Motivation: This chapter is intended to quantify steady state errors and transient performance of LPNI systems in the problem of tracking reference signals.

For linear systems, the steady state error in tracking deterministic references is characterized by the notion of system type, defined by the poles of the loop gain at the origin. This definition is not applicable to LPNI systems since, if the actuator saturates, controller poles at the origin play a different role than those of the plant and, thus, the loop gain does not define the system type. This motivates the first goal of this chapter, which is to introduce the notion of system type for LPNI systems.

Also in contrast with the linear case, LPNI systems with saturating actuators are not capable of tracking steps with arbitrary amplitudes. This motivates the second goal of this chapter: to introduce and quantify the notion of trackable domains, that is, the sets of step (or ramp, or parabolic) input magnitudes that can, in fact, be tracked by an LPNI system.

As far as the transients are concerned, the usual step tracking measures, such as overshoot, settling time, and so on, are clearly not appropriate for random references. The variance of tracking errors, as it turns out, is not a good measure either, since for the same variance, the nature of tracking errors can be qualitatively different.

A main objective of economics analysis consists in the derivation of demand and supply functions for the purpose of establishing market equilibria and predicting their response in relation to the displacement of some parameters (either prices or quantities). Such a variation of parameters corresponds to a comparative statics analysis common to theoretical investigations. When the available information allows it, the derivation of demand and supply functions can be obtained by econometric methods. In that case, the appropriate methodology consists of two phases: estimation and prediction. The estimation phase corresponds to a calibration of the econometric model and includes hypothesis testing. The prediction phase analyzes the behavior of the model outside the sample information and provides a measure of response induced by parameter variations.

When the available information is insufficient for a proper application of econometric procedures, mathematical programming admits the exploitation of limited information and a complete analysis of the given scenario. Problems of environmental economics, natural resources, economic development, and many others often defy econometric analysis because the necessary time series of information are too short for a meaningful employment of those procedures.

The outline of this chapter is articulated in various sections. The first development deals with the relations of comparative statics proper of the theory of the firm operating in conditions of certainty. Such conditions are known as the Hicksian relations of the competitive firm. The second section discusses the parametric programming of a general LP model.

This book formulates and discusses models of producers' economic behavior using the framework of mathematical programming. Furthermore, it introduces the Symmetry Principle in economics and demonstrates its analytical power in dealing with problems hitherto considered either difficult or intractable. It assumes that its readers have a beginner's knowledge of calculus and linear algebra and that, at least, they have taken an intermediate course in microeconomics.

The treatment of economic behavior expounded in this book acquires an operational character that, in general, is not present in a theory course. In a microeconomics course, for example – even at the graduate level – the treatment of a monopolist's behavior considers only one commodity. Often, however, a monopolist owns demand functions for three or more commodities (think of Microsoft) and her firm's equilibrium requires a careful analysis, especially in the case of perfectly discriminating behavior. Another example regards the analysis of risk and uncertainty in the presence of nonzero covariances between risky output market prices and input supplies. These and many other “realistic” economic scenarios require the introduction of a structure called the Equilibrium Problem. Although this specification is the logical representation of quantity and price equilibrium conditions for any commodity, economic theory courses privilege statements of economic models that involve a dual pair of maximizing and minimizing objective functions. This book makes it clear that these optimizing structures are only special cases of the class of Equilibrium Problems and that, often, they impose unnecessary restrictions on the specification of economic models.

Many economic agents display monopolistic behavior. In particular, we will discuss two variants of monopolistic behavior as represented by a pure monopolist and by a perfectly dicriminating monopolist. We will assume that the monopolist will produce and sell a vector of outputs, as final commodities.

The essential characteristic of a monopolist is that he “owns” the set of demand functions that are related to his outputs. The meaning of “owning” the demand functions must be understood in the sense that a monopolist “develops” the markets for his products and, in this sense, he is aware of the demand functions for such products that can be satisfied only by his outputs.

Pure Monopolist

A pure monopolist is a monopolist who charges the same price for all the units of his products sold on the market. We assume that he faces inverse demand functions for his products defined as p = c - Dx, where D is a (n × n) symmetric positive semidefinite matrix representing the slopes of the demand functions, p is a (n × 1) vector of output prices, c is a (n × 1) vector of intercepts of the demand functions, and x is a (n × 1) vector of output quantities sold on the market by the pure monopolist.

A second assumption is that the monopolist uses a linear technology for producing his outputs. Such a technology is represented by the matrix A of dimensions (m × n), m < n.

The notion of symmetric programming grew out of a gradual realization that symmetric structures – as defined in this book – provide the means for a wide ranging unification of economic problems. A conjecture immediately and naturally followed: symmetric structures are more general than asymmetric ones as long as the right approach to symmetry is embraced. There are, in fact, two ways to symmetrize asymmetric problems: a reductionist and an embedding approach. The reductionist strategy eliminates, by assumption, those elements that make the original problem asymmetric. This is the least interesting of the two approaches but one that is followed by the majority of researchers. The alternative strategy seeks to embed the original asymmetric problem into a larger symmetric structure. The way to execute this research program is never obvious but is always rewarding. This book is entirely devoted to the illustration of this second approach.

