competition and innovation: an inverted-u relationship* · 2016-01-29 · relationship between...

COMPETITION AND INNOVATIONAN INVERTED-U RELATIONSHIP

PHILIPPE AGHION

NICK BLOOM

RICHARD BLUNDELL

RACHEL GRIFFITH

PETER HOWITT

This paper investigates the relationship between product market competitionand innovation We find strong evidence of an inverted-U relationship using paneldata We develop a model where competition discourages laggard firms frominnovating but encourages neck-and-neck firms to innovate Together with theeffect of competition on the equilibrium industry structure these generate aninverted-U Two additional predictions of the modelmdashthat the average technologi-cal distance between leaders and followers increases with competition and thatthe inverted-U is steeper when industries are more neck-and-neckmdashare bothsupported by the data

I INTRODUCTION

Economists have long been interested in the relationshipbetween competition and innovation but economic theory seemsto be contradicted by the evidence Theories of industrial organi-zation typically predict that innovation should decline with com-petition1 while empirical work finds that it increases2 This paperreexamines this relationship using panel data and finds clearnonlinearities in the form of an inverted-U shape illustrated byFigure I which plots patents against the Lerner index with anexponential quadratic overlay The possibility of an inverted-Urelationship between competition and innovation was hinted atby Scherer [1967] who showed a positive relationship between

The authors would like to thank Daron Acemoglu William Baumol Timo-thy Bresnahan Jan Boone Wendy Carlin Paul David Janice Eberly EdwardGlaeser Dennis Ranque Mark Shankerman Robert Solow Manuel TrajtenbergAlwyn Young John Van Reenen two anonymous referees and participants atseminars including Canadian Institute of Advance Research Harvard Universityand Massachusetts Institute of Technology Financial support for this project wasprovided by the Economic and Social Research Council (ESRC) Centre for theMicroeconomic Analysis of Public Policy at the Institute for Fiscal Studies andthe ESRCEPSRC Advanced Institute of Management (AIM) initiative The datawere developed with funding from the Leverhulme Trust

1 See our discussion in Section III below However the replacement effect inArrow [1962] and the efficiency effects in Gilbert and Newbury [1982] and Rein-ganum [1983] go in the opposite direction

2 See Geroski [1995] Nickell [1996] and Blundell Griffith and Van Reenen[1999]

copy 2005 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnologyThe Quarterly Journal of Economics May 2005

701

patenting activity and firm size in the cross section with a di-minishing impact at larger sizes when allowing for nonlinearitiesTo our knowledge no existing model of product market competi-tion and innovation predicts an inverted-U pattern

An explanation for these results could be pieced together bycombining agency models3 with Schumpeterian models howeverthis seems unsatisfactory Instead we develop an extension ofAghion Harris and Vickers [1997]4 that can fit the entire curveIn this model both current technological leaders and their follow-ers in any industry can innovate and innovations by leaders andfollowers all occur step-by-step Innovation incentives depend notso much upon postinnovation rents as in previous endogenousgrowth models where all innovations are made by outsiders butupon the difference between postinnovation and preinnovationrents of incumbent firms In this case more competition mayfoster innovation and growth because it may reduce a firmrsquospreinnovation rents by more than it reduces its postinnovationrents In other words competition may increase the incrementalprofits from innovating and thereby encourage RampD investmentsaimed at ldquoescaping competitionrdquo This should be particularly truein sectors where incumbent firms are operating at similar tech-nological levels in these ldquoneck-and-neckrdquo sectors preinnovationrents should be especially reduced by product market competi-tion On the other hand in sectors where innovations are made bylaggard firms with already low initial profits product marketcompetition will mainly affect postinnovation rents and thereforethe Schumpeterian effect of competition should dominate

The essence of the inverted-U relationship between competi-tion and innovation is that the fraction of sectors with neck-and-neck competitors is itself endogenous and depends upon equilib-rium innovation intensities in the different types of sectors Morespecifically when competition is low a larger equilibrium frac-tion of sectors involve neck-and-neck competing incumbents sothat overall the escape-competition effect is more likely to domi-nate the Schumpeterian effect On the other hand when compe-tition is high the Schumpeterian effect is more likely to domi-nate because a larger fraction of sectors in equilibrium haveinnovation being performed by laggard firms with low initialprofits The inverted-U shape is confirmed by our U K panel

3 Hart [1983] Schmidt [1997] Aghion Dewatripont and Rey [1999]4 See also Aghion Harris Howitt and Vickers [2001]

702 QUARTERLY JOURNAL OF ECONOMICS

data and it is robust to a number of controls and experiments5

Our model provides additional testable predictions on the rela-tionship between competition and the composition of industriesand more specifically between competition and the average de-gree of ldquoneck-and-necknessrdquo in the economy which are also vin-dicated by the data

The rest of the paper is structured as follows Section IIdisplays the empirical evidence on the existence of an inverted-Urelationship between competition and innovation Section III ar-gues that existing models of competition and innovation cannotaccount for the inverted-U pattern We develop a theoreticalrationale for this relationship derive some additional empiricalpredictions and validate them with data Finally Section IVconcludes

II THE IMPACT OF COMPETITION ON INNOVATION

The early empirical literature inspired by Schumpeter[1943] estimated linear cross-sectional relationships and typi-cally found a negative relationship between competition and in-novation confirming the theoretical prejudices of the era Scherer[1967] developed this research by allowing for additional nonlin-earities and in a cross-sectional analysis of Fortune 500 firmsdiscovered a significant inverted-U shape with higher competi-tion initially increasing then decreasing the rate of innovationBut research since then has returned to estimating linear speci-fications for example Nickell [1996] and Blundell Griffith andVan Reenen [1999] both find a positive linear effect of competitionon innovation In this paper we allow for a nonmonotonicrelationship

IIA Measuring Innovation

There is a large literature on measuring innovation inten-sity with the most commonly used measures being RampD expen-diture and patenting activity We use the average number of

5 To deal with the possible endogeneity of competition we use U K data andexploit the major policy reforms undertaken over the 1970s and 1980s whichdramatically changed the nature and extent of competition across industries andover time The radical policies of the Thatcher administration the introduction ofthe European Single Market Program (SMP) and the reforms imposed by theMonopolies and Mergers Commission together provide a number of policy changesthat vary across time and industries and allow us to identify the causal impact ofcompetition on innovation

703COMPETITION AND INNOVATION

patents taken out by firms in an industry and to reflect theheterogeneous value of patents we weight each patent by thenumber of times it has been cited by another patent These dataare generated by matching the NBER patents database6 to ac-counting data on firms listed on the London Stock Exchange(from Datastream) Our sample includes all firms with namesbeginning ldquoArdquo to ldquoLrdquo plus all large RampD firms After removingfirms involved in large mergers or acquisitions and those withmissing data we have an unbalanced panel of 311 firms spanningseventeen two-digit SIC codes over the period 1973ndash1994 We alsohave information on citations to and from each patent whichenables us to construct a count of citation-weighted patents Wetake the average value of citation-weighted patents of firmswithin each industry (SIC code) in each year We do not observea sufficient number of firms in all industries in all years so ourresulting industry level panel is also unbalanced with 354industry-year observations Some descriptive statistics are pro-vided in Appendix 2

