Prices, Production, and Inventories over the Automotive Model Year

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Prices, Production and Inventories over the Automotive Model Year Adam Copeland, Wendy Dunn, and George Hall 2005-25 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Prices, Production and Inventories over the Automotive Model Year ∗ AdamCopeland†, WendyDunn‡, andGeorgeHall§. March2005 Abstract Thispaperstudiesthewithin-model-yearpricingandproductionofnewautomobiles. Usingnew monthly data on U.S. transaction prices, we document that for the typical new vehicle, prices typically fall over the model year at a 9.2 percent annual rate. Concurrently, both sales and inventories are hump shaped. To explain these time series, we formulate a market equilibrium model for new automobilesinwhichinventoryandpricingdecisionsaremadesimultaneously. Onthedemandside, we use micro-level data to estimate time-varying aggregate demand curves for each vehicle. On the supply side, we solve a dynamic programming model of an automaker that, while able to produce only one vintage of a product at a time, may accumulate inventories and consequently sell multiple vintagesofthesameproductsimultaneously. Theprofitmaximizingpricingandproductionstrategies underabuild-to-stockinventorypolicyimplydecliningpricesandhump-shapedsalesandinventories ofthemagnitudesobservedinthedata.Further,roughlyhalfofthepricedeclineisdrivenbyinventory controlconsiderations,asopposedtodecreasingdemand. Keywords: dynamicpricing,revenuemanagement,discrete-choicedemandestimation,build-to-stockinventorypolicy JELclassification: D21,D42,E22,L11,L62 ∗WewanttothankAnaAizcorbe,SteveBerry,AndrewCohen,GautamGowrisankaran,AmilPetrin,JohnRust,JohnStevens, andseminarparticipantsatnumerousconferencesandseminarsfortheirhelpfulcomments. WealsothankBobSchnorbusfor helpingusobtainandinterpretthedatafromJ.D.PowerandAssociates. GeorgeHallgratefullyacknowledgesfinancialsupport fromtheAlfredP.SloanFoundation.Theviewsexpressedinthispaperarethoseoftheauthorsanddonotnecessarilyreflectthe viewsofmembersoftheBoardofGovernorsortheviewsofothermembersofthestaffoftheFederalReserveSystem. †DivisionofResearchandStatistics,BoardofGovernorsoftheFederalReserveSystem,MailStop82,20thandCStreets, NW,WashingtonDC20551;e-mail:adam.copeland@gmail.com ‡DivisionofResearchandStatistics,BoardofGovernorsoftheFederalReserveSystem,MailStop82,20thandCStreets, NW,WashingtonDC20551;e-mail:wdunn@frb.gov §DepartmentofEconomics,YaleUniversity,28HillhouseAvenue,P.O.Box208268,NewHaven,CT06520-8268;phone: (203)432-3566;fax:(203)432-5779;e-mail:george.hall@yale.edu 1

Two common features of durable goods markets are high levels of inventories relative to sales and declining prices over the product cycle. The optimal pricing strategy and inventory management of a good over its product cycle are classic issues in the industrial organization, revenue management, and inventory literatures. However much of the work explicitly linking pricing and inventories policies has beentheoretical,whilemuchoftheempiricalliteraturehasstudiedeitherinventoriesorpricinginisolation. Inthispaperwejointlyconsidertheoptimalpricing,production,andinventorymanagementpoliciesfora durablegoodsproducerandquantifyempiricallytheroleinventorycontrolhasonthetimepathofprices withintheproductcycle. Weaccomplishthisbyanalyzingthepricingandproductiondecisionsofanautomakeroverthemodel year,takingintoaccountboththeannualintroductionofnewvintagesandtheneedtomaintainsufficient inventories to facilitate sales. We first document, by vintage, the within-model-year inventory, sales, and pricingbehaviorintheU.S.marketfornewautomobiles. Wethenformulateamarketequilibriummodel for new automobiles. We estimate “typical” within-model-year demand curves for each market segment andvintageandsolveadynamicprogrammingmodelofthefirmwithoverlappingvintages. Thedynamic productionandpricingrulesimpliedbythemodelincorporatenotonlythesaleslifecycleofeachvintage but also the competition across vintages in the new-vehicle market. We find that the interaction between inventoryandpricingpoliciesarekeytomatchingthemagnitudeofboththeobservedpricedeclinesand theratioofinventorytosales. Ourdataonprices,production,andsalesofautomobilesovertheproductcyclearetheresultofmergingtwodatasets. Wematchnewmodel-levelmonthlydataonU.S.transactionpriceswithwell-knowndata on production and sales. These new price data are of unusually high quality for the automobile industry because they not only record the actual transaction price (not the list or invoice price) but also take into accountrebatesandfinancingincentivesthecustomerreceived. Usingthesedata,wedocumentfivefacts: 1. For the typical new vehicle, the average monthly decline in retail prices (net of rebates and incentives)is9.2percentatanannualrate. 2. Althoughnewvintagesofavehicleareintroducedannually,theaveragevintageissoldfor16.7months. Thus,fornearlyhalfofeachcalendaryear,twovintagesofeachmodelaresoldsimultaneously. 3. Whentwomodelyearsofthesamemakearesellingsimultaneously,theoldvintagesellsforaprice 8.8percentless,onaverage,thanthepriceofthenewvintage. 2

4. Bothsalesandinventoriesarehumpshapedoverthemodelyear,wherethemeanratioofinventories tosalesis76days. 5. Allotherthingsbeingconstant,higherinventoriesareassociatedwithlowerretailprices. Facts 1 through 3 describe the price path of a typical model within and across model years. Price declines over the product cycle and the simultaneous sale of multiple vintages of the same product are notuniquetotheautomobileindustry,havingbeendocumentedforanumberofotherproducts,including textbooks, microprocessors, and consumer electronics. The sale of several vintages of the same good is apotentiallyimportantdimensiontothefirm’sproblem,becausethewithinnew-marketcompetitionisin additionto,andpotentiallymoreimportantthan,competitionfromtheusedmarket. Newgoodsofanolder vintageareusuallyquitesimilartothoseofthenewervintage,andtheydonotsufferfromtheasymmetric information problems inherent in the used good market. Most existing theories that seek to explain the pricepathsdescribedbyfacts1through3focusonintertemporalpricediscrimination(e.g. Stokey,1979) or fashion (e.g. Lazear, 1986; Pashigian, 1988; and Pesendorfer, 1995). These theories, however, place little emphasis on the firm’s production decisions. Perhaps not surprisingly, we find that with regard to the market for new automobiles, these theories alone cannot explain the contemporaneous comovements inprices,sales,andinventoriesovertheproductcycle. The last two stylized facts regarding sales and inventories are well known within the automobile industry. Automakers carry an extraordinary amount of inventory relative to sales, as lots at dealerships are kept fully stocked with vehicles. In combination with declining prices over the product cycle, these facts suggest that a model with either stable demand or stable supply will be unable to replicate the first three facts. During the first six months a vehicle is sold, prices are high but declining, quantities sold are lowbutrising,andinventoriesaccumulate. Thispatternsuggeststhatrightwardshiftsinthesupplycurve dominatechangesinthedemandcurveearlyinthemodelyear. Duringthelasttwelvemonthsthevehicle issold,however,bothpricesandsalesfall,anindicationthatleftwardshiftsinthedemandcurvenowplay themajorrole. To refine this intuition, we formulate a market equilibrium model for new cars that links the operations research literature on optimal inventory and revenue management with the economics literature on discrete-choicemodelsofproductdifferentiation. Inparticular,wesolveadynamicstructuralmodelwith overlappingvintagesinwhichanautomakercanadjustboththepriceandthequantityproducedwithinthe modelyear. Theautomakersellsavehiclethatisslightlymodified,orchangesvintage,everyyear. While 3

the automaker produces only the current vintage, the use of inventories allows the firm to sell more than one vintage of the product simultaneously. Each week, the firm must decide the number of units of the currentvintagetoproduceandtheoptimalpricesforthevintagesinstock. Eachweekthefirmfacesadifferentdownwardslopingdemandcurveforeachvintage. Wederivethese demand curves from estimates of consumer preferences for automobiles by employing the econometric methodologydevelopedinthediscrete-choiceliterature(forexample,Berry,Levinsohn,andPakes,1995; Goldberg, 1995; and Petrin, 2002; to name a few). Our approach differs from this standard approach in threemainways. First,wehaveabettermeasureofprices,asweusetransactionpricesinsteadoftheusual list prices. Second, we estimate our demand-side model at a quarterly, rather than an annual, frequency; thus, weestimatehowthedemandcurveshiftsthroughoutthemodelyear. Third, weallowconsumersto chooseamongmultiplevintageswithinandacrossmodels. Usingourestimatesofconsumers’preferences, we compute average demand curves for each automobile market segment (such as compact cars) and vintage over the automobile product cycle. An advantage of our approach is that we allow households to differ across quarters in their distaste for price, hence our model accommodates, at least in part, the possibilityofthefirmengaginginintertemporalpricediscrimination. Amainresultfromthedemandsideanalysisisthatdemandcurvesformostvehiclesshiftsignificantly leftward over the second half of the model year, and the slopes of the curves undergo small changes. Further, we estimate that cross-price elasticities between models of different vintages are quite small. Hence, despite the similarities between vintages of the same model, consumers view vehicles from one modelyearaspoorsubstitutesforvehiclesinanothermodelyear. Theseestimatesplayacentralroleinthe firm’sproblem,astheydirectlyaffectthefirm’srevenueflowsfromsellingmultiplevintagesofthesame model at the same time. Lastly, we find that households are slightly more price sensitive during periods oftheyearwhenmanufacturerstypicallyoffertwovintagesofamodelforsale,relativetothosequarters whereonlyonemodelyearisusuallyavailable. Takingthedemandcurvesasgiven,thefirmsolvesadynamicinventoryproblemtomaximizeprofits. Thejointproduction/pricingdecisionwemodelisaclassicissueintheoperationsresearchliteraturegoing backtoWhiten(1955)andKarlinandCarr(1962).1 Likemanypapersinthisliterature,weassumethatthe goodmustbesoldbyafixeddeadline, butweextendthetheorybyallowingthefirmtoselltwovintages simultaneouslyandtohaveacoststructureofproducingvehicleswithseveralnontrivialnonconvexities. 1FedergruenandHeching(1999)andElmaghrabyandKeskinocak(2003)provideaniceoverviewofthemorerecentrevenuemanagementliteraturewithinoperationsresearch. AlsoseeChan,Hall,andRust(2005)forananalysisofasimilarpricing/procurementdecision. 4

