Tom Phelan

What are the welfare implications of markup heterogeneity across firms? In standard monopolistic competition models, such heterogeneity implies inefficiency even in the presence of free entry. We enrich the standard model with heterogeneous firms so that preferences are non-separable in off-market time and market consumption and show that this changes the welfare implications of markup heterogeneity. In this context, homogeneity of markups is neither necessary nor sufficient for efficiency. The marginal cost of the marginal firm is weakly inefficiently high when off-market time and market consumption are complements and inefficiently low when they are substitutes, and the equilibrium allocation devotes weakly too few resources to firm creation. However, when off-market time and market consumption are perfect complements, markups are heterogeneous across firms and yet the equilibrium allocation is efficient.

A recent literature within quantitative macroeconomics has advocated the use of continuous-time methods for dynamic programming problems. In this paper we explore the relative merits of continuous-time and discrete-time methods within the context of stationary and nonstationary income fluctuation problems. For stationary problems in two dimensions, the continuous-time approach is both more stable and typically faster than the discrete-time approach for any given level of accuracy. In contrast, for convex lifecycle problems (in which age or time enters explicitly), simply iterating backwards from the terminal date in discrete time is superior to any continuous-time algorithm. However, we also show that the continuous-time framework can easily incorporate nonconvexities and multiple controls—complications that often require either problem-specific ingenuity or nonlinear root-finding in the discrete-time context. In general, neither approach unequivocally dominates the other, making the choice of one over the other an art, rather than an exact science.

How should a government balance risk-sharing and redistributive concerns with the need to provide incentives for investment? Should they tax firm profits or individual savings, or simply levy lump-sum transfers? I address these questions in an environment with entrepreneurs and workers in which output is subject to privately observed shocks and firm owners can both misreport profits and abscond with a fraction of assets. When frictions in financial markets restrict private risk-sharing, the stationary efficient allocation may be implemented in a competitive equilibrium with collateral constraints using (occupation-specific) linear taxes on savings and profits and lump-sum transfers to newborns. Further, the two taxes serve distinct roles and in general differ from one another. The savings tax affects consumption smoothing and may be positive or negative depending on the strength of general equilibrium effects, while the profits tax shares risk between the government and entrepreneurs, is unambiguously positive, and depends solely on the degree of frictions in financial markets.

In this paper we explore some benefits of using the finite-state Markov chain approximation (MCA) method of Kushner and Dupuis (2001) to solve continuous-time optimal control problems in economics. We first show that the implicit finite-difference scheme of Achdou et al. (2022) amounts to a limiting form of the MCA method for a certain choice of approximating chains and policy function iteration for the resulting system of equations. We then illustrate that, relative to the implicit finite-difference approach, using variations of modified policy function iteration to solve income fluctuation problems both with and without discrete choices can lead to an increase in the speed of convergence of more than an order of magnitude. Finally, we provide several consistent chain constructions for stationary portfolio problems with correlated state variables, and illustrate the flexibility of the MCA approach by using it to construct and compare two simple solution methods for a general equilibrium model with financial frictions.

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 contrast it with the efficient allocation subject to the same informational constraints. We identify two externalities, static (individual actions affect current risk of infection) and dynamic (individual actions affect future disease prevalence), and study how they are affected by limitations on testing and enforcement. We prove that a planner with imperfect enforcement will always wish to curtail activity, but that its incentives vanish as testing becomes perfect. When a vaccine arrives far into the future, the planner with perfect enforcement may encourage activity before herd immunity. We find that lockdown policies have modest welfare gains, whereas quarantine policies are effective even with imperfect testing.

Business owners in the United States are disproportionately represented among the wealthy and are exposed to substantial idiosyncratic risk. Further, recent evidence indicates that business income primarily reflects returns to the human capital of the owner. Motivated by these facts, this paper characterizes stationary efficient allocations and optimal linear taxes on income and wealth when business income depends on innate ability, luck, and the past effort of the owner. I first show that in stationary efficient allocations, more productive entrepreneurs typically bear more risk and the distributions of consumption and firm size are approximately Pareto, with the tail of the latter typically thicker than that of the former. I then characterize optimal linear taxes when owners may save in a risk-free bond and trade shares in their businesses. The optimal utilitarian policy calls for separate taxes on firm profits, capital income, and wealth, serving distinct purposes. The tax on profits plays a redistributive role, the tax on capital income affects the incentives to retain equity and exert effort, and the tax on wealth affects the degree of consumption smoothing over time.

