Tour operator revenue management – Competitive supply chain contracting

Abstract

Historically, revenue management (RM) has been focused on setting prices or allocations to maximize revenues from a fixed inventory. RM has expanded to include other facets of the selling problem, including competition, distribution channel management and selling mechanisms. We illustrate an example of RM converging with supply chain management. Under this setting an inventory supplier sets prices and inventory allocations for resellers, here tour operators; the resellers compete with each other by bundling the inventory with other services and selling to consumers. We illustrate optimal inventory allocations and prices under a two-period game where the resellers are competing for consumers. The model and formulation are motivated by work with a large North American tour operator.

Appendix

Proofs of analytical results

Proof of Result 1. It is straightforward to show that the service provider’s second-period decision is supermodular. From

it follows that

i≠k, so that π _SP ² has increasing differences in (w _i ², w _k ²), (w _i ², y _i ²), and (y _i ²,  y _k ²), providing the desired result. Then the service provider has an optimal wholesale price vector for this period, which we denoted w^2*. □

Proof of Result 2. This result was established in Bernstein and Federgruen (2005), and the approach for the proof is the same as for Result 2. As

has increasing differences in (p _i ², p _k ²), i≠k, π _i ² is log-supermodular. Here we are assuming the capacity is unconstrained. We denote the Nash equilibrium in prices and associated quantities by p^2u and y^2u. As in Result 2, multiple equilibria form a lattice, with greatest element preferred by all resellers. □

Proof of Result 3. Let second-period capacity be constrained, so that Q²<∞. For a given wholesale vector w², firms will take at least as much under Q² as they would were capacity unconstrained. The impetus for firms taking additional capacity is to restrict other’s action spaces in procurement amounts so that prices can be raised. Mathematically, this requires that revenue functions have increasing differences in all prices and quantities (not in prices only as for Result 4) for all firms up until the capacity constraint becomes tight. That is, we seek a condition for the revenue function

to have increasing differences in both (y _i ², p _k ²), for all k.

Increasing differences in own quantity and cross-price always holds, as

For payoffs to have increasing differences in own quantities and own price requires that

As (p _i ²/(d _i ²(p))²)g _i ²(y _i ²/d _i ²(p))⩾0, and ∂d _i ²(p)/∂p _i ²⩽0, we have that

implies that ∂²π _i ²/∂p _i ²∂y _i ²⩾0 holds. This is simply a restriction on firms’ own price elasticities. If the inequality in 25 holds for all firms as prices p _i ² and quantities y _i ² are increased from and until ∑ _i y _i ²=Q², a Pareto-improved Nash equilibrium is produced. And if this result holds for w^2* there is a constraint-induced Pareto benefit over the unconstrained revenues under this wholesale price. □

Proof of Result 4. The service provider has first-period revenue function

with decision vector w¹. As

∀i≠k, the condition

holds, and the revenue function is supermodular (Topkis, 1998) in (w _i ¹, w _k ¹), i≠k. Thus, the service provider has a wholesale price vector that maximizes first-period revenue myopically (that is with no consideration of the second period), which we denote w^1m. □

Proof of Result 5. From equation (19) we have that π _i ¹=(p _i ¹−w _i )d _i ¹(p¹)). This equation is log separable, as

The first term on the right-hand side depends on own price only, and in addition, because we have assumed that ((∂²d _i ^j(p))/(∂p _i ^j∂p _k ^j))⩾0, the resellers’ second-period revenues are log-supermodular (Vives, 1999) in prices (p _i ², p _k ²), i≠k. Hence, there is a Nash equilibrium in prices, which we denote by p^1m, that maximizes the resellers’ first-period revenues myopically. Multiple equilibria form a lattice (Topkis, 1998), so that equilibria are well ordered, and with greatest element preferred by all resellers. □