# Real debt burden

Many economists believe real interest payments-to-GDP is a superior measure of debt service burden than nominal interest payments-to-GDP. My question is why we divide real interest payments by nominal GDP . To recap the real interest burden on the LHS is typically defined as (see Furman and Summers, 2020):

$$(r_{t }* D_{t-1})/〖GDP〗_t=(i_t-π_t )*D_{t-1}/〖GDP〗_t$$

Where $$r_{t }$$ is the real interest rate in period t, $$D_{t-1}$$ is the gross national debt in the prior period and $$〖GDP〗_t$$ is the nominal gross domestic product, $$i_t$$ is the nominal interest rate, $$π_t$$ is the inflation rate.

To recap (as I understand) real debt burden (ignoring GDP for the moment) is superior to the nominal as it takes into account not only that the value of the interest payment is lower due to inflation, but so is the stock of debt. Take two economies, both with debt levels of 100 and both with interest rates of 5%, and both have 0 output growth. The first payment is due next year (t) on stock of debt in t-1. The total debt burden in both economies would at t be 105, just before the interest payment is made. However, the real value of the total burden in economy 2 would be 105/1.05 =100 so the real value of total payments due hasn’t changed, unlike in 1. It’s as if there were no interest payment due or it would be the same as economy 1 if economy 1 had 0 interest payments (and continued to have 0 inflation). So that:

$$r_{t }* D_{t-1}=(i_t-π_t )*D_{t-1}$$

Is indeed a sensible way to measure the real debt burden compared to just the nominal value of payments. So far so good. Bringing GDP into the fold, in order to measure real interest payments relative to output, as per first equation people divide by nominal GDP in period t. But I’m struggling to see the intuition behind dividing by the nominal GDP in t. Debt servicing is only 0 in real terms wrt prices in t-1. Therefore does it not make more sense to divide by real GDPt.
For instance, suppose instead we had inflation of only 4%. In economy 2, the total debt burden in initial period money would be 105/1.04 ≈ 101, which is an increase of 1. Or equivalently using equation 2: (5-4)*100 = 1. So the debt burden has increased by 1 unit in t-1 money. Using nominal GDP as per 1 we have real debt service of 1/104. But this seems misleading as the increase in debt burden is 1% the value of total output . That is, in real terms, it is 1/(104/1.04). What am I missing here? The only value I can see in using nominal is that there doesn’t need to be a base year, but it could be very misleading.

• I always did nominal interest payments divided by nominal GDP. Note: you would get the same result if you then divide the top and the bottom by the GDP deflator, thus getting real interest payments divided by real GDP. Commented Oct 18, 2023 at 15:19