easyjet Pricing Strategy: Should Low-Fare Airlines Offer Last-Minute Deals?
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1 easyjet Pricing trategy: hould ow-fare Airlines Offer ast-inute Deals? Aendi Proosition : he general otimization roblem is reresented by: Period : a ubject to 0 Period : a ubject to 0 and - > 0 hus, we can define agrangian in eriod with coefficient λ and in eriod with coefficients and μ o find the subgame erfect equilibrium, we solve the game backwards arge aacity: / / When caacity is large enough, the constraints in both eriods do not bind and thus their coefficients are equal to zero Otimal rices are given by h and h Intermediate aacity: < <, where is given above and is comuted in the last case We first derive the first-order condition of the second-eriod agrangian with resect to, which yields 0 ince in this intermediate case the constraint is binding, 0 and thus 0 herefore, the rice is reresented by ubstituting this rice into the agrangian of eriod yields the following: Differentiating with resect to and equating to zero yields the otimal solution for It is straightforward to check that if, then - > 0, thus = 0, and the otimal rice in eriod is reresented by
2 where is the caacity shortage; in other words, / / ubstituting this eression into the equation for yields Very ow aacity: /, > 0, and the constraint is binding; ie, - = 0 It follows that and thus the airline does not sell to the tourist segment ubstituting this equation into the eression for yields the following otimal second-rice-eriod for the lowcaacity case: ince the airline sells only to the business segment, it follows that ow aacity: < < where is given above We net derive the eression for ote that, when /, the airline sells only to business consumers at rice / his generates otimal unconstrained rofits from the business segment However, when the caacity eceeds that threshold, the airline continues to charge the monooly business rice / until caacity is large enough that the benefit revenue from the additional tourist travelers eceeds the losses from deviating from the otimal rice for the business traveler segment he loss in selling to business travelers is given by l m m m m m he revenue from selling to the tourist segment is given by Equating these two equations yields the threshold caacity level : hus for any, the airline charges, and for any, the airline charges and ote that / / for and that when 0 If this condition is not satisfied, then the airline moves directly from a olicy of low caacity to a olicy of intermediary caacity
3 Proosition : arge aacity: > = et be the tourist with the highest utility that will urchase the ticket in eriod at rice What we do net is find the values of,,, and that will constitute an equilibrium to the new game We begin at eriod, where rofits are reresented by / aimization with resect to the rice yields = / and / In eriod, the eected rofits deend on the business segment that did not urchase tickets in eriod, and the eected third-eriod rofits hus, second-eriod rofits can be reresented as First-order conditions imly that the rice in eriod is reresented by = _ / and rofits are reresented by In eriod, rofits are reresented by We now use the fact that is the marginal consumer for whom the utility from urchasing the ticket in eriod is eactly equal to the utility from waiting and buying the ticket in eriod his yields the following equation for : = β ubstituting = / and solving for yield: / ubstituting this eression for in the rofit function yields he first-order condition for otimality yields the following solution for rices: b and / b b ote that, since it follows that b b and, since, h b where the latter is the unconstrained rice of the two-eriod game ubstituting into the objective function yields Also, it is easy to see that the derivative of the rofits with resect to β yields 0 Demand of the unconstrained case: D D
4 D i i i D hus, for any caacity above this demand < the firm is unconstrained and follows the receding olicy Define ' =a{, } When ' < <, we have to solve the following roblem Intermediate aacity: ' < < et be the tourist with the highest utility that will urchase a ticket in eriod at rice We begin at eriod, where rofits are reresented by /, st 0 /, where is the caacity at the beginning of eriod When the constraint is binding, the rice is given by and the rofits by In eriod, the eected rofits deend on the business segment that did not urchase tickets in eriod, and the eected third-eriod rofits hus, second-eriod rofits can be reresented as First-order conditions imly that the rice in eriod is reresented by = _ / and rofits are reresented by ote that the remaining caacity at the end of the third eriod is In eriod, rofits are reresented by We now use the fact that is the marginal consumer for whom the utility from urchasing the ticket in eriod is eactly equal to the utility from waiting and buying the ticket in eriod his yields that: = β ubstituting and solving for yields note that this analysis is true when < α, which corresonds to ubstituting these eressions for and into the rofit function yields
5 5 First-order conditions yield the following solution for rices: c and c he rofits of the constrained case < < are given by where et, we comared these rofits to the rofits from the two-eriod game where the firm does not offer a last-minute deal and caacity is large < : his term is ositive if < < where, Define i i i={, } It is clear that, if we show that, then for the relevant range of caacity the rofit for the last-minute-deal is larger than that of the two-eriod game without a last-minute deal : It is straightforward to show that this eression is ositive since 0 is ositive for 0 < β < : It is straightforward to show that this eression is ositive since is ositive for 0 < β < Also, it is easy to see that the derivative of the rofits with resect to β yields
