An Implicit Törnqvist Index of Real GDP

Recently much attention has been devoted to superlative indexes in the context of the national accounts. In this paper we advocate the use of the implicit Törnqvist quantity index to measure real GDP. This index, which has been proposed by Diewert and Morrison (1986), has never received serious consideration in the literature. Yet, compared to the better-known Fisher index, the implicit Törnqvist index of real GDP has a number of advantages. Thus, it can be shown to be exact for the Translog GDP function, it allows for a complete multiplicative decomposition of nominal and real GDP, and it is consistent with state-of-the-art measures of total factor productivity that typically rely on the Törnqvist aggregation. Estimates for a sample of 26 countries are reported. We find that the Laspeyres quantity index still used by the statistical agencies of most countries tends to underestimate real growth. Over the 1960–1996 period, the cumulated shortfall was as much as 13.4% of GDP in the case of Japan.     

A superlative index number formula for the Hicks-Moorsteen productivity index

The Malmquist and Hicks-Moorsteen productivity indexes are the two most widely used theoretical indexes for measuring productivity growth. Since these productivity indexes are defined by unknown distance functions, it is necessary to estimate the distance functions to compute them in principle. On the other hand, the Törnqvist productivity index is an empirical index number formula that is directly computable from the prices and quantities of the inputs and outputs alone. Caves et al. (1982) imply that the Malmquist index coincides with the Törnqvist index under profit maximizing behaviour and constant returns to scale technology. The purpose of the present paper is to point out that the Hicks-Moorsteen productivity index coincides with the Törnqvist productivity index under the same condition. We emphasize that the condition of constant returns to scale is indispensable for deriving the equivalence between the two indexes. Moreover, even when this condition is relaxed to the α returns to scale, the equivalence between the Hicks-Moorsteen and Törnqvist productivity indexes is shown to hold true.     

Decomposing profit change: Konüs,Bennet and Luenberger indicators

We introduce complementary decompositions of profit change that, relying on the duality between the profit function and the directional distance function, shed light on the different sources of profit growth including measures of technical efficiency, allocative efficiency and technological change. Our decompositions extend the literature on Konüs and Bennet quantity and price indicators to profit change. The first decomposition is ‘exact’ in the sense of Diewert, by completely exhausting the sources of profit change into profit inefficiency change (including technical and allocative inefficiency change), technological change, and output and input price change. The second decomposition equates the Bennet quantity indicator to a productivity measure represented by the Luenberger indicator plus allocative inefficiency change. We deem it ‘complete’ because in contrast to the existing literature, it retains the information on allocative inefficiency change while preventing the existence of residual terms capturing price variations, whose meaningful interpretation has not been addressed until now. Our proposed solution takes advantage of the flexibility of the directional distance function when choosing a suitable directional vector. All decompositions have the same structural form and therefore their components can be compared to each other vis-à-vis, providing alternative measures of equivalent sources of profit growth.     

Revisiting the decomposition of cost efficiency for non-homothetic technologies: a directional distance function approach

In the early 1980’s Kopp and Diewert proposed a popular method to decompose cost efficiency into allocative and technical efficiency for parametric functional forms based on the radial approach initiated by Farrell. We show that, relying on recently proposed homogeneity and duality results, their approach is unnecessary for self-dual homothetic production functions, while it is inconsistent in the non-homothetic case. By stressing that for homothetic technologies the radial distance function can be correctly interpreted as a technical efficiency measure, since allocative efficiency is independent of the output level and radial input reductions leave it unchanged, we contend that for non-homothetic technologies this is not the case because optimal input demands depend on the output targeted by the firm, as does the inequality between marginal rates of substitution and market prices—allocative inefficiency. We demonstrate that a correct definition of technical efficiency corresponds to the directional distance function because its flexibility ensures that allocative efficiency is kept unchanged through movements in the input production possibility set when solving technical inefficiency, and therefore the associated cost reductions can be solely—and rightly—ascribed to technical-engineering-improvements. The new methodology allowing for a consistent decomposition of cost inefficiency is illustrated resorting to simple examples of non-homothetic production functions.     

