Chapter 15, Problem 3P

 (a)

To determine

The demand side of the market.

Explanation of Solution

The quantity demanded of cheese is ${\text{Q}}_{\text{c}}^{\text{d}}=30-{\text{P}}_{\text{C}}-{\text{P}}_{\text{W}}$ and the quantity demanded of wine is given as ${\text{Q}}_{\text{w}}^{\text{d}}=30-{\text{P}}_{\text{C}}-{\text{P}}_{\text{W}}$ where the quantity demanded of cheese moves in the same direction of the quantity demand of wine. Therefore, the wine and cheese are linked on the demand side of the market.

 (b)

To determine

The supply and demand in the cheese market.

Explanation of Solution

The quantity demanded of cheese is given by ${\text{Q}}_{\text{c}}^{\text{d}}=30-{\text{P}}_{\text{C}}-{\text{P}}_{\text{W}}$ 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 ${\text{P}}_{\text{C}}=15-0.5{\text{P}}_{\text{W}}$.

 (c)

To determine

The supply and demand in the cheese market. 

Explanation of Solution

The quantity demanded of wine market is ${\text{Q}}_{\text{w}}^{\text{d}}=30-{\text{P}}_{\text{C}}-{\text{P}}_{\text{W}}$ 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.

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

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

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

Therefore, the price of wine is ${\text{P}}_{\text{w}}=15-0.5{\text{P}}_{\text{c}}$.

 (d)

To determine

The price of cheese.

Explanation of Solution

Substitute the price function of wine (described in part (c)) in to the price function of cheese (described in part (b)), to get the value of the price of cheese.

$\begin{array}{c}{\text{P}}_{\text{C}}=15-0.5\left(15-0.5{\text{P}}_{\text{C}}\right)\\ =15-7.5+0.25{\text{P}}_{\text{C}}\\ {\text{P}}_{\text{C}}=7.5+0.25{\text{P}}_{\text{C}}\\ {\text{P}}_{\text{C}}-0.25{\text{P}}_{\text{C}}=7.5\\ 0.75{\text{P}}_{\text{C}}=7.5\\ {\text{P}}_{\text{C}}=\frac{7.5}{0.75}\\ =\$10\end{array}$

Therefore, the price of cheese is $10.

 (e)

To determine

The price of wine.

Explanation of Solution

The price function of wine (described in part (c)) is ${\text{P}}_{\text{w}}=15-0.5{\text{P}}_{\text{c}}$. Now, substitute the price of cheese (described in pat (d)) into this function.

$\begin{array}{c}{\text{P}}_{\text{w}}=15-\left(0.5\times 10\right)\\ =15-5\\ {\text{P}}_{\text{w}}=\$10\end{array}$

Therefore, the price of wine is $10.

 (f)

To determine

The equilibrium quantities of wine and cheese.

Explanation of Solution

Substitute the respective values described in part (d) and (e) into the demand function for wine.

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

Therefore, the quantity of wine produced is 10.

Substitute the respective values described in part (d), and (e) into the demand function of cheese.

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

Therefore, the quantity of cheese produced is 10.

Hence, the equilibrium quantities of wine and cheese is 10.