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What is the process used to distinguish the change in Gross Domestic Product (GDP) due to an increased output of goods and services from the change due to an increased prices? Why do we make it distinction?

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Real GDP (RGDP) is a measure of the value of goods and services produced in an economy over a period of time for a fixed set of prices. Nominal GDP (NGDP) is a measure of the value of goods and services produced in an economy over a period of time for the prices over that period in time. The way they calculate changes in RGDP is to first measure changes in NGDP, separately measure prices, and then subtract the difference. For NGDP in year $t$ and reference price year for RGDP of $k$ it works like this:

$$NGDP_t = RGDP_{t,k} \cdot \frac{P_t}{P_k}$$ Rearrange: $$\Rightarrow RGDP_{t,k} = NGDP_t \cdot \frac{P_k}{P_t}$$ Take logs: $$\Rightarrow \ln(RGDP_{t,k}) = \ln(NGDP_t) +\ln(P_k) - \ln(P_t)$$ Now we can calculate a change ($\Delta X_t = X_t - X_{t-1}$): $$ \Delta \ln(RGDP_{t,k}) = \Delta \ln(NGDP_t) + \Delta \ln(P_k) - \Delta \ln(P_t)$$ But by definition $\Delta \ln(P_k)=0$ because the reference year prices don't change so: $$ \Delta \ln(RGDP_{t,k}) = \Delta \ln(NGDP_t) - \Delta \ln(P_t)$$ So as claimed above, the percent change in real GDP ($\ln(RGDP_{t,k})$) equals the percent change in nominal GDP ($\Delta \ln(NGDP_t)$) less the percent change in prices ($\Delta \ln(P_t)$).

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