Fisher ideal output,input, and productivity indexes revisited

A productivity index for a firm is generally defined as an output index divided by an input index. The first part of the paper uses the test or axiomatic approach to index number theory in order to determine the appropriate functional form for the output and input indexes. It is found that the Fisher ideal index satisfies 21 reasonable tests and is uniquely characterized by a subset of these tests. In the remainder of the paper, the economic approach to productivity indexes introduced by Caves, Christensen, and Diewert is adopted and, again, a strong justification for the Fisher productivity index is provided.The editor of this paper was N.R. Adam.This research was supported by a Strategic Grant from the Social Science and Humanities Research Council of Canada. The author is indebted to Shelley Hey for typing assistance.     

Index Number Concepts,Measures and Decompositions of Productivity Growth

This paper explores the definitions and properties of total factor productivity growth (TFPG) indexes, focusing especially on the Paasche, Laspeyres, Fisher, Törnqvist, and implicit Törnqvist ones. These indexes can be evaluated from observable price and quantity data, and certain of these are shown to be measures of TFPG concepts and theoretical indexes that have been proposed in the literature. The mathematical relationships between these and quantity aggregates, financial measures, and price and quantity indexes are explored. Decompositions of the productivity growth indexes are also given. The paper concludes with a brief overview of some limitations on our analysis.     

Incorporating ancillary measures of process and quality change into a superlative productivity index

One of the persistent problems plaguing the measurement of productivity and output is accounting for changes in product quality. A similar problem arises in attempting to explain shifts in a production function using information on changes in the characteristics of the production process itself. We consider these problems under a behavioral model in which the firm chooses a profit-maximizing bundle of input/output/process characteristics as well as the profit maximizing levels of input and output. This view of quality change is similar to the endogenous design index advocated by Triplett [1983] for industrial prices and the endogenous quality indexes analyzed by Pollak [1983] for consumer prices. We show how a price-characteristics locus can be used to adjust the Tornqvist output- and input-oriented multifactor productivity indexes of Caves, Christensen and Diewert [1982] for changes in input, output and process characteristics. To show the applicability of the methodology to services, we apply the results in the framework of the commercial banking measurement of Fixler [1988] to measure the impact of bank branching on multifactor productivity.The refereeing process of this paper was handled through R. Färe.     

An aggregate quantity framework for measuring and decomposing productivity change

Total factor productivity (TFP) can be defined as the ratio of an aggregate output to an aggregate input. This definition naturally leads to TFP indexes that can be expressed as the ratio of an output quantity index to an input quantity index. If the aggregator functions satisfy certain regularity properties then these TFP indexes are said to be multiplicatively complete. This paper formally defines what is meant by completeness and reveals that (1) the class of multiplicatively complete TFP indexes includes Laspeyres, Paasche, Fisher, T?rnqvist and Hicks-Moorsteen indexes, (2) the popular Malmquist TFP index of Caves et al. (Econometrica 50(6):1393?C1414, 1982a) is incomplete, implying it cannot always be interpreted as a measure of productivity change, (3) all multiplicatively complete TFP indexes can be exhaustively decomposed into measures of technical change and efficiency change, and (4) the efficiency change component can be further decomposed into measures of technical, mix and scale efficiency change. Artificial data are used to illustrate the decomposition of Hicks-Moorsteen and Fisher TFP indexes.     

Productivity trends in U.S. manufacturing: Evidence from the NQ and AIM cost functions

In this paper, we take the econometric approach to productivity measurement in United States manufacturing, using KLEM data over the period from 1953 to 2001. We are also interested in technical change bias, price elasticities, and elasticities of substitution in the U.S. manufacturing industry. We present an empirical comparison and evaluation of the effectiveness of four well-known flexible cost functions—the locally flexible generalized Leontief (see Diewert [1971. An application of the Shephard duality theorem: a generalized Leontief production function. Journal of Political Economy 79, 481–507]), translog (see Christensen et al. [1975. Transendendal logarithmic utility functions. American Economic Review 65, 367–364]), and normalized quadratic (see Diewert and Wales [1987. Flexible functional forms and global curvature conditions. Econometrica 55, 43–68])—and the globally flexible asymptotically ideal model (see Barnett et al. [1991. Semi-nonparametric Bayesian estimation of the asymptotically ideal production model. Journal of Econometrics 49, 5–50]), the latter modified to introduce technical change by means of Thomsen's [2000. Short cuts to dynamic factor demand modelling. Journal of Econometrics 97, 1–23] factor-augmenting efficiency index approach.     

