MIT LIBRARIES DUPL 3 9080 02618 4496 Sum. ail Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium Member Libraries http://www.archive.org/details/laborcapitalaugmOOacem n DEv'siE) (^r^J Massachusetts Institute of Technology Department of Economics Working Paper Series Labor- and Capital-Augmenting Teclnnical Ctiange Daron Acemoglu Working Paper 03-02 December 2002 Room E52-251 50 Memorial Drive Cambridge, This MA 02142 paper can be downloaded without charge from the Network Paper Collection at Social Science Research http://papers.ssm. com/abstract_id=372820 December 2002 Labor- and Capital- Augmenting Technical Change^ Daron Acenioglu' Abstract ^s- I analyze an economy in which firms can imdertake both labor- and capital-augmenting economy resembles the stfrndard grow'th model with purely labor-augmenting technical change, and the share of labor in GDP is constant. Along the transition path, however, there is capital-augmenting technical change and factor shares change. Tax policy and changes in labor supply or savings typically change factor shares in the short nm. but have no or little effect on the long-nm technological improvements. In the long rim, the factor distribution of income. KeyM'ords: Economic Growth, Endogenous Growth, Factor Shares, Technical Change. JEL 'I Classification: 033, 014, 031, E25. thank Ruben Segura-Cayuela for excellent research assistance, and Manuel Amador, Abhijit Ba- Chang Hsieh, Steven Lin, Robert Lucas, Torsten Persson, Mathias Thoenig, Jaume Ventura, two anonymous referees, and seminar participants at the Canadian Institute for Advanced Research, Harvard, MIT, the NBER Economic Fluctuations meeting, and Rochester for useful comments. I also gratefully acknowledge financial help from The Canadian Institute for Advanced Research and the narjee, Olivier Blanchard, National Science Foundation Grant SES-009.52-53. t Massachusetts Institute of Technology-, Department of Economics, E52-.371, Cambridge, for Advanced Research; e-mail: daronQmit.edu and the Canadian Institute MA 02319, Introduction I. Figures and 1 show the shares 2 of GDP accrmng The over the past 80 years (with the remainder accrmng to capital).' well known, pattern significant capital is is years. For example, in Almost II. and exogenous movements all both countries, there run (despite all a large increase in the share of labor after is models of growth and capital accumulation, of both endogenous types, explain the stability of factor shares using technical change With an in the long but in the share of labor over periods as long as 10 or 20 either the elasticity of substitution or show no trend first striking, deepening during the same period). The second important observation that there are large World War that these factor shares and France to labor in the U.S. is between capital and labor one of two assumptions: is taken to be equal to assumed to be labor augmenting (Harrod elasticity of substitution between capital and labor 1, neutral).'' ecjual to 1, i.e., with a Cobb-Douglas production fvmction, the shares of capital and labor are pinned down by technology alone (as long as firms are along their factor suppose that the aggregate production function and L is labor. Then, the share of labor will is y = demand curves). AL'^K^~°' where reality, K is capital, always be ecjual to a. There are reasons to be skeptical that the Cobb-Douglas production function provides an approximation to For example, entirely satisfactory however. First, most estimates suggest that the aggregate elas- ticity of substitution is significantly less than 1.^ Second, a production fimction with an does not provide a framework for analyzing fluctuations in elasticity of substitution of 1 factor shares, such as those shown in Figures 1 and 2. 'The French data are from Piketty (2001), and the U.S. data are from Piketty and Saez (2001), who Income and Product Accounts data. -A third possibihty is that the aggregate production function is Y' = F{K,H) where H is human capital, accumulating at the same rate as A', so that there is no "capital deepening". However, the rate of accumulation of human capital appears to be substantially less than that of physical capital. For example, in the U.S., average schooling of the workforce increased by about one year in every decade in in turn use the National the postwar era, which translates roughly to a 6 percent increase in the human capital of the workforce Acemoglu and Angrist, 2000), compared to an approximately 4 percent a year growth in the capital stock between 1959 and 1998 (see The Economic Report of the President, 1999). ^For example, Nadiri (1970), Nerlove (1967) and Hamermesh (199.3) survey a range of early estimates of the elasticity of substitution, which are generally between 0.3 and 0.7. David and Van de Klundert (196.5), (e.g., similarly estimate this elasticity to be in the neighborhood of 0.3. Using the translog production function. and Gregory (1976) estimate elasticities of substitution for nine OECD economies between 0.06 and See also Eisner and Nadiri (1968) and Lucas (1969). Berndt (1976), on the other hand, estimates an elasticity of substitution equal to 1. but does not control for a time trend, creating a strong bias towards Griffin 0.52. Using more recent data, and various different specifications, Krusell, Ohanian, Rios-Rull, and Violante 1. Estimates implied by the response of in\'estment to the user cost of capital also typically yield an elcisticity of substitution between capital and labor significantly less than 1 (see, e.g., Chirinko, 1993, Chirinko, Fazzari and Mayer, 1999, and 2001. or Mairesse, Hall and Mulkay, 1999). 1. (2000) and Antras (2001) also find estimates of the elasticity significantly less than The sical patterns depicted in Figures 1 production function, but require and 2 are consistent with a more general neoclas- all technical change to be labor augmenting and to More take place exactly at the same rate as the rate of capital deepening. consider an aggregate production function of the form Y = F{MK, NL). specifically, The assump- tion of labor-augmenting technical change implies that technical progress only increases A'', and does not ital axis. A affect M—or in other words, it rotates the isoquants around the cap- neoclassical production function with (purely) labor-augmenting technical change provides an attractive framework for macroeconomic analysis, since tent not only with the long-run stability of factor shares, but also with why do It is in fact Barro and Sala-i-Martin, 1995). (e.g., However, this framework raises another important question: why labor-augmenting? Or equivalently, consis- medium-term swings in response to changes in capital stock, labor supply or technology. the starting point of graduate textbooks on growth it is technical change is all profit-maximizing firms choose innovations that only increase A^? Although starting with Romer's (1986, 1990) and Lucas' (1988) contributions a large literature has investigated the determinants of technological progress and growth, the direction of technical change of increases in A'^ —has received In this paper technical change. I I little —the reason why all progress takes the form attention. investigate the forces that push the economy towards labor-augmenting analyze an otherwise standard endogenous growth model where profit- maximizing firms can invest to increase both M and Y = F{MK,NL). that capital, The only asymmetry labor, L, cannot.^ I show that in this is economy A'^ in all terms of the production function K, can be accimiulated, while technical progress will be labor- augmenting along the balanced growth path. Hence, given the standard assimiptions endogenous gi-owt-h, the result that long-run technical change must be labor-augmenting follows fi'om profit-maximizing incentives. capital and the for interest rate remain to labor- augmenting technical change Conseciuently, in the long run, the share of stable, while the and wage rate capital deepening. In increases steadily due some therefore provides a microfoundation for the basic neoclassical growth sense, this paper model with labor- augmenting technical change. Notably, however, while the balanced growth path of this economy resembles the stan- dard neoclassical model, along the transition path there technical change. That is, pvirely is typically capital-augmenting labor-augmenting technical change is only a long-nm phenomenon. I also show that as long as capital and labor are gross complements, i.e., as long as the "•The important assumption is that (efficiency units of) labor cannot be accumulated asymptotically, which appears reasonable with finite lives, since individuals will have on a limited time to invest in human capital. See Jones (2002) on this, and also footnote 2. between these two factors elasticity of substitution with labor-augmenting technical change pvirely and eciuilibrivim, stable. it is The capital-augmenting techniques is complement or use than 1 , is increjising in the share of capital in capital increase the capital. Consequently, demand when the share less than new for new GDP —both a higher new of capital in be further capital-augmenting technical change. With the these the profitability of intuitive: will 1, the balanced growth path the unique asymptotic (non-cycling) is stability property and a larger supply of interest rate is less technologies that GDP is large, there elasticity of substitution technologies will reduce the share of capital, pushing the economy towards the BGP. In addition to providing an explanation why long-run ft)r technical change augmenting, the framework presented here also suggests a reason for the long-rim stabil- The shares despite major changes in taxes and labor market institutions. ity of factor neoclassical labor is growth model with labor-augmenting technical change predicts a constant long-run share of labor, but this share should respond to policies which affect the capitallabor ratio. In contrast, show that I framework here with both capital-augmenting in the and labor-augmenting technical change, a range of second-order effects) on long-nm factor shares: will also have an offsetting effect suggest that the framework here growth model, but tive statics that policies will they have no effect (or will affect capital-deepening, on capital-augmenting technical change. These is only but results not only useful as a microfoundation for the standard for policy analysis as well: it points out that a number of compara- assume capital-augmenting technical change away may give misleading answers. It is useful to briefly outline the intuition for long-run technical change is, the invention of new labor-intensive goods. ^ Similarly, capital- augmenting progress corresponds to the invention of new capital-intensive goods. In economy, new goods sale. When there are intensive good workers, and wage rate, w. is will be introduced because of future expected n labor-augmenting goods, the proportional to its profits profitability of profits are given by a markup over the marginal Similarly, w^hen there are m this from their an additional labor- — because each intermediate good producer will hire cost of production ^ — the capital-intensive goods, profits from further capital-augmenting progress are proportional to When will Suppose labor-augmenting progress takes the form of "labor- be labor augmenting. using" progress, that why — , where r is the rental rate of capital. technical progress relies on the use of scarce factors such as labor, long-run growth ^In principle, there are two ways to model labor-augmenting technical progress: as the introduction of new production methods that directly increase the productivity of labor, or as the introduction of new- goods and tasks that use labor. Here, I discuss the first formulation. Later, I will show that the same results appl\' when labor-augmenting progress takes the form of "lalsor-enhancing" progress. requires that further innovations build "upon the shoulders of giants" in n and m have to be proportional to their existing fLirther resoui'ces to labor-augmenting innovation the return to capital-augmenting innovation be balanced will is is levels.'