Mathematics Of The Budget Constraint (Again)

This article attempts to give a simpler mathematical discussion of the governmental budget constraint and transversality. After throwing my hands up in the air in my previous article, I run through the basic mathematics of the accounting identity for governments, and we can see that what is called "transversality" is just equivalent to making the assumption that the discounted primary surpluses converge to be equal to the initial stock of debt. However, household sector optimisation is nowhere in sight, which raises the question why it comes up in discussion of this topic in the first place.

Once again, the math-phobic may as well stay clear. I would also draw your attention to this article by Alex Douglas; he is jumping ahead to an extremely point about optimisations (in general, we have no reason to believe that the optimum exists when the set of solutions is not closed and finite).

The equations are being generated by MathJax, and they might not be rendered on some browsers. If you can read LaTex, you might be able to follow the argument anyway. Please note that I went nuts with the equations here, and so it may take some time for the equations to render. I will be reverting to low math content in the future, but I just wanted to underline how cumbersome it is to deal with infinite summations, a point that is glossed over in a lot of treatments I see.

As a disclaimer, this was relatively rushed; there's probably a few typos in here.

Preliminaries

Let $\cal T$ be the set of time series defined on $\mathbb Z_+$. That is, if $x \in {\cal T}$, then $x(t) \in {\mathbb R}$ for all $t \in {\mathbb Z_+}.$ (The set ${\mathbb Z}_+$ is the set of positive integers greater than or equal to $0$.)

(Note: I am unsure what is the formal name for $\cal T$; I have a bad feeling about its properties.)

Definitions associated with infinite sums and limits are described in the previous article.

Assumptions

• Money holdings are zero at all times; the only government liabilities are 1-period bills. (If we allow money to be held, we then get terms associated with money creation in the formulae. These added complexities offer little value-added.)
• We are starting at time $0$ for notational simplicity.
• The (expected) real discount rate is equal to $r$ for all times. (There is some embedded assumptions about deflation as a result of this. The household can get whatever real interest rate it wishes on money balances if there is sufficient deflation. This technicality is typically ignored elsewhere; I am following that assumption so that my equations align with the usual textbook ones. Otherwise, we need to start tracking nominal balances and the price level as well, and my treatment would bear no resemblance to what we see elsewhere.)
• Realised variables are equal to expectations at time $0$. (If perfect foresight is bothersome, pretend this is a simulation at $t=0$.) 

Variable definitions:

• Denote the real market value of government bills outstanding at time $t$ as $b(t)$. (That is, $b \in {\cal T}$.) The initial value of $b$ ($b(0)$) is a positive number.
• The primary fiscal surplus at time $t$ is $s(t)$. The variable $s$ is a fixed member of $\cal T$; that is, it is an exogenous variable. The initial value is fixed: $s(0) = 0.$

Definition The 1-period government accounting identity is given by (for $t>0$):

$$\begin{equation}
b(t+1) = (1+r) b(t) - s(t+1). \label{eq:accountident}
\end{equation}$$

Lemma We can relate $b(t)$ and $b(0)$ as 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 true by inspection for $t=0$, and by applying ($\ref{eq:accountident}$) for $t=1$.
• Assume true for $t$.
• Validate for $t+1$. Apply ($\ref{eq:accountident}$) and the induction assumption, we 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. $\fbox{}$

Lemma The following 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 well defined if and only if
$$\begin{equation}
\lim_{t-> \infty} \frac{b(t)}{(1+r)^t} = 0. \label{eq:limit}
\end{equation}$$
Proof: We first prove that ($\ref{eq:limit}$) implies ($\ref{eq:summation}$). Rearrange terms 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 any $\epsilon > 0$. By applying the definition of ($\ref{eq:limit}$), there 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 then apply the definition of an infinite summation to see that ($\ref{eq:summation}$) holds.

To validate that ($\ref{eq:summation}$) being well-posed implies ($\ref{eq:limit}$), we 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 any $\epsilon > 0$. By applying ($\ref{eq:summation}$), there exists an $M$ such that the right-hand side hand side has modulus less than $\epsilon$ for all $t \geq M$. We then apply the definition of the limit to see that the left-hand side converges to zero. $\fbox{}$

Remark This proof is plodding, but there still might be issues that it glosses over. In a journal article in applied mathematics, nobody would bother with the $\epsilon$ arguments (unless it was much more difficult). However, the proof text would have to be careful to indicate why the various summations and limits exist, That is, it is unacceptable to write down infinite summations and use them in other manipulations without ensuring that the summations exist.

Discussion

The theorem provided tells us that the condition that is called the "transversality condition" is a necessary and sufficient condition for the condition on the discounted sum of primary surpluses (equation ($\ref{eq:summation}$)).

This is what is asserted in various DSGE macro papers, and which caused me agony in my previous article. Since it is actually straightforward, why complain?

My complaint is this: this derivation was driven entirely by straightforward application of the 1-period accounting identity. There is no optimisation involved at any point during the derivation (the notion of transversality comes from optimisation theory). Very simply, the household sector has no choice with respect to this result, therefore it makes no sense to pretend that it is the result of microfoundations.

In other words, since the initial stock of household debt holdings ($b(0)$) is fixed, and the path of the primary surpluses was assumed to be exogenous (a crazy assumption, but standard for simple DSGE models), the future path of debt holdings ($b(t)$) is deterministic, and not the result of any optimisation result. This raises the obvious corollary: if household wealth is determined entirely by fiscal policy, in what sense does it even matter for the optimisation problem?

Correspondingly, there is no reason to believe that the condition must hold; it either holds or it does not. Since the nominal discount rate is quite often below the nominal growth rate of the economy, the expectation is that it will in general not hold.

I will return to the economic discussions in a later article (with less equations).

(c) Brian Romanchuk 2017