Build and criticize a fixed basket

Build and criticize a fixed basket

Start with the question

Can two households experience different inflation?

Inspect the teaching record

A fictional basket contains two food units and one transport trip. Base prices are 3 and 4 credits; later prices are 4 and 5. Quantities stay fixed for the classroom index.

This is fictional teaching evidence. It is not an observation about an actual community or economy.

Explain the mechanism

A price index compares the cost of a defined basket with a base-period cost. Inflation is a rate of change in the index; a lower inflation rate can still mean rising prices. Official consumer price statistics use extensive methods beyond this bounded example.

Follow the worked example

Base basket costs 2×3+4=10. Later it costs 2×4+5=13. With base 100, the index is 130 and inflation is 30%. The trip-heavy household changes from 19 to 24, about 26.3%.

Challenge the first explanation

A household using four trips and one food unit has a different spending pattern. Consumers might substitute goods; quality and new products can also change. The simple index is not the official CPI calculation.

Make and revise your evidence

Calculate both baskets and explain why one average cannot describe every household. Keep units, quantities and base year visible.

Revise your answer to “Can two households experience different inflation?” using both source IDs. Separate the checked calculation from a judgment, and identify evidence that could reverse your judgment.

Name the idea

Price index: A measure comparing prices with a specified reference period.

Assumption: A condition accepted for a model or argument; it needs justification before applying the result elsewhere.

Extend the investigation

Independently extend the investigation “Build and criticize a fixed basket” with a documented real dataset or scholarly source. Preserve definitions, recreate one calculation or evidence table, compare a rival explanation and report why the teaching example cannot establish the real-world conclusion.

Method reference: BLS: CPI questions and answers.

Keep the index level separate from the inflation rate

A price index of 115.5 means the specified basket costs 15.5% more than in the base period when the index was 100. It does not give the inflation rate since last year unless last year was that base period. To compare consecutive years, divide by the previous year’s index.

Track prices, pay and purchasing power

Original fictional annual records: price index 100, 110, 115.5, 113.19; hourly pay 20, 21, 22.05, 22.05 credits. Each price observation is one year after the previous one. Predict when a pay rise fails to increase purchasing power.

Wage ruleRecorded wagesHypothetical full price indexation

Use the stated wage record.

Solid circles: prices. Dashed squares: nominal wages. Dotted triangles: real wages. All are rebased to 100; the table gives hourly credit amounts. Lines connect annual observations, not an asserted within-year path. Under full indexation, price and nominal-wage lines overlap.

Same price path under either wage rule; displayed results rounded to two decimals

Year	Price index	Annual inflation	Nominal credits/hour	Base-year credits/hour
Base	100	Not available	20.00	20.00
Year 1	110	10.00%	21.00	19.09
Year 2	115.5	5.00%	22.05	19.09
Year 3	113.19	-2.00%	22.05	19.48

Recorded wages: real hourly pay falls in Year 1, stays unchanged in Year 2 and rises in Year 3.

Inflation = 100 × (new index ÷ prior index − 1). Real hourly wage = nominal hourly wage ÷ (price index ÷ 100). The base year has no preceding observation here, so its inflation is unavailable, not zero.

Explain why Year 2 has disinflation but rising prices, why Year 3 has deflation but prices still above the base, and why the chosen wage rule changes purchasing power without changing inflation. These are hourly comparisons; annual earnings also depend on hours worked.

Restore recorded wages

Record your comparison in the source notebook. This temporary control resets on reload and is not included in captures. The index is a fictional teaching series, not an official CPI calculation or a complete household wellbeing measure.

For Year 2, inflation is 100×(115.5−110)/110 = 5%, compared with 10% in Year 1. This is disinflation: prices continue rising, but at a slower rate. In Year 3 the index falls from 115.5 to 113.19, a 2% deflation. Prices are still 13.19% above the base. “Lower inflation,” “lower prices” and “prices back to the base” are three different statements.

Real wages express pay in base-period purchasing power. Let W be hourly nominal pay and P the price index with base 100. Then real hourly wage = W ÷ (P/100). For Year 1, 21/1.10 ≈ 19.09 base-year credits per hour, less than the original 20 even though money pay rose. In Year 2, pay and prices both rise 5%, leaving that real wage unchanged. In Year 3, unchanged pay buys more because prices fall. This measures purchasing power per hour, not annual earnings or wellbeing.

Test substitution rather than merely naming it

A separate fictional household needs four equally valued meal portions. Meal A and meal B are perfect substitutes by assumption, with identical nutrition, preparation cost and availability. Initially both cost 2 credits, and the household buys two of each. Later A costs 4 and B still costs 2.

