Return of an investment for a given period is by definition: $$r = \frac{P}{W_0} - 1$$ where $P$ is the price of the investment at the end of the period, and $W_0$ is the initial investment. I want to understand how much the return changes in terms of percentage, with respect to that of the price. This naturally is the definition of elasticity.
Elasticity can be calculated as: $$\epsilon = \frac{\partial{r}}{\partial{P}} \cdot \frac{P}{r} = \frac{P}{P-W_0}$$
Now if we have a leverage of 2x, we should have a elasticity of 2. However, I do not seem to be able to connect these two concepts. If $W_0$ is normalized to 1, then the formula becomes: $$\epsilon = \frac{P}{P-1}$$
When $\epsilon = 2$, this implies that P = 2. However, this says that only when $P$ is twice that initial investment $W_0$ that the leverage is 2. When price is higher, the leverage decreases.
Where did I get the logic wrong?