Problem 26 - Calculus Concepts

Question

The demand for a new type of kerosene lantern is as shown in the table. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Price } \\ \text { (dollars per lantern) } \end{array} & \begin{array}{c} \text { Quantity } \\ \text { (thousand lanterns) } \end{array} \\ \hline 21.52 & 1 \\ \hline 17.11 & 3 \\ \hline 14.00 & 5 \\ \hline 11.45 & 7 \\ \hline 9.23 & 9 \\ \hline 7.25 & 11 \\ \hline \end{array} $$ a. Find a model giving the average quantity demanded as a function of the price. b. How much are consumers willing and able to spend each day for these lanterns when the market price is \(\$ 12.34\) per lantern? c. Calculate the consumers' surplus when the equilibrium price for these lanterns is \(\$ 12.34\) per lantern.

Step-by-Step Solution

Answer

Consumers are willing to spend \($12.34 \times q\) where \(q\) is the quantity at \(p = 12.34\). Calculate the integral of the demand curve from 0 to \(q\) for the consumer surplus.

Step 1: Determine the demand curve

To model the average quantity demanded as a function of the price, first identify the relationship from the given price and quantity pairs. We're looking to fit a function, say a linear one initially, that represents this relationship. Since we have multiple data points, we'll find the line of best fit by assuming it takes a linear form: \( q = m \, p + b \), where \( q \) is the quantity and \( p \) is the price. We'll compute this using least squares regression or a similar method.

Step 2: Calculate the line of best fit

Using the data points given, we'll calculate the slope \( m \) and the y-intercept \( b \) of the line in the form \( q = m \, p + b \). Calculating these by inputting each price and quantity into the linear regression formula: \[ m = \frac{n(\sum(pq)) - (\sum p)(\sum q)}{n(\sum p^2) - (\sum p)^2} \] and \[ b = \frac{(\sum q) - m(\sum p)}{n} \] where \( n \) is the number of data points. Plug the values back into get the line equation.

Step 3: Use the demand function to find quantity at given price

Substitute \( p = 12.34 \) into the demand equation derived from the line of best fit to find the corresponding quantity \( q \). This is the quantity consumers are willing to purchase at this price. Multiply this quantity by the price to determine total spending.

Step 4: Calculate Consumer's Surplus

Consumer's surplus is calculated by finding the area between the demand curve and the equilibrium price line, up to the point of equilibrium quantity. Integrate the demand curve, \( q = m \, p + b \), from 0 to the quantity found in Step 3, and subtract the total amount spent calculated in Step 3 from this integral to find the consumer's surplus.

Key Concepts

Demand Curve

A demand curve is a graphical representation of the relationship between the price of a good and the quantity consumers are willing and able to purchase. In other words, it shows how much of a product people want to buy at different price levels. As price decreases, consumers typically buy more, which is why the demand curve usually slopes downward.

In the example of kerosene lanterns, we have several price points and corresponding quantities. The data points are (21.52, 1), (17.11, 3), (14.00, 5), (11.45, 7), (9.23, 9), and (7.25, 11). Each pair shows how price impacts demand. To create a demand curve from this data, we often use mathematical models. The most common model is a linear demand curve, which attempts to fit a straight line through the data points to best describe their relationship. Once established, this line can predict future demand at various prices.

Understanding the demand curve helps in setting pricing strategies and anticipating how changes in price could impact consumer behavior.

Linear Regression

Linear regression is a statistical method used to model the relationship between two variables by fitting a linear equation to observed data. The goal is to find the 'best fit' line through the data points, which allows us to predict the quantity demanded at any given price.

In the lantern example, a linear regression is used to determine the demand equation, typically in the form \( q = m \, p + b \), where \( q \) is the quantity, \( p \) is the price, \( m \) is the slope of the line, and \( b \) is the y-intercept. The slope \( m \) indicates how much quantity changes with a one-dollar change in price.

To calculate \( m \) and \( b \), use the formulas:

• \( m = \frac{n(\sum(pq)) - (\sum p)(\sum q)}{n(\sum p^2) - (\sum p)^2} \)
• \( b = \frac{(\sum q) - m(\sum p)}{n} \)

Here, \( n \) is the number of data points. By calculating these, we derive a line that best fits the historical demand data, helping us predict future demand scenarios.

Equilibrium Price

The equilibrium price is the price at which the quantity of a good demanded by consumers equals the quantity supplied by producers. At this price point, there is no surplus or shortage of the product in the market.

In our lantern example, if the market price is \( \\(12.34 \), we can assess how this price aligns with consumer demand and the supply available. By using the demand curve derived from linear regression, we can determine the quantity of lanterns consumers are willing to buy at \( \\)12.34 \).

Equilibrium is important because it represents a stable point in the market. It tells businesses at what price and quantity a stable marketplace for their product exists. Understanding this concept helps in strategic planning and ensuring that both supply and demand are balanced.