The Aggregate Implications of Innovative Investment in the Garcia-Macia, Hsieh, and Klenow Model (preliminary and incomplete)

Transcription

1 The Aggregae Implicaions of Innovaive Invesmen in he Garcia-Macia, Hsieh, and Klenow Model (preliminary and incomplee) Andy Akeson and Ariel Bursein December 2016 Absrac In his paper, we exend he model firm dynamics of Garcia-Macia, Klenow, and Hsieh (2016) (GHK) o include a descripion of he coss of innovaive invesmens as in he model of Klee and Korum (2004). In his model, aggregae produciviy (TFP) grows as a resul of innovaive invesmen by incumben and enering firms in improving coninuing producs and acquiring new producs o he firm. This model serves as a useful benchmark because i ness boh Qualiy-Ladders based Neo-Shumpeerian models and Expanding Varieies models commonly used in he lieraure and, a he same ime, i provides a rich model of firm dynamics as described in GHK. We show how daa on firm dynamics and firm value can be used o infer he elasiciies of aggregae produciviy growh wih respec o changes in incumben firms invesmens in improving heir incumben producs, incumben firms invesmens in acquiring producs new o he firm, and enering firms invesmens in acquiring new producs. As discussed in Akeson and Bursein (2015), hese elasiciies are a crucial inpu in evaluaing he exen o which i is possible o aler he medium erm growh pah of he macroeconomy hrough policies aimed a simulaing innovaive invesmens by firms. We use hese mehods o provide quaniaive esimaes of hese elasiciies of aggregae TFP growh wih respec o changes in each of he hree caegories of innovaive invesmen in he model as well as of he rae of social depreciaion of innovaion expendiures. We demonsrae ha hese quaniaive implicaions of he model are highly sensiive o one s esimae of he baseline research inensiy of he economy and o one s esimae of he baseline marke value of inangible capial wihin firms. Deparmen of Economics, Universiy of California Los Angeles, NBER, and Federal Reserve Bank of Minneapolis. Deparmen of Economics, Universiy of California Los Angeles, NBER. 1

2 1 Inroducion: Garcia-Macia, Klenow, and Hsieh (2016) (henceforh GHK) presen a racable model of firm dynamics ha capures many feaures of he daa on firm dynamics. Firms in his model are disinguished by he echnologies hey posses for producing inermediae producs. In his dimension, he model is a sraighforward exension of Klee and Korum (2004). This model allows for aggregae produciviy growh o arise hrough innovaion by incumben firms o improve heir own producs, innovaion by incumben firms o obain producs new o he firm, and innovaion by enering firms o obain new producs. Producs ha are new o a firm may be new o sociey or solen from oher firms. The goal of heir paper is o use daa on firm dynamics o esimae how much of he observed growh in aggregae produciviy comes from hese differen ypes of innovaion by firms. In his paper, we exend he GHK model of firm dynamics o include a descripion of he coss of innovaive invesmens ha are lef un-modelled in heir paper. We also exend his model o allow for wo simple forms of social depreciaion of innovaion expendiures: we allow incumben firms o lose producs due o exogenous exi of producs and we allow for he produciviy wih which incumben firms can produce producs o deeriorae over ime in he absence of innovaive invesmens by ha firm. Our exended version of he GHK model hen convenienly ness boh he canonical Expanding Varieies models analyzed in Lumer (2007), Lumer (2011) and Akeson and Bursein (2010) and he canonical Qualiy-Ladders based Neo-Shumpeerian models analyzed in Klee and Korum (2004) and he many models based on ha framework. As a resul, i incorporaes he increasing reurns due o increased variey as well as he ineremporal knowledge spillovers from one firm s success in innovaion o he social payoffs o anoher firm s innovaive invesmen. We hen use his exended version of he GHK model o consider he quesion of how an economis who has access o rich daa on firm dynamics and firm value migh idenify he social reurns o increased innovaive invesmen by firms. We measure he social reurns o innovaive invesmen by firms in erms of he increased growh of aggregae produciviy ha would resul from an increase in innovaive invesmen. We are able o measure hese social reurns o innovaive invesmen by firms due o several assumpions in our exended GHK model ha make he model implied relaionships beween firms innovaive invesmens and he dynamics of firm size and value paricularly racable. 2

3 Under hese assumpions, he curren producive capaciy of all firms corresponds o a measure M (z), where M (z) is he measure of inermediae producs ha can be produced wih physical capial and labor a produciviy exp(z). The corresponding measure of producs in his economy is M = z M (z). Under he assumpion ha markups are consan across producs, aggregae TFP in he economy is given as a geomerically weighed aggregae of produc level produciviies [ Z = exp((ρ 1)z)M (z) z ] 1/(ρ 1) In general, o measure he social reurns o firms innovaive invesmens, one mus consruc a model of how hose invesmens ranslae ino dynamics for he measure of produciviies wih which firms produce he producs consumers desire M (z). These dynamics mus be inferred from daa on he size of producs, which corresponds o he model implied measure of size s = exp((ρ 1)z) reflecing eiher he share of oupu of a produc in oal oupu, he share of producion labor devoed o a produc in oal producion labor, he share of physical capial devoed o a produc in oal physical capial, or he share of variable profis earned on a produc in oal variable profis. This inference is complicaed by he fac ha he growh in he number of producs of a given size may be genuinely new producs for sociey or simply a reallocaion due o business sealing. To make progress on his inference, we use wo key assumpions. Firs, we assume ha he equilibrium dynamics of z a he produc level obeys a srong form of Gibra s Law: boh he exi rae of producs and he incremen o z for coninuing producs are independen of produc size. This assumpion is quie close o he assumpions made in GHK o derive a simple ransiion law for aggregae TFP as a funcion of he arrival rae of differen ypes of innovaions o z a he produc level. (We do drop he assumpion in GHK of endogenous exi of small producs due o fixed operaing coss.) Second, we assume ha equilibrium innovaive invesmen a he produc level is proporional o produc size. This second assumpion is quie close o hose used in Klee and Korum (2004) and Akeson and Bursein (2010) o aggregae innovaive invesmen. In our model, hese assumpions lead us o he following simple form of he aggregae relaionship beween firms innovaive invesmens and TFP growh log Z +1 log Z = G(x ic, x im, x e ) 3

