Public Infrastructure And Private Productivity

CHARLES D. DELORME, JR. University of Georgia Athens, Georgia HERBERT G. THOMPSON, JR. Maryland Public Service Commission Baltirnore~Maryland RONALD S. WARREN, JR. University of Georgia Athens, Georgia

Public Infrastructure and Private Productivity: A Stochastic-Frontier Approach* We estimate an aggregate production frontier using time-series data for the private U.S. economy and show that estimated technical inefficiency is negatively correlated with the stock of public capital, or infrastructure. We then estimate a production frontier which includes a publiccapital variable and report that this variable has no statistically significant effect on private productivity. Our results suggest that public capital affects private productivity indirectly by reducing aggregate technical inefficiency rather than directly by increasing output. Consequently, previous evidence indicating a positive, direct effect of public capital on average privatesector output may have been obtained from stochastically misspecified models.

1. Introduction Since the classic analysis of public infrastructure by Arrow and Kurz (1970), a number of empirical studies have examined the role of public capital as an input in the production process. 1 Many of these studies have reported estimates of the marginal product of infrastructure that are positive and statistically significant, but often implausibly large. In particular, Aschaner (1989b) and Munnel] (1990) generated a lively debate on the ira*We thank two anonymous referees, the editor, participants of the Georgia Productivity Workshop II, Kevin Fox, Cliff Huang, and David Robinson for helpful comments on an earlier version of this paper. We also thank Dale Jorgenson and John Tatom for making their data available to us. 1For examples, see Ratner (1983), Aschauer (1989a, 1989b), Munnell (1990, 1991), Tatom (1991), Holtz-Eakin (1994), Nadiri and Manuneas (1994), Andrews and Swanson (1995), Garcia-Mil~, McGuire, and Porter (1996), Mullen, Wilfiams, and Moomaw (1996), and Otto and Voss (1996).

Journal of Macroeconomics, Summer 1999, Vol. 21, No. 3, pp. 56:3-576 Copyright © 1999 by Louisiana State University Press 0164-0704/99/$1.50

563

C. D. DeLorme, Jr., H. G. Thompson, Jr. and R. S. Warren, Jr. portance of public infrastructure by reporting output elasticities with respect to public capital of 0.39 and 0,34, respectively, which were higher than the consensus output elasticity for private capital of 0.30. These findings were then used to explain the decline in private-sector labor productivity in the 1970s and 1980s as a result, in part, of alleged disinvestment in the publiccapital stock. These initial studies have stimulated much additional research, some of which has been criticized for various econometric problems. For example, Gramlich (1994) points out that a number of time-series studies failed to consider the presence of common trends, which would bias upwards the estimated coefficient on public capital. Indeed, when Tatom (1991) firstdifferences his aggregate time-series observations, as a response to nonstationarity, he obtains much lower estimates of the marginal product of public capital, and some of them are not statistically significant or even positive. Another econometric problem noted by Gramlich (1994) involves the omission of potentially relevant variables, such as energy. However, when Tatom (1991) includes energy price in the production function, he confounds a production function and a cost function, for which he was properly criticized by Gramlich (1994). Finally, Holtz-Eakin (1994) argues that many paneldata studies do not include controls for unobserved, state-specific productivity effects. When such controls are included, he finds no role for public capital in directly affecting private-sector productivity. A standard assumption in this literature is that public infrastructure shifts an average production (cost) function upward (downward) in a neutral manner, z However, theoretical considerations suggest that at least some forms of public infrastructure are neither under the direct control of(shrinkable by) private-sector firms nor easily substitutable with private labor and capital. This argument implies that such infrastructure capital should not be modeled as an explicit input in a production function determining private output. An alternative view is that public infrastructure reduces the technical inefficiency of private-sector production, or the difference between actual and best-practice output. In this vein, Mullen, Williams, and Moomaw (1996) specify a translog stochastic production frontier, and employ panel data to estimate the direct (output) and indirect (efficiency) effects of public capital in manufacturing across U.S. states and over time. Their empirical ~Recent exceptions to this generalization are provided by Nadiri and Manuneas (1994) and by Morrison and Schwartz (1996). Both of these studies examine the effects of public capital on manufacturing-sector productivity in the context of a system of cost and input-demand functions. As a consequence, both the technical (neutral) and allocative (biased) efficiency effects on private output of public infrastructure could be estimated.

