Elasticity The demand function for a product is given by p=800-4 x, 0<x<200. (a) Find the pric

step 1 So given this demand function, we're trying to find the price or less sophisticated demand, give

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

Elasticity The demand function for a product is given by $p=800-4 x, \quad 0<x<200$. (a) Find the price elasticity of demand when $x=150$. (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 $p=800-4 x, \quad 0<x<200$.
(a) Find the price elasticity of demand when $x=150$.
(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

Unit Elasticity

Unit elasticity occurs when a percentage change in price leads to an equal percentage change in the quantity demanded, meaning the absolute value of the price elasticity of demand is one. This concept is critical because it signifies a balance point where changes in price do not affect total revenue.

Revenue Maximization

Revenue maximization involves finding the price and quantity at which total revenue, the product of price and quantity, is maximized. This is often done by formulating a revenue function from the demand function and then applying optimization techniques, usually involving calculus, to identify the critical points that yield maximum revenue.

Price Elasticity of Demand

Price elasticity of demand measures the responsiveness of the quantity demanded to a change in price. It is typically defined as the ratio of the percentage change in quantity demanded to the percentage change in price, often computed using derivatives. This concept helps determine whether demand is elastic (responsive), inelastic (less responsive), or unit elastic.

Demand Function

A demand function represents the relationship between the price of a product and the quantity demanded by consumers. It is used to model consumer behavior in response to changes in price, and in economic analysis, it serves as a starting point to determine how variables like price and quantity interact.

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