Exercise 6.5 - Chapter 6 - Application of Differentiation - 11th Business Maths Guide Samacheer Kalvi Solutions

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Exercise 6.5

Text Book Back Questions and Answers

Question 1.
Find the marginal productivities of capital (K) and labour (L) if P = 8L – 2K + 3K² – 2L² + 7KL when K = 3 and L = 1.
Solution:
P = 8L – 2K + 3K² – 2L² + 7KL
Marginal productivity of labour,   = 8 – 0 + 0 – 2(2L) + 7K(1)
= 8 – 4L + 7K
Marginal productivity of labour when K = 3 and L = 1 is

= 29 – 4
= 25
Marginal productivity of capital,

= -2 + 6K + 7L
Marginal productivity of capital when K = 3 and L = 1 is

= -2 + 18 + 7
= -2 + 25
= 23

 Question 2.
If the production of a firm is given by P = 4LK – L² + K², L > 0, K > 0, Prove that L  + K 

Solution:
P = 4LK – L² + K²
P(K, L) = 4LK – L² + K²
P(tK, tL) = 4(tL) (tK) – t²L² + t²K²
= t²(4LK – L² + K²)
= t²P
∴ P is a homogeneous function in L and K of degree 2.
∴ By Euler’s theorem, 

Question 3.
If the production function is z = 3x² – 4xy + 3y² where x is the labour and y is the capital, find the marginal productivities of x and y when x = 1, y = 2.
Solution:
Marginal productivity of labour,

Marginal productivity of labour when x = 1, y = 2 is

= 6 – 4
= 2
Marginal productivity of capital,

= -4 + 12
= 8

Question 4.
For the production function P = 3(L)^0.4 (K)^0.6, find the marginal productivities of labour (L) and capital (K) when L = 10 and K = 6. [use: (0.6)^0.6 = 0.736, (1.67)^0.4 = 1.2267]
Solution:
Given that P = 3(L)^0.4 (K)^0.6…….(1)
Differentiating partially with respect to L we get,

When L = 10, k = 6, 

= 1.2(0.6)^0.6
= 1.2(0.736)
i.e., the marginal productivity of labour = 0.8832
Again differentiating partially with respect to ‘k’ we get,
Marginal productivity of labour when L = 10, K = 6 is

Marginal productivity of capital when k = 10, k = 6

= 1.8(1.66666)^0.4
= 1.8(1.67)^0.4
= 1.8 × 1.2267
= 2.2081

Question 5.
The demand for a quantity A is  Find the partial elasticities   and  when p₁ = p₂ = 2.
Solution:

When p₁ = p₂ = 2,

When p₁ = p₂ = 2,

Question 6.
The demand for a commodity A is

 Find the partial elasticities  when p₁ = 2, p₂ = 1.
Solution:

When p₁ = 2, p₂ = 1,

When p₁ = 2, p₂ = 1,