That's a simplistic example with extreme inputs. If you gave the people earning the least, in a country like the UK, a sensible increase it doesn't follow that

That's a simplistic example with extreme inputs. If you gave the people earning the least, in a country like the UK, a sensible increase it doesn't follow that all prices will rise to match it, although they might rise a small amount. The reasons being that the total income and spending power of those on minimum wages and benefits is small compared to the total economy, and that each working person in the supply line of sandwiches is only a small proportion of the sandwich cost, i.e. once you get to a certain scale you can churn out a lot of sandwiches per worker. Sure you can cite silly and extreme examples like "give everyone 1 million dollars." But how about an analysis of "raise minimum wage by 5 dollars per hour and increase benefits by 50 dollars per week"?