# Alexis Akira TodaUniversity of California, San Diego | UCSD · Department of Economics

Alexis Akira Toda

PhD

## About

83

Publications

9,271

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572

Citations

Introduction

All of my papers are linked from my personal website at
https://alexisakira.github.io/research/

Additional affiliations

Education

August 2008 - May 2013

April 2006 - March 2008

April 1998 - March 2004

## Publications

Publications (83)

We provide evidence suggesting that the cross-sectional distributions of US consumption and its growth rate obey the power law in both the upper and lower tails, with exponents approximately equal to four. Consequently, high-order moments are unlikely to exist, and the generalized method of moments estimation of Euler equations that employs cross-s...

Approximating stochastic processes by finite-state Markov chains is useful for reducing computational complexity when solving dynamic economic models. We provide a new method for accurately discretizing general Markov processes by matching low order moments of the conditional distributions using maximum entropy. In contrast to existing methods, our...

It has been conjectured that canonical Bewley–Huggett–Aiyagari heterogeneous-agent models cannot explain the joint distribution of income and wealth. The results stated below verify this conjecture and clarify its implications under very general conditions. We show in particular that if (i) agents are infinitely-lived, (ii) saving is risk-free, and...

The first confirmed case of Coronavirus Disease 2019 (COVID-19) in the US was reported on January 21, 2020. By the end of March, 2020, there were more than 180000 confirmed cases in the US, distributed across more than 2000 counties. We find that the right tail of this distribution exhibits a power law, with Pareto exponent close to one. We investi...

We show that in a general equilibrium model with heterogeneity in risk aversion or belief, shifting wealth from an agent who holds comparatively fewer stocks to one who holds more reduces the equity premium. From an empirical view, the rich hold more stocks, so inequality should predict excess stock market returns. Consistent with our theory, we fi...

We study a game between $N$ job applicants who incur a cost $c$ (relative to the job value) to reveal their type during interviews and an administrator who seeks to maximize the probability of hiring the best. We define a full learning equilibrium and prove its existence, uniqueness, and optimality. In equilibrium, the administrator accepts the cur...

Accurately estimating income Pareto exponents is challenging due to limitations in data availability and the applicability of statistical methods. Using tabulated summaries of incomes from tax authorities and a recent estimation method, we estimate income Pareto exponents in U.S. for 1916-2019. We find that during the past three decades, the capita...

Existing nonparametric density estimators typically require individual-level data. For administrative data, individual-level information is difficult to access, but tabulated summaries are often publicly available. In this light, we propose a novel method of maximum entropy density estimation from tabulated summary data. We establish the strong uni...

We propose a new approach to solving dynamic decision problems with unbounded rewards based on the transformations used in Q-learning. In our case, however, the objective of the transform is not learning. Rather, it is to convert an unbounded dynamic program into a bounded one. The approach is general enough to handle problems for which existing me...

We study a general class of consumption-savings problems with recursive preferences. We characterize the sign of the consumption response to arbitrary shocks in terms of the product of two sufficient statistics: the elasticity of intertemporal substitution between contemporaneous consumption and continuation utility (EIS), and the relative elastici...

This article contains new tools for studying the shape of the stationary distribution of sizes in a dynamic economic system in which units experience random multiplicative shocks and are occasionally reset. Each unit has a Markov‐switching type, which influences their growth rate and reset probability. We show that the size distribution has a Paret...

We estimate capital and labor income Pareto exponents across 475 country‐year observations that span 52 countries over half a century (1967–2018). We document two stylized facts: (i) capital income is more unequally distributed than labor income in the tail; namely, the capital exponent (1–3, median 1.46) is smaller than labor (2–5, median 3.35), a...

This paper shows the usefulness of Perov's contraction principle, which generalizes Banach's contraction principle to a vector-valued metric, for studying dynamic programming problems in which the discount factor can be stochastic. The discounting condition β<1 is replaced by ρ(B)<1, where B is an appropriate nonnegative matrix and ρ denotes the sp...

