RAND Journal of Economics Vol. 40, No. 4, Winter 2009 pp. 673–687

Rockets and feathers: Understanding asymmetric pricing Mariano Tappata∗

Prices rise like rockets but fall like feathers. This stylized fact of many markets is confirmed by many empirical studies. In this article, I develop a model with competitive firms and rational partially informed consumers where the asymmetric response to costs by firms emerges naturally. In contrast to public opinion and past work, collusion is not necessary to explain such a result.

1. Introduction  Output prices do not react symmetrically to changes in input prices. According to Peltzman’s comprehensive study of 165 producer goods and 77 consumer goods, “In two out of three markets, output prices rise faster than they fall” (Peltzman, 2000; p. 480). This pattern is also known as rockets and feathers and has sometimes been used interchangeably with the term asymmetric pricing.1 Despite the abundance of empirical work confirming this stylized fact, there has not been much progress in terms of theoretical explanations for this widespread phenomenon.2 The first thing that comes to mind when talking about rockets and feathers is collusion. A classical example is gasoline retailing, a market operated by a handful of players with output and input prices easily observable by everyone. Asymmetric gas price adjustments are usually associated with collusive behavior by both government and the media.3 However, Peltzman finds that the rockets and feathers pattern is equally likely to be found in both concentrated and atomistic markets. In this article, I develop a consumer-search model that ∗ University of British Columbia; [email protected]. I am grateful to the editor, Mark Armstrong, and two anonymous referees for suggestions that have significantly improved the exposition of the article. This is a revised version of the first chapter of my dissertation. I would like to thank my advisors Hugo Hopenhayn and David K. Levine for great advice and support. I am also indebted to Florencia Jaureguiberry, Christine Hauser, Dan Ackerberg, Roberto Alvarez, David Rahman, and participants at the Midwest Economic Theory Meetings (Fall 2005) and seminars at various universities for helpful suggestions and discussion. Financial support from the University of California Energy Institute is greatly appreciated. 1 To the best of my knowledge, Bacon (1991) was the first to use the term rockets and feathers to describe the pattern of retail gasoline prices in the UK. This label suggests asymmetries in the immediate adjustment to a cost change as well as in the number of periods needed for a complete adjustment. The main focus of the literature—and this article—is on the former asymmetry. I return to this point in Section 3. 2 The majority of the empirical work is concentrated in the gasoline market (Bacon, 1991; Karrenbrock, 1991; Borenstein, Cameron and Gilbert, 1997; Lewis, 2005; Deltas, 2007; and Verlinda, 2008, among others). Other markets studied include banking (Hannan and Berger, 1991; Neumark and Sharpe 1992; and Arbatskaya and Baye, 2004), beef and pork (Boyd and Brorsen, 1998; and Goodwin and Holt, 1999) and fruits and vegetables (Ward, 1982). 3 See Karrenbrock (1991, p. 20) for media and government representative quotations about gasoline price gouging. C 2009, RAND. Copyright 



explains how an asymmetric response of prices to costs can arise in highly competitive markets. According to traditional economic theory, homogeneous firms that compete on prices earn zero profit, and cost shocks are completely transferred to final prices.4 The nature of this equilibrium changes drastically if consumers are imperfectly informed of market prices and a fraction of them has positive search costs. Firms now profit from ignorance in the market, and the equilibrium is characterized by price dispersion instead of a single price. Still, for any given level of production costs, firms’ optimal price markup is the same regardless of whether their cost shock was positive or negative. In order to obtain asymmetric pricing, the demand function faced by the firms must be sensitive to previous cost realizations. This is indeed what happens when consumers don’t observe the firms’ current production cost. I introduce uncertainty over production costs in a nonsequential search model similar to Varian (1980) and Burdett and Judd (1983). Given consumers’ search intensity, firms maximize profit by choosing prices that are less dispersed under high than under low production costs, since their scope to set prices—measured by the gap between marginal cost and the monopoly price—decreases. Rational consumers anticipate this and therefore the number of consumers that choose to search is lower when costs are expected to be high. Intuitively, when input cost shocks are not independent over time, consumers’ expectations differ depending on whether cost was high or low in the previous period. This translates into different demand elasticities faced by firms when cost falls or rises and therefore, prices react asymmetrically to cost shocks as the firms’ pass-through increases with the level of competition (search intensity) in the market. The rockets and feathers pattern emerges under persistent cost realizations. Suppose that the current marginal cost is high. Consumers expect it will remain high, so they expect little price dispersion and search very little. If in fact the unexpected occurs and marginal cost drops, firms have little incentive to lower their prices because consumers are not searching very much. On the other hand, if marginal cost is currently low, it is likely to stay low, so next period price dispersion is expected to be high, consumers’ search intensifies, and the response by firms to a positive cost shock is to raise prices significantly. The contribution of this article is in formalizing a model with rational agents that isolates the crucial features needed for asymmetric pricing to emerge in noncooperative markets. Costly consumer search stories have been suggested by some empirical studies as a possible explanation for the rockets and feathers pattern (Johnson, 2002; Borenstein, Cameron, and Gilbert, 1997; and Lewis, 2005). Lewis goes a step further and develops a reference-price search model whereby consumers form adaptive expectations about the current price distribution and firms use this myopic behavior to their advantage when setting prices. In this article consumers form rational expectations and use all information available to them. In that sense, the model is related to Dana’s (1994) study of the consequences of production cost uncertainty on consumer search strategies. Previous work on asymmetric pricing focuses on different features of oligopolistic markets. On the one hand, Borenstein, Cameron, and Gilbert (1997) suggest a model of tacit collusion with imperfect monitoring, as in Tirole (1988). With multiple equilibria, firms collude using the past-period price as a focal point. Decreases in production cost facilitate coordination on previous price, while if cost increases it is likely that the past price is unprofitable, collusion breaks down and a higher price emerges as a new equilibrium. On the other hand, Eckert (2002) uses a model of Edgeworth cycles to explain retail gasoline price movements that are observationally equivalent to rockets and feathers although independent of cost shocks. This pattern has been observed in some Canadian cities (Noel, 2007) and is different from the one this article tries to explain. More recently, alternatives to the model in this article were presented by Yang and Ye (2008) and Cabral and Fishman (2008). While I assume symmetric learning by consumers, Yang and Ye 4 Although firms with market power and costless consumer search don’t transfer all of their cost shocks to consumers, they still price symmetrically in that the price they optimally charge depends only on current cost realizations, not on previous costs. Therefore, the rate of change in prices is always the same (as a function of costs) regardless of previous prices, which eliminates the possibility of rockets and feathers.

