Airline networks, traffic densities, and value of links

Abstract

In airline networks, a link creates profits for its carrier in conjunction with the carrier’s other links. In other words, a link has “network” value. One prominent mechanism behind this network value is a hubbing effect: adding one single link to a hub creates many connecting routes. This paper studies a different and less explored mechanism of network value. It relies on the observation that passengers generally prefer a higher flight frequency (mainly because it provides more flexible options for travel times). Specifically, when a carrier adds a link, the created connecting routes will increase the traffic densities on adjacent links. As the carrier raises flight frequencies to meet the higher densities, there creates a positive effect on demand. By structurally estimating a model that incorporates this mechanism, I am able to quantify the density effect. It is found to be about 3.8 times as large as the hubbing effect. Furthermore, the model shows that the competitive impact of an airline entry (i.e., adding a new link) goes greatly beyond the local city-pair market where the entry happens.

Appendix

1.1 Uniqueness of the fixed point

Here I consider the mapping ${\Psi }(p,\xi ,\cdot )$ over the space of positive vectors. Below I present a parameter condition ($\theta _{1},\theta _{2}<\frac {\lambda }{2}$) which guarantees a unique fixed point. The uniqueness requires no restrictions on the network topology (e.g., whether it is a point-to-point or hub-spoke network). The condition is a sufficient one and the proof accounts for the worst case scenarios that could break the convergence to a unique fixed point. The downside of this is that the condition is stringent and estimated parameter values fail to satisfy it. Nevertheless, in my experiences, under the estimated parameter values, iteration of ${\Psi }(p,\xi ,\cdot )$ always converges to the same point no matter how I change the starting point. The main difficulty in extending the proof to include a larger range for $\theta$ is the complexity of competition effects. Suppose that one is willing to assume away the competition between routes, so that each route competes with the outside good separately (${\Psi }_{j}=M_{m}e^{v_{j}}/(1+e^{v_{j}})$). Then one can relax the condition to $\theta _{1},\theta _{2}<1$ by following essentially the same proof.

Proposition 1

If $\max \{\theta _{1},\theta _{2}\}<\frac {\lambda }{2}$ , then ${\Psi }(p,\xi ,\cdot )$ has a unique fixed point. Moreover, iteration of ${\Psi }(p,\xi ,\cdot )$ always converges to the fixed point.

Proof

First, let us define a “distance” measure${\Xi }$ betweenany two positive vectors of the same length. Note that${\Xi }$ is always greater thanor equal to 1, and ${\Xi }= 1$ implies$V=V^{^{*}}$.

$${\Xi}(V,V^{*})\equiv\max\left\{ \frac{V_{1}}{V_{1}^{*}},...,\frac{V_{n}}{V_{n}^{*}},\frac{V_{1}^{*}}{V_{1}},...,\frac{V_{n}^{*}}{V_{n}}\right\} . $$

Let D and$D^{*}$ betwo positive demand vectors. With the specification in Eq. 2.1, it is seenthat

$${\Xi}\left[F(D),F(D^{*})\right]\leq{\Xi}(D,D^{*}). $$

Let$f(D;\theta ,\sigma )$ be thevector that collects $f_{j}$,which is defined as in Eqs. 2.3and 2.4.Then,

$${\Xi}\left[e^{f(D;\theta,\sigma)},e^{f(D^{*};\theta,\sigma)}\right]\leq{\Xi}\left[F(D),F(D^{*})\right]^{\max\{\theta_{1},\theta_{2}\}}, $$

wherethe exponential of a vector refers to the vector collecting the exponential of each element. Let$v(p,\xi ,D)$ be thevector that collects $v_{j}$.Then,

$${\Xi}\left[e^{v(p,\xi,D)},e^{v(p,\xi,D^{*})}\right]={\Xi}\left[e^{f(D;\theta,\sigma)},e^{f(D^{*};\theta,\sigma)}\right]. $$

It isnot difficult (though not easy either) to see that the nested logit specification (2.6)implies

$${\Xi}\left[{\Psi}(p,\xi,D),{\Psi}(p,\xi,D^{*})\right]\leq{\Xi}\left[e^{v(p,\xi,D)},e^{v(p,\xi,D^{*})}\right]^{2/\lambda}. $$

Putting everything so far together, wehave

$${\Xi}\left[{\Psi}(p,\xi,D),{\Psi}(p,\xi,D^{*})\right]\leq{\Xi}(D,D^{*})^{2\max\{\theta_{1},\theta_{2}\}/\lambda}. $$

So${\Psi }$ shrinks thedistance between D and $D^{*}$. Asa result, ${\Psi }$ cannothave two distinct fixed points, and the iteration ofΨ leads to theunique fixed point. □

1.2 Monte Carlo experiment and bootstrapped standard errors

The model in this paper deviates from the standard framework for estimating differentiated goods markets by (i) introducing a fixed-point notion for the demand function and (ii) tying together all the city-pair markets by a network structure. The goal of the Monte Carlo experiment here is to check whether the estimator is still well-behaved under such conditions. Fix a network and a set of “true” parameter values. The Monte Carlo experiment first draws a set of the unobservables, $\xi$ and $\omega$, then computes a Nash-Bertrand equilibrium to produce a simulated data, and finally recovers the parameter values from the simulated data. The parameter recovery uses the estimator outlined in Section 3. I repeat the experiment many times to obtain a sample of recovered values for each parameter.

