I sketch a simple model where a shortage of collateral reduces the money supply, and makes the Cash-In-Advance constraint binding in an otherwise New Keynesian model. This post is a followup on my previous post on "negative money".
Start with an economy where the central bank issues green paper currency, and all goods must be bought for that green paper currency. There is a Cash-In-Advance constraint in this economy. If you do not have any green notes you will miss out on buying opportunites. If you want to sell goods, and the buyer does not have any green notes, you will miss out on selling opportunities. The CIA constraint will be binding.
Now suppose every agent keeps all his currency in a box at the central bank. When he buys something, the central bank takes currency out of his box and puts it in the seller's box. Nothing has changed (except for muggers). The currency boxes are just a simple form of ledger. It wouldn't matter if all the currency was destroyed in a fire, provided the ledger was saved.
Now suppose the central bank pays interest Rg on holdings of that currency, and can vary that rate of interest. (If you hold 100 green notes, the central bank gives you Rg more green notes every year.) An increase in Rg increases the demand for currency, and if the supply of currency is fixed, and if prices are sticky, this excess demand for currency would cause a recession.
Now suppose we add a second form of central bank currency. Red currency has negative value. If you buy $10 worth of apples, either you give the seller 10 green notes, or the seller gives you 10 red notes, or some combination of the two. Red currency pays a rate of interest Rr. (If you hold 100 red notes, the central bank gives you Rr more red notes every year.) If you want to sell something, and you do not have any red notes, and the buyer does not have any green notes, you will miss out on selling opportunities, and the buyer will miss out on buying opportunites. The CIA constraint will be binding. The gross money supply (green notes plus red notes) matters.
If we assume: the total stock of green notes is always equal to the total stock of red notes (the net money supply is always zero); the central bank will always give you one green note and one red note if you ask (the gross money supply is unlimited); the central bank will always destroy one green note and one red note if you ask; (these last two assumptions ensure the exchange rate between red and green notes stays fixed at minus one); the central bank always sets Rg=Rr (call it "Rm", for "interest on holding money"); Standard Euler equation IS curve; standard Calvo Phillips curve; then we have the standard New Keynesian model.
These assumptions ensure the CIA constraint is never binding, provided the central bank sets Rm just right so that the demand for green notes equals the demand for red notes. Because if neither the buyer nor seller of $10 worth of apples has any money, each goes to the central bank and asks for 5 green and 5 red notes, the buyer gives 5 green notes to the seller, the seller gives 5 red notes to the buyer, and they do the deal.
But if the central bank sets Rm too high, there is an excess demand for green money, an excess supply of red money, and so an excess demand for net money, and a standard New Keynesian recession.
But there is a problem with this New Keynesian model: who ensures that each agent's intertemporal budget constraint is satisfied? Who prevents an individual from always buying goods, never selling goods, accumulating an infinitely large stock of red currency, and so having a debt to the central bank he can never pay back? ("The economist who wrote the model, and who imposed the no-ponzi-condition" is not a satisfactory answer to this question.)
The simplest answer is that the central bank requires each agent to post collateral, and puts a ceiling on his holding of red notes equal to the value of that collateral (minus a haircut). If you do not post collateral you can only hold green money.
If collateral is scarce, so that collateral puts a binding constraint on the central bank's issuance of red notes, and if the central bank refuses to issue more green notes than red notes, then the scarcity of collateral puts a binding constraint on the total stock of green plus red notes (gross money supply). The CIA constraint becomes binding. If there is an exogenous drop in the quantity of collateral, or an exogenous drop in the quality of collateral so that the central bank increases haircuts, and if the central bank holds Rm constant, the result will be an excess demand for gross money, and so the CIA constraint bites more deeply, which causes a recession.
If the central bank will not or can not reduce Rm, the solution is for the central bank to issue more green money, and break the New Keynesian "rule" which says that the total stock of green money must always equal the total stock of red money. The central bank increases the stock of green money by buying assets outright. We call this "QE".
I think the ECB has a rule which says the total stock of green money must equal the total stock of red money. (JKH?) In other words, the ECB is a New Keynesian central bank. That's a problem for the ECB.
I'm not sure if I have got this right, but my head hurts, so I am going to post it.