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\begin{comment}
\title{Airline Fuel Costs: Hedging and Pass-Through\footnote{We would like to thank Yonghong An, Zhiqi Chen, Matt Weinberg and participants at Southern Economic Association Meeting (2016) and the Industrial Organization Workshop at University of Oklahoma for helpful comments and suggestions.}}

\author{Myongjin Kim\and Qihong Liu \and Long Shi\footnote{Address: Department of Economics, University of Oklahoma, 308 Cate Center Drive, Norman, OK 73019. E-mails: mjkim@ou.edu; qliu@ou.edu; Long.Shi-1@ou.edu.}}

\date{\today}
\end{comment}

\begin{document}
\setcounter{page}{49}

\begin{center}
\large{\textbf{Chapter 2: Airline Fuel Costs: Hedging and Pass-Through}}
\end{center}

\begin{abstract}
Airline industry is an important part of the U.S. economy, and it has been on a roller coaster in recent decades, due to various demand-side (e.g., terrorism, recession) and supply-side (e.g., fuel cost) shocks. Fuel cost, an important factor on the supply-side, has been volatile. In this paper, we analyze how fuel cost affects airlines' pricing decisions and how this impact varies with market structure. Our results show that a 10\% increase in fuel costs leads to a 1.2\% increase in airfare in the same quarter, and up to a total of 1.7\% increase in airfare in the next 4 quarters. We also construct fuel cost measures at the carrier-route-quarter level, and find that the fuel cost pass-through rate is more than 100\%. In particular, a \$1,000 increase in fuel cost will lead to a \$975 increase in fare revenue in the same quarter, and a further increase of \$970 in the next 4 quarters. The impact of fuel cost also varies with market structure. In more concentrated markets, airfare is more responsive to fuel cost changes and fuel cost pass-through rate is also higher. Drastic relative changes in airline specific fuel costs also allow us to test for sunk cost fallacy. This occurs when an airline gains or loses a significant lump sup amount from its fuel hedging contracts, which affects its reported fuel cost but not its true economic cost of using fuels. We find mixed evidence. When Southwest's fuel cost dips significantly below the level of other major airlines, we find that Southwest reduces its prices further. On the other hand, when Delta's fuel cost becomes significantly higher than other major airlines, it does not see to raise its fare relative to other major airlines except when its fuel cost disadvantage is very large.
\end{abstract}

\newpage

\section{Introduction}

%Cubic spline (first and second derivatives are smooth), Stata has existing command (something like "csinterpol")


Airline industry is an important part of the U.S. economy. In 2014, it generated about \$204 billion of total operating revenue and provided more than 11 million jobs.\footnote{http://www.statista.com/statistics/197680/total-operating-revenues-in-us-airline-industry-since-2004/ and http://airlines.org/industry/ respectively.} It has been quite profitable in recent years, after significant industry consolidation. The whole industry has been on a roller coaster in the new millennium, due to various demand-side shocks (such as terrorism, recession) and supply-side shocks. Fuel cost is one of the most important factor for the demands-side shocks. It is commonly thought that fuel costs represent about one-third of the airlines' total cost, with their weights slightly higher when fuel costs peaked and slightly lower in recent years when fuel cost has gone down dramatically. Fuel costs have had quite some runs in the new millennium.

%\begin{comment}
\begin{figure}[!hbt]
\centering
\includegraphics[width=5in,height=4in]{afc_wo_AFC.eps}
\caption{Airline fuel costs ($afc_{it}$)} \label{fig:fig1}
\end{figure}
%\end{comment}

In Figure \ref{fig:fig1}, we plot the monthly airline fuel prices for selected airlines.\footnote{We use the terms ``airline fuel prices" and ``airline fuel costs" interchangeably.} Two features are worth pointing out. First, fuel costs are volatile during the sample periods. It went from below \$1 per gallon in the beginning our sample period to almost \$4 per gallon at its peak. Then in a short few months, they dropped to slightly above \$1, only to rise above \$3 per gallon again. A natural question is how airlines respond to changes in fuel costs. In particular, in the short run when little can be done about product re-positioning, how do changes in fuel costs affect ticket prices? Are the effects uniform across markets?

A second prominent feature in Figure \ref{fig:fig1} is that the airlines' fuel prices match each other well in general. There are exceptions, however. Southwest clearly enjoyed lower fuel costs during the time periods when fuel prices experienced enormous increase.  On the other hand, for several months, Delta ended up paying higher fuel prices when the market price(s) were dropping quickly. What caused these discrepancies in fuel costs? It has been established that hedging is an important reason (see, for example, Carter, Rogers and Simkins (2004)). Southwest is well known to have benefited greatly due to its hedging contracts when fuel prices went up, and Delta's hedging loss has also been featured in the media.\footnote{See, for example, ``As Fuel Prices Soar, Southwest Airlines Protects Itself by Hedging Fuel Costs," \textit{Wall Street Journal}, January 16, 2001, ``Southwest Airlines gains advantage by hedging on long-term oil contracts," \textit{New York Times}, November 27, 2007, `` When Delta went gambling on jet fuel", \textit{Fortune}, June 2, 2014, and ``Delta CEO Admits To \$4 Billion Lost In Hedging Fuel Costs," \textit{Forbes}, June 2, 2016.}  An airline's fuel cost  can significantly differ from the cost of other airlines due to its unique hedging positions. For example, consider an airline which locked in fuel price at lower level when actual fuel prices skyrocketed. That airline would have a large financial gain from the hedging contract, which affects the accounting cost but not the true economic cost of fuel for that airline. If we think prices are determined by marginal cost, then a lump sum gain should not impact an airline's pricing decisions. However, a financial gain would allow the airline to ease on price increase and take away market share from its rivals.\footnote{It is noted that T-Mobile has been taking customers away from AT\&T and Verizon, facilitated by its competitive pricing which ironically was paid for by AT\&T in the form of a \$4 billion merger breakup fee.} The empirical question at hand is how airlines' ticket prices depend on its fuel hedging positions. If we consider hedging gain/loss as sunk when making ticket price decisions, are airlines prone to sunk cost fallacy?

We first investigate the question of how airline fuel costs affect ticket prices. Two types of estimation are employed. The first one estimates an elasticity measure, for example, if airline fuel cost goes up by 10\%, how much percentage will airfare go up by?  Our results show that if fuel cost (\$ per gallon) goes up by 10\%, then airfare will go up by 1.2\% in the same quarter. Evaluated at the average one way fare of about \$180, the price increase is about \$20 each way. We also allow fuel costs to have persistent impacts up for to 4 quarters later, and the cumulative impact of a 10\% increase in fuel costs is a total of about 2.9\% increase in airfare. We then analyze how the impact of fuel cost on airfare varies with market structure, and find that airfare is more responsive to the fuel costs in more concentrated markets. Our estimation include airline level controls such as merger and bankruptcy status. We find that airline raise fares in the quarter when merger is announced. They lower prices when filing for bankruptcy but make no additional price changes when they remain under bankruptcy protection.

Economists are also interested in cost pass-through rate, in particular, whether there is incomplete cost pass-through. The above elasticity analysis cannot answer this question. Instead, we need to bring fuel cost and airfare into the same unit. We do so by converting both fuel cost and airfare (revenue) into the same unit of carrier-route-year-quarter level. We find that the fuel cost pass-through rate is more than 100\%. Consider a market with $HHI=0.6$ (a measure for market competition levels). A \$1 increase in fuel cost on a carrier-route-year-quarter will lead to \$1.7 in ticket revenue ( $0.975+1.17*0.6$). If cumulative impacts are included, the fuel cost pass-through rate adds up to about 280\% (again for a market with $HHI=0.6$). Similar to the elasticity analysis, we find that fuel cost pass-through rate increases with HHI.

We then investigate the issue of sunk cost fallacy. We find sunk cost in the case of Southwest. Consistent with the media coverage, we find that Southwest lowers its prices (which are already lower than other major airlines' prices) further when hedging significantly reduces its fuel costs (to below 90\% level of other major airlines' average fuel costs). However, further reduction in its fuel cost (to be below 75\% of other major airlines' average fuel costs) does not lead to additional reduction in Southwest's airfares. On the other hand, we did not find evidence of sunk cost fallacy for Delta airlines. When Delta's fuel cost is significantly above the average of other legacy airlines' fuel costs, Delta actually reduced its airfare relative to other legacy airlines. This is rather counterintuitive, and we are in the process of exploring this puzzle further. Our result also shows that further increase in Delta's fuel costs relative to other legacy airlines raises Delta's price by about 2\%, which is consistent with sunk cost fallacy.

