Efectos colaterales

Previous price randomization models have shown that firms can deliberately vary prices as a means to partition customers with differing information. Those with more information of prices in the marketplace will on average pay a lower price than those with limited information. Striking in the models of Butters (1977), Varian (1980), and Burdett and Judd (1983) is that consumers differ only in the amount of information received in the marketplace; the willingness to pay of all consumers in this class of models is the same. Given the experience of certain markets – especially the airline market – this assumption seems quite unrealistic.

Besides heterogeneity in information, this model allows heterogeneity in

consumer valuation. The model finds that under the right mix of proportion of consumer types and valuations, that there will exist a price randomization equilibrium where firms sell to all consumers. The presence of the gap, g, is a good indication of how well price randomization works selling to consumers with different values. First, if g is greater than the spread between any price L in the support of FL and H, then price randomization

between consumers with different valuations is not a feasible equilibrium. In this case, firms will serve the low end of the market through price dispersion rather than serving

both groups of consumers. Charging the high types higher prices does not completely make up for the discrete loss of low types. On the other hand, too many uninformed high types may make steep discounting unprofitable for firms. In this case g is negative. The extra revenue generated selling to the low types will not outweigh the revenue lost by discounting to the uninformed high types. Only g values in the middle will be associated with equilibria with two different consumer types.

The comparative statics section suggests how price dispersion adjusts to changing consumer characteristics. Sometimes there will be a consistent change in the entire distribution of prices. Increasing both H and α lowers the value of both FH and FL at any

given price and concentrates the weight of the distribution functions at the high end. With higher willingness to pay of the business travelers or proportion of loyal business travelers, firms will discount less as they can make greater revenue from this smaller group of business travelers.

Changing other consumer characteristics only affects one part of the price distribution. β and θ have no effect on FH. Increasing θ decreases FL. θ has a big

impact on where FL is placed. Increasing β could either increase or decrease FL

depending on the proportion of uninformed high types θα and the spread between H and L. A large spread between H and L or high θα will result in ∂FL/∂β being negative. For

the parameter values used in this paper, FL does not change much with a change in β.

Proposition 9 provides good perspective of the how the two cumulative

probability distribution functions move when an exogenous variable changes. As each cumulative distribution function changes, the interval g connecting FL(L) and FH(M)

changes as well. Sometimes both functions change in a way that makes predicting g simple – such as when θ changes. As θ, the proportion of high types increases, the lower probability distribution function FL decreases - moving closer to a stationary FH thus

reducing g. Changing α - the proportion of uninformed high types – is not as simple. Here the proportion of uninformed low types β determines how the two functions move together when α changes. A high β generally implies that ∂g/∂α will generally be positive.

One important result is the case where the uninformed consumers approach one or zero. In the case where the uninformed α and β consumers approach zero, the

cumulative distribution collapses down to the lowest price which is near marginal cost. Near Bertrand pricing becomes the norm with very occasional price jumps to take in account the few uninformed consumers. In the case where the uninformed α and β consumers approach one, there is a spike in the cumulative distribution function around the monopoly high price for consumers and monopoly low price for consumers. The probability distribution for prices becomes very close to a bimodal one at the monopoly high and low prices.

Another important result in the model is in the area of increasing the number of firms and the resulting price distribution. This result continues some of the interesting result from previous price randomization models. Varian (1980), for instance, gives a clear answer about what happens to the lowest price when the number of firms increases. In Varian’s model, the lowest price decreases but weight around the highest price

increases when the number of firms increases. This model predicts the same result as Varian (1980).

The model developed allows some flexibility. Two counterintuitive results can occur under limited conditions. First, price randomization can occur even in cases not representing the usual demand case where there is a larger proportion of high types θ than low types (1 – θ). In this type of equilibrium, the two probability distributions FH and FL

are close together. g is small as the markup above L does not need to be very large to make up for the discrete loss of the low types. A small α of uninformed high types also helps make such equilibria possible.

Price randomization can also occur in cases where α is less than β. Normally we would expect α to be larger than β as there could be some correlation between value and the tendency to shop around. In this case, the lower cumulative distribution function FL

will become steeper. Under this case, ∂g/∂H changes signs to be negative.

There are a few main lessons learned from the graphs. First, the informed low types are important in determining equilibria with two different consumer types. Figure 12 provided an example of equilibrium with two different consumer types occurring despite the uninformed high types being so close to one. However, the shape of the distribution functions depend on the proportion of the uninformed high types. Figures 18 and 19, as well to a lesser extent Figures 16 – 17 show that the shape of the FL(ÿ) curve

does not change dramatically as β is changed, holding all other variables constant. Another conclusion that can be drawn is that equilibria do exist even when the proportion

of uninformed business and leisure consumers are quite low. This is important in

showing that this model is relevant even when loyalty by consumers is very low or when search costs are low. Figure 19 provides an example of this phenomenon. A third result is that there are equilibria with two different types of customers in cases when firms would appear to not have an incentive to run sales because the proportion of business travelers who are uninformed is high. These equilibria need the proportion of low types who are informed to be high and the overall proportion of business travelers to be

relatively low. As Figure 12 shows, firms mostly randomize prices around the monopoly price of the business type, but occasionally run extreme sales to capture the informed low types. Finally, as the pdf graphs indicate in Figures 24 – 25, fares in this model cluster around certain prices. This holds in the two valuation model, as well as a three valuation model depicted in Figures 26 and 27.

