Discounting the Distant Future: An Experimental Investigation

[Forthcoming, Environmental and Resource Economics, August 2013]

Therese C. Grijalva*

Weber State University

3807 University Circle, Ogden UT 84408-3807

Jayson L. Lusk

Oklahoma State University, 411 Ag Hall

Stillwater OK 74078

W. Douglass Shaw

Texas A&M University

College Station TX 77843-2124

Abstract

We use a laboratory experiment to elicit discount rates over a 20 time horizon using

government savings bonds as a payment vehicle. When using a constant (exponential) discount

rate function, we find an implied average discount rate of 4.9%, which is much lower than has

been found in previous experimental studies that used time horizons of days or months.

However, we also find strong support for non-constant, declining discount rates for longer time

horizons, with an extrapolated implied annual discount rate approaching 0.5% in 100 years.

There is heterogeneity in discount rates and risk preferences in that people with more optimistic

beliefs about technological progress have higher discount rates. These findings contribute to

the debate over the appropriate magnitude of the discount rate to use in comparing the long-

term benefits of climate change mitigation to the more immediate costs.

KeyWords: Discounting, declining discount rates, climate change, experiment

*Address correspondence to Jayson Lusk, Dept. Ag. Econ., 411 Ag Hall, Stillwater, OK 74078,

phone: (405)744-7465, fax: (405)744-8210; e-mail: jayson.lusk@okstate.edu

The authors would like to thank Glenn Harrison and Greg Parkhurst for comments on an initial

design of the experiment, and for comments by participants at seminars at the University of

Denver and Texas A&M University (Department of Psychology), Partha Dasgupta Andrea

Galeotti, and Edward Morey. Grijalva acknowledges funding from the Hemingway family.

Lusk and Shaw acknowledge funding from the USDA AFRI Hatch Grants.

1

Discounting the Distant Future: An Experimental Investigation

Abstract

We use a laboratory experiment to elicit discount rates over a 20 time horizon using

government savings bonds as a payment vehicle. When using a constant (exponential) discount

rate function, we find an implied average discount rate of 4.9%, which is much lower than has

been found in previous experimental studies that used time horizons of days or months.

However, we also find support for non-constant, declining discount rates for longer time

horizons, with an extrapolated implied annual discount rate approaching 0.5% in 100 years.

There is heterogeneity in discount rates and risk preferences in that people with more optimistic

beliefs about technological progress have higher discount rates. These findings contribute to

the debate over the appropriate magnitude of the discount rate to use in comparing the long-

term benefits of climate change mitigation to the more immediate costs.

Key words: discount rate, hyperbolic discount rates, time inconsistent preferences, climate

change, experiment

2

I. Introduction

Economists have long been interested in the extent to which people are willing to

sacrifice consumption in the present for rewards in the future. Pinning down the precise rate at

which people discount the future is key in determining whether people go college or invest for

retirement and whether governments should build roads or adopt carbon taxes. Although there

have been important conceptual and empirical advances made in recent decades, there remains

a lack of knowledge about how people discount the distant future (e.g., in 20 plus years). A key

challenge associated with estimating longer-term individual discount rates in a laboratory

setting has been the difficulty of credibly paying out future promised rewards – a challenge we

overcome in this research by having individuals make choices that include government savings

bonds. In this paper, we utilize data from a laboratory experiment to estimate 20-year discount

rates, and in so doing, contribute to the lively debate about the choice of a discount rate in

studying the effects of climate change mitigation policies (e.g. Moore et al. 2004; Weitzman

2007; Dasgupta 2008, 2011; Schilizzi 2006).

There are a number of gaps in the empirical literature on discount rates. First, few

empirical studies have differentiated between private-oriented and social discount rates,

primarily because elicitation of a social rate presents a number of practical and conceptual

hurdles. Although they offer no insights on how to empirically estimate the two rates, Goulder

and Williams (2012) argue that there are two distinct discounting concepts: a social-welfare-

equivalent discount rate is appropriate for evaluating climate change policies and its implication

for social welfare, while a finance-equivalent rate is useful in determining whether a policy

offers a Pareto improvement. Private discount rates may closely relate to expected rates of

return on private investment, and several authors have commented that private rates of return

3

may be inappropriate for social discounting of future impacts (e.g. see Moore, Boardman and

Vining 2013 and many references therein). Most empirical studies consider only private money

tradeoffs. However, other studies have considered non-monetary tradeoffs (see the review of

stated preference studies by Meyer, 2013) and have found a range in implied annual discount

rates from nearly zero (Alberini and Ščasný 2011) to rates as high as 35% (Kovacs and Larson

2008). Our study uses money to money tradeoffs and while our estimates relate primarily to

private-oriented concerns, we also explore whether individual attitudes and beliefs about

climate change and technological progress impact discount rates.

A second drawback of previous literature is the lack of knowledge about how people

discount impacts occurring in time periods far into the future (e.g., in 20 plus years). Examples

of where such information is relevant would be mid-life (50 years old or more) health outcomes

for individuals in their twenties today, investments in public projects that require a long period

of construction (e.g. light rail systems), issues related to storage of high-level nuclear wastes,

climate change mitigation, and investments in programs that yield benefits much later than

today such as environmental restoration projects. The barrier to estimating longer-term

individual discount rates in a laboratory setting has been the inability of delivering future

payoffs. Our study is primarily aimed at addressing this weakness in the literature.

Obtaining good estimates of longer-term discount rates, and learning how they differ

from shorter-term rates measured in previous studies is more than idle curiosity as evidenced by

climate change debates. Mitigating climate change imposes costs in the present with benefits

accruing anywhere from 10 to 100 years in the future. Indeed, one of the key normative

questions in the climate change debate is the rate at which future costs and benefits should be

discounted (e.g., see Becker, Murphy and Topel 2010; Dasgupta 2008; Nordhaus 2007; Stern

4

2006, 2008; and Weitzman 2007). There is perhaps little debate among economists that future

benefits and costs should be discounted at some rate greater than zero, but at what rate?

Moderate to high discount rates would not justify mitigation investments borne today to avoid

large damages predicted to occur in the distant future. Lower discount rates such as the one

recommended in the Stern Review (2006) might justify significant intervention at present.

When one surveys the literature on the social cost of carbon emissions and the urgency of

embarking on climate change mitigation, the recommendations often hinge critically on the

choice of the discount rate (Anthoff, Toll and Yohe 2008; Heal 2009; Nordhaus 1994; Stern

2006, 2008).

A common criticism of standard discounting in benefit-cost analysis is that the use of a

positive and constant discount rate (exponential discounting) for climate change actions today

implies that benefits accruing in the far-distant future are unimportant in present value terms; a

result that is intuitively unappealing for many observers (e.g. see Weitzman 1998). However, a

number of empirical studies of intertemporal choice have found that individual time preferences

are not constant over different time horizons (as is assumed by exponential discounting), but in

fact decline when the horizon lengthens (e.g., see Frederick, Loewenstein and O’Donoghue

2002; and many recent applications including Kan 2007, Prince and Shawan 2011). Thus, we

estimate models assuming a standard constant discount rate, and then relax the assumption of

constant discounting to allow for different forms of hyperbolic discounting (e.g., Herrnstein

1981; Mazur 1987; Loewenstein and Prelec 1992).

The social debate on climate change policy addresses the obvious trade-off between

certain mitigation costs imposed on present generations and risky or uncertain benefits to future

generations. Some individuals may put more weight on immediate consumption benefits,

5

whereas others might be more patient, or willing to sacrifice current benefits, and place greater

weight on consumption opportunities for future generations. Thus, it seems likely that different

beliefs and attitudes about climate change or technical progress would impact individual

discount rates. As part of the experimental exercises, our subjects answer a set of questions to

gauge their preferences, attitudes and knowledge about climate change. Responses to these

questions are used to explore heterogeneity in both time and risk preferences.

