Signature
D_Q = beta * delta^t; r_0 = (1 - beta * delta) / (beta * delta); r_L = (1 - delta) / delta
| Inputs | Definition | Unit |
|---|---|---|
beta | One-off discount applied to every outcome that is not immediate; 1 means no present bias | none, above zero and at most 1 |
delta | Per-period discount factor applied between any two future periods | none per period, above zero and below 1 |
t | Delay of the outcome from the present period | periods, a whole number of 1 or more |
D_Q | Weight placed on an outcome t periods ahead, for t of 1 or more, under beta-delta preferences | none |
|---|---|---|
r_0 | Per-period discount rate between now and the next period | a proportion per period |
r_L | Per-period discount rate between any two future periods | 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
Quasi-hyperbolic weight and per-period rates from named cells
The first formula returns the weight with the immediate period handled by IF. The second and third return the short-run and long-run per-period rates.
=IF(Delay=0,1,QHBeta*QHDelta^Delay); =(1-QHBeta*QHDelta)/(QHBeta*QHDelta); =(1-QHDelta)/QHDelta
Assumptions
Immediate outcome carries a quasi-hyperbolic weight of one
D(0) = 1 for the present period and the formula applies to whole delays of 1 or more, so the weight falls by the factor beta delta over the first period and by delta over each later period.
Period length chosen with the one-period conflict in mind
Gruber and Köszegi describe the quasi-hyperbolic model as a theoretical device that confines self-control conflict to exactly one period, so beta and delta are tied to the chosen period length and the period is set with that limit in view.
Worked examples
Article's quasi-hyperbolic smoker one month ahead
With beta = 0.7 and delta = 0.99 a month the weight one month ahead is 0.693. The rate is about 0.443 between now and next month and about 0.0101 between any two later months, as in the article's example (illustrative values).
beta = 0.7; delta = 0.99; t = 1; D_Q = 0.693; r_0 = 0.443; r_L = 0.0101
Quasi-hyperbolic weight at Wang and Sloan's estimates for adherence in diabetes
Wang and Sloan estimated beta = 0.38 and delta = 0.88 per period of their model for older people with diabetes. One period ahead the weight is about 0.334, so the short-run rate is about 1.99 against about 0.136 between later periods (computed here for illustration).
beta = 0.38; delta = 0.88; t = 1; D_Q = 0.3344; r_0 = 1.9904; r_L = 0.1364
Quasi-hyperbolic weight without present bias
With beta = 1 the two rates are equal, about 0.0101 a month for delta = 0.99, and the weight two months ahead is 0.9801, a limiting case that returns exponential discounting.
beta = 1; delta = 0.99; t = 2; D_Q = 0.9801; r_0 = 0.0101; r_L = 0.0101
Common errors
Present-bias factor compounded with delay
Writing (beta delta)^t applies beta once per period instead of once in total. With beta = 0.7 and delta = 0.99 the weight two periods ahead becomes about 0.480 instead of 0.686 (computed here for illustration), which is steep exponential discounting at the factor 0.693 and gives no preference reversal.
Present-bias factor applied to the immediate outcome
Weighting the present period by beta as well removes the bias. In the article's example smoking today would be worth 7 against 9.009 for quitting, so the smoker would quit at once instead of postponing (computed here for illustration).
Sources
Quasi-hyperbolic beta-delta form and its per-period rates
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 D(k) = 1 for k = 0 and beta delta to the power k for k above 0, first introduced by Phelps and Pollak and used from Laibson onwards, with per-period rates (1 minus beta delta)/beta delta between now and the next period and (1 minus delta)/delta between future periods.
Beta-delta preferences and present-bias estimates for diabetes care
Wang Y, Sloan FA. Present bias and health. Journal of Risk and Uncertainty. 2018;57(2) 177-198. Section 2, where a payoff t periods away is discounted by beta delta to the power t and the time-consistent model is the special case beta = 1, and section 5.3 with Table 4, which reports estimates of delta and beta of 0.88 and 0.38.
Quasi-hyperbolic model as a one-period theoretical device
Gruber J, Köszegi B. Is addiction "rational"? Theory and evidence. NBER Working Paper 7507. Cambridge, MA, National Bureau of Economic Research, January 2000. Page 35, which describes the quasi-hyperbolic model as a theoretical device that limits intrapersonal conflict to exactly one period and asks that the period length be chosen with that limitation in mind.
Canonical Identity
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