## 4.1 Optimization: introduction

*payoff function*(sometimes called

*utility function*), so that we can present the decision-maker's problem as one of choosing an action, among those feasible, that maximizes the value of this function. That is, we write the decision-maker's problem in the form

_{a}

*u*(

*a*) subject to

*a*∈

*S*,

*u*is the decision-maker's payoff function over her actions and

*S*is the set of her feasible actions.

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

_{a}

*u*(

*a*) subject to

*a*∈

*S*.

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.