The Review of Economic Studie Advance Acce publihed September 11, 2012 Review of Economic Studie 2012) 0, 1 28 doi:10.1093/retud/rd028 The Author 2012. Publihed by Oxford Univerity Pre on behalf of The Review of Economic Studie Limited. Inventorie, Markup, and Real Rigiditie in Menu Cot Model OLEKSIY KRYVTSOV Bank of Canada and VIRGILIU MIDRIGAN Federal Reerve Bank of Minneapoli and New York Univerity Firt verion received Augut 2010; final verion accepted May 2011 Ed.) 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 data on inventorie. We tudy an economy with nominal rigiditie in which good are torable. Our 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, if the data inventorie are fairly contant over the cycle and in repone to change in monetary policy. We how that cot mut increae and 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 fraction of the repone of real variable to monetary hock. Key word: Inventorie, markup, cot JEL Code: E31, F12. 1. INTRODUCTION A widely held view in macroeconomic i that change in monetary policy affect real economic activity becaue price are ticky. What give rie to price tickine in the aggregate, and the extent to which price are ticky i, however, a matter of coniderable debate. 1 Since price are equal to a markup time marginal cot, price tickine in the aggregate can arie via one of two channel. One channel i countercyclical variation in markup, due to menu cot of price adjutment that prevent firm from changing their price or other imperfection in the product market that make it optimal for firm to lower markup during boom. 2 A econd channel, often referred to a real rigiditie, i tickine in cot. Friction in the labour market that give rie to wage rigiditie, a well a the firm ability to flexibly vary the workweek 1. See, for example, Chari et al. 2000) and Woodford 2003). 2. See Goloov and Luca 2007) and Gertler and Leahy 2008) who tudy model with menu cot of price adjutment. Rotemberg and Woodford 1999) urvey everal alternative mechanim that generate countercyclical markup. 1
2 REVIEW OF ECONOMIC STUDIES of capital and labour, may imply that cot of production repond only gradually to monetary hock. 3 How trong each of thee channel are ha important implication for the trength of the monetary tranmiion mechanim, the role of nominal hock in accounting for buine cycle fluctuation, a well a the conduct of monetary and fical policy. 4 Our goal in thi article i to ue theory and data in order to meaure the extent to which markup and cot vary in repone to monetary policy hock. Our approach i to tudy data on inventorie through the len of a New Keyneian model in which we introduce demand uncertainty and thu a tockout-avoidance motive for holding inventorie. We focu on inventorie becaue theory predict a tight relationhip between markup, cot, and inventorie, a forcefully argued by Bil and Kahn 2000). The model predict that when markup decline, firm reduce their tock of inventorie relative to ale ince inventorie are le valuable when markup and profit are lower. Similarly, if cot are ticky after, ay, an expanionary monetary hock, firm take advantage of the temporarily low cot by ubtituting intertemporally and building up a larger tock of inventorie to run down in future period. A alient feature of the data i that the tock of inventorie react much le to monetary policy hock than ale do. The inventory ale ratio i thu countercyclical and decline after an expanionary monetary policy hock. For our model to reproduce thi fact, two condition mut be atified. Firt, cot of production mut increae immediately after an expanionary monetary hock in order for firm not to ubtitute intertemporally by inveting in inventorie. Second, markup mut decline to reduce the firm incentive to build up their tock of inventorie in repone to the lower interet rate that accompany epiode of monetary expanion. Overall, for the model to reproduce the behaviour of inventorie in the data, countercyclical markup, rather than cot rigiditie, mut account for the bulk of the real effect of monetary policy hock. 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. Studie of micro-price data find that input cot change infrequently and repond gradually to nominal hock, while conumer price tend to repond quickly to change in cot. 5 Moreover, Chritiano et al. 2005) find that cot rigiditie, a oppoed to countercyclical markup, are neceary to account for alient fact of key U.S. macroeconomic time-erie. Thee obervation led reearcher to conclude that wage rigiditie and tickine in input cot, rather than countercyclical markup, mut be the dominant ource of monetary non-neutrality. Our article reviit thee concluion. Although input cot are indeed ticky in the data, the relationhip between input cot and marginal cot i highly enitive to what one aume about the production technology, a Rotemberg and Woodford 1999) forcefully illutrate. In contrat, theory predict a very robut relationhip between the behaviour of inventorie and that of marginal cot, which we document and exploit in thi article. 3. See, for example, Chritiano et al. 2005) and Dotey and King 2006). 4. See Woodford 2003) and Galí 2008), who how how optimal monetary policy varie depending on whether the friction are in the good or labour market. Woodford 2003) and Dotey and King 2006) how how the ize of the real effect from monetary hock increae a one increae the degree of cot rigiditie. More recently, Hall 2009) argue that countercyclical markup greatly amplify the effect of change in government pending on output. 5. Bil and Klenow 2004); Klenow and Kryvtov 2008); Goldberg and Hellertein 2012); Eichenbaum et al. 2011). See alo Nekarda and Ramey 2010) who argue that markup are, in fact, procyclical.
