Price Adjustment in a Model with Multiple-Price Policies y

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1 Price Adjustment in a Model with Multiple-Price Policies y Luminita Stevens z Columbia University January 2012 Abstract Understanding the patterns of prices in the micro data is a key step towards settling the debates on the importance of nominal rigidities and the role of monetary policy. This paper presents the dynamic price-setting problem of a rm that cannot observe market conditions for free. If it pays a xed cost, the rm can obtain complete information on the state of the world and review its policy. In each period between policy reviews, the rm decides whether or not to pay the xed cost and review its policy and which price to charge from the current policy. The rm can purchase additional information in order to make each of these decisions, subject to a cost per unit of information. The paper shows that the rm optimally chooses to only infrequently undertake policy reviews, and that between reviews it implements a simple pricing policy that consists of a small set of prices. The model therefore matches the empirical evidence of discrete multiple-price regimes documented in Stevens (2011). Although prices change frequently, they are only partially related to market conditions at all times. Hence there is scope for signicant monetary non-neutrality. This project has beneted from the continued guidance and support of Mike Woodford. I am also especially greatful to Ricardo Reis for valuable suggestions. I would like to thank Stefania Albanesi, Ryan Chahrour, Christian Hellwig, Filip Matejka, Emi Nakamura, Jaromir Nosal, Ernesto Pasten, Bruce Preston, Dmitryi Sergeyev, Jon Steinsson, Heriberto Tapia, and seminar participants at Columbia University, The Center for European Economic Research, Mannheim, and Toulouse School of Economics for helpful comments. Part of this work was conducted while I was visiting the University of Mannheim. I thank Klaus Adam for his hospitality. y PRELIMINARY. New versions will be available at z Contact: lld2108@columbia.edu. 1

2 1 Introduction How do rms set prices in response to constantly changing market conditions? In particular, how quickly and accurately do they react to macroeconomic shocks? These questions have been central to monetary economics ever since nominal price rigidities were introduced as a key link in the monetary transmission mechanism. Dierent answers imply ascribing dierent roles to monetary policy, and yield a dierent account of what drives business cycles. Not surprisingly, there is a large literature on both the evidence and the theory of how prices are set. Starting with the seminal paper of Bils and Klenow (2004), recent empirical work 1 has focused on characterizing pricing patterns at the product level as a way to discipline theories of price-setting. This literature has documented broad-based high volatility of good-level prices, even under stable macroeconomic conditions. The evidence of high volatility at the micro level poses a challenge to popular monetary models, in which prices that remain unchanged for long periods of time are the key to obtaining signicant real eects of monetary policy. Specically, while monetary models often assume stickiness of one year or longer, Bils and Klenow (2004) show that in the US CPI, the prices of many products change at least every four months. Nakamura and Steinsson (2008) further document that much of this volatility is driven by short-lived price changes to and from a rigid price level. In response, recent theoretical work has sought to develop models of price-setting that distinguish between regular and transitory prices. 2 This paper proposes a dierent interpretation of product-level volatility, according to which rms choose pricing policies in which both regular and transitory prices are chosen to be jointly optimal. My approach is motivated by the stylized facts documented in Stevens (2011). That paper analyzes pricing patterns in Dominick's grocery store data, and nds that price series are characterized by infrequent breaks that identify \pricing regimes." Moreover, these regimes are dened by a small set of prices relative to both the duration of the regimes and the frequency of price changes within regimes. Approximately three quarters of products contain regimes in which a small number of prices are revisited over the life of the regime. The left panel of Figure 1 illustrates this pattern for the weekly price of frozen juice over a four-year period: prices change 1 The chapter by Klenow and Malin in the Handbook of Monetary Economics (2010) provides a comprehensive review of the recent empirical literature on price setting at the micro level. 2 See, for example, Kehoe and Midrigan (2010) and Guimaraes and Sheedy (2011). 2

3 Figure 1. Sample price series from Dominick's. very frequently and by large amounts, yet they alternate among a few distinct values over the life of each regime. Approximately one quarter of products consist entirely of regimes in which prices either do not change at all, or change very rarely (for example, around holidays), as in the right panel of Figure 1. While the pattern of single sticky prices can be accommodated by existing sticky price theories, the pattern of regimes with multiple rigid prices is inconsistent with existing theories of price-setting. This paper explains both of these patterns using a theory of dynamic price setting in which prices are determined by pricing policies that are sticky and simple, namely they are updated infrequently and consist of a small set of prices. Both the stickiness and the coarseness of the pricing policy are a result of costly information. I consider the problem of a monopolistically competitive rm that sets prices subject to uncertainty in its demand and its production technology. Obtaining any information about the state of the world is costly in two ways. First, both the rm's prices and its acquisition of information are determined by a policy that can be reviewed subject to a xed cost. As in Reis (2006), payment of this cost enables the rm to collect complete information about the state of the world at the time of the review. The xed cost represents the managerial resources required in order to acquire and process the information, design the policy, and communicate it within the rm, as documented by Zbaracki, Ritson, Levy, Dutta, and Bergen (2004). Due to the xed cost, the rm does not update its policy in every period. Second, in every period between policy reviews, the rm acquires additional information, based on which it makes two decisions: whether or not to review its policy and, if the policy consists of a menu of prices, which price to charge. The additional information acquired between policy reviews is subject to a cost per 3