To preview: our results indicate that implied discount rates are much lower than has

been found in previous studies that use short time horizons, even when assuming a conventional

or constant discount rate function. Our results persist even if we take into consideration the fact

that individuals can cash in 20 year savings bonds at a date that prior to maturation. Further, our

analysis of longer time horizons contributes to prior experimental studies which find evidence

for declining discount rates as the time horizon lengthens. Lastly, we find heterogeneity in

discount rates and risk preferences, which vary with the location of the experiments (the states

of Utah vs. Texas), beliefs about technological progress, and climate change attitudes.

II. Literature Review

A. Discount Rates

The consumption (M) discount rate (ρ) over time t is most often expressed by the

approximation to the Ramsey (1928) formulation:

(1) )( tt Mg

where δ is the pure rate of time preference, η is the elasticity of the marginal utility of

consumption, and g is the growth rate in consumption over time. With rare exceptions,

empirical estimates for individuals are of the overall consumption discount rate and do not

6

allow recovery of the separate components on the right-hand side of (1). However, it is worth

noting that the overall rate is comprised of the two terms on the right-hand side of equation (1),

and not just the pure rate of time preference.1

Generally, curvature in the utility function with respect to consumption, income or

wealth relates to η and covers the possibility that utility discount rates are different from money

discount rates, precisely because the marginal utility of income may not be constant. By

incorporating risk in the utility function using one of the conventional functional forms such as

constant relative risk aversion, it is possible to determine whether individuals exhibit

diminishing marginal utility of wealth. When individuals are risk neutral, utility is linear in

consumption, income or wealth, and the discount rate can be recovered by equating the present

value of two monetary outcomes, Mt and Mt+T, where t represents the time period, and T

represents some positive increment to t.

B. Revealed and Stated Preference Approaches

Empirical inferences about individual discount rates can be obtained from revealed

preference purchase decisions or investment choices (Warner and Pleeter 2001), but most often

are made on the basis of stated preference surveys or experimental data such as obtained from

the multiple price list (MPL) approach. The MPL is incentive compatible when actual payments

for choices are made (e.g. Coller and Williams 1999; Harrison, Lau and Williams 2002). Most

efforts to elicit discount rates in a non-hypothetical setting involve tradeoffs for periods of only

months, or even weeks, with some exceptions (e.g., Harrison, Lau and Williams (2002) utilized

a three year horizon).

1 See Dasgupta (2008) or Weitzman (2007) for some discussion on the interpretation of η and normative

arguments for its magnitude; or Stern (2006), Becker, Murphy and Topel (2010) , or Nordhaus (2007) for similar

discussions on δ.

7

A common finding from previous survey and experimental studies is high level of

impatience as compared to what is implied by conventional interest rates. Binary choices in

MPLs impose linear preferences over outcomes, thus resulting in an upward bias in discount

rate estimates (Andersen et al., 2008; Andreoni and Sprenger 2012a). Andersen et al. (2008)

find that the estimated discount rate falls from 25% to 10% when they adjust for risk aversion

(or non-linear utility). Andreoni and Sprenger propose a convex time budget approach, and

find aggregate discount rates of around 30%, lower than what is typically found in experimental

settings.

It is worth noting that some studies have considered non-monetary intertemporal

tradeoffs; however, choices are hypothetical and not incentive compatible. Examples include

value of reducing mortality or morbidity risk where illness/life-year changes occur at different

times (see Alberini et al. 2006; Alberini and Chiabai 2007; Alberini and Ščasný 2011; and early

studies by Horowitz and Carson 1990; Cropper, Aydede and Portney 1994) and various

willingness to pay studies for things like public open space (e.g. Kovacs and Larson 2008)

wildlife or aquatic habitat (Bond and Larson 2009; Meyer 2013), and recreational resources

(Viscusi, Huber and Bell 2008). These papers are too numerous to review extensively here, and

approaches and payment periods vary, but results indicate that implied annual discount rates

involving public goods vary with the context, from nearly zero to rates of over 30%. Generally,

the non-money studies appear to generate lower discount rates than typically found in non-

hypothetical laboratory experiments using money to money tradeoffs.

C. Hyperbolic Discounting

8

There has been a great deal of recent theoretical and empirical work on hyperbolic

discounting—discount rates that are not constant over periods of time. While Koopmans (1960)

long-ago laid out the assumptions and conditions that lead to constant utility discount rates over

time, there was a tone of some skepticism that these conditions would hold in practice. Several

economists deem people with hyperbolic discounting functions as “irrational” because, under

some assumptions, hyperbolic discounters will always optimal to postpone investment (see e.g.,

Winkler 2009). Because hyperbolic discounting implies time-inconsistent preferences (i.e.,

plans made today may not be carried out tomorrow), it has been used to explain perplexing

intertemporal decisions such as low savings rates (see Laibson 1994, 1997), drug addiction

(Gruber and Kőszegi 2001), or generally, in behaviors that endanger health (such as obesity or

drinking, e.g. see Richards and Hamilton 2012). Yet, Heal (2009) takes the position that it is

unnatural to require time consistency; individuals are different people at different points in time

and as such assuming time consistency is rather stringent.2

Although some authors such as Harrison, Lau and Williams (2002) argue that

hyperbolic-like discounting results from the peculiars of experimental protocol (i.e., the lack of

previous experimenters to use a front-end delay), it remains a common empirical finding for

individual discount rates to fall over longer time horizons, where the immediate future is

discounted more heavily than is the distant future (Frederick, Loewenstein and O’Donoghue

2002). As an example, Grijalva, Berrens and Shaw (2011) found that subjects in laboratory

experiments indicated significantly smaller discount rates for five year time horizon tradeoffs

than they did for shorter, one year tradeoffs.

2 The issue of hyperbolic discounting has been considered in the climate change debate by authors such as Duncan

and Papachristodoulou (2010), Groom et al. (2005), Karp (2005), Weitzman (1998)

9

Previous research has also explored whether there exists individual heterogeneity in

estimates of individual discount rates or in explaining time consistency (e.g., Harrison, Lau and

Williams 2002). For example, Harrison, Lau, and Rutstrom (2010) explored variation across

smokers and non-smokers. They found that male smokers have significantly higher discount

rates than male non-smokers, but this result did not hold for females. However, smokers were

not any less time consistent than non-smokers. To our knowledge, no previous study has

examined whether climate change preferences correlate with discount rates, similar to how

Harrison, Lau, and Rutstrom (2010) studied how time preferences varied across smokers and

non-smokers.

III. Methods

A. Experimental Design

The primary purposes of our study are to estimate discount rates over a longer-term

horizon of 20 years, compare these results with a shorter-term horizon (1 year), and determine

how the elicited discount rates vary with various behavioral attitudes towards climate change.

