and R&D productivity: Estimating production functions when productivity is endogenous
Ulrich Doraszelski Harvard University and CEPR Jordi Jaumandreu Universidad Carlos III and CEPR
April 2007 Preliminary and incomplete. Comments very welcome.
Abstract We develop a simple estimator for production functions in the presence of endogenous productivity change that allows us to retrieve productivity and its relationship with R&D at the rm level. Our dynamic investment model can be viewed as a generalization of the knowledge capital model (Griliches 1979) that has remained a cornerstone of the productivity literature for more than 25 years. We relax the assumptions on the R&D process and examine the impact of the investment in knowledge on the productivity of rms. We illustrate our approach on an unbalanced panel of more than 1800 Spanish manufacturing rms in nine industries during the 1990s. Our ndings indicate that the link between R&D and productivity is subject to a high degree of uncertainty, nonlinearity, and heterogeneity across rms. Abstracting from uncertainty and nonlinearity, as is done in the knowledge capital model, or assuming an exogenous process for productivity, as is done in the recent literature on structural estimation of production functions, overlooks some of its most interesting features.
1
Introduction
Firms invest in R&D and related activities to develop and introduce process and product innovations. By enhancing their productivity these investments in knowledge create longlived assets for rms, similar to their investments in physical capital. Our goal in this paper is to assess the role of R&D in determining the dierences in productivity across rms and
An earlier version of this paper was circulated as R&D and productivity: The knowledge capital model revisited. We thank Dan Ackerberg, Michaela Draganska, David Greenstreet, Wes Hartmann, Ken Judd, Jacques Mairesse, Harikesh Nair, Sridhar Narayanan, and Ariel Pakes for helpful discussions and Laia Castany for excellent research assistance. Department of Economics, Harvard University, Littauer Center, 1875 Cambridge Street, Cambridge, MA 02138, USA. E-mail: [e-mail]. Dpto. Econom Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe (Madrid), Spain. a, E-mail: [e-mail].
the evolution of rm-level productivity over time. To achieve this goal, we have to estimate the parameters of the production function and retrieve productivity at the level of the rm. Perhaps the major obstacle in production function estimation is that the decisions that a rm makes depend on its productivity. Because the productivity of the rm is unobserved by the econometrician, this gives rise to an endogeneity problem (Marschak & Andrews 1944). Intuitively, if a rm adjusts to a change in its productivity by expanding or contracting its production depending on whether the change is favorable or not, then unobserved productivity and input usage are correlated and biased estimates result. Recent advances in the structural estimation of production functions, starting with the dynamic investment model of Olley & Pakes (1996) (hereafter OP), tackle this issue. The insight of OP is that if (observed) investment is a monotone function of (unobserved) productivity, then this function can be inverted to back out productivity. Controlling for productivity resolves the endogeneity problem as well as, eventually, the selection problem that may arise if a rms decision to exit the industry depends on its productivity.1 In addition to OP, this line of research includes contributions by Levinsohn & Petrin (2003) (hereafter LP) and Ackerberg, Caves & Frazer (2005) (hereafter ACF) as well as a long list of applications. Common to the extant literature is the assumption that any changes in its productivity are exogenous to the rm. But if productivity is assumed to evolve independently of R&D, then this rules out that a rm invests in R&D in the rst place. This makes the available estimators ill-suited to study the link between R&D and productivity. Indeed, their foremost application has been the analysis of changes in productivity in response to exogenous shocks such as deregulation (e.g., OP) or trade liberalization (e.g., Pavcnik 2002, Topalova 2004). In this paper, we develop a dynamic model that accounts for investment in knowledge, thereby endogenizing productivity change, and derive a simple estimator for production functions in this setting. We use our approach to study the relationship between R&D and productivity in Spanish manufacturing rms during the 1990s. We particularly pay attention to the uncertainties and nonlinearities in the R&D process and their implications for heterogeneity across rms. We start by modeling a rm that can invest in R&D in order to improve its productivity over time in addition to carrying out a series of investments in physical capital. Both investment decisions depend on the current productivity and capital stock of the rm. The evolution of productivity is subject to random shocks. We interpret these innovations to productivity as representing the resolution over time of all uncertainties. They capture the factors that have a persistent inuence on productivity such as absorption of techniques, modication of processes, and gains and losses due to changes in labor composition and management abilities. R&D governs the evolution of productivity up to an unpredictable
See Griliches & Mairesse (1998) and Ackerberg, Benkard, Berry & Pakes (2005) for reviews of the problems involved in the estimation of production functions.
1
2
component. Hence, for rms that engage in R&D, the productivity innovations additionally capture the uncertainties inherent in the R&D process such as chance in discovery and success in implementation. Productivity thus follows a rst-order Markov process that can be shifted by R&D expenditures. Subsequently decisions on variable (or static) inputs such as labor and materials are taken according to the current productivity and capital stock of the rm. Next we develop a simple estimator for production functions that can accommodate the controlled Markov process that results from the impact of R&D on the evolution of productivity. Endogenizing the productivity process by incorporating R&D expenditures into the dynamic investment model of OP is dicult as Buettner (2005) has shown (see Section 3 for details). We use the fact that decisions on variable inputs are based on current productivity, similar to LP and ACF. These inputs are chosen with current productivity known and therefore contain information about it. The resulting input demands are invertible functions of unobserved productivity (as rst shown by LP). This enables us to control for productivity and obtain consistent estimates of the parameters of the production function. We dier from the previous literature in that we recognize that, given a parametric specication of the production function, the functional form of these inverse input demand functions is known. Because we make full use of the structural assumptions, we do not have to rely on nonparametric methods to estimate the inverse input demand function. This renders identication and estimation more tractable. It also yields eciency gains. Of course, it has long been recognized that the productivity process is endogenous. Griliches (1979), in particular, proposed to augment the production function with the stock of knowledge as proxied for by a rms past R&D expenditures. This knowledge capital model has remained a cornerstone of the productivity literature for more than 25 years and has been applied in hundreds of empirical studies on rm-level productivity and also extended to macroeconomic growth models (see Griliches (1995) for a comprehensive survey).2 While useful as a practical tool, the knowledge capital model has a long list of known drawbacks as explained, for example, in Griliches (2000). The critical (but implicit) assumptions of the basic model include the linear and certain accumulation of knowledge from period to period in proportion to R&D expenditures as well as the linear and certain depreciation. The link between R&D and productivity, however, is much more complex. The outcome of the R&D process is likely to be subject to a high degree of uncertainty. Discovery is, by its very nature, uncertain. Once discovered an idea has to be developed and applied, and there are the technical and commercial uncertainties linked to its practical implementation.
2 See Hall & Mairesse (1995) for a classic application. The knowledge capital model has evolved in many directions. Pakes & Schankerman (1984a) modeled the creation of knowledge by specifying a production function in terms of R&D capital and R&D labor. Jae (1986) initiated ways of accounting for the appropriability of the external ows of knowledge or spillovers. For recent examples see Grith, Redding & Van Reenen (2004) or Grith, Harrison & Van Reenen (2006).
3
In addition, current and past investments in knowledge are likely to interact with each other in many ways. For example, there is evidence of complementarities in the accumulation of knowledge (Klette 1996). In general, there is little reason to believe that this and other features such as economies of scale can be adequately captured by simple functional forms. Our dynamic investment model can be viewed as a generalization of the knowledge capital model. In particular, we recognize the uncertainties in the R&D process in the form of shocks to productivity. We model the interactions between current and past investments in knowledge in a exible fashion. Furthermore, we relax the assumption that the obsolescence of previously acquired knowledge can be described by a constant rate of depreciation. This allows us to more closely assess the impact of the investment in knowledge on the productivity of rms. We apply our estimator to an unbalanced panel of more than 1800 Spanish manufacturing rms in nine industries during the 1990s. The data refute the assumptions at the heart of the knowledge capital model. To begin with, the R&D process must be treated as inherently uncertain. We estimate that, depending on the industry, between 20% and 50% of the variance in actual productivity is explained by productivity innovations that cannot be predicted when decisions on R&D expenditures are made. Our estimates further imply that the return to R&D is often twice that of the return to investment in physical capital. This suggests that the uncertainties inherent in the R&D process are economically signicant and matter for rms investment decisions. While the relationship between current productivity, R&D expenditures, and future productivity takes a simple separable form in some cases, in most cases the impact of current R&D on future productivity depends crucially on current productivity. There is evidence of complementarities as well as increasing returns to R&D. Moreover, the data very clearly reject the functional form restrictions implied by the knowledge capital model, thus casting doubt on the linearity assumption in the accumulation and depreciation of knowledge. Capturing the uncertainties in the R&D process also paves the way for heterogeneity across rms. Whereas rms with the same time path of R&D expenditures have necessarily the same productivity in the knowledge capital model, in our setting this is no longer the case because we allow the shocks to productivity to accumulate over time. This gives us the ability to assess the role of R&D in determining the dierences in productivity across rms and the evolution of rm-level productivity over time. Despite the uncertainties in the R&D process, the expected productivity of rms that perform R&D is systematically more favorable in the sense that their distribution of expected productivity tends to stochastically dominate the distribution of rms that do not perform R&D. Assuming that the productivity process is exogenous takes a sort of average over rms with distinct innovative activities and hence blurs remarkable dierences in the impact of the investment in knowledge on the productivity of rms. In addition, we esti-
4
mate that the contribution of rms that perform R&D explains between 45% and 85% of productivity growth in the industries with intermediate or high innovative activity. R&D expenditures are thus a primary source of productivity growth. Our analysis further implies that productivity is considerably more uid than what the knowledge capital literature suggests. Our model allows us to recover the entire distribution of the elasticity of output with respect to R&D expendituresa measure of the return to R&Das well as that of the elasticity of output with respect to already attained productivitya measure of the degree of persistence in the productivity process. On average we obtain higher elasticities with respect to R&D expenditures than in the knowledge capital model and lower elasticities with respect to already attained productivity. Hidden behind these averages, however, is a substantial amount of heterogeneity across rms. Our ndings not only shed light on the link between R&D and productivity, but potentially also have implications for the design of R&D policy. While a fuller exploration is left to future research, we note here that in the knowledge capital model an extra dollar of R&D yields an extra unit of knowledge. Because this is no longer the case in the presence of a nonlinearity, the allocation of subsidies suddenly becomes important. Next, if uncertainty inhibits rms investments in R&D, then a case can be made for R&D policy to be directed towards providing insurance against particularly unfavorable outcomes. Finally, R&D policy has distributional consequences in the presence of heterogeneity as some rms gain while others lose. Overall, the link between R&D and productivity is subject to a high degree of uncertainty, nonlinearity, and heterogeneity across rms. Abstracting from uncertainty and nonlinearity, as is done in the knowledge capital model, or assuming an exogenous process for productivity, as is done in the literature following OP, overlooks some of its most interesting features.