4 where x ic is he aggregae invesmen incumben firms underake a o lower he marginal cos wih which hey produce heir curren producs, x im is he aggregae invesmen incumben firms underake a in acquiring new producs o he firm (boh hose new o sociey and hose solen from oher firms), and x e is he aggregae invesmen by enering firms a in acquiring new producs o produce a + 1 (boh hose new o sociey and hose solen from oher firms). These innovaive invesmens are in unis of a produced inpu o innovaion ha we erm he research good and are subjec o a resource consrain x ic + x im + x e = Y r where Y r is he aggregae oupu of he inpu ino innovaion and P r is he price of his research good relaive o consumpion. We refer o S r = P r Y r /GDP as he research inensiy of he economy. We seek o measure he social reurns of firms innovaive invesmens in erms of he derivaives of his equilibrium funcion G. 1 We do so a an assumed baseline level of aggregae TFP growh ḡ Z and unobserved baseline levels of invesmen. For example, we refer o G ic x ic as he elasiciy of aggregae TFP growh wih respec o a change in firms invesmens in lower he marginal cos of producion of heir currenly produced producs, and likewise for he oher wo forms of innovaive invesmen. The elasiciy of aggregae TFP growh wih respec o a proporional change in all hree forms of innovaive invesmen, is hen given by G ic x ic + G im x im + G e x e We also consider he elasiciy of changes in innovaive invesmen concenraed enirely on each one of he hree forms of innovaive invesmen G ic x ic Ȳ r x ic 1 As discussed in Akeson and Bursein (2015), hese elasiciies of aggregae TFP growh wih respec o changes in real innovaive invesmen are impac elasiciies. Tha is, hese elasiciies are useful for measuring he elasiciies of aggregae TFP growh wih respec o policy-induced changes in expendiures on innovaive invesmen on impac. As discussed in our earlier paper, he dynamics are shaped o a large exen by he degree of ineremporal knowledge spillovers. If hese are large enough o ensure ha he model displays fully endogenous growh, hen impac elasiciies are also useful in measuring he permanen change in he growh rae of aggregae TFP ha arises from a policy-induced permanen change in he innovaion inensiy of he economy. 4

5 and likewise for he oher wo forms of invesmen. Following Akeson and Bursein (2015), we firs show ha if one imposes he assumpion ha here is no social depreciaion of innovaion expendiures, as is done in Klee and Korum (2004) and GHK, hen he quaniaive implicaions of he model for he elasiciies of aggregae produciviy growh wih respec o changes in innovaive invesmens in economies wih low baseline levels of TFP growh are ighly resriced by ha low baseline level of TFP growh regardless of he fi of he model o he daa on firm dynamics and value. Specifically, he funcion G is concave in a proporional increase in is hree argumens so G ic x ic + G im x im + G e x e ḡ Z G(0, 0, 0) since ḡ Z corresponds o he growh of aggregae TFP a he baseline level of invesmens. We refer o he model s counerfacual implicaions for he growh rae of aggregae TFP if here were no invesmens by firms in innovaion (G(0, 0, 0)) as he social depreciaion of innovaion expendiures. Under he assumpion of no social depreciaion, G(0, 0, 0) = 0, and hence he social reurn we wish o measure is bounded by he baseline growh rae of TFP ḡ Z. We hen ake up he quesion of wha quaniaive social reurns o firms innovaive invesmens he exended GHK model wih he wo forms of social depreciaion can admi once i has been fi o daa on firm dynamics and firm value. We assume ha he economis conducing he measuremen has access o daa on he size and number of producs in enering and incumben firms and daa on he value of firms in excess of heir sock of physical capial. We show how o idenify he parameers of he funcion G and he levels of invesmen of each ype from hese daa up o wo unidenified parameers: he fracion of producs ha are new o enering and incumben firms ha are new o sociey as opposed o solen from oher firms, and he fracion of firm value ha is due o he variable profis earned on exising producs versus due o he added innovaive capaciy of he firm associaed wih he acquisiion of hese producs. We are able o use he model o impose upper and lower bounds on hese wo unobserved parameers. Our main resuls are presenaions of he model-implied relaionships beween hese daa on firm dynamics and firm value, he wo se-idenified parameers, and he elasiciies of aggregae TFP growh wih respec o changes in hese hree caegories of innovaive invesmen by firms. We nex use our heoreical resuls o deliver quaniaive esimaes of he social reurns o firms innovaive invesmens implied by our exended GHK model and he daa in he Business Dynamics Saisics daabase as well as calibraions of oal innovaive expendiure 5

6 from Nakamura (2009) and firm value from McGraan and Presco (Forhcoming). We find elasiciies of aggregae produciviy growh wih respec o individual caegories of invesmen ranging from 0 o 0.44 and wih respec o a proporional increase in all hree caegories of innovaive invesmen ranging from o 0.039, wih he alernaive esimaes driven primarily by differen calibraed values of he baseline marke value of inangible capial wihin firms. We find corresponding raes of social depreciaion of innovaion expendiures ranging from o per year. 2 Model and Equilibrium Properies: In his secion, we review he key resuls we derive from he dynamic equaions of he model. Deails of how hese dynamic equaions are derived from a fully specified version of he model are given in subsequen secions. 2.1 Aggregae Oupu and Toal Facor Produciviy: A each dae, aggregae oupu of he final good used for consumpion and invesmen in physical capial is given by Y = Z K α L 1 α p where K is he aggregae sock of physical capial, L p is he aggregae quaniy of labor used in producion of his final good, and Z is oal facor produciviy. As is sandard in his lieraure, we assume ha his final good Y is produced from a coninuum of inermediae producs hrough a CES aggregaor wih elasiciy ρ. These inermediae goods are each provided by monopolis producers who engage in monopolisic price compeiion. These inermediae producs are produced according o producion echnologies y (z) = exp(z)k (z) α l p (z) 1 α where z indexes he posiion of he marginal cos curve for he producer of he inermediae good and k (z) and l p (z) denoe he quaniies of physical capial and labor used in producion of he inermediae good wih index z a dae. To simplify our noaion, we assume ha he suppor of z is a counable grid wih z n = n for he inegers n. Assuming consan markups µ > 1 of prices o marginal coss across producs, we ge he resul ha in equilibrium oal facor produciviy is a CES aggregae of he indices z across he available inermediae 6

7 producs of he form [ Z = exp((ρ 1)z n )M (z n ) n ] 1/(ρ 1) where M (z n ) is he measure of inermediae producs wih index z n a dae. Noe ha he measure of producs available in he economy a dae is given by M = n M (z n ) Produc Size: In equilibrium we have ha s (z) = exp((ρ 1)z) = y (z) Y = k (z) K = l p(z) L p. Hence, we refer o s (z) as he size of a produc. In daa, his can be measured in erms of value added or profis or physical capial or producion labor. This measure is also addiive, so we can use i o refer o he size of caegories of producs as we do below Facor Shares: In equilibrium wih consan markups µ across producs, fracion α/µ of oupu Y is paid o physical capial, fracion (1 α)/µ o producion labor, and fracion (µ 1)/µ o he firms ha produce he inermediae producs are variable profis. 2.2 Conribuions o TFP by firm and produc caegories: As in Klee and Korum (2004), firms in his economy produce a number of producs j, where j is a naural number. Enering firms ener wih a single produc, so j = 1. Firms exi when he number of producs ha hey produce drops o zero. We say ha a firm is an incumben firm a if i also produced producs a 1. Oherwise, firms a are enering firms. We say ha a produc is an exising produc a if i was also produced a 1. Oherwise, producs a are new producs. New producs are new o sociey. No all producs ha a new o a firm a are new o sociey. Some producs ha are new o a firm a are incumben producs ha were produced by some oher firm a 1. We refer o exising producs ha are produced by a differen firm 7

