To obtain the national average elasticity of aggregate cost with respect to
highway capital, we weight the individual industry estimates by their respective
industry cost shares. For example, let us define the cost elasticity of highway
capital for industry *i* as . We obtain the "aggregated"
cost elasticity by using the expression

(10)

That is, the "aggregated" cost elasticity is a cost share weighted
average of individual industry elasticities. Using the envelope condition, the
output elasticity of highway capital is equivalent to the negative of the ratio
of the elasticity of cost over the cost elasticity of output.^{20}
Thus the output elasticity of highway capital for the economy is given by ,
where
is the cost weighted average of output cost elasticities of the industries in our sample.

**VIII. 1. Aggregate Cost and Output Elasticities**

The aggregate cost and output elasticities with respect to highway capital stock, the scale elasticities, and the output elasticities of factors of production–labor, materials and capital–are presented below in Table 11. These aggregate elasticities are derived from the individual elasticities reported in Tables 7 through 9.

The scale measures for the aggregate economy shown in panel A of Table 11 suggest a degree of scale with respect to private input of 1.06. This suggests a modest degree of increasing return to scale. However, the total scale measure is approximately 1.29. This measure reflects the contribution of both highway and other infrastructure capital as unpaid inputs in the production function.

Either the aggregate cost elasticity or the aggregate output elasticity shown in Table 11 can measure the productivity effect of highway capital. These elasticities imply the average cost reduction or output increase due to an increase in highway capital investment. For example, the magnitude of the cost elasticity suggests that a 1 % increase in highway capital ()leads to approximately a .08% average cost reduction for the whole economy. The magnitude of this elasticity is much

Table 11: Aggregate Elasticities* 1950-1991 |
||

Panel A |

-0.0797 | 1.0612 | 1.2910 | 0.0841 | 0.2943 |

Panel B |

Conditional:
Output Level Held Constant |
-0.0945 | 0.0996 | 0.0176 | ||

Total:
Output Level Varies |
-0.0189 | -0.0065 | 0.0627 |

Panel C |

0.3661 | 0.5307 | 0.1644 | 0.0849 |

*The bar indicates aggregate elasticity.

smaller than the elasticities reported in the literature reviewed in section II. In particular, our result contrasts markedly with the elasticity estimates originally reported in Aschauer (1989a) and Munnell (1990a, b). Their estimates are several times as large as our estimates for the aggregate economy. Specifically, our estimates of cost and output elasticities with respect to increases in highway capital are respectively 6 and 5 times lower than Aschauer's original estimate for total infrastructure capital. Our estimates are more comparable to the output elasticities of public capital reported for the highly disaggregated SMSAs in Duffy-Deno and Eberts (1991) and Eberts (1991).

The cost and output elasticities we estimate can be used to calculate the total
marginal benefit of an increase in highway capital for the entire economy. The
sum of the marginal benefits over all 35 industries, ,
is shown in last column of panel A of Table 11. This sum suggests that an increase
of 1% in net highway capital generates a total benefit of 0.29% at the aggregate
economy level. For example, a $10 billion (in 1987 prices) increase in net investment
in highway capital would approximately equal a 0.0128 increase in 1991 highway
capital stock. Using the average total marginal benefit for the entire period
of 0.294 shown in Table 11, and adjusting for the depreciation rate of approximately
0.08%, the total cost reduction would be 0.27%.^{21}
This is approximately 0.045% of the level of the GNP in 1991 in 1987 prices.

These estimates of the total benefit measure the externality benefits of highway capital network, exclusive of payments toward the construction and operation of highway capital already paid by the producers. The producer payments are included in our basic data set as the expenses for taxes and materials as part of factor costs. In addition, these externality benefits of highway capital network are exclusive of the benefits of highway capital in consumption sector of the economy which is probably very large and not accounted for here.

As noted above, we report that the total marginal benefit of highway capital is 0.29. This benefit is greater than the one reported in our 1996 report for total highway capital. However, our new estimate is close to the estimate reported in the 1996 study for the NLS capital stock. This occurs because in our present study, we do not have many positive signs for the cost elasticities in the non-manufacturing sector. Though the individual industry marginal benefits shown in Table 7 are much smaller in magnitude than in our previous report, we have more of them with the correct negative sign. These negative elasticities generate the relatively larger sum of marginal benefits of this paper.

Increased highway capital is labor and material saving at the aggregate economy level as well as at the industry level. That is, an increase in highway capital investment reduces the demand for both labor and materials. In addition, highway capital investment also has a strong positive effect on the demand for private capital. That is, private capital and public highway capital are complements. Highway capital investment leads to "crowding in" of private capital formation. Public highway capital seems to not only increase the marginal productivity of private capital but also may be in some degree a prerequisite for private investment. These conclusions hold, as shown in panel B of Table 11 whether the level of output is fixed or variable. In fact, the induced output expansion effect on demand for factors of production are all positive and fairly large. Increases in highway capital save on factors of production such as labor and materials, but they also increase demand for private capital investment. This pattern, as was noted earlier, also holds for the economy on a micro level.

