A few words on proportional response
Non Pareto-optimality along the way and a Cobb-Douglas perspective
Non Pareto-optimal allocation along the way
Consider the CES fisher market, that is, for divisible goods $j=1, \ldots, n$, we have a unit supply. Each agent has a budget $w_i \in \mathbb{R}_+$ and a utility coefficient vector $\mathbf{c}_i$, the utility reads,
$$u_i(\mathbf{x}_i) = \langle \mathbf{c}_i, \mathbf{x}_i^{r_i}\rangle^{\tfrac{1}{r_i}}$$
The best response is the following,
$$\mathbf{x}_i^\textrm{BR}(\mathbf{p}) = \mathbf{P}^{- 1} w_i \cdot \tfrac{\mathbf{P}^{-\tfrac{r_i}{1-r_i}} \mathbf{c}_i^{\tfrac{1}{1-r_i}}}{\langle \mathbf{P}^{-\tfrac{r_i}{1-r_i}} \mathbf{c}_i^{\tfrac{1}{1-r_i}}, \mathbf{1}\rangle}$$
which is used in a Tâtonnement Process. Cheung et al. [1] give the following update rule of a Proportional Response (PR) process.
If $r_i > 0$. it update the bids:
$$\begin{align*} b_{i j}^{t+1} &= w_i \cdot \tfrac{c_{i j}\left(x_{i j}^t\right)^{r_i}}{\textstyle\sum_k c_{i k}\left(x_{i k}^t\right)^{r_i}}=w_i \cdot \tfrac{c_{i j}\left(\tfrac{b_{i j}^t}{p_j^t}\right)^{r_i}}{\textstyle\sum_k c_{i k}\left(\tfrac{b_{i k}^t}{p_k^t}\right)^{r_i}} \\ p_{j}^{t+1} &= \textstyle\sum_i b_{i j}^{t+1} \\ x_{ij}^{t+1} &= \tfrac{b_{i j}^{t+1}}{p_j^{t+1}} \end{align*}$$
It is easy to see the market clears at $\mathbf{p}^{t+1}, \mathbf{x}^{t+1}$:
$$\textstyle\sum_i \mathbf{x}_i^{t+1} = \mathbf{1}.$$
If $r_i < 0$. it update the bids:
$$\begin{align*} b_{i j}^{t+1} & =w_i \cdot \tfrac{\left(\tfrac{c_{i j}}{\left(p_j^t\right)^{r_i}}\right)^{\tfrac{1}{1-r_i}}}{\textstyle\sum_k\left(\tfrac{c_{i k}}{\left(p_k^t\right)^{r_i}}\right)^{\tfrac{1}{1-r_i}}} \\ p_{j}^{t+1} &= \textstyle\sum_i b_{i j}^{t+1} \end{align*}$$
The nuance here is,
- if let $x^{t+1}_{ij} = \tfrac{b_{i j}^{t+1}}{p_j^{t}}$ then it is the best response at $\mathbf{p}^t$, but not at $\mathbf{p}^{t+1}$, and the market is not cleared;
- while if as usual allocating by the bids, $x^{t+1}_{ij} = \tfrac{b_{i j}^{t+1}}{p_j^{t+1}}$, then it is neither the best response at $\mathbf{p}^{t+1}$ nor at $\mathbf{p}^t$, but the market clears.
So in general, PR is a type of process that ensures the market clears, but the allocation is not necessarily Pareto-optimal along the way, unless it reaches the equilibrium price set.
PR from a Cobb-Douglas perspective
I one day found an unpublished paper by Prof. Michael Todd, that the PR process is iteratively approximating the economy by one with Cobb-Douglas utilities [2]. I have never seen it mentioned in the literature.
References
- [1] Dynamics of distributed updating in Fisher markets. arXiv preprint arXiv:1806.04746 (2018). http://arxiv.org/abs/1806.04746
- [2] Proportional response as iterated Cobb-Douglas. In: Equilibrium Computation, Dagstuhl Seminar Proceedings, vol. 10171, pp. 1–6. Schloss Dagstuhl (2010). https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10171.3