Laspeyres, Paasche and Fisher Index Numbers Made Simple

Imagine buying groceries for your family. Your bill has increased, but you also bought less rice and more milk this time. How much of the change came from prices, and how much came from the contents of your basket?

That is the puzzle behind weighted price index numbers. Comparing two bills directly mixes up two changes. To study prices, we need to hold the shopping basket still.

Laspeyres holds the old basket still. Paasche holds the new basket still. Fisher combines the two answers using their geometric mean.

Once you can picture those two baskets, the formulas become much easier to reconstruct. Let’s build them, use one complete numerical, and then tackle the questions that usually cause mistakes.

What do p0, p1, q0 and q1 mean?

The subscript tells you the period. In the notation below, 0 means the base period and 1 means the current period being compared with it.

Symbol	Meaning
p0	Price per unit in the base period
p1	Price per unit in the current period
q0	Quantity in the base period
q1	Quantity in the current period
Sum	Add the relevant products across all items

The current period is the period specified in the question. It does not have to be today’s calendar year.

For example, p1q0 means current price multiplied by base-period quantity. It answers a hypothetical question: what would the old quantity cost at the new price?

Every product such as p1q0 is calculated separately for each commodity before the products are added. Never multiply the sum of all prices by the sum of all quantities.

Assume throughout our examples that the goods are comparable across periods, their quality is unchanged, prices and quantities use matching units, and all required totals are positive.

The three price-index formulas

With the base period set equal to 100:

Laspeyres price index, L = [Sum(p1q0) / Sum(p0q0)] x 100

Paasche price index, P = [Sum(p1q1) / Sum(p0q1)] x 100

Fisher price index, F = square root of (L x P)

Here, L and P are already index numbers on the base-100 scale. Do not multiply their geometric mean by 100 again.

Method	Quantity basket held fixed	Question it answers
Laspeyres	Base-period quantities, q0	How much more or less would the old basket cost?
Paasche	Current-period quantities, q1	How much more or less does the new basket cost than it would at old prices?
Fisher	Combines both calculations	What is the geometric mean of those two price comparisons?

These are the standard weighted aggregate formulas presented in the price index theory guide from the Australian Bureau of Statistics.

Why are quantities called weights?

Suppose your family buys 20 kg of rice but only one small jar of jam. An equal percentage increase in their prices need not have an equal effect on your budget.

In an aggregate price formula, multiplying prices by quantities gives each item an appropriate place in the basket’s cost. The resulting influence depends on expenditure, not merely on how many physical units there are.

Do not compare 20 kg with one jar as though those units were interchangeable. Convert each item’s price and quantity into a money value first.

A complete solved numerical: one table, three answers

A household buys the following goods. All prices are in rupees per stated unit. The figures are illustrative, not actual market prices.

Item and unit	p0	p1	q0	q1
Rice, kg	20	30	10	6
Milk, litre	50	60	4	5
Soap, bar	40	44	5	6

Calculate Laspeyres, Paasche and Fisher price indices.

Step 1: Make the four product columns

Write the headings before doing the multiplication. That small habit prevents a surprisingly common error: calculating the right numbers under the wrong labels.

Item	p0q0	p1q0	p0q1	p1q1
Rice	200	300	120	180
Milk	200	240	250	300
Soap	200	220	240	264
Total	600	760	610	744

Read the totals aloud once:

• Rs 600 is the old basket at old prices.
• Rs 760 is the old basket at new prices.
• Rs 610 is the new basket at old prices.
• Rs 744 is the new basket at new prices.

The two middle descriptions are particularly useful. They are hypothetical bills that let us separate price movements from changes in purchases.

Step 2: Calculate Laspeyres

L = (760 / 600) x 100 = 126.67, approximately

The old basket now costs Rs 760 instead of Rs 600. Its cost has increased by Rs 160.

Percentage increase = (160 / 600) x 100 = 26.67%.

So the index of 126.67 means the cost of the base-period basket is about 26.67% higher. It does not mean prices rose by 126.67%.

Step 3: Calculate Paasche

P = (744 / 610) x 100 = 121.97, approximately

We now take the current basket: 6 kg of rice, 5 litres of milk and 6 bars of soap. It costs Rs 744 at current prices, compared with Rs 610 at base-period prices.