^ , that The return is, to allocating therefore proportional to proportional to m • ^. increases n • — These two returns for a specific factor distribution of income. Furthermore, when the elasticity of substitution between capital and labor than 1 , n a high level of m relative to implies that the share of capital to the share of labor. This will encourage increase while , m. The converse applies when m more capital-augmenting is is is less high compared technical progress and too high. Eciuilibrium technical progress will therefore stabilize factor shares. Finally, capital accumulation along the balanced growth path implies that technical n more than m.' progress will increase Intmtively, there are two ways to increase the production of capital-intensive goods, via capital-augmenting technical change and capiaccumulation, and only one way to increase the production of labor-intensive goods, tal through labor-augmenting technical change. Capital accumulation, therefore, implies that technical change has to be, on average, more labor-augmenting than capital- augmenting. In fact, the model implies a stronger result: with an elasticity of substitution between capital and labor less than 1, run there in the long change, rn will remain constant, and The all will be no net capital- augmenting technical technical change will be labor augmenting. ideas in this paper are closely related to the induced innovation literatm-e of the 1960s and to Hicks' discussion of the determinants of eciuilibrium bias of technical change in The Theory of Wages (1932). Hicks wrote: "A change in the of production is itself a spvir to invention, and to invention of a to economizing the use of a factor which has 124-5). Fellner (1961) expanded on relative prices of the factors become this argimient partici.ilar kind relatively expensive." —directed (1932, pp. and suggested that technical progress was more labor-augmenting because wages were growing, and were expected to grow, so technical change woi_ild try to save on this factor that was becoming more expensive. Li an important contribution, Kennedy (1964) argued that innovations should occur so as to keep the share of GDP ®Rivera-Batiz and work accruing to capital and labor constant. Samuelson (1965), inspired Romer (1991) refer to this case as the knowledge-based specification. in this area supports the notion of substantial spillovers from past research, e.g. and Henderson (1993). what n and m, correspond to in Empirical Caballero and Jaffee (1993), or Jaffee, Trajtenberg ^An important question is this question precisely within the context of a stylized model, Although it is difficult to answer seems plausible to think of many of the practice. it major inventions of the 20th century, including electricity, new chemicals and plastics, entertainment, and computers, as expanding the set of tasks that labor can perform and the types of goods that labor can produce. In contrast, some of the early important technological improvements, such as the introduction of coke, the hot blast, the Bessemer process, can be viewed as capital-augmenting advances, since they reduced the costs of capital and other nonlabor inputs, see Habakkuk (1962, pp. 1-57-159). by the contributions of Kennedy and firms chocjse M and in A'^ reduced form model where Fellner, constructed a terms of the production function above in order to maximize the He showed instantaneous rate of cost reduction. that under certain conditions, this would imply equalization of factor shares. Samuelson also noted that with capital tecfmical change would tend to be labor-augmenting. (1973), criticized this whole literature, however, because was not clear My who undertook paper change, as in, RfcD among others, Romer for the analysis here, I allow for both is lacked microfoundations: it priced. Anant and Dinopoulos (1990), and Aghion and Howitt (1992, 1998), where innova- (1991a,b), paper it and how they were financed and (1990), Segerstrom, by profit-maximizing tions are carried out rest of the activities, Others, for example Nordhaus but starts from a microeconomic model of technical revisits this territory, Grossman and Helpman The the accrrnivilation, firms. In contrast to these papers, la]:)or- and crucial and capital-augmenting innovations. organized as follows. The next section outlines the basic environment, and characterizes the asymptotic equilibria and the balanced growth path. Section III analyzes transitional dynamics, and shows that with an elasticity of substitu- tion less than the 1, economy tends to a balanced growth path with stable factor shares and labor-augmenting technical progress. Section IV analyzes the consequences of a range of policies on the factor distribution of income. Section V investigates the implications of alternative formulations of the "irmovation possibilities frontier," extends the model to allow for the production and same R&D sectors to compete for labor, and all Modeling The Direction of Technical Change A. The Environment consider an sector, and S scientists for workers. I economy consisting of L "scientists" is VI concludes, the proofs. II. and shows that the results obtain with different formulations of technical change. Section while the Appendix contains I also unskilled workers who perform RfcD. The who work distinction in the production between unskilled workers adopted to ensure that the production and RfcD sectors do not compete This is only to simplify the exposition, and will be relaxed in Section V. assume that the economy admits a representative consumer with the usual constant (CRRA) relative risk aversion preferences: oo c{t)'- / where C (t) When ^ neutral. when is -"'dt 1-e consumption at the time > and ^ t (i; is the elasticity of marginal utility. = 0, the utility function in (1) is linear, and the representative agent is risk When 9-^1, the utility function becomes logarithmic. I drop time arguments this causes The budget no confusion (I use the time arguments in the proofs in the Appendix). constraint of the representative consumer recjuires that consumption and investment expenditm'es are less than total income: C + 1 <ivL^ rK + uJsS + H, where / denotes investment, the capital stock, uis is w is the wage rate of labor, r the wage rate for scientists, and resource constraint of the Y is the interest rate, 11 is total profit The (3) an output aggregate produced from a labor-intensive and a capital-intensive For simplicity, I assume that there is no depreciation of f, K= labor-intensive where is < oo. change in the given by I. (4) (CES) production functions of labor-intensive and intensive intermediates, with elasticity z^ = 1/(1 — yi {l)^di capital- /3): 1//3 1//3 y(/)'s £ and capital-intensive goods are produced competitively from con- stant elasticity of substitution Yr < capital, so the capital stock (and in the representative consLmaer's asset level) where income. 7^^^ +(l-7)i^K good, respectively Yl and Yk, with elasticity of substitution The K denotes economy implies that wL + rK + LosS + n = y where is (2) and Yk denote the intermediate goods and intermediate goods are gross substitutes.^ Vkiifdi /? € (0, 1), so that v (5) > \ and different This formulation implies that there are two *The presence of two types of agents, scientists and workers, causes no problem for tlie representative consumer assumption since with CRRA utility functions these preferences can be aggregated into a CRRA representative consumer. See, for example, Caselli and Ventura (2000). ® Alternatively, preferences could be directly defined over the different varieties of ,y(i), with identical results. different sets of intermediate goods, of those that are produced with labor, 7i An that are produced using only capital. labor-intensive intermediates — corresponds m corresponds to an increase in n increase in — an and 7n expansion in the set of to labor- augmenting technical change, while capital- augmenting technical change. Intermediate goods are supplied by monopolists who hold the relevant patent, and k{il (6) are produced linearly from their respective factors: = rnii) where l{i) for labor and and = and y,CO and capital used k{i) are labor good in the production of /. Mai'ket clearing capital then requires: I / ^0 To l{i) close the model, di [t) =L and k / (?:) di = K. need to specify the innovation I (7) Jo possibilities frontier new the technological possibilities for transforming resources into Ijlueprints for of capital-intensive created by the There it is and labor-intensive intermediates. R&D free-entry into the R&D R&D n bi, b^ and varieties employed by firm invents a new R&D firms. intermediate, and becomes the perpetual monopolist of that firms have access to the following technologies for invention: - = where R&D is, assume that these blueprints are are, in turn, Once an sector. receives a perfectly enforced patent intermediate. who efforts of scientists, I — that bi(f) {Si ) 5, - 6 and - = b,0 6',- (5',) -6, (8) 777 5 are strictly positive constants and decreasing fimction such that d) (s) s is and ((>{) is a continuously differentiable always increasing, and 0(0) denote, respectively, the nimiber of scientists working to discover new < oo. Si and labor-intensive 5';- and capital-intensive intermediates, with the meirket clearing condition Si I also assume that the economy starts at + i S'k = = S. with (9) 77 > and 777 > 0. Eciuation (8) implies a nrrmber of important features: 1. Technical change faster is directed, in the sense that the society (researchers) improvements in one tj-pe of can generate intermediates than the other. This featiu-e -will enable the analysis of whether equilibrium technical change will be labor- or capital- augmenting. The 2. means that there are intra-temporal decreasing fact that 0(-) is decreasing R&D effort; returns to when more scientists are allocated to the invention of labor- This might be, for ex- intensive intermediates, the productivity of each declines. ample, because scientists crowd each other out in competing for the invention of similar intermediates. This decreasing returns assumption — when the analysis of transitional dynamics Sk is constant, the behavior of 5/ and discontinuous. Research 3. 