[Math] Intuition calculating average price change

average

Given two products p1 and p2, each with a price per single unit and an amount per year. The question is to calculate the average change in price for both products combined.

+-Product#-+-Amount_Y2011-+-Price_Y2011-+-Amount_Y2012-+-PriceY2012-+
| 1        |           10 |           2 |           20 |        2.5 |
| 2        |           60 |           3 |           80 |          4 |

I have two basic ideas how to calculate the average change:

  • Calculate the price change per product and use weighted average by amount.

\begin{aligned}
\frac{2.5-2}{2} &= 0.25 \text{(price change from 2011 to 2012 for p1)} \\
\frac{4-3}{3} &= 0.33333… \text{(price change from 2011 to 2012 for p2)} \\
0.25 * 10 + (1/3)*60 &= 22.5 \text{(sum price change weighted by amount 2011)} \\
\frac{22.5}{10+60} &= \frac{9}{28} = 0.32143…
\end{aligned}

  • Calculate the volume per product with respect to a base year and average that.

\begin{aligned}
2 * 10 &= 20 \text{(volume p1 for 2011 with price 2011)} \\
3 * 60 &= 180 \text{(volume p2 for 2011 with price 2011)} \\
2.5 * 10 &= 25 \text{(volume p1 for 2011 with price 2012)} \\
4 * 60 &= 240 \text{(volume p2 for 2011 with price 2012)} \\
\frac{(25+240)-(20+180)}{20+180} &= \frac{13}{40}=0.325
\end{aligned}

Obviously the percentages aren't the same. Unfortunately I don't really have an intuition why the result is different, or what the difference means. I most importantly which one is the 'right' solution.

During my quest I arrived at the following two abstract formulas for both cases, which didn't really help me but only confused matters further.

  • price change / weighted average
    \begin{aligned}
    \frac{\frac{12_{price1}-11_{price1}}{11_{price1}} * 11_{amount1} + \frac{12_{price2}-11_{price2}}{11_{price2}} * 11_{amount2}}{11_{amount1}+11_{amount2}} &=\\
    =\frac{12_{volume1}-11_{volume1}}{11_{volume1}+11_{price1}*11_{amount2}}+\frac{12_{volume2}-11_{volume2}}{11_{volume2}+11_{price2}*11_{amount1}}
    \end{aligned}

  • volume base year / average
    \begin{aligned}
    \frac{12_{volume1}+12_{volume2} – (11_{volume1} + 11_{volume2})}{11_{volume1} + 11_{volume2}} &= \\
    =\frac{12_{volume1}-11_{volume1}}{11_{volume1}+11_{price1}*11_{amount1}}+\frac{12_{volume2}-11_{volume2}}{11_{volume2}+11_{price2}*11_{amount2}}
    \end{aligned}

So mathematically the only difference is in the denominator. But I have no clue what this might mean, or how to interpret the different formulas.

So:

  • What am I missing here?
  • Why the difference?
  • Which one is 'correct'?

Best Answer

Using $p$ for prices, $q$ for quantities (amounts) the two formulae are $${{\sum_i (p_{2i}/p_{1i}) q_{1i}} \over {\sum_i q_{1i}}}-1$$ and $${{\sum_i p_{2i}q_{1i}} \over {\sum_i p_{1i}q_{1i}}}-1 ={{\sum_i (p_{2i}/p_{1i}) p_{1i}q_{1i}} \over {\sum_i p_{1i}q_{1i}}}-1$$

So both are weighted average of price changes with the weights being the first-period quantities in the former and the first-period volumes in the latter. There is no right answer since only the user can decide which system of weights better captures the relative importance of commodities.

However there is one argument against the first formula. The quantities depend on the choice of units. If you start measuring one commodity in grams rather than kilograms then its importance in the index would go up thousandfold. This is not a problem in the second case.

The second formula also has the advantage of being well known. It is called the Laspeyres index. The problem you are trying to solve has plagued economists for long. So much so that cutting edge macroeconomic texts of the early twentieth century would devote full chapters to it.

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