Alexis Akira Toda

Alexis Akira Toda

Mathematical economist

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A list of all the posts and pages found on the site. For you robots out there is an XML version available for digesting as well.

Pages

Posts

Website updated

less than 1 minute read

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After spending several days, I significantly updated my website. Below is a list of main changes.

  • Previously, there was a “Research” page that I created manually. But it was tedious to list all papers in a consistent style (like journal names in bold face, coauthors in italics, etc.) so I automated it.
  • I omitted publication information except the journal name and year, because I don’t think they are very important and one can easily find by clicking the link to the published paper.
  • I searched for all my paper drafts in my computer and added slides whenever I found them.
  • I merged “Conference proceedings” with “Other publications” because I don’t see a reason to keep proceedings as a separate section given that I don’t have many.

Talks

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Maybe it is not the best use of my time but I created a “Talks” page, which lists past and future talks and presentations. After spending all afternoon tweaking the code, I created something satisfactory. For now I included only talks in 2023 and 2024 but I might add past ones.

Nissan Leaf after 7 years (2)

less than 1 minute read

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I am still waiting for the replacement battery for my 2016 Nissan Leaf. It’s nearly 9 months wait now.

Nissan Leaf after 7 years

3 minute read

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In 2016 I purchased a Nissan Leaf. Given the technological progress in electric vehicles, now it’s almost a joke but it had a 30kWh battery with around 110 miles of range. Including everything (tax, registration, etc.), it cost 44,142 dollars. My employer subsidized 10,000 and I received 7,500 federal and 2,500 state tax credits, so at that time I thought it was not a bad deal. 110 miles of range seemed enough to do daily chores, and for road trips we had been renting a gas-powered vehicle. This was our only vehicle in the household until 2020, when we bought a Tesla Model Y.

biblatex

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For managing a bibliography for my papers, for a long time I have been using the natbib package with bibtex to compile. ``` \usepackage[authoryear,round]{natbib} \usepackage{doi}

Pitfall of electric bike

2 minute read

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When I moved from Connecticut to San Diego in 2013, one thing I paid attention to when searching for a house was to find one within reasonable distance from work. I knew that San Diego has nice weather and commuting by bicycle was feasible. So to save money and get some exercise, I cut the number of vehicles in my household from two to one and bought an electric bike to commute. (In 2020, I went back to two vehicles because we could no longer keep up with only one vehicle to manage kids’ activities.)

Investment and taxation

3 minute read

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Since I started working in 2013, I haven’t thought much about investment. Because I know that investing in an index fund is the theoretically right thing to do and I do not like paying taxes, in addition to participating in my employer’s mandatory defined benefit retirement plan, I have just been maxing out investment in 403(b), 457(b), and Roth IRA (all invested in low cost index funds).

Optimal use of sabbatical credit

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This post is completely boring for outsiders. I am just writing this down so that I do not forget.

Disabling Canvas Inbox

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My university uses Canvas for instructional purposes. Canvas has a functionality called “Inbox”, which allows you to send messages. I have long thought this functionality not only useless but even harmful. This is because I ignore everything sent to Canvas Inbox, but students who are not aware that I do not use Canvas Inbox may send me messages that gets ignored and may complain. Since I was not able to find information on how to disable Inbox, I contacted the university technical support.

Fun facts about tennis rule (3)

3 minute read

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These days I mostly play competitive tennis, so my opponents usually have sufficient understandings of the rules and we do not get into disputes. However, debates and disputes sometimes occur in recreational play, which could be annoying. So in a series of posts, let me talk about some rules that could be overlooked by recreational players. The official rules can be found in the “Friend at Court” here.

Fun facts about tennis rule (2)

1 minute read

Published:

These days I mostly play competitive tennis, so my opponents usually have sufficient understandings of the rules and we do not get into disputes. However, debates and disputes sometimes occur in recreational play, which could be annoying. So in a series of posts, let me talk about some rules that could be overlooked by recreational players. The official rules can be found in the “Friend at Court” here.

Fun facts about tennis rule (1)

3 minute read

Published:

These days I mostly play competitive tennis, so my opponents usually have sufficient understandings of the rules and we do not get into disputes. However, debates and disputes sometimes occur in recreational play, which could be annoying. So in a series of posts, let me talk about some rules that could be overlooked by recreational players. The official rules can be found in the “Friend at Court” here.

Capital and labor income Pareto exponents

1 minute read

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My paper “Capital and Labor Income Pareto Exponents across Time and Space” got accepted at Review of Income and Wealth. I have long felt that in the discussion of inequality, people often don’t make the distinction between income and wealth. In this paper we (my coauthor Tjeerd de Vries and I) estimate the Pareto exponents for capital and labor income separately for as many countries/years as possible. Using 475 country-year observations, we find that the median capital and labor income Pareto exponents are 1.46 and 3.35 respectively, so capital income (hence wealth) is more unequal than labor income. This conclusion is not surprising at all, but the point of the paper is to provide a systematic analysis, which was lacking in the earlier literature.