IIB Measuring Competition

Our main indicator of product market competition is theLerner Index or price cost margin following Nickell [1996] Thismeasure has several advantages over indicators such as marketshare or the Herfindahl concentration index These other mea-sures rely more directly on precise definitions of geographic andproduct markets which is particularly difficult in our applicationas many U K firms operate in international markets so thatmarket concentration measures based only on U K data may beextremely misleading

The price cost margin we use is measured by operating prof-its net of depreciation provisions and an estimated financial costof capital7 divided by sales

liit operating profit financial cost

sales

6 See Hall Jaffe and Trajtenberg [2000] The NBER database contains thepatents taken out in the U S patent office which is where innovations areeffectively patented internationally dated by the time of application

7 The cost of capital is assumed to be 0085 for all firms and time periods andthe capital stock is measured using the perpetual inventory method The invert-ed-U shape is robust to excluding this financial cost from the Lerner measureprincipally because it is relatively small and constant over time


Our competition measure is the average of this across firmswithin the industry

(1) cjt 1 1

Njtij

liit

where i indexes firms j indexes industry t indexes time and Njtis the number of firms in industry j in year t A value of 1indicates perfect competition (price equals marginal cost) whilevalues below 1 indicate some degree of market power In comput-ing this index we use the entire sample of Stock Market Listedfirms in each industry not only those in the patenting subsample

IIC A Nonlinear Relationship

We use flexible nonlinear estimators to investigate the basicshape of the relationship between competition and innovationDenoting n as the hazard rate and c as the measure of competi-tion we start by defining the competition innovation relationshipas

(2) n egc where g() is some unknown function

Suppose that the patent process has a Poisson distribution withhazard rate (2) then the expected number of patents satisfies

(3) E pc egc

Parametric models that study count data processes typicallybase their specification on the Poisson model with a parametric(linear) form for g(c) but they relax the strong assumptions onhigher moments8 We follow this approach in our empirical analy-sis basing our estimator on the first moment (3) We adopt aflexible specification for g(c) because we are particularly inter-ested in allowing the data to determine the shape of the relation-ship between innovation and product market competition

It is very likely that different industries will have observedlevels of patenting activity that have no direct causal relationshipwith product market competition but reflect other institutionalfeatures of the industry Consequently industry fixed effects areessential to remove any spurious correlation or ldquoendogeneityrdquo ofthis type Time effects are also included to remove common mac-roeconomic shocks Conditional on industry and time effects in-

8 See Hausman Hall and Griliches [1984] for example


dustry patent behavior is related to industry competition accord-ing to

(4) E pjtcjt xjt e gcjtxjt

where xjt represents a complete set of time and industry dummyvariables We use moment condition (4) to define a semiparamet-ric moment estimator and approximate g(c) with a spline follow-ing Ai and Chen [2003]

In Figure I we show the scatter of data points in between thetenth and ninetieth deciles of the citation-weighted patent distri-bution and overlay a fitted exponential quadratic curve Thesame exponential quadratic curve is plotted together with aspline approximation in Figure II It can be seen that the expo-nential quadratic specification provides a very reasonable approxi-mation to the nonparametric spline and that they both show aclear inverted-U shape The estimated coefficients for the expo-nential quadratic model are presented in Table I The first col-umn shows that both the linear and quadratic terms are individ-ually and jointly significant In the second column we reestimateincluding industry effects which is our preferred specification

FIGURE IScatter Plot of Innovation on Competition

The figure plots a measure of competition on the x-axis against citation-weighted patents on the y-axis Each point represents an industry-year Thescatter shows all data points that lie in between the tenth and ninetieth deciles inthe citation-weighted patents distribution The exponential quadratic curve thatis overlaid is reported in column (2) of Table I


and which removes the bias that results from correlation betweenpermanent levels of innovative activity and product market com-petition and again find a significant inverted-U shape

The underlying distribution of the data is shown by theintensity of the points on the estimated curves These indicatethat the peak of the inverted U lies near the median of thedistribution (the median is 095) so that industries are wellspread across the U-shape We can also see that a linear relation-ship would yield a positive slope confirming the results presentedin Nickell [1996] Before moving to the results using the policyinstruments we consider three robustness checks The first usesfive-year averages to estimate this relationship in view of thepotential lags in the relationship between competition and inno-vation As shown in the third column of Table I this also displaysa clear inverted-U shape9 The second robustness check notshown uses RampD expenditure as an alternative innovation mea-sure Again we find an inverted-U shape although due to a

9 The data span a 22-year period (1973 to 1994) Therefore the first periodwas set at seven years with the remainder at five years each

FIGURE IIInnovation and Competition Exponential Quadratic and the Semiparametric

Specifications with Year and Industry EffectsThe figure plots a measure of competition on the x-axis against citation-

weighted patents on the y-axis Each point represents an industry-year Thecircles show the exponential quadratic curve that is reported in column (2) ofTable I The triangles show a nonparametric spline


substantially smaller sample10 the coefficients are not statisti-cally significant Finally we fit the relationship for each of the topfour innovating industries in our sample and in each case thereis part or all of an inverted-U shape (see the earlier workingpaper version of this paper [Aghion et al 2004])

IID Using Policy Instruments

The major obstacle to empirical research in this area is thatcompetition and innovation are mutually endogenous Withoutaddressing this any results we find are likely to be biased toward

10 In the United Kingdom RampD was not a mandatory reporting item prior to1990 so it is not available for the majority of firms prior to this date This is oneof the reasons we use citation-weighted patents as our main innovation indicator

TABLE IEXPONENTIAL QUADRATIC BASIC SPECIFICATION

Dependent variable citation-weighted patents (1) (2) (3) (4)

Data frequency Annual Annual5-year

averages Annual

Competitionjt 15280 38746 81944 38513(5574) (6774) (26563) (6756)

Competition squaredjt 8099 20455 43443 20483(2961) (3617) (14143) (3606)

Significance of CompetitionjtCompetition squaredjt

760 3834 997 3259(002) (000) (001) (000)

Significance of policy instrumentsin reduced form

1011(0002)

Significance of other instrumentsin reduced form

500(0000)

Control functions in regression 438(404)

R2 of reduced form 0801Year effects Yes Yes Yes YesIndustry effects Yes Yes YesObservations 354 354 67 354

Competitionjt is measured by (1-Lerner index) in the industry-year All columns are estimated using anunbalanced panel of seventeen industries over the period 1973 to 1994 Estimates are from a Poissonregression Numbers in brackets are standard errors The standard errors in column (4) have not beencorrected for the inclusion of the control function Significance tests show likelihood ratio test-statistics andP-value from the F-test of joint significance The fourth column includes a control function The excludedvariables are policy instruments specified in Table II imports over value-added in the same industry-yearTFP in the same industry-year output minus variable costs over output in the same industry-year andestimates of markups from industry-country regression [Martins et al 1996] interacted with time trend allfor the United States and France


finding a more negative relationship between competition andinnovation if higher levels of innovation act for example toreduce competition11 We address this problem by using a set ofpolicy instruments that provide exogenous variation in the degreeof industrywide competition Since we are including industry andtime effects this approach identifies the competition effectthrough the differential timing of the introduction of policychanges across industries The three sets of policy instrumentused are the Thatcher era privatizations the EU Single MarketProgramme12 and the Monopoly and Merger Commission inves-tigations that resulted in structural or behavioral remedies beingimposed on the industry Table II lists the policy instrumentsthat are used the industries that are affected and the year(s) inwhich the policy changes occurred

