Imputed rent is the rental price an individual would pay for an asset they own.
Imputing housing rent is necessary to measure economic activity in national accounts.
Thus, imputed rent offers a way to compare homeowners' and tenants' economic decisions.
But if A and B are both owner-occupiers, no money changes hands even though the same economic relationships exists; there are still two owners and two occupiers, but the transactions between them no longer go through the market.
The amount that would have changed hands had the owner and occupier been different persons is the imputed rent.
In the United States, the Bureau of Labor Statistics uses a version of this approach to estimate changes in the price of owner-occupied housing.