Higher-curvature gravities as holographic toy models

3.2.1 Technical efficiency estimation

Technical efficiency refers to the ability of a decision-making unit (usually a firm) to minimize input used in the production of a given output vector, or the ability to obtain maximum output from a given input vector (Kumbhakar and Lovell, 2003). Consequently, a firm is fully technically efficient if it produces the maximum possible output from a fixed level of inputs (output orientation), or if it uses the minimum possible input to produce a       

3 Bird (1999) also found that most of the manufacturing sectors in Indonesia are characterized by a high degree of industrial concentration.

43 given level of output (input orientation). There are two well-known methods for estimating the technical efficiency: stochastic frontier analysis (SFA) and data envelopment analysis (DEA).

SFA is a parametric method for estimating firm-or sector-level technical efficiency scores by exploiting the skewness of the error in the specification of a production function (see Aigner et al., 1977; Meeusen and van den Broek, 1977 for details). The approach requires the specification of a functional form and assumptions about the distribution of the efficiency term. DEA (Charnes et al., 1978; Banker et al., 1984) is a non parametric approach to efficiency measurement and requires very few assumptions about the properties of the production possibilities set. DEA uses the frontier concept directly (by projecting the inefficient firm to this frontier) to calculate the technical efficiency score, without assuming a specific functional form for the relationship between inputs and outputs. In addition, DEA takes the most efficient decision-making units (to be 100% efficient) as the basis for calculating technical efficiency for other decision-making units.

Technical efficiency scores are estimated in this paper using DEA for two reasons. First, given that one of the objectives is to infer the direction of causality between efficiency and industrial concentration, time-varying estimates of technical are required. DEA can be applied separately in every subsector of the food and beverages industry and for every year of observed data to provide these estimates⁴. Second, DEA avoids imposing a common structure on the technology of transforming inputs into outputs across subsectors by assuming a common functional form for the production frontier.  

4 Although estimation of efficiency scores with SFA for every year and subsector separately is possible, such an approach would reduce the amount of information required for estimating the parameters of the production function in the dataset. On the other hand, imposing a single production function over the years would lead to autocorrelation of the efficiency score estimates. This autocorrelation could then affect the Granger-causality test that follows.

44 In general terms, DEA assumes that there are data on N inputs and M outputs for each of I firms. For the i-th firm these are represented by the column vectors xi and qi, respectively.

The NxI input matrix, X, and the MxI output matrix, Q represent the data for all I firms. This research uses the output-oriented DEA model⁵ by solving the mathematical programming problem as in Coelli et al. (2005):

where  represents a Farrel measure of technical efficiency (Farrel, 1957) with1 ∞, and

1

 is the proportional increase in outputs that could be achieved by the i-th firm, with input quantities held constant. λ is an Ix1 vector of constants and I1’λ=1 is a convexity constraint, with I1 being an Ix1 vector of ones. The convexity constraint is used to impose variable returns to scale (VRS), which ensures that an inefficient firm will only be compared to firms with a similar scale. The assumption of VRS technologies is also relevant because constant return to scale seems a too strong assumption for the Indonesian food and beverages sector, as this sector is characterized by many distortions. Regarding the calculation, if there are 100 firms in a subsector, the DEA frontier will be calculated by solving 100 linear programming problems of the form presented in (1), one for each of the 100 firms in the subsectors. We define ˆ(x,y)1/^ˆ as a measure of technical efficiency that assumes values in the unit interval so that the bootstrap method that follows is well defined.

5 We use output-oriented DEA to identify technical inefficiency as a proportional increase in output production, with input levels held fixed. This assumption might be relevant in the industry because small and medium firms find some difficulties to access the financial institution in order to expand their business in most of the periods.

45 To get robust estimates of the efficiency scores, this research uses the bootstrap technique of Simar and Wilson (Simar and Wilson, 1998). This technique is also expected to reduce the serial correlation problem in the efficiency scores among firms. The bootstraping method is repeated simulation of the data generating process, using a resampling method and applying it to the original estimator to the simulated sample so that the simulated estimates mimic the sampling distribution of the original estimator (Simar and Wilson, 1998). As the final result, we provide only the biased-corrected efficiency scores as accurate measure of efficiency, obtained using the formula:





3.2.2 Industrial concentration calculation

Industrial concentration is measured using both the Herfindahl-Hirshman Index (HHI) and the concentration ratio for n firms (CR_n), which is based on Pepall et al. (2008). Both indicators of industrial concentration are based on the market share of the firms and calculated by the following formulas:⁶

6 Considering stocks, this paper calculates industrial concentration based on sales data because sales seem to explain more about the market share than output.

46

where j = 1,2,…, m indexes the subsector, i = 1,2,…n indexes firms within a subsector, and MS_i is the market share of firm i in its respective subsector. CR4_j considers the collective share of the four largest firms in subsector j, while HHIj considers the inequality of distribution of market shares among all firms in subsector j. Both the CR4 and the Herfindahl-Hirschman Index (HHI) measures have limitations in their calculation, but they complement each other.⁷ Hence, it is necessary to use both concentration measures to clearly depict market structure in the industry.