8 a han a 1 as solen producs. We refer o incumben producs a ha are produced by he same firm a as a 1 as coninuing producs. Noe ha, by definiion, coninuing producs are produced by incumben firms. Wih his erminology, we decompose aggregae produciviy a ino five componens: Z ic = [ n exp((ρ 1)z n )M ic (z n ) ] 1/(ρ 1) where M ic (z n ) is he measure of coninuing producs wih index z n produced by incumben firms a dae, [ ] 1/(ρ 1) Z in = n exp((ρ 1)z n )M in (z n ) where M in (z n ) is he measure of new producs wih index z n produced by incumben firms a dae, [ ] 1/(ρ 1) Z is = n exp((ρ 1)z n )M is (z n ) where M is (z n ) is he measure of solen producs wih index z n produced by incumben firms a dae, [ ] 1/(ρ 1) Z en = n exp((ρ 1)z n )M en (z n ) where M en (z n ) is he measure of new producs wih index z n produced by enering firms a dae, and [ ] 1/(ρ 1) Z es = n exp((ρ 1)z n )M es (z n ) where M es (z n ) is he measure of solen producs wih index z n produced by enering firms a dae. Wih hese definiions, we have he following decomposiion of aggregae produciviy = ic + in + is + en + es As in GHK, his decomposiion can be done by produc caegories exising and new = ( ic + is ) ( + es + Z ρ 1 in ) + en and firm caegories, incumben and enering = ( ic + is ) ( + in + Z ρ 1 es ) + en 8

9 In using daa on firm dynamics o discipline he model, we use he resuls ha hese decomposiions of aggregae produciviy correspond o decomposiion of produc caegories caegories by caegory size ( Z ρ 1 ) ic 1 = + Zρ 1 is + Zρ 1 es + and likewise for firm caegories, incumben and enering 1 = ( Z ρ 1 ic + Zρ 1 is ) + Zρ 1 in + ( Z ρ 1 in ( Z ρ 1 es ) + Zρ 1 en ) + Zρ 1 en We also use noaion for a corresponding decomposiion of he measure of producs ino five caegories M ic = n and similarly for he caegories is, in, es, and en. Le M ic (z n ) S ic = Zρ 1 ic denoe he aggregae size of coninuing producs a incumben firms, and likewise le S in, S is, S en and S es denoe he sizes of he oher four produc caegories. Le S im = S in + S is denoe he size of hose producs ha are new o incumben firms a and S e = S en + S es he size of hose producs ha are new o enering firms a. F ic = M ic M denoe he fracion of producs ha are coninuing producs a incumben firms, and likewise le F in, F is, F en and F es denoe he sizes of he oher four produc caegories. Le F im = F in + F is denoe he fracion of producs ha are new o incumben firms a and F e = F en + F es he fracion of producs ha are new o enering firms a. producs in a caegory is given by he raio S/F. 2.3 Technology for Innovaive Invesmen: Noe ha he average size of We assume ha firms make hree ypes of invesmen in innovaion as in GHK: incumben firms inves o improve heir coninuing producs, incumben firms inves o acquire new producs o 9

10 he firm eiher hrough he creaion of a new produc or acquisiion of a solen produc, and enering firms inves o acquire new producs o he enering firm eiher hrough he creaion of a new produc or acquisiion of a solen produc. Innovaive invesmen is underaken using a second final good, which we erm he research good, as an inpu. The aggregae amoun of his research good produced a ime is Y r = A r Z φ 1 L r (1) where L r = L L p is he quaniy of labor devoed o producion of he research good, A r is he level of exogenous scienific progress, and he erm Z φ 1 for φ 1 reflecs ineremporal knowledge spillovers in he producion of he research good as in he model of Jones (2002) equaion 5. We denoe he aggregae quaniy of he research good ha incumben firms inves a in improving z for coninuing producs a + 1 by x ic. We denoe he aggregae quaniy of he research good ha incumben firms inves a in acquiring producs new o ha firm a + 1 by x im. We denoe he aggregae quaniy of he research good ha enering firms inves a in acquiring producs new o ha firm a + 1 by x e. The resource consrain for he research good is x ic + x im + x e = Y r (2) As in Klee and Korum (2004), we assume ha he innovaive invesmen underaken by a firm is simply he sum of he invesmens by ha firm a he produc level. As in Akeson and Bursein (2010), he equilibrium invesmen by incumben firms in improving each of heir coninuing producs is direcly proporional o he size of he produc. As in Klee and Korum (2004), he equilibrium invesmen by incumben firms in acquiring producs new o he firm is direcly proporional o he number of producs produced by he firm. Finally, we assume ha he amoun ha each enering firm invess is a parameer a ha we normalize o 1/M. The echnologies for innovaive invesmen ha we assume build in wo key spillovers. Firs, for incumben firms invesing in improving a coninuing produc, he cos of such invesmen falls wih aggregae TFP. Second, for incumben and enering firms invesing o acquire producs new o he firm, he produciviy index z of he acquired produc rises wih he average value of exp((ρ 1)z) across exising producs and he cos of he invesmen falls wih he measure of exising producs M. Furhermore, as in Klee and Korum (2004) and Akeson and Bursein (2010) and many oher papers in he lieraure, we assume ha hese invesmens resul in equilibrium dynamics 10

11 for he scale of exising producs consisen wih a srong form of Gibra s Law characerized by he following four properies of equilibrum. Firs, we assume ha a fracion δ 0 of exising producs a cease o be produced (for exogenous reasons) a + 1, wih his exogenous exi probabiliy δ 0 independen of he index z of he produc. Second, he probabiliy ha an exising produc is solen from an incumben firms is independen of he index z of he produc. Third, he incremen o he index z for exising producs ha are solen by incumben and enering firms beween and +1 is z +1 z = independen of he index z of he produc ha is solen. Fourh, he disribuion of he incremens o he index z beween and + 1 for hose producs ha are coninuing producs produced by incumben firms a + 1 is independen of he index z of hose producs a. We now describe he dynamics of he measures and conribuion o aggregae produciviy of each produc caegory, based on our assumpions on he echnologies for innovaive invesmen a he firm level. Furher deails are given in he Appendix (To be wrien) Measures by produc caegory The measure of producs produced a + 1 by enering firms is given as a funcion of enering firms innovaive invesmen a by M es+1 + M en+1 = x e M. This expression follows from he assumpion ha if an enering firm spends 1/M unis of he research good, i acquires 1 new produc. Thus if here are M e enering firms a ime hey spend M e /M unis of he research good and produce M e new producs a + 1. Furher deails are provided in he Appendix (o be be wrien). We assume ha fracion δ of hese producs produced by enering firms a + 1 are solen from incumben firms, wih he remaining fracion 1 δ being new o sociey. Thus, we assume ha M es+1 = δx e M We assume ha he measure of producs new o incumben firms a + 1 is given as a funcion of hese firms invesmens x im a. Specifically, his measure is M is+1 + M in+1 = h(x im )M where h( ) is a sricly increasing and concave funcion wih h(0) = 0 and h(x) < 1 for all x. This expression follows from he assumpion ha if each incumben firm spends x/m unis 11

