Chapter 3, Problem 16P

(a)

To determine

Equilibrium price and quantity of ice cream.

Explanation of Solution

The given information:

Demand equation for ice cream is ${\text{Q}}^{\text{D}}=20-2\text{P}$ (1)

Supply equation for ice cream is ${\text{Q}}^{\text{S}}=4\text{P}-10$ (2)

Calculation:

Rearrange Equation (1) in terms of price derive the inverse demand equation.

$\begin{array}{c}{\text{Q}}^{\text{D}}=20-2\text{P}\\ 2\text{P}=20-{\text{Q}}^{\text{D}}\\ \text{P}=\frac{20}{2}-\frac{{\text{Q}}^{\text{D}}}{2}\end{array}$

$\text{P}=10-0.5{\text{Q}}^{\text{D}}$ (3)

The inverse demand equation is $\text{P}=10-0.5{\text{Q}}^{\text{D}}$.

Rearrange Equation (2) in terms of price derive the inverse supply equation.

$\begin{array}{l}{\text{Q}}^{\text{S}}=4\text{P}-10\\ 4\text{P}=10+{\text{Q}}^{\text{S}}\\ \text{P}=\frac{10}{4}+\frac{{\text{Q}}^{\text{S}}}{4}\end{array}$

$\text{P}=2.5+0.25{\text{Q}}^{\text{S}}$ (4)

The inverse supply equation is$\text{P}=2.5+0.25{\text{Q}}^{\text{S}}$.

The intersecting point of the demand and supply curve is the equilibrium point. Calculation of equilibrium price is shown below:

$\begin{array}{c}{\text{Q}}^{\text{D}}={\text{Q}}^{\text{S}}\\ 20-2\text{P}=4\text{P}-10\\ 4\text{P}+2\text{P}=10+20\\ 6\text{P}=30\\ \text{P}=\frac{30}{6}\\ \text{P}=5\end{array}$

Equilibrium price is $5.

Substitute the price in the demand equation (Equation (1)) to calculate the equilibrium quantity.

$\begin{array}{l}{\text{Q}}^{\text{D}}=20-2\text{P}\\ {\text{Q}}^{\text{D}}=20-2\left(5\right)\\ {\text{Q}}^{\text{D}}=20-10\\ {\text{Q}}^{\text{D}}=10\end{array}$

Equilibrium quantity is 10 units (10 gallon of ice cream).

From the above information, the market for ice cream is shown below:

In Figure 1, the vertical axis measures the price of ice cream and the horizontal axis measures gallon of ice cream. The upward sloping curve is the supply curve and downward sloping curve is the demand curve of ice cream. The intersecting point of the demand and supply curve is the equilibrium point. Thus, the equilibrium price is $5 and quantity is 10 units.

(b)

To determine

New demand equation.

Explanation of Solution

The imposition of tax increases the price, when price increases, the demand for the goods will decrease, which shifts the demand curve leftward. The new demand curve is shown in below figure.

In Figure 2, the vertical axis measures the price of ice cream and the horizontal axis measures the gallon of ice cream. The upward sloping curve is the supply curve and downward sloping curve is the demand curve of ice cream. The demand curve shift inward by the imposition of tax by $1.

(c)

To determine

New price and quantity.

Explanation of Solution

The new demand equation can be written as follows:

$\begin{array}{c}\text{New demand}=20-2\left(\text{P}+\text{Increase in tax}\right)\\ =20-2\left(\text{P}+1\right)\\ =20-2\text{P}-2\end{array}$

The new demand equation is ${\text{Q}}^{\text{D}}{}_{\text{New}}=20-2\text{P}-2$ (5)

The new equilibrium price can be calculated as follows:

$\begin{array}{c}\text{New demand}=\text{Supply}\\ 20-2\text{P}-2=4\text{P}-10\\ 4\text{P}+2\text{P}=10+20-2\\ 6\text{P}=28\\ \text{P}=\frac{28}{6}\\ \text{P}=4.67\end{array}$

The new price is $4.67.

Substitute the price in the demand equation (Equation (2)) to calculate the equilibrium quantity.

$\begin{array}{l}{\text{Q}}^{\text{S}}=4\text{P}-10\\ {\text{Q}}^{\text{S}}=4\left(4.67\right)-10\\ {\text{Q}}^{\text{S}}=18.68-10\\ {\text{Q}}^{\text{S}}=8.68\end{array}$

New quantity is 8.68 units.

