# «Oleg Badunenko Michael Fritsch Andreas Stephan Allocative Efficiency Measurement Revisited – Do We Really Need Input Prices? FREIBERG WORKING ...»

## TECHNICAL UNIVERSITY BERGAKADEMIE FREIBERG

## TECHNISCHE UNIVERSITÄT BERGAKADEMIE FREIBERG

## FACULTY OF ECONOMICS AND BUSINESS ADMINISTRATION

## FAKULTÄT FÜR WIRTSCHAFTSWISSENSCHAFTEN

Oleg Badunenko

Michael Fritsch

Andreas Stephan

Allocative Efficiency Measurement

Revisited – Do We Really Need Input

Prices?

## FREIBERG WORKING PAPERS # 04

FREIBERGER ARBEITSPAPIERE 2006 The Faculty of Economics and Business Administration is an institution for teaching and research at the Technische Universität Bergakademie Freiberg (Saxony). For more detailed information about research and educational activities see our homepage in the World Wide Web (WWW): http://www.wiwi.tu-freiberg.de/index.html.**Addresses for correspondence:**

Oleg Badunenko European University Viadrina Department of Economics Große Scharrnstr. 59, D-15230 Frankfurt/Oder (Germany) Phone: ++49 / 335 55 34 29 46 Fax: ++49 / 335 55 34 29 59 E-mail: badunenko@uni-ffo.de (corresponding author) Prof. Dr. Michael Fritsch† Technical University Bergakademie Freiberg Faculty of Economics and Business Administration Lessingstraße 45, D-09596 Freiberg (Germany) Phone: ++49 / 3731 / 39 24 39 Fax: ++49 / 3731 / 39 36 90 E-mail: michael.fritsch@tu-freiberg.de Prof. Dr. Andreas Stephan European University Viadrina and German Institute for Economic Research (DIW–Berlin) Königin Luise Str. 5, D-14195, Berlin (Germany) E-mail: astephan@euv-ffo.de.

† German Institute for Economic Research (DIW) Berlin, and Max-Planck Institute for Research into Economic Systems, Jena, Germany.

The research on this project has benefited from the comments of participants of the Royal Economic Society Conference (2005), 3d International Industrial Organization Conference (2005), the 10th Spring Meeting of Young Economists (2005), and the IX European Workshop on Efficiency and Productivity Analysis (2005).

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ISSN 0949-9970 The Freiberg Working Paper is a copyrighted publication. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, translating, or otherwise without prior permission of the publishers.

Coordinator: Prof. Dr. Michael Fritsch All rights reserved.

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I Contents

**Abstract**

/ Zusammenfassung

1. Introduction

2. Allocative efficiency measurement

2.1 Traditional approach to allocative efficiency measurement

2.2 A new approach to allocative efficiency measurement

3. Monte-Carlo simulation

3.1 Empirical implementation of the traditional approach

3.2 Empirical implementation of the new approach

3.3 Design of the Monte-Carlo experiments

3.4 Results

4. Empirical illustration of the new approach

4.1 Data

4.2 Results

5. Conclusions

References

II Abstract The traditional approach to measuring allocative efficiency is based on input prices, which are rarely known at the firm level. This paper proposes a new approach to measure allocative efficiency which is based on the output-oriented distance to the frontier in a profit – technical efficiency space – and which does not require information on input prices. To validate the new approach, we perform a Monte-Carlo experiment which provides evidence that the estimates of the new and the traditional approach are highly correlated. Finally, as an illustration, we apply the new approach to a sample of about 900 enterprises from the chemical industry in Germany.

JEL-classification: D61, L23, L25, L65 Keywords: Allocative efficiency, data envelopment analysis, frontier analysis, technical efficiency, Monte-Carlo study, chemical industry.

