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Using the manufacturing productivity distribution to evaluate growth theories
Structural Change and Economic Dynamics 17 (2006) 248–258
Using the manufacturing productivity distribution to evaluate growth theories Jens J. Kr¨uger ∗ Department of Economics, Friedrich-Schiller-University Jena, Carl-Zeiss-Strasse 3, D-07743 Jena, Germany Received October 2004; received in revised form June 2005; accepted December 2005 Available online 23 February 2006
Abstract Some multi-sector endogenous growth models contain strong predictions about productivity differences across sectors in the form of a distribution or density function. In this paper it is demonstrated that this distribution is left-skewed for a wide range of plausible parameter values. This stands in contrast to the right-skewed shape of the respective empirical distribution estimated by kernel methods for a measure of relative productivity with data for more than 450 four-digit US manufacturing industries during 1958–1996. This difference can be traced back to the assumption of strong intertemporal technological spillover effects that are at work in these models. © 2006 Elsevier B.V. All rights reserved. JEL classification: D24; O47; L60 Keywords: Multi-sector growth models; Manufacturing productivity distribution; Skewness
1. Introduction Recent multi-sector Schumpeterian growth models are endowed with microfoundations that lead to specific theoretical predictions about differences of the productivity levels across sectors. These predictions imply that even on a balanced growth path there exits heterogeneity of the productivity levels across sectors, which can be expressed in the form of a stationary distribution. In the models of Aghion and Howitt (1998, chapter 3) and Aghion et al. (2001) the density function associated with this distribution is either explicitly derived or numerically computed ∗
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through simulation. In both models the shape and in particular the skewness of this distribution crucially depend on the size of an innovative step on the quality ladder, which is treated as an exogenous parameter. In this paper, the value of the step-size on the quality ladder that governs productivity growth or the associated cost reduction is calibrated using information from a variety of sources, such as the values used in numerical studies of the above-mentioned growth models. In addition, a plausible range of values is derived from the stylized result of the 80% learning curve that is representative for a wide variety of manufacturing production processes. All these values lead to a left-skewed theoretical distribution of the relative productivity levels. This finding stands in contrast to the distribution of relative productivity levels across the four-digit US manufacturing industries, which is estimated using kernel methods and appears to be consistently right-skewed during the years 1958–1996. The paper proceeds as follows: Section 2 provides a brief sketch of the derivation of the theoretical distribution of relative productivity levels and depicts the shape of the density function for various values of the innovation step-size. Section 3 calibrates the plausible range of values of the innovation step-size using the 80% learning curve result. Section 4 calculates relative productivity levels for the four-digit US manufacturing industries using nonparametric methods of efficiency measurement and presents the respective kernel density plots. The diametricallyopposed skewness properties of the theoretical and empirical distributions point to the need of paying more attention to the specification of the intertemporal technological spillover effects that are at work in these models. This issue is discussed in the concluding Section 5. 2. The theoretical distribution In most multi-sector growth models the sectors of the economy are treated symmetrically in every respect; Romer (1990) is a leading example. This implies that all sectors have the same productivity level and thus, the distribution of productivity across sectors is a one-point distribution. This feature makes these models useless for analyses concerning differential growth of industries and structural change in form of the change of the sectoral composition of an economy. Some other multi-sector growth models such as the model of Aghion and Howitt (1998, chapter 3) make explicit predictions about the shape of the distribution of productivity across sectors, while retaining the symmetry assumption in other respects. In this model the state of knowledge of the economy at time t is described by the level of the productivity parameter of the “leading-edge” technology Amax . Each sector of the economy t invests a certain amount of labor in research that determines the Poisson arrival rate λnit of new innovations, where nit denotes the amount of labor input invested in research in sector i at time t and λ is a positive parameter. An innovation in a particular sector i leads to a jump of the productivity level of this sector Ait to Amax . The growth of Amax is governed by the aggregate t t flow of innovations in the economy of λnt per unit of time, where nt is the total research labor input of the economy. The steady-state growth rate of aggregate productivity is then derived as d ln Amax /dt = λnt ln γ, which is identical to the steady-state growth rate of output. Therein, t γ > 1 denotes the step-size on the quality ladder in the form of a constant factor by which each innovation increases the productivity level. In the model, a strong intertemporal technological spillover effect is at work postulating that each innovation serves as the basis for discovering other innovations in other sectors of the economy, even though the current innovation can only be used by the generating sector. This spillover effect plays a key role for the explanation of the differences of the theoretical and empirical results to be discussed later in the paper.
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Fig. 1. Density functions of the theoretical distribution of relative productivity.
The different productivity levels over the whole continuum of sectors can be succinctly summarized by a distribution of the productivity levels across sectors on the support [0, Amax ]. t This support extends to the right as Amax grows and the distribution also becomes increasingly t left-skewed. Aghion and Howitt normalize all productivity levels by Amax to reach the relative t ∈ [0, 1] and derive the cross-sectional cumulative distribution productivity levels ait = Ait /Amax t function of these relative productivity levels, which is given by H(a) = a1/ln γ for a ∈ [0,1], irrespective of what happens to the aggregate rate of innovation over time.1 The density function associated with this theoretical distribution is h(a) = (a1/ln γ−1 )/ln γ for a ∈ [0,1] and zero otherwise. Fig. 1 shows this density function for various values of γ. Since the shape of the density function is exclusively determined by the step-size of the quality ladder, γ, we just have to find reasonable values for this parameter in order to compare the shape of this theoretical distribution with the shape of an empirical distribution that is estimated from productivity data of industries. It is immediate that the theoretical distribution is left-skewed for all 1 < γ < e, where, as usual, e = 2.71828. . . is the base of the natural logarithm. This range of values contains the benchmark value γ = 1.135 used in the numerical evaluations of the step-by-step innovation model of Aghion et al. (2001) in which the time unit is taken to be 1 year (see Aghion et al., 2001, p. 484). This magnitude implies that there are relatively more sectors with productivity levels close to the leading-edge productivity level and therefore small technology gaps. The numerical evaluations of the step-by-step innovation model in Hoernig (2003) use a “middle-of-the-road” parameter value of γ = 1.3, which is also contained in the interval of parameter values that are associated with a left-skewed distribution of relative productivity levels
1
See Appendix A on pages 115 and 116 in Aghion and Howitt (1998) for an elegant derivation of this result.
