advertisement

/ .§f ^f-- Y Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium Member Libraries http://www.archive.org/details/informativenessoOObena working paper department of economics THE INFORMATIVENESS OF PRICES: SEARCH WITH T.KAJINING AND INFLATION UNCERTAINTY Roland Benabou Robert Gertner No. 555 April 1990 massachusetts institute of technology 50 memorial drive Cambridge, mass. 02139 THE INFORMATIVENESS OF PRICES; SEARCH WITH LEARNING AND INFLATION UNCERTAINTY Roland Benabou Robert Gertner No. 555 April 1990 THE INFORMATIVENESS OF PRICES: SEARCH WITH LEARNING AND INFLATION UNCERTAINTY Roland Benabou M.I.T. Robert Gertner University of Chicago April 1990 This We is a much would revised like to and expanded version of "Inflation and Market Power in a Duopoly Model wiih Search." thank Olivier Blanchard, Dennis Carlton, Peter Diamond, Robert Lucas, Kevin Murphy, David Scharfstein, Jean Tirole, and Robert Vishny for beneficial discussions. THE INFORMATTV^ENESS OF PRICES: SEARCH \M1 H LEARNING AND INFLATION UNCERTAINTY' Roland Benabou M.I.T. Robert Gertner University of Chicago ABSTRACT Sources of aggregate cost uncertainty, such as real shocks or unanticipated reduce the information content of prices by making aggregate price variations. difficult to it may in fact lead them equilibrium, and prices will reflect increased competition. a model of search with learning, in whether is become To examine consumers' search inflation variabilit}' increases or thereby market efficiency, to for instance by better informed in this issue, we develop which consumers search optimally from an unknown price distribution, while firms price optimally given decisive factor in separate relative and But when agents can acquire information, searching, such increased noise inflation, rules. We show that the reduces the incentive to search, and the size of informational costs. I. Introduction Consider the problem of a consumer who She must estimate how much of station. is it suppliers, such as an oil shock or high inflation much of it is opposite case deemed more it manufacturer is relevant, may pay to do it is good behavior in turn deal, or will working when on new factors which are specific to this particular a truly to a its common factor affecting all other way through the economy; and how for this particular seller. If the first not worth looking for a better deal elsewhere; in the so. Similarly, offering large rebates due demand shock reflects a specific supply or explanation observes an unexpectedly high price at a gas this cars, brand whether the whole industry (e.g. is consumer observes that an automobile she must resolve whether this reflects excess inventories) and perhaps having a sale. make the offer Again, her search vary with the inferences she draws from observed prices. These inferences are based on her knowledge of the relative variability of idiosyncratic and aggregate shocks, inflation being one of the latter kind. This problem of search with learning from prices to us important on several counts. First, is the main focus of the paper; it seems from a microeconomic point of view, the inferences which buyers draw from prices underlie their demand functions, and should therefore play an important role in determining technological innovations, etc. implications for what is how markets react to oil shocks, weather variations, Secondly, from a macroeconomic point of view, it has often thought to be an important source of welfare losses from unanticipated inflation: a deterioration through increased relative price in variability. the information content of prices, operating ^ According to this view, stochastic inflation Several studies documenting the correlation between the variance of unanticipated and relative price variability (Vining and Elwertowski (1976), Parks (1978), Fischer (1981)) seem to lend support to this idea. Hercowitz (1982), on the other hand, finds that aggregate real shocks, not monetar)' shocks, explain relative price dispersion in the United States. As the average rate and the variabilit}' of inflation are also correlated (e.g. Taylor (1981). Pagan, Hall and Trivedi (1983)), high inflation is often seen as indirectly responsible for any informational costs of unanticipated inflation. The empirical literature on these issues is surveyed extensively in Fischer (1981) and Cukierman (1983). 1. inflation constitutes a source of aggregate noise in While decisions. this market well understood in is signals, leading to inefficient allocation models where the information structure is exogenous (Lucas (1973), Barro (1976), Cukierman (1979)), what happens when agents can decide to acquire additional information, by searching or otherwise, has not been explored. Finally, another common, but not yet formalized idea, is that sellers can "hide" behind aggregate or inflationary noise to charge higher real prices, taking advantage of consumers' reduced information to increase their markups. 2 In this paper we attempt to cost uncertainty, arising of efficiency from allocations ex'plore these issues real shocks or market a in by analyzing the effects of aggregate from unexpected general endogenous with inflation, on the and information-gathering price-setting.3 The literature traditionally on the effects of inflation on the information content of prices has focused on prices' role as signals for producers rather than consumers. Misperceptions by producers concerning aggregate and relative price movements (Lucas (1973), Barro (1976), movements relationship first in Cukierman (1979), Hercowitz (1981)) or (Cukierman (1982)) generate relative prices between the variance of inflation type of misperceptions availability of macroeconomic is in make transitory inefficiencies as well a as the price level and that of relative prices. price or monetary statistics should solve the problem. This models too literally, rather than as a the point that inflationary' noise interacts with the functioning of 2. For instance, whether or not oil companies and gasoline retailers engaged in such behavior following the shutting down of the Alaskan pipeline after the "Exxon Valdez" spill has been the subject of intense debate. 3. For a related (1988), (1989). The often dismissed as implausible on the grounds that the criticism takes the informational structure of these simplifying device to permanent and analysis of the effects of anticipated, or trend inflation, see Benabou oil the price system. A more and constructive restatement of valid, some pointing out that agents can indeed acquire information, but at example of this this criticism consists in cost; the clearest search. This in turn has two important consequences: is (i) Sellers generally have {}{) Information is market power. in is . endogenous: information acquisition and pricing strategies are determined jointly Our paper . equilibrium, and both in depend on inflation. the "misperceptions" or "learning" tradition but emphasizes the informational problems of consumers as well as producers, and embodies the above remarks, on which we now elaborate. First, informational costs realistically imply that sellers have market power. growing more sophisticated literature has in its treatment of signal-extraction, the "misperceptions" maintained the assumption of perfect competition, although with that of imperfect price information. asymmetry between informational In it modern economies, no market is more likely that both t}'pes it seems at odds Indeed, their coexistence relies on a drastic costs within a is While really cut off market (zero) and across markets from the others of costs are of the in (infinite). terms of information, and same order of magnitude. A realistic information-based model should therefore probably recognize the presence of some (possibly small) monopoly power . As a result, inflationary uncertainty these are not Walrasian to start with; the issue then will be worsened or Secondly, may alter relative prices, but becomes whether preexisting distortions alleviated. in the absence of fully adequate price statistics, buyers and sellers have an incentive to seek information, for instance by searching (within a market) or just by checking other prices (across markets). therefore embody For instance, we An equilibrium analysis of inflationary misperceptions should the effects of the inflation process on this incentive to acquire information. shall see that a deterioration in the quality of price signals due to increased inflationary noise can lead agents to acquire informed more in equilibrium. Naturally, the costs of information, making them actually better acquiring information (e.g. search costs) will play a determinant role. To take account of these points, we build a model with a stochastic structure somewhat Lucas (1973) and Barro (1976), but where information can be acquired similar to those of endogenously and the market structure is own very different. Duopolistic firms observe their production costs, then set prices. Since costs are correlated, buyers learn from one firm's price something about the price of the other. Given these inferences and their knowledge of firms' strategies, they decide whether or not to search. Conversely, when setting their prices firms take account of buyers' search rule, as well as of their cost and own inferences about their competitor's price. This requires addressing the To Bayesian learning. represent the first more general problem of search market equilibrium our knowledge this paper, and independent models which combine optimal adaptive search Not Rosenfield and Shapiro (1981)) with equilibrium pricing. quite complex, and this is why we have general equilibrium framework. microeconomic identify to focus attention The in inflation industry-wide cost shock. inter-industry costs, work by Dana (1990) (e.g. Rothschild (1973), surprisingly, the a single market, as question then arises of setting, the idea that inflation an increase on with how to problem opposed capture, is to a in reduces the informativeness of prices. a We uncertainty with an increase in the variance of an In effect, we take as given the fact that inflation affects and examine how such shocks affect intra-industry pricing behavior and market performance. Of course, this should affect the is only a partial and crude representation of inflation. First, inflation demand timing: realistically, side as well as the supply side. But this when consumers see prices change unexpectedly, is essentially an issue of their resources have not yet been fully and unambiguously affected by the inflationary shock change would not be unexpected, and there would be no Similarly, full when demand which consumers our model has no money. However, one takes if no substantial problem with is they had, the price difficulty in assessing relative