The Shape of Production Functions and the Direction of Technical Change

Transcription

1 The Shape f Prductin Functins and the Directin f Technical Change Preliminary Cmments appreciated Charles I. Jnes * Department f Ecnmics, U.C. Berkeley and NBER chad@ecn.berkeley.edu chad March 19, 2004 Versin 0.5 This paper views the standard prductin functin in macrecnmics as a reduced frm and derives its prperties frm micrfundatins. The shape f this prductin functin is gverned by the distributin f ideas. If that distributin is Paret, then tw results btain: the glbal prductin functin is Cbb-Duglas, and technical change in the lng run is labr-augmenting. Krtum (1997) shwed that Paret distributins were necessary if search-based idea mdels were t exhibit steady-state grwth. Here we shw that this same assumptin delivers the additinal results abut the shape f the prductin functin and the directin f technical change. Key Wrds: Cbb-Duglas, Paret, Pwer Law, Labr-Augmenting Technical Change, Steady State Grwth JEL Classificatin: O40, E10 * I am grateful t Susant Basu, Francesc Caselli, Hal Cle, Xavier Gabaix, Dug Gllin, Pete Klenw, Rbert Slw, Alwyn Yung, and seminar participants at Arizna State University, the Chicag GSB, the San Francisc Fed, Stanfrd, U.C. Berkeley, UCLA, U.C. Davis, and U.C. Santa Cruz fr cmments. Sam Krtum prvided especially useful insights, fr which I am mst appreciative. Dean Scrimgeur supplied excellent research assistance. This research is supprted by NSF grant SES

2 2 CHARLES I. JONES 1. INTRODUCTION Where d prductin functins cme frm? T take a cmmn example, ur mdels frequently specify a relatin y = f(k, ) that determines hw much utput per wrker y can be prduced with any quantity f capital per wrker k. We typically assume the ecnmy is endwed with this functin, but what micrfundatins can be prvided t tell us mre abut it? Suppse prductin techniques are discvered, i.e. they are ideas. One example f such an idea wuld be a Lentief technlgy that says, fr each unit f labr, take k units f capital. Fllw these instructins [mitted], and yu will get ut y units f utput. The values k and y are parameters f this prductin technique. If ne wants t prduce with a very different capital-labr rati frm k, the Lentief technique is nt particularly helpful, and ne needs t discver a new idea apprpriate t the higher capital-labr rati. 1 Ntice that ne can replace the Lentief structure with a prductin technlgy that exhibits a lw elasticity f substitutin, and this statement remains true: t take advantage f a substantially higher capital-labr rati, ne really needs a new technique targetted at that capital-labr rati. One needs a new idea. Accrding t this view, the standard prductin functin that we write dwn, mapping the entire range f capital-labr ratis int utputs, is a reduced frm. It is nt a single technlgy, but rather represents the substitutin pssibilities acrss different prductin techniques. The elasticity f substitutin fr this glbal prductin functin depends n the extent t which new techniques that are apprpriate at higher capital-labr ratis have been discvered. That is, it depends n the distributin f ideas. But frm what distributin are ideas drawn? Krtum (1997) examined a search mdel f grwth in which ideas are prductivity levels that are drawn frm a distributin. He shwed that the nly way t get expnential 1 This use f apprpriate technlgies is related t Basu and Weil (1998).

3 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE 3 grwth in such a mdel is if ideas are drawn frm a Paret distributin, at least in the upper tail. This same basic assumptin, that ideas are drawn frm a Paret distributin, yields tw additinal results in the framewrk cnsidered here. First, the glbal prductin functin is Cbb-Duglas. Secnd, technlgical change is purely labr-augmenting in the lng run. In ther wrds, an assumptin Krtum (1997) suggests we make if we want a mdel t exhibit steady-state grwth leads t imprtant predictins abut the shape f prductin functins and the directin f technical change. Sectin 2 f this paper presents a simple baseline mdel that illustrates all f the main results f this paper. In particular, that sectin shws hw a specific shape fr the technlgy frntier prduces a Cbb-Duglas prductin functin and labr-augmenting technical change. Sectin 3 develps the full mdel with richer micrfundatins and derives the Cbb-Duglas result, while Sectin 4 discusses the underlying assumptins and the relatinship between this mdel and Huthakker ( ). Sectin 5 develps the implicatins fr the directin f technical change. Sectin 6 prvides a numerical example f the mdel, and Sectin 7 cncludes. 2. A BASELINE MODEL 2.1. Preliminaries Let a particular prductin technique call it technique i be defined by tw parameters, a i and b i. With this technique, utput Y can be prduced with capital K and labr L accrding t a (lcal) prductin functin Y = F (b i K, a i L). (1) We assume that F (, ) exhibits a lw elasticity f substitutin between its inputs and cnstant returns t scale in K and L. In additin, we make the

4 4 CHARLES I. JONES usual neclassical assumptin that F pssesses psitive but diminishing marginal prducts and satisfies the Inada cnditins. This prductin functin can be rearranged t give ( ) Y = a i L F bi K a i L, 1, (2) s that in per wrker terms we have y = a i F ( ) bi k, 1, (3) a i where y Y/L and k K/L. Nw, define y i a i and k i a i /b i. Then the prductin technique can be written as ( ) k y = y i F, 1. (4) ki If we chse ur units s that F (1, 1) = 1, then we have the nice prperty that k = k i implies that y = y i. Therefre, we can think f technique i as being indexed by a i and b i, r, equivalently, by k i and y i. The shape f the glbal prductin functin is driven by the distributin f alternative prductin techniques rather than by the shape f the lcal prductin functin that applies fr a single technique. 2 T illustrate this, cnsider the example given in Figure 1. The circles in this figure dente different prductin techniques that are available the set f (k i, y i ) pairs. Fr a subset f these, we als plt the lcal prductin functin y = F (b i k, a i ). Finally, the heavy slid line shws the glbal prductin functin, given by the cnvex hull f the lcal prductin techniques. Fr any given level f k, the glbal prductin functin shws the maximum amunt f utput per wrker that can be prduced using the set f ideas that are available. 2 Other mdels in the literature feature a difference between the shrt-run and lng-run elasticities f substitutin, as ppsed t the lcal-glbal distinctin made here. These include the putty-clay mdels f Caballer and Hammur (1998) and Gilchrist and Williams (2000).