We use these policy variables to instrument the changes incompetition These policy changes were driven by a combinationof political orthodoxy desires for European integration and regu-latory responses to anticompetitive behavior with little obviouslink to industry-level innovation performance We specify a re-duced-form model for the competition measure

11 See for example the endogenous barrier to entry story in Sutton [1998]12 See the earlier working paper version for full details of all the policy

instruments [Aghion et al 2004] The Single Market Program differentially in-creased trade liberalization across U K industries in 1988 All tariff and nontariffbarriers were phased out with the increase in competition depending on theextent of pre-SMP tariff and nontariff barriers The European CommissionrsquosCechinni report ranked all three-digit SIC industries into low medium and highimpact and we use this ranking for our policy instrument identification

TABLE IIPOLICY INSTRUMENTS

Industry Year(s) Policy

All but differential impacts 1988 Single Market ProgramBrewing 1986 MMC actionCars 1984 1987 1988 MMC action PrivatizationCar parts 1982 1987 MMC action PrivatizationPeriodicals 1987 MMC actionRazors and blades 1990 MMC actionOrdnance 1987 PrivatizationSteel 1987 PrivatizationTelecoms 1981 1984 1989 MMC action PrivatizationTextiles 1989 MMC action


(5) cjt zjt xjt vjt

which assumes that E[vjtzjt xjt] 13 0 where zjt denote the policyinstruments The idea is then to use functions of the vjt ascontrols in an extended version of the moment condition to stripout any element of cjt which is endogenous with the error termThe control function assumption can be expressed as

(6) Eeujtcjt xjtvjt 1

so that controlling for vjt in the conditional moment condition issufficient to retrieve the conditional moment assumption13

The fourth column in Table I shows the estimates for ourexponential quadratic specification that control for endogeneityusing our set of instruments The coefficient estimates are similarto the second column In the bottom part of the table we presentsome diagnostic statistics They show that the instruments aresignificant in the reduced form that the policy instruments inparticular are significant and that they have explanatory powerThe relationship between innovation and product market compe-tition for the standard quadratic and also for the instrumentedquadratic display a similar inverted-U relationship

III EXPLAINING THE INVERTED U

IIIA Main Existing Theories of Competition and Innovation

In this subsection we briefly summarize what existing theo-ries have to say about the relationship between competition andinnovation or competition and productivity growth As it turnsout none of them can account for the inverted-U pattern de-scribed in the previous section

The leading IO models of product differentiation and monopo-listic competition namely Salop [1977]) and Dixit and Stiglitz[1977] deliver the prediction that more intense product marketcompetition14 reduces postentry rents and therefore reduces theequilibrium number of entrants Thus these models can onlyaccount for the decreasing part of the inverted-U curve increased

13 To recover the parameters of interest we integrate over the empiricaldistribution of v and recover the ldquoaverage structural functionrdquo (see Blundell andPowell [2003])

14 Increased product market competition is modeled as a reduction in unittransport cost in Salop [1977] or as an increase in the substitutability betweendifferentiated products in Dixit and Stiglitz [1977]


product market competition discourages innovation by reducingpostentry rents This prediction is shared by most existing modelsof endogenous growth (eg Romer [1990] Aghion and Howitt[1992] and Grossman and Helpman [1991]) where an increase inproduct market competition or in the rate of imitation has anegative effect on productivity growth by reducing the monopolyrents that reward new innovation15

In all the above-mentioned papers firms are simply profit-maximizing individuals Instead Hart [1983] considers the caseof firms run by so-called ldquosatisficingrdquo managers who do not valueprofits per se but yet draw private benefits from maintaining thefirm afloat and thereby keeping their job Then increased com-petition may induce otherwise reluctant managers to put moreeffort into reducing costs in order to avoid bankruptcy16 How-ever this does not generate an inverted-U relationship betweencompetition and managerial incentives either most firms havemanagers who are a sufficiently residual claimant over the firmrsquosmonetary revenue in which case we again obtain a negativecorrelation between competition and innovation or they are runby satisficing managers in which case competition is unambigu-ously good for innovation

IIIB A Theoretical Framework

There is a unit mass of identical consumers each supplyinga unit of labor inelastically with a constant rate of intertemporaldiscount r and a logarithmic instantaneous utility functionu( yt) 13 ln yt The consumption good y is produced at each date tusing input services from a continuum of intermediate sectorsaccording to the production function

(7) ln yt 0

1

ln xjtdj

in which each xj is an aggregate of two intermediate goods pro-duced by duopolists in sector j defined by the subutility function

15 In these models the reason why competition policy is unambiguouslydetrimental to growth is the same as the reason why patent protection is unam-biguously good for growth patent protection raises monopoly rents from innova-tion whereas increased product market competition destroys these rents

16 This positive effect of competition disappears if managers value monetarypayoffs sufficiently as shown by Scharfstein [1988] and more recently by Schmidt[1997]


xj xAj xBj

The logarithmic structure of (7) implies that in equilibriumindividuals spend the same amount on each basket xj We nor-malize this common amount to unity by using current expendi-ture as the numeraire for the prices pAj and pBj at each dateThus the representative household chooses each xAj and xBj tomaximize xAj xBj subject to the budget constraint pAjxAj pBjxBj 13 1

Each firm produces using labor as the only input accordingto a constant-returns production function and takes the wagerate as given Thus the unit costs of production cA and cB of thetwo firms in an industry are independent of the quantities pro-duced Now let k denote the technology level of duopoly firm i insome industry j that is one unit of labor currently employed byfirm i generates an output flow equal to

(8) Ai ki i AB

where 1 is a parameter that measures the size of a leading-edge innovation Equivalently it takes ki units of labor for firmi to produce one unit of output The state of an industry is thenfully characterized by a pair of integers (lm) where l is theleaderrsquos technology and m is the technology gap of the leader overthe follower We define m (respectively m) to be the equilib-rium profit flow of a firm m steps ahead of (respectively behind)its rival17

For expositional simplicity we assume that knowledge spill-overs between leader and follower in any intermediate industryare such that the maximum sustainable gap is m 13 1 That is ifa firm already one step ahead innovates the lagging firm willautomatically learn to copy the leaderrsquos previous technology andthereby remain only one step behind Thus at any point in timethere will be two kinds of intermediate sectors in the economy (i)leveled or neck-and-neck sectors where both firms are at techno-logical par with one another so that m 13 0 (ii) unleveled sectorswhere one firm (the leader) lies one step ahead of its competitor(the laggard or follower) in the same industry so that m 13 118

17 The above logarithmic technology along with the cost structure c( x) 13 x k implies that the profit in the industry depends only on the gap m between thetwo firms and not on absolute levels of technology

18 Aghion et al [2001] analyze the more general case where m is un-bounded However unlike in this section that paper provides no closed-formsolution for the equilibrium RampD levels and the steady-state industry structure


By spending the RampD cost (n) 13 n2 2 in units of labor aleader (or frontier) firm moves one technological step ahead witha Poisson hazard rate of n We call n the ldquoinnovation raterdquo orldquoRampD intensityrdquo of the firm We assume that a follower firm canmove one step ahead with hazard rate h even if it spends nothingon RampD by copying the leaderrsquos technology Thus n2 2 is theRampD cost of a follower firm moving ahead with a hazard rate n h Let n0 denote the RampD intensity of each firm in a neck-and-neck industry and let n1 denote the RampD intensity of a followerfirm in an unleveled industry if n1 denotes the RampD intensity ofthe leader in an unleveled industry note that n1 13 0 since ourassumption of automatic catch-up means that a leader cannotgain any further advantage by innovating

