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Let's say we have these data:

  • $p_{t-1}$, price index at time $t - 1$ = 100
  • $p_t$, price index at time $t$ = 120
  • $s_{t-1}$, sales at period $t - 1$ = 110
  • $s_t$, sales at period $t$ = 154

Then which is the better between the 2 following equations to know the true relation between the variation of the price index against the variation of the sales or to know the difference between what they say. This is about, how much do I earn or lost compared to/considering the inflation?

1. (variation of the sales) - (variation of the price index)

$$\frac{s_t - s_{t-1}}{s_{t-1}} - \frac{p_t - p_{t-1}}{p_{t-1}} = \frac{154 - 110}{110} - \frac{120 - 100}{100} = 0.40 - 0.20 = 0.20.$$

Then, we could say that the sales where 20 percentage points over the inflation

BUT...

2. DEFLATE (variation of sales adjusted by inflation)

That is to "carry" the sales from $t-1$ to $t$ by multiplying it by the inflation coefficient (indexation) of $p_2/p_1$.

$$\frac{s_t - s_{t-1}\cdot\frac{p_t}{p_{t-1}}}{s_{t-1}\cdot\frac{p_t}{p_{t-1}}} = \frac{154 - 110 \cdot ( 120 / 100 ) }{110 \cdot (120 / 100)} = \frac{ 154 - 132 }{ 132 } = 0.16.$$

Then... what should we say? Did I earn 20% or 16% over inflation?

3. The relation between the 2 equations is the following:

After some algebra:

equation 1 = equation 2 * $(p_t/p_{t-1})$

0.20 = 0.16 * (120 / 100)

0.20 = .20

Beyond the algebra, I can't give it a money-sense, economic interpretation to this equation. I mean, if 0.16 is already adjusted by inflation, why indexing it turns to be equation 1?

4. QUESTIONS:

I don't get what is the meaning and difference of the way 1) and 2) to calculate sales against inflation. Could anyone make it clear for me? When would be better to use equation 1 and when equation 2? Which are the different economic interpretations to them?

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