We follow the approach of Grochulski (2007), who determines the optimal income tax schedule when individuals have the possibility of avoiding paying income taxes. We however modify his model by considering a convex concealment cost function. This assumption violates the subadditivity property made by Grochulski (2007) and this has strong implications for the design of the tax schedule. This latter indeed shows that, with subadditivity, all individuals should declare their true income. Tax avoidance is thus not optimal. With a convex cost function, we find that a subset of individuals should be allowed to avoid taxes, provided that the marginal cost of avoiding the first euro is sufficiently small. We also provide a characterization of the optimal income tax curve.