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An Empirical Model of Dynamic Limit Pricing: The Airline Industry

An Empirical Model of Dynamic Limit Pricing: The Airline Industry

U(ɛ ijm , p jm , x jm ,

U(ɛ ijm , p jm , x jm , ξ jm ; θ) = x jm β − αp jm + ξ jm + ɛ ijm = δ jm + ɛ ijmwhere θ = (α, β) is the parameters of the system to be estimated.For convenience ofnotation the subscript m will be dropped except where its inclusion aids understanding.As is standard in the literature p j is the fare for product j, ξ j is a common (to allconsumers) vector of unobserved product attributes that acts to vertically differentiateairline-product pairs, e.g. upholstery quality, cabin attendant friendliness, and x j willinclude all observed product attributes, e.g. seat size, number of connections, distance,etc. Consumer i chooses product j if and only if,U(ɛ ij , p j , ξ j , x j ; θ) ≥ U(ɛ ik , p k , ξ k , x k ; θ) ∀ k = 0, 1, ....J (1)where product 0 is the outside option, to be thought of as taking an alternative mode oftransport.Heterogeneity in tastes will enter the model through ɛ, which is assumed to be independentlydistributed multivariate extreme value across consumers and products. Fromequation (1) these ɛ define the set of consumers who purchase product j. Formally, thisset is given by A j (δ) = {ɛ i : δ j + ɛ i,j ≥ δ k + ɛ i,k , ∀k ≠ j}. The market share of airline j isgiven by the probability that ɛ i is contained in A j ,∫s j (δ(x, p, ξ), x, θ) =A j (δ)f(ɛ, x, σ ɛ )dɛ =exp(δ j )∑k∈J exp(δ k)(2)where the second equality follows from the assumptions on the distribution of ɛ.The measure of consumers in market m is given by M.As with elsewhere in theliterature (e.g. Borenstein and Rose [1995], Ciliberto and Tamer [2009]) this is taken to bethe average of the population in the two endpoints of the route 3 . The observed output of3 The other popular definition (e.g. Berry and Jia [2008], Ciliberto and Williams [2010]) is to use thegeometric mean of the endpoints.8

firm j is thus,q j = Ms j (x, ξ, p; θ) (3)with q m = (q 1m , q 2m , ...q Jm ) denoting the vector of quantities on market m.3.2 Firm BehaviorWhilst the consumer side of the model is familiar the supply side will require more attention.As an overview, on each independent market there exists a monopolist incumbent anda potential entrant. At time t the incumbent has a private marginal cost of serving passengers,c I,t , and sets prices so as to maximize the flow of future profits. Importantly, priceis used as a strategic variable for two purposes. The first is as a mark-up over marginalcosts so as to choose the point on the demand curve that maximizes static profits as inmodels of complete information. The second is as a mechanism to deter entry, the ideabeing that the firm can set its price so as to signal its marginal cost. This second objectivemay well operate in conflict with the first so that price deviates from that of a staticprofit maximizing monopolist. The intuition is simple: by setting a price below that of amonopolist the entrant may infer that the marginal cost of the incumbent is sufficientlylow so as to make entry unattractive. If the signaled cost is insufficient to deter entry thenthe potential entrant can pay an entry cost κ t . Once the entrant is in the market the firmscompete as complete information price setting duopolists for the remainder of the game.The setup thus far described is similar to that of Milgrom and Roberts [1982]. Whereit will differ is that the marginal cost of the incumbent is assumed to be time varying andthat it will evolve according to a Markov process defined on continuous support. This generatesa repeated incentive to signal (Roddie [2012a,b]). Additionally, the entry cost willbe independently time varying and identically distributed. The advantage of this approachis that it can rationalize entry at different periods of the game, in contrast to the all ornothing outcome of the existing literature.9

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