Htykuut

This was line in my Econ notes that was "by definition". If anyone can clarify this I would really appreciate.

We have a relationship:

$APL = frac{Y}{L}$

Now, define $Delta x$ as "percentage change in variable x". Then by definition:

$Delta APL = Delta Y - Delta L$

I am not sure how this is claimed "by definition". When I think about it, if I define $alpha$ and $beta$ as % changes, I get:

$Delta APL = frac{frac{Y times (1+alpha)}{Ltimes (1+beta)} -frac{Y}{L}}{frac{Y}{L}}$

$= frac{1+alpha}{1+beta} -1$

Which is not equal to $alpha - beta$ as claimed by my notes.

Where am I going wrong here? Is there some limit stuff I am forgetting?

(If anyone is curious the particular relationship is about the average product of labor. But this is not important. The claim being made is general)

What they do is the following:

• They use the linear Taylor-approximation for the function $f(x,y) = frac{x}{y}$:

$$f(x+Delta x,y+Delta y)- f(x,y) stackrel{Taylor}{approx} frac{partial f}{partial x}Delta x + frac{partial f}{partial y}Delta y$$

So, you get
$$f(x,y) = frac{x}{y} Rightarrow frac{partial f}{partial x} =color{blue}{frac{1}{y}} , : frac{partial f}{partial y} = color{blue}{-frac{x}{y^2}}$$

For the percentage increase they divide by $f$:
begin{eqnarray*} frac{Delta f}{f}
& = & frac{color{blue}{frac{1}{y}}}{frac{x}{y}}Delta x + frac{color{blue}{-frac{x}{y^2}}}{frac{x}{y}}Delta y \
& = & frac{Delta x}{x} - frac{Delta y}{y}
end{eqnarray*}