PART II. A Summary of Empirical Work on the Public Capital Hypothesis*

Transcription

1 PART II A Summary of Empirical Work on the Public Capital Hypothesis* Raymond G. Batina Department of Economics Washington State University Pullman, WA USA Abstract: In this paper we summarize the empirical work that has been done on the public capital hypothesis. The early literature focused attention on estimating an aggregate production function and found that public capital had a large effect on output. Later research used the different approaches, e.g., cost function approach, flexible functional forms, a variety of different sources of data, and different statistical techniques. Most recent researchers have found a much smaller impact or no impact at all. * I am very grateful for the generous support of the Economic and Social Research Institute of Japan s Cabinet Office. I also thank the Kansai Institute for Social and Economic Research for its assistance. I am solely responsible for any errors that remain.

2 7. Empirical Work on the Public Capital Hypothesis 7.1 Introduction In the last two chapters we derived a number of interesting predictions from the various models considered. It would appear to be the case that if public infrastructure is productive, greater spending on it may raise private output, lower cost, and possibly increase the economy's growth rate. On the other hand, taxes tend to lower output and possibly lower the growth rate, especially if they distort economic decisions at the margin. There are different types of infrastructure and this complicates the basic issue. For example, one would expect highways and streets to have more of an impact on output than educational buildings. In addition to this, different levels of government make expenditures on infrastructure. So there are different dimensions of disaggregation to consider as well. We also described some of the theory behind the government's optimal choice of public capital investment. Certain conditions are required for the stock of capital to be chosen optimally and one can econometrically test to see whether or not those conditions hold. In this chapter we will present a summary of some of the various tests of the public capital hypothesis, namely, that public capital has a positive effect on output and growth. 1 Second, we will present tests of the impact of public capital on cost. Third, the extension of the hypothesis as an explanation of the productivity slowdown that occurred in the 1970's and 1980's will be examined. Finally, we will study whether the stock of public capital has been chosen optimally or not. Time series, cross sections, and panel data have all been used and data from a variety of countries have also been studied. The key to the hypothesis is the impact of public infrastructure capital on an individual firm's efficiency as we saw in previous chapters. In the next section we discuss the main empirical approaches to the study of the public capital hypothesis, the production function approach, the duality approach, and the dynamic 26

3 approach. In section 7.3 we present Aschauer's and Munnell's results and discuss them at length since this work generated a great deal of research activity. In section 7.4 we present several critical arguments against the work supporting the hypothesis. In particular, we discuss Tatom's (1991a, 1991b, 1993) results. Studies using disaggregated data are studied in section 7.5. There are essentially two levels of disaggregation, state level data for the US is used in several important studies and several studies use industry specific data. Some studies present results favorable to the hypothesis, e.g., Morrison and Schwartz (1996) and Nadiri and Mamuneas (1994), while others report the opposite, e.g., Hulten and Schwab (1991) and Holtz-Eakin (1994). Several additional studies using US time series are discussed in section 7.6. We report on the results of studies for other countries in section 7.7, as well as results using a cross section of OECD countries. Again, some studies support the hypothesis, e.g., Pereira (2000), while others reject it, e.g., Ford and Poret (1991). Section 7.8 concludes the chapter. Many of the studies we report on, especially in the early literature, use a production function approach. This requires data on output and the inputs, usually taken to be private capital, labor, and public capital. The actual data used will depend on the level of aggregation and there is some leeway on the proxies that are available for the theoretical variables. The early studies focused attention on the aggregate level of the economy and so tried to measure the impact of public capital on a broad measure of output that would most likely be affected by public infrastructure, e.g., industrial output. They also tended to use 'labor hours' as the proxy for the labor variable. State level studies, on the other hand, tended to use 'gross state product' as the proxy for output and 'employment' as the proxy for labor since labor hours is unavailable on a state by state basis for the US and other countries. Also, data on private and public capital is not available on a state by state basis either so construction of the capital variables is required under this approach. This is also true in other countries as well. 27

4 An alternative approach, the duality approach, typically involves posing the cost minimization problem of the firm and estimating the factor demands and cost function that stem from solving that decision problem. An alternative is to pose the profit maximization problem of the firm, solve it, and estimate the resulting factor demands and profit function. The two alternatives are dual to one another and dual to the production function in the sense that a production function can be recovered from information on cost using duality theory. The duality approach requires information on relative input prices, e.g., a proxy for the wage, the cost of private capital, and so on, and information on public capital stocks and private output. In some cases a system of equations including factor demand equations and pricing equations are estimated simultaneously. One possible advantage to the duality approach is that data on prices may be more readily available than data on factor inputs. The disadvantage to the duality approach is that it involves a more complicated hypothesis than the production function approach. Estimating the production function only requires that the firm be technologically efficient. However, in addition to this, estimating cost shares that stem from cost minimization, for example, involves the additional assumption that the firm is behaving optimally during the sample period by choosing inputs to minimize cost, or to maximize profit. Therefore, the duality approach requires a narrower hypothesis, that includes explicit optimizing behavior, than the production function approach. In addition, an assumption must be made about aggregation. When aggregate data is used, the researcher is implicitly assuming that firms are identical so that one can aggregate across all firms to obtain the aggregate production function, cost function, or profit function. There is also a great deal of aggregation in industry studies and in studies that use data aggregated to the state level. So part of the maintained hypothesis is that the behavior of firms in an industry, or in a state, or in the economy as a whole, is similar enough to allow 28

