The slope of the production function is the marginal product of capital MPKMPK. It represents how much extra output a worker produces when given an extra unit of capital:

MPK=f(k+1)f(k)MPK = f(k+1) - f(k)

We can examine how MPKMPK changes with kk:

kk

1.00
0123

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 α\alpha on the production function:

α\alpha

0.30
01

For a small α\alpha, 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 α=0\alpha = 0, output is independant of the amount of capital.

For a large α\alpha, 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 α=1\alpha = 1, output is a linear function of the amount of capital.


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