prices). firms discover unexpected changes in costs, they have not yet ex'perienced the increase in nominal there (if will eventually address to them. Secondly, nominal as given that inflation affects calling the numeraire good money. Since all costs, price changes are unex-pected by agents given their information (any inflationary trend has already been factored out), the dollar prices and may enter prices, Thus in spite their utility and which they observe are their best assessments of real profit calculations without implying of the model's obvious limitations, we feel that any money what we learn from illusion. it about the effects of real, aggregate cost shocks remains relevant for genuine, money-driven inflation. The reader who does not share these convictions can maintain a purely "real" reading of the paper; indeed the relationship between the stochastic structure of supply shocks and market efficiency is of interest independently of any possible link to inflation. A well-known discussion of the relationship macroeconomic shocks is contained in Okun repeat purchase behavior of consumers shocks. raises The reasoning its price. He argues informally that search and the may reduce the adjustment of prices to that a firm's previous customers will only search Lowering price can only therefore have high justification for is (1981). between consumer search behavior and demand elasticities. attract if demand the original firm consumers who are good shoppers and The argument, although consumers' search behavior. In particular, if intuitively plausible, lacks a inflation is expected, a nominal price increase at the inflation rate should not induce search. In our model, where search and pricing decisions are optimal given the stochastic structure of costs, the variability of the joint cost shock will affect consumers' signal-extraction problem, hence their search rules. This in turn will determine the elasticity of demand faced by each firm, hence between aggregate its pricing strategy and ultimately cost or inflation uncertainty, created by this mechanism is the The social welfare. relationship monopoly power and market efficiency main focus of the paper. We identify two major effects of an increase in the variance of inflationary shocks. We refer to the first one as the correlation an increase high, while if in effect. It operates so that if search costs are sufficiently power and the variability of inflation works to increase market real prices, search costs are sufficiently low, an increase in the variability of inflation works to decrease market power and real prices. Intuitively, the argument the following. is As the variance of the joint cost shock increases, the correlation between firms' costs increases. Since optimal prices depend positively upon cost, prices also Bayesian consumers to put more weight on the when forming their posterior belief the observed price is first become more correlated. This leads observed price and less on their prior, about the price distribution at the second firm. Thus, if high, greater variance of the joint cost shock implies a higher conditional expectation of the price at the second firm. This reduces the value of search for a given price at the first store, as the observed price it is becomes less likely that its high price is really a bad deal. Conversely, if low, greater variance of the joint cost shock implies a lower conditional expectation of the price at the second firm. This increases the value of search, as even this low price is price is it less likely to low, is with good deal. Now if search costs are low, buyers' reservation effect inflation variability tends to decrease making the market more competitive. Conversely if search costs are high, buyers' reservation price, and inflation variability will tend to increase it firms' An costs, This really and therefore through the correlation further, thereby so be a it even more, and market power. increase in the variance of the joint cost shock affects not only the correlation of but also their unconditional variance, and generally the conditional variance as well. in turn will tend to increase the conditional variance of prices, and thereby the option value of search, as we well known. This effect, to which is to raise the value of search and lower firms' refer as the variance effect, tends market power, as the variance of the inflationary shock increases. We identify several other channels through which aggregate cost shocks affect market performance, but the correlation and variance effects stand out as both more important and general. They formalize and fill the gaps in the intuition one generally has about the impact of inflation uncertainty on buyers' incentives to search. The main conclusion of the paper is that when is it recognized that informational imperfections generate imperfect competition and endogenous information gathering, the a from variable priori case for information-related welfare losses many Indeed, inflation has effects (which tend to improve market efficiency. crucially depends on how costly anticipated inflation and search, whether through efficiency cost its trend or its it is We we show whether Benabou (1989) comes still In analyzing the interaction of to quite similar conclusions. seems to The Thus, have no single identifiable which could be expected to swamp the benefits across inflationary regimes. much weaker. inflation lowers or raises welfare to acquire information. variance, inflation is on the price system, several of which identify) that inflation all market structures and strong aversion to inflation which people almost universally proclaim remains hard to reconcile with the consequences of the sophisticated behavior and information utilization with which economic models The paper proceeds as follows: Section the construction of equilibrium strategies. II The endow them. describes the effects of changes discussed in Section IV, both through analytic examples and a V concludes. model and Section III contains in inflation variability are number of simulations. Section The Model II. In this section and the next we present model of a duopolistic search market a equilibrium with Bayesian learning. Buyers' search decisions depend on the inferences they make from observed prices, taking into account firms' strategies. Conversely, firms' pricing inferences about their competitors' prices, decisions incorporate their ovkH knowledge of buyers' inference and search 1. Market and information structure costs [c~ , and Ci c*]'x[c' product of a observes i Cy . , C2 c*] own 0<c"<c*<-*-'». For cost c ^ , instance, each firm's cost c cost c its rival's j , although c , We denote by f on firm I's. ( c 2 I c 1 and / ( c 2 ) We assume that and C2 almost everywhere, and that I / (C2 c , 1 sum y, (real cost). or Firm provides information about is common the distribution and density of firm 2's ) c support could be the , and a private cost shock (e.g. inflation) but not joint distribution with 4 Buyers do not observe cost realizations but the joint distribution of costs cost, conditional ] , identical firms, with constant marginal which are drawn from a symmetric common cost shock its knowledge. c , their rules. There are two . and 1 ) is continuously differentiable in both costs are positively correlated in the following sense: dF(C2 |c,) dCi d < dCx 4. Suppose for instance C2 even though that c, = Q + y we assume Vcj , (2) . Ll-F(C2|Ci)J c,-c,-0 that firm 1 ^ , i= \ ,2; then c does not know , 1 contains information about but only c 1 . This assumption is not essential but fits well with the idea of aggregate uncertainty. Firms are themselves buyers of inputs, and do not have perfect information on whether high or low materials and labor prices are specific to their own suppliers or economy-wide. In a richer model, of course, firms could also decide to become informed at some cost. Condition (1) signals a higher c 2 is • just first-order stochastic To interpret it, would then say that the hazard rate decreasing in Cj monotone C2 ' > C2 . cost) But is need only hold (2) will it that such will Cg as Cj tends to c There turn to buyers. D(p) quasi-concave 2. , at ever}' ( c c , 1 2 ) It • to c' j , is /(C2 Ci) has the usual if I from below, so , that a higher c one moves from c" as be the case = S{p) is a continuum of increases in c it is in fact for 1 any quite weak. ^ its in -S'(p)is p for all c in c,letn^(c) denote measure identical consumers, with = j'°pD{r)dr be the surplus each of them derives from buying her demand function in the absence of search. standard assumption that a firm's profit per customer with cost were required /(C2'|ci)//(c2lci) i.e., if means ensure that a firm's equilibrium expected profit function (conditional on normalized to one. Let p, where that (2) also quasi-concave. We now at shown easily first for finding the true likelihood ratio property, Condition (2) own It is . suppose dominance; c [ " its Search and learning from prices c . * ]. Since maximum Initially, . 11 ( p , c ) n(p,c) is = We make the (p-c)D(p) is strictly the profit function of a monopolist value, achieved at Pmic). half the buyers observe firm one's price and half firm two's price, at no cost. Given the observed price, they must decide whether to purchase or to search and find out the other firm's price. consumer to buy at the recalled costlessly communication 5. Searching entails a cost but allows the cheapest of the two prices. The assumption that the means that a is a pure informational cost, rather first offer can be than a transportation or cost. Given the first Note also that (1) tends to observed price, say lower probability that c 2 is in make [c" , Ci p 1 , a (2) hold, ], i.e. consumer must and that if first infer an increase /2(C2|C))<0 the extent to which in c for C2 < Ci , 1 leads to a then (2) holds. 