My multiplier method

It is the change in total GDP for a unit change in G.

I assume Magic Walrus, we do not break Say's Law so the multiplier is never negative.  For example, if G took one unit of goods from the private sector, with a transaction cost great than one; not allowed. The good is no longer the same good and the smooth pricing assumption is broken..

Under normal circumstance, G takes my office desk, and puts it on the open market.  I buy it back,. pay transaction costs and recover the desk at some cost less than its normal replacement value.  Transaction  costs covered, Coase takes over. But the multiplier less than one.

Multiplier greater than one when G takes part of my property and charges me cement fees.  The sidewalk increases the utility of my remaining property.  In the latter case, who is G?  It is me, and my neighboring businesses, hammering out an agreement over beer.