Its cost is about 21.97% higher under that comparison.

Notice the denominator carefully. It is Rs 610, not the actual old bill of Rs 600. Paasche compares the same current basket under two price systems.

Step 4: Calculate Fisher

Use the unrounded values from the fractions:

F = square root of [(760 / 600 x 100) x (744 / 610 x 100)]

F = 124.29, approximately

Fisher’s method therefore indicates an overall price increase of approximately 24.29% relative to the base period for these goods.

The answer lies between 121.97 and 126.67, as the geometric mean of two positive numbers should.

Why do Laspeyres and Paasche give different answers?

In our example, rice rises from Rs 20 to Rs 30, a 50% increase. Milk rises by 20%, and soap by 10%.

The old basket contains 10 kg of rice. The new basket contains only 6 kg. The steepest price increase therefore receives less emphasis in the new basket, helping explain why Paasche is lower here.

This fits a familiar consumer response: people may buy less of a good whose relative price has increased and shift towards alternatives. The Office for National Statistics explains how this substitution affects the choice of weights.

But do not turn a common pattern into an unconditional rule.

Laspeyres is not always greater than Paasche. Quantities may change because of preferences, income, availability or other circumstances. A numerical can easily put greater current weight on the item with the larger price increase.

A quick counterexample

Suppose two goods both initially cost Rs 10 and have base quantities of one unit each. Their current prices are Rs 20 and Rs 10, while current quantities are three units and one unit.

Laspeyres = [(20 x 1 + 10 x 1) / (10 x 1 + 10 x 1)] x 100 = 150.

Paasche = [(20 x 3 + 10 x 1) / (10 x 3 + 10 x 1)] x 100 = 175.

Here Paasche is higher. Fisher is the square root of (150 x 175), or approximately 162.02, which still lies between them.

Use the between-the-two check for Fisher. Do not reject a correct answer merely because Paasche exceeds Laspeyres.

Price index versus value index: the bill is not the basket test

Return to our household. Actual expenditure rises from Rs 600 to Rs 744.

Value index = [Sum(p1q1) / Sum(p0q0)] x 100

Value index = (744 / 600) x 100 = 124.

Actual spending has increased by 24%. This differs from all three price-index results because the quantities purchased also changed.

The value index is useful, but it answers a different question. Calling it a price index would attribute the entire change in spending to prices.

A simple check makes this distinction memorable: if all prices stay the same but every quantity doubles, the price indices remain 100 while the value index becomes 200.

For the wider meaning of price measures, see our guide to CPI, WPI and the Index of Industrial Production.

Fisher and the extra-100 trap

There are two correct ways to write Fisher. Choose one and stay consistent.

Using index numbers:

F = square root of (L x P).

For example, if L = 144 and P = 100, then F = square root of 14,400 = 120.

Using raw ratios:

F = square root of {[Sum(p1q0) / Sum(p0q0)] x [Sum(p1q1) / Sum(p0q1)]} x 100.

The equivalent calculation is square root of (1.44 x 1.00) x 100 = 120.

The arithmetic mean, (144 + 100) / 2 = 122, is not Fisher’s index. Nor should you take the square root of 14,400 and then multiply by 100.

The IMF’s discussion of basic index number theory defines Fisher using this geometric mean and explains its symmetry between the two periods.

Finding a missing index

Because F squared = L x P, you can rearrange without rebuilding a commodity table.

If F = 126 and L = 132.3:

P = F squared / L = 15,876 / 132.3 = 120.

Check: square root of (132.3 x 120) = 126.

This shortcut is valid only when all three supplied indices use the same scale and refer to the same comparison.

What if the question gives expenditure instead of quantity?

A column headed expenditure is not a quantity column. Use the relationship:

Quantity = expenditure / price per unit.

Suppose base-period expenditure on an item is Rs 480 and its base price is Rs 40 per kg. Then q0 = 480 / 40 = 12 kg.

If current expenditure is Rs 550 and the current price is Rs 50 per kg, then q1 = 550 / 50 = 11 kg.

The four products for this item are then:

• p0q0 = Rs 480.
• p1q0 = Rs 600.
• p0q1 = Rs 440.
• p1q1 = Rs 550.

