Let $2<T\in \mathbb{N}$ and call $\left(y_{t}\right)_{t=1}^{T}$ a profile where $y_t\in \mathbb{R}_{++}$ for all $t \in \left\{ 1,...,T\right\}$. Call $y_t$ the income at time period $t$. Let $\succsim$ be a continuous, complete and transitive order of profiles. Consider the following properties:
- Monotonicity: $ \left(y_{t}\right)_{t=1}^{T} \succsim \left(y'_{t}\right)_{t=1}^{T}$ if $y_{t}\geq y'_{t}$ for all $t=1,...,T$ and $ \left(y_{t}\right)_{t=1}^{T} \succ \left(y'_{t}\right)_{t=1}^{T}$ if moreover $y_{t}>y'_{t}$ for some t;
- Comparability: for all numbers $\alpha\in\mathbb{R}$ and $\beta\in\mathbb{R}_{++}$, $\left(y_{t}\right)_{t=0}^{T}\succsim\left(y'_{t}\right)_{t=0}^{T}$ if and only if $\left(\alpha+\beta y_{t}\right)_{t=0}^{T}\succsim\left(\alpha+\beta y'_{t}\right)_{t=0}^{T}$;
- Time Separability: for all $M\subset\left\{ 0,...,T\right\}$ of consecutive periods, $\left( \left(y_{t}\right)_{t\in M},\left(y_{t}\right)_{t\notin M}\right)\succsim\left(\left(y_{t}\right)_{t\in M},\left(y_{t}\right)_{t\notin M}\right)$ if and only if $\left( \left(y_{t}\right)_{t\in M},\left(y'_{t}\right)_{t\notin M}\right)\succsim\left(\left(y_{t}\right)_{t\in M},\left(y'_{t}\right)_{t\notin M}\right)$.
I would like to make the following claim.
Claim If $\succsim$ satisfies Monotonicity, Comparability and Time Separability, there exist weights $\omega_{t}>0$ , $\sum_{t=0}^{T}\omega_{t}=1$ such that, for all $\left(y_{t}\right)_{t=1}^{T}, \left(y'_{t}\right)_{t=1}^{T}$, we have $ \left(y_{t}\right)_{t=1}^{T}\succsim \left(y'_{t}\right)_{t=1}^{T}$ if and only if $\sum_{t=0}^{T}\omega_{t}y_{t}\geq\sum_{t=0}^{T}\omega_{t}y'_{t}$.
"Ebert (1988). 'Rawls and Bentham reconciled.' Theory and Decision." proved something very similar to the previous result, in a setting where $t$ does not refer to time but individuals, and where:
- if $\left(y'_{t}\right)_{t=1}^{T}$ is a permutation of $\left(y_{t}\right)_{t=1}^{T}$, then the two profiles are equivalent (call this Anonimity), and
- the Time Separability above is replaced by Rank Separability, in which the set $M$ does not refer to consecutive periods but to individuals that occupy ranks positions in the (ordered) income profiles.
In Ebert (1988) Rank Separability implies Anonymity but this is not the case in my setting (or at least I don't want this to be the case). I think that despite this difference, my claim still holds. However, I'm afraid I'm missing something here. Can someone help me clarifying this?