Rockets and Feathers. Understanding Asymmetric Pricing. (Job Market paper)

Transcription

1 Rockets and Feathers. Understanding Asymmetric Pricing. (Job Market paper) Mariano Tappata UCLA October 005 [incomplete and preliminar] Abstract Prices rise like rockets but fall like feathers. This stylized fact of most markets has been con rmed by many empirical studies in the past. However, the advance on theoretical work trying to explain it has been rather low. I present a simple model of competitive rms and rational partially-informed consumers where the asymmetric response to costs by rms emerges naturally. In contrast to public opinion and past work, collusion is not needed for such result. JEL Codes: D, D40, L3. Keywords: Asymmetric Pricing, Rockets and Feathers, Consumer Search, Oligopoly.

2 Introduction Output prices do not react symmetrically to changes in input prices. This stylized fact has been getting the attention of both academics and the general public for a long time. As a result, an extensive empirical literature has developed to test for the existence of asymmetric pricing in speci c markets. Peltzman s comprehensive study of 65 producer goods and 77 consumer goods summarizes the results of this literature: "In two out of three markets, output prices rise faster than they fall" (Peltzman, 000; p. 480). This pattern is also known as rockets and feathers and has sometimes been used interchangeably with the term asymmetric pricing. But surprisingly, the research in the area has not developed symmetrically. Despite the abundance of empirical work available, there has not been much progress on theoretical models that support this common phenomenon. The rst thing that comes to mind when talking about the rockets and feathers pattern is gas prices and collusion. Input and output prices are easily observable by everyone and the market is composed by only a handful of players. When gas prices behave asymmetrically, the layperson (and the media) immediately associates it with collusion. This perception has in uenced the focus of economic research; most of the empirical work has concentrated around the retail markets for gasoline (Bacon, 99; Karrenbrock, 99; Borenstein, Cameron, and Gilbert, 997; Lewis, 003; Deltas, 004; and Verlinda, 005 among others). 3;4 However, Peltzman nds that the rockets and feathers pattern is equally likely to be found in both concentrated and atomistics markets. In this paper, I develop a consumer-search model that parses out the conditions under which asymmetric responses of prices to costs arise in non-collusive markets. According to traditional economic theory, homogeneous rms that compete in prices have zero pro ts and cost shocks are completely transferred to nal prices. 5 The nature of this equilibrium changes if consumers have imperfect information about market prices and a fraction of them has positive search costs. Firms now obtain positive pro ts (information rents) and the equilibrium is characterized by price dispersion -instead of a single price. Yet, for a given level of cost, the price margin is the same regardless of whether the cost shock was positive or negative. In order to account for asymmetric pricing, I introduce consumer uncertainty over production costs in a search model from Varian (980). Intuitively, when To the best of my knowledge, Bacon (99) was the rst one to use the term rockets and feathers to describe the pattern of retail gasoline prices in the U.K. See Karrenbrock (99, p. 0) for media and government representative quotations about gasoline price gouging. 3 The properties of the production process of retail gasoline -lack of input substitution and its main input representing 8% of the nal price net of taxes- make it also an attractive market for empirical tests. 4 Papers investigating asymmetric pricing in other markets include Neumark and Sharpe, 99 [Hannan and Berger, 99, [Pick et al, 99] in the banking sector and Ward, 98] Boyd and Brorsen., 988, Goodwin and Holt, 999 and Goodwin and Harper (000) in the food sector. 5 Firms with market power (product di erentiation or monopoly) don t transfer all the cost shocks to price, but still do it symmetrically.

3 input cost shocks are not independent over time, consumers expectations di er depending on whether costs were falling or rising in the previous periods. Asymmetric pricing then arises naturally as a consequence of changes in the perceived demand by rms; and the rockets and feathers pattern is supported under persistence in production cost realizations. Existing theories on asymmetric pricing are fully represented by the works of Borenstein et al. (997), Benabou and Gertner (993), and Lewis (003). The former tests the rockets and feathers pattern for the retailing gasoline market and suggests a model of collusion with imperfect monitoring to explain it (as in Tirole, 988; p. 64). With multiple equilibria, the focal point changes after a positive cost shock and it doesn t if the same shock is negative. The other two papers generate pricing asymmetries in duopoly markets without assuming collusive behavior. Benabou and Gertner analyze the e ect on e ciency of higher in ation uncertainty. In their model, each rm s production cost is composed of both an idiosyncratic (real) and a general (in ation) shock. Consumers update their beliefs about the general shock after observing the rst price and then decide whether to pay a cost to observe the price of the remaining rm. More in ation uncertainty can lead to lower or higher markups. Finally, Lewis develops a reference price search model with homogeneous rms and consumers that form expectations based on past-period prices. Therefore, consumers search strategy is optimal only with respect to the prior distribution of prices but not necessarily to the actual equilibrium prices adopted by the rms. 6 The rest of this paper is organized as follows. In the next section I describe the model and the static duopoly equilibrium. Next, I analyze the dynamic setting and show the rockets and feathers result. In section IV, I extend the result for markets with more than two rms. In section V, I show that rockets and feathers is not supported by models where search is done sequentially. Section VI concludes. The model Consider a duopoly where rms sell homogenous goods. Both rms have the same marginal and average production cost, and compete through prices. Consumers (normalized to the unit measure) have unit demands with a choke price v and obtain information about market prices by searching. A portion (0; ) has zero or negative search cost and they are referred to as shoppers (Stahl, 996). Shoppers can be interpreted as consumers who enjoy searching for prices or who obtained price information unintentionally through advertising or while shopping for other goods. The remaining ( ) consumers have positive search costs s i S = [0; s], s > drawn from a continuous and di erentiable cdf g (s i ). Firms decide their prices simultaneously with consumer s search strategies. The production cost can be high or low c C = fc L ; c H g and is only observed by the rms. Consumers search nonsequentially. That is, each consumer decides -before observing any 6 The work by Eckert (00) and Noel (00) use a model of Edgeworth Cycles to explain asymmetric gasoline prices not related to cost shocks that was observed in some canadian cities. 3

