Chapter 15, Problem 6P

 (a)

To determine

The general equilibrium price and quantity.

Explanation of Solution

The general equilibrium price and quantity is determined at the point where the quantity demand intersects the quantity supply in the market. Here, the quantity demanded of beans is ${\text{Q}}_{\text{b}}^{\text{d}}=30-{\text{P}}_{\text{b}}$ and the quantity supplied of beans is ${\text{Q}}_{\text{b}}^{\text{s}}=2{\text{P}}_{\text{b}}-{\text{P}}_{\text{C}}$.

Now, set the expressions of quantity demand and quantity supply as equal.

$30-{\text{P}}_{\text{b}}=2{\text{P}}_{\text{b}}-{\text{P}}_{\text{C}}$

Now set the price of beans as a function of the price of corn.

$\begin{array}{c}2{\text{P}}_{\text{b}}-{\text{P}}_{\text{c}}=30-{\text{P}}_{\text{b}}\\ {\text{3P}}_{\text{b}}=30+{\text{P}}_{\text{c}}\\ {\text{P}}_{\text{b}}=\frac{30}{3}+\frac{{\text{1P}}_{\text{c}}}{3}\\ {\text{P}}_{\text{b}}=10+\frac{1}{3}{\text{P}}_{\text{c}}\end{array}$

Therefore, the price function of beans is ${\text{P}}_{\text{b}}=10+\frac{1}{3}{\text{P}}_{\text{c}}$.

Now, the price function of corn can set using the similar method.

The quantity demanded of corn is ${\text{Q}}_{\text{b}}^{\text{d}}=30-{\text{P}}_{\text{c}}$ and the quantity supplied of corn is ${\text{Q}}_{\text{c}}^{\text{s}}=2{\text{P}}_{\text{c}}-{\text{P}}_{\text{b}}$.

Now, set the expressions of quantity demand and quantity supply as equal.

$30-{\text{P}}_{\text{c}}=2{\text{P}}_{\text{c}}-{\text{P}}_{\text{b}}$

Now, set the price of corn as a function of the price of beans.

$\begin{array}{c}2{\text{P}}_{\text{c}}-{\text{P}}_{\text{b}}=30-{\text{P}}_{\text{c}}\\ {\text{3P}}_{\text{c}}=30+{\text{P}}_{\text{b}}\\ {\text{P}}_{\text{c}}=\frac{30}{3}+\frac{{\text{1P}}_{\text{b}}}{3}\\ {\text{P}}_{\text{c}}=10+\frac{1}{3}{\text{P}}_{\text{b}}\end{array}$

Therefore, the price function of corn is ${\text{P}}_{\text{c}}=10+\frac{1}{3}{\text{P}}_{\text{b}}$.

Substitute the price function of corn in to the price function of beans for solving the price of corn.

$\begin{array}{c}{\text{P}}_{\text{b}}=10+\frac{1}{3}{\text{P}}_{\text{c}}\\ =10+\frac{1}{3}\left(10+\frac{1}{3}{\text{P}}_{\text{b}}\right)\\ =10+\frac{10}{3}+\frac{1}{9}{\text{P}}_{\text{b}}\end{array}$

Now, rearrange the equation.

$\begin{array}{c}{\text{P}}_{\text{b}}-\frac{1}{9}{\text{P}}_{\text{b}}=\frac{40}{3}\\ {\text{P}}_{\text{b}}\left(\frac{8}{9}\right)=\frac{40}{3}\\ {\text{P}}_{\text{b}}=\frac{45}{3}\\ {\text{P}}_{\text{b}}=15\end{array}$

Therefore, the price of beans is $15 per bushel.

Solve for the price of corn by substituting respective value into the price function of corn.

$\begin{array}{c}{\text{P}}_{\text{c}}=10+\left(\frac{1}{3}\times 15\right)\\ =15\end{array}$

Therefore, the price of corn is $15 per bushel.

Now, substitute the respective values into the demand function of beans to get the equilibrium quantity of beans.

$\begin{array}{c}{\text{Q}}_{\text{b}}^{\text{d}}=30-15\\ =15\end{array}$

Therefore, the equilibrium quantity of beans is 15 million bushels.

Substitute the respective values into the demand function of corn to get the equilibrium quantity of corn.

$\begin{array}{c}{\text{Q}}_{\text{c}}^{\text{d}}=30-15\\ =15\end{array}$

Therefore, the equilibrium quantity of corn is 15 million bushels.

