The link between economic growth, inflation, interest rates and fiscal deficit

Question

I want to ask a question about how economic growth, inflation, interest rates is linked to fiscal deficit.

I was reading a book on introductory economics and the following passage came up:

If a country is growing at around 2.5% per year, and there is 2% inflation and low interest rates, then a fiscal deficit of around 3% per year will probably maintain a stable level of national debt (the sum total of all outstanding government borrowing) as a percentage of GDP.

What I cannot understand is if there is a mathematical link or intuition that justifies this statement.

I understand that the 2.5% p/a refers to the economic growth, which has direct links to GDP. Inflation of 2% is a familiar target in macroeconomic objects to prevent excessive inflation but also avoid deflation. Low interest rates also promote Investment $I$, which in turn stimulates growth in GDP.

However, I do not understand where the figure of 3% national debt as a percentage of GDP is obtained.

Where is the 3% figure for national debt as a percentage of GDP obtained?

The extract is also provided in image form below:

Answer 1

First, note, that 3% is not debt to GDP but growth rate of debt (or in another words deficit). It’s not debt to GDP.

The statement in the textbook is based on a on a back of the envelope calculations rather than some elaborate model.

First, what matters is not absolute size of the debt but debt-to-GDP ratio $Z$ which is given by $\frac{D_t}{P_tY_t}$ where $D_t$ is the nominal value of the debt, $P$ is price level (change in which gives you inflation) and $Y_t$ is real GDP, so numerator $P_tY_t$ is the nominal GDP. Thus we have:

$$Z_t= \frac{D_t}{P_tY_t}$$

Now if you take logs of both sides to linearize this expression (where small letters indicate natural logs):

$$z_t = d_t - p_t - y_t$$

and then take time derivates of both sides you will arrive at:

$$\frac{\dot{z}}{z} =   \frac{\dot{d}}{d} -  \frac{\dot{p}}{p} -  \frac{\dot{y}}{y}$$

$\frac{\dot{x}}{x}$ is just continuous extension of growth rate.

Consequently, $\frac{\dot{d}}{d}$ is the growth rate of debt (given by yearly government budget deficit), $\frac{\dot{p}}{p} $ is inflation rate, and $\frac{\dot{y}}{y}$ is a growth rate.

Now, if we assume the parameters as in the textbook $\frac{\dot{d}}{d}=3\%$,  $\frac{\dot{p}}{p} = 2\% $ and $\frac{\dot{y}}{y}=2.5\%$   we get: $$\frac{\dot{z}}{z} =   0.03 -  0.02 - 0.025=-0.015 $$

Hence, we get that the debt to GDP ratio will, assuming these parameters, decrease at about $1.5\%$. Since, it is debt-to-GDP what really matters not absolute value of debt (e.g. whether \$ 1000 debt is large depends on your income if your income  if you earn \$100,000 per year then debt is negligible if you earn \$100 per year its disastrous), as the debt-to-GDP ratio declines debt is less problematic for a country regardless of how big the total debt is.

Also note the numbers (parameter values) $\frac{\dot{d}}{d}=3\%$,  $\frac{\dot{p}}{p} = 2\% $ and $\frac{\dot{y}}{y}=2.5\%$  are all assumed, they are not calculated within the problem you discuss.