# Comparing the Consumer Price Index and the Personal Consumption Expenditures Price Index

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3 28 Comparing the CPI and the PCE Price Index November 2007 Differences in Growth Rates This section describes four sources of the differences in growth rates between the CPI and the PCE price index. Although a large number of sources or effects may be important in explaining differences in growth rates for a given period, the four effects identified below were determined to be the most important for reconciling the CPI and PCE price measures from the first quarter of 2002 through the second quarter of It is important, however, to keep in mind that there is no best set of effects. Likewise, there is no best way of estimating a particular effect. These choices require weighing a variety of factors, including accuracy, transparency, and computational simplicity. Formula effect The CPI and the PCE price index are based on different price-index-number formulas. The CPI is based on a modified Laspeyres formula, while the PCE price index is based on a Fisher-Ideal formula. A Laspeyres price relative is defined as i i i i L t, t+1 = Σ i (p t + 1 q t ) Σ i (p t q t ), i where p is the price of item i in period t, and q i t t is the quantity of item i in period t. 2 If a specific base period is set to 0, then the resulting index is referred to as a fixed-weight Laspeyres price index. i i i i Fixed-weight L t =Σ i (p t q 0 ) Σ i (p0q 0 ) The CPI is based on the fixed-weight Laspeyres price index, where the base period is updated every 2 years. 3 The PCE price index is based on a Fisher-Ideal price-index-number formula. First, note that a Paasche price relative is defined as i i i i P = Σ i (p q ) Σ i (p q t, t+ 1 t + 1 t + 1 t t + 1). The Fisher-Ideal price relative is simply the geometric mean of the Laspeyres and Paasche price relatives, that is, ) 1 2 F = (L P. t, t+ 1 t, t + 1 t, t + 1 In general, the Paasche price relative is less than the Laspeyres price relative, implying that the Fisher-Ideal price relative is generally less than the Laspeyres price relative. Next, a Fisher-Ideal chain-type price index is obtained by multiplicatively chaining the Fisher- Ideal price relatives, that is, Chained F t + 1 = (Chained F) t (F t, t + 1). 2. In this article, the term price relative refers to an aggregate measure of price change between two adjacent periods. 3. Because the base period is necessarily updated with a time lag, the CPI is said to be based on a modified Laspeyres index. The PCE price index is based on the Fisher-Ideal chain-type price index. The fundamental difference between the fixedweight Laspeyres price index and the Fisher-Ideal chain-type price index involves the extent to which the two indexes reflect consumer substitution among detailed items as the relative prices of those items change. In general, consumers substitute away from those items whose prices rise most rapidly and toward those items whose prices rise less rapidly or decline. In theory, the Fisher-Ideal price index better reflects this substitution; in this regard, the Fisher-Ideal index is referred to as a superlative index. 4 In practice, the Fisher-Ideal index is difficult to implement because it requires expenditure data for the most recent period for which the index is being estimated. It is often the case that these data are not available. For example, data on household consumer expenditures that are used to estimate the CPI are not available for the most current period. Therefore, the Laspeyres index with regular base-period updates provides a practical alternative to the Fisher-Ideal index. It also should be noted that the BLS publishes the Chained Consumer Price Index for All Urban Consumers. This price index, like the PCE chain-type price index, is based on a superlative index-number formula that better reflects consumer substitution among item categories. 5 Because both the Chained CPI and the PCE chain-type price index are based on superlative index-number formulas, they tend to behave more similarly over time. The two indexes still diverge because of scope, weight, and other effects. To estimate the formula effect, the detailed price and quantity data used to estimate the PCE price index were reaggregated using the Laspeyres price-index formula. The base periods used in this calculation were consistent with the base periods used to estimate the CPI: Estimates for were prepared using a base period, estimates for used a base period, and estimates for 2006 used a base period. The formula effect was estimated as the percentage-point difference in growth rates between the PCE chain-type price index and the PCE fixed-weight price index. 6 From the outset, the formula effect was expected to 4. The Fisher-Ideal index is just one example of a superlative index; another example is the Tornqvist index. Both of these indexes possess a number of desirable properties that make them preferred over the fixedweight Laspeyres index. See Diewert (1976) for a discussion of superlative index numbers. 5. The Chained Consumer Price Index for All Urban Consumers is based on the Tornqvist chain-type price index. For more information on the Chained Consumer Price Index for All Urban Consumers, go to <www.bls.gov/cpi/super_paris.pdf>. 6. For the purposes of this analysis, BEA calculated a PCE fixed-weight price index.

