Table 2. Qualitative Hessian optimization outcomes for the eco-social model 3 and scenarios analysed in the current study.

D = the determinant of Hessian matrix, and i = number of variables. For more details see Supporting Information (S3 Appendix).

Model 3
Determinant of	MEABRs	OAAs	Sustainable scenario
Hessian matrixes
A. Qualitative analysis
Di = 1	(-) P, < 0	(+) L, > 0	(-) P, < 0
Di = 2	(+) PT, > 0	(+) LS, > 0	(+) PT, > 0
Di = 3	(-) PTϕ + 2Pg2e + 2Ta2e, < 0	(+) 2LSϕ—Lg2—Sa2, > 0	(-) 2PTϕ + Pg2 + Ta2, < 0
Di = 4	(-) 4PTϕe + 2Pg2e + 2Ta2e, < 0	(+) 4LSϕe -2Lg2e - 2Sa2e, > 0	(+) 4PTϕe - 2Pg2e - 2Ta2e, > 0
Di = 5	-	-	-
Hessian results for GFF	Saddle-point	Rel. Min.	Rel. Max.