Does inflation affect debt-to-gdp ratio?

Question

So consider this indicator https://fred.stlouisfed.org/series/GFDEGDQ188S I am wondering since Debt is nominal and GDP is also nominal, is it possible for inflation to have an effect on this ratio?

I'll make the story really short: $Debt\; stock = past\; debt + current\; nominal\;debt$ and $nominal\; GDP = P*Q$

Now, as the level of prices goes up, nominal gdp goes up, but what happens to the debt stock? The majority of debt is a past stock, and so the level of prices there wouldn't be an effect as strong as for the GDP. Is the change in nominal interest rate exactly mirroring the change in prices?

Why don't we directly use real quantities?

Answer 1

Does inflation affect debt-to-gdp ratio?

Yes, it does affect it.

Debt-to-GDP ratio is defined as:

$$DtGDP=\frac{D}{PY}$$

Where $D$ is nominal value of all debt (past and present - debt is not recorded at real value) and $PY$ is nominal GDP (e.g. real GDP $Y$ times price level $P$).

If country does not issue new debt $D$ then trivially inflation which by definition is positive change in $P$ would reduce DtGDP.

Of course, if country decides to issue new debt different things would happen. However, inflation only indirectly affects government borrowing. If inflation is high government might go for more debt to cover higher expenses, but government could equally decide to keep its current expenses same or even reduce them.

Hence inflation has only direct effect on reducing DtGDP. Indirectly, through changing government behavior it could make government borrow more so it could increase it, but then inflation is only one factor out of many that determines how much government borrows each year.

Is the change in nominal interest rate exactly mirroring the change in prices?

This is not relevant. Government uses fixed income securities (e.g. bonds) that do not pay more coupons when interest rate changes. Interest rate payments are also not part of D to begin with.

This being said nominal interest rate is given by: $i \approx r + \pi$ so indeed when inflation increases by 2% (ceteris paribus) nominal interest rate increases by approx 2% but that is only tangential to this issue since interest payments are not part of D, and government pays fixed interested on its old debt.

Why don't we directly use real quantities?

It does not matter the result would be literally the same. Real quantity is deflated quantity. Use P as a deflator for both debt and nominal output and you get:

$$rDtGDP = \frac{D/P}{PY/P} = \frac{D}{PY} \frac{P}{P} = \frac{D}{PY}1= \frac{D}{PY}$$

rDtGDP=DtGDP. Deflating variables here is pointless waste of time.

Answer 2

The nominal value of past debt is not affected by current inflation. Its real value decreases as inflation increases.

However, current debt interest rates are affected by inflation expectations. Even though inflation can devalue past debt, it can't devalue future debt because creditors will demand nominal interest rates higher than inflation.

If real interest rates are 2%, if you say inflation will be 5%, they will demand 7%. If you say inflation will be 10% they will demand 12%. If you catch on and start increasing inflation every year, creditors will also catch on and start writing debt whose interest rate increases every year. If you planned to devalue future debt using inflation, you would have to keep increasing inflation even faster than you did last time. Therefore it does make sense to speak of past debt nominally and future debt in real terms.