Regarding the Expenditure Function Underlying a Bliss Point
Usually the change in variables $X=x-\delta$ allows to write
$$ e(p,u) = \max_{x \geq 0}\{ p'x : U(x-\delta) \geq u \} =  \max_{X \geq -\delta} \{ p'X : U(X) \geq u \} + p'\delta $$
and so $ e(p,u) = E(p,u) + p'\delta.$