On the conditions that preclude the existence of the lerner paradox and the Metzler paradox

The Lerner paradox is the possibility that a tariff on an import good might worsen a country's terms of trade, and the Metzler paradox is the possibility that a tariff on an import good might reduce a country's import price. In a general equilibrium framework with multiple goods, this paper shows that the combination of the invertibility of the Slutsky matrix for the world economy and its similarity across countries will preclude both of the paradoxes, and that the combination of the gross-substitutes assumption for the world demand and the substitute assumption for the demand of an import country property of goods will preclude the Lerner paradox. A modified condition for the Slutsky matrix combined with the gross substitute for the world demand will do the same for the Metzler paradox. A concept of non-surpassed diagonal is used in deriving the result.