Measuring the Social Return to R&D

Measuring the Social Return to R&D CharlesI.Jones DepartmentofEconomics StanfordUniversity Stanford, CA94305 Chad.Jones@Stanford.edu and JohnC.Williams BoardofGovernorsoftheFederalReserveSystem Washington, DC20551 jwilliams@frb.gov February1997 Abstract AlargeempiricalliteraturereportsestimatesoftherateofreturntoR&D rangingfrom30%toover100%,supportingthenotionthatthereistoolittle private investment in research. This conclusion is challenged by the new growth theory. We derive analytically the relationship between the social rateofreturntoR&Dandthecoefficientestimatesoftheempiricalliterature. Weshowthattheseestimatesrepresentalowerboundonthetruesocialrate of return. Using a conservative estimate of the rate of return to R&D of about30%,optimalR&Dinvestmentisatleastfourtimeslargerthanactual investment. JELClassification:O32,O41 Keywords: Socialrateofreturn;researchanddevelopment;endogenous growth. Views expressed in this paper are those of the authors and do not necessarily represent those of the Board of Governors of the Federal Reserve Systemoritsstaff.

MeasuringtheSocialReturntoR&D 1 1 Introduction Do advanced economies engage in too much or too little R&D?1 By how much doesprivateinvestmentinresearchdifferfromoptimalinvestment? GiventhecentralroleofR&Dasanengineofgrowth,thesequestionshavespawnedalargetheoreticalandempiricalliterature. Theoryhasemphasizedtheimportanceofmarket failuressuchasimperfectcompetitionandexternalitiesindeterminingoutcomesin themarketfornewgoodsandideas.2 However,becausethereareincentivesworking to promote both over- and underinvestment in R&D,theory alone is unable to provide an unambiguous answer to the sign, much less the magnitude, of the net distortion to R&D. The empirical literature attempts to resolve this ambiguity by estimating directlytherateofreturntoR&Dinregressions ofproductivity growth on R&D-sales ratios.3 The findings of this literature are summarized by Griliches (1992, p. S43): In spite of [many] difficulties, there has been a significant number of reasonably welldone studies allpointing inthesamedirection: R&D spillovers are present, their magnitude may be quite large, and social ratesofreturnremainsignificantly aboveprivaterates. Theempiricalapproachseemstoprovideaclearanswertothequestionofwhether there is too much or too little private R&D; it does not, however indicate by how 0Apreviousversionofthispaperwascirculatedunderthetitle“TooMuchofaGoodThing?The Economics of Investment inR&D.”Wewould liketothank Roland Benabou, Ken Judd, Michael Horvath,SamKortum,ArielPakes,ScottStern,AlwynYoung,andparticipantsofseminarsatU.C. Berkeley, Chicago, U.C.Irvine, Michigan, N.Y.U., Penn, U.C.S.D.,Stanford, the NBERSummer Institute'95, theNBEREconomic Fluctuations meeting, theConference on Innovation inStrausborg,andtheHIIDGrowthmeeting.FinancialsupportfromtheNationalScienceFoundation(SBR- 9510916)isgratefullyacknowledged. 1Weshould emphasize fromthebeginning that thispaper isnot about basicsciencebut rather aboutappliedR&Dundertakenbyprofit-maximizingfirms.Ofcourse,werecognizethatthedistinctionissometimesdifficulttomakeinpractice. 2The theoretical literature includes contributions from the IO approach, as reviewed by Tirole (1988), aswellasthe general equilibrium approach exemplifiedby Romer (1990), Grossman and Helpman(1991),andAghionandHowitt(1992). 3Recent summaries of this literature include Cohen and Levin (1991), Griliches (1992), and Nadiri(1993).

