Problem 27 - Applied Calculus

Question

Table \(1.28\) gives data for the linear demand curve for a product, where \(p\) is the price of the product and \(q\) is the quantity sold every month at that price. Find formulas for the following functions. Interpret their slopes in terms of demand. (a) \(q\) as a function of \(p\). (b) \(p\) as a function of \(q\). $$ \begin{array}{l} \text { Table } 1.28\\\ \begin{array}{c|c|c|c|c|c} \hline p \text { (dollars) } & 16 & 18 & 20 & 22 & 24 \\ \hline q \text { (tons) } & 500 & 460 & 420 & 380 & 340 \\ \hline \end{array} \end{array} $$

Step-by-Step Solution

Answer

(a) \(q = -20p + 820\). (b) \(p = -\frac{1}{20}q + 41\). Slopes represent demand changes.

Step 1: Understand the Data Points

We have pairs of data points where \(p\) is the price and \(q\) is the quantity sold. These pairs are: \((16, 500), (18, 460), (20, 420), (22, 380), (24, 340)\). We need to form linear functions from these points.

Step 2: Find Slope for q as Function of p

To express \(q\) as a function of \(p\), find the slope \(m\) using two data points, such as \((16, 500)\) and \((18, 460)\). The formula for slope is \( m = \frac{q_2 - q_1}{p_2 - p_1} \). Substituting the values, \( m = \frac{460 - 500}{18 - 16} = \frac{-40}{2} = -20 \).

Step 3: Write Equation for q as Function of p

Using the slope \(-20\) and one point, such as \((16, 500)\), use the point-slope form to write the equation: \( q - 500 = -20(p - 16) \). Simplify it to \( q = -20p + 820 \).

Step 4: Find Slope for p as Function of q

To express \(p\) as a function of \(q\), find the slope by taking the reciprocal of the previous slope: \( m = \frac{-1}{-20} = -\frac{1}{20} \). This represents \(p\) as a function of \(q\).

Step 5: Write Equation for p as Function of q

Use the slope \(-\frac{1}{20}\) and the point \((500, 16)\) to write the equation: \( p - 16 = -\frac{1}{20}(q - 500) \). Simplify it to \( p = -\frac{1}{20}q + 41 \).

Step 6: Interpret Slopes

For \(q\) as a function of \(p\), each \(1 increase in price \(p\) reduces the quantity demanded \(q\) by 20 units. For \(p\) as a function of \(q\), each increase in quantity of 20 units results in a \)1 decrease in price.

Key Concepts

Slope Interpretation

When dealing with a linear demand curve, the slope is a crucial concept to understand. The slope of a line represents how much one variable changes when the other variable changes by one unit. In the context of the demand curve, we're considering how changes in price (\( p \)) affect the quantity sold (\( q \)), and vice versa.
In this exercise, we look at two functions: \( q \, \text{as a function of} \, p \) and \( p \, \text{as a function of} \, q \). First, consider \( q \, \text{as a function of} \, p \) with a slope of \( -20 \). This tells us that for every 1 dollar increase in price, the quantity demanded decreases by 20 tons. This negative slope is typical as higher prices often lead to lower demand.
On the other hand, when expressing \( p \, \text{as a function of} \, q \), the slope is \(-\frac{1}{20}\), meaning for every 20-unit increase in quantity, the price decreases by 1 dollar. Understanding these slopes helps businesses decide how a pricing change could affect sales volume.

Linear Functions

Linear functions are equations that describe a straight line when plotted on a graph. In economic contexts like ours, a linear demand curve connects changes in two different factors: price (\( p \)) and quantity (\( q \)).
A standard linear function has the form \( y = mx + b \), where \( m \) is the slope, indicating the rate of change, and \( b \) is the y-intercept, indicating where the line crosses the y-axis. For \( q \, \text{as a function of} \, p \), the linear equation is \( q = -20p + 820 \). Here, \( -20 \) is the slope, and \( 820 \) is the intercept.
For \( p \, \text{as a function of} \, q \), the linear equation becomes \( p = -\frac{1}{20}q + 41 \), demonstrating how different approaches to writing equations can show the same relationship from another angle. Grasping these forms helps with better predicting and understanding econometric trends.

Price and Quantity Relationship

The relationship between price and quantity is central to any discussion of supply and demand. In our case, the linear demand curve portrays a clear inverse relationship between price and quantity. This kind of relationship is often referred to as a 'negative correlation,' where an increase in one variable results in a decrease in the other.
For instance, as seen in the equation \( q = -20p + 820 \), as the price increases, the quantity sold decreases. Conversely, by looking at \( p = -\frac{1}{20}q + 41 \), it’s evident that if the quantity consumers want increases, sellers lower the price.
Businesses find it crucial to understand this connection because it affects pricing strategies, inventory management, and revenue forecasting. It's key to realizing how price incentives can be used to stimulate or suppress the demand for a product, allowing for more responsive and effective management of product supply.