Chapter01App-slides

Chapter01App-slides - Chapter 1 Appendix: Graphs &...

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Unformatted text preview: Chapter 1 Appendix: Graphs & Their Meaning Slide 1 Economists use graphs to portray relationships as pictures. The purpose of such picture-based portrayals of relationships is to help people visualize and understand economic relationships. Establishing a good fundamental knowledge of graphs is an essential first step in preparing to understand economic reasoning. Economics 17th Edition Campbell McConnell and Stan Brue Chapter #1 Appendix Graphs and Their Meaning Chapter 1A Slide 2 1 • Apprehensive concern regarding “all those graphs” is common for students to experience when completing their first course in the study of economics. • Our goal in this appendix is to erase this apprehension primarily by building your familiarity with, and comfort when working with graphs, and secondarily through helping you develop an essential set of basic reasoning skills to employ when examining and working with graphs. • Economists use graphs to tell a story. The more we seek to understand the story each graph is telling, the more successful we will be in working with graphs. The Purpose of Graphs It is common for introductory students in economics to feel apprehensive about all those graphs. • Our goal in this appendix is to diminish your apprehension and enable you to employ graphs constructively. • Economists use graphs to portray a situation or paint a picture of the relationship between variables such as income and consumption. • Most importantly, each graph tells a story and your challenge is to discover the story it is telling. Chapter 1A Slide 3 2 Construction of a Graph: The Table The content of most graphs in economics begins in tabular form. A graph is the visual representation between two variables. Table 1 lists income earned per week in one column and the corresponding level of consumption per week in the other, indicating a relationship exists between these variables. The Table 1 lists the relationship between income per week as one variable and consumption per week as the second variable. Chapter 1A 3 1 Chapter 1 Appendix: Graphs & Their Meaning Slide 4 • Each variable must be assigned to an axis on a graph in order to be illustrated in such a manner. • In this instance, income per week will be illustrated along the “x” or horizontal axis and consumption per week will be illustrated along the “y” or vertical axis. Assigning Variables to an Axis Each variable (income and consumption in our example) must represent either the “x” or “y” axis on a graph to be meaningfully portrayed. We have associated income per week with the “x” axis and consumption per week with the “y” axis. “y” axis “x” axis Chapter 1A Slide 5 4 The graph depicted on Slide 1 Appendix-5 features consumption (C) on the vertical axis and income (Y) on the horizontal axis. Axis Associations (x and y) The “y” axis is the vertical axis representing consumption (C). The “x” axis is the horizontal or base axis representing income (do not confuse the (Y) designation for income with the “x” axis). “x” axis Chapter 1A Slide 6 5 From Table to Graph Although the table conveys information about the relationship between income and consumption, a picture enables us to see the relationship more clearly. • In response to the desire to see this relationship we build a graph. • Building the graph merely involves plotting the x-y coordinates (listed as points a, b, c, d, and e in Table 1) on the graph and connecting the points with a line. Chapter 1A 6 This slide features both the content from Table 1 and the same information portrayed graphically. • Values from the graph were plotted on the graph as points a, b, c, d, and e. • Each point corresponds with an “x” axis value and a “y” axis value. • For example, the x value (representing income) for point “a” is zero and the “y” value (representing consumption) for point “a” is 50. • The “x” value for point “b” is $100 and the “y” value for point “b” is 100, so forth and so on. • Ultimately the linear relationship between these variables is divulged by connecting the points. 2 Chapter 1 Appendix: Graphs & Their Meaning Slide 7 A Direct Relationship Between Variables The relationship between income and consumption as depicted by Figure 1 is a positive or direct relationship. • Direct relationships represent a positive association between the variables indicating that each will respond in the same direction as the other. • For example, as income increases, consumption will increase. Alternatively, as income declines, consumption will also decline. • Positive relationships result in an upwardly sloping line. Direct (or positive) relationships between two variables means that the two variables change in the same direction. The relationship between consumption and income is direct in that as one increases the other does as well, resulting in an upwardly sloping line. Chapter 1A Slide 8 7 In contrast, some variables have inverse relationships relative to one another. • Table 2 and the graph labeled Figure 2 feature two variables with such a relationship. • In this instance ticket price (as the independent variable) and attendance (as the independent variable) are negatively related. • An increase in ticket price results in a decline in attendance. Alternatively, a decline in ticket price results in an increase in attendance. Inverse Relationships The relationship between ticket price and attendance in thousands as designated in Table 2 is featured graphically in Figure 2. In this instance the value of one variable increases as the other decreases and is referred to as an inverse relationship. Chapter 1A Slide 9 Figure 2 8 Inverse Relationships and Downward Sloping Lines • The points a, b, c, d, and e from Table 2 were plotted on graph 2 in precisely the same manner as similar points were plotted in our first example. • Graphical lines portraying inverse relationships slope downwardly. Figure 2 Figure 2 features an inverse (or indirect) relationship between the variables ticket price portrayed on the “y” axis and attendance portrayed on the “x” axis. Graphical lines portraying inverse relationships slope downwardly. Chapter 1A 9 3 Chapter 1 Appendix: Graphs & Their Meaning Slide 10 Independent and Dependent Variables Economics is concerned with cause and effect relationships. In our ticket price/attendance illustration, ticket price is the independent or cause variable and attendance is the dependent or effect variable. In this instance, an increase in the cause variable (ticket price) results in a decrease in the effect variable (attendance). Economics is concerned with “cause” and “effect” relationships. In our ticket price-attendance illustration ticket price is the independent (or cause) variable and attendance is the dependent (effect) variable. Chapter 1A Slide 11 In contrast, the cause variable in our first example is income and the effect variable is consumption. An increase in the independent variable (income) will result in an increase is the dependent variable (consumption). 