On a dynamical model of happiness

Eduardo Liz¹ ; Sergei Trofimchuk²

¹ Departamento de Matemática Aplicada II, Universidade de Vigo, 36310 Vigo, Spain
² Instituto de Matemáticas, Universidad de Talca, Casilla 747, Talca, Chile

Mathematical modelling of natural phenomena, Tome 18 (2023), article no. 10

Résumé

It is now recognized that the personal well-being of an individual can be evaluated numerically. The related utility (“happiness”) profile would give at each instant t the degree u(t) of happiness. The moment-based approach to the evaluation of happiness introduced by the Nobel laureate Daniel Kahneman establishes that the experienced utility of an episode can be derived from real-time measures of the pleasure and pain that the subject experienced during that episode. Since these evaluations consist of two types of utility concepts: instant utility and remembered utility, a dynamic model of happiness based on this approach must be defined by a delay differential equation. Furthermore, the application of the peak-end rule leads to a class of delay-differential equations called differential equations with maxima. We propose a dynamical model for happiness based on differential equations with maxima and provide results which shed some new light on important experimental observations. In particular, our model supports the U-shaped profile of the age-happiness curve, which is a widely observed pattern: well-being is high in youth, falls to a minimum in midlife (midlife crisis), and rises again in old age.

Détail

DOI : 10.1051/mmnp/2023008

Bibliographie

[1] F.M. Andrews and J.P. Robinson, Measures of subjective well-being, in Measures of Personality and Social Psychological Attitudes’, edited by J.P. Robinson, P.R. Shaver and L.S. Wrightsman. Academic Press, San Diego (1991), pp. 61–114.

[2] R. Amundson Quality of life, disability, and hedonic psychology 2010 374 392

[3] J.A.D. Appleby, H. Wu Exponential growth and Gaussian-like fluctuations of solutions of stochastic differential equations with maximum functional 2008 012002

[4] D.D. Bainov and S. Hristova, Differential Equations with Maxima. Chapman Hall/CRC, New York (2011).

[5] N.R. Bantsur, O.P. Trofimchuk Existence and stability of the periodic and almost periodic solutions of quasilinear systems with maxima 1998 847 856

[6] J. Bentham, An Introduction to the Principle of Morals and Legislations (1789), reprinted Blackwell, Oxford (1948).

[7] D.G. Blanchflower Is happiness U-shaped everywhere? Age and subjective well-being in 145 countries 2021 575 624

[8] D.G. Blanchflower, A.J. Oswald Is well-being U-shaped over the life cycle? 2008 1733 1749

[9] D.G. Blanchflower and A.J. Oswald, Do humans suffer a psychological low in midlife? Two approaches (with and without controls) in seven data sets, in The Economics of Happiness, edited by M. Rojas. Springer, Cham (2019), pp. 439–453.

[10] P. Brickman, D. Coates, R. Janoff-Bulman Lottery winners and accident victims: Is happiness relative? 1978 917 927

[11] I. Cojuharenco, D. Ryvkin Peak-End rule versus average utility: How utility aggregation affects evaluations of experiences 2008 326 335

[12] S. Dashkovskiy, S. Hristova, K. Sapozhnikova Stability properties of systems with maximum 2020 35 58

[13] A.M. Do, A.V. Rupert, G. Wolford Evaluations of pleasurable experiences: the peak-end rule 2008 96 98

[14] P. Dolan, Happiness by design: Change what you do, not how you think. Penguin Group, New York (2014).

[15] F.Y. Edgeworth, Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences (1881), reprinted M. Kelly, New York (1967).

[16] J. Elster and G. Loewenstein, Utility from memory and anticipation, in 'Choice over time. Russell Sage Foundation, New York (1992), pp. 213–234, Chapter 9.

[17] F. Feldman, What is this thing called happiness? Oxford University Press, New York (2010).

[18] B.L. Fredrickson Extracting meaning from past affective experiences: the importance of peaks, ends, and specific emotions 2000 577 606

[19] B.L. Fredrickson, D. Kahneman Duration neglect in retrospective evaluations of affective episodes 1993 45 55

[20] B.S. Frey, A. Stutzer What can economists learn from happiness research? 2002 402 435

[21] L. Graham, A.J. Oswald Hedonic capital, adaptation and resilience 2010 372 384

[22] A. Ivanov, E. Liz, S. Trofimchuk Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima 2002 277 295

[23] D. Kahneman, Objective happiness, in Well-Being: The Foundations of Hedonic Psychology, edited by D. Kahneman, E. Diener and N. Schwarz. The Russell Sage Foundation, New York (1999), pp. 3–25, Chapter 1.

[24] D. Kahneman, Experienced utility and objective happiness: A moment-based approach, in Choices, Values, and Frames, edited by D. Kahneman and A. Tversky. Cambridge University Press, New York (2000), Chapter 37.

[25] D. Kahneman, Evaluation by moments: Past and future, in Choices, Values, and Frames, edited by D. Kahneman and A. Tversky. Cambridge University Press, New York (2000), Chapter 38.

[26] D. Kahneman Maps of bounded rationality: psychology for behavioral economics 2003 1449 1475

[27] D. Kahneman, B.L. Fredrickson Ch. A. Schreiber and D.A. Redelmeier, When more pain is preferred to less: adding a better end 1993 401 405

[28] D. Kahneman, P.P. Wakker, R. Sarin Back to Bentham? Explorations of experienced utility 1997 375 406

[29] R. Larsen and B. Fredrickson, Measurement issues in emotion research, in Well-Being: The Foundations of Hedonic Psychology, edited by D. Kahneman, E. Diener and N. Schwarz. The Russell Sage Foundation, New York (1999), pp. 40–60, Chapter 3.

[30] M. Pinto, S. Trofimchuk Stability and existence of multiple periodic solutions for a quasilinear differential equation with maxima 2000 1103 1118

[31] A. Sherman, T. Shavit, G. Barokas A Dynamic model on happiness and exogenous wealth shock: the case of lottery winners 2020 117 137

[32] J.C. Sprott Dynamical models of happiness 2005 23 36

[33] J. Touboul, A. Romagnoni, R. Schwartz On the dynamic interplay between positive and negative affects 2017 897 936

[34] O. Trofymchuk, E. Liz, S. Trofimchuk The peak-end rule and its dynamic realization through differential equations with maxima 2023 507 536

[35] H.D. Voulov, D.D. Bainov On the asymptotic stability of differential equations with “maxima 1991 385 420

[36] A. Weiss, J.E. King, M. Inoue, T. Matsuzawa, A.J. Oswald Evidence for a midlife crisis in great apes consistent with the U-shape in human well-being 2012 19949 19952

[37] Yhappyliving, The Real Roots of Midlife Crisis (2016), document available in https://yhappiness.co.uk/post/136818075560/