Why Price a Fixed Basket?
Prices of individual goods and services do not necessarily move together. A household may face a higher price for one item and a lower price for another. A consumer price index summarizes the change in the cost of a specified collection of purchases by pricing the same quantities in each period. Holding quantities fixed helps isolate the effect of price changes from the effect of buying more or less.
The calculation begins with a basket: a list of goods and services, each with a stated quantity. For example, the basket might contain several kilograms of food, a number of transit rides, and a specified number of service visits. The basket is not necessarily what every household buys. It is the defined set of purchases used for the index calculation.
The quantities are held at their base-basket levels, \(q_j^0\), in both cost calculations. A price must be stated for the same item and unit in each period: for instance, the same quantity and grade of a product, or the same defined service. If an item’s unit changes, convert the price to a comparable unit before using it. The formula assumes that the listed prices are comparable across periods and that the basket contents and specifications have not changed.
The index is a relative measure, not a dollar amount. A base-period CPI of 100 means the basket costs \(C_0\) in that period. A comparison-period CPI above 100 means that the specified basket costs more than it did in the base period; an index below 100 means it costs less. The index value alone does not report the basket’s cost in dollars, so the cost calculations should remain visible in the working.
From Basket Quantities to an Index
For each item, multiply its price by the basket quantity to obtain that item’s cost in the period. Add those costs to find the basket cost. Do this once at base-period prices and again at comparison-period prices, always using the same quantities. Only after finding both totals should you divide and multiply by 100.
Write down each item and its fixed quantity \(q_j^0\), including the unit of measurement.
Multiply each base-period price by its basket quantity and sum the results to obtain \(C_0\).
Use the comparison-period prices with the unchanged basket quantities to obtain \(C_t\).
Calculate \(100(C_t/C_0)\). Check that the base-period cost is the denominator and that the index is 100 when \(t=0\).
Worked Example: Pricing a Small Household Basket
A hypothetical basket contains 10 kilograms of rice, 8 transit rides, and 3 laundry services. In the base period, their respective prices are $2.00 per kilogram, $2.50 per ride, and $4.00 per service. In the comparison period, the prices are $2.20, $2.75, and $4.50. Find the fixed-basket CPI for the comparison period. Report the index to two decimal places.
Method. Calculate the cost in each period using the stated quantities, then divide the comparison-period cost by the base-period cost. This method avoids substituting the comparison period’s quantities for the fixed basket.
| Item | Fixed quantity | Base price | Base-period cost | Comparison price | Comparison-period cost |
|---|---|---|---|---|---|
| Rice | 10 kg | $2.00/kg | $2.00 × 10 = $20.00 | $2.20/kg | $2.20 × 10 = $22.00 |
| Transit rides | 8 rides | $2.50/ride | $2.50 × 8 = $20.00 | $2.75/ride | $2.75 × 8 = $22.00 |
| Laundry services | 3 services | $4.00/service | $4.00 × 3 = $12.00 | $4.50/service | $4.50 × 3 = $13.50 |
| Total | $52.00 | $57.50 |
The base-period basket cost is \(C_0=\$20.00+\$20.00+\$12.00=\$52.00\). The comparison-period basket cost is \(C_t=\$22.00+\$22.00+\$13.50=\$57.50\). Therefore,
Rounded to two decimal places, the comparison-period CPI is 110.58. The specified basket costs more in the comparison period than in the base period. As a check, the base-period calculation would give \(100(52.00/52.00)=100\).
Why the Formula Has This Form
The index is designed to express the comparison-period basket cost as a percentage of the base-period basket cost. The ratio \(C_t/C_0\) compares the two dollar costs directly: a ratio of 1 means the costs are equal, and a ratio greater than 1 means the comparison-period cost is greater. Multiplication by 100 expresses that ratio on an index scale where the base period equals 100.
The same quantities appear in the numerator and denominator because the comparison is between the costs of one fixed basket, not between two different consumption patterns. Setting \(t=0\) makes the numerator and denominator identical, so \(100(C_0/C_0)=100\). This normalization is also a useful arithmetic check.
Worked Example: A Basket With Different Units
A hypothetical monthly basket contains 100 kilowatt-hours of electricity, 1 transit pass, 20 loaves of bread, and 2 clinic visits. The base-period prices are $0.12 per kilowatt-hour, $60 per pass, $3.00 per loaf, and $25.00 per visit. Comparison-period prices are $0.15 per kilowatt-hour, $66 per pass, $3.30 per loaf, and $27.50 per visit. Calculate the CPI to two decimal places.
Method. Keep the monthly quantities fixed, multiply each price by its corresponding quantity, and sum by period. The variety of units does not alter the calculation, provided each price is matched to the correct quantity and unit.
In the base period, electricity costs \(100\times\$0.12=\$12.00\); the transit pass costs \(1\times\$60.00=\$60.00\); bread costs \(20\times\$3.00=\$60.00\); and clinic visits cost \(2\times\$25.00=\$50.00\). Thus,
In the comparison period, electricity costs \(100\times\$0.15=\$15.00\); the transit pass costs \(1\times\$66.00=\$66.00\); bread costs \(20\times\$3.30=\$66.00\); and clinic visits cost \(2\times\$27.50=\$55.00\). Therefore,
The index is
Rounded to two decimal places, the CPI is 110.99. The check \(C_t>C_0\) agrees with an index above 100. Keep unrounded costs through the ratio in cases where the item costs do not sum to exact cents; round the reported index only at the final step.
