Smart Systems and Simple Agents: Industry Pricing by Parallel Rules

SMART SYSTEMS ANDSIMPLEAGENTS: IndustryPricingbyParallelRules RaymondBoardandP.A.Tinsley (cid:3) Version: December1996 Abstract: A standard macroeconomic specification is that the aggregate economy is directed by a single, smart representative agent using optimal decision rules. This paper explores an alternative conjecture that the dynamic behavior of markets is often better interpreted as the interactions of many heterogeneous, rule-of-thumb agents who are loosely-coupled in smart systems—much like the contrast of a single serial processor with global information versus parallel processors with limited communications. The illustration used in this paper is the contrast between a conventional macro model of sluggish adjustments in an aggregate producer price index and a model of delayed industry price adjustments in a distributed production systemundercostlyinter-firmcommunications. Keywords: Costlycommunications; parallel Jacobisolutions; producer pricing. (cid:3) Author addresses are: Federal Reserve Board, Washington, DC 20551, rboard@frb.govand ptinsley@frb.gov. A variant of this paper appears in Gilli (1996). Views expressed in this paper are those of the authors and do not necessarilyrepresentthoseoftheBoardofGovernorsoftheFederalReserveSystemoritsstaff.

1 1. Introduction Dynamicfirst-orderconditions,suchasEulerequations,areusefulabstractionsineconometric modeling to motivatedistributedlag responses to unanticipated shocks. However, the assumption that sectoral aggregates or entire economies may be viewed “as if” they are directed by a single, omniscientrepresentativeagent,whoissubjectedtocostsofadjustingactions,isanuncomfortable metaphor for macroeconomics. This is especially so in dynamic analysis of aggregate prices wherenot onlytheallocativeroleofprices isblurredbyaggregationbutsignificantdirect costsof adjustingprices are hardtoidentify. The effect of transforming the “invisible hand” into the “representative agent” is to replace system conditions for survival and inter-agent communication with the decision rules of a single, optimizingagent. A standardrationale for single-agent modelingis that profit-maximizingagents drive out all inferior strategies, and the dominant strategy is easily learned by all agents when shared in a simple feedback format. Objections to this Darwinian assertion of the approximate homogeneityofsurvivingstrategiesincludeeverydayobservationsofheterogeneousbehaviorand examplesinbiologyofthe“brittleness”ofsystemswithover-specializedgenetictraits. An intriguing alternative modeling design is suggested by the “zero-intelligence” agents of Becker (1962) and Gode and Sunder (1993) where local constraints, such as static or dynamic budget constraints, can cause random micro behavior to produce rational system results, such as downward-slopingaggregatedemandschedulesandefficient pricinginauctionmarkets. The analogous “smart system” conjecture explored in this paper is that parallel solution implementationsof optimizationalgorithms often require only local informationfor each parallel processor, but the system solution is the same as that reached by a serial processor with global information. Parallel Jacobi iterative solution of a linear equation system, such as B x (cid:0) b = 0 , appears to be a powerful example of this conjecture because solution of a system of linearized first-order conditions is the essential core of optimizations ranging from nonlinear searches of likelihood functions to companion systems of multivariate Euler equations. In the case of Jacobi solutions of the linear equation system, a number of accelerated methods exist for specialized structures of the B matrix. The case where B is nearly-decomposable into diagonal blocks is notablebecauseglobalinversionof B isnearlyachievedbyindependentinversionsofthediagonal blocks. Unfortunately, if the dimensions of B are sizable, it is very hard to determine the best near-blockpatternsin B .1 Producer pricing in a distributed production system is used to demonstrate the weak local information requirements of Jacobi algorithms. The relative sparseness of the US input-output system at the six-digit level of aggregation illustrates the contrast between aggregate information 1Notable exceptions are the patterns of banded matrices associated with boundary-value problems of dynamic systems;vid.StoerandBulirsch(1980)andFisher,Holly,andHughes-Hallett(1986).

2 and the industry-specific information that is required to maintain margins between revenues and costs in each industry. Sensible rules-of-thumb, such as sharing information with direct suppliers andcustomers,areshowntoaccelerate adjustmenttoequilibrium. 2. Producer Price Adjustments Deep divisions exist in macroeconomics regarding the dynamic adjustments of prices. In classical theories, markets are continuously cleared by flexible prices, including instantaneous adjustments of nominal prices to agents' perceptions of monetary policy. In contrast, Keynesian theories suggest non-auction prices are slow to adjust to equilibrium, and short-run clearing is achievedbychangesintransactedquantities. These theories are more than of academic interest, in part, due to differing implications for monetary policy. In classical models, inflation inertia is due only to anticipations of persistent inflationary policies, and costs of policy disinflations are negligible. In archetypal Keynesian models, announcement effects of policy are generally dominated by prolonged real effects of interestratesandbankcredit,anddisinflationsareinitiatedbyreducingthegrowthofrealactivity. A sufficient reason for the continuing existence of such disparate theories is that neither side is able to fully account for several stylized facts regarding actual price dynamics and so each can claimameasureofempiricalrelevance. There are three major stylized facts about producer price dynamics that a general theory of producerpricingmustexplain:2 (cid:15) Prices of manufactured goods are sticky. As shown later, the mean lag response of US manufacturingprices tounanticipatedshocksis aboutninemonths. (cid:15) Producer prices are adjusted infrequently. Although there are few systematic analyses of this characteristic, available estimates suggest that a typical U.S. producer price may be altered no morefrequentlythanonceayear.3 (cid:15) Adjustments of producer prices in inflationary episodes can be large and rapid. In a study of EuropeanhyperinflationsafterWWI,Sargent(1982)indicatesthatproducerpricesinGermany increased ontheorderof 1 0 7 inthetwelve-monthinterval,June1923-June1924. Although there are a number of theories of sticky nominal price adjustments, ranging from explicit costs of adjusting prices, Rotemberg (1982), to instrument uncertainty, Greenwald and Stiglitz(1989), none appear capable of explainingthe stylizedfacts listedabove– where nominal 2Additional macro and micro stylized facts regarding prices may be found in Gordon (1981, 1990) and Schmalensee(1989),respectively. 3In analyzingthe Stigler-Kindahldata, Carlton (1986) reportsan averageadjustmentfrequencyof aboutonce a year. Similarly,overhalfofthefirmsinterviewedbyBlinder(1991)indicatedpriceswerenotadjustedforayearor longer. Ofcourse,infrequentpriceadjustmentsmaybesufficientlylargesothatstaggeredpricechangesatthemicro levelmaybeconsistentwithpromptresponsesofaggregatepriceindexes,cf. CaplinandSpulbur(1987).

3 pricesaresometimesadjustedrapidlyandatothertimesappeartobenotwell-connectedtocurrent market conditions. After a brief review of a generic adjustment cost model of producer pricing and empirical estimates of selected characteristics of producer price adjustments, the remainder of this paper pursues a very different interpretation of dynamic price movements. An alternative to explicit adjustmentcostsisthatsluggishpricemovementsareduetolagsinobtainingrelevantinformation in a distributed production system. The consequent adjustment lags are not easily categorized as signalextractionorlearningbyarepresentativeagent butare ratheraresultofsysteminteractions among heterogeneous agents using reasonable rules-of-thumb to set margins between costs and revenues. Although the disaggregated pricing model is an extreme abstraction of information processing by real producers, it provides at least a benchmark of dynamic consequences of using affordablesolutionalgorithmsinacomplexenvironment. 2.1Models ofthe dynamicadjustment ofmanufacturing prices. Thepricingofastandardizedproductbyanindustrywith s identicalproducersmayberepresented by p (cid:3) = m + c ; (1) where p (cid:3) denotesthelogoftheoptimalor“target” price, m is thelogmarkupbyproducers,and c isthelogofmarginalcost. Ignoringstrategicconsiderations,themarkupis m = l o g 0 @ 1 (cid:0) 1 1 s (cid:17) 1 A ; (2) where (cid:17) isthepriceelasticityofdemand,and(2)displaysthemonopolyandcompetitivesolutions as either s ! 1 or s ! 1 . Gross productionis Cobb-Douglas inbothpurchasedmaterials andrented services of primary factors. Also, returns to scale are constant so that the log of marginal cost is proportional to the weightedaverageofloginputprices c / (cid:18) 1 p 1 + (cid:1) (cid:1) (cid:1) + (cid:18) m p m + ( 1 (cid:0) X i (cid:18) i ) v ; where (cid:18) i and p i are the cost share and log price of the i th materials input, and v denotes the log unitpriceofvalue-addedbytheproducingindustry. Now suppose the actual price is displaced from the target price, perhaps due to errors in estimatingcurrentcostordemandelasticity,butacostisincurredinadjustingpricefromitscurrent level. A tractable model of price dynamic adjustment is presented by Rotemberg (1982) where

