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II. 1. Production Function Methods

The basic method for incorporating public infrastructure in an aggregate production function is straightforward: expand a production function to include not only the private factors of production, labor and capital, but public capital as well. Specifically, redefine the production function Mathematical Formula as Mathematical Formula, where Mathematical Formula is the level of output, Mathematical Formula is the level of productivity, Mathematical Formula is the stock of private capital, Mathematical Formula is employment, Mathematical Formula is the government financed infrastructure capital stock and Mathematical Formula is total factor productivity purged of the influence of the government capital stock. A commonly used specification is the Cobb-Douglas production function, estimated by Aschauer (1989b) and others:

Mathematical Formula

Taking natural logarithms yields the equation typically estimated:

Mathematical Formula

Aschauer found Mathematical Formula, the elasticity of output Mathematical Formula with respect to public capital Mathematical Formula, to be positive, ranging from 0.39 to 0.56.2 The marginal product of public capital Mathematical Formula, defined as Mathematical Formula, implied by this result is 100 percent or more. The implication of this is that an increase in government capital pays for itself in terms of higher output within one year. Much of the subsequent research tries to explain this high rate of return to public capital.3

The literature examining the effect of public infrastructure capital on output growth and productivity using the production function framework is extensive. Production function studies can be grouped into two broad categories: (a) national level studies, and (b) regional or state level studies. Table 1 summarizes the characteristic features of a selected number of production function studies. Aschauer (1989a) stimulated an extensive discussion of the nature and magnitude of the impact of infrastructure capital on output and productivity growth.4 In his paper, he estimates an aggregate production function and argues that infrastructure capital financed by the public sector increases the productive capacity of the private sector, and that public infrastructure investment stimulates private sector investment by enhancing the rate of return to private sector investment. Munnell (1990a) extends this line of argument, and her results generally support the proposition that there is a strong and significant effect of public infrastructure capital on productivity growth.

Both Aschauer and Munnell employ aggregate time-series data of the United States to estimate the relationship between private output and the stock of nonmilitary public capital. Nonmilitary capital includes highways and streets, educational buildings, hospital buildings, sewer and water facilities, conservation and development facilities, gas, electric, and transit facilities, and other miscellaneous structures and equipment. As previously noted, Aschauer estimates the elasticity of output with respect to public capital in a range from 0.39 to 0.56. In a related study, Munnell finds an elasticity of 0.33 for output per man-hour with respect to public capital. She uses the estimated coefficients from the aggregate production function to calculate annual percentage changes in multifactor productivity and concludes: "The drop in labor productivity has not been due to a decline in the growth of some mystical concept of multifactor productivity or technical progress. Rather, it has been due to a decline in the growth of public infrastructure" (Munnell, 1990a, p.20).

These results generated a variety of criticisms:

Several production function studies address infrastructure and productivity relationships at the state level using time-series cross-section data for the 48 contiguous states. The cross-sectional aspect of these data has certain advantages that mitigate the possibility of spurious correlation over time. As a whole, studies based on state-level data support a relatively lower but still positive relationship between public infrastructure and productivity. Munnell's (1990b) elasticity estimates show that, while public capital has a positive effect on output productivity, it is only half the size of the effect of private capital. For example, a one percent increase in public capital results in a 0.15 percent increase in output, whereas a one percent increase in private capital results in a 0.31 percent increase in output. He also estimates the output elasticity of labor to be 0.59. Calculating the marginal product shows that an additional unit of public capital increases output by the same amount as an additional unit of private capital. The results remain plausible when public capital is split into three components—highways, water and sewer systems, and other. The first two, constituting the largest part of core infrastructure, have larger effects than the "other" category.

Using Munnell's data, Eisner (1991) suggests that for all functions considered, the coefficient of public capital in the estimated equation remains significant when the data are arranged to reflect cross-sectional variation, but becomes insignificant when the data are arranged to allow for time-series variation. This suggests that states with more public capital per capita have more output per capita. However, a state that increases its public capital in one year does not produce more output in that same year as a result. Therefore, Eisner regards the direction of causation between output and public capital as undetermined, and postulates that a lag structure is required to obtain a true time-series relationship between output and public capital.

By calculating manufacturing productivity growth rates for the years 1951 to 1978 for major regions of the United States, Hulten and Schwab (1984) test whether different rates of public capital growth correspond to different rates of productivity growth. They find that differences in output growth are not due to differences in the growth of public infrastructure, but rather to variation in the rates of growth of capital and labor. When they expand this analysis to include the years 1978 to 1986 (Hulten and Schwab (1991)) their conclusion remains the same: public infrastructure has had little impact on regional economic growth.

