units above (“north of”) the horizontal axis. Analogously, point
B
is de±ned by the coor-
dinates (8, 4). We call point
C
(0, 12), where the function cuts (or
intersects
) the vertical
axis, the
vertical intercept
, and point
H
(12, 0), where the function cuts (or
intersects
) the
horizontal axis, the
horizontal intercept
.
Figure M.1-2(b) gives the signs of the coordinates in all four quadrants of the plane
and along the two axes. In the southwest quadrant, for example, both
x
and
y
coordi-
nates are negative, while in the southeast quadrant, the
x
coordinate is positive and the
y
coordinate is negative. [Negative values can have economic signi±cance. For instance,
inputs into production are treated as “negative outputs” in economic activity analysis.
And a person could say, “I would be prepared to pay a price of
minus $2
for a Barry
Manilow CD,” meaning that she would accept it
only
if it were not only free but also
accompanied by a toonie.]
As an exercise, you may want to plot the following 6 points on a 4-quadrant graph,
and then compare them with the corresponding ones in Figure M.1-2(a): (4, –8), (–4, 8),
(–4, –8), (8, –4), (–8, 4), and (–8, –4).
1.3 TYPES OF FUNCTIONS
We can categorize the functions economists use in several ways.
1.3.1
One basic way relates to whether they are
increasing, decreasing, constant,
or
increas-
ing-and-decreasing
(or
decreasing-and-increasing
)
.
Figure M.1-3 shows examples of each
type.
Implicitly, when we speak of an “
increasing
function,” we mean that the value of the
function (
y
)
increases
as we move from
left to right
, to higher values of
x.
Figure M.1-3(a)
depicts what mathematicians call a “
monotonically strictly increasing
” function: English
translation, it has a
positive slope
throughout. In contrast, Figure M.1-3(b) depicts a
“
monotonically strictly decreasing
” function: it has a
negative slope
throughout. Suppose
that a function is never negatively sloped, but that it does have a
horizontal
portion.
We can call it a “
non-decreasing
function.” The
constant
(or horizontal) function in
Figure M.1-3(c) can be described as
simultaneously non-increasing and non-decreasing.
Figure M.1-3(d) shows a (non-monotonic)
decreasing-and-increasing
function.
You have already encountered all of these types of function in your introductory eco-
nomics course, and you will see them all again. Figure M.1-3(a) could represent a sup-
ply curve
or
a Total Cost function. Figure M.1-3(b) could be a demand curve,
or
a (not
very well-behaved) production possibilities frontier,
or
a marginal physical product
curve. Figure M.1-3(c) could be a perfectly elastic supply or demand curve,
or
the long-
run average cost curve for an industry characterized by constant costs. And Figure M.1-
3(d) could be a marginal, average variable or average total cost curve. In each case, the
mathematical
properties of the function remain the same, but the
economic
interpretations
will often be quite different.