Let q = f( p1, p2) be the demand for commodity A, which depends upon the prices

**Partial elasticity of demand**

Let *q* = *f*( *p*_{1}, *p*_{2}) be the demand for commodity *A*, which depends upon the prices

*p*_{1}* *and* p*_{2}* *of
commodities* A *and* B *respectively.

The partial elasticity of demand *q* with respect to *p*_{1} is defined to be

The partial elasticity of demand *q* with respect to *p*_{2} is defined to be

**Example 6.40**

Find the marginal productivities
of capital (*K*) and labour (*L*) if

=10*L* + 0.1*L*^{2} + 5*K* - 0.3*K*^{2} + 4*KL* when
K=L=10.

*Solution:*

We have *P* = 10*L* + 0.1*L*^{2} + 5*K* - 0.3*K*^{2} + 4*KL*

**Example 6.41**

The production function for a
commodity is *P*=10*L* + 0.1*L*^{2} + 15*K* - 0.2*K*^{2} + 2*KL* where
*L* is labour and *K* is Capital.

(i) Calculate
the marginal products of two inputs when 10 units of each of labour and Capital
are used

(ii) If 10 units of capital are
used, what is the upper limit for use of labour which a rational producer will
never exceed?

*Solution:*

(i) Given the production is *P* = 10*L* âˆ’ 0 . 1*L*^{2} +15*K* âˆ’ 0 .2*K*^{2} + 2*KL*

(ii) Upper limit for use of labour when *K*=10 is given by a ^{2}_{2}^{P}* _{L}* k â‰¥0

10 âˆ’ 0.2*L*+20 â‰¥0

30 â‰¥ 0.2*L*

i.e.,*L *â‰¤ 150

Hence the upper limit for the use
of labour will be 150 units

**Example 6.42**

**Exercise 6.5**

Tags : Applications of partial derivatives , 11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation

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11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation : Partial elasticity of demand | Applications of partial derivatives

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