Chapter 2, Problem 10P

(a)

To determine

Graph the inverse supply curve demand curve.

Explanation of Solution

Given information:

Demand function for towel:${\text{Q}}^{\text{D}}=\text{100}-\text{5P }$ (1)

Supply function for towel:${\text{Q}}^{\text{S}}=\text{10P }$ (2)

Calculation:

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

$\begin{array}{c}{\text{Q}}^{\text{D}}=\text{100}-\text{5P }\\ \text{5P}=\text{100}-{\text{Q}}^{\text{D}}\\ \text{P}=\frac{\text{100}}{5}-\frac{{\text{Q}}^{\text{D}}}{5}\end{array}$

$\text{P}=20-0.2{\text{Q}}^{\text{D}}$ (3)

The inverse demand equation is$\text{P}=20-0.2{\text{Q}}^{\text{D}}$.

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

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

The inverse supply function is$\text{P}=\frac{{\text{Q}}^{\text{S}}}{\text{10}}$.

Substitute quantity as zero in Equation (3) to calculate the price.

$\begin{array}{l}\text{P}=20-0.2{\text{Q}}^{\text{D}}\\ \text{P}=20-0.2\left(0\right)\\ \text{P}=20\end{array}$

The maximum willing price is $20.

Substitute price as zero in Equation (3) to calculate the quantity.

$\begin{array}{c}\text{P}=20-0.2{\text{Q}}^{\text{D}}\\ 0=20-0.2{\text{Q}}^{\text{D}}\\ 0.2{\text{Q}}^{\text{D}}=20-0\\ {\text{Q}}^{\text{D}}=\frac{20}{0.2}\\ {\text{Q}}^{\text{D}}=100\end{array}$

The maximum willing quantity is 100 units.

Substitute quantity as 0 in Equation (4) to calculate the price.

$\begin{array}{l}\text{P}=\frac{{\text{Q}}^{\text{S}}}{\text{10}}\\ \text{P}=\frac{0}{10}\\ \text{P}=0\end{array}$

The minimum willing price is 0.

Substitute quantity as 200 units in Equation (4) to calculate the price.

$\begin{array}{l}\text{P}=\frac{{\text{Q}}^{\text{S}}}{\text{10}}\\ \text{0}=\frac{\text{200}}{10}\\ {\text{Q}}^{\text{S}}=20\end{array}$

The minimum willing quantity is 20.

From these information, the inverse demand and supply curve are shown in Figure 1 below.

In Figure 1, the vertical axis measures the price of the towel and horizontal axis measures the quantity of the towel. The upward sloping curve “S” is the supply curve of the towel and downward sloping curve “D” is the demand curve for the towel.

Economics Concept Introduction

Demand curve: Demand curve shows the quantity demand at different price levels.

(b)

To determine

Equilibrium price and quantity.

Explanation of Solution

Intersecting point of the demand and supply curve is the equilibrium point. The corresponding price and quantity are the equilibrium price and quantity. Equilibrium price can be calculated as follows: 

$\begin{array}{c}{\text{Q}}^{\text{D}}={\text{Q}}^{\text{S}}\\ \text{100}-\text{5P}=\text{10P }\\ \text{5P}+10\text{P}=100\\ \text{P}=\frac{100}{15}\\ \text{P}=6.67\\ \end{array}$

Equilibrium price is $6.67.

Equilibrium price can be calculated by substituting the equilibrium price into supply equation (Equation (2)).

$\begin{array}{l}{\text{Q}}^{\text{S}}=\text{10P}\\ {\text{ Q}}^{\text{S}}=\text{10}\left(6.67\right)\\ {\text{Q}}^{\text{S}}=66.7\end{array}$

Equilibrium quantity is 66.7 units.

Economics Concept Introduction

Equilibrium price:  The equilibrium price is the market price determined by the interaction between the quantity demanded and the quantity supplied.

Equilibrium quantity: The equilibrium quantity is the point where the quantity demanded is equal to the quantity supplied.

(c)

To determine

New inverse supply curve. 

Explanation of Solution

Given information:

At each price, 20 fewer towels are offered for sale. 

