line, is equal to which means that for every one-unit
increase in E, P falls by 0.5 unit. We can thus state the equation of the line
as
where both P and E are expressed as thousands of units (tonnes and
dollars, respectively).
Non-linear Functions
Although it is sometimes convenient to simplify the situation by assuming
two variables to be linearly related, this is seldom the case over their
whole range. Non-linear relations are much more common than linear
ones. In the case of reducing pollution, it is usually quite cheap to
eliminate the first units of pollution. Then, as the environment gets
cleaner and cleaner, the cost of further clean-up tends to increase because
more and more sophisticated and expensive methods need to be used. As
a result, Figure 2-8 is more realistic than Figure 2-7. Inspection of
Figure 2-8 shows that as more and more is spent on reducing pollution,
the amount of pollution actually reduced for an additional $1 of
expenditure gets smaller and smaller. This is shown by the diminishing
slope of the curve as we move rightward along it. For example, as we
move from point A to point B, an increase in expenditure of $1000 is
required to reduce pollution by 1000 tonnes. Thus, each tonne of
pollution reduction costs $1. But as we move from point C (where we
have already reduced pollution considerably) to point D, an extra $6000
ΔP/ΔE, −0.5,P = 6 −(0.5)E