Problem 3 - 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. \(D(x)=(x-3)^{2}, \quad S(x)=x^{2}+2 x+1\)

Step-by-Step Solution

Answer

The equilibrium point is \((1, 4)\). The consumer surplus is \(\frac{2}{3}\) dollars, and the producer surplus is \(\frac{5}{3}\) dollars.

Step 1: Set Demand Equal to Supply

To find the equilibrium point, we need to set the demand function equal to the supply function: \[(x-3)^2 = x^2 + 2x + 1\] This will help us find the value of \(x\) where the price consumers are willing to pay equals the price producers are willing to accept.

Step 2: Solve for x in the Equilibrium Equation

Expand and simplify the equilibrium equation:\[(x-3)^2 = x^2 - 6x + 9\]Equate the equations:\[x^2 - 6x + 9 = x^2 + 2x + 1\]Subtract \(x^2\) from both sides:\[-6x + 9 = 2x + 1\]Rearrange and solve for \(x\):\[-8x = -8\]\[x = 1\]

Step 3: Find the Equilibrium Price

Substitute \(x = 1\) back into the demand function to find the equilibrium price:\[D(1) = (1-3)^2 = 4\]Thus, the equilibrium point is \((1,4)\).

Step 4: Calculate Consumer Surplus

The consumer surplus is found using the integral of the demand function from \(x = 0\) to \(x = 1\), minus the equilibrium quantity times price:\[\text{Consumer Surplus} = \int_0^1 (x-3)^2 \, dx - (1 \times 4)\]First, find the integral:\[\int_0^1 (x-3)^2 \, dx = \int_0^1 (x^2 - 6x + 9) \, dx\]\[= \left[ \frac{x^3}{3} - 3x^2 + 9x \right]_0^1 = \left(\frac{1}{3} - 3 + 9\right) - 0 = \frac{10}{3}\]Subtract the equilibrium quantity times price:\[\frac{10}{3} - 4 = \frac{2}{3}\]

Step 5: Calculate Producer Surplus

The producer surplus is found using the integral of the supply function from \(x = 0\) to \(x = 1\), minus the equilibrium quantity times price:\[\text{Producer Surplus} = (1 \times 4) - \int_0^1 (x^2 + 2x + 1) \, dx\]First, find the integral:\[\int_0^1 (x^2 + 2x + 1) \, dx = \left[ \frac{x^3}{3} + x^2 + x \right]_0^1 = \left(\frac{1}{3} + 1 + 1 \right) - 0 = \frac{7}{3}\]Subtract the integral from the equilibrium quantity times price:\[4 - \frac{7}{3} = \frac{5}{3}\]

Key Concepts

Consumer Surplus

Consumer surplus is a critical concept in economics, representing the difference between the highest price a consumer is willing to pay for a good and the actual price they pay. It provides insights into consumer satisfaction and economic welfare.

In the context of supply and demand, consumer surplus is visually depicted as the area above the equilibrium price and below the demand curve. This area signifies the benefit consumers receive by paying less than what they would be willing to pay.

• When calculating consumer surplus, you integrate the demand function from zero to the quantity at the equilibrium point.
• The equilibrium point occurs when the quantity demanded equals the quantity supplied.
• Subtract the total cost at equilibrium from the total benefit to find the surplus.

In the given exercise, by evaluating the integral of the demand function from 0 to 1, we find that the consumer surplus is \(\frac{2}{3}\). This means that consumers collectively benefit by that amount due to the price being lower than their willingness to pay at equilibrium.

Producer Surplus

Producer surplus is the financial gain producers receive, measured by the difference between the price at which they are willing to sell a good and the price they actually receive. It illustrates the benefit producers obtain from market transactions.

Graphically, producer surplus is the area below the equilibrium price and above the supply curve. This represents the difference between what producers are willing to accept and what they actually get.

• To find producer surplus, you integrate the supply function from zero to the equilibrium quantity.
• The equilibrium price and quantity are determined by where the demand and supply curves intersect.
• You then compare the total revenue at equilibrium with the area under the supply curve to find the surplus.

In the problem, the producer surplus calculated is \(\frac{5}{3}\), suggesting that producers benefit to that extent because they receive more than their minimum selling price at equilibrium.

Supply and Demand

The concepts of supply and demand are foundational to economics. They explain how prices and quantities of goods are determined in a market system.

**Equilibrium Point**The equilibrium point is where the supply of a good perfectly meets the demand. At this point, the market is said to be in balance.

• Supply reflects the quantity of a good producers are willing to sell at different price levels.
• Demand shows the quantity consumers are ready to purchase at various prices.
• Intersection of supply and demand curves determines the equilibrium price and quantity.

In the exercise, solving the equation \((x-3)^2 = x^2 + 2x + 1\) helps identify this critical point. The solution revealed that when \(x = 1\), both curves meet with an equilibrium price of 4, indicating a balanced market at that quantity and price.