Fun Amongst Cardinal Depository Fiscal Establishment Calvinball

In the comments to "Does the Governmental Budget Constraint Exist?" Nick Edmonds reminded nigh some other slice of mainstream macro logic that triggers me: the governmental budget constraint holds, because the inflation-targeting key banking concern says so. From the perspective of a mathematician, this is merely a variation of "because I desire to assume it to locomote true." In this article, I run through whatever logic appears to exist.

As a preamble, in that location powerfulness locomote an explanation for this that makes feel somewhere. I gave upwards after a twain of years of looking at this. I am certain some economic science professor tin give notice seek to explicate this; the alone interrogation is whether it tin give notice locomote translated into mathematics.

Fundamentalist Bourbakianism

As a preamble, I am a hardline fellow member of the Bourbaki faction: all mathematics is merely laid theory. Philosophers of mathematics powerfulness beg to differ amongst on me, but that is non my problem.

The persuasion is simple: everything inward mathematics is a set, or something related to a set. If it has zip to exercise amongst a set, it isn't mathematics.

If you lot are non familiar amongst this view, I volition inquire you lot to mean value nigh the next question: what is a function? (If you lot exercise non know the answer, I highly recommend that you lot mean value nigh it earlier running off to a search engine. The reply is at the bottom.)

We tin give notice straightaway plough to the number of key banks inward mainstream macro.

Position: The Budget Constraint Holds Because X Says So

If 1 is attempting to decipher mainstream DSGE macro, you lot tin give notice run into statements that run roughly similar this:

The budget constraint holds because of key banking concern policy [insert materials nigh inflation targeting, or whatever].

If nosotros interpret this into mathematics:

"The (budget constraint equation) holds because nosotros assume X, together with nosotros assume that X implies that the budget constraint holds."

Any of the next could locomote inserted every bit X inward that statement:

• inflation-targeting key banks;
• the Easter Bunny says so;
• Chuck Norris says so.

This is merely adding an extra grade of indirection when compared to merely saying: "We assume that the budget constraint holds."

Position: The Central Bank Theory of the Price Level

The side past times side declaration is that in that location is that the initial toll grade volition e'er movement 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 non the mortal sector doing this; rather, the key banking concern tin give notice e'er laid the toll grade at fourth dimension zero. There is no argue inside the mathematics why this is so: it merely is.

The interrogation nigh the key bank's command over the toll grade appears open.

• If it tin give notice alone laid the toll grade to the grade indicated past times the Fiscal Theory of the Price Level, thence it is merely "the Fiscal Theory of the Price Level, because key banks."
• If the key banking concern tin give notice laid the initial toll grade at whatsoever level, that implies that it has arbitrary command over the evolution of the toll level. We exercise non demand economic science models, nosotros merely inquire the key bankers what toll grade they desire that day. This seat raises many philosophical questions nigh the written report of economics.

Position: Set Elements Magically Changing

Yet some other option is the following:

• The Treasury picks an exogenous sequence of primary financial surpluses, s.
• If the budget constraint does non hold, the key banking concern says "you cannot exercise that." (How this translates into a laid performance is unclear.)
• Therefore, the Treasury is forced to choice a novel sequence of primary surpluses z, which follows the budget constraint.

Mathematically, this is the same affair every bit assuming the laid of viable primary surpluses are those that run into the budget constraint. Once again, in that location is no justification why this must hold. 

Position: Those Darned Infinite Rates

The side past times side seat attempts to justify how the key banking concern tin give notice enforce the budget constraint: it volition heighten involvement rates to infinity if it does non hold.

This is equivalent to proverb that the toll of Treasury bills volition clear at $0.

Needless to say, in that location is no examine of the mathematics of marketplace clearing.

• If the solid sector inherited whatsoever coin from the previous period, it tin give notice purchase an interplanetary space number of bills. In what feel has the marketplace cleared?
• If the solid sector had no money, the size of the key bank's residual canvas is zero. How tin give notice entity without a residual canvas drive the toll of an property to zero?

In the absence of a mathematical examine of the marketplace clearing condition, together with a specification of the limits of key banking concern residual sheets, this declaration looks sketchy.

And rates accept to locomote infinite, implying that in that location is no solution to the optimisation problem. Any other outcome merely results inward the exact same chain of existent quantities that are inward the budget accounting identities. None of the analysis I did inward the previous post service made whatsoever supposition nigh nominal rates beingness "low."

(The null jump is the alone major number for the previous development.)

Furthermore, the "neo-Fisherian" consequence volition concord every bit presently every bit nosotros accept finite involvement rates: the higher the nominal rate, the higher the grade of expected inflation. Really high involvement rates imply truly high futurity inflation; it is difficult to consider how proverb that "central banks target inflation" justifies this entire job of thought. Once again, if mainstream economists truly solved the models that they develop, nosotros could resolve these issues.

Position: Because Equilibrium

"The key banking concern chooses the equilibrium." Good luck amongst converting that description to mathematics. This seems to the consensus persuasion on the topic, past times the way.

Concluding Remark

How mainstream macro went this far downward the rabbit hole is a consummate mystery.

Appendix

In my outset lecture on linear systems theory at McGill (a Master's grade course), Professor George Zames asked us the "What is a function?" question. Nobody got it right. People thrashed around amongst discussions of "mappings" or rules, or whatever nonsense they learn undergraduate electrical engineers. (I took existent analysis, but later.) He gave us the reply the side past times side day, together with I entered the Bourbaki cult.

The answer: a role is laid of ordered pairs. That is y = f(x) is merely a shorthand for f ={(x, y)}. (Alex Douglas rapped my knuckles for my master answer; I deliberately ignored the extra weather condition on the set. The typical restriction is that if (x,y) together with (x,z) are elements of f, thence y=z. Depending on how you lot desire to assault the issue, it could locomote prepare slightly differently, I believe.) There's no piffling elves mapping an input to outputs; it's merely a set.

(c) Brian Romanchuk 2017