2 CPI, inflation, unemployment
2.1 CPI (Laspeyres)
The Consumer Price Index measures the cost of a fixed base-year basket of goods, evaluated at year-t prices, relative to the cost of that same basket at base-year prices.
\text{CPI}_t = \frac{\sum_i P_{i,t} \cdot Q_{i,\text{base}}}{\sum_i P_{i,\text{base}} \cdot Q_{i,\text{base}}} \cdot 100
In the base year, \text{CPI} = 100 by construction.
The basket is frozen. You only re-price it. If consumers in 2023 buy more chicken and less beef because beef got expensive, the CPI doesn’t notice — it still uses the base-year quantities. This is why CPI overestimates inflation (substitution bias).
2.1.1 Inflation
\pi_t = \frac{\text{CPI}_t - \text{CPI}_{t-1}}{\text{CPI}_{t-1}} \cdot 100
The denominator is the earlier year. The base year sets the level; it does not have to be the denominator.
2.1.2 IGDPDI (Paasche)
The implicit GDP deflator uses current quantities and base-year prices. It underestimates inflation. Reverse for deflation.
| index | quantities | prices | bias on inflation |
|---|---|---|---|
| CPI (Laspeyres) | base-year (frozen) | year-t | overestimates |
| IGDPDI (Paasche) | year-t | base-year | underestimates |
2.2 Unemployment
The labor force is everyone who is employed plus those actively looking for work:
LF = E + U
The unemployment rate is the share of the labor force who are looking but haven’t found work:
u = \frac{U}{E + U}
The labor force participation rate is the share of the working-age population in the labor force:
\text{LF participation} = \frac{LF}{\text{Pop}}
Discouraged workers are not in the labor force. Someone who has stopped looking is counted as out of the LF, so they don’t appear in u. This is why measured unemployment can fall during a recession (people give up) without the underlying labor market improving.
2.3 Types of unemployment
| type | definition |
|---|---|
| Frictional | Short-run job/skill matching. People between jobs. Always present in a healthy economy. |
| Structural | Whole industries decline. Workers’ skills no longer demanded. Painful and slow to resolve. |
| Cyclical | From recessions. Disappears in expansions. The Keynesian story. |
| Natural rate | Frictional + structural. Textbook value \approx 4\%. The level consistent with no accelerating inflation. |
The full-employment GDP (Y^{FE}) is the level of output the economy would produce at the natural rate of unemployment. Gaps from Y^{FE} are recessionary (if Y^* < Y^{FE}) or inflationary (if Y^* > Y^{FE}).
2.4 A worked example
Base year 2019, basket = 30 bread + 1 housing + 25 gas. Prices and quantities:
| 2019 price | 2020 price | |
|---|---|---|
| Bread | $1.50 | $2.00 |
| Housing | $1{,}000 | $950 |
| Gas | $3.50 | $5.75 |
Cost of base basket at 2019 prices: 30(1.50) + 1(1000) + 25(3.50) = 45 + 1000 + 87.5 = 1{,}132.5.
Cost at 2020 prices: 30(2.00) + 1(950) + 25(5.75) = 60 + 950 + 143.75 = 1{,}153.75.
CPI (2019) = 1{,}132.5/1{,}132.5 \cdot 100 = 100. CPI (2020) = 1{,}153.75/1{,}132.5 \cdot 100 = 101.88.
Inflation 2019 → 2020 = (101.88 - 100)/100 \cdot 100 = 1.88\%.