Krugman offers a response to a few critics, including I believe myself. His latter two points are on the macro model, his first point is trying to establish the relevance of the macro model for the minimum wage analysis:
1. Why did I go from minimum wages to overall wages? Clearly, a cut in minimum wages –which only apply to some workers – can raise the employment of those workers at the expense of other workers. But the advocates of a cut are claiming that they can raise overall employment. The only way that can happen is if a reduction in average wages raises employment.
There is a simple story here. Lower the minimum wage and firms with market power will in general hire more labor. (Sethi's critique refuses to consider that mechanism but simply shift the MC curve and watch it happen.) In the most straightforward setting the total wage bill increases, even if the average wage falls. With a higher total wage bill, there is no downward deflationary spiral. This general equilibrium point was emphasized by Jacob Viner in his very careful 1937 review of Keynes but it remains a neglected insight.
The negative scenario, namely the total lower wage bill, can possibly occur if employers use the lower legal minimum wage to lower wages for currently employed workers who were at the previous minimum. A few observations here:
1. Even then the net effect is indeterminate and not necessarily in the Keynesian direction. The total wage bill still could go up or even if the total wage bill goes down the total flow of purchasing power need not decline, given that employers just don't sit on their extra money. (This same point applies to all other second-best scenarios.)
2. The model already has assumed short-run wage stickiness, so it would be odd to suddenly relax that assumption as a way to get the total wage bill to fall.
3. Given that minimum wages don't cover so many workers, the AD effects are likely quite small in any case.
4. The new workers may well be collecting EITC, which will strengthen any aggregate demand effect from their employment.
5. The increase in aggregate supply — more work goes on! — itself has a positive effect on aggregate demand through subsequent Hutt-like, supply-side multipliers. It would be unusual if velocity shifts were completely neutralizing with respect to this increase in production.
6. The "then why don't we raise the minimum wage to $30 an hour" meme is an overrated "right-wing talking point" in a lot of policy debates. Still, in this context, it remains a good question from a purely analytical point of view. Such a change would not boost aggregate demand in most plausible models and from that admission you can work backwards.
Mixing up average wages and the wage bill is a common Keynesian confusion; they're not always moving in the same way, though they may seem to in some very simple models. Krugman's #1 is assuming a link between the micro and macro change that simply doesn't have to be there.
That all said, it's a fair enough point to note that changes in the minimum wage will likely bring only small positive effects in any case.