Strategy space and the theory of monetary policy Two

Industrial Organisation economists have known since 1883 that strategy space matters.

To say the same thing another way, Industrial Organisation economists have believed in fairies ever since 1883, when a French mathematician proved the existence of fairies in oligopoly theory.

Industrial Organisation economists, at least by stereotype, tend to be very practical down-to-earth people, quite different from Monetary economists. They like hard data, and other concrete things. But those practical people believe in fairies.

The French mathematician was Joseph Bertrand. Bertrand took Cournot's model of oligopoly. He kept all the assumptions about concrete stuff, like demand functions, cost functions, and profit-maximising firms, exactly the same. He even kept Cournot's assumption of what we would nowadays call "simultaneous moves Nash Equilibrium". Each firm makes its choice to maximise profits taking the other firm's choice as given, just like in Cournot. But Bertrand's model gave a totally different result from Cournot's model. Bertrand's model has lower prices and higher output than Cournot's. That's a concrete difference. And this result generalises even if we relax some of the common assumptions.

What caused the concrete difference between Bertrand's results and Cournot's results, if all the concrete assumptions were exactly the same? It had to be fairies. Bertrand proved that fairies can have causal powers.

Nobody calls them "fairies" of course. That would be too embarrassing for a scientific discipline with practical applications. Instead we call them "the strategy space".

We can think of firms as choosing quantities or choosing prices. Cournot thought of firms as choosing quantities; Bertrand thought of firms as choosing prices. Cournot and Bertrand got different results because they assumed a different strategy space for the Nash Equilibrium.

(The intuition is straightforward. The demand curve facing an individual firm will be more elastic if its competitors hold prices constant than if they hold quantities constant, and the more elastic a firm's demand curve the lower the price and/or the higher the quantity it will choose.)

But it doesn't matter what Cournot and Bertrand thought the strategy space is. What matters is what the firms think the strategy space is. If each firm thinks of other firms as choosing a quantity, we get Cournot's result. If each firm thinks of other firms as choosing a price, we get Bertrand's result. How firms think about what other firms are doing is what matters.

Might it also matter how we think about what it is that central banks are doing?

I can think of one example where it does matter, and it matters in much the same way that strategy space matters in oligopoly theory. A simultaneous move Nash Equilibrium two-person game between the fiscal authority and the central bank.

My Carleton colleague Simon Power and I once wrote a paper (pdf but open access) with a simple model like that, though we didn't explore the strategy space at all. That's what I'm going to do now.

We assumed a simple aggregate demand function NGDP = F + M , where NGDP is Nominal GDP, F is fiscal policy, and M is monetary policy. Both fiscal and monetary policy affect NGDP. We gave the government (in charge of F) and the central bank (in charge of M) preferences defined over NGDP and F. We assumed that the government wants a higher level of NGDP than the central bank wants. We assumed that, for a given level of NGDP, government and central bank want the same level of F. (For example, both government and central bank want a fiscal deficit of zero, because they don't want a burden on future generations, but the government wants a higher NGDP than the central bank wants.)

We looked at three equilibria:

1. Simultaneous Nash. It turns out that NGDP is exactly where the central bank wants it to be (and thus less than what the government wants), but the fiscal deficit is bigger than either government or central bank want it to be. (Hmm, maybe that's the US right now?) The intuition is that the central bank predicts that the government will choose a loose fiscal policy, so chooses a tight monetary policy to fully offset loose fiscal policy and get NGDP to where the central bank wants. But the government, knowing monetary policy will be tight, chooses a loose fiscal policy, but has to compromise between lower NGDP and higher deficit.

2. Stackelberg, where the central bank moves first, and the government moves last, after observing M. It turns out that NGDP is a compromise between what the two players wanted. The fiscal deficit is bigger than the two players wanted, but not as big as in the simultaneous Nash Equilibrium. The intuition is that the central bank is forced to compromise between low NGDP and high fiscal deficit, because it knows that if it keeps monetary policy tight the government will choose a bigger deficit.

3. Stackelberg, where the government moves first, and the central bank moves last, after observing F. It turns out that NGDP is exactly what the central bank wants (and thus less than the government wants), and the fiscal deficit is zero (which is what both players want). The intuition is that the government knows the central bank will fully offset any attempt to use fiscal policy to increase NGDP, so it doesn't even try, and sets the fiscal deficit to zero.

(The math is all there in the paper, if you are keen on that sort of thing. I could maybe show the whole model in one big picture, but my Paint skills aren't good enough).

(Our model, by the way, can be used to think about what Scott Sumner has been saying about the ineffectiveness of fiscal policy. Scott can perhaps be interpreted as saying that the central bank in fact moves last, so the government needs to recognise that it is playing game 3, and should stop thinking it is playing game 1.)

