econ101chainweighting

econ101chainweighting - Economics 101 A Quick Guide to...

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Unformatted text preview: Economics 101 A Quick Guide to Chain-Weighting 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 chain-weighting. Intuitively, a chain-weighted price index measures the change in prices over two periods using an average of quantities in those two periods; whereas, a chain-weighted 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 chain-weighted price and quantity indexes, follow the steps below: Step 1: Pick a base-year 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 base-period. Therefore: 100 $5 1 $1 10 $15 Step 2: Compute a chain-weighted (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 chain-weighted (Fisher) price index for 2008: For any t and t-1: 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 chain-weighting, the ratio of nominal GDP between any two periods is equal to the ratio of chain-weighted prices multiplied by the ratio of chain-weighted 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 chain-weighted 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 chain-weighted (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 chain-weighted (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 chain-weighting, the ratio of nominal GDP between any two periods is equal to the ratio of chain-weighted prices multiplied by the ratio of chain-weighted 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 chain-weighted price ratio and the chain-weighted 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, chain-weighted prices went up by a factor of 1.040063 and chain-weighted 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. Chain-weighting is used by the Bureau of Economic Analysis to compute chain-weighted real GDP and various price indices associated with GDP, including the chain-weighted 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 chain-weighted consumer price index for urban consumers (C-CPI-U) that will soon replace the standard fixed-weighted (Laspeyres) consumer price index for urban consumers (CPI-U). Because of the substitution bias, the fixed-weighted CPI-U likely overestimates changes in the true cost of living by about 1 percent per year. By chain-weighting the substitution bias is reduced significantly. 4 ...
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