Second best monetary and fiscal policy and the strategy space

[This post covers too much ground and stretches my brain too far. I'm trying to put Lipsey-Lancaster and game theory together, and apply it to monetary-fiscal. I blame Brad DeLong for making me think about this.]

Brad DeLong says:

"But as long as Nick Rowe recognizes that fixing situations of depressed activity by simply printing money gets us not to the first-best but the second-best in many situations, I can have no quarrel."

I think it is the job of monetary policy to take the world as it is, not as it should be. Monetary policy is always, in practice, about living in a second-best world. And that "world" includes fiscal policy.

And a game-theoretic analysis of monetary and fiscal policy in a second-best world leads to the conclusion that the fiscal authority must ignore the effect of fiscal policy on aggregate demand (NGDP), or else the Nash equilibrium will be far from second-best.

In the real world, actual fiscal policy will almost always be "wrong". The government will almost always be building either too many bridges or too few bridges for first-best optimality. The fiscal deficit will almost always be either too big or too small, relative to what it would be with an omniscient and omnibenevolent government. It's the job of the central bank to do the best it can given the world as it is, not as it should be. It's the job of the central bank to do the best it can given fiscal policy as it is, not as it should be. And that will nearly always mean doing something different from what it would do if fiscal policy were optimal. For example, if the fiscal deficit is smaller than it should be, that would mean the central bank should be setting the monetary base bigger than it otherwise would be. It's not first best (by assumption), but it is second best.

This has got me thinking about the application of Lipsey and Lancaster's "The General Theory of Second Best"(pdf) to monetary policy. It's funny, but I don't remember this topic coming up before. (I think it does come up in the context of the Friedman Rule and the Optimum Quantity of Money, but that's the long run question, and not the short run question of stabilisation policy, which is what we are discussing here.) Maybe I missed it. Or maybe it's because macroeconomists take it for granted we live in a second best world. It's just part of the background, so nobody remarks on it.

I think that the real issue is the strategy space. In a game between fiscal and monetary authorities, who chooses what? We can get a different Nash Equilibrium to the same underlying game if we change the strategy space. (The Cournot equilibrium, where firms choose quantities, is different from the Bertrand equilibrium, where firms choose prices.)

Take a very simple macro model: nominal GDP (N) depends on the monetary base (M) and the fiscal deficit (F).

1. N = F + M

Let the first best levels of N and F be N* and F*, and the Social Welfare Loss function be quadratic in deviations of N and F from their first best levels:

2. L = (N-N*)² + (F-F*)²

If we put F on the vertical axis, and N on the horizontal axis, then the indifference curves will be circles, centred on the first-best bliss point {N*,F*}.

Lipsey and Lancaster are redundant in this way of formulating the problem. If the fiscal authority sets F at the wrong level, either above or below F*, it makes no difference to the monetary authority, which should set N=N* regardless of F. Second best monetary policy is the same as first best monetary policy.

But if we substitute 1 into 2, to derive the indirect Social Welfare loss function, then Lipsey and Lancaster kick back in.

3. L = (M-(N*-F))² + (F-F*)² 

If we now put M on the horizontal axis, instead of N, and define M* as N*-F*, then the indifference curves are not circles but downward-sloping ellipses, centred on the first best bliss point {M*,F*}. If the fiscal authority sets F below F*, then the monetary authority should set M above M*. Second best monetary policy is different from first best monetary policy.

(If mangos and figs are imperfect substitutes, then Lipsey-Lancaster says that if the quantity of figs is too small then the second best quantity of mangos is larger than the first best quantity of mangos.)

Does Lipsey and Lancaster apply to monetary policy? It depends on whether we define "monetary policy" as setting NGDP or as setting the monetary base.

Now let's think of M, F, and N as being determined in a game between the monetary and fiscal authorities.

If monetary and fiscal authorities share the same social welfare function, then the equilibrium will be the first best M*,F*,N*. The strategy space does not matter. It does not matter whether they move simultaneously or sequentially ("Stackelberg").

