If there were no scarcity in the world, there would also be no economy. The reason for this is that without scarcity, there is no such thing as economics and there is no such thing as an economic system.
Economics can defined as the study of how people make choices to reconcile their unlimited wants with their limited resources. It can also be defined as the study of how society chooses to allocate its scarce resources when confronted with the fact that people have unlimited wants. Either way, scarcity is at the very core of economics. Without scarcity, there can be no study of the way in which people deal with scarcity.
In addition, there could be no economic systems as we know them. The economic systems (command, market, and traditional) are all ways of answering the fundamental economic questions. These questions are what to produce, how to produce it, and for whom to produce it. All of these assume that there are scarce resources. Thus, if there is no scarcity, there are no fundamental economic questions and no economic systems for answering them.
It would be possible that attempts could be made to use tradition or governments to try to regulate what people did with their unlimited resources if there were no scarcity. They could still, for example, ban drugs by law or try to discourage drug use through peer pressure. But this would be social/political action, not an economic system.
Scarcity is the basis of economics so without scarcity there is no type of an economy.