Inventories, Markups, and Real Rigidities in Menu Cost Models

Transcription

1 Inventorie, Markup, and Real Rigiditie in Menu Cot Model Olekiy Kryvtov Virgiliu Midrigan Bank of Canada New York Univerity Augut 2010 Abtract A growing conenu in New Keyneian macroeconomic i that nominal cot rigiditie, rather than countercyclical markup account for the bulk of the real effect of monetary policy hock. We reviit thee concluion uing theory and direct evidence on quantitie. We tudy an economy with nominal rigiditie in which good are torable. Theory predict that if cot of production are ticky and markup do not vary much in repone to, ay, expanionary monetary policy, firm react by exceively accumulating inventorie in anticipation of future cot increae. In contrat, in the data inventorie are fairly contant over the cycle and in repone to change in monetary policy. We how that markup mut decline ufficiently in time of a monetary expanion in order to reduce firm incentive to hold inventorie and thu bring the model inventory prediction in line with the data. Verion of the model conitent with the dynamic of inventorie in the data imply that countercyclical markup account for a izable 50-80%) fraction of the repone of real variable to monetary hock. JEL claification: E31, F12. Keyword: Inventorie, markup, cot. The view expreed here are thoe of the author and do not reflect thoe of the Bank of Canada. We thank George Aleandria, Ariel Burtein, Mike Dotey, Mark Getler, Boyan Jovanovic, and Huntley Schaller for ueful dicuion, a well a eminar participant at the NBER-SITE Price Dynamic Conference at Stanford, Duke Macroeconomic Jamboree, Québec à Montréal, Chicago, NYU, Yale, Federal Reerve Board, the Federal Reerve Bank of Atlanta, Philadelphia, Richmond and St. Loui, a well a at the Bank of Canada. Kryvtov: okryvtov@bank-banque-canada.ca. Midrigan: virgiliu.midrigan@nyu.edu.

2 1. Introduction The prediction of New Keyneian ticky price model, widely ued for buine cycle and policy analyi, are critically determined by the aumption reearcher make about the behavior of cot. If the real marginal cot of production repond trongly to monetary policy hock, thee model predict that uch hock have mall and hort-lived real effect, a in the work of Chari, Kehoe and McGrattan 2000). In contrat, if the real marginal cot repond lowly to monetary policy hock, a in the work of Woodford 2002), Chritiano, Eichenbaum and Evan 2005) and Dotey and King 2006), uch hock have much larger and more peritent real effect. Indeed, much of the recent debate about the effect of monetary policy and nominal rigiditie i really a debate about cot. Our paper reviit thi debate. We ak, How doe the cot of production repond to monetary policy hock? Since real marginal cot and markup are inverely related, an alternative way to poit our quetion i, How do markup repond to monetary policy hock? Are the real effect of monetary policy hock motly accounted for by nominal cot rigiditie or rather, by countercyclical variation in markup? We anwer thi quetion by tudying data on inventorie through the len of a New Keyneian model in which we embed a motive for inventory accumulation. We focu on inventorie becaue theory predict a tight relationhip between price, cot and inventorie, a forcefully argued by Bil and Kahn 2000). If good are torable, firm price are determined by the marginal valuation of inventorie. In turn, firm produce to the point at which the marginal valuation of inventorie i equal to the marginal cot. We exploit thee prediction of the theory to how that countercyclical markup account for a izable fraction of the real effect of monetary policy hock in verion of the model that replicate the behavior of inventorie in the data. Hence, a Bil and Kahn 2000) do, albeit uing a different methodology and for monetary-driven buine cycle, we find that markup are trongly countercyclical. Our reult tand in harp contrat to a number of finding in exiting work. A growing conenu in New Keyneian macroeconomic i that ticky nominal cot, rather than variable markup, account for the bulk of the repone of real activity to monetary policy hock. Two obervation have led reearcher to thi concluion. Firt, tudie of micro-price data find that input cot change infrequently and do not react well to nominal hock, while price tend to 1

3 repond quickly to change in cot 1. Second, the obervation that price change frequently in the data Bil and Klenow 2005) and Klenow and Kryvtov 2008)), ha led reearcher to conclude that wage rigiditie and tickine in input cot, rather than nominal price rigiditie, mut be the dominant ource of monetary non-neutrality ee for example Chritiano, Eichenbaum and Evan 2005)). Our paper reviit thee concluion. Although input cot are indeed ticky in the data, the marginal cot of production need not necearily be ticky a well. Rotemberg and Woodford 1999) illutrate, for example, how inference about marginal cot from data on input price i enitive to aumption about the production technology. Oberved factor price do not fully reflect production cot in the preence of factor adjutment friction or decreaing return. Moreover, a Stigler and Kindahl 1970), Barro 1977), and Hall 2006) have argued, factor price in long-term relationhip are not necearily allocative. Contract that tipulate quantity contraint, non-linear price chedule or other implicit arrangement may prevent buyer from taking advantage of, ay, a monetary expanion, by purchaing input at the cheaper real input price. Exiting New Keyneian model abtract from thee conideration. Intead, thee model aume that firm can eaily alter the cale of production in repone to nominal hock by varying the work-week of capital and labor and by freely hiring labor and intermediate input at preet wage and input price. Indeed, Dotey and King 2006) ue the term real flexibilitie to characterize the aumption made in thi cla of model. Our tarting point i the obervation that the debate about cot i eentially a debate about quantitie, that i, about the ability of firm to collectively increae the cale of production during expanion. Our approach i therefore to ue data on quantitie, and in particular data on the tock of inventorie, in order to learn about cot. Theory predict that if the nominal cot of production are indeed ticky and markup do not vary much, firm would take advantage of the tickine in cot and rapidly accumulate inventorie after a, ay, expanionary monetary hock when cot are low and expected to increae. In contrat, in the data inventorie react lowly to uch hock. Hence, cot of production mut be fairly enitive to monetary policy hock for the model to replicate 1 See the work of Goldberg and Hellertein 2008), Nakamura and Zerom 2010), Eichenbaum, Jaimovich and Rebelo 2010), Gopinath and Itkhoki 2010). See alo Nekarda and Ramey 2009) who argue that markup are, in fact, procyclical. 2