A significant aspect of the automotive market that we reflect in our model is the distribution of dealerships across the geographic market. Showrooms are instrumental in allowing consumers to learn about manufacturers’ products and to gauge products’ characteristics. In this industry, for example, consumers value the ability to observe the vehicle they are considering purchasing and to take possession of the vehicle without delay. Consequently, part of the automaker’s problem is ensuring that there are a sufficient level of inventories on dealer lots across the national market. The automotive trade press often mentions the necessity of a showroom presence when discussing manufacturers’ inventories. In a recent issue of Ward’s Automotive Reports (August 2, 2004), a Cadillac executive stated, “We have 1,000 dealers who sell less than 50 vehiclesa year. They’re holding 300 to 400 days’ supply [that is, inventoriesover sales] becausetheywanttodisplayallthemodels.” Weincorporatethisneedtobuild-to-stock(thatis, inventories are a prerequisite for sales) into the model by assuming that the firm faces a “revenue tax” that is a function of sales over inventory. We assume that increases in the sales-to-inventory ratio make it harder forthefirmtoconsummatesalesbyraisingthetaxthefirmpayspertransaction. Aftercalibratingthesupply-sideparametersofthemodel,weareabletoreplicatethedeclineinprices overthemodelyearalongwithhump-shapedsalesandinventoriesthatweobserveinthedata. Earlyinthe model year, the automaker sets the vehicle price high to dampen sales and thus accumulate a large stock of inventories. Building up inventories, or following a build-to-stock inventory management strategy, is optimal because it reduces the cost of carrying out a transaction (that is, it lowers the revenue tax). Over the remainder of the model year, our estimate of leftward-shifting demand lowers the shadow value of inventories, resulting in a 8.1 percent decline in the retail price of a vehicle over the entire product cycle andanaveragevintagepremiumof7.9percent. Wethensolvethemodelwithnorevenuetax,thusallowingthefirmtomanageinventoriesonabuildto-orderbasis. Inthiscase,retailpricesfallovertheproductcyclebutbyonlyone-halfofthemagnitudewe seeinthedata. Hence,unlikepreviousworkthatattributesfallingpricestofashionorpricediscrimination, our theory implies that build-to-stock inventory management is as important in driving price decline as theseotherforces. Further, underabuild-to-orderpolicythepathsofsalesandinventoriesdonotfeature the prominent hump-shaped patterns seen in the data. These results demonstrate the significance of the firm’sinventorystrategyontheoptimalpricingpath. Fallingdemandaloneexplainsabouthalfoftheprice declineinautomobilesandmissessignificantcomovementsamongprices,salesandinventories. 5

1 Data Sources and Empirical Observations Inthissection,weoutlinethesourcesofthedatausedinouranalysisanddocumentseveralstylizedfacts. 1.1 DataSources Toconstructadatasetwithinformationonprices, sales, production, andinventoriesby modelandmodel yearintheU.S., wecombineddatafromtwosources. Thefirstdatasourceincludesdetailedinformation on U.S. retail transactions collected from a sample of vehicle dealerships. It provides information on prices, by model and model year, and on the distribution of sales, also by model and model year. The second data source contains information on total sales in North America, by country and model, and on production,bymodelandmodelyear. ThefirstdatasetwasconstructedbyCorrado,Dunn,andOtoo(2004),whoobtainedthedatafromJ.D. Power and Associates (JDPA). JDPA collects daily transaction-level information from dealerships across the U.S., which it aggregates to a monthly frequency. Then, along the product space dimension, JDPA addsupthedatatoamodelandmodel-yearlevel. Thesampleoftransactionsweuserepresents70percent of the geographical markets in the U.S. and roughly 15 to 20 percent of national retail transactions. It contains monthly observations for almost all unique make, model, and model-year light motor vehicles (for example, 2000 Ford Escort) sold in the U.S. and covers the period from January 1999 to January 2004. Among other variables, the dataset includes information on the number of transactions recorded, the average transaction price, the average cash rebate, and details about the average financial package customersreceived. JDPAattemptsto precisely measurethe transactionprice of avehicle. This measure includes the price of accessories (such as roof racks) and transportation costs but excludes aftermarket options, taxes, title fees, and other documentary preparation costs. Further, JDPA adjusts this price to accountforinstanceswhenadealershipundervaluesorovervaluesacustomer’strade-invehicleaspartof anewvehiclesale. JDPA’stransactionpricedoesnotaccountforincentivesthecustomerreceivedtohelp finance the purchase of the car; hence, we define the average market price of a vehicle as the transaction priceminusthecashrebateminusameasureofthefinancialincentiveofferedbythemanufacturer. In the data, we observe the amount financed, interest rate, and loan term that the average customer received. Thefinancialdataarecapturedforloansthatcustomersobtainedfromanyfinancialinstitution, aslongasthefinancingwasarrangedthroughthedealership. Asamajorityofcarloansarrangedthrough dealershipsaremadebythefinancingarmsofmanufacturers,wetreatthefinancialdataasanapproxima- 6

tionoftheaveragefinancialpackagethatconsumersreceivedfrommanufacturers. Tomeasurethevalueof thesefinancialincentivestoconsumers,wecomparethefinancialpackageinthedataagainstabenchmark packageofferedbycommercialbanks. Wemakethiscomparisonbyfirstcomputingthenetpresentvalue (NPV)oftheaverageamountfinancedgiventheinterestrateandloanterminthedata. Wethencompute theNPVoffinancingthesameaverageamountattheaverageinterestratereportedfor48-monthnewcar loans at commercial banks.2 The value of the manufacturer’s financial incentive is then defined as the differencebetweenthetwoNPVamounts. Finally,weconvertthemarketpriceinto2000dollarsbyusing theBEA’spersonalconsumptiondeflator. Asstatedearlier,thedatafromJDPAprovidethemarketpricesofvehiclesandthedistributionofsales, by model and model year. Using the total number of transactions across model years in each period, we computethefractionofamodel’ssalesthatisaccountedforbyeachavailablemodelyear. WelinkedtheJDPAdatatoinformationontheU.S.salesandNorthAmericanproductionofGeneral Motors,Ford,andChrysler,whichweobtainedfromWard’sCommunications. Weexcludedforeignmanufacturers,asmeasuringoverseasproductionisdifficult. Thesalesdataforthesefirms(alsoknownasthe Big Three) are available only at the model level, not by model year. Therefore, we constructed estimates ofsalesbymodelandmodelyearonthebasisofthemonthlymodel-yeardistributionsintheJDPAsample. Using information from Ward’s on model changeover dates at North American assembly plants, we decomposed the production data by model into observations by model year. Finally, using the sales and production estimates by make, model, and model year, we constructed estimates of vehicle inventories over the sample period. All told, the work described here results in a dataset with monthly observations, by model year, on the real average market price, quantity sold, quantity produced, and inventory held for almostalllightvehiclemodelssoldbytheBigThreeintheU.S.from1999to2003. 1.2 EmpiricalObservations Asstatedearlier,byexaminingthesedatawecanobserveseveralstylizedfactsthatholdacrossmodelsand modelyears. Toprovideillustrativeexamples,weshowplotsoftheprice,sales,production,andinventory data for a midsize car and a pickup truck (figures 1-8). The steady decrease in price over the sales cycle isimmediatelyevidentforbothvehiclesshowninthefigures. Inthe2000modelyear,theaveragemarket priceforthemidsizecarfallsover$2,000,morethan10percentoftheinitialprice. Thedeclinesinprices forsubsequentmodelyearsarejustaspronounced. Forthepickup, thepricedeclinesaverageadramatic 2TheBoardofGovernorspublishesthesedatainitsG.19ConsumerCreditrelease 7

Market ModelYear All Segment 1999 2000 2001 2002 2003 Compact 7.7 5.9 8.1 9.4 17.5 9.5 (2.4) Midsize 9.1 6.7 6.1 9.0 16.4 9.2 (1.5) Fullsize 8.9 7.9 6.4 8.5 13.4 8.9 (2.1) Luxury 11.6 10.3 8.8 13.1 14.9 11.6 (1.2) Pickup 6.6 10.0 7.1 9.2 15.4 9.9 (2.2) SUV 7.0 6.7 7.2 5.2 13.6 8.2 (0.9) Sporty 2.3 6.2 0.4 6.1 10.9 5.1 (2.4) Vans 5.4 9.0 9.3 8.5 15.7 9.6 (1.4) Total 7.4 8.0 7.1 8.1 15.1 9.2 (0.6) Note: Standarderrorsareinparenthesis Table1: TheAverageMonthlyPriceDecline(annualrate)byMarketSegmentandModelYear $4,000 for the 2001 through 2003 model years. Both the midsize car and the pickup clearly exhibit the simultaneous sale of multiple vintages as well as the premium the newer model-year vehicle commands overtheoldermodel-yearvehicle. Werefertothisdifferenceinpriceasthe“newvintagepremium.”The size of this premium varies, but the average premium for the midsize car is almost 7 percent, while the premiumforthepickupaveragesabout9percent. Turningtothefiguresonsalesandinventories,wefind thatthesalesandinventoriesofboththemidsizecarandthepickupexhibitahump-shapedprofile. Table 1 provides a summary of the average monthly price decline, the first stylized fact, by market segmentandmodelyearandweightedbysales. Forthemidsizemarketsegment,themeanmonthlydecline in prices of 1999 model-year vehicles is 9.1 percent at an annual rate. On average, midsize automobiles fall 9.2 percent. Table 1 illustrates the wide range in average price declines both across market segments andmodelyears. Ingeneral,luxuryvehiclesdeclinethemostinprice,followedbypickuptrucks. Looking acrossmodelyears,2003vehiclesdeclinethemostinpricebyfar. Thisreflectsespeciallyhighincentives offered by manufacturers in the latter half of the product cycle. Overall, the monthly decline in price averages9.2percentatanannualrate. To observe the within-year price declines more generally, we illustrate the aggregate matched-model price indexes for successive model years as constructed by Corrado, Dunn, and Otoo (2004) (figure 9). This price index was constructed from the entire JDPA dataset and so includes price data on vehicles produced by European and Asian automakers. As can be seen, transaction prices for a given model year areattheirhighestlevelswheneachmodelisintroduced,andtheytrenddownwardinaconsistentpattern over the course of the sales cycle. The overlap of the various model-year price indexes highlights the 8

