# Integrating Sticky Prices and Sticky Information

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<ul><li><p>INTEGRATING STICKY PRICES AND STICKY INFORMATION</p><p>Bill Dupor, Tomiyuki Kitamura, and Takayuki Tsuruga*</p><p>AbstractUnderstanding the relationship between nominal and real vari-ables, most notably inflation and cyclical output, is one of the fundamentalquestions of economics. Toward this understanding, we develop a modelthat integrates sticky prices and sticky informationa dual-stickinessmodel. We find that both rigidities are present in U.S. data. We also showthat the dual-stickiness models closest competitor is the hybrid NewKeynesian model. For both models, current inflation depends in part onlast periods inflation. The former model achieves this dependence endo-genously through the interaction of the two rigidities rather than throughbackward-looking behavior. U.S. data support the dual-stickiness modelover the hybrid model because lagged expectations terms appear in theformers inflation Euler equation. Finally, we show that it is quantitativelyimportant to distinguish between the two by simulating a dynamic equi-librium model under each of the two inflation equations.</p><p>I. Introduction</p><p>THE interaction of real activity and inflation is a corner-stone issue of macroeconomics. As with every othermajor macro question, it has been placed under the lens ofrational expectations and microfounded dynamics in recentdecades. Approximately ten years ago, efforts to estimateexisting models of price rigidity intensified.1 These effortscenter on the aggregate Euler equation from a rationalexpectations sticky price model, often called the New Key-nesian Phillips curve (NKPC). A number of authors haveargued that the NKPC is empirically deficient. Fuhrer andMoore (1995) find that inflation is more persistent than themodels imply. Mankiw (2001) points out that the NKPC isinconsistent with the stylized fact of inflation inertia.Broadly speaking, experts have fallen into one of two campsin explaining the observed inflation inertia.</p><p>The first group contends that the original mechanism islargely successful once minor adjustments are made. Galand Gertler (1999) and Gal, Gertler, and Lopez-Salido(2001) state that the so-called hybrid NKPC, which assumesthe presence of backward-looking firms, matches U.S. andEuropean data very well. Another adjustment toward im-provement of the NKPC is to introduce real rigidities toreduce the sensitivity of prices to real marginal cost.2 Whilereal rigidities are useful in obtaining estimated frequencies</p><p>of price changes consistent with micro evidence, mostempirical studies continue to find that backward-lookingfirms play an important role in accounting for the observedinflation inertia. Thus, the hybrid NKPC has been stronglysupported by the data, even though the assumption ofbackward-looking firms might be unappealing from thetheoretical viewpoint.</p><p>The second group advocates a major overhaul of theNKPC to account for inflation inertia. A few of the alterna-tives include imperfect common knowledge (Woodford,2003) and sticky information (Mankiw & Reis, 2002; Reis,2006). In Mankiw and Reis, only a fraction of firms chooseprices attentively with currently available information.Their sticky information economy replicates inflation inertiaextremely well from the theoretical viewpoint. As such, theypropose to replace sticky prices with sticky information.Unfortunately, however, recent empirical studies find thatempirical comparisons favor the sticky price model ratherthan the sticky information model.3</p><p>This paper proposes an alternative model for explaininginflation inertia. We develop a dual-stickiness model thatintegrates price stickiness and information stickiness. In ourmodel, each firm has two adjustment probabilities everyperiod: a chance to reset its price and an independentlydistributed chance to update its information. Among firmsthat reset their prices, a fraction of the firms choose theirnominal prices with new information, and the remainingdetermine prices with old information.</p><p>Remarkably, the dual-stickiness models log-linearizedinflation equation has a lagged inflation term. It endog-enously arises through the integration of the two types ofstickiness. First, price stickiness makes current inflationproportional to the average of newly set relative prices.Then information stickiness makes some of todays price-setting firms behave similar to some of the last periodsprice-setting firms, creating dependence of the average ofnewly set relative prices on its own lag. The interaction ofthe two generates a lagged inflation term. In this sense, weargue that our dual-stickiness model provides a more plau-sible microfoundation for inflation inertia than the hybridmodel, which obtains a lagged inflation term from exog-enously assumed backward-looking firms.</p><p>The models log-linearized inflation equation also hascurrent and lagged expectations terms: forecasts of currentand future marginal cost based on current information andforecasts of current and future marginal cost growth andinflation based on old information. The lagged expectationsterm is empirically important in distinguishing the dual-stickiness from the hybrid model, because only the formerhas a lagged expectations term.