By formal method I mean a broadly known, standard and as objective as possible. For example, one of the major criteria for NSF grant are “broader impacts” which is not hugely objective(likewise with other funding agencies). In recent years, Altmetrics and similar tools and indexes have been devised to measure informal citations and calling them "social impact" – e.g. social media or non-traditional citations/mentions. In terms of economics and decision making for the policy making in academic research, is seems reasonable that should be some measurement method, performance index, or framework to evaluate “broader impacts” of the work we report in scientific publications or can refer to. Can you provide any references to such methods, indexes, or frameworks?

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There are interdisciplinary researchers who study such questions. It's labeled with various names: "Sociology of Science", "Science, Technology, and Society" (STS), and similar. You should look at journals in those fields. For example, Science and Public Policy has these articles:

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