Before checking, calculate the cost of the original basket and the least expensive four-portion replacement. Explain why these answer different questions. Then remove the equal-value assumption: what evidence would you need before calling the replacement equally good?

Check the substitution comparison

The original basket costs 8 initially and 12 later, giving a fixed-basket index of 150. Four B portions cost 8 later, so the least cost of the stipulated unchanged meal service has not risen. The fixed basket overstates the cost increase for this particular perfectly substitutable case. It does not prove that actual households can avoid inflation by switching goods: preferences, dietary needs, travel, time, availability and product quality may prevent that substitution.

The BLS CPI methodology explanation describes substitution provisions and why CPI is not a complete cost-of-living measure. Do not infer that the current official CPI simply freezes every quantity forever or that this two-meal result quantifies its bias.

For a quality problem, an old device costs 200 credits with 64 units of storage; a replacement costs 240 with 128 units. The sticker price rose 20%. Is that the price change for an unchanged service? List the missing evidence: which features changed, how consumers value them, and whether a comparable old model remains available. Doubling storage does not prove that utility doubled or justify dividing the price by two. A quality-adjusted comparison needs a defensible method, not an invented correction.

Transfer with a non-base prior year

A fictional hourly wage rises from 30 to 31.20 while its relevant price index rises from 120 to 126. Calculate inflation, both real wages and the percentage change in real pay. Predict whether rebasing both price indices changes that percentage result.

Check the real-wage transfer

Inflation is 5%, not 6%: 100×6/120. Initial real pay is 25 base-year credits; final real pay is approximately 24.7619. The real change is 100×(1.04/1.05−1) ≈ −0.9524%. Subtracting 5% from 4% gives −1%, a close approximation, not the exact result.

Rebase the old index to 100 and the new one to 105. The numerical real-wage levels become 30 and about 29.7143 in the new base units, but their percentage change stays −0.9524%. Changing the index’s reference scale does not change purchasing-power growth.

For high-school practice, show each denominator and compare the four annual records. AP practice should explain disinflation, deflation and real-pay changes in an unfamiliar table without prompts. Introductory-college practice should evaluate a documented index methodology and an authentic wage series with matching periods and population coverage. These open calculations support practice; independent evidence still needs review.

Did purchasing power rise? Match pay, prices and periods

Use the January 15, 2025 BLS release, Table A-1: all employees on private nonfarm payrolls, seasonally adjusted. November and December 2024 are preliminary in this frozen vintage. These are historical estimates, not current revised figures.

Predict: “Nominal hourly earnings rose in December, so workers' purchasing power rose. Real weekly earnings must change by the same percentage as real hourly earnings.” Write what evidence each claim needs.

Initial claim audit

Writing is temporary. Copy the report before leaving. This evidence practice does not alter saved lesson work or award independent mastery.

Matched comparisonNovember to December 2024December 2023 to December 2024 Try the year-over-year claim

Reconstruct the change before judging the headline

Hourly nominal pay: $35.59 → $35.69. Matched CPI-U: 316.441 → 317.685. Hourly-pay factor = 1.00281; price factor = 1.003931. A factor of 1 means no change.

Reconstructed real hourly change = 100 × [(35.69/35.59) ÷ (317.685/316.441) − 1] = -0.111705%. Dividing growth factors accounts for the changing purchasing-power unit; subtracting rounded percentage changes is only an approximation.

Nominal hourly change 0.280978%; CPI-U change 0.393122%; reconstructed real hourly change -0.111705%; reconstructed real weekly change -0.111706%.

Account for hours and preserve source precision

Average weekly hours: 34.3 → 34.3, a 0% change. Multiplying displayed hourly earnings by hours gives $1,220.737 → $1,224.167; the separately published weekly amounts are $1220.74 → $1224.17. Rounded cells need not multiply to the last cent.

Using the published nominal weekly amounts and the same CPI-U gives -0.111706% real weekly growth. Multiplying hourly-pay and hours factors before deflating gives -0.111705%; the small difference is retained. Weekly and hourly growth need not coincide when average hours change.

Published BLS real changes for this interval: hourly -0.2%, weekly -0.1%. These are separate source cells, not replaced by our reconstruction. In the monthly case, our displayed-input hourly calculation is about −0.112%, while BLS publishes −0.2%. This packet does not resolve that discrepancy or assign a specific cause; it preserves the source and its precision instead of forcing agreement.

Inspect the matched source table

All selected series are seasonally adjusted. CPI-U uses 1982–1984 = 100; published real amounts use 1982–1984 dollars. “p” marks preliminary figures in this release. Focus the table and use arrow keys on narrow screens.

