A Formula for Max Tax

Let:

R = the nominal tax rate and therefore the government component of GDP
P = Percentage of that government component which is productive
C = percentage cost of tax collection

Then the total productive GPD % is (1-R) + PxR

and so the total tax tield is R times that, ie R((1-R)+P) = R(P+1-R)

and the net yield after collection costs is R(1-C)(P+1-R)