A primal Divisia technical change index based on the output distance function

We derive a primal Divisia technical change index based on the output distance function and further show the validity of this index from both economic and axiomatic points of view. In particular, we derive the primal Divisia technical change index by total differentiation of the output distance function with respect to a time trend. We then show that this index is dual to the Jorgenson and Griliches (1967) dual Divisia total factor productivity growth (TFPG) index when both the output and input markets are competitive; dual to the Diewert and Fox (2008) markup-adjusted revenue-share-based dual Divisia technical change index when market power is limited to output markets; dual to the Denny et al. (1981) and Fuss (1994) cost-elasticity-share-based dual Divisia TFPG index when market power is limited to output markets and constant returns to scale is present; and also dual to a markup-and-markdown-adjusted Divisia technical change index when market power is present in both output and input markets. Finally, we show that the primal Divisia technical change index satisfies the properties of identity, commensurability, monotonicity, and time reversal. It also satisfies the property of proportionality in the presence of path independence, which in turn requires separability between inputs and outputs and homogeneity of subaggregator functions.     

Luenberger Productivity Indicators: Aggregation Across Firms

Two methods of aggregating Luenberger productivity indicators across firms are considered. One method makes use of some rather implausible allocative efficiency assumptions. The second method, a superlative index number approach, relies on more palatable assumptions and is judged to be the more promising of the two methods.     

A Decomposition of Total Factor Productivity Growth in Korean Manufacturing Industries: A Stochastic Frontier Approach

This paper applies a stochastic frontier production model to Korean manufacturing industries, to decompose the sources of total factor productivity (TFP) growth into technical progress, changes in technical efficiency, changes in allocative efficiency, and scale effects. Empirical results based on data from 1980–1994 show that productivity growth was driven mainly by technical progress, that changes in technical efficiency had a significant positive effect, and that allocative efficiency had a negative effect. This study suggests that specific guidelines are required to promote productivity in each industry, and provides additional insight into understanding the recent debate on TFP growth in Korean manufacturing.     

Theory and Application of Directional Distance Functions

In 1957 Farrell demonstrated how cost inefficiency could be decomposed into two mutually exclusive and exhaustive components: technical and allocative inefficiency. This result is consequence of the fact that—as shown by Shephard—the cost function and the input distance function (the reciprocal of Farrell's technical efficiency measure) are dual to each other. Similarly, the revenue function and the output distance function are dual providing the basis for the decomposition of revenue inefficiency into technical and allocative components (see for example, Färe, Grosskopf and Lovell (1994)). Here we extend those results to include the directional distance function and its dual, the profit function. This provides the basis for defining and decomposing profit efficiency. As we show, the output and input distance functions (reciprocals of Farrell efficiency measures) are special cases of the directional distance function. We also show how to use the directional distance function as a tool for measuring capacity utilization using DEA type techniques.     

Economic Efficiency and Productivity Growth in the Post-Privatization Chilean Hydroelectric Industry

Chiles hydroelectric industry was privatized in 1985, but required to operate within a regulatory framework designed to achieve a competitive outcome. A centralized dispatch center was established to ensure production at minimum cost, subject to constraints on minimum release and minimum reservoir stock. A reluctance to rapidly reduce the industry work force may also have existed. We develop a constrained cost-minimization model for thermal and hydro generation to obtain the shadow price of water and to determine the qualitative effect of these constraints on allocative efficiency. Using panel data from 1986–1997, we assess the economic efficiency of the hydro industry by estimating a stochastic distance frontier and price equations from the dual cost-minimization problem. We find dramatic increases in technical change and productivity change, with positive efficiency change for all years but the last. We also observe a dramatic decline in allocative inefficiencies over our sample period. The share of hydro generation from run-of-river and thermal plants relative to reservoir plants has increased, presumably in reaction to the water release and reservoir stock constraints, reducing the relative over-utilization of capital to water from the pre-1985 regime. Further, the over-utilization of labor to capital and water has fallen over time. However, considerable allocative inefficiencies remain, consistent with our finding of industry-wide scale economies. Substantial cost savings would result if technical and allocative efficiency were eliminated.JEL Classification: L94, D24     