4 the luggih dynamic of inventorie in the data. We thu find an important role for variation in markup in accounting for the behavior of conumer price. We begin our analyi by reviewing everal well-known fact about inventorie 2. In the data, inventorie are procyclical, but much le volatile then ale. The aggregate US tock of inventorie increae by about 0.16% for every 1% increae in ale during a buine cycle expanion. We reach a imilar concluion when conditioning fluctuation on identified meaure of monetary policy hock. In repone to an expanionary monetary policy hock, the tock of inventorie increae by about 0.33% for every 1% increae in ale. Hence, the aggregate tock of inventorie i relatively ticky and the aggregate inventory-ale ratio i countercyclical. We then build a model in which nominal price and wage change infrequently and firm hold inventorie. We conduct our baeline analyi uing a model in which inventorie arie due to a precautionary tockout-avoidance motive. We then extend the model to allow for non-convexitie in the form of fixed ordering cot that give rie to S, ) rule for inventory adjutment and how that our reult are robut to thi modification. We ue our model economy to tudy how the repone of inventorie to monetary policy hock depend on the aumption we make about the nature of cot. A key prediction of the model i that the tock of inventorie firm hold i very enitive to change in production cot and omewhat le enitive to change in markup. Thi feature, hared by the model with and without non-convexitie, i an outcome of the fact that in the model, a in the data, the cot of carrying inventorie i fairly low. The low cot of carrying inventorie make it eay for firm to ubtitute intertemporally by producing and toring good when production cot are relatively low and drawing down the tock of inventorie when production cot are relatively high. We briefly ummarize our finding. We firt tudy a verion of our model with nominal wage tickine and no price rigiditie, that i, in which markup are contant. We how that thi model account extremely poorly for the dynamic of inventorie in the data. Thi i true regardle of whether labor i the only factor of production and hence the marginal cot i proportional to the nominal wage, or whether we introduce capital or firm-level decreaing return that render marginal cot more volatile than wage. In all thee variation of the model inventorie increae much more 2 See, for example, Ramey and Wet 1999) and Bil and Khan 2000). 3

5 trongly in repone to an expanionary monetary hock then they do in the data. The reaon i that production cot are expected to increae after a monetary expanion a more and more union reet their nominal wage, thu making it optimal for firm to accumulate inventorie in anticipation of future cot increae. We then introduce nominal price rigiditie in addition to nominal wage tickine. Price rigiditie are important for our analyi ince they imply that markup decline during boom and thu reduce the firm incentive to hold inventorie. We how that thi verion of the model can indeed account for the dynamic of inventorie in the data, a long a production cot are ufficiently reponive to monetary hock, due to ufficiently trong diminihing return to labor. Production cot mut be ufficiently reponive to monetary hock in order to reduce the intertemporal ubtitution motive. Moreover, when cot are volatile, price rigiditie generate trongly countercyclical markup and further reduce the incentive to hold inventorie. Overall, we find that verion of our model that account for the dynamic of inventorie in the data imply that countercyclical variation in markup account for 50-80% of the repone of real variable to monetary policy hock. Thi tand in harp contrat to the finding of Chritiano, Eichenbaum and Evan 2005) who etimate parameter value that imply that markup play eentially no role in accounting for the real effect of monetary hock. Our work i related to a number of recent paper that tudy the behavior of inventorie, cot and markup over the buine cycle. Our tarting point i the obervation of Bil and Kahn 2000) that inventorie are cloely linked to markup and cot. The main difference between our work and that of Bil and Kahn i that they ue data on input price directly, together with a partial equilibrium model of inventorie, in order to meaure marginal cot. They find that the growth rate of marginal cot i acyclical and hence the intertemporal ubtitution motive i weak in the data. They therefore conclude that markup mut be countercyclical for the model to account for the fact that the inventory-ale ratio i countercyclical in the data. Khan and Thoma 2007) have recently argued that a countercyclical inventory-ale ratio i not necearily evidence of countercyclical markup. They tudy the dynamic of inventorie in a general equilibrium model driven by technology hock. Such a model account well for the behavior of inventorie in the data, depite the fact that markup are contant in their economy. Khan and Thoma 2007) how that general equilibrium conideration, and in particular capital 4

6 accumulation, are critical to thi reult. Diminihing return to labor low the repone of marginal cot to a technology hock and hence the incentive for inventory accumulation 3. A Khan and Thoma 2007) do, we explicitly tudy the dynamic of inventorie in a general equilibrium etting and find an important role for diminihing return to variable factor in accounting for the inventory fact. While their focu i on technology hock, our i on monetary hock in an economy with nominal rigiditie. We find that in uch an economy countercyclical markup play an important role: abent markup variation the model prediction are groly at odd with the data. The difference in our reult tem from the pecial nature of monetary hock in driving fluctuation in output. Unlike technology hock, monetary hock only affect real activity if nominal price do not change immediately with change in monetary policy. Since price are equal to a markup time cot, monetary hock only affect output if either i) markup vary or ii) nominal cot are ticky and do not react immediately to change in monetary policy. Hence, if markup are contant, monetary policy hock can only generate real effect if nominal cot are ticky. Thi, however, give rie to trong variability in inventorie due to intertemporal ubtitution in production which i at odd with the data. Alo related to our analyi i a paper by Jung and Yun 2005) who, a we do, tudy a ticky price model with inventorie. They find that high rate of depreciation and/or convex cot of deviating from a target inventory-ale ratio are neceary to reconcile the model prediction with the data 4. Thee two feature hut down the intertemporal ubtitution motive that i at the heart of our analyi. We argue below that uch feature are at odd with the micro data, ince inventoryale ratio are, in fact, very volatile at the firm level. In contrat, we preent evidence that firm-level decreaing return of the type that allow our model to match the aggregate behavior of inventorie alo allow the model to match alient feature of the micro-data regarding the comovement of price, order and inventorie. Finally, our work i cloely related to the quantitative tudie of Klenow and Willi 2006) and Burtein and Hellwig 2007) who alo meaure the trength of real rigiditie uing theory and 3 See alo Wen 2008) who tudie a tockout-avoidance model of inventorie and alo find that a real buine cycle model account well for the inventory fact. 4 See alo Chang, Horntein and Sarte 2007) who tudy the repone to a productivity hock in a ticky price model with inventorie. 5

