Based on the Cass-Koopmans Model with endogenous fertility given in the paper [1-4], the fertility dynamics is discussed in this paper. It is proved that the first quadrant of the k,c plane is separated into two parts by a increasingly differentiable curve. The fertility increases with time on the upper half and decreases with time on the lower half. The fertility decreases along one arm of the economic growth path on which the per capita capital and per consumption increase and increases along the other arm of the economic growth path on which the per capita capital and per person consumption decrease under the condition 0 <θ≤α. This confirms the empirical finds that negative relationship between the economic growth and population growth.
By introducing the logistic-like technology, the classical Solow-Swan model is extended to inquire the technological overflow and catch-up of the developing economy in this paper. The improved model is described by a two-dimensional dynamical system. It is proved that the model has a unique equilibrium which is a sink and the solution of the equation is globally asymptotically stable. And the classical Solow-Swan model is a special case of the model given here. The economic growth patterns are discussed by phase portrait analysis at the end of this paper.
The technological change as an exogenous variable is introduced into the RCK model with endogenous fertility in this paper. It is proved that the model has at least a nonzero steady state and the nonzero steady state is unique when the parameters satisfies some conditions. By phrase protrait analysis, the unique nonzero steady state is saddle and the economy has a unique optimal growth path. The results obtained implies that the relationship between the technological change and population growth rate determinated by the economic structure and the parental ethies. For the economy in which the parents is selfish, promoting the technological change rate decreases the fertility at the steady state. On the other hand, for the economy in which the parents is less selfish, the fertility increases as the technological change rate increases.
CAI Dong han College of Mathematics and Computer Science, Wuhan University, Wuhan 430072, China
In this paper,a C\|K model with solvable endogenous fertility under the strongly additive utility function is presented.The discrimination conditions of the existence of the nonzero steady states are given.Under a kind of utility function and production function,we prove that these conditions are satisfied and the economy at least has an optimal growth path.The positional relationship of the multiple steady states on the plane is discussed when multiple steady states and multiple growth paths exist.By numerical analysis,the fertility decreases with the per capita capital and per capita consumption increasing and increases with the per capita capital and per capita consumption decreasing on the economic growth path are obtained.
Cai DonghanCollege of Mathematics and Computer Science,Wuhan Univ.,Wuhan 430 0 72