Calculation:

At each price, 20 fewer towels are offered for sale. Thus, the new inverse supply curve is shown below:

${\text{Q}}^{\text{S}}=\text{10P}-\text{Decreasing quantity}$

${\text{Q}}^{\text{S}}=\text{10P}-20$ (5)

Substitute quantity as zero in Equation (5) to calculate the price (minimum willing price).

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

The minimum willing price is $2.

Substitute quantity as 110 in Equation (5) to calculate the price.

$\begin{array}{c}{\text{Q}}^{\text{S}}=\text{10P}-20\\ 200=\text{10P}-20\\ \text{10P}=20+20\text{0}\\ \text{P}=\frac{220}{10}\\ \text{P}=22\end{array}$

The minimum willing price is $2.

By using this information, change in supply curve of towel is shown in Figure 2 below.

In Figure 2, the vertical axis measures the price of the towel and horizontal axis measures the quantity of the towel. The upward sloping curve “S1” is the supply curve of the towel and downward sloping curve “D” is the demand curve for the towel. The reduction in 20 fewer towels shifts the supply curve to the left. Thus, curve “S2” is the new inversee supply curve, where the minimum willing price is 2.

Economics Concept Introduction

Supply curve: Supply curve shows the quantity supplied at different price levels.

(d)

To determine

Equilibrium price and quantity.

Explanation of Solution

The calculation of equilibrium price is shown below:

$\begin{array}{c}{\text{Q}}^{\text{D}}={\text{Q}}^{\text{S}}\\ \text{100}-\text{5P}=\text{10P}-20\\ \text{5P}+10\text{P}=100+20\\ \text{P}=\frac{120}{15}\\ \text{P}=8\\ \end{array}$

Equilibrium price is $8.

Equilibrium quantity can be calculated by substituting the equilibrium price in to new supply equation (Equation (5)).

$\begin{array}{l}{\text{Q}}^{\text{S}}=\text{10P}-20\\ {\text{Q}}^{\text{S}}=\text{10}\left(8\right)-20\\ {\text{Q}}^{\text{S}}=80-20\\ {\text{Q}}^{\text{S}}=60\end{array}$

Equilibrium quantity is 60 units.

Economics Concept Introduction

Equilibrium price:  The equilibrium price is the market price determined by the interaction between the quantity demanded and the quantity supplied.

Equilibrium quantity: The equilibrium quantity is the point where the quantity demanded is equal to the quantity supplied.

(e)

To determine

Equilibrium price and quantity.

Explanation of Solution

At each price, 25 fewer towels are offered for sale. Thus, the new inverse demand curve is shown below:

$\begin{array}{l}{\text{Q}}^{\text{D}}=\text{100}-\text{5P}-\text{Decreasing quantity}\\ {\text{Q}}^{\text{D}}=\text{100}-\text{5P}-25\end{array}$

${\text{Q}}^{\text{D}}=75-\text{5P}$ (6)

Equilibrium price can be calculated as follows:

$\begin{array}{c}{\text{Q}}^{\text{D}}={\text{Q}}^{\text{S}}\\ \text{75}-\text{5P}=\text{10P}\\ \text{5P}+\text{10P}=\text{75}\\ \text{15P}=\text{75}\\ \text{P}=\frac{75}{15}\\ \text{P}=5\end{array}$

Equilibrium price is $5.

Equilibrium quantity can be calculated by substituting the equilibrium price in to new demand equation (Equation (6)).

$\begin{array}{l}{\text{Q}}^{\text{S}}=75-5\text{P}\\ {\text{Q}}^{\text{S}}=75-5\left(5\right)\\ {\text{Q}}^{\text{S}}=50\end{array}$

Equilibrium quantity is 50 units. The equilibrium quantity and price are reducing. The new equilibrium price and quantity are less than the initial equilibrium quantity and price.  

Economics Concept Introduction

Equilibrium price:  The equilibrium price is the market price determined by the interaction between the quantity demanded and the quantity supplied.

Equilibrium quantity: The equilibrium quantity is the point where the quantity demanded is equal to the quantity supplied.