Public Infrastructure and Productivity in U.S. Manufacturing: Evidence from the Price Augmenting AIM Cost Function

Transcription

1 Public Infrastructure and Productivity in U.S. Manufacturing: Evidence from the Price Augmenting AIM Cost Function Guohua Feng Department of Econometrics Monash University Victoria 3800, Australia and Apostolos Serletis Department of Economics University of Calgary Calgary, Alberta, T2N 1N4 Canada March 6, 2009 Serletis gratefully acknowledges nancial support from the Social Sciences and Humanities Research Council (SSHRC). 1

2 Abstract In this paper, we take a cost function approach to the investigation of the e ects of public infrastructure on the performance of the U.S. manufacturing industry, using KLEMS data over the period from 1953 to We build on Duggal et al. (1999) and Feng and Serletis (2008) and propose a price augmenting Asymptotically Ideal Model (AIM) model where public infrastructure is incorporated into the cost function as part of the e ciency index rather than as a xed input. In doing so, we specify the growth rate of the e ciency index as a Box-Cox function of public infrastructure and a time trend, a proxy for other technology. The excellent exibility of our price augmenting AIM cost function o ers many insights regarding the e ects of infrastructure on the U.S. manufacturing sector. JEL classi cation: C30; D24. Keywords: Flexible functional forms; Asymptotically ideal model (AIM); Nonlinear cost function; Technological index; Public infrastructure. 2

3 1 Introduction The size and signi cance of the e ects of public infrastructure on the economic performance of the private sector has been a hot-debated topic since Aschauer (1989) raised the issue of productivity of infrastructure capital see, for example, surveys in Gramlich (1994), Sturm et al. (1998), and Romp and de Haan (2007). Early studies in this literature nd a close correlation between reductions in public capital investment and declining privatesector productivity in the United States and many other developed economies. Most of the subsequent studies have been aimed at reaching consensus on the extent of these e ects. Despite the voluminous literature, however, the issue remains contentious. The literature investigating the e ects of public infrastructure on the productivity of the private sector is dominated by two approaches the production function approach and the cost function approach. Introduced by Aschauer (1989), the basic idea of the production function approach is to expand an aggregate production function by specifying public infrastructure as a separate factor input. Early studies along this line [see, for example, Aschauer (1989) and Munnell (1990)] nd that public investment has a much greater return to private-sector economic performance than does private capital investment. For example, Aschauer (1989) nds the elasticity of output with respect to public capital to range from 0:39 to 0:56 and the marginal product of public capital implied by this result is 100% or more see Gramlich (1994). These ndings imply that policy measures designed to augment public infrastructure investment could dramatically enhance productivity. In questioning the robustness of the empirical results, subsequent studies using the production function approach focus more on re ning the econometric structures by incorporating state and time xed e ects and/or by solving potential econometric problems, such as nonstationarity, spurious correlation, and endogeneity. The estimated e ects of public infrastructure investment on private sector productivity vary across these later studies. In particular, some studies nd estimates similar to the earlier ones [see, for example, Munnell (1990)]; some nd estimates with reduced magnitude and signi cance [see, for example, Tatom (1991) and Kemmerling and Stephan (2002)]; and others even nd the estimated productivity e ects to disappear [see, for example, Holtz-Eakin (1994), Hulten and Schwab (1991), and Evans and Karras (1994)]. Although less frequently used than the production function approach, the cost function approach o ers a di erent perspective on the e ects of public infrastructure investment on the performance of the private sector. Pioneered by Berndt and Hansson (1992), Nadiri and Mamuneas (1994), and Morrison and Schwartz (1996), the cost function approach, in assuming that rms minimize cost subject to a given level of output, treats public infrastructure as an unpaid xed input. Compared with the production function approach, the cost function approach has some advantages see Berndt and Hansson (1992) for a detailed discussion of the merits of the cost function approach. It is less likely to su er from the endogeneity problem, since exogenous input prices, instead of endogenous input quantities, are used; it 3

4 usually employs exible functional forms in place of the restrictive Cobb-Douglas functional form as in the production function approach; and it enables one to assess whether the amount of public infrastructure is insu cient or excessive by comparing the shadow value of public infrastructure and its market price. Empirical results from such models suggest smaller, but statistically signi cant and more robust, estimates of infrastructure e ects on overall productivity growth than found in the studies employing the production function approach. Unfortunately, most of the above mentioned studies su er from a theoretical problem in that they violate the marginal productivity theory see Dugall et al. (1999, 2007). In particular, in treating public capital and other private inputs (i.e. private capital and labor) symmetrically, those studies using the production function approach implicitly assume that a market determined per unit cost of infrastructure is known to the individual rms and thus can be calculable to rms. However, since public sector capital formation is generally nanced through taxation or government debt, per unit costs of public capital are not market determined. In the case of the cost function approach, while public infrastructure capital is considered to be an external xed input, each individual rm must still decide the amount it should use. This is particularly clear from those studies that adjust the usage of public infrastructure by capacity utilization. This adjustment implies that a rm s usage of the available infrastructure is part of its optimization problem, which in turn leads to the necessity of a demand function for public infrastructure that must satisfy the conditions of standard marginal productivity theory see Dugall et al. (1999). Further, as Dugall et al. (1999) has argued there is no guarantee that the total cost of infrastructure to the rm is related to the amount it uses. In recognizing this theory violation problem, Dugall et al. (1999, 2007) innovatively incorporate public capital into a modi ed Cobb-Dougls production function as part of the technological constraint that determines total factor productivity. In particular, Dugall et al. (1999) specify the Hicks neutral technical change term (technological index) in the production function as a nonlinear function of public infrastructure and a time trend, which is used as a proxy for other technology. As such, instead of serving as a discretionary factor that will take away the contribution of the factor inputs, public infrastructure becomes a factor that shifts the production function upward, and thus enhances the marginal products of private factor inputs. In other words, rather than being an additional input factor, public infrastructure is now a determinant of productivity. Applying this framework to aggregate level data for the United States, over the period from 1960 to 1989, Dugall et al. (1999) nd an estimate of output elasticity with respect to public infrastructure similar to that in Aschauer (1989). Following Dugall et al. (1999, 2007) in spirit, this paper proposes a price augmenting Asymptotically Ideal Model [AIM] within a cost function framework [see Barnett et al. (1991)], where public infrastructure is treated as a determinant of productivity. In particular, public infrastructure is incorporated into the cost function as part of e ciency indexes, which are used to adjust the prices measured in conventional terms. In doing so, we assume that 4

