2.1 Exponential discounting (E, r)

The standard financial method of summarizing growth rates by which to translate present values into future values and vice versa is by a geometric mean taken on the ratio of increase. Thus, the compounding growth model for discrete time is:

(7)

with F (future value), P (present value), r (growth rate) and T (compounding periods) taking their usual meanings, so that

(8)

As an example, if $200 (P) becomes $400 (F) over a period of 4 years (T), then r = (400/200)^(1/4) – 1 = .19, or a 19% increase per annum. We can also compute P as:

(9)

where [1 / (1 + r)]^T is known as the discounting factor: it allows us to translate a future cash flow into its present value. When faced with a choice between:

1. (1) P, an instantaneous payoff now, and

2. (2) F at time T,

the decision maker, operating normatively as an exponential discounter, would apply the discounting factor to F using a given r, and choose the larger of P and the discounted F. Another way of describing this process is to focus on equation (8) and compute r for a given choice of P, F, and T. This is equivalent to the internal rate of return (IRR) as used by accountants, for instance to judge whether a planned project will meet a criterion level of return. If r exceeds a given criterion rate r₀, then the accountant would accept F. We posit that r₀ is not a commercially available rate of return, but a person-specific one. Each person uses their own r₀ as a criterion rate of return by which to judge whether r, computed from choice {P, F, T}, exceeds criterion (hence choose F) or not (hence choose P). We do not, however, claim that r₀ is impervious to context, manipulation, and simple fluctuation with time.

If we increase the number of compounding periods to be n per annum, equation (9) becomes:

(10)

and if n increases without limit so that compounding is continuous, this becomes

(11)

The discount factor therefore becomes e^−rT, hence the name “exponential discounting”, though the geometric form in (10) is often still used. The growth rate parameter r is determined from (11) as:

(12)

In practice, the exponential and geometric view of growth and discounting do not differ much in their computed values of r. In the above example, equation (12) gives r = 17%, rather than the previously calculated 19% from equation (8).

2.2 Hyperbolic discounting (H, h)

Accountants also frequently calculate the (arithmetic) average rate of return (Reference RyanRyan, 2007, p. 12). Similarly, if a quantity increases by x% over T years, then the arithmetic average of the rate of return should be (x/T)%. In the example cited above there was a 100% increase, which averages out to 25% per annum. Although frequently used to appraise investments and set a benchmark figure for project acceptance, this figure cannot be used in the compound growth formula. Instead of doubling the original as required by a 100% increase, its use would lead to a calculated increase of (1.25⁴ = 2.44) times the original. These arithmetic calculations are equivalent, in algebra, to computing a growth rate of h by:

(13)

In the example given in 2.1, h = (2 – 1) / 4 = .25. Equation (13) also implies:

(14)

According to equation (14), the discount factor [1 / (1 + hT)] has a hyperbolic form, hence the name “hyperbolic discounting.” Also, whereas E is underpinned by a model of compound interest, H is underpinned by a model of simple interest, with no compounding (Reference RachlinRachlin, 2006). This is also apparent if we let n → 1/T in equation (10) for a single compounding period, which reduces to the hyperbolic model. Thus discrete-time compounding is a half-way house between the extremes of E and H under the control of the frequency of compounding, n. When n is extremely large, the compounding periods become very small, until in the limit we have continuous compounding (therefore E). When n=1, the interest is accrued at the end of the investment period, so in effect we have simple interest (therefore H).

The fact that the 2.44 exceeds 2 (and 25% exceeds 19%) in the above example is no accident, and is due to the mathematical relationship that the arithmetic mean ≥ geometric mean for positive numbers. It follows that, in the period up to T, hyperbolic discounting (which uses the arithmetic mean) will always make a higher estimate of future growth than exponential discounting: therefore, the immediate future will be discounted more heavily in the hyperbolic model. After T, the situation reverses.

We can also use equation (13) to determine the hyperbolic discounting rate parameter h, implicit in any choice {P, F, T}, which we then use to compare against someone’s internal h₀. As with exponential discounting, if people apply a hyperbolic model, then if h > h₀ they will choose F; if h < h₀ they will choose P.

Although the hyperbolic discounting model in equation (14), given in Reference Mazur, Commons, Mazur, Nevin and RachlinMazur (1987), is the one most widely used in current research, it is founded on models that pre-date this formulation. As reviewed in Reference AinslieAinslie (1975), Reference Chung and HerrnsteinChung and Herrnstein’s (1967) found that the rate at which pigeons would peck was proportional to the reciprocal of the time delay, as an instance of the “matching law”. This led Ainslie to propose the first hyperbolic model, presented in Reference Mazur, Commons, Mazur, Nevin and RachlinMazur (1987) as: P = F / kT, where k is a constant of proportionality. The problem with this formulation is that as T → 0, the present value of any reward becomes infinite, but can be remedied by adding a constant to the denominator (Reference Herrnstein, Bradshaw and LoweHerrnstein, 1981), and it is then a short step to fix that constant as the value 1.

Finally, note that the word “hyperbolic” is often used loosely to cover any discounting in which the discount rate is higher at shorter delays than it is at longer delays. We use the term more precisely to refer to the particular model described in (13) and (14).

2.3 Arithmetic discounting (A, d)

In arithmetic discounting, future and present values are compared by constant increments d, exactly as in an arithmetic progression:

(15)

(16)

Instead of multiplying F by a discount factor to yield P, we subtract a discount decrement (Td). In one choice people are offered P: in the second they are offered P + (F – P). If the participant decomposes the choice in this way, then the decision is whether it is worth waiting for the excess (F – P). If they feel their time is worth d₀ per day to wait, then if (F – P) > Td₀ they chooses F. Whereas r and h are dimensionless, being essentially expressions of interest rates, d is measured in units of money per time (e.g. dollars per day).

