Start with the Fiscal Theory of the Price Level, where B is the nominal stock of government "bonds", and P is the price level:
(B/P) = PV(primary surpluses)
Government "bonds" are valued like a corporation's stocks, except they are the unit of account for the economy. Suppose the government does a 2-for-1 stock split, holding the Present Value of primary surpluses constant. B doubles, P doubles, and each "bond" is worth half as much in real terms.
Now suppose the government does a continuous stock split, by printing a flow of new "bonds" and using them to pay interest on the stock of existing "bonds". That increase in the nominal interest rate on "bonds" increases the growth rate in the stock of "bonds", and increases the inflation rate, all by an equal amount. We get the Neo-Fisherian result, just like in John Cochrane's paper. And notice that this policy has no fiscal consequences for the rest of the government budget constraint, just like a 2-for-1 stock split has no consequences for a firm's balance sheet.
Now suppose that the "bonds" are also used as the medium of exchange. We re-interpret "B" as "money", but money that pays interest. Because B is used as the medium of exchange, people will hold a stock of B even if the real rate of return on B is less than the rate of return on other assets. This means the government can "tax" the holding of B (collect seigniorage revenue) by paying a real rate of return less than the rate of return on other assets (or even a negative real rate of return). Which in turn means the government can run a smaller primary surplus (or run a primary deficit). Suppose the government does this.
Because holders of B are "taxed", (B/P) will now be smaller than socially optimal; there will be a liquidity premium on B, and so some otherwise mutually advantageous trades will not be made. The PV(primary surplus) is too small, or the PV(primary deficit) is too big, for the socially optimal allocation of resources.
Now let us introduce a second type of money, called "M", that is also issued by the government. There are two differences between M and B:
1. B can only be used in a subset of markets (called market 2). M can be used in all markets (both market 1 and market 2).
2. While nominal interest can be paid on B, the nominal interest rate on M is always 0%. So the real rate of return on M is minus the inflation rate, and the real rate of return on B is the nominal interest rate i minus the inflation rate.
So think of B as like a chequing account at the central bank, and M as central bank currency, and some markets (market 1) do not accept cheques. And people can swap M and B at the central bank, and so hold their desired mix of the two monies.
Suppose (though this is not what David and Stephen assume) the fiscal authority decides the total amount of real seigniorage revenue from M and B, but the central bank decides the composition of that seigniorage revenue between M and B. It can tax M more and tax B less, or vice versa, provided the total (real) tax revenue stays the same. And it varies the composition of the tax by varying the rate at which it prints M and B, using the newly-printed M and B to pay interest on the existing stock of B.
The Friedman rule says that M and B should yield the same real rate of return for optimality, which means that the central bank should set the nominal rate of interest on B equal to 0%. But the Ramsey rule of optimal taxation in a second best world where distorting taxes are needed to collect revenue says that if two goods are taxed, the one with the less elastic demand should pay the higher tax rate.
Empirically, the demand for government-issued bonds seems to me to be much more interest-elastic than the demand for government-issued currency. So the Ramsey rule says that central bank should set a higher tax rate on currency than on bonds, which means it should violate the Friedman rule, and ensure a strictly positive nominal interest rate on bonds. But that is not why the Friedman rule is violated in David and Stephen's model; because the demands for M and B are perfectly symmetric in their model, once the nominal interest rate on B is strictly positive.
It took me some time, plus some help from David, to figure out why their model violates the Friedman rule. They assume that B is a one-period Treasury bill, and the fiscal authority holds [M+B/(1+i)]/P constant when the central bank increases the nominal interest rate i, which means that [M+B]/P must increase, purely from the arithmetic. And we are in a second-best world, where M and B are taxed, so the amount of goods bought is sub-optimal. And the quantity of goods bought with M and B is equal to [M+B]/P, and so that quantity increases, and welfare increases, when the central bank increases i above 0%, purely as a matter of arithmetic. But if the central bank increases i too much, this reduces shopping in market 1 (where M is used) relative to market 2 (where B is used) and welfare decreases.
If the model had instead assumed that the fiscal authority held [M+B]/P constant (instead of holding [M+B/(1+i)]/P constant) when the central bank changes i, the results would be totally different. The Friedman rule would be optimal.
What we are talking about here is the optimal inflation target. Should the central bank target 2% inflation, or 4%, or 0%? Or should it target a negative inflation rate, low enough so that the nominal interest rate on bonds is typically 0%? I can think of many reasons for arguing why one inflation target is better than another. But the last thought on my mind would be whether the fiscal authority holds [M+B/(1+i)]/P constant or [M+B]/P constant when the central bank changes the inflation target.
I get uneasy when some of the key results of a model are extremely sensitive to very small changes in apparently arbitrary assumptions about government behaviour. (If the government issued 2-period Tbills, or God forbid perpetuities, the results would change a lot). (David worries about this too.) I would want to examine those assumptions very carefully indeed.
The Bank of Canada is a Crown Corporation. It operates mostly independently, but hands its profits over to the government. In my view, if the Bank of Canada did something that caused its profits to increase, the government would spend those extra profits, sooner or later, by increasing government expenditure or cutting taxes. I can imagine a world (Zimbabwe?) in which the government tells the central bank that it must deliver a fixed amount of profits each year, and adjust monetary policy to ensure it hits that profit target; but that world is not Canada. The Bank of Canada targets 2% inflation, not $2 billion in inflation-adjusted annual profits. But I find it very hard to imagine a world in which the government tells the central bank it can do what it likes provided the real market value of currency plus government bonds equals some annual target set by the government.
I am coming down on the same side as David in his recent post on the "dirty little secret".
"The starting point for answering the question of how a policy affects the economy is to be very clear what one means by policy. Most people do not get this very important point: a policy is not just an action, it is a set of rules. And because monetary and fiscal policy are tied together through a consolidated government budget constraint, a monetary policy is not completely specified without a corresponding (and consistent) fiscal policy."
I am of the view that the Bank of Canada targets 2% inflation (or NGDP or whatever), and it adjusts the nominal interest rate (or base money or whatever) to hit that target, and its actions affect its profits, and those profits affect the government's spending and taxation decisions. In the long run, the government adjusts its budget to be consistent with the Bank of Canada's actions. Not vice versa. We saw that adjustment in 1995.
Zimbabwe, under Mugabe, is different. The guy with the AK47 chooses how much currency you print for him to spend, even though he does give you a bond in return. And the higher the price level the more currency he wants you to print for him. Raising the nominal rate of interest on bonds lets you sell more bonds and withdraw currency from circulation, but also means you must print currency even faster to pay the higher interest. Because the guy with the AK47 is going to make you print enough additional currency to pay the interest on the bonds he issued. That's a Neo-Fisherite world.