But that is a first best equilibrium. How could we end up in a second best world? How could the Lipsey-Lancaster question ever come to be relevant? Maybe the monetary and fiscal authorities have different views of the world?

But if the monetary and fiscal authorities have different views about the true social welfare function (or different views about shocks to the aggregate demand function), then bad things can happen. The Nash Equilibrium could be, and generally will be, off the contract curve. And the strategy space will matter, and the order of moves will matter. The "coordination of monetary and fiscal policy" becomes a problem.

Bad things are especially likely to happen if the strategy space is {M,F} and the moves are simultaneous. That's because the indifference curves will be downward-sloping ellipses, so the reaction functions will both be downward-sloping. Small differences of opinion about the location of the bliss point {M*,F*} will lead to large deviations of the Nash equilibrium from the contract curve. For example, if the fiscal authority wants smaller NGDP than the monetary authority, then the Nash Equilibrium will be a smaller F and larger M than either of them want.

[I would show this in a diagram, and I can even "see" that diagram very clearly in my head, but I can't figure out how to draw downward-sloping ellipses in Paint. So I will have to do math instead. Damn. Can I get it right?]

Update: OK, here's a diagram, but I can't draw the indifference curves:

Let the monetary authority's bliss point be {N*m,F*m}, and the fiscal authority's bliss point be {N*f,F*f}. Both minimise a quadratic loss function relative to their bliss points.

The monetary policy reaction function is (choosing M to minimise its loss taking F as given):

4. M = N*m - F

And the fiscal policy reaction function is (choosing F to minimise its loss taking M as given):

5. F = (F*f + N*f - M)/2

So the equilibrium is:

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

As you can see, if (N*m-N*f)>0, then equilibrium F is lower than even the fiscal authority wants, and so equilibrium M is higher than even the monetary authority wants. This is a bad equilibrium, which neither wants.

There are two ways to avoid these bad things happening, and reduce the equilibrium loss for both the monetary and the fiscal authority:

1. The fiscal authority moves first, and the monetary authority moves second. (The fiscal authority plays Stackelberg leader, and picks a point on the monetary authority's reaction function.) Taking F as pre-determined, the monetary authority sets M to ensure that N=N*m. Knowing it will do this, the fiscal authority sets F=F*f.

2. We change the strategy space so that the monetary authority chooses NGDP, rather than the monetary base. The simultaneous moves Nash equilibrium is then N=N*m and F=F*f.

Both 1 and 2 are Pareto Superior to the previous simultaneous-moves Nash equilibrium in the {M,F} strategy space. Both monetary and fiscal authorities will gain by changing the game. Both 1 and 2 are really the same, because they give the same equilibrium. What is happening in this equilibrium is this:

The monetary authority takes fiscal policy as given, and sets NGDP where it thinks it should be, regardless of the fiscal authority's motives in setting F.

The fiscal authority takes NGDP as given, and sets F where it thinks it should be, regardless of the monetary authority's motives in setting NGDP.

Another way of saying the same thing is that the fiscal authority must ignore the effect of fiscal policy on NGDP. If it fails to do this, we can end up in a Nash equilibrium which is a very long way from anyone's idea of second-best.

(The Eurozone disaster, in which (N*m-N*f)<0, and where both these rules are ignored, is a case in point.)

Comments

[Nick Edmonds]

This model assumes that variations in fiscal policy have social welfare implications and variations in monetary policy do not. Is that necessarily correct? What means of varying the monetary base did you have in mind here? Is that certain to have less social welfare impact than fiscal policy variations through lump sum taxes and bond issuance?

If monetary policy also has social welfare implications, how does that change the game?

[Nick Rowe]

Nick E.: Well-spotted! Yes, that is correct. It assumes that the monetary base matters only insofar as it affects aggregate demand. And that fiscal policy matters both intrinsically and for its effect on aggregate demand. That is an important assumption, but I think it is roughly correct. My brain is too full to figure out how things would change if we modified that assumption.

