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. \(p=20-0.0002 x \quad x=30\)

Short answer

The elasticity of demand E is calculated using provided values and formula. The type of elasticity is determined based on the result. The intervals of elasticity and inelasticity are identified by graphing the revenue function and observing the slope that it takes.

Step-by-step solution

Understand the concepts

The price elasticity of demand (E) is calculated using the formula: E = (dx/dp)*(p/x), where dx/dp is the derivative of the demand function with respect to its price p, x is the quantity demanded and p is the price. The demand is elastic if E > 1, inelastic if E < 1, and of unit elasticity if E = 1. To identify the intervals of elasticity and inelasticity, we're going to graph the revenue function.

Differentiate the demand function

First, we differentiate the demand function with respect to p. The function is p = 20 - 0.0002x, so its derivative -0.0002.

Substitute into the formula and calculate elasticity

Substituting dx/dp = -0.0002, p = 20 - 0.0002*30, and x = 30 into the formula, we find E = (-0.0002) * (20 - 0.0002 * 30) / 30.

Analyze the elasticity of demand

Compute E to determine whether the demand is elastic, inelastic or of unit elasticity. If E > 1, the demand is elastic. If E < 1, it is inelastic. If E = 1, it is of unit elasticity.

Graph the revenue function and identify intervals

Using a graphing utility, graph the revenue function (R = p * x = (20 - 0.0002*x) * x). Look at the graph's slope to identify intervals: if it is positive, elasticity is greater than 1 (elastic); if it is negative, elasticity is less than 1 (inelastic). When the slope is zero, elasticity is unitary.

Key concepts

Demand Function

A demand function represents the relationship between the price of a good and the quantity demanded by consumers. In mathematical terms, it is expressed as a function where the price, denoted by \(p\), is a function of quantity \(x\), such as \(p = 20 - 0.0002x\). This equation shows how price changes as quantity changes. It's essential to understand that as the quantity increases, the price decreases, illustrating an inverse relationship.
In the given function \(p = 20 - 0.0002x\), the term \(-0.0002x\) represents the rate at which price decreases as more units of \(x\) are demanded. This linear demand function helps us determine the elasticity by assessing how sensitive the demand is to changes in price or vice versa.

Revenue Function

The revenue function calculates the total revenue earned by selling a certain number of units at a given price. It is derived by multiplying the price \(p\) by the quantity \(x\), giving \(R = p \, x\). For the demand function \(p = 20 - 0.0002x\), the revenue function becomes \(R = (20 - 0.0002x) \, x\).
This function essentially tracks how revenue changes as the quantity sold changes.

• If the graph of the revenue function is increasing, it suggests that selling more units increases total revenue, indicating elastic demand.
• If the revenue function graph is decreasing, it implies selling more units decreases total revenue, indicating inelastic demand.
• If the graph is flat, it implies unit elasticity, where revenue remains unchanged as sales change.

This graphical interpretation is crucial for identifying elastic or inelastic behaviors in demand.

Elasticity and Inelasticity

Elasticity in economics refers to how a change in one economic variable affects another. Specifically, price elasticity of demand measures how responsive the quantity demanded is to a price change. It is calculated as \(E = (dx/dp)\times(p/x)\), where \(dx/dp\) is the derivative of the demand function with respect to \(p\).

• Demand is elastic when \(E > 1\), meaning consumers are highly sensitive to price changes.
• Demand is inelastic when \(E < 1\), suggesting that price changes have little effect on the quantity demanded.
• Demand is of unit elasticity when \(E = 1\), meaning the percentage change in quantity is precisely equal to the percentage change in price.

In the context of our function, we evaluate the elasticity at given points, such as \(x=30\), to determine whether the demand is elastic, inelastic, or unitary. Using graphs and calculations, businesses learn how to set optimal prices for maximum revenue by understanding their elasticity intervals.