The consumer surplus is the difference between what consumers are willing to pay and what they actually pay. It is calculated by finding the area between the demand curve and the price level up to the equilibrium quantity.

Given the demand function:

$Q_d(p) = \max(10 - 4p, 0)$

This function implies that the quantity demanded decreases as price $p$ increases, and it reaches 0 when the price is high enough that $10 - 4p \leq 0$. Let’s break this down step by step for each price scenario.

a) Consumer Surplus with Market Clearing Price $p = 0$

For $p = 0$: $Q_d(0) = \max(10 - 4(0), 0) = 10$

To calculate consumer surplus, we find the area under the demand curve up to $Q_d(0)$. Since the demand curve is linear, the consumer surplus is the area of the triangle between the demand curve and the price axis.

The maximum willingness to pay occurs when $p = 2.5$, where $Q_d(p) = 0$ because: $10 - 4p = 0 \quad \Rightarrow \quad p = 2.5$

So the consumer surplus when $p = 0$ is: $\text{CS} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 10 \times 2.5 = 12.5$

b) Consumer Surplus with Market Clearing Price $p = 1$

For $p = 1$: $Q_d(1) = \max(10 - 4(1), 0) = 6$

The maximum willingness to pay is still $2.5$. The area under the demand curve up to $Q_d(1)$ is again a triangle: $\text{CS} = \frac{1}{2} \times 6 \times (2.5 - 1) = \frac{1}{2} \times 6 \times 1.5 = 4.5$

c) Consumer Surplus with Market Clearing Price $p = 2$

For $p = 2$: $Q_d(2) = \max(10 - 4(2), 0) = 2$

The maximum willingness to pay is still $2.5$, and the consumer surplus is: $\text{CS} = \frac{1}{2} \times 2 \times (2.5 - 2) = \frac{1}{2} \times 2 \times 0.5 = 0.5$

d) Consumer Surplus with Market Clearing Price $p = 3$

For $p = 3$: $Q_d(3) = \max(10 - 4(3), 0) = 0$

Since the quantity demanded is 0, the consumer surplus is also 0 because no consumer is willing to buy at this price.

Summary of Results:

• a) CS when $p = 0$: 12.5
• b) CS when $p = 1$: 4.5
• c) CS when $p = 2$: 0.5
• d) CS when $p = 3$: 0

Would you like more details on any of the steps?

Here are five related questions for further exploration:

1. How does consumer surplus change with respect to shifts in demand?
2. What is the interpretation of a consumer surplus being zero?
3. How would a tax on goods affect consumer surplus?
4. What happens to consumer surplus if the demand function is quadratic instead of linear?
5. How can we calculate the producer surplus in this scenario?

Tip: Consumer surplus is maximized when the price is lower, as it captures the highest difference between what consumers are willing to pay and what they actually pay.