Signature
rho_t = alpha / (1 + alpha * t)
| Inputs | Definition | Unit |
|---|---|---|
alpha | Parameter of the simple hyperbola, equal to the per-period rate between now and one period ahead | per period, above zero |
t | Delay at the start of the period over which the rate applies, measured from the moment of choice | periods, zero or above |
rho_t | Discount rate between delays t and t + 1 implied by the simple hyperbola | a proportion per period |
|---|
Function
Hyperbolic discount weight function with impatience that declines with delay
Maps a delay t, counted from the moment of choice, and the parameters of a chosen non-constant discount function to the weight placed on an outcome t periods ahead and to the per-period discount rate that the weights imply. Under the constant-rate factor of the Discount Rate page (HE-FM-DR-001), and in the stream totals of the Discounting page (HE-FN-DISC-001 and its formulas), the rate between two adjacent periods is the same at every delay, whereas the forms here let it fall as the delay grows. The package gives the simple hyperbola and its per-period rate, the generalised hyperbola of Loewenstein and Prelec, the quasi-hyperbolic beta-delta form, and a check for a preference reversal between a smaller-sooner and a larger-later outcome. Notation follows the Hyperbolic Discounting article.
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Implementations
Excel
Implied per-period rate of the simple hyperbola from named cells
The first formula gives the rate directly. The second builds it from the weights at Delay and Delay plus 1 and returns the same value, which checks a table of weights.
=HypAlpha/(1+HypAlpha*Delay); =(1/(1+HypAlpha*Delay))/(1/(1+HypAlpha*(Delay+1)))-1
Assumptions
Per-period rate defined from adjacent hyperbolic weights
The rate is the discrete per-period rate D(t)/D(t + 1) minus 1, the definition behind the per-period rates that Frederick and colleagues give for the quasi-hyperbolic form. For the simple hyperbola the instantaneous rate, the slope of minus log D at t, takes the same value alpha/(1 + alpha t), so the expression serves for both. That does not hold in general, since the constant-rate factor has a per-period rate of r but an instantaneous rate of log(1 + r).
Rate re-measured from each new present
The rate path is seen from the moment of choice. When the person reaches delay t, the next period again carries the first-period rate alpha, which is why a plan made in advance can be reversed.
Worked examples
First-period hyperbolic rate with alpha of 0.4
Between now and one month ahead the article's smoker discounts at 0.4 a month, the value of alpha itself.
alpha = 0.4; t = 0; rho_t = 0.4
Hyperbolic rate between months one and two with alpha of 0.4
Between one and two months ahead the rate has fallen to about 0.286, 0.4 divided by 1.4, matching the ratio of the article's weights 0.714 and 0.556 minus 1.
alpha = 0.4; t = 1; rho_t = 0.2857
Hyperbolic rate a year ahead with alpha of 0.4 a month
Between months 12 and 13 the rate is about 0.069 a month, 0.4 divided by 5.8, less than a fifth of the first-month rate (computed here for illustration).
alpha = 0.4; t = 12; rho_t = 0.069
Common errors
First-period hyperbolic rate quoted as the rate at every delay
Treating alpha = 0.4 a month as a constant rate puts a weight of about 0.0126 on an outcome 13 months ahead, 1.4 to the power minus 13, against about 0.161 under the simple hyperbola (computed here for illustration). Alpha is the rate for the first period only.
Sources
Hyperbolic discounting defined as a declining rate of time preference
Frederick S, Loewenstein G, O'Donoghue T. Time discounting and time preference: a critical review. Journal of Economic Literature. 2002;40(2) 351-401. Section 4.1, page 360, which adopts hyperbolic discounting to mean that a person has a declining rate of time preference, with footnote 13 listing the hyperbolic forms.
Per-period discount rates written from adjacent discount weights
Frederick S, Loewenstein G, O'Donoghue T. Time discounting and time preference: a critical review. Journal of Economic Literature. 2002;40(2) 351-401. Section 5.1, page 366, which gives the per-period discount rate between now and the next period and between two future periods for the quasi-hyperbolic form, the same ratio-of-weights definition applied here to the simple hyperbola.
Canonical Identity
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