Evolutionary Variational Inequalities and Long-Run Growth Equilibria with Transaction Costs

Abstract

We study a class of evolutionary variational inequalities in a Hilbert space that models the long-run balanced-growth equilibrium of a competitive economy with transaction costs, time-dependent production and infrastructure constraints, and exogenous price dynamics. The paper makes four contributions. First, we introduce a parametrized monotonicity functional μα(t;F;u,v;p) and prove an exact equivalence theorem: the inequality μα≥βu−v2 holds if and only if the operator F is strongly monotone with the explicitly computed constant m=β−α(1+p∞). This turns the growth parameter α and the price level into explicit terms of a single admissibility threshold and, for β<α(1+p∞), produces a scale of conditions that covers operators which are not monotone, i.e., economies with a bounded degree of increasing returns. Second, we prove well-posedness: for every admissible initial state there is exactly one Lipschitz equilibrium trajectory u*(·), obtained through Moreau’s catching-up algorithm for the associated perturbed sweeping process, together with the explicit velocity bound u˙*≤LK+2CF. Third, we derive one comparison estimate from which global exponential stability, the convergence rate u(t)−u*(t)≤r e−mt+Lpm−1supΔp+εm−1, and robustness with respect to perturbations of prices and of the operator all follow; we also show that, when the constraint sets stabilize, the trajectory converges to the stationary equilibrium of the limit problem. Fourth, we prove that strong monotonicity implies the c-covering property with c=m, so that the shock-absorbing capacity of the economy is governed by the same constant as the speed of convergence. Two examples—a two-resource system and an n-market network with nonlinear transaction costs—are worked out with a complete verification of every hypothesis and with explicit numerical constants.