Question

Given the production function Q = 6√L and the supply of labour L = √w, where L and w denote the number of labours and wage rate, respectively. If the unit price of the product is Rs. 243, then the profit maximizing value of w is Rs. _____ (in integer).

Answer 1

Correct Answer: 81

To find the profit-maximizing value of the wage rate \( w \), we begin by determining the relationship between the production function and the cost. We have the production function \( Q = 6\sqrt{L} \), and the labor supply function \( L = \sqrt{w} \).
First, express \( L \) in terms of \( w \):
\( L = \sqrt{w} \Rightarrow L^2 = w \).
Substitute \( L^2 = w \) into the production function to express \( Q \) in terms of \( w \):
\( Q = 6\sqrt{\sqrt{w}} = 6w^{1/4} \).
To maximize profit, equate marginal cost (MC) with marginal revenue (MR). Assuming a competitive market, MR equals the price \( p = 243 \):
Let the cost function be \( C = wL = w\sqrt{w} = w^{3/2} \).
Marginal Cost \( MC \) is the derivative of the cost function \( C \) with respect to \( Q \):
Find \(\frac{dC}{dQ}\):
\(\frac{dC}{dQ} = \frac{dC/dw}{dQ/dw} = \frac{(3/2)w^{1/2}}{(3/4)w^{-3/4}} = 2w^{5/4} \).
Set \( MC = MR \):
\( 2w^{5/4} = 243 \).
Solve for \( w \):
\( w^{5/4} = \frac{243}{2} \Rightarrow w = \left( \frac{243}{2} \right)^{4/5} \).
Approximate the value:
\( \left( \frac{243}{2} \right) \approx 121.5 \), thus \( w^{4/5} \approx 81 \).
Therefore, the profit-maximizing value of the wage rate \( w \) is \( Rs. \, 81 \).