theorem welfare_gap_full {J_high J_low ρ c : ℝ}
(hJl : 1 < J_low) (hJh : J_low < J_high)
(hρ0 : 0 < ρ) (hρ1 : ρ < 1) (hc : 0 < c) :
0 < (valueFunction J_high ρ c - valueFunction J_low ρ c) * J_low :=
welfare_gap_lower_bound hJl hJh hρ0 hρ1 hcPaper 1c, Section 4: Welfare and Coordination Failure