Non-constant discounting in finite horizon: The free terminal time case

Abstract

This paper derives the HJB (Hamilton-Jacobi-Bellman) equation for sophisticated agents in a finite horizon dynamic optimization problem with non-constant discounting in a continuous setting, by using a dynamic programming approach. A simple example is used in order to illustrate the applicability of this HJB equation, by suggesting a method for constructing the subgame perfect equilibrium solution to the problem.Conditions for the observational equivalence with an associated problem with constant discounting are analyzed. Special attention is paid to the case of free terminal time. Strotz's model (an eating cake problem of a nonrenewable resource with non-constant discounting) is revisited. JEL Classification: C61; D83; C73Keywords: Non-constant discounting, naive and sophisticated agents, free terminal time, observational equivalence Resumen: En este trabajo se deriva la ecuación de Hamilton-Jacobi-Bellman (HJB) para un agente sofisticado en un problema de optimización dinámica en tiempo continuo con horizonte finito, cuando la tasa de descuento de preferencia temporal es no constante, mediante la resolución de un problema de programación dinámica. Un sencillo ejemplo sirve para ilustrar la aplicabilidad de esta ecuación HJB. En particular, en el mismo se sugiere un método para construir el equilibrio perfecto en subjuegos solución del problema.El caso de tiempo final libre recibe una especial atención. Finalmente, se revisa el modelo de Strotz (un problema tipo "eating cake" de un recurso no renovable con descuento no constante).Palabras clave: Descuento no constante, agentes naive y sofisticados, tiempo final libre, equivalencia observacional

Heterogeneous discounting in economic problems

“…If we look for a time‐consistent decision rule under no commitment (the agent is sophisticated), we must analyze the game theoretic framework in which the final time is decided by the final agent, who is the one to decide whether or not to stop. We can adapt the proof in Marín‐Solano and Navas 12 for the case of non‐constant discounting to our framework. Since T * is optimum from the perspective of the T * ‐agent, if the optimal decision for the ( T * + ε)‐agent, ε>0, is to stop, then from the optimality of T * we get F ( x ( T * ), T * )⩾ V ( x ( T * ), T * ), where V denotes the value function of the T * ‐agent if the terminal time is T * + ε.…”

Time-consistent mean-variance reinsurance-investment strategy for insurers under CEV model

“…Pollak (1968) gave the right solution to the Strotz problem for both naive and sophisticated agents under a logarithmic function. Barro (1999), Karp (2007), Ekeland & Lazrak (2010), Ekeland & Pirvu (2008), Marin-Solano & Navas (2009, Ekeland et al (2012) further considered time-inconsistent preferences problems in different situations. Recently, there has been renewed interest in time-consistent mean-variance portfolio selection problem, see, for example, Björk & Murgoci (2009), Basak & Chabakauri (2010, Kryger & Steffensen (2010), Wang & Forsyth (2011), Czichowsky (2013, Björk et al (2014), Kronborg & Steffensen (2015).…”

Использование Двух Различных Ставок Дисконтирования Доходов И Расходов Для Стратегической Оценки Эффективности Проекта

“…Использование двух различных ставок дисконтирования доходов и расходов для стратегической оценки эффективности проекта. Российское предпринимательство, 16(13), 2053-2066.doi: 10.18334/rp.16.13.486 2055 Необходимо отметить, что дифференциация ставок дисконтирования не является общеупотребительной методологией (Агеев, 2011;Артемьев, Гергерт, Пономарева, 2013;Володин, 2013;Дорошенко, Авилова, 2007;Замбржицкая, Самохин, 2014;Золочевская, Кривошеева, 2014;Попова, 2013;Турыгин, 2007;Успенская, 2013;Черемушкин, 2013;Ahern, Leavy, Byrne, 2014;Bredillet, Tywoniak, Dwivedula, 2015;Caputo, 2013;Freeman, 2009;Grinblatt, Liu, 2008;Klein, Biesenthal, Dehlin, 2015;Marín-Solano, Navas, 2009;Price, 2011;Shimizu, Park, Choi, 2014;Singh, McAllister, Rinks, Jiang, 2010;Too, Weaver, 2014).…”