Question

Given the following market demand, \(Q^{D}=120-2 P\), find the consumers' surplus when \(P=£ 20\) and \(P=£ 15\). Calculate a demand schedule and then draw a graph showing consumer surplus.

Short answer

Consumer surplus is £1600 at £20 and £2025 at £15.

Step-by-step solution

Understanding Demand Function

The demand function given is \( Q^D = 120 - 2P \). Here, \( Q^D \) represents the quantity demanded, and \( P \) is the price. This equation shows how quantity demanded changes with price.

Calculate Quantity Demanded at Given Prices

Substitute \( P = 20 \) and \( P = 15 \) into the demand function to find the corresponding \( Q^D \). For \( P = 20 \): \[ Q^D = 120 - 2(20) = 120 - 40 = 80 \]For \( P = 15 \): \[ Q^D = 120 - 2(15) = 120 - 30 = 90 \]

Find Consumer Surplus for Each Price

Consumer surplus is the difference between what consumers are willing to pay versus what they actually pay. The maximum price consumers are willing to pay (\( P_{max} = 60 \)) is where \( Q^D = 0 \) (set \( 120 - 2P = 0 \) to find when consumers demand no units). Consumer surplus is given by:\[ \text{Consumer Surplus} = \frac{1}{2} \times (\text{Base}) \times (\text{Height}) \]For \( P = 20 \):\[ \text{Base} = 80, \text{Height} = 60 - 20 = 40 \]\[ \text{Consumer Surplus} = \frac{1}{2} \times 80 \times 40 = 1600 \]For \( P = 15 \):\[ \text{Base} = 90, \text{Height} = 60 - 15 = 45 \]\[ \text{Consumer Surplus} = \frac{1}{2} \times 90 \times 45 = 2025 \]

Create Demand Schedule

Create a demand schedule by listing price levels and corresponding quantities: | Price (P) | Quantity Demanded (Q^D) | |-----------|-----------------------| | 0 | 120 | | 10 | 100 | | 15 | 90 | | 20 | 80 | | 30 | 60 | | 40 | 40 | | 50 | 20 | | 60 | 0 |

Plot Demand Curve and Consumer Surplus

To plot the demand curve, use the demand schedule. The x-axis represents Quantity Demanded (\( Q^D \)) and the y-axis represents Price (\( P \)). The consumer surplus is the area under the demand curve and above the price level up to the quantity demanded.For \( P = 20 \), shade the area from \( Q = 0 \) to \( Q = 80 \) above the price line at 20.For \( P = 15 \), shade the area from \( Q = 0 \) to \( Q = 90 \) above the price line at 15. These represent the consumer surplus for the respective prices.

Key concepts

Demand Function

The demand function is a mathematical formula that shows the relationship between the quantity of a good demanded and its price. In this context, the given demand function is \( Q^D = 120 - 2P \), where \( Q^D \) represents the quantity demanded and \( P \) denotes the price. This function illustrates that as the price \( P \) rises, the quantity demanded \( Q^D \) decreases. This inverse relationship is a fundamental principle of the law of demand. Understanding the demand function helps predict how changes in price will affect consumer purchasing behavior.

Demand Schedule

A demand schedule lists various prices of a good or service alongside the corresponding quantities demanded. It provides a snapshot of how demand changes with price. To construct a demand schedule, we substitute different prices into the demand function to calculate the quantity demanded.

• Price £0: Quantity Demanded is 120
• Price £10: Quantity Demanded is 100
• Price £15: Quantity Demanded is 90
• Price £20: Quantity Demanded is 80
• Price £30: Quantity Demanded is 60
• Price £40: Quantity Demanded is 40
• Price £50: Quantity Demanded is 20
• Price £60: Quantity Demanded is 0

With the demand schedule, you can see how quantity demanded decreases as the price increases, confirming the foundational concept of demand in economics.

Market Demand

Market demand refers to the total quantity of a good or service that all consumers in a market are willing and able to purchase at various price levels. It aggregates individual demands and provides a comprehensive view of the market's reaction to price changes. In our example, the market demand is represented by the function \( Q^D = 120 - 2P \). This equation accumulates the demands of all consumers into a single function. By analyzing market demand, businesses can anticipate their total sales and adjust pricing strategies accordingly.

Demand Curve

The demand curve is a graphical representation of the demand schedule. It depicts the relationship between the price of a good and the quantity demanded. Typically, the curve slopes downwards from left to right, indicating the inverse relationship: as price decreases, demand increases. To plot the demand curve based on the demand schedule:

• Place the quantity demanded on the x-axis.
• Place the price on the y-axis.
• Plot each price and its corresponding quantity demanded.
• Connect these points to form the demand curve.

The area under the demand curve and above the horizontal line at the given price level represents the consumer surplus. The consumer surplus is the additional benefit consumers receive because they are paying less than what they are willing to pay. For the prices given, the area under the curve up to the respective quantities reflects the different levels of consumer surplus.