These disparate results are likely due to whether the estimation process controls for unobserved state-specific characteristics. Holtz-Eakin (1994) tests the hypothesis that the positive and strong effect of infrastructure reported in the literature will diminish or disappear if state-specific effects are accounted for. McGuire (1992) estimates four different specifications of a state-level production function with public capital as an input: Cobb-Douglas without state effects; Cobb-Douglas with fixed state effects; Cobb-Douglas with random state effects; and translog without state effects. The four specifications of the model yield broadly similar results, in which public capital has a positive and statistically significant effect on gross state product (GSP). Of the three component parts of public capital, highways, water and sewers, and other, highways has the strongest impact on GSP. Water and sewers has a much smaller but usually significant effect, and other public capital is not statistically significant or has a negative effect on private output. Indeed, some economists hypothesize that state-level data may systematically underestimate the productivity value of public capital, because such data cannot capture the aggregate effects of public capital as a system.

Similar findings have been reported in a number of production function studies that use even more disaggregated data. Studies by Eberts (1988), Eberts and Fogarty (1987), and Duffy-Deno and Eberts (1989) use data at the metropolitan level. They test the direction of causation between infrastructure capital and output and estimate the magnitude of the elasticity of output with respect to infrastructure capital. Their findings suggest that causation runs mostly from infrastructure capital to output growth and there is a positive but considerably smaller elasticity of output with respect to public capital than those based on the aggregate production function relationship between infrastructure and growth of output and productivity.

From a reading of the evidence based on these production function studies it is possible to draw the following conclusions:

(1) Early estimates based on aggregate production function analyses are likely to have overstated the magnitude of the effects of public infrastructure capital on output and productivity growth;

(2) Estimates based on state level data indicate a relatively smaller contribution of infrastructure and that the composition of infrastructure capital matters; some types of infrastructure may have a greater effect on productivity than others;

(3) There are serious estimation problems in both aggregate national level time series studies and state and regional level studies that lead to highly disparate results; and

(4) Overall, it seems that the recent studies report relatively smaller elasticity estimates for infrastructure than Aschauer's original study. The evidence points to a positive but lower elasticity of output with respect to public infrastructure capital of about 0.20 to 0.30 at the national level and possibly a lower range at the regional level.

One reason for the wide range of estimates of the elasticity of output with respect to infrastructure capital based on production function estimates may be due to the minimal structure imposed on the data. If sufficient structure is not imposed on the data, provided the underlying data are not subject to serious measurement problems, the parameter estimates of the underlying production structure are likely to be biased and the estimates are not likely to be robust. In estimating production functions, whether using national or state level data, the production function is treated as a purely technological relationship between output and inputs, and firms' optimization decisions with respect to how much output to produce and what mix of inputs to use in the production process is not considered specifically. In reality, inputs and output are simultaneously determined when firms maximize (minimize) their profit (costs). When firms' optimization is explicitly considered, the marginal productivity conditions for the inputs should be estimated jointly with the production function. If these conditions are not explicitly considered, the estimated production function parameters are likely to be seriously mismeasured.

II. 2. Cost (Profit) Function Methods

Although production function analyses provide a useful first look at linkages between infrastructure investment and productivity growth, they do not provide detailed consideration of the effects of public investment on the economic decisions and performance of the firm. Production function analyses invariably omit factor input prices that affect factor utilization, and can thereby lead to biased estimates of production function coefficients. The cost function approach offers detailed information on cost elasticity of output as well as specific effects of infrastructure capital on demand for private sector inputs. Using cost function methodologies, it is possible to trace, in considerable detail, the effect of infrastructure investment on a firm's production structure and performance including technical change, scale economies, and demand for employment, materials and private capital stock.

The cost- or profit-function approach takes explicit account of the firm's optimization behavior. It considers both inputs and outputs as endogenous variables, while deems prices, which are market determined and thus considered beyond the immediate control of the firm, as exogenous variables. In addition, most production function studies of infrastructure employ a Cobb-Douglas specification, which, a priori, imposes the restrictive condition of a unitary elasticity of substitution among inputs, including infrastructure capital. Rather than impose such restrictions at the outset, they should be tested within the framework of a more flexible cost function specification. To avoid shortcomings inherent in the Cobb-Douglas specification, most cost and profit function studies incorporate more flexible functional forms such as the translog or generalized Leontief functions. A further advantage of using cost functions is that they yield direct estimates of the various Allen-Uzawa elasticities of substitution. These parameters are the key to describing the pattern and degree of substitutability and complementarity among the factors of production.5 Furthermore, in cost models, the effect of public capital on the demand for inputs can be directly estimated. If the effect is positive, public capital and the private inputs are complements; if it is negative, public capital and private inputs are substitutes.

