MATHEMATICS AND ECONOMICS: A WORKSHOP TO SHARPEN YOUR SKILLS.

MATHEMATICS AND ECONOMICS:

A WORKSHOP TO SHARPEN YOUR SKILLS

OVERVIEWToday we will be looking at three different Mathematics & Economics topics for your classrooms. In an effort to reach out to a wide variety of educators, we’ll be looking at topics designed for students with mathematical competencies ranging from 8th grade math up to calculus.

TOPICSMiddle School Math:

Surviving on a Deserted Island

Math Objectives: Measures of Central Tendency, Range, Quartiles, Box-and-Whisker Plots

Econ Objectives: Operate within a specific budget to reach desired outcome, Make predictions about value of labor in the marketplace.

Algebra I:

The Slopes of Supply and Demand

Math Objectives: Plotting points in the first quadrant, slope, direct and inverse relationships

Econ Objectives: Demand curves, Law of Demand, Why quantity demanded depends on price, Connections to the Supply Activity and the Equilibrium activity

Algebra I/Algebra II

Linear Programming and Consumer & Producer Surplus

Math objectives: Graphing linear inequalities, solving a system of equations, Areas of geometric shapes

Econ Objectives: consumer surplus, producer surplus, efficiency, taxation, deadweight loss

Optional Calculus Activity (time permitting)

Math Objectives: graphing a cubic equation, calculating 1st and 2nd derivatives, economics connection between derivative and marginal cost/revenue

Econ Objectives: Calculating Total Profit, Maximizing Total Profit, mathematical connections of marginal cost/revenue and first derivative of cost/revenue graphs

SURVIVING ON A DESERTED ISLAND

Activity 9From: Mathematics & Economics

Connections for Life: Grades 6-8

We’re going to run this activity as you would with your own class. Please feel free to chime in at any time with questions.

WARM-UP

Median: The element of a set that is the central value (the element in the middle) when listed in order from least to greatest

Upper Quartile: The “new” median of just the values above the median of the entire set

Lower Quartile: The “new” median of just the values below the median of the entire set

When listed in order the Minimum, Lower Quartile, Median, Upper Quartile, and Maximum split the data set into 4 quartiles that each represent 25% of the entire data set.

A graph of these quartiles on a number-line and displayed horizontally or vertically is called a box-and-whisker plot or just a box plot.

WARM-UPGiven the set below, find all of the points required and create a Box Plot.

{2, 4, 5, 6, 7, 8, 11, 13, 14, 14, 14, 15, 19}

WARM-UPGiven the set below, find all of the points required and create a Box Plot.

{2, 4, 5, 6, 7, 8, 11, 13, 14, 14, 14, 15, 19}

Minimum: 2

Lower Quartile: 5.5

Median: 11

Upper Quartile: 14

Maximum: 195.5

2

11

14

19

If you were stranded on a deserted island, whom

would you want there with you?

ACTIVITY 9.1

Read the instructions and complete Activity 9.1

Make sure that the sum of the 5 bids your team is making does not exceed $150,000.

When you’ve finished your list, trade papers with another team.

9.1

Let’s create a list of the 6 most popular occupations, and the bids for each occupation:

Enter this information on the top of Activity 9.2 and find the Box and Whisker data for each of the 6 occupations.

Occupation Bid Data

1 Nurse 40000, 9998, 35000, 30000, 10000, 60000

2 Engineer 20000, 25000, 30000, 10000, 100000

3 Carpenter 25000, 10000, 10000, 13000

4 Farmer 50000, 40000, 40000, 1, 15000

5 Navy Seal 50000, 100000, 100001, 45000

6 Fishing guide 20000, 15000, 20000, 20000, 50000

9.2• Which of the top occupations had the highest bid?

• Which of the top occupations had the lowest bid?

• Which of the top occupations had the largest number of bids?

• Which of the top occupations had the greatest range of bids?

• Which of the top 6 occupations had the single highest interquartile range?

• Which of the top occupations seems to have the most variability in the data?

• Do each of the 6 box plots look the same? How are they similar? How are they different?

9.3 (CLOSURE)Let’s find out which occupations your team wound up with.

