Question

The weekly output (in units) of a factory depends on the amount of capital and labour it employs as follows: if it uses k units of capital (K) and l unit of labour (L) then its output is Q=L^0.5K^0.5 units. The cost to the firm of each unit of capital is $4, and the cost of each unit of labour is $1. Use the method of Lagrange multipliers to find the minimum weekly cost of producing a quantity of 200 units.

Answer 1

Given that: k units of capital and l units of labour

Q(k, l)=$\sqrt{k}\sqrt{l}$

The cost to the firm of each unit of capital is $4, and the cost of each unit of labour is $1.

Q(k, l)=$\sqrt{4}\sqrt{1}$

Q(k, l)=(2)(1)=2

$\sqrt{k}\sqrt{l}$ =2

P(k, l)=$\sqrt{k}\sqrt{l}$ -2...........................(say)

$\nabla$Q(k, l)=$\frac{1}{2\sqrt{k}},\frac{1}{2\sqrt{l}}$

$\nabla$P(k, l)=$\frac{1}{2\sqrt{k}},\frac{1}{2\sqrt{l}}$

By Lagrange multipliers;

$\nabla$Q=$\lambda\nabla$P

($\frac{1}{2\sqrt{k}},\frac{1}{2\sqrt{l}}$) =$\lambda$ ($\frac{1}{2\sqrt{k}},\frac{1}{2\sqrt{l}}$)

$\lambda$ =1, k=4 and l=1

Q(k, l)=$\sqrt{k}\sqrt{l}$ =2 (The minimum weekly cost of producing a quantity of 2 units)

The minimum weekly cost of producing 200 units will therefore be:

(200 units$\div$ 2 units)$\times 2\$$ =200$\$$ =200 dollars.