Policy rules and borrowing degrees of freedom.

In the olden days, as Peter Dorman notes, macro models would contain variables like M (or r), G, and T (or t) for monetary and fiscal policy. Nowadays they usually contain a monetary policy rule, and sometimes a fiscal policy rule too. Like a Taylor Rule, for example. [Update: but see my old post on fiscal policy rules.] We don't talk about policy; we talk about the policy rule. We don't choose the best policy; we choose the best policy rule. Why?

The short answer is that in most macro models what happens today depends not just on policy today but also on today's expectations of future policy. And if expected future policy matters today, policy is nearly always better if it follows a rule rather than discretion (Kydland and Prescott 1977 pdf). And since we want to use models to learn the best policy, we assume that policy follows a rule. Unless we think the policymaker can't make credible commitments, in which case we assume discretion.

But that also raises a puzzle. Why do real people and real policymakers vary the amount of rules vs discretion over time? (I blogged about this once before, and JW Mason raises a related puzzle in comments.)

Here's a simple "model":

Assume that every day I can: choose what I do today; choose whether to make a promise telling people what I will do tomorrow.

That sounds as if I have two choices every day. But if yesterday I made a promise, I no longer have any choice about what I will do today. Because if I don't do what I promised to do, people won't believe my promises, so I won't really be able to promise.

If I have to keep my promises, there are four types of day:

1. Days where I choose what to do today, and promise what I will do tomorrow. I use two degrees of freedom on days like that.

2. Days where I choose what I will do today, and make no promise about what I will do tomorrow. I use one degree of freedom on days like that.

3. Days where I do what I promised yesterday to do, and make a promise about what I will do tomorrow. I use one degree of freedom on days like that.

4. Days where I do what I promised yesterday to do, and make no promise about what I will do tomorrow. I use zero degrees of freedom on days like that.

So on any given day, I can either use two degrees of freedom, one degree of freedom, or zero degrees of freedom. But on average, I only have one degree of freedom per day. I can borrow one degree of freedom from the future, and so spend two degrees of freedom on one day, but that's my credit limit in this example. If I borrow one degree of freedom from the future, I can either roll over that debt, and continue to spend one degree of freedom per day, or I can repay the debt by spending zero degrees of freedom one day, so I can again spend two degrees of freedom at some future day.

I think you get the idea from that simple example. We could extend that example by assuming I could also make a promise today about what I will do the day after tomorrow. In which case my credit limit is two degrees of freedom. And so on.

There are only two steady states, where all days are the same, and I only use one degree of freedom each day. In the first steady state, all days are like the second type, where I never make a promise. That's "discretion". In the second steady state, all days are like the third type, where I always make a promise. That's "rules".

It is easy to prove that the second steady state ("rules") is almost always better (and never worse) for me than the first steady state ("discretion"), provided contingent promises are possible. The intuition is that my utility depends on others' actions, which depend on others' expectation of my future actions, and by making promises I can influence their expectations of my future actions, and thus influence their actions. [Math appendix, based on Kydland and Prescott 1977: If my utility is U(M,Y(M)), where M is my action and Y is your action, and where your action depends on your expectation of my future action, then choosing M by setting U₁ + U₂.Y' = 0 ("rules") is almost always better for me (and never worse) than choosing M by setting U₁ = 0 ("discretion"). But if I make no promises, then you have already chosen your action when I choose mine, so choosing M by setting U₁ = 0 is all I can do.]

That is why most macroeconomic models describe monetary policy in terms of policy rules which the policymaker has promised to follow. Because policy is better under rules than under discretion. Unless the macroeconomic modeller thinks that the policymaker's promises won't be credible, in which case the modeller assumes discretion, which is a steady state with no promises.

But that raises a puzzle. Why in the real world, both for policymakers and for regular folk, do we sometimes see the number of degrees of freedom we spend varying from day to day? Some days we increase our outstanding stock of promises-to-be-fulfilled and other days we reduce our outstanding stock of promises-to-be-fulfilled. Why?

Intuitively, just as we go deeper into debt on some days, and pay down that debt or go into credit on other days, it would seem to make sense to borrow and lend varying numbers of degrees of freedom too. Use more forward guidance in bad times, when you really need it, and less in good times, when you don't need it as much. But the math says we should always be borrowing degrees of freedom right up to our credit limits, or until extra degrees of freedom provide no marginal benefits.

