basic-math-and-graphing-for-econ2-3
- 2. 2 WHY DO I NEED TO KNOW THIS STUFF? You will find, in Economics, that math (yes, MATH)—specifically ALGEBRA—is the foundation for the most basic principles in this field of study. The vast majority of our topics will use graphs and graphing. Hopefully the figure below (figure 1) will bring back some memories from middle school or high school. figure 1 THE BASICS—THINGS TO REMEMBER FROM HIGH SCHOOL In the figure above, the red, highlighted numbers represent an ordered pair. Some key points to remember (from your days in High School): • The origin’s coordinates are (0, 0)-‐-‐-‐duh! • As mentioned above, points in figure 1 are named by an ordered pair. In the example (figure 1) above: o (3, 2) o 3 and 2 are ordered….read right then up. ! Go right 3 tick marks
- 3. 3 ! Go up 2 tick marks • The first number in the ordered pair is the x-‐coordinate, and the second number is the y-‐coordinate. o So, X = 3…… o and Y = 2 • Not to jump ahead of ourselves, but the X Axis (horizontal as in going left to right) coordinates are known as Independant Variables, while the Y Axis (vertical, as in going up) are known as Dependant Variables. This will be important as we move on. GRAPHING LINEAR EQUATIONS Back in the day, our high school teachers told us that Linear Equations can be graphed in the form: • Y = MX +B • Y is the Dependant Variable • X is the Independent Variable • M is the Slope (our teachers may have called it “rise over run”) • B is the point where the line crosses the Y axis. It is called the Y-Intercept. In Economics, you will hear it referred to as a Constant EXERCISE #1—fill in the table on your own Let’s put our review to good use. We are going to work through the following equation by hand. THE EQUATION: • Y = 2X + 1 WHAT’S THE EASIEST WAY TO START? • Take a look at the equation. It’s very simple, and it’s in the form Y = MX +B • We do not know what the Dependant Variable (Y) is. • We know M or Slope is 2 • The Constant is 1 Y IS STANDING ALL BY ITSELF, ALONE AND UNAFRAID….SO WE’LL “SOLVE FOR Y” • How?! • Easy! We’ll plug in an X-‐coordinate of 0. If you look at the equation, we’ll multiply 2 (or the slope) by 0 (X) to get…..ZERO (see below).
- 4. 4 • Y = 2 (0) + 1 • Y = 0 + 1 • Y = 1 • So, now that we’ve done that….let’s express the results as an ordered pair • X = 0 • Y = 1 • (0, 1) REPEAT THE ABOVE PROCESS WITH THE FOLLOWING TABLE (try to solve for Y without “cheating” from the figure below). Do this on your own, either in Excel or personal scratch paper. X Y -‐1 0 1 1 2 3 Table 1 IF YOU DID EVERYTHING CORRECTLY, THE GRAPH, DERVIED FROM THE TABLE ABOVE WILL LOOK LIKE THIS: figure 2
- 5. 5 EXERCISE #2 We’re going to make a simple graph using Microsoft Excel. This time, we’ll use a new equation THE EQUATION: • Y = -‐0.25X + 11 WHAT’S THE EASIEST WAY TO START THIS PROCESS OFF? • Look at the equation, the numbers are a little funky, but I did not throw you a curve ball! • Y = MX + B o Y= -‐MX + B (is just the same) • The Slope is a negative number! (and a fraction) • If the Slope is negative, it means the line will be downward sloping! o Remember what we talked about in class….Economics (as a social science) is all about relationships o A negative sign means that there is a negative relationship between Y and X • M (Slope) = -‐0.25 or -‐1/4 • The Y-‐Intercept is 11 • You could write the equation as Y = 11 – 0.25X and it would still adhere to the format. It’s a little easier to read that way. STEP 1: OPEN EXCEL STEP 2: IN CELL A1, PLUG IN THE LETTER ‘X’ (FOR YOUR INDEPENDENT VARIABLE) STEP 3: IN CELL B1, PLUG IN THE TITLE ‘Y’ (FOR YOUR DEPENDENT VARIABLE) STEP 4: ENTER THE FOLLOWING NUMBERS INTO THE SPREADSHEET (IN THE COLUMNS BELOW A1 AND B1 RESPECTIVELY):
- 6. 6 Table 2 STEP 5: GO INTO CHARTS (INSERT"CHART) On a Mac, it will look something like this: figure 3 STEP 6: SELECT X Y (Scatter)—see the green highlighted toolbar from figure 3 above. STEP 7: You will have several sub categories of X Y Scatter graphs. Go ahead and select the “Smooth Marked Scatter. A blank chart may pop up, that looks like this:
- 7. 7 figure 4 STEP 8: RIGHT CLICK ON THE BLANK CHART AND A DIALOG BOX OPENS. SELECT DATA. IT WILL LOOK LIKE THIS: figure 5
- 8. 8 STEP 9: ONCE YOU HAVE SELECTED DATA, A NEW DIALOG BOX WILL OPEN. FILL IN THE DATA RANGE BY MOUSE CLICKING IN A1, DRAW THE MOUSE TO B1 AND MOVE IT DOWN SO IT ENCOMPASSES ALL DATA. IT WILL LOOK LIKE THIS (EXCEL WILL HAVE PUT THE DATA (FROM THE CELLS) IN THE PROPER X AND Y VALUES : figure 6
- 9. 9 STEP 10: IF YOU HAVE DONE EVERYTHING CORRECTLY, YOUR CHART WILL LOOK LIKE THIS: figure 7 Looking at the graph, above (figure 7), you’ll note that it is downward sloping. In other words, Y is negatively related to X. Recall the original equation was this: • Y = -‐0.25X + 11 o The negative (-‐) in front of 0.25 (the slope) implies this. One more point. As mentioned in class, graphing and math, as it applies to Economics will focus on the northeast quadrant of the graph: figure 8