4 unit of information, which captures the cost of monitoring market conditions. The measurement of the amount of information acquired for each decision follows the rational inattention literature (Sims, 2003). The rm chooses what aspects of the state of the world to monitor and how much information to acquire for each decision. The signals that the rm chooses to acquire compress the state into a simpler representation, given the rm's objective, the xed and variable costs of information, and the market conditions that the rm expects to encounter under the current policy, until the next review. For each decision, the rm has access to no other information except that received through the corresponding signal: the review signal and the price signal act as the only interface between the rm and its environment at the time of each decision. I rst show that the rm's optimal policy consists of three elements: (1) a hazard function that species the probability of conducting a policy review conditional on the current state, (2) a set of prices, and (3) a conditional distribution that species which price to charge conditional on the current state. The optimal policy has the same form for all periods until the next review. Moreover, the optimal set of prices chosen at each review diers across reviews. Hence, every policy review starts a new regime, and every regime is identied by a new distribution of prices. I then characterize the properties of the solution. Prices vary stochastically with the current state, and policy reviews are stochastically state-dependent, as in a generalized Ss model (in the terminology of Caballero and Engel, 2007), and independent of the time elapsed since the last review. The random relationship between each of the two decisions and the current state is a result of the rm's need to economize on information. Obtaining more precise signals requires purchasing a larger quantity of information in each period. Hence, the rm faces a trade-o between economizing on information expenditure and pricing accuracy. The degree to which prices respond to concurrent market conditions depends on this trade-o. Moreover, the hazard function for policy reviews and the conditional distribution of prices depend on each other. If the cost of conducting a policy review is small, such that the rm can undertake frequent reviews, then it need not design a complex pricing policy to be implemented between reviews. If the cost of acquiring information for its pricing decision is small, then the rm can implement a pricing policy that discriminates more nely among dierent future states, which in turn implies that a policy review can be undertaken less frequently. 4

5 I derive conditions under which the rm chooses to implement a single-price versus a multiple-price policy: if the cost of information for the rm's pricing decision is higher than a certain threshold, the rm charges a single price between reviews. Below this threshold, the rm implements a multiple-price policy. This threshold is decreasing in the volatility of the shocks aecting the rm's prots, and also depends on the shape of the prot function, in particular, on the elasticity of demand. Hence, the theory can generate heterogeneity in the types of pricing policies pursued by dierent rms as a function of deep parameters that plausibly vary across rms. For information costs below this threshold, I show conditions under which the rm's pricing policy is a discrete set of prices. This allows me to generate simple pricing policies that consist of a small set of prices. I present the optimality conditions that determine how the set of prices increases as the cost per unit of information falls: given an asymmetric objective function (as is the case treated in this paper), new prices are added one by one as the support of the price distribution spreads out; for symmetric objective functions (and subject to shocks drawn from distributions that are also symmetric), a high and a low price are added symmetrically. These conditions also yield an algorithm that can be used to solve numerically for the optimal pricing policy. This novel algorithm makes it computationally feasible and fast to nd the optimal policy. I leave for future work the analysis of the degree of non-neutrality implied by the model in a general equilibrium framework. 3 Nonetheless, it is already clear that non-neutrality in this model will necessarily be higher than that implied by a model with the same frequency of price adjustment, but in which all price changes are based on full information regarding market conditions at the time of adjustment. Relative to the existing literature, this is the rst model to generate pricing regimes that consist of a small set of prices and that are infrequently updated. Moreover, consistent with the data, prices within regimes change frequently and by large amounts. The most commonly used models of pricing cannot generate these patterns. Full information exible price models, in which prices are continuously re-optimized, do not generate regimes except to the extent that there are regimes in the underlying shocks, and do not generate mass points in the distribution of prices observed over time, except to the extent that the underlying shocks are 3 Extending the model to general equilibrium requires keeping track of the entire distribution of prices, and hence increases the computational complexity of the numerical algorithm. The derivation of the algorithm in general equilibrium is in progress. 5

