Problem 10 - Applied Calculus

Question

The demand for a product is given by \(q=200-2 p^{2}\). Find the elasticity of demand when the price is \(\$ 5 .\) Is the demand inelastic or elastic, or neither?

Step-by-Step Solution

Answer

The elasticity of demand is -2/3, indicating inelastic demand at $5.

Step 1: Recall the Formula for Elasticity of Demand

The elasticity of demand, denoted as \(E\), is given by the formula:\[ E = \left( \frac{dq}{dp} \right) \left( \frac{p}{q} \right) \]where \( \frac{dq}{dp} \) is the derivative of the demand function with respect to price.

Step 2: Find the Derivative of the Demand Function

Given the demand function \( q = 200 - 2p^2 \), we find the derivative with respect to \( p \):\[ \frac{dq}{dp} = -4p \].This derivative indicates how the quantity demanded changes with a change in price.

Step 3: Calculate Quantity Demanded at Given Price

Evaluate the demand function at \( p = 5 \):\[ q = 200 - 2(5)^2 = 200 - 50 = 150 \].So the quantity demanded when the price is \( \$5 \) is 150 units.

Step 4: Compute Elasticity at Given Price

Substitute \( p = 5 \), \( q = 150 \), and \( \frac{dq}{dp} = -4p \) into the elasticity formula:\[ E = (-4 \times 5) \left( \frac{5}{150} \right) = -20 \times \frac{1}{30} = -\frac{20}{30} = -\frac{2}{3} \].

Step 5: Determine Elasticity Type

Since the elasticity is \( -\frac{2}{3} \), which is between -1 and 0, the demand is inelastic at the price of \( \$5 \).Inelastic means that the quantity demanded is not very responsive to price changes.

Key Concepts

Derivative of Demand

Understanding the derivative of a demand function is crucial in economics as it tells us how the quantity demanded reacts to changes in price. Let's break it down. For our function, the demand is represented by a mathematical relationship:

• The demand function given is \( q = 200 - 2p^2 \).
• To find how demand changes with price, we calculate the derivative of \( q \) with respect to \( p \), which is \( \frac{dq}{dp} = -4p \).

Here, the derivative \( -4p \) tells us the rate at which demand decreases as price increases. For every one unit increase in price, demand falls by \( 4p \) units.

In our problem, it serves as a stepping stone to calculate elasticity—showing the precise relationship between quantity and price.

Inelastic Demand

Inelastic demand is a concept where the quantity of a product demanded is relatively insensitive to price changes. We encounter inelastic demand in everyday life. Consider things like essential goods such as medications or basic food items.

For the given problem, we calculated the elasticity of demand at a price of \( \$5 \), and discovered it was \( -\frac{2}{3} \). In terms of elasticity:

• Elasticity values between -1 and 0 indicate inelastic demand.
• In our specific case, an elasticity of \(-\frac{2}{3}\) confirms the product is inelastic at this price.

This means minor price hikes do not significantly reduce the quantity demanded. Consumers might buy nearly the same amount, even if prices rise.

An inelastic demand can provide pricing power to producers since they know their customers aren't as sensitive to price changes.

Price Elasticity Calculation

Calculating price elasticity of demand involves a couple of straightforward steps. The aim is to understand how a percentage price change impacts the quantity demanded percentage:1. **Derivative Importance**: Start with finding the derivative of the demand function. We have \( \frac{dq}{dp} = -4p \).2. **Determine Quantities**: Identify current quantity and price. Here, when \( p = 5 \), \( q = 150 \).3. **Elasticity Formula**: Use the formula\[ E = \left( \frac{dq}{dp} \right) \left( \frac{p}{q} \right) \]4. **Substitute Values**: Apply the derived values in the formula:\[E = (-4 \times 5) \left( \frac{5}{150} \right) = -20 \times \frac{1}{30} = -\frac{20}{30} = -\frac{2}{3}\]This calculation tells us that for every 1% increase in price, the quantity demanded falls by about 0.67%. Understanding how to compute elasticity helps businesses determine pricing strategies and foresee consumer reactions.