This question is asked by Greg Mankiw on his blog. Mankiw points us to an article in the New York Times for discussion. Apparently, some government regulators think unpaid internships might violate minimum-wage laws. Mankiw also notes that the internship as the Council of Economic Advisers is unpaid! The CEA provides the US President with economic analysis and advice on the development and implementation of a wide range of domestic and international economic policy issues.
Why take an unpaid internship? The fun of it? May be, if you enjoy the work. But if it increases your human capital then it could make economic sense as well. An increase in human capital leads to better and more highly paid jobs in the future and so can increase the discounted value of your lifetime income stream.