Patience Auctions: The Impact of Time vs. Money Bidding on Elicited Discount Rates
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Patience Auctions
Patience Auctions:
The Impact of Time vs. Money Bidding on Elicited
Discount Rates
Christopher Y. Olivola
Warwick Business School
Stephanie W. Wang
California Institute of Technology
October 2010
Abstract
We introduce two auction-based methods for eliciting discount rates. In these “patience
auctions”, participants bid the smallest future sum they would prefer -or- the longest time
they would wait for a reward, rather than receive a smaller, immediate payoff. The most
patient bidder receives the delayed reward; all others receive the immediate payoff. These
auctions, in addition to offering certain potential methodological advantages, allow us to
compare discounting when participants’ attention is focused on the temporal vs. monetary
dimension of delayed rewards. We find that the best-fitting model and rate of discounting
differ between these two theoretically equivalent bidding methods.
Acknowledgements
We would like to thank Abhijit Banerji, Roland Benabou, Colin Camerer, John Doyle, Jens Großer, David
Hardisty, Joel Huber, Eric J. Johnson, Nobuo Koida, J.V. Meenakshi, Daniel Mochon, C. Dan Myers,
Thomas Palfrey, Uliana Popova Makarov, Eldar Shafir, Rohini Somanathan, Dean Spears, and an
anonymous reviewer from the Russell Sage Foundation for their helpful comments and suggestions. We
also received helpful feedback from participants of Yale Whitebox 2008, ESA 2008, SJDM 2008, BDRM
2008, ACR 2008, SCP 2009, SPUDM 2009, and the Center for Development Economics seminar at the
Delhi School of Economics .We are especially indebted to Chris Crabbe, who did a spectacular job
programming the experiment. This research was generously funded by a grant from the Russell Sage
Foundation’s small grants program in Behavioral Economics and an award from the Society for Judgment
and Decision Making.
1Patience Auctions
1. Introduction
How many years of schooling is someone willing to invest in order to obtain a
certain monetary reward (e.g., the salary associated with a medical degree)?
Alternatively, if one had to invest 4 years of schooling, what delayed reward would be
necessary at completion to not drop out (e.g., what is the expected salary for a college
graduate)? Many of our most consequential choices involve (at least implicit) tradeoffs
between present and future rewards (or punishments). Sometimes we make choices along
the time dimension (e.g., how many years to invest in schooling given the rewards
associated with various levels of education?) and sometimes we make choices along the
reward dimension (e.g., if four years are necessary to obtain a college degree, is the
expected salary increase sufficient to compensate for this investment in time?). Given its
undeniable impact on health, wealth, and happiness (e.g., Reimers et al., 2009), as well as
its relation to impulsivity and self-control, it is no surprise that intertemporal decision
making has attracted interest, not only within economics, but also among psychologists
(e.g., Killeen, 2009; Pronin et al., 2008; Scholten and Read, 2010), neuroscientists (e.g.,
Ersner-Hershfield et al., 2009; McClure et al., 2004), and even philosophers (e.g., Elster,
1984; Parfit, 1984).
Discount rates are important economic indicators of the way people trade off
consumption in the present and future. However, measuring discount rates is a
methodological challenge. The majority of current measurement procedures (e.g., choice
and matching) suffer from important weaknesses: either they only provide bounds on the
discount parameter (rather than a specific value) or they cannot realistically be carried out
with real money, limiting their ecological validity and giving participants no incentives to
provide accurate answers. In contrast, auction-based approaches can overcome these
limitations by providing incentive-compatible means to elicit discount rates. For example,
in second-price private-value auctions, the dominant strategy is for each bidder to bid her
true value for the good regardless of what the other bidders do. We introduce, test, and
compare two novel auction-based experimental methods for eliciting discount rates. In
these “patience auctions”, participants bid the smallest sum they would prefer receiving
in the future (money bid) -or- the longest time they would prefer waiting for a reward
2Patience Auctions
(time bid), rather than receive a smaller, immediate payoff. Only the most patient bidder
receives the delayed reward; all other bidders receive the smaller, immediate payoff.
These auctions offer important advantages over other incentive-compatible methods of
elicitation, including alternative auction designs and the Becker–DeGroot–Marschak
(BDM) procedure, which we discuss.
Despite their methodological advantages, however, our main interest in these
auctions has more to do with the fact that they allow us to examine important questions
about the determinants of discounting. As the old adage goes: “Time is money.” The
primary goal of this paper is to test this idea, to see whether people will indeed trade off
time and money in an economically consistent manner. Specifically, we compare how
discounting varies depending on whether the auction focuses participants’ attention on
the temporal or monetary dimension of delayed rewards. By having participants take part
in both the money-bid and time-bid auctions we can draw within-subject comparisons,
thereby controlling for bidder-specific fixed effects. We find that the best-fitting discount
function is often different in the money-bid vs. time-bid auctions. Furthermore, we find
that the estimated discount parameters in the three most commonly used discount
functions (exponential, hyperbolic, and quasi-hyperbolic) are significantly different
across the two bidding types. Finally, we compare the relative merits of using second-
price versus first-price auctions and conclude that, in practice (but contrary to auction
theory), first-price auctions provide more coherent estimates of the discount rates than
their second-price equivalents. Our paper is organized as follows. Section 2 reviews the
related literature on measuring discount rates, introduces our patience auction
mechanisms, and reviews evidence that people treat time and money very differently.
Section 3 describes the experimental design and procedures. We present the econometric
specifications in Section 4 and the results of our estimations in Section 5. Section 6
discusses the implications of our experimental results and future directions.
2. Measuring Discount Rates: Related Literature and Proposed
Mechanisms
3Patience Auctions
Originally introduced by Samuelson (1937) in his seminal paper on the discounted utility
model, discount rates describe how people trade off consumption in the present vs. future.
Discount rates can be thought of, essentially, as economic measures of impatience.