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(2008) focuses on rockets and feathers produced by asymmetric learning. In order to do this, the authors relax other assumptions that affect the equilibrium outcome generating different empirical implications from the model in this article.5 The price dispersion in Yang and Ye (2008) consists of a two-point distribution that includes the monopoly price. Additionally, the time needed for a complete adjustment of prices to cost shocks is asymmetric. In contrast, the price dispersion in this article consists of a finite number of prices drawn from a distribution with compact support. Also, the number of lags prices need to fully adjust to both positive and negative cost shocks is symmetric. That is, prices react initially faster to a positive cost shock than to a negative shock, but the remaining adjustment is done at a slower pace. Cabral and Fishman (2008) build a different model where firms have heterogeneous cost shocks and consumers search sequentially. Consumers’ willingness to search is low (high) when they observe small (large) price variations. Thus, firms’ incentives to pass cost shocks to final prices differ depending on the size of these shocks. When cost changes are small, sellers increase prices moderately without losing customers but do not reduce prices. The opposite asymmetry occurs when the cost change is large. Both the driving force of the asymmetric price adjustment and the empirical predictions differ from the ones presented here. The rest of the article is organized as follows. In Section 2, I describe the model and the equilibrium of the stage game. In Section 3 the dynamic setting is introduced together with the rockets and feathers result. Section 4 concludes and points the way for future research.

2. The stage game  In this section I lay out a static model in which firms compete choosing prices and consumers decide whether to search or not based on some prior over the firms’ production costs. The model is an extension of Varian’s model of sales (Varian, 1980) in which I endogenize consumers’ search decisions and incorporate uncertainty over production costs. The two main results of this section are the following. First, the market equilibrium involves price dispersion and a fraction of informed consumers (Proposition 2). Second, the search intensity in the market decreases with the expected production cost (Lemma 2).6 This static model serves as the stage game in a dynamic environment that is introduced in the next section. Consider a market with n firms selling a homogeneous good and with the same marginal and average cost. At the beginning of the period, Nature draws the cost for the industry, firms observe the cost realization and compete through prices. There is a continuum of consumers of measure 1 who know only the probability distribution of the production cost. They each have a unit demand with a choke price of υ, and can obtain information about market prices through all-or-nothing nonsequential search. A consumer that chooses to search observes all the prices in the market and, therefore, buys from the cheapest store. Alternatively, an uninformed consumer can only buy from a randomly selected store and faces no cost of observing its price.7 Nonsequential search protocols are especially appealing to consumers when there are economies of scale in price sampling. Products that are advertised in weekly newspapers are a classic example of such advantages. More recent examples include online shopping where specialized websites aggregate and compare all the relevant information across stores, saving consumers the trouble of a sequential search.8

5 The main differences between the two models reside on the assumptions about the number of firms and the distribution of consumers’ search costs. While the model in Section 2 assumes a finite set of firms and consumers with heterogeneous search costs, Yang and Ye assume a continuum of firms and restrict the critical consumers to have identical search cost. 6 As will be clear below, search intensity, the number of informed consumers, and the fraction of searchers are equivalent concepts. I use them interchangeably throughout the article. 7 All-or-nothing is a special case of nonsequential search. Burdett and Judd (1983) allow searchers to choose the number of prices sampled. 8 See Morgan and Manning (1985) for a discussion on the optimal search protocol.

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The cost of becoming informed is the search cost. Assume that a portion λ ∈ (0, 1) of the consumers has zero or negative search cost. I refer to them as shoppers. Shoppers can be interpreted as consumers who enjoy searching for prices or who have obtained price information unintentionally through advertising or while shopping for other goods. The remaining (1 − λ) consumers have positive search costs that are drawn from a continuous and differentiable probability function g(s i ), with si ∈ S = [0, s]. Given the nature of the search protocol, consumers and firms decide their actions simultaneously. The search/no search decision by consumers will be affected by the expected price dispersion in the market and their search costs. Thus, based on their priors about the marginal cost realization, consumers form rational expectations on firms’ pricing strategies to forecast the gains from search. At the same time, firms set their prices anticipating the search intensity in the market. More formally, firms and consumers play a simultaneous-move Bayesian game with a set N = {N F ∪ N D } of players, where j ∈ N F = {1, 2, . . . , n} denotes a firm and i ∈ N D = [0, 1] a consumer. Producers can be of either type cL or cH , where the probability of the high cost cH is α. The distribution of consumers’ search costs (or their types) s i ∈ S is public knowledge. Firms choose prices p j in the interval P = [c  1L , υ] and consumers choose actions a i ∈ A = {0, 1} = {don’t search, search}.9 Letting μ = 0 ai di represent the number of informed consumers, the profit of a firm j that charges a price p j and has production cost c is given by:   1−μ μ (1) + μI{ p j < p− j } + I{ p j = p− j } π j ( p j , p− j ; c) = ( p j − c) n k+1 where p −j represents the minimum price charged by firm j’s competitors, k ∈ {1, . . . , n − 1} is the number of competitors that chose p −j , and I is an indicator function. Meanwhile, the conditional utility of a consumer i with search cost s i is: u i (ai ; si ) = υ − ai (min{ p1 , . . . , pn } + si ) − (1 − ai )