4>{-) is adopted to simplify is devoted to the invention of labor-intensive intermediates, 4>{Si)Si, effort leads to a proportional increase in the supply of these intermediates at the rate while the same effort devoted to the discovery of capital-using intermediates leads to a proportional increase at the rate differ since more The parameters b^.. new the discovery of one type of difficult intermediate bi and b^ potentially may be "technically" than discovering the other type (the standard model with only labor- augmenting technical change can be thought as the special case with assume that the crowding by individual one more bi, R&D effect captured by the function firms, so each scientist to R&D The is = 0). I also not internalized firm takes the productivity of allocating each of the two sectors, bi4>{Si) or deciding which sector to enter. </)(•) 6^: results are identical as given when R&D firms act "non- 6^-0 when (S'fc), competitively" and form global research consortiums, internalizing these crowding- out effects. Each intermediate disappears 4. devoted to a particular type of intermediates, 6 = 0, its when no research effort stock declines exponentially. With at the rate 6, so that there is the results are similar, but there will exist multiple balanced growth paths (see Proposition 4 below). Notice that in knowledge (8) scientists spillovers are "standing on the shoulders of giants" from past research. This type of knowledge growth when technical change uses scarce —benefiting from spillover is necessary for factors, such as labor or scientists (see, e.g., Romer, 1990, Rivera-Batiz and Homer, 1991, or Barro and Sala-i-Martin, equation (8) is a direct generalization of the accumulation equation in the standard en- dogenous gi'owth model where we would have 7n (e.g., equation 1995). In fact, (3) of Romer, 1990). An = by assumption, and h/n = bi4> additional assi_miption implicit in (8) a higher stock of knowledge accumulated in one sector benefits only that sector higher n increases the productivity of scientists working in the n-sector). discussion of this assumption later. I is (S) S that (i.e., a retiu-n to a Finally, define S^ and S^ as the number of technology in each sector constant, Assumption 1: 5;*+ SI which implies that there in both is < i.e., bicp (Sf) scientists required to S'l = and 6 6j.0 (5^) S^ keep the state of = 6.1 impose: S, enough scientists in the society to enable technological progress sectors. Consumer and Firm Decisions B. An good equilibrium in this economy prices, w, saving decisions, scientists r, a;^, [pi{i)]"^o, between the two maximize the [Pk{'i)]iLo^ [^{^)\T=o^ [/(?')ir=0' given by time paths of factor, intermediate and is Pl ^"^d p^, emplo}Tnent, consumption and ^ [y/lOliLc {yk{^)]iLo^ sectors, 5'; and 5^, such that utility of the representative consumer given f^d 7, and the allocation of [y((?)]"=o' [yA-(0]ilo' factor, intermediate ^ '^^^ ^ and good and [/(OliLc [^(0]™ni [P;('0]r=o ^"^ LPfc(0]i^o maximize profits of intermediate imply zero-profits for all R&rD firms, and all markets clear. goods monopolists, 5/ and prices; S'/,. I start with the optimal satisfies consumption path of the representative consumer, which the familiar Euler equation: ''^ %-\[r-p), where recall that r is (10) The consumption sequence the rate of interest. [C'(i)]^ also satisfies the lifetime budget constraint of the representative agent (the no Ponzi lim (—»oo Consumer maximization K r{r)dv exp [t) game constraint): (11) 0. '0 gives the relative price of the capital-intensive good as: = i^(^)", P-^ \iLj PL 7 where I p/^- is the price of Y}^ and pi is the price of Yi. choose the price of the consumption aggregate, [i^pY^ + (1 - iYp^k']'^" PK = [7 V = + - 7)1 , To determine the in level of prices, each period as mmieraire, i.e., impfies that: 1, '^vhich (1 Y (12) ' ' and PL = [Y + (1 - ifP 1-.17^ ^"Equation (10) implies that when consumption grows at a constant rate, the interest rate will be conwhich is a well-known feature of CRRA preferences. It may therefore appear that these preferences ensure a stable interest rate in the long run. This is not the case, since there may not exist an ecjuilibrium with a constant growth rate of consumption. Conversely, if preferences were not CRRA, there could never stant, exist an equilibrium with a constant growth rate of consmnption and constant interest 9 rate. Next, consumer maximization and the lastic demand fyi PL will be fimctions in (5) yield the following isoe- cvirves for intermediates: Pi {i) Given these CES isoelastic {i) Pk and Pk Yr. demands, (14) maximization by the monopolists implies that prices profit markup over marginal set as a constant fVk («) Yu {i) cost (which is lu for the labor-intensive intermediates and r for the capital-intensive intermediates): 1 1 Pi{i) Since, from = ^« and -^ = pk{i) f 1 (15), all labor-intensive intermediates sell at = implies that yi{i) the same price, y^ for all yi, (i) = k and i, for all Then from the market T. = =- l{i) = ^ (15) the same price, equation (14) we obtain yi{i) ^ j since all capital-intensive intermediates also as well. i -- and yk{i) n = k{i) = sell at clearing equation (7), —K (16) m Substituting (16) into (5) and integrating gives the total supply of labor- and capitalintensive goods as: Yl These eciuations reiterate that technologies. Greater = n~T~L and Yk = mT'K. n and in correspond to labor- n enables the production of a (17) and capital-augmenting greater level of Yj, for a given quantity of labor, and similarly an increase in vi raises the productivity of capital. Equations (14), (15), (16) and (17) give the wage rate and the rental rate of capital as: w = [3n Finally, using (12) and = value of a monopolist r{t) is pl and r Pl£ 1-7 PL 7 who = Pm 1-3 i^ px- the interest rate at date t, is good is 1-3 m \ -0h Ln new (19) L J (r(u;) 6 (18) T^-l / invents a exp Vf{t) where " (17), the relative price of the capital intensive P The 1-3 /-intermediate, for + 6)duj Kf{v)dv, / = / or At, is: (20) the depreciation (obsolescence) rate of existing intermediates, and 1-pivL TTZ (3 n , and TTfc 1-prK m (3 10 (21) and are the flow profits from the sale of labor- wage ujs^ Scientists are paid a ensure that this wage find ecjual to the is competition between the two sectors and free-entry maximum of their contribution RA'D the two sectors. Recall that olists in so the marginal value of allocating one intermediates where and bi(f){Si)nVu is and 14 are given by Vi capital-intensive intermediate goods. firms more do not internalize the crowding effects, scientist to the invention of labor-intensive for capital-intensive intermediates, it is h);4>{Sk)mVk, (20). Therefore, free-entry reciuires: = max {6;0 {Si) nV], bkcf) {Sk) rnVk} ujs to the value of monop- Equation (22) implies zero expected (22) . profits for all firms at all point in time, so 11 = in (2). An equilibrium in this and satisfy (18) (22), economy good prices, therefore a set of factor prices, w, r is \pi{i)]^^Q, and u>s that that satisfy (15), intermediate b^k{'i)]^Q, production levels given by (16), output levels given by (17), sequences of aggregate consvimption and investment levels that satisfy (10) and (11), and sequences of Si and 6'^ that satisfy (22). Asymptotic and Balanced Growth Paths C. I to as define an asymptotic path — i hTiit^oo GO, > (- [t) / or \\int-,ooC possibly and does not include C (t) {t) = /C {t) oo, = i.e., at the same is we can have either consiimption grows more than exponentially (explodes), Qc, i.e., the rate of consimiption growth tends to a constant, defined as an finite In an AP, limit cycles." (including the case where limj^ooC'(i) growth path (BGP) grow (AP) as an equilibrium path that the economy tends AP constant rate, = as a special case). A balanced where output, consumption and the capital stock i.e., Vmit^^C {t) /C (t) = Yimt^^Y {t) /Y {t) — \imi^^k{t)/K{t)=g:^ This subsection is will show that with £ < 1 , only BGPs can be an AP, so if the economy going to tend to a non-cycling path, this has to be a BGP. In contrast, with ^ exists asymptotic paths different rate Tq than > 1, there where consmnption grows more than exponentially or grows at a capital. facilitate the analysis, it is useful at this point to define A =n "Unfortunately, I am 1-3 >^ and M=m 1-3 ^^ , unable to rule out limit cycles, except in the case with risk neutrality. See Section III. ^"This definition is con\'enient for the purposes here. consumption and capital grow at different rates as BGP. 11 Some authors also refer to growth paths where which in a with simplifies the notation below, and, together (17), allows me to write output more compact way: y = 7(iVL)^ + In addition, (23) define a normalized capital stock, I , k which (1-7)(MA')" ^ MK ^, (24) a direct generalization of the normalized capital stock defined in the standard is growth models as capital stock divided by the effective units of labor. Here the numerator contains the "effective imits of capital" as well, since there can be capital-augmenting technical change. Then, using (13), (18), (19) and (24), we can write the = R{M, k)=p{l-j) M jk-^ + r (1 - interest rate as: '^' 7) (25) • Also, define the "relative share of capital", a^-, as^^ = —r=pk = (TK ^-k— luL The relationship between the relative share of capital depends on s, which is if £ > 1 and labor , and Now we will is also the elasticity of substitution in this economy. In response to decrease 1: With ^ technical change, is capital stock if < £ an increase in k, gk between will also increase 1 can state (proof in the Appendix): Proposition This and the normalized the elasticity of substitution between capital-intensive and labor- intensive goods. Equation (26) shows that s capital 26 7 the first < i.e., 1, all APs are BGPs and they have limt_,oo M [t) important result of the paper. feature purely labor-augmenting /M [t) = It 0. demonstrates that with s < 1, i.e., with labor and capital as gross complements, the only asjmiptotic (non-cycling) paths feature purely labor-augmenting technical change. There will to the invention of capital-intensive intermediates, but this technology in that sector at a constant is be research ' ^ Note that 2: • With £ > 1 , devoted only to keep the state of level. For completeness, the next proposition covers the cases with £ Proposition effort will > 1 and s = 1. there are three APs: this "relative share of capital" leaves out the nominator. 12 income accruing to scientists from the de- 1. limt^oc, C (0 /C (t) = lim.^oo M /M (t) = [t) 3. = S; K (0 /!< {t) = liint-,o. Y {t) /Y (t) limt-^,^ k {t) / K = [t) l\mt^,^Y < g and cxi {t) /Y = [t] oo and and lim(_,oo Sk lim,^oo ^ (0 /'^ (0 = gc<oo. Ihnt^.y,. K (f) /A' (0 = g^ < (t) = 0; = \imt-,ooC{t)/C{t) 2. limt-,.x> g,, and lim,^oo S^ (i) 0. With • With the tion to the 5 = 1, there is a Linique elasticity of substitution BGP AP which is a BGP. between capital and labor greater than with purely labor- augmenting technical change, there is 1 and another equilibrium path where consumption grows stant finite rate greater than the rate of growth of the capital stock, change stable, is We labor augmenting. and the economy will below that the will in fact see but since ^ tV, purely at the con- all technical in this case not is = 1, Cobb-Douglcis, the type of technical change does is not matter, and the only possible asymptotic equilibrium path M and and is tend to one of the two other APs. In the case with £ the aggregate production function growth of both BGP in addi- an equilibrium path where consumption grows faster than exponentially, and technical change capital augmenting, , = 1, is a BGP both of these are neutral, (which features i.e., neither capital nor labor augmenting). D. Growth Path Characterization of Balanced We saw above that with change can be an AP. 5 < 1, Now I show BGP BGP as long as (5 > 0, this eqmlibriuni path. from the Euler equation, Moreover, since from Proposition constant. with pvirely lal^or-augmenting technical that there in fact exists a imicjue and characterize the properties of First note that only a (10), the 1 M /M BGP = rate of interest has to be 0, ecjuation (25) immediately implies that the price index for capital-intensive goods, pi^, and therefore, the relative price of capital-intensive goods, p, BGP, In addition, in gi'ow at a common output, rate, g. — (3) (or N Y , the wage rate, w, and the capital stock. A', will rate. Therefore, has to grow at the rate with g). We allowing for the depreciation of technologies at the rate M constant, n has to grow at the can then integrate equation 6, and the growth of w, to obtain the values of inventing labor- and capital-intensive goods T. 1 -P all Furthermore, for p to remain constant, (12) implies that Yi and Yk should grow at the same rate (5g/ (1 must remain constant. wL/n as: 1-p rK/m K (20), and n, Notice that these vakies also grow at a constant rate along the The denominator n are growing. growth rate the balanced growth path, while BGP, p and Recall that in remaining scientists is are constant, so there = S/bk, i.e., BGP grows along Sk is no net capital-augmenting = SI as defined above. The (10) p > 0. then gives the when the growth BGP interest rate as r* = p+9g*. The interest rate is higher in order to convince consumers to delay consumption, and the elasticity of marginal utility, 6, determines how strong this needs to be. Let k clear M tt;, its and remains constant. 