The ECB cannot move last

I think this diagram helps us understand the Eurozone problem in simple game-theoretic terms:

 

Let's start with a "normal" country, where there is one monetary authority that sets the money base M, and one fiscal authority that sets the fiscal deficit F. And aggregate demand (N for NGDP) depends on both M and F:

1. N = M + F

And suppose the monetary and fiscal authority agree on a quadratic Social Welfare loss Function, where the optimum N is N*, and the optimum F is F*:

2. Minimise L = (N-N*)² + (F-F*)²

The equilibrium, trivially, is N=N*, F=F*, and so M=M*=N*-F*

But what happens if the monetary authority wants smaller NGDP than the fiscal authority? So N*m < N*f. (They might disagree on the Social Welfare Function, or they might disagree about shocks hitting the aggregate demand function).

If monetary and fiscal authorities move simultaneously, the Nash equilibrium (as shown on the diagram) is the solution to the fiscal authority's reaction function (3), and the monetary authority's reaction function (4):

3. F = (F* + N*f - M)/2  (Set F to minimise L taking M as given)

4. M = N*m - F   (Set M to minimise L taking F as given)

And the equilibrium is:

5. F = F* + (N*f - N*m) and N = N*m and M = N*m - F* - (N*f-N*m)

This is a bad equilibrium. The monetary authority gets the level of NGDP it wants, but the policy mix is wrong. The fiscal deficit is too high, and the monetary base is too low.

If the monetary authority moves last, we get to a better (Stackelberg) equilibrium. The fiscal authority moves first, and picks a point on the monetary policy reaction function. The monetary authority takes F as given, so sets M to ensure that N=N*m. The fiscal authority recognises the monetary authority will do this, and so recognises it cannot affect N, and so sets F=F*.

Look at my diagram, to see the difference. If the monetary authority moves last, and takes F as given, it will set M to ensure that M+F=N*m. The fiscal authority knows it will face the vertical blue Aggregate Demand curve, where fiscal policy has no effect on NGDP. But if they move simultaneously, the fiscal authority takes M as given, and so faces the green upward-sloping Aggregate Demand curve, where fiscal policy affects NGDP.

[Notice, by the way, that my diagram is formally equivalent to the familiar Kydland-Prescott rules vs discretion diagram for inflation.]

In any normal country, with one fiscal authority, that fiscal authority will recognise that it will be on a higher indifference curve in the Stackelberg equilibrium than in the simultaneous Nash equilibrium. So it will let the central bank move last, to get to that Stackelberg equilibrium. It will ignore the effect of fiscal policy on NGDP, and let the central bank alone choose NGDP, even though it would prefer a higher level of NGDP than what it knows the central bank will choose.

But the Eurozone is not a normal country, because it has many fiscal authorities. The monetary authority can only set M and NGDP for the Eurozone as a whole. If each fiscal authority has its own region, it can use fiscal policy to increase NGDP for its own region, while recognising that the ECB will ensure that NGDP for the Eurozone as a whole is where the ECB wants it to be. Each individual Eurozone fiscal authority will face an upward-sloping Aggregate Demand curve, even though ex post the Aggregate Demand curve will be vertical for the Eurozone as a whole. (The exact slope of that Aggregate Demand curve will depend on the size of each individual country, and on how open its economy is.)

The normal country's Stackelberg equilibrium requires the monetary authority to be the Stackelberg follower and the fiscal authority to be the Stackelberg leader. The problem with the Eurozone is that you can't have 18 Stackelberg leaders. Each individual country, facing an upward-sloping Aggregate Demand curve at the Stackelberg equilibrium, has an incentive to defect from the leadership position, and become a follower.

The ECB cannot move last, because the 18 fiscal authorities have no individual incentive to move first. So the Eurozone is stuck at the undesirable simultaneous moves Nash equilibrium. That's why the Eurozone is a mess.

(This is a simple model, and like all simple models it leaves stuff out. That's the point of simple models. Please pay more attention to what it brings in, and less attention to what it leaves out.)

[P.S. Just a bit of Canadian content: To the best of my memory, the last time any Canadian province tried to move last, and use provincial fiscal policy to increase provincial aggregate demand, was Ontario under Bob Rae's NDP government, in the early 1990's. The attempt was recognised as a failure. Ontario is a very open economy, so it would face a fairly steep green AD curve.]

Comments

[dlr]

If each fiscal authority has its own region, it can use fiscal policy to increase NGDP for its own region, while recognising that the ECB will ensure that NGDP for the Eurozone as a whole is where the ECB wants it to be.

It's not done though, right? We need another game theoretic framework among Eurozone fiscal authorities. They all know the ECB will create X NGDP. They also know other fiscal authorities will somehow jockey for their preferred relative NGDP position and thus their absolute NGDP given X, but they all cannot maximize their relative position. Which fiscal authority acts last? This gets much more complicated when the fiscal authorities have varying NGDP preferences.

[Nick Rowe]

dlr: "This gets much more complicated when the fiscal authorities have varying NGDP preferences."