5 public infrastructure is factor augmenting. That is, like disembodied technological advances in the design of capital and increased formal education, public infrastructure can increase or decrease the e ciency level (e ectiveness) of private factor inputs. Under this assumption, we rst theoretically derive a general e ciency index augmented cost function, which is dual to a factor augmenting production function. We call this e ciency index augmented cost function price augmenting cost function to re ect the fact that e ciency adjusted private input prices, instead of e ciency adjusted private input quantities, are used. We further show that total factor productivity within this framework is an input cost share weighted average of the change in e ciency levels. As such, public infrastructure becomes a factor that shifts the cost function inward, rather than another input that is treated in the same way as the xed private factor inputs. We then assume that the general price augmenting cost function takes the AIM functional form, and that the e ciency index take the exponential Box-Cox functional form. Based on a linearly homogeneous multivariate Müntz-Szatz series expansion, the AIM cost function is globally exible in the sense that it is capable of approximating the underlying cost function at every point in the function s domain by increasing the order of the expansion. As shown in Feng and Serletis (2008), the AIM cost function has more exibility than locally exible functional forms (for example, the generalized Leontief, translog, and normalized quadratic) which theoretically can attain exibility only at a single point or in an in nitesimally small region. In modelling the e ciency index, the use of the Box-Cox function which nests exponential, logarithmic, and hyperbolic functions as special cases o ers an excellent insight into the temporal pattern of the e ects of public infrastructure on productivity and cost structures, which is missed in many previous studies. Finally, we apply this framework to the analysis of the e ects of public infrastructure on productivity in the U.S. manufacturing industries. This paper contributes to the literature in two ways. First, to the best of our knowledge, this is the rst paper that incorporates infrastructure into the cost function as part of the e ciency index, thereby treating infrastructure as a determinant of total factor productivity, thus avoiding the problem of violating the marginal productivity theory. In addition, the new approach pursued here yields results not obtained under the alternative models of parametric change studied earlier. In particular, unlike previous studies on public infrastructure, this approach enables us to measure input speci c spillover e ects from public infrastructure, as well as the contribution of each input to overall spillover e ects. More importantly, it yields more accurate productivity estimates which can not be obtained from the commonly used cost function approach, where public infrastructure is treated as an additional xed input. While our price augmenting cost function approach is proposed in the public infrastructure and productivity literature, it can also be used in the investigation of the impacts of other external factors (i.e. public R&D and R&D spillovers from other industries and countries) on the performance of the private sector, and thus provides an alternative to the commonly used cost function approach. 5

6 Second, this is the rst paper in this literature that applies a globally exible functional form. The most frequently used functional forms in this literature are the Cobb-Douglas, translog, and generalized Leontief. As is well known, however, the Cobb-Douglas function is very restrictive in that it has a constant elasticity of substitution for an excellent discussion of the Cobb-Douglas function, see Lloyd (2001). Regarding the generalized Leontief, Caves and Christensen (1980) have shown that it has satisfactory local properties only when technology is nearly homothetic and substitution is low. However, when technology is not homothetic and substitution increases, they show that the generalized Leontief has a rather small regularity region. In the case of the translog, Guilkey et al. (1983) show that it is globally regular if and only if technology is Cobb-Douglas. In other words, the translog performs well if substitution between all factors is close to unity. They also show that the regularity properties of the translog model deteriorate rapidly when substitution diverges from unity. Compared with these functional forms, the AIM model has larger regularity region and performs much better in modelling technical change and second order elasticities see Feng and Serletis (2008). The rest of the paper is organized as follows. In Section 2, we derive the general price augmenting cost function, where public infrastructure is incorporated as part of the e ciency indexes. In Section 3 we present the price augmenting AIM cost function, where the e ciency indexes are assumed to take the form of a Box-Cox function. Section 4 deals with the econometric speci cation and estimation issues. Section 5 provides a description of the U.S. manufacturing data and presents and discusses the empirical results. The nal section summarizes and concludes the paper. 2 The Price Augmenting Cost Function Factor augmenting technical change is usually modelled in the primal setup (production function), as follows y = f (X 1 ; X 2 ; ; X n ) = f (A 1 x 1 ; A 2 x 2 ; ; A n x n ), (1) where y is output, f is a continuous twice di erentiable nondecreasing and quasiconcave function of a vector of n e ective inputs (X 1 ; X 2 ; ; X n ) measured in e ciency units, rather than in conventional terms. In equation (1), X i (the ith e ective input) is related to x i (the ith actual input measured in conventional units) through the following functional relationship (for i = 1; ; n) X i = A i x i, (2) where A i is an e ciency index associated with the ith input. The multiplicative relationship between X i and x i in equation (2) is used in many macroeconomic studies, where A i is typically assumed to take an exponential form A i = A i;0 exp(# i t), 6