With the unification of problems there comes also the unification of methodologies. Rather than associating different algorithms to different problems, symmetric programming allows for the application of the same algorithm to a large family of problems.

Unification has always been one of the principal objectives of science. When different problems are unified under a new encompassing theory, a better understanding of those problems and of the theory itself is achieved. Paradoxically, unification leads to simplicity, albeit a kind of rarefied simplicity whose understanding requires long years of schooling.

In previous chapters, the information available to an economic agent was assumed to be certain. In general, this assumption is not realistic. Market prices of commodities, supplies of limiting inputs, and technical coefficients are types of information subject to uncertainty, to either a small or large degree. Consider a farmer who, in the fall season, must plant crops to be harvested in the spring and for which a market price is not known in the fall. On the basis of his past experience, he may be able to form some expectations about those prices and use these expected prices for making his planning decisions in the fall. On the limiting input side, the effective supply of family labor may depend on seasonal weather, which is a stochastic event. Therefore, in order to proceed to form a production plan, the farmer will also have to form expectations for the uncertain quantities of limiting inputs. Technical coefficients form a third category of information that, generally, is subject to uncertainty. In any given county, agricultural extension personnel knows the “average” input requirements for producing one unit of any crop. However, when that information is transferred to a given farm, the result may not be as suggested by the extension counselor. Again, therefore, a farmermay realistically regard technical coefficients with some uncertainty.

In discussing uncertain commodity market prices, limiting inputs and technical coefficients, we admit that all the information necessary for making rational decisions by an economic agent must be subject to expectation formation about these aleatory prospects.

A large majority of empirical studies using either an LP or a QP specification has neglected the consequences of one important aspect of mathematical programming. Simply stated, the polyhedral nature of the solution set in LP and QP models may be responsible for multiple optimal solutions, if some plausible conditions are realized. If and when an empirical problem possesses alternative optimal solutions, why is only one of them usually selected for presentation in final reports which, often, make efficiency judgements and prescribe significant policy changes? In this chapter, we discuss, separately, the existence, the computation, and the consequences of multiple optimal solutions (MOS)in linear programming and in quadratic programming models.

MOS in Linear Programming

When LP is used for analyzing empirical problems, an explicit comparison is usually made between the activities actually chosen and operated by the economic agent under scrutiny and the optimal level of activities suggested by the model's solution. In some instance (Wicks), the use of LP for policy planning has inspired the use of Theil's U-inequality coefficient to assess formally the discrepancy between actual and optimal (predicted) activities. Desired values of the U coefficient are those close to zero, attained when the squared distance between actual and LP optimally predicted activities is small. Implicitly, minimum distance criteria have been used by many authors to assess the plausibility and performance of their LP models. In the presence of multiple optimal solutions, however, the selection of a specific solution for final analysis is crucial and should not be left to computer codes, as probably has been the case in all reported studies.

The solution of linear and nonlinear programming problems involves the solution of a sequence of linear equation systems. In preparation for understanding the nature and the structure of algorithms for solving linear and nonlinear programming problems, we discuss in this chapter the fundamentals of a particular method of solving systems of linear equations that is known as the pivot method. We choose this algorithm because it articulates in great detail all the steps necessary to solve a system of equations with the opportunity of showing very clearly the structure of the linear system and of the steps leading to a solution, if it exists.

In the analysis and solution of a linear system of equations, the most important notion is that of a basis. A formal definition of a basis is given later. Here we discuss, in an informal way and borrowing from everyday life, the meaning of a basis. A basis is a system of measurement units. To measure temperature, we often use two systems of measurement units, the Fahrenheit and the Celsius systems. Equivalently, we could name them the Fahrenheit and the Celsius bases. Analogously, the decimal metric and the American systems are two different bases (systems of measurement units) for measuring length, surface, and volume. The essential role of a basis, therefore, is that of measuring objects. In an entirely similar way, the role of a basis in a given mathematical space is that of measuring objects (points, vectors) in that space.

Positive Mathematical Programming (PMP) is an approach to empirical analysis that uses all the available information, no matter how scarce. It uses sample and user-supplied information in the form of expert opinion. This approach is especially useful in situations where only short time series are available as, for example, in sectoral analyses of developing countries and environmental economics analyses. PMP is a policy-oriented approach. By this characterization we mean that, although the structure of the PMP specification assumes the form of a mathematical programming model, the ultimate objective of the analysis is to formulate policy recommendations. In this regard, PMP is not different from a traditional econometric analysis.

PMP grew out of two distinct dissatisfactions with current methodologies: first, the inability of standard econometrics to deal with limited and incomplete information and, second, the inability of linear programming (LP) to approximate, even roughly, realized farm production plans and, therefore, to become a useful methodology for policy analysis. The original work on PMP is due to Howitt. After the 1960s, a strange idea spread like a virus among empirical economists: only traditional econometric techniques were considered to be legitimate tools for economic analysis. In contrast, mathematical programming techniques, represented mainly by LP (which had flourished alongside traditional econometrics in the previous decade), were regarded as inadequate tools for interpreting economic behavior and for policy analysis. In reality, the emphasis on linear programming applications during the 1950s and 1960s provided ammunitions to the critics of mathematical programming who could not see the analytical potential of quadratic and, in general, nonlinear programming.

