Question

Given the following demand schedule for a monopolistic firm, plot the demand curve and the marginal revenue curve. $$ \begin{array}{|c|c|} \hline \text { Quantity } & \text { Price } \\ \hline 1 & \$ 30.00 \\ \hline 2 & \$ 26.75 \\ \hline 3 & \$ 23.50 \\ \hline 4 & \$ 20.25 \\ \hline 5 & \$ 17.00 \\ \hline 6 & \$ 13.75 \\ \hline \end{array} $$

Short answer

The demand equation is: \(P = -3.25Q +33.25\), and the marginal revenue equation is: \(MR = -6.5Q + 33.25\). To plot them, create a graph with Price (P) on the vertical axis and Quantity (Q) on the horizontal axis. Plot the demand curve using the data points from the demand schedule and the marginal revenue curve using the calculated MR equation.

Step-by-step solution

Calculate the demand equation#linear equation:

First, let's find the linear equation that represents the demand curve. We have the price (P) and quantity (Q) data from the demand schedule table. To derive the demand equation, we need to find the slope (m) and the intercept (b). Denote the demand equation as: \[P = mQ + b\] To find the slope, take two points from the table, say, (1, \$30.00) and (2, \$26.75). Apply the slope formula: \[m = \frac{P_2 - P_1}{Q_2 - Q_1} = \frac{26.75 - 30}{2-1} = -3.25\] Now, we can find the intercept (b) by substituting one of these points (let's use the first point, Quantity = 1 and Price = \$30.00) into the equation: \[30 = -3.25 * 1 + b\] \[b = 30 + 3.25 = 33.25\] Hence, the demand equation is: \[P = -3.25Q +33.25\]

Calculate the marginal revenue equation#linear equation:

The Marginal Revenue (MR) is the additional revenue a firm earns by selling one more unit. The marginal revenue curve has the same intercept as the demand curve but twice the slope: \[MR = m_MR Q + b_MR\] To find the slope of the marginal revenue curve, multiply the slope of the demand curve by 2: \[m_MR = 2m = 2(-3.25) = -6.5\] The intercept of the marginal revenue curve remains the same as that of the demand curve: \[b_MR = b = 33.25\] So, the marginal revenue equation is: \[MR = -6.5Q + 33.25\]

Plot the demand curve and the marginal revenue curve

Now that we have the equations for the demand curve and the marginal revenue curve, we can plot them on a graph. Create a graph with Price (P) on the vertical axis and Quantity (Q) on the horizontal axis. Plot the points from the demand schedule on the graph, connect them with a line, and label the line as the demand curve. Then, plot the marginal revenue curve using the equation \[MR = -6.5Q + 33.25\]. Keep the same vertical Price axis, and for each quantity level, calculate the corresponding marginal revenue using the marginal revenue equation, plot these points on the graph, and connect them with a line. Label this line as the marginal revenue curve. After completing these steps, the graph will display both the demand curve and the marginal revenue curve for the given monopolistic firm's demand schedule.

Key concepts

Demand Curve

In economics, a demand curve is a graphical representation that shows the relationship between the price of a good and the quantity demanded by consumers over a given period. For a monopolistic firm, this curve is crucial because it helps to determine the price at which the firm can maximize its profits.

For the given problem, the demand curve can be derived from the demand schedule, which lists different quantities of a good that consumers will buy at various prices. By plotting these points on a graph and connecting them, we obtain the demand curve line. The demand equation, which is a linear equation in this case, helps to express this relationship algebraically. It generally takes the form:
\[ P = mQ + b \]
where:

• \( P \) represents the price,
• \( Q \) is the quantity,
• \( m \) is the slope of the demand curve, and
• \( b \) is the y-intercept.

The slope indicates how much the price changes with a change in quantity, while the intercept is the price when quantity is zero.

Monopolistic Firm

A monopolistic firm is a single seller in a market, meaning it has significant control over the price and supply of a product. Unlike a perfectly competitive market, where firms are price takers, a monopolistic firm can influence market prices by adjusting its output level.

This control makes understanding demand and marginal revenue crucial. The monopolistic firm faces a downward-sloping demand curve, meaning that to sell more units, the firm must lower the price. This differs from perfect competition, where firms can sell any quantity at the market price.

• A monopolistic firm maximizes profit where its marginal revenue equals marginal cost rather than at the market equilibrium.
• The firm can also create product differentiation, leading to a demand curve that is not perfectly elastic.

It is essential to differentiate the marginal revenue curve from the demand curve since in a monopoly, MR declines faster than the demand curve.

Demand Schedule

A demand schedule is a tabular representation of the relationship between the price of a good and the quantity demanded at each price level. It provides the data necessary to plot a demand curve.

In our exercise, the demand schedule for the monopolistic firm is essential for plotting the demand curve and calculating the demand equation. The actual table lists the quantity of goods and the corresponding prices consumers are willing to pay:

• Quantity \( Q=1 \), Price \( P=\\(30.00 \)
• Quantity \( Q=2 \), Price \( P=\\)26.75 \)
• Quantity \( Q=3 \), Price \( P=\$23.50 \)
• ...

Each pair of price and quantity can be plotted to form a point on the demand curve. This schedule is simplistic but extremely valuable, especially in a classroom environment, as it provides fundamental insights into how changes in price affect consumer buying behavior.

Linear Equation

In mathematics and economics, a linear equation is a polynomial equation of the first degree. For demand and marginal revenue curves, linear equations help simplify the relationships of economic variables like price and quantity.

In determining the demand curve, the linear demand equation is derived using the slope formula, calculated from any two points on the table of the demand schedule. Once the slope \( m \) is known, the intercept \( b \) can be calculated to get the full equation:
\[ P = mQ + b \]
For marginal revenue, the equation is similar but the slope is doubled:
\[ MR = 2mQ + b \]
These linear equations not only help in plotting the graphical representation of economic relationships but also serve as tools for calculating precise data points such as where the marginal revenue equals marginal cost, aiding firms in deciding their optimal output level.