MeasuringtheSocialReturntoR&D 2 muchR&Dinvestmentneedstobeincreased. In fact, theory provides some reason to question the findings of the empirical productivity literature. The results of this literature are nearly all based on a neoclassical theory of growth in which R&D is simply an alternative form of capital investment. This simple capital-based approach ignores many of the distortions associated with research that are formalized by the new growth theory, including monopoly pricing, intertemporal knowledge spillovers, congestion externalities, and creative destruction. Because of these omissions, we may in fact have very littleinformation onthetruesocialrateofreturntoR&D. Themaincontribution ofthispaperistolinkthenewgrowththeorytotheempirical resultsintheproductivity literature. Wederiveanalytically therelationship betweenthesocialrateofreturntoR&Dandthecoefficientestimatesfromregressions of total factor productivity growth on R&D investment. In the process, we provideanintuitiveexplanationforthevariouscomponentsthatmakeupthesocial returntoR&D.Wealsoderivetherelationshipbetweenthemagnitudeofunder-or overinvestment inR&Dandtheestimatedrateofreturn. The results are rather surprising. Despite the methodological limitations of theproductivity literature—in particular itsomissionofdistortions thatmightlead to overinvestment—we show that the estimates in this literature represent lower bounds onthesocialrateofreturntoR&D.Thus,estimatesoftherateofreturnto R&Dfrom the productivity literature of 30 percent or higher imply that advanced economies like the U.S. substantially underinvest in R&D. Based on results from thenewgrowththeory, onemightbeinclined toquestion thebroadconclusions of theproductivityliterature; incontrasttothisintuition,weshowthatthefindingsof theproductivity literature areextremelyrobust. With an estimate of the social return to R&D in hand, a lower bound on the degreeofunderinvestmentinR&Dcanbecomputeddirectly. Usingaconservative estimate of the social return of 30% and a private rate of return to capital of 7%, optimalR&DspendingasashareofGDPismorethanfourtimeslargerthanactual spending.

MeasuringtheSocialReturntoR&D 3 The methodology developed in this paper allows one to derive these results directlyfromtheproductionpossibilitiesoftheeconomy—theproductionfunction for new ideas and the production function for the consumption/output good. It does not rely on anyparticular assumptions regarding market structure, the patent system, or distortionary taxes.4 More generally, this approach can be applied to a widevarietyofmodels. The remainder of this paper is organized as follows. We begin in Section 2 withageneral derivation ofthesocial rateofreturntoresearch. Section3reviews themethodology andresultsoftheempiricalproductivity literature andrelatesthe true social rate of return to R&D to the estimates in this literature. In section 4, the magnitude of over- or underinvestment in R&D is derived and related to the estimatedsocialreturn, andSection5concludes. 2 The Social Rate of Return to R&D What is the rate of return to society from performing an additional unit of R&D? Toanswerthisquestion, weconsiderthereturnassociated withthefollowingvariational argument. Suppose we reallocate one unit of output from consumption to R&D today, and then consume the proceeds tomorrow. In particular, we reduce R&D tomorrow to leave the subsequent stock of ideas unchanged. In the market equilibrium, an individual agent is indifferent to this deviation, but in the face of distortions and externalities, society as a whole generally will not be. We define thesocialrateofreturntoR&Dtobethegaininconsumption associated withthis variation. Thisparticulardefinitionturnsouttohaveanumberofusefulproperties thatwewillnowexplore. 4Inthissense,itisinterestingtocomparethisapproachtoStokey(1995). Stokey,andourown earlierwork,addresstheissueofinvestmentinR&DbycalibratinganR&D-basedgrowthmodel. Theresultsinthisapproachdependcriticallyonhowonecharacterizesthemarketeconomy.

