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Behavior of Slopes in Curved Graphs


Negative Positive
slope slope
Negative Positive
slope slope Negative

0 0 0 0
(a) (b) (c) (d)

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Part 1
16 Getting Acquainted with Economics

How to Measure Slope at a Point on a Curved Graph

The point at which a straight line cuts the vertical (Y)
axis is called the Y-intercept.
The Y-intercept of a line or a curve is the point at which
it touches the vertical axis (the Y-axis). The X-intercept
is defined similarly.
F For example, the Y-intercept of the line in Figure 3(a)
is a bit less than 8.
4 G r
Lines whose Y-intercept is zero have so many special
3 uses in economics and other disciplines that they have
been given a special name: a ray through the origin, or
a ray.
B Figure 6 shows three rays through the origin, and
X the slope of each is indicated in the diagram. The ray
0 1 2 3 4 5 6 7 8 9 10
in the center (whose slope is 1) is particularly useful in
many economic applications because it marks points
where X and Y are equal (as long as X and Y are meas-
ured in the same units). For example, at point A we
have X 5 3 and Y 5 3; at point B, X 5 4 and Y 5 4. A
curve at these two points by applying the definition. similar relation holds at any other point on that ray.
The calculation for point T, then, is the following: How do we know that this is always true for a ray
whose slope is 1? If we start from the origin (where
Slope at point T 5 Slope of line tt
both X and Y are zero) and the slope of the ray is 1, we
Distance BC
know from the definition of slope that
Distance BA
Vertical change
11 2 52 24 Slope 5 51
Horizontal change
5 5 5 22
13 2 12 2
This implies that the vertical change and the hori-
A similar calculation yields the slope of the curve at
zontal change are always equal, so the two variables
point R, which, as we can see from Figure 5, must be
smaller numerically. That is, the tangent line rr is less
steep than line tt:
F I GU R E 6
Slope at point R 5 Slope of line rr
Rays Through the Origin
15 2 72 22
5 5 5 21
18 2 62 2
Exercise Show that the slope of the curve at point G
is about 1.
Slope = + 2
What would happen if we tried to apply this graph-
ical technique to the high point in Figure 4(c) or to the Slope = + 1
low point in Figure 4(d)? Take a ruler and try it. The B
tangents that you construct should be horizontal,
meaning that they should have a slope exactly equal 3
to zero. It is always true that where the slope of a
smooth curve changes from positive to negative, or 2 1
K Slope = + “
vice versa, there will be at least one point whose slope 2
is zero. 1
Curves shaped like smooth hills, as in Figure 4(c),
have a zero slope at their highest point. Curves shaped D
like valleys, as in Figure 4(d), have a zero slope at their 0 1 2 3 4 5
lowest point.

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Chapter 1 17
What Is Economics?

must always remain equal. Any point along that ray Luckily, economists can use a well-known device for
(for example, point A) is exactly equal in distance from collapsing three dimensions into two”a contour map.
the horizontal and vertical axes (length DA = length Figure 7 is a contour map of the summit of the highest
CA)”the number on the X-axis (the abscissa) will be mountain in the world, Mt. Everest, on the border of
the same as the number on the Y-axis (the ordinate). Nepal and Tibet. On some of the irregularly shaped
“rings” on this map, we find numbers (like 8500) indi-
Rays through the origin with a slope of 1 are called 45°
cating the height (in meters) above sea level at that par-
lines because they form an angle of 45° with the hori-
ticular spot on the mountain. Thus, unlike the more
zontal axis. A 45° line marks off points where the vari-
usual sort of map, which gives only latitudes and lon-
ables measured on each axis have equal values.2
gitudes, this contour map (also called a topographical
If a point representing some data is above the 45° map) exhibits three pieces of information about each
line, we know that the value of Y exceeds the value of point: latitude, longitude, and altitude.
X. Similarly, whenever we find a point below the 45° Figure 8 looks more like the contour maps encoun-
line, we know that X is larger than Y. tered in economics. It shows how some third variable,
called Z (think of it as a firm™s output, for example),
varies as we change either variable X (think of it as a
SQUEEZING THREE DIMENSIONS firm™s employment of labor) or variable Y (think of it
INTO TWO: CONTOUR MAPS as the use of imported raw material). Just like the map
of Mt. Everest, any point on the diagram conveys
three pieces of data. At point A, we can read off the
Sometimes problems involve more than two vari-
values of X and Y in the conventional way (X is 30 and
ables, so two dimensions just are not enough to depict
Y is 40), and we can also note the value of Z by finding
them on a graph. This is unfortunate, because the
out on which contour line point A falls. (It is on the
surface of a sheet of paper is only two-dimensional.
Z 5 20 contour.) So point A is able to tell us that
When we study a business firm™s decision-making
process, for example, we may want to keep track si- 30 hours of labor and 40 yards of cloth produce
multaneously of three variables: how much labor it 20 units of output per day. The contour line that indi-
employs, how much raw material it imports from for- cates 20 units of output shows the various combina-
eign countries, and how much output it creates. tions of labor and cloth a manufacturer can use to

F I GU R E 7
A Geographic Contour Map SOURCE: Mount Everest. Alpenvereinskarte. Vienna: Kartographische Anstalt Freytag-
Berndt und Artaria, 1957, 1988.

The definition assumes that both variables are measured in the

same units.

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Licensed to:
Part 1
18 Getting Acquainted with Economics

produce 20 units of output. Economists call such maps
F I GU R E 8
production indifference maps.
An Economic Contour Map
A production indifference map is a graph whose axes
show the quantities of two inputs that are used to pro-
duce some output. A curve in the graph corresponds to
some given quantity of that output, and the different
points on that curve show the different quantities of the
two inputs that are just enough to produce the given
Yards of Cloth per Day

Although most of the analyses presented in this
book rely on the simpler two-variable diagrams, con-
A Z = 40 tour maps do find many applications in economics.

B Z = 30

Z = 20
Z = 10
0 10 20 30 40 50 60 70 80

Labor Hours per Day

1. Because graphs are used so often to portray economic 4. Often, the most important property of a line or curve
models, it is important for students to acquire some un- drawn on a diagram will be its slope, which is defined
derstanding of their construction and use. Fortunately, as the ratio of the “rise” over the “run,” or the vertical
the graphics used in economics are usually not very change divided by the horizontal change when one
complex. moves along the curve. Curves that go uphill as we
move to the right have positive slopes; curves that go
2. Most economic models are depicted in two-variable dia-
downhill have negative slopes.
grams. We read data from these diagrams just as we
read the latitude and longitude on a map: each point 5. By definition, a straight line has the same slope wher-
represents the values of two variables at the same time. ever we choose to measure it. The slope of a curved line
changes, but the slope at any point on the curve can be
3. In some instances, three variables must be shown at
calculated by measuring the slope of a straight line tan-
once. In these cases, economists use contour maps,
gent to the curve at that point.
which, as the name suggests, show “latitude,” “longi-
tude,” and “altitude” all at the same time.

Variable 13 Tangent to a curve 15 45° line 17
Origin (of a graph) 13 Y-intercept 16 Production indifference map 18
Slope of a straight (or curved) Ray through the origin,
line 15 or ray 16

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Licensed to:
Chapter 1 19
What Is Economics?


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