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Public Infrastructure and Private Productivity results suggest that public capital increases both output and productive efficiency in manufacturing. However, the estimated, direct output elasticity of public capital is quite small. In this paper, we also specify and implement a test of the role of public infrastructure in reducing technical inefficiency in the production of privatesector output. Our study differs from Mullen, Williams, and Moomaw (1996), however, by using aggregate time-series data. Consequently, our empirical results pertain to the entire U.S. private economy, rather than just to the manufacturing sector. The paper is organized in the following way. First, we replicate the findings of previous empirical studies by estimating an average production function that is augmented with public capital as an input. We then estimate a stochastic production frontier without public capital. These latter estimates are used to establish the presence of aggregate technical inefficiency. Subsequently, we show that the estimated level of technical inefficiency is negatively related to the stock of public infrastructure. Finally, we estimate a production frontier which includes public capital, and report that its estimated coefficient is not significantly different from zero. Therefore, we find that public capital affects U.S. private productivity only indirectly, by reducing aggregate technical inefficiency, rather than by directly increasing private-sector output. Our results, then, provide only mixed support for the empirical findings reported by Mullen, Williams, and Moomaw (1996). However, we affirm the conclusions of Tatom (1991), Evans and Karras (1994), and Holtz-Eakin (1994) that public capital does not directly increase private productivity.

2. Analysis The starting point for our analysis is the commonplace empirical fnding that public capital affects aggregate, real private output in the U.S. economy_ Following Ratner (1983), Aschaner (1989b), Munnell (1990), Tatom (1991), Gramlich (1994), Andrews and Swanson (1995), and Otto and Voss (1996), we represent aggregate technology by a constant-returns-to-scale Cobb-Douglas production function augmented by Hicks-neutral technical change. Because the dependent variable in this model (output per worker) varies procyclically over time, a control for the phase of the business cycle is included, a This also facilitates a comparison with studies such as Aschauer (1989b), which attach importance to declining capacity utilization in explainaFor an illustrationof the use of a control variablefor the phase of the business cyclein a time-seriesmodelof production,see Lovell,Sickles,and Warren (1988).Aschauer(1989b) and Tatom (1980) also employthe capacityutilizationrate to controlfor business-cycleeffects. 565

C. D. DeLorme, Jr., H. G. Thompson, Jr. and R. S. Warren, Jr. TABLE 1. Means and Definitions of Variables and Sources of Data Variable

Mean

Definition and Source

Q

2031.525

Real Gross Private Domestic Product, in billions of 1982 dollars; Jorgenson, p. 74, series C802.

L

1319.099

Private Domestic Labor Input Index (quality-adjusted private domestic man-hours scaled to labor income in 1982); Jorgenson, p. 34, series C394.

K

815.137

G

1003.413

GAP

82.215

Private Domestic Real Capital Input, in billions of 1982 dollars; Jorgenson, p. 52, series C700. State, Local, and Federal Government (non-military) Real Stock of Equipment and Structures, in billions of 1982 dollars; Bureau of Economic Analysis, U.S. Department of Commerce, from Tatom (1991); see also Musgrave (1988) and U.S. Department of Commerce (1993). Out-put as Percentage of Capacity in Manufacturing; Board of Governors of the Federal Reserve System. Economic Report of the President (1992), Table B-49, p. 403.

ing the productivity decline of the 1970s and 1980s. Annual data from the period 1948-1987 are used in the empirical analysis, and are described in Table 1. The model can be written In(Q/L)t = 130 + ~lln(K/L)t + 132 ln(G/L)~ + ~3aln(GAe) t

