物價指數(Price Index) 與 GDP deflator(GDP 平減指數

"Price index can be said to measure the economy's price level or a cost of living."
物價指數可用以衡量物價水準或者生活花費.

• Paasche price index (博氏指數? 敝人音譯: 怕什指數)
  
• Laspeyres price index (拉氏指數, 敝人音譯: 拉斯陪樂斯指數)
  
• Marshal-Edgeworth index
  
• Fisher index (費雪指數)
  
• GDP deflator?
  

$t_0$ 代表基期, 某個作為比較基礎的時間點
$t_n$ 代表$t_0$之後的的某個標的時間點

■ Paasche index
\[P_P=\frac{\sum\limits_c{p_{c,t_n}\centerdot q_{c,t_n}}} {\sum\limits_c{p_{c,t_0}\centerdot q_{c,t_n}}}\]


■ Laspeyres index
\[P_L=\frac{\sum\limits_c{p_{c,t_n}\centerdot q_{c,t_0}}} {\sum\limits_c{p_{c,t_0}\centerdot q_{c,t_0}}}\]


"...The Laspeyres index tends to overstate inflation (in a cost of living framework), while the Paasche index tends to understate it, ..."


■ Marshal-Edgeworth index: 可看成 Paasche index 與 Laspeyres index 的算術平均
\[P_{ME}=\frac{\sum\limits_c{p_{c,t_n}\centerdot (q_{c,t_0}+q_{c,t_n})}} {\sum\limits_c{p_{c,t_0}\centerdot (q_{c,t_0}+q_{c,t_n})}}\]


■ Fisher index: Paasche index 與 Laspeyres index 的幾何平均

\[P_F= \sqrt{P_P \centerdot P_L}\ (基期t_0, 標的時間t_n) \]

\[F_t= \sqrt{P_P \centerdot P_L} \vert_{t_0=t-1, t_n=t}\ (基期t-1, 標的時間t)\]

i.e.
\[F_t=\sqrt{
\frac{\sum\limits_c{p_{c,t}\centerdot q_{c,t}}} {\sum\limits_c{p_{c,t-1}\centerdot q_{c,t}}}
\cdot
\frac{\sum\limits_c{p_{c,t}\centerdot q_{c,t-1}}} {\sum\limits_c{p_{c,t-1}\centerdot q_{c,t-1}}}
}\]


■ PCE(Personal Consumption Expenditure)

■ PCE price index(PCE index for short)
\[P_{PCE,t}=F_1\centerdot F_2\dots F_t
\]


\[\therefore P_{PCE,n}=
\sqrt{
\frac{\sum\limits_c{p_{c,n}\centerdot q_{c,n}} }
{\sum\limits_c{p_{c,t_0}\centerdot q_{c,t_0}} }
}

\sqrt{
\prod_{t=1}^{t=n}
\frac{\sum\limits_c{p_{c,t}\centerdot q_{c,t-1}}} {\sum\limits_c{p_{c,t-1}\centerdot q_{c,t}}}
}
\]


■ GDP deflator(GDP 平減指數) $\equiv$
\[\frac{Nominal GDP(用當期價格計算)}{Real GDP(用基期價格計算)}\]


"...
In practice, the difference between the deflator and a price index like the Consumer price index (CPI) is often relatively small. On the other hand, with governments in developed countries increasingly utilizing price indexes for everything from fiscal and monetary planning to payments to social program recipients, the even small differences between inflation measures can shift budget revenues and expenses by millions or billions of dollars.
...
"

GDP與 Price Index 主要的差異在於取樣的產品範疇 ?



[ref]
http://en.wikipedia.org/wiki/Price_index#Paasche_and_Laspeyres_price_indices
http://en.wikipedia.org/wiki/GDP_deflator
http://en.wikipedia.org/wiki/Personal_consumption_expenditures_price_index

http://www.mathjax.org/docs/1.1/tex.html
http://en.wikipedia.org/wiki/Geometric_Shapes