5 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE 5 FIGURE 1. An Example f the Glbal Prductin Functin y Circles identify distinct prductin techniques; fr sme f these, the lcal prductin functin assciated with the technique has been drawn as a dashed line. The heavy slid line shws the cnvex hull f the lcal prductin functins, i.e. the glbal prductin functin. k The key questin we d like t answer is this: What is the shape f the glbal prductin functin? T make prgress, we nw turn t a simple baseline mdel The Baseline Mdel We begin with a simple mdel, really nt much mre than an example. Hwever this baseline mdel turns ut t be very useful: it is easy t analyze and captures the essence f the mdel with mre detailed micrfundatins that is presented in Sectin 3. At any given pint in time, a firm (r the ecnmy) has a stck f ideas frm which t chse. T characterize the glbal prductin functin,

6 6 CHARLES I. JONES ntice that fr each pint in the input space, the ecnmy will chse t prduce with the idea that yields the largest amunt f utput. T make this prblem mre precise, suppse N is the cumulative amunt f research the firm has undertaken, and suppse this research has generated a menu f technlgy chices given by b = H(a, N) (5) where H a < 0 and H N > 0. Frm its research effrt, N, the firm discvers a set f ideas. The frntier f this set is a technlgy menu that invlves a tradeff between ideas with a high level f a and ideas with a high level f b. As mre research is cnducted, N rises, the firm discvers mre ideas, and this menu shifts ut. The firm s prblem then is t chse a and b frm the technlgy menu t maximize the level f prductin fr any given set f inputs, and this defines the glbal prductin functin: Y = F (K, L; N) max b,a F (bk, al) (6) subject t (5), assuming K, L, and N are given. As discussed abve, we assume that F (, ) has an elasticity f substitutin less than ne and exhibits cnstant returns t scale in K and L. Ntice that the prblem here is t chse the levels f a and b. Related prblems appear in the literature n the directin f technical change; see Acemglu (2003a), Kennedy (1964), Samuelsn (1965) and Drandakis and Phelps (1966). Hwever, in these prblems the chice variables and the cnstraints are typically expressed in terms f the grwth rates f a and b rather than the levels, resulting in a cnceptually different prblem. There, the ntin is that ne can chse whether t discver ideas with a high a r a high b. Here, the ntin is that ne discvers ideas in undirected search, and that each idea is an (a, b) pair.

7 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE 7 FIGURE 2. Directin f Technical Change b b* Y = Y * b = H(a, N) a* a The slutin t this prblem is straightfrward. Graphically, it is shwn in Figure 2. Algebraically, an interir slutin equates the rati f the labr and capital shares t the elasticity f the H curve: 1 θ K θ K = H a a H η Ha, (7) where θ K (a, b; K, L) F 1 bk/y. In Figure 2, we drew the technlgy menu as cnvex t the rigin. Of curse, this is just fr illustrative purpses; we culd have drawn the curve as cncave r linear. Hwever, it turns ut that the cnstant elasticity versin f the cnvex curve delivers a particularly nice result. 3 In particular, suppse the cnstraint is given by b = Na η, η > 0. (8) 3 In this case, the assumptin that F has an elasticity f substitutin less than ne guarantees that the is-utput curves are mre sharply curved that the technlgy menu, prducing an interir slutin.

8 8 CHARLES I. JONES In this case, the elasticity η Ha = η is cnstant, s the chice f the technlgy levels leads t a first-rder cnditin that sets the capital share equal t the cnstant 1/1 + η. The cnstancy f the capital share then leads t tw useful and interesting results. First, the glbal prductin functin takes a Cbb-Duglas frm: fr any levels f the inputs K and L, and any lcatin f the technlgy frntier, N, the chice f technlgy leads the elasticity f utput with respect t capital and labr t be cnstant. In fact, it is easy t derive the exact frm f the glbal prductin functin by cmbining the lcal-glbal insights f Sectin 2.1 with the technlgy menu. Fr sme technique i, recall the equivalent ways we have f describing the technique: y i a i (9) k i a i b i (10) Frm the technlgy frntier in equatin (8), we knw that b i and a i are related by b i = Na η i. Simple algebra shws that y i and k i are therefre related by y i = (Nk i ) 1/1+η. (11) That is, given the cnstant elasticity frm f the technlgy frntier, a plt f the techniques in (k, y) space like that in Figure 1 yields a Cbb-Duglas prductin functin. With this cntinuus frmulatin fr the frntier, the glbal prductin functin is exactly equal t the technlgy frntier in (k, y) space. 4 Multiplying by L t get back t the standard frm, the glbal 4 Fr this t be true, we need the lcal prductin techniques t paste up smthly with the glbal prductin functin. Fr example, if F is a CES functin with a capital share parameter λ (see, fr example, equatin (33) belw), the glbal prductin functin is actually prprtinal t that in equatin (12). T make the factr f prprtinality equal t ne, we need λ = 1/1 + η, s that the factr share at k = k i is exactly 1/1 + η.

