David Cass

The structure and stability of competitive dynamical systems

Publisher Summary This chapter discusses the structure and stability of competitive dynamical systems. It presents a general framework that encompasses these and many other problems in the theory of economic growth, or more broadly, the theory of economic dynamics. It describes competitive dynamics as Hamiltonian dynamics, where the Hamiltonian can be written as a function of present output prices and current input stocks and can be interpreted as the present value of net national product (equal, by duality, to the present value of net national income). Such a Hamiltonian dynamical system is competitive in the sense that it derives from the perfect-foresight, zero-profit, asset-market clearing equations arising in descriptive growth theory and is consistent with efficiency pricing conditions developed in the Malinvaud tradition and Eulers conditions or, more generally, Pontryagins maximum principle, applying to production-maximal or consumption-optimal growth problems.

The role of money in supporting the pareto optimality of competitive equilibrium in consumption-loan type models

Perhaps the single most enduring theme in economics is that of the social desirability of the competitive mechanism. In its modern form, this theme occurs as the two basic theorems of welfare economics (see, in particular, Arrow). Our central concern in this paper is with the validity of the first of these two theorems—that every competitive equilibrium yields a Pareto optimal allocation—in idealized yet plausible models of intertemporal allocation in a market economy.

The structure of financial equilibrium with exogenous yields : The case of restricted participation

Abstract We analyze the competitive equilibria in a model in which there are two periods, with uncertainty in the second, and households exchange both physical commodities and financial instruments - or current and future credit - on the spot market in the first period. Future returns from the financial instruments are fixed in terms of units of account (exogenous yields), and households face linear homogeneous constraints on their opportunities for transacting in financial instruments (restricted participation). Our main result establishes that if enough households are subject to the same constraints on exchanging credit, then restricted participation leads to a degree of real indeterminacy comparable to that obtaining in a model where there are simply too few available financial instruments (incomplete markets).

Generic regularity of competitive equilibria with restricted participation

Abstract In this paper, we present a general version of the model of competitive equilibrium with restricted participation on financial markets. Our goal is to accommodate a wide range of portfolio constraints while at the same time still permitting (generically) differential analysis of the dependence of financial equilibria on “fundamental” parameters.

Market Participation and Sunspot Equilibria

We investigate the structure of competitive equilibria in an exchange economy parametrized by (i) endowments and (ii) restrictions on market participation. For arbitrary regular endowments, if few consumers are restricted, there are no sunspot equilibria. If endowments are allowed to vary, while restrictions on market participation are fixed, there is a generic set of preferences such that sunspot equilibria exist for a non-empty subset of endowments. Our analysis extends to the general case of an arbitrary number of restricted consumers the results of Cass and Shell for the polar cases in which either (i) no consumers are restricted or (ii) all consumers are restricted.

Sunspots and Incomplete Financial Markets: The General Case

SummaryThis paper examines the effects of extrinsic uncertainty or sunspots on competitive equilibrium when financial markets are incomplete. For the canonical two-period, pure-exchange model with bonds (or so-called “nominal assets”, yielding fixed overall returns specified in units of account, and including pure inside money), the following result is established: Generically in endowments, if there areS sunspot states in the second period, but only 0<I<S distinct types of bonds, then — corresponding to the inherent deficiency in the financial markets — sunspots will generateD=S−I dimensions of consumption allocation or real (as well as spot price or nominal) indeterminacy.

Indefinitely sustained consumption despite exhaustible natural resources

SummaryThis paper analyzes the feasibility of sustaining uniformly positive consumption forever — even when flows of exhaustible resources are an indispensable input. The main result is a characterization of an economys capability for sustaining such consumption — under quite general maintained assumptions on technology — in terms of a single, simple capital-resource substitution condition.

introduction to Hamiltonian Dynamics in Economics

Economics during the fifties and sixties was marked by a substantial resurgence of interest in the theory of capital. While the advances during this period were very impressive, there was also an uneveness in the development of the subject. One-good models were studied in detail, as were many-good models of production-maximal growth and many-good models of consumption-optimal growth for the special case in which there is no social impatience. When treating heterogeneous capital, the literatures on decentralized or descriptive growth and consumptionoptimal growth with positive time discounting were dominated by special cases and examples. Reliance on examples and special cases proved to have some unfortunate consequences. The Battle of the Two Cambridges, ostensibly an argument over approaches to modeling distribution and accumulation, often seemed to focus on the robustness (or lack of robustness) of certain “fundamental” properties of the one-sector model and other worked-out examples when extended to more general heterogeneous-capital models. Furthermore, in large part because growth theory appeared to be an enterprise based only on proliferating special cases, the attention of the young able minds in the profession turned elsewhere, for example, to theat least seemingly-more evenly-developed general equilibrium tradition. This is a shame. Intertemporal allocation and its relationship with the wealth of societies is one of the most important problems in our discipline. Growth models are natural vehicles for the study of what is called “temporary equilibrium.” Dynamic models of multi-asset accumulation provide the theoretically most satisfactory environment for modeling the macroeconomics of income determination, employment, and inflation. The papers in this volume can be thought of as attempts at providing some unification of the theory of heterogeneous capital. The major

Multiplicity in General Financial Equilibrium with Portfolio Constraints

This paper explores the role of portfolio constraints in generating multiplicity of equilibrium. We present a simple financial market economy with two goods and two households, households who face constraints on their ability to take unbounded positions in risky stocks. Absent such constraints, equilibrium allocation is unique and is Pareto efficient. With one portfolio constraint in place, the efficient equilibrium is still possible; however, additional inefficient equilibria in which the constraint is binding may emerge. We show further that with portfolio constraints cum incomplete markets, there may be a continuum of equilibria; adding incomplete markets may lead to real indeterminacy.

Convexity and sunspots: A remark

Abstract Economies in which competitive equilibrium allocations are optimal and individual preferences are strictly convex are immune to sunspots; this is so even in the presence of non-convexities in production.