# Sitemap

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.

## Backdoor Roth

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I have been investing in Roth IRA. Currenty the contribution limit is $7,000/year. I haven’t been paying attention much, but there are income limits for traditional and Roth IRA. For traditional, married couples filing jointly cannot deduct contributions if their modified adjusted gross income (MAGI) is above$136,000. For Roth, married couples filing jointly cannot contribute if their MAGI is above $240,000. Because I recently got a raise by changing employer, I can no longer contribute to Roth. However, there is a legal loophole called backdoor Roth. All you need is to contribute to a traditional IRA (which is not tax deductible due to the income limit but this is irrelevant) and then do a Roth conversion. This way, anybody can contribute to Roth regardless of their income. So from next year on, I can simply contribute$7,000 of cash to my traditional IRA account, do a Roth conversion, and invest the funds in whatever way I like.

However, for this year there is a problem because I have already contributed to Roth before I knew I would exceed the income limit. I did a bit of research and found this article. Reading it, I did the following.

1. First, I recharacterized this year’s Roth contributions to traditional.
2. Then, I contributed cash to traditional to hit the annual contribution limit.
3. Finally, I converted all funds from traditional to Roth.

## 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.).

## Letter of recommendation

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My policy for writing letters of recommendation

## Discretization

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Discretizing probability distributions and stochastic processes

## Applying to economics Ph.D. programs

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Advices on applying to economics Ph.D. programs

## Email writing tips

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Email writing tips for undergraduate students

## Coauthors

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List of my coauthors

## LaTeX

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Resources on LaTeX

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How this website was created

## My research philosophy

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My personal thoughts on how to do research

## Travel

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Places that I have visited

## How to string a racquet

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How to string a racquet

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

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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.

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.

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.

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## Visualizing the Contraction Mapping Theorem Permalink

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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.

## ‘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.

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## Existence of a Statistical Equilibrium for an Economy with Endogenous Offer Sets Permalink

Published in Economic Theory, 2010

(Theory) Generalize Foley’s (1994) statistical equilibrium model when offer sets are endogenous; my master thesis at U of Tokyo; further generalized in Toda (2015).

## Operator Reverse Monotonicity of the Inverse Permalink

Published in American Mathematical Monthly, 2011

(Mathematics) Simple proof of reverse monotonicity of the inverse of positive definite matrices based on convex conjugate functions.

## Income Dynamics with a Stationary Double Pareto Distribution Permalink

Published in Physical Review E, 2011

(Power law, Empirical) A certain mean-reverting income process generates a stationary double Pareto distribution; an abridged version of my third-year empirical paper at Yale.

## The Impact of Media Reports on the 2008 Outbreak of Hydrogen Sulfide Suicides in Japan Permalink

Published in International Journal of Psychiatry in Medicine, 2012

(Medicine, Empirical) The media coverage during the 2008 outbreak of hydrogen sulfide suicides in Japan caused more suicides.

## The Double Power Law in Income Distribution: Explanations and Evidence Permalink

Published in Journal of Economic Behavior and Organization, 2012

(Power law, Empirical) A certain mean-reverting income process generates a stationary double Pareto distribution; my third-year empirical paper at Yale.

## Discrete Approximations of Continuous Distributions by Maximum Entropy Permalink

Published in Economics Letters, 2013

(Numerical method) Simple maximum entropy method to discretize probability distributions.

## Learning Curve for Paramedic Endotracheal Intubation and Complications Permalink

Published in International Journal of Emergency Medicine, 2013

(Medicine, Empirical) You need to practice at least 30 times to intubate a patient consistently.

## 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.

## Incomplete Market Dynamics and Cross-Sectional Distributions Permalink

Published in Journal of Economic Theory, 2014

(Power law, Theory) Class of tractable dynamic general equilibrium models that generates power law in size distributions; one of my dissertation chapters at Yale.

Published in Economic Theory, 2015

(Theory) Walrasian equilibrium is a special limiting case of statistical equilibrium; extension of Toda (2010).

## Discretizing Distributions with Exact Moments: Error Estimate and Convergence Analysis Permalink

Published in SIAM Journal on Numerical Analysis, 2015

(Numerical method, Mathematics) Convergence and error analysis of maximum entropy discretization of Tanaka & Toda (2013).

## 👍The Double Power Law in Consumption and Implications for Testing Euler Equations Permalink

Published in Journal of Political Economy, 2015

👍(Power law, Econometrics, Empirical) Power law in cross-sectional household consumption data causes spurious inference.

## Asset Prices and Efficiency in a Krebs Economy Permalink

Published in Review of Economic Dynamics, 2015

(Theory, Macro, Finance) Asset pricing and optimal taxation in a class of tractable dynamic general equilibrium models; formerly a section of Toda (2014).

## Edgeworth Box Economies with Multiple Equilibria Permalink

Published in Economic Theory Bulletin, 2017

(Theory) Many general examples of multiple equilibria in Edgeworth box economies.

