Figure 8.

A numerical solution to the Lagrange problem. We observe that the line p₂ = (c − p₁N₁)/N₂ (dotted) is tangent only to the contour curve ℜ_v(p₁, p₂) = 2.2. They intersect at the point (p₁, p₂) = (0.66, 0.29), the optimal solution (marked with a red dot). The parameter c = 0.9 × N₁ with others the same as in figure 7. For some parameter values, the optimal solution might be outside the unit square in the p₁-p₂ plane.