Marginal Product of Capital
The slope of the production function is the marginal product of capital . It represents how much extra output a worker produces when given an extra unit of capital:
We can examine how changes with :
As each worker gets its hands on more capital, the production function becomes flatter, indicating a diminishing marginal product of capital. This is because with a small amount of capital, an extra unit of capital for the average worker is very useful and produces a lot of additional output. With a lot of capital, workers given additional units of capital can only increase output slightly.
We now examine the effect of on the production function:
For a small , aggregate output depends mostly on the number of workers. Hence marginal increases in capital do not lead to large increases in output both in aggregate and at the per-worker level. In the extreme case of , output is independant of the amount of capital.
For a large , aggregate output depends mostly on the amount of capital. Hence marginal increases in capital lead to large increases in output both in aggregate and at the per-worker level. In the extreme case of , output is a linear function of the amount of capital.