TEACHING CHAIN-WEIGHT REAL GDP MEASURES

TEACHING CHAIN-WEIGHT REAL GDP MEASURES Miles B. Cahill Associate Professor Department of Economics College of the Holy Cross One College Street Worcester, MA 01610 Presented at the 76 th annual Western Economic Association International Conference San Francisco, CA July 8, 2001 Abstract In 1996, the Bureau of Economic Analysis (BEA) changed the methodology used to calculate measures of real GDP. Previous to 1996, the BEA used a Laspeyres or Paasche index methodology to calculate real GDP. However, it is well-known that Laspeyres and Paasche indexes have consistent biases; the result is that real GDP growth was underestimated for years after the base year, and underestimated for years before the base year. This bias is analogous to the well-known substitution bias in the CPI. To correct this problem, the BEA revised its real GDP measures in 1996 to instead use a Fisher ideal index (the average of a Laspeyres and a Paasche index) to calculate real GDP growth rates. The resulting real GDP level data is commonly called a chain-weight measure because each year s value contains price information for a chain of years back to the base year. The effects of the revision are dramatic: using data obtained from the Federal Reserve Economic Data (FRED, http://www.stls.frb.org/fred/) from 1948-1973, the trend chain-weight growth rate for real GDP is 3.9%, rather than the oft-reported 3.6%. For the 1974-1996 period, the trend chain-weight growth rate of real GDP is 3.0%, rather than the oft-reported 2.3% (or more correctly, 2.6%; recently, the BEA revised the method of computing price indexes; this has resulted in an upward revision of real GDP growth rates of about 0.4% for this period). Yet, many intermediate-level textbooks treat this revision casually, if at all, and chain-weight methodology is rarely, if ever detailed. This paper will highlight some of the results of the GDP revision, including those results reported in the Survey of Current Business and more recently in Rossiter (2000), and suggest some ways chain-weight measures can be explored in undergraduate economics classes, including the use of spreadsheet-based exercises derived from those presented in Cahill and Kosicki (2000). JEL CODES: A22, A20, O47

1 Introduction Unfortunately, the method changes in the national income and product accounts have received little or no attention from economists in academia. However, the new approach has important implications for the measurement of economic growth and the interpretation of real-dollar estimates of gross domestic product and its components. (Rossiter 2000, p. 363) In 1996, The Bureau of Economic Analysis (BEA) introduced chain-weight (also known as Fisher ideal) measures of real gross domestic product (GDP) output and price indexes. The new measures were designed to correct for the substitution bias problem associated with the fixed-weight Laspeyres and Paasche indexes used previously. In recent decades, this substitution bias became a serious problem, especially after computer technology goods prices fell dramatically as sales increased substantially (Landefeld and Parker 1995). Despite the fact that chain-weight measures have been the standard way to measure real GDP for five years or so, many intermediate-level macroeconomic textbooks do not provide a thorough explanation of the chain-weight methodology. Table A1 in the appendix gives a survey of nine wellknown texts. In this list, three texts provide no explanation of chain-weight measures, two provide short overviews that minimize their importance, two provide reasonable descriptions in a sidebar or appendix, one provides a short description in the main text, and only one (Gordon 1998) provides a detailed description within the main text. At the same time, Gordon (1998) presents fixed-weight data in its data appendix. The purpose of this paper is to present ways to introduce chain weight measures into an intermediate theory course. After explaining the concept of chain-weight measures, this paper provides evidence that the introduction of chain-weight methodology have changed reported statistics in important and systematic ways. Then, this paper presents ways in which chain-weight concepts may be explored at the intermediate undergraduate level. In doing so, it will be shown that teaching chain-weight measures provides an opportunity to explore some fundamental economic concepts. These exercise make use of computer spreadsheets. 1 The original spreadsheets used in the examples may be downloaded at the paper s web page, http://sterling.holycross.edu/departments/economics/mcahill/wea01.html. 1 Cahill and Kosicki (2001) suggest different ways in which spreadsheets can be introduced into economics courses.

