Problem 21 - Calculus and its applications

Question

Graph each pair of demand and supply functions. Then: a) Find the equilibrium point using the INTERSECT feature or another feature that will allow you to find this point of intersection. b) Graph \(y=D\left(x_{\mathrm{E}}\right)\) and identify the regions of both consumer and producer surpluses. c) Find the consumer surplus. d) Find the producer surplus. \(D(x)=\frac{x+8}{x+1}, \quad S(x)=\frac{x^{2}+4}{20}\)

Step-by-Step Solution

Answer

Graph the demand and supply functions to find the intersection; this is the equilibrium point. Calculate the consumer surplus as the area between the demand curve and equilibrium line, and producer surplus as the area between the equilibrium line and supply curve.

Step 1: Plot the Demand and Supply Functions

Begin by graphing the demand function \(D(x) = \frac{x+8}{x+1}\) and the supply function \(S(x) = \frac{x^2+4}{20}\). Create a coordinate plane with \(x\) representing the quantity and \(y\) representing the price. You'll observe where these two curves intersect, which is crucial for finding the equilibrium.

Step 2: Determine the Equilibrium Point

At the equilibrium point, the quantity demanded equals the quantity supplied, meaning the graphs of the demand and supply functions intersect. Solve the equation \(\frac{x+8}{x+1} = \frac{x^2+4}{20}\) to find the equilibrium \(x_E\). Simplifying will provide the equilibrium quantity. Use the INTERSECT feature on your calculator or graphing tool to find \(x_E\) and the corresponding \(y_E = D(x_E) = S(x_E)\).

Step 3: Graph the Equilibrium Price Line

Once you have \(x_E\), graph the horizontal line \(y = D(x_E)\) across the range where demand and supply are defined. This line represents the equilibrium price at the equilibrium quantity.

Step 4: Identify Consumer & Producer Surplus Regions

The consumer surplus region is above the equilibrium price line \(y = D(x_E)\) and below the demand curve \(y = D(x)\) up to \(x_E\). The producer surplus region is below the equilibrium price line and above the supply curve \(y = S(x)\), up to \(x_E\). Highlight these regions on the graph.

Step 5: Calculate Consumer Surplus

The consumer surplus is the area between the demand curve and the equilibrium price line up to \(x_E\). This is calculated as \(\int_0^{x_E} D(x) \, dx - (x_E \cdot D(x_E))\). Evaluate the integral from 0 to \(x_E\) and subtract the area of the rectangle formed by \(x_E\) and \(D(x_E)\).

Step 6: Calculate Producer Surplus

The producer surplus is the area between the equilibrium price line \(y = D(x_E)\) and the supply curve \(y = S(x)\) up to \(x_E\). Calculate this as \((x_E \cdot D(x_E)) - \int_0^{x_E} S(x) \, dx\). Evaluate the integral from 0 to \(x_E\) and find the difference.

Key Concepts

Demand Function

A demand function shows the relationship between the quantity demanded of a product and its price. In simple terms, it tells us how much of a product consumers are willing and able to purchase at different price levels. The demand function in this exercise is represented by \[ D(x) = \frac{x+8}{x+1} \] where \(x\) represents the quantity demanded, and \(D(x)\) represents the price consumers are willing to pay.### Understanding the Demand CurveThink of the demand curve as a visual tool:

• Slopes downward, illustrating the law of demand.
• As price decreases, the quantity demanded increases.

This downward slope signifies that as the product becomes cheaper, more consumers are inclined to buy it. Using the demand function graph patiently helps in determining how much the demand changes with price fluctuations.

Supply Function

The supply function visually represents the relationship between the quantity of a product that producers are willing to sell and its price. The exercise's supply function is:\[ S(x) = \frac{x^2 + 4}{20} \] Here, \(x\) stands for the quantity supplied, and \(S(x)\) gives the minimum price at which producers are willing to sell.### Characteristics of the Supply CurveThe supply curve generally slopes upwards. This visually depicts the law of supply:

• As the price rises, producers are more willing to sell more.
• The graph displays higher selling prices at greater quantity levels.

This positive relationship indicates that producers are incentivized to increase production as the price rises. Understanding this function helps in grasping how the supply operates amidst varying prices.

Consumer Surplus

Consumer surplus is a key concept in understanding market efficiency. It represents the difference between what consumers are willing to pay and what they actually pay at equilibrium. In simple terms, it's the extra satisfaction consumers get because they pay less than what they are willing to pay.### Calculating Consumer SurplusConsumer surplus is the area located above the price consumers pay (equilibrium price) and below the demand curve, up to the equilibrium quantity \(x_E\). Mathematically, it is computed as:\[ \int_0^{x_E} D(x) \, dx - (x_E \cdot D(x_E)) \]

• The first term expresses the total willingness to pay over all quantities sold up to \(x_E\).
• The second term represents the actual spending by the consumers at the equilibrium price.

Consumer surplus vividly highlights the value consumers receive due to market transactions.

Producer Surplus

Producer surplus reflects the difference between what producers are willing to accept for a good versus what they actually receive. It represents additional benefit gained by producers when selling at the market price.### Calculating Producer SurplusTo find producer surplus, calculate the area below the price line and above the supply curve up to the equilibrium quantity \(x_E\). The calculation is as follows:\[ (x_E \cdot D(x_E)) - \int_0^{x_E} S(x) \, dx \]

• The first component, \((x_E \cdot D(x_E))\), shows the revenue received by producers at the equilibrium price.
• The integral indicates the sellers' willingness to accept for each unit sold.

This concept emphasizes the gain producers experience through market exchanges, showing the benefits in efficient resource allocation.