To calculate the consumer surplus ($CS$) for the given demand equation $p = 23 - 2q^{\frac{1}{3}}$ at a unit price $\bar{p} = 9$, we use the formula for consumer surplus:

$CS = \int_0^{\bar{q}} \left[ D(q) - \bar{p} \right] dq$

Here:

• $D(q)$ is the demand function (in terms of price),
• $\bar{p} = 9$,
• $\bar{q}$ is the quantity where $p = \bar{p}$.

Step 1: Find $\bar{q}$

Substitute $\bar{p} = 9$ into the demand equation to find $\bar{q}$: $9 = 23 - 2q^{\frac{1}{3}}$ $2q^{\frac{1}{3}} = 23 - 9$ $2q^{\frac{1}{3}} = 14$ $q^{\frac{1}{3}} = 7$ $q = 7^3 = 343$

So, $\bar{q} = 343$.

Step 2: Set up the integral

The consumer surplus formula becomes: $CS = \int_0^{343} \left[ (23 - 2q^{\frac{1}{3}}) - 9 \right] dq$ $CS = \int_0^{343} \left[ 14 - 2q^{\frac{1}{3}} \right] dq$

Step 3: Solve the integral

Separate the integral: $CS = \int_0^{343} 14 \, dq - \int_0^{343} 2q^{\frac{1}{3}} dq$

First term:

$\int_0^{343} 14 \, dq = 14q \Big|_0^{343} = 14(343) - 14(0) = 4802$

Second term:

$\int_0^{343} 2q^{\frac{1}{3}} dq = 2 \int_0^{343} q^{\frac{1}{3}} dq$ Use the power rule for integration ($\int q^n \, dq = \frac{q^{n+1}}{n+1}$): $\int q^{\frac{1}{3}} dq = \frac{q^{\frac{4}{3}}}{\frac{4}{3}} = \frac{3}{4} q^{\frac{4}{3}}$ $\int_0^{343} q^{\frac{1}{3}} dq = \frac{3}{4} q^{\frac{4}{3}} \Big|_0^{343} = \frac{3}{4} (343^{\frac{4}{3}} - 0^{\frac{4}{3}})$ Since $343^{\frac{1}{3}} = 7$: $343^{\frac{4}{3}} = (343^{\frac{1}{3}})^4 = 7^4 = 2401$ $\int_0^{343} q^{\frac{1}{3}} dq = \frac{3}{4} (2401) = 1800.75$ $\int_0^{343} 2q^{\frac{1}{3}} dq = 2(1800.75) = 3601.5$

Step 4: Compute the consumer surplus

$CS = 4802 - 3601.5 = 1200.5$

Final Answer:

The consumer surplus is $1200.50.

Would you like additional details or further assistance?

Related Questions:

1. How does the consumer surplus change if the price is increased?
2. What is the maximum possible consumer surplus for this demand curve?
3. How would producer surplus be calculated in this case?
4. How does consumer surplus relate to elasticity of demand?
5. Can the formula for $D(q)$ represent inelastic or elastic demand?

Tip: Always verify that your units (e.g., dollars, quantity) are consistent across calculations.