Question

Suppose a firm has the following production function:
\[ f(x_1, x_2, x_3, x_4) = \min{x_1, x_2} + \min{x_3, x_4} \] The per unit cost of input \( x_i \) is given by \( w_i \), where \( i = 1, 2, 3, 4 \). Suppose that \( w_1 = 1, w_2 = 5, w_3 = 3, \) and \( w_4 = 6 \). If the firm is minimizing cost, which of the following input choices by the firm can be observed?

• A. \( x_1>0, x_2 = 0, x_3>0, x_4 = 0 \)
• B. \( x_1>0, x_2>0, x_3 = 0, x_4 = 0 \)
• C. \( x_1>0, x_2 = 0, x_3 = 0, x_4>0 \)
• D. \( x_1 = 0, x_2 = 0, x_3>0, x_4>0 \)

Hint:

To minimize costs, choose the inputs that yield the minimum cost in production functions involving minimums.

Answer 1

The Correct Option is B

Step 1: Understanding the production function.
The production function involves two minimum functions. The firm needs to minimize its cost by choosing input quantities that optimize this function. Since \( w_1 = 1 \), \( w_2 = 5 \), \( w_3 = 3 \), and \( w_4 = 6 \), the firm will choose inputs where the cost is minimized. Step 2: Cost minimization analysis.
The firm should use \( x_1 \) and \( x_2 \) together (since their combined cost is lower than using \( x_3 \) and \( x_4 \)). For this, \( x_1>0 \) and \( x_2>0 \) should be chosen. However, for the second minimum, \( x_3 = 0 \) and \( x_4 = 0 \), as using both \( x_3 \) and \( x_4 \) would be too costly. Step 3: Conclusion.
The correct answer is (B), as it minimizes the cost by using inputs \( x_1 \) and \( x_2 \) while setting \( x_3 \) and \( x_4 \) to zero.