7 where A i;0 is the initial e ciency level of input i and # i is the rate of growth in e ciency of input i and the time trend, t, representing technology see, for example, David and van de Klundert (1965), Acemoglu (2003), and Jones and Scrimgeour (2008). According to equation (2), the e ciency index A i is used to augment the actual quantities x i into e ective quantities X i. For example, the expanded Cobb-Douglas production function used by Dugall et al. (1999, 2007) can be considered to be a special case of equation (1), where A = A 1 = A 2 = = A n is a nonlinear function of public infrastructure and the time trend. Within this primal setup, technical change (or total factor productivity growth, or T F P G for short), de ned as a shift in the production frontier, can be easily shown to be T F P G ln = nx i=1 A i i A i, where i ln f=@ ln x i is the elasticity of output with respect to input x i, and the dot above A i denotes the change in A i over time. However, the estimation of the production function is likely to su er from several problems, including endogeneity, lack of exibility, lack of economic content, and exclusion of intermediate materials when they are substitutable to private capital and labor see, for example, Berndt and Hansson (1992), Caves et al. (1992), and Morrison and Siegel (1999). To avoid these problems, in what follows we derive a general price augmenting cost function which is dual to the factor augmenting production function in equation (1). In doing so, we assume that a rm is minimizing its total cost, as follows min fx 1 ;;x ng C = nx p i x i (3) i=1 subject to y = f (A 1 x 1 ; A 2 x 2 ; ; A n x n ), where C is the total cost and p i (i = 1; ; n) represents the price for the ith input in conventional units. As in equation (2), the actual quantities x i in (3) can be augmented into e ective quantities X i by the e ciency index A i. Previous studies using the factor augmenting production function approach in the macroeconomics literature are usually vogue about the determinants of A i, typically assuming that A i is a function of the time trend, t. We depart from that approach and in this paper we explicitly assume that A i is a function of public infrastructure, g, and the time trend, t, the latter used as a proxy for other technology. Formally, we assume that A i = A i (g; t), (4) for i = 1; ; n. Hence, in our formulation, public infrastructure, g, plays the role of increasing the e ciency of private factor inputs (i.e. capital, labor, energy, and materials). 7

8 De ne P i p i =A i, for i = 1; ; n, to be the e ective price of X i so that p i x i = P i X i. The inputs and their corresponding prices in conventional units can then be written as x i = X i A i ; (5) p i = A i P i, (6) respectively, for i = 1; ; n. Substituting equations (5) and (6) into (3) gives a minimization problem equivalent to that in (3), min fx 1 ;;X ng C = nx P i X i (7) i=1 subject to y = f (X 1 ; X 2 ; ; X n ). The optimal solution to the minimization problem (7), Xi, for i = 1; 2; ; n, is a function of prices and y; formally, Xi = X i (P 1 ; ; P n ; y). Substituting Xi into the objective function in (7) gives the following (general) e ciency index augmented cost function, which is dual to the factor augmenting production function in (1), C = C (y; P 1 ; P 2 ; ; P n ) = C y; p 1 ; p 2 ; ; p n, (8) A 1 A 2 A n where the second equality is obtained by using (6), and the asterisk superscript indicating the optimal cost is dropped for simplicity. According to equation (8), the e ciency index augmented cost function is a function of output, y, and e ective input prices, p i =A i (i = 1; ; n). We call a cost function with e ective input prices the price augmenting cost function, to re ect the fact that e ciency adjusted private input prices are used. Using the (A 1 x 1 ; A 2 x 2 ; ; A n x n technical change (or total factor productivity growth) can then be measured from the cost 8

9 function as follows T F P G ln f (A 1x 1 ; A 2 x 2 ; ; A n x n = 1 (A 1 x 1 ; A 2 x 2 ; ; A n x n y@c=@y ln ln C=@ ln y = ct 1 cy, (9) where cy ln C=@ ln y and ct ln C=@t. Using Shephard s lemma, we can also obtain (see Appendix A1) x i, i = 1; ; n, (10) ct ln C=@t = nx i=1 A i s i A i, (11) where s i = p i x i =C is the cost share of input i. According to equation (9), total factor productivity growth is the negative of the product of the dual rate of cost diminution, ct, which is the average of the growth rates of e ciency levels weighted by their respective input cost shares, s i, and the dual rate of returns to scale, cy 1. Under constant returns to scale, cy = 1, and total factor productivity is the negative of the dual rate of cost diminution, meaning that a 1% upward shift in the production function is equal to a 1% decrease in the cost of production. In our formulation, as part of A i, public infrastructure, g, is a factor that determines productivity growth, rather than being a public xed input that is symmetric to xed private inputs, as in previous studies. 3 Model Speci cation The parametric analysis of the e ect of public infrastructure on private sector performance within the framework of the price augmenting cost function in (8) requires the speci cation of two elements the cost function, C(y; p 1 =A 1 ; ; p n =A n ), and the e ciency index, A i (i = 1; ; n). 9

