you today would make you indifferent to $10 paid to you in 1 month. State Indifference Amount ______________. Questions 5: What amount of money, $Y, if paid to you today would make you indifferent to $10 paid to you in 3 months. State Indifference Amount ______________. Questions 6: What amount of money, $Y, if paid to you today would make you indifferent to $10 paid to you in 6 months. Questions 7: What amount of money, $Y, if paid to you today would make you indifferent to $20 paid to you in 3 days. State Indifference Amount ______________. Questions 8: What amount of money, $Y, if paid to you today would make you indifferent to $20 paid to you in 1 week. State Indifference Amount ______________.

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Questions 9: What amount of money, $y, if paid to you today would make you indifferent to $20 paid to you in 2 weeks. State Indifference Amount ______________. Questions 10: What amount of money, $y, if paid to you today would make you indifferent to $20 paid to you in 1 month. State Indifference Amount ______________. Questions 11: What amount of money, $y, if paid to you today would make you indifferent to $20 paid to you in 3 months. State Indifference Amount ______________. Questions 12: What amount of money, $y, if paid to you today would make you indifferent to $20 paid to you in 6 months. Questions 13: What amount of money, $y, if paid to you today would make you indifferent to $30 paid to you in 3 days. State Indifference Amount ______________. Questions 14: What amount of money, $y, if paid to you today would make you indifferent to $30 paid to you in 1 week. State Indifference Amount ______________. Questions 15: What amount of money, $y, if paid to you today would make you indifferent to $30 paid to you in 2 weeks. State Indifference Amount ______________. Questions 16: What amount of money, $y, if paid to you today would make you indifferent to $30 paid to you in 1 month. State Indifference Amount ______________.

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Questions 17: What amount of money, $y, if paid to you today would make you indifferent to $30 paid to you in 3 months. State Indifference Amount ______________. Questions 18: What amount of money, $y, if paid to you today would make you indifferent to $30 paid to you in 6 months. Questions 19: What amount of money, $y, if paid to you today would make you indifferent to $50 paid to you in 3 days. State Indifference Amount ______________. Questions 20: What amount of money, $y, if paid to you today would make you indifferent to $50 paid to you in 1 week. State Indifference Amount ______________. Questions 21: What amount of money, $y, if paid to you today would make you indifferent to $50 paid to you in 2 weeks. State Indifference Amount ______________. Questions 22: What amount of money, $y, if paid to you today would make you indifferent to $10 paid to you in 1 month. State Indifference Amount ______________. Questions 23: What amount of money, $y, if paid to you today would make you indifferent to $10 paid to you in 3 months. State Indifference Amount ______________. Questions 24: What amount of money, $y, if paid to you today would make you indifferent to $10 paid to you in 6 months. Questions 25:

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What amount of money, $y, if paid to you today would make you indifferent to $100 paid to you in 3 days. State Indifference Amount ______________. Questions 26: What amount of money, $y, if paid to you today would make you indifferent to $100 paid to you in 1 week. State Indifference Amount ______________. Questions 27: What amount of money, $y, if paid to you today would make you indifferent to $100 paid to you in 2 weeks. State Indifference Amount ______________. Questions 28: What amount of money, $y, if paid to you today would make you indifferent to $100 paid to you in 1 month. State Indifference Amount ______________. Questions 29: What amount of money, $y, if paid to you today would make you indifferent to $100 paid to you in 3 months. State Indifference Amount ______________. Questions 30: What amount of money, $y, if paid to you today would make you indifferent to $100 paid to you in 6 months. State Indifference Amount ______________.

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