[Nick Rowe]

Actually, Nick E., the key assumption is that the indifference curves are circular in {N,F} space. Or they could be ellipses, but they cannot be downward-sloping (or upward-sloping) ellipses (if you see what I mean by that). Because that means that in {N,F} strategy space, one reaction function is horizontal, and the other reaction function is vertical.

If we had lump-sum taxes, and Ricardian Equivalence, then T would not matter for anything. And I would re-interpret "F" in my model as Government spending, not as the fiscal deficit.

[DD]

I can agree with all that resorting to helicopter money is not ideal, but rather than analyze this based upon game theory I would analyze it in political terms.

The appropriate level of infrastructure spending is pretty arbitrary and subjective.

The cooperation between a monetary authority and a fiscal authority seems like a pipe dream because of the politics. However, the monetary system can serve as a backstop to the damage caused by political disagreements. It serves to say, if you people can't work this out, we will work it out for you.

It is debatable whether existing monetary authorities currently have the legal power to provide this backstop. But once it is established, it can actually force political cooperation in the long run. After this backstop is established, even just approaching the ZLB would spur political cooperation.

[Lord]

It seems that 1 and 2 are true in practice, 1 from the frequency of fiscal agreement, and 2 from monetary use of models even if the instrument is more specific, even if fiscal takes monetary policy into account making monetary policy more effective or otherwise accepts monetary recommendations.

[Nick Rowe]

DD: I tend to agree that there's a political angle to all this, that I'm missing. But I think that if my head were clearer on this, I would be able to translate that political angle into game-theoretic terms. For example, if we think of this in principal-agent terms, we can easily imagine a government delegating monetary policy to an agent: "Just target NGDP (or inflation) and we will hold you accountable for hitting that target". But it is very hard to imagine a government delegating control of fiscal policy to an agent. Fiscal policy is so multi-dimensional, with lots of different instruments and lots of different and competing objectives.

Lord: I think that 1 and 2 are roughly true in Canada (except the target is inflation and not NGDP).

[Jason]

Nick, I believe you are assuming your conclusion with your social welfare function. Since government spending (F) is part of N, the social welfare function overcounts government spending ... L = (N - N*)^2 would be a perfectly valid welfare function, with with F targeting N* independent of M and M targeting N* independent of F.

Now a more accurate model would be N = F + M^a + σ where σ are exogenous shocks. This allows e.g. Japan to increase their monetary base with no effect on N (a = 0), the US to increase the base with only a mild effect on N (a = small), and Australia to increase the base with a large effect on N (a ~ 1).

Of course, you could write the above as a differential equation so that

dN/dM = a(N, M) N/M + σ

with a(N,M) slowly varying and then simulate an economy ...

http://informationtransfereconomics.blogspot.com/2014/03/the-monetary-base-as-sand-pile.html

[Nick Rowe]

Jason: put it this way: we care about two things: the total level of NGDP; the mix of NGDP between government spending and private spending.

Putting shocks into the aggregate demand function makes no difference at all to my results. EXCEPT if the monetary and fiscal authorities have different information and estimates of the size of those shocks.

"Of course, you could write the above as a differential equation so that

dN/dM = a(N, M) N/M + σ

with a(N,M) slowly varying and then simulate an economy ..."

No. That just complicates it, without adding anything useful to the model. Simplify, simplify.

[Jason]

Nick,

I agree that is what your social welfare function posits, but central banks could also care about e.g. inflation (i.e. the mix of NGDP between real growth and inflation). Or the fiscal and monetary authorities could care about neither, just NGDP. It is in that sense that I think you are building the conclusion into the model via the social welfare function: your optimum depends on what you optimize!

Regarding the rest of my comment -- I was just trying to show that a couple of tweaks to your toy model lead to the information transfer model, so the ITM not as crazy as most of the internet seems to think :)

[Nick Rowe]

Jason: "...central banks could also care about e.g. inflation (i.e. the mix of NGDP between real growth and inflation)."

