<p>At every time period, we have:
<span class="math-container">$$
y_t = c_t + i_t + g_t
$$</span>
Long run steady state gives:
<span class="math-container">$$
y = c + i + g
$$</span>
so, taking differences we have:
<span class="math-container">$$
\begin{align*}
&amp;y_t - y = c_t - c + i_t - i + g_t - g,\\
\iff &amp;\frac{y_t - y}{y} = \frac{c}{y}\frac{c_t - c}{c} + \frac{i}{y}\frac{i_t - i}{i}+ \frac{g}{y} \frac{g_t - g}{g}
\end{align*}
$$</span>
Set <span class="math-container">$s_c = \dfrac{c}{y}, s_i = \dfrac{i}{y}$</span> and <span class="math-container">$s_g = \dfrac{g}{y}$</span> to be the shares of consumption, investment and government expenditures.</p>
<p>This gives:
<span class="math-container">$$
\frac{y_t - y}{y} = s_c \frac{c_t - c}{c} + s_i \frac{i_t - i}{i} + s_g \frac{g_t - g}{g}
$$</span>
Now, we approximate the growth rates. For a variable <span class="math-container">$x_t$</span>, consider a Taylor expansion of <span class="math-container">$\ln(x_t)$</span> around the steady state <span class="math-container">$x$</span>
<span class="math-container">$$
\ln(x_t) \approx \ln(x) + \frac{(x_t - x)}{x}
$$</span>
This gives <span class="math-container">$\dfrac{x_t - x}{x} \approx \ln(x_t) - \ln(x) = \ln \dfrac{x_t}{x}$</span>. Substituting gives:
<span class="math-container">$$
\begin{align*}
&amp;\ln\frac{y_t}{y} \approx s_c \ln \frac{c_t}{c} + s_i \ln \frac{i_t}{i} + s_g \ln \frac{g_t}{g}.\\
\iff &amp;\hat y = s_c \hat c + s_i \hat i + s_g \hat g.
\end{align*}
$$</span></p>

CC BY-SA 4.0
2021-10-16T08:27:13.860Z
2021-10-16T08:27:13.860Z
tdm
<p>Define <span class="math-container">$\hat{y}=\log\frac{y_t}{y}$</span> which means percentage deviation from steady state.<br />
Then <span class="math-container">$\hat{y}=s_c \hat{c}+s_i \hat{i} + s_g \hat{g}$</span> where <span class="math-container">$s_c +s_i +s_g =1$</span>.<br />
I can't derive that result.<br />
Is that <span class="math-container">$\log s_c \approx s_c$</span>?</p>

<macroeconomics>
Simple taylor approximation in macroeconomic model
guest
# Simple taylor approximation in macroeconomic model

(Asked by: username_0 on 2021-10-16)

Define $\hat{y}=\log\frac{y\_t}{y}$ which means percentage deviation from steady state.  
Then $\hat{y}=s\_c \hat{c}+s\_i \hat{i} + s\_g \hat{g}$ where $s\_c +s\_i +s\_g =1$.  
I can't derive that result.  
Is that $\log s\_c \approx s\_c$?

**Tags:** macroeconomics


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## 1 answers

### Answer by username_1 on 2021-10-16. Score: 2 | Accepted answer

At every time period, we have:
$$
y\_t = c\_t + i\_t + g\_t
$$
Long run steady state gives:
$$
y = c + i + g
$$
so, taking differences we have:
$$
\begin{align\*}
&y\_t - y = c\_t - c + i\_t - i + g\_t - g,\\
\iff &\frac{y\_t - y}{y} = \frac{c}{y}\frac{c\_t - c}{c} + \frac{i}{y}\frac{i\_t - i}{i}+ \frac{g}{y} \frac{g\_t - g}{g}
\end{align\*}
$$
Set $s\_c = \dfrac{c}{y}, s\_i = \dfrac{i}{y}$ and $s\_g = \dfrac{g}{y}$ to be the shares of consumption, investment and government expenditures.

This gives:
$$
\frac{y\_t - y}{y} = s\_c \frac{c\_t - c}{c} + s\_i \frac{i\_t - i}{i} + s\_g \frac{g\_t - g}{g}
$$
Now, we approximate the growth rates. For a variable $x\_t$, consider a Taylor expansion of $\ln(x\_t)$ around the steady state $x$
$$
\ln(x\_t) \approx \ln(x) + \frac{(x\_t - x)}{x}
$$
This gives $\dfrac{x\_t - x}{x} \approx \ln(x\_t) - \ln(x) = \ln \dfrac{x\_t}{x}$. Substituting gives:
$$
\begin{align\*}
&\ln\frac{y\_t}{y} \approx s\_c \ln \frac{c\_t}{c} + s\_i \ln \frac{i\_t}{i} + s\_g \ln \frac{g\_t}{g}.\\
\iff &\hat y = s\_c \hat c + s\_i \hat i + s\_g \hat g.
\end{align\*}
$$


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