Hyperbolic preference:
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Exponential preference:
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Reversal at t =
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Hyperbolic: V(R,t) = R/(1+kt)
Exponential: V(R,t) = R·e-δt
Preference reversal:
A/(1+kt_A) = B/(1+kt_B) → t*
β-δ quasi-hyperbolic:
V(t) = β·δ^t·R (β<1 present bias)
Exponential: V(R,t) = R·e-δt
Preference reversal:
A/(1+kt_A) = B/(1+kt_B) → t*
β-δ quasi-hyperbolic:
V(t) = β·δ^t·R (β<1 present bias)
Hyperbolic discounting explains why people prefer $10 now over $20 in a month, yet prefer $20 in 13 months over $10 in 12 months — the same gap, but different proximity.