Estimating DEA technical and allocative inefficiency using aggregate cost or revenue data

In this paper, we address the question of Data Envelopment Analysis (DEA) evaluation of efficiency when aggregate cost or revenue data must be used. We show that the DEA technical inefficiency measure using total revenues as the single output variable or total costs as the single input variable equals the aggregate technical and allocative inefficiency. We employ this result to estimate allocative inefficiency and construct statistical tests of the null hypothesis of no allocative inefficiency analogous to those of the null hypothesis of no scale inefficiency. We illustrate our method using revenue and personnel data for the top U.S. public accounting firms over 1995–1998. Our empirical results indicate the existence of statistically significant allocative inefficiency in the public accounting industry.

A general approach to intertemporal and interspatial productivity comparisons

The purpose of this paper is to introduce a general methodology for analysing the sources of intertemporal or interspatial differences in outputs and costs, general in the sense that our methodology allows the productivity analyst to ‘break out’ of the quadratic ‘straightjacket’ imposed by the class of superlative index number comparisons. Starting fromTaylor's series expansions about the two points to be compared, we develop a general growth accounting equation which can be approximated to any desired degree of accuracy, depending on the information available.The theoretical framework is applied to two recent examples of interspatial comparisons which use the Tornqvist superlative index. In the first example, we show that the biases in regional Canadian total manufacturing cost-efficiency comparisons which result from the use of this index are negligible. However, in the second example, it is shown that the Tornqvist index imparts a substantial bias in United States-Japan total domestic economy productivity comparisons. The index consistently overestimates the relative productivitylevel of the U.S. economy and misses the turning point, when the Japanese economy becomes more efficient, by two years.     

On the Determinants of School District Efficiency: Competition and Monitoring

A number of researchers have asserted that inefficiency in the U.S. school system arises from a lack of incentives for public schools to behave efficiently. This paper uses a Shephard input distance function to model educational production, and a switching-regressions estimation to explore the relationship between school district efficiency and two existing incentive mechanisms—competition and voter monitoring. We find evidence that ease of monitoring enhances both technical and allocative efficiency of urban school districts, and that increased competition reduces allocative inefficiency in communities above a competitive threshold. We find no evidence that competition is related to technical inefficiency.     

Econometric estimation of distance functions and associated measures of productivity and efficiency change

Multi-input multi-output production technologies can be represented using distance functions. Econometric estimation of these functions typically involves factoring out one of the outputs or inputs and estimating the resulting equation using maximum likelihood methods. A problem with this approach is that the outputs or inputs that are not factored out may be correlated with the composite error term. Fernandez et al. (J Econ 98:47–79, 2000) show how to solve this so-called ‘endogeneity problem’ using Bayesian methods. In this paper I use the approach to estimate an output distance function and an associated index of total factor productivity (TFP) change. The TFP index is a new index that satisfies most, if not all, economically-relevant axioms from index number theory. It can also be exhaustively decomposed into a measure of technical change and various measures of efficiency change. I illustrate the methodology using state-level data on U.S. agricultural input and output quantities (no prices are needed). Results are summarized in terms of the characteristics (e.g., means) of estimated probability density functions for measures of TFP change, technical change and efficiency change.     

A note on the decomposition of cost efficiency into technical and allocative components

In this note a simpler method of cost efficiency decomposition than that of Kopp and Diewert (1982) is proposed. The method is obtained by application of a duality theorem for cost and distance functions.     

Eliminating chain drift in price indexes based on scanner data

The use of scanner data in the CPI makes it possible to compile superlative price indexes at detailed aggregation levels since prices and quantities are available. A potential drawback is the high attrition rate of items. The usual solution to handle this problem, high-frequency chaining, can create drift in the index series due to price and quantity bouncing arising from sales. Ivancic, Diewert and Fox (2009) have recently proposed an approach that provides drift free, superlative-type indexes through adapting multilateral index number theory. In this paper we apply their proposal to seven product groups and find promising results. We compare the results with those obtained by using the Dutch method to deal with supermarket scanner data.