arXiv subjects
Publications and source records attributed to John Stachurski.
Eisenberg and Noe (2001) analyze systemic risk for financial institutions linked by a network of liabilities. They show that the solution to their model is unique when the financial system is satisfies a regularity condition involving risk orbits. We show that this condition is not needed: a unique solution always exists.
We show that competitive equilibria in a range of models related to production networks can be recovered as solutions to dynamic programs. Although these programs fail to be contractive, we prove that they are tractable. As an illustration, we treat Coase's theory of the firm, equilibria in production chains with transaction costs, and equilibria in production networks with multiple partners. We then show how the same techniques extend to other equilibrium and decision problems, such as the distribution of management layers within firms and the spatial distribution of cities.
We propose a new approach to solving dynamic decision problems with unbounded rewards based on the transformations used in Q-learning. In our case, the objective of the transform is to convert an unbounded dynamic program into a bounded one. The approach is general enough to handle problems for which existing methods struggle, and yet simple relative to other techniques and accessible for applied work. We show by example that many common decision problems satisfy our conditions.
We obtain an exact necessary and sufficient condition for the existence and uniqueness of equilibrium asset prices in infinite horizon, discrete-time, arbitrage free environments. Through several applications we show how the condition sharpens and improves on previous results. We connect the condition, and hence the problem of existence and uniqueness of asset prices, with the recent literature on stochastic discount factor decompositions. Finally, we discuss computation of the test value associated with our condition, providing a Monte Carlo method that is naturally parallelizable.
This paper extends the core results of discrete time infinite horizon dynamic programming to the case of state-dependent discounting. We obtain a condition on the discount factor process under which all of the standard optimality results can be recovered. We also show that the condition cannot be significantly weakened. Our framework is general enough to handle complications such as recursive preferences and unbounded rewards. Economic and financial applications are discussed.
This paper provides new conditions for dynamic optimality in discrete time and uses them to establish fundamental dynamic programming results for several commonly used recursive preference specifications. These include Epstein-Zin preferences, risk-sensitive preferences, narrow framing models and recursive preferences with sensitivity to ambiguity. The results obtained for these applications include (i) existence of optimal policies, (ii) uniqueness of solutions to the Bellman equation, (iii) a complete characterization of optimal policies via Bellman's principle of optimality, and (iv) a globally convergent method of computation via value function iteration.
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 earlier literature, we determine conditions under which (a) solutions exist, are unique and are globally computable, (b) the resulting wealth dynamics are stationary, ergodic and geometrically mixing, and (c) the wealth distribution has a Pareto tail. We show how these results can be used to extend recent studies of the wealth distribution. Our conditions have natural economic interpretations in terms of asymptotic growth rates for discounting and return on savings.
Some approaches to solving challenging dynamic programming problems, such as Q-learning, begin by transforming the Bellman equation into an alternative functional equation, in order to open up a new line of attack. Our paper studies this idea systematically, with a focus on boosting computational efficiency. We provide a characterization of the set of valid transformations of the Bellman equation, where validity means that the transformed Bellman equation maintains the link to optimality held by the original Bellman equation. We then examine the solutions of the transformed Bellman equations and analyze correspondingly transformed versions of the algorithms used to solve for optimal policies. These investigations yield new approaches to a variety of discrete time dynamic programming problems, including those with features such as recursive preferences or desire for robustness. Increased computational efficiency is demonstrated via time complexity arguments and numerical experiments.
We propose a new approach to solving dynamic decision problems with rewards that are unbounded below. The approach involves transforming the Bellman equation in order to convert an unbounded problem into a bounded one. The major advantage is that, when the conditions stated below are satisfied, the transformed problem can be solved by iterating with a contraction mapping. While the method is not universal, we show by example that many common decision problems do satisfy our conditions.
We study existence, uniqueness and computability of solutions for a class of discrete time recursive utilities models. By combining two streams of the recent literature on recursive preferences---one that analyzes principal eigenvalues of valuation operators and another that exploits the theory of monotone concave operators---we obtain conditions that are both necessary and sufficient for existence and uniqueness of solutions. We also show that the natural iterative algorithm is convergent if and only if a solution exists. Consumption processes are allowed to be nonstationary.
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 (iii) agents have constant discount factors, then the wealth distribution inherits the tail behavior of income shocks (e.g., light-tailedness or the Pareto exponent). Our restrictions on utility require only that relative risk aversion is bounded, and a large variety of income processes are admitted. Our results show conclusively that it is necessary to go beyond standard models to explain the empirical fact that wealth is heavier-tailed than income. We demonstrate through examples that relaxing any of the above three conditions can generate Pareto tails.
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 uniqueness of solutions, stochastic stability of the model economy, as well as efficient computation of the ergodic wealth distribution. A variety of applications are discussed. Quantitative analysis shows that both stochastic volatility and mean persistence in wealth returns have nontrivial impact on wealth inequality.
Building on insights of Jovanovic (1982) and subsequent authors, we develop a comprehensive theory of optimal timing of decisions based around continuation value functions and operators that act on them. Optimality results are provided under general settings, with bounded or unbounded reward functions. This approach has several intrinsic advantages that we exploit in developing the theory. One is that continuation value functions are smoother than value functions, allowing for sharper analysis of optimal policies and more efficient computation. Another is that, for a range of problems, the continuation value function exists in a lower dimensional space than the value function, mitigating the curse of dimensionality. In one typical experiment, this reduces the computation time from over a week to less than three minutes.