Optimization
 6 minutes read  1078 wordsDecision problems in economics are predominantly described as optimization problems. For example, suppose that one has to decide among a finite number of alternatives \(\alpha_{1}, \alpha_{2}, \dots, \alpha_{n}\). Alternatives are evaluated based on a payoff function \(u\). The payoff she receives by choosing alternative \(\alpha_{j}\) is given by \(u(\alpha_{j})\). One way to model the agent’s decision is to assume that she chooses the alternative that maximizes the payoff she gains.
We say that \(\alpha_{j}\) maximizes \(u\) or \(\alpha_{j}\) is a maximizer of \(u\) if \(u(a_{j}) \ge u(a_{i})\) for all \(i = 1, \dots, n\). Then, we also say that \(u(a_{j})\) is the maximum of \(u\) or that \(u\) attains its maximum at \(\alpha_{j}\). Analogously we can define the minimizer and minimum of \(u\) based on the condition \(u(\alpha_{j}) \le u(a_{i})\) for all \(i = 1, \dots, n\).
In our small example, optimization reduces to comparing (sorting) all the values \(u(\alpha_{i})\) for \(i = 1, \dots, n\) and picking the greatest or smaller values. Nevertheless, in many business applications, the set of alternatives from which the agent is choosing is either infinite or so large that it is practically impossible to perform all required comparisons for detecting maxima and minima. Luckily, calculus comes to the rescue in such cases.
Unconstrained Optimization
Suppose that we are given a decision problem described by a function \(f(c, p)\), where \(c\) is a decision variable under the agent’s control and \(p\) is a parameter variable that the agent does not affect. The goal of the decision problem is to maximize \(f\). The unconstrained optimization problem corresponding to this decision setting is
\begin{align*} \max_{c} f(c, p). \end{align*}
The function \(f\) is known as the objective and the optimization variable \(c\) is the control variable of the optimization problem.
Using Fermat’s theorem, we know that if \(f\) attains its maximum at \(c_{0}\) given \(p\), then its first (partial) derivative should necessarily be equal to zero, namely \(f_{c}(c_{0}, p) = 0\). We can use this idea to locate any local maximizers. By imposing the condition
\begin{align*} f_{c}(c, p) \overset{!}{=} 0, \end{align*}
and solving it for \(c\), we can locate potential local maximizers of the optimization problem. This condition is so commonly used that it is named. It is called the firstorder or necessary condition. The candidate maximizers are called critical points.
Thus, we have a way of obtaining candidate maximizers, but how do we know which candidates are actually maximizers? Some of the candidates might as well be minimizers, as those too satisfy the firstorder condition. Whether a candidate is a local maximizer or minimizer depends on the (local) curvature of the function. If the function is concave in a small neighborhood around the candidate (curvature opening down), then the candidate is a local maximizer. Instead, if the function is convex in a small neighborhood around the candidate (curvature opening up), then the candidate is a local minimizer.
Therefore, the problem’s maximizers are all the candidates that satisfy the condition
\begin{align*} f_{cc}(c, p) \overset{!}{<} 0. \end{align*}
This condition is called the secondorder or sufficient condition.
We have kept the parameter value \(p\) fixed in the preceding discussion. Thus, any maximizer that we have obtained depends on the fixed value of \(p\). We can change the value of \(p\) to \(p’\) and repeat the process. The new maximizer will be based on the new parameter value \(p’\) in this case. In this manner, we can define a function such that for each parameter value \(p\), we get an optimizer \(c(p)\). This mapping is called the optimal control function. Substituting the optimal control in the objective function we get
\begin{align*} v(p) = f(c(p), p) = \max_{c} f(c, p). \end{align*}
In this way, we have eliminated the dependence of the objective on the control variable and expressed the maximized values exclusively as a function of the parameter \(p\). The function \(v\) gives the maximum value of \(f\) for each parameter \(p\), and it is called the value function.

Consider the function \(f(c, p) = p  (cp)^{2}\) for \(c, p \in \mathbb{R}\).

Calculate the maximization candidates using the firstorder condition for a fixed value of \(p\).

Show that these candidates are indeed maximizers using the secondorder condition.

What is the optimal control function of this problem?

What is the value function of this problem?

Let \(f\) be defined by \(f(x) = 2  \frac{4x}{x^{2} + 3}\)

Find \(f’\) and \(f’’\).

Find all the critical points and classify them into maxima and minima.
Constrained Optimization
In some decision problems, agents do not have access to the full universe of alternatives. For instance, firms are subject to technological constraints, and they can only choose to produce technologically feasible quantities. Similarly, households can only buy commodities and services that are affordable for their budgets. How can we describe such decision problems?
Suppose that we are given a decision problem described again by a function \(f(c, p)\), where \(c\) is the control variable and \(p\) is a parameter. The goal of the decision problem is to maximize \(f\). However, the agent cannot freely choose \(c\). The agent’s choices have to satisfy the constraint \(g(c, I) = 0\), where \(I\) is another parameter that the agent does not control. The constrained optimization problem corresponding to this decision setting is
\begin{align*} \max_{c} \, & f(c, p) \\ s.t. \quad &g(c, I) = 0. \end{align*}
The general approach with which one solves constraint optimization problems is by forming the Lagrangian function
\begin{align*} \mathcal{L}(c, \lambda, p, I) = f(c, p)  \lambda g(c, I). \end{align*}
The Lagrangian introduces an additional variable \(\lambda\), which is called the Lagrange multiplier. The Lagrange multiplier takes many useful interpretations in economics and finance decision problems. Specifically, the Lagrange multiplier gives us the shadow price of the constraint, i.e., the change in the value function when the constraint is infinitesimally relaxed.
Under some conditions, the maximizers or minimizers of the constrained maximization problem are obtained by solving the unconstrained optimization problem with \(\mathcal{L}(c, \lambda, p, I)\) as the objective and \(c\), \(\lambda\) as control variables. Therefore, we can use the Lagrangian to reduce a constrained optimization problem to the relatively simpler unconstrained problem
\begin{align*} \max_{c, \lambda} \mathcal{L}(c, \lambda, p, I), \end{align*}
We can get the maximizers or minimizers of the original constrained problem using the unconstrained problem’s first and secondorder conditions.
Topic's Concepts
 lagrange multiplier
 lagrangian
 constrained optimization problem
 constraint
 value function
 optimal control function
 sufficient
 second order
 critical points
 necessary
 first order
 control variable
 objective
 unconstrained optimization problem
 maximum
 maximizer
 maximizes