There is another justification for equations (15) and (16). Reference KilleenKilleen (2009) started from the assumption that the marginal change in utility with respect to time follows a power law; also, that utility and value are themselves related by a power law. Taken together these were shown to yield an equation of the form: (Reference KilleenKilleen, 2009, equation 6, p.605). We examine the general form of this equation in a later section (4.1). But for now, notice that if c = 1, subjective time is objective time; and that if k = 1, the utility of money is just its face value, giving equation (16). Reference Doyle and ChenDoyle and Chen (2012) found empirical support for A, relative to E and H, both in other researchers’ past data, and in their own. They surmised that people were treating delay discounting by analogy with “wages for waiting” rather than by analogy with “investment growth” as implied by E and H.

3.1 Hyperboloid models

3.1.1 Green and Myerson (H_m, h_m; m)

Reference Myerson and GreenMyerson and Green (1995) proposed a model which generalizes the hyperbolic. Whereas in this and other publications these authors consistently use the symbol s for the exponent, we use 1/m for consistency with the rest of this article. Their model is:

(17)

This model is a special case of Loewenstein and Prelec’s (1992) generalized hyperbola formulation. Further simplifying, if m = 1, we have the hyperbolic. When 1/m (=s) < 1 the tendency of H to discount more steeply than E at short durations (but less so over longer durations) is exaggerated. Equation (17) may be re-arranged to give a rate parameter that is the equivalent of r, h, or d:

(18)

The subscript m on the rate parameter emphasizes that money may be treated subjectively, and requires the researcher to estimate m (=1/s) empirically. Typical values that are reported in the literature for s (=1/m) are in the range [.45, .78] (Reference McKerchar, Green, Myerson, Pickford, Hill and StoutMcKerchar, Green, Myerson, Pickford, Hill, & Stout, 2009; Reference McKerchar, Green and MyersonMcKerchar, Green & Myerson, 2010), and these authors find that modeling the additional parameter leads to statistically significant improvements in fit over the hyperbolic. Notably, s is typically less than 1, which means that the ratio (F/R) in equation (18) is being raised to a power greater than 1, thus acting to exaggerate the relative difference between F and P in calculating h _m. At the other extreme, when s becomes large, the exponent acts to shrink the ratio (F/R) towards 1, and in so doing (18) begins to approximate the exponential model.

To see why, we use the mathematical relationship that as c → 0. Here x is (F/P) and c is m. The smaller m becomes, the closer (F/P)^m -1 comes to m.log(F/P). Once the value of m is fixed at some small value, all calculations in (18) are multiplied by the same m, which in this way just acts as a scaling constant, and can be ignored. It follows that the exponential model is a special case of (18) when s gets to be very large. The take-home point is that since m > 1 in empirical data, actual behavioral discounting is not a compromise between E and H, but lies even further from E than H had suggested.

3.1.2 Rachlin (H_τ, h_τ; τ)

An alternative way to generalize the hyperbolic was proposed by Reference RachlinRachlin (2006), and also presented in Reference Mazur, Commons, Mazur, Nevin and RachlinMazur (1987):

(19)

Here, the exponent τ applies specifically to T rather than the whole brackets. This minor modification leads to a subtly different model. Rearranging (19) to isolate the rate parameter, we get:

(20)

In this version of the hyperboloid, the numerator (the treatment of money) is the same as for the hyperbolic. However, unlike in the previous four models, time may be treated subjectively by means of a Stevens-like power law. Similar to before, if τ =1 we have the hyperbolic. But once again, τ < 1 in empirical data. Thus, subjective time is increasingly compressed at longer periods. The typical range for τ is [.67, .90] (Reference McKerchar, Green, Myerson, Pickford, Hill and StoutMcKerchar et al. 2009, Reference McKerchar, Green and MyersonMcKerchar et al. 2010) which is just slightly lower than other estimates of the subjective time exponent (Reference EislerEisler, 1976). This is the first model we have considered in which the rate parameter cannot be expressed in straightforward units of measurement.

3.2 Present bias / present premium models

3.2.1 Quasi-hyperbolic (β-δ) discounting  (qH, q; β)

Quasi-hyperbolic discounting (qH) is rarely used in psychological research, though it is used extensively by economists hoping to preserve as much of the exponential model as possible. In the discrete time version of the exponential model each factor discounts the previous one by δ (= 1/(1+r)) to form a geometric progression of discount factors: {1, δ, δ², δ³,…δⁿ} for t = 0, 1, 2, 3…n, respectively, with δ < 1. As explained in Section 2.1, when the compounding periods become very small, and n becomes correspondingly large, we have continuous compounding, and therefore E itself. Contrasting with discrete-time E, the quasi-hyperbolic model posits discount factors: {1, βδ, βδ², βδ³, …βδⁿ} for t = 0, 1, 2, 3…n, respectively, with 0 < β, δ < 1 (Reference LaibsonLaibson, 1997). If β = 1 we would have a straightforward compounding model, as described in equation (9). But if instead, β < 1, this parameter acts to give a one-off boost to discounting over the first compounding period. The purpose of β is to help tick the box of unexpectedly steep discounting at short durations, which is the hallmark of actual behavioral data, and which first motivated the use of hyperbolic discounting. Thereafter, the model assumes that discounting follows the standard normative model. Paralleling the development in Section 2.1, the quasi-hyperbolic can also be used for continuous discounting (e.g., Reference Benhabib, Bisin and SchotterBenhabib, Bisin, & Schotter, 2010). The discount factor D is:

(21)

The rate parameter for quasi-hyperbolic discounting is therefore:

(22)

All F first get scaled by the additional parameter β before being treated by the exponential model. Obviously, if β = 1, then q = r as in equation (12). It is thus clear that qH discounting is hyperbolic only in its intent to mimic the steep initial discounting of the hyperbolic model, but in every other sense it is an exponential model.