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(see Hoernig, 2003, p. 251).2 In their related work on step-by-step innovation models, Aghion et al. (2001, Fig. 5 on p. 486) also report the steady-state distribution of the technological leads, which essentially are technology gaps, measured in discrete units of innovation steps. The distribution depicted there is right-skewed, implying a high frequency of small technology gaps and correspondingly a low frequency of large technology gaps. Since the measure of technology gaps is inversely related to the measure of relative productivity levels, it is clear that the step-by-step innovation model also generates a left-skewed distribution for the relative productivity levels. 3. Calibration from the learning curve As the previous section has shown, the value of γ is crucial for the shape of the distribution of productivity across sectors. In this section additional evidence about the magnitude of this parameter is generated by relating it to the basic origins of learning-by-doing: the widely found result of the 80% learning curve. This stylized result states that in many production processes each doubling of the cumulative output is associated with a unit cost reduction of 20% to about 80% of the previous unit cost level, because the workers are becoming increasingly familiar with their tasks. Cooper and Johri (2002) and Jovanovic and Nyarko (1995) contain brief surveys and further references to the learning curve literature. Formally, we start from the assumption of a constant growth rate of the output of a particular sector, g = d ln y(t)/dt, expressed in continuous time. This is consistent with the focus on steadystate dynamics and balanced growth paths in growth theory. The growth path of output is thus y(t) = y0 egt with y(0) = y0 . Integrating output over time gives the cumulative output t t egt y(j)dj = y0 egj dj = y0 . s(t) = g −∞ −∞ Taking logarithms results in ln s(t) = ln y0 − ln g + gt from which the growth rate of the cumulative output can be calculated as d ln s(t)/dt = g. This is identical to the growth rate of output. The time it takes to double the cumulative output td can then be obtained from s(t + td ) = 2s(t) ⇔ y0
eg(t+td ) egt = 2y0 , g g
which implies that egtd = 2 and therefore the doubling time td = ln 2/g. Given that each doubling of cumulative output is associated with a 20% reduction of unit costs, the unit cost dynamics can be stated as c(t + td ) = 0.8c(t). The annual rate of cost reduction z can be deduced from 0.8 = e−ztd and using the previous expression for the doubling time we reach z = − g(ln 0.8/ln 2). Assuming a balanced output growth rate of about 3%, g = 0.03, the doubling time is approximately 23 years and the annual rate of cost reduction is approximately z = 0.01. Alternatively assuming a balanced growth rate of 2% reduces this rate even more. 2
Although the step-by-step innovation model expresses the effect of technological progress in terms of cost reductions, the step-size parameter of the quality ladder is the same as in models where technological progress is expressed directly in terms of productivity improvements. One can say that after k innovations production increases to γ k units of output per unit of labor employed or can say that γ −k units of labor are required to produce one unit of output implying that unit costs are proportional to γ −k . Both statements are equivalent.
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At this stage of analysis, the problem arises that the numerical value for the annual rate of cost reduction cannot be used directly since the model of Aghion et al. (2001) is not formulated in real time but instead uses an index for the innovation sequence. Thus, unit cost dynamics in this model are given by cτ+1 = γ −1 cτ , where γ denotes the step-size of the quality ladder as above and τ = 0, 1, 2, . . . denotes the number of innovations. When innovations arrive with a Poisson rate of λn this innovation count sequence can be translated into real time. Given that ε(t) innovations arrive between t and t + 1 in real time, unit cost dynamics can equivalently be expressed as ln c(t + 1) = ln c(t) − ε(t) ln γ,
with
ε(t) ∼ Po(λn),
where Po(λn) denotes a Poisson distribution with expectation λn. Thus the expected rate of decline of unit costs is E(ln c(t + 1) − ln c(t)) = −λn ln γ (see Aghion and Howitt (1998, p. 59) for this kind of argument). Collecting results and ignoring the expectations operator (or invoking a law of large numbers) we can state that ln c(t + 1) − ln c(t) = −z ≈ −0.01 ≈ −λn ln γ and thereby that γ ≈ e0.01/λn . The latter magnitude is smaller than e if 0.01/λn < 1 and consequently if λn > 0.01. Thus, whenever there is a minimal probability of innovation that leads to a Poisson arrival rate which is larger than 0.01, it follows that γ < e and therefore that the distribution of relative productivity levels is left-skewed. In summary, all exercises performed to deduce γ agree on a range of values that is associated with a left-skewed cross-industry distribution of relative productivity levels. As will be shown in the next section, this result stands in strong contradiction to the right-skewed shape of the empirical distribution of relative productivity levels in the US manufacturing sector. 4. The empirical distribution In this section, the theoretical distribution of relative productivity is confronted with the distribution of an empirically computed relative measure of total factor productivity. To quantify total factor productivity a nonparametric frontier function approach is used that closely corresponds to the definition of the relative productivity variable of the theoretical model. The specific method used here is the Andersen–Petersen variant of data envelopment analysis (Andersen and Petersen, 1993). This is a nonparametric method that calculates an index of total factor productivity by the distance of the input–output combinations of the N industries to a piece-wise linear frontier production function that is determined from quantity data alone without requiring any assumptions about the functional form of the production relationship and without having to rely on price data. The Andersen–Petersen model calculates productivity by computing an index that indicates to which level the output of an industry has to be increased in order to reach a point on the frontier production function that is determined by the observations of the other N − 1 industries, excluding the industry for which productivity is computed. Mathematically, the distance measure to the frontier of industry i in year t is the solution φit of the following linear programming problem ⎫ ⎧ ⎬ ⎨ λh yht ; λh xht ≤ xit ; −i ≥ 0 . max φit : φit yit ≤ ⎭ ⎩ hε{1,...,N}\i
h ∈ {1,...,N}\i
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In the present application yit denotes the single output variable and xit is the 6 × 1-vector of input variables of industry i in year t.3 λ−i denotes the (N − 1)-vector of the weight factors λh , omitting the i-th element.4 The radial distance measure φit essentially quantifies the size of the technological gap towards the frontier function from which the measure of total factor productivity can be obtained by inversion, i.e. φit−1 . The computations of the Andersen–Petersen model for each year separately results in a balanced panel of total factor productivity scores φit−1 (i = 1, . . . , N, t = 1, . . . , T ) for N = 454 four-digit industries5 over the T = 39 years covering the period 1958–1996. The majority of the observations that do not determine the frontier function get assigned a productivity score smaller than unity by the solution of the linear programming problem. Those observations that determine the frontier function get assigned a productivity score equal or larger than unity. To be consistent with the definition of relative productivity in the theoretical model these productivity scores are normalized −1 −1 by ait = φit−1 / max{φ1t , . . . , φNt } for each period and are thus bounded in the interval [0,1] as required for the comparison with the theoretical results. The density functions for each period are nonparametrically estimated by the univariate kernel density estimator (Wand and Jones, 1995, chapter 2) for the normalized relative productivity scores on a grid of points x ∈ [0,1]
N 1 x − ait , fˆ t (x) = K Nht ht i=1
where the standard normal density is used as kernel function K(·) and the bandwidth ht is chosen for each period separately by the Sheather–Jones second generation bandwidth estimator (Sheather and Jones, 1991). This bandwidth estimator has been advocated as the preferred method for one-dimensional kernel density estimation in the comparison of Jones et al. (1996). The upper panel of Fig. 2 shows the kernel density estimates of the cross-section productivity distribution for the years 1958, 1969, 1985 and 1996, which are chosen according to Jorgenson (1990, 2001) and provide an approximately equidistant division of the sample period. The productivity distributions show no clear pattern of change during the sampling period. Part of this diversity may be attributed to the relativity of the productivity measures towards the frontier functions, which are themselves subject to change from year to year. Despite this, it is apparent that the density functions are consistently right-skewed with the bulk of the industries showing relatively low productivity scores and a few industries having substantially larger productivity scores. This stands in contradiction to the theoretical distribution, which is left-skewed for a wide range of reasonable parameter values of the innovation step-size.