(d)

To determine

Burden of tax.

Explanation of Solution

After imposition of tax ($1), the buyer would pay $5.67$\left(\$1+4.67\right)$ for one gallon of ice cream but the seller only end up with $4.67, thus $1 goes to government.

(e)

To determine

Consumer surplus.

Explanation of Solution

To find out the consumer surplus, the choke price has to be calculated. The calculation of choke price is shown below:

Substitute the value of quantity as zero in Equation (1).

$\begin{array}{c}{\text{Q}}^{\text{D}}=20-2\text{P}\\ \text{0}=20-2\text{P}\\ \text{P}=\frac{20}{2}\\ \text{P}=10\end{array}$

The demand choke price (maximum willing price) is $10.

Consumer surplus before tax imposition is calculated as follows:

$\begin{array}{c}\text{Consumer surplus}=\frac{1}{2}\times \left(\text{Maximum willing to pay}-\text{Actual pay}\right)\times \left(\text{Quantity}\right)\\ =\frac{1}{2}\times \left(10-5\right)\times \left(10\right)\\ =\frac{1}{2}\times 5\times 10\\ =\frac{1}{2}\times 50\\ =25\end{array}$

Consumer surplus is $25.

Consumer surplus after tax imposition is calculated as follows:

To find out the consumer surplus after tax, the choke price has to be calculated. The calculation of choke price is shown below:

Substitute the value of quantity as zero in Equation (5).

$\begin{array}{c}{\text{Q}}^{\text{D}}=20-2\text{P}-\text{2}\\ \text{0}=18-2\text{P}\\ \text{P}=\frac{18}{2}\\ \text{P}=9\end{array}$

The demand choke price (maximum willing price) is $9.

Consumer surplus after tax imposition is calculated as follows:

$\begin{array}{c}\text{Consumer surplus}=\frac{1}{2}\times \left(\begin{array}{l}\text{Maximum willing to pay}\\ -\text{New price}\end{array}\right)\times \left(\text{New quantity}\right)\\ =\frac{1}{2}\times \left(9-4.67\right)\times \left(8.68\right)\\ =18.79\end{array}$

Consumer surplus is $18.79.

Economics Concept Introduction

Producer surplus: Producer surplus is the difference between the lowest willing price accepted by a producer and the actual price received by a producer.

(f)

To determine

Producer surplus.

Explanation of Solution

To find out the producer surplus, the choke price has to be calculated. The calculation of choke price is shown below:

Substitute the value of quantity as zero in Equation (2).

$\begin{array}{c}{\text{Q}}^{\text{S}}=4\text{P}-10\\ 4\text{P}=10\\ \text{P}=\frac{10}{4}\\ \text{P}=2.5\end{array}$

The supply choke price (minimum acceptable price) is $2.5.

Producer surplus before tax imposition is calculated as follows:

$\begin{array}{c}\text{Producer surplus}=\frac{1}{2}\times \left(\begin{array}{l}{\text{Price}}_{\text{Initial}}\\ -\text{Minimum acceptable price}\end{array}\right)\times \left(\text{Quantity}\right)\\ =\frac{1}{2}\times \left(5-2.5\right)\times \left(10\right)\\ =12.5\end{array}$

Producer surplus is $12.50.

Producer surplus after tax imposition is calculated as follows”

$\begin{array}{c}\text{Producer surplus}=\frac{1}{2}\times \left(\begin{array}{l}\text{New price}\\ -\text{Minimum acceptable price}\end{array}\right)\times \left(\text{New quantity}\right)\\ =\frac{1}{2}\times \left(4.67-2.5\right)\times \left(8.68\right)\\ =9.42\end{array}$

Producer surplus is $9.42.

(g)

To determine

Total tax revenue.

Explanation of Solution

Total tax revenue can be calculated as follows:

$\begin{array}{c}\text{Tax revenue}=\text{Tax}\times \left(\text{New quantity}\right)\\ =1\times \left(8.68\right)\\ =8.68\end{array}$

Tax revenue is $8.68.

(h)

To determine

Deadweight loss.

Explanation of Solution

Deadweight loss can be calculated as follows:

$\begin{array}{c}\text{Deadweight loss}=\frac{1}{2}\times \text{Tax}\times \left(\text{Initial quantity}-\text{New quantity}\right)\\ =\frac{1}{2}\times 1\times \left(10-8.68\right)\\ =0.66\end{array}$

Deadweight loss is $0.66.