6 themselves drawn from distributions with mass points. By disregarding the substantial rigidity in price levels apparent in Figure 1, and documented more broadly in Stevens (2011), these models may overstate the degree of exibility in the pricing data. Sticky price models, such as time-dependent models (Taylor, 1980 or Calvo, 1983) or state-dependent models (Sheshinski and Weiss, 1977, Golosov and Lucas, 2007), generate single-price regimes. As in the case of exible price models, there is no reason for past prices to be revisited once the rm re-optimizes its policy, hence these models cannot explain the discreteness of prices observed in the data. Moreover, sticky price models that abstract from transitory price changes within regimes may overstate the degree of rigidity in the pricing data. As others have documented, a signicant portion of rms' revenues is derived from sales at the non-modal prices, which suggests that rms should have a strong incentive to tie transitory prices to concurrent market conditions, at least partially. Klenow and Willis (2007) further document that transitory prices have macro content that does not wash out with aggregation. It is important to note that in the model proposed here there are no physical costs of price adjustment; in fact, prices can change all the time in this model. Rigidity arises because they are always drawn from a xed set of prices over the life of the regime, and are based on noisy information about market conditions. There are also no a priori constraints on the rm's ability to change "regular" versus "temporary" prices, thus distinguishing this model from those proposed by Kehoe and Midrigan (2010) and Guimaraes and Sheedy (2011). The model brings together dierent features of the growing literature on imperfect information in price setting. In particular, the introduction of both xed and variable costs of information combines two competing approaches to modeling information acquisition. However, the model departs from both literatures by generating simple pricing policies that consist of a small set of prices. First, as in the inattentiveness model of Reis (2006), the rm faces a xed cost of conducting a policy review and the strategy that is used to decide when to conduct the next review is itself part of the policy that is chosen at the time of a review. In the model of Reis (2006), the policy species the path of prices to be charged until the next review, and the date of the next review. Between reviews, the rm cannot obtain any information about market conditions, other than information regarding the passage of time, which is 6

7 available for free. In contrast, I allow the rm to acquire information between reviews, but all information, including knowledge about the number of periods since the last review, is subject to the same cost per unit of information. The resulting timing of reviews and the price charged in each period are stochastically state-dependent rather than time-dependent. In my model, a perfectly precise review signal would generate the triggers in an Ss model of policy reviews, as in the model of Burstein (2006). Conversely, if the rm acquired no information through its review signal, the timing of policy reviews would be completely random, as in the model of Mankiw and Reis (2002). Second, as in the rational inattention literature, the acquisition of information between reviews is subject to a cost per unit of information, using entropy as a measure of information. Allowing the rm to occasionally review its policy, subject to a cost, enables me to generate regime changes, distinguishing this setup from other rational inattention papers, such as those of Sims (2003, 2006), Mackowiak and Wiederholt (2009), or Matejka (2011). In those models, the rm species the optimal policy once and for all at some initial date, and then receives signals in accordance with that policy. In contrast, I model both the decision to change the price and the decision to change the overall policy, and hence to move to a new regime. The fact that the rm can occasionally review its policy means that it can implement simple policies between reviews. Moreover, other rational inattention models assume that the cost per unit of information only applies to current market conditions, whereas the full history of past signals and knowledge about the number of periods since the policy was rst chosen are both available for free. In contrast, I assume that all information, including memory and knowledge of the number of periods elapsed since the last review, is subject to the same cost per unit of information. This assumption identies the information friction directly with the limited attention of the decision-maker processing the information from a particular signal. This assumption is critical in generating regimes that are identied by a single distribution of prices: without it, the rm would charge prices from a dierent pricing policy in every period; moreover, the optimal policy would not generate a discrete distribution of prices, except in the special case of i.i.d. variations in market conditions, as assumed in Matejka (2011). This treatment of time and memory is the same as in Woodford (2009), who also models policy reviews that are subject to a xed cost and whose timing is determined by a stochastically state-dependent hazard 7

8 function. The present model diers from Woodford (2009) along two dimensions. Firstly, I relax that model's assumption that between policy reviews the rm charges a single price. Introducing the price signal generates price volatility between policy reviews, consistent with the empirical evidence of multiple-price regimes documented in Stevens (2011). Secondly, I allow the rm to redesign its signals at each review, whereas in Woodford (2009), the rm's information acquisition policy is chosen once and for all at some initial date. Although in my innite horizon model, the optimal prices of individual rms eventually drift innitely far apart, the occasional policy reviews enable rms to choose from a small set of prices in each regime (under certain assumptions). Hence within regimes, the model maintains the pattern of discrete prices obtained in a static context by of Matejka (2011). That paper, instead, obtains a discrete set of prices only under the additional assumption that the range over which the rm's prices can vary remains forever bounded. However, such an assumption is inconsistent with evidence that both individual and relative prices exhibit unit roots. Section 2 presents the setup and introduces the information costs, starting from the full-information frictionless benchmark. Section 3 presents the acquisition of information between reviews and denes the rm's problem. Section 4 derives and discusses the optimal policy. Section 5 maps a standard monopolistic competition model with Dixit-Stiglitz preferences into the problem introduced in section 2. Section 6 presents numerical results. Section 7 concludes. 2 The Setup A monopolistic rm producing a non-durable good must choose the price to charge for its output in every period, subject to a demand function and a production technology that vary stochastically. The rm's per-period prot in units of marginal utility, (p x), is a function of the rm's actual log-price, p, and its target log-price, x. The prot function is a smooth real-valued function with a unique global maximum at p = x. All the information about rm-specic and aggregate market conditions that the rm needs in order to choose its optimal price is summarized in the target price, x t. This target is a linear combination of the 8

