Markups, Aggregation, and Inventory Adjustmentcoen/research/inventories.pdf · The omission of...

Markups Aggregation and Inventory Adjustment

By DANIELE COEN-PIRANI

In this paper I suggest a unified explanation for two puzzles in the inventoryliterature first estimates of inventory speeds of adjustment in aggregate data arevery small relative to the apparent rapid reaction of stocks to unanticipatedvariations in sales Second estimates of inventory speeds of adjustment in firm-leveldata are significantly higher than in aggregate data The paper develops a multi-sector model where inventories are held to avoid stockouts and price markups varyalong the business cycle The omission of countercyclical markup variations frominventory targets introduces a downward bias in estimates of adjustment speedsobtained from partial adjustment models When the cyclicality of markups differsacross sectors this downward bias is shown to be more severe with aggregaterather than firm-level data Similar results apply not only to inventories but also tolabor and prices Montercarlo simulations of a calibrated version of the modelsuggest that these biases are quantitatively significant (JEL E22 E32)

Finished goods inventories to expected salesratios are countercyclical and persistent over thebusiness cycle (Valerie A Ramey and KennethD West 1999 Mark Bils and James A Kahn2000) While this behavior suggests that inven-tories adjust slowly to changes in sales tradi-tional partial adjustment models of inventorieshave not been able to provide a satisfactoryexplanation for these facts In particular begin-ning with Martin S Feldstein and Alan Auer-bach (1976) researchers have found it difficultto reconcile the small estimated adjustmentspeeds of inventory stocks toward target withtheir apparent rapid reaction to unanticipatedvariations in sales1 For example according to

Feldstein and Auerbachrsquos estimates in a typicalquarter firms eliminate less than 6 percent of thegap between current and desired inventorieswhile being able to correct within the quartermore than 95 percent of current sales surprises2

Some recent papers have added a seconddimension to this inventory adjustment puzzleThe stock-adjustment model has been tradition-ally estimated with aggregate or two-digit man-ufacturing data In a comprehensive study ofUS manufacturing firms Scott Schuh (1996)has shown how inventory speeds of adjustmentestimated using firm-level data are significantly

Tepper School of Business Carnegie Mellon UniversityPittsburgh PA 15213 (e-mail coenpandrewcmuedu)Thanks to Mark Bils Rui Castro Aubhik Khan Jim KahnDavid Orbison Marla Ripoll Tony Smith Scott Schuh andAmir Yaron for helpful conversations and suggestionsThanks also to the audiences at the 2002 Society for Eco-nomic Dynamics Meeting in New York City the 2004 AEAMeetings in San Diego Carnegie Mellon University theUniversity of Pittsburgh and The Wharton School at theUniversity of Pennsylvania for helpful comments The usualdisclaimer applies

1 The conceptual framework underlying most of theseempirical exercises is represented by Michael C Lovellrsquos(1961) reduced-form stock-adjustment model and CharlesC Holt et alrsquos (1960) linear-quadratic model where firmssolve an explicit dynamic optimization problem Undersome conditions (see Ramey and West 1999) these two

models give rise to observationally equivalent equations forinventories In the linear-quadratic model the adjustmentspeed coefficient decreases with the slope of marginal costIn turn higher slopes of marginal cost induce firms tosmooth production more relative to sales However theempirical literature (see eg West 1986 Jeffrey A Mironand Stephen P Zeldes 1988) has not found significantevidence of production smoothing behavior

2 Similar results are reported by among others Louis JMaccini and Robert J Rossana (1984) Alan S Blinder(1986a) and John C Haltiwanger and Maccini (1989)Using monthly finished-goods inventory data from the De-partment of Commerce for the period 196701ndash199712 Iobtain estimated speeds of adjustment equal to approxi-mately 2 and 3 percent for respectively durable and non-durable goods industries The estimates of the sales surpriseparameter imply that durable and nondurable goods firmscorrect respectively 95 and 86 percent of a sales surprisewithin a month

1328

larger than their counterparts estimated usingaggregate data Specifically he finds that ac-cording to the empirical specification of themodel the weighted average of adjustmentspeeds estimated using firm-level data is 67 to105 percent larger than the one obtained usingaggregated data constructed from the samepanel of firms Despite this evidence the effectof aggregation across heterogeneous firms is notwell understood yet3

I suggest a unified explanation for Feldsteinand Auerbachrsquos and Schuhrsquos inventory puzzlesThe explanation relies on the idea that firmsrsquoinventory targets might change systematicallyover the business cycle for reasons other thanvariations in expected sales as usually assumedin the literature Following the empirical evi-dence presented by Bils and Kahn (2000) thespecific mechanism leading to variations in in-ventory targets is countercyclical movements inprice markups The paper shows qualitativelyand quantitatively how the omission of mark-ups from standard partial adjustment equationsresults in downward-biased estimates of adjust-ment speeds The bias is more severe in regres-sions that use aggregate rather than firm-leveldata Moreover while inventories are an illus-trative special case of these mechanisms theirapplicability extends to other aspects of firmsrsquobehavior such as labor demand and price set-ting of interest to macroeconomists

To make these points I consider a finishedgoods inventory model in the spirit of Bils andKahn (2000) The key feature distinguishing

this model from the linear-quadratic model ofHolt et al (1960) is that in the former higherinventories contribute to increase firmsrsquo salesand revenues at a given price Thus in thismodel higher price markups induce firms toceteris paribus hold more inventories Theeconomy is composed of two sectors with dif-ferent cyclical volatilities of markups The equi-librium law of motion for the modelrsquos sectoraland aggregate inventories can be written in thestandard partial adjustment form with the inven-tory target being a function of expected salesand price markups The model is calibrated andits simulated firm-level and aggregate inventoryand sales data are then used to estimate partialadjustment equations in which price markupsare omitted from the inventory target The Mon-tecarlo exercise reveals that this omission re-sults in large downward biases in estimatedspeeds of adjustment of aggregate inventoriesrelative to their ldquotruerdquo value implied by themodel For example when sectors displayingthe most volatile markup movements accounton average for only 10 percent of aggregateinventories the ldquotruerdquo speed of adjustment ofinventories is 12 percent higher than theweighted average of firm-level estimates and327 percent higher than the estimate obtainedfrom aggregate data (ie Schuhrsquos puzzle) Theaverage value of the latter implies that in onequarter firms eliminate only 30 percent of thegap between current and desired inventories Incontrast the estimates of the effects of salessurprises on inventory investment suggest thatfirms are able to correct completely unantici-pated sales shocks within the quarter (ie Feld-stein and Auerbachrsquos puzzle)

The intuition behind these results is as fol-lows If price markups are countercyclical theiromission from inventory targets leads standardpartial adjustment models to overpredict the gapbetween desired and current stocks during peri-ods of economic expansion and to underpredictit in recessions Therefore these models tend toattribute the discrepancy between the desiredand observed changes in inventories to the factthat firms are adjusting slowly In contrast inthis model countercyclical variations in pricemarkups induce firms to reduce their targetinventories relative to sales in an expansion andincrease them in a recession According to thisview the desired stock of inventories should

3 Other authors have also argued that estimated speeds ofadjustment tend to be higher at lower levels of aggregationUsing data on German manufacturing firms Helmut Seitz(1993) estimates average adjustment speeds at the firm-levelthat are more than twice as large as the one obtained byrunning the regression with aggregated data from the samepanel Blinder (1986a) finds that adjustment speeds esti-mated with aggregate data for the durable and nondurablegoods sectors tend to be lower than the ones estimated withdata from their constituent industries Lovell (1993) showshow aggregation may bias adjustment speedsrsquo estimatesdownward by means of a simulation approach He uses areduced-form model however where firms donrsquot optimizeand parameters are not calibrated Moreover he doesnot provide an explanation for this result John A Carlsonand William C Dunkelberg (1989) using firm-level dataon small businesses in the United States find that firms onaverage fully adjust their inventories to target within aquarter

1329VOL 94 NO 5 COEN-PIRANI MARKUPS AGGREGATION AND INVENTORY ADJUSTMENT

track the data more closely than what is usuallypredicted by estimated inventory targets

These points can be illustrated with referenceto Figure 1 The figure plots three variables (i)investment in finished goods inventories in theUS manufacturing sector (dashed line)4 (ii)the gap between the desired and current inven-tory stocks (solid line) where the former isestimated as a function of expected sales and aconstant and (iii) the gap between desired andcurrent inventories (bold line) where the formeris just a smoothed version of the latter it issupposed to illustrate the effect of countercycli-cal variations in markups on desired inventorystocks5 In comparing the first and second vari-

ables notice that while inventory investment isnot closely correlated with the business cyclethe gap between desired and current inventoriespredicted by traditional partial adjustment re-gressions (second variable) increases systemat-ically in expansions and decreases duringrecessions (the shaded areas in the figure repre-sent recessions as defined by the National Bu-reau of Economic Research [NBER]) Itsassociated speed of adjustment suggests that ina quarter firms eliminate less than 10 percent ofthe gap between desired and current inventoriesThe bold line in Figure 1 illustrates the effect ofcountercyclical variations in markups on de-sired inventories As sales increase markupsfall reducing firmsrsquo incentives to increase theirdesired inventories during expansions There-

4 The real finished goods inventory and sales data used toconstruct Figure 1 are from the Department of Commercefor the sample period 19671ndash199712 They are seasonallyadjusted and expressed in chain-weighted 1996 dollarsBoth inventories and sales have been linearly detrended

5 To construct this third series I have Hodrick-Prescottfiltered the ratio of inventories to expected sales to extract asmooth trend The smoothing parameter is equal to 140 and

is chosen so that the implied estimated speed of adjustmentof inventories is equal to 047 This value corresponds to theldquotruerdquo adjustment speed of aggregate inventories in thecalibrated model of this paper The resulting trend is thenmultiplied by expected sales in order to obtain the measureof desired inventory stocks

FIGURE 1 INVENTORY INVESTMENT AND INVENTORY GAPS

Notes This figure represents three variables The first one (ndash ndash) is inventory investment in finished goods in aggregatemanufacturing in the period 19671ndash199712 The second (mdash) is the gap between desired and actual inventories where theformer is estimated as function of a constant and expected sales only The third one (mdash) is the gap between desired and actualinventories where the former is obtained by interpolating the latter with the Hodrick-Prescott filter The inventory and salesdata used to construct the figure have been linearly detrended Shaded areas denote recessions as defined by the NBER

1330 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

fore the gap between desired and current inven-tory stocks is always relatively small The speedof adjustment associated with this measure ofthe inventory gap is such that it would takefirms only one quarter to close about 85 percentof the gap between current and desiredinventories

The paper also shows that while aggregateinventory and sales data are just averages offirm-level data the difference between the esti-mated and ldquotruerdquo speed of adjustment of aggre-gate inventories depends disproportionately onthe properties of inventories in sectors with themost volatile markups Therefore using aggre-gate data tends to give rise to smaller estimatesof adjustment speeds than what is obtainedwhen firm-level data are used The apparent fastadjustment of inventories to ldquounanticipatedrdquosales shocks is explained by the fact that thelatter are assumed to be such only to the econo-metrician not to firms (Blinder 1986b)

The implication of these results is that aggre-gate inventories are significantly more respon-sive to changes in target stocks than usuallyfound in the literature They are more respon-sive because countercyclical variations in pricemarkups make target stocks less responsive tochanges in expected sales Therefore the resultsof the paper are consistent with the basic featureof the data cited at the beginning inventories doadjust slowly to variations in expected sales

This is not the first paper arguing that tradi-tionally specified inventory targets might beomitting important variables In particular sev-eral authors (see eg Maccini and Rossana1984 Miron and Zeldes 1988 Martin SEichenbaum 1989) have tested the production-smoothing model by allowing for both observ-able and unobservable cost shifters in theinventory target The latter group of variablesincluding real wages material prices and inter-est rates has usually been found to be insignif-icant for explaining inventory behavior (seeTable 11 in Ramey and West 1999 for a sum-mary of the evidence) Unobservable and per-sistent cost shocks instead seem to improvesignificantly the modelrsquos fit Notice howeverthat in order to explain why finished-goods in-ventories to sales ratios fall systematically dur-ing economic expansions one might need toargue that these unobservable costs are procy-clical This would then be inconsistent with the

usual interpretation of these shocks as technol-ogy shocks (Eichenbaum 1989)6 Bils andKahn (2000) instead have recently providedempirical support in favor of the hypothesis thatinventory targets are significantly affected bycyclical variations in price markups Using datafor six manufacturing industries they estimate arepresentative-agent version of the modeladopted here and show how allowing for coun-tercyclical markups significantly improves thefit of the model Differently from Bils and Kahn(2000) though the issue of aggregation acrossfirms plays a central role here in reconciling therelatively high estimates of inventory speeds ofadjustment obtained using firm-level data withthe relatively small estimates obtained usingaggregate data7

