# IV. ANALYTICAL FRAMEWORK

IV. 1. Model Specification

The basic methodology employed in this study and in the 1996 report consists of estimating industry demand and cost functions. Using these industry parameter estimates, we deduce the corresponding estimates for the aggregate economy. We evaluate the demand and cost equations separately and then we use their estimated parameters to decompose TFP growth and to calculate the marginal benefits of highway capital for each industry. Based on the underlying parameter estimates we calculate the rate of return on highway capital infrastructure.

The critical parameter estimates for the decomposition of total productivity growth are the price and income elasticities, obtained by estimating the output demand function, and the degree of scale and input substitutions, obtained by estimating the cost function. The estimates of marginal benefits, the impact of highway capital on demand for labor, private capital, and materials for the entire 1950-1991 and for the recent period 1981-1991, are based on estimating the model to the data for the 35 industries and sectors for the sample period, 1950-1991.

IV. 2. Demand Function Estimates

We estimate separate demand equations for each industry. The output demand equation for each industry, i, is specified as a log linear function

(1)

The demand function is estimated in growth rates. According to equation (1), the output growth rate in each industry is regressed on a constant, the growth rate of its output price normalized by the GNP deflator and the growth rate of real GNP per capita. Thus, changes in quantity demanded in an industry are related to its own price movement in comparison to the GNP deflator and changes in the level of aggregate income and population of the economy.

We are interested in two of the estimated parameters of demand function (1). They are:

• The price elasticity of output demand, which is measured by the coefficient ( implies demand is perfectly inelastic; implies demand is unitary elastic; and implies demand is elastic); and
• The per capita income elasticity of output demand, which is measured by the coefficient (same definitions as for a) .

IV. 3. The Cost Function Specification and Estimation

We write the cost function for the th industry as

(2)

where is a twice continuously differentiable, normalized cost function; is an n - 1 dimensional vector of relative variable factor prices, is the quantity of output, tis an index of time representing disembodied technical change, and is an m-dimensional vector of public capital services.

Public capital services affect the cost structure of an industry in several ways. First, an increase in quantity (or better quality) of public capital services shifts the cost per unit of output downward in an industry. We call this the "productivity effect". Second, firms will adjust their demand for labor, intermediate inputs, and physical capital stock if public sector capital services are either substitutes for, or complements to, the factors of production in the private sector. That is, the effects of public sector services may not be neutral with respect to private sector input demand decisions. We call this the "factor demand effect." Third, the cost reduction induced by the increase in public capital investment may lead to a reduction in the price of output which in turn may result in an increase in the demand for output. We call this the "output expansion effect." This output expansion effect is an indirect effect of public capital investment. The increased capital investment leads to greater output production, which in turn leads to an increase in the demand for employment materials and private capital investment. The net effect of public capital investment on total cost and its structure will be the combination of the productivity, factor demand, and output expansion effects.

We assume that the technology of the industry is represented by a translog cost function of the following form.7

(3)

and the share equations

(4)

where is total cost normalized by the price of materials, . and are the relative prices of labor and capital, respectively. The production cost is given by , where . denotes the acquisition price of capital, is the interest rate, is the rate of depreciation and is the price change of capital goods. is the level of output, and and are the level of total highway capital and other infrastructure capital, respectively. is an index of technical change. Finally, and are the cost shares of labor and capital.

As stated above, and are the stocks of highway capital and other infrastructure capital, respectively. These are both public goods, and as a result, no market prices can be related to the services provided by these two types of infrastructure capital. However, it is possible to determine the shadow price or willingness to pay for each of these services as the in private production cost savings associated with and . The marginal benefit (MB1) of highway capital, , is measured as8

(5)

That is, an additional unit of highway capital, result in a cost reduction, . For example, a better network of highways reduces driving time and saves labor, fuel and other operating costs.,

Aside from the direct productivity effect of highway capital indicated by (5), there are factor demand adjustment effects that arise out of the complementarity and substitutability of private inputs (such as labor, private capital and materials) for highway capital. These effects can be calculated as

(6)

where is the quantity of input . If the expression in (6) is greater than, equal to or less than zero, then highway capital has a negative, zero, or positive effect on the demand for that particular factor of production.

Finally, the effect of an increase in highway infrastructure capital on the rate of technical change can be calculated using the following expression,

(7)

which indicates that an additional unit of highway capital results in a productivity increase or cost decrease due to technical change.