9 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE 9 prductin functin is given by where θ 1/1 + η. Y = N θ K θ L 1 θ, (12) The secnd key result is related t the directin f technical change. T see this, cnsider embedding this prductin setup in a standard neclassical grwth mdel. 5 The fact that the glbal prductin functin is Cbb- Duglas implies immediately that such a mdel will exhibit a balanced grwth path with psitive grwth prvided N grws expnentially. The fact that such a mdel pssesses a balanced grwth path turns ut t have a strng implicatin fr the directin f technical change. This is just an applicatin f the Steady-State Grwth Therem: If a neclassical grwth mdel exhibits steady-state grwth with a nnzer capital share, then either the prductin functin is Cbb-Duglas r technical change is labr augmenting. 6 Since the lcal prductin functin is nt Cbb-Duglas (and prductin always ccurs with sme lcal prductin functin), this must mean that in the lng run, technical change is purely labr augmenting and b is cnstant. Mrever, the fact that the capital share equals 1/1 + η implies that the level f b is chsen s that the capital share is invariant t the capital-utput rati, ne f the key results in Acemglu (2003b). What is the intuitin fr the result that technical change is purely labr augmenting? Ntice that in steady state K/Y must be cnstant. T have a psitive capital share, then, the marginal prduct f capital must als be 5 By this we mean the usual Ramsey-Cass-Kpmans mdel with iselastic utility, cnstant ppulatin grwth, and cnstant grwth in N. 6 Fr a discussin and prf f this therem, see Barr and Sala-i-Martin (1995), Chapter 2. The nnzer capital share qualificatin is imprtant. Fr example, a neclassical grwth mdel with a CES prductin functin Y = F (BK, AL) with an elasticity f substitutin less than ne and with expnential grwth in A and B des have a balanced grwth path. Asympttically, prductin behaves as if there were an infinite amunt f effective capital s that Y = AL.

10 10 CHARLES I. JONES psitive and cnstant. An increase in b has tw effects: first, it raises the marginal prduct f capital directly. But secnd, it increases the amunt f effective capital, driving dwn the marginal prduct. In the Cbb-Duglas case, these tw effects cancel, but therwise ne r the ther dminates, imparting a trend t the marginal prduct unless b is cnstant. 7 The insight frm this simple example is that if the technlgy frntier i.e. the way in which the levels f a and b trade ff exhibits a cnstant elasticity, then the glbal prductin functin will be Cbb-Duglas and technlgical change will be labr-augmenting in the lng run. But is there any reasn t think that the technlgy frntier takes this particular shape? 3. MICROFOUNDATIONS: PARETO DISTRIBUTIONS The baseline mdel is straightfrward and yields strng predictins. Hwever, it invlves a very particular specificatin f the technlgy menu. It turns ut that this specificatin can be derived frm a mdel f ideas with substantially richer micrfundatins. This is the subject f the current sectin Setup Let i = 0, 1,... index the prductin techniques the ideas that are available at a given pint in time. The prductin technique assciated 7 Anther intuitin is the fllwing. Ntice that Y = F (bk, al) has an elasticity f substitutin between effective capital and effective labr that is less than ne. Therefre, K and a are relative cmplements while b and K are relative substitutes (the elasticity f substitutin between b and K is equal t ne). Capital accumulatin substitutes fr grwth in b while it cmplements and reinfrces grwth in a. 8 I we a large debt t Sam Krtum in this sectin. A previus versin f this paper cntained a much mre cumbersme derivatin f the Cbb-Duglas result that applied asympttically as research effrt gets large. Krtum, in discussing this earlier versin at a cnference, develped the mre elegant and useful Pissn apprach that fllws. The asympttic apprach gets by withut making the Pissn assumptin, while the Pissn apprach is mre tractable and delivers a result that applies fr finite research effrt. The asympttic result that drps the Pissn assumptin is develped in the Appendix.

11 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE 11 with idea i is F (b i K, a i L). Because it results in a mre tractable prblem that yields analytic results, we make the extreme assumptin that this lcal prductin technlgy is Lentief: Y = F (b i K, a i L) = min{b i K, a i L}. (13) Of curse, the intuitin regarding the glbal prductin functin suggests that it is determined by the distributin f ideas, nt by the shape f the lcal prductin functin. In later simulatin results, we cnfirm that the Lentief assumptin can be relaxed. New ideas fr prductin are discvered thrugh research. A single research endeavr yields a number f ideas drawn frm a Pissn distributin with a parameter nrmalized t ne. In expectatin, then, each research endeavr yields ne idea. Let N dente the cumulative number f research endeavrs that have been undertaken. Then the number f ideas, n that have been discvered as a result f these N attempts is a randm variable drawn frm a Pissn distributin with parameter N. The discvery f an idea results in a new prductin technique, described by its labr-augmenting and capital-augmenting parameters a i and b i. These parameters are drawn frm a jint distributin given by Prb [b i > b, a i > a] = G(b, a), (14) where the supprt fr this distributin is a i γ a > 0 and b i γ b > 0. We specify this distributin in its cmplementary frm because this simplifies sme f the equatins that fllw Deriving the Prductin Functin The glbal prductin functin describes, as a functin f inputs, the maximum amunt f utput that can be prduced using any cmbinatin f existing prductin techniques. We have already made ne simplificatin

12 12 CHARLES I. JONES in ur setup by limiting cnsideratin t Lentief techniques. Nw we make anther by ignring cmbinatins f techniques and allwing nly a single technique t be used at each pint in time. Again, this is a simplifying assumptin that allws fr an analytic result, but it will be relaxed later in the numerical simulatins. Definitin 3.1. as The glbal prductin functin F (K, L; n) is given F (K, L; n) max F (b i K, a i L) (15) i {0,...,n 1} We build up t characterizing this bject thrugh several steps. First, cnsider particular levels f the inputs, K and L. Let Y i (K, L) F (b i K, a i L) dente utput using technique i. Then, since F is Lentief, the distributin f Y i is given by 9 Prb [Y i > ỹ] = Prb [b i K > ỹ, a i L > ỹ] = G(ỹ/K, ỹ/l). (16) Next, suppse there are n ideas that have been discvered. In this case, the utput level assciated with the glbal prductin functin is distributed as Prb [max{y i } ỹ] = (1 G(ỹ/K, ỹ/l)) n (17) i At this pint, we can use the nice prperties f the Pissn distributin t make further prgress. Recall that n P issn(n), s as a functin f the ttal number f research attempts, N, we have 9 Since b i γ b and a i γ a, the supprt fr this distributin is ỹ max{γ b K, γ al}.

