Yt,Τ = Year T Outputs Valued at Year Τ Prices

Chaining to Real GDP

This year’s “Real GDP” is this year’s outputs valued at base year prices. If 2005 is our base year, for example, Real GDP in 2010 is everything produced in 2010 valued at its price in 2005. In symbols, year t Real GDP, i.e., year toutputs in constant year 0 prices, is

(1)Yt,0 = P0Yt

where

Yt,τ = year t outputs valued at year τ prices

Pτ = prices in year τ, τ= 0 in the base year

Yt = output in year t = “Real GDP” in year t

We can also write this as

(2)Yt,0 = P0Yt= P0Yt(Pt/Pt) =PtYt/(Pt/P0)=Yt,t/(Pt/P0)

Thus, year t’s outputs valued in year 0 prices [year t “Real GDP,” Yt,0]is the same as year t’s outputs (Yt) valued at their actual prices, Pt [the year’s nominal GDP, Yt,t]deflated by prices in year t relative to prices in base year t = 0, Pt/P0. For the base year itself, Pt/P0 =P0/P0 = 1; for each year t, Pt/P0is the year’s price index with year 0 as the base.

If we knew what things produced in year t sold for in the base year, the computation of real GDP using equation (1) would be straightforward. But we obviously can’t know the 2005 price of an iPad produced in 2010; there were no iPads in 2005. And it makes little sense to apply the price of a visit to a doctor in 2005 to doctor visits in 1930 when we compute real GDP in 1930; medical knowledge and capabilities were very much greater in 2005 than in 1930, making a “visit” different and more productive. We can, however, value 2004 doctor visits and other outputs at their 2005 prices and 2003 outputs at their 2004 prices. We can then “chain” year-to-year relative prices to compute a 1930 price index, i.e., the ratio of 1930 prices relative to 2005 prices, and use this index to deflate 1930’s nominal GDP to measure 1930 real GDP as in equation (2). In terms of our notation, for any year n years before base year 0,

(3a)P-n/P0 = (P-1/P0)(P-2/P-1)… (P-n+1/P-n+2)(P-n/P-n+1).

For any year N years after base year0,

(3b)PN/P0 = (P1/P0)(P2/P1)…(PN-1/PN-2)(PN/PN-1).

It remains to compute the ratios of each year’s prices relative to prices in the year before or prices in the year after, i.e, Pt/Pτ. We can either compute i) what year t’s output actually cost in year t, i.e., Yt,t , relative to what it would have cost in year τ, Yt,τ; or we can compute ii) what year τ’s output would have cost in year t’s prices, Yτ,t, relative to what it actually cost in year τ, Yτ,τ. Algebraically,

(4a)Pt/Pτ =Yt,t /Yt,τ = Year t nominal GDP relative to its “real” value in “base year” τ prices.

This tells us how much the prices of year t’soutput changed from year τ to year t. Alternatively,

(4b) Pt/Pτ = Yτ,t /Yτ,τ = Year τ’s“real” GDP in “base year” t prices relative to its actual value in yearτ , i.e., Year τ nominal GDP.

This tells us how much prices of year τ’s output changed from year τ to year t.

Which calculation ofPt/Pτ should we choose? Irving Fisher, the leading American economist of the first half of the 20th century, suggests we compute both and use the geometric mean.[1],[2] Chaining the ratios of Pt/Pτ computed this way, we get the price indexes we need to deflate any year’s nominal GDP to its real GDP in constant base year dollars.

(5)Yt = Yt,0 = Yt,t /(Pt/P0).

[1] In addition to his work on index numbers, Fisher is noted for the “Fisher equation” which adjusts nominal interest rates to their “real” values by netting out price inflation. Thus, if you receive 10% on your savings but if prices rise by 5%, you’ll really be able to buy only 5% more than before. More importantly for our current economic situation, Fisher studied and warned against the vicious circle of debt deflation in the 1930s: falling prices increase the burdens of debts leading to defaults and bankruptcies and further price declines. The heroic measures taken by the Federal Reserve to oppose deflation in the recent crisis reflects lessons learned from Fisher.

[2]The geometric mean of Pt/Pτ computed first according to (4a) and then according to (4b) is preferred to the arithmetic mean. When we chain two years of Pt/Pτ computed this way, for example, we get

SQRT[{Y1,1 /Y1,2}{{Y2,1 /Y2,2} ]x SQRT[{Y2,2/Y2,3}{Y3,2/Y3,3}]

= SQRT[{P1/P2}{P1/P2}{P2/P3}[{P2/P3}] = SQRT[P12/P32] = P1/P3.

When we chain two years of Pt/Pτ computed as the arithmetic mean of (4a) and (4b), we get something ugly which is left as an exercise to the reader.