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Model of Exponential Discounting

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Roman Sheremeta, Ph.D.

Professor, Weatherhead School of Management

Case Western Reserve University

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The “standard” model�

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Model of Exponential Discounting

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Intertemporal choices�

  • Intertemporal choices: decisions involving tradeoffs among costs and benefits occurring at different times

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  • Why intertemporal choices important?
    • Wealth
    • Savings
    • Investments
    • Health
    • Happiness

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Influencing factors�

  • What are the psychological and economic motives (positive and negative) underlining intertemporal decision-making?
    • Bequest and other-regarding preferences (positive motive)
    • Self-control (positive motive)
    • Uncertainty of human life (negative motive)
    • Temptation of immediate consumption (negative motive)

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The marshmallow test�

  • The marshmallow experiment:

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Exponential discounting�

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Graphical view of discounting 𝛿�

  • Exponential discounting function:

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What do we know about 𝛿?�

  • Frederick et al. (2002) reviewed published articles, concluding:
    • There is large variability in the estimates of 𝛿
    • There is no evidence of methodological progress over time
    • High discounting is more dominant

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Example 1�

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Example 2�

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Example 3�

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Example 3�

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Foundation of exponential discounting�

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Properties of exponential discounting�

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(1) Stationarity�

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Experiment #7 and #8 results�

  • Two questions:
    • Do you prefer $x today or $200 in 3 months?
    • Do you prefer $x in 12 months or $200 in 15 months?

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Today

In 3 months

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$155

$200

2

$160

$200

3

$165

$200

4

$170

$200

5

$175

$200

6

$180

$200

7

$185

$200

8

$190

$200

9

$195

$200

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$200

$200

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In 12 months

In 15 months

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$155

$200

2

$160

$200

3

$165

$200

4

$170

$200

5

$175

$200

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$180

$200

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$185

$200

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$190

$200

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$195

$200

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$200

$200

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(2) Constant discounting�

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Experiment #7 and #9 results�

  • Two questions:
    • Do you prefer $x today or $200 in 3 months?
    • Do you prefer $x today or $2400 in 1 year?

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Today

In 1 year

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$600

$2400

2

$800

$2400

3

$1000

$2400

4

$1200

$2400

5

$1400

$2400

6

$1600

$2400

7

$1800

$2400

8

$2000

$2400

9

$2200

$2400

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$2400

$2400

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Today

In 3 months

1

$155

$200

2

$160

$200

3

$165

$200

4

$170

$200

5

$175

$200

6

$180

$200

7

$185

$200

8

$190

$200

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$195

$200

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$200

$200

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(2) Constant discounting�

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(3) Time consistency�

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(3) Time consistency�

  • Homer's classic “The Odyssey”
    • What famous scene is illustrated here?

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(3) Time consistency�

  • More recent examples:
    • Real news: “Man locks his head in a cage in an attempt to quit smoking. Wife has the key and only opens it for meals.”
    • Many web and phone apps allow you to limit the time you spend on time-wasting sites each day

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(3) Time consistency�

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References�

  • Dhami, S. (2016). The Foundations of Behavioral Economic Analysis. Oxford University Press.
  • Frederick, S., Loewenstein, G., & O'donoghue, T. (2002). Time discounting and time preference: A critical review. Journal of Economic Literature, 40, 351-401.
  • Koopmans, T.C. (1960). Stationary ordinal utility and impatience. Econometrica, 28, 287-309.
  • Thaler, R. (1981). Some empirical evidence on dynamic inconsistency. Economics letters, 8, 201-207.

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