INTERTEMPORAL PRICE DIFFERENTIATION
3. Equilibrium
We now examine the properties of optimal dynamic pricing. Starting in the next subsection, we first characterize equilibrium prices in the early market, while taking the shadow cost of capacity as given. The value of capacity is then derived subsequently in Subsection 3.2 by solving for the equilibria in the last-minute markets. The analysis of how competition effects the dispersion of prices over time is deferred to Section 4.
3.1. The early market
First note that, sinceµ1<X¯1, there can be no symmetric equilibrium where air- lines are constrained by their capacity in the early market. Accordingly, consider a candidate equilibrium where no airline is constrained by its capacity int =1
and suppose that airlines set prices such that every customer finds it optimal to buy a ticket. Just like in the case of Salop competition, we can derive the demand for any airline by finding the customerθ who is indifferent between purchasing from airline iand airline jfor any j ≠ i. Solving the indifference condition yields that demand for airlineion this particular segment is given by qi
1 = (1−α{p
i
1− p
j
1}) ×d/2. Hence, in a symmetric equilibrium with p
j
1 = p1 for all j, total demand per airline is given byQ1i = (1−α{pi1 −p1})/N, after aggregating over allN−1 segments of the market in which airlineioperates.
In order to characterize equilibrium prices in the early market, we also need to account for how expected profits Πi2(x¯2i, ¯x2)in the late market change as a function of airlinei’s capacity, given that all other airlinesj≠iare equipped with the symmetric equilibrium capacity ¯x2j=x¯2=x¯1−Q
j
1. Taking the shadow value of capacity into account, airlines choose their pricep1ito maximizeQ1i(pi1,p1) ×
(pi
1−c) +Π2i(x¯2i, ¯x2). It follows that as long as expected future profits Πi2are concave in their first argument, airlines’ best response functions are given by8
pi 1(p1) = 1 2[ 1 α +c+p1+ dΠ2i d ¯x2i ].
Hence, in a symmetric equilibrium all airlines set the equilibrium price,
pi1= p1= 1 α +c+ dΠi2 d ¯x2i ,
and share the early market, Qi
1 =Q1=1/N.
Hence, because ¯x1 > 1/N, capacity constraints are indeed not binding in the early market. Also, as long as the value of capacity in period 2 is not too large, firms do set prices such that all customers want to buy a ticket and the period 1 market is covered. We therefore conclude that, as long as Π2i satisfies the technical condition discussed above and prices in period 1 are sufficiently low, any symmetric equilibrium in the early market will be uniquely pinned down by the late markets.
8 To keep the presentation comprehensible, we do not require airlines to set positive prices in the early market. In light of our results, it will become clear that this simplification is without loss of generality and no airline will ever charge negative prices in equilibrium.
Lemma 1: Fix anyΠi2∶ [0, 2]
2→R
+and suppose thatΠi2is concave in its first
argument and thatdΠi2(x¯2, ¯x2)/d ¯x2i < (v−αc− 3
2)/α. Then, in any symmetric
equilibrium, airlines split the early market equally and charge a price of p1= 1 α +c+ dΠi2 d ¯x2i (x¯2, ¯x2), (1) wherex¯2=1/N.
Lemma 1 implies that airlines always enter the late market with the same residual capacity ¯x2=1/N, irrespective of how valuable capacity is in period 2. Nevertheless, as the marginal value of capacity dΠ2i/d ¯x2i increases, airlines have smaller incentives to underbid each other in the early market, leading to higher equilibrium prices in period 1.
3.2. The late market
In order to fix ideas, consider the symmetric equilibrium situation, in which all airlines have the same residual capacity ¯x2i =x¯2available at the beginning of period 2. Airlines then end up to be capacity constrained in period 2 whenever the last minute demandµ2turns out to exceed the available market capacity ¯X2=Nx¯2. In this case, airlines can set monopoly prices and serve as many customers as they have free seats available, without interfering with any of their competitors. By Assumptions 1 and 2 airlines then find it optimal to sell off exactly their remaining capacity and to extract the full surplus of the marginal customer by charging p2(µ2) = (v−X¯2/2)/α.