KRYVTSOV & MIDRIGAN MENU COST MODELS 3 We begin our analyi by reviewing everal well-known fact about inventorie. 6 In the data, inventorie are procyclical but much le volatile than ale. The aggregate U.S. 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 0.34% 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 turn to the model. Our baeline model i characterized by price and wage rigiditie, decreaing return to cale, a well a convex adjutment cot that limit the firm ability to rapidly change their tock of capital. Thee feature imply that marginal cot are very reponive to monetary policy hock: decreaing return to labour make it cotly for firm to change production very much, preventing them from varying their tock of inventorie. Moreover, ince price are ticky, markup are countercyclical. Thee two feature of the model, trongly procyclical marginal cot and countercyclical markup, imply that inventorie are much le volatile than ale, a in the data. Importantly, countercyclical markup account for the bulk of the real effect of monetary policy hock in our baeline model. We then demontrate that eliminating thee two key ingredient of the baeline model countercyclical markup and volatile marginal cot viibly woren the model ability to account for the inventory fact. When we eliminate the decreaing return to labour, marginal cot increae much more gradually in repone to expanionary monetary hock. Since the cot of carrying inventorie i low, both in the model and in the data, firm find it optimal to invet in inventorie in anticipation of future increae in production cot. The model thu predict that inventorie are much more volatile than in the data. Similarly, when we eliminate price rigiditie and thu the countercyclical variation in markup, firm find it optimal to take advantage of the lower interet rate that accompany monetary expanion and exceively build up their tock of inventorie. Abent a decline in markup to reduce the profit from holding inventorie, the model fail to account for the inventory data. We have tudied a number of extenion of our model and have found that thee reult are robut to the exact model of inventorie: the S,) model with fixed ordering cot a oppoed to a tockout-avoidance model, detail about the monetary policy rule, the rate at which inventorie depreciate, the degree of demand uncertainty, a well a allowing inventorie to be held at multiple tage of production. In all of thee experiment, we found that markup mut account for the bulk of the real effect of monetary hock for the model to be able to account for the behaviour of inventorie in the data. Our reult thu tand in harp contrat to the finding of Chritiano et al. 2005), who etimate parameter value that imply eentially no role for markup in accounting for the real effect of monetary policy hock. Our work i related to a number of recent paper that tudy the behaviour 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 countercyclical inventory ale ratio 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 6. See, for example, Ramey and Wet 1999) and Bil and Kahn 2000).
4 REVIEW OF ECONOMIC STUDIES in a general equilibrium model driven by productivity hock. Their model account well for the behaviour of inventorie in the data, depite the fact that markup are contant. Thee author how that general equilibrium conideration, and in particular capital accumulation, are critical to thi reult. Diminihing return to labour reduce the repone of marginal cot to a productivity hock and hence the incentive for inventory accumulation. 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 productivity hock, our i on monetary hock in an economy with nominal rigiditie. We find that in our 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 productivity hock, monetary hock affect real activity only if nominal price are ticky and do not react immediately to change in monetary policy. Since price are equal to a markup time cot, monetary hock can affect output only if either markup vary or if nominal cot are ticky. Hence, if markup are contant, monetary policy hock can generate real effect only if nominal cot are ticky. Cot tickine give rie, however, to trong variability in inventorie due to intertemporal ubtitution in production, which i at odd with the data on inventorie. 7 Finally, our article i cloely related to the work of Klenow and Willi 2006) and Burtein and Hellwig 2007), who alo meaure the trength of real rigiditie 8 uing theory and microprice data. Thee reearcher focu on an alternative type of real rigidity, 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 behaviour of inventorie. Thee fact are well known from earlier work. 9 We dicu them briefly for completene, a they are central to our quantitative analyi below. We ue data from the Bureau of Economic Analyi National Income and Product Account NIPA) on monthly final ale and inventorie for the U.S. Manufacturing and Trade ector from January 1967 to December 2009. 10 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 i real final dometic ale. We define production 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 detrend all erie uing a Hodrick Precott filter with a moothing parameter equal to 14,400. We alo ue a meaure of identified monetary policy hock to report tatitic conditional on identified exogenou monetary policy hock. Figure 1A preent the time erie of ale and the inventory ale ratio for the manufacturing and trade ector. The figure how that the two erie are trongly negatively correlated and are 7. See Jung and Yun 2005), Chang et al. 2006), and Wen 2011), who alo tudy the buine cycle prediction of inventory model. 8. See Ball and Romer 1990). 9. See Ramey and Wet 1999) and Bil and Kahn 2000). 10. 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).
KRYVTSOV & MIDRIGAN MENU COST MODELS 5 A B Figure 1 Inventorie and ale, U.S. manufacturing and trade. A) Unconditional HP-filtered erie and B) Conditional on monetary policy hock almot equally volatile. Receion are aociated with a decline in ale and a imilarly ized increae in the inventory ale ratio. Likewie, expanion are 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. The column labelled Unconditional report unconditional tatitic for thee erie. We focu on the erie for the entire manufacturing and trade ector and later briefly dicu the tatitic for the retail ector in iolation. Notice in the firt column of Table 1 that the correlation between the inventory ale ratio and ale in manufacturing and trade i equal to 0.82. The tandard deviation of the inventory ale ratio i about a large a the tandard deviation of ale. Conequently, the elaticity of the inventory ale ratio with repect to ale i equal to 0.84. 11 In other word, the inventory ale ratio decline by about 0.84% for every 1% increae in ale. The tock of inventorie i thu fairly contant over the cycle, increaing by only 0.16% = 0.84+1) for every 1% increae in ale. Note alo that the inventory ale ratio i very peritent: it autocorrelation i equal to 0.87. The fact that the tock of inventorie i fairly contant over time 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. 12 There i, in fact, no contradiction, ince inventory invetment 11. 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 the log inventory ale ratio on log ale. 12. See, for example, Ramey and Wet 1999).