10 it reflects a firm-specific From shock or a joint shock. about the other firm's price, with distribution G price support [ p" p* , p2 finding out = 1^(P,) ] • ( p2 p and density ) i I '[-5(P2)-S(p,)]g(p2|p,)dp2 search. otherwise she ; distribution, a higher found, and this tends to increase Ii^'Cp p 1 ) As . distribution is 1 G(p2 p news" about the distribution U'' ( I ) 1 ; ) p \' > p \ I effect is Pi) accepted. ) To ( P 1 ) effects more away at p , worth (3) . on the expected return from likely that a better deal firms' prices are correlated, a high p 1 can be is "bad of the other firm's price, and this tends to reduce and on consumer preferences from above. where a price make assumptions on not too strong, so that U^(Pi) = it is p is 1 (1981)) this rejected but a higher price preser\'e the reservation price property, Rothschild (1973) impose the condition that the return '^'' it right (1970), Rothschild (1973), Rosenfield and Shapiro Rosenfield and Shapiro (1979) G^(P2 on the well-known from the literature on optimal search from an exogenous unknov^n (DeGroot is if buy two makes 1 but learning effect can result in search strategies p that ) 1 'D(p2)G(p2|p,)dp2, r = will just (3) that observing a high price has For a given I p' larger than the search cost a Note from p2 p ( the expected benefit from search, if p' is g consumer will decide Finally, given these beliefs, the before buying r inference she then forms posteriors this to is ( D(p) the distribution of prices (equivalently = 1 ) which ensure that the learning monotone. Alternatively, Rosenfield and Shapiro an additional search never cross the horizontal an equilibrium model, however, C In and ( p2 p I 1 ) line incorporates firms' optimal pricing rule, therefore no such assumptions can be made. Moreover, the pricing rule itself depends on buyers' inferences, making the problem quite complicated. We shall nonetheless follow similar lines of reasoning, but with respect to the exogenous distribution of costs f (cj I c 1 ) . We mainly focus on equilibria where buyers' search rules . 11 have the reservation price property (reservation price equilibria), and show that a pure strategy reservation price equilibrium exists provided that either: i) Firm's costs are not too correlated ii) Buyers' search cost The will first assumption W ensure that The model ( p 1 ) is will f2 is not too large falls W ( p , ) is below a once it monotonic. " Alternatively, the second has risen above. on Reinganum (1979). The two essential differences are that we also builds analyze a duopoly while she assumes a continuum of firms, and she has independent costs. ), relatively small. ensure that never ( we have correlated These two features allow respectively costs while for search to take place in We expand on these remarks in Section equilibrium, and for consumers to learn from prices. III. 3. Technical conditions Since they offer First, in intuition, little some readers might want f (c that this implies Turning now 6. to go technical assumptions. directly to Section to the | c) > demand > , side, for 0. all Wc£[c',c'], c> (4) c' we assume that -Z)'(p) = S"(p) Consider a parameterized family of conditional distributions F derivatives F\ = f^ and Then (i) 7. We define / holds for X f2 limits in >^, and such (C2 I that for c >^ is 1 = ) bounded on with partial , f2 - not too large. c ( which are continuous " | c " ) and /(c'|c')aslim/(c|c) and c->c' These III. addition to (1) and (2), the distribution of costs must satisfy:' /(c|c) Note more This paragraph gathers additional, . 1 1 m/ ( c | c) , respectively. c->c' can be positive even with /(c"|c,) = Oor/(c*|Ci) = , Vcie(s",c*). • 12 [c\c^] strict . i-e. A - sup {- Z)'(p) c-<p<c'><c». This may quasi-concavity of n ( • , c ) c->0- This also holds in the limit at on the compact Finally, the implies that: P(p.c)-—V^ continuity require I p^(c) set A' = > , Vp. 0. since c<p<p^ic). <0 U pp^p^^c) ,c) . Therefore, by uniform {(c,p)|c"<c<c*,c<p<p^(c)>: 0<m = min{p(p,c),(c,p)e/v}<max{p(p,c),(c,p)G/^} = M<oo. (5) We shall assume: n„(c-) > S(c-)-S(c ) + AV. ..2 . A^|^-J (c -C-) This basically requires that the monopoly profit functions Fl (p . , c) be neither too (6) flat nor too spiked, that monopoly profits for the most efficient firm be sufficiently large, and that the range of possible costs (c' ,c^) not be too wide. Condition (6) will ensure the existence of a solution to the differential equation defining the optimal price strategy of firms with relatively high costs. : 14 We intuitive conditions. exists, do not have conditions exists when Nor do we or a very general existence theorem. parameter values there for more than one the mixed strategy equilibrium rule out the possibility that for some of the four equilibrium types, or even some other, less intuitive type.° The first proposition shows that all equilibria share charge their monopoly price. Proposition c£[c' 1 In an , any equilibrium of the game, there ,c' + e] sets price equal to p„(c) intuitive feature: low cost firms . exists an e > such that a firm with cost . Proof: See appendix. The intuition is quite simple. Since search costs are observe a price sufficiently close to the lowest price charged the lowest price charged in equilibrium deviate and raise argument if its is less strictly positive, consumers who equilibrium not search. in will than Pmi^^') the firm charging price without losing any customer, thereby increasing the lowest price is above pm( c " ) is its this price profits. even simpler. Given Proposition If can The 1, it will be useful to define: V. Sc)^j\s(ip^Cc^)-S(ip„ic))]fic^\c)dc^ 8. For instance, one can not even exclude decreases with its here because c and p2 Pic) . is i cost c i = j[Dip^(ic,))p^'ic,)Fic,\c)dc,. where a pCci) The usual revealed preference argument fails demand function through its correlation with C2 equilibria firm's price over some range. affects firm I's ex^pected We shall, however, restrict attention throughout the paper to equilibria where non-decreasing. (7) 13 III. 1. Equilibrium General properties . We between firms and buyers, with a consistency (see Fudemberg and restriction on beliefs off the equilibrium Tirole [1989]): a consumer's observation of firm directly affects her beliefs about firms Ts cost c . i Thus, not to search, she must use these arbitrary beliefs about c p if there are no restrictions on the consumer's beliefs about c ; i i is i I's p~ if a consumer observes played with positive probability, she must to form firm's having a price less than p " Moreover, these each firm will . This beliefs will be than the lowest put zero probability on the other . We identify four different types of equilibria to the model. high, only about c 2 which beliefs at firm 1 a price less still i off the equilibrium path, about C2 and the equilibrium strategy must be used to form beliefs about P2 important below; for example, p price path but in determining whether or are consistent with the joint distribution of costs and Bayes' rule. price game look for a symmetric, perfect Bayesian equilibrium of the be able to charge its monopoly If search costs are sufficiently price without triggering search costs are not quite large enough to support this any search. outcome, there may still If be an equilibrium without search, where firms with higher costs bunch at consumers' reservation price to prevent search. Reinganum model. These two equilibria are qualitatively similar to those of the A new type of equilibrium arises for lower search costs: a pure strategy, reservation price equilibrium with buyers searching at higher prices and firms' decreasing to zero as their costs increase toward the maximum c* . The fourth possible type of equilibrium involves mixed strategies. High cost firms charge prices which indifferent price between buying and searching, and the makes We this pricing rule fraction of markups make consumers consumers which search at any optimal for firms. prove the existence of the first, second or third types of equilibria, under quite 15 l/„(c) is the value of search W(p) when p observing a price = Pm(c), with cost below c charge their monopoly price, and no firm with cost above than Pmic) • We now move 1 . When 2: can cover all cost realizations, firm charges its and consumers o for all monopoly price p^(c), and consumers never c in [ c " , c * ] , attract is quite intuitive. search. more than .- . . No firm can earn half of the customers, since V^ is optimal given the definition of 3. No-search equilibrium with bunching the equilibrium of Proposition 2, not search, there exists an equilibrium in which each Proof: See appendix. This will still first store. K„(c) < If ... search costs are large enough, the range of monopolistic independently of the price observed at the Proposition c charges less . Monopolistic equilibrium pricing of Proposition firms to a characterization of equilibrium, starting with the case of large search costs. 2. if all more per customer by .. - • '. deviating, nor can it none of them search. Conversely, not searching • there . When may search costs are not large enough to support still be an equilibrium in which no consumers search. Define c' as the smallest solution to: I/^(c')- f D(p„(c2))p„-(c2)f-(c2lc')dC2 = o. (8) 16 and n' " let charging less p ' p , (c*) if all A consumer . than p* (so that p' reveals c' p Firms with c< c' are firm with cost just above c up i * . to We ). p charge ° between search and purchasing p^ic) and no firm with c>c' cost charges but reject higher ones.9 charges it at a store on reservation price equilibria where shall focus able to charge their still If indifferent c<c' firms with cost consumers accept prices is its monopoly monopoly price, will it Consider, however, a price. induce search; rather than accept the resulting first-order loss in customers (they search and find a lower price with a probability of at least f (c and only a second-order is an equilibrium in | c) effect which ajl on p' prefers charging ), it profits per customer. In firms with cost above which causes no , Proposition 3: If search costs o are such that which consumers have reservation price p* c< c' and p(c) = p* for p' > and we show that if p fact, c* charge consumers do not search but prices are constrained by the loss of p' * customers > c* , there In this equilibrium, . possibility of search. c* > c' , there exists an equilibrium in firms' pricing rule is: p(c) = p^ic) for c*<c<c*. Proof: See appendix. To sustain this equilibrium, search, in response to any price p we can i > p * ; simply assign beliefs which for instance, c such a price then earns zero profits, while 9. It it i = c' . Any make it profitable to firm which deviates to could earn positive profits by playing its can be shown that any reservation price equilibrium with p(c) non-decreasing must have the same form as those we examine < p* necessarily (with p ' simply replaced by p < p * ); on very implausible out-of-equilibrium beliefs. p Since the basic features of the equilibrium and the spirit of the results remain unchanged, we do not think it worthwhile to go into the complexities of equilibrium refinement, and moreover those with simply concentrate on the more rest natural reservation price equilibrium where p = p\ 17 equilibrium strategy. Therefore such deviations will not occur, and this justifies the imposed beliefs. Each firm chooses instead the price and prevents search. Finally, p< p' which maximizes its profit by definition of c', accepting offers below p' per customer optimal for is consumers. The and 3 are analogous equilibria in Propositions 2 model, except that consumers' reservation price p' to those of the Reinganum (1979) depends here on the learning which occurs in equilibrium. 