Panel C of Table 11 presents the output elasticities of various factors of production and the output elasticity of highway capital. As expected, the output elasticity of materials is the largest followed by that of labor and private capital. Furthermore, these elasticities correspond to the shares of these inputs in total output. A one- percent increase in highway capital leads to an average of a 0.08 percent increase in total output over the period 1950-91. It is also important to note that the output elasticity of private sector capital is clearly larger than the output elasticity of highway capital. The results indicate that the contribution to the economy's growth of a one- percent increase in private capital stock is more than twice that of a similar increase in the highway capital stock.

However, it is important to note that the output elasticity with respect to highway capital has been declining over time. The average elasticity of output with respect to highway capital, , starts out relatively high in 1950–about 0.15 -- but steadily declines thereafter. The average output elasticity with respect to highway capital for the years , 1981-91is about 0.039; in 1991, the value of is about 0.026. The reduction in the value of during this period to a great extent reflects the fall in the ratio of total highway capital to total cost and output of the economy, i.e., . In contrast, the aggregate output elasticity with respect to private capital, , is relatively stable over the entire sample period. It ranges from about 0.14 to 0.18.

An important public policy question raised in the literature focuses on whether public capital is over- or under-supplied. The optimal provision of public capital services (highway capital) can be derived by the well-known Samuelson condition, as modified by Kaizuka (1965). This condition requires that the level of public capital provided is at the point where the sum of marginal benefits of producers and consumers is equal to the marginal cost of providing an additional unit of public capital. Ignoring the consumption sector, an alternative method for determining whether public capital is provided optimally is to compute the rate of return to highway capital and compare it with the rate of return to private capital for the whole economy. The optimal provision of public capital requires that the rates of return to both types of capital are equal. Thus, if the rate of return of highway capital is higher than that of private capital, highway capital is under-supplied and an increase of public investment is necessary.

Nadiri and Mamuneas (1993) find that the rate of return of public capital in the manufacturing sector is about 7 percent, and the rate of return of private capital is about 9 percent. Morrison and Schwartz (1991) compare the shadow price of public capital with the "user cost" of public capital, and find that Tobin's q ratio of public investment exceeds one. This result suggests that infrastructure investment is lower than the socially optimal amount in their sample of manufacturing sectors. Similarly Shah (1992) estimates a Tobin's q equal to 1.04 for the Mexican manufacturing sector, and concludes there is under-investment in public capital. Berndt and Hansson (1992) use the following methodology. First, they equate the marginal benefit of public capital with its ex-ante rental price. Then, they solve for the optimal capital stock, and calculate the ratio of the optimal level of the public capital stock to the actual public capital. They find that this ratio in Sweden is above one for the period 1960 to 1970, and below one for the period 1970-1990, which suggests under- and then over-investment.

In our 1996 study we assume that the government chooses the amount of highway capital by minimizing the present value of the costs of all the resources in the economy. That is, the government selects the level of public capital such that the sum of the industry marginal benefits equals the user cost of public capital, i.e.,

(11)

where is the acquisition price, is the discount factor approximated by the long-term interest rate and is the depreciation rate of highway capital. Solving equation (11) for yields the optimal amount of highway capital.

From equation (11), we derive the net social rate of return from public capital as the ratio of the sum of marginal benefits to cost minus the depreciation of public capital, i.e.,

(12)

This rate of return on highway capital is calculated assuming the user cost of highway capital is where is the government capital price deflator, and the government long-term bond rate is used to measure . is the depreciation rate of highway capital and is the price distortion effect of taxes levied to finance highway capital ( is set to 0.46; see Jorgenson and Yun (1990)). This distortion effect arises because no country relies extensively on head taxes to finance infrastructure capital. Distortionary taxes (e.g., an income tax) are often used to fund public investments. Therefore, the social cost of additional public capital is the sum of the direct burden of the taxes needed to pay for the infrastructure and the dead weight cost associated with these taxes.

The results reported in Table 12 are much lower than
previously reported in the literature., Fernald (1992) estimates the rate of
return to investment in roads using essentially the same set of data as we use
in this study. He concludes that "a conservative statement -- is that the
data strongly supports the view that roads investments are highly productive,
offering rates of return of 50% to100%, perhaps more." ^{22}
Our results suggest rates of return well below Fernald's lower bound estimates.
Our average rate of return for the period 1950 - 1991 is 29 percent, about half
of his rate of return of 50 percent. Even so, the rate of return over the postwar
period has been quite impressive. However, in recent years the returns to highway
capital have converged to those estimated for private capital stock.

Table 12 lists for several sub-periods the net social rate of return to total highway capital , the net rate of return to private capital stock and the interest rates on government bonds. The net rate of return on total highway capital, , during the 1950's and 1960's is very high. This reflects the shortage of highway capital stock during the 1950's when the Interstate Highway System was under construction. This rate has declined continuously since the late 1960's and in 1991, it is about the same level as the long term interest rate.

