Write down an individual’s optimization problem regarding consumption of good and bad behaviors using the Bellman equation representation of lifetime utility (as we did in class). Be sure to include the budget and time constraints and any constructs (i.e., probabilities) that capture uncertainty about the future. In words, what is uncertain in the future to an individual today?

Using the framework above, suppose the individual also values health (i.e., receives utility from health) and that the good and bad behaviors impact health evolution (i.e., how health evolves from one period to the next)? How would you show this in the Bellman formulation and what “constraint” would you have to add (i.e., what function or process needs to be defined)?