11DIFFERENTIATION - Department of Mathematics

742 11 DIFFERENTIATION
Elasticity of
Demand
Suppose the unit price of a commodity is increased by h dollars from p
dollars to (p h) dollars (Figure 11.11b). Then the quantity demanded drops
from f(p) units to f(p h) units, a change of [ f(p h) f(p)] units. The
percentage change in the unit price is
h
p (100) Change in unit price (100)
Price p
and the corresponding percentage change in the quantity demanded is
f(p h) f(p) Change in quantity demanded
100 (100)
f(p)
Quantity demanded at price p
Now, one good way to measure the effect that a percentage change in price
has on the percentage change in the quantity demanded is to look at the ratio
of the latter to the former. We find
Percentage change in the quantity demanded
Percentage change in the unit price
100
f(p h) f(p)
f(p)
100h f(p h) f(p)
h
f(p)
p
If f is differentiable at p, then
f(p h) f(p)
f(p)
h
when h is small. Therefore, if h is small, then the ratio is approximately equal to
f(p)
f(p)
p
pf(p)
f(p)
Economists call the negative of this quantity the elasticity of demand.
If f is a differentiable demand function defined by x f(p), then the elasticity
of demand at price p is given by
E(p) pf(p)
(7)
f(p)
REMARK It will be shown later (Section 12.1) that if f is decreasing on an
interval, then f(p) 0 for p in that interval. In light of this, we see that since
both p and f(p) are positive, the quantity pf(p)
is negative. Because econof(p)
p