Why don't we do this? Some possible answers:

1. Some sort of uncertainty that we can't get around by making contingent promises.

2. It would take too long to figure out all the contingencies and make an optimal contingent promise. Our brains are the scarce resource.

3. We are always borrowing up to our maximum credit limit of degrees of freedom, but that credit (credibility) limit varies over time.

4. There is some sort of counterpart to a rate of interest charged on the outstanding stock of promises-to-be-fulfilled.

5. Dunno.

Comments

[Nick Edmonds]

"Why do real people and real policymakers vary the amount of rules vs discretion over time?"

Because when the rules end up not working, we get rid of the people who like rules and bring in those who like discretion. And then when discretion fails, we bring in people who like rules again. And when that starts to run into problems, we give the discretion guys a go again. And so on.

[Nick Rowe]

Nick Edmonds: I rather like that. (Or we change the rules). In principle we could maybe make a meta-rule, telling us when we will change rules. But it would be very hard to write down that meta-rule.

[Odie]

I think that sentence describes why we are not adhering to rules all the time:

"The intuition is that my utility depends on others' actions, which depend on others' expectation of my future actions, and by making promises I can influence their expectations of my future actions, and thus influence their actions."

The expectation is complete continuity in externalities (not captured under the rule; somewhat case 2) and not just influencing the other decisions but determining their actions (kind of case 1). If other people do not act as I want to have them influenced through my promise I will need to do a choice today as the environment will be different than what I had tried to accomplish. If I keep my promise I may get an outcome I definitively do not want or I change my choice to get the desired result but break my promise.

In the end, that model relies on people acting completely rational and predictable which is often assumed in economics but easily as often violated in the real world.

[Nick Rowe]

Odie: No. Do you really think economists haven't thought about uncertainty???!!! Give me a goddam break.

As I said in the post, the promises I make can be contingent promises. They can be contingent on exogenous events (please don't call them "externalities" on an economics blog, because that word has a very specific meaning here), like "I promise to take you swimming if the sun shines", or contingent on other people's actions, like "I promise to give you an apple if you give me a banana".

[Odie]

Nick,

You are right; externality was certainly not the right word to use here. I will be more careful in the future. However, I cannot help myself when I see human behavior put into equations to think that something is really off.

Back to topic: How believable is a promise that needs a bunch of contingencies around it? At the end, the other person will be either clueless what you actually promised or not believe that you will follow through. In principle, what you said in 1 and 2.

In addition:"That is why most macroeconomic models describe monetary policy in terms of policy rules which the policymaker has promised to follow. Because policy is better under rules than under discretion."

Would that not assume that those models are at least equally as good in describing the future (for making the promise) than discretion in making a decision during the current state?

[Achim]

Nick, these questions are, I think, very much related to the question "why is debt nearly never indexed to anything in the real world?"
Contingent contracts are optimal, but we do not even see credit contracts based on real interest rates (instead of nominal rates) very often. Why?

[scepticus]

Probably all of your points 1,2,3 at least.

point 1: not sure. probably yes in the real world but maybe not in model-land

point 2: yes: My utility depends on exogenous events as well as reactions of others to an extent I can't predict up front and which may have unpredictable effects on my contingent promises. If I have quite a number of contingent promises outstanding predicated on different variables then it rapidly gets too complex to process.

Anecdote: "To remember what complexity is, consider the number of possible “relation structures” for a set of points with one line or no line between any two of them. How many points are needed to let the
number of such possible structures surpass the number of baryons in our observed universe? The answer is: 24 points!" - Peter Kafka

Further, the FED is less constrained by exogenous events than an individual is, which is why forward guidance works reasonably for them but not me or some small corporation, relatively speaking.

point 3: Not sure, but it seems to me your point 3 is really two questions:

3a: Can additional degrees of freedom (credibility) be suddenly conferred upon me (or maybe on all or some subset of participants) exogenously? If so then that very much refers back to your point 1.

