Optmal Growth Through Product Innovaton Rasmus Lentz Unversty of Wscons-Madson and CAM Dale T. Mortensen Northwestern Unversty, IZA, and NBER Aprl 26, 26 Abstract In Lentz and Mortensen (25), we formulate and estmate a market equlbrum model of endogenous growth through product nnovaton n the sprt of Klette and Kortum (24). In ths paper, we provde a quanttatve soluton to the socal planner s problem n the modeled envronment. We fnd that the optmal growth rate s over three tmes larger than ts value n market equlbrum and that the assocated welfare gans along a transton path from the market equlbrum soluton to the optmal steady state s equvalent to about 47% of consumpton. JEL Classfcaton: E22, E24, J23, J24, L11, L25 Keywords: Optmal growth, planner s problem, product nnovaton, nnovaton spll overs, creatve-destructon externalty. 1
1 Introducton In Lentz and Mortensen (26), we formulate and estmate a structural market equlbrum model of growth through product nnovaton. The model s an extended verson of that proposed by Klette and Kortum (24) orgnally desgned to explan the relatonshps between nnovaton nvestment and the sze dstrbuton of frms. Ther framework n turn s an elaboraton of the Grossman and Helpman (1991) model of endogenous growth through creatve-destructon. In our verson of the model, frms dffer wth respect to the qualty of the ntermedate products they create as a consequence of nvestment n research and development (R&D). We fnd that heterogenety n ths sense s needed to explan the sze dstrbuton of frms and the dstrbuton of labor productvty observed n our panel data of Dansh frms. One mportant mplcaton of ths form of heterogenety s that labor reallocaton to faster growng frmsthatcreatemoreproftable hgher qualty ntermedate products plays an mportant role n determnng the aggregate growth rate. The purpose of ths paper s to explore the model s quanttatve welfare mplcatons. Namely, we formulate and compute the socally optmal R&D strategy for the modeled envronment. The market equlbrum soluton need not be socally optmal for three dfferent reasons spelled out n Grossman and Helpman (1991). Frst, product nnovators have monopoly power and use t to set prces above the margnal cost of producton. Second, every nnovaton replace an older verson of some product and by dong so truncates the stream of quas rents accrung to ts creator, whch adversely affects the ncentve to nnovate. Fnally, because each new mprovement bulds on past technology, nnovaton has a postve "spll-over" effect on future productvty whchsnotfullycapturedbythennovatornamarketequlbrum. Thenet devaton of the equlbrum growth rate from that whch s socally optmal s unclear. One of the contrbutons of a quanttatve equlbrum model s ts ablty to reflectlghtontherelatvemagntudesoftheseeffects. In ths paper, we begn by formulatng and characterzng the market soluton to the estmated model and the soluton to the obvous planner s problem n the envronment modelled. We fnd that the planner s optmal nnovaton nvestment strategy dffers sgnfcantly from that observed n equlbrum. Although frms that produce better products nvest more n R&D, all types have an ncentve to make a postve nvestment n market equlbrum. In contrast, almost all of the R&D nvestment n the optmal soluton s make 2
by an elte few frms, those that can create products of the hghest qualty. The result s the consequence of the fact that the "spll over" externalty offset the "product destructon" externalty only f the qualty mprovement emboded n an nnovaton s suffcently large. The optmal growth rate s over three tmes larger than the market equlbrum value. The welfare gans attrbutable to the optmal entry and R&D nvestment strateges are also large. Indeed, the typcal household would be wllng to forego up to almost 47% of consumpton along the optmal path n order to adopt the optmal plan. Although the frstbestsolutonrequres margnal cost prcng, we also show that there are substantal welfare gans to be had even f nnovators are granted patent rghts and allowed to set monopoly lmt prces. Indeed, only about 5 percentage ponts of the total welfare gan attrbutable to the optmal R&D nvestment strategy can be attrbuted to margnal cost prcng. 2 Growth Through Product Innovaton Frms come n an amazng range of shapes and szes. Ths fact cannot be gnored n any analyss of the relatonshp between frm sze and factor productvty. Furthermore, an adequate theory must account for entry, ext, and frm evoluton n order to explan observed sze dstrbutons. Klette and Kortum (24) construct a stochastc model of frm product nnovaton and growth that s consstent wth stylzed facts regardng the frm sze evoluton and dstrbuton. In Lentz and Mortensen (25), we fnd that an extenson of ther model that allows for cross frm heterogenety n the qualty of nnovatons s needed to explan our Dansh data. 2.1 Preferences and Technology The utlty of the representatve household at tme t s gven by U t = Z t ln C s e ρ(s t) ds (1) where ln C t denotes the nstantaneous utlty of aggregate consumpton at date t and ρ represents the pure rate of tme dscount. Each household s free to borrow or lend at nterest rate r t. Nomnal household expendture at date t s E t = P t C t. Optmal consumpton expendture must solve the 3