2 Chain-weight indexes vs. fixed-weight indexes Chain-weight indexes are explained thoroughly in several sources. Landefeld and Parker (1995) explain the concepts in the Survey of Current Business article that introduced the chain-weight methodology. This article contains a sidebar note that provides a clear two-good numerical example that illustrates the shortcomings of the fixed-weight methodologies and shows how the chain-weight methodology corrects for the problems. This example is repeated in the Gordon (1998) intermediate macroeconomics text. The BEA s national accounts articles web page (http://www.bea.doc.gov/bea/an1.htm) makes available a series of articles from the Survey of Current Business concerning the adoption of the chain-weight methodology. Other descriptions may be found in the Rossiter (2000) Journal of Economic Education article, and the Abel and Bernanke (2000), Barro (1997), and Blanchard (2000) intermediate-level macroeconomics texts. Real GDP is an index of the quantity of production for a given year relative to a base year. While with a price index, the base year s index value is usually set to 100, the real GDP index base year value is usually scaled to nominal GDP in that year. The traditional fixed-weight method of computing real GDP involves using the base year s prices as weights in the index to compute GDP for every other year. While the fixed-weight method makes intuitive sense, it creates a substitution effect bias. When relative prices change, households substitute towards purchasing relatively cheaper goods, causing relative production to rise for those products. As a result, goods whose relative prices fall over time tend to see relative increases in production, and vice versa. With a fixed-weighting scheme, this means that for years after the base year, the index weight (e.g. the base year s price) for goods with the largest increases in production tend to have relatively high base year price weights, and vice versa. This means that GDP growth is overstated for years after the base year. Furthermore, when the base year is adjusted forward, historical real GDP growth figures fall. For years before the base year, goods with the largest increases in production have relatively low base year price weights (compared to contemporaneous prices), and so GDP growth is understated. This substitution bias problem is similar to a problem inherent in the Consumer Price Index (CPI); the CPI overstates the percentage change in the price level for the same reason fixed-weight GDP overstates the percentage change in the quantity of production. The chain-weight methodology corrects for the bias by taking the geometric average of two fixedweight statistics: the GDP growth rate calculated with the current year s prices i.e. ( the end of period

3 prices) and the previous year s prices (i.e. beginning of period prices). Thus, it is the average of a Laspeyres and a Paasche growth quantity index. This type of index is also known as a Fisher ideal index. When applied to price indexes, the Fisher ideal index has been shown to provide a good approximation of an ideal cost of living index. 2 One of the examples below demonstrates this fact. Because it has been shown to limit the substitution bias, the chain-weight method is an obvious way to calculate the GDP quantity index. While national income accountants have been long aware of this fact, the substitution bias did not become a problem until the precipitous fall in computer prices combined with a dramatic increase in computer sales (Landefeld and Parker 1995). Differences between chain-weight and fixed-weight statistics Figure 1 below depicts fixed-weight and chain-weight GDP measures for all quarterly data available following the most recent comprehensive NIPA revision. 3 Note that with the exception of the base year, the fixed-weight measure overstates real GDP, and the bias worsens farther in time from the base year. This is to be expected as relative price changes are more extreme farther from the base year. Note also that with the exception of the base year, chain-weight real GDP is less than its fixed-weight counterpart. This is a consequence of the fact that fixed-weight and chain-weight real GDP values are equal in the base year, but chain-weight GDP grows slower after the base year and faster before the base year. 2 See Diewert (1976) and Stone and Paris (1952). 3 Both the data for all figures and the figures themselves may be found in the spreadsheet posted on the paper s web site, The original spreadsheets used in the examples may be downloaded at the paper s web page, http://sterling.holycross.edu/departments/economics/mcahill/wea01.html.

4 Figure 1 Chain vs. fixed-weight GDP level (base = 1996) 10000 9000 8000 GDP 7000 6000 5000 Fixed weight 4000 3000 Chain weight 2000 1968 1972 1976 1980 1984 1988 1992 1996 2000 Year Data Source: Federal Reserve Bank of St. Louis (2001) Figure 2 depicts the errors associated with the fixed-weight measure. About thirty-five years ago, the error was approximately 10% of GDP; currently, with the base year five years in the past, the error is about 3%. Figure 2 also depicts the error in the GDP growth rate. As expected, this error is mostly negative before the base year, and positive after the base year. The average of the absolute value of the error is 0.5 percentage points, and the error ranges from 1.5 percentage points to 1.27 percentage points. This is a significant error in a single quarter s GDP figure.