10 3.1 The Price Augmenting AIM Cost Function There are many alternatives for the functional form of the cost function in equation (8). For example, Feng and Serletis (2008) present an empirical comparison and evaluation of the e ectiveness of four well-known exible cost functions the locally exible generalized Leontief [see Diewert (1971)], translog [see Christensen et al. (1975)], and normalized quadratic [see Diewert and Wales (1987)], and one globally exible cost function, the Asymptotically Ideal Model [see Barnett et al. (1991)]. Another globally exible functional form is Gallant s (1982) Fourier exible functional cost form, based on the Fourier series expansion, recently used by Feng and Serletis (2009). Both, the Fourier and AIM globally exible forms, are capable of approximating the underlying cost function at every point in the function s domain by increasing the order of the expansion, and thus have more exibility than most of the locally exible functional forms which theoretically can attain exibility only at a single point or in an in nitesimally small region. As noted, however, by Barnett and Serletis (2008), in this literature, there is no a priori view as to which exible functional forms are appropriate, once they satisfy the theoretical regularity conditions of neoclassical microeconomic theory positivity, monotonicity, and curvature. With this in mind, in this study we employ the AIM cost functional form, in an e ort to extend our earlier work in this area [see Feng and Serletis (2008)]. By assuming that the (general) price augmenting cost function in (8) is a linearly homogeneous multivariate Müntz Szatz series expansion, we get the price augmenting AIM total cost function " X 2Y # " # 2 pij X 2Y 2 C = C (p; y; g; t) = y = y P ij, (12) z2a b z j=1 A ij z2a b z where p > 0 is a vector of input prices (in conventional units), is the order of the expansion, b z the unknown parameters, n the number of production factors, B = f(i 1 ; i 2 ; ; i 2 ) : i 1 ; i 2 ; ; i 2 2 f1; 2; ; n); i 1 i 2 ; i 2 g, A the e ciency index, and P is used to denote e ective prices, as de ned above. It should be noted that (12) is an extension of the Barnett et al. (1991) AIM cost function without technical change, and also a generalization of the AIM cost function with technical change proposed by Feng and Serletis (2008). Moreover, returns to scale can be easily incorporated in the AIM cost function by modifying the e ciency index, as noted in Feng and Serletis (2008). The assumption of constant returns to scale in this paper is consistent with the data used (see the description of the data in Section 5) and can also alleviate the problem of multicolliearity between y, g and t. Regardless of the speci cation of the e ciency index, A ij, in (12), our new AIM total cost function with price augmenting technical change retains all the theoretical properties of the Barnett et al. (1991) AIM cost function without technical change. In particular, it is still globally exible in the sense that it is capable of approximating the underlying cost function at every point in the function s domain by increasing the order of expansion. Moreover, 10 j=1

11 the sum of the exponents of prices in each term in (12) is still 2 2 = 1, thus satisfying the property of global linear homogeneity. 3.2 The Price Augmenting E ciency Index We also need to specify a functional form for the e ciency index, A i, i = 1; ; n. We may assume an exponential form with its argument being linear in public infrastructure, g, and a time trend, t, as follows 1 A i = exp [h i (t; g)] = exp (# i t + i ln g), (13) for i = 1; ; n, where # i is the constant growth rate of e ciency due to other technology for input i and i is the constant elasticity of the total cost with respect to public infrastructure for input i. This speci cation is quite similar to that used in the macroeconomics growth literature where the factor augmenting e ciency index is commonly speci ed as an exponential function of the time trend, for example, A i = exp (# i t). While simple and elegant, the speci cation in (13) lacks enough exibility in modelling the e ects of public infrastructure and other technology (t) on the e ciency levels of private factor inputs, since it is not clear whether these growth rates (# i and i ) should exhibit constant, logarithmic, or hyperbolic patterns over time. To allow for more exibility in modeling both the e ect of public infrastructure and that of the time trend on the e ciency level of private inputs, we instead use a Box-Cox functional form for both the growth rate of the time trend and the cost elasticity with respect to public infrastructure. Formally, A i = exp [h i (t; g)] = exp ( " " i # i t 0 t 1# + i i ln g 0 ln g 1#), (14) i t 0 i ln g 0 where i is the curvature parameter of the Box-Cox function for the time trend, t, and i that for the log of public i infrastructure, ln g. Note that when i = 1, i = 0, or i < 0, then # i t 0 h(t=t 0 ) i 1 = i is a linear, log-linear, or hyperbolic, respectively, function in t. i Similarly, when i = 1, i = 0, or i < 0, then i ln g 0 h(ln g= ln g 0 ) i 1 = i is a linear, log-linear, or hyperbolic, respectively, function in ln g. It should also be noted that ln g is scaled by its initial value (i.e. ln g 0 ), and thus i can be interpreted as the cost elasticity with respect to public infrastructure for input i at the beginning of the sample period. Similarly, # i can be interpreted as the growth rate of e ciency due to other technology for input i at 1 A i;0, the initial e ciency level of input i, is a constant and does not a ect the calculation of elasticities and productivity growth, and thus is dropped from the e ciency index hereafter for notational simplicity. 11

