Fruit Stand Lab
Supplies Needed: Color Pencils Skills Covered: Solving for a variable, finding the intersection of two lines, plotting lines Vocabulary words: supply, demand, slope-intercept form
We are starting an after-school fruit stand at Sayre High School. The goal is to sell healthy snacks (fruits and vegetables) to Sayre students. What do we need to think about before we can start selling our healthy snacks?
There are many things to consider when starting a business. Once you've decided on what you want to sell, one of the most basic decisions is how much to make, and at what price you should sell your goods or services. The two phenomena that we need to keep in mind are supply and demand. The demand for a good or service is the amount of said good or service that people are interested in buying, as influenced by price and other factors. The supply of a good or service is the amount of said good or service that manufacturers want to make, as influenced by price and other factors. Though there are many complex influences that actually affect the supply and demand for an object in the real world, we will focus here on the single most preeminent of those – price.
Think about a favorite candy, an electronic device, or an everyday good. We'll use basketballs here. To determine the supply of basketballs at a given price, we are asking the following question to basketball producers: “How many basketballs are you willing to make at this price?” Suppose it costs $10 to make a basketball. Then if you asked a basketball company how many they would be willing to make if the price was $5, they would answer zero since they would be losing money at that price. To put this in mathematical terms, we'll call S(x) the supply function, where x represents the price, and S(x) is the supply at price x. As we mentioned above, S(5)= 0. Now, if we asked the basketball company, “How many basketballs would you make if you could sell them for $10,000?”, they would tell you that they'd make as many as they possibly could, because they'd be making a tremendous amount of money. They'd hire more people, expand their manufacturing plants, and do whatever they could to keep the public supplied with basketballs that they were buying at such a high price.
To determine the demand of basketballs at a given price, you ask the same question of consumers (including you!): “How many basketballs would you buy at this price?” How many people in the class would buy basketballs if they were only $1 (assume quality doesn't vary with price; these are good basketballs)? It wouldn't only be people who played a whole lot. For $1, someone who only occasionally used it might be inclined to buy a basketball. Serious players might want two or three. The demand would be pretty high. Let's say that we're looking at the population of your class, and suppose that your class would buy 30 basketballs if they were $1 each. Let's name our demand function D(x). We would know then that D(1) = 30.
Class exercise: determine the demand of basketballs in your class as a function of price. By a show of hands, ask who would buy a basketball for $1. If someone would buy two, mark that person down twice. Then see how many people would pay $10 for a basketball. Then $20, $30, and $50. Ask if anyone would pay $100 for a basketball.
1 As you might expect, as the price of an object goes up, the demand for it goes down. For supply, the case is exactly the opposite. How do we balance these two phenomena to determine what price we should actually use in our fruit stand? How do stores everywhere figure out the price at which they should sell their goods?
The Case of Pineapples: Growers Vs. Buyers
Two groups determine the price of pineapples: the growers and the buyers. Both the growers and the buyers want the best possible outcome for themselves. The growers are happiest when the price they sell their pineapples for is highest. The buyers are happiest when the price of pineapples is the lowest. So what happens? How do they reach a compromise? Let’s find out!
Data from all the pineapples growers in the world has been gathered and analyzed. The data is sorted into two rows. Row 1 shows the price of each pineapple and row 2 shows the number of pineapples that the grower is willing to sell for that price.
Price per Pineapple $0 $1 $1.50 $2 $2.50 $3 $4 Number of Pineapples the Grower Will Make 0 1 2 3 4 5 6
1. How many pineapples will the growers sell for $4? ______
2. The growers are willing to sell three pineapples for $______/each
3. How many pineapples will the growers sell for $1.50? ______
Just like the grower will make a decision about the number of pineapples she’s willing to sell for a certain price, the buyer makes a similar decision. For each price the buyer sees at the grocery store, he’s willing to buy a certain amount of fruit. Data from pineapples buyers has been collected and is represented in the chart below. The first row represents the price per pineapple, the second row represents how many pineapples the buyer is willing to purchase for that price.