4. Reservation price equilibrium with search more difficult case, in p" cost exceeds which consumers' reservation price p" can then not avoid search, unless contained a continuum of firms (as because its Smaller search costs lead to a . in all still expect positive profits a firm find a lower price. firms in the market (more generally a finite number), induces search, but when than c*. makes negative it Reinganum's model), such consumers who searched would less is it is one's new but much A firm whose If the market would have to stay out, profits. However, with only two possible to charge a price which rival, who follows the same strategy, has an even higher cost, hence an even higher price. We now characterize a pure strategy reservation price equilibrium in which search actually takes place. realizations, it is Firms' pricing strategy p' = p^{c'). Afirm with higher all all consumers who visit this firm We now derive Pfic). If consumers on Figure similar to that of Proposition 3: a firm v-ith cost and a firm with cost between c' and some c^ > so illustrated is if its rival first firm i For low cost below c' charges Pm(c) charges consumers' reservation price c' cost realization c 1. , however, charges a price Pf{c)>p', search. charges a price has a higher price, and to none p , if its which induces search, rival it will sell to has a lower price. Therefore 18 EQUILIBRIUM PRICING STRATEGY WITH SEARCH p(c) Pf(c) P' PmiC) Figure 1 . . 19 firm probability of making sales i's on firm i's true cost c, , and its Y(p,.c,) where p( • ) is differentiable = is the probability that firm j has a higher price, conditional expected profits are: n(p.,c,)-[l-/^ro6[p(c,)<p,|c,))]. Assume the equilibrium pricing strateg)'. on [c' ,c'], with p(c')> p' for this will ; now that p ( be (9) ) is increasing and We verified below. can then rewrite: Prob[picp<p,\c,] and = F[p;\p,)\c,] differentiate (9) with respect to p, to obtain the first-order condition which p(c) must solve:10 • n(P(c).c) p^Cc^^ ^ ^ i-f(cic) np(p(c).c) ^^^'^^ . _ . This differential equation must be satisfied along the price path [c\c^] which lead to search. the highest possible cost, c to find a lower price. would lower its sales at a price does not Instead, * . The boundar)' Such a firm makes no Therefore, price by a condition little bit it price c we sales since consumers search and are sure ( ' , c ; otherwise the firm positive expected profits, since its rival's satisfy Lipschitz conditions at the region of costs found by considering a firm with must be the case that p{c') = c' and make above cost whenever is in ' ) ,1 ^ was higher. it would make Because p(c') = c^ is (10) so standard theorems are not applicable. construct the solution as the fixed-point of a contraction mapping; this condition (6) , is where needed. Since charging p> p' leads consumers become informed, it is not surprising that this pricing rule is quite similar to the optimal bidding rule in an auction with correlated values (Milgrom and Weber [19S2]). The difference, and source of difficult}', is that n ( p c ) cannot be expressed as a function U (p- c) 10, , 11. Nor at points (Pm(c), c) to fully . 20 Lemma The 1: differential equation (10) has a unique solution Moreover, p f '( c )> Pf(c) , for on all c in [c" ,c^) c*] , with terminal condition p^^*)^^,- » c< p f(c) < satisfying , (^c' Prrt(.c) , with equality only at c* . . Proof: See appendix. While Pf{c) solves the characterize optimal prices for we show that first-order condition (10), c> c' equilibrium profits are First, . level it really using condition (2) on the distribution of costs, quasiconcave. strictly Lemma 2: Assume that buyers have reservation price p' some remains to prove that it = p^(c') If firms with cost above c''>c* charge Pf(c), while firms with cost c<c'' charge min(p^(c),p'), p^ then a firm's profits from charging any price p' . are: w(p,c)^ n(p,c).[i-F(p;'(p)ic)]. They are strictly quasiconcave and maximized over pe[p^,c*] at (II) p = p/r(c). Proof: See appendix. The final step in characterizing firms' strategies separating those which prefer to charge The first ' to find the threshold cost those which prefer to charge p f(c) strategy prevents search but the second yields greater profit per consumer, come back. At c ^ a firm is l-^-F(c'\c') The all p from is indifferent between the two, so c ^ •n(p*,c^) = visit it first (1/2) and to all p ' . > p' they if defined by: [l-F{c'\c')]-n(pF{c'),c'). left-hand side represents profits from charging consumers who is c^ (12) A firm which does this sells to consumers who visit the other firm first and . 21 observe a price above p Given the symmetry of the equilibrium pricing strategy, * 1/2 times the probability that the other firm has cost The right-hand represents profits from charging customers who visit find a lower price, which Lemma c' 3: Assume £ic' ,c^] , ^-[l p ^(c'). A firm which does and also sells to customers It is also -[ 1 - /^ ( c "" I which is search, find a higher price it first, -[l-fCc'^lc^)] customers. and above c' c ^ )] that buyers have reservation price , - Fie' just c')] \ this sells to all the other firm visit is which given symmetry, return, who this is first, search, exists a unique customers. p' = p ^^c') There . such that the following strategies are mutual best responses for firms: (i) Force (ii) For c e [c* (iii) For c e p(c) [c",c"]. = p„(c); c^) , p(c) = p'; [c\c"] , p(c) = pp(ic). , Moreover, c^ and p' are continuous and non-decreasing in c" . Proof: See appendix. This result concludes the characterization of firms' strategies: strategy equilibrium c' . P f(') ' where consumers have reservation price and c' only remains to , which are the unique solutions to verify that, given firms' characterized by the reservation price p* is where , strategies, i.e. that p' = P mi^' ) first Next, p^<p'\ if p 1 = p then Ii''(Pi ' , ) = ^'^(Ci consumers only ' it is , (8), (10), and there exists a pure uniquely defined by (12), respectively. optimal buyer search U'(Pi)>a issues associated v.ith equilibrium learning will Consider p if if and only if p, is >p' It indeed . This be most important. ) < Km(c') = a since by definition , c infer that c , 6 [ ' , c ^ ) , so, 22 l^(Pl)= D(p„.(C2))P^'(C2)f(C2|C°<C,<C^)dC2 1 < [' D(p„(C2))p^-(C2)/'(C2lC,=C*)G!C2 = K„(C°) = 0. and they still do not want to search. For out-of-equilibrium prices they lead to the belief that c P i = c ° (4) p^), assume that Dip,)Gip,\p,)dp,-[Sip')-Sip,)]Gip'\p,) J^ I l/„(c')-[5(p')-S(p,)]F(c'|c')>a. and weight on c , r p , \ due to ^ so that: , I^(P>) = /_'-D(p2)G(p2lp,)dp2 D{P2)C{P2 Pi)dp2== = p e(p° i c' > c' . More generally, any out-of-equilibrium beliefs being closer to c ' than to c ^ Finally, buyers will search at all strategies, this is equivalent to K(c i ) will > a for if all c , and only > c ^ if I-/ ( p sufficient pie(p*,p^).12 lead to search at prices pi> p' which put , ) > o ; given firms' where: l/(ci)- f' D(p^(c2))p„'(c2)/^(c2|c,)dc2-[S(p*)-5(pO]/^(c^|c,) • 1 ^(Pf(C2))Pf'(C2)^(C2lCi)dC2. (13) c 12. The fact that consumers' beliefs do not remain monotonic in the observed price as it moves off the equilibrium path is admittedly unappealing. But one has to choose between such monotonicity and the reservation price property. Indeed, due to bunching, consumers who observe their reservation price p strictly do not want to search; if observing p + e did not lead them to infer a lower c hence a lower Cj and p 2 they would not want to ' i search, a contradition. , ' , 23 Using the definition (8) of c", the condition for search [5(p')-S(p^)]f(c^|c,) 'D(p,(c2))p/(c2)f(c2|c,)dc2 r + becomes: [' Z)(p^(c2))p^'(C2)[/^(C2|c')-f(C2lc,)]a!c2, The P2 p G * [ . hand left p 1 ] ; side denote C] rather than c* , it the is Wc,>c' expected return, conditional on as ^{c^ ,g) The . right hand side rule, x(c i . , (14) . from finding , the "bad news" from inferring P2<p"; about the likelihood of finding a price For buyers to have a reservation price is c > denote o) must not be too large. x(Ci,a). as it From (14) we see that such will be the case in two intuitive cases, which extend the insights of Rosenfield and Shapiro [1981] and Rothschild [1973] to an equilibrium context. The small, so that c' is close to c' , making the one is when a is small; the following proposition x integral in first formalizes this intuition. Proposition 4: If search costs are relatively low, there exists a pure-strategy, reservation-price equilibrium characterized by p' = p„{c'). have reservation price c 6 [ c * c , ^ ) , and critical cost levels p f{c) if cg[c^c*] c' and c^ Firms charge , with , with c' <c' p,n(c) if <c^ <c' c£[c' .c'] . Buyers , p' if is when a c<p;r(c)<Pm(c). Proof: See appendix. The second case firm's cost ^2(^2 Ci) I in which a reservation price rule will be optimal for buyers does not reveal too much information about that of is small, so that the two distribution functions in x its competitor, i.e., when are close to one another. : 24 Proposition 5: If firms' costs are not too correlated, i.e., if, F2'^niax{F2(c2 Cj) |c"<C2^Ci <c*} I is not too large, there exists a pure-strategy reservation price equilibrium. For a above level ct' , whether it involves V mCc*} is no search and corresponds larger or smaller than a . to that of Proposition 2 or 3, For a < o ° , it some depending on involves search and corresponds to the equilibrium in Proposition 4. Proof: 5. See appendix. Mixed strategy equilibrium hold, there may be no . If the assumptions of low search costs or low correlation do not reservation price equilibrium, as one should expect. Indeed, simulations suggest that there exists an intermediate range of search costs in which none of the three types of equilibria discussed above exists (see Section IV). briefly to a fourth type of equilibrium, We must therefore turn which involves mixed strategies by consumers and generally does not have the reservation price property. In such p(ic) = Pm(c) indifferent an equilibrium, pricing , for c<ic'. From between searching and c' not, low levels of costs remains unchanged, at the pricing rule to c* and they will randomize i.e., Pr(c) makes consumers this decision. Thus for all c, e [c' ,c^] f •^ [S(p„(c,))-S(p,(c,))]/(c2|c,)dc2+ r -^ c' '[S(p,(c2))-S(p,(c,))]/(c,|c,)dc2 Differentiating this expression with respect to Cj yields Pr(c2) ,forc2<Ci ; = o. (15) c' this allows the function p /? ( ) to Pr'(c^) as a function of all be constructed, moving up from the : 25 initial condition p>p' p rc'\ = p' . The must then be such that fraction it makes the it that in this type of equilibrium because not all its would then make zero price equilibria. It is pii{c')> optimal; for all p>p c' if it . The raised highest-cost firm its price they all > makes (16) positive would search, and profits. also is somewhat less appealing intuitively than the previous