Unformatted text preview: Chapter 1 Appendix: Graphs & Their Meaning
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
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
• 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
• Our goal in this appendix is to diminish your
apprehension and enable you to employ graphs
• 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
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
income per week as one
consumption per week as
the second variable.
Chapter 1A 3 1 Chapter 1 Appendix: Graphs & Their Meaning
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.
“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
The “x” axis is the
horizontal or base axis
representing income (do
not confuse the (Y)
designation for income
with the “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
7 A Direct Relationship
Between Variables The relationship between income and consumption
as depicted by Figure 1 is a positive or direct
• Direct relationships represent a positive
association between the variables indicating that
each will respond in the same direction as the
• 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
The relationship between
consumption and income is
direct in that as one increases
the other does as well,
resulting in an upwardly
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
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
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
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
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
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
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
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
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
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
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
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
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
Here, consumption remains
the same no matter what
happens to the divorce rate
therefore these two variables
are totally unrelated to one
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
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
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
C = $50 + .5 ($250)
C = $175 7 Chapter 1 Appendix: Graphs & Their Meaning
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
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
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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- Spring '08
- Slope, Euclidean geometry, Elementary mathematics, Changing Slopes