This paper deals with efficient computation of a certain type of Nash-Cournot equilibria in multi-stage oligopolies. The generic structure analyzed herein represents the case when at any one stage a single homogeneous output is produced that serves as the only input to the subsequent stage. Thus at each intermediate level in the production process several non-cooperative firms competitively buy appropriate quantities of one single input, transform these quantities, and then, à la Cournot, sell their homogeneous output further on. Eventually, at the final stage a finished product emerges to satisfy competitive demand. For such interrelated markets, suffering from imperfect competition, we propose to find non-cooperative equilibria by a dynamic programming approach. When cost are quadratic and final demand is linear it is shown that one backward recursion followed by a forward sweep, produced equilibrium solution. This provides a facility to analyze issues on integration and mergers.