Chapter 17, Problem 3P

(a)

To determine

The graphical representations of the demand, marginal cost (MC), and external marginal cost (EMC).

Explanation of Solution

Figure 1 illustrates the demand curve, marginal cost (MC) curve, and the external marginal cost (EMC) curve.

In Figure 1, the vertical axis measures the prices of marching bands and the horizontal axis measures the quantity of marching bands where curve D shows the inverse demand function, $\text{P}=\$1,000-\text{Q}$. Thus, the demand curve intercept at 1,000 in the price line have the slope of 1. The curve MC shows the MC function, $\text{P}=\$0.75\text{Q}$. Thus, the curve that starts from the origin has the slope of 0.75. The EMC curve shows the EMC function, $\text{EMC}=\text{Q}.25\text{Q}$. Thus, it starts at the origin and has the slope of

Economics Concept Introduction

External marginal cost: The external marginal cost is the change in cost imposed on a third party when an additional unit of a good is produced or consumed.

Marginal cost (MC): The marginal cost refers to the amount of additional cost incurred in the process of increasing one more unit of output.

 (b)

To determine

The consumer surplus, producer surplus, total surplus, and the total damage.

Explanation of Solution

The optimal quantity of marching bands is determined at the point where the marginal revenue (demand) equals MC.

Now, set the expressions of demand and MC functions to solve the optimal amount of marching bands.

$\begin{array}{c}1,000-\text{Q}=0.75\text{Q}\\ \text{Q}=\frac{1,000}{1.75}\\ =571\end{array}$

Therefore, the optimal quantity of marching band is approximately 571 bands.

Now, substitute the respective values in the demand function to get the optimal price of marching bands.

$1,000-571=429$

Therefore, the optimal price of marching band is $429.

Substitute the respective values in the EMC function.

$\begin{array}{c}\text{EMC}=0.25\times 571\\ =142.75\end{array}$

Therefore, the EMC is 142.75.

The consumer surplus can calculated by using the following formula:

$\text{Consumer surplus}=\frac{1}{2}\times \left(\begin{array}{l}\text{Maximum willing price}\\ -\text{Equilibrium price}\end{array}\right)\times \text{Equilibrium quantity}$ (1)

Substitute the respective values (from Figure 1) in Equation (1) to get the value of consumer surplus.

$\begin{array}{c}\text{Consumer surplus}=\frac{1}{2}\times \left(1,000-429\right)\times 571\\ =\$163,020.50\end{array}$

Therefore, the consumer surplus is $163,020.50.

The producer surplus can calculated by using the following formula:

$\text{Producer surplus}=\frac{1}{2}\times \left(\begin{array}{l}\text{Equilibrium price}\\ -\text{Minimum willing price}\end{array}\right)\times \text{ Equilibrium quantity}$ (2)

Substitute the respective values (from Figure 1) in Equation (2) to get the value of producer surplus.

$\begin{array}{c}\text{Producer surplus}=\frac{1}{2}\times \left(429-0\right)\times 571\\ =\$163,020.50\\ =122,479.50\end{array}$

Therefore, the producer surplus is $122,479.50.

The total surplus to market participants is equal to the sum total of producer surplus and the consumer surplus.

$\begin{array}{c}\text{Total surplus}=\text{Consumer surplus}+\text{Producer surpls}\\ =\$163,020.50+\$122,479.50\\ =\$285,500\end{array}$

Therefore, the total surplus to market participants is $285,500.

In Figure 1, the area under EMC shows the total damaged to harmed people. It can be calculated as follows:

$\begin{array}{c}\text{Total damage}=\frac{1}{2}\times \left(\text{EMC}-\text{Minimum willing price}\right)\times \text{Equilibrium quantity}\\ =\frac{1}{2}\times 142.75\times 571\\ =\$40,755.13\end{array}$

Therefore, the total damage to harmed people is $40,755.13.

The net value created for society is the difference between the total surplus to market participants and the total damage to harmed people. 

$\begin{array}{c}\text{Net value}=\text{Total surplus}-\text{Total damage}\\ =\$258,500-\$40,755.13\\ =\$244,744.87\end{array}$

Therefore, the net value created for society by marching band music is $244,744.87.

Economics Concept Introduction

Consumer Surplus: Consumer surplus is defined as the difference between the maximum amount a person is willing to pay for consuming a commodity and the actual price he pays for it.

Producer Surplus: Producer surplus is defined as the difference between the actual market price for which a commodity is sold and the minimum cost at which the producer is willing to sell the commodity. This minimum accepted price is usually the cost of production of the commodity.

 (c)

To determine

The social marginal cost (SMC).

Explanation of Solution

The social marginal cost (SMC) is equal to the sum total of marginal cost and the external marginal cost. The marginal cost function is $0.75\text{Q}$. Here, the external marginal cost function is $0.25\text{Q}$.

