If the decision-maker is a classical consumer, for example, then a is a consumption bundle, u is the consumer's utility function, and S is the set of bundles of goods the consumer can afford. If the decision-maker is a classical firm then a is an input-output vector, u(a) is the profit the action a generates, and S is the set of all feasible input-output vectors (as determined by the firm's technology).
Even outside the classical theory, the actions chosen by decision-makers are often modeled as solutions of maximization problems. A firm, for example, may be assumed to maximize its sales, rather than its profit; a consumer may care not only about the bundle of goods she consumes, but also about the bundles of goods the other members of her family consumes, maximizing a function that includes these bundles as well as her own; a government may choose policies to maximize its chance of reelection.
In economic theory we sometimes need to solve a minimization problem of the form
The next three parts of the tutorial develop tools for solving maximization and minimization problems, which are collectively known as optimization problems.
- This part discusses some basic definitions and a fundamental result regarding the existence of an optimum.
- The next part, on interior optima, focuses on conditions for solutions that are strictly inside the constraint set S—“interior” solutions.
- The third part, on equality constraints, discusses the key technique developed by Lagrange for finding the solutions of problems in which S is the set of points that satisfy a set of equations.
- The last part, on the Kuhn-Tucker conditions for problems with inequality constraints, discusses a set of conditions that may be used to find the solutions of any problem in which S is the set of points that satisfy a set of inequalities. These conditions encompass both the conditions for interior optima and those developed by Lagrange. The last section is a summary of the conditions in the previous parts.