5 Figure 2 Fixed weight measurement errors 10.0% 2.0% 8.0% level % error (left scale) 1.0% 6.0% 0.0% Error 4.0% grth rate error (right scale) -1.0% 2.0% -2.0% 0.0% 1968 1972 1976 1980 1984 1988 1992 1996 2000-3.0% Year Data Source: Federal Reserve Bank of St. Louis (2001) Landefeld and Parker (1995, 1997) provide a detailed description of some of the other problems associated with fixed-weight measures, including the fact that the growth rate is overestimated during expansions and underestimated during contractions. Rossiter (2000) details some important aspects of chain-weight data, including the fact that for technical reasons, the (weighted) sum of the growth rates of the GDP components do not add up to the overall GDP growth rate, necessitating the inclusion of a statistical discrepancy. Teaching chain-weight measures There are two main aspects of chain-weight methodology to present to students, and I believe that each is facilitated by the use of a computer and spreadsheet program like Microsoft Excel. Each of these approaches are based on examples appearing in Cahill and Kosicki (2000). As noted earlier, relevant spreadsheets may be downloaded from the paper s web page, http://sterling.holycross.edu/departments/ economics/mcahill/wea01.html. The first approach uses utility function analysis to compare cost of living adjustments based on Laspeyres and Paasche indexes to an ideal cost of living adjustment that makes the

6 consumer exactly as well-off (in terms of utility) after the price change. The chain-weight price index is the (geometric) average of the Laspeyres and Paasche indexes, and is shown to well approximate the ideal price index. This example is useful because it clearly depicts the substitution bias at work from firstprinciples, and demonstrates the desired feature of the chain-weight index. The second example is a more practical example of a chain weight GDP calculation: it constructs a chain-weight quantity (real GDP) index from data using a multi-step procedure. Understanding the substitution bias and the Fisher Ideal Index with utility theory This example, depicted in Figure 3 below, follows the set-up of section 3 of Cahill and Kosicki (2000). Consider a consumer with a Cobb-Douglas utility function over two goods (x and y), U = x a y b where a>0, b>0. Denote the prices of the goods with p x and p y and the consumer s level of income with I. An individual maximizing utility will have demand functions x * = [a/(a+b)](i/p x ) and y * = [b/(a+b)](i/p y ). The top left-hand corner of Figure 3 depicts this information set up on a spreadsheet where parameter values are typed into column B (a=b=1, p x = $4.00, p y =$1.00, I=$400), and Excel formulas are used to calculate the values of x *, y * and U. Note the relative price p x /p y =4.

7 Figure 3 Utility analysis of chain-weight index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 A B C D E Original values copy & paste, New values a 1 then 1 b 1 update prices 1 p x $4.00 $4.20 p y $1.00 $1.40 I $400 $400 x* 50 47.62 y* 200 142.86 U 10,000 6,802.72 Laspeyres price index 1.225 Paasche price index 1.200 Chain weight (Fisher Ideal) price index 1.212 Cost of living raise Laspeyres Paasche Holding U constant a 1 1 1 b 1 1 1 p x $4.20 $4.20 $4.20 p y $1.40 $1.40 $1.40 I $490.00 $480.00 $484.97 by adjusting income x* 58.33 57.14 57.74 y* 175.00 171.43 173.21 Goal Seek drives U 10,208.33 9,795.92 10,000 this cell to original U Ideal price index 1.212 Column D depicts the same information copied, with both prices increased in such a way so that the price ratio is changed to p x /p y =3. With constant nominal income, note that the demand for goods x and y have fallen, but the demand for y has fallen by more in percentage terms as the price of y has increased by more in percentage terms. Rows 11-13 calculate the Laspeyres, Paasche, and chain-weight price indexes for this example. On the spreadsheet, the Laspeyres price index is computed by finding the cost of the original consumption bundle at the new prices and dividing by the cost of the original bundle at the original prices. The Paasche price index is computed by finding the cost of the new consumption bundle at the new prices and dividing by the cost of the new bundle at the original prices. The chain-weight index is the geometric average of the Laspeyres and Paasche price indexes. To understand that the Laspeyres overstates inflation and the Paasche underst ates inflation, columns B and C in rows 16-23 compute the utility level of the consumer when given a nominal income raise equal to the Laspeyres and Paasche indexes, respectively. Note that the utility level of the Laspeyres raise is above the original pre-inflation 10,000 and the utility level of the Paasche raise is less than

