The elasticity $\eta_{f,x}(x)$ of the function $f(x)$ at point $x$ is:
$\eta_{f,x}(x) = \dfrac{f(x+1)-f(x)}{1} \cdot \dfrac{x}{f(x)}$
The elasticity $\eta_{f,x}(x)$ of the function $f(x)$ at point $x$ is:
$\eta_{f,x}(x) = \dfrac{f'(x)}{1} \cdot \dfrac{x}{f(x)}$
Another way to define the elasticity is as the ratio of the percentage change in $f$ to the percentage change in $x$.
For two values of $x$, $x_0$ and $x_1$, define: $\Delta x = x_1 - x_0$, $\Delta f = f(x_1) - f(x_0)$, $\%\Delta x = \frac{\Delta x}{x_0}$, and $\%\Delta f = \frac{\Delta f}{f(x_0)}$. Thus:
$\eta_{f,x}(x) = \dfrac{\%\Delta f}{\%\Delta x}$
Consider two price levels $P_0$ and $P_1$ at period 0 and period 1 respectively.
The price elasticity $\eta_{Q^d,P}$ of the demand $Q^d$ curve is:
$\eta_{Q^d,P} = \dfrac{Q^d(P_1) - Q^d(P_0)}{P_1 - P_0} \cdot \dfrac{P_0}{Q^d(P_0)}$
The price elasticity $\eta_{Q^d,P}$ of the demand $Q^d$ curve at price $P$ is:
$\eta_{Q^d,P} = \frac{\partial Q^d}{\partial P}(P) \cdot \dfrac{P}{Q^d(P)}$
Knowing the values of the elasticity allow us to estimate the impact of price changes since the percentage change in quantities will be the percentage change in price times the value of the elasticity:
$\%\Delta Q^d = \eta_{Q^d,P} \cdot \%\Delta P$
| Elasticity | Range | Classification |
|---|---|---|
| Own-Price Elasticity of Demand | ||
| $\eta_{Q_x^D, P_x}$ | $> 0$ | Giffen Good |
| $\eta_{Q_x^D, P_x}$ | $< 0$ | Ordinary Good |
| Cross-Price Elasticity of Demand | ||
| $\eta_{Q_x^D, P_y}$ | $> 0$ | Substitutes |
| $\eta_{Q_x^D, P_y}$ | $= 0$ | Unrelated Goods |
| $\eta_{Q_x^D, P_y}$ | $< 0$ | Complements |
| Income Elasticity of Demand | ||
| $\eta_{Q_x^D, I}$ | $< 0$ | Inferior Good |
| $\eta_{Q_x^D, I}$ | $\in (0,1)$ | Normal Good (Necessity) |
| $\eta_{Q_x^D, I}$ | $> 1$ | Normal Good (Luxury) |