Question

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

Short answer

The formula for the elasticity of savings with respect to interest rates is given by: \[ E = \frac{r}{S} \cdot f'(r) \] Here, \(E\) is the elasticity, \(r\) represents interest rates, \(S\) denotes savings, and \(f'(r)\) is the derivative of the savings function with respect to interest rates.

Step-by-step solution

Define the formulas for savings and interest rates

The first step is to define the formulas for the savings and interest rates. Let's denote the savings as \(S\), interest rates as \(r\), and the initial level of savings (before any change in interest rates) as \(S_0\). The formula for savings as a function of interest rates can be written as: \[ S = f(r) \]

Define elasticity

Elasticity is a measure of the responsiveness of a dependent variable to changes in an independent variable. In our case, we are looking at the responsiveness of savings to changes in interest rates. The formula for the elasticity of savings with respect to interest rates is given by: \[ E = \frac{\% \text{ change in savings}}{\% \text{ change in interest rate}} \]

Derive elasticity using the formula

To find the elasticity of savings with respect to interest rates, we need to find the percentage change in savings with respect to the percentage change in interest rates, i.e., the derivative of \(S\) with respect to \(r\). We can rewrite the elasticity formula and express percentage changes as derivatives: \[ E = \frac{\frac{\Delta S}{S}}{\frac{\Delta r}{r}} = \frac{r}{S} \cdot \frac{\Delta S}{\Delta r} \]

Calculate the derivative

We can find \(\frac{\Delta S}{\Delta r}\) by differentiating the function \(S=f(r)\) with respect to the interest rate \(r\). Using the differentiation rules, we have: \[ \frac{\Delta S}{\Delta r} = f'(r) \]

Plug the derivative back into the elasticity formula

Now, substitute the derivative back into the formula for elasticity, which will give us our final answer: \[ E = \frac{r}{S} \cdot f'(r) \] This formula represents the elasticity of savings with respect to interest rates. The elasticity tells us how sensitive the savings level is to changes in interest rates.