2
A model for investment in knowledge
A rm carries out two types of investments, one in physical capital and another in knowledge through R&D expenditures. The investment decisions are made in a discrete time setting with the goal of maximizing the expected net present value of future cash ows. The rm has the Cobb-Douglas production function yjt = 0 + l ljt + k kjt + jt + ejt , where yjt is the log of output of rm j in period t, ljt the log of labor, and kjt the log of capital. We follow the convention that lower case letters denote logs and upper case letters levels and focus on a value-added specication to simplify the exposition. Capital is the only xed (or dynamic) input among the conventional factors of production, and accumulates according to Kjt = (1 )Kjt1 + Ijt1 . This law of motion implies that 5
investment Ijt1 chosen in period t 1 becomes productive in period t. The productivity of rm j in period t is jt . We follow OP and often refer to jt as unobserved productivity since it is unobserved from the point of view of the econometrician (but known to the rm). Productivity is presumably highly correlated over time and perhaps also across rms. In contrast, ejt is a mean zero random shock that is uncorrelated over time and across rms. The rm does not know the value of ejt at the time it makes its decisions for period t. The assumption usually made about productivity (see OP, LP, and the subsequent literature) is that it follows an exogenous rst-order Markov process with transition probabilities P (jt |jt1 ). This rules out that the rm spends on R&D and related activities. However, investment in knowledge has always been thought of as aimed at modifying productivity for given conventional factors of production (see, e.g., the tradition started by Griliches (1979)). Our goal is thus to assess the role of R&D in determining the dierences in productivity across rms and the evolution of rm-level productivity over time. We therefore consider productivity to be governed by a controlled rst-order Markov process with transition probabilities P (jt |jt1 , rjt1 ), where rjt1 is the log of R&D expenditures. The Bellman equation for the rms dynamic programming problem is V (kjt , jt ) = max (kjt , jt ) ci (ijt ) cr (rjt ) +
ijt ,rjt
1 E [V (kjt+1 , jt+1 )|kjt , jt , ijt , rjt ] , 1+
where () denotes per-period prots and is the discount rate. In the simplest case the cost functions ci () and cr () just transform logs into levels, but their exact forms are irrelevant for our purposes. The dynamic problem gives rise to two policy functions, i(kjt , jt ) and r(kjt , jt ) for the investments in physical capital and knowledge, respectively. The main dierence between the two types of investments is that they aect the evolution of dierent state variables, i.e., either the capital stock kjt or the productivity jt of the rm. When the decision about investment in knowledge is made in period t 1, the rm is only able to anticipate the expected eect of R&D on productivity in period t. The Markovian assumption implies jt = E [jt |jt1 , rjt1 ] + jt = g(jt1 , rjt1 ) + jt . That is, actual productivity jt in period t can be decomposed into expected productivity g(jt1 , rjt1 ) and a random shock jt . Our key assumption is that the impact of R&D on productivity can be expressed through the dependence of the conditional expectation function g() on R&D expenditures. In contrast, jt does not depend on R&D expenditures: by construction jt is mean independent (although not necessarily fully independent) of rjt1 . This productivity innovation may be thought of as the realization of the uncertainties that are naturally linked to productivity plus the uncertainties inherent in the R&D process (e.g., chance in discovery, degree of applicability, success in implementation). It is important 6
to stress the timing of decisions in this context: When the decision about investment in knowledge is made in period t 1, the rm is only able to anticipate the expected eect of R&D on productivity in period t as given by g(jt1 , rjt1 ) while its actual eect also depends on the realization of the productivity innovation jt that occurs after the investment has been completely carried out. Of course, the conditional expectation function g() is unobserved from the point of view of the econometrician (but known to the rm) and must be estimated nonparametrically. If we consider a ceteris paribus increase in R&D expenditures that changes jt to jt , then jt jt approximates the eect of this change in productivity on output in percentage terms, i.e., (Yjt Yjt )/Yjt = exp( jt jt ) 1 jt jt . That is, the change in jt shifts the production function and hence measures the change in total factor productivity. Also g() and jt can be interpreted in percentage terms and decompose the change in total factor productivity. Finally, expenditures. Our setting encompasses as a particular case the knowledge capital model (see Griliches (1979, 2000)). In this model, a conventional Cobb-Douglas production function is augmented by including the log of knowledge capital cjt as an extra input yielding yjt = 0 + l ljt + k kjt + cjt + ejt , (1)
jt rjt1
=
g(jt1 ,rjt1 ) rjt1
is the elasticity of output with respect to R&D
where is the elasticity of output with respect to knowledge capital. Knowledge capital is assumed to accumulate with R&D expenditures and to depreciate from period to period at a rate . Hence, its law of motion can be written as Cjt = (1 )Cjt1 + Rjt1 = Cjt1 1 + Taking logs we have cjt where
Rjt1 Cjt1
Rjt1 Cjt1
.
cjt1 +
Rjt1 , Cjt1
is the rate of investment in knowledge. Letting jt = cjt it is easy to see that jt jt1 + exp(rjt1 ) exp(jt1 /) (2)
and hence jt = g(jt1 , rjt1 ). That is, the classical accumulation of knowledge capital induces a particular expression for the conditional expectation function g() that depends on both productivity and R&D expenditures in the previous period. The knowledge capital model ignores that the accumulation of improvements to productivity is likely to be subjected to shocks. To capture this assume that the eect of the rate of investment in knowledge has an unpredictable component jt . The law of motion becomes
7
Cjt = Cjt1 1 +
Rjt1 Cjt1
+ 1 jt . This simple extension causes the law of motion of pro
ductivity to be jt = g(jt1 , rjt1 ) + jt , which turns out to be our controlled rst-order Markov process. Therefore, a useful way to think of our setting is as a generalization of the knowledge capital model to the more realistic situation of uncertainty in the R&D process.3 In addition, our setting overcomes other problems of the knowledge capital model, in particular the linear accumulation of knowledge from period to period in proportion to R&D expenditures and the linear depreciation. The absence of functional form restrictions on the combined impact of R&D and already attained productivity on future productivity is an important step in the direction of relaxing all these assumptions. Of course, there is a basic dierence between the two models. In the case of the knowledge capital model, given data on R&D and a guess for the initial condition, one must be able to construct the stock of knowledge capital at all times and with it control for the impact of R&D on productivity. In our setting, in contrast, the random nature of accumulation and the unspecied form of the law of motion prevents the construction of the stock of productivity, which remains unobserved. Consequently, no guess for the initial condition is required. Moreover, our empirical strategy takes into account that the endogeneity problem in production function estimation may not be completely resolved by adding the stock of knowledge capital to the conventional factors of production.
3
Empirical strategy
Our model relaxes the assumption of an exogenous Markov process for productivity. As emphasized in Ackerberg, Benkard, Berry & Pakes (2005), endogenizing this process is problematic for the standard estimation procedures. First, it tends to invalidate the usual instrumental variables approaches. Given an exogenous Markov process, input prices are natural instruments for input quantities. This is, however, no longer the case if the transitions from current to future productivity are aected by the choice of an additional unobserved input such as R&D because all quantities depend on all prices. Second, the absence of data on R&D implies that a critical determinant of the probability distribution of jt given jt1 is missing. Recovering jt from kjt , ijt , and their lags, the key step in OP, may thus be dicult.
We note that there are ways of introducing uncertainty into the knowledge capital model, although there are few such attempts in the literature. Borrowing from the dynamic investment model of Hall & Hayashi (1989), let the law of motion for the log of knowledge capital be cjt = (1 )cjt1 + Rjt1 + jt . Then cjt = (1)t c0 + t =1 (1)t Rj 1 + t =1 (1)t j can be split into a deterministic and a stochastic part that is incorporated into the error term of the estimation equation. In this case, however, using R&D expenditures as a proxy for the stock of knowledge gives rise to an endogeneity problem that invalidates the traditional estimation strategies such as running OLS on rst-dierences of logs. A further problem is that the ability to split the log of knowledge capital into a deterministic and a stochastic part relies heavily on functional form. In particular, it is no longer possible if, as is customary in the literature, the law of motion for the level of knowledge capital is assumed to be linear.
3
8
Buettner (2005) extends the OP approach by studying a model similar to ours while assuming transition probabilities for unobserved productivity of the form P (jt |t ), where t = (jt1 , rjt1 ) is an index that orders the probability distributions for jt . The restriction to an index excludes the possibility that current productivity and R&D expenditures aect future productivity in qualitatively dierent ways. Under certain assumptions it ensures that the policy function for investment in physical capital is still invertible and that unobserved productivity can hence still be written as an unknown function of the capital stock and the investment as jt = h(kjt , ijt ). Buettner (2005) further notes, however, that there are problems with identication even when data on R&D is available. Our estimation procedure solves entirely the identication problem when there is data on R&D by using a known function h() that is derived from the demand for variable inputs such as labor and materials in order to recover unobserved productivity. These variable inputs are chosen with current productivity known, and therefore contain information about it. This allows us to back out productivity without making assumptions on the rms dynamic investment problem. In particular, our approach does not rely on an index and frees up the relationship between current productivity, R&D expenditures, and future productivity. It can also solve potentially the identication problem when there is no data on R&D but this point needs further research.4 While our approach pertains to production functions that are written in terms of either gross output or value added, in what follows we focus on the value added case for the sake of simplicity. The extension to the gross output case is straightforward. Given the Cobb-Douglas production function yjt = 0 + l ljt + k kjt + jt + ejt , the assumption that the rm chooses labor based on the expectation E(ejt ) = 0 gives the demand for labor as ljt = 1 (0 + ln l + k kjt + jt (wjt pjt )) . 1 l (3)
Solving for jt we obtain the inverse labor demand function h(ljt, kjt, wjt pjt ) = 0 + (1 l )ljt k kjt + (wjt pjt ), where 0 combines the constant terms 0 and ln l and (wjt pjt ) is the relative wage (homogeneity of degree zero in prices). From hereon we call h() the inverse labor demand function and use hjt as shorthand for its value h(ljt, kjt, wjt pjt ). Substituting the inverse labor demand function h() for jt in the production function cancels out parameters of interest and leaves us with the marginal productivity condition for prot maximization, i.e., ln l + (yjt ljt ) = wjt pjt + ejt . Using its value in period
Muendler (2005) suggests to use investment in physical capital interacted with industry-specic competition variables to proxy for endogenously evolving productivity. His rationale is that rms make R&D decisions in light of their expectations about future market prospects. Hence, in the absence of data on R&D, these competition variables should to some extent capture the drivers of R&D decisions.
