Solving Problems using Hyperbolic Discounting

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Roman Sheremeta, Ph.D.

Professor, Weatherhead School of Management

Case Western Reserve University

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Hyperbolic discounting�

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Foundations of hyperbolic discounting�

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Naive versus Sophisticated�

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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

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• 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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Solving Problems using Hyperbolic Discounting

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Solving hyperbolic discounting problems�

• Naïve decision makers:
• (1) start at the beginning
• (2) solve for the optimal plan, assuming future selves will follow the plan
• (3) the person takes the first step in that plan
• (4) go to the next period, and return to step (2)

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• Sophisticated decision makers:
• (1) start at the end
• (2) solve for optimal action
• (3) go back to the previous period
• (4) solve for the optimal action, taking into account what happens in the next period
• (5) return to step (4)

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Homework�

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Grading is boring�This makes you a hero�But that's all it does –

You still earned a zero

Problem 1: Homework�

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Problem 1: Homework�

• Period 1:
• 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

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• Period 2:
• If the problem set is done in period 2, the utility is -2.5

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• Naïve student:
• Period 0: plans to do in period 1, so doesn’t do it
• Period 1: plans to do in period 2, so doesn’t do it
• Period 2: does the homework and receives -2.5

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Problem 1: Homework�

• Sophisticated student:
• Period 2: does the homework if she hasn’t done so far
• Period 1: plans to do in period 2, so doesn’t do it
• Period 0: realizes that if she doesn’t do the problem set in period 0, she will end up doing it in period 2, so she chooses to do in period 0 (-1) instead of period 2 (-1.25), and receives -1

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• Who is better off from the perspective of self 0, a sophisticated or a naïve student?
• The sophisticated student receives the discounted utility of -1 (since the student does the problem set in period 0)
• The naïve student receives the discounted utility of 0.5×(-2.5) = -1.25 (since the student does the problem set in period 2)
• So, the sophisticated student is better off than a naïve student

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Problem 1: Homework�

• What is the maximum amount a sophisticated student would pay in period 0 to commit herself to doing the problem set in period 1?
• A sophisticate student knows that the problem set will not be done in period 1, in which case the discounted utility would be 0.5×(-1.5) = -0.75
• Instead the student chooses to do the problem set in period 0, receiving the utility of -1
• So, she could pay up to = (-0.75) - (-1) = 0.25

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• What is the maximum amount a naive student would pay in period 0 to commit herself to doing it in period 1?
• Nothing, a naïve student doesn’t realize the problem!

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Choosing a movie�

• BBC One:
• “Picking a Christmas film”
• https://www.youtube.com/watch?v=c7nnHe7KBlA

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Problem 2: Choosing a movie�

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Problem 2: Choosing a movie�

• Naïve student:
• Period 0: plans to go in period 3, so doesn’t go
• Period 1: plans to go in period 3, so doesn’t go
• Period 2: plans to go in period 2, so goes

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• Sophisticated student:
• Period 2: goes if she hasn’t gone so far
• Period 1: realizes she won’t wait until period 3, so she goes
• Period 0: realizes she won’t wait until period 2 or 3, so she goes

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Problem 2: Choosing a movie�

• Behavior of a naïve student:
• Goes in period 2, and receives a payoff of 9/4

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• Behavior of a sophisticated student:
• Goes in period 0, and receives a payoff of 1

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• Who is better off from the perspective of self 0, a sophisticated or a naïve student?
• The sophisticated student receives the utility of 1 (since the student goes to the movie in period 0)
• The naïve student receives the discounted utility 0.5×(9/4) = 9/8 (since the student goes to the movie in period 2)
• So, the sophisticated student is worse off than a naïve student

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Problem 2: Choosing a movie�

• The problem with sophistication:
• The student is realistically pessimistic about her future behavior: she knows that she won’t have the patience to wait until the best movie, so she goes immediately
• The same pessimism that leads her to do worse with movies (Problem 2) leads her to do better with problem sets (Problem 1)

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• The general lesson:
• Sophistication is not always beneficial

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Budgeting�

• Theo's budget:
• “Bill Cosby Economics 101”
• https://www.youtube.com/watch?v=TZ8QvafmUcw

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Problem 3: Budgeting�

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Problem 3: Budgeting�

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Problem 3: Budgeting�

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Problem 3: Budgeting�

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References�

• Dhami, S. (2016). The Foundations of Behavioral Economic Analysis. Oxford University Press.

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