4 price- between becoming informed about all the prices in the market (and buying from the store with the lowest price) or remaining uninformed, in which case they buy costlessly from a random store. If a consumer were to search sequentially, after visiting a store (observe its price) she would decide whether to sample for another price or shop at the lowest price observed at that moment. 7 I will postpone the analysis of sequential search protocol to section V. The early literature on consumer-search models focused on nonsequential search protocols (Salop and Stiglitz, 977; Braverman, 980; and Varian, 980) while more recently sequential search models dominated the scene (Stahl, 989 and 996; and Benabou and Gertner, 993). Both sequential and nonsequential search protocols can be optimal depending on the context of the decision problem (Morgan and Manning, 985). When price quotes are not obtained instantaneously (insurance quotes, repair estimates, etc.) and the opportunity cost of time is important, nonsequential search dominates sequential search protocols. Also, the advance of technology might make nonsequential search optimal. For example, consider online shopping. Consumers have the alternative to sample prices sequentially by visiting each store s websites, or conduct a one-time search in an independent website that compares prices across online stores. 8;9 Nonsequential search dominates sequential search not only because it saves time, but also because it makes the comparison easier by organizing the relevant information (quality, seller s reputation, shipping cost, taxes, etc.) from each seller. In cases where price quotes are obtained easily and there are no economies of scale in the size of the price sample, sequential search dominates nonsequential search protocols since it allows consumers to stop searching as soon as they nd a good bargain. Summarizing, rms and consumers play a simultaneous-move Bayesian game. There are N = N F [ N D players, where j N F = fi; IIg represents a rm and i N D = [0; ]. Producers can be of type c j = c C for all j N F and nature (player 0) moves with probability = P r (c = c H ) : Consumer types (search cost) s i S are common knowledge. Firms choose prices p j P = [c L ; ] and consumers choose actions a i A = f0; g = R fdon t search, searchg : 0 Letting = + ( ) a i di represent the number of informed consumers, the pro ts of a rm j that charges price p j and has production cost c are: + j (p j ; p j ; a; c) = (p j c) fp j < p j g + fp j = p j g + fp j > p j g () 7 This is the case of sequential search with perfect recall. In the case of no recall, if the consumer stops searching, she has to shop at the last price observed. 8 This is also true for the case of o ine shopping where prices are posted online by the stores. 9 Some popular portals that compare online prices are pricegrabber.com, pricescan.com, pricingcentral.com, kelkoo.com and bizrate.com. 0 I ignore the decision between buying or not for the consumer by setting as the upper bound for p j: This simpli es notation and does not a ect any result. 0 4

5 R where = + ( ) 0 a i di represents the number of informed consumers. The expected utility of a consumer i with search cost s i is given by: u i (a i ; a i ; p) = a i (Min [p] s i ) ( a i ) P p j () Firm j 0 s strategy pro les are represented by all possible price distributions for a given cost realization: f j (; c) = (f j (p j ; c)) pj P with f j (p j ; c) 0 for all p j P and R P f j (p; c) dp = : Consumers on the other hand have strategy pro les q i (; s i ) (A). The interaction between consumers and rms is summarized by the number of informed consumers : Any pro le D = fq i (; s i )g in D implies a value of [; ]: De ne a Nash best response N (; c) as a symmetric Nash Equilibrium strategy of the game = [N J ; P; jnj ] where j is de ned in (): A Symmetric Bayesian Nash Equilibrium (SBNE) or market equilibrium from now on is a strategy pro le = D ; F such that, i) D is a best response to F = (f (p; c; )) pp and ii) F is a NR D. Let s start analyzing the supply side of the model. Given the number of informed consumers, a rm with cost c that sets a price p can either fail or succeed in capturing the informed consumers. Its pro ts are: f (p; c) = ( ) j (p c) (3) s ( + ) (p; c) = (p c) (4) The assumption of an atom of shoppers ( > 0) rules out the monopoly price as a Single Price Equilibrium (SPE). Otherwise, a store would capture the informed consumers by undercutting slightly. On the other hand, = implies perfect information and p = c is the unique SPE. As it will be shown below, the assumption of s > is su cient for < in equilibrium: In what follows, consider to be less than. By charging the highest possible price a rm obtains half of the uninformed consumers. The expected pro t is ( ) (; c) = ( c) > 0 De ne p as the minimum price a rm is willing to charge, p = s ( ) ((; c)) = c + ( c) (5) ( + ) Even capturing all the informed consumers, at prices below p ; pro ts are lower than charging the monopoly price. For any potential SPE bp (p ; ]; the presence of shoppers This is consistent with the de nition of shoppers given above. If shoppers are thought of as consumers with zero search cost, I break the indi erence in () by assuming they always search. 5

