Update post-rebuttal: Please include explanation / references related to "Cons" 1 and 2 below. Summary: The authors study a variant of a standard economic model of production economies, wherein agents may produce and consume goods according to simple rules. It is shown that a simple decentralized dynamic yields economic growth whenever such growth is possible according to the production rules. Other facets of the model, such as inequality, are also addressed. Pros: The results presented are strong and compelling: a simple distributed trade dynamic "finds" a growth opportunity and naturally rewards the participants multiplicatively, leading to growth of the economy as a whole. The insights about inequality are also interesting and natural, and several other natural questions, such as the basic dynamics in terms of periodic points, are also addressed. Cons: Firstly, I would like the authors to give better intuition or rationale behind the choice of dynamic. Why should bids be updated this way? Is there a natural economic or game-theoretic reason from the agents' point of view? (I.e., is this mechanism "rational" for the agents?) Secondly, and more importantly, I struggle to see this paper as a good fit for NIPS. What is the connection to core NIPS topics? A brief glance at the bibliography shows one JAIR paper (I may have miss some) which I believe is a symptom of the authors not taking the time to make the paper of interest to more than a tiny fraction of NIPS attendees. I would invite the authors to respond to this point (and the previous). Other comments and questions: Line 289: The linearity of the proportional update mechanism seems crucial in this result; have the authors thought about how robust this result is, in the sense that "nearby" dynamics yield the same result? Line 325: Use ~, as in Lemma~\ref{...} Line 395: viceversa -> vice versa

The paper studies a model of production economies with linear production functions, and a dynamics based on trading posts and proportional updates, to show that (i) growth is guaranteed whenever it is possible and (ii) inequality grows under this model. The paper also provides several numerical experiments. The paper is well written and pleasant to read. The setting and main results are illustrated with simple examples in the introduction, then further developed in the main text. The proofs rely on elementary arguments that are easy to follow. The literature review is extensive and the results are placed in the context of the relevant literature. I would like to see further motivation of the model. It is mentioned in the paper that most of the existing work focuses on studying convergence of dynamics to equilibria, while this paper is interested in diverging trajectories, and I do not see a good justification for this model. Diverging growth does not seem to be a realistic model of a production economy. The paper has some formatting issues: references to the appendix are broken, the orgnization paragraph (line 185) refers to inexisting sections, and the paper ends abruptly at the end of Section 5. It is evident that the paper was cut down to eight pages from a longer version given in the supplement. Further efforts need to be put into polishing the main paper. The statements of the results in the introduction are informal, especially Theorem 1.2. For example, "most efficient production cycle", and "inequality gap" are not properly defined until later sections. The definition of universal mechanism is also informal, as "mechanism" is not properly defined until much later in the paper. How can one define a universal mechanism before defining a mechanism? Given the need for additional space (see my previous remark), and the issues caused by stating results before definitions, I strongly recommend reorganizing the introduction to move statements of main results to later sections. The numerical simulations of the Gini index make for interesting plots, but they are simply given without discussion. What is the purpose of these figures? Is it to highlight the dependence on initial conditions? Or to show the existence of certain phase transitions? What insights can we draw from these plots? * Post rebuttal * Thank you for the responses and clarifications. I agree with the argument that the quantities produced in production economies do not converge, but this does not prevent other quantities (such as production rates, prices, etc.) from converging, and the study of a model which does not guarantee convergence must be more carefully motivated and justified. The second argument made in the rebuttal, that there is a "conceptual problem with equilibrium based analysis", namely that "market equilibrium solution does not explain how players would trade when the market is out-of-equilibrium" conflates equilibrium and adjustment dynamics: the study of adjustment dynamics (such as tatonnement, proportional response, etc.) answers the question of how players can reach equilibrium. My question on the purpose of the Gini index plots was not addressed.