6 REVIEW OF ECONOMIC STUDIES TABLE 1 Inventory fact, U.S. NIPA, January 1967 December 2009 Unconditional Conditional on monetary hock Manufacturing Retail Manufacturing Retail and Trade and Trade ρis t,s t ) 0.82 0.65 0.71 0.57 σ IS t )/σ S t ) 1.03 1.17 0.93 1.28 elat. IS t w.r.t. S t 0.84 0.76 0.66 0.73 elat. I t w.r.t. S t 0.16 0.24 0.34 0.27 ρis t,is t 1 ) 0.87 0.72 0.88 0.74 ρy t,s t ) 0.98 0.89 0.98 0.84 ρy t )/σ S t ) 1.12 1.14 1.11 1.17 ρy t, I t ) 0.55 0.47 0.63 0.50 ρ Y t )/σ Y t ) 0.23 0.46 0.20 0.51 Note: All erie are real, monthly. IS t, I t,s t,y t denote real inventory ale ratio, inventory invetment, and final ale, repectively. The column labelled Unconditional report tatitic for HP 14 400)-filtered data. The column labelled Conditional on monetary hock report tatitic computed uing data projected on current and 36 lag of Chritiano et al. 1999) meaure of monetary policy hock etimated uing a VAR for 1960:01 2000:12. i mall relative to the entire tock of inventorie: the average monthly inventory invetment i equal to 0.22% of the inventory tock in the manufacturing and trade ector. We next report the fact on inventory invetment. One way to do o i by exploiting the following accounting identity: Y t =S t + I t, where Y t i production, S t i ale and I t i inventory invetment. To meaure the volatility of inventory invetment, 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. Alternatively, the tandard deviation of inventory invetment expreed, a i typical in the inventory literature, a a fraction of production, I t /Y t ) i equal to 0.23 time the tandard deviation of production. Alo note that inventory invetment i alo trongly procyclical: it correlation with production i equal to 0.55. Thee lat two tatitic jointly imply that inventory invetment expreed a a fraction of production) increae by 0.13% =0.55 0.23) for every 1% increae in production. The other column of Table 1 preent everal additional robutne check. We note that the fact above alo characterize the behaviour of firm in the retail ector: the elaticity of inventorie to ale i equal to 0.24 and production i 1.14 a volatile a ale. 13 Thee fact alo hold when we condition on meaure of monetary policy hock. To ee thi, we project the data erie on current and 36 lag of Chritiano et al. 1999) meaure of monetary policy hock and recompute thee tatitic. 14 We plot the reulting erie in Figure 1B. Although monetary hock account for a maller fraction of the buine cycle the tandard deviation of thee erie i about one-third a large when conditioning on meaure of monetary hock), the main pattern we documented above are evident now a well. A Table 1 how, the elaticity of inventorie to ale i now 13. Iacoviello et al. 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). Thee reult thu reinforce our concluion that the tock of inventorie i fairly contant over the cycle. 14. Our reult are robut to uing an alternative meaure of monetary policy hock, due to Romer and Romer 2004).
KRYVTSOV & MIDRIGAN MENU COST MODELS 7 equal to 0.34 0.27 for retail) and production i 1.11 time more volatile than ale 1.17 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. The evidence in thi ection i robut to the detrending method, the level of aggregation and the tage of fabrication of inventorie. We have alo excluded the lat five year of the ample to enure that our reult are not driven by the large recent receion and have found imilar reult. See our Supplementary Appendix for more detail. 3. MODEL We tudy a monetary economy populated by a large number of infinitely lived houehold and three type of firm: producer of intermediate good, ditributor, and final good firm. Intermediate good producer and final good firm are perfectly competitive. Ditributor are monopolitically competitive, have ticky price, and hold inventorie. In each period the commoditie are differentiated varietie of labour ervice, a final labour ervice, money, intermediate good, a continuum of differentiated varietie of good old by ditributor, and a final good. The final good i ued for conumption and invetment. In each period t, thi economy experience one of infinitely many event t. We denote by t = 0,..., t ) the hitory or tate) of event up to 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. The hock in thi economy are aggregate hock to the money upply and idioyncratic demand hock. We decribe the idioyncratic hock below. We aume that the upply of money follow a random-walk proce of the form logm t )=logm t 1 )+g m t ), 1) where g m t ) i money growth, a normally ditributed i.i.d. random variable with mean 0 and tandard deviation σ m. 3.1. Houehold Houehold conume, trade money and bond, and work. We aume friction in the labour market in the form of ticky wage. Houehold are organized in monopolitically competitive union, indexed by j. Each union upplie a differentiated variety of labour ervice, l j t ), that aggregate into a final labour ervice, l t), according to l t) 1 = l j t ) ) ϑ ϑ 1 ϑ 1 ϑ dj, 2) 0 where ϑ i the elaticity of ubtitution acro different type of labour ervice. Each union chooe it wage, W j t ), and face demand for it labour ervice given by l j t ) Wj t ) ) ϑ = W t) l t), 3) where l t) i the amount of final labour hired by firm, and W t) i the aggregate wage rate: W t )= W j t ) 1 ϑ dj ) 1 1 ϑ. 4)
