Why the Choice of Basket Matters
A price index compares the cost of goods or services across periods. The comparison depends not only on the prices, but also on which quantities are used to form the basket. The Laspeyres and Paasche price indexes use the same price observations but different baskets: Laspeyres holds the base-period quantities fixed, while Paasche uses current-period quantities.
This distinction is useful when measured prices change and the quantities purchased also change. The two indexes answer related but different questions, so they need not have the same value. These measures are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
For \(n\) items, the formulas are:
Here, \(P_L\) is the Laspeyres index and \(P_P\) is the Paasche index. The numerator in each formula is the cost of its specified basket at current prices; the denominator is the cost of that same basket at base prices. The quantities must be the same in the numerator and denominator of a particular index. The basket changes between the two indexes, not between the two cost calculations within one index.
These formulas require comparable prices for the same set of items in both periods, quantities measured in consistent units, and a positive denominator. If the set of items changes or a price is not meaningfully comparable across periods, the simple formulas alone do not resolve that measurement problem. In the calculations below, assume the listed items are comparable and every required price and quantity is available.
Deriving the Index Formulas
The formulas follow directly from the definition of a price comparison. Choose a basket of items, calculate its total cost at current prices, and divide by its total cost at base prices. Multiplying the cost ratio by 100 makes the base-period value 100 when the basket is priced at base-period prices.
For Laspeyres, use the base quantities \(q_0\). For Paasche, use the current quantities \(q_1\).
For a chosen quantity \(q_i\), its total cost at prices \(p_i\) is the sum \(\sum_i p_iq_i\).
Divide the current-price cost by the base-price cost for the same basket. Multiplication by 100 puts the index on a base-100 scale.
For the Laspeyres index, the basket is \(q_0\). Its current-price cost is \(\sum_i p_{1i}q_{0i}\), and its base-price cost is \(\sum_i p_{0i}q_{0i}\), giving the Laspeyres formula. Replacing that fixed basket with \(q_1\) gives the Paasche formula. As a check, if all prices are unchanged, then \(p_{1i}=p_{0i}\) for every item; the numerator equals the denominator in each index, so both equal 100.
Worked Calculations
Use a consistent order when calculating: identify the quantities required for the index, calculate each basket’s total cost at both sets of prices, then form the ratio. Keeping the basket visible helps prevent a common substitution error.
Worked Example: Comparing Two Indexes for a Three-Item Basket
A hypothetical household buys rice, transit rides, and notebooks. Rice costs $2 per unit in the base period and $3 currently; the household’s quantities are 10 and 8 units, respectively. A transit ride costs $4 in the base period and $5 currently, with quantities 5 and 6. A notebook costs $1 in the base period and $0.80 currently, with quantities 20 and 25. Calculate both indexes, using a base of 100.
Method. Use base quantities throughout the Laspeyres calculation and current quantities throughout the Paasche calculation. The trap is to use current quantities in only one part of a Laspeyres ratio, or base quantities in only one part of a Paasche ratio.
Laspeyres index. The base-period basket costs \(2(10)+4(5)+1(20)=20+20+20=60\) at base prices. Valued at current prices, the same quantities cost \(3(10)+5(5)+0.80(20)=30+25+16=71\). Therefore,
Paasche index. The current-period basket costs \(3(8)+5(6)+0.80(25)=24+30+20=74\) at current prices. At base prices, those same current quantities cost \(2(8)+4(6)+1(25)=16+24+25=65\). Therefore,
Rounded to one decimal place, the Laspeyres index is 118.3 and the Paasche index is 113.8. The base basket costs 18.3% more at current prices than at base prices; the current basket costs 13.8% more at current prices than the same current quantities would have cost at base prices. Both results are checked against their cost ratios: each is above 100 because its current-price basket cost exceeds its base-price cost.
Worked Example: A Paasche Index Below 100
A hypothetical workshop purchases two supplies. Supply A costs $6 in the base period and $9 currently; quantities are 4 in the base period and 3 currently. Supply B costs $10 in the base period and $8 currently; quantities are 2 in the base period and 5 currently. Find both indexes on a base of 100.
Method. Calculate each index from its own basket. Do not assume that both indexes must be above 100 just because one item’s price rose.
For Laspeyres, use the base quantities. The base-price cost is \(6(4)+10(2)=24+20=44\). The same basket at current prices costs \(9(4)+8(2)=36+16=52\). Thus,
For Paasche, use the current quantities. Their current-price cost is \(9(3)+8(5)=27+40=67\). Their base-price cost is \(6(3)+10(5)=18+50=68\). Thus,
Rounded to one decimal place, the indexes are 118.2 and 98.5, respectively. Paasche is below 100 because, for the current basket, the total cost at current prices is 67, less than its base-price cost of 68. This is consistent with the current basket containing more of B, whose price fell, and less of A, whose price rose. The result is not an arithmetic contradiction: the two indexes price different baskets.
Worked Example: Reading the Basket Weights
A hypothetical catering service buys three ingredients. In the base period, ingredient A costs $4 and the service buys 8 units; ingredient B costs $5 and it buys 4 units; ingredient C costs $2 and it buys 10 units. Current prices are $5, $6, and $1.50, respectively, and current quantities are 6, 6, and 12. Calculate both indexes and describe the basket comparison.
Method. First total the cost of each basket at each period’s prices. The quantities show how each item enters the basket; the prices determine its cost.
