## Papers

- The Existence of Competitive Equilibrium over an Infinite Horizon with Production and General Consumption Sets
- The Existence of Ramsey Equilibrium
- The Existence of Steady States in Multisector Capital Accumulation Models with Recursive Preferences
- Fundamental Nonconvexities in Arrovian Markets and a Coasian Solution to the Problem of Externalities, with John Conley
- Recursive Utility and Optimal Capital Accumulation, I: Existence
- Recursive Utility and Optimal Capital Accumulation II: Sensitivity and Duality Theory
- Recursive Utility and the Ramsey Problem
- Recursive Utility: Discrete-Time Theory
- Symmetries, Dynamic Equilibria and the Value Function

## The Existence of Competitive Equilibrium over an Infinite Horizon with Production and General Consumption Sets

### by John H. Boyd III and Lionel W. McKenzie

*International Economic Review*, vol. 34 (1993), 1-20.

Although many theorems have been proved on the existence of competitive equilibrium in production economies with an infinite set of goods and a finite set of consumers, nearly all suffer from a major defect. The consumption possibility sets are required to equal the positive orthant. This rules out trade in personal services and it does not allow for substitutions between goods on the subsistence boundary. Using methods similar to Peleg and Yaari, we show both equilibrium existence and core equivalence for economies with production and general consumption sets.

## The Existence of Ramsey Equilibrium

### by Robert A. Becker, John H. Boyd III, and Ciprian Foias

*Econometrica*, vol. 59 (1991), 441-460.

We demonstrate existence of a perfect foresight equilibrium under borrowing constraints in a one-sector model with infinitely-lived heterogeneous agents. The class of admissible preferences includes, but is not limited to, recursive preferences. Existence is proven using a tatonnement argument under appropriate conditions on preferences and technology. A new measure of discounting, the norm of marginal impatience, is used to determine which technologies are admissible. Depending on the norm of marginal impatience, the admissible technology may either allow for permanent growth, or have a maximum sustainable stock.

## The Existence of Steady States in Multisector Capital Accumulation Models with Recursive Preferences

### by John H. Boyd III

*Journal of Economic Theory*, vol. 71 (1996), 289-297.

This paper proves the existence of a non-trivial stationary optimal path in a multisectoral capital accumulation model with recursive preferences. The reduced-form recursive preferences are represented by an aggregator function. I introduce a new form of delta normality that is appropriate for use with recursive preferences. Under some mild conditions on the aggregator, non-trivial steady states exist when the technology is bounded and delta normal.

## Fundamental Nonconvexities in Arrovian Markets and a Coasian Solution to the Problem of Externalities

### by John H. Boyd III and John Conley

*Journal of Economic Theory*, vol. 72 (1997), 388-407.

Starrett (1972) argues that the presence of externalities implies fundamental nonconvexities which cause Arrow markets to fail. While this is true, we argue this failure is due to the structure of the Arrovian markets that Starrett uses, and not to the presence of externalities as such. We provide an extension of a general equilibrium public goods model in which property rights are explicitly treated. Nonconvexities are not fundamental in this framework. We define a notion of Coasian equilibrium for this economy, and show first and second welfare theorems. In this context, the first welfare theorem is a type of Coase theorem.

## Recursive Utility and Optimal Capital Accumulation, I: Existence

### by Robert A. Becker, John H. Boyd III and Bom Yong Sung

*Journal of Economic Theory*, vol. 47 (1989), 326-345.

This paper demonstrates existence of optimal capital accumulation paths when the planner's preferences are represented by a recursive objective functional. Time preference is flexible. We employ a general multiple capital good reduced-form model. Existence of optimal paths is addressed via the classical Weierstrass theorem. The topology is uniform convergence of capital stocks on compact subsets, which is equivalent to weak convergence of investment flows under our maintained hypotheses. An improved version of a lemma due to Varaiya proves compactness of the feasible set. A monotonicity argument is combined with a powerful theorem of Cesari to demonstrate upper semicontinuity.

## Recursive Utility and Optimal Capital Accumulation II: Sensitivity and Duality Theory

### by Robert A. Becker and John H. Boyd III

*Economic Theory*, vol. 2 (1992), 547-563.

This paper provides sensitivity and duality results for continuous-time optimal capital accumulation models where preferences belong to a class of recursive objectives. We combine the topology used by Becker, Boyd and Sung (1989) with a controllability condition to demonstrate that optimal paths are continuous with respect to changes in both the initial capital stock, and the rate of time preference. Under convexity and an interiority condition, we find the value function is differentiable, and derive a multiplier equation for the supporting prices. Finally, under some mild additional conditions, we show that supporting prices obeying the transversality and multiplier equations are both necessary and sufficient for an optimum.

## Recursive Utility and the Ramsey Problem

### by John H. Boyd III

*Journal of Economic Theory*, April 1990

This paper examines existence, continuity and characterization of optimal paths under "recursive" preferences. Current utility is a fixed (aggregator) function of current consumption and future utility. For suitable aggregators, a useful refinement of the Contraction Mapping Theorem generates the utility function, as in Lucas and Stokey. A broader class of aggregators is handled via a limiting argument analogous to partial summation. The Weierstrass theorem yields the existence of optimal paths. Under somewhat more stringent conditions on the aggregator and technology, optimal paths are continuous in initial capital stocks, and are characterized by generalized Euler equations and a transversality condition.

## Recursive Utility: Discrete-Time Theory

### by Robert A. Becker and John H. Boyd III

*Hitotsubashi Journal of Economics*, vol. 34 Special Issue (1993), 49-98. (English).

*Cuadernos Economicos*, #46 (1990), 103-160. (Spanish).

Most of the modern literature on capital theory and optimal growth has proceeded on the assumption that preferences are represented by a functional which is additive over time and discounts future rewards at a constant rate. Recent research in the study of preference orders and utility functions has led to advances in intertemporal allocation theory on the basis of weaker hypotheses. The class of recursive utility functions has been proposed as a generalization of the additive utility family. The recursive utility functions share many of the important characteristics of the additive class. Notably, recursive utility functions enjoy a time consistency property that permits dynamic programming analysis of optimal growth and competitive equilibrium models. The purpose of this paper is to survey the discrete time theory of recursive utility functions and their applications in optimal growth theory.

## Symmetries, Dynamic Equilibria and the Value Function

### by John H. Boyd III

in "Conservation Laws and Symmetry: Applications to Economics and Finance",
ed. by R. Ramachandran and R. Sato, Kluwer, Boston, 1990.

This paper presents a geometric approach (symmetries) to dynamic economic problems that integrates the solution procedure with the economics of the problem. Techniques for using symmetries are developed in the context of portfolio choice, optimal growth, and dynamic equilibria. Information on preferences, budget sets, and technology is combined to explicitly compute the solution. By focusing on the geometry of the underlying economic structure, the symmetry method can handle many types of problems with equal ease. Given an appropriate economic structure, it is immaterial whether the problem is in continuous or discrete time, is deterministic or stochastic with a Brownian, Poisson or other process, uses a finite or infinite time horizon, or even whether the rate of time preference is fixed or variable. These details are unimportant as long as the geometry is unchanged. All cases are treated in a unified manner. A major strength of the symmetry technique is its ability to ferret out the solutions to complex models with simple underlying economic structures. For example, a previously unsolved optimal growth model with both time-varying discount rates and technology is easily solved via symmetries.