+

~4T + [35T2 + v t ,

(1)

where Q, L, K, and G are defined in Table 1, and GAP is the manufacturingsector capacity utilization rate, which is our measure of the phase of the business cycle. We follow Aschauer (1989b) and Munnell (1991) and assume

566

Public Infrastructure and Private Productivity that the flow of public-capital services is proportional to the stock of public capital, G. 4 The variables T and T 2 are, respectively, linear and quadratic time-trends which, following Darby (1984) and Tatom (1991), proxy nonlinear technical change, vt is a normally-distributed error term, and t indexes the N observations. It is well known that aggregate time-series variables are frequently nonstationary in level form. This feature of such data may result in a spurious-regression bias that would lead one to conclude that there are causal relations among the variables when in fact no such relations exist (see Granger and Newbold 1974). In the present context, if nonstationarity exists one might infer from estimates of the parameters of the model in (1) that public infrastructure affects private output even if it does not. To determine the integrated properties of the data, we performed augmented Dickey-Fuller (D-F) tests for unit roots in levels and in first differences of each variable in Equation (1) using two lagged differences. Each augmented D-F regression was estimated with and without a constant and a linear time trend. The evidence from the D-F tests is consistent with a unit root in levels but no unit root in first differences. We are therefore justified in assuming that the levels of the variables are integrated of order 1, which means that these variables must be differenced once to achieve stationarity. To examine the relationships among the variables, we used the cointegration tests developed by Johansen (1988) and Johansen and Jusefius (1990). We conducted both the trace test and the maximum eigenvalue test to determine the presence of cointegrating vectors in the nonstationary timeseries data. As with the unit-root tests, two lagged differences were used for the cointegration tests. The critical values for the appropriate test statistics are provided by Johansen and ]uselius (1990, 208, Table A2). The results of these tests support the null hypothesis of no cointegration, since the values of the relevant test statistics are less than the 5% critical values. Thus, the production function to be estimated is correctly specified in first differences since the variables in Equation (1) are integrated of order 1, but are not cointegrated. 5 Consequently, we estimated a first-differenced version of (1):

5Detailed results of the D-F unit-root tests and the Johansen cointegration tests are available from the authors. A discussion of the Johansen test procedure can be found in Dickey, Jansen, and Thornton (1991, 73-75). 4This assumption explains why, in Table 1, the mean of G (the stock ofpubfic capital) exceeds the mean of K (our measure of the flow of public-capital services). Because the estimating Equations (2) and (3) are specified in logarithmic first differences, this method for measuring public capital does not affect the estimated coefficients, which are interpreted as elasticities.

567

C. D. DeLorme, Jr., H. G. Thompson, Jr. and R. S. Warren, Jr. TABLE 2. Variable

Average Production Function a Equation (2) Coefficient b

G=G~

G=G,_I

Intercept

0.0024 (0.345)

0.0016 (0.225)

Aln(K/L)

0.6099 (3.763)

0.7482 (4.695)

Aln(G/L)

0.2762 (2.246)

0.1759 (2.348)

Aln(GAP)

0.4122 (5.069)

0.3371 (4.290)

T

0.0001 (0.450)

- 0.0001 (0.327)

D.W.

1.8407

1.7433

/]2

0.49

0.48

NOTE: aOrdinary least-squares estimates, bAbsolute values of estimated t-statistics are in parentheses under the coefficient estimates.

