TheIronWarrior
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The _power_ output of an engine is the only thing that matters to performance (acceleration as well as top speed).
I will begin by assuming by "power output" you are referring to the entire power curve and not just the reported peak power figure so as to not get in to the finer details of engine performance. We will also ignore the effects of wind resistance, friction, weight, etc. for simplicity's sake, even though these have a huge impact on performance (acceleration as well as top speed).
Your statement above is not accurate. Take two identical motorcycles and change the final drive gear ratios on one of them and one will undoubtedly accelerate faster than the other. This comes as a trade-off as you will lose out on top speed. Identical engines with different gear ratios WILL have different acceleration and top speeds. This is a fact. Many other things can have an impact on performance as we are talking not about the performance of the engine as a separate unit, but of the entire system of a motorcycle.
Providing his determination that he "needs more power" was made based on lack of acceleration and he is willing to lower the top speed of the machine proportionally, changing the final-drive gearing could provide the answer he is looking without increasing the performance characteristics of the engine. From a strictly technical point of view, this changes the relationship between the motorcycle's speed and the power output of the engine, as you are moving the desired speed range up into the higher power band of the engine. With this in mind, you could argue (semantically) that this is an increase in 'power' but that's more a language-based argument than physics-based.