Determine The Point Elasticity \[$\eta\$\] Of The Demand Equation $p=-3q+120$, Where $p \ \textgreater \ 0$ And $q \ \textgreater \ 0$.

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Introduction

In economics, the demand equation is a fundamental concept that describes the relationship between the price of a product and the quantity demanded. The demand equation is often 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 demand equation p=−3q+120p=-3q+120, where p>0p > 0 and q>0q > 0.

Understanding Point Elasticity

Point elasticity, denoted by η\eta, is a measure of how responsive the quantity demanded is to a change in price. It is a ratio of the percentage change in quantity demanded to the percentage change in price. Mathematically, point elasticity can be represented as:

η=% change in q% change in p\eta = \frac{\% \text{ change in } q}{\% \text{ change in } p}

Calculating Point Elasticity

To calculate the point elasticity of demand, we need to find the derivative of the demand equation with respect to price (p). The derivative of the demand equation p=−3q+120p=-3q+120 with respect to price (p) is:

dpdq=−3\frac{dp}{dq} = -3

The point elasticity of demand can be calculated using the formula:

η=pq⋅dqdp\eta = \frac{p}{q} \cdot \frac{dq}{dp}

Substituting the values of pp and qq from the demand equation, we get:

η=−3q+120q⋅1−3\eta = \frac{-3q+120}{q} \cdot \frac{1}{-3}

Simplifying the expression, we get:

η=120q\eta = \frac{120}{q}

Interpreting the Results

The point elasticity of demand is a measure of how responsive the quantity demanded is to a change in price. A value of η\eta greater than 1 indicates that the quantity demanded is highly responsive to changes in price, while a value of η\eta less than 1 indicates that the quantity demanded is less responsive to changes in price.

In this case, the point elasticity of demand is 120q\frac{120}{q}. This means that the quantity demanded is highly responsive to changes in price, especially at low values of qq. As the value of qq increases, the point elasticity of demand decreases, indicating that the quantity demanded becomes less responsive to changes in price.

Conclusion

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 price. It is a ratio of the percentage change in quantity demanded to the percentage change in price.

Q: How is point elasticity of demand calculated?

A: Point elasticity of demand can be calculated using the formula:

η=pq⋅dqdp\eta = \frac{p}{q} \cdot \frac{dq}{dp}

Where pp is the price, qq is the quantity demanded, and dqdp\frac{dq}{dp} is the derivative of the demand equation with respect to price.

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

A: Point elasticity of demand is significant because it helps businesses and policymakers understand how changes in price will affect the quantity demanded of a product. This information can be used to make informed decisions about pricing strategies, production levels, and investment in marketing and advertising.

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

A: There are three types of elasticity of demand:

  1. Price elasticity of demand: This measures how responsive the quantity demanded is to a change in price.
  2. Income elasticity of demand: This measures how responsive the quantity demanded is to a change in income.
  3. Cross-price elasticity of demand: This measures how responsive the quantity demanded of one product is to a change in the price of another product.

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

A: Point elasticity of demand measures the responsiveness of the quantity demanded to a change in price at a specific point on the demand curve. Arc elasticity of demand, on the other hand, measures the responsiveness of the quantity demanded to a change in price over a range of prices.

Q: How can businesses use point elasticity of demand to inform their pricing strategies?

A: Businesses can use point elasticity of demand to inform their pricing strategies by analyzing how changes in price will affect the quantity demanded of their product. For example, if the point elasticity of demand is high, it may be more profitable to charge a higher price, as the quantity demanded will be less responsive to changes in price.

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

A: Point elasticity of demand has a wide range of applications in business, economics, and policy-making. Some common applications include:

  1. Pricing strategies: Businesses use point elasticity of demand to determine the optimal price for their product.
  2. Production planning: Businesses use point elasticity of demand to determine the optimal level of production.
  3. Marketing and advertising: Businesses use point elasticity of demand to determine the effectiveness of their marketing and advertising campaigns.
  4. Policy-making: Governments use point elasticity of demand to inform their policy decisions, such as taxation and regulation.

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

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

  1. Ignoring the derivative: Failing to calculate the derivative of the demand equation with respect to price can lead to incorrect results.
  2. Using the wrong formula: Using the wrong formula for point elasticity of demand can lead to incorrect results.
  3. Not considering the context: Failing to consider the context in which the demand equation is being used can lead to incorrect results.