<p>$$\frac {\Delta d_{t+1}}{d_t} = \frac {\frac {D_{t+1}}{Y_{t+1}}-\frac {D_{t}}{Y_{t}}}{\frac  {D_{t}}{Y_{t}}} = \frac {D_{t+1}Y_{t}-D_{t}Y_{t+1}}{D_{t}Y_{t+1}}$$</p>

<p>$$=\frac {D_{t+1}Y_{t}}{{D_{t}Y_{t+1}}}-1 = \frac{Y_{t}}{Y_{t+1}}\cdot \left[{\frac {D_{t+1}}{D_{t}}   -\frac {Y_{t+1}}{{Y_{t}}}}\right]$$</p>

<p>$$=\frac{Y_{t}}{Y_{t+1}}\cdot \left[{\frac {\Delta D_{t+1}}{D_{t}}   -\frac {\Delta Y_{t+1}}{{Y_{t}}}}\right]$$</p>

<p>So the "rule of thumb" deviates from the above exact expression by (multiplicatively) $Y_{t}/Y_{t+1}$. </p>

<p>Now, we are talking about the growth <em>rate</em> of the Debt/GDP <em>ratio</em>. Say the ratio was $120\text{%}$ and it went to what? in one year? Say it went to $D_{t+1} /Y_{t+1} =  130\text{%}$. Then its exact growth rate is $130/120 -1 = 8.33\text{%}$ Assume also that GDP grew at 5% yearly rate, so $Y_{t}/Y_{t+1} = 0.95238$.  </p>

<p>In this numerical example, the "rule of thumb will give as Debt/GDP ratio growth rate $8.33\text{%} \times 0.95238 = 7.933 \text{%}$, or less than half percentage point underestimation. This is considered a small inaccuracy in the real world, <em>when one discusses broadly these magnitudes</em>. And in general the above numerical example exaggerates, in most cases the growth rate of the GDP and of the Debt/GDP ratio are smaller than above, so the inaccuracy will be even smaller. Hence, the "rule of thumb" becomes acceptable in general also in theoretical models. </p>

CC BY-SA 3.0
2018-02-15T13:55:43.387Z
Thanks! It's kind of weird that they (+ another undergraduate macro textbook I was poring over) use a straight-up equality sign instead of an 'approximately equal' sign.
Jayden Watson
CC BY-SA 3.0
2018-02-15T14:06:53.090Z
Also, do you know if there is a mathematical name for this approximation? (e.g. was this approximation named after a mathematician or does it have a name like, idk, the 'ratio approximation' etc.)
Jayden Watson
CC BY-SA 3.0
2018-02-15T14:20:34.773Z
Discrete approximation of a continuous relation
Alecos Papadopoulos
CC BY-SA 3.0
2018-02-14T18:07:26.070Z
2018-02-15T10:07:11.553Z
2018-02-15T10:07:11.553Z
Alecos Papadopoulos
<p><a href="https://i.sstatic.net/ZnIVu.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ZnIVu.png" alt="enter image description here"></a></p>

<p>Above is an image of what I’m referring to. The top ‘rule of thumb’ is derived using calculus, and hence is only valid for infinitesimal changes in t. However, this equation is considered over the period of a year, which is a large unit of time. How can this equation be justified?</p>

<p>I've put a derivation of the 'rule of thumb' below (in case you weren't sure how it was derived). I've derived the product case, but it can be used to show the quotient case, which is what I put at the very bottom. Please excuse my naff handwriting!!</p>

<p><a href="https://i.sstatic.net/E2Na1.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/E2Na1.png" alt="enter image description here"></a></p>

<macroeconomics><mathematical-economics><fiscal-policy>
Debt accumulation equation derivation: valid given that it rests on calculus and the time period is a year? (nowhere near infinitesimal?)
Jayden Watson
# Debt accumulation equation derivation: valid given that it rests on calculus and the time period is a year? (nowhere near infinitesimal?)

(Asked by: username_0 on 2018-02-14)

[![enter image description here](https://i.sstatic.net/ZnIVu.png)](https://i.sstatic.net/ZnIVu.png)

Above is an image of what I’m referring to. The top ‘rule of thumb’ is derived using calculus, and hence is only valid for infinitesimal changes in t. However, this equation is considered over the period of a year, which is a large unit of time. How can this equation be justified?

I've put a derivation of the 'rule of thumb' below (in case you weren't sure how it was derived). I've derived the product case, but it can be used to show the quotient case, which is what I put at the very bottom. Please excuse my naff handwriting!!

[![enter image description here](https://i.sstatic.net/E2Na1.png)](https://i.sstatic.net/E2Na1.png)

**Tags:** macroeconomics, mathematical-economics, fiscal-policy

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## 1 answers

### Answer by username_1 on 2018-02-14. Score: 2 | Accepted answer

$$\frac {\Delta d\_{t+1}}{d\_t} = \frac {\frac {D\_{t+1}}{Y\_{t+1}}-\frac {D\_{t}}{Y\_{t}}}{\frac {D\_{t}}{Y\_{t}}} = \frac {D\_{t+1}Y\_{t}-D\_{t}Y\_{t+1}}{D\_{t}Y\_{t+1}}$$

$$=\frac {D\_{t+1}Y\_{t}}{{D\_{t}Y\_{t+1}}}-1 = \frac{Y\_{t}}{Y\_{t+1}}\cdot \left[{\frac {D\_{t+1}}{D\_{t}} -\frac {Y\_{t+1}}{{Y\_{t}}}}\right]$$

$$=\frac{Y\_{t}}{Y\_{t+1}}\cdot \left[{\frac {\Delta D\_{t+1}}{D\_{t}} -\frac {\Delta Y\_{t+1}}{{Y\_{t}}}}\right]$$

So the "rule of thumb" deviates from the above exact expression by (multiplicatively) $Y\_{t}/Y\_{t+1}$.

Now, we are talking about the growth *rate* of the Debt/GDP *ratio*. Say the ratio was $120\text{%}$ and it went to what? in one year? Say it went to $D\_{t+1} /Y\_{t+1} = 130\text{%}$. Then its exact growth rate is $130/120 -1 = 8.33\text{%}$ Assume also that GDP grew at 5% yearly rate, so $Y\_{t}/Y\_{t+1} = 0.95238$.

In this numerical example, the "rule of thumb will give as Debt/GDP ratio growth rate $8.33\text{%} \times 0.95238 = 7.933 \text{%}$, or less than half percentage point underestimation. This is considered a small inaccuracy in the real world, *when one discusses broadly these magnitudes*. And in general the above numerical example exaggerates, in most cases the growth rate of the GDP and of the Debt/GDP ratio are smaller than above, so the inaccuracy will be even smaller. Hence, the "rule of thumb" becomes acceptable in general also in theoretical models.

> **Comments:**
> **username_0**: Thanks! It's kind of weird that they (+ another undergraduate macro textbook I was poring over) use a straight-up equality sign instead of an 'approximately equal' sign.  
> **username_0**: Also, do you know if there is a mathematical name for this approximation? (e.g. was this approximation named after a mathematician or does it have a name like, idk, the 'ratio approximation' etc.)  
> **username_1**: Discrete approximation of a continuous relation  

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