Interactive modelBehavioral Economicsintermediate

Hyperbolic Discounting & Present Bias

Why we plan patiently and act impulsively, beta-delta preferences and commitment devices.

Equilibrium: t* = 16.2, V* = 43.5

Current equations

Exponential δᵗ: V = 100·0.95^tHyperbolic 1/(1+kt): V = 100/(1 + 0.08·t)

$100 today beats $110 next week, but $110 in 53 weeks beats $100 in 52. Nothing changed except the delay to the first option. The pink curve shows why: it dives steeply at first, then flattens.

Shift the curves

Present bias (k): how sharply now beats soon0

Higher k means tomorrow feels worthless next to today, but far-future days feel almost identical.

The Hyperbolic Discounting & Present Bias model, in writing

Definition

Present bias: people discount the near future steeply and the far future gently, so preferences reverse as a payoff approaches and today's self overrules yesterday's plan.

Value = β·δ^t·u (quasi-hyperbolic: β < 1 hits everything not-now)

The intuition

Choosing between $100 in a year and $110 in a year-plus-a-week, everyone waits; move the same choice to today versus next week and many grab the $100. That reversal cannot happen with exponential discounting. It predicts procrastination, gym memberships unused, and the demand for commitment devices that tie our future hands.

Exam tip

The signature is PREFERENCE REVERSAL over time; exponential discounters never reverse. β is the present-bias parameter, δ ordinary patience: keep them separate.

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