The model does not impose any distributional assumption on $\xi$ and $\omega$. However, for Monte Carlo experiment, such a distribution needs to be specified. One possibility is to use i.i.d. normal distributions. However, it is reasonable to expect a positive correlation between ξ_j and $\xi _{k}$ if j and k belong to the same market. It is also reasonable to expect a positive correlation between $\omega _{j}$ and $\omega _{k}$ if j and k share a link. In fact, these correlations are present in the backed-out values of $\xi$ and $\omega$. To replicate the data generating process as much as possible, I add these correlations in the Monte Carlo experiment.

As to the choice of the “true” parameter values, I use the benchmark point estimates in Tables 4 and 5. As to the choice of network, it may be ideal to simply use the observed network in the data. However, it is very costly to compute the equilibrium for such a large network and much more costly to repeat the experiment for many times. In principle, there are many alternative choices for the network; one may simulate a random network and then fix it throughout the experiment. To mimic the observed network structure as much as possible, I choose to use a downsized version of the observed network. Specifically, I exclude a link if the observed number of passengers on the link is no more than 2.5% of the size of the associated city-pair market. The resulting “backbone” network has 602 links and $n^{b}= 7,353$ routes across all the carriers. For a network of this size, it is possible to compute $\partial D(\xi ,p)/\partial p$ with implicit function theorem (see Section 3.1), which significantly speeds up the computation. The downsized network effectively reduces the data size used to recover the parameters, which should inflate the variances of the estimator. To compensate, I set the standard deviations of the distributions for $\xi$ and $\omega$ at $\sqrt {n^{b}/n}$ of their levels in the real data.

Figure 6 illustrates the results by showing the kernel densities of the recovered values for four parameters: $\theta _{1}$, 𝜃₂, the intercept in $\gamma$, and the coefficient in γ for the point-to-point distance. Keep in mind that the estimator is sequential (i.e., it first recovers the demand parameters then supply parameters), so the dispersions in $\gamma$ partly come from the estimation errors on the demand parameters. It is seen that each density is centered around the true parameter value and well-behaved. The same result also applies to the other parameters not shown in Fig. 6. The standard deviations of the recovered values are used as bootstrapped standard errors, reported in Tables 4, 5, and 6.

Fig. 6

Monte Carlo results. Notes: The experiment is repeated for 500 times. Each plot shows the kernel density (with optimal bandwidth) of the recovered values for a particular parameter. The vertical line shows the “true” parameter value

1.3 Equilibrium computation

Below shows the algorithm that I use to compute a Bertrand-Nash equilibrium. The algorithm basically solves for the prices that satisfy the first-order conditions (2.8). The inputs include the network structure, ξ, and marginal costs mc. The outputs include the equilibrium prices, demand, and densities.

1 :

Choose a convergence criterion $\tau >0$ and an initial value for the price vector $p^{*}$.

2 :

Iterate ${\Psi }(p^{*},\xi ,\cdot )$ to find its fixed point $D(p^{*},\xi )$.

3 :

Compute $\partial D(p^{*},\xi )/\partial p$ and ${\Delta }$ defined as in Eq. 3.2. Let $p^{**}=mc-{\Delta }^{-1}D(p^{*},\xi )$.

4 :

If $\left \Vert p^{**}-p^{*}\right \Vert _{\infty }<\tau$, exit; otherwise, update $p^{*}$ by moving it closer to $p^{**}$.

5 :

Repeat step 2-5.

The most time-consuming part of the algorithm is step 3. There are mainly two complexities. First, this step needs to differentiate an implicitly defined demand function w.r.t. tens of thousands of prices; the differentiation w.r.t. each price requires fixed-point iterations of ${\Psi }$ (The iterations may be avoided for smaller-size networks, see Section 3.1). Second, ${\Delta }^{-1}D(p^{*},\xi )$ involves solving a large linear system. (As noted in Section 3.1, ${\Delta }$ can be organized into a block-diagonal matrix, but the blocks corresponding to the major carriers are still of significant sizes.)

Given the computational complexity, it is important to reduce error propagation. In my experience, it is particularly important to keep the numerical errors small at two places. First, I use a very stringent stop criterion for the fixed point iteration of ${\Psi }$. Second, I use the two-sided formula (also called the central difference method) when computing the derivative $\partial D(p^{*},\xi )/\partial p.$