Chapter 13, Problem 21C

Subpart (a):

To determine

Marginal revenue.

Explanation of Solution

In case A, part (I):

Marginal revenue equation can be derived as follows:

$\begin{array}{c}\text{Marginal revenue}=\frac{\partial \left(\text{<x-custom-btb-me data-me-id="2095" class="microExplainerHighlight">Price</x-custom-btb-me>}\times \text{Quantity}\right)}{\partial \text{Quantity}}\\ =\frac{\partial \left(\left(\text{50}-\text{Q}\right)\times \text{Q}\right)}{\partial \text{Q}}\\ =\frac{\partial \left(\left(\text{50Q}-{\text{Q}}^2\right)\right)}{\partial \text{Q}}\\ =50-2\text{Q}\end{array}$

Marginal revenue equation is 50-2Q.

When the quantity is 0 units, the marginal revenue can be calculated by substituting the respective values in the marginal revenue equation.

$\begin{array}{c}\text{Marginal revenue}=50-2\text{Q}\\ =50-2\left(0\right)\\ =50-0\\ =50\end{array}$

Marginal revenue is 50 at the point where the quantity is 0 units.

When the quantity is 25 units, the marginal revenue can be calculated by substituting the respective values in the marginal revenue equation.

$\begin{array}{c}\text{Marginal revenue}=50-2\text{Q}\\ =50-2\left(25\right)\\ =50-50\\ =0\end{array}$

Marginal revenue is 0 at the point where the quantity is 50 units.

Figure 1 illustrates the marginal revenue curve for 50-2Q.

In Figure 1, the vertical axis measures price and the horizontal axis measures quantity; when the quantity is zero, the price would be 50. At price zero, the quantity demanded is 25; thus, this joins the two points the marginal revenue curve gets.

In case A, part (II):

The profit-maximizing output can be calculated by equating the marginal revenue to the marginal cost. This can be done as follows:

$\begin{array}{c}\text{MR}=\text{MC}\\ \text{50}-\text{2Q}=10\\ 2\text{Q}=50-10\\ \text{Q}=\frac{40}{2}\\ =20\end{array}$

Profit-maximizing output is 20 units.

In case A, part (III):

Substitute the profit-maximizing output in the demand equation to calculate the profit-maximizing price.

$\begin{array}{c}\text{Price}=50-\text{Q}\\ =50-20\\ =30\end{array}$

Profit-maximizing price is $30.

In case A, part (IV):

Total revenue can be obtained by multiplying the profit-maximizing price with the profit-maximizing quantity. This can be done as follows:

$\begin{array}{c}\text{Total revenue}=\text{Price}\times \text{Quantity}\\ =\text{30}\times \text{20}\\ =600\end{array}$

Total revenue is $600.

Total cost can be calculated as follows:

$\begin{array}{c}\text{Total cost}=\left(\text{Marginal cost}\times \text{Quantity}\right)+\text{Fixed cost}\\ =\left(10\times 20\right)+100\\ =200+100\\ =300\end{array}$

Total cost is $300.

In case A, part (V):

Profit can be calculated as follows:

$\begin{array}{c}\text{Profit}=\text{Total revenue}-\text{Total cost}\\ =600-300\\ =300\end{array}$

Profit is $300.

In case B, part (I):

Marginal revenue equation can be derived as follows:

$\begin{array}{c}\text{Marginal revenue}=\frac{\partial \left(\text{Price}\times \text{Quantity}\right)}{\partial \text{Quantity}}\\ =\frac{\partial \left(\left(\text{100}-2\text{Q}\right)\times \text{Q}\right)}{\partial \text{Q}}\\ =\frac{\partial \left(\left(\text{100Q}-2{\text{Q}}^2\right)\right)}{\partial \text{Q}}\\ =100-4\text{Q}\end{array}$

Marginal revenue equation is 100-4Q.