dfferental equaton Ė/E = r t ρ. Followng Grossman and Helpman (1991),wechoosethenumeraresothatE t =1for all t wthout loss of generalty, whch mples r t = r = ρ for all t. Note that ths choce of the numerare also mples that prce of the consumpton good, P t, falls over tme at a rate equal to the rate of growth n consumpton. The quantty of consumpton produced s determned by the quantty and qualty of the economy s ntermedate nputs. Specfcally, there s a unt contnuum of nputs and consumpton s determned by the producton functon ln C t = Z 1 ln(a t (j)x t (j))dj =lna t + Z 1 ln x t (j)dj (2) where x t (j) s the quantty of nput j [, 1] at tme t, A t (j) s the productvty of nput j at tme t, anda t represent aggregate productvty. The level of productvty of each nput and aggregate productvty are determned by the number of techncal mprovements made n the past. Specfcally, A t (j) =Π J t(j) =1 q (j) and ln A t Z 1 ln A t (j)dj. (3) where J t (j) sthenumberofnnovatonsmadennputj up to date t and q (j) > 1 denotes the quanttatve mprovement (step sze) n productvty attrbutable to the th nnovaton n product j. Innovatons arrve at rate δ whch s endogenous but the same for all ntermedate products. The model s constructed so that a steady state growth path exsts wth the property that consumpton output grows at a constant rate, equal to the rate of productvty growth, whle the ntermedate good quanttes produced and the nnovaton frequences are statonary. As a consequence of the law of large numbers, the assumpton that the number of nnovatons to date s Posson wth arrval frequency δ for all ntermedate goods mples ln C t = lna t + Z 1 = δe ln(q)t + ln x(j)dj = Z 1 ln x(j)dj. Z 1 J t (j) X ln q (j)dj + =1 Z 1 ln x(j)dj (4) where EJ t (j) =δt for all j s the expected number of nnovatons per ntermedate product over a tme perod of length t and E ln(q) R 1 P 1 Jt (j) J t(j) =1 ln q (j)dj 4
s the expected mprovement n productvty per nnovaton. In other words, consumpton grows at the rate of growth n productvty whch s the product of the creatve-destructon rate and the expected log of the sze of an mprovement n productvty nduced by an nnovaton. 2.2 The Behavor of a Frm As a consequence of enforceable patents, each ndvdual frm s the sole suppler of the products t created n the past that have survved to the present. The prce charged for each s lmted by the ablty of supplers of prevous versons to provde a substtute. In Nash-Bertrand equlbrum, any successful nnovator takes over the market for ts good type by settng the prce just below that at whch fnal good producers are ndfferent between the new more productve product suppled by the nnovator and the alternatve suppled by the prevous provder. The prce charged s the product of the relatve qualty of the nnovaton and the prevous producer s margnal cost of producton. Gven the symmetry of demands for the dfferent good types and the assumpton that future qualty mprovements are ndependent of the type of good, one can drop the good subscrpt wthout confuson. Because quanttes along the equlbrum growth path are constant, the tme subscrpt can be dropped as well. Labor and captal, n fxed proportons, are used n the producton of ntermedate nputs to the fnal goods producton process. Labor productvty s the same across all ntermedate products and s set equal to unty. The requred captal expressed n unts of output, a constant κ, s also the same for all products. The operatng proft per unt obtaned from supplyng an ntermedate product s p(1 κ) w whch mples that the lowest prce that the prevous suppler s wllng to charge, that whch yelds no proft, s w/(1 κ). The qualty leader wll charge p = qw/(1 κ) because consumers are exactly ndfferent between buyng from the qualty leader at ths prce and the zero proft prce of the prevous suppler. Hence, product output supply and employment demand are both equal to x = 1 p = 1 κ wq. (5) and the gross proft assocated wth supplyng the good s π(q) =p(1 κ)x wx =(1 κ) 1 q 1. (6) 5
