Why price signals fail for by-products: an intrinsic inelasticity risk metric for companion metals

Critical metals underpin key technologies, yet many are produced primarily as by-products (or "companion") of "host" metals. This paper highlights an intrinsic supply-risk effect -companionality-driven inelasticity -that is distinct from exogenous access risks such as country concentration, trade restrictions, or governance shocks. In such companion-metal markets, higher prices may not translate into higher output because production is constrained by the host-metal value chain. This paper proposes a practical metric to quantify the additional economic and financial risk induced by companionality. Starting from observed production shares and relative value contributions within host operations, we define a perceived supply elasticity of a companion metal with respect to its own price. The associated companionality risk index is the inverse of this perceived elasticity, enabling consistent cross-element screening. In theory, the perceived elasticities satisfy an eigenvector condition implied by the joint-production structure; in practice, real-world price and production data are imperfect and make this condition ill-posed. We therefore estimate elasticities using an L 2 -regularized formulation and select the regularization strength via an L-curve trade-off, yielding stable and unique estimates. We illustrate the approach with a periodic-table heatmap that visualizes companionality-driven exposure and discuss decision implications for technology choice and procurement, including the roles of long-term offtake contracts, strategic inventories, and how increased recycling of host metals may reduce by-product availability. To align mitigation with the dominant mechanism, we further combine the intrinsic inelasticity signal ( R c ) with geographic concentration Herfindahl-Hirschman Index (HHI) and scale actions by Economic Importance (EI) in a decision map. Highlights• By-product dependence can make "pay more" ineffective for raising supply.• We define a perceived supply elasticity implied by a joint production structure.• Companionality risk is the inverse of perceived elasticity (screening metric).• L 2 regularization fixes ill-posed inference under imperfect price data.