\subsection{Literature review}

Our paper is related to the extensive literature on pricing in the airline industry, ranging from hub-and-spoke system and the related hub premium (Brueckner et. al. 1992, Borenstein 1989), price discrimination and price dispersion (Borenstein and Rose (1994), Gerardi and Shapiro (2009), Dai et. al. (2014)), to how prices are impacted by merger (e.g., Prince and Simon (2017), Liu, Ghosh and Liu (2017)), airline financial conditions (e.g., Ciliberto and Schenone (2012 IJIO)) or macroeconomic conditions and business cycle (e.g., Cornia et. al. (2011)). We control the numerous factors that were analyzed in these studies, including competition intensity, airline financial conditions, merger and macroeconomic economic conditions (which are captured by time fixed effects). Our focus, however, is on how airline fuel costs affect ticket prices, and how this impact varies with market characteristics such as competition intensity. Our paper contributes to this extensive airline pricing literature by offering a new angle to analyze airlines' pricing decisions.

Our paper is closely related to the cost pass-through literature. It is easy to see that under perfect competition, pricing at marginal cost means a 100\% pass-through rate. One would expect that for firms with market power, when cost goes up, they will raise prices but not by as much, i.e., \textit{incomplete pass-through}. The issue of cost pass-through rate has been studied in standard microeconomic theory. The answer is relatively clear cut: cost pass-through rate depends on  the curvature of demand curve. In particular, if demand is log-concave, then the cost pass-through rate will be less than 100\%. Since Log-concavity is a relatively weak assumption for demand functions, one would expect to find incomplete cost pass-through in most markets.

Pass-through rate is an important question in at least two areas of economics.\footnote{See Fabra and Reguant (2013) for an excellent discussion of the cost pass-through literature.} In international trade, one major question of interest to researchers is exchange pass-through (Goldberg and Knetter (1997)). Even if the exporter experiences no cost or price change in its own currency, its price may change when denominated in the importer's currency. A natural question then is how changes in exchange rate affects the final price. Exchange rate pass-through is also used to infer firms' market power, as in the pricing-to-market literature.\footnote{See Lavoie and Liu (2007) and papers cited there.} In particular, if the product market is perfectly competitive, then marginal cost pricing indicates a complete exchange rate pass-through. On the other hand, if firms have market power (such as in the U.S. airline industry), then we would expect them to price above marginal cost and may absorb cost increase partially, suggesting an incomplete cost pass-through. In public economics, pass-through is key to the question of tax incidence (e.g., Marion and Muehlegger (2011), Colon and Rao (2016)). When a tax is levied on a product/service, how much will it be borne by consumers and how much by firms? The answer depends on cost pass-through rate, if we view tax as an increase in cost. Correspondingly, consumers will bear more of the cost increase if and only if the cost pass-through rate is more than 50\%. The natural question then is whether this is confirmed in empirical studies.

Results of empirical studies of cost pass-through has been mixed. For example, Gron and Swenson (1999) find that the level of pass-through in US automobile market is significantly higher when accounting for firms' decisions in upstream factor market.  Kim and Cotterill (2008) estimate the demand in the U.S. processed cheese market. They allow both Nash-Bertrand price competition as well as collusion between the firms. Their estimates suggest incomplete pass-through in most cases. Fabra and Reguant (2013) use rich micro-level data from the Spanish wholesale electricity, and see how much of the introduction in emissions costs is passed through to wholesale prices. They find almost complete pass-through, with average pass-through rate above 80\% which goes up close to 100\% during peak time. Linn et. al. (2015) also look at the cost shock to wholesale electricity price, but in the U.S. market where the cost shock is due to drastic reduction in delivered price of natural gas. They find that incomplete pass-through can occur due to productions shifting across firms and analyze the corresponding environmental impacts. It has been found in some studies that the rate of cost pass-through can be more than 100\%. That is, a \$1 increase in cost will lead to a more than \$1 increase in price. For example, Miller et. al. (2015) find more than 100\% cost pass-through in the cement industry and use their result to estimate possible welfare effect of policies. We also find more than 100\% cost pass-through if one includes the cumulative impact of fuel cost change on ticket price.

Our paper is also related to studies on fuel efficiency (e.g., An and Zhao (2016)), fuel hedging (Lim and Hong (2014)) and in particular, sunk cost fallacy. Sunk cost fallacy is featured in various undergraduate economics textbooks. The idea is that, any cost that is sunk should not affect your decision. Some studies (e.g., Al-Najjar et. al. (2005) and McAfee et. al. (2007)) introduce new features into the setting and show that it may be rational for individuals or firms to condition behavior on sunk costs. Friedman et. al. (2007) conduct experiments to check for sunk cost fallacy and Augenblick (2015) empirically analyze sunk cost fallacy in penny auctions. Ho et. al. (2015) take advantage of policies in Singapore which substantially raised the sunk cost of buying cars. They find that clear evidence of sunk cost fallacy: a higher sunk cost of buying cars leads to significantly more driving time. We have conflicting evidence of sunk cost fallacy. On one hand, when Southwest's fuel cost goes down relative to other major airlines due to hedging contracts, we find that Southwest further reduces its airfare, consistent with sunk cost fallacy. On the other hand, when wrong hedging raised Delta's fuel costs significantly above other major airlines, we do not always observe Delta raising its airfare.

The rest of the paper is organized as follows. In Section \ref{sec:data} we describe the data and report summary stats. Section \ref{sec:pass-through} and \ref{sec:fallacy} present the results on fuel cost pass-through and sunk cost fallacy respectively. We conclude in Section \ref{sec:conclusion}.

%Nathan Miller paper on cement: (i) more than 100\% pass-through; (ii) policy experiments (we can also conduct similar experiments using our log-log specifications).

\section{Data and descriptive stats}\label{sec:data}

\subsection{Data}

%\footnote{Information for exclusion, including the number of stops and fare class (first / business / coach), is also from DB1B dataset.}

We use two main sources of data. The first one is DB1B which report market fare data from the Airline Origin and Destination Survey. DB1B data is a 10\% random sample of all domestic flight tickets published by Bureau of Transportation Statistics (hereafter BTS) on quarterly basis. Following what is standard in the literature, we include only direct, coach class itineraries, both one-way and round-trip fares.\footnote{DB1B data do not distinguish non-stop flights from direct flights with a stop but no plane change.} A round-trip itinerary is split into two one-way itineraries, each with fare at half of the round-trip airfare. Several additional cleaning is done. First, we drop carrier-route-quarter combinations which only have charter/freighter flights.\footnote{The identification is through T-100 data. If T-100 data show that in a given route-quarter, a carrier only uses aircrafts in freight configuration or aircrafts with no more than 30 seats, or if none of its flights takes more than 15 passengers then we drop this carrier-route-quarter combination.}  Next, we drop the top 2\% fares at the high end, and drop fares below \$15 at the end.\footnote{We do this to avoid observations with potential punch errors and/or tickets from reward travel.} What's more, we drop the return portion of roundtrip itineraries to avoid double counting. While we include code-shared itineraries (in which ticketing and operating carriers are not the same) when calculating market share and HHI (Herfindahl-Hirschman Index), we drop them when constructing price variables. The reason is that we are not sure how revenue and cost are allocated between ticketing and operating carriers. Individual tickets are then aggregated to carrier-route-quarter cells.

The other primary dataset is an airline fuel cost. There are two types of fuel cost measures: (1) airline-specific fuel prices and (2) spot market jet fuel price common to the industry. Airline-specific fuel cost data come from  Schedule P-12(a) by BTS, a monthly report of airlines' fuel consumption and expenditure.\footnote{http://www.transtats.bts.gov/Fields.asp?Table\_ID=294} Since our ticket price data (from DB1B) are at the quarterly level, we aggregate the fuel consumption and expenditure from monthly to quarterly level as well, to calculate the airline fuel cost variable $afc_{it}$ at the airline-quarter level. Another fuel cost measure is the monthly spot price of Kerosene-Type Jet Fuel in the U.S. (we denote it $AFC_t$).\footnote{It is reported by the U.S. Energy Information Administration, http://www.transtats.bts.gov/Fields.asp?Table\_ID=294} Note that $AFC_t$ does not vary across airlines at the monthly level. If we aggregate it from monthly to quarterly level, the resulting $AFC_{it}$ varies across carriers, but this is purely due to airlines having different consumption weights across months within a quarter. The  only meaningful changes of spot market fuel prices -- in the form of intertemporal changes -- do not vary across airlines, and as such would be absorbed by our time fixed effects. Correspondingly, we do not use $AFC_{it}$ in our empirical analysis.

Next, we include a set of controls to capture route level and airline level characteristics. Route level controls first include competition measures, starting with HHI. We also include average household income at both end points of all the routes. The CBSA (Core Based Statistical Area) level income data is available on the website of Bureau of Economic Analysis (BEA).\footnote{While population is another important route-level characteristics, we use it to construct instruments for HHI following the literature. As a result, we do not include population as a separate control.} HHI is often viewed as endogenous in airline pricing literature, and researchers have suggested various instrument variables for HHI. We adopt the commonly used instruments following this literature. They include market distance of a route, average population at both end points and an indicator of airlines' average loading share at the end points. All of them were used in Gerardi and Shapiro (2009) and Dai et al (2014). The population variable is constructed using yearly population estimate from US Census Bureau, and the share of boarding carriers for each airline is calculated using T-100 dataset.