We model the degree of product market competition inverselyby the degree to which the two firms in a neck-and-neck industryare able to collude They do not collude when the industry isunlevel Thus the laggard in an unlevel industry makes zeroprofit while the leader makes a profit equal to the differencebetween its revenue which we have normalized to unity and itscost which is 1 times its revenue given that its price is timesits unit cost

1 0 and 1 1 1

Each firm in a level industry earns a profit of 0 if the firms areunable to collude since they are in Bertrand competition withidentical products and identical unit costs and 12 if there ismaximum collusion More generally we assume that

0 ε1 0 ε 1 2

and we parameterize product market competition by 13 1 εthat is one minus the fraction of a leaderrsquos profits that the levelfirm can attain through collusion Note that is also the incre-mental profit of an innovator in a neck-and-neck industry nor-malized by the leaderrsquos profit

We next analyze how the equilibrium research intensities n0and n1 and consequently the aggregate innovation rate varywith our measure of competition

and therefore cannot formally establish qualitative results such as the existenceof an inverted-U relationship between competition and innovation or characterizethe relationship between competition and the distribution of technological gaps


IIIC The Schumpeterian and ldquoEscape-Competitionrdquo Effects

We assume that the equilibrium innovation rates n0 and n1are determined by the necessary conditions for a symmetricMarkov-stationary equilibrium in which each firm seeks to maxi-mize expected discounted profits with a discount rate r 13 019 InAppendix 1 we establish

PROPOSITION 1 The equilibrium research intensity by each neck-and-neck firm is

n0 h2 21 h

which increases with higher product market competition whereas the equilibrium research intensity of a laggard firmis

n1 h2 n02 21 h n0

which decreases with higher product market competition

The latter effect (on n1) is the basic Schumpeterian effectthat results from reducing the rents that can be captured by afollower who succeeds in catching up with its rival by innovatingThe former effect (on n0) we refer to as an ldquoescape-competitioneffectrdquo namely that more competition induces neck-and-neckfirms to innovate in order to escape competition since the incre-mental value of getting ahead is increased with higher PMC Onaverage an increase in product market competition will thushave an ambiguous effect on growth It induces faster productiv-ity growth in currently neck-and-neck sectors and slower growthin currently unleveled sectors The overall effect on growth willthus depend on the (steady-state) fraction of leveled versus un-leveled sectors But this steady-state fraction is itself endogenoussince it depends upon equilibrium RampD intensities in both typesof sectors We proceed to show under which condition this overalleffect is an inverted U and at the same time derive additionalpredictions for further empirical testing

Let 1 (respectively 0) denote the steady-state probabilityof being an unleveled (respectively neck-and-neck) industryDuring any unit time interval the steady-state probability that asector moves from being unleveled to leveled is 1(n1 h) and

19 We have established by numerical simulation that all of our results holdwith r 0


the probability that it moves in the opposite direction is 20n0 Insteady state these two probabilities must be equal

(9) 1n1 h 20n0

This together with the fact that 1 0 13 1 implies that theaggregate flow of innovations is

(10) I 20n0 1n1 h 21n1 h 4n0n1 h

2n0 n1 h

We now analyze how this flow varies with product marketcompetition and establish the possibility of an inverted-U pat-tern Since by Proposition 1 n0 is an increasing function of wecan use n0 as our proximate measure of product market compe-tition which in turn will prove convenient when deriving ourmain predictions below Note that n0 takes values on the interval[ x 13 h2 1 h x 13 h2 21 h] with x 13 xcorresponding to maximum collusion (0 13 12) and x 13 xcorresponding to maximum competition (0 13 0) We have

PROPOSITION 2 Whenever the value x 13 [h2 21]3 is inte-rior to the interval [ x x ] the aggregate innovation rate (n0)follows an inverted-U pattern it increases with competitionn0 for all n0 [ x x) and decreases for all n0 ( x x ] If x x then the aggregate innovation rate increases with n0 forall n0 [ x x ] so that the escape-competition effect alwaysdominates If x x then it decreases with n0 for all n0 [ x x ] so that the Schumpeterian effect always dominatesMoreover each of these cases arises for a nonempty subset ofparameter values

Proof See Appendix 1

The inverted-U shape can be explained as follows Whenthere is not much product market competition there is hardlyany incentive for neck-and-neck firms to innovate and thereforethe overall innovation rate will be highest when the sector isunleveled Thus the industry will be quick to leave the unleveledstate (which it does as soon as the laggard innovates) and slow toleave the leveled state (which will not happen until one of theneck-and-neck firms innovates) As a result the industry willspend most of the time in the leveled state where the escape-competition effect dominates (n0 is decreasing in 0) In otherwords if the degree of competition is very low to begin with an


increase in competition should result in a faster average innova-tion rate

On the other hand when competition is initially very highthere is relatively little incentive for the laggard in an unleveledstate to innovate Thus the industry will be relatively slow toleave the unleveled state Meanwhile the large incrementalprofit 1 0 gives firms in the leveled state a relatively largeincentive to innovate so that the industry will be relatively quickto leave the leveled state As a result the industry will spendmost of the time in the unleveled state where the Schumpeterianeffect is at work on the laggard while the leader never innovatesIn other words if the degree of competition is very high to beginwith an increase in competition should result in a slower averageinnovation rate20

Note that according to this model the empirical measure c ofcompetition that we used in the previous section has an expectedvalue of

(11) c 1 1

1

2 21 11 1

That is in accordance with equation (1) above the industryaverage Lerner index in an uneven industry is 12 since thelaggard has a Lerner of zero whereas the industry average Ler-ner index in a leveled industry is the Lerner index of each firmequal to its profits divided by its revenue 0(12) 13 2(1 )1We show in Appendix 1 that in our model

PROPOSITION 3 The empirical measure c defined by (11) is a mono-tonically increasing function of the theoretical measure used in the other propositions

IIID Additional Predictions on Technological Spreadand Competition

In addition to providing a rationale for the inverted-U pat-tern uncovered in the previous section the model delivers twoadditional predictions which are summarized in the followingtwo propositions and which we prove in Appendix 1

20 Thus the reason why the escape-competition effect dominates whencompetition is low whereas the Schumpeterian effect on laggards dominateswhen competition is intense is the ldquocomposition effectrdquo of competition on thesteady-state distribution of technology gaps across sectors


PROPOSITION 4 The expected technological gap in an industryincreases with product market competition

The intuition is simple we know that the higher the degreeof product market competition the higher the research inten-sity in neck-and-neck sectors and the lower the research in-tensity in unleveled sectors This in turn implies that anysector will spend a higher fraction of its time being unleveledso that on average over time the technological gap betweenfirms in that sector will be higher By the law of large numbersthe same will be true for the economy as a whole at any pointin time

The next proposition is equally intuitive it states the exis-tence of a positive interaction between the escape-competitioneffect and the average distance of the industry to its frontier inthe sense that in industries where firms are closer to their tech-nological frontier over time the escape-competition effect tends tobe stronger (that is the increasing part of the inverted U tends tobe steeper) More precisely suppose that there are industrieswith large spillover parameter h and industries with small hThose with large h will tend to be more neck-and-neck on averageover time21 Now we can compare the magnitude of the escape-competition effect across industries with different values of h andestablish that