4
9
t 1 in the controlled Markov process, however, we have yjt = 0 + l ljt + k kjt + g(h(ljt1 , kjt1 , wjt1 pjt1 ), rjt1 ) + jt + ejt . (4)
Both kjt , whose value is determined in period t 1 by it1 , and rjt1 are uncorrelated with jt by virtue of our timing assumptions. Only ljt is correlated with jt (since jt is part of jt and ljt is a function of jt ). Nonlinear functions of the other variables can be used as instruments for ljt , as can be lagged values of ljt and the other variables. If rms can be assumed to be perfectly competitive, then current wages and prices are exogenous and constitute the most adequate instruments (since demand for labor is directly a function of current wages and prices). As noted by LP and ACF, backing out unobserved productivity from the demand for either labor or materials is a convenient alternative to backing out unobserved productivity from investment as in OP. In the tradition of OP, however, LP and ACF use nonparametric methods to estimate the inverse input demand function. This forces them either to rely on a two-stage procedure or to jointly estimate a system of equations as suggested by Wooldridge (2004). The drawback of the two-stage approach is a loss of eciency whereas the joint estimation of a system of equations is numerically more demanding (see Ackerberg, Benkard, Berry & Pakes (2005) for a discussion of the relative merits of the two approaches). We dier from the previous literature in that we recognize that the parametric specication of the production function implies a known form for the inverse labor demand function h() that can be used to control for unobserved productivity. As a consequence, only the conditional expectation function g() is unknown and must be estimated nonparametrically. This yields eciency gains (see Section 5 for details). In addition, because we make full use of the structural assumptions, our approach is immune to the collinearity problem that hinders identication in the OP/LP framework (as shown by ACF). We are also able to relax the assumption of perfect competition that LP invoke in proving that labor demand is an invertible function of unobserved productivity. We will come back to these points below. Finally, unlike OP, LP, and ACF, have but a single equation to estimate, thus easing the computational burden. Apart from the presence of R&D expenditures, our estimation equation (4) is similar in structure to the second equation of OP and LP when viewed through the lens of Wooldridges (2004) GMM framework. In our setting the rst equation of OP and LP is the marginal productivity condition for prot maximization. Combining it with our estimating equation (4) may help to estimate the labor coecient, but this point needs further research. A drawback of our approach is that, in principle, it requires rm-level wage and price data to estimate the model. The model remains identied, however, if the log of relative wage is replaced by a set of dummies.5
5
This may be an appropriate solution in the absence of wage and price data if the industry can be
10
Our model nests, as a particular case, the dynamic panel model proposed by Blundell & Bond (2000). Suppose the Markov process is simply an autoregressive process that does not depend on R&D expenditures so that we have g(jt1 ) = jt1 . Using the marginal productivity condition for prot maximization to substitute yjt1 for ( ln l + (wjt1 pjt1 ) + ljt1 ), we are in the Blundell & Bond (2000) specication. Hence, the dierences between their and our approach lie in the generality of the assumption on the Markov process and the strategy of estimation. In the tradition of OP and LP our method basically proposes the replacement of unobservable autocorrelated productivity by an expression in terms of observed variables and an unpredictable component, whereas their method models the same term through the use of lags of the dependent variable (see ACF for a detailed description of these two literatures). Below we discuss how imperfect competition can be taken into account and the likelihood of sample selection. Then we turn to identication, estimation, and testing. Imperfect competition. Until now we have assumed a perfectly competitive environment. But when rms have some market power, say because products are dierentiated, then output demand enters the specication of the inverse input demand functions (see, e.g., Jaumandreu & Mairesse 2005). Consider rms facing a downward sloping demand function that depends on the price of the output Pjt and the demand shifters Zjt . Prot maximization requires that rms set the price that equates marginal cost to marginal revenue Pjt 1
1 (pjt ,zjt )
, where () is the absolute value of the elasticity of demand evaluated at
the equilibrium price and the particular value of the demand shifter and, for convenience, is written as a function of pjt = ln Pjt and zjt = ln Zjt . With rms minimizing costs, marginal cost and conditional labor demand can be determined from the cost function and combined with marginal revenue to give the inverse labor demand function hIC (ljt , kjt , wjt pjt , pjt , zjt ) = 0 + (1 l )ljt k kjt + (wjt pjt ) ln 1 Thus, the estimation equation is yjt = 0 + l ljt + k kjt + g hjt1 ln 1 1 (pjt1 , zjt1 ) , rjt1 + jt + ejt . (5) 1 (pjt , zjt ) .
As both pjt and zjt enter the equations lagged they are expected to be uncorrelated with the productivity innovation jt .6
considered perfectly competitive. 6 Note that this setting yields an estimate of the average elasticity of demand. The reason by which this is possible is the same by which correcting the Solow residual for imperfect competition allows for estimating margins and elasticities (see, e.g., Hall 1990).
11
Sample selection. A potential problem in the estimation of production functions is sample selection. If a rms dynamic programming problem generates an optimal exit decision, based on the comparison between the sell-o value of the rm and its expected protability in the future, then this decision is a function of current productivity. The simplest model, based on an exogenous Markov process, predicts that if an adversely enough shock to productivity is followed immediately by exit, then there will be a negative correlation between the shocks and the capital stocks of the rms that remain in the industry. Hence, sample selection will lead to biased estimates. Accounting for R&D expenditures in the Markov process complicates matters. On the one hand, a rm now has an instrument to try to rectify an adverse shock and the optimal exit decision is likely to become more complicated. To begin with, there are many more relevant decisions such as beginning, continuing, or stopping innovative activities whilst remaining in the industry, and exiting in any of the dierent positions. On the other hand, a rm now is more likely to remain in the industry despite an adverse shock. Innovative activities often imply large sunk cost which will make the rm more reluctant to exit the industry or at least to exit it immediately. This will tend to alleviate the selection problem. At this stage we do not model any of these decisions. Instead, we simply explore whether there is a link between exit decisions and estimated productivity.
3.1
Identication
Our estimation equation (4) is a semiparametric, so-called partially-linear, model with the additional restriction that the inverse labor demand function h() is of known form. To see how this restriction aids identication, suppose to the contrary that h() were of unknown form. In this case, the composition of h() and g() is another function of unknown form. The fundamental condition for identication is that the variables in the parametric part of the model are not perfectly predictable (in the least squares sense) by the variables in the nonparametric part (Robinson 1988). In other words, there cannot be a functional relationship between the variables in the parametric and nonparametric parts (see Newey, Powell & Vella (1999) and also ACF for an application to the OP/LP framework). To see that this condition is violated, recall that Kjt = (1 )Kjt1 + exp(i(kjt1 , jt1 )) by the law of motion and the policy function for investment in physical capital. But kjt1 is one of the arguments of h() and jt1 is by construction a function of all arguments of h(), thereby making kjt perfectly predictable from the variables in the nonparametric part. Of course, in our setting the inverse labor demand function h() is of known form. The central question thus becomes whether kjt is perfectly predictable from the value of h() (as opposed to its arguments) and rjt1 . Since hjt1 is identical to jt1 , we have to ask if kjt1 and hence kjt (via i(kjt1 , jt1 )) can be inferred from rjt1 . This may indeed be possible. Recall that rjt1 = r(kjt1 , jt1 ) by the policy function for investment in knowledge. Hence, if its R&D expenditures happen to be increasing in the capital stock of 12
the rm, then r() can be inverted to back out kjt1 . Fortunately, there is little reason to believe that this is the case. In fact, even under the fairly stringent assumptions in Buettner (2005), it is not clear that r() is invertible. Moreover, there is empirical evidence that invertibility may fail even for investment in physical capital (Greenstreet 2005) and it seems clear that R&D expenditures are even more ckle. Even if r() happens to be an invertible function of kjt1 , anything that shifts the costs of the investments in physical capital and knowledge over time guarantees identication. The price of equipment goods is likely to vary, for example, and the marginal cost of investment in knowledge depends greatly on the nature of the undertaken project. Using xjt to denote these shifters, the policy functions become i(kjt , jt , xjt ) and r(kjt , jt , xjt ). Obviously, xjt cannot be perfectly predicted from hjt1 and rjt1 . This breaks the functional relationship between Kjt = (1 )Kjt1 + exp(i(kjt1 , jt1 , xjt )) and hjt1 and rjt1 .7 In closing we note that the previous arguments carry over to the OP/LP framework with an exogenous Markov process for productivity. As ACF argue, the estimators in LP and OP suer from a collinearity problem that hinders identication. Our approach diers in that it exploits the known form of the inverse labor demand function. Consequently, if the productivity process can indeed be taken as exogenous, then the model is identied because kjt is not perfectly predictable from the value of h(). Remarkably this is the case even in the absence of cost shifters for investment in physical capital and knowledge.
3.2
Estimation
The estimation problem can be cast in the nonlinear GMM framework E zjt (jt + ejt ) = E zjt vjt () = 0, where zjt is a vector of instruments and we write the error term vjt () as a function of the parameters to be estimated. The objective function is min
1 N
j
j
zj vj () ,
1 zj vj () AN N
where zj and vj () are L Tj and Tj 1 vectors, respectively, with L being the number of instruments, Tj being the number of observations of rm j, and N the number of rms. We rst use the weighting matrix AN =
1 N j zj zj 1
to obtain a consistent
estimator of and then we compute the optimal estimator which uses weighting matrix
7 Depending on the construction of the capital stock in the data, we may also be able to account for uncertainty in the impact of investment in physical capital. But once an error term is added to the law of motion for physical capital, kjt can no longer be written as a function of hjt1 and rjt1 , and identication is restored.
