Does The Governmental Budget Constraint Exist?

This article wraps up my discussion of the transversality condition and the governmental budget constraint. In summary, the governmental budget constraint used within mainstream macro has very serious flaws. I would have liked to use the title "The Governmental Budget Constraint Does Not Exist," but we need to take into account the rather curious Fiscal Theory of the Price Level. Furthermore, there are unsettling implications for the entire Dynamic Stochastic General Equilibrium (DSGE) model approach that relies upon optimisation. I abandoned looking at these models for this reason, and this article suggests why I believe the problems run much deeper. Unlike the previous articles, this article is largely free of mathematics, but I start out listing the various equations I refer to.

(I have been involved in discussions on this topic on Twitter, starting from an initial contact by Alex Douglas. I have just run across the work of C Trombley, who has written an article on similar lines here -- I Am The Very Model Of A Modern Macro Textbook. It's interesting, but I had been stuck writing out my own chain of logic, and could not respond to his points.)

Can't Keep Your Equations Straight Without A Program

This section lists the various equations discussed here, and how to interpret them. They all refer to simplified mathematical frameworks that are seen in a variety of textbook DSGE models.

The notation is defined in the previous article. In all cases, the variable $b(t)$ refers to the stock of government debt in real terms, $r$ is the real interest rate, and $s(t)$ is the primary fiscal surplus.

What I refer to as the governmental budget constraint is the following equation.
$$\begin{equation}
b(0) = \sum_{i=1}^{\infty} \frac{s(i)}{(1+r)^i} \label{eq:summation}
\end{equation}$$
This equation says that the summation of the discounted real primary surpluses is finite, and equals the initial stock of debt. The validity of this expression is what we are interested in.

The next equation is the 1-period accounting identity:

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

This just says that the debt at time $t+1$ is equal to $(1+r)$ times the debt level at time $t$, minus the surplus in the current period ($t+1$). That is, debt will compound by the discount rate, less the primary surplus. As an accounting identity, this has to hold.* One annoying habit of some mainstream economists is to also refer to this as a "governmental budget constraint," conflating the dubious ($\ref{eq:summation}$) with the true-by-definition ($\ref{eq:accountident}$).

The next equation is the transversality condition:
 $$\begin{equation}
\lim_{t-> \infty} \frac{b(t)}{(1+r)^t} = 0. \label{eq:limit}
\end{equation}$$
This equation says that the stock of debt outstanding cannot grow faster than the discount rate $r$. Transversality is a term that comes from optimisation theory. Since we do not actually apply it, the exact definition does not matter.

Finally, there are couple of workhorse accounting identities that relate the stock of debt at time $0$ and time $t$.

The reasonable-looking forward relation, which tells us the level of future debt based on the initial debt level, and intervening primary surpluses.
$$\begin{equation}
b(t) = (1+r)^t b(0) - \sum_{i=1}^t (1+r)^{t-i} s(i). \label{eq:fwdsum}
\end{equation}$$

There is also the unusual backward relation, which tells us the current level of debt based on a future level. This equation is just an algebraic restatement of the previous. (My first article had a deliberately obtuse attempt to decipher this equation.)
$$\begin{equation}
b(0) = \sum_{i=1}^t \frac{s(i)}{(1+r)^i} + \frac{b(t)}{(1+r)^t}. \label{eq:bkwdsum}
\end{equation}$$

Attempting to Follow Mainstream Logic

For a variety of reasons, mainstream economists want to use the governmental budget constraint ($\ref{eq:summation}$). Within models, households face a budget constraint, and it would be unfair if governments did not have one (beyond the accounting identity ($\ref{eq:accountident}$)). As I discussed in "On Being Pelted With Peanuts: Part I," they could just assume it to be true.

Of course, just assuming something to be true is not too useful. Mexico will pay for that wall, if we assume that they will do so. As such, mainstream economists searched for a reason for it to be true, The usual logic appears to work as follows. (I have never seen a coherent description of the logic behind this, so I had to use guesswork.)

1. Starting with the backward relation ($\ref{eq:bkwdsum}$), we can manipulate equations to show that the transversality condition ($\ref{eq:limit}$) implies the accounting identity ($\ref{eq:summation}$). (I did the proof in the previous article.)
2. When households search for the optimal solution to their utility maximisation problem in the DSGE model, the optimal solution (allegedly) displays the transversality condition ($\ref{eq:limit}$).
3. Therefore, household optimisation preferences will imply that ($\ref{eq:summation}$) holds.

The mainstream economists then go on to wave their hands about future surpluses cancelling out the effect of "debt-financed" fiscal stimulus, and so fiscal policy is ineffective, etc.

Not so fast.

If the backward relation ($\ref{eq:bkwdsum}$) holds, so does the forward relation ($\ref{eq:fwdsum}$). For the sake of argument, assume that the initial real stock of debt is fixed. (We return to this assumption later.) If we look at that equation, we see that the future debt level is pinned down by an accounting identity: the household sector in aggregate cannot alter the future trajectory of the debt by one (real) penny, no matter what optimisation choices it takes. (The models we are discussing feature fiscal policy that is completely unrelated to the state of the economy.) That is, the premises behind logical steps 1 and 2 above are inconsistent. The confusion in "Being Pelted With Peanuts" was the result of the inconsistency in the logic being used.

The next line of defence is to argue that since households have future money ("at infinity") that they will not need, they will spend it now. In other words, they cannot stop the chain of single-period accounting identities which determine the ratio of future debt to the initial level, they can (somehow) change the starting point.