71 ensures that g* rate has to be higher is because denominator of in the is Vfc K therefore The Euler equation effect from that of because w, work on labor-augmenting technical change. The growth rate of will P 1 m m This implies (f){Sk)Sk technical change. Assumption different is lower than that of 14: n, which is the economy for V/ BGP and = G{M) such that from (25) that G' This k. is so capital has to > M and k are consistent with BGP r* = R{M, —that there a increasing relationship between k)). (i.e., strictly is is, It because a greater k implies a lower price of capital-intensive goods, become more productive, i.e., M has to increase in order to keep the interest rate at r*. Next, be the let k* stock and at M/M technical change, = i.e., level of normalized capital such that at this normalized capital R&D firms are indifferent between capital- and labor-augmenting 0, bi4>{S — Sl,)7iVi = bk4>{Sl)7nVk, or from equation (27), bi^{S-Sl)xoL r* This implies that, at k ^ with g* given by +6-{l-2p)g*/{l-p) = M* technology that + r* k*, the relative share of capital, 5 (29) - g* ax, must satisfy: k4>{S-Sl){l-p){p + 6 + {e-l)g*) -bk(t>{si){{i-p){p + 6) + i{i-p){e-i) + p)g*y _.._ (28). In = k*^ be such that k* is ^ ' other words, using ecjuation (26), we have: k Finally, let bk4>{Sl)r*K = ' (i?^) G{M*), i.e., ^=^ M* ''^' is = ^*- the level of capital-augmenting consistent with the equilibrium interest rate taking 14 ^^^^ its BGP value when k = r* and the when result, = A: MM = 0, N /N while M~ and k* relative share of capital will BGP, In gies, As a k*. be h* > 0. — < have to make equal which 0. new we need conditions profits, so M = M* can therefore state (proof Proposition = k 3: labor-intensive ^ by M (31), = so that r (29) and This proposition is (30) to hold, and I > 6 0. = M* = Then there exists a unique = G''^ {k*), r r* = p + Oy*, A; BGP there the second main result of the paper. It In this BGP, most is just is = equal to aj^ RfeD b*, firms ai'e Intuitively, re- enough — that result, despite gi^owth capital deepening, factor shares remain constant in the long run. relative share of capital where characterizes the unique features purely labor-augmenting technical change. no net capital-augmenting technical change. As a is A;*, and output, capital-augmenting technical change to keep the productivity of capital constant is, = (28). devoted to the invention of labor-intensive intermediates. There is i.e.. r*. consumption and wages grow at the rate g* given by search we in the text): Suppose that £ k* as given BGP, which and capital-intensive This implies that firms working to invent both types of goods in turn requires that We the interest rate will be equal to there were no research directed at capital-intensive intermediates, if M/M would have , Because of the depreciation of technolo- there must be both research to invent intermediates M* and when the just indifferent between in- venting capital-intensive and labor-intensive intermediates; so in equilibrium they allocate their effort We between the two sectors precisely to keep the relative share of capital have already seen that when change is 5 < different initial conditions, the CRRA Given the and growth — change a BGP BGP with purely labor-augmenting technical the only possible asymptotic equilibriimr path. below that, i_mder certain conditions, est rate the 1. is this economy BGP will is In addition, we need not surprising. Wliat is M tend towards = M* — i.e., we will see also dynamically stable, so starting from this preferences, the conclusion that for a rate, at b*. growth path. BGP with constant inter- no net capital-augmenting technical important (perhaps siu-prising), however, is that such exists despite the possibility of capital-augmenting technical change.^'' The results but there are g* (given We saw by are similar in spirit now many there is balcinced growth paths. (28) evaluated at (5 = in Proposition 2 that with £ We when 0), > 1, no technological depreciation, i.e., 5 These paths have the same growth = 0, rate, but different factor distributions of income. This there are other equihbrium paths with capital-augmenting and V that the equilibrium path with purely laboraugmenting technical change is unstable and that for other formulations of the innovation possibilities frontier such a balanced growth path typically fails to exist. technical change. will also will see in Sections III 15 reflects that the equilibrium in 6 at 6 = correspondence Summarizing (proof 0. Proposition M > M* = G-i (k*), lower-hemi continuous, but not continuous, in the text): ^ Suppose that £ 4: is 1 where k* and is 6 = BGP Then, there exists a 0. given by (30) and (31) with <5 = 0. In for all each BGPs, output, consumption, wages, and the capital stock grow at the same rate g* given by (28) with 6 = 0, and the share of labor normalized capital stock, k The is = G (M), intuition for the multiplicity of reqmred BGP for a is and a BGPs is constant. Each BGP has a different different relative share of capital, aj^. is simple: without depreciation, that all that labor-augmenting improvements should be more profitable than capital-augmenting improvements, T4 i.e. ^ ^^nd this ^'^h can happen range of for a capital (labor) shares. Dynamics Transitonal III. The previous (when 6 > section established the existence of a unique balanced growth path with a constant interest rate, stable factor shares and purely labor- 0) augmenting technical change, very much resembling the textbook growth model. Nevertheless, balanced growt.h would be of limited interest no other APs are in Proposition 1 that cient to establish stability, since there dynamics in I to this by possible, but this, can also be limit cycles. I now BGP. itself, is I will establish analyze local stability, already not suffi- discuss transitional economy. Unforti_mately, the transitional dynamics are rather this to analyze. So starting from an arbitrary capital economy did not tend stock and factor distribution of income, the showed if, and then prove global difficult stability in a special case. The key is less than unique result 1, i.e., BGP when the that when s < 1, elasticity of substitution dynamics transitional will between capital and labor take the economy towards the with purely labor- augmenting technical change. Along the transition path, however, there change. is will also In contrast, —that M be net capital-augmenting technical change when £ > 1, the economy will tend to an is, AP that is will also not a (explosive growth or different asymptotic growth rates of consumption and capital) A. BGP . Local Stability The key Proposition stable result in this section 5: when Suppose 6 > £ < 1, 0. is that: Then the BGP and vmstable when s 16 > characterized above 1. is locally saddle-path This proposition proved in the Appendix. The argument is BGP, the equihbriuni behavior k = MK/NL, latter and SV, two are control c approximated by four hnear is = C/K. The variables. show I is standard: around the differential two of those are state first Appendix in the that, with r equations in variables, while the < 1, the set of linear has two positive and two negative eigenvalues, and differential equations M, is thus locally saddle-path stable. The recjuires (Ta' > b* intuition for local stability can cr^- = then , < 6^-0 {SD mVk > With b*. £ the relative share of capital, aj^, I, {S bi(f) technical change than along the (Ta' is be obtained from equations — S'l) and there nVi, BGP. This when £ > arbitrarily close to Finally, when 1 k. If be more capital-augmenting will BGP. BGP (31). decreasing in Clearly, this But because argument applies and the economy moves away from the BGP, even when , it starts it. £ = 1, the economy is identical to a standard with a CobbDouglas production fimction, jind the BGP endogenous growth model locally (and globally) stable. is Global Stability with Risk Neutrality B. I next characterize the global stability properties in the special case where ^ where the representative agent is and implies that k will increase.'^ decreasing in k, the economy will approach the in reverse is (30) is risk neutral. I also = 0, i.e., assume that negative consimrption allowed. This immediately implies that the interest rate, r, always has to be equal to the discoimt rate p, and removes the Euler ecjuation of the representative consumer, (10), and the that at becomes a control capital stock (and therefore k) also all points in time p = R {k, M) where R {k, M) is variable. This ensures given by (25). In other words, the relationship k = G [M) has to hold at all points in time, and as before, G is strictly increasing in M, with k* = G (A/*). These properties imply (proof in the Appendix); LemiTia 1: With 6* = 0, the transitional d3Tiamics of the ^= where V [M*) = i'{M) I This lemma < 1, i' {M) = (32) for all M = M* and when 5 > 1, implies that transitional dynamics can be represented by Figures 3 and '^Unfortunately, this 0, and when s (A/) M = M*. for all 4 for the cases with £ m/m > 0. i' economy are given by < is 1 and £ > I, respectively. Inspection of these figures immediately not true globally, since, in general, k because we can have K /K < 0. may Hence, the argument here 17 not increase despite the fact that is a local one. implies that the BGP The intmtion the same as in the last subsection: is is when globally stable e < and globally 1, with s < ^instable when 1, when M s > 1. and k are BGP levels, there will be capital-augmenting technical change, reducing both towards their BGP levels. Because the dynamics of k are pinned down by the behavior of M via the equation k = G (M), we can also rule out limit cycles, and the result is one of global stability. In contrast, with £ > 1, levels of M and k greater than M* and k* above their lead to further increases, taking the economy towards the asymptotic path with augmenting technical change and explosive growth, while capital- M < M* leads to the AP with purely labor-augmenting technical change, and consumption growing faster than capital. Finally, as noted above, the economy with s The next Proposition IV. With 9 = when s > 0, BGP the always converges to the imique BGP. In this section, when £ < 1, Dynamics analyze the effect of policy on the factor distribution of income I income The main II. is result of this analysis is that the long-run independent of fiscal policy and labor market approximately independent of the discount rate (the savings with the implications of the standard gi'owt.h rate). policy, countries, tax and This result contrasts model with only labor- augmenting technical change, where such policies would affect the long-run factor distribution of income. many OECD and 1. model of Section factor distribution of globally (saddle-path) stable is Policy and Comparative in the basic 1 proposition simimarizes these results (proof in the text): 6: unstable = and labor market policies In have changed substantially over the past 100 years (see Persson and Tabellini, 2003, on tax policies, and Saint-Paul, 2000, on labor market policies). Figures distribution of income, but is difficult to reconcile change, but is in line Changes and 2 show large medium-run changes no long-run changes. The long-rim in the factor stability of factor shares with the standard model with only labor-augmenting technical with the predictions of the framework outlined the discussion, in this section, A. 1 I focus on the case with in Capital e: < here.