To blame "income inequality" and all the rest of the world's ills on "unchecked capitalism" that hasn't existed in most people's lifetimes just plain fails on every front, given the ever increasing interference, meddling and "regulation" of industry and economy by the government, especially as "income inequality" has largely only worsened ever since the government started meddling more and more with the economy, introducing more and more regulation, giving unions patently anti-liberty powers they should never have to fulfill a purpose that shouldn't even be necessary, and in return, businesses use the regulation and legal process to bring about ever more and more laws that are patently anti-capitalist in order to protect them from competition and to game the economy.
The failure of this analysis becomes only more evident when you consider the fact that when you compare America - arguably the closest thing to "unrestricted capitalism" you'll find among First World nations today, regardless of how far it is from being completely true - to the rest of the world, our "income inequality" gap is as narrow - or narrower - here than almost anywhere else on the planet, and certainly among nations with a population exceeding 20 million.
What, does Piketty think that there was no "income inequality" anywhere else in the world, prior to the existence of Capitalism as we know it today? That it is somehow worse today than it was in the 18th century? That there's no "income inequality" in nations with heavily "managed" (i.e., meddled-with) economies? That all government will always work towards the betterment of "the people" by default, more so than private industry will?
Sorry, but Piketty's analysis is just plain flawed on too many levels to be worth taking seriously, as it is premised entirely on the falsehood that "unrestricted capitalism" has ever even existed recently enough in this country - or anywhere else in the world - recently enough to be a cause of income inequality, when it is readily shown that the ever-increasing regulations imposed by government have led to ever increasing meddling by that government in the operation of business, paving the way for other businesses to then abuse the political process to create ever more anti-capitalist laws and regulations, and that a great deal of that very same regulation, taxation, and interference is very often itself a factor in income inequality.
In a truly "unrestricted, free-market capitalist" economy, under a strong, just and ethical system of reasonable laws based on liberty and freedom that are equally enforced regardless of wealth, influence or social station, the only restriction on wealth and income (aside, perhaps, from physical/mental health) is one's own industry and willingness to work, learn, and create industry where none exists.
"The policy of the American government is to leave their citizens free, neither restraining nor aiding them in their pursuits."
--Thomas Jefferson to M. L'Hommande, 1787
"[...] a wise and frugal Government, which shall restrain men from injuring one another, shall leave them otherwise free to regulate their own pursuits of industry and improvement, and shall not take from the mouth of labor the bread it has earned. This is the sum of good government..."
--Thomas Jefferson, First Inaugural Address, Washington, D.C. March 4, 1801
"[Ours is a] policy of not embarking the public in enterprises better managed by individuals, and which might occupy as much of our time as those political duties for which the public functionaries are particularly instituted."
--Thomas Jefferson to W. C. C. Claiborne, 1808
"Taxes on consumption, like those on capital or income, to be just, must be uniform."
--Thomas Jefferson to Samuel Smith, 1823
"And I sincerely believe, with you, that banking establishments are more dangerous than standing armies; and that the principle of spending money to be paid by posterity, under the name of funding, is but swindling futurity on a large scale."
--Thomas Jefferson to John Taylor, 1816