Elasticity The demand function for a product is given by...

Elasticity The demand function for a product is given by $p=20-0.02 x, \quad 0<x<1000$. (a) Find the price elasticity of demand when $x=560$. (b) Find the values of $x$ and $p$ that maximize the total revenue. (c) For the value of $x$ found in part (b), show that the price elasticity of demand has unit elasticity.

Elasticity The demand function for a product is given by $p=20-0.02 x, \quad 0<x<1000$.
(a) Find the price elasticity of demand when $x=560$.
(b) Find the values of $x$ and $p$ that maximize the total revenue.
(c) For the value of $x$ found in part (b), show that the price elasticity of demand has unit elasticity.

Key Concepts

Price Elasticity of Demand

This concept measures the responsiveness of the quantity demanded to a change in price. It is calculated as the ratio of the percentage change in quantity demanded to the percentage change in price, often using the derivative of the demand function to capture small changes. Understanding this concept helps in analyzing how price changes can impact sales volumes.

Revenue Maximization

Revenue maximization involves finding the price and quantity combination that yields the highest total revenue, where total revenue is defined as the product of price and quantity sold. This typically requires formulating the revenue function from the demand function and then using calculus, such as finding the derivative, to determine the optimal point.

Differentiation and Optimization

Differentiation is the mathematical process used to determine the rate at which a function changes, and it is essential for solving optimization problems. In economic contexts, differentiation is used to find critical points where functions like revenue are maximized or minimized, by setting the derivative equal to zero and verifying the nature of these points.

Unit Elasticity

Unit elasticity occurs when the absolute value of the elasticity measure is equal to one, meaning that the percentage change in quantity demanded is exactly equal to the percentage change in price. This point is significant as it often corresponds to the maximum total revenue in a market, where any small change in price leads to an equivalent change in quantity sold.

Transcript

Please give Ace some feedback