13
AN =
1 N
j zj vj ()vj () zj
1
.
Production function. Our preliminary estimates indicate that in some industries it is useful to add a time trend to the production function. One can say that there is an observable trend in the evolution of productivity that is treated separately from jt but of course taken into account when substituting hjt for jt . Our goal is thus to estimate the gross-output production function yjt = 0 + t t + l ljt + k kjt + m mjt + g(hjt1 , rjt1 ) + jt + ejt . where hjt = 0 t t + (1 l m )ljt k kjt + (1 m )(wjt pjt ) + m (pM jt pjt ) and (pM jt pjt ) is the relative price of materials. Series estimator. As suggested by Wooldridge (2004) when modeling an unknown function q(v, u) of two variables v and u we use a series estimator made of a complete set of polynomials of degree Q (see Judd 1998), i.e., all polynomials of the form v j uk , where j and k are nonnegative integers such that j +k Q. When the unknown function q() has a single argument, we use a polynomial of degree Q to model it, i.e., q(v) = 0 + 1 v + . . . + Q v Q . Taking into account that there are rms that do not perform R&D, the most general formulation is yjt = 0 + t t + l ljt + k kjt + m mjt +1(Rjt1 = 0)g0 (hjt1 ) + 1(Rjt1 > 0)g1 (hjt1 , rjt1 ) + jt + ejt . (6)
This allows for a dierent unknown function when the rm adopts the corner solution of zero R&D expenditures and when it chooses positive R&D expenditures. It is important to note that any constant that its arguments may have will be subsumed in the constant of the unknown function. Our specication is therefore g0 (hjt1 ) = g00 + g01 (hjt1 0 ), g1 (hjt1 , rjt1 ) = g10 + g11 (hjt1 0 , rjt1 ), where in g00 and g10 we collapse the constants of the unknown functions g0 () and g1 () and the constant of hjt1 . The constants g00 , g10 , and 0 cannot be estimated separately. We thus estimate the constant for nonperformers g00 together with the constant of the production function 0 and include a dummy for performers to measure the dierence between constants 0 + g10 (0 + g00 ) = g10 g00 .
14
In the case of imperfect competition, where we have to nonparametrically estimate the absolute value of the elasticity of demand, we impose the theoretical restriction that () > 1 by using the specication (pjt1 , zjt1 ) = 1 + exp(q(pjt1 , zjt1 )), where q() is modeled as described above. Instrumental variables. As discussed before, kjt is always a valid instrument because it is not correlated with jt because the latter is unpredictable when it1 is chosen. Labor and materials, however, are contemporaneously correlated with the innovation to productivity. The lags of these variables are valid instruments but when the demand for one of these inputs is being used to substitute for jt it appears itself in hjt1 . We can use the lag of the other input. Constant and trend are valid instruments. Therefore, we have four instruments to estimate the constant and the coecients for the trend, capital, labor, and materials. This leaves us with the need for at least one more instrument. We use as instruments for the whole equation the complete set of polynomials of degree Q in the variables which enter hjt1 , the powers up to degree Q of rjt1 , and the interactions up to degree Q of the variables which enter hjt1 and rjt1 . The nonlinear functions of all exogenous variables included in these polynomials provide enough instruments. We set Q = 3 and use polynomials of order three. Hence, when there are four variables in the inverse input demand function h(), say ljt1 , kjt1 , wjt1 pjt1 , and pM jt1 pjt1 , we use as instruments the polynomials which result from the complete set of polynomials of degree 3 corresponding to the third power of hjt1 (34 instruments), plus 3 terms which correspond to the powers of rjt1 (3 instruments) and 12 interactions formed from the
2 products hjt1 rjt1 , h2 rjt1 , and hjt1 rjt1 (12 instruments). In fact when we enter jt1
pjt1 linearly we use it detached from the other prices and we also need a dummy for the rms that perform R&D (2 more instruments). In addition, when there are enough degrees of freedom we instrument separately hjt1 for nonperformers and hjt1 and rjt1 for performers by interacting the instruments with the dummy for performers. In addition, we have the exogenous variables included in the equation: constant, trend, current capital and lagged materials (4 instruments). This gives a total of 34 + 34 + 3 + 12 + 1 + 2 + 4 = 90 instruments. When we combine the demands for labor and materials, both equations have the same number of instruments (recall that not all are equal) and hence we have a total of 190 instruments. *** UPDATE/DELETE LAST SENTENCE. *** Given these instruments, our estimator has exactly the form of the GMM version of Ai & Chens (2003) sieve minimum distance estimator, a nonparametric least squares technique (see Newey & Powell 2003). This means that, if the conditional expectation function g() is specied in terms of variables which are correlated with the error term of the estimation equation, we still obtain a consistent and asymptotically normal estimator of the parameters by specifying the instrumenting polynomials in terms of exogenous conditioning variables.
15
Productivity estimates. Once the model is estimated we can compute jt , hjt , and g() up to a constant. We can also obtain an estimate of jt up to a constant as the dierence between the estimates of jt and g(). Recall that the productivity of rm j in period t is given by t t + jt = t t + g(jt1 , rjt1 ) + jt with jt = hjt . Using the notational convention that jt , hjt , and g() represent the estimates up to a constant, we have jt = hjt = t t + (1 l m )ljt k kjt + (1 m )(wjt pjt ) + m (pM jt pjt ) and g(hjt1 , rjt1 ) = 1(Rjt1 = 0)g01 (hjt1 ) +1(Rjt1 > 0)[(g10 g00 ) + g11 (hjt1 , rt1 )]. We can also estimate the random shocks ejt . When we combine multiple input demands, we use an average of the input-specic estimates. *** UPDATE/DELETE LAST SENTENCE. ***
3.3
Testing
The value of the GMM objective function for the optimal estimator, multiplied by N , has a limiting 2 distribution with LP degrees of freedom, where L is the number of instruments and P the number of parameters to be estimated.8 We use it as a test for overidentifying restrictions or validity of the moment conditions based on the instruments. We test whether the model satises certain restrictions by computing the restricted estimator using the weighting matrix for the optimal estimator and then comparing the values of the properly scaled objective functions. The dierence has a limiting 2 distribution with degrees of freedom equal to the number of restrictions. We also test whether the conditional expectation function is consistent with the knowledge capital model. Recall from Section 2 that the knowledge capital model implies that g1 (hjt1 , rjt1 ) = g10 + g11 (hjt1 0 , rjt1 ) has a particular functional form: jt = jt1 + exp(rjt1 ) exp(hjt1 /) exp(rjt1 ) = 0 + (hjt1 0 ) + exp(0 /) exp((hjt1 0 )/) exp(rjt1 ) = (0 ) + (hjt1 0 ) + exp((hjt1 0 )/) = g10 + g11 (hjt1 0 , rjt1 ), = hjt1 + exp(rjt1 ) exp(jt1 /)
where is a parameter to be estimated. We apply the Rivers & Vuong (2002) test for
Our baseline specication has 18 parameters: constant, trend, three production function coecients, and thirteen coecients in the series approximations.
8
16
model selection among nonnested models. After multiplying the dierence between the GMM objective functions by N , the test statistic has an asymptotic normal distribution with variance 2 = 4 (
j
zj vj ()) AN (
j
zj vj ()vj () zj )AN (
j
zj vj ()) ) zj )AN (
j
+(
j
zj vj (
KCM
)) AN (
j
zj vj (
KCM
)vj (
KCM
zj vj (KCM )) )) ,
2(
j
zj vj ()) AN (
j
zj vj ()vj (
KCM
) zj )AN (
j
zj vj (
KCM
where and KCM are the unrestricted and restricted parameter estimates, respectively, the instruments in zj are kept the same, and AN is a common rst-step weighting matrix.
4
Data
We use an unbalanced panel of Spanish manufacturing rms in nine industries during the 1990s. This broad coverage of industries is unusual, and it allows us to examine the link between R&D and productivity in a variety of settings that potentially dier in the importance of R&D. Our data come from the ESEE (Encuesta Sobre Estrategias Empresariales) survey, a rm-level survey of Spanish manufacturing sponsored by the Ministry of Industry.9 The unit surveyed is the rm, not the plant or the establishment. At the beginning of this survey in 1990, 5% of rms with up to 200 workers were sampled randomly by industry and size strata. All rms with more than 200 workers were asked to participate, and 70% of all rms of this size chose to respond. Some rms vanish from the sample, due to both exit and attrition. The two reasons can be distinguished, and attrition remained within acceptable limits. In what follows we reserve the word exit to characterize shutdown by death or abandonment of activity. To preserve representativeness, samples of newly created rms were added to the initial sample every year. We account for the survey design as follows. First, to compare the productivities of rms that perform R&D to those of rms that do not perform R&D we conduct separate tests on the subsamples of small and large rms. Second, to be able to interpret some of our descriptive statistics as aggregates that are representative for an industry as a whole, we replicate the subsample of small rms
70 5
= 14 times before merging it with the subsample
of large rms. Details on industry and variable denitions can be found in Appendix A. Given that our estimation procedure requires a lag of one year, we restrict the sample to rms with at least two years of data. The resulting sample covers a total of 1879 rms
This data has been used elsewhere, e.g., in Gonzalez, Jaumandreu & Pazo (2005) to study the eect of subsidies to R&D and in Delgado, Farinas & Ruano (2002) to study the productivity of exporting rms.