All of the subjects were university students who participated in one of seven

experimental sessions: five sessions at Weber State University (WSU) and two sessions at

Texas A&M University (TAMU). All sessions were conducted in late April and May of 2011,

prior to the downgrading of the U.S. Federal Government’s credit rating by Standard and Poor

from AAA (outstanding) to AA+ (excellent) on August 5, 2011. Two university locations were

chosen due to concerns about seemingly homogenous student populations at each respective

university. For instance, at WSU a large majority of the students are from Utah and are of the

same religious affiliation (predominantly Mormon). At TAMU, students are predominantly

Texans, affluent, and of a certain Christian affiliation. At WSU, students were recruited via

10

University electronic bulletins where a student would complete an online participation request

form. Researchers utilized online sign-up made available through TAMU’s Economic

Research Laboratory as well as by posting flyers throughout several buildings at TAMU. Our

data includes choices made by 119 total subjects, consisting of 48 from TAMU and 71 from

WSU. The experiment was administered through individual stations in a computer laboratory,

and took approximately 45 minutes to complete.3

The introduction to the experiment explained that each subject would be making

intertemporal choices and their earnings would be based on the choices they would make

between receiving money sooner (e.g., today) versus a larger amount later (e.g., tomorrow). In

total, each subject was guaranteed to earn between $17-$18; a $10 guarantee payment plus an

additional $7-$8 based on two short-term delay discount rate exercises where subjects made

choices between an immediate cash amount today of $5 and $2 and higher cash amount

tomorrow $5 + x and $2 + x, where 0< x ≤ $0.50, respectively. Given concerns in the literature

(e.g., see Harrison, Lau and Williams 2002) about subjects choosing today, i.e. “immediate”

payment amounts rather than the tomorrow amount, we divided the $10 guarantee payment into

two payments, $5 today and $5 tomorrow, so that no matter what choices were made in the

short-term intertemporal choice exercises (described above), students would have to come back

to get the remainder of their earnings, made available at the Economics Department at each

respective university. One possible advantage of the short-term exercises is that they provide

an opportunity for subjects to learn, and thus potentially minimize a learning effect bias that

would otherwise be present in the long-term delay exercises (e.g., see Andersen et al. 2006).

3 A copy of the full set of experimental instructions is available at

http://faculty.weber.edu/tgrijalva/Experiment%20V1.pdf.

11

Subjects were also told that they would participate in 23 additional choice exercises,

where one subject would be randomly selected, and depending on a randomly selected choice

made, would earn a larger amount ranging from an additional $C (C included $5, $25, $50, $75,

or $95) tomorrow, $100 in 1 year, or a $200, $500 or $1000 face value 20 year U.S. Treasury

Savings Bond (Series EE, paper bonds).4 Our choice of the Series EE paper bonds was

motivated by our desire to use a security that was the closest instrument to offering cash in

twenty years, while remaining easy to deliver to the subject. We found that we could purchase

a bond in a subject’s name and then have it mailed directly to the subject, avoiding the need to

keep track of where the subject would be in twenty years.5

Upon completion of the experiment, the randomly selected subject completed a form

prepared by either the Treasury Department or WSU Accounting Services. Both forms

requested the subject’s name, address, social security number and a signature. All payments

(including the bond) were processed (purchased) by WSU Accounting Services and delivered

via U.S. post, except for the 1 day payment where the subject could have the payment mailed

by WSU Accounting Services or he/she could pick up a cash payment tomorrow from the

researchers. In addition, the student received a letter for their records from the researchers.

The letter provided information about their participation and earnings and who they should

4The payouts for the randomly selected subjects included: a $25 payment tomorrow, a $95 payment tomorrow, two

$100 payments in 1 year, and 3 savings bonds, two $1000 bonds and one $500 bond. Further, a $1000 bond was

purchased for one subject who participated in the pretests. 5 The Series EE paper bonds are stated as 30 year bonds, but are guaranteed to be worth the face value at 20 years.

It should be noted that we explored a number of security and savings instruments with varying maturity dates.

Advantages of Series EE bonds include the fact that this is among the longest-term securities in the market and is

relatively risk-free, the ability to name a beneficiary different than the purchaser (i.e., gift), and the fact that the

paper bonds have a stated face value, where they are purchased at a discount (half face value). Not all of these

features are true of other treasury securities or electronic Series EE bonds. Similarly, we did not use certificates of

deposit due to the fact that they have shorter maturity dates (typically ranging from 3 months to 5 years) with

future values being a function of the fixed interest rate offered by banks and early withdrawal penalties. For

information on treasury securities and programs go to http://treasurydirect.gov/indiv/products/products.htm.

12

contact if they had questions. In addition, the researchers placed reminders on their calendars

to follow-up with recipients in one year to ensure there had been no change in address.6

The next section of the experiment commenced with a set of questions to assess each

subject’s knowledge of financial markets; we also provided current interest rate information on

savings accounts, certificates of deposit, US treasury Savings Bonds, and mortgage rates. The

purpose of providing the interest rate information is to ensure that all subjects had the same

information regarding arbitrage possibilities before answering the set of 23 choice exercises

about receiving money tomorrow, in 1 year or in 20 years (e.g., see Coller and Williams 1999).

Following others (e.g. Ida and Goto 2009; Viscusi, Huber and Bell 2008), a choice

exercise (CE) approach was used as the basis for estimating utility discount rates over 1 and 20

year time horizons, however, a notable difference between our approach and others that use

hypothetical CE data is that we randomly selected one of the choices for actual payout to ensure

incentive compatibility (Lusk and Schroeder 2004; Vossler, Doyon and Rondeau 2012).

Further, our approach is different than the standard MPL approach because each exercise was

presented one at a time. .

To ensure understandability of the long-term CEs, to minimize learning or order effects

(Andersen et al. 2006), 7

and to demonstrate the random draw process via a drawing from a

bingo cage, subjects first worked through two practice exercises and watched a mock bingo

cage drawing. All subjects then answered the same set of 23 binary CEs in the same order

6 Andeoni and Sprenger (2012a) stress the importance ensuring that all aspects of the choices must be equivalent

except for the timing of the payment. Researchers should equalize payment transaction costs across all time

periods and take steps to ensure confidence or trust in receipt of payment. 7 Andersen et al. (2006) randomize the sequence of time horizon MPL tables to test for order and learning effects.

They find that the 2nd

task is associated with slightly higher, and statistically significant, discount rates, which can

be attributed to pure order effects (although, they note that these order effects are not very large). Further, they are

unable to reject they hypothesis that the average rates for task 2 and 3 are the same. This suggests that there is

some learning, but only after the initial task.

13

between receiving differing amounts of money paid out at different times. The full set of

choices is shown in Table 1. The choice tasks that we used were structured to provide

reasonable choices to subjects, while maintaining our study objectives. One drawback is that

the risk neutral discount rates (annual percentage rates) that would be implied by a choice of

option A over B are considerably higher (and with greater variance) in the first five choices

overall (with the one year delay), as compared to the choices involving the twenty year delay.

Recall that the main objective of our study is to estimate the long-term 20 year discount rate,

and it is here that our design provides more precision.8

Because of concerns associated with trust or transaction costs in receiving future

payments, the intertemporal choice exercises were followed by a set of questions to assess a

subject’s reasoning for their choices. The subject was instructed to indicate the importance of

each of the following in making their choices: (1) I thought I might be able to turn around and

invest the money I get tomorrow, at a better rate than you offered with the Savings Bond; (2) I

thought I really needed the money tomorrow, because I have an immediate need for it; (3)

Twenty years is too long in the future for me to think about; (4) I thought I really didn’t trust

being able to get the money from the treasury security in 20 years; and (5) I thought of

giving the proceeds from the bond in 20 years to my children, or to someone else. The

breakdown of the responses are included in Table 2.

Subjects were randomly assigned to one of three treatments. One treatment consisted of

providing subjects with the following information about US Treasury Savings Bonds (Series

EE):

8 A note in regards to the longer-term intertemporal choice exercises of 1 year or 20 years: while this might imply

more noise surrounding our estimates of the 1-year discount rates than that surrounding the 20-year discount rate,

both discount rates are estimated without bias. For future research, one might design the variation in monetary

outcomes across time horizons in any number of different ways, testing for the robustness of the results we obtain.

14

Paper bonds are sold at half the face value. For example, a $100 bond is

purchased for $50. Series EE Savings bonds earn a fixed rate of interest

for the life of the bond (30 years), but at 20 years there may be a one

time interest adjustment as Series EE bonds are guaranteed to be worth

double the purchase price at that time. These bonds may be redeemed

early. The minimum holding period is 1 year, where the owner will

receive the original purchasing price plus any earned interest. If the bond

is redeemed before 5 years, the owner forfeits the 3 most recent months’

interest.