However, measuring discount rates has proven to be difficult and the method of
measurement itself has been shown to affect reported discount rates (Frederick et al.,
2002). Frederick et al. (2002), in a critical review of the literature on discounting,
compared several experimental methods commonly used to elicit discount rates and
argued that the two most popular procedures, choice and matching, suffer from important
weaknesses - either they only provide bounds on the discount parameter (rather than a
specific value) or they cannot realistically be carried out with real money, limiting their
ecological validity and giving participants no incentives to provide accurate answers. In
the choice procedure, participants choose between pairs of options: a smaller, more
immediate payoff and a larger, delayed payoff. While this method can be realistically
implemented using real money, it only identifies upper or lower bounds on a person’s
discount rate, and thus fails to provide point estimates. Furthermore, even though
participants could be given choices involving real payoffs and delays, this procedure can
become expensive and complicated to carry out, especially when payoffs and delays are
large or many participants are recruited (e.g., imagine the case where 75 out of 100
recruited participants choose the larger, delayed option when given a choice between $10
now and $100 in 1 year). Although the Multiple Price List method (Andersen et al., 2006;
Meier and Sprenger, 2010) can be used to measure point estimates, it is, like the choice
procedure, expensive to carry out if every participant is paid (especially if several receive
the larger, delayed payoff). More importantly, it does not allow us to compare the effects
of eliciting preferences along the time vs. money dimension.
In the matching procedure, participants are given a pair of options like the ones
used in the choice task, except that one of the parameters -either length of time or amount
of payoff- in one of the options is missing. Participants must then indicate the size of the
missing delay or payoff that would make them indifferent between the two options. If
participants respond honestly then in theory this method should provide a point estimate
of the discount rate for each person. However, participants have no incentives to
deliberate or to respond truthfully using this method. In fact, the opposite is true when
4Patience Auctions
real money is involved and participants are given a chance to receive one of the options.
In this case, they have an incentive to distort their discount rates in order to maximize the
present value of their earnings. Participants who only care about their own discounted
payoffs should report being infinitely impatient (i.e., by choosing an arbitrarily large
payment or an arbitrarily small delay). For this reason, the matching procedure is neither
incentive compatible nor feasible to carry out with real money.
Despite their important shortcomings, choice and matching are the most
frequently used methods of discount rate elicitation. Frederick et al. (2002) provided a
table summarizing the methods used to measure discount rates in articles published
between 1978 and 2002 (see their Table 1, p. 378). Using this table, we measured the
relative popularity of various discount elicitation procedures. The results of this analysis
are summarized in (our own) Table 1. Out of the 42 papers surveyed, 22 (52%) used the
choice procedure and another 13 (31%) used the matching procedure. The choice and
matching procedures were by far the most popular approaches, and together they
accounted for 83% of the methods used (the third most popular approach, pricing, was
only used in 3 papers, or 7% of the sample). The inhibitive costs associated with these
methods are evident when we look at the proportion of studies that involved real (rather
than hypothetical) payoffs. Across the entire sample, only 17 studies (40%) had
participants making decisions about real goods. Of the 22 studies using the choice
procedure, 13 (59%) involved real choices. Not surprisingly, the proportion of matching
studies using actual payoffs was much lower: only 3 out of 13 studies (23%) involved
real tradeoffs. In fact, of all the papers summarized in this table, only one (Kirby, 1997)
involved a procedure that simultaneously provided point estimates, offered real payoffs,
and used an incentive-compatible design. We discuss this paper below.
5Patience Auctions
Table 1. Methods used to elicit discount rates (19782002)
Note: This table was created using data from Table 1 in Frederick, Loewenstein, and
O’Donoghue (2002, p. 378). Numbers outside the brackets are the proportion of papers
that used a specific method. Numbers inside the brackets are the proportion of papers that
used a specific method and involved real (rather than hypothetical) payoffs. The right
column focuses on studies in which money was the only payoff under consideration.
The general failure, in the empirical literature, to use elicitation methods that
satisfy these conditions should make economists skeptical about drawing serious
conclusions concerning the distribution of individual discount rates. Indeed, as Frederick
et al. (2002) found, the mean discount rates that have been observed in these studies span
a very large range, including values that are hard to explain (e.g., values that imply
negative discounting or infinite impatience). Furthermore, they found that this range has
not gotten any smaller over the years, leading them to conclude that there has been “no
evidence of methodological progress” (p. 377) with regard to the measurement of
discount rates.
2.1 Auction Mechanisms
To overcome the limitations of current methods, we suggest an auction-based approach
designed to elicit discount rates in an incentive-compatible way (i.e., giving participants
an incentive to accurately report their discount rates). Vickrey (1961) showed that in a
second-price private-value auction, it is a dominant strategy for each bidder to bid her
true value for the object regardless of what the other bidders do. In our experiment, the
assumption of a private-value auction is a valid one, especially given the lack of a
6Patience Auctions
secondary market for the future reward. Each bidder’s valuation of the future reward
relative to the immediate gain depends on her discount rate only and not those of the
other bidders. Therefore the bidders should, through their bids, provide accurate and
exact measurements of their willingness to trade off smaller immediate gains with larger
delayed ones. Nevertheless, overbidding has been documented in laboratory second-price
auctions (Heyman et al., 2004; Kagel et al., 1987; Kagel and Levin, 1993). This
phenomenon could be due to a myriad of factors ranging from a failure to understand the
dominant bidding strategy to “auction fever” over the possibility of “winning” (Delgado
et al., 2008). In the Experimental Design and Procedures section, we discuss the steps we
take to minimize these factors.