10 Economists as Mathematical Nonconformists Mathematicians always place independent variables along the horizontal axis and dependent variables along the vertical axis. • Economists are not as rigid in that we choose such associations arbitrarily. • As a result, it becomes important to learn how to distinguish between cause and effect variables. • We revisit this issue a bit later in this section. You may recall from high school courses that mathematicians always associate the independent variable with the horizontal “x” axis and the dependent variable with the “y” axis. Economists are less tidy; their graphing of independent and dependent variables is more arbitrary. Chapter 1A Slide 12 11 Assuming all else equal or ceteris paribus is essential when portraying economic relationships in graphical format in that we are seeking to examine the most probable response that will occur between two variables without the clutter of less closelyrelated variables and their reactions. Ceteris Paribus Ceteris Paribus translates to “other things equal” and economists employ this assumption when portraying the relationship between two variables in graphical format. In reality, factors other than income can cause changes in consumption and factors other than ticket price can cause changes in attendance to events but for the sake clarifying the relationships under the variables studied in cause and effect relationships, economists assume “all else remains equal.” Chapter 1A As an example, in reality factors other than income cause changes in consumption to occur and factors other than ticket price can cause changes in attendance at events to occur. For the sake of clarifying the relationships between our initial set of variables, we assume all else remains the same. 12 4 Chapter 1 Appendix: Graphs & Their Meaning Slide 13 Determining and understanding the relevance of the slope of lines becomes an important tool to employ when studying economics because the value revealed by the slope inherently explains some aspects of the relationship between two variables. Slope of a Line The slope of a straight line is the ratio of vertical change to horizontal change between any two points of the line. Slope is more simply characterized as the change in the rise divided by change in the run as we move from left to right along a graphical illustration. We measure slope as the rise/run or stated differently vertical change/horizontal change. vertical change Slope ________________________________________________ = horizontal change Chapter 1A Slide 14 13 Using our example from Table 1, the rise from points “b to c” is positive $50 and the horizontal run or horizontal change over this range is positive $100. • Therefore the slope of the curve in Figure 1 is 50/100 = .50. • Note that this value is positive. • When applied, the slope also indicates that $1 of consumption will occur as a result of a $2 increase in income. Positive Slope Between point b and point c in figure 1 the rise or vertical change is +50 and the run or horizontal change is +100. • Therefore the slope is: 50/100 = .5 • Note that the value of the slope is positive. When applied, the slope also points to the fact that $1 of consumption will occur for every $2 increase in income. Chapter 1A Slide 15 14 Alternatively, the slope of the line illustrated in Figure 2 is negative. • Between any two points along the line the vertical change is negative ten (- 10) and the horizontal change over the same range is positive four (+ 4). • Therefore the slope is: - 10/+4= -2.5. • This slope is negative because ticket price and attendance share an inverse relationship. Negative Slope Between any two points along the line in Figure 2 (c and d for example) the vertical change is 10 (the drop) and the horizontal change is + 4 (the run). Figure 2 Therefore the slope is: -10/+ 4 = - 2.5 This slope is negative because ticket price and attendance have an inverse relationship. Chapter 1A 15 5 Chapter 1 Appendix: Graphs & Their Meaning Slide 16 The concept of slope is important for use in the study of economics for another reason as well. The slope of a line reflects the marginal changes between two variables. Slopes and Marginal Analysis The concept of slope is important in economics because it reflects marginal changes—those involving 1 more (or 1 less) unit. For example, in Figure 1 the slope of .5 reveals that $.50 extra (or marginal) consumption is associated with each $1 increase in income. For example, in Figure 1 the .5 slope reveals that $.50 extra or marginal consumption is associated with each $1 increase in income. Chapter 1A Slide 17 Infinite Slopes In Figure 3 (a) we represent the price of bananas on the vertical axis and the quantity of watches demanded on the horizontal axis. The graph of their relationship is parallel to the vertical axis indicating that the same quantity of watches is purchased no matter what the price of bananas therefore no relationship exists. The slope of such a line is infinite. Figure 3 (a) Chapter 1A Slide 18 16 17 The other slope extreme is zero slope. Figure 3 (b) features an illustration of zero slope. • In this instance, consumption is depicted along the vertical axis and the divorce rate is depicted along the horizontal axis. • In this instance, consumption rates continue to remain