In the introduction chapter it was shown that the Euler-Legendre transformation defines duality relations for differentiable concave/convex problems without constraints. When constraints of any kind are introduced into the specification, the Lagrangean function represents the link between primal and dual problems.

At first, it would appear that the duality established by the new function has little to do with that of the Euler-Legendre transformation. As illustrated in Chapter 1, however, this is not the case. It is in fact possible to reestablish the connection by applying the Euler-Legendre transformation to the Lagrangean function and obtain a symmetric specification of dual nonlinear programs. The relationship among the various specifications of duality is illustrated in the flow chart of Figure 3.1.

Starting with unconstrained problems, we have seen in Chapter 1 that these models can, under certain conditions such as concavity and convexity, be analyzed by means of the Euler-Legendre transformation that exhibits a symmetry between the components of the original and the transformation functions. There follow problems subject to equality constraints that must be analyzed by means of the traditional Lagrange approach that presents an asymmetric structure as illustrated in Figure 2.1 of Chapter 2. When problems are constrained by inequality relations, the Karush-Kuhn- Tucker theory must be invoked. This theory uses the Lagrangean function, which is usually an asymmetric structure. However, we have already seen in Chapter 1 that the Lagrangean function can be subject to the Euler- Legendre transformation that provides a symmetrization of the original structure.

An economic agent who is the sole buyer of all the available supply of an input is called a monopsonist. We discuss two categories of monopsonists: the pure monopsonist and the perfectly discriminating monopsonist.

Pure Monopsonist

In analogy to the monopolist treatment, we discuss the pure monopsonist's behavior assuming that she will be the sole buyer of a vector of inputs. Let ps = g + Gs be such a vector of inverse linear supply functions, where s is a (m × 1) vector of quantities of inputs purchased by (supplied to) the monopsonist, the matrix G is a (m × m) matrix of price/quantity slopes in the input supply functions, the vector g contains intercept coefficients and ps is a (m × 1) vector of input prices. We assume that the matrix G is symmetric and positive definite. In analogy to the monopolist, the monopsonist “owns” the input supply functions.

In order to concentrate on the pure monopsonist's behavior, we assume that this economic agent is a price taker on the output markets and produces her outputs by means of a linear technology. The decision of the pure monopsonist is to find the optimal quantities of inputs to purchase on the market and to find the optimal quantity of outputs to produce in such a way to maximize profit. Total revenue of the price-taking entrepreneur is defined as TR = c′x, where c is a (n × 1) vector of market prices for the outputs, which are represented by the vector x, conformable to the vector c.

Since moving from UC Davis in 2003, I have come to appreciate even more what Quirino has to offer colleagues and students. Take, for example, the symmetry paradigm he expounds in the opening chapter. Quirino effectively demonstrates how the application of the symmetry principle, when combined with the implementation of well-known mathematical programming theory, can be used not only to numerically solve various microeconomic models, but also to gain deeper economic understanding of them. Although this is exciting to those of us who work in this field, it might come off as a bit abstract and technical to students. No problem, as Quirino motivated the symmetry principle quite differently to the students by bringing a symmetric vegetable to class! As I recall, it was broccoli Romanesco, whose symmetry was with respect to scale, that is, it is a fractal veggie. Neither faculty nor students of that era will forget that day, or more generally, the passion and unique point of view that Quirino brought to every conversation, lecture, and seminar. This book embodies all of that fervor and zeal, and more, as it is the culmination of his lifelong research in the application of mathematical programming to economics. What's more, Quirino's guiding concept of symmetry is peppered all throughout the text, and this fact alone separates it from the multitude of books on mathematical programming and applications. It really does not get much better than this for a user of these methods.

The purpose of this chapter is to outline an overview of the basic criteria for characterizing the solution of static optimization problems. Its intent is neither rigor nor generality but a sufficient understanding for the logic leading to the conditions of optimization and a sufficient degree of practicality. An immediate issue is whether we are satisfied with relative optima or require global optima as the solution of our problems. Any further elaboration necessitates a definition of these two notions.

Definition: A point x* ∈ Rn is a relative (or local) maximum point of a function f defined over Rn if there exists an ε > 0 such that f(x*) ≥ f(x) for all x ∈ Rn and |x - x*| < ε.

A point x* ∈ Rn is a strict relative (or local) maximumpoint of a function f defined over Rn if there exists an ε > 0 such that f(x*) > f(x) for all x ∈ Rn and |x - x*| < ε.

In general, we would like to deal with solution points that correspond to global optima.

Definition: A point x* ∈ Rn is a global maximum point of a function f defined over Rn if f (x*) ≥ f (x) for all x ∈ Rn.

Apoint x* ∈ Rn is a strict global maximum point of a function f defined over Rn if f (x*) > f (x) for all x ∈ Rn.