> If someone has no job the government pays the full $x. So the employer is only helping the government.
Better to let the government pay the full $X, rather than keeping companies around that only succeed because they depend on the government to keep their employees alive. And think of a company like WalMart—by paying its employees very little, and depending on the government to take over things it doesn't want to pay for (like security, leaving it to the police instead, at enormous cost to them, but no cost to WalMart), they can undercut other companies that cost the government far less, leaving the whole economy less healthy.
> However, the subsidies are to the worker, not the company
The subsidies are to the worker, yes. GP was asking why the government has any say in a business relationship between a worker and a company, and the answer is: because they're involved as well, through the need to subsidize whatever the company doesn't pay. They're automatically part of the equation, and thus have a say in what the worker & company can settle for.
> To see this, imagine a hypothetical world without the subsidies, and try to figure out if the employer would be richer or poorer. (Hint: richer, since they don't need to compete with the "stay home and watch TV while collecting welfare" option.)
You mean, if companies were unable to pay less than a living wage? They would have to pay the worker more money, so they would be poorer. However, since all companies were paying more, everyone would probably do better in the end.
Or do you mean, if they could pay people as little as they wanted? They'd be richer, but less and less money would end up moving around in the economy, and everything would slowly grind to a halt. And lots of people would starve.