9 exogenous disturbances in the economy, both transitory and permanent. It evolves over time according to: x t = ex t + t (1) ex t = ex t 1 + e t (2) where the permanent and transitory innovations, e t and t, are drawn independently from known distributions g e and g, with continuous, but bounded support. After both e t and t have been realized, the period t price is set and orders are fullled. Section 5 maps a standard monopolistic competition model with Dixit-Stiglitz preferences into this specication. 2.1 The Setup Under Full Information In the frictionless benchmark, the rm chooses a pricing policy that species what price to charge in each period and state of the world, to maximize its discounted prot stream, X 1 E 0 t=0 t (p t x t ) (3) where 2 (0; 1) is the discount factor. In the absence of information costs, the rm perfectly observes the realization of x t in each period. If there are no other frictions, such as physical costs of price adjustment, the rm's optimal policy is to charge p t = x t, 8t (4) 2.2 The Setup Under Costly Information This paper departs from the frictionless benchmark by assuming that although complete information about the state of the economy is available in principle, the rm must expend resources to receive any information. The rm chooses how much information about market conditions to acquire. Acquiring a larger quantity of information in turn translates into more precise information about market conditions. The measurement of the quantity of information is based on the literature on rational inattention (Sims, 2003). 9

10 In an important departure from most models of rational inattention, I assume that both the rm's prices and its acquisition of information are determined by a policy that can be occasionally reviewed. Reviewing this policy entails payment of a xed cost, as further discussed below. The rm's ability to review its policy implies that in every period it must make two decisions: (1) whether or not to undertake a policy review, and (2) what price to charge from the current policy. Each of these decisions is based on information acquired in the form of signals. The information acquired in order to make the review decision is subject to a unit cost r. Letting I r t denote the quantity of information acquired for this decision in period t, the information expenditure associated with the review decision in period t is r It r. Similarly, the information acquired in order to decide which price to charge is subject to a cost per unit of information, p, and the expenditure associated with the pricing decision in period t is p I p t. Modeling the acquisition of information using a unit cost of information is equivalent to assuming that the decision maker processing the information acquired for each decision has a capacity limit on the quantity of information that he or she can process. The xed cost of conducting a policy review, denoted by, is also a type of information cost. It represents the managerial resources associated with acquisition of the information necessary to design a new policy and with the decision-making and communication of the new policy. As documented by Zbaracki et al (2004), rms spend a signicant amount of resources acquiring information and deciding what type of policy to implement. Payment of this cost allows the rm to acquire extensive information about the state of the world, on the basis of which it designs its new policy. For simplicity, I assume that it enables the rm to receive complete information about the state of the world at the time of the review. The assumption that the cost is xed can be rationalized via economies of scale in the review technology. This assumption follows Reis (2006). 4 It is important to note that except for the information received at the time of the policy review, and reected in the policy adopted at that time, the rm has no additional information that can guide it in making its two decisions for free. 4 The assumption of a xed cost of policy reviews is also similar to that of Burstein (2006), except for the fact that in that model, the rm has full information at all times for free, and the xed cost represents only the resources required to design and communicate the new policy. 10

11 The rm's objective under costly information is to maximize X 1 E 0 t=0 t [(p t x t ) r t r I r t p I p t ] (5) where is the xed cost of a policy review, r t is an indicator function that takes the value 1 if the rm reviews its policy in period t and 0 otherwise, r is the unit cost of the information acquired in order to make the review decision, I r t is the quantity of information that informs the review decision in period t, p is the unit cost of the information acquired in order to make the pricing decision, and I p t is the quantity of information that informs the pricing decision in period t. The two unit costs, r and p, are not necessarily equal. For instance, it may be the case that two individuals with dierent costs of acquiring information make the two decisions within the rm. However, for each decision, the same unit cost applies to all types of information that may be relevant to that decision. The types of information potentially relevant to each decision include information about the current state, the history of signals previously received, and the number of periods that have elapsed since the last review. The equal-cost assumption identies the information friction with the limited attention of the decision-maker processing the information from a particular signal. Moreover, this information is assumed to require a small fraction of the decision-maker's overall capacity to process information. Hence, regardless of the degrees of complexity of dierent types of information, the eort per unit of information required of the decision-maker receiving a particular signal is taken as xed. As a result, the rm's problem involves a single signalling mechanism for each of the two decisions. This equal-cost assumption follows Woodford (2009), and its consequences are discussed in more detail in section The Sequence of Events The sequence of events that occur in each period t is as follows: 5 An alternative approach, implemented by Sims (2003), and other rational inattention papers, is to assume that the entire history of past signals is available to the decision-maker for free. One consequence of that assumption is that it makes dynamic problems stationary. However, that assumption is no longer necessary once I allow the rm to occasionally review its policy, as in the current paper. Yet another approach, left for future work, would be to assume that some types of information are easier to process than others. This approach would increase the complexity of the model, since it would require modeling a dierent information acquisition strategy for each type of information. 11

