Abstract: In the work of Arrow et al. (2007, Proc. Natl. Acad. Sci. U.S.A.), they studied a macroeconomic growth model so that the population dynamic was involved in both the total utility (objective function) of the whole population and in the capital investment process. In essence, they assumed the deterministic evolution for both dynamics, such that the labour force of the population is also incurred through the Cobb-Douglas production function. In this talk, we will first introduce an extension of their problem, particularly over a finite time horizon, in which we also allow more realistic and generic population growth and incorporate a stochastic environment for both the demography and capital investment. For the corresponding Hamilton-Jacobi-Bellman equation, we show the existence and uniqueness of classical solutions over any arbitrarily large time horizon by using a hybrid approach that combines techniques in both partial differential equations and stochastic analysis. We believe that the mathematical analysis and methodology developed in this work can also apply to various sophisticated models arising from macroeconomics and mathematical finance. 

Reference: Arrow, K., Bensoussan, A., Feng, Q., and Sethi, S.P. (2007). Optimal savings and the value of population, Proceedings of the National Academy of Sciences, 47: 18421-6.

Acknowledgement: This work is supported by the Hong Kong General Research Fund (GRF) “Controlling the Growth of Classical Solutions of a Class of Parabolic Differential Equations with Singular Coefficients: Resolutions for Some Lasting Problems from Economics” with project number 17302521.