There have been very few auction-based elicitations of discount rates (Horowitz
1991; Kirby 1997; Kirby and Marakovic 1995; Manzini et al., 2008). Horowitz (1991)
used a sealed, first-rejected-price auction in which students bid their own money on $50
bonds that matured in 34 or 64 days. While this auction mechanism does theoretically
eliminate under- and over-bidding, the design does suffer, as Kirby (1997) points out,
from a number of limitations. For example, Horowitz reports that both the mean and
median willingness-to-pay bid were higher for the bond with the larger delay –a curious
result that may be due to the group average rather than individual approach to estimating
the discount rates (see Kirby, 1997). Kirby and Marakovic’s (1995) (simulated) sealed,
low-bid “wins” auction only reduced rather than eliminated incentives to overbid, so the
bids that they observed still cannot be taken as reliable estimates of the subjects’
discounted value of delayed rewards. It should also be noted that the subjects in their
study were led to believe that they were bidding against other human subjects, when in
fact they were (unbeknownst to them) actually bidding against a computer1.
To our knowledge, Kirby’s (1997) paper is the only published one to use the
second-price auction mechanism. However, his auction design contained several potential
flaws, the most significant being that subjects were asked to bid their own money for a
delayed reward that varied in amount. Such a design limits the size of the delayed payoffs
that subjects can be asked to bid for, as they cannot bid as much as they would like for a
1
The computer randomly determined whether the bidder “won” the auction with a probability inversely
related to bid size.
7Patience Auctions
delayed payoff whose private value exceeds their personal budget. Asking subjects to bid
their own money may also engender suspicion about the nature of the task, leading some
to be overly conservative with their bidding. In fact, Kirby (1997) reported that one of his
subjects explicitly declared that he thought the experiment was a scam (i.e., an attempt by
the experimenter to extract money from the subjects) and refused to bid more than $0 on
any trial. Kirby’s auction design also likely engendered a different mindset than our
auctions. Participants in his auctions were paying for a larger, delayed reward, rather than
explicitly trading-off a smaller immediate reward with a larger delayed one (as they do in
our auctions). Although this distinction should be irrelevant from a standard economic
perspective, research shows that it does matter in reality (Carmon and Ariely, 2000;
Kahneman et al., 1990). Participants in Kirby’s study might have been hesitant to part
with their own money, and their bids could thus have been distorted by loss aversion
(Kahneman and Tversky, 1979; Tversky and Kahneman, 1991). Finally, the longest delay
used in his experiment was only 30 days (1 month).
The only other paper we are aware of that uses second-price auctions is an
unpublished study by Manzini, Mariotti, and Mittone (2008). Their method, like Kirby’s,
differs from ours in several ways. First, whereas our mechanism involves bidding on the
larger, later payoff, theirs involves bidding on the smaller, sooner payoff. Specifically,
the subjects in their study bid the lowest amount they would be willing to receive in a day
(i.e., tomorrow) instead of a larger amount (either 20 or 50) in the more distant future
(either 1, 2, or 4 months). Second, their auctions only had two levels of delayed payoffs
(20 and 50) and three levels of delays (1, 2, and 4 months). The study we report
involves a larger range of payoffs and delays. Third, only half of their subjects (drawn at
random) were actually paid according to their bids and had to incur a delay (one day for
the winning bidder and 1-4 months for all other randomly drawn bidders). This was likely
due to the fact that paying all subjects (except the winning bidder) the higher amount and
keeping track of them over a period of 1-4 months is rather costly in terms of time and
money. In contrast, our elicitation method allows us to pay all of our subjects according
to their bids, without incurring a large cost.
In the next section, we introduce a pair of auction designs that overcome the
limitations of previous auction-based studies and those associated with standard methods
8Patience Auctions
of discount rate elicitation. To avoid problems with unobserved reference points, we
clearly induce the tradeoff between immediate gain and delayed reward (see the
Experimental Design and Procedures section for details). Secondly, while Kirby (1997)
focuses on fitting individual bids to exponential vs. hyperbolic discounting functions and
Manzini et al. (2008) compare different elicitation mechanisms, our study mainly aims to
determine whether individuals are consistent in discounting the future when bidding with
time vs. money. To the best of our knowledge, no published research has directly
compared (im)patience when people focus on the temporal vs. monetary dimension,
particularly under incentive-compatible conditions.
2.2 The Money-bid and Time-bid Auctions
There are four parameters of interest in standard experimental tests of discounting: the
size of the earlier, smaller payoff (m0), the size of the later, larger payoff (md), the length
of the delay until the earlier payoff (t0), and the length of the delay until the later payoff
(d). Proper elicitation of the discount rates requires that we have m0 < md and t0 < d. Like
many previous studies, we make the earlier payoff option available at the end of the
experimental session (t0 0)2. This then leaves us with 3 parameters that we can
manipulate: m0 , md , and d. We propose a pair of novel sealed-bid single-round (first-
price or second-price) auction-based methods for eliciting discount rates in which
participants bid for the value of one of the latter two parameters (md or d). In the “money-
bid auction”, m0 and d are known beforehand, and participants bid for md, the size of the
later option. The person providing the lowest bid “wins” the auction and receives, after a
delay of d, a payoff equal to her bid (first-price auction) or the second-lowest bid
(second-price auction). All other participants receive m0 at the end of the experiment.
Thus, according to standard auction theory, the person who is willing to accept the
smallest delayed payoff should “win” the auction (barring heterogeneity of beliefs about
the distribution of preferences in the first-price auction). In the “time-bid auction”, m0
and md are known beforehand, and participants bid for d, the length of the delay. The
person bidding the longest delay obtains md after a delay equal to her bid or the second-
2
Our auctions also allow us to set t0 > 0, in which case subjects would be trading off two delayed payoffs.
However, doing so would make the auctions costlier to run since we would need to keep track of all
subjects over an extended period.
9Patience Auctions
largest bid. All other participants receive m0 at the end of the experiment. Here, the
person willing to wait the longest to receive the delayed payoff should “win” the auction
(with a similar caveat concerning beliefs about other bidders for the first-price auction).
