## Capital Theory, Equilibrium Analysis and Recursive Utility

Capital theory provides the foundation for economic dynamics. Even a casual glance at the modern literature on macroeconomic dynamics and growth theory shows the ways in which dynamic optimization and equilibrium principles rooted in capital theory provide the theoretical underpinnings for those theories.

Much work in capital theory, growth theory, macroeconomic dynamics, and resource economics derives from Ramsey's classic paper on optimal growth. Such models typically specify a planner's objective which is time additive separable (TAS). Properties of these models, or their equilibrium counterparts, are then found by exploiting this additional structural assumption. The additive separability hypothesis has long been recognized as special, but has been regarded as relatively harmless in the optimal growth context. The TAS model was preferred primarily for its technical convenience over Fisher's theory of interest. He assumed the rate of impatience depended on the underlying consumption stream, even in a steady state. In the TAS case the steady state rate of impatience is independent of the consumption profile.

The Walrasian theory of competitive equilibrium has influenced the structure of capital theoretic models ever since the publication of von Neumann's seminal paper on balanced growth in a multisector economy. Not only did his model and the methods used to demonstrate the existence of a solution foreshadow many of the developments of modern general equilibrium theory, but they also exerted a profound influence on the development of capital theory. Hicks's program of replacing static equilibrium with dynamic equilibrium served to further intensify the linkage between capital theory and Walrasian equilibrium theory.

The Hicksian program, marrying capital theory and equilibrium analysis, has been extraordinarily fruitful. In spite of this success, it suffers from a fundamental weakness. The restrictions imposed by additive separability are an obstacle to the proper analysis of long-run phenomena, and to the analysis of economies where different agents evaluate the future differently.

Koopmans's theory of recursive utility, first formulated in the early 1960s, provides a way out of this dilemma. Koopmans suggested that we broaden the class of dynamic preferences by relaxing the separability requirement, allowing more general specifications of intertemporal preferences. The resulting recursive utility functions enjoy a time consistency property that still permits a dynamic programming analysis of optimal growth. They also allow the steady state rate of impatience to vary with the consumption level. This permits enough flexibility to avoid the long-run and heterogeneity problems of additive separable dynamic equilibrium models. Moreover, recursive utility permits relatively comprehensive analytical and computational analysis of dynamic models.

The development of both recursive utility theory and equilibrium analysis has now reached the point where they can be joined on Hicksian lines. Our book presents a synthesis of capital theory and equilibrium analysis within a recursive utility framework.

Our goal is to combine the ideas of Fisher, Ramsey, and Koopmans with those of Walras, von Neumann, and Hicks. We have integrated related research by others into our framework, and drawn on our own previously published and unpublished work as well as that of our students. The theory we present is cast in a unified framework where infinitely lived agents act out their economic decisions in a discrete time setting devoid of any uncertainty.

We chose the deterministic discrete time setup in part because its technical demands seemed to intrude less on the economic arguments than would otherwise be the case. This choice also allows us to present our topics in a common framework, and address a greater variety of issues.

This book is intended as both a research monograph and an overview of capital theory. It can be used by researchers, or as a textbook in graduate courses on growth theory, general equilibrium theory, or economic dynamics. In addition, the book may be of interest to mathematicians seeking applications of functional analysis and Riesz spaces.

We weave existing theory together within a unified framework. In doing so we show how the pieces fit together. This process yielded a number of connections between different literatures. The knowledgeable reader will notice these scattered throughout the text. For instance, we relate biconvergence and myopia in Chapter 3, and treat the overtaking criterion as a variety of recursive utility. We reworked results from the literature, as in our presentation of turnpike theory in Chapter 5. We also found significant gaps in the published record. To fill these gaps, we worked out multisector results for the existence and characterization of optima in Chapter 4 and built Chapter 8 upon recent developments in infinite dimensional general equilibrium analysis.

The numerous examples throughout the book are an essential element of our presentation. They help provide intuition, delineate the scope of the results, clarify the role of the assumptions, and illustrate the type of models that our work encompasses. Besides the 84 named and numbered examples, the reader will discover many other small examples tucked away in the text.

Our treatment begins with an introductory chapter that illustrates some of the problems that occur with additive utility, and shows how they may be resolved by using recursive models. It also contains a number of examples illustrating the variety of different models that a general theory of dynamic economics should encompass.

The basic tools we use are drawn from functional analysis, dynamical systems theory, and lattice programming. The deterministic discrete time framework is based on sequences of vectors of capital, consumption and prices. Chapter 2 provides most of the necessary functional analytic background, including the theory of ordered vector spaces, for examining optimal growth and equilibrium models based on such sequences.

