Question

Use the table to answer the questions below. $$ \begin{array}{|rc|rc|} \hline \begin{array}{c} \text { Quantity } \\ \text { produced } \\ \text { and sold } \\ (Q) \end{array} & \begin{array}{c} \text { Price } \\ (p) \end{array} & \begin{array}{c} \text { Total } \\ \text { revenue } \\ (T R) \end{array} & \begin{array}{c} \text { Marginal } \\ \text { revenue } \\ (M R) \end{array} \\ \hline 0 & 160 & 0 & \- \\ 2 & 140 & 280 & 130 \\ 4 & 120 & 480 & 90 \\ 6 & 100 & 600 & 50 \\ 8 & 80 & 640 & 10 \\ 10 & 60 & 600 & -30 \\ \hline \end{array} $$ (a) Use the regression feature of a graphing utility to find a quadratic model that relates the total revenue \((T R)\) to the quantity produced and sold \((Q)\). (b) Using derivatives, find a model for marginal revenue from the model you found in part (a). (c) Calculate the marginal revenue for all values of \(Q\) using your model in part (b), and compare these values with the actual values given. How good is your model?

Short answer

The short answer would be the quadratic regression equation found in Step 1, the derived equation for marginal revenue found in Step 2 and finally a judgement statement on how accurately the regression model predicts the marginal revenue based on the comparison made in Step 4.

Step-by-step solution

Find a Quadratic model for total revenue 'TR' and quantity produced and sold 'Q'

Use a graphing utility and plot the data points for Total Revenue (TR) against Quantity (Q). From these data points, perform a quadratic regression to find the best-fitting quadratic equation. This equation will be of the form \(TR = aQ^2 + bQ + c\), where a, b, and c are the regression coefficients.

Derive the Quadratic model to find the Marginal Revenue 'MR'

The marginal revenue (MR) is the rate of change of total revenue with respect to quantity. This is given by the derivative of the total revenue with respect to quantity. So, calculate the derivative of the regression equation found in Step 1. This derivative will give the mathematical model for the marginal revenue.

Calculate the Marginal Revenue for all values of Q

Use the Marginal Revenue model from Step 2, plug in the values of Q from the table and calculate the Marginal Revenue (MR).

Compare the Calculated and Actual values

Compare the calculated values of Marginal Revenue from Step 3 with the actual values given in the table. Check how close the calculated values are to the actual ones. If the calculated values match the actual values closely, then the quadratic model is a good fit for the data.

Key concepts

Total Revenue

Total Revenue (TR) is a fundamental concept in economics and business. It represents the total income a firm receives from selling its goods or services. In mathematical terms, it is calculated as the product of the quantity of goods sold (Q) and the price per unit (p). The formula for Total Revenue is:

• TR = Q × p

In the context of the exercise, Total Revenue changes with the quantity produced and sold. As the quantity sold increases, the Total Revenue can either increase or decrease, depending on how the price per unit changes.
In the table provided in the exercise, you can see how the Total Revenue peaks and then decreases as more units are sold, showing the importance of finding the optimal quantity that maximizes revenue.
When doing quadratic regression, we use data to fit a quadratic model that predicts Total Revenue based on the quantity produced and sold. This model is typically of the form:

• TR = aQ^2 + bQ + c

Here, \(a\), \(b\), and \(c\) are constants that are determined through regression, showing how good the quadratic equation describes the actual data. This helps businesses in decision making for production and pricing strategies.

Marginal Revenue

Marginal Revenue (MR) is the additional revenue a company earns by selling one more unit of a product. It is crucial because it helps determine the price and output level that maximizes profit.
In other words, it shows how Total Revenue changes as the quantity increases by one unit.
Mathematically, Marginal Revenue is the derivative of the Total Revenue (TR) with respect to Quantity (Q). If the Total Revenue is expressed as a quadratic function, \(TR = aQ^2 + bQ + c\), the Marginal Revenue can be found by differentiating with respect to \(Q\):

• MR = \(d(TR)/dQ = 2aQ + b\)

This formula allows us to calculate MR at any given quantity.
In practice, businesses use Marginal Revenue to decide the optimal level of production. If the MR is greater than the cost of producing an additional unit, it makes sense to increase production.
For this exercise, after finding the MR using a quadratic regression, you compare these theoretical MR values to the actual ones provided, assessing the effectiveness of the model used for estimation.

Derivatives

Derivatives are a key concept in calculus and are used to analyze and calculate rates of change.
In the context of business and economics, they are often used to determine how one variable changes with respect to another, such as the change in revenue concerning changes in sales quantity.
In this particular exercise, we use derivatives to calculate Marginal Revenue from a quadratic Total Revenue function.

• If \(TR\) is given by the quadratic equation \(aQ^2 + bQ + c\), the derivative \(d(TR)/dQ\) gives us the Marginal Revenue \(MR = 2aQ + b\).

The derivative, \(d(TR)/dQ\), measures the slope of the tangent line at any point on the graph of \(TR\).
This slope represents the instantaneous rate of change of Total Revenue with respect to Quantity, giving valuable insight into how additional units sold affect income.
Understanding and applying derivatives enable businesses to effectively strategize their production and pricing by showing the relationship between different economic variables.