3 The data used to calculate the productivity scores are taken from the NBER-CES manufacturing industry database which is described in detail by Bartelsman and Gray (1996). This unique database provides consistent annual time series over the period 1958–1996 for quantity and price data of more than 450 manufacturing industries on the four-digit level of aggregation. The specification used takes the real value of shipments as the output variable. The two labor input variables are the number of non-production workers and the production worker hours. Capital input is represented by the real equipment capital stock and the real structures capital stock. Finally, the two variables that represent the input of materials and energy are the real cost of non-energy materials and the real expenditures on fuels and electricity, respectively. 4 This procedure is completely deterministic. There exists an alternative econometric approach to the estimation of frontier functions that promises to be able to divide measurement error from the productivity measure (see e.g. Greene, 1993). However, the Monte Carlo studies of Banker et al. (1993) and Ruggiero (1999) show that this advantage of the econometric approach is absent in small- to medium-sized samples. 5 For the productivity computations five industries had to be eliminated from the database.
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Fig. 2. Density functions of the empirical distribution of relative productivity.
The years chosen in the upper panel of Fig. 2 are in no way exceptional as the lower panel shows in which the kernel density estimates for all years between 1958 and 1996 are plotted together. The findings regarding the diversity and skewness of the productivity distributions are also robust to variations of the input–output specification used in the productivity computations (these mostly consist of merging certain input variables). In some of these variations right-skewness appeared to be even more pronounced. Altogether, the outcome of these variations is very assuring for the robustness of the findings reported here. This can be further sharpened by computing the empirical skewness measure visual impression 3 N −1 N (a − a ¯ ) for each year separately, where a¯ t denotes the arithmetic mean of the norit t i=1
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Fig. 3. Skewness of the theoretical distribution.
malized productivity scores of period t. Empirical skewness ranges between 0.0017 and 0.0033 and is thus positive in each year, consistent with the visual impression of the density estimates. With this range at hand we can gain further evidence regarding the step-size parameter γ of the theoretical model. Rewriting the density function as h(a) = cac−1 , a ∈ [0,1] with c = 1/ln γ, the first three moments of the theoretical distribution are easily computed as E(a) = c/(c + 1), E(a2 ) = c/(c + 2) and E(a3 ) = c/(c + 3). This allows to express the skewness of the theoretical distribution by E((a − E(a))3 ) = E(a3 ) − 3E(a2 )E(a)2E(a)3 = −
2c(c − 1) . (c + 1)3 (c + 2)(c + 3)
Equating this formula to the minimum and maximum values of empirical skewness, solving numerically for c and translating back to the γ scale results in the interval [2.947, 3.185] for γ which is completely in the range of values larger than e. This finding is illustrated in Fig. 3, where the solid line is the skewness of the theoretical distribution depending on γ. The horizontal and vertical dashed lines mark the zero line and the position of e, respectively. Consistent with the previous results, skewness is negative for γ < e and the distribution is therefore left-skewed in this range. For γ > e skewness becomes positive and the distribution is right-skewed accordingly. The empirically estimated range of skewness and the associated values of γ are indicated by the dotted lines. These values are in the range of positive skewness, consistent with the right-skewed density found in the productivity data and plotted in Fig. 2 above. 5. Discussion and conclusion The analysis in this paper has shown that the distribution of relative productivity generated by the models of Aghion and Howitt (1998, chapter 3) and Aghion et al. (2001) is fundamentally different from the distribution of a relative measure of total factor productivity for the four-digit
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industries of the US manufacturing sector. Both distributions are consistently unimodal, but the striking difference is that the theoretical cross-industry distribution is left-skewed for a wide range of plausible values for the innovation step-size parameter on which it exclusively depends, whereas its empirical counterpart is consistently right-skewed in each single year during the period 1958–1996. These differences are indeed important because they shed light on the validity of a specific assumption in the multi-sector endogenous growth models considered here. The most compelling explanation for the left-skewed shape of the theoretical distribution can be identified with the force of the strong technological spillover effect that is at work in the multi-sector growth models. To reiterate, this spillover effect postulates that the leading-edge technology discovered by any one industry is the basis for further productivity improvements in all other industries of the economy. As a consequence it is unlikely that the productivity levels differ by much unless the innovation step-size becomes very large. So the productivity levels of all sectors are tied to that of the leading-edge technology, causing the distribution of the normalized productivity levels to be left-skewed. Beyond doubt, technological spillover effects across industries and firms exist (Mohnen, 1996), but it seems to be the case that these spillover effects are much more limited than assumed in the multi-sector growth models. If the spillover effects are limited to some degree, the ties to the leading-edge productivity level are weaker and it is quite plausible that a few industries forge ahead in terms of productivity with the rest lagging behind persistently. Moreover, the shape characteristics of the empirical productivity distribution are not the only piece of evidence against the strong spillover effects across industries and the rapid catchingup toward the leading-edge productivity level. The spillover effects are also at odds with the finding of persistence of the relative productivity levels analyzed in Kr¨uger (2004, 2005b). This persistence implies that industries with relatively high productivity levels tend to sustain high productivity levels in the future and likewise for industries with relatively low productivity levels. If the spillover effects are as strong as assumed in the multi-sector growth models, persistence should only be observed at the upper end of the productivity distribution, however. Thus, not only the shape of the productivity distribution is in conflict with the theoretical prediction as shown in this paper, but also are the changes within the distribution in conflict with the spillover mechanism that is supposed in the multi-sector growth models. This is reinforced by the treatment of the industries as symmetric implying that all industries devote the same amount of resources to research and thus have the same probability of catching-up to the leading-edge technology. Therefore, other models such as that of Metcalfe (1994), Metcalfe et al. (2002) and Montobbio (2002) which are concerned with the explanation of aggregate growth driven by differential rates of productivity growth of the industries together with changes of the structural composition of the economy deserve more attention. The replicator dynamics mechanism used there potentially leads to more divergent developments across industries and sectors. Accordingly, industries grow or shrink in terms of their output shares if their productivity levels are above or below the average, respectively. The most likely outcome of this selection mechanism is that some industries are forging ahead with the rest being eventually marginalized in terms of their output shares. This implies a right-skewed shape of the output share distribution as observed in Kr¨uger (2005a) for exactly the same industry sample that is analyzed in the present paper. Related to that, Harberger (1998) discusses empirical evidence that illustrates the very diverse growth experiences and productivity developments within and across industries. Based on that he points to the need of devoting more attention to the process of differential growth of industries and sectors. In Harberger’s “yeast versus mushrooms” story the view that all sectors expand evenly (like yeast) is confronted with the view that different sectors grow in unpredictable ways (like mushrooms) caused by a multitude of
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influences. He concludes that “the ‘mushrooms’ story prevails just as much among firms within an industry as it does among industries within a sector or broader aggregate” (Harberger, 1998, p. 11). For growth theory this implies that it is important to devote more attention to the differential developments at the level of industries and sectors, not at least because structural change as a medium-run phenomenon is very important for economic policy. All this implies not only that the strong intertemporal technological spillover effect is at odds with the measured productivity differences across industries and sectors, but also that it is difficult to reconcile the equality and invariance of the sector shares that result from the symmetry assumption for all sectors with the empirical evidence. Although important to keep the models of Aghion and Howitt (1998, chapter 3) and Aghion et al. (2001) tractable, both assumptions of multi-sector growth models should be weakened or abandoned to reach a more realistic micro-economically based account of economic growth. The analyses of the interaction of differential productivity growth and structural change in the models of Metcalfe et al. (2002) and Montobbio (2002) show that this is indeed possible without the recourse to explicit equilibrium solutions. However, this critique of a neglected concern with the issues of differential productivity developments and structural change at the sectoral level in growth theory needs not necessarily affect the validity of the assumption of balanced growth of the aggregate magnitudes. At least in the case of the US economy the time series of log GDP per capita over the period 1870–1994 fluctuates remarkably close around a linear time trend (important exceptions being the years of the great depression and the second world war) as demonstrated in Fig. 1 of Jones (2002). Thus, the long-run aggregate output dynamics of the US economy are indeed well-described by a balanced growth path. If structural change occurs at the sectoral level as a result of nonhomothetic preferences, the concept of a generalized balanced growth path introduced by Kongsamut et al. (2001) and Meckl (2002) guarantees for the specifications analyzed in these papers that structural change and aggregate balanced growth are compatible with each other. Acknowledgments I am grateful for the comments of two anonymous referees that substantially improved the paper. All remaining errors are mine. References Aghion, P., Harris, C., Howitt, P., Vickers, J., 2001. Competition, imitation and growth with step-by-step innovation. Review of Economic Studies 68, 467–492. Aghion, P., Howitt, P., 1998. Endogenous Growth Theory. MIT Press, Cambridge, MA. Andersen, P., Petersen, N.C., 1993. A procedure for ranking efficient units in data envelopment analysis. Management Science 39, 1261–1264. Banker, R.D., Gadh, V.M., Gorr, W.L., 1993. A monte carlo comparison of two production frontier estimation methods: corrected ordinary least squares and data envelopment analysis. European Journal of Operational Research 67, 332–343. Bartelsman, E.J., Gray, W., 1996. The NBER manufacturing productivity data base. NBER Technical Working Paper 205. Cooper, R., Johri, A., 2002. Learning-by-doing and aggregate fluctuations. Journal of Monetary Economics 49, 1539–1566. Greene, W.H., 1993. The econometric approach to efficiency analysis. In: Fried, H.O., Lovell, C.A.K., Schmidt, S.S. (Eds.), The Measurement of Productive Efficiency. Oxford University Press, Oxford, pp. 68–119. Harberger, A.C., 1998. A vision of the growth process. American Economic Review 88, 1–32. Hoernig, S.H., 2003. Asymmetry, stability and growth in a step-by-step R&D-race. European Economic Review 47, 245–257.
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Jorgenson, D.W., 1990. Productivity and economic growth. In: Berndt, E.R., Triplett, J.E. (Eds.), Fifty Years of Economic Measurement. University of Chicago Press, Chicago, pp. 19–118. Jorgenson, D.W., 2001. Information technology and the U.S. economy. American Economic Review 91, 1–32. Jones, C.I., 2002. Sources of U.S. economic growth in a world of ideas. American Economic Review 92, 220–239. Jones, M.C., Marron, J.S., Sheather, S.J., 1996. A brief survey of bandwidth selection for density estimation. Journal of the American Statistical Association 91, 401–407. Jovanovic, B., Nyarko, Y., 1995. A Bayesian learning model fitted to a variety of empirical learning curves. Brookings Paper on Economic Activity: Microeconomics, 247–305. Kongsamut, P., Rebelo, S., Xie, D., 2001. Beyond balanced growth. Review of Economic Studies 68, 869–882. Kr¨uger, J.J., 2004. Productivity transitions in the U.S. manufacturing sector. Applied Economics Letters 11, 935–937. Kr¨uger, J.J., 2005a. Structural change in U.S. manufacturing: stationarity and intra-distributional changes. Economics Letters 87, 387–392. Kr¨uger, J.J., 2005b. Productivity and structural change: an empirical investigation of the U.S. manufacturing sector. Unpublished book manuscript. Meckl, J., 2002. Structural change and generalized balanced growth. Journal of Economics 77, 241–266. Metcalfe, J.S., 1994. Competition, Fisher’s principle and increasing returns in the selection process. Journal of Evolutionary Economics 4, 327–346. Metcalfe, J.S., Foster, J., Ramlogan, R., 2002. Adaptive economic growth. CRIC Discussion Paper No. 59. Mohnen, P., 1996. R&D externalities and productivity growth. STI Review 18, 39–66. Montobbio, F., 2002. An evolutionary model of industrial growth and structural change. Structural Change and Economic Dynamics 13, 387–414. Romer, P.M., 1990. Endogenous technological change. Journal of Political Economy 98, S71–S102. Ruggiero, J., 1999. Efficiency estimation and error decomposition in the stochastic frontier model: a monte carlo analysis. European Journal of Operational Research 115, 555–563. Sheather, S.J., Jones, M.C., 1991. A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society 53 Series B, 683–690. Wand, M.P., Jones, M.C., 1995. Kernel Smoothing. Chapman&Hall, London.
