Convergence in a Dynamic Model of Tax Compliance

We study a dynamic model of tax compliance. The tax authority adjusts the fraction of firms it is going to audit based on a naive rule, which depends on a fixed target on tax collection. If the tax collection is less than the target, the fraction of the firms audited by the authority increases and vice-versa. The replicator equation determines the fraction of firms complying with tax in each period. The dynamical system generated based on the decisions of the firms and the tax authority has a unique interior equilibrium. The unique interior equilibrium is locally stable. We rule out the presence of the closed orbits in a bounded region which contains the interior equilibrium point using the Dulac condition. Then, we get that the trajectories converge at the interior equilibrium point as a corollary of Poincare Bendixson theorem.