The auction-based methods we propose offer some important advantages over
standard procedures for eliciting discount rates. First, they provide incentive-compatible
ways to measure individual discount rates. The subjects’ bids, combined with the value of
the predetermined parameters, can be used to calculate their discount rates3. Second,
these methods are both time and cost-efficient. Using these auctions, we can
simultaneously measure the discount rates of multiple participants in a single session
without having to incur large costs (in contrast, for example, to the auctions used by
Manzini et al., 2008). Although every person bids, and thus reveals his or her discount
rate (or a close approximation), only one participant in each session receives the larger,
delayed amount, with all other participants receiving the earlier smaller amount. Finally,
and in contrast to Kirby’s (1997) auction method, the participants in our auctions do not
bid their own money, so the size of the delayed payoff is constrained only by the
experimenter’s budget, not the participants’.
This method also offers several important advantages over another popular
incentive-compatible method for eliciting preferences: the Becker-DeGroot-Marschak, or
BDM, mechanism (Becker et al., 1964)4. As typically carried out, BDM requires that the
seller (i.e., the experimenter) specify, beforehand, a continuous range of plausible values
(e.g., time or money) that subjects can choose from to indicate a buying price for a good.
In the case of intertemporal choice, subjects could choose a value from this range to
indicate the delay or payoff that they are willing to accept for the later reward. A value
would then be randomly drawn from this range and participants would receive the
delayed payoff if they were willing to (i) wait longer than the randomly selected delay
length or (ii) receive a payoff smaller than the randomly selected monetary amount. The
first disadvantage of this approach is that the range imposes an upper bound on the
3
In the first-price auction, bidders will, in theory, have an incentive to over-bid in money-bid auctions and
under-bid in time-bid auctions. However, as the number of bidders increases, the tendency to bid away
from one’s true value should decrease until the difference between “true” preferences and actual bids
approaches zero. Therefore, with enough bidders, first-price patience auctions should provide a close
approximation of the “true” implied discount rate.
4
For other discussions on the limitations of BDM, see Bohm et al. (1997) and Noussair et al. (2004).
10Patience Auctions
amount of time or money that participants can bid. Thus, participants who would be
willing to bid above the specified range would not reveal their “true” preferences.
Second, specifying a range of values might send a strong signal to subjects that there is
an “acceptable” or “rational” range of bid values. This may lead them to adjust their bids
accordingly, thus distorting their revealed preferences (Asch, 1956; Gneezy and
Rustichini, 2000). Indeed, Bohm et al. (1997) found that seller prices elicited by the
BDM procedure are sensitive to the upper bound chosen for the randomly generated
buyout prices, thereby hindering its incentive compatibility. The presence of a range may
also make participants more likely to rely on naïve heuristics when choosing bid values,
such as anchoring their bid on the top of the range (Tversky and Kahneman, 1974) or
picking a value that falls in the middle of the range. Numerous studies have shown that
the particular range of a response scale can have an important impact on the responses
provided (Sudman et al., 1996; Tourangeau et al., 2000). In a direct comparison of BDM
and second-price auctions, Noussair et al. (2004) found the latter to be a more effective
mechanism for value elicitation. In particular they found that “the BDM is subject to
more severe biases, greater dispersion of bids, and slower convergence to truthful
revelation than the Vickrey auction.” (p. 739).
One solution to these problems is to modify the standard BDM procedure5 so that,
instead of specifying a range, the seller chooses a secret selling price, beforehand, that is
not revealed to the buyer (e.g., by writing it on a sheet of paper that the buyer cannot see)
until she offers a buying price. Once the bid is offered, the value of the secret selling
price is revealed and the buyer receives the later reward only if her time (money) bid is
higher (lower) than or equal to the secret value chosen by the seller, in which case the
delay (payoff) of the later reward is the time (money) previously chosen by the seller.
Otherwise, she receives the immediate, smaller payoff. This procedure provides an
incentive-compatible and range-free approach to eliciting discount rates, as the optimal
strategy (barring peculiar beliefs about the seller’s intentions or preferences6) is to bid
5
We thank Daniel Mochon for bringing this modification of the standard BDM procedure to our attention.
6
One (more) potential limitation of this modified BDM procedure is the possibility that participants will
expend mental effort trying to infer what secret selling value the seller (or experimenter) is likely to
propose, rather than focusing on their own private values. As a result, they may modify their bids (away
from their “true” private values) to the extent that they have preconceptions about the seller’s motivations
or preferences.
11Patience Auctions
one’s “true” willingness to trade off time and money. However, this approach also shares
BDM’s third methodological weakness: it is much more costly --in terms of both the
experimenter’s time and her money-- to carry out than the patience auctions. Patience
auctions allow experimenters to simultaneously elicit discount rates from multiple
participants while only offering the delayed payoff option --which is more costly for the
experimenter, in terms of time spent tracking participants and money paid to participants-
- to a single bidder. In contrast, both the standard and modified BDM procedures allow
multiple participants to obtain the delayed payoff, which means experimenters will have
to allocate more resources to keep track of participants who obtained the delayed option
(in order to make sure that they are paid at the end of the delay) and pay larger sums to
compensate these participants. Compare, for example, the expected costs of using
patience auctions versus BDM procedures to obtain data from 50 participants, each
bidding for a later reward that either has to be paid in a year or, alternatively, whose
payoff amount is $100. The BDM procedure is also quite difficult to explain and harder
still for participants to understand (e.g., Bohm et al., 1997). First-price auction rules are,
by contrast, familiar to most participants and relatively easy to comprehend. Finally, it
should be noted that the modified BDM procedure is, in many ways, like a special case of
an auction mechanism; namely, a two-bidder auction. Indeed, Manzini et al. (2008) found
no significant differences between discount rates elicited with a BDM vs. second-price
auction mechanism. Thus, there are empirical priors suggesting that our results would
replicate with a BDM mechanism (once its methodological limitations are overcome).
