theorem pigouvian_subsidy_decomposition {J ρ c : ℝ} :
J * valueFunctionDeriv J ρ c =
J * marginalK J ρ * perCapitaBenefit J ρ c +
J * curvatureKReal J ρ * ((1 / ρ - 1) * J ^ (1 / ρ - 2) * c) := by
simp only [valueFunctionDeriv]; ringPaper 1c: Formal Calculus on K(J) and V(J)