4 producers are assumed to minimize the discounted sum of the square of the distance between the price and the moving target price and the cost of squared changes in the level of the price. The required equationofmotionforthepriceisthefamiliarsecond-orderEulerequation E t f A ( L ) A ( B F ) p t (cid:0) A ( 1 ) A ( B ) p (cid:3) t g = 0 ; where E t f : g denotes the expectation given information through t (cid:0) 1 ; A ( L ) is a first-order polynomialinthelagoperator L ; A ( L ) = ( 1 (cid:0) (cid:21) L ) ; F isthelead operator; and B is a(quarterly) discountfactor. Toobtainabenchmarkestimateofthemeanadjustmentlagofproducerprices,let p denotethe log of the price of U.S. manufacturinggoods. Producer prices are generally difference-stationary, containingone unit root. This, in turn, permits two empirical simplifications. First, the target gap or log distance between the current price and target price can be established by a cointegration regressionintheloglevels p t = = (cid:12) p ~x (cid:3) t t + + ( (cid:15) p t t ; (cid:0) p (cid:3) t ) ; where ~x t is a vector of the arguments of the target price as defined above, including the prices of non-manufacturing inputs, the wage rate, and trend rate of labor productivity. Note that the cointegratingdiscrepancyis anestimateoftheprice“gap”ordistancetothetarget pricetrend. Also, using the fact that both p and p (cid:3) are I(1), the decision rule for p can be expressed in a “rational”error-correctionformat as (cid:1) p t = (cid:0) ( 1 (cid:0) (cid:21) ) ( p (cid:0)t 1 (cid:0) p (cid:3) (cid:0)t 1 ) + S t ( (cid:21) B ; (cid:1) p (cid:3) ) ; (3) wherethesecondtermisa present-valueeffect offorecasts offuturechanges inthetarget path, S t ( : ) = ( 1 (cid:0) (cid:21) ) 1 X =i 0 ( (cid:21) B ) i E t f (cid:1) p (cid:3) +t i g : FollowingTinsley(1993),weuseatwo-stepestimationprocedurewhereforecastsofthetarget path, p (cid:3) , are providedby a VAR model in the arguments of ~x t . Also, the quarterly discount factor is set to B = 0 : 9 8 , consistent with the annual postwar real return to equity of about 8 percent. Givenestimatesofthediscountfactor B and oftheVARforecast modelof p (cid:3) ,theonlyremaining unknownparameterinthedecisionrule ( 3 ) is thecoefficient oftheerror-correction term, ( 1 (cid:0) (cid:21) ) . Estimates of the dynamic decision rule for the US manufacturing price using the sample span 1957Q1-1991Q4are listedintable1.

5 Table1: Estimatedpricing rules forU.S.manufacturing. Eqna p (cid:0)t 1 (cid:0) p (cid:3) (cid:0)t 1 (cid:1) p (cid:0)t 1 ( p (cid:0)t 1 (cid:0) p (cid:3) (cid:0)t 1 ) + R 2 B G ( 1 2 ) Meanlagb 1 (cid:0) ( 0 2 : 1 : 6 0 ) 0 : 4 0 0 : 0 0 M L = 1 0 2 (cid:0) ( 0 3 : 0 : 4 8 ) ( 0 1 : 5 6 : 9 2 ) 0 : 7 2 0 : 2 0 M L = 2 : 9 3 (cid:0) ( 0 2 : 1 : 5 1 ) ( 0 1 : 3 6 : 5 3 ) (cid:0) ( 0 0 : 0 : 7 4 ) 0 : 7 3 0 : 1 7 M M L L (cid:0) + = = 2 7 : : 2 8 at-ratiosinparentheses. bMeanlaginquarters. The first line of table 1 presents statistics of the error-correction decision rule that follows from the simple two-root Euler equation. Considering it has only one free parameter, 1 (cid:0) (cid:21) , the R 2 of this equation is respectable but the zero p -value of the Breusch-Godfrey statistic, B G ( 1 2 ) , indicates strong residual autocorrelations. Also, the estimate of a ten-quarter mean lag, M L , of priceresponsestounanticipatedshocksis implausiblylong. The second equation reported in table 1 uses an extension developed in Tinsley (1993) where adjustment costs are generalized to include quadratic smoothing penalties not only of changes in the current price but of changes in moving averages of the price, such as might be associated with the survey findings noted earlier that some firms adjust every quarter but many adjust less frequently. One result of this change in the specification of dynamic costs is that lags in the dependent variable are added to the decision rule in (3). With this addition4, as shown for the second equation, the R 2 is much higher, the p -value of the BG statistic does not indicate residual autocorrelationatstandardsignificancelevels,andthemeanlagisnow2.9quarters,oraroundnine months. 4Alterationsoftheestimateddecisionruleareabitmorecomplicated:Inthecaseofthesecondandthirdequations intable1,theunderlyingEulerequationisfourth-orderandtwocharacteristicrootsareusedtodiscountfuturechanges ofthetargetpricein S t ( :) . Anotherimportantempiricaldifferenceisthatrationalexpectationsrestrictionsimposed bytheVARforecastmodelof p (cid:3) arerejectedbythe firstdecisionrulein table1, butnotbytheremainingdecision rulesinthetable. SeeadditionaldiscussioninTinsley(1993).

6 Finally,thethirdequationintable1exhibitsanothernotablecharacteristicofpostwarproducer prices. The term in the third column replaces the error-correction term (listed in the first column) when the lagged price is above the lagged target price, p (cid:0)t 1 (cid:0) p (cid:3) (cid:0)t 1 > 0 . As indicated, error-correction towards the target path is much smaller (and statistically insignificant from zero) whenthepriceisabovethetarget. Theresultisastrongpositiveasymmetryindynamicadjustment ofmanufacturingprices,wherepositivecorrectionsarefasterthannegativecorrections. Themean adjustment lag in raising prices to catch up to higher target prices, M L (cid:0) , is about 2 quarters but themeanlaginreducingprices toapproachlowertarget prices, M L + ,isnearlytwoyears. A finding of positive asymmetry in price adjustments helps reconcile two of the stylized facts regarding prices noted earlier. In inflationary periods, producers are not reluctant to raise nominalpricestocoverrisingnominalcosts,aspredictedbyclassicalanalyses. Ontheotherhand, resistancetodownwardadjustmentsofnominalpricesisconsistentwiththeKeynesiannotionthat pricesmaygetstucktemporarilyatlevelsthataretoohighformarketclearing,suchaswhenoutput contractionsarea preferredresponsetocyclical reductionsindemand. Even in the case of symmetric price responses, the adjustment cost rationale of gradual price adjustment seems to imply that the costs of price frictions exceed the costs of adjusting rates of production. Theaimintheremainderofthispaperistoillustrateanalternativeconjecturethatthe gradual adjustments of producer prices may be due to transmission lags in information required for formulation of equilibrium prices. We suggest also reasons why system adjustments may be slowerforpricereductionsthanforpriceincreases. 3. Price Adjustment as Message Passing in Distributed Production. One drawback of macroeconomic analysis of pricing is that it ignores essential differences in pricing within the stages of production. It was widely recognized in early empirical studies, such as Mitchell (1913) and Means (1935), that prices tend to be less flexible as they advance from basiccommodityinputstospecializedfinaldemandoutputs. Thissectionpresentsapricingmodel where the only dynamics are due to lags in the transmission of industry-specific information in a large-scale input-outputsystem. 3.1Historicalviewsofpricing indistributed production. In his extensivestudy of five U.S. business cycles from 1890-1910, Mitchell (1913, p. 102) noted that “the prices of raw materials respond more promptly and in larger measure to changes in businessconditionsthandothepricesoftheirproducts. Sincethe...partlymanufacturedproducts pursueacourseintermediatebetweentheirrawmaterialsandfinishedgoods,itseemsthatthemore manufacturingcostshavebeenbestoweduponmaterials thesteadierdotheirprices become.” Table 2, drawn from Mitchell (1913, p. 101), contrasts average reductions during business