1 1 4 5 1 . . 5 5 5 2000 model year → ← 2001 model year ← 2002 model year ← 2003 model year 14 13.5 13 12.5 12 11.5 Jan 1 1 9 1 99 Jul 1999 Jan 2000 Jul 2000 Jan 2001 Jul 2001 Jan 2002 Jul 2002 Jan 2003 Jul 2003 Jan 2004 )srallod fo sdnasuoht ni( secirp 10 8 9 2000 model year ↓ ← total sales ← 2001 model year ← 2002 model year 7 5 6 ↓ 2003 model year 4 3 2 1 Jan 19 0 99 Jul 1999 Jan 2000 Jul 2000 Jan 2001 Jul 2001 Jan 2002 Jul 2002 Jan 2003 Jul 2003 Jan 2004 Figure1: AverageTransactionPrices. )sdnasuoht ni( dlos selcihev Figure2: MonthlySales. Thedashedlineisthesumofsalesacrossmodelyears. 15 ← 2000 model year ← 2001 model year ← 2002 model year 10 ← 2003 model year 5 Jan 19 0 99 Jul 1999 Jan 2000 Jul 2000 Jan 2001 Jul 2001 Jan 2002 Jul 2002 Jan 2003 Jul 2003 Jan 2004 )sdnasuoht ni( decudorp selcihev 40 3 3 0 5 2000 model year ↓ ← 2001 model ye ← a r 2002 model year ↓ 2003 model year 25 20 15 10 5 Jan 19 0 99 Jul 1999 Jan 2000 Jul 2000 Jan 2001 Jul 2001 Jan 2002 Jul 2002 Jan 2003 Jul 2003 Jan 2004 Figure3: MonthlyProduction. )sdnasuoht ni( yrotnevni ni selcihev Figure4: MonthlyInventories. Prices,Sales,Production,andInventoriesforaMidsizeCarbyModelYear Source:J.D.PowerandAssociates,Ward’sCommunicationsandauthors’calculations 9

23 2 2 1 2 ← 2001 model year ← 2002 model year 20 2000 model year → ← 2003 model year 19 18 17 Jan 1 1 9 6 99 Jul 1999 Jan 2000 Jul 2000 Jan 2001 Jul 2001 Jan 2002 Jul 2002 Jan 2003 Jul 2003 Jan 2004 )srallod fo sdnasuoht ni( secirp 20 1 1 1 4 6 8 2000 model year → 2001 model year → ← to 2 ta 0 l 0 s 2 a → les 12 2003 → 10 8 6 4 2 Jan 19 0 99 Jul 1999 Jan 2000 Jul 2000 Jan 2001 Jul 2001 Jan 2002 Jul 2002 Jan 2003 Jul 2003 Jan 2004 Figure5: AverageTransactionPrices. )sdnasuoht ni( dlos selcihev Figure6: MonthlySales. Thedashedlineisthesumofsalesacrossmodelyears. 25 1 2 5 0 ← 2000 model year ← 2001 model year ← 2002 m ← o d 2 e 0 l 0 y 3 e a m r odel year 10 5 Jan 19 0 99 Jul 1999 Jan 2000 Jul 2000 Jan 2001 Jul 2001 Jan 2002 Jul 2002 Jan 2003 Jul 2003 Jan 2004 )sdnasuoht ni( decudorp selcihev 60 50 ← 2000 m ← od 2 e 0 l 0 y 1 e m a o r del yea ↓ r 2002 model year ← 2003 model year 40 30 20 10 Jan 19 0 99 Jul 1999 Jan 2000 Jul 2000 Jan 2001 Jul 2001 Jan 2002 Jul 2002 Jan 2003 Jul 2003 Jan 2004 Figure7: MonthlyProduction. )sdnasuoht ni( yrotnevni ni selcihev Figure8: MonthlyInventories. Prices,Sales,Production,andInventoriesforaPickupTruckbyModelYear Source:J.D.PowerandAssociates,Ward’sCommunicationsandauthors’calculations 10

Matched−model Price Indexes by Model Year 1.02 1 ← 2000 ← 1998 ← 1999 ← 2001 ← 2002 ← 2003 0.98 2004 → 0.96 0.94 0.92 0.9 0.88 Jan 1998 Jan 1999 Jan 2000 Jan 2001 Jan 2002 Jan 2003 Jan 2004 Figure9: Matched-ModelPriceIndexesbyModelYear. second stylized fact–that multiple vintages of vehicles are simultaneously sold for a significant portion of the model year. In our database of transactions, the mean length of time a vehicle is on the market is 16.7 months. The number of months sold varies little across vehicles; the mean length of the automobile productcyclehasastandarderrorofonly0.02. Turningtothethirdstylizedfact,wereporttheaveragenewvintagepremiumbymarketsegmentand modelyear,weightedbysales(table2). Thetableillustratesthatthenewvintagepremiumof2000modelyearmidsizecarsovertheir1999model-yearcounterpartsis10.0percent. Inthesample,thenewvintage premiumis8.5percent,onaverage,formidsizecars,andthestandarderroris0.4. Although the new vintage premium varies quite a bit across market segments and time, overall it amounts to 8.8 percent, on average, in our sample. This premium is highest for luxury cars and pickup trucksandlowestforcompactcars,sportutilityvehicles,andsportycars,wherethedifferenceinpremiums between luxury and compacts is 4.5 percent. Across model years, the average new vintage premium is typically between 7 and 9 percent, though the premium during the 2003 to 2004 changeover is 13.1 percent. This large premium is related to the steep decline in within model-year prices for 2003 modelyear vehicles, shown in table 1. Both numbers reflect the unusually high incentives the Big Three placed on2003model-yearvehiclesoverthesecondhalfoftheproductcycle. One might argue that the new vintage premium simply reflects improvements in quality or additional 11

Market ModelYear All Segment 2000 2001 2002 2003 2004 Compact 5.9 6.9 6.7 6.9 11.0 7.1 (0.5) Midsize 10.0 5.8 6.1 7.5 11.9 8.5 (0.4) Fullsize 9.9 6.2 7.8 8.2 9.2 8.3 (0.6) Luxury 11.0 11.6 9.8 14.1 11.1 11.6 (0.4) Pickup 10.7 9.8 6.3 8.6 20.0 10.6 (0.7) SUV 5.4 0.4 10.1 8.8 10.9 7.2 (0.4) Sporty 2.6 7.5 3.3 28.9 -7.8 7.2 (0.8) Van 7.7 11.8 3.6 9.1 12.4 8.6 (0.4) All 8.6 7.2 7.1 9.1 13.1 8.8 (0.2) Note: Standarderrorsareinparenthesis Table2: TheAverage‘NewVintagePremium’byMarketSegmentandModelYear features. For example, the 20 percent new vintage premium recorded for 2004 model-year pickup trucks reflects, in part, a quality improvement made to Ford’s F-series pickup truck.3 For many of the vehicles in our sample, however, changes in the observable characteristics from one model year to the next were minimal,andevenforvehicleswithsuchchanges,thedownward-slopingpricepatternwasstillapparent. To further investigate this fact, we looked at the new vintage premium for a subsample of vehicles that had not undergone a major redesign. We determined when a model received a major redesign by using data from Ward’s Communications on the vehicle’s platform. Given that the platform choice designates the basic structure of the vehicle, we take a manufacturer’s decision to change a vehicle’s platform as a sufficient,thoughnotnecessary,conditionthatthevehiclehasundergoneamajorredesign. Werecomputed the new vintage premium for these vehicles and found that the average premium differs little from the figuresreportedintable2. The fourth stylized fact states that both sales and inventories for a particular model and model year exhibitahumped-shapedpattern. WhenweplottheBigThree’saggregatesalesbymodelyear,weseethe distinctivehumpshapethatsalesfollowovertheproductcycle(figure10). Thecontourofaggregatesales, however,confoundstheevolutionofsalesovertheproductcyclewithcalendereffects,becausevehiclesof agivenmodelyeararenotallintroducedinthesamemonth. Toseparateoutthesetwoeffects,wedefine the dummy variables 1 for t =1,2,...,14 as indicators of how many months the model year has been t sold. If 1 is equal to 1, then at this date the associated vehicle is in its first month of sales. For model t=1 3Alternatively,ifacheaperbasemodelisintroduced,thevehiclepremiummaybebiaseddownwards(see,forexample,the -7.8premiumfor2004sportycars). 12

1200 1000 800 600 400 200 0 )sdnasuoht( selaS ← 1999 model yea ← r 2000 model year ← 2001 model yea ← r 2002 model yea ← r 2003 model year Jan 1999 Jul 1999 Jan 2000 Jul 2000 Jan 2001 Jul 2001 Jan 2002 Jul 2002 Jan 2003 Jul 2003 Jan 2004 Figure10: BigThreeSalesbyModelYear years that are sold for more than fourteen months, we define a dummy variable1 , which is equal to 1 15+ forsalesthattakeplacemorethanfourteenmonthsaftertheintroductionofthevehicletothemarketplace. Wethenrunaregressionoflogsalesonthesedummyvariables,includingfixedeffectsforeachmodeland controlling for the calender month. Examining the coefficients on these dummy variables then provides anestimateoftheshapeofthetypicalsalespathforavehicle. Usingasimilarapproach, butsubstituting inventoriesforsales,wecanalsostudytheshapeofthetypicalinventorypathforavehicle. Wereportthe estimatedcoefficientsfromrunningthesetworegressions(table3). Regardingthesalesregression,theestimatedcoefficientsimplythatsalesdoindeedfollowahumpedshapedpath. Afterrapidsalesgrowthinthefirstthreesellingmonths,thelevelofsalesthenslowlyclimbs to its peak six to eight months after the vehicle’s debut before declining. This same pattern can be seen intheestimatedcoefficientsfromtheregressionusinginventories. Inventoriespeakslightlylater,roughly ninemonthsafteravehicle’sdebut,beforerapidlyfalling. To better analyze the relationship between sales and inventories, we consider the ratio of inventories to sales, also known as days’ supply. We use the stock of inventories at the beginning of the period and sales for the current month. Hence, this ratio measures the number of days the firm could continue to sell cars if it used only the stock of inventories available at the start of the month, assuming future sales are equal to the current month’s flow. A major focus of the days’ supply calculation is the large stock of 13