MeasuringtheSocialReturntoR&D 4 2.1 General Derivation We begin with a general derivation of the social rate of return. The first useful result related toourdefinition isthatthesocialrateofreturn canbederived solely from the production possibilities of the economy. Typically, only two equations areneeded, theproduction function for ideas andtheproduction function forfinal output. Let A denote the stock of ideas inthe economy. Newideas, thechange in A , are produced by foregoing consumption of the final output good Y , according tosomeproduction function G : A +t 1 (cid:0) A t = G ( R t ; A t ) ; (1) where R represents resources devoted toresearch. Weassumethat G isincreasing in its first argument: more research leads to more ideas. G might be increasing or decreasing in its second argument, depending on the way past ideas affect the current productivity of research. If @ G = @ A > 0 , then past inventions raise the productivity of research today, a case that corresponds to “knowledge spillovers” in research. On the other hand, if the best ideas are discovered first, G might be decreasing in A . Theconsumption/final outputgoodisproduced usingideasandacollection of privateinputs X according totheproduction function F : Y t = F ( A t ; X t ) : (2) Weassumethat F isincreasingineachofitsarguments. FollowingRomer(1990), one would expect F to exhibit constant returns to X and therefore increasing returnstoscaleoverall. We will assume the existence of a balanced growth path in which all variablesaregrowingatconstant ratesovertime. Thismayentailsomerestrictions on the shapes of G and F ; we will specialize to the Cobb-Douglas functional forms shortly. Our use of a more general notation is not necessarily intended to suggest generality; rather, itilluminates whereeachterminthesocial rateofreturn comes from.

MeasuringtheSocialReturntoR&D 5 ThesocialrateofreturntoR&Discomputedusingthefollowingdiscretetime variational argument. Supposewereallocate oneunitofoutput fromconsumption toR&Dattime t ,andthenconsumetheproceeds inthenextperiod, t + 1 . Moreover,wereduceR&Dattime t + 1 soastoleavethestockofknowledgeunchanged from time t + 2 onward. The total gain in consumption at time t + 1 associated withthisvariationisthesocialrateofreturntoR&D. Theincreasein A +t 1 associated withasmallchangein R t is r A +t 1 = (cid:18) @ @ G R (cid:19) t where r is used to denote the change relative to the steady state path. The additionalknowledge r A +t 1 increasesoutputattime t + 1 by (cid:16) @ @ Y A (cid:17) +t 1 . Anadditional increaseinconsumptionattime t + 1 occursbecause R +t 1 canbereducedtoleave thepathofknowledgeunchanged. Todeterminehowmuchconsumption isgained fromreducing R&D,notethat A +t 2 = A +t 1 + G ( R +t 1 ; A +t 1 ) : Considering thedeviation fromthebalanced growthpath, r A +t 2 = r A +t 1 + (cid:18) @ @ G R (cid:19) +t 1 r R +t 1 + (cid:18) @ @ G A (cid:19) +t 1 r A +t 1 : Thedeviation inR&D, r R +t 1 ,thatwillreturn thestockofknowledge toitsoriginalpathisfoundbysetting r A +t 2 = 0 : r R +t 1 = (cid:0) ( ( @ @ G G = = @ @ R R ) ) t +t 1 ( 1 + (cid:18) @ @ G A (cid:19) +t 1 ) : Thetotalgaintoconsumptioninperiod t + 1 isthesumoftheadditionaloutput producedandthereductioninR&Dthatismadepossible. Thesocialrateofreturn, ~r ,isthusgivenby 1 + ~r = (cid:18) @ @ G R (cid:19) t (cid:18) @ @ Y A (cid:19) +t 1 + ( ( @ @ G G = = @ @ R R ) ) t +t 1 ( 1 + (cid:18) @ @ G A (cid:19) +t 1 ) : (3) The intuition behind this equation becomes more transparent if one thinks of knowledge as an asset “purchased” by society, held for a short period of time in

