<p>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.</p>
<p>The statement in the textbook is based on a on a back of the envelope calculations rather than some elaborate model.</p>
<p>First, what matters is not absolute size of the debt but debt-to-GDP ratio <span class="math-container">$Z$</span> which is given by <span class="math-container">$\frac{D_t}{P_tY_t}$</span> where <span class="math-container">$D_t$</span> is the nominal value of the debt, <span class="math-container">$P$</span> is price level (change in which gives you inflation) and <span class="math-container">$Y_t$</span> is real GDP, so numerator <span class="math-container">$P_tY_t$</span> is the nominal GDP. Thus we have:</p>
<p><span class="math-container">$$Z_t= \frac{D_t}{P_tY_t}$$</span></p>
<p>Now if you take logs of both sides to linearize this expression (where small letters indicate natural logs):</p>
<p><span class="math-container">$$z_t = d_t - p_t - y_t$$</span></p>
<p>and then take time derivates of both sides you will arrive at:</p>
<p><span class="math-container">$$\frac{\dot{z}}{z} =   \frac{\dot{d}}{d} -  \frac{\dot{p}}{p} -  \frac{\dot{y}}{y}$$</span></p>
<p><span class="math-container">$\frac{\dot{x}}{x}$</span> is just continuous extension of growth rate.</p>
<p>Consequently,  <span class="math-container">$\frac{\dot{d}}{d}$</span> is the growth rate of debt (given by yearly government budget deficit), <span class="math-container">$\frac{\dot{p}}{p} $</span> is inflation rate, and <span class="math-container">$\frac{\dot{y}}{y}$</span> is a growth rate.</p>
<p>Now, if we assume the parameters as in the textbook <span class="math-container">$\frac{\dot{d}}{d}=3\%$</span>,  <span class="math-container">$\frac{\dot{p}}{p} = 2\% $</span> and <span class="math-container">$\frac{\dot{y}}{y}=2.5\%$</span>   we get:
<span class="math-container">$$\frac{\dot{z}}{z} =   0.03 -  0.02 - 0.025=-0.015 $$</span></p>
<p>Hence, we get that the debt to GDP ratio will, assuming these parameters, decrease at about <span class="math-container">$1.5\%$</span>. 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.</p>
<p>Also note the numbers (parameter values) <span class="math-container">$\frac{\dot{d}}{d}=3\%$</span>,  <span class="math-container">$\frac{\dot{p}}{p} = 2\% $</span> and <span class="math-container">$\frac{\dot{y}}{y}=2.5\%$</span>  are all assumed, they are not calculated within the problem you discuss.</p>

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2021-08-13T21:00:35.050Z
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2021-08-13T21:06:30.480Z
1muflon1
<p>I want to ask a question about how economic growth, inflation, interest rates is linked to fiscal deficit.</p>
<p>I was reading a book on introductory economics and the following passage came up:</p>
<blockquote>
<p>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.</p>
</blockquote>
<p>What I cannot understand is if there is a mathematical link or intuition that justifies this statement.</p>
<p>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 <span class="math-container">$I$</span>, which in turn stimulates growth in GDP.</p>
<p>However, I do not understand where the figure of 3% national debt as a percentage of GDP is obtained.</p>
<p><strong>Where is the 3% figure for national debt as a percentage of GDP obtained?</strong></p>
<p>The extract is also provided in image form below:</p>
<p><a href="https://i.sstatic.net/bccgv.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/bccgv.png" alt="Anderton, 2015 p. 196 Theme 2 Unit 37" /></a></p>

<macroeconomics><inflation><government-debt><fiscal-policy>
The link between economic growth, inflation, interest rates and fiscal deficit
vik1245
# The link between economic growth, inflation, interest rates and fiscal deficit

(Asked by: username_0 on 2021-08-13)

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:

[![Anderton, 2015 p. 196 Theme 2 Unit 37](https://i.sstatic.net/bccgv.png)](https://i.sstatic.net/bccgv.png)

**Tags:** macroeconomics, inflation, government-debt, fiscal-policy

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## 1 answers

### Answer by username_1 on 2021-08-13. Score: 2

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.


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