Followng Klette and Kortum (24), the dscrete number of products suppled by a frm, denoted as k, s defned on the ntegers and ts value evolves over tme as a brth-death process reflectng product creaton and destructon. In ther nterpretaton, k reflects the frm s past successes n the product nnovaton process as well as current frm sze. New products are generated by R&D nvestment. The frm s R&D nvestment flow generates new product arrvals at frequency γk. ThetotalR&Dnvestment costs wc(γ)k where c(γ)k represents the labor nput requred n the research and development process. The functon c(γ) s assumed to be strctly ncreasng and convex. Accordng to the authors, the mpled assumpton that the total cost of R&D nvestment s lnearly homogenous n the new product arrval rate and the number of exstng product, "captures the dea that a frm s knowledge captal facltates nnovaton." In any case, ths cost structure s needed to obtan frm growth rates that are ndependent of sze as typcally observed n the data. The market for any current product suppled by the frm s destroyed by the creaton of a new verson by some other frm, whch occurs at the rate δ. Belowwerefertoγ as the frm s product nnovaton rate and to δ as the aggregate creatve-destructon rate faced by all frms. The frm chooses the creaton rate γ tomaxmzetheexpectedpresentvalueoftsfuturenetproft flow. Frms dffer wth the respect to the expected qualty mprovement that ther products offer. Specfcally, there are =1, 2,...n types of frms and the dstrbuton (c.d.f.) of qualty, denoted as F (q), s stochastcally decreasngnthetypendex;thats,f (q) F +1 (q) for all q 1. Hence, the ndex reflects the rank order of the types by the expected qualty of ther nnovatons. The value of the frm of type that currently markets k products s the soluton to the asset prcng equaton ½ ¾ [E {π(q)} wc(γ)] k + γk[v rv k () =max k+1 () V k ()] γ (7) +δk[v k 1 () V k ()] where E {π(q)} = R π(q)df (q) s the expected gross proft flow obtaned per product lne a frm of type. Hence, the frsttermontherghtsdeof(7)s the total net proft flow from supplyng t current products. The second term s the expected captal gan assocated wth the arrval of a new product lne. Fnally, the last term represents the expected captal loss assocated wth the possblty that one among the exstng product lnes wll be destroyed. 6
The unque soluton to (7) s proportonal to the number of product lnes. Formally, ½ ¾ E {π(q)} wc(γ) V k () =k max (8) γ r + δ γ as one can verfy by substtuton. Consequently, any postve optmal choce of the product creaton rate for a type q frm must satsfy ½ ¾ wc E {π(q)} wc(γ) (γ )=V k+1 () V k () =max. (9) γ r + δ γ The second order condton, c (γ) >,thefactthatproft π s ncreasng n q, and the assumpton that the qualty c.d.f. F (q) s stochastcally decreasng n mply that the a frm s creaton rate ncreases wth ts product qualty rank. That s γ γ +1, =1,..., n. 2.3 Frm Entry and Labor Market Clearng The entry of a new frm requres a successful nnovaton. Suppose that there are a constant measure m of potental entrants. The rate at whch any one of them generates a new product s γ and the total cost s wc(γ ) where the cost functon s the same as that faced by an ncumbent. The frm s type s unknown ex ante but s realzed mmedately after the arrval of an nnovaton. However, entry requres that the nnovaton provde a postve expected future ncome stream ex post. Snce the aggregate entry rate s η = mγ, the entry rate satsfes the followng free entry condton ³ η wc m = = nx µz =1 nx =1 q max γ max hv 1 (), df (q) φ (1) ½ E {π(q)} wc(γ) r + δ γ ¾ φ where φ s the probablty that the entrant wll turn out to be of type and the second equalty s mpled by the fact that the value of adopton, V 1 (q) as defned by equaton (8), s always non-negatve and lnear n gross proft per product. Obvously, n ths formulaton learnng one s type takes no tme, whch s unrealstc but a useful abstracton for the purposes of ths paper. 7
There s a fxed measure of avalable workers, denoted by, seekng employment at any postve wage. In equlbrum, these are allocated across producton and R&D actvtes, those performed by both ncumbent frms and potental entrants. Snce the number of workers employed for producton purposes per product of qualty q s x =1/p =(1 κ)/qw from equatons (5) and (6), the number of workers demanded for producton of a product of qualty q s x (q) =(1 κ)/qw. The number of R&D workers employed per product by ncumbent frms of type s R () =c(γ ). Because each potental entrant nnovates at frequency η/m, the aggregate number of workers engaged by all m n R&D s E = mc(η/m). Hence, the equlbrum wage satsfes the labor market clearng condton = = nx =1 =1 X ([E x (q)k} + R ()k] M (k)) φ + E (11) k=1 Ã nx ½ ¾! 1 κ X E + c(γ()) km (k) φ qw + mc(η/m) k=1 where M (k) represents the mass of frms of type that supply k products. 2.4 The Dstrbuton of Frm Sze Once a frm enters, ts sze as reflected n the number of product lnes suppled, evolves as a brth-death process. As the set of frms wth k products at a pont n tme must ether have had k products already and nether lost nor ganed another, have had k 1 and nnovated, or have had k +1 and lost one to destructon over any suffcently short tme perod, the equalty of the flows nto and out of the set of frms of type wth k>1 products requres γ (k 1)M (k 1) + δ(k +1)M (k +1)=(γ + δ)km (k) where M k () s the steady state mass of frms of type that supply k products. Because an ncumbent des when ts last product s destroyed by assumpton but entrants flow nto the set of frms wth a sngle product at rate η, φ η +2δM (2) = (γ + δ)m (1) where φ s the fracton of the new entrants who are of type. Brths must equal deaths n steady state and only frms wth one product are subject to 8
