1970s anxieties about inflation [are being substituted by] today's concerns about the emergence of the plutocratic rich and their impact on economy and society. [Economist Thomas] Piketty is in no doubt, as he indicates in an interview in today's Observer New Review, that the current level of rising wealth inequality, set to grow still further, now imperils the very future of capitalism. He has proved it.
via www.theguardian.com
The situation that French economist Thomas Piketty describes in his best-seller Capital in the Twenty-First Century might be outlined more or less like this:
Wealth inequality rises as
1) return on capital rises faster than both workers' wages and general economic growth (see chart #1, click to enlarge, and for more on the related issue of decreasing income mobility in the U.S. see "Inequality Is Not the Problem," Jeff Madrick, NYR Blog, 2014),
2) super-high-income workers (e.g., CEOs) reward each other with mega-salaries to "keep up with the other rich" (see chart #2 from "We're More Unequal Than You Think," The New York Review of Books, 2012),
3) inherited wealth and corporate gains aren't greatly taxed (compared to the early post-WWII era especially),
4) tax-reduction/-avoidance schemes abound especially for the rich who can afford the experts to manage their money globally, and
5) the cultural and societal insularity of the wealthy, their disconnectedness from the vast majority of those who are not exceedingly wealthy, combines with their money-driven power (e.g., campaign contributions and armies of lobbyists) to keep the system in place. Such power puts me in mind of the old "golden rule"—he who has the gold makes the rules.

Importantly, due largely to #4 above, the middle class ends up with a disproportionate share of the tax burden to keep the social safety net, defense, services (sanitation, policing, fire fighting), education, and transportation infrastructure in place, even though the services, education, and infrastructure benefit the mega-wealthy, too, directly or indirectly.

The result: it becomes more important who you're born to than what job you have or even what business you create. In the situation Piketty describes, not even typical entrepreneurs can ever expect to gain close to the kind of wealth that the rentier class will enjoy, will see grow (faster than will grow wages or the general economy), and will pass on to offspring...largely untaxed.
It might be noted, too, that with the middle-class's retirement funds so tired up in stocks due to the financial innovation of the 401k, the super-rich can use political rhetoric that suggests they and simple shareholders are on the same team, which they are not.
Another key point of Piketty's book is that the mid-20th-century period of reduced wealth inequality and reduced income inequality is the exception, not the rule, because, as a friend of mine summarized, the disruptions of two world wars and the Great Depressions hugely reduced the capital controlled by the upper classes both through direct destruction and by making very high taxation politically possible. (See that plunge in the rate of return on chart #1 above, 1913–1950.)
Stating that capitalism isn't working is not the same as stating that capitalism doesn't work. Piketty seems to promote a mixed economy. As I think is better understood by the voting public in much of Europe (perhaps especially Germany and the Scandinavian countries) than in the U.S., there is no strict, binary choice between socialism and capitalism. There are myriad gradations in between the two. Capitalism's tendency toward a final winner-take-all result can be curtailed and social unrest kept at bay by policies such as more progressive taxation, global wealth taxes, etc. However, these tactics are not practicable now. Outrage among the voting majority simply isn't great enough to precipitate change. And all of this is hard to tweet, so good luck getting anyone under the age of 30 to give a damn.
Piketty's book stems from many years of work and research. It will take some time for challenges to emerge robustly, but some are already published. Examples include these considerations via Forbes.com. (Scroll down on the linked-to page for additional Forbes posts by Tom Worstall and Scott Winship about possible problems with some of Piketty's ideas. For instance, Worstall suggests that taxation on consumption is a better approach than Piketty's suggestion to tax capital.)
Necessarily more thorough than the Guardian article about Piketty's book is Paul Krugman's review of Capital for the New York Review of Books.
The Scandinavian Model
Slightly off-topic but not entirely unrelated: As others have pointed out, one of the factors driving the Scottish independence referendum (September 18, 2014), which I think will pass by a very slim margin, is a laudable consensus among the Scots that they do not want the kind of radical wealth inequality seen in England and the U.S.A. (See, "Scottish nationalists look to Nordic model for independence," Financial Times.) However, whether independence is the best course for lessening or protecting again wealth inequality is arguable. (I side with the Better Together campaign.) The Conservatives who support continued union with Scotland may go down in history as the party that led the Government that lost the 307-year-old union between Scotland and the rest of Britain and [Northern] Ireland, despite their strenuous rhetorical efforts to preserve it. We'll find out in less that 5 months' time.
The Koch Brothers
Also, it is interesting to look in light of Piketty's book at the efforts of the billionaire Koch brothers, who are deeply tied to the fossil fuel industries, to have surcharges put upon users of solar power. Piketty notes that the very wealthy will engage in various efforts to maintain the status quo. As Piketty writes, "The experience of France in the Belle Époque proves, if proof were needed, that no hypocrisy is too great when economic and financial elites are obliged to defend their interest."
The assessment of two-time Pulitzer Prize-winning editorial cartoonist and columnist David Horsey in his regular "Top of the Ticket" column in the Los Angeles Times is that
the Koch brothers have a new ploy to protect the traditional energy business that helped make them the planet’s fifth- and sixth-richest humans. They are funding a campaign to shackle solar energy consumers who have escaped the grip of big electric utilities.
Of all the pro-business, anti-government causes they have funded with their billions, this may be the most cynical and self-serving.
(Click any image to enlarge it.)
It seems to me that the mega-wealthy, like the Koch brothers, will happily and doggedly seek to further game the system and entrench their wealth through tax havens, low tax rates, falsehoods widely disseminated by their media operations—complete with crocodile tears for the middle class—and the political and societal influence that they buy through campaign contributions, armadas of lobbyists, and, frankly, their "charitable" giving, too. (To give a large donation to an arts institution or medical facility not only offers tax advantages but in a sense puts those entities' workers in your pocket; they dare not speak out or too obviously work to reform the status quo lest they lose a big-money donor, patron, lord.)
Those with great wealth will work both to rig the system and to keep the masses' outrage at bay by fueling the narrative of the government as being the only true enemy, by fueling misinformation against whatever hurts their interests, including—in the case of the Koch brothers especially—climate change, and by fueling media coverage of and political focus on non-economic issues.
In the U.S., with the (probably temporary) decrease in the public intensity of religious conservatives' concerns about social issues and, arguably, social conservatives dwindling numbers (for now), the economic right-wing (and the self-described libertarian wing) of the U.S. will increasingly attack government in all its forms and experiment with new distractions. Old distractions like gay marriage or the war on drugs are losing their appeal. But new ones will be found. You can count on it.
I suspect that the super-massively rich, the top 0.01%, like the Koch brothers, convince themselves that they are patriots. But I believe that the Koch brothers' efforts subvert the general welfare, and in that regard they are really more like oligarchic monarchists, insiders in the lordly court of plutocracy, than they are like true champions of liberty and equality before the law.
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