When quantity is 0 units, the marginal revenue can be calculated by substituting the respective values in the marginal revenue equation.

$\begin{array}{c}\text{Marginal revenue}=100-4\text{Q}\\ =100-4\left(0\right)\\ =100-0\\ =100\end{array}$

Marginal revenue is 100 at the point where the quantity is 0 units.

When quantity is 25 units, the marginal revenue can be calculated by substituting the respective values in the marginal revenue equation.

$\begin{array}{c}\text{Marginal revenue}=100-4\text{Q}\\ =100-4\left(25\right)\\ =100-100\\ =0\end{array}$

Marginal revenue is 0 at the point where the quantity is 50 units.

Figure 2 illustrates the marginal revenue curve for 100-4Q.

In Figure 2, the vertical axis measures price and the horizontal axis measures quantity. When the quantity is zero, the price would be 100, and at price zero, the quantity demanded is 25. Thus, joining the two points can help obtain the marginal revenue curve.

In case B, part (II):

Profit-maximizing output can be calculated by equating the marginal revenue to the marginal cost. This can be done as follows:

$\begin{array}{c}\text{MR}=\text{MC}\\ \text{100}-\text{4Q}=10\\ 4\text{Q}=100-10\\ \text{Q}=\frac{90}{4}\\ =22.5\end{array}$

Profit-maximizing output is 22.5 units.

In case B, part (III):

Substitute the profit-maximizing output in the demand equation to calculate the profit maximizing price.

$\begin{array}{c}\text{Price}=100-2\text{Q}\\ =100-2\left(22.5\right)\\ =100-45\\ =55\end{array}$

Profit-maximizing price is $55.

In case B, part (IV):

Total revenue can be obtained by multiplying the profit-maximizing price with profit-maximizing quantity. This can be done as follows:

$\begin{array}{c}\text{Total revenue}=\text{Price}\times \text{Quantity}\\ =\text{55}\times \text{22}\text{.5}\\ =1,237.5\end{array}$

Total revenue is $1,237.5.

Total cost can be calculated as follows:

$\begin{array}{c}\text{Total cost}=\left(\text{Marginal cost}\times \text{Quantity}\right)+\text{Fixed cost}\\ =\left(10\times 22.5\right)+100\\ =225+100\\ =325\end{array}$

Total cost is $325.

In case B, part (V):

Profit can be calculated as follows:

$\begin{array}{c}\text{Profit}=\text{Total revenue}-\text{Total cost}\\ =1,237.5-325\\ =912.5\end{array}$

Profit is $912.5.

In case C, part (I):

Marginal revenue equation can be derived as follows:

$\begin{array}{c}\text{Marginal revenue}=\frac{\partial \left(\text{Price}\times \text{Quantity}\right)}{\partial \text{Quantity}}\\ =\frac{\partial \left(\left(\text{100}-2\text{Q}\right)\times \text{Q}\right)}{\partial \text{Q}}\\ =\frac{\partial \left(\left(\text{100Q}-2{\text{Q}}^2\right)\right)}{\partial \text{Q}}\\ =100-4\text{Q}\end{array}$

Marginal revenue equation is 100-4Q.

When the quantity is 0 units, the marginal revenue can be calculated by substituting the respective values in the marginal revenue equation.

$\begin{array}{c}\text{Marginal revenue}=100-4\text{Q}\\ =100-4\left(0\right)\\ =100-0\\ =100\end{array}$

Marginal revenue is 100 at the point where the quantity is 0 units.

When the quantity is 25 units, the marginal revenue can be calculated by substituting the respective values in the marginal revenue equation.

$\begin{array}{c}\text{Marginal revenue}=100-4\text{Q}\\ =100-4\left(25\right)\\ =100-100\\ =0\end{array}$

Marginal revenue is 0 at the point where the quantity is 50 units.

Figure 3 illustrates the marginal revenue curve for100-4Q.

