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Chapter 9: Problem 43

Boing Hobbies is willing to produce 100 yo-yo's at \(\$ 2.00\) each and 500 yo- yo's at \(\$ 8.00\) each. Research indicates that the public will buy 500 yo- yo's at \(\$ 1.00\) each and 100 yo-yo's at \(\$ 9.00\) each. Find the equilibrium point. (IMAGE CAN'T COPY)

Short Answer

Expert verified
The equilibrium point is 300 yo-yos at a price of $5.00 each.

Step by step solution

01

Define Variables

Let the price per yo-yo be denoted by p and the quantity of yo-yos be denoted by q.
02

Supply and Demand Equations

The supply equation and the demand equation are both linear. Based on the given data, we can set up two points for each equation and find their respective slopes.- Supply: (100, 2.00) and (500, 8.00)- Demand: (500, 1.00) and (100, 9.00)
03

Finding the Supply Equation

Use the two points for supply to find the slope (m) with the formula: \[ m = \frac{(p_2 - p_1)}{(q_2 - q_1)} \]Substitute the values: \[ m = \frac{8.00 - 2.00}{500 - 100} = \frac{6.00}{400} = 0.015 \] Now use the point-slope form to find the supply equation: \[ p = 0.015q + b \] Substituting one of the points (100, 2.00) to solve for b: \[ 2.00 = 0.015(100) + b \] \[ b = 2.00 - 1.50 \] \[ b = 0.50 \] The supply equation is: \[ p = 0.015q + 0.50 \]
04

Finding the Demand Equation

Now use the two points for demand to find the slope (m) with the formula: \[ m = \frac{(p_2 - p_1)}{(q_2 - q_1)} \] Substituting the values: \[ m = \frac{9.00 - 1.00}{100 - 500} = \frac{8.00}{-400} = -0.02 \] Now use the point-slope form to find the demand equation: \[ p = -0.02q + b \] Substituting one of the points (500, 1.00) to solve for b: \[ 1.00 = -0.02(500) + b \] \[ b = 1.00 + 10.00 \] \[ b = 11.00 \] The demand equation is: \[ p = -0.02q + 11.00 \]
05

Finding the Equilibrium Point

To find the equilibrium point, set the supply equation equal to the demand equation: \[ 0.015q + 0.50 = -0.02q + 11.00 \] Combine like terms: \[ 0.035q = 10.50 \] Solve for q: \[ q = \frac{10.50}{0.035} = 300 \] Substitute q into either the supply or demand equation to find p: \[ p = 0.015(300) + 0.50 = 4.50 + 0.50 = 5.00 \] The equilibrium point is (300, 5.00).

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Key Concepts

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

linear equations
Linear equations are algebraic equations where the variables are of the first degree. This means that the variables do not have exponents and are not multiplied together. In the context of the supply and demand exercise, linear equations represent the relationship between the price (\( p \)) and the quantity (\( q \)) of yo-yos either supplied or demanded. These equations are characterized by a constant rate of change, which is represented by the slope. For example, in our supply equation (\( p = 0.015q + 0.50 \)), the slope (\( 0.015 \)) indicates how much the price changes when the quantity changes by one unit.
supply and demand
The concepts of supply and demand are foundational in economics.
  • Supply refers to how much of a product or service is available at different price points.
  • Demand is about how much of a product consumers are willing to buy at various prices.
In our exercise, Boing Hobbies supplies more yo-yos as the price increases, which is reflected in the supply equation. Conversely, consumer demand decreases as the price rises, shown in the demand equation. The equilibrium point is where supply meets demand. This means that the quantity supplied equals the quantity demanded at a particular price, ensuring that the market is in balance.
algebraic equations
Algebraic equations use symbols and numbers to express relationships between quantities. They can be solved to find specific values of unknowns, like the price (\( p \)) and quantity (\( q \)) in our exercise. For finding the equilibrium point, we use two algebraic equations: one for supply and one for demand.
To solve these, we equate them and solve for one variable. This provides us with the equilibrium quantity, which can then be used to find the equilibrium price by substituting it back into one of the original equations. These steps help us understand how algebraic equations are instrumental in solving real-world economic problems.
slope-intercept form
The slope-intercept form of a linear equation is \( p = mq + b \), where:
  • \( m \) is the slope
  • \( b \) is the y-intercept,
In this form, the slope (\( m \)) quantifies the change in the dependent variable (\( p \)) per unit change in the independent variable (\( q \)).
For the supply equation (\( p = 0.015q + 0.50 \)), the slope 0.015 indicates that for each additional yo-yo supplied, the price increases by 0.015.
Meanwhile, the y-intercept 0.50 represents the price when no yo-yos are produced. Similarly, the demand equation (\( p = -0.02q + 11.00 \)) has a negative slope, showing that price decreases as quantity demanded increases.

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