At the airline level, labor cost is an important part of operational cost besides fuel cost. In all our specifications we include average salary at the airline-year-quarter level, measured in \$1000. The data source is BTS airline financial reports. Previous studies have shown that airline's pricing decisions are likely affected by their financial/operational conditions. We include two such conditions: bankruptcy and merger, all at the carrier-quarter-year level. For bankruptcy, we first introduce a dummy variable $Bankruptcy_{it}$ which takes value of 1 in the quarter-year, $t$, when bankruptcy is announced for an airline $i$ and 0 otherwise. We also use $Bankruptcy\_Duration$ which takes the value of 1 if the carrier is under bankruptcy in that quarter-year and 0 otherwise. The dummy variable $Merger$ captures the transition period from when a merger is announced till the merger is closed. In particular, it takes the value of 1 if the carrier experiences this merger transition in the period and 0 otherwise.


\subsection{Definition of market and variables}

Below we summarize the variables we use in the estimation. $i$, $j$ and $t$ refers to carrier, route and year-quarter respectively.

\bi

\item $roundtrip\_sh_{ijt}$: The share of roundtrip tickets, taking some value between 0 and 1. (Roundtrip tickets are counted twice, but in different directions. Given that we did distinguish origins from destinations when defining routes, that is not a problem.)

\item $mktfare_{ijt}$: average (one-way) fare (Although we are free from the double-counting issue, we are aware of that it could be distorting the price measure to compute the airfare for the directional flight by dividing the round-trip ticket price by two. For example, a round-trip ticket consists of two components: the fare from an airport A to an airport B and the fare from B to A. It is well-known that these fares are not the same but the fare ratio is bigger than 1 in many cases as shown in Kim and Shen (2016). In order to alleviate this concern, we perform a robustness checks using only one-way fares later on.)

\item $HHI_{ijt}$: sum of squared market shares based on ticketing carrier. Take value between 0 and 1.

%Share of passengers buying tickets from the ticketing carrier on the specific route.

\item $afc_{it}$: airline specific fuel cost (\$/gallon).

\item $Merger_{it}$: a dummy variable which takes value 1 if the carrier is in merger transition, i.e.,  after the merger is confirmed (either announced or approved) but before merger is completed.

\item $Bankruptcy_{it}$: a dummy variable which takes value 1 in the quarter when the carrier files for bankruptcy and 0 otherwise.

\item $Bankruptcy\_Duration_{it}$: a dummy variable that takes value 1 when the carrier is under bankruptcy and 0 otherwise.

\item $salary_{it}$: average salary. We use its natural log in the regressions.

\item $market_j$: We employ the directional market definition, i.e. the route A to B is a different market from the route B to A, as in Gerardi and Shapiro (2009). This raises the concern that our standard errors in the estimation might not be robust. We perform a robustness check by dropping the returning flight.)

\item $acquisition_{it}$: This is a variable to be collected for a robustness check. Aircraft purchase is also one of the important cost that airlines pay for.

\ei

\subsection{Descriptives stats}

Our sample period runs from 2001 to 2014, and the summary statistics for the main variables are presented in Table 1.

\begin{comment}
Note that BTS only requires airlines with annual revenue over $\$20$ million to report their fuel and labor cost. We compare the summary statistics in Table 1 with collapsed DB1B data before merging with cost variables, and it shows that dropping smaller airlines does not significantly change market- and carrier- level characteristics in our study. This is understandable as only a limited number of passengers choose to travel with those small airlines. Besides, most small airlines primarily offers charter service, which has already been dropped from our observations.
\end{comment}

%And as there is anecdotal evidence that the operating carrier bears all the cost in codesharing agreements, we drop codesharing itineraries as well to make the pass-through process less ambiguous.
\begin{comment}
\begin{table}[!htbp]\centering
\caption{Summary statistics} \label{table:sumstat1}
\medskip

\begin{tabular}{l c c c c }\hline\hline
\multicolumn{1}{c}{\textbf{Variable}} & \textbf{Mean}
 & \textbf{Std. Dev.}& \textbf{Min.} &  \textbf{Max.} \\ \hline
average\_fare & 178.637 & 76.609 & 15 & 1033.33 \\
number\_of\_passengers & 1275.077 & 2272.718 & 1 & 65221 \\
share\_of\_roundtrip\_tickets & 0.803 & 0.205 & 0 & 1 \\
HHI & 0.699 & 0.287 & 0.151 & 1 \\
afc & 2.076 & 0.939 & 0.553 & 6.736 \\
average\_salary & 14.97 & 3.408 & 6.511 & 33.686 \\
merger & 0.071 & 0.256 & 0 & 1 \\
bankruptcy & 0.012 & 0.108 & 0 & 1 \\
bankruptcy\_duration & 0.091 & 0.288 & 0 & 1 \\\hline
\multicolumn{1}{c}{N} & \multicolumn{4}{c}{196,053}\\ \hline
\end{tabular}
\end{table}
\end{comment}

From Table 1, we can see that the average ticket price is about $\$179$ with a standard deviation of $\$76.6$. Recall that this is for one-way fare and roundtrip share is about $80\%$ on average. Average number of passengers on a carrier-route-quarter is 12,750 (DB1B is a 10\% random sample). While some markets are fairly competitive -- the lowest HHI is only about 0.15, competition is more limited in most markets. The average HHI is almost $0.7$, slightly higher than the HHI in other studies (e.g., Borenstein and Rose, Dai et. al. (2014)). There are 12 airline mergers and 13 bankruptcies in our sample, but less than 10\% of our observations is affected by either one respectively.

\begin{comment}
\begin{table}[htbp]\centering \caption{Summary statistics\_Non-monotonic Routes \label{nonmonsumstat}}
\begin{tabular}{l c c c c }\hline\hline
\multicolumn{1}{c}{\textbf{Variable}} & \textbf{Mean}
 & \textbf{Std. Dev.}& \textbf{Min.} &  \textbf{Max.} \\ \hline
mktfare & 169.443 & 66.961 & 17.667 & 1033.33 \\
passengers & 1833.524 & 2830.819 & 1 & 65221 \\
roundtrip & 0.827 & 0.153 & 0 & 1 \\
HHI & 0.48 & 0.173 & 0.151 & 1 \\
afc & 2.066 & 0.919 & 0.553 & 6.736 \\
merger & 0.075 & 0.263 & 0 & 1 \\
salary & 14.862 & 3.318 & 6.511 & 33.686 \\
Bankruptcy & 0.01 & 0.101 & 0 & 1 \\
Bankruptcy\_Duration & 0.092 & 0.29 & 0 & 1 \\
comp\_afc & 2.071 & 0.904 & 0.553 & 6.736 \\
\multicolumn{1}{c}{N} & \multicolumn{4}{c}{111213}\\ \hline
\end{tabular}
\end{table}
\end{comment}

%Next, we move on to fuel costs data. 86 carriers reported their fuel cost and consumption for at least a month. 80 of them reported for at least one complete quarter (all 3 months in a quarter). Their summary stats are presented at the bottom of Table 1.

\section{Fuel cost pass-through}\label{sec:pass-through}

In this section, we analyze how fuel costs affect airfares and how the impacts vary with market structure. We first pick a sample route ($NYC$ to $LAX$) and two airlines operating on this route ($DL$ and $AA$) to illustrate whether/how ticket prices may vary with fuel prices.

\begin{figure}[h!]
\centering

\includegraphics[width=3in,height=2.5in]{DL_Dallas_to_Atlanta}
\includegraphics[width=3in,height=2.5in]{AA_Dallas_to_Atlanta}

\caption{Fuel costs vs. ticket prices on a sample route}\label{fig:fig2}
\end{figure}

The left panel shows Delta Airline's prices which seem match well with its fuel prices. The right panel is for United Airlines and its ticket prices match well with fuel prices except a sudden drop in ticket prices late in the sample period.

\subsection{Elasticity}

We first regress ticket prices on airline specific fuel costs ($afc$). Our basic econometric model take the following form:
$$\ln fare_{ijt}= \alpha + \beta \ln afc_{it} +\sum_{k=1}^4 \beta_k \left( lag_k \ln afc_{it} \right) + \left[\gamma X_{ijt} +\lambda Y_{jt} + \tau Z_{it} \right]+ \theta_{ij} + \sigma_t,$$

where $lag_k \ln afc_{it}$ is the $k_{th}$ lag of $\ln afc_{it}$, $k=1,\cdots, 4$, terms in the square brackets are various carrier, route and time controls. The impact of fuel cost on airfare may depend on market structure. To allow for this possibility, we also interact $HHI$ with the log of airline fuel cost. $\theta_{ij}$ are combined carrier-route fixed effects and $\sigma_t$ are year-quarter fixed effects.

