Massachusetts Institute of Technology This is to certify that Arthur Augusto Alves da Silva has successfully completed Microeconomics Issued: May 9, 2026
Jonathan Gruber

Professor

Massachusetts Institute of Technology

Dimitris Bertsimas

Vice Provost for Open Learning

Associate Dean, Online Education and Artificial Intelligence

Boeing Professor of Operations Research

Massachusetts Institute of Technology

Valid Certificate ID: 9b26d5a6-e261-4d44-8979-84ebca94c5bb

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