Sales Inventories Parameter Coefficient StandardError Coefficient StandardError 1 0.76 0.038 0.32 0.058 t=2 1 1.09 0.039 0.50 0.060 t=3 1 1.24 0.039 0.62 0.061 t=4 1 1.24 0.039 0.79 0.061 t=5 1 1.25 0.040 0.94 0.062 t=6 1 1.26 0.041 1.03 0.066 t=7 1 1.25 0.041 1.05 0.067 t=8 1 1.18 0.041 1.11 0.068 t=9 1 0.98 0.042 1.09 0.067 t=10 1 0.93 0.042 0.84 0.066 t=11 1 0.95 0.042 0.38 0.064 t=12 1 0.82 0.041 -0.07 0.063 t=13 1 0.60 0.043 -0.45 0.066 t=14 1 -0.11 0.034 -0.80 0.047 t=15+ Table3: TheShapeofSalesandInventoriesbyModelandModelYearOvertheProductCycle MarketSegment Compact Midsize Fullsize Luxury Pickup SUV Sporty Vans All DaysSupply 73 60 75 80 84 75 83 85 76 (2.1) Note: Standarderrorsareinparenthesis Table4: TheAverageDaysSupplybyMarketSegment inventories that automakers carry relative to sales. On average, automakers carry 76 days’ supply in our data–implyingautomakerstypicallycarryenoughinventoriestosellvehiclesforovertwomonthswithout any additional production. Table 4 provides a breakdown of the average days supply by market segment andillustratesthesubstantialvariationindayssupplyacrossdifferenttypesofvehicles. Turning to the last stylized fact, we look at the correlation between inventories and prices, by model and model year. To analyze the relationship between prices and moments when inventories are above or below trend, we first need to accurately measure when inventories are ample or lean. The residuals fromtheinventoryregressiondescribedearlierprovideameasureofthedeviationsfromtheusualcontour of inventories over the product cycle. We measure whether prices are correlated with these inventory fluctuations by regressing the log of price on a lag of these inventory residuals. We find the expected significant negative relationship between lagged inventory residuals and price: The estimated coefficient onthelaggedresidualsis-0.02,andtheassociatedstandarderroris0.003. The decline in an automobile’s price over the model year and the resulting new vintage premium has 14

been studied by Pashigian, Bowen, and Gould (1995). They hypothesize that the new vintage premium reflects optimal pricing behavior in an environment in which demand is driven by fashion. They use monthlydatafromtheconsumerpriceindextoshowthatattheaggregatelevelpricesfornewcarsdecline between December and September of the model year. Although the magnitude of the within-model-year pricedeclineshavefallenbetween1954and1989,Pashigian,Bowen,andGouldfindthesepricedeclines are larger for luxury and speciality cars than for compacts and subcompacts. They argue that the larger the changes in styling and quality improvements between model years, the larger the within-model-year pricedeclines. Hence,thenew-carmarketbehaveslikethemarketforfashionableapparel. Incontrast,we find only marginal evidence of a particular pattern in the new vintage premium across market segments. As shown in table 2, although luxury cars command the highest vintage premium, other fashion-oriented vehicle types such as SUVs and sporty automobiles command premiums nearly equal to, or markedly below,thoseofthemoreplaincompactandmidsizeautomobiles. Giventheevidence,wepositthatwithin-model-yearpricedeclinesaredrivenmorebytheused-vehicle marketthanbyfashion. Considerthecaseofa2000model-yearvehicleproducedattheendoftheproduct cycleanda2001model-yearvehicleproducedatthebeginningoftheproductcycle. Althoughthesetwo vehicles may have been produced just a few weeks apart, we expect that in the used-vehicle market, the 2000vintagewillbeperceivedtohavebeenontheroadtwelvemonthslongerthanthe2001vintage. Toprovideevidenceinsupportofthishypothesis,weestimatedapriceregressiononaseparateJDPA dataset of used-vehicle transactions from 2001-2003. The left-hand-side variable is the log of the transaction price for a given model and vintage of a used vehicle. The explanatory variables in the regression include time and model dummies as well as vehicle characteristics such as engine size. As a proxy for thevehicle’sphysicaldepreciation,weincludethevehicle’sodometerreadingwhensold. Finally,wealso addameasureofthevehicle’smodelage,whichequalsthecalendaryearminusthemodelyearplusone.4 Weshowtheresultingcoefficientsonageandodometerreading,bothofwhicharestatisticallysignificant with greater than 99 percent confidence (table 5). As expected, the coefficient on the odometer reading is negative and implies a price decline of about 0.4 percent for each additional 1,000 miles on a given vehicle. Notably, the coefficient on age implies that, even after controlling for the odometer reading and othervehiclecharacteristics,ahighermodelage(thatis,anoldermodelyear)impliesalowerpriceinthe used vehicle market. All other things constant, increasing the age (as defined by the model year) by one 4Becausewehavealimitedsetofphysicalcharacteristicstocontrolforchangesinvehiclequalityacrossvintagesofthesame model,werestrictthesampletovehiclesofagefourorunder. Thisrestrictionreducesthevariationinpriceacrossvintagesof thesamemodelduetochangesinunobservedcharacteristics. 15

Variable Coefficient StandardError Age -0.093 0.004 Odometer(thousandsofmiles) -0.004 0.000 Table5: CoefficientsonAgeandOdometerfromtheUsed-vehiclePriceRegression yeardecreasesthevalueofausedvehicleby9.3percent,afigureonlyslightlygreaterthanourestimateof thenewvintagepremium. Thisstronglysuggeststhatthenewvintagepremiumisdrivenbythedifference inthenewvehicles’valuesintheused-vehiclemarket. 2 A Market Equilibrium Model with Overlapping Model Years Inthissection,wepresentamarketequilibriummodeldesignedtocapturetheempiricalregularitiesdocumented earlier. We first describe the firm’s problem. We assume the automaker takes market demand curves as given and solves a dynamic profit maximization problem. As the automaker is able to hold inventories, at certain times the automaker is able to sell two vehicles, the current year’s vintage and the previousyear’svintage. Thefirm’smodelparametersarecalibratedtomatchthekeyfeaturesofthefirm’s cost structure and the means of prices, output and inventories. We derive decision rules that govern the production and pricing of vehicles over the model year. Through numerical simulations, we demonstrate that the empirical regularities documented earlier are consistent with our derived decision rules under a build-to-stockinventorypolicy. An essential feature of the firm’s problem is the market demand curve. We posit a semi-log demand curvewhoseparametersarepricesemi-elasticities. Wethendrawupontheexistingdemand-choiceliteraturetoestimatethesesemi-elasticitiesandtheirchangeovertheproductcycle. 2.1 TheAutomaker’sProblem In the interest of tractability, we make several strong simplifying assumptions on the supply side. First, we assume that each vehicle line within the firm can be considered a separate, independent subfirm or profit center. Hence, an automaker is modelled as a collection of dynamic programs that can be solved independentlyofeachother. Second,weintegratethedealershipintotheautomakerandconsideraunified pricing decision. Third, we abstract from issues of bargaining and price discrimination by assuming that allcustomerswhopurchaseduringaparticularperiodpaythesameretailprice. Ofcourse,therearemany interestingquestionsabouthowtheautomakersactuallydecentralizetheiroperationsbothacrossproducts 16

andbetweentheproductionandmarketingsidesofthebusiness. Butbecausetheseissuesarenotcentral tounderstandingthefactspresentedearlier,wedeferfurtherconsiderationtootherpapers.5 Theautomakersellstwoproducts: thisyear’svintageandlastyear’svintage. Thedecisionperiodisa week. ThereareT weeksinamodelyear,andanewmodelyearbeginstheweekaftertheoldmodelyear ends. So the automaker solves an infinite horizon problem by repeatedly solving a T-week model-year problem. Successivemodelyearsarelinkedbecausethisyear’svintagebecomeslastyear’svintageatthe endoftheTth week. Eachweekthefirmmustdecide(1)thenumberofvehiclesofthecurrentmodelyear to produce, q ; (2) the number of days to operate the plant, D , the number of shifts to run, S , and the t t t numberofhourspershift,h ;(3)theretailpriceofthecurrentvintage, pthis;and(4)theretailpriceoflast t t year’svintage, plast (ifanyarestillinstock). t j Weassumethatweeklysales,s ,foreachofthetwovintagesdependoneachvintage’sownpriceand t thepriceoftheothervintage: s j =µ j −h j log(p j )+f ji log(pi) for j,i={this,last}andi(cid:54)= j,} (1) t t t t t t where µ j is a constant term, h j is the own-price semi-elasticity, and f ji is the cross-price semi-elasticity. t t t Thedemandparametersmayvaryacrossthe52weeksoftheyear. j Unsold vehicles can be inventoried without depreciation. Let I be the stock of vintage j vehicles t+1 that are inventoried at the end of period t and carried over into period t+1. Current production is not availableforimmediatesale,sosalescanbemadeonlyfromthebeginning-of-periodinventories: s j ≤I j. (2) t t Further,salescannotbebacklogged. Inventoriesforthecurrentvintagefollowthestandardlawofmotion: Ithis=Ithis+q −sthis. (3) t+1 t t t Becausenovehiclesforthelastmodelyearareproducedduringthecurrentyear,inventoriesforlastyear’s vintageevolveaccordingto Ilast =Ilast−slast. (4) t+1 t t At the conclusion of the current model year, any unsold vehicles of last year’s vintage are scrapped at a zeroprice,andthisyear’svintagebecomeslastyear’svintage: Ilast =Ithis+q −sthis. (5) 1 T T T 5Forexample,BresnahanandReiss(1985)modelandestimatethedivisionofmarkupsbetweenautomobilemanufacturers anddealers. FordiscussionsofbargainingandpricediscriminationintheretailautomarketseeAyresandSiegelman(1995), Goldberg(1996),andZettelmeyer,ScottMorton,andSilva-Risso(2001). 17

We assume the vehicle is assembled at a single plant. Each period, the firm must decide how many vehiclesofthecurrentvintagetoproduceandhowtoorganizeproductiontominimizecosts. Asistypical inmostmanufacturingindustries,assemblyplantmanagersincreaseordecreaseproductionbyalteringthe workweekratherthantherateofproduction. TheplantcanoperateDdaysaweek. Itcanrunoneortwo shifts,S,eachday,andbothshiftsare hhourslong. Weassumethenumberofemployeespershift,n,and thelinespeed,LS,arefixed. Sothefirm’sproductionfunctionislinearinhours: q =D ×S ×h ×LS. (6) t t t t Although the production function is linear, the firm faces several important non-convexities because of its labor contract.6 The average straight-time, day-shift wage at these plants is about $27 an hour plus benefits. Workersonthesecond(evening)shiftreceivea5percentpremium. Anyworkinexcessofeight hoursaday,andallSaturdaywork,arepaidatarateoftimeandahalf. Employeeswhoworkfewerthan 40 hours per week must be paid 85 percent of their hourly wage times the difference between 40 and the number of hours worked. This “short week compensation” is in addition to the wages a worker receives for the hours actually worked. If the firm chooses to not operate a plant for a week, the workers are laid off. Laid-offworkersreceive95centsonthedollaroftheir40hourpayinunemploymentcompensation. Ofthese95cents,thefirmpaysabout65cents. Given such a labor contract, if the firm decides to produceq vehicles, it must then choose how many days to operate the plant, how many shifts to run, and how many hours to run each shift to minimize its costofproduction. Giventhesechoices,thefirm’sweekt costfunctionisexpressedas c(D ,S ,h |q ) = g q + (w +I(S =2)w )×(D h n+max[0,0.85(40−D h )n] (7) t t t t t 1 t 2 t t t t +max[0,0.5D (h −8)n]+max[0,0.5(D −5)8n])+0.65w 40(2−S )n, t t t 1 t where g is the per vehicle material cost, n is the number of employees per shift, and w and w are the 1 2 hourly wage rates paid to the first-shift and second-shift workers, respectively. The first term is the pervehicle cost; it incorporates all costs (such as materials, energy, transaction) that do not depend on the allocation of production over the week. The first term within the brackets represents the straight-time wages paid to the production workers. The subsequent terms within the brackets capture the 85 percent ruleforshortweeksandtherequiredovertimepremium. Thelasttermistheunemploymentcompensation 6For further discussion of the institutional details of labor contracts in automobile manufacturing, see Bresnahan and Ramey(1994),Hall(2000),orRameyandVine(2004). 18

