Some New Results on the Local Stability of the Process of Capital Accumulation

In the “Mecanique Analytique” (1788) Lagrange proved the following theorem. If the potential function of a conservative dynamical system attains a minimum (maximum) at a position of equilibrium then the motion in a neighborhood of this equilibrium point is stable (unstabZe).l In 1885 Poincare showed that if the potential function is made a function not only of the state of the system but also of an exogenous parameter and if the equilibria induced by the potential function are considered as functions of the parameter we obtain an equilibrium surface in the parameter-state space for which the stable and unstable branches are separated by bifurcation equilibria [50, pp. 43-551. If we combine these two ideas for a class of dynamical systems that arises in economic theory we obtain the beginings of a rich and interesting theory of economic dynamics. Such a theory has two parts, a short-run dynamics and a long-run dynamics. For a fixed value of the exogenous parameter the short-run dynamics classifies the equilibria associated with this parameter value into stable and unstable equilibria, and shows the local nature of the motion in the neighborhood of each equilibrium point when viewed in the state space. Such an analysis carried out for all feasible values of the parameter leads to a classification of the equilibrium manifold into stable and unstable submanifolds. A system with only stable equilibria will in general have associated with it a confinlrous long-run equilibrium manifold. When the exogenous parameter, taken as fixed in the short-run dynamics, is allowed to vary in a slow and systematic way, the system will trace out a trajectory along the equilibrium