8 10,000. The ideal price index answers the question: How much extra income is needed to leave the consumer with the same level of utility as in the original price regime? To derive an ideal price index, the new price values in rows 2-9 in column D are copied to a new area of the spreadsheet (rows 16-23). The Tools/Goal Seek command is used to adjust the income level so that utility returns to 10,000: specifically, click on the utility cell (D23), click on the Tools menu then Goal Seek, and in the window (see Figure 4 below), set the to value to 10,000 by changing the cell with the income (D20). Figure 4 The Tools/Goal Seek Window The ideal price index is the ratio of this income level to the original income level. Note that it gives exactly the same value as the chain-weight index. This suggests that the chain-weight index can indeed approximate the ideal price index. The example above constructed a series of indexes for prices. However, a quantity index could be computed by using base year prices rather than quantities. The next example more explicitly calculates chain-weight measures for real GDP growth, real GDP level, and a deflator. Chain-weight GDP calculations This section will present a step-by-step process for computing the chain-weight GDP measures. The step-by-step process is necessary because the equation defining chain-weight GDP is difficult for students to understand: Y t = t t p p t 1 t 1 q q t t 1 t t p q t t p q t t 1 Y t 1

9 Further, the method used by the BEA to calculate real GDP growth is probably intractable at the intermediate level, as it uses a chain-weight price index to deflate nominal figures, and the price index is difficult to calculate. The process outlined below gives equivalent answers, uses the same basic methodology, but is more intuitive. It is also the method suggested in numerical examples provided by the BEA (Landefeld and Parker (1995). While these steps may be easier for students to follow than those used by the BEA, they do require tedious calculations. A spreadsheet can help to keep the calculations organized, increase accuracy, and make the assignment less time consuming. The spreadsheet example is based on the web supplement to Cahill and Kosicki (2000). As an additional aid, Appendix B contains a handout and example which can be used as a reference for students. The steps are outlined in the spreadsheet depicted in Figure 5, below. Column A labels the rows, and columns B-F display five years of data. Rows 3-8 display price and quantity data for two goods. Row 9 calculates nominal GDP. The first step is to first calculate the chain-weight growth rate of real GDP for each year. To accomplish this, it is first necessary to calculate the annual growth rate of real output using the current year s prices as weights (row 10) and then the annual growth rate of real output using the previous year s prices as weights (row 11). The chain-weight real growth rate is then computed as the geometric mean of these two fixed-weight real growth rates (row 12). The next step is to choose a base year and calculate real GDP figures for each year by using the chain-weight real growth rates. In Figure 5, year 3 is chosen as the base year, so nominal GDP in year 3 is declared chain-weight real GDP. Real chain-weight GDP in year 4 is calculated by adding the chain-weight growth to the base year s real GDP, and chain-weight GDP in year 5 is similarly calculated using its chain-weight growth rate and year 4 s real GDP value. Year 2 s real chain-weight GDP level is calculated by dividing year 3 s real GDP figure by year 3 s growth rate plus one, and year 1 s real GDP is calculated similarly. It is here that the term chain-weight shows its relevancy: year 5 s real GDP contains price information on years 5 and 4 (through the chain-weight growth rate calculation) and years 4 and 3 (though using year 4 as a base). Any given year s real GDP figure contains a chain of prices back to the base year. Finally, the chain-weight GDP deflator (called the chain-weight price index) can be found by dividing the nominal GDP figure by the chain-weight real GDP figure (row 14). Note that because it uses chain-weight real GDP data, it also contains price information chaining to the base year.