8 REVIEW OF ECONOMIC STUDIES We aume that 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) =Q t+1) /Q t), where Q t) i the date 0 price of a ecurity that pay one unit of money if hitory t i realized. The problem of union j i to chooe it member money holding, M j t ), conumption, c j t ), tate-contingent bond, B j t+1 ), a well a a wage, W j t ), to maximize the houehold utility: β t π t)[ u c j t )) v l j t ))] d t 5) t=0 t ubject to the budget contraint a cah-in-advance contraint, M j t )+ Q t+1 t) B j t+1) d t+1 6) t+1 M j t 1) P t 1) c j t 1) +W j t ) l j t ) + j t ) +B j t ), P t) c j t ) M j t ), 7) and ubject to the demand for labour given by 3) a well a the friction on wage etting we decribe later. We aume that utility i eparable between conumption and leiure. Here, P t ) i the price of the final good and j t ) are firm dividend. 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 ), labour income, dividend, a well a return from bond holding. The houehold divide thee balance into money holding and purchae of tate-contingent bond. We aume Calvo 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 labour they upply, the preence of a complete et of ecuritie and the eparability between conumption and leiure implie that they make identical conumption and aving choice in equilibrium. Since thee deciion rule are well undertood, we imply drop the j ubcript and note that the bond price atify Q t+1 t) =βπ t+1 t) u c c t+1 )) u c c t )) P t ) t+1), 8) P where π t+1 t) =π t+1) /π t) i the conditional probability of t+1 given t. Similarly, the date 0 price atify Q t) =β t π t) u c c t )) P t). 9)
KRYVTSOV & MIDRIGAN MENU COST MODELS 9 3.2. Final good firm The final good ector conit of a unit ma of identical and perfectly competitive firm. The final good i produced by combining the good old by ditributor we refer to thee good a varietie) according to q t) 1 = 0 v i t ) 1 θ qi t ) θ 1 θ di ) θ θ 1, 10) where q i t ) i the amount of variety i purchaed by a final good firm, v i t ) i a varietypecific hock and θ i the elaticity of ubtitution acro varietie. We aume that v i t ) i an i.i.d. lognormal random variable. We adopt the i.i.d. aumption for implicity, a it allow u to characterize the firm inventory deciion in cloed-form, although, a we how below, thi aumption counterfactually implie a negative correlation between ale and inventory invetment at the firm level. The data, in contrat, how little correlation between ale and inventory invetment, a feature that the S,) model we tudy in the Robutne ection can account for. At the beginning of period t, ditributor have z i t ) unit of the good available for ale. We decribe how the ditributor chooe z i t ) below. Given the price and inventory adjutment friction we aume, ditributor will occaionally be unable to meet all demand and will thu tock out. We aume, in cae of a tockout, a rationing rule under which all final good firm are able to purchae an equal hare of that ditributor good. Since the ma of final good firm i equal to 1, z i t ) i both the amount of good the ditributor ha available for ale, a well a the amount of good that any particular final good firm can purchae. The problem of a firm in the final good ector i therefore max P t) q t) 1 P i t ) q i t ) di, 11) {q i t )} 0 ubject to the contraint q i t ) z i t ) i 12) and the final good production technology 10). 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), 13) where μ i t ) i the multiplier on 12). Notice here that the hock, v i t ), act a demand hock for a ditributor. Perfect competition implie that the price of the final good, P t), i equal to P [ t) 1 = v i t ) [ P i t ) +μ i t )] ] 1 1 θ 1 θ di. 14) 0 Alo note that if the inventory contraint bind, then μ i t ) atifie P i t ) +μ i t ) z i t ) ) 1 θ = v i t )P t) θ q t ). 15) The left-hand ide of thi expreion i the price that a ditributor that tock out would have choen abent the price adjutment friction. Since uch a ditributor face inelatic demand, it
10 REVIEW OF ECONOMIC STUDIES would like to increae it price to the point at which final good firm demand exactly all of it available good. Together with the inventory friction we decribe later, price adjutment friction give rie to tockout in the equilibrium of thi economy, ince they prevent ditributor from increaing their price. 3.3. Intermediate good firm There i a continuum of intermediate good firm that ell a homogeneou good to ditributor. Any uch firm own it capital tock, hire labour and ell the good to the ditributor. Intermediate good firm augment their capital tock by purchaing invetment good from final good producer at price P t). We aume that invetment i ubject to a convex cot of intalling new capital. The production function take the Cobb Dougla form y t) = l t) α k t 1) 1 α ) γ, 16) where y t) i output, k t 1) i the amount of capital a producer own at date t, l t) i the amount of labour it hire, and γ i the degree of return to cale. Let t) be the price of the intermediate good. Recall that the price of the final good, ued for invetment, i P t) and the wage rate i W t). The intermediate good producer olve ubject to max Q t)[ t) y t) P t) x t) +φ t)) W t) l t)] d t 17) y t ),x t ),l t ) t=0 t k t) =1 δ)k t 1) +x t) and the production function 16). Here, δ i the depreciation rate of capital and φ t) i the adjutment cot: φ t) = ξ x t ) ) 2 2 k t 1) δ k t 1). 