5 aggregation to occur. Therefore, if the public capital hypothesis fails, it is possible that it may be due to a failure of the aggregation assumption. Third, some authors undertake a dynamic system approach by estimating a system of equations designed to capture direct and indirect interaction among several variables over time or use advanced techniques to control for certain problems like simultaneity bias or autocorrelation. The so-called Vector Autoregression Model (VARM) and the Error Correction Model (ECM) are examples of this type of analysis. There are some common econometric problems across many of the studies. First, and foremost, is the simultaneity problem. Many of the variables used to proxy for capital, labor, and public capital, or price variables under the cost approach, will be econometrically endogenous to the system, thus, there may be a simultaneity bias. There are techniques which can be used to deal with this issue, e.g., instrumental variables, Phillips and Perron's fully modified estimator (FME), or Stock and Watson's dynamic OLS (DOLS). A second problem that arises in a time series context or with panel data is the stationarity of the data. Two time series that tend to drift together may yield a positive correlation that is entirely spurious. Techniques are now readily available for dealing with this issue but are sometimes not used, e.g., unit root testing and tests for cointegration. Indeed, this issue may arise even in the context of a panel data set, and yet, is typically ignored in that context. 2 We will point out the various issues and problems along the way as we survey the literature. We also summarize several papers that attempt to calculate whether the government has actually invested in the optimal amount of public capital using data from the US, Japan, Sweden, and Mexico. 7.2 Empirical approaches Under the production function approach, a production function is posited according to Y = F(K,L,G,t), (7.1) 29

6 where an output (Y) is produced using inputs of private capital (K), labor (L), public capital (G), and t is time, which represents the possibility that the technology may change over time. More specifically, G may represent the services that flow from the stock of public infrastructure. The first derivative of the production function, F( ), measure the marginal productivity of the input. The impact of public capital on output at the margin is given by Y / G = F(K, L,G) / G F G. An example of a production function is provided by the Cobb-Douglas technology, Y = F(K,L,G) = AK α 1 Lα 2 Gα 3 eηt +u, (7.2a) where A is a constant representing the technology that may evolve over time due to technological change, T represents a time trend, and u is an error term which may capture random differences across firms in their technology. So the marginal product of public capital is F G (K, L,G) = α 3 AK α 1 L α 2 G α 3 e η T +u = α 3 Y /G. If α 3 > 0, then public capital is productive. On the other hand, if α 3 = 0, then it is not. Furthermore, if α 1 + α 2 = 1, there is constant returns to scale in private inputs, and if in addition α 3 > 0, then there is increasing returns in all three inputs. If α 1 + α 2 + α 3 = 1 and α 3 > 0, then there is constant returns to scale in all three inputs and decreasing returns to scale in the private inputs. The Cobb-Douglas functional form is very restrictive. A generalization of it is the Translog production function. Essentially, it is a second order Taylor's series approximation of an unknown production function that nests the Cobb-Douglas function as a special case. One version of the Translog function is Y Y = α 0 + α 1 (L L ) + α 2 (K K ) + α 3 (G G ) + β 1 (L L ) 2 +β 2 (K K ) 2 + β 3 (G G ) 2 + β 4 (L L )(K K ) +β 5 (L L )(G G ) + β 6 (K K )(G G ), (7.2b) 30

7 where the variables are in natural logarithms and a bar over a variable denotes the mean of the variable. The function is homogeneous if Σβ j = 0, and we have the Cobb-Douglas form. Thus, the translog function generalizes the Cobb-Douglas form when the restriction on the β coefficients is not imposed. Constant returns to scale in private inputs requires Σβ j = 0 and α 1 +α 2 = 1, while constant returns in all three inputs requires Σβ j = 0 and α 1 +α 2 +α 3 = 1. We can define an elasticity that captures the impact of public capital on output in the following natural way, E G = (G/ Y)( Y / G) = (G / Y)F G. (7.3) Similarly, we can define an elasticity that measures the impact of private capital and labor on output in the same manner. It follows from this that we can attempt to measure the contribution of public capital to output by estimating the elasticity in equation (7.3). Under the Cobb-Douglas form E G = α 3, and similarly for the other inputs, E K = α 1 and E L = α 2. The simplest way to measure these elasticities is to take natural logs of equation (7.2a) and estimate the resulting equation directly, or a more general version of it like the translog function. This calculation can be done for an individual firm, an industry, a region of a country, the economy as a whole, or a group of economies, depending on the data that is available. We would expect the marginal productivity of public capital to be positive, F G = Y / G 0. Since the ratio G/Y is positive, it follows that the elasticity in equation (7.2a), for example, E G, will also be positive, although we leave open the possibility that it can be zero. The larger it is in magnitude, however, the greater the impact of public capital on output. An alternative approach is to use duality theory and optimizing behavior on the part of the firm to derive input demand functions that in part depend on the services of the public capital stock. We will focus on the cost version of the duality approach rather than profit. 31