10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Figure 5 Chain-weight GDP deflator calculation A B C D E F Year 1 2 3 4 5 Prices Good 1 $20 $19 $20 $22 $21 Good 2 $60 $64 $66 $67 $77 Quantities Good 1 75 80 79 78 92 Good 2 25 24 27 29 26 Nominal GDP 3000 3056 3362 3659 3934 Real growth rate of output (current year's prices used as weights) 1.02% 5.59% 3.16% 1.63% Real growth rate of output (previous year's prices used as weights) 1.33% 5.66% 3.33% 2.92% Real growth rate of output (chain-weight) 1.17% 5.63% 3.24% 2.18% Real chain-weight GDP ( Year 3 = base year) 3146.2 3182.9 3362.0 3471.0 3546.8 Chain-weight GDP deflator 0.954 0.960 1.000 1.054 1.109 Real fixed-weight GDP (Year 3 = base year) 3150 3184 3362 3474 3556 Real growth rate of production (fixed base year weight) 1.08% 5.59% 3.33% 2.36% Fixed-weight GDP deflator 0.952 0.960 1.000 1.053 1.106 Inflation rate (fixed weight) 0.78% 4.19% 5.33% 5.04% Inflation rate (chain weight) 0.69% 4.15% 5.42% 5.22% Growth difference (chain - fixed) 0.09% 0.04% -0.09% -0.18% Inflation difference (chain - fixed) -0.09% -0.03% 0.09% 0.18% An interesting follow-up example is to calculate real GDP, real GDP growth, and the GDP deflator using the prices in the base year as fixed-weights, and then compare the results to the chainweight results. If the figures are chosen carefully (so that the good whose relative price falls has the highest increase in production growth), the figures will display the substitution bias issue detailed in section 3. Other rows have been added to calculate inflation rates implied by the fixed-weight and chainweight deflator, and the difference between growth and inflation rates in the two methodologies. Conclusion The chain-weight methodology is now used by the BEA to calculate official national income and product account statistics. While it is more complex than the standard Laspeyres or Paasche fixed-weight

11 methodologies, it is important for undergraduate students to understand how GDP and related statistics are calculated. 4 While the chain-weight methodology is certainly more echnical, t teaching it creates an opportunity to explore other important economic concepts, and indeed to see them at work. Most obviously, the chain weight calculations demonstrates the importance of the substitution effect, both theoretically and empirically. It also is a good application to use to show both rudimentary and fairly sophisticated spreadsheet applications, and students are likely to use spreadsheets both in other courses and in future employment. Finally, it highlights the importance of computers in the current economy generally, as it was indeed falling computer prices that led to the switch to chain-weight measures. 4 Chain weight figures gained new prominence last year when the Federal Open Market Committee announced it uses the personal consumption expenditures (PCE) chain-weight price index as its measure of inflation, rather than the CPI.

12 Appendix A Table A1: Coverage of chain-weight measures in intermediate-level macroeconomics texts Textbook Abel and Bernanke (2001) Barro (1997) Blanchard (2000) Dornbusch, Fischer and Startz (2001) Farmer (1999) Froyen (1998) Gordon (1998) Hall and Taylor (1997) Mankiw (1999) Chain-weight material pretty good overview in sidebar short overview pretty good overview in appendix one footnote reference to Survey of Current Business articles none one small section: The new chain-weighted measures do not differ greatly from previous measures gives good overview, but reports fixed-weight data in appendix none one small section, brief description: For most purposes, however, the differences are not important.

