Accounting for the sources of growth
Accounting for growth
From the prior section we know that given a production function of $Y = K^{\alpha}(AL)^{1-\alpha}$ the growth rate of GDP per capita can be written as
\[g_y = \alpha (g_K - g_A - g_L) + g_A.\]This divides up growth in GDP per capita into two parts. $\alpha(g_K - g_A - g_L)$ is the contribution of net capital growth. The “net” refers to the fact that we care about how fast capital grows relative to the growth rate of productivity and labor. Why? Because capital is itself something that is produced, so if it is growing quickly that may only indicate that the economy is growing for other reasons. Capital contributes to growth in living standards to the extent that it grows faster than the pace of productivity and labor; in that case we are actually accumulating more capital relative to our ability to use it.
The second part is just $g_A$ itself, productivity growth. This contributes directly because it enhances our production given the stock of capital and labor available.
The question here is how big these two contributions to growth are in the real world. Some of this is straightforward to find. We can look up (or calculate) the growth rate of GDP per capita, $g_y$, the growth rate of capital, $g_K$, and the growth rate of labor, $g_L$. Given what we learned about cost shares, we can infer a value of $\alpha$ of about 0.3.
The one thing that isn’t immediately clear here is what the growth rate of productivity will be, $g_A$. Productivity isn’t something we can measure directly at all. But since we have all this other information, we can back out what $g_A$ must be to ensure this equation holds.
First, re-arrange the above equation to be
\[g_y = \alpha (g_K - g_L) + (1-\alpha) g_A.\]Second, re-arrange that again to put $g_A$ by itself
\[g_A = \frac{1}{1-\alpha} g_y - \frac{\alpha}{1-\alpha} (g_K - g_L).\]We have all the information for the right-hand side, and it is easy to calculate the values for $g_A$ in any given year. The growth rate in productivity is just the growth rate of GDP per capita minus whatever is accounted for by the growth rate of capital relative to labor. Productivity growth is just the residual explanation for growth beyond what we can explain using $g_K$ and $g_L$.
The data to calculate these values for the USA, Japan, and South Korea is available at This link.
Once you’ve got $g_A$, you can calculate size of the two contributions to growth: $g_A$ (productivity) and $\alpha(g_K - g_A - g_L)$ (capital accumulation).
I’ve done this for a couple of countries already. In each case, the growth rates I’ve calculated are 10-year annualized rates, to avoid wild swings associated with business cycles. Here’s Japan:
The contribution to growth of productivity (which remember, is everything we cannot explain with capital and labor, basically) is huge at the outset, and remains relatively large throughout. The contribution from capital accumulation (relative to productivity and labor) looks a lot smaller. Note that there is a period where Japan benefits a lot from capital accumulation, but by the end this has run out completely.
If you look at the US, the story is slightly different.
Here, productivity growth is never as high as in Japan. But also note that the capital accumulation term is never that large. There is no burst like in Japan. For a country like the US that has experienced stable growth, the story is that the $\alpha(g_K - g_A - g_L)$ does not contribute a whole lot to the party. That won’t mean that capital is unimportant, as we will see going forward. It’s just that, on net, capital growth over and above productivity and labor growth is not a big contributor.
For Japan, there is a period in which capital grows very fast (faster than $g_A$ and $g_L$), which boosts growth. But note that this is temporary. We’ll also study why that burst cannot last forever, and how it came to be that Japan got this boost. The short version is that when you start with very little capital, as after World War II, then there is the potential for a spike in growth as you catch up.