With the log-log specification, the coefficients for airline fuel price can be interpreted as elasticities. That is, if fuel cost goes up by 1\%, how much percentage will ticket price go up? Note that these coefficients indicate whether there is fuel cost pass-through but do not give information about pass-through rate, in particular, whether there is incomplete pass-through. This is because airfare and fuel cost have different units. Unit for airfare is dollar per ticket, while the unit for fuel cost is dollar per gallon. They are not directly comparable without knowing fuel consumption per passenger. We construct fuel consumption per passenger measures and check the pass-through rate in the next subsection.

The results are presented in Table 2. In model (1), we include $lnafc$ and its lags without controlling the fixed effects. We can see that fares increase with both the current  and lagged fuel costs. In particular, a 10\% one time increase in fuel cost will lead to a 2\% increase in fuel price in the same quarter (ignoring the interaction term $HHI\times \ln afc$), 1\% increase in price in the next quarter, but with little subsequent impacts. These estimates also illustrate the cumulative impact of fuel cost on airfares. Adding the coefficients together, a 10\% increase in fuel cost will lead to 3.0\% cumulative increase in airfares over 5 quarters. We also look at the impact of HHI. Our results suggest that fares will increase by about 2.9\% if HHI increases by 10\% (e.g., from 0.6 to 0.66), ignoring the impact through the interaction term $lnafc\times HHI$. The coefficient of interaction term $lnafc\times HHI$ is negative, meaning that airfare is less responsive to fuel costs in more concentrated markets, everything else the same.
\begin{comment}
\begin{table}[!htbp]
\centering
\caption{Fuel cost pass-through (afc)}\label{table:afc}
\medskip

\begin{tabular}{lcccccc} \hline
 & (1) & (2) & (3) & (4) & (5) & (6) \\
VARIABLES & lnfare & lnfare & lnfare & lnfare & lnfare & lnfare \\ \hline
$\ln afc$        &       0.199*** &       0.124***&       0.383***&       0.119***&       0.114***&       0.105***\\
&    (0.0149)         &    (0.0101)         &    (0.0211)         &    (0.0109)         &    (0.0209)         &    (0.0206)         \\
$lag \ln afc$    &      0.0899***&      0.0604***&      0.0620***&      0.0592***&      0.0542***&      0.0533***\\
&   (0.00628)         &   (0.00668)         &   (0.00889)         &   (0.00668)         &   (0.00707)         &   (0.00716)         \\
$lag2 \ln afc$    &    0.000544         &      0.0205**&      0.0486***&      0.0201***&      0.0174** &      0.0160* \\
  &   (0.00553)         &   (0.00602)         &   (0.00881)         &   (0.00605)         &   (0.00722)         &   (0.00720)         \\
$lag3 \ln afc$    &    -0.00958*  &      0.0488** &       0.101*** &      0.0479*** &      0.0621*** &      0.0606*** \\
            &   (0.00577)         &   (0.00587)         &   (0.00806)         &   (0.00588)         &   (0.00737)         &   (0.00736)         \\
$lag4 \ln afc$   &      0.0134*   &      0.0421*** &       0.119*** &      0.0406*** &      0.0348*** &      0.0322*** \\
            &   (0.00692)         &   (0.00736)         &    (0.0147)         &   (0.00735)         &   (0.00912)         &   (0.00903)         \\
lnsalary    &       0.272*** &       0.195*** &       0.314*** &       0.190*** &       0.164*** &       0.165*** \\
            &    (0.0141)         &   (0.00675)         &    (0.0162)         &   (0.00667)         &   (0.00887)         &   (0.00890)         \\
roundtrip\_share   &       0.319*** &      -0.143*** &       0.313*** &      -0.143*** &      -0.160*** &      -0.160*** \\
            &    (0.0205)         &   (0.00999)         &    (0.0214)         &   (0.00995)         &    (0.0150)         &    (0.0150)         \\
merger      &       0.105*** &      0.0633*** &       0.176*** &      0.0649*** &      0.0899*** &      0.0857*** \\
            &    (0.0103)         &   (0.00711)         &    (0.0104)         &   (0.00712)         &   (0.00782)         &   (0.00789)         \\
$merger\times \ln afc$&     -0.0279**  &     -0.0708*** &     -0.0941*** &     -0.0718*** &      -0.104*** &     -0.0992*** \\
            &    (0.0112)         &   (0.00710)         &    (0.0123)         &   (0.00711)         &   (0.00786)         &   (0.00793)         \\
Bankruptcy  &     -0.0778*** &     -0.0413*** &     -0.0522*** &     -0.0411*** &     -0.0535*** &     -0.0498*** \\
            &   (0.00644)         &   (0.00473)         &   (0.00721)         &   (0.00471)         &   (0.00576)         &   (0.00573)         \\
Bankruptcy\_Duration&       0.161*** &      0.0117**  &       0.152*** &      0.0103*   &     0.00857         &     0.00735         \\
            &    (0.0105)         &   (0.00578)         &    (0.0110)         &   (0.00577)         &   (0.00625)         &   (0.00626)         \\
{\footnotesize{Bankruptcy\_Duration}}$\times \ln afc$ &     -0.0136         &      0.0197*** &      0.0136         &      0.0225*** &      0.0196**  &      0.0221*** \\
            &    (0.0140)         &   (0.00668)         &    (0.0147)         &   (0.00670)         &   (0.00763)         &   (0.00769)         \\
HHI         &       0.291*** &       0.270*** &       0.279*** &       0.360*** &       0.669*** &       0.696*** \\
            &    (0.0261)         &    (0.0141)         &    (0.0292)         &    (0.0199)         &    (0.0622)         &    (0.0644)         \\
interaction&      -0.214*** &      0.0313*** &      -0.230*** &      0.0367*** &       0.126*** &       0.115*** \\
            &    (0.0185)         &    (0.0105)         &    (0.0202)         &    (0.0120)         &    (0.0385)         &    (0.0387)         \\
lnafc\_competitor&                     &                     &                     &                     &       0.110*** &                     \\
            &                     &                     &                     &                     &    (0.0161)         &                     \\
            &                     &                     &                     &                     &    (0.0161)         &                     \\
Observations  &      190600         &      190600         &      190600         &      188869         &      106726         &      106726  \\
R-squared & 0.11 & 0.33 & 0.14 & 0.33 & 0.34 & 0.35 \\
%clustered se & y & y & y & y & y & y \\
route-carrier FE & n & y & n & y & y & y \\
year-quarter FE & n & y & y & y & y & y \\
IV & n & n & y & y & y & y \\
Number of carrier\_route &  & 8,691 &  & 8,691 & 5,223 & 5,223 \\ \hline
\multicolumn{7}{c}{ Clustered standard errors by route in parentheses} \\
\multicolumn{7}{c}{ *** p$<$0.01, ** p$<$0.05, * p$<$0.1} \\
\end{tabular}
\end{table}
\end{comment}
Moving onto other variables, the coefficient for $Bankruptcy$ is negative, suggesting that carriers charge lower fares in the quarter they file for bankruptcy. On the other hand, the coefficient for $Bankruptcy\_Duration$ is positive, implying that carriers raise their prices in subsequent periods when they remain under bankruptcy. When an airline is in a merger which was approved but not completed, their prices are higher by about 10\%. The coefficient of $merger\times \ln afc$ is negative, implying that for an airline going through merger transition, its ticket price is less responsive to fuel price changes.

It is intuitive that a round trip fare is cheaper than two one-way fares, suggesting that the coefficient for $roundtrip$ should be negative. This is not what we found in model (1) which may suffer from two problems. First is the endogeneity problem. In particular, market structure variables ($HHI$) may be endogenous. Second, we are not controlling for year-quarter fixed effects and carrier-route fixed effects. Next, we first control for these fixed effects in model (2). We can see that the coefficient for $roundtrip$ becomes negative. Another change relative to model (1) is that fuel cost has smaller immediate impact but similar cumulative impacts on airfares. In particular, the current afc and all 4 lags have positive significant impacts on airfare. The cumulative impact is similar as in model (1), a 10\% increase in fuel cost leads to about 2.9\% in ticket price over 5 quarters, but the immediate impact is smaller - a mere 1.2\% increase in airfare in the same quarter.


In model (3) we use an IV approach but do not control for route-carrier fixed effects. Hausman test result suggests that HHI should be instrumented. Following the literature (e.g., Borenstein and Rose (1994), Geradi and Shapiro (2009)), we use the logarithms of market distance, average population and carriers' average share of enplanements at the endpoints as instruments. These instruments pass both weak IV test \footnote{Kleibergen-Paap F statistic: 1134.44 v.s. Stock-Yogo critical value of 13.97 for 5\% maximal IV relative bias} and over identification test\footnote{Hansen\_J Statistic:1.80; p-value: 0.62}, indicating they are both relevant and valid. The coefficient for $roundtrip$ is back to being positive, suggesting the importance of controlling for route-carrier fixed effects. Model (4) includes both fixed effects controls and IV control. The results are quite comparable to those in model (2).

