International economics has overwhelmingly relied on Samuelson's (1954) assumption that trade costs are proportional to value. We develop a quantitative analytical framework that features both additive and multiplicative (iceberg) trade costs, building on a model of international trade with heterogeneous firms and demand heterogeneity. We structurally estimate the magnitude of additive trade costs, for every product and destination available in our firm-level data of Norwegian exporters. Identification is aided by the theoretical finding that the elasticity of demand to producer price is dampened, in absolute value, when prices are low, and this mechanism is magnified when additive trade costs are high. This magnification mechanism becomes useful in the subsequent econometric analysis. Estimated additive trade costs are substantial. On average, additive costs are 33 percent, expressed relative to the median price. This leads us to reject the pure iceberg cost assumption. We assess the importance of these costs in shaping global trade flows. Our micro estimates of additive trade costs explain most of the geographical variation in aggregate trade. An implication of our work is that inferring trade costs from standard gravity models suffers from specification bias, since these models assume away the role of additive trade costs.