Anecdotes from "Why Richistan? Why Now?"
Richistan is a book by Robert Frank, a WSJ writer, covering ever increasing and separating super rich in the US. His article in the WSJ gives a brief introduction to the book and includes some interesting data:

"They were wealthier than most nations, with the top 1% controlling $17 trillion in wealth. And they were increasingly building a self-contained world, with its own health-care system (concierge doctors), travel system (private jets, destination clubs) and language. (”Who’s your household manager?”) They had created their own breakaway republic — one I called Richistan."
Gini Coefficient and the World state of income discrepancy
From Wikipedia: "The Gini coefficient is a measure of statistical dispersion most prominently used as a measure of inequality of income distribution or inequality of wealth distribution. It is defined as a ratio with values between 0 and 1: A low Gini coefficient indicates more equal income or wealth distribution, while a high Gini coefficient indicates more unequal distribution."
Gini coefficient around the world (source, original data):

As you can see, the highest Gini coefficients (areas of most wealth discrepancy) lies in Africa and South America. Of the developed nations, US is higher than all of Europe and Japan/Australia. China also has high wealth inequality, somewhat surprising for the socialist nature of the country. Here is the historical change of Gini coefficients after WWII (source):

While it's hard to see, you can see the wealth inequality slowly growing in the US and UK, and a sudden jump for both China and India starting around 1980, 1990 respectively. Below is the income share of the top 0.1% of the population (source):

Red is the US, Green is Canada, Blue is the UK, Purple is France, and Black is Japan. You can clearly see a recent trend of income inequality after a relatively stable period after WWII.
Wealth disparity versus GDP per capita for countries with more than 30 million people (PPP adjusted) (source) (I'm translating the notable countries):

The y-axis is the ratio of wealth of the top 20% versus the bottom 20% (So in the US, the top 20% on average makes about 7 times as much as the bottom 20%).
The countries with the most wealth discrepancy are the mid income countries in Latin America while the wealthiest and poorest countries have less discrepancy. This phenomenon is well documented by the economist Kuznets and the Kuznets curve.
However, as seen by the time based plots above, the wealth discrepancy of some of the wealthiest countries are also increasing too. Japanese economist Toshiaki Tachibanaki has tackled this issue and has some hypothesis beyond the Kuznets curve.
Kuznets curve

It should be noted that Kuznets's work on inequality was done in the 50s and 60s, an economic world much different to ours.
From Wikipedia: "Kuznets curve is the graphical representation of Simon Kuznets's theory ('Kuznets hypothesis') that economic inequality increases over time while a country is developing, then after a critical average income is attained, begins to decrease."
Kuznets's hypothesis on the two phases of the curve (and I'm not a fan of redistributing this kind of information from Wikipedia, but I can't find any other source that seems credible either):
- When a country moves from a primarily agricultural society to an industrial society, inequality increases as the variation of jobs increases. (This can explain what is happening in China and India currently, but not US or UK)
- As a country industrializes, mass education takes hold and closes the wealth gap amongst the populace. Also new social policies redistribute the wealth and lessens the inequality in the countries. (This does not seem to have happened in many Latin America countries, which could explain their large wealth discrepancy)
Increasing inequality in the wealthiest countries
Going beyond the Kuznets curve, Dr. Tachibana shows the increasing inequality in developed nations and tries to explain the underlying mechanics. Most of what I learned here came from this paper published in 1998 (in Japanese).
- The economic resurgence of US in the late 80s and early 90s was stimulated by relaxing regulations and privatization, corporate restructuring for efficiency, and reducing taxes and social welfare (Reaganomics). Similar trends were seen in the UK while Japan and Germany (along with Scandenavia) kept a more socialist economy.
- This in turn increased inequality in several ways: drive for efficiency shifted production to foreign countries which in turn decreased the number of blue collar manufacturing jobs, ultimately decreasing their wage; with technology becoming a primary produce in the US, demand and wage for highly educated people increased while relaxed labor laws increased the number of low wage part-time and temp workers. (Similar trends have happened in Japan in the last 8 years initiated by Prime Minister Koizumi, and the number of "irregular employees" have skyrocketed making inequality a large social issue in Japan)
- There is an inherent economic trade off between efficiency (low taxes) and equality (social welfare). By pursuing one, you naturally give up the other.
Conclusion
This, of course is a very high level overview on inequality, and there are plenty of country-specific reasons for wealth discrepancy. Nevertheless, there are clearly marked trends towards inequality in many of the relevant countries and plausible explanations for them. Now the question is, what's different about these new generation of rich people? Is this going to continue or is there something that will change course?
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