Before attending graduate school, Courant served as a research assistant at the Brookings Institution. After receiving his PhD, he joined the faculty at the University of Michigan, where he has remained. In 1979 and 1980 he was a Senior Staff Economist at the Council of Economic Advisers. From 2002-2005 he served as Provost and Executive Vice-President for Academic Affairs, the chief academic officer and the chief budget officer of the University, He has also served as the Associate Provost for Academic and Budgetary Affairs, Chair of the Department of Economics and Director of the Institute of Public Policy Studies (which is now the Gerald R. Ford School of Public Policy).
Courant is Arthur F. Thurnau Professor, Harold T. Shapiro Collegiate Professor of Public Policy, Professor of Economics, Professor of Information, and Faculty Associate in the Institute for Social Research at the University of Michigan.
Courant has authored half a dozen books and over seventy papers covering a broad range of topics in economics and public policy, including tax policy, local economic development, gender differences in pay, housing, radon and public health, relationships between economic growth and environmental policy, and university budgeting systems.
In November 2007, Courant started a blog in which he defended the University of Michigan's participation in the Google Library Project. His remarks joined an ongoing debate about the project, specifically with regard to the role of libraries in large scale digitization efforts.