This question came out of a discussion with a colleague from economics about price indices. Here is MattF's formulation of the question which differs somehow from the original problem.


Let $Y=({\mathbf R}^+)^3$, where an element $y=(r,v,w)$ is interpreted as the inflation rate $r$ of an item (price at time 1 / price at time 0) and the expenditures $v$ and $w$ in the two periods.


Let $\pi\_1$ be the first projection from $Y$ or any other space.


A price index for $Y$ is a sequence of symmetric functions $P\_n:Y^n\to {\mathbf R}^+$ with $P\_1=\pi\_1$. E.g.:


* The Laspeyres index is $P\_n((r\_i,v\_i,w\_i)\_{i\le n})=\sum v\_ir\_i /\sum v\_i$.
* The Fisher index is the geometric mean of two weighted arithmetic
means, $$ P\_n((r\_i,v\_i,w\_i)\_{i\le n})= \left(\sum v\_i r\_i/\sum
 v\_i\right)^{1/2} \left(\sum w\_i r\_i/\sum w\_i\right)^{1/2}.$$


We want to know if $P$ arises naturally from some type of aggregation.


Let us call $P$ consistent-in-aggregation (CIA) if there is a transformation $h:({\mathbf R}^+)^3\rightarrow {\mathbf R}^+\times H$ with $\pi\_1\circ h=\pi\_1$, $H \subset {\mathbf R}^k$, and an associative aggregator $\oplus:({\mathbf R}^+\times H)^2\rightarrow({\mathbf R}^+\times H)$, such that $P\_n(y\_1,\ldots,y\_n)=\pi\_1(h(y\_1)\oplus\ldots\oplus h(y\_n))$. Then:


* The Laspeyres index is CIA with $h(r,v,w)=(r,v)$ and
$(r,v)\oplus(s,u)=((vr+us)/(v+u),v+u)$.
* Every price index is CIA for some transformation, using the axiom of
choice. (The proof uses a Hamel basis $\lbrace e\_y:y\in Y\rbrace$ of
$\mathbb R$ considered as a $\mathbb Q$-vector space and
$h(y)=(\pi\_1(y), e\_y)$. The sum of transformations is enough to
reconstruct $y\_1,\ldots,y\_n$ up to permutation.)
* Can we find a *continuous* transformation $h$ and *continuous*
aggregator $\oplus$ which make the Fisher index CIA?



Consistent price index
Let's think of it as making a Markov process out of a function of two Markov processes, by possibly increasing the dimension.


That is, let $h(r,v\_1,v\_2)=(r,r,v\_1,r,v\_2)\in\mathbb R^5$ and define $\oplus\_{\text{Fisher}}$ by
$$
(r,r\_1,v\_1,r\_2,v\_2)\oplus\_{Fisher} (r',r'\_1,v'\_1,r'\_2,v'\_2):=(({\pi\_1(\vec\alpha\_1)\cdot\pi\_1(\vec\alpha\_2)})^{1/2},\vec\alpha\_1,\vec\alpha\_2)$$
where
$$\vec\alpha\_i=(r\_i,v\_i)\oplus\_{\text{Laspeyres}} (r\_i',v\_i')\in\mathbb R^2,\quad i=1,2$$
with $\oplus\_{\text{Laspeyres}}$ being the $\oplus$ that you defined for the Laspeyres index.