What skills seem to be highly valued? Why is that?

Activity 9.3 could be distributed as homework or in-class closure.

SLOPES OF SUPPLY AND DEMAND

Activity 1From: Mathematics & Economics


For this example, we’ll complete Activity 1 and discuss Activities 2 & 3.

WARM-UP

HS Book: pg 8 Document Camera

DEMAND

How does the price of a certain CD change the number of CD’s that will be bought?

Let’s do an activity to find out!

DEMAND SCHEDULEDemand Schedule

Price of CD in $ Quantity demanded of CD Ordered Pair

(Independent Variable) (Dependent Variable) (Dep. Variable, Indep. Variable)32 0 (0, 32)30 1 (1, 30)28 2 26 3 24 4 22 5 20 6 18 7 16 8 14 9 12 10 10 11 8 12 6 13 4 14 2 15

DEMAND CURVE

CLOSURE: WRITING THE EQUATIONActivity 1.3 an be used as homework or as a closure activity.

However, it is very important to talk about the difference between graphing in economics and in math and other sciences.

In mathematics, the slope of a line is typically viewed as:

Because economists put the independent variable on the vertical axis, and the dependent variable on the horizontal

axis, we must refer to slope as:

So for the example we just completed, the slope would be:

WHAT DOES THE CHANGE IN AXES DO TO OUR EQUATIONS?Consider this graph from Activity 3. (pg 45 of the HS Book)

It is used to show that the equilibrium between supply and demand occurs at the intersection of those two graphs. What do you notice about the equations of the graphs?

LINEAR PROGRAMMING AND CONSUMER & PRODUCER SURPLUS Mathematical Supplement & Activity 5From: Mathematics & Economics


Now that we can graph in both a mathematical and economics setting, what can we use those graphs to find? Let’s look at two ideas that use the graphs of inequalities.

LINEAR PROGRAMMING• Linear programming has nothing to do with computer

programming.

• The use of the word “programming” here means “choosing a course of action.”

• Linear programming involves choosing a course of action when the mathematical model of the problem contains only linear functions.

• The maximization or minimization of some quantity is the objective in all linear programming problems.

• All LP problems have constraints that limit the degree to which the objective can be pursued.

• A feasible solution satisfies all the problem's constraints.

• An optimal solution is a feasible solution that results in the largest possible objective function value when maximizing (or smallest when minimizing).

• A graphical solution method can be used to solve a linear program with two variables.

LINEAR PROGRAMMINGSteps for Solving a Linear Programming Question

1. Graph the constraints.

2. Locate the ordered pairs of the vertices of the feasible region.

• If the feasible region is bounded (or closed), it will have a minimum & a maximum.

• If the region is unbounded (or open), it will have only one (a minimum OR a maximum).

3. Plug the vertices into the two variable linear equation to find the min. and/or max.

A farmer has 10 acres to plant in wheat and barley. He has to plant at least 7 acres. However, he has only $1200 to spend and each acre of wheat costs $200 to plant and each acre of barley costs $100 to plant. Moreover, the farmer has to get the planting done in 12 hours and it takes an hour to plant an acre of wheat and 2 hours to plant an acre of barley. If the profit is $500 per acre of wheat and $300 per acre of barley how many acres of each should be planted to maximize profit?

Let w = the number of acres of wheat planted

Let b = the number of acres of barley planted

Constraint Functions:

Function to be maximized:

STEP 1: GRAPH THE CONSTRAINTS

Let’s graph the constraints together.

We can find the vertices by solving systems of equations.

Now, let’s plug the vertices found from the feasible region into the profit equation to find the maximum and minimum profit possibilities.

STEP 2: IDENTIFY THE VERTICES OF THE FEASIBLE REGION

STEP 3: PLUG THE VERTICES INTO THE TWO VARIABLE LINEAR EQUATION TO FIND THE MIN. AND/OR MAX.

Because both w and b must be greater than or equal to zero, only the first quadrant would supply feasible answers.


Plot the constraint function

By first graphing and then testing a point to see which region defined by the line should be shaded.


Plotting the line

Plot the constraint function

Test point (0,0) makes the inequality true:

So the region defined by the line which contains (0,0) is shaded.