13 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE 13 Prb [max{ Y i } ỹ] e N N n = (1 G(ỹ/K, ỹ/l)) n n! n=0 = e N (N(1 G(ỹ/K, ỹ/l))) n n! n=0 = e N e N(1 G( )) = e NG(ỹ/K,ỹ/L). (18) Fr a general jint distributin functin G, this last equatin describes the distributin f the glbal prductin functin when cumulative research effrt is N. 10 T g further, we nw make a key assumptin abut the distributin f ideas: Assumptin 3.1. Ideas are drawn frm independent Paret distributins: ( ) a α Prb [a i a] = 1, a γ a > 0 (19) γa ( ) b β Prb [b i b] = 1, b γ b > 0, (20) γb where α > 0, β > 0, and α + β > See Prpsitin 2.1 in Krtum (1997) fr this style f reasning, i.e. fr an apprach that uses a Pissn prcess t get an exact extreme value distributin that is easy t wrk with rather than an asympttic result. Als Jhnsn, Ktz, and Balakrishnan, Cntinuus Univariate Distributins, Vlume 2, pp. 11 and This last cnditin that the sum f the tw parameters be greater than ne is needed s that the mean f the Fréchet distributin belw exists. On a related pint, recall that fr a Paret distributin, the kth mment exists nly if the shape parameter (e.g. α r β) is larger than k.

14 14 CHARLES I. JONES With this assumptin, the jint distributin f a i and b i satisfies ( ) b β ( ) a α Prb [b i > b, a i > a] = G(b, a) =. (21) γb γa And therefre, Prb [Y i > ỹ] = G(ỹ/K, ỹ/l) = γk β L α ỹ (α+β) (22) where γ γ α a γ β b. That is, the distributin f Y i is itself Paret. The final step is t cmbine the Paret assumptin with the glbal prductin functin result derived in equatin (18). It is straightfrward t shw that the distributin f the utput that can be prduced with the glbal prductin functin, given inputs f K and L, is Prb [max{y i } ỹ] = e γnkβ L α ỹ (α+β). (23) This distributin is knwn as a Fréchet distributin. 12 Finally, taking expectatins ver this distributin, ne sees that expected utput, given N cumulative research draws and inputs K and L, is given by E[Y ] E[max Y i ] = µ (γnk β L α) 1 α+β (24) where µ Γ(1 1/(α + β)) is a cnstant that depends n Euler s factrial functin. 13 One can als use the distributin in equatin (23) t write the level f utput as a randm variable: 14 Y = (γnk β L α) 1 α+β ɛ (25) 12 See Krtum (1997), Galambs (1978), and Castill (1988) fr mre. 13 Surprisingly few f the reference bks n extreme value thery actually reprt the mean f the Fréchet distributin. Fr a distributin functin F (x) = exp( ((x λ)/δ) β ), Castill (1988) reprts that the mean is λ + δγ(1 1/β) fr β > In particular, ntice that where z (γk β L α ) 1/α+β. Prb [Y/z ỹ] = exp( ỹ (α+β) )

15 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE 15 where ɛ is a randm variable drawn frm a Fréchet distributin with shape parameter α + β and a scale parameter equal t unity. The Pissn structure fr the arrival f ideas is a cnvenient device fr simplifying the presentatin f the Cbb-Duglas result, but it is nt necessary. The Appendix derives the Cbb-Duglas prductin functin as an asympttic result when the number f ideas gets large. 4. DISCUSSION The result given in equatin (25) is ne f the main results in the paper. If ideas are drawn frm Paret distributins, then the glbal prductin functin takes, at least in expectatin, the Cbb-Duglas frm. Fr any given prductin technique, a firm may find it difficult t substitute capital fr labr and vice versa, leading the curvature f the prductin functin t set in quickly. Hwever, when firms are allwed t switch between prductin technlgies, the glbal prductin functin depends n the distributin f ideas. If that distributin happens t be a Paret distributin, then the prductin functin is Cbb-Duglas. We can nw make a number f remarks. First, the expnent in the Cbb- Duglas functin depends directly n the parameters f the Paret search distributins. The easier it is t find ideas that augment a particular factr, the lwer is the relevant Paret parameter (e.g. α r β), and the lwer is the expnent n that factr. Intuitively, better ideas n average reduce factr shares because the elasticity f substitutin is less than ne. Sme additinal remarks fllw Relatinship t the Baseline Mdel The simple baseline mdel given at the beginning f this paper pstulated a technlgy menu and shwed that if this menu exhibited a cnstant elasticity, then ne culd derive a Cbb-Duglas glbal prductin functin.

16 16 CHARLES I. JONES The mdel with micrfundatins based n Paret distributins turns ut t deliver a stchastic versin f this technlgy menu. In the mdel, the stchastic versin f this menu can be seen by cnsidering an is-prbability curve Prb [b i > b, a i > a] G(b, a) = C, where C > 0 is sme cnstant. With the jint Paret distributin, this is-prbability curve is given by ( ) γ 1/β b = a α/β. (26) C This is-curve has a cnstant elasticity equal t α/β and shifts up as the prbability C is lwered. In terms f the baseline mdel, the Paret distributin therefre delivers η Ha = α/β. The first-rder cnditin in equatin (7) then implies that technlgies will be chsen s that the capital share is equal t β/(α + β), which is exactly the expnent in the glbal Cbb-Duglas prductin functin f the Paret mdel, as shwn in equatin (25) Huthakker ( ) The ntin that Paret distributins, apprpriately kicked, can deliver a Cbb-Duglas prductin functin is a classic result by Huthakker ( ). Huthakker cnsiders a wrld f prductin units (e.g. firms) that prduce with Lentief technlgies where the Lentief cefficients are distributed acrss firms accrding t a Paret distributin. Imprtantly, each firm has limited capacity, s that the nly way t expand utput is t use additinal firms. Huthakker then shws that the aggregate prductin functin acrss these units is Cbb-Duglas. The result here bviusly builds directly n Huthakker s insight that Paret distributins can generate Cbb-Duglas prductin functins. The result differs frm Huthakker s in several ways, hwever. First, Huthakker s result is an aggregatin result. Here, in cntrast, the result applies at the

