Why Have These Economies Converged?
By and large economies which have converged are those which belong to OECD: the Organization for Economic Cooperation and Development that was started back in first post-WWII years in the days of Marshall Plan as a club of countries which received (or gave) Marshall Plan aid to help reconstruct and rebuild after World War II. Countries which received Marshall Plan aid adopted a common set of economic policies: large private sectors freed of government regulation of prices, investment with its direction determined by profit-seeking businesses, large social insurance systems to redistribute income and governments committed to avoiding mass unemployment.
The original OECD members all wound up with mixed economies. Markets direct the flow of resources whereas governments stabilize the economy, provide social-insurance safety nets as well as encourage enterprise and entrepreneurship. They arrived at this institutional setup largely because of good luck partially because of the Cold War and partially as a result of post-World War II institutional reforms.
This post-World War II institutional configuration was basically the price countries had to pay for receiving Marshall Plan aid. U.S. executive was unwilling to send much aid to countries which it thought were likely to involve in destructive economic policies, largely because it did not believe that it could win funding from Republican-dominated congress for a Marshall Plan that didn't impose such strict conditionality upon recipients. By contrast nations which were relatively rich after World War II however that didn't adopt OECD-style institutional arrangements-such as Venezuela and Argentina --have lost relative ground.
As OECD economies became richer they completed their demographic transitions: population growth rates fell. The policy lay emphasis on entrepreneurship and enterprise boosted national investment rates so OECD economies all had healthy investment rates as well. These factors boosted their steady-state capital-output ratios. And diffusion of technology from U.S. did rest of the job in bringing OECD standards of economic productivity close to U.S. level.