Echoing Friedman (who said that the social responsibility of firms was to make profits), one could say that the social responsibility of economists is just to write and publish academic papers. However, that’s not the view taken by many economists, from Paul Krugman to Thomas Piketty -who have also published many academic papers.
There’s a very interesting video with a recent debate at the UCL policy lab about the role of economic expertise in a world of populism and post-truth politics, where the participants suggest that economists can play a positive role, but that this role is difficult and its effectiveness does not only depend on politicians listening.
There are many examples of economists finding difficulties to convince the public opinion about their lessons. Italian economic technocrats have been more successful reaching power by default than by election. Several scholars have studied precisely the difficulties of convincing citizens of complex ideas, such as indirect or unintended effects of simplistic policies. The mistakes made by many economists in the global financial crisis do not help. People may have a clear and natural understanding of ideas that promote fairness (although some may take a very local and narrow view of it), but may have more difficulties understanding efficiency-enhancing policies.
In some contexts, economists have played an important role in convincing people of realistic policies, without losing values and social objectives from sight. For example, in the recent Chilean presidential election (especially in the second round) and in the Constitutional process, center-left economists played an influential and in my view positive and successful role. Several of the left-wing presidents in Latin America (Chile, Colombia) have center-left economists in government as a moderating factor. Others may believe that this is not necessarily a positive development. In experiments, it is not difficult to show that well argued conventional economic ideas may be more powerful than populist arguments.
Other economists may exert their social responsibility in different ways. Nobel Prize winner Daniel Kahneman and very prestigious Israeli economists recently signed a manifesto rejecting the illiberal reforms of the current far right coalition government. Projects to reform the teaching of economics have been promoted by economists that got involved into that in a way that went beyond their professional (paid) duties, for example in the CORE Project.
There is a recent article by Jean Tirole on the social responsibility of independent agencies, where he calls for caution about the involvement of Central Banks and antitrust authorities on objectives that go beyond their original remit. The same caution should apply to economists individually when they get involved in social, political or civic issues. But in my opinion there are two wrong attitudes in front of the necessary engagement of the profession with social problems. One is to reject it, because “it is not our job,” which would have a social cost when this contribution may result into socially desirable policies. Another is to be involved only in a marginal way, rejecting the scientific method and creating a straw man about the rest of the profession.
Sometimes the social responsibility is to be partisan (see the trajectory of Paul Krugman), and sometimes it is to defend one’s independence, when this is not disguised as artificial balance (perhaps the BBC economic approach with Brexit during the referendum campaign). Economists should be more than dentists and technocrats, but they should also be as effective as good dentists and technocrats who use the scientific method. As social scientists, we are scientists but also part of society.