7 contractionsinprices ofrawmaterials,intermediatematerials,and final goods. Table2: Pricedeclines in1893-4and1907-8 contractions. Raw Intermediate Finished materials materials goods Pricefall (%) (cid:0) 1 2 (cid:0) 9 (cid:0) 5 More than twenty years later, essentially the same phenomenon was observed by Gardiner Means (1935),who furtherillustratedthat theextentofassociatedoutputcontractionsis inversely associated with the degree of price responsiveness. Extracts from a table in Means (1935, p. 405) are shownintable3. Table3: Drop inprices and production, 1929-1933. Agricultural Textile Iron& Agricultural commodities Oil products steel machinery Price fall (%) (cid:0) 6 3 (cid:0) 5 6 (cid:0) 4 5 (cid:0) 2 0 (cid:0) 6 Outputfall (%) (cid:0) 6 (cid:0) 2 0 (cid:0) 3 0 (cid:0) 8 3 (cid:0) 8 0 Among subsequent analyses of the increasing “downstream” inflexibility of manufacturing prices,manyinterpretationsreducetooneoftwoexplanationsadvancedbyMitchellandMeans. First, as noted in the quotation above, Mitchell suggested that rigid costs in the successive value-added stages of manufacturing, especially of labor, may account for much of the reduced price responsiveness. Wage rigidity,especially downward inflexibility,is also discussed by Pigou (1927), Keynes (1936), and such postwar interpretations as Tobin (1972). To gloss over a large empiricalliterature,thestylizedfact appearstobethatrealwagesappeartobeslightlyprocyclical and price margins over unit costs are countercyclical. In other words, available evidence suggests that manufacturingprices are somewhatless cyclicallyresponsivethanunitcosts ofproductionor wages. Second, in addition to the unsurprising implication that downward sloping demand schedules implythatreductionsindemandmaybepartiallyoffsetbypricereductions,Means(1935)indicted

8 the noncompetitive structure of many manufacturing industries. An extensive literature has exploredtherelationshipofproducerpricemarginstovariousmeasuresofindustrymarketpower, suchastheHerfindahlconcentrationindex. IncontrasttoMeans' thesis,modelsofstrategicpricing by oligopolies, such as Green and Porter (1984) and Rotemberg and Saloner (1986), suggest that high concentration may lead to episodes of sharply moving prices as individual oligopolists are induced, at various stages of the business cycle, to defect from implicit collusive agreements to maintainacommonprice. Althoughmanyempirical studiesindicatethatprice marginsare higherinhighlyconcentrated markets,evidence is inconclusiveonthe responsivenessof producer prices tochanges in demand, Domowitz, Hubbard, and Peterson (1987). Interesting exceptions are Qualls (1979) and Eckard (1982), who suggest concentrated industries may respond faster due to better market information and inter-firm communications—an interpretation consistent with the costly communications modelexploredinthispaper. 3.2Aparallel Jacobimodel ofprice adjustment. Bycontrastwithstandardadjustmentcostinterpretationsofstickyproducerpricing,theconjecture explored here is that the demand and cost information relevant to each producer is not instantaneously accessible and is transmitted through specific directed links between transacting agents. In the case of the downstream flow of input materials costs, it is not implausible that producers in each stage of production respond to received ripples in input prices with at least a one-cyclelag. The industry-to-industry transmission of current cost information is described by Gordon (1990, p. 1150-1), “the typical firm has no idea of the identity of its full set of suppliers when all the indirect links within the input-output table are considered. Because the informational problem of trying to anticipate the effect of a currently perceived nominal demand change on the weighted-average cost of all these suppliers is difficult to formulate and probably impossible to solve, ...the sensible firm waits by the mailbox for news of cost increases and then ...passes themonas priceincreases.”5 A simple description of this “waiting by the mailbox” transmission of cost increases is a one-cycle lagged version of the open-Leontief pricing system adapted for Cobb-Douglas production p t = A 0 p (cid:0)t 1 + v t ; (4) where p t denotesthe n (cid:2) 1 vectoroflogpricesofoutputsproducedin n industries; A isthe n (cid:2) n 5Differencesbetweenthepricelagresponsesofindividualproducersandofaggregatepriceindexesarediscussed inGordon(1981,1990)andBlanchard(1987).

9 Leontiefinput-outputmatrix,and v t isan n (cid:2) 1 forcingvector. Consistentwiththebasicmodelin section 2 ,eachindustryforcingterm v i isthesumofthelogoftheindustrymarkup m i andthelog ofunitlaborcosts, w i (cid:0) (cid:26) i ,where w i is theindustrylogwagerateand (cid:26) i isloglaborproductivity. The i th column of A contains the cost shares of purchased material inputs; using earlier notation, A 0 :;i = [ (cid:18) 1 ;i ; : : : ; (cid:18) n ;i ] .6 Of course, the dynamics of this simple staggered pricing system are equivalent to parallel Jacobi solutions of a linear system. Given a fixed unit cost impulse v (cid:3) , the desired price response is p (cid:3) = = = [ [ p I I 1 (cid:0) + + A A ( 0 (cid:0) ] 0 + 2 p (cid:0) v (cid:3) A p 02 1 + ) + A ( 03 p 3 + (cid:0) : : p : 1 ] v ) (cid:3) + : : : The iteration expansion on the third line has two implications for successive price adjustments. First, the “error-correction” adjustment rates of this system are determined by the characteristic rootsof A . Thatis,thereductionofdistancesbetweenthecurrentpriceandthetargetpricevectors is7 p t (cid:0) p (cid:3) = A 0 ( p (cid:0)t 1 (cid:0) p (cid:3) ) : (5) Second, each stage of successive price revisions adjusts for the change in costs of preceding stages of production. That is, p 1 adjusts for the initial change in unit value-added costs in each industry. After these costs are passed on to the next downstream stage of processing, p 2 (cid:0) p 1 denotes the adjustment in each industry to the consequent change in costs of direct suppliers — that is, the possible n suppliers of inputs to each industry represented by the columns of A . The next roundofrevisions, p 3 (cid:0) p 2 ,incorporates responses tochanges inthecosts ofthe suppliersof the direct suppliers, i.e. the n 2 suppliers once-removed. The next revision, p 4 (cid:0) p 3 , accounts for price adjustments due to changes in the costs of the possible n 3 suppliers twice-removed, and so on. 6Because the pricing system is open, the sum of nonzeroelements in each column of A is nonnegativeand less thanunity;consequently, I[ (cid:0) A 0 ] isstrictlydiagonallydominant,invertible,anditsspectralradiusis lessthanone, vid. HornandJohnson(1985). 7Here,wefocusonalternativeinterpretationsoftheerror-correctionadjustmenttermintheoptimaldecisionrule (3)ofthe“representative”manufacturingsystem. Theadditionalforward-lookingterminequation(3)isonlyrequired ifthereareperceivedadjustmentlags.Althoughwesuspectthattheeffectofthissecondtermisnotdifficulttoreplicate aftertheformofsystemlagsisestablished,perhapsthroughmulti-periodpricingcontractsinverticalagreements,this refinementisnotexploredinthispaper.