The remainder of the paper is organized asfollows Section I describes the model econ-omy Section II discusses from a qualitativepoint of view the effects of omitting cyclicalmarkups from target stocks on the estimates ofinventory speeds of adjustment Section III con-tains the quantitative results of the paper Sec-tion IV extends the analysis by considering firstthe adjustment speeds of labor and prices andthen a version of the model where sectors differin the slope of marginal cost rather than mark-ups Section V concludes

I The Model Economy

The economy I examine extends Bils andKahnrsquos (2000) model of a representative firm to

6 Aubhik Khan and Julia K Thomas (2002) develop ageneral equilibrium model of inventories with lumpy ad-justment at the firm level due to fixed costs In their modeltechnology shocks generate fluctuations in output and in-ventory-sales ratios are countercyclical They focus on in-put however rather than output inventories

7 A related literature reviewed by Ricardo J Caballero(1999) has instead explored the aggregate implications ofinfrequent and discrete microeconomic adjustment In thisliterature the slow adjustment of aggregates is due to het-erogeneity in the timing of adjustment at the micro levelSince in this case firms either adjust or donrsquot adjust the realdichotomy is between lumpy microeconomic adjustmentand smooth aggregate dynamics rather than between fastmicro adjustment and slow macro adjustment Caballeroand Edwardo Engel (2003) show how failure to account formicroeconomic lumpiness might erroneously induce aneconometrician to conclude that aggregates adjust moreslowly than individual units


an economy with heterogeneous sectors Sec-tors are assumed to differ in terms of the cycli-cal properties of their markups8 The marketstructure is the simplest that captures the fol-lowing two key elements (i) firms have somedegree of market power so that it is meaningfulto discuss the effects of changes in price mark-ups and (ii) firms are ex ante heterogeneouswhich is necessary to analyze the effects ofaggregation

A Setup

To keep the model as simple as possible Iwork with a two-sector economy Each sectorindexed by k 1 2 produces a continuum ofvarieties of a product Each variety is producedby one and only one monopolistically compet-itive firm Firms within a sector are otherwisehomogeneous as the key dimension of hetero-geneity in the model is across sectors There isa measure of firms in sector 1 and 1 insector 2

The objective of each firm is to maximize thepresent discounted value of its profits ex-pressed in units of an arbitrary numeraire goodTime is discrete and infinite and future profitsare discounted at a rate 1 assumed to beconstant relative to the numeraire

Sales and InventoriesmdashThe distinguishingfeature of Bils and Kahnrsquos (2000) inventory modelis that unlike the standard linear-quadratic modelinventories contribute to increased sales at agiven price by reducing the likelihood of stock-outs9 This revenue role of inventories is impor-tant in order to analyze the effect of variationsin price markups on a firmrsquos decision to hold

stocks The building block of the model is rep-resented by the relationship between a firmrsquosfinished goods inventories and its sales Denoteby akt

j the sum of firm jrsquos beginning-of-the-period output inventories ikt

j and current produc-tion ykt

j Then sales for firm j in sector k at timet are given by

(1)

sktj t pkt

j

Pktkt

aktj 0 1 kt 1

The term (aktj ) in this equation captures the

revenue-generating role of inventories The pa-rameter determines the extent to which ahigher stock of goods contributes to generatehigher sales at a given price Period t sales in allsectors are affected by an aggregate shock tobserved at the beginning of the period Thelatter evolves over time according to the process

(2) t t 1 ut 0 1

The random variable ut is identically and inde-pendently distributed over time according tosome distribution with positive support With-out loss of generality I normalize its uncondi-tional mean to one

In equation (1) a firmrsquos sales in sector k arealso assumed to depend on this firmrsquos pricerelative to a measure Pkt of the price level insector k The only restriction imposed on Pkt isthat when all firms in sector k charge the sameprice p then Pkt p10 The elasticity of demandfaced by firms operating in sector k is denotedby kt and is allowed to change stochasticallyover time Cyclical variations in the elasticity ofdemand give rise to cyclical variations in mark-ups As will become clear in the next section aconstant elasticity kt in (1) implies thatfirms choose a constant markup of price overexpected discounted marginal cost To generate

8 In Section IV B I discuss the case where the slope ofthe marginal cost function differs across sectors

9 For a discussion of this point in relation to the linear-quadratic model instead see Ramey and West (1999 page885 footnote 11) Bils and Kahnrsquos approach to modelingthe role of inventories acknowledges that in reality firmsmight stock out even if their observed inventory stocks arenot zero because goods come in different colors sizes etcand consumers have preferences about these characteristicsTherefore having higher inventories decreases the chancesof a mismatch between the available stock and the prefer-ences of consumers Kahn (1987 1992) develops and testsa structural model of the stock-out avoidance motive forholding inventories

10 Notice that a firmrsquos sales do not depend on the relativeprice of goods in the two sectors The demand function (1)can be easily generalized to allow for a dependence of skt

j onthe ratio P1tP2t This generalization does not howeversignificantly affect the quantitative properties of the modelso it is omitted for simplicity


cyclical variations in markups in a simple wayI allow this elasticity to change over time withthe aggregate state of the economy11

(3)

kt 1 11 kt 1 1ut13k 1

When 13k 0 positive sales shocks tend tomake demand relatively more elastic and de-crease firmsrsquo desired price markups during eco-nomic expansions12

ProductionmdashWith the exception of Ramey(1991) researchers in the inventory literature havemostly postulated that marginal production cost isupward sloping While estimates of this slope varygreatly across studies some researchers (see inparticular Jeffrey C Fuhrer et al 1995) havefound strong evidence in support of this hypothe-sis13 Bils and Kahn (2000) have instead empha-sized variations in measures of hourly wages as animportant reason for the procyclicality of real mar-ginal cost According to their measured wagesthough the growth rate of marginal cost is toopersistent at a monthly frequency to give rise tosignificant intertemporal substitution in produc-tion Given these results and the partial equilib-rium nature of this model in this paper I allow forincreasing marginal cost of production resultingfrom diminishing returns to labor rather than cy-clical variations in input prices The latter aretherefore held constant in terms of the numeraire

good14 A firmrsquos cost function takes the standardlinear-quadratic form

(4) cyktj ykt

j

2ykt

j 2

where denotes the slope of the marginal costcurve

A firm j that starts period t with inventoriesiktj produces output ykt

j and sells sktj units of the

good begins period t 1 with inventories equalto

(5) ikt 1j akt

j sktj

with aktj ikt

j yktj Since inventories cannot be

negative it must be the case that aktj skt

j Tosimplify the notation in the rest of the paper Iignore this non-negativity constraint on inven-tories In the simulations presented below thisconstraint never binds

Heterogeneity across SectorsmdashMarkups insectors one and two are assumed to be counter-cyclical (131 0 and 132 0) but more so insector one than in sector two (131 132) Theempirical evidence largely supports the assump-tion that the cyclical properties of markups varyacross sectors15 Bils (1987) and Rotembergand Woodford (1991) study two-digit-SIC man-ufacturing industries and report that price mark-ups over marginal costs are countercyclical inalmost all of these industries Moreover theirresults point to a wide dispersion across indus-tries in the degree of cyclicality of markups (seeespecially Bilsrsquo Table 5 and Rotemberg andWoodfordrsquos Table 8) Evidence of differencesacross sectors in the cyclical properties of mark-ups can also be found in more highly disaggre-

11 As Julio J Rotemberg and Michael Woodford (1999page 1119) observe ldquothe simplest and most familiar modelof desired markup variations attributes them to changes inthe elasticity of demand faced by the representative firmrdquoHere I assume for simplicity that variations in the elasticityof demand are exogenous Bils (1989) and Jordi Gali (1994)show how when purchasers differ in their elasticity ofdemand cyclical changes in the composition of demand cangenerate endogenous variations in its elasticity Alterna-tively it is possible to obtain countercyclical variations inmarkups through some degree of price stickiness combinedwith increasing marginal cost of production This approachwhich I pursued in a previous version of the paper givesrise to qualitative and quantitative results that are similar tothe ones presented here

12 The specification in equation (3) guarantees that kt 1 at all times and that in the steady state the elasticity kt isthe same across sectors kt

13 The usual motivation for assuming decreasing returnsto scale in production is the fixity of capital in the short run

14 I have experimented with versions of the model inwhich the cost function (4) is affected by a procyclical costshock that can be interpreted as resulting from cyclicalvariations in real wages As long as this shock is sufficientlypersistent though the quantitative results of the paper arenot affected

15 One reason why the cyclical properties of price mark-ups vary across sectors is represented by differences inlevels of market power The latter can be introduced in themodel by assuming that the steady state elasticity of demanddiffers across sectors This generalization is ignored forsimplicity


gated data16 For example Binder (1995)analyzes business cycles across four-digit-SICmanufacturing industries and concludes (page27) that in light of his results ldquofindings of auniform cyclical variation of markups in pro-ducer goods manufacturing industries may haveto be reconsideredrdquo Ian Domowitz et al (1987)consider 57 four-digit-SIC manufacturing in-dustries from 1958 to 1981 and report a widedispersion in the yearly standard deviation ofmarkups across industries

B Firmsrsquo Optimization and Equilibrium

When making its pricing and productionchoices firm j in sector k takes as given thestochastic process for the price index Pkt theelasticity kt and the demand shocks t Itsoptimization problem in sequence form is

maxpkt

j aktj

E0t 0

tpktj skt

j aktj akt 1

j skt 1j

2akt

j akt 1j skt 1

j 2subject to skt

j tpktj

Pktkt

aktj i0 0 0 0

The first order conditions with respect to aktj

and pktj are respectively

(6) aktj akt 1

j skt 1j

pktj skt

j

aktj 1

sktj

aktj

Et akt 1j akt

j sktj

(7) kt 1pktj

ktEt akt 1j akt

j sktj

The first-order conditions (6) and (7) have astraightforward interpretation Consider (6)first The marginal cost of increasing akt

j isrepresented by the left-hand side of (6) Itsmarginal benefit is represented by the right-hand side of this equation and is composed oftwo terms First an extra unit of akt

j contrib-utes to increase current sales at the price pkt

j second to the extent that it does not contrib-ute to current sales it increases future inven-tories An extra unit of the good held ininventory at the beginning of period t 1allows the firm to save marginal cost of pro-duction in that period

Consider now the first-order condition withrespect to pkt

j A marginally higher pricecauses a loss of current revenue representedby the left-hand side of equation (7) Givenakt

j this reduction in current sales translatesinto a higher stock of inventories at the be-ginning of t 1 which allows the firm tosave on production costs in that period Thismarginal benefit of a higher price is repre-sented by the right-hand side of equation (7)and at the margin must compensate the firmexactly for the corresponding loss of currentrevenue

Define firm jrsquos price markup as the ratiobetween its price and expected discounted mar-ginal cost minus one

(8) Mkt pkt

j

Et akt1j akt

j sktj

1

1

kt 1

where the second equality follows from equa-tion (7)

In the following I focus on a symmetricequilibrium in which all firms in a givensector make the same production and pricingdecisions Thus in equilibrium pkt

j equals Pktfor all firms j in sector k I denote by AktIkt 1 Skt and Ykt the equilibrium values ofakt

j ikt1j skt

j and yktj

16 It might be necessary to consider this more disaggre-gated evidence because Schuh (1996) suggests that industryaffiliation as measured by two-digit-SIC codes explains a

relatively small fraction of the cross-sectional variance ofhis firm-level estimates of adjustment speeds


The model does not admit a closed-formsolution except when marginal costs areconstant over time ( 0)17 I obtain a linearapproximation to the solution of the model bylinearizing the first order conditions (6) and(7) around the non-stochastic steady state andthen applying the method of undeterminedcoefficients (see eg Lawrence Christiano2002)18 The linearized decision rule for thelevel of the stock of goods available for saletakes the form

(9) Akt 0 1 Akt 1 2 Mkt

3t 4t 1 k 1 2

where the coefficients are nonlinear functionsof the structural parameters of the model

Notice that the coefficients are the samefor firms of the two sectors This follows fromcertainty equivalence because in this modelthe only source of sectoral heterogeneity isrepresented by the forcing process ut

13k in (3)This property of the linearized solution im-plies that it is possible to aggregate the firm-level decision rules (9) into a decision rule forthe aggregate stock of goods available forsale At

At 0 1 At 1 2 Mt 3t 4t 1

where At and the average markup Mt are definedas follows

At A1t 1 A2t

Mt M1t 1 M2t

This aggregation result will prove useful inproviding intuition about the effects of aggre-gation on the estimated speeds of adjustment ofaggregate inventories

II Mechanisms

In this section I discuss the qualitative im-plications of cyclical variations in target in-ventories for the adjustment speeds estimatedusing standard partial adjustment models Inthe first subsection I show how the linearizeddecision rule for inventories implied by themodel can be written in a standard partialadjustment form with the markup variableMkt included in the target equation for inven-tories The second and third subsections de-rive the implications of omitting this markupvariable for the estimates of inventory speedsof adjustment In particular the second sub-section considers an individual sector in iso-lation while the third one analyzes the effectof aggregation across sectors with differentcyclical properties of markups The fourthsubsection discusses Feldstein and Auer-bachrsquos inventory adjustment puzzle