6 allows for a positive gain to deviators if they set p = bp " (by de nition, bp = p cannot be a SPE). Thus, a rm NR consists of mixing strategies over [p ; ] : By the same argument, all mixing strategies that involve a positive mass over any price is ruled out. Denote the cumulative distribution implied by a particular F with F (; c; ). A rm is indi erent between charging any price that generates an expected pro t () : s (p)( F ()) + f (p)f () = () (6) High prices increase mark-ups per unit sold but decrease the expected market share by reducing the likelihood of being the lowest price in the market. This trade-o (surplusappropiation vs. business-stealing) faced by rms induces price dispersion. Proposition There is a unique Nash Response F. Given and c the cumulative distribution of market prices is pr ( )( p) F (p; c; ) = f (x; c; ) dx = (7) p (p c) for all p h ( ) c + (+) ( c); i As the number of informed consumers increases, the pricing distribution expands its support while it rst-order stochastically dominates distributions for lower values. Firms place more density on low prices than high prices since the size of the captive market decreases and the gains from being the lowest price in the market increase. In the limit (! ) ; expected pro ts and prices approach the competitive outcome. On the demand side, consumers decide between becoming informed about the market prices (and pay cost s i ) or buy from a random store. The demand in the market is composed by many consumers and their individual choices i cannot in uence the total number of informed consumers in the pool. The expected bene t of being informed -taking as given the rm Nash Response F is measured by the di erence between the expected price and the expected minimum price in the market. The expected gains from search for consumers are: Z v E [p p min j] = E c 4 p [ [ F (p; c; )]] df (; c; ) 5 = (8) p ( ) + = ( E [c]) log where the last equation is obtained using (7) and integrating by parts. 3 6

7 Figure : Nonsequential search equilibrium with uniformly distributed search costs Price dispersion in the market is drives consumer to search. As! 0 and! the gains from search disappear. Since there is an atom of shoppers, E [p p min j = ] > 0 and as increases, E [p] and E [p min ] decrease. They do it at di erent rates and eventually the maximum gains from searching are achieved at b. For > b, adding informed consumers reduces the gains from search since the rms are placing too much probability in low prices. The following lemma characterizes (8). Lemma The expected gain from search is a strictly concave function of the number of informed consumers. Furthermore, it has a maximum at = < b < : Shoppers always choose to know the actual prices while consumers with search cost higher than their valuation of the good will never search. 3 In equilibrium, rms face at ( g())( ) least informed and captive consumers. The rest of the consumers optimal search strategies are q i (s i < es) = and q i (s i > es) = 0 where es is the search cost of the indi erent consumer: E[p p min j = + ( ) g (es)]] es = 0 (9) Figure represents the unique market equilibrium when search costs are uniformly distributed. This result holds for any search cost distribution as long as the number of shoppers is bigger than b. In that case, g crosses E [] only once. When the number of shoppers is low, a su cient condition for a unique equilibrium is that the slope of g is steeper than the slope of E [] : Proposition There is a unique market equilibrium if: a) > b, or h i b) 0 < < b and g > E[p p min] over ; b : E [p] decreases at an increasing rate for any while E [p min] decreases at an increasing rate for < 0:7834 and then at a decreasing rate. 3 See footnote 9. 7

8 As long as the demand is composed by informed and uninformed consumers, a market equilibrium implies price dispersion. This is not the consequence of heterogeneous search costs. Consider the case of a degenerate g () with S = f0; sg ; 0 < s <. Intuitively, when search cost is su ciently high, nonshoppers don t search and = : This always happens if s > E [p p min j = b] : For very low search costs, gains from search are higher than its cost and everyone would search. But the negative externality generated by > b on the gains from search imply that = cannot be an equilibrium. Imagine =. Firms NR implies p = c and no price dispersion. But then q = is not optimal for nonshoppers. The number of shoppers relative to the search cost of the nonshoppers can be thought of as a measure of competition in the market. Expected prices decrease with. As mentioned before, when search cost is high, only shoppers search and determines the spread between expected prices and marginal cost. If s < E [p p min j = b] ; there is a such that gains from search are higher than its cost (E [p p min j = ] > s). The nal equilibrium could exhibit more informed consumers than shoppers ( > ) and lower mark-ups. In order to analyze the equilibrium properties better, let the number of shoppers be high or low; and the search costs level be high, moderate or low: De nition The number of shoppers are low (high) if is (>) than b. Given ; search costs are de ned to be low, moderate, or high if: s < E [p p min j = ] ; E [p p min j = ] s E [p p min j = b] ; or s > E [p p min j = b] respectively. Figure shows all possible equilibria. 4 There is always a market equilibrium with = if E [p p min j = ] < s: The rest of the equilibria imply > and is de ned jointly by the roots q (if they exist) of E [p p min j = + ( )q] = s (0) There is a unique equilibrium when the number of shoppers is high, or if the search cost is either low or high. But there are three possible equilibria if there is a low number of shoppers and the search cost is moderate. One in which only shoppers search, and two where > : Note that in this latter case, the equilibrium with the smaller root q is unstable, while the other is locally stable (as well as the one with q = 0). Corollary and Table summarize the equilibrium results. Corollary There can be one, two or three possible market equilibria when there are shoppers and ( ) consumers have positive search cost (s > 0) If search cost is high: equilibrium is unique and = If search cost is low: equilibrium is unique and > If search cost is moderate and the number of shoppers is high: equilibrium is unique and = and the number of shoppers is low, there are three equilibria: i) =, ii) > and iii) 3 > > : 4 Exept for the case where is such that E [p p minj = ] = s or s = E [p p minj = b]. 8

9 (a) High search cost (b) Low search cost (c) Moderate search cost Figure : Nonsequential search equilibria with constant search cost Shoppers () Search Cost (s) low high low > > moderate = ; > ; 3 > > = high = = Table : Nonsequential search equilibria with constant search cost 9

