Technological improvement is the most important cause of long-term economic growth, but the factors that drive it are still not fully understood. In standard growth models technology is treated in the aggregate, and a main goal has been to understand how growth depends on factors such as knowledge production. But an economy can also be viewed as a network, in which producers purchase goods, convert them to new goods, and sell them to households or other producers. Here we develop a simple theory that shows how the network properties of an economy can amplify the effects of technological improvements as they propagate along chains of production. A key property of an industry is its output multiplier, which can be understood as the average number of production steps required to make a good. The model predicts that the output multiplier of an industry predicts future changes in prices, and that the average output multiplier of a country predicts future economic growth. We test these predictions using data from the World Input Output Database and find results in good agreement with the model. The results show how purely structural properties of an economy, that have nothing to do with innovation or human creativity, can exert an important influence on long-term growth.