## Dynamic Optimization

To better understand the behavior of complex, dynamic economic systems, mathematical economists often apply the mathematical tools of optimal control theory, dynamic programming, and dynamic optimization. These tools help the analyst to maximize the effectiveness, performance, or functionality of a system or minimize undesirable characteristics through the use of advanced mathematical techniques. Although the complex variable set, the probabilistic nature of many variables, and the tendency for economic systems to change over time makes much economic modeling and decision making complex in nature, these tools can be of great use to help decision makers make pragmatic, empirically based decisions that will lead to optimal decisions.

Keywords: Dynamic Programming; Dynamic System; Forecasting; Mathematical Economics; Model; Optimal Control Theory; Probability; Scientific Method

### Dynamic Optimization

### Overview

There is a reason that the English language has the concept of the "armchair" philosopher or theorist. It is relatively easy to sit back in a comfortable chair and postulate theories about how the real world works. However, although armchair theories may seem reasonable (at least at first glance), they also tend not to be testable through the application of the scientific method. In addition, such theories tend to be based only on the armchair theorist's personal observations. Therefore, they may be circumscribed and not widely generalizable. For example, I recently received a survey from a local politician asking what county services we wished improved and whether we wanted to pay for the improvements through an increase in property taxes or other taxes. I wrote back explaining that since we are in the middle of an economic crisis, the county needs to cut back on spending rather than increase services (and, concomitantly, taxes). The politician simply had not taken into account all the variables affecting his constituents. Although such inductive reasoning is an important first step in better understanding the world around us, unless the theory is testable and tested, it does not do the scientist or the practitioner much good.

In economics — as is true in the other social sciences, as well — the scientific method and concomitant mathematical tools and techniques are applied to test, validate, and revise theories so that they better reflect the reality of the world around us. Mathematical economics is a branch of economics that focuses on the application of mathematical tools and techniques in the modeling and testing of economic theories and to analyze problems posed by theoretical economics. Mathematical economics is concerned with the application of scientific method to economic data in order to advance the state of understanding about economic issues.

### Theory Testing in the Social Sciences

In some sciences, the testing of theories can be easily done through experimentation. For example, to determine at what temperature a metal melts, one could systematically apply greater temperatures to a sample of the metal in a controlled setting, making careful observations as to the temperature at which the metal melts. In the social sciences, however, this is often not possible. For example, if one desired to determine the effects of a financial loss on different income levels, it would be morally reprehensible to intentionally bring about financial loss on others even if it were logistically possible.

As a result, economists and other social scientists often need to rely on secondary analysis in order to test their theories. This limitation is further complicated by the fact that human behavior is apt to fluctuate in apparently unpredictable ways, making the development of a workable model difficult at best. In addition, many theories with which economists are interested evolve over time. For example, in the illustration above concerning the local politician, before the economic crisis of 2008, the question regarding raising taxes might have been considered to be a more reasonable one. In the aftermath of the crisis, however, the question is less so. Such systems that evolve over time are referred to as dynamic systems. As the recent economic crisis — not only in the United States but across the globe — amply illustrates, because the real world is so complex, in most cases the model also needs to be complex. Economic policy decisions that were made in good faith and that seemed harmless enough in isolation snowballed over time into an economic crisis of global proportions and implications.

### Mathematical Modeling

Mathematical modeling can be a useful tool to economists interested in forecasting the implications of economic policy or other economic decisions. One of the essential aspects of model building is to determine which of the potentially innumerable variables that could be built into the model are important and which are not. As a result, model building is often an iterative process in which the analyst or theorist repeatedly tries to develop a model that adequately and accurately models the real world without including extraneous variables that do not account for a significant proportion of the variation. Mathematical modeling techniques are often used for this process to better take into consideration all the important variables that occur in the real world.

Although theoretically taking into account every variable has intuitive appeal, in actuality, if a model is too mathematically complex, it can become useless from a practical point of view, and result in a model that is only applicable to a very restricted situation and not sufficiently generalizable to real world situations to be of pragmatic use to the economist or other analyst. Even when a large number of variables are considered, there is almost always some degree of uncertainty even when using a model. Therefore, models need to be simple to use and understand, adaptable to other situations or products, and be complete with the salient factors of the situation. Determining this mix can be a difficult process because of such factors as a lack of an adequate theory on which to base the model or lack of sufficient data to build and test the model. In addition, many problems in economics are often concerned with changes of output variables that occur over time. So, while it theoretically may be relatively easy to model behavior in static systems, developing dynamic models where the situation changes over time can be much more daunting.

### Optimal Control Theory

To better understand the behavior of such systems, mathematical economists often apply the mathematical tools of optimal control theory and dynamic programming. Both these tools attempt to optimize the model; maximize the effectiveness, performance, or functionality of a system using mathematical techniques to include desired factors and exclude undesired ones. Optimal control theory is an extension of the calculus of variations that is used by mathematical economists and others to understand dynamic systems with one independent variable (usually time). Various input variables are determined to maximize or minimize an output variable for the system within predetermined constraints. Control (output) variables can be determined as functions of time for a specified initial state of the system or as functions of the current state of the system. Optimal control theory requires a working knowledge of differential and integral calculus, matrix algebra, and vector algebra.

Dynamic programming is a recursive method used to find competitive equilibria in dynamic economic models and solve other multi-step optimal control problems. Dynamic programming is based on the assumption that decisions must be made under conditions of uncertainty and are often both probabilistic and sequential in nature. Dynamic programming attempts to find a control that gives a maximum (or minimum, as the case may be) control value of an objective function. Dynamic programming uses a multistage process consisting of several steps, or subdivides the control into sequential stages or steps corresponding to different moments in time. "Dynamic" in this context refers to the fact that time is an important factor in the model and "programming" refers to the planning and decision making associated with optimizing the model.

### Applications

The following sections give some non-mathematical examples of how dynamic optimization techniques can be applied to economic problems. The first of these regards an intraseasonal dynamic optimization model to allocate...

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