18) Letting R t) =1 α)γ t) y t) /k t 1) denote the marginal product of capital, perfect competition and optimization by intermediate good firm imply the following relationhip between the price of intermediate input, t ), the wage rate and the firm tock of capital: t) = 1 γ 1 α) 1 α) α α W t) α R t ) 1 α ) y t) 1 γ 1. 19) 3.4. Ditributor Ditributor purchae good from intermediate good firm at a price t), convert thee to ditributor-pecific varietie and ell thee varietie to final good firm. A key aumption we make i that the ditributor chooe how much to order prior to learning the value of v i t ), it demand hock but after learning the realization of all other hock, including the monetary hock). Thi aumption introduce a precautionary motive for holding inventorie, the tockout-avoidance motive. Ditributor face two friction. Firt, they mut chooe how much to order, y i t ), and the price to et, P i t ), prior to learning their demand hock, v i t ). Second, they change price
KRYVTSOV & MIDRIGAN MENU COST MODELS 11 infrequently, in a Calvo fahion. A fraction 1 λ p of randomly choen ditributor are allowed to reet their nominal price in any given period; the remaining λ p of ditributor leave their price unchanged. Let n i t 1 ) denote the tock of inventorie ditributor i ha at the beginning of date t. If the firm order y i t ) additional unit, the amount it ha available for ale i equal to z i t ) =n i t 1) +y i t ). 20) Given a price, P i t ), and an amount of good available for ale, z i t ), the firm ale are q i t ) =min v i t ) P i t ) ) θ P t) q t), z i t ). 21) The firm problem i therefore to chooe P i t ) and z i t ) o a to maximize it objective given by Q t) P i t ) q i t ) t) y i t )) d t, 22) t=0 t where n i 0 ) i given. The contraint are the demand function in equation 21), the retriction that z i t ) and P i t ) are not meaurable with repect to v i t ) but meaurable with repect to the money growth hock), 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: where δ z i the rate at which inventorie depreciate. n i t ) =1 δ z ) z i t ) q i t )), 23) 3.5. Equilibrium Conider now thi economy market-clearing condition and the definition of the equilibrium. The market-clearing condition for labour tate that the amount of the final labour ervice upplied by houehold i equal to the amount hired by the intermediate good firm: 1 l j t ) ) ϑ ϑ 1 ϑ 1 ϑ dj =l t ). 24) 0 Similarly, the market-clearing condition for the final good tate that conumption and invetment um up to the total amount of the final good produced: c t) +x t) +φ t) =q t) 25) The market-clearing condition for intermediate input i y t) 1 = y i t ) di. 26) 0 We can rewrite thi further a y t) 1 = q i t ) di+ 1 1 n i t ) 1 δ z )n i t 1)) di, 27) 0 1 δ z 0 which ay that total production i equal to ale plu inventory invetment.
12 REVIEW OF ECONOMIC STUDIES An equilibrium for thi economy i a collection of allocation for houehold c t ), M t ), B t+1 ), l j t ), and wage W j t ); price and allocation for firm P i t ), q i t ), y i t ), z i t ), n i t ), y t), l t), x t), k t) ; and aggregate price W t ), P t ), t), and Q t ), all of which atify conumer and firm maximization and the market-clearing condition. We olve for the equilibrium numerically, uing firt-order perturbation method that are tandard in the New Keyneian literature. We how, in the Supplementary Appendix, that perturbation method are very accurate in our context: global projection method yield nearly identical reult. 3.6. Deciion rule To build intuition for our reult, we next dicu the deciion rule in thi economy and the determinant of the inventory ale ratio. Combining equation 22) and 23), the firm problem of how many good to make available for ale, z i t ), reduce to [ max P i t ) Q t+1 ) 1 δ z ) z i t ) t+1 Q t) t+1) d ]D t+1 P i t ),z i t ), t) 28) [ t) Q t+1 ) 1 δ z ) t+1 Q t) t+1) d ]z t+1 i t ), where D P i t ),z i t ), t) = Pi t ) ) θ min v P t) q t),z i t ) df v) 29) are the firm expected ale and F v) i the lognormal) ditribution of demand hock, v. Equation 28) how that on the one hand, a higher z i t ) increae expected ale by reducing the probability of a tockout the firt term of the equation). On the other hand, the firm loe t) 1 δ z ) Q t+1) t+1 Q t ) t+1) in inventory carrying cot: the difference between the cot of ordering in the current period and the dicounted cot of ordering in the next period the econd term of the equation). The olution to the problem in 28) i where 1 F vi t )) = 1 ri t) b i t ) r I t), 30) vi t ) =z i t ) Pi t ) ) θ / P t) q t) 31) i the amount of good available for ale caled by a term that capture demand, while r I t) Q t+1 ) =1 δ z ) t+1 Q t+1 ) t) t) dt+1 32) i the return to holding inventorie and i the markup. b i t ) =P i t ) / t) 33)