9
17
(before replication). Columns (1) and (2) of Table 1 show the number of observations and rms by industry. The samples are of moderate size. Firms tend to remain in the sample for short periods, ranging from a minimum of two years to a maximum of 10 years between 1990 and 1999. The descriptive statistics in Table 1 are computed for the period from 1991 to 1999 and exclude the rst observation for each rm.10 The small size of the samples is compensated for by the quality of the data, which seems to keep noise coming from errors in variables at relatively low levels. Entry and exit reported in columns (3) and (4) of Table 1 refer to the incorporation of newly created rms and to exit. Newly created rms are a large share of the total number of rms, ranging from 15% to one third in the dierent industries. In each industry there is a signicant proportion of exiting rms (from 5% to above 10% in a few cases). Columns (5)(9) of Table 1 show that the 1990s were a period of rapid output growth, coupled with stagnant or at best slightly increasing employment and intense investment in physical capital. The growth of prices, averaged from the growth of prices as reported individually by each rm, is moderate. The R&D intensity of Spanish manufacturing rms is low by European standards, but R&D became increasingly important during the 1990s (see, e.g., European Commission 2001).11 The manufacturing sector consists partly of transnational companies with production facilities in Spain and huge R&D expenditures and partly of small and medium-sized companies that invested heavily in R&D in a struggle to increase their competitiveness in a growing and already very open economy. Government funded R&D in the form of subsidies and other forms of support amounts to 7.7% of rms total R&D expenditures in the EU-15, 9.3% in the US, and 0.9% in Japan (European Commission 2004a). In Spain at most a small fraction of the rms that engaged in R&D received subsidies. The typical subsidy covers between 20% and 50% of R&D expenditures and its magnitude is inversely related to the size of the rm. Subsidies are used eciently without crowding out private funds and even stimulate some projects. Their eect is mostly limited to the amount that they add to the project (see Gonzalez et al. 2005). This suggests that R&D expenditures irrespective of their origin are the relevant variable for explaining productivity.12 Columns (10)(13) of Table 1 reveal that the nine industries are rather dierent when it comes to innovative activities of rms. This can be seen along three dimensions: the share of rms that perform R&D (columns (11) and (12)), the degree of persistence in performing R&D over time, and R&D intensity among performers dened as the ratio of
10 Since R&D expenditures appear lagged in our estimation equation (4), we report them for the period 1990 to 1998. 11 R&D intensities for manufacturing rms are 2.1% in France, 2.6% in Germany, and 2.2% in the UK as compared to 0.6% in Spain (European Commission 2004b). 12 While some R&D expenditures were tax deductible during the 1990s, the schedule was not overly generous and most rms simply ignored it. A big reform that introduced some real stimulus took place towards the end of our sample period in 1999.
18
R&D expenditures to output (column (13)). Three industries are highly active: Chemical products (3), agricultural and industrial machinery (4), and transport equipment (6). The share of rms that perform R&D during at least one year in the sample period is two thirds, with slightly more than 40% of stable performers that engage in R&D in all years and slightly more than 20% of occasional performers that engage in R&D in some (but not all) years. Dividing the share of stable performers by the combined share of stable and occasional performers yields the conditional share of stable performers and gives an indication of the persistence in performing R&D over time. With about 65% the degree of persistence is is very high. Finally, the average R&D intensity among performers ranges from 2.2% to 2.7%. Four industries are in an intermediate position: Metals and metal products (1), nonmetallic minerals (2), food, drink and tobacco (7), and textile, leather and shoes (8). The share of performers is lower than one half, but it is near one half in the rst two industries. With a conditional share of stable performers of about 40% the degree of persistence tends to be lower. The average R&D intensity among performers is between 1.1% and 1.5% with a much lower value of 0.7% in industry 7. Two industries, timber and furniture (9) and paper and printing products (10), exhibit low innovative activity. The rst industry is weak in the share of performers (below 20%) and degree of persistence. In the second industry the degree of persistence is somewhat higher with a conditional share of stable performers of 46% but the share of performers remains below 30%. The average R&D intensity is 1.4% in both industries. This heterogeneity in the three dimensions of innovative activities makes it dicult to t a single model to explain the impact of R&D on productivity. In addition, the standard deviation of R&D intensity is of substantial magnitude in the nine industries. This suggests that that heterogeneity across rms within industries is important, partly because rms engage in R&D to various degrees and partly because the level of aggregation used in dening these industries encompasses many dierent specic innovative activities.
5
Estimation results
We rst present our estimates of the production function and the Markov process that governs the evolution of productivity and test the linearity and certainty assumptions of the knowledge capital model. Next we turn to the link between R&D and productivity. In order to assess the role of R&D in determining the dierences in productivity across rms and the evolution of rm-level productivity over time, we examine ve aspects of this link in more detail: productivity levels and growth, the return to R&D, the persistence in productivity, and the rate of return.
19
5.1
Production function and Markov process
Table 2 summarizes dierent production function estimates. Columns (1)(3) report the coecients estimated from OLS regressions of the log of output on the logs of inputs. The coecients are reasonable as usual when running OLS on logs (but not when running OLS on rst-dierences of logs), and returns to scale are close to constant. The share of capital in value added, as given by the capital coecient scaled by the sum of the labor and capital coecients, is between 0.15 and 0.35 as expected. Columns (4)(9) of Table 2 report the coecients estimated when we use the demand for labor to back out unobserved productivity. Treating labor as a variable input is appropriate because Spain greatly enhanced the possibilities for hiring and ring temporary workers during the 1980s. By the beginning of the 1990s the share of temporary workers in the manufacturing sector had stabilized in excess of a quarter, one of the highest shares in Europe. Rapid expansion and contraction of the number of temporary workers became common (Dolado, Garcia-Serrano & Jimeno 2002). In addition, we measure labor as hours worked (see Appendix A for details). At this margin at least rms enjoy a high degree of exibility in determining the demand for labor. Specifying the law of motion of productivity to be an exogenous Markov process that does not depend on R&D expenditures yields the coecients reported in columns (4) (6). Compared to the OLS regressions, the changes go in the direction that is expected from theory. The labor coecients decrease considerably in all industries while the capital coecients increase somewhat in 7 industries. The materials coecients show no particular pattern. Changes are as expected not huge because we are comparing estimates in logs (as opposed to rst-dierences of logs). All this matches the results in OP and LP. Columns (7)(9) show the coecients obtained when specifying a controlled Markov process. Again, compared to the OLS regressions, the changes go in the expected direction. The labor coecients decrease in 8 cases, the capital coecients increase in 5 cases and are virtually the same in 2 more cases. In fact, changes from the exogenous to the controlled Markov process do not exhibit a distinct pattern. This leaves open the question whether it is possible to obtain consistent estimates of the parameters of the production function in the absence of data on R&D, although it is clear that omitting R&D expenditures from the Markov process substantially distorts the retrieved productivities (see Section 5.2 for details). To check the validity of our estimates we have conducted a series of tests as reported in Table 3. We rst test for overidentifying restrictions or validity of the moment conditions based on the instruments as described in Section 3.3 (columns (1) and (2)). The test statistic is too high for the usual signicance levels in only the case of industry 1. The other values indicate the validity of the moment conditions by a wide margin. Since the orthogonality of lagged labor and lagged materials plays a key role in the estimation, it is important to verify this assumption particularly carefully. Olley & Pakes 20
(1996) and Levinsohn & Petrin (2003) do so by testing the absence of correlation between the lagged inputs and the productivity innovation. In our case, the above test for overidentifying restrictions is already informing us of the closeness to zero of the set of all moment conditions. To more explicitly assess the validity of lagged labor and lagged materials as instruments, we compute the dierence in the value of the objective function when all moments are included to its value when the moments involving either lagged labor or lagged materials are excluded. As columns (3)(6) of Table 3 show, the validity of lagged labor and lagged materials as instruments cannot be rejected with the possible exception of lagged labor in industry 6 and lagged materials in industry 4. We also test the subset of moments involving capital and lagged capital. As columns (7) and (8) of Table 3 show, the exogeneity assumption on capital and lagged capital is only rejected at the usual signicance levels for industry 1. Taken together, our overidentifying tests also support our choice of the functional form for the production function: Had the assumed linearity in the log of inputs been violated, then at least part of the nonlinearity would have been pushed into the productivity innovation, thereby resulting in high values of the overidentifying test statistics. Our nal specication test validates more directly the structure of the model. Recall that the production function parameters appear both in the production function and in the inverse labor demand function. If the inverse labor demand function is misspecied (e.g., because labor is not a variable input), then this causes l and k in the inverse labor demand function to diverge from their counterparts in the production function. By testing the null hypothesis that the structural parameters in the two parts of the model are equal, we may thus rule out that our model is misspecied. Fortunately, as columns (9) and (10) of Table 3 show, while we must reject the null hypothesis of equality in industries 1, 7, and 10, in the remaining industries the test suggests by a wide margin that we may rule out that our model is misspecied. Imperfect competition. We test for imperfect competition by adding an unknown function in the equilibrium price pjt1 and the demand shifter zjt1 to hjt1 inside the conditional expectation function g() in equation (5). Under the null hypothesis of perfect competition pjt1 and zjt1 play no role. The data very clearly reject the assumption of a perfectly competitive environment, see columns (11) and (12) of Table 3. Our estimates of the average elasticity of demand are around 2 (column (13)). *** ALL THE ESTIMATES REPORTED IN THIS DRAFT OF THE PAPER ARE DONE ASSUMING PERFECT COMPETITION. PRELIMINARY ESTIMATES SHOW THAT BASIC RESULTS DO NOT CHANGE WITH IMPERFECT COMPETITION. *** Sample selection. *** SHOW THAT THE SELECTION PROBLEM IS NOT OVERLY SEVERE ON OUR SAMPLE BY COMPARING THE PRODUCTIVITY OF EXITORS 21
