Superlative Index Research Articles

An alternative estimate of real inventory change for national economic accounts

In 1996, the Bureau of Economic Analysis adopted the Fisher index (a particular superlative index) for the national income and product accounts of the United States (NIPAs). This change was, on balance, a major improvement in the US national accounts, for it greatly improved the measurement of rates of growth of real economic aggregates. However, the change posed a special problem for inventories: the Fisher index is not suitable for time series that, like inventory change, exhibit changes in sign. The Fisher quantity relative is the geometric mean of Laspeyres and Paasche quantity relatives. When some detailed components of inventory change are negative, the Laspeyres and Paasche can easily differ in sign, in which case the Fisher index is undefined. In the NIPAs, this problem is avoided by measuring real-inventory change as the period-to-period change in inventory stocks. However, attempts to approximate real GDP by two-stage aggregation, combining this measure of inventory change and other major GDP subaggregates, show that the NIPA measures of real inventory change and of real GDP are inconsistent. This paper attempts to resolve the problem by investigating an alternative measure of inventory change.The alternative measure of inventory change considered is the difference between a Fisher index of inventory acquisitions and a Fisher index of inventory disposals. The consistency of this measure with real GDP is first examined analytically. It is shown that, for inventory changes likely to be observed, the approximation error in the aggregation of GDP components due to measuring inventory change as the difference between Fisher indexes of inventory acquisitions and disposals should be small. Second, the consistency of the alternative measure with real GDP is tested over the period 1977–98. The tests compare the actual approximation errors in the two-stage aggregation of GDP using inventory change based on the two methods: the present NIPA methodology and the alternative, Fisher difference method. These tests provide further support for the alternative method of estimating real inventory change.

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 Tornqvist 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 Tornqvist 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 Tornqvist 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.

Chained, exact and superlative hedonic price changes: estimates from microdata

Scanner data are used to calculate chained, exact (and superlative) hedonic price indexes for television sets. The data source is available for a wide range of goods, the application providing an example of how this method can be more widely applied. The indexes correspond to constant utility, hedonic cost-of-living indexes. The approach improves on the existing direct method, which takes its estimates directly from the coefficients on time dummies in a hedonic regression. It also improves on the matched model method used by statistical agencies. The differences between actual price changes and exact hedonic quality-adjusted price changes are found to be substantial. Base-period and current-period weighted exact hedonic indexes are similar, thus providing good approximations to a superlative index. Estimates from the direct, dummy variable approach were compared to the superlative indexes. The disparities between the results argue for caution in the use of the direct, dummy variable approach to estimating quality-adjusted price changes.

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.

The quadratic approximation lemma and decompositions of superlative indexes

It was shown in 1976 that a difference in a quadratic function of N variables evaluated at two points is exactly equal to the sum of the arithmetic average of the first order partial derivatives of the function evaluated at the two points times the differences in the independent variables. In the present paper, this result is generalized and the resulting generalized quadratic approximation lemma is used to establish all of the superlative index number formulae that were derived in Diewert [4]. In addition, some new exact decompositions of the percentage change in the Fisher and Walsh superlative indexes into N components are derived. Each component in this decomposition represents the contribution of a change in a single independent variable to the overall percentage change in the index. Finally, these components are given economic interpretations.

The measurement of inflation: Aggregation at the basic level

The advent of bar-code, retail scanner data provides an alternative data source for the compilation of consumer price indexes. This paper outlines the nature and merits of such data for this purpose. The data allows aggregation via superlative index number formulae at a very elementary level and provides superior coverage to conventional sources. However, it has been argued that aggregation bias may still arise if the unit values which feed into the superlative indexes are defined for product items, as opposed to product items for given outlets. We explore for six products the nature and extent of unit value bias by outlet type. For three products we found the level of aggregation did not matter. However, we also found it can matter and established the extent of its effects which were always the aggregation by model only, falling less than by model and outlet. This should help provide a basis for an understanding of the issues at stake in the use of scanner data for the compilation of consumer price indexes.

Evaluating Imperfections and Biases in Price Indexes During Transition

The rapidly changing environment of the transition may create special problems for calculation of index numbers that require a fixed basket of goods and retail outlets. Using referent-level data we find that fixed-weight Laspeyres index on average overstated cost of living increases by approximately 5 per cent a year when compared with a superlative index in the Czech Republic. This difference is smaller than might be expected given the large changes in relative prices that occurred during transition and suggests that consumer substitution impacts may have been largely offset by other factors, especially rising prices combined with increased consumption of some goods as artificial shortages under communism were removed. Indeed, in the period of greatest supply response to price liberalization, the Laspeyres index appears to understate increases in the cost of living.