MeasuringtheSocialReturntoR&D 6 order to reap a dividend, and then sold. The return can then be thought of as the sumofadividendandacapitalgain(orloss). Let P A ;t denotethecosttosocietyof anewideainunitsofconsumption (thenumeraire). Then,becauseasmallchange inR&Dleadsto @ @ G R newideas, P A isgivenby P A ;t = (cid:18) @ @ G R (cid:19) (cid:0) t 1 : Therateofchangeinthecostofproducing newideas,denoted g P A ;t equals g P A ;t = (cid:16) @ @ (cid:16) G R @ @ (cid:17) G R (cid:0) t t (cid:17) 1 (cid:0) 1 ; whichisconstantalongabalanced growthpath. Afterrearrangement andsubstitution, thesocialrateofreturnequals ~r = P d A + g P A (4) where d = @ @ Y A + @ @ G A P A : (5) In equation (4), d is the “dividend” to society and g P A is the “capital gain.”5 The dividend associated withanadditional idea consists oftwocomponents. First, the additional knowledge directly raises the productivity of capital and labor in the economy. Second, the additional knowledge changes the productivity of future R&D investment because of either knowledge spillovers or because subsequent ideasaremoredifficulttodiscover. Finally,thereisacapitalgainorlossassociated withanychangeinthecostofproducing newideas,denoted g P A . 2.2 ASpecific Model Thepreceding derivation ispurposefully abstract. Tomake theideas concrete, we now derive the social rate of return to R&D using Cobb-Douglas specifications for the final goods and research technologies. Forease ofpresentation, weswitch 5Thesecond-ordercrossterm g P A @ @ G A hasbeensuppressed.

MeasuringtheSocialReturntoR&D 7 to continuous time. In this generalized version of Romer's (1990) variety-based endogenous growthmodel,thefinalgoodstechnology isgivenby Y = A (cid:27) K (cid:11) L 1 (cid:0) (cid:11) ; (6) where L islaborinput,and K isthe(aggregated) capitalstock.6 Theproduction function fornewideastakestheform ( 1 + ) A _ = ~(cid:14) R = (cid:14) R (cid:21) A (cid:30) : (7) Individual researchers take the productivity of research ~(cid:14) as given. Because they are small relative to the total number of researchers, they view the production of new ideas as taking place with constant returns to research effort R . Economywide, however, production of ideas need not be characterized by constant returns. For example, the presence of 0 < (cid:21) (cid:20) 1 may reflect duplication of effort in the research process: the social marginal product of R may be less than the private marginal product, a classic congestion externality. The parameter (cid:30) measures the neteffectofknowledgespilloversand“fishingout”effectsinresearch, bothexternal to atomistic research firms. If the net effect is such that (cid:30) > 0 , we might call this the standing on shoulders effect. The duplication externality associated with (cid:21) < 1 mightbecalledthestepping ontoeseffect. A third distortion inthe research process, highlighted byGrossman and Helpman(1991)andAghionandHowitt(1992),isassociatedwithcreativedestruction. That is, new ideas may replace old ideas. Creative destruction can provide an incentive for overinvestment in research in that some innovators earn rents on ideas that arenot entirely new. Inthe market economy, creative destruction affects who gets compensated forwhichidea, andonehas tobecareful indescribing how this process works. However, in terms of the production possibilities of the model (which arerelevant forcalculating thesocial return), introducing creative destruction involves only aminor change. In equation (7), weassume that for every new ideacreated, “upgrades” areproduced thatreplaceexistingideas. 6Weassumethat L growsexogenouslyatrate n > 0 ,andcapitalisaccumulatedinthestandard way,byforegoingconsumption.

MeasuringtheSocialReturntoR&D 8 Figure1: TheSocialReturntoResearchFunction ~ r ~ r* ~ r(s) s s* With these functional form assumptions, the social rate of return to R&D impliedbyequation (4)is ~r ( s ) = (cid:21) (cid:27) g s A + (cid:30) g A + ( g Y (cid:0) g A ) (8) where ~r ( s ) represents the steady state social rate of return to R&D,evaluated at a givensteadystateR&Dshareoftotaloutput, s . Thenotation g x isusedtoindicate thesteady state growthrateoftheplace-holder x . Thefirstterm ontheright-hand sideofequation(8)isthedividendassociatedwithextraoutput,thesecondtermis the dividend associated with knowledge spillovers, and the last term is the capital gainassociated withthechanging relativevalueofideas. Equation (8) identifies the functional relationship between the social return to research and the share of output invested in research by the economy. This functional relationship is plotted inFigure 1. This figure, together with the analysis in theprevioussection,motivatesourfirstkeyresultconcerningsocialratesofreturn:

MeasuringtheSocialReturntoR&D 9 The functional relationship between the social rate of return and the shareofresourcesdevotedtoresearchdependsonlyontheproduction possibilities of the economy. Features of the market economy affect the allocation of resources, which determines the point on the social returnfunction. The attractiveness of this result is that one does not need to make additional assumptions about the nature of the market economy (market structure, patent arrangements, taxes, etc.) to determine the social return to R&D. Provided one can writedownanaccuraterepresentationofproductionpossibilities,includingknowledgeoftheparameters, oneknowsthefunction plotted inFigure1. Then,onecan lookattheallocation ofresources —the s —actually chosen intheeconomy and “readoff”thesocialrateofreturnfromthefigure. Why is the functional relationship between the social rate of return and the allocation of resources independent of any market features of the economy? Intuitively, the answer is that no allocative decisions are involved in computing ~r — we force the economy to do one more unit of R&D and then calculate the total amount of output than can be consumed with this variation. In this sense, the productionpossibilitiesofthemodel(e.g. theequationsdescribingthesocialplanner's problem) determine the social rate of return function. The market economy and whatever distortions are present determine the allocation of resources — i.e. thepointonthesocialrateofreturnfunction.7 7Withthisinmind,itisinterestingtocompareourmeasureofthesocialrateofreturntoanalternativecalculation,thechangeinthe“valuefunction”ofthedecentralizedeconomy. Thisalternative providesthechangeinwelfareassociatedwithadditionalR&Dtakingintoaccountthedynamicresponseofagentstothevariation. However, calculatingthisalternativerequiressubstantiallymore structure and effort. First, the value function approach depends criticallyon the assumptions one makesaboutmarketstructureandthedistortionspresentinthedecentralizedeconomy. Asalready emphasized,oneadvantageofourapproachisthatwedonotneedthisadditionalstructure.Second, analytical solutions arenot availablewiththisapproach. Finally, in thecontext of thispaper, our calculation is extremely relevant because it is directly related to the estimates in the productivity literature.

MeasuringtheSocialReturntoR&D 10 3 Estimating the Social Rate of Return Now consider the following question: how can the available data on productivity andR&Dexpenditures beusedtoestimatethesocialreturntoR&D?Onewidelyused approach found in the literature is to treat R&D investment simply as an alternative capital investment in a standard neoclassical model.8 The R&D “stock” is included in the production function, and the partial derivative of output with respect to that stock is treated as the rate of return to R&D.9 The analogy to the marginal product of physical capital is clear. This basic relationship is described bythefollowingtwoequations: Y = e (cid:22) t Z (cid:24) K (cid:11) L 1 (cid:0) (cid:11) ; (9) Z _ = R ; (10) where Z is the measured R&Dstock and we assume no depreciation of the R&D capital.10 In this approach, the marginal product of the R&D stock, @ @ Y Z , is interpreted as the rate of return to R&D; let's call this marginal product ~r P L . By standard growth accounting logic, estimated TFP growth accounted for by R&D is then ~r P L R = Y . This motivates the following empirical specification for estimating the rateofreturntoR&D: (cid:1) l o g T F P = (cid:22) + ~r P L R Y + " : (11) That is, total factor productivity growth is regressed on the R&D share of output (andperhapsothercontrolvariablesaswell). 8Asecondapproach,pursuedbyBernsteinandNadiri(1989),istocomputethereturntoR&D usingestimatedcostfunctions.ThesetwoapproachesyieldsimilarresultsastothereturntoR&D. 9Thisbasicapproachisextendedinseveraldirectionsintheproductivityliterature.Forexample, Jaffe (1986) makes progress by incorporating R&D from other industries into the R&D stock to estimatethegainsfrominter-industryspillovers. 10This assumption of zero depreciation is somewhat standard in the productivity literature. In generalresearchershavefoundthatregressionestimatesarenotsensitivetoalternativeassumptions aboutthedepreciationrate.