The qH model may have been motivated pragmatically to preserve exponential discounting and all the useful mathematics that goes with it, while also modeling initial steep discounting. Nonetheless, the idea that F (but not P) gets a special “tax” simply because it is not received right now does have a more constructive justification. There is a qualitative difference between now and any time to come, which we all appreciate intuitively, and which language mirrors in verb tenses. From this perspective, the discontinuity in qH between t = 0 and t > 0 is not an ugly kludge to workaround bothersome behavioral evidence, but a clever modeling device that allows what happens now (P) to be treated differently from all future events (F). In this way qH, a model devised by economists for economists, has the capacity to capture a psychological distinction that the other models cannot.

To help rescue this insight from the limitation of being identified exclusively with the exponential model, let γ = 1 / β . Whereas β captures the idea that F is less than it should be, γ captures the idea that P is more than it should be, and we call γ the present premium. Analogous ideas are met in risky decision, where a certain reward (p = 1) are valued at a premium over rewards with p < 1; and also in the premium due to mere possession or the endowment effect (Reference McKerchar, Green and MyersonThaler, 1980; Reference Sen and JohnsonSen & Johnson, 1997). Rewards that are certain, mine, and now are all over-valued. Potentially, the models we have considered or are about to consider, could incorporate a present-premium parameter by simply replacing P with γ P (or F with β F) in all calculations. Reference Benhabib, Bisin and SchotterBenhabib, Bisin and Schotter (2004) formulate exactly such a hybrid between H _m and qH in this way (see also Section 4.4, and Reference Tanaka, Camerer and NguyenTanaka, Camerer & Nguyen, 2010).

Nonetheless, there are still issues to be confronted with qH. In particular, if β < P/F in a particular choice, the model predicts P should always be preferred to F, no matter how short the delay in F—unless someone’s internal rate parameter is negative. Reference LaibsonLaibson (2003) suggested “calibrating” qH with β ≃ .5, meaning that 21 of Reference Kirby, Petry and BickelKirby, Petry, and Bickel’s (1999) 27 choice questions would be in the negative rate parameter category. Even β = .90 would put 8 questions into that category. Summarizing several empirical studies in economics, Reference Van de Ven and Wealevan de Ven and Weale (2010) noted β s of .296, .308, .674, .687, .846, .942; and .825 in their own work; and in Reference Albrecht, Volz, Sutter, Laibson and von CramonAlbrecht, Volz, Sutter, Laibson and von Cramon (2011) the median β from individual experimental data was .86, with 24 of 27 participants having β < 1. If these estimates are close to what individuals use in binary choice, then β < P/F would occur more than just occasionally in typical data.

The quasi-hyperbolic is popular with behavioral economists. It lends itself to convenient testing against the normative model E, by testing whether β is significantly different from (less than) 1. Unfortunately, in the literature qH has rarely been tested against the kinds of models surveyed here which are more potent challengers in that they themselves have consistently proven better than E. Furthermore, Reference Kable and GlimcherKable and Glimcher (2010) argued that the present bias is more strictly a soon-as-possible bias, implying that a deflationary β should be applied to all f (t > 0), and not just P (t = 0). Finally, as has already been pointed out in Section 1.3.3, nothing ever actually happens at t = 0, in that a reward offered “now” is actually received at t > 0. Therefore, the special status of t = 0 in qH can be questioned on the grounds that it may never occur in practice.

3.2.2 Benhabib et al.’s fixed cost model (X, x; χ)

As an alternative to the percentage decrement for non-present rewards, Reference Benhabib, Bisin and SchotterBenhabib, Bisin and Schottter (2010) suggested using a fixed cost to model present bias—that is, an absolute decrement: use (F – χ ) in place of F. Thereafter use the exponential model, exactly as in qH.

(23)

They found a present bias / present premium of about $4 among amounts of $10 through $100. The fixed cost model better fitted their data than the qH model. A different, though very similar model is obtained if χ is added to P rather than subtracted from F (see Section 4.4). Reference Hardisty, Appelt and WeberHardisty, Appelt, and Weber (2012) found that χ was positive for both gains and losses, suggesting a general “want-it-now” bias.

3.3 Read’s interval model (B, b; θ)

Reference ReadRead (2001) presented his model in terms of a discount fraction D that occurs within the time window [t, T]. D is the fraction by which F is reduced by being at the end of the interval (T), rather than the start (t): namely, f / F. The exponent θ applies to the time interval, rather than time itself (as would be the case with the exponent τ, which is not used here). The algebraic steps to isolate the rate parameter b are:

(24)

From equation (24) we see that Read’s is an exponential model performed on the subjectively (because θ ≠ 1) perceived time interval (T – t). It nests the standard exponential model E when the interval of inspection starts from t=0 (at which point f is to be known as P), and when subjective time is assumed to be linear with objective time (θ = 1).

While this model seeks to understand cognitive processes through the use of interval data (t > 0), it still must compete with other models at the particular case of t = 0, even if a given competitor model has not (yet) been generalized to the case of t > 0.

