Chapter 3.1, Problem 40AYU

Supply and Demand Suppose that the quantity supplied S and the 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?

To determine

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

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

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

Answer to Problem 40AYU

Solution:

a. The equilibrium price is $\$3$ and the equilibrium quantity is 7000.

b. If $D(p)<S(p)$, then the price will be more than $\$3$.

c. When the quantity demanded is less than the quantity supplied, the price of the hot dogs becomes more than $\$3$.

Explanation of Solution

Given:

The quantity supplied is $S(p)=-2000+3000p$.

The quantity demanded is $D(p)=10000-1000p$.

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

Therefore,

$S(p)=D(p)$

$\Rightarrow -2000+3000p=10000-1000p$

$\Rightarrow 3000p+1000p=2000+10000$

$\Rightarrow p=\frac{12000}{4000}=3$

Therefore, the equilibrium price for the hot dogs is $\$3$.

Now, we have to find the equilibrium quantity.

Let us just substitute $p=3$ to any of the given functions.

Therefore, we get $D(3)=10000-1000(3)=7000$.

Therefore, the equilibrium quantity of the hot dogs to be produced is 7000.

b. When $D(p)<S(p)$, we get

$D(p)<S(p)$

$\Rightarrow 10000-1000p<-2000+3000p$

$\Rightarrow 2000+10000<3000p+1000p$

$\Rightarrow \frac{12000}{4000}<p$

$\Rightarrow 3<p$

$\Rightarrow p>3$

c. Therefore, when the quantity demanded is less than the quantity supplied, the price of the hot dogs becomes greater than $\$3$.