Now, in the paper we don't actually say what "M" is, other than some monetary policy instrument. We don't say which instrument. And the one thing we say in the paper which I would most like to retract, because it's just plain wrong, is: "With little loss of generality, we define units for [NGDP], F, and M so that their relationship is linear [NGDP] = F + M". (My fault, not Simon's, for letting that through.)

Actually, it's with a lot of loss of generality.

For example, take a standard ISLM model of aggregate demand. It is well-understood that a given change in the fiscal deficit (given rightward shift in IS) will have a bigger effect on aggregate demand if the central bank holds the rate of interest constant (horizontal LM) than if the central bank holds the stock of money constant (upward-sloping LM). So the slope of the trade-off the government faces between deficits and NGDP changes depending on whether the central bank sets the rate of interest or sets the stock of money. The steeper the LM curve, the less incentive the government has to use fiscal policy to increase NGDP, and the smaller the equilibrium deficit would be in our model. Except in 3, the Stackelberg equilibrium where the central bank moves last, where the deficit is zero anyway.

In the limit, if the LM curve became vertical, by choice of a monetary policy instrument that made it vertical, all three equilibria converge to the third Stackelberg equilibrium in which the central bank moves last. At that limit, if we had modelled the central bank as choosing NGDP, then all three of the equilibria in our model would have collapsed to the third (rather trivially, and we wouldn't have had a publishable paper).

Except in the case where a player moves last, that player's strategy space matters for the equilibrium of the game. And a player's strategy space is not what that player actually chooses, but what the other players think it chooses.

Unless the central bank always moves last, and the fiscal authority knows that the central bank always moves last, it matters how the fiscal authority thinks of monetary policy. Is monetary policy best understood as setting a nominal rate of interest, a monetary aggregate, the price of gold, an index for stock prices, a rate of inflation, NGDP, or what?

Our way of thinking about what monetary policy is will affect what actually happens.

Comments

[Jciconsult]

It is interesting that in modern times such as these when our central banks speak with such calm authority of the virtue of their decisions that we can have intelligent debate about the nature of policy interaction that might question some of their decisions or at least their timing. My own sense is that our bank always will act last because F is relatively well known.
P.S. That you for indicating the source of your article. I will explore further there.

[Nick Rowe]

Jci: "My own sense is that our bank always will act last because F is relatively well known."

That is my sense too. But it's not totally obvious. You could think of monetary and fiscal authorities as taking turns to move in a never-ending repeated game, rather than moving just once like in my one-shot game here. And it depends on what each has promised about how long it will maintain its current choice, or whether each is free to move again whenever it feels like it.

[Linda]

Nick: It never occurred to me that fairies played a role in IO. However, maybe I was implicitly biased towards accepting them, having spent several years as a monetary economist convinced that money was like pixie dust.

[Nick Rowe]

So Linda: do you think the stereotype of IO types is correct?

Yep, it sounds strange to hear the difference between Cournot and Bertrand as being due to "fairies". "Those aren't magical fairies; they are just the little people who are all around us and define the strategy space, which we take for granted." It's all just "expectations".

[Alex Godofsky]

Nick:

You mention that re: Bertrand or Cournot competition it matters what the actual firms believe about the strategy space.

Observably, firms (or the people who run them) actually do have models in their head about what will happen if they do X. Business schools teach class after class on it. Many firms have employees whose only job is to think about that strategy space.

Just as observably, Congress has no coherent model. Most of the individual members don't really have any model at all for what the Fed does, and the rest disagree with each other! Treating them as an optimizer seems clearly wrong, even as the roughest of approximations. You can't even treat them as optimizing their chances of reelection, because different parts of Congress are actually enemies!

Finally, if Congress ever perceived that the Fed really was playing this sort of game with them, they would probably be furious and immediately amend its charter to force it to stop sabotaging them.

[Determinant]

These aren't fairies, these are choices and behaviours.

Is the question different if a country has parliamentary government where the budget, as proposed by the Minister of Finance, is substantially what will be enacted because the continued life of the current parliament clearly depends on passing a budget?

In Canada, the UK, Australia, France, Germany and most other countries, the Minister of Finance is clearly responsible for fiscal policy and has the backing of his party to enact that policy. Thus fiscal policy in parliamentary countries is far, far more coherent than in the US.

[Britmouse]

Very interesting post.

Is it fair to say that denying central banks "goal independence" is done specifically to avoid playing this game entirely, or at least to always reach your (3) equilibrium? At least in countries where the central bank is a subordinate of the fiscal authority, and the central bank's nominal target derives from the same Treasury which sets fiscal policy. I just read this Charlie Bean paper where he talks all this from the BoE perspective, with specific mention of (3).

