Both Bitcoin and LETS are monetary exchange systems.
There are many differences between Bitcoin and LETS. I am going to ignore all of those differences except the one I want to focus on. Those other differences are off-topic.
[Yes, this post is not really about Bitcoin vs LETS. I am just using Bitcoin and LETS as two real-world examples to illustrate extreme versions of one aspect of alternative monetary systems that I think is important.]
Bitcoin is a pure green money system. LETS is a pure red=green money system. Let me explain what I mean by that.
In LETS, everyone starts with a zero balance. If I sell something to you, and we agree on a price of L$100, then I now have plus L$100 in my account ("green money") and you now have minus L$100 in your account ("red money"). [I am using "L$" to refer to the LETS monetary unit.] The value of the green money assets is always equal to the value of the red money liabilities. In aggregate, the LETS account holders have zero net wealth in their accounts. LETS is like a commercial bank, where some people have positive balances and others have negative balances (overdrafts) in their chequing accounts, where the positive and negative balances sum to zero, and the bank itself has no other assets or liabilities. Except that commercial banks fix the exchange rate of those balances to central bank money (by making them redeemable at par), and LETS does not.
In Bitcoin, everyone starts with a balance that is either positive ("green money") or zero. There are no negative balances ("red money"). In aggregate, the Bitcoin account holders have positive net wealth in their accounts.
Bitcoin is like a fixed quantity of green bits of paper with positive value that fell from the sky into people's pockets. (Yes, I know about mining, and I'm ignoring it.) The buyer gives green paper to the seller in exchange for goods received.
In LETS there is always an equal number of red and green bits of paper, where each red bit of paper is worth minus one green bit of paper. The buyer gives green paper to the seller in exchange for goods received, or the buyer accepts red paper from the seller in exchange for goods received.
[The real world is a mixture of both; It has more green than red. We could say it's a red < green system.]
Green Bitcoin vs red=green LETS. Which system is better?
LETS expands and contracts automatically to meet the Needs of Trade. Bitcoin does not. That is both an advantage and a disadvantage.
We hold stocks of inventory ("buffer stocks") when it is costly to synchronise inflows and outflows. There is an old literature on the inventory-theoretic approach to the demand for money.
In LETS, everyone starts with a zero balance. If each individual always had perfectly synchronised payments and receipts of money, each individual's balance would remain at zero forever. Because if I pay someone L$100 at exactly the same time that I receive L$100 from someone else, my balance stays the same. If there is some exogenous change that causes individuals' payments and receipts to become less synchronised, some will have positive balances and others will have negative balances. The net money stock (green minus red) always stays at zero; but the gross money stock (the absolute value of green plus red) adjusts automatically. Nobody runs out of cash and is unable to pay if somebody else delays payment. Negative inventories are allowed.
Bitcoin is just the opposite. The net money stock and gross money stock are always equal, both are positive, and fixed exogenously in nominal units. You cannot hold negative inventory. If someone else delays payment, you may run out of cash and be unable to pay, unless you can borrow Bitcoin from someone else. And that may be difficult to do, especially if you want to do it quickly, and especially if everyone else is trying to do the same thing. There is a risk of liquidity shortages, and self-fulfilling liquidity crises, and a risk of recessions, where people won't buy because they can't sell.
Advantage LETS. But that same advantage is also a disadvantage. Because the Needs of Trade are endogenous.
The Needs of Trade depends both on the degree of synchronisation of payments and receipts, and on the nominal volume of trade (measured in L$ or B$ per year). [I am using L$ and B$ for the LETS and Bitcoin currency units.] The nominal volume of trade is like NGDP, except it includes all trade, not just trade in newly-produced final goods and services.
Bitcoin provides an anchor for the nominal volume of trade. There is a negative feedback process that helps equilibrate the nominal volume of trade. If the nominal volume of trade rises above equilibrium, there is an insufficient stock of money for the Needs of Trade, so people want to buy less goods and sell more goods, and in the long run prices fall. If the nominal volume of trade falls below equilibrium, there is a surplus of money for the Needs of Trade, so people want to buy more goods and sell less goods, and in the long run prices rise.
I don't see how that would happen with LETS. Imagine a mischievous fairy halves the prices of everything one night. If people keep on trading the same volume of goods as before, the nominal volume of trade halves, and the Needs of Trade halve too. In real (inflation-adjusted) terms, it is exactly as if the mischievous fairy had doubled everyone's balance, both positive and negative. Those with a positive balance are richer in real terms; those with a negative balance are poorer in real terms. Where is the equilibrating process that would bring the price level back up again? Would those positive and negative balances just shrink over time to half their original size, so in real terms back to where they were before the fairy intervened? If so, there is no equilibrating process that would bring the price level back up again.
We could ask exactly the same question about a second mischievous fairy who makes everyone believe that everyone else will only buy half as many real goods as before. What is there to prevent a new equilibrium with half the real volume of trade as before? A permanent recession?
The New Keynesian (Neo-Wicksellian) answer to this question is that LETS lacks an automatic equilibrating mechanism that would create a nominal anchor, and would need to be actively managed. And the way to actively manage it would be to adjust the rate of interest on LETS balances. The correct response to those two mischievous fairies would be for the managers of LETS to cut the rate of interest paid on positive balances (or charged on negative balances), to encourage each individual to want to buy more goods than he sells. Whether this would work is another question.
[Thanks to my commenters on other red/green posts, especially JP Koning for this one.]