3.4 Roelofsma—exponential time (V, v; υ)

Arguing from the Weber-Fechner Law of logarithmic sensitivity to psychophysical stimuli, Roelofsma (1996) presented his model as:

(25)

If we take U(F), the utility of F, as itself being logarithmic as in the exponential model E, and move from discrete to continuous time, we derive the rate parameter:

(26)

Whether log(υ) is an intrinsic part of this model is not really discussed by the author. If it is not, the obvious choice is to set υ = 1, so that this model becomes a simple rate parameter model in log money and log time, with no additional parameters to be estimated. In treating both money and time logarithmically, among the models I have surveyed, Roelofsma’s is the most enthusiastic adherent of the Weber-Fechner law. Assuming υ = 1, and writing (26) as: v = [log(F) – log(P) ] / log(T), we see that Roelofsma is structurally similar to the arithmetic model except that, instead of using raw values of F, P, and T in the rate parameter equation, each is replaced with its logged equivalent.

3.5 Synopsis of the models so far

E is the normatively correct model to value investments. It assumes a continuously compounding interest model of growth. But E fails to capture important aspects of how laypeople actually evaluate intertemporal choices. qH extends E by making a distinction between present choices and future choices. In qH, all choices in the future are devalued by a constant scalar β before treating them as in E, which itself is the special case of qH when β = 1. Model X, like qH, maintains that there is a special difference between P and all future values F, but whereas qH treats the present premium multiplicatively, X does so additively.

For over two decades, H has been the psychologists’ default alternative to E in that it consistently estimates people’s actual discounting behaviors better than E. H is equivalent to a simple interest model of growth. The hyperboloid models are extensions of H that treat money (H_m) and time (H_τ) as non-linear. H_m is equivalent to H when its additional parameter m = 1, and E when 1/m → 0. But past evidence estimates m to be less than 1, which might imply increasing utility to money, contrary to classical theory and common intuition. Also, H_τ is equivalent to H when its additional parameter τ = 1. The mental operations involved in A are simpler than those in all other models, which all start by forming a ratio between (possibly transformed) F and P. Model A starts by forming a difference between F and P.

The special case of t = 0 in Read’s model B (all time intervals start from now; hence f = P) presents as a hybrid between E and H_τ. It is an exponential model (logarithmic money) calculated on subjective (power law) time. Of course, in modeling time intervals in general, not just ones that start from the present, Read’s model aims to examine phenomena that the others cannot. Roelefsma’s model V treats both money and time logarithmically, which makes it equivalent to A, where all monies and times have first been logged.

Finally, because of the reciprocal T in calculations of r, h, d, h_m, q, and x for a given P and F, all of these rate parameters decline hyperbolically with time. In fact, one might distinguish models E, H, A, H_m, qH, and X from each other only by their treatment of money, and once that is determined in the numerators of their respective rate parameter equations, r, h, d, h_m, q, and x all decline identically with objective time. This neglected perspective suggests the essential difference between the five models of intertemporal choice, A, A_z, H, E, H_τ, qH, and X has nothing to do with time per se, and could even be tested against each other by considering a single fixed time gap between P and F.

3.6 Time (im)patience models

From the point of view of modeling rate parameters, most of the above models have focused on their treatment of money, making few inroads into modeling the subjectivity of time. In this Section we collect together research that takes a serious look at how time is perceived—while incidentally accepting the log(F/P) of model E as a given.

3.6.1 Ebert & Prelec’s constant sensitivity (CS) function (C, a; b)

“Given that unit elasticity defines compounding discounting [E], it is natural to interpret lower-than-unit elasticities as indicating insufficient time-sensitivity relative to compound discounting. If elasticity is constant and equal to b > 0, the discount function has the constant-sensitivity form” (Reference Ebert and PrelecEbert & Prelec, 2007, p.1425), which is:

(27)

As b → 0 we have diminishing time sensitivity. If b ≫ 1, then the CS function begins to appear as a step function, meaning that time is ever more clearly dichotomized into “near future” and “far future” (at the boundary 1/α ). See their Figure 1. We can isolate the rate parameter associated with the CS model from equation (27) as:

(28)

Clearly, when b = 1 (unit elasticity), we have model E as the special case. Ebert and Prelec note, but avoid, the similar discount function:

(29)

leading to the rate parameter:

(30)

which is a special case of the generalized hyperboloid to be examined in Section 4.3, with m → 0.

3.6.2 Bleichrodt et al. Constant Relative Decreasing Impatience (CRDI) functions (M, ρ; ψ, (β =1))

CRDI is explicitly described as an analog of constant absolute risk averse functions (CARA) that appear in risky choice models. Reference Bleichrodt, Rohde and WakkerBleichrodt, Rhode, and Wakker (2009) observed that hyperbolic discount functions were developed to account for decreasing impatience. That is, relative to exponential discounting, people consistently show greater impatience (for P) in the short term, but greater patience in the longer term. They also noted that such models failed to accommodate “increasing impatience or strongly decreasing impatience” (p. 27), particularly given the attempt to model individuals’ discounting behavior: hence their search for discounting functions to fulfill this need. Let D(T) be the discount function (i.e., D(T) = P/F). Then their CRDI function is / are:

1. (i) D(T) =

2. (ii) D(T) =

3. (iii) D(T) =

Examining each in turn, we have for ψ > 0,

, thus:

(31)

The following points should be noted. First, money is treated as in qH, with β fulfilling the same role in both models. Second, there is a power law on time; if β = 1, the model is exactly the route not taken by Reference Ebert and PrelecEbert and Prelec (2007), as in equation (30). Third, if ψ = 1, we have objective clock time, and thus “constant impatience”; if 0 <ψ < 1, we have the conventional view that time becomes increasingly contracted the more distant in the future it is, and thus we have “decreasing impatience”; but if ψ > 1, the more distant the time, the more stretched it becomes, which therefore models “increasing impatience”.