reservation much more difficult to must show the existence of solutions p^(c) which are extremely complicated. We construct for a general specification: one to (15) and ouy^ (p) g [0, 1 ] to (16), both of have not established general conditions under which equilibrium exists, but report below on simulations using simple functional forms which indicate that do. search at any price Pr{c) t«,(p^(C2))/(C2|C,)dC2 consumers search; yet Such an equilibrium this pricing rule who . Pg{c^)£argmax{ nCp.c,) 1-CU,(P)+| /:: profits of consumers - Cj e (c', c*] Note uo^(p) it does exists in the intermediate range where none of the other three equilibria ' 26 The IV. EfTects of Inflation Uncertainty We are now ready to analyze changes in how equilibrium pricing and search rules are affected by We the underlying distribution of costs. are interested in particular in increase in inflation uncertainty impacts firms' market power, the and consumer surplus. Again, what view detailed its it interactions with price information we demonstrate correlation effect c ' , is also the cost at and the variance effect. and market efficiency, rise to similar issues This and she will which firms are first is power in the model; this specification (costs are the which we report in is we done by analyzing the comparative partial but generally . in c' is the statics of demand p^^c") , and sum of uniform private and joint cost shocks and demand is linear), the second part of this section. which are equal up prices are again equal However, the call good indicator of monopoly to c' 3, it is clear that an increase in c' to c' and greater above. The effects of on the search equilibrium of Proposition 4 are more complicated. Since , of confirmed by a variety of simulations using an alternative In the no-search equilibrium of Proposition results in prices some share constrained by search from charging their monopoly As explained below, c' provide a p' we have effects. characterizes consumer's reservation price p' = price. ' speaking subscribe to our interpretation can thus the two most important intuitions, regarding what Recall that c' iso-elastic. c^ strictly using a particularly convenient specification where costs are jointly log-linear and is in is search, profits as arising from real shocks (raw materials, weather). Hopefully, once our conviction that nominal inflation gives First label inflation uncertainty The reader who does not aggregate cost uncertainty. safely we amount of how an effect ambiguous. it an increase in c' implies an increase from c' to c' and greater from c* to the lower of c" and on c^ of a change We know that if the in the cost distribution which causes an increase differential equation (10) which determines prices ; 27 Pp(c) above c" from c^to prices p' since is less than Thus by no means do the comparative show .^^ Pf{c) (10) will generally be affected in a complicated simulations (Lemma 3). This will imply lower But of course the differential equation c ^ remains unchanged, c * will also be greater statics manner by changes of c' tell Nonetheless our the entire story. do provide robust intuitions about the that they in the inflation process. full equilibrium effects of inflationary uncertainty in this model. Case 1 : Log-Normal We costs, Costs, Iso-Elastic begin the analysis by making the following distributional assumptions about firms' now denoted C, = , i (i) C, = er, ; (ii) log (iii) log r, ^ Yi (jv) Yi The Demand ^ G . Y2 • 1 ,2: c.^logC, is distributed normally with is distributed normally with log-costs c 1 (1 - 9^){'-'q + v^) and C2 . mean and variance lie , on where c p - We shall examine how is , c and variance fy therefore normal, with ^-^ i/g uncertaint)', affects the conditional distribution , is mean makes it + - p ( 1 ) c F(c, Icj) and consumers' return inflation to search. does not it < + °° This to prove the existence of an equilibrium with search, largely because we this difficult 1 which measures aggregate cost or however, that a firm earns greater profits at this lower price since induce search, while it could have charged p f(c) and permitted search. Note that p c the correlation coefficient of the two 13. Note, 14. ; 3nd 9 are independent. 14 distribution of c 2 conditional and variance mean example does not satisfy the assumption of the model that can not use Pfic') = c' as a terminal condition for (10). However, this c * . formulation provides the clearest way to analyze the effects of inflation uncertainty on c' closed-form ex^pressions can be obtained. , because 28 Assume that demand is iso-elastic, D{P) priceis: P;„(c) - log /'^(c) = log(;^) + n p^iog M(P) + c; Ti-1 The unconditional The distribution of variance s^ moment . To distribution of Pm(c2) . V[> 1 . so that the log of the monopoly Define: PP^(1-P)P: Pm(c) is s'-(l-p')(fye^i^v): mean p and normal, with conditional on Ci = c , , , is consider p and s^, rather than at the 5(p^(c)) monopoly = Therefore the return to search variance Ug normal with mean |-i(Pm(c)) i^e and v^, as the parameters of interest. 1 1 We can ^P„(c)'-'^ = ^exp[(l-ri)p^(c)]. in the region of monopoly prices is: •{exp[(l -ri)P2]-exp[(l - Ti)p„(c)]}exp (P2-^)' dp-. 2s^ with |j. = |iCPm(c)) . Rewriting in terms of the distribution and density ^ of a standard 4> normal: i^.(c) -exp ^(Ti-l)[2p + 2p(p„(c)-p)-s'(Ti-l)] exp[-(Ti- l)p„(c)]- * (l-p)(p^(c)-p)+S^(Tl-l) 4> (l-p)(p„(c)-p) (18) Tl-1 Hence: cp • and price as: •p„(o l^.(c) = (1^)' better demonstrate the two effects of inflation uncertainty, let us for the consumer surplus write = c = P"^ (p„(c)-p)-exp-^(ri-l)[2p + 2p(p^(c)-p)-s'(Tl-l)] ^ * (i-p)(p„(c)-p)-^s2(Tn-i) , 29 = ds \\-9){Pm{c)-p)^s\^-\) + s(r|- (l-p)(p„(c)-p) + s^(ri-l) 1 )4> <J) exp--(n-l)[2p^2p(p„(c)-p)-s^(Ti-l)] So finally: sg^ — sgnip- p^{c)) = ; dp > sgnic-c) = (19) (20) . ds The first result (19) shows what we call the distribution of C2 conditional on 0^ = the unconditional mean c , this conditional distribution p , the effect of first to lower it at c uncertainty > is increasing in p in v^ is if mean of a weighted average of the observation c and - p 1 c> c , . respectively. Therefore, the and decreasing if c < c . By mean of increasing therefore to raise the value of search at c < c , and Xhjs correlation effect captures the idea that inflation or aggregate cost makes people search things are just as is with weights p and an increase c .15 the correlation effect. Recall that the less when bad elsewhere. However, they see a high price because they think that , it makes them search more when they see also a low price because they think that there are other, possibly better bargains to be found. , The second return to the 15. first result (20) shows what we really determines search but the conditional distribution of surplus involves the equilibrium pricing rule ) . the variance effect. Given that buyers can store costlesslv, an increase in the variance s Of course, what S(p call p{c) -S ( is ^ of the conditional distribution not the conditional distribution of cost c 2 p2) ; see (3). The difference between the two as well as the convexit)' of the surplus function This discussion, and the one which follows, are meant to give the main qualitative intuitions. 30 increases the option value of search. But note that an increase in Vq , the unconditional variance of the joint cost shock, does lead to such an increase in the conditional variance: vl ds^ ye + ^Y. thereby raising the value of search and reducing the market power of firms. We now examine how these two effects impact consumers' reservation price and firms' pricing. We focus on the case where search matters, i.e. where c* = inf{c|K„(c) = o><oo. Then K„'(c')>0 , so (19) and (20) imply: sgn—- = sgn[p^ic )-p] dp — sgn(c -c) = (21) dc' < 0. (22) ds If exceeds c' c , monopoly markups over opposite direction. With a relatively costly. this By the correlation effect tends to increase it further, resulting in a wider range of costs; the variance effect, however, definition, such a configuration with relatively case both the variance and the correlation effects reduce when c' > c occurs low cost of search, on the other hand, c' it works is less in the search is than c; in even more, and the market becomes more competitive. While log-normal case this is variance effects seem quite robust. of any common shock to firms' very specific, the intuitions behind the correlation and One can costs, as think in general of the variability of inflation, or having two effects on the conditional distribution of by making costs more correlated, costs (and through equilibrium pricing, of surplus). First, shifts it is F(c2 I c 1 ) and its conditional mean, up for high c a source of additional uncertainty, it i , down for low c j ; it secondly, since causes a mean-preserving spread in the shifted 31 The distribution. generality of these intuitions is also supported by the simulations, using a very different specification, reported below. In addition to these two central, information-related effects, inflationary uncertainty has other consequences in our model. First, even in the absence of search, an increase variance of prices can be beneficial to consumers, because consumer surplus price. This of income that it is is is clearly a result of our partial equilibrium constant; 1" we positively affected because they framework, where the marginal in the simulations). depend in part on Similarly, equilibrium profits monopoly profits, is raise more robust and each interesting. firm's elasticit)' of Even though an increase demand, an increase consumers and purchases toward the firm with the lower profitable, this tends to increase expected profits cost. and efficiency in cost. The do with information, since and consumers always bought at the cheapest in it price. the value of shifts Since these firms are more same time. would remain even 16. Also, this effect has little to third search activity in at the was costless utility' that firms face both ex-ante and ex-post uncertainty, because they set their additional effect may in may be which are convex price after learning their cost, but before learning that of their competitor. search convex thus tend to de-emphasize this effect, but one should be aware present (for instance Note however is in the if search ; 32 Case 2 Uniform : Let us Costs. Linear now assume and that demand is Demand that costs are the sum of a joint cost shock and a private cost shock, linear: (i) c, = c (ii) 9 is uniformly distributed on (iii) Y is uniformly distributed on [-6,6]; (iv) Yi (v) D{p) We assume that represents the i , + e + Y,; Y2 a< • 3rid [ -a , a ] 9 are independent; A-p. = 6 which reduces the , volatilit}' of inflation. The number of cases to analyze but not essential; is unconditional density of cost, /z(c) is the sum a of two uniform distributions and has the familiar trapeze shape: h(c) = ^ ^ h(c) = The 4ab ^ h(c) = ^ c-c+a+6 — 26 if ^ c- if ^ c c+a+6~c on from a price observation causes a consumer c to , < c< is also the update a- b c->- a-b <c<c- a + c-a + 6<c<c + a + sum his beliefs is Q~U[-a 6] the posterior of 9 is Q~U[-a,a] 6,c + a + 6] the posterior of 9 is Q~U[ci- c- e[c-a-6,c + a-6] if Ci e[c + if Ci G[c-a a-6,c-a + + 1 about the joint cost shock 9 the posterior of 9 Ci 6. of two uniforms; inferring c as follows: if b - - if 4a6 distribution of Cg conditional + a-b ,Ci - c + b] ; b ,a] . ; 33 Note in the , if c the intermediate region there falls in , lowest region, the conditional expectation of c 2 variability c that a while , and increasing if c a in falls in 1 log-normal case. The same always widens as a is is no learning. However, if , works here effect c than c and decreasing less the highest region, the conditional expectation of c Thus the correlation . is in a way 2 is falls i in the above similar to the true of the variance effect, since the support of 6 , given c 1 , increases. We now look at a number of simulations of this example, in order to get some feeling for the relative size of the various effects of an increase in aggregate uncertaint}'. simulations, Z?