lecturenotes

To get the mindset: 

Imagine that from a stock of (positive) size x, you get a proportional yield bx which can be reinvested for higher stock next period, or consumed now. b>0 is a constant.

You reinvest a fraction u (in the unit interval [0,1]) of bx, and consume (1-u)bx at log utility. There is capital depreciation at constant rate ? so we assume that this will give you a stock tomorrow of g(x,u) = (1-?)x + bux = (1-?+bu)x. There is no discounting.

To get a solution, we shall assume b>1-?. (And, ? in [0,1].)