Problem 5 In each \(D(x)\) is the price, i... [FREE SOLUTION] (2024)

Short Answer

Consumer Surplus Calculation

Consumer surplus represents the difference between what consumers are willing to pay for a good and what they actually pay at the equilibrium price. This surplus is found in the area below the demand curve but above the equilibrium price line, up to the equilibrium quantity.
In our exercise, the demand function is \(D(x) = (x - 6)^2\).
Here’s how we calculate it step-by-step:
1. Find the maximum price consumers are willing to pay when \(x = 0\):
\(D(0) = (0-6)^2 = 36\).
2. Identify the equilibrium price, \(P = 9\).
3. Determine the height difference: \(36 - 9 = 27\).
4. Calculate the consumer surplus as a triangle's area with base \(3\) (the equilibrium quantity) and height \(27\):
\( \text{Consumer Surplus} = \frac{1}{2} \times 27 \times 3 = 40.5\).
Therefore, the consumer surplus at the equilibrium point is \(40.5\).

Producer Surplus Calculation

Producer surplus is the difference between what producers are willing to accept for a good and what they actually receive at the equilibrium price. This surplus is found in the area above the supply curve but below the equilibrium price line, up to the equilibrium quantity.
In the given exercise, the supply function is \(S(x) = x^2\).
Let’s break down how we calculate this step-by-step:
1. Find the minimum price producers are willing to accept when \(x = 0\):
\(S(0) = (0)^2 = 0\).
2. Identify the equilibrium price, \(P = 9\).
3. Determine the height difference: \(9 - 0 = 9\).
4. Calculate the producer surplus as a triangle's area with base \(3\) (the equilibrium quantity) and height \(9\):
\( \text{Producer Surplus} = \frac{1}{2} \times 9 \times 3 = 13.5\).
Hence, the producer surplus at the equilibrium point is \(13.5\).

Demand and Supply Functions

Equilibrium Price

The equilibrium price is the price at which the quantity of goods demanded by consumers equals the quantity supplied by producers.
To find the equilibrium price, we set the demand function equal to the supply function and solve for \(x\):\[(x-6)^2 = x^2\].
Solving this equation:
1. Expand and simplify the equation: \(x^2 - 12x + 36 = x^2\).
2. Subtract \(x^2\) from both sides: \(-12x + 36 = 0\).
3. Solve for \(x\): \(x = 3\).
Plugging \(x\) back into either function to find price: \(P = (3-6)^2 = 9\).
Thus, the equilibrium price is \(9\) dollars per unit, and the equilibrium quantity is 3 units.
At this point, the demand and supply match perfectly, balancing the market.