  We study the question of dividing a collection of indivisible goods amongst a
set of agents. The main objective of research in the area is to achieve one of
two goals: fairness or efficiency. On the fairness side, envy-freeness is the
central fairness criterion in economics, but envy-free allocations typically do
not exist when the goods are indivisible. A recent line of research shows that
envy-freeness can be achieved if a small quantity of a homogeneous divisible
good (money) is introduced into the system, or equivalently, if transfer
payments are allowed between the agents. A natural question to explore, then,
is whether transfer payments can be used to provide high welfare in addition to
envy-freeness, and if so, how much money is needed to be transferred.
  We show that for general monotone valuations, there always exists an
allocation with transfers that is envy-free and whose Nash social welfare (NSW)
is at least an $e^{-1/e}$-fraction of the optimal Nash social welfare.
Additionally, when the agents have additive valuations, an envy-free allocation
with negligible transfers and whose NSW is within a constant factor of optimal
can be found in polynomial time. Consequently, we demonstrate that the
seemingly incompatible objectives of fairness and high welfare can be achieved
simultaneously via transfer payments, even for general valuations, when the
welfare objective is NSW. On the other hand, we show that a similar result is
impossible for utilitarian social welfare: any envy-freeable allocation that
achieves a constant fraction of the optimal welfare requires non-negligible
transfers. To complement this result we present algorithms that compute an
envy-free allocation with a given target welfare and with bounded transfers.