Because there was likely heterogeneity across subjects in their understanding of savings bonds,

this treatment was meant to study the effect of homogenizing expectations and to test for

information and context effects, potentially an issue of concern in laboratory settings (e.g., see

discussion in Harrison and List 2004). After making their choices, and to control for the fact

that some subjects may intend to cash in their bonds prior to maturation, subjects were asked to

indicate when they would likely cash in the 20 year bond using a slider scale that showed the

expected value of the bond at each potential cash-in date. The question was answered for bond

amounts equal to $100, $500, and $1000. In estimating discount rates, we will compare models

with the “objective” payout amounts and dates with the subjects’ stated cash-in date and

implied payout amount to explore how this might impact the estimates.

A second treatment focused on determining whether a subject’s choice on each of the 23

choice exercises would remain stable with feedback on the consequences of their choices

assuming the payouts were scaled to larger dollar amounts (henceforth referred to as the follow-

up feedback treatment). In particular, following each of the 23 choices, subjects were asked

whether they would make the same choice if the future dollar amount was $1 million. For

example, following Exercise 4, the follow-up question read:

15

[For those answering A] If you have typical attitudes that most people

have about the risk that is associated with waiting to receive money in

the future, then your choice implies you might prefer to receive:

$869,000 tomorrow rather than $1 million in 1 year.

Think carefully: Does the above statement reflect your true preferences

in regards to waiting 1 year to receive $1 million?

[For those answering B] If you have typical attitudes that most people

have about the risk that is associated with waiting to receive money in

the future, then your choice implies you might prefer to receive:

$1 million in 1 year rather than $869,000 tomorrow.

Think carefully: Does the above statement reflect your true preferences

in regards to waiting 1 year to receive $1 million?

The purpose of the follow-up feedback treatment was to attempt to mimic a market-like process

that has more feedback than is present in isolated discrete choice exercises. Plott (1996) has

argued that rational choice is a process that occurs when people gain experience and receive

feedback from a particular market environment, and although we did not implement a market

explicitly, we attempted to adopt an environment which helps people discover their preferences.

Finally, a control treatment was conducted that did not provide information about savings

bonds or the follow-up feedback treatment.

Because a focus of our experimental design is to determine how discount rates may vary

for various behavioral attitudes towards climate change, subjects answered several questions

gauging their preferences, attitudes, knowledge and beliefs about the environment and climate

change. This included utilizing several survey techniques including: (1) allocating points

according to one’s value over a set of future goods including future salary or future health of

the environment; (2) using scales (0 to 100) to reflect attitudes and beliefs about climate change

and climate change policy; and (3) answering true/false questions about climate change and

16

greenhouse gases. The experiment concluded with several questions eliciting personal, or

socio-economic and demographic information.

B. Econometric Models

Let individual i’s utility for payout xi occurring in time t be given by Vit = D(t)U(xi),

where D(t) is a discount function, which can take on any functional form such as exponential or

hyperbolic, and U(xit) is the utility of receiving payout xit. As pointed out by Andersen et al.

(2008), one must control for the level of risk aversion to obtain an unbiased measure of time

preference. In the analysis that follows, we assume a constant relative risk aversion (CRRA)

specification where U(xit) = xit(1-r)

/(1-r) and r is the coefficient of relative risk aversion. Risk

neutrality, risk aversion and risk loving behavior are exhibited when r =0, r > 0, and r < 0,

respectively.9

Recall that individuals in the experiment made a series of choices between two payout

options occurring in differing time periods. If xAic is the payoff of option A for individual i in

choice c, then the random utility for the option can be written as:

(2) Aic

r

Aic

AicAicr

xtDU

1)(

1

where tAic in equation (3) is the length of time until option A pays out in years, and εAic is an

overall error term meant to capture the fact that the analyst cannot perfectly observe an

individual’s preferences. Choice between option A and B is explained by the difference in

9 Our experimental tasks did not actually involve any choices over risky outcomes; however, such choices are not

required to estimate the parameter r, which simply provides an estimate of the extent of diminishing marginal

utility. The use of multiple payout amounts ranging from $5 to $1000 provides ample variation in payouts to

detect curvature in the utility function.

17

random utilities of options A and B. Accordingly, the difference in the overall error terms of

options A and B in choice c is:

(3) )(1

)(1

)(11

BicAic

r

Bic

Bic

r

Aic

Aicicr

xtD

r

xtDU

If the difference in error terms is normally distributed with standard deviation σ, a likelihood

function can be specified as:

(4)

where N is the number of participants, C is the number of choices per participant, yic = 1 if

individual i chose option A in choice c, yic = 0 if individual i chose option B in choice c, σe is

the standard deviation of the overall error term, and Φ is the standard normal cumulative

density function. The term γi was added to the likelihood function in equation (5) to account for

the panel nature of the data (i.e., each individual made a series of choices between option A and

option B), and the random effect is assumed to be distributed Normal with mean zero and

standard deviation σi. Gaussian quadrature is used to integrate the individual-specific random

effect, γi, out of the likelihood function. Standard errors are corrected to account for the

repeated nature of the data. The likelihood function is maximized with respect to the

parameters of D(t), r, σe, and σi.

A few comments about stochastic specification of the model are in order. Our model

set-up follows in the tradition of McFadden’s (1974) random utility models, which posit that

the errors arise from the fact that we cannot perfectly observe preferences. Estimates of σ

provide an estimate of the sample error, and as σ approaches zero, there is no error and the

model would provide precise predictions of which option would be chosen. Fechner (1860)

suggested a specification that is functionally equivalent to the one outlined above except that

18

many analysts interpret σ as a measure of behavioral errors, i.e., the inability of the model to

adequately describe behavior. Some recent papers such as Holt and Laury (2002) and Andersen

et al. (2008) use a version of the Luce (1959) model, where instead of assuming the difference

in utility errors are distributed normally, an index of the utilities is used. In such models, the

term in equation (5) is instead replaced with the index, )/( /1/1/1 BicAicAic UUU

,

where μ is a noise parameter. As μ approaches zero, the model becomes one of deterministic

choice, and as μ approaches positive infinity the model is unable to distinguish preferences

between the two options. In our analysis, we used the specification shown in equation (5) for

two reasons. First, it provided a better fit to the data (according to AIC and BIC measures of

fit) than the Luce-type model. Second, our set-up provided a more natural way to incorporate

the individual-specific random effect, γi, which controls for the panel nature of the data, and a

way to test the null hypothesis of no heterogeneity across people.

We considered several alternative specifications for the discount function D(t). The first

is the conventional time-consistent discount function where D(t) = (1+d)-t, where d is the annual

discount rate which is constant for all periods. To allow for the possibility of hyperbolic

discounting, we also used the Loewenstein and Prelec (1992) discount function:

(5) D(t) = (1+at)-b/a

,

where a and b are parameters to be estimated. If a = 0, the model reverts to one of constant

discounting, conveniently allowing a simple testable hypothesis of the assumption of constant

discounting. When a and b take positive values, discount rates decline over time, i.e., the

discount function is negatively sloped. Larger positive values of a and smaller positive values

of b flatten out the discount function over time, so that the discount factor gets smaller at a

slower rate. If a is positive but b takes on negative values, the discount function actually

19

increases over time. Finally, for a range of negative values of the a parameter, with b positive,

the discount function over time (negatively sloped) becomes quite steep, so that people

essentially put zero weight on future impacts only a short time out. We also consider a simpler

hyperbolic specification originating in studies such as Mazur (1987) and Herrnstein (1981),

where D(t) = (1+gt)-1

, where g is a parameter to be estimated.

In our current study, we use the 23 choice exercises (presented in Table 1) and use three

payoff dates, a one day delay (t=1/365), a one year delay (t=1), and a 20 year delay (t=20).