n 1 pj. n j=1


A strategy profile for the firms is represented by all possible price distributions given a  cost realization: σ F = { f j (·, p)} j∈N F with f j ( p;c) ≥ 0 for all p ∈ P and P f j ( p; c)dp = 1. Consumers, on the other hand, have strategies q i (· ; s i ) ∈  (A) that include the possibility of randomizing between search and no search. The interaction between consumers and firms can be summarized by the proportion of informed consumers μ. Any strategy profile for the consumers σ D = {qi (·; si )}i∈N D implies a value of μ ∈ [λ, 1].10 Define a Nash Best Response NBR(μ, c) as a symmetric Nash Equilibrium of the game played by the firms given the search intensity and production cost. A Symmetric Bayesian Nash Equilibrium (SBNE) or market equilibrium is composed of consumers’ consistent beliefs about the marginal cost (α) and a strategy profile σ = (σ D , σ F ) such that i) σ D is a best response to σ F and ii) σ F is a NBR(μ (σ D ), c). In words, a market equilibrium is characterized by consumers that search optimally given the pricing strategies of the firms, and firms that set prices optimally given the number of consumers that decided to search. Start analyzing the supply side of the model by obtaining the firms’ NBR. A given number of informed consumers μ can be related to the expected elasticity of demand faced by each firm. This is clear when we examine the extreme cases of μ = 0 and μ = 1. The former corresponds to n separate monopolies. Each firm faces a completely inelastic demand and maximizes profits by extracting all the consumer surplus ( p = υ). On the other hand, when all consumers are informed about the market prices (μ = 1), firms face perfectly elastic demands that leave them 9 I ignore the consumer’s decision between buying or not by setting υ as the upper bound for p j . This simplifies notation and does not affect any result. 10 This is consistent with the definition of shoppers given above. If shoppers are thought of as consumers with zero search cost, I break any potential indifference in (2) by assuming they always search.

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no option but to price at marginal cost. In the rest of the cases (0 < μ < 1), each firm faces an expected downward sloping demand. It is easy to verify that there is no single price equilibrium (SPE) since a store would capture the informed consumers μ by slightly undercutting its competitor.11 Denote the cumulative distribution implied by a particular equilibrium strategy profile σ F by F(·, μ; c). By the same argument used to rule out SPE, all mixing strategies that involve a positive mass over any price can be ignored. A firm is indifferent between charging the monopoly price υ and any price above or equal to a lower bound p ∗ > c: High prices increase the markup per unit sold but decrease the expected market share by reducing the likelihood of being the cheapest firm in the market. The reverse occurs when low prices are chosen. These surplus-appropriation and business-stealing effects characterize the trade-off faced by firms and, as Varian (1980) shows, induces price dispersion (sales) in equilibrium. Proposition 1. There is a unique Nash Best Response σ F . Given μ and c the cumulative distribution of market prices is  1  (1 − μ)(υ − p) n−1 F( p, μ; c) = 1 − (3) μn( p − c) for all p ∈ [ p ∗ = c +

(1−μ)(υ−c) 1+(n−1)μ

, υ].

Proof . Varian (1980). The share of informed consumers affects the pricing strategies of the firms in two ways. First, as μ increases, there is a smaller captive market for each firm and the profit made by charging the monopoly price decreases. This increases the range of prices that keep the firms indifferent, as they are willing to set lower prices in order to attract the informed consumers. At the same time, a larger proportion of informed consumers makes the business-stealing effect more attractive, hence relatively more weight is placed on low prices. This can be seen in (3) as F(·, μ ) first-order stochastically dominates F(·, μ) for μ < μ. Notice that the presence of more stores in the market increases the likelihood of observing prices in the extremes of the distribution. This is because the chances of being the lowest price in the market decrease with n and middle-range prices will most likely not be enough to capture the informed consumers. But the strengthening of the business-stealing and surplus-appropriation effects is not symmetric. As n increases, the probability of being the lowest price in the market decreases exponentially while the benefits from charging high prices decrease at a rate 1/n. Thus, the surplus-appropriation effect becomes relatively more important than the business-stealing effect and firms prefer to increase the likelihood with which they set prices close to the monopoly price.12 Figure 1 shows the price distributions for different values of n. On the demand side, consumers decide between becoming informed about the market prices (at a cost s i ) or buying from a random store. The market demand is composed of consumers whose individual choices a i do not influence the search intensity in the market. Given the firms’ NBR, the expected benefit for each consumer of being informed is measured by the difference between the expected price and the expected minimum price in the market (price dispersion):  v  p[1 − n[1 − F( p, μ; c)]n−1 ] dF(·, μ; c) . (4) E[ p − pmin | μ] = E p∗

Alternatively, letting z = 1 − F( p, μ ; c) and changing variables, the gains from search can be restated as


Note that SPE and pure strategies equilibrium are the same since an NBR is defined to be a symmetric NE. Even though in the limit the price distribution F converges weakly to the monopoly price (Stahl, 1989; and Janssen and Moraga-Gonz´alez, 2004), its support widens with n and the lower bound approaches the marginal cost. 12

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F(p) 1


p* n=100

p* n=10


p* n=3 p* n=2


 1 E[ p − pmin | μ] = 0


p* n=100

p* n=10

υ(1 − μ) + E [c] μnz n−1 μnz n−1 + 1 − μ

p* n=3 p* n=2

p v

(1 − nz n−1 ) dz


where E[c] = αcH + (1 − α) cL . An individual is more likely to search if the expected price dispersion is large. At the same time, the price dispersion depends on the number of informed consumers. Starting from a monopoly situation with no price dispersion (μ = 0 and p = υ), as μ increases and the profits fade away, the firms start sampling prices from a wider support and placing relatively more weight on low prices. This has the effect that both the expected price and the expected minimum price decrease. But they do so at different rates and there exists a number of informed consumers μ at which the consumers’ gains from search are maximized. For μ > μ, adding informed consumers generates a positive externality on those that remain uninformed because the spread between the average price and minimum expected price is reduced. The following lemma characterizes the benefits of becoming informed. Lemma 1. Consumers’ expected gains from search is a strictly concave function of the number of informed consumers. Furthermore, the gains from search increase with the number of firms in the market:     1+μ (1 − μ) log − 2μ . (6) E[ p − pmin | μ, n] ≥ E[ p − pmin | μ, n = 2] = (υ − E [c]) 2μ2 1−μ Proof . See the Appendix. A consumer decides to search if the benefit of becoming informed is greater than her search cost. Thus, shoppers always search for low prices, while consumers with search cost higher than υ − p ∗ never search. Hence, in any market equilibrium at least λ consumers are informed and (1 − λ) (1 − g(υ − p ∗ )) are uninformed. For the remaining consumers, the optimal search s) = 1 and q(si > s) = 0 where s is the search cost of the indifferent strategies are q(si < consumer: s )] − s = 0. E[ p − pmin | μ = λ + (1 − λ) g(


There is potentially more than one solution to (7) depending on the number of shoppers, the distribution of search costs and the parameters that affect the gains from search. Figure 2 illustrates the case in which search costs are uniformly distributed across nonshoppers. The proportion of informed consumers is measured on the horizontal axis, while the search costs and gains from search are shown on the vertical axis. The dashed and solid concave curve depict the gains from search to consumers. Each consumer compares her search cost with the gains  C RAND 2009.