(AND ENTRANTS) TO THAT OF CONTINUING FIRMS. *** Alternative estimators and eciency gains. *** ADD COMPARISON TO ALTERNATIVE SPECIFICATIONS USING THE DEMAND FOR MATERIALS OR THE DEMANDS FOR BOTH LABOR AND MATERIALS TO BACK OUT UNOBSERVED PRODUCTIVITY. NONPARAMETRIC METHODS. ADD COMPARISON TO OP METHOD PUT TABLES IN APPENDIX. *** Nonlinearity. We next turn to the conditional expectation function g() that describes the Markov process of unobserved productivity. We assess the role of R&D by comparing the controlled with the exogenous Markov process. To this end, we test whether all terms in rjt1 can be excluded from the conditional expectation function g11 (hjt1 0 , rjt1 ) for performers plus the equality of the common part of the conditional expectation functions for performers and nonperformers, i.e., g11 (hjt1 0 , rjt1 ) = g01 (hjt1 0 ) for all rjt1 . As columns (1) and (2) of Table 4 shows, the result is overwhelming: In all cases the constraints imposed by the model with the exogenous Markov process are clearly rejected. We use a standard growth decomposition to get a sense of the importance of R&D. Roughly two thirds of the growth in output is explained by the growth in inputs, with the glaring exception of industry 8 where output is growing while inputs are shrinking. While there are considerable dierences across industries, about one half of the year-to-year variation in expected productivity due is the variation in R&D expenditures. While these numbers already hint at the major role played by R&D, they have to be interpreted as lower bounds because a part of the impact of current R&D expenditures persists and is carried forward into future productivity. We will come back to the persistence in productivity in Section 5.4. Next we test whether the conditional expectation function g() is separable in current productivity and R&D expenditures, i.e., whether g11 (hjt1 0 , rjt1 ) for rms that perform R&D can be broken up into two additively separable functions g11 (hjt1 0 ) and g12 (rjt1 ). The test statistics indicate that this is only the case in industries 1, 4, and perhaps 9 (columns (3) and (4) of Table 4). From hereon we impose separability on industry 4, where it slightly improves the estimates, but we keep nonseparability in industry 1, where separability does not seem to change anything. Given the limited number of rms that perform R&D in industries 9 and 10, we also impose separability in the interest of parsimony. The main result, however, is that the R&D process can hardly be considered separable. From the economic point of view this stresses that the impact of current R&D 22
on future productivity depends crucially on current productivity, and that current and past investments in knowledge interact in a complex fashion. We further illustrate the economic signicance of these interactions in columns (5)(8) of Table 4. We list the percentage of observations where
2 g(jt1 ,rjt1 ) jt1 Rjt1
=
2 1 g(jt1 ,rjt1 ) Rjt1 jt1 rjt1
is
signicantly positive (negative) so that current productivity and (the level of) R&D expenditures are, at least locally, complements (substitutes) in the accumulation of productivity. There is evidence of complementarities in industries 2, 3, and 6 whereas in industry 7 current productivity and R&D expenditures appear to be largely substitutes. We also list the percentage of observations where
2 g(jt1 ,rjt1 ) 2 Rjt1
=
2 Rjt1
1
2 g(jt1 ,rjt1 ) 2 rjt1
g(jt1 ,rjt1 ) rjt1
is signicantly positive (negative) so that there are locally increasing (decreasing) returns to R&D. There is evidence of increasing returns to R&D in industries 1, 2, 6, 7, 8, and 9. We nally test whether the conditional expectation function is consistent with the knowledge capital model. Our estimates of the elasticity of output with respect to knowledge capital are between 0.32 and 0.67 for the dierent industries, see column (11) of Table 4. Nevertheless, the data very clearly reject the functional form restrictions implied by the knowledge capital model (columns (9) and (10)).13 This suggests that the linearity assumption in the accumulation and depreciation of knowledge that underlies the knowledge capital model may have to be relaxed in order to fully assess the impact of the investment in knowledge on the productivity of rms. Uncertainty. We nally turn from the certainty to the linearity assumption in the knowledge capital model. Column (12) of Table 4 tells us the ratio of the variance of the random shock ejt to the variance of unobserved productivity jt . Despite dierences among industries, the variances are quite similar in magnitude. This suggests that unobserved productivity is at least as important in explaining the data as the host of other factors that are embedded in the random shock. Column (13) of Table 4 gives the ratio of the variance of the productivity innovation jt to the variance of actual productivity jt . The ratio shows that the unpredictable component accounts for a large part of attained productivity, between 20% and 50%, thereby casting doubt on the certainty assumption of the knowledge capital model.14 Interestingly enough, a high degree of uncertainty in the R&D process seems to be characteristic for both some
We continue to reject when we base the test on the exact form for the law of motion implied by the knowledge capital model rather than the approximate form in equation (2). 14 Further scrutiny shows as expected that the degree of uncertainty as measured by the ratio of the variance of jt to the variance of jt is at least as large for observations with positive R&D expenditures than for those without (with the exception of industry 8), although this is sometimes due to a smaller denominator rather than a larger numerator. Note that, to the extent that uncertainty inhibits rms investments in R&D, we underestimate the degree of uncertainty for observations with positive R&D expenditures, and that this may also explain why the variance of jt is smaller for observations with positive R&D expenditures in some industries. The degree of uncertainty tends to be smaller for observations with positive investment in physical capital than for those without (with the exceptions of industries 4 and 10). There does not seem to be a relationship with rm size.
13
23
of the most and some of the least R&D intensive industries. We will come back to the economic signicance of the uncertainties inherent in the R&D process in Section 5.5.
5.2
Productivity levels
To describe dierences in expected productivity between rms that perform R&D and rms that do not perform R&D, we employ kernels to estimate the density and the distribution functions associated with the subsamples of observations with R&D and without R&D. To be able to interpret these descriptive measures as representative aggregates, we proceed as described in Section 4. Figure 1 shows the density and distribution functions for performers (solid line) and nonperformers (dashed line) for each industry. In all industries but 4, 9, and 10, the distribution for performers is to the right of the distribution for nonperformers. This strongly suggests stochastic dominance. In contrast, in industries 4 and 10 the distribution functions openly cross: Attaining the highest levels seems more likely for the nonperformers than for the performers. In industry 9 the distribution for nonperformers dominates the one for performers. Before formally comparing the means and variances of the distributions and the distributions themselves, we illustrate the impact of omitting R&D expenditures from the Markov process of unobserved productivity. We have added the so-obtained density and distribution functions to Figure 1 (dotted line). Comparing them to the density and distribution functions for a controlled Markov process reveals that the exogenous process takes a sort of average over rms with distinct innovative activities and hence blurs remarkable dierences in the impact of the investment in knowledge on the productivity of rms. *** WHEN USING ALTERNATIVE ESTIMATORS SHOW THAT THE PRODUCTIVITY OF PERFORMERS AND NONPERFORMERS IS NOT SYSTEMATICALLY DIFFERENT, JUST LIKE WHEN WE ASSUME THAT PRODUCTIVITY IS EXOGENOUS (PUT TABLES/GRAPHS IN APPENDIX). *** Mean and variance. Turning to the moments of the distributions, the dierence in means is computed as g0 g1 = 1 N T0 1 N T1 1(rjt1 = 0)g01 (hjt1 )
t
j j
t
1(rjt1 > 0)[(g10 g00 ) + g11 (hjt1 , rt1 )],
where N T0 and N T1 are the size of the subsamples of observations without and with R&D, respectively. We compare the means using the test statistic t= g0 g1 V ar(g01 )/(N T0 1) + V ar(g11 )/(N T1 1)
24
which follows a t distribution with min(N T0 , N T1 ) 1 degrees of freedom and the variances using F = V ar(g01 ) V ar(g11 )
which follows an F distribution with N T0 1 and N T1 1 degrees of freedom. Column (4) of Table 5 reports the dierence in means g 1 g 0 (with the opposite sign of the test statistic for the sake of intuition) and columns (5)(8) report the standard deviations and the test statistics along with their probability values separately for the subsamples of small and large rms. The dierence in means is positive for rms of all sizes in all industries that exhibit medium or high innovative activity, with the striking exception of industry 4. The dierences are sizable, with many values between 4% and 5% and up to 9%. They are often larger for the smaller rms. In the two industries that exhibit low innovative activity, however, one size group shows a lower mean of expected productivity than the other: The small rms in industry 9 and the large rms in industry 10. The formal statistical test duly rejects, at the usual signicance levels, the hypothesis of a higher mean of expected productivity among performers than among nonperformers in these two cases and in both size groups in industry 4. The hypothesis of greater variability for performers than for nonperformers is rejected in many cases, although there does not seem to be a recognizable pattern. As can be seen in columns (9) and (10) of Table 5, it is rejected for both size groups in industries 4, 6, 7, and 10, for small rms in industries 2, 3 and 9, and for large rms in industries 1 and 8. Distribution. The above results suggest to compare the distributions themselves. We use a Kolmogorov-Smirnov test to compare the empirical distributions of two independent samples (see Barret & Donald (2003) and Delgado et al. (2002) for similar applications). Since this test requires that the observations in each sample are independent, we consider as the variable of interest the average of expected productivity for each rm, where for occasional performers we average only over the years with R&D (and discard the years without R&D). This avoids dependent observations and sets the sample sizes equal to the number of nonperformers and performers, N0 and N1 , respectively. Let FN0 () and GN1 () be the empirical cumulative distribution functions of nonperformers and performers, respectively. We apply the two-sided test of the hypothesis FN0 (g) GN1 (g) = 0 for all g, i.e., the distributions of expected productivity are equal, and the one-sided test of the hypothesis FN0 (g) GN1 (g) 0 for all g, i.e., the distribution GN1 () of expected productivity of performers stochastically dominates the distribution FN0 () of expected productivity of nonperformers. The test statistics are S1 = N0 N1 max {|FN0 (g) GN1 (g)|} , N0 + N1 g S2 = N0 N1 max {FN0 (g) GN1 (g)} , N0 + N1 g
respectively, and the probability values can be computed using the limiting distributions 25
P (S 1 > c) = 2
k k=1 (1)
exp(2k 2 c2 ) and P (S 2 > c) = exp(2c2 ).
Because the test tends to be inconclusive when the number of rms is small, we limit it to cases in which we have at least 20 performers and 20 nonperformers. This allows us to carry out the tests for the small rms in 8 industries and for the large rms in industries 7 and 8. The results are reported in columns (11)(14) of Table 5. Equality of distributions is rejected in six out of ten cases. Stochastic dominance can hardly be rejected anywhere with the exception of industry 4. To further illustrate the consequences of omitting R&D expenditures from the Markov process of unobserved productivity, we have redone the above tests for the case of an exogenous Markov process. The results are striking: We can no longer reject the equality of the productivity distributions of performers and nonperformers in eight out of ten cases. This once more makes apparent that omitting R&D expenditures substantially distorts the retrieved productivities. In sum, comparing expected productivity across rms that perform R&D and rms that do not perform R&D we nd strong evidence of stochastic dominance in most industries. It remains to be explained why expected productivity for performers appears eventually lower than for nonperformers in some industries. One possible explanation is heterogeneity across rms within industries, i.e., stochastic dominance may hold if we were able to split these industries into more homogeneous innovative activities.
