Increasing the minimum wage wont cost the government anything as it will be paid by employers (who will also have more tax deductions themselves but this wont be as substantial a cost of lost tax revenue). It will however reduce employment prospects and squeeze a lot of small business such that they may just shut down. It may also squeeze service levels in such sectors as care industry where many staff are on zero hour, minimum wage deals. Thus pushing up costs for the most vulnerable in society. Swings and roundabouts.

Its a delicate balance.

Decreasing the incentive not to work has been in progress a long time, but doesn't seem to get some idlers off their arses. With the flight of much of the eastern european workforce post Brexit and pandemic, you'd have thought there would be queues of the unemployed stepping up to take on the roles that brexit was supposed to free up for them, but it seems to me that workforce UK consider such roles beneath them - which is why EU source, african and caribbean immigrant workforces came in in the first place.

It truly is insoluble in my view