8 However, the same techniques can be applied to the profit function version of the duality approach. Under the assumption of cost minimization, as an example of the duality approch, it is posited that firms choose their inputs to minimize the cost of producing a given output level subject to the technology. This yields the optimal input demand functions which when substituted back into the cost equation gives us the cost function for firm i, C i (p, y i, G, t), where p is a vector of input prices for private capital, labor, and so on, y i is the output the firm wishes to produce, and G is either the stock of public infrastructure, or the service flow of public infrastructure capital. The derivatives C i / p j are the input demands for firm i. In addition, the shadow value of public infrastructure to the firm is minus the amount of cost savings due to the provision of public capital at the margin, which is captured by the derivative C i / G. For the Cobb-Douglas technology in (7.1) the unit cost function is given by C i = {(1/ A)[(α 2 /α 1 ) α 1 + (α 1 / α 2 )α 2 ]wα 2 r α 1 G α 3 e (η T+u) } 1/(α 1+α 2 ) (7.4) where p = (w, r). Cost is increasing in output and each input price, but decreasing in public capital and the time trend if α 3 > 0 and η > 0. Firm i's unit input demands for labor and capital are given by l i = {(1/ A)(α 2 /α 1 ) α 1 y(r /w)α 1 G α 3 e (η T+u) } 1/(α 1+α 2 ) k i ={(1/ A)(α 1 / α 2 ) α 2 y(w / r)α 2 G α 3 e (η T+u) } 1 /(α 1+α 2 ). (7.5a) (7.5b) The demand for labor and capital are both increasing in output, but decreasing in both public capital and the time trend if α 3 > 0 and η > 0. Thus, private capital and labor are both substitutes for public capital, holding output constant. In addition, if the isoquants are shifting in over time with technological progress over time (η > 0), then less of both private inputs will be needed to produce the same output. Finally, the demand for labor (private capital) is 32

9 increasing (decreasing) in the price ratio r/w. One can also test for constant returns in private inputs, α 1 +α 2 = 1, or all inputs, α 1 +α 2 +α 3 = 1. If all firms are identical we can aggregate up to the industry, state, or economy-wide level, depending on the data that is available and estimate (7.4) and (7.5). This allows us to recover the parameters of the technology and essentially recover the production function empirically. Although, it should be remembered that there is more to the duality approach than the production function approach since it is assumed that agents are choosing optimally under the former but not the latter approach. One might reject the public capital hypothesis under the duality approach if choices are fixed by contract, agents are not all minimizing cost, or the aggregate functions do not adequately reflect what is going on at the micro level during the sample period. Once again, the Cobb-Douglas form is somewhat restrictive. For example, it implies that both inputs are necessarily substitutes for public capital. Yet, there may be some inputs that are complementary to public capital. Estimation of (7.4) and (7.5) will not uncover such complementarity by assumption. Second, it is easy to show that (G/ l)( l / G) = (G / k)( k / G) = α 3, i.e., the input elasticities with respect to public capital are equal. To see this take natural logs of (7.5) and differentiate. The effect of public capital may differ empirically across inputs, however. Again, (7.5) will not allow for this possibility and the joint hypothesis could be rejected because of the form chosen and not necessarily because public capital is unproductive. The translog cost function generalizes the Cobb-Douglas form and is similar in spirit to (7.2). It allows for differential effects across inputs and for complementarity across inputs to occur. Finally, there is the dynamic system approach. Let y represent an nx1 vector of endogenous variables, let x be an mx1 vector of exogenous variables, and let e be an nx1 33

