Bifurcation Routes and Economic Stability

There are two basic tools to analyse fundamental issues in dynamical macroeconomics. One of them is a model of optimal growth describing savings behaviour. The second one is the Solow-Swan model with a constant aggregate propensity to save out of income. A steady state of the dynamical economic system corresponds to a growth path satisfying a stationary solution of a properly defined differential system and thus exhibiting certain conditions of constant growth rate in a single-sector model. One of possible mechanisms for regulating growth path is a savings rate. A regulation through the savings rate is made possible by distinguishing two types of income, two social classes or by introducing money and financial assets (see Henin(1986)). The standard neoclassical growth theory, Kaldor(1956), Pasinetti(1962), Samuelson and Modigliani (1966) were investigating the question to what extent different saving behaviour of the two income groups (labour and capital) might influence the growth path. The case that each agent is able to save by accumulating capital but not to borrow from the other is solved by Bewley(1998). Woodford(1990) went beyond BewleyÌs results both showing that equilibrium cycles are possible and exhibiting conditions under which equilibrium dynamics are chaotic. The aggregate savings function need no longer be concave, so that multiple and unstable steady states can occur. The role of differential simple savings behaviour and distribution effects for stability of stationary states was investigated in Bñhm and Kaas(2000). They demonstrated that the economy exhibits unstable steady states and fluctuations if the income distribution varies sufficiently and if shareholders save more than workers. A central question of this paper is whether using and/or un-using of a foreign investment change the qualitative properties of a growth path of the dynamical economic system. The analysis will be provided in the two steps: First, a stability the economic system without a foreign financing will be analysed by the behaviour of the eigenvalues of the Jacobian matrix; Second, the economic system with a foreign financing will be analysed by the Hopf bifurcation.