3b: "We are always borrowing up to our maximum credit limit of degrees of freedom" - can this be more reasonably be stated as 'in aggregate we are always borrowing up to our maximum aggregate credit limit of degrees of freedom'? Obviously this restatement tells us nothing about whether any given individual subject or subsystem we select at random (which could be the FED or could be joe bloggs) has minimised its own degrees of freedom or not, all it gives is a probability.

Finally, wrt degrees of freedom, making contingent promises presumably restricts future degrees of freedom with respect to some variable, but increases current degrees of freedom with regard to the same variable.

Also what about the theoretical treatment of other side of the trade:

should all participants, according to your math and logic set out in your post, maximise or minimise their acceptance of contingent promises from others?

[Nick Rowe]

Achim: very good comment. I think you are right. I hadn't seen that connection before.

scepticus: I had to Google "baryon". I figure there's a very large number of them in the universe!

3b: I was thinking that each of us is always borrowing up to our individual credit limit.

"Also what about the theoretical treatment of other side of the trade:"

Not sure. Sometimes it's to my advantage that others make promises (like when they trade with me) but sometimes it isn't (like when they threaten me). Not sure. I need to think about that one.

[Scott Sumner]

Perhaps we have only a very rough idea as the the correct model of the economy. So we make some fairly vague promises based on that estimated model. Let's say those promises constitute "flexible inflation targeting," where the "flexible" part is sort of vague. Then new information comes in. It now looks like we should have done NGDP targeting, or maybe price level targeting. So we change our minds a little bit about the correct model of the economy, and make new promises (let's say the Evan's Rule.)

Just a thought.

[Nick Rowe]

Scott: yep. And it would be very hard to describe in full detail the contingency under which we would want to change from IT to (say) NGDPLT.

[Kevin Donoghue]

In games, both in theory and in practice, it's common for mixed strategies to dominate pure (hence predictable) strategies. Is policy really better under rules than under discretion, when an "opponent" has an incentive to game the system? Policy is often worked out under adversarial conditions: Thatcher v Scargill, IMF v debtor nations etc.

Representative-agent models have their uses but we don't live in one.

[genauer]

Achim,

in the US you can buy TIPS, which reduces, but does not completely eliminate your interest rate risk.

http://www.bloomberg.com/markets/rates-bonds/government-bonds/us/

at present 0.42% over 10 years, with a presumed inflation rate of 2.0%, and a federal tax rate of 25%, http://www.treasurydirect.gov/indiv/products/prod_tips_glance.htm

you pay only a storage fee of 0.19% per anno + tax rate change & sovereign default risk, which is still pretty theoretical, despite all shut down nonsense.

[genauer]

Achim,

given your German name, the german equivalents are
http://www.deutsche-finanzagentur.de/fileadmin/Material_Deutsche_Finanzagentur/PDF/Aktuelle_Informationen/kredit_renditetabelle.pdf

and http://www.abgeltungsteuer.de/artikel/1-Anleihen/163-Inflationsanleihen+sch%C3%BCtzen+vor+Preisschub_1.html

The present calculation is (0.12% nominal yield + presumed 2.0% inflation) * (1- 26.75% total tax (incl. soli)) - 2.0% inflation = 0.45%/anno storage fee, you have to pay, given the higher quality(triple-A DE vs double-A US) of the sovereign, who grants you the privilege of safe storage.

Since at least 100 years, there is enough capital sloshing around, so that no premium has to be paid beyond principal minus storage fees, but only for risk taken.

[Alexander]

The explanation in Scott Sumner's comment is intuitively appealing, although what he describes is not really an example of commitment. A 'promise' that is abandoned as soon as new information comes along is not a promise at all. At best, it is a prediction of future behavior.

In formal terms, I think Nick's response was right on the mark. We want to explain why a policymaker might prefer not to commit. If the policymaker is choosing M to maximize U(M,Y(M)), a preference for discretion would mean that the policymaker prefers to set U1 = 0 instead of taking account of the effect of M on Y. This seems very difficult to rationalize unless the policymaker faces some additional constraint which is relaxed by discretion. A restriction on her ability to make commitments contingent on all future histories (perhaps because some notion of Knightian uncertainty makes it impossible) might be such a constraint.