So far we have only used firms' own fuel costs as explanatory variables. One would expect that a firm's price decisions may also depend on its rivals' costs. In model (5), we introduce a new variable $\ln afc\_competitor$, calculated as the weighted average of its rivals' fuel costs at the carrier-route-quarter level. Our result suggests significant and positive spillover -- a firm would raise its price by 1.1\% if its rival's fuel costs go up by 10\% -- same impact as its own current fuel cost. Results for the other variables are qualitatively the same as those in model (4). One exception is $HHI$ - we see the coefficient almost doubles. It turns out that this is not directly due to adding rival's cost, but rather because for any carrier-route-quarter where the rival has no competitor in our final sample, this rivals' fuel costs variable cannot be calculated and this observation will be dropped.\footnote{We use all airlines to calculate $HHI$ first. Afterwards, we drop the small airlines, and focus on the major airlines including low-cost carriers.} To confirm this, we use the same observations as in model (5), and re-run the regression without including rivals' fuel costs, the results, as presented in model (6), are quite comparable to those in model (5).

\subsection{Pass-through rate}

Our analysis so far has used $\ln-\ln$ for both ticket prices and fuel costs. The corresponding estimate for fuel cost can be interpreted as an elasticity measure. This elasticity measure does not directly tell us about the fuel cost pass-through rate. For the latter, we need to use level-level specification. The problem is, ticket price and fuel costs have different units: ticket price is measured as \$ per passenger while fuel cost is measured as \$ per gallon. We need to convert the measures to have the same unit.

One option is to transfer fuel price from \$ per gallon to \$ per passenger so they have the same units. But it is rather difficult to construct a measure of how much it costs to transport a single passenger. We pursue an alternative route, which is to convert ticket revenue and fuel cost both to carrier-route-quarter level. Aggregating individual fares to the carrier-route-quarter ($ijt$) level is straightforward., but calculating fuel cost at the $ijt$ level is much less so. As mentioned before, we only include the non-stop itineraries in this study. However, many flights nowadays carry both direct and connecting passengers, which makes it difficult to estimate the fuel cost spent only on the group of our concern. Besides, the aggregated fuel cost reported by airlines combines costs both for passenger and freighter flights, and the bias in passenger-cost relationship in the latter prevents us from recovering fuel cost based on regressions. In the Appendix, we explain in detail how we construct $Cost_{ijt}$, which measures ailrine $i$'s fuel cost of transporting passengers on route $j$ in quarter $t$.

In constructing the $Cost$ measure, some observations in the sample used in the previous section are dropped. We first report summary stats for the new sample. We can see that they are quite comparable to the summary stats for the whole sample in Table 1.

\begin{comment}
\begin{table}[htbp]\centering \caption{Summary statistics\_Pass-through Rate \label{table:sumstat2}}
\begin{tabular}{l c c c c }\hline\hline
\multicolumn{1}{c}{\textbf{Variable}} & \textbf{Mean}
& \textbf{Std. Dev.}& \textbf{Min.} &  \textbf{Max.} \\ \hline
Revenue & 186779.839 & 293711.597 & 15 & 7573022.5 \\
Cost & 43064.203 & 74144.051 & 1.063 & 1361936.25 \\\hline

average\_fare & 191.155 & 77.898 & 15 & 844.04 \\
number\_of\_passengers & 1113.366 & 1908.071 & 1 & 57085 \\
share\_of\_roundtrip\_tickets & 0.782 & 0.217 & 0 & 1 \\
HHI & 0.655 & 0.291 & 0.151 & 1 \\
afc & 2.348 & 0.853 & 0.553 & 6.736 \\
merger & 0.073 & 0.26 & 0 & 1 \\
average\_salary & 15.172 & 3.559 & 6.511 & 33.686 \\
Bankruptcy & 0.009 & 0.095 & 0 & 1 \\
Bankruptcy\_Duration & 0.077 & 0.267 & 0 & 1 \\\hline
\multicolumn{1}{c}{N} & \multicolumn{4}{c}{76633}\\ \hline
\end{tabular}
\end{table}

\begin{comment}
\begin{table}[htbp]\centering \caption{Summary statistics\_Pass-through Rate \label{sumstat}}
\begin{tabular}{l c c c c }\hline\hline
\multicolumn{1}{c}{\textbf{Variable}} & \textbf{Mean}
 & \textbf{Std. Dev.}& \textbf{Min.} &  \textbf{Max.} \\ \hline
CR\_Revenue & 186779.839 & 293711.597 & 15 & 7573022.5 \\
CR\_Cost & 35351.669 & 62028.656 & 0.608 & 923531.938 \\\hline

roundtrip & 0.782 & 0.217 & 0 & 1 \\
passengers & 1113.366 & 1908.071 & 1 & 57085 \\
HHI & 0.655 & 0.291 & 0.151 & 1 \\

merger & 0.073 & 0.26 & 0 & 1 \\
salary & 15.172 & 3.559 & 6.511 & 33.686 \\
Bankruptcy & 0.009 & 0.095 & 0 & 1 \\
Bankruptcy\_Duration & 0.077 & 0.267 & 0 & 1 \\\hline
\multicolumn{1}{c}{N} & \multicolumn{4}{c}{76633}\\ \hline
\end{tabular}
\end{table}
\end{comment}

Using $Cost_{ijt}$ and its various lags as explanatory variables, we estimate the fuel cost pass-through rate and see how it varies with the market characteristics variables. The results are presented in Table 3. Results for the baseline model are presented in column (1).We can see that the pass-through rate is more than 100\%. In particular, when fuel cost goes up by \$1 in one quarter, total fare in that quarter on average increases by \$1.42. If we take into account the interaction terms between $Cost$ and $Merger$, $BRD$ and $HHI$, the contemporaneous pass-through rate will be even higher.  To look at the cumulative impact of a change in fuel cost over time, we need to add up the coefficients of $Cost$ and the lagged cost variables. From column (1), they add up to almost 300\%, suggesting that a \$1 increase in fuel cost would lead to almost a \$3 cumulative increase in total fare revenue.

%\textbf{(QL: Discuss why not use rival's cost as an explanatory variable.)}\textbf{(LS: I am not the person who first suggested adding competitors' fuel price into log-log specification. According to my understanding, price is within the control of the ticketing carrier and the intuition behind the specification is that airlines will take competitors' fuel price into consideration when setting their prices. It makes sense to me. However, I doubt whether there is a clear channel linking the total revenue of a carrier to the average total (fuel) cost of ALL its competitors. To me, that specification is more acceptable if we are exploring the relationship between revenue and cost relationship at per-passenger level, which is more comparable across carrier types and passenger volumes.)}
\begin{comment}
\begin{table}
\centering
\caption{Pass-through rate}\label{table:rate}
\medskip

\begin{tabular}{lcccc} \hline
 & (1) & (2) & (3) & (4)  \\
VARIABLES & Revenue & Revenue & Revenue & Revenue \\ \hline
 &  &  &  & \\
Cost    &       1.31***&       1.06***&       0.96***&       0.90***\\
            &     (0.28)         &     (0.12)         &     (0.32)         &     (0.19)           \\
lag\_Cost   &       0.29*  &       0.25***&       0.32*  &       0.26***      \\
            &     (0.16)         &    (0.09)         &     (0.17)         &    (0.10)            \\
lag2\_Cost  &       0.12         &    -0.01         &      0.05         &     -0.01             \\
            &     (0.13)         &    (0.09)         &     (0.14)         &    (0.09)          \\
lag3\_Cost  &       0.22*  &       0.26***&       0.39***&       0.25***    \\
            &     (0.12)         &    (0.08)         &     (0.14)         &    (0.08)              \\
lag4\_Cost  &       1.01***&       0.45***&       0.93***&       0.45***\\
            &     (0.16)         &    (0.06)         &     (0.18)         &    (0.06)             \\
roundtrip\_share   &     33174.2***&     43604.5***&      -374.4         &     42214.0***\\
            &    (9201.9)         &    (4824.6)         &   (12924.5)         &    (5826.2)             \\
merger      &     -9358.1** &     -7406.7***&       792.9         &     -6783.7***\\
            &    (3732.4)         &    (1606.6)         &    (4842.0)         &    (1634.9)          \\
$ Merger\times Cost$ &       0.37***&      0.05         &       0.26*  &      0.03              \\
            &     (0.14)         &    (0.05)         &     (0.15)         &    (0.06)            \\
Bankruptcy  &      6416.6*  &     -4409.9** &     -3202.6         &     -5175.5**  \\
            &    (3560.5)         &    (2085.6)         &    (3530.9)         &    (2106.2)                 \\
Bankruptcy\_Duration&     11065.9** &     20261.3***&      7084.0         &     20526.4***\\
            &    (4550.2)         &    (1998.2)         &    (4920.3)         &    (2145.7)              \\
\footnotesize{Bankruptcy Duration} $\times Cost$&      -0.67***&      -0.19***&      -0.53***&      -0.16** \\
            &     (0.16)         &    (0.06)         &     (0.18)         &    (0.07)            \\
HHI         &    -72675.7***&     22921.2*  &    -42221.3***&    114457.8***\\
            &   (15943.5)         &   (12258.5)         &   (14622.3)         &   (19306.8)            \\
$HHI\times Cost$ &       1.11*  &       0.71*  &       2.38***&       1.17**  \\
            &     (0.57)         &     (0.37)         &     (0.71)         &     (0.51)           \\
 &  &  &  & \\
Observations & 76,633 &  74,739  & 76,633  & 74,739  \\
R-squared & 0.71 & 0.63 & 0.71 & 0.62 \\
%clustered se & y & y & y & y   \\
route-carrier FE & n & y & n & y   \\
year-quarter FE & n & y & y & y \\
IV & n & n & y & y  \\
Number of carrier-routes &  & 5,806 &  & 5,806  \\ \hline
\multicolumn{5}{c}{ Clustered standard errors by route in parentheses} \\
\multicolumn{5}{c}{ *** p$<$0.01, ** p$<$0.05, * p$<$0.1} \\
\end{tabular}
\end{table}