</p><p>Received for publication October 11, 2006. Revision accepted forpublication August 22, 2008.</p><p>* Dupor: Ohio State University; Kitamura: Bank of Japan; Tsuruga:Kansai University.</p><p>We thank two anonymous referees, Paul Evans, Oleg Korenok, EijiOkano, Simon Price, Ricardo Reis, John M. Roberts, Julio Rotemberg,Mototsugu Shintani, Peter Sinclair, and seminar participants at variousinstitutions and conferences for helpful comments and discussions. Anearlier version of this paper was circulated as Do Sticky Prices Need toBe Replaced with Sticky Information? The views expressed in the paperare those of the authors and are not reflective of those of the Bank ofJapan.</p><p>1 The existing models at that time included Calvo (1983), Rotemberg(1982), and Taylor (1980).</p><p>2 Sbordone (2002) assumes firm-specific marginal cost under the as-sumption of inflexible capital movement. Christiano, Eichenbaum, andEvans (2005) and Tsuruga (2007) emphasize the importance of variablecapital utilization (and nominal wage rigidities) in reducing the sensitivityof prices to real marginal cost. 3 Examples include Coibion (2010), Kiley (2007), and Laforte (2007).</p><p>The Review of Economics and Statistics, August 2010, 92(3): 657669</p></li><li><p>We take the model to U.S. data and simultaneouslyestimate the importance of price and information stickinessin a nested framework.4 We find that both rigidities arepresent in U.S. data. Hence, our empirical results contra-vene a wholesale replacement of sticky prices with stickyinformation. Instead, our results suggest that the integrationof price and information stickiness is extremely helpful inaccounting for U.S. inflation dynamics.</p><p>Our benchmark estimates are that in each quarter, 14% offirms reset prices and 42% update information. When weallow for a typical degree of strategic complementarity,these probabilities become 28% and 70%, respectively. Wealso measure the relative importance of each nominal rigid-ity and find that sticky prices are more important than stickyinformation in fitting U.S. inflation dynamics.</p><p>We then empirically compare the dual-stickiness andhybrid models. To distinguish the dual-stickiness from hy-brid model, we first focus on the correlations betweeninflation and lagged expectations, which feature the role ofsticky information. While both models account for inflationquite well in terms of goodness of fit, we find that laggedexpectations matter for inflation in a statistically significantway. Next, we present a further generalized specificationthat nests both the dual-stickiness and hybrid models. Wefind that the data support the dual-stickiness model over thehybrid model and thus argue that the latter may be mis-specified in explaining U.S. inflation dynamics. Finally,using a simple general equilibrium analysis, we show thatimpulse responses to a cost-push shock can be qualitativelydifferent between the two pricing frictions. This implies thatit is important to distinguish the two in understandingmacroeconomic dynamics.</p><p>The findings of this paper are broadly in line with thoseof recent papers by Klenow and Willis (2007) and Knotek(2006). They introduce sticky information into a state-dependent pricing model in general equilibrium and showcalibrations and estimation results emphasizing the role ofsticky information. Our time-dependent approach has anadvantage over the state-dependent approach in that we caneasily include dual-stickiness pricing into a wide class ofdynamic general equilibrium models with many state vari-ables.</p><p>An outline of the rest of the paper is as follows. SectionII describes the dual-stickiness and hybrid models. SectionsIII and IV present our empirical method and findings.Section V shows general equilibrium comparisons betweenthe two models. The final section concludes.</p><p>II. Two Pricing Problems</p><p>This section describes a firms problem under two differ-ent sets of frictions. After aggregation, we characterize twoinflation equationsone for each set of frictions. The dual-</p><p>stickiness model has both sticky prices and sticky informa-tion. This model also nests both the pure sticky price andpure sticky information cases. Our second set of frictions issticky prices and backward-looking firms, that is, the hybridmodel. This latter model has become a workhorse in em-pirical monetary economics.</p><p>A. The Dual-Stickiness Model</p><p>Consider a firm that is the monopolist producer and sellerof a particular good. The firm infrequently changes itsnominal price and also infrequently updates its information.With probability 1 , the firm may change its price;otherwise, its current period price equals its previous periodprice. With probability 1 , the firm updates its informa-tion set to include all current variables; otherwise, the firmsinformation is the same as its previous periods information.For tractability, these two random eventsthe opportunitiesto change a price and to update informationare uncorre-lated over time and with each other.