Mathematics Of The Budget Constraint (Again)
Once again, the math-phobic may equally good remain clear. I would likewise clit your attending to this article past times Alex Douglas; he is jumping ahead to an extremely dot most optimisations (in general, nosotros receive got no argue to believe that the optimum exists when the laid of solutions is non unopen as well as finite).
The equations are existence generated past times MathJax, as well as they mightiness non move rendered on to a greater extent than or less browsers. If you lot tin read LaTex, you lot mightiness move able to follow the declaration anyway. Please annotation that I went nuts alongside the equations here, as well as so it may convey to a greater extent than or less fourth dimension for the equations to render. I volition move reverting to depression math content inwards the future, but I exactly wanted to underline how cumbersome it is to bargain alongside interplanetary space summations, a dot that is glossed over inwards a lot of treatments I see.
As a disclaimer, this was relatively rushed; there's in all probability a few typos inwards here.
Preliminaries
Let $\cal T$ move the laid of fourth dimension serial defined on $\mathbb Z_+$. That is, if $x \in {\cal T}$, as well as then $x(t) \in {\mathbb R}$ for all $t \in {\mathbb Z_+}.$ (The laid ${\mathbb Z}_+$ is the laid of positive integers greater than or equal to $0$.)(Note: I am unsure what is the formal cite for $\cal T$; I receive got a bad feeling most its properties.)
Definitions associated alongside interplanetary space sums as well as limits are described inwards the previous article.
Assumptions
- Money holdings are null at all times; the only regime liabilities are 1-period bills. (If nosotros let coin to move held, nosotros as well as then larn price associated alongside coin creation inwards the formulae. These added complexities offering piffling value-added.)
- We are starting at fourth dimension $0$ for notational simplicity.
- The (expected) existent discount charge per unit of measurement is equal to $r$ for all times. (There is to a greater extent than or less embedded assumptions most deflation equally a number of this. The work solid tin larn whatever existent involvement charge per unit of measurement it wishes on coin balances if at that spot is sufficient deflation. This technicality is typically ignored elsewhere; I am next that supposition so that my equations align alongside the green textbook ones. Otherwise, nosotros require to start tracking nominal balances as well as the toll grade equally well, as well as my handling would deport no resemblance to what nosotros encounter elsewhere.)
- Realised variables are equal to expectations at fourth dimension $0$. (If perfect foresight is bothersome, pretend this is a simulation at $t=0$.)
Variable definitions:
- Denote the existent marketplace position value of regime bills outstanding at fourth dimension $t$ equally $b(t)$. (That is, $b \in {\cal T}$.) The initial value of $b$ ($b(0)$) is a positive number.
- The principal financial surplus at fourth dimension $t$ is $s(t)$. The variable $s$ is a fixed fellow member of $\cal T$; that is, it is an exogenous variable. The initial value is fixed: $s(0) = 0.$
Definition The 1-period regime accounting identity is given past times (for $t>0$):
$$\begin{equation}
b(t+1) = (1+r) b(t) - s(t+1). \label{eq:accountident}
\end{equation}$$
Lemma We tin relate $b(t)$ as well as $b(0)$ equally follows, for all $t \in {\mathbb Z}_+$:
$$\begin{equation}
b(t) = (1+r)^t b(0) - \sum_{i=1}^t (1+r)^{t-i} s(i). \label{eq:fwdsum}
\end{equation}$$
Proof Use induction.
- Equation ($\ref{eq:fwdsum}$) is truthful past times inspection for $t=0$, as well as past times applying ($\ref{eq:accountident}$) for $t=1$.
- Assume truthful for $t$.
- Validate for $t+1$. Apply ($\ref{eq:accountident}$) as well as the induction assumption, nosotros get:
$$\begin{eqnarray}
b(t+1) & = & (1+r)b(t) - s(t+1), \\
& = & (1+r) \left( (1+r)^t b(0) - \sum_{i=1}^t (1+r)^{t-i} s(i) \right) - s(t+1), \\
&= & (1+r)^{t+1} b(0) - \sum_{i=1}^{t+1} (1+r)^{(t+1)-i} s(t).
\end{eqnarray}$$
Validating the induction assumption. $\fbox{}$
b(t+1) & = & (1+r)b(t) - s(t+1), \\
& = & (1+r) \left( (1+r)^t b(0) - \sum_{i=1}^t (1+r)^{t-i} s(i) \right) - s(t+1), \\
&= & (1+r)^{t+1} b(0) - \sum_{i=1}^{t+1} (1+r)^{(t+1)-i} s(t).
