Good times, bad times, and vertical upstream integration

We consider a set of downstream firms each of which has a stochastic requirement for a particular input. Downstream firms can produce the input themselves yet do not trade it. Upstream firms produce the input to sell it through a Walrasian market to downstream firms. Efficient pooling of capacities requires the input to be produced by upstream firms and traded in the market. Yet, downstream firms will always vertically integrate. By producing some of its own input needs, a downstream firm cuts down on aggregate input demand thus depressing prices in the market.