There are relatively few studies using the cost (profit) function approach to analyze the effect of infrastructure capital or other types of publicly financed capital on output and productivity growth. Table 2 summarizes several of these studies and their more important features.

It is difficult to find exact agreement on the influence of publicly financed capital on productivity among the existing cost function studies. This is due to a general heterogeneity of these studies. Cost functions are estimated using diverse sets of data at the national and international level, state and metropolitan level, and industry level. Differences also exist with respect to assumptions about the optimizing behavior of firms, and the specification of the cost function, although many use either the translog or generalized Leontief functional forms. In addition, different authors use different notions of public infrastructure. Some focus on core infrastructure, while others use the total stock of public capital. Even though a single estimate cannot be provided for the effect of public infrastructure on total cost or on its contribution to productivity, all available cost (profit) function studies reach the general conclusion that publicly financed capital contributes positively to productivity by generating cost savings.

Lynde and Richmond (1992) estimate a translog cost function using aggregate US nonfinancial corporate business sector data for the period 1958 to 1989. They impose constant returns to scale on all inputs, public capital included, and assume firms behave competitively. Their findings suggest that publicly financed infrastructure reduces the costs of production in the nonfinancial corporate business sector.

Nadiri and Mamuneas (1993) estimate a translog cost function for 12 manufacturing sector industries for the period 1955 to 1986. Their findings indicate that increases in public infrastructure and publicly financed R&D reduce the cost to the industries in their sample. The magnitudes of the cost elasticities of infrastructure capital with respect to public infrastructure vary across the 12 industries ranging from -0.05 to -0.21.

Keeler and Ying (1988) estimate a translog cost function for regional trucking firms in the US road freight transport industry, during the period 1950 to 1973. They find that highway infrastructure has a significant effect on the productivity growth of the trucking industry, generating benefits that would justify about half of the cost of the Federal-aid Highway System.

Morrison and Schwartz (1991) estimate a variable cost function using state level data for the entire manufacturing sector over the period 1971 to 1987. They specify a generalized Leontief cost function, treating private and public capital as exogenous variables. They estimate a system of input-output equations for production labor, non-production labor and energy, and a short-run output price equation Mathematical Formula to incorporate profit maximization. They separately define and estimate equations for four U.S. regions -- Northeast, North-Central, South and West. Their results suggest that an increase of one percent of public capital reduces manufacturing costs between 0.15 percent in the Northeast and 0.25 percent in the West. In addition, the authors calculate the contribution of infrastructure to productivity growth in each region and for various states.

Deno (1988) estimates a translog profit function for the manufacturing industries from 1970 to 1978 using data from 36 SMSAs. He estimates the effects of highway, sewer and water capital on output supply and capital and labor demand. . In order to take into account the collective nature of public capital, he multiplies the public capital stocks by the percentage of the metropolitan population employed in the manufacturing sector. His findings suggest that all types of public capital contribute positively to output growth, but that highway and sewer capital contribute the most to output growth, capital formation and employment. He finds that output supply responds strongly to total public capital with an elasticity of 0.69. The corresponding elasticities for specific types of capital are 0.31 for highway capital, 0.30 for sewer capital, and 0.07 for water capital.

Berndt and Hansson (1992) estimate a short-run (variable) cost function using aggregate data from the Swedish private sector, by specifying a labor requirement function and assuming that private and public capital are fixed in the short run. They find that public infrastructure and labor inputs are complements during the 1960's and 1980's, but are substitutes in the 1970's. The authors conclude that an increase in public infrastructure reduces private costs. In addition, the authors estimate the optimal ratio of the amount of infrastructure capital to the existing capital stock and conclude that for the period 1970 to 1988 there was excess infrastructure for Swedish private production needs.

Lynde and Richmond (1993) estimate a translog cost function for U.K. manufacturing using quarterly data for the period 1966-1990. In their study, the elasticity of output with respect to public capital averages 0.20. They attribute approximately 40 percent of the UK's observed productivity slowdown to the decline in the public capital to manufacturing labor ratio. Their estimates indicate a significant role for public capital in the production of value-added output of the manufacturing sector.

