# Simple taylor approximation in macroeconomic model

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 $$\log s_c \approx s_c$$?

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*}