Asymmetric Price Adjustments in Airlines

Transcription

1 Department of Economics & Finance The University of Texas-Pan American University of Alabama December 3, 2010

2 Outline 1 Introduction Motivation Dynamic pricing in airlines Contribution 2 Construction Realized demand 3 Decomposition of Price t Transitory component Price T t Markov-switching with FTP and TVTP Asymmetries specifications: Sign t, Size t, Wknd t, and Summ t 4 Significance of the regime-switch Maximum Likelihood Estimation Transitory component of price: Price T t State dependent Impulse Response Functions Transition probabilities 5

3 Motivation Introduction Motivation Dynamic pricing in airlines Contribution Large price dispersion in airlines. Borenstein and Rose (JPE 1994), Gerardi and Shapiro (JPE 2009). Positive demand shifts drive prices up: Basic prediction of economic theory. Common view among travelers and the media: The inevitable outcome of limited seats and stronger demand will be higher fares. New York Times, May 31, 2010 Do prices fall as a response to negative demand shifts?

4 Motivation Introduction Motivation Dynamic pricing in airlines Contribution Potential explanations for the asymmetric response: Simple convex supply curve. Models with costly capacity and demand uncertainty: Prescott (JPE 1976), Eden (JPE 1990), Dana (Rand 1999) Menu costs. Mankiw (QJE 1985), Ball and Mankiw (1994) Asymmetric pricing. (changes in costs) Peltzman (JPE 2000) Collusion / Market power. Uninformed consumers. Tappata (Rand 2009) Inventories. Borenstein and Shepard (Rand 2002)

5 Motivation Dynamic pricing in airlines Contribution Motivation: Dynamic pricing in airlines Key characteristics: Fixed capacity. Perishable good. Aggregate demand uncertainty. Advance sales. Carriers exploit fences such as: Saturday-night-stayover. Advance purchase discounts. Minimum- and maximum-stay. Refundable tickets. Frequent flier miles. Blackouts. Volume discounts. Fare classes (e.g. coach, first class) Airlines have the most sophisticated pricing systems in the world.

6 Contribution of the current paper Motivation Dynamic pricing in airlines Contribution First to explain price variation over different departure dates. Finds strong evidence of an asymmetric response. Combines different sources of asymmetries. Positive cost shifts have a positive effect, but negative demand shifts have no effect on prices. Cost shifts have larger effect on prices during summer travel. Less evidence that the shifts are related to the size of the demand shift or weekend and holiday travel. Importance of capacity constraints as a source of asymmetric pricing. Importance of asymmetric pricing to stabilize demand fluctuations.

7 Construction of the Construction Realized demand Minimum available non-refundable fare from expedia.com. Controls for more expensive refundable fares. 126 days (18 weeks) for 48 flights departing between Tuesday June 9 and Monday October 12, Keeping the same flight-number. Controls for time-invariant specific characteristics. One-way, non-stop, economy-class. Connecting passengers / sophisticated itineraries / legs. Uncertainty in the return portion of the ticket. Saturday-night-stayover / min- and max-stay. Fare classes (e.g. coach, first class). American, Alaska, Continental, Delta, United and US Airways.

8 Introduction Construction Realized demand Tue, Jun 09 Fri, Jun 12 Mon, Jun 15 Thu, Jun 18 Sun, Jun 21 Wed, Jun 24 Sat, Jun 27 Tue, Jun 30 Fri, Jul 03 Mon, Jul 06 Thu, Jul 09 Sun, Jul 12 Wed, Jul 15 Sat, Jul 18 Tue, Jul 21 Fri, Jul 24 Mon, Jul 27 Thu, Jul 30 Sun, Aug 02 Wed, Aug 05 Sat, Aug 08 Tue, Aug 11 Fri, Aug 14 Mon, Aug 17 Thu, Aug 20 Sun, Aug 23 Wed, Aug 26 Sat, Aug 29 Tue, Sep 01 Fri, Sep 04 Mon, Sep 07 Thu, Sep 10 Sun, Sep 13 Wed, Sep 16 Sat, Sep 19 Tue, Sep 22 Fri, Sep 25 Mon, Sep 28 Thu, Oct 01 Sun, Oct 04 Wed, Oct 07 Sat, Oct 10 Figure: Realized demand. Shaded areas: Weekends and holidays

