Many key economic decisions involve implicit trade-offs over time: how much to save or invest today affects how much to spend both today and tomorrow, and individuals will differ in their preferences for satisfaction today versus delayed satisfaction tomorrow. Economists call the relative preference (or disfavor) for the present over the future a discount rate (i.e. the rate at which we discount the future for the present), and the discount rate is a core parameter in economic models of choice and behavior. The discount rate can also be an important factor in understanding the impact heterogeneity of an intervention, or perhaps even the targeted outcome of an intervention, hence our interest here on this blog.

Various methods have been developed to measure time preferences in the population of study, the most common of which is called the multiple price list. This method presents the study subject with a series of binary choices between a payment now and a payment at some point in the future, such as in four weeks. The future payment stays fixed at, say, $20 while the value of the present payment decreases through subsequent choices. For example the price list may start at $19 – and if you would prefer the immediate $19 to $20 in four weeks you would then be asked to compare $18 today versus $20 in the future and so on until the point at which you switch from preferring the sooner payment of lesser value to the larger later payment. This switch point identifies your particular discount rate, at least for the small monetary values and short periods of time considered.

Unfortunately, however, time preference measures from this approach will be biased if the underlying utility function is not linear – in economic theory we generally don’t assume utility is linear (i.e. the benefit from the 20

More recently, two groups of researchers have put forward “corrections” to this standard methodology so that time preferences can be more accurately measured through survey or in the lab. Andersen and co-authors propose the addition of what they call the Double Multiple Price List (DMPL). Andreoni and Sprenger propose a device they name the Convex Time Budget (CTB). Both methods appear relatively easy to implement – which is great news for us practitioners – but how do they perform relative to each other?

First let’s review the two approaches. The DMPL attempts to correct for this bias by presenting a second series of questions that compare safe and risky gambles with different payoffs and different odds, including some gambles that payout with certainty. In theory, your response to these questions – which identifies your attitude towards risk by the point in the question series where you switch from a safe payout to a risky gamble – will identify the degree of utility function curvature, and this information can be used in turn to correct the discount rate estimate from the multiple price list.

The CTB method takes a different approach to identifying the degree of utility function curvature through estimating the respondent’s sensitivity to changing interest rates. This is done by the presentation of a series of pay-off options to choose between and not just a binary choice. For example you are asked to select between $15 today and zero in the future versus zero today and $20 in the future or additional various combinations of non-zero payouts both today and in the future. The key is to vary the implicit interest rate in the options presented across subsequent sets of options. The expressed sensitivity to changing interest rates across the question sequence identifies the utility curvature; the time preference is identified through the stated preference over the timing of payments. (If this is confusing see the Andreoni et al. paper I link in the next paragraph for a very clear example).

A very recent paper by Andreoni, Kuhn, and Sprenger explores the performance of both the DMPL and the CTB methods – they enlist university students to answer both types of questions at various payout amounts – and find that both methods identify a high degree of curvature in the utility function. First off by just assuming linear utility, as in the standard multiple price list, they estimate an average annual discount rate of 102% among the students in the sample. The DMPL in turn estimates a high degree of curvature (and hence a high degree of upward bias in the old linear utility method) and corrects the discount rate down to 47%. The CTB method also estimates some curvature, but to a lesser degree, and corrects the discount rate to the range of 63%-74% (depending on exact estimation method).

Without the ability to benchmark either experimental measure against a known parameter (since we don’t observe the discount rate directly) it makes sense to look at the out-of-sample predictive ability of each approach. This is exactly what the authors do by using the individual parameter estimates from each method to predict responses to subsequent time preference scenarios. It turns out that, while both methods are somewhat predictive of subsequent choices, the CTB method is significantly more predictive (at least for this study sample and this selected range of monetary values).

It’s important to note that both methods find utility curvature to be a key factor, even with the relatively low quantities of money used in these studies. While the CTB method does a better job in predicting out-of-sample predictions in this study, it doesn’t necessarily mean that the CTB method is the more “correct” method. But in absence of verification against known parameters, predictive validity counts for something.

Various methods have been developed to measure time preferences in the population of study, the most common of which is called the multiple price list. This method presents the study subject with a series of binary choices between a payment now and a payment at some point in the future, such as in four weeks. The future payment stays fixed at, say, $20 while the value of the present payment decreases through subsequent choices. For example the price list may start at $19 – and if you would prefer the immediate $19 to $20 in four weeks you would then be asked to compare $18 today versus $20 in the future and so on until the point at which you switch from preferring the sooner payment of lesser value to the larger later payment. This switch point identifies your particular discount rate, at least for the small monetary values and short periods of time considered.

Unfortunately, however, time preference measures from this approach will be biased if the underlying utility function is not linear – in economic theory we generally don’t assume utility is linear (i.e. the benefit from the 20

^{th}piece of chocolate in the day may still be great, but not as great as the first piece). For the sake of convenience the multiple price list assumes linear utility over the range of the small stakes of this measurement, yet assuming linear utility when it is in fact concave will bias upwards the estimate of the discount rate.More recently, two groups of researchers have put forward “corrections” to this standard methodology so that time preferences can be more accurately measured through survey or in the lab. Andersen and co-authors propose the addition of what they call the Double Multiple Price List (DMPL). Andreoni and Sprenger propose a device they name the Convex Time Budget (CTB). Both methods appear relatively easy to implement – which is great news for us practitioners – but how do they perform relative to each other?

