Page 324 - DMTH201_Basic Mathematics-1

Unit 12: Successive Differentiation

Notes

Notes

(i) The above formula gives the elasticity of demand at a point on the demand curve
and hence is also referred to as point-elasticity formula.
(ii) Since is a ratio, it is a pure number.
d
(iii) As per the convention in economics, the elasticity of demand is defined as the
negative of the ratio of proportional change in quantity demanded to proportionate
change in price.

Income Demand (Engel Function)
Assuming price of a commodity and prices of other commodities etc., as constant, we can say
that quantity demanded (x) of a commodity is a function of consumer’s income (Y). Symbolically,
dx
we can write this as x = g(Y). Note here that can be positive or negative.
dY

dx
If 0 (< 0), the commodity is said to be normal (inferior).
dY
Income Elasticity of Demand

The income elasticity of demand, , is defined as the ratio of proportionate change in quantity
Y
demanded to proportionate change in price.
d log x d log x dY dx Y
=
Y d logY dY d log Y dY x
We note that if < 0, the commodity is inferior.
Y
Cross Demand

Let there be two commodities A and B. Assuming other things as constant, we can write demand
for A, denoted as x , as a function of the price of B(P ); and also the demand for B(x ) as a function
A B B
of the price of A(P ). Using symbols, we can write
A
x = (P ) and x = (P )
A B B A
dx dx B
Such functions are termed as cross demand functions. We note here that if A 0 and 0,
dp dp A
B
then A and B are termed as substitutes. If both the derivatives are negative, the two commodities
are termed as compliments. Nothing can be said about the relationship between A and B, if these
derivative are of opposite signs.

Cross Elasticity of Demand

The cross elasticity of demand of commodity A as compared with price of B, denoted by , is
AB
the ratio of proportionate change in quantity demanded of A to proportionate change in price of
B. Symbolically, we can write
d log x dx p d log x dx p
A A B B B A .
AB . Similarly, BA
d log p B dp B x A d log p A dP A x B

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