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The "$x$-elasticity of $y$" is defined as $\epsilon = \dfrac{\partial y/y}{\partial x/x}$.

The $x$ goes down, and the $y$ goes up.

But we have two definitions of semi-elasticity:

$\epsilon_1 = \dfrac{\partial y/y}{\partial x}$

Which is the percentage change of $y$ in response to a (non percentage) change of x, and:

$\epsilon_2 = \dfrac{\partial y}{\partial x/x}$

Which is the (non percentage) change in $y$ in response to a percentage change in $x$.

How do we distinguish between the two?

That is, what are they called?

When we say "the $x$-semi-elasticity of $y$", what do we refer to?

REFERENCES

STATA manual for rmargins, page 26.


Note: The actual definition is on page 47.

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