As far as the assumptions go, the ones you presented are fairly standard in economic models. And really, it looks like the basic infinite-horizon problem one sees (a generalization of the standard intertemporal consumption model).
Exponential discounting is typical -- the question is usually to what extent people discount the future. Though researchers, especially behavior economists, are questioning the constant discounting rate assumption, and are beginning to prefer hyperbolic discounting models.
(One of the things that you lose with hyperbolic discounting, by the way, is the idea of "time consistency" -- in this case, your consumption/savings decision you planned for a future time t
is not necessarily the same one you would prefer (and thus carry out) once you hit t
. There's been a lot of work done on this, especially with regards to policy rules.)
Log utility functions are also a standard assumption. They're fairly nice because the derivatives are simple, it's a special case of the Cobb-Douglas function
(a standard functional form in econ for utility and production), and they're a good and plausible-ish example of a function coming from risk-averse preferences.
This is a really basic model: no borrowing, a single asset, no taxes, no inflation, no exogenous shocks, a constant discount rate, a single agent with log utility, and so forth. Consumption smoothing would dictate that the best strategy would be to consume a fixed amount in each period, saving the rest. It is a little odd, though, that the interest rate wouldn't matter -- naively, I would expect that by consumption smoothing, I would consume more today by a factor of the interest rate, so that I could consume the same amount tomorrow.
(I need to think about this some more -- this is the kind of stuff, actually, that they don't teach in PhD programs (at least at Northwestern's).)