For one item, both price indices equal (50 / 40) x 100 = 125. Actual expenditure rises by only about 14.58%, because quantity falls.

When many items are given, repeat this reconstruction row by row. Do not divide total expenditure across all goods by an average price to invent a combined quantity.

Are expenditure weights the same as quantity weights?

No. The form of the formula matters.

In a weighted aggregate formula, multiply prices by the selected quantities. If the question instead supplies price relatives, R = (p1 / p0) x 100, then a weighted arithmetic average using base expenditures gives Laspeyres:

L = Sum(R x p0q0) / Sum(p0q0).

You can verify this by cancelling p0 within each numerator term. For our main example the price relatives are 150, 120 and 110. Base expenditures are all Rs 200, so:

L = (150 x 200 + 120 x 200 + 110 x 200) / 600 = 126.67.

Do not automatically take an arithmetic average with current expenditures and label it Paasche. With current expenditures as weights, Paasche has a weighted harmonic-mean form. For routine aggregate numericals, the four-column method avoids this extra complication.

Why is Fisher called the ideal index?

In the usual textbook treatment, Fisher is called ideal because it satisfies the time reversal and factor reversal tests. These are consistency checks, not a promise that every dataset or real-world measurement is perfect.

The ICAI lesson on index numbers provides further practice with these formulas and index-number relationships.

Time reversal: what happens when the periods swap?

On a ratio scale, a forward Fisher price index multiplied by its reverse equals 1. On the base-100 scale used here:

Forward Fisher index x reverse Fisher index = 10,000.

For our original example, the reverse Fisher index is approximately 80.45. Using the unrounded figures, 124.2947317 x 80.4539329 is approximately 10,000.

A forward increase of 24.29% does not imply an equal percentage decrease on reversing the comparison. The denominator changes.

When actually recalculating the reverse index, swap the periods throughout, including both prices and quantities. Do not reverse just one price column.

Factor reversal: separating price and quantity changes

The quantity-index formulas exchange the roles of prices and quantities:

Laspeyres quantity index, QL = [Sum(p0q1) / Sum(p0q0)] x 100

Paasche quantity index, QP = [Sum(p1q1) / Sum(p1q0)] x 100

Fisher quantity index, QF = square root of (QL x QP)

Using our four totals:

QL = (610 / 600) x 100 = 101.67, approximately.

QP = (744 / 760) x 100 = 97.89, approximately.

QF = square root of [(610 / 600 x 100) x (744 / 760 x 100)] = 99.76, approximately.

On a base-100 scale, the factor reversal relationship is:

Fisher price index x Fisher quantity index / 100 = value index.

Using unrounded answers, 124.2947317 x 99.7628768 / 100 = 124, matching our value index.

The quantity index is a weighted measure of volume. It is not obtained by adding kilograms, litres and bars together. A value below 100 can therefore coexist with increases in the physical quantities of some goods.

When do all three price indices agree?

Two useful sufficient conditions help you check a numerical.

If q1 = q0 for every item, the two baskets are identical. Laspeyres and Paasche become the same fraction, and Fisher equals them.

More generally, they also agree when current quantities are the same positive multiple of base quantities for every item. That common multiplier cancels from Paasche.

If every price changes by the same proportion, both baskets experience the same proportional cost change, even if their contents differ.

For example, if every price rises by 15%, all three indices equal 115. If every price is unchanged, all three equal 100.

These are sufficient conditions, not the only possible ways numerical equality can occur.

Common mistakes and the repair

Mistake	What to do instead
Use q0 above and q1 below in a price index	Keep one basket fixed within each ratio
Put Sum(p0q0) below Paasche	Use Sum(p0q1), the current basket at old prices
Call current spending divided by old spending a price index	Identify it as the value index
Average L and P by adding and dividing by two	Take the square root of their product
Multiply Fisher by 100 twice	Check whether L and P already use base 100
Write that an index of 124 means a 124% increase	Subtract 100 to get the increase from the base
Treat Laspeyres as always the largest	Calculate from the given data; ordering can reverse
Round every product or intermediate ratio too early	Retain precision until the final answer

One more interpretation trap: if a comparable index series moves from 120 to 132, that is a rise of 12 index points, but the percentage rise is (132 / 120 - 1) x 100 = 10%.