PROPOSITION 5 The peak of the inverted U is larger and occurs ata higher degree of competition in more neck-and-neck indus-tries More formally let be the incremental profit at whichx 13 x 13 [h2 21]3 then both and ( x) are increasingin h


IIIE Testing the Technological Spread Predictions

To empirically assess the predictions on the technologicalspread and competition relationship we first need a measure ofthe size of the technology gap between firms within an industryWe capture this by the proportional distance a firm is from thetechnological frontier as measured by total factor productivityMore formally we let

21 This is formally established in Appendix 1 in the remark following theproof of Proposition 4


(12) mit TFPFt TFPitTFPFt

where F denotes the frontier firm (with the highest TFP) and idenotes nonfrontier firms So at the firm level mit 0 andmFt 13 0 We use an industry level measure mjt that is theaverage across firms in the industry A lower value of mjtindicates that firms in industry j are technologically closer tothe frontier (and therefore more like the neck-and-neck firmsin our theoretical section) while a high value of mjt indicates alarge technological gap with the frontier (so that firms in thoseindustries are more like laggard firms in an unleveledindustry)

The first key prediction on the technological spread is that inequilibrium the average technology gap between leaders andfollowers should be an increasing function of the overall level ofindustrywide competition (so that average neck-and-necknessshould be a decreasing function of competition) This is tested andconfirmed in Table III which reports the results from regressingthe industry average technology gap on the Lerner index with afull set of year dummies (first column) and a full set of year andindustry dummies (second column) In both cases there is a sig-nificantly positive coefficient suggesting that industries withgreater levels of competition have a higher average spread intechnology within the industry and a lower degree of technologyneck-and-neckness

This result is perhaps surprising because the static intui-tion is that higher competition should reduce the spread of TFPby increasing the exit rate of low TFP firms truncating thelower tail of the distribution But empirically we find this staticeffect of competition appears to be dominated by a more pow-erful dynamic effect to increase the rate of innovation As firmsinnovate to try to escape competition they increase the spreadof TFP within the industry dominating any potential selectiveexit effect

The second theoretical prediction is that the inverted-U-shaped relationship between competition and growth should besteeper in more neck-and-neck industries To assess this pre-diction we consider a subsample of our datamdashindustries withbelow median technological gapmdashwhich are the more neck-and-neck industries Figure III shows our baseline fitted expo-nential quadratic curve as well as an exponential quadraticcurve fitted to the subsample of high neck-and-neck industries


Two features stand out clearly First more neck-and-neck in-dustries show a higher level of innovation activity for any levelof product market competition Second the inverted-U curve issteeper for the more neck-and-neck industries which accordswell with our theoretical predictions These differences arestatistically significant as shown in columns 3 and 4 of TableII which reports the quadratic coefficients for the whole sam-ple and the high neck-and-neck industry subsample includinga full set of year dummies (third column) and a full set of yearand industry dummies (fourth column) The interaction termsare jointly significant in both cases

TABLE IIITECHNOLOGY GAP AND EXPONENTIAL QUADRATIC WITH NECK-AND-NECK SPLIT

(1) (2) (3) (4)Dependent variable Technology

gapTechnology

gapCitation-weightedpatents

Citation-weightedpatents

Estimation procedureLinear

regressionLinear

regression Poisson Poisson

Competitionjt 2858 0942 18381 42446(0400) (0419) (5899) (695)

Competition squaredjt 9635 2229(3101) (369)

Competitionjt Technology 143 382gapjt (248) (266)

Competition squaredjt 130 384Technology gapjt (259) (278)

Significance ofCompetitionjt 1659 3921Competition squaredjt (000) (000)

Significance ofCompetitionjt Technology

gapjt974 793

Competition squaredjt

Technology gapjt

(001) (002)

Year effects Yes Yes Yes YesIndustry effects Yes Yes

Competitionjt is measured by (1-Lerner index) in the industry-year Technology gapjt is measured by theaverage distance to the TFP frontier firm across all firms in the industry-year so it is an inverse measure ofneck-and-neckness All columns estimated using an unbalanced panel of 354 yearly observations on seven-teen industries over the period 1973 to 1994 Significance tests show likelihood ratio test-statistics andP-value from the F-test of joint significance Numbers in brackets are standard errors The standard errorsin columns 3 and 4 have not been corrected for the inclusion of the control function


IV CONCLUSIONS

This paper investigates the relationship between productmarket competition (PMC) and innovation using a flexiblenonlinear estimator We find evidence that the competition-innovation relationship takes the form of an inverted-U shapewith industries distributed across both the increasing and de-creasing sections of the U-shape This inverted-U is robust to anumber of alternative specifications including identifying thecausal impact of competition by exploiting a series of major policyreforms in the United Kingdom

To understand what is driving this inverted-U shape weextend the current theoretical literature on step-by-step innova-tion to produce a model that delivers an inverted-U prediction Inthis model competition may increase the incremental profit frominnovating labeled the ldquoescape-competition effectrdquo but competi-tion may also reduce innovation incentives for laggards labeledthe ldquoSchumpeterian effectrdquo The balance between these two ef-

FIGURE IIIInnovation and Competition The Neck-and-Neck Split

The figure plots a measure of competition on the x-axis against citation-weighted patents on the y-axis Each point represents an industry-year Thecircles show the exponential quadratic curve that is reported in column (2) ofTable I The triangles show the exponential quadratic curve estimated only onneck-and-neck industries that is reported in column (4) of Table III


fects changes between low and high levels of competition gener-ating an inverted-U relationship In addition this extension ofthe theory provides two new predictions First the equilibriumdegree of technological neck-and-neckness among firms shoulddecrease with PMC and second the higher the average degree ofneck-and-neckness in an industry the steeper the inverted-Urelationship between PMC and innovation We take these twoadditional predictions to the data and find them to be consistentwith the data This dual empirical and theoretical approach pro-vides useful results on the impact of competition and closeness intechnology space on innovation and also a model to understandthis and experiment with potential policy reforms

APPENDIX 1 PROOFS

Proof of Proposition 1

To solve for the equilibrium research intensities n0 and n1of neck-and-neck and laggard firms we use Bellman equationsMore precisely let Vm (respectively Vm) denote the steady statevalue of being currently a leader (respectively a follower) in anindustry with technology gap m and let r denote the rate of timediscount We have the following Bellman equations

(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2

In words the annuity value rV1 of currently being a techno-logical leader in an industry with gap m 13 1 at date t equals thecurrent profit flow 1 minus the expected capital loss (n1 h)(V0 V1) from having the follower catch up by one step withthe leader The annuity value rV1 of currently being a laggardis equal to the current profit flow 1 plus the expected capitalgain (n1 h)(V0 V1) from catching up with the leaderminus the RampD cost22 (n1)2 2 Finally in the Bellman equationfor a neck-and-neck firm there is no help factor h because thereis no leader and n0 denotes the RampD intensity by the other firm

22 It follows by the same argument as used in Aghion et al [2001] that theequilibrium real wage rate will equal unity given our choice to normalize expen-ditures equal to unity


in the same sector in a symmetric Nash equilibrium both firmsrsquoRampD intensities are equal that is

n0 n0

Now using the fact that each firm chooses its own RampDintensity to maximize its current value that is to maximize theright-hand side of the corresponding Bellman equation we obtainthe first-order conditions