10 3 Dynamics and asymmetric pricing Asymmetric pricing refers to the case where output prices react di erently to positive and negative changes in input prices There is an abundant empirical literature that suggests asymmetric pricing is more the norm than an anomaly. 5 In particular, those studies nd that prices react faster to positive than to negative cost shocks (rockets and feathers pattern). In general, most tests of asymmetric pricing are done estimating a dynamic error correction model of the type y t = m+ P i=0 + i (x t i ) + + m P i (x t i ) + (y t 0 x t ) + " t () i=0 where y t is the change in the output price in period t and x t is the change in input price and " t is iid. The last term in parenthesis is the error correction term that accounts for the current deviations from the long-run equilibrium price level. The parameter is expected to be negative and = (sometimes imposed and not estimated). The rockets and feathers pattern is supported by the data when the cumulative response function (CRF) P to positive shocks is above the one for negative cost shocks ( m b + P i > m b i for m Min [m + ; m ]). In general, data restrictions prevent the econometrician from including a su cient number of lags in () such that the CRF is estimated for all the periods where the price accommodates to the shock. 6 A simple dynamic environment where the static game presented in the previous section is repeated over time can be used to explain the rockets and feathers pattern. At the beginning of each period, nature chooses a high or low production cost with probabilities and ( ). Firms set their prices and consumers their searching rules simultaneously. After the market clears, nature draws another production cost shock while consumers learn the previous cost realization. Since the main motivation for this model is to explain asymmetric pricing in competitive markets with atomistics rms, I do not consider the possibility of collusion. There are two sources of price variation over time. Production cost changes and/or changes in consumers expectations. For simplicity, let the probability of high costs follow a Markov process = h(c t ) with h (c H ) = and h (c L ) = ( ). The state of the economy is represented by the past and current cost realizations k t = (c t ; c t ) with k t 5 See footnote 4 in the introduction for empirical work on asymmetric pricing. 6 Although the complete adjustmet to a one-time cost change should imply a common nal price level regardless of the sign of the shock (otherwise mark-ups will grow inde nitely with very volatile inputs), the number of periods to complete the adjustment could be di erent. Thus, there are two possible type of rockets and feathers. A vertical asymmetry refers to higher CRF for the rst periods of adjustment while a horizontal asymmetry implies that the adjustment to a positive shock is completed in less periods than to a negative shock. For the reasons said above, there is no evidence of horizontal rockets and feathers. i=0 i=0 0

11 K = fll; LH; HL; HHg. The transition matrix is P = () where P ij = Pr (s t = K (j) js t = K (i)) : The unique invariant distribution for K is = f=; ( ) =; ( ) =; =g : In this simpli ed model, it takes only two periods for prices to fully adjust to an isolated cost shock. After a cost shock, rms increase (decrease) prices reacting to bigger (lower) production costs and in the following period, they adjust prices to be consistent with consumers updated prior. We are interested in nding the conditions such that + 0 > 0 : Denote p k the average price in the market when state is k. The expected change in prices to a positive and negative cost shock are respectively: 4p E 4c + = Pr (HL) Pr (LHjHL) [p LH p HL ] + Pr (LL) Pr (LHjLL) [p LH p LL ] (3) 4p E = Pr (LH) Pr (HLjLH) [p LH p HL ]+Pr (HH) Pr (HLjHH) [p HH p HL ] (4) 4c and the di erence is 4p 4p E 4c + E = 4c ( ) [(p LH p LL ) (p HH p HL )] ( ) p t c t (5) Prices will be set asymmetrically as long as the reaction of prices to cost shocks is sensitive to the prior held by consumers. Clearly, there would be no asymmetry if cost shocks were iid ( = =) or if consumers held constant priors over time. A change in consumer s priors a ects the gains from search and therefore the equilibrium pool of informed consumers. This in turn a ects the pricing decision of the rm. In general, an increase in the expected production cost increases the expected minimum price more than the expected price in the market. This can be seen in (8). The gains from search E [p p min j] become atter as increases. Thus, the indi erent consumer has a lower search cost (see equation 9) and the equilibrium search intensity decreases with. When consumers expect higher production costs, they search less anticipating lower price dispersion: 0: The analysis is similar for the case of a degenerate g () : When an equilibrium has the property of = ; a change in does not have an e ect in the search intensity. 7 It is necessary that the equilibrium involves searching from consumers with positive search 7 For expositional purposes, I am assuming that the change in does not change the equilibrium type. c t

12 Figure 3: Gains from search and higher expected costs cost ( > ). Ignoring the instance of the unstable equilibrium, an increase in decreases consumer s search intensity. The probability with which consumers search is lower: q (s > 0; 0 ) > q (s > 0; 00 ) when 0 < 00 : Figure 3 shows this e ect for the case of low number of shoppers and moderate search costs. The expected market price for a given cost realization c and prior is given by E [pjc] = Z p F (p; c)dp (6) where the price distribution F (; c) is the market equilibrium distribution (F (; c; ) in (7) with = + ( ) g (es) from (9)). Integrating by parts and deriving: E (pjc) ( ) = + c Log > 0 (7) + it is easy to check that price adjustments to marginal cost is indeed sensible to the amount of search by consumers (hence ). The more search there is in equilibrium the bigger the passthrough. If = ; the pass-through is complete while if = 0 there is no price adjustment. It is important to note that the response of prices to production costs doesn t depend on the reserve price : This will become important when analyzing sequential search. Given (7) and the fact that higher priors imply less search, the sign of the asymmetry is determined by the process that determines : Under cost persistence consumers search more if past cost was low. Firms know that and compete more intensively. As a result, the pass-through is higher when previous cost is low than when it is high. The term in brackets in (5) is positive. The next proposition summarizes the results. Proposition 3 Prices rise faster than they fall when consumers search nonsequentially and there is cost persistence ( > =).