12 of he research good per produc i produces, i acquires h(x) new producs per produc if x = x im is he same for each produc hen x im is spen on aggregae and h(x im )M new producs are creaed a + 1. We again assume ha fracion δ of hese producs are solen producs, so M is+1 = δh(x im )M The measure of coninuing producs produced by incumben firms a +1 are hose producs a ha do no cease o exis a + 1 plus hose producs a ha are no solen a + 1. Thus M ic+1 = (1 δ 0 )M M is+1 M es+1 = (1 δ M )M where δ M δ 0 + δ (h(x im ) + x e ) (3) Dynamics of he measure of producs These assumpions imply dynamics for he oal measure of producs M +1 M = (1 δ M ) + h(x im ) + x e (4) Invesmen and growh of Z by produc caegory In his subsecion, we make assumpions abou he relaionship beween invesmen and he growh of Z by produc caegory. These assumpions deliver our model s predicions for he relaionship beween invesmen and he growh of size by produc caegory as well as for he relaionship beween invesmen and he growh rae of aggregae TFP. We assume ha, due o spillovers as in Lumer (2007), new producs in enering firms a + 1 have average exp((ρ 1)z) indexed by η en /M so ha heir oal conribuion o aggregae produciviy is where we used he assumpion ha en+1 = η en (1 δ)x e. M en+1 M = (1 δ)x e New producs obained by incumben firms have average exp((ρ 1)z) indexed by η in /M so ha in+1 = η in (1 δ)h(x im ). 12

13 where we use he assumpion ha M in+1 M = (1 δ)h(x im ) Producs ha are solen each have incremen of size o heir index z from o + 1. Thus, hey have average exp((ρ 1)z) indexed by exp((ρ 1) ) /M so es+1 = exp((ρ 1) ) δx e, and is+1 = exp((ρ 1) )Zρ 1 δh(x im ), Those producs ha are coninuing producs in incumben firms a + 1 experience, on average, incremen ζ(x ic ) o heir exp((ρ 1)z) from o + 1. Thus ic+1 = (1 δ M) ζ(x ic ) This expression follows from he assumpions ha if an incumben firm wih a coninuing produc wih produciviy z spends x of he research good on improving ha produc, i draws a new produciviy, condiional on survival, ha is k seps from z wih probabiliy ( ) Z ζ k, x ρ 1. In equilibrium, each incumben firm chooses he same invesmen per uni exp((ρ 1)z) exp((ρ 1)z) size x (z) = x ic, so for all producs he probabiliy of drawing a new produciviy ha is k seps from he curren level is ζ (k, x ic ). We define he expecaion over k as ζ (x ic ). Adding over all incumben producs, he aggregae resources spen in his aciviy are x ic. We assume ha ζ(x ic ) is a sricly increasing and concave funcion, and ha 0 < ζ(0) < Dynamics of TFP These assumpions imply dynamics for aggregae produciviy where and +1 = (1 δ 0 δ(h(x im ) + x e )) ζ(x ic ) + η i h(x m ) + η e x e (5) η i = δ exp((ρ 1) ) + (1 δ) η in η e = δ exp((ρ 1) ) + (1 δ) η en 13

14 2.3.5 Parameers and he lengh of he ime period The parameers δ 0, δ, η in,and η en all represen raios of level variables and hence are se independen of he lengh of he ime period. The variables x ic, x im and x e represen flows, and hence are proporional o he lengh of he ime period. Likewise, since h(x im ) reflecs an arrival rae of new producs, his rae should also be proporional o he lengh of he ime period. The sep size for solen producs is independen of he lengh of he ime period. Since log( ζ(x)) corresponds o he growh rae of z for coninuing producs in incumben firms, we do require ha log( ζ(x)) converges o zero as he ime inerval beween periods and + 1 shrinks o zero. Thus, wihou loss of generaliy, we can assume ha exp((ρ 1) ) ζ(x). This corresponds o he requiremen ha a produc ha is solen from incumben firms is produced wih a higher z a + 1 in is new firm han i would have had as a coninuing produc in he firm ha previously produced i. As we discuss below, his assumpion also hus corresponds o he assumpion ha solen producs have larger average size han coninuing producs in incumben firms. 2.4 Nesed Models This model nes five commonly used models in he lieraure: hree ypes of Expanding Varieies models and wo ypes of Neo-Schumpeerian models. If δ = 0, hen here is no business sealing and hence all new producs acquired by incumben and enering firms are new producs for sociey, expanding he measure of producs M. This is he assumpion ypically made in an Expanding Varieies model. Lumer (2007) is an example of an expanding varieies model in which here is only innovaive invesmen in enry. (Noe ha we do no consider he endogenous exi of producs due o fixed operaing coss feaured in ha paper and in GHK). Akeson and Bursein (2010) is an example of an expanding varieies model in which here is innovaive invesmen in enry and by incumben firms in coninuing producs. Lumer (2011) is an example of an expanding varieies model in which here is innovaive invesmen in enry and in he acquisiion of new producs by incumben firms. Neo-Schumpeerian models based on he Qualiy-Ladders framework ypically assume δ = 1 and δ 0 = 0. The simples versions of hese models do no accommodae growh in he measure of varieies M. Grossman and Helpman (1991) and Aghion and Howi (1992) are examples 14

