On Beingness Pelted Past Times Peanuts: Purpose I
(He embeds some mathematics inwards his post, in addition to therefore I am using MathJax to format an answer. The equations may non live properly rendered on some browsers. People who are allergic to equations may desire to skip this one... I did this quickly, in addition to already squashed a few typos.)
Update: I direct keep written out the involved mathematics inwards a straightforward mode in "Mathematics of Budget Constraint (Again)". This article is a deliberately obtuse reading of some equations that look inwards many introductory mainstream macro textbooks. (To live clear, I am targeting the textbook that Alex Douglas linked to; other introductory texts direct keep really similar developments. I did non await at the linked textbook that carefully; it may direct keep explained things amend than a reader powerfulness conclude from my comments here.) If yous tin follow the mathematics, the other article follows a straightforward (semi-)rigourous approach in addition to makes everything await truly simple. (Pure mathematicians could in all probability detect something to yelp about.) The sting inwards the tail, however, is what this agency from the interpretation of economical models, which is what everyone hither is presumably interested in. I give a brief version of my interpretation inwards that article, but I volition afterward write a non-mathematical article explaining what problems I see.
Transversality
Transversality refers to a status associated amongst the governmental budget constraint of mainstream macro. I wrote nearly this inwards before articles, such every bit "If r < g, DSGE Model Assumptions Break Down." From the perspective of mainstream macro, my logic inwards that article is non roofing the mainstream microeconomic arguments involved. I volition motility to rest closer to the mainstream logic inwards the give-and-take here.Professor Douglas refers to this text, which offers a fairly criterion handling of the governmental budget constraint. (For those of yous who are novel to this topic, mainstream economists refer to 2 concepts are existence the governmental budget constraint. The showtime is an accounting identity that links the electrical flow menstruation to the previous, which is non-controversial. The instant is the deportment at infinity, which is what I am discussing here.)
The fundamental claim is that if $r$ is a existent discount rate, $b_t$ represents (real) regime bond outstanding at fourth dimension $t$, in addition to $s_t$ is the government's (real) financial surplus at fourth dimension $t$, in addition to then nosotros direct keep the next relationship:
\[
b_t = \sum_{i=1}^N \frac{s_{t+i}}{(1 + r)^i} + \frac{b_{t+N}}{(1+r)^N}.
\]
We tin allegedly allow $N$ "go to infinity," allow the "second term larn to zero", in addition to and then nosotros larn the interplanetary space horizon budget constraint:
\[
b_t = \sum_{i=1}^{\infty} \frac{s_{t+i}}{(1+r)^i}.
\]
To Infinity, in addition to Beyond!
(OK, the header was cheesy.)Professor Douglas discussed interplanetary space fourth dimension inwards his article, using damage that makes my caput hurt. ($\aleph_0$, seriously?) I am going to approach interplanetary space fourth dimension the way it is usually done inwards applied mathematics: it does non truly exist. From the perspective of existent analysis, $\infty$ is simply a short-hand.
For discrete fourth dimension models, the fourth dimension axis is unremarkably taken to live the laid of positive integers (including zero), denoted ${\mathbb Z}_+$ (${\mathbb Z}$ is all integers. We tin write ${\mathbb Z_+} = [0, 1, 2, ... \infty),$ every bit a shorthand. Importantly nosotros closed the sequence Definition amongst ")" to announce that the chemical constituent $\infty$ (whatever that is!) is non an chemical constituent of ${\mathbb Z}_+$.
If nosotros write interplanetary space addition of the form:
\[
x = \sum_{i=0}^{\infty}a_i,
\]
where $a_i$ is a sequence inwards $\mathbb R$ defined on the back upward ${\mathbb Z}$, what it translates into:
\[
x = \lim_{N -> \infty} \sum_{i=0}^N a_i,
\]
which is itself a short-hand. What the to a higher house equation says: if $x$ exists, $x \in {\mathbb R}$, amongst the property: for whatever $\epsilon > 0$, at that topographic point exists an $M(\epsilon)$ amongst the belongings that:
\[
\left| x- \sum_{i=0}^{N} a_i \right| < \epsilon, \forall north > M(\epsilon).