17 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE 17 level f a single prductin unit (be it a firm, industry, r cuntry). Secnd, the Lentief restrictin in Huthakker s paper is imprtant fr the result; it allws the aggregatin t be a functin nly f the Paret distributins. Here, in cntrast, the result is really abut the shape f the glbal prductin functin, lking acrss techniques. The lcal shape f the prductin functin des nt really matter. This was apparent in the simple baseline mdel given earlier, and it will be cnfirmed numerically in Sectin 6. Finally, Huthakker s result relies n the presence f capacity cnstraints. If ne wants t expand utput, ne has t add additinal prductin units, essentially f lwer quality. Because f these capacity cnstraints, his aggregate prductin functin is characterized by decreasing returns t scale. In the cntext f an idea mdel, such cnstraints are undesirable: ne wuld like t allw the firm t take its best idea and use it fr every unit f prductin. That is, ne wuld like the setup t respect the nnrivalry f ideas and the replicatin argument fr cnstant returns, as is true in the frmulatin here Evidence fr Paret Distributins? The next main cmment is that Paret distributins are crucial t the result. Is it plausible that the distributins fr ideas are Paret? In terms f direct evidence, there are a number f references related t patents, prfitability, and citatins. First, it is wrth nting that many f the tests in this literature are abut whether r nt the relevant variable beys a Paret distributin. That is, Paret serves as a benchmark. In terms f findings, this literature either supprts the Paret distributin r 15 Lags (2004) embeds the Huthakker frmulatin in a Mrtensn-Pissarides search mdel t prvide a thery f ttal factr prductivity differences. In his setup, firms (capital) match with labr and have a match quality that is drawn frm a Paret distributin. Capital is the quasi-fixed factr s that the setup generates cnstant returns t scale in capital and labr. Nevertheless, because each unit f capital gets its wn Paret draw, a firm cannt expand prductin by increasing its size at its best match quality.

18 18 CHARLES I. JONES finds that it is difficult t distinguish between the Paret and the lgnrmal distributins. Fr example, Harhff, Scherer and Vpel (1997) examine the distributin f the value f patents in Germany and the United States. Fr patents wrth mre than $500,000 r mre than 100,000 Deutsche Marks, a Paret distributin accurately describes patent values, althugh fr the entire range f patent values a lgnrmal seems t fit better. Bertran (2003) finds evidence f a Paret distributin fr ideas by using patent citatin data t value patents. Grabwski (2002) prduces a graph f the present discunted value f prfits fr new chemical entities by decile in the pharmaceutical industry fr that supprts a highly-skewed distributin. Srnette and Zajdenweber (2000) interpret earlier similar wrk by Grabwski and Vernn as evidence in favr f a Paret distributin. Ltka (1926), a classic reference n scientific prductivity, shws that the distributin f scientific publicatins per authr is Paret. This result appears t have std the test f time acrss a range f disciplines, even in ecnmics, as shwn by Cx and Chung (1991). Huber (1998) lks fr this result amng inventrs and finds sme evidence that the distributin f patents per inventr is als Paret, althugh the sample is small. Smewhat further remved but still related, evidence f Paret distributins is fund by Srnette and Zajdenweber (1999) fr wrld mvie revenues and by Chevalier and Glsbee (2004) fr bk sales. In additin t the direct evidence, there are als cnceptual reasns t think ideas might cme frm Paret distributins. T begin, cnsider a simple example. Imagine drawing scial security numbers fr the U.S. ppulatin at randm, and fr each persn drawn, recrd their incme and their height. Als keep track f the largest draw t date: let yz max dente the maximum incme and let h max z dente the maximum height after z draws have been made. Nw cnsider the fllwing cnditinal prbability: Prb {X γx max X x max } fr γ > 1, where x stands

19 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE 19 fr either incme r height. This prbability answers the questin: Given that the tallest persn bserved s far is 6 feet 6 inches tall and given that we just fund smene even taller, what is the prbability that this new persn is mre than 5 percent taller than ur 6 ft 6 inch persn? Clearly as h max gets larger and larger, this cnditinal prbability gets smaller and smaller there is n ne in the wrld taller than ten feet. In cntrast, cnsider the incme draws. Nw, the prbability answers the questin: Given that the highest-earning persn bserved s far has an annual incme f $240,000 and given that we just fund smene wh earns even mre, what is the prbability that this new persn s earnings exceed the previus maximum by mre than 5 percent? It turns ut empirically that in the case f incmes, this prbability des nt depend n the level f y max being cnsidered. Indeed, it was exactly this bservatin n incmes that led Paret t frmulate the distributin that bears his name: the defining characteristic f the Paret distributin is that the cnditinal prbability given abve is invariant t x max. 16 In applying this example t grwth mdels, ne is led t ask whether the distributin f ideas is mre like the distributin f heights r the distributin f incmes. An imprtant insight int this questin was develped by Krtum (1997). Krtum frmulates a grwth mdel where prductivity levels (ideas) are draws frm a distributin. He shws that this mdel generates steady-state grwth nly if the distributin has Paret tails. That is, what the mdel requires is that the prbability f finding an idea that is 5 percent better than the current best idea is invariant t the level f prductivity embdied in the current best idea. Of curse, this is almst the very definitin f a steady state: the prbability f imprving ecnmywide prductivity by 5 percent can t depend n the level f prductivity. 16 Saez (2001) shws this invariance fr the United States in 1992 and 1993 fr incmes between $100,000 and $30 millin.

