## Thursday, April 27, 2017

### Fun With Central Bank Calvinball

In the comments to "Does the Governmental Budget Constraint Exist?" Nick Edmonds reminded about another piece of mainstream macro logic that triggers me: the governmental budget constraint holds, because the inflation-targeting central bank says so. From the perspective of a mathematician, this is just a variation of "because I want to assume it to be true." In this article, I run through whatever logic appears to exist.

As a preamble, there might be an explanation for this that makes sense somewhere. I gave up after a couple of years of looking at this. I am sure some economics professor can try to explain this; the only question is whether it can be translated into mathematics.

## Fundamentalist Bourbakianism

As a preamble, I am a hardline member of the Bourbaki faction: all mathematics is just set theory. Philosophers of mathematics might beg to differ with on me, but that is not my problem.

The view is simple: everything in mathematics is a set, or something related to a set. If it has nothing to do with a set, it isn't mathematics.

If you are not familiar with this view, I will ask you to think about the following question: what is a function? (If you do not know the answer, I highly recommend that you think about it before running off to a search engine. The answer is at the bottom.)

We can now turn to the issue of central banks in mainstream macro.

## Position: The Budget Constraint Holds Because X Says So

If one is attempting to decipher mainstream DSGE macro, you can run into statements that run roughly like this:
The budget constraint holds because of central bank policy [insert stuff about inflation targeting, or whatever].

If we translate this into mathematics:

"The (budget constraint equation) holds because we assume X, and we assume that X implies that the budget constraint holds."

Any of the following could be inserted as X in that statement:

• inflation-targeting central banks;
• the Easter Bunny says so;
• Chuck Norris says so.
This is just adding an extra level of indirection when compared to just saying: "We assume that the budget constraint holds."

## Position: The Central Bank Theory of the Price Level

The next argument is that there is that the initial price level will always move to ensure that the budget constraint holds.

From a mathematical perspective, this seems indistinguishable from the Fiscal Theory of the Price Level. However, it is not the private sector doing this; rather, the central bank can always set the price level at time zero. There is no reason within the mathematics why this is so: it just is.

The question about the central bank's control over the price level appears open.
• If it can only set the price level to the level indicated by the Fiscal Theory of the Price Level, then it is just "the Fiscal Theory of the Price Level, because central banks."
• If the central bank can set the initial price level at any level, that implies that it has arbitrary control over the evolution of the price level. We do not need economics models, we just ask the central bankers what price level they want that day. This position raises many philosophical questions about the study of economics.

## Position: Set Elements Magically Changing

Yet another alternative is the following:
• The Treasury picks an exogenous sequence of primary fiscal surpluses, s.
• If the budget constraint does not hold, the central bank says "you cannot do that." (How this translates into a set operation is unclear.)
• Therefore, the Treasury is forced to pick a new sequence of primary surpluses z, which follows the budget constraint.
Mathematically, this is the same thing as assuming the set of feasible primary surpluses are those that meet the budget constraint. Once again, there is no justification why this must hold.

## Position: Those Darned Infinite Rates

The next position attempts to justify how the central bank can enforce the budget constraint: it will raise interest rates to infinity if it does not hold.

This is equivalent to saying that the price of Treasury bills will clear at \$0.

Needless to say, there is no examination of the mathematics of market clearing.
• If the household sector inherited any money from the previous period, it can buy an infinite number of bills. In what sense has the market cleared?
• If the household sector had no money, the size of the central bank's balance sheet is zero. How can entity without a balance sheet drive the price of an asset to zero?
In the absence of a mathematical examination of the market clearing condition, and a specification of the limits of central bank balance sheets, this argument looks sketchy.

And rates have to be infinite, implying that there is no solution to the optimisation problem. Any other outcome just results in the exact same chain of real quantities that are in the budget accounting identities. None of the analysis I did in the previous post made any assumption about nominal rates being "low."

(The zero bound is the only major issue for the previous development.)

Furthermore, the "neo-Fisherian" effect will hold as soon as we have finite interest rates: the higher the nominal rate, the higher the level of expected inflation. Really high interest rates imply really high future inflation; it is hard to see how saying that "central banks target inflation" justifies this entire line of thought. Once again, if mainstream economists actually solved the models that they develop, we could resolve these issues.

## Position: Because Equilibrium

"The central bank chooses the equilibrium." Good luck with converting that description to mathematics. This seems to the consensus view on the topic, by the way.

## Concluding Remark

How mainstream macro went this far down the rabbit hole is a complete mystery.

## Appendix

In my first lecture on linear systems theory at McGill (a Master's level course), Professor George Zames asked us the "What is a function?" question. Nobody got it right. People thrashed around with discussions of "mappings" or rules, or whatever nonsense they teach undergraduate electrical engineers. (I took real analysis, but later.) He gave us the answer the next day, and I entered the Bourbaki cult.

The answer: a function is set of ordered pairs. That is y = f(x) is just a shorthand for f ={(x, y)}. (Alex Douglas rapped my knuckles for my original answer; I deliberately ignored the extra conditions on the set. The typical restriction is that if (x,y) and (x,z) are elements of f, then y=z. Depending on how you want to attack the issue, it could be set up slightly differently, I believe.) There's no little elves mapping an input to outputs; it's just a set.

(c) Brian Romanchuk 2017