\begin{comment}
\begin{table}
\centering
\caption{Pass-through rate}\label{table:rate}
\medskip

\begin{tabular}{lccccc} \hline
 & (1) & (2) & (3) & (4) & (5)  \\
VARIABLES & revenue & revenue & revenue & revenue & revenue \\ \hline
 &  &  &  & &  \\
Cost    &       1.31***&       1.06***&       0.96***&       0.90***&       1.21***\\
            &     (0.28)         &     (0.12)         &     (0.32)         &     (0.19)         &     (0.15)         \\
lag\_Cost   &       0.29*  &       0.25***&       0.32*  &       0.26***&       0.15         \\
            &     (0.16)         &    (0.09)         &     (0.17)         &    (0.10)         &    (0.10)         \\
lag2\_Cost  &       0.12         &    -0.01         &      0.05         &     -0.01         &       0.13         \\
            &     (0.13)         &    (0.09)         &     (0.14)         &    (0.09)         &     (0.11)         \\
lag3\_Cost  &       0.22*  &       0.26***&       0.39***&       0.25***&       0.21***\\
            &     (0.12)         &    (0.08)         &     (0.14)         &    (0.08)         &    (0.08)         \\
lag4\_Cost  &       1.01***&       0.45***&       0.93***&       0.45***&       0.49***\\
            &     (0.16)         &    (0.06)         &     (0.18)         &    (0.06)         &    (0.06)         \\
roundtrip   &     33174.2***&     43604.5***&      -374.4         &     42214.0***&     41736.8***\\
            &    (9201.9)         &    (4824.6)         &   (12924.5)         &    (5826.2)         &    (5814.1)         \\
merger      &     -9358.1** &     -7406.7***&       792.9         &     -6783.7***&     -3754.7** \\
            &    (3732.4)         &    (1606.6)         &    (4842.0)         &    (1634.9)         &    (1714.6)         \\
$ Merger\times Cost$ &       0.37***&      0.05         &       0.26*  &      0.03         &      0.03         \\
            &     (0.14)         &    (0.05)         &     (0.15)         &    (0.06)         &    (0.06)         \\
Bankruptcy  &      6416.6*  &     -4409.9** &     -3202.6         &     -5175.5** &     -4915.8** \\
            &    (3560.5)         &    (2085.6)         &    (3530.9)         &    (2106.2)         &    (2035.7)         \\
Bankruptcy\_Duration&     11065.9** &     20261.3***&      7084.0         &     20526.4***&     12686.5***\\
            &    (4550.2)         &    (1998.2)         &    (4920.3)         &    (2145.7)         &    (1787.2)         \\
\footnotesize{Bankruptcy Duration} $\times Cost$&      -0.67***&      -0.19***&      -0.53***&      -0.16** &      -0.23***\\
            &     (0.16)         &    (0.06)         &     (0.18)         &    (0.07)         &    (0.05)         \\
HHI         &    -72675.7***&     22921.2*  &    -42221.3***&    114457.8***&    134520.8***\\
            &   (15943.5)         &   (12258.5)         &   (14622.3)         &   (19306.8)         &   (18789.8)         \\
$HHI\times Cost$ &       1.11*  &       0.71*  &       2.38***&       1.17** &       0.79*  \\
            &     (0.57)         &     (0.37)         &     (0.71)         &     (0.51)         &     (0.48)         \\
 &  &  &  & &  \\
Observations & 76,633 &  74,739  & 76,633  & 74,739  & 73,406  \\
R-squared & 0.71 & 0.63 & 0.71 & 0.62 & 0.66  \\
%clustered se & y & y & y & y & y  \\
route-carrier FE & n & y & n & y & y  \\
year-quarter FE & n & y & y & y & y  \\
IV & n & n & y & y & y  \\
Number of carrier-routes &  & 5,806 &  & 5,806 & 5,693 \\ \hline
\multicolumn{6}{c}{ Clustered standard errors by route in parentheses} \\
\multicolumn{6}{c}{ *** p$<$0.01, ** p$<$0.05, * p$<$0.1} \\
\end{tabular}
\end{table}
\end{comment}

In column (2), we control for carrier-route and year-quarter fixed effects. The results are qualitatively the same. We can see that the fuel cost pass-through rate is now lower, for both contemporaneous and cumulative impacts on revenue. After controlling for route fixed effects, coefficient for the interaction term $HHI\times Cost$ remains positive and significant, but goes down slightly to about 0.7. Consider a market with $HHI=0.6$. A \$1 increase in fuel cost leads to an additional $0.7\times 0.6\times 1=\$0.42$ increase in fare revenue. In addition, the coefficient for $HHI$ has also become positive. Combined, revenue at the $ijt$ level must increase with $HHI$, implying that fare revenue goes up when the route becomes more concentrated, after controlling for carrier-route and year-quarter fixed effects.

Column (3) uses IV to deal with the potential endogeneity of $HHI$, but do not control for carrier-route fixed effects.  In Column (4), we use IV and control for carrier-route and year-quarter fixed effects. We use the logarithm of market distance and carriers' average share of enplanements at the endpoints as instruments for this specification. The results are qualitatively the same as those in column (2). In particular, there is more than 100\% fuel cost pass-through and pass-through rate increases with market concentration (measured by $HHI$). Looking and bankruptcy and bankruptcy duration, the estimates suggest that revenue goes down when the airline files for bankruptcy, but goes back up in subsequent quarters when the airline remains under bankruptcy protection. Our instrument variables also pass both the weak identification test \footnote{Kleibergen-Paap F statistic: 927.45 v.s. Stock-Yogo critical value of 13.43 for 5\% maximal IV relative bias} and the over identification test\footnote{Hansen\_J Statistic:2.18; p-value: 0.14}.

Coefficient for the interaction term Bankruptcy Duration $\times Cost$ is negative, suggesting that pass-through rate is lower when the carrier is under bankruptcy protection. Pass-through rate does not seem to change when the carrier is in merger transition. The positive impact of $Merger \times Cost$ in column (1) disappears once we control for carrier-route and year-quarter fixed effects.

As briefly discussed in the variable description section, we would run the same specifications including the newly constructed variable, $acquisition_{it}$, as robustness checks. There could be omitted variables that we do not observe and control that affect the main cost variable and the price variable at the same time. This concern is more likely raised because we do not observe all the carrier-time specific variables that could leave a room for endogeneity, and unfortunately, we cannot include carrier-time specific fixed effect because the variable of interest is carrier-time specific in our regressions. Hence, we try to include most carrier-time specific variables that are observable in the data and that we could think of. Also, we provide several robustness checks later on and discuss further what IVs we could make use of to handle the potential endogeneity.

\begin{comment}
\textbf{Discussions:}

\bi

\item Assumption that fuel consumption is linear in distance and payload, discussed briefly earlier.