4 November 2007 SURVEY OF CURRENT BUSINESS 29 be negative because the Fisher-Ideal price relative (on which the PCE chain-type price index is based) is generally less than the Laspeyres price relative (on which the PCE fixed-weight price index is based). In addition, the formula effect was expected to be more important for the categories in which there were large relative price changes, such as gasoline and computers. Weight effect The relative weights assigned to comparable item prices in the CPI and in the PCE price index differ because these weights are based on different data sources. The relative weights used in the CPI are based on the Consumer Expenditure Survey, a household survey conducted for BLS by the Census Bureau. The relative weights used in the PCE price index are based primarily on business surveys, such as the Census Bureau s annual and monthly retail trade surveys, the Service Annual Survey, and the Quarterly Services Survey. The differences in response rates and response quality between household and business surveys are well known. 7 These issues, such as recall bias, may play a key role in explaining differences in the resulting weights. In addition, there are differences in total consumer spending that may also affect the relative weights. 8 To estimate the weight effect, the differences in the relative weights between comparable items in the PCE fixed-weight price index and the CPI are calculated. These differences are then multiplied by the growth rates in the corresponding item price indexes to yield percentage-point contributions; that is, the weight effect is computed as W i W i i i ( PCE CPI ) [(p t + 1 p t ) 1], wherewpce i is the average relative weight for item i in the PCE fixed-weight price index; WCPI i is the average relative weight for item i in the CPI; and p i t is the price for item i used in both the CPI and PCE price index calculations. 9 It is important to note that weight differences resulting from different base periods are not accounted for in the weight effect; these differences are accounted for in the formula effect. 7. See Garner, et al. (2006) for a thorough examination of the differences between the Consumer Expenditure Survey and the expenditure surveys used to estimate the PCE price index. 8. See Appendix Table A in McCully, Moyer, and Stewart (2007) for a detailed list of weight differences. 9. Symmetric to the weight effect is a price effect. It is estimated as W i [(( p i p i ) 1) (( p i p i PCE, t + 1 PCE, t CPI, t + 1 CPI, t ) 1 )], where W i is the average relative weight used for item i in the CPI calculapi tion; PCE, t is the price for item i used in the PCE price index calculation; and p CPI i, t is the price for item i used in the CPI calculation. Empirically, the price effect is small; for this analysis, it is included in other effects. Scope effect The CPI measures the out-of-pocket expenditures of all urban households, while the PCE price index measures the goods and services purchased by individuals and nonprofit institutions within the framework of the NIPAs. As such, there are items in the CPI that are not included in the PCE price index, and there are items in the PCE price index that are not included in the CPI. For example, medical care services in the CPI consist only of those services directly purchased by consumers. In the PCE price index, medical care services include services directly purchased by consumers and services paid for on behalf of consumers for example, medical care services paid for by employers through employer-provided health insurance and medical care services paid for by governments through programs such as Medicare and Medicaid. 10 The scope effect adjusts for items that are not comparable between the CPI and the PCE price index using a two-stage approach. First, the percentage-point contributions to the growth in the PCE fixed-weight price index for those items that are not included in the CPI are calculated. Second, the percentage-point contributions to growth in the CPI for those items that are not included in the PCE price index are calculated. As will be discussed in the next section, the reconciliation algorithm begins with the PCE price index and ends with the CPI; therefore, the contributions for PCE items that are out-of-scope for the CPI enter the reconciliation with negative signs, and the contributions for CPI items that are out-of-scope for the PCE price index enter with positive signs. A net scope effect is also calculated as the difference between the contributions for CPI items that are out-of-scope for the PCE price index and the contributions for PCE items that are out-of-scope for the CPI. Other effects There are a variety of remaining differences including seasonal adjustment differences, price differences, and residual differences that must be taken into account for a complete reconciliation of the CPI and the PCE price index. Seasonal adjustment differences result from two factors: Differences in revision cycles and differences in aggregation. First, the BLS and the BEA revision cycles differ. For example, in February 2007, BLS revised the CPI seasonal factors for In July 2007, BEA revised the NIPAs for ; however, because of BEA s revision policy of only revising the 3 most recent years during an annual revision, the revised seasonal 10. See Appendix Table A in McCully, Moyer, and Stewart (2007) for a detailed list of scope differences.