In Figure 3, the vertical axis measures price and the horizontal axis measures quantity. When the quantity is zero, the price would be 100, and at price zero, the quantity demanded is 25. Thus, joining the two points can help obtain the marginal revenue curve.

In case C, part (II):

Profit-maximizing output can be calculated by equating marginal revenue to the marginal cost. This can be done as follows:

$\begin{array}{c}\text{MR}=\text{MC}\\ \text{100}-\text{4Q}=20\\ 4\text{Q}=100-20\\ \text{Q}=\frac{80}{4}\\ =20\end{array}$

Profit-maximizing output is 20 units.

In case C, part (III):

Substitute the profit-maximizing output in demand equation to calculate the profit-maximizing price.

$\begin{array}{c}\text{Price}=100-2\text{Q}\\ =100-2\left(20\right)\\ =100-20\\ =60\end{array}$

Profit-maximizing price is $60.

In case C, part (IV):

Total revenue can be obtained by multiplying the profit-maximizing price with profit-maximizing quantity. This can be done as follows:

$\begin{array}{c}\text{Total revenue}=\text{Price}\times \text{Quantity}\\ =\text{60}\times \text{20}\\ =1,200\end{array}$

Total revenue is $1,200.

Total cost can be calculated as follows:

$\begin{array}{c}\text{Total cost}=\left(\text{Marginal cost}\times \text{Quantity}\right)+\text{Fixed cost}\\ =\left(20\times 20\right)+100\\ =400+100\\ =500\end{array}$

Total cost is $500.

In case C, part (V):

Profit can be calculated as follows:

$\begin{array}{c}\text{Profit}=\text{Total revenue}-\text{Total cost}\\ =1,200-500\\ =700\end{array}$

Profit is $700.

Economics Concept Introduction

Concept introduction:

Marginal revenue: The change in total revenue from selling an additional unit is known as marginal revenue.

Mark-up: Mark-up refers to the amount that is added by the seller to the cost of the goods to determine the selling price.

Sub part (b):

To determine

Marginal revenue.

Explanation of Solution

The markup refers to the amount that is added by the seller to the cost of the goods to determine the selling price. For calculating the percentage markup, the following equation can be used:

$\text{Markup= }\left[\text{100×}\left(\frac{\text{P}-\text{MC}}{\text{MC}}\right)\right]$ (1)

Case A:

Using equation (1), the percentage mark-up in case A can be calculated as follows:

$\begin{array}{c}\text{Markup= }\left[\text{100×}\left(\frac{30-10}{10}\right)\right]\\ \text{=}\left(100\times \frac{20}{10}\right)\\ =200\%\end{array}$

The percentage mark-up in price $20 is 200%.

Case B:

To calculate the percentage mark-up in case B, substitute the values in equation (1).

   $\begin{array}{c}\text{Markup= }\left[\text{100×}\left(\frac{55-10}{10}\right)\right]\\ \text{=}\left(100\times \frac{45}{10}\right)\\ =450\%\end{array}$

The percentage mark-up in price $45 is 450%.

Case C:

To calculate the percentage mark-up in case C, substitute the values in equation (1).

$\begin{array}{c}\text{Markup= }\left[\text{100×}\left(\frac{60-20}{20}\right)\right]\\ \text{=}\left(100\times \frac{40}{20}\right)\\ =200\%\end{array}$

The percentage mark-up in price $40 is 200%.

Economics Concept Introduction

Concept introduction:

Marginal revenue: The change in total revenue from selling an additional unit is known as marginal revenue.

Mark-up: Mark-up refers to the amount that is added by the seller to the cost of the goods to determine the selling price.

Sub part (c):

To determine

Marginal revenue.

Explanation of Solution

If monopoly firms raise the price, they lose their customers.  From case A to B to C, the percentage mark-up increases. When monopoly firms increase the price, the consumers move to other close substitutes; this affects the profit maximisation of the monopoly company.