13 Appendix B HANDOUT: COMPUTING TRADITIONAL AND CHAIN-WEIGHT GDP MEASURES Overview As discussed in class, fixed-weight measures tend to give consistently wrong numbers. This handout will discuss how to compute chain-weight real GDP, chain-weight real GDP growth rate, and chainweight price indexes as well as their fixed-weight counterparts. Answers to practice problems are found at the end of the handout; the necessary spreadsheet may be found as a ply on spreadsheet file available on the paper s web page, http://sterling.holycross.edu/departments/economics/mcahill/wea01.html. Notation We will use the following notation: - Y means real GDP - P means a good s price or the price index - Q means a good s quantity produced - y& means real GDP growth rate - Σ means summation - a subscript defines the year For example, y& t means real GDP growth rate in year t. ΣP b Q t means sum up the product of the prices and quantities for all goods, using year b s prices, and year t s quantities. Fixed-weight measures Real GDP The fixed-weight real GDP figure uses a chosen base year s prices to calculate real GDP for every year. To figure out the fixed-weight real GDP figure for any year t, you must do the following: 1. Choose a base year. (We will call this base year year b ) 2. Real GDP for any year t equals the sum of year t s quantities of production multiplied by the base year s prices: Y t = ΣP b Q t 3. For year t+1, real GDP is: Y t+1 = Σ P b Q t+1 ; for year t-1, real GDP is: Y t-1 = Σ P b Q t-1, etc. Real GDP growth rate To figure out the real GDP growth rate (y& ) for year t, figure out the percent increase in real GDP from year t-1 to year t: y& t = (Y t - Y t-1 )/Y t-1 Real GDP deflator (price index) To figure out the fixed-weight GDP deflator (price index) (P), divide real GDP (computed above) into nominal GDP (ΣP t Q t ): P t = ΣP t Q t / Y t

14 Chain-weight measures Calculating real GDP with the chain-weight method is significantly more complicated. The following brings you through the steps in calculating the chain-weight GDP price level index. Along the way, the chain-weight real GDP growth rate and chain-weight real GDP figures will also be determined. Chain-weight real GDP growth rate To figure out the chain-weight real growth rate for year t, you take the geometric average of 2 measures of the real GDP growth rate: one using the current year, and one using the previous year. Specifically, complete the following 3 steps for every year: 1. Calculate the real GDP growth rate for a year t using that year t as the base year. That is, use the prices in year t for both year t s and year (t-1) s GDP. The formula is: (ΣP t Q t - ΣP t Q t-1 )/ ΣP t Q t-1. 2. Calculate the real GDP growth rate for year t using the previous year (t-1) as the base year. That is, use the prices in year (t-1) for both year t s and year (t-1) s GDP. The formula is: (ΣP t-1 Q t - ΣP t-1 Q t-1 )/ ΣP t-1 Q t-1 3. Geometrically average your answers in steps 1 and 2 to get the chain-weight growth rate of real GDP ( y& ). To geometrically average two numbers (say a and b), multiply them and take the square root of the resulting product ((a b) 1/2 ). Note: if you have 2 negative growth rates from steps 1 and 2, make sure the final answer is negative. If you have 1 negative and 1 positive growth rate, use an arithmetic average instead. Repeat steps 1-3 for each year. Note that this growth rate is not dependent on a base year. Each year s real growth rate uses its year and the previous year as the base year. Fixed-weight growth rate statistics use a single year as the base year. This means that fixed-weight growth rate statistics vary depending on which base year is chosen; when the base year is changed, so does history! Chain-weight real GDP To figure out actual real GDP (denoted with Y), continue from step 3 above. Essentially, you choose a base year, and declare real GDP equal to nominal GDP in the base year. Then, impute chain-weight real GDP after and before the base year by using the chain-weight growth rates calculated above. Specifically, complete the following series of steps: 4. Choose a base year b. By definition, in the base year, nominal GDP is real GDP. Thus, chain-weight real GDP is equal to nominal GDP in the base year: Y b =ΣP t Q t. 5. To compute chain-weight real GDP for the following year (b+1), use the chain-weight growth rate for year b+1 computed above in step 3 to add this growth to the base year s (b) real GDP. The formula is: (Y b+1 ) = (Y b (computed in step 4)) (1+ y& b+1 (computed in step 3)) 6. Chain-weight real GDP for year (b+2) and after is calculated similarly; add the growth for any given year to the previous year s real GDP: Y b+2 = Y b+1 (1+ y& b+2 ), Y b+3 = Y b+2 (1+ y& b+3 ) etc. 7. Chain-weight real GDP for years before the base year is computed a little differently. Essentially, you start with the base year s real GDP and subtract the growth that occurred in the base year. In this way, you move backwards in time. Mathematically, start with the formula for the base year s real GDP, except written like the formulas in steps 5 and 6: Y b = Y b-1 (1+ y& b ). If you rearrange this equation to solve for Y b-1, you get: Y b-1 = Y b /(1+ y& b ). 8. Chain-weight real GDP in years (b-2) and before can be computed similarly divide the following year s real GDP by (1+ the following year s growth rate): Y b-2 = Y b-1 /(1+ y& b-1 ), Y b-3 = Y b-2 /(1+ y& b-2 ), etc. Note that the calculations form a chain to the base year: to figure out any year s real GDP after the base year, you need to know the previous year s figure to figure out that previous year s figure, you need to know the year before, etc. all the way to the base year. Chain-weight price index (deflator) Continuing from step 8 above, to calculate the chain-weight price index P, divide the chain-weight real GDP figure by nominal GDP for each year: 9. The formula is: P t = ΣP t Q t /Y t

