# A New Approach to Intertemporal Choice: The Delay Function

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Delay Function

**Definition**

**1.**

**Example**

**1.**

- (i)
- ${\Phi}_{s}(s,t)=t$ (in particular, ${\Phi}_{s}(s,0)=0$).
- (ii)
- ${\Phi}_{l}(s,t)$ is strictly increasing with respect to t.
- (iii)
- ${\Phi}_{l}(s,t)$ is strictly increasing with respect to l.
- (iv)
- ${\Phi}_{l}(s,t)$ is strictly decreasing with respect to s.

**Definition**

**2.**

- (i)
- $F(m,0)=m$, for every $m\in \mathcal{M}\cup \left\{0\right\}$.
- (ii)
- $F(0,z)=0$, for every $z\in \mathcal{T}$.
- (iii)
- $F(m,z)$ is strictly decreasing with respect to z.
- (iv)
- $F(m,z)$ is strictly increasing with respect to m.

## 3. Discount and Delay Functions

#### 3.1. Discount from Delay Functions

**Remark**

**1.**

**Proposition**

**1.**

**Proof.**

- (i)
- (ii)
- (iii)
- Assume $t<{t}^{\prime}$. By condition (iv) of Definition 1, ${\Phi}_{m}^{-1}(\xb7,0)\left(z\right)$ is strictly decreasing with respect to z and then$${\Phi}_{m}^{-1}(\xb7,0)\left(t\right)>{\Phi}_{m}^{-1}(\xb7,0)\left({t}^{\prime}\right)$$$$F(m,t)>F(m,{t}^{\prime}).$$Consequently, $F(m,z)$ is strictly decreasing with respect to z.
- (iv)
- Assume $s<l$. By condition (iii) of Definition 1, ${\Phi}_{m}^{-1}(\xb7,0)\left(z\right)$ is strictly increasing with respect to m and then$${\Phi}_{s}^{-1}(\xb7,0)\left(z\right)<{\Phi}_{l}^{-1}(\xb7,0)\left(z\right)$$$$F(s,z)<F(l,z).$$Consequently, $F(m,z)$ is strictly increasing with respect to m.

**Example**

**2.**

**Example**

**3.**

**Proposition**

**2.**

**Proof.**

**Remark**

**2.**

**Example**

**4.**

**Definition**

**3.**

**Proposition**

**3.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Example**

**5.**

#### 3.2. Delay from Discount Functions

**Definition**

**4.**

**Remark**

**3.**

## 4. Measures of Inconsistency with Delay Functions

#### 4.1. The Instantaneous Variation Rate

#### 4.2. Prelec’s Measure of Inconsistency

**Example**

**6.**

- $\frac{\partial {\Phi}_{l}(s,t)}{\partial l}=\frac{1+kt}{ks}=\frac{\partial {\Phi}_{l}(s,t)}{\partial l}{|}_{l=s}$.
- $\frac{\partial}{\partial t}ln\frac{\partial {\Phi}_{l}(s,t)}{\partial l}{|}_{l=s}=\frac{k}{1+kt}$.
- $P\left(t\right)=\frac{k}{1+kt}$.

- $\frac{\partial {\Phi}_{l}(s,t)}{\partial l}{|}_{l=s}=\frac{\partial {\Phi}_{1}\left(\right)open="("\; close=")">\frac{s}{l},t}{}\partial \left(\right)open="("\; close=")">\frac{s}{l}{|}_{l=s}\frac{\partial \left(\right)open="("\; close=")">\frac{s}{l}}{}\partial l$.
- $ln\frac{\partial {\Phi}_{l}(s,t)}{\partial l}{|}_{l=s}=ln\frac{\partial {\Phi}_{1}(m,t)}{\partial m}{|}_{m=1}+lns$.
- $P\left(t\right)=\frac{\partial}{\partial t}ln\frac{\partial {\Phi}_{l}(s,t)}{\partial l}{|}_{l=s}=\frac{\partial}{\partial t}ln\frac{\partial {\Phi}_{1}(m,t)}{\partial m}{|}_{m=1}$.

**Example**

**7.**

- $\frac{\partial {\Phi}_{1}(m,t)}{\partial m}=-\frac{1}{m}[1+{tan}^{2}(arctant-lnm)]$.
- $\frac{\partial {\Phi}_{1}(m,t)}{\partial m}{|}_{m=1}=1+{t}^{2}$.
- $\frac{\partial}{\partial t}ln\frac{\partial {\Phi}_{1}(m,t)}{\partial m}{|}_{m=1}=\frac{2t}{1+{t}^{2}}$.
- $P\left(t\right)=\frac{2t}{1+{t}^{2}}$.

