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?

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