We derive formulas for the optimal tariff rate in four theoretical models. We start with a model in which industries are competitive and then successively allow for: monopoly pricing by export industries, revenue-replacement costs and cold-shower effects. The theoretical formulas accurately explain results from MONASH, a detailed computable general equilibrium model. A critical parameter in determining the optimal tariff is the export-demand elasticity. Modellers are often reluctant to adopt empirically justifiable values for export-demand elasticities because such values generate embarrassingly large optimal tariff rates. A way out of this dilemma is the adoption of a non-linear cold-shower specification.