Question

Find the price elasticity of demand for the demand function at the indicated \(x\) -value. Is the demand elastic, inelastic, or of unit elasticity at the indicated \(x\) -value? Use a graphing utility to graph the revenue function, and identify the intervals of elasticity and inelasticity. \(p=5-0.03 x \quad x=100\)

Short answer

The price elasticity of demand at \(x=100\) for the demand function \(p=5-0.03x\) is -0.09. The demand is inelastic because the absolute value of the price elasticity is less than 1. The revenue function is given by \(R = p * x = (5 - 0.03x) * x\). The demand is elastic on the intervals where the revenue function is increasing and inelastic on the intervals where the revenue function is decreasing.

Step-by-step solution

Calculate The Derivative Of The Demand Function

Differentiate the given demand function \(p=5-0.03x\) with respect to \(x\). This gives the slope or rate of change of \(p\) in terms of \(x\). Using the power rule, the derivative is \(-0.03\).

Compute The Price Elasticity Of Demand

The price elasticity of demand is computed using the formula \(\epsilon = \frac{p}{x} * \frac{dp}{dx}\). Substitute \(p=5-0.03x\), \(\frac{dp}{dx} = -0.03\), and \(x=100\) into the formula. This gives \(\epsilon = \frac{(5-0.03*100)}{100} * -0.03 = -0.09\). The demand is inelastic because the absolute value of \(\epsilon\) is less than 1.

Plot The Revenue Function And Identify The Intervals Of Elasticity And Inelasticity

The revenue function is given by \(R = p * x = (5 - 0.03x) * x\). Graph this function using a graphing utility. The intervals of elasticity and inelasticity can be identified from the graph. The demand is elastic on the intervals where the revenue function is increasing and inelastic on the intervals where the revenue function is decreasing.

Key concepts

Revenue Function

The revenue function is an integral part of understanding how businesses make money given different pricing and demand situations. It is represented as the product of price

• Revenue = Price × Quantity
• In the problem given, the revenue function is represented as: \[R = (5 - 0.03x) \times x\]

Visually representing this function on a graph can provide insights into how changes in quantity sold (x) can affect total revenue (R).
By graphing this function, you can identify intervals where the revenue increases, indicating elasticity, and where it decreases, indicating inelasticity.
The understanding of when revenue increases or decreases depending on pricing and quantities is essential in making informed business decisions to maximize profit.

Demand Function

The demand function shows the relationship between price and quantity demanded. In mathematical terms, it represents how much of a product consumers are willing to purchase at different price levels.

• Formalized, it’s given by: \[p = 5 - 0.03x\] where p is the price and x is the quantity.

This particular function is linear, meaning it suggests a steady relationship between the increase in quantity (x) and the decrease in price (p).
Understanding the demand function is crucial for grasping how market demand changes with price fluctuations. You can visualize this using graphs, making it easier to predict market behaviors for given scenarios, and plan accordingly for pricing strategies.

Elasticity and Inelasticity

Elasticity and inelasticity describe how quantity demanded responds to changes in price.

• Price elasticity of demand is a measure of the sensitivity of quantity demanded to a change in price.
• If the absolute value of elasticity (\(\epsilon\)) is greater than 1, demand is elastic; if it is less than 1, demand is inelastic; and if exactly 1, it's unit elastic.

In the given exercise, the elasticity was calculated as: \[\frac{(5 - 0.03 \times 100)}{100} \times (-0.03) = -0.09\]The result here implies inelastic demand, as the absolute value is less than 1.
Understanding elasticity helps in determining whether it would be beneficial for a business to increase or decrease prices. For inelastic goods, price increases might not significantly reduce quantity sold and could increase overall revenue.

Derivative

The derivative provides insight into the rate of change of one variable concerning another, and is crucial in determining elasticity.

• In calculating price elasticity of demand, the derivative of the demand function provides the change rate in price with respect to quantity, \(\frac{dp}{dx}\).
• For \(p = 5 - 0.03x\)
• The derivative is constant: \(-0.03\), indicating a steady rate of price decline as more units are supplied.

The derivative helps in constructing graphs that visualize how changes in price affect demand, and ultimately, revenue.
Understanding derivatives and their application in elasticity is vital for predicting market trends and making decisions based on expected changes in consumer demand relative to pricing shifts.