(16) n1 V0 V1

(17) n0 V1 V0

Eliminating the Vrsquos between the Bellman equations andfirst-order conditions (13) to (17) yield the reduced form RampDequations

(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0

This system is recursive as the first equation solves for n0 andthen given n0 the second equation solves for n1 For the specialcase where r 13 0 we use the relationship 0 13 (1 )1 to obtain

(20) n0 h h2 21

(21) n1 h n0 h2 n02 21

We immediately see that n0 increases whereas n1 de-creases with higher product market competition23 This estab-lishes Proposition 1


Let

x n0

23 From equation (20)n0

1

n0 h 0

From this and equation (21)n1

n0

1 n0

n1 h n0 0


According to equation (21) above

n1 x2 B x h

where

B h2 21

Thus we can reexpress the aggregate innovation rate (10) as

x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B

is decreasing in x with a unique value

x B3

at which f( x) 13 0 Therefore ( x) is quasi-concave with ( x) 0 as x x Therefore the inverted-U pattern will obtain when-ever x ( x x ) the escape-competition effect will dominate if x x and the Schumpeterian effect will always dominate if x x

Now let 13 h1 One can easily establish that

xx

2 2

2 23and

xx

2 1

2 23

Thus the inverted-U pattern will obtain whenever

2 1 2 23 2 2

the escape-competition effect will strictly dominate over thewhole interval [ x x ] whenever

2 23 2 2

finally the Schumpeterian effect will dominate over the wholeinterval [ x x ] whenever


2 23 2 1

Each of the corresponding three regions is nonempty which es-tablishes Proposition 2


From equations (9) and (21) we have

1 2n0

n02 B n0

where B is defined in the proof of Proposition 2 above From thisand (11)

c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0

where ε 13 1 From this and Proposition 1

c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h

so we need only show that

n02 Bn0 h 21

4 ε1

This clearly holds when ε 14 So suppose that ε 14 Then weneed only show that

(22) n02 Bn0 h2 41

4 ε 2

12

It follows from equations (20) and (21) that

n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2

which implies condition (22)


Note that the expected technological gap is given by

G 0 0 1 1 1 2n0

2n0 n1 h

which can be reexpressed as

G 1 n0

2 B n0

2n01

This latter expression is clearly increasing in n0 and thereforewith product market competition This establishes Proposition 4

Remark The expected technological gap

G 1 n1 h

2n01

is decreasing in h This stems from the fact that n0 is decreasingin h whereas n1 h is increasing in h To see the former notethat from equation (20)

n0

h n0

n0 h 10

whereas the latter follows from this and equation (21)

n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21

therefore h will affect x 13 B3 and ( x) via its positive effecton B Assume that x is interior to the interval ( x x ) From theenvelope theorem the marginal effect of B on

x maxx x x

x

is just equal to the direct effect

E

B xx2 B x

x2 B x

which is unambiguously positive The marginal effect of h is E (Bh) which is therefore also positive Therefore more neck-and-neck industries (those with larger h) have a higher peak inthe inverted U Moreover the peak occurs at the value of suchthat x 13 n0 13 B3 or equivalently

(24) 0 h2 21 h h2 213

The peak lies farther to the right on the line in more neck-and-neck industries if ddh 0 where is implicitly defined by(24) Applying the implicit functions theorem to (24) we get

d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21

Therefore ddh 0 This establishes Proposition 5

APPENDIX 2 SUMMARY STATISTICS

Table IV presents the descriptive statistics on our unbal-anced panel spanning 17 two-digit industries from 1973 to 1994


This is taken from an underlying firm level panel of 311 firmswhich remain after matching the accounting and innovation dataand cleaning the data (removing firms with missing observationsfirms involved in major mergers or firms with less than threeyears of consecutive data see Bloom and Van Reenen [2002] forfull details) The Lerner and technology gap measures are gener-ated from the entire population of U K firms From the data wecan see even the industry level patenting count is highly skewedwith most industries taking out no patents in any given yearPatenting levels also vary strongly across industries in part dueto different patenting intensities although in estimation the in-dustry dummies will control for this The Lerner averages 4percent and ranges from 13 percent in Office and ComputingMachinery in 1973 to less than 1 percent in Motor Vehicles in1982

HARVARD UNIVERSITY AND INSTITUTE FOR FISCAL STUDIES

CENTRE FOR ECONOMIC PERFORMANCE LONDON SCHOOL OF ECONOMICS

INSTITUTE FOR FISCAL STUDIES AND UNIVERSITY COLLEGE LONDON


BROWN UNIVERSITY

REFERENCES

Aghion Philippe Nick Bloom Richard Blundell Rachel Griffith and PeterHowitt ldquoCompetition and Innovation An Inverted-U Relationshiprdquo UCLWorking Paper No 0406 July 2004

Aghion Philippe Mathias Dewatripont and Patrick Rey ldquoCompetition FinancialDiscipline and Growthrdquo Review of Economic Studies LXVI (1999) 825ndash852

TABLE IVDESCRIPTIVE STATISTICS

Mean(sd) Median Min Max

Patents 659 35 0 54(852)

Cite weighted patents 665 335 0 45(843)

1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)

The sample is an unbalanced panel of 354 yearly observations on seventeen industries over the period1973 to 1994


Aghion Philippe Christopher Harris and John Vickers ldquoCompetition andGrowth with Step-by-Step Innovation An Examplerdquo European EconomicReview Papers and Proceedings XLI (1997) 771ndash782

Aghion Philippe Christopher Harris Peter Howitt and John Vickers ldquoCompe-tition Imitation and Growth with Step-by-Step Innovationrdquo Review of Eco-nomic Studies LXVIII (2001) 467ndash492

Aghion Philippe and Peter Howitt ldquoA Model of Growth through Creative De-structionrdquo Econometrica LX (1992) 323ndash351

Ai Chunrong and Xiaohong Chen ldquoEfficient Estimation of Models with Condi-tional Moment Restrictions Containing Unknown Functionsrdquo EconometricaLXXI (2003) 1795ndash1843

Arrow Kenneth ldquoEconomic Welfare and the Allocation of Resources for Inven-tionrdquo in R Nelson ed The Rate and Direction of Inventive Activity (Prince-ton NJ Princeton University Press 1962) pp 609ndash625

Bloom Nicholas and John Van Reenen ldquoPatents Real Options and Firm Perfor-mancerdquo Economic Journal CXII (2002) C97ndashC116

Blundell Richard Rachel Griffith and John Van Reenen ldquoMarket Share MarketValue and Innovation in a Panel of British Manufacturing Firmsrdquo Review ofEconomic Studies LXVI (1999) 529ndash554

Blundell Richard and James Powell ldquoEndogeneity in Nonparametric and Semi-parametric Regression Modelsrdquo Chapter 8 in M Dewatripont L P Hanseneds Advances in Economics and Econometrics Vol II Econometric Mono-graph Series 36 (Cambridge Cambridge University Press 2003)

Dixit Avinash and Joseph Stiglitz ldquoMonopolistic Competition and OptimumProduct Diversityrdquo American Economic Review LXVII (1977) 297ndash308

Geroski Paul Market Structure Corporate Performance and Innovative Activity(Oxford UK Oxford University Press 1995)

