This solution explains how to calculate the consumer surplus given a specific demand function and market equilibrium point.

Understanding Consumer Surplus

Consumer surplus represents the economic gain consumers receive when they are willing to pay more for a product than the actual market price. It's calculated as the area below the demand curve and above the market price line, up to the quantity consumed.

Given Information

• Demand Function: $P = 110 - Q^2$
• Market Equilibrium Price: $P_0 = 29$
• Market Equilibrium Quantity: $Q_0 = 9$

Calculating Consumer Surplus

To find the consumer surplus, we need to calculate the area between the demand curve and the equilibrium price level ($P_0$) from quantity zero ($Q=0$) up to the equilibrium quantity ($Q_0=9$).

Step 1: Integrate the Demand Function

The area under the demand curve is found by integrating the demand function with respect to quantity ($Q$) from 0 to $Q_0$.

Area under demand curve = $\int_{0}^{Q_0} P \,dQ$

Substitute the demand function $P = 110 - Q^2$ and the equilibrium quantity $Q_0 = 9$:

Area = $\int_{0}^{9} (110 - Q^2) \,dQ$

Step 2: Evaluate the Integral

Find the antiderivative of the demand function:

Antiderivative = $110Q - \frac{Q^3}{3}$

Now, evaluate this antiderivative at the limits of integration (from $Q=0$ to $Q=9$):

Area = $\left[ 110Q - \frac{Q^3}{3} \right]_{0}^{9}$

Area = $\left( 110(9) - \frac{(9)^3}{3} \right) - \left( 110(0) - \frac{(0)^3}{3} \right)$

Area = $\left( 990 - \frac{729}{3} \right) - (0 - 0)$

Area = $990 - 243$

Area = $747$

This value ($747$) represents the total value consumers place on the first 9 units of the good.

Step 3: Calculate Total Expenditure

The total amount consumers actually pay is the equilibrium price multiplied by the equilibrium quantity:

Total Expenditure = $P_0 \times Q_0$

Total Expenditure = $29 \times 9$

Total Expenditure = $261$

Step 4: Calculate Consumer Surplus

Consumer Surplus is the difference between the total value consumers are willing to pay (Area under demand curve) and the total amount they actually pay (Total Expenditure).

Consumer Surplus = Area under demand curve - Total Expenditure

Consumer Surplus = $747 - 261$

Consumer Surplus = $486$

Conclusion

The consumer surplus at the market equilibrium of $P_0 = 29$ and $Q_0 = 9$, with the demand function $P = 110 - Q^2$, is $486$.