Paper accepted

less than 1 minute read

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My paper “Asymptotic Linearity of Consumption Functions and Computational Efficiency” with Qingyin Ma got accepted at Journal of Mathematical Economics. The main result is that when the marginal utility function is regularly varying (behave like a power function), the consumption function in optimal savings problem becomes asymptotically linear and we characterize the asymptotic slopes. Initially this paper was part of a bigger project with Ma & Toda (2021), but we split the paper in two to keep them focused and at manageable lengths. The JET paper treats only the case with CRRA (constant relative risk aversion) utility but has an economic application. The JME paper assumes regular variation plus some technical condition and discusses computational efficiency.

Paper accepted

less than 1 minute read

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My paper “Perov’s Contraction Principle and Dynamic Programming with Stochastic Discounting” got accepted at Operations Research Letters. I found a generalization of Banach’s contraction mapping theorem when I was studying a certain dynamic programming problem with stochstic discounting. It turned out that the fixed point theorem was due to Perov (1964) but I thought the application was interesting, so I wrote a short paper.

Good pusher, bad pusher

2 minute read

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In tennis, a pusher is a player who can consistently hit the ball back inside the opponent’s court. They have good footwork and run to every ball. Michael Chang and Rafael Nadal are legendary professional players that have perfected this valuable skill.

Fun drill to improve consistency in tennis

2 minute read

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I have been a recreational tennis player for 6 or 7 years. Since fall 2020, I have started to play some matches. (You can see my statistics here.) After starting to play competitively, I have read a lot about tennis strategies because obviously it is more fun to win than to lose. I soon learned (both from theory and experience) that one of the easiest ways to improve your match results is to be more consistent, that is, to reduce unforced errors. So I have been paying attention to consistency and my games have improved.

Full-text articles on ResearchGate

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I don’t remember when I opened a ResearchGate account; I guess I did so to increase the visibility of my research. I don’t really use their service, but I find it annoying that I get requests to upload full-text articles. This is a waste of time because most of my papers are available online as working paper versions. (These days I upload my working papers exclusively to arXiv.) So I decided to write a short document stating that all of my papers are linked from my personal website (and provided the link), and uploaded it as a full-text article.

GPIF Award

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It is a great honor to announce that I have been awarded the 4th Government Pension Investment Fund (GPIF) Finance Award under the auspices of Ministry of Health, Labor and Welfare and Ministry of Education, Culture, Sports, Science and Technology in Japan. I am very happy that my research has been recognized.

Homothetic theory of the saving rate of the rich

1 minute read

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My paper “A Theory of the Saving Rate of the Rich” was accepted at Journal of Economic Theory. I started this project in early 2020. Initially, the paper was about rigorously establishing the asymptotic linearity of policy functions when preferences are homothetic and the constraint is asymptotically homogeneous of degree 1. This is not surprising but the proof is difficult. As we worked on the proof, we (my coauthor and I) discovered that the asymptotic slope of the policy function can be zero, which was surprising. When the asymptotic marginal propensity to consume (MPC) is zero, an infinitely wealthy agent saves 100% of the wealth, which can explain the empirical puzzle that wealthy people save a lot although it seems unnecessary. So instead of focusing on just a mathematical fact, we decided to frame the paper as a new theory of the saving rate of the rich.

Financial lesson from 2020

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I have been investing for over 20 years. After learning about the capital asset pricing model and the mutual fund theorem and reading “A Random Walk Down Wall Street” and “Stocks for the Long Run”, I have been more or less consistently investing in low-cost index ETFs such as VTI and VXUS. This allowed me to stay in the market during the bottom in March 2009 and not to miss the bull market since then despite some of my colleagues advising me that stocks are overpriced. My kids’ 529 funds have grown about 3 times in nominal value. I have been maxing out my 403b, 457b, and Roth IRA contributions and joking I could retire if I choose to. Based on theory and experience, I preach the importance of passive investing to students in my finance class.

Necessity of HARA for the concavity of consumption functions

less than 1 minute read

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My paper “Necessity of Hyperbolic Absolute Risk Aversion for the Concavity of Consumption Functions” was accepted at Journal of Mathematical Economics. The publication process was quite efficient. I came up with the idea in late September 2020 and wrote a short paper. After getting rejected from a different journal, I sent to JME. I am very happy that it came out in less than three months after I have started the project. Journals in economics tend to be very slow in the review process, perhaps because many papers tend to be long and unfocused. We should all write concise and focused papers.