In this paper we explore some of the benefits of using the finite-state Markov chain approximation (MCA) method of Kushner and Dupuis (2001) to solve continuous-time optimal control problems. We first show that the implicit finite-difference scheme of Achdou et al. (2017) amounts to a limiting form of the MCA method for a certain choice of approximating chains and policy function iteration for the resulting system of equations. We then illustrate the benefits of departing from policy function iteration by showing that using variations of modified policy function iteration to solve income fluctuation problems in two and three dimensions can lead to an increase in the speed of convergence of more than an order of magnitude. We then show that the MCA method is also well-suited to solving portfolio problems with highly correlated state variables, a setting that commonly occurs within general equilibrium models with financial frictions and for which it is difficult to construct monotone (and hence convergent) finite-difference schemes.

Business owners in the United States are disproportionately represented among the very wealthy and are exposed to substantial idiosyncratic risk. Further, recent evidence indicates business income primarily reflects returns to the human (rather than financial) capital of the owner. Motivated by these facts, this paper characterizes the optimal taxation of income and wealth in an environment where business income depends jointly on innate ability, luck, and the accumulated past effort exerted by the owner. I show that in (constrained) efficient allocations, more productive entrepreneurs typically bear more risk and that the associated stationary distributions of income, wealth, and firm size exhibit the thick right (Pareto) tails observed in the data. Finally, when owners may save in a risk-free bond and trade shares of their business, I show that the optimal linear taxes in this environment call for positive taxes on firm profits and risk-free savings, and for a tax/subsidy on wealth that may assume either sign. [Note: The final sentence of the abstract was revised for clarity two days after the paper was initially posted.]

How should a utilitarian government balance redistributive concerns with the need to provide incentives for business creation and investment? Should they tax business profits, the (risk-free) savings of owners, or some combination of both? To address this question, this paper presents a model in which the desirability of differential asset taxation emerges endogenously from the presence of agency frictions. I consider an environment in which entrepreneurs hire workers and rent capital to produce output subject to privately observed shocks and have the ability to both divert capital to private consumption and abscond with a fraction of assets. To provide incentives to invest, the wealth of an agent must depend on the performance of his/her firm, leading to ex-post inequality in all efficient allocations. I show that the efficient stationary distribution of wealth exhibits a thick right (Pareto) tail, with the degree of inequality monotonically increasing in the number of workers per entrepreneur. The efficient allocation is then implemented in a general equilibrium model using history-independent linear taxes on risk-free savings and (reported) business profits. The tax on entrepreneurs’ savings may be positive or negative, while the tax on business profits depends solely upon the degree of private information and is independent of all technological and preference parameters.

Researchers study student debt cancellation and the challenges in using different measures to determine impact on inequality.

We compare two types of models used to predict the spread of the coronavirus, both of which have been used by government officials and agencies. We describe the nature of the difference between the two approaches and their advantages and limitations. We compare examples of each type of model—the University of Washington IHME or “Murray” model, which follows a curve-fitting approach, and the Ohio State University model, which follows a structural approach.

Many models have been developed to forecast the spread of the COVID-19 virus. We present one that is enhanced to allow individuals to alter their behavior in response to the virus. We show how adding this feature to the model both changes the resulting forecast and informs our understanding of the appropriate policy response. We find that when left to their own devices, individuals do curb their social activity in the face of risk, but not as much as a government planner would. The planner fully internalizes the effect of all individuals’ actions on others in society, while individuals do not. Further, our simulations suggest that government intervention may be particularly important in the middle and later stages of a pandemic.

Many of the models used to track, forecast, and inform the response to epidemics such as COVID-19 assume that everyone has an equal chance of encountering those who are infected with a disease. But this assumption does not reflect the fact that individuals interact mostly within much narrower groups. We argue that incorporating a network perspective, which accounts for patterns of real-world interactions, into epidemiological models provides useful insights into the spread of infectious diseases.