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I'll use the US as an example. I have three data series

  1. nominal GDP $(Y)$
  2. real GDP in 2005 USD $(\bar{Y})$
  3. the GDP deflator $(d)$, with 2005 as the base year, so $d_{2005} = 100$

I want to change the base year to 2000. Are these calculations accurate? I use the notation $\$_{t}$ for USD in year $t$ prices to help myself keep the units straight.

My goal is $\bar{Y}_{t} \ \$_{2000}$. \begin{align} \frac{d_{2000}}{d_{t}} \cdot Y_{t} \ \$_t &= \frac{Y_{2000} \ \$_{2000}}{\bar{Y}_{2000} \ \$_{2005}} \cdot \frac{\bar{Y}_{t} \ \$_{2005}}{Y_{t} \ \$_{t}} \cdot Y_{t} \ \$_t \\ &= \frac{Y_{2000} \ \$_{2000}}{\bar{Y}_{2000}} \cdot \bar{Y}_{t} \\ &= \bar{Y}_{t} \ \$_{2000} \cdot \frac{Y_{2000}}{\bar{Y}_{2000}} \end{align}

I think those are all the correct unit cancellations, but now I'm stuck with the unitless quantity $\frac{Y_{2000}}{\bar{Y}_{2000}}$, so I don't know how to complete the conversion.

Am I doing this right?

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It's just nominal GDP in year t, times (deflator in year 2000 / deflator in year t).

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Deflator in 2000 should be 100 as it your base year.

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