MeasuringtheSocialReturntoR&D 11 Theempiricalliterature distinguishes betweentheprivatereturnandthesocial return to R&D.Theformer refers to the estimate of ~r P L using a firm'sownR&D share as the explanatory variable. The latter attempts to mitigate measurement problemsandtocaptureinterfirmtechnologyspilloversbyfocusingontheindustry level.11 Table1providesapartialreviewofestimatesofso-defined“social”ratesof returnfromtheproductivityliterature. Estimatesofthesocialreturnaverageabout 28percentwhenonlyR&Dfromone'sownindustryisincludedandaveragenearly 100 percent when the broadest concept of return (the sum of the two columns in thetable)isemployed.12 The framework used in the empirical approach outlined above places two important restrictions on the R&D stock accumulation process. First, no explicit allowance is made for congestion effects. Second, this approach does not explicitly allow for intertemporal knowledge spillovers or diminishing technological opportunities. Assuming these restrictions on the R&D technology are violated, the modelismisspecified. Inthiscaseitisnotpossibletorelateexactlytheparameters estimatedintheproductivityliteraturetoourmodelparameters. Itispossible,however,toobtainalinearapproximationtotherelationship, accurateinthevicinityof thesteadystateequilibrium. Supposetheeconomyconsistsofanumberofindustries,eachdescribedbythe production possibilities outlined inSection2. Considerrunning the(misspecified) regression of the productivity literature in this economy. To determine what this regression will produce, we linearize the production function for ideas given in 11Inregressionswithfirm-leveldata,measurementissuesareparticularlyacute. Forexample,the developmentofanewhigh-speedcomputermaynotbereflectedinthedevelopingfirm'stotalfactor productivity;someofthemeasuredproductivitygainmayshowupdownstream. Totheextentthat productinnovationsarecreatedandusedinthesameindustry,aggregationtotheindustrylevelhelps mitigatetheseproblems. 12The“used”columnreportstheadditionaleffectofR&Dconductedinanupstreamindustryon ownproductivity.TheseestimatesmaybebiaseddownwardsduetodoublecountingofR&Dinputs asbothR&Dandcapitalandlabor.Schankerman(1981)estimatesthatadjustingfordouble-counting raisestheestimatedrateofreturnbyabout0.1,whileHallandMairesse(1995)findabiasofonly 0.03to0.04.

MeasuringtheSocialReturntoR&D 12 Table1: EstimatedRatesofReturntoR&D (1) (2) Study ~r (own) ~r (used) (1)+(2) Sveikauskas (1981) 0.17(.06) - - Hall(1995) 0.33(.07) - - GrilichesandLichtenberg (1984b) 0.34(.04) - - Terleckyj(1980) 0.25(.08) 0.82(.21) 1.07 Scherer(1982) 0.29(.14) 0.74(.39) 1.03 GrilichesandLichtenberg (1984a) 0.30(.09) 0.41(.20) 0.71 Notes: Theleft-handsidevariableistypicallythegrowthrateofTFP.Therighthand side variables include the industry's R&D intensity (typically R&D/Sales) and, where indicated, a measure of R&D intensity of used inputs, along with a constantandothervariables. RepresentativepointestimatesandassociatedstandarderrorsoftheR&Dintensitycoefficientsaregiveninthetable. equation (7)aroundthebalanced growthpath. Thelinearapproximation is A A _ t t ’ c + (cid:21) g A s (cid:22)s t + (cid:21) g A l n (cid:16) Y (cid:22)Y t t (cid:17) + ( (cid:30) (cid:0) 1 ) g A l n (cid:16) A (cid:22)A t t (cid:17) ; (12) where c isaconstant. Multiplying by (cid:27) , d l n T d F t P t ’ c + (cid:21) g T (cid:22)s F P s t + (cid:21) g T F P l n (cid:16) Y (cid:22)Y t t (cid:17) + ( (cid:30) (cid:0) 1 ) g A l n (cid:16) T T F F P P t t (cid:17) : (13) Regression of the TFP growth rate on the R&D share of output should yield a coefficientgivenby ~r P L = (cid:21) (cid:22)g T (cid:22)s F P : (14) Therearetwopotentialsourcesofomittedvariablebias,representedbythepercentdeviationsofoutputandTFPfromtheirrespectivesteadystatelevels. Thesign of the potential bias of not controlling for these terms is ambiguous. In practice, industry and time dummies, capacity utilization rates, and measures of technical