## 5. Types of Impatience with Delay Functions

- Decreasing impatience holds if $\sigma <\tau $.
- Increasing impatience holds if $\sigma >\tau $.
- Constant impatience (stationarity) holds if $\sigma =\tau $.

- (i)
- Decreasing impatience holds if $\sigma <\tau $. In this case,$$\sigma <\tau ={\Phi}_{l}(s,t+\sigma )-{\Phi}_{l}(s,t).$$So,$$\sigma +{\Phi}_{l}(s,t)<{\Phi}_{l}(s,t+\sigma ).$$
- -
- More specifically, moderately decreasing impatience holds if also $t\tau <{t}^{\prime}\sigma $. Simple algebra shows that this condition is equivalent to$${\Phi}_{l}(s,t+\sigma )<\frac{\sigma +t}{t}{\Phi}_{l}(s,t).$$
- -
- On the other hand, strongly decreasing impatience holds if also $t\tau \ge {t}^{\prime}\sigma $. Now, this condition is equivalent to$${\Phi}_{l}(s,t+\sigma )\ge \frac{\sigma +t}{t}{\Phi}_{l}(s,t).$$

- (ii)
- Increasing impatience holds if$$\sigma +{\Phi}_{l}(s,t)>{\Phi}_{l}(s,t+\sigma ).$$
- (iii)
- Finally, constant impatience (stationarity) holds if$$\sigma +{\Phi}_{l}(s,t)={\Phi}_{l}(s,t+\sigma ).$$

#### 5.1. Characterizing Constant, Decreasing, and Increasing Impatience

- $\psi (s,s)=\psi (l,l)=0$.
- $\psi (s,l)$ is strictly increasing with respect to l.
- $\psi (s,l)$ is strictly decreasing with respect to s.

- ${\Phi}_{s}(s,t)=t+\psi (s,s)=t$, since $\psi (s,s)=0$.
- ${\Phi}_{l}(s,t)$ is strictly increasing with respect to t, since obviously ${t}_{1}<{t}_{2}$ implies$${t}_{1}+\psi (s,l)<{t}_{2}+\psi (s,l).$$
- ${\Phi}_{l}(s,t)$ is strictly increasing with respect to l, since ${l}_{1}<{l}_{2}$ implies$$\psi (s,{l}_{1})<\psi (s,{l}_{2})$$
- ${\Phi}_{l}(s,t)$ is strictly decreasing with respect to s, since ${s}_{1}<{s}_{2}$ implies$$\psi ({s}_{1},l)>\psi ({s}_{2},l)$$

**Example**

**8.**

#### 5.2. Particular Cases

- $\psi (s,l)=g\left(l\right)-g\left(s\right)$, where we obtain ${\Phi}_{l}(s,t)=t+g\left(l\right)-g\left(s\right)$. In this case, if $t=0$, then$${t}^{\prime}=0+g\left(l\right)-g\left(F(l,{t}^{\prime})\right),$$$$g\left(F(l,{t}^{\prime})\right)=g\left(l\right)-{t}^{\prime}$$$$F(l,{t}^{\prime})={g}^{-1}[g\left(l\right)-{t}^{\prime}],$$
- Another way to generate delay functions is by considering$$\psi (s,l)=\frac{g\left(l\right)}{g\left(s\right)}-1,$$$${\Phi}_{l}(s,t)=t+\frac{g\left(l\right)}{g\left(s\right)}-1.$$If $t=0$, then$${t}^{\prime}=0+\frac{g\left(l\right)}{g\left(F(l,{t}^{\prime})\right)}-1,$$$$g\left(F(l,{t}^{\prime})\right)=\frac{g\left(l\right)}{t+1}$$$$F(l,{t}^{\prime})={g}^{-1}\left(\right)open="["\; close="]">\frac{g\left(l\right)}{t+1}$$
- Another way to generate delay functions is$${\Phi}_{l}(s,t)={f}^{-1}[\psi (s,l)+f\left(t\right)],$$$${t}^{\prime}={f}^{-1}[\psi (l,F(l,{t}^{\prime}))+f\left(0\right)],$$$$f\left({t}^{\prime}\right)=\psi (l,F(l,{t}^{\prime}))+f\left(0\right),$$$$F(l,{t}^{\prime})={\psi}^{-1}(l,\xb7)[f\left({t}^{\prime}\right)-f\left(0\right)].$$

