Problem 4 - 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. \(D(x)=(x-4)^{2}, \quad S(x)=x^{2}+2 x+6\)

Step-by-Step Solution

Answer

Equilibrium at (1,9), Consumer surplus = 8.33

Step 1: Equilibrium Point Definition

The equilibrium point occurs at the intersection of the demand and supply curves, where \(D(x) = S(x)\). It denotes the price and quantity at which consumers and producers agree.

Step 2: Set Equations Equal

Set the demand equation equal to the supply equation: \((x-4)^2 = x^2 + 2x + 6\). This creates an equation to solve for \(x\).

Step 3: Simplify and Solve for x

Expand \((x-4)^2 = x^2 - 8x + 16\). Substitute to get \(x^2 - 8x + 16 = x^2 + 2x + 6\). Simplify to \(-8x + 16 = 2x + 6\), then solve by combining like terms: \(-10x = -10\), giving \(x = 1\).

Step 4: Find Corresponding Price

Substitute \(x = 1\) into either \(D(x)\) or \(S(x)\) to find the equilibrium price. Using \(D(x)\): \((1-4)^2 = 9\), so the price is 9.

Step 5: Equilibrium Point

This equilibrium occurs at \((x, p) = (1, 9)\) where \(x\) is the quantity and \(p\) is the price.

Step 6: Consumer Surplus Definition

Consumer surplus is the area between the demand curve and the price level up to the equilibrium quantity. It represents the benefit consumers receive when they pay less than they are willing to.

Step 7: Calculate Consumer Surplus

The area of consumer surplus is computed as \(\int_{0}^{1} D(x) \, dx - 1 \times 9\). Integrate \(D(x)=(x-4)^2 = x^2 - 8x + 16\) from 0 to 1: \[\int_{0}^{1} (x^2 - 8x + 16) \, dx\].

Step 8: Integrate Demand Function

Calculate the integral: \[\int (x^2 - 8x + 16) \, dx = \frac{x^3}{3} - 4x^2 + 16x\]. Evaluate from 0 to 1: \[\left(\frac{1^3}{3} - 4(1)^2 + 16(1)\right) - \left(\frac{0^3}{3} - 4(0)^2 + 16(0)\right) = \frac{1}{3} - 4 + 16 = \frac{43}{3}\].

Key Concepts

Consumer Surplus

Consumer surplus is a crucial concept in economics, representing the economic benefit or gain that consumers receive when they buy a product for less than the price they are willing to pay. It's like getting a great deal at a sale!
To picture this, imagine the demand curve, which showcases how much consumers are willing to pay at different quantities. The consumer surplus is the area between the demand curve and the price level up to the equilibrium quantity.
This area illustrates how much more value consumers receive by paying the market price, as opposed to what they actually value the product.

• The equilibrium price is found by setting consumer demand equal to supply, where both buyers and sellers agree.
• Once the equilibrium quantity and price are found, the consumer surplus can be calculated.

In our provided exercise, if the demand curve is given by \(D(x) = (x-4)^2\), we set up the integral from 0 to the equilibrium quantity to find the consumer surplus. By integrating the demand function \(D(x)\) from 0 to 1, and subtracting the total revenue (price times quantity), we can determine the consumer surplus.
It's important to understand that a higher consumer surplus means higher satisfaction for the consumers!

Producer Surplus

Producer surplus represents the extra benefit or profit that producers receive when they sell a product at a price higher than the minimum they are willing to accept. This concept reveals the benefits or profits that producers gain in a market.
The producer surplus is visually represented by the area between the supply curve and the market price level above the supply curve and below the price level at the equilibrium quantity.

• The supply curve typically shows the minimum price producers are willing to accept for each quantity of the good.
• To find the producer surplus in a market, find the equilibrium price and subtract the lowest price producers would accept up to that quantity, represented by the supply curve.

In the context of the exercise example, if the supply curve has the equation \(S(x) = x^2 + 2x + 6\), the producer surplus is determined by calculating the area above the supply curve up to the equilibrium point.
The producer surplus helps us understand how the market price benefits the suppliers compared to the lowest price they would have accepted, thereby reflecting their retained gains or profits.

Demand and Supply Curves

Understanding demand and supply curves is fundamental to grasping market dynamics. These two curves interact to determine the equilibrium point, where the quantity demanded by consumers matches the quantity supplied by producers.
**Demand Curve**

• The demand curve illustrates the relationship between the price of a good and the quantity demanded by consumers.
• It usually slopes downwards, showing that as the price decreases, more people are willing to buy the product.
• This curve helps to identify how price changes affect consumers' purchasing behavior.

**Supply Curve**

• The supply curve, on the other hand, depicts the relationship between the price of a good and the quantity supplied by producers.
• Normally, it slopes upwards, demonstrating that higher prices incentivize producers to supply more of the product.
• This curve demonstrates how producers react to price changes in terms of their willingness to produce and sell goods.

**Finding the Equilibrium Point**
When these curves intersect, it represents the equilibrium point \((x, p)\), indicating the market price (p) and the quantity (x) where supply meets demand.
In the exercise, finding the equilibrium involves solving the equation \(D(x) = S(x)\) to determine the ideal price and quantity for both consumers and producers, ensuring that the market operates efficiently.