Just check the statistics, Bill D. Family size really does go down when people get secure. That's just as true for cultures where there has been a tradition of big families, made up of people belonging to religions that discourage birth control. For example Ireland and Italy.

The trouble is that, even when family size goes down in rich countries, pollution and over-use of resources by the people of those countries tends to continue to stay high and even to rise - that's especially true for the United States of course, but in a lesser degree true of many other prosperous places.