Arbitrage and nonlinear tax scales

We look at the theory of arbitrage with taxation under certainty. The tax scale in our model is not linear. Under the premise that tax scale is convex, we analyze prices that do not exhibit arbitrage opportunities. It turns out that there are two kinds of arbitrage: unbounded as well as bounded arbitrage. With bounded arbitrage, the gain from forming an arbitrage portfolio is bounded from above and cannot increase infinitely. In a model with a linear tax scale such a bounded arbitrage cannot exist, all arbitrage portfolios will generate an infinite gain from trade. In contrast to earlier research, we are able to give a complete characterization (i.e., if and only if) whether bounded as well as unbounded arbitrage opportunities will occur only relying on market prices and properties of the tax scale. This characterization relies on so-called implicit tax rates that are defined by a simple relation copied from the case of linear tax scales.