From Basket Prices to CPI and Inflation
A consumer price index calculation links two steps: first, calculate what the same basket of items costs in different periods; then compare those costs. Keeping the basket quantities fixed makes the price change interpretable: the calculation asks how the cost of a specified basket changes as its item prices change.
The CPI is an index, not a dollar amount. Its base-period value is conventionally set to 100, so later values show the basket cost relative to that base. Inflation between two periods is the percentage change in the index, not the difference in index points alone.
Here, \(q_j\) is the quantity of item \(j\) in the specified basket, and \(p_{j,t}\) is its price in period \(t\). The quantities remain the same for every period in this fixed-basket calculation. The base-period prices and quantities determine \(C_0\); later-period prices are applied to those same quantities. If the CPI rises from 100 to 108, that is an 8% increase since the base period. If it rises from 108 to 112, the period-to-period inflation rate is not 4%; it is \(100(112/108-1)\), or about 3.70%.
Why the CPI Formula Uses a Cost Ratio
The CPI expresses the cost of a basket in one period as a proportion of its cost in the base period, with that proportion multiplied by 100. This normalization gives the base period the convenient index value 100. For example, a basket costing 105 dollars in a later period when it cost 100 dollars in the base period has a CPI of 105.
The inflation formula follows from comparing the two index values. Because each CPI uses the same base cost \(C_0\), that base cost cancels:
Thus, when the basket and its quantities are unchanged, inflation can be calculated either from consecutive CPI values or from consecutive basket costs. The result is the same. The CPI ratio also explains why the percentage change in CPI is the relevant rate: index-point changes are measured relative to the earlier index, just as a percentage change in any quantity is measured relative to its starting value.
A Reliable Calculation Sequence
For a problem that supplies item quantities and prices, organize the work before calculating. This reduces the risk of using a current-period quantity by mistake, omitting an item, or dividing by the wrong period’s cost.
Record each item’s quantity and the period whose CPI is defined as 100.
For each item, multiply its price in that period by the basket quantity, then add the item costs.
Divide the period’s basket cost by the base-period basket cost and multiply by 100.
Divide the later CPI by the earlier CPI, subtract 1, and multiply by 100 to express the rate as a percentage.
Worked Calculations
Worked Example: Constructing a Three-Period CPI Series
A hypothetical basket contains 8 kg of rice, 2 transit passes, and 5 bars of soap. The base-period prices are $3.00 per kg, $25.00 per pass, and $4.00 per bar. In the next period, the prices are $3.30, $27.00, and $4.40; in the following period, they are $3.60, $28.00, and $4.50. Calculate both later CPI values and the inflation rate in each interval. Round CPI values and inflation rates to two decimal places.
Method. Use the fixed quantities to calculate all three basket costs, normalize each later cost to the base cost, and then compare consecutive CPI values.
Base-period basket cost. Rice costs \(8\times \$3.00=\$24.00\), transit passes cost \(2\times \$25.00=\$50.00\), and soap costs \(5\times \$4.00=\$20.00\). Therefore, \(C_0=\$24.00+\$50.00+\$20.00=\$94.00\), and the base CPI is 100.
First later-period basket cost and CPI. The item costs are \(8\times \$3.30=\$26.40\), \(2\times \$27.00=\$54.00\), and \(5\times \$4.40=\$22.00\). Thus, \(C_1=\$26.40+\$54.00+\$22.00=\$102.40\), and \(\mathrm{CPI}_1=100(102.40/94.00)=108.94\).
Second later-period basket cost and CPI. The item costs are \(8\times \$3.60=\$28.80\), \(2\times \$28.00=\$56.00\), and \(5\times \$4.50=\$22.50\). Thus, \(C_2=\$28.80+\$56.00+\$22.50=\$107.30\), and \(\mathrm{CPI}_2=100(107.30/94.00)=114.15\).
Inflation in the first interval. The rate is \(100(108.93617/100-1)=8.94\%\). The unrounded CPI was used in the calculation.
Inflation in the second interval. The rate is \(100(107.30/102.40-1)=100(4.90/102.40)=4.79\%\). It is the change relative to the first later-period cost, not relative to the base-period cost.
Check and common mistake. The overall CPI increase from the base to the second later period is \(114.14894-100=14.15\%\). This agrees with the basket-cost comparison \(100(107.30/94.00-1)=14.15\%\). A common error is to report the second interval’s 4.90-dollar cost increase as 4.90% inflation. The correct rate divides $4.90 by the earlier interval’s $102.40 basket cost, giving 4.79%.
Worked Example: Price Increases and a Price Decrease in One Series
A fixed basket contains 5 units of item A, 2 units of item B, and 4 units of item C. In the base period, their prices are $2.00, $10.00, and $3.00. In period 1, they cost $2.40, $9.00, and $3.30. In period 2, they cost $2.50, $10.50, and $3.00. In period 3, they cost $2.25, $12.00, and $3.60. Find the CPI for each later period, inflation in periods 2 and 3, and the cumulative inflation from the base to period 3. Round to two decimals.
Method. Reprice the same quantities in each period. A price decline for one item is included just like a price increase; its effect depends on that item’s quantity and price change.
Base cost. The three costs are \(5\times \$2.00=\$10.00\), \(2\times \$10.00=\$20.00\), and \(4\times \$3.00=\$12.00\). Therefore, \(C_0=\$42.00\).
Period 1. The costs are \(5\times \$2.40=\$12.00\), \(2\times \$9.00=\$18.00\), and \(4\times \$3.30=\$13.20\). Their sum is \(C_1=\$43.20\), so \(\mathrm{CPI}_1=100(43.20/42.00)=102.86\).
