The Bellman equation is a fundamental equation in economics that describes the optimal way to make decisions over time. The equation is named after American economist Richard Bellman, who first developed it in the 1950s.
The Bellman equation is important because it provides a framework for thinking about how to make optimal decisions when there are multiple decision points and uncertainty about the future. The equation can be used to analyze a wide range of economic problems, from investment decisions to consumption choices.
The basic idea behind the Bellman equation is that the best decision at any given point in time is the one that will maximize your expected utility over all future periods. This means taking into account not only the immediate benefits or costs of a particular decision, but also the impacts that decision is likely to have on your future welfare.
To make this concrete, consider a simple example of deciding whether to buy a new car or keep your old one. The decision might come down to a trade-off between the upfront cost of buying a new car and the monthly payments you’ll need to make. But it’s also important to think about how long you expect the car to last and how much money you’ll save (or spend) on fuel and maintenance over that time. By taking all of these factors into account, you can arrive at an optimal decision that maximizes your expected utility.
The Bellman equation can be used to solve for the optimal decision in this and other similar situations. In general, the equation takes the form of a differential equation, which can be solved using mathematical methods. Once the optimal decision is known, it can be applied to real-world situations to help make better economic decisions.
While the Bellman equation is a powerful tool, it is important to keep in mind that it only provides a framework for thinking about decisions. The optimal decision in any given situation will depend on the specific details of the problem at hand. As such, the Bellman equation should be used as a starting point for making economic decisions, not as a definitive guide.