Between $2012$ and early $2021,$ the average rate of inflation as measured by the core personal consumption expenditures $($PCE$)$ price index was $1\mathrm{.}6$ percent$$below the Federal Reserve's target of $2$ percent.

a$.$ Briefly explain why using the consumer price index $($CPI$)$ might yield a rate of inflation different from that found using the core PCE price index.

The price index includes all goods and services that are in the consumption component of GDP$,$ while the price index includes only the goods and services that are in the market basket that the Bureau of Labor Statistics uses in calculating it$.$