The time profile of the net social rate of return for total highway capital indicates that since 1955 the net rate of return has declined from 0.46 to an average of about 0.16 for the period 1981-91. The rate of return for 1991 stands close to less than 10%. We can compare the rate of return for highway capital to the net rate of return for private capital, , and the long-term interest on government bond, . The net rate of return on private capital is calculated as where is the user cost of private capital . When and are compared with the interest rate over the period 1960 - 1991, the gaps between these rates and are very large over the period 1960-91. The gap between the net rates of return for highway capital and private capital significantly large in earlier period. In 1981-91, the gap narrows considerably, and almost disappears.

Highway capital constitutes a network of roads and facilities that serves all of the industries in the economy. This network has the characteristics of a public good that cannot and probably will not be provided by the private sector. If every industry attempted to provide its own road system, the costs of duplication, management disputes, etc., would be prohibitively high for the private sector. Industry and society would be better off if the participants pooled their efforts and established a network of highways to serve all. The cost saving of such a system is enormous.

Consider the case where highway capital is not publicly provided. If a private industry had to provide the highway capital, it would provide a level , at a point where the marginal benefit and marginal cost of highway capital are equal:

(13)

Based on our estimates, we could solve equation (13) for and calculate, for each industry, the highway capital that satisfies equation (13). However, it is well known via the Samuelson condition, that the level of highway capital chosen by each industry will be below the social optimum because private industry does not take into account the benefits that accrue to the other industries. In addition, the private sector will be unwilling to provide highway services since the individual benefit of an additional unit of highway services will be close to zero.

Consider now the hypothetical case in which each industry builds capital stock. Each industry bears the whole cost of investing in highway capital , and the net rate of return to highway capital for industry , evaluated at the actual level of capital , is given by

(14)

may be negative if the real gross private marginal benefit is less than the depreciation rate of . Individual industries may not invest in highway capital since its cost will be prohibitive. However, by sharing the cost of highway capital , the economy can achieve the maximum benefit with the minimum cost. Comparing equations (12) and (14), the following relationship exists between the social and private rates of return on highway capital:

(15)

where *F* is the total number of industries sharing the cost and benefits
of highway capital. If each industry had to build its own highway capital ,
the cost of the duplicated network of highways would be too high for the economy
due to the total depreciation .
By sharing highway infrastructure, the economy is saving depreciation costs
of .
Using our estimated marginal benefit functions, the sum of net private rates
of return ,
under our hypothetical case is equal to -2.61 on average and the savings to
the economy
is 2.90. Thus, the social net rate of return for total highway capital is equal to 0.29.

Note that the real *gross* private benefit , in terms of private
cost reduction will be the same whether highway capital
is built and owned by individual industries or by industries sharing the benefits
and costs together. The *net* private benefit will be higher through sharing.
In the simple case where all industries have the same demand for highway capital,
the cost of infrastructure will be equally shared by all industries and the
cost per unit of for each industry will be equal
to . Then the net private rate of return
for each industry will be ;
clearly
is less than .

The industry marginal benefits of highway capital shown in Table 9 are gross rates of return, inclusive of the depreciation rate . The marginal benefit in each of the industries is much smaller than the actual value of the depreciation rate , which is, on average, about 0.08. It is only through a shared network of highways that each industry avoids the duplicative cost of individual highway systems, each with a separate depreciation rate.

**VIII. 4. Decomposition of Aggregate TFP Growth**

A central issue in the debate on the role of infrastructure centers on the question of its contribution to the growth of aggregate TFP and to the deceleration of TFP growth in the period 1973-1979. Aschauer (1989a), Munnell (1990a) and others claim that the decline in this period is mainly, if not exclusively, due to the decline in growth of infrastructure capital. Hulten and Schwab (1991), Gramlich (1994) and others have argued for no or minimal contribution of infrastructure capital to productivity slowdown.

To calculate the sources of productivity growth at the aggregate economy level, we first aggregate the industry decomposition shown in Table 10 using the industry's share in total output as weights. The second step is to use the cost function elasticities shown in Table 7 with equation (8) to decompose aggregate TFP growth into its component parts. That is, we calculate the effect of exogenous demand, relative prices, highway capital stock, and technical change on the growth rate of TFP in the US economy during the period 1950-1991.

As indicated in Table 13, the main source of TFP growth over the period 1950-91 is the changes in exogenous demand, . It accounts for about 50% of TFP growth over the period. Input price changes contribute less than 1 percent to TFP growth. Highway capital's contribution to total factor productivity growth is about 15%. This pattern generally persists across sample periods, although some of the magnitudes fluctuate. change according to the . Highway capital's contribution to TFP was high in the period before 1965. Since then, it has contributed much less.

These results stands in contrast to those reported by Aschauer, Munnell and other proponents of large contributions of infrastructure. Furthermore, our results also differ from the results reported by those who deny any role for infrastructure in enhancing the growth rate of productivity. Our analysis suggests a middle course. That is, increases in highway capital stock contribute to the expansion of the productive capacity of the economy. However, the magnitude of its contributions to growth of output and productivity are modest in comparison to the contribution of exogenous demand. Most of the contribution of highway capital to productivity growth occurred in the 1950s and 1960s. Since 1973, highway capital has made a small contribution to trend TFP.