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Article Excerpt
In many retail markets, prices rise faster than they fall. We develop a model of search with learning to explain this phenomenon of asymmetric price adjustments. By extending our static game analysis to the dynamic setting, we demonstrate that asymmetric price adjustments arise naturally. When a positive cost shock occurs, all the searchers immediately learn the true state; the search intensity, and hence the prices, fully adjust in the next period. When a negative cost shock occurs, it takes longer for nonsearchers to learn the true state, and the search intensity increases gradually, leading to slow falling of prices.
Introduction
* Firms are quick to raise prices in response to their cost increases, but not so keen to reduce prices when their costs fall. This widespread phenomenon is known as asymmetric price adjustment, or the rockets and feathers. This pattern of asymmetric price adjustment has been reported in a broad range of product markets. In fact, a growing empirical literature documents asymmetric price adjustment in various markets, including gasoline (Bacon, 1991; Karrenbrock, 1991; Duffy-Deno, 1996; Borenstein et al., 1997; Eckert, 2002; Deltas, 2004), fruit and vegetables (Pick et al., 1991; Ward, 1982), beef and pork (Boyd and Brorsen, 1988; Goodwin and Holt, 1999; Goodwin and Harper, 2000), and banking (Hannan and Berger, 1991; Neumark and Sharpe, 1992; O'Brien, 2000). (1) According to Peltzman (2000), asymmetric price adjustment is found in more than two of every three markets examined in a large sample with 77 consumer goods and 165 producer goods.
Despite these extensive empirical studies confirming the general pattern of asymmetric price adjustment, there is little theoretical work examining this phenomenon. In fact, asymmetric price adjustment first appeared to be inconsistent with conventional microeconomic theory, which usually suggests that an increase or decrease of input prices should affect marginal costs, and hence move prices up or down in a symmetric, rather than asymmetric, way. As Peltzman (2000) puts it, the "stylized fact" of asymmetric price adjustment "poses a challenge to theory." This article attempts to help bridge such a gap in the literature.
More specifically, we develop a model of search with learning in a dynamic framework. We start with a description of the static game. There are a continuum of consumers and a continuum of firms with capacity constraints. Firms have a common unit production cost (either high or low). Although known to the firms, the cost is unknown to the consumers. There are three types of consumers: the low search cost consumers who always search, the high search cost consumers who never search, and critical consumers whose search cost is intermediate. The decision for a critical consumer to search or not depends on whether the expected benefit of searching outweighs her search cost, so the percentage of consumers who search (the search intensity) will be endogenously determined. We adopt the protocol of nonsequential search, that is, consumers who search observe the prices charged by all firms, so searchers always shop at firms with the lowest price available (unless they are rationed due to firms' capacity constraint, in which case they will shop at the firms with the next lowest price, and so forth). On the other hand, nonsearchers shop randomly and only observe one price.
In the static game, we show that there is a unique equilibrium. Critical consumers hold heterogeneous beliefs regarding the firms' production cost (the state), and the equilibrium search intensity only depends on critical consumers' distribution of initial beliefs. As more consumers' initial beliefs about the high-cost state lie below some cut off level, the equilibrium search intensity increases. This is because prices are more dispersed when the cost is low due to competition among firms, leading to a higher expected gain from search. The equilibrium price distribution depends on the search intensity and the actual cost state. Specifically, the equilibrium prices are increasing in the actual cost, and are decreasing in search intensity, because each firm's demand becomes more elastic as more consumers search. Thus, the full adjustment of equilibrium prices requires the adjustment of search intensity, which solely depends on the critical consumers' belief-updating process.
We then extend our static game analysis to a dynamic setting where the cost evolves according to a Markov process with positive persistence. Because consumers never observe the cost realizations, each consumer updates her belief based on the history of prices she observed. Thus consumers have heterogeneous beliefs. In equilibrium, searchers and nonsearchers have different belief-updating processes. Searchers always correctly learn the true state, because they always observe the lowest price that fully reveals the true state. But nonsearchers do not always learn the true state.
Asymmetric price adjustment thus arises naturally. In the event of positive cost shocks, all the searchers among the critical consumers immediately learn the true state and stop searching. In the following period, no critical consumers search and the search intensity is the lowest possible. Thus, the search intensity and hence the prices fully adjust within two periods. In the event of negative cost shocks, it takes longer for critical consumers who do not search originally to learn the true state and start searching, thus the search intensity increases gradually, leading to slow falling of prices. To sum up, asymmetric price adjustment is caused by learning asymmetry between searchers and nonsearchers, which is closely related to the evolution of search intensity.
