Mathematical methods for economic theory

4.2 Optimization: definitions

The optimization problems we study take the form where X is a set of n-vectors, f is a function defined on X, x is an n-vector (which we can also write as (x₁, ..., x_n)), and S is a subset of X. We call f the objective function, x the choice variable, and S the constraint set.

To be very clear about what we mean by the “max” operator, here is a precise definition.

A minimizer is defined analogously. An optimization problem may have many maximizers and minimizers, but has at most one maximum and at most one minimum.

Consider the function f of one variable shown in the following figure, with S equal to the blue line segment. Both x* and x** are maximizers of f(x) subject to the constraint x ∈ S, and x" is a minimizer. (The graph of f is horizontal between x₃ and x₄ and between x₅ and x₆.) The number M is the maximum of f(x) subject to x ∈ S and the number m is the minimum.

What about the point x'? This point is decidedly not a maximizer, because f(x*) > f(x'), for example. But it is a maximum among the points close to it. We call such a point a “local maximizer”. Recall that the distance between two n-vectors x and y is the square root of ∑n
i=1(x_i − y_i)² (the length of the straight line joining x and y).

A local minimizer is defined analogously. Sometimes we refer to a maximizer as a global maximizer to emphasize that it is not only a local maximizer. Every global maximizer is, in particular, a local maximizer (ε can take any value), and every global minimizer is a local minimizer.

For the function f in the figure above we have f(x) ≤ f(x') for all x between x₁ and x₂ (for example), so that x' is a local maximizer of f(x); because x' − x₁ = x₂ − x', we can take ε = x' − x₁ in the definition.

But note that the point x" in this figure is also a local maximizer of f(x), even though it is a global minimizer. The function is constant between x₃ and x₄. The point x₄ is (slightly) closer to x" than is the point x₃, so we can take the ε in the definition of a local maximizer to be x₄ − x". For every point x within the distance ε of x", we have f(x) = f(x"), so that in particular f(x) ≤ f(x"). The point is that the definition of a local maximizer does not require the value of the function at nearby points to be less than f(x*), only less than or equal to f(x*).

In economic theory we are almost always interested in global maximizers, not merely local maximizers. For example, the standard theory is that a consumer chooses the bundle she most prefers among all those that are available; she is not satisfied by a bundle that is merely better than the other bundles that are nearby. Similarly, a firm is assumed to choose the input-output vector that maximizes its profit among all those that are feasible; it is not satisfied by an input-output vector that merely yields a higher profit than does similar vectors.

Transforming the objective function

Let g be an increasing function of a single variable. (That is, if z' > z then g(z') > g(z).) Then the set of solutions to the problem is identical to the set of solutions to the problem Why? If x* is a solution to the first problem then by definition But if f(x) ≤ f(x*) then g(f(x)) ≤ g(f(x*)), so that Hence x* is a solution of the second problem.

This fact is sometimes useful: for example, if the objective function is f(x₁, x₂) = xα
1xβ
2 and x₁ and x₂ are positive for all points in the constraint set, it may be easier to work with the logarithm of this function, namely αln x₁ + βln x₂. Since log z is an increasing function, the set of solutions of the problem in which xα
1xβ
2 is the objective function is the same as the set of solutions of the problem is which αln x₁ + βln x₂ is the objective function.

Minimization problems

Throughout the tutorial, I concentrate on maximization problems. What about minimization problems? Any discussion of maximization problems encompasses minimization problems because of a simple fact: any minimization problem can be turned into a maximization problem by multiplying the objective function by minus one. That is, the problem is equivalent to—in particular, has the same set of solutions as—the problem Thus we can solve any minimization problem by multiplying the objective function by minus one applying the results for maximization problems.

Although I emphasize maximization problems, I usually state results separately both for maximization problems and for minimization problems.