91Ó°ÊÓ

Elasticity of demand measures how responsive the quantity demanded of a good is to changes in its price. Essentially, it tells us how much the quantity demanded will change in response to a change in price. Here are some important points to remember:

• Elastic demand: If the quantity demanded changes significantly when the price changes, the demand is elastic.
• Inelastic demand: If the quantity demanded changes little when the price changes, the demand is inelastic.

Mathematically, elasticity of demand ( \epsilon ) is calculated using the formula: \[ \epsilon = \frac{d q}{d p} \cdot \frac{p}{q} \] where \frac{d q}{d p} is the derivative of the quantity demanded with respect to price, p is the price, and q is the quantity demanded.
This tells us whether the demand is elastic or inelastic.

To find the elasticity of demand for a given demand function, we need to use differentiation. Differentiation is a calculus technique that allows us to find the rate at which one quantity changes with respect to another.

In this exercise, we are working with the demand function \[ q = \frac{a}{p^m} \] where q is the quantity demanded and p is the price. To find the rate at which q changes with respect to p , we differentiate the function: \[ \frac{dq}{dp} = \frac{d}{dp} (a \cdot p^{-m}) = -m a \cdot p^{-(m+1)} \] This result will be used to calculate the elasticity of demand.

A demand function shows the relationship between the quantity demanded of a good and its price. It is usually represented in the form \[ q = f(p) \] where q is the quantity demanded and p is the price.
In the exercise, the demand function given is \[ q = \frac{a}{p^m} \] This function indicates that the quantity demanded decreases as the price increases, which is typical for most goods.

To verify that this demand function has a constant elasticity m , we use its specific form to calculate the elasticity of demand. Substituting the differentiated expression into the elasticity formula, we demonstrate that the elasticity of demand indeed remains constant at -m .