Unformatted text preview: Economics 101 A Quick Guide to ChainWeighting Department of Economics UC Davis Professor Siegler Summer 2009 The index number problem is how to combine the relative changes in the prices and quantities of various products into (i) a single measure of the relative change of the overall price level and (ii) a single measure of the relative change of the overall quantity level. There is no unique way to combine prices and quantities. In general, a price index measure the level of aggregate prices using quantity weights, while a quantity index measure the level of aggregate quantities using price weights. Typically, relative prices and relative quantities move in the opposite direction. If a price of a good gets relatively more expensive, people buy less of it. When this happens, the choice of weights has large implications. One way to minimize the importance of weighting is to use chainweighting. Intuitively, a chainweighted price index measures the change in prices over two periods using an average of quantities in those two periods; whereas, a chainweighted quantity index measures the change in quantities over two periods using an average of prices in those two periods. Consider the example of a hypothetical economy that produces two goods from lecture on June 23. Table 1 Hypothetical Economy Year 2007 2008 2009 P1 $5 $4 $3 Q1 1 3 6 P2 $1 $2 $3 Q2 10 7 4 To compute chainweighted price and quantity indexes, follow the steps below: Step 1: Pick a baseyear to serve as a reference value. With price indices, the base period is typically set equal to 100. With quantity indices, like real GDP, nominal GDP equals real GDP in the base period. In this case, let 2007 be the baseperiod. Therefore: 100 $5 1 $1 10 $15 Step 2: Compute a chainweighted (Fisher) quantity index for 2008 (real GDP for 2008): 1 $5 3 $5 1 $1 7 $1 10 $4 3 $4 1 $2 7 $2 10 $15 $22 $26 $15 $15 $24 $18.90767 $18.91 Step 3: Compute a chainweighted (Fisher) price index for 2008: For any t and t1: In this case, $4 1 $5 1 $2 10 $1 10 $4 3 $5 3 $2 7 $1 7 100 $24 $26 100 $15 $22 137.5103 Step 4: Check your work. With chainweighting, the ratio of nominal GDP between any two periods is equal to the ratio of chainweighted prices multiplied by the ratio of chainweighted quantities. Nominal GDP in 2008 is $4 3 $2 7 $26, while nominal GDP for 2007 is $15 as computed above. The ratio of nominal GDP for these two years is: $26 $15 1.733 137.5103 $18.90767 100 $15 1.733 Repeat Steps 2 through 4 to compute the chainweighted price and quantity indexes for 2009. The answers are on the following pages. 2 Table 1 Again Year 2007 2008 2009 P1 $5 $4 $3 Q1 1 3 6 P2 $1 $2 $3 Q2 10 7 4 Step 2: Compute a chainweighted (Fisher) quantity index for 2009: $4 6 $4 3 $2 4 $2 7 $3 6 $3 3 $3 4 $3 7 $18.90767 $32 $30 $18.90767 $26 $30 1.109400392 $18.90767 $20.97617652 $20.98 Step 3: Compute a chainweighted (Fisher) price index for 2009: For 2009 and 2008: $3 3 $4 3 $3 7 $2 7 $3 6 $4 6 $3 4 $2 4 137.5103 $30 $30 137.5103 $26 $32 1.040063 137.5103 143.0193751 Step 4: Check your work. With chainweighting, the ratio of nominal GDP between any two periods is equal to the ratio of chainweighted prices multiplied by the ratio of chainweighted quantities. Nominal GDP in 2009 is $3 6 $3 4 $30, while nominal GDP for 2008 is $4 3 $2 7 $26. The ratio of nominal GDP for these two years is: 3 $30 $26 1.153846 If we've done the math correctly, this should also be equal to the product of the chainweighted price ratio and the chainweighted quantity ratio: 143.0193751 $20.97617652 137.5103 $18.90767 1.153846 The math works out so I must have done it right. Between 2008 and 2009 nominal GDP went up by a factor of 1.153846, chainweighted prices went up by a factor of 1.040063 and chainweighted quantities went up by a factor of 1.109400393. Therefore, the roughly 15 percent increase in nominal GDP is the product of the roughly 4 percent increase in prices and the nearly 11 percent increase in quantities. Chainweighting is used by the Bureau of Economic Analysis to compute chainweighted real GDP and various price indices associated with GDP, including the chainweighted Personal Consumption Expenditures (PCE) price index that is used by the Federal Reserve to measure inflation in making monetary policy. In addition, the Bureau of Labor Statistics is also publishing a chainweighted consumer price index for urban consumers (CCPIU) that will soon replace the standard fixedweighted (Laspeyres) consumer price index for urban consumers (CPIU). Because of the substitution bias, the fixedweighted CPIU likely overestimates changes in the true cost of living by about 1 percent per year. By chainweighting the substitution bias is reduced significantly. 4 ...
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This note was uploaded on 06/28/2009 for the course ARE 100A taught by Professor Constantine during the Winter '08 term at UC Davis.
 Winter '08
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