Optimal tariffs when production is fixed

The effects of tariff wars on welfare are analysed for the case of trade between two countries with fixed outputs of the traded good. Assuming mild conditions, it is shown that if there are non-zero tariffs for which welfare-maximizing equilibrium holds, then free trade is not strictly preferable when the coutries' output are equal, and if there are not equal is strictly disadvantageous to the country with the smaller output. It is also shown that welfare-maximizing equilibria do exist if the demand function is linear.