Beyond their methodological advantages, these auctions allow us to gain
important insights into some of the cognitive factors that affect intertemporal decision
making. Specifically, by comparing the level of discounting that occurs when bids are in
terms of delay versus payoff, we can see whether people consistently trade off time and
money.
2.3 Evidence of Differential Thinking about Time vs. Money
Benjamin Franklin famously pointed out that “time is money.” This wise adage has been
formalized in economics, where time and money are often considered exchangeable
goods. For example, Becker (1965) proposed a theory of time-allocation in which the
12Patience Auctions
value of time was equal to its opportunity cost, which could be approximated by a
person’s after-tax wage rate. Thus, according to most economic models, decision makers
should think of time and money as equivalent and be able to easily trade off these two
resources. As a result, the decisions we make about a given length of time should be
identical to our choices concerning an equivalent amount of money.
However, several studies have found differences in the way people think and
make decisions about time versus money. Leclerc et al. (1995) found that, whereas
people tend to be risk-seeking with respect to decisions involving monetary losses (see
Kahneman and Tversky, 1979), they tend to be risk-averse with respect to decisions
about waiting (a loss of time). Soman (2001) found that people do not show a sunk-cost
effect7 for past investments of time, as they tend to do with past monetary investments
(although, see Navarro and Fantino, 2009, for evidence that people do show a sunk-cost
effect of time). Okada and Hoch (2004) found that, although people are less willing to
invest money in high-risk, high-return options (compared to low-risk, low-return
options), they are willing to invest more time in high-variance than in low-variance
options. Zauberman and Lynch (2005) found that future investments of time are
discounted more steeply than similarly delayed monetary investments. Duxbury et al.
(2005) replicated the previously shown tendency for people to value relative, rather than
absolute, savings, when they must spend time to save money, but found that this tendency
disappears when they must spend money to save time. Saini and Monga (2008) show that
decisions about spending time are more prone to certain cognitive biases than decisions
about spending money. Finally, Liu and Aaker (2008) found that thinking about donating
time to a charity increases subsequent willingness to donate (either time or money), while
thinking about donating money decreases subsequent willingness to donate.
A number of accounts have been proposed to explain why we treat time and
money differently. Some authors (e.g., Leclerc et al., 1995) have stressed the fact that
time is much less fungible than money (unlike money, we cannot easily save, borrow,
lend, or otherwise transfer time). Others have pointed to a greater ambiguity in the value
of time, compared to money, for most people (Okada and Hoch, 2004; Saini and Monga,
7
The “sunk-cost effect” is the irrational tendency for people to base their current and future decisions on
past, non-recoverable costs. This tendency has been demonstrated widely in the psychology and economic
literature (e.g., Arkes and Blumer, 1985; Garland, 1990; Thaler, 1980).
13Patience Auctions
2008; Soman, 2001; see also DeVoe and Pfeffer, 2007) or to different expectations about
access to time or money, in the present and future (Zauberman and Lynch, 2005).
Given the existing evidence, there may be reasons to expect money-bid-elicited
discount rates to differ from time-bid-elicited discount rates, although exactly how is not
clear, since none of the preceding studies have addressed this specific question. While
eliciting discount rates through matching (i.e., measuring indifference points) along the
payoff dimension is a relatively common procedure in the intertemporal choice literature,
only a handful of studies have elicited discount rates through matching along the time
dimension (e.g., Attema et al., 2009; Chesson and Viscusi, 2000; Green et al., 1994;
Mazur, 1987; Takeuchi, 2008). In fact, we only know of one previous attempt to directly
compare the discount rates obtained when subjects focus on the monetary vs. temporal
dimension of the delayed payoff: an unpublished dissertation by Roelofsma (1994, cited
in Roelofsma, 1996 and in Frederick et al., 2002). Roelofsma’s (1994) results suggest
that discount rates are much higher with time matching than with payoff matching (a
result that we replicate when fitting the standard exponential discounting model).
However, as far as we know, he neither used an incentive-compatible elicitation
mechanism, nor fitted models other than standard exponential discounting. Therefore, it
is still an open question whether (and how) focusing on the time vs. money dimension
will impact discounting under incentive-compatible elicitation conditions.
3. Experimental Design and Procedures
We conducted 4 experimental sessions with 15 participants in each session and
administered surveys to 30 additional participants, all in the Fall of 2007. The
participants were all registered undergraduate students at Princeton University who were
recruited via email announcements. All sessions were conducted in a computer lab and all
interactions took place through a multistage (http://multistage.ssel.caltech.edu) interface
on networked computers. No one participated in more than one session. The treatment
variables in this 22 design were auction mechanism: first-price vs. second-price auction,
and bid type: bidding money vs. bidding time.
14Patience Auctions
We ran single round (one-shot) sealed bid (concealed response) auctions of two
bid types in each session: the money-bid auction and the time-bid auction. In each of the
four sessions, participants bid in eight money-bid auctions and eight time-bid auctions. In
two of the sessions all the auctions were first-price auctions and in the other two sessions
they were all second-price auctions. The difference between these two auction rules will
be explained below. Furthermore, we varied the ordering of the bid type: half the sessions
involved eight money-bid auctions followed by eight time-bid auctions, while the other
half involved eight time-bid auctions followed by eight money-bid auctions. Bid type
order was manipulated in this way for both first-price and second-price auctions, so as to
control for possible order effects. Thus, the bid type (money vs. time) was a within-
subject manipulation, whereas the auction mechanism (first-price vs. second-price) was a
between-subjects manipulation. All other conditions were kept constant across the four
sessions. Table 2 summarizes the experimental design.
