The reason that 220v welders are cheaper to run:

A welder is basically a step down converter. This means that it takes a high voltage (110/220v) and low amperage (15A-50A) and converts it to low voltage (17v-45v) and high amperage(190A-590A).

The power company charges us by the amount of kilowatts of electricity (1000 watts) that we use.

If I'm welding using 110 Amps I'd be using 1 watt of power (110A/110v = 1w). However if I'm using a 220v welder then the equation changes to 110A/220v = 0.5w

That's why 220v welders are cheaper to run.