Economic Measurement: Progress and Challenges

The national income accounts, together with the source data which they use, form the core of our economic measurement system. The development of the concepts and measurements of national income are among the most important achievements of modern economics, replacing the older notions of national wealth of the physiocrats and mercantilists, for example. Is national income and product, confined to market activity, the appropriate fundamental concept? Yes, but we ask a lot of national income and product accounting. It is used, most importantly, to gauge economic progress. How much better off are we than our parents or grandparents? Is the economy growing adequately, slowly, or rapidly in recent years? And, are we at a cyclical turning point? The information requirements, of course, will differ depending on the question. Gaps in inventory measurement may be an appropriate resource claimant to those interested in cyclical issues whereas finding, identifying, and incorporating new products and quality change more rapidly may be far more significant to long-term economic growth. We also tend to have a survivor selection bias in our concern with the speed with which new ideas and concepts are incorporated into the national accounts and other data series. Superlative indexes, hedonic regression methods for quality change, and many others were much more obvious after some years of diffusion throughout the profession than at their inculcation. As a result, the gap between major suggested improvements and their adoption by statistical agencies can be extensive, often decades (Erwin Diewert, 1999). Should national accounts be extended to nonmarket activity? Nonmarket work time is perhaps the most important area for supplemental information. Natural resources and the environment are close behind (William D. Nordhaus and Edward Kokkenlenberg, 1999). R&D capital, environmental capital, human capital, and other intangibles are high-priority items but should be dealt with in the form of satellite accounting, so I applaud the BEA’s recent initiatives in this direction. I come to this conclusion for a variety of reasons, not to suggest that technological capital and human capital are less important than tangible capital. In fact, perhaps quite the opposite, they are all complementary (Boskin and Lawrence Lau, 2000), but the issues in definition, classification, and measurements are still more problematic for these items than for traditional items, and therefore I believe they should not be added directly to the National Income and Product Accounts (NIPA). In short, my own view is that substantial improvements can and should be made in the traditional domain of the national accounts. Improvements in the core national accounts should not be sacrificed to a proliferation of satellite accounts. In principle, we should do both, and sufficient resources should be provided to accomplish these tasks.

An Overview of Experimental U.S. Consumer Price Indexes

The Consumer Price Index (CPI) has come under intense scrutiny in recent years. To help assess potential differences between what the CPI measures and how it is used for various purposes, the Bureau of Labor Statistics has created a series of experimental price indexes. This article contrasts how the CPI is currently constructed with those experimental indexes. Specifically, experimental CPPs have been constructed for two population groups, older Americans and poor Americans. In addition, experimental geometric-mean and superlative indexes have been developed that move the CPI away from its traditional “fixed-market-basket” concept.

Re-examining the Cost-of-Living Index and the Biases of Price Indices

The U.S. CPI is based on the Laspeyres price index, an index type that has an upward Thus, the CPI tends to overstate increases in the cost of living. To address this bias, the Advisory Commission to Study the Consumer Price Index recommended adopting for the CPI a price index, e.g., the Fisher or Tornqvist indices. Under the assumption of homothetic preferences, superlative indices always have smaller substitution biases--hence, are closer to the cost-of-living index (COLI)--than the Laspeyres index, but this assumption implies that: all elasticities equal 1, the true COLI is independent of the utility level (standard of living), and expenditure shares are unaffected by changes in income. These implications are contradicted, however, by all known household budget studies. Therefore, superlative indices are not necessarily closer to the true COLI than the Laspeyres index except in the unrealistic case of homothetic preferences. Under more realistic non-homothetic preferences, expenditure shares vary with and, thus, income is introduced into the superlative indices. This, in turn, could result in biases larger than the Laspeyres substitution bias in the CPI. The Commission did not, however, address this possibility. The Laspeyres index has a larger substitution bias but no bias because it uses fixed expenditure shares. Under plausible conditions, by using a non-homothetic almost ideal demand system (AIDS) model, we carry out empirical simulations that show that the combined substitution and biases of either the Fisher or the Tornqvist index could be either positive or negative--that is, a superlative index could differ even more from the true COLI than is the case for the present CPI. Thus, adjustments resulting from a CPI based on a superlative index could exceed those from using the current CPI, when the combined bias is positive, or could fall below those warranted by the true COLI in the case of negative combined bias. Therefore, the Commission's recommendation for a superlative index CPI formula needs further examination. We propose a theoretically rigorous and practical procedure to determine a COLI free from substitution and bias, using estimated ordinary demand functions without postulating a specific structure of consumer preferences.

