Problem 2 - Calculus and its applications

Question

\(D(x)\) is the price, in dollars per unit, that consumers will pay for \(x\) units of an item, and \(S(x)\) is the price, in dollars per unit, that producers will accept for \(x\) units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. \(D(x)=-\frac{5}{6} x+9, \quad S(x)=\frac{1}{2} x+1\)

Step-by-Step Solution

Answer

Equilibrium point: (6, 4). Consumer surplus: 15. Producer surplus: 9.

Step 1: Set Equations Equal

To find the equilibrium point, set the demand equation equal to the supply equation: \(-\frac{5}{6}x + 9 = \frac{1}{2}x + 1\)

Step 2: Clear Fractions and Solve for x

Multiply all terms by 6 to eliminate fractions: \(-5x + 54 = 3x + 6\). Rearrange to solve for \(x\): \(-5x - 3x = 6 - 54\) leading to \(-8x = -48\). Solve for \(x\): \(x = 6\).

Step 3: Find Equilibrium Price

Substitute \(x = 6\) back into either \(D(x)\) or \(S(x)\) to find the equilibrium price. Using \(D(x)\): \(D(6) = -\frac{5}{6}(6) + 9 = -5 + 9 = 4\).So, the equilibrium point is (6, 4).

Step 4: Calculate Consumer Surplus

Consumer surplus is the area under the demand curve above the equilibrium price. Integrate the demand function from \(0\) to \(6\) and subtract \(4 \times 6\):\[ \int_{0}^{6} \left(-\frac{5}{6}x + 9\right)\,dx - 4 \times 6. \]Calculate the integral: \[\left[ -\frac{5}{12}x^2 + 9x \right]_0^6 = -\frac{5}{12} (6)^2 + 9(6) = -15 + 54 = 39.\]the consumer surplus = \(39 - 24 = 15\).

Step 5: Calculate Producer Surplus

Producer surplus is the area above the supply curve and below the equilibrium price. Integrate the supply function from \(0\) to \(6\) and subtract from \(4 \times 6\):\[4 \times 6 - \int_{0}^{6} \left(\frac{1}{2}x + 1\right)\, dx.\]Calculate the integral: \[ \left[ \frac{1}{4}x^2 + x \right]_0^6 = \frac{1}{4} (6)^2 + 6 = 9 + 6 = 15.\] Producer surplus = \(24 - 15 = 9\).

Key Concepts

Consumer Surplus

Consumer surplus is an important economic concept that describes the difference between what consumers are willing to pay for a good or service, and what they actually pay. It's essentially the extra satisfaction or benefit that consumers receive from purchasing a product at a market price that is less than the highest price they would be willing to pay.

• This surplus is illustrated graphically as the area below the demand curve and above the price level up to the point of equilibrium.
• In mathematical terms, it's calculated using the integral of the demand function minus the total revenue, which is the price at equilibrium times quantity.

For our specific exercise, the demand function is given by: \(D(x) = -\frac{5}{6}x + 9\). At equilibrium, the price is 4 and the quantity is 6.
Calculate the consumer surplus by integrating the demand function from 0 to 6 and subtracting the payment at equilibrium: \[ \int_{0}^{6} \left( -\frac{5}{6}x + 9 \right) \, dx - 4 \times 6.= 15\].
Therefore, the consumer surplus is $15.

Producer Surplus

Producer surplus is a measure of producer benefit, defined as the difference between what producers are willing to accept for a good or service and what they actually receive. It is akin to profit in that it captures the extra benefit producers get from selling at a higher market price than the minimum they would accept.

• The producer surplus is depicted diagrammatically as the area above the supply curve and below the equilibrium price up to the equilibrium quantity.
• Mathematically, it is found by integrating the supply function and then subtracting this from the area of the rectangle formed by the equilibrium price and quantity.

In the given exercise, the supply function is: \(S(x) = \frac{1}{2}x + 1\).
The equilibrium point results in a price of 4 and a quantity of 6. Calculate the producer surplus by subtracting the integral of the supply function from the total revenue: \[ 4 \times 6 - \int_{0}^{6} \left( \frac{1}{2}x + 1 \right) \, dx = 9\].
Thus, the producer surplus is $9.

Demand and Supply Functions

In economics, demand and supply functions are fundamental to understanding market dynamics. They collectively determine the equilibrium price and quantity in a market.

• The **demand function** describes how much of a product consumers are willing to purchase at different prices. It typically slopes downward, reflecting that higher prices lead to lower demand.
• The **supply function** shows how much of a product producers are willing to supply at different prices. This usually has an upward slope, indicating that higher prices incentivize producers to supply more.

For our practical example, the demand is represented by \(D(x) = -\frac{5}{6}x + 9\), decreasing as the number of units, x, increases.
The supply equation is \(S(x) = \frac{1}{2}x + 1\), indicating an increase in price with more units supplied.
To find the **equilibrium** point where supply equals demand, set both functions equal: \(-\frac{5}{6}x + 9 = \frac{1}{2}x + 1\).
Solving gives the equilibrium quantity and price, pinpointing where market demand aligns perfectly with market supply. For this problem, the equilibrium occurs at 6 units and $4.