<p>The first one is more correct in discrete time and in general the second one is a good continuous time approximation.</p>
<p>We can decompose nominal GDP as the product of real GDP (<span class="math-container">$Y$</span>) and the price level (<span class="math-container">$P$</span>):
<span class="math-container">$$
GDP_t = P_t Y_t
$$</span></p>
<h3>Discrete time</h3>
<p>For a variable <span class="math-container">$X_t$</span> let <span class="math-container">$\Delta X = \dfrac{X_{t+1} - X_t}{X_t}= \dfrac{X_{t+1}}{X_t} - 1$</span> be the growth rate of <span class="math-container">$X$</span>.</p>
<p>Then:
<span class="math-container">$$
\begin{align*}
\Delta GDP + 1 &amp;= \frac{GDP_{t+1}}{GDP_t}\\
&amp;= \frac{P_{t+1}Y_{t+1}}{P_t Y_t},\\
&amp;= \frac{P_{t+1}}{P_t} \frac{Y_{t+1}}{Y_t}\\
&amp;= \left(1+ \Delta P_t\right)\left(1+ \Delta Y\right)\\
\end{align*}
$$</span>
Therefore:
<span class="math-container">$$
1 + \Delta Y = \frac{1 + \Delta GDP}{1 + \Delta P}
$$</span>
Which is the first formula that you have.</p>
<h3>Continuous time approximation</h3>
<p>Now if you would work in continuous time then growth rate of <span class="math-container">$X$</span> can be approximated by:
<span class="math-container">$$
\frac{d \ln(X(t))}{d t} = \frac{1}{X(t)} \dfrac{d X(t)}{d t} \approx \frac{X(t+1) - X(t)}{X_t}
$$</span>
Then the rate of of GDP growth would be:
<span class="math-container">$$
\begin{align*}
\Delta GDP &amp;= \frac{d \ln(GDP(t))}{d t} = \frac{d \ln(P(t))}{dt } + \frac{d \ln(Y(t))}{d t},\\
&amp;= \Delta P + \Delta Y,\\
\end{align*}
$$</span>
As such we can use the approximation:
<span class="math-container">$$
\Delta Y = \Delta GDP - \Delta P
$$</span></p>

CC BY-SA 4.0
2022-03-12T09:34:45.267Z
2022-03-12T09:34:45.267Z
tdm
<p>I have nominal GDP growth (%) and inflation (CPI price level growth, %). Which is the correct way to find real GDP growth (%)?</p>
<p>=((1+[nominal GDP growth%])/(1+[inflation%]))-1</p>
<p>or simply:</p>
<p>=[nominal GDP growth%] - [inflation%]</p>
<p>or something else?</p>
<p>Thanks.</p>

<inflation><gdp>
Correct formula for real GDP growth?
cells interlinked
# Correct formula for real GDP growth?

(Asked by: username_0 on 2022-03-11)

I have nominal GDP growth (%) and inflation (CPI price level growth, %). Which is the correct way to find real GDP growth (%)?

=((1+[nominal GDP growth%])/(1+[inflation%]))-1

or simply:

=[nominal GDP growth%] - [inflation%]

or something else?

Thanks.

**Tags:** inflation, gdp

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## 1 answers

### Answer by username_1 on 2022-03-12. Score: 4

The first one is more correct in discrete time and in general the second one is a good continuous time approximation.

We can decompose nominal GDP as the product of real GDP ($Y$) and the price level ($P$):
$$
GDP\_t = P\_t Y\_t
$$

### Discrete time

For a variable $X\_t$ let $\Delta X = \dfrac{X\_{t+1} - X\_t}{X\_t}= \dfrac{X\_{t+1}}{X\_t} - 1$ be the growth rate of $X$.

Then:
$$
\begin{align\*}
\Delta GDP + 1 &= \frac{GDP\_{t+1}}{GDP\_t}\\
&= \frac{P\_{t+1}Y\_{t+1}}{P\_t Y\_t},\\
&= \frac{P\_{t+1}}{P\_t} \frac{Y\_{t+1}}{Y\_t}\\
&= \left(1+ \Delta P\_t\right)\left(1+ \Delta Y\right)\\
\end{align\*}
$$
Therefore:
$$
1 + \Delta Y = \frac{1 + \Delta GDP}{1 + \Delta P}
$$
Which is the first formula that you have.

### Continuous time approximation

Now if you would work in continuous time then growth rate of $X$ can be approximated by:
$$
\frac{d \ln(X(t))}{d t} = \frac{1}{X(t)} \dfrac{d X(t)}{d t} \approx \frac{X(t+1) - X(t)}{X\_t}
$$
Then the rate of of GDP growth would be:
$$
\begin{align\*}
\Delta GDP &= \frac{d \ln(GDP(t))}{d t} = \frac{d \ln(P(t))}{dt } + \frac{d \ln(Y(t))}{d t},\\
&= \Delta P + \Delta Y,\\
\end{align\*}
$$
As such we can use the approximation:
$$
\Delta Y = \Delta GDP - \Delta P
$$


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