Stochastic income and wealth

As is well known, the deterministic version of the maximum principle allows us to state rigorously the relationship between deterministic income (essentially the Hamiltonian) and deterministic wealth (essentially the state evaluation function). As a background motivating point of departure, consider the simple story of an infinitely long-lived individual whose sole wealth consists of a bank deposit paying a constant rate of interest. For this simple parable, whether we define and measure income in the spirit of Fisher as being the return on wealth, or in the spirit of Lindahl as being consumption plus net value of investment, or in the spirit of Hicks as being the largest permanently maintainable level of consumption, we get the same answer to the question ‘‘what is income?’’. The theory of the deterministic maximum principle shows is that this fundamental identity of the three seemingly different definitions of income is much broader and goes much deeper than the simple bank-account parable. We now seek to extend the investigation to cover uncertainty by dealing with the case where net capital accumulation is described by a stochastic diffusion equation in place of a deterministic differential equation. Stochastic diffusion equations introduce mathematical subtleties and complexities, which we will only deal with casually in this treatment. The introduction of uncertainty ratchets up the required level of analysis another several notches on the scale of mathematical complexity. To treat the subject of this paper fully rigorously and also at the level of generality of a multi-dimensional economic growth problem could easily turn this paper into a book and would be out of all proportion to its www.elsevier.com/locate/econbase Japan and the World Economy 16 (2004) 277–301