Stochastic One-Sector and Two-Sector Models

This chapter extends the basic deterministic one-sector and two-sector growth models to a stochastic context in continuous time, using Wiener processes for the description of various sources of uncertainties in the growth rate of the labor force, the rate of the capital depreciation and the saving rate. The drift and diffusion coefficients of the stochastic dynamic systems are homogeneous of degree 1 in the two state variables: labor and capital—which allows a reduction into the stochastic dynamics of the capital-labor ratio. The crucial issue of hitting absorbing boundaries for the stochastic growth models are rigorously examined, and simple criteria (sufficient conditions) for inaccessible boundaries are established, a subject on which existing literature has been deficient. From the Kolmogorov forward equation is derived the steady state probability distribution of the capital-labor ratios. For both stochastic one- and two-sector growth models, the sample paths—of transition to steady states or for persistent endogenous growth—of particular state variables are simulated for many parametric specifications of the Cobb-Douglas (CD) and constant elasticity of substitution (CES) sector technologies involved. Impacts of technology shocks are similarly demonstrated. All relevant sample paths are exhibited on both shorter and longer time horizons.