In a recent post, Nick Rowe has been playing around with how to think about money when measured money can be both posive and negative.
It is worth distinguishing two models of money, which on the face of it are very different.
In the first model, money is a type of good. Monetary exchange means that every purchase involves the buyer delivering the agreed amount of money to the seller in return for goods and services. It's useful to imagine this as a physical delivery - the buyer hands over some gold coins or paper notes to the seller.
In the second model, monetary exchange takes place through a series of accounts kept by a central registry (the Bank). Each agent holds an account, which is nothing more than a record of a balance at the close of each day. When a purchase takes place, the buyer requests the Bank to debit the purchase price from his own account and credit it to that of the seller.
In each case there is a limit on how much people can spend. In the first model, this is a physical limit. The buyer cannot hand over more paper notes than he actually has - you can't have negative notes.
In the second model, there is no physical limit. If the buyer spends more than he holds, his balance simply becomes negative. However, the Bank will want to impose some limits. The more an account goes negative, the greater the risk to the Bank if the accountholder fails to get it positive again. So the Bank will decline payment requests which will make accounts more negative than it wants.
So in the first model, the limit is necessarily zero. In the second model, it may be any non-positive number and will depend on the Bank's view on credit risk.
This question of limits is the essential difference between the two models, rather than any question of whether money is in bearer or registered form. Of course, the form money takes may dictate the limit - you can't have negative bearer notes. But it's the limits that matter.
This allows us to see that the first model is simply a special case of the second model. In the second model, the limit need only be non-positive. In the first model, it must be zero.
Negative account balances in the second model are debts of the accountholder. We can have debts in the first model as well, where the Bank (or any other agent) lends notes to someone. However, there is a clear distinction here between debts and money. Although the paper notes may constitute debts of the Bank, as far as the non-banking sector is concerned debts and money are quite different things.
This makes it easy to isolate a particular thing - in this case the collection of paper notes - and identify it as money or the medium of exchange. Helpfully, such a thing can be captured in a single aggregate measure. So, it's often easier to think through monetary mechanics using the first model, and often this is OK.
However, in the general case the distinction between medium of exchange and debt becomes more blurred. We can no longer think of money as a simple aggregate; we need to start to think about the general liquidity position. This manifests itself in the way counterparties to monetary contracts - both banks and non-banks - agree the terms under which those contracts may be settled on demand.
The second model is a closer reflection of the real world. We can learn useful things from thinking about models where money is a good with a strictly positive value. But we should not be misled into believing that everything we see in those models will translate neatly to the real world.