We prove that the consumption functions in optimal savings problems are asymptotically linear if the marginal utility is regularly varying, or loosely speaking, behaves like a power function for wealthy agents. We also analytically characterize the asymptotic marginal propensities to consume (MPCs) out of wealth. Our results are useful for obtainin...

This article concerns the tail probabilities of a light-tailed Markov-modulated Lévy process stopped at a state-dependent Poisson rate. The tails are shown to decay exponentially at rates given by the unique positive and negative roots of the spectral abscissa of a certain matrix-valued function. We illustrate the use of our results with an applica...

How does social distancing affect the reach of an epidemic in social networks? We present Monte Carlo simulation results of a susceptible–infected–removed with social distancing model. The key feature of the model is that individuals are limited in the number of acquaintances that they can interact with, thereby constraining disease transmission to...

Using Forbes 400 data together with historical data on tax rates and macroeconomic indicators, we study the relationship between the maximum marginal income tax rate and wealth inequality. We find through a novel tail regression model that a higher maximum tax rate is associated with a higher wealth Pareto exponent. Setting the maximum rate to 0.30...

We analyze equilibrium behavior and optimal policy within a Susceptible-Infected-Recovered epidemic model augmented with potentially undiagnosed agents who infer their health status and a social planner with imperfect enforcement of social distancing. We define and prove the existence of a perfect Bayesian Markov competitive equilibrium and contras...

Although using non-Gaussian distributions in economic models has become increasingly popular, currently there is no systematic way for calibrating a discrete distribution from the data without imposing parametric assumptions. This paper proposes a simple nonparametric calibration method based on the Golub-Welsch algorithm (Golub and Welsch in Math...

This paper extends the classical contraction mapping theorem to the Cartesian product of complete metric spaces. The modulus $\beta<1$ of a contraction mapping is replaced by the condition $\rho(B)<1$, where $B$ is an appropriate nonnegative matrix and $\rho$ denotes the spectral radius. Blackwell's sufficient condition is also generalized in this...

Empirical evidence suggests that the rich have higher propensity to save than do the poor. While this observation may appear to contradict the homotheticity of preferences, we theoretically show that that is not the case. Specifically, we consider an income fluctuation problem with homothetic preferences and general shocks and prove that consumptio...

Carroll and Kimball (1996) have shown that, in the class of utility functions that are strictly increasing, strictly concave, and have nonnegative third derivatives, hyperbolic absolute risk aversion (HARA) is sufficient for the concavity of consumption functions in general consumption-saving problems. This paper shows that HARA is necessary, imply...

We propose a new approach to solving dynamic decision problems with unbounded rewards, based on an application of the Q-learning transform to the Bellman equation. Q-learning is a technique from the reinforcement learning literature with strong convergence properties. In our case, the objective of the transform is to convert an unbounded dynamic pr...

In the seminal book “Les Inégalités Économiques,” Gibrat (Les Inégalités Économiques, Librairie du Recueil Sirey, Paris, 2013) proposed the law of proportional effect and claimed that a variety of empirical size distributions—such as income, wealth, firm size, and city size—obey the lognormal distribution. Gibrat’s law went on to become a stylized...

Carroll and Kimball (1996) have shown that, in the class of utility functions that are strictly increasing, strictly concave, and have nonnegative third derivatives, hyperbolic absolute risk aversion (HARA) is sufficient for the concavity of consumption functions in general consumption-saving problems. This paper shows that HARA is necessary, imply...

This article concerns the tail probabilities of a light-tailed Markov-modulated L\'evy process stopped at a state-dependent Poisson rate. The tails are shown to decay exponentially at rates given by the unique positive and negative roots of the spectral abscissa of a certain matrix-valued function. We illustrate the use of our results with an appli...

This note corrects the proof of Proposition 5 in Stachurski and Toda (2019), which shows that the consumption function has an explicit linear lower bound and is used to prove their main result that wealth inherits the tail behavior of income in Bewley–Huggett–Aiyagari models.

We estimate capital and labor income Pareto exponents across 348 country-year observations that span 51 countries over half a century. We document two stylized facts: (i) capital income is more unequally distributed than labor income; namely, the capital exponent (1-3) is smaller than labor (2-5), and (ii) capital and labor exponents are nearly unc...

