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Chapter 4: Problem 21

For each demand function \(D(p)\) : a. Find the elasticity of demand \(E(p)\). b. Determine whether the demand is elastic, inelastic, or unit-elastic at the given price \(p\). $$ D(p)=\frac{300}{p}, \quad p=4 $$

Short Answer

Expert verified
The demand is unit-elastic at \( p = 4 \), as the elasticity \( E(4) \) is \(-1\).

Step by step solution

01

Understand the Formula for Elasticity of Demand

The elasticity of demand \( E(p) \) is given by the formula \( E(p) = \frac{dD(p)}{dp} \times \frac{p}{D(p)} \). This formula determines how sensitive the quantity demanded \( D(p) \) is to a change in price \( p \).
02

Differentiate the Demand Function

To find \( \frac{dD(p)}{dp} \), differentiate \( D(p) = \frac{300}{p} \) with respect to \( p \). This gives \( \frac{dD(p)}{dp} = -\frac{300}{p^2} \).
03

Substitute Values into the Elasticity Formula

Substitute \( \frac{dD(p)}{dp} = -\frac{300}{p^2} \), \( D(p) = \frac{300}{p} \), and \( p=4 \) into the elasticity formula. We get:\[E(4) = \left(-\frac{300}{4^2}\right) \times \frac{4}{\frac{300}{4}} = \left(-\frac{300}{16}\right) \times \frac{4}{75} = -1.\]
04

Interpret the Result

The elasticity of demand \( E(4) \) is \(-1\), which means the demand is unit-elastic at \( p = 4 \). This signifies that the percentage change in quantity demanded is exactly equal to the percentage change in price.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Demand Function
A demand function is a mathematical expression that describes the relationship between the price of a product and the quantity demanded by consumers. In our example, the demand function is represented by \[ D(p) = \frac{300}{p} \] where \( D(p) \) is the quantity demanded, and \( p \) is the price of the product.
This specific function indicates an inverse relationship between price and demand:
  • As price decreases, the quantity demanded increases.
  • As price increases, the quantity demanded decreases.
This reciprocal relationship is typical in economics, as consumers usually purchase more of a product when its price falls.
Differentiation
The process of differentiation allows us to find the rate at which one quantity changes with respect to another. In the context of a demand function, we differentiate to find how sensitive the quantity demanded is to changes in price.
For the demand function \[ D(p) = \frac{300}{p} \], when we differentiate this with respect to \( p \), we get: \[ \frac{dD(p)}{dp} = -\frac{300}{p^2} \].
Here's what this derivative tells us:
  • The negative sign indicates an inverse relationship; as price increases, demand decreases at that rate.
  • The term \( \frac{300}{p^2} \) denotes the magnitude of the change in demand concerning a change in price.
Differentiation provides the necessary component for calculating elasticity, which measures demand responsiveness.
Unit-elastic Demand
Unit-elastic demand is a concept where the percentage change in quantity demanded is exactly equal to the percentage change in price. This is found when the elasticity of demand \( E(p) \) equals \(-1\).
In our example, we calculated the elasticity at \( p = 4 \): \[ E(4) = -1 \].This outcome shows that the demand at this price is unit-elastic.
Understanding demand as unit-elastic helps businesses and economists make pricing decisions. Some important notes include:
  • Revenue remains constant if prices change because the percentage fall in demand offsets the percentage rise in price.
  • This is a critical point for maximizing revenue since any deviation in price could alter total sales revenue.
Thus, correctly identifying when demand is unit-elastic can effectively guide pricing strategies and economic predictions.

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