This article is a longer reflection on the use of the phrase
“unsustainable debt trajectory”, triggered by this article by Professor Brad DeLong. I am not attempting to address the deeper questions he poses, just the
question whether we can define what an “unsustainable debt trajectory” really
means.

I will immediately state that my comments are only
applicable for economies with free-floating exchange rates and a welfare state,
and for the central government that controls the central bank. Think Canada and
the United States, and not Greece, Argentina, Orange County or the Weimar
Republic.

The phrase “unsustainable debt trajectory” is inherently
vague, so I assume that such a condition implies that debt will rise in the
future to such a level that it “forces” a debt default by the government. In
other words, there has to be some serious consequence of the debt being
unsustainable. For example, a lot of people have said that Japanese debt levels have been “unsustainable” for
years, but with no apparent ill effect generated by
that condition of “unsustainability”.

I will note that Warren Mosler (one of the founders of MMT)
has suggested that government interest rates be locked at zero. This has the
effect of creating sustainable debt trajectories, no matter what else happens.
However, this would effectively abolish the government bond market, which would
seriously impair the value of my newly acquired

**bondeconomics.com**domain name. So I will not follow up this line of thought within this article; for more details see Mosler’s paper.
There would be two ways to define whether a debt trajectory
is unsustainable:

- Operationally: We can define a rule based on observable economic data that determines whether a future debt default is certain, independent of a model. For example, imagine that there is an “upper limit” to the debt/GDP ratio (no, not my view).
- Model-based: Define a rule which determines whether a debt growth path is sustainable within the dynamics of that model.

As for operational definitions, I can only assert that I am
unaware of any rule that works unconditionally (and since I worked as a fixed
income analyst for 15 years, this either means that I was particularly
inattentive or else no such rule exists in the public domain). One difficulty with the operational definition
is that it cannot have been triggered by the realised data for the
developed countries with floating currencies, given their absence of debt
defaults based on “explosive debt trajectories”. Given the wide variance of
fiscal ratios for this data set, the triggers for the rules would have to be
fairly extreme.

As for model-based rules, you end up in a quagmire. You end
up risking

*reification*: mistaking your model for reality. For example, I could create a model where the private sector adds an accelerating risk premium on interest rates as the debt/GDP ratio rises above 10%. If you simulated this model, it should eventually create some really cool explosive behaviour as the debt/GDP rises above 10%. But so what? My proposed model generates predictions which are wildly away from observed reality. The model used to generate the sustainability condition has to have some contact with empirical observations.
I will start with what I see as the correct methodology
(stock-flow consistent modelling), and then turn to the modelling methodologies
that I see as being problematic.

In a Stock-Flow Consistent (SFC) model with stock-flow norms, sectors in the economy are assumed to spend out of income flows received
during a time period, as well as selling financial assets to other sectors if
the net value of those assets is very high relative to the sector’s income
flows. In other words, there is a behavioural rule relating income to desired asset
holdings. For example, if I were handed $100 million in Treasury Bills, there
is a very good probability that my aggregate spending would rise at least
slightly afterwards. Also, it assumed that the model reflects a modern welfare
state: there are monetary transfers towards the unemployed, and that taxes are
imposed based on a percentage of nominal income and revenue flows (e.g., income
taxes, VAT).

Under such a modelling assumption, what happens is that
rising government debt corresponds to rising net asset values for other
sectors, and there is tendency for spending to “melt upwards”. In other words,
accelerating spending and hence income. As the economy moves to full
employment, welfare spending drops, taxes rise as nominal spending and incomes
rise.

I will assert that if we take reasonable parameter values
for the behavioural norms and policy settings, fiscal ratios (debt/GDP,
deficit/GDP) will converge towards a finite steady state no matter what. (Yes,
the debt levels increase without bound. The ratios remain stubbornly finite.)

*If this assertion is correct, every single realised debt trajectory is sustainable, as these fiscal ratio trajectories never grow in an unbounded fashion.*
I do not have the
space to justify this assertion; the
reader is referred to the paper “

*Fiscal Policy in a Stock-Flow Consistent (SFC) Model*” by Wynne Godley and Marc Lavoie (my copy is in this book (non-affiliate link); a free version may be available elsewhere). To be clear, their statements are less extreme than my assertion above; and they actually use equations in their exposition.
I will also have to throw in a caveat: I am assuming the
absence of hyperinflation. I do not think hyperinflation is possible under the
economic conditions I am assuming, but once again that is beyond the scope of
this article.

This framework also gives one explanation as to why debt/GDP
ratios have shifted higher in the developed economies. We now have an
increasingly important sector within the economy that does not appear to obey
stock-flow norms: foreign central banks. These banks have been accumulating
government bonds relentlessly as they are targeting the support of mercantilist
trade policies. However, stock-flow norms are taking their revenge: even these
policymakers have figured out that there are limits to their accumulation of
reserve assets. Therefore, there are still limits as to how far the debt ratios
will expand.