12 1. The value of e t is realized. 2. The rm receives the review signal, based on which it decides whether or not to undertake a review, in accordance with its current policy: (a) if it decides to undertake a review, it acquires complete information about the current state of the world, and chooses a new policy that consists of a strategy for its review decision, to be implemented starting in period t + 1, and a strategy for its pricing decision, to be used starting in period t; (b) otherwise, the existing policy is maintained. 3. The value of t is realized. 4. The rm receives the price signal, based on which it decides what price to charge in the current period, in accordance with its current policy. 5. Period-t demand is met and prots are realized. The assumption that in each period the rm makes its review decision before that period's transitory shocks are realized, and hence that this decision cannot depend on these shocks, is a simplication that reduces the state space relevant for this decision, while only having small quantitative implications. If, instead, both permanent and transitory shocks were realized at the beginning of the period, the review decision would depend on both types of shocks. However, the extent to which the transitory shock would impact the review decision would be small: only particularly large transitory shocks would justify triggering a review, despite the transient character of the shock. The timing assumption abstracts from this complication by eliminating the possibility of such an eect. 3 The Firm's Problem This section reformulates the rm's objective (5) in terms of the choices that the rm makes each time it undertakes a review, and denes the rm's complete optimization problem. 12

13 First, I dene the signals that inform the rm's two decisions, and the quantities of information required by each signal, starting from general denitions for each signalling mechanism. A crucial determinant of the optimal signals is the way in which we measure the quantity of information required by each signal, I r t and I p t. Following the rational inattention literature, I use a measure derived from information theory (Shannon, 1948), which quanties the reduction in the agent's uncertainty about a random variable. I then simplify the denition of each signalling mechanism, using some preliminary results that exploit the information theoretic framework. In particular, the optimal signalling mechanism for each of the two decisions is shown to generate signals that specify the action that the rm should take. In the case of the review decision, the review signal directly indicates whether or not the rm should undertake a review in the current period, and in the case of the pricing decision, the price signal directly tells the rm what price to charge in the current period. These results ensure that the rm implements the most ecient signal structure. They also allow me to redene the rm's objective (5) in terms of a more tractable set of rm choices. Finally, I employ a normalization that makes the rm's problem stationary. 3.1 The Review Signal Let e! t denote the state of the world at the time of the receipt of the review signal in period t. It includes the realization of the permanent shock in the current period, e t, as well as the full history of shocks and the full history of signals and the decisions made by the rm in response to these signals, through the end of period t 1. Suppose that the rm conducts a policy review in an arbitrary state e! t in period t. As noted in the previous section, the review policy is implemented starting in period t + 1, since there can only be one review per period. Denition 1 The review signal, implemented following a policy review in an arbitrary state e! t in period t, is dened by a signalling mechanism that species 1. R t, the set of possible review signals; 2. t+ (rje! t+ ), the sequence of conditional probabilities of receiving the review signal r for all > 0, all r 2 R t, and all e! t+ that follow the policy review in state e! t, period t, until the next review; 13

14 3. t (r), the overall frequency with which each review signal is received, until the next review, for all r 2 R t ; 4. t (r) : R t! [0; 1], the decision rule for conducting a policy review, which species the probability of conducting a policy review when the review signal r is received, for all r 2 R t. At the time of each policy review, part of the rm's problem is to choose these four objects, which make up the rm's optimal review policy. Before the receipt of the review signal in each period, the rm has no additional information, except for that information received at the time of the policy review, and reected in the current policy. Hence, the overall frequency of the review signals, t (r), and the decision rule for each signal, t (r), can depend only on e! t, the state at the time of the policy review in period t. Therefore, these two objects are only indexed by t, the period in which the last review took place. In particular, because awareness of the passage of time is not treated as free information, the signals received in all periods in which the policy may apply must be treated as part of a single information structure. Conversely, if the rm had independent knowledge of before receiving the review signal in each period, either for free, as in Reis (2006), or through a separate signalling mechanism on the passage of time, the rm would design separate signalling mechanisms for each period, with period-specic anticipated frequencies of review signals, t+ (r), and period-specic decision rules, t+ (r). The information transmitted by this signalling mechanism is the average amount by which receipt of the signal reduces the decision-maker's uncertainty about the state. As proposed by Shannon, uncertainty of a random variable is measured by entropy. For convenience, both in the denition of the review signal and in the later denition of the price signal, I employ an equivalent denition, namely the average amount by which uncertainty about the optimal signal would be reduced by observing the state. 6 This leads to the following denition: Denition 2 For any state e! in which a review signal is received, the amount of information that is required 6 These two denitions are equivalent since the information about a state variable contained in the signal is equal to the information about the signal contained in the state variable. Exploiting this symmetry simplies the exposition. 14