\end{eqnarray}$$
Validating the induction assumption. $\fbox{}$
Lemma The next human relationship holds:
$$\begin{equation}
b(0) = \sum_{i=1}^t \frac{s(i)}{(1+r)^i} + \frac{b(t)}{(1+r)^t}. \label{eq:bkwdsum}
\end{equation}$$
Proof By inspection (apply ($\ref{eq:fwdsum}$)). $\fbox{}$
Theorem The equation
$$\begin{equation}
b(0) = \sum_{i=1}^{\infty} \frac{s(i)}{(1+r)^i} \label{eq:summation}
\end{equation}$$
is good defined if as well as only if
$$\begin{equation}
\lim_{t-> \infty} \frac{b(t)}{(1+r)^t} = 0. \label{eq:limit}
\end{equation}$$
Proof: We start attempt out that ($\ref{eq:limit}$) implies ($\ref{eq:summation}$). Rearrange price of ($\ref{eq:bkwdsum}$) to give:
\[
b(0) - \sum_{i=1}^t \frac{s(i)}{(1+r)^i} = \frac{b(t)}{(1+r)^t}.
\]
This implies that
$$\begin{equation}
\left| b(0) - \sum_{i=1}^t \frac{s(i)}{(1+r)^i} \right|= \left| \frac{b(t)}{(1+r)^t} \right|.
\label{eq:absval} \end{equation}$$
Fix whatever $\epsilon > 0$. By applying the definition of ($\ref{eq:limit}$), at that spot exists an $M$ such that
\[
\left| \frac{b(t)}{(1+r)^n} \right| < \epsilon, \forall n \geq M.
\]
Apply to ($\ref{eq:absval}$):
\[
\left| b(0) - \sum_{i=1}^n \frac{s(i)}{(1+r)^i} \right| < \epsilon, \forall n \geq M
\]
We as well as then apply the definition of an interplanetary space addition to encounter that ($\ref{eq:summation}$) holds.
To validate that ($\ref{eq:summation}$) existence well-posed implies ($\ref{eq:limit}$), nosotros rearrange ($\ref{eq:bkwdsum}$) to give:
\[
\frac{b(t)}{(1+r)^t} = b(0) - \sum_{i=1}^t \frac{s(i)}{(1+r)^i}.
\]
Fix whatever $\epsilon > 0$. By applying ($\ref{eq:summation}$), at that spot exists an $M$ such that the right-hand side manus side has modulus less than $\epsilon$ for all $t \geq M$. We as well as then apply the definition of the bound to encounter that the left-hand side converges to zero. $\fbox{}$
Remark This proof is plodding, but at that spot nevertheless mightiness move issues that it glosses over. In a magazine article inwards applied mathematics, nobody would bother alongside the $\epsilon$ arguments (unless it was much to a greater extent than difficult). However, the proof text would receive got to move careful to dot why the diverse summations as well as limits exist, That is, it is unacceptable to write downwards interplanetary space summations as well as role them inwards other manipulations without ensuring that the summations exist.
Discussion
The theorem provided tells us that the status that is called the "transversality condition" is a necessary as well as sufficient status for the status on the discounted amount of principal surpluses (equation ($\ref{eq:summation}$)).This is what is asserted inwards diverse DSGE macro papers, as well as which caused me agony inwards my previous article. Since it is truly straightforward, why complain?
My electrical load is this: this derivation was driven exclusively past times straightforward application of the 1-period accounting identity. There is no optimisation involved at whatever dot during the derivation (the notion of transversality comes from optimisation theory). Very simply, the work solid sector has no pick alongside honour to this result, thus it makes no sense to pretend that it is the number of microfoundations.
In other words, since the initial stock of work solid debt holdings ($b(0)$) is fixed, as well as the path of the principal surpluses was assumed to move exogenous (a crazy assumption, but criterion for unproblematic DSGE models), the futurity path of debt holdings ($b(t)$) is deterministic, as well as non the number of whatever optimisation result. This raises the obvious corollary: if work solid wealth is determined exclusively past times financial policy, inwards what sense does it fifty-fifty thing for the optimisation problem?
Correspondingly, at that spot is no argue to believe that the condition must hold; it either holds or it does not. Since the nominal discount charge per unit of measurement is quite oftentimes below the nominal increment charge per unit of measurement of the economy, the expectation is that it volition inwards full general non hold.
I volition homecoming to the economical discussions inwards a afterwards article (with less equations).
(c) Brian Romanchuk 2017
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