Proof of % Δ Real GDP Formula

May I know how the following formula is derived:

% Δ Real GDP = % Δ Nominal GDPInflation 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:

  1. The quotient rule for differentiation is utilised here; and
  2. 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.

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