May I know how the following formula is derived:
% Δ Real GDP = % Δ Nominal GDP – Inflation Rate
First, we know that Real GDP is derived from Nominal GDP, by multiplying it against the GDP Deflator:
Real GDP = Nominal GDP * ( CPI Base / CPI Current )
But before we move on, due to the long-ish math expressions that will follow, some truncation would be needed here:
Real GDP = R
Nominal GDP = N
CPIBase = B
CPI Current = C
So then we can re-write our earlier expression as:
R = N * ( B / C )
We then differentiate the expression to derive:
δR = B * ( δN * C – δC * N ) / C 2
Notice that:
- The quotient rule for differentiation is utilised here; and
- B is a constant here, and therefore not subject to the differentiation.
Since:
% Δ Real GDP = ( Δ Real GDP / Real GDP ) * 100
We can plug our differentiation result from earlier into the expression:
% Δ R ≈ [ B * ( δN * C – δC * N ) / C 2 ] / R * 100
Since R = N * ( B / C ), we can further transform the expression:
% Δ R ≈ [ B * ( δN * C – δC * N ) / C 2] / [ N * ( B / C ) ] * 100
It looks complicated! But with some patience, you should be able to simplify the expression to derive:
% Δ R ≈ ( δN / N – δC / C ) / * 100
And therefore:
% Δ R ≈ % Δ N – % Δ C
Since the inflation rate is defined mathematically as % Δ C, therefore:
% Δ R ≈ % Δ N – Inflation Rate
And voilà! Proof completed! If you have queries or comments, feel free to comment below.