15 in order to implement a signalling mechanism dened by the conditional probabilities (rje!) is dened as I r (; ) X r2r (rje!) [log (rje!) log (r)] (6) This quantity is the relative entropy between the conditional distribution (je!) and the default distribution (r). More precise information about e! implies a bigger dierence between the distribution of the signal conditional on e!, (rje!), and the frequency with which the rm anticipates receiving the signal r prior to the realization of the state, (r). For expository purposes, R is a countable set, though the denition can be extended to allow for continuous signal distributions. It will be established below that the optimal set of review signals is not only countable, but nite. The quantity of information expected, at the time of the review in an arbitrary state e! t and period t, to be required by the implementation of the signalling mechanism dened in Denition 1, in each period t +, > 0, over all states e! t+ that follow e! t with positive probability, is given by I r t+ E t I r t+ (rje! t+ ) ; t (r) (7) where E t [] is always dened as expectations conditional on e! t and on a policy review having taken place in state e! t, period t. I simplify the problem using a preliminary result from Woodford (2008), who proves that the optimal review signal is binary and directly species whether or not a review should be undertaken. Hence, without loss of generality, the review signal received in any period and state of the world is drawn from the set f0; 1g, and the decision rule is simply t (r) = r, such that the rm conducts a policy review upon receiving the signal r = 1, and continues with its existing policy upon receiving the signal r = 0. The quantity of information conveyed by this signalling mechanism in a particular state, relative to the probability of a policy review anticipated before receiving the review signal, is expressed as I r ; log log + (1 ) log (1 ) log 1 (8) 15

16 This quantity is the relative entropy between two binary random variables for which the probabilities of observing the signal r = 1 are and respectively. As a result, I can express (7) in terms of a sequence of hazard functions, in the terminology of Caballero and Engel (2007), f t+ (e! t+ )g, and an anticipated frequency of policy reviews, t : I r t+ = E t I r t+ (e! t+ ) ; t (9) The signalling mechanism for the review decision is accordingly redened in the following lemma: Lemma 1 The optimal review signal, implemented following a policy review in an arbitrary state e! t in period t, is always drawn from the set f0; 1g, and is described by a signalling mechanism that species 1. f t+ (e! t+ )g, the sequence of conditional probabilities of receiving the review signal r t+ = 1 (conducting a policy review) for all > 0 and all e! t+ that follow the policy review in state e! t, period t, until the next review; 2. t, the anticipated frequency with which the review signal r t+ = 1 is received, over all states and periods, until the next review. The quantity of information I r t+ required by this signalling mechanism in each period t +, > 0, is given by equation (9). This result is not only intuitive, but it also formally denes the cheapest signalling mechanism that the rm can employ in order to make its review decision. Any other signal structure would require a quantity of information weakly greater than (9). Reformulating the signalling mechanism in this way also leads to a simplication in solving for the rm's review decision: rather than choosing the four objects dened in Denition 1, the rm chooses the sequence of hazard functions, f t+ (e! t+ )g, and the anticipated frequency of policy reviews, t. 16

17 3.2 The Price Signal The price signal in each period t is received after the review signal, and after the realization of the transitory shock, t. As above, suppose that the rm conducts a policy review in an arbitrary state e! t in period t. The pricing policy applies starting in period t. For any 0, let! t+ = fe! t+ ; r t+ ; t+ g denote the state of the world at the time of the receipt of the price signal in period t +. Denition 3 The price signal, implemented following a policy review in an arbitrary state e! t in period t, is dened by a signalling mechanism that species 1. S t, the set of possible price signals; 2. ff t+ (sj! t+ )g, the sequence of conditional probabilities of receiving the price signal s for all 0, all s 2 S t, and all! t+ that follow the policy review in state e! t, period t, until the next review; 3. f t (s), the overall frequency with which each price signal is received, until the next review, for all s 2 S t ; 4. t (pjs) : S t R +! [0; 1], the decision rule for price-setting, which species the probability of charging price p 2 R + when the price signal s is received, for all s 2 S t. As in the case of the review signal, discussed above, both f t (s) and the decision rule for price-setting can depend only on the state at the time of the policy review in period t. The quantity of information required to implement a particular signalling mechanism for the rm's pricing decision is dened below: Denition 4 For any state! in which a price signal is received, the amount of information that is required in order to implement a signalling mechanism dened by the conditional probability f (sj!) is dened as I p f; f X s2s f (sj!) log f (sj!) log f (s) (10) For expository purposes, S is a countable set, although the denition can be extended to allow for continuous signal distributions. 17