^*"' To simplify 1. Income Taxation and Discount Rates First consider taxation of capital income at some rate t. Assume that the proceeds from capital income taxation are distributed lumpsum to consumers. This implies that ^^ Obviously, long-run stability of factor shares in response to policy changes is consistent with the Cobb-Douglas production function, but with such a production function, we caiinot explain/analyze short-run and medium-run swings in factor shares as those shown in Figures 1 and 2. 18 the budget constraint of the representative agent changes to C + I < wL + where r is {1 - that pre-tax interest rate and t) T rK + usS + H + T, is the hrmpsiim redistribution to consLuners from the proceeds of taxation. The government budget constraint implies that Clearly, the resource constraint of the the representative consumer after-tax one, (1 - which at g, M = will also equation (10), the and be the BGP - r) r - is of the p)/e. consider the case of exogenous labor-augmenting technical first 0, ((1 rrK. The Euler equation identical to (3). is similar to (10), except that the relevant interest rate C/C = r) r, thus For comparison, change, where is economy T= N = BGP e^'. The BGP growth rate now exogenously is given growth rate of consumption. Therefore, from the Euler after-tax interest rate has to satisfy r* still = {p + 6g) / (1 — r). Since the pre-tax interest rate must equal the marginal product of capital, this also implies: /3(l-7)A/k-^ + (l-7)l^ = J L where k is the BGP ^, — 1 value of the normalized capital stock in this case. immediately implies a decreasing relationship between r and k there is only labor-augmenting technical change, so elasticity of substitution, c, is less reduces the BGP = than 1, an increase M is This equation (recall that, by assumption, As long constant). as the income taxation in the rate of capital value of the normalized capital, eind through this channel, increases the share of capital income in case where £ r 1, i.e., GDP > (the case with s \ when the production function would give the reverse). Only in the takes the Cobb-Douglas form, is the long-run factor distribution of income independent of the rate of capital income taxation. Now cal consider the case where both capital-augmenting and labor-augmenting techni- change are allowed and endogenous. Equation (10) of consumption, but for balanced growth we need still determines the rate of growth to have both M = and N > 0, which implies that there should be both capital-augmenting and labor-augTnenting innovations (otherwise we would have interest rate, equilibrium factor distribution of BGP still income M/M < requires a^is 0). = Since firm profits depend on the pre-tax b* <==^ k = k* Therefore, the long-nm . unaffected by capital income taxation. In addition, in consiimption must grow at the rate g* as given by (28), and so the Euler equation (10) implies that the pre-tax interest rate has to satisfy r = (p + 6g*) / (1 therefore an increasing function of the rate of capital income taxation. to ensure both capitalinterest rate, K/NL, also and labor-augmenting research, and since k falls. = MK/NL, is M — t), Since k and = is A;*, has to increase to raise the also implies that capital to effective labor ratio, Therefore, with endogenoios capital- and labor-augmenting technical 19 change, capital income taxation reduces the capital-labor ratio, but creates an exactly and offsetting capital- augmenting technical change, leaves the long-rim factor distribution of income vmchanged. Next, consider a change in the discount rate The p. analysis is standard model with only labor-augmenting technical progress, this ings rate, the capital-labor ratio R&D will requires still gk = towards both types of technologies. b*. so that on the factor distribution of income because the change BGP interest rate faced and through r, in p BGP growth rate zero. is when Similarly, Therefore changes in the discount rate on the g* is this channel, it will be an change b* the and when the be second order. have small or second-order effects small, this effect will will generally will will also influence Inspection of ecjuation (30) immediately shows that this effect disappears k*. in the remains profitable However, now there effect by consumers, it In the change the sav- and the factor distribution of income. In contrast, framework here, long-run equilibrium to imdertake analogous. factor distribution of income. Labor Market Policy B. Next to analyze labor market policy imposes a (binding) level of in a simple way, minimum wage w, and suppose that the government moreover, indexes this minimum wage to the income. In particular, assume that w= x~^ {fK + wL) , > ^^^ L is the level of employment, which is now determined endogenously.^^ the minimum wage is binding, the eciuilibrium wage rate has to be lu = w at all where x Since points in time. Multiplying both sides of this equation by L, and rearranging we obtain the quasi-labor supply curve, relating employment, L, to the relative share of capital, ax- L = X"' I refer to (33) as the quasi-labor (1 + (33) 'JK-r' supply cm-ve of the economy, since any equilibrium has to be along this curve. As before, BGP requires long-run share of capital in of employment. An M/M GDP increase in will \- = 0, thus a^ = be imchanged will via this channel raise k (recall that k b* and k = k*. Therefore, the —irrespective of the equilibrium level immediately reduce employment, however, and = MK/NL). In the case where the elasticity of ^ This expression makes the minimum wage proportional to the sum of capital and labor income rather than total income, which also includes scientists' earnings. This is only to simplify the expressions, without any substantive implications. ' 20 substitution between capital and labor, BGP Subsequently, the economy adjusts back to process, the share of labor in relates this GDP falls, employment to the share than £, is less 1, the labor share will also increase. starting with k and returns to k* Thi-oughout this . its initial level in GDP, employment of labor in > BGP. Since (33) also falls steadily during adjustment process. This result is interesting in light of the developments in many European kets over the past several decades. For example, Blanchard (1997) unemployment and the labor share in a number labor mar- documents that both of continental European economies rose Hammour sharply starting in the late 1960s. Both Blanchard and Caballero and (1998) interpret this as the response of these economies to a wage-push; the militancy and/or the bargaining power of workers increased because of changes in labor market regulations taking place over this time period, or because of the ideological effects of 1968. This wage-push translated into higher wages and lower employment. During the 1980s, we see a different pattern: unemployment in these countries continues to increase, but the labor share falls sharply. Blanchard documents that the decline explained by capital-labor substitution, and conjectures that ased technical change''. The framework presented here wage-push shock, in response to a and unemployment and employment increase. Then The falls further. i.e., is it may have been due to "bi- consistent with these patterns: GDP an increase in Xi both the share of labor in as technology adjusts, fall cannot be in the labor share in k is k returns to accompanied by an which corresponds to capital-biased technical change. BGP value, A;*, offsetting decline in M, its ^^ Discussion and Extensions V. The analysis so far has established that in a natural capital-augmenting technical change, there is model with potentially labor- and a unique balanced growth path equilibrium with no net capital-augmenting technical change, stable factor shares and a constant long-rim interest rate. Moreover, as long as capital and labor are gross complements the elasticity of substitution is less than the economy converges to this 1), (i.e., BGP. This analysis relied on a number of assumptions. For example, technical change took the form of invention of new R&D goods; sectors did not RfcD compete w£is carried for labor; on the number of existing goods, with out by scientists so that the production and and productivity spillovers in the from past research. of these assumptions are importeint for the substantive results. '"Because the elasticity of substitution, technical change. and See Acenioglu (2002) s, is less for than 1. R&:D a decline in We I sector now will see depended clarify which that the only M corresponds to "capital-biased" a discussion of the relationship between factor-augmenting factor-bicised technical change. 21 important assumption (or the form of spillovers Possibilities Frontier There are two important assumptions embedded in tier (8): first, Second, there R&D possibilities frontier from past research). The Innovation A. form of the innovation for the results is the this innovation possibilities fron- uses a scarce factor (scientists, or labor as in the next subsection). a specific form of spillovers from past research; an increase in n raises is the productivity of R&D in the n-sector, but not in the 7n-sector. dependence, since the relative productivities of R&D in the I refer to this as state- two sectors depend on the state of the system, (n, m). Let us now relax each of these two assumptions on the form of the innovation possi- The bilities frontier. an alternative to R&D sector using scarce labor what Romer and is Rivera-Batiz (1991) refer to as the lab-eqmpment model where the final good (or capital) is used for R&D. For example, we coLild have (implicitly setting (/){) = 1 in terms of (8) to simplify the notation) n where Xi and Xk are the R&D m = bkX^ — 5m, — 6n and biXi two sectors expenditvires in the good, and the resource constraint needs to be modified to important point is that long-rim growtrh from past research, because Consecjuently, there in the two sectors is is also biVi or 6fc 1 and bkVk = 1, is BGP rK = is we need m= 0, condition (35) implies that good. R&D R&D now requires used to invent 6/ labor-intensive recjuires: wL bi m. (35) . n and w, n and K final remains unchanged. not consistent with balanced gi'owth, however. to remain constant, the BGP final Y. The relative productivity of rest of the setup since one unit of final output h This condition —only the so far applies, but the free-entry condition into capital-intensive goods. Therefore, terms of the possible without knowledge spillovers no state-dependence, since the The in C + / + A'; + Xk = does not use any scarce factors always constant. ^^ Much of the analysis = R&D now is (34) and 7n will K For the interest rate to grow at the same rate. But grow together. Therefore, with the innovation possibilities frontier given as in the lab-equipment specification, there