bill charged to the firm. LetD =0 if and only ifS =0. Thus, the cost function is piecewise linear with t t kinksatoneshiftrunning40hoursperweekandtwoshiftsrunning40hoursperweek. Thisimpliesthat thefirmwillminimizeaveragecostsbyoperatingtheplantwitheitheroneshiftortwoshiftsfor40hours perweek. The firm’s objective is to maximize the present value of the discounted stream of profits. For each modelyeartheautomaker’sproblemistomaximize (cid:181) (cid:182) (cid:229) T 1 t−1(cid:110) plastslast(1−t (slast/ilast) y )+pthissthis(1−t (sthis/ithis) y )−c(D ,S ,h |q ) (cid:111) 1+r t t t t t t t t t t t t t=1 (cid:181) (cid:182) T 1 + V(Ilast,0,1) (8) 1+r 1 subject to (1)-(6) and where c(D,S,h|q) is given by (7). The terms t (slast/ilast) y and t (sthis/ithis) y are t t t t revenue taxes the automaker must pay if the sales-to-inventory ratio is large. This term captures the distributionalcoststheautomakerfaces,asdescribedpreviouslyintheintroduction. Wheninventoriesare low, it is harder for potential customers to observe and gauge a vehicle (that is, to test-drive it and view the choice set), and thus it is more costly to consummate a sale. The tax effectively disappears when the sales-to-inventoryratioissmall.7 ThetermV(Ilast ,0,1)isacontinuationvalue,whichwenowdefine. T+1 LetV(Ilast,Ithis,t)betheoptimalvalueatweekt forthefirmthatholdsininventoryIlast oflastyear’s vintageandIthis ofthisyear’svintage. Thenthefirm’svaluefunctioncanbewritten: (cid:189) V(Ilast,Ithis,t)= max plastslast(1−t (slast/ilast) y ) + pthissthis(1−t (sthis/ithis) y )− minc(D,S,h|q) pthis,plast,q D,S,h (cid:190) 1 + V(Ilast−slast, Ithis+q−sthis,t+1) fort =1,...,T−1 (9) 1+r subjectto(1),(2),and(6)andwherec(D,S,h|q)isgivenby(7). AtweekT,thisyear’svintagebecomeslastyear’svintage,andsothevaluefunctionis (cid:189) V(Ilast,Ithis,T)= max plastslast(1−t (slast/ilast) y )+pthissthis(1−t (sthis/ithis) y )−minc(D,S,h|q) pthis,plast,q D,S,h (cid:190) 1 + V(Ithis+q−sthis,0,1) .(10) 1+r 7We tried specifying demand as an increasing function of the level of inventories; that is, customers are willing to pay a higherpriceiftheyaremorelikelytobematchedtotheiridealvehicle. However,inthiscase,themodelhasthecounterfactual implicationthat,allotherthingsbeingequal,higherinventoriesareassociatedwithhigherprices. Fact5suggeststhat,evenif demandisincreasingininventories,thesupply-sideinventoryeffectdominatesthedemand-sideeffect. Wealsotriedreplacing therevenuetaxwithanexplicitinventory-to-salestarget;thisyieldssimilarresults,butthetaxiscomputationallymorerobust. 19

2.2 ParameterizingtheModel Thereanumberofparametersinthismodel. Ourapproachistochoosethesupply-sideparametersinthe firm’sproblembasedonthefirstmomentsinthedataandfrompublishedinformationonassemblyplants. For demand parameters, we employ a discrete-choice methodology to estimate consumer’s preferences over automobiles. We then use these estimates to compute the intercepts, own-price semi-elasticities and cross-pricesemi-elasticitiesthatareparametersinthemarketdemandfunction,equation(1). 2.2.1 Demand-sideparameters Overview: The demand for automobiles is modelled within a discrete-choice framework. Following Berry, Levinsohn, and Pakes (1995), henceforth BLP, we construct the demand system by aggregating over the discrete choices of heterogeneous individuals. The utility derived from choosing an automobile depends on the interaction between a consumer’s characteristics and a product’s characteristics. Consumers are heterogeneous in income as well as in their tastes for certain product characteristics. We distinguish between two types of product characteristics: those that are observed by the econometrician (such as horsepower and miles per gallon), which are denoted by X; and those that are unobserved by the econometrician (such as styling or prestige), which are denoted by x . Drawing from the nested logit literature, we also incorporate a correlation in the consumer’s tastes for vehicles of the same model year. We divide vehicles into G+1 mutually exclusive groups (that is, model years)–g=0,1,2,...,G–where theoutsidegoodisthesolememberofgroup0. Wealsoallowhouseholds’distasteforprice,denotedby a ,tovaryfromquartertoquarter. Thiscapturesthepossibilitythatdifferenttypesofhouseholdsshowup to purchase a new automobile at different times of the year. We specify the indirect utility derived from consumeripurchasingproduct j,droppingthetimesubscript,as u =X b +x −a p + (cid:229) s n x +z +(1−r )e , (11) ijq j j iq j k ik jk ig ij k where p denotes the price of product j and x ∈ X is the kth observable characteristic of product j. j jk j The term X b +x , where b are parameters to be estimated, represents the utility from product j that is j j commontoallconsumers,orameanlevelofutility. Consumersthenhaveadistributionoftastesforeach observable characteristic. For each characteristic k, consumer i has a taste n , which is drawn from an ik independentlyand identically distributed(i.i.d.) standard normal distribution. The parameters captures k thevarianceinconsumertastes. Theterma measuresaconsumer’sdistasteforpriceincreasesinquarter iq q={1,2,3,4}. Following Berry, Levinsohn, and Pakes (1999), we assume that a = a q, where a is a iq yi q 20

parameter to be estimated and y is a draw from the income distribution. We assume the distribution of i householdincomeislognormal,and,foreachyearinoursample,weestimateitsmeanandvariancefrom the Current Population Survey (CPS). The second-to-last term in equation (11) captures correlations in a consumer’s tastes for products within the same group. For consumer i, the variable z is common to all ig productsingroupgandhasadistributionthatdependsuponr . Finally,e isani.i.d. extremevalue. ij Consumerschooseamongthe j=1,2,...,J automobilesinoursampleandtheoutsidegood(denoted j=0), which represents the choice not to buy a new automobile from the Big Three. Consumers choose theproduct jthatmaximizesutility,andmarketsharesareobtainedbyaggregatingoverconsumers. Implementation: As described in section 1, our sample includes data for the Big Three firms over the five-year period from February 1999 to January 2004. There are 638 observations of unique model and model-yearvehicles. Weaggregatesalesandpricestothequarterlyfrequencybecausethereisasignificant amount of volatility in monthly sales due, in part, to intertemporal substitution. Moreover, BLP’s static utilitymaximizationapproachisbettersuitedtoanalyzingquarterlydata. Wedonotestimatethemodelat anannualfrequencybecausethevariationinpriceandintheconsumer’schoicesetfromquarter-to-quarter isasignificantsourceofidentificationintheBLPframework. Weusevehiclecharacteristicsthatincludeasetofmodeldummies,ameasureofacceleration,vehicle dimensions,ameasureofsafety,andfuelefficiency.8 Aswasdoneinpreviousresearch,welinkquantity soldandtransactionpricetothecharacteristicsofthebasemodeltoproduceavehicle-quarterobservation. Tothisstandardsetofcharacteristics,wealsoaddameasureofhowlongaproducthasbeensold–thenumber of quarters since the vehicle was first introduced. The model dummies, the number of quarters since introduction, and a quadratic time trend make up the vector of observable characteristic used to compute the mean utility of a product, X b . The measures of acceleration, dimension, safety, and fuel efficiency, j alongwiththenumberofquarterssinceintroduction,areincludedinthevectorofobservablecharacteristicsusedtomeasureheterogeneityinhouseholds’preferences,(cid:229) s n x . Inessence,themodeldummies k k ik jk helpexplainthemeanutilitylevelofaproduct,whilethemeasuresofacceleration,dimension,safety,fuel efficiency,andnumberofquarterssinceintroductiondrivethesubstitutionpatternsamongvehicles. Unlike the approach taken in BLP, we also incorporate (through z ) correlation in consumers’ tastes acrossadiscretecharacteristic, modelyears. ThisnestedlogitapproachisfoldedintotheBLPalgorithm inthenaturalway. FollowingBLP,weusethenumberofhouseholdsintheU.S.asreportedintheCPSas 8InformationonvehiclecharacteristicsweretakenfromAutomotiveNews’sMarketDataBook(variousyears). 21

a measure of market size for the year. We assume that one-fourth of all households in a given year show upeachquarter. Our estimation strategy follows the generalized method of moments approach taken by BLP. Given the vector of parameters q , we solve for the unique vector of mean utilities such that the model’s predictedmarketsharesequalactualmarketshares.9 Wethenmatchthemomentsrelatedtothemarket-level disturbance,x ,usingtheassumptionthatx isuncorrelatedwiththevehiclecharacteristics,X,or j E[x (q )|X]=0. (12) Asinthetypicalnestedlogitexercise,theparameterr isestimatedthroughthedecompositionofthemean utilityofaproduct, d =X b −x +r ln(s ), (13) j j j j/g where d is equal to the difference between the market share of product j and the market share of the j outside good, and s is the market share of product j relative to the market share of the group to which j/g product jbelongs. As x is correlated with both price and s , an endogeneity problem arises. Berry (1994) provides a j/g methodologythatallowsustouseinstrumentalvariables. WefollowBLP’sapproximationoftheoptimal instruments,thoughinoursettingtheyhaveadiminishedeffect. Theseinstrumentsarebasedoncompeting products’ characteristics, which change at the model-year frequency, though our price and quantity data vary at the quarterly level. Accordingly, we augment the set of instruments to include indicator variables forwhethermultiplevintagesofamodelarebeingsoldsimultaneously. Whetherornotanothervintageis beingsoldatthesametimehasaneffectonprice,andthiseffectcanvaryataquarterlyfrequency. Further, there is little reason to suspect that the sale of multiple vintages of a model is related to the unobserved characteristicofavehicle. Newvintagesaremostoftenintroducedatanannualfrequency,andautomakers facelargecoststoalteringthescheduledintroductionofanewvintage. Results: We present a subset of the parameter estimates in table 6. Given their large number, we do not report our fixed-effects estimates. Instead, we show those estimates that measure the heterogeneity in consumers’ tastes (s ) along with estimates of the substitutability of vehicles across model years (r ) and estimates of a consumer’s distaste for price (a ). The coefficients on miles per dollar (measure of 9WemodifiedtheprogramsprovidedinNevo(2000)toestimatethedemandsystem.Anotableadditiontothissetofprograms istheimportancesamplingsimulatordescribedinBLP,whichisusedtointegrateovern toobtainapredictionofmarketshare. 22

