When examining the cost-minimizing combination of labor and capital, which condition must hold for a firm to be producing a given level of output at the lowest possible cost?

• A. The marginal product per dollar spent on labor equals the marginal product per dollar spent on capital.
• B. The total cost of labor equals the total cost of capital.
• C. The average product of labor equals the average product of capital.
• D. The firm uses only labor and no capital.

Answer: (A)

Verified step by step guidance

1. Understand that the problem is about cost minimization in production, where a firm wants to produce a given output level at the lowest possible cost by choosing the right combination of labor and capital.
2. Recall the concept of the marginal product (MP) of an input, which is the additional output produced by using one more unit of that input, holding other inputs constant.
3. Recognize that the cost of each input depends on its price: the price of labor (wage rate, denoted as \(w\)) and the price of capital (rental rate, denoted as \(r\)).
4. The cost-minimizing condition requires that the firm equalizes the marginal product per dollar spent on each input. Mathematically, this is expressed as: \[\frac{MP_L}{w} = \frac{MP_K}{r}\] where \(MP_L\) is the marginal product of labor and \(MP_K\) is the marginal product of capital.
5. This condition ensures that the last dollar spent on labor produces the same additional output as the last dollar spent on capital, so the firm cannot reduce costs by reallocating spending between labor and capital.