What Must Stay Fixed—and What May Change
The defining feature of this calculation is not that prices stay fixed; prices are what the index is intended to compare. It is the basket quantities that stay fixed. If a household buys fewer units of an item in the comparison period, that change in actual behavior does not change \(q_j^0\) for this calculation. Similarly, an item’s comparison-period price must not be multiplied by a new quantity simply because that quantity appears in later-period spending data.
A fixed basket can include goods and services in different units. Add their costs as monetary amounts after multiplying each price by the matching quantity. Do not add prices alone: a price per kilogram and a price per visit cannot be meaningfully summed into a basket cost until each has been multiplied by its quantity.
The calculation is only as clear as the item definitions. If the base-period price refers to one size or quality and the comparison-period price refers to another, the price comparison may not isolate a change in the price of the same item. For a basic fixed-basket exercise, assume that the products and services are comparable in both periods, as stated in the problem.
Worked Example: Solving for a Missing Comparison Price
A hypothetical fixed basket contains 5 notebooks, 2 haircuts, and 10 cups of coffee. Base-period prices are $4.00 per notebook, $30.00 per haircut, and $2.00 per cup. A comparison-period CPI is reported as 112.00, and comparison-period prices are $4.40 per notebook and $33.00 per haircut. What must the comparison-period coffee price be? Show the basket-cost calculation.
Method. First find the base-period basket cost. Then use the index definition to find the comparison-period basket cost implied by the stated CPI. Subtract the known comparison-period item costs and divide the remainder by the fixed coffee quantity.
The base-period cost is \(5\times\$4.00=\$20.00\) for notebooks, \(2\times\$30.00=\$60.00\) for haircuts, and \(10\times\$2.00=\$20.00\) for coffee. Therefore, \(C_0=\$20.00+\$60.00+\$20.00=\$100.00\).
Rearrange the index formula to obtain the comparison-period cost:
At comparison-period prices, notebooks cost \(5\times\$4.40=\$22.00\), and haircuts cost \(2\times\$33.00=\$66.00\). Let \(p\) be the comparison-period price per cup of coffee. Its basket cost is \(10p\), so
The required coffee price is $2.40 per cup. Checking the result, coffee costs \(10\times\$2.40=\$24.00\), making the comparison-period total \(\$22.00+\$66.00+\$24.00=\$112.00\). Its index is \(100(112/100)=112.00\), as required.
Common Mistakes and Exam Traps
A particularly tempting error is to reverse the ratio. In the first worked example, the correct comparison-period calculation is \(100(\$57.50/\$52.00)=110.58\). Reversing it gives \(100(\$52.00/\$57.50)=90.43\), which is generally incorrect for the comparison-period index. At the base period, however, reversing the ratio still gives \(100(C_0/C_0)=100\). Thus, checking only the base-period normalization will not reveal a reversed ratio; check that the comparison-period cost is the numerator.
Another trap is to calculate the cost of the quantities actually purchased in the comparison period. For example, suppose a household in the second example instead used 90 kilowatt-hours, bought 18 loaves, kept 1 transit pass, and had 2 clinic visits. Multiplying the comparison prices by those changed quantities gives \(90(\$0.15)=\$13.50\), \(18(\$3.30)=\$59.40\), \(1(\$66)=\$66.00\), and \(2(\$27.50)=\$55.00\), for a total of \(\$193.90\). That is not the cost of the specified fixed basket, which is \(\$202.00\). Use the listed basket quantities in both periods.
- Adding prices instead of item costs. Multiply each price by its quantity before adding; the resulting costs, not the unweighted prices, form the basket total.
- Mixing units. Confirm that a price per unit is multiplied by a quantity in that same unit.
- Rounding too early. Retain the period totals and ratio at full available precision, then round the final index as requested.
- Assuming an index is a dollar cost. An index of 110.58 is an index number, not a basket cost of $110.58.
Check Your Understanding
For each question, use the stated quantities as the fixed basket in both periods. Report index values to two decimal places unless the result is exact.
- A basket contains 3 bags of grain and 2 cooking-oil bottles. Base-period prices are $4.00 per bag and $10.00 per bottle; comparison-period prices are $4.40 and $11.00. Find the comparison-period CPI. (Answer: 110.00)
- A basket contains 12 kilograms of produce, 3 monthly passes, and 5 repair services. Base-period prices are $0.50 per kilogram, $8.00 per pass, and $4.00 per service. Comparison-period prices are $0.60, $9.00, and $3.80. Find the CPI. (Answer: 106.40)
- A basket costs $240.00 in the base period. Its CPI in a comparison period is 125.00. What is its comparison-period cost? (Answer: $300.00)
- A basket contains 4 units of one good and 2 units of another. Base-period prices are $5 and $10; comparison-period prices are $6 and $12. A household actually buys 3 and 4 units in the comparison period. Find the fixed-basket CPI, and state whether those actual quantities affect it. (Answer: 120.00; no, the fixed quantities remain 4 and 2)
- A fixed basket contains 2 service visits and 10 supply packs. Base-period prices are $40 per visit and $3 per pack. Comparison-period visits cost $44 each, and the CPI is 110.00. Find the comparison-period price per supply pack. (Answer: $3.30 per pack)