Parameters Coefficient StandardError TasteforVariety s Constant 0.61 0.968 Horsepower/Weight 3.78 0.905 Height 3.91 1.086 Size 2.07 0.752 Miles/dollar 0.50 2.806 Airbag 0.93 0.631 Intro -0.04 0.154 Model-YearNest r 0.58 0.014 DistasteforPrice(Q1) a 42.31 7.684 1 DistasteforPrice(Q2) a 36.64 7.998 2 DistasteforPrice(Q3) a 34.99 6.823 3 DistasteforPrice(Q4) a 32.95 4.707 4 Table6: ParameterEstimates fuel efficiency), air bag (measure of safety), and intro (number of quarters since introduction) are not statistically significant. However, we estimate that consumers are quite heterogeneous in their tastes for acceleration(horsepoweroverweight),height,andsize(lengthtimeswidthofthevehicle). The model-year nest and the price coefficients are precisely estimated. The estimate ofr reflects the substitutabilityofmodelsfromdifferentmodelyears,wherer =0indicatesnosubstitutabilityandr =1 implies perfect substitutability. Our estimate of r is 0.58, which suggests that consumers do not find vehicles from different model years to be close substitutes. The estimated value of household’s distaste forpricefallsfromquarter1toquarter4. Thequartersdifferfromcalendarquarters. Wedefinedthefirst quarterasthefirstthreemonthsofatypicalvehicle’sproductcycle: August,September,andOctober. We then defined the second through fourth quarters on the basis of this new grouping of months. Although thedifferencesamongthea ’sarenotsignificant,theyimplythathouseholdsaremoresensitivetopricein quarterswhenautomakerstypicallyoffermultiplevintagesofvehicles,relativetoquarterswhenonlyone vintageisavailable. Theestimateofr andthemagnitudeofthepricecoefficientsaremoreeasilyinterpretedbyexamining the implied own-price and cross-price elasticities. These elasticities provide the clearest picture of the valuesofthesemi-elasticitiesthatweuseinourspecifieddemandfunction. Wereporttheown-priceelasticitiesofindividualvehiclesaveragedacrossmarketsegments, quarters, andvintages, wherethevintage labelsignifieswhetherthevehicleisthenewestmodelyearavailableornot(table7). 23

Vintage MarketSegment 1stQuarter 2ndQuarter 3rdQuarter 4thQuarter New Compact 8.2 9.2 7.9 8.6 Full 10.3 11.3 9.1 9.6 Luxury 9.8 11.4 8.4 9.0 Midsize 9.4 10.5 8.9 9.7 Pickup 9.7 10.8 9.1 9.0 SUV 9.7 10.7 8.6 8.6 Sporty 10.8 10.9 9.1 10.0 Van 10.2 11.6 9.7 10.1 All 9.8 10.8 8.8 9.3 Old Compact 7.8 8.3 8.8 8.8 Full 10.0 11.4 10.5 10.6 Luxury 9.9 11.0 9.2 9.2 Midsize 9.4 10.0 8.9 8.1 Pickup 9.8 10.8 8.9 11.6 SUV 9.9 10.8 9.6 8.6 Sporty 9.8 11.7 9.3 8.6 Van 10.2 10.9 9.6 5.8 All 9.6 10.6 9.3 8.9 Table7: TheAbsoluteValueofOwn-PriceElasticitiesbyMarketSegment,Quarter,andVintage Theown-priceelasticitiesgeneratedbyourparameterestimatesrangebetween6and12,anindication that manufacturers face quite elastic demand. In the first quarter a car is sold, our results imply that a 1 percentpriceincreaseforatypicalcompactcar(roughly$140)causesan8.2percentfallinsales,holding everything else equal. The average own-price elasticity across all vehicles is reported in the “All” row, andillustratesthatelasticitiesinthefirstandsecondquarteraretypicallyhigherthanthoseinthethirdand fourthquarterforbothnewandoldvintages. Thisislikelydrivenbythehighera ’sandlargerchoicesetsin thesequarters. Ingeneral,ourestimatedelasticitiesarehigherthanthosefoundinthepreviousliterature; BLP, for example, report a range of elasticities between 3 and 6. It is not surprising, however, that our elasticity estimates are higher than some observed elsewhere because previous research estimated ownpriceelasticitiesamongmodels–thatis,atalevelofaggregationhigherthanthatofourdata. Indeed,when we re-estimate the parameters from data aggregated to the model level, the implied own-priceelasticities fallwithintherangeofthosereportedinBLP. Giventhatautomakersselltwovintagesofthesamemodelsimultaneouslyforalmosthalfofthemodel year,thecross-priceelasticitybetweenvintagesofthesamemodelisofparticularinteresttothefirm. We 24

Vintage MarketSegment 1stQuarter 2ndQuarter 3rdQuarter 4thQuarter NewtoOld Compact 0.02 0.01 0.01 0.04 Full 0.02 0.00 0.00 0.03 Luxury 0.02 0.01 0.00 0.01 Midsize 0.02 0.01 0.00 0.01 Pickup 0.10 0.02 0.03 0.12 SUV 0.04 0.01 0.02 0.04 Sporty 0.02 0.01 0.01 0.01 Van 0.02 0.01 0.00 0.00 OldtoNew Compact 0.01 0.03 0.03 0.01 Full 0.01 0.02 0.00 0.02 Luxury 0.01 0.03 0.01 0.00 Midsize 0.01 0.03 0.01 0.00 Pickup 0.06 0.18 0.02 0.05 SUV 0.02 0.05 0.02 0.05 Sporty 0.01 0.03 0.03 0.00 Van 0.01 0.04 0.05 0.01 Notes:“NewtoOld”indicatesthepercentagechangeinthemarketshareofthenewervintageofamodelgiven apercentagechangeinthepriceoftheoldervintage.“OldtoNew”indicatestheoppositerelationship. Table8: Cross-PriceElasticitiesBetweenVintagesoftheSameModelbyMarketSegmentandQuarter reportourestimatesofthecross-priceelasticitiesbetweentwovintagesofthesamemodel,averagedover quartersandacrossmarketsegments(table8). Formostofthevehiclesinoursample, theoldandnewvintagesofthesamemodelaresoldsimultaneouslyduringthefirstandsecondquarters(AugustthroughJanuary). However,afairnumberofvehicles are introduced at other times in the year, and so we can compute cross-price elasticities throughout the year. The upper portion of table 8 displays the percentage change in the market share of the newer vintage of a model given a percentage change in the price of the older vintage. The bottom portion of the table shows the opposite relationship–the percentage change in market share for the old vintage given a percentage change in the price of the newer vintage. Generally, the estimated cross-price elasticities are quitesmallrelativetotheown-priceelasticities. Thisimpliesthatvariousvintagesofthesamemodelare typicallyquiteimperfectsubstitutes.10 Tochecktherobustnessoftheseresults,weestimatedseveraldifferentspecificationsoftheconsumer’s 10AnaAizcorbesuggestedthatgeographicalfactorsmayexplainourlowcross-priceelasticityestimates. Ifdifferentvintages ofthesamemodelarerarelyofferedforsaleatthesamelocation,thenthedegreetowhichconsumerscansubstitutebetween vintagesmaybelimited. 25

utilityfunction. Werejectedaspecificationofutilityforwhichgroupsaredefinedatthemodellevel(that is, thecorrelationinconsumertastesisacrossallavailablevintagesofaparticularmodel)becauser was estimatedtobegreaterthan1.11 Inaddition, boththehighown-priceandthesmallcross-priceestimated elasticitiesareconsistentacrossseveraldifferentutilityspecifications.12 2.2.2 Supply-sideparameters For the parameters in the firm’s problem, we set T, the number of weeks in a model year, to 52 and the time-invariant interest rate such that (1+r)−52 =0.95. The interest rate is the only cost of holding inventories in the model. To parameterize the cost function, we set the line speed, workers per shift, and wage rates to values typically observed at assembly plants. We set the remaining three parameters, g , t andy tomatchtheaverageretailprice,averagerateofproduction,andaveragedays-supplyofinventories observed in the data. Although we would have preferred to estimate these parameters econometrically, computational issues made such estimation infeasible. The line speed at most North American assembly plants is set between 40 and 60 cars per hour; thus, we fix the line speed to 50 cars per hour. Using the employmentdatafromHall(2000),wesetnto1300workerspershift,sothefirmemploys2600workers. Wereadthewagesofftheunioncontract: w =$27.00perhour, andw =$28.35perhour. Wesetg , the 1 2 per vehicle cost, to the average retail price observed minus $1500; g effectively scales the cost function linearly, and thus we allow g to differ across market segments. We set the revenue tax parameters t to 1 andy to1.75suchthatwematchtheaveragedays-supplyofinventoriesobservedinthedata. 2.3 ModelResults Using these parameter values, we solve the dynamic program given by (9) and (10) via an algorithm suggestedbyJohnRust. Specifically,wemergetheT valuefunctionsintoasingletime-invariantBellman equation: (cid:40) (cid:181) (cid:182) V(Ilast,0,1)= max (cid:229) T 1 t−1(cid:179) plastslast(1−t (slast/ilast) y ) {pt t his,p t last,qt,Dt,St,ht} t=1 1+r t t (cid:41) (cid:181) (cid:182) (cid:180) T 1 +pthissthis(1−t (sthis/ithis) y )− c(D ,S ,h |q ) + V(Ithis+q −sthis,0,1) . t t t t t t 1+r T T T 11Tobeconsistentwiththehypothesisofrandomutilitymaximization,itmustbethatr ∈[0,1]. 12In general, we were unable to estimate alternate utility specifications that had more parameters than did the specification describedinthepaper(suchasrandomcoefficientsonasetofmodeldummies). Thesealternativespecificationsdemandedtoo muchofthedata. 26

