Output & Productivity Growth Decomposition: A Panel Study of Manufacturing Industries in India.

AuthorRoy, Prasanta Kumar


In studying the technological change in the US economy, Solow (1957) differentiated the movements along the production function and the shifts of the production function. The former is caused by the input growth while the latter is defined as technological progress. Assuming constant returns to scale and perfect competition in the product market, he showed that the growth of output per unit of labor can be decomposed into technological progress and the weighted growth of capital per unit of labor. Alternatively, technological progress can be estimated with the time series data of output per unit of labor, capital per unit of labor, and the share of capital. This measure of technological progress is known as "Solow residual" (1). The residual is calculated by subtracting the growth of the primary inputs (weighted by their respective shares in nominal output) from the growth of output. Under "Solow residual approach", technological progress is usually considered to be the unique source of total factor productivity (TFP) growth. Under this approach TFP growth can be defined as the residual of output growth after the contributions of labor and capital inputs are subtracted from total output growth. This approach is based on the assumption that the economies are producing along the production possibility frontier with full technical efficiency (it does not allow inefficiency). Thus under "Solow residual" approach TFP growth is shown by solely shifting the production possibility frontier.

While the neo-classical approach of TFP analysis often assumes optimality in production capacity, the output-oriented stochastic frontier production approach (Aigner et al., 1977) argues that, with given sets of factor inputs and due to possible technical inefficiency, there can be difference between actual and optimal output. The measure of technical inefficiency can thus be added to the analysis of TFP growth by using the stochastic frontier model. This study extends the production function approach of TFP measure and follows Denny et al. (1981), Bauer (1990), and Kumbhakar & Lovell (2000) to examine the theoretical foundations of the decomposition of output and productivity growth. We relax the assumption of constant returns to scale and consider technical inefficiency in a stochastic frontier model. The output growth is then decomposed into: input growth and TFP growth and once again TFP growth is decomposed into: adjusted scale effect (ASE), technological progress (TP) and the technical efficiency growth (TE) (Kumbhakar & Lovell, 2000).

In our study we decomposed output and TFP growth of the thirteen 2-digit manufacturing industries in India namely, manufacture of food products (20-21), manufacture of beverages and tobacco products (22), manufacture of textile and textile products (23+24+25+26), manufacture of wood and wood products; furniture and fixtures (27), manufacture of paper and paper products (28), manufacture of leather and leather products (29), manufacture of chemicals and chemical products (30), manufacture of rubber, petroleum and coal products (31), manufacture of non-metallic mineral products (32), manufacture of basic metals and alloys (33), manufacture of metal products (34), manufacture of machinery and transport equipments (35+36+37), other manufacturing industries (38) and total manufacturing industry assuming that the afore-mentioned industries are not able to fully utilize the existing resources and technology because of various non-price and organizational factors that might have led to technical inefficiencies in production. Using panel data over a period from 1981-82 to 2010-11[during the entire period, pre-reform period (1981-82 to 1990-91), post-reform-period (199192 to 2010-11) and during two different decades of the post-reform period, i.e., during 1991-91 to 2000-01 and during 2001-02 to 2010-11], we have decomposed output growth of the organized manufacturing industries into input and TFP growth and again TFP growth into adjusted scale effect, technological progress and technical efficiency effect. This decomposition of output and TFP growth of the organized manufacturing industries has also been made for the pre-and post reform periods, and also for different decades in order to examine the trend and variations in the TFPG and its different components, during these sub-periods. The following section discusses the theoretical foundations of the decomposition of output and productivity growth (2).

Decomposing Output & Productivity Growth

Neo-classical growth models assume that there exists always technical efficiency and production occurs on the production frontier. But the existence of technical inefficiency cannot be ruled out altogether. The stochastic frontier model (Aigner et al., 1977; Battese & Coelli, 1988; 1992; Greene, 2005) can be used to check whether there exists technical inefficiency in production. The model is given by:

[Please download the PDF to view the mathematical expression] (1)

where Y is the actual level of output; F is the potential production function with 'n' inputs; [X.sub.it] is ith input; and 'u' is a half-normally distributed random variable with a positive mean. The inclusion of 't' in 'F' allows for the production function to shift over time due to technological progress. The last term [Please download the PDF to view the mathematical expression] measures technical inefficiency.

