René Caldentey

Commissioned Paper: An Overview of Pricing Models for Revenue Management

This publication contains reprint articles for which IEEE does not hold copyright. Full text is not available on IEEE Xplore for these articles.

Dynamic Pricing for Nonperishable Products with Demand Learning

A retailer is endowed with a finite inventory of a nonperishable product. Demand for this product is driven by a price-sensitive Poisson process that depends on an unknown parameter that is a proxy for the market size. The retailer has a prior belief on the value of this parameter that he updates as time and available information (prices and sales) evolve. The retailers objective is to maximize the discounted long-term average profits of his operation using dynamic pricing policies. We consider two cases. In the first case, the retailer is constrained to sell the entire initial stock of the nonperishable product before a different assortment is considered. In the second case, the retailer is able to stop selling the nonperishable product at any time and switch to a different menu of products. For both cases, we formulate the retailers problem as a (Poisson) intensity control problem and derive structural properties of an optimal solution, and suggest a simple and efficient approximated solution. We use numerical computations, together with asymptotic analysis, to evaluate the performance of our proposed policy.

Supply Contracts with Financial Hedging

We study the performance of a stylized supply chain where two firms, a retailer and a producer, compete in a Stackelberg game. The retailer purchases a single product from the producer and afterward sells it in the retail market at a stochastic clearance price. The retailer, however, is budget constrained and is therefore limited in the number of units that he may purchase from the producer. We also assume that the retailers profit depends in part on the realized path or terminal value of some observable stochastic process. We interpret this process as a financial process such as a foreign exchange rate or interest rate. More generally, the process can be interpreted as any relevant economic index. We consider a variation (the flexible contract) of the traditional wholesale price contract that is offered by the producer to the retailer. Under this flexible contract, at t = 0 the producer offers a menu of wholesale prices to the retailer, one for each realization of the financial process up to a future time τ. The retailer then commits to purchasing at time τ a variable number of units, with the specific quantity depending on the realization of the process up to time τ. Because of the retailers budget constraint, the supply chain might be more profitable if the retailer was able to shift some of the budget from states where the constraint is not binding to states where it is binding. We therefore consider a variation of the flexible contract, where we assume that the retailer is able to trade dynamically between zero and τ in the financial market. We refer to this variation as the flexible contract with hedging. We compare the decentralized competitive solution for the two contracts with the solutions obtained by a central planner. We also compare the supply chains performance across the two contracts. We find, for example, that the producer always prefers the flexible contract with hedging to the flexible contract without hedging. Depending on model parameters, however, the retailer might or might not prefer the flexible contract with hedging.

Optimal Control and Hedging of Operations in the Presence of Financial Markets

We consider the problem of dynamically hedging the profits of a corporation when these profits are correlated with returns in the financial markets. In particular, we consider the general problem of simultaneously optimizing over both the operating policy and the hedging strategy of the corporation. We discuss how different informational assumptions give rise to different types of hedging and solution techniques. Finally, we solve some problems commonly encountered in operations management to demonstrate the methodology.

Online Auction and List Price Revenue Management

We analyze a revenue management problem in which a seller facing a Poisson arrival stream of consumers operates an online multiunit auction. Consumers can get the product from an alternative list price channel. We consider two variants of this problem: In the first variant, the list price is an external channel run by another firm. In the second one, the seller manages both the auction and the list price channels. Each consumer, trying to maximize his own surplus, must decide either to buy at the posted price and get the item at no risk, or to join the auction and wait until its end, when the winners are revealed and the auction price is disclosed. Our approach consists of two parts. First, we study structural properties of the problem, and show that the equilibrium strategy for both versions of this game is of the threshold type, meaning that a consumer will join the auction only if his arrival time is above a function of his own valuation. This consumers strategy can be computed using an iterative algorithm in a function space, provably convergent under some conditions. Unfortunately, this procedure is computationally intensive. Second, and to overcome this limitation, we formulate an asymptotic version of the problem, in which the demand rate and the initial number of units grow proportionally large. We obtain a simple closed-form expression for the equilibrium strategy in this regime, which is then used as an approximate solution to the original problem. Numerical computations show that this heuristic is very accurate. The asymptotic solution culminates in simple and precise recipes of how bidders should behave, as well as how the seller should structure the auction, and price the product in the dual-channel case.