Using the manufacturing productivity distribution to evaluate growth theories Jens J. Kr¨uger ∗ Department of Economics, Friedrich-Schiller-University Jena, Carl-Zeiss-Strasse 3, D-07743 Jena, Germany Received October 2004; received in revised form June 2005; accepted December 2005 Available online 23 February 2006
Abstract Some multi-sector endogenous growth models contain strong predictions about productivity differences across sectors in the form of a distribution or density function. In this paper it is demonstrated that this distribution is left-skewed for a wide range of plausible parameter values. This stands in contrast to the right-skewed shape of the respective empirical distribution estimated by kernel methods for a measure of relative productivity with data for more than 450 four-digit US manufacturing industries during 1958–1996. This difference can be traced back to the assumption of strong intertemporal technological spillover effects that are at work in these models. © 2006 Elsevier B.V. All rights reserved. JEL classification: D24; O47; L60 Keywords: Multi-sector growth models; Manufacturing productivity distribution; Skewness
1. Introduction Recent multi-sector Schumpeterian growth models are endowed with microfoundations that lead to specific theoretical predictions about differences of the productivity levels across sectors. These predictions imply that even on a balanced growth path there exits heterogeneity of the productivity levels across sectors, which can be expressed in the form of a stationary distribution. In the models of Aghion and Howitt (1998, chapter 3) and Aghion et al. (2001) the density function associated with this distribution is either explicitly derived or numerically computed ∗
Tel.: +49 3641 943203; fax: +49 3641 943202. E-mail address: [email protected].
0954-349X/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.strueco.2005.12.002
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through simulation. In both models the shape and in particular the skewness of this distribution crucially depend on the size of an innovative step on the quality ladder, which is treated as an exogenous parameter. In this paper, the value of the step-size on the quality ladder that governs productivity growth or the associated cost reduction is calibrated using information from a variety of sources, such as the values used in numerical studies of the above-mentioned growth models. In addition, a plausible range of values is derived from the stylized result of the 80% learning curve that is representative for a wide variety of manufacturing production processes. All these values lead to a left-skewed theoretical distribution of the relative productivity levels. This finding stands in contrast to the distribution of relative productivity levels across the four-digit US manufacturing industries, which is estimated using kernel methods and appears to be consistently right-skewed during the years 1958–1996. The paper proceeds as follows: Section 2 provides a brief sketch of the derivation of the theoretical distribution of relative productivity levels and depicts the shape of the density function for various values of the innovation step-size. Section 3 calibrates the plausible range of values of the innovation step-size using the 80% learning curve result. Section 4 calculates relative productivity levels for the four-digit US manufacturing industries using nonparametric methods of efficiency measurement and presents the respective kernel density plots. The diametricallyopposed skewness properties of the theoretical and empirical distributions point to the need of paying more attention to the specification of the intertemporal technological spillover effects that are at work in these models. This issue is discussed in the concluding Section 5. 2. The theoretical distribution In most multi-sector growth models the sectors of the economy are treated symmetrically in every respect; Romer (1990) is a leading example. This implies that all sectors have the same productivity level and thus, the distribution of productivity across sectors is a one-point distribution. This feature makes these models useless for analyses concerning differential growth of industries and structural change in form of the change of the sectoral composition of an economy. Some other multi-sector growth models such as the model of Aghion and Howitt (1998, chapter 3) make explicit predictions about the shape of the distribution of productivity across sectors, while retaining the symmetry assumption in other respects. In this model the state of knowledge of the economy at time t is described by the level of the productivity parameter of the “leading-edge” technology Amax . Each sector of the economy t invests a certain amount of labor in research that determines the Poisson arrival rate λnit of new innovations, where nit denotes the amount of labor input invested in research in sector i at time t and λ is a positive parameter. An innovation in a particular sector i leads to a jump of the productivity level of this sector Ait to Amax . The growth of Amax is governed by the aggregate t t flow of innovations in the economy of λnt per unit of time, where nt is the total research labor input of the economy. The steady-state growth rate of aggregate productivity is then derived as d ln Amax /dt = λnt ln γ, which is identical to the steady-state growth rate of output. Therein, t γ > 1 denotes the step-size on the quality ladder in the form of a constant factor by which each innovation increases the productivity level. In the model, a strong intertemporal technological spillover effect is at work postulating that each innovation serves as the basis for discovering other innovations in other sectors of the economy, even though the current innovation can only be used by the generating sector. This spillover effect plays a key role for the explanation of the differences of the theoretical and empirical results to be discussed later in the paper.
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Fig. 1. Density functions of the theoretical distribution of relative productivity.