Examining ψ = 0, we have:

(32)

If β = 1, we have Roelofsma’s model (presuming υ =1), described in Section 3.4.

Finally, if ψ < 0, we have:

(33)

In this case the denominator asymptotes to zero, but from below. It is designed to model impatience that decreases more rapidly than for the first part with 0 < ψ < 1. In order for ρ to be positive (Bleichrot et al, 2009, Definition 4.1, p. 31), we must have that β < P/F so that the log in the numerator is negative allowing the minus signs to cancel.

As ψ moves through zero from above, for a given T > 0, the denominators in parts (31), (32) and (33) become 1, logT and –1, respectively. This means that any algorithm that hopes to recover the impatience parameter ψ from data has a discontinuity to negotiate at ψ = 0. It may therefore be better to think of (i), (ii) and (iii), or equations (31), (32) and (33), as three distinct sub-models, rather than a single model.

3.6.3 Bleichrodt et al. Constant Absolute Decreasing Impatience (CADI) functions (N, σ; η, (β = 1))

Their analog of a CARA Constant Relative Risk Averse function is the CADI function given by:

“[β] is a scaling factor without empirical meaning in the sense that it does not affect preferences”; “[η ] is the constant that indicates the convexity of log(D)”; and the rate parameter σ “determines the degree of discounting.”

Looking first at the η > 0 part:

(34)

The denominator is a convex function, indicating increasing impatience. It is the analogous equation to the first part of 3.6.2, in the region ψ > 1.

For η = 0, we have:

(35)

which is the Exponential model, and for η < 0 we have:

(36)

is a decaying function with time if η < 0, but because of the negative sign, the denominator becomes a concave increasing term, and is therefore analogous with 3.6.2, part(iii) and equation (33).

The same kind of remarks made for the CRDI functions, concerning algorithms and betas can be repeated here.

3.7 Size-sensitivity in the arithmetic model (A _z, d _z; z)

Returning to the arithmetic model, one shortcoming is its insensitivity to size. That is, if someone prefers to receive $11 in 2 days rather than $1 today, model A assumes that person would also prefer $1,000,011 in 2 days to $1,000,001 today. Both imply d = 5 so that if d exceeds the internal d₀ in the first scenario, implying that F should be preferred to P, d will also exceed d₀ in the second. However, in the second scenario, just contemplating the impact of becoming a millionaire right now is likely to affect the internal criterion d₀ in the sense of requiring greater “wages for waiting.” A more flexible model is therefore to suggest that the rate parameter may vary with the size of P. For instance, d may be given by a power law scaling of a rate parameter d_z which is stable and does not vary depending on the size of the choices offered,. Hence, d = P^zd_z. Therefore, we have:

(37)

If z = 0, d_z is not size-sensitive at all, and we have model A. If z = 1, d_z is highly size-sensitive and in fact we have the simple hyperbolic model H. Thus the size parameter z links A with H.

An alternative is to scale d by F^−z (Reference KirbyKirby, 1997), though see Section 5.1 for arguments why P is preferred to F.

4.1 Killeen’s additive utility model (K, k; m, τ)

Reference KilleenKilleen (2009) started from the assumption that the marginal change in utility with respect to time follows a power law; also, that utility and value are themselves related by a power law. Taken together these were shown to yield an equation of the form: (Reference KilleenKilleen, 2009, equation 6, p. 605), where his v_t, v, α, β, t, and λ are respectively our P, F, m, τ, T,and k. He explained as follows: “this additive utility discount function is the central contribution of this article. It is additive because the (negative) utility of a delay is added to the nominal utility of the deferred good.” (p. 605). Some rearrangement isolates the rate parameter to give our canonical form:

(38)

Clearly, when both m = 1 and τ = 1, we have the simple arithmetic model. The numerator can be written as:

So that as m → 0, it becomes:

m.log(F) – m.log(P) = m.log(F/P)

But since m is a common factor, it can be treated as a scaling constant, meaning that K nests E when m → 0 and τ = 1. It also nests the particular t = 0 case of Read’s model, which occurs in K when m → 0 and τ is a free parameter. Killeen’s used his model to survey aggregated data from past research, finding that m ≃ .15, and τ ≃ .53. Both parameters are smaller than other researchers have estimated, and in particular m < τ , which is contrary to evidence collected by researchers in connection with H_m and H_τ. Note, however, that within the datasets that Killeen examined, the number of observations used to fit the model was never greater than eight, which is rather small for non-linear estimation.

4.2 Killeen, intervals, and started time (K_τ, k_τ; m, τ)

Killeen’s additive utility model can be generalized to time intervals that start in the future (at time t ≠ 0), either with a power law on the interval, as modeled in Reference ReadRead (2001) and Section 3.3:

(39)

or with a power on each point in time itself:

(40)

There is an important distinction between these models for time exponents θ or τ if they are ≪ 1, which exaggeratedly condense distant time. In our first generalization of Killeen, as θ → 0, the denominator (T − t)^θ → 1, which means that people would be completely time-insensitive, and therefore presumably always choose F over f. However, in the second generalization, as τ → 0, the denominator (T^τ – t^τ) → τ .log(T/t), meaning that at this extreme people become only logarithmically (in)sensitive to time. Started time does not affect (39) since the start is added to both T and t, which cancel out.

