Q. 39 - Precalculus Enhanced with Graphing Utilities

Question

Supply and Demand Suppose that the quantity supplied S and quantity demanded D of T-shirts at a concert are given by the following functions:

$S\left(p\right)=-200+50p$

$D\left(p\right)=1000-25p$

where p is the price of a T-shirt. 

(a) Find the equilibrium price for T-shirts at this concert. What is the equilibrium quantity? 

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

(c) What do you think will eventually happen to the price of T-shirts if quantity demanded is greater than quantity supplied? 

Step-by-Step Solution

Answer

Part (a)

• The equilibrium price for T-shirts are 48.
• The equilibrium quantity are 2200 for T-shirts.

Part (b) The prices of T-shirts will be less than 16.

Part (c) There will be increase in prices of T-shirts

Part (a) Step 1 . Given Information

The quantity of T-shirt supplied at a concert is $S\left(p\right)=-200+50p$

The quantity of T-shirts demanded  at a concert is

$D\left(p\right)=1000 - 25p$

where p is the price of a T-shirt. 

Part (a). Step 2. Explanation.

To find the equilibrium price for T-shirts, equate supply function and demand function.

$$\begin{gathered} S\left(p\right)=D\left(p\right) \\ -200+50p=1000-25p \\ 25p=1200 \\ p=48 \end{gathered}$$

Substitute $p=48$ in supply function to find supplied T-shirt.

$$\begin{gathered} S\left(p\right)=-200+50p \\ S\left(48\right)=-200+50\left(48\right) \\ S\left(48\right)=2200 \end{gathered}$$

Part (a). Step 3. Conclusion.

• The equilibrium price for T-shirts are 48.
• The equilibrium quantity are 2200 for T-shirts.

Part (b). Step 1. Explanation.

Find the prices, if  demanded quantity is greater than quantity supplied. 

$D\left(p\right)>S\left(p\right)$

Find p.

$$\begin{gathered} D\left(p\right)>S\left(p\right) \\ 1000-25p>-200+50p \\ 1200>75p \\ 16>p \end{gathered}$$

The prices of T-shirts will be  $p<16$.

Part (c). Step 1. Explanation.

If quantity demanded is greater than quantity supplied, there will shortages of items that lead to increase in prices of items.