I use a 7.5kW propane burner as I am cast out onto the patio by my better half who cannot stand the smell. Also the consequences of a major wort spill in my wife's pride & joy kitchen would not be survivable with all parts still attached!
I had in mind that I use about 1kg of gas for a typical heat up to boil and 60 minute vigorous boil. I don't have any notes of the calculation but I think that's about right, next brew I will check properly. This fits the maths roughly:
LPG energy density is 49MJ/kg. A 7.5kW burner will consume around 0.55kg per hour from these figures. So allowing for heat up time etc this fits roughly with the estimate of 1kg per brew. The cost of 1kg of propane based on a 19kg refill from my local Calor Gas centre is £2.85. For a 35 minute warm up followed by 60 minute boil the calculations indicate 0.87kg and a price of £2.48.
Calculations for a 3kW electrical input: Assuming wort is collected at +60C (allowing for losses) and assuming 30 litres then 3kW input will raise this to 100C in 28 minutes assuming no losses, so at a guess allowing 25% lets say 35 minutes followed by a 60 minute boil at full power. The total usage would be 4.75kWh. Googling an average price per kWh give 16.3p so a total price of £0.78
Obviously very back of envelope stuff, but calculations fit what I'd expect and point towards a gas heat efficiency of about 50% (i.e. gas & electric heater giving roughly similar heat energy per second into the liquid). My suspicion would be that this is a little pessimistic on gas and optimistic on electricity. But say assume gas would be at least 2.5x the price of electricity then that gives a decent working assumption.
I guess the other thing to check is whether your container actually work well on the induction hobb. We have one and some vessels do not work very well. Good quality stainless steel will not work at all. Most pans intended for induction hobbs include a layer of netall which is good at generating heat when eddy currents are induced in it.
But overall LPG does look expensive & next brew I'll collect some accurate measurements. However gas is convenient as it delivers a good reliable heat and removes the need to clean any heater elements, but the apparent difference does make me think whether I should be looking for a good immersible 3kW heater. I don't know if such things are available, I would not want to cut holes in my 50 litre kettle to fit one permanently.