Mingze Huang
2021-08-05
| Year | Weight (Cars) | Price (cars) | Weight (Gas) | Price (Gas) |
|---|---|---|---|---|
| 2019 | 0.020 | 25 | 0.25 | 2.0 |
| 2020 | 0.035 | 20 | 0.20 | 1.5 |
| 2021 | 0.013 | 30 | 0.20 | 3.0 |
Take 2020 as base year, assume a representative (average) consumer spend 70% of money on cars, 30% of money on gasoline on 2020. By construction we have Cost of (base year) market basket for 2019~2021 is:
\[ C_{2019}=P_{2019}^{C}W_{2020}^{C}+P_{2019}^{G}W_{2020}^{G}=25\times0.035+2\times0.2=1.275\\ C_{2020}=P_{2020}^{C}W_{2020}^{C}+P_{2020}^{G}W_{2020}^{G}=20\times0.035+1.5\times0.2=1\\ C_{2021}=P_{2021}^{C}W_{2020}^{C}+P_{2021}^{G}W_{2020}^{G}=30\times0.035+3\times0.2=1.65 \]
By definition, CPI for 2019~2021 is: \[ CPI_{2019}=\frac{C_{2019}}{C_{2020}}\times100=127.5\\ CPI_{2020}=\frac{C_{2020}}{C_{2020}}\times100=100\\ CPI_{2021}=\frac{C_{2021}}{C_{2020}}\times100=165 \] \(\$1650\)’s purchase power in 2021, probably equivalent to \(\$1650\times\frac{127.5}{165}=\$1270\) in 2019!
CPI inflation for 2020~2021 is:
\[ \pi_{2020}=\frac{CPI_{2020}-CPI_{2019}}{CPI_{2019}}\times100\%=\frac{100-127.5}{127.5}\times100\%=-21.57\%\\ \pi_{2021}=\frac{CPI_{2021}-CPI_{2020}}{CPI_{2020}}\times100\%=65\% \]
In practice, BLS use market basket 24 months ago to calculate CPI.
| Year | Weight (Cars) | Price (cars) | Weight (Gas) | Price (Gas) |
|---|---|---|---|---|
| 2019 | 0.020 | 25 | 0.25 | 2.0 |
| 2020 | 0.035 | 20 | 0.20 | 1.5 |
| 2021 | 0.013 | 30 | 0.20 | 3.0 |
Alternatively take 2019 as base year. Note that we assume a representative consumer spend half of money on cars and half on gasoline on 2019. By construction we have Cost of (base year) market basket for 2019~2021 is: \[ C_{2019}=P_{2019}^{C}W_{2019}^{C}+P_{2019}^{G}W_{2019}^{G}=25\times0.02+2\times0.25=1\\ C_{2020}=P_{2020}^{C}W_{2019}^{C}+P_{2020}^{G}W_{2019}^{G}=20\times0.02+1.5\times0.25=0.775\\ C_{2021}=P_{2021}^{C}W_{2019}^{C}+P_{2021}^{G}W_{2019}^{G}=30\times0.02+3\times0.25=1.35 \]
CPI for 2019~2021 is: \[ CPI_{2019}=\frac{C_{2019}}{C_{2019}}\times100=100\\ CPI_{2020}=\frac{C_{2020}}{C_{2019}}\times100=77.5\\ CPI_{2021}=\frac{C_{2021}}{C_{2020}}\times100=135 \]
\(\$1650\)’s purchase power in 2021, probably equivalent to \(\$1650\times\frac{100}{135}=\$1222.22\) in 2019!
Base year matters because your market basket changes!
CPI inflation for 2020~2021 is:
\[ \pi_{2020}=\frac{CPI_{2020}-CPI_{2019}}{CPI_{2019}}\times100\%=\frac{77.5-100}{100}=-22.5\%\\ \pi_{2021}=\frac{CPI_{2021}-CPI_{2020}}{CPI_{2020}}\times100\%=\frac{135-77.5}{77.5}=74.19\% \]