Quasihomogeneous Dynamics in the Plane

This chapter generalizes the theory of homogeneous dynamics to a wider class of differential equations. As homogeneous production functions gave the impetus to study the solutions to homogeneous dynamic systems, Chap. 28 , it was also quasi (also named, “almost” or “pseudo”)-homogeneous production functions that lead us to study the qualitative properties of quasihomogeneous dynamics. While the trajectories for homogeneous dynamics are obtained from each other by a multiplication along the rays through the origin, the trajectories for quasihomogeneous dynamics are obtained from each other by a multiplication along the family of power curves y = ( αx ) β , x > 0 $$y=(\alpha x)^\beta ,\;x>0$$ . The stability properties of family coordinate solutions are summarized in Theorems 30.5 and 30.6. With the class of quasihomogeneous governing functions, growth models, with the total factor endowments ( L , K ) $$(L,K)$$ as state variables, will be able to exhibit also a permanent increasing capital-labor ratio. Many economic (allocational, distributional, etc.) implications of the quasihomogeneous technologies need to be further and fully explored, when such technologies are admitted to enter the dynamics of economic growth models. Lau (1972) initially introduced quasi (almost)-homogeneous functions in economics.