I think this is an important research avenue because policymakers do often seem to have a preference for discretion, especially in uncertain times. See, for example, this recent interview with Fed governor Jeremy Stein:

http://www.cfr.org/united-states/conversation-jeremy-stein/p31044

[Alexander]

Another point: Nick got at the notion of credibility when he wrote, "[T]he macroeconomic modeller [may think] that the policymaker's promises won't be credible, in which case the modeller assumes discretion."

When there is a time inconsistency problem, it is by definition true that the optimal policy under commitment is not credible. When we conduct policy experiments under optimal policy with commitment, we are implicitly assuming that the policymaker has access to some costless and effective technology for enforcing her promises even though she will want to break them in the future.

So a good approach to modeling behavior that looks like commitment may be to explicitly model the commitment technologies available to policymakers. This approach would explicitly explain when it is optimal for the policymaker to keep her promises and when it is optimal to break them. I think there is already a literature on this, though I am not that familiar with it.

My own feeling is that real-world policymakers' ability to make credible promises is extremely limited. The available commitment technologies (e.g. costs to professional reputation for deviating too far from conventional wisdom) are not that powerful.

[Squarely Rooted]

My answer to this question is "meteors."

That is - implicit in all "rules" are a model of what the world is like. If f(x) = y, the rule is f() and the desired outcome is y, and both of those are explicit; the x is implicit.

Meteors are unanticipated and perhaps unanticipateable changes to x. Sometimes they are really like meteors, in which they happen overnight; sometimes they happen a little bit at a time over a long period of time and people don't always notice. Sometimes people DO notice but they can't convince a critical mass of others because the rule continues to produce y's roughly equal to y-hat. It's only when f(x') starts consistently producing unexpected and undesirable y's that x, and by extension f(), is revised.

Meteors can be lots of things, in addition to actual meteors that vaporize actual cities. They can be technological change, social change, outbreaks of the plague, loopy demographics *cough cough see recent discussion on the Great Inflation cough cough*, etc etc etc. All they have to do is change x in ways rulemakers failed to anticipate.

[Nick Rowe]

Kevin: dunno. Can't we have a commitment to play a mixed strategy? "I promise I will toss a coin to decide which of these two questions to put on the exam."

Alexander: "When there is a time inconsistency problem, it is by definition true that the optimal policy under commitment is not credible. When we conduct policy experiments under optimal policy with commitment, we are implicitly assuming that the policymaker has access to some costless and effective technology for enforcing her promises even though she will want to break them in the future."

Well, it depends on whether you view a reputation for keeping promises, and the benefits to maintaining that reputation, as a "technology". Which gets us deep into the theory of repeated games....

[Kevin Donoghue]

"I promise I will toss a coin to decide which of these two questions to put on the exam."

Arguably this strategy can be improved: candidates are advised that the coin is biased and the bias is undisclosed. Confront them with uncertainty rather than risk.

[Tom Brown]

The last time I saw Kydland and Prescott here was when Mark A. Sandowski wrote:

"I've wagered my whole economic life on the defeat of RBC."

and

"Kydland and Prescott's work is a tough sell to me. You don't seem to get it. As far as I'm concerned they are the dark side. I'll combat with every ounce of energy I have into the darkest corners of hell."

http://worthwhile.typepad.com/worthwhile_canadian_initi/2012/04/monetary-policy-is-just-one-damn-interest-rate-after-another.html

... Ha... well perhaps they're not ALL bad. I assume that's why you mention them Nick... or do you largely feel the same as Mark here? Or it it just their RBC work which is questionable (or not)?

[Tom Brown]

... shoot! I meant Sadowski, not Sandowski! Sorry Mark.

[Tom Brown]

Nick, BTW, nice post. I've seen you mention these DoF before on JP's site. It's nice to see more details.

[Nick Rowe]

Tom Brown: I like K & P's work on rules vs discretion. I'm not keen on RBC theory. You can agree with someone on some things, and disagree on other things.

[Philippe]

Nick,

Totally unrelated question.

I remember a while ago you wrote (regarding government debt), that if the interest rate was consistently below the GDP growth rate, that would mean the economy was dynamically inefficient. Could you possibly expand on that, and explain more precisely what you meant? Thanks!

[Nick Rowe]

Philippe: here's one of my old posts on the subject