MeasuringtheSocialReturntoR&D 13 opportunities are typically included in regressions of this type (see, for example, ClarkandGriliches(1984)). Thisshouldmitigatetheextenttowhichomittedvariable bias enters. In the following, we assume there is no omitted variable bias in estimatesof ~r P L .13 Now compare the coefficient estimated in the productivity literature with the truesocialrateofreturngivenbyequation(8). Theproductivityliteraturecaptures onlythebasicoutputdividendandignoresthedynamiceffectsassociated withthe intertemporal knowledgespilloverandthecapitalgainorloss. Mathematically, ~r ( (cid:22)s ) = ~r P L + ( (cid:30) g A + g Y (cid:0) g A ) : (15) As written, it appears that the term determining the difference between the true social return and the estimate from the productivity literature could be either positive ornegative. However,thistermcanberewrittentorevealthatitisalways positive, at least in steady state. Along the balanced growth path, (cid:30) ) g A (cid:21) g Y = ( 1 (cid:0) .14 Therefore, ~r ( (cid:22)s ) = ~r P L + ( 1 (cid:0) (cid:21) ) g Y : (16) Thisequationrevealsarathersurprising result: ~r P L represents an underestimate of the true social rate of return to R&D with a maximum downward bias equal to the rate of growth of output. The general conclusion from this literature that the social rate of return to R&D isvery large evidently survives rigorous analysis inthe context ofthenew growth theory. Howdoestheproductivity literature nearlygettherightanswer? Theexplanationinvolvestwodifferenterrorsthatnearlyoffset. First,theproductivityliterature focuses on @ @ Y Z as the rate of return to R&D. This focus captures the basic output 13MonteCarlosimulationevidencebasedonreasonablysizedi.i.d. demandshocksconfirmsthis approximaterelationship. 14Thiscanbeseenbydividingbothsidesofequation(7)by A andlog-differentiating.

MeasuringtheSocialReturntoR&D 14 effect but ignores two dynamic factors that determine the social rate of return to R&D in equation (8): intertemporal knowledge spillovers and the “capital gain” (orloss)duetochanges intherelativevalueofknowledge creationovertime. The empirical productivity literature implicitly assumes these termsequal zero; infact bothtermsmaybelargeinmagnitude, buttheirsumislimitedto ( 1 (cid:0) (cid:21) ) g Y .15 The intuition for why the sum of the knowledge spillover and the capital gain terms is bounded is seen by noting that the capital gain reflects the change in the value of ideas. This value equals the cost in terms of consumption goods of producing a new idea, R = A _ . From the production function for ideas, one sees that this cost is proportional to R 1 (cid:0) (cid:21) A (cid:0) (cid:30) . Thereturn tosociety due to the knowledge spillover, (cid:30) g A , exactly offsets the capital loss due to the fall in value of ideas as ideas become less costly to generate over time due to the accumulation of knowledge. What remains is the capital gain due to the increase in the value of designs resulting fromthegrowthinR&Dand (cid:21) < 1 ,reflectedbytheterm ( 1 (cid:0) (cid:21) ) g Y . Onemightexpectthatthemethodusedintheproductivity literaturewouldnot correctly incorporate the distortions associated with creative destruction and the monopolypricingofcapitalgoods. However,theresultsindicate thatthesefactors enter the rate of return calculation directly through s . More generally, distortions associated withthemarket economy that do notaffect theproduction possibilities of the economy do not affect either the rate of return calculation or the optimal amount of R&D.Thus, adjustments to estimates of ~r P L to reflect monopoly pricing,imitation,orcreativedestruction, assometimessuggestedintheliterature, are unnecessary andinappropriate. 4 The Extent of Underinvestment in R&D One drawback to discussing underinvestment in terms of social rates of return is thattheextentofunderinvestment isnotreadilyapparent. Fortunately,theanalytic 15Wehavemaintainedtheassumptionthat (cid:21) (cid:20) 1 ,i.e.therearecongestionexternalities.Ifinstead (cid:21) > 1 ,indicatingcomplementaritybetweenresearchtodayapartfromknowledgespillovers,thenthe productivityliteraturewouldunderestimatetherateofreturn. Notice,however,thatthemagnitude oftheerrorissmallbecauseofthemultiplicationby g Y .