This paper studies the evolution of a growing production economy under a certain dynamic model of participant behavior. The model is a minor variation on a classic model of von Neumann. There are n firms, each producing their own type of good, and each with a budget of money. Each firm i also has the ability to convert units of any other good j into units of their own good, at some given conversion rate a_{ij}. Each round, the firms bid on the goods produced by other firms. This results in exchanges of goods for money (more on this below). After this, the firms convert any acquired goods into their own type of good, and the process repeats. The economy is said to expand if the total quantity of goods increases without bound. Rather than study the optimal equilibrium growth, the paper studies the following dynamics. In each round, firms bid on goods in proportion to how much the goods contributed to their production last round. Also, the bidding is resolved by proportional share: each firm's output is divided in proportion to the bids received, and every firm pays their bids. The main result of the paper is that if there is any production path such that the economy expands, then this proportional dynamics will also expand. Furthermore, the rate of expansion converges to the optimal rate of expansion. This result is complemented with a variety of simulations and structural observations. E.g., growth can oscillate, and some firms will grow more quickly than others in the limit. The authors relate this last observation to notions of income inequality, and simulate the evolution of the GINI index in simple economies. Comments I like the idea of studying growth in pure production economies. This is a classic topic in market theory that has not received much (or any) attention in the TCS literature. The model studied here is a simple and natural starting point. I also like the agenda of understanding natural dynamics by which firms learn to bid and bring the economy toward equilibrium. This has been a successful line of inquiry for markets with consumption, so it makes sense to apply it here as well. The main result --- that the proposed dynamics leads to optimal growth --- is clean and appealing. The proof is short and slick, following a potential-function argument. While similar arguments have been used for other market types, the setting here is different enough that new ideas were needed. The strength of the main result depends on how much the reader likes the "trading-post proportional-bidding" dynamics being studied. Personally I think the model is reasonable, as it blends together two ideas (proportional sharing of outcomes and proportional bidding) that have been well-studied in related contexts. I do find myself wondering if there is evidence that this dynamics is predictive. Do we expect that markets actually behave this way? Or is the point of the paper that a market-maker should encourage production firms to behave this way? More motivation would be welcome. I found the other content in the paper less exciting. For instance: - I find it natural that some firms grow more quickly than others, since there is no consumption, taxation, or limited labor pool. Since those forces are missing here, it's unsurprising that firms on the "efficient" production path will extract much of the gains. Because of this, I don't view the paper as saying too much about inequality in practical modern economies. - The presence of phase transitions in growth vs shrinkage of a firm is not particularly exciting, as this essentially amounts to a certain coefficient being larger or smaller than 1, which naturally occurs at a transition point. - The presence of oscillation patterns is to be expected, as the discrete nature of time coupled with the fact that an efficient cycle might contract at some points and expand at others leads directly to non-monotonicities. One thing I would have liked to see is more comparison between the dynamics and the equilibrium (i.e., optimal) growth. E.g., rate of convergence, and whether the set of growing vs vanishing firms differs between the dynamic and equilibrium outcomes in the long run. A more complete analysis of equilibria in this particular variant of von Neumann's model would help to better appreciate the outcome of the learning dynamics. Overall, I think the model being studied is natural, and the main result provides a crisp result for a natural set of local dynamics. The fact that one very specific set of dynamics is studied does limit the significance of the work, but I believe there is a reasonable chance that this model could generate follow-up work. On the other hand, the paper is padded out with a significant amount of extra content that I think detracts from the overall message, and the clarity of the paper suffers as a result. The topic of "firms learning to bid in an economy" fits within the scope of NIPS, if perhaps tangentially. Comments for the authors Abstract, line 7: the comment about "unbounded inequality" makes it sound as though this is an outcome specific to the dynamics. Would a social planner designing an optimal rate of growth not also generate unbounded inequality? I'd clarify and/or rephrase both here and in the introduction. Page 3, line 96: Should we take these oscillation patterns as evidence that the model is not predictive? Or should we expect these outcomes in practice? Consider commenting on this here or elsewhere. Page 7, line 270: please comment on the strength of this assumption of a unique best cycle Page 7, line 274: broken reference to an Appendix Response to Author Comments: The authors make a good point about adversarial initial states, and the challenge that they pose. I still feel that many of the "extra" results are unsurprising at a qualitative/conceptual level, but I certainly don't mean to imply that they are obvious or trivial on a technical level. I still feel very positively about the "main" (in my opinion) contribution, which is the convergence and optimal growth properties of the studied dynamics.