12 the beginning of the sample period. Substituting (14) into (12) gives the price augmenting AIM cost function used in this paper. According to our theoretical discussion in Section 2, there must exist a production function dual to our price augmenting AIM cost function de ned by equations (12) and (14). This production function can be written as y = F (A 1 x 1 ; A 2 x 2 ; ; A n x n ), where F () is a speci c functional form taken by f() in equation (1) and A i is de ned in equation (14). Under the assumption of Hicksian neutrality (that is, A = A i, for i = 1; ; n), this production function reduces to y = AF (x 1 ; x 2 ; ; x n ) = A 0 exp where A = A i ( " #t 0 t = exp t 0 #t 0 1# + ln g 0 t t ln g 0 " ln g 1#) F (x 1 ; x 2 ; ; x n ), (15) ln g 0 ln g ln g 0 1 in (15) is essentially the standard Hicks neutral technical change in the Cobb-Douglas production function, and F (x 1 ; x 2 ; ; x n ) is dual to the AIM cost function without technical change. This further con rms the validity of A i in equation (14) as a measure of technical change. Our price augmenting cost function approach, based on our new price augmenting AIM cost function de ned by equations (12) and (14), possesses a number of advantages over the simple Cobb-Douglas production function approach. First, it takes explicit account of the rm s cost optimization behavior, by considering input quantities as endogenous variables, while treating input prices, which are more likely to be market determined, as exogenous variables. As such, it is less likely to su er from the problem of endogeneity. Second, it is globally exible in that it is capable of approximating the underlying cost function at every point in the function s domain by increasing the order of the expansion,, whereas the Cobb-Douglas production function is very restrictive in the sense that it imposes a priori the condition of a constant elasticity of substitution among inputs. Third, it allows the measurement of input speci c cost elasticity with respect to public infrastructure, as well as the contribution of each input to overall cost elasticity, since an e ciency index is speci ed for each of the n inputs. Last but not least, the speci cation of the e ciency index as a Box- Cox function enables us to investigate the time pattern of the e ect of public infrastructure on cost structure and productivity, which is unfortunately missed in most of the previous studies. Our price augmenting cost function approach provides a rich framework in investigating the e ects of public infrastructure on private sector performance. This can be accomplished using three measures the cost elasticity with respect to public infrastructure, the output elasticity with respect to public infrastructure, and the contribution of public infrastructure to total factor productivity growth. In addition, we can also obtain the social rate of return 12

13 to public infrastructure, which can help answer the important policy question of whether public infrastructure is over-supplied or under-supplied. These measures are discussed in detail in what follows. 3.3 Cost Elasticity with Respect to Public Infrastructure The spillover e ect of public infrastructure on total cost is captured by the magnitude and sign of the cost elasticity with respect to public ln C=@ ln g, which represents the percentage change in total cost due to a one percent change in public infrastructure. In our particular case, the cost elasticity can be obtained by (see Appendix A2) " nx s ln ln g = i=1 # i 1 ln g i. (16) ln g 0 According to (16), the cost elasticity with respect to public infrastructure is an input costshare weighted average of the input speci c cost elasticities with respect to public infrastructure, i (ln g= ln g 0 ) i 1. An advantage of (16), unlike previous studies on public infrastructure, is that we can measure input speci c cost elasticity with respect to public infrastructure, as well as the contribution of each input to overall cost elasticity, which can be written as cg i = s i i (ln g= ln g 0 ) i 1 nx h i. (17) s i i (ln g= ln g 0 ) i 1 i=1 3.4 Output Elasticity with Respect to Public Infrastructure Commonly used in the production function approach, the output elasticity with respect to public ln f=@ ln g, is another important measure in evaluating the e ects of public infrastructure on private sector performance. While it can not be directly obtained within a cost function framework, it can be derived indirectly from the cost elasticity with respect to public infrastructure in (17), by exploiting the duality between the cost function and the production function. In fact, the implied output elasticity is the negative of the cost elasticity in (17) under the assumption of constant returns to scale. To see this, we apply the envelope theorem to the cost minimization problem in (3) ln @f (A 1 x 1 ; ; A n x n ln g The implied output elasticity of public ln f (A 1 x 1 ; ; A n x n ) =@ ln g, can 13

14 then be measured from the cost function as ln f (A 1 x 1 ; ; A n x n ln g = 1 (A 1 x 1 ; ; A n x n ln g = ln g ln C=@ ln ln C=@ ln y = c ln g 1 cy, where c ln g ln C=@ ln g and cy ln C=@ ln y is returns to scale. Hence, under constant returns to scale, cy = 1, and the implied output elasticity of public infrastructure is the negative of its cost elasticity, that ln f (A 1 x 1 ; ; A n x n ) =@ ln g = c ln g. 3.5 Total Factor Productivity Growth and Public Infrastructure The contribution of public infrastructure to total factor productivity growth, when coupled with that of other technology (t), provides a third perspective regarding the e ects of public infrastructure on private sector performance. Applying (9) to the cost function de ned by (12) and (14) within a discrete time framework, we can obtain the total factor productivity growth at time t as nx Ai;t+1 T F P G t = s i 1. (18) A i;t i=1 We can also decompose T F P G t into two important components, the productivity growth due to public infrastructure, T F P G g t, and the productivity growth due to other technology, T F P G t t. For this purpose, we rst decompose A i in (14) into two components, A g i and A t i, de ned as ( " i A g i = exp i ln g 0 ln g 1#) i ln g 0 and A t i = exp ( " i # i t 0 t 1#), i t 0 respectively. The productivity growth due to public infrastructure at time t, T F P G g t, is 14

