Question

Suppose that the quantity supplied \(S\) and the quantity demanded \(D\) of hot dogs at a baseball game are given by the following functions:$$\begin{array}{l}S(p)=-2000+3000 p \\\D(p)=10,000-1000 p\end{array}$$ 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?

Short answer

The equilibrium price is \(\$3\) and the equilibrium quantity is 7000. Quantity demanded is less than quantity supplied when \(p>3\). Prices will likely decrease if quantity demanded is less than quantity supplied.

Step-by-step solution

Set up the equilibrium equation

In equilibrium, the quantity supplied equals the quantity demanded. Therefore, set the functions equal to each other: \[\begin{equation} onumber \-2000+3000p=10000-1000p\end{equation}\]

Solve for the equilibrium price

Combine like terms by adding 1000p to both sides and adding 2000 to both sides: \[\begin{equation}onumber \3000p+1000p=10000+2000\ 4000p=12000 \ p=3 \end{equation}\]Therefore, the equilibrium price is \(\$3\).

Find the equilibrium quantity

Substitute the equilibrium price back into either of the original functions. Using the supply function: \[\begin{equation} \ onumber S(3)=-2000+3000(3)\ S(3)=-2000+9000\ S(3)=7000 \end{equation}\] Therefore, the equilibrium quantity is 7000.

Determine prices for which quantity demanded is less than quantity supplied

Set up the inequality \[\begin{equation} \onumber-2000+3000p>10000-1000p\end{equation}\] and solve for \(p\): \[\begin{equation} \ 3000p+1000p>10000+2000\ 4000p>12000\ p>3 \end{equation}\] Therefore, quantity demanded is less than quantity supplied when \(p>3\).

Discuss the effect on price when quantity demanded is less than quantity supplied

If quantity demanded is less than quantity supplied, it means there are more hot dogs available than people want to buy at that price. Therefore, to sell the extra hot dogs, the price will likely decrease.

Key concepts

Quantity supplied

In economics, the term 'quantity supplied' refers to the number of goods or services that producers are willing and able to sell at a given price. The concept is derived from the supply function, which is a mathematical representation showing the relationship between the price of a good and the quantity of that good producers are willing to supply. For example, in our exercise, the supply function for hot dogs is given by \(S(p) = -2000 + 3000p\).

This function indicates that at a price of zero dollars, the quantity supplied would actually be negative 2000 hot dogs, which doesn't make practical sense. This shows producers need some minimum price to start supplying hot dogs. As the price increases, the quantity supplied increases as well. This positive relationship between price and quantity supplied is known as the law of supply.

Quantity demanded

Quantity demanded is the amount of a good or service that consumers are willing and able to purchase at a given price. This is described by the demand function, which, similar to the supply function, mathematically represents the relationship between price and quantity demanded. In the exercise, the demand function for hot dogs is \(D(p) = 10,000 - 1000p\).

According to this function, if the price is zero dollars, consumers would demand 10,000 hot dogs. Conversely, as the price increases, the quantity demanded decreases. This negative relationship between price and quantity demanded is known as the law of demand. The interplay between supply and demand functions helps us find important market metrics, like the equilibrium price and quantity.

Inequalities in economics

Inequalities in economics can help determine various market conditions, such as when the quantity demanded is less than the quantity supplied. In our exercise, we set up such an inequality to find out when quantity supplied exceeds quantity demanded. We wrote:
\( S(p) > D(p) \),
and plugged in our functions:
\(-2000 + 3000p > 10000 - 1000p\).

Solving this inequality, we found the critical price point: \( p > 3 \). This tells us that when the price of hot dogs is greater than $3, the quantity demanded becomes less than the quantity supplied. This condition often leads sellers to lower the price to balance supply and demand again, thus bringing the market back to equilibrium.

Supply and demand functions

Supply and demand functions are crucial for analyzing how markets work. They consist of mathematical representations that show the relationship between price and quantity for given goods and services. For example, our exercise uses two functions:
- Supply function: \( S(p) = -2000 + 3000p \)
- Demand function: \ (D(p) = 10,000 - 1000p) \.

In these functions, 'p' represents the price of a hot dog. The supply function shows that as prices rise, producers supply more hot dogs. Conversely, the demand function shows that as prices rise, consumers demand fewer hot dogs.

These functions help us find the equilibrium price where the quantity supplied equals the quantity demanded. Additionally, we can derive insights into market behaviors, like what happens when supply exceeds demand or vice versa. Understanding these functions is key to analyzing and predicting market outcomes efficiently.