Mathematical methods for economic theory

4.1 Optimization: introduction

Decision-makers (e.g. consumers, firms, governments) in standard economic theory are assumed to be “rational”. That is, each decision-maker is assumed to have a preference ordering over the outcomes to which her actions lead and to choose an action, among those feasible, that is most preferred according to this ordering. We usually make assumptions that guarantee that a decision-maker's preference ordering is represented by a 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 where 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

We assume, for example, that firms choose input bundles to minimize the cost of producing any given output; an analysis of the problem of minimizing the cost of achieving a certain payoff greatly facilitates the study of a payoff-maximizing consumer.

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.