# đź‘ŤAn Impossibility Theorem for Wealth in Heterogeneous-agent Models with Limited Heterogeneity

Published in *Journal of Economic Theory*, 2019

On January 7, 2017, I attended the AEA session â€śHeterogeneity and Power Laws in Macroeconomicsâ€ť chaired by Xavier Gabaix. I was the discussant of Benhabib, Bisin, & Luo (2017) â€śEarnings Inequality and Other Determinants of Wealth Inequalityâ€ť. This was a very interesting paper, because the authors used a stylized linear consumption function to argue that the wealth Pareto exponent is either equal to the income Pareto exponent or is entirely determined by the return on wealth (and unrelated to income). In my discussion, I praised the authors, but commented

it would be even better if authors have a truly micro-founded model

and suggested that they may use CARA utility for analytical tractability. (Interestingly, in my discussion I already argued that applying the results of Beare & Toda (2022) may explain why Krusell & Smith (1998) needed random discount factors for generating realistic wealth distributions, which is a point I later published in Toda (2019).)

In June 11-13, 2018, I organized a session â€śPower Law in Economics and Financeâ€ť at the SAET conference in Taipei. One of the participants was Dan Cao. During a dinner conversation with him, the papers Quadrini (2000) and Cagetti & De Nardi (2006), which are often cited in the context of wealth inequality because their models feature idiosyncratic investment risks, came up on the topic, and we wondered whether these models actually generate Pareto tails as they are numerical. Until that point I had written many models that generate Pareto tails (*possibility*), but I had not thought about classes of models that cannot generate Pareto tails (*impossibility*).

After giving talks in Hong Kong, I visited Australian National University Research School of Economics from June 17 to July 24. I spent most of my time working on the Pareto extrapolation paper with Ă‰milien, but John and Qingyin had already invited me to join the optimal savings project, so I studied relevant papers such as Li & Stachurski (2014). On July 16, 2018, using the concavity and asymptotic linearity of consumption functions in what is now Example 2.2 of Ma et al. (2020), I proved the boundedness of the wealth distribution in Bewley-type models. I showed my notes to John, and he pushed me to generalize, so I came up with an idea of using the properties of the moment generation functions and the stylized model in what is now Section 2 of Stachurski & Toda (2019) to prove the impossibility of Bewley-Huggett-Aiyagari models to generate heavier-tailed wealth than income. Initially I proved it only for constant relative risk aversion (CRRA) utility, but John generalized it for arbitrary bounded relative risk aversion utility, so we wrote a paper. The draft was essentially complete by the following day, July 17.

I think this is one of the best papers I have ever written. Since Huggett (1993) and Aiyagari (1994), thousands of quantitative macroeconomics papers have tried to explain the wealth distribution and failed. In the Handbook of Macroeconomics, Krueger, Mitman, & Perri (2016) state

idiosyncratic unemployment risk and incomplete financial markets alone are insufficient to generate a sufficiently dispersed model-based cross-sectional wealth distribution

but there have been no theoretical explanations for a quarter of century, only numerical experiments and intuitive discussions. My paper with John resolved this open question and proved the impossibility of canonical Bewley-Huggett-Aiyagari models to generate heavier-tailed wealth than income in a micro-founded general equilibrium setting.

John and I thought this paper was a scientific breakthrough, and we submitted it to AER: Insights on July 26, 2018 after polishing for a week. It was rejected on September 5, one referee recommending acceptance, one on the fence, and one recommending rejection with the following report:

This is an interesting insight but it is not new, as it appears in a series of papers by Benhabib and co-authors, which the authors cite. The underlying math result pertains to a class of linear stochastic difference equation which can be mapped into several macroeconomic models of wealth dynamics.

This paper characterizes precisely a set of sufficient conditions on preferences to obtain this mapping. On the other hand, the class of economies is restricted to consider only constant rate of return on wealth, which is a severe restriction in the wealth distribution literature.

The method of proof, based on previous results of one of the authors on policy function iteration, is also a useful contribution.

For this reason I believe the paper is useful and should be published. But itâ€™s a technical note rather than a AERI paper, in my opinion. I would have no reservations to suggest acceptance e.g., in Economics Letters.

You can see the full report here. Many of these claims are false, for instance â€śnot newâ€ť (it is), â€śappears in a series of papers by Benhabibâ€ť (the BBL paper above is not micro-founded), â€ślinear stochastic difference equationâ€ť (it is nonlinear), and â€śsevere restrictionâ€ť (the canonical model features constant returns by definition). The last paragraph â€śtechnical note [â€¦] Economics Lettersâ€ť is insulting, but of course that is a subjective judgement that cannot be refuted. We were outraged and submitted the paper to Econometrica by attaching the AERI reports and a detailed cover letter with rebuttals, but the paper got desk-rejected with this decision letter. The editor seemed to have consulted the same individual (and not to value theorems). We then submitted the paper to REStud with an even clearer cover letter that explained the conflict of interest and the unfair treatment at previous journals, but we again got a rejection. In the decision letter, the editor acknowledged that he had consulted a fifth referee, who seemed to have provided the same false information.

Within the current peer-review system, where experts are supposed to evaluate peersâ€™ research fairly and professionally, when we work on a small and technical topic, it is already challenging to draw qualified referees. But what can we do if in addition an â€śexpertâ€ť abuses power to manipulate editors of important journals? I was traumatized. In January 2019, we submitted the paper to JET, this time just attaching the REStud decision letter because we didnâ€™t want to draw attention to that referee again. We received only one report, whose writing style resembled previous reports and what previous editors quoted (like â€śGreyâ€ť, which refers to this paper but not really related), but this time with full of praise. There are â€śexpertsâ€ť who cannot tolerate accepting competitorsâ€™ papers at top journals, but are happy to support them after making sure they failed. The JET editor accepted our paper as is.

During this ordeal, it seems that nobody read the proofs. We later found an error in Proposition 5, and published a corrigendum.