Period 2. The costs are \(5\times \$2.50=\$12.50\), \(2\times \$10.50=\$21.00\), and \(4\times \$3.00=\$12.00\). Their sum is \(C_2=\$45.50\), so \(\mathrm{CPI}_2=100(45.50/42.00)=108.33\).
Period 3. The costs are \(5\times \$2.25=\$11.25\), \(2\times \$12.00=\$24.00\), and \(4\times \$3.60=\$14.40\). Their sum is \(C_3=\$49.65\), so \(\mathrm{CPI}_3=100(49.65/42.00)=118.21\).
Period 2 inflation. Use consecutive basket costs (equivalently, consecutive CPI values): \(100(45.50/43.20-1)=100(2.30/43.20)=5.32\%\).
Period 3 inflation. The rate is \(100(49.65/45.50-1)=100(4.15/45.50)=9.12\%\). Although item A becomes cheaper in period 3, the combined fixed basket cost still rises.
Cumulative check and common mistake. From the base to period 3, cumulative inflation is \(100(49.65/42.00-1)=18.21\%\). Do not add the period 2 and period 3 rates to obtain cumulative inflation: \(5.32\%+9.12\%=14.44\%\), which is not the total. Consecutive rates compound, and their combined factor is \(45.50/43.20\times49.65/45.50=49.65/43.20\); comparing the final cost with the base gives the stated 18.21% cumulative change since the base.
Worked Example: Keeping the Basket Fixed When Quantities Change
A specified CPI basket contains 2 units of item A and 3 units of item B. Base-period prices are $4.00 and $10.00. In period 1, prices are $5.00 and $12.00. In period 2, prices are $5.50 and $11.50. Separately, suppose a shopper’s purchases in period 1 were 1 unit of A and 5 units of B. Calculate the two later CPI values and period 2 inflation using the specified basket. Explain why the shopper’s quantities do not replace it. Round to two decimals.
Method. Apply the defined CPI basket in every period, regardless of the shopper’s separate purchase pattern.
Base cost. The specified basket costs \(2\times \$4.00+3\times \$10.00=\$8.00+\$30.00=\$38.00\).
Period 1. Its cost is \(2\times \$5.00+3\times \$12.00=\$10.00+\$36.00=\$46.00\). Therefore, \(\mathrm{CPI}_1=100(46.00/38.00)=121.05\).
Period 2. Its cost is \(2\times \$5.50+3\times \$11.50=\$11.00+\$34.50=\$45.50\). Therefore, \(\mathrm{CPI}_2=100(45.50/38.00)=119.74\).
Period 2 inflation. The rate is \(100(45.50/46.00-1)=100(-0.50/46.00)=-1.09\%\). The negative sign indicates that the specified basket’s cost fell between periods 1 and 2.
Common mistake and check. Using the shopper’s period 1 quantities for period 1 would produce a cost of \(1\times \$5.00+5\times \$12.00=\$65.00\). Comparing that cost with the specified basket’s $38.00 base cost would mix two different baskets, so it would not calculate this CPI. With the fixed basket, the final CPI of 119.74 is above 100, meaning the basket still costs 19.74% more than in the base period, even though its cost declined by 1.09% in the latest interval.
Checks That Prevent Common Errors
Before accepting a result, confirm that the quantities used in every period match the basket specified in the question. Then check the denominator: a CPI uses the base-period cost, while a period-to-period inflation rate uses the earlier period’s CPI or cost. Keep extra digits during the calculations and round only the requested final answers.
- Index points are not percentage points. A rise from 110 to 115 is 5 index points, but the inflation rate is \(100(115/110-1)=4.55\%\).
- The base period is not always the comparison period. To find inflation between two later periods, divide the later index by the earlier one, not by 100.
- Do not add consecutive inflation rates to find cumulative inflation. Percentage changes apply to changing starting values, so they compound.
- Keep the basket consistent. In a fixed-basket calculation, use the stated quantities in every period.
- Include every item’s cost. A price fall still contributes to the total basket cost; it does not mean inflation must be negative overall.
These calculations are useful whenever a problem supplies a CPI basket and a sequence of item prices. The arithmetic produces an index series and its rates of change for that specified basket. Interpreting how well an index represents different households is a separate question from carrying out these calculations.
Check Your Understanding
For each question, keep full precision through the calculations and round final CPI values and rates to two decimals.
- A fixed basket contains 3 units of item P and 2 units of item Q. Base-period prices are $4.00 and $7.00; later prices are $5.00 and $8.00. Find the later CPI. (Answer: 119.23)
- A basket costs $90.00 in the base period and $97.20 in the next period. Its CPI then is 108.00. If the following CPI is 113.40, find inflation from the next period to the following period. (Answer: 5.00%)
- A basket contains 2 units of item R and 3 units of item S. Base prices are $6.00 and $4.00. In period 1, prices are $7.00 and $5.00; in period 2, they are $6.50 and $6.00. Find both later CPI values and inflation from period 1 to period 2. (Answer: CPI values 120.83 and 129.17; inflation 6.90%)
- The CPI is 124.80 in one year and 130.416 in the next. Find the inflation rate between those years. (Answer: 4.50%)
- A CPI series has values 100 in the base period, 96 in period 1, 102 in period 2, and 99 in period 3. Find inflation from period 1 to period 2, inflation from period 2 to period 3, and cumulative inflation from the base to period 3. (Answer: 6.25%; -2.94%; -1.00%)