Problem 9 - Applied Calculus

Question

The demand curve for a quantity \(q\) of a product is \(q=\) \(5500-100 p\) where \(p\) is price in dollars. Interpret the 5500 and the 100 in terms of demand. Give units.

Step-by-Step Solution

Answer

5500 is the maximum quantity at $0, and -100 indicates a 100-unit decrease per $1 price increase.

Step 1: Understanding the Demand Equation

The demand equation given is \( q = 5500 - 100p \). This is a linear equation where \( q \) represents the quantity demanded and \( p \) represents the price of the product in dollars.

Step 2: Interpreting the '5500'

The constant term '5500' represents the intercept of the demand curve. This means that when the price \( p \) is zero, the quantity demanded \( q \) is 5500 units. In terms of demand, 5500 is the maximum quantity demanded when the price drops to zero.

Step 3: Analyzing the Coefficient '-100'

The coefficient of \( p \) is '-100'. This represents the slope of the demand curve. It indicates the rate at which the quantity demanded decreases per unit increase in price. In other words, for each additional dollar increase in price, the quantity demanded decreases by 100 units.

Key Concepts

Linear Demand Equation

A linear demand equation is a straightforward mathematical model expressing how quantity demanded varies with price. In a typical scenario, the equation is formulated as \( q = a - bp \). Here, \( q \) is the quantity demanded, \( p \) is the price, and \( a \) and \( b \) are constants. The constant \( a \) represents the intercept. This is the quantity demanded when price is zero. In our example, it's 5500, meaning at zero price, 5500 units are demanded. The coefficient \( b \) defines the slope. It tells us how much the quantity changes when the price changes by one unit.
Using the equation, we can predict how changes in price will likely affect demand in a linear fashion. A negative \( b \) suggests that as price increases, demand decreases - highlighting the basic law of demand. Such equations help businesses in pricing products by understanding consumer behavior patterns.

Price Elasticity

Price elasticity measures how sensitive the quantity demanded is to a change in price. This concept is pivotal in economics as it helps determine how a change in price might influence total revenue.

• Elastic Demand: When a small change in price causes a large change in demand.
• Inelastic Demand: When a significant change in price results in a minimal change in demand.

We calculate price elasticity by dividing the percentage change in quantity demanded by the percentage change in price. If the result is greater than one, demand is elastic. If less, demand is inelastic. In the case of the demand curve \( q = 5500 - 100p \), the elasticity can vary along the curve. Understanding elasticity helps businesses make informed pricing strategies, maximizing revenue while considering consumer reactions.

Quantity Demanded

The term "quantity demanded" refers to the total amount of a product that consumers are willing and able to purchase at a given price. It represents a point on the demand curve.
The demand equation \( q = 5500 - 100p \) demonstrates how quantity demanded differs according to price changes:

• At zero price, demand is at its maximum, 5500 units.
• As price increases, quantity demanded decreases at a constant rate.

This concept is crucial to understanding consumer purchasing behavior. By analyzing how quantity demanded shifts with price changes, businesses can adjust their supply chain, inventory management, and pricing policies to better match consumer demand.