Der traditionelle Ansatz zur Messung allokativer Effizienz erfordert Informationen über Input-Preise der Unternehmen, die allerdings nur selten vorliegen. In diesem Aufsatz schlagen wir eine neue Methode zur Bestimmung der allokativen Effizienz vor, der als wesentliche Information den Abstand eines Unternehmens von der Effizienz-Grenze nutzt und keine Information über Input-Preise erfordert. Ein Monte-Carlo Experiment zur Überprüfung der Tragfähigkeit dieses Ansatzes zeigt, dass die Schätzwerte nach der traditionellen Methode und dem von uns vorgeschlagenen Verfahren eng miteinander korreliert sind. Zur Illustration wenden wir den neuen Ansatz auf ein Sample von 900 Unternehmen der Chemischen Industrie in Deutschland an.

JEL-Klassifikation: D61, L23, L25, L65 Schlagworte: Allokative Effizienz, Data Envelopment Analysis, Frontier Analysis, Technische Effizienz, Monte-Carlo Methode, Chemische Industrie.

1. Introduction A significant number of empirical studies have investigated the extent and determinants of technical efficiency within and across industries (see Alvarez and Crespi (2003), GumbauAlbert and Maudos (2002), Caves and Barton (1990), Green and Mayes (1991), Fritsch and Stephan (2004a)). Comprehensive literature reviews of the variety of empirical applications are made by Lovell (1993) and Seiford (1996, 1997). Compared to this literature, attempts to quantify the extent and distribution of allocative efficiency are relatively rare (for a survey, see Greene (1997)).1 This is quite surprising since allocative efficiency has traditionally attracted the attention of economists: what is the optimal combination of inputs so that output is produced at minimal cost? How much could the profits be increased by simply reallocating resources? To what extent does competitive pressure reduce the heterogeneity of allocative inefficiency within industries?2 A firm is said to have realized allocative efficiency if it is operating with the optimal combination of inputs. The traditional approach to measuring allocative efficiency requires input prices (see Atkinson and Cornwell (1994), Green (1997), Kumbhakar (1991), Kumbhakar and Tsionas (2005), Oum and Zhang (1995)) which are hardly available in reality.3 This explains why empirical studies of allocative efficiency are highly concentrated on certain industries, particularly banking, because information on input price can be obtained for these industries.

This paper introduces a new approach to estimating allocative efficiency, which is solely based on quantities and profits and does not require information on input prices. An indicator for allocative efficiency is derived as the output-oriented distance to a frontier in a profit-technical efficiency space. What is, however, needed is an assessment of input-saving technical efficiency; i.e., how less input could be used to produce given outputs.

For studies in the financial sector, see the review by Berger and Humphrey (1997) and also Topuz et al. (2005), Färe et al.

(2004), Isik and Hassan (2002). Some studies have been performed for the agricultural sector (e.g., Coelli et al., (2002), Chavas et al., (1993, 2005), Grazhdaninova (2005)). Studies for manufacturing sector are relatively rare (e.g., Burki (1997), Kim and Han (2001)).

Moreover, allocative efficiency is also import for the analysis of the production process; e.g., to estimate the bias of (i) the cost function parameters, (ii) returns to scale, (iii) input price elasticities, and (iv) cost-inefficiency (Kumbhakar and Wang, forthcoming) or to validate the aggregation of productivity index (Raa (2005)).

This includes retrieving allocative efficiency using shadow prices (see Green (1997), Lovell (1993)).

The paper proceeds as follows: section 2 theoretically derives a new method for estimating allocative efficiency and introduces a theoretical framework for activity analysis models.

Section 3 presents the results of the Monte-Carlo experiment on comparison of allocative efficiency scores calculated using both traditional and new approaches. Section 4 provides a rationale and a simple illustration using the new approach; section 5 concludes.

2. Allocative efficiency measurement

2.1 Traditional approach to allocative efficiency measurement A definition of technical and allocative efficiency was made by Farrell (1957). According to this definition, a firm is technically efficient if it uses the minimal possible combination of inputs for producing a certain output (input orientation). Allocative efficiency, or as Farrell called it price efficiency, refers to the ability of a firm to choose the optimal combination of inputs given input prices. If a firm has realized both technical and allocative efficiency, it is then cost efficient (overall efficient).

