{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T16:46:04Z","timestamp":1769273164267,"version":"3.49.0"},"reference-count":36,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,13]],"date-time":"2020-05-13T00:00:00Z","timestamp":1589328000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00eda y Competitividad","doi-asserted-by":"publisher","award":["DER2016-76053-R"],"award-info":[{"award-number":["DER2016-76053-R"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The framework of this paper is intertemporal choice, which traditionally has been studied with preference relations and discount functions. However, the interest of econophysics in this topic makes time become a central magnitude. Therefore, the aim of this paper is to introduce the concept of delay function and, by using this tool, to analyze the concept of impatience and the different types of inconsistency. In behavioral finance, consistency is correlated with the concept of symmetry because, in this case, the indifference between two rewards does not change when the same delay is added to their respective availability dates. Moreover, we have shown the way to derive a discount (respectively, delay) function starting from the expression of its corresponding delay (respectively, discount) function by requiring some suitable conditions for this construction. Finally, we have deduced the concept of instantaneous variation rate and Prelec\u2019s measure of inconsistency in terms of the delay function.<\/jats:p>","DOI":"10.3390\/sym12050807","type":"journal-article","created":{"date-parts":[[2020,5,14]],"date-time":"2020-05-14T02:55:41Z","timestamp":1589424941000},"page":"807","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A New Approach to Intertemporal Choice: The Delay Function"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1318-1221","authenticated-orcid":false,"given":"Salvador","family":"Cruz Rambaud","sequence":"first","affiliation":[{"name":"Departamento de Econom\u00eda y Empresa, Universidad de Almer\u00eda, La Ca\u00f1ada de San Urbano, s\/n, 04120 Almer\u00eda, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Isabel","family":"Gonz\u00e1lez Fern\u00e1ndez","sequence":"additional","affiliation":[{"name":"Departamento de Econom\u00eda y Empresa, Universidad de Almer\u00eda, La Ca\u00f1ada de San Urbano, s\/n, 04120 Almer\u00eda, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"155","DOI":"10.2307\/2967612","article-title":"A note on measurement of utility","volume":"4","author":"Samuelson","year":"1937","journal-title":"Rev. Econ. Stud."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"991","DOI":"10.1080\/14697688.2010.539248","article-title":"Econophysics review 1: Empirical facts","volume":"11","author":"Chakraborti","year":"2011","journal-title":"Quant. Financ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.1080\/14697688.2010.539249","article-title":"Econophysics review 2: Agent-based models","volume":"11","author":"Chakraborti","year":"2011","journal-title":"Quant. Financ."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Mantegna, R.N., and Stanley, H.E. (1999). Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press.","DOI":"10.1017\/CBO9780511755767"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1086\/378617","article-title":"The intertemporal dynamics of consumer lock-in","volume":"30","author":"Zauberman","year":"2003","journal-title":"J. Consum. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1037\/a0017686","article-title":"Perception of anticipatory time in temporal discounting","volume":"2","author":"Kim","year":"2009","journal-title":"J. Neurosci. Psychol. Econ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1509\/jmkr.46.4.543","article-title":"Discounting time and time discounting: Subjective time perception and intertemporal preferences","volume":"46","author":"Zauberman","year":"2009","journal-title":"J. Mark. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1016\/j.physa.2005.08.056","article-title":"A note on the relevance of the q-exponential function in the context of intertemporal choices","volume":"364","author":"Cajueiro","year":"2006","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_9","first-page":"10","article-title":"Time discounting: Psychophysics of intertemporal and probabilistic choices","volume":"5","author":"Takahashi","year":"2012","journal-title":"J. Behav. Econ. Financ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"666","DOI":"10.2307\/2232929","article-title":"Anticipation and the valuation of delayed consumption","volume":"97","author":"Loewenstein","year":"1987","journal-title":"Econ. J."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Koehler, D., and Harvey, N. (2004). Intertemporal choice. Blackwell Handbook of Judgment and Decision Making, Blackwell.","DOI":"10.1002\/9780470752937"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1037\/0021-843X.112.3.382","article-title":"Delay discounting in current and never-before cigarette smokers: Similarities and differences across commodity, sign, and magnitude","volume":"112","author":"Baker","year":"2003","journal-title":"J. Abnorm. Psychol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/0165-1765(81)90067-7","article-title":"Some empirical evidence on dynamic inconsistency","volume":"8","author":"Thaler","year":"1981","journal-title":"Econ. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.brainres.2008.12.061","article-title":"Neural mechanism of intertemporal choice: From discounting future gains to future losses","volume":"1261","author":"Xu","year":"2009","journal-title":"Brain Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1111\/j.0347-0520.2004.00375.x","article-title":"Decreasing impatience: A criterion for non-stationary time preference and hyperbolic discounting","volume":"106","author":"Prelec","year":"2004","journal-title":"Scand. J. Econ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s11166-010-9100-2","article-title":"The hyperbolic factor: A measure of time inconsistency","volume":"41","author":"Rohde","year":"2010","journal-title":"J. Risk Uncertain."},{"key":"ref_17","unstructured":"Rohde, K.I.M. (2015, January 1\u20133). An Index to Measure Decreasing Impatience. Proceedings of the 2015 Risk, Uncertainty and Decision Conference, Milano, Italy."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.mathsocsci.2019.05.004","article-title":"Mixing discount functions: Implications for collective time preferences","volume":"102","author":"Anchugina","year":"2019","journal-title":"Math. Soc. Sci."},{"key":"ref_19","unstructured":"Mandelbrot, B., Fisher, A., and Calvet, L. (1997). A Multifractal Model of Asset Returns, Yale University. Cowles Foundation Discussion Paper #1164."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Slanina, F. (2013). Essentials of Econophysics Modelling, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780199299683.001.0001"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1016\/j.physa.2007.07.020","article-title":"A comparison of intertemporal choices for oneself versus someone else based on Tsallis\u2019 statistics","volume":"385","author":"Takahashi","year":"2007","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_22","first-page":"236","article-title":"The influence of time perception on intertemporal preference and its psychological mechanism","volume":"8","author":"Jin","year":"2020","journal-title":"Open J. Soc. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"042025","DOI":"10.1088\/1742-6596\/1302\/4\/042025","article-title":"Psychophysics of consumer\u2019s sequence preference of e-commerce loan repayments","volume":"1302","author":"Lu","year":"2019","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/j.physa.2007.03.038","article-title":"Empirical estimation of consistency parameter in intertemporal choice based on Tsallis\u2019 statistics","volume":"381","author":"Takahashi","year":"2007","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3045","DOI":"10.1016\/j.physa.2013.03.009","article-title":"A generalization of the q-exponential discounting function","volume":"392","year":"2013","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1002\/int.21842","article-title":"Deforming time in a nonadditive discount function","volume":"32","author":"Ventre","year":"2017","journal-title":"Int. J. Intell. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/s10436-018-0318-3","article-title":"Modeling the inconsistency in intertemporal choice: The generalized Weibull discount function and its extension","volume":"14","author":"Ventre","year":"2018","journal-title":"Ann. Financ."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s11238-019-09710-x","article-title":"Trichotomic discounted utility","volume":"87","author":"Webb","year":"2019","journal-title":"Theory Decis."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"831","DOI":"10.1287\/mnsc.1110.1450","article-title":"Probability and time trade-off","volume":"58","author":"Baucells","year":"2012","journal-title":"Manag. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"677","DOI":"10.2307\/2526382","article-title":"Time preference","volume":"23","author":"Fishburn","year":"1982","journal-title":"Int. Econ. Rev."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Ericson, K.M., and Noor, J. (2015). Delay Functions as the Foundations of Time Preference: Testing for Separable Discounted Utility, Boston University. Working paper of the Boston University.","DOI":"10.3386\/w21095"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"456","DOI":"10.1016\/j.geb.2010.05.005","article-title":"Non-parametric test of time consistency: Present bias and future bias","volume":"71","author":"Takeuchi","year":"2011","journal-title":"Games Econ. Behav."},{"key":"ref_33","unstructured":"Cruz Rambaud, S., and Mu\u00f1oz Torrecillas, M.J. (July, January 28). Capitalization Speed of a Financial Law. Proceedings of the Fourth Italian-Spanish Conference on Financial Mathematics, Alghero, Italy."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1700","DOI":"10.1287\/mnsc.2017.3015","article-title":"Measuring decreasing and increasing impatience","volume":"65","author":"Rohde","year":"2019","journal-title":"Manag. Sci."},{"key":"ref_35","first-page":"19","article-title":"Su un\u2019equazione funzionale collegata alla scindibilit\u00e0 delle leggi finanziarie","volume":"XLII","author":"Lisei","year":"1979","journal-title":"Giornalle dell\u2019Istituto Italiano Degli Attuari"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Cruz Rambaud, S., and Gonz\u00e1lez Fern\u00e1ndez, I. (2019). A measure of inconsistencies in intertemporal choice. 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