Measuring Effective Protection as a Superlative Index Number: An Application to European Agriculture

AbstractIn this study duality and the theory of superlative index numbers are used to obtain a theoretically consistent Effective Protection Rate (EPR) index that imposes no a priori restrictions on input‐output substitution possibilities. Within this framework, EPRs are estimated for European agriculture over the period 1973 to 1989. The empirical results demonstrate that maintaining zero input/output substitutability—a restriction embedded in all common protection measures—can lead to false results about the level of protection both across countries and time.

Productivity growth and price trends in the North American sawmilling industries: an inter-regional comparison

This paper applies nonparametric superlative index techniques to measure productivity growth in the sawmilling industries of the United States and Canada. Six geographic regions are examined: British Columbia (Coast and Interior), Ontario, Quebec, U.S. South, and U.S. West. The results indicate significant adjustment of resources both within and across regional industries over time. Over the long-term, labor has been the input that has experienced the highest growth in productivity: 3–4% per annum in the commodity-oriented regions. This result likely reflects the significant increases in capital stock throughout most of the North American industry. From 1980 to 1988 there have been significant differences in the annualized growth rates in total factor productivity across regions: U.S. West (3.3%), B.C. Interior (2.7%), Quebec (1.9%), U.S. South (1.4%), B.C. Coast (1.3%), and Ontario (1.1%). However, growth in total factor productivity over the 24-year period from 1965 to 1988 is relatively uniform across most regions (1.2% per annum).

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.

PROMISES, PROMISES: THE STATES' EXPERIENCE WITH INCOME TAX INDEXING

The paper discusses five early approaches to the price (and quantity) index number problem. The five approaches are: (1) the fixed basket approach; (ii) the statistical approach; (iii) the test or axiomatic approach; (iv) the Divisia approach and (v) the economic approach. The economic approach makes use of the assumption of optimizing behavior under constraint and the approach is discussed under four subtopics: (i) basic theoretical definitions; (ii) the theory of bounds; (iii) exact index numbers and (iv) econometric estimation of preferences. The paper also discusses several topics raised by Jack Triplett in a recent paper, including: (i) the merits of the test approach to index number theory, (ii) the chain principle and alternatives to it; (iii) the substitution bias and (iv) the new good bias. Although the paper is for the most part an extensive historical survey, there are a few new results in section 8 on multilateral alternatives to the chain principle. Also in section 6.3, it is shown that the Paasche, Laspeyres and all superlative indexes will satisfy the circularity test to the first order.

An Analysis of Substitution Bias in Measuring Inflation, 1959-85

We analyze the substitution bias in Laspeyres type price indexes, such as the CPI, using 1959-1985 NIPA consumption data for 101 commodities. We employ two methodological approaches to calculation of the bias. First, we construct the tightest theoretical bounds on the COL by applying nonparametric methods, using algorithms developed by Afriat and by Varian. Second, following Diewert, we construct superlative price index formulas, namely Tomqvist and Fisher's Ideal indexes, under a chain as well as a fixed-base specification. Homothetic preferences are found to be consistent with the data, and sensitivity tests indicate that this result is not vacuous. If the hypothesis of homotheticity is maintained, the COL bias has upper and lower limits of 0.22 and 0.14 percent per year, respectively. Using the superlative indexes as the measure of the COL, the bias is about 0.18 percent per year for the period 1959-85. This estimate is somewhat larger than found in earlier studies. Our use of more disaggregated data is found to be responsible for part of the difference, with the size of the estimated bias directly related to the extent of the aggregation.

Constant-market-shares analysis and index number theory

This paper examines the constant market-shares (CMS) analysis of a country's export growth within the context of index number theory and is aimed at finding a satisfactory solution to the problems encountered by the traditional CMS decomposition procedures. The last ones perform the task of disentangling the variations of each accounting factor only partially, because they are based on linear approximations to non-linear functions. The residual “interaction” term left out in the current CMS analyses does not appear if a more flexible CMS decomposition based on the so-called superlative index numbers is used. Moreover, the discussion of the basic identities has clarified the difference in meaning and levels of the accounting components between the alternative versions of CMS analysis.