Method # of Choices N
First-Price Auction 8 money bid, then 8 time bid 15
First-Price Auction 8 time bid, then 8 money bid 15
Second-Price Auction 8 money bid, then 8 time bid 15
Second-Price Auction 8 time bid, then 8 money bid 15
Hypothetical Matching 8 money maching, then 8 time matching 15
Hypothetical Matching 8 time matching, then 8 money matching 15
Table 2. Summary of Experimental Design
After entering the laboratory, participants were given an instruction packet and an auction
agreement form before being seated at workstations with dividers (seating assignment
was randomized). Unless otherwise specified, subjects were informed about all the
features of the auctions in which they participated. Sample instructions, sample
screenshots, the auction agreement form, and survey questions can be found in the
Appendix. Before the instruction period, participants first signed the agreement form,
stating that they understood the possibility of receiving a delayed payoff in the future
rather than a payoff at the end of the session, depending on the auction outcome. Next
came the instruction period in which the instruction packet was presented verbally with
15Patience Auctions
the assistance of a projected visual presentation. In order to make sure that subjects
understood the rules and payoff structure of each auction, they were given both verbal
and written instructions. Participants were first given a complete description of single
round sealed bid auctions in general and told how the outcome is determined by the
auction mechanism, either first-price or second-price depending on the session. They
were also told exactly what information feedback they would receive and how their
overall earnings would be determined.
They were given the rules for the first block of eight auctions they would be
taking part in, either money-bid auctions or time-bid auctions depending on the session.
They were also familiarized with the computer interface through which they would make
all of their decisions by viewing sample screens. After the instructions for this first block
of eight auctions were covered, participants had to complete a short comprehension quiz
and they could not proceed to the actual auctions until all answers were correct. After the
eight auctions of one bid type concluded, participants were instructed on the rules of the
next eight auctions with the other bid type and again familiarized with the corresponding
computer interface. Participants again completed a short comprehension quiz before they
could proceed to the actual auctions. In each auction, a participant could either receive
$10 at the end of the session or a delayed payoff sometime in the future, as determined by
the bidding process.
In each money-bid auction, the length of delay for the future payoff was pre-set
and participants simultaneously bid the monetary amount for that payoff. They did so by
entering numbers into respective boxes for dollars and cents then clicking the “submit”
button (see sample screenshot in the Appendix). The lowest bidder obtained the bid-
determined payoff at the end of the pre-set delay period and all other bidders received
$10 at the end of the session. In the first-price auction, the bid-determined payoff was the
lowest bid whereas in the second-price auction, the bid-determined payoff was the second
lowest bid.
In each time-bid auction, the monetary amount for the future payoff was pre-set
and participants simultaneously bid the length of delay for that payoff. They did so by
entering numbers into the respective boxes for years, months, weeks, and days, then
pressing the “submit” button (see sample screenshot in the Appendix). The participants
16Patience Auctions
were allowed to use any combination of the boxes to place their bids. The only restriction
was that they had to use at least one of the boxes to submit their bid. However, which box
or boxes they chose to use and how many boxes they chose to fill-in was entirely up to
them. We implemented this flexibility to minimize possible anchoring or framing effects
(Kahneman and Tversky, 1979; Tversky and Kahneman, 1974). The highest bidder
obtained the pre-set payoff at the end of the bid-determined delay period and all other
bidders received $10 at the end of the session. In the first-price auction, the bid-
determined length of the delay was the highest bid, whereas in the second-price auction it
was the second highest bid.
Table 3 contains all the pre-set values that were used in each session. We
randomized the order of the eight pre-set delay lengths for the money-bid auctions and
the order of the eight pre-set delayed payoff amounts for the time-bid auctions. This order
was held constant across the four sessions. The parameters are presented from lowest to
highest in Table 3, which does not correspond to the order used in the sessions.
Money Bid: Time Bid:
Pre-set delay d Pre-set delayed payoff md
1 day $15
3 days $20
1 week $25
2 weeks $30
1 month $50
3 months $75
6 months $90
1 year $100
Table 3. Experimental parameter values for money-bid and time-bid auctions
Each participant submitted a single bid in each auction. They were given no
feedback about other participants’ bids or the outcome of each auction (neither the
winning bid nor the identity of the winning bidder was revealed at any point, except to
the winning bidder). All subjects were informed of this strict anonymity policy at the
very beginning of the experiment. This precaution was taken in order to minimize
“auction fever” (i.e., the tendency to overbid in time-bid auctions or to underbid in
17Patience Auctions
money-bid auctions). They were told that they would be paid based on the outcome of
one of the sixteen auctions chosen at random, after all auctions were completed. Given
this structure, they were told that the optimal strategy was to treat each auction as if (i) it
was the only auction and (ii) its outcome would determine their real earnings. After
completing all 16 auctions, all participants completed a short survey with demographic
questions and questions about their decision-making strategy during the experiment. All
participants were paid one at a time, in private, on their way out of the lab. They were
each given a sealed envelope that contained $10, except the one who received the delayed
payoff. This participant instead received a sheet of paper explaining the payoff amount
he/she would receive and the length of the delay until he/she received that payoff, as well
as instructions for receiving the delayed payoff (along with the experimenters’ contact
information). The participant would then contact the experimenters to specify how and
where he/she would prefer to be paid.
A few things are worth noting about the payments in our auctions. First, they were
not trivial, ranging from $10 to $100 (or potentially higher in the money-bid auctions).
Given that the distributions of prices, monetary gains, and monetary losses that people
experience are approximately power-law shaped (Stewart et al., 2006), this range
represents the vast majority of payments that most people (especially undergraduates)
experience most of the time. Thus, most financial transactions fall within this range (at
the time these auctions were conducted). Second, the incentives we offered could not be
too low for participants to take the auctions seriously since they willingly chose to sign-
up knowing that most of them would earn $10 (the lower bound of the range). Finally,
our auction design guards against a “house-money effect” (Thaler and Johnson, 1990),
since participants were neither paid a show-up fee nor given a starting budget with which
to bid. This is important because the house-money effect has been shown to distort
preferences (e.g., Thaler and Johnson, 1990).