13 The result does not depend on the discrete nature of the cost level. Indeed, assume that every period production cost can go up or down with probability = Pr ct c t > j c t c t > = Pr ct c t < j c t c t < = > =: The states of the economy and transition matrix are equivalent to the c = fc L ; c H g case with H representing c > 0 and L = c < 0. 4 More sellers In this section, I extend the previous results to more atomistics markets. The setup of the model is the same as before with the only exception that n. The reason to present the results in a separate section is that the rms trade-o between low and high prices (business-stealing and surplus-appropiation e ects) changes in a way that the problem becomes less tractable and simulations are needed to characterize the gains from search for consumers. As before, I rst obtain the Nash best responses for the rms and then nd the market equilibrium of the stage game. Dynamics and the rockets and feathers result follow. Equations (3) to (6) as well as Proposition can be reexpressed to account for n > (see proof in Appendix and (Varian, 980)). The rms NR to a given number of informed consumers and production cost c implies a cdf : n ( )( p) F (p; c; ) = (8) n(p c) h i ( )( c) with support c + +(n ) ; : Again, the relationship between and prices implies F (; 0 ) > F (; 00 ) if 0 > 00. That is, rms set lower prices (mark-ups) as expected demand becomes more elastic. However, the traditional oligopoly model predictions on the relationship between number of rms and mark-up is not met. The presence of more stores in the market increases the likelihood of setting prices in the extremes of the distribution. But the strenghtening of the business-stealing and surplus-appropriation e ects is not symmetric. The changes in F () can be appreciated in Figure 4. As n increases, the probability of being the lowest price in the market decreases exponentially while the bene ts from charging high prices decrease at a rate =n. Thus, the surplus-appropriation e ect becomes relatively more important. In other words, even though the captive market becomes smaller with greater n, becoming the supplier of the informed consumers is harder. Fixing, the expected price in the market increases with the number of rms and in the limit the price distribution converges weakly to monopoly pricing (Stahl, 989; and Janssen and Moraga-González, 004). 8 still show that property. Since this is true for any ; the market equilibrium will 8 Monopoly SPE in a search model without shoppers was rst found by Diamond (Diamond, 97). 3

14 Figure 4: Equilibrium price distribution and number of stores (c=0,u=,ţ=0.) The gains from search for a consumer are E c [E [p p min jc; ; n]] = E c Z v p p ( c) n [ F (p)] n [ F (p)] dp (n ) (p c) ( p) (9) The e ect of n on the equilibrium search intensity is the combination of the expected prices and minimum prices. c, hence, decrease the search intensity. When n changes, it, the support of the price distribution increases and the E [p min ] is smaller. Thus, the gains from search are bigger with n and a market equilibrium will have consumers searching more. Proposition 4 Search intensity increases with n : E [p p min jc; ; n + ] > E [p p min jc; ; n] As in the duopoly case, E [p p min jc; n] is continuous in and increases as is away from the extremes. The conditions for uniqueness of Proposition as well as its corollary depend on the concavity of E [] in (9). From simulations (Figure 5.a and Table ), it can be shown that the expected spread has the same shape as in the duopoly case. For any n; expected prices decrease with, but E [p; n] and E [p min ; n] do it at di erent rates. This makes the gains from search increase at a decreasing rate until a maximum b (n) is reached. Then, E [p; n] starts converging to E [p min ; n] : Also, b (n) increases with n independent of and c: Similarly Figure 5.b shows the e ects of higher costs on gains from search. This leads us to the following conjecture and claim Conjecture E[p p minjc;n] c 0; E[p p min jc;n] 0 and has a maximum b (n) with b 0 > 0: Claim Proposition and its corollary hold when n > As in the duopoly case, higher production costs increase expected prices but expected minimum price increase as well. The latter e ect compensates the former and consumers nd less attractive to search when they expect higher costs. < 0 Assuming cost 4

15 (a) Number of rms (=c = ) (b) Cost variation (n = 00; = ) Figure 5: Conditional Gains from search n> =c = =c = 5 =c = 0 jn : 0: :68 0:7555 0:4958 :005 :0408 0:48377 :3076 : : 0: : : :6345 :9046 :347 0:7385 :33967 : :3 0: : : : :75007 : :85046 : : :4 0:57 0:8740 0: :83634 :3384 :4339 0:94088 :4893 :6337 0:5 0: : :9094 0:89045 :35558 : :00758 :54553 : :6 0: : : :9688 :37607 : :0496 :54803 :656 0:7 0: : :9363 0:94699 : :47778 : :5638 : :8 0: :8759 0:9676 0: : :4885 :0674 :57055 : :9 0: : :9638 0:9079 :39604 :4808 :03534 : : Table : Simulated E[p p min jc; n] n b (n) 0:6349 0:847 0:866 0:8704 0:8755 0:879 0:88 0:8844 0:8863 0:888 0:8894 Table 3: Maximum E[p p min jc; n] and n 5