With three payoff dates we can also estimate a non-parametric specification in which D(t) =

d0*I(=1 if t is1/365) + d1*I(=1 if t is 1) + d20*I(=1 if t is 20), where I is an indicator variable

taking the value of 1 if the condition in parentheses are met, d1 and d20 are coefficients to be

estimated indicating the extent to which payoffs in 1 and 20 years are discounted relative to the

one day delay, where d0 is normalized to 1 for identification purposes. Although the non-

parametric specification is apparently more general than the others, it is observationally

equivalent to the Loewenstein and Prelec (1992) specification given the fact that only three

payout dates were used in our study; nevertheless, we report the results for completeness and

robustness check. We select between models using AIC and BIC measures of model fit.

The above discussion assumes individuals will hold the 20 year bond until maturation

and receive the face value. However, 58% (69 subjects) of our sample knew how the bonds

actually work (i.e. knew of ability to take early rewards), and of those, 74% (51 subjects), said

this knowledge would influence their decision.10

To control for the fact that some individuals

might cash in their bond prior to maturation, we can re-estimate any of the above models

(except the non-parametric specification) and replace the “objective” or terminal payout

10

Interestingly, a much smaller percentage of those who did not know about the early redemption option said it

would have influenced their decision (19 of 50, or about 38%).

20

amounts and dates with the subjects’ stated cash-in date and implied payout amount. For

example, if an individual indicated that they would cash in the $1000 face-value bond in 10

years, we would use t = 10 rather than t = 20 in the discount function and would use x =

$557.97 rather than x = $1000 in the utility function, where the lower amount was the amount

implied by interest rates on the days of the experiment (note: it was also the amount shown to

the individual when making this choice). In practice, we find that the coefficients from the

models that control for these early cash-in dates were very similar to the estimates using the

“objective” values, although the degree of decline in the discount rate over time was less

pronounced than in the original specifications (refer to Appendix A for a comparison of

results).

One of the primary goals of this study is to analyze how discount rates change with

beliefs about climate change and technological progress. To study these issues, we specify the

aforementioned coefficients r, d1, and d20 as linear functions of demographics and climate

change and technological progress beliefs. For example, the coefficient r is replaced with the

function r = r0 + r1*info + r2*TAMU + r3*followup + r4*male + r5*climate + r6*tech, where

info is a dummy variable indicating treatments where subjects were given information about the

workings of U.S. savings bonds, TAMU is a dummy variable indicating subjects at Texas A&M

University, followup is a dummy variable indicating the follow-up feedback treatment where

subjects were given information about the implications of their choices in terms of $1 million,

male is a dummy variable for gender, and climate and tech are beliefs about climate change and

technology. These later two variables were measured as continuous variables by asking people

to respond to the following questions on a 100-point scale [Subjects dragged a cursor along the

scale to indicate their level of belief]:

21

On a scale of 0 to 100, how likely is it that you believe climate

change will have a substantially negative influence on the U.S.

economy in 20 years?

No negative effect [0…….100] Certain to have a negative

effect

On a scale of 0 to 100 indicate the degree to which you believe

technological progress will enable future generations to solve

environmental problems? Technology…

Will NOT solve Will solve

environmental [0…….100] environmental

problems problems

A higher score on the second question is associated with a more prominent belief that

technological progress will enable future generations to solve environmental problems.

IV. Results and Discussion

Table 3 reports the results of the simple econometric specifications that do not control

for differences in demographics or beliefs across subjects. Overall, the results are reasonable.

To begin, the term γi is statistically significant across all models and therefore shows the

random effect is needed (i.e., the null of no heterogeneity across people is rejected). Next, the

estimated coefficient of relative risk aversion is around 0.40 consistent with a moderate degree

of risk aversion or diminishing marginal utility of income. This finding is very similar to

previous results from other authors (e.g., Holt and Laury 2002): a common finding in empirical

work is at least some degree of risk aversion, as opposed to risk neutrality or risk loving

attitudes, although we note that the usual assumption is homogeneity across individuals. There

are exceptions to this assumption, allowing gender or schooling to affect risk attitudes.

22

Aside from the usual considerations of the effects of demographic variables on risk

attitudes, more recent papers also explore the possible effect of the nature of the experiment.

These include the individual’s search process during tasks (Schunk and Winter 2009) and

possible selection bias in samples used in experiments (Harrison, Lau, and Rutsröm 2009).

Schunk and Winter (2009) find that search heuristics are not related to measures of risk

aversion, however, the loss versus gain domain does influence results: 63 percent of subjects

are risk averse when there is a potential gain at stake, while only 23 percent are risk averse

when facing a loss. Using the CRRA form, Harrison, Lau, and Rutsröm (2009) find evidence of

a fairly wide range on the CRRA coefficient for individuals from samples, and their estimate of

the coefficient with no sample selection correction is 0.45 and falls to 0.23 with an appropriate

correction. They state that the sample they have is more risk averse than the population. Thus,

even these careful considerations of sampling and experimental design demonstrate some

support for risk aversion.

Lastly, the discount rates implied by the estimated parameters (or functions depending

on the model) are also very reasonable. In the constant discount model, for example, the

estimates imply an annual discount rate of 4.9%. Note that this is much lower than most

empirical studies obtain. For example, Harrison, Lau and Williams (2002) use field experiments

in Denmark and estimate average discount rates for three year horizons to be about 28%; and in

his review of the rational addiction smoking literature, Laux (2000) also reports some typically

high estimates of implied discount rates, one being as high as 90% per year. Bond et al. (2009)

find a range of implied annual discount rates from 23% to 80%, when payment programs in

support of a public good (Sea Lion habitat) are contemplated for as long as 15 years.

23

Our finding of a relatively low annual discount rate is probably due to our use of a very

long time horizon, as opposed to only very short ones used in previous literature. Support for

this conjecture can be found in the models that allow for non-constant discounting. For

example, the Loewenstein and Prelec (1992) specification implies a discount rate of 19.9%

when moving from year 0 to year 1, a result very similar to that found in previous literature.

Only when we move out to the 20 year range does the Loewenstein and Prelec (1992)

specification imply annual discount rates of about 2.5%.

Even our estimate of 4.9% from the constant-rate specification is much closer to what

most economists intuitively believe a discount rate perhaps should be. Still, the rates are higher

than the rate used in the Stern Review, which is close to zero. Our estimate is, in fact,

reasonably close to the value of 4% Nordhaus assumed in his climate change benefit-cost

analysis. While 4.9% leads to higher net present values 100 years from now than a rate of 20%

implied by other empirical studies, it is still consistent with the Nordhaus policy prescription of

ramping up climate change protection only very slowly, if at all.

Comparing across models in Table 3, there is evidence of hyperbolic discounting with

discount rates falling over time. As noted in equation (5), the test of whether

a = 0 in the Loewenstein and Prelec model is a direct test of whether there is constant

discounting. As can be seen in Table 3, we reject a = 0 in favor of hyperbolic discounting.

Moreover, the a and b parameters were estimated to be positive, indicating declining discount

rates. The Loewenstein and Prelec model performs equally as well as the non-parametric model

for best fit according to the AIC and BIC criteria. In fact, the identical likelihood function

values imply the two models are functionally equivalent (this can be clearly seen in the graph in

Figure 1).

24

Perhaps an easier way to interpret the results is to see what they imply in terms of

discount factors (how the present weighs future benefits and costs) and discount rates, both of

which are plotted in Figures 1 and 2, respectively. Clearly, when t=0, then D(t=0)=1; that is,

individuals of course do not discount the present. Figure 1 illustrates what the different models

imply for discount factors.

