Is an increasing capital share under capitalism inevitable?

•An increasing capital share of income is not inevitable under r > g.•This inevitability argument of Piketty and Milanovic has been very influential.•A specific counter-example, an intuition, and a logical argument are provided.•A necessary and sufficient condition for the capital share to increase is given.

Piketty's influential book Capital in the Twenty-First Century and its prominent review by Milanovic in the Journal of Economic Literature both assert the inevitability of an increasing share of capital in total income, given a higher rate of return to capital than the rate of growth in income. This paper shows by a specific example, a logical argument and its intuition that the alleged inevitability is not valid. Even just for capital to grow faster than income, we need an additional requirement that saving of non-capital income is larger than consumption of capital income. Even if this is satisfied, the capital share may not increase as the rate of return may fall and non-capital incomes may increase with capital accumulation.