Krugman and the Minimum Wage

Krugman slams the idea of lowering the minimum wage to boost employment:

Here’s how the fallacy works: if some subset of the work force accepts lower wages, it can gain jobs. If workers in the widget industry take a pay cut, this will lead to lower prices of widgets relative to other things, so people will buy more widgets, hence more employment.

But if everyone takes a pay cut, that logic no longer applies. The only way a general cut in wages can increase employment is if it leads people to buy more across the board. And why should it do that?

Well, the textbook argument — illustrated in this little writeup — runs like this: lower wages lead to a lower overall price level. This increases the real money supply, and therefore liquidity. As people try to make use of their excess liquidity, interest rates go down, leading to an overall rise in demand.

Even in this case, it’s hard to see the point of cutting wages: you could achieve the same effect, much more easily, simply by having the Fed increase the money supply.

But what if we’re in a liquidity trap, with short-run interest rates at zero? Then the Fed can’t achieve anything by increasing the money supply; but by the same token, wage cuts do nothing to increase demand.

The problems at a zero interest rate are complicated and Krugman may be right on this point (he won the Nobel prize after all), but I'm confused. As I understand it, a minimum wage law is only redistributing money, not creating new money. This redistribution increases the cost of a unit of labor. The higher the cost of a unit of labor, the fewer units will be demanded. This argument has nothing to do with the demand for the widgets they produce, like Krugman was talking about.

In theory, at least during normal economic conditions, lower wages create demand for my units of labor. It essentially, frees up money within a firm to be distributed to additional workers. That's the theory anyhow. Obviously, there is no guarantee, if a firm had low demand for new labor they could always just save the money and keep their staffing the same.

One more question on Krugman: in his notes he writes:

But in liquidity trap conditions, the interest rate isn’t affected at the margin by either the supply
or the demand for money – it’s hard up against the zero bound. And as a result the usual
explanation for the downward slope of the AD curve doesn’t work. You can appeal to the Pigou effect, I guess – but against that you have to put Fisherian debt deflation. In a liquidity trap, the AD curve is at least as likely to be upward-sloping as it is to be downward-sloping.

Again, I'm not an economist, but how can an AD curve ever be upward sloping? Vertical, I get, but upward sloping?