10 3.3An empirical input-output example. Toprovideconcreteillustrationsofthetimingofinformationflowsunderdifferentcommunication structures among industries, a 356-industry input-output system is constructed from the US DepartmentofCommerce(1991)industryuseandmaketables for1982. Also,toobtaindata-based estimatesof discrepancies between actual prices p andtarget prices p (cid:3) , monthly target prices for industries are estimated using a procedure similar to that outlined earlier.8 Cointegration regressions are used to construct estimates of the trend or target level of value-addedcostsinthe i thindustry, v (cid:3) i;t , p i;t (cid:0) X j a j ;i p j ;t = = b v 0 ;j (cid:3) i;t + (cid:0) b ~v 1 w ;i (cid:3) : i;t i;t + b 2 ;i t + e i;t where the second line indicates that the residual, e i;t , is an estimate of the discrepancy between the target unit cost of value-added and the construction implied by current prices. Denoting ~v (cid:3) t as the n (cid:2) 1 vector of deviations in value-added price for the 356 industries, estimates of the target deviationsinindustryproducerprices isprovidedby ~p (cid:3) t = [ I (cid:0) A 0 ] (cid:0) ~v (cid:3) t : (6) A convenient estimate of the average size of the log gaps in value-added and final prices is provided by the sample means of the absolute values, j ~v (cid:3) j and j ~p (cid:3) j . Averaged over all 356 industries, the mean absolute values of the value-added and producer price gaps are 0.022 and 0.025,respectively. Hereafter,becausewewilloftendiscussresponsestopricechanges,unlessotherwiseindicated, it will be notationally convenient to drop the absolute value “ j j ” and gap “ ~v ” notation and simply use p and v todenote j ~p (cid:3) j and j ~v (cid:3) j . Under this notational convention, the industry mean lag responses by parallel Jacobi ( P J ) iterationstoaninitialcostdiscrepancy, v ,are M L ( P J ) = = [ [ I I + (cid:0) 2 A A 0 0 (cid:0) ] + [ I 3 (cid:0) A 02 A + 0 ] (cid:0) 4 v A = 03 [ I + (cid:0) : A : : ] v 0 (cid:0) ] = v [ ; I (cid:0) A 0 ] (cid:0) v 8Althoughsomepriceswereavailableforthefullpostwarperiod,acommonsampleofthe356industryproducer priceswasavailableonlyforthe1986.01-1994.02span. Consequently,“cointegration”regressionsdiscussedbelow areonlysuggestiveofwhatmightbeobtainedforalongersample.

11 where “/” denotes element-by-element divisionof the numerator and denominator vectors. Using theempiricalestimatesof v ,theaverageindustrymeanlagis1.93iterationcycles. The first two lines in table 4 compare empirical estimates of industry price mean lags (in months) and parallel Jacobi mean lags (in iteration cycles) for approximately the same industry groupingsusedbyMeans (1935). Table4: Characteristics ofrepresentative production stages. Agricultural Textile Iron& steel commodities Oil products forgings Machinery M L a 5.3 6.2 6.9 7.9 11.3 M L ( P J ) b 1.4 1.0 2.0 2.0 1.9 Materialsinput share(%) 38 6 58 41 37 Averagenumber ofsuppliersc 48 34 52 68 93 aMeanlaginmonths. bMeanlaginiterationcycles. cBasedonindustrydirectsupplierswithinputsharesgreaterthan0.0001. Although the correspondence is not exact, both mean lag estimates suggest a tendency for slower adjustment speeds of prices in succeeding stages of production. The third line suggests that this is not merely due to the amount of purchased materials used in production. Except for extractiveindustries, the average share of output due to purchased materials from other industries in this input-output system is about 40%. As shown in the fourth line of the table, a more appropriateindicatorof meanlags inprice adjustmentappears tobe thecomplexityofproduction and coordination of inter-firm communications, as measured here by the average number of industrydirect suppliers. Ofcourse,thereisnoobviouswaytotranslateiterationcyclesintocalendartime. Nevertheless, if we assume surveys are correct in suggesting that the average firm adjusts its price about once a year, then a mean lag estimate of 1.9 cycles would translate to a calendar mean lag of about 23 months,considerably longer than the average mean lag of about 9 months indicated earlier for historicalmanufacturingprices.

12 Obviously,there can be manyoverstatementsof inter-industryinformationlags inthe stylized parallelJacobi example. Forexample,firmsmaybeawareofkeypressuresontheprices ofinputs purchasedfromtheirdirectsuppliers. Toobtainacrudedecompositionofthecyclesofinformation embeddedinthemeanlagestimate,notethatanapproximatelineardecompositionis M L ( P J ) = [ I + 2 A 0 + 3 A 02 + 4 A 03 + : : : ] [ I (cid:0) A 0 ] (cid:0) v = [ I (cid:0) A 0 ] (cid:0) v (7) ’ (cid:17) [ ~1 I + + 2 2 M A 0 L + ( 2 3 ) A + 02 3 + M 4 L A ( 03 3 ) + + : : : : : ] [ : p = p ] where ~1 isa 3 5 6 (cid:2) 1 unitvector. WewouldnotexpectthesimplemodelofparallelJacobiiterationstoprovidegoodpredictions of cross-industryvariations inmean lag adjustments. Thus,it is not surprisingthat thecorrelation between industry estimates of historical mean lags, M L , and the corresponding parallel Jacobi mean lag measures, M L ( P J ) , is only .10. A more revealing insight into the timing of inter-industry flows is obtained by regressing the 356 industry estimates of historical mean lags, M L , on the parallel Jacobi mean lags, M L ( P J ) , and initial terms of the approximate expansion inequation(7). Table5: Regressionofindustrymean lags,ML,oniteration mean lags,ML(PJ), anditerationstages. M L ( P J ) M L ( 2 ) M L ( 3 ) M L ( 4 ) R 2 16.8 (cid:0) 3 2 : 7 0.12 (6.7) (6.4) 29.0 (cid:0) 2 4 : 7 (cid:0) 5 5 : 4 0.20 (9.4) (4.9) (6.1) 27.5 (cid:0) 2 1 : 1 (cid:0) 6 6 : 2 24.1 0.20 (8.1) (3.6) (5.1) (1.1) UsingtheparallelJacobimeanlagestimates, M L ( P J ) ,asabenchmark,thenegativecontributions of initial iteration responses suggest that historical mean lags are somewhat faster in response to the average information contained in the initial rounds of information transmission in the chain of production. However, this faster information transmission does not seem to penetrate much

13 deeper than early receipt of information on the input costs of direct suppliers, as summarized in the M L ( 3 ) terms. Thus,itappearsthatamorerealisticmodelofempiricalpriceadjustmentlagsshouldallowfor industry differences in speeds of processing information relative to the fixed lags depicted in the simpleparallel Jacobi solution. The remainder of this section discusses three modifications of the parallel Jacobi solutionthat maybe sources of different speeds of price adjustmentsbyindividual industries. 3.4Acceleration through communication groups. The input-output system highlights only the structure of disaggregated production of heterogeneouscommoditiesanddoesnotaddressthevaryingdegreesofinter-industryinformation that may be available through alternative organization of corporate control. For example, in contrast to theparallel Jacobi model of isolatedindustriesat each stage of production,all relevant production stages might be controlled by the management of a single vertical monopoly. In this case, it would seem unlikely that relevant informationon altered costs in any phase of production wouldnotberapidlydisseminatedtoallplantscontrolledbythevertical organization. AsreviewedbyPerry(1989),therearealsomanyformsofvertical“quasi-integration”ranging fromproductioncontracts,leasingagreements,andmarketingfranchisestoequityinvestments,all ofwhichareaimedatobtainingsomeoftheinformationandcontrolbenefitsofverticalintegration. Even at the level of least intrusion on corporate control, trade organizations provide a forum for collectingandsharinginformationonrecent trendsindemandandcostsofpurchasedmaterials. We assume that information on current shocks in various horizontal and vertical stages of production is effectively processed within the relevant group of industries before submitting revised prices to the general round of inter-industry price communications defined by the global parallel Jacobiiteration. Thatis,industriesorganizedintocommunicationgroupsuseall available intra-groupinformationpriortosubmittingrevisedprices. Fromtheperspectiveofthe m industriesinagivencommunicationgroup,theglobalproduction systemisdividedintomemberandnon-membergroupings: p p 1 2 ;t ;t = = A A 1 2 1 1 p p 1 1 ;t (cid:0) ;t + 1 A + 1 2 A p 2 2 2 (cid:0) ;t 2 p 1 (cid:0) ;t + 1 v + 1 v 2 where A i;j denotespartitionsofthe n (cid:2) n input-outputmatrix, A 0 ; p 1 is m (cid:2) 1 ;and p 2 is ( n (cid:0) m ) (cid:2) 1 . The first equation describes the response by the m member industries in the p 1 communication group to current information within the group. The second equation is a conjecture regarding adjustmentofthe n (cid:0) m pricesin p 2 ofthenon-memberindustries. Forthemoment,weassumethis