10 vector of white noise error terms. The Vector Autogression Model (VARM) can be represented by y t = A 0 + A 1 y t + A 2 y t 1 + Bx t + e t, where A 0 is a vector of constants, A j is an nxn matrix for j = 1, 2, and B is an nxm matrix. Sims' (1980) goal in putting forth the VARM was to get away from some of the arbitrary assumptions used in the macroeconomics literature to identify an empirical macro model. In a closed economy most variables will be endogenous. Furthermore, if the government chooses its policies in response to economic conditions, even variables controlled by the government will be endogenous. We can thus consider the special case of y t = A 0 + A 1 y t + A 2 y t 1 + e t, where there are no exogenous variables. For example, y = (Y, K, L, G)', where a prime denotes transpose. As it turns out, n(n-1)/2 restrictions are required for the system to be econometrically identified. Solve the last equation to obtain y t = (I A 1 ) 1 A 0 + (I A 1 ) 1 A 2 y t 1 + (I A 1 ) 1 e t = B 0 + B 1 y t 1 + u t, (7.6) where I is the identity matrix, B 0 = (I-A 1 ) -1 A 0, and so on. Notice that the error term in the equation to be estimated, u, is a linear combination of the white noise terms and is thus correlated across equations. Following Sims (1980) researchers usually use the so-called Choleski decomposition where A 1 is assumed to have a triangular structure. However, this is somewhat arbitrary. For example, consider the ordering y = (Y, K, L, G). In that case I - A 1 is given by 1 0 I A 1 = 0 0 a a 13 a a 14 a 24 a

11 under the Choleski decomposition and the system is exactly identified. Under a different ordering of the variables the form of the matrix I - A 1 will differ in an obvious way. Once the model has been identified and estimated various dynamic simulation exercises can be carried out. For example, suppose the variables are ordered in the following way and we assume a Choleski decomposition, y = (Y, K, L, G)'. Then current values of K, L, and G will affect output Y. However, for this ordering only past values of Y, K, and L will affect the current value of G, not current values. Given the time to build public infrastructure, this seems like a natural ordering to study. We can then simulate the effect over time of an innovation in one component of the e vector, which yields the so-called impulse response function. For example, an innovation in G by one standard deviation, should increase output over time and have a lagged effect that dissipates slowly over time. Several analysts have estimated a system like (7.6) in the public capital literature. Alternatively, by subtracting y t-1 from both sides of (7.6) we obtain y t = B 0 + (B 1 I)y t 1 + u t. With a longer lag structure we have instead, h y t = B 0 + (B 1 I)y t 1 + C s y t s + u t. (7.6') s=1 This is the so-called Error Correction model. The terms in the first differences capture the short run effects. If the vector y t is cointegrated, the term (B 1 I)y t-1 captures the long run relationship among the variables. This is also convenient if the data series are not stationary but cointegrated. (See the discussion in Hamilton (1994).) Clearly, the system approach can characterize a wide range of direct and indirect effects among a set of variables. 7.3 Early results Ratner (1983) first included a public capital variable in an aggregate production function. He estimated a log-linear version of the function using annual, aggregate, US time series data for 35

12 His estimate of the public capital elasticity E G was only It increased substantially to 27.7% when revised data was used by Tatom (1991b) to estimate the elasticity for the same time period. However, when Tatom corrected for serial correlation, the elasticity was not statistically different from zero when the revised data was used. It was really Aschauer's (1989) paper that sparked the current interest in the literature. He used annual, aggregate, time series data covering the period from 1949 to 1985 for the US and estimated a Cobb-Douglas version of the production function in equation (7.1). The early results are collected in Table 7.1. Suppose there is constant returns to scale in all three inputs and suppose the technology is given by Y = AK 1 α 2 α 3 L α 2 G α 3 e α 4 ln(cu)+α 5 T+u, where CU is capacity utilization, T is a time trend, and u is an i.i.d. random error term that is uncorrelated with any of the other variables. Rearranging and taking natural logs, we obtain ln(y/k) = ln(a) + α 2 ln(l/k) + α 3 ln(g/k) + α 4 ln(cu) + α 5 T + u. (7.7) An estimate of α 1 can easily be obtained from α 1 = 1 - α 2 - α 3 once the parameters α 2 and α 3 have been estimated. Aschauer estimated equation (7.7) using various techniques. His preferred equation was the following, Y* = T L* G* CU, (7.8) where Y* = ln(y/k), L* = ln(l/k), and G* = ln(g/k) and each variable was highly statistically significant. 3 The output elasticity with respect to public capital in (7.8) is 0.39, almost seven times larger than Ratner's estimate. The interpretation of this elasticity is that if the government increases the public capital stock relative to the private capital stock by say 10%, then output per unit of private capital will increase by 3.9%. This is quite large, much larger than most other estimates. Second, the output elasticity of labor is 0.35, and the output elasticity of private capital can be calculated according to = So it would 36

