Chapter 13, Problem 2C

To determine

Calculation of deadweight loss.

Explanation of Solution

Case A:

In case “A”, profit-maximizing quantity (PMQ) can be calculated as follows:

$\begin{array}{c}\text{Marginal revenue}=\text{Marginal cost}\\ \frac{\partial \left(\text{Price}\times \text{Quantity}\right)}{\partial \text{Quantity}}=10\\ \frac{\partial \left(\left(\text{50}-\text{Q}\right)\times \text{Q}\right)}{\partial \text{Q}}=10\\ \frac{\partial \left(\left(\text{50Q}-{\text{Q}}^2\right)\right)}{\partial \text{Q}}=10\end{array}$

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

The profit-maximizing output is 20 units.

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

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

The profit-maximizing price is $30.

Socially-optimal quantity (SOQ) can be calculated as follows:

$\begin{array}{c}\text{Demand}=\text{Marginal cost}\\ \text{50}-\text{Q}=10\\ \text{Q}=\text{50}-\text{10}\\ =40\end{array}$

The socially-optimal quantity is 40.

Substitute the sociality-optimal output in demand equation to calculate the socially-optimal price (SOP).

$\begin{array}{c}\text{Price}=50-\text{Q}\\ =50-40\\ =10\end{array}$

The socially-optimal price is $10.

The deadweight loss is calculated as follows:

$\begin{array}{c}\text{Deadweight}\text{loss=}\frac{\text{1}}{\text{2}}\left(\text{PMP}-\text{SOP}\right)\text{×}\left(\text{SOQ}-\text{PMQ}\right)\\ \text{=}\frac{1}{2}\left(30-10\right)\times \left(40-20\right)\\ =0.5\times \left(20\times 20\right)\\ =200\end{array}$

The deadweight loss is $200.

Case B:

In case “B”, the profit-maximizing quantity (PMQ) can be calculated as follows:

$\begin{array}{c}\text{Marginal revenue}=\text{Marginal cost}\\ \frac{\partial \left(\text{Price}\times \text{Quantity}\right)}{\partial \text{Quantity}}=10\\ \frac{\partial \left(\left(100-2\text{Q}\right)\times \text{Q}\right)}{\partial \text{Q}}=10\\ \frac{\partial \left(\left(\text{100Q}-2{\text{Q}}^2\right)\right)}{\partial \text{Q}}=10\end{array}$

$\begin{array}{c}100-2\times 2\text{Q}=10\\ 4\text{Q}=100-10\\ \text{Q}=\frac{90}{4}\\ =22.5\end{array}$

The profit-maximizing output is 22.5 units.

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

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

The profit-maximizing price is $55.

The socially-optimal quantity (SOQ) can be calculated as follows:

$\begin{array}{c}\text{Demand}=\text{Marginal cost}\\ 100-2\text{Q}=10\\ \text{2Q}=100-\text{10}\\ \text{Q}=\frac{90}{2}\\ =45\end{array}$

The socially-optimal quantity is 45.

Substitute the sociality-optimal output in demand equation to calculate the socially optimal price (SOP).

$\begin{array}{c}\text{Price}=100-2\text{Q}\\ =100-2\left(45\right)\\ =100-90\\ =10\end{array}$

The socially-optimal price is $10.

The deadweight loss is calculated as follows:

$\begin{array}{c}\text{Deadweight}\text{loss=}\frac{\text{1}}{\text{2}}\left(\text{PMP}-\text{SOP}\right)\text{×}\left(\text{SOQ}-\text{PMQ}\right)\\ \text{=}\frac{1}{2}\left(55-10\right)\times \left(45-22.5\right)\\ =0.5\times \left(45\times 22.5\right)\\ =506.25\end{array}$

The deadweight loss is 506.25.

Case C:

In case “C”, the profit-maximizing quantity (PMQ) can be calculated as follows:

$\begin{array}{c}\text{Marginal revenue}=\text{Marginal cost}\\ \frac{\partial \left(\text{Price}\times \text{Quantity}\right)}{\partial \text{Quantity}}=20\\ \frac{\partial \left(\left(100-2\text{Q}\right)\times \text{Q}\right)}{\partial \text{Q}}=20\\ \frac{\partial \left(\left(\text{100Q}-2{\text{Q}}^2\right)\right)}{\partial \text{Q}}=20\end{array}$

$\begin{array}{c}100-2\times 2\text{Q}=20\\ \text{4Q}=100-20\\ \text{Q}=\frac{80}{4}\\ =20\end{array}$

The profit-maximizing output is 20 units.

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

$\begin{array}{c}\text{Price}=100-2\text{Q}\\ =100-2\left(20\right)\\ =100-40\\ =60\end{array}$

The profit-maximizing price is $60.

The socially-optimal quantity (SOQ) can be calculated as follows:

$\begin{array}{c}\text{Demand}=\text{Marginal cost}\\ 100-2\text{Q}=10\\ \text{2Q}=100-2\text{0}\\ \text{Q}=\frac{80}{2}\\ =40\end{array}$

The socially-optimal quantity is 40.

Substitute the socially-optimal output in demand equation to calculate the socially optimal price (SOP).

$\begin{array}{c}\text{Price}=100-2\text{Q}\\ =100-2\left(40\right)\\ =100-80\\ =20\end{array}$

The socially-optimal price is $20.

The deadweight loss is calculated as follows:

$\begin{array}{c}\text{Deadweight}\text{loss=}\frac{\text{1}}{\text{2}}\left(\text{PMP}-\text{SOP}\right)\text{×}\left(\text{SOQ}-\text{PMQ}\right)\\ \text{=}\frac{1}{2}\left(60-20\right)\times \left(40-20\right)\\ =0.5\times \left(40\times 20\right)\\ =400\end{array}$

The deadweight loss is $400.

Economics Concept Introduction

Concept introduction:

Deadweight loss:  The fall in total surplus that results from a market distortion is termed as deadweight loss.