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Chapter 3: Problem 24

Find the equilibrium point for the following pairs of demand and supply functions. \(D(p)=8800-30 p\) \(S(p)=7000+15 p\)

Short Answer

Expert verified
The equilibrium price is 40, and the equilibrium quantity is 7600.

Step by step solution

01

Understand the Equilibrium Point

The equilibrium point occurs where the demand function (D(p)) equals the supply function (S(p)). Mathematically, this means solving the equation: \[ D(p) = S(p) \]
02

Set the Functions Equal to Each Other

Set the given demand function equal to the given supply function: \[ 8800 - 30p = 7000 + 15p \]
03

Solve for p

Rearrange the equation to solve for the price (p):Subtract 7000 from both sides: \[ 8800 - 7000 - 30p = 15p \]This simplifies to: \[ 1800 - 30p = 15p \]Combine like terms by adding 30p to both sides: \[ 1800 = 45p \]Solve for p by dividing both sides by 45: \[ p = \frac{1800}{45} \]Thus, \[ p = 40 \]
04

Find the Quantity at Equilibrium

Substitute the price (p) back into either the demand or supply function to find the equilibrium quantity. Using the demand function:\[ D(40) = 8800 - 30(40) \]This simplifies to: \[ D(40) = 8800 - 1200 \]So, \[ D(40) = 7600 \]Therefore, the equilibrium quantity is 7600.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

demand function
The demand function shows how the quantity demanded of a good or service changes with its price. It's usually written as a function of price, denoted as \( D(p) \). In our example, the demand function is \( D(p) = 8800 - 30p \). Here are some important points to grasp:\
\
    \
  • The coefficient of \( p \) (price) is -30, which represents how demand decreases as price increases.\
    \
  • The constant term is 8800, indicating the quantity demanded when the price is zero.\
    \
\The negative sign before the coefficient means there's an inverse relationship between price and quantity demanded. Simply put, as price goes up, demand goes down.
supply function
The supply function represents how the quantity supplied varies with price. It's typically expressed as \( S(p) \). In our scenario, the supply function is \( S(p) = 7000 + 15p \). Here's what you need to know:\
\
    \
  • The coefficient of \( p \) (price) is 15, which shows how supply increases with increasing price.\
    \
  • The constant term is 7000, indicating the quantity supplied when the price is zero.\
    \
\The positive sign before the coefficient indicates a direct relationship between price and quantity supplied. That means, as the price goes up, the supply also goes up.
solving equations
To find the equilibrium point where the demand equals supply, we need to solve the equation \( D(p) = S(p) \). Here鈥檚 how you can solve this step-by-step:\
\
    \
  • Set the functions equal: \( 8800 - 30p = 7000 + 15p \).\
    \
  • Simplify the equation by moving terms involving \( p \) to one side and constants to the other: \[ 8800 - 7000 - 30p = 15p \] which simplifies to \[ 1800 - 30p = 15p \].\
    \
  • Combine like terms: \[ 1800 = 45p \].\
    \
  • Divide both sides by 45 to solve for \( p \): \[ p = \frac{1800}{45} = 40 \].\
    \
\Thus, the equilibrium price \( p \) is 40.
intermediate algebra
To work on these equations, you need knowledge of intermediate algebra. This includes:\
    \
  • Understanding variables and constants.\
    \
  • Performing basic operations like addition, subtraction, multiplication, and division on both sides of an equation.\
    \
  • Simplifying expressions by combining like terms.\
    \
\Knowing these basics allows you to manipulate and solve equations effectively. For instance, to find the equilibrium quantity:\
\
    \
  • Substituting \( p = 40 \) back into the demand function: \[ D(40) = 8800 - 30(40) \].\
    \
  • This simplifies to: \[ D(40) = 8800 - 1200 = 7600 \].\
    \
\So, the equilibrium quantity is 7600 units.

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