On Being Pelted By Peanuts: Part I

Alexander Douglas (a lecturer in philosophy) wrote an interesting article on two subjects recently: "Macroeconomics -- A view from the peanut gallery." He covers two diverging topics: the transversality condition from mainstream macro, and the question of welfare functions in Stock-Flow Consistent models. I will take a stab at these topics over the coming days. In this article, I am expressing my deep displeasure with what is supposed to be trivial mathematics used by the mainstream: the transversality condition.

(He embeds some mathematics in his post, and so I am using MathJax to format an answer. The equations may not be properly rendered on some browsers. People who are allergic to equations may want to skip this one... I did this quickly, and already squashed a few typos.)
Update: I have written out the involved mathematics in a straightforward manner in "Mathematics of Budget Constraint (Again)". This article is a deliberately obtuse reading of some equations that appear in many introductory mainstream macro textbooks. (To be clear, I am targeting the textbook that Alex Douglas linked to; other introductory texts have very similar developments. I did not look at the linked textbook that carefully; it may have explained things better than a reader might conclude from my comments here.) If you can follow the mathematics, the other article follows a straightforward (semi-)rigourous approach and makes everything look really simple. (Pure mathematicians could probably find something to yelp about.) The sting in the tail, however, is what this means from the interpretation of economic models, which is what everyone here is presumably interested in. I give a brief version of my interpretation in that article, but I will later write a non-mathematical article explaining what problems I see. 

Transversality

Transversality refers to a condition associated with the governmental budget constraint of mainstream macro. I wrote about this in earlier articles, such as "If r < g, DSGE Model Assumptions Break Down." From the perspective of mainstream macro, my logic in that article is not covering the mainstream microeconomic arguments involved. I will try to stay closer to the mainstream logic in the discussion here.

Professor Douglas refers to this text, which offers a fairly standard treatment of the governmental budget constraint. (For those of you who are new to this topic, mainstream economists refer to two concepts are being the governmental budget constraint. The first is an accounting identity that links the current period to the previous, which is non-controversial. The second is the behaviour at infinity, which is what I am discussing here.)

The key claim is that if $r$ is a real discount rate, $b_t$ represents (real) government bond outstanding at time $t$, and $s_t$ is the government's (real) fiscal surplus at time $t$, then we have the following relationship:
\[
b_t = \sum_{i=1}^N \frac{s_{t+i}}{(1 + r)^i} + \frac{b_{t+N}}{(1+r)^N}.
\]
We can allegedly let $N$ "go to infinity," let the "second term go to zero", and then we get the infinite horizon budget constraint:
\[
b_t = \sum_{i=1}^{\infty} \frac{s_{t+i}}{(1+r)^i}.
\]

To Infinity, and Beyond!

(OK, the header was cheesy.)

Professor Douglas discussed infinite time in his article, using terms that makes my head hurt. ($\aleph_0$, seriously?) I am going to approach infinite time the way it is usually done in applied mathematics: it does not really exist. From the perspective of real analysis, $\infty$ is just a short-hand.

For discrete time models, the time axis is normally taken to be the set of positive integers (including zero), denoted ${\mathbb Z}_+$ (${\mathbb Z}$ is all integers. We can write ${\mathbb Z_+} = [0, 1, 2, ... \infty),$ as a shorthand. Importantly we close the sequence definition with ")" to denote that the element $\infty$ (whatever that is!) is not an element of ${\mathbb Z}_+$.

If we write infinite summation of the form:
\[
 x = \sum_{i=0}^{\infty}a_i,
\]
where $a_i$ is a sequence in $\mathbb R$ defined on the support ${\mathbb Z}$, what it translates into:
\[
x = \lim_{N -> \infty} \sum_{i=0}^N a_i,
\]
which is itself a short-hand. What the above equation says: if $x$ exists, $x \in {\mathbb R}$, with the property: for any $\epsilon > 0$, there exists an $M(\epsilon)$ with the property that:
\[
\left| x- \sum_{i=0}^{N} a_i \right| < \epsilon, \forall N > M(\epsilon).
\]

In plain English, for any non-zero error tolerance (normally denoted $\epsilon$), we can guarantee that all sufficiently long summations lie within that error bound of $x$. Note that "$\infty$" appears no where in the underlying mathematical statements.