death rsk. Therefore, φ η = δm (1) and M (k) = k 1 k γ M (k 1) = ηφ δk ³ γ k 1 (12) δ by nducton. The sze dstrbuton of frms condtonal on type can be derved usng equaton (12). Specfcally, the total mass of frms of type s M = X M (k) = φ η δ k=1 = η δ ln µ δ δ γ X k=1 δφ γ. 1 k ³ γ k 1 (13) δ where convergence requres that the aggregate rate of creatve destructon exceed the creaton rate of every ncumbent type,.e., δ>γ. Hence, the fracton of type frm wth k product s M (k) M = 1 k ³ ln γ k δ. (14) δ δ γ Ths s the logarthmc dstrbuton wth parameter γ(q)/δ. 1 Consstent wth the observatons on frm sze dstrbutons, that mpled by the model s hghly skewed to the rght. By equaton (14), the mean of the frm sze dstrbuton condtonal on product proftablty s E{k } = X k=1 km (k) M = γ δ γ ³, (15) δ ln δ γ whchonecanshowsanncreasngfunctonofγ.asthefrst order condton for optmal nvestment n R&D, equaton (9), mples that the product creaton rate γ ncreases wth expected proftablty E {π(q)}, the expected number of products suppled ncreases wth expected productvty rank. That s E{k } E{k +1}. 1 Ths result s n Klette and Kortum (1992). We nclude the dervaton here smply for completeness. 9
2.5 Creatve-Destructon and Growth Because the number of products s fxed, the rate of creatve-destructon s the sum of the entry rate and the creaton rates of all the ncumbents. As thenewproductarrvalrateofafrm of type q wth k products s γ(q)k and the measure of such frms s M k (q), δ = η + nx X γ =1 k=1 km (k)φ. (16) Fnally, the contrbuton to growth of a new entrant and a current ncumbent s ln q, the growth rate n consumpton s g = δe {ln(q)} = η nx E {ln(q)}φ + =1 nx X γ E {ln(q)} km (k)φ (17) =1 k=1 2.6 Market Equlbrum A steady state market equlbrum s a trple composed of a labor market clearng wage w, entry rate η, and creatve destructon rate δ together wth an optmal creaton rate γ(q) and a steady state sze dstrbuton M k (q) for each type that satsfy equatons (9), (1), (11), (12), and (16) provded that γ(q) < δ,for every q n the support of the entry dstrbuton. Lentz and Mortensen (24) provde a proof of exstence. 3 The Socal Planner s Problem 3.1 Formulaton As already stated, a frm s type s not known pror to entry but s revealed and common knowledge afterward. Although ths assumpton abstracts from the obvous learnng problem, dong so allows for a much less complcated analyss. The socal planner, then, chooses non-negatve tme paths for the producton rate, x, and the rate of new product creaton, γ, per product lne for each frm type =1,..., n, the rate of product nnovaton, γ,andtheaggregate creatve-destructon rate δ to maxmze the present dscounted utlty of the representatve household s consumpton subject to the fact that there are a fxed number of ntermedate products, a labor resource constrant, and 1
laws of moton for the state varables. Under symmetrc nformaton, both the frm and the planner observes the realzed productvty of any nnovaton. Gven ths nformaton, the planner fnds t n her nterest to screen nnovatons before adopton. Let Φ (q) represent an ndcator functon whch takes on the value unty f the planner adopts an nnovaton of realzed qualty q created by a frm of type. Of course, the ndcator s zero f the decson s not to adopt. Specfcally, the planner s strategy determned (x,γ, Φ (q)) for =1,..., n, γ.andδateach date. It maxmzes the expected present value of the representatve consumer s utlty stream subject to a set of constrants. Formally, the crteron s Z Z " ln C t e rt dt = ln A t + X # ln (x (t)) K (t) e rt dt where A represents aggregate productvty, K = P k=1 km (k) s the mass of products suppled by type frms, and x s the quantty of each product suppled by a frm of type and the t ndex ndcates the date of each. The constrants follow: Employment cannot exceed the avalable labor supply, X [x + c(γ )] K + mc (γ ). As entry requres both nnovaton and adopton, the aggregate entry rate s X η = mγ E {Φ (q)}φ where mγ s the frequency wth whch the mass of potental entrants nnovated and E {Φ (q)} represents the fracton of the nnovatons by frm s of type that are adopted. The assumpton that there s a contnuum of potental nnovators of each type and the usual appeal to the law of large numbers justfes the substtuton of the expected fracton for the realzed one. The creatve-destructon constrant s δ = η + X γ E {Φ (q)}k where K s the fracton of products suppled by frms of type.the law of moton for aggregate productvty s d ln A g = γ dt m X E {ln (q) Φ (q)}φ + X γ E {ln (q) Φ (q)}k, 11