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_ s



~ λ+ (1-λ )g (s)




from search given the total number of informed consumers. The straight line with positive slope represents the search cost of the marginal consumer that decides to search. There is a unique equilibrium represented by the intersection of the two curves. Consumers with search cost lower than s search and those with higher cost choose to remain uninformed. The following proposition states the conditions required for a unique market equilibrium. Proposition 2. There is a unique market equilibrium if: a) λ > μ, or b) 0 < λ < μ and

∂ g −1 ∂μ


∂ E[ p− pmin ] ∂μ

over μ ∈ [λ, μ].

Proof . See the Appendix. The market equilibrium is characterized by price dispersion and a proportion of informed consumers.13 The equilibrium search intensity is a measure of how strongly the firms compete in prices and is influenced by the expected production cost through its effect on the price dispersion. Even though the level of the production cost does not affect the trade-offs faced by the firms when setting prices, it alters the range over which firms can choose those prices. In other words, the relative benefits and costs of attracting the informed consumers are the same under low and high costs, but as production cost increases, the gap between the monopoly price and the minimum profitable price p∗ decreases (the extreme case being c = υ). This negative relationship between price dispersion and production cost induces consumers to search less when they expect high costs. As can be seen in the duopoly case (6), the gains from search E[ p − p min | μ] become flatter as the probability of high cost increases. Thus, the indifferent consumer has a lower search cost and the equilibrium search intensity decreases with α. Lemma 2. The number of informed consumers decreases with the expected production cost. Proof . See the Appendix. The relevance of this comparative static will be more apparent in the next section, where I present a dynamic setup in which consumers’ priors are based on past cost realizations. Changes in their priors imply changes in the search intensity which in turn affects the way firms pass cost 13 An example of multiple equilibria can be obtained when there are few shoppers and all nonshoppers have the same search cost (degenerate g). Nevertheless, given that λ > 0, any of the market equilibria implies price dispersion and μ ∈ (0, 1) (Tappata, 2006).

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shocks to final prices. Before turning to that, it is important to note that the results in this section should not be restricted to unit demands. If consumers have multiunit demands, the firms’ pricing strategies take into account the search intensity as well as the price elasticity of consumers’ demand, and the gains from search are a function of the price dispersion and the price level. The extension to the linear demand case is straightforward, and it can be shown that, in the case of duopolies, the results of this section hold for the concave and convex demand functions of the form q( p) = (1 − p)β , with β > 0 (Tappata, 2006). In all these cases, consumers search less when they expect higher production costs for two reasons: First, the gain from search per unit (price dispersion) decreases and second, the incentives to search are lower because consumers expect to buy fewer units.

3. Dynamics and asymmetric pricing  In this section, I present a simple dynamic model that sets out the conditions under which asymmetric pricing in highly competitive markets holds. The main result is captured by Proposition 3. Firms react differently to positive cost shocks than to negative shocks as long as those shocks are not i.i.d. Given that consumers do not know the actual production cost, their search decisions are linked to past cost realizations and firms face demands with different elasticities, depending on whether the cost dropped or rose in the past period. This demand change implies asymmetric cost pass-through by the firms. Before getting into the setup of the model, it is important to characterize more precisely the rockets and feathers pattern. The rockets and feathers and asymmetric pricing labels have been used by the empirical literature to describe the immediate reaction of output prices to input cost shocks. These studies find that the immediate partial adjustments of output prices to a positive cost shock are larger than after a negative cost shock.14 Interestingly enough, there is no clear evidence that the time taken for output prices to fully adjust to cost changes is asymmetric (See, e.g., Borenstein, Cameron, and Gilbert, 1997; and Peltzman, 2000). Therefore, it must be the case that output prices initially react faster when the input cost shock is positive but the reverse occurs as the total adjustment is near completion. The majority of the asymmetric pricing studies estimate a dynamic model of the following type: m+ m−   + + βi (xt−i ) + βi− (xt−i )− + εt (8) yt = i=0


where y t and x t represent output and input prices, and  their change with respect to the levels in the previous period. Abusing of notation, the superscripts + and − act as indicators for positive and negative cost changes. The model in (8) allows for different effects of positive and negative cost shocks on prices and it assumes that output prices adjust completely to a cost shock after m + or m − periods.15,16 By separating the effects of positive and negative cost changes, a cumulative response functions CRF(k) can be constructed for each type of shock. These functions predict the cumulative price adjustment after k periods from a one-time cost shock and are used to test for asymmetric pricing.17 The rockets and feathers evidence consists of the CRF + being (statistically) larger than CRF − for low values of k and their difference disappearing as the periods away from the cost shock approach min{m + , m − }.18 That is the pattern that the model below generates. 14

See footnote 2 for references on empirical work. The number of periods is also estimated and there is no clear evidence on whether m − = m + . In many cases, data restrictions prevent the econometrician from including a sufficient number of lags and therefore only partial adjustments are studied (Peltzman, 2000). 16 The last term in (8) includes a white noise error term as well as an error-correction term that accounts for the current deviations from a long-run equilibrium relationship between the output and input prices. 17 See Borenstein, Cameron, and Gilbert (1997) for a description of the CRF construction from the estimated parameters in (8). 18 An example of this pattern for retail gasoline prices can be seen in Borenstein, Cameron, and Gilbert (1997). 15