Because of the symmetry in the treatment of money and time in the k_τ generalizationFootnote ³, and to allow the possibility that subjective time is logarithmic, as argued in recent research (Zaubermann et al, 2009), we now focus attention on k_τ. But immediately we hit a problem, which is that if t = 0 (as it usually is in intertemporal choice questionnaires), equation (39) cannot be distinguished from (40), so that we end up with the same time insensitivity as in k_θ when τ → 0. Persisting with the logarithmic limit idea doesn’t help either because of divide-by-zero problems.

As argued earlier, because of log(0) problems a useful generalization of Killeen is to use started timeFootnote ⁴:

(41)

where T_* = T + ε, and t_* = t + ε , and ε is small. This form is flexible enough to capture many possible states of the world, as reflected in the recovered exponents m and τ . If m → 0, subjective money (F, f) would be logarithmic. If τ → 0 started time (T_*, t_*) would be logarithmic. If m and τ were close to 1, money and (started) time would be linear (objective), while intermediate values of m and τ would indicate that subjective time and money have less-than-logarithmic curvature. It is also possible that both m and / or τ could be > 1.

4.3 Dual hyperboloid with subjective money and time (H_mτ, h_mτ; m, τ)

As a robustness analysis, Reference Doyle and ChenDoyle and Chen (2012) used power exponents to model time and money for A, which then becomes model K, and for the hyperbolic, which then becomes:

(42)

This nests a number of models already considered. The rate parameter h_m for Green and Myerson’s hyperboloid model is the special case when τ = 1. The rate parameter h_τ for Rachlin’s hyperboloid occurs when m = 1; and the hyperbolic rate parameter h occurs when both m = 1 and τ = 1. Furthermore, when m → 0 we have:

(43)

Which is model B (Read), and b = h _{m τ}. Note also that the subjective-time exponential only has one additional parameter, because any attempt to introduce a new exponent for money results in , so that m just becomes a scaling constant. The model Reference Ebert and PrelecEbert and Prelec (2007) reject, equation (30), occurs when m → 0.

In conclusion, most of the models considered so far can be generated from equation (42) as special cases. However, the CS model that Elbert and Prelec do propose, equation (28), has the exponent acting on all of log(F/P), not just what is in the brackets. Therefore, CS cannot be derived.

Similarly, the general hyperbola (Reference HarveyHarvey, 1986) cannot be derived as a special case of equation (42). It has the form . As Reference Loewenstein and PrelecLoewenstein and Prelec (1992) state: “The α -coefficient determines how much the function departs from constant discounting; the limiting case, as α goes to zero, is the exponential discount function, e^−βT.” When β=α , it is the simple hyperbolic, which we can write as P = F / (1 + βT), as in Section 2.2. Therefore β may be treated as a rate parameter for the special case of both hyperbolic and exponential, and in the more general case, . Finally, as α becomes large, the discounting function becomes increasingly step-like.

4.4 Benhabib et al.’s generalization of E, H, qH, and X (G, g; β, χ)

Reference Benhabib, Bisin and SchotterBenhabib, Bisin and Schotter (2004) presented their model as a discount function:

(44)

Substituting D = P/F and re-arranging, we have:

When n → 0, we have:

Thus:

(45)

which is E if the quasi-hyperbolic multiplier β = 1, and the analogous additive term χ = 0. If β = 1, and χ = 0, and n = –1, we have H, the simple hyperbolic.

We begin to see here how elements of different models may be combined to good effect. In this example the authors were able to test whether the multiplier β or additive element χ was more predictive of behavior. In their analyses, it was the latter. They also noted that their particular data had little power to distinguish between functional forms H (n = -1) and E (n = 0), though the main point to note here is that their generalization model does allow for testing this possibility.

4.5 Hyperboloid over intervals (H_mω, h_mω; m, ω)

Reference Green, Myerson and MacauxGreen, Myerson and Macaux (2005) extended Green and Myerson’s model described in Section 3.1.1 to include time intervals that do not begin at t=0. Starting from their statement of the model in their equation A16, we derive the following equation for the rate parameter h_mw. Letting M = (F/f)^m – 1 be the numerator, exactly as in Section 3.1.1, then:

(46)

Clearly, when t = 0, this model reduces to H_m. But when t > 0, the authors identified different models according to the values of ω and m.

Equation (46)a: ω = 0 is their elimination by aspects model. One aspect that is common to the two choices (f, t) and (F, T) is the wait to the first choice, and so it is eliminated, reducing the time component to (T – t). We should also note that f is common to the monetary choices, but this aspect is left untouched in 37a.

Equation (46)b: m = ω = 1 is their present value comparison model. In this model, people are assumed to discount both F and f to present values, using the simple hyperbolic model in Section 2.2 (i.e., m = 1).

Equation (46)c: m = 1 is their common aspect attenuation model. Once again, F and f are assumed to be hyperbolically discounted to a present value, but this model assumes that an extra attenuation factor, operationalized by the weight ω , may act on f.

One point to note from the general statement in (46) is that although a money-only component can be identified in the numerator, the subjective time component in the denominator is not independent of money, because the denominator in (46) has the additional term ω tM, and M has terms involving F and f. This contrasts with all other models presented thus far. Finally, equation (46) implies that if (F ≫ f), M could become large enough so that (T – t) < ω tM, thus turning the rate parameter negative.

