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Maybe you are interested in the following paper, which deals with similar questions as yours:
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Interesting, if as you say a bit unrealistic. If I'm interpreting your graph correctly (although I feel like I am probably not; I'm definitely not an economist), you end up describing an endowment-like structure, where if you're going to live forever, you'll want to end up giving away a constant amount of money each year (your b=0 line in the chart), or maybe an amount of money that represents something like a constant fraction of the growing world economy?? (Your b = (r-p)/r line?) It might be helpful for you to provide a layman-accessible summary, if I'm getting this all wrong.
In your conclusion, you talk about a couple ways that your model could be extended:
Of these, personally I'd be most interested in hearing about the last two, as they seem perhaps the most answerable with a pure mathematical approach. It would be interesting to know what ideal investment strategies look like under different combinations of assumptions:
And other things like that.
In fact, the optimal spending rate is not constant but starts growing (approximately and asymptotically precisely) linearly once funds exceed a certain threshold. The solid lines in the hand-drawn phase diagram you are refering to are the points where the growth rate of funds (b_dot) is the same. The optimal spending policy is the one starting at the threshold b_hat approaching the line where the growth rate of the budget is constant (at (r-rho)/r). Although I do not prove that this is the optimal policy, what I do prove is that the time trajectory of the spending rate is asymptotically linear. I edited the post to make this more clear.
In case this led to confusion: By spending rate, I am not referring to a proportion of available funds but to the rate of change of how much money has been spent at a given point in time. Since the model is in continuous rather than discrete time, I talk about a spending rate at a certain point in time rather than spending in a certain time period.