In addition to the auctions, we conducted surveys that were designed to be the
hypothetical matching task equivalents of the auctions: the parameter values used and
their orderings were identical to the ones used in the auctions. For this matching task
survey, we recruited another 30 Princeton University undergraduate students (no
monetary compensation given) to fill-out a short paper-pencil questionnaire examining
18Patience Auctions
their willingness to trade off an imaginary $10 in the present with a hypothetical payoff
in the future. These hypothetical value-matching tasks asked subjects to indicate the sizes
or delays of future payoffs that would make them indifferent between receiving those
later payments and receiving the $10 dollars now. In each money-matching task, they
were asked to report what value of the delayed payoff would make them indifferent
between receiving $10 now and that delayed payoff after a pre-set length of delay. In
each time-matching task, they were asked to report the delay length that would make
them indifferent between receiving $10 now and a pre-set payoff amount at the end of
that delay. In order to further parallel the design of the first-price and second-price
auctions, half the participants (N = 15) were assigned to a survey in which they first
answered money-matching questions, followed by time-matching questions. For these
participants, the first page contained 8 matching questions asking them to match on future
payoff and the second page contained 8 matching questions asking them to match on
delay. This ordering was reversed for the other participants. In both groups, the last page
of the survey asked participants to report the same demographic characteristics that we
asked auction subjects. Participants completed the survey by themselves, usually as part
of a larger set of unrelated studies.
4. Econometric Specification
We follow the Benhabib, Bisin, and Schotter (BBS 2010) specification for the discount
function but without a fixed cost component of present bias. This flexible specification
nests both the exponential and hyperbolic functions in a generalized hyperbola and
includes a quasi-hyperbolic term for present bias as well. Let m0 be the immediate
reward, which is $10 in our experiments. Let d be the length of delay in days, which is
pre-determined in the money-bid auctions and determined by the bids in the time-bid
auctions. md is the delayed reward after d days, where the amount is determined by the
bids in the money-bid auctions and pre-determined in the time-bid auctions. Let r be the
19Patience Auctions
daily discount rate. The immediate reward is equal to the delayed reward multiplied by
the discount function8 D:
m0 = Dmd (1)
where D is:
(2)
specifies the curvature of the function. The exponential and hyperbolic discount
functions without present bias are two special cases in which = 1 and = 2 respectively
and is held at 1. To account for heterogeneity in discounting across individuals and to
test whether individuals discount differently in the money- vs. time-bid auctions, we
estimate the parameters of the discount function separately for each individual i (90 in
total across first-price, second-price, and hypothetical survey treatments) and each type of
auction (money-bid and time-bid).
Money-bid Auctions
In the money-bid auctions, md is the choice variable (bid). Following BBS, we do
nonlinear least squares (NLS) estimation of the discount function parameters for the
money-bid auctions by individual using the econometric specification:
m0i = Di (mdi (m0i ,d),d;r,, )mdi (m0i ,d) i (m0i ,d) (3)
where i (m0i ,d) , the error associated with the money bids, is distributed log-normally and
i.i.d. with respect to individuals and the pre-determined parameters in Equation 3.
Time-bid Auctions
8
Risk neutrality is assumed throughout. We cannot reject the possibility that the differences we observe in
intertemporal decision making across the two domains are at least partially due to shifts in risk attitudes
(Andersen et al., 2008) between time bidding and money bidding.
20Patience Auctions
In the time-bid auctions, d is the choice variable (bid), thus unlike the estimations for the
money-bid auctions, the length of delay instead of the amount of delayed reward is now
the dependent variable measured with error:
m0i = Di (mdi ,d(m0i ,mdi );r,, )mdi i (m0i ,mdi ) (4)
where i (m0i ,mdi ) , the error associated with the time bids, is distributed log-normally and
i.i.d. with respect to individuals and the pre-determined parameters in Equation 4.
We implement five specifications of both Equations 3 and 4 in which one, two, or all
three parameters of the discount function were estimated.
Exponential Discounting
We set both and to be 1 so the discount function is the canonical exponential function
without any present bias.
D(d;r, = 1, = 1) = erd (5)
We estimate r for each individual using NLS where md is a function of m0, d, and r in the
money-bid auctions and d is a function of m0, md, and r in the time-bid auctions.
Hyperbolic Discounting
We also set both equal to 1 and equal to 2 to estimate r using the hyperbolic function
specified by Equation 6 that has been compared to the exponential function in terms of
empirical fit (Kirby, 1997; Kirby and Marakovic, 1995)
1
D(d;r, = 2, = 1) =
1+ rd (6)
-Hyperbolic Discounting
21Patience Auctions
In another specification, we estimate both and r while holding constant at 1 to
determine the curvature of the hyperbola that best fits the data along with the discount
rate.
(7)
Quasi-hyperbolic Discounting ( - Model)
Next we consider another specification, Equation 8, that allows for present bias by
estimating rather than assuming it to be 1 (and setting equal to 1).
(8)
An individual demonstrates present-bias, or overweighs the current period, if < 1.
However, all future periods are still discounted exponentially at rate r (or in the
common notation) in this model (Laibson, 1997; Phelps and Pollak, 1968).
Full Specification
Finally, we do the unrestricted estimation of all three parameters (the curvature of the
hyperbola, present bias, and the discount rate) in Equation 2 following Equations 3 and 4.
5. Results
5.1 Bidding as a Function of Elicitation Method and Parameter Values
First, we examine how bids vary as a function of the elicitation mechanism and
the parameter values. Bidding values are quite volatile, highly skewed, and the
distribution of bids is necessarily censored at zero (no one could bid negative payoffs or
negative delays). We therefore examine median bids, as these are more robust to outliers.