5.1 Size-sensitive J model (J, j; z, m, τ)

If we combine the size-sensitivity component described in 3.7 with the started-time version of Killeen’s model extended in Section 4.2 to cope with discounting over non-present time intervals, we get a general model that generates many of the above as special cases:

(47)

First, we use the triplet (z, m, τ), each taking values of 0 or 1 to describe whether money and time are modeled as linear (m = 1, τ = 1) or logarithmic (m → 0, τ → 0); and whether there is size-sensitivity (if z = 0 there is not). J nests the following simpler models: Exponential is (0,0,1); Hyperbolic is (1,1,1); Arithmetic is (0,1,1); Roelofsma, if υ is take to be 1, is (0, 0, 0). Moreover, other models can be stated as partially constrained versions of J: Green and Myerson is (m, m, 1), i.e. z = m; Rachlin is (1, 1, τ ); Read with t = 0 and t_* = 1 is (0, 0, τ ); Size-sensitive Arithmetic is (z, 1, 1); the general hyperboloid H_mτ in 4.3 is (m, m, τ ); and Killeen is (0, m, τ ). Also nested in J are several of the memory trace models described in Reference Yi, Landes and BickelYi, Landes, and Bickel (2009), namely “power” (0, 0, 0)—see also Roelofsma; “exponential power” (0, 0, .5), and “hyperbolic power” (1, 1, .5).

In being able to generate these alternative models, J has a number of useful properties. First, there is the simple matter of parsimony. We need hold in mind a single equation from which many others may be generated. Second, the generation process encourages us to consider and investigate other models that could be generated from the triplets, but have not. For instance, there are eight possible fully-constrained, simple rate parameter models. Each is a combination of the 0 or 1 for each of the triplet values z, m, and τ . Only four of these models have been considered: what of the other four? Similarly, though less mechanically, new models may be considered with partial constraints. Last, but not least, one may use model J with no constraints to recover parameters for z, m, and τ . The fact that so many models may be generated from this form hints that the best fitting model may be a compromise between of all of them, as should be evident if the parameters z, m, and τ were all to fall between 0 and 1.

This model scales by P^−z rather than F^−z because the latter is less productive in generating other models as special cases, and also because it is more likely that P, will be used as the given situation, against which F will be compared, rather than vice versa. After all, P is mentioned first in the typical question frame (e.g. “Would you prefer to receive $70 now, or $100 in 20 days?”), and is on offer right now. Thus F will be evaluated in terms of P. Nonetheless, it may be possible to prime F to fulfill the role of given information, as in the question frame: “Assuming you can receive $100 in 20 days’ time, would you accept $70 now instead?” Thus biasing P to be evaluated in terms of F. Manipulating the salience of P relative F, or vice versa, can be taken a stage further in matching tasks where people estimate their indifference points between P and F. Thus if people are required to adjust P for a fixed F they will be biased towards scaling by F^−z; whereas if they adjust F for a given P, they will be biased towards scaling by P^−z. Using F^−z gives rise to a parallel set of models to those presented here. But the degree to which P^−z outperforms F^−z as a scaling component, or vice versa, and their susceptibility to manipulation are all matters for future empirical investigation.

5.2 Discounting by intervals, DBI (I, I; m, τ, θ)

Like Read’s model, Reference Scholten and ReadScholten and Read’s (2006) discounting by intervals (DBI) model is designed to highlight phenomena which occur over several intervals that do not necessarily include the special one bounded at t=0. Nonetheless, as a viable model it must be able to stand comparison with other models when t=0. The rate parameter for DBI is:

(48)

This model extends Read’s in two ways. First, money is modeled as in Green and Myerson with a to-be-estimated exponent m, rather than with log(F/f), as in Read. This is more flexible because the power form nests the logarithmic money form. Second, although there is still an exponent on the time interval, the times in the interval are themselves subjective. Therefore, Read’s model is nested in DBI as the special case where m → 0, and τ = 1. Note that if t = 0, the denominator becomes T^τθ. Given that τ θ only occur together, they can be treated as a single parameter, and DBI becomes the generalized hyperboloid H_mτ. Also, if θ = 1, we approximate J (m, m, τ ) if τ is sufficiently far from 0 so that started and objective time are effectively equivalent.

Finally, the nesting of time and time-interval exponents in the denominator (T^τ – t^τ)^θ means that care must be taken in experimentally estimating the separate parameters.

6.1 Two rate parameters—two exponentials discounting (E_βδ, r_β, r_δ; w)

This model is related to the quasi-hyperbolic model in its intent to separate short-term from long-term processes. But, whereas qH separates events at t=0 into a qualitatively distinct category from those at t>0, this model deals with the same short/long-term issue in a more graded manner. Carrying over the same terminology as used in qH, Reference McClure, Laibson, Loewensein and CohenMcClure, Laibson, Loewenstein, and Cohen (2007) suggested a β -system associated with limbic areas of the brain, that is impulsive, myopic, and discounts at a high rate; and a δ -system, associated with prefrontal and parietal cortical regions (“higher man”), which discounts at lower rates. Each of these sub-systems is assumed to discount according to the exponential model, with the overall discount rate being a weighted sum of each sub-system:

(49)

where if the first term represents the β-system, then r_β > r_δ. Quite explicitly, there are two rate parameters.

