No one is proposing that the United States in its current form be transitioned to a monarchy, Michael Anissimov notes:
It is too large and culturally/politically disconnected. It is more likely that a part of the United States would have to be broken off and converted into a monarchy. As a concrete example, consider a monarchy composed of northern California, Oregon, Washington, and Idaho, which could be called “Greater Cascadia”.
To start off, the economic system of the new country would be identical to the way it was before. No one wants to scare away business. In fact, a prudent move might be to make the economy more libertarian, less regulated than it was before. A monarchical government is minarchist, and traditionally composed only of an administration and the military. There would be no alphabet soup of agencies designed to control every aspect of business and the economy.
Eliminating income tax and taxing land is a possibility. See Georgism, but apply salt.
Might I suggest divvying the country into much smaller parcels than states — say, counties — and having one CEO for each such parcel — called, oh, I don’t know, count?