In the constant discounting model, people discount the utility received in twenty years at

about 40% (i.e., D(t=20)=0.38 relative to the present in which D(t=0)=1). Stated differently,

outcomes in 20 years are given 38% the weight of outcomes today. The hyperbolic models

imply steeper initial discounting, but the functions flatten out over time. The discount factors

become smaller over time as they would in a conventional exponential framework, but they do

so at a much slower rate far into the future. In the best-fitting models, for example, the discount

factors are almost 30% in year 20 as compared to the present (i.e., D(t=20)=0.27). So, while

individuals are still impatient with near term tradeoffs, hyperbolic discounting implies that

people consider each increment into the distant future more than they would if we assume

constant discounting. Referring to Figure 1, note that the triangles for the non-parametric model

align with the Loewenstein and Prelec model – again showing the two are equivalent given our

experiment data.

The discount rates implied by the estimated models are shown in Figure 2. The constant

model of course means a flat rate over time of 4.9% annually. The best-fitting hyperbolic

model (Loewenstein and Prelec) starts out initially with high annual discount rates of around

20%, indicating a high degree of impatience early on, but by the end of the period the implied

rate is also below 5% and is near 2.5%. The conventional hyperbolic specification has a similar

pattern but has a lower initial implied rate early on. Indeed, extrapolating out to 100 years, the

25

implied annual discount rate from the Loewenstein and Prelec model approaches 0.5% which is

nearing the value assumed in the Stern Report. The result may suggest a strategy in which

commitments are made today to incur costs at some future time period when discount rates are

lower. This of course presumes that scaling up mitigation activities later, as opposed to now, is

effective climate change policy, the reality of which is in the domain of the physical sciences.

A. Treatment Effects, Climate Change Attitudes and Demographics

A motivation for the current study relates to connections between the discount rate and

climate change protection. To further explore this connection, we investigate the relationship

between discount rates and beliefs/perceptions about climate change and the future. Though any

effect of the latter might pertain to the pure rate of time preference term in the Ramsey

approximation, we cannot separate out an effect on the separate components here. To do the

primary investigation on the overall consumption discount rate, we work with the non-

parametric model which has the best fit and is the easiest of the models to interpret since the

coefficients are just estimates of the discount factor (See Appendix A, Table A2, for a

comparison of results using the Constant Discounting Model). We interact the key shape

coefficients or parameters of interest (discount factor and coefficient of relative risk aversion)

with climate questions, experimental treatments and demographic variables. Table 4 shows the

key results of this investigation.

In Model 1 of Table 4, the discount factors are a function of the demographic variables,

but the demographics are not assumed to affect the risk aversion coefficient. [Note that Prince

and Shawan (2011) found evidence that males were time inconsistent in their experiment, while

females were not.] In Models 2 and 3, we then allow the climate change questions to affect the

26

risk parameter. People who think climate change will be a problem in 20 years are more risk

averse than those that don’t. Referring to Model 3, which adds in belief questions, we also see

that people who believe that technology will enable the future to solve environmental problems

are less risk averse.

In regards to the treatments, 93 subjects received the $1 million follow-up questions to

the 23 choice exercises, and 52 subjects received information about the Savings Bonds. First

we investigate whether subjects made consistent choices in the follow-up feedback exercises.

On average, subjects choosing option B were more likely than those selecting option A to make

a consistent choice in the follow-up. It could be the case that subjects are less impatient with

the larger $1 million reward at a future date at stake; this corresponds well with theory of

diminishing marginal utility of wealth represented by a concave utility function. Again

referring to Models 2 and 3 in Table 4, the estimated parameter on the follow-up feedback

treatment is not statistically significant, thus we are unable to reject the null hypothesis that the

follow-up choices do not influence risk. However, providing additional information on savings

bonds (provided in one of the treatments) lowers risk aversion. In addition, the TAMU subjects

were less risk averse than the subjects from WSU. This result may be due to any number of

potential regional or cultural differences between the student populations or differences due to

the nature of these school attendees.

Referring to Models 2 and 3 in Table 4, the TAMU subjects also discounted the future

less than the WSU subjects, suggesting that TAMU students placed greater weight on future

consumption than the latter. The TAMU student body largely consists of in-state residents

(students coming from within the State of Texas) and WSU students are largely residents of the

State of Utah; thus, the university location may serve as a proxy for preferences that relate to

27

state residence and indicate differences between educated, young residents in the two states.

This again may have to do with any number of different aspects of the two subject pools.11

We also investigate whether beliefs about technology’s role in solving future

environmental problems affected discount rates. This attitude might pertain to embedded

subjective estimates that would relate to the elasticity of marginal utility in the Ramsey

approximation (1), and could also relate to optimism regarding future consumption growth.

Referring to Model 3 in Table 4, those who are optimistic about technology improving living

conditions in the future are more apt to opt for sooner payout amounts than those more

pessimistic. That is, optimism about the future of technology results in less concern for the

future as reflected in a higher discount factor. The magnitude of the coefficient shows that for

someone with perfect optimism (score=1) will discount the payouts in 20 years vs. now 21.4%

more than someone with no technological optimism (score=0). The results are also suggestive

of the idea that debates about the appropriate discount rate to use in a benefit-cost analysis of

climate change might also hinge on deeper, more implicit assumptions about the possible

benefits of technological development. Those who believe a lower discount rate should be used

in discounting future benefits of climate change mitigation are likely the same who have less

faith in technology to solve future environmental problems.

V. Conclusions and Suggestions for Future Research

Debate over what is an appropriate rate of discount relate to considerations about the

potential size of future beneficial or costly impacts from the current generation’s perspective.

Some have argued that the social rate of discount might not be found by simply observing

11

Income across the subject pool does not vary enough to allow the use of this variable in estimation of the model,

and many students provide misleading answers to income questions, as they do not consider all sources if income

in their responses.

28

market rates of return on investment and that ethical considerations are involved (Dasgupta

2008; Stern 2006), while others largely believe that the rate should be tied to opportunity costs

of investment, as would be true in much conventional benefit-cost analysis (e.g. Becker,

Murphy and Topel 2010). An underlying assumption of this latter view is operation of

perfectly functioning economies.

In this study we have begun the empirical study of long-term discount rates to help

inform researchers and policy makers about preferences for long-term decisions, such as long-

term environmental improvement programs or climate change protection. We employed choice

exercises in an experimental setting with real monetary payouts to elicit utility discount rates

over short and longer-term horizons, 1 year and 20 years. We overcome the key difficulty of

credibly paying out future promised rewards by having individuals make choices that include

government savings bonds, the closest instrument to offering cash in twenty years, while

remaining easy to deliver to the subject.12

When using a constant discount rate function, we

estimate a discount rate of 4.9%, which is much lower than has been found in previous studies

that use short time horizons for the delays, such as weeks or months. However, we also find

strong support for declining discount rates when the time horizon lengthens, with an implied

annual discount rate (based on extrapolation) approaching 0.5% in 100 years. It is not easy to

compare our results with non-money tradeoff stated preference studies that base results on

purely hypothetical choices, but we note that some of these studies find low implied discount

rates for tradeoffs such as saving lives (reduced risks) or for different time periods for

12

An anonymous reviewer raised an interesting question in regards to the bond, “Is there a difference between

possessing the good and not possessing the good?” While we do not have an a priori expectation, the reviewer did

note that from a pure rate of time preference standpoint, it should not matter because the bond removes some of the

uncertainty or transaction costs associated with a cash payment in 20 years. It is worth considering this question as

laboratory experiments are refined to handle long-term horizons and possible confounds.

29

environmental restoration. Further, it has been noted and found in other studies that money

tradeoffs and non-money tradeoffs need not lead to the same discount rate. For example,

Alberini and Chiabai (2007) found the implied discount rate based on future mortality risk

reduction and money to be much lower (only 0.3 to 1.7%) than that based on money to money.

