Chapter 15, Problem 4P

 (a)

To determine

The partial equilibrium effects of the change in the demand for wine.  

Explanation of Solution

The new quantity demanded of wine can represented as follows:

 ${\text{Q}}_{\text{c}}^{\text{d}}=20-{\text{P}}_{\text{C}}-{\text{P}}_{\text{W}}$

 The supply of wine is given as ${\text{Q}}_{\text{w}}^{\text{S}}={\text{P}}_{\text{w}}$. Now, equate the supply and demand in the wine market.

$20-{\text{P}}_{\text{C}}-{\text{P}}_{\text{W}}={\text{P}}_{\text{w}}$

Now, set the price of wine as a function of the price of cheese where the price of cheese is constant.

$\begin{array}{c}20-{\text{P}}_{\text{C}}-{\text{P}}_{\text{W}}={\text{P}}_{\text{w}}\\ 2{\text{P}}_{\text{W}}=20-{\text{P}}_{\text{C}}\\ {\text{P}}_{\text{w}}=\frac{20}{2}-\frac{{\text{1P}}_{\text{C}}}{2}\\ {\text{P}}_{\text{w}}=10-0.5\overline{{\text{P}}_{\text{C}}}\end{array}$

Therefore, the price function of wine is ${\text{P}}_{\text{w}}=10-0.5{\overline{\text{P}}}_{\text{C}}$ where the price of cheese keeps it as the same $10.

Now, substitute the value of cheese into the price function of wine.

$\begin{array}{c}{\text{P}}_{\text{w}}=10-\left(0.5\times 10\right)\\ =5\end{array}$

Therefore, the partial equilibrium effect of the change in the demand for wine causes to decrease the price by $5.

 (b)

To determine

The demand in the cheese market.

Explanation of Solution

The quantity demanded of cheese is given by ${\text{Q}}_{\text{c}}^{\text{d}}=30-{\text{P}}_{\text{C}}-{\text{P}}_{\text{W}}$ where the price of wine is $10. Therefore, the demand function of cheese can represented as follows:

$\begin{array}{c}{\text{Q}}_{\text{c}}^{\text{d}}=30-{\text{P}}_{\text{C}}-10\\ {\text{Q}}_{\text{c}}^{\text{d}}=20-{\text{P}}_{\text{C}}\end{array}$

If the price of wine falls to $5 due to the demand changes, the new demand function of cheese can be represented as follows:

$\begin{array}{c}{\text{Q}}_{\text{c}}^{\text{d}}=30-{\text{P}}_{\text{C}}-5\\ {\text{Q}}_{\text{c}}^{\text{d}}=25-{\text{P}}_{\text{C}}\end{array}$

Here, the new demand function of cheese shows that a fall in the price of wine causes the demand for cheese to increase.

 (c)

To determine

The effects of demand changes.

Explanation of Solution

The new quantity demanded of cheese market is ${\text{Q}}_{\text{c}}^{\text{d}}=25-{\text{P}}_{\text{C}}$ and supply of cheese is given by ${\text{Q}}_{\text{c}}^{\text{S}}={\text{P}}_{\text{c}}$. Now, equate the supply and demand in the cheese market.

$25-{\text{P}}_{\text{C}}={\text{P}}_{\text{c}}$

Now, set the price of cheese as a function of the price of wine.

$\begin{array}{c}25-{\text{P}}_{\text{C}}={\text{P}}_{\text{c}}\\ 2{\text{P}}_{\text{c}}=25\\ {\text{P}}_{\text{c}}=\frac{25}{2}\\ {\text{P}}_{\text{c}}=12.5\end{array}$

Therefore, the price of cheese is $12.5.

Now, substitute the respective value into the supply function.

${\text{Q}}_{\text{c}}^{\text{S}}=12.5$

Therefore, the quantity supply of cheese is12.5 units.

Hence, due to the change in demand for cheese, the price of cheese increases by $2.5$\left(\$12.5-\$10\right)$ and the quantity also increases by $2.5$\left(\$12.5-\$10\right)$.

 (d)

To determine

The price of wine.

Explanation of Solution

Substitute the respective value of the price of cheese into the price function of wine.

$\begin{array}{c}{\text{P}}_{\text{W}}=10-\left(0.5\times \$12.5\right)\\ =\$3.75\end{array}$

Therefore, the new price of wine is $3.75 which means the demand for wine decreases due to the higher price of cheese.

The quantity supplied of wine can be represented as follows:

Substitute the respective values into the demand function for wine.

$\begin{array}{c}{\text{Q}}_{\text{w}}^{\text{d}}=20-12.5-3.75\\ =3.75\end{array}$

Therefore, the quantity of wine produced is 3.75.

Hence, both price and quantity of wine falls by 3.75. Therefore, the wine market pushed farther from its initial equilibrium. Therefore, this decrease in the price of wine market causes to increase the demand for cheese.