7 micro-price data. Thee reearcher focu on an alternative type of real rigidity 5, in the form of trategic complementaritie in price etting, and find weak evidence of uch complementaritie. 2. Data In thi ection we review everal alient fact regarding the cyclical behavior of inventorie. Thee fact are well-known from earlier work 6. We dicu them briefly for completene, a they are central to our quantitative analyi below. We ue data from the Bureau of Economic Analyi NIPA) on monthly final ale, inventorie, and inventory-ale ratio for the Manufacturing and Trade ector from January 1967 to December Thee two ector of the economy account for mot 85%) of the U.S. inventory tock; the ret of the tock i in mining, utilitie, and contruction. All erie are real. Our meaure of ale are real final dometic ale. Production i defined a the um of final ale and the change in the end-of-period inventory tock. We contruct the inventory-ale ratio a the ratio of the end-of-period inventory tock to final ale in that period. When reporting unconditional buine cycle moment, we HP filter all erie with a moothing parameter equal to Below we alo ue a meaure of identified monetary policy hock to report tatitic conditional on monetary policy diturbance. Panel A of Figure 1 preent the time-erie of ale and the inventory-ale ratio for Manufacturing and Trade. The Figure how that the two erie are trongly negatively correlated and are almot equally volatile. Every receion i aociated with a decline in ale and a imilarly-ized increae in the inventory-ale ratio. Likewie, every expanion i aociated with an increae in ale and a decline in the inventory-ale ratio of a imilar magnitude. Table 1 quantifie what i evident in the Figure. Panel A report unconditional tatitic for thee erie. We focu on the erie for the entire Manufacturing and Trade ector and briefly dicu the Retail ector to gauge the robutne of thee fact. Notice in the firt column of Panel A that the correlation between the inventory-ale ratio 5 See Ball and Romer 1990). 6 Ramey and Wet 1999), Bil and Kahn 2000). 7 The Bureau of Economic Analyi ue an inventory valuation adjutment to revalue inventory holding reported by variou companie uing potentially different accounting method) to replacement cot. Thee adjutment are baed on urvey that report the accounting valuation ued in an indutry and from information on how long good are held in inventorie. See Ribarky 2004). 6

8 and ale for the entire Manufacturing and Trade ector i equal to The tandard deviation of the inventory-ale ratio i almot a large 1.03 time larger) a the tandard deviation of ale. Conequently, the elaticity of the inventory-ale ratio with repect to ale i equal to In other word, for every 1% increae in ale at buine cycle frequencie, the inventory-to-ale ratio decline by about 0.84%. The tock of inventorie i thu fairly contant over the cycle, increaing by only 0.16% = ) for every 1% increae in ale. Thi feature of the data may eem to contradict the well-known fact that inventory invetment i trongly procyclical and account for a izable proportion of the volatility of GDP. 9 There i, in fact, no contradiction, ince inventory invetment i mall relative to the entire tock of inventorie: monthly inventory invetment i equal to 0.22% of the inventory tock in Manufacturing and Trade and 0.29% of the inventory tock in Retail. To ee thi, we alo report the fact on inventory invetment. We find it ueful to do o by exploiting the following accounting identity: Y t = S t + I t where Y t i production, S t are ale and I t i inventory invetment. For a meaure of how volatile inventory invetment i, we compare the tandard deviation of production to that of ale both expreed a log-deviation from an HP trend). Notice in Table 1 that production and ale are trongly correlated and that production i 1.12 time more volatile than ale. We will ue thi fact, in addition to the fact on the tock of inventorie, in order to evaluate the model. The other column of Table 1 preent everal additional robutne check. We note that the fact above hold if we focu eparately on the Retail ector: the elaticity of inventorie to ale i equal to 0.24 and production i 1.14 a volatile a ale 10. Thee fact alo hold conditional on meaure of monetary policy hock. To ee thi we project the data erie on current and 36 lag of Chritiano, Eichenbaum and Evan 2002) meaure of monetary policy hock and recompute 8 Thi elaticity i defined a the product of the correlation and the ratio of the tandard deviation, or equivalently, a the lope coefficient in a regreion of log inventory-ale ratio on log ale. 9 See, e.g. Ramey and Wet 1999). 10 Iachoviello, Schiantarelli and Schuh 2007) report that the inventory-ale ratio in Retail i acyclical a they find a low correlation between the inventory-ale ratio in Retail and aggregate GDP. Their reult are conitent with our, ince at the monthly frequency aggregate GDP i imperfectly correlated with Retail ale the correlation of 0.10) and reinforce our concluion that the tock of inventorie i fairly contant over the cycle. 7