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Consider a dynamic environment in which the static game presented in the previous section is repeated over time. Assume that at the beginning of each period, Nature chooses a high or low production cost with probabilities α and (1 − α). After that, each firm observes the cost realization and sets prices while consumers observe the previous period cost realization and decide whether to search or not. The next period starts after the market clears, and the process is repeated. I concentrate on the analysis of a Markov Perfect Equilibrium. Since the main motivation for this model is to explain asymmetric pricing in atomistic markets, I ignore the possibility of collusion among firms. There are two sources of average-price variation over time in this setup. On the one hand, expected prices can change as a reaction to a change in the production cost. All else equal, a higher production cost implies higher prices. On the other hand, prices can vary as a result of a change in consumers’ priors. This is an indirect effect on expected prices that materializes through the variations in the search intensity μ. Firms can anticipate this change and adjust their prices accordingly.19 The expected market prices are completely characterized by the current production cost level and the amount of search in the market. For simplicity, let the probability of high costs follow a first-order Markov process α = h(ct−1 ) where h(cH ) = ρ and h(cL ) = (1 − ρ) with 0 < ρ < 1. It then follows that there is a one-to-one map between the previous period cost and the actual search intensity. Therefore, the state of the economy can be represented by past and current cost realizations. Denote the current state by k = (ct−1 , ct ). Because production costs can only be low or high, there are four possible states denoted by the set K = {L L, L H , H L, H H } with k i = K (i). Given a current state k i , the probability of moving to a new state k j next period is denoted by the element P ij in the following transition matrix: ⎤ ⎡ ρ 1−ρ 0 0 ⎥ ⎢ ⎢0 0 1−ρ ρ⎥ ⎥ ⎢ P=⎢ ⎥. ⎥ ⎢ρ 1 − ρ 0 0 ⎦ ⎣ 0 0 1−ρ ρ Thus, if the current state involves low actual and low past cost realizations (k 1 = L L), it can never happen that the next state indicates high as the previous cost (P 13 = P 14 = 0). Last, there is a unique invariant distribution for K and that is represented by π = {ρ/2, (1 − ρ)/2, (1 − ρ)/2, ρ/2}. In this simplified world, it takes only two periods for prices to fully adjust to an isolated cost change.20 After a shock, firms increase (decrease) prices reacting to bigger (lower) production costs. In the following period, assuming production cost does not change, firms adjust prices to be consistent with the new updated prior held by consumers. After two periods, the prices are in line with the new cost level, and the size of the price adjustment is the same, independent of the sign of the cost shock. Therefore, asymmetric pricing, if any, has to be observed in the first period of adjustment to a cost shock. We are interested in finding the conditions such that β +0 > β −0 in (8). First, consider the estimated β +0 that this model would generate. Let p k be the average market price when the state of the economy is k. For a positive cost shock to occur, the previous cost realization has to be low. Thus, the previous state was either LL or HL and the new state is LH. Similarly, for β −0 the state of the period in which the cost drops can only be HL while the previous state could have been either HH or LH. The expected change in prices to a positive and negative cost shock are, respectively: 19 Note that firms set new—dispersed—prices in every period regardless of changes in the production cost or consumers’ prior. The changes mentioned above refer to adjustments to the equilibrium distribution function from which firms draw their prices. 20 Note that this is just a simplification. The length of the adjustment can be easily increased if the learning by consumers in the period following the cost shock was not perfect.

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p c+

p E c−

 = Pr(HL) Pr(LH | HL) [ p LH − p HL ] + Pr (LL) Pr(LH | LL) [ p LH − p LL ]  = Pr(LH) Pr(HL | LH) [ p LH − p HL ] + Pr (HH) Pr(HL | HH) [ p HH − p HL ] ,

and using the transition and unconditional probabilities (P and π ), the difference becomes     p p −1 E −E = ρ (1 − ρ) [( p HH − p HL ) − ( p LH − p LL )] . + − c c 2


This last equation summarizes the conditions for asymmetric pricing. Note that the economy cannot move from a state HL to a state H H , so p HH − p HL represents the change in expected prices after an increase in production cost holding consumers’ priors constant at α = ρ. Likewise, p LH − p LL represents the increase in prices if consumers’ priors are α = 1 − ρ . It is straightforward to anticipate that β +0 = β −0 if the cost shocks are not iid. The following proposition shows that the rockets and feathers arise if ρ > 1/2. Proposition 3. Output prices rise faster than they fall when input prices show persistence:     p p −E > 0. E c+ c− Proof . See the Appendix. Intuitively, the rockets and feathers outcome is a consequence of firms facing more inelastic demands when the marginal cost drops than when it goes up. Suppose that marginal cost is currently high; consumers expect it will remain high, so they expect little price dispersion and very few choose to search. If, in fact, marginal cost drops, firms have few incentives to lower their prices because consumers are not searching very much. On the other hand, if marginal cost is currently low, it is likely to stay low, so next period’s price dispersion is expected to be high, the proportion of consumers searching increases, and the response to a positive cost shock is to pass most of it to prices. One way of identifying the drivers behind asymmetric pricing is by decomposing the terms in the square bracket of (9) into: (i) The effect of previous cost realization on consumers’ priors, (ii) the effect of those priors on the equilibrium search intensity, and (iii) the effect of the search intensity on the cost pass-through. That is,   p  p  [( p HH − p HL ) − ( p LH − p LL )] = − ct ct−1 =c H ct ct−1 =cL     p  μ α p  = − , ct μ=μ H ct μ=μL α ct−1 with c αt−1 = c2ρ−1 , and μH and μL are the equilibrium search intensities when consumer priors H −c L about the probability of a high cost are h(cH ) = ρ and h(cL ) = 1 − ρ, respectively, ( μ = μH − μL ). Hence,   

 p ∂ 2 p ∂μ α p − E = sign − . sign E c+ c− ∂c∂μ ∂α ct−1 In the previous section, Lemma 2 showed that a higher expected production cost implies a lower search intensity. Lower gains from search are associated with higher costs since, as the gap between the marginal cost and the monopoly price is reduced, price dispersion decreases. Thus, the equilibrium pool of consumers that want to become informed decreases with α( ∂μ < 0). Now ∂α turn to the effect of search on the cost pass-through. The limiting cases of perfect competition and monopoly are useful benchmarks. In a perfectly competitive environment, prices are driven  C RAND 2009.