20 20 CHARLES I. JONES This requirement is satisfied nly if the upper tail f the distributin is a pwer functin, i.e. nly if the upper tail is Paret. 17 A literature in physics n scale invariance suggests that if a stchastic prcess is t be invariant t scale, it must invlve Paret distributins. Steady-state grwth is simply a grwth rate that is invariant t scale (defined in this cntext as the initial level f prductivity). Whether incmes are at 100 r 1000, steady-state grwth requires the grwth rate t be the same in bth cases. Additinal insight int this issue emerges frm Gabaix (1999). Whereas Krtum shws that Paret distributins lead t steady-state grwth, Gabaix essentially shws the reverse in his explanatin f Zipf s Law fr the size f cities. He assumes that city sizes grw at a cmmn expnential rate plus an idisynchratic shck. He then shws that this expnential grwth generates a Paret distributin fr city sizes. 18 These papers by Krtum and Gabaix suggest that Paret distributins and expnential grwth are really just tw sides f the same cin. The result in the present paper draws ut this cnnectin further and highlights the additinal implicatin fr the shape f prductin functins. Nt nly are Paret distributins necessary fr expnential grwth, but they als imply that the glbal prductin functin takes a Cbb-Duglas frm. 17 Krtum als shws that if the tails f the distributin are thinner than Paret, as is the case fr the lg nrmal r expnential distributins, then expnential grwth rates decline t zer. If the tails are thicker, then presumably grwth rates rise ver time, but this case is nt analyzed by Krtum. 18 An imprtant additinal requirement in the Gabaix paper is that there be sme psitive lwer bund t city sizes that functins as a reflecting barrier. Otherwise, fr example, nrmally distributed randm shcks results in a lg-nrmal distributin f city cizes. Alternatively, if the length f time that has passed since each city was created is a randm variable with an expnential distributin, then n lwer bund is needed and ne recvers the Paret result. See Mitzenmacher (2003) fr a direct discussin f these alternatives, as well as Crdba (2003) and Rssi-Hansberg and Wright (2004).

21 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE THE DIRECTION OF TECHNICAL CHANGE The secnd main result f the paper is related t the directin f technical change. It turns ut that this same setup, when embedded in a standard neclassical grwth mdel, delivers the result that technlgical change is purely labr augmenting in the lng run. That is, even thugh the largest value f b i assciated with any idea ges t infinity, this Paret-based grwth mdel delivers the result that a i (t) grws n average while b i (t) is statinary. T see this result, we first embed ur existing setup in a standard neclassical grwth mdel. The prductin side f the mdel is exactly as specified in Sectin 3. Capital accumulates in the usual way, and we assume the investment rate s is a cnstant: K t+1 = (1 δ)k t + sy t, δ, s (0, 1). (27) Finally, we assume that the cumulative number f research endeavrs as f date t, N t, grws exgenusly at rate g > 0: N t = N 0 e gt. (28) As in Jnes (1995) and Krtum (1997), ne natural interpretatin f this assumptin is that research endeavrs are undertaken by researchers, and g is prprtinal t ppulatin grwth. 19 Fr this mdel, we have already shwn that the glbal prductin functin is given by Y t = ( ) γn t K β 1 t Lα α+β t ɛ t. (29) 19 Fr example, ne culd have N t+1 = R λ t N φ t, where R t represents the number f researchers wrking in perid t. In this case, if the number f researchers grws at a cnstant expnential rate, then the grwth rate f N cnverges t a cnstant that is prprtinal t this ppulatin grwth rate. Of curse, ne lgical case t cnsider is where λ = 1 and φ = 0, s that the number f research endeavrs in a perid is just prprtinal t the number f researchers.

22 22 CHARLES I. JONES It is then straightfrward t shw that the average grwth rate f utput per wrker y in the mdel in a statinary steady state is given by 20 E[lg y t+1 y t ] g/α. (31) The grwth rate f utput per wrker is prprtinal t the rate f grwth f research effrt. The factr f prprtinality depends nly n the search parameter f the Paret distributin fr the labr-augmenting ideas. In particular, the easier it is t find higher a i, the faster is the average rate f ecnmic grwth. The fact that this grwth rate depends n α but nt n β is the first clue that there is smething further t explre here: if it is easier t find better labr-augmenting ideas, the average grwth rate is higher, but if it is easier t find better capital-augmenting ideas, the average grwth rate is unaffected. T understand this fact, it is helpful t lk back at the lcal prductin functin. Even thugh the glbal prductin functin is Cbb-Duglas, prductin at sme date t always ccurs with sme technique i(t): Y t = F (b i(t) K t, a i(t) L t ). (32) Nw recall the Steady-State Grwth Therem discussed earlier: If a neclassical grwth mdel exhibits steady-state grwth with a nnzer capital share, then either the prductin functin is Cbb-Duglas r technical change is labr augmenting. In this case, the (lcal) prductin functin is nt Cbb-Duglas and we d have a (statinary) steady state. The implicatin is that technical change must be labr-augmenting in the mdel. 20 Rewriting the prductin functin in per wrker terms, ne has lg yt+1 = 1 Nt+1 lg + β kt+1 lg + lg ɛt+1. (30) y t α + β N t α + β k t ɛ t Taking expectatins f this equatin and equating the grwth rates f y and k yields the desired result.

23 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE 23 That is, despite the fact that max i b i as t, the time path fr b i(t) i.e. the time path f the b i s assciated with the ideas that are actually used must have an average grwth rate equal t zer in the limit. The intuitin is the same as in the simple baseline mdel: K and b are relative substitutes, while K and a are relative cmplements in prductin. This means that capital accumulatin leads the ecnmy t increase a at the expense f a stable b SIMULATION RESULTS We nw turn t a full simulatin based n the Paret mdel. In additin t prviding an illustratin f the results, we take this pprtunity t relax the Lentief assumptin n the lcal prductin functin. Instead, we assume the lcal prductin functin takes the CES frm: Y t = F (b i K t, a i L t ) = (λ(b i K t ) ρ + (1 λ)(a i L t ) ρ ) 1/ρ, (33) where ρ < 0 s that the elasticity f substitutin is σ 1 1 ρ < 1. We als allw prductin units t use tw prductin techniques at a time in rder t cnvexify the prductin set, analgus t the picture given at the beginning f the paper in Figure 1. The remainder f the mdel is as specified befre. Apart frm the change t the CES functin, the prductin setup is the same as that given in Sectin 3 and the rest f the mdel fllws the cnstant saving setup f Sectin 5. We begin by shwing that the CES setup still delivers a Cbb-Duglas glbal prductin functin, at least n average. Fr this result, we repeat 21 This result leads t an imprtant bservatin related t extending the mdel. Recall that with the Paret assumptin, γ b is the smallest value f b that can be drawn, and similarly γ a is the smallest value f a that can be drawn. Nw cnsider allwing these distributins t shift. There seems t be n bstacle t allwing fr expnential shifts in γ a ver time. Hwever, increases in γ b turn ut t lwer the capital share in the mdel. If γ b were t rise expnentially, the capital share wuld be driven tward zer, n average.

