Innovation and the adoption of new ideas is fundamental to economic progress, and so is free trade of goods and services. Here we examine the underlying economics of the market for ideas and its implications for trade. From a positive perspective, we examine how such markets function and how international trade interact with them. From a normative perspective, we examine the pitfalls of current intellectual property regulations, and how might they be improved. We highlight recent research by ourselves and others challenging the notion that government awards of monopoly through patents and copyright are "the way" to provide appropriate incentives for innovation.