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n=50 n=100 0



entirely by costs and a complete pass-through is expected after a cost shock. On the other hand, the monopolist pass-through is zero since the monopoly price is not affected by cost shocks. An increase in the number of informed consumers is similar to an increase in the elasticity of the expected demand faced by each firm. In other words, as more consumers become informed, the market gets more competitive and the link between costs and prices is stronger. Firms compete more fiercely for the increasing mass of informed consumers by setting prices closer to marginal ∂2 p > 0). cost. As a result, the cost pass-through increases with μ( ∂c∂μ Figure 3 shows the cost pass-through and search intensity relationship for different number of firms. Note that the convergence to the complete pass-through happens at a lower pace in markets with more sellers. Starting from μ = 0, as consumers become informed, the surplusappropriation effect is stronger in more atomistic markets. That is, for a given μ > 0, firms in markets with more sellers sample high prices more often than low prices and average prices are further away from the marginal cost. Therefore, holding the number of informed consumers fixed, each firm is more concentrated on its captive consumers and—like a monopolist—has fewer incentives to adjust prices to cost shocks. Finally, the sign of (9) is determined by the process behind α. It is important to note that the result obtained above is related to the average price chosen by the firms in a given market. While this is consistent with the empirical tests of rockets and feathers, one might argue that the relevant question is whether we observe this asymmetric pattern in the prices paid by consumers. To answer this question, the expected prices in (9) need to incorporate the fact that the firm with the lowest price in the market has a high market share and than the remaining firms sell only to uninformed consumers. Denote this expected price by p = μE[ pmin | c] + (1 − μ)E[ p | c]. It can be shown that     p −1 p E − E = ρ (1 − ρ) (c H − c L )(μ H − μ L ). c+ c− 2 This expression is positive when there is cost persistence because consumers assign low probability to a change in the production cost and (by Lemma 2)μH < μL . The intuition from Proposition 3 might be helpful when thinking about asymmetric pricing in other environments. Although a complete analysis is outside the scope of this article, some insights can be obtained for the case where consumers search sequentially. Under this protocol, consumers’ optimal search strategy comprises searching for an additional price only when the lowest observed price is above some reservation value. The analysis with an unknown and  C RAND 2009.


endogenous price distribution is more involved than in the case of nonsequential search and the usual simplifying assumption adopted in the literature is that nonshoppers have a common search cost (Rothschild, 1974; Reinganum, 1979; Stahl, 1989; and Benabou and Gertner, 1993). As a consequence, firms never choose a price above the unique reservation value and shoppers are the only consumers that engage in active search. Firms draw prices from a distribution as in (3) where υ is replaced by nonshoppers’ reservation value (Stahl, 1989). The analysis does not change considerably when there is production cost uncertainty. Consumers’ reservation value is based on the previous cost realization. The current cost information is updated from the observed prices, but consumers still find it suboptimal to search for a new price quote (Dana,1994; and Tappata, 2006). Let υ H and υ L denote the reservation values for nonshoppers associated with a high and low past cost realization. It can be shown that the expected price in the market is a linear combination of the production cost and the reservation value (equation (A1) in the Appendix). Therefore, p HH − p HL = p LH − p LL in (9) and prices react symetrically to cost shocks. From the previous analysis, we know that the rockets and feathers pattern occurs as a consequence of the search intensity changing with consumers’ priors. Thus, the nonsequential model suggests that a necessary condition for asymmetric pricing in sequential search environments would be to relax the assumption of homogeneous search costs for nonshoppers. The main point of this article is that consumer search can play an important role in explaining the rockets and feathers. In general, decentralized markets share the feature that consumers do not know all the prices quoted by the firms. This is the case of gasoline retailing, probably the most well known case of asymmetric pricing.21 Furthermore, the additional modeling assumptions fit well with the characteristics of the industry. Gas stations in a same geographic market sell a fairly homogeneous good and, since the technology does not allow for input substitution, the retail price is mainly driven by the wholesale price of gasoline.22,23 Also, capacity constraints appear to be irrelevant at the retail level (Borenstein, Cameron, and Gilbert, 1997; and Noel, 2007). On the demand side, consumers are generally aware of the latest changes in the oil price (a good proxy for the movements in the wholesale price) and can use this information to update their gains from search. Evidence that the price dispersion in the gasoline market is consistent with a model of nonsequential consumer search is found by Chandra and Tappata (2008). In addition, they show that different measures of price dispersion decrease with the price level of gasoline. Although search intensity is not observed, this provides indirect evidence that consumers search less when they expect higher production costs, as predicted in Lemma 2.24

4. Conclusion  The contribution of this article lies in developing a theoretical explanation for the unexplained but widely observed rockets and feathers pattern. The model links the firms’ asymmetric response to cost shocks to the fact that consumers are imperfectly informed about market prices and the industry’s production cost. Consumers’ search decisions affect the firms’ elasticity of the demand and therefore their cost pass-through. If production cost shows serial correlation, the number of informed consumers in the market depends on the previous cost realization and as a result the cost pass-through exercised by the firms is different when the cost drops than when it raises. 21

See footnote 2 and Geweke (2004) for more references on rockets and feathers in the retail gasoline market. Gas prices can differ significantly between markets due to differences in local and state taxes, fuel requirements and supply conditions in the local upstream market. 23 The wholesale cost paid by each gas station could differ according to the degree of vertical integration between refineries and stations. However, evidence shows that the vertical contracts are motivated by agency conflicts (Shepard, 1993) and do not affect the stations’ pricing strategies (Hastings, 2004). 24 Another industry that is heavily represented in the rockets and feathers literature is the banking sector. In this case, the evidence indicates that consumer deposit interest rates react slower to positive shocks to the open market interest rate than to negative shocks (Hannan and Berger, 1991; and Neumark and Sharpe, 1992). The consumer deposits market shares the same characteristics as the gasoline retailing market captured by the model in Section 2. 22