9 thee tatitic. We report the reulting erie in Panel B of Figure 1. Although monetary hock account for a mall fraction of the buine cycle the tandard deviation of thee erie i about half a large when conditioning on meaure of monetary hock), the main pattern i evident in thi Panel a well. In particular, we again find that the inventory-ale ratio i countercyclical. A Table 1 how, the elaticity of inventorie to ale i now equal to for Retail) and production i 1.10 time more volatile than ale 1.15 in Retail). Thu, in repone to an expanionary monetary policy hock, both ale and inventory invetment increae, but inventory invetment increae much le than ale, and o the inventory-ale ratio decline 11. The evidence in thi Section i robut to the detrending method, the level of aggregation and tage-of-fabrication of inventorie. We alo checked the robutne of thee fact uing data from the NBER manufacturing productivity databae and meaure of monetary hock from Romer and Romer 2007). See Kryvtov and Midrigan 2009) for detail. 3. Model We tudy a monetary economy populated by a large number of infinitely lived houehold, a continuum of monopolitically competitive firm that produce differentiated intermediate good, a continuum of perfectly competitive firm that produce a final good, and a government. In each period t the commoditie are differentiated varietie of labor ervice, a final labor ervice, money, a continuum of intermediate good indexed by i [0, 1], and a final good. The final good i ued for conumption and invetment. In each period t, thi economy experience on of infinitely many event t. We denote by t = 0,..., t ) the hitory or tate) of event up through and including period t. The probability denity, a of period 0, of any particular hitory t i π t ). The initial realization 0 i given. In the model, we have aggregate hock to the money upply and idioyncratic demand hock. We decribe the idioyncratic hock below. In term of the money upply hock, we aume, throughout mot of the paper, that the upply of money follow a random-walk proce of the form log M t ) = log M t 1 ) + log µ t ), 1) 11 See Jung and Yun 2005) who provide imilar evidence. 8

10 where log µ t) i money growth, a normally ditributed i.i.d. random variable with mean 0 and tandard deviation σ µ. We conider alternative pecification of monetary policy in a robutne ection below. A. Houehold Houehold conume, trade bond, and work. They alo own the capital tock and rent it to intermediate good producer. We aume friction in the labor market in the form of ticky wage. In particular, we aume that houehold are organized in monopolitically competitive union, indexed by j. Each union upplie a differentiated variety of labor ervice, l j t ), that aggregate into a final labor ervice, l t) according to l t) = l j t ) ϑ 1 ϑ ) ϑ ϑ 1 dj where ϑ i the elaticity of ubtitution acro different type of labor ervice. Each union et it wage W j t ) and therefore face demand for it ervice given by Wj t ) ) ϑ l t) 2) l j t ) = W t ) where l t) i the amount of labor hired by firm, and W t) i the aggregate wage rate: W t ) = W j t ) 1 ϑ dj ) 1 1 ϑ. In thi economy the market for tate-contingent money claim are complete. We repreent the aet tructure by having complete, tate-contingent, one-period nominal bond. Let B j t+1 ) denote the conumer holding of uch a bond purchaed in period t and tate t with payoff contingent on a particular tate t+1 at date t + 1. One unit of thi bond pay one unit of money at date t + 1 if the particular tate t+1 occur and 0 otherwie. Let Q t+1 t) denote the price of thi bond in period t and tate t. Clearly, Q t+1 t) = Qt+1 ) Q t ) where Q t) i the date 0 price of a ecurity that pay one unit if hitory t i realized. The problem of union j i to chooe it member money holding M j t ), conumption 9

11 c j t ), invetment x j t ), tate-contingent bond B j t+1 ), a well a a wage W j t ), to maximize the houehold utility: t=0 t β t π t) [ u c j t )) v l j t ))] d t 3) ubject to the budget contraint P t) x j t ) + ξ xj t ) ) 2 2 k j t 1 ) δ k j ) t 1 + Q t+1 t) B j t+1 ) d t+1 + M j t ) t+1 M j t 1 ) P t 1) c j t 1 ) + W t) l j t ) + Π j t ) + B j t ) + R t) k j t ), a cah-in-advance contraint, P t) c j t ) M j t ), and ubject to the demand for labor given by 2) a well a ubject to the friction on wage etting. We aume that utility i eparable between conumption and leiure. Here P t ) i the price of the final good, x j t ) = k j t ) 1 δ) k j t 1 ) i invetment, W t ) i the nominal wage, Π j t ) are firm dividend, and R t) i the rental rate of capital. Invetment i ubject to capital adjutment cot, the ize of which i governed by ξ. The budget contraint ay that the houehold beginning-of-period balance are equal to unpent money from the previou period, M j t 1 ) P t 1) c j t 1 ), labor income, dividend, a well a return from aet market activity and from rental of the capital tock to firm. The houehold divide thee balance into money holding, M j t ), finance invetment pending, a well a purchae of tatecontingent bond. We aume Calvo-type friction on wage etting. The probability that any given union i allowed to reet it wage at date t i contant and equal to 1 λ w. A meaure λ w of the union leave their nominal wage unchanged. We chooe the initial bond holding of union o that each union ha the ame preent dicounted value of income. Even though union differ in the wage they et and hence the amount of labor they upply, the preence of a complete et of ecuritie and the eparability between conumption and leiure implie that they make identical conumption and 10