6 November 2007 SURVEY OF CURRENT BUSINESS The weight effect is estimated. Line 13 shows the percentage-point contribution of the weight effect to the growth rate in the PCE fixed-weight price index. Lines show the percentage-point contributions of selected categories to the growth rate in the PCE fixed-weight price index. 4. The PCE portion of the scope effect is estimated. Line 18 shows the percentage-point contribution to the growth rate in the PCE fixed-weight price index for those items that are out-of-scope for the CPI. Lines show the percentage-point contributions for selected components. 5. The CPI portion of the scope effect is estimated. Line 24 shows the percentage-point contribution to the growth rate in the CPI for those items that are out-of-scope for the PCE price index. Lines show the percentage-point contributions for selected components. 6. Other effects are estimated. Line 28 shows the percentage-point contributions of seasonal-adjustment differences, price differences, and other differences. 7. Finally, the growth rate in the CPI equals the growth rate in the PCE chain-type price index less the formula effect, less the weight effect, less the PCE portion of the scope effect, plus the CPI portion of the scope effect, less other effects. Highlights of the reconciliation can be seen in table 2. This table presents average annual estimates. From the first quarter of 2002 through the second quarter of 2007, the PCE chain-type price index (line 1) increased at an average annual rate of 2.5 percent, while the CPI (line 32) increased at an average annual rate of 2.9 percent. Over that period, the formula effect (line 2) contributed 0.17 percentage point to the 2.5-percent growth rate in the PCE chain-type price index. 13 As expected, the formula effect was negative, reflecting differences between the fixed-weight Laspeyres price index and the Fisher-Ideal chain-type price index. 14 Also as expected, the formula effect had a larger impact for those categories with large relative price changes for example, gasoline and oil (line 3) and computers, peripherals, and software (line 4). After adjusting for the formula effect, the PCE fixed-weight price index increased at an average annual rate of 2.7 percent (line 12). 13. Recall that the base periods used to estimate the formula effect are consistent with the base periods used to construct the CPI. 14. Recall that BLS publishes another measure of price change, called the Chained Consumer Price Index for All Urban Consumers (C CPI U), which employs a superlative index-number formula. The C CPI U moves more similarly to the PCE chain-type price index. For example, from the fourth quarter of 2001 through the fourth quarter of 2005 (the last date for which final C CPI U data have been published), both the C CPI U and the PCE chain-type price index increased 2.5 percent at an annual rate. The weight effect (line 13) contributed 0.67 percentage point to the 2.7-percent growth in the PCE fixed-weight price index. Within the weight effect, the rent of shelter category (line 14) which includes owners equivalent rent made the largest contribution. The relative weight for rent of shelter in the CPI is consistently and significantly higher than its relative weight in the PCE price index. 15 The scope effect consists of two parts. First, items in the PCE price index that are out-of-scope for the CPI (line 18) contributed 0.76 percentage point to the 2.7 percent growth in the PCE fixed-weight price index. Second, items in the CPI that are out-of-scope for the PCE price index (line 24) contributed 0.24 percentage point to the 2.9-percent growth in the CPI. Because the first part of the scope effect enters the reconciliation with a negative sign and the second part enters with a positive sign, the net contribution of the scope effect 15. The relative weight for rent of shelter in the CPI is about 32 percent; its relative weight in the PCE price index is about 15 percent. While the data sources for expenditure estimates for rent of shelter and for owners equivalent rent differ in the CPI and the PCE, the aggregate estimates for expenditures in each are similar. The relative weight of rent of shelter is larger in the CPI than in the PCE because total consumer expenditures (for nonrent of shelter components) reported in the Consumer Expenditure Survey are less than those estimated from business surveys used to prepare the PCE price index. Differences in total expenditure estimates can be attributed to both the different expenditure survey methods used as well as differences in scope. See Garner et al. (2006) for a detailed description of the differences between expenditure estimates derived from the Consumer Expenditure Survey and those estimated for the PCE price index. Table 2. Reconciliation of Quarterly Percent Changes Between the CPI and the PCE Price Index [2002 to Second Quarter of 2007] PCE chain-type price index (average annual percent change) Less: Formula effect (percentage points) Gasoline and oil Computers peripherals and software Video and audio goods Tobacco products Medical care services Electricity, gas, fuel oil, and other household fuels Housing Food Other Equals: PCE fixed-weight price index (average annual percent change) Less: Weight effect (percentage points) Rent of shelter Gasoline and oil Electricity, gas, fuel oil, and other household fuels Other Less: Scope effect PCE items out-of-scope of the CPI (percentage points) Physicians Hospitals and nursing homes Services furnished without payment by financial intermediaries except life insurance and pension plans Foreign travel by U.S. residents Other Plus: Scope effect CPI items out-of-scope of the PCE price index (percentage points) Physicians Hospitals and related services Other Less: Other effects Seasonal adjustment Price All other Equals: CPI (average annual percent change) Line