KRYVTSOV & MIDRIGAN MENU COST MODELS 13 The left-hand ide of equation 30) i the probability of a tockout: a higher inventory tock, relative to demand, lower thi probability. The right-hand ide of thi expreion i decreaing in the return to holding inventorie and the firm markup. A in Bil and Kahn 2000), a higher return to holding inventorie make it optimal for firm to increae the amount of good available for ale and lower the probability of a tockout. Similarly, a higher markup make tockout epecially cotly, ince the profit lot by failing to make a ale i greater. Higher markup thu lead firm to lower the probability of a tockout by increaing the amount offered for ale. We can gain ome inight about how the amount of good available for ale varie over time by log linearizing equation 30) around the teady tate: ˆv i t ) 1 [ 1 = b β 1 δ z ) ) F ) b ˆP F i t ) ˆ t)) +β 1 δ z ) F ˆr i t )], where hat denote log deviation from the teady tate, b i the teady-tate markup, and 1 F i the teady-tate tockout probability. Notice that when the teady-tate cot of carrying inventorie, δ z, i lower, the amount of good available for ale i more enitive to fluctuation in the return to holding inventorie. Intuitively, if the cot of carrying inventorie i low, firm are able to intertemporally ubtitute order to take advantage of temporarily lower input price. 15 Alo note that the return to holding inventorie, r I t), can be written up to a firt-order approximation a r I t) 1 δ z ) t+1 t+1) / t) d t+1 1+i t), 34) where i t) i the nominal rik-free interet rate: i [ t) Q t+1 ) ] 1 = t+1 Q dt+1 t) 1. Equation 34) tate that the return to holding inventorie increae with the expected change in cot, t+1) / t), and decreae with the nominal interet rate. Hence, the model implication for how inventorie react to monetary policy hock depend on the expected change in cot, a well a the repone of interet rate and markup. So far we have dicued the model implication for the amount of good the firm make available for ale, vi t ). Thi object, on it own, i not ueful to evaluate the model empirically becaue we do not directly oberve it in the data. Notice, however, that the model predict a monotone relationhip between the expected end-of-period inventory ale ratio and vi t ). In particular, uing equation 21) and 23) and integrating over the ditribution of demand hock, we have that the expected end-of-period inventory ale ratio i equal to t )) e σ v 2/2 F t ) ) e σ v 2 IS i t ) v i t ) F vi = e σ v 2/2 F vi t ) ) e σ v 2 +vi vi t ) 1 F vi ))), 35) t and increaing in vi t ) for a lognormal ditribution F. 15. See Houe 2008), who make a imilar argument in the context of a model with invetment.
14 REVIEW OF ECONOMIC STUDIES TABLE 2 Parameterization Calibrated parameter Target Data Model δ z Inventory depreciation 0.011 I/S ratio 1.4 1.4 σ v.d. demand hock 0.626 Frequency tockout 0.05 0.05 ξ Invetm. adj. cot 46.2 0.05% of x) σ x t )/σ c t ) 4 4 Aigned parameter period 1 month θ elat. ubt. good 5 ϑ elat. ubt. labour 5 1 λ w freq. wage change 1/12 1 λ p Freq. price change 1/12 α Labor hare 2/3 δ Capital depreciation 0.01 γ Return to cale 0.9 β Dicount factor 0.96 1/12 σ 1/IES 2 χ Invere Frich elaticity 0.40 We alo briefly dicu how firm chooe price in thi economy. Abent price rigiditie, when λ p =0, the firm price i imply a markup over it hadow valuation of inventorie, P i t ) = ε i t ) ε i t ) 1 1 δ Q t+1 ) z) t+1 Q t) t+1), 36) where the markup depend on price elaticity of expected ale, ε i t ). Thi elaticity i equal to θ the elaticity of ubtitution acro varietie) time the hare of ale in the tate in which the firm doe not tock out. Markup thu decreae with the ditributor inventory tock, ince a greater tock of inventorie lower the probability of a tockout and raie the demand elaticity. With price rigiditie, the firm price, conditional on the firm being able to reet it, i imply a weighted average of future frictionle price in 36). 4. QUANTITATIVE RESULTS We next how that our model account well for the dynamic of inventorie, production and ale in the data. We then trace thi reult to our aumption on pricing and technology that imply that markup are countercyclical. In the aftermath of expanionary monetary policy hock, cot of production rie quickly while price do not, o that firm find it cotly and le valuable to build up their tock of inventorie. A a reult, the inventory ale ratio fall. We then how that verion of the model that imply that cot increae much more gradually and that markup are le countercyclical predict, counterfactually, that the inventory ale ratio increae harply after monetary policy expanion. 4.1. Parameterization Table 2 report the parameter value we ued in our quantitative analyi. We et the length of the period equal to one month and therefore chooe a dicount factor of β =0.96 1/12. We aume preference of the form uc) vl)=c 1 σ /1 σ ) l 1+χ /1+χ). We et σ =2, a commonly
KRYVTSOV & MIDRIGAN MENU COST MODELS 15 ued value in the buine cycle literature. A we how below, our baeline parameterization require an intertemporal elaticity of ubtitution, 1/σ, le than unity in order to account for the drop in nominal interet rate after a monetary policy hock. We et χ =0.4, implying a Frich elaticity of labour upply of 2.5, conitent with the etimate of Rogeron and Walleniu 2009). We aume that the growth rate of the money upply i erially uncorrelated. We chooe the tandard deviation of money growth, σ m, equal to 0.23% o that the model matche the tandard deviation of the exogenou component identified uing the Chritiano et al. 1999) meaure of monetary policy hock) of the U.S. monthly growth rate of the money upply. We et the rate at which capital depreciate, δ, equal to 0.01 and the hare of capital in production equal to α =1/3, tandard