Q. 40 - Precalculus Enhanced with Graphing Utilities

Question

Supply and Demand Suppose that the quantity supplied S and quantity demanded D of hot dogs at a baseball game are given by the following functions:

 $S\left(p\right)=-2000+3000p$

$D\left(p\right)=10,000-1000p$

where p is the price of a hot dog. 

(a) Find the equilibrium price for hot dogs at the baseball game. What is the equilibrium quantity? 

(b) Determine the prices for which quantity demanded is less than quantity supplied. 

(c) What do you think will eventually happen to the price of hot dogs if quantity demanded is less than quantity supplied? 

Step-by-Step Solution

Answer

Part (a)

• The equilibrium price for hot dog are 3.
• The equilibrium quantity are 7000 for hot dog.
• 
  

Part  (b) The prices of hot dog will be less than 3 when quantity demanded is less than quantity supplied.  

Part  (c) If quantity demanded of hot dog is less than quantity supplied, then there will be excess hot dogs that lead in decrease in price.

Part (a) Step 1 . Given Information

The hot dogs quantity supplied at a baseball game are 

$S\left(p\right)=-2000+3000p$.

The hot dogs quantity demanded at a baseball game are

 $D\left(p\right)=10,000-1000p$

where p is the price of a hot dog. 

Part (a). Step 2. Explanation.

To find the equilibrium price for hot dogs at the baseball game, equate supply function and demand function. 

$$\begin{gathered} S\left(p\right)=D\left(p\right) \\ -2000+3000p=10000-1000p \\ 4000p=12000 \\ p=3 \end{gathered}$$

Find the equilibrium quantity by substitute $p=3$ in supply function. $S\left(p\right)=-2000+3000p$

$$\begin{gathered} S\left(3\right)=-2000+3000\left(3\right) \\ =7000 \end{gathered}$$

Part (a). Step 3. Conclusion.

• The equilibrium price for hot dog are 3.
• The equilibrium quantity are 7000 for hot dog.

Part (b). Step 1. Explanation.

Find the prices for which quantity demanded is less than quantity supplied. 

$$\begin{gathered} D\left(p\right)>S\left(p\right) \\ 10000-1000p>-2000+3000p \\ 12000>4000p \\ 3>p \end{gathered}$$

Price of hot dog is $p<3$

Part (c). Step 1. Explanation.

If quantity demanded of hot dog is less than quantity supplied, then there will be excess hot dogs that lead in decrease in price.