other things) their initial wealth. Therefore, wealth would be one of their state variables, but there would probably be others. The variables chosen at any given point in time are often called the control variables. For instance, given their current wealth, people might decide how much to consume now. Choosing the control variables now may be equivalent to choosing the next state; more generally, the next state is affected by other factors in addition to the current control. For example, in the simplest case, today's wealth (the state) and consumption (the control) might exactly determine tomorrow's wealth (the new state), though typically other factors will affect tomorrow's wealth too. The dynamic programming approach describes the optimal plan by finding a rule that tells what the controls should be, given any possible value of the state. For example, if consumption (c) depends only on wealth (W), we would seek a rule that gives consumption as a function of wealth. Such a rule, determining the controls as a function of the states, is called a policy function (See Bellman, 1957, Ch. III.2). Finally, by definition, the optimal decision rule is the one that achieves the best possible value of the objective. For example, if someone chooses consumption, given wealth, in order to maximize happiness (assuming happiness H can be represented by a mathematical function, such as a utility function and is something defined by wealth), then each level of wealth will be associated with some highest possible level of happiness, . The best possible value of the objective, written as a function of the state, is called the value function. Bellman showed that a dynamic optimization problem in discrete time can be stated in a recursive, step-by-step form known as backward induction by writing down the relationship between the value function in one period and the value function in the next period. The relationship between these two value functions is called the "Bellman equation". In this approach, the optimal policy in the last time period is specified in advance as a function of the state variable's value at that time, and the resulting optimal value of the objective function is thus expressed in terms of that value of the state variable. Next, the next-to-last period's optimization involves maximizing the sum of that period's period-specific objective function and the optimal value of the future objective function, giving that period's optimal policy contingent upon