choice in exiting work. We follow Khan and Thoma 2008) and et the value of the pan-of-control parameter, γ, equal to 0.9, in the range of value ued in model of firm dynamic. We et the elaticity of ubtitution acro intermediate good θ) equal to 5, implying a 25% markup. Thi elaticity i omewhat higher than etimate of price elaticitie reported in IO tudie around 3, but lower than elaticitie that are conitent with etimate of markup from production function etimate e.g. Bau and Fernald 1997)) around 10. There i little agreement about the value of ϑ, the elaticity of ubtitution acro varietie of labour. Chritiano et al. 2005) et ϑ =21, implying a 5% markup, while Smet and Wouter 2007) et ϑ =3, implying a 50% markup. We et ϑ =5, implying a markup of 25%, in the middle of the value ued in thee two tudie. We chooe the ize of the capital adjutment cot, ξ, to enure that the model reproduce the fact that the tandard deviation of invetment i four time greater than the tandard deviation of conumption. The value of ξ i equal to 46.2, implying that reource conumed by capital adjutment cot account for about 0.05% of the total amount of invetment in imulation of our economy. The ize of thee adjutment cot i omewhat maller than that ued by Chari, Kehoe, and McGrattan 2000), who find that adjutment cot range between 0.09% and 0.40% of the total invetment. We aume that wage and price change on average once every 12 month λ w =λ p =1 1/12), conitent with what i typically aumed in exiting tudie. Thi degree of price tickine i omewhat higher than what Nakamura and Steinon 2008) and Klenow and Kryvtov 2008) report, but i conitent with the finding of Kehoe and Midrigan 2010). The parameter that are pecific to our inventory model are the rate at which inventorie depreciate, δ z, and the volatility of demand hock, σ v. Thee two parameter jointly determine the teady-tate inventory ale ratio and the frequency of tockout. We thu chooe value for thee parameter o that the model can reproduce the 1.4 ratio of inventorie to monthly ale in the U.S. manufacturing and trade ector and the 5% frequency of tockout that Bil 2004) report uing micro-price data from the Bureau of Labor Statitic BLS). A Table 2 how, matching an inventory ale ratio of 1.4 and a frequency of tockout of 5% require a volatility of demand hock of σ v =0.63 and a depreciation rate of 1.1%. 4.2. Aggregate implication Figure 2 report the impule repone of nominal and real variable to a 1% expanionary hock to the money upply in our model with capital, decreaing return and ticky price. We refer to thi parameterization a our baeline. Figure 2A how that both wage and price repond gradually to the increae in the money upply: a year after the hock, price and wage increae by about one-half of a percent. In contrat, the marginal cot of production, t), i quite flexible and repond eentially one-for-one to the monetary hock, depite the wage tickine. A equation 19) make clear, the marginal cot rie becaue the marginal product of capital increae harply after the hock, a well a becaue of the decreaing return to cale in the
16 REVIEW OF ECONOMIC STUDIES A 1.2 Nominal variable B 1.5 Inventorie and ale % change 1 0.8 0.6 0.4 0.2 Money Price level Wage Marginal Cot 0 0 5 10 15 20 25 % deviation from teady tate 1 0.5 Inventory Sale 0 0 5 10 15 20 25 C % deviation from teady tate 1.5 1 0.5 Production, ale and conumption Production Sale Conumption 0 0 5 10 15 20 25 month D percentage point Figure 2 0 0.5 1 1.5 Impule repone to monetary hock. Baeline model. Interet rate, annualized Nominal Real 2 0 5 10 15 20 25 month A) Nominal variable, B) Inventorie and ale, C) Production, ale and conumption, and D) Interet rate, annualized production of intermediate good. Since the marginal cot i fairly flexible, but the price level i ticky, the average level of markup, a meaured by P t) / t), decline, thu reducing the ditributor incentive to hold inventorie. Figure 2B how the repone of the tock of inventorie and ale. We report, a in the data, the repone of real ale and the aggregate end-of-period real inventory tock defined a I t) = 1[ 0 zi t ) q i t )] di. Notice that ale increae by about 1.4% and gradually decline, while the tock of inventorie increae gradually and reache it peak of about 0.6% about a year after the hock. Since inventorie increae much more gradually than ale do, the inventory ale ratio decline. Figure 2C how the repone of production, ale, and aggregate conumption. Recall that production i equal to the um of all order made by ditributor and i therefore equal to total ale plu inventory invetment. Since production increae by about 1.6% in repone to the monetary hock and ale by only 1.4%, the exce production contribute to an increae in inventory invetment. Thi increae in inventory invetment i mall, however, relative to the total tock of inventorie in the economy, and hence the increae in the inventory tock i gradual. Finally, Figure 2D how that both nominal and real interet rate decline in repone to the monetary policy hock, a in the data. 16 The nominal interet rate decline by le than the real rate doe, becaue of expected inflation, but neverthele decline becaue of the large drop 16. See the Supplementary Appendix for evidence on how conumption and interet rate repond to monetary hock in the data.
