A Mathematical Review of a Dynamical Growth Model With Endogenous Savings and Subsistence Consumption

We provide formal mathematical support for an endogenous saving rate extension, as suggested by King and Rebelo in a Solow-Swan model context. The saving rates st are determined by the workers’ decisions through a Stone-Geary utility function over subsistence income ψ and livelihood consumption ˜c. This endogenous saving rate gives rise to two critical points: one lower equilibrium kψ analogous to a poverty trap and a higher long-term equilibrium k∗. The lower equilibrium is repulsive and unstable, and the higher equilibrium is proven to be unique and stable.