15 of Neo-Schumpeerian models in which here is only innovaive invesmen in enry. Klee and Korum (2004) is an example of a Neo-Schumpeerian model in which here is innovaive invesmen in enry and by incumben firms in acquiring new producs. 3 Disciplining he Model wih Daa on Firm Dynamics We consider an economis who has daa on he growh rae of he measure of producs M +1 /M as well as daa on he fracion of producs ha are coninuing producs in incumben firms F ic+1, he fracion of producs ha are new o incumben firms measured as he sum of hose ha are new o sociey and solen F im+1, and he fracion of producs ha are produced in enering firms measured as he sum of hose ha are new o sociey and solen F e+1. We also assume ha his economis has daa on he growh rae of aggregae TFP Z +1 /Z (a leas for some baseline value) and daa on he aggregae size of coninuing producs in incumben firms S ic+1, he aggregae size of producs ha are new o incumben firms measured as he sum of hose ha are new producs and hose ha are solen S im+1, and he aggregae size of producs ha are new o enering firms measured as he sum of hose ha are new producs an hose ha are solen S em+1. We use he noaion g Z+1 = log(z +1 ) log(z ) o denoe he growh rae of aggregae TFP and g M+1 = log(m +1 ) log(m ) o denoe he growh rae of he measure of producs. 3.1 Implicaions of daa on he dynamics of he measure of producs Daa on he dynamics of he measure of producs allow one o idenify he values of he following on a balanced growh pah (BGP) (1 δ M ) = F ic exp(ḡ M ) h( x im ) = F im exp(ḡ M ) x e = F e exp(ḡ M ) 3.2 Implicaions of daa on size and aggregae TFP growh Daa on size and aggregae TFP growh, ogeher wih he daa on he measures of producs discussed above, allow one o idenify he following model parameers. 15

16 The parameers η e and η i are idenified from daa on he average size of producs ha are new o enering and incumben firms on a balanced growh pah (hese BGP average sizes are denoed wih a bar): η e = S e F e exp((ρ 1)ḡ Z ) exp(ḡ M ) η i = S im F im exp((ρ 1)ḡ Z ) exp(ḡ M ) The value of ζ(x ic ) is idenified from daa on he average size of coninuing producs in incumben firms ζ( x ic ) = S ic F ic exp((ρ 1)ḡ Z ) exp(ḡ M ) 3.3 Parameers ha are no idenified The parameers δ 0 and δ governing he share of producs new o incumben and enering firms ha are solen from oher incumben firms ogeher wih he parameers η in and η en governing he average value of exp((ρ 1)z)/ for ha fracion (1 δ) of producs new o incumben and enering firms ha are new o sociey are no pinned down by he daa on firm dynamics we have discussed. Insead, we have only idenified δ M as defined in equaion (3) and η i and η e defined in equaion (5). Our model does impose bounds on hese parameers δ 0 and δ. Boh of hese parameers mus be non-negaive. We mus have δ 0 δ M as observed in he daa ( F ic exp(ḡ M )). We mus have δ below he minimum of four upper bounds. The firs of hese is δ 1. The second of hese corresponds o he value of δ implied by equaion (3) wih δ 0 = 0 (since δ 0 canno be negaive) and he daa on he exi rae of incumben producs and he fracion of new producs in incumben and enering firms. Specifically δ M δ = 1 F ic exp(ḡ M ) h( x im ) + x e (1 F ic ) exp(ḡ M ) The hird and fourh bound corresponds o he requiremen ha solen producs in incumben firms have higher z (on average) han he producs ha hey replace, i.e. exp((ρ 1) ) ζ( x ic ). Noe ha our model s implicaions for he average size of solen producs in incumben and new firms is S is = S es exp((ρ 1) ) = F is F es exp(ρ 1)ḡ Z ) exp(ḡ M) and ha he average size of new producs in incumben firms is he soluion o S im = δ S is + (1 δ) S in F im F is F in 16

17 The requiremen ha solen producs be larger han hose ha hey replace, i.e. S is F is S ic F ic (or, similarly, exp((ρ 1) ) ζ( x ic )) ogeher wih he requiremen ha new producs have non-negaive average size ( η in > 0) implies ha we mus have δ S im F im / Sic F ic Similar argumens for enering firms give he fourh bound δ S e F e / Sic F ic The minimum of hese bounds binds when new producs in incumben or enering firms are smaller han coninuing producs on average in he daa. We have imposed direcly by assumpion ha h(0) = 0 and ha he measure of producs in enering firms a + 1 is equal o zero when x e = 0. The parameer ζ(0) corresponding o he counerfacual growh rae of exp((ρ 1)z) for coninuing producs in he absence of invesmen in improving hese producs is also no idenified. 4 Elasiciies of TFP growh wih respec o innovaive invesmen From equaion (5), we can wrie he growh rae of TFP as a funcion of innovaive invesmens as where log(z +1 ) log(z ) = G(x ic, x im, x e ) G(x ic, x im, x e ) = 1 ρ 1 log ( (1 δ 0 δ(h(x im ) + x e )) ζ(x ic ) + η i h(x m ) + η e x e ) We now consider he elasiciies of TFP growh wih respec o he hree ypes of innovaive invesmen. To do so, we evaluae derivaives of G a a poin ( x ic, x im, x e ) and Ȳr ha saisfies equaion (2). Define ḡ Z o be he growh rae of TFP a hose levels of invesmen. Define ˆx = log(x) log( x) 17

18 Then, o a firs order approximaion we have where ĝ Z G ic x icˆx ic + G im x imˆx im + G e x eˆx e ĝ Z log(z ) log(z) ḡ Z and and G ic = 1 (1 δ 0 δ(h( x im ) + x e )) ζ ( x ic ) ρ 1 exp((ρ 1)ḡ Z ) G im = 1 ( ) δ ζ( xic ) + η i h ( x m ) ρ 1 exp((ρ 1)ḡ Z ) G e = 1 ( ) δ ζ( xic ) + η e ρ 1 exp((ρ 1)ḡ Z ) (6) (7) (8) 4.1 Idenifying or bounding elasiciies In wha follows, we look o use daa on firm dynamics and he value of firms o idenify precisely or bound he model-implied impac elasiciy ĝ Z /Ŷr compued from derivaives (6), (7), and (8), baseline invesmen levels Ȳr, x ic, x im and x e, and alernaive perurbaions o invesmen ˆx ic, ˆx im, ˆx e. We consider wo ypes of perurbaions o invesmen. The firs ype is a proporional change in all caegories of invesmen ˆx ic = ˆx im = ˆx e = Ŷr The second ype of perurbaion is concenraed on a single form of invesmen or ˆx ic = Ȳr x ic Ŷ r ˆx im = Ȳr x im Ŷ r ˆx e = Ȳr x e Ŷ r We sar our analysis wih an applicaion of he resuls of Akeson and Bursein (2015) bounding he impac elasiciy ĝ Z /Ŷr wih respec o a proporional change in all hree caegories of invesmen by he difference beween he baseline growh rae of TFP and he 18

