Distributional dynamics of cautious economic adjustment processes

A class of discrete dynamical processes defined by xt+1=min⁡{αxt,θ(xt)}xt+1=min⁡{αxt,θ(xt)}, where α>1α>1 and θθ is a monotonically continuous decreasing function, has frequently appeared in economic analysis. The process is studied theoretically and numerically. The types of invariant density that can be generated by such processes are identified and the approach for the construction of such processes analytically with specified invariant densities are provided. Finally, the relationship between the second-order derivative of the right-branch θθ and the shape of invariant density is addressed.