The government, through its Department of Trade, is analyzing the market for a new product. The demand and supply curves for the product are linear. At a price of $71 per unit, the quantity demanded is 542 units, and the quantity supplied is 360 units. At a price of $92 per unit, the quantity demanded is 458 units, and the quantity supplied is 486 units.
Given this information, calculate the consumer surplus (in $) at a market price of $60 per unit.
Note: Round your answer to 2-decimal places.

To calculate the consumer surplus at a market price of $60 per unit, we'll first need to determine the demand curve's equation. The demand curve is linear, so it can be represented as:

\[ Q_d = a - bP \]

Where:
- $ Q_d $ is the quantity demanded,
- $ P $ is the price,
- $ a $ and $ b $ are constants.

### Step 1: Find the Demand Curve
We are given two points on the demand curve:
1. At $ P = 71 $, $ Q_d = 542 $
2. At $ P = 92 $, $ Q_d = 458 $

Using these points, we can solve for $ a $ and $ b $.

First, calculate the slope $ b $:
\[
b = \frac{\Delta Q_d}{\Delta P} = \frac{458 - 542}{92 - 71} = \frac{-84}{21} = -4
\]

Now that we have $ b = -4 $, we can use one of the points to solve for $ a $. Using the point $ (P = 71, Q_d = 542) $:

\[
542 = a - 4 \times 71
\]
\[
542 = a - 284
\]
\[
a = 542 + 284 = 826
\]

So, the demand equation is:
\[
Q_d = 826 - 4P
\]

### Step 2: Find the Quantity Demanded at a Price of $60
Substitute $ P = 60 $ into the demand equation:
\[
Q_d = 826 - 4(60) = 826 - 240 = 586
\]

### Step 3: Determine the Maximum Willingness to Pay (Intercept of Demand Curve)
The maximum willingness to pay (where $ Q_d = 0 $) occurs when:
\[
0 = 826 - 4P
\]
\[
P = \frac{826}{4} = 206.5
\]

### Step 4: Calculate Consumer Surplus
Consumer surplus is the area of the triangle formed by the price, the demand curve, and the vertical axis. The formula for the area of a triangle is:
\[
\text{Consumer Surplus} = \frac{1}{2} \times \text{Base} \times \text{Height}
\]
Here, the base is the difference between the maximum willingness to pay (\$206.5) and the actual price (\$60), and the height is the quantity demanded at \$60 (586 units):
\[
\text{Consumer Surplus} = \frac{1}{2} \times (206.5 - 60) \times 586
\]
\[
\text{Consumer Surplus} = \frac{1}{2} \times 146.5 \times 586 = 42914.5
\]

Thus, the consumer surplus at a market price of $60 per unit is **$42,914.50**.

Would you like further details or have any questions?

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Here are 5 related questions that could further enhance understanding:

1. How would the consumer surplus change if the market price was set at $71 instead of $60?
2. What is the producer surplus at the price of $60, and how is it calculated?
3. Can you derive the supply curve using the same method as we did for the demand curve?
4. How does the concept of price elasticity relate to the linear demand curve?
5. How would government intervention in setting a price ceiling or floor affect consumer surplus?

**Tip:** When working with linear demand and supply curves, always check if the slopes (coefficients of $P$) are intuitive; for example, a negative slope for demand indicates that as price increases, quantity demanded decreases, which aligns with typical economic behavior.

Learn how to calculate consumer surplus in economics using demand and supply analysis. Solve for consumer surplus at a market price of $60 per unit with a linear demand curve equation. Understand the economic implications of consumer surplus and apply mathematical methods to real-world market scenarios.

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