16 persistence E(p) c > 0: c > 0 ; the only result needed to prove the rockets and feathers for n > is Proposition 5 Given conjecture, the Rockets and Feathers result holds for n>. 5 Sequential search In this section I explore the implications of changing the search protocol used by consumers to sequential search. A consumer that searches sequentially decides whether to search for lower prices after visiting the rst store or buy at the observed price. Thus, compared with nonsequential search, there is a gain in the sense that a consumer might not need to search if she nds a good deal in the rst price sampled. Sequential search can be optimal when the search technology does not allow for signi cant economies of scale in the sample size. All else equal, consumers uncertainty over production cost makes sequential searching more desirable than in the certainty case since consumers are better informed about the cost realization (hence, the market price distribution) after observing a price. The change in the search protocol used by consumers modi es the stage game presented in section II from simultaneous to a sequential-move game. After nature draws a production cost c C; rms choose their prices simultaneously and commit to them for period. Once prices are set, each consumer visits a random store and decides between buying at the price observed p 0 or searching (with a cost s i ) for a better price in the remaining store. A consumer that decided to search buys from the store with the lowest price of the two observed (perfect recall). Assume that the distribution of search costs g () is degenerate at s >> 0. The market demand is composed by shoppers and ( ) consumers with search cost s: A consumer i that visits a store j (p j = p 0 ) has conditional utility u ij (a i ; a i ; p) = a i [Min [p] s i ] + ( a i ) p j (0) Similarly, a rm j visited by the set of [0; ( ) =] nonshoppers has a payo 8 >< j (p j ; p j ; a; c) = (p j c) >: + R! a i d i fp j < p j g + fp j = p j g + s i =s () A consumer strategy is a function q i (; p 0 ; s i ) (A) of the observed price p 0 P and search cost. Since shoppers enjoy being informed, q i (; p 0 ; s = 0) = (search) for any p 0 : Firm j s strategy pro le is represented by all possible price distributions conditional on the cost realization: f jc () = (f jc (p)) pp;cc : 9 A SBNE (market equilibrium) consists 9 For expositional reasons, the cost realization now is expressed as a subindex of the strategy pro le instead of an argument of it. Z 0 9 >= ( a i ) d i fp j > p j g >; 6

17 on a strategy pro le = D ; F and system of beliefs 0 p 0 ; F p 0 such that i) P D = (q i (; 0) ; q i (; s)) in D is a best response to F = f c p; D pp; cc ; ii) F is a Nash best response (NR) to D, and iii) 0 = Pr c H jp 0 ; ; F is the Bayesian update of prior for each observed price consistent with F : An intuitive and desired property for the optimal search rule is that it preserves the reservation price property (RPP). In such a case, q would be a step function with q (p 0 ev; s) = 0 (don t search) and q (p 0 > ev; s) = (search) where ev is the reservation price. A feature of models that involve sequential search from unknown price distributions is that consumers search rules may not preserve the RPP. For example, a consumer that observes a low price p 0 might consider that the cost realization for the rms was low and continue searching for lower prices, while if he had observed a high price, the fact that high cost is more probable might make it a bargain. Assume that the price distributions chosen by rms for each production cost level are represented by F H (p; r) p[p H ;r] and F L (p; r) p[p H ;r] with F L (p; r) > F H (p; r) for all p and p H > p L The expected bene t of sampling one more price for a consumer that observed p 0 is the expected price from the other store, conditional on it being lower than p 0 : Z EG(p 0 ; r) = 0 (p 0 ) p 0 p H F H (p; r) dp + 0 (p 0 ) Z p0 p L F L (p; r) dp () A monotonic EG implies a unique (if there is one) reservation price ev such that EG (ev; r) = s: Two e ects take place when p 0 is observed: a direct and a learning e ect. The former is the standard e ect when sampling from a known distribution. High observed prices mean a higher probability of sampling a lower price next time. The learning e ect isolates the new information about the unknown sampling distribution obtained with p 0. When the learning e ect is opposite to the direct e ect EG might be non-monotonic and the RPP can be violated. As it is usual in the literature of optimal sampling from unknown distributions, assumptions are made to preserve the reservation price property. When the price distribution is exogenous, assumptions rely on F (Rothschild, 974). If the price distribution is endogenously determined, assumptions on either production cost structure or the number of shoppers guarantee a weak learning e ect. 0 I will assume that (for a given search cost s) the gap between the two possible production costs (c H c L ) is su ciently 0 For example, Benabou et. al. (993) assume low correlation in rm s marginal costs. In their model, a consumer that observes a low price knows that the rm has a low cost and needs to infer the cost of the other producer in order to decide to sample another price. When costs have low correlation, the learning e ect is dominated by the direct e ect and EG is monotonic. 7

18 low that guarantees the RPP. Thus, restrict consumer strategy pro les to the type: D = (q i (; 0) ; q i (; s)) in D (3) q i (p 0 ; 0) = for all p for p0 > ev q i (p 0 ; s) = 0 otherwise Shoppers search at any price below while nonshoppers search if the price is above ev: As in the case of nonsequential search, the existence of an atom of shoppers eliminates the possibility of a SPE (Stahl, 989). Furthermore, the upper bound for any price distribution is ev instead of the monopoly price. Lemma Any rm NR implies p ev and f c (p = ev) > 0. A consequence of this lemma is that in any market equilibrium, consumers with positive search cost never search ( = ). As in Proposition, there is no pure strategy equilibrium for the rms and the price distribution does not present any mass points or gaps on the support [p c; ev]. A rm that sets a price ev sells to captive consumers while if it charges lower prices, the likelihood of capturing the informed consumers increases. In equilibrium, the rm is indi erent between the reserve price and the prices that generate the same expected pro t: j (p j ; c; ev) = ( F c (p j ; ev)) + ( ) ( ) (p j c) = (ev c) = j (ev; c) ; c fc L ; c H g Lemma 3 Given the number of shoppers and consumers search rule D from (3); there is a unique rm Nash best response that implies: ( )(ev p) F c (p; ev) = p [p (p c) c; ev] ; c fc L ; c H g (5) with p c = c + ( )(ev c) + : Consumers choose their optimal search strategies by maximizing their expected utility () given F and prior Using (5) and Bayes rule, the posterior probability of high cost becomes (4) 0 = ( 0 p0 < p H FH 0 FH 0 +( )F L 0 p 0 p H (6) Below, I show that there is no rm NR if the search rule does not satisfy the RPP. 8