True. But I think it would be a fairly straightforward extension of the model to build that in. For example, if we use lowercase for individual countries, and uppercase for the Eurozone as a whole, the AD function could be something like:

n = N + a(f-F) = M + F + a(f-F) (with everything normalised to keep it simple, so we don't have to keep multiplying and dividing by 28).

Then solve for each country's reaction function.

"Which fiscal authority acts last?"

What matters is that they won't all move first, before the ECB moves. I would model it as all 29 (the 28 plus the ECB) moving simultaneously.

[Lorenzo from Oz]

But the Eurozone was not always a mess, it seemed to be working for a while. It was only when a monetary shock hit that things went bad. (Taking the GFC as not being the problem, but the unrequited upward shift in monetary demand it created being the problem.)

[Nick Rowe]

Lorenzo: good critique!

*If* this model is right, we would have to explain it as follows: for a long time, N*m and N*f were equal to each other, so the simultaneous Nash equilibrium was at the bliss point, and all was well with the Eurozone. Then *for some reason*, M*m fell relative to N*f, and we got the bad equilibrium shown above.

Plausible?

[Min]

How do the players know when the game ends? If the game does not end, who moves last?

[Min]

But the Euro was a predictable mess, wasn't it?

And what about the USA, with 50 state fiscal authorities and one federal fiscal authority? Why isn't it as much of a mess as the Eurozone?

[Nick Rowe]

Min: the easiest way to think about it: the game is only played once. But before the game begins, the fiscal authority decides whether to move first, or whether they will move simultaneously.

Maybe because there is greater labour mobility in the US.

[Lorenzo from Oz]

Yes, it is plausible. Especially if we take it that, in entering the Eurozone, the fiscal authorities were setting N*f so as to conform to the requisite monetary setting. So, they were at a "bliss point". The problem was dealing with shock(s), when coordination broke down (somewhat predictably).
Of course, if Tom Sargent is correct, they were not as much on it as it appeared.
http://www.minneapolisfed.org/publications_papers/pub_display.cfm?id=4526&

On the "why not the US?" question, lots of US states have balanced budget amendments, which suggest not much in the way of fiscal "movement". Also, federal spending provides a stabiliser and labour mobility is, indeed, higher. Both Krugman, in his discussion of OCA's revenge,
http://krugman.blogs.nytimes.com/2012/06/24/revenge-of-the-optimum-currency-area/?_php=true&_type=blogs&_r=0
and Sargent, in his Nobel Memorial Lecture, have discussed this, though Sargent more about the early history of the US.

[Min]

Nick Rowe: "the easiest way to think about it: the game is only played once."

Well, as we have known for more than 30 years, iterated games can have different qualities from single games, and that seems likely to be the case here.

[Nick Rowe]

Min: suppose we literally have a one-shot game. And suppose the monetary and fiscal players literally must move at the same time. Then it would be impossible for a normal country to escape the simultaneous Nash equilibrium.

But in a repeated game, the fiscal authority in a normal country could invest in a reputation for acting like a Stackelberg leader. It could cheat once, by playing simultaneous Nash, but then the monetary authority would not trust it again. So a normal country could escape the simultaneous Nash equilibrium, in a repeated game, even if the moves in each round of the game are literally simultaneous. But in the Eurozone there is a free-rider problem, since the monetary authority responds only to the average fiscal policy. So it can't escape simultaneous Nash.

That's the other way to look at it.

[J.V. Dubois]

Excellent blog and also good point by Lorenzo. For me this is just another way of saying why fiscal transfers may play important role in any optimal currency area. Or why in such a circumstances it is good if central bank has sufficiently large NGDP goal so that the situation wher CB wants smaller NGDP than particular fiscal authority is less likely to arise.

So to sum it up - the best course of action for Europe is strong fiscal discipline rules accompanied by easier monetary policy acompanied by some sort of fiscal transfer policy in case of very large asymmetric shock.

[Nick Rowe]

JV: thanks!

Definitely easier (and better) monetary policy. I'm not sure about the fiscal "discipline" bit, because running large deficits isn't *always* undisciplined. It's hard to distinguish deficits run for micro grounds, from deficits that are run because monetary policy isn't what it should be and fiscal/monetary coordination problems.

Breaking up the Eurozone would be better, I think.

[Jacques René Giguère]

This where "CB independance" leads us: instead of having clear messages about a common goal, a guessing game. CB independance, in the sense of oacting as a separate unit from fiscal authorities is not needed in the competent countries west of the 1054 schism borders. And impossible anyway elsewhere ( why would the CB be more competent that the fisacal authority?
Which is why I always taught my students that, monetarily, what Canada and Québec needed wasn't two countries with the same currency but one country with two (or more ) currencies, coordinated by the same CB.
"Oh what fun it is to ride in a wild horses open sleigh..."

[Nick Rowe]

Jacques Rene: I think the problem is not CB independence, but fake independence. True independence is you let the CB control aggregate demand (NGDP, inflation, whatever) and don't try to use fiscal policy as well as monetary policy to control AD. The fiscal authority ignores AD. Fake independence is like when you tell your kids they are financially independent and must make their own earning and spending decisions, but keep on bailing them out.