Aln(Q/L)t = 7o + 71Aln(K/L)t + 7zAln(G/L)t + 13Aln(GAP)t + 74T + v ' ,

(2)

where v; ~- Avt is a normally distributed random variable. 6 Ordinary least squares (OLS) estimates of the production function in first-differenced form are reported in Table 2. Column 1 contains estimates obtained by using the contemporaneous value of public infrastructure. Column 2 presents estimates with G lagged one period, as suggested by Munnell (1992, 194), to reduce the possibility of any feedback effect of private-sector output on the stock of public capital. Estimates of the coefficients on the private-capital variable and the capacity-utilization rate are positive, as expected, and significantly different from zero, while the coefficients on the trend variable are positive but statistically insignificant. 7 The estimated co6The first difference (v') of a normally distributed random variable (v) is also normally distributed. 7The estimate of 3'1 in Table 2 is somewhat higher than typicallyreported° but nonetheless falls within the range of estimates recently provided by Evans and Karras (1994)_There are two 568

Public Infrastructure and Private Productivity efficients on the public-capital variable are positive and significantlydifferent from zero, thus replicating the widely reported empirical finding that public capital increases average private-sector real output in the U.S. economy. The estimated coefficient on private capital is more than twice the size of the estimated coefficient on public capital, and the null hypothesis of equality of the two coefficients can be rejected at the 7% level of significance (t = 1.454). The foregoing results, as well as those from previous empirical studies, were obtained by estimating an average production function, rather than the best-practice or frontier technology. Suppose, however, that public infrastructure reduces technical inefficiency, but no allowance has been made for the latter in the specification of the production function. Then it is possible to infer from estimates of an average production function that public capital should be included as a conventional input when, in fact, the way in which public capital affects real output is by reducing the gap between actual output and potential output. Schmidt (1986) has discussed a type of specification test which can, in this context, be used to determine whether or not public capital has a statistically significant effect on aggregate private output because it reduces technical inefficiency in production. Suppose we estimate the stochastic production frontier Aln(Q/L)t = 60 + 61Aln(K/L)t + 62Aln(GAP)t + 6aT + v[ - gt,

(3)

where v' is a normally distributed error term, and I-t > 0 is a one-sided error which represents technical inefficiency in production, s The argument discussed above, that public capital reduces technical inefficiency, implies that there is an auxiliary relation ~tt = ~o + ~IAIn(G/L)t + w t ,

(4)

in which ~1 < 0 is predicted, and where w -> - [~0 + ~IA ln(G/L)] since the distribution of the dependent variable, g, is truncated at zero.

possible explanationswhy the estimatedcoefficientson T in Table 2 are not significantlydifferent fromzero. First, the labor-inputvariableL is adjustedfor improvementsin laborquality so (Hicks-neutral)technicalchange may be embodiedin that series. Second, the estimating equationsare specifiedand estimatedin first-differenceform,implyingthat productivitygrowth maybe unaffectedby neutraltechnicalchange. SAmongthe admissibleforms for the distributionof g, the half-normalis the most widely used and is employedhere. Aigner,Lovell,and Schmidt(1977) introducedthis stochasticspecificationof the productionfrontier. 569

C. D. DeLorme, Jr., H. G. Thompson, Jr. and R_ S. Warren, Jr.

The specification test proceeds in two stages. First, we estimate the production frontier by maximizing the log-likelihood function ln~ = Nln(~/2/]-n) + Nlncr -1 + ZlnN[1 -- F*(q)~¢r-1)] (1/2a 2) Z [ A l n ( Q / L ) t - 60 - 5~Aln(K/L)t - 52Aln(GAP)t - 5aT], -

(5)