 (e)

To determine

The equilibrium quantities of wine and cheese.

Explanation of Solution

The quantity demanded of cheese is given by ${\text{Q}}_{\text{c}}^{\text{d}}=30-{\text{P}}_{\text{C}}-{\text{P}}_{\text{W}}$ and supply of cheese is given as ${\text{Q}}_{\text{C}}^{\text{S}}={\text{P}}_{\text{C}}$. Now, equate the supply and demand in the cheese market.

$30-{\text{P}}_{\text{C}}-{\text{P}}_{\text{W}}={\text{P}}_{\text{C}}$

Now, set the price of cheese as a function of the price of wine.

$\begin{array}{c}30-{\text{P}}_{\text{C}}-{\text{P}}_{\text{W}}={\text{P}}_{\text{C}}\\ 2{\text{P}}_{\text{C}}=30-{\text{P}}_{\text{W}}\\ {\text{P}}_{\text{C}}=\frac{30}{2}-\frac{{\text{1P}}_{\text{W}}}{2}\\ {\text{P}}_{\text{C}}=15-0.5{\text{P}}_{\text{W}}\end{array}$

Therefore, the price of cheese is ${\text{P}}_{\text{C}}=15-0.5{\text{P}}_{\text{W}}$

The quantity demanded of wine market is ${\text{Q}}_{\text{w}}^{\text{d}}=20-{\text{P}}_{\text{C}}-{\text{P}}_{\text{W}}$ and supply of wine is given by  ${\text{Q}}_{\text{w}}^{\text{S}}={\text{P}}_{\text{w}}$. Now, equate the supply and demand in the wine market.

$20-{\text{P}}_{\text{C}}-{\text{P}}_{\text{W}}={\text{P}}_{\text{w}}$

Now, set the price of wine as a function of price of cheese.

$\begin{array}{c}20-{\text{P}}_{\text{C}}-{\text{P}}_{\text{W}}={\text{P}}_{\text{w}}\\ 2{\text{P}}_{\text{w}}=20-{\text{P}}_{\text{c}}\\ {\text{P}}_{\text{w}}=\frac{20}{2}-\frac{{\text{1P}}_{\text{c}}}{2}\\ {\text{P}}_{\text{w}}=10-0.5{\text{P}}_{\text{c}}\end{array}$

Therefore, the price of wine is ${\text{P}}_{\text{w}}=10-0.5{\text{P}}_{\text{c}}$.

Substitute the price function of wine in to the price function of cheese to get the value of the price of cheese.

$\begin{array}{c}{\text{P}}_{\text{C}}=15-0.5\left(10-0.5{\text{P}}_{\text{C}}\right)\\ =15-5+0.25{\text{P}}_{\text{C}}\\ {\text{P}}_{\text{C}}=10+0.25{\text{P}}_{\text{C}}\\ \left(\text{1}-0.25\right){\text{P}}_{\text{C}}=10\\ 0.75{\text{P}}_{\text{C}}=\$10\\ {\text{P}}_{\text{C}}=\frac{10}{0.75}\\ =\$\mathrm{13..33}\end{array}$

Therefore, the price of cheese is $13.33.

Since the price function of wine is ${\text{P}}_{\text{w}}=10-0.5{\text{P}}_{\text{C}}$, substitute the price of cheese into this function.

$\begin{array}{c}{\text{P}}_{\text{w}}=10-\left(0.5\times 13.33\right)\\ =10-6.66\\ {\text{P}}_{\text{w}}=\$3.33\end{array}$

Therefore, the price of wine is $3.33.

Substitute the respective values into the demand function for wine.

$\begin{array}{c}{\text{Q}}_{\text{w}}^{\text{d}}=20-13.33-3.33\\ =3.33\end{array}$

Therefore, the quantity of wine produced is 3.33.

Substitute the respective values into the demand function of cheese.

$\begin{array}{c}{\text{Q}}_{\text{c}}^{\text{d}}=30-13.33-3.33\\ =13.33\end{array}$

Therefore, the quantity of cheese produced is 13.33.

Hence, the equilibrium quantities of wine is 3.33 units and cheese is 13.33 units.

 (f)

To determine

The general equilibrium analysis of price and quantities of wine compared with the partial equilibrium analysis.

Explanation of Solution

As part (a) describes, the partial equilibrium analysis shows that the changes in demand causes to reduce the price and quantity of wine by more than five from the initial value. In part (e), the general equilibrium analysis describes that the price and quantity of wine decreases by 6.66 (10-3.33). Hence, the general equilibrium analysis shows a higher value of price and quantity of wine than the partial equilibrium analysis.

Economics Concept Introduction

General equilibrium analysis: The general equilibrium analysis is the study of market behaviour that accounts for cross-market influences and is concerned with conditions present when all markets are simultaneously in equilibrium.

Partial equilibrium analysis: The Partial equilibrium analysis is the determination of the equilibrium in a particular market that assumes there are no cross-market spill overs.