9 Decomposition of Pricet Transitory component Price T t Markov-switching with FTP and TVTP Asymmetries specifications: Signt, Sizet, Wkndt, and Summt Decomposition of Price t Decompose Price t into: Price t = Price P t + Price T t. Permanent component as a random walk with time-varying drift: ν t is a normally distributed i.i.d. r.v. Price P t = µ + Price P t 1 + ν t,

10 Decomposition of Pricet Transitory component Price T t Markov-switching with FTP and TVTP Asymmetries specifications: Signt, Sizet, Wkndt, and Summt Transitory component Price T t Transitory component as an autoregressive process: φ(l) Price T t = γ 0 (L) Cost t + γ 1 (L) Cost t S t + ε t, φ(l) = K φ k L k ; φ 0 = 1; γ i (L) = k=0 J γ j,i L j. S t indicator variable equal to 0 or 1 to capture the regime switches in the response. ε t normally distributed i.i.d. r.v. Lo and Piger (2005) j=0

11 Fixed transition probabilities (FTP) Decomposition of Pricet Transitory component Price T t Markov-switching with FTP and TVTP Asymmetries specifications: Signt, Sizet, Wkndt, and Summt First-order Markov-switching fixed transitions probabilities: P(S t = 0 S t 1 = 0) = exp(c 0 ) (1 + exp(c 0 )) P(S t = 1 S t 1 = 0) = 1 P(S t = 0 S t 1 = 0) P(S t = 1 S t 1 = 1) = exp(c 1 ) (1 + exp(c 1 )) P(S t = 0 S t 1 = 1) = 1 P(S t = 1 S t 1 = 1)

12 Decomposition of Pricet Transitory component Price T t Markov-switching with FTP and TVTP Asymmetries specifications: Signt, Sizet, Wkndt, and Summt Time-varying transition probabilities (TVTP) First-order Markov-switching time-varying transitions probabilities: P(S t = 0 S t 1 = 0) = P(S t = 1 S t 1 = 1) = exp(c 0 + z t a 0 ) (1 + exp(c 0 + z t a 0 )) exp(c 1 + z t a 1 ) (1 + exp(c 1 + z t a 1 )) Four specifications for the vector z t : z t = (z 1t, z 2t,..., z qt ) : Vector of state variables. a 0 = (a 01, a 02,..., a 0q ) : Vector of coefficients. ( low response) a 1 = (a 11, a 12,..., a 1q ) : Vector of coefficients. ( high response)

13 Asymmetries specifications Decomposition of Pricet Transitory component Price T t Markov-switching with FTP and TVTP Asymmetries specifications: Signt, Sizet, Wkndt, and Summt Four specifications for the vector z t : Sign t = 1, if the demand shift is positive. Size t = 1, if the demand shift is more than one standard deviation away from its mean. Wknd t = 1, if the departure date is during a weekend or holiday. Summ t = 1, if the departure date is during the summer.

14 Significance of the regime-switch Significance of the regime-switch Maximum Likelihood Estimation Transitory component of price: Price T t State dependent Impulse Response Functions Transition probabilities Markov-switching state-space representation: Kim (1994) filter. Significance of the regime-switch. Hansen (1992): H 0: γ j,0 = γ j,1 for all j p-value of Table: Model Selection Elements of z t SIC AIC Log likelihood LR test a FTP None TVTP Sign Size Wknd Summ Note: a p-value for a test of the null of the FTP.

15 Table: MLE Parameter Estimates Parameter FTP Sign Size Wknd Summ σ ν (0.0005) (0.0037) (0.0064) (0.0038) (0.0049) σ ɛ (0.0041) (0.0038) (0.0039) (0.0039) (0.0037) φ (0.1173) (0.0935) (0.1228) (0.0137) (0.0858) φ (0.0117) (0.0058) (0.0117) (0.0598) (0.0080) γ 0, (0.1150) (0.0966) (0.1073) (0.0882) (0.0971) γ 1, (0.0954) (0.0879) (0.0948) (0.0929) (0.0913) γ 0, (0.5238) (0.5869) (0.6688) (0.4653) (0.5563) γ 1, (0.4517) (0.6347) (0.4614) (0.4390) (0.3893) c (0.5620) ( b ) (0.8459) (0.5647) (1.0222) c (0.4481) (0.9417) (1.9404) (0.9612) (0.7516) a (0.5538) (1.2950) (1.6140) (1.2852) a ( b ) (1.2449) ( b ) ( b ) Log likelihood Note: The ML estimate of c 0 appears on the boundary, violating regularity conditions. Hence, to calculate the standard errors, c 0 = 0 was imposed to calculate the second derivatives of the log likelihood.