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The simplicity of the model helps to identify the main forces that might drive asymmetric pricing in highly competitive markets. This comes at a price, and extensions to the model would clearly improve its capacity to predict the observed price patterns. The search protocol, the consumers’ learning process, as well as the production cost stochastic process, are dimensions worth exploring in future research. In many markets, sequential rather than nonsequential search is the optimal protocol for consumers, and even though the analysis of sequential search from unknown (and endogenous) distributions is notably more complicated, the insights from the nonsequential search model can be useful. Also, consumers in real markets do not get to learn uniformly the cost realization on the previous period. This simplifying assumption could be replaced by one in which the learning process is not homogeneous or perfect. It is more likely that consumers who decided to search in the past are better informed than those who did not. That way, the search intensity adjustment to a cost shock would take more than just one period and the adjustment of output prices to input cost shocks would be smoother. Contrary to public opinion and previous work suggesting collusive behavior as the cause of asymmetric pricing, this article shows that it can well be the outcome in noncooperative markets. This finding reinforces the importance of consumer search models in explaining actual markets’ functioning.25 The extent to which price dispersion is explained by consumer search models also has significant policy implications. Dispersed prices have different effects on welfare when there is product differentiation than when consumer search is costly. Under product differentiation, less variety (hence lower price dispersion) in the market is associated with lower welfare. In contrast, if consumer search is costly, policies oriented to eliminate search costs and price dispersion can be welfare improving. Appendix Proof of Lemma 1. Concavity can be shown directly from (5): ∂ E[ p − pmin | μ] = −(υ − E[c]) ∂μ and ∂ 2 E[ p − pmin | μ] = − (υ − E [c]) ∂μ2





nz n−1 (1 − nz n−1 ) dz (μnz n−1 + 1 − μ)2

2nz n−1 (1 − nz n−1 )2 dz < 0. (μnz n−1 + 1 − μ)3

The gains from search when n = 2 are obtained using (3) and integrating by parts (4). To show that E[ p − p min |μ] increase with n, first normalize the marginal cost to 0 and adjust υ to υ  in (5):  1 1 − nz n−1 dz. E[ p − pmin | n] = υ  μ 0 1+ nz n−1 (1 − μ) Define An+1 = 1 + E[ p − = = ≥

μ (1−μ)

(n + 1) z n and An = 1 +

μ (1−μ)

nz n−1 :   1 1 − (n + 1) z n 1 − nz n−1 dz − pmin | n + 1] − E[ p − pmin | n] = υ  An+1 An 0  1 [n − (n + 1) z] z n−1 υ dz (1 − μ) 0 An+1 An    1 n/(n+1) [n − (n + 1) z] z n−1 [(n + 1) z − n] z n−1 υ dz − dz (1 − μ) 0 An+1 An An+1 An n/(n+1)  1 υ  /(1 − μ) z n−1 [n − (n + 1) z] dz = 0. 


n−1  0 n n μ μ (n + 1) 1+ 1+ n (1 − μ) n+1 (1 − μ) n+1 Q.E.D.

25 An immediate application of these type of models is in macroeconomics where recent findings show that product differentiation alone does not do a good job in explaining the observed price rigidities and high frequency of changes on intra-industry relative prices (Klenow and Willis, 2006).

 C RAND 2009.

686 / THE RAND JOURNAL OF ECONOMICS Proof of Proposition 2. Reexpress (7) using (4)

  v μ−λ . p[1 − n[1 − F( p, μ; c)]n−1 ] dF(·, μ; c) = g −1 E 1−λ p∗ At μ = λ + (1 − λ) g(0), the RHS = 0 while the LHS > 0. The gains from search (LHS) is a strictly concave function of μ and equals 0 at μ = 1. Thus, g −1 cuts from below the expected gains from search at least once. If λ > μ, it is easy to see that there is a unique solution to (7). If λ < μ, the possibility of multiple solutions is eliminated under the Q.E.D. assumption that g −1 has steeper slope than the gains from search for any value of μ in the range (λ, μ). Proof of Lemma 2. Let the equation in (7) be represented by G: G = E[ p − pmin | μ] − g −1

μ−λ 1−λ

where μ = λ + (1 − λ)g( s ). By the IFT, ∂G ∂ s = − ∂α . ∂G ∂α ∂ s The denominator is

μ] ∂ E[ p − pmin | ∂ g −1 ∂ g ∂G = (1 − λ) − <0 ∂ s ∂μ ∂ μ ∂ s

because at s, g −1 cuts the expected price differential from below (the term in parenthesis is negative). The numerator is negative as long as the gains from search decrease with α. Using (4),  1 ∂ E[ p − pmin | μ] μnz n−1 = (c H − c L ) (1 − nz n−1 ) dz n−1 + 1 − μ ∂α 0 μnz   z  1  n−1  μnz n−1 μnz n−1 n−1 = (c H − c L ) (1 − nz nz ) dz − − 1 dz n−1 + 1 − μ n−1 + 1 − μ 0 μnz z μnz 1  1  n−1 where z = n . Thus,  (c H − c L ) μnz n−1 z ∂ E[ p − pmin | μ] (1 − nz n−1 ) dz = 0. < ∂α μnz n−1 + 1 − μ 0 As α increases, the search cost of the indifferent consumer (hence μ) decreases.


Proof of Proposition 3. The expected market price for a given cost realization c and search intensity μ is given by  υ  1 υ(1 − μ) + cμnz n−1 dz (A1) p = E[ p | c] = pd F( p, μ; c) = μnz n−1 + 1 − μ ∗ p 0 where the last equation by changing variables with z = 1 − F(·).  1 is obtained n−1 Let (μ) = 0 μnzμnz n−1 +1−μ dz. For a given previous period cost realization i, the difference in the expected prices associated with high and low current cost is pi H − pi L = (c H − c L ) (μi ) where μi is the search intensity associated with past cost i. We can reexpress (9)     p −1 p −E = ρ (1 − ρ) (c H − c L ) [ (μ H ) − (μ L )] . E + − c c 2 The expression above is positive as (μ) is an increasing function of μ and μH < μL (Lemma 1).