</p><p>The economy is made up of the above types of firms witha measure of one, each producing and selling a distinctgood. Each faces the above probabilities of price changesand information updates.</p><p>We are interested in the behavior of inflation in thiseconomy. Let us define two nominal price indices. First, ptdenotes the log aggregate nominal price level in period t.Second, qt is a nominal price index for all newly set pricesin period t. We will say more about qt below.</p><p>Because a measure 1 of firms resets its price in eachperiod,</p><p>pt pt1 1 qt.</p><p>Or equivalently, subtracting pt from both sides and rearrang-ing yield</p><p>t 1 </p><p>qt pt, (1)</p><p>where t is inflation rate. Intuitively, equation (1) states thatinflation is positive when the newly set prices are higherthan the overall price level. It also states that inflation isproportional to newly set relative prices qt pt. Figure 1shows this proportionality diagrammatically. Note thatpt1 pt is the average relative prices for firms that are notallowed to change prices. Because the weighted sum of alllog relative prices must be 0 by definition, the two shadedareas in the figure must be equal.</p><p>Due to the sticky price assumption, a firm with zeroperiod old information (that is, current information) and theability to change its price chooses</p><p>ptf 1 </p><p>j0</p><p>jEtmctjn , (2)4 Note that our nested framework contrasts with the previous model</p><p>selection studies by Kiley (2007), Korenok (2008), and Laforte (2007).</p><p>THE REVIEW OF ECONOMICS AND STATISTICS658</p></li><li><p>where ptf is the full information optimal price and mct</p><p>n isnominal marginal cost in period t.5 Intuitively, equation (2)states that the firm sets its nominal price to the weightedaverage of current and future nominal marginal costs. Thisdecision is forward looking because of infrequent opportu-nities for price changes.</p><p>The decision of a price-resetting firm in period t with oneperiod old information is similar, except that this firm isrestricted to conditioning its optimal price on Et1, in-stead of Et. Newly set prices based on older vintages ofinformation are similarly restricted.</p><p>Next, consider qt, the nominal price index for newly setprices. There will be newly set prices in period t based oninformation sets of various vintages: current, one period old,two period old, and so on. Thus, given the probability ofinformation updating 1 , qt is given by</p><p>qt 1 k0</p><p>kEtk ptf. (3)</p><p>Thus, the formulation of the price index is identical to thesticky information model by Mankiw and Reis (2002),except that each individual price is determined in a forward-looking manner.</p><p>This equation can be rewritten as a first-order differenceequation. Using the fact that pt</p><p>f ptf pt1</p><p>f ,</p><p>qt qt1 1 ptf</p><p>(4)</p><p>1 k0</p><p>kEtk1ptf.</p><p>The intuition behind this structure is that some firms con-tinue to hold the same information between periods due toinformation stickiness, and so a similarity arises in thenewly set prices between two periods. To explore theintuition further, suppose that prices were initially stabilizedat 0 and that a positive shock occurs at period 0. The leftdiagram of figure 2 depicts a hypothetical path of qt as a</p><p>thick line. In period t 1, some firms are inattentive to theshock. They set prices to 0 since they do not know that theshock occurred. In this sense, they stick to the initial state,and this stickiness is depicted in the diagram as an arrowmoving from qt1 to q1. In the next period t, some firmsremain inattentive to the shock (with a probability). They settheir prices to 0 by sticking to the initial state, as the arrowmoving from qt to q1 indicates. As a result of the commonstickiness to the initial state, persistence of qt arises endo-genously, as the dotted arrow in the diagram shows.</p><p>The persistence of qt is carried over to its relative levelqt pt. Using an identity pt pt1 t (1 ) pt, we can express qt pt as a first-order differenceequation of the form</p><p>qt pt qt1 pt1 t 1 ptf pt</p><p>(5)</p><p>1 k0</p><p>kEtk1ptf.</p><p>Note that qt pt is more persistent as increases.Combining equations (2) and (5) with equation (1), we</p><p>can derive</p><p>t Dt1 1</p><p>D1 j0</p><p>jEtmctjn pt</p><p>2D1 </p><p>k0</p><p>k (6)</p><p>1 j0</p><p>jEtk1mctj tj,</p><p>where D /( ), 1D (1 )(1 </p><p>)/( ), and 2D (1 )/( ).</p><p>Also, mct is real marginal cost given by mct mctn pt.</p><p>In the inflation equation (6), lagged inflation appearsendogenously. As equation (1) suggests, the sticky priceassumption generates a one-to-one relationship between tand qt pt. As equation (5) suggests, the sticky informa-tion assumption generates persistent dynamics of qt pt.This newly reset relative price persistence is transformedinto inflation persistence through the one-to-one relation-ship. Therefore, the combination of price and informationstickiness endogenously generates lagged inflation in theinflation equation.</p><p>Besides the lagged inflation, there are two other terms inequation (6). The second term of...</p></li></ul>

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