Signature
D_G = (1 + alpha * t)^(-beta / alpha)
| Inputs | Definition | Unit |
|---|---|---|
alpha | Parameter that controls how quickly the rate declines with delay; values near zero give nearly constant discounting | per period, above zero |
t | Delay of the outcome measured from the moment of choice | periods, zero or above |
beta | Parameter that sets the overall level of discounting, the continuous rate in the limit as alpha approaches zero | per period, above zero |
D_G | Weight placed on an outcome t periods ahead under the generalised hyperbola, relative to the same outcome now | none |
|---|
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.
Try this function
Implementations
Excel
Generalised hyperbolic discount factor from named cells
With the parameters in GenAlpha and GenBeta and the delay in Delay, the cell returns the weight.
=(1+GenAlpha*Delay)^(-GenBeta/GenAlpha)
Assumptions
Positive parameters of the generalised hyperbola
Both alpha and beta are above zero, so the weight is 1 at t = 0 and falls with delay. The formula divides by alpha, so a constant-rate analysis is written with exp(minus beta t) directly, not with alpha set to zero.
Generalised hyperbola parameters in the time unit of the delay
Alpha and beta are per period in the unit of t. Restating a monthly delay in years multiplies both by 12, which leaves the ratio beta over alpha unchanged.
Worked examples
Generalised hyperbola with beta half of alpha at two periods
With alpha = 0.4 and beta = 0.2 the weight two periods ahead is 1.8 to the power minus 0.5, about 0.745 (illustrative parameters, computed here for illustration).
alpha = 0.4; beta = 0.2; t = 2; D_G = 0.7454
Generalised hyperbola close to constant discounting with a very small alpha
With alpha = 0.0001 and beta = 0.1 the weight ten periods ahead is about 0.3681, close to exp(minus 1), about 0.3679, the constant-rate weight at a continuous rate of 0.1 (computed here for illustration).
alpha = 0.0001; beta = 0.1; t = 10; D_G = 0.3681
Common errors
Division by alpha dropped from the generalised hyperbola exponent
Writing (1 + alpha t) to the power minus beta gives about 0.889 two periods ahead with alpha = 0.4 and beta = 0.2, instead of 0.745 (computed here for illustration), and loses the constant-rate limit.
Generalised hyperbola beta confused with the quasi-hyperbolic present-bias factor
The article warns that the two betas differ. Entering a present-bias estimate of 0.7 as the generalised beta with alpha = 0.4 gives about 0.555 one period ahead, against 0.693 for the quasi-hyperbolic form with beta = 0.7 and delta = 0.99 (computed here for illustration).
Sources
Generalised hyperbola of Loewenstein and Prelec as listed by Frederick and colleagues
Frederick S, Loewenstein G, O'Donoghue T. Time discounting and time preference: a critical review. Journal of Economic Literature. 2002;40(2) 351-401. Page 360, footnote 13, which states that Loewenstein and Prelec (1992) suggested D(t) = 1/(1 + alpha t) to the power beta over alpha.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0