Buyout Price Optimization in the Rent-to-Own Business

Abstract

Problem definition: We study the multidimensional price optimization problem faced by a rent-to-own (RTO) firm, which rents a product for a periodic fee and during the rental repeatedly offers it for sale to the renter at a sequence of prices forming the buyout price path.

Methodology/results: We first employ calculus of variations to obtain optimal buyout prices in closed form for a special case. Next, to overcome the nonconcavity of the profit in the general problem, we formulate an equivalent bilevel optimization over the resource utilization and price path. We then transform the inner pricing problem into a deterministic dynamic program (DP) with a one-dimensional state. Leveraging this transformation, we develop an efficient algorithm to find the optimal price path. We also apply our methodology to jointly optimize buyout prices and inventory levels.

Managerial implications: Standard practice in the RTO industry is to use a price path that decreases steeply early in the agreement and gradually later. However, we prove for the special case that the optimal prices are in contrast concave decreasing; prices should optimally decrease gradually early in the agreement and steeply later. In the joint inventory and pricing case, our results reveal, perhaps counterintuitively, that higher inventory levels tend to entail higher optimal prices. Applying our algorithm in a case study with parameters calibrated from our discussions with an RTO firm, we again find that prices should optimally decrease gradually early in the agreement and steeply later, validating our insights from the special case. Moreover, our methodology yields approximately a 22\% increase in profit relative to industry prices.

Keywords: rent-to-own, price path and inventory optimization, calculus of variations, dynamic programming, practical algorithms