Percentage change of a quotient

Question

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:

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

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

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)

Answer 1

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*}