15 given by T F P G g t = nx i=1 s i A g i;t+1 A g i;t 1, (19) and the productivity growth due to other technology at time t, T F P G t t, is given by 0 1 nx T F P G t A t t i i A i=1 A t i = = nx i=1 ln C s i " nx (s #) i 1 t i i. (20) t 0 A comparison can thus be made between T F P G g t and T F P G t t to see whether public infrastructure is a signi cant contributor to total factor productivity growth. It should be noted that T F P G is not equal to the sum of T F P G g and T F P G t. In fact, it is easy to show, as follows, that it is equal to the sum of T F P G g t and T F P G t, plus a term representing the interaction between T F P G g and T F P G t T F P G = (1 + T F P G g ) 1 + T F P G t 1 = T F P G g + T F P G t + T F P G g T F P G t. (21) 3.6 The Social Rate of Return to Public Infrastructure An important public policy question in this literature is whether public capital is over- or under-supplied. This question can be answered by resorting to the well known Samuelson condition [see Samuelson (1954)] which requires that public capital (under the assumption of lump-sum taxation) be provided up to the point where the sum of marginal bene ts to producers and consumers is equal to the marginal cost of providing an additional unit of public capital see Kaizuka (1965). In calculating the marginal bene t and marginal cost, we follow the previous literature and ignore the bene ts to consumers and complications resulting from the absence of lump-sum taxation. In other words, we can determine the marginal bene t and marginal cost of providing an additional unit of public capital based only on the production sector of the economy. 15

16 We assume that the government chooses the amount of public infrastructure by minimizing the present value of the costs of all the resources in the economy see Nadiri and Mamuneas (1998). That is, the government selects the level of public infrastructure such that the sum of the industry marginal bene ts equals the user cost of public capital, i.e., HX m h;g (p; y; t; g) = p g (r + g ), (22) h=1 where m h;g (p; y; t; g) is the marginal bene t (or shadow value) of public infrastructure, which re ects the reduction in costs due to an incremental addition to the stock and can be obtained using the cost elasticity in (16); h indicates industry; H is the total number of industries (which is equal to one when only aggregate manufacturing industry is considered); r is the discount factor; g is the depreciation rate of public infrastructure; and p g is the acquisition price. Since public sector capital formation is generally nanced through taxation and has signi cant distortive e ects on private sector decisions, p g is the sum of the direct burden of the taxes needed to pay for the infrastructure and the dead weight cost associated with these taxes for the last dollar of public investment. Solving equation (22) for g yields the optimal amount of public infrastructure. Let MB = P H h=1 m h;g (p; y; t; g) denote the marginal bene t and MC = p g (r + ) denote the marginal cost. Then the Samuelson condition, together with (22), implies that Public infrastructure is 8 < : optimally supplied MB = MC under supplied if MB > MC over supplied MB < MC The Samuelson condition can be also stated in terms of the net social rate of return to public infrastructure. To see this, we can rearrange (22) to obtain P H h=1 m h;g (p; y; t; g) g = r (24) p g Letting s denote the left side of equation (24) (the net social rate of return to public infrastructure), then equation (23), the Samuelson condition, can be alternatively written as Public infrastructure is 8 < : optimally supplied s = r under supplied if s > r over supplied s < r where r and g are de ned as above. Equation (25) is the the Samuelson condition we will use in this paper in answering the question of whether public capital is over- or under-supplied. However, as can be seen from (24), the application of (25) requires the speci cation of p g, g, and r, which will be discussed in Section (23) (25)

17 4 Econometric Speci cation and Estimation Issues In empirical applications, the approximation of the AIM cost function must be truncated at some nite value (i.e. nite partial sums). The order of approximation is usually determined empirically and stops when the elasticity estimates and the covariance matrix of the disturbances converge. In this paper, because of degree of freedom problems, we set = 2. Hence, with n = 3 (the case in this paper) and = 2 in equation (12), we get the following price augmenting AIM(2) cost function C =2 (p; y; g; t) = y b 1 P 1 + b 2 P 2 + b 3 P 3 + b 5 P 1=2 1 P 1=2 3 + b 6 P 1=2 2 P 1=2 3 + b 7 P 3=4 1 P 1=4 2 + b 8 P 1=4 1 P 3=4 2 + b 9 P 3=4 1 P 1=4 3 + b 10 P 1=4 1 P 3=4 3 + b 11 P 3=4 2 P 1=4 3 + b 12 P 1=4 2 P 3=4 3 + b 13 P 1=2 1 P 1=4 2 P 1=4 +b 15 P 1=4 1 P 1=4 2 P 1= b 24 P 1=4 1 P 1=2 2 P 1=4 3. (26) Applying (10) to (26) yields the following system of factor demand equations for the AIM(2) model with n = 3 x 1 y = 1 b A 1 2 b 4P 1=2 1 P 1= b 5P 1=2 1 P 1= b 7P 1=4 1 P 1= b 8P 3=4 1 P 3= b 9P 1=4 1 P 1= b 10P 3=4 1 P 3= b 13P 1= b 15P 3=4 1 P 1=4 2 P 1=2 3 1 P 1=4 2 P 1= b 14P 3=4 1 P 1=2 2 P 1=4 3 ; (27) x 2 y = 1 b A 2 2 b 4P 1=2 1 P 1= b 6P 1=2 2 P 1= b 7P 3=4 1 P 3= b 8P 1=4 1 P 1= b 11P 1=4 2 P 1= b 12P 3=4 2 P 3= b 13P 1= b 15P 1=4 1 P 3=4 2 P 1=2 3 1 P 3=4 2 P 1= b 14P 1=4 1 P 1=2 2 P 1=4 3 ; (28) 17