5.3
Productivity growth
We explore productivity growth from the point of view of what a rm expects when it makes its decisions in period t 1. Because jt1 is known to the rm at the time it decides on rjt1 , we compute the expectation of productivity growth as t + E(jt jt1 |jt1 , rjt1, ) = t + g(jt1 , rjt1 ) jt1 . (7)
Using the fact that the innovation to productivity has mean zero, i.e., E(jt1 |jt2 , rjt2 ) =
1 0, we estimate the average of the expectation of productivity growth as t + N j 1 t Tj [g(hjt1 , rjt1 )
g(hjt2 , rjt2 )]. Columns (1)(3) of Table 6 report the results for the entire sample and for the subsamples of observations with and without R&D. In what follows we drop 2.5% of observations at each tail of the distribution to guard against outliers. We also compute a weighted version to be able to interpret the expectation of productivity growth as representative for an industry as a whole. The weights jt = Yjt2 / mate the average as t + with and without R&D.
1 T t j j
Yjt2 are given by the share
of output of a rm two periods ago. Assuming that E(jt jt1 |jt2 , rjt2 ) = 0, we estijt [g(hjt1 , rjt1 ) g(hjt2 , rjt2 )]. Columns (4)(6) of Table 6 report the results along with a decomposition into the contributions of observations
26
Productivity growth is higher for performers than for nonperformers in 5 industries, sometimes considerably so. Taken together these industries account for two thirds of manufacturing output. The industries in which the relationship is reversed coincide again with industries 4, 9, and 10 to which we must now add industry 8. The standard deviations indicate that there are considerable dierences in productivity growth within rms that engage in R&D as well as within those that do not. Productivity growth is more variable for performers than for nonperformers in six out of nine industries, including industries 4, 9, and 10. This indicates that the productivity of at least some performers tends to grow much faster than the productivity of nonperformers, even though on average performers exhibits slower productivity growth than nonperformers in these industries. A comparison of unweighted and weighted productivity growth shows that there is no denite pattern in productivity growth by size group: The productivity of small rms grows more rapidly in some industries and less in others. What is clear, however, is that productivity growth is highest in some of the industries with high innovative activity (above 2% in industries 3 and 6) followed by some of the industries with intermediate innovative activity (above 1.5% in industries 1 and 2). Columns (5) and (6) of Table 6 are particularly important. The contribution to productivity growth of rms that perform R&D is estimated to explain between 70% and 85% of productivity growth in the industries with high innovative activity and between 45% and 65% in the industries with intermediate innovative activity (with the exception of industry 8). This is all the more remarkable since in these industries between 35% and 45% and between 10% and 20% of rms engage in R&D. While these rms manufacture between 70% and 75% of output in the industries with high innovative activity and between 45% and 55% in the industries with intermediate innovative activity, their contribution to productivity growth exceeds their share of output by between 5% and 15%. That is, rms that engage in R&D tend not only to be larger than those that do not but also to grow even larger over time. R&D expenditures are thus indeed a primary source of productivity growth. Decomposition. The growth in expected productivity in equation (7) can be decomposed (excluding the trend) as g(jt1 , rjt1 ) jt1 = [g(jt1 , rjt1 ) g(jt1 , r)] + [g(jt1 , r) jt1 ] , (8)
where r denotes a negligible amount of R&D expenditures.15 The rst term in brackets reects the change in expected productivity that is attributable to R&D expenditures rjt1 , the second the change that takes place in the absence of investment in knowledge. That is, the second term in brackets is attributable to depreciation of already attained productivity
Recall that we allow the conditional expectation function g() to be dierent when the rm adopts the corner solution of zero R&D expenditures and when it chooses positive R&D expenditures. We take g(jt1 , r) to be either g0 (jt1 ) or g1 (jt1 , r), where r is the fth percentile of R&D expenditures, to avoid the discontinuity. Both approaches failed to produce sensible results in case of industry 9.
15
27
and, consequently, is expected to be negative. The net eect of R&D is thus the sum of its gross eect (rst term) and the impact of depreciation (second term). Columns (7)(9) of Table 6 report unweighted averages. The results are striking given the scarce structure that our model imposes on the data. The impact of depreciation is, in fact, negative with the exception of industry 7. Its magnitude is substantial, ranging from 50% to 85% of the gross eect of R&D. As a consequence, the gross eect of R&D considerably exceeds its net eect. In sum, a large part of rms R&D expenditures is devoted to maintaining already attained productivity rather than to advancing it.
5.4
Return to R&D and persistence in productivity
To more closely assess how hard a rm must work to maintain and advance its productivity, recall that a change in the conditional expectation function g() can be interpreted as the expected percentage change in total factor productivity. Hence, Similarly,
jt g(jt1 ,rjt1 ) is jt1 = jt1 g(jt1 ,rjt1 ) is the productivity. jt1 jt rjt1
=
g(jt1 ,rjt1 ) rjt1
is the
elasticity of output with respect to R&D expenditures or a measure of the return to R&D. the elasticity of output with respect to already attained degree of persistence in the productivity process or a
measure of inertia. It tells us the fraction of past productivity that is carried forward into current productivity. Note that the elasticities of output with respect to R&D expenditures and already attained productivity vary from rm to rm with already attained productivity and R&D expenditures. Our model thus allows us to recover the distribution of these elasticities and to describe the heterogeneity across rms. Columns (1)(4) of Table 7 present the quartiles of the distribution of the elasticity with respect to R&D expenditures along with a weighted average computed as where the weights jt = Yjt /
j 1 T t j
jt
g(jt1 ,rjt1 ) , rjt1
Yjt are given by the share of output of a rm. There is a
considerable amount of variation across industries and the rms within an industry. The returns to R&D at the rst, second, and third quartile range between 0.032 and 0.009, 0.010 and 0.015, and 0.007 and 0.029, respectively. Their average is close to 0.015, varying from 0.002 to 0.028 across industries. Note that negative returns to R&D are legitimate and meaningful in our setting, although some of them may be an artifact of the nonparametric estimation of g() at the boundaries of the support. A negative return at the margin is consistent with an overall positive impact of R&D expenditures on output. A rm may invest in R&D to the point of driving returns below zero for a number of reasons including indivisibilities and strategic considerations such as a loss of an early-mover advantage. This type of eect is excluded by the functional form restrictions of the knowledge capital model, in particular the assumption that the stock of knowledge capital depreciates at a constant rate. More generally, it is plausible that investments in knowledge take place in response to existing knowledge becoming obsolete or vice versa that investments render existing knowledge obsolete. Our model captures this interplay between adding new knowledge and keeping 28
old knowledge. The degree of persistence can be computed separately for performers using the conditional expectation function g1 () that depends both on already attained productivity and R&D expenditures and for nonperformers using g0 () that depends solely on already attained productivity. Columns (5)(10) of Table 7 summarize the distributions for performers and nonperformers. Again there is a considerable amount of variation across industries and the rms within an industry. Nevertheless, nonperformers systematically demonstrate a higher degree of persistence than performers (with the exception of industry 8). An intuitive explanation for this nding is that nonperformers learn from performers, but by the time this happens the transferred knowledge is already entrenched in the industry and therefore more persistent. Put dierently, common practice may be stickier than best practice. The degree of persistence for performers is negatively related to the degree of uncertainty in the productivity process as measured by the ratio of the variance of the productivity innovation jt to the variance of actual productivity jt . That is, productivity is less persistent in an industry where a large part of its variance is due to random shocks that represent the uncertainties inherent in the R&D process. Figure 2 illustrates this relationship between persistence and uncertainty at the level of the industry. To facilitate the comparison with the existing literature, we have estimated the knowledge capital model as given in equation (1). Proceeding along the lines of Hall & Mairesse (1995), we construct Cjt , the stock of knowledge capital of rm j in period t, from R&D expenditures using the perpetual inventory method. We assume that the rate of depreciation is 0.15 per period and estimate the initial capital from the date of birth of the rm by extrapolating its average R&D expenditures during the time that it is observed.16 Column (11) of Table 7 presents the estimate of the elasticity of output with respect to the stock of knowledge capital from the knowledge capital model. In addition to the gross-output version in equation (1) we have also estimated a value-added version of the knowledge capital model (column (13)). In contrast to our model, the knowledge capital model yields one numberan average elasticityper industry. The elasticity of output with respect to the stock of knowledge capital tends to be small and rarely signicant in the gross-output version but becomes larger in the value-added version. The estimates turn out to be on the low side for this type of exercise. One possible reason may be the non self-selected character of the sample, but perhaps this is the magnitude of estimates that one should expect given the low R&D intensity of Spanish manufacturing rms. Beneito (2001) and Ornaghi (2006), for example, estimate aggregate elasticities ranging from 0.04 to 0.10. To convert the elasticity with respect to the stock of knowledge capital into an elasticity
We drop the term cjt from equation (1) for nonperformers and specify a dierent constant and time trend for performers and nonperformers. To facilitate estimation we impose the widely accepted constraint of constant returns to scale in the conventional inputs.
16
29
with respect to R&D expenditures that is comparable to our model, we multiply the former by Rjt1 /Cjt . Columns (12) and (14) of Table 7 show a weighted average of the so-obtained elasticities. The elasticities with respect to R&D expenditures from our model are higher than the highest elasticities from the knowledge capital model in ve industries and lower but very close in three more industries. In addition, the elasticities obtained with our model have a non-normal, fairly spread out distribution. This sharply contrasts with the fact that the dispersion of elasticities in the knowledge capital model is purely driven by the distribution of the ratio Rjt1 /Cjt (since, recall, the knowledge capital model yields just an average of the elasticity with respect to the stock of knowledge capital). Turning to persistence in productivity, note that the degree of persistence is 1 0.15 = 0.85 by assumption in the knowledge capital model. In contrast, the degree of persistence in our model is much lower (see also Pakes & Schankerman 1984b). Moreover, we nd that there are substantial dierences between rms in the degree of persistence. The degree of persistence is expected to be lower when process innovations are rapidly spread or when product innovations are quickly imitated or superseded. (Since output is measured in dollars, we are unable to distinguish between product and process innovations, similar to the knowledge capital literature.) On other other hand, the demand advantage of a product innovation may be oset by a productivity disadvantage if newer products are costlier to produce, thereby lessening the impact of product innovations on persistence.17 The heterogeneity across rms and industries in the degree of persistence points to an interesting avenue for future research that explores the link between the dynamics of productivity and the nature of product market competition. One could also argue that the lower degree of persistence is a result of the substantial variability in the R&D expenditures that drive the evolution of productivity. The knowledge capital model constructs the stock of knowledge capital that is much smoother and less variable than R&D expenditures. Our view is that the variability in R&D expenditures across rms and periods is likely to contain useful information on the impact of R&D on productivity, but we acknowledge that some of the variability in the R&D expenditures is an artifact of accounting conventions. In sum, it appears that old knowledge is hard to keep but new knowledge is easy to add. Productivity is therefore considerably more uid than what the knowledge capital literature suggests.
