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Nick, for the past several years the major central banks have been "coordinating" to remain far below target!

marcus: are they *coordinating* on a stupid equilibrium? Or just being stupid in their own individual ways?

More likely the latter.

I love it when you work in game theory!

I agree with your general point. But since I'm a stickler for matters involving utility NE and stuff like that I want to point out that swapping the maximization of a utility function for an NGDP target is not technically a big difference, a utility function is just a more general approach. If the CBs are trying to hit an NGDP target and that is all that they care about it just means that they have a particular set of utility functions that happen to be maximized wherever NGDP is equal to the target. So basically what you are doing is taking a general model which may or may not result in an efficient equilibrium and imposing restrictions on the utility functions that happen to be sufficient to insure such an equilibrium.

It's important to point out that the tendency to expect an inefficient equilibrium here is probably driven by (perhaps erroneously) projecting a set of restrictions which we are all familiar with from a consumer maximization model onto the utility functions, specifically that "more is preferred to less" (or less is preferred to more would also work). This is what leads to this conclusion:

"The Nash Equilibrium is a point in {M,Y} space where one indifference curve is vertical and the other is horizontal (so neither wants to move unilaterally); the set of Pareto Optimal points is where the two indifference curves are tangent (so making one better off makes the other worse off), so both curves have the same slope."

However, if their goal is simply to hit an NGDP target, and that is an interior solution for any policy choice by the other party, then this is actually not the case. The budget constraints that they face are horizontal and vertical respectively, but the condition of tangency with the ICs comes from the assumption that you are not at an internal maximum (the assumption that it is possible to improve by moving in SOME direction but the constraint will not allow you to move that direction). To see this, imagine a set of utility functions with a "ridge" somewhere in the middle that is the same height all along the top (in other words the top of the ridge is every combination of strategies where NGDP is on target). There will be one ridge for each and this will be their best response function. Where they cross, will be the NE. The ridge will be an indifference curve but note that it doesn't have to be horizontal/vertical at any point NOR do the two need to be tangent at the NE.

P.S.

What this really does is highlight the difference in thinking about issues of monetary policy with a single policy goal in mind rather than multiple contradictory goals. For instance, if you had some kind of tradeoff in mind, for instance between output/unemployment and inflation and you believed in somehow boosting output and exporting inflation, then you have utility functions that would lead more to the inefficient equilibrium conclusion. I don't think that's a good way to think about monetary policy but some people seem to have something like that in mind though I'm not sure exactly how they would explain it. (The hardest thing to do in economics is to characterize accurately a view that you think is dumb....)

Is there really a difference between "maximizing" and "satisficing"? Define u(M,Y)=U(M,Y)-U* and assume they maximize that (sort of like in your "Irrelevance of Basic Income" post). Isn't that still the same thing?

I think the "standard" thinking goes something like this: two central banks are trying to pick a point on a Phillips Curve. But the choice of one shifts the position of the Phillips curve of the other. Since each one only cares about its own inflation and unemployment they both shift each other's curves to the "wrong place".

Mike and notsneaky: I think you are right about the utility-maximising vs satisficing thing. The simplest way to think about it would be to define a Loss function (which is just a negative utility function) equal to the squared (say) deviation of NGDP (or whatever) from target.

But I'm trying to think what the indifference maps of the two players would look like in {M,Y} space, if we did that. Normally we draw indifference curves a circles (or ovals) centered on a bliss point. (That's the picture I had in mind for my general model of a game between two agents.) But in this case I think the ovals get very elongated, so the bliss point is not a point in {M,Y} space, but a line (or curve). Nash Equilibrium is where the two bliss lines/curves cross.

Which maybe isn't quite as intuitive?

Tell me what you think.

I'm still thinking about the two conflicting/trade-off goals question.

I *think* that when I'm talking about elongated oval bliss points, I'm saying the same thing as Mike is saying with his ridges?