14 equation is inaccessible to members of the p 1 communicationgroup because the elements of A 2 1 , A 2 2 , and v 2 are not known to these industries. Under this information structure, communications withinthe m -industrygroupleadtoa p 1 adjustmentinthecurrent iterationof p 1 ;t = [ I (cid:0) A 1 1 ] (cid:0) [ A 1 2 p 2 (cid:0) ;t 1 + v 1 ] : (8) which makes more effective use of available information in p 2 (cid:0) ;t 1 and v 1 than does the parallel Jacobisolution,sothemeanlags ofprice adjustmentsinthe p 1 groupwillbereduced. Of course, contrary to the uninformedview of the m member industriesof the p 1 group, there may be one or more information groups among the n (cid:0) m non-member industries controlling p 2 . Indeed, if all groups are separable ( A 1 2 = A 2 1 = 0 ), then full price adjustments to the news in v 1 and v 2 could be completed in a single block-Jacobi iteration, with an associated mean lag of one iteration. Note also that intra-group communications can lead to system acceleration spillovers because responses to information within the p 1 group that might have taken several iterations to reach non-member industries will now be disclosed by transmitting the accelerated priceadjustmentsinthecurrent systemiterationcycle, p 1 ;t . The solution in equation (8) also suggests a rough estimate of the cost of communications among the m industries within the p 1 communication group. Inversion of the m (cid:2) m matrix I (cid:0) A 1 1 can be approximated by the familiar expansion I + A 1 1 + A 2 1 1 + A 3 1 1 + : : : . Each matrix multiplicationrequires m 2 message transmissions, and O ( l o g m ) terms are required in the expansion approximation of the inversion.9 Assuming the costs of intra-group communications are shared equally, each industry's share of intra-group communication costs is approximately m O ( l o g m ) . 3.5Additional acceleration byfeedback conjectures. Givenconvex communicationcost pressures to contain the size of the communicationgroup, it is likely that some group members are aware of important suppliers who are not members. In turn, other group industries may be suppliers of these excluded suppliers. By pooling this information, group members can approximate the roundtrip effects of current changes in the p 1 prices on the pricesofexternalindustrieswhoareexpectedtosubsequentlyalterinputpricestomembersofthe p 1 group. However, to remain consistent with our original assertion that industries have very limited knowledge outside explicit communication groups, we assume the industries who control p 1 have only an aggregated view of industries in the external p 2 group. Under this restriction, the 9SeeJa´Ja´ (1992),p.409,where O ( r ) denotesthatthereexistpositiveconstants k and r 0 ,suchthatforall r > r 0 thenumberofexpansiontermsisboundedby k r .

15 conjecturedfeedbackresponsetoacurrent iterationchangein p 1 prices is A 1 2 g (cid:0) g [ I (cid:0) A 2 2 ] (cid:0) A 2 1 ( p 1 ;t (cid:0) p 1 (cid:0) ;t 1 ) ; where g is a k (cid:2) ( n (cid:0) m ) aggregation matrix with k (cid:28) ( n (cid:0) m ) .10 A later section will illustrate the case where the external non-member group is aggregated to a single industry, reducing g to a 1 (cid:2) ( n (cid:0) m ) aggregationvector. 3.6Retardation ofindustry adjustments through non-transmissions. Althoughthefocusofthispaperisprimarilyonorganizationofinter-industrycommunicationsthat canacceleratesimpleparallelJacobitransmissions,weshouldnotethereisalonglistofconditions thatmayslowtheadjustmentofproducerprices. Stale quotes may be a problem in large communication groups if the transmission chain of requiredmessagingwithinthegroupislengthy. Althoughwelargelyneglecttheroleofhorizontal firms within an industry, there may be strategic reasons for firms to slow communications of key informationortotransmitmisinformation. If price transmissions are costly, it seems likely that firms may decide to internalize small discrepancies from target prices and transmit only sizable changes in prices. Under this interpretation,“menucosts”ofpostingpricechanges maybeasourceofstalequotations.11 Therearelikelytobeasymmetricdifferencesinresponsestodownstreamflowsofcostchanges and reverse flows of information on changes in final demand. Downstream flows of actual cost changes are visiblecommitmentsbysupplierstochanges ininputprices. Receivingfirms havean incentiveto pass on highercosts of intermediatematerials because to not do sorisks lowerprofits orevenbankruptcy. Reverse transmissions of responses to changes in final demand are more problematic and depend on the competitive structure of the stream of producing industries. Remember that the effective value-added component, v , at each production stage contains both the marginal cost of the value-added in that industry, c , and the industrymarkup, m , which is a decreasing function of the perceived price elasticity of demand, (cid:17) . Although changes in demand can alter both marginal cost and demand elasticities, often the former is relatively flat and invariant to moderate changes 10By contrast, if all elements in A 0 partitions associated with the p 2 group were known to industries in the p 1 group, any “aggregation” conjecture should fully preserve this information, g (cid:0) g = I n (cid:0) m . Use of conjectural aggregationwithin communication blocks and global Jacobi iterations of the full system is similar to the recursive aggregate/disaggregatealgorithms discussed in Vakhutinsky, Dudkin, and Ryvkin (1979) and Tsai, Huang, and Lu (1994). 11AsexploredbyTsiddon(1993)andBallandMankiw(1992),thecombinationofmenucostsandpositivetrends intargetpricescaninducepositivepricingasymmetries.

16 in the level of output. The elasticity of demand may be a more likely source of countercyclical movementsthat inducepro-cyclicaladjustmentsoftarget prices. Inthecaseofanominaldemandshift,anindustry'sshareoffinaldemandcouldbemaintained if all prices in the chain of production move proportionately. In inflationary periods, independent industries at each stage of production may increase prices with alacrity since the consequence of moving before suppliers is temporary profits due to higher margins over costs. However, in the case of price reductions, a failure to coordinate significant price reductions with suppliers can be severeilliquidityorbankruptcy. Thus,downstreamfirmsmaybe“stuck”forsometimewithlower sales untillowercost agreementsare struckwithupstream suppliers. It is especially likely that independent upstream suppliers, who are not bound by vertical agreements, may be relatively insensitive to movements in final demand elasticity. The elasticity of deriveddemandforthe industryproducing,say,the i thmaterials inputis (cid:17) i = (cid:18) i (cid:17) + ( 1 (cid:0) (cid:18) i ) (cid:27) , where (cid:18) i isthecostshareofthe i thinput, (cid:17) isthepriceelasticityoffinaldemand,and (cid:27) isthefactor elasticity of substitution in production ( (cid:27) = 1 in Cobb-Douglas production), Waterson (1982). Substitutioninproductionis prominent inthisexpressionbecause a consequenceof a stand-alone inputpriceincreaseistoshiftthecompositionoffinaldemandtowardslessexpensiveinputs. Thus, if the contribution of the supplying industry to final product, (cid:18) i , is small then the derived demand elasticityoftheupstreamproducerislikelytobeinsensitivetomoderatevariationsintheelasticity offinal demand, (cid:17) . Thus, apart from explicit coordination arrangements such as vertical mergers, it seems likely thatbackwardorupstreameffectsofshiftsinfinaldemandarelikelytobesmallerandhaveslower transmission rates than forward or downstream flows of rising costs. Also, it appears that less inter-industry coordination is required to pass along incurred cost increases than desired price reductions. 4. Searching for Block Patterns in a Large-Scale System Considerthepricingsystemdescribedinequation(4), p t = A 0 p (cid:0)t 1 + v t ; where A istheinput-output(I/O)matrix, v isaforcingvector,and p istheindustrypricevector. We considertheeffectsofindustriescoalescingintogroupsinordertoshareinformation;eachindustry belongs to exactly one group. A group, or block, of industries is represented by a subset of the rows of A 0 . The industries in a group solve their I/O subsystem to get new prices at time t , using currentpriceinformationfromotherindustrieswithinthegroup,andlagged t (cid:0) 1 priceinformation from industries outside the group. Each communicationgroup is solved simultaneously,and then