(p)=15-p, c dispersion of the joint cost shock costs are given in Tables 1. 2. and ' to a c' near c, and high search first 6, to vary. 3. first at We 5 = 3. and allow search costs The results for low, respectively. well below the unconditional costs lead to a c Looking a = We c = 6 above costs lead to a c' well we all and the intermediate, and high search define these terms so that low search mean of the effects of search costs, a In , intermediate search costs lead c. see that as they increase, the equilibrium involves reservation price strategies and search, then mixed strategies, then monopolistic pricing plus bunching at p support the intuitive way in ' , and finally unconstrained monopolistic pricing. These results which we associated each t\'pe of equilibrium to a different range of search costs. Next we turn to our main subject of interest: the effects of inflation uncertainty on monopoly power and on the components of welfare For low search costs, which are reported the variance and correlation effects make equilibrium pricing strategies with increases gains in consumer surplus. in in in Table search equilibrium. 1 . c ' is more decreasing valuable. in a , because both This reduction in the variance of the joint cost shock leads to 34 LOW SEARCH COSTS g = 0.1: * cs ps n CS Welfare E(c) E(P) 3.6413 ~ - 18.863 16.227 35.090 6.0000 9.3034 3 3.5719 9.3189 9.3789 18.827 16.425 35.252 5.9919 9.2688 1.00 3 3.0871 8.9957 9.3204 18.814 17.779 36.593 5.8663 9.0151 1.50 3 2.5647 8.6442 9.2282 18.834 19.270 38.104 5.6789 8.6989 2.00 3 2.0438 8.2318 9.0783 18.899 20.692 39.591 5.4329 8.3351 2.50 3 1.5240 7.7904 8.8921 18.995 21.966 40.961 5.1407 7.9309 cs ps n CS Welfare E(c) E(P) a Type 0.10 2 0.50 = c 0.5: * a Type 0.10 2 4.4747 — ~ 19.791 14.248 34.039 6.0000 9.6466 0.50 2 4.4435 - — 19.691 14.430 34.121 6.0000 9.6315 1.00 3 4.3420 9.6651 9.7756 19.687 14.697 34.384 5.9860 9.5816 1.50 3 3.9877 9.4448 9.7844 19.745 15.550 35.295 5.9004 9.4066 2.00 3 3.4317 9.0605 9.6659 19.793 16.923 36.716 5.7316 9.0955 2.50 3 2.8796 8.6373 9.5031 19.880 18.281 38.161 5.5029 8.7325 c Table 1 NOTES: In all simulations Type D(p) = = 15-p, c 6, b = 3. refers to the equilibrium pricing strategy in the following way: 1= p(c)-p„(c) 2= p(c)- p„{c) Vc . Nosearch. p(c)- p' Voc'. Nosearch. 3=p(c)-p„(c) Vc<c'. p(c)-p' yc£[c'.c'). p(c)-pf(c) Vc>c'. Search above p" 4=p(c)-p„(c) Vc <c' p(c)- p,(c) Vc>c'. Consumerssearch with a mixed slraleg)' above Max {c', 'Vc<c' . . . n and CS are, respectively, E(p) and E(c) c- a* b> expected industry profits and consumer surplus per customer (net of search costs). are, respectively, the expected price that a customer pays and the expected cost at the store she purchases from. . 35 INTERMEDIATE SEARCH COSTS a =15: * cs ps n CS Welfare E(c) E(P) -- - 20.790 11.476 32.266 6.0000 10.2448 6.5098 - - 20.798 11.510 32.308 6.0000 10.2381 2 6.4592 - - 20.820 11.782 32.602 6.0000 10.2171 1.50 2 6.3740 -- -- 20.854 11.801 32.655 6.0000 10.1814 2.00 4 6.2524 - -- 21.358 11.581 32.939 5.9482 10.2333 2.50 4 6.0917 -- ~ 21.584 11.573 33.157 5.9431 10.2504 CS p^ n CS Welfare E(c) E(p) -- -- 20.854 ]].170 32.024 6.0000 10.319 - ~ 20.886 11.200 32.086 6.0000 10.313 20.916 11.297 32.213 6.0000 10.294 a Type 0.10 2 6.5259 0.50 2 1.00 = c 3.0: a Type 0.10 2 6.9156 0.50 2 6.9004 1.00 2 6.85 28 -- 1.50 2 6.7727 -- - 20.964 11.461 32.425 6.0000 10.262 2.00 4 6.6590 - - 21.396 11.165 32.561 5.9645 10.342 2.50 4 6.5462 - — 21.588 11.074 32.662 5.9695 10.389 c Table 2 NOTES: In all simulalions "Ape D(p) = = 15-p, c 6. b = 3. refers to the equilibrium pricing stralcg\- in the following way: 1= p(,c)- p„(,c) Vc . Noscarch. 2=p(c)-p„(c) Vc<c'. p{z)- p' 'Vc>c'. Nosearch. 3=p(c)-p„(c) Vc<c', p(c)-p' Vce[c'.c'). p(c)-pf(c) Wc>c' Search above p" 4=p(c)-p„(c) Vc <c' p(c)- p,(c) ydc' Consumerssearch with a mixed sirategT,' above Max{c'. c-a* . . . . n and CSare. respectively, expected industrj' profits E(p) and E(c) are, respectively, the expected price that a and consumer surplus per customer (net of search customer pays and the expected cost b} costs). at the store she purchases from. . . 36 HIGH SEARCH COSTS g = 4.0: * CS ps n CS Welfare E(c) E(P) 7.6430 — ~ 20.966 10.766 31.732 6.0000 10.423 2 7.6289 ~ — 20.980 10.792 31.772 6.0000 10.418 1.00 2 7.5849 ~ ~ 21.024 10.873 31.897 6.0000 10.403 1.50 2 7.5435 — ~ 21.100 10.996 32.096 6.0000 10.380 2.00 2 8.7607 ~ ~ 21.312 10.777 32.089 6.0000 10.461 2.50 2 10.448 ~ — 21.520 10.767 32.287 6.0000 10.497 CS ps n CS Welfare E(c) E(P) a Type 0.10 2 0.50 = c 5.0: * a Type 0.10 2 8.3300 - - 20.986 10.561 31.547 6.0000 10.481 0.50 2 8.3165 — — 21.014 10.584 31.598 6.0000 10.477 1.00 2 9.1327 — — 21.082 10.554 31.636 6.0000 10.496 1.50 1 ~ ~ 21.188 10.594 31.782 6.0000 10.500 2.00 1 2.50 1 c ~ ~ - 21.333 10.667 32.000 6.0000 10.500 — - 21.520 10.760 32.280 6.0000 10.500 Table 3 NOTES: In all simulations Type D(p) = 15-p, c = 6. b = 3. refers to the equilibrium pricing strategy in the following way: l=p(c)-p„(c) Vc. Nosearch. 2=p(c)-p„(c) Vc<c'. p(c)- p' '</c>c'. Nosearch. 3=p(c)-p„(c) Vc<c', p(c}- p' Vcc[c'.c'). p(c)-Pf(c) 'Vc>c'. Search above p' 4=p(c)-p„(c) 'Vc<c' p{c)- p,(_c) Vc>c'. Consumerssearch with a mixed strategy above Max{c'. c-a* . n and CS are, resf)ectively, expected industry profits and consumer surplus per customer (net of search E(p) and E(c) are, respectively, the ex-pected price that a customer pays and the expected cost b> costs). at the store she purchases from. . 37 Another interesting aspect of these simulations increases in a two reasons for this. consumers . a In fact, for high levels of a First as is that profits do not decrease much with profits are actually increasing in a increases, search increases. This raises the likelihood that better matching of consumers to low-cost firms actually tends to increase profits. and the is offset prices two columns of Table 1, by the decrease is The second in cost. expectation w^th the unconditional variability Table 2 reports simulation the correlation effect affects These simulations because in some ranges Table is , ver}' little or c' is dominant. In profits are a is not at is when this range, convex in cost, all. The variance is decreasing, yet profits increase and when c" is a is WTien c' . 1. all is made c . reduce not the entire story, consumer surplus decreases. . In all equilibria, there cost specification. less When then becomes than profits increase. The of the correlation effect. the simulations is that social welfare increasing or almost constant in the variance of inflation. In those cases adversely affected, producers are near Similarly, the total sufficiently large; the correlation effect Indeed, one noteworthy feature of in small relative to the previous demand, uniform some of the harm is effect alone acts to significantly greater than c consumer surplus decreases, but by increased variance effect mitigates and thus increase . also indicate that the determination of c" ' the decrease in price , very limited compared to Table partially a result of the linear increases a levels of on consumer surplus and profits 3 reports the results no search. This = 4 c At higher results for intermediate values of a increases, but this reduction effect of inflation variability case. it see factor which offsets the deleterious effecfeof lower monopoly the fact that One can which report the price paid by the average consumer cost incurred for the average customer. on producers c ' as a There are purchase at a low-cost firm, which has higher profits per customer. Thus, the will this in the last . is always where consumer are better off by at least an equal magnitude. 38 The is lesson to take away from these simulations, however, is not that stochastic inflation welfare-improving. This result clearly depends a great deal on the exact specification of the model. Rather, we interpret them as saying that one needs to know a great deal more, particular about the size of informational costs, before inflation effects are at play, and one cannot say a into equilibrium price setting, several priori what be without simultaneously making a statement about market that it is that increases in stochastic reduce the informativeness of prices and thereby decrease welfare. In an economy where endogenous information gathering feeds back complex one can say in their net structure. impact on welfare will The simulations show conceivable that the benefits of increases in the variance of joint cost shocks outweigh the losses. 