Universal Growth in Production Economies

We consider a simple variant of the von Neumann model of an expanding economy, in which multiple producers make goods according to their production function. The players trade their goods at the market and then use the bundles acquired for the production in the next round. We study a simple decentralized dynamic---known as proportional response---in which players update their bids proportionally to how useful the investments were in the past round. We show this dynamic leads to growth of the economy in the long term (whenever growth is possible) but also creates unbounded inequality, i.e. very rich and very poor players emerge. We analyze several other phenomena, such as how the relation of a player with others influences its development and the Gini index of the system. One of the key technical findings is that the players learn a global feature of the network (the optimal cycle) in a decentralized way, while interacting locally with their direct neighbors. We obtain this by studying the volume in the resulting dynamical system and showing that the volume of each cycle expands or contracts by a constant factor in each round.