**Example**

**9.**

#### 5.3. Characterizing Strongly and Moderately Decreasing Impatience

- $\xi (s,s)=\xi (l,l)=1$.
- $\xi (s,l)$ is strictly increasing with respect to l.
- $\xi (s,l)$ is strictly decreasing with respect to s.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Samuelson, P.A. A note on measurement of utility. Rev. Econ. Stud.
**1937**, 4, 155–161. [Google Scholar] [CrossRef] - Chakraborti, A.; Muni Toke, I.; Patriarca, M.; Abergel, F. Econophysics review 1: Empirical facts. Quant. Financ.
**2011**, 11, 991–1012. [Google Scholar] [CrossRef][Green Version] - Chakraborti, A.; Muni Toke, I.; Patriarca, M.; Abergel, F. Econophysics review 2: Agent-based models. Quant. Financ.
**2011**, 11, 1013–1041. [Google Scholar] [CrossRef][Green Version] - Mantegna, R.N.; Stanley, H.E. Introduction to Econophysics: Correlations and Complexity in Finance; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Zauberman, G. The intertemporal dynamics of consumer lock-in. J. Consum. Res.
**2003**, 30, 405–419. [Google Scholar] [CrossRef] - Kim, B.K.; Zauberman, G. Perception of anticipatory time in temporal discounting. J. Neurosci. Psychol. Econ.
**2009**, 2, 91–101. [Google Scholar] [CrossRef][Green Version] - Zauberman, G.; Kim, B.K.; Malkoc, S.A.; Bettman, J.R. Discounting time and time discounting: Subjective time perception and intertemporal preferences. J. Mark. Res.
**2009**, 46, 543–556. [Google Scholar] [CrossRef] - Cajueiro, D. A note on the relevance of the q-exponential function in the context of intertemporal choices. Phys. A Stat. Mech. Appl.
**2006**, 364, 385–388. [Google Scholar] [CrossRef] - Takahashi, T.; Han, R.; Nakamura, F. Time discounting: Psychophysics of intertemporal and probabilistic choices. J. Behav. Econ. Financ.
**2012**, 5, 10–14. [Google Scholar] [CrossRef][Green Version] - Loewenstein, G. Anticipation and the valuation of delayed consumption. Econ. J.
**1987**, 97, 666–684. [Google Scholar] [CrossRef][Green Version] - Read, D. Intertemporal choice. In Blackwell Handbook of Judgment and Decision Making; Koehler, D., Harvey, N., Eds.; Blackwell: Oxford, UK, 2004; pp. 424–443. [Google Scholar]
- Baker, F.; Johnson, M.W.; Bickel, W.K. Delay discounting in current and never-before cigarette smokers: Similarities and differences across commodity, sign, and magnitude. J. Abnorm. Psychol.
**2003**, 112, 382–392. [Google Scholar] [CrossRef] - Thaler, R. Some empirical evidence on dynamic inconsistency. Econ. Lett.
**1981**, 8, 201–207. [Google Scholar] [CrossRef] - Xu, L.; Liang, Z.Y.; Wang, K.; Li, S.; Jiang, T. Neural mechanism of intertemporal choice: From discounting future gains to future losses. Brain Res.
**2009**, 1261, 65–74. [Google Scholar] [CrossRef] - Prelec, D. Decreasing impatience: A criterion for non-stationary time preference and hyperbolic discounting. Scand. J. Econ.
**2004**, 106, 511–532. [Google Scholar] [CrossRef] - Rohde, K.I.M. The hyperbolic factor: A measure of time inconsistency. J. Risk Uncertain.
**2010**, 41, 125–140. [Google Scholar] [CrossRef][Green Version] - Rohde, K.I.M. An Index to Measure Decreasing Impatience. In Proceedings of the 2015 Risk, Uncertainty and Decision Conference, Milano, Italy, 1–3 June 2015. [Google Scholar]
- Anchugina, N.; Ryan, M.; Slinkoc, A. Mixing discount functions: Implications for collective time preferences. Math. Soc. Sci.
**2019**, 102, 1–14. [Google Scholar] [CrossRef] - Mandelbrot, B.; Fisher, A.; Calvet, L. A Multifractal Model of Asset Returns; Cowles Foundation Discussion Paper #1164; Yale University: New Haven, CT, USA, 1997. [Google Scholar]
- Slanina, F. Essentials of Econophysics Modelling; Oxford University Press: Oxford, UK, 2013. [Google Scholar]
