theorem underentry_proved {J ρ c κ : ℝ}
(hJ : 1 < J) (hρ0 : 0 < ρ) (hρ1 : ρ < 1) (hc : 0 < c)
(hFOC : valueFunction J ρ c = κ) :
κ < valueFunction J ρ c + J * valueFunctionDeriv J ρ c :=
underentry_at_private_optimum hJ hρ0 hρ1 hc hFOCPaper 1c, Section 4: Welfare and Coordination Failure