First let’s review the two approaches. The DMPL attempts to correct for this bias by presenting a second series of questions that compare safe and risky gambles with different payoffs and different odds, including some gambles that payout with certainty. In theory, your response to these questions – which identifies your attitude towards risk by the point in the question series where you switch from a safe payout to a risky gamble – will identify the degree of utility function curvature, and this information can be used in turn to correct the discount rate estimate from the multiple price list.

The CTB method takes a different approach to identifying the degree of utility function curvature through estimating the respondent’s sensitivity to changing interest rates. This is done by the presentation of a series of pay-off options to choose between and not just a binary choice. For example you are asked to select between $15 today and zero in the future versus zero today and $20 in the future or additional various combinations of non-zero payouts both today and in the future. The key is to vary the implicit interest rate in the options presented across subsequent sets of options. The expressed sensitivity to changing interest rates across the question sequence identifies the utility curvature; the time preference is identified through the stated preference over the timing of payments. (If this is confusing see the Andreoni et al. paper I link in the next paragraph for a very clear example).

A very recent paper by Andreoni, Kuhn, and Sprenger explores the performance of both the DMPL and the CTB methods – they enlist university students to answer both types of questions at various payout amounts – and find that both methods identify a high degree of curvature in the utility function. First off by just assuming linear utility, as in the standard multiple price list, they estimate an average annual discount rate of 102% among the students in the sample. The DMPL in turn estimates a high degree of curvature (and hence a high degree of upward bias in the old linear utility method) and corrects the discount rate down to 47%. The CTB method also estimates some curvature, but to a lesser degree, and corrects the discount rate to the range of 63%-74% (depending on exact estimation method).

Without the ability to benchmark either experimental measure against a known parameter (since we don’t observe the discount rate directly) it makes sense to look at the out-of-sample predictive ability of each approach. This is exactly what the authors do by using the individual parameter estimates from each method to predict responses to subsequent time preference scenarios. It turns out that, while both methods are somewhat predictive of subsequent choices, the CTB method is significantly more predictive (at least for this study sample and this selected range of monetary values).

It’s important to note that both methods find utility curvature to be a key factor, even with the relatively low quantities of money used in these studies. While the CTB method does a better job in predicting out-of-sample predictions in this study, it doesn’t necessarily mean that the CTB method is the more “correct” method. But in absence of verification against known parameters, predictive validity counts for something.

## Comments

## Do you know development

Do you know development papers that have used DMPL in the field? I know CTB has been done in Malawi (Gine et al. 2013), but haven't seen DMPL, or really any Holt-Laury risk aversion measures that ask people for their preferences between two gambles.

I ask because I'm interested in respondent comprehension and preference expression. We all have a tendency to choose simple options over complex ones, and this fact has the potential to confound either one of these time preference measurements. CTB could be affected because people may choose corner allocations instead of interior ones simply because "All Now" or "All Later" is easier to think about than "35% now AND 65% later." Indeed, Andreoni and Sprenger (2012), Andreoni, Kuhn, and Sprenger (2013), Gine et al. (2013), and Kuhn, Kuhn, and Villeval (2013) do document many corner allocations. Are less educated, more hungry, or more cognitively fatigued people more likely to choose corner allocations, and how does that affect our measurement of treatment effects? Open question.

DMPL could be problematic because those Holt-Laury questions are hard to think about. Do I prefer a 10% chance of 2000 kroner AND a 90% chance of 1600 kroner OR a 10% chance at 3850 kroner AND a 90% chance at 100 kroner? I have no idea. I would love to see these two methods tested side-by-side in a development context, with special emphasis on respondent understanding and qualitative reports of whether respondents just throw up their hands and pick something at random.

p.s. the Andersen et al. camp may object to the term "Double Multiple Price List method." In their comments on this debate (Harrison et al. 2013), they call their method "Binary Choice."

Giné, Xavier, Jessica Goldberg, Dan Silverman, and Dean Yang. 2013. “Revising Commitments : Field Evidence on the Adjustment of Prior Choices” (August).

Harrison, GW, MI Lau, and EE Rutström. 2013. “Identifying Time Preferences with Experiments: Comment” (February).

Kuhn, Michael A, Peter Kuhn, and Marie-Claire Villeval. 2013. “The Importance of Cognitive Environment for Intertemporal Choice.”

## Hi Jessica, thanks very much

## It is an interesting problem

It is an interesting problem to estimate the discount rate we need to stay below to avoid dooming all future generations beyond our grandchildren. Assuming a generation of 20 years, then the discount rate per generation needs to be held at less than 18%. That puts more weight on the interests of generations beyond the present and next two. Otherwise, we doom them (assuming their fate depends on our decisions). No society has ever come close to that rate.

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