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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes Appendix Statistical Definition and Estimation of Price Indexes This report addresses foundational economic concepts for cost-of-living or price indexes. In the panel’s view, the concepts must reflect the reality of the marketplace; they must capture the change in real prices paid by real consumers. The concepts must be measurable in the context of a system of surveys and other data collection activities that the Bureau of Labor Statistics (BLS) can feasibly implement. An important step in assessing the measurability and reality of a particular price index concept is to express the concept statistically in the form of a population parameter to be estimated. If one can write down the parameter, one can examine the feasibility of surveys and other data collection activities necessary to produce accurate statistical estimators of the parameters. One can also examine whether the parameter is defined in terms of the prices actually paid by consumers. In what follows, we translate our concepts into explicit population parameters. We define the price indexes motivated by our concepts and demonstrate briefly the survey data required to estimate the indexes. To begin, consider a simple world in which there is only one good and two time periods, base and comparison and a static universe of households (HH), denoted by the set H. For cases in which it would be better to work in terms of subgroups within HHs called consumer units (CU), let H denote the universe of CUs.
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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes Next, let us introduce the bulk of the requisite notation. Let i signify the HH (i = 1, . . . N), j the purchase occasion, Ji0 the set of purchase occasions by the ith HH in base period 0, Jit the set of purchase occasions by the ith HH in comparison period t, Qgij0 the number of units of good g purchased by the ith HH, jth purchase occasion, during base period 0, Qgijt the number of units of good g purchased by the ith HH, jth purchase occasion, during comparison period t, N the number of households in the universe pgij0 price per unit (of good g) paid by the ith HH, jth purchase occasion, during base period 0, and pgijt price per unit (of good g) paid by the ith HH, jth purchase occasion, during comparison period t. In these definitions, we use the convention for nonbuyers in the base period and for nonbuyers in the comparison period. We assume there is at least one buyer, Q0 > 0 and Qt > 0, in each period. Average unit volumes, and , and average prices per unit, and , are defined in the obvious way. The decomposition of the period-to-period trend in total dollar volume is now given by where TN is the trend in the total HH count, Tq is again the trend in average units per HH, and Tp is again the trend in average price per unit. As above, Tp may be called the price index and Tq the unit volume index. We can next further extend the work to a still more realistic world in which a static set of goods is available in the market at both time periods. Let subscript g signify a good, and to simplify the notation let G represent both the set and the number of goods. Total dollar volumes are now defined by
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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes for base and comparison periods, respectively. Average units volumes, and , and average prices per unit, and , are defined in the obvious way. Also, define the G × 1 vectors of average unit volumes and average prices per unit and Then, the period-to-period trend in total dollar volume is given by where TN is again the trend in the total HH count, TPq is the trend in average units per HH, and TLp is the trend in average price per unit. The trend in average units is weighted by comparison prices, and thus one might view TPq as a Paasche index of unit volume. Since the trend in average prices is weighted by base units volume, one might thus view TLp as a Laspeyres price index. An alternative decomposition of the trend is where TLq is a Laspeyres index of units volume and TPp is a Paasche price index. A second alternative decomposition of the trend is
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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes where TFq and TFp are Fisher indexes of unit volume and prices, respectively. Finally, we reach the real world in which both the sets of goods marketed, G0 and Gt, and the sets of households, H0 and Ht, vary by period. Partition the set of goods marketed at the base period by G0=G0Q ∪G and partition the set of goods marketed at the comparison period by Gt = G ∪ GtE, where G0Q denotes exiting goods, GtE denotes entering goods, and G denotes both continuing and linkable goods. Continuing goods are marketed in both time periods, while exiting goods appear in the base period but not in the comparison, and entering goods appear in the comparison but not in the base. There is gray area we have called linkable goods. These are goods for which there is no exact match between the predecessor good and the successor good, but for which economic theory nevertheless accepts the link for purposes of index number construction. BLS has some linkage rules or criteria which it uses currently in producing the monthly CPI. Period-to-period trend in total dollar volume is now where
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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes is the continuing and linkable volume as a proportion of the total base volume, and is the continuing and linkable volume as a proportion of the total comparison volume. Let be the trend in the proportion continuing or linkable. Then building on the above, the trend in total dollar volume can be decomposed as Alternative price indexes based on continuing and linkable goods are given byTLp is Laspeyres price index, TPp is the Paasche price index, and TFp is the Fisher price index. All price indexes discussed here extend naturally to a time series of comparison periods. In the first two formulations, we faced a simple world with only one good. In this world, the price index is both plutocratic and democratic. In the third formulation, we faced a limited world in which a static set of goods is available in the marketplace. In this world, the plutocratic Laspeyres price index can be rewritten as where the plutocratic weight applied to the simple trend in average price
