Investment in time preference and long-run distribution

This paper presents a simple dynamic general equilibrium model in which each household can make a costly investment in patience capital at each time. We show that the interior long-run steady state is unstable, in the sense that per household, there is a one-dimensional curve lying in the two-dimensional space of its patience capital and physical capital amounts, and convergence happens only when its initial pair falls exactly on the curve. Households with the initial vectors falling in the upper side of the curve invest more in patience capital, which leads themselves to save more, and hence, the consumption level grows in the long run. Households with the initial vectors falling in the lower side opt out from investing in patience capital, leading to a decay of patience level, which leads themselves to save less and hence they perish in the long run. We also show a possibility that there is an expanding swing between the two classes.