Question

A firm has a production function that is homogenous of degree one given by $𝑄 = 2\sqrt{𝐿𝐾}$, where 𝑄 is quantity, 𝐿 is labour and 𝐾 is capital. The unit price of 𝐿 is Rs. 4 and the unit price of 𝐾 is Rs. 16. Assuming that there is zero fixed cost, the total cost (long run) of producing 10 units of 𝑄 is Rs. (in integer).

Answer 1

Correct Answer: 80

Cost Minimization for the Production Function

Step 1: Given Production Function

The production function is: 

\[ Q = 2\sqrt{L K} \]

For \( Q = 10 \), we set:

\[ 10 = 2\sqrt{L K} \Rightarrow 5 = \sqrt{L K} \]

Squaring both sides:

\[ L K = 25 \]

Step 2: Expressing Cost Function

Let \( L = x \) and \( K = \frac{25}{x} \), then the cost function is:

\[ C = 4L + 16K = 4x + 16 \times \frac{25}{x} \]

Simplifying:

\[ C = 4x + \frac{400}{x} \]

Step 3: Minimizing the Cost Function

Differentiate \( C \) with respect to \( x \) and set it to zero:

\[ \frac{dC}{dx} = 4 - \frac{400}{x^2} = 0 \]

Solving for \( x \):

\[ 4 = \frac{400}{x^2} \Rightarrow x^2 = 100 \Rightarrow x = 10 \]

Step 4: Calculating the Minimum Cost

Substituting \( x = 10 \) into the cost function:

\[ C = 4(10) + 16 \times \frac{25}{10} \]

\[ C = 40 + 40 = 80 \]

Final Answer:

The minimum cost is 80.