19 model implied counerfacual growh rae of TFP in he absence of any innovaive invesmen, ḡ Z G(0, 0, 0). In he original GHK model, i is assumed ha G(0, 0, 0) = 0, so his proposiion imposes a igh bound for advanced economies wih low baseline ḡ Z. We hen show ha in our specificaion of he GHK model, G(0, 0, 0) can be less han zero, relaxing his bound. Once we allow for he possibiliy ha G(0, 0, 0) < 0, he model admis for a large value of he impac elasiciy ĝ Z /Ŷr even in an advanced economy. As we have discussed above, daa on firm dynamics allows us o idenify some erms in he derivaives (6), (7), and (8), bu we are no able o idenify he parameer δ governing he exen of business sealing, he derivaives of he innovaion echnologies ζ ( x ic ) and h ( x m ), and he baseline invesmen levels Ȳr, x ic, x im and x e needed o compue hese elasiciies. In secion 5 we discuss how o use daa on he value of inangible capial in firms ogeher wih he condiion ha firms choose invesmen o maximize ha privae value o idenify he derivaives of he innovaion echnologies ζ ( x ic ) and h ( x m ) and he breakdown of expendiure on innovaion ino he hree caegories of innovaive invesmen. In secion 6 we hen repor on he model s quaniaive implicaions for hese elasiciies for various assumed values of he parameer δ governing he exen of business sealing. 4.2 Elasiciy wih respec o a proporional change in all innovaive invesmen Following Akeson and Bursein (2015), we are able o bound he elasiciy of TFP growh wih respec o a proporional change in all innovaive invesmens as follows. Proposiion 1. If ˆx ic = ˆx im = ˆx e = Ŷr, hen he elasiciy of TFP growh wih respec o innovaive invesmen is bounded by he difference beween he baseline growh rae of TFP and he growh rae of TFP when all invesmen is zero, i.e. ĝ Z (ḡ Z G(0, 0, 0)) Ŷr Proof. The proof follows from he concaviy of he funcion H(a) (see appendix) defined as H(a) G(a x ic, a x im, a x e ) Specifically, if ˆx ic = ˆx im = ˆx e = Ŷr, hen ĝ Z = H (1)Ŷr The resul follows from he fac ha for concave funcions H (1)1 H(1) H(0). 19

20 4.3 Social Depreciaion of Innovaion in he GHK model Noe ha he bound on he elasiciy ĝ Z /Ŷr esablished in proposiion 1 is independen of parameers ouside of hose ha deermine he model s implicaions for ḡ Z and G(0, 0, 0). We refer o he growh rae of TFP ha would arise if all innovaive invesmen were se o zero, G(0, 0, 0), as he rae of social depreciaion of innovaion. In boh Klee and Korum (2004) and GHK, i is assumed ha he rae of social depreciaion of innovaion G(0, 0, 0) = 0. As a resul, he elasiciy ĝ Z /Ŷr is bounded by ḡ Z. In our implemenaion of he GHK model, we do no make his assumpion ha here is no social depreciaion of innovaion. Because we allow for exogenous exi of exising varieies, denoed by δ 0, and for deerioraion of he index z of coninuing varieies in incumben firms, denoed by ζ(0) 1, we have social depreciaion of innovaion given by G(0, 0, 0) = 1 ρ 1 log ( (1 δ 0 ) ζ(0) ) 0 (9) (recall ha since h(x im ) denoes a rae a which incumben firms acquire new producs, we impose ha h(0) = 0). Thus, our version of he GHK model poenially allows for a higher elasiciy ĝ Z /Ŷr > ḡ Z if we allow for social depreciaion by calibraing δ 0 > 0 and/or ζ(0) < Bounding elasiciies wih respec o componens of invesmen and social depreciaion In he experimen in which we consider proporional changes in invesmen, we have ha he elasiciy of TFP growh wih respec o oal innovaive expendiure is equal o he sum of hree individual elasiciies: ĝ Z = [G ic x ic + G im x im + G e x e ] Ŷr Using he same logic based on concaviy, in his model, we can bound hese elasiciies of TFP growh wih respec o each componen of innovaive invesmen as follows. Corollary 2. The individual elasiciies of TFP growh wih respec o he componens of invesmen are bounded by G ic x ic ḡ Z G(0, x im, x e ) G im x im ḡ Z G( x ic, 0, x e ) G e x e ḡ Z G( x ic, x im, 0) 20

21 Proof. These bounds follow as a corollary o proposiion 1 from he concaviy of he funcions H ic (a) = G(a x ic, x im, x e ) H im (a) = G( x ic, a x im, x e ) and H e (a) = G( x ic, x im, a x e ) and he observaion ha he desired elasiciies are given by H ic(1), H im(1), and H e(1) respecively. The bounds on he elasiciies of TFP growh wih respec o changes in he componens of innovaive invesmen presened in Proposiion 2 offer a procedure for bounding he change in TFP growh ĝ Z ha would arise from a change in oal innovaive invesmen Ŷr focused enirely on one of he caegories of innovaive invesmen. invesmen of he form (ˆx ic, 0, 0), (0, ˆx im, 0), and (0, 0, ˆx e ). Then we have ˆx ic = Ȳr x ic Ŷ r Specifically, consider changes in and similarly for ˆx im and ˆx e. This observaion gives us ha he change in TFP growh ĝ Z ha would arise from a change in oal innovaive invesmen Ŷr focused enirely on an individual caegory of innovaive invesmen (ˆx ic, 0, 0) is bounded by and similarly for ˆx im and ˆx e. ĝ Z (ḡ Z G(0, x im, x e )) Ȳr x ic Ŷ r We look o calculae hese bounds and he corresponding exac elasiciies as funcions of observables below. 4.5 Exac Elasiciies Given he funcional form of he ransiion law for TFP in equaion (5), we can obain formulas for hese individual elasiciies ha show he relaionship beween he exac elasiciies and hese bounds. Specifically, hese elasiciies are given by G ic x ic = 1 exp((ρ 1)ḡ Z ) exp((ρ 1)G(0, x im, x e )) ζ ( x ic ) x ic ρ 1 exp((ρ 1)ḡ Z ) ζ( x ic ) ζ(0) G im x im = 1 ρ 1 exp((ρ 1)ḡ Z ) exp((ρ 1)G( x ic, 0, x e )) h ( x im ) x im exp((ρ 1)ḡ Z ) h( x im ) (10) (11) 21

