A Welfare Assessment of Revenue Management Systems

Appendix A: Notes on the Theoretical Model

Our model of revenue management is in the spirit of Talluri and Van Ryzin (2004) who study revenue management in the transport industry. In their model one product consists of a seat and a price, meaning that two seats with similar comfort but at different prices constitute two different products. In contrast, we consider that the price is a separate variable.^[18] In our case, the revenue manager never proposes simultaneously the exact same product at two different prices. When we allow for some randomness in the preferences of consumers, e.g. when demand is simulated using a multinomial logit approach, consumers cannot choose the highest price.

An additional difference is that we allow for different products to have different capacity constraints. Taking the example of two flights departing the same day, once all the tickets for the first one have been sold, the revenue manager can only propose tickets for the second one. Introducing physically constrained characteristics has an impact on the revenue manager’s strategy. When there is only one capacity constraint, the revenue manager only proposes menus on the efficient frontier (see Talluri and Van Ryzin 2004). A menu of tariffs is efficient if linear combinations of other menus are less profitable except when they increase the total probability of purchase. This is not necessarily the case in our model. We take the example of two types of products, each limited to 200 units. Possible prices for each product are P={50,70,90,110}. Consumer preferences are defined as in Section 4. In Figure 5, we plot all combinations of products and prices according to their total probability of purchase and expected revenue. Product-price pairs that are efficient according to Talluri and Van Ryzin (2004) are represented by coloured dots. When there is only one capacity constraint, other combinations should not be proposed by the revenue manager. When we allow for several capacity constraints, the revenue manager proposes an inefficient menu of prices, (90,50), 12.9% of the time when λ=0.6. This percentage corresponds to an average over 1000 simulations.

Figure 5:

Counterexample to the efficient frontier argument. Note: Total probability of purchase (horizontal axis) and the associated expected revenue (vertical axis) of the different possible menus of prices. Efficient frontiers are represented by plain lines. Menu A, which is not on the efficient frontier, is played 12.9% of the time when the arrival rate of the consumer is λ=0.6.

B: Values of Parameters

Table 5 summarizes the values of parameters for consumer preferences and train characteristics.

C: Discussion of Revenue Management Vs. Optimal Fixed Price

Compared to the fixed price strategy, the increase in consumer surplus is driven by higher load rates.^[19] The finer choice set of the revenue manager allows her to post higher or lower prices depending on past sales. For λ=0.9, when the spread around the optimal price is equal to 5, the revenue manager posts the lowest and highest prices with a relatively high frequency: respectively 21% and 17% of the time. The larger the spread, the smaller the frequencies at which the lowest and highest prices are posted since they become dominated by the fixed price.

Our results seem to indicate that revenue management is especially useful for a large demand to the total capacity (at least when consumers are homogeneous) because products may sell out with a substantial probability before the end of the booking period and by changing the optimal fixed price, the revenue manager can increase substantially her profit. Conversely, for a low demand, the capacity is too large whatever the price posted by the revenue manager. She can hope to sell all the products by proposing very low prices, but it would not increase her profit. In that case, the capacity constraint does not matter and the optimal fixed price is always optimal.

Figure 6 displays the average posted price as a function of the booking period for different intensities of demand and a small spread, ξ=7. The average level of posted prices increases with the arrival rate of consumers. The optimal fixed price is played with probability 1 at the beginning of the booking period. Consistently with our previous observation, deviations from the optimal fixed price occur if the market is sufficiently large, and these deviations are centred around the optimal fixed price, i.e. the revenue manager has on average equal incentives to lower or increase the price. At the end of the booking period, the revenue manager tends to post the lowest price more often. This is consistent with the incentives derived from the theoretical model.

Figure 6:

Change in posted average prices when the price set is P={po−7,po,po+7}.

Figure 8 of Appendix G shows similar qualitative results in that case.

