Appendix 2--Total Factor Productivity (TFP) Decompositiona

Assume that the production function of an industry is given by

(1) Mathematical Formula

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) Mathematical Formula

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

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

(3) Mathematical Formula

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

(4) Mathematical Formula Mathematical Formula

and Mathematical Formula Mathematical Formula

where Mathematical Formula is the Lagrangian multiplier, together with the envelope conditions

(5) Mathematical Formula and Mathematical Formula

where Mathematical Formula 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) Mathematical Formula

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 Mathematical Formula. Then the marginal benefit of an increase of public capital at equilibrium will be given by

(7) Mathematical Formula

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

Mathematical Formula where

Mathematical FormulaMathematical Formula

and Mathematical Formula 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

Mathematical Formula

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 Mathematical Formula.

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) Mathematical Formula

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

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:

Mathematical Formula

where Mathematical Formula 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, Mathematical Formula, along with the private cost function, we obtain

(9) Mathematical Formula

After time differentiating (9), the pricing rule implies

(10) Mathematical Formula

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

(11) Mathematical Formula

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

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) Mathematical Formula

where Mathematical Formula and Mathematical Formula are real aggregate income and population, respectively, and Mathematical Formula reflects a demand time trend, and Mathematical Formula 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) Mathematical Formula

where Mathematical Formula.

Equation (13) decomposes TFP growth 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 public capital and disembodied technical change effects can be further decomposed into direct and indirect effects. The direct effect of infrastructure Mathematical Formula, for instance, 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 (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 (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. Also, if there are constant returns to scale including public inputs, Mathematical Formula, then (13) collapses to Mathematical Formula.

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

Appendix 1 | References


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