My unpublished COVID-19 paper is now my most cited paper

1 minute read

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Recently, my working paper Susceptible-Infected-Recovered (SIR) Dynamics of COVID-19 and Economic Impact has surpassed my JEBO paper in terms of citation counts, and has become my most cited paper. My COVID-19 paper is one of the very first written by an economist on this topic, and it appeared in the first issue of the working paper series Covid Economics. Although I am no longer working on this paper since the situation with COVID-19 has been changing too quickly (especially when I wrote the paper in March 2020) to keep up with, I am glad that this paper has made some impact. In fact, it was featured in VoxEU and Fortune articles.

Difference between $$..., \[..., and \begin{equation*}...

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I have been using \(\LaTeX\) for over 20 years now. When I write a displayed equation without numbering on a single line, I have been using $$...$$ because it was simple. I didn’t understand why some people use \[...\], because the latter takes more time to type and is not necessarily easy to read. Today I read this article and learned that $$...$$ is incorrect. From now on, I will switch to \begin{equation*}...\end{equation*} because it is easy to read and we can add equation numbering by deleting * if we change our mind.

Inequality on spectral abscissa

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Let \(A, B\) be square (complex) matrices such that \(|B| \le A\). Then it is well known that \(\rho(B) \le \rho(|B|) \le \rho(A)\), where \(\rho\) denotes the spectral radius (largest absolute value of all eigenvalues). See, for example, Theorem 8.4.5 of Horn and Johnson (2013). In my recent paper with Brendan Beare and Won-Ki Seo, we needed to use the spectral abscissa (largest real part of all eigenvalues) instead of the spectral radius. By analogy, we can make the following conjecture: if \(A, B\) are square complex matrices such that \(\mathrm{Re} b_{nn} \le a_{nn}\) for all \(n\) and \(|b_{nn’}| \le a_{nn’}\) for all \(n \neq n’\), then is it true that \(\zeta(B) \le \zeta(A)\), where \(\zeta\) denotes the spectral abscissa?

White rice and brown rice

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There is a restaurant called “The Bistro” on UCSD campus. Although I don’t like this restaurant because it’s basically a fusion American Asian place, sometimes I have to eat there when we take seminar speakers for lunch. Once I ordered some fried cod (neither quite fish and chips nor tempura). On the menu it said the dish comes with brown rice, so I asked the server to substitute white rice for brown rice. (Although I don’t have statistics to quote, I would say most Japanese people eat white rice - only those that are health-conscious and opinionated eat brown rice, though obviously the latter is healthier.) When the dish arrived, I was stunned that the rice, though white, was sushi rice (i.e., vinegared rice). I asked the server to bring proper white rice but she didn’t know the difference. Since then, whenever I organize the seminar lunch, I choose a different place.

Created new website

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I have created a new website. I have been using Google Sites to create my old website. I like the classic Google Sites because it allows the user to take control of the structure by programming in html. However, the new Google Sites no longer have this feature, and since the classic Google Sites will be discontinued in 2021, I had to do something else. After a bit of Google search, I found this template, which is exactly what I wanted (ability to take full control, free, no advertisements, etc.).

misc

Discretization

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Discretizing probability distributions and stochastic processes

Coauthors

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List of my coauthors

LaTeX

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Resources on LaTeX

Travel

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Places that I have visited

other

Susceptible-Infected-Recovered (SIR) Dynamics of COVID-19 and Economic Impact Permalink

Published:

After returning from my UK trip in early March 2020, just a few days before the closure of the border, I got interested in COVID-19, like many of us. I played around with some models and I derived a system of differential equations that turned out to be identical to the Kermack & McKendrick (1927) susceptible-infected-recovered (SIR) model. I searched the literature and found the closed-form solution by Harko et al. (2014). At that time, little was known about COVID-19, so I decided to estimate the epidemiological parameters from data to predict the epidemic dynamics.

In mid March 2020, the state of California imposed lockdown, my kids’ school suddenly closed (teachers were not yet used to online teaching, so education was laissez faire to parents), tennis courts closed, and we could no longer eat at restaurants. I thought this was all wrong. I have a medical degree and knew about herd immunity, so just shutting down everything would simply delay the problem without solving it, while imposing significant economic costs to the society. So for the first time in my career as a researcher, I worked on a project not just driven by scientific curiosity but also by a sense of civic duty to inform the public and policy makers.

According to my analysis in the figure below, imposing an early (and necessarily temporary) lockdown would simply delay the spread of the epidemic without affecting the ultimate toll.

Early mitigation

However, if the government delays intervention until about 1% of the population is infected, it would significantly reduce the total cumulative number of infections due to the build up of herd immunity, as the following figure shows.

