Does gear ratio affect Power? Let me preface by saying that this may not be the right SE. I considered asking on the Physics SE, but I thought I might try here first. If it is wrong, I am not against it being migrated.

From basic principles of Physics, power is computed as Work/time. So consider a rider and bike system climbing a hill. The work done is the difference in potential from bottom to top, and obviously the time will be climb-time. 

Now, my question is:


  Given the same rider, same bike weight, and same climb-time, does your gearing affect power? Also assume that the climb is efficient, no slipping tires, normal pedaling, etc.


From a physical perspective, I expect the answer is no. Same difference in potentials, same time, same power. However, from the rider perspective, I know that it sure feels like a lot more power is being used to climb with a harder ratio.

I expect that the answer is that the disparity comes from idealizing the system. If we consider the bike a closed system, we expect that all the energy put into the bike carries it up the hill, but this is not the case. Moreover I think the inefficiency of the human body will be relevant. However, I still cannot resolve the question.
I suspect you mean efficiency rather than power.

In my opinion the main trade-off is between increased biomechanical losses at higher rpm (basically muscle friction) and decreased blood flow with higher forces at lower rpm. The balance depends both on the rider and the duration.

In the IHPVA Journal of Human Power, Issue 45(pdf, index here) is a paper called Maximum Human Power where they talk about Tyler Hamilton winning the Mt Washinton climb in 51 minutes:


  "he rode much of the climb, however, in the 23-tooth cog, and made several surges in the 21." If he had 700- mm wheels, as seems likely, his average cadence would have been 63 RPM.


The whole article is worth reading, and it might pay to browse the index for similar papers.

The flip side is that top sprinters often go well over 150rpm in the final sprint. At that point they're trading biomechanical efficiency for peak power. I used to peak at over 900W for 10 seconds (>8W/kg) at around 130rpm, but my hour performance of around 350W used a cadence of about 80-90rpm.

The real answer is specific to you. It will depend on your body shape, muscle type, fitness and more transient factors. It's also a question best answered by experiment, and should be part of your tr4aining schedule if you're competing. If not, I suggest finding a climb you ride regularly and keep a training diary.

There has also been much discussion over the hydration for long climbs. Is it better to hydrate and start heavier, or run slightly dehydrated so you weigh less? IIRC the conclusion was that hydration was better, but I can't find the reference.