I think it might be a # of instruments vs # of targets kind of thing. If all you care about is the level of aggregate demand, PY, then you got two targets (two nominal GDPs) and two instruments (two money supplies or two interest rates). But if you also care about the composition of aggregate demand, P and Y, then you got four targets and two instruments. Even under the best of circumstances you're one instrument short but now the international spillovers can make the possible trade offs even worse. I think this also corresponds to the line vs oval presentation.

"I *think* that when I'm talking about elongated oval bliss points, I'm saying the same thing as Mike is saying with his ridges?"

That's correct!

You can imagine the two ridges running through M,Y space and where they cross is the equilibrium (if they cross). Your special cases are basically ways of saying that they don't cross. Note that at you "bliss point" the indifference curve is not tangent to the budget constraint because the indifference curve is a point. Same thing is the case when it's technically a curve but the curve represents a local maximum. (i.e. a "ridge")

Incidentally, the coordination crowd must have some more specific idea in mind for why coordination is necessary....right? What sneaky said is basically what I imagine them thinking. Is that how you see it?

BTW, if you reason via Phillips curves (which I don't think you do, but some people might) you can think of the individual maximization problem with a standard utility function in u, pi space (except that less is preferred to more and the indifference curves are concave) and the Phillips curve is the budget constraint. So if another CB's choices shift your Phillips curve, it's just like changing the budget constraint and may make you better or worse off. Then you have something like what you describe initially. (Note, for the record, that this utility function is different from one over the strategy space--M,Y)

I think the way to summarize it is this: if all you care about is the level of PY then coordination is not an issue. But if you care about the combination of PY (a (P1)*(Y1) can be preferred to (P2)*(Y2), even when P1*Y1=P2*Y2) then coordination is potentially an issue.

I have basically nothing to add to the discussion, but I shall say that I've really enjoyed this post. It's both very interesting in itself and also it's helped me have a bit more of a handle on game theory by seeing it applied in a more familiar context.

Sometimes, central bank coordination seems to be code for trying to obviate the Impossible Trinity by getting other countries to do the work. When I hear, from some academics, that "The Fed should take other countries into account when setting its monetary policy", what they seem to mean is "Other countries should be able to both have fixed exchange rates vs. the dollar and control over domestic AD". What they don't mention is that this puts the burden of price adjustment solely on the US domestic economy, as far as I can tell.

Mike: I'm not sure what the coordination people have in mind. Before digging into the nuts and bolts of it, I thought I would step back and try to present the issue in a very abstract way, in a 2-player game, without any specific macro model. To try to get some more general insights into the question.

W Peden: Thanks!

I reckon you added to the conversation with your Impossible Trinity point. Take the Gold Standard world, for example. I can sorta see how each central bank might want to accumulate gold reserves to help preserve its fixed exchange rate. But collectively, in a global recession, they would all be better off if they sold gold reserves. (Short history of the Depression?)

I think your point here must (somehow) tie in with what notsneaky and Mike are saying about number of targets and instruments. But I'm still trying to see this more clearly.

Is it just as simple as getting another instrument? Suppose you have a Dictator Central Bank, which can adjust other countries' monetary policies at will. The DCB thereby escapes the Impossible Trinity, because it has a number of instruments (the monetary base of other countries) equal to a domestic target plus targets for every exchange rate.

Capital controls and monetary policy coordination (not, in practice, via dictatorial methods) mean that monetary authorities can add to their repetoire of instruments, and therefore their list of targets.

I have never taken the time to understand how the Gold Standard worked in detail, in any of its versions, so I can't add on that point, but it would be interesting to see a David Glasner-type analysis of the Great Depression (focusing on the Insane Bank of France's choices setting off an international crisis) modelled using game-theory.

