A productive line of recent research has shown there's much less social mobility in the United States than we like to think. That same research has tended to find much more mobility in the Nordic countries where, for example, a father's earnings are much less statistically predictive of a son's earnings. But UC Davis economic historian Gregory Clark asks what happens on a longer time frame and finds that even in Sweden there's not that much mobility.
The research is a bit complicated, but the chart above illustrates a key finding. What Clark is doing is an analysis of surnames. Sweden stopped minting new noble houses in the seventeenth century, so we can be sure that anyone with one of those noble surnames in the 21st century is patrilineally descended from a high status man from hundreds of years ago. By contrast, low status Swedes tended to end up with patronymic surnames ending in "sson" illustrative of a lack of noble status or noteworthy occupational skill. So take the super-common Swedish surname Andersson and you're looking at people who are patrilineally descended from families that were low status hundreds of years ago. You can see that to this day people with aristocratic surnames are much likelier to be rich than people named Andersson. Clark repeats this kind of analysis for wealth, for presence in high status occupation groups like doctors and lawyers, and also compares lowly "sson" names to middling names. Over and over again you see the same thing. Despite the ups and downs of fortune, despite the possibility of marrying up and marrying down, and despite the fact that there's probably some false paternity down the line somewhere surnames are meaningfully predictive of 21st social status.
That's not to say that the earlier line of research about Swedish mobility is mistaken.
Precisely as other people had found, because Sweden has much less overall income inequality than the United States it really is true that your dad's earnings matter less and random fluctuations matter more in driving your own earnings. But Clark shows that this tends to wash out in the longer term. Presumably the issue is that privilege is a much broader phenomenon than income, and tax and redistribution policies don't succeed in eradicating its other elements.