# Evolution of GDP to debt ratio

Is it possible for an entity (say a state, e.g. USA) to keep a constant GDP/debt ratio with a monotonically increasing debt.

I should probably mention that I am not an economist (physics and c.s background). In order to answer my question above, I want to know what are the governing equations for the gdp/debt ratio evolution (in time). From the simple analysis I've encountered, the evolution is treated relative to a single entity, where the controlling factor is the relation between the interest rate $$r$$ (treated as a scalar) and the GDP growth factor $$g$$. If $$g>r$$, then we can increase our debt while keeping the ratio constant.

My problem is that this seems counter intuitive, especially if international relations are taken into account (a low ratio in one country has to mean a high ratio in another). Even if I allow the interest to somehow grow with the debt, the simple analysis does not assume any restriction on the GDP growth (which seems to me, has to be affected by the other factors). What models should I look into in this regard. Additionally, as a mere amature in these matters, I wonder why are all the magnitudes scalar and all the equations algebraic (where are the differential equations? show me some dynamics).

• "a low ratio in one country has to mean a high ratio in another" How do you figure that? GDP growth is not zero sum. And while there is international competition for interest rates, there's nothing saying that all countries couldn't have the same risk profile and interest rate. – Brythan Oct 29 '18 at 23:25
• Suppose there are only two countries, then one has to be able to provide the loan. If the loaning country also takes a debt, then this is equivalent to the first country taking a smaller debt. – Ariel Oct 30 '18 at 5:02
• Welcome to the forum! Loans don't have to be provided by another country. Many government bonds are held domestically, so the government owes money to its own people (who produce GDP). It's possible to have low debt-to-GDP ratio in all countries or high debt-to-GDP ratio in all countries. There is no need for this to balance across countries. (PS: As a physicist you'll be shocked how useless these models are compared to physics models - be warned!) – M3RS Oct 30 '18 at 10:24

I believe you're trying to approach this to theoretically, when this is an empirical question

The debt to GDP ratio is the total government debt divided by the GDP These are, of course, real numbers

The government controls the budget, sets a tax policy and sets an annual deficit goal, that should, in turn, control the total debt. Interest rates are a part of the budget, and aren't included in the debt to GDP ratio

Regarding you remark on exchange rates: A debt can be raised in a foreign currency, when raised abroad, or to back up any import /export activity, etc. In which case the effect is direct

Another exchange rate impact can occur when the debt isn't managed well by the state, foreign investors might pull out and raise the exchange rates. This will also mean that the country will either raise its interest, to reflect the risk, or it might suffer inflation

The US case is unique in a sense that it used the fact that it is the world's largest economy, the best alternative for risk-wise, and the government behind the dollar

Yes, all these numbers are real scalar series. However, they are sampled at the end of accounting time periods, they are difference equations, not differential equations. Some people like trying to force economic data into differential equations, and end up ignoring the theoretical issues of such a conversion. (My training is in electrical engineering, and there is a very well developed theory for conversions between discrete time and continuous time models.)

The dynamics for government debt in a simplified model are simple. GDP is more complex, as it is the aggregate of underlying variables.

Based on my experience, the easiest textbook for a non-economist to grasp economics is “Monetary Economics” by Wynne Godley and Marc Lavoie. It is an introduction to stock-flow consistent (SFC) models. It illustrates how models of varying complexity build up.

It should be noted that SFC models are from outside the economic mainstream. The mainstream prefers models in which outcomes are the result of optimisation problems. However, it is difficult to see how things like the debt-GDP ratio evolve using those models. The Godley-Lavoie text does explain differences in views.

To return to your question, it is not that difficult to create SFC models where all countries enter a steady state where they grow at the same rate in real terms (growth excluding inflation). Nominal growth rates (in local currency terms) could be different as a result of different inflation rates. Debt levels would grow at the same pace as GDP.

Of course, economies are not always going to be in steady state. That said, during an expansion, it is fairly typical for growth rates of variables to be relatively steady, and so debt/GDP ratios move at a steady pace. We could only predict all the wiggles in the ratio if we could make perfect economic forecasts, which is unlikely. (That would be an entirely different question.)

Finally, discussions of debt ratio evolution using fixed $$r$$ and $$g$$ are seen as defective from the SFC modelling tradition. If one looks at those models, the behaviour of the private sector will drive the steady state debt ratios. (See Godley and Lavoie, Chapter 3.)