We propose an efficient estimation method for the income Pareto exponent when only certain top income shares are observable. Our estimator is based on the asymptotic theory of weighted sums of order statistics and the efficient minimum distance estimator. Simulations show that our estimator has excellent finite sample properties. We apply our estim...

How does social distancing affect the reach of an epidemic in social networks? We present Monte Carlo simulation results of a Susceptible- Infected-Removed (SIR) model on a network, where individuals are limited in the number of other people they can interact with. While increased social distancing always reduces the spread of an infectious disease...

Empirical evidence suggests that the rich have higher propensity to save than do the poor. While this observation may appear to contradict the homotheticity of preferences, we theoretically show that that is not the case. Specifically, we consider an income fluctuation problem with homothetic preferences and general shocks and prove that consumptio...

The first confirmed case of Coronavirus Disease 2019 (COVID-19) in the US was reported on January 21, 2020. By the end of March, 2020, there were more than 350000 confirmed cases in the US, distributed across more than 2000 counties. We find that the upper tail of this distribution exhibits a power law, with Pareto exponent close to one. We investi...

I estimate the Susceptible-Infected-Recovered (SIR) epidemic model using cross-country daily data on Coronavirus Disease 2019 (COVID-19) cases. The transmission rate is heterogeneous across countries and far exceeds the recovery rate, which enables a fast spread. The model suggests that around 30 percent of the population may be simultaneously infe...

We prove that the consumption functions in income fluctuation problems are asymptotically linear if the marginal utility is regularly varying. We also analytically characterize the asymptotic marginal propensities to consume (MPCs) out of wealth and derive necessary and sufficient conditions under which they are 0, 1, or are somewhere in between. W...

We analyze the household savings problem in a general setting where returns on assets, non-financial income and impatience are all state dependent and fluctuate over time. All three processes can be serially correlated and mutually dependent. Rewards can be bounded or unbounded, and wealth can be arbitrarily large. Extending classic results from an...

We analyze the household savings problem in a general setting where returns on assets, non-financial income and impatience are all state dependent and fluctuate over time. All three processes can be serially correlated and mutually dependent. Rewards can be bounded or unbounded and wealth can be arbitrarily large. Extending classic results from an...

We propose an efficient estimation method for the income Pareto exponent when only certain top income shares are observable. Our estimator is based on the asymptotic theory of weighted sums of order statistics and the efficient minimum distance estimator. Simulations show that our estimator has excellent finite sample properties. We apply our estim...

This paper studies the income fluctuation problem with capital income risk (i.e., dispersion in the rate of return to wealth). Wealth returns and labor earnings are allowed to be serially correlated and mutually dependent. Rewards can be bounded or unbounded. Under rather general conditions, we develop a set of new results on the existence and uniq...

We study the effect of collateralized lending and securitization on international capital flows and welfare in a two-country general equilibrium model with idiosyncratic investment risk. The low-margin country (Home) endogenously supplies more safe assets and enables more risk sharing. Upon financial integration, capital flows from Foreign (high-ma...

To explain the Pareto tail behavior empirically observed in wealth distributions, the quantitative macro literature has occasionally assumed that agents have random discount factors. This paper formally proves that the stationary wealth distribution in a simple Huggett model with random discounting has power law tails and characterizes the Pareto e...

We show that in heterogeneous-agent dynamic general equilibrium models that feature only idiosyncratic income risk, the wealth distribution inherits the tail behavior of income shocks such as light-tailedness and the Pareto exponent. Consequently, in this class of models, (i) it is impossible to generate heavy-tailed wealth distributions from light...

Although using non-Gaussian distributions in economic models has become increasingly popular, currently there is no systematic way for calibrating a discrete distribution from the data without imposing parametric assumptions. This paper proposes a simple nonparametric calibration method that combines the kernel density estimator and Gaussian quadra...

Many empirical studies document power law behavior in size distributions of economic interest such as cities, firms, income, and wealth. One mechanism for generating such behavior combines independent and identically distributed Gaussian additive shocks to log-size with a geometric age distribution. We generalize this mechanism by allowing the shoc...