\]
In patently English, for whatever non-zero fault tolerance (normally denoted $\epsilon$), nosotros tin guarantee that all sufficiently long summations prevarication inside that fault jump of $x$. Note that "$\infty$" appears no where inwards the underlying mathematical statements.
We tin ever write downward an interplanetary space summation, but nosotros involve to validate that it converges. Otherwise, the addition is equivalent to the laid $\{x: x \in {\mathbb R}, x = x + 1\},$ which is simply a fancy way of proverb $\{\emptyset\}$ (empty set). In most fields of applied mathematics, the showtime affair to exercise when faced amongst interplanetary space summations is to validate convergence; mainstream economics, non therefore much.
Back to Transversality
The expression:\[
b_t = \sum_{i=1}^{\infty} \frac{s_{t+i}}{(1+r)^i}.
\]
is reasonable plenty from the quest of mathematics; the entirely number is convergence. (Why it must concur volition direct keep to hold back for some other time; which is truly what Professor Douglas wanted to discuss. But nosotros can't larn at that topographic point from here.)
If mainstream economists started from there, things would live fine. Unfortunately, they wanted to link to optimisation theory somehow, in addition to wanted to link to the next facial expression (which would live the upshot 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
Going the other way is fine. Starting from the interplanetary space addition 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, yous cannot larn the other way. The equation:
\[
b_t = \sum_{i=1}^N \frac{s_{t+i}}{(1 + r)^i} + \frac{b_{t+N}}{(1+r)^N},
\]
truly defines $b(t)$ every bit a role of
\[
b(t) = f(t, s, r, N, b(t+N, s, r, N_N)).
\]
This is a recursive Definition that does non look to brand sense, every bit nosotros are defining $b(t)$ based on a futurity value of $b(t)$, which tin entirely live defined inwards damage of some other futurity value ($N_N$, whatever that is). When nosotros define a recursive relationship, nosotros unremarkably involve to define the initial value of the sequence.
Update: As pointed out yesteryear C Trombley (@C_Trombley1 on Twitter), it powerfulness live possible to larn such a backwards recursive Definition to piece of work somehow. I tried to arrive clear that it was non impossible, but I direct keep no thought how such a proof tin live constructed without assuming that the interplanetary space addition converges -- which is what nosotros are trying to prove. In whatever event, although it is acceptable to skip some steps inwards proofs inwards published mathematics, relying on readers to justice what non-standard proof method exists is beyond the pale. If it were non necessary to provide missing steps, Fermat truly proved his lastly theorem.
We could motility to pretend that the next 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, at that topographic point is no guarantee that
\[
\alpha_N = \frac{b_{t+N}}{(1+r)^N}.
\]
As far every bit I tin tell, the thought is that nosotros are supposed to larn the regime debt holdings from some optimisation job somewhere. However, on a finite horizon optimisation, at that topographic point is notion of optimisation over the menstruation beyond the horizon. If nosotros terminate the optimisation at menstruation $N$, at that topographic point is no menstruation $N+1$ inwards the optimisation, in addition to the menage should dump all of its financial assets in addition to direct keep a concluding blowout inwards menstruation $N$. ("Party similar its 1999!") But the optimal solution presumably changes if inwards fact nosotros create upward one's heed that nosotros volition arrive to the yr 2000, in addition to instead assume that the footing blows upward inwards 2001.
Concluding Remarks
The fact that nosotros cannot larn to a greater extent than than a duo of lines inwards the mathematics of DSGE macro without raising existential questions similar this is a sign that the mathematics inwards DSGE macro has long departed accepted mathematical norms.It should live noted that at that topographic point is a simpler version of this analysis; the query is trying it to align to the way it is described yesteryear DSGE macro papers.
(c) Brian Romanchuk 2017
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