where agan the expectaton s used by an appeal to the law of large numbers. The mass of products suppled by each frm type evolves accordng to dk dt = K = mγ E {Φ (q)}φ +(γ E {Φ (q)} δ)k,=1,...,n. (18) The planner s problem s a relatvely standard one n dynamc control. The constrant augmented present value Hamltonan for the problem can be wrtten as H = lna + X ln x K à +ω X! [x + c(γ )] K mc (γ ) à X +τ δ mγ E {Φ (q)}φ X! γ E {Φ (q)}k à X +λ mγ E {ln (q) Φ (q)}φ + X! γ E {ln (q) Φ (q)}k + X v [mγ E {Φ (q)φ + γ E {Φ (q)}k δk ] where ω and τ are multplers assocated respectvely wth the labor supply constrant and the creatve-destructon constant whle λ s the shadow prce of the state varable ln A and v s the shadow prce or co-state varable assocated wth K, =1,...,n. In addton, the necessary transversalty condtons requre that λe rt and v e rt converge to zero as t. As the optmal controls maxmze the Hamltonan gven state and costate varables, the frst order necessary condtons for all the contnuous choce varables are µ H 1 = ω K (wth = f x > ), =1,...,n. x x H γ = [λe {ln (q) Φ (q)} +(v τ) E {Φ (q)} ωc (γ )] K (wth = f γ > ), =1,..,n. 12
H γ = " # X (λe {ln (q) Φ (q)} +(v τ)) E {Φ (q)}φ ωc (γ ) m (wth = f η>) H δ = τ X v K = (wth = f δ>) TheassumptonthatthecostofR&Dsconvexnthennovatonratessuffcent to guarantee that the second order necessary condtons are satsfed. Fnally, the optmalty condton for the dscrete choce of whether or not to adopt an nnovaton can be represent as Φ (q) = 1 f and only f H Φ (q) = [λ ln(q)+v τ][mγ φ + γ K ] F (q) for all and q. Theco-state(Euler)equatonsare H ln A = 1 = rλ λ. H K = lnx ω [x + c(γ )] + λγ E {ln (q) Φ (q)} +(v τ)e {Φ (q)}γ δv = rv v, =1,...,n. The fact that the crteron, laws of moton, and constrants are all lnear n the states s suffcent to guarantee that a unque soluton to the problem exsts. (See Kamen and Schwartz (1991).) The transversalty condton lm t λe rt =requres that the shadow prce of log productvty equal the nverse of the dscount rate at all dates, 1 = r. (19) λ The optmal strategy for screenng new nnovatons reflects the externaltes present. As v s the shadow value of a product developed by a frm of type and τ s the expected value of the product t replaced, the optmal adopton polcy requres that sum of the value of t contrbuton to future 13
productvty growth, represented by λe {ln (q)}, plus the present value of the nventor s future proft from both producton and nnovaton actvty must exceed the expected value of the product that t wll replace. As a consequence, the optmal adopton strategy has the followng reservaton property where Φ (q) =1f and only f ln q τ v λ τ = X = r(τ v ) (2) v K (21) Snce τ v > for at least one s n>1, t follows that some nnovatons by frm types wth low product value v should not be adopted even when they represent mprovements n qualty n the sense that ln(q) >. The screenng polcy mples that the aggregate entry, creatve-destructons and growth rates are X Z η = mγ Φ (q)df (q)φ (22) δ = η + nx =1 1 Z 1 Φ (q)df (q)γ K. Z g = X (γ mφ + γ K ) ln (q) Φ (q)df (q). 1 Under the reasonable regularty condtons c () = c() =, one can rewrte the frst order necessary condtons for a soluton as ωx = 1, =1,...,.. (23) Z ωc (γ ) = 1 max hln q r(τ v ), df (q), =1,..., n. r 1 ωc (γ ) = 1 X Z max hln q r(τ v ), df (q)φ r = X ωc (γ )φ 1 because E {λ ln (q) Φ (q)} +(v τ) E {Φ (q)} (24) = 1 r E {Φ (q)} [E{ln q ln q r(τ v )} r(τ v )] = 1 r Z 1 max hln q r(τ v ), df (q) 14
Ths term, the product of the probablty of adopton and the expected present value of the future socal gan condtonal on adopton, s the expected gan n socal surplus attrbutable to an nnovaton created by a frm of type. Hence, the second equaton of (23) requres that the margnal cost of nvestment n R&D by an ncumbent of type equal the present value of expected future socal surplus condtonal on type and the thrd requres that the analogous condton hold for potental entrants under the assumpton that frm type s not yet known. As a corollary, t follows that the margnal cost of R&D nvestment by a potental entrant s equal to the expected margnal cost of nvestment by ncumbents taken wth respect to the ntal dstrbuton of types at entry, whch s the nterpretaton of the last equalty. The frst equaton of (23) mples that all nputs should be suppled at the same rate. From the labor supply constrant t follows that the common rate of producton s x = x = 1 ω = X ³ η c(γ )K mc for all. (25) m The optmalty of equal producton rates reflects the fact that the utlty functon s symmetrc across product types and that the margnal costs of producton are dentcal. By mplcaton, hgher qualty ntermedate nputs are generally under produced relatve to lower qualty nputs because demands are equal whle supplers of hgher qualty products charge a hgher lmt prce. Under the assumpton that c () = c() =, the co-state equaton for the value of product suppled by each frm type can be wrtten as v =(r + δ)v +1 ln(x) ω (γ c (γ ) c(γ )), =1,...,n (26) by usng the second equaton of (23) and equaton (21) to substtute the margnal cost the nnovaton rate for the expected return to an nnovaton. Because ln(x) ωx =ln(x) 1 s the proft earned by supplyng a product expressed n utlty terms and ω (γ c (γ ) c(γ )) s the net flow return to R&D actvty expressed n terms of utlty, the nstantaneous utlty of a product s the sum of these two sources of producer surplus and v s the present value of ths future utlty stream dscounted as the rate r + δ. The planner s problem can be decentralzed as follows: The planner controls product adopton by offerng to buy the rghts to each nnovaton created 15