The different productivity levels over the whole continuum of sectors can be succinctly summarized by a distribution of the productivity levels across sectors on the support [0, Amax ]. t This support extends to the right as Amax grows and the distribution also becomes increasingly t left-skewed. Aghion and Howitt normalize all productivity levels by Amax to reach the relative t ∈ [0, 1] and derive the cross-sectional cumulative distribution productivity levels ait = Ait /Amax t function of these relative productivity levels, which is given by H(a) = a1/ln γ for a ∈ [0,1], irrespective of what happens to the aggregate rate of innovation over time.1 The density function associated with this theoretical distribution is h(a) = (a1/ln γ−1 )/ln γ for a ∈ [0,1] and zero otherwise. Fig. 1 shows this density function for various values of γ. Since the shape of the density function is exclusively determined by the step-size of the quality ladder, γ, we just have to find reasonable values for this parameter in order to compare the shape of this theoretical distribution with the shape of an empirical distribution that is estimated from productivity data of industries. It is immediate that the theoretical distribution is left-skewed for all 1 < γ < e, where, as usual, e = 2.71828. . . is the base of the natural logarithm. This range of values contains the benchmark value γ = 1.135 used in the numerical evaluations of the step-by-step innovation model of Aghion et al. (2001) in which the time unit is taken to be 1 year (see Aghion et al., 2001, p. 484). This magnitude implies that there are relatively more sectors with productivity levels close to the leading-edge productivity level and therefore small technology gaps. The numerical evaluations of the step-by-step innovation model in Hoernig (2003) use a “middle-of-the-road” parameter value of γ = 1.3, which is also contained in the interval of parameter values that are associated with a left-skewed distribution of relative productivity levels
1
See Appendix A on pages 115 and 116 in Aghion and Howitt (1998) for an elegant derivation of this result.
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(see Hoernig, 2003, p. 251).2 In their related work on step-by-step innovation models, Aghion et al. (2001, Fig. 5 on p. 486) also report the steady-state distribution of the technological leads, which essentially are technology gaps, measured in discrete units of innovation steps. The distribution depicted there is right-skewed, implying a high frequency of small technology gaps and correspondingly a low frequency of large technology gaps. Since the measure of technology gaps is inversely related to the measure of relative productivity levels, it is clear that the step-by-step innovation model also generates a left-skewed distribution for the relative productivity levels. 3. Calibration from the learning curve As the previous section has shown, the value of γ is crucial for the shape of the distribution of productivity across sectors. In this section additional evidence about the magnitude of this parameter is generated by relating it to the basic origins of learning-by-doing: the widely found result of the 80% learning curve. This stylized result states that in many production processes each doubling of the cumulative output is associated with a unit cost reduction of 20% to about 80% of the previous unit cost level, because the workers are becoming increasingly familiar with their tasks. Cooper and Johri (2002) and Jovanovic and Nyarko (1995) contain brief surveys and further references to the learning curve literature. Formally, we start from the assumption of a constant growth rate of the output of a particular sector, g = d ln y(t)/dt, expressed in continuous time. This is consistent with the focus on steadystate dynamics and balanced growth paths in growth theory. The growth path of output is thus y(t) = y0 egt with y(0) = y0 . Integrating output over time gives the cumulative output t t egt y(j)dj = y0 egj dj = y0 . s(t) = g −∞ −∞ Taking logarithms results in ln s(t) = ln y0 − ln g + gt from which the growth rate of the cumulative output can be calculated as d ln s(t)/dt = g. This is identical to the growth rate of output. The time it takes to double the cumulative output td can then be obtained from s(t + td ) = 2s(t) ⇔ y0
eg(t+td ) egt = 2y0 , g g
which implies that egtd = 2 and therefore the doubling time td = ln 2/g. Given that each doubling of cumulative output is associated with a 20% reduction of unit costs, the unit cost dynamics can be stated as c(t + td ) = 0.8c(t). The annual rate of cost reduction z can be deduced from 0.8 = e−ztd and using the previous expression for the doubling time we reach z = − g(ln 0.8/ln 2). Assuming a balanced output growth rate of about 3%, g = 0.03, the doubling time is approximately 23 years and the annual rate of cost reduction is approximately z = 0.01. Alternatively assuming a balanced growth rate of 2% reduces this rate even more. 2
Although the step-by-step innovation model expresses the effect of technological progress in terms of cost reductions, the step-size parameter of the quality ladder is the same as in models where technological progress is expressed directly in terms of productivity improvements. One can say that after k innovations production increases to γ k units of output per unit of labor employed or can say that γ −k units of labor are required to produce one unit of output implying that unit costs are proportional to γ −k . Both statements are equivalent.
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At this stage of analysis, the problem arises that the numerical value for the annual rate of cost reduction cannot be used directly since the model of Aghion et al. (2001) is not formulated in real time but instead uses an index for the innovation sequence. Thus, unit cost dynamics in this model are given by cτ+1 = γ −1 cτ , where γ denotes the step-size of the quality ladder as above and τ = 0, 1, 2, . . . denotes the number of innovations. When innovations arrive with a Poisson rate of λn this innovation count sequence can be translated into real time. Given that ε(t) innovations arrive between t and t + 1 in real time, unit cost dynamics can equivalently be expressed as ln c(t + 1) = ln c(t) − ε(t) ln γ,
with
ε(t) ∼ Po(λn),
