Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.
Find the elasticity function for this demand function.

검증된 단계별 안내

Step 1: Understand the concept of elasticity. Elasticity of demand measures how the quantity demanded of a good responds to a change in price. It is defined as E(p) = - (p / D(p)) * (dD/dp), where D(p) is the demand function and dD/dp is its derivative.

Step 2: Identify the given demand function. The demand function provided is D(p) = 40 - 2p.

Step 3: Differentiate the demand function with respect to p. Find dD/dp by differentiating D(p) = 40 - 2p. The derivative, dD/dp, is the rate of change of demand with respect to price.

Step 4: Substitute D(p) and dD/dp into the elasticity formula. Use the formula E(p) = - (p / D(p)) * (dD/dp) and substitute the expressions for D(p) and dD/dp.

Step 5: Simplify the expression for E(p). Simplify the expression obtained in Step 4 to get the elasticity function in terms of p.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Demand Function

A demand function expresses the relationship between the price of a good and the quantity demanded by consumers. In this case, D(p) = 40 - 2p indicates that as the price (p) increases, the quantity of DVDs sold (D) decreases. Understanding this function is crucial for analyzing how price changes affect consumer behavior.

Elasticity of Demand

Elasticity of demand measures how responsive the quantity demanded is to a change in price. It is calculated as the percentage change in quantity demanded divided by the percentage change in price. This concept helps determine whether demand is elastic (sensitive to price changes) or inelastic (less sensitive), which is essential for pricing strategies.

Calculating Elasticity

To find the elasticity function for the given demand function, we use the formula E(p) = (D'(p) * p) / D(p), where D'(p) is the derivative of the demand function with respect to price. This calculation provides a function that indicates how elasticity varies with price, allowing the store owner to make informed decisions about pricing and inventory.

Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.Find the elasticity function for this demand function.