We can always write down an infinite summation, but we need to validate that it converges. Otherwise, the summation is equivalent to the set $\{x: x \in {\mathbb R}, x = x + 1\},$ which is just a fancy way of saying $\{\emptyset\}$ (empty set). In most fields of applied mathematics, the first thing to do when faced with infinite summations is to validate convergence; mainstream economics, not so much.

Back to Transversality

The expression:
\[
b_t = \sum_{i=1}^{\infty} \frac{s_{t+i}}{(1+r)^i}.
\]
is reasonable enough from the point of mathematics; the only issue is convergence. (Why it must hold will have to wait for another time; which is actually what Professor Douglas wanted to discuss. But we can't get there from here.)

If mainstream economists started from there, things would be fine. Unfortunately, they wanted to link to optimisation theory somehow, and wanted to link to the following expression (which would be the result of a finite horizon optimisation).
\[
b_t = \sum_{i=1}^N \frac{s_{t+i}}{(1 + r)^i} + \frac{b_{t+N}}{(1+r)^N}.
\]
As a starting point, this expression is gibberish (to use the technical mathematical term) expression leaves a lot of open questions. (I got my knuckles wrapped by a mathematician on Twitter for that, it was a joke, honest!)

Going the other way is fine. Starting from the infinite summation expression.
$$\begin{eqnarray}
 b_t  & = & \sum_{i=1}^{\infty} \frac{s_{t+i}}{(1+r)^i}\\
 & = & \sum_{i=1}^{N} \frac{s_{t+i}}{(1+r)^i} + \sum_{j=N+1}^{\infty} \frac{s_{t+j}}{(1+r)^j},\\
& = & \sum_{i=1}^{N} \frac{s_{t+i}}{(1+r)^i} + \frac{1}{(1+r)^N}\sum_{k=1}^{\infty} \frac{s_{(t+N)+k}}{(1+r)^k},\\
& = & \sum_{i=1}^N \frac{s_{t+i}}{(1 + r)^i} + \frac{b_{t+N}}{(1+r)^N}.
\end{eqnarray}$$

However, you cannot go the other way. The equation:
\[
b_t = \sum_{i=1}^N \frac{s_{t+i}}{(1 + r)^i} + \frac{b_{t+N}}{(1+r)^N},
\]
actually defines $b(t)$ as a function of four five variables: $t$, the sequence $s$ (which is fixed), the summation termination $N$, the discount rate $r$, and $b(t+N,...)$ (there's a variable hidden in $b(t+N,...)$). That is,
\[
b(t) = f(t, s, r, N, b(t+N, s, r, N_N)).
\]
This is a recursive definition that does not appear to make sense, as we are defining $b(t)$ based on a future value of $b(t)$, which can only be defined in terms of another future value ($N_N$, whatever that is). When we define a recursive relationship, we normally need to define the initial value of the sequence.

Update: As pointed out by C Trombley (@C_Trombley1 on Twitter), it might be possible to get such a backwards recursive definition to work somehow.  I tried to make it clear that it was not impossible, but I have no idea how such a proof can be constructed without assuming that the infinite summation converges -- which is what we are trying to prove. In any event, although it is acceptable to skip some steps in proofs in published mathematics, relying on readers to guess what non-standard proof method exists is beyond the pale. If it were not necessary to supply missing steps, Fermat actually proved his last theorem.

We could try to pretend that the following works:
\[
b(t,r,N,s,\alpha_N) = \sum_{i=1}^N \frac{s_{t+i}}{(1 + r)^i} + \alpha_N,
\]
where $\alpha_N \in {\mathbb R}$. This gives us a well-defined result. However, there is no guarantee that
\[
\alpha_N = \frac{b_{t+N}}{(1+r)^N}.
\]

As far as I can tell, the idea is that we are supposed to get the government debt holdings from some optimisation problem somewhere. However, on a finite horizon optimisation, there is notion of optimisation over the period beyond the horizon. If we terminate the optimisation at period $N$, there is no period $N+1$ in the optimisation, and the household should dump all of its financial assets and have a final blowout in period $N$. ("Party like its 1999!") But the optimal solution presumably changes if in fact we decide that we will make it to the year 2000, and instead assume that the world blows up in 2001.

Concluding Remarks

The fact that we cannot get more than a couple of lines in the mathematics of DSGE macro without raising existential questions like this is a sign that the mathematics in DSGE macro has long departed accepted mathematical norms.

It should be noted that there is a simpler version of this analysis; the question is trying it to align to the way it is described by DSGE macro papers.

(c) Brian Romanchuk 2017