This post meanders and doesn't really come to any clear conclusion. Read at your own risk.
Imagine, just imagine, that every dollar I spent was attached to a very long rubber band; and sooner or later it would return to my wallet to buy something I wanted to sell. Or that every dollar I spent had my name on it, and would eventually return to me to be redeemed for my goods and services. Think of it as "boomerang money", that always returns to the sender.
Within a currency area for a real world (non-boomerang) money, "income=expediture" is always true at the aggregate level, but is not true for each individual. My expenditure is someone else's income. If there are n people in my currency area, and I increase my expenditure by $1, my income will increase by (1/n) times the multiplier. If n is millions of people, (1/n) will be a very small number, so I will ignore the possibility that any extra dollar I spend will return to me as extra demand for the goods and services I want to sell.
Barter is different. If I barter my apple for your banana, it is as if I offered you the following deal: "If will spend $1 to buy your banana if and only if you agree to spend $1 to buy my apple". In barter, my expenditure of an extra $1 creates an extra $1 income for me, and if you accept my offer, your expenditure of an extra $1 creates an extra $1 income for you. With barter, income=expenditure is true at the individual level.
Barter is like boomerang money. Boomerang money is like barter. My expenditure of an extra $1 of boomerang money will create an extra $1 of income for me. With boomerang money, income=expenditure is true at the individual level. (OK, with maybe a time-lag).
A "keynesian" (it's not just keynesian) deficient-demand underemployment "equilibrium" (let's avoid arguments about whether it's really an equilibrium) is a consequence of the fact that real-world money is not boomerang money. If the underemployed individual spends an extra $1 of real world money, only a fraction (1/n) of the extra income that creates returns to that individual. The remaining (n-1)/n, which is nearly all the benefits, goes to other individuals. It's like the common proprty resource problem; if I pick up litter in the park, where n people use the park, only (1/n) of the benefits return to me. If people in a keynesian deficient-demand underemployment equilibrium could use boomerang money, this internalises the externality. Each underemployed individual would immediately increase his spending, since he knows his income will increase dollar for dollar, until he is fully-employed.
Personal debt is not like boomerang money. I can pay for goods with credit, and my IOU will sooner or later return to me for redemption. But if my IOU can only be redeemed in money that I myself cannot produce, it does not create a demand for the goods and services that I can produce or want to sell. Only if I paid for goods with an IOU for the apples that I myself produce and want to sell, would my IOU be boomerang money.
Perhaps Local Exchange Trading Systems are an attempt to create something that approximates boomerang money. With a smaller number of participants, (1/n) is a larger number. The individual reasons that sooner or later an extra $1 he spends will return to him as demand for the goods and services he wants to sell. "The money stays in the community". But there's a tension between having a currency area with a small number of participants to internalise the externality, and having a currency area with a large number of participants so that it's more likely you will find a trading partner who wants to sell what you want to buy or buy what you want to sell.
As I have argued in previous posts: an excess demand for the existing medium of exchange is what causes deficient-demand recessions; and at the same time it creates an incentive for people to engage in barter, and to start setting up and using new monetary exchange systems. And this is what we seem to observe.
I think that the concept of boomerang money might prove useful in helping us understand how these new monetary systems can get started. But my head is not 100% clear on it yet.