Elasticity of log-log regression

Question

In the log-log regression case,

$$\log(Y) = B_0 + B_1 \log(X) + U \>,$$

can you show $B_1$ is the elasticity of $Y$ with respect to $X$, i.e., that $E_{yx} = \frac{dY}{dX}(\frac{Y}{X})$?

Answer 1

whuber has made the point in the comment.

If $\log_e(Y) = B_0 + B_1\log_e(X) + U$ and $U$ is independent of $X$ then taking the partial derivative with respect to $X$ gives $\frac{\partial Y}{\partial X}\cdot\frac{1}{Y} = B_1\frac{1}{X}$, i.e. $B_1 = \frac{\partial Y}{\partial X}\cdot\frac{X}{Y}$.

$E_{y,x} = \lim_{X \rightarrow x} \frac { \Delta Y} { y} / \frac { \Delta X} { x}$, which is the same thing. Take absolute values if you want to avoid negative elasticities.