where F*() is the standard-normal distribution function, or2 = cyn2 = ry~ z + cy~ is the variance of the composite error q = v' - ~, and = ry~/~ is the ratio of the standard error of technical inefficiency to the standard error of statistical noise. Estimates of the one-sided error components gt are calculated as the expectation of the gt conditional on the fitted values of v; - g,, according to the formula provided by Jondrow, et al. (1982). These estimates are then used as the values of the dependent variable in the auxiliary equation given in (4) above. Equation (4) is specifed as a truncated-normal regression model, which is then estimated by maximum likelihood.9 If the estimate of {t is not significantly different from zero, then we accept the null hypothesis that public capital does not affect technical inefficiency. On the other hand, if the estimate of {1 is significantly less than zero, we reject the null hypothesis and conclude that public capital indeed reduces technical inefficiency, thereby narrowing the difference between potential output and actual output in the private sector for any given levels of private capital and labor. Table 3 reports maximum-likelihood estimates of the stochastic production frontier. As before, the estimated coefficient on private capital is positive and significantly different from zero. The estimated coefficient on the capacity-utilization variable is also positive and significantly different from zero, reflecting the procyclical behavior of output per worker. One indicator of the presence or absence of technical ineffieiencyin the aggregate private economy is given by the estimate of )~. Under the null hypothesis of no inefficiency, )~ = 0 and all of the variance in the estimated equation would be attributed to statistical noise. In this ease, E(gt) = 0 and there would be no gain relative to ordinary least squares from modeling the production function as a frontier and estimating it by maximum likelihood. According to conventional test criteria, the estimated t-statistic on )~ implies that there is no technical inefficiency, at least at the 5% level of significance. This inference is suspect, however, since, as Greene (1993b) has noted, the usual t-test of statistical significance is invalid in this ease because the value

9For a discussionof the truncated-normalregressionmodel, see Greene (1993a, 687-90). 570

Public Infrastructure and Private Productivity TABLE 3. Variable

Production Frontier a Equation (3) Coefficientb (1)

(2)

Intercept

0.0216 (2.773)

0.0146 (1.738)

Aln(K/L)

0.6777 (4.041)

0.6270 (3.592)

Aln(G/L)

--

0.2127

(1.6o8) AIn(GAP)

0.3201 (3.579)

0.3835 (4.093)

T

0.0001 (0.245)

0.0001 (0.404)

)~

2.4879 (1.387)

2.0150 (1_284)

0.0185

0.0167

(5.227)

(4.670)

NOTE: aMaximum-likelihoodestimates, bAbsolutevalues of estimatedt-statistics are in parentheses under the coefficientestimates.

of L under the null hypothesis (zero) is at the boundary of the admissible parameter space. An alternative test for the presence of technical inefficiency, discussed by Schmidt and Lin (1984, 351), can be constructed as follows. If there is no inefficiency, then the distribution of the disturbances in the production frontier is symmetric. Consequently, deviations of the sample estimates of the disturbances (the OLS residuals) from symmetry can be used to infer the extent of technical inefficiency. Specifically, symmetry implies that the third moment of the disturbances is zero. Deviations of the OLS residuals from symmetry can be measured by the sample estimate of the skewness coefficient

~11 = m3/m3/2 , where m2 and ma are, respectively, the second and third moments of the 571

C. D. DeLorme, Jr., H. G. Thompson, Jr. and R. S. Warren, Jr.

TABLE 4. Variable

Technical Inefficiency Equation a

Equation (4) Coefficientb G = Gt

G = Gt-1

Intercept

0.0141 (9.429)

0.0153 (10.618)

Aln(G/L)

- 0.0479 (2.114)

- 0.1182 (5.494)

0.0089 (8.944)

0.0083 (8.832)

Ow

NOTE: aMaximum-likelihood estimates, bAbsolute values of estimated asymptotic t-statistics are in parentheses under the coefficient estimates.

OLS residuals. The distribution of ~/~ is widely tabulated, and Bowman and Shenton (1975) provide the 10%, 5%, and 1% critical values for various (small) sample sizes. For the problem at hand ~ = 0.623, which exceeds the 5% critical value for the present sample size. As a consequence, we reject the null hypothesis of symmetry of the OLS disturbances and from this result infer the presence of technical inefficiency in the U.S, economy. The results of estimating (4) by the truncated-normal regression procedure, in which the one-sided residuals from the estimated production frontier were regressed against the "omitted variable," the ratio of public capital to labor, are presented in Table 4. As before, estimates were obtained using both contemporaneous G and G lagged one period. The estimated coefficient on public capital is negative in each case and significantly different from zero at the 5% level. These results show that the one-sided residuals (the estimates of private-sector technical inefficiency) are negatively related to public capital, regardless of whether public capital is defined contemporaneously or with a one-year lag. This finding is consistent with the hypothesis that public capital facilitates the production of private-sector output by lowering technical inefficiency. These results provide an explanation for the statistical significance of public infrastructure in estimated average production functions, which assume no technical inefficiency. If private-sector technical inefficiency is assumed to be zero when in fact it exists, then the stochastic structure of the production function is misspecified. In the case, then, in which technical inefficiency is omitted from the model but is negatively correlated with public capital, the latter will be positively related to output if it is included as an explanatory variable in an estimated average production function. 572

