Chapter 3.1, Problem 39AYU

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

$S\left(p\right)=-600\text{+}\text{50p}$

$D\left(p\right)=1200-25p$

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?

To determine

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

b. Determine the price for which the quantity demanded is greater than the quantity supplied.

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

Answer to Problem 39AYU

Solution:

a. The equilibrium price is $\$24$ and the equilibrium quantity is 600.

b. If $D(p)>S(p)$, then the price will become less than $\$24$.

c. When the quantity demanded is greater than the quantity supplied, the price of the T-shirts becomes lesser than $\$24$ and it becomes very cheap.

Explanation of Solution

Given:

The quantity supplied is $S(p)=-600+50p$

The quantity demanded is $D(p)=1200-25p$

a.

The equilibrium point is when the quantity supplied is equal to the quantity demanded.

Therefore,

$S(p)=D(p)$

$\Rightarrow -600+50p=1200-25p$

$\Rightarrow 50p+25p=1200+600$

$\Rightarrow p=\frac{1800}{75}=24$

Therefore, the equilibrium price for the T-shirt is $\$24$.

Now, we have to find the equilibrium quantity.

Let us just substitute $p=24$ to any of the given functions.

Therefore, we get

$S(p)=-600+50(24)=600$

Therefore, the equilibrium quantity of the T-shirts to be produced is 600.

b.

When $D(p)>S(p)$, we get

$D(p)>S(p)$

$\Rightarrow 1200-25p>-600+50p$

$\Rightarrow 1200+600>50p+25p$

$\Rightarrow \frac{1800}{75}>p$

$\Rightarrow 24>p$

$\Rightarrow p<24$

c.

Therefore, when the quantity demanded is greater than the quantity supplied, the price of the T-shirts becomes lesser than $\$24$ and it becomes very cheap.