Frozen Table A-1 levels; dollar amounts per hour or week as labeled

Period	Hourly $	Hours/week	Weekly $	CPI-U	Published real $/hour	Published real $/week
2023-12	34.34	34.4	1181.30	308.742	11.12	382.62
2024-10	35.46	34.3	1216.28	315.454	11.24	385.56
2024-11 (p)	35.59	34.3	1220.74	316.441	11.25	385.77
2024-12 (p)	35.69	34.3	1224.17	317.685	11.23	385.34

Do not join Table A-1 all-employee pay to Table A-2's CPI-W deflator for production/nonsupervisory employees. Do not combine an annual pay change with a one-month price change. A seasonally adjusted series is also distinct from a not-seasonally-adjusted series.

Rebase the units, then test transfer

Predict whether changing only the index reference changes purchasing-power growth.

Express reconstructed purchasing power in the starting month's dollars

Reconstructed real hourly levels: 11.246962 → 11.234399 1982–1984 dollars/hour. The reference index is 100. Real hourly growth stays -0.111705%; rebasing changes units, not growth. These reconstructed levels are not substituted for the separately published real-dollar cells.

Transfer: switch to year-over-year. Explain why the monthly and yearly real-hourly changes have opposite signs, why weekly growth differs from hourly growth, and why neither result establishes what happened to every employee. Add one source or observation you would need to evaluate a particular household's experience, without supplying personal financial information.

Evidence brief: supported claim, matched calculation, contrary period and limitations

Compare with the reasoning guide

The monthly reconstruction is negative despite nominal growth; the yearly reconstruction is about +1.0056% hourly and +0.7118% weekly. Average hours fall from 34.4 to 34.3 over the year. Period selection can change the answer without either period being fabricated. State both dates and keep source-published changes separate from reconstructed changes.

These are aggregate means across jobs, not a median, a tracked employee's raise, annual household income or a complete welfare measure. Changes in the mix of high- and low-paid jobs can change the average. A suitable brief names population, period, adjustment and deflator, retains preliminary/vintage limitations, and avoids a causal policy claim. Open guidance is practice feedback, not independent assessment.

Source and method boundaries

Archived BLS January 15, 2025 Real Earnings, Table A-1 supplies all values. Subsequent revisions are not imported. BLS technical note (accessed October 28, 2026; methodological context rather than the frozen numerical vintage) explains CES payroll/hour/job aggregation, CPI-U versus CPI-W and composition limits. Job averages include part-time and multiple-jobholding; they are not individual incomes. No sampling intervals or significance tests are computed. Source factual cells are public domain with attribution.

Copy your earnings evidence report

Read-only JSON report{ "activity": "MAC-02.b: matched earnings and price evidence", "release": "2025-01-15", "source": "https://www.bls.gov/news.release/archives/realer_01152025.htm", "comparison": { "before": { "id": "2024-11", "preliminary": true, "publishedRealHourly": 11.25, "publishedRealWeekly": 385.77, "cpi": 316.441, "hourly": 35.59, "hours": 34.3, "weekly": 1220.74 }, "after": { "id": "2024-12", "preliminary": true, "publishedRealHourly": 11.23, "publishedRealWeekly": 385.34, "cpi": 317.685, "hourly": 35.69, "hours": 34.3, "weekly": 1224.17 }, "reference": 100, "rebase": false, "period": "monthly", "published": { "publishedRealHourly": -0.2, "publishedRealWeekly": -0.1, "cpi": 0.4, "hourly": 0.3, "hours": 0, "weekly": 0.3 }, "levels": { "beforeHourly": 11.246962308929628, "afterHourly": 11.234398854210932, "beforeWeekly": 385.7717552403134, "afterWeekly": 385.34082503108425 }, "factors": { "hourly": 1.0028097780275356, "price": 1.003931222565976, "hours": 1, "weekly": 1.002809771122434 }, "changes": { "nominalHourly": 0.280977802753557, "prices": 0.39312225659760003, "hours": 0, "realHourly": -0.11170531538745587, "realWeekly": -0.11170600319369006 }, "weeklyProduct": { "before": 1220.737, "after": 1224.167 }, "realWeeklyFromHourlyHours": -0.11170531538745587 }, "worksheet": { "prediction": "", "revision": "" }, "feedbackSeen": false, "limits": "Frozen January 2025 vintage; displayed-source reconstruction differs from some published real changes. Aggregate means do not establish individual wage changes. Temporary session; exposure retained by reset, cleared on reload. Learner writing is not graded or independently verified." }

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