This paper extends the classical consumption and portfolio rules model in continuous time (Merton 1969(Merton , 1971 to the framework of decision-makers with time-inconsistent preferences. The model is solved for different utility functions for both, naive and sophisticated agents, and the results are compared. In order to solve the problem for sophisticated agents, we derive a modified HJB (Hamilton-Jacobi-Bellman) equation. It is illustrated how for CRRA functions within the family of HARA functions (logarithmic and potential cases) the optimal portfolio rule does not depend on the discount rate, but this is not the case for a general utility function, such as the exponential (CARA) utility function.
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
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