39 V. Conclusion Aggregate cost uncertainty, whether due to inflation, input prices or to stochastic reduces the information content of prices by making and aggregate price in common variations. In this in a single The difficuh to separate relative paper we have explored how an environment where agents can act to enhance how the stochastic structure it this mechanism operates their information via search. of shocks, consumer search, and non-competitive pricing interact product market. v results indicate that the a priori case for significant welfare losses associated with reduced informativeness of prices endogenous information acquisition and agents to seek and prices more will We studied from inflation becomes much weaker when one allows for Indeed, inflationar}' noise can lead price-setting. information, so that in equilibrium they will in fact be better informed, The reflect increased competition. whether decisive factor in inflation uncertainty improves or deteriorates market efficiency will be the size of informational costs. Another contribution of this paper to the theoretical literature analysis will be useful search behavior rules of develops an consumers. We hope that this between pricing and in general. to be done before we can information-related welfare losses from stochastic inflation. market structure and information technology, incentive to search and on pricing It it price distribution, for attaining a better understanding of the relation There remains of course much structure. that unknown equilibrium model in which consumers search optimally from an and firms price optimally given the learning and search is would be useful may ver)' well in fully assess the possibility of Our model has a very specific which the effect of noisy inflation on the offset the reduced informativeness of the cost to extend the analysis to other market structures and information technologies, so as to ascertain the robustness of our results. In particular, making the model 40 dynamic by incorporating repeat purchases seems quite desirable, although quite Finally, one should also introduce general inflationary interpretation more precise. equilibrium effects, in order to difficult. make the 41 A ppendix: Proof of Proposition p' > p^(,c') price. This is . If Let the lowest price charged in a given equilibrium of the game be p" 1: firm i has cost c" then i, since it consumers who all will not search since there is profitable. Assume p' < p^(^c~) arbitrarily small positive . Thus p' < p^(_c') No consumer will wish sure to find the lowest price possible, the gain of profits, a firm with cost c ' earns many customers by is any other as at zero probability of finding is i's customer are maximized profits per S(p")-o to search if 5-S(p" + p he observes insufficient to cover the search costs. more per customer at deviating. Therefore, the deviation than at p" profitable and p" p is ' + e , + " + e,), e where since even , number, will charge profits per customer. Proof of Proposition first its monopoly price, since it is of buyers is he were will get = p^(_c'). is a small Q.E.D. 2: Km(c,) < equilibrium, the an Given the quasi-concavity Furthermore, the firm . is maximizes the number of customers as well as the A consumer's search decision for any price played in equilibrium store and not search. Given the equilibrium pricing strategy to search 5 if Since no consumers search at prices sufficiently close to p" a firm with cost c" + e where e positive at . Let us implicitly define e, by number. assume go to the other store and decide to search will purchase from has the lowest price charged in equilibrium. Since firm p„(c"), the deviation as first First, . many customers charges p„(c"),it gets at least as if it because a consumer who observes P;n(c") a lower price. In addition, firm Proofs of Propositions max I'^Cc) < o Thus, . it maximum number of customers not sensitive to price, (except if if is to purchase from the at the first store, the he observes p, value does not pay to search. Given that no consumer searches in the firm can attract is half of all customers. Since the a firm charges a price greater than number Pm(c') the number of customers could be lower) a firm charges a price which maximizes profit per customer which is p„ (c) . This is the optimal price to charge independent of off-the-equilibrium-path behaxior of consumers. Q.E.D. Proof of Proposition price exceeds p' consumer knows . is 3: In this equilibrium, consumers' search rule By the definition of c that c^ £[c' .c']. ', no consumer wishes By the definition of c ' he is to search if and only to search at prices is indifferent if the below p' . first observed At p' , between searching and not all if a he 42 p° and knows observes Given the positive correlation knewc, = search if c°. c is in this equilibrium, observing p' only reveals that c, > c° consumer's beliefs of Cj are at least as great as in costs this implies that the he observes p° must be such that sufficient to he proposed equilibrium. This shows that consumer search decisions are optimal in the it all below p^{c'). prices are If consumer observes a price above p' a pays for him to search. Believing that c , = c ', or more generally that If any firm deviates to a price greater than p° since they search and find a lower price. Thus, so long as a firm charges a price the consumers. Thus, the optimal price of profits) the optimal price Proof of Lemma c is , his , close to ensure that he does wish to search. Pricing rules are clearly optimal. 1: is p '. is p„(c) unless it gets zero consumers, no greater than p ° , it gets half exceeds p'in which case (given the quasi-concavity it Q.E.D. Since the differential equation: not applicable. Instead, p f{c) will ^^-'^ rT7(^-n,(p(c),c) = (p(c),c)^ satisfy Lipschitz conditions at n(p(c).c) /(c|c) ,, ^^^), does not if This combined with the (weak) monotonicity of the pricing rule implies that he does not wish to on the equilibrium path, where beliefs However, that c, = c'- (c' ,c') nor at any {p„(^c),c) , standard theorems are be constructed as the fixed point of a contraction mapping. Lxt Co be the spaceof continuous functions on [0,c*],and: C {p(-)GColp(c)e[c,p„(c)] - Denote by v(c) = ^^tt^ the hazard rate entering (Al). p(- ) Vc}. solves (A. 1) if (/1. and only if /(c)-n(p(c),c) 2) the function (.'1.3) obeys the differential equation: /'(c) = W^{p{c),c)p'{c)-D{p{c)) with terminal condition /(c') = n(c',c*) = 0. Integrating /(c) This integral is = Z)(p(x))e J' convergent at c", since /(c) p{x^>x < {' '' (A 4) ""'"'dx v{c)I{c)-D{p{c)) = back-wards: - J(p(-).c). implies: D{x)dx = {AA) S(c)-S(c*). (^.5) 43 We have transformed the differential equation (A,l) into an equiv-alent integral equation: By assumption n(-,c) is fixed point formulation (clearly / Pic) endowed with the sup norm: I iA.6) [0,n„(c)] ) sup |p(c)| pi - ( ) e [ c , n „ ( c ) ]) into [c.p„,{c)]. is: n(-.c)-'(J(p(-).c)). = mapping T:p(-)-* T p{- the Vc. quasi-concave, hence can be ins-erted from strictly Hence, with obvious notation, the We now show that n(p(c),c). - J(pi-).c) . where T p{c) Let p(- . e ) is (A.7) the r.h.s. of (A.7), is C; byconsiruction, 7p(c) a contraction e [c, p„(c)] on C Vc. , ctlO.c") Moreover, it is ne[0.n„(c)]. Tp{-) £ C . n(-,c)"'(n) easily verified that J{pi-).c)- I{c) Since Consider X>y denote , n(- .c)"'(A') = y- . n(- ,c)''(>) ^ < ^ definition of m and .X>Z>y. .\f • or z - n(- . n„(c).x- = is (Z) for all then continuous, hence We have: unbounded. For is all .V, )' e (y!.8) [0, n„(c)] .with Weclaim: ^ < . x-> p„(c). >-- = .V. < < Pm(.c)-z y<2< x<p„(c). x-y m p„(c)-x , ^ ^, (A-9) \ z> y such that . ^- 11^(2. c) . But, by (4), the , Finally, replace (A.10) in the M . Nexn, apply the first inequality in (A.9) with y-=.v: n„(c)-A- [p^ic)-x]'>^^ n„(c)-A' or p^ic)-x>^^^ (/1. 10) second part of (A.9), to obtain: ., ., n(-.c) '(A-)-n(-.c) '(y) = Therefore, Tp and : Inequality (A.9) then follows from = . x-y c)"' m J- . c. c'] . |n(-,c)-'(J(p(-).c)-n(-,c)-'(J(q(-).c))|. x-y M p„(c)-y — Indeed there exists Z clearly continuous in l/np(n(-,c)''.c). which c)"' has derivative x- jointly continuous in (c. fl), for all c e [c' now (p.q)£ C>^C and any c£[c' .c'] |7p(c)-7q(c)! Note that n(- is is sfJi X-y x--y<^ m vn^(c)-A' (^.ii) . 44 ./; v(y)rty i: \Tp(c)-Tq(c)\<^-^- [D(p(.v))-Z)(g(x-))]dx Vn„(c)-J(p(-).c) 'c' ,,. e , v(y)dy -J ' |p(x)-g(.v)|d>r <^.Ar^^ m , Vn^(c)-J(p(-).c) <lp-q|Thus T will be a contraction AVM m (^.12) Vn„(c)-J(p(-).c) for all c e if, [ , c * ] . - n„(c)-J(p(-).c)>M ic'-c-) Indeed.uniformcontinuitywill then imply that (A.13) holds with so that I 7p- 7g |< (J I p- g . I p(c)>c. Because, G n„(c)-S(c) is remains to show that p^'(c)> while /(c The I c) > by assumption case of the denominator in (Al) go left (4), , for some pe(0, 1) , , = n„(c)-s(c) + s(c-), increasing, (A.13) then holds by assumption (6). This concludes Pf{c) . Wc^[c',c"\. hence the derivative D(x)dx z)(p(x))dx>n„(c)-J^ '' the proof of the existence and uniqueness of It replaced by M/%j\i But note that we have, n„(c)-j(p(-).c) = n„(c)-j^ since M (A. 13) result, Pf'{c') to zero as c goes to c* . is For For ce [c',c*) T\{pp{c),c)> , by (A.5) by (A.1) more complicated because both all e > the numerator and , /(c--€|c--e)n(p,(c--e),c--e)-[l-F(c--e|c--e)]n^(p,(c--£).c--e)p/(c--e) = 0. (/1. 14) But, f(c*-e|c--e)-f(cn(Pf(c*).c-)-n(p,(c--e).c*-e) Since f(c*|c')=l and n(p,(c*), |c-) = [/(c- | c* ) + f ^Cc* c* )](-e) + o(e) . = [n^(p,(c*-e),c'-e)p,-(c--€)-^n,(p,(c*-e).c'-e)]e = [n^(c-|c-)p/(c--e)+n,(c-,c')]e + o(e) + o(e). c*) = 0, (A.14) becomes: {/(c- |c-)[-n^(c-lc*)p/(c--e)+D(c-)]-[/(c-|c-)+f2(cor | | c-)]n^(c* . c' )p, '(c* - e)>e = o(e) . 45 p/(c*-e)[2/(c* |c")+f2(c' |c*)]n^(c-,c')=/(c* c-)D(c-)+ | p/(c*-e) which means that the limit Pf'ic')- lim exists and, since o( 1 ) OpCc' (c')- Z)(c') c-O ^I /(c-|c-) ^' ^^ 2f(c-\c-)*F,{c-\c-) 2- This proves the result. Q.E.D. Proof of Lemma firm has Pz> p if and only if 2: Consumers observe p ,> p' Given i Thus it ^ p{p.c) = p C2< c' and p fic^) is Y „p p ( ^^ ( c ) . c ) < , , . . The the other this occurs first-order which p^(c)is the only ct c') But by the implicit function theorem, . . the second-order condition becomes Yp;.(p/^(c),c)>0. Y,(p.c) ^ {p .c) if implying that equilibrium profits (for -Vpe(Pf(c).c) Pf ic)= ^' H'^/p,(c).c) Given that p^' > only increasing in Cj for Cz ^ c' precisely the differential equation (A.1), to is only remains to show that are strictly quasiconcave in it come back given that they search, they will has probability /^(p}'(Pi)|c); hence the form of Pf(c2)>p, and condition for maximization solution. PriCi) < p' that : But, -D(p)[i-f(p;'(p)ic)]-n(p.c)f2(p;'(p)|c). = Ypc(PA(c).c) = -D-(p,(c))[l-/^(c|c)]*D(p,(c))-:^^^7^-n(p.c)-:^^^^-np(p(c).c)f2(clc). Pf The first two terms are positive. Pf (.c; Replacing Pr'(c) from (A.1), the sum of the (.c; last two terms has the sign of: -np(p,(c),c)[/,(cic)(l-f(c|c))*f,(cic)/(c|c)]>0. from assumption Proof of Lemma (4) and the We 3: first np(p,(c).c)>0. fact that find c' Q.E.D. the point of indifference between . p' and Pf(,c) (when c'<c'). by examining: 5(c)-n(p",c) We first show that if 5(c) = . then 6'(c) < n(p'.c) Then: l-^f(clc) = . n(p,(c).c)[l-f(c|c)]. Indeed, 5(c) = ifandonly (/1. 15) if, -!-^^^^i^n(p,(c),c)<n(p,(c).c). l-;f(c|c) (^.i6) ' 46 5'(c)--D(p ^(P.