A duopoly model with switching and transport costs María Martín UC3M April 13, 2011 María Martín (UC3M ) Switching and transport costs April 13, 2011 1 / 27
Introduction María Martín (UC3M ) Switching and transport costs April 13, 2011 2 / 27
Introduction Theory: what is a switching cost? We do not have a unique de…nition for this concept: Thompson and Cats-Baril (2002): associated with switching supplier. Farrell and Klemperer (2007): when an investment speci…c to his current seller must be duplicated for a new seller. Three types of switching costs have been deeply analyzed in the literature: learning costs 1 transaction costs 2 loyalty rewards or contractual switching costs 3 María Martín (UC3M ) Switching and transport costs April 13, 2011 3 / 27
Introduction Theory: e¤ects For all the previous types of switching costs, products that are ex-ante homogeneous become ex-post heterogeneous. E¤ect in a two-period framework, in terms of prices: reduce elasticity of …rm’s demand in the second period + competition for market shares in the …rst period is more intense, due to the expected higher market power over their segments. María Martín (UC3M ) Switching and transport costs April 13, 2011 4 / 27
Introduction What about data? "How do experience and shopping frequency a¤ect consumers’ brand choice?" (T. Farina, 2010) Experience good: orange juice. Large supermarket chain in Brazil. All else equal, the ratio the experience coe¢cient and the price coe¢cient shows consumer’s willigness to pay for knowledge about a brand. Result: consumers are willing to pay roughly R$9.50 ( � US$6) per liter more for a brand of orange juice they have already purchased. María Martín (UC3M ) Switching and transport costs April 13, 2011 5 / 27
Introduction Endogenous vs. exogenous switching costs Let us think about the switching costs as a matter of information: It seems reasonable to assume that each signal provides me more information about the quality of the good when it is uncertain (Bayesian updating). So, if consumers are risk-averse, increasing the number of consumptions increases the information about the brand, and therefore the switching cost is also higher. But... is it reasonable to assume that consumers do not know with certainty the quality of the good after experiencing it? Environment: juice tastes better when you are thirsty. Memory is not perfect. María Martín (UC3M ) Switching and transport costs April 13, 2011 6 / 27
Review of the literature Bayesian updating: Vives: "Information and learning in markets" (2008) Caminal & Vives: "Why market shares matter" (1996) Competition in markets with switching costs: Farrell & Shapiro: "Dynamic competition with switching costs" (1988) Klemperer: "Markets with consumer switching costs" (1987) Endogenous switching costs: Villas-Boas: "Dynamic competition with experience goods" (2004) Loss-aversion: Köszegi & Rabin: "A model of reference-dependent preferences" (2006) María Martín (UC3M ) Switching and transport costs April 13, 2011 7 / 27
Model Experience good Two periods: t = 1 and t = 2 Risk-averse consumers, who are uniformly distributed into [0,1] Demand is a dichotomic variable 2 generations of consumers: parents and children Standard quadratic transport costs Consumers do not know they quality of the brands: signals ) Bayesian updating Firms A [ 0 ] and B [ 1 ] live both periods ) prices Firms also face uncertainty: they do not know the realization of a random shock when …xing prices María Martín (UC3M ) Switching and transport costs April 13, 2011 8 / 27
Notation Firms’ qualities (r.v.): θ A and θ B "Public" signals (r.v.): s 0 A , s 0 B , s 2 A , s 2 B Signals after tasting the good (r.v.): s 1 A , s 2 B Prices: p 1 A , p 1 B , p 2 A , p 2 B First-period market shares: x 1 and 1 � x 1 Random shocks (r.v.): q 1 and q 2 María Martín (UC3M ) Switching and transport costs April 13, 2011 9 / 27
Timing θ A , θ B � N ( θ , σ 2 θ ) s tf = θ f + ε tf ; ε tf � N ( 0 , σ 2 ε ) q t � N ( 0 , 1 ) María Martín (UC3M ) Switching and transport costs April 13, 2011 10 / 27
Information sets All the information related to the distribution of random variables is common knowledge. Consumers: t = 1: C 1 = f s 0 A , s 0 B , p 1 A , p 1 B , q 1 g t = 2: C 2 = f s 0 A , s 0 B , s 1 E , s 2 A , s 2 B , p 2 A , p 2 B , q 2 g Firms: t = 1: F 1 = f ? g t = 2: F 2 = f x 1 , q 1 , p 1 A , p 1 B g María Martín (UC3M ) Switching and transport costs April 13, 2011 11 / 27
