Elasticity

The elasticity of a function $f(x)$ with respect to the variable $x$ is denoted by:

$$e_{f}(x)=f^{\prime}(x) \times \frac{x}{f(x)}$$

Partial elasticity

The partial elasticity of a function $f(x,y)$ with respect to the variable $x$ is denoted by:

$$e_{f / x}(x, y)=\dfrac{\partial f(x, y)}{\partial x} \times \dfrac{x}{f(x, y)}$$

The partial elasticity of a function $f(x,y)$ with respect to the variable y is denoted by:

$$e_{f / y}(x, y)=\frac{\partial f(x, y)}{\partial y} \times \frac{y}{f(x, y)}$$

Elasticity of demand

The formula of Price Elasticity of Demand (PED)

$$\rm \underline{PED}=\frac{\text { Percentage change in quantity demanded }}{\text { Percentage change in price }}$$ $$\rm =\dfrac{\frac{Q 2-Q 1}{Q 1} \times 100}{\frac{P 2-P 1}{P 1} \times 100}=\dfrac{\frac{Q 2-Q 1}{Q 1}}{\frac{P 2-P 1}{P 1}}$$
$$\rm =\dfrac{Q 2-Q 1}{Q 1} \times \dfrac{P 1}{P 2-P 1}$$

The different cases:

Elastic demand:

Demand in which the price change causes a large change in the quantity demanded.

$\rm PED > 1$

Inelastic demand:

Demand in which the price change causes a small change in the quantity demanded.

$\rm PED < 1$

Unitary elasticity:

The elasticity coefficient is equal to $1$.

$\rm PED = 1$

Perfectly inelastic:

$\rm PED = 0$

Perfectly elastic:

$\rm PED = infinity$

Elasticity of supply

The formula of Price Elasticity of Supply (PES)

$$\rm \underline{P E S}=\frac{\text { Percentage change in quantity supplied }}{\text { Percentage change in price }}=\frac{\frac{Q 2-Q 1}{Q 1} \times 100}{\frac{P 2-P 1}{P 1} \times 100}=\frac{\frac{Q 2-Q 1}{Q 1}}{\frac{P 2-P 1}{P 1}}$$ $$\rm =\dfrac{Q 2-Q 1}{Q 1} \times \dfrac{P 1}{P 2-P 1}$$

The different cases:

Elastic supply:

Supply in which the price change causes a large change in the quantity supplied. The elasticity coefficient is greater than 1. $\rm PES > 1$

Inelastic supply:

Supply in which the price change causes a small change in the quantity supplied. The elasticity coefficient is smaller than 1. $\rm PES < 1$

Unitary elasticity:

The elasticity coefficient is equal to $1$.

Perfectly inelastic:

$\rm PES = 0$

Perfectly elastic:

$\rm PES = infinity$