13 appear that public capital is more productive than either private capital or labor alone in equation (7.8). This result was quite astonishing when first published. Aschauer also discovered that other forms of government spending did not increase private output. General government consumption, non-military spending, military spending, and compensation for government employees, had no effect on output when the public capital variable was included in the regression equation. Second, almost the entire effect of public capital was due to structures, not equipment. Third, core infrastructure, which includes highways, mass transit, airports, electrical and gas facilities, water, and sewers, had an output elasticity of 0.24 and was highly significant, although this estimate is about 37% smaller in magnitude than Ashauer's estimate for total public capital. Other buildings and hospitals had an output elasticity of only 0.04 and 0.06, respectively, and were not highly significant. Conservation and development capital and educational buildings were not significant at all. Finally, a cross-country comparison of G7 countries including the United States, the United Kingdom, Canada, Italy, France, West Germany, and Japan, revealed a positive correlation between growth in labor productivity and the ratio of public investment to GDP from 1973 to However, this was only part of the story. When the model is simulated and the impact of the decline in public infrastructure spending in the late 1960's and early 1970's is calculated, the model can explain most (about 57%) of the slowdown in the growth rate that occurred in the 1970's and 1980's. Researchers began to notice a decline in the growth rate of GDP and the rate of growth in labor productivity in the 1970's in several countries including the US. Several explanations were offered and there were supporters and critics of each one. 4 However, Aschauer offered a novel explanation; the slowdown in growth could be explained as a response to the marked decline in spending on public infrastructure. The policy implication is fairly obvious; growth would improve if more were spent on infrastructure. 37

14 Aschauer recognized there were potential problems with his statistical model and tried to correct for them. First, he checked the stability of the parameters across different time periods. (See Table 2, pp. 186). The estimates of the output elasticity of public capital range from to Second, despite the fact that serial correlation did not appear to exist in his original equation, he also used lagged variables in the regression equation without affecting his main result. Third, he recognized that causation may not go from public capital to private output, but the other way around; public capital might not be exogenous but might respond to output instead. To handle this he used two-stage least squares (TSLS) with lagged public capital as an instrument. His main result was again unaffected by this. Finally, Aschauer presented results on a specific industry, namely trucking, that also indicated that public capital, in this case, highway capital, has a positive effect on output in the industry. It is somewhat unlikely that reverse causality would hold for a single industry. In a follow up study, Munnell (1990a) also found some supporting evidence using similar data. 5 For the period , she estimated an aggregate output elasticity for public capital of E G = 0.31, without any constraint on the estimation, E G = 0.37 when constant returns to scale is imposed for (K, L), and E G = 0.33 when constant returns to (K, L, G) is imposed. For core infrastructure she obtained 0.37, 0.21, and 0.39 when there were no constraints imposed, when constant returns was imposed on private inputs, and when constant Table 7.1: Early time series results Author Data Approach Main result Aschauer (1989) aggregate production function E G = 0.39 (total public capital) E G = 0.24 (core infrastructure) E G = 0.00 (govt consumption) E G = 0.00 (military capital) Munnell (1990a) aggregate production function E G = 0.31 (total public capital) E G = 0.37 (core infrastructure) Tatom (1991b) aggregate production function E G = (with energy price) E G = 0.00 (first differences) Lynde and Richmond aggregate translog cost public capital is productive. (1992) Lynde and Richmond (1993) aggregate function profit function public capital is productive 38

15 returns was imposed on all three inputs. The main problem with her results is that the labor coefficient is implausibly low and sometimes even negative. (See Table 7, equation (1), pp. 18, for example.) Another problem is that when CRS is imposed on private capital and labor and competition prevails, private capital's share is also implausibly high. Aschauer (1990) provided additional evidence in support of the public capital hypothesis. He used a panel of data on the fifty states in the US for and time averaged the data to estimate a production function that captures the long run effect of public capital on output. He estimated the output elasticity of core infrastructure spending to be about Since this is well above the share of core infrastructure spending to output ratio, 0.025, he further argued that this strongly suggests that too little has been spent on core infrastructure. 6 Of course, one general criticism of this work is that the other possible reasons that have been put forth as explanations of the productivity slowdown have not been included in the estimation strategy. Therefore, a test of the different explanations, and how well the public capital hypothesis does relative to other explanations, has not been undertaken. It is certainly possible that the public capital hypothesis may dominate the other explanations. Of course, it is also possible that the public capital hypothesis may not explain any of the slowdown once some of the other reasons have been included in the model. 7.4 Criticism of the early work Other researchers were highly critical of the public capital hypothesis. In particular, Tatom (1991a) argued that most of the decline in spending on infrastructure in the US during the early 1970's was due to a decline in spending on highways, streets, and educational buildings by state and local governments. Federal spending was relatively constant in this period. Next, he pointed out that the number of school age children and the amount of miles driven per 39

16 person were both decreasing during this same period. Thus, he argued that it was rational for state and local governments to spend less on highways, streets, and educational buildings since people seemed to have less demand for such investments. Furthermore, Tatom (1991b, 1993) argued that there were statistical problems with the earlier studies. First, the early work did not take into account the energy crisis and this could lead to a serious bias in the results. If energy is not somehow taken into account in the estimation, reduced growth in productivity that would have been attributed to energy related problems, especially during the 1970's, would naturally be attributed to a slowdown in public capital investment. Second, significant breaks in the time trend were omitted. Trends are included in these studies to account for technological change. More specifically, Tatom argued there was a break in the trend around Ignoring this would again bias the public capital productivity elasticity upwards. Third, the data may not be stationary. 7 When two data series, such as output and public capital, are both drifting upwards over time, a regression of one series on the other can lead to a statistically significant positive coefficient even if there is no real relationship between the two variables. This is known as the spurious regression problem. Tatom (1991b) presented evidence from the Augmented Dickey Fuller unit root test that the data is not stationary in levels. 8 This strongly suggests that the early results may possibly be spurious. When these various problems are taken into account, Tatom found no real evidence that public capital has a large effect on output or growth. For example, he obtained the following result using data for the US for the period , ln(y/k) = ln(l/k) ln(g/k) ln(p) T, where T is a time trend, p represents energy prices, and X is the first difference of the variable X. All of the variables were highly significant except the public capital variable. 40

