Why Two Inflation Measures Can Diverge
The CPI and the GDP deflator both summarize price change, but they do not price the same collection of items in the same way. The CPI focuses on a household consumption basket. The GDP deflator covers final goods and services produced domestically, including items such as investment output, government services, and exports. As a result, their inflation rates can differ even when both are calculated correctly.
The comparison is useful only when the measures are interpreted by their coverage and weights. For example, a sharp increase in the price of an imported household product may raise the CPI, but the import is not itself part of domestic production and therefore does not enter the GDP deflator directly. A change in the price of domestically produced equipment, by contrast, can affect the GDP deflator even if households do not buy that equipment.
For index values \(I_0\) and \(I_1\) in two periods, the percentage inflation rate over the interval is
This calculation applies to either index. Compare the resulting rates, not the index levels themselves: a CPI of 150 and a GDP deflator of 125 do not imply that CPI inflation is higher. The base values and reference baskets differ, so each index must first be compared with its own earlier value.
How Coverage and Weights Affect the Comparison
A household basket can include imported consumer goods when households buy them. The CPI is therefore exposed to price changes in imported consumption, as well as to changes in prices of domestic consumer goods and services. The GDP deflator does not include imported final goods as domestic output. It does include prices of domestically produced output that is not part of household consumption, such as capital goods and exports.
Weights matter as well. A large price increase in a category that has a small weight may contribute less to an index than a modest increase in a heavily weighted category. In the fixed-basket CPI, the base-period expenditure shares determine the direct contributions. In a Paasche GDP deflator, current-period output quantities determine the comparison. Actual published measures may use chain-weighted methods; the exact weighting procedure should be identified before interpreting a numerical decomposition.
A price increase in an imported input may indirectly affect the prices of domestic output if producers pass higher costs through. That indirect effect can appear in the GDP deflator through domestic output prices. The distinction is that the imported input’s own price is not directly counted as the price of domestic final output.
Worked Example: Comparing Inflation Rates From Index Values
In a hypothetical economy, the CPI rises from 142.0 to 149.1 over a year, while the GDP deflator rises from 128.0 to 131.2. Calculate each inflation rate and the difference between them. State what that difference does and does not tell us.
Method. Calculate each percentage change using that index’s own starting value, then subtract the rates. This avoids treating index-point changes as percentage inflation.
CPI inflation. The index-point increase is \(149.1-142.0=7.1\). Relative to the starting CPI, \(7.1/142.0=0.05\). Thus CPI inflation is \(0.05\times100\%=5.00\%\).
GDP-deflator inflation. The index-point increase is \(131.2-128.0=3.2\). The proportional change is \(3.2/128.0=0.025\). Thus GDP-deflator inflation is \(0.025\times100\%=2.50\%\).
Comparison. CPI inflation exceeds GDP-deflator inflation by \(5.00\%-2.50\%=2.50\) percentage points. This is a difference between percentage rates, not a difference in the index levels.
The comparison alone does not identify the cause of the gap. It is consistent with different coverage, weights, and price movements, but the index values by themselves do not reveal which categories drove the difference. The category-level data are needed for that diagnosis.
Common mistake avoided: Subtracting the index-point increases, \(7.1-3.2=3.9\), and calling the result the inflation-rate gap is incorrect. The correct rate gap is 2.50 percentage points because each index-point change must be divided by its own starting index.
A Short Derivation: Fixed-Basket CPI Contributions
The fixed-basket calculation explains why CPI expenditure weights provide a direct way to trace price changes. Let \(p_{i0}\) and \(p_{i1}\) be the base- and current-period prices of item \(i\), and let \(q_{i0}\) be its base-period quantity. The fixed-basket CPI relative to its base period is the cost of the basket now divided by its cost then:
Define the base-period expenditure share \(s_{i0}=p_{i0}q_{i0}/\sum_j p_{j0}q_{j0}\), and the item’s price relative \(r_i=p_{i1}/p_{i0}\). Since \(p_{i1}q_{i0}=p_{i0}q_{i0}r_i\), substitute into the numerator and divide each term by the base-basket cost:
The shares sum to one. Subtracting one from both sides gives the exact fixed-basket inflation identity:
Thus, under this fixed-basket model, each item’s price-change contribution is its base expenditure share multiplied by its proportional price change. This identity does not mean that the GDP deflator can always be decomposed using the same base-period shares: the Paasche calculation uses current output quantities, and chain-weighted measures can introduce further changes in weights.