Market Data t =1 t =0 Segment Product Price D-S Product Price D-S Product Price D-S Compact 14,524 $13,622 73 16,208 $13,594 70 16,000 $13,642 12 Midsize 10,886 19,193 60 12,000 19,273 75 16,333 19,075 11 Fullsize 7,184 23,772 75 6,167 24,107 77 9,833 23,708 13 Luxury 3,106 36,032 80 2,167 36,430 80 2,479 36,765 9 Pickup 35,114 23,662 84 17,812 24,121 67 22,900 24,224 9 SUV 11,532 28,660 75 9,117 29,140 74 14,500 28,741 11 Sporty 5,721 27,227 83 3,167 27,331 82 6,500 27,081 21 Van 8,396 22,716 85 9,667 22,868 80 13,667 22,581 10 Average 76 76 12 Table9: AverageMonthlyProduction,RetailPrices,andDays-SupplybyMarketSegment Tosolveforthefixedpoint,wecarriedoutthefollowingsteps: (1)GuessaninitialvalueforV(Ilast,0,1); (2)solvetheT Bellmanequationsin(9)and(10)throughbackwardrecursions; (3)computeanewvalue forV(Ilast,0,1)throughpolicyiteration;and(4)repeatsteps2and3untilafixedpointisreached. Becauseofthenon-convexitiesinthecostfunction,wesolveforboththeoptimallevelofoutputand the cost minimizing production schedule through grid search. We allow weekly production,q, to take on valuesbetween0and6000inincrementsof50. ThegridsforD andS aresetfrom1to6andfrom0to t t 2,respectively,inincrementsof1. TheplantisclosedfortheweekwheneverS =0. Theshiftlength,h , t t cantakeonvaluesof7,8,9or10. Sothereareupto72feasibleproductionschedulestoevaluateforeach 121possiblelevelsofproduction. We discretize each inventory grid into 26 points from 0 to 60,000. The distance between grid points increaseswiththelevelofinventories. Thus,thegridpointsaremoredenselyspacedintheregionwhere the value function has more curvature. For each of the 676 inventory pairs, we maximize the right hand side of equations (9) and (10) over each sales price and level of output. Points off the two inventory gridsareapproximatedusingbi-linearinterpolation. Thetwosalesprices, plast and pthis,maytakeonany positivevaluesuchthatquantitydemandedremainspositive. Finally,weimposeonlymildseasonalityon production,assumingthattheplantclosesfortwoweeksinJuly(weeks51and52)foramodelchangeover. Intable9wereporttheaveragemonthlyrateofproduction,retailprice,anddays-supplyofinventories for a typical vehicle in each market segment. We then report the corresponding averages implied by the model for the case with the revenue tax (t =1) and the case without the revenue tax (t =0). In general, the model replicates the average price and quantities produced for each market segment. This should not 27

18.7 18.6 18.5 18.4 18.3 18.2 18.1 18 17.9 0 20 50 60 40 40 20 30 10 60 0 This Model Year Inventory (in 1000 vehicles) Last Model Year Inventory (in 1000 vehicles) srallod 0001 ni eulaV wodahS 20 19.9 19.8 19.7 19.6 19.5 19.4 19.3 19.2 19.1 0 20 50 60 40 40 20 30 10 60 0 Last Model Year Inventory (in 1000 vehicles) This Model Year Inventory (in 1000 vehicles) Figure11: Week27ShadowValueofInventories forThisYear’sVintage. srallod 0001 ni ecirP lamitpO Figure12: Week26OptimalPricingRuleforThis Year’sVintage be terribly surprising because, given the estimated demand curves, the parameter, g , was set to match these moments.13 With the revenue tax, the model matches the average observed day-supply. Without the revenue tax, however, the average level of inventories are 1/6 the level observed in the data. As is well-understoodintheinventoryliterature,itisdifficulttomatchthehighlevelofinventoriesobservedin manyindustrieswithoutanadhocinventory-to-salestargetinthefirm’sobjectivefunction. To illustrate the dynamics of the model, we first consider the firm’s pricing decision for a typical midsizecarsettingt =1. Laterwereportresultsforwhichweshutdowntherevenuetaxbysettingt =0. Infigure11,weplotthepartialderivativeofthevaluefunctionforweek27withrespecttoinventoriesof the current model year (other weeks are qualitatively similar). This figure illustrates the shadow value of inventories,ormarginalincreaseinthefirm’snetworthfromanadditionalunitofinventory,ateachpoint in the state space for week 27. The shadow value of inventories is a decreasing function of the level of inventories. Whenthelevelofinventoryforthisyear’smodelisclosetozero,anadditionalvehicleofthis year’smodelisworth$18,600tothefirm; however, attheupperboundoftheinventorygrid, theshadow value of an additional vehicle from this model year is worth only about $17,960 to the firm–roughly the average cost of producing a vehicle when the plant operates two forty-hour shifts per week. Given the non-convexitiesinthecostfunction,computingandreportingthemarginalcostofanadditionalvehicleis bitmoreinvolved,butthevalueforg (thematerialcostpervehicle)of$17,693($19,193-$1,500)provides alowerboundonthemarginalcost. 13Themodelhasdifficultymatchingthequantitiesofpickuptrucks. Whileweassumeeachvehicleismadeatasingleplant, severalpopularpickuptrucks(e.g.FordF-series,ChevySilverado,andDodgeRam)areproducedatfourorfiveplants. 28

Because the automaker faces a downward-sloping demand curve, the profit-maximizing price sets marginal revenue equal to the shadow value of inventories next period. If we set the cross-price semielasticitiesequaltozeroandignoretherevenuetax,theoptimalpriceforthisyear’smodelis −sthis(p ) 1 pthis= t t + V (Ilast−slast,Ithis+qthis−sthis,t+1), t ¶ sthis(p )/¶ q 1+r 2 t t t t t t t t where V denotes the derivative of the value function with respect to the second argument. This is the 2 standardconditionformonopolypricing,butinthiscasemarginalcostistheshadowvalueofanadditional unitofinventorynextperiod. Wethenplotthepricingruleforthisyear’svintageforweek26infigure12. Thepricingruleisalmost the shape of the shadow value of inventories. Holding all other things constant, the optimal price is a decreasingfunctionofthelevelofinventory—ourfifthfact. Infigure13,weplotslicesofthepricingrules for different weeks in the product cycle, holding the inventory of the competing vintage fixed. Prices are a decreasing function of the level of inventory and the pricing curve shifts down over time. These price rulesareconsistentwiththefindingsofZettelmeyer,ScottMorton,andSilva-Risso(2003)thattheaverage retailpriceatadealershipwithampleinventoryisabout$230percarlessthanthatatadealershipwithlow inventory. These pricing rules do not guarantee that prices fall over the product cycle; after week 52, for example, inventories are monotonically decreasing over time (since there is no replenishment), therefore pricesmaygoupordowndependingontheevolutionofinventories. Infigures14-17, weplotasimulationfromthemodelforfive52-weekmodelyears, time-aggregated toamonthlyfrequency. Becausethemodelisdeterministic,eachofthesesimulationsisidentical. These graphsaredesignedtobeanalogoustofigurespresentedinsection1;howevernotethatfigures1-4arefor aparticularmidsizecarwhileweparameterizethemodelforanaveragemidsizecar. Theimpliedtimeseriesfromthemodelareconsistentwiththefivefactsputforthintheintroduction. As these figures illustrate, the model generates both downward sloping price paths and hump-shaped inventory and sales. The revenue tax term plays a key role in this. Early on in the model year, inventories are naturally low, so it is expensive to sell a lot of vehicles. In order to reduce this tax in the future, the automaker needs to build up inventories. Hence the automaker sets prices high early on in the year to dampen down sales and allow inventories to accumulate. Once inventories are high (inventories peak in the seven month of the model year), the tax effectively disappears and the firm lowers prices in order to stimulate sales. Further exacerbating the fall in prices, demand for the vehicle starts to decrease as the model year progresses. For forty-one weeks (about three-quarters of the year) the automaker sell both 29

21 20.5 20 19.5 19 18.5 18 17.5 17 16.5 0 10 20 30 40 50 60 Inventory (in 1000 vehicles) )srallod 0001 ni( ecirP liateR ← Week 2 → Week 35 → Week 50 → Week 70 → Week 60 Figure 13: Optimal Price Rule Over the Product Cycle, Holding the Inventory Level of the Competing VintageFixed. vintagessimultaneouslyandpricesfallfrom$20,061to$17,470: adropof12.9percentovertheproduct cycle or an average monthly price decline of 8.4 percent at an annual rate. This results in a new vintage premium of 8.1 percent, which is within a single standard error of the 8.5 percent new vintage premium foratypicalmidsizecar,asreportedintables2and10. More generally, as reported in table 10, the model with t set to 1 is able to match the observed price declinesandvintagepremiaforallmarketsegments. Forsevenoftheeightmarketsegments,theimplied pricedeclinesarewithinasinglestandarderroroftheaveragedeclinesseeninthedata;andtheremaining market segment (luxury cars) is well within the two-standard error band. The model underestimates the averagevintagepremiabynine-tenthsofapercent(8.8versus7.9). Whilethisisoutsidethetwo-standard errorband,themodelgetsthemagnituderight. Webelievethatrelaxingourassumptionthatnewvintages arrivestrictlyevery52weekswouldenablethemodeltobettermatchthismoment. Formostoftheyearinthemidsizecarcase,theautomakerproduces4000vehiclesperweekrunning twoeight-hourshiftsforfivedaysperweek. Forthelasttwomonthsofthemodelyear,theplantrunsjust a single shift before shutting down and changing vintages. Comparing figures 3 and 7 to figure 16, it is clear the model predicts production to be quite smooth relative to what is observed in the data. At most 30

20.5 20 19.5 19 18.5 18 17.5 17 10 15 20 25 30 35 40 45 50 55 60 month )srallod 000,1 ni( ecirp liater 20 18 16 14 12 10 8 6 4 2 0 10 15 20 25 30 35 40 45 50 55 60 month Figure14: MonthlyPrices. )sdnasuoht ni( selcihev fo selas Figure15: MonthlySales. 20 18 16 14 12 10 8 6 4 2 0 15 20 25 30 35 40 45 50 55 60 month )sdnasuoht ni( selcihev fo noitcudorp 50 45 40 35 30 25 20 15 10 5 0 10 15 20 25 30 35 40 45 50 55 60 month Figure16: MonthlyProduction. )sdnasuoht ni( selcihev fo yrotnevni Figure17: MonthlyInventories. SimulatedPrices,Sales,Production,andInventoriesforaTypicalMidsizeCarByModelYear: t =1. 31