by a type frm wth a payment equal to ln q/r + v less a "destructon tax" equal to τ. If ths dfference s negatve, the nnovator can choose not to sell but cannot adopt by law. The producton rghts to any nnovaton of postve value are then resold at aucton to many dfferent frms n order to nduce competton n the supply of each good. These frms sell at prces equal to margnal cost, ω, n compettve equlbrum. Under ths scheme, potental entrants and ncumbents have the correct ncentves for an optmal nvestment n R&D choce. 4 Numercal Solutons In ths secton, we use the model s parameter estmates reported by Lentz and Mortensen (25) to compare the quanttatve propertes of the market and the planner s solutons. We conclude the secton by computng the welfare gan that could be acheved by a swtch to the optmal soluton when the market steady state soluton characterzes the ntal condtons. 4.1 The Market Equlbrum Soluton A steady state compettve equlbrum soluton to the model satsfes the followng condtons: The nnovaton rate for each frm type solves the FOC ½ ¾ wc E {π(q)} wc(γ) (γ )=max (27) γ r + δ γ where the proft rate for each type s π(q) =(1 κ)(1 q 1 ), (28) the market wage w satsfes the labor market clearng condtons L Z = X ½ ¾ 1 κ E + X ³ η c(γ qw )+mc m = X 1 κ E {π(q)} + X ³ η c(γ w )+mc m (29) the entry rate η satsfes ts own FOC ³ η wc = X ½ ¾ E {π(q)} wc(γ) max φ m γ r + δ γ = X q wc (γ )φ. (3) 16
The steady state number of products suppled by each type s gven by K = ηφ. (31) δ γ The rate of creatve-destructon and the growth rate are δ = η + X γ(q )K. (32) and g = η X E {ln(q)}φ + X γ E {ln(q)}k. (33) The parameter estmates obtaned by Lentz and Mortensen (25) derved from Dansh frm data usng a smulated method of moments are reported n Table 1. In the verson of the model estmated, the R&D cost functon s assumed to take the power form c(γ) =c γ 1+c 1. The parameter Z s the average real value added per product lne and L s the total labor force. Snce real value added per product s normalzed at unty and the total number of product s set to unty as well n the model, the total labor supply per product lne s = L/Z. These estmates are condtonal on only three types (n =3)of frms. The ftted dstrbutons of product qualty condtonal on type s a three-parameter Webull. The mpled condtonal mean proft per product lne and log qualty are reported n the table as well as the parameter values. The case of four frm types was consdered but added lttle to the ft. Table 1: Model Parameter Values Cost scale parameter c 9.34 Interest rate r.5 Cost curvature parameter c 1 2.892 Wage rate w 19.24 Creatve-destructon rate δ.798 Labor supply L 44.89 Captal cost per product κ.425 Value added Z 1581.14 Incumbent mass M.689 Entrant mass m 1.358 Frm Types φ E {π(q)} E {ln q} 1.958.2254.6964 2.321.162.32 3.5841.54.97 Webull Parameters gamma beta eta 1 1.116.4228.7224 2 1.52.4228.97 3 1.17.4228.29 17
The extreme rght skew n the dstrbuton of frm types at entry reflects the fact few entrants create products of the hghest qualty. Indeed, over 9% of the entrants can be expected to create nnovaton of only margnal value. These frms mght be nterpreted as smply "mtators." The assocated vector of equlbrum nnovaton rates per year and vector of steady state shares of products by type mpled by the parameter values n Table 1 and equatons (27) - (31) are γ =.6736.241 and K =.3455.25375. (34).1622.457 Fnally, the growth rate obtaned usng equaton (33) s 1.985%, The dfferences between the dstrbuton of frm types at entry reported n Table 1 and the steady state dstrbuton of products across frm types reported n equaton (34) reflect the fact that frms that create hgher qualty products are more proftable and grow faster as a consequence. Specfcally, those that are expected to create hgher qualty products eventually supply relatvely more product lnes than at entry because they nnovate more frequently. Ths selecton process reflects a contnual reallocaton of employment from dyng frms to younger fast growng frms that create products of hgh qualty. The mpact of ths reallocaton process on growth can be measured usng the followng decomposton of the growth rate: g = η X ln(q )φ + X γ ln(q )φ + X γ ln(q )(K γ ). (35) Obvously, the frst term reflects the contrbuton of entry to the growth rate. The second term represents the contrbuton to the growth rate of ncumbent frms f there were no change n the relatve number of products suppled and, consequently, no need to reallocate workers across ncumbent frms. Fnally, the last term, the contrbuton to growth attrbutable to thefactthatmoreproftable frms grow faster, s assocated wth worker reallocaton. The values of the parameters and the steady state values of the nnovaton rates and the dstrbuton of products across frm types mply that entry accounts for 18.26% of the overall rate of growth whle the magntude of the last term reflects the fact that the process of reallocaton accounts for 57.46% of growth. In sum, over three-quarters of the growth rate s the consequence of entry and reallocaton. 18