In our study we can only tangentially bring in preferences related to environmental or

public goods. As a convenient example of a long-term horizon decision we explore whether

different beliefs or attitudes about climate change impact individual discount rates. We do not

find consistent significant effects of climate change attitudes on estimated individual discount

rates, but do find evidence that individual attitudes about technology’s ability to solve future

environmental problems matter. In particular, those who believe a lower discount rate should

be used in discounting future benefits are likely the same who have less faith in technology to

solve future environmental problems.

This debate suggests that empirical studies of discount rates over long-term time

horizons might be appropriate and important in shedding light on how people really do value

the more distant future. Our finding of declining discount rates support other recent research

that has indicated that individuals may have time-inconsistent preferences, and declining

discount rates. As such, declining rates will give greater weight to benefits received in the

distant future of investments and costs incurred today. If, however, the ideal strategy for

making investments in climate change mitigation is to scale up mitigation overtime, then

lessons from the literature on hyperbolic discounting suggests the importance of making

commitments so as to prevent planners from succumbing to temptations to reevaluate plans as

time passes.

30

Finally, there is a growing consensus that for utility-theoretic reasons, risk and

intertemporal choice should not be examined independently (Andersen et al. 2008; Booij and

van Praag 2009; Gerber and Rohde 2010). The future is, by its very nature, unknown and

therefore risky. Thus, while we have included an estimate of the extent of diminishing marginal

utility in our modeling, the interface between risk and time can be further pursued, as the

possibility that both risk and impatience may affect intertemporal choices has become

increasingly evident, and risk perceptions may also play an important role in such choices (e.g.

Gerber and Rohde 2010).

Future research is warranted, as we do not tackle several thorny issues that might relate

to discounting climate change or other long-term environmental impacts. First, we note that

discounting or long-term investment experiments might yield different results for older aged

subjects than we used in our experiments, so future experiments might recruit from non-student

populations. Second, Hardisty et al. (2012) highlight the importance of differences in discount

rates that arise because of different methods to elicit and estimate them, and this is also featured

in Andreoni in Sprenger (2012a). Lastly, Dasgupta (2008) and Atkinson et al. (2009) note the

important components of an optimal consumption discount rate and how these relate to climate

change. They break apart concerns about equity between generations, following the Ramsey

(1928) definition of the rate, and specifically focus on the elasticity of the marginal utility of

consumption, as opposed to the pure rate of time preference. Future research might consider

structuring experiments that provide for such a decomposition. In addition, one might design

the variation in monetary outcomes across time horizons in any number of different ways,

testing for the robustness of the result we have obtained.

31

Acknowledgements

The authors thank Glenn Harrison and Greg Parkhurst for their careful and constructive

comments on an initial design of the experiment instrument, participants at seminars at the

University of Denver and Texas A&M University (Department of Psychology) for feedback,

Partha Dasgupta for his comments and sending a forthcoming paper of his, and Andrea

Galeotti, David Hardisty, Anthony Kwasnica, and Edward Morey for their comments on an

earlier draft of the manuscript. Two anonymous reviewers for this journal made very helpful

comments which have led to improvements in this paper. Grijalva acknowledges funding from

the Hemingway family. Shaw acknowledges funding from a U.S.D.A. Hatch Grant Project on

risk and uncertainty.

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38

Table 1: Twenty Three Choice Exercises

Choice

No.

Option A Option B Annual

Percentage

Rate

Percent

choosing A

1 $5 tomorrow $100 in 1 year 1900.00 9%

2 $25 tomorrow $100 in 1 year 300.00 29%

3 $50 tomorrow $100 in 1 year 100.00 53%

4 $75 tomorrow $100 in 1 year 33.33 71%

5 $95 tomorrow $100 in 1 year 5.26 86%

6 $5 tomorrow $100 in 20 years 16.16 61%

7 $25 tomorrow $100 in 20 years 7.18 71%

8 $50 tomorrow $100 in 20 years 3.53 79%

9 $75 tomorrow $100 in 20 years 1.45 88%

10 $95 tomorrow $100 in 20 years 0.26 92%

11 $5 tomorrow $500 in 20 years 25.89 29%

12 $25 tomorrow $500 in 20 years 16.16 41%

13 $50 tomorrow $500 in 20 years 12.20 52%

14 $75 tomorrow $500 in 20 years 9.95 62%

15 $95 tomorrow $500 in 20 years 8.66 70%

16 $5 tomorrow $1000 in 20 years 30.33 13%

17 $25 tomorrow $1000 in 20 years 20.25 21%

18 $50 tomorrow $1000 in 20 years 16.16 32%

19 $75 tomorrow $1000 in 20 years 13.83 49%

20 $95 tomorrow $1000 in 20 years 12.49 55%

21 $100 in 1 year $1000 in 20 years 12.20 45%

22 $100 in 1 year $500 in 20 years 8.38 71%

23 $100 in 1 year $200 in 20 years 3.53 87%

39

Table 2. Subject Responses to Their Reasoning for Intertemporal Choices

Not important at

all

Somewhat

important

Very

important

I thought I might be able to turn around and

invest the money I get tomorrow, at a better

rate than you offered with the Savings

Bond.

45a

(38%)

43

(36%)

31

(26%)

I thought I really needed the money

tomorrow, because I have an immediate

need for it.

49

(41%)

32

(27%)

38

(32%)

Twenty years is too long in the future for me

to think about.

54

(45%)

38

(32%)

27

(23%)

I thought I really didn’t trust being able to

get the money from the treasury security in

20 years

67

(56%)

40

(34%)

12

(10%)

I thought of giving the proceeds from the

bond in 20 years to my children, or to

someone else

48

(40%)

39

(33%)

32

(27%)

a The number of subjects selecting this indicator (percentages in parentheses)

40

Table 3. Basic Model Results

Parameters

Constant

Discounting

Model

Hyperbolic

Discounting

(Loewenstein,

and Prelec)

Hyperbolic

Discounting

(e.g., Mazur

and

Herrnstein)

Non-

Parametric

Discounting

Coefficient of Relative Risk Aversion, r,

where U(x) = x(1-r)

/(1-r)

0.510*a

(0.067)b

0.417*

(0.067)

0.445*

(0.071)

0.417*

(0.067)

Annual discount rate d, where D(t)= (1+d)-t

0.049*

(0.008)

a, where D(t)= (1+at)-b/a

0.492*

(0.144)

b, where D(t)= (1+at)-b/a

0.271*

(0.067)

g, where D(t)= (1+gt)-1

0.109*

(0.028)

D(t=1/365) 1

D(t=1) 0.803*

(0.036)

D(t=20) 0.270*

(0.046)

γi 110.300*

(56.556)

150.070*

(72.530)

150.320*

(73.788)

150.070*

(72.511)

γe 6.329*

(1.597)

7.943*

(1.921)

7.695*

(1.951)

7.943*

(1.921)

-2*LLF 2130 2110.2 2120.6 2110.2

LLF -1065 -1055.1 -1060.3 -1055.1

AIC 2138 2120.2 2128.6 2120.2

BIC 2149.1 2134.1 2139.7 2134.1

Number of People 119 119 119 119

Number of Choice Observations 2737 2737 2737 2737 a *implies statistical significance at the 0.05 level.

b Standard errors in parentheses.

