Inflation: audit the index, then the trade-off
導入済み| English | 意味 |
|---|---|
| inflation/ɪnˈfleɪʃn/ | A sustained increase in the general price level of goods and services. |
| consumer price index/kənˈsuːmə praɪs ˈɪndeks/ | An index tracking changes in the cost of a representative basket of goods and services purchased by consumers. |
| Fisher equation/ˈfɪʃə ɪˈkweɪʒn/ | Fisher equation |
| unemployment rate/ʌnemˈplɔɪmənt reɪt/ | The percentage of the labour force that is unemployed under the measure being used. |
| Phillips curve/ˈfɪlɪps kɜːv/ | A relationship between inflation and unemployment, often represented as an inverse association in the short run. |
The tax nobody voted for
- Aurora's 2021 headline read "inflation 6.0%". A tenant heard "rent up 6%". A saver watched a 5% deposit rate lose money. All three readings were about the same number — and only one was right.
- This lesson audits the index, then meets the oldest trade-off in macro.
Price a fixed basket
- The consumer price index 消费者价格指数 prices an unchanged basket: 10 loaves, one month of rent, 40 bus rides.
- The basket cost £920 last year and £975 this year: index $975/920 \times 100 = 106.0$, so inflation 通货膨胀 is 6.0% ✓.
Basket cost 920 last year, 975 this year. What is the CPI (base 100, one decimal)?
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$975/920 \times 100 = 106.0$, so inflation was 6.0%.
Saying "the CPI went up 106%" is the correct reading of an index that moved from 100 to 106.
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106 is the LEVEL; the change was 6%. Confusing level with rate turns modest inflation into catastrophe on paper.
Weights turn items into a headline
- Contributions are weighted item rates: bread $2.17\% \times 10.0 = 0.22$ points; rent $86.96\% \times 6.0 = 5.22$; bus $10.87\% \times 5.0 = 0.54$ — total $5.98 \approx 6.0$ ✓.
- So rent was 5.2 of the 6 points: the tenant's rent did NOT rise 6% "like everything" — the basket did, mostly BECAUSE of rent.
Read the number correctly
Distinguish index levels from rates, naive from Fisher real rates, and honest unemployment falls from discouragement. Each card is one published number.
Rent (weight 86.96%) rose 6.0%. How many points of the 6.0% headline did rent contribute (two decimals)?
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$0.8696 \times 6.0 = 5.22$ points - most of the headline was one item.
Real rates: the Fisher equation
- The Fisher equation 费雪方程: $(1 + i) = (1 + r)(1 + \pi)$, so $r = (1+i)/(1+\pi) - 1$.
- At 5% nominal and 3% inflation: $r = 1.05/1.03 - 1 = \mathbf{1.94}$% — not the naive 2%. If inflation surprises at 6%: $1.05/1.06 - 1 = -0.94$% — the saver paid for the surprise.
Nominal 5%, inflation 3%. The Fisher real rate, in % (two decimals)?
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$1.05/1.03 - 1 = 1.94$% - not the naive 2.0. The subtraction error grows with the level of rates.
The saver took 5% expecting 3% inflation; inflation landed at 6%. Ex-post real rate, in % (two decimals)?
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$1.05/1.06 - 1 = -0.94$% - the surprise redistributed from lender to borrower.
Jobs: read the rate properly
- The unemployment rate 失业率 = unemployed ÷ labour force, and "unemployed" requires active search.
- A falling rate can mean jobs — or discouraged workers leaving the force. Always ask what participation did.
A falling unemployment rate always means more people are working.
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Discouraged workers leave the labour force: the numerator and denominator fall together, the rate falls, and no job was created. Always read participation beside the rate.
The trade-off that expires
- The Phillips curve 菲利普斯曲线: $\pi = \pi^e - 0.8(u - 5)$ with natural rate 5%. With $\pi^e = 4$: u = 7 → 2.4; u = 5 → 4.0; u = 3 → 5.6 ✓.
- Each point of unemployment below 5 buys 0.8 of inflation — until expectations catch up and the whole line shifts. The trade-off is a lease, not a deed.
With pi-e = 4 and the curve pi = 4 - 0.8(u - 5), what inflation comes with u = 3, in % (one decimal)?
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$4 - 0.8 \times (3-5) = 4 + 1.6 = 5.6$%. Each point below the natural rate buys 0.8 of inflation.
What is special about unemployment of 5% on this curve?
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At u = 5 inflation equals pi-e. Push below and inflation exceeds expectations - until expectations adjust and the trade-off expires.
Three index slips
- "The CPI went up 106%." The index LEVEL went 100 → 106; inflation was 6.
- Subtracting inflation from the nominal rate. Divide instead: the gap grows with the rates.
- Reading unemployment without participation. Discouragement shrinks the rate without creating a job.
Recap
- Index = fixed basket priced twice; contributions are weighted item rates.
- $1 + i = (1 + r)(1 + \pi)$ — divide, never subtract.
- Phillips trades inflation for unemployment only while expectations stand still.