Price per Pineapple $0 $1 $1.50 $2 $2.50 $3 $4 Number of Pineapples the Buyer Will Buy 8 6 5 4 3 2 0
1. How many pineapples will the buyer purchase for $2.50? ______
2. How many pineapples will the buyer purchase for $0? ______
3. How many pineapples will the buyer purchase for $4? ______
Our Goal: Determine the final price of a pineapple in this free market exchange. Approach 1: Plotting the Supply and Demand lines from the given data points.
In the same coordinate plane, plot the pairs of points from the grower and the buyer. Use two different colored pencils to plot to lines: one color for the grower line and one color for the buyer line.
2 1. Start by labeling the x and y axis on the graph: x: this is the independent variable, in our case the number of pineapples. y: this is the dependent variable, in our case the price of a pineapple.
2. Refer back to the grower and buyer charts from above to label the coordinate points on the x-axis and the y-axis.
3. Using different colored pencils, plot each point for the grower line and each point for the buyer line onto the graph (all buyer points should be the same color, and all grower points the same color).
4. Finally, connect the dots using the appropriate color.
1. What is the approximate coordinate point where the two lines intersect? What do you think this point means?
Approach 2: Graphing the Supply and Demand lines as equations.
Background
Now, we'll graph the same lines as linear equations with slopes and intercepts.
The formula we will use is called the Slope-Intercept Form: y = mx + b
Let's break this down and understand what each letter in the equation means. x: this is the independent variable, in our case the number of pineapples. y: this is the dependent variable, in our case the price of a pineapple. m: this is the slope of the line. We think of slope as "rise over run"
What does "rise over run" actually mean? It represents how much the line is changing in height divided by how much it's growing horizontally. Think of a ball rolling down a hill. The steeper the hill, the faster the ball rolls.
In other words, slope is
Remember that when dealing with linear equations, "change" just means "subtraction". So "change in Y" means: final value of Y - beginning value of Y
So in our case, the slope is :
3 b: this is the y-intercept of the line. In other words, this is the coordinate where our line meets the Y-axis.
So with this information, let’s find the equations for Supply (the Seller) and Demand (the Buyer)
Supply Line Start with the Seller (grower).
Where does the grower line start?______Where does the grower line end? ______
What does this mean?
(3, 2) means that ______pineapples are sold when the price is ______.
Step 1) Calculate the slope for the Supply Line:
Change in Price =
Change in Number of Pineapples =
Step 2) Find the Y-intercept for the Supply Line (ask yourself the question: "When the number of pineapples, what is the corresponding price?"
Y-intercept =
Step 3) Plug in the slope and y-intercept you found into the Slope-Intercept formula.
This is the Supply line.
Demand Line
Where does the buyer line start? ______Where does the buyer line end? ______
(4,0) means that ______pineapples are bought when the price is ______
Step 4) Calculate the slope for the Demand Line
Change in Price =
Change in Number of Pineapples =
** Be careful with the subtractions here! **
Step 5) Find the Y-intercept for the Demand Line (ask yourself the question: "When the number of pineapples is 0, what is the corresponding price?"
4 Y-intercept =
Step 6) Plug in the slope and y-intercept you found into the Slope-Intercept formula.
This is the Demand line.
