Variational methods for equilibrium problems involving quasi-concave utility functions

Existence of General Competitive Equilibria: A Variational Approach

“…Finally, we point out that our main result contains, as special cases, some recent existence results of competitive equilibria for pure exchange economics established in [24,25]. Probably, our techniques also work if we consider the dynamic version of pure exchange economics introduced in [16,17] (see also [26,27]) and so the existing results of these papers could be improved by replacing the concavity condition with the quasiconcavity condition.…”

“…, ), then E becomes a pure exchange economy. In this case Problem 8 assumes the form of the one in [24,25].…”

“…, ℓ, is concave if ∑ ℎ=1 ℎ < 1, but it is only quasiconcave (and not necessarily concave) regardless of the sum of exponents ℎ . Following the same arguments at page 178 of [25], Theorems 16 and 19 can be applied to prove the existence of competitive equilibrium for Problem 8 with given by (66) for arbitrary positive exponents ℎ . The same result also applies to CES utility function defined by ( …”

On a competitive economic equilibrium referred to a continuous time period

“…Among the novel techniques, which had unified different research areas, the variational approach is that more recent: an economic equilibrium is characterized as a solution to a suitable quasi-variational inequality. Following this line of investigation, among the most significant papers, we can cite [1,2,8,11,15,16].…”

“…For the goals we walk the way traced by Anello et al in [1,2] (which now we label as static case) and each time we carry out the necessary economical and mathematical adjustments due to the introduction of the time in the considered objects.…”