12 invetment choice in equilibrium. Since thee deciion rule are well-undertood, we imply note that the bond price atify Q t+1 t) = βπ t+1 t) u c c t+1 )) P t) u c c t )) P t+1 ) where π t+1 t) i the conditional probability of t+1 given t and we have dropped the j ubcript. Similarly, the date 0 price atify: Q t) = β t π t) u c c t )) P t. ) B. Final good producer The final good ector conit of a unit ma of identical and perfectly competitive firm. The final good i produced by combining the good produced by intermediate good firm we refer to thee good a varietie) according to: q t) 1 = 0 ) θ v i t ) 1 θ qi t ) θ 1 θ 1 θ di where q i t ) i the amount of variety i purchae by a final good firm, v i t ) i a variety-pecific hock and θ i the elaticity of ubtitution acro varietie. For implicity we aume that v i t ) i an iid log-normal random variable. In thi economy, intermediate good firm ell out of their exiting tock of inventorie, z i t ). We decribe the evolution of a firm tock of inventorie below. Given the price and inventory adjutment friction we aume, thi tock of inventorie will occaionally be inufficient to meet all demand and intermediate good firm will tockout. In uch a cae, we aume a rationing rule under which all final good firm are allowed to purchae an equal hare of that intermediate good tock of inventorie. Since the ma of final good firm i equal to 1, z i t ) i both the amount of inventorie the intermediate good firm ha available for ale, a well a the amount of inventorie that any particular final good firm can purchae. The problem of a firm in the final good ector i therefore: 11

13 ubject to the inventory contraint max P t) q t) q i t ) 1 0 P i t ) q i t ) di,.t. q i t ) z i t ) i and the final good production technology. Cot minimization by the final good firm implie the following demand for each variety: q i t ) = v i t Pi t ) + µ ) i t ) ) θ P t q t) ) where µ i t ) i the multiplier on the inventory contraint. Notice here that the hock v i t ), act a a demand hock for an intermediate good firm. We will thu refer to uch hock a demand hock. Perfect competition implie that the price of the final good, P t), i equal to P t) [ 1 = 0 v i t ) [ P i t ) + µ i t )] 1 θ di ] 1 1 θ. Alo note that if µ i t ) > 0 o that the inventory contraint bind, then it atifie: P i t ) + µ i t ) = z i t ) v i t )P t ) θ q t ) ) 1 θ. The left hand ide of thi expreion i the price that a firm that tock out would have choen abent price adjutment friction. Since uch a firm face an inelatic demand curve, it would like to increae it price to the point at which final good firm demand exactly all of it tock of inventorie. Together with the inventory friction we decribe below, price adjutment friction give rie to tockout in the equilibrium of thi economy ince they prevent firm from increaing their price. 12

14 C. Intermediate good firm The intermediate good firm are monopolitically competitive. Any given uch firm ell a ingle variety i. Such a firm rent capital from conumer, hire labor and produce the intermediate good. It then ell the good to final good firm. The critical aumption we make i that the firm make the deciion of how much to produce, q i t ), prior to learning the value of v i t ), the demand hock. Thi aumption introduce a precautionary motive for holding inventorie, the tockout-avoidance motive. We aume a production function y i t ) = l i t ) α ki t ) 1 α ) γ, where y i t ) i output, k i t ) i the amount of capital firm i rent and l i i the amount of labor it hire, while γ 1 determine the degree of return to cale. Letting R t) and W t) denote the rental rate of capital and the aggregate nominal wage rate, repectively, thi production function implie that the minimum cot of producing y i t ) unit of the intermediate good i given by Ω t) y i t ) 1 γ, where Ω t) = χw t) α R t ) 1 α, and χ i a contant. Intermediate good firm face two friction. Firt, they mut chooe how much to produce, y i t ), and the price to et, P i t ), prior to learning their demand hock, v i t ). Second, they change price infrequently, in a Calvo fahion. An exogenouly choen faction 1 λ p of firm are allowed to reet their nominal price in any given period; the remaining λ p of firm leave their price unchanged. Let m i t 1 ) denote the tock of inventorie firm i ha at the beginning of date t. If the firm produce y i t ) additional unit, the amount it ha available for ale i equal to z i t ) = m i t 1 ) + y i t ). Recall that, given a price P i t ) and tock z i t ), the firm ale are equal to: q i t ) = min v i t ) P i t ) ) θ P t q t), z i t ) 4) ) 13

15 The firm problem i therefore to chooe p i t ) and z i t ) m i t 1 ), o a to maximize it objective given by max p i t ), z i t ) m i t 1 ) t=0 t Q t) [ P i t ) q i t ) Ω t) [ z i t ) m i t 1 )] ] 1 γ d t where recall Q t) i the date 0 price of one unit of currency to be delivered in tate t and m i 0 ) i given. The contraint are the demand function in 4), the retriction that z i t ) and P i t ) are not meaurable with repect to v i t ), a well a the contraint that P i t ) = P i t 1 ) in the abence of a price adjutment opportunity, a well a the law of motion for inventorie : m i t ) = 1 δ z ) z i t ) q i t )) where δ z i the rate at which inventorie depreciate. D. Equilibrium Conider now thi economy market-clearing condition and the definition of equilibrium. The market-clearing condition on labor are: l j t ) ϑ 1 ϑ ) ϑ ϑ 1 dj = l t ) and l t) = l i t )di i The firt expreion i the production function for producing final labor ervice l t) out of the differentiated ervice upplied by each union. The econd expreion ay that the total amount of the final labor ervice mut be equal to the amount of labor hired by each intermediate good firm. Similarly, the market clearing condition for the final good are: q t) 1 = 0 ) θ v i t ) 1 θ qi t ) θ 1 θ 1 θ di 14

