The Aggregate Production Function
Assuming a fixed stock of labor and other resources construct a total product of capital curve (TPk) for the economy as a whole. It will rise at a decreasing rate because the marginal product of capital (MPk) is positive and decreasing as the capital stock (K) grows. The product of the capital stock is the GDP (Y).

The Savings Function
If we assume that the proportion of income saved (APS) is constant, the savings function will be similar to the total product of capital function. Thus the size of the capital stock (K) will determine both the level of GDP (Y) and the level of Savings (S).

Per Worker Basis
These graphs can be put on a per worker basis by dividing the capital stock (K), GDP (Y) and Savings (S) by the umber of workers (N).
The shapes of the curves will not change if we assume that there is constant returns to scale.

The Capital Requirements Curve
In order to maintain a constant capital-worker ratio (K/N), it is necessary to replace depreciation and to equip new workers with the same amount of capital per worker as the old workers. (This is called capital widening.) The amount of new capital needed to maintain a constant capital-labor ratio is shown by the red line. The slope of the red line is equal to the sum of the population growth rate (n) and the depreciation rate (d).

The Current Situation
At present all nations find themselves in a situation where the capital stock per worker is below the equilibrium level. The result is that gross savings exceeds the amount needed for depreciation and capital widening. The net savings or net investment can go towards increasing the capital-labor ratio. (That is called capital deepening.)

The Steady State
The process of capital deepening will continue every year until at some future time, all of gross savings goes to depreciation and capital widening. At that point, there is no more capital deepening and the system is in equilibrium. There will be no further growth in living standards. This is referred to as a “Steady State”.

Time Path of Income Per-worker
As the economy moves toward a steady state level of capital per worker, output per worker grows until it reaches its “Steady State” level. After that, it remains constant. There are no further improvements in living standards.

A Lower Population Growth Rate
If the population grows more slowly, less saving and investment will be required for capital widening, more will be available for capital deepening. As a result the “Steady State” level of capital per worker will be higher and the steady state level of output per worker will also be higher.

The Effect of a Fall in the Population Growth Rate
In the long run, the rate of growth of per worker income is zero, no matter what the rate of population growth. But at lower rates of population growth, the long run standard of living will be higher. A fall in the population growth rate after a nation is in a “Steady State” would allow living standards to begin growing again until it reached the new “steady State”.