Tax Relief is Not a Subsidy
They also point out that such a tax relief would in fact likely increase tax revenues as the loss in tax revenues from the few domestic services that are performed in the formal economy today would be more than upweighed by the increase in domestic services performed and the shift from the black market to the formal market. Or as supply-siders would have put it, current tax rates are above the Laffer point (the tax rate which maximizes tax revenues).When Social Democrats claim that it costs a crtain amount of money for the state they're wrong thus.
They also make the important point that the current high tax rates prevent the division of labor that could have occurred where people able to hire someone else to do cleaning etc. rather than having to spend time doing it yourself. Taxes thus works as a kind of internal tariffs, with the same kind of negative effects as tariffs on international trade.
I would also add that it is wrong to regard tax relief as a subsidy since domestic services would still pay a lot of money to the state even after the tax relief. Subsidy is when a activity receieves more money from the state then it pays in taxes, such as they do with government consumption services like government schools, day care centers etc., not when it pays a smaller share of value added to the state.
While I would prefer that the tax cuts where made across-the board and not just on domestic services there is a case for starting with domestic services since they are likely more price sensitive and for that reason it is costless to reduce its taxation even more than other taxes even if radical tax cuts on other products can't be secured.