16 and 1 0 c j t ) + x j t ) + ξ xj t ) ) 2 2 k j t 1 ) δ k j ) t 1 dj = q t) The firt expreion i the final good production function and the econd ay that the total conumption and invetment of the different houehold mut um up to the total amount of the final good produced. Since all houehold make identical conumption and invetment deciion, we can write the reource contraint for final good a: x t ) c t) + x t) + ξ 2 k t 1 ) δ ) 2 k t 1 ) = q t) Next, the market-clearing condition on bond i B t ) = 0 and the cah-in-advance contraint require P t) c t) = M t). Finally, the market clearing condition for capital i k i t ) di = k t 1) i An equilibrium for thi economy i a collection of allocation for houehold c t ), M t ), B t+1 ), k t), x t), l j t ) and W j t ); price and allocation for firm p i t ), q i t ), y i t ), l i t ), k i t ), z i t ) ; and aggregate price W t ), P t ), R t) and Q t+1 t ), all of which atify the following condition: i) the conumer allocation olve the conumer problem; ii) the price and allocation of firm olve their maximization problem; iii) the market-clearing condition hold; and iv) the money upply proce atifie the pecification above. E. Recurive Formulation and Solution Method We next recat the problem recurively. At the beginning of period t, after the realization of the money hock, µ t), but before the realization of the demand hock, v i t ), the tate of an individual firm i i characterized by it price in the preceding period, P i t 1 ) and it inventory tock m i t 1 ). It i convenient to normalize all nominal price and wage by the current money upply. Specifically, let p i t ) = P i t 1 )/M t ) and ω t ) = Ω t )/M t ) and ue imilar notation for other price. Let p t) = Pt ) M t ) denote the normalized aggregate price level. With thi normalization, we can write the tate of an individual firm i in t a [p i, m i ]. Let λ t) denote the meaure of firm 15

17 over thee variable. Let w t) denote the ditribution of wage et by union. The only ource of aggregate uncertainty i the growth rate of the money upply, µ t). However, ince money growth i iid, = λ, w) fully characterize the aggregate tate of thi economy. Let d p, z, m, ) denote the expected dividend of the firm that charge a price p and ha z unit of inventorie after production take place, evaluated at date 0 price: d p, z, m, ) = uc p p min v ) ) p θ q, z) ω z m) 1 γ dφ v), p where Φ v) i the ditribution of demand hock. We can write the firm problem recurively a follow: V a m, ) = max d p,z m p, z, m, ) + V p V n p, m, ) = max z m d p, z, m, ) + V ) µ, m, df µ ) dφ v), ) p µ, m, df µ ) dφ v). Here V a i the value of a firm that i allowed to reet it price, V n the value of a firm that cannot reet it price and mut ell at it old price p, and V = 1 λ p ) V a + λ p V n i the expected value of a firm prior to learning whether it can adjut it price. We ue F µ) to denote the ditribution of money growth hock. The olution to thi problem give deciion rule p m, ) and z a m, ) for firm that reet their price and z n p, m, ) for firm that do not reet price, which, together with the equilibrium condition, are ued to compute all other equilibrium object. For example, the aggregate price level i equal to: [ p ) = 1 λ p ) p m, ) 1 θ + λ p p 1 θ] ) 1 1 θ dλ p, m) Since λ and w are large object, we approximate the olution to thi problem uing the Kruell-Smith approach. That i, we potulate that aggregate price and quantitie are a function of a mall number of moment of the ditribution λ.and w. For wage, we follow the tandard approach in the New Keyneian literature and imply log-linearize the union wage deciion. Hence the pat aggregate wage, w t 1, i the only relevant variable that characterize the dynamic of wage. We approximate λ uing the pat aggregate price level, p t 1, a well a the aggregate tock of inventorie, m t 1. We then potulate that all aggregate variable are log-linear function of w t 1, p t 1, m t 1 a 16

18 well a the aggregate tock of capital, k t 1. For example, we potulate the aggregate price level evolve according to: log p t = α 0 + α 1 log p t 1 log µ t ) + α 2 log w t 1 log µ t ) + α 3 log m t 1 + α 4 log k t 1. Notice that we mut detrend the pat price and wage level by the growth rate of the money upply o a to expre thee tate variable relative to the current period money upply, M t. Given a gue for the coefficient on thee aggregate law of motion, we olve the firm problem uing pline and projection method to approximate the firm value function and deciion rule) and union problem uing, a tandard in Calvo-type model, a log-linear approximation around the teady-tate), imulate a time-erie of thi economy and update the gue for the unknown coefficient α i uing OLS regreion. A Kruell and Smith 1998) and Khan and Thoma 2007) do, we found that thee few moment provide a very good approximation, in that the mean quared error between the predicted variable and the actual one i extremely mall le than 0.01% of the time-erie variance of thee erie). F. The Working of the Model We next dicu the deciion rule in thi economy. We begin by tudying a verion of the model with contant return at the firm level γ = 1). We then characterize the optimal pricing and inventory deciion in the preence of decreaing return γ < 1) and provide ome empirical evidence for decreaing return uing micro-level data. Firm-level Contant Return To build intuition, aume away the irreveribility contraint z i t ) m i t 1 ). Thi contraint turn out not to bind for mot of the experiment we decribe here, with the exception of the economy with non-convexitie we decribe later on. Moreover, aume that price are flexible, λ p = 0. Recall that v i t ), the demand hock, are iid and log-normal. Let Φ denote the cdf and 17

19 σ 2 v the variance of thee hock. Then we can write the firm expected ale a = R P i t ), z i t )) = 0 min v Pi t ) ) θ P t q t), z i t ) dφ v) = ) Pi t ) ) θ P t q t) ) σ 2 exp v Φ log v i t ) σ 2 ) v + zi t ) 1 Φ log v i t ))), 5) ) 2 where v i t ) = z i t ) ) Pi t θ ) P t ) q t ) i the highet value of the productivity hock for which the firm doe not tockout. To undertand expreion 5), notice that the firt term i the expected value of ale in thoe tate in which the firm doe not tockout, while the econd term i the amount of inventorie the firm ha, z i t ), time the probability of a tockout. Clearly, R z = 1 Φ log v i t ))) > 0 : an increae in it tock of inventorie allow the firm to ell in thoe tate in which it would otherwie tockout 12. With contant return and no irreveribility, the value of a firm i linear in the tock of inventorie it ha inherited from the previou period: an unold unit of inventorie at date t depreciate to 1 δ z ) unit next period and ave the firm production cot equal to 1 δ z ) Q t+1 t) Ω t+1) evaluated at date t price. Let Ω t) = 1 δ z ) Q t+1 t) Ω t+1) d t+1 t+1 denote the expected value of thee aving. The problem of the firm thu reduce to: max Pi t ) Ω t)) R P i t ), z i t )) Ω t) Ω t)) z i t ), P i t ), z i t ) where, recall, Ω t ) i the marginal cot of production. To undertand thi expreion, notice that the choice of price i imilar to that in the tandard problem of a monopolit, except that R P i t ), z i t )) i the demand function and Ω t) 12 To derive thi expreion, notice that z enter 5) in three place, but two of thee term cancel out. 18