Chapter 3 focuses on preferences and recursive utility. We examine recursive preferences from three vantage points. The first, in the spirit of Koopmans's original work, seeks to axiomatically characterize recursive preferences. The second viewpoint asks about attitudes toward the future, how much patience or impatience do preferences exhibit? The final vantage point sees intertemporal preferences as deriving from Fisherian preferences over present consumption and future utility. It focuses on the construction of recursive preference orderings from these Fisherian primitives.

Questions of existence and characterization of optimal paths take center stage in Chapter 4. We start with one-sector TAS models in order to illustrate the basic principles of existence theory, with and without discounting. These ideas are used in showing existence and continuity of optimal paths in multisector recursive models. Dynamic programming gives us another perspective on these issues. It also provides some key information for showing the necessity of the transversality condition. We end the chapter by completely characterizing optimal paths via Euler equations and a transversality condition.

Once paths are characterized, we are ready to start studying their dynamics in Chapter 5. Monotonicity of optimal paths, existence of steady states, convergence to steady states, and the presence of cycles and chaos are all examined here. Much more could be written about the dynamics of optimal paths. To do it justice would probably require another volume, and we have confined our attention to a few key results and tools.

Chapter 6 takes a first stab at dynamic equilibrium theory. We focus on representative agent economies, and show the interrelationship between optimal growth theories and their equilibrium counterparts when each agent has recursive preferences. Under a wide range of conditions, equilibria of representative agent economies correspond to optimal growth problems, and vice versa. This connection even holds in certain economies with tax distortions, where the equilibrium solves an artificially constructed optimal growth problem.

This equivalence between equilibria and optimal paths facilitates the study of comparative dynamics in Chapter 7. The economy is subjected to a parametric change, resulting in a new optimal path. This section draws heavily on recent developments in lattice programming, and we have included the necessary background material.

Chapter 8 ties together threads from the previous chapters. It considers dynamic equilibria when agents are heterogeneous. We show the existence of equilibria, prove the welfare theorems, examine the equivalence between welfare maxima and equilibria, and provide a core equivalence theorem.

Neither uncertainly nor continuous time are examined here. We have excluded them because they require significantly more technical prerequisites, and because each would require an additional volume if treated on an equal footing with the discrete time deterministic models. The literature in both cases is voluminous and our references contain only a few key citations that fit into our current treatment's scope.

Most of the notation is introduced as needed. Many of the more specialized symbols also appear at the start of the index, or indexed under “spaces.” Section numbers have the form chapter.section or chapter.section.subsection. Examples, theorems, etc. are numbered consecutively within each chapter. We refer to them by number within their chapter. When the reference is in another chapter, we prepend the chapter number, e.g., Example 4.2 is in Chapter 4. The named and numbered examples are set off within starting ($\blacktriangleright$) and ending ($\blacktriangleleft$) triangles. The ends of proofs are denoted by $\square$.

As usual, we have accumulated a long list of debts to our
colleagues, collaborators, and peers. We would like to single out Buz
Brock, Edwin Burmeister, Lionel McKenzie, and the late Trout Rader for
their intellectual inspiration, and Ciprian Foias, Lionel McKenzie,
and Mukul Majumdar for the joint work that underlies portions of the
book. Thanks go to the following for their encouragement and comments
during this long process: Roko Aliprantis, Buz Brock, Subir
Chakrabarti, Fwu-Ranq Chang, Roy Gardner, Michael Kaganovich, Tom
Kniesner, and Itzhak Zilcha, with special thanks to Nick Spulber for
first suggesting we write this book and providing his usual insightful
comments on our manuscript, and to Robert Lucas for stressing the
advantages of a unified technical framework. Robert Becker thanks
Dean Morton Lowengrub and Jay Wilson, former chairman of the economics
department, for making various resources available that helped us
complete the manuscript. John Boyd also thanks Academic Press for
permission to utilize portions of papers he previously published in
the *Journal of Economic Theory* (Boyd,
1990a;
1996).

This book has its immediate origins in our survey (Becker and Boyd, 1990; 1993) and our lectures on capital theory at Indiana University and the University of Rochester. We would like to thank the various generations of capital theory students at Indiana and Rochester who have suffered through many preliminary versions of portions of this book. Among the current and former students, special thanks go to Hajime Kubota, Eiichi Miyagawa, Tomoichi Shinotsuka, and Danyang Xie (all Rochester) for their helpful comments on recent drafts. Although they are implicitly thanked in the references, we also explicitly thank those former students whose dissertations and later work is used in our text: James Dolmas, Christophe Faugère, Alejandro Hernández D., Mark Hertzendorf, and Tomoichi Shinotsuka (all Rochester) and Sumit Joshi (Indiana).

We would also like to thank Donald E. Knuth for creating the mathematical typesetting program TeX, Eberhard Mattes for his excellent OS/2 implementation, emTeX, and the American Mathematical Society for the use of their AMS-TeX macros.

Finally, Robert Becker thanks his family, Karen and Adam, for their encouragement and support.