\[ b_t = \frac{b_{t+N}}{(1+r)^N}+ \sum_{i=1}^{N} \frac{s_{t+i}}{(1+r)^i}\]

\[ b_t = \sum_{i=1}^{\infty} \frac{s_{t+i}}{(1+r)^i}\]

where \( b_t \) is the value of outlaying government bonds, \( r \) is the real interest rate (assumed given & constant) and \( s_t \) is the primary surplus. These equations say*, respectively:

The current value of outlaying government bonds (of a finitely lived government) is exactly the present value of the lump of remaining government bonds at the end of its life plus the present value of government surplus over that time.

The current value of outlaying government bonds is exactly the present value of its surplus over time.

The book treats these as more or less accounting terms, though since they involve value they aren't really accounting. Douglas asks how the second equation relates to the first. I give my answer in the paragraph with the underlined sentence. What seems to have given Douglas fits is the following sentence:

"The governments solvency ... condition can therefore be seen to be ... that ... debt must be backed at all times be backed by primary surpluses with a present value equal to the debt's face value."

This sentence in it's full version refers to a "transversality condition", a name that comes from optimization models but is a meaningless meme in the textbook's approach.

Economist Nick Rowe gave a very nice response - his point of view was that too answer Douglas's question the best thing to do is to throw out the book situation and give an example tailored to Douglas's question. Brian Romanchuck have another nice response, it seems to be his idea also to throw out the book's description but with more force.

My own initial response was displeasure that Douglas (and Romanchuck) seem to criticize the very idea of infinite mathematical objects. In these modern days of lazy evaluation, even constructive finitists shouldn't do that! But Douglas and Romanchuck have had time to lay out their objections in more detail and I don't think pure math is the relevant issue.

I think the relevant issue is economics.

There is an implicit causality to the story in the textbook that doesn't come through in the equations. The causality - in both equations - is from left to right. The sequence \( s_t \) is the cause and the sequence \( b_t \) is the effect. The desired story is "Governments can only increase debt by promising surplus in the future.". The story of how \( s_t \) evolves is the model, if it fits the equations then we accept it - otherwise reject it.

There's an implicit institutional structure there, that \( r \) is determined by an independent and fierce central bank. In the US and Europe, of course, \( r_t \) is actually determined by cybernetic feedback process by a politically constrained central bank (the constraints seem weaker in Europe for some reason).

On the bottom of page 185, the story suddenly stops being about \( r \), \( s_t \) and \( b_t \) and suddenly starts being about \( r - g \). Implicitly \( g \) is an independent variable set by technological process, as in the Solow models**. This is a strong, but common assumption. In reality, even "just" technological growth is a complex process that involves discovery, distributional concerns, government backing of science and lots and lots of complex non-linear feedback.

The textbook implicitly assumes we're looking at a 'long run' in which both these feedback processes have settled down.

In the finite story, \( N \) is outside the model - the government implodes on a predetermined date. The implicit institutional structure sets \( r > g \) so that \( b_{t+N} \) will have a positive value - but its present value is proportional to \( r - g \). The implicit Central Bank chooses that variable. If \(N = \infty \), the implicit Central Bank is constrained to choose \( r \) such that the second equation comes out of the first.

__This is the answer to Douglas's question.__

Romanchuck has a further objection - there's no optimization in this model (it's not just macro, it's

**very**macro). That doesn't matter to this textbook - it doesn't care because \( s_t \) and \( b_t \) are assumed to be

*measurable*. If you have a microfounded theory that gives the wrong values (such as infinity), who cares, that just means that either the theory is wrong - or that the implicit theories of the equations are wrong. Both are valid answers.

Is this good economics? Well, it's definitely

**standard**economics. The ideas were invented by Irving Fisher and reinvented many times after him. This model isn't perfectly general. There's an implicit institutional structure patterned on the US and Europe. Maybe in the previous six chapters, the textbook went over all these implicit assumptions. In the immortal words of Dom Mazzetti "Doubt it".

One helpful question is "What would it mean for it to be wrong?". Well, for one, bonds can be worth more than the present value of government surplus. This is probably true but due to risk and uncertainty, which are left out of this model.

This is all that I can get out of these poor equations. But I'll end on one slightly more "philosophical" (maybe I should be careful with that word around a philosophy professor!) observation:

**In macroeconomics, there is no data, only models.***There's no uncertainty or randomness in these equations, so I'm not gonna keep saying "under rational expectations", "assuming expected value exists", "certainty equivalent", "common knowledge", blah blah blah. That's all under the rug.

** Fish don't see the water they swim in. To many economists, Solow is just accounting. C'est la vie.

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