17 theprocessis repeateduntiltheentirepricearray converges. This solution process can be thought of as modeling the exchange of cost and demand information among firms and their customers and suppliers. The goal of firms is to be able to adjust to the optimal price, p (cid:3) , as fast as possible. We measure the speed of convergence by the meaniterationlag,definedinthefirstlineofequation(7),toestimatehowquicklypricesconverge totheoptimalpriceinresponsetoexternalshocks. The 356-industry example that we consider is highly abstracted from the dimensions of a problem faced by an actual firm. A firm in the auto industry, for example, may have a chain of production involving thousands of suppliers. However, even in our 356-industry example, the computational problem of finding optimal communication groups is a daunting one, from the perspective of either an individual firm or a central planner. The scope of the possible search space is enormous; the number of different groups that an individual industry might join is 2 3 5 5 , an unimaginably large number. The related feedback vertex set problem was shown to be NP-completebyKarp(1972);see alsoGareyandJohnson(1979).12 Notethatthefastestconvergencewilloccurwheneachindustryhastherecentpriceinformation oneveryotherindustry—i.e.,whentheentire 3 5 6 (cid:2) 3 5 6 matrixissolvedasasingleblock. Inreal life, however, there is a cost associated with gathering current information that mitigates against this sort of arrangement. Consequently, we define a cost function that charges industries for the information they gain through communication with members of their own group. The cost is related to the computational complexity of solving the subsystem defined by the group. The cost per group member increases with the size of the group; thus, very large groups are not attractive sincethecost ofcommunicationexceeds thebenefits tobegained. 4.1Estimatingthe costofcommunication. Wederiveanestimateofcommunicationcostthatiscalibratedinsystemiterationor“cycle”units as follows. In a standard parallel Jacobian iteration, the column vector of industry mean lags is computedas m l = ( I + 2 A 0 + 3 ( A 0 ) 2 ( I + (cid:0) 4 ( A A 0 ) 0 (cid:0) ) 3 v + 5 ( A 0 ) 4 + : : : ) v : Note that the first time this criterion addresses the cost of contacting unknown suppliers (i.e., industries that are upstream in the flow of production, but are not direct suppliers) is in the third term, ( I 3 ( (cid:0) A 0 ) A 2 0 ) v (cid:0) v ; 12AproblemthatisNP-completehasbeenprovedtobeequivalentindifficultytoalargenumberofotherproblems widelyregardedbycomputerscientiststohavenopolynomial-timesolutionalgorithm,andthustobeintractablefor largeprobleminstances.

18 which is proportional to the relative price adjustment due to changes in the costs of the suppliers ofdirect suppliers. Supposeeach industryhas k suppliers;then each industrymustcontact (inthe worstcase) k 2 suppliersinordertolearnaboutthesecostsoneperiodbeforetheyfilterthroughthe costsofthedirectsuppliers. We assume that industries want to minimizemean lag responses. By contacting all k 2 of their suppliers' suppliers,theycouldreducetheirmeanlagto m l a = ( I + 2 ( A 0 + ( A 0 ) 2 ) ( I + (cid:0) 4 ( A A 0 ) 0 ) (cid:0) 3 v + 5 ( A 0 ) 4 + : : : ) v : Thisresultsina reductioninthemeanlagof m l (cid:0) m l a = ( I ( A (cid:0) 0 ) A 2 v 0 ) (cid:0) v : InthecaseofparallelJacobianresponses,theaveragevalueof m l is1.9294cyclesandtheaverage of ( m l (cid:0) m l a ) is 0.1675 cycles. The average number of suppliers per industry is around 25; thus an estimate of the mean lag reduction per supplier contact is c = mean 0 : 0 0 0 2 6 8 ( m l (cid:0) m l a ) = ( 2 5 (cid:2) 2 5 ) = . Under the assumption of parallel Jacobi responses, this is a lower bound estimate of the cost of a contact with an individual supplier, in the sense that the perceived cost of contact musthaveexceededthereductioninthemeanlagthatcouldhavebeenobtainedbycontactingthe additionalsuppliers. As described in section 3 , each industry's share of the number of communications required to solve an m -industry group is about m 2 l o g m . The logarithmic factor is an estimate of the number of terms required in the expansion of the inversion; our experience with matrices of this size suggests that 3 l o g m is a more appropriate approximation. Consequently,our estimate of the communicationcosts associated witha group of m industries is 3 c m 2 l o g m (where c = 0 : 0 0 0 2 6 8 as derived above); the per-industry share of this cost is thus 3 c m l o g m . This cost is added to the average mean lag to determine the total performance measure of a partition of industries into groups. Obviously,thesmallertheresultingnumber,thebetter. 4.2Computing environment. The computations described in this paper were run on Sun workstations in a Unix environment. TheapplicationprogramswerewritteninC++,usingtheSPARCOMPILER C++3.0compilerfrom SunPro. Extensive use was made of the MatClass C++ matrix libraries, written by Birchenhall (1993). While these libraries were reliable and simplified the programming a great deal, their performance was at times somewhat slow. Consequently, portions of the code that were to be executed many times were written as iterated scalar operations, rather than as matrix operations

19 using MatClass library functions. The code was written in a style that combines elements of the object-oriented and procedural paradigms. Industries, groups of industries, and collections of groupswere eachencapsulatedas C++ classes. Note that the “parallel” Jacobian iterations were actually performed sequentially, though independently so as to simulate parallel computation. Since we were not measuring performance by elapsed time and running times were (usually) not excessive, there was little need to actually run them in parallel. Exceptions to the latter were the genetic algorithms, discussed later, which were particularly computation-intensive. Future work may implement a parallel version of this code,usingthePVM libraries writtenbyGeistet al.(1993). 4.3Forminggroups byrules-of-thumb. We first consider several simple algorithms for forming groups. Each is a rule-of-thumb for clustering into communication groups which requires only local information on the part of individualindustries. Theseincludesuchbasicideasasindustriesjoininggroupsthatcontaintheir largest suppliers or customers, as well as forming random groups; the latter is included primarily as abaseline. Thealgorithmswehavestudiedareas follows: k-Largest-suppliers Each industry's group merges with the groups containing its k largest suppliers. k-Largest-customers Each industry's group merges with the groups containing its k largest customers. Suppliers-over-f Each industry's group merges with all groups that contain an industry that suppliesat least thefraction f ofitsinputs. Customers-over-f Each industry's group merges with all groups that contain an industry that purchases atleast thefraction f ofitsoutputs. Random-p Groups are formed randomly. In particular, with probability p each industry forms a new group. With probability 1 (cid:0) p it joins an existing group, with the particular group selectedequiprobablyfrom allexistinggroups. The table below shows some statistics on the collections of groups generated by these algorithms, as well as the performance of the collections as measured by their mean lags and communicationcosts.