And doesn't flexible IT hence somewhat undermine the cause of goal independence? Because flexible IT involves discretion over the weighting of inflation and the "output gap", and the CB's judgement of the gap and the weighting are neither directly observable nor (necessarily) disclosed or agreed between CB and fiscal authority. The central bank's next choice again becomes unknown to the fiscal authority, even if the judgements in the last choice are known.

[Nick Rowe]

Alex: "Observably, firms (or the people who run them) actually do have models in their head about what will happen if they do X"

What matters here is what model they hold in their heads about what the other firms are doing. "Are the other firms choosing a price, or a level of output?"

Determinant: "These aren't fairies, these are choices and behaviours."

My choices include: how many bank shares to own, and how much to hold in my chequing account. Those two variables are part of my strategy space. That strategy space would be incoherent for someone living 2,000 years ago.

200 years ago (is my history correct?) how many slaves to own would have been part of my strategy space. Today that strategy space is incoherent.

We don't think the same way as we used to about the set of things we can do, and that others can do. What makes conceptual sense has changed.

The history of civilisation is the history of changes in the strategy space, which change the equilibrium of how we interact with each other.

Must do a post on this.

Britmouse: Thanks!

"Is it fair to say that denying central banks "goal independence" is done specifically to avoid playing this game entirely, or at least to always reach your (3) equilibrium?"

Hmm. I can see how governments telling central banks they have to target 2% inflation (which is what the govt wants, rather than the 0% target central bank wants) would solve the problem. Good point.

"And doesn't flexible IT hence somewhat undermine the cause of goal independence?"

Hmm. Yes.

[Frances Woolley]

Good post - I'm wondering if the lack of comments simply indicates that what you're saying is so sensible that no one could possibly disagree with it? Perhaps people are away on holiday.

[Scott Sumner]

Another excellent post. One reason you have this problem is that the fiscal and monetary authority don't realize they are playing the same game. Especially the fiscal authority. The government thinks it's targeting real growth with fiscal policy, and it thinks the central bank is targeting inflation.

Alex says the Congress would be furious if they understood the Fed was sabotaging fiscal policy. That's probably true, but what's so amusing is that Congress INSTRUCTED THE FED to sabotage fiscal policy--when they gave the Fed the dual mandate.

I'm sure you know what I'm going to say next. If both the government and the central bank thought in terms of NGDP targets, the game would immediately become transparent, and budget deficits might well become smaller (I agree with you that many fiscal authorities (wrongly) believe they are playing game one.)

[Nick Rowe]

Frances: you know game theory. Your thesis had a lot of game theory. You were taught by Ken Binmore, IIRC? The intersection of game theory and monetary policy is a narrow one. Most readers' eyes are probably glazing over. I should have titled it: "A mathematical proof of the existence of fairies". Then people would think what I am saying is so silly they couldn't possibly agree with it. And I would get lots of comments.

Thanks Scott! Hmmm. I am assuming common knowledge of the Aggregate Demand function. Yes, I think your point about people thinking that fiscal policy affects Y and monetary policy affects P probably has a lot of truth in it. "After all, it's obvious when you look at Y=C+I+G+NX and P=MV/Y"

[Lord]

Continuing after Scott, what if one doesn't even realize it is playing the game? What if the other is deciding both ngdp and the deficit wanting neither larger but is faced with either one or both larger? We will have just as large a deficit as the Fed wants.

[Determinant]

If values, institutions, laws and other social choices are "fairies", so be it, but if I bounce a cheque there are real consequences, and I don't have to run the risk of losing my total earthly wealth by getting mugged because it's all in the bank.

Maybe I'm being too practical. D**m Engineering degree.

[Lorenzo from Oz]

This expectations matter on steroids: I like it.

[Peter N]

I think a more realistic model is one where each side has a herd of cats. The conference room has different doors each standing for a strategy. You meet once a month and have thirty minutes to herd your cats through the door of your choice using only verbal encouragement.

Now you have two considerations monetary strategy and feline strategy. If you have the more well behaved cats, the likelihood is that you'll get to choose first, or, if you want to move second, you'll be able to do so without the risk lack of lack of feline consensus.

Without a consensus, the result is defined to be keep on doing nothing concrete. You can, of course, drop hints about the superior quality of your cats, which enables you to act in a timely manner when the need arises, which need has not yet arisen, but your cats stand ready.

As the source of monetary policy, you get to meet in a boardroom. The fiscal side meets in Grand Central Station at rush hour. Sometimes this encourages the cats to move quickly, but it makes it hard to control their direction. Particularly as any number of interfering bystanders are calling "here, kitty, kitty" and brandishing cat treats.

The point, of course, is that a strategy has to persuade as well as work. You have to persuade your side to adopt it and the public to believe in it. Some policies are popular and persuasive. Others are not.