13 - Advanced Material: Part I. Continuous Time and the Calculus of Variations

Summary

Learning Objectives

In this chapter we offer a (more) mathematical formulation of the main ideas behind strategic financial planning for individuals. The intended audience for this chapter is upper level or graduate students with sufficient calculus training. We start by reviewing some earlier results – which we presented back in Chapters 2 to 5 without much justification – and then move on to obtain some newer ones. Some of the concepts might seem familiar (and perhaps even repetitive), but we urge you to pay special attention to the new context in which they are being presented. As this chapter (and then Chapter 14) progress, the material will get exceedingly mathematical and involve some advanced concepts from differential equations to stochastic processes. Hang in there (or skip to the numerical examples). Our objective is to provide a complete representation of the model for those who want to derive the equations themselves.

Wages and Salary in Continuous Time

In this section we introduce the notation we will be using in continuous time. Start by assuming that you are x years old, still in school (i.e., not earning any money) and you plan to graduate, start your working life, join the labor force, and earn an income at an age denoted by I. From a modeling perspective, it helps to consider the remainder of your lifecycle as consisting of three distinct stages: study, work, and retirement, although we do acknowledge that in practice some people alternate between and merge these stages over their life.