20 Table 6: Performances ofalternativecommunicationgroupings. Algorithm Number Largest Average Average usedfor of group meanlag communication forminggroups groups size (AML) cost (ACC) AML+ ACC Random-0.2 78 168 1.6239 0.3307 1.9546 Random-0.4 158 59 1.9050 0.0356 1.9407 Random-0.6 215 21 1.9151 0.0052 1.9203 1-Largest-customer 160 20 1.6739 0.0085 1.6824 2-Largest-customers 24 324 1.1171 1.3706 2.4876 1-Largest-supplier 52 74 1.4514 0.1069 1.5583 2-Largest-suppliers 1 356 1.0000 1.6816 2.6816 Customers-over-0.075 155 174 1.3372 0.3534 1.6905 Customers-over-0.10 207 21 1.5193 0.0392 1.5585 Customers-over-0.125 236 25 1.5775 0.0104 1.5879 Suppliers-over-0.075 135 194 1.2990 0.4484 1.7474 Suppliers-over-0.10 195 83 1.4830 0.0738 1.5568 Suppliers-over-0.125 224 29 1.5529 0.0159 1.5688 Using the mean lag adjusted for the costs of inter-industry communication, randomly-formed groupsofindustriesresult inaperformance measureofabout 1.92–1.96cycles,withperformance varying little according to the size and number of groups. This is similar to the value of 1.93 cyclesobtainedwheneachindustrycomprisesitsowngroup;i.e.,whentheentirematrixissolved byasingleparallelJacobianalgorithm.13 Thus, there appears to be no benefit to forming groups at random. At the other end of the spectrum,all 356industriesare placedintoasinglegroupwhenthe2-Largest-suppliersalgorithm is run. The average mean lag then takes on its best possible value, which is one iteration cycle. However, the cost of communication among such a large number of industries pushes the performancemeasureupto2.68cycles –theworstamongall algorithmstested. The best results are achieved by the 1-Largest-supplier, Suppliers-over-0.10, and Customers-over-0.10 algorithms. When the thresholds for the latter two are increased to 0.125, 13Weoptimisticallysuggestthatperhapsthefactthatavarietyofrandomgroupcollectionsresultinperformances similar to each other, and to the case where all groups have size one, indicates that our performance measure is well-calibrated.

21 they continue to perform well. Note that the number of groups generated by these algorithms varies widely. Although the performances of the 1-Largest-supplier and Customers-over-0.10 algorithmsarenearlyidentical,thenumberofgroupsformedbythelatteris207,whilethenumber formed by the former is only 52. Along the same lines, compare the results for Random-0.6 and Customers-over-0.10. The number of groups generated by these algorithms, as well as the size of the largest group, are quite similar. However, note the difference in the average mean lags — 1.92 for the random algorithm versus 1.52 for the threshold algorithm. This sizable difference graphically illustrates the increased efficiency of a collection of groups formed by a reasonable rule-of-thumb,as comparedtotheperformanceofrandomgroupingswithacomparabledegreeof clustering. Althoughtheinput-outputmatrix, A ,isrelativelysparse,thegroupingsinthetablealsoindicate that all elements or industries are eventually connected through a transmission chain of common inputs or customers. Note that the 2-Largest-suppliers algorithm clusters all 356 industries into a single group. That is, there is no proper subset of industries such that for each industry in the subset, the subset contains its two largest suppliers. The 2-Largest-customers algorithm places 324 of the 356 industries into a single group, and extending this to the 3-Largest-customers algorithm results in one 356-industry group. We also note that connectivity is somewhat greater in the upstream direction, i.e. in the direction of suppliers. For example, the 1-Largest-supplier algorithm results in fewer groups — as well as a larger dominant group — than does the 1-Largest-customer algorithm. The same is true of the 2-Largest-suppliers algorithm when comparedwith2-Largest-customers,andeachofSuppliers-over-0.075/0.10/0.125whencompared withtheircustomer-basedcounterparts.14 If we assume, as suggested earlier, that a representative interval between price adjustments is on the order of a year, then the difference between a value of about 1.9 (as might be obtained from eitherrandomgroupsortheabsenceof anygroupsat all)andavalueofabout1.5(as results from the best of the rules-of-thumb we tested) may correspond to a difference in average price adjustment frequency of around five months. Thus, the way in which firms in different industries shareinformationmaysignificantlyaffectthespeedwithwhichtheyupdatetheirpricesinresponse toexternal shocks. 14Thisasymmetrymaycontributetoasymmetricpricemovementsdiscussedearliersuchasgreaterresponsiveness toupstreaminputcoststhantochangesindownstreamfinaldemands.

22 5. Variations Thissectionexploreshowtheperformancesofcommunicationgroupsmayalterunderdifferent informational assumptions. First, we consider the effect of industries taking into account the effects of their own price adjustments, not just on other members of their group, but also on the aggregation of all industries outside their communicationgroup. Next, we investigatewhat effect occasional communications failures — both within and between groups — have on the speed of priceadjustments. 5.1An aggregatedviewoftherest oftheworld. In our basic model, industries exchange current price information with members of their own group until they have solved the corresponding subblock of the I/O pricing system. During this communication period, they use the previous period's price information from all non-member industries outside the group. Thus, intra-group price iterations are performed only on rows of A 0 that correspond to members of the group. Members of a group do not attempt to solve rows of the A 0 matrix that correspond to industries in other groups, such as would be required to anticipate the effects of member price changes on non-member industries and, in turn, the effects of non-member induced price changes back on the production costs of members. This exclusion restriction seems reasonable because calculation of non-member feedback effects would require that member industries have detailed knowledge of the current cost structure of the complete US manufacturingsector. However,wedonotdismissthepossibilitythatmemberindustriesformapproximateestimates of the effects of member price changes on non-member industries. To illustrate, we introduce a variation of the original model where industries use a simplified, aggregated view of the “rest of theworld”(i.e.,industriesoutsideoftheirgroup)insettingprices. Let n be the total number of industries, and m be the number of industries in a group G . Without loss of generality, we reorder the rows of A 0 such that the rows corresponding to the m industriesingroup G comefirst,andpartition A 0 as follows:15 A 0 = 2 4 A A 1 2 1 1 A A 1 2 2 2 3 5 ; where A 1 1 is m (cid:2) m , A 1 2 is m (cid:2) ( n (cid:0) m ) , A 2 1 is ( n (cid:0) m ) (cid:2) m , and A 2 2 is ( n (cid:0) m ) (cid:2) ( n (cid:0) m ) . 15Notethatthepartitionsindicatedarethoseof A 0 ,thetransposeof A .

23 TheI/Osystemcanthenbepartitionedas 2 4 p p 1 2 3 5 = 2 4 A A 1 2 1 1 A A 1 2 2 2 3 5 2 4 p p 1 2 3 5 + 2 4 v v 1 2 3 5 ; (9) where p 1 and v 1 are m (cid:2) 1 and p 2 and v 2 are ( n (cid:0) m ) (cid:2) 1 . Theinputrowsfor p 1 containwithin-group relations( A 1 1 )andinputweightsdueto p 2 prices( A 1 2 ) . Both A 1 1 and A 1 2 areknowntoindustries in G . However, we assume that industries in G know only the sums of columns in A 2 1 , where these totals are the ratios of input shipmentsfrom industries in G to theproductionby non- G ( p 2 ) industries. Bycontrast,weassumeindustriesin G knownothingusefulabout A 2 2 ;16 theparticular defaultusedheretorepresent thislackofinformationisthatindustriesintheGgroupassume A 2 2 iszero. Undertheseinformationassumptions,ateachparallelJacobianiteration,pricesingroup G are solvedby p 1 ( t ) = A 1 1 p 1 ( t ) + A 1 2 p 2 ( t (cid:0) 1 ) + v 1 + B ( p 1 ( t ) (cid:0) p 1 ( t (cid:0) 1 ) ) where p 1 ( t ) is iterated to a solution but p 2 is held at its last known ( t (cid:0) 1 ) value. B is the feedback adjustmentwhichaccelerates anychange in p 1 from itslast knownvalueby“rationally” anticipating the eventual response of p 2 to the revision in p 1 . The solution for B obtained by substitutionfrom equation 9 is B = A 1 2 ( I (cid:0) A 2 2 ) (cid:0) A 2 1 ; whichis nowapproximatedbymemberindustriesas A 1 2 g (cid:0) g A 2 1 ; where g is the 1 (cid:2) ( n (cid:0) m ) aggregation vector (cid:19) 0 = ( n (cid:0) m ) ; g (cid:0) = (cid:19) ; and (cid:19) is the ( n (cid:0) m ) (cid:2) 1 unit column vector. That is, g is simply a row-averaging operator, and g (cid:0) is a column-summing operator. Thus, member industries approximate the ( n (cid:0) m ) non-member industries as a single external“industry”. Table 7 illustrates the effect of incorporating this restrictive aggregated view of the outside world on the performance of three of the algorithms described in section 4 . Each shows only a modestimprovement.17 16Inputspurchasedfromotherindustriesdonotsumtoonebecauseofinputsintheforcingvector, v ,suchaslabor. 17Althoughwedonotpursuetheeffectsofnonzeroconjecturesregarding A 2 2 ,notethatguessesaboutrowsumsof A 2 2 areequivalenttooverrelaxationaccelerationofthefeedbackconjecture, B .