## 👍Discretizing Nonlinear, Non-Gaussian Markov Processes with Exact Conditional Moments Permalink

Published in Quantitative Economics, 2017

👍(Numerical method, Finance) Approximate Markov processes by finite-state Markov chains using maximum entropy discretization of Tanaka & Toda (2013, 2015); applications to solving asset pricing models.

## Fat Tails and Spurious Estimation of Consumption-Based Asset Pricing Models Permalink

Published in Journal of Applied Econometrics, 2017

(Power law, Econometrics, Finance) Monte Carlo study of spurious inference caused by power law; formerly a section of Toda & Walsh (2015).

## A Note on the Size Distribution of Consumption: More Double Pareto than Lognormal Permalink

Published in Macroeconomic Dynamics, 2017

(Power law, Empirical) Cross-sectional household consumption is well-approximated by double Pareto-lognormal distribution; formerly a section of Toda & Walsh (2015).

## Growth Effects of Annuities and Government Transfers in Perpetual Youth Models Permalink

Published in Journal of Mathematical Economics, 2017

(Theory, Macro) In perpetual youth models, introduction of government transfer crowds out annuity market and increases growth.

## Huggett Economies with Multiple Stationary Equilibria Permalink

Published in Journal of Economic Dynamics and Control, 2017

(Theory, Macro) Closed-form solution to a Huggett (1993) economy with non-Gaussian VAR(1) dynamics and general examples of multiple stationary equilibria.

## Securitized Markets, International Capital Flows, and Global Welfare Permalink

Published in Journal of Financial Economics, 2019

(Finance, Theory) With collateral constraints, financial integration may hurt the less constrained country.

## Wealth Distribution with Random Discount Factors Permalink

Published in Journal of Monetary Economics, 2019

(Power law, Theory, Macro) Formal proof that the Krusell & Smith (1998) random discount factor trick generates power law tails; Pareto exponent is sensitive to the calibration of discount factor process.

## 👍An Impossibility Theorem for Wealth in Heterogeneous-agent Models with Limited Heterogeneity Permalink

Published in Journal of Economic Theory, 2019

👍(Power law, Theory, Macro) Prove the impossibility for the canonical Bewley-Huggett-Aiyagari model to generate heavier-tailed wealth than income.

## Publications, Citations, Position, and Compensation of Economics Professors Permalink

Published in Econ Journal Watch, 2019

(Empirical) Publications (Top 5/Non-top 5) and job rank explains over 80% of variations in salaries among economics professors in the UC system; no evidence of gender gap

## The Income Fluctuation Problem and the Evolution of Wealth Permalink

Published in Journal of Economic Theory, 2020

(Power law, Theory) Establish existence and uniqueness of a solution to a general income fluctuation problem; characterize tail behavior of stationary wealth distribution.

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.

## Is Gibrat’s ‘Economic Inequality’ Lognormal? Permalink

Published in Empirical Economics, 2020

(Power law, Empirical) The majority of data sets analyzed by Gibrat and claimed to be lognormal are actually closer to Pareto-type distributions.

## 👍On the Emergence of a Power Law in the Distribution of COVID-19 Cases Permalink

Published in Physica D: Nonlinear Phenomena, 2020

👍(Power law, Empirical) Size distribution of COVID-19 cases across US counties as of March 2020 obeys the power law; empirical Pareto exponent is consistent with the estimated growth rate and age distributions.

## Efficient Minimum Distance Estimation of Pareto Exponent from Top Income Shares Permalink

Published in Journal of Applied Econometrics, 2021

(Power law, Econometrics) Efficient estimation of Pareto exponents when only certain top income shares are observable.

## The Effect of Social Distancing on the Reach of an Epidemic in Social Networks Permalink

Published in Journal of Economic Interaction and Coordination, 2021

(Network) Simulation study of the evolution of an epidemic disease on social networks; the effectiveness of social distancing greatly depends on network structure.

## 👍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.

## Data-based Automatic Discretization of Nonparametric Distributions Permalink

Published in Computational Economics, 2021

(Numerical method) Automatic discretization method of nonparametric distributions using Gaussian quadrature.

## 👍Necessity of Hyperbolic Absolute Risk Aversion for the Concavity of Consumption Functions Permalink

Published in Journal of Mathematical Economics, 2021

👍(Theory) That HARA utility implies concave consumption functions is well-known, but the converse is also true.

## Perov’s Contraction Principle and Dynamic Programming with Stochastic Discounting Permalink

Published in Operations Research Letters, 2021

(Mathematics) Show the usefulness of Perov’s contraction principle (which is a generalization of Banach’s contraction principle) for solving certain dynamic programming problems.

## Asymptotic Linearity of Consumption Functions and Computational Efficiency Permalink

Published in Journal of Mathematical Economics, 2022

(Theory, Numerical method) Prove asymptotic linearity of policy functions when marginal utility is regularly varying; follow-up of Ma & Toda (2021).