Gilbert Richard and David Newbery ldquoPreemptive Patenting and the Persistenceof Monopolyrdquo American Economic Review LXXII (1982) 514ndash526

Grossman Gene and Elhanan Helpman Innovation and Growth in the GlobalEconomy (Cambridge MA MIT Press 1991)

Hall Bronwyn Adam Jaffe and Manuel Trajtenberg ldquoMarket Value and PatentCitations A First Lookrdquo NBER working paper No 7741 June 2000

Hart Oliver ldquoThe Market Mechanism as an Incentive Schemerdquo Bell Journal ofEconomics XIV (1983) 366ndash382

Hausman Jerry Bronwyn Hall and Zvi Griliches ldquoEconometric Models for CountData and an Application to the Patents-RampD Relationshiprdquo EconometricaLII (1984) 909ndash938

Martins Joaquim Oliveira Stephano Scarpetta and Dirk Pilat ldquoMarkup-upRatios in Manufacturing Industries Estimates for 14 OECD CountriesrdquoOECD Working Paper No 162 1996

Nickell Steven ldquoCompetition and Corporate Performancerdquo Journal of PoliticalEconomy CIV (1996) 724ndash746

Reinganum Jennifer ldquoUncertain Innovation and the Persistence of MonopolyrdquoAmerican Economic Review LXXIII (1983) 61ndash66

Romer Paul ldquoEndogenous Technological Changerdquo Journal of Political EconomyXCVIII (1990) 71ndash102

Salop Steven ldquoThe Noisy Monopolist Imperfect Information Price Dispersionand Price Discriminationrdquo Review of Economic Studies XLIV (1977)393ndash406

Scharfstein David ldquoProduct Market Competition and Managerial Slackrdquo RANDJournal of Economics XIX (1988) 147ndash155

Scherer Fredrick ldquoMarket Structure and the Employment of Scientists andEngineersrdquo American Economic Review LVII (1967) 524ndash531

Schmidt Klaus ldquoManagerial Incentives and Product Market Competitionrdquo Re-view of Economic Studies LXIV (1997) 191ndash213

Schumpeter Joseph Capitalism Socialism and Democracy (London Allen Un-win 1943)

Sutton John Technology and Market Structure (Cambridge MA MIT Press1998)


patenting activity and firm size in the cross section with a di-minishing impact at larger sizes when allowing for nonlinearitiesTo our knowledge no existing model of product market competi-tion and innovation predicts an inverted-U pattern

An explanation for these results could be pieced together bycombining agency models3 with Schumpeterian models howeverthis seems unsatisfactory Instead we develop an extension ofAghion Harris and Vickers [1997]4 that can fit the entire curveIn this model both current technological leaders and their follow-ers in any industry can innovate and innovations by leaders andfollowers all occur step-by-step Innovation incentives depend notso much upon postinnovation rents as in previous endogenousgrowth models where all innovations are made by outsiders butupon the difference between postinnovation and preinnovationrents of incumbent firms In this case more competition mayfoster innovation and growth because it may reduce a firmrsquospreinnovation rents by more than it reduces its postinnovationrents In other words competition may increase the incrementalprofits from innovating and thereby encourage RampD investmentsaimed at ldquoescaping competitionrdquo This should be particularly truein sectors where incumbent firms are operating at similar tech-nological levels in these ldquoneck-and-neckrdquo sectors preinnovationrents should be especially reduced by product market competi-tion On the other hand in sectors where innovations are made bylaggard firms with already low initial profits product marketcompetition will mainly affect postinnovation rents and thereforethe Schumpeterian effect of competition should dominate

The essence of the inverted-U relationship between competi-tion and innovation is that the fraction of sectors with neck-and-neck competitors is itself endogenous and depends upon equilib-rium innovation intensities in the different types of sectors Morespecifically when competition is low a larger equilibrium frac-tion of sectors involve neck-and-neck competing incumbents sothat overall the escape-competition effect is more likely to domi-nate the Schumpeterian effect On the other hand when compe-tition is high the Schumpeterian effect is more likely to domi-nate because a larger fraction of sectors in equilibrium haveinnovation being performed by laggard firms with low initialprofits The inverted-U shape is confirmed by our U K panel

3 Hart [1983] Schmidt [1997] Aghion Dewatripont and Rey [1999]4 See also Aghion Harris Howitt and Vickers [2001]
















sales





(1) cjt 1 1

Njtij

liit






(3) E pc egc


























averages Annual

Competitionjt 15280 38746 81944 38513(5574) (6774) (26563) (6756)



760 3834 997 3259(002) (000) (001) (000)


1011(0002)


500(0000)













(5) cjt zjt xjt vjt
















(7) ln yt 0

1

ln xjtdj





xj xAj xBj



(8) Ai ki i AB








1 0 and 1 1 1


0 ε1 0 ε 1 2








n0 h2 21 h


n1 h2 n02 21 h n0







(9) 1n1 h 20n0


(10) I 20n0 1n1 h 21n1 h 4n0n1 h

2n0 n1 h









(11) c 1 1

1

2 21 11 1

























gapTechnology




regressionLinear


Competitionjt 2858 0942 18381 42446(0400) (0419) (5899) (695)






gapjt974 793


Technology gapjt

(001) (002)




IV CONCLUSIONS







APPENDIX 1 PROOFS



(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2





n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)











































sales





(1) cjt 1 1

Njtij

liit






(3) E pc egc


























averages Annual

Competitionjt 15280 38746 81944 38513(5574) (6774) (26563) (6756)



760 3834 997 3259(002) (000) (001) (000)


1011(0002)


500(0000)













(5) cjt zjt xjt vjt
















(7) ln yt 0

1

ln xjtdj





xj xAj xBj



(8) Ai ki i AB








1 0 and 1 1 1


0 ε1 0 ε 1 2








n0 h2 21 h


n1 h2 n02 21 h n0







(9) 1n1 h 20n0


(10) I 20n0 1n1 h 21n1 h 4n0n1 h

2n0 n1 h









(11) c 1 1

1

2 21 11 1

























gapTechnology




regressionLinear


Competitionjt 2858 0942 18381 42446(0400) (0419) (5899) (695)






gapjt974 793


Technology gapjt

(001) (002)




IV CONCLUSIONS







APPENDIX 1 PROOFS



(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2





n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)






























(1) cjt 1 1

Njtij

liit






(3) E pc egc


























averages Annual

Competitionjt 15280 38746 81944 38513(5574) (6774) (26563) (6756)



760 3834 997 3259(002) (000) (001) (000)


1011(0002)


500(0000)













(5) cjt zjt xjt vjt
















(7) ln yt 0

1

ln xjtdj





xj xAj xBj



(8) Ai ki i AB








1 0 and 1 1 1


0 ε1 0 ε 1 2








n0 h2 21 h


n1 h2 n02 21 h n0







(9) 1n1 h 20n0


(10) I 20n0 1n1 h 21n1 h 4n0n1 h

2n0 n1 h









(11) c 1 1

1

2 21 11 1

























gapTechnology




regressionLinear


Competitionjt 2858 0942 18381 42446(0400) (0419) (5899) (695)






gapjt974 793


Technology gapjt

(001) (002)




IV CONCLUSIONS







APPENDIX 1 PROOFS



(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2





n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)


















































averages Annual

Competitionjt 15280 38746 81944 38513(5574) (6774) (26563) (6756)



