Equilibrium in Production Chains with Multiple Upstream Partners

TL;DR

This paper advances the understanding of production chain equilibria by extending existing models to multiple upstream partners, proving key properties, and introducing a faster computational algorithm, with a stochastic generalization for broader applicability.

Contribution

It improves the production chain model by proving equilibrium properties and developing a significantly faster algorithm, also extending the model to stochastic environments.

Findings

Proved existence, uniqueness, and stability of equilibrium prices.

Developed an algorithm over ten times faster than previous methods.

Generalized the model to stochastic production networks.

Abstract

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.