In a more realistic framework, the revenue manager has to optimize her profit over two types of products from which consumers can choose. We want to test whether adding more flexibility in the choices of consumers affects the previous results. We suppose that the choice set P of prices available to the revenue manager remains identical between the two types of products. In the case of a fixed price strategy, we also assume that the proposed price is identical for both types. In transports, this could correspond to a situation in which the revenue manager is required to set an identical price for all transports running on a particular origin-destination leg. The parameters used in this section can be found in Table 5 of Appendix B, column (5). α=−0.5 means that consumers now prefer product 1 over 2. In the case of two different flights the same day on the same origin-destination leg, we interpret this as one flight being more convenient than the other.

We compute the optimal fixed price po and the simulated average profit and consumer surplus associated with this price. Then, we construct the choice set P of the revenue manager to include the optimal fixed price and some small variations around it:

Figure 9 of Appendix G shows that revenue management strongly increases both profits and consumer surplus. All our previous results extend to this case and are even more pronounced. Depending on spread values and arrival rates, profits rise from 4% up to 12% and consumer surplus increases between 20% and 30%.

We explain the impact on profit by the lack of flexibility induced by the unique fixed price for the two types of products. The revenue manager cannot price discriminate between the more attractive product and the other. The greater consumer surplus is driven by higher load rates. The optimal fixed price is too high for the less attractive product. Lowering the price for this product generates higher sales and a higher surplus. For λ=0.5, 88% of the seats of the less attractive product are sold under revenue management with a spread ξ=15 whereas only 61% of the seats are sold under a fixed price strategy.

D: Extension to Heterogeneous Consumers

E: Mixed Equilibria

In a mixed equilibrium, players randomize over their action set, here: P˜={p¯i,p¯i,+∞},i=R,A. We have shown that in this special case, p˜i=+∞ is always dominated by the other prices unless Xi=0. Therefore, any mixed equilibrium of a continuation game at t is a randomization over {p¯i,p¯i} for i=R,A. In the following, we denote σi=P(p˜i=p¯i) and by abuse of notation σi also denotes the action of player i when she randomizes.^[20]

Finding an equilibrium in mixed strategies of the continuation game at t amounts to find σi for i=R,A:

Suppose i plays σi. Then, −i needs to be indifferent between p¯−i and p¯−i to play a mixed strategy, which is summarized by the following condition:

where:

Equation (E.1) therefore yields:

F: Multinomial Logit Approach and Horizontal Differentiation

Here are more detailed explanations about why overall demand for the proposed products increases if we introduce an additional vertically undifferentiated choice. Assume that given a vector of price p, the utility of buying any product is given by u whether we are in the monopolistic or the duopolistic case.

Then, in the monopolistic case, the probability of buying a product is:

In the duopolistic case, the probability of taking of choosing the product of one firm (e.g. firm 1) is given by:

which is of course lower than the probability of buying a product in the monopoly. However, the overall probability of buying in the duopoly is given by 1−12eu+1, which is higher than in the monopoly.

To give some economic intuition to this result, we say that adding an additional type of products, even vertically undifferentiated, can create horizontal differentiation.

G: Simulation Results

Figure 7:

The welfare impact of revenue management for 1 train and 3 possible prices. Note: Average change in profit and consumer surplus between optimal fixed pricing and revenue management for different intensities of demand for homogeneous products.

Figure 8:

The welfare impact of revenue management for 1 train and rich price sets. Note: Average change in profit and consumer surplus between optimal fixed pricing and revenue management for different intensities of demand for homogeneous products and various sizes of choice sets.

Figure 9:

The welfare impact of RM for 2 train and both share 1 optimal fixed price.

Figure 10:

The welfare impact of RM for 2 train and an optimal fixed price for each.

Figure 11:

The welfare impact of revenue management with increasing willingness to purchase.

Figure 12:

The welfare impact of revenue management with noisy arrival rates.

Figure 13:

Distribution of the RM’s price choices when consumers are heterogeneous. Note: The dotted lines represent the optimal fixed prices for each train.

Figure 14:

Change in posted average prices in the case of heterogeneous consumers.