 (b)

To determine

The new general equilibrium price and quantity after the price increase. 

Explanation of Solution

The new quantity demanded of beans is ${\text{Q}}_{\text{b}}^{\text{d}}=38-{\text{P}}_{\text{b}}$ and the quantity supplied of beans remained as the same ${\text{Q}}_{\text{b}}^{\text{s}}=2{\text{P}}_{\text{b}}-{\text{P}}_{\text{C}}$.

Now, set the expressions of quantity demand and quantity supply as equal.

$38-{\text{P}}_{\text{b}}=2{\text{P}}_{\text{b}}-{\text{P}}_{\text{C}}$

Now, set the price of beans as a function of the price of corn.

$\begin{array}{c}2{\text{P}}_{\text{b}}-{\text{P}}_{\text{c}}=38-{\text{P}}_{\text{b}}\\ {\text{3P}}_{\text{b}}=38+{\text{P}}_{\text{c}}\\ {\text{P}}_{\text{b}}=\frac{38}{3}+\frac{{\text{1P}}_{\text{c}}}{3}\\ {\text{P}}_{\text{b}}=12.67+\frac{1}{3}{\text{P}}_{\text{c}}\end{array}$

Therefore, the new price function of beans is ${\text{P}}_{\text{b}}=12.67+\frac{1}{3}{\text{P}}_{\text{c}}$.

There is no change in the demand for and supply of corn. Therefore, the price function of corn is ${\text{P}}_{\text{c}}=10+\frac{1}{3}{\text{P}}_{\text{b}}$ (described in part (a)).

Substitute the price function of corn in to the price function of beans for solving the new price of corn.

$\begin{array}{c}{\text{P}}_{\text{b}}=12.67+\frac{1}{3}{\text{P}}_{\text{c}}\\ =12.67+\frac{1}{3}\left(10+\frac{1}{3}{\text{P}}_{\text{b}}\right)\\ =12.67+\frac{10}{3}+\frac{1}{9}{\text{P}}_{\text{b}}\end{array}$

Now, rearrange the equation.

$\begin{array}{c}{\text{P}}_{\text{b}}-\frac{1}{9}{\text{P}}_{\text{b}}=\frac{48.01}{3}\\ {\text{P}}_{\text{b}}\left(\frac{8}{9}\right)=\frac{54}{3}\\ {\text{P}}_{\text{b}}=18\end{array}$

Therefore, the new price of beans is $18 per bushel.

Solve for the price of corn by substituting the respective value into the price function of corn

$\begin{array}{c}{\text{P}}_{\text{c}}=10+\frac{1}{3}{\text{P}}_{\text{b}}\\ =10+\left(\frac{1}{3}\times 18\right)\\ =\$16\end{array}$

Therefore, the new price of corn is $16 per bushel.

Substitute the respective values into the demand function of beans to get the new equilibrium quantity of beans.

$\begin{array}{c}{\text{Q}}_{\text{b}}^{\text{d}}=38-18\\ =20\end{array}$

Therefore, the new equilibrium quantity of beans is 20 million bushels.

Substitute the respective values into the demand function of corn to get the equilibrium quantity of corn.

$\begin{array}{c}{\text{Q}}_{\text{c}}^{\text{d}}=30-16\\ =14\end{array}$

Therefore, the equilibrium quantity of corn is 14 million bushels.

 (c)

To determine

The demand effect on the price and quantity of beans.

Explanation of Solution

The initial price of beans was $15 per bushel and the equilibrium quantity was also 15 million bushels (described in part (a)). After the increases of demand, the price shift to $18 per bushel and the quantity equilibrium becomes 20 million bushels (described in part (b)). Therefore, due to the demand increase, the price of beans increases by $3 per bushel $\left(\$18-\$15\right)$ and the quantity increases by 5$\left(\$20-\$15\right)$ million bushels.

 (d)

To determine

The effect of increasing demand of beans on the price and quantity of corn.

Explanation of Solution

The initial price of corn was $15 per bushel and the equilibrium quantity was also 15 million bushels (described in part (a)). After the increases of demand for beans, the price of corn shifts to $16 per bushel and the quantity equilibrium becomes 14 million bushels (described in part (b)). Therefore, due to the demand increases of beans, the price of corn increases by $1 per bushel $\left(\$18-\$15\right)$ and the quantity decreases by 1$\left(\$20-\$15\right)$ million bushels.