If only the rich were allowed to reproduce, and the trend went on for decades, there would be ramifications.  Where would the labor come from; either directly or indirectly; to produce the goods and services the rich would want to/have to consume?  Who would pay the taxes for which the state could ultimately protect the interests of the rich?

You could never have a society where everybody is rich.  It's not possible.  Wealth would be divided.  The most saavy, the strongest, the smartest and those with special state protection would remain among the rich.  While everybody else in that society (or from adjacent societies to fill the gap) would be poor.

At it's best, a society could develop a sizable middle class.  This is what happened with the United States in the 20th. century.  But as the economy globalized, it was not possible to sustain a very common middle class when labor could be had in countries without labor laws or widgets could be produced by robots.  The 1950's United States was charactetrized with a middle class who enjoyed a good and increasing standard of living.  Most products consumed by Americans were made in America.  The wealth disparity (and wage disparity) between the richest and the poorest was not as wide as it is today.  Sure in modern America, we have more conveniences such as microwaves and fast food outlets.  But Americans have become dependent on these because most households now require two incomes (meaning two working parents) in order to make ends meet.  It's a kind of conundrum.

Now countries like Monaco, Kuwait and Dubai come to mind.  These are very small countries.  Supposedly the citizens of these countries are all rich.  (I've only read articles, not performed extensive research into this.)  The citizens of these countries may all be rich, but they are dependent on imported labor from poor countries such as the Philippines, India, Bangladesh and Sri Lanka.  These rich citizens need somebody to clean the mansions, prepare the food, take care of their children, etc.

Another thing to look at is the demographics of developed countries; particularly European countries and Japan, and to a lesser extent, the United States.  These countries have aging native populations.  With rising standards of living, the desire to have better standards of living, and the availability of birth control, these countries saw declines in the birth rates.  To make up for the gap for necessary labor, immigrants from poor countries aspiring for better lives come in.  Macroeconomically speaking, there needs to be some kind of working, tax paying base in order to support the aging members of these societies.  In these societies, the working immigrant poor may contribute very little individually; but collectively they support the elite (in this case, the aged natives) with their labor and taxes collected on their earnings.  This is far from an ideal model if the immigrants themselves (or their offspring) do not value good education, advancement of skills, innovation, etc.  (Of which certainly paved the way for the emergence of the American middle class).

In Israel, Jews have far lower birthrates than Palestinians.  One time I read that the Palestinians need not necessarily eliminate Israel as a sovereign nation through violence.  All they have to do is keep having babies and eventually there will be no more Jews (or the few that remain will be very insignificant, for there is power in numbers) to occupy what is now known as Israel.