Migration wage differential, Borjas (2013) textbook Basic calculus question here! I started reading the textbook Borjas, G. J. (2014). Immigration economics. Harvard University Press and am confused already with the first result.
Background: He frames the migration decision in terms of wage differences between two countries $i \in \{0,1\}$ in each country (home is 0)
$$ 
log (w_i) = \mu_i + v_i 
$$
where $\mu_i$ is the average wage in home and potential destination and $v_i $ is an i.i.d normally distributed random variable with mean 0 and variance $\sigma^{2}$. If $i=1$ is destination country, then migrant will pay a cost $C$ to move there.
Calculation: Okay, the simple algebra is then to combine these elements into an index
$$ 
I = log \left(\frac{w_1}{w_0+C} \right) \approx (\mu_1 - \mu_0 - \pi) + (v_1 - v_0) 
$$
with $\pi = C/w_0$.
Steps: So I did some log manipulations (see below) and think I arrive at this if the $log\left(1+\frac{C}{w_0}\right) \approx log\left(\frac{C}{w_0}\right)$. Is this an approximation one does? If yes, do you background at to why? Or is the simplification different?
$$ 
log \left(\frac{w_1}{w_0+C} \right) = log (w_1) - log(w_0+C) + log(w_0) - log(w_0) = log (w_1)  - log(w_0) - log\left(1+\frac{C}{w_0}\right) = e^{log(\mu_1 + v_1)} - e^{log(\mu_0 + v_0)} - e^{log\left(1+\frac{C}{w_0}\right)} \approx (\mu_i - \mu_i - \pi) + (v_1 - v_0) 
$$
Yes, that is an approximation one does. You have stumbled upon the log-approximation, which is very commonly used in Economics.
It states that $log(1+x)≈x$, for very small $x$. Hence, typically $x$ is a fraction. It is also used in macro to approximate growth rates.
There are lots of resources out there that explain it, e.g. here.
The way I remember it is that $log(1) =0$ so $log(1+x)≈x$ for small $x$. This is mathematically totally wrong, but I find it to be a useful memory device personally.