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

Step-by-Step Solution

Answer

(a) Equilibrium point: (3, 9)\n(b) Consumer surplus: 36\n(c) Producer surplus: 18

Step 1: Find the Equilibrium Point

To find the equilibrium point, we need to set the demand function equal to the supply function, i.e., \(D(x) = S(x)\). So, we solve \((x-6)^2 = x^2\).First, expand \((x-6)^2\): \((x-6)^2 = x^2 - 12x + 36\).So the equation becomes:\[x^2 - 12x + 36 = x^2\]Subtract \(x^2\) from both sides:\[-12x + 36 = 0\]Solve for \(x\):\[12x = 36\]\[x = 3\].Substitute \(x = 3\) back into either \(D(x)\) or \(S(x)\) to find the equilibrium price. Using \(S(x)\):\[S(3) = 3^2 = 9\].Thus, the equilibrium point is \((3, 9)\).

Step 2: Calculate Consumer Surplus

Consumer surplus is the area between the demand curve and the equilibrium price level, up to the equilibrium quantity.The consumer surplus is given by the integral:\[\text{Consumer Surplus} = \int_{0}^{3} ((x-6)^2 - 9) \, dx\]In order to find this integral:\(\int ((x-6)^2 - 9) \, dx = \int (x^2 - 12x + 36 - 9) \, dx \ = \int (x^2 - 12x + 27) \, dx = \left[ \frac{x^3}{3} - 6x^2 + 27x \right]_0^3\).Plug in the limits:\[= \left( \frac{3^3}{3} - 6(3)^2 + 27(3) \right) - \left( 0 \right)= \left( 9 - 54 + 81 \right)= 36\].Therefore, the consumer surplus is 36.

Step 3: Calculate Producer Surplus

Producer surplus is the area between the supply curve and the equilibrium price level, up to the equilibrium quantity.The producer surplus is given by the integral:\[\text{Producer Surplus} = \int_{0}^{3} (9 - x^2) \, dx\]In order to find this integral:\(\int (9 - x^2) \, dx = \left[ 9x - \frac{x^3}{3} \right]_0^3\).Plug in the limits:\[= \left(9(3) - \frac{3^3}{3} \right) - \left(0\right)= \left( 27 - 9 \right)= 18\].Therefore, the producer surplus is 18.

Key Concepts

Consumer Surplus

Consumer surplus is an important concept in economics, representing the difference between what consumers are willing to pay and what they actually pay for a product. It’s like getting more value than what you paid for.
In our exercise, this surplus is found at the equilibrium point, where the demand and supply functions meet. When we calculated the consumer surplus, we used the demand curve and the equilibrium price. The formula \[\text{Consumer Surplus} = \int_{0}^{3} ((x-6)^2 - 9) \, dx\]helps in determining this area under the demand curve and above the equilibrium price line.
By solving the integral, we found a consumer surplus of 36. This means consumers are collectively saving or gaining 36 units of value beyond what they spent. This area represents the potential benefit to consumers because they are buying the product for less than the maximum price they are willing to pay.

Producer Surplus

Producer surplus is the flip side of consumer surplus, related to sellers. It shows how much more money producers are making compared to the minimum price they’re willing to accept. This extra money is their reward for producing and selling goods.
In this exercise, the producer surplus is calculated as the area between the supply curve and the equilibrium price, up to the equilibrium quantity. We determine it using the simple integral\[\text{Producer Surplus} = \int_{0}^{3} (9 - x^2) \, dx\]The producer surplus here is found to be 18, showing us the additional value producers receive. This extra amount signifies that producers are benefiting from selling their product for more than the lowest price they would be willing to sell it.
Understanding producer surplus helps in recognizing how well producers are doing in the market and how much profit they might be making over the break-even price.

Demand and Supply Functions

In economics, demand and supply functions are crucial to understanding how a market operates. They show the relationship between the price of a product and the quantity demanded or supplied.
The demand function \(D(x)\) in our problem is \((x-6)^{2}\), and it shows the price that consumers will pay for \(x\) units. On the other hand, the supply function \(S(x)\), represented by \(x^{2}\), reflects the price producers need for supplying \(x\) units.
When these two functions intersect, we find the equilibrium point, where the quantity demanded by consumers equals the quantity supplied by producers. It's basically the sweet spot in the market where buyers and sellers agree. In our example, the intersection gives an equilibrium point of \((3, 9)\). This means at 3 units, the market is in balance with the price set at 9 dollars.
The demand and supply functions and their intersection at equilibrium help us understand consumer habits and production strategies, revealing key insights into market behavior.