I think what you meant was 220v*5amps = 1100 watts. All 1k ballast draw 1100 watts, the extra 100 is lost as heat and in inefficiency. But we already know its 1k ballast.

At 110v*10amps=1100 watts the same at 220. But because the cable only carries 5 amps instead of 10 it is more efficient.

I don't think your math is right breeder and what do you mean by ballast and lamp should be 1=1? 1v does not equal 1amp! Volt is potential difference and amps is how much juice is being drawn.