W P: Simplest way to model it:

Assume that gold is not used as money. Money is paper currency that is convertible at a fixed exchange rate into gold. CBs hold a mix of gold and bonds on the asset side of their balance sheets. There is a fixed stock of gold, and two demands for gold: industrial demand (jewelry); CBs' demand. CB's care about: AD; having enough gold reserves to maintain their exchange rates.

We can get a PD equilibrium where no central bank will increase its stock of currency by buying bonds (because it would lose gold reserves if it did so), but all CBs would cooperate to increase their stocks of currency at the same time.

I *think* that's David Glasner's model, massively oversimplified. France defected from the cooperative solution.

You guys are going off topic. The question is "when is there a coordination problem" for countries' central banks? I think Nick's point is actually more general than he realizes. The simple case is where the central banks care only about the level of aggregate demand but not about its composition. Then there's no coordination problem. But even if central banks care about the levels of P and Y, not just PY, there still could be no coordination problem. If P is a function of Y, P(Y), which is basically your standard Phillips Curve (possibly with some other stuff thrown in there). In that case P(Y)*Y is just a function of whatever the instrument that Central Bank has at its disposal. So whatever the foreign central bank chooses, as long as you can hit the Y you want it doesn't matter. You choose different nominal interest rates (or money supply) but the outcome is the same. For there to really be a coordination problem, the other country's choices have to affect your constraint *directly* (not just through the choice of your aggregate demand). That argument is much harder to make unless you add a lot of other stuff to the model - different price stickiness, different long run real interest rates, different inflation targets - basically you need all those phenomena that make International Economics a field of its own rather than just a two-sector Macro model (which is how Woodford and Benigno frame it) So my take on this is that yes, if you start with some standard Macro model (whether old school Keynesian, monetarist, or New Keynesian) and you think through the logic then you get that there is no coordination problem. You pick whatever level of instrument (interest rate or money supply) you need. There's only this vague, "intuitive", sense in which you have a coordination problem because in the back of your head you realize that your model does not capture all the relevant features so you wave some hands and say "coordination problem!". But those "intuitive" features are not really part of your model so you can't actually say that.

Nick's right in theory, but that's only because everyone's wrong in theory. In practice there actually is a coordination problem but we don't have a good model of why it's there.

To put it another way, suppose you got one instrument, R, and two targets Y and P. Your instrument allows you to control Y but Y is also affected by someone else's choice R*. So Y(R,R*). P is just a function of Y. The Phillips curve. Whatever your objective is, and whatever the other person's choice of R*, you can always pick the Y you want. So you can always pick the P you want. There's no problem here. Or more precisely, whatever problem you have, is the same problem you'd have even if R* wasn't an issue. You might pick a different R because someone else is choosing some particular value of R* but you can still get the same Y, hence the same P.

For there to be a problem here, we need that P itself is a function of the other guy's choice. We need not just P(Y) but P(Y,R*). And that requires some extra justification.

@notsneaky:

Y is not an admissible central bank target, because it is a real and not a nominal quantity. If Y could be controlled by monetary policy, there would be no need to control P at all – we'd have full employment and flexible prices.

The most common central bank target at the moment is dP/dt, or the inflation rate. A price level target would of course target P, and an NGDP target would act on PY. You are perhaps confused because Phillips curve thinking (or a Taylor rule) would use (Y-Y*) as a proxy for either dP/dt (classic Phillips curve) or d^2P/dt^2 (expectations-augmented).

In the basic New Keynesian model we have C=C*+rho-r where r is the real interest rate and rho is the "natural rate". r is nominal minus expected inflation. And in equilibrium Y=C. Inflation is equal to expected inflation + part which depends on Y. So by controlling the nominal rate the central bank controls the real rate which determines consumption, which determines output. Which then determines inflation, given expectations. Which are rational.

Y is not a target but the Central Bank can control it. That does not mean that prices are flexible. The Central Bank might have to accept a particular level of inflation or price level to get it.

I'm not sure I understand what you mean by that Y-Y* is a "proxy" for inflation or acceleration (not sure where this one comes from either). It's Y-Y* in there.

