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In this module, we looked at constrained optimization. For what models in economics is this their fundamental feature?
This module concludes the unit on optimization. In the previous module, we introduced some of the calculus of n-variable functions. In this module, we apply what we’ve learned to optimization. Optimization in this module comes in three different flavours. First, there is unconstrained optimization. We are given a function and we first need to identify for what values of the variables the function is stationary, that is, for values of the function’s variables in the neighbourhood of the point the function lies on a (hyper)plane. We know from the study of functions of one variable that a point where the function’s derivative is zero, the one-dimensional version of a stationary point, can be a minima or maxima or neither. Again, in the case of a function of n-variables, there are similar conditions involving the partial derivatives of the function which we already know can be arrayed in a Hessian matrix. Similar to optimizing a function of one variable, if the function looks like a hill, that is, it has a concave shape, it can have a maximum. Conversely, if it looks like a bowl, then it will have a minimum. If at a stationary point, the function’s Hessian matrix of second partial derivatives is negative semidefinite, then the function is concave, and if the Hessian is negative definite, strictly so. Some of the conditions under which this holds are reviewed. Next, we will discuss the constrained optimization setting in which the function, also known as the objective function, is subject to constraints. Lagrange multipliers solve this problem. Optimization using Lagrange multipliers is a key technique in economics as many common situations – consumers optimizing their consumption choices subject to a budget constraint, firms maximizing profits subject to a cost function – lend themselves to its application. The Lagrange multipliers will turn out to be the marginal value to the objective function of relaxing the constraints. Finally, maximizing a concave function of n-variables over a convex set, a general and common problem is introduced. The technique to solve that problem involves the Karush-Kuhn-Tucker conditions. Again, Lagrange multipliers will be used and their interpretation in this setting is similar.
In this module, we discussed the conditions for a function of n-variables to have an optimum, which is a maximum or a minimum in various situations. We initially reviewed the situation in which the function is unconstrained. We then examined the situation when the constraints took the form of equalities and used Lagrangean functions to find the optimum. We then examined the situation where the function possessed some form of concavity and the constraint set was convex. The Karhunen-Kuhn-Tucker Theorem provided necessary and sufficient conditions for determining an optimum.