In a Cobb-Douglas production function, what happens to the marginal product of labor if the quantity of capital increases?

In a Cobb-Douglas production function, which commonly takes the form ( Y = A K^\alpha L^\beta ), where ( Y ) represents output, ( K ) represents capital, ( L ) represents labor, and ( A ), ( \alpha ), and ( \beta ) are constants, the relationship between the inputs and output is particularly noteworthy.

When the quantity of capital increases, the marginal product of labor experiences an increase due to the nature of diminishing marginal returns, particularly in a scenario where both capital and labor are used in production. Initially, as capital increases, labor can be used more effectively. For example, with more machines (capital), workers (labor) can produce more output because they have more tools at their disposal, increasing their productivity.

This enhancement in the productivity of labor occurs because each worker can now operate more capital effectively, leading to a higher marginal product of labor. Consequently, the additional output produced by an extra unit of labor rises as the quantity of capital expands.

In summary, in a Cobb-Douglas production function, an increase in the amount of capital leads to an increase in the marginal product of labor, illustrating the synergetic effect of capital and labor