Finally, the ten percent airline sample data shows that fares cluster around certain prices, indicating a place where the theory matches the data. Sometimes they cluster only around a leisure price – such as in the Los Angeles to Honolulu market. Other times they mainly cluster around a business price – such as in the Minneapolis to Chicago O’Hare market. Sometimes they cluster in more than one place – sometimes almost

symmetrically as in the Minneapolis to Atlanta market or non- symmetrically as in the Austin – Providence market. Sometimes there are several modes as there is in the Washington National to New York LaGuardia market. This clustering of fares around certain prices might suggest that there are regions in the distribution of prices that airlines find it profitable to target customers while regions where there are dips are places in the

distribution of prices where it is not profitable to target consumers. This would be a place where the theory matches the data.

Even with price discrimination providing an explanation behind differences between airline fares, the price dispersion story with different consumer types still applies in the face of price discrimination. This clustering of fares observed in overall markets occurs even controlling for price discrimination between fare types coach Y and coach discount YD. As the Minneapolis - Atlanta and New York City - Orlando markets indicate, there is fare clustering at the YD level for many individual airlines. Clustering occurs for the full coach Y fare for the largest carriers in terms of market passenger share - Continental, Delta, and US Airways - on the New York - Orlando market. Price discrimination cannot explain these clustering of fares on the subclasses of coach fares.

This model begins an important step in the direction of including valuation as part of the analysis behind price randomization. Future research can now include linking up the literature on price discrimination and price randomization since consumers in both types of models have heterogeneous valuations. Adding asymmetry in the model also is good area for future research. How does dispersion change if one firm has more of the uninformed types or if another firm has lower costs than the existing firms? Could there be a cost to keep a certain group of customers loyal, or not willing to compare with other firms? Does entry by new firms become deterred with firms with large shares of loyal high value customers? Answering these questions could provide further predictions on how electronic commerce will continue to shape markets.

2:“Capacity and Random Prices” 2-1. Introduction

Airlines are characterized by rapidly changing prices and mobile but limited capacity. Prices are notorious for changing frequently and at the last minute. Price dispersion is large. Firms offering heterogeneous prices for the same product, based on imperfect consumer information, have been modeled with such price dispersion models such as Butters (1977), Varian (1980), and Burdett and Judd (1983). Consumers with the same valuation of the good are segmented on the basis of the amount of information that they receive about the price from each of the firms in the market. The more information they receive, the greater their ability to shop between offers, and thus the lower price on average they pay for the product or service. Those shoppers with little information do not have this luxury of comparing between firms so they go with the offer that they receive. These models are formally identical to models in which some consumers are “locked-in” not by loyalty but by preference, and others are indifferent between suppliers.

Understanding the difference in the information that shoppers are facing, firms face a tradeoff between capturing these “shoppers” (consumers with more information) by pricing lower versus pricing higher and getting more revenue from the consumers with little information in the marketplace. The nature of this tradeoff induces randomization in equilibrium. The reason is that a pure strategy is easily exploited by rivals, by a price just below the pure strategy price. In such a circumstance, the firm is better off with a monopoly price on the loyal customers. The overall randomization of pricing depends on

the distribution of consumer information, the number of firms competing in the marketplace, and the profits of the firm in serving customers with few options. Randomization works as a best response in this type of situation because firms can balance the risk of going for the group of informed consumers against the steady revenue of the less informed shoppers. Standard models provide a satisfactory account of the behavior of airlines with regard to their prices – randomization induces a dispersion that is observed.

Besides randomization of prices, limited capacity is important within the airline industry. Schedules are set ahead of time, with many flights departing either with excess demand or with many excess seats that are left empty at the time of departure. Since it is costly to shift around schedules more than a few times a year, airlines have to price within the confines of a fixed seating allotment on routes by the flight, day, week, and month. This problem of capacity is not unique to the airline industry, as hotels, stadiums, heavy manufacturing (whose lines cannot be flexible to be changed into another product quickly) industries all grapple with the same issues of pricing within the same capacity within the same period of time.

Kreps and Scheinkman (1983) provided a two-stage model that allowed two firms to build capacities in the first period and then announce prices in the second period. If each firm built capacities in the first period below the best response function of the other firm’s reaction curve, then both firms announced Cournot prices with probability one in the second period, provided the consumers are allocated using the efficient rationing rule.

However, if one firm has a larger capacity than the others, but the sum of the capacities doesn’t clear the market at a price of zero, price randomization will result. (This is the mixed strategy solution to the classic Bertrand Paradox; see Baye and

Morgan, 1997.) The smaller firm will have a higher probability density at each price than the larger firm until the cumulative density function equals one and will earn a smaller portion of profits due to the lower capacity size. Yet the large firm benefits as well since part of the time it will place an atom on the highest price in the probability distribution and earns more profits due to its larger capacity size.. The big assumption in this price randomization model is that the smaller firm did not build enough capacity to have the option of forcing prices to zero (assuming zero marginal cost of production), regardless of what action the larger firm takes.

Both information and capacity models lead to a similar result of price

randomization. I will show that the similar outcomes of these two distinct models is not an accident. In the equilibrium formula for the pricing distributions, there is a

multiplicative term involving capacity or the amount of information that consumers are receiving. This term is important in the actual calculation of both the price distribution and ultimately the profits that firms will earn in each of the models. Given that these type of models are similar in how they solve for random prices, how does a model that

encompasses both kinds of features – limited information of consumers and limited capacity of firms – work? What does pricing look like in that type of model? Would that validate these two different types of models or would something different be said about price randomization? This chapter answers that question by synthesizing the two models.