Announcing the appointments on Truth Social on Tuesday night, Trump said that DoGE "will become, potentially the 'Manhattan Project' of our time" - a reference to the US effort during World War II to develop atomic weapons in record time and profoundly affect the world.
Musk said, "This will send shockwaves through the system, and anyone involved in Government waste, which is a lot of people!"
He wrote on X: "All actions of the Department of Government Efficiency will be posted online for maximum transparency. Anytime the public thinks we are cutting something important or not cutting something wasteful, just let us know! We will also have a leaderboard for most insanely dumb spending of your tax dollars. This will be both extremely tragic and extremely entertaining (with a smiley emoji)."
Ramaswamy posted his reaction on X, "We will not go gently, @elonmusk."
The structure of the department and its leadership is vague because Trump said, "The Great Elon Musk, working in conjunction with American Patriot Vivek Ramaswamy, will lead the Department of Government Efficiency."
Since it is a new department, it is not clear if the leaders would be in the Cabinet, have to be confirmed by the Senate, and what their titles would be.
"Together, these two wonderful Americans will pave the way for my Administration to dismantle Government bureaucracy, slash excess regulations, cut wasteful expenditures, and restructure Federal Agencies," Trump said.
About its role, he said, "To drive this kind of drastic change, the Department of Government Efficiency will provide advice and guidance from outside of Government and will partner with the White House and Office of Management & Budget to drive large-scale structural reform and create an entrepreneurial approach to Government never seen before."
Musk said during the campaign that he could cut $2 trillion from the federal budget of $6.5 trillion.
Both Ramaswamy and Musk campaigned for Trump, and Musk ran a parallel campaign for Trump in Pennsylvania to turn out the vote in the state that he won.
The world's richest person, Musk heads the electric car company, Tesla, and the rocket company Space X, and owns X, which was known as Twitter before he bought it.
He is an immigrant from South Africa.
Ramaswamy, whose parents are from India, is a multimillionaire pharmaceutical entrepreneur.
He ran against Trump for the Republican presidential nomination, but dropped out and endorsed him.
While they can be expected to bring their business expertise to make government, there are questions about conflict of interest, especially in the case of Musk, whose SpaceX has government contracts.
He also has extensive business in China, where he makes Tesla vehicles and sells and exports them.