A A Correctly Specified Partial AdjustmentEquation

In this section I show how the equilibriumlaw of motion for inventories in a given sectorand for the economy as a whole can be rewrittenin the traditional partial adjustment form To doso consider the linearized version of equation(1)

(10) Skt St S

AAkt A

and use it to replace the unobservable shockst and t 1 in equation (9) Then use theidentity Akt Ikt 1 Skt to obtain a versionof equation (9) that depends on the inventory

17 In this case it is easy to show that the solution whereinventories are always strictly positive is

Akt t

kt 11 11

and

Pkt c

1 kt 1

provided that demand is sufficiently inelastic ie kt 1 (1 )1

18 Linearization provides an accurate approximation tothe decision rules of the model despite the fact that themarginal benefit of increasing inventories is not linear as inthe standard linear-quadratic model but rather convex in theinventory stock This is because the calibration of the modelis such that the steady state value of Akt is quite far fromzero and the magnitude of the shocks to the economy isrelatively small Appendix A contains expressions for thecoefficients of the linearized decision rules as function ofthe structural parameters of the model


stock Ikt 1 rather than the stock available forsale Akt

(11) Ikt 1 0 1 Ikt 2 Mkt 3 Skt

where the coefficients depend on the onesand are reported in Appendix B19 Now denoteby Skt

e the expectation of period t sales condi-tional on the information available in the previ-ous periods Then add and subtract 3Skt

e fromequation (11) and rewrite the latter in the fol-lowing partial adjustment form20

(12)

Ikt 1 Ikt Ikt 1 Ikt Skt Skte

where the inventory target is defined as

(13) Ikt 1 Skte Mkt

The parameters in (12) and (13) are a functionof the structural parameters of the model and aredefined as follows

1 1 3 1 1 10

1 1 13 1 1 12

The parameter in equation (12) is usuallyreferred to as the ldquospeed of adjustmentrdquo of in-ventory stocks toward their target Ikt1 be-cause it denotes the fraction of the gap betweencurrent and desired inventories that firms fill ina period In optimizing models such as thelinear-quadratic model and the model of thispaper depends on the curvature of the mar-ginal cost function with 1 corresponding tothe case of constant returns to scale21 The termrepresenting the discrepancy between actualand expected sales in (12) captures the effect ofldquounanticipatedrdquo changes in sales on end-of-the-period inventory stocks

Equation (12) together with the target (13)represent the reduced-form representation of thelaw of motion for the inventory stock in sectork implied by the (linearized) version of Bils andKahnrsquos (2000) model adopted here Moreovergiven that the coefficients are the same acrosssectors these two equations also hold whensectoral inventory stocks sales and markupsare replaced by their aggregate counterpartsdefined as22

It I1t 1 I2t

St S1t 1 S2t

Ste S1t

e 1 S2te

Thus the parameter also denotes the ldquotruerdquoadjustment speed of aggregate inventories

It is interesting to notice that the partial ad-justment equation (12) has the same form as thestandard stock-adjustment equation estimated inthe empirical literature (see for exampleRamey and West 1999) However the latterusually specifies the inventory target Ikt1 onlyas a function of expected sales and possiblysome measure of wages and interest rates Bilsand Kahnrsquos model implies that the inventorytarget should also depend on the markup vari-able Mkt Failure to incorporate a time-varyingprice markup in the inventory target might re-sult in a downward bias in the estimate of theadjustment speed parameter This point isdeveloped in the following section in regard toone sector in isolation

B Partial Adjustment Equations and CyclicalMarkups

Consider first sector k in isolation Supposethat an econometrician would try to estimateequation (12) but did not include the markupvariable Mkt in the target equation (13) For-

19 Notice that Ikt1 does not depend on Skt120 For the purpose of the Montecarlo experiment I will

specify the expectation variable Skte as an affine function of

lagged sales Skt1 (see Section III C)21 As shown in Appendix B the parameter 1 depends

linearly on 1 and 4 and is equal to zero when 0

22 Aggregate sales and inventories are constructed asweighted sums of the corresponding firm-level data usingsale prices in the modelrsquos steady state as weights Sincefirms in all sectors charge the same price in the steadystate this implies that aggregate inventories and sales canbe obtained as simple sums of firm-level variables


mally he would estimate the following equa-tion

(15) Zk Xk k

where the t-th row of Xk is the vector Xkt [1Ikt Skt

e Skt Skte ] the t-th element of Zk is

Zkt Ikt1 Ikt and k is an error term Thevector [ ] denotes the param-eters to be estimated23 The ordinary leastsquares estimator of obtained using sector krsquosdata is denoted by k and is given by

(16) k X13kXk1X13kZk

If price markups change over the businesscycle Zk does not evolve according to (15) butrather according to equation (12) The latter canbe rewritten as

(17) Zk Xk Mk2

where Mk denotes the vector whose t-th elementis Mkt Replacing (17) into (16) yields

(18) k qk2

where qk represents the 4 1 vector of esti-mated coefficients in a regression of Mk on Xk

qk(X13kXk)1X13kMk

Denote by qk2 the second element of qk iethe estimator of the coefficient on Ikt Thenthe estimator of obtained using sector krsquosdata is

(19) k qk22

The parameter 2 is a positive number as highermarkups ceteris paribus result in higher desiredstocks of inventories Thus k is a downward-biased estimator of if E[qk2] 0 In turn qk2has the same sign as the partial correlation co-efficient between Mk and Ik after controlling

for sales and expected sales in sector k Sincethe model predicts that larger markups forgiven sales are associated with higher inven-tory stocks the sign of E[qk2] is likely to benegative Thus omitting the markup variableMk from standard partial adjustment regressionswill induce a downward bias in the estimates ofinventory speeds of adjustment

C Cyclical Markups and Aggregation

In this section I consider the estimate of obtained using aggregate data Let X and Zdenote the aggregate counterparts of Xk and ZkAlso M represents the vector whose t-th ele-ment is the average markup Mt

24 Then theordinary least squares estimator of obtainedusing aggregate data is

(20) X13X1X13Z

Following the same steps as in the previoussection to replace Z with the weighted averageof inventory investment in the two sectors it iseasy to show that

(21) q22

where q2 denotes the estimator of the coefficienton It in a regression of Mt on aggregate salesSt and expected sales St

e Equations (21) and(19) together imply that the expected differencebetween the weighted average of the estimatorsof obtained using firm-level data and the oneobtained with aggregate data is given by

(22) E1 1 2

2Eq12 1 q22 q2

To obtain some intuition about this expres-sion consider the extreme case where markupsin sector two do not depend on the state of theeconomy and the volatility of markups in sectorone is sufficiently large (132 0 and 131ldquolargerdquo) In this circumstance 132 0 impliesthat q22 0 Moreover a sufficiently large 131implies that the dynamics of aggregate invento-

23 Of course from it is possible to identify separately and

24 Since markups in sector two are constant by assump-tion M2 is just a constant vector


ries conditional on aggregate sales is domi-nated by variations in sector one inventories sothat q2 q1225 It follows that (22) simplifies to

(23) E1 1 2

21 Eq12

The term on the right-hand side of this equationis positive because 2 is positive and as argued inthe previous section E[q12] tends to be negative

To analyze the general case where markupsin sector two are allowed to vary over time andalso to provide a quantitative estimate of theexpression in (22) it is necessary to calibrateand simulate the model This task is undertakenin Section III

D Feldstein and Auerbachrsquos Puzzle

In order to address Feldstein and Auerbachrsquosoriginal inventory ldquopuzzlerdquo it is not sufficientto show that the estimated speed of adjustmentof aggregate inventories is ldquosmallrdquo It needs tobe ldquosmallrdquo relative to the apparent ease withwhich firms adjust their inventories in responseto sales surprises Formally this means that theestimates of the sales surprise coefficient mustalso be either negative and relatively small orpositive This would suggest that firms react tounexpectedly high sales by increasing withinperiod production and either drawing down oreven increasing their inventory stocks

In the Montecarlo experiments that followthe key to generating a relatively quick ad-justment of inventories to sales surprises isthe fact that the sales shock ut is assumed tobe observed by the firm prior to making itsperiod t production decisions Thus any dis-crepancy between current and expected sales iscorrected within the same period since the vari-able St St

e represents a surprise only to theeconometrician not to the firm This argumentwas first advanced by Blinder (1986b) who alsoprovided some empirical evidence to support it

The ldquotruerdquo coefficient on the sales surprisevariable in the correctly specified partial adjust-ment equation is just 3 ie the coefficient oncurrent sales in the law of motion (11) forinventories In turn the sign and magnitude of3 is determined by two opposite forces Firstincreasing marginal costs of production andcountercyclical variations in markups tend tomake inventories countercyclical so that highersales would lead to lower inventory stocks InBils and Kahnrsquos model however inventoriescontribute to increase a firmrsquos revenue at agiven price by reducing stockouts This secondeffect tends to make inventories procyclical rel-ative to sales In aggregate data inventory in-vestment is moderately procyclical which isconsistent with a positive though relativelysmall value of 3

It follows that if the estimated value of 3 isnot significantly biased by the omission of themarkup variable Mt from the regression aneconometrician would estimate small positive(or possibly negative) values for this parameterInterpreting the estimated parameter as the ef-fect of sales surprises on end-of-the-period in-ventories would then lead to the erroneousconclusion that firms respond quickly to unex-pected sales while possibly adjusting slowly tobridge the gap between current and targetinventories

III Quantitative Results

The following two subsections respectivelydescribe the calibration of the model of SectionI and verify that it can account for the aggregatemoments of sales production and inventoriesthat have been emphasized in the inventoryliterature (see for example Blinder and Mac-cini 1991) The third subsection reports resultson the Montecarlo experiment where artificialdata on inventories and sales are first generatedfrom the calibrated version of the model andthen used to estimate the speed of adjustmentparameter and the ldquosales-surpriserdquo parameter

A Calibration

Calibration of the model requires choosingvalues for the following parameters 131 132 as well as a choice for thedistribution of the forcing variable ut Table

25 To see this notice that q12 represents the coefficient onI1t in a regression of M1t on I1t S1t and S1t

e Moreoverup to a constant term Mt M1t and since there is only oneaggregate demand shock in the model the correlation be-tween St and S1t is very close to one If 131 is large and 132 0 after controlling for variations in St I1t and It tend tomove together This is equivalent to q12 q2


1 summarizes the benchmark values of the mod-elrsquos parameters

The model is calibrated at a monthly frequencyThe discount factor is set equal to 098 im-plying a real (in terms of the numerairegood) interest rate of about 2 percent per monthThis includes storage and goodsrsquo depreciationcosts for the firm The distribution of the ran-dom variable ut is taken to be lognormal withmean one and standard deviation The autocor-relation parameter and the standard deviation are set by estimating the following autoregressiveprocess for the logarithm of linearly detrendedsales in the manufacturing sector

ln St 1 ln St vt 1 stdvt 1

Estimating this equation for the period 196701ndash199712 with US manufacturing data yieldspoint estimates of 094 for nondurablesectors and 096 for durables I thus set 095 The estimate of the monthly standarddeviation of the shock is approximately 002 fordurables and 001 for nondurables Here Ichoose a value 0015

In order to calibrate the parameters and it is convenient to use a steady-state version ofthe first order conditions (6) and (7) Equation(8) implies that the steady-state markup of priceover discounted marginal cost in the two sectors is

M 1

1

I set the latter equal to 0065 so that the corre-sponding price elasticity of demand is 1638 This choice for the average markup isconsistent with the available empirical esti-mates of price markups (see eg Morrison1992 Norrbin 1993)26

To calibrate the elasticity of sales with re-spect to goods available for sale use the steady-state expression for the ratio AS and solve it interms of the parameter

A

S

11

The value for derived above together with aratio AS 15 implies that 051 Thechoice of AS yields a steady-state inventory-sales ratio of 05 This is consistent with the datafrom the manufacturing sector where the aver-age ratio of nominal inventories to nominalsales is equal to 046 for durables and to 057 fornondurables in the period 196701ndash19971227

The parameter that determines the slope ofmarginal cost is set equal to 005 The scaleparameter is chosen to normalize steady-statemarginal costs S to one in both sectorsGiven this normalization can be directlycompared with the estimates of the slope ofmarginal cost obtained by Bils and Kahn (2000Table 6) In only two out of the six sectors theystudy the estimated slope of marginal cost isabove 010 and in two of the remaining four theslope is slightly negative indicating increasingreturns to scale

The elasticity kt is assumed to comove withaggregate demand over the business cycle Theparameters 131 and 132 determine the extent towhich innovations to aggregate sales affectkt and consequently the markup variableMkt The parameter instead determines the

26 Estimates of markup ratios in the US manufacturingindustry tend to be higher when value-added rather than

gross-output data are used (see eg Hall 1988 Norrbin1993) The benchmark value of M selected in this paperreflects estimates of markups based on gross output dataThe quantitative results of Sections III A III B and IVhowever depend on the different cyclical properties of pricemarkups across sectors rather than on their steady-statevalues They are therefore robust to significant variations in M

27 The nominal data used to compute this ratio are pub-lished by the US Census Bureau

TABLE 1mdashBENCHMARK CALIBRATION

Parameter 131 132

Value 098 0015 095 051 1638 097 005 092 010 076 0063

Note This table reports the parameters used in the benchmark calibration of the model with heterogeneous markups andhomogeneous cost


first-order autocorrelation of kt and Mkt28 A

value of corresponds to the case where Sktand Mkt are almost linearly related In this spe-cific circumstance the omission of Mkt from thepartial adjustment equation (12) would not leadto any bias in the estimate of the adjustmentspeed parameter Even small differences be-tween and imply however that the omis-sion of Mkt from (12) can lead to relatively largedownward biases in the estimates of Here Ichoose a value of equal to 09729

The parameter determines the relative sizeof the two sectors and thus determines theextent by which aggregate data reflect relativelymore the behavior of sector-one or sector-twofirms For completeness in sections III C andIV I report the Montecarlo results for differentvalues of this parameter in (01) In order toimpose more discipline on this quantitative ex-periment though it is convenient to set abenchmark value for Specifically to do so Ichoose the parameters 131 and 132 jointly inorder to replicate some of the estimates of firm-level speeds of adjustment obtained by Schuh(1996) In particular I set 010 131 076and 132 0063 The values for 13k imply thatthe average speeds of adjustment for sector oneand sector two firms are respectively 013 and045 These figures are consistent with the re-sults reported by Schuh (1996 Table 3) Hefinds that for 10 percent of the divisions in hissample the estimated inventory speeds of ad-justment were equal to or less than 013 whilefor the next 80 percent of firms they were be-tween 013 and 103 with a median value of040 The choice of 131 gives rise to cyclicalmarkup variations that are empirically plausi-ble when sector-one sales are 1 percent abovetheir steady state the markup of price overmarginal cost for sector-one firms is about 01percent below its steady-state value of 1065For comparison Rotemberg and Woodford

(1999) present estimates of this elasticity ashigh as 04

Before undertaking the Montecarlo experi-ment it is useful to verify that the calibratedversion of the model is consistent with the ob-served business cycle dynamics of aggregateinventories sales and output The next sectionanalyzes the cyclical implications of the modelfor these variables

B Business Cycle Implications of the Model

In order to better understand the working ofthe model it is useful to consider a graph de-picting the aggregate variables produced by themodel economy Figure 2 presents aggregatesales production and the inventory stock inpercentage deviation from their steady state val-ues over 360 months Aggregate production andsales move quite closely together and are basi-cally indistinguishable in the figure The stockof inventories moves procyclically but it issmooth enough for the inventory-sales ratio tomove countercyclically

Tables 2 and 3 present some key statisticsabout these artificially generated aggregate vari-ables and compare them to the correspondingones for the durables and nondurables industriesof the US manufacturing sector Table 2 showsthat the benchmark version of the model isconsistent with the main features of aggregateinventories production and sales data In par-ticular the model correctly predicts that thevariance of production is roughly the same asthe variance of sales and that inventory invest-ment is procyclical It also correctly predicts thevolatility of the stock-sales ratios ItSt and AtStand their countercyclical nature due to the pos-itive slope of marginal cost and countercyclicalvariations in price markups Table 3 displaystheir autocorrelation structure in the data and inthe model at a one- three- and six-month lagTable 3 captures the persistency of these ratioseven if it tends to predict a slightly faster con-vergence toward their long-run values than whatis observed in the data Taken together thesetwo tables suggest that the model consideredhere calibrated with reasonable degrees of cy-clical variations in markups and marginal costdoes a good job in accounting for the mainstylized facts about finished goods inventoriesproduction and sales

28 Notice that equations (3) and (8) jointly imply that themarkup variable Mkt follows the law of motion

Mkt M1 Mkt 1 ut

13k29 The qualitative results of sections II B and II C on the

estimated adjustment speeds of inventories are robust todifferent choices of the parameter both above and below The quantitative results tend to vary but the differencebetween estimates of firm-level and aggregate speeds ofadjustment is always close to the one reported in section III C


C Inventory Adjustment Speeds Results fromSimulated Data

Given the calibration of the model of Sec-tion III A the ldquotruerdquo speed of adjustment ofboth firm-level and aggregate inventories im-plied by the modelrsquos parameters is 047

This section presents the estimates of ob-tained using artificial firm-level and aggregatedata generated by the calibrated version of themodel The estimated equation has the partialadjustment form (12) Instead of including amarkup variable as in (13) however theseregressions specify the inventory target in the

TABLE 2mdashAGGREGATE STATISTICS

varY

varS c(I S) c(AS Y) c(IS Y) cv(AS) cv(IS)

Benchmark model 101 019 099 099 002 006(000) (006) (000) (000) (000) (001)

US manufacturingDurables 102 021 086 087 004 009Nondurables 100 006 071 069 002 005

Notes Y output S sales I beginning-of-the-period inventory stock I inventory investment A I Y cv coefficient ofvariation c correlation var variance The statistics for the US economy have been computed using chain-weighted data onfinished-goods inventories and sales from the Department of Commerce for the period 196701ndash199712 The Y and S serieshave been linearly detrended Model statistics have been obtained by simulating the benchmark economy for 372 periods for1000 times and then computing averages The standard deviation across simulations is reported in parenthesis An asteriskindicates that the correlation is significant at the 1-percent level

FIGURE 2 AGGREGATE SALES OUTPUT AND INVENTORIES OVER TIME

Notes This figure represents time series for aggregate sales (mdash) aggregate output (ndash ) and the aggregate inventory stock(ndash ndash) expressed as percentage deviations from their steady state Notice that the output and sales series overlap almostperfectly Data have been generated by simulating the benchmark version of the model for 360 periods


traditional form adopted in the empirical lit-erature

(24) Ikt 1 Skte

Both the regressions that use firm-level data andthe ones that use aggregate data therefore suf-fer from an omitted variable problem The pur-pose of the Montercarlo exercise is to assess thequantitative significance of the difference be-tween the firm-level estimates of and theaggregate estimate

To estimate these partial adjustment equa-tions it is necessary to specify the sales expec-tation variable Skt

e The latter is taken to be anaffine function of lagged sales30

(25) Skte S1 Skt 1

The model is simulated 1000 times and foreach set of data the firm-level and aggregatepartial adjustment equations with the inventorytarget as in (24) are estimated by ordinary leastsquares The length of the data series in eachsimulation is 100 periods which correspondsapproximately to the one in Schuh (1996) The

results are reported in Table 4 for differentvalues of the parameter determining the rel-ative size of the two sectors

The second column of Table 4 shows theaverage (across simulations) estimate of ob-tained using aggregate data The third andfourth columns show the average speed of ad-justment estimated for sector one and sector twofirms The fifth column represents the weighted(by ) sum of firm-level estimates of adjust-ment speeds Finally the last column shows theamount by which the aggregate adjustmentspeed computed using firm-level data exceedsthe one computed using aggregate data

Table 4 shows several interesting resultsFirst it confirms the qualitative predictions de-veloped in Section II countercyclical changesin markups tend to bias the inventory adjust-ment speeds estimated from partial adjustmentequations downward The bias gets larger asmarkups become more cyclical as can be in-ferred from comparing columns (3) and (4)Moreover countercyclical changes in markupsresult in an econometric aggregation bias in thesense that the adjustment speeds estimated fromaggregate data are significantly smaller than theweighted average of firm-level speeds of adjust-ment In interpreting these results it is useful tokeep in mind that if the markup variables Mktwere included in the partial adjustment regres-sions the estimated speeds of adjustment atboth the firm and aggregate levels would al-ways be equal to 047

Second the quantitative effects of time-varyingmarkups on inventory adjustment speeds arequite large When sector one firms account on

30 Equation (25) is obtained by manipulating equation(10) and noticing that when the calibrated is relativelysmall Skt is approximately given by

Skt S1 Skt 1 Sut 1

The parameters of this equation are for simplicity spec-ified a priori rather than estimated but the results thatfollow do not change when sales expectations are esti-mated instead

TABLE 3mdashAUTOCORRELATION PROPERTIES OF INVENTORY-SALES RATIOS AT DIFFERENT LAGS

c(AS (AS)k) c(IS (IS)k)

k 1 k 3 k 6 k 1 k 3 k 6

Benchmark model 093 082 069 092 080 067(002) (006) (010) (002) (006) (010)


Notes S sales I beginning-of-the-period inventory stock A I Y where Y is output c correlation The statistics for theUS economy have been computed using chain-weighted data on finished-goods inventories and sales from the Departmentof Commerce for the period 196701ndash199712 Model statistics have been obtained by simulating the benchmark economyfor 372 periods for 1000 times and then computing averages The standard deviation across simulations is reported inparentheses An asterisk indicates that the correlation is significant at the 1-percent level


average for only 10 percent of aggregate inven-tories and sales the weighted average of esti-mated firm-level speeds of adjustment is almostfour times higher than the aggregate estimateThis aggregation bias is large for a wide rangeof values of and declines as the share ofsector-one firms in the economy increases Thedecline is due to two forces On the one handmechanically a higher gives rise to a lowerweighted average of firm-level speeds of adjust-ment (column [5] in Table 4) On the otherhand a higher has ambiguous effects on q2 inequation (21)31 The numerical results suggestthat if is sufficiently large E[] increases with

In the spirit of the stock-adjustment model itis instructive to compute the number of monthsT that are required to close 95 percent of the gapbetween current and target inventories accord-ing to the ldquotruerdquo and the average estimate of obtained with aggregate data32 The ldquotruerdquospeed of adjustment of aggregate inventories 047 suggests that approximately T 5months Using the average speed of adjustment

estimated with aggregate data generated by thebenchmark version of the model (011) yieldsinstead T 25 Thus failure to control forchanges in markups would induce one to con-clude erroneously that the aggregate economytakes more than 2 years rather than only 5months to bridge 95 percent of the gap betweentarget and desired inventories Notice howeverthat as observed in the introduction this resultdoes not imply that aggregate inventory stocksare more responsive to variations in expectedsales than previously found They are simplymore responsive to a target that tends to movecountercyclically relative to sales due to varia-tions in markups

Last notice that the results of Table 4 arebroadly consistent with the ones reported bySchuh (1996) In particular the benchmark cal-ibration of the model (ie 010) impliesthat the weighted average of firm-level adjust-ment speeds is 042 while Schuh (1996 Table5) reports a value of 045 for a balanced panel ofdivisions in the M3 Longitudinal Research Da-tabase He also estimates an adjustment speedof 027 based on aggregate data constructedfrom that same balanced panel The latter figureis somewhat higher than 011 the average esti-mate of based on aggregate data obtainedhere

Table 5 presents the estimates of the salessurprise parameter whose ldquotruerdquo value is010 The estimate of obtained using aggre-gate data from the benchmark version of themodel is on average equal to 004 As increases

31 Notice that the partial regression coefficient q2 tends toincrease with the covariance between Mt and It and todecrease with the variance of It after netting out from thesetwo variables the effects of St and St

e A higher makes thecovariance and variance larger with ambiguous effects on q2

32 If x is the relevant adjustment speed then

T ln 005

ln1 x

TABLE 4mdashESTIMATES OF AGGREGATE AND FIRM-LEVEL ADJUSTMENT SPEEDS FOR INVENTORIES (BENCHMARK MODEL)

True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)

Notes ldquotruerdquo speed of adjustment of inventory stocks implied by the benchmark model The estimates of have beenobtained by simulating the benchmark model and estimating a partial adjustment equation for inventories for 1000 times Thesample size of each regression is 100 E[] average estimate of obtained using aggregate data E[k] average estimate of obtained using data on a sector krsquos firm k 1 2 The standard deviation of the estimates across simulations is reportedin parentheses


the average values of become negative butthey remain relatively small in absolute valueThis result could lead to the incorrect conclu-sion that in the aggregate firms respond ex-tremely fast to sales surprises by increasingproduction and even in the benchmark econ-omy end-of-the-period inventory stocks Thisinterpretation would then lead to the puzzleoriginally identified by Feldstein and Auerbach(1976) because the estimates of in Table4 suggest that it takes a few months for firms tocorrect the imbalance between current and tar-get inventories

IV Extensions

In this section I consider two extensions ofthe analysis33 First I ask whether the mecha-nism emphasized in Section II tends to affectthe estimated speeds of adjustment of othervariables of the model in the same way as itbiases the estimates for inventories Second Iconsider an extension of the model where theslope of marginal cost is different across sectorsand ask whether this kind of heterogeneity can

also give rise to the aggregation bias results ofSection III C

A Speed of Adjustment of Labor and Prices

This model focuses on the adjustment speedof finished goods inventories It is interesting toask though whether cyclical changes in mark-ups will also generate an econometric aggrega-tion bias of the kind described in this paper inthe estimated adjustment speeds of other vari-ables of interest to macroeconomists Extendingthe analysis to variables other than inventoriesalso represents a further consistency check forthe mechanism emphasized in this paper Infact for many macroeconomic variables suchas aggregate employment and prices speeds ofadjustment estimated using aggregate data tendto be relatively small (see eg Robert HTopel 1982 Oliver J Blanchard 1987) It alsoappears that considering more disaggregatedunits such as firm and sectors leads to higherestimates of adjustment speeds for these vari-ables than what is obtained with aggregate data(see Caballero and Engel 2003 Blanchard1987)34

