The Supermodularity of the Tax Competition Game

We determine su cient conditions to establish the supermodularity of a n asymmetric country tax competition, when countries maximize their tax revenue. Using a notion of generalized concavity, namelyρ−concavity, we establish that the tax competition game is supermodular when the marginal production function of each country is log-concave and 1/2-convex. Alternative su cient conditions for the supermodularity rely on the log-concavity of the marginal production function and a limit on the di rence between elasticities of net and gross return of capital. These conditions allow us to bound the degree of curvature of the inverse demand for capital. Applying some results from supermodular games, we deduce the existence and uniqueness of the Nash equilibrium. We show also that any increase in the number of tax-competing jurisdictions decreases tax rates and tax revenues and improves the net return of capital. Establishing similar su cient conditions for the supermodularity of the tax competition game with welfare maximizers raises multiple issues. Besides the question of the nature of public spending (complement or substitute of private consumption), we discuss the role of capital by considering successively an elastic worldwide stock of capital and capital ownership.