Consumers Consumer located at x will buy one unit of brand A i¤ CE A � CE B ; and will purchase one unit of brand B otherwise. t = 1 : CE A = E [ θ A j s 0 A ] � 1 2 ρ Var [ θ A j s 0 A ] � p 1 A � γ x 2 + q 1 CE B = E [ θ B j s 0 B ] � 1 2 ρ Var [ θ B j s 0 B ] � p 1 B � γ ( 1 � x ) 2 María Martín (UC3M ) Switching and transport costs April 13, 2011 12 / 27
Consumers t = 2 : If consumer located at x consumed brand A in the previous period: CE A = E [ θ A j s 0 A , s 1 A , s 2 A ] � 1 2 ρ Var [ θ A j s 0 A , s 1 A , s 2 A ] � p 2 A � γ x 2 + q 2 If consumer located at x consumed brand B in the previous period: CE A = E [ θ A j s 0 A , s 2 A ] � 1 2 ρ Var [ θ A j s 0 A , s 2 A ] � p 2 A � γ x 2 + q 2 (Similar when …nding the CE B ) María Martín (UC3M ) Switching and transport costs April 13, 2011 13 / 27
Consumers Lemma 1: in the …rst period, we only have one marginal consumer located at � � x 1 = 1 τ ε γ + τ θ + τ ε ( s 0 A � s 0 B ) + q 1 � p 1 A + p 1 B 2 γ If x 1 � 0 (or x 1 � 1) ! corner solution If x 1 2 ( 0 , 1 ) ! interior solution Corollary: in the second period, for the interior solution we have two di¤erentiated markets: fraction of customers who have three signals for brand A and two for brand B ; and fraction of customers who have two signals for brand A and three for brand B . María Martín (UC3M ) Switching and transport costs April 13, 2011 14 / 27
Consumers Due to the di¤erence in information, the switching cost appears. Let us describe the case of the marginal captive customer of brand A (although for the marginal captive customer of B is analogous): CE A j A = CE B j A 1 ρ � e 2 γ ( q 2 + γ � p 2 A + p 2 B + e x 2 CA = θ + � 1 � � 1 � 3 τ ε 2 τ ε + 3 ( s 0 A + s 1 A + s 2 A ) � 2 ( s 0 B + s 2 B ) ) τ θ + 3 τ ε τ θ + 2 τ ε María Martín (UC3M ) Switching and transport costs April 13, 2011 15 / 27
Consumers ρ = 1 τ ε e ( τ θ + 2 τ ε ) ( τ θ + 3 τ ε ) ρ is the switching cost 2 Only appears in the problem of the second period. Endogenous. As customer has more signals for the most repeated product, the conditional variance is always less (and this variance is not dependent on the signal values). When the customer chooses the brand he tasted before, the switching cost shows that the individual enjoys a positive e¤ect due to the smaller uncertainty. We are NOT saying that CE should be bigger because of this fact. María Martín (UC3M ) Switching and transport costs April 13, 2011 16 / 27
Firms in t=2 In t = 2, …rms can infer some information from the outcome of the previous period. As in t = 2 , x 1 and q 1 are known, s 0 A � s 0 B = τ θ + τ ε ( 2 γ x 1 � q 1 � γ + p 1 A � p 1 B ) τ ε s 0 A = s 0 B + τ θ + τ ε ( 2 γ x 1 � q 1 � γ + p 1 A � p 1 B ) τ ε María Martín (UC3M ) Switching and transport costs April 13, 2011 17 / 27
Firms in t=2 Firms plug this relationship into the expressions of the marginal consumers of each market, and then apply expectation: 0 0 1 1 @ ρ + τ θ + τ ε @ 2 γ x 1 � q 1 + p 1 A � p 1 B A A x E 2 CA = 1 2 τ ε e τ θ + 3 τ ε γ � p 2 A + p 2 B + e 2 γ τ θ + 3 τ ε | {z } v 1 0 0 1 1 @ τ ε ρ + τ θ + τ ε @ 2 γ x 1 � q 1 + p 1 A � p 1 B A A x E 2 CB = 1 e τ θ + 2 τ ε γ � p 2 A + p 2 B � e 2 γ τ θ + 2 τ ε | {z } v 1 *Notice that, when the market was completely polarized in the …rst period, we only have one marginal consumer. María Martín (UC3M ) Switching and transport costs April 13, 2011 18 / 27
Equilibrium in t=2 x E x E Notice that, depending on the relative position of e 2 CA and e 2 CB , we have di¤erent possibilities. For instance, María Martín (UC3M ) Switching and transport costs April 13, 2011 19 / 27
Equilibrium in t=2 Lemma 2: if x 1 2 ( 0 , 1 ) , only two possibilities are sustainable in equilibrium in pure strategies; in particular, both …rms cannot exploit their captive customers at the same time. x E x E (a) e 2 CA 2 ( 0 , x 1 ) , e 2 CB � x 1 1 ρ + 2 τ θ + 4 τ ε γ + 1 τ θ + τ ε 3 e p 2 A = v 1 τ θ + 3 τ ε τ θ + 3 τ ε 3 3 � 1 ρ + 2 2 τ θ + 5 τ ε γ � 1 τ θ + τ ε 3 e p 2 B = v 1 τ θ + 3 τ ε τ θ + 3 τ ε 3 3 x E x E (b) e 2 CA � x 1 , e 2 CB 2 ( x 1 , 1 ) * "Exploit": to …x higher prices taking advantage of the risk-aversion of consumers. María Martín (UC3M ) Switching and transport costs April 13, 2011 20 / 27
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