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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes is simply market share expressed in dollars calculated across all HHs in the base period population with respect to the total market basket, G. Given the same assumptions, the democratic Laspeyres price index is defined by where the democratic weight applied to the simple trend in average price is the unweighted population mean across all HHs in the base period population of the HH specific market shares. Thus, plutocratic weights are ratios of means and democratic weights are means of ratios. Similar weighting yields IPp, a democratic Paasche price index, and IFp, a democratic Fisher price index. It is straightforward to establish the following relationship between plutocratic and democratic weights: where is market share within the ith household, is total consumption volume by the ith HH in the base period, and is the population mean per HH of total consumption volume. Thus, the plutocratic weight exceeds (is exceeded by) the democratic weight for any good that displays a positive (negative) correlation between total HH consumption and HH market share. The weights are equal in the event of zero correlation. For example,
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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes let g = automobiles. If there is positive correlation between total HH consumption and the share of HH consumption on automobiles, the plutocratic weight will exceed the democratic weight. Across all goods, one can now conclude the following relationship between price indexes: The difference between the price indexes is determined by the pattern of covariances and price trends across goods. If goods for which the covariance is positive experience relatively large increases in average price, plutocratic price indexes may exceed their democratic counterparts. In general, however, the direction of the difference between the price indexes is far from certain for a given comparison period, t, let alone across periods. This matter is ripe for empirical investigation. The democratic price indexes, and the relationship between plutocratic and democratic price indexes just discussed, extend naturally to the real-world situation described above where the domain of goods varies by period. There are at least two approaches to estimating the price indexes: household (HH) survey data and store survey data. In this section, we explore the first approach; in the next section we look at the second. Let s0 and st denote probability samples drawn from the universe of HHs at times 0 and t, respectively. At each time period, assume that BLS collects unit volume and prices for all buying occasions for all goods from each HH, i, in the sample. Comprehensive data of this kind are not currently collected by any BLS survey. It might be feasible—using scanning technology or other approaches—to design surveys to collect such data. Let Wit and Wi0 denote survey weights such that and are essentially design-unbiased estimators of the totals Qgt and Qg0, respectively. Similarly, define the estimated totals
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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes and The latter two estimators reflect total dollar volume across all continuing and linkable goods. From these basic estimated totals, one can consistently estimate the ratios and
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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes One can estimate the price indexes: and Estimators for the other trends and indexes— and —are defined in the obvious way. Next, consider the possibility of estimating the price indexes exclusively using store-level data. Let s0 and st denote probability samples of stores, let the subscript k index the store, and let Wk0 and Wkt denote survey weights corresponding to the unbiased estimator of a population total at times 0 and t, respectively. It is easy to imagine estimators of total dollar volume and of total unit volume. These estimators obviously require data on prices and unit volume by good at the store level. Current BLS surveys do not collect such data, but surveys based upon scanning technology could produce these data, at least for a subset of goods in a subdomain of stores. Given and , it is possible to estimate the plutocratic price indexes. The question is whether the price indexes estimated on the basis of store data really estimate TLp, TPp, and TFp. One would anticipate some biases due to such factors as goods purchased from stores for business use, not for home consumption;
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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes shrinkage due to breakage and pilferage (this component of bias would depend on the mode of data collection); and coverage errors in the store sampling frame (i.e., missing stores in which consumers shop and including stores in which they do not). Regrettably, it is not possible to estimate the democratic index ILp exclusively from store-level data, at least not without additional assumptions. The democratic weights, , are population means per HH, and HH data are necessary to estimate the means unbiasedly; such data are not usually available from stores (some store chains have adopted ID card programs that allow tracking of purchases by consumer). It may be possible to approximate the democratic index from store-level data with periodic adjustment of the weights. This possibility exploits the relationship between plutocratic and democratic weights set forth above. From store-level data, one can construct an estimator of the plutocratic weights Then we define the estimator of the democratic weights as where the adjustment factor is the second term on the right side, developed from an independent HH survey, such as the Consumer Expenditure Survey. In this factor, c(Dgi0,Y+i0) is an estimator of the covariance between HH share and total HH consumption, and is an estimator of mean total consumption per HH in the base period. It does not seem necessary to estimate the adjustment factor for each time period (month) the price index is produced. Perhaps it might be acceptable to maintain the adjustment factor only on an infrequent basis. Without question, one can imagine other hybrid schemes for estimating plutocratic or democratic price indexes. BLS’s current method is an outstanding example, with quantity weights coming from one survey and monthly prices from another.
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