Taking logarithm on both sides of (1) yields

log[Y.sub.t] = log F ([X.sub.1t], [X.sub.2t], ..., [X..sub.nt], t) -[u.sub.t] ...... (2)

Technical inefficiency occurs when [u.sub.t] > 0 and the level of log[Y.sub.t] is less than the level of log F. Differentiating Equation (2) with respect to time yields the following output growth equation:

[Please download the PDF to view the mathematical expression] (3)

where [[??].sub.t] = [partial derivative][Y.sub.t]/[partial derivative]t/[Y.sub.t] is the growth of

output and [[??].sub.it] = [partial derivative][X.sub.it]/[partial derivative]t/[X.sub.it] is the growth of input [X.sub.it].

Define [e.sub.it] = [partial derivative] F/[partial derivative][X/sub.it] [X.sub.it]/F as the output

elasticity for input [X.sub.it] Let [e.sub.t] = [summation over (i)] [e.sub.it] it (the sum of the elasticity to each input). It can be shown that e is a measure of returns to scale. The production shows increasing (constant, decreasing) returns to scale when [e.sub.t] > 1 (= 1

Define the technical efficiency (TE) as the ratio of the actual output and the potential output, [Please download the PDF to view the mathematical expression]. Then, the growth of the technical efficiency,

T[[??].sub.t] = - [partial derivative][u.sub.t]/[partial derivative]t ..........(4)

The output growth can therefore be presented as

[Please download the PDF to view the mathematical expression] (5)

Consider the following cost minimization problem under perfect competition in the factor markets, but not necessary in the product market.

[Please download the PDF to view the mathematical expression] (6)

We express the objective function and the constraint in the following Lagrangian form,

[Please download the PDF to view the mathematical expression] (7)

where [lambda] is the Lagrange multiplier. The first-order condition for minimization requires,

[Please download the PDF to view the mathematical expression] (8)

Multiplying both sides by [X.sub.it],

[W.sub.it] [X.sub.it] = [lambda][e.sub.it] [Y.sub.t] .......... (9)

Taking the sum of all inputs, the total cost is

[summation over (t)] [w.sub.it] [X.sub.it] = [summation over (i)] [lambda][e.sub.it] [Y.sub.t] .......... (10)

or Ct = [lambda] [e.sub.t] [Y.sub.t] .......... (11)

where [C.sub.t] is the total cost of production at time 't'

Denote the cost share of input [X.sub.it] as [S.sub.it]. Dividing equation (9) by equation (11), the cost share is

[S.sub.it] = [w.sub.it] [X.sub.it]/[C.sub.t] = [e.sub.it]/[e.sub.t] .......... (12)

This shows that the cost share is always equal to the relative output elasticity in the case of cost minimization (3). We can rewrite the output growth Equation (5) as

[Please download the PDF to view the mathematical expression] (13)

By adding and subtracting term,

[Please download the PDF to view the mathematical expression] (14)

Using Equation (12),

[Please download the PDF to view the mathematical expression] (15)

Equation (14) shows the decomposition without cost information (w) and can be used for the empirical estimation of the sources of output growth, if the parameters of the production function are known. Equation (15) shows that output growth can be decomposed into four components: weighted sum of input growth, adjusted scale effect, technological progress, and growth of technical efficiency.

For the first term in equation (14), the weight for each input growth is equal to the cost share of each input. The second term represents the adjusted scale effect. When the returns to scale are constant, this term is zero. For the production with increasing returns to scale, [e.sub.t] > 1, a part of returns to scale ([e.sub.t] - 1) contributes to the output growth if aggregate input growth is positive. The contribution from returns to scale ([e.sub.t] - 1) is weighted by the aggregate input growth [summation over (t)][s.sub.it][[??].sub.it]. If the aggregate input growth is zero, then the scale effect is zero.

The first two terms in equation (15) show that input growth has two impacts on output growth. One is the direct impact through its growth and the other is the indirect impact through scale effect. The decomposition in equations (14) and (15) has relaxed a major assumption in Solow's (1957) decomposition of economic growth, as equation (15) does not require the constant returns to scale assumption. Indeed, the growth decomposition as shown by equations (14) and (15) can be applied to any type of production function as long as output elasticity for each input can be derived. This implies...

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