An overview of pricing models for revenue management

In this paper, we examine the research and results of dynamic pricing policies and their relation to Revenue Management. The survey is based on a generic Revenue Management problem in which a perishable and non-renewable set of resources satisfy stochastic price-sensitive demand processes over a finite period of time. In this class of problems, the owner (or the seller) of these resources uses them to produce and offer a menu of final products to the end customers. Within this context, we formulate the stochastic control problem of capacity that the seller faces: how to dynamically set the menu and the quantity of products and their corresponding prices in order to maximize the total revenue over the selling horizon.

Revenue Sharing in Airline Alliances

We propose a two-stage game-theoretic approach to study the operations of an airline alliance in which independent carriers, managing different reservation and information systems, can collaboratively market and operate codeshare and interline itineraries. In the first-stage game, airlines negotiate fixed proration rates to share the revenues generated by such itineraries. In the second-stage game, airlines operate independent inventory control systems to maximize their own expected revenues. We derive a revenue-sharing rule that is (i) an admissible outcome of the first-stage negotiation, in the sense that no airline coalition has enough incentives to secede from the grand alliance, and (ii) efficient for the second-stage game, in the sense that the decentralized system can achieve the same revenues as a central planner managing the global alliance network. Our numerical study shows that the proposed proration rates can lead to a significant increase in revenues with respect to other rules commonly used in practice. Finally, because our proposal requires the disclosure of private demand information, we introduce a simple alternative rule that is based on public information. This heuristic performs remarkably well, becoming an interesting candidate to be pursued in practice. This paper was accepted by Martin Lariviere, operations management.

Insider Trading With a Random Deadline

We consider a model of strategic trading with asymmetric information of an asset whose value follows a Brownian motion. An insider continuously observes a signal that tracks the evolution of the assets fundamental value. The value of the asset is publicly revealed at a random time. The equilibrium has two regimes separated by an endogenously determined time T. In [0, T), the insider gradually transfers her information to the market. By time T, all her information has been transferred and the price agrees with the market value of the asset. In the interval [T, ∞), the insider trades large volumes and reveals her information immediately, so market prices track the market value perfectly. Despite this market efficiency, the insider is able to collect strictly positive rents after T. Copyright 2010 The Econometric Society.

Fcfs infinite bipartite matching of servers and customers

We consider an infinite sequence of customers of types and an infinite sequence of servers of types where a server of type j can serve a subset of customer types C(j) and where a customer of type i can be served by a subset of server types S(i). We assume that the types of customers and servers in the infinite sequences are random, independent, and identically distributed, and that customers and servers are matched according to their order in the sequence, on a first-come–first-served (FCFS) basis. We investigate this process of infinite bipartite matching. In particular, we are interested in the rate r i,j that customers of type i are assigned to servers of type j. We present a countable state Markov chain to describe this process, and for some previously unsolved instances, we prove ergodicity and existence of limiting rates, and calculate r i,j .

Policy Model for Pollution Control in the Copper Industry, Including a Model for the Sulfuric Acid Market

In this paper we develop a policy model for pollution control investment and operational decisions in the copper industry. The system consists of (i) a nonlinear integer model to optimize smelter operations, including the investment decisions relating to smelting capacity and pollution control plants that complies with environmental regulations, and (ii) a network flow model to describe the economic behavior of the sulfuric acid market, which considers the sulfuric acid produced at the pollution abatement stages in the smelting process. This second model solves for an equilibrium among spatially separated markets, that determines the price and distribution of acid in each demand and supply region. The two models interact through the input each receives from the other. Thus, the smelter model uses the sulfuric acid price at each smelter to find optimal operational and investment decisions, whereas the sulfuric acid market model considers sulfuric acid output at the smelters as part of the supply input to find the price of this product at each smelter location. The solution given by the policy model is the global equilibrium obtained when this iterative process between the two models converges. Thus, the price of the sulfuric acid, which is the central component when deciding when and where to locate a sulfuric acid plant, is determined endogenously, rather than assumed exogenous as in most models of this type. Computational experiments show that expected profits associated with the copper industry can increase significantly when the problem is solved in aggregate, as compared with the smelters making their decisions independently. Several applications of the policy model are described.