Page 291 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes
Example
Consumer price index of a certain group of workers increases by 15% per year and their quantity
index by 6%. What is the annual growth of their expenditure.
Solution:
Let P denotes price index, Q the quantity index and E the expenditure index. We can write E = P
× Q.
Taking log of both sides, we get log E = log P + log Q
Differentiating w.r.t. t, we get
d log E d log P d logQ
=
dt dt dt

d log E d log P
Let us denote , the rate of growth of E, by r . Similarly, we denote r P and
E
dt dt
d logQ
r Q .
dt
Thus, we can write r = r + r = 0.15 + 0.06 = 0.21.
E P Q
Hence the rate of growth of expenditure index is 21%.

Example
bt
Agricultural output is the following function of time: X = K × a , where K, a and b are all positive
constants with a < 1 and b < 1.
(i) Show that, starting from an initial level of output X , the output is always increasing but
0
is subject to a ceiling which is never, exceeded.
(ii) Show that proportional rate of growth of output is always positive, but declines over
time.
Solution:
(i) Note that initial output X = Ka
0
dX
t
To find we take log of both sides i.e. log X = log K + b log a
dt
d log X 1 dX
Differentiating w.r.t. t, we get b t log a log b
dt X dt
dX b t t
or K a b log a × log b > 0 (Since log a, log b < 0)
dt
To determine the ceiling on output, we find

K
X lim .a b t K lim a b t K
t t

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