More formally, we show that given the evolution of the underlying cost states, there is a unique equilibrium in the dynamic game, with the evolution of the distribution of beliefs, the search intensity, and the prices uniquely determined. We demonstrate that as long as the cost shocks are persistent, the pattern of asymmetric price adjustments emerges in statistical sense on the equilibrium path of the dynamic game. Moreover, as the cost shocks become more persistent, the pattern of asymmetric price adjustments becomes more prominent, because the downward price adjustment on average spreads over longer periods of time.
Several recent papers (Lewis, 2005; Tappata, 2006; Cabral and Fishman, 2006) also attempt to explain asymmetric price adjustment based on search models, (2) Lewis (2005) develops a reference price search model in which the expected distribution of prices is exogenously given, rather than endogenously determined; consumers have adaptive expectations and thus are not rational. On the other hand, consumers are rational in our model, because they form expectations of prices based on all the information available.
Cabral and Fishman (2006) develop a search model in which the cost changes are positively but not perfectly correlated across firms. (3) They show that consumers have a greater incentive to search in the case of large price increases or small price decreases, but little incentive to search when prices increase a little or decrease by a lot. This implies that firms are reluctant to change prices when costs decrease by a little bit or increase by a lot, but quick to change prices as costs increase by a little bit or decrease by a lot. In other words, when the cost change is small, the price adjustment exhibits downward rigidity and upward flexibility; when the cost change is big, the asymmetry is reversed: prices exhibit downward flexibility and upward rigidity. These implications are quite different from ours.
The paper that is most closely related to ours is Tappata (2006). Our article differs from Tappata in an important aspect in terms of modelling. That is, although Tappata assumes that the firms' past costs are known to the consumers, we do not impose this assumption in our analysis. Because consumers know past costs, there is no learning in Tappata. In contrast, the learning asymmetry between searchers and nonsearchers about the underlying cost is the driving force in our analysis. This modelling difference leads to very different empirical implications. In Tappata's setting, because there is no learning, it takes exactly two periods for prices to fully adjust to both the positive and negative cost shocks. Moreover, in Tappata, the asymmetry in price adjustments is only present in the first period after a cost shock occurs: the magnitude of price adjustment in the first period is bigger in the case of positive cost shocks than in the case of negative cost shocks. In contrast, by endogenizing the time periods that are needed for prices to fully adjust to cost shocks, we are able to show that asymmetric price adjustment goes beyond the first period after a cost shock occurs: although it takes two periods for prices to fully adjust to positive cost shocks, it takes much longer periods for prices to fully adjust in response to negative cost shocks.
Our article also contributes to the literature on consumer search, in that we develop a dynamic search model with consumers, in a heterogeneous fashion, learning about the underlying states based on the personal histories of prices they observed. (4) Several previous papers have studied equilibrium search with learning (Benabou and Gertner, 1993; Dana, 1994; Fishman, 1996). Benabou and Gertner (1993) study how the correlations among firms' cost shocks affect consumers' incentive to search and the equilibrium prices. In a static model, Dana (1994) shows that if consumers are uncertain about firms' costs, then the response of prices to cost shocks will be limited. In a dynamic framework, Fishman (1996) shows that cost shocks have different short-run and long-run effects on prices. (5) None of these papers study asymmetric price adjustments. The article is organized as follows. Section 2 presents the static game and characterizes the unique static game equilibrium. In Section 3, we extend the static game analysis to the dynamic setting and show that asymmetric price adjustment arises naturally on the equilibrium path. In Section 4, we discuss the restrictions of our key assumptions and the robustness of our results. Section 5 concludes.
2. Static game
* The model. We consider a market with a continuum of firms producing a homogeneous good. The total measure of firms is normalized to be 1. All the firms have the same cost c in producing each unit of the good (firms have common cost shocks). Ex ante, c (i.e., the state of the world) can take value either [C.sub.H] or [C.sub.L], where [C.sub.L] < [C.sub.H.] At the beginning of the period, firms observe the realization of the cost and then compete in prices. We also assume that each firm has a capacity constraint k (finite), that is, no firm can sell more than k units of the good. (6)
There is a continuum of consumers with total measure [beta] > 1. The parameter [beta] can also be interpreted as the number of consumers per firm in the market. Each consumer has a unit demand with a choke price of v > [c.sub.H.] We assume that [beta] < k, that is, the number of consumers per firm in the market is less than each firm's capacity constraint. (7) Consumers do not observe the realization of c. Instead, consumers...
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