18 x 3 y = 1 b A3 2 b 5P 1=2 1 P 1= b 6P 1=2 2 P 1= b 9P 3=4 1 P 3= b 10P 1=4 1 P 1= b 11P 3=4 2 P 3= b 12P 1=4 2 P 1= b 13P 1= b 15P 1=4 1 P 1=4 2 P 1=2 3 1 P 1=4 2 P 3= b 14P 1=4 1 P 1=2 2 P 3=4 3. (29) Concavity (in prices) requires that the Hessian matrix of the second derivatives of the cost function with respect to prices, r pi p j C (p; y; g; t), is negative semide nite. In practice, concavity of the cost function may not be satis ed. In that case, we impose concavity fully (that is, at every data point in the sample) on the AIM model using methods suggested by Gallant and Golub (1984), to which we now turn. 4.1 Semi-Nonparametric Estimation The AIM(2) factor demand system, equations (27)-(29), can be written as z t = (p; y; g; t; ) + t, (30) where z = (z 1 ; ; z n ) 0 is the vector of input-output ratios, (p; y; g; t; ) is given by the the right side of equations (27)-(29), and = (b 1 ; b 2 ; ; b n 2 ; # 1 ; # 2 ; # 3 ; 1 ; 2 ; 3 ; 1 ; 2 ; 3 ; 1 ; 2 ; 3 ). t is a vector of stochastic errors and we assume that N (0; ), where 0 is a null matrix and is the n n symmetric positive de nite error covariance matrix. The same assumption about the error term,, has also been made by Nadiri and Mamuneas (1994), Berndt and Hansson (1992), and Morrison and Schwartz (1996) in the public infrastructure and productivity literature, and by Diewert and Fox (2008), among many others, in the broader literature of demand systems. As Gallant and Golub (1984, p. 298) put it, all statistical estimation procedures that are commonly used in econometric research can be formulated as an optimization problem of the following type [Burguete, Gallant, and Souza (1982)] with ' () twice continuously di erentiable in. b minimizes ' () over (31) Notice that (p; y; g; t; ) is nonlinear in # 1 ; # 2 ; # 3 ; 1 ; 2, and 3, and therefore the AIM(2) factor demand system in (30) can be tted using Gallant s (1975) seemingly un- 18

19 related nonlinear regression method to estimate. Hence, ' () has the form ' () = 1 T 0 t t = 1 T TX (z t ()) 0 b 1 (z t ()), (32) t=1 where b is an estimate of the error variance-covariance matrix,. In minimizing (32), we use the TOMLAB/NPSOL tool box with MATLAB. NPSOL uses a sequential quadratic programming algorithm and is suitable for both unconstrained and constrained optimization of smooth (that is, at least twice-continuously di erentiable) nonlinear functions. 4.2 Endogeneity One issue concerning our stochastic speci cation is that of endogeneity. At the individual rm level, it may be reasonably assumed that input prices on the right-hand side of (30) are exogenous. However, at the more aggregated industry level (like U.S. manufacturing), input prices are less likely to be exogenous. In this literature, the possibility of endogeneity has been addressed by using iterative three-stage least squares (3SLS), but the results generally have been about the same as those with iterative Zellner estimation see, for example, Barnett et al. (1991). Diewert and Fox (2004) also argue that instrumental variables estimation may be more biased, since the instruments may not be completely exogenous, and Burnside (1996) shows that results can vary markedly depending on the set of instruments used. In this paper, we choose to use the more commonly used iterative Zellner method of estimation. 4.3 Econometric Regularity Another issue is that of nonstationarity. If the errors are nonstationary, then there is no theory linking the left hand side to the right hand side variables in equation (30) or, equivalently, no evidence for the theoretical models in level form. In such cases, some important nonstationary variables might have been omitted. Allowing for rst order serial correlation, as is usually done in the literature, is almost the same as taking rst di erences of the data if the autocorrelation coe cient is close to unity. In that case, the equation errors become stationary, but there is no theory for the models in rst di erences. Munnell (1992), in the discussion about infrastructure investment and economic growth, has argued that rst di erencing destroys the long-term relationships in the data and therefore it does not make economic sense to use equations in this form. In fact, previous studies in this literature that have estimated equations in rst di erences have found private capital and labor to be insigni cant see, for example, Hulten and Schwab (1991) and Sturm and 19

20 de Haan (1995). While the contribution of infrastructure can be questionable, the role played by labor and private capital is doubtless. Dugall et al. (1999) have argued that the fact that rst di erenced equations generate insigni cant estimates for the labor and capital coe cients is enough to question the validity of using rst di erences of the data. 4.4 Theoretical Regularity Finally, in the estimation of (30), we pay special attention to the theoretical regularity conditions of positivity, monotonicity, and curvature. The regularity conditions are checked as in Feng and Serletis (2008), as follows: Positivity is checked by checking if the estimated cost is positive, C (p; y; g; t) > 0. Monotonicity is checked by direct computation of the values of the rst gradient vector of the estimated cost function with respect to p. It is satis ed if r p C (p; y; g; t) > 0. Curvature requires the Hessian matrix of the cost function to be negative semide nite and is checked by performing a Cholesky factorization of that matrix and checking whether the Cholesky values are nonpositive [since a matrix is negative semide nite if its Cholesky factors are nonpositive see Lau (1978, Theorem 3.2)]. Curvature can also be checked by examining the eigenvalues of the Hessian matrix provided that the monotonicity condition holds. It requires that these eigenvalues be negative or zero. We rst run an unconstrained optimization using (31). If theoretical regularity is not attained then we impose the theoretical regularity conditions. 5 Data and Empirical Evidence 5.1 The Data We use annual data for capital, labor, and intermediate materials for the total manufacturing industry in the United States over the period from 1953 to We also use annual data for capital, labor, and intermediate materials for each of the 12 two digit manufacturing industries to check the robustness of our results. The 12 two digit manufacturing industries chosen are exactly the same as in Nadiri and Mamuneas (1994) and are listed in Table 1. It is to be noted that we constructed the price and quantity series for intermediate materials by aggregating energy, materials, and purchased business services, using the Fisher ideal index. All data on quantities and prices were obtained from the Bureau of Labor Statistics 20