Shading the region


Now, the feasible region exists only for those points in both the green and red regions. This defines an even more constrained feasible region.

→

Following the same steps as used for the first constraining function, graph the next inequality to further constrain the feasible region


Feasible region after graphing the constraint fuction

Following the same steps as used for the previous constraining functions, graph the next inequality to further constrain the feasible region


Feasible region after graphing the constraint function

Following the same steps as used for the previous constraining functions, graph the final inequality to further constrain the feasible region


Feasible region after graphing the constraint function

Now that the final feasible region has been found, the vertices of this bounded polygon must be found by solving systems of equations.

The intersection of the orange and purple lines can be found by solving the system:

And now, because

The point must be (5,2)

STEP 2: LOCATE THE ORDERED PAIRS OF THE VERTICES OF THE FEASIBLE REGION.

In the same way the other two systems can be solved to find the remaining vertices:

The 2nd point must be (2,5)

And the final point must be (4,4)

STEP 2: LOCATE THE ORDERED PAIRS OF THE VERTICES OF THE FEASIBLE REGION.

(5,2)

(2,5) (4,4)

STEP 3: PLUG THE VERTICES INTO THE TWO VARIABLE LINEAR EQUATION TO FIND THE MIN. AND/OR MAX.

Now that the three vertices of the feasible region are known, let’s plug each of them into the equation to calculate profit.

The profit in planting 2 acres of wheat and 5 acres of barley is $2500.



We have shown mathematically, that of all of the possible combinations of acres of wheat and barley to be planted, the famer will have a maximum

profit of $3200 when planting 4 acres each of wheat and barley, and a minimum profit of $2500 when planting 2 acres of wheat and 5 acres of

barley.

CONSUMER & PRODUCER SURPLUS

Now let’s take a look at another use of graphing inequalities, but this time from an economics point of view.

We’ll be looking at Activity 5 from the HS book: The Gains from Trade

We will skip the Warm-up since we’ve just had practice graphing linear inequalities.

CONSUMER & PRODUCER SURPLUS

Let’s start by graphing the supply and demand curves for Activity 5.1.

Please note that the equations are given in the forms:

Where P (price) is independent variable on the vertical axis, and Q is the dependent variable on the horizontal axis. Does that make these equation follow the form of

or ???

After you’ve graphed both lines, draw a horizontal line through the equilibrium point where the supply and demand curves cross.

CONSUMER & PRODUCER SURPLUSThis shaded triangular region, above the horizontal line but below the demand curve is a representation of Consumer Surplus. It indicates the total amount of money that consumers were willing to pay for the product, but didn’t have to spend.

CONSUMER & PRODUCER SURPLUSSimilarly, the shaded triangular region below the horizontal line but above the supply curve is a representation of Producer Surplus. It indicates the total amount of money that producers to earned that is more than the minimum amount they were willing to earn.

CONSUMER & PRODUCER SURPLUSCombined together, these two right triangles form the larger (non-right) triangle shown to the right. This represents total surplus or gains from trade. It is the combined benefits to both consumers and producers.

TAXATIONWhat happens to the benefit to consumer and producer when the government (or some other outside entity) imposes a tax?

Let’s look at Activity 5.2 together and see what happens to the consumer and producer surplus.

TAXATIONThe original consumer surplus was , after a tax is introduced to the smaller triangle .

The original producer surplus was , after a tax is introduced to the smaller triangle .

The benefit to the government or outside entity is represented by the rectangle ______.

What part of the original gains from trade (represented by the largest triangle ) no longer benefits anyone?

DEADWEIGHT LOSS

The small triangle is the deadweight loss. This is the inefficiency of taxation. The area of the deadweight loss represents the amount of money that benefits nobody (consumer, producer, or government.) It arises because buyers pay more than producers receive. This inefficiency manifests as deadweight loss.

Please don’t hesitate to send an email with an further math questions you

might have. I can be reached at:

[email protected]

QUESTIONS

mailto:[email protected]

MATHEMATICS AND ECONOMICS: A WORKSHOP TO SHARPEN YOUR SKILLS.

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