15 Practice problems 1. Using the data below for an economy that only produces stuffed animals, calculate for each year possible: (a) nominal GDP (b) real GDP using the fixed-weight measure and 1999 as the base year (c) the real GDP growth rate using the fixed-weight measure (and 1999 as the base year) (d) the GDP deflator using the fixed-weight method (and 1999 as the base year) (e) real GDP growth rate using the chain-weight measure (f) real GDP using the chain-weight measure (using 1999 as the base year) (g) the GDP price index using the chain-weight method (and 1999 as the base year) Four decimal places should be enough for percentages (e. g. 0.0452 = 4.52%). Year 1997 1998 1999 2000 2001 Prices lions $20.00 $19.50 $19.00 $18.50 $18.00 elephants $60.00 $64.00 $68.00 $72.00 $76.00 Quantities lions 75 85 96 107 118 elephants 25 23 21 19 18 Once you get the hang of these calculations, you will find it easier to use Excel to do your calculations. Set up the formulas in Excel, and copy the formulas where needed. You may wish to model your Excel worksheet on the following: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 A B C D E F Year 1997 1998 1999 2000 2001 Prices lions elephants Quantities lions elephants Answers (a) Nominal GDP (b) Real GDP, fixed weight (1999=base) (c) Real GDP growth, fixed weight (1999=base) --- (d) GDP deflator (1999 = base) (e) Growth rate (current = base) --- (e) Growth rate (previous = base) --- (e) GDP growth rate (chain) --- (f) Real chain GDP (1999 = base) (g) Chain weight price index (1999 = base) 2. Are there consistent differences between the fixed-weight and chain-weight statistics? If so, explain what may be causing them. (Hint: by what percentage did each of the prices change? By what percentage did each of the quantities change?)

