"It is hardly necessary to say that Post Keynesians reject the ‘old classical’, ‘Bastard Keynesian’ and ‘New Keynesian’ argument that unemployment is due to the existence of a real wage above the equilibrium or ‘market clearing’ level owing to the trade union or government interference in the operation of the free market for labour. They also dismiss the ‘New Classical’ notion that unemployment is the (voluntary) result of intertemporal income-leisure choices by individual workers. As was demonstrated above, neither claim is supported by micro foundations; and neither has any macro foundation whatsoever. A Post Keynesian theory of unemployment would instead start from the proposition that in aggregate the level of employment depends on the level of output, which is itself determined by aggregate demand and therefore heavily influenced by macroeconomic policy. Unemployment is simply the difference between the level of employment and the aggregate supply of labour, which may - as explained earlier - safely be regarded as invariant in the short run with respect to the real wage, but variable with respect to the number of job opportunities.” (King 2002: 84)."I don't think I quite agree with King, at least on the New Keynesian part (or the Post Keynesian part, for that matter) - or at least I think it's misleading. I think the importance of price stickiness for New Keynesianism is often misunderstood. People have this image in their heads of a non-clearing labor market when they hear "sticky prices". The market's just broken for a little while and eventually it'll get back on track. Austrian types don't like the implication that the market's broken. Post Keynesian types don't like the implication that it's just a hiccup.
But actually the price stickiness point has little to do with labor market directly. As I've pointed out in the past, the principle function is to explain why the short run aggregate supply curve is upward sloping and not vertical (that's why it's called New Keynesian and not New Classical - emphasizing non-vertical aggregate supply is very non-Classical). You might know the short run aggregate supply curve in more modern models as the Phillips Curve. Just rearrange things, switch from prices to inflation, and tack on Okun's Law and you've got the Phillips Curve. So price stickiness gives us the Phillips Curve tradeoff which, with the IS curve, can pin down an output level below a full employment level of output. [This is related, btw, to why I think the substantive difference between New Keynesianism and Old Keynesianism (not to be confused with Post Keynesianism) is overblown... but that's another discussion].
This is a lot less neoliberalish than Post Keynesians often make it out to be. This sentence by King: "A Post Keynesian theory of unemployment would instead start from the proposition that in aggregate the level of employment depends on the level of output, which is itself determined by aggregate demand" could just as accurately be said of unemployment in a New Keynesian model.
There are of course lingering questions about tendency to move to the natural rate of unemployment and the stability of the natural rate of unemployment. It is true that New Keynesians are more likely to suggest that there is a strong tendency towards a stable (albeit changing) natural rate of unemployment. And there's probably good reason to say that - outside of liquidity trap situations the economy has seemed to exhibit that sort of behavior (and we've got good liquidity trap models when it doesn't). There's great discussion by New Keynesians in good standing like Mankiw, Ball, Moffitt, Akerlof, etc. about less conventional NAIRUs too.
The lack of a (necessarily) stark difference between New Keynesians and Post Keynesians on all these points is made forcefully by Stockhammer (2011)*. Of course, not all Post Keynesians agree. LK, in his post, also cites Lavoie and Davidson. It just so happens that Lavoie and Davidson are two Post Keynesians that are NOT inclined to agree with Stockhammer on this sort of point. It would be a mistake, though, to conclude that they speak for all Post Keynesians. LK is just sharing one side of the Post Keynesian position.
OK, so I say there's no real difference between New Keynesians and Post Keynesians on this particular issue raised by King. But surely there are differences between them, so what are they?
Leaving aside the point that someone like Stockhammer might specify or derive the Phillips Curve/IS Curve/NAIRU/MP curve nexus differently than someone like Mankiw I would say that a principle difference is the other contraptions they have that are often completely neglected by New Keynesians. These include:
1. Mark up pricing
2. Dividing the population into classical classes (with rentiers replacing landowners)
3. Various conflict theories of inflation
4. Explicit modeling of capacity utilization for inclusion in the investment function (capacity utilization is of course not much different from an output gap - but the inclusion in the investment function introduces accelerator effects which is interesting and [I think?] dropped from most modern analyses)
5. Modeling it with wage shares front-and-center so we can talk about distributional issues
There's plenty of heterodoxy in all that, chiefly through the distributional facets and the departure from marginalism. That, it seems to me, distinguishes Post Keynesianism. It's less a matter of one side talking about inadequate demand and the other not. In other words, to the extent that I'm a Post Keynesian (and I do like a lot of what they do) I'm a Stockhammer and (I guess? I should be much better read:) Pollin/Blecker/Dutt sort of Post Keynesian. I am less of a Davidson/Lavoie sort.
* "The Macroeconomics of Unemployment" in A Modern Guide to Keynesian Macroeconomics and Economic Policies, ed. Eckhard Hein and Engelbert Stockhammer