22 G e x e = 1 exp((ρ 1)ḡ Z ) exp((ρ 1)G( x ic, x im, 0)) ρ 1 exp((ρ 1)ḡ Z ) Thus, our assumpions ha ζ(x) and h(x) are concave and ha h(0) = 0 give us igher bounds on hese elasiciies han hose in proposiion 2. Tha is, for example, since 1 exp((ρ 1)ḡ Z ) exp((ρ 1)G(0, x im, x e )) ρ 1 exp((ρ 1)ḡ Z ) < ḡ Z G(0, x im, x e ) for ρ > 1, hese elasiciies lie sricly below he bounds above unless we se ρ = 1. As ρ, hese elasiciies converge o zero. 4.6 Exac Elasiciies and Counerfacual Size of Producs To calculae hese exac individual elasiciies, we need o idenify he concaviy of he funcions ζ(x ic ) and h(x im ). We can bound hese individual elasiciies, however, by expressing he erms 1 exp((ρ 1)ḡ Z ) exp((ρ 1)G(0, x im, x e )) ρ 1 exp((ρ 1)ḡ Z ) ec. in erms of baseline and counerfacual sizes of produc caegories. We do so in he nex proposiion. Define S ic (0) (1 δ M ) ζ(0) exp((ρ 1)ḡ Z ) = F ic exp(ḡ M ) exp((ρ 1)ḡ Z ) ζ(0) where he second equaliy follows from he idenificaion of parameers in daa on firm dynamics as described above. Proposiion 3. We have he following bounds on individual elasiciies. G ic x ic 1 ( Sic ρ 1 S ic (0)) ( S im δ S ) ic Fim F ic G im x im 1 ρ 1 G e x e = 1 ρ 1 ( S e δ S ic F ic Fe ) Proof. These bounds follow immediaely from equaions (10), (11), and (12), he concaviy of h(x) and ζ(x), he assumpion ha h(0) = 0, and he definiion of S ic (0). Noe ha equaion (15) is an equaliy because we have a linear enry echnology. (12) (13) (14) (15) 22

23 Noe ha if we impose he resricion as in GHK ha incumben firms do no suffer regress of z on coninuing producs even if hey do no inves in improving hese producs (i.e. ζ(0) = 1), we hen have from (13) he bound ha G ic x ic 1 ( S ic F ) ic exp(ḡ M ) ρ 1 exp((ρ 1)ḡ Z ) (16) This bound can be evaluaed given he daa we assume is available o he economis. Alernaively, as we argue in Corollary 6, if he economis is able o esimae he elasiciy G ic x ic using oher daa, one can hen use (13) o pu an upper bound on he value of ζ(0). We do so in our calibraion secion below. The resul in equaion (15) shows ha he only source of uncerainy in idenifying he elasiciy of TFP growh wih respec o enry given he daa we have assumed he economis o have is he parameer δ. Likewise, he resul in equaion (14) allows us o bound he elasiciy of TFP growh wih respec o invesmen by incumbens in acquiring new producs given knowledge of δ. If we se δ a is minimum value of 0, so ha here is no business sealing as in an Expanding Varieies model, we have G e x e = 1 ρ 1 S e (17) This means ha a model wih δ = 0 ha is calibraed o a baseline growh TFP growh rae of ḡ Z =.01, ρ = 4 and aggregae size of enering firms of S e >.03 implies an elasiciy a leas as large as 0.01 as long as he oher wo derivaives are non-negaive. Given he bound in proposiion 1, a reasonably calibraed Expanding Varieies model mus imply ha here is some social depreciaion of innovaion expendiures Noe ha he corresponding elasiciy of aggregae TFP o a change in innovaive invesmen devoed enirely o invesmen in enry when δ = 0 is hen given by G e x e Ȳ r x e = 1 ρ 1 S e Ȳ r x e (18) where x e /Ȳr is he share of innovaive invesmen underaken by enering firms. We use daa on he value of firms inangible capial o esimae his share of innovaive invesmen by enering firms below. In he nex secion, we urn o using daa on he value of firms o idenify he levels of invesmen x ic /Ȳr, x im /Ȳr, and x e /Ȳr. We hen use equilibrium condiions o idenify ζ ( x ic ) and h ( x im ), which allows us o fully idenify he desired elasiciies up o uncerainy over δ. 23

24 5 Firm Values and he Equilibrium Levels of Innovaive Invesmen We now consider he feaures of he model ha can be idenified if he economis requires ha he model also mach daa on he allocaion of facor paymens observed in GDP and daa on he value of he inangible capial in firms relaive o GDP. We assume hroughou ha here are no axes or subsidies in he observed equilibrium. A full calibraion exercise would have o ake ino accoun he impac of axes and subsidies on measures of firm value. We inend his secion o demonsrae he mehod for using daa on firm value o discipline he model-implied impac elasiciies of aggregae TFP growh wih respec o changes in innovaive invesmen). Here we consider he older noion of GDP as equal o he sum of expendiures on consumpion and invesmen in physical capial. This noion of GDP corresponds o C + K +1 (1 δ K )K = Y = Z K α L 1 α p Sandard argumens imply ha fracion α/µ of GDP is paid o physical capial as eiher explici or impued renals, fracion (1 α)/µ is paid as compensaion o producion labor L p. The remaining fracion (µ 1)/µ of oupu is available o owners of firms in he form of variable profis. A porion of hese variable profis are spen on firms innovaive invesmens. The cos of hese innovaive invesmen is paid o research labor. Noe ha o mach he model implicaions for compensaion of research labor o daa from he Naional Income and Produc Accouns, one mus impue compensaion of research labor in he form of swea equiy as in McGraan and Presco (2010). We assume ha he compensaion of research labor W L r is equal o he oal value of research oupu P r Y r where P r represens he price of he research good in erms of consumpion. Thus, he oal compensaion of labor is given by W L p + W L r. The aggregae flow of variable profis paid o firm owners in excess of aggregae coss of innovaive invesmens is hen [ ] µ 1 S r Y (19) µ where S r = P ry r Y denoes he research inensiy of he economy. pah. We now urn o he equaions ha we use o mach daa on firm value on a balanced growh 24

25 5.1 The value of a coninuing produc for an incumben firm We guess ha a produc of size s a adds value V s o he firm ha produces his produc a dae. We also guess ha in equilibrium, invesmen by incumbens in improving heir own producs is direcly proporional o produc size s = exp((ρ 1)z)/, or x z (s) = x z s for all s. We hen have he following recursion for his value of a produc in erms of is variable profis o an incumben V s = µ 1 µ Y s P r x ic s + 1 V +1 (1 δ M ) ζ(x ic ) Zρ R +1 where R denoes he consumpion ineres rae beween and + 1. Using he noaion v V Y s exp(g Y +1 ) = Y +1 Y we have v = µ 1 µ P rx ic Y + exp(g Y +1) 1 + R v +1 S ic+1 (20) and in a BGP P r x ic Y = µ 1 µ [ 1 exp(ḡ ] Y ) 1 + R S ic v (21) When an incumben firm chooses invesmen x ic opimally, is choice saisfies he firs order condiion which, in a BGP is equivalen o P r = 1 V +1 (1 δ M ) ζ (x ic ) Zρ R +1 P r x ic Y = exp(ḡ Y ) 1 + R v (1 δ M ) ζ ( x ic ) x ic exp((ρ 1)ḡ Z ) (22) 5.2 The value of a new produc in faciliaing furher acquisiion of producs We guess ha each incumben firm invess a consan amoun of he research good per produc ha i owns a in acquiring new producs for ha incumben firm. Thus, if x im is he oal invesmen of his ype by incumbens, hen his invesmen per producs is x im (s) = x im /M independen of he size s of he produc. 25