17 Finally, it is possible for two variables that are not stationary in levels to be stationary when taken in linear combination. Suppose y and x are both stationary in first differences. Then the two variables are said to be cointegrated if the error term u t in the following regresion equation is stationary, y t = α + βx t + u t. One can apply a unit root test to the residual of this equation. If the unit root test rejects the hypothesis of a unit root, then the residual is stationary and the variables y and x are cointegrated. In that case, an estimate of β does capture the appropriate correlation between y and x in a long run sense. A statistical procedure known as Dynamic OLS (DOLS), suggested by Stock and Watson (1989), can be used to estimate the parameters of a cointegrated long run relationship. Tatom applied the DOLS procedure and once again found that public capital does not have a statistically significant effect on output. One criticism of Tatom is that the evidence he presented on stationarity is decidedly mixed. It appears from the evidence presented in his paper that ln(y/k) is trend stationary and that ln(l/k) is actually stationary in levels. If this is correct, then the equation he estimated in first differences is inappropriate; the variables must be integrated of the same order. It is also not clear why the time trend was in the equation if it is in first differences. Third, labor's productivity coefficient was only in the cointegrated equation he estimated. This seems too low to be capturing the elasticity of labor's marginal productivity properly. This suggests there is something fundamentally wrong with the approach altogether. Finally, it is not immediately clear that including a price in a production function is appropriate. 9 Other critics argued that causality could go in the opposite direction. (See Aaron, 1990 and Eisner (1991), among others.) Estimating a production function implicitly assumes that public capital "causes" output or growth. However, the direction of causality could be the other way around. As a society becomes wealthier it may spend more on public infrastructure 41

18 so an increase in output, which leads to greater wealth, may in fact cause greater spending on public capital. More to the point, a regression equation is predicated on the assumption that the right hand variables, like L/K and G/K, are exogenous. However, in a regression equation like equation (7.8), variables like private capital, labor, and even public capital, may be endogenous. For example, during a recession when output is low, the government may undertake a public infrastructure project to boost output and employment. In that case, low output is causing an increase in infrastructure spending and hence the causality is "reversed." This sort of "simultaneity problem" can bias the estimation of the coefficients. Both Munnell and Aschauer have revised their view on the effect of public capital on the economy in light of this criticism. Munnell (1992) stated that, "the numbers emerging from the aggregate time-series studies are not credible." Similarly, Aschauer (1993) revised his estimates downward as well. He suggested that about 10% of the productivity slowdown can be explained by the decline in public infrastructure spending, which is much lower than his original estimate. However, Lynde and Richmond (1992, 1993) provided further evidence supporting the hypothesis. In their first paper they estimated the cost share equations stemming from a cost function using aggregate US data for the period They found that state and local public capital and private capital were complements, while labor and state and local public capital were substitutes. Second, public capital appeared to have a positive marginal product throughout the estimation period. Third, federal non-military public capital did not appear to have much of an effect on the cost shares at the aggregate level. In their second paper, Lynde and Richmond (1993) estimated a profit function for aggregate US data for the period and specifically corrected for stationarity problems. They tested for unit roots and discovered that the hypothesis of a unit root could not be rejected for each data series in their study. They used an estimation procedure that is 42

19 asymptotically equivalent to DOLS, the Fully Modified (bias eliminated) estimator due to Phillips and Hansen, which produces consistent estimates of the parameters of the empirical model even if there are endogenous variables on the right-hand side of the regression equation. In addition, the estimates of the parameters can be interpreted as estimating the long run relationships in the data. Their estimates suggest that approximately 40% of the productivity slowdown could be explained by reduced spending on public capital in the US. While lower than Aschauer's original assessment, this is still quite large. 7.5 Studies using disaggregated US data Regional and state studies Regional economists were especially interested in the public capital hypothesis since much infrastructure is locally determined in many countries and local governments would need to know if infrastructure investments have a beneficial impact on the local economy. The results using disaggregated US data are collected in Table 7.2 below. Eberts (1986) used data on 38 US metropolitan areas to estimate the elasticity of public capital, which included highways, water, and sewer capital. The estimated output elasticity of public capital was only Furthermore, he found that private capital and public capital were complements. An increase in public capital in that case would possibly attract private capital to the local economy. Costa, Ellson, and Martin (1987) estimated a translog production function using a cross section of the 48 contiguous states in the US for They found that public capital had a significant effect on state manufacturing output. The elasticities for labor, private capital, and public capital were 0.77, 0.11, and 0.19, respectively. Labor and public capital were complements, while the two capital stocks were unrelated in their data. However, for 10 states the output elasticity of public capital was actually negative. 43

