Why Express Prices as an Index?
A price measure may be recorded in dollars, but its dollar value alone does not reveal how much it has changed relative to an earlier period. Expressing the measure as an index makes that comparison direct: the chosen base period is assigned a value of 100, and other periods are shown relative to it. The index is a scale for comparison, not a price quoted in dollars.
Price indexes are used to summarize movements in defined sets of prices. The precise meaning of an index depends on which prices it covers and how they are combined. Here the focus is the general idea of relative measurement and base-year normalization; constructing a particular consumer price index is a separate task.
A base period is the reference period used for comparison. It may be a year or another stated period. In a normalized index, the measure for that period is assigned 100 by convention. This does not mean that prices in the base period were $100; it means later index values are scaled relative to that period.
The Normalization Formula
Let \(P_t\) denote a specified measure of the price level in period \(t\), and let \(P_b\) denote the same measure in base period \(b\). Both measures must be defined consistently: they must refer to the same scope and be calculated on a comparable basis. The normalized price index is
The ratio \(P_t/P_b\) expresses the current measure as a fraction of the base-period measure. Multiplication by 100 puts that ratio on the familiar scale where the base equals 100. Because the numerator and denominator have the same units, the ratio—and therefore the index—is dimensionless.
This formula also gives a direct interpretation of an index value. If \(I_t=100\), the measured price level equals its base-period value. If \(I_t>100\), it is above the base-period value; if \(I_t<100\), it is below. The percentage difference from the base is \(I_t-100\) percent. This is a comparison with the base period, not necessarily a change over one year.
Worked Examples
Worked Example: Normalizing a Price Measure
A hypothetical equipment-repair cost measure is 80 units in its chosen base year and 110 units four years later. The units are an internal measure for this example, not dollars. Find the later normalized index and interpret it. Assume the measure has been defined consistently in both years.
Method. Divide the later measure by the base measure and multiply by 100:
The base-year index is 100 by construction. The later index is 137.5, so the measure is \(137.5-100=37.5\%\) above its base-year value.
Check. Reversing the index formula gives \(P_t=(I_t/100)P_b=1.375(80)=110\), the stated later measure. The common mistake is to say the measure increased by 137.5%; the index is 137.5, but the increase relative to the base is 37.5%.
Worked Example: Changing the Reference Period
A hypothetical regional materials index is reported with 2018 as its base period. Its values are 100 in 2018, 124 in 2022, and 150 in 2024. Express the 2024 value on a scale where 2022 equals 100, and interpret the result.
Method. Treat the old 2022 index value, 124, as the reference measure and compare the 2024 index value, 150, with it:
To check the conversion, the 2022 value on the new scale is \(100(124/124)=100\). The 2024 value is about 121.0, meaning the measured level in 2024 is about 21.0% above its 2022 level. If the same conversion is applied to 2018, its value becomes \(100(100/124)=80.6452\ldots\), or about 80.6 on the new scale.
Check. The ratio \(150/124\) is about 1.2097, so the converted index must be just over 120. The common mistake is to subtract 124 from 150 and report 26 as the new index. That difference is in old index points; it does not set 2022 equal to 100.
Worked Example: Comparing Relative Movements
A hypothetical analyst creates two consistently measured price indexes for the same two periods, both with 2020 as the base. Index A is 126 in 2024, and Index B is 122 in 2024. Compare their changes from the base period. Do not interpret the indexes as dollar prices.
Method. For each index, subtract the base value of 100. Index A is \(126-100=26\%\) above its base-period measure. Index B is \(122-100=22\%\) above its base-period measure.
The measured increase for A exceeds the measured increase for B by \(26-22=4\) percentage points over the shared base-to-2024 comparison. Equivalently, their index values differ by four index points in 2024. These statements describe relative movements; they do not establish that A's prices are four dollars higher than B's.
Check. Both indexes use 2020 as the base and equal 100 then, so comparing their percentage changes from that base is meaningful for the stated purpose. The common mistake is to call 126 a price that is 126% higher than B's price. It means Index A's own measured level is 26% above its own base-period level.
Worked Example: Recovering a Price Measure
A hypothetical service-cost index equals 118.4 when its base-period cost measure is $250. Find the current measure in dollars and state its percentage difference from the base.
Method. Rearrange the normalization formula to recover the measure:
The measure is \(118.4-100=18.4\%\) above its base-period value. The calculation is consistent because \(100(296/250)=118.4\). The common mistake is to read 118.4 as a dollar amount; the index is dimensionless, while the recovered measure is in dollars.
How to Read an Index Carefully
An index is meaningful only alongside information about its base period and the price measure it represents. Two indexes may both equal 100 in their own base periods yet cover different goods, services, or populations. Their numerical values should not be treated as directly comparable price levels merely because both use the same normalization convention.
When an index is converted to a different base, its numerical scale changes, but the underlying relative comparisons do not. For any two periods \(s\) and \(t\), the ratio of their index values remains \(I_t/I_s=P_t/P_s\), provided the values come from the same consistently defined series. Thus, changing the reference period changes where the scale reads 100, not the underlying comparison between periods.
This cancellation is why an index can be rebased without changing the relative movement within a series. It does not mean that indexes with different coverage or construction can be freely combined; the underlying measures must remain comparable.
An index level also does not, on its own, state how quickly prices changed. An index of 120 says the measured level is 20% above its stated base-period level. It does not say whether that movement took one year or ten years. To describe a change between particular periods, compare those periods' index values; a rate for a specified interval requires attention to the interval's length.
Common Exam Traps
- Confusing index level with percentage increase. An index of 108 is 8% above a base of 100, not 108% above it.
- Confusing index points with dollars. An index of 125 is not a price of $125. The index has no dollar unit.
- Using the wrong denominator. For a base-normalized index, divide by the base-period measure, not by the current measure.
- Ignoring the reference period. “The index is 115” is incomplete without knowing what period equals 100 and what the index measures.
- Calling a base comparison an annual change. An index 15% above its base is 15% above that reference level, regardless of how much time has passed.
A disciplined interpretation names the measure, the reference period, and the direction and size of the difference. This avoids treating the index as a raw price or as a rate. The normalization and interpretation principles in this tutorial are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
Check Your Understanding
For each problem, identify the base period or comparison being used before interpreting the index.
- A hypothetical agricultural input measure is 64 units in 2017 and 80 units in 2022. Calculate the 2022 index with 2017 as the base, then state the percentage difference from the base. (Answer: Index 125; 25% above the 2017 measure.)
- A service-price index has a base of 100 in 2021 and a value of 109.5 in 2024. State what the 2024 value means relative to 2021, and whether it gives an annual change. (Answer: 9.5% above the 2021 level; it does not by itself give an annual change.)
- A series has values 100 in 2010, 115 in 2015, and 138 in 2020, with 2010 as the original base. Convert the 2015 and 2020 values to a scale where 2015 equals 100. (Answer: 2015 = 100; 2020 = 120.)
- Two hypothetical indexes share a base period and measure comparable price movements. Their values in the later period are 112 and 108. State each percentage difference from the base and the difference between those changes in percentage points. (Answer: 12% and 8% above base; a 4-percentage-point difference.)
- A hypothetical cost measure has a base-period value of $750 and a current index of 96. Recover the current measure and interpret it relative to the base. (Answer: $720.00; 4% below the base-period measure.)