This is really interesting stuff, and thanks for the references.
A few comments:
It’d be nice to clarify what:
“finite intergenerational equity over [0,1]^N”
means (specifically, the “over [0,1]^N” bit).
Why isn’t the sequence 1,1,1,… a counter-example to Thm4.8 (dictatorship of the present)? I’m imagining exponential discounting, e.g. of 1⁄2 so the welfare function of this should return 2 (but a different number if u_t is changed, for any t).
Regarding your second question: the idea is that if x is better than y, then there is a point in time after which improvements to y, no matter how great, will never make y better than x.
So in your example where there is a constant discount rate of one half: (1, 1, 1, (something)) will always be preferred to (0, 0, 0, (something else)), no matter what we put in for (something) and (something else). In this sense, the first three generations “dictate” the utility function.
As you point out, there is no single time at which dictatorship kicks in, it will depend on the two vectors you are comparing and the discount rate.
This is really interesting stuff, and thanks for the references.
A few comments:
It’d be nice to clarify what: “finite intergenerational equity over [0,1]^N” means (specifically, the “over [0,1]^N” bit).
Why isn’t the sequence 1,1,1,… a counter-example to Thm4.8 (dictatorship of the present)? I’m imagining exponential discounting, e.g. of 1⁄2 so the welfare function of this should return 2 (but a different number if u_t is changed, for any t).
Thanks for the comments!
Regarding your second question: the idea is that if x is better than y, then there is a point in time after which improvements to y, no matter how great, will never make y better than x.
So in your example where there is a constant discount rate of one half: (1, 1, 1, (something)) will always be preferred to (0, 0, 0, (something else)), no matter what we put in for (something) and (something else). In this sense, the first three generations “dictate” the utility function.
As you point out, there is no single time at which dictatorship kicks in, it will depend on the two vectors you are comparing and the discount rate.