19 Figure 6: Unique reservation price Thus, there is no learning e ect over p 0 < p H and EG (p 0 < p H ; ev) increases with observed prices. At the same time, for p 0 p H ; 0 > and decreases with p 0 : That is, when observed prices are supported by both, high and low production costs, the learning e ect reinforces the direct e ect and EG (p 0 p H ; ev) is also monotonic in p 0: To characterize the market equilibrium, we need to nd an optimal reserve price given that rms price accordingly. Uninformed consumers are indi erent between searching or not when the observed price is equal to the reservation price. At the same time, rms anticipate ev correctly and price accordingly. Thus, an equilibrium is characterized by EG(p 0 = ev; ev) s (7) Figure 6 shows the expected gains from search for the case where { = ; = =; s = 0:5; c H = =; = =5g: EG is not monotonic in p 0 but, the assumption of low c H guarantees a unique reservation price. Intuitively, the closer c H is to c L, the closer p H is to p L. Therefore, the array of possible observed prices that reveal low cost realization are never high enough to justify searching. From the point of view of the rms, the relevant reservation price is any root in (7) below :The next proposition characterizes the market equilibrium. Proposition 6 Under sequential search and assuming RPP (c H c L su ciently small), there is a unique market equilibrium given by (3), (5) and (6) with ev = Min farg solve(7); g If the assumption on low c H c L is relaxed, the possibility of non-monotonic search rules arises. Consider the case where ( ) uninformed consumers have a search rule q consisting of two reservation prices (ev L ; ev H ): 8 0 p 0 ev L < ep >< ev q = L < p 0 < ep 0 ep p >: 0 ev H ev H p 0 with ep > c H. A rm chooses prices given (; q) and -when cost realization is low- takes into account that the ( ) = potentially captive consumers search if p (ev L ; ep) are chosen. 9

20 Proposition 7 There is no (symmetric) market equilibrium when consumers have nonmonotonic search rules with two reservation prices. Even though in a market equilibrium, only shoppers search for low prices, the search rule chosen by the uninformed consumers in uences nal prices by setting an upper bound to them. As the number of shoppers increases, the expected prices decline for two reasons. First, holding the search strategies xed, more shoppers imply more competition and expected prices decrease since p c shifts down and Fc > 0. Second, lower expected prices imply lower equilibrium reservation price in (7) thus, lower prices charged by the rms. Similarly, the amount of informed consumers is not a ected by. The e ect of higher expected production costs occur through the reservation price chosen by the uninformed consumers. An increase in doesn t a ect the search gains when low prices are observed EG (p 0 < p H ; ev) but decreases EG (ev; ev). Assuming c L = 0, EG(ev; ev) = c H EG (ev; ev) < 0 ev c H ( ) Therefore the reservation price will be higher (lower search intensity) when a higher production cost is expected. As in the case of nonsequential search, the search intensity (through reservation price) is lower when a higher production cost is expected. Nevertheless, there is no room for asymmetric pricing with sequential searching in this setting. The equivalent to equation (5) would be: 4p E 4c + 4p E = 4c ( ) p t ev (8) c t ev c t As can be seen from (7), p c only depends on the number of informed consumers and not on their reserve price. Proposition 8 There is no price asymmetry when consumers have search cost s f0; sg and search sequentially. 6 Conclusion 0

21 7 Appendix Proof. [Proposition ] This proof is done for the n rms case since it is also used in Section III. Therefore, p ( )( c) = c + +(n ) in (5). To show that F (; c; ) is a unique symmetric NR the proof is divided in three steps (to simplify notation, ignore the fact that F is conditional on (c; )). First, it shows that there are no point masses in the equilibrium pdf. Second, for " > 0; F (p + ") > 0 and F ( ") <. Last, there are no gaps in the support of F (p). Assume there exist a price bp (p ; ] such that Pr (p = bp) F (fbpg) > 0 (by de nition, F (fp g) = 0). Then, there is an arbitrary small " such that F (fbp "g) = 0:A rm could deviate from F () by applying F d () similar to F () with the exception that F d (fbpg) = 0 and F d (fbp "g) = F (fbpg) : The expected gains for the deviator can be decomposed to four scenarios, depending on the prices charged by the other rms. Let p l be their lowest of the n prices in the market. If p l < bp " : If p l > bp : When p l = bp : nx n j j= nx n F (bp ") j [ F (bp ")] n j ( ) " j n j= Lastly, if p l (bp F (fbpg) j [ F (fbpg)] n j (9) ( ) " + [ F (bp)] n (30) n "; bp) ; the expected gains are: (p c) j ( ) nx n [F (bp) F (fbpg) F (bp ")] j [ F (bp)] n j (p c) j j= n + " ( ) n (3) + " (3) As "! 0, (9) and (30) go to zero while (3) and (3) remain positive.. Suppose F ( ") =. Then at setting p = generates an increase in pro ts (with respect to ") and no loss in customers. Similarly, if F (bp + ") = 0; it has to be that (bp + ") = () : By charging p = bp+"=; pro ts are bigger: (bp + "=) > (p ) = () : 3. Suppose there exists an interval (p ; p ) such that F (p ) = F (p ) : Then, by placing some density on bp (p ; p ) ; a rm will gain by increasing its markup. There is no expected loss since by part of the proof, there are no ties at p. Given,,, and 3 above, the only function that satis es s (p)( F (p)) n + f (p)f (p) = ()