where Po(λn) denotes a Poisson distribution with expectation λn. Thus the expected rate of decline of unit costs is E(ln c(t + 1) − ln c(t)) = −λn ln γ (see Aghion and Howitt (1998, p. 59) for this kind of argument). Collecting results and ignoring the expectations operator (or invoking a law of large numbers) we can state that ln c(t + 1) − ln c(t) = −z ≈ −0.01 ≈ −λn ln γ and thereby that γ ≈ e0.01/λn . The latter magnitude is smaller than e if 0.01/λn < 1 and consequently if λn > 0.01. Thus, whenever there is a minimal probability of innovation that leads to a Poisson arrival rate which is larger than 0.01, it follows that γ < e and therefore that the distribution of relative productivity levels is left-skewed. In summary, all exercises performed to deduce γ agree on a range of values that is associated with a left-skewed cross-industry distribution of relative productivity levels. As will be shown in the next section, this result stands in strong contradiction to the right-skewed shape of the empirical distribution of relative productivity levels in the US manufacturing sector. 4. The empirical distribution In this section, the theoretical distribution of relative productivity is confronted with the distribution of an empirically computed relative measure of total factor productivity. To quantify total factor productivity a nonparametric frontier function approach is used that closely corresponds to the definition of the relative productivity variable of the theoretical model. The specific method used here is the Andersen–Petersen variant of data envelopment analysis (Andersen and Petersen, 1993). This is a nonparametric method that calculates an index of total factor productivity by the distance of the input–output combinations of the N industries to a piece-wise linear frontier production function that is determined from quantity data alone without requiring any assumptions about the functional form of the production relationship and without having to rely on price data. The Andersen–Petersen model calculates productivity by computing an index that indicates to which level the output of an industry has to be increased in order to reach a point on the frontier production function that is determined by the observations of the other N − 1 industries, excluding the industry for which productivity is computed. Mathematically, the distance measure to the frontier of industry i in year t is the solution φit of the following linear programming problem ⎫ ⎧ ⎬ ⎨ λh yht ; λh xht ≤ xit ; −i ≥ 0 . max φit : φit yit ≤ ⎭ ⎩ hε{1,...,N}\i
h ∈ {1,...,N}\i
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In the present application yit denotes the single output variable and xit is the 6 × 1-vector of input variables of industry i in year t.3 λ−i denotes the (N − 1)-vector of the weight factors λh , omitting the i-th element.4 The radial distance measure φit essentially quantifies the size of the technological gap towards the frontier function from which the measure of total factor productivity can be obtained by inversion, i.e. φit−1 . The computations of the Andersen–Petersen model for each year separately results in a balanced panel of total factor productivity scores φit−1 (i = 1, . . . , N, t = 1, . . . , T ) for N = 454 four-digit industries5 over the T = 39 years covering the period 1958–1996. The majority of the observations that do not determine the frontier function get assigned a productivity score smaller than unity by the solution of the linear programming problem. Those observations that determine the frontier function get assigned a productivity score equal or larger than unity. To be consistent with the definition of relative productivity in the theoretical model these productivity scores are normalized −1 −1 by ait = φit−1 / max{φ1t , . . . , φNt } for each period and are thus bounded in the interval [0,1] as required for the comparison with the theoretical results. The density functions for each period are nonparametrically estimated by the univariate kernel density estimator (Wand and Jones, 1995, chapter 2) for the normalized relative productivity scores on a grid of points x ∈ [0,1]
N 1 x − ait , fˆ t (x) = K Nht ht i=1
where the standard normal density is used as kernel function K(·) and the bandwidth ht is chosen for each period separately by the Sheather–Jones second generation bandwidth estimator (Sheather and Jones, 1991). This bandwidth estimator has been advocated as the preferred method for one-dimensional kernel density estimation in the comparison of Jones et al. (1996). The upper panel of Fig. 2 shows the kernel density estimates of the cross-section productivity distribution for the years 1958, 1969, 1985 and 1996, which are chosen according to Jorgenson (1990, 2001) and provide an approximately equidistant division of the sample period. The productivity distributions show no clear pattern of change during the sampling period. Part of this diversity may be attributed to the relativity of the productivity measures towards the frontier functions, which are themselves subject to change from year to year. Despite this, it is apparent that the density functions are consistently right-skewed with the bulk of the industries showing relatively low productivity scores and a few industries having substantially larger productivity scores. This stands in contradiction to the theoretical distribution, which is left-skewed for a wide range of reasonable parameter values of the innovation step-size.
3 The data used to calculate the productivity scores are taken from the NBER-CES manufacturing industry database which is described in detail by Bartelsman and Gray (1996). This unique database provides consistent annual time series over the period 1958–1996 for quantity and price data of more than 450 manufacturing industries on the four-digit level of aggregation. The specification used takes the real value of shipments as the output variable. The two labor input variables are the number of non-production workers and the production worker hours. Capital input is represented by the real equipment capital stock and the real structures capital stock. Finally, the two variables that represent the input of materials and energy are the real cost of non-energy materials and the real expenditures on fuels and electricity, respectively. 4 This procedure is completely deterministic. There exists an alternative econometric approach to the estimation of frontier functions that promises to be able to divide measurement error from the productivity measure (see e.g. Greene, 1993). However, the Monte Carlo studies of Banker et al. (1993) and Ruggiero (1999) show that this advantage of the econometric approach is absent in small- to medium-sized samples. 5 For the productivity computations five industries had to be eliminated from the database.
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Fig. 2. Density functions of the empirical distribution of relative productivity.
The years chosen in the upper panel of Fig. 2 are in no way exceptional as the lower panel shows in which the kernel density estimates for all years between 1958 and 1996 are plotted together. The findings regarding the diversity and skewness of the productivity distributions are also robust to variations of the input–output specification used in the productivity computations (these mostly consist of merging certain input variables). In some of these variations right-skewness appeared to be even more pronounced. Altogether, the outcome of these variations is very assuring for the robustness of the findings reported here. This can be further sharpened by computing the empirical skewness measure visual impression 3 N −1 N (a − a ¯ ) for each year separately, where a¯ t denotes the arithmetic mean of the norit t i=1
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Fig. 3. Skewness of the theoretical distribution.
