Competition in multiechelon assembly supply chains

In this paper, we study competition in multiechelon supply chains with an assembly structure. Firms in the supply chain are grouped into homogenous sectors (nodes) that contain identical firms with identical production capabilities that all produce exactly one undifferentiated product (that may itself be a "kit" of components). Each sector may use several inputs to produce its product, and these inputs are supplied by different sectors. The production process within any sector is taken to be pure assembly in fixed proportions. The number of firms in each sector is known. The demand curve for the final product is assumed to be linear, as are production costs in all sectors. Competition is modeled via a "coordinated successive Cournot" model in which firms choose production quantities for their downstream market so as to maximize profits, given prices for all inputs and all complementary products. Production quantities for sectors supplying the same successor are coordinated through pricing mechanisms, so that complementary products are produced in the right proportions. Under these assumptions, equilibrium prices for any multiechelon assembly network are characterized by a system of linear equations. We derive closed-form expressions for equilibrium quantities and prices in any two-stage system (i.e., a system with multiple input sectors and a single assembly sector). We show that any assembly structure can be converted to an equivalent (larger) structure in which no more than two components are assembled at any node. Finally, large structures can be solved either by direct solution of the characteristic linear equations or through an iterative reduction (compression) to smaller structures.