exists no BGP (though there exist other '^Equation (34) is APs with constant gi'owth of consumption). equivalent to the formulation of the innovation possibility frontier I used in Acemoglu (1998) in the context of technical change directed at skilled and unskilled labor. See Acemoglu (2002) for a more detailed discussion of the implications of different forms of the innovation possibilities 22 frontier. Next, us retmii to the formulation with scientists undertaking let from past research, but modify In (36), there are But there case. is remove state-dependence (and again (8) to n = hirfm}'''*'Si still spillovers — 6n and 7h ^ R&D in one of the sectors which leads to equation (35) as a uJs- also, there is = 1): (36) affects in this both sectors This contrasts with (8) where current research for the invention of R&D for labor-intensive goods in the = R&D now requires 6;V/ BGP condition. As a result, in this case future, but not for capital-intensive goods. Free-entry into bkVf; set 0(-) — 6m. bf;n'^Tn^~'^Sk labor-intensive goods increases the producti\ity of and spillovers from past research to ensure long-run growth no state-dependence: ecjually in the future. = RfcD and cvg no BGP. Therefore, the BGP with purely labor-augmenting technical change is only consis- tent with an innovation possibilities frontier with a strong degree of state-dependence. Intuitively, venting ing new balanced growth with capital accumulation requires the profitability of new in- capital-intensive goods not to increase faster than the profitability of invent- labor-intensive goods —so that in equilibrium firms are happy to imdertake only labor-augmenting improvements. Since capital accumulation increases the profitability of research towards capital-augmenting technologies, a strong form of state-dependence in the R&D technology, whereby labor-augmenting technical change raises the profitability of further research towards labor-augmenting technologies, necessary to balance this is eS-ect.20 Competition For Labor Between Production and B. In the baseline model, there are two tj^pes of workers, unskilled labor R&D and scientists, with scientists specializing in R&:D and thus no feedback from the relative price of labor to growth. This assumption was made an innovation possibilities frontier '-'Is Unfortunately, analyzed (1992) by, I am among in -Jaffe (1993), the model to (8), plausible? The data on patent citations and Henderson (1993), Trajtenberg, Henderson and may be A now modify with a strong degree of state-dependence, like others, Jaffe, Trajtenberg citations of patents by other innovations. rele\-ant in this context. Tliese citation of a previous patent is .JafFe papers study subsequent interpreted as evidence that a by the previous invention. This corresponds to some from past research. One can therefore use patent citations data to in\'estigate whether is degree of spillo\er I not aware of anj- direct im'estigation of this issue. and Caballero and current in\-ention only for simplicity, and e.xploiting information generated state-dependence at the industry level. Industrj' level state-dependence corresponds to patents which they originated. Results reported in Table 1 in Trajtenberg, Henderson and .Jaffe (1992) suggest that there is some amount of industry- state-dependence. For example, patents are likely to be cited in the same three-digit industry from which they originated. Nevertheless, it is currently impossible to investigate state-dependence at the factor level. This is because, although we have infoniiation about the industry' for which the patent was developed, we do not know which factor the inno\'ation was directed at. there is being cited in the same industrs- in 23 R&D allow the production and let sectors to us again focus on the case with ix - = where L; employed in production. condition is employed in the L+ R&D R&D of the analysis eciuation (22) is 1. from Section replaced by w= — 2P) g = bik, R&D 1, is compete sectors condition binVi where f and g are the =w wage workers for workers. R&D now for scientists), M needs to remain new to inventing binVi is by workers are invented Taax{binVi,bkmVk}. In BGP, we need L^ the labor market clearing (rather than the some research devoted The bkTnVk given by (27) above. and L but the free-entry condition into wage rate intermediates to balance depreciation. Thus, (1 for labor-intensive goods, new goods production and to the (37) for capital-intensive goods, II applies, new innovation constant, so there has to be f + (5— R&D In this framework, sector, so the simplify the discussion, 6, m Normalizing total labor supply to + L^ = L; relates the value of a and in — = hkLk - and 6 To for labor. Ecjuation (8) then changes to 1. the number of workers employed in is the number of workers employed in Most - hiLi n = () compete = bkmVk = capital-intensive lu, with binVi and immediately implies that in BGP: BGP interest and growt.h Now using rates. the Euler equation for consumption, (10), we have {2p Furthermore, in pg/ {1 labor, — P), and BGP ., = ^ (1 + pg/ {1 — - P)Pb,k - rest of the analysis is and stable factor 6/bi (38), the long-rim ^ The m/m = we again have also L; and equation + 6-iyg + shares. 6 +p= 0, bi^. (38) which implies Lk P)bi. Using the 6/bk, is = and h/n market clearing condition growth rate of the economy for therefore given by: + - (1 - P)bi6 {i-P)pb,e + p6-{i-P)P{i-2p)bk (1 - P)Pbk = (p rmchanged. In particular, As (5) BGP before, along the balanced reqi.iires m = 0, hence ax = b* growth path, technical change is pm-ely labor-augmenting, with research towards capital-augmenting goods only to keep the net productivity of capital constant. complicated, however, because both the In this case, transitional dynamics are more number of production workers and the speed of technical progress change along the transition path. C. Different Forms of Technical Progress Labor- augmenting technical change has so far been interpreted as "labor-using" change, that is, the introduction of new goods and 24 tasks that use labor. I now show that the results of the above analysis generalize to different formulations of the techno- change process, including a model of technological progress with new varieties of logical machines, and one where technical change takes the form of Grossman and Helpman (1991a,b) and Aghion and Howitt run technical change cjuality improvements as in In both cases, long- (1992). be labor-augmenting, with no net capital-augmenting technical will change. To new discuss the consequences of technical change resulting from the invention of machines, let me modify the basic framework such that the two goods have the following production functions: where z; (j) this quantity of the the quantity of the j-th machine complementing labor, and Zk [j) j-ih. machine complementing capital. This is two factors are not accumulable, and more solution can be found there. Notice that n and types of machines complementing these two factors. monopolists, while producers of Y^ and depreciate at the rate ^ 1 in terms of the from and profit [j) X';. final zi {j) = for these Xk U) = (^ + + 5, ^) / (1 ~ is r z^. ., /?)-^' (j) {pk/v) . = (1 is K, where a constant n and Pr r = ^ Xi (j) depend on the we can _„ (1^)""^ n \r + interest rate. Next, see that The monopolist BGP will originally markup over Therefore, p,'- m. profits of technolog}' monopolists are: and ... Ml -« (i-l)'"^. m \r + assuming the innovation recjuires a^- produce 2; (39) J profits are identical to those in (21), except for the constant depreciated. normalized to straightforward to derive is — is Next, h'om market clearing, factor prices are J (8), assume that machines the interest rate plus the depreciation rate. These equations imply that the by machines L and {pl/Xi U)) I new machine profit-maximizing monopoly price of machines w= These These machines are supplied by cost of producing a The demand on the denote the user cost of machines. Since the demand curves for machines are marginal cost, which = and the 0, good. maximization: isoelastic, the Xi [j) > details the numbers of different competitive. cire V/^- now are the the model used in Acemoglu (2002), for the case where the m is = b* , and the fact that they possibilities frontier is given and we obtain exactly the same (or Zk) units, results and then replace the machines that have and as in Sections II the introduction of This demonstrates that whether technical change III. new labor-intensive and capital-intensive goods or as the invention of labor-enhancing and capital-enhancing machines The second up the possibility now produced = yL ^ where zl and zk are Ql and Qk -|- (1 {qW-') L' and Yk = j^ {qW-') K^ machines that complement labor and ciuantities of capital, and > 0, or a new vintage of capital-complement the machines, + X)Qk- This R&D firm would be the monopoly supplier of and at the flow rate Q'j^ it = would dominate the market scientist 6; (1 who works until a to discover a (or bk). Notice that this new new, and better, vintage Ql vintage of BGP the resulting improvement in the "level" of productivity loss of generality, and normalize the marginal I Qk) before: research leads to proportionately better machines, so the greater Without {or arrives. successful is assumption already builds in knowledge- based were required for the existence of a spillovers that is (40) \)Ql, where A assume that a Ql competitively with the production functions vintage of labor-complementary machines, with productivity this vintage, quality replace old ones. Suppose new with productivity I new goods/machines two formulations denote the qualities of these machines. Technical progress results when an firm discovers a = immaterial. one where technical progress takes the form of firms moving features creative destruction: it the two goods are Q'l^ is quality ladder (vertical innovations), which differs from the other because R&D is modeled as is is on a vintage of Qi, the greater XQl)- (i.e. assume here that machines depreciate cost of producing z to 1/(1 is + A). I fully after use, assume that A also small enough that the leading monopolist will set a limit price to ensvire that the next best vintage breaks even (see, for the marginal cost of production Zl = is 1/(1 QlL and zk = p^ QkK- Pl example, Grossman and Helpman, 1991b). Since -f- A), the limit price /3(1 — BGP values of inventing new V, = ^"^ . (l and (5;,- Vi > P{1 14 is + A)(r + ., — P)~^Pjl- bi = (l <5,) fej.