Identifying the effective cost ratio and a model-consistent shadow carbon price in deteriorating item inventory systems: a deterministic inverse optimisation approach

Firms in perishable goods supply chains leave a detailed trail of replenishment decisions, yet the cost structure behind those decisions, and the value a firm place on its own carbon emissions, stay hidden. This paper develops a deterministic inverse optimisation framework that reads part of that hidden structure back out of observed behaviour. A forward inventory model for a deteriorating item shows that ordering, holding, spoilage and carbon costs collapse into an effective setup term and an effective holding term, so behaviour identifies only a single cost ratio and the deterioration rate; individual accounting costs are not recoverable from decisions alone. A two-stage estimator recovers the deterioration rate from the inventory balance linking orders to sales and the cost ratio from the optimality condition, then backs out a model-consistent shadow carbon price from the ratio and the firm’s known accounting costs. We prove which quantities are identified, characterise a confounding valley in which holding, spoilage and emission costs cannot be separated from decisions alone, and show the shadow price is recoverable only when the emission signature differs from the cost signature. A numerical study separates finite-sample dispersion from asymptotic bias, documents a bias floor that persists as the sample grows at fixed noise, tests recovery under realistic measurement error and heterogeneous products, validates a bootstrap bias correction, reports confidence interval coverage, and quantifies how often the recovered price is negative. The resulting claims about consistency, identification, and practical reliability are stated in the form that the theory and data jointly support.