Karl Shell

The overlapping-generations model, I: The case of pure exchange without money

The overlapping-generations model’ was introduced by Samuelson [ 171 in 1958. Samuelson’s seminal paper has several important themes. He provides examples with nicely behaved agents, complete and costless markets, full information, and no externalities, in which Walrasian equilibria are not Pareto-optimal. This failure of the First Theorem of Welfare Economics was clarified in Shell [ 191. It is shown that the “double infinity” (of consumers and commodities)-in particular, the assumption of an unbounded time horizon along with (dated) commodities for each period-suffices to render the First Theorem inapplicable to the general overlapping-generations model. A second theme in [ 171 is that the limited opportunities for intertemporal exchange are a possible cause of inoptimality. Cass and Yaari [8] study this aspect, stressing that this source of inoptimality is not exclusive to dynamic models. A third theme of the Samuelson article is that paper assets (e.g., money) created in consequence of the government’s deficit can cure--or, at least, reduce-inoptimality. This theme has been explored in several articles, including 16, 19, 21, 221. The tools of this research have been picked up by macroeconomic theorists and used to address the more traditional questions about government policy. Since the overlapping-generations model is the

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 Allocation of Investment in a Dynamic Economy

I. Introduction and summary, 592. — II. The one-sector two-capital model, 593. — III. Dynamic analysis, 597. — IV. Market structure, 603.— V. Rationality and irrationality, 607. — VI. Concluding comments, 609.

Public debt, taxation, and capital intensiveness

This chapter provides an overview of the steady-state behavior and certain propositions in comparative dynamics, in particular, that across steady-state sign ( d Δ/ dk ) = signer( sr − n ). The chapter presents a full dynamic analysis. A certain stability analysis and a closely related assumption about uniqueness of the balanced growth state seems to be fundamental to Diamonds claim that d Δ/ dk d Δ/ dk d Δ/ dc > 0.

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

The overlapping-generations model. III. The case of log-linear utility functions

Existence of competitive equilibria in the overlapping-generations model has been studied in some detail; see [4-6,9]. We need to know more about the properties of these equilibria. How many are there? How do they depend on the basic parameters of the economy? We also need to compare the set of monetary competitive equilibria with the set of nonmonetary competitive equilibria. Are there “vastly more” monetary equilibria than nonmonetary equilibria? Analysis of the properties of competitive equilibria is for the overlappinggenerations model an important, but difficult, task. In his seminal article [8], Samuelson put it this way: “... if we take any finite stretch of time and write out the equilibrium conditions, we always find them containing discount rates from before the finite period and discount rates from afterward. We never seem to get enough equations; lengthening our time periods turns out always to add as many new unknowns as it supplies equations . ..” Samuelson suggested that, rather than facing this difficulty directly, “We can try to cut the Gordian knot by our special assumption of stationariness . ..”

The Theory of Hamiltonian Dynamical Systems, and an Application to Economics

A Hamiltonian dynamical system (HDS) naturally arises in the standard control problem involving optimization over time. On this ground alone, a systematic study of the basic structure of the general HDS should be extremely useful for mathematical control theory. Applications of HDS theory extend beyond models involving optimization. The classic studies of such systems were motivated by problems in celestial mechanics. While much of the analysis of my lecture will be motivated by the normative (optimizing) model of macroeconomic growth, I will show in passing how HDS theory may be useful in analyzing many positive (“nonoptimizing”) models of macroeconomic growth.

ESSAY I – Introduction to Hamiltonian Dynamics in Economics

Publisher Summary This chapter presents an introduction to Hamiltonian dynamics in economics. There is an alternative representation of static technological opportunities that is more congenial to dynamic analysis, that is, the representation of the technology by its Hamiltonian function. The contribution by Cass and Shell treats optimal growth and decentralized or descriptive growth models in both continuous and discrete time as applications of Hamiltonian dynamics. While optimal growth with time-discounting yields a very simple perturbation of a Hamiltonian dynamical system, there are other perturbed Hamiltonian dynamical systems that arise in economic theory. Cass and Shell discuss the general problem of decentralized growth with instantaneously adjusted expectations about price changes. In its general form, the model allows for the interpretation of competitive growth with utility-maximizing agents.

ESSAY III – 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.