The growers are happiest when the price they sell their pineapples at is highest. The buyers are happiest when the price of pineapples is lowest. This creates a dynamic market exchange, where two opposing forces have to balance each other out. This "balancing" is what drives us to market equilibrium. The tables below present market data sorted into two rows:
Price per Pineapple 1.5 2.1 2.7 3.5 3.9 4.5 5.1 Number of Pineapples the Grower Sells 0 1 2 3 4 5 6
Price per Pineapple 0 1 1.5 2 2.5 3 4 Number of Pineapples the Buyer Buys 8 6 5 4 3 2 0
Our Goal: Determine the final price of a pineapple in this free market exchange. Approach 1 : Plotting the Supply and Demand lines from the given data points. In the same coordinate plane, plot the pairs of points from the grower and the buyer. Connect the dots when you're done. Note that we will have two axes: Price and Quantity of Pineapples Just like we use Y as the dependent variable and X as the independent variable, we'll use Price as our dependent variable and quantity as our independent variable. i r e P c
Note: Quantity of pineapples Make sure you keep track of which coordinate points belong to which person. (Using two
5 Also, note the approximate coordinate point where the two lines intersect. Approach 2: Graphing the Supply and Demand lines as equations. Background Now, we'll graph the same lines as linear equations with slopes and intercepts. Slope-Intercept Form: y = mx + b Let's break this down and understand what each letter in the equation means. y: this is the dependent variable, in our case the Price of a pineapple. x: this is the independent variable, in our case the Number of pineapples. m: this is the slope of the line. We think of slope as "rise over run"
rise run
What does "rise over run" actually mean? It represents how much the line is changing in height divided by how much it's growing horizontally. Think of a ball rolling down a hill. The steeper the hill, the faster the ball rolls.
In other words, slope is Change in Y Change in X
Remember that when dealing with linear equations, "change" just means "subtraction". So "change in Y" means: final value of Y - beginning value of Y
So in our case, the slope is Change in Price Change in Number of Pineapples
b: this is the y-intercept of the line. In other words, this is the coordinate where our line meets the Y-axis.
y-intercept
So with this information, let’s find the equations for Supply (the Seller) and Demand (the Buyer) Directions: 2. Break up into groups, and together as a team find the equations for the two lines 3. After you find the equations, you will be asked to find the intersection of the two lines. 4. Compare your group’s finds with the rest of the class!
6 Group Activity Supply Line
Step 1) Calculate the slope for the Supply Line
Change in Price =
Change in Number of Pineapples =
Step 2) Find the Y-intercept for the Supply Line
Y-intercept =
Step 3) Plug in the slope and y-intercept you found into the Slope-Intercept formula. This is the Supply line.
Y = _____ X + ______
Demand Line
Step 4) Calculate the slope for the Demand Line
Change in Price =
Change in Number of Pineapples =
** Be careful with the subtractions here! If demand goes down as price increases **
Step 5) Find the Y-intercept for the Demand Line
Y-intercept =
Step 6) Plug in the slope and y-intercept you found into the Slope-Intercept formula. This is the Demand line. Y = _____ X + ______
7 The Equilibrium Price We'll find this by setting the Supply equation equal to the Demand equation.
mD XD + bD = mS XS + bS
Solve for X (We'll call the X value you found X*) :
Check the equilibrium result you got by looking at the lines you plotted at the beginning. Are the coordinates of the intersection the same as X*, Y*? This point, (X*, Y*), should be the same as the one you found by solving the equations.
So what does this mean? (X*, Y*) represents the equilibrium quantity and price of pineapples sold. At the price X*, the suppliers and buyers agree to exchange Y* pineapples. This same idea applies to the stock market, real estate market, and just about any other form of economic exchange. This is an important concept in economics and it's an algebra problem at heart!
Part 2: What happens if Demand increases for pineapples, but Supply stays the same?
Here’s our new Demand equation: Y = -0.5 X + 6 And our same Supply equation: Y = 0.6 X + 1.5
Solve for the intersection (X**, Y**) of these lines like you did before.
-0.5 X** + 6 = 0.6 X** + 1.5
Recall that, once you have found X**, you can find Y** by substituting into either of the equations (since it satisfies them both).
8 (X**, Y**) =
How does this new value of (X**, Y**) compare to the old value you found? X** is ______than X*
Y** is ______than Y*
Consider the picture representing what you found algebraically.
P
New Demand 6 Supply
4 Old Demand
(X**,Y**) (X*,Y*) 1.5
Q 2.3 4.1 8 12
What does this mean in terms of economics?
What happens to the equilibrium quantity and price of a good if the demand grows while the supply stays the same?
Can you think of everyday examples of this phenomenon?
Also, what do you think happens if Demand falls while Supply stays fixed?
9 What about if Supply increases while Demand stays fixed?
10