26 We hen have he following recursion for he value of a produc o an incumben in erms of he added research capaciy i gives o ha incumben U = P rx im V +1 h(x im ) η i M 1 + R M +1 + Using he noaion and muliplying by M gives R (h(x im ) + (1 δ M )) U +1 u U M Y u = P rx im Y + exp(g Y +1) 1 + R [v +1 S im+1 + u +1 F i+1 ] where F i+1 F ic+1 + F im+1 is he fracion of producs ha produced by incumben firms a + 1. These include producs ha are new o he firm and producs ha are coninuing. or In a BGP, we hen have P r x im Y ū = P r x im Y + exp(ḡ Y ) 1 + R [ v Sim + ū F i ] = exp(ḡ [ Y ) 1 + R v S im 1 exp(g ] Y ) 1 + R F i ū (23) When an incumben firm chooses invesmen x im opimally, is choice saisfies he firs order condiion In a BGP, his is equivalen o P r x im Y [ ] P r = 1 V +1 η i 1 + R + U +1 M h (x im ) (24) +1 = exp(ḡ Y ) 1 + R [ v( S in + S is ) + ū M ] h(x im ) h ( x im ) x im M +1 h( x im ) or P r x im Y = exp(ḡ Y ) 1 + R [ v Sim + ū F im ] h ( x im ) x im h( x im ) (25) 26

27 5.3 Value of a new firm Finally, we have free enry condiion P r = 1 M 1 + R Muliplying by x e M and dividing by Y gives [ η e V +1 M +1 + U +1 ] (26) P r x e Y = exp(g Y +1) 1 + R [v +1 S e+1 + u +1 F e+1 ] In a BGP, his equaion becomes P r x e Y = exp(ḡ Y ) 1 + R [ v Se + ū F e ] (27) 5.4 Aggregae Value of Firms Summing ogeher equaions (21), (23), and (27) gives he sandard formula [ µ 1 [ v + ū] = µ ] [ S r 1 exp(ḡ ] 1 Y ) 1 + R (28) where v + ū is oal firm value relaive o GDP and, from equaion (19), µ 1 µ S r is he fracion of GDP ha is paid as profis in excess of innovaive invesmens o owners of firms. This formula values he aggregae sock of inangible capial of firms in he economy as a perpeuiy ha grows a he same rae as consumpion in he economy. Under he sandard decenralizaion in which firms manage he sock of physical capial, we hen have ha he oal value of all firms in he economy relaive o GDP is given by K Y + v + u equal o he sum of he values of he angible and inangible capial wihin firms. 5.5 Implicaions of daa on firm value for he level of each caegory of invesmen We now consider he model s implicaions for he division of firm value ino componens v and u and he division of innovaive invesmen ino componens S r = P rx ic Y + P rx im Y + P rx e Y 27

28 To do so, assume ha one has daa on he division of firm inangible value ino componens v/( v + ū) and ū/( v + ū). This spli of oal firm value may be difficul o idenify in daa. We are able o bound his fracion v/( v + ū) above and below by he requiremen ha boh x ic and x im are non-negaive. Proposiion 4. In equilibrium, on a balanced growh pah, he fracion of inangible firm value v/( v + ū) mus lie in he inerval [ 1 exp(g Y ) 1+ R F i ] [ 1 exp(g Y ) 1+ R ( F i S im ) ] v µ 1 v + ū µ [ µ 1 µ S r ] [ ] 1 exp(g Y ) 1+ [ R 1 exp(ḡ Y ) S ] (29) 1+ R ic Proof. If one has such daa on he division of firm value, hen equaion (21) deermines P r x ic /Y. To ensure ha he implied value of P r x ic /Y 0, we mus have 0 µ 1 [ 1 exp(g ] Y ) µ 1 + R S ic [ 1 exp(g ] Y ) 1 + R S ic v ( v + ū) v + ū v µ 1 ( v + ū) v + ū µ µ 1 v v + ū 1 µ [ 1 exp(g Y ) S ] v + ū 1+ R ic and hus he second inequaliy above follows from equaion (28). Likewise, given a division of firm value, hen equaion (23) deermines P r x im /Y. To ensure ha P r x im /Y 0, we mus have 0 exp(g Y ) 1 + R S v im v + ū [ 1 exp(g ] Y ) 1 + R F i which implies he firs inequaliy above. [ 1 exp(g ] ( Y ) 1 + R F i 1 v [ 1 exp(g Y ) 1 + R ( F i S im ) ] v + ū v v + ū ) This proposiion implies ha any calibraion of oal inangible value of firms relaive o GDP, v + ū, ogeher wih a division of ha value ha lies in he bounds se in proposiion 4, we have P r x ic /Ȳr given from equaion (21) and P r x im /Ȳr given from equaion (23). The value of P r x e /Ȳr hen follows from equaion (27) and is immediaely non-negaive. 28

29 5.6 Implicaions of opimaliy of invesmen for he elasiciy of TFP growh wih respec o changes in invesmen in improving own producs From equaion (6), we see ha we mus idenify he derivaive ζ ( x ic ) o idenify he elasiciies ha we desire. We do so using our model s implicaions for seady-sae firm value and he assumpion ha invesmen in coninuing producs is chosen opimally in seady-sae given in equaions (21) and (22). Specifically, equaion (22) idenifies he desired derivaive up o knowledge of v and he scale of his caegory of innovaive invesmen from he condiion ha he privae reurn o innovaive invesmen in coninuing producs is equal o he ineres rae. We hen use our calibraion of he scale of firms inangible value v + ū and is division beween v and ū o deermine v and he valuaion equaion (21) o deermine he scale of his caegory of innovaive invesmen. These argumens give he following proposiion. Proposiion 5. Daa on firm value in seady-sae ogeher wih he condiion ha innovaive invesmen in coninuing producs be privaely opimal gives G ic x ic = 1 P r x ic Y ρ 1 exp(ḡ Y ) 1+ R v (30) Thus, he elasiciy of aggregae TFP growh wih respec o a change in aggregae innovaive invesmen direced enirely a invesmen by incumben firms in improving heir own producs is given by [ Ȳ r G ic x ic = 1 x ic ρ 1 S r exp(ḡ Y ) 1+ R Proof. Equaion (30) follows direcly from equaions (6) and (22). We have Ȳ r Y = S r x ic P r x ic Wih equaion (30), we hen have he resul (31). Noe ha his procedure allows us o idenify his elasiciy G ic x ic wihou knowledge of he exen of business sealing. Noe as well ha we have he following corollary ha allows us o use firm value o place an upper bound on he counerfacual value ζ(0), i.e., he modelimplied deerioraion of produciviy for coninuing producs ha would occur in he absence of innovaive invesmen by incumben firms in hose producs. 29 ] 1 v (31)