18 The quantity of information expected at the time of the review in period t to be required by the implementation of the price signal in each period t +, 0, is then given by I p t+ E t I p f t+ (sj! t+ ) ; f t (s) (11) As in the case of the review signal, the rm's choices for the optimal pricing policy are simplied by showing that the optimal price signal directly species the price that the rm should charge in each period. In order to obtain this result, I begin by noting that the optimal decision rule assigns a deterministic price to each signal s, and hence can be represented by a function p t (s) : S t! R +. Otherwise, the rm could economize on its information expenditure in (10) by acquiring a less informative (more random) signal, and arranging for the action, namely the price to be charged, to be a deterministic function of the signal. 7 Second, the rm always uses the price signal to decide which price to charge, such that the function p t (s) is onto, p t (s) : S t! P t, where P t is the subset of R + of prices that are charged with positive probability. Third, the optimal signal does not dierentiate between states in which the same action is taken, since doing so would increase the quantity of information that the rm must acquire, without improving the rm's decision. Hence the function p t (s) is one-to-one. Finally, receiving a signal directly on the price p requires the same quantity of information as does receiving a signal s that is subsequently transformed into a price using the bijection p t (s). This result ensures that the rm implements the cheapest signal structure in order to make its pricing decision. The rm's pricing policy can be more succinctly described by the set of prices P t, the sequence of conditional probabilities ff t+ (pj! t+ )g, and the anticipated frequency with which each price is charged, f t (p), for all p 2 P t, all 0, and all states! t+ that follow e! t. 8 Equations (10) and (11) are replaced by: I p f; f = X p2p f (pj!) log f (pj!) log f (p) (12) I p t+ = E t I p f t+ (pj! t+ ) ; f t (p) (13) 7 The state,!, the signal, s, and the price p, form a Markov chain, in that order, so that prices are distributed independently of states conditional on signals. As a result, the mutual information between prices and states is less than or equal to the mutual information between signals and states. If prices are a random function of signals, then the inequality is strict. 8 With a slight abuse of notation, I continue to use \f," which from now on denotes the distribution of the prices. 18

19 The following lemma summarizes the results of this sub-section: Lemma 2 The optimal price signal, implemented following a policy review in an arbitrary state e! t in period t, is described by a signalling mechanism that species 1. P t, the set of prices charged with positive probability; 2. ff t+ (pj! t+ )g, the sequence of conditional probabilities of charging price p for all 0, all p 2 P t, and all! t+ that follow the policy review in state e! t, period t, until the next review; 3. f t (p), the anticipated frequency with which each price is charged over all states and periods until the next review, for all p 2 P t. The quantity of information I p t+ required by this signalling mechanism in each period t +, 0, is given by equation (13). Following a policy review in period t, the rm's choices when solving for the optimal pricing policy are therefore reduced to choosing three objects: the set of prices P t, the sequence of conditional probabilities,ff t+ (pj! t+ )g, and the overall frequency f t (p). Any other signal structure would require a quantity of information weakly greater than (13). 3.3 The Firm's Problem The choices dened in Lemma 1 and Lemma 2 provide a mathematical representation of the rm's policy. Using these choices, the continuation value of the rm's objective (5), looking forward from the time of a policy review in an arbitrary state e! t in period t, is given by E t 8 >< >: 1X = r I r P p2p t+ (p x t+ )f t+ (pj! t+ ) t++1 (e! t++1 ) t++1 (e! t++1 ) ; t++1 p I p f t+ (pj! t+ ) ; f t+ (p) 39 >= 7 5 >; (14) Here, t+, t+ (e! t+ ), P t+, f t+ (p), and f t+ (pj! t+ ) are the policy choices that are in eect in each period t+ and in each state of the world, regardless of whether they were adopted at the time of the review 19

20 in period t or in some subsequent policy review. Hence, in this equation, I make no explicit reference to the period and state in which the policy that applies in each t + was chosen. The rm's continuation value can be written more compactly by collecting all of the terms in the objective that depend on the pricing policy in eect in a particular period. Using (12), I dene t+ (! t+ ) as the rm's per-period prot in state! t+, period t +, expected under the pricing policy in eect in that state, net of the cost of the price signal only: t+ (! t+ ) X f t+ (pj! t+ ) (p xt+ ) p log f t+ (pj! t+ ) log f t+ (p) (15) p2p t+ As in (14), here I continue to make no explicit reference to the period and state in which the pricing policy that species P t+, f t+ (pj! t+ ) and f t+ (p) was chosen. The continuation value dened in (14) can then be written as 1X E t ( t (! t ) + t+ (! t+ ) t+ (e! t+ ) r I r ) t+ (e! t+ ) ; t+ (16) =1 As above, this equation makes no explicit reference to the period and state in which the policy that applies in each t + was chosen. In order to formulate the rm's objective in terms of its choices at the time of a particular policy review, it is convenient to dene a survival probability, t+ (e! t+ 1 ), which denotes the probability, expected at the time of the review, that the review policy chosen in state e! t, period t, continues to apply periods later, as a function of the history of states. Since there can only be one review per period, t+1 (e! t ) 1 for all e! t ; and for > 1, t+ (e! t+ 1 ) Y 1 k=1 [1 t+k (e! t+k )] (17) Let V t (e! t ) denote the maximum attainable value of the rm's continuation value (16). Under the assumption that an optimal policy will be chosen in all future policy reviews, (16), written in terms of the 20