On the other hand, in states of the world where market capacity ¯X2exceeds ag- gregate demandµ2, airlines can no longer commit to charging monopoly prices and enter into price competition. Because airlines now have an incentive to marginally undercut their competitors in order to sell additional seats, prices are determined by the same logic as in the early market and profits drop discontinu- ously. In the unique equilibrium of the subgame, each airline sellsµ2/Nseats at a price of p2(µ) =µ2/α+ceach.
It follows that in any situation where all firms have the same capacity available expected profits in the late market are given by
Πi2(x¯2, ¯x2) =
∫
¯ X2 0 µ2 αNd(µ/2) +∫
2 ¯ X2(v−αc− ¯ X2 2 ) ¯ X2 αNd(µ/2), (2)where the first term reflects states where airlines are in competition, and the second term reflects states where industry demand exceeds industry supply and airlines charge monopoly prices.
From Lemma 1, we know that the incentive to shift capacity from the early to the late market is an important determinant of equilibrium prices in the first period. Intuitively, the main benefit from selling an extra seat in period 1 lies in restricting the overall capacity, which increases the probability of ending up with monopoly power in period 2. Standing against this effect is that conditionally on having monopoly power, having an extra seat available in the late market is valuable since it allows airlines to sell this seat at a mark-up.
Formally, this tradeoff corresponds to the incentive for an airlineito unilat- erally deviate from a symmetric equilibrium by selling one more seat in period 1. Because this incentive depends on the expected continuation payoff from all reachable subgamesoff the equilibrium path, we can no longer restrict ourselves to symmetric situations in period 2. In Appendix A, we show that (i) the expected payoff Πi2(x¯2i, ¯x2) resulting from the asymmetric equilibria in these subgames is differentiable around Πi2(x¯2i, ¯x2)∣x¯2i=x¯2 as given by equation (2); and (ii) that Π2i(x¯2i, ¯x2) is globally concave in ¯x2i. Accordingly, we can characterize the in- centives to deviate from the symmetric equilibrium by differentiating (2) with respect to its first argument. The following lemma summarizes the discussion and states the relevant conclusion.
Lemma 2: In any symmetric equilibrium,Πi2is globally concave in its first argu-
ment, and marginal returns to an unilateral increase in second period capacityx¯2i are given by dΠ2i(x¯ i 2, ¯x2) d ¯x2i ∣x¯i 2=x¯2 = − (v−αc− 3 ¯X2 2 ) ¯ X2 2αN +
∫
2 ¯ X2α −1(v−αc−X¯ 2)d(µ/2). (3) Equation (3) reflects the previously discussed tradeoff. The first term stands for the strategic benefit fromrestrictingcapacity in order to avoid price competition; the second term defines the expected value of reserving capacity in order to serve last minute customers in high demand states.99 There is another case, not reflected in (3), in which demand is so low that supply always exceeds demand. In this case, however, having extra capacity available on the last minute market is worthless, because airlines are already unable to sell their full capacity and any additional seats will go unsold.
3.3. Equilibrium prices and quantities
Equipped with Lemmas 1 and 2, we can now state the equilibrium predictions for the full game. In particular, by Lemma 2, dΠi2/d ¯x2i satisfies the conditions of Lemma 1, so that we can characterize the equilibrium by substituting (3) into (1).
Proposition 1: There exists a unique symmetric equilibrium in pure strategies. In this equilibrium, prices in the early market are given by
p1= ( 1 α +c) − v−αc−3/2 2αN + v−αc−1 2α , (4)
and airlines share the market, selling1/N tickets each. In the late market, prices are given by p2(µ2) =⎧⎪⎪⎨⎪⎪ ⎩ µ2/α+c if µ2≤1 (v−1/2)/α if µ2>1, (5)
and airlines sellmin{1,µ2}/N tickets each.
In equilibrium, first period prices ensure that airlines are indifferent between selling tickets on the early and on the late market. The value of capacity in period 2 is determined by the tradeoff discussed above and is captured by the second and third term in equation (4). That is, as reserving capacity for the period 2 market becomes more attractive, selling tickets on the early market becomes less attractive, which increases the equilibrium price in period 1.