I've been thinking about a simplified problem involving farming:
Imagine that you're growing a crop on an infinite amount of farmland, starting with a certain number of seeds A (natural number), with average yield per seed y>1 (that is, if you plant A seeds, you'll most likely get yA seeds for the next harvest). Each season, you save a certain portion of seeds r (0<r<1) for the next season and eat the rest. Somehow, the yield per seed does not change as a result of practices like overusing the same plot of soil or tilling the rest of the plant back in (because only the seeds have value), and the present value of the seeds depreciates by a factor of k (0<k<1) per season, so the average present value of a seed's yield is v=ky.
If N is the number of seeds available per harvest, and t is the number of seasons since the beginning (whole number), then N=A*(yr)^t, so to keep this enterprise going, r>=1/y.
If H is the total number of seeds harvested after T seasons (whole number), then
H=sum(A*(1-r)(yr)^t,t,0,T)=A*(1-r)(1-(yr)^(1+T))/(1-yr)
If r>=1/y then H grows without bound as T->infinity, otherwise H->(1-r)A/(1-yr).
For this case, dH/dr=(y-1)A/(1-yr)^2, which is positive because y>1, so the total harvest grows without bound as r->1/y.
Now, let V be the net present value of the harvests, then
V=sum(A*(1-r)(vr)^t,t,0,T)=A*(1-r)(1-(vr)^(1+T))/(1-vr).
What is important to note here is that although it is possible for v>=1, this is not guaranteed, but what is guaranteed is that 1/y<1/v; if v<1, then r<1<1/v, so V=(1-r)A/(1-vr). Then dV/dr=(v-1)A/(1-vr)^2<0, so the ideal strategy to maximize net present value is to save no seed (r=0) and consume it all without planting.
If v>=1 then it is possible for r=1/v, and then V=(1+T)(1-1/v)A, which grows without bound as T->infinity.
The curious thing happened when I tried to maximize the net present value over limited timescales; in this case,
dV/dr=(v*(1-r)(vr-1)(T+1)(vr)^T-(1-v)(1-(vr)^(1+T)))/(1-vr)^2
so dV/dr=0 when
v*(1-r)(vr-1)(T+1)(vr)^T=(1-v)(1-(vr)^(1+T))
and although I was able to see that this means r=1/v for the cases T=0, T=1, T=2, T=3, and T=4, (after eliminating the extraneous roots), and that r=1/v is a root in the general case, it's curious that WolframAlpha wasn't able to solve this equation for r or at least figure out that r=1/v is always a root, not even with the sophisticated non-elementary functions that it knows about.