21 rm's choices at the time of the review in period t, becomes: E t 8 >< t (! t ) + >: 1X t+ (e! t+ 1 ) 6 4 =1 2 (1 t+ (e! t+ )) t+ (! t+ ) + t+ (e! t+ ) V t+ (e! t+ ) r I r t+ (e! t+ ) ; t 39 >= 7 5 >; (18) This objective states that conditional on reaching a particular period t + and state e! t+ under the current policy (which is given by the survival probability), the rm pays the cost of the review signal, r I r t+ (e! t+ ) ; t. It then continues to apply the current policy with probability 1 t+ (e! t+ ), in which case it expects a per-period prot equal to t+ (e! t+ ). On the other hand, if it undertakes a policy review and pays the review cost, which occurs with probability t+ (e! t+ ), it expects the maximum attainable value from that state onward, V t+ (e! t+ ). I can now formally state the rm's problem at the time of a particular policy review: Problem 1 If the rm undertakes a policy review in an arbitrary state e! t and period t, it chooses: 1. a review policy that species t and f t+ (e! t+ )g for all > 0 and all e! t+ that follow the policy review in state e! t, period t, until the next review; 2. a pricing policy that species P t, f t (p), and ff t+ (pj! t+ )g for all p 2 P t, all 0, and all! t+ that follow the policy review in state e! t, period t, until the next review. The two policies are chosen to maximize (18) subject to the denitions (17), (15), and (8). 3.4 The Stationary Formulation Employing a normalization allows me to rewrite Problem 1 in a stationary form. The normalization involves introducing a set Q of normalized prices, and a normalization of the state variables e! t+ and! t+. Using this simplication, I derive the optimal policy independent of the time and state in which a policy review is conducted. The problem is reformulated in terms of the following normalized variables: q, ey, y, 21

22 ew, and w. First, I dene q p ex t (19) ey ex t+ ex t (20) y ey + (21) As a result, the rm's prot function, (p x t+ ), is replaced by the normalized function (q y ). Second, for any state e! t+, ew is the part of e! t+ that represents news since the state of the world at the time of the last review, e! t, for all > 0; similarly, w represents the part of! t+ that is news since e! t, for all 0. Given the laws of motion (1)-(2), all of these variables are distributed independently of the state e! t at the time of the policy review in period t. The optimal pricing policy chosen in any review can be written as the choice of the set of normalized prices q 2 Q, anticipated to occur with frequencies f (q), and the sequence of conditional distributions ff (qjw )g. The expected value, t+ (! t+ ), dened in (15), can be replaced by its normalized counterpart (w ), (w ) X q2q f (qjw ) (q y ) p log f (qjw ) log f (q) (22) Finally, the rm's optimal review policy can also be written in normalized terms as the choice of a sequence of hazard functions f (ew )g and an anticipated frequency of reviews. The survival probability (17) becomes (ew 1 ) Y 1 k=1 [1 k (ew k )] ; 8 > 1 (23) with 1 (ew 0 ) 1, since there is only one policy review per period. Therefore, the rm's choices can be written without any reference to either the date t or the state e! t in which a policy review takes place. In normalized terms, the rm's continuation value conditional on a policy review in an arbitrary state and period is ( 1X E 0 (w 0 ) + (ew 1 ) (1 (ew )) (w ) + (ew ) V r I r (ew ) ; ) (24) =1 22

23 where V is the maximum attainable value of the rm's continuation value at the time of a policy review in any state and period. The expectations operator E [] integrates over all future news states following the policy review in any arbitrary state and period. Below I state the stationary formulation of the rm's problem: Problem 2 If the rm undertakes a policy review in any arbitrary state and period, it chooses: 1. a review policy that species and f (ew )g for all > 0 and all news states ew until the next review, and 2. a pricing policy that species Q, f (q), and ff (qjw )g for all normalized prices q 2 Q, all 0, and all news states w until the next review. The two policies are chosen to maximize (24) subject to the denitions in (23), (22), and (8). Moreover, V in (24) is the maximized value of the quantity dened by (24). 4 The Optimal Policy I obtain the solution to Problem 2 in steps, deriving each element of the optimal policy taking the other elements as given. I rst solve for the sequence of conditional price distributions, ff (qjw )g. Using this result, I then nd the optimal sequence of hazard functions, f (ew )g. The set of optimality conditions simplify the rm's policy further: the optimal policy will be shown to involve a single conditional price distribution, f (qj) for all q 2 Q, and a single hazard function, (), which are dened for all periods and all states until the next policy review. I then solve for the anticipated frequency of each normalized price, f (q), for each q 2 Q, the optimal set of normalized prices, Q, and the anticipated frequency of policy reviews,. Although I allow the rm to condition its policy on the complete state, including the number of periods elapsed since the last review and the history of past signals, the optimal signalling mechanisms for both the review decision and the pricing decision will be shown to allocate the entire information capacity to learning about the change in market conditions since the last review directly, rather than paying any attention to past events or to the passage of time directly. 23