malized productivity scores of period t. Empirical skewness ranges between 0.0017 and 0.0033 and is thus positive in each year, consistent with the visual impression of the density estimates. With this range at hand we can gain further evidence regarding the step-size parameter γ of the theoretical model. Rewriting the density function as h(a) = cac−1 , a ∈ [0,1] with c = 1/ln γ, the first three moments of the theoretical distribution are easily computed as E(a) = c/(c + 1), E(a2 ) = c/(c + 2) and E(a3 ) = c/(c + 3). This allows to express the skewness of the theoretical distribution by E((a − E(a))3 ) = E(a3 ) − 3E(a2 )E(a)2E(a)3 = −
2c(c − 1) . (c + 1)3 (c + 2)(c + 3)
Equating this formula to the minimum and maximum values of empirical skewness, solving numerically for c and translating back to the γ scale results in the interval [2.947, 3.185] for γ which is completely in the range of values larger than e. This finding is illustrated in Fig. 3, where the solid line is the skewness of the theoretical distribution depending on γ. The horizontal and vertical dashed lines mark the zero line and the position of e, respectively. Consistent with the previous results, skewness is negative for γ < e and the distribution is therefore left-skewed in this range. For γ > e skewness becomes positive and the distribution is right-skewed accordingly. The empirically estimated range of skewness and the associated values of γ are indicated by the dotted lines. These values are in the range of positive skewness, consistent with the right-skewed density found in the productivity data and plotted in Fig. 2 above. 5. Discussion and conclusion The analysis in this paper has shown that the distribution of relative productivity generated by the models of Aghion and Howitt (1998, chapter 3) and Aghion et al. (2001) is fundamentally different from the distribution of a relative measure of total factor productivity for the four-digit
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industries of the US manufacturing sector. Both distributions are consistently unimodal, but the striking difference is that the theoretical cross-industry distribution is left-skewed for a wide range of plausible values for the innovation step-size parameter on which it exclusively depends, whereas its empirical counterpart is consistently right-skewed in each single year during the period 1958–1996. These differences are indeed important because they shed light on the validity of a specific assumption in the multi-sector endogenous growth models considered here. The most compelling explanation for the left-skewed shape of the theoretical distribution can be identified with the force of the strong technological spillover effect that is at work in the multi-sector growth models. To reiterate, this spillover effect postulates that the leading-edge technology discovered by any one industry is the basis for further productivity improvements in all other industries of the economy. As a consequence it is unlikely that the productivity levels differ by much unless the innovation step-size becomes very large. So the productivity levels of all sectors are tied to that of the leading-edge technology, causing the distribution of the normalized productivity levels to be left-skewed. Beyond doubt, technological spillover effects across industries and firms exist (Mohnen, 1996), but it seems to be the case that these spillover effects are much more limited than assumed in the multi-sector growth models. If the spillover effects are limited to some degree, the ties to the leading-edge productivity level are weaker and it is quite plausible that a few industries forge ahead in terms of productivity with the rest lagging behind persistently. Moreover, the shape characteristics of the empirical productivity distribution are not the only piece of evidence against the strong spillover effects across industries and the rapid catchingup toward the leading-edge productivity level. The spillover effects are also at odds with the finding of persistence of the relative productivity levels analyzed in Kr¨uger (2004, 2005b). This persistence implies that industries with relatively high productivity levels tend to sustain high productivity levels in the future and likewise for industries with relatively low productivity levels. If the spillover effects are as strong as assumed in the multi-sector growth models, persistence should only be observed at the upper end of the productivity distribution, however. Thus, not only the shape of the productivity distribution is in conflict with the theoretical prediction as shown in this paper, but also are the changes within the distribution in conflict with the spillover mechanism that is supposed in the multi-sector growth models. This is reinforced by the treatment of the industries as symmetric implying that all industries devote the same amount of resources to research and thus have the same probability of catching-up to the leading-edge technology. Therefore, other models such as that of Metcalfe (1994), Metcalfe et al. (2002) and Montobbio (2002) which are concerned with the explanation of aggregate growth driven by differential rates of productivity growth of the industries together with changes of the structural composition of the economy deserve more attention. The replicator dynamics mechanism used there potentially leads to more divergent developments across industries and sectors. Accordingly, industries grow or shrink in terms of their output shares if their productivity levels are above or below the average, respectively. The most likely outcome of this selection mechanism is that some industries are forging ahead with the rest being eventually marginalized in terms of their output shares. This implies a right-skewed shape of the output share distribution as observed in Kr¨uger (2005a) for exactly the same industry sample that is analyzed in the present paper. Related to that, Harberger (1998) discusses empirical evidence that illustrates the very diverse growth experiences and productivity developments within and across industries. Based on that he points to the need of devoting more attention to the process of differential growth of industries and sectors. In Harberger’s “yeast versus mushrooms” story the view that all sectors expand evenly (like yeast) is confronted with the view that different sectors grow in unpredictable ways (like mushrooms) caused by a multitude of
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influences. He concludes that “the ‘mushrooms’ story prevails just as much among firms within an industry as it does among industries within a sector or broader aggregate” (Harberger, 1998, p. 11). For growth theory this implies that it is important to devote more attention to the differential developments at the level of industries and sectors, not at least because structural change as a medium-run phenomenon is very important for economic policy. All this implies not only that the strong intertemporal technological spillover effect is at odds with the measured productivity differences across industries and sectors, but also that it is difficult to reconcile the equality and invariance of the sector shares that result from the symmetry assumption for all sectors with the empirical evidence. Although important to keep the models of Aghion and Howitt (1998, chapter 3) and Aghion et al. (2001) tractable, both assumptions of multi-sector growth models should be weakened or abandoned to reach a more realistic micro-economically based account of economic growth. The analyses of the interaction of differential productivity growth and structural change in the models of Metcalfe et al. (2002) and Montobbio (2002) show that this is indeed possible without the recourse to explicit equilibrium solutions. However, this critique of a neglected concern with the issues of differential productivity developments and structural change at the sectoral level in growth theory needs not necessarily affect the validity of the assumption of balanced growth of the aggregate magnitudes. At least in the case of the US economy the time series of log GDP per capita over the period 1870–1994 fluctuates remarkably close around a linear time trend (important exceptions being the years of the great depression and the second world war) as demonstrated in Fig. 1 of Jones (2002). Thus, the long-run aggregate output dynamics of the US economy are indeed well-described by a balanced growth path. If structural change occurs at the sectoral level as a result of nonhomothetic preferences, the concept of a generalized balanced growth path introduced by Kongsamut et al. (2001) and Meckl (2002) guarantees for the specifications analyzed in these papers that structural change and aggregate balanced growth are compatible with each other. Acknowledgments I am grateful for the comments of two anonymous referees that substantially improved the paper. All remaining errors are mine. References Aghion, P., Harris, C., Howitt, P., Vickers, J., 2001. Competition, imitation and growth with step-by-step innovation. Review of Economic Studies 68, 467–492. Aghion, P., Howitt, P., 1998. Endogenous Growth Theory. MIT Press, Cambridge, MA. Andersen, P., Petersen, N.C., 1993. A procedure for ranking efficient units in data envelopment analysis. Management Science 39, 1261–1264. Banker, R.D., Gadh, V.M., Gorr, W.L., 1993. A monte carlo comparison of two production frontier estimation methods: corrected ordinary least squares and data envelopment analysis. European Journal of Operational Research 67, 332–343. Bartelsman, E.J., Gray, W., 1996. The NBER manufacturing productivity data base. NBER Technical Working Paper 205. Cooper, R., Johri, A., 2002. Learning-by-doing and aggregate fluctuations. Journal of Monetary Economics 49, 1539–1566. Greene, W.H., 1993. The econometric approach to efficiency analysis. In: Fried, H.O., Lovell, C.A.K., Schmidt, S.S. (Eds.), The Measurement of Productive Efficiency. Oxford University Press, Oxford, pp. 68–119. Harberger, A.C., 1998. A vision of the growth process. American Economic Review 88, 1–32. Hoernig, S.H., 2003. Asymmetry, stability and growth in a step-by-step R&D-race. European Economic Review 47, 245–257.
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