Fishing effort: Its testing, specification, and internal structure in fisheries economics and management

The concept of fishing effort is central to fisheries economics and management. However, effort is an aggregate index of inputs which can be consistently formed only under the condition on production technology of homothetic separability of inputs. This paper develops the conditions under which effort can be consistently formed. It then provides the first empirical test for effort and jointness in inputs in a fishery by estimating a multiproduct function for the New England otter trawl fleet. After not rejecting input-output separability and rejecting nonjointness in inputs, the construction of a superlative index for effort is demonstrated through estimating a translog production function. The implications of effort's internal structure for fisheries management are then considered.

Analysis of Productivity at the Firm Level: An Application to Life Insurers

Analysis of at the Firm Level: An Application to Life Insurers: Comment The non-parametric measurement of total factor productivity (TFP) has become widely accepted and applied to many industries [2], and more recently to life insurers. A limitation of this approach is that differences in TFP may arise due to factors other than production efficiency. Denny, Fuss and Waverman [5], for example, identify two additional attributes to differences in TFP across time: exploitation of economies of scale, and changes in deviation from marginal cost pricing. To help overcome these limitations, various researchers employed the parametric technique of neoclassical cost estimation [10]. The parametric approach is a more general and detailed specification of the production structure of an industry or a company than the TFP approach. Further, in most recent studies of this type, researchers have utilized flexible functional forms in order to avoid imposing unnecessary restrictions on the production technology (see for example Berndt and Khaled [1] among many others). This approach is appealing but is often expensive to undertake and requires substantial time series observations. For practitioners, however, TFP analysis has various advantages. TFP is a relative measure showing how the ratio of total output to total input changes from one period to the other. It is relatively inexpensive to perform and since the data are displayed in an index number form, it is easy to identify anomalies in the data. In addition, TFP and the variables used in its construction may reveal valuable information on trends and changes in an industry. Moreover, Diewert and Morrison [7] have added another factor, the terms of trade effect, in explaining productivity. This Comment extends the study of Weiss [13] in which new techniques for measuring output of life insurers are developed and used in computing divisia and exact indexes of TFP for one stock and one mutual insurer over a five-year interval. Weiss [13] maintains that because the rate of technological change calculated using the Tornqvist approximation (TFP) equals the exact measure of the shift of the variable cost function due to technological change ( t) (see Diewert [6]), the nature of returns to scale is constant. In her words, Productivity theory suggests that superlative indexes such as the exact index and the Tornqvist-Theil approximation to the divisia index yield similar results if the production function reflects constant returns to scale, regardless of whether the insurer is acting competitively... [13, p. 74]. Given the result obtained by Weiss, which indicates that the divisia and exact indexes vary directly with each other, it was concluded that ...the sample insurers' production functions exhibited constant returns to scale over the sample periods [13, p. 74]. While this is true if all inputs are variable, if some of the inputs are fixed (e.g. number of square feet of home office building and constant dollar capital input in Weiss), the statement is in error, primarily because the shift in the variable cost function due to technological change ( t) is greater than the negative value of the overall production relationship's rate of technological change ( t). that is, t at t where at is the mean share of variable relative to total costs (at The latter can be proven with the aid of two equations: t = 1/2 (at + at - 1) t + R ( ) where t is the shift in total cost function due to technological change and R ( ) is a remainder term of first differences which are at least of second order. t = - t + R* ( ) where t is the rate of technological change corresponding to the joint production function (of all inputs), and R* ( ) is a remainder term of finite differences which are at least of second order.(1) Now suppose there is a constant mark-up, over marginal cost pricing structure. …

Flexible Consumer Demand Functions and Linear Estimation: Reply

We are grateful to Wohlgenant for pointing out that the approximation of utility by real income requires that the underlying (flexible) utility function be homothetic. Indeed Diewert's proof of correspondence between utility changes and real income changes (as measured by superlative index numbers) relies on homotheticity. We thus concur with Wohlgenant that the three functional forms proposed by us are best regarded as statistical demand functions integrable into an arbitrary neoclassical utility function only at some base point. However, it is not clear that this is the reason for implausible expenditure elasticities obtained in the TL model as we move from sample means. This functional form (and the other two used in the study) provide a flexible approximation to some true consumption-expenditure relationship. It is unlikely to give reasonable estimates of elasticities at expenditure levels which are less than half the mean or nearly twice as high. What is interesting is that elasticities for all the food grains decline as expenditure levels rise.

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.