Graphs and Their Meaning Chapter 1A This appendix helps in understanding graphs, curves, and slopes as they relate to economics. It demonstrates how relationships between variables can be examined and how an equation that expresses that relationship can be derived. Graphs and Their Meaning Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Construction of a Graph A visual representation of the relationship between two variables Horizontal axis Vertical axis Independent variable Dependent variable Ceteris paribus The two dimensional graphs used in this textbook consist of a horizontal axis and a vertical axis. The point where the two meet is called the origin. As one moves away from the origin either up or to the right, the plotted values increase. Often the independent variable which is the variable that changes first, is placed on the horizontal axis and the dependent variable which is the variable that changes in response to the first variable changing, is placed on the vertical axis. However, in economics it is customary to always place price or cost data on the vertical axis even if it is the independent variable. Economist invoke the ceteris paribus assumption in economics in order to be able to analyze just the two variables that are labeled on the axis. Ceteris paribus means to hold other things equal or constant and assumes that the only changes in the model are brought about by the dependent variable changing in response to the change in the independent variable. LO8

Direct and Inverse Relationships Direct relationship Both variables move in the same direction Direct or positive relationships occur as the two variables change in the same direction. In other words, when the independent variable increases (decreases), the dependent variable also increases (decreases). This graphs as an upsloping line. LO8

Construction of a Graph Table Income Consumption Point $ 0 100 200 300 400 $ 50 100 150 200 250 a b c d e $400 300 200 100 e Consumption (C) Each pair of values in the table represents one point on the graph. This data represents variables that are positively or directly related, such as income and consumption, which will graph as an upsloping line. d c b a $100 200 300 400 LO8 Income (Y)

Direct and Inverse Relationships Variables move in opposite directions Inverse or negative relationships are those where the two variables move in opposite directions. So when the independent variable increases (decreases), the dependent variable decreases (increases). An inverse relationship graphs as a downsloping line. LO8

Construction of a Graph Table Graph a Ticket Price Attendance, Thousands $ 50 40 30 20 10 Point $ 50 40 30 20 10 4 8 12 16 20 a b c d e f b c Ticket price (P) d Each pair of values in the table represents one point on the graph. This data represents variables that are negatively or inversely related, such as ticket price and attendance at basketball games, which will graph as a downsloping line. e f 4 8 12 16 20 LO8 Attendance in thousands (Q)

Slope of a Line Slope Slopes and measurement units Slopes and marginal analysis Infinite and zero slopes Vertical intercept The slope of a line can vary based on the units that are used to measure the slope. If in the ticket price and attendance example, if the number 4000 had been used instead of 4 for the horizontal change, the slope would have been -0.0025 instead of -2.5. Slope is also important in economics when analyzing marginal changes. In the income example, a slope of 0.5 means that an increase of $1 in income will result in a $.50 increase in consumption spending. If there is no relationship between the two variables, the line will graph as a vertical line parallel to the vertical axis which reflects an infinite slope or it will graph as a horizontal line that is parallel to the horizontal axis which would exhibit a slope of zero. The vertical intercept is the point where the line meets the vertical axis. LO8

Positive Slope of a Line vertical change +50 1 2 Slope = = = = 0.5 horizontal change +100 $400 300 200 100 As the independent variable, income, increases, the dependent variable, consumption, increases; this results in a positively sloped line. Consumption (C) 50 Vertical change 100 Horizontal change LO8 Income (Y)

Negative Slope of a Line vertical change -10 1 2 Slope = = -2 -2.5 = = 50 4030 20 10 horizontal change +4 Ticket Price $ Vertical change -10 As the independent variable, ticket price increases, attendance decreases. Here the variables are moving in opposite directions illustrating a negative slope. Note: though generally the independent variable is placed on the horizontal axis, when price is one of the variables, in economics it is customary to always place it on the vertical axis. 4 Horizontal change 4 8 12 16 20 Attendance LO8

Slope of a Line Slope = zero Slope = infinite Price of Bananas Consumption The graphs above show that the variables are unrelated. No matter what the price of bananas might be, the number of watches purchased remains the same. This is because the price of bananas does not affect the number of watches purchased. There is no relationship between consumption and the divorce rate. Consumption remains the same no matter how high or low the divorce rate is. Purchases of watches Divorce rate LO8

Equation of a Linear Relationship y = a + bx, where y is the dependent variable a is the vertical intercept b is the slope of the line x is the independent variable This is the general equation of a line and expresses the relationship between the two variables y and x. If you know the slope and the vertical intercept, you can solve for y or x. LO8

Equation of a Line Y = 50 + .5C $400 300 Consumption (C) 200 100 Consumption (C) We calculated the slope of this line earlier, and we found that the slope is +.5. We can see from the graph that the vertical intercept is 50. Therefore the equation of this line is: Y (income) = 50 (vertical intercept) + .5C (where .5 is the slope and C is the consumption). $100 200 300 400 Income (Y) LO8

Equation of a Line P = 50 – 2.5Q 50 4030 20 10 Ticket price We calculated the slope of this line earlier, and we found that the slope is -2.5. We can see from the graph that the vertical intercept is 50. Therefore the equation of this line is: P (price) = 50 (vertical intercept) - 2.5Q (where -2.5 is the slope and Q is the quantity). In this case, the slope is being subtracted because the slope is negative. 4 8 12 16 20 Attendance LO8

Slope of a Nonlinear Curve Slope always changes Use a line tangent to the curve to find slope at that point When calculating the slope of a nonlinear curve, the slope is always changing. This means the slope will be different at every point on the curve. In this case to find the slope, you must draw a line tangent to the curve and calculate the slope of the line by finding the change between two points along the tangent line. LO8

Slope of a Nonlinear Curve 20 15 10 5 The slopes of a nonlinear curve will change along the curve. Here, the slope at these two points on the curve will be different from each other. To find the slope of a nonlinear curve, draw a straight line tangent to the curve and find the slope of that line by dividing the vertical change by the horizontal change of that line. 5 10 15 20 LO8