20 Deno (1987) estimated an aggregate translog profit function using data from 36 SMSA's for Three types of public capital were included in the study, highway, sewer, and water. He found that all three types had a positive impact on output supply. Second, all three types of public capital tended to be complementary to both private capital and labor. This would tend to support Eberts' conclusion; local policy makers may be able to attract more firms and hence private inputs by investing in public infrastructure. Munnell (1990b) constructed a panel data set consisting of annual observations on gross state product, private capital, non-agricultural employment, and state and local public capital for the 48 contiguous states of the US for Her main result was that public capital had a positive effect on gross state product, although the elasticity was much smaller than the time series estimates. It was only 0.15 when there were no constraints imposed, 0.06 when constant returns in private inputs was imposed, and 0.08 when constant returns in all three inputs was imposed. When public capital is disaggregated the estimated elasticities of highways, water, and sewer systems were 0.06, 0.12, and 0.01, respectively, which are quite small relative to the early time series estimates. Third, evidence was put forth that public infrastructure had a positive impact when the data was aggregated to the regional level. The productivity elasticity for public capital in the Northeast, North Central, South, and West regions were 0.07, 0.12, 0.36, and 0.08, respectively, only one of which supports the early estimates. Of course, these results are subject to some of the same criticism as the other early work. Some of the variables on the right hand side of the regression equations are possibly endogenous. For example, Munnell used the state unemployment rate in her equations. Most likely the unemployment rate will be correlated with public capital if states specifically increase public capital spending in a recession when unemployment is especially high. 44

21 Second, the data may not be stationary, yet the typical estimation technique (OLS) ignores this problem. 10 Hulten and Schwab (1984, 1991) used a multi-factor productivity analysis to show that public infrastructure had little explanatory power for growth in multi-factor productivity. Suppose output is given by Y t = A t F(K t,l t ), where A is capturing Hick's-neutral technological progress. Differentiate and divide through by the technology to obtain dy t / Y t = da t / A t + ( AF K K / Y )(dk t / K t ) +( AF L L/ Y)(dL t / L t ). If firms maximize profit and competition prevails, then factors will receive their marginal product as payment. In that case, after rearranging, the last equation becomes da t / A t = dy t / Y t S K (dk t / K t ) S L (dl t / L t )), where S K = rk/y is capital's share in output, r = F K, and so on. With data on the shares of the individual factors, and the growth rates of output and the various inputs, one can calculate the growth of so-called multi-factor productivity, da/a. Hulten and Schwab extended this framework by incorporating public capital into the model, Y t = H(G t, A t )F(K t, L t,g t ), where H( ) captures the Hick's neutral technological change that now depends on two components, the pure component of Hick's neutral technological change (A) and the effect of public capital (G). They calculated the sources of regional growth in the US for for manufacturing and showed that productivity actually grew faster in the so-called snowbelt than in the sunbelt despite the fact that infrastructure grew at a faster rate in the sunbelt. Indeed, during the sub-period the stock of infrastructure grew faster in the Sunbelt while multi-factor productivity grew faster in the snowbelt. Holtz-Eakin (1994) used a panel of state data on gross state product, employment, Munnell's private capital series, and his own estimates of state and local public capital for the US for the period to estimate several versions of a state production function. The 45

22 extension here is that he controlled for various fixed effects in an attempt to reconcile the various disparate findings appearing in the literature. The main equation of his model is an aggregate state production function, Y st = α 0 + α 1 K st +α 3 L st +α 4 G st +ε st, (7.9) where ε is an error term. The s subscript indexes states while the t subscript indexes time. Holtz-Eakin considered the following specification of the error term, ε t = f s +γ t + µ st, (7.10) where f is a state-specific effect, γ is a time-specific effect, and µ is an i.i.d. random variable. The state-specific component of the error term picks up any effects that are invariant over time but may vary across states, while the time-specific component picks up effects that are the same across states but may differ over time, where both effects are unobservable by the econometrician. This is important because other studies using state data did not differentiate these various effects and Holtz-Eakin argued this was the reason for the disparate results. There are different ways of estimating the statistical model composed of equations (7.9) and (7.10), depending on the assumption made regarding the components of the error term. First, one can assume the state-specific effect f s and the time-specific effect γ t each period are both fixed. State dummy variables can then be used to capture the state-specific effects. The estimation results are then contingent on the state-specific effects actually present in the data and the focus is on the time variation within each state, not cross sectional variation across the states. Alternatively, the state-specific effect can be eliminated by first differencing the data. Second, one can assume the state-specific fixed effect is random. However, this may introduce a bias in the estimation because of the error terms that may be common across states. In that case, GLS will be efficient, but inconsistent and biased. 11 Holtz-Eakin estimated various versions of the model controlling for time effects, state effects, and several econometric problems. Only in the simplest model did public capital have 46

