## Monday, February 13, 2017

### Real and Nominal Debt-to-GDP Ratios

A question from the econ section of Stack Exchange:
Are debt/GDP ratios calculated with real or nominal GDP as the denominator?

As the title suggests, I would like to know whether the numbers are generally calculated with real or nominal GDP. Besides that I would also like to know whether it matters and how significant of a difference this could make.

There is an existing response that includes these remarks:
If you use real gdp as denominator and nominal debt as numerator, you end up with a number which is less clear to interpret. It is more relevant to have either both variables in nominal terms or both in real terms.

I agree. You don't want to mix nominal and real. You want either a ratio of reals or a ratio of nominals. My response:
In my experience the ratio of nominals is far more common than the ratio of reals. Nominals are useful when the topic is current spending. Reals are useful when the topic is growth apart from changes in the value of money.

GDP is the sum of one year's final spending. Given annual data for GDP and prices, there is only one year's price level embedded in any one year's GDP number.

Debt is the accumulation of borrowing (less repayment) over many years. So the price levels of many years come into play even when only one year's nominal debt is converted to a real (inflation-adjusted) number.

Because many years' price levels are embedded in one year's debt, the calculation of real debt is not the same as the calculation of real GDP. Because the calculations differ, the ratio of reals is not identical to the ratio of nominals.

People often assume that the calculation used for real GDP can also be used for real debt. Using the same calculation gives a ratio of reals that is identical to the ratio of nominals. When the topic is current value ("how much is it now") no harm is done by this mistake. But when the topic is growth ("how much bigger is it now") the wrong calculation gives incorrect information ("The public debt remained fairly constant from the late 1940s through 1981") and may lead to serious or fatal errors in policy.

Jerry said...

I guess you could be interested in one of a couple different things:

1. If you devoted the whole GDP toward paying off the debt at any given year, how long would it take?
this is what people usually draw a picture of -- debt / gdp, or [ (the integral of nominal debt) divided by each year's nominal gdp]

2. How much did each year add to the debt in that year's current dollars?
that's the picture you draw with the sort of [integral of (real deficit / real gdp)]

I guess #2 would be useful if you're asking something like, "how bad was the policy at any given year" or "which president/congress ran up the most debt", which I guess is what many people are asking when they show #1. (an error)

But, #1 could be useful too, right?
e.g., an "Arthurian" question to ask is something like: at each year, how much portion of the federal budget goes to paying interest on debt? The budget is probably roughly proportional to the GDP, and the interest is probably roughly proportional to the total-debt-in-current-year-dollars, so I guess this graph would look something like #1?
I don't know, but, my point is maybe sometimes #1 is not an error, depending on what you're trying to find out.

The Arthurian said...

Hey Jerry. Sure, #1 is useful, debt to GDP, nominal to nominal. As you say, "If you devoted the whole GDP toward paying off the debt at any given year, how long would it take?"

So if debt-to-GDP was 3.0 it would take 3 years to pay it off. But any such plan would require that all borrowing cease immediately, or the additions to debt would lengthen the payback period. Case in point: the debt-to-GDP ratio was 3.0 in 2003. Three years later, in 2006, the debt-to-GDP ratio was 3.3.

Rather than asking how long it would take to pay off the debt, it might be better to ask "How far are we behind?" In 2003 we were three years behind. In 2006 we were 3.3 years behind.

As far as #2 goes, consider real GDP. Real GDP is useful for comparing the sizes of output in different years. Similarly, to my mind, "real debt" is useful for comparing the sizes of output purchased on credit in different years. As debt (or, really, the use of credit) is a way to increase the size of GDP over and above what income alone allows, we may find ourselves talking about the growth that results from debt.

As we know from experience with GDP, when evaluating growth (as opposed to the increase in prices attributable to changes in the value of money over time) it is useful to work with real or inflation-adjusted data.

When considering only "current" values (as I say) or only the values of "any given year" (as you say), the ratio of nominals is the appropriate measure.

When considering the changes that occur between one year and another, the ratio of reals is the appropriate measure -- remembering that the calculation of real debt is not the same as the calculation of "real GDP".

The nominal-to-nominal ratio is correct for some applications, and the real-to-real ratio is correct for some applications, but the calculation that gives "real GDP" is never the correct calculation to use when figuring "real debt".