91Ó°ÊÓ

In economics, understanding the derivative of a demand function is crucial. It tells us how quantity demanded changes as price changes. Consider the function given by the exercise, where the demand function is given by:

• \( D(p) = \frac{300}{p} \).

The derivative of this function, noted as \( D'(p) \), is a measure of the rate of change of demand with respect to price change. To find this derivative, apply standard calculus rules. The derivative of \( \frac{300}{p} \) with respect to \( p \) is:

• \( D'(p) = -\frac{300}{p^2} \).

This negative sign indicates that as price increases, the quantity demanded decreases, which aligns with the typical behavior of most goods in economic theory where there is an inverse relationship between price and quantity demanded. Calculating derivatives helps us define elasticity and analyze demand behavior.

Elasticity in economics measures how much one variable responds to changes in another variable. When we talk about the elasticity of demand, specifically, we are analyzing how the quantity demanded responds to changes in price. There are typically three kinds of elasticity:

• Elastic Demand: When the elasticity measure is greater than 1, it indicates elastic demand. This means quantity demanded is very responsive to changes in price.
• Inelastic Demand: When the elasticity is less than 1. Here, quantity demanded is not very responsive to price changes.
• Unit Elastic Demand: Occurs when elasticity is exactly 1. This implies that percentage changes in price and quantity demanded are equal and offset each other.

Understanding these types ensures that businesses and policymakers can make informed decisions about pricing and production. Knowing whether demand is elastic, inelastic, or unit elastic can help predict how changes in price will affect overall sales and revenue.

Unit elastic demand is an interesting concept in economics. It occurs when a change in price leads to an exactly proportional change in the quantity demanded. In mathematical terms, the elasticity measure, \( E(p) \), is equal to 1.
In the exercise provided, the elasticity at price \( p = 4 \) was calculated and simplified to be \( E(p) = -1 \), which in terms of absolute value is 1. This demonstrates unit elasticity, meaning the percentage change in price equates to an equal percentage change in demand.
What makes unit elastic demand unique is its representation of a balance between price and demand fluctuations. Unlike elastic or inelastic, where demand varies significantly or barely, unit elasticity suggests considerable precision in how demand adjusts to price changes. Adjustments in price do not affect total revenue because the increases or decreases in quantity sold perfectly compensate for the change in price per unit. It is crucial when aiming for stable revenue amidst price adjustments.