4.2 The Steady State Soluton to the Planner s Problem A steady state soluton to the planner s problem s a vector of product values and a dstrbuton of products across types that satsfy the steady state condtons v = ln(x) 1+ω (γ c (γ ) c(γ )),=1,..,n. (36) r + δ and K = mγ φ,=1,..., n. (37) δ γ Of course, the producton rate, the nnovaton rates, the adopton ndcator, and the "destructon tax" satsfy equatons (23), (2) and (21). Table 2 provdes a comparson of the market equlbrum and the planner s steady state solutons. Because low qualty nnovatons are not adopted n the optmal soluton, the entry rate s much smaller than n the market equlbrum and most of the enterng frms are those that are expected to create hgh qualty products n the future. The large dfference between the expected qualty of product created by the hghest qualty frm type and the others mples that the nnovaton frequency of the hghest qualty type s hgher than n equlbrum whle the nnovaton frequences of the other two types s trval. As a consequence, the hghest qualty frm type, type 1, supples vrtually all the products n steady state and account for almost all of the growth. 2 The rate of entry n the planner s soluton s much lower than n market equlbrum because few potental entrants can create hgh qualty products and because potental entrants that create low qualty products are not allowed to enter. Although the optmal rate of creatve-destructon s larger thantsmarketequlbrumvalue,mostoftsaccountedforbyther&d nvestment of ncumbent frms. Fnally, the optmal steady state growth rate s over three tmes larger than ts equlbrum value because almost all nnovatons are by the hghest qualty type frms whch have a larger mpact on productvty growth. As total aggregate nvestment n R&D s also larger under the optmal polcy, the output rate per product lne s necessarly smaller than ts equlbrum value as a consequence of the labor supply constrant. 2 The contrbuton to the growth rate s 8.74 1 8, 19
Table 2: A Comparson of Equlbrum and Planner s SS Solutons Innovaton Rates by Type Equlbrum Planner s γ 1.6736.8579 γ 2.241.111 γ 3.1622.286 Product Fractons by Type K 1.3455.99812 K 2.25375.198 K 3.457.12 Entry rate η.4415.513 Creatve-destructon rate δ.7979.976 Growth Rate g.1985.6316 Producton rate x.273.215 5 Transton Dynamcs The comparatve results reported n Table 2 suggest the adopton of the planner s nnovaton strategy wll yeld large welfare gans. To obtan a quanttatve measure of the gan, one must take account of the entre transton path from the market equlbrum to the optmal steady state. The transton dynamc s lkely to be mportant for two reasons. Frst, an ntal consumpton sacrfce wll be requred to acheve the hgher steady state growth promsed bytheoptmalr&dnvestmentstrategy. Second,thefactthatonlyasmall fracton of new frm brths have the ablty to create products of the hghest qualty and that frmsofthstypesupplyalmostalltheproductsnsteady state, suggest that the transton may take a long tme. Hence, suffcent patence s requred to realze a sgnfcant welfare gan from adoptng the entry and R&D nvestment polces. 5.1 The ODE System In general, the transton path s a soluton to the system of ordnary dfferental equatons K = mγ φ +(γ δ) K (38) (r + δ)v v = lnx 1+ω (γ c (γ ) c(γ )) 2
=1,..,n. The boundary condtons are the ntal values of the state varables, whch we wll take to be the equlbrum dstrbuton of product shares across frm types and the steady state values of the co-states. As we know, only frms able to create products of the hghest qualty contrbute to the supply of products and growth n the steady state. If thesamestruentranston,thenthedynamcscanbeapproxmatedby the soluton to a system of only two dfferental equatons nvolvng a sngle decson relevant state, the measure of products of the hghest qualty, K 1, and the dfference between ts value and that of any product suppled by one of the other types. As the soluton for v s the same for all >1 under the assumpton that γ =, τ = P v K = v n K n + v (1 K n ) where v = v for all >1. Hence, gven the defnton y 1 v 1 v, (39) the transtory dynamcs can be descrbed by the ODE system K 1 = mγ φ 1 +(γ 1 δ) K 1 = δ(1 K 1 ) (4) ẏ 1 = (r + δ)y 1 ω (γ 1 c (γ 1 ) c(γ 1 )) where γ 1,γ,δ,x,and ω are determned by the followng equatons: ωc (γ 1 ) = lnq 1 /r + y 1 (1 K 1 ) (41) c (γ ) = c (γ 1 )φ 1 δ = mγ φ 1 + γ 1 K 1 x = 1 ω = c(γ 1)K 1 mc (γ ). Because the state of the system, the fracton of products suppled, K 1, converges to unty, ts unque postve steady state, and because the other characterstc roots of the system exceed the nterest rate r, the steady state soluton s a saddle and the transversalty condtons mply that the soluton ofntereststhestablemanfold,theunquetrajectorythatconvergestothe steady state assocated wth the gven ntal value of K 1. One can easly fnd the numercal soluton to the problem usng any ordnary dfferental equaton solver. 5.2 The Transtory Soluton The tme paths of the creatve-destructon and growth rates for the soluton are all plotted n Fgure 1 and the assocated tme paths of the fracton of 21