- 10. 10 CALCULATING OR FINDING THE SLOPE In Economics, graphing is very important. As you will see, when we discuss Supply and Demand, the slope will be key. The slope will describe the relationship between the Dependant Variable and the Independent Variable. In high school Algebra, our teachers often expressed the slope as “rise over run”. In reality, the slope of the straight line is the ratio of the vertical change (rise or drop of the Dependant Variable) to the horizontal change (the run or distance of the Independent Variable). Let’s take a look at figure 9 below. The data come from Table 2 and Exercise 2 that we did earlier. Recall that the equation for that example is: • Y = -‐0.25X + 11 figure 9
- 11. 11 Between two of the points in figure 9 (point A and point B), the vertical change (Dependent Variable) is -‐0.25 (10.25 down to 10). The horizontal change (Independent Variable) is +1 (increase from 3 to 4). • Slope = Vertical Change (Dependant Variable)/Horizontal Change (Independent Variable) o 0.25 = (10-‐10.25)/(4-‐3) o 0.25 = -‐0.25/1 o Remember, in order to find the rate of change (slope), we subtract the coordinates in Point A from those in Point B—figure 9) MANKIW’S PRINCIPLE #3 THINKING AT THE MARGIN As I touched on briefly in class, Mankiw has a principle that “rational people think at the margin”. Economics, among other things, is concerned with change—especially from the status quo. The concept of slope is important, because it reflects marginal changes. In other words, what happens when we add one (1) more or one (1) less unit? Think about it this way, although I don’t want to get too far ahead (don’t let this freak you out if you don’t understand it quite yet): • The equation Y = -‐0.25(X) +11 is a “fine” example of a demand curve. • The negative coefficient (sign) in front of the slope (0.25) tells us that the curve is downward sloping. • If the Dependent Variable (Y) was price, and the Independent Variable (X) was quantity, the equation would read something like this: o PRICE = -‐0.25(QUANTITY) + 11 ! -‐0.25 (slope) and 11 are constants o The equation states that for every decrease of $0.25 (twenty five cents), customers will ask for one additional item (quantity). o Looking at figure 9 above, the movement from A to B tells us just that!
- 12. 12 PICTORIAL EXAMPLES OF SLOPES Figure 10, below, represents an example of an infinite slope: figure 10
- 13. 13 In Figure 11, below, there is virtually no slope, or ZERO slope: figure 11
- 14. 14 PARTING SHOTS • Graphs represent relationships – especially Economic relationships! • The dependant variable is positively related to the independent variable when their values change in the same direction. In other words, a positive slope will have a “+” before the appropriate constant coefficient, while a negative or downward slope will have a “-‐“ before the appropriate constant coefficient. o Y = mX + b (slope goes up) or….. o Y = -‐mX + b (slope goes down) • The slope of a perfectly horizontal line is zero • The slope of a perfectly vertical line is infinite Good luck on the application exercises on the following pages….
- 15. 15 The following practical application problems are due on Tuesday night, 2359 pm to my email box: timothy.c.fawcett@gmail.com 1. Use Microsoft Excel to solve this problem. The following table contains data on the relationship between savings and income. Savings is the Dependant Variable. In other words, Savings will go on the vertical (or Y) axis. Savings (Y) Income (X) $1,000 $15,000 -‐$500 0 $500 $10,000 0 $5,000 $1,500 $20,000 a. Rearrange the data in appropriate order in Microsoft Excel. In other words, CELLS A1 and B1 should be: ‘Savings (Y)’ and ‘Income (X)’ respectively. Cells A2 and B2 should be: ‘-‐$500’ and 0’ respectively. Arrange the values in the Savings column (from lowest to highest) which correspond to the values in the income column—in other words do not arrange each column independently. b. Once arranged, use the CHART function in EXCEL to graph the data. Post the graph in the worksheet where it is visible. c. Write the equation that represents this relationship, or line that EXCEL plots out. Put the d. equation in EXCEL in plain text. -‐-‐HINT: SAVINGS = m(INCOME) + b -‐-‐Look at the graph. When INCOME is zero, our SAVINGS (or Y INTERCEPT) is -‐$500 (negative $500 dollars). Why in the world is that?? Well, it is because we have daily needs—food, water, shelter, rent. We would have to borrow money to survive, even though we have no income! SAVINGS =m(INCOME) -‐ $500
- 16. 16 -‐-‐Remember how we solved for slope. Pick the savings change from $1,000 to $1,500 (and the corresponding Income changes). It would look something like this: ($1,500-‐$1,000)/($20,000-‐$15,000) = $500/$5,000 d. What would you predict Savings to be when income is $12,500? -‐-‐HINT: SAVINGS = m(INCOME) -‐ $500 SAVINGS = ($500/$5000)($12,500) -‐ $500 2. NO HINTS THIS TIME!! Turn this portion of your homework in on MS word. Refer to the graph below and construct a table. (BTW, this graph will not necessarily apply to your study efforts in this class) From the table you construct, based on the graph above, determine: a. What is the Dependant Variable? Is the Study time in Hours b. What is the Independent Variable? The test scores c. Make an equation that summarizes the relationship. Y= mx+b d. From your equation, point out the constants (slope and Vertical or Y intercept). not really sure how to go about doing things backwards.