Public Infrastructure and Private Productivity Further evidence in support of the model proposed here, and against the standard approach, is provided by estimating a production frontier which includes the public-capital variable. Such an estimating equation incorporates aggregate technical inefficiency, but otherwise follows previous literature by specifying public capital explicitly as an input_ Maximum likelihood estimates are reported in column (2) of Table 3 and reveal that the estimated coefficient on public capital is not significantly different from zero. This result contrasts with the finding of Mullen, Williams, and Moomaw (1996) that public capital increases both technical efficiency and output in U.S. manufacturing. This finding, coupled with the estimate of the statistically significant effect of public capital on technical inefficiency reported in Table 4, leads us to the following conclusions. Estimates of a positive, direct effect of public capital on (average) private-sector productivity reported in previous studies may have been obtained from a stochastically misspecified model. When the possibility of aggregate technical inefficiency in production is incorporated through the specification and estimation of a stochastic-frontier model, the estimated coefficient on public capital is no longer significantly different from zero. Instead, public capital is found to affect private productivity only indirectly, by reducing technical inefficiency in an aggregate production frontier which omits public capital as a direct input. 3. Conclusion

The stochastic-frontier approach to modeling aggregate production was employed in this paper to test the hypothesis that public infrastructure reduces technical inefficiency in the private economy. Most previous empirical studies of the effect of public capital on aggregate real private output have reported estimates of average production functions. A number of those studies concluded that public capital increases real private output but in the context of a model in which public capital is included as a conventional input in the production function, along with private capital and labor. With the exception of Mullen, Williams, and Moomaw (1996), however, none of these studies considered the potential role for public capital in reducing the technical inefficiency of private-sector production. Consequently, the stochastic structure of these earlier models may well have been misspecified, thereby calling into question the conclusions based on their estimates. To determine whether public capital reduces aggregate technical inefficiency in the private sector, we employed a specification test originally proposed by Schmidt (1986). A stochastic production frontier was estimated in first-differenced form, and the one-sided residuals were regressed against the change in the ratio of public capital to labor. The empirical results are 573

C. D. DeLorme, Jr., H. G. Thompson, Jr. and R. S. Warren, Jr. consistent with the view that public capital reduces private-sector technical inefficiency, but does not directly affect private output. A policy implication of these results is that additions to the public-capital stock cannot be used countercyclically to replace periodic shortfalls in private-capital accumulation or reductions in employment. Instead, infrastructure investment can contribute to a steady-state policy of reducing the technical inefficiency of private-sector production. The results reported in this paper are, of course, conditional on the particular specifications of technology and technical inefficiency that were employed. A constant-returns-to-scale, Cobb-Douglas production function was specified in order to provide comparability with the preponderance of past empirical studies of public capital as a productive input. A more flexible representation of technology (the translog) has been used by Mullen, Williams, and Moomaw (1996) in previous research on this topic. However, Moroney (1992) showed that, with aggregate U.S. time-series data (of the sort used in this paper), the translog function generates severe multicollinearity and is conclusively rejected in favor of a constant-returns-to-scale, Cobb-Douglas representation of technology. Finally, technical inefficiency was modeled by a random disturbance that has a half-normal distribution. Other stochastic specifications of inefficiency (like the exponential distribution) are feasible, and it may be worthwhile to examine the sensitivity of these results to alternative parameterizations. Received: February 1997 Final version: September 1998

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