Figure 1, similarly to Kumbhakar and Lovell (2000), shows firm A producing output yA represented by the isoquant L(yA). Dotted lines are the isocosts which show level of expenditures for a certain combination of inputs. The slope of the isocosts is equal to the ratio of input prices, w(w1,w2). If the firm is producing output yA with the factor combination xA (a in Figure 1), it is operating technically inefficient. Potentially, it could produce the same output contracting both inputs x1 and x2 (available at prices w), proportionally (radial approach); the smallest possible contraction is in point b, representing (θxA) a factor combination. Having reached this point, the firm is considered to be technically efficient.

Formally, technical efficiency is measured by the ratio of the current input level to the lowest attainable input level for producing a given amount of output. In terms of Figure 1, technical inefficiency of unit xA is given by

or geometrically by oc/oa. Thus, cost inefficiency is the ratio of expenditures at xE to expenditures at xA while technical efficiency is the ratio of expenditures at (θxA) to expenditures at xA. The remaining portion of the cost efficiency is given by the ratio of expenditures at xE to expenditures at (θxA). It is attributable to the misallocation of inputs

**given input prices and is known as allocative efficiency:**

or in terms of Figure 1 is given by oc/ob.

2.2 A new approach to allocative efficiency measurement When input prices are available, allocative efficiency in the pure Farrell sense can be calculated using, for example, a non-parametric frontier approach (Färe et al., 1994) or a parametric one (Greene (1997) among others). However, if input prices are not available these approaches are not applicable. In contrast to this, the new approach we propose allows measuring allocative efficiency without information on input prices. An estimate of allocative efficiency can be obtained with the new approach that is solely based on information on input and output quantities and on profits.

The first step of this new approach involves the estimation of technical efficiency; whereby, in the second step allocative efficiency is estimated as an output-oriented distance to the frontier in a profit-technical efficiency space.

Proposition 1: Existence of the frontier in profit-technical efficiency space. A profit maximum exists for any level of technical efficiency.

In Figure 2, three firms, A, B, and C using inputs xA, xB, and xC, available at prices w,4 produce output yA, which is measured by the isoquant L(yA). For the sake of argument, firms A, B, and C are all equally technically efficient (the level of technical efficiency θ, however, is Let us assume that the ratios of input prices are equal for each firm. This assumption is needed to have the isocosts parallel

arbitrarily chosen) which is read from expenditure levels at (θxA), (θxB), and at (θxC), respectively. In geometrical terms obA/oaA = obB/oaB = obC/oaC. The costs of these three firms are determined by wxA, wxB, and by wxC. The isocost corresponding to expenditures at xC is the closest possible to the origin o for this level of technical efficiency and, therefore, implies the lowest level of cost. This is because xC is the combination of inputs lying on the ray from origin and going through the tangent point of the isocost (corresponding to expenditure level of wxE) to the isoquant L(yA). This implies that for θ-level of technical efficiency costs have a lower bound and using the fact that firms are producing the same output yA, profits have an upper bound. Without loss of generality, for each level θ of technical efficiency there is a profit maximum, which proves the existence of a frontier in profit—technical efficiency space.

Remark 1: Frontier in profit—technical efficiency space is sloped upwards.

Figure 3: Relationship between technical efficiency and profit In Figure 3, two firms, C and D, use inputs xC and xD to produce output yA, which is measured by the isoquant L(yA). Both firms are allocatively efficient because they lie on the same ray from the origin that goes through the tangent point xE; thus, in terms of proposition 1 we only look at the frontier points. These firms operate, however, at different levels of technical efficiency θC and θD, respectively. Since the isocost representing the level of expenditure wxC is closer to the origin than that of the expenditure level wxD, costs of firm C are smaller than those of firm D and firm C is more profitable than firm D. Since obC/oaCobC/oaD, θC θD, larger technical efficiency is associated with larger profits for points forming the frontier in profit-technical efficiency space. This proves that such frontier is upward sloping.

Proposition 2: The higher the allocative efficiency the higher the profit. For any arbitrarily chosen level of technical efficiency, the closer the input combination to the optimal one (i.e., the larger the allocative efficiency) the larger the profit will be.