Early mitigation

The analysis was of course not perfect, but because I thought the problem was urgent, I submitted this paper to AER: Insights on March 26, 2020. I presented the paper at the UCSD Econometrics seminar on March 31, 2020, and the paper was included in the first issue of the working paper series Covid Economics. It was also featured in the April 2020 VoxEU and May 2020 Fortune articles. This paper is by far my most cited paper. Unfortunately, AER: Insights rejected my paper in May 2020, by which time so many things have changed and there was so much competition, so I gave up the project.

After this experience, I started to question the relevance of traditional economics research. I felt that the long review process at typical economics journals prevents researchers from working on pressing issues. Of course, one could write a sophisticated paper after the pandemic is long gone (instead of writing a preliminary paper in a few weeks), but what is the benefit to the society? I also felt that most COVID papers written by economists encouraged lockdown and sided with increased government control, perhaps because researchers wanted to play the good guy. Later, I published a more sophisticated paper in JET.

Visualizing the Contraction Mapping Theorem Permalink

Published:

After my 2022 ECMA paper got accepted in January 2022, I got burnt out. I stopped doing research and spent most of my time playing tennis. I became a captain of a USTA 7.0 mixed doubles team (I had just become 4.0), and my wife and I recruited strong players, organized practices, and advanced to sectionals twice. In one of the Southern California sectionals, we narrowly lost in the semifinal.

My coauthor Jim Rauch was the recruitment chair when I got hired at UCSD in 2013. He was also the department chair from 2013 to 2016, so I had a lot of interaction with him. When we chatted in June 2022, he mentioned he was interested in a project to animate the contraction mapping theorem to help build intuition. We knew it had zero career benefit but it sounded fun, so I wrote up some pedagogical material and Matlab codes and Jim made the video.

We tried to publish this in journals on economic education but we got desk-rejected each time and the paper became dormant. Later, I was asked to review a paper at Qeios. Because I had never heard of that journal, I thought it was a predatory journal, but upon inspection its business model seemed interesting: they publish anything, but reviews are open and (to prevent abuse) not anonymous. So we posted our paper there, and we are happy that our paper and video have been well received.

Convergence of value functions

‘Ergodicity Economics’ is Pseudoscience Permalink

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In May 2023, I read something about Ergodicity Economics, and I thought it was a completely nonsense pseudoscience promoted by failed self-proclaimed physicists. Although it had zero career benefit, to contribute to the public good to prevent the spread of pseudoscience, I spent a few days writing this critique. The paper got desk-rejected from Physical Review Letters, Physical Review E, and Chaos and became dormant. After my pleasant experience at Qeios, I published the paper there.

portfolio

publications

Radii of the Inscribed and Escribed Spheres of a Simplex Permalink

Published in International Journal of Geometry, 2014

(Mathematics) High-dimensional generalization of the fact that the sum of the reciprocals of the radii of escribed circles of a triangle equals the reciprocal of the radius of the inscribed circle; obtained those results in 1998 when I was freshman.

👍The Equity Premium and the One Percent Permalink

Published in Review of Financial Studies, 2020

👍(Finance, Theory, Empirical) In general equilibrium model with heterogeneous risk aversion and/or beliefs, the wealth distribution predicts excess stock returns, which we confirm in data using estate tax rate change as instrument.

👍A Theory of the Saving Rate of the Rich Permalink

Published in Journal of Economic Theory, 2021

👍(Theory, Macro) Prove asymptotic linearity of policy functions when preferences are homothetic; show that asymptotic marginal propensities to consume can be zero, implying a large saving rate of the rich.

👍Optimal Epidemic Control in Equilibrium with Imperfect Testing and Enforcement Permalink

Published in Journal of Economic Theory, 2022

👍(Theory) Study a behavioral SIR model with imperfect testing and government enforcement and show that equilibrium action is approximately static efficient in the sense that the laissez faire equilibrium allocation is close to the optimal short-term lockdown policy, implying that short-term lockdown policies are redundant.

Capital and Labor Income Pareto Exponents across Time and Space Permalink

Published in Review of Income and Wealth, 2022

(Power law, Empirical) Estimate capital and labor income Pareto exponents across 475 country-year observations and document that capital income inequality is higher than labor income inequality (median Pareto exponents 1.46 and 3.35 respectively) and the two inequalities are uncorrelated, suggesting importance of distinguishing the two.

👍Bubble Economics Permalink

Published in Journal of Mathematical Economics, 2024

👍(Theory, Macro, Finance) Self-contained review of the theory of asset price bubbles.

talks

teaching

Operations Research (Econ 172B)

Undergraduate, UCSD, 2015

This course covers some topics in operations research, such as convex analysis, nonlinear programming, and dynamic programming. I do not currently teach this course.

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