What is the formula for elasticity of savings with respect to interest rates?

Elasticity is a measure of the responsiveness of one variable to changes in another variable. In this case, we are looking at the responsiveness of savings (S) to changes in interest rates (r). The formula for elasticity (E) can be generalised as: \[E = \frac{\% \text{change in dependent variable}}{\% \text{change in independent variable}}\] In our context, the dependent variable is savings (S) and the independent variable is the interest rate (r).

To derive the formula for the elasticity of savings with respect to interest rates, we will use the general formula for elasticity. In our case, the elasticity of savings (ES) with respect to interest rates (r) can be represented as: \[ES = \frac{\% \text{change in S}}{\% \text{change in r}}\] To express the percentage change in each variable, we can use the derivative. The percentage change in savings is the derivative of savings with respect to interest rates (dS/dr) divided by the savings (S) itself: \[\frac{dS}{dr} \div S\] Similarly, the percentage change in interest rates is the change in interest rates (dr) divided by the interest rate (r) itself: \[dr \div r\] Now we can plug these percentage changes back into the elasticity formula: \[ES = \frac{\frac{dS}{dr} \div S}{dr \div r}\] By simplifying the expression and multiplying the numerator and denominator by the inverse of the denominator which is \(\frac{r}{dr}\), we get: \[ES = \frac{r \cdot \frac{dS}{dr}}{S}\] This is the formula for the elasticity of savings with respect to interest rates.