\item Robustness test: use alternative measures of fuel cost at the $ijt$ level. Also get similar results: pass-through rate is more than 100\% and increases with $HHI$.
\ei
\end{comment}

\section{Sunk cost fallacy}\label{sec:fallacy}

It is well known that some airlines sign hedging contracts to stabilize their fuel costs over time, which in turn allows them to smooth profits over time. A common form of hedging contracts is as follows (see Carter, Rogers and Simkins for more details and examples). Airline $A$ signs a contract with a company $B$ where $A$ agrees to buy $x$ amount of fuel from $B$ at price $p_0$ at time $t$ in the future. Suppose that when time $t$ arrives the market fuel price is $p_1$. There is no need for companies $A$ and $B$ to actually exchange fuel, only money changing hands. Assume that $p_1<p_0$, then company $A$ will pay $B$ a lump sum in the amount of $(p_0-p_1)\cdot x$. Such hedging contracts affect airline $A$'s finanical position (due to the lump sum payment) and its reported fuel costs since its $afc$ is now $p_0$ instead of $p_1$. However, such a hedging contract should have no impact on the eocnomic cost of fuel airline $A$ actually buys and consumes (from company $B$ or not). That is, if an airline made a wrong bet, the money it will lose due to the wrong bet is a sunk cost, which should not affects its true fuel cost and in turn should not affect its price decisions.

To test for sunk cost fallacy, we look at airlines' $afc$ over our sample period and look for incidence where some airlines report significantly different $afc$ due to hedging. We find two such cases.

\subsection{Sunk cost fallacy -- Southwest}

The first case involves Southwest airlines from year 2003 to 2008. During this time period, Southwest made the right hedging bets and locked in lower fuel prices. We introduce two dummy variables $Low\_WN90_t$ which takes the value of 1 if Southwest's $afc$ is 90\% or less of the average $afc$ of major airlines, and zero otherwise.\footnote{Southwest's competitors vary across routes, so their average fuel costs also vary across routes. An alternative is to calculate $Low\_WN90_{jt}$ (at the carrier-year-quarter level) which takes value 1 if Southwest's fuel price is lower than the average of major airlines. However, we treat price decisions as being made a single decision maker across routes and sunk cost fallacy, if exists, will occur across routes. Moreover, we already control for carrier-route and year-quarter fixed effects. So we use $Low\_WN90_t$ which varies across year-quarter $t$ but not across routes $j$}  We also introduce another dummy to indicate the periods where Southwest enjoys an even larger advantage -- $Low\_WN75_t$ takes the value of 1 if Southwest's $afc$ is 75\% or less of the average $afc$ of major airlines, and zero otherwise.\footnote{Note that when $Low\_WN75_t=1$, $Low\_WN90_t=1$ automatically holds.}

Next, we run a regression with $ln\_fare_{ijt}$ as the dependent variable and we want to see how the four dummy variables affect prices. Since we control for year-quarter fixed effects, these dummy variables, if included, will be absorbed. Instead, we introduce their interaction with their corresponding airline dummies, for example, $WN\times Low\_WN90_t$. The results are presented in Table 4. Standard errors are clustered by carrier-route $ij$ throughout the columns. Instruments \footnote{the logarithms of market distance, average population and carriers' average share of enplanements at the endpoints} are also used in this section to deal with the endogeneity of HHI, and they easily pass all the tests. From model (1), the coefficient for $WN\times Low\_WN90_t$ is negative and significant. We treat this as evidence that Southwest is subject to sunk cost fallacy.

Negative coefficient for the interaction term implies that during the periods where Southwest's $afc$ is in less than 90\% of major airlines' average $afc$, Southwest further reduces its fares relative to other major airlines. Note that this does not just say that Southwest charges lower price in general, which would be controlled by the carrier-route fixed effects. But instead, if Southwest's fare is lower on average, then when Southwest's $afc$ is lower, Southwest reduces its price further relative to the other major airlines. Model (2) include $WN\times Low\_WN75_t$ and the results are similar. In model (3), we combine include both  $WN\times Low\_WN90_t$  and  $WN\times Low\_WN75_t$. We can see that the impact mostly comes from  $WN\times Low\_WN90_t$, suggesting that once Southwest's $afc$ is less than 90\% of other major airlines' average $afc$, a further reduction in Southwest's $afc$ does not lead to further reduction in Southwest's airfares.
\begin{comment}
\begin{table}[!htbp]
\centering
\caption{Sunk Cost Fallacy - Southwest Airlines}\label{table:fallacy1}
\medskip

\def\sym#1{\ifmmode^{#1}\else\(^{#1}\)\fi}
\begin{tabular}{l*{3}{c}}
\hline
            &\multicolumn{1}{c}{(1)}&\multicolumn{1}{c}{(2)}&\multicolumn{1}{c}{(3)}\\
            &\multicolumn{1}{c}{lnfare}&\multicolumn{1}{c}{lnfare}&\multicolumn{1}{c}{lnfare}\\
\hline
$Low\_WN90_t \times WN$&     -0.0602\sym{***}&                     &     -0.0589\sym{***}\\
            &   (0.00395)         &                     &   (0.00409)         \\
$Low\_WN75_t \times WN$&                     &     -0.0432\sym{***}&    -0.00276         \\
            &                     &   (0.00330)         &   (0.00261)         \\
HHI         &       0.375\sym{***}&       0.384\sym{***}&       0.375\sym{***}\\
            &    (0.0190)         &    (0.0191)         &    (0.0190)         \\
lnsalary    &       0.205\sym{***}&       0.204\sym{***}&       0.205\sym{***}\\
            &   (0.00805)         &   (0.00802)         &   (0.00804)         \\
roundtrip\_share   &      -0.129\sym{***}&      -0.129\sym{***}&      -0.129\sym{***}\\
            &    (0.0121)         &    (0.0121)         &    (0.0121)         \\
merger      &      0.0165\sym{***}&      0.0171\sym{***}&      0.0164\sym{***}\\
            &   (0.00289)         &   (0.00289)         &   (0.00289)         \\
Bankruptcy  &     -0.0362\sym{***}&     -0.0422\sym{***}&     -0.0365\sym{***}\\
            &   (0.00461)         &   (0.00464)         &   (0.00463)         \\
Bankruptcy\_Period&      0.0273\sym{***}&      0.0314\sym{***}&      0.0273\sym{***}\\
            &   (0.00382)         &   (0.00386)         &   (0.00382)         \\\hline
Carrier-route FE       &         Yes         &         Yes         &         Yes         \\
Year-quarter FE       &         Yes         &         Yes         &         Yes         \\
IV &    Yes         &         Yes         &         Yes         \\
\hline
N           &      160907         &      160907         &      160907         \\
carrier\_route&                     &                     &                     \\
\hline\hline
\multicolumn{4}{l}{\footnotesize Standard errors in parentheses}\\
\multicolumn{4}{l}{\footnotesize \sym{*} \(p<0.1\), \sym{**} \(p<0.05\), \sym{***} \(p<0.01\)}\\
\end{tabular}
\end{table}
\end{comment}
\subsection{Sunk cost fallacy -- Delta}

The other case involves Delta for the time period from 2008 to 2009. Delta locked in fuel price when fuel price dropped significantly during this period. We define two similar dummy variables $High\_DL90$ and $High\_DL75$ if the average $afc$ of major airlines (other than Delta itself \footnote{Northwest observations are also dropped after its merger with Delta.}) is less than 90\% and 75\% of Delta's $afc$ respectively. We run similar regressions as those for Southwest. The results are presented in Table 6. The results are quite puzzling. From model (1), we see that when Delta's $afc$ is above other major airlines' average $afc$, Delta actually lowers its price, which is in the opposite direction as sunk cost fallacy would suggest. When Delta's $afc$ is further lower (model (2)), Delta does not lower its fares lower relative to other major airlines, but keep the same level ($WN\times High\_DL75_t$ is both small and insignificant). Similar results are obtained in model (3) where we include both $WN\times High\_DL90_t$ and $WN\times High\_DL75_t$

\begin{comment}
\begin{table}
\centering
\caption{Sunk Cost Fallacy - Delta Airlines}\label{table:fallacy2}
\medskip