7 32 Comparing the CPI and the PCE Price Index November 2007 was 0.52 percentage point. Within the scope effect, the components that made the largest contributions were related to medical care services. These included physician services (line 19 and line 25), hospitals and nursing homes (line 20), and hospitals and related services (line 26). A large portion of the total scope effect is accounted for by the differing concepts of medical care services. Recall that for the CPI, medical care services include only those services directly purchased by consumers. For the PCE price index, medical care services include services purchased directly by consumers and services paid for on behalf of consumers for example, medical care services paid for by employers through employer-provided health insurance and medical care services paid for by governments through programs such as Medicare and Medicaid. Finally, other effects (line 28) was negligible over the period, making only a small negative contribution to the 2.7-percent growth rate in the PCE fixed-weight price index. As mentioned above, this effect includes seasonal adjustment differences, price differences, and residual differences. In summary, the PCE price index increased at an average annual rate of 2.5 percent over the period, and the CPI increased at an average annual rate of 2.9 percent a difference of 0.4 percentage point. The formula effect explains almost half of this growth-rate difference. After adjusting for formula differences, the weight effect more than accounted for the remaining difference in growth rates between the CPI and the PCE fixed-weight price index. The large weight effect was primarily explained by the larger relative weight for rent of shelter in the CPI than in the PCE price index. The scope effect, in contrast, partly offset the weight effect. Other effects had only a minor impact over the period. It is important to keep in mind that this reconciliation is not unique; using a different reconciliation framework will lead to different results. For example, by comparing the CPI and the Chained Consumer Price Index for All Urban Consumers, the formula effect could be estimated last rather than first. In this case, using data from the fourth quarter of 2001 through the fourth quarter of 2005 (the last year for which final data for the chained CPI have been published) the formula effect would be 0.33 percentage point (2.46 percent minus 2.79 percent), compared with 0.17 percentage point when the formula effect is estimated first. Upcoming Work Beginning in early 2008, BEA plans to begin preparing tables that reconcile the differences in growth rates between the CPI and the PCE price index on an ongoing basis. These tables will be similar to table 1 and will be available on both the BLS and the BEA Web sites. Quarterly and monthly reconciliation tables will be available. Quarterly tables will be available after the release of the advance estimates of gross domestic product (GDP) and will be revised after the release of the preliminary and final estimates of GDP. Monthly tables will be available after the release of the monthly personal income and outlays. BLS and BEA will also continue to review differences between the CPI and the PCE price index and will adjust the reconciliation procedures as needed. The reconciliation framework presented in this article was determined to be the most relevant for reconciling the CPI and PCE price measures over the specified period. It is possible that sources of difference will change over time, either because of structural changes in the economy or because of changes in data sources and estimation procedures. In July 2009, BEA plans to introduce a new classification structure for the PCE estimates as part of the 2009 comprehensive revision of the NIPAs. The new PCE classification structure will be more closely aligned with the Classification of Individual Consumption by Purpose (COICOP) and with the Classification of the Purposes of NonProfit Institutions (COPNI). Both the COICOP and the COPNI are recommended by the 1993 System of National Accounts. This new PCE classification structure will improve the consistency between detailed PCE items and detailed CPIs used for deflation. While the impact of this new classification on the reconciliation process is not yet known, it is expected that small differences currently being captured as part of weight effects and other effects will be reduced.

8 November 2007 SURVEY OF CURRENT BUSINESS 33 References Bureau of Economic Analysis Reconciliation of Quarterly Changes in Measures of Prices Paid by Consumers. SURVEY OF CURRENT BUSINESS 58 (March). Cage, Robert, John Greenlees, and Patrick Jackman Introducing the Chained Consumer Price Index. Presented at the 7 th meeting of the International Working Group on Price Indices in Paris in May. Clark, Todd E A Comparison of the CPI and the PCE Price Index. Economic Review: Third Quarter Federal Reserve Bank of Kansas City, Diewert, W. Erwin Exact and Superlative Index Numbers. Journal of Econometrics 4 (May): Fixler, Dennis, and Ted Jaditz An Examination of the Difference Between the CPI and the PCE Deflator. Bureau of Labor Statistics Working Paper no. 361, June. Garner, Thesia I., George Janini, William Passero, Laura Paszkiewicz, and Mark Vendemia The CE and the PCE: A Comparison. Monthly Labor Review (September): Johnson, David S Comparison of Movements in the CPI and PCE Price Indexes. Presented at the Federal Economic Statistics Advisory Committee Meeting at the Bureau of Labor Statistics in Washington, DC, on March 21. Lebow, David E., and Jeremy B. Rudd Measurement Error in the CPI: Where Do We Stand? Journal of Economic Literature (March): McCully, Clinton P., Brian C. Moyer, and Kenneth J. Stewart A Reconciliation Between the Consumer Price Index and the Personal Consumption Expenditures Price Index ; <www.bea.gov/papers/pdf/ cpi_pce.pdf>. Schultze, Charles L., and Christopher Mackie, editors At What Price? Conceptualizing and Measuring Cost-of-Living and Price Indexes. In National Research Council Panel on Conceptual, Measurement, and Other Statistical Issues in Developing Cost-of- Living Indexes. Washington, DC: National Academy Press. Triplett, Jack E Reconciling the CPI and the PCE Deflator. Monthly Labor Review (September): 3 15.

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