Worked Comparisons With Basket Data
Worked Example: A Consumer Basket With an Imported Good
A hypothetical household basket contains 2 units of a domestically produced household item and 4 units of imported fruit. In the base period, their prices are $10 and $5 per unit. In the current period, the prices are $11 and $6. Calculate the fixed-basket CPI inflation rate, using a base index of 100.
Method. Price the same base-period quantities in both periods. This preserves the CPI basket rather than substituting current quantities.
Base-period basket cost. The domestic item costs \(2\times \$10=\$20\). The imported fruit costs \(4\times \$5=\$20\). Total base cost is \(\$20+\$20=\$40\).
Current cost of the base basket. The domestic item costs \(2\times \$11=\$22\). The imported fruit costs \(4\times \$6=\$24\). Total current cost is \(\$22+\$24=\$46\).
The CPI is \(100\times \$46/\$40=115\). With a base index of 100, the inflation rate is \((115/100-1)\times100\%=15.00\%\). As a check, the domestic item’s price rose 10% and the imported fruit’s price rose 20%; their equal base expenditure shares give \(0.5(10\%)+0.5(20\%)=15.00\%\).
The imported product contributes to the household CPI because it is in the household basket. Its price is not, for that reason alone, part of the GDP deflator: the deflator concerns domestic production.
Common mistake avoided: Replacing the base quantities with current quantities changes the fixed basket. For a CPI fixed-basket calculation, use the stated base-period quantities in both cost totals.
Worked Example: CPI and GDP-Deflator Inflation in One Economy
Use the household basket from the preceding example, so its fixed-basket CPI rises from 100 to 115. Separately, a simplified domestic-output account has three final-output categories. Their current quantities and prices are shown below. Calculate the GDP deflator and compare its inflation rate with CPI inflation.
| Domestic final output | Base price | Current quantity | Current price |
|---|---|---|---|
| Transport equipment | $20 | 2 | $22 |
| Construction services | $4 | 10 | $4.40 |
| Public services | $10 | 3 | $10.50 |
Method. For the GDP deflator, value the same current domestic output once at current prices and once at base prices. The imported fruit in the CPI basket is not added as domestic output.
Current value of domestic output. Transport equipment contributes \(2\times \$22=\$44\). Construction services contribute \(10\times \$4.40=\$44\). Public services contribute \(3\times \$10.50=\$31.50\). The total is \(\$44+\$44+\$31.50=\$119.50\).
Base-price value of current output. Transport equipment contributes \(2\times \$20=\$40\). Construction services contribute \(10\times \$4=\$40\). Public services contribute \(3\times \$10=\$30\). The total is \(\$40+\$40+\$30=\$110\).
The GDP deflator is \(100\times \$119.50/\$110=108.6364\), rounded to four decimal places. Its inflation rate from a base of 100 is \(8.6364\%\). CPI inflation is \(15.00\%\), so CPI inflation exceeds GDP-deflator inflation by \(15.00\%-8.6364\%=6.3636\) percentage points, rounded to four decimal places.
The result reflects both different coverage and different weights. The CPI includes the imported fruit, whose price rose substantially. The GDP deflator excludes that imported good directly and weights current domestic output, including categories that need not receive large weights in a household basket.
Common mistake avoided: Putting the imported fruit into the GDP-deflator numerator just because it appears in the CPI is wrong. Use domestic final output for the deflator, and use the same current output quantities at current and base prices.
Worked Example: Separating an Import-Price Effect From Domestic Output Prices
In a hypothetical economy, imported goods make up 40% of base household expenditure and their prices rise 18%. Domestic household goods make up the other 60% and their prices rise 2%. The domestic-output data for the GDP deflator are listed below. Calculate both inflation rates and explain the role of the import price increase.
| Domestic output category | Base price | Current quantity | Current price |
|---|---|---|---|
| Domestic household output | $10 | 4 | $10.20 |
| Equipment | $25 | 2 | $27 |
| Government services | $5 | 10 | $5.10 |
| Exports | $8 | 5 | $8.40 |
CPI calculation. For this fixed-basket illustration, base expenditure shares and category inflation rates give CPI inflation of \(0.40(18\%)+0.60(2\%)=7.20\%+1.20\%=8.40\%\). The import category contributes 7.20 percentage points; domestic household goods contribute 1.20 percentage points.