You can't satisfy both conditions at once, except in weird cases.

Hmmm, I would have thought that all zero-sum games that have a Nash Equilibrium would automatically also be Pareto Optimal at equilibrium. In many cases the optimal solution might be "don't play the game" but based on the Austrian perspective on central banking that's probably an entirely reasonable response.

At any rate, I don't think you could say "except in weird cases" when you are talking about a fairly well known and broad class of games here.

As for your Keynesian "Aggregate Demand" with monetary stimulus and all that print yourself wealthy, I doubt it will work any better when coordinated than it has done when uncoordinated.

Just to show I'm not cynical about everything all the time. I recommend a listen to this podcast. I don't agree with him, but I sort of see what he is getting at. He brings up a lot of the practical issues (although avoiding inflation in the first place would help dodge the issues he puts forward, but we can't seriously expect bankers to do that).

Raghuram Rajan, on the Global Financial Safety Net

In this Podcast, we speak with Raghu Rajan, India’s central bank governor about the Global Financial Safety Net. Rajan is charged with securing monetary stability in one of the world’s largest Emerging Market economies, at a time when others are stumbling in the face of difficult global economic trends. Rajan participated in a seminar entitled Fortifying the Global Financial Safety Net, during the IMF-World Bank Spring meetings.

Nick,

Minor quibble. Does technological upheaval (either rapid progress or decline) preclude both central banks from hitting their NGDP target?

Suppose the Earth went through a prolonged phase of solar flare activity / weakened magnetic field that rendered a lot of electrical equipment useless. Presumably there would be a collapse in both real and nominal GDP while alternative technologies were developed and older technologies were brought back into mainstream. I think that central banks would have a tough time reconstituting nominal GDP by themselves. And yes, I used electrical failures in my example because a lot of monetary transactions are handled electronically.

Likewise, I can see the case where progress happens so quickly that measuring new technological additions to nominal GDP becomes problematic.

"Hmmm, I would have thought that all zero-sum games that have a Nash Equilibrium would automatically also be Pareto Optimal at equilibrium."

This is true (they don't even need to have NE, any outcome is automatically PO) but this isn't a zero sum game, it's a positive sum game. Sometimes - often - people confuse "bad outcome" with "zero sum" or "prisoner's dilemma" (PD is not zero sum either)

The second section is just minimizing U' = (U-U*)^2 and V' = (V-V*)^2. Just rewrite the first part with these utility functions and you get coordination back, na?

One definition of "helicopter money":

http://www.bloomberg.com/news/articles/2016-07-14/bernanke-floated-japan-perpetual-bonds-idea-to-abe-adviser-honda

"Etsuro Honda, who has emerged as a matchmaker for Abe in corralling foreign economic experts to offer policy guidance, said that during an hour-long discussion with Bernanke in April the former Federal Reserve chief warned there was a risk Japan at any time could return to deflation. He noted that helicopter money -- in which the government issues non-marketable perpetual bonds with no maturity date and the Bank of Japan directly buys them -- could work as the strongest tool to overcome deflation, according to Honda. Bernanke noted it was an option, he said."

It looks to me like this definition involves a new bond being issued.

It seems to me that time should be considered more carefully in this game-theory explanation. Do both central banks meet at the same time (same day, same hour)? If so, there could be coordination between them utilizing modern communications.

On the other hand, with a time lag between CB meetings, the later-meeting-bank will always have the knowledge of the actions-of-the-first as factors in it's decision. The presence of time lag makes coordination a qualified choice at every CB meeting.

Yes, the CB's can meet in secret but actions can not be kept secret for long. Actions intended to move markets would be ineffective if markets failed to move, leaving only market moving actions as decision drivers for later meeting central banks.

My conclusion: Central Banks base policy changes on the sum of current conditions and the sum of political-will-in-attendance at each meeting.

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