the same no matter what the divorce rate happens to be. • Therefore no relationship exists between these two variables, either. Zero Slope A line parallel to the horizontal axis has a slope of zero. Figure 3 (b) In Figure 3 (b) consumption is portrayed on the vertical axis and the divorce rate is portrayed on the horizontal axis. Here, consumption remains the same no matter what happens to the divorce rate therefore these two variables are totally unrelated to one another. Chapter 1A In certain instances we encounter the extremes of slopes. One extreme is referred to as infinite slope and the other is referred to as zero slope. • Figure 3 (a) illustrates an example of infinite slope in which the price of bananas is featured along the vertical axis and purchases of watches is featured along the horizontal axis. • The graph of the relationships between the two variables is parallel to the vertical axis and indicates that the same quantity of watches will be purchased no matter what the price of bananas happens to be. • Therefore no relationship exists between the two variables. 18 6 Chapter 1 Appendix: Graphs & Their Meaning Slide 19 In another interesting facet of graphs, a line can be located on a graph (without plotting points) if we know its slope and its vertical intercept value. • The vertical intercept of a line is the point where the line meets the vertical axis. • In Figure 1, the intercept is $50 which means that if current income were zero, consumers would still spend $50 for consumption. Vertical Intercept A line representing the relationship between two variables can be located on a graph (without plotting points) if we know its slope and its vertical (y) intercept. The vertical intercept of a line is the point where the line meets the vertical axis. In Figure 1, the intercept is $50 which means that if current income were zero, consumers would still spend $50 for consumption. Chapter 1A Slide 20 19 Equation of a Linear Relationship The equation of a straight line is y = a + bx where: The equation of a straight line is y = a + bx where: y = the dependent variable a = the vertical intercept b = the slope of the line x = the independent variable • y = the dependent variable • a = the vertical intercept • b = the slope of the line • x = the independent variable Chapter 1A Slide 21 20 Determining Values Using the Equation of a Linear Relationship In our income-consumption example from Figure 1, C represents the dependent variable and Y represents the independent variable. We know the vertical intercept value is $50 and we formerly calculated the slope value as .50 therefore the equation for this linear relation ship is: C = 50 + .5y From our income-consumption example from Figure 1, C represents the dependent variable and Y represents the independent variable. We already know that the vertical intercept is 50 and the slope is .5 therefore our equation for this line is: C = 50 + .5Y Using this equation we can determine any value of consumption (C) by inserting a value of income (Y). For example, assume income is $250: C = 50 + .5($250) = $175 Therefore we have determined that at an income level of $250, consumption spending is $175. Chapter 1A 21 Using this equation we can determine any value of consumption (C) by inserting a value of income (Y) (and vice versa). In the event we assume income is $250, we can determine the corresponding value of consumption. C = $50 + .5 ($250) C = $175 7 Chapter 1 Appendix: Graphs & Their Meaning Slide 22 Equation of a Linear Relationship Example 2 The variables portrayed in Figure 2 are reversed in terms of independent and dependent variable association according to mathematical convention. We mentioned this as a possibility earlier in this section. Figure 2 variables are reversed from mathematical convention in terms of independent and dependent variable association with the vertical axis and horizontal axis. In our second example from Figure 2, we deduce that the vertical (y) intercept is 50 in this instance as well and the slope as calculated earlier is – 2.5. Therefore we must be able to distinguish that P (ticket price) is the independent or cause variable and Q (attendance) is the dependent or effect variable. Therefore in this instance the equation is structured as: Therefore our equation for this linear relationship is: P = 50 – 2.5Q P = 50 - 2.5Q Chapter 1A Slide 23 22 The slope of a straight line is the same at all points along the line whereas the slope of a line representing a nonlinear relationship (relationships illustrated with curved lines) changes from one point to another. Slope of a Nonlinear Curve The slope of a straight line is the same at all points along the line whereas the slope of a line representing nonlinear relationships (relationships illustrated with curved lines) changes from one point to another. In the study of economics we often see these kinds of lines and refer to them as curvilinear relationships or simply “curves.” • We refer to such lines as curves in economics. Chapter 1A Slide 24 23 The downwardly sloping curve depicted in Figure 4 has a negative slope throughout, but flattens as we move down along it. The Changing Slopes of a Curve The downwardly sloping curve in Figure 4 has a negative slope throughout, but flattens as we move down along it. Figure 4 Thus the slope constantly changes and the curve has a different slope at each point. Thus the slope constantly changes and has a different slope at each point. Chapter 1A 24 8 Chapter 1 Appendix: Graphs & Their Meaning Slide 25 Determining the Slopes of Curves To measure the slope at a specific point, we draw a straight line tangent to the curve at that point such as point A in Figure 4. The slope of the curve at point A is equal to the slope of the straight tangent line: - 20 ÷ 5 = - 4 So the slope of the curve at point A is - 4. Chapter 1A Figure 4 To measure the slope at a specific point along a curve, we draw a straight line tangent to the curve at the point (such as point A in Figure 4) and calculate the slope of the tangential straight line. The slope of the line tangent to point A is: - 20/5 = - 4 Therefore the slope at point A is – 4. 25 9 ...
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