GDP-deflator numerator. Value current output at current prices. Domestic household output contributes \(4\times \$10.20=\$40.80\); equipment contributes \(2\times \$27=\$54\); government services contribute \(10\times \$5.10=\$51\); and exports contribute \(5\times \$8.40=\$42\). The total is \(\$40.80+\$54+\$51+\$42=\$187.80\).
GDP-deflator denominator. Value those same current quantities at base prices: \(4\times \$10=\$40\), \(2\times \$25=\$50\), \(10\times \$5=\$50\), and \(5\times \$8=\$40\). The total is \(\$40+\$50+\$50+\$40=\$180\).
The GDP deflator is \(100\times \$187.80/\$180=104.3333\), rounded to four decimal places, so GDP-deflator inflation is \(4.3333\%\). CPI inflation exceeds it by \(8.40\%-4.3333\%=4.0667\) percentage points, rounded to four decimal places.
The imported-price increase explains a direct CPI contribution of 7.20 percentage points in this illustration, but it is not itself a direct GDP-deflator contribution. The deflator instead reflects prices of domestic output, including equipment, government services, and exports. The arithmetic does not claim that imports can never influence domestic prices indirectly.
Common mistake avoided: Treating the 18% import-price rise as an 18% increase in the GDP deflator confuses the price of imported goods with the price of domestic production.
Interpreting the Gap and Avoiding Traps
A higher CPI inflation rate than GDP-deflator inflation is plausible when imported consumer prices rise quickly, especially if imported goods have substantial household expenditure weight. It can also arise because the CPI gives weight to household purchases while the deflator includes a wider range of domestic output. Conversely, the GDP deflator can rise faster if prices increase rapidly in domestically produced investment goods, government output, or exports while consumer-basket prices rise less.
The size of the difference cannot be attributed to imports alone without more information. A useful comparison asks three questions: which items are covered by each measure, what weights are being used, and how did prices change in each relevant category? For a chain-weighted index, also identify the index method and period weights before interpreting a simple weighted contribution.
- Do not compare index levels as inflation rates. Calculate each rate from its own initial index.
- Do not call an index-point difference a percentage-point difference. First calculate the two percentage inflation rates.
- Do not assume imports are excluded from CPI. Imported consumer goods can be included when households purchase them.
- Do not assume an imported good is domestic output. Its price is not directly included in the GDP deflator, though imported input costs may affect domestic prices indirectly.
- Do not use one set of weights automatically for both measures. A fixed-basket CPI and a Paasche GDP deflator weight prices differently.
Check Your Understanding
For each problem, calculate the requested measure before interpreting the result.
- A hypothetical CPI rises from 240 to 250.8, and a GDP deflator rises from 180 to 185.4. Calculate both inflation rates and their difference in percentage points. (Answer: CPI inflation 4.50%; GDP-deflator inflation 3.00%; gap 1.50 percentage points.)
- A fixed household basket has base expenditure of $40 on domestic goods and $40 on imported goods. Their respective price increases are 3% and 17.5%. Calculate fixed-basket CPI inflation and each category’s contribution. (Answer: CPI inflation 10.25%; domestic contribution 1.50 percentage points; import contribution 8.75 percentage points.)
- A domestic producer makes two final goods. At current quantities, good A has base price $3, current quantity 10, and current price $3.30; good B has base price $8, current quantity 5, and current price $8.80. Calculate the GDP deflator using base 100. (Answer: 110.00, so GDP-deflator inflation is 10.00%.)
- Imports are 30% of a fixed household basket and rise in price by 15%; domestic consumer goods are 70% and rise by 3%. Calculate CPI inflation, then calculate the counterfactual rate if imported prices had not changed and all other data stayed fixed. (Answer: CPI inflation 6.60%; counterfactual inflation 2.10%; import-price effect 4.50 percentage points.)
- CPI inflation is 7% and GDP-deflator inflation is 4% in a year when imported consumer goods rise sharply in price. Can the full 3-percentage-point gap be attributed to imports from this information alone? Give the reason. (Answer: No; the gap may also reflect differing coverage, weights, domestic output prices, and index methods, so category-level data are needed.)