Rapid economic growth in the large developing countries collectively known as BRICSAM (Brazil, Russia, India, China, South Africa and Mexico) has the potential to change the balance of economic power in the world. This paper analyzes this potential building on developments in these economies over the past four decades in the context of the evolution of the world economy. This evolution has two significant features: increasing economic integration and a hiatus in growth. Increasing integration can be observed in the almost universal rise in the share of the exports of goods and services in GDP, and the increase in private capital flows. There has been a hiatus in growth since the 1973-1974 increase in the price of oil. This paper's analysis of various scenarios concerning the relative size of major economies until 2050 based on incomes measured at official exchange rates, a preference explained in the paper, suggests a sharp increase in the share of world income of China and partly India over the next four decades. Coupled with the expected increase in the share of China in world exports during this period, the prospect is of a substantial increase in China's influence over world economic management. This influence will be even more substantial if China is able to build alliances with the other BRICSAM countries. It is in the common interest of the BRICSAM countries to ensure that the workings of the major international economic institutions contribute to maintaining the free flow of goods, services and capital necessary for these economies to achieve their potential.