KRYVTSOV & MIDRIGAN MENU COST MODELS 17 TABLE 3 Buine cycle prediction of the model 1) Data 2) Baeline 3) Labour only 4) Flexible price Impule repone of conumption to monetary hock Average repone 0.55 0.45 0.60 0.25 Maximum repone 1.01 0.90 1.02 0.33 Half-life, month 16.20 10.5 13.1 26.2 Markup contribution 0.89 0.36 0.08 Inventory tatitic ρis t,s t ) 0.71 0.91 0.99 0.52 ρis t )/σ S t ) 0.93 0.82 3.17 1.15 Elat. IS t to S t 0.66 0.75 3.15 0.60 Elat. I t to S t 0.34 0.25 4.15 1.60 ρis t,is t 1 ) 0.88 0.74 0.82 0.94 ρy t )/σ S t ) 1.11 1.10 3.82 1.47 ρy t, I t ) 0.63 0.89 0.97 0.85 ρ I t )/σ Y t ) 0.20 0.10 0.88 0.41 Elat. It to Y t 0.12 0.09 0.85 0.35 Invetment tatitic σ x t )/σ c t ) 4 4 4 Note: All variable are HP-filtered with moothing parameter 14,400. The average conumption repone i computed for firt 24 month after hock. in the real interet rate induced by the monetary policy expanion and price rigiditie. One key aumption that we have made that enure thi drop in nominal interet rate i that the intertemporal elaticity of ubtitution i le than 1, o that σ>1. To ee thi, note that the binding cah-in-advance contraint implie that the nominal interet rate in our etup i equal to 1+i t) = β 1 g m t+1 ) c t+1 ) ) 1 σ 1 c t) π t+1 t) d t+1. 37) Since money growth, g m t ), i erially uncorrelated and conumption i expected to decline toward the teady tate, the nominal interet rate decline a long a σ>1. In Table 3, we report the buine cycle propertie of the model and compare them to the data. We report two et of tatitic. The firt et are meaure of the real effect of money which ummarize the impule repone of aggregate conumption, c t), to a monetary hock. Alternatively, ince the cah-in-advance contraint, logc t) =logm t) logp t), bind in our model, thee tatitic ummarize the degree of aggregate price tickine: the extent to which price trail the change in the money upply. Column 2 of Table 3 how that the average repone of conumption in the firt two year after the hock i equal to 0.45% in our baeline parameterization. The maximum conumption repone i equal to 0.90%. The half-life of the conumption repone, our meaure of the peritence, i equal to 10.5 month. Comparion with the Data column how that the model predict omewhat maller real effect of monetary policy hock. In the data, the average conumption repone to identified monetary hock i about 0.55% and conumption i more peritent, with a half-life of about 16 month. 17 Our reult are thu conitent with thoe of Chari et al. 2000), 17. We compute thee impule repone in the data uing the Chritiano et al. 1999) meaure of monetary policy hock. See the Supplementary Appendix for detail.
18 REVIEW OF ECONOMIC STUDIES who find that ticky price model are unable, abent additional ource of rigiditie, to account for the peritence of output and conumption in the data. We next ak: what i the role of markup variation in accounting for the real effect of monetary hock?a noted above, the effect of an increae in the money tock on conumption in our model i determined olely by the degree of aggregate price tickine. The aggregate price level i ticky for two reaon: cot of production are ticky becaue of wage rigidity, and markup decline becaue of price rigidity. To decompoe the role of thee two effect, note that the cah-in-advance contraint implie that lnc t) )= [ lnm t) ) ln t) ) ] + [ ln t) ) lnp t) ) ]. 38) }{{}}{{} cot rigidity term markup term The repone of conumption to a monetary hock i thu equal to the um of two term. The firt term capture the extent to which cot, t), decline relative to the money tock. The econd term capture the extent to which price decline relative to cot. Table 3 report the average repone of the markup term relative to the average repone of conumption. Thi ratio i equal to the area between the price and cot impule repone relative to the area between the money and price repone in Figure 2 and i our meaure of the fraction of the real effect accounted for by countercyclical markup. A the row labelled markup contribution in Table 3 how, markup account for almot 90% of the repone of conumption to the monetary hock. 18 The econd et of tatitic we report are thoe that characterize the behaviour of inventorie, ale, and production. To compute thee tatitic, we HP-filter imulated time erie for thee variable, a we have done in the data, with a moothing parameter of 14,400. We then contrat the model prediction with thoe in the data for the manufacturing and trade ector. We focu on the tatitic that are conditional on identified monetary policy hock, although recall from Table 1 that the unconditional tatitic are very imilar. The model doe a very good job in accounting for the behaviour of inventorie in the data. It predict a countercyclical inventory ale ratio the correlation with ale i 0.91 in the model veru 0.71 in the data. The inventory ale ratio i about 0.82 time more volatile than ale in the model 0.93 in the data). Conequently, the elaticity of the inventory ale ratio to ale i equal to 0.75 in the model 0.66 in the data). The elaticity of inventorie with repect to ale i equal to 0.25, o that a 1% increae in ale i aociated with an increae in the inventory tock of only 0.25%. Alo note that the inventory ale ratio i omewhat le peritent than in the data. It autocorrelation i equal to 0.74 in the model, compared to 0.88 in the data. Thi lack of peritence i partly due to the lack of peritence in ale in the model the autocorrelation of ale i equal to 0.80 in the model and 0.89 in the data). What account for the model implication that the inventory ale ratio decreae in period with expanionary monetary hock? Recall that the inventory ale ratio in thi economy increae with the expected change in cot and the markup, and depend negatively on the nominal interet rate. Since cot increae immediately in repone to an unanticipated increae in the money upply, the expected change in cot i mall. If anything, a Figure 2 how, cot are expected to decline in the firt few period after the hock a the economy accumulate more capital. The decline in nominal interet rate counteract thi effect, and the return to holding inventorie i eentially contant. The drop in markup thu make it optimal for firm to lower 18. A variance decompoition baed on equation 38) uing imulated data from the model yield a very imilar reult to that uing impule repone.