Lecture Notes #3 Page 1 of 15

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1 Lecture Notes #3 Page 1 of 15 PbAf 499 Lecture Notes #3: Graphing Graphing is cool and leads to great insights. Graphing Points in a Plane A point in the (x,y) plane is graphed simply by moving horizontally along the x-axis the proper distance and then moving vertically the proper distance. Here we see the points (3,5) and (-1,-2) graphed: Graphing Functions A function is a relationship between two or more variables. To keep things relatively simple, we'll focus on graphing functions of two variables here, but the principles apply to functions of more than two variables, they're just a bit more difficult to graph. The graph of a function is the set of points which satisfy the function. Here are some examples. The first is a linear function, the second is non-linear. As advertised, the first is a straight line and the second is a curve.

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3 Lecture Notes #3 Page 3 of 15 Try graphing the following functions for the given values of x: y = 5x + 2, x=0,1,2,3,4,5

4 Lecture Notes #3 Page 4 of 15 y = x 2 x=0,1,2,3 y = 2-4x + x 2, x=0, 1, 2

5 Lecture Notes #3 Page 5 of 15 Slope of a Line Graphing a line requires only two points, call them (x 1, y 1 ) and (x 2, y 2 ). The slope of the line is given by the rise over the run: slope rise run y x 2 2 y x 1 1 If the line is upward sloping, slope > 0. If the line is downward sloping, slope < 0. If the line is flat, slope = 0. If the line is vertical, slope is undefined, or -.

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7 Lecture Notes #3 Page 7 of 15 If you have a non-linear function, you can calculate the average slope over different ranges. If the function is non-linear, the slope will change as you move from one range to another. Going back to the example from above: You can calculate the average slope of the function over the ranges: RISE 1 0 X=0 to X=1: SLOPE 1 RUN 1 0 X=1 to X=4: SLOPE RISE RUN X=4 to X=9: SLOPE RISE RUN Because the curve flattens out as X increases, the slope keeps decreasing, although it will always remain positive.

8 Lecture Notes #3 Page 8 of 15 Interpretation of Graphs A graph tells you the value of the dependent variable for different values of the explanatory variable(s). A graph also tells you how the value of the dependent variable will change as the value of the explanatory variable(s) change. Imagine that you observed a previous movement from X 1 to X 2 and, as a result, the value of the dependent variable changed from Y 1 to Y 2. Now, there are forecasts that the value of X will decrease by the same amount again to the new value X 3 and people are wondering what will happen. Some people are predicting the same change as previously happened. If the function shown above is a reasonably accurate description of the relationship between X and Y, how will this change compare with the previous change? Ex: Consider the function y=5x-x 2. How does the value of y change as the value of x changes from 3 to 5 to 7? Graph this.

9 Lecture Notes #3 Page 9 of 15 Interpreting Simultaneous Equations The graph of an equation or a function is really just the set of combinations of values of x and y which satisfy the equation. For example, each point on a graph of y=2+x 2 contains a value of x and a corresponding value of y which equals 2+x 2. The graph is a picture of the equation or the function. It is all the points at which the equation is satisfied. If there are two functions or equations graphed together, the point or points of intersection between the two curves are combinations of x's and y's which satisfy both equations. For example, if you have two functions, y=f 1 (x) and y=f 2 (x), the point at which they cross, (x*,y*) has the characteristic that y*=f 1 (x*) and y*=f 2 (x*). That is, both equations are true for that value of (x,y). Here are some examples:

10 Lecture Notes #3 Page 10 of 15 Try the following Supply and Demand Example: Supply: P = Q Demand: P = Q

11 Lecture Notes #3 Page 11 of 15 Note: The solution shown here is arrived at in a bit of an odd way. Usually, to solve an equation of the form ax 2 + bx + c = 0 you need to use the quadratic equation x b b 2a 2 4ac solve it in the following way:. However, if a, b and c are integers, you may be able to x 2 + bx + c = 0 can be rewritten as (x + g) (x + h) = 0 where gh=c and g - h = b. As shown above, x 2 + 4x - 77 = 0 can be written as (x + 11) (x - 7) = 0 because 11 x -7 = -77 and 11-7 = 4. If we know that (x + 11) (x - 7) = 0, then the solutions are x=-11 and x=7.

12 Lecture Notes #3 Page 12 of 15 Can you solve for the points of intersection on the following graph? Graphs and Utility Maximization In microeconomics, consumer theory is based on utility maximization. That is, a person makes themselves as happy as they can be given their income and the prices at which they can buy things. It being economics, happiness is derived from the stuff you consume. The diagram behind this has two functions graphed on it. One is the budget constraint, or what the person can afford to buy. The other is the indifference curve, or how happy a person is with each combination of goods. As an example, imagine a person who consumes only wine (W) and bread (B). Budget Constraint (a.k.a. Budget Line) Imagine now that this person has income of $1200 and that the price of wine P W =$20 and that the price of bread P B =$10. If this person consumes only wine, they will be able to $1200 consume 60 units of wine. If this person consumes only bread, they will be $20

13 Lecture Notes #3 Page 13 of 15 $1200 able to consume 120 units of bread. They could also afford a combination $10 of the two, such as 30W and 60B. To graph this: In an equation, you can think of the total amount of income, $1200, being spent on bread at $10/unit and on wine at $20/unit: $1200 = $10xB + $20xW The total amount spent on bread is $10 multiplied by the quantity of bread, B. The total amount spent on wine is $20 multiplied by the quantity of wine, W. These add up to $1200. In the above graph, however, wine is on the x-axis and bread is on the y-axis. To write the budget equation in the usual linear form, we need to play with it a bit and solve for B in terms of W:

14 Lecture Notes #3 Page 14 of B B OR B B W W 20W W The vertical intercept, 120, is the quantity of bread that could be consumed if this person buys no wine. The slope, -2, is the ratio of the price of wine to the price of bread (20/10). Indifference Curve (based on a Utility Function) In an economic model, you get happiness (called utility) from the combination of goods you consume. In this case, we would say that utility is a function of the quantity of wine and the quantity of bread you consume, or U=U(W,B) For simplicity of calculation, this will generally take a form similar to U(W,B) = WB Or utility is just the quantity of wine consumed multiplied by the quantity of bread consumed. To graph this, we can solve for B as a function of W to get B U(W, B) W The trick to doing this is to pick some level of utility. As an example, we might graph the utility function when U=1800, so the graph would be B 1800 W The graph of this would represent all of the combinations of bread and wine that give this person a utility level of 1800:

15 Lecture Notes #3 Page 15 of 15 All the points on this curve have the same utility level, so this person is indifferent between them. To put this another way, if offered the choice of one point on this curve (90 units of wine and 20 units of bread, for example) and another point on this curve (30 units of wine and 60 units of bread) this person would be indifferent between them. Higher indifference curves are better, so the goal is to get onto the highest indifference curve you possible can given that you have to stay on your budget line: