Chapter 2, Problem 3P

(a)

To determine

Graph the inverse demand curve.

Explanation of Solution

Given information:

Demand function for organic carrots:${\text{Q}}^{\text{D}}{}_{\text{O}}=75-5{\text{P}}_{\text{O}}+{\text{P}}_{\text{C}}+2\text{I}$ (1)

In Equation (1), “P_O” is the price of organic carrots, “P_C” is the price of conventional carrots, and “I” is the average consumer income.

Price of conventional carrots (P_C): 5

Average consumer income (I): 10

Calculation:

Demand equation can be simplified (with the price of the particular good alone) by substituting the respective values in Equation (1). This is done because when calculating the change in demand due to changes in the price of that good, all other factors are considered constant.

${\text{Q}}^{\text{D}}{}_{\text{O}}=75-5{\text{P}}_{\text{O}}+\text{1}\left(5\right)+2\left(10\right)$

${\text{Q}}^{\text{D}}{}_{\text{O}}=100-5{\text{P}}_{\text{O}}$ (2)

The demand equation when other things remain the same is ${\text{Q}}^{\text{D}}{}_{\text{O}}=100-5{\text{P}}_{\text{O}}$.

Rearrange Equation (2) in terms of price of the goods to derive the inverse demand equation.

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

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

Substitute quantity as zero in Equation (3) to calculate the price of organic carrots (maximum willing price).

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

Price of organic carrots is $20.

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

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

Quantity of organic carrots is 100 units.

By using this information, the demand curve of organic carrots is illustrated below in Figure 1.

In Figure 1, the vertical axis measures the price of organic carrots and the horizontal axis measures the quantity of organic carrots. The downward sloping curve “D” is the demand curve of organic carrots.

Economics Concept Introduction

Demand curve: A demand curve is a graph which shows the quantities of a commodity that the consumers will buy at different price levels.

(b)

To determine

The quantity demanded.

Explanation of Solution

At price $5, the quantity demanded for organic carrots $\left({\text{Q}}^{\text{D}}{}_{\text{O}}\right)$ is calculated by substituting the respective values in Equation (2).

$\begin{array}{l}{\text{Q}}^{\text{D}}{}_{\text{O}}=100-5{\text{P}}_{\text{O}}\\ {\text{Q}}^{\text{D}}{}_{\text{O}}=100-5\left(5\right)\\ {\text{Q}}^{\text{D}}{}_{\text{O}}=75\end{array}$

Quantity demanded for organic carrots at price $5 is 75.

At price $10, the quantity demanded for organic carrots $\left({\text{Q}}^{\text{D}}{}_{\text{O}}\right)$ is calculated by substituting the respective values in the inverse demand equation (Equation (2)).

$\begin{array}{l}{\text{Q}}^{\text{D}}{}_{\text{O}}=100-5{\text{P}}_{\text{O}}\\ {\text{Q}}^{\text{D}}{}_{\text{O}}=100-5\left(10\right)\\ {\text{Q}}^{\text{D}}{}_{\text{O}}=50\end{array}$

Quantity demanded for organic carrots at price $10 is 50.

(c)

To determine

Changes in demand for organic carrots.

Explanation of Solution

Given information:

Price of conventional carrots (P_C): 15

Average consumer income (I): 10

Calculation:

Demand equation can be simplified (with the price of the particular good alone) by substituting the respective values in Equation (1). This is done because when calculating the change in demand due to changes in the price of the good, all other factors are considered constant.

$\begin{array}{l}{\text{Q}}^{\text{D}}{}_{\text{O}}=75-5{\text{P}}_{\text{O}}+{\text{P}}_{\text{C}}+2\text{I}\\ {\text{Q}}^{\text{D}}{}_{\text{O}}=75-5{\text{P}}_{\text{O}}+1\left(15\right)+2\left(10\right)\end{array}$

${\text{Q}}^{\text{D}}{}_{\text{O}}=110-5{\text{P}}_{\text{O}}$ (4)

The new demand equation when other things remain the same is ${\text{Q}}^{\text{D}}{}_{\text{O}}=110-5{\text{P}}_{\text{O}}$.

Rearrange Equation (4) in terms of price of the goods to derive the inverse demand equation.

$\begin{array}{c}{\text{Q}}^{\text{D}}{}_{\text{O}}=110-5{\text{P}}_{\text{O}}\\ 5{\text{P}}_{\text{O}}=110-{\text{Q}}^{\text{D}}{}_{\text{O}}\\ {\text{P}}_{\text{O}}=\frac{110}{5}-\frac{{\text{Q}}^{\text{D}}{}_{\text{O}}}{5}\end{array}$

${\text{P}}_{\text{O}}=22-0.2{\text{Q}}^{\text{D}}{}_{\text{O}}$ (5)

Substitute quantity as zero in Equation (5) to calculate the price of organic carrots (maximum willing price).

$\begin{array}{l}{\text{P}}_{\text{O}}=22-0.2{\text{Q}}^{\text{D}}{}_{\text{O}}\\ {\text{P}}_{\text{O}}=22-\left(0.2\times 0\right)\\ {\text{P}}_{\text{O}}=22\end{array}$

Price of organic carrots is $22.

Substitute price as zero in Equation (5) to calculate the maximum quantity demanded.

$\begin{array}{c}\text{0}=22-0.2{\text{Q}}^{\text{D}}{}_{\text{O}}\\ 0.2{\text{Q}}^{\text{D}}{}_{\text{O}}=22\\ {\text{Q}}^{\text{D}}{}_{\text{O}}=\frac{22}{0.2}\\ =110\end{array}$

Quantity of organic carrots is 110 units.

Substitute the value of price ($10) to calculate the quantity demanded of organic carrots at this particular price. This is done to know the change in quantity due to change in the price of P_C.

$\begin{array}{c}{\text{P}}_{\text{O}}=22-0.2{\text{Q}}^{\text{D}}{}_{\text{O}}\\ 10=22-0.2{\text{Q}}^{\text{D}}{}_{\text{O}}\\ 0.2{\text{Q}}^{\text{D}}{}_{\text{O}}=22-10\\ {\text{Q}}^{\text{D}}{}_{\text{O}}=\frac{12}{0.2}\\ {\text{Q}}^{\text{D}}{}_{\text{O}}=60\end{array}$

Quantity demand of organic carrots at price $10 is 60. When the price of P_C increases from $5 to $15, the quantity increases from $50 to $60.

By using this information, the change in demand curve of organic carrots is shown below in Figure 2.

In Figure 2, the vertical axis measures the price of organic carrots and the horizontal axis measures the quantity of organic carrots demanded. D1 is the initial demand curve of the organic carrots. The increase in the quantity demand for organic carrots shifts the demand curve from D1 to D2. It leads to increase the price from $20 to $22. At the initial demand curve, people demand 50 at the price $10; then it increases to 60 after the change in demand.

Economics Concept Introduction

Demand curve: A demand curve is a graph which shows the quantities of a commodity that the consumers will buy at different price levels.

(d)

To determine

Demand for organic carrots.

Explanation of Solution

The demand for organic carrot increases. This increase in demand shifts the demand curve rightward. This rightward shift shows the two goods are substitute goods because increase in the price of conventional carrots increases the demand for organic carrots.

Economics Concept Introduction

Substitute goods: Substitute goods are those goods that can be used for the same purpose.

(e)

To determine

Demand for organic carrots.

Explanation of Solution

The demand for organic carrot increases when the average income of consumer increases. Thus, it is clear that the organic carrots are normal goods.

Economics Concept Introduction

Normal good: It is the good whose quantity demanded increases when the income of the consumer increases and vice versa.