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Model of Hyperbolic Discounting

Roman Sheremeta, Ph.D.

Professor, Weatherhead School of Management

Case Western Reserve University

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Exponential discounting�

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Graphical view of discounting 𝛿�

Exponential discounting function:

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Foundation of exponential discounting�

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Properties of exponential discounting�

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Model of Hyperbolic Discounting

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Evidence against exponential discounting�

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Evidence against exponential discounting�

What they find?

Those who are arrested just before 18 have about a 20% chance of being formally prosecuted as an adult, while those arrested just after their 18 have a 70% chance
Little indication of a systematic drop in arrests at the age of 18

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Evidence against exponential discounting�

Conclusions:

Perhaps criminal offenders have much lower discount rates, although the estimated discount rates are extremely small and unlikely
Perhaps young criminal offenders are very myopic and they put too much emphasis on today versus the future: “… It may well be that criminal offending is better described by a hyperbolic discounting model, or some alternative model of time preferences, rather than time-consistent exponential discounting…” (Lee and McCrary 2017)
An important policy implication is that longer sentencing may not be as effective as more immediate and salient punishment

How do we model such behavior?

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Hyperbolic discounting�

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Graphical view of discounting 𝛽,𝛿�

Hyperbolic versus exponential discounting function:

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Foundations of hyperbolic discounting�

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Benefits of hyperbolic discounting�

Hyperbolic discounting model offers tree improvements over the standard exponential model, in terms of better explaining human behavior

(1) It explains violations of stationarity: people put more emphasis on today (“present bias”)
(2) It explains violations of constant discounting: people look more impatient over current trade-offs than future trade-offs
(3) It explains violation of time consistency: we plan to do one thing in the future, but when the future comes we do not follow up with our plan (broken commitments, New Year’s Resolutions, etc.)

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Example�

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Example�

When the actual period 1 comes, what will the student (self 1) do?

If the problem set is done in period 1, the discounted utility is -1.5
If it is done in period 2, the discounted utility is 0.5×(-2.5) = -1.25
So, self 1 would like to do the problem set in period 2

Hence, the student’s preference are dynamically inconsistent

And this would explain why many students procrastinate!

Given the conflict demonstrated by this example, when should the students actually do the problem?

The answer depends on an additional consideration: whether the student is aware of her time inconsistency (sophisticated) or not (naive)

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Naive versus Sophisticated�

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Example�

Behavior of a naïve student:

Self 0 prefers to do the problem set in period 1, believing that she will actually do it in period 1, so she doesn't do it in period 0
When period 1 comes, self 1 doesn't want to do the problem set in period 1, so the naive student ends up doing it in period 2 and receives -2.5 (or -1.25 = 0.5×(-2.5) from self 0's perspective)

Behavior of a sophisticated student:

A sophisticated student recognizes that if she doesn’t do the problem set in period 0, her period 1 self will end up doing it in period 2, so her period 0 self needs to decide between period 0 (-1) and period 2 (-1.25)
Since she (period 0 self) doesn't like doing the problem set in period 2, she reluctantly does it in period 0, and receives -1
Consequently, a sophisticated student is better off than a naïve student (but this is not always the case!)

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Commitment device�

Commitment device: A way to lock yourself into following a plan of action that you might not want to do but you know is good for you

The use of commitment devices is “proof” of sophistication. An effort to force your future self to stick to your current self wishes
To a naive person, commitment device is unnecessary and could only cause harm

Systematic misprediction of future behavior indicates that most people are naïve:

Gruber (2001) finds that teenage smokers who predict that they will not be smoking in five years have essentially the same (but actually slightly higher) probability of smoking in five years (74%) than those who predict that they will still be smoking then (72%)
The same awareness issue doesn't arise with exponential discounting, because there is no time inconsistency to begin with

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References�

Dhami, S. (2016). The Foundations of Behavioral Economic Analysis. Oxford University Press.
Lee, D.S., & McCrary, J. (2017). The deterrence effect of prison: Dynamic theory and evidence. In Regression Discontinuity Designs: Theory and Applications, pp. 73-146.
Montiel Olea, J.L., & Strzalecki, T. (2014). Axiomatization and measurement of quasi-hyperbolic discounting. Quarterly Journal of Economics, 129, 1449-1499.
Gruber, J. (2001). Youth smoking in the 1990's: Why did it rise and what are the long-run implications? American Economic Review, 91, 85-90.