References ARBATSKAYA, M. AND BAYE, M.R. “Are Prices ‘Sticky’ Online? Market Structure Effects and Asymmetric Responses to Cost Shocks in Online Mortgage Markets.” International Journal of Industrial Organization, Vol. 22 (2004), pp. 1443–1462. BACON, R.W. “Rockets and Feathers: The Asymmetric Speed of Adjustment of UK Retail Gasoline Prices to Cost Changes.” Energy Economics, Vol. 13 (1991), pp. 211–218. BENABOU, R. AND GERTNER, R. “Search with Learning from Prices: Does Increased Inflationary Uncertainty Lead to Higher Markups?” Review of Economic Studies, Vol. 60 (1993), pp. 69–93.  C RAND 2009.


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BORENSTEIN, S., CAMERON, A.C., AND GILBERT, R. “Do Gasoline Prices Respond Asymmetrically to Crude Oil Price Changes?” Quarterly Journal of Economics, Vol. 112 (1997), pp. 305–339. BOYD, M.S. AND BRORSEN, B.W. “Price Asymmetry in the U.S. Pork Marketing Channel.” North Central Journal of Agricultural Economics, Vol. 10 (1988), pp. 103–110. BURDETT, K. AND JUDD, K.L. “Equilibrium Price Dispersion.” Econometrica, Vol. 51 (1983), pp. 955–970. CABRAL, L. AND FISHMAN, A. “Business as Usual: A Consumer Search Theory of Sticky Prices and Asymmetric Price Adjustment.” Department of Economics, Stem School of Business, New York University, 2008. CHANDRA, A. AND TAPPATA, M. “Consumer Search and Dynamic Price Dispersion: An Application to Gasoline Markets.” Sauder School of Business, University of British Columbia, 2008. DANA, J.D. “Learning in an Equilibrium Search Model.” International Economic Review, Vol. 35 (1994), pp. 745–771. DELTAS, G. “Assymetries in Retail Gasoline Price Dynamics and Local Market Power.” Department of Economics, University of Illinois, Urbana-Champaign, 2007. ECKERT, A. “Retail Price Cycles and Response Asymmetry.” Canadian Journal of Economics, Vol. 35 (2002), pp. 52–77. GEWEKE, J.F. “Issues in the Rockets and Feathers Gasoline Price Literature.” Report to Federal Trade Commission, 2004. GOODWIN, B. AND HOLT, M.T. “Price Transmission and Asymmetric Adjustment in the U.S. Beef Sector.” American Journal of Agricultural Economics, Vol. 81 (1999), pp. 630–637. HANNAN, T.H. AND BERGER, A.N. “The Rigidity of Prices: Evidence from the Banking Industry.” American Economic Review, Vol. 84 (1991), pp. 938–945. HASTINGS, J. “Vertical Relationships and Competition in Retail Gasoline Markets: Empirical Evidence from Contract Changes in Southern California.” American Economic Review, Vol. 94 (2004), pp. 317–328. ´ , J.L. “Strategic Pricing, Consumer Search and the Number of Firms.” The JANSSEN, M.C.W. AND MORAGA-GONZALEZ Review of Economic Studies, Vol. 71 (2004), pp. 1089–1118. JOHNSON, R.N. “Search Costs, Lags and Prices at the Pump.” Review of Industrial Organization, Vol. 20 (2002), pp. 33–50. KARRENBROCK, J.D. “The Behavior of Retail Gasoline Prices: Symmetric or Not?” Federal Reserve Bank of St. Louis Review, Vol. 73(4), July/August (1991), pp. 19–29. KLENOW, P.J. AND WILLIS J. “Real Rigidities and Nominal Price Changes.” Working Paper No. RWP06-03, Federal Reserve Board of Kansas City, 2006. LEWIS, M. “Asymmetric Price Adjustment and Consumer Search: An Examination of the Retail Gasoline Market.” Department of Economics, The Ohio State University, 2005. MORGAN, P. AND MANNING, R. “Optimal Search.” Econometrica, Vol. 53 (1985), pp. 923–944. NEUMARK, D. AND SHARPE, S.A. “Market Structure and the Nature of Price Rigidity: Evidence from the Market for Consumer Deposits.” Quarterly Journal of Economics, Vol. 107 (1992), pp. 657–680. NOEL, M. “Edgeworth Price Cycles: Evidence form the Toronto Retail Gasoline Market.” Journal of Industrial Economics, Vol. 55 (2007), pp. 69–92. PELTZMAN, S. “Prices Rise Faster Than They Fall.” Journal of Political Economy, Vol. 108 (2000), pp. 466–502. REINGANUM, J.F. “A Simple Model of Equilibrium Price Dispersion.” Journal of Political Economy, Vol. 87 (1979), pp. 851–858. ROTHSCHILD, M. “Searching for the Lowest Price When the Distribution of Prices Is Unknown.” Journal of Political Economy, Vol. 82 (1974), pp. 689–711. SHEPARD, A. “Contractual Form, Retail Price, and Asset Characteristics in Gasoline Retailing.” The RAND Journal of Economics, Vol. 24 (1993), pp. 58–77. STAHL, D.O. “Oligopolistic Pricing with Sequential Consumer Search.” American Economic Review, Vol. 79 (1989), pp. 700–712. TAPPATA, M. “Consumer Search, Price Dispersion, and Asymmetric Pricing.” Ph.D. dissertation, UCLA, 2006. TIROLE, J. The Theory of Industrial Organization. Cambridge, Mass.: MIT Press, 1988. VARIAN, H.R. “A Model of Sales.” American Economic Review, Vol. 70 (1980), pp. 651–659. VERLINDA, J. “Do Rockets Rise Faster and Feathers Fall Slower in an Atmosphere of Local Market Power? Evidence from the Retail Gasoline Market.” Journal of Industrial Economics, Vol. 56 (2008), pp. 581–612. WARD, R.W. “Asymmetry in Retail, Wholesale and Shipping Point Pricing for Fresh Vegetables.” American Agricultural Economics Association, Vol. 14 (1982), pp. 205–212. YANG, H. AND YE, L. “Search with Learning: Understanding Asymmetric Price Adjustments.” The RAND Journal of Economics, Vol. 39 (2008), pp. 547–564.

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