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VI. MODEL ESTIMATION

The analytical framework of the present study is similar to that of our 1996 study. The main difference is that in this study we adapt a translog functional form instead of a Normalized Symmetric MacFadden functional form to approximate an industry's cost function. After some experimentation it became clear that the translog functional form permitted to more easily incorporate some of the new features of the model, such as the inclusion of two public infrastructure capital stocks as separate arguments of the cost function. When we compared the results of the cost models with translog and Normalized Symmetric MacFadden functional forms, the overall results were not very different.

We estimated the demand function (1) for each industry. Initial estimation revealed that in a few industries the price elasticities had incorrect signs. A different formulation of the demand functions was attempted by estimating the model with the industry panel data; we also formulated alternative specifications of equation (1), introducing other variables in the demand function such as interest rate, unemployment rate, and the price of imports. The results of these alternative specifications did not differ much from those reported in Table 5. The results indicate that the price elasticities of demand and the per-capita income elasticities of demand vary across industries. The price elasticity of output demand is negative, less than one, and statistically significant in almost all industries. In two industries, SIC 34 and SIC 35, the price elasticities are positive but statistically insignificant. We then set these coefficients to zero and t re-estimated the demand equation. The magnitude of the price elasticity varies across industries, but generally they are very small. Unfortunately, there are not many recent studies available to provide a basis for comparison. Houthakker and Taylor (1966) calculated price elasticities for different industries based on product classification, as opposed to the industry classification used here. However, in cases where our frameworks are comparable, their estimated price elasticities are similar to ours.

The parameters of the underlying cost function are estimated using the cost function (3) and the share equations (4). These equations depend on private input prices, the level of industry output Mathematical Formula, the time trend Mathematical Formula, the level of total highway capital stock Mathematical Formula, and the level of other infrastructure capital Mathematical Formula. Hulten (1990) argues that the intensity of public capital usage fluctuates over time. There are variations in the utilization of highways, evidenced for example, by the ratio of vehicle miles traveled to the capital stock of roads. Also, public capital is a collective input which firms must share with others and therefore is subject to congestion (see Deno (1988)). Firms might have some control over the use of the public stock (see Shah (1992) and Fernald (1992)). For instance, a firm may have no influence on the level of highways provided by the government, but it can vary its use of existing highways by choosing routes. Adjustment for utilization of highway capital could indeed affect both the magnitudes of industry marginal benefits and the time pattern of the rates of return to highway capital stock. However, it is difficult at present to obtain a reliable and appropriate measure of highway usage by each industry over the period 1950-91.15 Therefore, at this stage of our analysis, we have not made any adjustment for utilization of the public capital stocks Mathematical Formula and Mathematical Formula.

The sample consists of pooled time-series cross-section data for 35 two-digit industries during the period 1950 - 1991. In order to capture industry specific effects, we assume the parameters Mathematical Formula are industry specific. Thus, we assume Mathematical Formula, where the parameters are normalized with respect to the k-th industry Mathematical Formula, Mathematical Formula is an industry dummy variable taking values either 1 or 0, and Mathematical Formula is an industry identification index. We estimate the model using an iterative seemingly unrelated regression approach (ISUR). Initial estimation revealed serial correlation in the residuals. Therefore, the equations were re-estimated with a correction for first order serial autocorrelation.

Moreover, in order to account for the effect of infrastructure capital other than total highway capital in the model, we assume that the two types of public capital are complements. Therefore, we add an auxiliary equation of the form Mathematical FormulaMathematical Formula where Mathematical Formula and Mathematical Formula are highway capital and other public capital, respectively. The equation fit extremely well and the coefficient Mathematical Formula was statistically significant and robust with a magnitude of about 0.37. Other, more complicated forms of this equation were also estimated but the estimates did not change much. We have estimated the overall model assuming different values for Mathematical Formula, but the most sensible results were obtained when Mathematical Formula was close to the estimated value of 0.37.