16 Handout answers to practice problems Below, it is an explanation of how to calculate answers for one or two years for each part. For a full list of answers, see the table. My original spreadsheet can be downloaded from the ECON 256 home page. (a) Use each year s quantities and prices. For 1998, your answer should be: (P. of lions in 99 Quan. of lions in 98) + (P. of elephants in 99 Quan. of elephants in 98) = $19.50 85 + $64.00 23 = $3,129.50 (b) Same as in (a), except use each year s quantities and 1999 s prices. For 1998, your formula should be: (P. of lions in 99 Quan. of lions in 98) + (P. of elephants in 99 Quan. of elephants in 98) = $3,179.00 (c) Find the percent change in real GDP for each year, using your answers in (b). For 1998: (Real GDP in 1998 - Real GDP in 1997)/(Real GDP in 1997)=($3,179-$3,125)/ $3,125 =.0173 = 1.73% (d) Divide your answer in (a) by your answer in (b). For 1998: Nominal GDP in 1998/Real GDP in 1998 = $3,129.50/$3,179.00 = 0.984 (e) Calculate the real growth rate of GDP two ways: first using the current year s prices, then the previous year s prices. Then, take the geometric average of the results. For 1998: [(Real GDP growth rate using 1998 prices) (Real GDP growth rate using 1997 prices)] 1/2 = [(0.0219) (0.0267)] 1/2 = 0.0242 = 2.42% (f) The base year s (1999) real GDP is equal to nominal GDP. For 1999: Nominal GDP = Real GDP = $3,252.00. For years after the base year, multiply the previous year s chain-weight real GDP by (1 + the chain-weight growth rate of real GDP for that year). For 2000: Real GDP in year 2000 = (Real GDP in 1999) (1+chain-weight growth rate of GDP in 2000) = $3,252 (1+0.0202) = $3,317.54. For years before the base year, divide the next year s chain-weight real GDP by (1 + the next year s real chain-weight growth rate of GDP). For 1998: Real GDP in 1998 = (Real GDP in 1999)/(1 + chain-weight growth rate of GDP in 1999) = $3,252/(1+.0242) = $3,172.08. (g) Like in (c), divide nominal GDP (in (a)) by real GDP, except use the chain-weight real GDP figure found in part (e). For 1998: (Nominal GDP)/(Real chain-weight GDP) = $3,129.50/$3,172.08 = 0.987 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 A B C D E F Year 1997 1998 1999 2000 2001 Prices lions $20.00 $19.50 $19.00 $18.50 $18.00 elephants $60.00 $64.00 $68.00 $72.00 $76.00 Quantities lions 75 85 96 107 118 elephants 25 23 21 19 18 Answers (a) Nominal GDP $3,000.00 $3,129.50 $3,252.00 $3,347.50 $3,492.00 (b) Real GDP, fixed weight (1999=base) $3,125.00 $3,179.00 $3,252.00 $3,325.00 $3,466.00 (c) Real GDP growth, fixed weight (1999=base) --- 1.73% 2.30% 2.24% 4.24% (d) GDP deflator (1999 = base) 0.960 0.984 1.000 1.007 1.008 (e) Growth rate (current = base) --- 2.19% 2.30% 1.81% 3.62% (e) Growth rate (previous = base) --- 2.67% 2.76% 2.24% 3.93% (e) GDP growth rate (chain) --- 2.42% 2.52% 2.02% 3.77% (f) Real chain GDP (1999 = base) $3,097.27 $3,172.08 $3,252.00 $3,317.54 $3,442.65 (g) Chain weight price index (1999 = base) 0.969 0.987 1.000 1.009 1.014

2. Are there consistent differences between the fixed-weight and chain-weight statistics? If so, explain what may be causing them. Answer: For years after the 1999 base year, fixed-weight real GDP numbers are larger than chainweight GDP numbers. For years before the base year, fixed-weight GDP figures are smaller than chainweight numbers. (By consequence, fixed-weight price level numbers are lower after the base year, and higher before the base year that chain-weight numbers). The reason for this is that the good whose price fell lions, at about a 2½% drop in price each year saw an increase in demand, and hence production, of about 10-13% per year; elephants, with about 5½ -6½ % increase in prices each year saw a decrease in demand, and hence production of 5-9½ % per year. For years after the base year, the fixed-weight measure uses the relatively high 1999 lion prices to compute real GDP this use of a historically high price exaggerates the growth in lions. At the same time, the fixed-weight measure uses the relatively low 1999 price for elephants, so the fall in elephant production doesn t subtract from GDP as much. You also see this effect comparing each year s GDP growth using the current price as a base vs. the previous year as base: the previous base year figure is always higher. Note that the opposite is true for years before the base; the fixed-weight measure gives a lower figure for GDP growth because the growing lion production is given a historically low weight when 1999 is used (and negative growth in elephants is given a high 1999 weight); lion prices are higher in 1997 and 1998. Essentially what is happening is a substitution effect; as lions become cheaper, households are substituting towards them, and away from elephants. By fixing base year prices, in much the same way the CPI fixes base year quantities, GDP tends to be overstated after the base year, and understated before. The chain-weight method fixes this problem for GDP, and would fix the problem for the CPI as well, if it was used. One final note: this substitution bias exists even if both goods see an increase in prices and quantities of production; if one good s price is rising faster than another s, it will probably have a slower growth in production, and fixed-weight GDP figures will yield too high a GDP figure for years after the base year. 17

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