The following two sections extend the anal-ysis of Sections II and III to consider the ad-justment speeds of labor demand and prices Inwhat follows I show that the omission of mark-ups from partial adjustment regressions for pricesand the labor input might lead to downward-biased estimates of adjustment speeds particularlywhen using aggregate data

Labor DemandmdashIn this simple model thedemand for labor as a function of output is inlinearized form given by

(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y

where L denotes the steady state value of labor

33 I thank an anonymous referee for suggesting theseextensions

34 Caballero and Engel (2003) show how failure to ac-count for (S s)-type of adjustment policies used by firmswhen estimating partial adjustment models at the firm levelresults in an upward bias in the estimates of adjustmentspeeds for employment and prices In their model aggre-gation across firms tends to mitigate this bias

TABLE 5mdashESTIMATES OF THE SALES SURPRISE PARAMETER

(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)

Notes ldquotruerdquo sales-surprise parameter implied by thebenchmark model in the partial adjustment equation forinventories The estimates of have been obtained bysimulating the benchmark model and estimating a partialadjustment equation for inventories for 1000 times Thesample size of each regression is 100 E[] average esti-mate of obtained using aggregate data E[k] averageestimate of obtained using data on a sector krsquos firm k 1 2 The standard deviation of the estimates across simu-lations is reported in parentheses


and Akt1 Ikt1 is simply output at t 135

Using equations (9) and (11) to replace Akt1and Ikt1 into equation (26) it is possible torewrite (26) in partial adjustment form

(27) Lkt 1 Lkt Lkt 1 Lkt

where the labor target Lkt1 is defined as36

(28) Lkt 1 l lSkt 1 lSkt

lMkt 1 Mkt

The parameters in equation (27) are functionsof the structural parameters of the model and arereported in Appendix B It is easy to show thatthe ldquotruerdquo speed of adjustment of labor towardits target is equal to ie the ldquotruerdquo speed ofadjustment of inventories toward their targetThis is not a coincidence Firms adjust theirinventory stocks by changing production and inthis model labor is approximately a linear func-tion of production Therefore the speed of ad-justment of labor and output coincides with thespeed of adjustment of inventories The target

equation for labor depends on sales and mark-ups in two consecutive periods37

The omission of markups from the targetfor labor (28) generates similar biases to theones discussed in Section II in relation toinventories In Table 6 I report the estimatesof generated by a version of equation (27)that ignores variations in price markups38 Asthe table shows countercyclical changes inmarkups tend to generate relatively small es-timates of adjustment speeds for labor de-mand in sector one (column [3]) Moreoverfor higher values of the adjustment speedsestimated from aggregate data (column [2])are generally lower than the weighted averageof their firm-level counterparts (column [5])As a reference for comparison if markupswere constant in both sectors the estimatedspeeds of adjustment in all columns of Table

35 This expression for labor demand is obtained by treat-ing the cost function (4) as a linear-quadratic approximationof the cost function implied by a production function of thetype Y L| for some | 1

36 Notice that for simplicity I have not distinguishedbetween expected and unexpected variations in sales andmarkups

37 Notice that while the beginning-of-the-period inven-tory stock Ikt1 does not appear explicitly in equation (28)its effect on L

kt1 is captured in part by Skt It is possibleto show in fact that the parameter l is negative higherperiod t sales lead to higher end-of-the-period inventorystocks Ikt1 which generates a lower target for labor inperiod t 1 This dependence has been directly explored byTopel (1982) John C Haltiwanger and Maccini (1989) andRamey (1989) among others with mixed evidence Signif-icant effects have been found by Haltiwanger and Maccini(1989 page 342) who estimate that higher initial invento-ries lead to lower hours per worker and more layoffsespecially in the nondurables sector

38 The description of the different columns of this tablematches exactly the one for Table 4 so it is omitted

TABLE 6mdashESTIMATES OF AGGREGATE AND FIRM-LEVEL ADJUSTMENT SPEEDS FOR LABOR (BENCHMARK MODEL)

True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)

Notes ldquotruerdquo speed of adjustment of labor implied by the model The estimates of have been obtained by simulating thebenchmark model and estimating a partial adjustment equation for labor for 1000 times The sample size of each regressionis 100 E[] average estimate of obtained using aggregate data E[k] average estimate of obtained using data on a sectorkrsquos firm k 1 2 The standard deviation of the estimates across simulations is reported in parentheses


6 would be equal to 047 and the aggregationbias would disappear

From a quantitative point of view the re-sults concerning the aggregation bias for la-bor are not as large as those for inventoriesTherefore inventories seem to adjust moreslowly than labor to a target that dependsexclusively on sales Empirically the rela-tionship between the relative speeds of adjust-ment of finished goods inventories and laborover the business cycle has been explored byseveral authors39 The general motivation ofthese papers is the observation that firms mayadjust to a cyclical decline in demand for theirproduct by choosing some combination oflower labor and higher inventories It seemsthat this literature has not reached a consensuson this issue with some authors (see egHaltiwanger and Maccini 1989) arguing thatthe labor input adjusts faster than inventorystocks to demand shocks and others (egEichenbaum 1984) arriving at the oppositeconclusion The goal of this section is not toprovide new evidence on this controversy asboth the estimated partial adjustment regres-sions for inventories and labor are affected bythe omission of markups Instead this sectionpoints out how the omission of markups frompartial adjustment equations for labor mightalso lead to a downward bias in estimatedspeeds of adjustment of this input

PricesmdashIn this model cyclical variations inthe elasticity of demand induce firms to reducetheir price relative to expected future marginalcost in an expansion and increase it in a reces-sion Failure to account for these cyclical changesin the elasticity of demand when estimatingpartial adjustment equations for prices tends to

give rise to the same type of results alreadyemphasized in the previous section with regardto labor

The linearized first-order condition for pricesis given by

(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1

The second term on the right-hand side of thisequation captures the effect of changes in theelasticity of demand on prices while the thirdterm reflects the effect of variations in marginalcosts of production It is easy but lengthy toshow that equation (29) can be rewritten inpartial adjustment form as

Pkt 1 Pkt Pkt 1 Pkt

where the price target is defined as40

Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt

As in the previous section the ldquotruerdquo speed ofadjustment of prices toward a target that de-pends on sales and markups is equal to theldquotruerdquo speed of adjustment of inventories andlabor This is not surprising as both marginalcost of productionmdashon which prices are basedmdashand labor depend linearly on output

For the same reason illustrated above for thelabor input if countercyclical markups areomitted from the partial adjustment regressionthe estimated speed of adjustment of Pkt1 to-ward target will tend to be smaller than Table7 reports the average estimates for sectors oneand two as well as the speed of adjustment ofthe aggregate price index Pt where Pt is definedas Pt P1t

P2t1 In this case the quantita-

tive results are quite large In particular if ishigh enough the speed of adjustment of theaggregate price index tends to be smaller than

39 See the references in footnote 37 In the data for bothdurable and nondurable goods sectors aggregate hours tendto lead the aggregate inventory stock over the businesscycle In particular for nondurable (durable) goods themaximum correlation between the Index of AggregateWeekly Hours (published by the Bureau of Labor Statistics)in month t and the inventory stock at the beginning of montht k is equal to 037 (047) and occurs at k 6 (k 12)In the benchmark version of the model considered here thecorrelation between Lt and Itk for k 6 is 049 which isclose to the data In the model however the maximumcorrelation between Lt and Itk occurs at k 1 and is equalto 079

40 The parameters of this equation are a function of thestructural parameters of the model The exact expressionsare available from the author upon request


the weighted average of the firm-level adjust-ment speeds

This result is consistent with the evidencepresented by Blanchard (1987) He first esti-mated separate price adjustment equations forseven two-digit manufacturing industries Hethen aggregated the sectoral data to constructaggregate price and wage indices and usedthem to estimate an aggregate price adjustmentequation The sectoral speeds of price adjust-ment to a wage shock found by Blanchard weresignificantly faster than the aggregate one withthe former exceeding the latter by about 30percent on average41

B Heterogeneous Slopes of Marginal Cost

This section discusses an alternative interpre-tation of the evidence that inventory speeds ofadjustment estimated using aggregate data tendto be smaller than the ones obtained using microdata Instead of focusing on the omission of themarkup variable Mkt from standard partial ad-justment regressions I ask whether the sameevidence might be explained by differences inthe slopes of marginal cost across sectors Thisis a natural alternative explanation as produc-tion technologies or the cyclical properties ofinput prices might differ across sectors It turnsout that the answer to this question is positive

This alternative interpretation has different im-plications however for the use of aggregatemodels of inventories with respect to the oneoffered thus far

Consider a simple extension of the modeldeveloped in Section I that allows for heteroge-neity across sectors in the parameter govern-ing the slope of the marginal cost functionAssume that sector-one firms are characterizedby steeper marginal cost functions than sectortwo firms 1 2 For simplicity supposealso that markups in the two sectors are constantover the business cycle (13k 0 for k 1 2)In this case the firm-level partial adjustmentregressions (15) are correctly specified andtherefore the least square estimators k of firm-level adjustment speeds are always equal to theldquotruerdquo sectoral adjustment speeds k 1 k1for k 1 242 Moreover 1 2 because theslope of marginal cost is higher in sector onethan in sector two

Notice that in this economy the coefficientsin the equilibrium law of motion for inventories(see equation [12]) are sector-specific There-fore the firm-level partial adjustment equationscannot be aggregated into an aggregate partialadjustment equation of the form

41 Blanchard (1987) explained these results by suggest-ing that individual price setters adjust their prices quicklybut that interactions among them leads to slow adjustment atthe aggregate level

42 The parameter k1 generalizes the parameter 1 inequation (11) to the case where 1 2 It can be obtainedby replacing with k in the expressions in Appendix B Inthis section markups are assumed to be constant in order tohighlight the effects of heterogeneous slopes of marginalcosts on the estimates of aggregate adjustment speeds In-troducing cyclical variations in markups across sectors doesnot significantly affect the quantitative results of Table 8

TABLE 7mdashESTIMATES OF AGGREGATE AND FIRM-LEVEL ADJUSTMENT SPEEDS FOR PRICES (BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)

Notes ldquotruerdquo speed of adjustment of prices implied by the model The estimates of have been obtained by simulatingthe benchmark model and estimating a partial adjustment equation for prices for 1000 times The sample size of eachregression is 100 E[] average estimate of obtained using aggregate data E[k] average estimate of obtained using dataon a sector krsquos firm k 1 2 The standard deviation of the estimates across simulations is reported in parentheses


(30) It 1 It Ste It

St Ste t

This means that while it is still possible tointerpret the firm-level equations (11) in termsof a traditional partial adjustment model with aconstant adjustment speed no such interpreta-tion can be made in regard to the equilibriumequation for aggregate inventories obtained bysumming equations (11) across firms43

It is of course still possible to try to estimatethe parameter in equation (30) As shown byHenri Theil (1954) the estimator is aweighted average of all the firm-level parame-ters of the model not only the adjustmentspeeds k but also the parameters k and k Inparticular it is easy to show that the differencebetween the weighted average of firm-levelspeeds of adjustment and the expectation of isgiven by

(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1

where x x and x denote the coefficient onaggregate inventories It in a regression of (re-spectively) I1t S1t

e and (S1t S1te ) on the three

aggregate variables It Ste and St The economet-

ric aggregation bias (the left-hand side of [31])can then be attributed using Theilrsquos languageto the effect of ldquocorrespondingrdquo and ldquononcorre-spondingrdquo microparameters The former is rep-resented by the first term on the right-hand sideof equation (31) while the latter is representedby the second and third terms While the bias

due to noncorresponding microparameters isdifficult to interpret the bias due to correspond-ing microparameters is relatively straightfor-ward As an example according to equation(31) the weighted average of firm-level speedsof adjustment tends to exceed E[] if ceterisparibus E[ x] 1 This condition is morelikely to be verified when after conditioning onaggregate sales and sales expectations the in-ventory stock in sector one is more volatile thanthe average inventory stock A sufficient condi-tion for this is that sector-one firms smoothproduction through accumulation and decumu-lation of inventories more than sector-two firmsie 1 2

To make the calibrated version of the modelcomparable to the one in Section I I set 1 and2 so that the speeds of adjustment in the twosectors are the same as the ones estimated forthe benchmark model and reported in Table4 In order to obtain 1 013 and 2 045the slopes of marginal cost in the two sectorshave to be such that 1 105 and 2 0057The other parameters of the model are the sameas in Section IIIA The aggregate statistics im-plied by this version of the model are remark-ably similar to those of the benchmark economy(Tables 2 and 3) and are omitted for simplicityThe results of the Montecarlo experiment forthis version of the model are reported in Table8 This table also reports Theilrsquos decompositionof the bias on the left-hand side of equation (31)into the effects of corresponding and noncorre-sponding microparameters They are repre-sented by the columns (C) and (NC) in Table 8

