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...inequality is likely to escalate. This says absolutely nothing about whether the standard of living of the poorest will improve or decline.I believe you are making the same mistake as the Bloomberg reviewer in separating inequality from the fate of the poorest.
To see the connection, you have to look deeper. That is, you must consider the causes of inequality that Piketty offers. You might argue that inequality in itself doesn't produce a negative outcome for the poorest, but that the causes of our current inequality produce those negative outcomes as a side effect. I would say that's semantics, but in any case, what Piketty lays out does support the notion of a rising inequality along with a comcomitant stagnation/slippage for the poorest/others.
So, let's say it the other way, if you prefer: some of the primary reasons (increased productivity, automation, etc.) for the rate of wealth-accumulation for the rich that we're seeing also produces negative outcomes for the poorest (unemployment, downward wage pressure, etc.). This has led us to (or, rather, is by definition) the growing inequality that we're seeing.
Finally, these arguments of the ilk that "inequality must be OK because the standard of living for the poor has risen even in the face of historical inequality" fall flat. It really is a false equivalence.
Firstly, such arguments imply that degree is irrelevant and that no degree of inequality would produce (or represent) a negative outcome for the poor. In fact, there is some point at which capture by one party must come at the expense of others to the extent that the latter parties slip backwards. Secondly--and to the first point--Piketty outlines that the degree of inequality was historically kept in check by various forces in other examples where the outcomes for the poor weren't detrimental.
In those cases, the threshold for negative outcomes was simply not crossed. However, this is certainly not proof that they cannot be.