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Chapter 7: Problem 46

In Exercises 45 - 48, find the equilibrium point of the demand and supply equations. The equilibrium point is the price and number of units that satisfy both the demand and supply equations. Demand \( p = 100 - 0.05x \) Supply \( p = 25 + 0.1x \)

Short Answer

Expert verified
The equilibrium point, where demand equals supply, occurs at \( p = 50 \) units price and \( x = 500 \) units quantity.

Step by step solution

01

Set the Demand Equation Equal to the Supply Equation

In order to find the equilibrium point, first set the demand equation (\( p = 100 - 0.05x \)) equal to the supply equation (\( p = 25 + 0.1x \)). Equating these gives: \n\n\( 100 - 0.05x = 25 + 0.1x \)
02

Solve for \( x \)

Next, gather the terms with \( x \) on one side of the equation and the constants to the other. This requires adding \( 0.05x \) to both sides and subtracting 25 from both sides. This results in: \n\n\( 0.05x + 0.1x = 100 - 25 \)\n\nOn further simplification, it gives:\n\n\( 0.15x = 75 \)\n\nDividing both sides by 0.15, we get \( x \) as: \n\n\( x = 500 \)
03

Substitute \( x \) into one of the original equations

Plugging this value for \( x \) back into either the demand or supply equation allows for solving for \( p \). So, substituting \( x = 500 \) into the supply equation,\n\n \( p = 25 + 0.1*500 \)\n\nSimplifying, we find:\n\n\( p = 50 \)

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Key Concepts

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

Demand Equation
The demand equation represents the relationship between the price of a good and the quantity that consumers are willing and able to purchase. In our exercise, the demand equation is given by \( p = 100 - 0.05x \). This equation tells us that as the price \( p \) decreases, the quantity \( x \), or the number of units demanded, increases. It is a linear equation with a negative slope, indicating an inverse relationship between price and demand.


This type of equation can be used to forecast consumer behavior. For example:
  • The intercept \( 100 \) shows the price at which the demand would be zero, assuming the trend continues.
  • The slope \(-0.05\) indicates the rate at which demand changes with price. Specifically, for every 1-unit decrease in price, demand increases by 20 units since \( \frac{1}{0.05} = 20 \).
Understanding the demand equation is crucial for determining how changing conditions affect consumer purchasing.
Supply Equation
Conversely, the supply equation demonstrates the relationship between the price of a good and the quantity that producers are prepared to sell. Given by the equation \( p = 25 + 0.1x \), it suggests that the supply increases with an increase in the price, showcasing a positive correlation.


A closer look into the supply equation reveals valuable insights:
  • The intercept \( 25 \) implies that at this price, suppliers are unwilling to sell any products.
  • The slope \( 0.1 \) means that for every additional unit sold, the price increases by 0.1 units, which represents the responsiveness of how much supply changes with price.
By studying the supply equation, businesses can better understand how supply and pricing are interconnected, assisting in making informed production decisions.
Solving Linear Equations
To find where the quantity demanded equals the quantity supplied, you need to find the equilibrium point, which involves solving linear equations. Setting the demand equation equal to the supply equation allows us to find this point of balance. Here is how you solve:

  • First, equate the two equations: \( 100 - 0.05x = 25 + 0.1x \).
  • Next, gather the terms with \( x \) on one side and the constants on the other: \( 0.05x + 0.1x = 75 \).
  • By adding and simplifying, you solve for \( x \): \( x = 500 \).

After identifying \( x \), substitute it back into either the supply or demand equation to find \( p \), the price at equilibrium. For instance:
  • Substituting \( x = 500 \) into the supply equation, you find: \( p = 25 + 0.1 \times 500 = 50 \).
The equilibrium point where demand meets supply is \( x = 500 \) units at \( p = 50 \) in price. This approach using linear equations is an essential skill in economics to balance various market scenarios.

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