Caputo, Michael R. Foundations of Dynamic Economic Analysis: Optimal Control Theory and Applications.

Caputo, Michael R. Foundations of Dynamic Economic Analysis: Optimal Control Theory and Applications. Cambridge University Press, 2005, xii + 579 pp., $100 (hardcover), $48 (soft)

Since the publication of Dynamic Optimization in 1981 by Morton Kamien and Nancy Schwartz, a number of books have arrived on the scene with the specific intent of developing the foundations of dynamic optimization in a context of economic problem structures. The optimal control approach has been de rigueur with other authors paying varying degrees of attention to the calculus of variations and the dynamic programming approach. The trend commenced from the development of models of dynamic interactions replete with analytical and qualitative characterizations of equilibria. As computing power increased, the interest migrated to econometric-based approaches to primal- or dual-problem structures and computational solutions that cared little for closed-form solutions.

Michael Caputo has prepared a book that is a throw-back in the sense of returning to the basics of the mathematics of dynamic optimization, with an eye on motivation for the economist. Professor Caputo is legendary as an excellent and demanding lecturer and I have seen him present a lecture where students walk away wanting to know more about the mathematics of open balls. He fully subscribes to the notion that it is our interest in the economic issues that motivates the interest in studying the mathematics. His book is a masterpiece in this sense. I view the book in four parts where each part scaffolds the reader to the next part.

The first four chapters encompass Part I of the book, starting off with an outstanding introductory chapter that clearly sets the stage for the rest of the text. There is a mixture of rigor and motivation throughout relating the mathematics and the reader's intuition about static optimization to the notions of time and decisions. There are four different applied problem settings introduced in this chapter to keep the mix between theory and economic context well matched. The mathematics is the context of continuous time, as is the setting for the entire text, and ends with a set of exercises, as is the case in all chapters. The chapters that follow pay equal attention to necessity and sufficiency. Applied economists often take the conditions revolving around necessity more seriously, being willing to assume that solutions exist and take a passing nod to sufficiency. Chapter 2 focuses entirely on the necessary conditions that an optimal solution path/trajectory must obey. The notions of state equation and transversality are introduced here and the use of a fairly general notion is used, as well as the motivation and development of the Pontryagin conditions. Chapter 4 takes explicit aim at interpretations of the Maximum Principle starting off with both rigor and intuitive motivation for the backward nature of the optimal solution paths. The economic interpretations follow in some detail along with a set of examples that build on the presentation of the same problems in earlier chapters. The sufficiency conditions are also treated to the interpretations including consideration of inequality constraints.

Chapters 5 through 8 encompass the next major part of the book dealing with the classification of optimal control problems by the structure of the objective. The most common starting point is the linear control problem addressed in Chapter 5. It is rare to see an entire chapter addressing this class of problem that addresses the bang-bang and singular control solutions. The jump discontinuities require care in interpretation and characterization for the sufficiency of a solution, with care being taken to point out that linearity of an optimal control problem in the control variable alone is not sufficient for bang-bang control to be optimal. Chapter 6 broadens the discussion of necessity and sufficiency to condition the set of constraints that involve both the state and control variables (or mixed constraints) ending with the generalization of the Mangasarian version of sufficiency (which assumes the concavity of the Lagrangian in states and controls) to the Arrow sufficiency version (which assumes that the optimized Hamiltonian is concave in states). Chapter 7 takes the next step in the generalization to consider an infrequently addressed case of isoperimetric problems, which involve an integral expression as constraints, and develops the necessary and sufficient conditions to solve these problems directly. The principal-agent problem is considered in some detail and this is the first opportunity Prof. Caputo takes to articulate the Dynamic Envelope Theorem, which he pioneered. Chapter 8 takes care to characterize the reciprocal or inverse problems associated with an optimization that included the duality relations. Prof. Caputo takes care to make sure the reader is particularly clear on his distinction between reciprocal and dual relationships. The economic cognates to static theory are drawn as he is working through these conditions, and the abstract characterizations presented are then definitized in the context of a nonrenewable resource-extracting firm problem. This all leads to a characterization of comparative dynamics and the properties of its analysis. This chapter is an example of the mixture of rigor, motivation, and interpretation.

Part III of the book encompasses chapters 9 through 14, which take close aim on the analysis of economic adjustment over time and the characterization of microeconomic dynamics. Chapters 9 and 10, respectively, deal with economic interpretations and the transversality relationships associated with the Dynamic Envelope Theorem. This is tied together nicely with comparative dynamics in a primal-dual framework in the following chapter, with Chapter 12 addressing the notions of discounting, the current value Hamiltonian, and the time inconsistency problem. The qualitative characterization of optimal solutions is introduced in Chapter 12 with the presentation of autonomous (not explicitly time-dependent) differential equations and the phase plane portraits they imply. There are many exercises to encourage practice and this is arguably the best presentation in an economics text of such analysis. Chapter 13 returns to optimal control problems with the presentation of the necessary and sufficient conditions of the infinite horizon optimal control problem and points out where the finite horizon results differ.

The final part of the book is the focused consideration of some classic archetype dynamic economic models. Chapter 15 returns us to neoclassical growth that I recall slogging through in Intrigator's mathematical economics text, while Chapter 16 addresses dynamic limit pricing. The adjustment costs model of the firm is addressed in a subsequent chapter. Each presentation is unified with the discussion of the necessary and sufficiency conditions and phase plane analysis presented in the chapters in Part III. Chapter 18 takes a step back to considering a classic structure of the infinite horizon optimal control with one state and one control, which seems to be included for the sake of completeness for the discussion of the properties and character of classic dynamic economic problem structures. The penultimate chapter introduces an alternative to optimal control with the dynamic programming approach leading to the Hamilton-Jacobi-Bellman equation. The value of this chapter for this text is for the final chapter on the dynamic duality theory of the adjustment cost firm.

As promoted in the title, Prof. Caputo indeed lays a foundation with the heart of teacher as he presents the goals of each chapter in the outset and offers exercises that are designed to both stimulate and exercise the reader's training. The mixture of mathematical reasoning, economic intuition, and examples is well balanced in every single chapter. The index is impressive, allowing the reader to search quickly for the problem of interest. This is an impressive book that should find its way to the shelves of those interested in the core understanding of dynamic economic relationships and their structure, analysis, and interpretation.

Spiro E. Stefanou

Pennsylvania State University