20 i the marginal valuation of the good the firm ell. The choice of inventorie, z i t ), i alo traightforward: on one hand a higher z i t ) increae expected ale, but the firm expect to loe Ω t ) Ω t)) z i t ) in inventory carrying cot. The firm optimal price i then a markup over it hadow valuation of inventorie: P i t ) = ε i t ) ε i t ) 1 Ω t). Here ε i t ) i the price elaticity of expected ale and i equal to θ the elaticity of ubtitution acro varietie) time the hare of ale in the tate in which the firm doe not tockout: ε i t ) = θ ) exp σ 2 v 2 Φ log vi t ) σ 2 v) exp σ 2 v 2 ) Φ log v i t ) σ 2 v) + v i t ) 1 Φ log v i t ))) We next turn to the inventory accumulation deciion. The choice of z i t ) atifie. 1 Φ log v i t )) = 1 r i t ) P i t ) /Ω t ) r i t ). 6) The left-hand ide of thi expreion i the probability that the firm tock out. A in Bil and Kahn 2000), the firm chooe a higher tock of inventorie a lower tockout probability) the higher the markup P i t ) /Ω t), and the higher the return to holding inventorie, r i t ), where r i t ) = Ω t) Ω t ) = 1 δ z) Q t+1 t) Ω t+1 ) t+1 Ω t ) dt+1. 7) Stockout are epecially cotly for firm that have higher markup ince the profit lot by failing to make a ale i greater. Similarly, a higher return to holding inventorie converely, a lower carrying cot) make it optimal to increae the tock of inventorie available for ale. One important implication of the model i that inventorie are much more enitive to change in the return to holding inventorie, rather than change in markup. To ee thi, we find it ueful 19

21 to log-linearize 6) around the teady-tate: vˆv i t ) = 1 [ 1 [ Φ [ b β 1 δz ) ] Φ) b ˆPi t ) ˆΩ t)] + β 1 δ z ) Φˆr i t )], where hat denote log-deviation from the teady tate, b i the teady-tate markup, v i the pre-ale teady-tate inventory-ale ratio, and 1 Φ i the teady-tate probability of a tockout. If the tockout probability and markup are low, a in the data, and 1 Φ ) b 0, then inventorie are relatively inenitive to markup. In contrat, a long a the cot of carrying inventorie a determined by δ z ) i ufficiently mall, inventorie are much more enitive to fluctuation in the return to holding inventorie. Intuitively, if the cot of carrying inventorie i ufficiently low, firm find it optimal to intertemporally ubtitute production in order to react to expected change in the marginal cot of production and/or change in the interet rate 13. Hence, the dynamic of inventorie i cloely related to the dynamic of cot but alo influenced by the behavior of markup. In the next ection we exploit thi key feature of the model to draw implication for the dynamic of cot in repone to monetary policy hock. So far we have dicued the model implication for v i t ). Thi object, on it own, i not ueful to evaluate the model empirically a we do not directly oberve it in the data. Notice however that there i a monotonic relationhip between the aggregate inventory-to-ale ratio and v i t ). In particular, integrating the ditribution of demand hock, and noting that all firm make the ame v i t ) choice, it follow that the end-of-period inventory-ale ratio, which we do oberve in the data, i equal to: IS t) = v t) Φ log v t)) ) exp σ 2 v 2 Φ log v t) σ 2 ) v exp σ 2 v 2 ) Φ log v t ) σ 2 v) + v t 1 Φ log v t ))). 8) 13 See Houe 2008) who make a imilar argument in the context of a model with invetment. 20

22 Firm-level Decreaing Return With decreaing return, γ < 1, a firm marginal cot of producing y i t ) unit of output i increaing in y i t ). In particular, thi cot i now equal to Ω i t ) = Ω t) y i t ) 1 γ 1. Price and production deciion are imilar to thoe in the economy with contant return, except that now the hadow valuation of inventorie depend on how much the firm expect to produce next period: Ω i t ) = 1 δ z ) Q t+1 t) Ω t+1) 1 [ zi t+1 ) m i t )] 1 γ 1 d t+1. t+1 γ The deciion rule under firm-level decreaing return differ from thoe under contant return along two important dimenion. Firt, with decreaing return inventory holding at the firm level are peritent the tock of inventorie after order are made, z i t ), i increaing in the initial tock, m i t 1 ). Second, the firm price i negatively correlated with it initial inventory tock. Thee two prediction follow from the fact that a firm marginal cot of production i decreaing in it initial inventory tock: firm-level decreaing return act much like a convex inventory-adjutment cot. We illutrate thee prediction of the model in Figure 2 in which we contrat the deciion rule in an economy with decreaing return with thoe in an economy with contant return. Panel A how that in the economy with decreaing return the pre-ale tock, z i t ), i increaing in the initial tock, m i t 1 ). Panel B how that firm that tart with more inventorie charge lower price in the economy with decreaing return. In contrat, price and inventorie are independent of the initial tock in the economy with contant return. Empirical Evidence on Firm-level Decreaing Return Since decreaing return play an important role in our analyi below, we briefly preent ome empirical evidence for thee uing micro-level data. We find thee reult of independent interet in light of the increaing role they play in amplifying the real effect of monetary policy hock in 21

