Costly Multi-Unit Search
39 Pages Posted: 27 Oct 2021 Last revised: 28 Nov 2022
Date Written: November 29, 2022
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 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 searching or not. We set this as a dynamic programming problem and solve it inductively by characterizing optimal search rules and reservation prices. Our inductive logic does not rely in the use of the first order conditions for optimal pricing and optimal search decisions.
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 sophis- ticated ones that endogenously combine purchase premiums, when inventory is large, with clearance sales and discounts, when inventory is low. Instead, a vari- able search cost always yields an increasing marginal selling cost and thus can only make sense of pricing strategies that employ purchase premiums.
Our model extends search theory by explicitly accounting for the effects of search costs on optimal multi-unit pricing strategies, pushing it into a richer class of problems and offering solutions that extend beyond optimal stopping rules.
Keywords: Search Theory, Multi Unit Search, Inventory
JEL Classification: C61, D80
Suggested Citation: Suggested Citation