24 24 CHARLES I. JONES FIGURE 3. The Cbb-Duglas Result OLS Slpe = Std. Err. = R 2 = 0.73 lg k lg y Nte: The figure shws 1000 capital-utput cmbinatins frm the glbal prductin functin. The parameter values used in the simulatin are N = 500, α = 5, β = 2.5, γ a = 1, γ b = 0.2, and ρ = 1. the fllwing set f steps t btain 1000 capital-utput pairs: We first set cumulative research effrt N t 500, s that n average there are 500 ideas in each iteratin. We cmpute the cnvex hull f the CES functins assciated with these ideas t get a glbal prductin functin. 22 Next, we chse a level f capital per wrker k randmly frm a unifrm distributin between the smallest value f k i and the largest value f k i fr the iteratin. Finally, we recrd the utput f the glbal prductin functin assciated with this input. Fllwing this prcedure yields a graph like that shwn in Figure 3. The key parameter values in this simulatin are α = 5 and β = 2.5, s that 22 Cmputing the cnvex hull f the verlapping CES prductin functins is a cmputatinally intensive prblem, especially when the number f ideas gets large. T simplify, we first cmpute the cnvex hull f the (k i, y i) pints. Then, we cmpute the cnvex hull f the CES functins assciated with this limited set f pints. T apprximate the CES curve, we divide the capital interval int 100 equally-spaced pints.

25 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE 25 the thery suggests we shuld expect a Cbb-Duglas prductin functin with a capital expnent f β/α + β = 1/3. 23 As the figure shws, the relatin between lg y and lg k is linear, with a slpe that is very clse t this value. We next cnsider a simulatin run fr the full dynamic time path f the Paret mdel. Cntinuing with the parameter chices already made, we additinally assume g =.10, which implies an annual grwth rate f 2 percent fr utput per wrker in the steady state. We simulate this mdel fr 100 years and plt the results in several figures. 24 Figure 4 shws a subset f the mre than 1 millin techniques that are discvered ver these 100 perids. In particular, we plt nly the 300 pints with the highest values f y (these are shwn with circles ). Withut this truncatin, the lwer triangle in the figure that is currently blank but fr the plus signs is filled in as slid black. In additin, the capital-utput cmbinatins that are actually used in each perid are pltted with a plus sign ( + ). When a single technique is used fr a large number f perids, the pints trace ut the CES prductin functin. Alternatively, if the ecnmy is cnvexifying by using tw techniques, the pints trace ut a line. Finally, when the ecnmy switches t a new technique, the capital-utput cmbinatins jump upward. Figure 5 shws utput per wrker ver time, pltted n a lg scale. The 23 The standard capital share pins dwn the rati f α/β, but it des nt tell us the basic scale f these parameters. The studies cited earlier related t patent values and scientific prductivity typically find Paret parameters that are in the range f 0.5 t 1.5. We have chsen higher values here fr illustratin. The fllwing exercise is helpful in thinking abut this: What is the median value f a prductivity draw, cnditinal n that draw being larger than sme value, x? If α is the Paret parameter, then the answer t this questin turns ut t be 2 1/α x ( /α)x. Fr example, if α = 2, then the median value, cnditinal n a draw being higher than x, is abut 1.4x. This says that the average idea that exceeds the frntier exceeds it by 40 percent. This implies very large jumps, which might be plausible at the micr level but seem t large at the macr level. A value f α = 5 instead gives an average jump f abut 14 percent, which is still smewhat large. Aggregatin wuld, ne suspects, smth these jumps ut. 24 Additinal parameter values used in the simulatin are listed in the ntes t Figure 4.

26 26 CHARLES I. JONES FIGURE 4. Prductin in the Simulated Ecnmy Output per Wrker, y Capital per Wrker, k Nte: Circles indicate ideas, plus signs indicate capital-utput cmbinatins that are actually used. The mdel is simulated fr 100 perids with N 0 = 50, α = 5, β = 2.5, g =.10, γ a = 1, γ b = 0.2, k 0 = 2.5, s = 0.2, δ =.05, and ρ = 1. average grwth rate f utput per wrker in this particular simulatin is 1.48 percent, as cmpared t the theretical value f 2 percent implied by the parameter values, given by g/α. 25 Figure 6 plts the capital share F K K/Y ver time. Even thugh the ecnmy grws at a stable average rate, the capital share exhibits fairly large mvements. When the ecnmy is using a single prductin technique, the accumulatin f capital leads the capital share t decline. Alternatively, when the ecnmy is using tw techniques t cnvexify the prductin set, the marginal prduct f capital is cnstant, s the capital share rises smthly. 25 We cmpute the average grwth rate by drpping the first 20 bservatins (t minimize the effect f initial cnditins) and then regressing the lg f utput per wrker n a cnstant and a time trend.

27 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE 27 FIGURE 5. Output per Wrker ver Time Output per Wrker (lg scale) Time Nte: See ntes t Figure 4. It is interesting t cmpare the behavir f the capital share in the Paret mdel with the behavir that ccurs in the simple baseline mdel. In the simple mdel, the ecnmy equates the capital share t a functin f the elasticity f the technlgy menu. If this elasticity is cnstant, then the capital share wuld be cnstant ver time. Here, the technlgy menu exhibits a cnstant elasticity n average, but the menu is nt a smth, cntinuus functin. Quite the ppsite: the extreme value nature f this prblem means the frntier is sparse, as the example back in Figure 1 suggests. This means the capital share will be statinary, but that it can mve arund, bth as the ecnmy accumulates capital and as it switches techniques. Figure 7 shws the technlgy chices that ccur in this simulatin. As in Figure 4, the 300 ideas with the highest level f y i = a i are pltted. This time, hwever, the (a i, b i ) pair crrespnding t each idea is pltted. The

28 28 CHARLES I. JONES FIGURE 6. The Capital Share ver Time Capital Share Time Nte: See ntes t Figure 4. graph therefre shws the stchastic versin f the technlgy menu. In additin, the figure plts with a + the idea cmbinatins that are actually used as the ecnmy grws ver time. Crrespnding t the theretical finding earlier, ne sees that the level f b i appears statinary, while the level f a i trends upward. On average, technlgical change is labr augmenting. 7. CONCLUSION This paper prvides micrfundatins fr the standard prductin functin that serves as a building blck fr many ecnmic mdels. An idea is a set f instructins that tells hw t prduce with a given cllectin f inputs. It can be used with a different mix f inputs, but it is nt especially effective with the different mix; the elasticity f substitutin in prductin is lw fr a given prductin technique. Instead, prducing with a different input mix typically leads the prductin unit t switch t a new technique.