MeasuringtheSocialReturntoR&D 15 framework we've used to interpret the estimates from the productivity literature provides this translation. This is apparent from Figure 1: intuitively, in order to findtheoptimalrateofR&Dinvestment, allweneedtodoisinvert thesocial rate ofreturnfunction. First, notice that the actual rate of investment in research by the economy, s a c tu a l ,satisfiestherelation ~r P L = (cid:21) s g a T c F tu P a l : (17) Second, the optimal amount of research is given by the condition that the social rate of return is equal to the real interest rate, r .16 Using this condition and equation (11), the optimal rate of investment in R&D along a balanced growth path is s o p tim a l = r (cid:0) (cid:21) ( g 1 T (cid:0) F P (cid:21) ) g Y : (18) Combining this equation with equation (17) gives the ratio of optimal investmenttoactualinvestment inresearch:17 o p s a s tim c tu a l a l = r (cid:0) ( ~r 1 P (cid:0) L (cid:21) ) g Y : (19) Withestimatesof ~r P L inmind,wecancomputeaconservative “lowerbound” on this ratio. First, notice that the denominator is no greater than the real rate of return for the economy. Hence, it is no larger than a number like 7 percent, the average real return on the stock market for the last century (Mehra and Prescott, (1985)). Pickingavaluefor ~r P L of28or30percent, towardsthelowerendofthe estimatesinTable1,equation(19)impliesaconservativeestimateof s o p tim a l = s a c tu a l of about 4. That is, the optimal share of resources to invest in research is conservatively estimatedtobe4timeslarger thantheactualamountinvested bytheU.S. economy. Theextentofunderinvestment issubstantial, andcouldbemuchlarger. 16Thisissimplyanalternativewaytoderivethesolutiontotherelevantsocialplannerproblem. 17Implicitinthisderivationisthatsteadystategrowthratesarethesameinthedecentralized(market)andplannedeconomies. Thisistruebecausethemodelconsideredhereisasemi-endogenous growthmodellikethatofJones(1995).Inthismodel, g A isgivenby (cid:21) n = ( 1 (cid:0) (cid:30) (cid:0) (cid:21) (cid:27) = ( 1 (cid:0) (cid:11) ) ) .A standardEulerequationforconsumptionimpliesthatthesteadystateinterestrateisalsothesamein thetwoeconomies.

MeasuringtheSocialReturntoR&D 16 5 Conclusion Recent endogenous growth models have emphasized the importance of R&D and the production of knowledge for understanding long run growth. A key issue is whether the economy undertakes too little or too much R&D, and by how much. In exploring these questions, we uncover several findings. First, we provide a methodological contribution inshowinghowtocomputesocialratesofreturn. For the case of R&D, we establish that the functional relationship between the social rateofreturnandtheshareofresources devoted toR&Ddepends onlyontheproduction possibilities of the economy. Market distortions such as patents, taxes, and monopoly power affect the allocation of resources to R&D, but not the functional relationship itself. Everything we need to know about the market economy issummarizedintheobserved allocation ofresources. Second, we examine the answer to these questions provided by the empirical productivity literature. A number of studies in that literature purport to find large ratesofreturntoR&D,suggestingsubstantialunderinvestment. Weshowthatthese estimates should be interpreted as a lower bound on the true social rate of return, eveninlightofthedistortions toR&Dhighlighted bythetheoretical literature. Finally,theapproachdevelopedhereallowsustogobeyondmeasuringratesof return. Knowledge of the social rate of return function provides a ready mapping between social rates of return and the extent of underinvestment. A conservative estimateindicatesthatoptimalinvestmentinresearchismorethanfourtimesactual investment.

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