Chapter 17, Problem 17.10P

a)

To determine

To calculate: The producer surplus, equilibrium quantity, price, consumer surplus, and social welfare.

Explanation of Solution

Given the following demand and supply function of the potato chips

  $\begin{array}{l}{Q}_S=\frac{P}{2}\\ Q_D=100-2P\end{array}$

The equilibrium price is calculated by equating the supply and demand function. Thus, in equilibrium

  $\begin{array}{c}{Q}_S=Q_D\\ \frac{P}{2}=100-2P\\ P=\left(100-2P\right)\times 2\\ P+4P=200\\ 5P=200\\ P=\frac{200}{5}\\ =\$40\end{array}$

To calculate equilibrium, we need to substitute $P=\$40$ into either the demand or supply function. Thus, equilibrium quantity is

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

Hence, the equilibrium quantity and price with respect to given demand and supply function is Q = 20 and P = $40.

In order to calculate conumer surplus, producer surplus and social welfare we need to plot the above demand and supply function.

  

In the graph, the x axis depicts the quantity of potato chips and the y axis depicts the price per potato chips (in $), $Q_D$ is the demand curve for potato chips and $Q_S$ is the supply curve for potato chips. The equilibrium, E, is at Q = 20 and P = $40.

Consumer surplus in the graph is represented by triangle A, so to compute consumer surplus, we need to calculate the area of the triangle. Thus, consumer surplus (CS) is

  $\begin{array}{c}\text{CS}\left(\text{Area}\text{of}\text{triangle}\text{A}\right)\text{=}\frac{\text{1}}{\text{2}}\left(\text{b}\right)\left(\text{h}\right)\\ \text{=}\frac{\text{1}}{\text{2}}\left(\text{20}\right)\left(\text{50-40}\right)\\ \text{=}\left(\text{10}\right)\left(\text{10}\right)\\ \text{=100}\end{array}$

Producer surplus in the graph is represented by triangle B, so to compute producer surplus, we need to calculate the area of the triangle. Thus, producer surplus (PS) is

  $\begin{array}{c}\text{PS}\left(\text{area}\text{of}\text{triangle}\text{B}\right)\text{=}\frac{\text{1}}{\text{2}}\left(\text{b}\right)\left(\text{h}\right)\\ \text{=}\frac{\text{1}}{\text{2}}\left(\text{20}\right)\left(\text{40}\right)\\ \text{=}\left(\text{10}\right)\left(\text{40}\right)\\ \text{=400}\end{array}$

Social welfare (SW) is the sum of consumer surplus and producer surplus. Thus,

  $\begin{array}{c}\text{SW=100+400}\\ \text{=500}\end{array}$

Economics Concept Introduction

Introduction:

The equilibrium price is the defined as the price level at which both consumer and producer satisfies. It is the price where both want to exchange commodities and services.

b)

To determine

To compute: The equilibrium quantity, price, and deadweight loss triangle on the diagram.

Explanation of Solution

Given information:

Perceived demand function, ${\text{Q}}_{\text{D}}^{\text{*}}\text{=200-2P}$

Supply function, ${\text{Q}}_{\text{S}}\text{=}\frac{\text{P}}{\text{2}}$

The equilibrium price is calculated by equating the supply and demand function. Thus, in equilibrium

  $\begin{array}{c}{\text{Q}}_{\text{S}}{\text{=Q}}_{\text{D}}^{\text{*}}\\ \frac{\text{P}}{\text{2}}\text{=200-2P}\\ \text{P=}\left(\text{200-2P}\right)\text{×2}\\ \text{P=400-4P}\\ \text{P+4P=400}\\ \text{5P=400}\\ \text{P=}\frac{\text{400}}{\text{5}}\\ \text{=\$80}\end{array}$

In order to calculate equilibrium quantity, we need to substitute P = $80 into either the demand or supply function. Thus, equilibrium quantity is

  $\begin{array}{c}{\text{Q}}_{\text{D}}^{\text{*}}\text{=200-2P}\\ \text{=200-2}\left(\text{80}\right)\\ \text{=200-160}\\ \text{=40}\end{array}$

Hence, the equilibrium quantity and price with respect to given demand and supply function is

  $\text{Q= 40 and}\text{P= \$80}$

In order to calculate deadweight loss we need to plot the above demand function and supply function. This has been done below:

  

In the graph, the x axis measures the quantity of potato chips and the y axis measures the price per potato chips (in $), $Q_D$ is the demand curve for potato chips and $Q_S$ is the supply curve of potato chips. The equilibrium, E, is at Q = 20 and P = $40 when it assumed that consumer behavior is not limited by his or her cognitive ability.

When it assumed that consumer behavious is influenced by his or her limited cognitive ability, then his percieved demand curve is ${\text{Q}}_{\text{D}}^{\text{*}}$ wherein it can be seen that the consumer ends up buying extra 100 units of potato chips. Given the supply curve, ${\text{Q}}_{\text{S}}$ , the new equilibrium is E* where the equilibrium quantity and price Q = 40 and P = $80.

The deadweight loss indicating loss in social welfare due to extra purchase of potato chips than required is represented by triangle C and D. to compute the deadweight loss we need to calculate the area of triangle C and D. Thus

  $\begin{array}{c}\text{Area}\text{of}\text{triangle}C=\frac{1}{2}\times 20\times 40\\ =10\times 40\\ =400\end{array}$

  $\begin{array}{c}\text{Area}\text{of}\text{triangle}D=\frac{1}{2}\times 10\times 20\\ =5\times 20\\ =100\end{array}$

Thus, deadweight loss is $400+100=500$

Economics Concept Introduction

Introduction: A deadweight loss is an expense to economy that is caused by market failure, when supply and demand are out of equilibrium. Primarily used in economics, deadweight loss can be extended to any deficit caused by inadequate resource allocation.

c)

To determine

To compute: The per unit tax that the government could impose to correct the deadweight loss problem.

Explanation of Solution

In order to correct the deadweight loss of 500 due to extra purchase of potato chips, government could impose a specific tax, $\text{t= \$50 per unit}\text{.}$

As a result, the consumer will reduce its purchase of potato chips from 40 units to 20 units on account of increased price of potato chips to $90.

Economics Concept Introduction

Introduction: A deadweight loss is an expense to economy that is caused by market failure, when supply and demand are out of equilibrium. Primarily used in economics, the loss of deadweight can be extended to any deficit caused by inadequate resource allocation.

d)

To determine

To compute: The deadweight loss the governement has introduced with the tax if instead the government made a mistake and the second demand is actually the true demand stemming from rational decisions.

Explanation of Solution

Dead weight loss happens when there is no equilibrium between supply and demand which leads to market failure. Market inefficiency happens when products are either overvalued or undervalued inside the industry. Although the disparity will favor some people in society, a change from equilibrium would negatively affect others.

If it is assumed that $Q_D^{*}$ is the true demand curve, then government by imposing a specific tax, t = $50 per unit, has created deadweight loss of 500.

Economics Concept Introduction

Introduction: A deadweight loss is an expense to economy that is caused by market failure, when supply and demand are out of equilibrium. Primarily used in economics, the loss of deadweight can be extended to any deficit caused by inadequate resource allocation.