16 Significance of the regime-switch Maximum Likelihood Estimation Transitory component of price: Price T t State dependent Impulse Response Functions Transition probabilities Price T t Positively skewed Price T t Sat, Jun 27 Tue, Jun 30 Fri, Jul 03 Mon, Jul 06 Thu, Jul 09 Sun, Jul 12 Wed, Jul 15 Sat, Jul 18 Tue, Jul 21 Fri, Jul 24 Mon, Jul 27 Thu, Jul 30 Sun, Aug 02 Wed, Aug 05 Sat, Aug 08 Tue, Aug 11 Fri, Aug 14 Mon, Aug 17 Thu, Aug 20 Sun, Aug 23 Wed, Aug 26 Sat, Aug 29 Tue, Sep 01 Fri, Sep 04 Mon, Sep 07 Thu, Sep 10 Sun, Sep 13 Wed, Sep 16 Sat, Sep 19 Tue, Sep 22 Fri, Sep 25 Mon, Sep 28 Thu, Oct 01 Sun, Oct 04 Wed, Oct 07 Sat, Oct 10 Figure: Price T t. Shaded areas: Sign t = 1

17 Significance of the regime-switch Maximum Likelihood Estimation Transitory component of price: Price T t State dependent Impulse Response Functions Transition probabilities State dependent Impulse Response Functions Simulate the path of Price T t+j as captured by: ˆγ 0,0, ˆγ 1,0, ˆγ 0,1, and ˆγ 1,1. Price T t 1 = Price T t 2 = 0, ε t+j = 0, j and Cost t j = 0, j 0. Cost t = 5.5% 14% 10% S(t)=1, S(t+1)=1 S(t)=0, S(t+1)=0 S(t)=1, S(t+1)=0 S(t)=0, S(t+1)=1 6% 2% % Figure: State dependent IRF of Price T t. TVTP: Sign

18 Transition Probabilities Introduction Significance of the regime-switch Maximum Likelihood Estimation Transitory component of price: Price T t State dependent Impulse Response Functions Transition probabilities Determine the transition probabilities as captured by: ĉ 0, ĉ 1, â 01 and â 02 When Sign t 1 = Sign t = 0: P(S t = 0 S t 1 = 0) = exp(ĉ 0)/(1 + exp(ĉ 0)) = 1 P(S t = 1 S t 1 = 0) = 0 When Sign t 1 = Sign t = 1: P(S t = 0 S t 1 = 0) = exp(ĉ 0 + â 01 + â 02)/(1 + exp(ĉ 0 + â 01 + â 02)) = P(S t = 1 S t 1 = 0) = Positive demand shifts are more likely to have a large effect on prices than negative demand shifts.

19 Significance of the regime-switch Maximum Likelihood Estimation Transitory component of price: Price T t State dependent Impulse Response Functions Transition probabilities Filtered probability, P(S t = 1 t). TVTP: Sign Sat, Jun 27 Tue, Jun 30 Fri, Jul 03 Mon, Jul 06 Thu, Jul 09 Sun, Jul 12 Wed, Jul 15 Sat, Jul 18 Tue, Jul 21 Fri, Jul 24 Mon, Jul 27 Thu, Jul 30 Sun, Aug 02 Wed, Aug 05 Sat, Aug 08 Tue, Aug 11 Fri, Aug 14 Mon, Aug 17 Thu, Aug 20 Sun, Aug 23 Wed, Aug 26 Sat, Aug 29 Tue, Sep 01 Fri, Sep 04 Mon, Sep 07 Thu, Sep 10 Sun, Sep 13 Wed, Sep 16 Sat, Sep 19 Tue, Sep 22 Fri, Sep 25 Mon, Sep 28 Thu, Oct 01 Sun, Oct 04 Wed, Oct 07 Sat, Oct 10 Figure: Filtered probability, P(S t = 1 t). TVTP: Sign. Shaded areas: Sign t = 1

