When the Industrial Revolution broke out of the Malthusian trap, economies began to accumulate capital goods. In order to accumulate capital, you have to save. This means that you cannot consume all of your output.
[1] S = 5%K
Let us define the savings rate "s" to be the ratio of savings to output, s=S/Y.
and let us write equation [1] with "s", we have:
[2] s = S/Y = 5%K/Y
Equation [2] says, that to maintain output constant, we need a savings rate "s" that equals the rate of depreciation (5%) times the capital/output ratio K/Y.
Now, if we want to have high labor productivity, we need a high ratio of capital to output, and therefore we need a high savings rate "s". Thus, we would expect to find a relationship between countries with high savings rates and countries with high productivity, and this is indeed what Brad DeLong found when he indicated that countries with high productivity tend to have saving rates over 20%.
Suppose that we want the capital stock to grow at a rate of 2%. In that case, we need
[3] S = 5%K + 2%K = 7%K
Or, in terms of s and K/Y,
[4] s = 5%K/Y + 2%K/Y = 7%K/Y
Let "d" be depreciation, "k" be capital-output ratio K/Y, and "x" be growth rate of capital DK/K, then we can write
[5] s = dk + xk
To see how the saving rate affects the growth rate of capital, we can solve equation [5] for x, the growth rate of capital.
[6] x = DK/K = s/k - d
If the labor force is growing at a rate n%, then the capital/labor ratio will grow at the rate of x%-n%. For example, if:
Then we have
[7] x - n = s/k - d - n = 25%/2.5 - 5% -1% = 4%
which says that capital/labor ratio grows at 4 percent per year. In a moment, when I discuss balanced growth, I will argue that this is not a reasonable long-term growth rate for the capital/labor ratio.