The base basket costs \(4(8)+5(4)+2(10)=32+20+20=72\) at base prices. At current prices, it costs \(5(8)+6(4)+1.50(10)=40+24+15=79\). Thus,
The current basket costs \(5(6)+6(6)+1.50(12)=30+36+18=84\) at current prices. At base prices, the same current quantities cost \(4(6)+5(6)+2(12)=24+30+24=78\). Thus,
Rounded to one decimal place, Laspeyres is 109.7 and Paasche is 107.7. Both are above 100 because each basket costs more at current prices. Their difference reflects the changed quantities as well as the price changes: for example, current purchases include fewer units of A and more of C than the base basket. It would be incorrect to attribute the entire difference to a different price change; the prices are identical in the two calculations, but the baskets differ.
How the Indexes Weight Price Changes
The quantities in each index determine the relative influence of each item’s price change. For Laspeyres, the base-period expenditure on item \(i\) is \(p_{0i}q_{0i}\). Dividing it by total base-period expenditure gives the item’s base expenditure share \(w_{0i}\). Because these shares sum to 1, the Laspeyres index can also be written as a weighted average of the individual price relatives:
To see why, multiply each price relative \(p_{1i}/p_{0i}\) by its base expenditure share. The product is \(p_{1i}q_{0i}/\sum_jp_{0j}q_{0j}\); summing across items and multiplying by 100 gives the Laspeyres formula. Thus, an item with a larger share of base-period expenditure has more influence on the Laspeyres index.
Paasche uses current quantities instead. Its current-period expenditure shares are \(w_{1i}=p_{1i}q_{1i}/\sum_jp_{1j}q_{1j}\). Equivalently, its price ratio can be written as the reciprocal of a current-share-weighted average of inverse price relatives:
This form follows because \(w_{1i}(p_{0i}/p_{1i})=p_{0i}q_{1i}/\sum_jp_{1j}q_{1j}\). Summing gives the cost of the current basket at base prices divided by its cost at current prices; taking the reciprocal and multiplying by 100 gives Paasche. This representation makes clear that Paasche’s price comparison is tied to current-period expenditure shares, while its direct formula prices the current-quantity basket in both periods.
| Feature | Laspeyres | Paasche |
|---|---|---|
| Basket quantities | Base-period quantities \(q_0\) | Current-period quantities \(q_1\) |
| Current-price cost in numerator | \(\sum_i p_{1i}q_{0i}\) | \(\sum_i p_{1i}q_{1i}\) |
| Base-price cost in denominator | \(\sum_i p_{0i}q_{0i}\) | \(\sum_i p_{0i}q_{1i}\) |
| Price-change weighting | Base-period expenditure shares | Current-period quantities; equivalently, current shares applied to inverse price relatives and reciprocated |
Interpreting Differences and Avoiding Traps
Neither index is guaranteed to be larger than the other. Their ordering depends on the price and quantity patterns. If quantities shift toward items with relatively smaller price increases, the current basket may produce a lower Paasche index than Laspeyres, as in the first worked example. Other patterns can reverse the ordering. The calculation itself establishes the result; a general rule that one index must always exceed the other is not justified.
- Wrong quantities in the formula. Laspeyres uses \(q_0\) in both cost totals; Paasche uses \(q_1\) in both. Mixing the periods does not calculate either index.
- Confusing an index value with a percentage change. An index of 118.3 on a base of 100 means the specified basket’s cost is 18.3% above its base-period cost. It does not mean the cost is 118.3% above the base.
- Comparing different denominators as if they were the same basket. The denominators differ when \(q_0\) and \(q_1\) differ, so interpret each ratio using its own basket.
- Assuming every item’s price change has equal influence. The index weights depend on quantities and expenditure shares, not simply on the number of items.
A fast exam check is to write the quantity vector beside each index before doing arithmetic: Laspeyres \(q_0\), Paasche \(q_1\). Then verify that each quantity vector appears in both the numerator and denominator of its ratio. Finally, check that an index of 100 results when all prices are unchanged, and explain any value above or below 100 by comparing the two costs of that index’s basket.
Check Your Understanding
For each problem, calculate the requested index or indexes on a base of 100. Keep the quantities fixed within each index calculation.
- A workshop buys 5 units of material A and 3 of material B in the base period. Base prices are $2 and $4; current prices are $3 and $5. Current quantities are 4 units of A and 4 of B. Find both indexes. (Answer: Laspeyres \(136.3636\ldots\); Paasche \(133.3333\ldots\).)
- A food stall buys 2 units of A and 6 of B in the base period, when their prices are $10 and $5. Current prices are $12 and $4, and current quantities are 3 of A and 8 of B. Find both indexes and state which is below 100. (Answer: Laspeyres \(96.0\); Paasche \(97.1429\); both are below 100.)
- A studio’s base-period quantities are 4 lamps and 2 screens; base prices are $3 and $8. Current prices are $6 and $10, and current quantities are 2 lamps and 3 screens. Find both indexes, rounded to one decimal place. (Answer: Laspeyres \(157.1\); Paasche \(140.0\).)
- Two items have base-period expenditure shares of 0.25 and 0.75. Their current-to-base price relatives are 1.20 and 0.90. Find the Laspeyres index and interpret it relative to 100. (Answer: \(97.5\); the base basket’s cost is 2.5% below its base-period cost.)
- For a current basket, current-price cost is $360 and base-price cost is $300. For the base basket, current-price cost is $440 and base-price cost is $400. Calculate both indexes and identify which basket shows the greater measured increase. (Answer: Paasche \(120.0\), Laspeyres \(110.0\); the current basket shows the greater increase.)