Understanding Point Elasticity of Demand

Oct 13, 2025 by miodeya.com 149 views

Introduction

In economics, the demand equation is a fundamental concept that helps businesses understand the relationship between the price of a product and the quantity demanded. The demand equation is typically represented as a linear function, where the price (p) is a function of the quantity demanded (q). In this article, we will determine the point elasticity of demand for the given equation p = -3q + 120, where p > 0 and q > 0.

Understanding Point Elasticity of Demand

Point elasticity of demand, denoted by η, measures the responsiveness of the quantity demanded to a change in the price of a product. It is a measure of how much the quantity demanded changes in response to a change in the price. The point elasticity of demand is calculated as the percentage change in quantity demanded in response to a 1% change in price.

Calculating Point Elasticity of Demand

To calculate the point elasticity of demand, we need to follow these steps:

Take the derivative of the demand equation: The first step is to take the derivative of the demand equation with respect to q. This will give us the marginal revenue function, which represents the rate of change of revenue with respect to the quantity demanded.
Calculate the marginal revenue function: The marginal revenue function is given by the derivative of the demand equation. In this case, the demand equation is p = -3q + 120. Taking the derivative of this equation with respect to q, we get:
$\frac{\partial p}{\partial q} = -3$
This represents the marginal revenue function, which is the rate of change of revenue with respect to the quantity demanded.
Calculate the point elasticity of demand: The point elasticity of demand is calculated using the following formula:
$\eta = \frac{p}{q} \cdot \frac{1}{\frac{\partial p}{\partial q}}$
Substituting the values from the demand equation and the marginal revenue function, we get:
$\eta = \frac{p}{q} \cdot \frac{1}{-3}$
Simplifying this expression, we get:
$\eta = -\frac{p}{3q}$

Solving for Point Elasticity of Demand

Now that we have the formula for point elasticity of demand, we can substitute the values from the demand equation to solve for η.

The demand equation is p = -3q + 120. We can rearrange this equation to solve for p:

p = -3q + 120

p - 120 = -3q

\frac{p - 120}{-3} = q

q = \frac{120 - p}{3}

Now, we can substitute this expression for q into the formula for point elasticity of demand:

\eta = -\frac{p}{3q}

\eta = -\frac{p}{3 \left(\frac{120 - p}{3}\right)}

\eta = -\frac{p}{\frac{120 - p}{3}}

\eta = -\frac{3p}{120 - p}

Interpreting the Results

The point elasticity of demand is a measure of how responsive the quantity demanded is to a change in the price. A negative value for η indicates that the quantity demanded decreases as the price increases. The magnitude of η indicates the degree of responsiveness.

In this case, the point elasticity of demand is given by the formula:

\eta = -\frac{3p}{120 - p}

This formula indicates that the point elasticity of demand is a function of the price. As the price increases, the point elasticity of demand decreases.

Conclusion

In this article, we determined the point elasticity of demand for the given equation p = -3q + 120, where p > 0 and q > 0. We followed a step-by-step approach to calculate the point elasticity of demand, including taking the derivative of the demand equation, calculating the marginal revenue function, and solving for the point elasticity of demand. The results indicate that the point elasticity of demand is a function of the price and decreases as the price increases.

References

Microeconomics: Theory and Applications by R. Preston McAfee and Michael D. Whinston
Economics by Gregory Mankiw
Managerial Economics by William F. Samuelson and Stephen G. Marks

Frequently Asked Questions

What is point elasticity of demand?

Point elasticity of demand is a measure of how responsive the quantity demanded is to a change in the price of a product.
How is point elasticity of demand calculated?

Point elasticity of demand is calculated using the formula: η = (p/q) * (1/∂p/∂q), where p is the price, q is the quantity demanded, and ∂p/∂q is the marginal revenue function.
What is the significance of point elasticity of demand?

Point elasticity of demand is a measure of how responsive the quantity demanded is to a change in the price. It is used to determine the optimal price and quantity to maximize revenue.
Frequently Asked Questions: Point Elasticity of Demand =====================================================

Q: What is point elasticity of demand?

A: Point elasticity of demand is a measure of how responsive the quantity demanded is to a change in the price of a product. It is a measure of how much the quantity demanded changes in response to a change in the price.

Q: How is point elasticity of demand calculated?

A: Point elasticity of demand is calculated using the formula: η = (p/q) * (1/∂p/∂q), where p is the price, q is the quantity demanded, and ∂p/∂q is the marginal revenue function.

Q: What is the significance of point elasticity of demand?

A: Point elasticity of demand is a measure of how responsive the quantity demanded is to a change in the price. It is used to determine the optimal price and quantity to maximize revenue.

Q: What are the different types of elasticity of demand?

A: There are three types of elasticity of demand:

Price elasticity of demand: This measures the responsiveness of the quantity demanded to a change in the price.
Income elasticity of demand: This measures the responsiveness of the quantity demanded to a change in the consumer's income.
Cross-price elasticity of demand: This measures the responsiveness of the quantity demanded of one product to a change in the price of another product.

Q: What is the difference between point elasticity of demand and arc elasticity of demand?

A: Point elasticity of demand measures the responsiveness of the quantity demanded to a change in the price at a single point, while arc elasticity of demand measures the responsiveness of the quantity demanded to a change in the price over a range of prices.

Q: How is point elasticity of demand used in business?

A: Point elasticity of demand is used in business to determine the optimal price and quantity to maximize revenue. It is also used to determine the impact of changes in the price on the quantity demanded.

Q: What are the limitations of point elasticity of demand?

A: The limitations of point elasticity of demand include:

Assumes linear demand curve: Point elasticity of demand assumes a linear demand curve, which may not be the case in reality.
Does not account for other factors: Point elasticity of demand does not account for other factors that may affect the quantity demanded, such as changes in consumer preferences or the introduction of new products.

Q: How can point elasticity of demand be improved?

A: Point elasticity of demand can be improved by:

Using more accurate demand curves: Using more accurate demand curves, such as non-linear demand curves, can improve the accuracy of point elasticity of demand.
Accounting for other factors: Accounting for other factors that may affect the quantity demanded, such as changes in consumer preferences or the introduction of new products, can improve the accuracy of point elasticity of demand.

Q: What are some common applications of point elasticity of demand?

A: Some common applications of point elasticity of demand include:

Pricing strategy: Point elasticity of demand is used to determine the optimal price to maximize revenue.
Quantity determination: Point elasticity of demand is used to determine the optimal quantity to produce and sell.
Market research: Point elasticity of demand is used to understand the behavior of consumers and the demand for a product.

Q: What are some common mistakes to avoid when using point elasticity of demand?

A: Some common mistakes to avoid when using point elasticity of demand include:

Assuming a linear demand curve: Assuming a linear demand curve when the demand curve is non-linear can lead to inaccurate results.
Not accounting for other factors: Not accounting for other factors that may affect the quantity demanded, such as changes in consumer preferences or the introduction of new products, can lead to inaccurate results.

Conclusion

Point elasticity of demand is a measure of how responsive the quantity demanded is to a change in the price of a product. It is a useful tool for businesses to determine the optimal price and quantity to maximize revenue. However, it has its limitations and should be used in conjunction with other tools and techniques to ensure accurate results.