Costly Multi-Unit Search

We formulate and solve a costly multi-unit search problem for the optimal selling of a stock of goods. Our showcase application is an inventory liquidation problem with fixed holding costs, such as warehousing, salaries or floor planning. A seller faces a stream of buyers periodically arriving with random capped demands. At each decision point, he decides how to price each unit and also whether to stop search or not. We set this as a dynamic programming problem and solve it inductively by characterizing optimal search rules and reservation prices. We show that combining multiple units with a fixed per period search cost might translate into non-monotone selling costs and reservation prices. This lack of monotonicity naturally leads to discontinuities of the pricing strategy. In particular, the seller optimally employs strategies such as bundling, and more sophisticated ones that endogenously combine purchase premiums, when inventory is large, with clearance sales and discounts, when inventory is low. Our model extends search theory by explicitly accounting for the effects of fixed costs on optimal multi-unit pricing strategies, pushing it into a richer class of problems and offering solutions that extend beyond optimal stopping rules.