760 3834 997 3259(002) (000) (001) (000)


1011(0002)


500(0000)













(5) cjt zjt xjt vjt
















(7) ln yt 0

1

ln xjtdj





xj xAj xBj



(8) Ai ki i AB








1 0 and 1 1 1


0 ε1 0 ε 1 2








n0 h2 21 h


n1 h2 n02 21 h n0







(9) 1n1 h 20n0


(10) I 20n0 1n1 h 21n1 h 4n0n1 h

2n0 n1 h









(11) c 1 1

1

2 21 11 1

























gapTechnology




regressionLinear


Competitionjt 2858 0942 18381 42446(0400) (0419) (5899) (695)






gapjt974 793


Technology gapjt

(001) (002)




IV CONCLUSIONS







APPENDIX 1 PROOFS



(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2





n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)





































(5) cjt zjt xjt vjt
















(7) ln yt 0

1

ln xjtdj





xj xAj xBj



(8) Ai ki i AB








1 0 and 1 1 1


0 ε1 0 ε 1 2








n0 h2 21 h


n1 h2 n02 21 h n0







(9) 1n1 h 20n0


(10) I 20n0 1n1 h 21n1 h 4n0n1 h

2n0 n1 h









(11) c 1 1

1

2 21 11 1

























gapTechnology




regressionLinear


Competitionjt 2858 0942 18381 42446(0400) (0419) (5899) (695)






gapjt974 793


Technology gapjt

(001) (002)




IV CONCLUSIONS







APPENDIX 1 PROOFS



(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2





n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)

































(7) ln yt 0

1

ln xjtdj





xj xAj xBj



(8) Ai ki i AB








1 0 and 1 1 1


0 ε1 0 ε 1 2








n0 h2 21 h


n1 h2 n02 21 h n0







(9) 1n1 h 20n0


(10) I 20n0 1n1 h 21n1 h 4n0n1 h

2n0 n1 h









(11) c 1 1

1

2 21 11 1

























gapTechnology




regressionLinear


Competitionjt 2858 0942 18381 42446(0400) (0419) (5899) (695)






gapjt974 793


Technology gapjt

(001) (002)




IV CONCLUSIONS







APPENDIX 1 PROOFS



(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2





n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)





























xj xAj xBj



(8) Ai ki i AB








1 0 and 1 1 1


0 ε1 0 ε 1 2








n0 h2 21 h


n1 h2 n02 21 h n0







(9) 1n1 h 20n0


(10) I 20n0 1n1 h 21n1 h 4n0n1 h

2n0 n1 h









(11) c 1 1

1

2 21 11 1

























gapTechnology




regressionLinear


Competitionjt 2858 0942 18381 42446(0400) (0419) (5899) (695)






gapjt974 793


Technology gapjt

(001) (002)




IV CONCLUSIONS







APPENDIX 1 PROOFS



(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2





n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)































1 0 and 1 1 1


0 ε1 0 ε 1 2








n0 h2 21 h


n1 h2 n02 21 h n0







(9) 1n1 h 20n0


(10) I 20n0 1n1 h 21n1 h 4n0n1 h

2n0 n1 h









(11) c 1 1

1

2 21 11 1

























gapTechnology




regressionLinear


Competitionjt 2858 0942 18381 42446(0400) (0419) (5899) (695)






gapjt974 793


Technology gapjt

(001) (002)




IV CONCLUSIONS







APPENDIX 1 PROOFS



(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2





n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)
































n0 h2 21 h


n1 h2 n02 21 h n0







(9) 1n1 h 20n0


(10) I 20n0 1n1 h 21n1 h 4n0n1 h

2n0 n1 h









(11) c 1 1

1

2 21 11 1

























gapTechnology




regressionLinear


Competitionjt 2858 0942 18381 42446(0400) (0419) (5899) (695)






gapjt974 793


Technology gapjt

(001) (002)




IV CONCLUSIONS







APPENDIX 1 PROOFS



(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2





n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)






























(9) 1n1 h 20n0


(10) I 20n0 1n1 h 21n1 h 4n0n1 h

2n0 n1 h









(11) c 1 1

1

2 21 11 1

























gapTechnology




regressionLinear


Competitionjt 2858 0942 18381 42446(0400) (0419) (5899) (695)






gapjt974 793


Technology gapjt

(001) (002)




IV CONCLUSIONS







APPENDIX 1 PROOFS



(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2





n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)
































(11) c 1 1

1

2 21 11 1

























gapTechnology




regressionLinear


Competitionjt 2858 0942 18381 42446(0400) (0419) (5899) (695)






gapjt974 793


Technology gapjt

(001) (002)




IV CONCLUSIONS







APPENDIX 1 PROOFS



(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2





n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)















































gapTechnology




regressionLinear


Competitionjt 2858 0942 18381 42446(0400) (0419) (5899) (695)






gapjt974 793


Technology gapjt

(001) (002)




IV CONCLUSIONS







APPENDIX 1 PROOFS



(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2





n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)





























IV CONCLUSIONS







APPENDIX 1 PROOFS



(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2





n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)






























APPENDIX 1 PROOFS



(13) rV1 1 n1 hV0 V1

(14) rV1 1 n1 hV0 V1 n12 2

(15) rV0 0 n0V1 V0 n0V1 V0 n02 2





n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)






























n0 n0


(16) n1 V0 V1

(17) n0 V1 V0


(18)n0

2

2 r hn0 1 0 0

(19)n1

2

2 r h n0n1 0 n0

2

2 0


(20) n0 h h2 21

(21) n1 h n0 h2 n02 21



Let

x n0


1

n0 h 0


n0

1 n0

n1 h n0 0



n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)






























n1 x2 B x h

where

B h2 21


x 4xx2 B x

x2 B x

with

x 4B 1

x2 B x21

2x

x2 B

The expression

f x 1 2x

x2 B


x B3



xx

2 2

2 23and

xx

2 1

2 23


2 1 2 23 2 2


2 23 2 2



2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)





























2 23 2 1




1 2n0

n02 B n0


c

21 1 1 14 ε 1

21 n0

2 B n0

n02 B n0

14 ε 1

21 n0

2 B n0

n02 B n0

1 14 ε 2

n02 B n0


c

21 n02 B n0

n02 B n0

1 14 ε 21

n02 B 1

n0 h


n02 Bn0 h 21

4 ε1


(22) n02 Bn0 h2 41

4 ε 2

12


n02 B n1 n0 h n0 h

so

n02 B n0 h2 h2 21 ε1 21 ε1


Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)





























Therefore

n02 Bn0 h2 41 ε21

2




G 0 0 1 1 1 2n0

2n0 n1 h


G 1 n0

2 B n0

2n01



G 1 n1 h

2n01


n0

h n0

n0 h 10


n1

h h

h2 n02 21

1 n0

h2 n02 21

1 n0

h 1

since

n0 h2 n02 21

and

n0

h 0



Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)






























Since

B h2 21


x maxx x x

x


E

B xx2 B x

x2 B x


(24) 0 h2 21 h h2 213


d

dh h2 21

1 F

where

F h

h2 21 1

h3

h2 213 0

since

h h2 21










BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)


































BROWN UNIVERSITY

REFERENCES





Patents 659 35 0 54(852)


1-Lerner 095 095 087 099(0023)

Technology gap (m) 049 051 0080 081(0155)





























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