6 0 First eriod rice can be greater or less than the rice of the unconstrained two-eriod game: c c h Define i as rices of eriod i in the game we solved in roosition and as rices in eriod i of the two-eriod unconstrained game roosition, large caacity c h It is straightforward to show that if > c h We still must show that if ote that c h, which is ositive if c h ote that, when then it is always true that However, if, there then eists a range of caacity such c h that It is easy to show that 0, which is ositive for 0 < = / and negative for 6 0 < he solution for the game where the firm chooses the otimal β in the first stage: In this section, we solve the game where the airline decides the frequency of the last-minute deal ie, the robability that it introduces the third-eriod β We start the analysis with the case of large caacity and continue with the case of smaller caacity arge aacity: > = et be the tourist with the highest utility that will urchase the ticket in eriod at rice What we do net is find the values of,,,, and β that will constitute an equilibrium to the new game We begin at eriod where rofits are reresented by / aimization with resect to the rice yields = / and / In eriod, eected rofits deend on the business segment that did not urchase tickets in eriod, and eected third-eriod rofits hus, second-eriod rofits can be reresented as First-order conditions imly that the rice in eriod is reresented by = _ / and rofits by In eriod, rofits are reresented by 6
7 We now use the fact that is the marginal consumer for whom the utility from urchasing the ticket in eriod is eactly equal to the utility from waiting and buying the ticket in eriod his fact yields the following equation for : = β ubstituting = / and solving for yields / ubstituting this eression for in the rofit function yields he first-order conditions with resect to and β yield the following solution for rices: b and β = / b ubstituting β into the terms of the ricing terms yields and b b b It is obvious to see that 9 Demand of the unconstrained case: D D D Di i i hus, for any caacity above the demand <, the firm is unconstrained and follows the ' ' receding olicy Define =a{, } When < <, we have to solve the following roblem ' Intermediate aacity: < < et be the tourist with the highest utility that will urchase a ticket in eriod at rice What we do net is find the values of,,, and that will constitute an equilibrium to the new game We begin at eriod, where rofits are reresented by /, st / 0, where is the caacity at the beginning of eriod When the constraint is binding, rice is given by and rofits by In eriod, eected rofits deend on the business segment that did not urchase tickets in eriod, and rofits eected in eriod hus, second-eriod rofits can be reresented as 7
8 8 First-order conditions imly that the rice in eriod is reresented by = _ / and rofits are reresented by ote that the remaining caacity at the end of eriod is In eriod, rofits are reresented by We now use the fact that is the marginal consumer for whom the utility from urchasing the ticket in eriod is eactly equal to the utility from waiting and buying the ticket in eriod his fact yields the following equation for : = β ubstituting and solving for yields ubstituting these eressions for and into the rofit function and deriving with resect to yields c ubstituting this eression into the rofits function and deriving it with resect to β yields 0 c c As the rofit decreases with β, the firm chooses β = 0 Proosition : Define the caacity at the beginning of the i th eriod as i Recall that, due to uncertainty about business consumers arrival, the initial caacity at the beginning of the third eriod can be = if business consumers do not arrive or = if business consumers do arrive We also define to be the tourist with the highest utility that will urchase a ticket in eriod at rice Recall that this consumer is indifferent between urchasing a ticket in eriod at rice and urchasing a ticket during eriod at rice We net solve the game starting with the highest caacity case
9 arge aacity: > = o find the subgame erfect equilibrium, we solve the game backward starting at eriod, where rofits are reresented by / aimization with resect to rice yields = / and / In eriod, the eected rofits are given by aimization with resect to rice yields = / and We now use the fact that is the marginal consumer for whom the utility from urchasing the ticket in eriod is eactly equal to the utility from waiting and buying the ticket in eriod his fact yields the following equation for : = θ + θ ubstituting = / and solving for yields = In eriod, the eected rofits are given by aimization with resect to rice yields = α/ and = α his indicates that tourist consumers do not urchase during the first eriod and thus the firm sells to business consumers during eriod at rice = / and to tourist consumers in eriod at rice = α / his in this case there is no rice discrimination within the tourist segment High-Intermediate aacity: = < < his is the case at which the firm is unconstrained in eriod, and if the business consumers do arrive, the firm will be constrained in eriod We can consider two cases in eriod : If business consumers do not arrive during the second eriod, the initial caacity is given by = In this case, the firm is unconstrained during the third eriod and rofits are reresented by / aimization with resect to rice yields = / and / If business consumers arrive during the second eriod, the initial caacity is given by = In this case, the firm is constrained during the third eriod and has to adjust the third-eriod rice accordingly hus, demand at rice is = olving this equality, we determine that the third-eriod rice is and 9
