Figure 8. The French DEBT/GDP ratio projected forward. Here the growth in the GDP was assumed 1.5% -- the 2011 rate -- and the starting DEBT/GDP ratio is assumed 97.2% -- the rate at the end of 2011. These results are for the recursion model: "The Recursive Problem -- II" presented subsequently.  The line is at the paradoxical 7% value -- where rates are suggested to lead to "unsustaniable" run-away debt condition.

(2/7/2012)

This set of examples applies to the question of national debt.  In this case we are interested in looking at the DEBT/GDP ratio for selected countries in the US, Germany, France and Italy.  These examples use the following simplified recursive equations, where D_T is the DEBT at year T, D_T-1 is the DEBT the year earlier (T-1 ); G_T is the GDP at year T, and G_T-1 is the GDP one year earlier.  The basic equations are:

                                                G_T = G_T-1 [1 + DG],                                       (2)

And

                                                D_T = D_T-1 [1 + DD_T].                                     (3)

with 

                                                [DEBT/GDP] (%) = 100 [G_T / D_T ] .                (4)

The quantity DG  is the average growth in the GDP (fraction), which we assume to be dependent on the country in question (this is indicated on the figure); The quantity DD_T is the weighted sovereign bond interest paid (the yield to investors at year T as a fraction).  The output in the graph is the DEBT/GDP ratio (see color scale indicated in the color-bar).  We also indicate the starting DEBT/GDP ratio on the figure. 

It is clear that the DEBT/GDP ratios can grow fairly rapidly over a 10-year period, provided the “weighted sovereign yields” become large relative to the growth rate of the GDP.  We were handicapped in that we fixed this growth rate in GDP at the 2011 values.  Even so, the plots are instructive.

Consider the plots for Germany and France (Figures 6 and 8).  The current German ratio of 81.7% tends to be quite sustainable for the next few years when one considers the relatively low bond yields characterized by this economy (see, for example, Figure 3).   The same can be said for France, where typically the rates are below 4% for the past two years.

Discussion

The assumptions for generating the figures are as follows:

1. The Debt is assumed to accrue from a build-up caused solely by paying interest on this Debt (a lower bound).  This corresponds to a non-zero annual country DEFICIT that is at least caused by the pay-out of this interest.
2. The DEFICIT represents an accumulation due to paying this interest, as indicated above.
3. The Debt indicated represents the addition of the annual DEFICIT amount due to interest payments only.
4. We start with the DEBT/GDP (%) fixed at its level on 1/2012.
5. We allow the GDP to change at a fixed amount set by its value on 1/2012 – across “Future Years” indicated (Year 0 = 2012):

                                       DG = DG (%) / 100.                                      (5)

In general it is clear that for “positive” annual DEFICIT amounts the country’s Debt increases.  This results in higher interest paid on next year’s Debt.  Such a feed-back eventually results in a “run-away” condition – the recursive nature of the phenomena.  In the figures it is clear that using an “idealized” model the DEBT/GDP ratio will increase substantially during the next 10 years, if countries continue to run DEFICITS.