21 (BLS), at We normalized all the price series to be equal to 1 in 1953, and obtained the quantity series for each of output, capital, labor, energy, materials, and purchased business services by dividing the value of production or factor cost by the corresponding normalized price series. This BLS (KLEMS) dataset has been used previously by Nadiri and Mamuneas (1994), Diewert and Fox (2008), and Feng and Serletis (2008), among many others. A major feature of the BLS data set is that constant returns to scale is built in by constructing input factor payments in such a way that they add up to the value of output. Thus, tests of returns to scale and scale bias are inappropriate, as are some tests of imperfect competition. Another feature of the BLS data set is that it provides the price and quantity series for purchased business services inputs. Directly collected data on purchased business services are relatively scant, and for that reason they have been ignored by similar studies in the past. However, there is ample evidence of an increased use of purchased business services by industries over the post-war period and there are two important issues to consider. The rst is that a sizable and growing input should not be ignored in productivity measurement, if aggregate inputs are not to be underestimated and mismeasured. The other is the possibility of substitution between capital, labor, and services purchased from outside. Examples of the latter are the substitution of leased equipment for owned capital and purchased accounting for services performed by payroll employees. As in Dugall et al. (1999, 2007), we restrict our analysis to core public infrastructure, which consists of the following three categories: (i) Highways and streets; (ii) Other buildings (which includes police, re stations, court houses, auditoriums, and passenger terminals); and (iii) Other structures (which includes electric and gas facilities, transit systems, and air elds). g represents the net capital stock (net of depreciation) held by the federal, state, and local governments expressed in billions of 1953 dollars. The data on the stock of public infrastructure is obtained from the Bureau of Economic Analysis at Theoretical Regularity Tests In the rst column of Table 2 we present the parameter estimates and theoretical regularity violations for the unconstrained AIM(2) system for the U.S. total manufacturing industry. As results in nonlinear optimization are sensitive to the initial parameter values, to achieve global convergence, we randomly generated 500 sets of initial parameter values and chose the starting that led to the lowest value of the objective function. It is also to be noted that a parametric bootstrapping method is usually used in constrained optimization to obtain statistical inference for the estimated parameters ( ) b or nonlinear transformations of these parameters (( ), b i.e. elasticities) see Gallant and Golub (1984). This involves the use of Monte Carlo methods, generating a sample from the distribution of the inequality constrained 21

22 estimator ( ), b large enough to obtain a reliable estimate of the sampling distributions of ( ) b and ( ). b However, for computational reasons this is una ordable at present. Therefore, only point estimates are provided for the estimated parameters ( ) b in Table 2. As can be seen in the rst column of Table 2, while positivity and monotonicity are satis ed at all sample observations, curvature is violated at 12 data points. Because regularity hasn t been attained, we follow Feng and Serletis (2008, 2009) and use the NPSOL nonlinear programming program to minimize ' () subject to the constraint that the three eigenvalues of the Hessian matrix, H, are non-positive. This is because a necessary and su cient condition for the concavity of H is that all its eigenvalues are nonpositive see, for example, Morey (1986). Thus, our constrained optimization problem is written as min ' () subject to ' i (p; y; g; t; ) < 0, for i = 1; ; n, where ' i (p; y; g; t; ), i = 1; ; n, are the eigenvalues of the Hessian matrix of the AIM(2) cost function. With the constrained optimization method, we can impose curvature restrictions at any arbitrary set of points at a single data point, over a region of data points, or fully (at every data point in the sample). We minimize ' () subject to the constraint that the cost function is locally concave at 1977 and also subject to the constraint that it is fully concave (concave at every data point). The results are reported in the second and third columns of Table 2 the second column shows the results when the curvature constraint is imposed locally (at 1977) and the third column shows the results when the constraint is imposed at every data point in the sample. Clearly, the e ect of imposing the curvature constraint locally is unsatisfactory, as the number of curvature violations drops from 12 to 4. However, as we expect, the imposition of the curvature constraint at every data point in the sample has reduced the number of curvature violations to zero, producing parameter estimates that are consistent with all three theoretical regularity conditions, at every data point in the sample; that is, fully. In what follows we focus on results from the AIM(2) cost function with the curvature conditions imposed fully. As seen from the third column of Table 2, the input speci c cost elasticities at the beginning of the sample period are all positive (i.e., 1 = 0:4396, 2 = 0:1683, and 3 = 0:0720), suggesting that public infrastructure does increase the e ciency level of private inputs. Apparently, physical capital is most favored by public infrastructure, followed by labor and materials, at the beginning of the sample period (this is also true for the whole sample period, as can seen from Figure 3). However, as indicated by the negative sign of the curvature parameters for public infrastructure (i.e., 1 = 0:4220, 2 = 2:9598, and 3 = 6:0000), the ability of public infrastructure to increase the e ciency level of private 22