10 In eriod, the eected rofits are given by aimization with resect to rice yields = / and We now use the fact that is the marginal consumer for whom the utility from urchasing the ticket in eriod is eactly equal to the utility from waiting and buying the ticket in eriod his fact yields the following equation for : = θ + - θ ubstituting and for and solving for yield: In eriod, the eected rofits are given by aimization with resect to rice yields and = α his indicates that tourist consumers do not urchase during the first eriod and thus the firm sells to business consumers during eriod at rice = / and to tourist consumers in eriod at rice = α / Just as in the revious case, there is no rice discrimination within the tourist segment in this case as well ow-intermediate aacity: = < < his is the case at which the firm is constrained in eriod, and if the business consumers do arrive, the firm will not have any seats left for the third eriod We can consider two cases in eriod : If business consumers do not arrive during the second eriod, the initial caacity is given by = In this case, the firm is constrained during the third eriod and has to adjust the third-eriod rice accordingly hus, demand at rice is = olving this equality, we determine that the third-eriod rice is and If business consumers arrive during the second eriod, there is no caacity left for the third eriod; in other words, 0 In eriod, the firm is also restricted by the initial caacity and the demand at rice, = olving this equality, we determine that the second-eriod rice is Recall that he second-eriod eected rofits are, which yields 0
11 We now use the fact that is the marginal consumer for whom the utility from urchasing the ticket in eriod is eactly equal to the utility from waiting and buying the ticket in eriod his fact yields the following equation for : = - θ ubstituting and solving for yields In eriod, the eected rofits are given by aimization with resect to rice yields It is easy to verify that < α hus, under this caacity range, the firm sells to some tourist consumers during the first eriod and to other tourist consumers during the third eriod o find out what is, note that this case is valid only if is smaller than the unconstrained demand in eriod he latter is equal to Recall that and substituting into this equation yields = ow aacity: < If caacity is lower than the unconstrained demand in eriod,, the firm has no incentive to sell tickets in eriod ince if the business consumers arrive in eriod it sells in higher rice to all of them and if they do not arrive it sells the same seats to tourists in eriod hus, in this case, no rice discrimination within the tourist segment is ossible Proosition : We assume that the tourist segment is slit into two sub-segments A fraction -δ of the consumers arrives at eriod 0 and has a valuation for the flight that is drawn from a uniform distribution 0,, where < / he rest of the tourist segment δ arrives in eriod one with valuation for the flight that is drawn from a uniform distribution, We resent only the case which is equivalent to the low-intermediate caacity of the revious case Proosition 5;, where U and U Define the caacity at the beginning of the i th eriod as i Recall that, due to uncertainty about business consumers arrival, the initial caacity at the beginning of the third eriod can be = if
12 business consumers do not arrive or = if business consumers do arrive We also define to be the tourist with the highest utility that will urchase a ticket in eriod at rice Recall that this consumer is indifferent between urchasing a ticket in eriod 0 at rice 0 and urchasing a ticket during eriod at rice We only consider the case where <, since if > the firm does not sell in eriod 0 and the roblem is reduced to the roblem we solved in Proosition 5 We consider two cases in eriod : If business consumers do not arrive during the second eriod, the initial caacity is given by = In this case, the firm is constrained during the third eriod and has to adjust the third-eriod rice accordingly Also, as we are interested in the case were <, then it means that the firm sets the monooly rice in eriod hus, demand at rice is = olving this equality, we determine that the third-eriod rice is and If business consumers arrive during the second eriod, there is no caacity left for the third eriod; in other words, 0 In eriod, the firm sells only for business consumers and maimizes the following eected rofits; with resect to If business consumers arrive during the second eriod, the initial caacity is given by; In this case, the firm is constrained during the second eriod and has to adjust the second-eriod rice accordingly hus In eriod the firm maimizes the following eected rofits; his otimization roblem yields In eriod 0, we use the fact that is the marginal consumer for whom the utility from urchasing the ticket in eriod 0 is eactly equal to the utility from waiting and buying the ticket in eriod his fact yields the following equation for : 0 = - θ - ubstituting and solving for yields; 0 In eriod 0, the eected rofits are given 0 by 0 aimization with resect to rice yields that < 0 It can be shown he eressions for and 0 can be retrieved from the authors by request
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