23 a positive effect on gross state product. In every other specification the effect was either zero or negative. In some cases it was negative and statistically significant. For example, when the state effect is fixed, the time effect is included, and constant returns to scale is imposed, the output elasticity of public capital is and the t-statistic is about Holtz-Eakin concluded that the effect of state and local public capital, "is essentially zero." (pp. 20.) Garcia-Mila and McGuire (1992) estimated a Cobb-Douglas production function using a panel data set for the 48 contiguous states in the US for In their model output, as measured by state GDP, is a function of private capital structures, private equipment, labor, as measured by employment, and public capital as measured by highways and total state and local educational expenditures. They found that the elasticity of output with respect to highways was and the elasticity with respect to educational expenditures was This was more evidence in support of public capital having a small effect. More recently, Garcia-Mila, McGuire, and Porter (1996) came to the opposite conclusion, namely, that public capital does not affect output. They estimated several Cobb- Douglas functions and found that when the function was estimated in levels the output elasticities for highways, water and sewer systems, and other public capital, were 0.37, 0.069, and 0.01, respectively, although the last coefficient was not significant. However, when either random state effects or fixed state effects were taken into account, the elasticities either drop considerably in magnitude, or are negative. For example, when fixed state effects are taken into account, the elasticities were 0.127, 0.064, and 0.071, all statistically significant. However, they also noted that stationarity may be a problem and estimated an equation in first difference form. The public capital variables were not statistically significant. One criticism of the first differencing approach is that it ignores the possibility that a long run cointegrated relationship might exist. An error correction model might be more appropriate in estimating the long run relationships. Second, the paucity of data in terms of 47

24 the length of the panel used in some of these studies may also be a problem. This will affect unit root testing and tests for cointegration, and may not provide enough time variation to allow for precise estimates Industry studies There are industry studies that focus on individual industries over time or a cross section of industries. Typically, these studies at the micro level tend to find that public capital has a significant effect on output or cost. Nadiri and Mamuneas (1994) derived share equations from the representative firm's cost minimization problem. They then estimated the cost shares after imposing certain regularity conditions on the model, e.g., concavity in prices. One important issue they raised was whether the public capital variable should be adjusted by the utilization rate. Aschauer, among others, included the capacity utilization rate for manufacturing as a separate variable in his estimated equation in order to capture certain cyclical elements not captured by the other variables in the model. Other researchers take the product of the utilization rate and the public capital stock to calculate "capital services," which then enters the production function rather than the stock itself. This will typically increase the variation in the capital series and improve the fit of the model. 12 Nadiri and Mamuneas used data on twelve two digit-manufacturing industries in the US for The public capital infrastructure stock was adjusted by the utilization rate of a given industry to obtain "public capital infrastructure services" for that industry, although their main results were not affected by this adjustment. Public capital was measured in two ways, total, non-military net public capital stock, (federal, state and local), and R&D public capital. The latter was constructed from data on total R&D expenditures by the government. They also allowed the public capital stocks and the utilization rates to enter the share equations separately without affecting the results. 48

25 Their main result was that both public capital variables exerted a negative effect on cost for most of the industries in their sample. The magnitude of the cost elasticities for public infrastructure services ranged from to , and the range for public R&D capital services was to The marginal benefits were generally higher in durable manufacturing industries. In addition, public infrastructure was found to be a substitute for private inputs. However, public R&D capital was a substitute for private capital and material inputs but complementary to labor for most industries. Finally, they estimated the marginal benefit from the two public capital stocks and a social rate of return for public infrastructure, public R&D capital, and private capital. The marginal benefit, as measured by the cost reduction given a small increase in a public capital stock, i.e., - C h / G i, is positive for most industries but small in magnitude, ranging from to for infrastructures and from to 0.01 for public R&D capital. Estimates of the average social return across industries for public infrastructure and public R&D were and , respectively. The social rate of return to private capital was In addition, the return to both public infrastructure and private capital increased throughout the sample period while the return to R&D capital fell. Morrison and Schwartz (1996) employed a cost function approach using US data for for public capital (highways, water and sewer systems), private capital, production and non-production labor, and energy. Of course, additional data on relative prices is required under this approach and it is especially difficult to find a proxy for the shadow price of public capital. They defined the shadow value of public capital as Z G = C/ G. If P G is the price of a unit of public capital, c G in the last chapter, the value of Z G - P G is very informative. They estimated a system of equations that included the demands for the variable inputs (labor and energy), the cost function (A Leontief function that allows for non-constant returns to scale and fixed inputs.), and a pricing equation to capture profit maximization. The 49