.12.12.9 Annual Rates γ 1t, δ t g t.6.3 1 2 3 4 5 6 1 t Years 6 Fgure 1: Optmal Innovaton, Creatve-Destructon, and Growth Rates products suppled by the three frm types are llustrated n Fgure 2. In Fgure 1, γ 1t represents the nnovaton frequency of type 1 frms. The annual nnovaton frequency of the hghest qualty frm type jumps up to over 1% ntally. As llustrated n Fgure 2, the effect of the hgh nnovaton rate early n the transton s a contnual ncrease n the fracton of the products suppled by type 1 frms. Although the nnovaton rate falls toward ts steady state value throughout the transton phase, the rate of creatve-destructon and the growth rate both rse n response to the rsng product share of type 1 frms. Fnally, note that the transton to the steady state takes 6 years wth about half of the adjustment takng place n the frst 15 years. 5.3 Welfare Gan The paths of nstantaneous utlty for a market equlbrum and the planner s soluton are llustrated n Fgure 3. Specfcally, ln C t represents the tme path of the nstantaneous utlty, the log of consumpton, assocated wth a market equlbrum soluton whle ln C 1t s the utlty obtaned n the dynamc soluton to the planner s problem where ln C jt = g j t +ln(x jt ) (42) 22
1 1.8 Fracton of Products K 1t, K 2t, K 3t,.6.4.2 1 2 3 4 5 6 1 t Years 6 Fgure 2: The Dstrbuton of Products Suppled Over Frm Types from equatons (4), (17), and (25). Note that t takes about ten years for the utlty provded by the planner s strategy to domnate that of the steady state market equlbrum soluton. Stll, because the optmal growth rate converges to a value over three tmes that n the market equlbrum soluton mples, the eventual welfare gan s consderable. Of course, a contemporaneous comparson between the two utlty paths gnores tme dscountng. The fracton of consumpton that the typcal household would be wllng to forego at every pont along the optmal transton path n order to adopt the planner s strategy s a standard way to measure the welfare gan. Gven that θ represents the fracton of consumpton foregone, the resultng compensated utlty path realzed n each perod s ln C 2t = g 1 t +ln(x 1t )+ln(1 θ) (43) where θ s chosen so that Z ln C 2t e rt = Z ln C 1t e rt + ln(1 θ) r = Z ln C t e rt. (44) 23
1 1 Log Consumpton lnc t, lnc 1t, lnc 2t, 3 5 7 7 1 2 3 4 5 6 t Years 6 Fgure 3: Equlbrum (ln C ),Optmal (ln C 1 ),andcompensated(ln C 1 ) Utlty The computed value of the fracton s θ =.4687 and the assocated compensated utlty path s that represented as C 2t n Fgure 4. In other words, the welfare gan attrbutable to the optmal R&D strategy s equvalent to an addtonal 46.87% of consumpton per perod along the optmal transton path. Fgure 4 mples that the utlty path equvalent n present value to that obtaned n equlbrum yelds more consumpton only after the 24 th year nto the transton to steady state! In sum, the welfare gan from adoptng the optmal entry and R&D nvestment polcy s huge. 6 Concluson In our companon paper, Lentz and Mortensen (25), we show that a fully artculated structural model of frm evoluton through creatve destructon can explan the observed dstrbutons of value added, wage bll, and employment across frms found n Dansh panel data and ther shfts over tme. Here we show that the parameter estmates and aggregate equlbrum condtons have mportant mplcatons for growth. The mpled equlbrum growth rate s 1.98% annually. Because frms that create ntermedate good nnovatons 24
of hgher qualty grow faster, reallocaton of labor resources to the relatvely more rapdly growng frms accounts for 57% of aggregate growth n market equlbrum. However, the optmal growth rate s much larger (6.32%) because the socal planner weeds out "mtatons," nnovatons wth values that do not compensate for the destructon of the products they replace. Even after accountng for the transton from the market equlbrum to the new steady state assocated wth the optmal strategy, the mpled welfare gans, 47% of optmal consumpton, are large. References [1] Grossman, G. and E. Helpman (1991). Innovaton and Growth n the Global Economy. Cambrdge, Ma: MIT Press. [2] Klette, J., and S. Kortum (24). Innovatng Frms and Aggregate Innovaton, Journal of Poltcal Economy 112(5), 986-118. [3] Kamen, M.I., and N.L. Schwartz (1991). Dynamc Optmzaton: The Calculus of Varatons and Optmal Control n Economcs and Management, 2d edton. North-Holland. [4] Lentz, R., and D.T. Mortensen (25) An Emprcal Model of Productvty Growth Though Product Innovaton. IZA Dscusson Paper #1685 and NBER Workng Paper #111546. Revsed May 26. 25