# Suppose r is less than g

Earlier today, I wrote a post about Justin Wolfers' incorrect understanding of Piketty. I recommend you read that post before this one. The point I am making in it, which I have made before, is that Piketty's argument that the wealth-to-income ratio will increase does not rely upon the famous r > g (the rate of return on capital is greater than growth).

I keep pointing out in these posts that it is solely a function of the savings rate divided by the growth rate (s/g) and Piketty's forecast that growth will slow. To drive this point home though, let's actually create a hypothetical world where r is less than g (r < g) to show that in fact the wealth-to-income ratio can increase in this world and can lead to greater income inequality.

Suppose that the savings rate is 10 percent (meaning 10 percent of the national income is added to the capital stock each year). Suppose that the growth rate is 4 percent (meaning national income grows 4 percent each year). Suppose that right now the wealth-to-income ratio is 2.5 (which is 10 divided by 4). Suppose finally that the rate of return on capital is 1 percent. So to sum up: s=10, g=4, K/Y=2.5, r=1.

Now watch this. Suppose growth goes from 4 percent to 2 percent, but everything else stays the same. What happens?

Firstly, note that 2% growth is still higher than the 1% rate of return on capital (1 < 2, r < g). The wealth-to-income ratio (K/Y) will eventually move from 2.5 to 5 (10/4 became 10/2). Since the rate of return is sitll at 1 percent, capital's share of the national income will double from 2.5 percent to 5 percent. If capital is unevenly distributed (which it is), then a doubling of capital's share like this will increase income inequality.

This is what I mean when I say Piketty's Capital Share Effect argument has nothing to do with r > g. I've just made it happen above when r < g. The r > g stuff only pertains to Piketty's Capital Concentration Argument.

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