Paasche Price Index

The Paasche price index is a way to measure price change over time using current-period quantities as weights. It answers: “How much more (or less) would today’s basket cost at today’s prices compared with base-period prices?”

Formula

Let (p_t) be current prices, (p_0) base-period prices, and (q_t) current-period quantities. The Paasche price index is:

[ P_P(t,0) = \frac{\sum_i p_{t,i} q_{t,i}}{\sum_i p_{0,i} q_{t,i}}. ]

The numerator is the cost of the current basket at current prices.
The denominator is the cost of the same current basket at base prices.

Interpretation

Because it weights by (q_t), the Paasche index reflects the fact that consumers (or firms) may substitute toward relatively cheaper goods over time. In many settings, this makes Paasche growth rates lower than Laspeyres growth rates.

Comparison with Laspeyres and Fisher

Laspeyres index: weights by base-period quantities (q_0). It can overstate inflation when substitution is important.
Paasche index: weights by current quantities (q_t). It can understate inflation for similar reasons.
Fisher “ideal” index: the geometric mean of Laspeyres and Paasche, often used as a compromise:

[ P_F = \sqrt{P_L,P_P}. ]

Simple example

If consumers shift away from a good whose price rises sharply, (q_t) puts less weight on that good. The Paasche index therefore embeds some substitution and can show a smaller measured price increase than a fixed-base basket.

Knowledge Check

### The Paasche price index uses which quantities as weights? - [x] Current-period quantities \(q_t\) - [ ] Base-period quantities \(q_0\) - [ ] Equal weights for every good - [ ] No quantity weights at all > **Explanation:** Paasche compares prices using the current basket \(q_t\), not the base basket. ### Why can the Paasche index be lower than the Laspeyres index? - [x] It reflects substitution toward relatively cheaper goods in the current period - [ ] It always ignores services prices - [ ] It is computed in real terms rather than nominal terms - [ ] It uses a larger sample size by definition > **Explanation:** Weighting by \(q_t\) tends to put less weight on goods whose relative prices rose, embedding some substitution. ### The Fisher “ideal” price index is commonly defined as: - [x] The geometric mean of Laspeyres and Paasche: \(P_F=\sqrt{P_L P_P}\) - [ ] The arithmetic mean of Laspeyres and Paasche - [ ] The maximum of Laspeyres and Paasche - [ ] The Paasche index scaled by GDP > **Explanation:** Fisher combines the two fixed-basket measures to reduce bias from using only \(q_0\) or only \(q_t\).