Consumption II: Laspeyres and Paasche price indices

Price indices are used to monitor changes in prices levels over time. This is useful when separating real income from nominal income, as inflation is a drain on purchasing power. The two most basic indices are the Laspeyres index (named after Etienne Laspeyres) and the Paasche index (named after Hermann Paasche).

Both indices are very similar:

They work by dividing expense on a specific basket in the current period (the sum of p*q for each product in the basket considered when calculating the index) by how much the same basket would cost in the base period (period 0). The main difference is the quantities used: the Laspeyres index uses q₀ quantities, whereas the Paasche index uses period n quantities.

What this translates to is that a Laspeyres index of 1 means that, as the nominator is the same as the denominator, an individual can afford the same basket of goods in the current period as he did in the base period. As the quantities are the same, this just leaves price as a variable, which must remain unchanged. This translates to the concept of compensated variation (CV): by how much do we need to increase an individual’s income in order to offset inflation? This is, at the new level of prices, how much is required to compensate the effect of the price increase? In the top diagram, we see that an increase in the price of good x₁ means a move to a lower utility curve. The compensated variation is the theoretical amount of money the individual would need to maintain their level of utility, putting them back on their original utility curve.

What we can also see is that the Laspeyres index overestimates this CV: it assumes that inflation has a greater effect than it does. This can be more clearly seen in the bottom diagram. The change in price from P₀ to P₁ leads to a change in the quantity of X consumed from X₀ to X₁. The green rectangle shows the Laspeyres effect associated with this, which is greater than the CV effect. The CS effect, the dark blue trapezoid, shows the loss of consumer surplus associated with the price variation.

A Paasche index of 1 means that the consumer could have afforded the same bundle of goods in the base period as they can now. This can be translated to the concept of equivalent variation (EV): how much income would we have to take away from an individual, at the base price level, to have the same impact on their utility as the inflation between the base period and period 1? That is, if we took the individual’s utility in 2009 at that price level, how much would we have to take away from it to have the same utility as a person on the same income with 2012’s level of prices? Applying the same dynamic we applied to the Laspeyres index, we can see from the top diagram that the Paasche index underestimates the equivalent variation.

We can see this more clearly in the bottom diagram: the consumer surplus is greater than the variation effect, which is in turn greater than the Paasche effect.

If we put this all together, it may be easier to understand each of the effects and understand their downisdes:

These diagramas are simply a combination of the two we saw individually. They help us, however see the differences between both indices. The main downside to these indices is the fact that they do not take into effect substitution effects. When the price of something rises, we tend to consume less of it. Because the Laspeyres index uses base period quantities, it tends to overestimate inflation by assuming that individuals’ income expense is still distributed in the same way. The opposite is true of the Paasche index: because it uses current period quantities, it underestimates inflation.

Therefore, for normal goods, if inflation exists: