Just trying to get my head clearer on some related stuff.

I have a weird thought-experiment, that I think helps us understand fractional reserve banking better. Even though, paradoxically, there are no commercial banks in my thought-experiment. There is just One Big Bank, owned and controlled by the government, that issues the only form of money, that is used as the only medium of exchange and unit of account.

There are two parallel worlds:

In one world the Bank holds assets equal in value to its monetary liabilities.

In the second world, the Bank holds no assets at all. It has M on the liability side of its balance sheet, and absolutely nothing on the asset side. So if the Bank wants to expand or contract the money supply, it cannot use Open Market Operations, but can only use helicopter (lump sum transfers) and vacuum cleaner (lump sum taxes) operations.

I'm going to ask what happens, in both of those parallel worlds, in an extreme case – when we go to Milton Friedman's Optimum Quantity of Money (aka ultra-liquidity).

To keep it simple, so I don't have to differentiate between real and nominal interest rates, assume the Bank successfully targets 0% inflation. Assume that money pays a rate of interest Rm, that is set by the Bank. And it adjusts the quantity of money to keep aggregate demand at the right level to keep inflation at 0% and the price level constant.

People willingly hold some money (there is a demand for money), even if the rate of return on other assets, R, exceeds Rm, because those other assets are not used as media of exchange, and it is very inconvenient to buy and sell things unless you hold a positive average stock of money. And the Bank adjusts the stock of money, using purchases/sales of assets (world one), or helicopter/vacuum cleaner operations (world two), until it is just willingly held at the target price level, and there is neither upward or downward pressure on that price level. [Previous sentence edited for clarity.] (It adjusts M to adjust Aggregate Demand to ensure the price level target is hit).

The cost to people of holding the liquid asset M, rather than less liquid assets, is the interest rate differential R-Rm. Suppose we start out with Rm a long way below R. Then suppose the Bank decides to reduce that interest rate differential, to reduce the price of liquidity. So the Bank decides to increase Rm.

[If paying interest on currency is impractical, then reducing the inflation target below 0% is an alternative policy that is equivalent to increasing Rm above 0%.]

But if the Bank sets a higher Rm, while holding M constant, the demand for money (the stock of money people want to hold) will increase and exceed the supply of money. This excess demand for money will cause a fall in aggregate demand, which in turn will cause the inflation rate to fall below target. So when it increases Rm, the Bank must also increase the supply of money to match the increased demand for money, to prevent the price level falling.

In the first parallel world, the Bank would use Open Market Operations to buy assets to increase the supply of money. In the second parallel world the Bank would use helicopters to increase the supply of money. What is the economic difference between these two different ways of increasing the supply of money to match the increased demand for money when the Bank increases Rm?

Sometimes it is easier to answer a question if we take an extreme case. Let's do that here. Let's suppose the Bank increases Rm all the way up, until Rm equals R. The Bank eliminates the interest rate differential between money and all other assets. It makes the price of liquidity zero. (This is the "Friedman Rule", for the optimum quantity of money, if the marginal cost to the Bank of providing liquidity is zero. The government-owned monopolist should set price = marginal cost. [And if it impractical to pay interest on currency, reducing the inflation target until nominal interest rates on other assets equals 0% is equivalent to increasing Rm until it equals R.])

If we assume that people are never satiated in liquidity, because there is always the possibility, however remote, that an individual will want to spend all his wealth immediately, we get some very extreme conclusions:

1. In the first parallel world, the Bank would own all the non-money assets. People prefer holding money to other assets, because it's more liquid, so would sell all the non-money assets to the Bank, if R=Rm.

2. In the second parallel world, money would be the only asset. People prefer holding money to other assets, because it's more liquid, so nobody would want to hold any other assets, if R=Rm. Physical capital would disappear, because it would not be replaced when it depreciates. But land cannot disappear. And if land pays positive rents, but people always prefer holding money to owning land, the price of land will drop to zero, which means R and Rm will rise to infinity.

Weird, huh?

"But Nick," I hear you ask, "what does this have to do with fractional reserve banking? You don't even have any commercial banks in your "model"!"

My answer:

My first parallel world is like a world in which commercial banks hold 0% reserves, and the central bank is very small relative to the size of the commercial banking system. Because (solvent) commercial banks do hold assets equal in value to their monetary liabilities. The only difference between my first world with One Big Bank is that the central bank has contracted out the management of its assets and liabilities to the private sector.

My second parallel world is like a world in which commercial banks are required to hold 100% reserves, and the central bank only holds government bonds. Because if the government owns the central bank, then the central bank owning government bonds is like my left pocket having a debt to my right pocket. It's a wash. The whole banking system only holds intrinsically worthless bits of paper as assets, so really holds no real assets at all.

What my weird thought-experiment shows is that there are trade-offs. If you want 100% reserves you can have it. But you must give up something else in exchange. Either:

1. Moving away from the Optimum Quantity of Money, so that liquidity is priced above the marginal cost of producing liquidity.

2. Moving towards communism, where the government-owned central bank owns all the assets.

3. Moving towards a world where real interest rates are infinite, capital disappears and money is the only valuable asset.

Your call.

(I have ducked the question of the optimum mix of two different types of money: chequing account money (normally provided by commercial banks), and currency (normally provided by central banks). Because it is too hard for me to get my head around that question too.)

[Update: OK, maybe it's clearer this way. If the marginal cost of producing producing money/liquidity is zero, then Friedman says we should price liquidity at zero, so that people are satiated in liquidity. But it is reasonable to assume that people never are satiated in liquidity, and will always prefer a more liquid to a less liquid asset, if they have the same rate of return. So people will hold only money, and no other assets. But if people hold only money, who is holding all the real assets like capital and land? Under 0% reserve banking, banks will hold those real assets. But under 100% reserve banking, commercial banks only hold central bank money. If the government-owned central bank holds all the real assets, we have communism. And if the central bank doesn't hold any real assets, then either there are no real assets, or those real assets are worthless. But land always exists. Therefore land must be worthless. But if worthless land pays positive rents, the real interest rate must be infinite. Which means money must pay an infinite real interest rate too.]