24 Table7: Effects ofconjectured nonmember feedbacks. Algorithm Number Largest Average Average usedfor of group meanlag communication forminggroups groups size (AML) cost (ACC) AML+ACC Random-0.6 215 21 1.9151 0.0052 1.9203 w/rest-of-worldaggregation 1.9047 0.0052 1.9099 1-Largest-customer 160 20 1.6739 0.0085 1.6824 w/rest-of-worldaggregation 1.6623 0.0085 1.6708 1-Largest-supplier 52 74 1.4514 0.1069 1.5583 w/rest-of-worldaggregation 1.4325 0.1069 1.5394 5.2Effects ofimperfect communication. In experiments thus far, we have assumed that communication between industries is always reliable and noiseless. We now relax that assumption, and specify that the communication of price information is stochastic. A new price will be communicated from one industry to another only with a fixed probability; if the price is not communicated, then the prospective recipient of the information will use its most recent available value instead. Thus, the recipient is never fed incorrect priceinformation,merelyoutdatedpriceinformation(witha nonzeroprobability). Probabilistic communication represents several circumstances. First it reflects the effects of occasional “stale quotes”, i.e., industries being forced to use outdated price estimates because of delays intransmittingpriceinformation. This couldbe duetoimperfectionsinthemechanismfor transmitting information, or due to “freeriders” in the group, hoping to take advantage of other industries' information disclosures, while not revealing their own. In a rough way, probabilistic communicationmayreflectalsotheeffectof“menucosts”. Bythis,wemeanthephenomenonthat industries may feel that the fixed expense of adjusting their prices exceeds the value to be gained by doing so. Refraining from small adjustments causes inaccurate price signals to be transmitted, andmayimpairtheefficiencyoftheentiresystem. Inourmodeltherearenothresholds,butadding the element of randomization may suggest the general effect of industries failing to consistently transmitcurrent priceinformation. We first consider the case when communication within groups is imperfect. Table 8 shows results on three of the algorithms from section 4 when each intragroup communication fails with probability 0.10 and 0.20. All price information communicated between industries in different groupsistransmittedaccurately,thoughwithaoneperiodlagas before.

25 Table8: Imperfect communications withingroups. Algorithm Number Largest Average Average usedfor of group meanlag communication forminggroups groups size (AML) cost (ACC) AML+ACC Random-0.6 215 21 1.9151 0.0052 1.9203 90%reliabilitya 2.0418 0.0052 2.0470 80%reliability a 2.2193 0.0052 2.2245 1-Largest-customer 160 20 1.6739 0.0085 1.6824 90%reliability a 1.7104 0.0085 1.7189 80%reliability a 1.7664 0.0085 1.7749 1-Largest-supplier 52 74 1.4514 0.1069 1.5583 90%reliability a 1.4607 0.1069 1.5676 80%reliability a 1.4749 0.1069 1.5817 aCommunicationreliabilitywithingroups. The performance degradations of stochastic intra-group communications are small (below 7%) in every case except when random groups communicate with a 20% failure rate. Note that groups formed randomly suffer more from faulty intragroup communications than groups based on customer/supplierrelationships. In the latter case, there is more interconnectivityamong firms in the group and, thus, more opportunity for information missing due to transmission failures to besuppliedfrom anothersourcewithinthecommunicationgroup. Now,considerthereversecaseofstochasticinter-groupcommunications,introducing10%and 20% error rates for transmissions between groups and restoring perfect communications within groups. Theeffects onthethreealgorithmsappearintable9.

26 Table 9: Imperfect communications between groups. Algorithm Number Largest Average Average usedfor of group meanlag communication forminggroups groups size (AML) cost (ACC) AML+ACC Random-0.6 215 21 1.9151 0.0052 1.9203 90%reliabilitya 2.0931 0.0052 2.0982 80%reliability a 2.4469 0.0052 2.4520 1-Largest-customer 160 20 1.6739 0.0085 1.6824 90%reliability a 1.8247 0.0085 1.8332 80%reliability a 2.0227 0.0085 2.0312 1-Largest-supplier 52 74 1.4514 0.1069 1.5583 90%reliability a 1.5770 0.1069 1.6838 80%reliability a 1.7479 0.1069 1.8548 aCommunicationreliabilitybetweengroups. Imperfect communications between groups is much more disruptive than faulty intragroup communications. At a 10% failure rate, the performance degrades by 9% for each of the original three algorithms; when the rate is increased to 20%, the performance is at least 20% worse than the case of perfect communications for all three of the algorithms. The reason that flawed communication among different groups is more problematic is probably that there are fewer opportunities for communicating prices between industries in different groups. Thus if an out-of-date price is transmitted, a longer period of time elapses before it will be updated. Consequently, more computations are performed using the inaccurate price information, and convergencetotheoptimalprices is delayedforlongerintervals. 5.3Genetic algorithms. We also explored the use of genetic algorithms to find effective industry communication groups. A genetic algorithm is a randomized search procedure in which the goal is to “evolve” good algorithmsthroughaprocessanalogoustonaturalselection. Apopulationofstringsismaintained; eachstringrepresentsasetofparametersthatdefinesaparticularalgorithm. Thegeneticalgorithm seeks to improve the performance of the strings in its population through successive generations. In each generation, the strings in the population are evaluated as to their performance on the

27 problem at hand. Some of these strings are then selected randomly according to a probability distribution that weights good performers more heavily. The selected strings are then “bred” to formthenextgeneration. “Breeding” isachievedbypairingoffstringsandhavingthemexchange selected attributes; random mutations are also added to diversify the population. The resulting strings comprise the next generation. Genetic algorithms have been applied in a wide range of fields, including biology, operations research, integrated circuit design, and artificial intelligence. Foranintroductiontogeneticalgorithmssee,forexample,Goldberg(1989). Due tothelimitedsuccess wehavethus far achievedusinggeneticalgorithms,discussionwill be brief. In order to apply genetic algorithms to the problem of finding good industry groupings, we designed a string encoding to represent a collection of groups. Each string is an array of 356 integers,wherethe i thelementofthearrayisthenameofthegrouptowhichindustry i belongs. A pairofstringsis“bred”byrandomlychoosingaposition p between1and356. Eachgroupisthen splitat p ,and theprep groupfragments from thefirst stringare splicedontothepostp fragments from the second string, and vice-versa. Operations of this type are typically known as crossovers inthegeneticalgorithmsliterature. 18 Strings are alsomutatedbyrandomlysplittinggroups,and byrandomlycrossingoverpairs of groups withinasinglestring,bya process similartothat describedabovefor pairs ofstrings. The probability of each of these operations being carried out is controlled by parameters supplied to thealgorithm. We conductedtestsof from 500to2500generations,withpopulationsizes varying from 20to40. Unfortunately, results thus far with genetic algorithms have not been very promising. The best performing string produced by such an algorithm has achieved a performance measure of only about 1.77. While this is clearly better than randomly-selected groups, it does not approach the results of the better rule-of-thumb algorithms described earlier. The reason for this is almost certainly that we have not incorporated any domain knowledge into the genetic algorithm; it is essentially a random search. Since the search space of possible groupings is so huge, as noted earlier,algorithmsthatarenotguidedbyheuristicsthattakeintoaccounttheinput-outputstructure ofthe A matrixareunlikelytobesuccessful. 18Notethatthistypeofoperationisverysensitivetotheorderingoftheinformationcontainedinthestring,since adjacent string bits will almost never be separated, while bits far apart on the string will frequently be split up by crossovers. Thus, bestresultsare likely to beachievedwhenthestring canbeencodedso thatrelatedattributesare positioned close together. We tried to achievethis by orderingthe industries along the string so that an industry is closetoitscustomersandsuppliers,somewhatfurtherfromitssuppliers' suppliersanditscustomers' customers,and quite distant from those industries with which it has virtually no contact. Unfortunately, our results improvedvery littleevenafterthisreordering.

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