(c))[l-f(c|c)]-n/p,(c).c)[l-f(c|c)]p/(c) l--f(clc) ) -n(p*.c)-n(p,(c).c) [f(c\c)^F,(c\c)] -Dip l--f(c|c) ) -T^^iP .C)/(C|C) 2 Pf(c)-cJ n(p,(c).c)-in(p".c) f^Cdc). where we used both (A.1) and (A.16). Now p° < Pf(c) since n ( on that the first , • c ) increasing is 6(c*) = jn(p',c*) = If (i) therefore, also (by j(p'-c')D(p') p price for c > c' that c<c° it is . c * 6 , 2) to is its * ) Q.E.D. . This and also in that turn implies ' ( c ) < is c ° . has the sign of p' - c' , so . , p' all , Thus two cases can . Thus p' maximizes while arise: p to ' Pf{c) and, profits. preferred price 6(c)<0so those which do not induce search. By the firm would rather charge p'<p„(c) p^Cc) among those which induce search. Thus Pf(c) . Note Moreover, for c>c' ' . c°) > n(p;r(c'), c"), firms with cost in [c° ,c"] prefer charging is its among no search below increases in c n(p', has a unique zero c^£{c',c'). and a firm with cost above c^ prefers ) c>c' For ce[c' .c') continuous in > preferred price since there "" and p f{c) negative, always negative, we have showed that 6 is so , any price above p" 6(- since ; ( clearly will prefer charging Therefore, c c" . > p'<c*, charging Pf(c) to p' Lemma 2, Pf(c) c Lemma If (ii) p is l-^f(c°|c') -n(p,(c°).c')[l-f(c"|c-)]>0 p' = p„{c')> Pf{c') because . which contains , term can have at most one zero. Moreover, ) • 6(c°) = n(p'.c") p p„{c)] last . Therefore, 5( with [c, term above is also negative. Since the second term whenever 6(c) = any p < (A.16) and (2) imply that the . is the globally optimal p' than Pfic) and n(-,c) increases on [c, p^(c)] , . and also than Finally, a firm that the uniqueness of the solution c'^to 5(c) = Censures p'<p„(c) p and so does p' = Pf(c') , since so n(p°.c) and 6(c) p,( ) • is increase in increasing and does not ' or c ' . depend on , . 47 Proof of Proposition c] ,cl a for small ^ make . when a (14) holds !''„() if not monotonic on is because V„'(c")>0. interval, i.e., Wc£[c'.c'] for a < from a" K„(c) > K^ > V^(c)>V^ . of [c' all , c' ( • . c ' on o explicit, by denoting them as ' if and only decreases to c" p\<c' if if c>c". and only . In particular, J is defined by (see ) is the limit: strictly o" Let c> if Lemma 3, K^C ) to is some (maximal) ' and cl . r„(c") - c" e[c' = F^ this such that: .c^] for all a<a", namely c^- max{ce[c'.c"]lI'„(c)<o} to the cost The Then . c ' , and not above. Clearly, we shall assume that c'c<Cn(c') . c^ £ (^c' .c'] bounded away from zero on there exists a unique differential , equation (10) and i.e, its as o decreases p"- p„(^cl)< solution . Pf{) c' . do Lemma 3): i-|^(c:ic:) n(p,(cn.c:)[i-f(c:ic:)] ensures that a unique solution c\£{c'„.c') p„,(c") so by in closely the determination of c monotonic up it is . K„(c")-0. Since . n(p;.c:) otherwise ] monopoly pricing can be sustained up to zero, c (Recall that we examine more Moreover, l'„(c)>Oin [c',c*]. so not depend on a, while c n etc., . D(p„(c3))p„'(c2)f(c2|c')dc,-o. Let us turn next to the determination of c^ po- c' c' 3lc^e[c".c"]Vce[c'c"], I„,(c)>o Thus small enough, is f' •^ to , Recall that V^Cc')Even the dependence of c' etc. , To show that c We shall 4: c^ decreases to a limit el's. c~ exists). . n(p„(c).c')= W{pf{c').c'). which is As o decreases In fact c J to zero, p\ decreases remains bounded away from c", impossible since c' < Pf{c')< p^{c~) and quasiconcave. We are now ready to examine (14) for low values of o . First, for all c^t c\: X(c,.o)< [' Z)(p„(c2))p„-(c2)f(c2|c')dc,-o (,-1.17) c" SO x(ci.o) goes to zero (uniformly in cjuith o . On the other hand we show §(o)-min{^(o,c,).c,e[c:.c-]}>0 for o low enough. First note, that for all o<a"and c,>cJ>Co: that: (/1. 18) 48 I^(0.c,)-^(0.c,)|<|5(p:)-5(p;)|*|S(p;)-5(pJ)|+ |5(p:)-S(p;)|+2 1S(p:)-S(p')|. = SO ^(a.c^) converges to ^(0 c , if |(0)> , i.e., , uniformly in c ) g[co.c*] forall c, D(p,(c2))p/(Cj)dc2 / , as o goes (^.19) to zero. Therefore, by continuity (A. 18) will hold : ^(0.c,)-[S(p;)-S(p,(c^))]f(cS|c,)+J^'D(p,(C2))p/(c2)f(c2|c,)dc2>0. (A. 20) Since pS = P/^(Co)>Po''Pm(c'). c,>Co. requires The the first second term can only be zero term in (A.20) f (co c,) if 1 then can continuous and non-negative, is D(P/:(ci))>£)(c*)>0 by Lemma p,'(ci)>0, and by assumption ; for c 2 near c 1, is , be only D(.PF(c2))Pf'ic2)F(c2\ci), which = which by assumption zero identically zero on Proof of Proposition We consider conditional distributions 5: this , ] But . cannot be f (cj I Q.E.D. c,) satisfying assumptions (1),(2), and (4), for which: Fj " sup{f 2(^2 c,) c'<C2^c,<c*} . I is c co and (A.20) must therefore hold. Thus (A18) holds, which, with (A17), implies that for alow , I enough (below some o'e(0,o"]), y(C|)-o-=^(o,C|)-x(o.C|)>0,Vc,£[c°,c*]. and [ f(c, |c,)>0 ,so (4), integrand the if (4) small. More precisely, we consider a family of distributions indexed by continuously. For instance, ifc, = c + e + Yi .i' (i) This defines c 3; if p^{c°)< (ii) (iii) l^ ', m( c 1 ) c' , we can l^ ( c F(c 1 ' ) ) , p^(c')>c' if construct monopoly < a < 6 This will prove the theorem. , , it can be shown that /"g ^ — . pricing at p " p^(c) . is increasing. the equilibrium corresponds to either Proposition 2 or Proposition p f{-) and define c' £{c' ,c'] the equilibrium return to search at > fj low enough: the returns to search under . uniquely; is affects 1.2, where G is uniformly distributed on[-a,a]andY,,Y2 independently drawn from a uniform distribution on [-6,6]. with We show that for distributions for which F 2 some parameter which so that searching above p ^ is . Then for F 2 low enough, we show: p=p,:(c).c>c'. is optimal. increasing; : 49 As before with a c'f . c'f , , we shall make the dependence of c' c' . on e'c., . F explicit by denoting them as etc. Since, K„-(c,)-Z)(p„(c,))p„-(c,)f(c, |c,)+ Dip^(c,))p„-(c2)F,(c^\c,)dc^. r c' (i) will hold if for c all ,€( c ', c ] _ ^Z)(p„(c,))p„-(c,)^(c,|c,) F,< (yl.21) We can impose (A,21) directly because the r.h.s. is a continuous, positive function of c away from zero. Indeed, the /(c"|c')>0 limit r.h.s. at c, - c' . of (A.21) is strictly positive for c D(Pr,ic')) similarly the term in brackets For c^t c'f > Alternatively, using the convexity of ^ ,i>(P^(c-)) ^2^ where , is c " , . and therefore bounded and apphing L'Hopital's , S(p) (A.21) . is rule, it has implied by, Pm'(C|)f (c, |c,) min iA.22) Pm(c^)- p^ic-) c,c:c-.c-]L bounded away from zero on [c" . c " ] , due to (3) and L'Hopital's rule. we have: , K'(c,) = r 'D(p„,(c,))p„'(c,)f2(c,|c,)dc3 D(p,(c,))p/(c,)f(c,lc,)+ c' -[S(p;)-S(p=)]f3(c'Jc,)* [ ' Dip,ic,))p/{c,)F,(c,\c,)dc^ >Z?(p,(c,))p/(c,)f(c,|c,)-f,[S(p„(c-))-S(p„(c;))-S(p;)-S(p^)-S(pn-S(p,(c,))] >Z)(p,(c,))p/(c,)f(c,|c,)-f2[S(p„(c-))-S(c,)]. But, ., /(c,ic,) , l-f(c, |c,) n(p,(c,).c,) /(c,ic,) n„(p,(c,).c,) l-f(c,|c,) /(c,|c,) n(p„(c,).c,) Dipfic,)) n„(c,) l-f(c,|c,) D(p,(c,)) SO; l"(c,)> Thus, (ii) will hold /(Ci , I Ci)f (C| ' I ' , Ci) " - n„(C|)-f,[5(p )-S(c,)]. if: F 2- min ce c,.c n„(c-) /(c|c)f(c|c) l-f(clc) . S(p„(c-))-5(c-) (^.23) . 50 By assumption if we (4), the but both sides of (A.23) involve the distribution F. Nonetheless, r.h.s. is strictly positive, assumptions to zero, so (1), (2) does the and (4), and such that F*" of (A.23) whereas the l.h.s. Finally, . Therefore, (A.23) holds for by (14), (iii) will hold ; in the r.h.s. satisfy then as K goes converges to (by L'Hopital's rule): f\c\c)F'(c\c) ^^ >0, \-F\c\c) cclCc.c > c~ \ e[0,X] which , minimum . c'f° , depend continuously on \ with F° - j'" . min since F'"(c2\c,) consider a one parameter family of conditional distributions F'' with X small enough. if: l^(cn = [S(p;)-S(p^)]F(c'Jc=)> [ '/)(p„(c,))p„-(c,)[f(c,|c;)-f(c2|c=)]dc2. for which it suffices that: F,< r^^-^ Again, this condition involves as to find c } small This X the is F on both sides; Sip-)-S(p'f) moreover, Nonetheless, given a family of distribution l.h.s. of (A.24) v.ill be small, while the because the equality (12) defining Cf = p' = p^f; but c = Pf(.c) excluded by focusing (as we is c^ only possible at r.h.s. it requires that the function P/r( f will '' ( c 2 1 have) on the case where c' . c , ) Thus p^o ) be computed, so with the properties described above, for be close to the always requires c= (/1. 24) . c}-Cf Pf> = p):o finite p'f. value corresponding to f"° unless n{p' .Cf) would require p^o = c' /^D(p„(c2))p„'(c^)f°(c2lc)dc2>o. = , i.e., which can be Q.E.D. 51 References Barro, R. (1976), "Rational Expectation and the Role of Monetary Policy," Journal of Monetary Economics , 2, 1-32. Benabou, R. (1988), "Search, Price Setting and Inflation," Review of Economic Studies , 55, 353-376. Benabou, R. (1989), "Some Welfare Effects of Inflation," mimeo, MIT, September. Cukierman, A. (1979), "The Relationship Between Relative Prices and the General Price Level: A Suggested Interpretation," American Economic Review, 69, 444-447. Cukierman, A. (1982), "Relative Price the Price System," Journal of Variability, Inflation, Monetary Economics, Cukierman, A. (1983), "Relative Price Variability and Results," Carnegie-Rochester Series Cukierman, A. and P. Variability of the Dana, J. (1989), on Public 9, and the AJlocative Efficiency of 131-162. Inflation: Policy, Survey and Further Autumn. Wachtel (1979), "Differential Inflationary Expectations and the Rate of Inflation," American Economic Re\iew "A Search-Theoretic Model of Price DeGroot, M. (1970), Optimal Statistical Decisions . Stickiness," New York: Papers on Economic 69, 595-609. McGraw Hill. and Inflation," Brookings Activir\\ 2, 381-441. "The Benefits of Price Federal Reserve Bank of Kansas Fudenberg, D. and Tirole, . mimeo, Dartmouth College. Fischer, S. (1981), "Relative Shocks, Relative Price Variability, Fischer, S. (1984), A J. Stabilit)'," in Cit}', Price Stability and Public Policy , The 33-49. (1989) "Perfect Bayesian and Sequential Equilibrium," Harvard University Discussion Paper No. 1436, May. Hercowitz, Z. (1981), "Money and the Dispersion of Relative Prices," Journal of Political Economy . 89, 328-356. 52 Lucas, R. (1973), "Some Economic Review , International Evidence on Output-Inflation Tradeoffs," American 63, 326-334. Milgrom, P. and Weber, R. (1982) "A Theory of Auctions and Competitive Bidding," ' Econometrica . vol. 50, No. 5, 1089-1122. Okun, A. (1981), Prices and Quantities: Brookings A Macroeconomic Analysis . Washington, D.C.: The Institution, 134-181. Pagan, A., A. Hall, and P. Trivedi (1983), "Assessing the Variability of Inflation," Review of Economic Studies , 50, 585-596. Parks, R. (1978), "Inflation and Relative Price Variability," Journal of Political Economy , 86, 79-85. Reinganum, J. "A Simple Model of Equilibrium (1979), Economy . Price Dispersion," Journal of Political 87, 851-858. Rosenfield, D. and R. Shapiro, (1981), "Optimal Adaptive Search," Journal of Theory Rothschild, . 25, 1-20. M. (1973), "Searching for the Lowest price Unknown," Journal of Political Economy, Taylor, J. (1981), "On the Relation Inflation Rate," in K. Series Economic on Public the Distribution of Prices is 81, 689-711. Between the Variability of Inflation Brunner and A. Metzler, Policy: When eds., and the Average Carnegie -Rochester Conference The Costs and Consequences of Inflation 15, 57-85. , Vining, D. and T. Elwertowski (1976), "The Relationship between Relative Prices and the General Price Level," American Economic Review, 1181 021 66, 699-708. Date Due MIT LIBRARIES 3 TOflO 00bSa3T0 7