22 is: F (p) = ( )( p) n(p c) n Proof. [Lemma ] I rst show that there exists a unique global maximum b for E [p p min ] and strict local concavity around E [p p min j = b] : Then, concavity everywhere is provided. From (8), E[p p min ] ( c) ( ) ( + ) + = 3 log ( ) ( + ) E[] with Lim!0! c E[] 3 and Lim!! : The term in curly brackets determines the sign of this expression. Critical points are at = b 6= f0; g; log + = ( + ) ( ) ( + ) At = 0, LHS = RHS: The di erence in slopes between RHS and LHS is: (33) LHS RHS = 4 ( ) ( ) ( + ( )) which is positive (negative) for < (>) =. Since at = ; LHS > RHS; there is a unique critical point at b > 0:5: The second derivative of (8) is: E[p p min ] = ( c) ( ) ( + ) 4 Using (33) and rearranging, at b; E[p p min ] = (3 + ( (3 + ))) + (3 ) ( ) ( + ) log ( c) b 3 ( b) ( b) ( b) ( + b) b 4 < 0 For concavity everywhere, (3 + ( (3 + ))) + (3 ) ( ) ( + ) log At = 0; both expressions are equal to zero. For > 0; it can be veri ed that LHS > RHS > 0 Numerically, the maximum can be shown to be b t 0:63486

23 Proof. [Corollary ] Reexpress (9) using (8) ( E [c]) log ( ) + = g () At = g (0) ; the RHS is positive while the LHS is zero. By Lemma, LHS is concave and lower than. Thus, g cuts from below the expected gains from search at least once. If g(0) > b; it is easy to see that there is a unique solution to (9). If g(0) < b; the possibility of multiple solutions is eliminated under the assumptions of concave g (s) on the relevant range Proof. [Proposition 3] Let the equation in (9) be represented by G: Then es = G G es < 0 From (8), the numerator is negative. the denominator is G es = g g es + E [p p minjc; b] < 0 Since at es the inverse cdf cuts the expected price di erential from below, the term in parenthesis is negative. From (7) E [p] c = + + ( ) Log > 0 ( + ) Therefore, E[p] c < 0 and equation (5) is negative when h0 > 0 ( < =). The same argument applies to the case of degenerate g (). E [p p min j = + ( ) q] = s could have one or two roots q depending on the size of and s: The stable equilibrium has E [] cutting s from above. As increases, E [] gets atter and q (hence ) decreases.: Proof. [Proposition 4] As long as the conditional gains from search increase with n; es q n > 0 in (9) and n 0 in a stable equilibrium of (0). The gains from search are: E [p p min jc; n] = Z v p pn [ F (p)] n f (p) dp = De ne z = F (p). Then, p = Changing variables, E [p p min jc; n] = Z 0 Z v p p(v c) (n ) (p c) (v p) n [ F (p)]n dp ( )+cnzn (n )(p c)( p) and dp = ( )n(n )( c)zn dz: (z( )+z n ) ( ) + cnz nz n n Z ( ) + nz n dz = nz n + dz ( ) nzn

24 wlg, the marginal cost can be normalized to 0 and adjusted to 0. De ne A n+ = + ( ) (n + ) zn and A n = + ( ) nzn : Z E [p p min jn + ] E [p p min jn] = 0 Z = 0 = 0 0 z n = ( ) [n (n + ) z] A n+ A n dz = n=(n+) Z 0 z n = ( ) [n (n + ) z] A n+ A n dz 0 h + ( ) (n + ) n n+ 0 n i + 0 n ( ) n n n+ (n + ) z n nz n dz = A n+ A n Z n=(n+) Z 0 z n = ( ) [(n + ) z n] dz A n+ A n z n [n (n + ) z] dz = 0 ( ) Proof. [Lemma ] Given non-shoppers reservation price ev; assume that rms set prices according to a cdf F c () with upper bound bp: It is clear that bp < ev:is not possible since a rm could increase revenues and keep the same number of consumers by setting p = ev: Suppose bp > ev. A rm that charges p = bp would not capture any customers while by ( ) charging p = ev the customers are at least : Thus, the only possibility left is bp = ev Proof. [Lemma 3] Directly from Lemma and Proposition. Proof. [Proposition 6] Assume wlg c L = 0: From (), the upper bound for EG (p 0 < p H ; ev) is given by Lim EG (p 0 < p H; ev) = ev ( ) c H + ev ( ) Log p 0!p H ev ( ) + c H This limit grows with c H. Assuming c H su ciently low, EG (p 0 < p H ; r) < s and thus ev > p H : For observed prices (p 0 > p H ), using p 0 = ev in (), EG (ev; ev) = ev (ev c H ) [ev c H ( )] + ( ) Log + there are two roots to EG (ev; r) = s:and the smaller one can be ignored since EG ev 0 for ev c H : h h ii EG (ev; ev) + ( ) Log + = ev (ev ( ) ch ) + ( ) c ev [ev ( ) c H ] H 4 (34)

25 Proof. [Proposition 7] By Lemma, p ev H : Expected pro ts for any p have to satisfy (p) (ev H c) ( ) =: As in Proposition, an equilibrium can not consist on pure strategies nor the price distribution have mass points or gaps over [p ; ev L ] [ (ep; ev H ]. Also, F (ev H ") < as well as F p c H + " > 0 Assume rst that F (p 0 ) = F (ev L ) for p 0 (ev L ; ep) : The expected pro ts are (ev H ) = (ev H c) ( ) = (35) (p) = (p c) f( ) = + [ F (p)]g p [p ; ev L ] [ (ep; ev H ] (36) (ep) = (ep c) f( ) = + [ F (ep)] + F (fepg) =g (37) and there is no F (p) that satis es (35), (36) and (37) : If p 0 (ev L ; ep) are considered, expected pro t is p 0 = p 0 c + F p 0 + F (ep) F p 0 Clearly, Lim p 0!ep F (p0 ) < F (ep) and thus an atom of probability is needed on ep 5

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