- Takahashi, T. A comparison of intertemporal choices for oneself versus someone else based on Tsallis’ statistics. Phys. A Stat. Mech. Appl.
**2007**, 385, 637–644. [Google Scholar] [CrossRef] - Jin, S. The influence of time perception on intertemporal preference and its psychological mechanism. Open J. Soc. Sci.
**2020**, 8, 236–249. [Google Scholar] [CrossRef][Green Version] - Lu, Y.; Li, Y. Psychophysics of consumer’s sequence preference of e-commerce loan repayments. J. Phys. Conf. Ser.
**2019**, 1302, 042025. [Google Scholar] [CrossRef][Green Version] - Takahashi, T.; Oono, H.; Radford, M. Empirical estimation of consistency parameter in intertemporal choice based on Tsallis’ statistics. Phys. A Stat. Mech. Appl.
**2007**, 381, 338–342. [Google Scholar] [CrossRef] - Cruz Rambaud, S.; Muñoz Torrecillas, M.J. A generalization of the q-exponential discounting function. Phys. A Stat. Mech. Appl.
**2013**, 392, 3045–3050. [Google Scholar] [CrossRef] - Cruz Rambaud, S.; Ventre, V. Deforming time in a nonadditive discount function. Int. J. Intell. Syst.
**2017**, 32, 467–480. [Google Scholar] [CrossRef] - Cruz Rambaud, S.; González Fernández, I.; Ventre, V. Modeling the inconsistency in intertemporal choice: The generalized Weibull discount function and its extension. Ann. Financ.
**2018**, 14, 415–426. [Google Scholar] [CrossRef] - Webb, C.S. Trichotomic discounted utility. Theory Decis.
**2019**, 87, 321–339. [Google Scholar] [CrossRef][Green Version] - Baucells, M.; Heukamp, F.H. Probability and time trade-off. Manag. Sci.
**2012**, 58, 831–842. [Google Scholar] [CrossRef] - Fishburn, P.C.; Rubinstein, A. Time preference. Int. Econ. Rev.
**1982**, 23, 677–694. [Google Scholar] [CrossRef] - Ericson, K.M.; Noor, J. Delay Functions as the Foundations of Time Preference: Testing for Separable Discounted Utility; Working paper of the Boston University; Boston University: Boston, MA, USA, 2015. [Google Scholar]
- Takeuchi, K. Non-parametric test of time consistency: Present bias and future bias. Games Econ. Behav.
**2011**, 71, 456–478. [Google Scholar] [CrossRef] - Cruz Rambaud, S.; Muñoz Torrecillas, M.J. Capitalization Speed of a Financial Law. In Proceedings of the Fourth Italian-Spanish Conference on Financial Mathematics, Alghero, Italy, 28 June–1 July 2001. [Google Scholar]
- Rohde, K.I.M. Measuring decreasing and increasing impatience. Manag. Sci.
**2019**, 65, 1700–1716. [Google Scholar] [CrossRef][Green Version] - Lisei, G. Su un’equazione funzionale collegata alla scindibilità delle leggi finanziarie. Giornalle dell’Istituto Italiano Degli Attuari
**1979**, XLII, 19–24. [Google Scholar] - Cruz Rambaud, S.; González Fernández, I. A measure of inconsistencies in intertemporal choice. PLoS ONE
**2019**, 14, e0224242. [Google Scholar] [CrossRef]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cruz Rambaud, S.; González Fernández, I.
A New Approach to Intertemporal Choice: The Delay Function. *Symmetry* **2020**, *12*, 807.
https://doi.org/10.3390/sym12050807

**AMA Style**

Cruz Rambaud S, González Fernández I.
A New Approach to Intertemporal Choice: The Delay Function. *Symmetry*. 2020; 12(5):807.
https://doi.org/10.3390/sym12050807

**Chicago/Turabian Style**

Cruz Rambaud, Salvador, and Isabel González Fernández.
2020. "A New Approach to Intertemporal Choice: The Delay Function" *Symmetry* 12, no. 5: 807.
https://doi.org/10.3390/sym12050807