J.W. Mason and Arjun Jayadev have a paper making a new (to me) point about the assignment of targets to instruments.
First I'm going to present an (over-?) simplified version of their model, to explain the gist of it. [I think I've got the gist of it, but I'm not 100% sure, and I know my simplification isn't 100% accurate.] Then I'm going to attack it (because it's a good paper and so worth attacking).
There are two instruments: M and F. There are two target variables: A and D, with targets A* and D* respectively. Which instrument do you assign to which target?
Suppose the picture looks like this:
You want to set M and F to get to the point where the red and green lines cross, so you hit both targets at the same time.
What happens if the two instruments are set sequentially, with each instrument assigned to one of the two targets, taking the other instrument as given? (Like a sorta sequential repeated but myopic Nash game)?
If F is assigned to target A, and M is assigned to target D, then (if we start at some point away from where the red and green lines cross) we follow a counter-clockwise spiral that approaches the desired point. Like the blue spiral followed counter-clockwise.
But under the opposite assignment, where F is assigned to D, and M assigned to A, we follow a clockwise spiral that gets further and further away from the desired point. Like the blue spiral followed clockwise.
The first assignment leads to a stable equilibrium; the second assignment leads to an unstable equilibrium.
If I had drawn a different diagram, with the red line flatter and the green line steeper, I could have got the opposite result. It all depends on the relative slopes of the two lines. To say the same thing another way: to avoid instability you need to assign targets to instruments according to their comparative advantages (like the relative slopes of the PPFs in the textbook Ricardian trade model). Which could go either way, depending on the parameters of the model (which is what J.W. and Arjun explore in their paper).
Attack 1. Stackelberg. If a government tells the central bank to target inflation (or NGDP) then the government is ipso facto telling the central bank to offset any changes that fiscal policy might have on inflation (or NGDP). In other words, the government is telling the central bank to do what is needed to ensure that the fiscal policy multiplier is always zero. It would be self-contradictory for a government to play Nash (acting as if it expected the central bank would hold its instrument constant) when it has told the central bank to adjust its instrument in response to any shocks including fiscal policy. A government that tells its central bank to target inflation (or NGDP), and that expects the central bank to do so, will instead be playing Stackelberg leader, picking a point on the central bank's reaction function. Under the standard assignment, if "A" represents Aggregate Demand, the government sets F by picking a point on the green line which represents the central bank's reaction function.
(Though, I gotta admit, sometimes I wonder if the current Canadian government and central bank totally get this point.)
Attack 2. Micro. There are always more targets than instruments. That's why economics is about trade-offs. Public Finance (which is what microeconomists call "fiscal policy" is as much about micro targets as macro targets. If the level of government spending, and the level of taxes, were both totally irrelevant, except for their effects on Aggregate Demand and long run fiscal sustainability (the things macroeconomists worry about) then it would be perfectly OK for macroeconomists to use fiscal policy for hitting macroeconomic targets, since microeconomists wouldn't care one way or the other. But microeconomists do care about the level of government spending and taxes, quite apart from their effects on Aggregate Demand. The whole point of assigning Aggregate Demand to monetary policy is so that microeconomists, and governments, can ignore Aggregate Demand when they worry about all the other trade-offs in public finance.
Take a very simplified view of micro public finance. The composition of spending between government and private spending matters. There are diminishing Marginal Benefits to government spending (for a given level of total spending), and that MB curve may shift over time (it shifts right in various emergencies, or when there's lots of kids needing new schools, etc.). And there are increasing Marginal Costs for tax revenue (because taxes are distorting, and the deadweight cost triangle rises with the square of the tax rate), and that MC curve may shift over time. So if the MB curve temporarily shifts right, with no shift in the MC curve, good microeconomic policy will be to run a temporary deficit, so the baby boom kids get new schools now but the taxes that pay for them are smoothed out over time for efficiency and equity.
The whole problem with the Functional Finance assignment of targets to instruments is that economists would say "Sorry kids, but you can't have the new schools you need, because the economy doesn't need any more Aggregate Demand just now". Or it would cause tax rates to predictably rise and fall over time in violation of the tax-smoothing motive of minimising the present value of deadweight losses (Jensen's Inequality).
Micro public finance matters. (And I need to write another post attacking that so-called "New View" of fiscal policy for totally ignoring micro public finance.)