3.7 - Elasticity of Demand

This paragraph introduces the concept of demand in relation to pricing. It explains the inverse relationship between product pricing and the quantity consumers are willing to purchase. It highlights the interest of producers in analyzing how price changes, particularly increases, affect the quantity demanded. The paragraph also touches on the idea of elasticity of demand, which measures the sensitivity of demand to price changes, and the importance of finding a pricing 'sweet spot' to maximize revenue without deterring consumers.

The second paragraph delves into the specifics of how a company, using a hypothetical Blu-ray rental service as an example, can assess the impact of price changes on revenue. It outlines the calculation of demand and revenue at a price point of two dollars, and then examines the effects of a 10 percent price increase on both the quantity demanded and the resulting revenue. The summary emphasizes the importance of balancing price increases with the potential decrease in rentals to maintain or increase overall revenue.

This paragraph explores a scenario where the rental price is set at four dollars and the impact of a subsequent 10 percent increase is analyzed. It demonstrates that while doubling the price from two to four dollars halves the expected rentals, the revenue remains the same due to the price increase. However, the paragraph reveals that further increasing the price from four dollars results in a more significant drop in rentals, leading to decreased revenue. This illustrates the concept of demand elasticity and how it can influence business decisions.

The fourth paragraph focuses on the creation of an elasticity function to generalize the sensitivity of demand to price changes across various prices. It defines elasticity as a ratio involving the derivative of the demand function and uses the given demand function to derive the elasticity function. The paragraph explains the significance of this function in predicting how changes in price will affect the quantity demanded, which is crucial for a producer's pricing strategy.

In this paragraph, the concept of unit elasticity is introduced as the point where the change in price is matched by an equal change in quantity demanded. It describes how to find this balance point by setting the elasticity function equal to one and solving for the price. The paragraph also discusses the revenue function, which is derived by multiplying the price by the demand function, and emphasizes the importance of this function in determining optimal pricing for maximizing revenue.

The sixth paragraph establishes the relationship between elasticity and revenue. It explains that revenue increases when demand is inelastic (elasticity < 1) and decreases when demand is elastic (elasticity > 1). The paragraph identifies the point of unit elasticity as the optimal pricing strategy, where changes in price and quantity demanded are balanced, resulting in maximized revenue. It also discusses the process of confirming that the critical value found from the revenue function indeed represents a maximum.

The seventh paragraph applies the concept of elasticity to a new example with a given demand function involving a square root expression. It outlines the steps to find the elasticity function, determine if demand is elastic or inelastic at a specific price point, and calculate the optimal pricing for maximizing revenue. The summary highlights the practical implications of pricing strategies on customer retention and overall revenue.

The final paragraph provides a detailed calculation of the elasticity function for a given demand function and evaluates the elasticity at a price of $250. It determines that demand is elastic at this price, suggesting that the product is likely overpriced. The paragraph then demonstrates how to find the optimal pricing that results in unit elasticity and maximized revenue, concluding that $200 is the optimal price for the product based on the given demand function.