Two observations are in order First noticethat in Table 8 the average estimate of ob-tained with aggregate data is lower than theweighted average of the firm-level speeds ofadjustment The size of the bias is large even ifsmaller than the one reported in Table 4 for thebenchmark version of the model44 This result isvery sensitive however to small variations inthe slope parameter 2 For example if 010 when 2 goes from 0057 to 0051 (0059)the adjustment speed 2 goes from 045 to 046(045) but the average value of increases(decreases) dramatically from 027 to 066

43 It is still possible to organize the aggregate data inpartial adjustment form with time-varying ldquoparametersrdquo Inthis case the ldquotruerdquo adjustment speed at the aggregate levelwould be a weighted average of 1 and 2 with time-varyingweights represented respectively by the shares (at each pointin time) of sector one and tworsquos inventories in the aggregateSchuh (1996) pursues this point further and shows how amodel of this sort can fit the data better than the estimatedaggregate model (30)

44 The estimates of the ldquosales surpriserdquo parameter areclose to the ones in Table 5


(013) Second Theilrsquos decomposition revealsthat the bias on the left-hand side of (31) whileclose to the empirical evidence contained inSchuh (1996) is the sum of a large positive biasdue to corresponding microparameters and alarge negative bias due to noncorrespondingmicroparameters Therefore while the bias dueto corresponding microparameters is indeedpositive as suggested above the overall bias tendsto be of the correct magnitude only because of thecompensating effect of noncorresponding micro-parameters on

More generally the results of Table 8 suggestthat the relatively small estimates of aggregateinventoriesrsquo speeds of adjustment are consistentwith two alternative stories According to thefirst one the target equation for inventories instandard partial adjustment regressions is mis-specified because it fails to control for varia-tions in price markups Heterogeneity in thecyclical variations of price markups gives rise torelatively small estimates of aggregate speeds ofadjustment and on average larger estimates offirm-level speeds of adjustment According tothe second story the explanation for small es-timated speeds of adjustment at the aggregatelevel is not misspecification of target stocks butstructural heterogeneity across firms The latterimplies that there is no ldquotruerdquo aggregate adjust-ment speed parameter and the average value of is a complex combination of all structuralparameters of the model

Given the lack of significant evidence of pro-

duction-smoothing behavior in the data and thefact that the properties of the production-smoothing model are relatively well under-stood this paper has emphasized more theexplanation based on heterogeneous markupsFuture empirical work will have to differentiateamong these explanations by direct measure-ment of marginal costs45 Despite their differ-ences however these two stories share oneunderlying theme a small subset of sectors inthe economy might exert a disproportionate im-pact on the estimates of the parameters of ag-gregate partial-adjustment models

V Concluding Comments

In this paper I suggest a common explanationfor two puzzles in the inventory literature Theexplanation reconciles the small estimates ofadjustment speeds of aggregate inventorystocks with the apparent rapid response of firmsto aggregate sales surprises It also reconcilesthese small estimates with the larger ones ob-

45 In this regard it is interesting to note that this kind ofidentification problem between variations in price markupsand intertemporal substitution in production is also presentin Bils and Kahn (2000) Since they measure marginal costsdirectly they are able to resolve this issue Interestinglythey conclude that cyclical variations in price markupsrather than intertemporal substitution in production ex-plains the behavior of inventories for the six manufacturingindustries they consider

TABLE 8mdashESTIMATES OF AGGREGATE AND FIRM-LEVEL ADJUSTMENT SPEEDS FOR INVENTORIES

(HETEROGENEOUS COST MODEL)

Aggregatedata

Firm-level data Bias

1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)

Notes Estimates have been obtained by simulating the heterogeneous cost model and estimating a partial adjustment equationfor inventories for 1000 times The sample size of each regression is 100 E[] average estimate of the adjustment speed ofinventories obtained using aggregate data E[k] average estimate of k obtained using data on a sector krsquos firm k 1 2Notice that for each simulation k k because the firm-level equations are correctly specified The standard deviation ofthe estimates across simulations is reported in parentheses


tained with firm-level data I show qualitativelyand quantitatively how these results canemerge if countercyclical variations in pricemarkups are omitted from standard partial ad-justment regressions and firms belong to sectorswith different cyclical variations in markupsThe first condition explains the downward biasof estimated speeds of adjustment with respectto their ldquotruerdquo value The second one explainswhy this bias is more severe when aggregateinstead of firm-level data are used

These arguments rationalize the relativelysmall estimates of adjustment speeds of inven-tories to their target found in the literature Theydo not imply however that either aggregate orfirm-level inventories adjust faster to changes insales over the business cycle than previouslythought Countercyclical variations in markupsand procyclical variations in marginal cost giverise to the sluggish movement of inventoriesrelative to sales over the business cycle

It is important to notice that the approach toaggregation developed here differs from otherstudies that have addressed the dichotomy be-tween fast but smooth adjustment of microeco-nomic units to shocks and gradual adjustment atthe aggregate level46 In these studies stagger-ing of decisions and production-chain interac-tions among fast-adjusting micro units leads togradual adjustment at the aggregate level Inthis paper instead interaction among firms

within a sector is of a very limited nature andadjustment decisions are taken simultaneouslyThe advantage of this simple formulation is toallow for aggregation across firms so that thelaws of motion of aggregate and firm-level vari-ables have the same form The upshot is thatdespite this similar structure estimated speedsof adjustment of aggregate and firm-level vari-ables might differ significantly if some key vari-ables are omitted from the empiricalspecification of partial adjustment equations

As far as the inventory literature is concernedthis study confirms the importance of using infor-mation contained in prices to improve the fit oftraditional inventory models (Bils and Kahn2000) To implement this idea it is necessary toallow inventories to affect firmsrsquo revenue moreexplicitly than in the standard linear-quadraticmodel where stockout costs do not depend onmarkups

While inventories are an illustrative specialcase of the ideas emphasized in this papercountercyclical variations in price markupsshould affect all aspects of a firmrsquos behaviorover the business cycle Moreover heteroge-neous variations in price markups across sectorsare likely to give rise to different sectoral re-sponses to common shocks This paper showsthat this heterogeneity might have a profoundinfluence on the inferences made by macro-economists about the behavior not only of ag-gregate inventories but also of labor and pricesPursuing this issue further for these and othervariables is an interesting avenue for futureresearch

APPENDIX A DECISION RULE FOR Akt

In this appendix I report the coefficients of the linearized decision rules for the variable Akt Thecoefficients of equation (9) are found using the method of undetermined coefficients (see egChristiano 2002) The relationship between these coefficients and the structural parameters of themodel is as follows Define

3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A

46 See for example the discussion in Blanchard (1987)and the references therein


where A and S denote the steady state values of Akt and Skt and are given by

A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2

Then the coefficients in equation (9) are

0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A

In particular notice that if production occurs under constant returns to scale (3 0) then 13 With lrsquoHopitalrsquos rule it is easy to show that 1 3 0 Thus when 0 1 0 and 4 0

APPENDIX B PARTIAL ADJUSTMENT EQUATIONS IMPLIED BY THE MODEL

Inventories

The coefficients in the inventory equation (11) are

0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1

In particular notice that if 0 since 1 0 4 0 the adjustment speed coefficient 1 isalso equal to zero (ie adjustment is instantaneous)

Labor

Define

0 L

S1 S 1 LS

where the steady state level of labor is given by L S1S The coefficients of the partial adjustmentequation for labor (equation 27) are

1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES

Bils Mark ldquoThe Cyclical Behavior of MarginalCost and Pricerdquo American Economic Review1987 77(5) pp 838ndash55

Bils Mark ldquoPricing in a Customer MarketrdquoQuarterly Journal of Economics 1989104(4) pp 699ndash718

Bils Mark and Kahn James A ldquoWhat InventoryBehavior Tells Us about Business CyclesrdquoAmerican Economic Review 2000 90(3) pp458ndash81

Binder Michael ldquoCyclical Fluctuations in Oli-gopolistic Industries under HeterogeneousInformation An Empirical Analysisrdquo Un-published Paper 1995

Blanchard Olivier Jean ldquoAggregate and Indi-vidual Price Adjustmentrdquo Brookings Paperson Economic Activity 1987 0(1) pp 57ndash109

Blinder Alan S ldquoMore on the Speed of Adjust-ment in Inventory Modelsrdquo Journal ofMoney Credit and Banking 1986a 18(3)pp 355ndash65

Blinder Alan S ldquoCan the Production SmoothingModel of Inventory Behavior Be SavedrdquoQuarterly Journal of Economics 1986b101(3) pp 431ndash53

Blinder Alan S and Maccini Louis J ldquoTakingStock A Critical Assessment of Recent Re-search on Inventoriesrdquo Journal of EconomicPerspectives 1991 5(1) pp 73ndash96

Caballero Ricardo J ldquoAggregate Investmentrdquoin J B Taylor and M Woodford eds Hand-book of macroeconomics Volume 1b Amster-dam New York and Oxford ElsevierScience North-Holland 1999 pp 813ndash62

Caballero Ricardo J and Engel Eduardo ldquoAd-justment Is Much Slower Than You ThinkrdquoUnpublished Paper 2003

Carlson John A and Dunkelberg William CldquoMarket Perceptions and Inventory-Price-Employment Plansrdquo Review of Economicsand Statistics 1989 71(2) pp 318ndash24

Christiano Lawrence J ldquoSolving DynamicEquilibrium Models by a Method of Unde-termined Coefficientsrdquo Computational Eco-nomics 2002 20(1ndash2)

Domowitz Ian Hubbard R Glenn and PetersenBruce C ldquoOligopoly Supergames Some Em-pirical Evidence on Prices and MarginsrdquoJournal of Industrial Economics 198735(4) pp 379ndash98

Eichenbaum Martin S ldquoRational Expectationsand the Smoothing Properties of Inventoriesof Finished Goodsrdquo Journal of MonetaryEconomics 1984 14(1) pp 71ndash96

Eichenbaum Martin S ldquoSome Empirical Evi-dence on the Production Level and Produc-tion Cost Smoothing Models of InventoryInvestmentrdquo American Economic Review1989 79(4) pp 853ndash64

Feldstein Martin S and Auerbach Alan ldquoInven-tory Behavior in Durable-Goods Manufactur-ing The Target-Adjustment Modelrdquo BrookingsPapers on Economic Activity 1976 2(76) pp351ndash96

Fuhrer Jeffrey C Moore George R and SchuhScott D ldquoEstimating the Linear-Quadratic In-ventory Model Maximum Likelihood versusGeneralized Method of Momentsrdquo Journalof Monetary Economics 1995 35(1) pp115ndash57

Gali Jordi ldquoMonopolistic Competition Busi-ness Cycles and the Competition of Aggre-gate Demandrdquo Journal of Economic Theory1994 63(1) pp 73ndash96

Hall Robert E ldquoThe Relation between Price andMarginal Cost in US Industryrdquo Journal ofPolitical Economy 1988 96(5) pp 921ndash47

Haltiwanger John C and Maccini Louis J ldquoIn-ventories Orders Temporary and PermanentLayoffs An Econometric Analysisrdquo Carnegie-Rochester Conference Series on Public Policy1989 30(0) pp 301ndash66

Holt Charles Modigliani Franco Murth JohnF and Simon Herbert A Planing productioninventories and work force EnglewoodCliffs Prentice Hall 1960

Kahn James A ldquoInventories and the Volatilityof Productionrdquo American Economic Review1987 77(4) pp 667ndash79

Kahn James A ldquoWhy Is Production More Vol-atile Than Sales Theory and Evidence on theStockout-Avoidance Motive for Inventory-Holdingrdquo Quarterly Journal of Economics1992 107(2) pp 481ndash510

Khan Aubhik and Thomas Julia K ldquoInventoriesand the Business Cycle An EquilibriumAnalysis of (S s) Policiesrdquo Federal ReserveBank of Philadelphia Working Papers 02-20 2002

Lovell Michael C ldquoManufacturersrsquo InventoriesSales Expectations and the Acceleration


Principlerdquo Econometrica 1961 29(3) pp147ndash79

Lovell Michael C ldquoSimulating the InventoryCyclerdquo Journal of Economic Behavior andOrganization 1961 21(2) pp 147ndash79

Maccini Louis J and Rossana Robert J ldquoInvest-ment in Finished Goods Inventories AnAnalysis of Adjustment Speedsrdquo AmericanEconomic Review 1981 71(2) pp 17ndash22

Miron Jeffrey A and Zeldes Stephen P ldquoSea-sonality Cost Shocks and the ProductionSmoothing Models of Inventoriesrdquo Econo-metrica 1988 56(4) pp 877ndash908

