"ZMP" stands for "Zero Marginal Product". It should really stand for "Zero Value Marginal Product". Can an additional worker produce no additional goods of any value? Is that why they are unemployed? Yes, but.
There are three types of ZMP. Two of them are based on two different types of coordination failure. The disagreement between macroeconomists is on what type of coordination failure is causing the ZMP of unemployed workers.
It is possible that some potential workers, perhaps because of severe disabilities, simply cannot produce anything useful, even in a perfectly functioning economy. Even the mythical central planner, who could allocate resources perfectly, wouldn't be able to think of anything useful they could work at, and would leave them idle. They have a Zero Value Marginal Product. There's no coordination failure. The problem simply can't be solved, given existing technology, tastes, and resources.
Nobody (I think) is talking about ZMP1.
ZMP2. Coordination failure 1. (Patterns of Sustainable Specialisation and Trade). (Thanks David and Wonks Anonymous).
The unemployed plumber wants his house re-wired. The unemployed electrician wants his pipes replaced. But the plumber doesn't know that there's an electrician in town; and the electrician doesn't know that there's a plumber in town. Given the current state of knowledge, both workers have Zero Value Marginal Product. Nobody is willing to pay for their labour. They would be willing to pay, if they knew of each other's existence, but they don't. The mythical central planner, who knows everything that anybody knows, would sort this one out immediately. All he has to do is introduce the plumber and the electrician, and both jobs are done.
Arnold Kling, I think (I'm not sure if I am representing his view correctly), is talking about ZMP2. (Of course, I have drastically oversimplified the coordination problem, just so we can understand it simply. In the real world it is much more complicated, and a real world entrepreneur might have to bring together many more people to interact in complex ways to resolve the coordination problem.)
There's an excess supply of plumbers' and electricians' services, and an excess demand for plumbers' and electricians' services, all at the same time. Both want to sell more of their own labour, and buy more of the other's labour.
ZMP3. Coordination failure 2. (Monetary Disequilibrium).
The unemployed hairdresser wants her nails done. The unemployed manicurist wants a massage. The unemployed masseuse wants a haircut. The hairdresser knows where the manicurist can be found. The manicurist knows where the masseuse can be found. The masseuse knows where the hairdresser can be found. But all three women are short of money, and won't spend any until after she earns some. Given the unwillingness of each to spend money, each of the three has a Zero Value Marginal Product. Nobody is willing to pay for their labour.
This one's a bit trickier for the mythical central planner to solve. Introductions alone won't do the trick. Each woman already knows where to find the service she wants. That's not the problem.
The central planner could simply order them all to provide the desired service, for no cash. That would make all three women better off. But if the central planner is mythical, and can only give mythical orders, that won't work.
He could try to get all three women together, and try for a three-way barter deal. But it's hard to cut someone's hair while your customer is giving a massage and you are having a manicure. Who's going to go first, and how will she know the other two won't break the chain?
The easiest solution is for the central planner to print more money and give each of the three women enough so they buy what they want.
ZMP3 is what the rest of us Monetarists and Keynesians (quasi or otherwise) are talking about.
There's an excess supply of each woman's labour. Each one wants to sell her labour and can't. In one sense (in Robert Clower's sense) there's also an excess notional demand for each woman's labour, because each woman would like to buy labour from one of the others. But she would only actually buy it if she could sell her own labour, and she can't. So there's no excess constrained or effective demand for labour.