"which theory posits that the postcolonial world can become like the countries of the developed world if they follow the lead of the developed world economically, politically, and socially? "

Modernization theory

Explanation:

Modernization theory is known to be utilized in order to describe modernization process within a society. Modernization tends to refer to the state of continuous transition from a traditional to contemporary society. Modernization theory was originated from notion of Max Weber who was a  German sociologist, he provided the ground work for modernization paradigm which was further developed Talcott Parsons.

the above statement/question is (true);