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Optimal Allocation of Spending on Existential Risk Reduction over an Infinite Time Horizon (in a too simplistic model)

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.