To determine how many additional labor hours (ΔL) are required for each unit of capital (ΔK) that the firm reduces, we need to use the concept of the marginal rate of technical substitution (MRTS), which measures the rate at which one input can be substituted for another while keeping output constant.

The MRTS is calculated as the ratio of the marginal products of capital and labor, given by:

$\text{MRTS} = -\frac{MP_K}{MP_L}$

where:

• $MP_K$ is the marginal product of capital,
• $MP_L$ is the marginal product of labor.

Since we are not provided with the specific production function, let’s proceed with the assumption that the MRTS is given by the ratio of capital to labor, which is a common assumption for Cobb-Douglas production functions. In that case:

$\text{MRTS} = \frac{K}{L}$

Solution:

1. Substitute the values of $K = 50$ and $L = 100$ into the MRTS formula:

   $\text{MRTS} = \frac{50}{100} = 0.5$

2. Interpret the MRTS: An MRTS of 0.5 means that for each unit of capital (machine hour) the firm cuts, it needs to add approximately 0.5 units of labor (labor hours) to maintain the same output level.

Answer:

The firm will need to add approximately 0.5 labor hours for each machine hour it cuts to stay on the same isoquant.

Would you like more detailed steps or explanations on this? Here are some questions that build on this topic:

1. How does the MRTS change if the ratio of capital to labor changes?
2. What happens if the production function is not Cobb-Douglas?
3. How would a higher level of capital impact the marginal rate of technical substitution?
4. Can we use MRTS to find the optimal combination of labor and capital inputs for cost minimization?
5. How would increasing returns to scale affect this substitution between labor and capital?

Tip: When working with isoquants, remember that MRTS helps in understanding the trade-offs between different input combinations for a given output level.