41

Table 4. Model with Additional Demographic Explanatory Variables

Parameters Model 1 Model 2 Model 3

Coefficient of Relative Risk Aversion, r, where

U(x) = x(1-r)

/(1-r)

constant 0.416*a

(0.067)b

0.387*

(0.077)

0.468*

(0.09)

1=information; 0=no information -0.063*

(0.024)

-0.056*

(0.024)

1=TAMU; 0=WSU -0.106*

(0.03)

-0.127*

(0.03)

1=followup; 0=followup 0.061

(0.039)

0.048

(0.038)

1=male; 0=female -0.029

(0.027)

-0.018

(0.023)

Climate change will have negative

impact in 20 years (scale/100)

0.134*

(0.046)

0.151*

(0.04)

Technology will enable the future to

solve environmental problems

(scale/100)

-0.416*

(0.192)

Discount Function

D(t=1/365) 1 1 1

D(t=1)

constant 0.907*

(0.078)

0.873*

(0.084)

1.077*

(0.121)

1=information; 0=no information 0.007

(0.049)

-0.093

(0.054)

-0.075

(0.053)

1=TAMU; 0=WSU -0.023

(0.056)

-0.179*

(0.071)

-0.211*

(0.071)

1=followup; 0=followup 0.002

(0.065)

0.122

(0.089)

0.109

(0.090)

1=male; 0=female 0.026

(0.048)

-0.014

(0.058)

0.017

(0.054)

Climate change will have negative

impact (scale/100)

-0.219*

(0.083)

-0.058

(0.093)

-0.058

(0.087)

Technology will enable the future to

solve environmental problems

(scale/100)

0.288*

(0.121)

D(t=20)

constant 0.323*

(0.058)

0.282*

(0.062)

0.431*

(0.094)

42

1=information; 0=no information 0.028

(0.021)

-0.05

(0.027)

-0.035

(0.026)

1=TAMU; 0=WSU -0.010

(0.023)

-0.144*

(0.049)

-0.170*

(0.050)

1=followup; 0=followup -0.018

(0.027)

0.083

(0.058)

0.078

(0.058)

1=male; 0=female 0.018

(0.019)

-0.015

(0.032)

0.006

(0.026)

Climate change will have negative

impact (scale/100)

-0.108*

(0.036)

0.056

(0.055)

0.064

(0.045)

Technology will enable the future to

solve environmental problems

(scale/100)

0.214*

(0.071)

γi 150.07*

(73.077)

150.07

(78.91)

150.06

(81.409)

γe 7.891*

(1.909)

7.949*

(2.068)

8.094*

(2.17)

-2*LLF 2092.8 2061.4 2048.9

LLF -1046.4 -1030.7 -1024.45

AIC 2122.8 2101.7 2094.9

BIC 2164.4 2157 2158.8

Number of People 119 119 119

Number of Choice Observations 2737 2737 2737 a *implies statistical significance at the 0.05 level.

b Standard errors in parentheses.

43

Figure 1: Discount Factors

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20

Dis

cou

nt

Fact

or

Time (years)

Constant Discounting Model

Hyperbolic Discounting (Loewenstein and Prelec)

Hyperbolic Discounting (Mazur and Herrnstein)

Non-Parametric Discounting

44

Figure 2: Discount Rates

0%

5%

10%

15%

20%

0 5 10 15 20

An

nu

al D

isco

un

t R

ate

Time (years)

Constant Discounting Model

Hyperbolic Discounting (Loewenstein and Prelec)

Hyperbolic Discounting (Mazur and Herrnstein)

45

Appendix A

Table A1 presents models using stated bond cash-in dates and amounts (to be contrasted

with Table 3 which uses terminal, “objective” cash in dates and amounts for bond). Figures A1

and A2 present the discount factors and rates when using the stated bond cash-in dates and

amounts (to be contrasted with Figures 1 and 2 which are based on the terminal cash in dates).

In Table A2, we present the results for the constant discounting model, where we

interact the key shape coefficients or parameters of interest (discount factor and coefficient of

relative risk aversion) with climate questions, experimental treatments and demographic

variables. This is presented as a comparison to Table 4.

46

Table A1: Model Results Using Stated Bond Cash-in Dates and Amounts

Parameters Constant

Discounting

Model

Hyperbolic

Discounting

(Loewenstein,

and Prelec)

Hyperbolic

Discounting

(e.g., Mazur

and

Herrnstein)

Coefficient of Relative Risk Aversion, r, where

U(x) = x(1-r)

/(1-r)

0.485*

(0.064)

0.467*

(0.057)

0.471*

(0.054)

Annual discount rate d, where D(t)= (1+d)-t

0.044*

(0.007)

a, where D(t)= (1+at)-b/a

0.258+c

(0.141)

b, where D(t)= (1+at)-b/a

0.147*

(0.050)

g, where D(t)= (1+gt)-1

0.086*

(0.019)

D(t=1/365)

D(t=1)

D(t=20)

σi 183.860*

(89.338)

150.13*

(61.015)

150.13*

(56.069)

σe 6.341*

(1.519)

6.267*

(1.297)

6.251*

(1.218)

-2*LLF 1459.9 1441.3 1444.4

LLF -729.95 -720.65 -722.20

AIC 1467.9 1451.3 1452.4

BIC 1479.1 1465.2 1463.5

Number of People 119 119 119

Number of Choice Observations 2737 2737 2737 a *implies statistical significance at the 0.05 level;

+ implies statistical significance at the 0.10 level

b Standard errors in parentheses.

cNote: in the hyperbolic discounting model, the coefficient a is significantly different from zero at the

p=0.07 level; if a is zero, the model reverts back to constant discounting.

47

Figure A1: Discount Factor Using Stated Bond Cash-in Dates and Amounts

Figure A2: Discount Rates Using Stated Bond Cash-in Dates and Amounts

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20

Dis

cou

nt

Fact

or

Time (years)

Constant Discounting Model

Hyperbolic Discounting (Loewenstein and Prelec)

Hyperbolic Discounting (Mazur and Herrnstein)

0%

5%

10%

15%

20%

0 5 10 15 20

An

nu

al D

isco

un

t R

ate

Time (years)

Constant Discounting Model

Hyperbolic Discounting (Loewenstein and Prelec)

Hyperbolic Discounting (Mazur and Herrnstein)

48

Table A2: Constant Discounting Model with climate questions, experimental treatments and

demographic variables

Parameters

Model 1 Model 2 Model 3

Coefficient of Relative Risk Aversion, r, where

U(x) = x(1-r)

/(1-r)

constant 0.520*a

(0.066)b

0.477*

(0.079)

0.469*

(0.086)

1=information; 0=no information -0.028

(0.023)

-0.028

(0.023)

1=TAMU; 0=WSU -0.054*

(0.026)

-0.056*

(0.026)

1=followup; 0=followup 0.021

(0.032)

0.022

(0.032)

1=male; 0=female -0.014

(0.022)

-0.015

(0.023)

Climate change will have negative

impact in 20 years (scale/100)

0.142*

(0.04)

0.145*

(0.041)

Technology will enable the future to

solve environmental problems

(scale/100)

0.010

(0.051)

Discount Function

constant 0.044*

(0.009)

0.051*

(0.01)

0.049*

(0.011)

1=information; 0=no information -0.005

(0.003)

0.001

(0.004)

-0.001

(0.004)

1=TAMU; 0=WSU 0.002

(0.003)

0.010*

(0.005)

0.010*

(0.005)

1=followup; 0=followup 0.002

(0.004)

-0.001

(0.006)

-0.001

(0.006)

1=male; 0=female -0.002

(0.003)

0.001

(0.004)

-0.001

(0.004)

Climate change will have negative

impact (scale/100)

0.009

(0.005)

-0.013+

(0.007)

-0.013+

(0.007)

Technology will enable the future to

solve environmental problems

(scale/100)

0.002

(0.006)

0.003

(0.009)

σi 100.19

(50.838)

100.19

(56.776)

100.19

(56.784)

49

σe 6.085*

(1.516)

5.934*

(1.642)

5.942*

(1.643)

-2*LLF 2120.9 2095.8 2095.2

LLF -1060.45 -1047.9 -1047.6

AIC 2140.9 2123.8 2127.2

BIC 2168.7 2162.8 2171.7

Number of People 119 119 119

Number of Choice Observations 2737 2737 2737 a *implies statistical significance at the 0.05 level; + implies significance at the 0.10 level.

b Standard errors in parentheses.