In Table 6, we present parameter estimates for the translog cost function (3). The estimated factor demand system satisfies all the required regularity conditions: the estimated cost function is shown to be nondecreasing in output, linearly homogeneous in input prices, and concave in factor prices. The results show that the cost model is well specified and that the parameter estimates are statistically significant. Coefficients of the industry dummy variables, not shown in Table A in Appendix 1, are also statistically, which suggests significant differences in the cost structure across industries. The squares of the correlation coefficients between the actual and predicted values are high, and the standard errors of each equation are small in both versions of the model. The estimated value of the autocorrelation parameter, Mathematical Formula, was very high–about 0.9. Since the estimate of Mathematical Formula was so close to unity we re-estimated the model, setting Mathematical Formula=1. Furthermore, we carried out a number of tests, similar to those in our 1996 report to test the sensitivity of our results.

To estimate the effect of an increase in highway capital stock on industry cost and demand for factors of production, we need estimates of the cost elasticity with respect to highway capital, Mathematical Formula, and the cost elasticity of output for each industry. The indirect or "factor bias effect" of Mathematical Formulacan be measured by its impact on private sector input demand functions. Furthermore, we can derive the "induced" output effect of cost reduction due to an increase in highway capital. This occurs because the reduction in cost (due to an increase in highway capital investment) induces a price decline, which in turn leads to an output increase. To produce the extra output, however, will require demand for inputs to increase, thus leading to higher costs of production.

VI. 1. Cost Reduction, Scale Elasticities and Output Expansion

From the estimates of the demand and cost functions, we derive the critical demand and cost elasticities that are necessary to measure the impact of highway capital on the cost structure and productivity growth of each industry. The critical elasticities are the following:16

VI. 2. Productivity Decomposition

Total factor productivity growth, Mathematical Formula, can be decomposed as17

(8) Mathematical Formula

where Mathematical Formula.

We obtain the relevant parameter estimates from the estimates of the industry demand and the cost functions. The parameters of equation (8) are defined as follows. Mathematical Formula is the output elasticity of demand, Mathematical Formula is the income elasticity of output demand, Mathematical Formula is the markup over cost, Mathematical Formula and Mathematical Formula are respectively the growth of GNP and population, and Mathematical Formula and Mathematical Formula are respectively the growth rates of the industry output prices and the GNP price deflator. Mathematical Formula is the change in degree of scale, Mathematical Formula is the ratio of output price, Mathematical Formula, to average cost. Finally, the Mathematical Formula are the cost elasticities with respect to highway capital and other infrastructure capital, Mathematical Formula the changes in these two types of public capital and Mathematical Formula is the change in level of technology.

According to equation (8) TFP growth can be decomposed into the following components:

(i) an exogenous demand effect,Mathematical Formula;

(ii) a factor price effect, Mathematical Formula;

(iii) a public capital effect, Mathematical Formula; and

(iv) disembodied technical change Mathematical Formula.

The parameter estimates of both the demand function (1) and the cost function (3) are critical for the decomposition of Mathematical Formula. In particular, Mathematical Formula, the price and income elasticities of demand, and Mathematical Formula and Mathematical Formula, the output cost elasticity and elasticity of cost with respect to an increase in highway capital,Mathematical Formula, play critical roles in the decomposition of total factor productivity growth.

The public capital and disembodied technical change effects can be decomposed further into direct and indirect effects. For example, the direct effect of infrastructure Mathematical Formula, is given by Mathematical Formula while its indirect effect is given by Mathematical Formula. Thus, an increase in public infrastructure initially increases total factor productivity by reducing the private cost of production, which in turn leads to a lower output price and higher output growth. Changes in output growth in turn lead to changes in TFP growth.

The important parameters in (8) are the price and income elasticities of demand and the cost elasticities of the private cost function. Note that if the demand function is completely inelastic (Mathematical Formula= 0), then shifts in the cost function due to real factor price changes, public capital, or disembodied technical change have no effect on output and hence no indirect effect on TFP. In addition, if the technology exhibits constant returns to scale with respect to all inputs, including public capital inputs, (i.e.Mathematical Formula), then equation (8) collapses to Mathematical Formula.

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