Morrison Catherine J ldquoMarkups in US andJapanese Manufacturing A Short-Run Econo-metric Analysisrdquo Journal of Business and Eco-nomic Statistics 1992 10(1) pp 51ndash63

Norrbin Stefan C ldquoThe Relation between Priceand Marginal Cost in US Industry A Con-tradictionrdquo Journal of Political Economy1993 101(6) pp 1149ndash64

Ramey Valerie A ldquoInventories as Factors ofProduction and Economic FluctuationsrdquoAmerican Economic Review 1989 79(3) pp338ndash54

Ramey Valerie A ldquoNonconvex Costs and theBehavior of Inventoriesrdquo Journal of PoliticalEconomy 1991 99(2) pp 306ndash34

Ramey Valerie A and West Kenneth D ldquoInven-toriesrdquo in J B Taylor and M Woodfordeds Handbook of macroeconomics Volume1b Amsterdam New York and Oxford

Elsevier Science North-Holland 1999 pp863ndash923

Rotemberg Julio J and Woodford MichaelldquoMarkups and the Business Cyclerdquo in O JBlanchard and S Fischer eds NBER mac-roeconomics annual 1991 Cambridge andLondon MIT Press 1991 pp 63ndash129

Rotemberg Julio J and Woodford MichaelldquoThe Cyclical Behavior of Prices and Costsrdquoin J B Taylor and M Woodford eds Hand-book of macroeconomics Volume 1b Amster-dam New York and Oxford Elsevier ScienceNorth-Holland 1999 pp 1051ndash135

Schuh Scott ldquoEvidence on the Link betweenFirm-Level and Aggregate Inventory Behav-iorrdquo Board of Governors of the Federal Re-serve System (US) Finance and EconomicsDiscussion Series 96-46 1996

Seitz Helmut ldquoStill More on the Speed of Ad-justment in Inventory Models A Lesson inAggregationrdquo Empirical Economics 199318(1) pp 103ndash27

Theil Henri Linear aggregation of economicrelations Amsterdam North-Holland 1954

Topel Robert H ldquoInventories Layoffs and theShort-Run Demand for Laborrdquo AmericanEconomic Review 1982 72(4) pp 769ndash87

West Kenneth D ldquoA Variance Bounds Test ofthe Linear Quadratic Inventory ModelrdquoJournal of Political Economy 1986 94(2)pp 374ndash401


larger than their counterparts estimated usingaggregate data Specifically he finds that ac-cording to the empirical specification of themodel the weighted average of adjustmentspeeds estimated using firm-level data is 67 to105 percent larger than the one obtained usingaggregated data constructed from the samepanel of firms Despite this evidence the effectof aggregation across heterogeneous firms is notwell understood yet3

I suggest a unified explanation for Feldsteinand Auerbachrsquos and Schuhrsquos inventory puzzlesThe explanation relies on the idea that firmsrsquoinventory targets might change systematicallyover the business cycle for reasons other thanvariations in expected sales as usually assumedin the literature Following the empirical evi-dence presented by Bils and Kahn (2000) thespecific mechanism leading to variations in in-ventory targets is countercyclical movements inprice markups The paper shows qualitativelyand quantitatively how the omission of mark-ups from standard partial adjustment equationsresults in downward-biased estimates of adjust-ment speeds The bias is more severe in regres-sions that use aggregate rather than firm-leveldata Moreover while inventories are an illus-trative special case of these mechanisms theirapplicability extends to other aspects of firmsrsquobehavior such as labor demand and price set-ting of interest to macroeconomists

To make these points I consider a finishedgoods inventory model in the spirit of Bils andKahn (2000) The key feature distinguishing

this model from the linear-quadratic model ofHolt et al (1960) is that in the former higherinventories contribute to increase firmsrsquo salesand revenues at a given price Thus in thismodel higher price markups induce firms toceteris paribus hold more inventories Theeconomy is composed of two sectors with dif-ferent cyclical volatilities of markups The equi-librium law of motion for the modelrsquos sectoraland aggregate inventories can be written in thestandard partial adjustment form with the inven-tory target being a function of expected salesand price markups The model is calibrated andits simulated firm-level and aggregate inventoryand sales data are then used to estimate partialadjustment equations in which price markupsare omitted from the inventory target The Mon-tecarlo exercise reveals that this omission re-sults in large downward biases in estimatedspeeds of adjustment of aggregate inventoriesrelative to their ldquotruerdquo value implied by themodel For example when sectors displayingthe most volatile markup movements accounton average for only 10 percent of aggregateinventories the ldquotruerdquo speed of adjustment ofinventories is 12 percent higher than theweighted average of firm-level estimates and327 percent higher than the estimate obtainedfrom aggregate data (ie Schuhrsquos puzzle) Theaverage value of the latter implies that in onequarter firms eliminate only 30 percent of thegap between current and desired inventories Incontrast the estimates of the effects of salessurprises on inventory investment suggest thatfirms are able to correct completely unantici-pated sales shocks within the quarter (ie Feld-stein and Auerbachrsquos puzzle)

The intuition behind these results is as fol-lows If price markups are countercyclical theiromission from inventory targets leads standardpartial adjustment models to overpredict the gapbetween desired and current stocks during peri-ods of economic expansion and to underpredictit in recessions Therefore these models tend toattribute the discrepancy between the desiredand observed changes in inventories to the factthat firms are adjusting slowly In contrast inthis model countercyclical variations in pricemarkups induce firms to reduce their targetinventories relative to sales in an expansion andincrease them in a recession According to thisview the desired stock of inventories should

3 Other authors have also argued that estimated speeds ofadjustment tend to be higher at lower levels of aggregationUsing data on German manufacturing firms Helmut Seitz(1993) estimates average adjustment speeds at the firm-levelthat are more than twice as large as the one obtained byrunning the regression with aggregated data from the samepanel Blinder (1986a) finds that adjustment speeds esti-mated with aggregate data for the durable and nondurablegoods sectors tend to be lower than the ones estimated withdata from their constituent industries Lovell (1993) showshow aggregation may bias adjustment speedsrsquo estimatesdownward by means of a simulation approach He uses areduced-form model however where firms donrsquot optimizeand parameters are not calibrated Moreover he doesnot provide an explanation for this result John A Carlsonand William C Dunkelberg (1989) using firm-level dataon small businesses in the United States find that firms onaverage fully adjust their inventories to target within aquarter

















I The Model Economy






A Setup








(1)

sktj t pkt

j

Pktkt

aktj 0 1 kt 1



(2) t t 1 ut 0 1









(3)

kt 1 11 kt 1 1ut13k 1




(4) cyktj ykt

j

2ykt

j 2





(5) ikt 1j akt

j sktj

with aktj ikt














maxpkt

j aktj

E0t 0

tpktj skt

j aktj akt 1

j skt 1j

2akt

j akt 1j skt 1

j 2subject to skt

j tpktj

Pktkt

aktj i0 0 0 0



(6) aktj akt 1

j skt 1j

pktj skt

j

aktj 1

sktj

aktj

Et akt 1j akt

j sktj

(7) kt 1pktj

ktEt akt 1j akt

j sktj









(8) Mkt pkt

j

Et akt1j akt

j sktj

1

1

kt 1




j ikt1j skt

j and yktj





(9) Akt 0 1 Akt 1 2 Mkt

3t 4t 1 k 1 2




At 0 1 At 1 2 Mt 3t 4t 1


At A1t 1 A2t

Mt M1t 1 M2t


II Mechanisms




(10) Skt St S

AAkt A



Akt t

kt 11 11

and

Pkt c

1 kt 1









(12)



(13) Ikt 1 Skte Mkt


1 1 3 1 1 10

1 1 13 1 1 12



It I1t 1 I2t

St S1t 1 S2t

Ste S1t

e 1 S2te












(15) Zk Xk k






(17) Zk Xk Mk2


(18) k qk2


qk(X13kXk)1X13kMk


(19) k qk22






(20) X13X1X13Z


(21) q22



(22) E1 1 2

2Eq12 1 q22 q2






(23) E1 1 2

21 Eq12











A Calibration










M 1

1



A

S

11








Parameter 131 132

Value 098 0015 095 051 1638 097 005 092 010 076 0063












Mkt M1 Mkt 1 ut








varY


Benchmark model 101 019 099 099 002 006(000) (006) (000) (000) (000) (001)







(24) Ikt 1 Skte




(25) Skte S1 Skt 1







Skt S1 Skt 1 Sut 1




k 1 k 3 k 6 k 1 k 3 k 6

Benchmark model 093 082 069 092 080 067(002) (006) (010) (002) (006) (010)












T ln 005

ln1 x


True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)




IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES




























































I The Model Economy






A Setup








(1)

sktj t pkt

j

Pktkt

aktj 0 1 kt 1



(2) t t 1 ut 0 1









(3)

kt 1 11 kt 1 1ut13k 1




(4) cyktj ykt

j

2ykt

j 2





(5) ikt 1j akt

j sktj

with aktj ikt














maxpkt

j aktj

E0t 0

tpktj skt

j aktj akt 1

j skt 1j

2akt

j akt 1j skt 1

j 2subject to skt

j tpktj

Pktkt

aktj i0 0 0 0



(6) aktj akt 1

j skt 1j

pktj skt

j

aktj 1

sktj

aktj

Et akt 1j akt

j sktj

(7) kt 1pktj

ktEt akt 1j akt

j sktj









(8) Mkt pkt

j

Et akt1j akt

j sktj

1

1

kt 1




j ikt1j skt

j and yktj





(9) Akt 0 1 Akt 1 2 Mkt

3t 4t 1 k 1 2




At 0 1 At 1 2 Mt 3t 4t 1


At A1t 1 A2t

Mt M1t 1 M2t


II Mechanisms




(10) Skt St S

AAkt A



Akt t

kt 11 11

and

Pkt c

1 kt 1









(12)



(13) Ikt 1 Skte Mkt


1 1 3 1 1 10

1 1 13 1 1 12



It I1t 1 I2t

St S1t 1 S2t

Ste S1t

e 1 S2te












(15) Zk Xk k






(17) Zk Xk Mk2


(18) k qk2


qk(X13kXk)1X13kMk


(19) k qk22






(20) X13X1X13Z


(21) q22



(22) E1 1 2

2Eq12 1 q22 q2






(23) E1 1 2

21 Eq12











A Calibration










M 1

1



A

S

11








Parameter 131 132

Value 098 0015 095 051 1638 097 005 092 010 076 0063












Mkt M1 Mkt 1 ut








varY


Benchmark model 101 019 099 099 002 006(000) (006) (000) (000) (000) (001)







(24) Ikt 1 Skte




(25) Skte S1 Skt 1







Skt S1 Skt 1 Sut 1




k 1 k 3 k 6 k 1 k 3 k 6

Benchmark model 093 082 069 092 080 067(002) (006) (010) (002) (006) (010)












T ln 005

ln1 x


True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)




IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES














































A Setup








(1)

sktj t pkt

j

Pktkt

aktj 0 1 kt 1



(2) t t 1 ut 0 1









(3)

kt 1 11 kt 1 1ut13k 1




(4) cyktj ykt

j

2ykt

j 2





(5) ikt 1j akt

j sktj

with aktj ikt














maxpkt

j aktj

E0t 0

tpktj skt

j aktj akt 1

j skt 1j

2akt

j akt 1j skt 1

j 2subject to skt

j tpktj

Pktkt

aktj i0 0 0 0



(6) aktj akt 1

j skt 1j

pktj skt

j

aktj 1

sktj

aktj

Et akt 1j akt

j sktj

(7) kt 1pktj

ktEt akt 1j akt

j sktj









(8) Mkt pkt

j

Et akt1j akt

j sktj

1

1

kt 1




j ikt1j skt

j and yktj





(9) Akt 0 1 Akt 1 2 Mkt

3t 4t 1 k 1 2




At 0 1 At 1 2 Mt 3t 4t 1


At A1t 1 A2t

Mt M1t 1 M2t


II Mechanisms




(10) Skt St S

AAkt A



Akt t

kt 11 11

and

Pkt c

1 kt 1









(12)



(13) Ikt 1 Skte Mkt


1 1 3 1 1 10

1 1 13 1 1 12



It I1t 1 I2t

St S1t 1 S2t

Ste S1t

e 1 S2te












(15) Zk Xk k






(17) Zk Xk Mk2


(18) k qk2


qk(X13kXk)1X13kMk


(19) k qk22






(20) X13X1X13Z


(21) q22



(22) E1 1 2

2Eq12 1 q22 q2






(23) E1 1 2

21 Eq12











A Calibration










M 1

1



A

S

11








Parameter 131 132

Value 098 0015 095 051 1638 097 005 092 010 076 0063












Mkt M1 Mkt 1 ut








varY


Benchmark model 101 019 099 099 002 006(000) (006) (000) (000) (000) (001)







(24) Ikt 1 Skte




(25) Skte S1 Skt 1







Skt S1 Skt 1 Sut 1




k 1 k 3 k 6 k 1 k 3 k 6

Benchmark model 093 082 069 092 080 067(002) (006) (010) (002) (006) (010)












T ln 005

ln1 x


True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)




IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES














































(3)

kt 1 11 kt 1 1ut13k 1




(4) cyktj ykt

j

2ykt

j 2





(5) ikt 1j akt

j sktj

with aktj ikt














maxpkt

j aktj

E0t 0

tpktj skt

j aktj akt 1

j skt 1j

2akt

j akt 1j skt 1

j 2subject to skt

j tpktj

Pktkt

aktj i0 0 0 0



(6) aktj akt 1

j skt 1j

pktj skt

j

aktj 1

sktj

aktj

Et akt 1j akt

j sktj

(7) kt 1pktj

ktEt akt 1j akt

j sktj









(8) Mkt pkt

j

Et akt1j akt

j sktj

1

1

kt 1




j ikt1j skt

j and yktj





(9) Akt 0 1 Akt 1 2 Mkt

3t 4t 1 k 1 2




At 0 1 At 1 2 Mt 3t 4t 1


At A1t 1 A2t

Mt M1t 1 M2t


II Mechanisms




(10) Skt St S

AAkt A



Akt t

kt 11 11

and

Pkt c

1 kt 1









(12)



(13) Ikt 1 Skte Mkt


1 1 3 1 1 10

1 1 13 1 1 12



It I1t 1 I2t

St S1t 1 S2t

Ste S1t

e 1 S2te












(15) Zk Xk k






(17) Zk Xk Mk2


(18) k qk2


qk(X13kXk)1X13kMk


(19) k qk22






(20) X13X1X13Z


(21) q22



(22) E1 1 2

2Eq12 1 q22 q2






(23) E1 1 2

21 Eq12











A Calibration










M 1

1



A

S

11








Parameter 131 132

Value 098 0015 095 051 1638 097 005 092 010 076 0063












Mkt M1 Mkt 1 ut








varY


Benchmark model 101 019 099 099 002 006(000) (006) (000) (000) (000) (001)







(24) Ikt 1 Skte




(25) Skte S1 Skt 1







Skt S1 Skt 1 Sut 1




k 1 k 3 k 6 k 1 k 3 k 6

Benchmark model 093 082 069 092 080 067(002) (006) (010) (002) (006) (010)












T ln 005

ln1 x


True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)




IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES
















































maxpkt

j aktj

E0t 0

tpktj skt

j aktj akt 1

j skt 1j

2akt

j akt 1j skt 1

j 2subject to skt

j tpktj

Pktkt

aktj i0 0 0 0



(6) aktj akt 1

j skt 1j

pktj skt

j

aktj 1

sktj

aktj

Et akt 1j akt

j sktj

(7) kt 1pktj

ktEt akt 1j akt

j sktj









(8) Mkt pkt

j

Et akt1j akt

j sktj

1

1

kt 1




j ikt1j skt

j and yktj





(9) Akt 0 1 Akt 1 2 Mkt

3t 4t 1 k 1 2




At 0 1 At 1 2 Mt 3t 4t 1


At A1t 1 A2t

Mt M1t 1 M2t


II Mechanisms




(10) Skt St S

AAkt A



Akt t

kt 11 11

and

Pkt c

1 kt 1









(12)



(13) Ikt 1 Skte Mkt


1 1 3 1 1 10

1 1 13 1 1 12



It I1t 1 I2t

St S1t 1 S2t

Ste S1t

e 1 S2te












(15) Zk Xk k






(17) Zk Xk Mk2


(18) k qk2


qk(X13kXk)1X13kMk


(19) k qk22






(20) X13X1X13Z


(21) q22



(22) E1 1 2

2Eq12 1 q22 q2






(23) E1 1 2

21 Eq12











A Calibration










M 1

1



A

S

11








Parameter 131 132

Value 098 0015 095 051 1638 097 005 092 010 076 0063












Mkt M1 Mkt 1 ut








varY


Benchmark model 101 019 099 099 002 006(000) (006) (000) (000) (000) (001)







(24) Ikt 1 Skte




(25) Skte S1 Skt 1







Skt S1 Skt 1 Sut 1




k 1 k 3 k 6 k 1 k 3 k 6

Benchmark model 093 082 069 092 080 067(002) (006) (010) (002) (006) (010)












T ln 005

ln1 x


True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)




IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES














































(9) Akt 0 1 Akt 1 2 Mkt

3t 4t 1 k 1 2




At 0 1 At 1 2 Mt 3t 4t 1


At A1t 1 A2t

Mt M1t 1 M2t


II Mechanisms




(10) Skt St S

AAkt A



Akt t

kt 11 11

and

Pkt c

1 kt 1









(12)



(13) Ikt 1 Skte Mkt


1 1 3 1 1 10

1 1 13 1 1 12



It I1t 1 I2t

St S1t 1 S2t

Ste S1t

e 1 S2te












(15) Zk Xk k






(17) Zk Xk Mk2


(18) k qk2


qk(X13kXk)1X13kMk


(19) k qk22






(20) X13X1X13Z


(21) q22



(22) E1 1 2

2Eq12 1 q22 q2






(23) E1 1 2

21 Eq12











A Calibration










M 1

1



A

S

11








Parameter 131 132

Value 098 0015 095 051 1638 097 005 092 010 076 0063












Mkt M1 Mkt 1 ut








varY


Benchmark model 101 019 099 099 002 006(000) (006) (000) (000) (000) (001)







(24) Ikt 1 Skte




(25) Skte S1 Skt 1







Skt S1 Skt 1 Sut 1




k 1 k 3 k 6 k 1 k 3 k 6

Benchmark model 093 082 069 092 080 067(002) (006) (010) (002) (006) (010)












T ln 005

ln1 x


True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)




IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES


















































(12)



(13) Ikt 1 Skte Mkt


1 1 3 1 1 10

1 1 13 1 1 12



It I1t 1 I2t

St S1t 1 S2t

Ste S1t

e 1 S2te












(15) Zk Xk k






(17) Zk Xk Mk2


(18) k qk2


qk(X13kXk)1X13kMk


(19) k qk22






(20) X13X1X13Z


(21) q22



(22) E1 1 2

2Eq12 1 q22 q2






(23) E1 1 2

21 Eq12











A Calibration










M 1

1



A

S

11








Parameter 131 132

Value 098 0015 095 051 1638 097 005 092 010 076 0063












Mkt M1 Mkt 1 ut








varY


Benchmark model 101 019 099 099 002 006(000) (006) (000) (000) (000) (001)







(24) Ikt 1 Skte




(25) Skte S1 Skt 1







Skt S1 Skt 1 Sut 1




k 1 k 3 k 6 k 1 k 3 k 6

Benchmark model 093 082 069 092 080 067(002) (006) (010) (002) (006) (010)












T ln 005

ln1 x


True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)




IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES














































(15) Zk Xk k






(17) Zk Xk Mk2


(18) k qk2


qk(X13kXk)1X13kMk


(19) k qk22






(20) X13X1X13Z


(21) q22



(22) E1 1 2

2Eq12 1 q22 q2






(23) E1 1 2

21 Eq12











A Calibration










M 1

1



A

S

11








Parameter 131 132

Value 098 0015 095 051 1638 097 005 092 010 076 0063












Mkt M1 Mkt 1 ut








varY


Benchmark model 101 019 099 099 002 006(000) (006) (000) (000) (000) (001)







(24) Ikt 1 Skte




(25) Skte S1 Skt 1







Skt S1 Skt 1 Sut 1




k 1 k 3 k 6 k 1 k 3 k 6

Benchmark model 093 082 069 092 080 067(002) (006) (010) (002) (006) (010)












T ln 005

ln1 x


True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)




IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES














































(23) E1 1 2

21 Eq12











A Calibration










M 1

1



A

S

11








Parameter 131 132

Value 098 0015 095 051 1638 097 005 092 010 076 0063












Mkt M1 Mkt 1 ut








varY


Benchmark model 101 019 099 099 002 006(000) (006) (000) (000) (000) (001)







(24) Ikt 1 Skte




(25) Skte S1 Skt 1







Skt S1 Skt 1 Sut 1




k 1 k 3 k 6 k 1 k 3 k 6

Benchmark model 093 082 069 092 080 067(002) (006) (010) (002) (006) (010)












T ln 005

ln1 x


True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)




IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES


















































M 1

1



A

S

11








Parameter 131 132

Value 098 0015 095 051 1638 097 005 092 010 076 0063












Mkt M1 Mkt 1 ut








varY


Benchmark model 101 019 099 099 002 006(000) (006) (000) (000) (000) (001)







(24) Ikt 1 Skte




(25) Skte S1 Skt 1







Skt S1 Skt 1 Sut 1




k 1 k 3 k 6 k 1 k 3 k 6

Benchmark model 093 082 069 092 080 067(002) (006) (010) (002) (006) (010)












T ln 005

ln1 x


True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)




IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES






















































Mkt M1 Mkt 1 ut








varY


Benchmark model 101 019 099 099 002 006(000) (006) (000) (000) (000) (001)







(24) Ikt 1 Skte




(25) Skte S1 Skt 1







Skt S1 Skt 1 Sut 1




k 1 k 3 k 6 k 1 k 3 k 6

Benchmark model 093 082 069 092 080 067(002) (006) (010) (002) (006) (010)












T ln 005

ln1 x


True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)




IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES

















































varY


Benchmark model 101 019 099 099 002 006(000) (006) (000) (000) (000) (001)







(24) Ikt 1 Skte




(25) Skte S1 Skt 1







Skt S1 Skt 1 Sut 1




k 1 k 3 k 6 k 1 k 3 k 6

Benchmark model 093 082 069 092 080 067(002) (006) (010) (002) (006) (010)












T ln 005

ln1 x


True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)




IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES














































(24) Ikt 1 Skte




(25) Skte S1 Skt 1







Skt S1 Skt 1 Sut 1




k 1 k 3 k 6 k 1 k 3 k 6

Benchmark model 093 082 069 092 080 067(002) (006) (010) (002) (006) (010)












T ln 005

ln1 x


True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)




IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES





















































T ln 005

ln1 x


True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 011 013 045 042 031(011) (004) (005) (005) (009)

030 047 007 013 045 036 029(002) (004) (005) (004) (003)

060 047 013 013 045 026 013(004) (004) (005) (004) (001)

090 047 013 013 045 016 003(004) (004) (005) (004) (000)




IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES














































IV Extensions







(26)

Lkt 1 L L

S1 SAkt 1 Ikt 1 Y





(BENCHMARK MODEL)

True

Aggregate dataE[]

Firm-level data

E[1] E[2]

010 010 004 032 007(000) (001) (000)

030 010 003 032 007(000) (001) (000)

060 010 015 032 007(001) (001) (000)

090 010 027 032 007(001) (001) (000)








lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES


















































lMkt 1 Mkt










True

Aggregatedata E[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[1] E[1 (1 )2]

010 047 046 035 046 045 001(000) (001) (000) (000) (000)

030 047 038 035 046 043 004(002) (001) (000) (000) (002)

060 047 039 035 046 040 001(001) (001) (000) (000) (000)

090 047 036 035 046 036 000(001) (001) (000) (000) (000)








(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES


















































(29) Pkt 1 1 1 MS Mkt

1 MEtYkt 1


Pkt 1 Pkt Pkt 1 Pkt


Pkt 1 p pSkt 1 pSkt

pMkt 1 pMkt


















True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES























































True

Aggregate dataE[]

Firm-level dataBias

E[1 (1 )2 ]E[1] E[2] E[1 (1 )2]

010 047 020 001 038 034 014(005) (002) (003) (003) (003)

030 047 001 001 038 026 025(001) (002) (003) (003) (003)

060 047 003 001 038 015 012(005) (002) (003) (002) (003)

090 047 001 001 038 004 003(003) (002) (003) (002) (001)



(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES













































(30) It 1 It Ste It

St Ste t



(31) E1 1 2

1 Ex1 2

Ex22 11

Ex2 1




















Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES






















































Aggregatedata


1 2 1 (1 )2 1 (1 )2 C NC

010 027 013 045 042 015 049 034(010) (000) (000) (000) (010) (030) (021)

030 026 013 045 035 009 033 024(000) (000) (000) (000) (000) (000) (000)

060 022 013 045 026 004 016 012(000) (000) (000) (000) (000) (000) (000)

090 016 013 045 016 000 003 003(000) (000) (000) (000) (000) (000) (000)











3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES





















































3 S

A

SM

A SM

1 13 2 S

A

2 1 S

A4 3

S

A5

S

A




A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES














































A M

1 11

S A

Also define

1 1 1

2 42

22

2

11

3

3 2

111

5

2 4 4 1

5

2


0 A1 1 2 3 4

1 1 2 2

A

M3 3 A 4 4 A



Inventories


0 1 3

A1

0 3 4 1 1 3

A11

4

A2 1

3

A1

2 3 1 3

A1

3

1

S 1


Labor

Define

0 L

S1 S 1 LS


1 1 l 0 l 1 1 11 1 3

l 1 1113 1 l 1 1112


REFERENCES













































REFERENCES













































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