29 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE 29 FIGURE 7. Technlgy Chices Capital Aug. Technlgy, b Labr Augmenting Technlgy, a Nte: Frm mre than 1 millin ideas generated, the 300 with the highest level f a are pltted as circles. The figure als plts with a + the (a i, b i) cmbinatins that are used at each date and links them with a line. When tw ideas are used simultaneusly, the idea with the higher level f utput is pltted. See als ntes t Figure 4.

30 30 CHARLES I. JONES This suggests that the shape f the glbal prductin functin hinges n the distributin f available techniques. Krtum (1997) examined a mdel in which prductivity levels are draws frm a distributin and shwed that nly distributins in which the upper tail is a pwer functin are cnsistent with expnential grwth. If ne wants a mdel in which steady-state grwth ccurs, then ne needs t build in a Paret distributin fr ideas. What we shw here is that this assumptin delivers tw additinal results. Paret distributins lead the glbal prductin functin t take a Cbb-Duglas frm and prduce a setup where technlgical change, in the lng run, is entirely labr augmenting. There are several additinal directins fr research suggested by this apprach. First, ur standard ways f intrducing skilled and unskilled labr int prductin invlve prductin functins with an elasticity f substitutin bigger than ne, cnsistent with the bservatin that unskilled labr s share f incme seems t be falling. 26 Hw can this view be recnciled with the reasning here? Secnd, the large declines in the prices f durable investment gds are ften interpreted as investment-specific technlgical change. That is, they are thught f as increases in b rather than increases in a. 27 This is the case in Greenwd, Hercwitz and Krusell (1997) and Whelan (2001), and it is als implicitly the way the hednic pricing f cmputers wrks in the Natinal Incme and Prduct Accunts: better cmputers are interpreted as mre cmputers. The mdel in this paper suggests instead that b might be statinary, s there is a tensin with this ther wrk. Of curse, it is nt at all bvius that better cmputers are equivalent t mre cmputers. 26 See Katz and Murphy (1992) and Krusell, Ohanian, Ris-Rull and Vilante (2000), fr example. 27 This is lse. In fact, they are thught f as increases in a term that multiplies investment in the capital accumulatin equatin. Of curse, fr many purpses this is like an increase in b.

31 PRODUCTION FUNCTIONS AND TECHNICAL CHANGE 31 Perhaps a better cmputer is like having tw peple wrking with a single cmputer (as in extreme prgramming). In this case, better cmputers might be thught f as increases in a instead. This remains an pen questin. Alternatively, it might be desirable t have micrfundatins fr a Cbb- Duglas prductin functin that permits capital-augmenting technlgical change t ccur in the steady state. Finally, ne might ask hw the mdel relates t recent discussins abut the behavir f capital shares. The literature is in smething f a flux. Fr a lng time, f curse, the stylized fact has been that capital s share is relatively stable. This turns ut t be true at the aggregate level fr the United States and Great Britain, but it is nt true at the disaggregated level in the U.S. r in the aggregate fr many ther cuntries. Rather, the mre accurate versin f the fact appears t be that capital s share can exhibit large medium term mvements and even trends ver perids lnger than 20 years in sme cuntries and industries. 28 This paper is smewhat agnstic abut factr shares. As shwn in Figure 6, the Paret mdel predicts the capital share may vary ver time, while f curse the baseline mdel implied a cnstant capital share. Hwever, there are many ther determinants f capital shares left ut f this mdel, including aggregatin issues and wedges between marginal prducts and prices, s care shuld be taken in interpreting the mdel alng this particular dimensin. 28 The recent papers by Blanchard (1997), Bentlila and Saint-Paul (2003), and Harrisn (2003) discuss in detail the facts abut capital and labr shares and hw they vary. Gllin (2002) is als related; that paper argues the in the crss-sectin f cuntries, labr shares are mre similar than rugh data n emplyee cmpensatin as a share f GDP suggests because f the very high levels f self-emplyment in many pr cuntries.

32 32 CHARLES I. JONES APPENDIX: AN ALTERNATIVE DERIVATION OF THE COBB-DOUGLAS RESULT Here we drp the assumptin that ideas arrive as a Pissn prcess and shw that ne still recvers the Cbb-Duglas result (and therefre the labr-augmenting technical change). The difference is that the result nw hlds asympttically, as the number f ideas ges t infinity, and the prf invlves the use f extreme value thery. Let N nw dente the ttal number f ideas that have been discvered and drp the Pissn prcess. This is the nly change t the mdel in Sectin 3. As befre, let Y i dente prductin using technique i with a given amunt f capital and labr. Then H(ỹ) Prb [Y i > ỹ] = Prb [b i K > ỹ, a i L > ỹ] ( ỹ = G K, ỹ ) L That is, the distributin f Y i is Paret. 1 = γk β L α ỹ (α+β). (A.1) The glbal prductin functin is the maximum amunt f utput that can be prduced with a single technique. Frmally, it is defined as Y = F (K, L, N) = Since the N draws are independent, max Y i. i=1,...,n (A.2) Prb [Y ỹ] = (1 H(ỹ)) N. ( = 1 γk β L α ỹ (α+β)) N. (A.3) Of curse, as the number f ideas N gets large, this prbability fr any given level f ỹ ges t zer. S t get a stable distributin, we need t nrmalize ur randm variable smehw, in a manner analgus t that used in the Central Limit Therem. 1 Since b i γ b and a i γ a, the supprt fr this distributin is ỹ max{γ b K, γ al}.