Recent trade data exhibit the following four empirical regularities: (i) countries import only a small fraction of all traded varieties (ii) per capita income and the number of imported varieties correlate positively (iii) per capita income and trade shares correlate positively and, finally, (iv) world trade shares have increased substantially. The present paper argues that standard theories fail to explain at least some of these patterns and subsequently shows that a small and reasonable change in the demand structure can reconcile the New Trade model with the data. Its key assumption imposes an upper bound on consumers' marginal utility from varieties. This implies that consumers purchase only the cheaper share of varieties, while expensive foreign varieties bearing high transport costs are not consumed. Technological progress that increases per capita consumption of those varieties in the consumption basket decreases marginal utility derived from them and induces consumers to extend their consumption to more expensive varieties produced at distant locations. Through this additional margin trade shares increase as productivity grows. Productivity growth is thus identified as a joint determinant of trade shares, the number imported varieties, and per capita income.