5.5
Rate of return
We nally compute an alternativeand perhaps more intuitivemeasure of the return to R&D. Recall from the decomposition of the growth in expected productivity in equation (8)
The ESEE survey asks rms whether they have introduced a new product or process over the course of the survey year. This data suggests that, at the level of the industry, the degree of persistence is negatively related to the prevalence of both product and process innovations.
17
30
that g(jt1 , rjt1 ) g(jt1 , r) is the change in expected productivity that is attributable to R&D expenditures. Multiplying it by a measure of expected value added, say Vjt , gives the rent that the rm can expect from this investment at the time it makes its decisions. Dividing it further by R&D expenditures Rjt1 gives an estimate of the gross rate of return, or dollars obtained by spending one dollar on R&D.18 Note that we compute the gross rate of return on R&D using value added instead of gross output both to make it comparable to the existing literature (e.g., Nadiri 1993, Griliches & Regev 1995, Griliches 2000) and because value added is closer to prots than gross output. We further decompose the gross rate of return to R&D into a net rate of return and a compensation for depreciation. To do so, we rearrange the growth decomposition in equation (8) to yield g(jt1 , rjt1 ) g(jt1 , r) = [g(jt1 , rjt1 ) g(jt2, rjt2 )] + [g(jt2, rjt2 ) g(jt1 , r)], (9)
where we also have replaced jt1 by g(jt2, rjt2 ) + jt1 and canceled jt1 from the equation to reduce the impact of uncertainty. Multiplying and dividing through by Vjt and Rjt1 , respectively, we obtain the net rate of return to R&D as the rst term in brackets and the compensation for depreciation as the second term. Columns (1)(3) of Table 8 summarize the gross rate of return to R&D and its decomposition into the net rate and the compensation for depreciation. We report weighted averages where the weights jt = Rjt2 /
j
Rjt2 are given by the share of R&D expenditures of a
rm two periods ago.19 The gross rate of return to R&D exceeds the net rate in line with our previous nding that a large part of rms R&D expenditures is devoted to maintaining already attained productivity rather than to advancing it. The net rates of return to R&D dier across industries, ranging from very modest values near zero to 35%. Interestingly enough, the net rate of return to R&D is higher in an industry where a large part of the variance in productivity is due to random shocks, as can be seen in Figure 3. This suggests that the net rate of return to R&D includes a compensation for the uncertainties inherent in the R&D process. Column (4) of Table 8 reports the gross rate of return on investment in physical capital as a point of comparison and column (5) the ratio of the gross rates of return to R&D and investment in physical capital.20 Returns to R&D are clearly higher than returns to
The average rate that we compute is close to the marginal rate of return to R&D. To see this, ling(jt1 ,ln Rjt1 ) 1 early approximate g(jt1 , ln R) g(jt1 , ln Rjt1 ) + (R Rjt1 ). If R 0, then rjt1 Rjt1
jt1 jt1 g(jt1 , rjt1 ) g(jt1 , r) g(jt1 , ln Rjt1 ) g(jt1 , ln R) . rjt1 19 As before we take g(jt1 , r) to be either g0 (jt1 ) or g1 (jt1 , r), where r is the fth percentile of R&D expenditures. We calculate the rst two terms of the decomposition in equation (9) and infer the third term. We either trim 2.5% of observations at each tail of both of these distributions and 3.5% at both distributions of rates (obtained by multiplying the dierences in the conditional expectation functions by Vjt /Rjt1 ) or 5% at both distributions of rates. As a result we always employ around of 80% of the data. 20 The gross rate of return on investment in physical capital is computed as k Vjt /Kjt . We report averages
18
g(
,r
)
31
investment in physical capital. The gross rate of return to R&D is often twice that of the gross rate of return to investment in physical capital, with the ratio of the gross rates being as large as 3.5 and 3.9 in industries 6 and 3, respectively. This reects the fact that knowledge depreciates faster than physical capital,21 but also that investment in knowledge is systematically more uncertain than investment in physical capital. In fact, the large ratios suggest that the uncertainties inherent in the R&D process are economically signicant and matter for rms investment decisions. To facilitate the comparison with the existing literature, we have used the value-added version of the knowledge capital model to estimate the gross rate of return to R&D by regressing the rst-dierence of the log of value added on the rst-dierences of the logs of conventional inputs and the ratio Rjt1 /Vjt1 of R&D expenditures to value added. The estimated coecient of this ratio can be interpreted as the rate of return to R&D.22 As can be seen from column (6) of Table 8, while the gross rates are imprecisely estimated in the knowledge capital model, they tend to be higher than the gross rates in our model. The question is then whether and why our rates of return to R&D should be considered more reliable and whether this justies the extra eort of pursuing the more structural approach. Our rates are computed from more reliable coecient estimates than what the knowledge capital model provides because our estimator takes into account the possibility of endogeneity bias in assessing the role of R&D. Because our model is structural we are more condent in the causality of the estimated relationship between expected productivity, current productivity, and R&D expenditures. The drawback of our approach is that it depends on the informational and timing assumptions that we make. These assumptions, however, appear to be broadly accepted in the literature following OP. More generally, the knowledge capital literature has had limited success in estimating the rate of return to R&D. Griliches (2000) contends that [e]arly studies of this topic were happy to get the sign of the R&D variable right and to show that it matters, that it is a signicant variable, contributing to productivity growth (p. 51). Estimates of the rate of return to R&D tended to be high, often implausibly high: our current quantitative understanding of this whole process remains seriously awed ... [T]he size of the eects we have estimated may be seriously o, perhaps by an order of magnitude (Griliches 1995, p. 83). Our estimates, by contrast, are more modest. R&D policy. While the knowledge capital model yields an industry-wide average rate of return to R&D, our model allows us to recover the entire distribution of rates. The fact that
after trimming 2.5% of observations at each tail of the distribution. 21 The rate of depreciation that is assumed in computing the stock of physical capital is around 0.1 but diers across industries and groups of rms within industries. 22 Recall that is the elasticity of value added with respect to knowledge capital. Since cjt = V Cjt1 V Cjt cjt and Rjt1 approximates Cjt (by the law of motion for knowledge capital), C Vjt1 C Vjt1 the estimated coecient is V . Since spending one dollar on R&D adds one unit of knowledge capital C is, in turn, equal to V or the rate of return to R&D. R
V C
32
rates diers across rms makes our model potentially useful in determining the allocation of subsidies, a major issue in R&D policy. To illustrate, we have run a reduced-form regression of the gross rate of return on the characteristics of the rm that are observable to a policy maker, including polynomials in the size of the rm and its R&D expenditures, the nature of innovation (process vs. product), the R&D employment of the rm, the proportion of R&D subsidies that it receives, the age of the rm, and its investment in physical capital. We have also run similar regressions for the net rate of return and the compensation for depreciation. These regressions indicate a systematic relationship with rm size and R&D expenditures but not with the other variables. Our conclusions are twofold. First, R&D expenditures and the gross rate of return are related, although this relationship can go in either direction (positive in industries 2, 8, and 10 and negative in industries 3, 4, and 6). The relationship with the net rate of return tends to be positive (industries 1, 2, 3, and 8): As expected, rms with higher returns also invest more. Finally, there is a negative relationship between R&D expenditures and the compensation for depreciation (industries 1, 3, and 4) suggesting that depreciation holds back investment in knowledge. Second, given the positive relationship between rm size and the gross rate of return (industries 1, 3, 4, 6, 7, and 10), we tentatively conclude that, if the goal is to maximize returns, then larger rms should be subsidized more extensively than smaller rms. Of course, this is just the gross rate of return whereas the net rate bears a much weaker relationship with rm size (positive in industries 6 and 10). In fact, rm size tends to be positively associated with the compensation for depreciation (industries 1, 3, 4, and 7), i.e., the wedge between the gross and net rate of return to R&D. A fuller exploration of the implications of the heterogeneity across rms for R&D policy is left to future research.
6
Concluding remarks
In this paper we develop a simple estimator for production functions. The basic idea is to exploit the fact that decisions on variable inputs such as labor and materials are based on current productivity. This results in input demands that are invertible functions and thus can be used to control for unobserved productivity in the estimation. Moreover, the parametric specication of the production function implies a known form for these functions. This renders identication and estimation more tractable. As a result, we are able to accommodate a controlled Markov process, thereby capturing the impact of R&D on the evolution of productivity. We illustrate our approach to production function estimation on an unbalanced panel of more than 1800 Spanish manufacturing rms in nine industries during the 1990s. We obtain sensible parameters estimates. Our estimator thus appears to work well. *** COMMENT ON EFFICIENCY GAINS. *** Overall, we show that the link between R&D and productivity is subject to a high
33
degree of uncertainty, nonlinearity, and heterogeneity. By accounting for uncertainty and nonlinearity, our approach extends the knowledge capital model. In fact, the knowledge capital model is a special case of our model, albeit one that is rejected by the data. Our model is richer, in particular with regard to the treatment of heterogeneity, thereby allowing us to show that R&D is a major determinant of the dierences in productivity across rms and the evolution of rm-level productivity over time. Productivity appears to be considerably more uid than what the knowledge capital literature suggests. Our approach also appears to provide us with more plausible answers to questions regarding the rate of return to R&D. The rate of return includes a compensation for the uncertainties inherent in the R&D process. Moreover, the large gap between the rates of return to R&D and investment in physical capital suggests that these uncertainties are economically signicant and matter for rms investment decisions. Our method can be applied to other contexts, for example, to model and test for two types of technological progress in production functions: Hicks-neutral technological progress that shifts the production function in its entirety and labor-saving technological progress that shifts the ratio of labor to capital. Economists have been for a long time interested in disentangling these eects. In ongoing work we have begun to explore how ...