Data t =1 t =0 Market Price Vintage Price Vintage Price Vintage Segment Decline Premium Decline Premium Decline Premium Compact 9.5 (2.4) 7.1 (0.5) 10.0 9.3 4.6 5.0 Midsize 9.2 (1.5) 8.5 (0.4) 8.4 8.1 3.1 4.3 Fullsize 8.9 (2.1) 8.3 (0.6) 7.1 6.4 2.2 4.1 Luxury 11.6 (1.2) 11.6 (0.4) 10.1 10.7 5.1 5.7 Pickup 9.9 (2.2) 10.6 (0.7) 9.5 9.1 5.3 6.1 SUV 8.2 (0.9) 7.2 (0.4) 7.4 6.9 3.4 4.3 Sporty 5.1 (2.4) 7.2 (0.8) 4.3 4.1 1.7 2.8 Van 9.6 (1.4) 8.6 (0.4) 8.3 8.7 4.3 5.2 Average 9.2 (0.6) 8.8 (0.2) 8.1 7.9 3.7 4.7 Note: Thepercentagepricedeclinesareatannualrates. Standarderrorsare inparentheses. Table10: AveragePriceDeclinesandVintagePremia(inpercent) assemblyplants,productionduringthefirstseveralweeksofthemodelyearistypicallybelowaverageas theplant“rampsup”output;furtherthroughouttheyear,plantsregularlycloseforholidaysandinresponse to both supply and demand shocks. While these features can be easily incorporated into the model and wouldincreasethevolatilityofproduction,theyobscuretheintuitionofthemodel. The model simulation can also be viewed in price-quantity space. In figure 18 we plot the time path ofthequantity-pricepairsoverasinglemodelyearforatypicalmidsizecar. Thestarsdenotetheweekly realizations from a single simulation from the model. The crosses denote the average observations from the data for the midsize sector. These quantity-price pairs are interpolated from monthly observations to obtain weekly points. As the figure illustrates, prices fall over the model year while sales initially start small, grow, and then decrease. Any model with a stable supply or demand curve will be unable to match the price and sales patterns in the data; however, the current model successfully replicates the basichorseshoepattern. Forthefirsthalfofthemodelyear,demandisrelativelystablewhilethemarginal revenuecurverotatescounter-clockwisetowardthedemandcurveasthefirmaccumulatesinventoriesand reduces the revenue tax. The shadow value of inventories (i.e. the supply curve) also shifts right. This generatesdecreasingpricesandincreasingsales. Afterabout30weeks,thedemandcurvestartstoshiftto theleftandcontinuesfallingfortheremainderofthemodelyear. Thiscausesbothpricesandsalestofall duringthesecondhalfoftheproductcycle. To isolate the role inventory policy has on the optimal price paths, we resolve the model removing 32

20.5 20 19.5 19 18.5 18 17.5 0 500 1000 1500 2000 2500 3000 3500 4000 quantity sold )srallod fo sdnasuoht ni( ecirp Figure18: Weekly(quantity,price)pairsoverthemodelyear. Thestarsarerealizationfromonesimulation from the model. The blue stars denote this year’s model. The green stars denote last year’s model. The redcrosses(thisyear’smodel)andpurplecrosses(lastyear’smodel)aretheobservationsinthedata. the revenue tax (i.e. setting t =0). This simulation can be interpreted as allowing the firm to engage in a build-to-order rather than a build-to-stock inventory policy. In figures 19-22 we plot the price, sales, production, and inventory pathes under a build-to-order policy while still allowing for multiple vintages. Toeasecomparison,wesetthescalesofthefiguresconsistentwiththoseinfigures14-17. Underabuildto-order policy, the firm sets a downward sloping price path that is far less dramatic than the price path with the revenue tax. Prices fall only $838 (from $19,369 to $18,531) or 4.3 percent (3.1 percent at an annual rate) over the product cycle. For the first 40 weeks the firm essentially builds to order, producing andsellingroughly4000carsperweek(correspondingtooperatingtwo40hoursshifts). Asthefirmnears the52ndweek,thefirmmoderatesthepricedeclinestodampensalesandaccumulatesmodestinventories to carry over into the next model year. The firm then sells last year’s vintage for only fourteen weeks into the following model year with an average vintage premium of 4.3 percent. Hence a model with just declining demand yields within-product-cycle price declines and vintage premia that are half of the declines generated by a model with a build-to-stock inventory motive. Furthermore, under the build-toorderpolicyneithersalesnorinventorieshavethepronouncedhump-shapedpatternseeninthedata. 33

20.5 20 19.5 19 18.5 18 17.5 17 10 15 20 25 30 35 40 45 50 55 60 month )srallod 000,1 ni( ecirp liater 20 18 16 14 12 10 8 6 4 2 0 10 15 20 25 30 35 40 45 50 55 60 month Figure19: MonthlyPrices. )sdnasuoht ni( selcihev fo selas Figure20: MonthlySales. 20 18 16 14 12 10 8 6 4 2 0 15 20 25 30 35 40 45 50 55 60 month )sdnasuoht ni( selcihev fo noitcudorp 50 45 40 35 30 25 20 15 10 5 0 10 15 20 25 30 35 40 45 50 55 60 month Figure21: MonthlyProduction. )sdnasuoht ni( selcihev fo yrotnevni Figure22: MonthlyInventories. SimulatedPrices,Sales,Production,andInventoriesforaTypicalMidsizeCarByModelYear: t =0. 34

These patterns hold across the other market segments as well. As can be seen in table 10, under the build-to-order policy the average price decline is only 3.7. This is less than half of the average price decline implied by the build-to-stock models (8.1) or seen in the data (9.2). Furthermore the implied vintagepremiawitht =0isonly60percentonaverageofthevintagepremiawitht =1(4.7versus7.9). While it is certainly the case that the demand for a particular vintage of new vehicle falls over the model year, this alone explains about half of the price decline in automobiles and misses salient comovements amongprices,salesandinventories. 3 Conclusion Wehavedocumentedasetofstylizedfactsforthewithin-model-yearpricingandsalesofnewautomobiles. Pricesdeclinesteadilyoverthemodelyearwhilesalesandinventoriesarehump-shaped. Itisnotthecase thatpricesonlyfallduringtheoverlapperiodbetweenvintageswhendealersshoutovertheradio“Weare slashingpricestomakeroomforthenewmodelyear!”Tounderstandthesefactsweformulateandsolvea marketequilibriummodelforasinglevehicleline. Whilefallingdemandovertheproductcycleexplains about one half these price declines, our model suggests that the other half of these declines is driven by build-to-stockinventorymanagement. Advances in production and information technology have made it easier to implement build-to-order policies. For example, the computer maker Dell, has been successful in selling built-to-order computers. Itisourunderstandingfromdiscussionswithindustryexecutivesthattheautomakerswouldliketomove toward an inventory policy in which a larger fraction of consumers order their new vehicles rather than buywhateverisonthedealer’slot. Ouranalysissuggeststhatenactingsuchapolicywilldampenwithinmodel-yearpricedeclinesandreducetheperiodinwhichconsecutivevintagescompetewitheachother. Finallyweviewthispaperaspartofalargerresearchagendatounderstandhowfirmsadjustpricesand productionwhenfacedwithdemandshocks. Whiletheanalysispresentedaboveisentirelydeterministic, in future work we plan to add persistent demand shocks and a richer cost structure to the model. We can then ask whether a negative demand shock will result in lower prices (e.g. rebates or other financial incentives)orinshuttingdowntheplant. Giventheimportantnon-convexitiesinproductionschedulingat automobileassembly,itmaybeoptimalattimestocutpricestokeeptheplantsrunningwhileatothertimes it may be optimal to shut plants down for an “inventory adjustment”. Ultimately, we want to understand theshapeofthesupplycurvefornewvehiclesandhowitvariesoverthemodelyear. 35

References [1] Ayres,I.,SiegelmanP.,1995.Raceandgenderdiscriminationinbargainingforanewcar.American EconomicReview85,304-321. [2] Berry, S., 1994. Estimating discrete-choice models of product differentiation. Rand Journal of Economics25,242-261. [3] Berry,S.,LevinsohnJ.,PakesA.,1995.Automobilepricesinmarketequilibrium.Econometrica63, 841-890. [4] Berry, S., Levinsohn J., Pakes A., 1999. Voluntary export restraints on automobiles: Evaluating a tradepolicy.AmericanEconomicReview89,400-430. [5] Bresnahan,T.,Reiss,P.,1985.Dealerandmanufactermargins.RandJournalofEconomics16,253- 268. [6] Bresnahan, T., Ramey, V., 1994. Output fluctuations at the plant level. Quarterly Journal of Economics109,593-624. [7] Chan,H.,Hall,G.,Rust,J.,2005.Pricediscriminationinthesteelmarket.manuscript,Universityof Maryland. [8] Corrado, C., Dunn W., Otoo M., 2004. An initial look at incentives and prices for motor vehicles: Whathasbeenhappeninginrecentyears? manuscript,BoardofGovernorsoftheFederalReserve. [9] Federgruen,A.,HechingA.,1999.Combinedpricingandinventorycontrolunderuncertainty.OperationsResearch47,454-475. [10] Elmaghraby W., Keskinocak P. 2003. Dynamic pricing in the presence of inventory considerations: Researchoverview,currentpractices,andfuturedirections.ManagmentScience49,1287-1309. [11] Goldberg, P., 1995. Product differentiation and oliogopoly in international markets: The case of the U.S.automobileindustry.Econometrica63,891-951. [12] Goldberg, P., 1996. Dealer price discrimination in new car purchases: Evidence from the consumer expendituresurvey.JournalofPoliticalEconomy104,622-654. 36

[13] Hall G., 2000. Non-convex costs and capital utilization: A study of production scheduling at automobileassemblyplants.JournalofMonetaryEconomics45,681-716. [14] Karlin,S.,CarrR.,1962.Pricesandoptimalinventorypolicy.inStudiesinAppliedProbabilityand ManagementScience,K.Arrow,S.Karlin,H.Scarf(ed.),Stanford,CA:StanfordUniversityPress. [15] Lazear,E.,1986.Retailpricingandclearencesales.AmericanEconomicReview76,14-31. [16] Nevo,A.,2000.Apractitioner’sguidetoestimationofrandomcoefficientslogitmodelsofdemand. JournalofEconomics&ManagementStrategy9,513-548. [17] Pashigian,P.,1988.Demanduncertaintyandsales: Astudyoffashionandmarkdownpricing.AmericanEconomicReview78,936-953. [18] Pashigian,P.,BowenB.,GouldE.,1995.Fashion,styling,andthewithinseasondeclineinautomobileprices.JournalofLawandEconomics38,281-310. [19] Pesendorfer,W.,1995.Designinnovationandfashioncycles.AmericanEconomicReview85,771- 792. [20] Petrin, A., 2002. Quantifying the benefits of new products: The case of the minivan. Journal of PoliticalEconomy110,705-729. [21] Ramey, V., Vine D., 2004. Tracking the source of the decline in GDP volatility: An analysis of the automobileindustry.NBERworkingpaper10384. [22] Stokey,N.,1979.Intertemporalpricediscrimination.QuarterlyJournalofEconomics93,355-371. [23] Whiten,T.,1955.Inventorycontrolandpricetheory.ManagmentScience2,61-80. [24] Zettelmeyer, F., Scott Morton F., Silva-Risso J., 2001. Cowboys or cowards: Why are internet car priceslower? manuscript,YaleSchoolofManagement. [25] Zettelmeyer, F., Scott Morton F., Silva-Risso J., 2003. Inventory fluctuations and price discrimination: Thedeterminantsofpricevariationincarretailing.manuscript,YaleSchoolofManagement. 37