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: $$ϵ=\frac{dS}{S}\cdot \frac{r}{dr}$$

Step-by-step solution

Identify the Variables

In this exercise, we are interested in two variables: the interest rate (denoted by $r$) and the level of savings (denoted by $S$). The goal is to find the elasticity of savings with respect to interest rates. Elasticity is typically denoted by $ϵ$.

Calculate the Percentage Change in Savings

To determine the elasticity of savings with respect to interest rates, we first need to find the percentage change in savings. This is given by the formula: $$\frac{dS}{S}$$ Where $dS$ represents the change in savings and $S$ represents the initial level of savings.

Calculate the Percentage Change in Interest Rate

Next, we need to calculate the percentage change in interest rates. This is given by the formula: $$\frac{dr}{r}$$ Where $dr$ represents the change in interest rate, and $r$ represents the initial interest rate.

Determine the Elasticity (Formula) of Savings with Respect to Interest Rates

Now that we have both the percentage change in savings and the percentage change in interest rates, we can determine the elasticity of savings with respect to interest rates. The formula for the elasticity of savings with respect to interest rates is as follows: $$ϵ=\frac{\left(\frac{dS}{S}\right)}{\left(\frac{dr}{r}\right)}$$ Simplifying the formula, we get: $$ϵ=\frac{dS}{S}\cdot \frac{r}{dr}$$ This is the formula for the elasticity of savings with respect to interest rates. It shows how the percentage change in savings relates to the percentage change in interest rates. If the elasticity is greater than 1, it means that savings are relatively responsive to changes in interest rates. If it is less than 1, savings are relatively unresponsive to changes in interest rates.