Although that might work for an individual household, that cannot work in aggregate. All purchases made by households in the crippled DSGE modelling frameworks flow right back to the household sector, and there is no way of the household sector reducing its aggregate holdings of government-issued liabilities (other than voluntarily destroying money or bill holdings, which is not optimising behaviour). After all, we were only able to argue that household preferences influenced government debt outstanding because the only sector that held government debt in the model was the household sector. Furthermore, since we are assuming that all households act the same ("representative household"), they would all try to buy at the same time, without any extra supply forthcoming. There is no way of affecting the nominal debt level at time zero.

The only thing that can adjust is the price level at time zero. This is not "inflation," as that is the rise in the price level in future periods versus time zero. Instead, the entire price level has to shift instantly, which destroys the real value of financial assets held during the previous period. (Raising interest rates in time zero does nothing to stop this, as this only protects the real value of government bills against the inflation from time period $0$ to $1$.) This is how the assumption that the initial real debt level is fixed is relaxed: by changing the initial price level (which is the only thing that can move).

This adjustment mechanism appears implausible, but it reflects the general under-determination of the price level at $t=0$. Almost all attention is paid on the relative price between current prices and the future, but there is little discussion why the initial price level has to be at any particular level. The only variables with nominal scaling are the inherited financial assets from the previous period. If those debt ratios are "too high," we just scale nominal GDP instantly so that the ratio hits the correct level. (Calvo pricing does not help; firms that are unable to adjust prices to the new starting point get squashed like bugs.)

This effect is the Fiscal Theory of the Price Level (FTPL). The implications of the FTPL are stark: the price level is entirely driven by the state of expectations about fiscal policy. The price level at $t=0$ is entirely determined by fiscal policy expectations at $t=0$. The price level at $t=1$ is entirely determined by fiscal policy expectations at $t=1$. This means that monetary policy settings at $t=0$ is utterly irrelevant for the level of inflation at $t=0$.

The FTPL justifies the governmental budget constraint by saying that the private sector will raise (or lower) the price level -- changing the real value of existing debt -- if it ever looks like the budget constraint will not hold. This has nothing to do with "transversality" in optimisations. However, it is once again an assumption about economic behaviour that holds only because we assume that is true. If we assume that other factors influence the initial determination of the price level. the governmental budget constraint disappears.

The FTPL appears to be the only internally coherent class of representative household DSGE models. Unfortunately, the models are fairly degenerate, in that nothing else will really matter for inflation. Furthermore, it seems that their empirical usefulness is highly questionable. (Do we have plausible infinite horizon fiscal forecasts?)

Further Optimisation Ugliness

Even if we want to ignore the Fiscal Theory of the Price Level, the interaction between the macro constraints and household constraints are worrisome. I have not wasted much of my time looking at microeconomics, but I have severe doubts about how its "laws" have been applied to DSGE macro. We cannot assume that households are "infinitely small"; we need to model $N$ households, and see how they interact with an aggregate macro budget identity. In other words, we cannot use theorem statements that are cherry-picked from micro textbooks without ensuring that all the conditions required by those theorems apply to the model in question.

The household sector's financial assets are held in a vise, which is the governmental fiscal surplus. There is no action that can be taken to change the end-of-period holdings, no matter what level of production takes place.

If your financial asset holdings is completely outside of your control, how do they matter in an optimisation? It seems that the optimal strategy is to completely ignore financial asset holdings in the utility maximisation problem.**

However, such a step destroys the entire premise of inter-temporal optimisation. Unless the model features real investment, there is nothing (other than the irrelevant financial balances) that link period $t$ and $t+1$. This suggests that the optimal solution to these problems is just to pick the naive point-in-time utility maximisation at every time point. (Insert the production function into the 1-period component of the utility function; find the maximising output.) The fact that optimisation is carried out on an infinite horizon is just a smoke screen; what happens in period 1 has no effect on the solution in period 0.

Finally, since financial asset balances do not matter in these models, the rate of interest does not matter. Once again, the actions of the central bank are entirely irrelevant to the model outcome.

Concluding Remarks

It is unacceptable that we have to speculate about the solutions to these models. The usefulness of mathematics is that it forces you to think clearly, and crystallise your logic in equations. However, once we lose the discipline of properly solving the equations, we are back to literary speculation.

Footnotes:

* In some treatments of the topic, this accounting identity has been turned into an inequality, based on logic from financial mathematics. You know a field has completely lost any shred of common sense when the only things that we know hold with equality are turned into inequalities.

** If money does not appear in the utility function, the construction appears straightforward. Assume we have a feasible trajectory $x$, with utility $U(x)$. We then construct $x^*$ with all state variables equal to $x$, other than the primary surplus sequence $s$ and the affected financial asset holdings (normally bills and money). The only restriction on $s^*$ is that it does not somehow bind the household sector's financial constraint by running too large surpluses. This set is non-empty; the time series $s^*(t) = s(t) - 1$ is one such primary surplus series. Importantly, the series $s^*$ is created by only adjusting the taxes imposed; real government consumption is fixed. Since the financial constraints would never bind, $x^*$ is also feasible. Moreover, $U(x^*) = U(x)$. Therefore, we can see that the optimal trajectory is (somewhat) indifferent to the path of financial variables. Of course, there is the problem of attaining a maximum of an optimisation where the set of feasible solutions is not closed and finite, see this article by Alex Douglas.  If money appears in the utility function, the set of exogenous primary surplus sequences that we can use is limited to the set that allow the household sector to match the optimal money balance in each period (assuming that we do not allow the household sector to run negative bill holdings, or borrowings from the government).

(c) Brian Romanchuk 2017