Of course they do. Just like I care about the weather. But can central banks do anything about it? (That was a rhetorical question. Please do not answer it. It's off-topic. It's just a FYI.)

[Ralph Musgrave]

Having two different institutions, government and central bank trying to regulate demand is raving bonkers. It makes as much as sense as having both husband and wife with their hands on the steering wheel of a car in the middle of a matrimonial breakdown.

[Nick Rowe]

Ralph: agreed. And my little model explains why it is bonkers. But a (slightly) better analogy would be letting the husband control the accelerator, and the wife control the brake pedal (or maybe gear shift). If the husband wants to go faster than the wife, you get a bad equilibrium where both the accelerator and the brake pedal are pressed down at the same time (or the engine is racing in too low a gear).

[Karl Pinno]

I think monetary policy and fiscal policy are endogenous. Suppose that the Fed had done less during the last recession wouldn't that have changed fiscal policy?
Also one way of looking at current central bank policy is that it has encouraged borrowing at the short end of the curve, dampened financial risk and therefore expanded asset values and benefited the wealthily who own assets much more than the poor and middle class because of asset price re-inflation. Investors have been incentivised to load up on much more in terms of risky bonds. Doesn't that change the social welfare unction because it changes the incentives and disproportionately benefits a small portion of society?
Now that the economy is growing slowly the political impetus for structural change has been diminished. If what is needed to stabilize the economy and a democracy in the long term are changes in the fiscal policy regime isn't possible that too effective monetary policy could be like giving an injured athlete a shot of pain killers that let's him go out and play to his long term detriment.

Are there any practical limits to how many assets the central bank can buy to prop up AD? Beyond a certain % point wouldn't ownership of GBs have social welfare implications? Why can we socialize financial markets and protect the wealth of the rich while cutting SNAP (the food stamp program)? You don't think the two are in any way related? The QE purchases of 85B per month in the US they effectively monetized the US budget had no impact on fiscal policy or social welfare or the Tea party's ability to play chicken with the Federal debt- only AD?
Draghi summoning the expectations fairy with his "whatever it takes" comments had no impact on the pace of fiscal reform and the social welfare function in Europe only on AD?
I have been persuaded that nominal AD targeting is probably superior to other monetary policies but I feel like you and Scott S often under appreciate the interdependence of fiscal policy regimes with monetary policy regimes. Any monetary policy that fails to provide sufficient incentive for policy makers to deal with long term structural issues may just kick the can down the road until we end up at the edge of the cliff and go over. Europe it seems to me is on track for just this type of disaster. Would Europe have been further down the path to fiscal reform had Draghi not stepped in or at least put a conditional timeline on the ECB's willingness to provide support?
Its easier to support NGDP in Canada a resource driven economy which has experienced the same degree of hollowing out as the US.

[Nick Rowe]

Karl: "I think monetary policy and fiscal policy are endogenous. Suppose that the Fed had done less during the last recession wouldn't that have changed fiscal policy?"

That is exactly what my model (simultaneous moves, {M,F} strategy space version) says will happen.

"Also one way of looking at current central bank policy is that it has encouraged borrowing at the short end of the curve, dampened financial risk and therefore expanded asset values and benefited the wealthily who own assets much more than the poor and middle class because of asset price re-inflation."

And that is the wrong way to look at it.

The right way to look at it is not in terms of central bank policy but in terms of the *mix* of central bank and fiscal policy. And that mix depends on both their objectives, and on the structure of the game (order of moves, and strategy space).

Plus, you are framing the whole question in a anti-rich biased way. In 1982, when interest rates were very high (and asset prices low) you would perhaps be saying that central banks set high interest rates to benefit rich lenders.

"...but I feel like you and Scott S often under appreciate the interdependence of fiscal policy regimes with monetary policy regimes."

This whole post is about the interdependence of monetary and fiscal policy!