20 Significance of the regime-switch Maximum Likelihood Estimation Transitory component of price: Price T t State dependent Impulse Response Functions Transition probabilities Filtered probability, P(S t = 1 t). TVTP: Summ Sat, Jun 27 Tue, Jun 30 Fri, Jul 03 Mon, Jul 06 Thu, Jul 09 Sun, Jul 12 Wed, Jul 15 Sat, Jul 18 Tue, Jul 21 Fri, Jul 24 Mon, Jul 27 Thu, Jul 30 Sun, Aug 02 Wed, Aug 05 Sat, Aug 08 Tue, Aug 11 Fri, Aug 14 Mon, Aug 17 Thu, Aug 20 Sun, Aug 23 Wed, Aug 26 Sat, Aug 29 Tue, Sep 01 Fri, Sep 04 Mon, Sep 07 Thu, Sep 10 Sun, Sep 13 Wed, Sep 16 Sat, Sep 19 Tue, Sep 22 Fri, Sep 25 Mon, Sep 28 Thu, Oct 01 Sun, Oct 04 Wed, Oct 07 Sat, Oct 10 Figure: Filtered probability, P(S t = 1 t). TVTP: Summ. Shaded areas: Summ t = 1

21 Combined Asymmetries Introduction Significance of the regime-switch Maximum Likelihood Estimation Transitory component of price: Price T t State dependent Impulse Response Functions Transition probabilities Combined asymmetries. Robustness of the Summ specification. Variation within the summer departure dates. Table: Model Selection Elements of z t SIC AIC Log likelihood LR test a LR test b TVTP Summ, Sign Summ, Size Summ, Wknd Summ, Summ Sign Summ, Summ Size Summ, Summ Wknd Note: a p-value for a test of the null of the FTP model. b p-value for a test of the null of the Summ model.

22 Significance of the regime-switch Maximum Likelihood Estimation Transitory component of price: Price T t State dependent Impulse Response Functions Transition probabilities State Dependent Impulse Response Functions State dependent IRF. 10% S(t)=1, S(t+1)=1 S(t)=0, S(t+1)=0 S(t)=1, S(t+1)=0 S(t)=0, S(t+1)=1 6% 2% % Figure: State dependent IRF of Price T t. TVTP: Summ, Sign Summ

23 Filtered probability, P(S t = 1 t) Significance of the regime-switch Maximum Likelihood Estimation Transitory component of price: Price T t State dependent Impulse Response Functions Transition probabilities Sat, Jun 27 Tue, Jun 30 Fri, Jul 03 Mon, Jul 06 Thu, Jul 09 Sun, Jul 12 Wed, Jul 15 Sat, Jul 18 Tue, Jul 21 Fri, Jul 24 Mon, Jul 27 Thu, Jul 30 Sun, Aug 02 Wed, Aug 05 Sat, Aug 08 Tue, Aug 11 Fri, Aug 14 Mon, Aug 17 Thu, Aug 20 Sun, Aug 23 Wed, Aug 26 Sat, Aug 29 Tue, Sep 01 Fri, Sep 04 Mon, Sep 07 Thu, Sep 10 Sun, Sep 13 Wed, Sep 16 Sat, Sep 19 Tue, Sep 22 Fri, Sep 25 Mon, Sep 28 Thu, Oct 01 Sun, Oct 04 Wed, Oct 07 Sat, Oct 10 Figure: Filtered probability, P(S t = 1 t). TVTP: Summ, Sign Summ. Shaded areas: Sign t Summ t = 1

24 Introduction Strong evidence of response asymmetries. Prices are more sensitive to demand fluctuations during summer. Positive demand shifts are more likely to have a positive effect. Importance of capacity constraints as a source of asymmetric pricing. Importance of pricing to stabilize demand fluctuations. are consistent with: Simple convex supply curve. Models with costly capacity and demand uncertainty: Prescott (1976), Eden (1990), Dana (1999) Menu cost models. Ball and Mankiw (1994) Asymmetric pricing. Peltzman (2000)