^; and 5^ = ij-^fc. + A)(r + introduced technological obsolescence i.e.. By From the above as- Similar reasoning to before implies that Si =S (in addition to 26 (39). and 5fc)' consistent with stable factor shares. Therefore, along the only be labor-augmenting technical change, Q ^'^ and V," are the endogenous rates of creative destruction. sumptions, we have only = straightforward (higher) quality intermediate goods are bi r it is Hence, ^- P)~^Pl Ql, and profits are given by an ecjuation similar to (21) or standard arguments, the where Xk — Xl ~ Substituting these into (40), to verify that the eciuilibriimi interest and wage rates are: w= is and Sk = 0. BGP, Because there will I have not the endogenous creative destruction already present in these models), now BGP requires V; a range of labor shares consistent with V^ rather than V] = so there Vj., is BGP. Conclusion VI. Almost all existing models of economic growth rely the production function labor-augmenting. Much on one of two assumptions: either supposed to be Cobb-Douglas (an is between capital and labor exactly equal to 1. > 1), or all evidence suggests that the elasticity of substitution technical change eleisticity augmenting technical change more is factor distribution of A It is less ecjual to 1 than does not model with purely labor- attractive, but poses the question of no capital-augmenting technological improvements. assumed to be of substitution Moreover, a framework with an elasticity of substitution exactly enable an analysis of medium-run changes in factor shares. is why there are also suggests that the long-rim income should be a fimction of tax and labor market policies and of the savings rate, while in the data, the long-run factor distribution of income appears to be stable despite changes in these variables. This paper studied the determinants of the direction of technical change in a model where the invention of new production methods is a purposeful activitj^ Profit-maximizing firms can introduce capital- and/or labor-augmenting technological improvements. major result is that, with the standard long-rrm technical change path, the economy looks will like The assumptions used to generate endogenous growth, be purely labor-augmenting. Along the balance growth the standard model with a steadily increasing wage rate and a constant interest rate. Therefore, the framework here offers a microfoundation for the standard neoclassical growth model with (exogenous or endogenous) labor-augmenting technical change. But, typically it also shows that away from the balanced growth path, there be capital-augmenting technical change. Furthermore, policies that affect the long-run factor distribution of no long-term It effects in this income I will showed that a range of in the standard model have model. has to be noted that the results here hold i.mder a very specific form of the innova- tion possibilities frontier, and the discussion in subsection V.A indicated that with other forms, a balanced growth path with purely labor-augmenting technical change typically fails or to exist. why Work on why technical change area for future research. this may be form of the innovation possibilities frontier labor-augmenting with other plausible forms is is plausible, a fruitful Appendix: Proofs VII. Throughout this appendix, changeably. In addition, I first = x and x [t) use the notation 11™^.,^^ x{t) I —> x inter- some state the following resvilt which will be useful in of the proofs: Lemma Al: Let < Suppose that s 1. ,._ 7n{t)V,{t) ^' n{t)mt)- Then, Umt^oo k = {t) =^^ hmt^oo A [t) = M oo, limf^r» [t) /M {t) = S-6) //?, and lim^^oo N {t) /N {t) = - {I - (3) 6/15. And limt^^ k{t) = oo=^ \imt-,^ A [t) = 0, limt_oo M [t) jM it) = -(!-/?) ^//?, and limi^oo N (t) /N [t) = {I - (3) {S) S - b) //?. (1 - {h(t> [S) P) {bicf) > Suppose that e \. = Then, limf^oo k[t) => limt_,oo A (i) = oo oo, limt^oo S-6) //?, and Wmt^^ N (t) /N (i) = -(!-/?) 6/p. And limt^oo k{t) = 0=> fimt^oo A (t) = 0, hm^^oo M (i) /M {t) =. and limt^oo N (i) /N (f) = (!-/?) (6^0 {S) S - 6) /p. (1 - M {t) /M [t) {hcj^ {S) P) Lemma Proof of We Al: (1 - /3) 6/p, have from (27) that E-l ^' w{v)L(v) y, As k{t) -^ 0, therefore Si {t) (1 - all p) {h(p {S) k ^ {s) —> exp[-/;(r(w) + «)<ia,],o(tOiH 7, and Sk S-6)/p > for all s {t) -^ S. and lim.^oo When t. s < I, this impHes /N (i) = - [t) {I - p) ^ A (i) M oo, and {t) /M (t) = The other cases fol- limt_,oo C" (i) /C (t) = This immediately gives limt_,oo N ^' 7, 6 /p. low analogously. Proof of Proposition oo cannot be be a BGP limi_,oo K First, eciuilibria, 1: and that any equilibrium path with where limt_,oo with pm-ely labor-augmenting technical change, /K {t) = (t) I will /M (f = 0. limf^oo C {t) /C [t) = ^ and prove that a contradiction, suppose this lim(_,oo The proof will show that paths with y [t) /Y {t) = linit^oo k {t) = oo, oo, Y {t) = i—»oo lim lil (t) is which (^) /^ (0 = show N{t)L 7 + t—KX (1 oo cannot be an equilibrium. To derive Then from the budget is and note that, from lim lim^^oo^ ) the case. I will i.e., C [i) /C [t) = g must - constraint, (2), not possible. To start with, take the case (23), output can be written <r-l j)k {t)~ = lim t—*oo 28 we need A {t) as: j—L = since, with 5 < 1, k (t)^ -. 7 + /N (1 {t) - < -> oo. Therefore, hm;^,^ {t) '^' is -> ^ (0 /^^ (0 = lim^^oo Y so limt_oo 0, Y {t) /Y =constant. then clearly. [t) where k oo. Finally, consider the case j)k (f)^ neither case A: where hnii^oo k oo. Next, take the case A'^ [t) as {t) /Y (f —f ) °o possible, so [t) which case, 0. in limt_,oo N {t) /N ^(0/^(0 = lim(_,,3o N {t) /N {t) < {t) = [t) < As a oo. C (t) jC (f) = resrUt, in oo cannot be an equilibrium. This implies that Xvaxi^^.^C limt^oo [i) C {t) APs must have all C {t) jC (f = g. Next will show liTat^oo M [t) /M {t) = 0, and that in this limt_oo jC {t) = g also necessitates /C {t) = limt^^ Y (t) /Y [t) - I ) \\mt-.^ K [t) jK [t) = g. which \^ill that case complete the proof. consumption to grow at the constant First, recall that for Euler ecjuation (10), the interest rate r (i) = /3M {t) p^- {t) to rate, we need, from the remain constant. Recall also that 1 PK it) = j{i-jr'k{tr E-l +(i-7r ^ (Al) J which implies j{i-jf-^k{ty PK it) 1 PK [t) ^j{l-^f~'k[t)- Constant interest rate, r(i) i.e., = 0, then M{t) To derive a contradiction, first (A2) .^(0 (1 - 7)'- ^ \^J +(1-7) reciuires: PK{ty suppose that M (i) /M {t) < 0, then pK (i) /px {t) > 0, < 0, or k{t) —^ 0. But then from Lemma Al, > 0, giving a contradiction with A/ (t) /M [t) < 0. which, from (A2), implies k{t)/k{t) M Next suppose that M ^ k{t) ^ /M {t) (i) /M [t) > 0, which reciuires p^ {t) /pk {t) > 0, which, from (A2), implies k (t) /k {t) > 0, or (t) — oo. But then, from Lemma Al, we have A {t) -^ 0. Thus, M (0 /M (i) < 0, contradicting the supposition that M (i) /M {t) > 0. Therefore, we must have M (i) /M {t) = 0, and pK (t) /pk (0 = 0, and thus, k {t) jk [t) k {t) -> implies limt_,oo (i) A- 0. by This also implies that limt^oo (28) in the text. limt^oo (^ {t) {t) K jK [t) [t] = limj^oo N [t) /N Then, we immediately have that limi^oo /C (i) = g*. /C (t) = g* {t) Y {t) = g* with g* as given /Y (0 — f/*i ^^i^l thus completing the proof. Proof of Proposition Wmt—ryoC" > 2: Th.e case and llmt^oo J^-f with s [t) > I. /M [t) = 29 First, it is outHned clear that the BGP in Proposition 1, with where < limt_^oo k {t) /k Proposition 1 — (t) Moreover, the same argument as in the proof of 0, is still possible. implies that there exists Second, consider the case where limi^oo k £>1 Al with and limj^oo M N [t) immediately implies that limj^oo {t) /N = - (t) - (1 6/0. Next /3) AP no other with limi^oo k /k M [t] > jM (t) = {t) > (23) with s /M {t)+K (i) /K (t), and r {t) -^ r (t) = (3M (f) (1 limt^oo y (0 /Y (i) = limt^oo K {t) /K (i) = 00 is an AP. (t) Finally, consider the case Al gives M hm,^oo {t) (23), in turn, together lim Y [t) t—>oo Thus limt_oo 00 /M (t) = with lim t—oo Y (i) /Y (t) = impossible from is where ^ and e N{t)L 7 + - {I A; /k (i) {t) < -(!-/?) 6/f3 and lim^^oo (f) /c lim(_,oo P) (bicf) > k 0, i.e., e/(e-l) {bk(p {S) S- {t) — 0. [t) /N (f) = {I > 6) //?, Y {t) /Y Thus limf_oo C 00. k 0, i.e., Lemma {t) (i) /C (i) = Then Lemma - P) {bi<P (5) S- implies that \, lim t—00 5- 0. —> oo, then p) = 7) N = {t) implies that limj^oo 1 (l-7)A;(t)- (5) {t) - {1 /k (i) 5) //? < 00, but Nit)^^^ then limt_.^ C (t) jC [t) - (2). So we must have limj^oo C [t) /C [t] = Qc < oo. Then, for consiimption to have a constant growth rate, the Euler equation (10) requires the interest rate to remain constant. Prom (A2) with linif^oo i^ {t) k{t) -^ /K (t) = {1 0, this — requires k {S) {bi4> p) rate of consimiption growt.h, which S— is, [t) /k sS) = sM {t) //?, /M (i) = (t) -e {1 - P) djp, or giving us another AP, with a constant however, not a BGP, since limf_,oo K {t) /K [t) < \imt^^C{t)/C{t). = 1. As £ — 1, the aggregate production fimction becomes CobbDouglas, Y = B {NLy {MK) ~^, and the equilibrium relative share of capital is always ax — (1 — 7) /7- Therefore, the equilibriimi allocation of research effort is given by The case with £ s- 1-7 kcP (5 all S,) {p points in time, where g^ = — lim.^oo C (0 /C {t) = lim,^oo Y [t) /Y (i) = g* By standard arguments, Proof of Proposition using k k (2), (8), _ ~ K_ and (17), we 5: 5 + eg*- +e gj) -9n) bi<j>{S-S,){S-S,) bi4> {S - Sk) {S - Sk) -6,gn 7i/K+(l 7) g^. + bk(t>{s,){p^6 7 at - = hk(j) [Sk) S^ - 6, and the unique eciuilibrium path has limj^oo First, exploiting the definition of K {t) M N c {bkcP{Sk)Sk-bi(P{Si)Si] K 30 / K k given by (24), and obtain: K~^Ji~N' j{NL)~ + {l-j){MKy = g* [t) 6) /(3. M = jk-'^ + il-^) +^ where {b,4> (5',) recall that c - 5, = C/K. (A3) bicp - {S - 5,) [S S,)) Linearizing around the BGP. we have the first linear differ- k*) (A4) ential equation of the four-equation system: k - = with aks > 0, aks {Sk 7 Ofr,, using the definition c - + SI) ' (A:*) = C/K, akm {M - M*) + + - (1 > ^ 7) akc {c 0, the Euler equation - c*) + = -1 < a^.f. and an- < and equations (8), - a^- {k (10) Similarly, 0. and (25): K C (A5) C~^K 1 1 = -(/9(l-7)M 7A;-^ + (l-7) M Linearizing around the BGP a-cm with a^c ac^ where = = ^/5 a^.^ > 1 0, 7^-^+(l-7) is P c gives the second linear differential equation: {M - M*) + ace (c - c*) + a^- (A; - (A6) A;*) and (1-7) 7(^-*)"~ + (l-7) > + - defined above, and In addition, a^h is > a' Q.m - ^kr, 0. yk-^ + the derivative of = 7 (A:*)-— + (1-7; (1 -7) { where both terms have negative derivatives, hence < Qca- '-' i/?(l -7) -7A;-^ - (1 0. Next, recall that (8) gives: M M Linearizing this relationship, 1 - 9 M(5V)SV.-<5] p we obtain the M— M with a^, > third differential equation: 0,ms {Sk 0. 31 — SI) , (A7) -7) Finally, recall that in BGP, we hict) Define = (pi 4> — {S Sk) and {S have: - = 7nVk bk<t> (S'fc) {Sk) to simplify the notation, 4> 0^, Sk) nVi and differentiate the above equation to obtain: where efc = ating (20), 0' {Sk) Sk n Vi Sk Sk n Vi Sk < Ski 4) {Sk) and TT/ and jVk - {S Sk) (A8) Vk Sk/(f> {S - Sk) < 0. Next differenti- and rVk 5Vi - F, = - iVk (A9) <514, given by (21). Therefore, Sk - [Gk + m m ei\ Sk - [ek + a constant proportional to the C = (pilp + eg* + 6 - - 2(3) BGP g*/ fom-th differential equation (1 — 1 n /? (5 - fact that is {I h 4>kSk ei] where the second step uses the oiu- 0' Vk we have rVi-Vi = 7Vtwith = e; m m , <Pi {S - we are cus/wL - p)r\ f wL rK \nVi mVk Sk) + in the [b* ^-j^<Pk - ok) neighborhood of the BGP, and C ratio given by, Also recall that [e^ + e;]"^ < 0. Therefore, is: Sk ass {Sk - SI) + a^k {k - k*) (AlO) , Sk with Uss > and Usk Expressing these > 0. ecjuations together, fotu- / Sk/Sk \ 1 M/M Since M and A: \ aks I are state variables \ 1 ^^ \ M ams r-^ k/k ask ass c/c \ we have acm ace ack akm akc a.kk and Sk and \ k 1 J c are control variables, requires the determinant in (All) to have two positive a.ss - ams det A ask V aks \ -A acm akm = dec akc 32 '^ ack akk saddle-path stability and two negative eigenvalues. To find these eigenvalues, write: / (All) c - ^ / 0, which gives the following forth-order polynomial: - A^ + [uss — + ace The four roots, Aj, A2, — [amsO-kmO-sk -]-ams + a^k] A' OsA- " — CLkm - akcO-ck A' aksO-sk] =0. + UssO-ccClkk] ^ CLsslkcIck [O^cc • - [assUkk ^kc ^cm) ' and A4 give the eigenvalues. By standard arguments we have A.3 that these four roots satisfy: Ai Now • A2 • using the fact that Occ > Since a^m > 0- Qsi- 0, and A3 = A4 • = Ums • {Cikm 0.sk o^c = — 1 and '^l ^^2 A3 a/.,„ > 1 , A4 • ^ ams " acm) have , Qctti- Ai either foiu- positive roots, or four negative roots, or O-kc — akm we cl'^^ dsk we have that 0. — O-cc = Ccm ? ' • A2 A3 • > A4 So we must have 0. two positive and two negative roots. Next also note that Aj A2 + Aj • A3 + Ai • A4 + A2 • A3 + A2 • + A4 A3 = A4 • Uss —dkc Qkk —CLks O-ck -(-)•(-) (+)•(-) -(+)(+) which means that we can have neither four positive roots nor four negative roots, establishing that there must be some must be two positive and two negative With c > 1, we have agk and some negative positive roots, < 0, roots, so the thus Aj A2 • • A3 system A4 • is BGP have three positive and one negative eigenvalues, and the roots. Therefore, there locally saddle-path stable. = — Gms ' ' ^sA- < a'^^^ equilibrimn 0. and we not locally is stable. Proof of Lemma 1: I will lemma prove this that if bi(p*n{t)Vi[t) < bk0lm{t)Vk{t) where > bi(()*in{tf)Vi{if) < bk(plm{t')Vk{t') for all (f>*i t' dynamics can be represented by (32), where ti' = t. (A/*) two in (p{S Next = 0, steps. First, = - S*^) and I uill show that and (A/) (pl = I I will show 0(5^), then transitional for all M =| M* 0 (M) | for all M | M* for £ > 1. Recall that k{t) = G{M{t)) at all i, with G'(-) strictly increasing. Next recall that bi4)ln [t) Vi [t) = bk^lm (t) Vk («), when aj^- = 6*, or when M = M* and k = k*. Also in this case, M = and N/N = g*. Now I will show that these properties imply that, for £ < 1, Gji [t) > h* if and only if 6;0^n (i) V; (f) < bkCplrn {t) \4 {t). Moreover, these imply > {t') for all bi(p*n {t') V; (f < bkdlm {t') for £ < 1 and ) V'l, t' t. Clsk < Suppose that a^ > biifiJi (i) Vi (i) and thus m m< bkcplm Vk bk<P {Sk (i)) {t) + bi(f>*in{t [t) > bk(plin (t) \4 (i) Vk First {t). some S^ for {t) and to derive a contradiction, suppose b*, if this is the case, < (t) Now {t) if bi(/)*in (t) At)Vi{t + > Vi (i) fefe^^m (f) 14 = At) + At)Vk{t + = At) therefore, limt^oo k (i) ''™ {t) Vk (0 /n (t) Vi (0 — bk<f> (0) m and 6/0^n At -^ + 6i<^;n(i)VH0 \4 {t) (t) V] (t) {t)), > (i) V; {t) 0, we have bi(t,*n {t) Vi {t) > > bk4>lm{t)Vk{t) bu<t)lm{t)Vk{t) bk<t)lm{t)Vk[t) M °°! giving a contradiction. Therefore, bi(j)*n{t)Vi {t) = G (M (t)), we have establishing that to b* t' M addition, recall that < > n {t)) > A; (i) < < bkcplm k* and therefore {t) Vk {t) But whenever . M (t) < M*. 1, M aK{t) when {M*) cta- (i) < M*, we (t) < M = M* ip b*, . > = = 0. (i) 6*, M\ The arguments Finally, ax it (0 = b*, M (i) < M* ^ < b* {t) b*, from /M (t) = V' (M (t)). In [t) = M* and k (t) = k*, (t) and 34 0, Since 0, > And (Ta' (i) M[t) /M[t) > we have M[t) /M{t) < for a^- (f) > cr^' (t) > ax has already been shown that in the case with we have M[t) /M{t) > have that M{t) > requires whenever This estabhshes that the M therefore M law of motion of the economy can be represented simply by £ S^ > bk(t>lm (t) \4 {t) implies m < 0, which in tvirn, from implies k (t) < 0. So a^ [t) > 0. Therefore, we must have > t, and consequently (i) /M {t) < 0, and bk4>lm [t') 14 {t') at all /k (t) < 0, and hmt_>oo k (i) = 0. But Lemma Al implies that in this case we must have (t) {t) Vi {t) ax (0 > + - have i;0 (5 recall that bi(t)*n [i] Vi [t) A; k will Third, note that for [t). bi(l)*n{t)Vi{t) {t) = G {M (i)), bi(p*in {t') Vi {if) > b*, n (5) bi<t> Second, from (A9), we can only have 0. > bk(f>lm{t 5* (or we also that 0. proving that for the case where £ > ?/; 1 b* is equivalent similarly, (M) = when for all are analogous. VIII. References "Why Do New Acemoglu, Daron, Technologies Complement Skills? Directed Tech- Change and Wage Inequality" Quarterly Journal of Economics, CXIII nical (1998), 1055- 1090. Acemoglu, Daron, "Directed Tectmical Change" Review of Economic Studies, LXIX (2002), 781-810. Acemoglu, Daron and Joshua Angrist, "How Large Argument Capital Externalities? Evidence From Compulsory Schooling Laws" Bernanke and Kenneth Rogoff, Aghion, Philippe and tion" Econom.etrica, LX "A Model Howitt, MacroAnnual, 2000, edited by Ben Cambridge, pp. JN/nT Press, F'eter NBER of 9-58. Growth Thi'ough Creative Destruc- (1992), 323-351. Aghion, Philippe and Peter Howitt, Endogenous Growth. Theory, Cambridge, MIT MA, Press, 1998. Ahmad, Syed, "On The Theory of Induced Invention," Economic Journal LXXVI, (1966), 344-357. Antras, Pol, "Is the U.S. Aggregate Production Function Cobb-Douglas? mates of the Elasticity of Substitution" MIT mimeo New Esti- (2001). Barro, Robert and Xavier Sala-i-Martin. Economic Growth, McGraw Hill, New York (1995). Medium Run" Blanchard, Olivier, "The Brookings Papers on Economic Activity, (1998). Berndt, Ernst, "Reconciling Alternative Estimates of the Elasticity of Substitution," Review of Economics and Statistics 58, No. l.,(1976) pp. 59-68. Boyce, William E. and Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 3rd Edition, John Wiley and Sons, New York, Caballero, Ricardo ity, and J. Mohammad 1977. Hamnioiu" "Jobless Growth: Appropriabil- Factor Substitution and Unemployment" Carnegie- Rochester Conference Proceedings (1998) vol. 48, pp. 51-94. Caballero, Ricardo J. and Adeim B. Jaife, "How High are the Giants' Shoulders: An Empirical Assessment of Knowledge Spillovers and Creative Destruction in a Model of Economic Growth" Caselli, tribution" NBER Macroeconomics Annual 1993, Francesco and Jaume Ventura "A Representative Consiimer Theory American Economic Review Chirinko, Robert S. 15-74. (2000). "Business Fixed Investment: Strategies. Empirical Results of Dis- a Critical Survey of Modeling and Policy Implications" Journal of Economic 35 Literature, XXXI, (1993), 1875-1911. Chirinko, Robert S., Steven M. Fezari and Business Capital Formation to Its Andrew User Cost?" Mayer, P. Journal of Public "How Responsive Is Economics, LXXV, (1999), 53-80. Chirinko, Robert A S., Steven M. Fezari and Andi-ew P. Mayer, "That Elusive Elasticity: Long- Panel Approach to Estimating the Price Sensitivity of Business Capital" Federal Reserve Bank of Louis, working paper (2001). St. David, Paul, Technical Choice, Innovation and Economic Growth: Essays on American and British Experience in the Nineteenth Century, London, Cambridge University Press, 1975. David, Paul, and Th. Van de Klundert, "Biased Efficiency Growth and Capital-Labor Substitution in the U.S., 1899-1960," American Economic Review, LV, (1965), 357-393. Edmund Phelps, "A Model of Induced Economic Journal, CXXVI (1965), 823-840. Drandakis, E. and Distribution" Eisner, Robert, and M. Review of Economics and I. Nadiri, "Investment Behavior Invention, Growth and and the Neoclassical Theory," Statistics, (1968) pp. 369-382. William Fellner, "Two Propositions in the Theory of Induced Innovations The Eco- nomic Journal", LXXI, (1961), 305-308. Griffin, James M. and Paul R. Gregory, "An Intercountry Translog Model Substitution Responses", American Economic Review, LXVI LUX in the MA, MIT Habakkuk, H. J., Theory of Growth" (1991a), 43-61. Grossman, Gene and Elhanan Helpman, Innovation and Growth omy, Cambridge, Energy (1976), 845-857. Grossman, Gene and Elhanan Helpman, "Quality Ladders Review of Economic Studies, of in the Global Econ- Press, 1991b. American and British Technology in the Nineteenth Century: Search for Labor Saving Inventions, Cambridge LIniversity Press, 1962. Hamermesh, David S., Labor Dem.and, Princeton Llniversity Press, Princeton 1993. Hayami, Yujiro and Vernon Ruttan, "Factor Prices and Technical Change ti.iral Development: the LI.S. in Agricul- and Japan, 1880-1960" Journal of Political Economy, LXXII (1970), 1115-1141. Hicks, .John Jaffe, izaion of Adam The Theory of Wages, Macmillan, London, 1932. B., Knowledge Manuel Trajtenberg, and Rebecca Henderson, "Geographic LocalSpillovers as Evidenced by Patent Citations" Quarterly Journal of Economics, CVIII (1993), 577-598. Jones, Charles I. "Sources of U.S. Economic Growth in a World of Ideas" American 36 Econom.ic Review, XCII (2002), 220-239. Jones, Larry E. and Rodolfo E. Manuelli, Theory and Policy Implications" Joum.al of "A Convex Model Political of Equilibrium Growth: Economy. IIC (1990), 1008-1038. Jorgenson, Dale W., Frank M. Gollop, and Barbara M. Fraumeni. Productivity and U. S. Economic Growth, Cambridge, MA, Harvard University Kennedy, Charles, "Induced Bias LXXIV Economic Journal Innovation and the Theory of Distribution" in (1964), 541-547. Ohanian and Victor Rios-Rull and Giovanni Violante, "Capital Krusell, Per; Lee Complementary and Press. 1987. Skill LXIIX, Econometrica, (2000), 1029-1053. Inequality," Lucas, Robert E., "Labor-Capital Substitution in U.S. Manufacturing," in The Taxation of Income ington, from DC: The Brookings Lucas, Robert E., tary Economics, Bailey, Wash- the mechanics of economic development." Journal of Mone- Capital, edited XXII Mairesse, .Jacques, "On (1988), 3-42. Bronwyn H. Years" Annales d'Economie M. I. An Hall and Benoit Mulkay, "Firm-Level Investment Exploration over "Some Approaches to What We Have Returned (editor) Twenty Theory and Measurement of Total Factor Pro- Economic Literature, VIII 8, (1970), pp. 1117-77. Nerlove, Mark, "Recent Empirical Studies of the M. Brown in in LV, (1999), 27-67. et Statistiques, ductivity: a Survey" Journal of tions" in J. Institution, (1969) pp. 223-274. France and the United States: Nadiri, by Arnold Harberger and Martin CES and Related Production Func- The Theory and Empirical Analysis of ProductionNew York (1967) Nordhaus, William, "Some Skeptical Thoughts on the Theory of Induced Innovation" Quarterly Journal of Economics, LXXXXTI. (1973), 208-219. Perrson, Torsten and Guido Tabellini, The Do the Data Say? forthcoming Piketty, Thomas. "Income MIT Press, Economic Effects of Constitutions: What Cambridge, (2003). Inecjuality in France, 1901 - 1998." CEPREMAP mimeo, (2001). Piketty, - 1998." Thomas and Ernannuel CEPREMAP Saez, "Income Inequality in the United States, 1913 and Harvard mimeo, 2001. Rebelo, Sergio, "Long-Run Policy Analysis and litical Economy 99 Long-Run Growth" Journal of Po- (1991). 500-521. Rivera-Batiz, Luis A. and Paul M. Romer, "Economic Integration and Endogenous Growth" Quarterly Journal of Economics. 106 (1991), 531-555. Romer, Paul M., "Increasing Returns and Long-Run Growth" Journal of 37 Political Economy 94 (1986), 1002-1037. Romer, Paul M., "Endogenous Technological Change" Journal of Political Economy, lie (1990), S71-S102. Saint- Paul, Gilles, The Political versity Press, (2000) Oxford, United Salter W. Economy of Labor Market Institutions, Oxford Uni- Kingdom. E.G., Productivity and Technical Change, 2nd edition, Cambridge Univer- sity Press, (1966) Samuelson, Cambridge, United Kingdom. Paxil, "A Theory of Induced Innovations Along Kennedy- Weisacker Lines" Review of Economics and Statistics, XLVII (1965), 444-464. Schmookler, Jacob, Invention and Economic Growth, Cambridge, Harvard University Press, 1966. Segerstrom, Paul S., T. C. A. Anant, and Elias Dinopoulos, "A Schumpeterian Model of the Product Life Cycle" American Economic Review 80 (1990), 1077-1092. Trajtenberg, Manuel, Rebecca Henderson, and Jaffe, Corporate Lab: Adam B., "Ivory Tower Vs. an Empirical Study of Basic Research and Appropriatability" working paper 4146,1992. 38 NBER 90% oa 85% •a •a o 80% "5 > A TO 2 75% 'I o \/\ / m A ^/-\N t\ V^ / S 70% A 65% ^^ A^ K^ vy^ / / "\ V 60% 3> n CO r* Figiu-e 1: Labor share 1 a> CO 1 CD in total value lO o added Saez (2001). Source: National Accounts, CD o CO to 00 00 in the U.S. corporate sector NIPA Table 39 1,16. 00 CO a> 05 from Piketty and 90% M V a> Figure 2: ^ CM o> Labor share « n on r- n m in total value T— t o> lO t a> added o> Tt o> in the (2001) based on French National Accounts. 40 ^ (O o> lO to at o> U5 o> CO ho> French corporate hha> ^ CO o> sector. lO 00 O) a> CO o» to o> a> t~. o> O) From Piketty p {b,(/>{S)S-d^) p M Figure 3: Transitional dynamics uith risk neutrality 41 and £ < I. M_ M \-p 5 Figure 3837 4: Transitional d}Tiainics with risk neutrality .37 42 and ^ > 1. 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