23 recent tudie 14. We ue a dataet of price, inventorie, and order for a Spanih upermarket ued by Aguirregabiria 1999) in hi tudy of markup and inventorie in retail firm. See the original tudy for a detailed decription of the data. We ue a panel of monthly obervation on inventorie, price and order for 534 product motly non-perihable food and houehold upplie) old by the upermarket chain for a period of 29 month from January 1990 to May We ue the data to etablih empirical evidence for the two prediction of the model we dicu above about the relationhip between price, order and the initial inventory tock. To do o, we etimate the following fixed-effect regreion: log p it = α p i + βp log m it 1 + ε p it, log z it = α z i + β z log m it 1 + ε z it where m it 1 i the tock of inventorie at the end of period t 1 for good i, p it i the price for the good at date t, and z it i the amount of good available for ale the initial tock plu order). Uing the data we etimate an elaticity of price to inventorie, β p, equal to the tandard error i equal to 0.001) and an elaticity of the pre-ale tock to inventorie, β z, equal to 0.45 the tandard error i equal to 0.002). The ign of thee two elaticitie are conitent with the prediction of the theory. We how in our quantitative analyi below that a value of γ of about 2/3 bet fit thee two moment of the data. We thu conclude that firm-level decreaing return or other form of convex inventory adjutment cot) are neceary to account for the pattern of price, order and inventorie in the micro data. A we alo how below, uch decreaing return alo improve the model ability to account for the behavior of inventorie at the aggregate level, epecially in an economy with ticky price. 14 See for example Altig et. al 2005) and Burtein and Hellwig 2008) who meaure the ize of decreaing return uing macro and micro-level evidence, repectively. 22

24 G. Parametrization We next decribe how we have choen parameter to evaluate the model quantitative implication. We et the length of the period a one month and therefore chooe a dicount factor of β =.96 1/12. We aume preference of the form u c) v n) = log c n. Thee preference imply an infinite Frich labor upply elaticity, and can be interpreted a the outcome of indiviibilitie combined with Hanen 1985) and Rogeron 1988) type lotterie. We focu on thee preference ince they imply, in a verion of the model without capital and nominal wage rigiditie and with contant return at the firm level, that the marginal cot of production increae one-for-one with the monetary hock 15. Below we conider the implication of changing the aumption we make about preference. Table 2 report the parameter value we ued in our quantitative analyi. We et the rate at which capital depreciate, δ, equal to We et the elaticity of ubtitution acro intermediate good and varietie of labor, θ = ϑ = 5, implying a 25% markup, in the range of etimate in exiting work. Finally, we aume a frequency of wage change of once a year, 1 λ w = 1/12, conitent with what i typically aumed in exiting tudie. We calibrate the inventory parameter, namely, the rate at which inventorie depreciate, δ z, and the volatility of demand hock, σ v, to enure that the model account for two fact about inventorie and tockout in the data. Firt, a can be een from the deciion rule 6)-7) above, δ z directly affect the frequency of tockout: a higher cot of carrying inventorie make it optimal for firm to tockout more often. Bil 2004) ue micro CPI data from the BLS and report a frequency of tockout of 5%. 16 In all of the experiment we conider below we chooe δ z o that the model generate a 5% frequency of tockout. Second, σ v, the volatility of demand hock, directly map into the average inventory-ale ratio in the model, a 8) how. We thu chooe thi parameter o a to match an end-of-period) inventory-ale ratio of 1.4 month, a in the US Manufacturing and Trade ector. For example, a Panel A. I. of Table 2 how, in the economy with contant firm-level return, 15 Thi particular parametrization ha been widely ued in the menu cot literature. See for example Goloov and Luca 2007). 16 Thi tatitic exclude obervation that are three month or more prior to a product becoming permanently or eaonally unavailable. The overall frequency of tockout i approximately 8% when focuing on all obervation. Thi number i alo conitent with the finding of Aguirregabiria 1999, 2003). 23

25 the value of σ v neceary to match thee two fact i equal to 0.63, while the rate of depreciation i equal to δ z = 0.91%. We how below that uch a high volatility of demand i alo neceary to reconcile the model with the comovement of price and quantitie in the micro-data. Our etimate of the depreciation rate i in the range of the inventory-carrying cot meaured directly in the logitic literature, ee for example Richardon 1995). For the economy with firm-level decreaing return we mut alo calibrate the degree of return to cale, γ. We do o by requiring the model to match the two elaticitie of price and order with repect to the initial inventory tock in the Aguirregabiria 1999) data. It turn out that a value of γ of 2/3 bet fit thee two moment of the data. Thi value of γ implie an elaticity of price to inventorie β p = and an elaticity of the pot-production tock to inventorie β z = Panel A.II. of Table 2 report the parameter value and moment in thi economy with firm-level decreaing return. A omewhat maller volatility of demand hock 0.57 v earlier) and rate of depreciation 0.75% v. 0.91% earlier) are required to match the inventory-ale ratio of 1.4 and frequency of tockout of 0.05 in the data. 4. Quantitative Invetigation We ue the model to make two related point. Firt, verion of the model with flexible price that imply nearly contant markup) predict a much tronger repone of inventorie to a monetary policy hock than in the data. Second, model with countercyclical markup ticky price) can account for the dynamic of inventorie in the data, but only if markup decline real marginal cot increae) ufficiently in repone to an expanionary monetary hock. To make our firt point we tudy a verion of our economy with ticky wage and flexible price. We then introduce nominal price rigiditie and how that if the marginal cot i ufficiently reponive to monetary hock, the model can indeed account for the dynamic of inventorie in the data. A. Economy with flexible price We tart by tudying an economy with contant return to labor. We then allow for decreaing return to labor, either by auming decreaing return to cale at the firm level or by introducing capital in the production function. 24