6.2 Scholten and Read’s intertemporal tradeoff model—7 parameters (Y, γ, τ, ε_m, ε_τ, and three of {a₁, a₂, b₁, b₂})

Reference Scholten and ReadScholten and Read (2010) provide a densely argued justification for a complex model which they present in terms of an indifference point between the perk of additional compensation, and the irk of waiting for it. They define the money value function on a future payment F to be:

(50)

and a time-weighting function to be:

(51)

They define an “effective compensation” to be:

(52)

and an “effective interval” between t and T to be:

(53)

The point of indifference is when:

(54)

such that if then the irk of waiting is greater than the perk of compensation, so DM chooses f at t, rather than F at T, and vice versa if However, Q_τ and Q_m are not simple scaling factors, but two-part linear functions:

(55)

ε_τ is a threshold below which effective time intervals (here just x) are weighted more steeply than above-threshold. Note the similarity of intent with the two exponentials model. Similarly,

(56)

If the indifference equation is written out in full, it is possible to see that a discount function of the form f/F cannot be extracted: nor does limiting analysis to t=0 overcome the problem. Similarly, no simple rate parameter emerges from this model. It follows that the indifference equation is the only simple way to present the model. The model requires three parameters to specify the two-part linear function for time (a₁, a₂, ε _τ), three more to specify the equivalent parameters for money (b₁, b₂, ε _m), as well as τ and γ to specify the time-weighting and value functions, respectively. However, the authors note that “if only the relative magnitude of scaled effective differences is of interest, one of these scaling constants {a₁, a₂, b₁, b₂} can be set to unity.” (p. 934). This model therefore requires seven parameters to be estimated. So clearly, one of the biggest problems in testing this model is the number of parameters to be recovered, which means designing choice problems over which the parameters are sufficiently independent of each other, and the number of observations sufficiently large in order to estimate the parameters adequately.

The tradeoff model treats time and money symmetrically, which we argued in 1.3.4 should be the default position for a quasi-psychophysical theory of time and money. Indeed, the authors explicitly deny that people are discounting in any way that an accountant would recognize, espousing instead a thoroughly psychological perspective on intertemporal choice.

7.1 Models have similar forms

The world is simpler than a list of twenty or so models might suggest. Our presentation of models via their rate parameters emphasizes the following common structure:

(57)

where s(.) and S(.) are functions that return subjective perceptions of the money and time aspects, respectively, and ⊗ is an operation by which these become combined. According to this view, if someone operates according to the exponential model it is because his/her subjective perception of money is logarithmic, while his/her subjective perception of time is linear. In (57), subjective perceptions then get combined by the operation ⊗. In most cases, ⊗ is arithmetic division, but can be subtraction, or conceivably something else. The separability of s and S in the mathematics implies that subjective perceptions of time and money should take place independently of each other. In most models, objective time and money are transformed into subjective equivalents by power laws, or logs, but other transformations are possible.

To further simplify the map of the models surveyed here, note that half are special cases of model J, and the present-bias models of qH and X could also easily be incorporated into J. Two further themes that lie outside J are: the time impatience models of Section 3.6 that go to town on the modeling of time itself; and models that place power laws on time intervals rather than on points of time—Sections 3.3 and 5.2. Finally, we are left with just four models that are not captured by these themes. First there is the two-exponentials model—which is just a doubling up of the reference model E, as if we had two normative discounters in the same head: one patient (δ ), the other impatient (β ). Then there is Reference Scholten and ReadSchotten and Read’s (2010) trade-off model I, which for all its complexity is still recognisable as an assemblage of parameter-based treatments of subjective money, time, and intervals. H_mω, although an innocuous extension of the hyperboloid to time intervals turns out to be the one model for which time and money cannot be cleanly separated.

7.2 Speculative models

We end with a sample of speculative models (newborns) that have clearly been assembled from the components of existing models. Being able to mix and match almost indefinitely emphasizes the family resemblances that exist between models, and helps to compare and contrast treatments of time and money. Some of these examples are semi-serious; others are intended to provoke thought.

7.2.1 Arithmetic interval model

One of the innovations in B and I is that intervals and subjective intervals are treated as if psychophysical elements in their own right, which may then be investigated using power laws, and so on. We cross this with Killeen (extended) to give the hybrid:

(58)

For mnemonic value note that µ is “mu” Greek m; and θ “theta” sounds vaguely like T.

7.2.2 Fully additive model

Instead of treating ⊗ as a division in 7.2.1, we could treat it as a subtraction, echoing the difference models examined in Stevenson (1986):

(59)

The weighting coefficient w cannot simply be interpreted as the perceived relative importance of time, because it also rescales for m, µ, τ, θ, and units of time and money. However, Stevenson found little support for this way of combining time and money, compared with multiplicative (ratio) forms.

7.2.3 Fully multiplicative hyperbolic model

Analogously, if H treats money as a percentage increase (that gets averaged over time), then a model that also treated time as a percentage increase would look like:

(60)

This example highlights the point that the default setting in models is to treat time as additive, whereas money is treated as multiplicative. From the point of view of Reference StevensStevens’ (1946) levels of measurement, both are ratio scale measurement, with properly defined zeros. So if in theory we can treat time and money symmetrically, what is to stop the naïve behavioral theorist thinking likewise?

7.2.4 Time-size and money-size sensitive model

Model J only treated money as size-sensitive. A symmetrical treatment of time and money is instead:

(61)

where g is the time analog of z for money, implying that someone’s criterion rate parameter may vary, not just with the size of f, but also how far in the future f is offered.

7.2.5 Arithmetic present premium model

A present premium can be incorporated into other models. Using both the multiplicative and additive components of Reference Benhabib, Bisin and SchotterBenhabib et al’s (2004) generalizing model in Section 4.4, the arithmetic model would become, for instance:

(62)