The cross-sectional distribution of consumption is commonly approximated by the lognormal distribution. This note shows that consumption is better described by the double Pareto-lognormal distribution (dPlN), which has a lognormal body with two Pareto tails and arises as the stationary distribution in recently proposed dynamic general equilibrium m...

I obtain a closed-form solution to a Huggett economy with constant absolute risk aversion (CARA) utility when the vector of individual state variables follows a VAR(1) process with an arbitrary shock distribution. The stationary equilibrium is unique if the income process is AR(1), but not necessarily so otherwise. With Gaussian shocks, I provide g...

We derive sufficient conditions for the existence of multiple equilibria in two-good, two-agent pure exchange economies with heterogeneous but symmetric preferences with identical Bernoulli utilities. When preferences are non-homothetic (for example, quadratic, quasi-linear, or HARA), multiple equilibria are possible even with identical endowments.

The standard generalized method of moments (GMM) estimation of Euler equations in heterogeneous-agent consumption-based asset pricing models is inconsistent under fat tails because the GMM criterion is asymptotically random. To illustrate this, we generate asset returns and consumption data from an incomplete-market dynamic general equilibrium mode...

I obtain a closed-form solution to a Huggett economy with constant absolute risk aversion (CARA) utility when the vector of individual state variables follows a VAR(1) process with an arbitrary shock distribution. The stationary equilibrium is unique if the income process is AR(1), but not necessarily so otherwise. With Gaussian shocks, I provide g...

I study the asset pricing implications and the efficiency of a tractable dynamic stochastic general equilibrium model with heterogeneous agents and incomplete markets along the lines of Krebs (2003a). Contrary to previous applications of these types of models, I find that generically the distribution of idiosyncratic shocks affects the risk premia...

The maximum entropy principle is a powerful tool for solving underdetermined inverse problems. This paper considers the problem of discretizing a continuous distribution, which arises in various applied fields. We obtain the approximating distribution by minimizing the Kullback--Leibler information (relative entropy) of the unknown discrete distrib...

I introduce a general equilibrium model of non-optimizing agents that respond to aggregate variables (prices and the average demand profile of agent types) by putting a “prior” on their demand. An interim equilibrium is defined by the posterior demand distribution of agent types conditional on market clearing. A Bayesian general equilibrium (BGE) i...

We study the effect of collateralized lending and securitization on international capital flows, growth, and welfare in a two country general equilibrium model. We find that capital flows from the high- to low-margin country, leading to high investment levels and economic growth in the latter. Despite low growth, the high-margin country substantial...

Approximating stochastic processes by finite-state Markov chains is useful for reducing computational complexity when solving dynamic economic models. We provide a new method for accurately discretizing general Markov processes by matching low order moments of the conditional distributions using maximum entropy. In contrast to existing methods, our...

We define the elasticity of intertemporal substitution (EIS) for general recursive preferences and identify a sharp comparative static from a general dynamic portfolio choice problem. In the homothetic case, if the EIS is smaller (larger) than 1, an investor will increase (decrease) current consumption in response to bad news about the future. Exam...

We propose a numerical method to approximate a given continuous distribution by a discrete distribution with prescribed moments. The approximation is achieved by minimizing the Kullback-Leibler information of the unknown discrete distribution relative to the known continuous distribution (evaluated at given discrete points) subject to some moment c...

The size distributions of many economic variables seem to obey the double power law, that is, the power law holds in both the upper and the lower tails. I explain this emergence of the double power law—which has important economic, econometric, and social implications—using a tractable dynamic stochastic general equilibrium model with heterogeneous...

It is well-known that the reciprocal of the radius of the inscribed circle of a triangle is equal to the sum of the reciprocals of the radii of the three escribed circles. I generalize this theorem for an n-dimensional
simplex. In three or higher dimensions, there are many types of spheres tangent to a simplex. I characterize the existence and uniq...

We find that when the income share of the top 1% income earners in the U.S. rises above trend by one percentage point, subsequent one year market excess returns decline on average by 5.6%. This negative relation remains strong and significant even when controlling for classic return predictors such as the price-dividend and the consumption-wealth r...

