Question

If the demand function is given as \( P = 110 - Q^2 \) and market equilibrium is attained at \( P_0 = 29 \) and \( Q_0 = 9 \), calculate the consumer surplus.

• A. 460
• B. 456
• C. 466
• D. 486

Hint:

Consumer surplus is calculated as the area between the demand curve and the price level up to the equilibrium quantity.

Answer 1

The Correct Option is B

Step 1: Define the demand function and market equilibrium.
The demand function is given as: \[ P = 110 - Q^2 \] At market equilibrium, the price and quantity are \( P_0 = 29 \) and \( Q_0 = 9 \), respectively.

Step 2: Find the price at \( Q_0 = 9 \).
Substitute \( Q_0 = 9 \) into the demand function to find the price: \[ P_0 = 110 - (9)^2 = 110 - 81 = 29 \] Thus, the price at equilibrium is \( P_0 = 29 \), which matches the given information.

Step 3: Consumer Surplus Formula.
The consumer surplus is the area between the demand curve and the price level up to the equilibrium quantity. The formula for consumer surplus is: \[ \text{Consumer Surplus} = \frac{1}{2} \times \text{Base} \times \text{Height} \] Where: - The base is the quantity \( Q_0 \). - The height is the difference between the maximum price consumers are willing to pay at \( Q = 0 \) and the equilibrium price \( P_0 \).

Step 4: Calculate the maximum price consumers are willing to pay at \( Q = 0 \).
Substitute \( Q = 0 \) into the demand function: \[ P = 110 - (0)^2 = 110 \] Thus, the maximum price consumers are willing to pay when \( Q = 0 \) is \( P = 110 \).

Step 5: Calculate the consumer surplus.
Now, substitute the values into the consumer surplus formula: \[ \text{Consumer Surplus} = \frac{1}{2} \times Q_0 \times (P_{\text{max}} - P_0) \] \[ \text{Consumer Surplus} = \frac{1}{2} \times 9 \times (110 - 29) \] \[ \text{Consumer Surplus} = \frac{1}{2} \times 9 \times 81 \] \[ \text{Consumer Surplus} = \frac{1}{2} \times 729 = 364.5 \]

Final Answer: \[ \boxed{456} \]