\def\sym#1{\ifmmode^{#1}\else\(^{#1}\)\fi}
\begin{tabular}{l*{3}{c}}
\hline
            &\multicolumn{1}{c}{(1)}&\multicolumn{1}{c}{(2)}&\multicolumn{1}{c}{(3)}\\
            &\multicolumn{1}{c}{lnfare}&\multicolumn{1}{c}{lnfare}&\multicolumn{1}{c}{lnfare}\\
\hline
$High\_DL90_t \times DL$&     -0.0205\sym{**} &                     &     -0.0285\sym{***}\\
            &    (0.0101)         &                     &    (0.0106)         \\
$High\_DL75_t \times DL$&                      &    -0.00260         &      0.0198\sym{*}  \\
            &                     &    (0.0117)         &    (0.0114)         \\
HHI         &       0.408\sym{***}&       0.408\sym{***}&       0.408\sym{***}\\
            &    (0.0233)         &    (0.0233)         &    (0.0233)         \\
lnsalary    &      0.0571\sym{***}&      0.0580\sym{***}&      0.0569\sym{***}\\
            &    (0.0104)         &    (0.0103)         &    (0.0104)         \\
roundtrip\_share   &      -0.131\sym{***}&      -0.131\sym{***}&      -0.131\sym{***}\\
            &    (0.0151)         &    (0.0151)         &    (0.0151)         \\
merger      &      0.0218\sym{***}&      0.0174\sym{***}&      0.0218\sym{***}\\
            &   (0.00415)         &   (0.00387)         &   (0.00415)         \\
Bankruptcy  &     -0.0282\sym{***}&     -0.0290\sym{***}&     -0.0282\sym{***}\\
            &   (0.00489)         &   (0.00488)         &   (0.00489)         \\
Bankruptcy\_Period&      0.0332\sym{***}&      0.0339\sym{***}&      0.0333\sym{***}\\
            &   (0.00390)         &   (0.00387)         &   (0.00390)         \\\hline
Carrier-route FE       &         Yes         &         Yes         &         Yes         \\
Year-quarter FE       &         Yes         &         Yes         &         Yes         \\
IV &    Yes         &         Yes         &         Yes         \\
\hline
N           &      112979         &      112979         &      112979         \\
carrier\_route&                     &                     &                     \\
\hline
\multicolumn{4}{l}{\footnotesize Standard errors in parentheses}\\
\multicolumn{4}{l}{\footnotesize \sym{*} \(p<0.1\), \sym{**} \(p<0.05\), \sym{***} \(p<0.01\)}\\
\end{tabular}
\end{table}
\end{comment}

\section{Conclusion}\label{sec:conclusion}

Airline fuel cost is a significant component of airline operating cost and fuel cost has been volatile in recent decades. In this paper, we analyze how fuel costs affect airlines' pricing decisions and how this impact varies with market structure. We find significant and lasting impact of fuel cost on fuel prices. We also find that fuel cost pass-through is more than 100\%, realized over multiple periods. We then investigate the issue of sunk cost fallacy, in particular, when fuel hedging nets the airline a lump sum financial gain or loss without affecting its true fuel cost, whether its price decision will depend on its hedging position. We find evidence of sunk cost fallacy in the case of Southwest -- it lowers its price further relative to other major airlines when its fuel cost dips significantly below other major airlines (due to hedging). However, the evidence is mixed for Delta when its fuel cost is significantly higher than other major airlines due to hedging.

There are a few directions which we can explore next. First, we want to refine the construction of fuel cost measures at the carrier-route-year-quarter level. We have made the simplifying assumption that fuel consumption is linear in payload and distance. If this is not the case, then our fuel cost measure will be biased and the direction of bias may depend on payload (which will depend on load factor) and distance. This is a tricky issue. For example, ticket price affects passengers' choice among flights, so prices will affect load factor which in turn affects fuel costs. As a result, we have fuel price affecting fuel cost (on the right hand side) as well as fare revenue (on the left hand side) - a simultaneity problem. Second, we are interested in looking at asymmetric adjustment, distinguishing between the case of fuel cost increase and decrease. Since we have micro-level airfare data, we can also analyze how fuel cost changes affect the distribution (not just the mean) of airfare. This would require careful construction of the price distribution measure. Given that one does not observe how exactly the round trip fare would break for each component, we could interpolate the break of the round-trip fare based on one-way fare within a route and measure the price distribution.

As robustness checks, we further investigate the specifications in use by alternating the sample data. That is, we would drop the returning flight to adjust the standard errors as in Gerardi and Shapiro (2009). In addition, we would redefine the market as the non-directional as in Kim and Shen (2016) and confirm if our findings are consistent.

\begin{comment}\textit{First}, the BTS record of energy intensity of certificated air carriers, which we use to calculate the yearly fuel efficiency level, was no longer updated after 2012. As a result, we lose observations in the most recent 2 years in this study.\textbf{(QL: I am not sure why. If they don't update, some aircraft models remain in use, right? Or are we using energy efficiency measures that vary across time for the same aircraft type? But earlier we just said that they remain constant over time.)} \end{comment}

\newpage

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\newpage
\begin{comment}
\begin{center}
\textbf{\large{Appendix B: Constructing fuel cost measures}}
\end{center}

\medskip

We construct fuel cost measures, using data from multiple sources.
\medskip

\textbf{From external source:  https://www.aircraftcompare.com}

%\textbf{(QL: Can we say something about the reliability of this data? Also, what about the case where an aircraft with the same model-series has better efficiency over time?)} \textbf{(LS: The website does not leave direct information to show the reliability of their data, but there's contact information for inquiries and commercial cooperation. That's possibly better than nothing. Unfortunately, we may not be able to find it if an aircraft with the same model-series improves its efficiency over time.)}

\bi

\item Fuel efficiency ($FE^k$) at the aircraft ($k$) level

For each aircraft type $k$, we collect its fuel efficiency data (at gallon per nautical mile level) from the Internet. The website AircraftCompare.com reports basic characteristics like size, weight and fuel economy for each aircraft type, which remain constant over time. Aircraft type is at the model-series level, for example, Boeing 737- 600 / 700 / 800 / 900 are separately reported. The same source also gives payload data (in thousand pound) for each aircraft type. We divide the gallon/mile value by the payload to obtain fuel efficiency (at the unit of gallon per thousand pound per mile) to obtain fuel efficiency for aircraft type $k$, $FE^k$. Note that this variable depends only on aircraft type $k$, not on $i$, $j$ or $t$.
\ei

\textbf{From T-100 data}

\bi

\item Passenger share at the ($ijt,k$) level

For T-100 data, for each $ijt$, we count the number of passengers transported under each aircraft type, $passenger_{ijt}^k$.  Let $K$ denote the set of $k$'s for which we have $FE^k$ data. If
$$\frac{\sum_{k\in K} passenger_{ijt}^k}{\sum_k passenger_{ijt}^k}\ge 80\%,$$

then we keep this $ijt$ (otherwise we drop it). We then calculate the passenger share among all known aircraft types
\be psg\_share_{ijt}^k=\frac{passenger_{ijt}^k}{\sum_{l\in K} passenger_{ijt}^l}.\ee

\item Average fuel efficiency at the $ijt$ level

We then calculate the average fuel efficiency
$$AFE_{ijt}=\sum_{k\in K} \left(psg\_share_{ijt}^k\times FE^k\right),$$

still at the gallon (of fuel) per thousand pound per mile unit.\footnote{For $k\notin K$, we do not have fuel efficiency data for these aircraft types. When calculating average fuel efficiency at the $ijt$ level, we drop these aircraft types. If smaller aircraft types are both less likely to report fuel efficiency data and less efficient, then the actual $AFE_{ijt}$ may be larger since we are dropping the least efficient aircraft types. We want to control this bias somewhat, which is why we impose the constraint that only $ijt$'s for which at least 80\% of the passengers are transported by aircraft type $k\in K$.}

\item Average payload per passenger at the $ijt$ level

T-100 reports total payload and freight payload at the $ijt,k$ level. We treat the difference between total payload and freight as passenger payload. Dividing it by the total numer of passengers, we obtain average payload per passenger ($APP_{ijt}^k$) at the $ijt,k$ level. Aggregating over $k$'s on $ijt$, we have
$$APP_{ijt}=\sum_{k\in K} \left(psg\_share_{ijt}^k\times APP_{ijt}^k\right).$$

\ei

\textbf{From DB1B data}

\bi

\item Number of passengers at the $ijt$ level, $Passenger_{ijt}$.

DB1B data does not report aircraft type information and $Passenger_{ijt}$ includes passengers transported through all aircraft types. This is not to be confused with $passenger_{ijt}^k$ (with superscript $k$) from the T-100 data.

\item Sum over all passenger fares at the $ijt$ level, we obtain $Revenue_{ijt}$.

\ei

\textbf{Combining data}

\bi

\item Combining fuel efficiency, T-100 and DB1B data to calculate fuel consumption a the $ijt$ level
$$fuel\_consmption_{ijt}=Passenger_{ijt}\times APP_{ijt} \times AFE_{ijt} \times Distance_{ij},$$

gives gallons of fuel consumed to transport the passengers on $ijt$.

We have made a simplifying assumption that fuel consumption is linear in payload and distance. In practice, take off and landing are more like fixed cost so we would expect fuel consumption to increase with distance slower and slower. On the other hand, the longer the distance, the more fuel will be needed to travel that distance and carrying the extra fuel requires more fuel as well. This suggests the longer distance is less fuel efficient. We are implicitly assuming that the two opposite impacts cancel out each other.

\item Fuel cost is the product of airline fuel cost ($afc_{it}$) and fuel consumption
$$Cost_{ijt}=afc_{it}\times fuel\_consmption_{ijt}.$$

We also introduce one quarter lag of the fuel cost variable as follows:
$$Lag\_Cost_{ijt}= afc_{i,t-1}\times fuel\_consmption_{ijt}.$$

Other lags (2, 3 or 4-quarter lags respectively) are constructed similarly.

\ei

\end{comment}

\end{document}

