Inventory management under price-based and stockout-based substitution

We show that the demand function is not continuous in price. By decomposing the profit function into different price regimes, we are able to derive closed-form expressions for the stockout-based substitution rates (upward and downward substitution) and the optimal inventory levels under exogenous pricing. Under endogenous pricing, we find that the profit function is not necessarily unimodal. However, we show that a unique solution exists for the integrated price and inventory problem under price-based substitution only. Numerical results reveal that not considering stockout-based substitution (i) leads to lower profit margins for high-quality products and (ii) may cause severe supply-demand mismatches throughout the entire assortment. Finally, we show the performance of two approximated pricing policies.