Figures 1-2 plot median bids for each elicitation method, as a function of the parameter
value. These figures suggest that the elicitation method had an impact on the level of
patience that bidders revealed. In particular, we see that for both time-bid and money-bid
22Patience Auctions
auctions, bidders seemed to be most patient under a second-price auction mechanism, and
least patient when responding to a hypothetical survey. This difference cannot be easily
accounted for by the possible bid shading in the first-price auctions. The theoretically
optimal bid is to shade the true value by a factor of (N-1)/N which is close to 1 when N =
15. Another possible reason for this disparity is non-truthful bidding in the second-price
auctions. The prediction for the hypothetical survey is less clear since participants had no
incentive to distort their revealed preferences in either direction, although these results
suggest that participants had an initial tendency to bid too impatiently (i.e., when there
was no incentive for them to carefully consider their responses).
"#$!" %$
!" %$
'! $$ #%"&'
" $" ''#
Figure 1a. Median money bids as a function of delay and elicitation method (full range of
delays).
23Patience Auctions
!
!
# ! "#
##
Figure 1b. Median money bids as a function of delay and elicitation method (lower range
of delays).
24Patience Auctions
#$%"# &% !
! "# &% !
("!% % $&#'(
($
# %# ("(!
Figure 2. Median time bids as a function of delayed payoff and elicitation method (full
range of delayed payoffs).
These figures also reveal a monotonically increasing relationship between bid size
and parameter value, as we would expect. After all, money bids should increase with the
delay of the future payment and time bids should increase with the size of the future
payment. The main exception to this monotonic relationship seems to be for the highest
($100) payoff in the second-price time-bid auction, which saw a noticeable drop in the
median delay bid. Although median bids are mostly coherent and rational in that they are
sensitive to parameter values (and adjust in the correct direction), they only provide us
with an aggregate picture. In contrast, they say little about the coherence and rationality
of individual bidders –a question that we now turn to.
5.2 Rank-Correlation Between Bids and Parameter Values
25Patience Auctions
As a measure of the consistency and rationality of participants’ bids, we examine the
individual-level rank-correlations between their bids and the auction parameter values.
As we mentioned above, money bids should increase with the delay of the future
payment. Similarly, time bids should increase with the size of the future payment. The
rank-correlations between bids and auction parameter values should therefore provide a
test of bidder coherence and rationality. In fact, this correlation should be equal to 1 for
rational bidders. The mean rank-correlations for each bid-type are presented in Figure 3.
Figure 3. Mean within-subject correlation between bid-rank and parameter-rank (N =
8/subject) as a function of bid-type and elicitation method. (Error bars represent standard
error)
A pair of Kruskal-Wallis tests reveal a significant difference in the bid-parameter
rank-correlations produced by the different elicitation methods, for both money bids
(2(2) = 12.86, p < .002) and time bids (2(2) = 11.41, p < .004). A series of Mann-
Whitney U tests further reveal that these bid-parameter rank-correlations are significantly
lower in the second-price auction (regardless of bid-type) than they are in the first-price
auction (both ps < .03) or the hypothetical survey (both ps < .002). In contrast, the first-
26Patience Auctions
price auction and the hypothetical survey do not differ in terms of the bid-parameter rank-
correlations they produce (both ps > .2).
These results suggest that the second-price auction did not promote coherent or,
for that matter, optimal bidding on the part of the subjects. Indeed, as Figure 3 makes
visually clear, the average level of bidding coherence was noticeably lower for
participants in the second-price auction than it was for those in the first-price auction or
the hypothetical survey. In an effort to increase the coherence of bids in the second-price
auction we conducted pilot sessions in which we either gave participants round-by-round
feedback on auction outcomes, had them participate in a practice auction beforehand with
feedback so that they could see the consequences of over-or-underbidding, or even told
them outright that the optimal strategy in second-price auctions was to bid one’s true
value. Unfortunately, all three of these approaches seemed to aggravate, rather than
reduce, eccentric bidding.
5.3 Model Comparison
Next we examine how well the five main discounting models found in the literature fit
the bidding data. Figure 4 shows the proportion of subjects whose bidding data were best
fit by a given model, as determined by the Bayesian Information Criterion (BIC –
Schwarz, 1978). We can see that, for money bids, most individuals’ bidding patterns
were best fit by the more complex models, in particular the -hyperbolic and the full-
specification (3-parameter) model. With time bids, in contrast, relatively more
individuals seemed to follow the simpler single-parameter (exponential or hyperbolic)
discounting models. This difference, between time bids and money bids, in the proportion
of subjects falling into the various models, was significant for the first-price auction
(2(4) = 10.72, p = .03), significant for the second-price auction (2(4) = 10.06, p = .04),
and marginally significant for the hypothetical survey (2(4) = 9.18, p = .057). While the
standard exponential discount function was rarely the best-fitting model for the money
bids, it produced the best fit for around 30% of participants in both the first-price and
second-price time-bid auctions. Figure 4 also shows that higher proportions of subjects in
the hypothetical survey group were best fit by the -hyperbolic model than those in the
auctions.
27Patience Auctions
Figure 4. The proportion of subjects whose bidding data were best fit by a given model as
a function of bid-type and elicitation method.
We also calculated the median value of each model parameter9, for each bid-type,
and for each elicitation procedure. These values are presented in Table 4. As this table
shows, the model parameters consistently differed between money bids and time bids, for
all but the full (3-parameter) model. Furthermore, as Table 4 shows, most of these
differences were statistically significant. Under the most commonly used exponential,
hyperbolic, and quasi-hyperbolic functions, we find that bidders were more patient when
they bid money than when they bid time (r is lower for money bids than for time bids).
Furthermore, the parameter representing present-bias is closer to 1 ( = 1 indicates no
present-bias) for the time bids in both types of auctions and the hypothetical matching
tasks. These results are consistent with bidders valuing time more when they are bidding
time (i.e., when time is more salient). A number of theoretical and empirical studies have
shown that greater attention to a particular attribute or good will lead to greater valuation
of that attribute or good (Armel et al., 2008; Bordalo et al., 2010; Huber et al., 2002).
9
These model parameters were estimated using data from all subjects with estimable parameters, not just
those for whom the model represented the best fit according to the BIC.
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