Subtracting 100 works for a comparison with the base of 100. For two other index levels, divide the later value by the earlier value first.

A practical answer-writing routine

Start by identifying whether the question asks for a price, quantity or value index. Then mark the base and current periods and label the supplied columns.

For all three price indices, prepare the four product columns, total them and write each formula before substituting. Keep calculator precision for Fisher and finish with one sentence interpreting the answer relative to the base period.

Before moving on, ask: have I compared the same basket at two sets of prices? Does Fisher lie between the two positive indices? Did I distinguish the index level from the percentage change?

Laspeyres needs base quantities and both sets of prices. Paasche needs current quantities and both sets of prices. Fisher generally needs the information for both. If current quantities are missing, do not silently substitute base quantities unless the question explicitly says the quantities are unchanged.

Try these questions yourself

1. Sum(p0q0) = 800 and Sum(p1q0) = 920. Calculate Laspeyres and interpret it.
2. Sum(p0q1) = 900 and Sum(p1q1) = 990. Calculate Paasche and interpret it.
3. Laspeyres is 125 and Paasche is 115. Find Fisher to two decimal places.
4. Fisher is 120 and Paasche is 112.5. Find Laspeyres.
5. Every price falls by 8%, while quantities change. What are the three price indices?
6. Actual expenditure doubles while all prices stay unchanged. What happens to the price indices and value index?
7. A comparable price index rises from 125 to 140. Find the index-point rise and percentage rise.
8. Only p0, p1 and q0 are supplied. Which of the three price indices can you normally calculate?

Answers with short workings

1. (920 / 800) x 100 = 115. The base basket costs 15% more.
2. (990 / 900) x 100 = 110. The current basket costs 10% more than at base prices.
3. Square root of (125 x 115) = 119.90, approximately.
4. L = 120 squared / 112.5 = 128.
5. 92 for all three, assuming comparable goods and valid positive totals.
6. 100 for each price index; 200 for the value index. The spending increase comes from quantities.
7. 15 index points; 12%, because (140 / 125 - 1) x 100 = 12%.
8. Laspeyres. Paasche and Fisher generally require current quantities too.

Frequently asked questions

What is the main difference between Laspeyres and Paasche?

Laspeyres holds base-period quantities fixed while comparing prices. Paasche holds current-period quantities fixed. Their different baskets can produce different measures of overall price change.

What is Fisher’s index-number formula?

Fisher’s price index is the square root of the product of Laspeyres and Paasche price indices. If both are already on a base-100 scale, their geometric mean is also on that scale.

Why does Paasche use p0q1 in the denominator?

It measures what the current basket would have cost at base-period prices. Using that denominator keeps current quantities fixed, so the ratio isolates the price comparison for that basket.

Is Fisher the arithmetic average of Laspeyres and Paasche?

No. It is their geometric mean. Multiply the two indices and take the square root; do not add them and divide by two.

Is Laspeyres always higher than Paasche?

No. It is often higher in consumer examples involving substitution away from goods with larger relative price increases, but the ordering depends on the data. Fisher lies between the two positive indices whichever is larger.

What does a price index of 87 mean?

On a base-100 scale, it indicates that the measured price level is 13% lower than in the base period. It does not mean that every individual item’s price fell by 13%.

Can I calculate Fisher without current quantities?

Generally, you cannot calculate it from base quantities and prices alone. You need enough information to obtain both Laspeyres and Paasche. Given values of those indices, or an explicit assumption that quantities are unchanged, can make a calculation possible.

Why is Fisher called an ideal index?

It satisfies the time reversal and factor reversal tests and treats the two periods symmetrically. The label does not remove practical problems such as poor data, changing product quality or unrepresentative goods.

Can I use total spending to measure the price increase?

Not directly when quantities change. Current spending divided by base spending gives a value comparison, reflecting both prices and quantities. A price index holds a basket fixed or combines such comparisons.

How do I remember the formulas without mixing them up?

Picture two baskets. Laspeyres reprices the old one, so both quantity subscripts are 0. Paasche reprices the current one, so both are 1. Fisher then takes the geometric mean of their answers.