While teaching a guided-reading course on the economics of inequality in the PEP program (philosophy, econ, poli-sci) at the Hebrew University, I drafted this flow chart to help students make sense of the big picture made by the many sub-arguments that Piketty is making in Capital in the 21st Century. I felt that it worked as a helpful digestive enzyme. I taught it using this PDF slideshow that adds the pieces element-by-element, and I suggest economics professors to adopt it for their own teaching purposes.
To be sure, the book says much more than this chart suggests. There are many arguments supporting each of the arrows portrayed here, and there are numerous side discussions, as well as additional big arguments that are closely related. The focus here is primarily on the bare-bones argument that leads from the prediction that future growth will be slower than in the past to the conclusion that there are forces pushing the income distribution toward greater inequality throughout the course of the twenty-first century.
There are two main mechanisms at work here. According to the first, on the left side of the chart, given the prediction that the net saving rate is not expected to decrease as much as growth, the capital-income ratio is expected to rise. Since, as Piketty argues, the income elasticity of substitution is greater than one, this would lead to an increase in the capital share of income. Next, because income from capital is the more unequal component of total income, as historically income from labor is consistently more equally distributed than it, this composition effect pushes the total income distribution toward greater inequality. While the book is quite readable (though, in my opinion, it could have benefited from a gentle touch of a team of hard working copy-editors and a translator that does not stop half-way between French and English), I would refer readers to Piketty and Zucman (QJE 2014), where this leg of the argument is more succinctly described.
The second mechanism, on the right, is the famous r-g. As Piketty has it, the rate of return to capital is not expected to decrease as much as the total rate of growth. As a result, the r-g gap widdens. Since this gap affects the long run distribution of wealth in the population, its increase is a force pushing the wealth distribution toward a more unequal steady-state. Hence, an increase in the dispersion of income from capital causes an increase in inequality in the combined income of all sources put together. This mechanism is discussed in length in the book (see Ch. 10). However, its mathematical underpinnings are never shown there; instead they could be found in Piketty and Zucman (Handbook of Economic Inequality, 2015), Section 15.5.4.
I hope that readers would find this slideshow a useful teaching tool. I welcome feedback and suggestions that would help me improving it.