Now, the equation for SMC can set as follows:

$\text{SMC}=\text{MC}+\text{EMC}$

$\begin{array}{c}\text{SMC}=0.75+0.25\text{Q}\\ =\text{Q}\end{array}$

Therefore, the SMC is equal to the quantity of marching band music.

Economics Concept Introduction

External marginal cost: The external marginal cost is the change in cost imposed on a third party when an additional unit of a good is produced or consumed.  

Marginal cost (MC): The marginal cost refers to the amount of additional cost incurred in the process of increasing one more unit of output.

 (d)

To determine

The new optimal quantity of marching band music.

Explanation of Solution

The new optimal quantity of marching band is determined at the point where the demand function equals the SMC.

$\begin{array}{c}1,000-\text{Q}=\text{Q}\\ \text{1,000=2Q}\\ \text{Q}=\frac{1,000}{2}\\ =500\end{array}$

Therefore, if the marching bands were forced to consider the costs that they imposed on others, then the quantity of marching band music will be 500.

Now, substitute the respective values in the demand function to get the new optimal price of marching bands.

$1,000-500=\$500$

Therefore, the new optimal price of marching band is $500.

Substitute the respective values in the MC function.

$\begin{array}{c}\text{MC}=0.75\times 500\\ =375\end{array}$

Therefore, the new MC is $375.

Substitute the respective value in the EMC function.

$\begin{array}{c}\text{EMC}=0.25\times 500\\ =125\end{array}$

Therefore, the new EMC is 125.

Figure 2 illustrates the new optimal quantity of marching band.

In Figure 2, the vertical axis measures the price of marching bands and the horizontal axis measures the quantity of marching bands where the interaction of curve D with the SMC determines the new optimal quantity and price of marching bands.

Economics Concept Introduction

External marginal cost: The external marginal cost is the change in cost imposed on a third party when an additional unit of a good is produced or consumed.  

Marginal cost (MC): The marginal cost refers to the amount of additional cost incurred in the process of increasing one more unit of output.

 (e)

To determine

The new price of marching band.

Explanation of Solution

As described in part (d), the new optimal quantity of marching band is 500, which means that the price of band increases by $71 $\left(\$500-\$429\right)$.

 (f)

To determine

The new values for consumer surplus, producer surplus, total surplus, and total damage.

Explanation of Solution

The consumer surplus can calculated by using the following formula:

$\text{Consumer surplus}=\frac{1}{2}\times \left(\text{Equilibrium price}\right)\times \text{Equilibrium quantity}$ (3)

Substitute the respective values (from Figure 2) in Equation (3) to get the value of consumer surplus.

$\begin{array}{c}\text{Consumer surplus}=\frac{1}{2}\times 500\times 500\\ =\$125,000\end{array}$

Therefore, the consumer surplus is $125,000. Hence, the consumer surplus decreases from $163,020.50 to $125,000.

The producer surplus is the area between the private marginal cost and the price.

The private marginal cost (PMC) is calculated as follows:

$\begin{array}{c}\text{PMC}=\text{EMC}\times \text{Equilibrium quantity}\\ =125\times 500\\ =62,500\end{array}$

Therefore, the private marginal cost is 62,500.

Now, the value of producer surplus can be calculated as follows:

$\begin{array}{c}\text{Producer surplus}=\text{PMC}+\frac{1}{2}\times \text{MC}\times \text{Quantity}\\ =62,500+\frac{1}{2}\times 375\times \text{ 500}\\ =\$156,250\end{array}$

Therefore, the producer surplus is $156,250. Hence, the producer surplus increases from $122,479.50 to $156,250.

The total damage can be calculated as follows:

$\begin{array}{c}\text{Total damage}=\frac{1}{2}\times \left(\text{Equilibriumprice}-\text{MC}\right)\times \text{Equilibrium quantity}\\ =\frac{1}{2}\times \left(500-375\right)\times 500\\ =\$31,250\end{array}$

Therefore, the total damage to harmed people is $31,250. Hence, the total damage decreases from $40,755.13 to $31,250.

The total surplus is equal to the sum total of producer surplus and consumer surplus. Thus, it is $\$281,250\left(\$125,000+\$156,250\right)$. Therefore, the total surplus decreases from $285,500 to $281,250.

The net value created for society is the difference between the total surplus to market participants and the total damage to harmed people.

$\begin{array}{c}\text{Net value}=\text{Total surplus}-\text{Total damage}\\ =\$281,250-\$31,250\\ =\$250,000\end{array}$

Therefore, the net value created for society by marching band music increases from $244,744.87 to $250,000.

Economics Concept Introduction

Consumer Surplus: Consumer surplus is defined as the difference between the maximum amount a person is willing to pay for consuming a commodity and the actual price he pays for it.

Producer Surplus: Producer surplus is defined as the difference between the actual market price for which a commodity is sold and the minimum cost at which the producer is willing to sell the commodity. This minimum accepted price is usually the cost of production of the commodity.