the negative of the corresponding terms in Equation M.2.1, and so the two negatives
cancel.
Having a feel for the meaning of the
slope
of a line is an extremely important part of
developing not only your
mathematical
but also your
economic
intuition. The slope is a
ratio of
changes
in 2 variables. If the slope of the graph of an equation is
positive
, then an
increase in one variable is associated with an
increase
in the other (and similarly, a
decrease in one variable is associated with a decrease in the other). If the slope is
nega
tive
, then an
increase
in one variable is associated with a
decrease
in the other (and simi
larly, a
decrease
in that variable is associated with an
increase
in the other).
If the slope is relatively
low
in absolute value and the curve is hence relatively
ﬂat
,
then a unit change in the horizontal variable is associated with a relatively
small
change
in the vertical variable. In contrast, if the slope is relatively
high
in absolute value and
the curve is hence relatively
steep
, then a unit change in the horizontal variable is asso
ciated with a relatively
large
change in the vertical variable.
The two limiting cases occur when the line is horizontal or vertical. If the line is hor
izontal and its slope ∆
y/
∆
x
is zero (that is, ∆
y
= 0), then no matter how great a change
occurs in the horizontal variable, there is
no
associated change in the vertical variable:
it is
constant
. If in contrast the line is vertical and its slope ∆
y/
∆
x
is infinite or undefined
(that is, ∆
x
= 0), then no matter how great a change occurs in the vertical variable, there
is
no
associated change in the horizontal variable:
it
is now the one that is constant.
The importance of the slope of an equation in an economic model is thus that it
encapsulates our assumptions both about the
nature
of the correlation between two eco
nomic variables (positive or negative)
and
about the
magnitude
of the association, or
about the degree of responsiveness of changes in one variable to changes in the other.
In this context, we can proceed to review the four ways of defining a linear equation.
1.
“Standard” (SlopeIntercept) Form
: The equation
y
= 3 + 2
x
in Figure M.21 has the
“standard” form of a linear equation:
y
=
a
+
b
x.
(M.2.3)
Here
a
is the
vertical intercept
, the point at which the graph intersects or cuts the ver
tical axis, with coordinates (0,
a
), and
b
is the slope of the function, ∆
y/
∆
x.
Simply
looking at the equation, we can visualize the straight line passing through the verti
cal axis 3 units above the origin (since when
x
= 0 in the equation,
y
= 3). It slopes
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 Fall '12
 Danvo
 Slope, Supply And Demand, Euclidean geometry

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