Even if SFC models are not the “true” model of the economy,
the implication is that if a model generates these unbounded fiscal ratio
trajectories, sectors cannot have behaviour consistent with stock-flow norms.
We then have to ask – is this behaviour plausible?

I will now turn to some other methodologies that I have
seen. My list is fairly general, but probably non-exhaustive.

The simplest analyses are too crude to really be viewed as
models. They are accounting simulations based on values of the “primary deficit”
(deficit ex-interest spending), interest rates, GDP growth and the existing
stock of debt. And they invariably find that economies are just on the verge of
“explosive” debt behaviour.

Their problem is that the distinction between “primary
deficit” and the “deficit” is dubious: interest spending is a form of welfare
payment to bond holders, and thus has a “multiplier effect” on the economy. It
is therefore impossible for interest payments to explode upward without any
impact on other economic variables, importantly GDP growth. The explosive
behaviour within these models is essentially based on the assumption that all
bond holders resemble foreign central banks during the past years: they will
accumulate nominal holdings of bonds without adjusting their spending or saving
behaviour.

Despite the obvious problems with these models, they have
been heavily used by policymakers (in the euro crisis, for example).

The next class of models are the modern Dynamic Stochastic
General Equilibrium (DSGE) models. These models represent the state of the art
in academic and central bank theoretical macroeconomics. I have considerable
reservations about them in this context, but it would take a fair amount of
mathematical blah-blah to get there. I will thus give some simple verbal
critiques of their use in the discussion of what “sustainable” debt
trajectories are.

One small problem one hits is that the models used in
practice tend to be “log-linearisations”. In plain English, they use a linear
model to model the small deviations around a reference trajectory, which is generated
by a complex non-linear model. However, the dynamics of the “debt” variable in the linear
model is not the model for the actual debt level; the linear system can be
unstable around the reference trajectory, but the debt trajectory could converge
towards another “sustainable” trajectory. You have to stick with the ultra-complex
nonlinear specifications of the model to see how fiscal ratios actually evolve.

A second problem is that the simplest DSGE models treat
fiscal policy as exogenous: spending and tax rates are determined outside the
model. In the worst case, tax revenues are independent of the level of income,
and “welfare spending” does not drop as the economy moves towards full
employment. This broken description of fiscal policy has been acceptable to
modellers, as they have been focussed almost completely on monetary policy. To
properly analyse fiscal policy, a model would have to have the tax and revenue
functions correspond to real-world behaviour. (I know that there have been
steps in this direction; but I do not know the literature enough to state that
the representation of fiscal policy has been completely fixed. It is fairly
difficult to change the mathematical framework to handle fiscal policy
properly, as far as I can tell.)

Even if the modeller has a reasonable description of fiscal
policy (which in principle can be done), there are still questions as to what “unsustainable”
means in the context of DSGE models. These models use the Inter-temporal
Governmental Budget Constraint, which is a combination of (a) an accounting
identity for finite time and (b)

**assumed**behaviour as time goes to infinity. (An astute reader could guess that I have some quibbles with the behaviour at infinity. Once again, I will have to discuss this elsewhere.)
If we take as given the inter-temporal budget constraint, we
see that it in fact is a mathematical constraint. It has to hold if there is going
to be a solution to agent’s optimising decision at time

*t*. If there is a solution to the optimising problem agent’s face within the model, the debt trajectory has to be well defined for all time in the future, which means that the debt trajectory has to be “sustainable”. In other words, as long as there is an economy, the debt trajectory is sustainable.*If the government somehow decides to violate the inter-temporal governmental budget constraint, the whole economy disappears in a flash of mathematical contradiction.*I would humbly suggest that this is not an empirically observed phenomenon, other than the sudden disappearance of Norway during the budget speech in 2012.
Somewhat more seriously, the models do not appear to offer a
guideline to determine whether debt trajectories are unsustainable. Such a
determination would require access to unobservable deep preference parameters.
Given the absence of debt defaults in relevant countries, I see no obvious way
to calibrate such models to determine these parameters. This is problematic as
the calibrations are typically done for the linearised models; there is an
infinite number of nonlinear models that can give rise to the same
linearisation. Therefore, we cannot distinguish which is the correct nonlinear
model within this family of models. Moreover, the calibrated models will have
to show behaviour which violates stock-flow norms, and so it remains to be seen
how well they approximate actual economic dynamics.

As such, it is
difficult to use DSGE models to declare a debt trajectory as being “sustainable”,
even if we are willing to accept DSGE models' mathematical treatment of time at
infinity.

(c) Brian Romanchuk 2013

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