Question

If the market demand curve is \(D(p)=100-.5 p,\) what is the inverse demand curve?

Short answer

The inverse demand curve is \( p = 200 - 2Q \).

Step-by-step solution

Understand the Demand Curve Equation

The given demand curve represents quantity demanded as a function of price, denoted by \( D(p) = 100 - 0.5p \), where \( Q \) is the quantity demanded and \( p \) is the price.

Swap Variables for Inverse Demand

To find the inverse demand curve, you'll need to express price as a function of quantity demanded. Set \( Q = 100 - 0.5p \) and solve for \( p \).

Rearrange the Equation

Start by rearranging the equation to isolate terms involving \( p \) on one side. So, add \( 0.5p \) to both sides to get \( Q + 0.5p = 100 \).

Solve for Price

Isolate \( p \) by subtracting \( Q \) from both sides of the equation: \( 0.5p = 100 - Q \). Then, divide both sides by 0.5 to solve for \( p \). You get \( p = 200 - 2Q \).

State the Inverse Demand Curve

The inverse demand curve is the equation for price in terms of quantity demanded, which is \( p = 200 - 2Q \).

Key concepts

Market Demand Curve

A market demand curve is a graphical representation that illustrates the relationship between the price of a good or service and the quantity that consumers are willing and able to purchase at each price level. This curve is typically downward sloping, indicating that as the price decreases, the quantity demanded increases, and vice versa.

• This is derived from the law of demand, which states that, all else being equal, an increase in price results in a decrease in quantity demanded.
• The curve is assumed to reflect consumer behavior and serves as a vital tool for businesses and economists to anticipate the impacts of pricing changes.

In the given exercise, the market demand curve is modeled by the equation \( D(p) = 100 - 0.5p \), which helps in predicting how many units will be sold at different prices. This linear equation represents a straight-line demand curve on a graph with price \( p \) on the vertical axis and quantity demanded \( Q \) on the horizontal axis.

Quantity Demanded

Quantity demanded refers to the total number of units of a good or service that consumers are willing to purchase at a given price. It's an essential concept in determining how market dynamics work. The quantity demanded depends on a variety of factors, such as:

• Price Changes: Typically, a lower price increases the quantity demanded, while a higher price decreases it, reflecting a movement along the demand curve.
• Substitute Goods: Availability and pricing of substitutes can affect quantity demanded for a particular good.
• Income Levels: A higher disposable income usually boosts the ability and willingness to buy more goods.

In the formula \( D(p) = 100 - 0.5p \), every price \( p \) has a corresponding quantity demanded \( Q \). For example, if the price is \( 20 \), the quantity demanded would be \( 100 - 0.5 imes 20 = 90 \). Understanding quantity demanded helps businesses set appropriate prices for their goods.

Price Function

The price function, also known as the inverse demand curve, expresses price as a function of the quantity demanded, reversing the typical demand curve relationship. This concept is useful for companies when determining how much they can charge based on expected sales volume. In this exercise, by finding the inverse demand curve from the original demand equation, we get \( p = 200 - 2Q \). Here’s why this process matters:

• The inverse demand curve allows businesses to adapt pricing strategies by understanding how price levels must adjust to meet a specific quantity demanded.
• This can be particularly valuable in maximizing revenue by identifying the optimal price and quantity balance.

By rearranging the demand equation \( Q = 100 - 0.5p \) to isolate price, we convert it to \( p = 200 - 2Q \). This conversion emphasizes the adaptability of price in response to demand changes, a valuable insight in pricing strategy development.