  In this paper, we extend and improve the production chain model introduced by
Kikuchi et al. (2018). Utilizing the theory of monotone concave operators, we
prove the existence, uniqueness, and global stability of equilibrium price,
hence improving their results on production networks with multiple upstream
partners. We propose an algorithm for computing the equilibrium price function
that is more than ten times faster than successive evaluations of the operator.
The model is then generalized to a stochastic setting that offers richer
implications for the distribution of firms in a production network.
