A typical proof of the existence of a perfectly competitive market equilibrium employs an appropriately continuous price-to-price mapping that depends on excess demands. If trade tax-subsidy distortions are introduced into the model, the excess demand mappings may have disconnected image sets and destroy the continuity of the price-to-price mapping. This difficulty is overcome by developing a technique which explicitly takes account of the dependence of demand on both income and prices and simultaneously solves for equilibrium prices and income levels for all agents. This technique is then applied to establish two existence theorems for models of international trade with trade tax-subsidy distortions.