Question

For the production function \( Q = F(K, L) = \sqrt{KL} \) with \( P_K = 4 \) and \( P_L = 2 \), find the values of \( K \) and \( L \) that will minimize the cost of producing 2 units of output.

• A. \( K = 2\sqrt{3}; L = 3\sqrt{2} \)
• B. \( K = 2\sqrt{2}; L = \sqrt{2} \)
• C. \( K = \sqrt{2}; L = 2\sqrt{2} \)
• D. \( K = 2; L = 2 \)

Hint:

When minimizing cost functions in production, use the Lagrangian multiplier method to incorporate constraints like output level.

Answer 1

The Correct Option is C

The objective is to minimize the cost function \( C = P_K K + P_L L \), subject to the constraint \( Q = \sqrt{KL} = 2 \). From the constraint, we can square both sides to get \( KL = 4 \). Substituting this into the cost function, we get: \[ C = 4K + 2L. \] Now, substitute \( L = \frac{4}{K} \) into the cost function to get: \[ C = 4K + 2\left(\frac{4}{K}\right) = 4K + \frac{8}{K}. \] Taking the derivative of \( C \) with respect to \( K \) and setting it to zero to minimize the cost, we find: \[ \frac{dC}{dK} = 4 - \frac{8}{K^2} = 0. \] Solving for \( K \), we get \( K = \sqrt{2} \). Substituting this into the constraint \( KL = 4 \), we find \( L = 2\sqrt{2} \). Final Answer: \boxed{K = \sqrt{2}; L = 2\sqrt{2}}