Yes and no, it depends on which context in which they are used. If you are evaluating systems where the power output is only heat (ie space heater) and at a constant voltage then that formula is simply interchangeable with the others.

However if you are evaluating a system where not all the energy put into the system is converted to heat (ie lighting system or electric car) then current x voltage is the sum of I^2 x line resistance + useable power output + non-thermal losses .

This is why it is more efficient to transfer power at high voltage/low current then the other way around.

Anyways the answer to my original question depends on how the DC restistance of the MH ballast changes with the different voltage taps on it. If it is constant and the power input is constant there should be less heat from the ballast and more energy to the lamp (heat and/or light), as miniscule as it may be. When I do my upgrade Ill break out the multimeter and find out...

Ooooo, formulae! Did you know that the standard deviation of a normal curve is equal to the square root of the mean square within groups in a one-way analysis of variance?
Can you tell I'm jumping between surfing the reefing boards and doing a stats assignment?

double post

because there is less line loss at a higher voltage and lower current.

if you pass 1000 watts at 100 volts you are using 10 amps but if you do 1000 watts at 1000 volts you are only pushing 1 amp and this will cause less line loss. this is why they send 10000 volts and more on the high voltage power lins instead of 220 volts and megga amps.

I do agree it is better and slightly more efficient to run ballast on 220volt power if you can as it will use less current so you can put more on your breaker and you will have a small percentage less line loss.

Steve