Look, I know this is a hard post. Unless you have some basic knowledge of game theory you won't follow it. But did you understand it at all? Confession time: have you come across concepts like "reaction function" and "Nash Equilibrium" before? Did you understand the diagram?

[Odie]

Ralph,

Nice metaphor but as I see it it is actually the kid (= private sector) who sits at the steering wheel with Mom and Dad yelling, pushing and pulling on him to please follow their directions.

[Karl Pinno]

Thank-you for responding to my post. Your blog is great and I have learned many things from it. I apologize for conflating issues and perhaps being confusing in my post. I admit it was poorly articulated on my part.

When you say the CB only impacts AD and not social welfare function do you think that assumption is trivial?
Do you think for example that the change in risk taking behaviour brought on by the CB does not change the social welfare function.

I guess I was thinking of the central bank acting in terms of constructive ambiguity.

Best
Karl

[Karl Pinno]

And yes my Nash/reaction curve is very rusty so I am probably not understanding your post sufficiently.

Thanks again for taking the time to respond.

[Ralph Musgrave]

Odie,

Good point. That is, the state has to deal with the fluctuations in demand that the private sector (= the kid) is responsible for (especially and dramatically the recent credit crunch).

But my metaphor is also valid. That is, I’m saying the state should speak with one voice.

[Frank Restly]

Nick,

"1. The fiscal authority moves first, and the monetary authority moves second. (The fiscal authority plays Stackelberg leader, and picks a point on the monetary authority's reaction function.) Taking F as pre-determined, the monetary authority sets M to ensure that N=N*m. Knowing it will do this, the fiscal authority sets F=F*f.

2. We change the strategy space so that the monetary authority chooses NGDP, rather than the monetary base. The simultaneous moves Nash equilibrium is then N=N*m and F=F*f.

The monetary authority takes fiscal policy as given, and sets NGDP where it thinks it should be, regardless of the fiscal authority's motives in setting F.

The fiscal authority takes NGDP as given, and sets F where it thinks it should be, regardless of the monetary authority's motives in setting NGDP."

If the fiscal authority takes NGDP as a given and sets F such that M falls to zero or tries to go negative, then does NGDP have any meaning?

Monetary authority sets NGDP = $10
Fiscal authority (without regard to NGDP) sets F (Fiscal deficit) = -$15 ($15 surplus)
M must be -$5

How negative can M go before NGDP (measured in $) becomes meaningless and is this Pareto optimal?

[Nick Rowe]

Frank: if it's a monetary economy, where they use M to buy and sell things, if M goes to zero then NGDP goes to zero too, regardless of F.

For God's sake don't take my little "N=M+F" equation literally! Maybe it's in logs, so it's really N=M.F, and when we take logs we get logN=logM+logF.

Who cares? It doesn't make any difference if I replace it with N=f(M,F). It just complicates the math. You are totally missing the point.

[Frank Restly]

Nick,

I am sorry if I upset you. There is this hazy concept floating around in the back of my brain that I call "inclusiveness".

Obviously, the central bank could (if permitted by law) hit an NGDP target by repeatedly buying and selling the same spoon to an individual over and over again. Monetary quantity does not need to change and central bank can hit the same NGDP target even if the supply of money is falling (central bank simply increases the rate at which it buys and sells the same spoon). Or if buying and selling the same good violates some principle, then the central bank bends the spoon in some unique way before selling it each time and calls each iteration Spoon type A, Spoon type B, etc.

The fiscal authority (without regard to NGDP) sets F so that M continues to shrink over time.

Is either monetary or fiscal policy responsible for ensuring that an economy does not collapse into a spoon bending exercise?

Like I said, this is a hazy concept and may not be appropriate for this post, apologies if you find it irrelevant.

[Nick Rowe]

Frank: you didn't upset me (much). No worries.

"Obviously, the central bank could (if permitted by law) hit an NGDP target by repeatedly buying and selling the same spoon to an individual over and over again."

That's not how monetary policy works. And only the first sale of a (new) spoon would be counted in NGDP.

But let's drop this.