The standard generalized method of moments (GMM) estimation of Euler equations in heterogeneous-agent consumption-based asset pricing models is inconsistent under fat tails because the GMM criterion is asymptotically random. To illustrate this, we generate asset returns and consumption data from an incomplete-market dynamic general equilibrium mode...

We propose a numerical method to approximate a given continuous distribution by a discrete distribution with prescribed moments. The approximation is achieved by minimizing the Kullback-Leibler information of the unknown discrete distribution relative to the known continuous distribution (evaluated at given discrete points) subject to some moment c...

Pre-hospital laryngoscopic endotracheal intubation (ETI) is potentially a life-saving procedure but is a technique difficult to acquire. This study aimed to obtain a recommendation for the number of times ETI should be practiced by constructing the learning curve for endotracheal intubation by paramedics, as well as to report the change in the freq...

In a recent paper, we proposed a numerical method to approximate a given
continuous distribution by a discrete distribution with prescribed moments.
This approximation is achieved by minimizing the Kullback-Leibler information
of the unknown discrete distribution relative to the known continuous
distribution (evaluated at given discrete points) sub...

I develop a highly tractable dynamic general equilibrium model with collateralized lending and securitization in which asset-backed securities (ABS) function as a mechanism for risk sharing. Entrepreneurs who face aggregate and idiosyncratic investment risks can borrow from a menu of non-recourse loans offered by financial intermediaries by putting...

Japan experienced a nationwide outbreak of hydrogen sulfide suicides (HSS) between April and May 2008. The annual number of HSS skyrocketed from 19 in 2007 to 1,056 in 2008. However, the factors affecting this enormous increase remain unknown. The present study aimed to examine the effect of media coverage of the incidents on the subsequent epidemi...

I axiomatize the maximum entropy principle and the minimum Kullback-Leibler information principle (mKLIP) without using the Bayes rule or probabilistic concepts such as independence. Combined with Caticha and Giffin's result that mKLIP implies Bayesian inference, I show that mKLIP is a fundamental method of inductive inference when a decision maker...

I obtain a closed-form solution for a general equilibrium model with incomplete markets and heterogeneous agents that allow for an arbitrary number of assets, an arbitrary number of aggregate states, and arbitrary shock distributions for asset returns. Agent heterogeneity has non-trivial implications for asset prices and risk premia. In a stationar...

We show that when $\set{X_j}$ is a sequence of independent (but not
necessarily identically distributed) random variables which satisfies a
condition similar to the Lindeberg condition, the properly normalized geometric
sum $\sum_{j=1}^{\nu_p}X_j$ (where $\nu_p$ is a geometric random variable with
mean $1/p$) converges in distribution to a Laplace...

Once controlled for the trend, the distribution of personal income appears to be double Pareto, a distribution that obeys the power law exactly in both the upper and the lower tails. I propose a model of income dynamics with a stationary distribution that is consistent with this fact. Using US male wage data for 1970-1993, I estimate the power law...

This paper applies the recently axiomatized Optimum Information Principle
(minimize the Kullback-Leibler information subject to all relevant information)
to nonparametric density estimation, which provides a theoretical foundation as
well as a computational algorithm for maximum entropy density estimation. The
estimator, called optimum information...

This paper modifies Jaynes's axioms of plausible reasoning and derives the
minimum relative entropy principle, Bayes's rule, as well as maximum likelihood
from first principles. The new axioms, which I call the Optimum Information
Principle, is applicable whenever the decision maker is given the data and the
relevant background information. These a...

In statistics and econometrics, the equivalence between matrix inequalities A ≥ B ⇔ B−1 ≥ A−1 is used to obtain a lower bound on the variance matrix, where A, B are symmetric and positive definite. The same property holds for linear operators on Hilbert spaces that are bijective, self-adjoint, and positive definite. I give a short and elementary pr...

A statistical equilibrium consists of distributions over offer sets of agents that can be achieved in the greatest number of ways. Although it is known that a unique equilibrium exists when the offer sets are finite and exogenous, the general case has been an open question. This paper (1) generalizes the concept of statistical equilibrium to the ca...