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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'?
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It seems to me that in both your calculations you are using the volume of sales in 2011 but not the volume of sales in 2012. Is this what you want? Does the average change in price depend on the volume in 2011, but not on the volume in 2012? –  Gerry Myerson Jan 20 '13 at 23:31
    
Note also that $(4-3)/3$ isn't $0.33$, it is $0.33333333\dots$. I wonder whether the two answers you got would be equal, if you hadn't introduced this roundoff error. –  Gerry Myerson Jan 20 '13 at 23:37
    
I actually used the exact value 1/3 for the calculation. I fixed the question to clarify this –  ewolter Jan 21 '13 at 10:25
    
I'm not quite sure what you mean exactly. I need to use something as the base value to calculate the change? I'm interested in the combined price increase/decrease percentage. How would you even include the year 2012, and what would this implicate? Again my intuition is simply missing for this calculation... –  ewolter Jan 21 '13 at 10:29
    
The simplest thing in problems like this is to stop trying to be "clever" -- actually write down the algebraic problem you're trying to solve (which can include many new variables, such as a variable for "the average price in 2012") rather than trying to jump straight to a formula for its solution. –  Hurkyl Jan 21 '13 at 10:58
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1 Answer

up vote 1 down vote accepted

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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Thanks for giving me a view into the greater context of my question! I really helped a lot! –  ewolter Jan 21 '13 at 12:12
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