Problem 21 - Elementary and Intermediate Algebra

Question

Find the equilibrium point for each of the following pairs of demand and supply functions. $$ \begin{array}{l} {D(p)=760-13 p} \\ {S(p)=430+2 p} \end{array} $$

Step-by-Step Solution

Answer

The equilibrium price is 22, and the equilibrium quantity is 474.

Step 1: Understand the Demand and Supply Functions

The demand function is represented as \( D(p) = 760 - 13p \) and the supply function is given by \( S(p) = 430 + 2p \). The equilibrium point occurs where demand equals supply, i.e., \( D(p) = S(p) \).

Step 2: Set Up the Equation

To find the equilibrium point, set the demand function equal to the supply function: \[ 760 - 13p = 430 + 2p \]

Step 3: Solve for the Price \( p \)

Combine like terms to solve for \( p \): \[ 760 - 430 = 13p + 2p \] \[ 330 = 15p \] \[ p = \frac{330}{15} \] \[ p = 22 \]

Step 4: Find the Equilibrium Quantity

Plug the equilibrium price back into either the demand or supply function to find the equilibrium quantity. Using the demand function: \[ D(22) = 760 - 13(22) \] \[ D(22) = 760 - 286 \] \[ D(22) = 474 \] Therefore, the equilibrium quantity is 474.

Key Concepts

Demand and Supply Functions

In economics, demand and supply functions help us understand how different factors influence the market. The **demand function** shows the relationship between the price of a good and the quantity demanded by consumers. For example, the function given is: \( D(p) = 760 - 13p \). This means that as the price \( p \), increases, the quantity demanded decreases.

The **supply function** represents how much of a good producers are willing to sell at different prices. Here, it is given by: \( S(p) = 430 + 2p \). This implies that as the price \( p \), rises, the quantity supplied also increases.

The equilibrium point occurs where the demand function equals the supply function, indicating a price and quantity where the market is balanced. Understanding these functions is key to analyzing real-world economic problems and predicting market behavior.

Solving Equations

To find the equilibrium point, we need to solve the equation that sets the demand function equal to the supply function. This involves basic algebra skills:

• First, write down the demand and supply functions: \( 760 - 13p = 430 + 2p \).
• Combine like terms to isolate the variable \( p \):
  \[ 760 - 430 = 13p + 2p \]
  \[ 330 = 15p \].
• Solve for the price \( p \) by dividing both sides by 15:
  \[ p = \frac{330}{15} \]
  which simplifies to \[ p = 22 \].

After finding the equilibrium price, it’s important to double-check by substituting this price back into the original functions.

Equilibrium Quantity

Once the equilibrium price \( p \) is known, finding the equilibrium quantity is straightforward. This is done by substituting the price back into either of the original functions.

For instance, using the demand function:

• \