Shah (1992) estimates a translog variable cost function using data on twenty-six Mexican three-digit level manufacturing industries. He treats labor and materials as variable inputs and private and public capital as fixed inputs. The short run effect of increased public capital is to reduce variable costs. He argues that there is underinvestment in public capital.

Conrad and Seitz (1994) estimate a translog cost function that imposes marginal revenue equal to marginal cost for the manufacturing, construction and trade, and transport sectors of the West German economy for the period 1960 to 1988. They find substantial cost reductions in these sectors due to infrastructure investment. Seitz (1992, 1994) reports similar results for the effect of core and total public capital stock on the production cost of 31 two-digit industries of the West German manufacturing sector from 1970 to 1987.

In general, evidence gathered from cost (profit) function studies suggests that the contribution of infrastructure on output growth is positive, but its magnitude is relatively smaller than those suggested by production function efforts. Also, there is evidence of an important influence of infrastructure capital on the demand for private sector inputs such labor, materials and capital. Most of the studies suggest, as noted later, a substitutional relationship between infrastructure capital and private inputs, holding the level of output constant.

From this brief review of literature on the linkage and magnitude of the contribution of infrastructure capital to growth in output and productivity, we can make a few tentative statements.

  1. There is a preponderance of evidence that suggests that infrastructure capital contributes significantly to growth in output, reductions in cost and increases in profitability. The magnitudes of these contributions, however, vary considerably from one study to another because of differences in econometric methodology and level of data aggregation.
  2. There appears to be a consensus on a much lower estimate of the magnitude of the contribution of infrastructure capital to output and productivity growth than suggested in the original Aschauer work. Output elasticity estimates of infrastructure capital at the national level should probably be in the range of 0.16 to 0.25. Estimates based on state and metropolitan level data suggest elasticities of approximately 0.06 to 0.20.
  3. Most studies indicate an underinvestment in public infrastructure capital, the degree of which varies among different studies. Most of the cost function studies suggest that private capital and infrastructure capital are substitutes, although some studies report that they are complements.
  4. The available studies are either too global or partial in their coverage of the economy. Most of these studies, particularly those at the national level, use real GDP, a value added measure, as the dependent variable. However, the appropriate measure for an analysis of the contribution of infrastructure (highway) capital is gross output. Gross output includes purchases of intermediate inputs, along with the primary inputs of private capital and labor. Because highways are used to transport intermediate inputs, the relationship between public capital and intermediate purchases can be taken into account.
  5. Using value-added data is justified if there is no substitution between intermediate inputs, such as materials and energy, and primary factors of production, such as capital and labor. If intermediate input prices are relatively stable, the use of value added in productivity analysis can be justified on practical grounds. However, oil price shocks substantially affected the course of the U.S. economy in the 1970's and 1980's. Similar effects to a lesser extent were associated with price increases in other intermediate inputs during this time period. Therefore, it is important to include explicitly energy and material inputs in the productivity analysis.
  6. Studies at the industry level are generally confined to the manufacturing sector or a specific subset of this sector. Infrastructure capital, however, may have important effects on other industries outside the manufacturing sector as well. It is very important to undertake a comprehensive study that includes all sectors of an economy in order to study the role and degree of externalities generated by publicly financed infrastructure capital such as highway capital.

Moreover, all of these studies have been challenged on conceptual and econometric grounds. Hulten and Schwab (1994) proposed a number of considerations to guide future research:

In this study, as we did in our 1996 report, we attempt to take into account explicitly these considerations. We consider a comprehensive set of industry data that cover the entire economy and we obtain the aggregate results for the total economy from the individual industry estimates. We introduce explicitly highway capital and other infrastructure capital as unpaid factors into the cost function and allow for the interaction of these varieties with each other and with other inputs such as labor, materials, and capital. We do not impose a priori any restrictions, such as constant returns to scale, on the parameters of the cost function-- rather, we test for such restrictions. The issue of simultaneity is addressed by estimating the model using appropriate econometric techniques. We estimate the demand function for each industry separately and we use the estimated output price and income elasticities with the cost function estimates to decompose the sources of output and productivity growth at both the industry and aggregate levels. We define a general analytical model that identifies the sources of TFP growth and the contribution of highway capital to TFP growth is evaluated in the context of competing determinants in each industry. Finally, we combine the individual industry estimates of the demand and cost parameters to obtain the corresponding "aggregate" parameter estimates. These "aggregated" parameter estimates are then employed to calculate the rate of return on investment in highway capital as well as its contribution to the output and TFP growth for the overall US economy.

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