In a diverse economy with extensive division of labor, it is inefficient to pay people in the form of output. If I am the accountant for a bakery, it is not my comparative advantage to sell bread. Paying me in bread would lead to a waste of resources as I try to go out and exchange it for what I wish to consume. To take the point even further, picture me as an accountant in a hospital--am I to be paid in surgeries?
I think that to take the view that there would be full employment in a barter economy is to fall into an intellectual trap. That trap is set in part by thinking in terms of a representative agent and in part by putting too much emphasis on money as a store of value.
Although Keynesian economics has these traps, Keynesians do not need to fall into them. Robert Solow is quite good on this. He understands the problems with representative-agent models.
The intellectually useful line of thought to pursue is this:
1. We know from trade theory that there always exists comparative advantage.
2. When there is unemployment, people are not exploiting comparative advantage.
3. Why not?
The Keynesian/monetarist story has two components: deficient aggregate demand; and sticky wages. Both components are needed. You can have deficient demand in the worst way and still clear the labor market if wages are flexible*.
The recalculation story is that patterns of trade that exploit comparative advantage have to be discovered by entrepreneurs, through trial and error. These patterns are constantly shifting. When the changes required are incremental, you get low unemployment. A sudden lurch gives you high unemployment.**
*There may be some exotic stories of unemployment with perfectly flexible wages, but they are nowhere near mainstream.
**Probably the recalculation story requires some stickiness in wages as well. But I have learned from Garett Jones and Tyler Cowen that employers can view workers as capital, and sometimes the marginal product of a specific unit of capital can be close to zero. The worker has value somewhere in the economy, but at this particular firm the worker's value can drop to almost nothing. So "sticky wages" need be little more than "I refuse to work for nothing."