# Appendix 2--Total Factor Productivity (TFP) Decompositiona

Assume that the production function of an industry is given by

(1)

where Y is the output of the industry, X is an n-dimensional vector of traditional private inputs, S is an m-dimensional vector of infrastructure capital services, and T denotes the level of disembodied technology. The traditional measure of total factor productivity growth is defined by the path-independent Divisia index:

(2)

where the dot denotes rate of growth, for example, ; and is the revenue share of the ith private input.

Differentiating (1) with respect to time, and dividing by output, we obtain

(3)

Assuming cost minimization of all inputs, public capital included, and letting be the price of the ith private input and the shadow price of public input k, we obtain the following first-order conditions:

(4)

and

where is the Lagrangian multiplier, together with the envelope conditions

(5) and

where is the total cost function including the shadow cost of public capital. Eliminating m from (4) and (5) and substituting (4) and (5) in (3), we obtain:

(6)

Firms, however, do not adjust the public capital stocks - they are exogenously given. What actually is observed is that firms minimize their private production cost subject to the production function (1). Let the optimal private cost of production, given the output level and public capital, be . Then the marginal benefit of an increase of public capital at equilibrium will be given by

(7)

It is not difficult to show using comparative statics that the total cost elasticity, , is given by

where

and is the private cost elasticity with respect to public inputs, and h is the private cost elasticity. The cost diminution due to technical change is

Following Caves et al. (1981), total returns to scale of the production function is defined as the proportional increase in output due to an equiproportional increase of all inputs (private and public, holding technology fixed), and is given by the inverse of .

Private returns to scale, i.e., the proportional increase in output due to an equiproportional increase in private inputs, holding public inputs and technology fixed, is given by the inverse of h. Thus, we identify two scale effects in our study, one internal and the other total, which is the sum of internal and external scale effects. Substituting (7) in (6) and then in (2) we have

(8)

where is the ratio of output price, , to average total cost, .

According to equation (8), TFP growth is decomposed into three components: a gross total scale effect given by the first term; a public capital stock effect given by the second term; and the technological change effect given by the last term.

The next step is to further decompose the scale effect. We assume the output price is related to private marginal cost in the following manner:

where is a markup over marginal cost. The markup depends on the elasticity of demand as well as on the conjectural variations held by the firms within an industry. Using the definition of output elasticity, , along with the private cost function, we obtain

(9)

After time differentiating (9), the pricing rule implies

(10)

Differentiating the private cost function with respect to time and using Shephard's lemma yields

(11)

where is the share of the ith input in private cost, .

In order to obtain the equilibrium of output growth we assume a log linear demand function (see Nadiri and Schankerman (1981a)) in growth rate form:

(12)

where and are real aggregate income and population, respectively, and reflects a demand time trend, and is the GNP deflator. Substituting (11) in (10) and the result in (12), we obtain the reduced form function for the growth rate of total factor productivity:

(13)

where .

Equation (13) decomposes TFP growth into the following components:

(i) an exogenous demand effect ;

(ii) a factor price effect ;

(iii) a public capital effect ; and

(iv) disembodied technical change .

The public capital and disembodied technical change effects can be further decomposed into direct and indirect effects. The direct effect of infrastructure , for instance, is given by while its indirect effect is given by . 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 (13) 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 ( = 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. Also, if there are constant returns to scale including public inputs, , then (13) collapses to .

aFor further details of this approach to decomposition of , see Nadiri and Schankerman (1981a, b) and Nadiri and Mamuneas (1993).