## Unbounded Dynamic Programming via the Q-Transform Permalink

Published in Journal of Mathematical Economics, 2022

(Theory) Simple operation that often transforms an unbounded dynamic programming problem into a bounded one.

## 👍Determination of Pareto Exponents in Economic Models Driven by Markov Multiplicative Processes Permalink

Published in Econometrica, 2022

👍(Power law, Mathematics) Provide simple formula and mathematical foundation for Pareto exponents of stationary Markov multiplicative processes.

## Tail Behavior of Stopped Lévy Processes with Markov Modulation Permalink

Published in Econometric Theory, 2022

This is a follow up paper of Beare & Toda (2022), where we characterize the tail behavior of Markov-modulated Lévy processes that are stopped at state-dependent Poisson rates. In 2018, Brendan gave the project to Won-Ki, who was his student. I joined Won-Ki’s dissertation committee to advise on this project, and we essentially translated the discrete-time results in Beare & Toda (2022) to continuous-time. The CARA-Huggett economy example in Section 4 was recycled from an earlier version of Beare & Toda (2022).

## 👍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.

## 👍Pareto Extrapolation: An Analytical Framework for Studying Tail Inequality Permalink

Published in Quantitative Economics, 2023

👍(Power law, Numerical method, Macro) Analytical framework designed to solve and analyze heterogeneous-agent models that endogenously generate fat-tailed wealth distributions.

## Robust Comparative Statics for the Elasticity of Intertemporal Substitution Permalink

Published in Theoretical Economics, 2023

(Theory) Robust comparative statics for the elasticity of intertemporal substitution; sign- and point-identification of EIS minus 1.

## Tuning Parameter-Free Nonparametric Density Estimation from Tabulated Summary Data Permalink

Published in Journal of Econometrics, 2024

(Econometrics) Just like what the title says, for example estimation of income distributions from tax returns data.

Published in Journal of Mathematical Economics, 2024

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

## Unbounded Markov Dynamic Programming with Weighted Supremum Norm Perov Contractions Permalink

Published in Economic Theory Bulletin, 2024

(Theory) Combine weighted supremum norm and Perov contraction theorem for solving unbounded dynamic programming problems.

## 👍Bubble Necessity Theorem

Published in Accepted at Journal of Political Economy, 2024

## On Equilibrium Determinacy in Overlapping Generations Models with Money Permalink

Published in Economics Letters, 2024

I started to study models of bubble and money in late 2022 and learned the usefulness of the local stable manifold theorem. During my studies, I noticed that there is often hand-waving in applied works. For instance, Blanchard & Fishcer (1989, p. 268, Endnote 16) state

Care must be taken in using a phase diagram to analyze the dynamics of a difference equation system. […] Thus we must check in this case whether the system is indeed saddle point stable […]. This check is left to the reader.

Furthermore, during the review process of our JPE paper on the Bubble Necessity Theorem, we learned about Scheinkman (1980)’s sufficient condition for local determinacy that involves the curvature of the utility function, which was a bit mysterious. So after resubmitting our paper to JPE at the end of December 2023, we started to work on a complete analysis of the local determinacy of equilibria in Tirole (1985)’s model. It turns out that Scheinkman’s sufficient condition does not generalize to production economies: there are robust examples with arbitrary utility functions in which the nonmonetary steady state is locally determinate or indeterminate. In contrast, the monetary steady state is locally determinate under fairly weak conditions.

Published in Journal of Mathematical Economics, 2024

The Journal of Mathematical Economics, which was founded in 1974, was running a 50th anniversary issue and Kieran (associate editor at JME) was invited to contribute a review article. As we have worked on equilibrium uniqueness, he invited me to write an article together.

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## Operations Research (Econ 172B)

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

## Third Year Research Paper/Presentation (Econ 285/296)

This course trains third year Ph.D. students to conduct research, write papers, and make presentations.

## Intertemporal Asset Pricing Theory (Econ 272)

This course covers topics in finance theory. I do not currently teach this course.

## Mathematics for Economists (Econ 205)

This course covers mathematical topics that are essential for economics, very quickly but rigorously.

## Mathematical Economics/General Equilibrium (Econ 113/200A)

This course covers the classical Arrow-Debreu theory of general equilibrium. The undergraduate course (Econ 113) meets 3 hours per week for 10 weeks and covers about 2/3 of the lecure notes. The graduate course (Econ 200A) meets 3 hours per week for 5 weeks and covers the entire lecture notes plus additional topics on mathematical economics.

## Financial Markets (Econ 173A)

This course covers some institutional details on the financial markets, bond pricing (including duration analysis), optimal portfolio problem, mutual fund theorem, Capital Asset Pricing Model, and option pricing (including bounds on option prices, suboptimality of early exercise of American call options, put-call parity, and binomial option pricing).

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